43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Physics [16 ed.] 8194767733, 9788194767732
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Table of contents :
Contents
1. Physical World, Units
and Measurements
2. Motion in a Straight
Line
3. Motion in a Plane
4. Laws of Motion
5. Work, Energy and Power
6. System of Particles and
Rotational Motion
7. Gravitation
8. Mechanical Properties
of Solids
9. Mechanical Properties
of Fluids
10. Thermal Properties
of Matter
11. Thermodynamics
12. Kinetic Theory
13. Oscillations
14. Waves
15. Electric Charges
and Fields
16. Electrostatic Potential
and Capacitance
17. Current Electricity
18. Moving Charges
and Magnetism
19. Magnetism and Matter
20. Electromagnetic
Induction
21. Alternating Current
22. Electromagnetic
Waves
23. Ray Optics and
Optical Instruments
24. Wave Optics
25. Dual Nature of
Radiation and Matter
26. Atoms
27. Nuclei
28. Semiconductor Electronics:
Materials, Devices and
Simple Circuits
29. Communication
Systems
Citation preview
EBD_8341
EBD_8341
EBD_8341
EBD_8341
P-1
Physical World, Units and Measurements
1
Topic-1 : Unit of Physical Quantities 1 ҈
MCQs with One Correct Answer
矄�泐ᨨ 鼀�� 和ᔼ盐 � �矄҈├�鼀 ᔼ
ᆠ� 盐 �矄� 犨 �ᆠ 盐 � ᨨ � �矄�ᆠ鼀 盐 �ᨨ 鼀矄矄�矄�� 盐鼀�ᆠ盐ᨨ 矄�泐 ᨨ 鼀��矄 㝐� ҈ ҈黨 ᔼ㡰桌 㡰
㡰
ᆠ㡰
㡰 ҈
㡰 ҈
꾀鼀矄鼀ᆠ泐� �ᆠ盐 ᨨ �㜠 �矄犨� 矄�泐 �矄矄盐ᆠ ᨨ鼀� �ᆠ � ᨨ � 㜠 �矄㡰 ᆠ�ᆠ � 쟼 泐 ᨨ ᨨ � Ӹ㜠 �矄 �矄 �ᆠ泐 ᨨᨨ�ᨨ ᨨ 矄 犨 矄 ᆠᨨ 矄 ᨨ ᨨᨨ�ᨨ �泐 矄和ᔼ盐 �
ᔴ㡰
㡰
ᆠ㡰 ᨨ盐⏌ 鼀㡰 7
q B
F ᨨᆠ
GM e G盐 �矄
꾀㡰 GMeMs , G 盐 �矄���ᨨ ᆠᨨ 矄
�
㡰 ᨨ 鼀㜠矄ᆠᨨ
ᔴᨨ盐鼀 ᔼ 꾀㡰 ᔴᆠ� ᆠ 㡰 ᔼ 盐ᆠ ᆠ ᔴ㡰 㝐� �ᆠᔼ 盐ᆠ�ᨨ
Me鼀矄矄ᨨ Ms鼀矄矄ᨨ 和盐
ᔴᨨ盐鼀 ᔼᔼ
矄㡰 㡰 ᨨ㡰 ├�㡰҈
�㡰 ��㡰 ���㡰 �㡰 㡰
ᔴᨨ盐鼀 ᔼᔼ ᨨ 鼀㜠矄ᆠᨨ ᆠᨨ盐ᨨ鼀ᨨ盐҈ ᆠᨨ盐ᨨ鼀 ᨨ㡰҈
犨ᨨ 鼀㜠鼀㡰҈ ᨨ㜠矄ᆠᨨ 鼀㡰҈ ҈黨黨㜠㝐├矄桌
10 Subjective Problems
㡰 ᨨ㡰 ᆠᨨ盐ᨨ鼀㡰 鼀㡰 䰘桌
㡰 鼀㡰 矄ᆠᨨ 㡰
Re �盐矄ᨨ
Match the Following ᔴᨨ盐鼀 ᔼ
矄ᆠᨨ 㡰
Me鼀矄矄ᨨ
㡰
F
⼄㡰 ├�ᨨ�鼀㡰 鼀㡰
qᆠ � B鼀� �ᆠ �
㝐� 졠 � 꾀��҈桌 As m m m 㡰 ᆠ㡰 㡰 As As m A SI盐 �ᨨ � 盐ᆠ ᆠ 泐ᆠ 犨� 矄 ҈黨黨㜠㝐├矄桌 㡰 犨⏌鼀 㡰 ᨨ㜠矄ᆠ⏌鼀
RT R盐 �矄 M �矄 ᆠᨨ 矄 T矄ᨨ盐鼀盐 M鼀ᨨ鼀矄矄
� 㝐⧤和盐 �矄 ᨨᆠ ᨨ ᨨᨨ犨� �⼄盐 ��矄 꾀㡰 Ӹᨨ盐 �矄鼀ᨨ 盐盐矄
҈黨桌
㡰 㝐� �ᆠᔼ 盐ᆠ�ᨨ
ᔴ㡰 ᨨ犨ᨨ 矄
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EBD_8341 P-2
Physics
Topic-2 : Dimensions of Physical Quantities 1
MCQs with One Correct Answer 黨
҈
�
䰘
㡰
҈
E ҈ z = B CR m e � 犨 C㜠ᆠᆠ� ᆠR㜠 矄�矄 ᆠl㜠 � E㜠쟼ᆠ�ᆠ � B㜠鼀� �ᆠ � e m 㜠 矄ᆠ 鼀���泐 鼀��泐矄ᆠ�泐 㝐� 和� ᔼᔼ㡰桌 㡰 x, y z 矄鼀 �鼀 矄�ᨨ 㡰 졠 泐x z 矄鼀 �鼀 矄�ᨨ ᆠ㡰 졠 泐x y 矄鼀 �鼀 矄�ᨨ 㡰 졠 泐y z 矄鼀 �鼀 矄�ᨨ �鼀 矄�ᨨ ᨨ鼀盐 ᨨ 鼀ᆠᨨ 盐ᆠ��泐�矄 K ᨨ矄 鼀盐 㝐� 和 ᔼ㡰桌 㡰 㝐 ⧤ 㡰 㝐 ⧤ ᆠ㡰 㝐 ⧤ 㡰 㝐 ⧤҈ 꾀⼄盐 �泐x�矄�� 泐 IFv⏌WL㡰� 鼀矄ᨨ 鼀ᨨ鼀 ᨨ � �I ᨨᆠFᨨᆠ�泐v犨ᨨ├W � L �鼀 矄�ᨨ ᨨ鼀盐 ᨨx�矄矄鼀矄 ᨨ 㝐� 和 ᔼᔼ㡰桌 㡰 ᆠ├矄ᆠᨨ 矄 㡰 ᨨᆠ ᆠᨨ 矄 ᆠ㡰 �泐 矄�泐 㡰 ᆠᨨ �ᆠ� ᨨ �矄ᆠᨨ矄�泐 꾀鼀ᨨ盐 ᨨ 矄ᨨ �泐ᆠ� ᨨ Ἐ矄矄盐 ᆠ 盐 � 盐 ��鼀�矄 � 矄ᨨ ᆠᨨ 矄 �鼀 矄�ᨨ ᨨ 矄ᨨᆠᨨ 矄 �矄㝐� 和 ᔼᔼ㡰桌 㡰 㝐 㡰 㝐 ᆠ㡰 㝐 ҈ 㡰 㝐 ᔼ 矄 꾀 ᨨᆠ ᆠ ᨨ矄 矄 盐 鼀 盐 �矄 �鼀 矄�ᨨ ᨨ Ӹᨨ盐 �Ἐ矄鼀ᨨ 盐盐矄犨� 㝐� 和 ᔼ㡰桌 ҈ ᆠ㡰 꾀 㡰 꾀҈ 㡰 꾀 㡰 꾀 ᔼ 鼀ᨨ鼀 盐鼀 㡰 꾀㡰 �鼀 㡰├ ᨨ 盐 鼀 ⼄盐 ��矄 �鼀 矄�ᨨ ᨨ鼀盐 ᨨ �泐�矄 㝐� 和 ᔼᔼ㡰桌 㡰 꾀 桌 㡰 꾀҈ 桌 ᆠ㡰 꾀҈⏌ -҈ 桌 㡰 ҈⏌ 꾀 -҈桌 ␔ �ᆠ ᨨ ᨨᨨ犨� �ᆠᨨ鼀� �ᨨ 矄 矄 �鼀 矄�ᨨ
ᨨ ᆠ�ᆠ矄�矄 ᆠ Î0�矄 鼀���泐ᨨ ᆠ盐盐鼀 m�矄 鼀��泐ᨨ ᆠ盐盐鼀㡰䁸 㝐� ҈꾀�҈黨 ᔼ㡰桌
⼄盐 ��矄 x =
m m 㡰 e e
y=
ᆠ㡰
e m
e 㡰 m
ᔼ ᨨ鼀盐㦨�Ӹ �鼀 矄�ᨨ 矄ᨨ ᆠᆠ� ᆠ 鼀� �ᆠ � 矄ᆠ�泐 ␔ �鼀 矄�ᨨ 矄ᨨ Ӹ� 和ᔼ盐 �矄䁸 㝐� ҈꾀�҈黨 ᔼᔼ㡰桌 㡰 㝐҈ 꾀桌 㡰 㝐 꾀桌 ᆠ㡰 㝐 꾀桌 㡰 㝐 꾀桌
ᔼ 和ᔼ盐 �矄 �鼀 矄�ᨨ 矄ᨨ
Î �矄 m
㝐� 꾀�҈黨 ᔼ㡰桌 㡰 꾀 㝐҈҈ 㡰 꾀 㝐 ⏌ ᆠ㡰 꾀 㝐 㡰 꾀 㝐҈ ҈ l r c v 矄 � 盐ᆠ ᆠ 矄�矄 ᆠ ᆠᆠ� ᆠ ᨨ�矄ᆠ�泐 �鼀 矄�ᨨ ᨨ
҈
l � 和ᔼ盐 �矄犨� 㝐� ҈ ҈黨 ᔼᔼ㡰桌 rcv 㡰 꾀桌 㡰 꾀҈ 桌 ᆠ㡰 꾀桌 㡰 桌 ᨨᆠᨨ � ᆠ�ᨨ 犨 犨ᨨᨨ鼀矄�矄�� 泐 ҈҈ æ x ö F = ab ç ç akT ÷÷ 錐 犨 x �矄 �矄 ᆠ ├�矄 è ø ᨨ 鼀
ᆠᨨ 矄 �矄鼀盐 a b 犨ᨨᆠᨨ 矄 矄 �鼀 矄�ᨨ 矄ᨨ b�矄 㝐� ҈҈ ҈黨 ᔼ㡰桌 㡰 㝐 㡰 㝐 ᆠ㡰 㝐 㡰 㝐 ҈ ᔼ 矄 㡰ᆠᆠ�ᨨ 꾀㡰 ᨨᆠ 㡰ᆠᨨ 矄� 矄 盐 鼀 盐 �矄 �鼀 矄�ᨨ ᨨ Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 犨� 㝐� ҈҈ ҈黨 ᔼᔼ㡰桌 㡰 꾀 㡰 꾀 ᆠ㡰 꾀 㡰 꾀 hc� 犨 ᆠ�矄矄 ᨨ G �� 盐 �矄���ᨨ ᆠᨨ 矄 h�矄 ᆠ├矄 ᆠᨨ 矄 �鼀 矄�ᨨ ᨨ f�矄 ᨨ 㝐� 黨 ҈黨 ᔼ㡰桌 㡰 㡰 �泐 ᆠ㡰 鼀ᨨ鼀 盐鼀 㡰 ᨨ盐鼀 ҈ 쟼 矄矄�ᨨ ᨨ�鼀� 鼀矄ᨨ 盐 �矄���ᨨ ᆠᨨ 矄 㡰 ᆠ├Ἐ矄ᆠᨨ 矄 㡰 ᆠ 矄 ᨨ �� 㡰�矄 ᨨᨨ�ᨨ ᨨ 㝐� 黨 ҈黨 ᔼᔼ㡰桌
҈ 꾀⼄盐 �泐f�矄�� 泐f㦨
㡰
ᆠ�
㡰
ᆠ
ᆠ㡰
㡰
ᆠ ᆠ ҈� �鼀 矄�ᨨ 矄ᨨ 矄ᨨ� �ᨨ �V� ᨨᨨᆠ�ᆠ ᆠ� 盐 �矄ᨨ ᆠ├sᆠᨨ 矄 �h矄 ᨨ �� �c ��ᨨ ᆠᨨ 矄 �G 鼀A�矄 㝐� ҈黨 ᔼ㡰桌 㡰 h⏌c�⏌G҈⏌A҈ 㡰 h⏌G⏌c⏌A҈ ᆠ㡰 h⏌e҈⏌G⏌A҈ 㡰 hG⏌C҈⏌A҈ ҈
�
B 犨 B�矄鼀� �ᆠ � m0 m �矄 鼀� �ᆠ鼀��泐ᨨ ᆠ盐盐鼀�矄 㝐� ҈黨 ᔼᔼ㡰桌 㡰 㝐 㡰 㝐 ҈ ᆠ㡰 㝐 㡰㝐҈ �鼀 矄�ᨨ 矄ᨨ
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P-3
Physical World, Units and Measurements
҈䰘
ᆠ ᆠ�矄�ᆠ �矄 ᆠ犨 �ᆠ ⼄盐 盐鼀���ᨨ ᆠ矄矄�� � �ᆠ ᆠ├ � ᆠ 鼀� ᨨ鼀矄盐�ᆠᨨ鼀� �ᨨ ᨨ 盐 鼀 泐矄�ᆠ ᆠᨨ 矄 矄 ᆠ␔ �ᆠ ᨨ ᨨᨨ犨� �ᆠᨨᆠ泐 ��矄 ᆠ├ � 䁸 㝐� 졠 � 꾀�҈�҈桌 ҈
҈
҈黨
҈
҈ æ ö 㡰 ᆠ 㡰 ç ÷ ᆠ㡰 ᆠ 㡰 ᆠ èᆠ ø
�鼀 㡰 ᨨᆠ�泐 ᔴ㡰 �盐 鼀ᨨ鼀 盐鼀 㡰 ᆠ ᨨ矄 矄 盐 鼀 ⼄盐 ��矄� 矄 ᨨ 鼀矄矄 � �鼀ᔼ 鼀矄ᨨ 矄 �鼀 矄�ᨨ 矄ᨨ 鼀矄矄犨ᨨ盐 㝐� 졠 � 꾀�҈䰘桌 㡰 㝐桌㦨 ҈ᔴ 桌 㡰 㝐桌㦨 ҈ᔴ 桌 ᆠ㡰 㝐桌㦨 ҈ᔴ ҈桌 㡰 㝐桌㦨 ᔴ 桌 ᔼ ᨨᨨ犨� �ἘᔼἘ 矄ᨨᆠ盐 ᨨ 矄泐鼀ᨨ矄 � 盐矄盐 鼀 � � ᔴ ᨨᨨ矄 ᨨ�ᨨ ᆠᨨ矄ᨨ 矄ᨨ �鼀 矄�ᨨ 矄ᨨ ᆠ�ᆠᆠᨨ 盐ᆠ��泐 㝐� 졠 � 꾀�黨҈桌 㡰 㝐҈ ᔼ 㡰 㝐҈ ᔼ ҈ ᆠ㡰 㝐 ᔼ 㡰 㝐 ᔼ ᔼ ᆠᨨ �ᆠᆠ �ᆠᨨ 鼀矄矄鼀矄 ᨨ �� �
ᆠ盐盐鼀ᆠ ᆠ├矄ᆠᨨ 矄 ├ 矄 盐 鼀 ⼄盐 ��矄 鼀��泐ᨨ ᆠ盐盐鼀mᆠ 矄矄 � 盐 �矄ᨨ 㝐� 졠 � 꾀�҈҈҈�桌
æ 鼀ᆠ ö æ ᆠ ö æ ö æ ö 㡰 ç ÷ 㡰 ç ÷ ᆠ㡰 ç ÷ 㡰 çç ÷÷ è 鼀 ø è ᆠ ø è ø è 鼀 ø ᔼ ᆠᆠ� ᆠᨨ ᨨᆠᆠ�ᨨ�矄鼀矄盐 � 鼀矄 ᨨ 盐 ��u鼀 泐ᆠᨨ鼀� � � ᆠ�ᆠᆠ ��e ᨨ �盐矄� ᆠ├矄ᆠᨨ 矄 � 矄 ᨨ �� �c 㝐� 졠 � 꾀�҈҈�桌 ᆠ ᆠ 㡰 盐 = ᆠ㡰 盐 = 㡰 盐 = ᆠ ᨨ鼀 ᨨᨨ犨� �ᆠᨨ鼀� �ᨨ 矄ᨨ 泐矄�ᆠᆠᨨ 矄 矄 矄矄 ᨨ盐� � 盐矄盐 矄泐鼀ᨨ矄㡰 ᨨ 泐 ᆠᨨ鼀� �ᨨ 犨ᨨ盐 矄鼀盐� � 矄泐矄鼀矄ᨨ 盐 �矄�矄 㝐� 졠 � 꾀�҈҈桌
Î 桌 ᨨ �鼀 矄�ᨨ ᨨ鼀盐ᨨ 鼀���泐 ᨨ ᆠ盐盐鼀 ᔼ 㝐 㦨 鼀矄矄 㦨 � 㦨 �鼀 꾀㦨ᆠ�ᆠᆠ盐 㝐� ҈桌 㡰 Î 㦨㝐҈
꾀桌 㡰 Î 㦨㝐҈
꾀桌 ᆠ㡰 Î 㦨㝐҈
҈꾀桌 㡰 Î 㦨㝐҈
҈꾀桌 � �鼀 矄�ᨨ 矄ᨨ �盐鼀ᨨ鼀 盐鼀 ᆠᆠ� ᆠ矄ᆠ�泐 㝐� 졠 � 꾀�҈桌 㡰 㝐 ҈꾀 - 㝐 -҈- 㡰 㝐 - 㝐 -҈- 꾀 ᆠ㡰 㝐 -҈ - 㝐 꾀
䰘
㡰 盐 =
㡰
㡰
ᆠ㡰
2
peᨨ
pe ᨨ 鼀
鼀㦨鼀矄矄ᨨ ᆠᨨ 㡰
m ᨨ eᨨ
ᆠ
p m ᨨ eᨨ ᆠ ᔼ 鼀矄ᨨ 矄�矄 ᆠ �鼀 �鼀 矄�ᨨ 矄ᨨ �ᨨ m ᨨ 鼀��泐m 鼀���泐e�矄 e 㝐� 졠 � 꾀�҈҈҈桌 㡰 桌 㡰 ҈桌 ᆠ㡰 桌 㡰 桌 㡰
ᆠ
黨
㡰 㝐 -҈ - 㝐 -҈- 꾀 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄 � �鼀 矄�ᨨ 矄䁸 㡰 矄矄盐Ӹᨨ盐 �矄鼀ᨨ 盐盐矄和矄矄 �和桌 㡰 쟼㝐 ᨨ � � ᆠ쟼ᆠ�ᆠᨨ � ᆠ㡰 ␔ᨨ├ ᨨ 쟼 �泐 㡰 �ᨨ鼀ᨨ鼀 쟼ᆠ�ᆠ 盐 쟼ᆠ�ᆠ � a az 矄矄盐 矄 ᨨ �矄 ᆠ 矄 P = æç - ö÷ è kq ø b 犨 abᆠᨨ 矄 矄z�矄 �矄 ᆠk�矄ᨨ 鼀 矄 ᆠᨨ 矄 q�矄鼀盐 �鼀 矄�ᨨ ᨨ b 和桌 㡰 MLT 㡰 M҈L҈T ҈ ᆠ㡰 MLT 㡰 M҈L҈T DV 꾀 ⼄盐 �泐 X �矄 �� 泐 e L 犨 Î �矄 Dt 鼀���泐ᨨ 矄ᆠL �矄 � DV�矄ᨨ � � ᆠ Dt�矄�鼀� �鼀 矄�ᨨ ᨨ鼀盐 ᨨX �矄 矄鼀矄 ᨨ ҈和桌 㡰 矄�矄 ᆠ 㡰 ᆠ � ᆠ㡰 ᨨ� 㡰 ᆠ盐 æ ҈ö
�鼀 矄�ᨨ ᨨ ç ÷ e E e 鼀���泐ᨨ è ø 矄ᆠEᆠ�ᆠ � 㡰 和桌 㡰 MLT҈ 㡰 MLT ᆠ㡰 ML҈T 㡰 MLT҈ Integer Value Answer
ᨨ � �矄 ᆠd ᨨ犨 �ᆠ 矄�� ᆠ 矄
ᆠ泐 � ᨨ��泐 ᆠᨨ ��ᨨ 矄 �犨泐矄㜠 �� 盐矄矄 �鼀 矄�ᨨ 矄 矄矄盐鼀矄 �矄 ᆠ 矄 ᨨ
鼀矄矄 矄�泐rᨨ ᨨ�� 矄�泐 ᨨ犨⏌㡰Sᨨ �� ᨨ鼀 矄�� �矄 ⼄盐 ᆠ泐f �� � 矄 d�矄ᨨᨨ�ᨨ ᨨS҈⏌n 盐ᨨ n�矄 꾀 ҈桌 4
Fill in the Blanks
҈ �鼀 矄�ᨨ ᨨ ᆠ�ᆠᆠᨨ 盐ᆠ��泐�矄힘힘힘힘힘힘힘힘힘 ҈黨黨䰘㜠҈㝐├桌 ⼄盐�ᨨ ᨨ 矄 ᨨ �矄 �矄 �� 泐 æ a ö çè P + ÷ø V - b㡰 = RT �鼀 矄�ᨨ 矄ᨨ ᆠᨨ 矄 V a�矄힘힘힘힘힘힘힘힘힘 ҈黨黨䰘㜠㝐├矄桌
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EBD_8341 P-4
Physics
ᔼ ᨨ鼀盐X㦨YZ X Z �鼀 矄�ᨨ 矄ᨨ ᆠᆠ� ᆠ 鼀� �ᆠ � 盐ᆠ�ᨨ 矄ᆠ�泐 �鼀 矄�ᨨ 矄 ᨨ Y � MKSQ 矄泐鼀 힘힘힘힘힘힘힘힘힘 힘힘힘힘힘힘힘힘 ҈黨㜠㝐├矄桌 ᆠ├矄ᆠᨨ 矄 矄 �鼀 矄�ᨨ 힘힘힘힘힘힘힘힘 ҈黨�㜠㝐├矄桌 6
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MCQs with One or More Than One Correct
和ᨨ鼀�鼀矄��矄ᆠᨨ � ᨨᆠᨨ 矄盐ᆠ矄泐矄鼀ᨨ 盐 �矄 矄ᨨ ⼄盐 ��矄ᆠ 矄矄 � 鼀矄ᨨ ᨨ 泐ᨨ 泐矄�ᆠ ⼄盐 �泐 ᔼ ᨨ 矄盐ᆠ 矄泐矄鼀 �鼀 矄�ᨨ 矄 ᨨ � ⼄盐 ��矄�� � 鼀矄ᨨ ⼄盐 �泐X矄 ᨨᨨ犨矄 ᨨ矄��ᨨ 桌㦨Xa桌錐矄 桌 㦨Xb桌錐ᆠᆠ�ᨨ 桌㦨Xp桌錐 � 鼀ᨨ鼀 盐鼀桌㦨Xq桌錐 ᨨᆠ桌㦨Xr桌 꾀 桌 㡰 a + p = b
䰘
黨
ᆠ㡰 p - q + r = a 㡰 p + q + r = b 盐矄 ᆠᨨ 矄� 矄泐矄鼀 ᨨ 盐 �矄 � 犨 �ᆠ 鼀矄矄 �盐 鼀ᨨ鼀 盐鼀 �鼀 矄�ᨨ 矄矄 ᔼ � 矄 �鼀 矄�ᨨ 矄ᨨ 犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄⏌ ᆠᨨᆠ䁸 꾀 ҈黨桌 㡰 �鼀 矄�ᨨ ᨨ ᨨᆠ�矄L 㡰 �鼀 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀�矄L҈ ᆠ㡰 �鼀 矄�ᨨ ᨨ �泐�矄L 㡰 �鼀 矄�ᨨ ᨨ ᨨ犨�矄L� 꾀 � 矄ᆠ l㡰 矄ᨨ 鼀���泐 e㡰 ᨨ �ᆠ�ᆠ鼀�ᨨ 鼀
ᆠᨨ 矄 ├㡰 矄ᨨ盐 鼀盐 㡰 盐鼀盐 �ᨨ盐鼀 㡰ᨨ ᆠ�
ᆠ � �ᆠ矄 ᆠ � ⼄㡰ᆠ� 泐ᆠ ᨨ �ᆠ矄␔ �ᆠ ᨨ ᨨᨨ犨� � 矄矄�ᨨ 矄㡰 ᨨI �矄 㡰 �鼀 矄�ᨨ 泐ᆠᨨᆠ䁸 꾀 ҈桌
æ ⼄ ö l = ç ÷ 㡰 ç e├ ÷ è ø
æ e├ ö ÷ 㡰 l = çç ÷ è ⼄ ø
æ ö ⼄ ÷ ᆠ㡰 l = çç ⏌ ÷ è e ├ ø
æ ö ⼄ ÷ 㡰 l = çç ҈⏌ ÷ è e ├ ø
ᔼ 鼀矄 ᨨ ᨨ � � ᆠ V ᆠ�ᆠ ᆠ盐 I 鼀���泐e鼀��泐m 矄 ᨨ �� c �鼀 矄�ᨨ 泐ᆠᨨᆠ⼄盐�ᨨ 矄㡰�矄 㡰 꾀 ҈�桌 㡰 mᔼ㦨e 㡰 mᔼ㦨m ᆠ㡰 ᔼ㦨eᆠ 㡰 mᆠᔼ㦨e ᆠ├矄ᆠᨨ 矄 h 矄 ᨨ �� c ���ᨨ ᆠᨨ 矄 G盐矄 ᨨ ᨨ鼀盐 �ᨨ � L 盐 �ᨨ 鼀矄矄M ᆠᨨᆠᨨ�ᨨ 矄㡰�矄 㡰 꾀 ҈�桌 㡰 Mµ c 㡰 Mµ G ᆠ㡰 Lµ h
㡰 p + q - r = b
㡰 Lµ G
[e 0 ] ᨨ �鼀 矄�ᨨ ᨨ鼀盐ᨨ 鼀���泐 ᨨ ᆠ盐盐鼀 [m 0 ] ᨨ 鼀��泐ᨨ ᆠ盐盐鼀ᔼ M㦨鼀矄矄L㦨 � T㦨�鼀 I㦨ᆠ�ᆠ ᆠ盐 ҈黨黨㜠㝐├矄桌
㡰 e 桌 = M -҈ L- T Ι 㡰 e 桌 = M -҈ L- T Ι ᆠ㡰 m 桌 = M LT - Ι - 㡰 m 桌 = M LT -҈ Ι ҈ �矄 ᨨ 泐矄�ᆠ ⼄盐 ��矄 矄鼀 �鼀 矄�ᨨ 矄�矄 㡰 ҈黨黨�和桌 㡰 泐 ᨨ 矄 盐鼀 ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ
㡰 ᔴ盐� ⼄盐 ᆠ泐ᨨ �� 犨 ᆠ㡰 ���ᨨ ᨨ � 㡰 ᆠ├矄ᆠᨨ 矄 ᨨ⼄盐 �鼀 矄�ᨨ 矄ᨨ ⼄盐 ��矄� ᨨ ᨨ鼀ᨨ㡰ᨨ ᨨᨨ犨� ��矄 矄鼀ᔼ � 泐 � 矄㡰 㡰 ᨨ⼄盐 ␔ᨨ├ ҈黨㜠㝐├矄桌 㡰 꾀 �盐鼀ᨨ鼀 盐鼀 ␔ᨨ├ ᆠ㡰 쟼 �泐 Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 㡰 �� 泐 ␔ � 7
Match the Following
㝐ᆠ �矄 ᔼ犨� �矄 ᔼᔼ 矄ᆠ ᆠᨨᆠ 矄犨 盐矄� � ᆠᨨ 矄�� ᨨ犨 �矄矄 꾀 ҈桌 �矄ᔼ �矄ᔼᔼ ᨨ 鼀
ᆠᨨ 矄 ҈ 㝐 㜠҈桌 턼 ᔴᨨ �ᆠ� ᨨ �矄ᆠᨨ矄�泐 㝐҈ ҈桌 ᆠ├ᆠᨨ 矄 㝐 ⧤҈桌 和 鼀ᆠᨨ 盐ᆠ��泐 㝐 ⧤҈桌 ᔴᨨ 矄 턼 和 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ 㝐ᆠ 泐矄�ᆠ ⼄盐 ��矄 �� � ᆠᨨ盐鼀 ᔼ 犨� �鼀 矄�ᨨ 矄 矄矄 � 鼀矄ᨨ 鼀矄矄 M㡰 � L㡰 �鼀 T㡰 ᆠ � Q㡰�� � ᆠᨨ盐鼀 ᔼᔼ 犨� ᆠᨨᆠ 矄犨�� 矄 鼀ᆠ ⼄盐 �泐� 盐 ᨨ鼀� 泐ᨨ盐 矄犨ᨨᨨ├ ҈黨㜠㝐├矄桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀 �盐鼀ᨨ鼀 盐鼀 MLT MLQ
ᨨ⼄盐 MLT҈ ᔴᆠ� ᆠ MLT҈Q ᔼ 盐ᆠ ᆠ M ҈ L T Q 矄�矄��泐 LT 8
Comprehension/Passage Based Questions
矄矄� ᔼ ᆠᨨ鼀� �ᆠ ᨨ泐 ᆠ�ᆠ 鼀� �ᆠ ᨨ鼀 ᨨᆠ ᨨ ᨨ �鼀 矄�ᨨ 矄ᨨ ᆠ�ᆠ 鼀� �ᆠ⼄盐 ��矄鼀盐矄矄ᨨ ᨨᆠ ᨨ ᔼ
⼄盐矄�ᨨ 矄ᨨ犨E桌 B桌矄 ᨨ �鼀 矄�ᨨ 矄ᨨ ᆠ�ᆠ 鼀� �ᆠ � 矄矄ᆠ�泐犨 �e桌 m桌矄 ᨨ �鼀 矄�ᨨ 矄ᨨ 鼀���泐 鼀��泐ᨨ 矄ᆠ 矄ᆠ�泐 L桌 T桌 �鼀 矄�ᨨ 矄 ᨨ � �鼀 矄ᆠ�泐꾀 ⼄盐 ��矄�� � 和ᔼ盐 �矄 꾀 ҈桌 � �ᨨ 犨 E桌 B桌�矄 㡰 E桌㦨B桌L桌҈T桌 㡰 E桌㦨B桌L桌T桌 ᆠ㡰 E桌㦨B桌L桌T桌҈ 㡰 E桌㦨B桌L桌҈T桌҈
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P-5
Physical World, Units and Measurements
�ᨨ 犨 e桌 m桌�矄 㡰 m桌㦨e桌L桌T桌 㡰 m桌㦨e桌L桌T桌 ҈ ᆠ㡰 m桌㦨e桌 L桌 T桌 㡰 m桌㦨e桌҈L桌T桌
10 Subjective Problems 䰘
␔� �鼀 矄�ᨨ 矄ᨨ ᨨᨨ犨� �� 鼀矄ᨨ 鼀矄矄 �鼀 � ᆠ � ҈黨㜠㝐├矄桌
�㡰 鼀� �ᆠ 盐 ��㡰 ��� �泐鼀ᨨ 盐盐矄 꾀�矄盐 ᨨ鼀 ᨨ矄�ᨨ 盐 犨ᨨ矄ᆠ�矄 犨� �ᨨ Tᨨᨨ�ᨨ ᨨp a d b E c␔ �P�矄 矄�ᆠ矄矄盐�d�矄 矄�泐ᨨ 犨 �E�矄 ᨨ �泐ᨨ ᨨ矄�ᨨ � 盐矄ᨨ ab c ҈黨҈㜠㝐├矄桌
Topic-3 : Errors in Measurements & Experimental Physics 1 ҈
MCQs with One Correct Answer
꾀矄ᆠ犨�盐� 矄� ��矄�ᨨ 矄ᨨ �矄ᆠ�ᆠ盐矄ᆠ ᆠ�ᆠ盐矄ᆠ�矄盐 �矄 ᨨ �ᆠ 矄ᆠ鼀├� � �ᨨᨨ盐矄ᨨ ᨨ ᆠᨨ鼀ᨨ�ᨨ ᨨ ᆠ�ᆠ盐 矄ᆠ �矄ᆠ鼀 ᨨ �鼀鼀�矄 ᨨ�ᆠ ᨨ �ᆠ 矄ᆠ 盐 ᨨ ᨨᨨ � ᨨ 矄 ᆠᨨ盐 ᨨ 矄ᆠ犨�盐�矄ᆠ�泐 㝐� 和 ᔼ㡰桌 㡰 ��m鼀 㡰 ᨨ矄��҈m鼀 ᆠ㡰 ᨨ矄��҈鼀鼀 㡰 ᨨ矄��҈m鼀 꾀矄盐 鼀矄盐� � �鼀ᨨ ᆠ�ᨨ ᆠ�ᆠ盐 ᆠᨨ矄矄㜠矄ᆠ�ᨨ 犨� ᨨ �矄ᆠᆠᨨ 矄 ᨨᨨ犨� � ᨨ盐 � �矄��鼀鼀���鼀鼀 ��鼀鼀 ��鼀鼀 �ᨨ 矄 ᨨ盐 � ��矄��䰘�鼀鼀 矄 ��ᨨ ᨨ �矄䰘黨�鼀鼀 � �鼀ᨨ ᆠ�矄 ᨨ盐 ᨨᆠᨨ 㝐� 和 ᔼᔼ㡰桌 矄 㡰 ��䰘�±䰘黨㡰鼀鼀 㡰 ��䰘�±䰘㡰鼀鼀 ᆠ㡰 ��±䰘㡰鼀鼀 㡰 ��±䰘㡰鼀鼀 꾀 泐矄�ᆠ⼄盐 �泐z 矄ᨨ ᨨ盐ᨨ矄矄abc d矄z㦨
�
a b
ᆠ �矄ᨨ ᨨ� 鼀㜠 cd 矄盐鼀 ᨨ abc d奠҈�奠奠 �奠㜠 矄ᆠ�泐 ᆠ �ᨨ ᨨ� z�矄 㝐� 和� ᔼ㡰桌 㡰 ҈�奠 㡰 ҈�奠 ᆠ㡰 ҈�奠 㡰 ҈�奠 矄� �矄ᆠ犨�盐�ᨨ �ᆠ ҈ᆠ鼀 � ��矄�ᨨ 矄ᨨ �矄 ᆠ�ᆠ盐矄ᆠ �ᆠ├ 矄矄ᨨ ᨨᆠ�矄 鼀矄盐 ᔼ 矄 ᨨ盐 ᆠᨨᆠ泐ᆠᨨ 矄[㝐� Sep. 03, 2020 (I)] 㡰 ҈҈ᆠ鼀 㡰 ҈ᆠ鼀 ᆠ㡰 ҈�ᆠ鼀 㡰 ҈ᆠ鼀
矄ᆠᨨ盐 ᨨ 鼀� 矄ᆠᨨ �ᆠ�矄�矄 ҈鼀鼀ᔼ矄 �矄ᆠ�矄 �� � ᨨ҈ ��矄�ᨨ 矄 ᆠᨨ� ᆠ� 犨� 黨 ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠ␔ 犨矄 ᨨ盐ᆠ � � ᆠ ᨨ 䰘 ��矄�ᨨ ᨨ � 矄ᆠ ᆠᨨ� ᆠ� 矄犨� ��矄�ᨨ ᨨ 鼀� 矄ᆠ ᨨᨨ �矄ᆠ�矄泐� ��� 矄� ᨨ ᨨᨨ 鼀� 矄ᆠ ␔ �矄 ��矄盐矄 ᨨ鼀矄盐 � ᨨ ᆠ泐� ᨨᨨ �矄ᆠ犨 ҈ᆠ鼀 ᆠ鼀 和ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ ��矄�ᨨ � ᨨ ᆠ泐� �矄 和�矄 �矄ᆠ ��矄�ᨨ 㡰 [㝐� Sep.02, 2020 (I)]
㡰 ᆠ鼀 㡰 ҈ᆠ鼀 ᆠ㡰 䰘ᆠ鼀 㡰 黨黨ᆠ鼀 ᔼ 矄ᆠ犨ᨨ 矄ᆠ犨㜠�盐��矄�� 矄� ᨨ�ᨨ 矄�鼀ᨨ矄 泐鼀鼀ᨨ 鼀� 矄ᆠᔼ � ��矄�ᨨ 矄ᨨ ᆠ�ᆠ盐矄ᆠ 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐��矄 㝐� 黨 ᔼ㡰桌 㡰 ҈ᆠ鼀 㡰 鼀鼀 ᆠ㡰 ҈ᆠ鼀 㡰 ҈鼀鼀 䰘 ᨨ ᨨ盐矄矄ᨨ 鼀矄盐 泐矄�ᆠ⼄盐 ��矄矄 �� ᨨ犨␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄�矄ᆠᨨᆠ䁸 㝐� 黨 ᔼᔼ㡰桌 꾀㡰 꾀҈㦨҈㦨䰘ᔴ҈㦨� 㡰 꾀㦨㦨҈ᔴ㦨 ᔴ㡰 꾀㦨�㦨҈黨҈ᔴ㦨҈ 㡰 꾀㦨�㦨҈黨҈ᔴ㦨҈黨� 㡰 꾀졔졔ᔴ꾀҈졔҈졔ᔴ҈꾀졔졔ᔴ꾀졔졔ᔴ 㡰 꾀҈졔҈졔ᔴ҈㦨꾀졔졔ᔴ㦨꾀졔졔ᔴ㦨꾀졔졔ᔴ ᆠ㡰 꾀졔졔ᔴ꾀҈졔҈졔ᔴ҈㦨꾀졔졔ᔴ㦨꾀졔졔ᔴ 㡰 꾀҈졔҈졔ᔴ҈꾀졔졔ᔴ꾀졔졔ᔴ꾀졔졔ᔴ 꾀矄�鼀 盐盐鼀�矄� �盐矄 ᨨ 鼀� 盐 ᨨ ���ᨨ ᆠᆠ�ᨨ gᆠ� ᆠ � ᨨ 盐盐鼀�矄�ᆠ鼀 矄ᨨ犨ᆠ 犨� ҈矄 矄ᨨ盐�ᨨ 鼀矄盐矄 �鼀├ ᨨᨨ矄ᆠ��ᨨ 矄ᨨ �矄 ᆠᆠ盐ᆠ泐� g�矄 㝐� ᔼᔼ㡰桌 㡰 �奠 㡰 奠 ᆠ㡰 奠 㡰 奠 黨 ᔼ 矄�泐鼀矄盐鼀 ᨨ ᆠ盐 鼀矄矄 � � 鼀矄盐 矄 ҈᠀҈㡰├� ҈᠀҈㡰 鼀矄ᆠ�泐 ᨨ� 鼀矄盐鼀 ᨨ 矄�泐 �矄 㝐� 黨꾀�҈黨 ᔼ㡰桌 㡰 ҈├�⏌鼀 㡰 ҈├�⏌鼀 ᆠ㡰 ҈├�⏌鼀 㡰 䰘├�⏌鼀 ҈ ᨨ 矄⼄盐�矄�黨ᆠ鼀 ᨨ 䰘矄盐ᆠ 矄⼄盐矄├� �� ᨨᆠᆠᨨ盐 矄�� � �ᆠ ��盐矄�矄 㝐� 黨꾀�҈黨 ᔼᔼ㡰桌 㡰 䰘ᆠ鼀 㡰 䰘ᆠ鼀 ᆠ㡰 䰘ᆠ鼀 㡰 䰘ᆠ鼀 ҈҈ ᔼ 矄�鼀 盐盐鼀 �鼀 ᨨ 鼀� �ᨨ ᨨ ᆠᆠ�ᨨ 盐 ᨨ ��泐 g㡰 �鼀 ├ ᨨ ᨨ矄ᆠ��ᨨ 矄 �矄 鼀矄盐 泐盐矄� � 犨ᆠ ᨨ ҈ 矄ᆠᨨ 矄ᆠᨨ盐 鼀 盐ᨨ �鼀├ ᆠᨨ鼀矄ᨨ盐ᨨ 矄 � ᨨ 盐盐鼀 �矄 鼀矄盐 泐盐矄� � 鼀矄ᆠᨨ 矄ᆠᨨ盐 ҈鼀鼀 盐ᨨ� �矄 ��ᆠ鼀 ᆠ �ᨨ� 鼀� �ᨨ ᨨ g�矄 ᆠᨨ矄ᨨ 㝐� 꾀�҈黨 ᔼᔼ㡰桌 㡰 䰘奠 㡰 奠 ᆠ㡰 �奠 㡰 奠
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EBD_8341 P-6
҈
҈
҈
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Physics
矄ᆠᨨ盐 ᨨ 鼀� 矄ᆠᨨ 矄ᆠ犨�盐��矄҈鼀鼀
鼀� �鼀盐鼀 盐鼀ᨨ ��矄�ᨨ 矄ᨨ �矄ᆠ�ᆠ盐矄ᆠ ⼄盐� ᨨ鼀矄盐�Ҙ鼀 �鼀ᨨ 犨��矄 㝐� ҈ ҈黨 ᔼ㡰桌 㡰 � 㡰 ᆠ㡰 ҈ 㡰 �
�ᆠ 盐鼀ᨨ ��矄�ᨨ 矄 ᨨ ᆠ�ᆠ盐 矄ᆠ ᨨ �� 矄ᆠ犨 �盐� � 鼀鼀 ҈ 矄ᆠ�泐 ␔ 矄ᆠ犨 �盐� �矄 盐泐 �� 犨� ᨨ盐 泐ᨨᆠ ᨨᨨ �矄 ᆠ�ᆠ盐 矄ᆠ �矄 ��矄�ᨨ ᨨ犨 鼀 �
� �矄ᨨ 鼀� 矄ᆠ ᆠ�ᆠ盐矄ᆠ ᨨ � 矄 �� 鼀鼀 矄ᆠ�泐 �ᆠ├ 矄矄 ᨨ 矄 �矄 㝐� 黨 ҈黨 ᔼᔼ㡰桌 㡰 �䰘��鼀鼀 㡰 �黨�鼀鼀 ᆠ㡰 �䰘�鼀鼀 㡰 �䰘鼀鼀
矄�泐ᨨ 鼀� � 矄 ᨨ ᆠ盐 �矄 鼀� 泐鼀矄盐� � 矄� 矄ᨨ ᆠ盐 �矄 鼀矄矄ᔼ �ᨨ矄� 鼀矄盐� � 鼀矄矄 � 矄ᆠ�泐҈�奠 ҈奠 鼀 �鼀盐鼀 ᨨ �
鼀� � � 矄�泐�矄 㝐� ҈桌 㡰 �奠 㡰 �奠 ᆠ㡰 �奠 㡰 奠
ᆠ �ᨨ矄� ⼄盐 ��矄턼 和�奠 ҈奠奠 ҈�奠矄ᆠ�泐� 鼀矄盐鼀 ᨨ 턼 和
鼀 �鼀盐鼀ᆠ �ᨨ� 盐ᨨ 꾀犨� 㝐� 졠 � 꾀�҈҈桌 㡰 �奠 㡰 奠 ᆠ㡰 䰘�奠 㡰 �奠 ᔼ 矄ᆠ犨�盐��ᆠᨨ鼀ᨨ�ᨨ 矄ᨨ 矄ᆠ犨ᆠ盐矄 �ᨨ鼀ᨨ� �矄 ᆠᨨ �ᆠ鼀 ҈ᆠ�ᆠ盐 矄ᆠ ��矄�ᨨ 矄 �ᆠ├ 矄矄ᨨ 犨�鼀矄盐 泐 �矄 矄ᆠ犨�盐���矄 � �ᨨ 鼀� 矄ᆠ ��矄�ᨨ 矄 ᆠ�ᆠ盐矄ᆠ ��矄�ᨨ 矄꾀矄矄盐鼀� � ���� ᨨᨨ �ᆠ├ 矄矄ᨨ 犨��矄 㝐� 졠 � 꾀�҈�҈桌 㡰 ᆠ鼀 㡰 ҈�ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ҈�ᆠ鼀
ᨨᨨ犨� �ᨨ矄�ᨨ 矄犨├ ᨨ 鼀� � � 矄盐 ᆠ 矄�ᨨ ᨨ 犨泐ᆠ�泐鼀 ᨨ �鼀ᨨ ᆠ�泐㦨҈�맔҈鼀 �矄ᨨ 犨 㦨҈�맔҈鼀 矄� ��㦨黨鼀⏌矄 矄�鼀� � �ᨨ
泐矄�ᆠ⼄盐 �泐 꾀 =
҈
҈䰘
rhg ´҈ ⏌鼀 ᨨ矄矄�ᨨ� 矄盐 ᆠ 矄�ᨨ
�矄ᆠᨨ矄矄ᨨ 㝐� ҈䰘桌 㡰 奠 㡰 ҈奠 ᆠ㡰 ҈�奠 㡰 ҈�奠 꾀 泐矄�ᆠ⼄盐 �泐�矄 矄ᆠ� 泐 �ᨨ
㦨a҈⏌bᆠd ᔼ �ᨨ矄� 鼀矄盐鼀 ᨨ a, b, c d 矄ᆠ�泐奠҈奠奠 �奠 � ᨨ� 犨� 㝐� 졠 � 꾀�黨҈䰘桌 㡰 奠 㡰 ҈奠 ᆠ㡰 奠 㡰 �奠 T=
҈
҈黨 꾀矄ᨨ 鼀矄盐矄 ᨨ 犨泐鼀矄盐� � �鼀� 犨 ᨨ� �矄ᨨ ᆠ�� � 矄ᨨ盐 ᨨ �鼀ᆠ犨� ᨨᨨ鼀ᨨ 犨 ᨨ�
�矄鼀矄盐鼀 ᨨ �鼀�矄d 㦨 ҈ 矄ᆠᨨ 矄 鼀矄盐矄 ᨨ 犨ᨨ㦨鼀矄 ├ ᆠᆠ�ᨨ 盐ᨨ��泐�㦨҈鼀矄 ᨨᆠ�泐 ᨨ 矄ᨨ盐 �矄 鼀矄҈ ᆠ�ᨨ ᨨ � 꾀 ҈䰘桌 鼀矄盐鼀 d⏌�矄ᆠᨨ矄矄ᨨ 㡰 奠 㡰 ҈奠 ᆠ㡰 奠 㡰 �奠 꾀矄ᆠ犨�盐�犨� �ᆠ ᨨ �鼀鼀 ᆠ�ᆠ盐矄ᆠ 犨� � ��矄�ᨨ 矄�矄盐矄 ᨨ鼀矄盐 �ᆠ├ 矄矄ᨨ � 矄 ᨨ 꾀盐鼀� �盐鼀 ᨨ矄� � 鼀矄盐鼀 ��矄 ᨨ盐 犨 犨ᨨ犨矄ᨨ 矄ᆠ犨�盐� ᨨ盐� � ᆠᨨ ᆠ � ��矄�ᨨ ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ� ᨨᨨ 鼀� 矄ᆠ�矄泐 �矄�␔ �矄 �ᆠ├ 矄矄ᨨ 矄 � 鼀� 矄ᆠ � ��矄�鼀鼀 � ��矄�ᨨ ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ� 䁸 㝐� ҈桌 㡰 䰘鼀鼀 㡰 �鼀鼀 ᆠ㡰 䰘�鼀鼀 㡰 鼀鼀 ҈ 꾀矄盐 鼀矄盐矄 �鼀�ᨨ ᨨ ҈ᨨ矄ᆠ��ᨨ 矄ᨨ 矄�鼀 盐盐鼀 ᨨ盐�鼀矄 矄�矄黨矄黨҈矄黨� 矄 黨矄ᔼ 鼀� �鼀盐鼀 ��矄�ᨨ � 鼀矄盐� �ᆠᨨᆠ├ �矄҈矄 ᨨ 鼀 �鼀矄 ᨨ盐 㝐� ҈桌 㡰 黨 ᠀ ҈矄 㡰 黨 ᠀ 矄 ᆠ㡰 黨 ᠀ ҈�矄 㡰 黨 ᠀ �矄 犨ᨨ �ᆠ�矄ᨨ ᨨ 犨 �ᆠ ҈ᆠ鼀 �� � ᨨ҈⼄盐 ��矄�ᨨ 矄 ᨨ 鼀� 矄ᆠ � 矄ᆠ ᨨ ᨨ ᨨ ᆠ�矄 ᔴ҈㡰 矄 ҈ ⼄盐 ��矄�ᨨ 矄 ᆠᨨ矄ᨨ ᨨ黨鼀� 矄ᆠ ��矄�ᨨ 矄 �矄ᆠᨨ ᨨ ᆠ� ᔴ㡰 矄҈⼄盐 ��矄�ᨨ 矄 ᆠᨨ矄ᨨ ᨨ҈҈鼀� 矄ᆠ ��矄�ᨨ 矄 � �矄 ᨨ 犨ᨨᆠ�矄矄 ᨨ犨 � ��盐 鼀矄盐 盐矄 � ᆠ鼀㡰泐ᆠ�矄ᔴ҈ ᔴ矄ᆠ�泐 꾀 ҈桌
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㡰 䰘 㡰 䰘 䰘
�ᨨ ᨨ ᨨ矄ᆠ��ᨨ ᨨ 矄�鼀 盐盐鼀�矄 㝐矄盐 盐ᨨ �矄ᆠ鼀├ ᨨ犨 ᨨ҈鼀鼀 � ᆠᆠ盐ᆠ泐 �鼀 ᨨ҈ᨨ矄ᆠ��ᨨ 矄ᨨ 盐盐鼀�矄 ᨨ盐 ᨨ黨矄盐矄� �犨�矄犨ᆠ ᨨ ҈矄矄ᨨ盐�ᨨ ᆠᆠ盐ᆠ泐� 鼀� �ᨨ ᨨ ��矄 㝐� ҈�桌 㡰 ҈奠 㡰 �奠 ᆠ㡰 奠 㡰 奠
㦨 p
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P-7
Physical World, Units and Measurements
�鼀ᨨ 矄�矄鼀矄盐 盐矄� � �ᆠ�矄 犨 �ᆠ 矄 ��矄�ᨨ 矄ᨨ ҈ᆠ鼀ᨨ �矄鼀� 矄ᆠ 㝐和㡰 ҈ ��矄�ᨨ 矄ᨨ �矄 �矄ᆠ 和㡰鼀ᆠ 黨 ��矄�ᨨ 矄ᨨ
鼀� 矄ᆠ 矄盐ᆠ 鼀矄盐鼀 矄 ᨨ �� 矄 㝐� 졠 � 꾀�҈҈�桌 和ᨨ 㝐和 ᆠ鼀㡰 和 ��矄�ᨨ 矄
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ᔼ ᨨᨨ�矄ᆠ鼀 鼀 ᆠᨨᆠ �鼀 �矄 㡰 �ᆠ鼀 㡰 �黨ᆠ鼀 ᆠ㡰 �ᆠ鼀 㡰 �ᆠ鼀
ᆠ盐 ᨨ� �ᨨ ᨨ �ᨨ �矄 �� 泐 ᔼ = ( ҈ - ҈) 鼀꾀 犨 � ᨨ��矄 �
䰘
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ᨨ矄 鼀盐 �矄� �├� ᔼ 矄盐 鼀├矄 ᨨ鼀矄盐� � ±҈ 犨 �鼀矄盐� � ᆠ盐 ᨨ �鼀꾀⧤犨 犨� ᨨ� 盐 ᨨ ᆠ盐 � 鼀꾀䁸 㝐� ҈桌 㡰 鼀꾀 㡰 鼀꾀 ᆠ㡰 �鼀꾀 㡰 �鼀꾀 꾀矄盐 鼀矄盐 � ᨨ ᨨ 犨ᨨ�矄� ᆠ鼀␔ �ᆠ � 矄盐鼀 � 盐矄ᨨ鼀矄盐�䁸 㝐� ҈桌 㡰 꾀鼀矄ᆠ 㡰 꾀 �ᆠ�犨 ҈ ��矄�ᨨ 矄� � 矄ᆠ鼀ᆠ 矄犨� 黨 ��矄�ᨨ � 鼀� 矄ᆠ 鼀�
矄ᆠ 矄҈ ��矄�ᨨ 矄� ҈ᆠ鼀 ᆠ㡰 꾀矄ᆠ犨�盐� � �҈ ��矄�ᨨ 矄� ᆠ�ᆠ盐 矄ᆠ �ᆠ 矄҈鼀鼀 㡰 꾀矄ᆠ犨�盐� � �� ��矄�ᨨ 矄� ᆠ�ᆠ盐矄ᆠ �ᆠ 矄҈鼀鼀 ᔼ �鼀 ᨨ ᆠ��ᨨ ᨨ ᨨ鼀矄 ᆠ ᨨ 鼀 ҈҈ᨨ�矄 ᆠ �� 矄ᆠ鼀ᨨ ᨨ �犨�
ᨨ � � ᆠ ᆠᨨ矄矄 矄 ᨨ 矄�矄 ᆠ �矄 ᨨ盐 ᨨ ᆠ �� 矄 ᆠ鼀 ᨨ 犨� ᆠᨨ矄ᨨ � � � �ᨨ ᨨ鼀�矄�ᨨ ᨨ�
� �ᨨ ᨨ鼀犨� 㝐� 졠 � 꾀�҈҈桌 㡰 ҈�ᨨ 㡰 ҈�ᨨ ᆠ㡰 �ᨨ 㡰 �ᨨ 꾀 �鼀 �矄 ᨨ鼀 ᨨ ᨨ� 盐 ᨨ ᆠᆠ�ᨨ 盐ᨨ��泐�泐盐矄� �矄�鼀 盐盐鼀ᨨ � ᔼ �矄 �鼀 �鼀 ᨨ ҈ᨨ矄ᆠ��ᨨ 矄 �矄 鼀矄盐 泐盐矄� �犨ᆠ ᨨ ҈矄ᆠᨨ 矄ᆠᨨ盐 盐�矄黨矄ᆠᨨ 矄 � �矄鼀矄盐 泐盐矄� � 鼀矄ᆠᨨ 矄ᆠᨨ盐 ҈鼀鼀 盐�矄ᆠ鼀 ᨨ� 鼀� �ᨨ ᨨ �犨ᨨ盐 㝐� 졠 � 꾀�黨҈桌 㡰 ҈䰘奠 㡰 䰘奠 ᆠ㡰 奠 㡰 䰘奠
�鼀ᨨ ᆠ泐� �矄鼀矄盐 盐矄� � � ᆠ�矄犨� ᨨ ᨨᨨᔼ�矄 ᨨ盐 ᨨᨨ �矄ᆠ�矄犨 �҈ᆠ鼀 �҈�ᆠ鼀ᨨ 鼀�
矄ᆠ �矄ᆠ 矄� ��矄�ᨨ 矄⼄盐� ᨨ�
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ᆠ鼀 ��矄�ᨨ ᨨ �矄ᆠ ᆠ泐ᆠᨨ� ᆠ� 矄 犨� ᨨ ᨨ 鼀� 矄ᆠ ��矄�ᨨ 矄 �鼀ᨨ ᆠ泐� �矄 꾀 ҈桌 㡰 �҈҈ᆠ鼀 㡰 �҈ᆠ鼀 ᆠ㡰 �҈ᆠ鼀 㡰 �҈ᆠ鼀 MLg ö æ ᔼ 鼀� �ᨨ ᨨ Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 ç Y = ÷ è pld ø 泐盐矄� �和矄鼀 ᨨ 犨�ᨨ � L㦨鼀 �鼀d㦨�鼀鼀�矄盐矄 ᨨᨨ M㦨�├�
矄�ᨨ l㦨�鼀鼀� � ᨨ 犨��矄ᨨ矄 턼盐 ��矄d l鼀矄盐 盐矄� �矄ᆠ犨�盐� 鼀�ᆠᨨ鼀矄ᆠ�泐 泐 矄鼀�ᆠ ᨨ �鼀鼀 盐鼀ᨨ ��矄�ᨨ 矄ᨨ �ᆠ�ᆠ盐矄ᆠ�矄 ҈ ᆠᨨ �盐�ᨨ 矄ᨨ 鼀 �鼀盐鼀ᨨᨨᨨ Y鼀矄盐鼀 ҈桌 㡰 盐 ᨨ ᨨ矄 � 鼀矄盐鼀 矄 ᨨ d l 矄鼀 㡰 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ d�矄犨�ᆠ 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ l ᆠ㡰 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ l�矄犨�ᆠ 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ d 㡰 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ d�矄 ᨨ盐�鼀矄 盐ᨨ ᨨ� 鼀矄盐鼀 ᨨ l
矄�泐ᨨ 矄ᨨ� �矄 ᨨ 鼀� �
�鼀 �鼀ᨨ �矄鼀矄盐 犨� 矄ᆠ犨�盐� 犨 ᨨ矄 �ᆠ �矄 � 鼀鼀 � ��矄�ᨨ 矄ᨨ ᆠ�ᆠ盐矄ᆠ � �ᨨ 鼀�
矄ᆠ�矄�鼀鼀 ᨨ ᆠ�ᆠ盐矄ᆠ�矄 ��矄�ᨨ 矄 ᔼ 鼀矄盐 鼀矄矄ᨨ 矄�ᨨᨨ 奠 �ᆠ �ᨨ� 矄�泐�矄 ҈҈桌 㡰 黨奠 㡰 奠 ᆠ㡰 ҈奠 㡰 奠 꾀 �ᆠ�矄 矄҈鼀鼀鼀├矄ᨨ 鼀� 矄ᆠᔼ 矄 ⼄盐 ��矄�ᨨ 矄ᨨ �矄ᆠ犨 �ᆠ 鼀ᆠ 犨� ҈ 鼀� 矄ᆠ ��矄�ᨨ 矄 ᨨ �矄 �ᆠ�矄 矄ᆠᨨ盐 ҈桌 �矄 㡰 鼀鼀 㡰 �鼀鼀 ᆠ㡰 ҈鼀鼀 㡰 鼀鼀 和盐 矄ᔼᔼᔼ ᔼᔼᔼ ᨨ鼀 �鼀 ᨨ鼀矄盐� � ᆠᆠ�ᨨ 盐ᨨ��泐 g㡰盐矄� �矄�鼀 盐㜠 盐鼀 泐盐矄 � � 矄ᨨ 盐盐鼀 ⏌ᨨ ᆠᨨ �鼀 ᨨ � 盐鼀ᨨ ᨨ矄ᆠ��ᨨ 矄 ᨨ㜠 矄�ᨨ 矄矄 ᨨ犨 � 桌 矄ᆠᨨ盐 ᨨ � 㦨҈ᆠ鼀 矄ᆠᨨ盐 ᨨ�鼀㦨҈矄
和盐 � ᨨ ᨨᨨ
ᨨ�鼀 盐盐鼀 ᨨ矄ᆠ��ᨨ 矄 ᨨ 㡰 ᆠ鼀㡰 㡰 ᨨ矄ᆠ��ᨨ 矄 矄㡰 ᔼ ҈ ᔼᔼ ᔼᔼᔼ
�鼀 �ᨨ 矄㡰 ҈ ҈ 黨
ᔼ EIEII EIII ᆠ �ᨨ矄� g� æ Dg ö çè g ´ ҈÷ø ᨨ矄盐 矄ᔼᔼᔼ ᔼᔼᔼ矄ᆠ�泐
㡰 EI㦨 ᆠ㡰 EI㦨EII
㡰 EI�矄鼀� �鼀盐鼀 㡰 EII�矄鼀 �鼀盐鼀
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EBD_8341 P-8
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Physics
꾀矄盐 ᨨ鼀矄 �鼀 ᨨ 鼀� Ӹᨨ盐 �Ἐ矄 鼀ᨨ 盐盐矄ᨨ 犨� ᆠ泐鼀ᨨ �泐和Ἐ矄鼀 ᨨ ᔼ
�ᆠ盐 � � 矄盐 鼀矄盐矄 矄�ᨨ �
� ᨨ 犨�ᨨ鼀鼀犨� 盐 ᆠ� 泐 ᨨ ±�鼀鼀ᨨ ᨨ ᆠ泐҈├� 矄盐 矄ᨨ 鼀矄盐矄 �鼀ᨨ 犨�ᨨ鼀鼀犨�
盐 ᆠ� 泐ᨨ ±҈鼀鼀 ├g㦨黨鼀⏌矄 ᆠ㡰 Ӹᨨ盐 �Ἐ矄鼀ᨨ 盐盐矄ᨨ� ᨨ鼀 � ��矄 䰘桌 㡰 ±㡰맔҈҈҈⏌鼀 㡰 ±㡰맔҈҈҈⏌鼀 ᆠ㡰 ±҈㡰맔҈҈҈⏌鼀 㡰 ±�㡰맔҈҈҈⏌鼀 꾀矄盐 ᨨ鼀矄 �鼀 ᨨ 鼀� �ᨨ ᨨ æ p l ö g ç = ÷ ᨨ� � l�矄Dl � �鼀T�矄DT è T ø n �矄 盐鼀 ᨨ �鼀矄 � � �矄 ├ 鼀矄盐鼀 ᨨ g�矄鼀ᨨ矄ᆠᆠ盐 ᨨ 㜠㝐҈桌 Dl DT n 㡰 �鼀鼀 矄ᆠ ҈ 㡰 �鼀鼀 矄ᆠ ᆠ㡰 �鼀鼀 ҈矄ᆠ ҈ 㡰 ҈鼀鼀 ҈矄ᆠ � ᔼ 矄ᆠ犨�盐� ᨨᨨ 鼀� 矄ᆠᆠᨨ� ᆠ� 矄犨� � ��矄�ᨨ ᨨ ᆠ�ᆠ盐矄ᆠ� ��盐 �㡰 ᆠ�ᆠ盐 ��矄�ᨨ ᨨ 矄ᆠ犨�盐��ᔼ鼀ᨨ矄�鼀鼀ᨨ 鼀�
矄ᆠ� ᨨ ᨨ�ᨨ �鼀ᨨ � ��盐 ��㡰 �矄 㜠㝐҈桌
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��盐 ��㡰
䰘
㡰 �鼀鼀 㡰 鼀鼀 ᆠ㡰 ҈鼀鼀 㡰 ҈�鼀鼀 꾀犨�ᨨ � l㦨±ᆠ鼀 �盐矄r㦨�±�ᆠ鼀 鼀矄矄m㦨±�鼀㝐 �鼀盐鼀ᆠ �ᨨ�
矄�泐�矄 和桌 㡰 㡰 ᆠ㡰 ҈ 㡰 꾀ᆠ盐 矄矄� ᨨ � ҈맔҈鼀ᔴᆠ盐�矄ᨨ盐鼀 㡰 ҈䰘맔҈鼀 㡰 ҈䰘맔҈鼀 和桌 ᆠ㡰 ҈䰘맔҈ 鼀 㡰 ҈䰘맔҈鼀 2
黨
Integer Value Answer
�泐ᨨ 矄泐矄鼀矄 盐 ᆠ�ᨨ ᨨ �鼀�矄�� 矄 E t㡰㦨A a㡰犨 a㦨矄҈ 鼀矄盐鼀 ᨨ A 矄 ᨨᨨ ҈�奠ᔼ ᨨ� 鼀矄盐鼀 ᨨ �鼀�矄҈�奠 ᆠ �ᨨ� 盐ᨨ E t㡰 t㦨�矄�矄 꾀 ҈�桌
盐� �和矄 �鼀 ᨨᨨ �矄ᆠ�矄 犨 맔҈鼀 �맔҈鼀ᨨ 鼀� 矄ᆠ
��矄�ᨨ ᨨ �矄ᆠ ᆠ泐ᆠᨨ� ᆠ� 矄 犨� ᨨ ᨨ 鼀� 矄ᆠ ��矄�ᨨ 矄␔ ��ᨨ ᨨ ᨨ ├��矄� ᨨ 犨� ᨨᨨ � 矄ᆠ矄��矄犨 맔҈鼀 �맔҈鼀ᨨ 鼀� 矄ᆠ盐 ᨨ犨 � ��矄�ᨨ ᨨ �矄ᆠ ᆠᨨ� ᆠ� 矄犨� ᨨ ᨨ 鼀� 矄ᆠ ��矄�ᨨ 矄 � ᨨ � 鼀�ᆠ犨��矄鼀 �矄ᆠᨨ矄矄㜠矄ᆠ�ᨨ �矄맔҈䰘鼀 矄ᆠᨨ盐 ᨨ �矄ᆠ�矄҈맔 ҈�鼀 鼀 �鼀盐鼀ᆠ �ᨨ� Ӹᨨ盐 �矄 鼀ᨨ 盐盐矄ᨨ 犨��矄 꾀 ҈桌 3
Numeric Answer
҈ 矄�泐ᨨ 矄ᨨ� 鼀 矄 �矄 鼀� 泐 鼀矄盐� ��矄鼀矄矄 �矄 �鼀 鼀 �鼀盐鼀ᨨ�
æ x ö 矄�泐ᨨ 矄 �矄 çè ÷ 奠ᔼ �ᨨ矄 ҈ ø � 鼀矄盐� � 鼀矄矄 �鼀奠 ҈�奠 矄ᆠ�泐 盐ᨨ x�矄힘힘힘힘힘힘 㝐� 和 ᔼ㡰桌 犨ᨨᆠᆠ�ᨨ矄犨� ᆠᆠ� ᆠ盐矄C҈㦨᠀҈ C㦨᠀҈� ᆠᨨ
ᆠ � 矄�矄 ᨨ� � ᆠᨨ矄矄 �矄ᆠᨨ鼀� �ᨨ �矄V㦨�᠀ ᆠ �ᨨ� ᆠᆠ盐�ᨨ ᨨ �泐矄ᨨ �
�矄ᆠᨨ鼀� �ᨨ ᨨ ᆠᆠ�ᨨ矄�矄힘힘힘힘힘힘힘꾀 桌 꾀 ᨨ�ᆠ ᆠ 矄҈�鼀ᨨ �矄ᆠ � � ᨨ盐⼄盐 ��矄�ᨨ 矄� ᆠ ᆠ鼀␔ �鼀矄盐� � ᨨᆠ � ᨨ ᆠᨨ 矄 矄�矄├䰘�ᆠ鼀鼀├ᨨ 矄ᆠ ᨨᆠ� �矄├�ᆠ鼀鼀├ �鼀�ᨨ ᨨᆠ � ᨨ ᨨ 矄� ᨨ 矄 ᨨ矄 犨� �鼀�� �矄├҈�ᆠ鼀鼀├ᔼ �矄 �鼀 ᆠ �ᨨ� 鼀矄盐鼀 ᨨ ᨨᆠ � ᨨ 矄�矄힘힘힘 꾀 ҈黨桌 꾀矄 犨�ᨨ �鼀�mm Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 맔҈҈҈N mᆠ�矄ᨨ ᨨ 鼀矄矄M � ᨨ 犨�犨� ᨨ �矄҈m꾀 �矄ᆠ犨� ҈ ��矄�ᨨ 矄 �矄ᆠ ᨨ ᨨ �矄犨� ᨨ 矄犨��矄 ᆠ犨�ᨨ犨 �ᆠ 鼀� 矄ᆠᨨ 矄ᆠᨨ盐 ҈ mm �矄ᆠ ҈ ��矄�ᨨ 矄 ᨨ � 矄ᆠ ᆠᨨ矄ᨨ ᨨ黨 ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠᔼ ��泐 ᨨᨨ �矄ᆠᆠᨨ� ᆠ� 矄犨� ᨨᨨ 鼀� 矄ᆠ ᔼ ᨨ ᨨ 矄犨��矄� ᆠ矄 泐҈kg �矄ᆠ ��矄�ᨨ 犨 �ᆠ ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ ��矄�ᨨ �矄힘힘힘힘힘힘힘힘힘힘 ├�㦨҈m s p 㦨 꾀 ҈桌 � 矄� ᨨ ᆠ盐�矄鼀矄盐 泐 �ᆠ�矄 ҈ ��矄�ᨨ 矄ᨨ �矄ᆠᆠᨨ� ᆠ� 犨� 黨 ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠ犨 ҈ ��矄�ᨨ ᨨ 鼀� 矄ᆠ�矄҈鼀鼀㡰 鼀� 矄ᆠ 矄҈鼀鼀 �矄 ��矄�ᨨ ᨨ �矄ᆠ ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ㝐矄矄ᨨ ᆠ盐�矄䰘� � 矄�泐ᨨ ᆠ盐� ᨨ�矄�� � �ᆠ �㜠㝐├矄桌 ��盐矄 ᔼ 和矄 �鼀 犨 �ᆠ �矄 盐矄 ᨨ � Ӹᨨ盐 �矄 㝐ᨨ 盐盐矄ᨨ 矄�ᆠ�泐 �鼀ᨨ �鼀 犨��矄 D㦨�ᆠ鼀 鼀矄盐 泐矄ᆠᨨ 矄ᆠᨨ盐 ҈ᆠ鼀㡰 � �矄L㦨҈҈ᆠ鼀 鼀矄盐 泐矄ᆠᨨ 矄
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P-9
Physical World, Units and Measurements
䰘
ᆠᨨ盐 ҈ᆠ鼀㡰꾀犨�� ᨨ �Nᆠ盐矄矄 矄�ᨨ ᨨ X 㦨 ҈�ᆠ鼀 鼀矄盐 泐 鼀�ᆠᨨ鼀 ᨨ 矄 ᆠᨨ盐 ҈ᆠ鼀㡰 � 鼀 �鼀盐鼀ᨨ矄矄�ᨨ� 盐矄ᨨ Ӹᨨ盐 �矄鼀ᨨ 盐盐矄和ᆠ犨�盐� 鼀矄ᆠ ᨨ鼀ᨨ 㜠㝐├矄桌 꾀矄ᆠ犨�盐� � �҈⼄盐 ��矄�ᨨ 矄 �ᆠ ᨨ � ҈鼀鼀�矄盐矄 ᨨ鼀矄盐 �鼀ᨨ 犨�ᨨ � �ᆠ鼀 鼀� 矄ᆠ � ��矄҈鼀鼀 䰘 ᆠ�ᆠ盐 ��矄�ᨨ ᆠᨨ� ᆠ� 矄犨� 鼀� 矄ᆠ � ᆠ盐 矄盐 ᆠᨨ 犨�� ᆠ鼀ᨨᨨ�矄�� � �ᆠ ��盐 盐矄 p = 6
㡰 䰘
MCQs with One or More than One Correct Answer
ᔼ �鼀 ᨨ 鼀� ᆠᆠ�ᨨ 盐ᨨ�㜠 �泐� ᨨ鼀盐盐矄 ᨨ �鼀�ᨨ ᨨ �ᨨ �ᆠ 鼀ᨨ�ᨨ �矄 = p
黨
�
㜠㝐├矄桌
䰘 - 㡰 盐矄ᨨ ��
鼀矄盐 ᨨ ᠀҈㡰鼀鼀 ҈᠀҈㡰鼀鼀矄ᆠ�泐 ᔼ �矄盐ᆠᆠ矄矄�鼀矄盐鼀 矄 �鼀�ᨨ �矄 ᨨ盐 ᨨ�矄�矄�䰘矄�矄 �黨矄 矄ᆠᨨ盐 ᨨ 犨ᆠ 盐矄 ᨨ 鼀矄盐鼀 ᨨ �鼀�ᨨ �矄҈矄 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌 㡰 ᨨ� 鼀矄盐鼀 ᨨ �矄҈奠 㡰 ᨨ� 鼀矄盐鼀 ᨨ �矄䰘�奠 ᆠ㡰 ᨨ� 鼀矄盐鼀 ᨨ �矄奠 㡰 ᨨ� 鼀� 盐ᨨ ��矄҈҈奠 ᔴᨨ 矄� �ᆠ�矄� 犨 �ᆠ ᆠ ҈ᆠ鼀ᨨ 鼀� 矄ᆠ�矄 �� � ᨨ⼄盐 ��矄�ᨨ 矄 矄ᆠ犨 �盐�犨� ҈ ��矄�ᨨ 矄ᨨ �矄ᆠ�ᆠ盐矄ᆠᔼ � ᆠ�矄� ��矄�ᨨ 矄ᨨ �矄ᆠᆠᨨ� ᆠ� 犨� ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠ � 矄ᆠ犨�盐�ᨨ ᆠᨨ鼀ᨨ�ᨨ ᨨ ᆠ�ᆠ盐矄ᆠ鼀ᨨ矄�泐犨ᨨ ��矄�ᨨ 矄ᨨ � 矄ᆠ 꾀 ҈�桌 㡰 ᔼ �ᆠ ᨨ 矄ᆠ犨�盐��矄犨�ᆠ 矄ᆠᨨ盐 ᨨ �ᆠ�矄 矄ᆠᨨ盐 ᨨ 矄ᆠ犨 �盐��矄҈鼀鼀 㡰 ᔼ �ᆠ ᨨ 矄ᆠ犨�盐��矄犨�ᆠ 矄ᆠᨨ盐 ᨨ �ᆠ�矄 矄ᆠᨨ盐 ᨨ 矄ᆠ犨 �盐��矄�鼀鼀 ᆠ㡰 ᔼ 矄ᆠᨨ盐 ᨨ � 矄ᆠᨨ 矄ᆠ犨�盐� �矄犨�ᆠ 矄ᆠᨨ盐 ᨨ �ᆠ�矄 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐��矄҈鼀鼀 㡰 ᔼ 矄ᆠᨨ盐 ᨨ � 矄ᆠᨨ 矄ᆠ犨�盐� �矄犨�ᆠ 矄ᆠᨨ盐 ᨨ �ᆠ�矄 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐��矄�鼀鼀 矄� � 矄矄�ᨨ 矄� q㦨lᨨ ᆠᆠ盐矄 盐矄 ᨨ 泐鼀矄盐� � ᆠᨨ矄ᨨ � � �矄q� � ᨨ黨 犨 � l�矄 ᆠ泐├ ᨨ犨 ᨨ � q�矄ᆠᨨ 矄 ᨨ盐矄ᨨ q꾀矄q� ᆠ矄矄 ᨨ鼀 꾀 ҈桌
㡰 矄ᨨ盐ᨨ� d鼀� 矄ᆠᨨ 矄 㡰 矄ᨨ盐ᨨ� d� ᆠ矄矄 ᆠ㡰 ᆠ�ᨨ ᨨ� d鼀� 矄ᆠᨨ 矄 㡰 ᆠ�ᨨ ᨨ� d ᆠ矄矄 �҈ 꾀矄盐 盐矄矄矄�鼀 盐盐鼀ᨨ ᆠ泐҈鼀 � ᨨ 鼀� g ᆠᆠ�ᨨ 盐ᨨ��泐 盐矄矄 矄ᨨ犨ᆠ 犨� 矄ᆠᨨ盐 ᨨ ҈矄ᆠ ᨨ �矄 ᆠᨨ 矄 矄ᆠᨨ 矄 ᨨᨨ矄ᆠ��ᨨ 矄 ᨨ �矄ᨨ矄�ᨨ 犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 ҈桌 㡰 쟼ᨨDT� 鼀矄盐� �T �鼀�ᨨ �矄�矄ᆠᨨ 矄 㡰 쟼ᨨDT� 鼀矄盐� �T �鼀�ᨨ �矄҈矄ᆠᨨ ᆠ㡰 ᆠ �ᨨ� 鼀� �ᨨ ᨨ g�矄�奠 㡰 ᆠ �ᨨ� 鼀� �ᨨ ᨨ g�矄�奠 8
Comprehension/Passage Based Questions
矄矄� ᔼ 鼀矄盐鼀 ᨨ矄� � ⼄盐 ��矄 ├ ᨨ犨 � �矄 ᨨ矄矄� ᨨ 鼀� ᨨ � 泐 ⼄盐 �泐 �矄�矄 ᨨ 泐 盐矄ᨨ 矄�矄 矄�ᨨ
盐 ᆠ� � 矄�ᨨ �矄ᨨ犨ᨨ ᨨ ᨨ 鼀ᆠᨨ 矄� �ᨨ z㦨x⏌yᔼ ᨨ矄� xy z DxDy Dz矄ᆠ�泐
҈
x ± Dx x æ Dx ö æ Dy ö z± Dz㦨 㦨 ç҈ ± ÷ ÷ ç҈ ± y ± Dy yè x øè y ø ҈
æ Dy ö
矄�矄 矄�ᨨ ᨨ ç ҈ ± ÷ ᨨ �矄ᨨ犨� Dy⏌y�矄 y ø è ҈ m Dy⏌y㡰 �ᨨ矄� � �矄 犨泐矄 和ᨨ ᨨ� z犨� æ Dx Dy ö + Dz㦨 z ç ÷ y ø è x
ᨨ ��ᨨ 鼀├矄 矄矄盐鼀�ᨨ Dx⏌x = ҈ Dy⏌y = ҈ ᨨ �� ᨨ犨矄ᨨ 矄⼄盐 ��矄
�ᆠ 꾀 ҈桌 ҈ a 㡰 � ᔴᨨ 矄� �ᨨ r = ᨨ 鼀� 泐 ҈ + a 㡰 鼀矄盐� � �鼀 矄�ᨨ 矄矄⼄盐 �泐aᔼ ᨨ� 鼀矄盐鼀 ᨨ a�矄Da Da⏌a = ҈ 犨 �矄 ᨨ Dr� 鼀� � �r䁸 Da D a D a a D a 㡰 㡰 ᆠ㡰 㡰 ҈ + a㡰 ҈ + a 㡰 ҈ a 㡰 ҈ a 㡰 � ᔼ �鼀 � �� 盐鼀ᨨ �ᨨᆠ� 盐ᆠ��矄 ᔼ�矄 ᨨ盐 ҈᠀ 盐ᆠ� ᆠ泐 � �矄 ҈矄 ᨨx = ҈ ҈졔x㡰㦨x盐ᨨ �矄ᨨ犨� x ᨨDl� 鼀� �ᨨ ᨨ ᆠ泐ᆠᨨ 矄 l�
s҈�矄 㡰 㡰 ᆠ㡰 㡰 ҈
10 Subjective Problems � ᔼ n ��矄�ᨨ ᨨ 鼀� 矄ᆠᆠᨨ� ᆠ� 矄犨� n졔҈㡰 ��矄�ᨨ 矄 ᨨ �矄ᆠ� ᨨ 鼀� 矄ᆠ ��矄�ᨨ �矄⼄盐ᨨ �a盐 �矄 � 矄ᆠᨨ盐 ᨨ � 㜠㝐├矄桌
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EBD_8341 P-10
Physics
Topic-4 : Miscellaneous (Mixed Concepts) Problems 8
Comprehension/Passage Based Questions
矄矄� 꾀 矄ᆠᨨᆠ�ᨨ ᨨ ⼄盐 盐鼀ᨨ ᆠᨨ 矄 ᨨ矄�� �ᨨ 矄�矄ᆠ 盐矄鼀ᔴ� 矄ᨨ� 矄ᆠᨨ � � � � ᨨ矄�� �ᨨ 矄 矄盐ᨨ盐 泐 ᆠᨨ 矄 ᆠ 矄
盐矄鼀� 盐鼀 矄�泐ᨨ ᆠᨨ 矄 ᆠ ᨨ 鼀矄矄�鼀␔ ᆠᨨ 矄矄盐ᆠ ᨨ ᆠ�ᆠ � 泐 �矄ᆠ �泐犨泐 ᨨ鼀 泐ᨨ矄�� �ᨨ 矄ᔼ ᆠ�ᆠ � ᆠᨨ鼀矄 ᨨ ᆠᨨ 矄�� ᨨ ᨨ矄ᆠ�ᨨ盐 ᨨ矄���ᨨ 矄犨� 盐 �盐 ⼄盐 ᆠ泐 �w犨 �ᆠ �矄 ᆠ 矄鼀 ⼄盐 ᆠ泐 ᨨ 矄盐矄� ᨨ矄ᆠ��ᨨ 矄�鼀泐� �ᆠ�ᆠ � 矄ᨨ� 矄 �盐 ⼄盐 ᆠ泐 w 犨 ᨨ �泐 �矄 矄ᨨ ᨨ ��矄 ᆠ 꾀矄w ᨨᆠ 矄w
ᆠᨨ 矄矄ᨨ矄ᨨ ᆠᨨ� �泐�矄 ᆠ �矄�矄 �ᨨ ᨨ �� ᆠ��泐ᨨ 鼀矄 ҈҈桌 ҈
├� � ᆠᨨ �ᆠᆠ �矄� 鼀���泐矄 �e矄 �鼀 矄�ᨨ 泐矄�矄ᨨ 鼀� ᆠᨨᆠ 矄矄�ᨨ ᨨw 㡰
Ne 㡰 me
me Ne
ᆠ㡰
Ne me
㡰
Ne me
쟼矄�鼀 犨 � 犨 �ᆠ 矄鼀 ᆠ�ᨨ 犨� ᨨᆠᆠ盐 ᨨ 鼀 � � 矄�泐 ᨨ ᆠᨨ 矄 » ´ ҈䰘鼀 ├� �e㦨҈҈҈ 鼀矄矄鼀» ҈ 犨 矄⼄盐 ��矄� ᨨ和ᔼ盐 �矄 㡰 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀
Answer Key Topic-1 : Unit of Physical Quantities ҈ 㡰
㡰
ᆠ 㡰 Topic-2 : Dimensions of Physical Quantities
҈ ҈҈ ҈ ҈ 䰘
㡰 㡰 㡰 ҈ 㡰 㡰 㡰 ҈ 㝐 꾀 桌 㡰 ᆠ㡰
҈ 黨
ᆠ㡰 㡰 㡰 ҈ ᆠ㡰 ᆠ㡰 㡰 ҈ 㝐 꾀 桌 ᆠ 㡰 ᆠ㡰
� ҈� � ҈
㡰 ᆠ㡰 ᨨ 㡰҈ 㡰 㡰 㡰 㝐 턼 桌 ᆠ㡰
䰘 ҈䰘 䰘
㡰 㡰 ҈ ᆠ㡰 ҈ 㝐 桌 㡰
㡰 㡰 㡰 ᆠ㡰
黨 ҈黨 黨 � �
㡰 ҈ 㡰 ᆠ㡰 㡰 ᆠ㡰
㡰 ᆠ㡰 㡰 ᆠ㡰 㡰
Topic-3 : Errors in Measurements & Experimental Physics ҈ ҈
㡰 㡰 㡰 㡰 㡰
҈҈ ҈ ҈ ҈
㡰 㡰 ᆠ㡰 ᆠ㡰 ҈�㡰
䰘 ᆠ鼀㡰
҈
㡰 㡰 㡰 㡰 ҈㡰
҈
㡰 ᆠ㡰 㡰 㡰 ҈黨㡰
� ҈
ᆠ㡰 ᆠ㡰 㡰 㡰 㡰
㡰 黨 ᆠ㡰 � 㡰
҈� � � �
㡰 䰘 ᨨ 盐矄㡰 㡰 ҈ 㡰 ҈䰘 㡰 㡰 䰘 㡰 ᆠ㡰 䰘 �ᆠ鼀㡰
�҈ ᆠ㡰 � 㡰
ᆠ㡰 黨 ᨨ 盐矄㡰 㡰 ҈ ᆠ㡰 ҈黨 㡰 㡰 㡰 黨 㡰 㡰 㡰 黨 㡰 ҈黨맔҈҈⏌鼀㡰 æ ö � ᆠ㡰 � ç ÷ è +҈ø
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈ ᆠ㡰
㡰
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P-11
Motion in a Straight Line
Motion in a Straight Line
2
Topic-1 : Distance, Displacement & Uniform Motion 1 ҈
꾀�ᆠ�矄鼀ᨨ� �犨� 矄 v = b x ᨨ �ᨨ矄�� x㜠 �矄ᔴᆠ盐 矄 ᨨ �ᆠ�鼀t㦨 t 矄矄盐鼀 �ᆠ�矄ᨨ��� 㦨㡰㝐� ҈꾀҈黨ᔼᔼ桌 㡰
b t
㡰
ᆠ㡰
b t
㡰
b t
㡰
�
b t
㡰
ᆠ㡰
㡰
�鼀
꾀�ᆠ�矄鼀ᨨ� �矄犨 矄犨� ᨨᆠ�泐ᨨ �鼀⏌矄ᔼ
҈矄 ᨨᆠ�泐ᆠ �矄ᨨ�鼀⏌矄 ᨨ 犨 矄 � ᆠᆠ�ᨨ � �矄�鼀�矄⏌ ҈黨㜠㝐├矄桌
㡰 ҈⏌Ö鼀⏌矄ᨨ犨 矄 ᨨ 㜠犨矄
�鼀
ᆠ㡰 ҈⏌ 鼀⏌矄ᨨ犨 矄 ᨨ 㜠矄
ᨨᆠ�泐
ᨨ矄��ᨨ
㡰
ᔼ ҈矄�ᆠ�ᨨ矄 ᨨ鼀ᨨ� Aᨨᨨ� B鼀ᨨ� ��
矄鼀�ᆠ�ᆠᨨ �盐矄҈鼀 矄 ��盐㡰 鼀� �盐 ᨨ �ᨨᆠ�泐 A ҈黨黨黨和㜠㝐├矄桌
҈ m ⏌ s ᨨ犨 矄 ᨨ
10 Subjective Problems
㡰 ҈鼀⏌矄
҈ m ⏌ s ᨨ犨 矄 ᨨ 㜠犨矄
�鼀
㡰
MCQs with One or More than One Correct Answer
㡰 ᨨ
�矄 ᆠ
ᨨ矄��ᨨ
꾀�ᆠ鼀ᨨ矄� ᆠ�ᆠᨨ �盐矄Rᔼ �ᨨ ᨨ ᨨ盐�ᨨ �矄 �矄ᆠ鼀 �矄힘힘힘힘힘힘힘힘힘힘힘힘 �矄 ᆠ ᆠᨨ �矄힘힘힘힘힘힘힘힘 ҈黨㜠㝐├矄桌 6
꾀 � 矄ᨨ犨� ᨨ矄 矄鼀 鼀ᨨ�ᨨ 졠 ᨨ 鼀 ᨨ矄�� ᆠᨨᆠ泐�ᆠ├�盐 㝐� ҈桌 ᨨᆠ�泐
Fill in the Blanks
4
MCQs with One Correct Answer
꾀 矄犨 ᨨᨨ犨� ���� � 矄ᨨ 矄� �
㡰 鼀⏌矄
1.0m
ᔼ矄 �鼀 ��ᨨ ᨨ ᨨ矄��ᨨ 矄 ᨨ犨 � ��盐
ᆠ㡰 ҈鼀⏌矄
ᨨ矄 � 盐䁸҈黨䰘黨桌 㡰 ᨨ B
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EBD_8341 P-12
Physics
Topic-2 : Non-uniform Motion
҈
�
MCQs with One Correct Answer
ᨨᆠ�泐 v㡰 �鼀 t㡰� ᨨ ᨨ 泐� 矄�� � 鼀ᨨ�ᨨ �矄矄 ᨨ犨 � ��盐 ᨨ� S�矄 矄ᆠᨨ 矄 ᨨ �矄 ᆠᆠᨨ 泐 ᨨ 泐� 矄�矄 㝐� �和 ᔼᔼ㡰桌 v 鼀⏌矄㡰
A
B S
D
t � 矄㡰
҈ � C
䰘 黨 鼀 㡰 ҈鼀 ᆠ㡰 ҈҈鼀 㡰 鼀 ҈ 꾀盐ᨨ 鼀矄矄� 矄 � ��矄 ᨨ ҈鼀矄 盐矄 ᨨ�矄矄 � �鼀盐 犨ᨨ �ᆠ├ 矄矄ᆠ鼀 ᔼ 犨ᨨ 矄鼀 矄�矄 ᆠᨨ �맔҈ 矄 ᨨ 盐 鼀�� � ᨨ鼀 ᨨ 矄� ᨨ 犨 �矄ᆠᨨ矄ᨨ 㝐� ҈꾀҈黨 ᔼᔼ㡰桌 㡰 ҈鼀矄҈ 㡰 䰘鼀矄҈ ᆠ㡰 鼀矄҈ 㡰 鼀矄҈
ᨨ矄��ᨨ ᨨ �ᆠ矄 盐 ᆠ�ᨨ ᨨ �鼀t�矄�� 泐 x t㡰㦨at졔btct 犨 ab cᆠᨨ 矄 矄␔ �ᆠ� 矄 ᨨᆠᆠ�ᨨ �矄ᨨᆠ�泐犨� 㝐� 黨꾀҈黨 ᔼᔼ㡰桌
㡰
b 㡰 a + c
b 㡰 a + c
㡰 䰘�鼀 �矄
㡰 ��鼀 ҈矄 ᆠ㡰 ҈҈�鼀 �矄
䰘
b b 㡰 a + c c 꾀�ᆠ矄矄 ᨨ鼀ᨨ��� 졠 ᨨ鼀矄 鼀ᨨ矄犨� 盐 � ᨨ鼀ᆠᆠ�ᨨ ᨨ � ᨨ矄��x㜠 �矄ᔼ � 泐 ��盐矄 ᆠᨨᆠ泐矄 矄 鼀ᨨ�ᨨ ⼄盐��泐 a㦨ᆠᆠ�ᨨ v㦨ᨨᆠ�泐x㦨 �矄ᆠ鼀 t㦨�鼀㡰 㝐� 꾀҈黨 ᔼᔼ㡰桌
҈�
졠
�
ᔴ ҈ ҈� �
�鼀� 矄㡰®
㡰
㝐
㡰 ᆠ㡰 㡰 ␔ �ᆠ � ᆠᨨ矄ᨨ 矄ᨨ ᨨᆠ 鼀ᨨ� � 犨� ᆠᨨ 矄 ��ᆠᆠ�ᨨ ᨨ矄��ᨨᆠ�泐䁸 㝐� 졠 � 꾀�҈䰘桌
㡰
㡰
㡰
㡰 ᨨᆠ�泐
㡰 㡰 ᔴ㡰 ᆠ㡰 꾀㡰 㡰 ᔴ㡰
和ᆠᨨᨨ
쟼
ᔴ㡰
꾀鼀 � ᆠᨨᆠ�ᨨ 턼ᨨ 矄�� �� 犨泐�矄鼀ᨨ� � 犨� 矄 ᆠ� 矄ᨨᆠ ᨨ� � � �矄 ᆠ ᨨ鼀 �� 犨泐 ᨨ� 㝐㡰矄矄 ᨨ犨 � ��盐 和 ᨨ ᆠ� � �矄 ᨨ ᨨ �� 犨泐 ␔ 矄 ᨨ盐 �矄 ᆠ 㝐矄ᨨ �鼀├ ᨨ ᆠ �矄鼀� �鼀盐鼀䁸㝐� 졠 � 꾀�҈�҈桌
턼
꾀㡰
꾀 ᔴ
�
㡰 ҈҈�鼀 ҈�矄
ᆠ㡰 a +
ᔼ ᆠᆠᨨ 矄�� ᨨ ᆠ꾀├矄�鼀矄矄 ᆠ � �矄 矄矄矄 � �矄 � �ᨨ� 犨� 矄 ἘvἘ鼀ᨨ ᨨ ᆠᨨ ᆠ矄矄 ᨨ鼀 矄 犨� ᆠᨨ 矄 ᆠᆠ�ᨨ ҈ 矄ᆠ�泐 ἘvἘ �矄 ⼄盐 ᨨ 㝐� 黨 ҈黨 ᔼᔼ㡰桌 ҈ 㡰 㡰 ҈ ҈ + ҈ + ҈ ᆠ㡰 㡰
ᨨᆠ�泐㜠�鼀� 矄ᨨ ᆠ 矄ᆠᨨᨨ矄 ᨨ犨
� ��盐 �㡰 � ᆠ 犨 �矄 ᆠ 泐 ᆠ 矄ᆠᨨᨨ� ҈�矄 ��㡰 �鼀 犨 �ᆠ ᆠ 犨� ᆠᆠ 盐 犨� 矄ᆠᨨᨨ 矄ᆠ�泐 㝐� 졠 � 꾀�҈�҈桌
ᨨᆠ�泐 鼀矄҈㡰®
1
㡰 꾀㡰 㡰 꾀㡰 㡰 㡰
ᨨᆠ�泐
�鼀
�鼀
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P-13
Motion in a Straight Line
ᆠ㡰
㡰
2 ᨨᆠ�泐
ᨨᆠ�泐
�矄 ᆠ
�矄 ᆠ
黨
�矄 ᆠ 泐ᨨ 泐鼀ᨨ� �ᨨ �� �
�鼀t�矄ᨨᨨ�ᨨ ᨨt
ᆠᆠ�ᨨ 㜠�鼀 at㡰� ᨨ 鼀ᨨ�ᨨ ᨨ ᨨ 泐犨� 㝐� 졠 � 㝐y҈҈桌
Integer Value Answer
҈ 꾀ᨨᆠ├�矄鼀ᨨ� �� ��泐 矄ᆠ犨� ᆠᨨ 矄 ᆠᆠ�ᨨ ᨨ 鼀⏌矄ᨨ �졔 �ᆠ�ᨨ 矄 ��盐㡰 � ᨨ ᆠ 鼀� 矄� ᨨᆠ├�矄鼀A�矄 ᨨ犨 ᨨ鼀 ᨨ ᆠ 鼀� 졔x �ᆠ�ᨨ
犨� 矄 ᨨ 鼀⏌矄�ᨨ ᨨᆠ├꾀 矄鼀 �鼀 ᨨ �矄 ᨨ犨 � x �ᆠ�ᨨ 犨� 矄 ᨨ 鼀⏌矄 ᨨ鼀�矄�� �ᨨ ᨨᆠ├ �鼀�
矄ᆠᨨ 矄犨 犨ᨨ矄 �ᆠ ᨨ �矄 꾀 ҈桌 鼀⏌矄 㦨鼀⏌矄
鼀⏌矄 a
a
㡰
鼀
㡰 3 t
t
a ᆠ㡰
a
Numeric Answer
҈ 矄 矄矄�鼀� ᨨ�ᆠ�矄矄 ᨨ犨 � ��盐 �矄 ᆠ � 鼀㡰泐 �ᆠ 盐� � �鼀� t㦨ᨨt㦨�矄犨�힘힘힘힘힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌
㡰
҈ u ҈ 鼀矄 㡰
t t ҈ ᨨᆠ�泐㜠 �矄ᆠ鼀 � ᨨ �ᆠ鼀ᨨ� � ᨨ �矄�� � �矄矄 ᨨ犨
�和桌
҈ � �鼀 s㡰
鼀ᨨ矄矄盐�ᆠᆠ�ᨨ 㜠 �矄ᆠ鼀 � 犨�
҈ �矄 ᆠ ᆠᨨ 泐�ᆠ� ᨨ �鼀 矄�ᨨ 鼀ᨨ�ᨨ �矄犨� �鼀t矄 x㦨 at졔 bt졔 cᔼ ᆠᆠ�ᨨ ᨨ �ᆠ 矄ᨨ x矄xn犨 n �矄 � � 盐ᨨ n�矄힘힘힘힘힘힘 㝐� 黨 ᔼ㡰桌 6
҈҈
㡰
㡰
ᆠ㡰
㡰
҈� 꾀�ᆠᨨ 鼀矄矄m鼀ᨨ矄ᨨ x㜠 �矄矄 ᨨᨨ犨矄� 矄矄 ᨨ鼀矄t㦨 ᨨ鼀 ᨨ� x㦨 ᆠᨨ鼀矄ᨨ 矄t㦨҈ ᨨ� x㦨҈졠ᨨ � ᨨ鼀�ᨨ �矄 �ᨨ盐�矄鼀ᨨ�ᨨ � 鼀 ��鼀矄 ҈㡰ᔼ a ᨨ矄 � 矄 ᨨ盐矄ᆠᆠ�ᨨ ᨨ �ᆠ ҈黨黨㜠㝐├矄桌 㡰 a ᆠ
ᨨ 鼀� ᨨ矄�� ᨨ t � �
£ t £ ҈
꾀ᨨ 泐矄矄 ᨨ鼀矄�鼀t㦨 ᆠᆠ�ᨨ �鼀 � �矄 矄 ᨨ犨 � ��盐 鼀 �鼀盐鼀 ᨨᆠ�泐 � 泐 ᨨ 泐犨� 和桌 㡰 ҈҈鼀⏌矄
꾀ᆠᆠ�ᨨ
鼀⏌矄 㡰 ҈
㡰 ��鼀⏌矄 ᆠ㡰 �鼀⏌矄 㡰 ��鼀⏌矄
MCQs with One or More than One Correct Answer
҈҈ �鼀 矄ᆠ㡰
㡰 aᆠ
ᨨ ᆠ 泐ᨨ� � �矄 ᆠ㡰 a| 鼀盐矄 ³ 矄ᨨ鼀ᨨ� ᨨᨨ� 矄� �矄 㡰 a鼀盐矄ᆠ �矄�� 盐� � 鼀ᨨ�ᨨ 盐 ᨨᨨ 矄矄�ᨨ ᆠ 鼀 犨� � ᨨ鼀�ᨨ �� 10 Subjective Problems ҈ 꾀ᆠᆠᆠ矄 ᨨ鼀矄ᆠᨨ 矄 a ᨨ矄ᨨ鼀 �鼀 犨 �ᆠ � ᆠ矄ᆠᨨ 矄 bᨨᆠᨨ鼀 ᨨ矄ᔼ ᨨ�鼀矄�矄t矄ᆠᨨ 矄盐 ҈黨䰘桌 �㡰 鼀 �鼀盐鼀ᨨᆠ�泐ᆠ ��㡰 ᨨ �矄 ᆠ
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EBD_8341 P-14
Physics
Topic-3 : Relative Velocity in One Dimension 1 ҈
�
MCQs with One Correct Answer
� A � B盐
� �ᨨ ᆠ├矄� ᨨᨨ矄� �ᆠ�ᨨ 矄犨� 矄 矄ᨨ ├鼀⏌ ᨨ盐 䰘├鼀⏌ ᨨ盐矄ᆠ�泐꾀矄ᨨ �矄犨├� �� � A� �ᆠ�ᨨ ᨨᨨ矄�ᨨ�矄鼀ᨨ�ᨨ 犨� 矄 ᨨ ҈├鼀⏌ ᨨ盐和 � 鼀矄҈㡰ᨨ �矄矄ᨨ 矄ᨨ矄 ᨨ鼀�
B犨�ᆠᨨ矄ᨨ ├ �矄 ᆠ犨 ᆠ├矄矄
����㡰 [Main 2 Sep. 2020 (I)] ҈ ҈ 㡰 黨�鼀矄 㡰 �鼀矄 ᆠ㡰 ҈�鼀矄҈⼄ 㡰 �鼀矄҈ 꾀矄ᨨ 矄 � �ᨨ ᨨ �ᨨ盐 矄 矄ᨨ盐 ᨨ ᨨ ᆠᨨ鼀� � ᨨ鼀 ᨨ �º犨� �ᨨ盐 盐 � 矄 ᨨ �� �ᆠ泐 ᨨ �矄ᨨ矄��ᨨ ᔼ v�矄 矄 ᨨ 矄ᨨ盐 矄 ᨨ �矄 㝐� ҈ ҈黨 ᔼᔼ㡰桌
v v 㡰 v 㡰 ᆠ㡰 v 㡰 꾀ᆠ�矄矄 � �鼀 � 盐矄犨 �ᆠ �矄矄ᨨ矄
犨ᨨ矄鼀ᨨ� � 矄鼀� 矄 盐犨� � 㜠 ᨨ犨 ᆠᆠ�ᨨ 矄 盐矄 矄ᆠᆠ�ᨨ 鼀⏌矄 ᆠ 矄ᆠᆠ�ᨨ 鼀⏌矄 ᆠ犨�ᆠᆠ 盐 犨� 盐矄 �鼀ᨨ 㝐� 졠 � 꾀�黨҈䰘桌 㡰 ҈҈ 矄 㡰 ҈ 矄 ᆠ㡰 ҈ 矄 㡰 ҈�矄 꾀矄ᨨ ᆠ�鼀矄盐矄 矄ᆠᨨ� 矄ᔼ 矄 � � ᨨ 矄鼀盐矄ᆠᨨ盐
� �犨� ᆠᨨ 矄 ᨨᆠ�泐 ├矄矄ᨨ犨鼀盐ᆠ �鼀�矄├ 泐 矄ᨨ ᨨ 犨├盐 鼀ᨨ� �矄ᆠᨨ䁸 㝐� 졠 � 꾀�҈҈桌 㡰 䰘矄 㡰 䰘矄 ᆠ㡰 矄 㡰 �矄
2 Integer Value Answer 꾀� 矄A B 泐� �犨� ᆠᨨ 矄 ᨨᆠ�泐� 矄鼀�ᆠ �矄 犨� 矄ᆠᨨ ᨨ� ᨨ 矄ᆠ�泐矄矄 ᨨ犨 � ��盐 矄 ᨨ A�矄 ҈ 鼀⏌矄꾀�鼀t㦨矄 ᨨ矄� A � 矄 B �矄 ᆠᨨ �鼀 ᨨ矄矄矄B鼀ᨨ� �犨� ᆠᨨ 矄 ᨨᆠ�泐 �ᆠ盐ᨨ � ᨨ 鼀ᨨ�ᨨ ᨨ Aᔼ t㦨t꾀盐矄矄ᆠ矄� � �泐Bt� 矄ᆠᨨ 矄 �矄 꾀 ҈桌 A
B
䰘
黨
4 Fill in the Blanks ᨨ盐矄ᨨ 矄KLMN� ��泐 ᨨ盐ᆠᨨ 矄ᨨ 矄⼄盐ᨨ 矄� d쟼ᆠ 矄ᨨ ᨨ犨鼀ᨨ矄犨� 盐 � ᨨ鼀 矄 v� 矄盐ᆠ 犨泐 ⧤犨泐矄鼀ᨨ矄 �ᆠ泐ᨨ犨 矄 L L �ᆠ泐ᨨ犨 矄 M M �ᆠ泐ᨨ犨 矄N N �ᆠ泐ᨨ犨 矄K ᨨ盐矄ᨨ 矄犨�鼀�鼀 ҈黨㜠㝐├矄桌 5 True / False
犨ᨨ� �ᆠ� 矄鼀ᨨ� �ᨨ �矄ᨨ � ⼄盐ᨨ ᨨ � ᨨᨨ矄� �ᆠ�ᨨ 矄犨� 矄鼀矄
泐犨� 矄鼀矄矄盐ᨨ �矄 ҈黨�㜠㝐├矄桌 9 Assertion and Reason Type Questions 和T꾀TE㝐ENT㜠҈ ᨨ ᨨ矄ᨨᨨ├� �ᨨ盐 ᨨ盐� 犨� ᨨ犨ᨨ 矄鼀ᨨ� �� 泐ᨨᆠ矄 ᨨ鼀ᨨ� ᨨᨨ矄� �ᆠ�ᨨ ᨨ � 犨 � �矄 ᨨᆠ矄ᨨ矄�ᨨ 泐 和T꾀TE㝐ENT㜠ᔼ ᨨ矄 ᨨᆠ鼀ᨨ� � r r ᨨᆠ��矄 v ҈ v 矄ᆠ�泐犨� ᆠᨨ ᨨᨨ泐 鼀 ᨨᆠ�泐ᨨ ᨨᆠ犨� 矄ᆠᨨ r r 桌 ᨨ矄�矄 v v ҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 ҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐 10 Subjective Problems 졠 �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ矄矄盐鼀 ᨨ x㜠 y 矄鼀ᨨ泐A�矄鼀ᨨ� �ᨨ �矄�� � ᨨ y㜠 �矄 矄 ��盐㡰犨� ᆠᨨ 矄 ᨨᆠ�泐 ᨨ - ҈㡰 鼀⏌矄꾀 꾀�ᆠ盐� 矄 犨 � OA 鼀├矄 �ᨨ � 犨� x㜠 �矄�矄 ᨨ犨
ᨨ � 矄盐 ᆠ ᨨ鼀 ᨨ��� Oᔼ矄ᨨᆠ�泐鼀├矄
�f犨� x㜠 �矄 � �矄 ᨨ泐 泐 㡰 鼀ᨨ�ᨨ ᨨ �矄 꾀 ᨨ矄 ᨨ鼀 鼀ᨨ ᨨ泐 ᔴᆠ盐 � q 鼀 泐 ᨨᆠ�泐ᆠᨨᨨ 犨� x㜠 �矄� �矄 鼀 � 㡰 � 矄 ᨨ 졠 犨� 矄ᆠᨨ 矄盐 ᆠ � f㦨q⏌ 㜠�㝐├矄桌
Topic-4 : Motion Under Gravity 1 ҈
MCQs with One Correct Answer
꾀 �ᆠᨨ�矄矄 ᨨ鼀矄ᨨ �ᨨ盐 �ᆠ泐盐㜠 犨 矄犨� ᆠᨨ 矄 ᆠᆠ�ᨨ g 꾀 ᨨᨨ ᆠ├ �矄
ᨨ ᨨ鼀 �ᆠᨨ犨 ��矄 �� h �鼀├ 泐 ᆠ├ᨨᆠ �ᨨ盐 �矄ᆠᨨ矄ᨨg �矄 ᆠᆠ�ᨨ 盐ᨨ��泐桌 㝐� �和 ᔼ㡰桌
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P-15
Motion in a Straight Line
(a) t =
2.
2 æhö ç ÷ 3 ègø
(The figures are schematic and not drawn to scale)
h g
(b) t = 1.8
240
æhö 2h (c) t = 3.4 ç ÷ (d) t = 3g ègø A Tennis ball is released from a height h and after freely falling on a wooden floor it rebounds and reaches height
(a)
h . The velocity versus height of the ball during its motion 2
(c)
may be represented graphically by : (graph are drawn schematically and on not to scale) [Main 4 Sep. 2020 (I)] h/2
h(v)
(b)
(c)
h h/2
h(v)
(d)
h h/2
h(v)
A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time? [Main 2017]
3.
(b)
(a)
240
8 (y2 – y1) m
12
t(s)
(d) t® 8
6.
v
v
t(s)
12
(y2 – y1 ) m
t(s)
12
12
t(s)
From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle, to hit the ground, is n times that taken by it to reach the highest point of its path. The relation between H, u and n is: [Main 2014] (a) 2gH = n2u2 (b) gH = (n – 2)2 u2 d (c) 2gH = nu2 (n – 2) (d) gH = (n – 2)u2 A ball is dropped vertically from a height d above the ground. It hits the ground and bounces up vertically to a height d/2. Neglecting subsequent motion and air resistance, its velocity v varies with the height h above the ground as [2000S]
5.
h(v)
h
h
(y2 – y1 ) m
(b)
240
h/2
(a)
240
8
v
v
(y2 – y1) m
(a)
(b)
(c)
(d)
3 Numeric Answer A ball is dropped from the top of a 100 m high tower on a
7.
1 s before hitting the ground, it covers a 2 distance of 19 m. Acceleration due to gravity (in ms–2) near the surface on that planet is _______. [Main 8 Jan. 2020 (II)]
planet. In the last
(c)
(d)
Two stones are thrown up simultaneously from the edge of a cliff 240 m high with initial speed of 10 m/s and 40 m/s respectively. Which of the following graph best represents the time variation of relative position of the second stone with respect to the first ? (Assume stones do not rebound after hitting the ground and neglect air resistance, take g = 10 m/ s2) [Main 2015]
4.
5 True / False Two balls of different masses are thrown vertically upwards with the same speed. They pass through the point of projection in their downward motion with the same speed (Neglect air resistance). [1983 - 2 Marks]
8.
Answer Key Topic-1 : Distance, Displacement & Uniform Motion 4. (2R, pR) 5. (b) Topic-2 : Non-uniform Motion 3. (b) 4. (d) 5. (c) 6. (c) 7. (a) 13. (20) 14. (3) 15. (a, c, d) Topic-3 : Relative Velocity in one Dimension
1.
(b)
2. (b)
3. (b)
1. 11.
(a) (b)
2. (b) 12. (8)
1.
(a)
2. (d)
3. (c)
1.
(c)
2. (c)
3. (a)
8. (c)
9. (b)
10. (a)
ædö ç v ÷ 7. (False) 8. (b) è ø Topic-4 : Motion Under Gravity 4. (b) 5. (c) 6. (a) 7. (8.00) 8. (True) 4. (c)
5. (5)
6.
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EBD_8341 P-16
Physics
3
Motion in a Plane
Topic-1 : Vectors MCQs with One Correct Answer uur uuur uur uuur 꾀҈ 㦨 꾀 㦨� 꾀҈ + 꾀 㦨� 盐ᨨ uur uuur uur uuur 꾀҈ + 꾀 · 꾀҈ - 꾀 �矄㝐� 꾀�҈黨 ᔼᔼ㡰桌 㡰 ҈� 㡰 黨黨� ᆠ㡰 ҈҈� 㡰҈҈� ᔼ ᆠ盐ᨨ 矄� � 矄 ᨨ犨 � ��盐 ᆠᨨ ᨨ鼀 ᆠ ᨨ� ᨨ ᆠ꾀졠ᨨ ᆠ ᨨ� ᨨ ᆠ쟼 졠犨�㝐� ҈ ҈黨 ᔼ㡰桌 1
҈
(
)(
)
� ҈ 黨 㡰 ᆠ㡰 㡰 黨 黨 黨 ҈ r r r
ᆠᨨ矄 턼 矄 ᨨ犨 � ��盐和 r 泐ᨨ� ᨨ ᆠᨨ �矄 ᆠ犨 r ᨨ� 矄 和�矄 � �ᨨ 鼀ᨨ �ᆠᨨ矄 r r r 꾀 ҈䰘桌 턼 和 �矄
㡰 �
Ӹ
和
졠
r r r 㡰 和 = ҈ - 㡰 + 턼 r r r ᆠ㡰 和 = ҈ - 㡰 + 턼
(
)
(
)
( (
) )
҈ ҈ ˆ ˆ � -k kˆ - �ˆ 㡰 ҈ ҈ ˆ - kˆ ˆ - �ˆ ᆠ㡰 㡰 ur ur
犨ᨨ ᆠᨨ矄 꾀 ur ur ⼄盐 鼀� �盐 矄 鼀� �盐 ᨨ 꾀 + �矄 � �鼀矄 鼀� �盐 ᨨ ur ur ur ur 꾀 - � 犨 꾀 �矄 㝐� ҈ ҈黨 ᔼᔼ㡰桌
㡰
(
(
)
3
䰘
和 턼
= 턼- 턼
r r r 㡰 和 = - ҈㡰 + 턼 r r r 㡰 和 = ҈ - 㡰 + 턼
Numeric Answer ®
)
é - ҈ù é - ҈ù 㡰 ᆠᨨ矄 -҈ ê ú 㡰 ᆠᨨ矄 -҈ ê ë + ҈úû ë + ҈û é -҈ù é -҈ù ᆠ㡰 矄� -҈ ê ú 㡰 矄� -҈ ê
+ ҈ ë + ҈úû ë û r r 꾀 = �ˆ + ˆ㡰 = �ˆ - ˆ㡰 鼀� �盐 ᨨ r r r r r r r ᆠᨨ ᆠᨨ ᔴ 矄盐ᆠ 꾀ᔴ = ᔴ = 꾀 �矄��
泐 㝐� 졠 � 꾀�҈҈桌
꾀 ᨨᆠ F = �$ + $ + k$ 㡰 ᆠ矄ᨨ� $� + $ - k$ 㡰 鼀
鼀� �盐 ᨨ ᨨ⼄盐ᨨ盐 ᨨ� $� + $ + k$ 㡰 鼀 犨� x 㜠鼀 盐ᨨ x�矄힘힘힘힘힘힘 㝐� 和� ᔼ㡰桌 r r r r r
矄盐鼀ᨨ 犨ᨨ ᨨᆠ矄 P Q �矄 R 矄盐ᆠ R 㦨 P r r
�q � �矄㡰 矄盐 ᨨ P Q r 犨�鼀├犨� Q �矄_______ 㝐� 䰘 ᔼᔼ桌 r r r r
犨ᨨᆠᨨ矄 A B � 矄 A 㦨 a�ˆ B 㦨 a ᆠᨨ矄 wt �ˆ 졔矄� wt ˆ 㡰犨 a�矄ᆠᨨ 矄 w㦨p⏌ r r r r s҈ᔼ A + B = A B �鼀t㦨t ᨨ �矄�鼀 盐ᨨ t� secᨨnds�矄힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌
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P-17
Motion in a Plane
Topic-2 : Motion in a Plane with Constant Acceleration 1 ҈
MCQs with One Correct Answer
꾀ᨨᨨ �矄鼀ᨨ� �盐� ��ᆠ泐ᨨᨨ� 꾀ᨨ
�ᨨ盐 ␔ ��矄 �� h҈��矄 � � �矄 ᆠd ᨨ� 㡰 ᨨ鼀꾀 矄 ��盐㡰矄矄� � �º犨� 矄ᆠᨨ �ᆠ␔ ᨨᨨ ᆠ�鼀矄盐 盐 �� h��矄矄 �º犨� 矄ᆠᨨ �ᆠ� ��鼀ᨨ矄 盐 泐 �矄 ᆠd ᨨ� ᔴ㡰 �� h�矄 �� º㦨�䰘䰘㡰 㝐� 和� ᔼ㡰桌
�°
°
꾀 ᔴ 㡰 ҈d 㡰 䰘d ᆠ㡰 d 㡰 d 和� � ᨨ鼀 ᨨ��� �鼀t㦨犨� � ��ᨨᆠ�泐 � ˆ 鼀矄҈�ᆠ鼀ᨨ矄� xy 犨� ᆠᨨ 矄 ᆠᆠ�ᨨ ᨨ ҈�ˆ + ˆ㡰 鼀矄꾀�鼀t�矄ᆠᨨᨨ � 矄 鼀y鼀㡰 盐矄ᨨ t y矄ᆠ�泐 [Main Sep.04, 2020 (I)] 㡰 矄 ҈鼀 㡰 矄 �鼀 ᆠ㡰 矄 鼀 㡰 �矄 �鼀 꾀�ᆠ矄矄 ᨨ鼀 ᨨ��� t㦨犨� � �� ᨨᆠ�泐ᨨ �ˆ 鼀⏌矄 鼀ᨨ矄� x㜠y 犨� ᆠᨨ 矄 ᆠᆠ�ᨨ �ˆ + ˆ㡰 鼀⏌ 矄 x㜠
ᆠᨨᨨ � ᨨ �ᆠ � 矄 犨 �矄 y㜠 ᆠᨨᨨ � �矄鼀�矄鼀矄 盐ᨨ �矄 㝐� 黨 ᔼᔼ㡰桌 㡰 㡰 � ᆠ㡰 㡰 r 꾀�ᆠ鼀ᨨ矄矄盐ᆠ �矄ᨨ矄��ᨨ ᆠᨨ r t㡰㦨ᆠᨨ矄 w �ˆ 졔矄� w ˆ 犨 w�矄ᆠᨨ 矄 t�矄�鼀
�
犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄盐 ᨨ ᨨᆠ�泐 r r v t㡰 ᆠᆠ�ᨨ a t㡰ᨨ �ᆠ 㝐� ᔼᔼ㡰桌 r r r 㡰 v �矄 �ᆠ盐ᨨ r a �矄 �ᆠ 犨泐 ᨨ鼀 ᨨ���
r r r 㡰 v a ᨨ �ᆠ盐ᨨ r r r r ᆠ㡰 v a ᨨ ᨨ r r r r 㡰 v �矄 �ᆠ盐ᨨ r a �矄 �ᆠ ᨨ犨 矄 ᨨ���
ᨨ矄��ᨨ ᆠᨨ ᨨ �ᆠ ᆠ �矄 犨� �鼀 r ᆠᆠᨨ � � ᨨ �ᨨ r 㡰 = ҈� $� + - 㡰 $
(
(
䰘
҈
␔ �矄 鼀� �盐 ᨨ ᆠᆠ�ᨨ t㦨҈䁸 㝐� 黨꾀�҈黨 ᔼᔼ㡰桌 㡰 㡰 � ᆠ㡰 ҈ 㡰 � ˆ ˆ 鼀 㦨 � + 꾀�ᆠ鼀ᨨ矄 ᨨ鼀 ᨨ� ҈ 犨� � ��ᨨᆠ�泐 ��ˆ + ˆ 鼀矄 ᔼ�矄ᆠ 盐ᨨ
泐ᆠᨨ 矄 ᨨᆠ犨 �ᆠ ᨨ 盐ᆠ矄ᆠᨨ 矄 ᆠᆠ�ᨨ
�ˆ + ˆ 鼀矄 - ␔ �矄 �矄 ᆠᨨ �ᆠ ᨨ鼀 ᨨ��� �鼀矄䁸㝐� ҈҈ ҈黨 ᔼᔼ㡰桌 㡰 ҈�鼀 㡰 鼀 ᆠ㡰 �鼀 㡰 ҈ 鼀 r 꾀�ᆠ�矄鼀ᨨ� �犨� ᨨᆠ�泐 v 㦨⧤ y �ˆ 졔x ˆ 㡰 犨 ⧤�矄ᆠᨨ 矄 � ⼄盐�ᨨ ᨨ�矄 �矄 㝐� 黨 ҈黨 ᔼ㡰桌 㡰 y㦨x졔ᆠᨨ 矄 㡰 y㦨x졔ᆠᨨ 矄 ᆠ㡰 y㦨x졔ᆠᨨ 矄 㡰 xy㦨ᆠᨨ 矄 Numeric Answer
꾀�ᆠ�矄鼀ᨨ� �ᨨ � x㜠ax�矄犨� �矄ᆠᨨᨨ � 犨� �鼀�t�� 泐x t㡰㦨҈졔tt꾀 ᨨ �ᆠ �矄鼀ᨨ� �ᨨ � y㜠 �矄犨� �矄ᆠᨨᨨ � 矄 盐 ᆠ�ᨨ
ᨨ �鼀�� 泐y t㡰㦨�t꾀t㦨҈矄 矄 ᨨ 矄ᆠᨨ �ᆠ矄鼀矄盐 � 鼀ᨨ �矄�ᆠ �矄�� 矄 v v � 鼀⏌矄㡰�矄힘힘힘힘 㝐� ᔼ㡰桌 6
黨
)
)
)
3
(
MCQs with One or More than One Correct Answer
和� ��鼀t 㦨 ᨨ鼀 ᨨ��� 犨� 矄 ҈鼀矄҈ �ᆠ ᨨᨨ犨矄犨ᨨ㜠 �鼀 矄�ᨨ ᆠᨨ泐� x㜠y 矄ᨨ �矄 ᆠᨨᨨ � 矄 泐
x x y ᆠᨨ鼀ᨨ 矄 ᨨ �矄 ᆠᆠ�ᨨ ᨨ 泐ax ay矄ᆠ�泐
꾀 桌 㡰 ax㦨҈鼀矄�鼀�矄 犨 �ᆠ�矄 ᨨ��� ay 㦨҈鼀矄 㡰 ax 㦨�鼀�矄ay 㦨҈鼀矄�鼀矄 ᆠ㡰 t 㦨 �ᆠ矄ᨨᆠ�泐ᨨ� 矄� x㜠 �ᆠ�ᨨ
㡰 ax㦨�鼀�矄 t 㦨҈矄 �犨 �ᆠ矄ᨨᆠ�泐 x �矄�矄� ҈ ᆠᨨᨨ � 矄ᨨ �ᆠ鼀ᨨ� �� ��
泐x t㡰㦨aᆠᨨ矄 㡰 y t㡰㦨b矄� 㡰犨 ab a㡰 pᨨ矄��ᆠᨨ 矄 矄ᨨ ᨨ� �鼀 矄�ᨨ 矄
҈黨黨黨和㜠㝐├矄桌 㡰 ᨨ �ᆠ�矄 �矄 㡰 ᨨᆠ�泐 ᆠᆠ�ᨨ ᨨ �ᆠ
ᨨ鼀ᨨᆠ ᨨ t㦨p⏌ p㡰 ᆠ㡰 ᆠᆠ�ᨨ ᨨ �ᆠ�矄 犨泐矄 �ᆠ ᨨ犨 矄 ᨨᆠ盐矄 㡰 �矄 ᆠ 泐 �ᆠ� �鼀� t㦨ᨨt㦨p⏌ p㡰�矄a ⼄盐�ᨨ y =
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EBD_8341 P-18
Physics
Topic-3 : Projectile Motion 1 1.
A particle of mass m is projected with a speed u from the p ground at an angle q = w.r.t. horizontal (x-axis). When 3 it has reached its maximum height, it collides completely inelastically with another particle of the same mass and velocity uiˆ. The horizontal distance covered by the combined mass before reaching the ground is: [Main 9 Jan. 2020 (II)] (a)
2.
3 3 u2 8 g
(b)
3 2 u2 4 g
u2 5 u2 2 2 (c) (d) 8 g g The trajectory of a projectile near the surface of the earth is given as y = 2x – 9x2. If it were launched at an angle q0 with speed v0 then (g = 10 ms–2): [Main 12 April 2019 (I)] 5 1 (a) q0 = sin –1 and v0 = ms–1 3 5 3 æ 2 ö ms–1 (b) q0 = cos–1 ç ÷ and v0 = 5 è 5ø
3.
4.
5.
9 æ 1 ö (c) q0 = cos–1 ç ÷ and v0 = ms–1 3 è 5ø 3 æ 2 ö (d) q0 = sin –1 ç ÷ and v0 = ms–1 5 è 5ø A shell is fired from a fixed artillery gun with an initial speed u such that it hits the target on the ground at a distance R from it. If t1 and t2 are the values of the time taken by it to hit the target in two possible ways, the product t1t2 is : [Main 12 April 2019 (I)] (a) R/4g (b) R/g (c) R/2g (d) 2R/g Two particles are projected from the same point with the same speed u such that they have the same range R, but different maximum heights, h1 and h2. Which of the following is correct ? [Main 12 April 2019 (II)] (a) R2 = 4 h1h2 (b) R2 =16 h1h2 (c) R2 = 2 h1h2 (d) R2 = h1h2 A plane is inclined at an angle a = 30o with respect to the horizontal. A particle is projected with a speed u = 2 ms–1, from the base of the plane, as shown in figure. The distance from the base, at which the particle hits the plane is close to: (Take g=10 ms–2) [Main 10 April 2019 (II)]
(a) 20 cm (b) 18 cm
(c) 26 cm
A body is projected at t = 0 with a velocity 10 ms–1 at an angle of 60° with the horizontal. The radius of curvature of its trajectory at t = 1s is R. Neglecting air resistance and taking acceleration due to gravity g = 10 ms–2, the value of R is: [Main 11 Jan. 2019 (I)] (a) 10.3 m (b) 2.8 m (c) 2.5 m (d) 5.1 m 7. Two guns A and B can fire bullets at speeds 1 km/s and 2 km/s respectively. From a point on a horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the bullets fired by the two guns, on the ground is: [Main 10 Jan. 2019 (I)] (a) 1 : 16 (b) 1 : 2 (c) 1 : 4 (d) 1 : 8 8. The position of a projectile launched from the origin at r t = 0 is given by r = 40iˆ + 50 ˆj m at t = 2s. If the projectile was launched at an angle q from the horizontal, then q is (take g = 10 ms–2) [Main Online April 9, 2014] 2 -1 -1 4 -1 3 -1 7 (a) tan (c) tan (d) tan (b) tan 3 5 2 4 9. A projectile is given an initial velocity of (iˆ + 2 ˆj ) m/s, where iˆ is along the ground and ˆj is along the vertical. If g = 10 m/s2 , the equation of its trajectory is [Main 2013] (a) y = x - 5 x 2 (b) y = 2 x - 5 x 2 (c) 4 y = 2 x - 5 x 2 (d) 4 y = 2 x - 25 x 2 10. A ball projected from ground at an angle of 45° just clears a wall in front. If point of projection is 4 m from the foot of wall and ball strikes the ground at a distance of 6 m on the other side of the wall, the height of the wall is : [Main Online April 22, 2013] (a) 4.4 m (b) 2.4 m (c) 3.6 m (d) 1.6 m 6.
MCQs with One Correct Answer
(d) 14 cm
(
2
)
Integer Value Answer
11. A ball is thrown from ground at angle q with horizontal and with an initial speed u0. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V1. After hitting the ground, the ball rebounds at the same angle q but with a reduced speed of u0/a. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V1, the value of a is ____ [Adv. 2019]
12. A train is moving along a straight line with a constant acceleration ‘a’. A boy standing in the train throws a ball forward with a speed of 10 m/s, at an angle of 60° to the horizontal. The boy has to move forward by 1.15 m inside the train to catch the ball back at the initial height. The acceleration of the train, in m/s2, is [Adv. 2011]
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P-19
Motion in a Plane
5 ҈
True / False
꾀ᨨᆠ� � ᨨ鼀 �ᨨ盐 ᨨᨨ犨矄ᨨ�ᆠ
矄 ᨨ ᨨᆠ��矄鼀� �鼀盐鼀 ᨨᨨ �矄 ҈黨㜠㝐├矄桌 10 Subjective Problems
҈
꾀 ᨨᆠA�矄├ � ᨨ� x㦨鼀 y㦨҈�鼀 ᨨ ├P�矄 ᨨ �ᨨ盐 꾀�鼀t㦨 ├ 矄矄鼀ᨨ� � ᨨ � 졔 x �ᆠ�ᨨ 犨�
ᆠᆠ�ᨨ ҈� 鼀⏌矄 꾀 矄鼀 � 矄 矄ᨨ �矄 r ᨨᆠ ᨨ鼀 ᨨ��� 犨� ᨨᆠ�泐 u 矄矄 ᨨ犨 꾀 꾀 矄�ᨨ 泐矄ᨨ ᨨ �ᨨ盐 ᨨ矄矄 矄ᨨ �� � ᨨᆠ 盐� ��矄 ᨨ犨 犨 鼀ᨨ�ᨨ �ᨨ � ᨨ ᨨ� ᨨ 꾀 鼀ᨨ�ᨨ 矄� XY r � u �鼀 犨 �ᆠ 矄ᨨ �矄 ᨨᆠ
├g㦨҈鼀⏌矄 㜠҈㝐├矄桌
҈�
꾀� 泐ᨨ �矄矄� � �犨� ᨨ盐 �ᆠ�ᨨ ᨨ犨 矄鼀ᨨᨨ ᨨ � ᆠ� �ᨨ q ᨨ鼀ᨨ� Pᨨ ᨨᨨ鼀ᨨ ᨨ �ᆠ�矄ᨨᆠ � 矄� ᨨ � ��矄 ᨨ �ᆠ犨� 矄ᆠᨨ ᨨ �矄u �ᆠ�ᨨ
ᨨ ᨨᆠ�ᨨ 鼀├矄 �a犨� ᨨᨨ鼀矄矄 ᨨ犨
� ��盐 ҈黨黨㜠㝐├矄桌
a
҈ 犨ᨨ�盐 矄矄�盐 ᨨ ᨨᨨ �ᨨ �� ҈鼀 � ᨨ 矄 ᨨᆠ 犨� 矄鼀矄 � 鼀 矄҈ 矄ᨨ鼀 � ᨨ �鼀졠 �盐 �矄 ᨨ� ᨨ 泐 ᨨ �矄 盐犨 矄 �ᨨ 犨� ᨨ� ᨨ 矄 ᨨ矄 ᆠᨨ� � �ᨨ� P � �㡰 �鼀㜠� 犨
�� �矄 ��㡰 ᆠᨨᨨ � 矄ᨨ ᨨ� P ├ᨨ���
ᨨ ᆠᨨᨨ � 矄泐矄鼀 ᨨᨨᨨ ��� ᨨ犨 鼀盐 ᆠᨨ�矄� x㜠y ҈黨黨㜠�㝐├矄桌 ҈䰘 犨ᨨᨨ犨矄AB CD矄�盐 �矄 ᆠ 矄 矄 ᨨ犨 � ��盐 AB�矄鼀 �� CD�矄鼀 �� ᨨ鼀 �ᨨ盐 꾀
ᨨᆠᨨ 鼀矄矄鼀�矄 ᨨ犨 ᨨ鼀 ᨨᨨ AB ᨨ� ᨨ 泐 犨� ᨨᆠ�泐ᨨ ҈鼀⏌矄ᨨ犨 矄CD ҈黨黨㜠㝐├矄桌 和�鼀盐 ᨨ盐矄泐 ᨨ ᨨᆠᨨ 鼀矄矄鼀�矄 ᨨ犨 ᨨ鼀 ᨨᨨ CD �ᨨ ᨨ ᨨ� ᨨ ᨨ犨 矄 AB犨� 矄鼀鼀� �盐 ᨨ � ��ᨨᆠ�泐矄 ᨨ �矄ᨨᆠ 犨ᨨᨨᆠ矄鼀ᨨ� 矄鼀�ᆠ ᆠᨨ� � 鼀� 㜠� 矄�ᆠ├ᨨᆠ ᨨ
�㡰 ᔴᆠ盐 �矄 ᆠ�d犨 ᨨ犨矄 ��㡰 � ᨨ矄��ᨨ 犨 ᨨᆠ矄 � �ᨨ盐 ҈ �ᆠ矄P Qᨨ 鼀矄矄�鼀 �鼀矄ᆠ�泐 矄�鼀盐 ᨨ盐矄泐 ᨨᆠ ᨨ鼀 ᨨ� 矄 A B ᨨ �ᨨ盐 � ��ᨨᆠ��矄ᨨ P Q鼀├� ҈� �矄矄ᆠ�泐犨� ᨨ� ᨨ AB矄矄 ᨨ犨 �
��盐쟼ᆠ �ᆠ 矄 � ��矄 ᨨ 黨鼀⏌矄 矄�ᨨ AB�矄�鼀 ҈黨㜠㝐├矄桌
턼
q
㡰 � �矄 ᆠᨨ � ᨨᨨ鼀ᨨ ᨨ 犨
ᨨ� ᨨ ᨨᆠ�ᨨ P ᨨ� Q犨 �ᆠ 矄 꾀矄矄盐鼀 �ᆠ ᨨ矄 ᨨ � 泐ᨨ 矄盐 ᆠᨨ ᨨ �ᆠ�矄�矄 ᆠ㡰 㡰 ᔼ ᨨ� ᨨ �矄ᆠ鼀 ᨨ �ᆠ矄矄
泐 ᨨ矄ᨨ �ᨨ盐 �矄 ᨨ � 矄 ᨨ ᨨ 犨� 矄ᆠᨨ �ᨨ盐 � 矄 犨 �ᆠ犨矄ᨨᆠ
ᨨ �ᆠ� 矄鼀�ᆠ 盐 �ᨨ ᆠᨨ�矄�ᨨ 꾀 ᆠᨨ�矄�ᨨ Pᆠ矄�矄 鼀� ᨨ矄��ᨨ ᨨ Q犨 � �矄 �ᨨ盐 ᨨ犨鼀盐ᆠ �鼀 ᆠᨨ�矄�ᨨ ᨨ矄 �ᆠQ├ᨨᆠ �ᨨ盐 䁸 ├g㦨黨鼀⏌矄
Topic 4 : Relative Velocity in Two Dimensions & Uniform Circular Motion 1 ҈
MCQs with One Correct Answer
꾀ᆠᨨᆠ├ 矄ᆠᨨ � 盐ᨨ盐矄泐鼀ᨨ� �矄ᆠᨨ Ἐ矄 ᨨ ҈ 鼀 � �ᆠᆠ�ᨨ ᨨ �ᨨ
� 盐 �矄ᨨ 鼀矄㡰�矄ᨨ ᨨ ᨨ 㝐� 和 ᔼ㡰桌 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈҈
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EBD_8341 P-20
Physics
When a car sit at rest, its driver sees raindrops falling on it vertically. When driving the car with speed v, he sees that raindrops are coming at an angle 60º from the horizontal. On furter increasing the speed of the car to (1 + b)v, this angle changes to 45º. The value of b is close to: [Main Sep. 06, 2020 (II)] (a) 0.50 (b) 0.41 (c) 0.37 (d) 0.73 The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight? [Main 9 April 2019 (I)] (a) 90° (b) 150° (c) 120° (d) 60° Ship A is sailing towards north-east with velocity km/hr where points east and , north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. A will be at minimum distance from B in: [Main 8 April 2019 (I)] (a) 4.2 hrs.(b) 2.6 hrs. (c) 3.2 hrs. (d)2.2 hrs. Two particles A, B are moving on two concentric circles of radii R1 and R2 with equal angular speed w. At t = 0, their positions and direction of motion are shown in the
2.
3.
4.
5.
p ö æ figure : ç t = 2w ÷ø è
6.
7.
a
(c)
(d)
r r A boat which has a speed of 5 km/hr in still water crosses a river of width 1 km along the shortest possible path in 15 minutes. The velocity of the river water in km/hr is [1988 - 1 Mark]
(a) 1
X
9.
B
® ® The relative velocity vA - vB is given by: (a) w(R1 + R2) iˆ (b) –w(R1 + R2) iˆ (c) w(R2 – R1) iˆ (d) w(R1 – R2) iˆ
r
a
A
R2
(b) r
8.
R1
a
a
(a)
[Main 12 Jan. 2019 (II)] Y
A particle is moving along a circular path with a constant speed of 10 ms–1. What is the magnitude of the change in velocity of the particle, when it moves through an angle of 60° around the centre of the circle? [Main Online April 10, 2015] (a) 10 3m/s (b) zero (d) 10 m/s (c) 10 2m/s If a body moving in circular path maintains constant speed of 10 ms–1, then which of the following correctly describes relation between acceleration and radius? [Main Online April 10, 2015]
(b) 3
(c) 4
(d) 41 A river is flowing from west to east at a speed of 5 metres per minute. A man on the south bank of the river, capable of swimming at 10 metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction [1983 - 1 Mark] (a) due north (b) 30° east of north (c) 30° west of north (d) 60° east of north
Answer Key Topic-1 : Vectors 1.
(d)
2. (c)
3. (a)
4. (a)
5. (a)
6. (195)
7. (90)
8. (2.00)
Topic-2 : Motion in a Plane with Constant Acceleration 1. 10.
(d) 2. (a) (a, b, c)
3. (c)
4. (d)
5. (d)
6. (b)
7. (c)
8. (580)
9. (a, b, c, d)
7. (a)
8. (c)
9. (b)
Topic-3 : Projectile Motion 1. 11.
(a) 2. (c) (4.00) 12. (5)
3. (d) 13. (True)
4. (b)
5. (a)
6. (b)
10. (b)
Topic-4 : Relative Velocity in two Dimensions & Uniform Circular Motion 1.
(a)
2. (d)
3. (c)
4. (b)
5. (c)
6. (d)
7. (c)
8. (b)
9. (a)
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P-21
Laws of Motion
Laws of Motion
4
Topic-1 : Ist, IInd & IIIrd Laws of Motion MCQs with One Correct Answer
1 ҈
꾀�ᆠ鼀ᨨ� �� xy � ᆠ矄ᨨᆠ�泐 r ᨨᆠ F = k v y �$ + v x $ 㡰 犨 vx vyx r r yᆠᨨ鼀ᨨ 矄ᨨ �矄ᨨᆠ�泐 v � a �矄 ᆠᆠ�ᨨ
ᨨ �ᆠ 犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 盐 ᨨ �ᆠ䁸 㝐� 和 ᔼᔼ㡰桌 r r 㡰 ⼄盐 �泐 v ´ a �矄ᆠᨨ 矄 � �鼀 r 㡰 F �矄矄 盐ᨨ鼀� �ᆠ � ᆠ㡰 ├� �ᆠ �泐ᨨ �ᆠ�矄ᆠᨨ 矄 � �鼀 r r 㡰 ⼄盐 �泐 v × a �矄ᆠᨨ 矄 � �鼀 꾀矄ᆠ矄 �� 矄ᆠ矄犨矄矄�ᨨ 泐� 泐 盐矄
꾀�矄 ᨨ犨 盐犨 犨� � ��ᨨᆠ�泐 ᨨ鼀 矄盐 ᆠᨨ 鼀ᨨ�ᨨ ᨨ �矄 ᆠ 泐 � ᨨᆠ⼄盐ᨨ鼀g 犨 鼀�矄鼀矄矄ᨨ �矄 �矄� 矄 ᨨ盐矄ᨨᆠ�泐 g �矄ᆠᨨ 矄 㡰 �鼀├
泐 ᨨ�矄ᨨ�矄 � �矄 㝐� ҈꾀�҈黨ᔼ桌 æ g ö æ g ö ҈ -҈ ç ÷ 矄� -҈ ç 㡰 ç ÷ ç � ÷÷ g� g� è � ø è ø æ æ g ö ҈ g ö ҈ ÷ 㡰 -҈ ç l ç҈ + ᆠ㡰 ç ÷ ç � ÷÷ � ø g� è g � è ø 꾀�矄 ᨨ犨 �ᆠ泐盐 ├ 矄졔 㜠 �矄㡰 ᨨ鼀 �ᨨ盐 ᆠᨨᆠ鼀ᨨ鼀 盐鼀㜠 �� 㜠 㡰 ��鼀�矄
㡰
�
҈
㝐� 黨꾀�҈黨ᔼ桌
dM t 㡰 = bv t 㡰 dt 犨 v t㡰�矄�矄� 矄 ᨨ盐矄ᨨᆠ�泐 � 矄 ᨨ盐矄 ᆠᆠ�ᨨ ᨨ 矄��矄 㝐� 和� ᔼᔼ㡰桌
꾀矄矄盐�矄鼀矄矄� ᆠ矄矄
㡰 -bv t 㡰
㡰 -
bv M t 㡰
bv M t 㡰
㡰 -
bv M t 㡰
ᆠ㡰
꾀矄鼀ᨨ 鼀矄矄m�矄 ᨨ犨 盐犨 犨� ᨨᆠ�泐u ᨨ鼀 �ᨨ盐 � ᆠ矄矄�矄� ᨨᆠmkv 犨 v�矄�矄矄 鼀 �鼀盐鼀 �� � 泐 �矄 㝐� 和 ᔼᔼ㡰桌
㡰
㡰
ᆠ㡰
㡰
꾀�ᆠᨨ 鼀矄矄鼀�矄鼀ᨨ� �� 矄�� � 犨� 鼀ᨨ鼀 盐鼀和� ��鼀㦨 ᨨᆠ 㦨kᆠ矄� 矄鼀 �ᆠ�ᨨ ᨨ 鼀ᨨ� ��ᆠ 盐� ��鼀�
矄ᨨ �矄鼀ᨨ鼀 盐鼀ᆠ �矄 ᨨ鼀ᨨk�矄 ᆠᨨ 矄 盐ᨨ �矄 㝐� ҈҈ ҈黨ᔼᔼ桌
㡰
ku ҈ -҈ g k
㡰
҈ æ ku ö ҈ + g ÷ø k çè
㡰
ᆠ㡰
ku ҈ -҈ k g
㡰
҈ æ ku ö ҈ + k çè g ÷ø
ᆠ㡰
k
㡰
k
㡰
k p ├
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EBD_8341 P-22
䰘
Physics
ABBC CD矄�� � 矄�鼀 矄 鼀� �盐 ᨨ ᨨ�鼀盐矄ᨨ ᨨᆠᨨ ᨨ 泐 ᨨ鼀t㦨m矄 ᨨt㦨҈m矄�矄矄 ҈黨黨㜠㝐├矄桌 ᔴ
꾀�ᆠᨨ 鼀矄矄鼀�矄ᆠ 盐ᨨ 泐 ᨨᆠ �� 泐
ᨨᆠ 㡰
鼀��ᆠ犨 = 㡰ᔼ �矄犨�矄ᨨ矄 �鼀 泐泐ᨨ矄� � 鼀ᨨ�ᨨ 矄� � ᨨ鼀矄 矄犨泐�矄ᨨᨨ㝐� 졠 � 꾀�҈҈桌 㡰 㡰�� 矄
ᆠ㡰
҈ ᨨ� 㡰�� 矄
㡰 ᨨ� 㡰�� 矄
㡰
҈ ᨨ� 㡰�� 矄
꾀
꾀� 盐鼀 㡰ᨨ � �ᆠ 矄ᆠ ᨨ 鼀矄矄鼀 �盐矄矄盐 �ᨨ � 矄鼀ᨨᨨ ��� ᨨ� ᨨ ᨨ ᨨ � >> 㡰 矄 ᨨ鼀 ᨨ矄��ᨨ 矄 ᨨ �矄 鼀ᨨ盐 犨 犨ᨨ ��� 矄盐ᨨ矄 矄 ��盐㡰 ᔼ ᨨ ᨨ 矄 �矄 ᨨ犨��
矄 � ᨨᆠ � ᆠ 泐 ᆠ 矄盐ᨨ ᨨ � �鼀 �矄 矄矄盐鼀 ᆠᨨ�矄�ᨨ 矄 矄�ᆠ㡰㝐� 졠 � 꾀�҈҈ ҈�桌
5
쟼
҈
�鼀 m 矄㡰
҈
҈ ҈
True / False
҈ 꾀ᨨᆠ├鼀ᨨ矄 ᨨ犨 泐盐矄 � � 矄盐ᨨ盐 � �� ᆠ├犨 矄 ҈黨桌 6
MCQs with One or More than One Correct Answer
҈ 꾀 ᆠ 鼀ᆠ ᨨ ҈黨㜠㝐├矄桌 㡰 �矄 � � 鼀泐 � ��ᨨ 㡰 ᆠ
ᨨ � � 鼀 ᆠ盐矄 �矄 ᨨ� �ᨨ盐 矄盐 ᆠ㡰 �矄 � � 鼀 ᆠ盐矄 犨ᨨ 矄 犨矄 �ᆠ� �矄 鼀 㡰 ᆠ
ᨨ � � 鼀 ᆠ盐矄 �矄 ᨨ� �ᨨ盐�矄ᨨ犨 �矄
㡰
mv 2 2(L - nr)
㡰
mv 2 L - 2nr
9
mv 2 㡰 ᨨ L - nr 꾀ᨨ 泐ᨨ 鼀矄矄�├�盐 ᆠ�ᨨ ᨨ ᆠᨨ 矄 ᨨᆠ r r = ˆ� + ˆ 矄ᨨᆠ�泐㦨矄矄 = �ˆ - ˆ 鼀⏌矄
ᆠ㡰 黨
(
泐
)
r r 㦨҈矄矄 = +ˆ 鼀 ⏌ 矄 ᨨᆠ �矄 㝐� 졠 � 꾀�҈҈҈桌
҈
㡰
( -ˆ + ˆ)
㡰
æ ˆ ˆö ç - � + ÷ è � � ø
ᆠ㡰
( �ˆ - ˆ)
㡰
æ ˆ ˆö ç � - ÷ è� � ø
꾀�ᆠ鼀ᨨ矄� X㜠Y 盐 � 盐 ᆠᨨ ᨨᆠ 矄盐ᆠ �矄 � 鼀ᨨ鼀 盐鼀 �矄 r p t 㡰 = A �ˆ ᆠᨨ矄 kt 㡰 - ˆ 矄� kt 㡰桌 犨 A k ᆠᨨ 矄 矄 �犨 ᨨᆠ 鼀ᨨ鼀 盐鼀 �矄 䰘桌 㡰 㡰 ᆠ㡰 � 㡰 黨 4
҈҈
Fill in the Blanks
Assertion and Reason Type Questions
҈ 和鼀 㜠҈ᔼ�矄矄�ᨨ盐 泐ᨨᆠ ᨨ 盐矄 �ᨨ �ᨨ盐 和鼀 㜠 鼀� �盐 ᨨ �ᆠ�ᨨ ᨨᆠ 矄 ᨨ 盐ᨨ 犨ᨨ矄盐 ᆠ矄� ᆠᨨ ᆠ 桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐 ҈� 和T꾀TE㝐ENT㜠҈꾀ᆠᨨ ᆠᨨ矄和ᨨ鼀 �矄 矄 ├ᨨ � ᆠᨨ ᆠ 盐 ᨨ盐犨� ᨨ盐 �矄ᨨ �� � �矄 矄 ᨨ鼀 和T꾀TE㝐ENT㜠 ᨨ泐ᆠ�ᨨ �矄 ⼄盐 ᨨᨨ矄�ᆠ�ᨨ 䰘桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和㜠 鼀 㜠�矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和㜠 鼀 㜠�矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐
鼀� �盐 ᨨ ᨨᆠ � 犨ᨨ 矄㡰ᆠ� �ᨨ ᨨ 泐 �矄犨� �鼀t � 鼀�ᆠᨨ矄ᆠᨨ 矄㡰矄矄 ᨨ犨 � ��
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P-23
Laws of Motion
Topic-2 : Motion of Connected Bodies, Pulley & Equilibrium of Forces
1 ҈
㡰 ҈
MCQs with One Correct Answer
㝐҈
꾀鼀矄矄ᨨ ҈├��矄矄盐矄 泐ᨨᨨ � 鼀 ᨨ鼀 ᆠ�� �꾀 ᨨᆠ �矄� ᨨ� ᨨ 泐 鼀� 㜠 ᨨ� ᨨ ᨨ矄盐ᆠ ᨨ ᨨ ᨨ鼀├矄 �ᨨ � 犨� �ᆠ F⼄盐矄 ├g 㦨҈鼀矄 ᨨᨨ鼀矄矄矄矄㡰 㝐� 䰘 ᔼᔼ桌 㡰 ҈ 㡰 黨 ᆠ㡰 䰘 㡰 䰘� 꾀 ᨨ � 盐� � � ᆠ ᆠ泐 鼀 �鼀盐鼀 ᨨ ҈ 矄ᨨ 矄犨� �鼀矄矄ᨨ ᆠ 矄ᨨ � � ├� 鼀矄矄ᨨ ᨨ�矄 �矄黨├� �鼀ᨨ矄 犨� ᆠᨨ 矄 矄 ᨨ 鼀⏌矄 �ᆠ�ᨨ ᨨᆠᨨᨨ矄� � 鼀ᨨ�ᨨ �矄ᔼ ᨨ�矄鼀ᨨ� �盐犨� �矄 盐 ᆠᆠ�泐 ᨨ犨 � 泐 鼀ᨨᨨ ᨨ ᨨ g㦨҈鼀⏌矄㡰鼀盐矄矄 㝐� 䰘 ᔼᔼ桌 㡰 �␔ 㡰 ␔ ᆠ㡰 ␔ 㡰 ␔ 꾀鼀矄矄ᨨ ҈├��矄矄盐矄 �ᆠ泐泐ᨨ ᨨ鼀 ᨨᨨ ␔ ᨨ� ᨨ ᨨᆠ�矄� ᨨ ᨨ 矄ᨨ鼀ᨨ� ᨨ � �ᨨ � ᨨᨨ ᨨ� ᔼ 矄盐矄 鼀矄矄�矄⼄盐���盐鼀 鼀� �盐 ᨨ ᨨᆠ� �矄 �㦨҈鼀矄㡰 㝐� 黨 ҈黨ᔼᔼ桌 㡰 㡰 ҈ ᆠ㡰 䰘 㡰 ҈ 꾀鼀矄矄�鼀�矄矄盐ᨨ 泐鼀矄矄矄矄矄� �犨ᨨ盐 ᨨ盐 盐 � ᨨ鼀 ᨨᨨ犨ᆠ泐� ᨨ 鼀矄矄鼀 �盐矄 ᔼ 矄� � ᨨ矄 ᨨ矄�ᨨ ᆠ泐� 犨� 犨 ᆠᆠ�ᨨ
犨� 鼀矄矄 ᨨ矄䁸 㝐� ҈桌 㡰
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㡰
�
ᆠ㡰
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㡰 黨 ᆠ㡰 㡰 ҈黨
㝐
犨ᨨ�ᆠ矄ᨨ 鼀矄矄mᆠ � 矄ᨨ �� 矄� �ᨨ � a 犨 ᨨ矄泐矄鼀�矄├ᨨ �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ犨� 矄� � �� 矄ᨨ ᆠ 鼀矄矄�矄 �矄 ᆠἘaἘ ᨨ鼀 ᆠ P 矄矄 ᨨ犨 � ��盐㡰 ᨨ犨 鼀� 㜠ᨨ� ᨨ 矄� ��矄盐 �ᆠ泐盐犨 矄 犨� 矄鼀盐ᆠᨨ 矄 ᨨᆠF꾀矄矄盐 �ᆠ矄 鼀ᨨᨨ犨 矄ᆠ ᨨ ᨨ 矄盐 ᆠ 鼀� �盐 ᨨ ᆠᆠ�ᨨ 犨 矄�ᨨ 犨 鼀 ᆠᨨ鼀矄x�矄 䰘桌 㡰
F a m a 㜠 x
㡰
F x m a 㜠 x
ᆠ㡰
F x m a
㡰
F a 㜠 x m x
F
m
a
䰘
鼀
m
P
a
矄� �犨 ᨨᆠ├矄ᨨ 鼀矄矄m 鼀�矄鼀矄矄矄矄 � 矄� 矄泐矄鼀 �矄 矄盐矄 泐 鼀矄矄矄矄 矄� �矄矄 ᨨ犨 ᔼ 矄� � �矄ᆠ盐 � 鼀� �盐 矄ᨨ ᆠᆠ�ᨨ 矄ᨨ 鼀矄矄m m �鼀鼀 �泐 ᆠ盐� �㡰 㜠㝐҈桌
m m
�
鼀 㡰 �
犨ᨨ ᨨᆠ├矄 ᨨ 鼀矄矄 㝐҈ 㦨 ├� 㝐 㦨 ҈ ├� ᆠᨨ
ᆠ 泐 鼀 ᨨ ᨨ 鼀矄矄 ├� 矄泐矄鼀 �矄 盐 �ᆠ泐盐泐泐� � ᨨᆠᨨ 矄矄 ᨨ犨
矄�ᨨ 鼀� 㜠ᨨ� ᨨ ᨨ �矄 㝐� 졠 � 꾀�҈桌
㡰 gg ᆠ㡰
㡰 g
g �
㡰
g
g g
꾀矄� �ᨨ ����鼀矄矄�ᨨ� �ᨨᆠ鼀 盐泐ᨨ 鼀矄矄m 矄盐ᨨ矄ᨨᆠ├ᨨ 鼀矄矄M 矄矄 ᨨ犨 � ��盐
ᨨᆠᨨ 盐泐泐 ᆠ鼀�矄�� 泐҈和桌
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EBD_8341 P-24
Physics
M
黨
㡰
2 㝐�
㡰
2 鼀�
ᆠ㡰
( M + m) 2 + m 2 g
㡰
(M + m) + M g
m
2
a҈
M
a
2
a M
盐泐矄 矄� �矄矄 ᨨ犨 � ��盐矄鼀ᨨᨨ ᨨ ���� 鼀矄矄 ᨨ 矄泐矄鼀 ᨨ鼀� �
⼄盐���盐鼀 �q矄 ᨨ盐 ҈和桌
M
㡰 ᨨ q
㡰 ᨨ
Öm
ᆠ㡰 �ᨨ m
㡰 ᨨ 5 ҈
x ᨨ 矄� � 鼀� �盐 ᨨ ᆠᆠ�ᨨ ᨨ ᨨᆠ├ᆠᨨ
ᆠ ᨨ 矄� � g �矄 ҈ Mg 㡰 x = k x ᆠ㡰 ␔ 矄� �ᆠ �矄 矄�ᨨ ᨨ ᨨ �矄�鼀 矄 ᨨ ᨨᆠ├ᆠᨨ
ᆠ ᨨ
㡰 꾀 矄�ᨨ ᨨ
m
True / False
盐泐 �鼀 矄ᨨ ��矄 㡰 㡰� �ᆠ
鼀矄矄ᨨ ᨨ�矄 ����ᔼ 㡰 鼀矄矄m�矄 � 盐泐ᆠ � �鼀矄矄mᨨ ᨨ ᨨ ᨨᔼ 㡰m�矄� 盐泐盐� � ᨨ ᨨ ᨨ 犨� ᆠᨨ 矄 ᨨ犨 犨 ᨨᆠ F 㦨 mg ᆠᆠ�ᨨ ᨨ m�矄 矄鼀� ᨨ ᆠ矄矄 ҈黨㜠㝐├矄桌
M �k 㡰 aa҈㦨a҈a ҈ ᔼ �鼀 矄 ᨨ犨 � �� 矄P Qᨨ 盐 矄ᆠ 矄� �鼀ᨨ ᨨ犨 犨 矄犨� 盐 � ᨨ鼀 矄 U盐泐矄A B � ҈黨㜠㝐├矄桌 㝐矄矄M鼀ᨨ矄盐犨 矄犨� 矄
矄� ��矄�
㡰 ᆠᨨ矄q
A
㡰 ⏌ᆠᨨ矄 q ᆠ㡰 ⏌ᆠᨨ矄q 㦨mg
m m
6 ҈҈
m 㡰
㡰
MCQs with One or More than One Correct Answer
꾀ᨨᆠ├ᨨ 鼀矄矄㝐�矄ᆠ ᨨ鼀矄矄矄矄矄� �犨� 矄� �㜠ᆠᨨ 矄 k �矄ᨨᆠ├�矄ᆠᨨ
ᆠ ᨨ犨ᨨᨨ ᨨᆠ├矄ᨨ 鼀矄矄矄㝐 㝐盐矄� �犨ᨨ鼀矄矄矄矄盐泐矄 矄� �矄 ᆠᆠ�ᨨ ᨨ ᨨᆠ├矄a҈㜠a a矄矄 ᨨ犨 � ��盐 矄泐矄鼀�矄矄 ᨨ鼀矄 犨� 矄� �� �矄盐 矄ᆠ 矄 鼀 �鼀盐鼀 矄�ᨨ ᨨ 矄� � �矄 x ␔ �ᆠ ᨨ ᨨᨨ犨� � ᨨ�ᨨ 矄㡰 �矄⏌ ᆠᨨᆠ䁸 g �矄 ᆠᆠ�ᨨ 盐 ᨨ ��泐�ᆠ �ᆠ�ᨨ 桌 꾀 ҈黨桌
B
q q P
M
Q
㡰 ᆠᨨ矄q 10 Subjective Problems ҈ 犨ᨨᨨᆠ├矄ᨨ 鼀矄矄黨├� ҈黨├� 矄盐矄 ᨨ鼀 ��� 矄盐ᨨ S 泐 犨ᨨ � 矄�犨�矄ᆠ ᨨ � ҈鼀 矄 �� 盐犨� 矄 ����鼀矄矄 ᨨ犨犨� 矄盐 � ᨨ鼀鼀矄矄ᨨ ├�⏌鼀 犨 ᨨ矄泐矄鼀ᨨ ᨨᆠ├矄犨�矄 矄盐ᨨ 盐犨 ᆠᆠ�ᨨ ᨨ 鼀⏌矄꾀ᆠᆠ�ᨨ 盐ᨨ��泐�矄黨鼀⏌ 矄 ҈黨黨㜠㝐├矄桌 �㡰 � 矄�ᨨ 鼀� 㜠ᨨ� ᨨ ᨨ犨犨� ��㡰 � 矄�ᨨ 鼀� 㜠ᨨ� ᨨ 盐犨�
S
黨├�
҈黨├�
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P-25
Laws of Motion
Topic-3 : Friction
An insect is at the bottom of a hemispherical ditch of radius 1 m. It crawls up the ditch but starts slipping after it is at height h from the bottom. If the coefficient of friction between the ground and the insect is 0.75, then h is : (g = 10 ms–2) [Main Sep. 06, 2020 (I)]
Two masses m1 = 5 kg and m2 = 10 kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15. The minimum weight m that should be put on top of m2 to stop the motion is: [Main 2018] T m (a) 18.3 kg m2 m 2
(a) 0.20 m
(b) 27.3 kg
1 1.
2.
(b) 0.45 m
(c) 0.60 m (d) 0.80 m A block of mass 5 kg is (i) pushed in case (A) and (ii) pulled in case (B), by a force F=20 N, making an angle of 30 o with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is m = 0.2. The difference between the accelerations of the block, in case (B) and case (A) will be : (g =10 ms–2) [Main 12 April 2019 (II)]
–2
3.
5.
MCQs with One Correct Answer
m1
(d) 10.3 kg 6.
7.
8. (a) 8 N (b) 16 N (c) 40 N (d) 12 N A block of mass 10 kg is kept on a rough inclined plane as shown in the figure. A force of 3 N is applied on the block. The coefficient of static friction between the plane and the block is 0.6. What should be the minimum value of force P, such that the block doesnot move downward? (take g = 10 ms–2) [Main 9 Jan. 2019 (I)]
1
(d)
2 - n2
1 1 - n2
A rocket is fired vertically from the earth with an acceleration of 2g, where g is the gravitational acceleration. On an inclined plane inside the rocket, making an angle q with the horizontal, a point object of mass m is kept. The minimum coefficient of friction mmin between the mass and the inclined surface such that the mass does not move is : [Main Online April 9, 2016] (a) tan2q (b) tanq (c) 3 tanq (d) 2 tan q Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is: [Main 2015] F
A
B
P g
N
k 10
3
4.
m1g
A given object takes n times more time to slide down a 45° rough inclined plane as it takes to slide down a perfectly smooth 45° incline. The coefficient of kinetic friction between the object and the incline is : [Main Online April 15, 2018] 1 1 1- 2 (b) 1 - 2 (a) n n (c)
–2
(a) 0.4 ms (b) 3.2 ms (c) 0.8 ms–2 (d) 0 ms–2 Two blocks A and B masses mA=1 kg and mB = 3 kg are kept on the table as shown in figure. The coefficient of friction between A and B is 0.2 and between B and the surface of the table is also 0.2. The maximum force F that can be applied on B horizontally, so that the block A does not slide over the block B is : [Take g = 10 m/s2] [Main 10 April 2019 (II)]
T
(c) 43.3 kg
(a) 32 N
45°
(b) 18 N
(c) 23 N
(d)
25 N
(a) 120 N (c) 100 N
(b) 150 N (d) 80 N
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EBD_8341 P-26
Physics
꾀ᨨᆠ├ᨨ 鼀矄矄鼀㦨҈├�矄矄ᨨ ᨨ� ᨨ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ �矄 �␔ �泐盐ᨨ 鼀矄矄��鼀ᨨ� �犨� 矄 n �矄鼀 � � ᨨᆠ├鼀ᨨ矄 ᆠᨨ鼀矄ᨨ 矄ᨨ 鼀ᨨ� � �矄 ᆠᨨ 鼀ᨨ ᔼ 泐 n � �ᨨᆠ犨ᨨᆠ⼄盐�矄 � � ᨨ ҈ ᨨ鼀 �� �ᆠ� � �泐ᨨ矄矄矄 ├� �� 㦨҈鼀矄 盐ᨨ �矄ᆠᨨ矄ᨨ 㝐� 졠 � 꾀�҈҈�桌 㡰 �├鼀 㡰 ├鼀 ᆠ㡰 ├鼀 㡰 ├鼀 ҈ 꾀ᨨᆠ├ᨨ 鼀矄矄鼀�矄ᆠ ᨨ 矄盐 ᆠ犨� �ᆠ 黨
ᔼ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ
�矄� 鼀 �鼀盐鼀 �� ᨨ �ᨨ盐 犨 �ᆠ ᨨᆠ├ᆠ ᆠ 犨� ᨨ盐矄�� ��矄 㝐� ҈桌
ᆠᨨ矄矄矄ᆠ�ᨨ �� 泐 泐 =
҈ ҈ ҈ 鼀 鼀 鼀 鼀 㡰 ᆠ㡰 㡰 ᔴᨨ 矄� ᆠ泐� ᨨ 鼀矄矄㝐矄� �ᨨ ᨨ盐� ᨨ� ᨨ 盐� �矄盐 ᨨ盐 ᨨ鼀盐 �犨� ᆠᆠ�ᨨ � �ᆠ盐ᨨ �矄ᨨ ᆠ泐� ␔ �矄 �ᆠ�ᨨ
ᨨ� 䁸ᔼ�矄矄矄盐鼀 ᆠ泐� ᨨ矄 ᨨ矄� 㝐� 졠 � 꾀�҈黨҈桌
㡰 ҈҈
a
ᆠ㡰
㡰 졠
졠 t t ҈ 꾀ᨨ 泐矄矄 ᨨ鼀矄ᨨ ᨨ �� ᆠ� ᨨ 矄ᨨ � ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨ 泐 �矄矄m㦨x犨 x�矄 �矄 ᆠ ᨨ犨 ᨨ 泐犨� 鼀 �鼀盐鼀矄 ᨨg㦨҈鼀⏌矄㡰犨 x 㦨㝐� 졠 � 꾀�҈桌 ᆠ㡰 ҈鼀 㡰 鼀 㡰 黨鼀 㡰 䰘鼀 ҈� 꾀ᨨᆠ├ᨨ 鼀矄矄鼀�矄ᨨ � ᆠ� ᨨ �q ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ �矄m q>m ᨨᆠ├�矄 矄�ᨨ 泐泐泐� � ᨨᆠᨨ �ᆠ�ᨨ ᨨ ᨨᆠᨨ� � � 盐 �矄├ ᨨᨨ矄��꾀矄�矄� ᨨ鼀 ҈㦨鼀� 矄� qm ᆠᨨ矄q㡰ᨨ㦨鼀� 矄� q졔m ᆠᨨ矄q㡰 �ᆠ�ᨨ ᨨᆠf矄盐矄� 犨�ᨨᨨ├�├ ҈桌 f
f
㡰
㡰
f
犨
a
졠
ᆠ㡰
f
㡰
�ᆠ�ᨨ
㝐 㝐 㡰 꾀ᨨᆠ├꾀ᨨ 鼀矄矄├��矄ᆠ ᨨ ᨨ ᨨᆠ├ᨨ 鼀矄矄�├� ᨨᆠ├矄矄ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ ᔼ 鼀� �鼀盐鼀 ᨨᆠ ᆠ � ᨨ 꾀矄ᨨ ᨨ ᨨᆠ├矄鼀ᨨᨨ� �矄҈ 鼀 �鼀盐鼀 ᨨᆠ ᆠ � ᨨ ᨨ ᨨᆠ├矄 ᨨ 鼀ᨨ 㝐� 졠 � 꾀�黨҈桌 ᨨ� 犨� 㡰 㡰 � ᆠ㡰 䰘 㡰 꾀ᨨᆠ├�矄ᆠ ᨨ ᨨ盐� ᨨ� ᨨ 꾀�鼀 ᨨ� ᨨ ᨨᆠ 㦨├ᆠ矄ᨨ ᨨᆠ├犨 ├�矄ᨨ矄��ᆠᨨ 矄 ᆠᆠ�ᨨ 㜠�鼀� ᨨ ᨨᆠ├�矄 㝐� 졠 � 꾀��҈桌
㡰 㝐� ҈
҈
㡰 㝐
ᆠ㡰
a
a
㡰
㡰
졠
҈ 꾀ᨨᆠ├ᨨ 矄҈ᆠ鼀맔҈ᆠ鼀 �� ҈�ᆠ鼀�矄├ᨨ
� ᆠ� ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 鼀 �矄 � ᆠ� �ᨨ qᨨ �矄� ᆠ� ᨨ鼀 ᨨ� ᨨ �矄� 盐泐� ᆠ矄 ᨨ鼀
黨桌 㡰 q㦨 ᨨᆠ├犨�矄矄� � � ᨨ犨 㡰 ᨨᆠ├犨�鼀� 矄ᨨ 盐ᨨᆠ�
q �犨�ᨨ ᆠ㡰 q㦨 ᨨᆠ├犨�矄矄� � � ᨨ犨 ᆠᨨ � 盐ᨨ ᨨ矄ᨨ �� �矄 㡰 q㦨 ᨨᆠ├犨�矄矄� � � ᨨ犨 ᨨ 盐 � ᆠ矄� �q�犨�ᨨᆠ� q. ҈䰘 ␔ �矄 鼀 �鼀盐鼀盐ᨨ ᨨᆠF矄盐ᆠ ᨨᆠ├矄 ᨨ犨 � �鼀 ᨨ矄 ᨨ鼀ᨨ䁸和桌 ҈ m= 鼀 = ├�
t
졠
t
㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈� ҈ 꾀 � 矄ᆠᆠ犨矄盐 鼀�矄 �ᆠ矄盐 ᆠ泐矄ᨨ犨泐 矄 ��㡰 ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 � 矄ᆠ 矄盐 ᆠ�矄҈⏌ᔼ � ᨨ� � � ᆠ ᨨ 鼀�矄 �ᆠ矄盐 ᆠᨨ � 矄ᆠ鼀├矄 �a犨� �ᆠ 鼀 �鼀盐鼀ᨨ矄矄�盐ᨨ a�矄�� 泐 ҈和桌
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P-27
Laws of Motion
a
҈黨
㡰 ᆠᨨa㦨 㡰 a㦨 ᆠ㡰 矄ᆠa㦨 㡰 ᆠᨨ矄ᆠa㦨 꾀 ᨨᆠ├ᨨ 鼀矄矄҈ �矄 �� 矄 犨 泐� � ᨨ� ᨨ ᨨᆠᨨ �ᨨ ᨨᆠ├ᔼ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ 犨�矄� 鼀� �盐 ᨨ �ᆠ�ᨨ ᨨᆠᆠ� �ᨨ ᨨᆠ├�矄 ҈黨黨㜠҈㝐├桌 㡰 � 㡰 黨 ᆠ㡰 黨 㡰 黨 꾀ᨨᆠ├ᨨ 鼀矄矄├�矄矄ᨨ ᨨ盐� � ᆠ� 鼀├� � �ᨨ 犨� ᨨ� ᨨ ᆠᨨ �ᆠ� ᨨ 矄�ᆠ �ᆠ�ᨨ 犨 ᨨᆠ├ �矄䰘 �ᆠ�ᨨ ᨨᆠᨨ ᨨᆠ├�矄 ҈黨桌 㡰 黨 㡰 䰘 ´ 黨 ´ ᆠ㡰 黨 ´ 㡰 䰘맔黨 2
҈
꾀ᨨᆠ├�矄鼀ᨨ� �ᨨ � ᆠ� 鼀├� � � � 犨� ᨨ� ᨨ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ �矄m
ᨨᆠ⼄盐� ᨨ盐矄盐矄 �盐 � ᆠ� �矄 �鼀矄 ᨨᆠ⼄盐� ᨨ盐矄 � ᨨ鼀矄� � � ᨨ犨 ᔼ 犨 � 㦨҈m �矄 ҈҈桌 꾀ᨨᆠ├ᨨ 鼀矄矄҈├��矄ᨨ ᨨ� ᨨ 矄盐 ᆠ� 盐ᆠ├
ᆠᨨ �ᆠ� ᨨ 矄�ᆠ �ᆠ�ᨨ 犨 ᨨᆠ├ 矄盐 ᆠ�矄ᔼ ᆠᆠ�ᨨ ᨨ 盐ᆠ├�矄�鼀⏌矄 �ᆠ�ᨨ ᨨᆠᆠ� �ᨨ ᨨᆠ├�矄 犨ᨨ 矄 ҈黨㜠㝐├矄桌 3
True / False
␔ 矄ᨨ 犨├矄ᨨ ᨨ盐� 矄盐 ᆠ �ᆠ�ᨨ ᨨᆠ 泐 矄盐 ᆠᨨ 矄ᨨ �矄ᨨᨨ矄�ᨨ �ᆠ�ᨨ ᨨ �矄鼀ᨨ�ᨨ ҈黨҈㜠㝐├矄桌 6
�
Numeric Answer
꾀ᨨᆠ├矄矄鼀ᨨ� �盐 � ᆠ� ᨨ � ᆠ� �ᨨ
犨� � ��ᨨᆠ�泐ᨨ vᔼᆠᨨ鼀矄ᆠ├ᨨ�矄� �� v ᨨ矄��ᨨ 犨� ᨨᆠ�泐 盐ᨨ ᆠᨨ �ᆠ� ᨨ ├� �ᆠ �ᆠ�ᨨ 犨 ᨨᆠ├ � ᆠ� �矄 I 盐ᨨ I�矄힘힘힘힘힘힘힘힘힘 ᆠᨨ矄ᨨ ҈ 㝐� 和 ᔼᔼ㡰桌 5
Integer Value Answer
MCQs with One or More than One Correct Answer
꾀矄鼀ᨨᆠ├ᨨ 鼀矄矄ᨨ ҈├��矄ᨨ � � ᆠ� 턼犨 �ᆠ 鼀├矄 �q 犨� ᨨ� ᨨ 꾀 ᨨ� ᨨ ᨨᆠᨨ ҈ᆠ矄ᨨ ᨨᆠ├ ᨨ盐� �矄ᆠ ᨨ 鼀矄矄矄矄 ᨨ犨 � ��盐 ҈桌
ᨨᆠ├鼀� 矄矄�ᨨ 泐� ├g㦨҈鼀⏌矄㡰 㡰 q 㦨�° 㡰 q >�° �ᆠ�ᨨ ᨨᆠᆠ矄ᨨ ᨨᆠ├ᨨ犨 矄 ᆠ㡰 q >�° �ᆠ�ᨨ ᨨᆠᆠ矄ᨨ ᨨᆠ├ᨨ犨 矄턼 㡰 q �° �ᆠ�ᨨ ᨨᆠᆠ矄ᨨ ᨨᆠ├ᨨ犨 矄턼
꾀�ᆠP�矄矄� � � ᨨ犨 �ᆠ�ᨨ 矄矄 鼀�矄 �ᆠ ᨨ犨ᔼ矄矄矄 ᨨ� At㦨꾀 �矄� 矄 ᨨ �鼀 ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ �矄ᨨᆠ�泐�矄v꾀 Qᨨ 矄鼀鼀矄矄矄P�矄ᆠ ᨨ鼀At㦨ᨨ � ᨨ� ᨨ 矄� �AB犨� 矄 v �ᆠ�ᨨ 犨
矄� �鼀泐 �ᆠ tP tQ 矄ᆠ��鼀矄├ 泐P Qᨨᆠ ᨨ� B
҈黨黨㜠㝐├矄桌 㡰
t P < tQ
㡰
t P = tQ
ᆠ㡰
t P > tQ
㡰
tP � ᨨ ᆠ ACB = � ᨨ ᆠ AB tQ
7
턼
A
B ᔴ
Match the Following
䰘 꾀ᨨᆠ├ᨨ 鼀矄矄m҈㦨҈├� ᨨ 鼀矄矄m㦨├� ᆠ ᨨ� 矄 ��盐㡰ᨨ � ᆠ� 犨� � ᨨ � ᆠ� �ᨨ q�ᨨ盐矄盐矄ᨨ q�� � �矄㜠ᔼ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├m҈ �矄 犨泐矄 ᨨ ᆠᨨ �ᆠ� ᨨ 矄�ᆠ 泐 鼀�ᆠ �ᆠ�ᨨ
犨 ᨨᆠ├m ⼄盐ᨨm㦨ᔼ
�矄㜠ᔼᔼ 矄矄�ᨨ 矄 ᨨ �ᆠ�ᨨ ᨨ ᨨᆠ├m�� 㝐ᆠ ᆠᨨᆠ 矄矄�ᨨ ᨨ �ᆠ�ᨨ � �矄㜠ᔼᔼ犨� �矄�� � �矄㜠ᔼ ᆠ ᨨᨨ矄 ᆠᨨᆠᨨ�ᨨ
ᆠᆠ�ᨨ 盐ᨨ��泐�矄 ᨨ 泐g 矄 盐� ᨨ鼀�ᨨ �� 㡰 » ҈錐 ҈҈� 㡰 » 錐
҈� 㡰 » 꾀 ҈桌 �矄㜠ᔼ �矄㜠ᔼᔼ m҈ m q㦨� ҈ mg矄� q 턼 q㦨҈ m҈졔m㡰g矄� q q㦨҈� mmgᆠᨨ矄q 和 q㦨 m m҈졔m㡰gᆠᨨ矄q q ᔴᨨ 㡰 㜠҈턼㜠҈ 㜠҈和㜠 㡰 㜠턼㜠 㜠和㜠 ᆠ㡰 㜠턼㜠 㜠和㜠 㡰 㜠턼㜠 㜠和㜠 10 Subjective Problems 꾀ᆠ�ᆠ盐 �矄ᆠ犨� �ᨨᨨᨨ ��矄 �鼀�矄ᆠ ᨨ� ᨨ 泐ᨨ ᨨ盐� 矄盐 ᆠ꾀ᨨᆠ├ᨨ 鼀矄矄҈├��矄 ᆠ 矄矄 ᨨ犨 ᆠᨨ㜠 �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ 矄盐 ᆠ矄ᨨ �ᨨᨨ ᨨ� ᨨ 矄盐 ᆠ�
ᆠᨨ ᆠ�矄m㦨 �矄ᆠ 矄 ᆠᆠ�ᨨ ᨨ �鼀⏌矄 � ᨨ犨 矄 � ᆠᆠ�ᨨ ᨨ ᨨᆠ├犨� 矄ᆠ 㜠㝐桌 ᨨ �矄ᆠ� ᆠᨨ矄 q = 矄� q = � �
�鼀⏌矄
q
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EBD_8341 P-28
黨
Physics
犨ᨨᨨᆠ├A Bᨨ ⼄盐鼀矄矄矄ᆠ ᨨ ᨨ盐� � ᆠ� 矄矄 ᨨ犨 � ��盐␔ 犨 犨� 犨ᨨᨨᆠ├矄ᆠᨨ鼀ᨨ 矄鼀� ᨨ � ᆠ� � 泐矄 矄�鼀盐 ᨨ盐矄泐䁸ᔼ ��泐 ᨨᆠ├A �矄 m � ᨨᆠ├ B ᔴᨨ㜠 �ᆠ� ᨨ ├� �ᆠ �ᆠ�ᨨ ᨨ ᨨᆠ├矄A B 矄ᆠ�泐 g㦨҈鼀⏌矄㡰 㜠㝐├矄桌 鼀
�º
p
҈
p
҈
꾀
ᆠ ᆠᨨ �ᆠ� ᨨ ├� �ᆠ �ᆠ�ᨨ 犨 鼀矄矄矄 矄盐 ᆠ矄�矄� � ᆠ� 鼀├矄 �ᨨ 䰘 犨� ᨨ� ᨨ ҈黨҈㜠㝐├矄桌
䰘
꾀
ᔼ ��盐鼀矄矄矄m҈m M├��├� �├� 矄ᆠ�泐 ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 M �ᨨ盐 �矄 ᨨ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 m҈ M 犨 m �ᨨ盐 �矄 盐泐矄 矄� �矄鼀矄矄矄矄 矄� ��矄 ᆠ泐 ᨨ� ᨨ 犨 P҈ 鼀҈ 矄ᨨ犨 P m 矄� � �矄 ᆠ泐 �ᆠ 犨 P҈ P꾀 ᨨ� ᨨ ᨨᆠ F �矄 � ᨨ 鼀矄矄 M ├ g㦨҈鼀⏌矄 㜠҈㝐├矄桌
ᔼ 鼀矄矄M҈鼀ᨨ矄 ᨨ犨 犨 矄犨� 盐 � ᨨ鼀ᨨᆠ�泐 � �㡰 鼀矄矄ᨨ M҈ ��㡰 矄�ᨨ � ᨨ� ᨨ ᨨ�ᨨ ᨨ 矄� � g㦨黨鼀⏌矄ᆠ矄� 䰘 ~ - ⏌�㡰 꾀 ᨨ� ᨨ 盐 � ᨨ鼀 ᨨ ᨨ � L 矄� � ᨨ �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ�矄盐 ᨨ 泐 ᨨᆠ F␔ �矄 矄�ᨨ � ᨨ �矄 ᆠl ᨨ鼀 ҈黨䰘桌 犨 ᨨᆠ�矄� 䁸 犨ᨨᆠ盐矄ᨨ 鼀矄矄矄m҈ mᨨ 犨ᨨ �ᆠ�ᨨ 矄矄 矄ᨨ矄ᨨ ᨨᆠ├A犨 �ᆠ 矄矄ᨨ ᨨ� ᨨ ᆠ盐矄ᆠᨨ
ᆠ 泐矄� �犨 �ᆠ 矄矄矄ᨨ盐泐 矄矄 ᨨ犨 � ��盐 ᨨ犨 ᨨ� ᨨ ᆠᆠ�ᨨ
m҈
P҈
m P
M
F
b
f
m
m҈ A
㡰 犨 ᨨ 泐 ��鼀 ᨨ鼀矄矄Mᆠ泐矄 ᨨ犨� � ᨨᆠ矄
҈
㡰 鼀� �盐 ᨨ ᨨᆠᨨ �ᆠ�ᨨ 犨 m҈ Mf҈ 犨 m �ᨨ盐 f ᨨ�ᆠ盐F��矄 ᨨ盐 f҈㦨f � f҈ f ␔�⼄盐�ᨨ 矄ᨨ 鼀ᨨ�ᨨ ᨨ 鼀矄矄矄 � F 矄�ᨨ � 矄� � ᆠᆠ�ᨨ ᨨ 鼀矄矄矄 꾀�ᆠᨨ 鼀矄矄m矄矄ᨨ ᨨ� ᨨ ᨨᨨ犨� 犨 �ᆠ � 矄ᆠᨨ �ᆠ� ᨨ 矄�ᆠ �ᆠ�ᨨ m ᔼ�矄 矄� ᨨ鼀├ ᨨ 泐鼀ᨨ泐泐� � 鼀� �鼀盐鼀ᨨ矄矄� ᨨᆠF � 鼀�盐�盐 ᨨ F �ᆠ�ᨨ � 犨 �ᆠ � 矄 ᨨ� ҈黨䰘㜠䰘㝐├矄桌 㝐矄矄矄 M҈ M M ᆠᨨ
ᆠ 泐 矄� �矄 ᨨ
����鼀矄矄犨 �ᆠ 矄矄ᨨ鼀矄矄矄矄 �ᆠ�ᨨ 矄矄 盐泐矄P҈ P矄矄 ᨨ犨 � �� 鼀矄矄矄鼀ᨨ矄盐ᆠ ᨨ�ᨨ ᨨ 矄� �犨 P҈ P� ᨨ � ᆠ� ᨨ�ᨨ ᨨ 矄� �犨
P M�矄 ᨨ� ᨨ 鼀矄矄矄M M├�
a
f矄 ᨨ盐 犨 ᨨ矄泐矄鼀 �矄ᨨᆠ├矄 ᆠ盐矄㡰 矄盐ᆠ 矄ᨨ ᆠ盐矄 ᨨ ᨨ矄� ᨨ犨 矄 ␔ �矄 矄�ᨨ ᨨ 矄� �� �矄矄�盐�ᨨ 䁸҈黨䰘桌 � ᔼ ��鼀矄 ᨨ犨 ᨨᆠ├矄AB C A 犨�� 矄├�├� B �├� 矄ᆠ�泐 C f ᆠᨨ �ᆠ� ᨨ 矄� � � �ᆠ�ᨨ 犨 泐 犨ᨨ矄盐 ᆠ�矄�A�矄 矄泐鼀矄矄矄矄��� ᨨ � ᨨ 犨犨 �B Cᆠᨨ
ᆠ 泐�� �ᆠᨨ 矄矄� �ᨨ盐 �ᆠ�ᨨ 矄矄盐泐 � ᨨᆠF ᆠ矄矄泐ᨨ �Cᨨ � ᨨ� ᨨ 矄盐 ᆠ ᨨ ᆠᨨ 矄 矄 꾀矄矄盐鼀 �鼀 矄 ᨨ犨 � ��鼀Bᨨ C Aᨨ B�矄鼀� � ᨨ盐� �㦨黨鼀⏌矄㡰 ҈黨䰘桌
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P-29
Laws of Motion
Topic-4 : Circular Motion & Banking of Road 1 1.
2.
5.
MCQs with One Correct Answer
A disc rotates about its axis of symmetry in a hoizontal plane at a steady rate of 3.5 revolutions per second. A coin placed at a distance of 1.25cm from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is (g = 10m/s2) [Main Online April 15, 2018] (a) 0.5 (b) 0.7 (c) 0.3 (d) 0.6 A conical pendulum of length 1 m makes an angle q = 45° w.r.t. Z-axis and moves in a circle in the XY plane.The radius of the circle is 0.4 m and its centre is vertically below O. The speed of the pendulum, in its circular path, will be : [Main Online April 9, 2017] (Take g = 10 ms–2) Z (a) 0.4 m/s
(a)
(c) 5 6.
O q
(b) 4 m/s (c) 0.2 m/s 3.
7.
X Y
q f O
4.
(b)
1 sin f = cos q 2 3 sin f = cos q 4
2 (c) sin f = cos q (d) 3 A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of anguar velocity of ball (in radian/s) m is [2011] (a) 9 (b) 18 (c) 27
L
8.
v
v
(b)
(d)
v
v
True / False
A simple pendulum with a bob of mass m swings with an angular amplitude of 40°. When its angular displacement is 20°, the tension in the string is greater than mg cos 20°. [1984 - 2 Marks] 6
C
(d) 2 m/s A particle is released on a vertical smooth semicircular track from point X so that OX makes angle q from the vertical (see figure). The normal reaction of the track on the particle vanishes at point Y where OY makes angle f with the horizontal. Then: [Main Online April 19, 2014]
(a) sin f = cos f
A small block is shot into each of the four tracks as shown below. Each of the tracks rises to the same height. The speed with which the block enters the track is the same in all cases. At the highest point of the track, the normal reaction is maximum in [2001S]
MCQs with One or More than One Correct Answer
A wire, which passes through the hole in a small bead, is bent in the form of quarter of a A circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire 90° without friction. As the bead B moves from A to B, the force it applies on the wire is [Adv. 2014] (a) always radially outwards (b) always radially inwards (c) radially outwards initially and radially inwards later (d) radially inwards initially and radially outwards later A simple pendulum of length L and mass (bob) M is oscillating in a plane about a vertical line between angular limit – f and + f . For an angular displacement q ( | q | < f ), the tension in the string and the velocity of the bob are T and V respectively. The following relations hold good under the above conditions : [1986 - 2 Marks] (a) T cos q = Mg. MV 2 L (c) The magnitude of the tangenial acceleration of the bob | aT | = g sin q (d) T = Mg cos q
(b) T – Mg cos q =
(d) 36
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EBD_8341 P-30
Physics
10 Subject Problems 꾀矄鼀ᨨᨨ 矄鼀�ᆠ�ᆠ盐犨�㜠ᆠ├ᨨ �盐矄R�矄 � � �ᆠ 졠 ᨨ 鼀矄矄矄矄矄� �ᨨ 盐 � R⏌�矄ᆠ ᨨ ᨨ犨矄ᨨ� Oᨨ 犨�㜠 ᆠ├꾀矄鼀� �ᨨ 鼀矄矄m犨 �ᆠ ᆠ 矄� ᨨ ᆠ├ �矄ᆠ ᨨ ᨨ ᨨ 矄� � � � �矄 矄�ᨨ 泐ᨨ� P矄盐ᆠ 矄� �鼀├矄 �ᨨ 犨� �ᆠ 矄� � ᆠᨨ 矄 K㦨 mg⏌R ᔴᨨ 矄� � 矄 犨 � ��矄矄 �㡰 犨 ᨨ 泐 ��鼀ᨨ � � ��㡰 鼀� � �ᆠᆠ�ᨨ ᨨ � � ᨨ鼀ᆠ�ᨨ ҈黨黨㜠�㝐├矄桌
黨
Answer Key Topic-1 : Ist, IInd & IIIrd Laws of Motion ҈ ҈҈
㡰 㡰 �㡰 ҈ 矄
㡰 ҈ 㡰
㡰 ҈ 㡰
� 㡰 ҈� 㡰
㡰
䰘 㡰
㡰
黨 㡰
҈ 㡰
҈ 矄
Topic-2 : Motion of Connected Bodies, Pulley & Equilibrium of Forces ҈ ҈҈
㡰 㡰
㡰 ҈ 㡰
㡰
㡰
� 㡰
㡰
䰘 ᆠ㡰
㡰
黨 ᆠ㡰
䰘 㡰 ҈䰘 㡰 䰘 㡰
㡰 ҈ 㡰
黨 㡰 ҈黨 㡰
Topic-3 : Friction ҈ 㡰 ҈҈ 㡰 ҈ �㡰
ᆠ㡰 ҈ ᆠ㡰 �㡰
㡰 ҈ 㡰 㡰
㡰 � 㡰 㡰 ҈ 㡰 ҈� 㡰 ҈ 㡰 矄 � ᆠ㡰 㡰
҈ 㡰 㡰
Topic-4 : Circular Motion & Banking of Road ҈
㡰
㡰
ᆠ㡰
㡰
� 㡰
矄
䰘 㡰
ᆠ㡰
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P-31
Work, Energy and Power
5
Work, Energy and Power
Topic-1 : Work Done by Constant & Variable Force 1 ҈
MCQs with One Correct Answer
꾀矄ᨨ 盐矄 矄 ᨨ ᨨ ᨨ盐� ᨨ� ᨨ ᨨ鼀 矄盐 ᆠ �矄 ᨨᆠ ᨨ ᨨ �矄 ᆠ ᨨ ҈�鼀 �矄ᨨ�矄矄�泐� �矄� ᨨᆠ 盐ᆠ矄� 泐犨� �矄 ᆠᨨ҈ ᨨ �矄 ᆠ ᨨ盐� 犨 �ᆠ ᨨ 矄 鼀ᨨ �矄鼀 ␔ �矄 犨ᨨ├ ᨨ 泐 矄ᨨ 盐� � ᨨ 鼀ᨨ鼀 ᨨ ᨨ 䁸 㝐� 和 ᔼᔼ㡰桌 㡰 㡰 䰘 ᆠ㡰 �黨 㡰 �� B C
q A 꾀矄鼀ᨨᆠ├矄矄矄�� � ᨨ犨 ᨨ鼀ᨨ� Bᨨ
� ᆠ� AB犨 �ᆠ �矄鼀├� � �q犨� ᨨ� ᨨ 矄ᆠ�ᨨ BC�矄矄鼀ᨨᨨ 鼀� � �矄ᆠ�ᨨ
CA�矄ᨨ盐� 犨� ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ mᔼ�矄 ᨨ盐 ᨨᆠ├ᆠᨨ鼀矄ᨨ矄矄�ᆠ 矄 ᨨᨨ鼀 ᨨ� A㡰ᨨ � ᆠ� ᔼ BC = AC ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ �矄 �� 泐m㦨k q 盐ᨨ k�矄[㝐� 2 Sep.2020 (I)] 㡰 ҈ 㡰 ᆠ㡰 㡰 r ᔴᨨ 矄� ᨨᆠ F = - x�ˆ + yˆ 犨ᨨ├ ᨨ 泐 �矄 ᨨᆠ� 鼀ᨨ� ��ᆠ ᨨ鼀ᨨ� 꾀 ҈㡰ᨨ ҈㡰 ᨨ � � 矄�鼀 �矄 ⼄盐 ��矄� 和ᔼ盐 �矄㡰 㝐� 黨 ᔼ㡰桌
҈ ᆠ㡰 ҈ 㡰 꾀 ᨨᆠ├ ᨨ 鼀矄矄鼀 �矄 ├ ᨨ ᨨ鼀 犨 �ᆠ 矄矄 ᨨ鼀 矄 犨� ᆠᨨ 矄 ᆠᆠ�ᨨ �⏌ 盐犨 矄 矄 ᨨ犨 � ��犨ᨨ├ ᨨ 泐 ᨨ鼀ᆠ�ᨨ ᨨ ᨨᆠ├�
�鼀�矄 㝐� ҈ ҈黨 ᔼ㡰桌
㡰
㡰 -
�
㡰
鼀 �
㡰
鼀 �
鼀 � 꾀ᨨ 泐ᨨ 鼀矄矄矄矄鼀ᨨ� � ᨨ鼀矄ᨨ � 㜠 �矄矄ᨨ �矄ᨨᆠ�泐�矄矄 = 矄 犨 �矄ᆠᨨ 矄 矄 �矄 �矄 ᆠᆠᨨ 泐 ᨨ 泐 ᨨ犨ᨨ├ ᨨ 泐 ᨨᆠ矄ᆠ� �ᨨ ᨨ 泐� �矄矄ᆠᨨ 矄ᨨ 鼀ᨨ�ᨨ �矄㝐� 졠 � 꾀�҈҈桌 ҈ ҈ 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀 㡰 ␔ 盐㜠 �矄矄ᆠ 泐 �矄 ᆠ � 矄 矄ᨨ� � ᨨᆠᨨ 鼀� �盐 㦨 졔 犨 ᆠᨨ 矄 矄 犨ᨨ├ ᨨ � 矄ᆠ � � 盐 矄ᆠ 盐㜠 泐�矄 㝐� ҈桌 ᆠ㡰
㡰
㡰 +
㡰
+ ᆠ㡰
҈ æ ö 㡰 çç + ÷÷ è ø
(
҈ +
)
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EBD_8341 P-32
7.
8.
Physics
The work done on a particle of mass m by a force, é ù x y ˆj ú Kê iˆ + 2 2 32 2 2 32 (x + y ) êë ( x + y ) úû (K being a constant of appropriate dimensions), when the particle is taken from the point (a, 0) to the point (0, a) along a circular path of radius a about the origin in the x – y plane is [Adv. 2013] Kp Kp 2K p (a) (b) (c) (d) 0 a a 2a If W1, W2 and W3 represent the work done in moving a particle from A to B along three different paths 1,2 and 3 respectively (as shown) in the gravitational field of a point mass m, find the correct relation between W1, W2 and W3 [2003S] B
(a) W1> W2 > W3 (b) W1= W2 = W3
9.
2
1
Integer Value Answer
11. A light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses 0.36 kg and 0.72 kg. Taking g = 10 m/s2, find the work done (in joules) by the string on the block of mass 0.36 kg during the first second after the system is released from rest. [2009] 3
m
Numeric Answer
12. A particle is moved along a path AB-BC-CD-DE-EF-FA, as ur shown in figure, in presence of a forced F = (ayi$ + 2ax $j ) N , where x and y are in meter and a = –1Nm–1. The work done on the particle by this force ur [Adv. 2019] F will be ___ Joule.
2 3
(c) W1< W2 < W3 (d) W2> W1 > W3
10. A uniform chain of length L and mass M is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If g is acceleration due to gravity, the work required to pull the hanging part on to the table is [1985 - 2 Marks] (a) MgL (b) MgL/3 (c) MgL/9 (d) MgL/18
A
y
A force F = - K ( yiˆ + xjˆ) (where K is a positive constant) acts on a particle moving in the xy plane. Starting from the origin, the particle is taken along the positive x axis to the point (a, 0), and then parallel to the y axis to the point (a, a), The total work done by the force F on the particle is [1998S - 2 Marks] (a) – 2Ka2 (b) 2Ka2 (c) – Ka2 (d) Ka 2
A
1.0
B
D
.5 F
C
E 0.5
0
x
1.0
Topic-2 : Energy 1 1.
MCQs with One Correct Answer
A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is : [Main 7 Jan. 2020 II]
3.
4.
1
1 (d) n 2 n n A body of mass 1 kg falls freely from a height of 100m, on a platform of mass 3 kg which is mounted on a spring having spring constant k = 1.25 × 106 N/m. The body sticks to the platform and the spring’s maximum compression is found to be x. Given that g = 10 ms–2, the value of x will be close to : [Main 11 April 2019 (I)] (a) 40 cm (b) 4 cm (c) 80 cm (d) 8 cm A uniform cable of mass ‘M’ and length ‘L’ is placed on a
(a) n
(b)
2
(c)
th
2.
(a) 4 J (b) 2.5 J (c) 6.5 J (d) 5 J A spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths 11 and l2 where, l1 = nl2 and n is an integer. The ratio k1/k2 of the corresponding force constants, k1 and k2 will be: [Main 12 April 2019 (II)]
æ 1ö horizontal surface such that its ç ÷ part is hanging è nø below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be: [Main 9 April 2019 (I)] 2 MgL MgL MgL (a) (b) (c) (d) nMgL 2 2 2n n n2
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P-33
Work, Energy and Power
5.
A particle moves in one dimension from rest under the influence of a force that varies with the distance travelled by the particle as shown in the figure. The kinetic energy of the particle after it has travelled 3 m is : [Main 8 April 2019 (I)]
11. A point particle of mass m, moves long the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals m. The particle is released, from rest from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR. The value of the coefficient of friction m and the distance x (= QR), are, respectively close to : [Main 2016] P
6.
(a) 4 J (b) 2.5 J (c) 6.5 J (d) 5 J A block of mass m, lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant k. The other end of the spring is fixed, as shown in the figure. The block is initally at rest in its equilibrium position. If now the block is pulled with a constant force F, the maximum speed of the block is:[Main 9 Jan. 2019 (I)] m
(a) 7.
2F mk
(b)
F p mk
(c)
F
pF mk
(a) -
k 4a
(c) zero
2
(b)
10.
mk
k 2r 2
. Its total energy is: [Main 2018]
k
2a 2 3 k (d) 2 a2
A body of mass m = 10–2 kg is moving in a medium and experiences a frictional force F = –kv2. Its intial speed is v0 = 10 ms–1. If, after 10 s, its energy is
9.
F
A particle is moving in a circular path of radius a under the action of an attractive potential U = -
8.
(d)
1 2 mv0 , the value of k 8
will be: [Main 2017] (a) 10–4 kg m–1 (b) 10–1 kg m–1 s–1 (c) 10–3 kg m–1 (d) 10–3 kg s–1 An object is dropped from a height h from the ground. Every time it hits the ground it looses 50奠 of its kinetic energy. The total distance covered as t ® ¥ is [Main Online April 8, 2017] 8 5 h (a) 3h (b) ¥ (c) (d) h 3 3 A time dependent force F = 6t acts on a particle of mass 1 kg. If the particle starts from rest, the work done by the [Main 2017] force during the first 1 second will be (a) 9 J (b) 18 J (c) 4.5 J (d) 22 J
h=2m
30° Horizontal Surface
R Q
(a) 0.29 and 3.5 m (b) 0.29 and 6.5 m (c) 0.2 and 6.5 m (d) 0.2 and 3.5 m 12. A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up? Fat supplies 3.8 × 107 J of energy per kg which is converted to mechanical energy with a 20奠 efficiency rate. Take g = 9.8 ms–2 : [Main 2016] –3 –3 (a) 9.89 × 10 kg (b) 12.89 × 10 kg (c) 2.45 × 10–3 kg (d) 6.45 × 10–3 kg 13. A particle is moving in a circle of radius r under the action of a force F = ar2 which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0) : [Main Online April 11, 2015] 5 3 4 3 1 3 αr ar (b) ar (c) (d) ar3 6 3 2 14. A block of mass m = 0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance x from its equilibrium position and released from rest. After approaching half the distance æç x ö÷ from equilibrium è 2ø position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms–1. The total initial energy of the spring is : [Main Online April 10, 2015] (a) 0.3 J (b) 0.6 J (c) 0.8 J (d) 1.5 J
(a)
th
æ1ö 15. A bullet looses ç ÷ of its velocity passing through ènø one plank. The number of such planks that are required to stop the bullet can be: [Main Online April 19, 2014] (a)
n2 2n 2 (b) 2n - 1 n -1
(c) infinite
(d) n
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EBD_8341 P-34
҈
҈䰘
Physics
犨ᨨ 矄� �矄 ᨨ ᨨᆠ ᆠᨨ 矄 矄 ⏌鼀 和� �꾀㡰 ⏌鼀 和� �㡰ᨨ� ᨨ� �
矄�矄 ᆠᨨ鼀� �ᨨ �矄ᆠᨨ鼀矄矄 泐䰘�ᆠ鼀 E E �ᨨᨨ �泐矄ᨨ � 꾀 �矄 A A �矄 EB EB ⼄盐 㝐� 졠 � 꾀�黨҈桌 ᨨ ҈ 黨 㡰 㡰 ᆠ㡰 㡰 黨 ҈ 꾀ᨨᆠ├ᨨ 鼀矄矄├��矄 ᨨ鼀ᨨᨨ � 㜠 �矄ᔼ�矄 矄 ᨨ鼀㦨ᨨ 犨 矄��矄矄盐ᆠ ᨨ�鼀㜠 ᨨᆠF t㡰� x �ᆠ�ᨨ ᨨᆠF t㡰�矄 犨� t矄矄 ᨨ犨 � ��盐 ├� �ᆠ �泐ᨨ ᨨᆠ├ �矄ᆠᨨ 矄�矄 ҈桌 F t㡰
꾀 � 矄� �犨� 矄� �㜠ᆠᨨ 矄 k�矄 盐 � ᨨ鼀 ᆠ�� � ᨨᆠ├ᨨ 鼀矄矄M�矄ᆠ ᨨ�矄ᨨ犨
鼀矄矄�矄矄 犨� 矄� �� ��泐盐 矄ᆠ
鼀 �鼀盐鼀 矄�ᨨ � 矄� ��矄 和桌 Mg Mg Mg Mg 㡰 ᆠ㡰 㡰 k k k k ҈ 꾀�ᆠ犨 �ᆠ �矄ᆠᨨ 矄� ᨨ鼀ᨨᨨ � x㜠 �矄 �矄 矄盐ᆠ ᨨ ᨨᆠ � 矄鼀 �ᆠ�ᨨ 犨 �ᆠ �矄犨� �矄 ᆠxᨨ �ᆠ ᨨ鼀 ᨨ��� 矄 F x㡰㦨kx졔axk aᨨ矄��ᆠᨨ 矄 矄 ᨨ x ³ 盐 ᆠ�ᨨ ᨨ鼀ᨨ ᨨ � �泐U x㡰ᨨ �ᆠ�矄 和桌
㡰
U(x)
U(x)
㡰
㡰 �
X
㡰
X
㡰
X
N
㡰 䰘� �s
ᆠ㡰 �
s
O
t U(x)
҈
㡰 ҈ 꾀ᨨᆠ├ B㡰�矄ᆠ ᨨ犨ᨨ盐 矄ᆠ 矄� �矄S҈ S犨� 矄� �ᆠᨨ 矄 矄k k矄ᆠ�泐 矄 ��ᔼ㡰
ᨨ 矄ᆠ ᨨ� �ᆠ矄盐ᨨ矄M҈ M ᨨᆠ ᨨ 犨矄 矄� �矄 矄盐ᨨ矄 ����鼀矄矄 �矄 ᨨ �ᆠ�ᨨ 泐犨 ᨨᆠ├B�矄 �矄ᆠ ᨨ犨 矄犨҈泐矄鼀 �矄 ᆠx ��盐ᔼᔼ㡰 矄 ᨨᆠ├盐 矄 鼀ᨨ矄 鼀 �鼀盐鼀 �矄 ᆠyᨨ犨 矄犨�矄ᆠ鼀 矄x y 鼀矄盐 犨� 矄ᆠᨨ ⼄盐���盐鼀ᨨ矄��ᨨ ᨨ ᨨᆠ├B �ᨨy⏌x�矄 桌 ҈ M҈ M S S҈ ᔼ B x
M
҈黨
S
҈ S҈
B
M҈
ᔼᔼ
x 㡰 㡰 ᆠ㡰 ҈⏌ 㡰 ҈⏌ 꾀�ᆠ�矄ᆠ 泐 ᨨᆠF㦨kx犨 k�矄졔 ᆠᨨ 矄 ᔼ矄ᨨ � �泐x㦨�矄 ᨨ␔ �ᆠ ᆠ盐 ᆠᨨᆠ泐矄 矄 ��ᨨ ᨨ ᨨ � �泐ᨨ ᨨᆠ├犨� 矄ᆠᨨx 和桌 U
U(x)
ᆠ㡰
X
꾀矄� �ᨨ ᨨᆠ㜠ᆠᨨ 矄 k�矄ᆠ盐� ᨨ犨ᨨ�ᆠ矄矄盐ᆠ ᨨ �ᆠ�矄 ᨨ盐 � ᨨ ᨨ ᨨ ��ᆠ犨� ᨨᆠ㜠ᆠᨨ 矄 ᨨ ҈黨黨黨和㜠㝐├矄桌 㡰 ⏌㡰k 㡰 ⏌㡰k ᆠ㡰 k 㡰 k 犨ᨨ鼀矄矄矄ᨨ ҈�鼀 �鼀鼀ᨨ� �犨� ⼄盐├� �ᆠ ��矄 �ᨨ ᨨ 鼀� �盐 矄 ᨨ � � 鼀ᨨ鼀 �矄 ҈黨桌 㡰 ҈ 㡰 㡰 ҈҈ ҈ ᆠ㡰 ҈ 2
Integer Value Answer
ᔴᨨ 矄� ��ᆠ矄 �PQ � �ᆠ 犨� OP㦨鼀 OQ㦨鼀꾀ᨨᆠ├ᨨ 鼀矄矄҈├��矄盐 ᨨ � � ᨨ鼀PᨨQ犨� ᨨᆠᨨ ҈犨 �ᆠ �矄 犨泐矄ᨨ� PQ 矄 ��盐�� 㡰꾀矄矄盐鼀� �
ᨨ �ᆠ�ᨨ 矄矄ᨨ矄矄矄 ├� �ᆠ �泐ᨨ ᨨᆠ├犨
� ᆠ 矄Q�矄 n 맔҈㡰 ᨨ盐矄 盐ᨨ n �矄 ├ ᆠᆠ�ᨨ 盐ᨨ��泐㦨҈鼀矄㡰 꾀 ҈桌 Q
U
鼀
㡰
x
㡰
x
黨 U
ᆠ㡰
O
U x
㡰
x
鼀
P
� 꾀ᨨᆠ├ᨨ 鼀矄矄҈├��矄ᆠ ᨨ矄� �ᨨ ᨨᆠ㜠 ᆠᨨ 矄 ⏌鼀 ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ ᨨᨨ�矄҈ᔼ ��泐 ᨨᆠ├�矄矄 矄� ��矄盐 㜠矄ᆠ 꾀 �鼀盐矄�矄�� ᨨ ᨨᆠ├
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P-35
Work, Energy and Power
矄矄 ᨨ犨 � ��盐 ᨨᆠ├矄� 矄 �矄 ᆠᨨ 鼀 ᆠᨨ鼀矄ᨨ矄 ᨨ �矄�鼀 � ��ᨨᆠ�泐ᨨ ᨨᆠ├� 鼀⏌矄�矄㦨⏌҈ �矄 ҈҈桌
3
䰘
Numeric Answer
꾀ᆠ�ᆠ├ᨨ 鼀矄矄҈�├��矄 ᨨ犨 �ᆠ泐盐泐 ᨨ犨� �鼀ᆠ � 矄ᨨ ��矄矄ᨨ鼀 �鼀盐鼀 �� ᨨ 鼀 � � 鼀ᆠ � ᔼ 盐矄 � � �矄 ᆠᨨ 矄 ᨨᆠ F ᨨ 鼀ᨨ矄 ᨨ� ᨨ 泐 �矄 ᆠᨨ 鼀犨 �盐 ᆠ � � 盐ᨨ F � 㡰�矄 g㦨҈鼀矄㡰힘힘힘힘힘힘힘힘힘힘 [Main 3 Sep. 2020 (I)] 꾀�ᆠ m 㦨├�㡰矄� 矄 ᨨ犨 �ᆠ�ᨨ 矄矄ᆠ├ 꾀졠ᔴ㡰 矄� � ᨨ鼀矄ᨨ� A �� 鼀㡰꾀 ᆠ � � C �ᆠᆠᨨ � 盐矄ᨨ鼀ᨨ 泐� �矄ᨨᆠ� ␔ �ᆠ � ��矄 �� 矄ᨨ� P �� ҈鼀㡰 ├� �ᆠ �泐ᨨ �ᆠ � J㡰�矄 ��盐 犨 �矄 矄ᆠ 鼀�ᆠ ᨨᨨ矄ᆠ錐├g 㦨҈鼀矄㡰¾¾¾ . Main 䰘 ᔼ㡰桌
㡰 v - gh =
҈ gR
gR 㡰 v - gh = ᆠ㡰 ᆠ � ᨨᆠ⼄盐� ᨨ� 矄x z�矄 ᨨ 㡰 ᆠ � ᨨᆠ⼄盐� �矄鼀 �鼀盐鼀ᨨ� 矄x z 黨 꾀�ᆠᨨ 鼀矄矄m�矄� ��泐矄 ᨨ��� ᔼ�矄 矄盐ᆠ ᨨ ᨨᆠ 矄矄鼀ᨨ� �ᨨ � x㜠 �矄ᔼ矄 ├� �ᆠ �泐Kᆠ �矄犨� �鼀矄 ⧤⏌ 㦨g犨 g �矄ᨨ矄��ᆠᨨ 矄 ᨨ ᨨ� �鼀 矄�ᨨ 矄␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 㡰盐䁸 꾀 ҈桌 㡰 ᨨᆠ� ᨨ �ᆠ�矄ᆠᨨ 矄 㡰 矄 ᨨ �ᆠ�矄ᨨᨨ�ᨨ ᨨ�鼀 ᆠ㡰 �矄 ᆠᨨ �ᆠ ᨨ鼀 ᨨ��� � ᆠ矄矄 � 泐犨� �鼀 㡰 ᨨᆠ�矄ᆠᨨ 矄� 꾀矄鼀矄矄鼀ᨨ� � ᨨ鼀Aᨨ � ᆠ├矄矄 ᨨ犨
� ��盐和盐 ᆠAB 矄 �ᆠ�ᨨ ᨨ鼀AᨨB ᨨ矄犨� ᨨ盐矄�� �和盐 ᆠBC�矄 �ᆠ�ᨨ 矄矄KAKB KC├� �ᆠ ��矄ᨨ AB C矄ᆠ�泐
㜠�㝐҈桌 C
��
A
A
P C
鼀
hC
hA
B
O
6
㡰
hA > hC 錐 K B > KC
㡰
hA > hC 錐 KC > K A
ᆠ㡰
hA = hC 錐 K B = KC
㡰
hA < hC 錐 K B > KC
MCQs with One or More than One Correct Answer
꾀矄盐 矄├矄盐鼀 鼀├矄 � 犨� ᨨ� ᨨ ⏌矄 矄矄 矄矄 ᨨ犨 � ��盐㡰 ᨨᨨ鼀ᨨ 鼀犨� 矄 v 犨 矄ᨨ盐 ᨨ盐 ᨨ矄鼀�ᆠ�ᆠ盐 xyz ᨨ �盐矄R 盐� �犨 �ᆠ ⏌ 矄 ᆠ 矄鼀 �鼀盐鼀 �� h ᨨ� y㡰 ᨨ鼀 �ᨨ盐 矄矄 ᨨ犨 � ��盐꾀矄矄盐鼀 �泐ᨨ矄矄 �矄
���� ᨨᆠ⼄盐� ᨨ �矄盐 �� 矄 ᨨ� �矄ᨨ� 泐 �矄⏌ 犨�� ᨨ 泐 g�矄 ᆠᆠ�ᨨ 盐ᨨ��泐㡰 꾀 桌 z R
y
x h
҈ 꾀矄ᨨ � ᨨ矄� �ᨨ � �矄犨 � � �ᆠ ᆠ�ᆠ犨� ᨨ ᨨ 矄� � ᆠ 꾀 ᆠ� � 矄 ᨨ �鼀 矄ᨨ �矄�矄ᨨ犨矄ᨨ矄��ᨨ 矄矄 u 鼀� �盐 ᨨ ᆠ �� �矄 ᨨᆠ�泐矄 � ᆠ 矄 ᨨ矄��ᨨ 犨 矄� � �矄 ᨨ� ᨨ �矄 ҈黨黨和㜠㝐├矄桌 㡰
u - gL
㡰
gL
ᆠ㡰
u - gL
㡰
u - gL㡰
꾀�ᆠ�矄ᆠ 盐ᨨ 泐 ᨨᆠᨨ ᆠᨨ 矄 鼀� �盐 犨 �ᆠ �矄 犨泐矄 �ᆠ盐 ᨨ ᨨᆠ�泐 ᨨ �ᆠ 鼀ᨨ�ᨨ ᨨ �ᆠ├矄ᆠ� ᔼ ᨨᨨ犨矄 ҈黨䰘㜠㝐├矄桌 㡰 �矄ᨨᆠ�泐�矄ᆠᨨ 矄 㡰 �矄ᆠᆠ�ᨨ �矄ᆠᨨ 矄 ᆠ㡰 �矄├� �ᆠ �泐�矄ᆠᨨ 矄 㡰 �鼀ᨨ矄� ᆠ�ᆠ盐
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EBD_8341 P-36
Physics
Match the Following
7
꾀�ᆠᨨ 盐 �鼀矄矄�矄鼀ᨨ� �ᨨ � x㜠 �矄盐 � 盐 ᆠᨨ ᨨᆠ �矄ᨨ �泐�矄ᆠᨨ 矄 ᨨ盐ᨨ矄矄� ᨨ鼀矄ᨨ ᨨ � �泐ᨨ �ᆠ�� � ᆠᨨ盐鼀 ᔼ a Uᆠᨨ 矄 矄㡰㝐ᆠ ᨨ � ��矄� ᆠᨨ盐鼀
ᔼᨨ ᆠᨨ矄ᨨ � �矄鼀 矄㡰� ᆠᨨ盐鼀 ᔼᔼ ᔴᨨ盐鼀 ᔼ U 꾀㡰 U҈ x㡰㦨
㡰
U x㡰㦨
ᔴᨨ盐鼀 ᔼᔼ é æ x ö ù ê҈ - ç ÷ ú êë è a ø úû
U æ x ö ç ÷ èaø
ù é æ x ö ê- æ x ö ú ç ÷ ᔴ㡰 ç ÷ êë è a ø úû èaø ù é ê x ҈ æ x ö ú ç ÷ 㡰 U x㡰㦨 êë a è a ø úû
U U x㡰㦨
㡰
ᨨᆠᆠ� �ᨨ �ᆠ�矄 ᨨx㦨a
⼄㡰
ᨨᆠᆠ� �ᨨ �ᆠ�矄 ᨨx㦨
㡰
ᨨᆠᆠ� �ᨨ �ᆠ�矄 ᨨx㦨a
矄㡰
�ᆠ � ᆠ矄 ᆠ� ᨨᆠᨨ犨 矄x㦨� ��ᨨ x
㡰
�ᆠ犨� ᨨ �泐
Comprehension/Passage Based Questions
8
꾀和和꾀E 꾀矄鼀ᨨᆠ├ᨨ 鼀矄矄҈├��矄矄 ᨨ鼀矄 ᨨᨨ ᨨ盐� ᆠ├ ᆠ├�矄ᆠ�ᆠ盐ᆠᨨ �盐矄鼀 ᨨᆠ├ 矄� 矄ᨨ � ᆠ├犨� ᨨ盐ᨨ� � �ᆠ�ᨨ ᨨᆠ ᆠ矄ᨨ �� �ᆠ�ᨨ ᨨᨨ矄�ᨨ � 矄 ᨨ盐矄ᨨᆠ�泐
犨ᨨ├ ᨨ � ᨨᆠᨨ鼀� � �ᆠ�ᨨ 盐ᨨ ᨨ� 턼矄 矄 ᨨ犨 � ��盐ᨨ犨�矄҈� ├ ᆠᆠ�ᨨ 盐ᨨ��泐�㦨҈鼀矄㡰 泐
꾀 ҈桌
턼
졠
鼀� �盐 ᨨ ᨨ鼀ᆠ�ᨨ ᆠ矄ᨨ ᨨᆠ├ ᨨ� 턼�矄 㡰 䰘�
㡰
ᆠ㡰 ҈҈�
㡰 �
� 矄 ᨨ ᨨᆠ├犨 �ᆠ 矄 ᨨ� 턼�矄 㡰 �鼀矄҈ ᆠ㡰
҈ 鼀矄 -҈
㡰 ҈鼀矄҈ 㡰 鼀矄҈
9
U ᆠ ᨨ矄ᆠ�ᨨ盐 ᨨ� x㦨
Assertion and Reason Type Questions
和T꾀TE㝐ENT҈꾀ᨨᆠ├ᨨ 鼀矄矄m矄矄鼀ᨨ� �ᨨ ᨨ盐� ᨨ� ᨨ 矄盐 ᆠ犨� ᨨᆠ�泐vᔼ矄ᨨ矄 盐ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ 矄盐 ᆠ 鼀ᨨ� � ᨨ盐� ᆠ� �矄 ᆠ 矄盐 ᆠ�矄 ᨨ犨� ᨨ
�ᨨ 犨� ᨨ� ᨨ 矄鼀ᨨᆠ├�矄 鼀 ᨨ�ᨨ盐ᨨ 矄盐 ᆠ犨� 矄鼀� ��ᨨᆠ�泐 v ᆠ矄� 鼀ᆠ �ᆠ �泐� 矄ᆠᨨ 矄�盐�ᨨ �矄矄鼀 � �矄矄�盐�ᨨ 和T꾀TE㝐ENT ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨᆠ├ 矄盐 ᆠ ᆠ矄矄犨� � ᆠ矄� �ᨨ � ᆠ� �ᨨ 䰘桌 㡰 和鼀 ҈�矄 盐和鼀 �矄 盐錐和鼀 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 ҈ 㡰 和鼀 ҈�矄 盐和鼀 �矄 盐錐和鼀 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 ҈ ᆠ㡰 和鼀 ҈�矄 盐和鼀 �矄 矄 㡰 和鼀 ҈�矄 矄和鼀 �矄 盐 10 Subjective Problems 䰘 꾀矄� �犨� ᨨ � ᨨ ��� 犨矄矄� �ᨨ � �ᆠ�ᨨ 矄矄盐泐 �矄 ᆠᨨ 鼀 ᨨ鼀 犨 矄ᨨ� 鼀矄矄M㦨├�ᆠ ᨨ� �矄 ᆠᨨ ҈鼀 ᨨ鼀 犨꾀鼀矄矄m 㦨�├�ᆠ M �矄 矄矄ᨨ 矄� ��矄 ᨨ� ᨨ 犨 犨 盐泐 �ᆠ泐ᨨ 盐泐␔ 犨� 矄 m 犨� 犨 �ᆠ 鼀矄矄M犨� � 犨犨 鼀矄矄m�矄矄 䁸 ҈黨�㜠㝐├矄桌
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P-37
Work, Energy and Power
꾀� ├�ᨨᆠ├矄� 矄 ᨨ鼀 ᨨ� A 矄 ��㡰ᨨ ᨨ� ᨨ ᆠ├犨� � ��矄 ᨨ 鼀⏌矄ᨨ犨 矄 犨�� 矄矄 ᨨ� ᨨ 矄� � ᨨ � ҈ 鼀 ᨨᆠ ᆠᨨ 矄 犨ᨨ ⏌鼀 AB ᨨ ᆠ├ �矄 �ᆠ�ᨨ 矄矄 BC 矄 ᆠᨨ �ᆠ� 矄ᨨ 矄�ᆠ ├� �ᆠ �ᆠ�ᨨ 矄 矄ᆠ�泐ᔼ �矄 ᆠ矄AB BD鼀 ҈鼀矄ᆠ�泐 � ᨨ �矄 ᆠ ᨨ盐� 犨 �ᆠ ᨨᆠ├鼀ᨨ矄 ᨨ �ᆠᨨ鼀矄ᨨ矄ᆠᨨ鼀泐 ├�㦨҈鼀⏌矄㡰 ҈黨㜠䰘㝐├矄桌
A
黨
B
C
D
犨ᨨ ᨨᆠ├矄 A B ᆠᨨ
ᆠ ᨨᆠ ᨨ 泐 B 矄� � 矄� �錐 矄� � 矄矄矄 ᨨ �ᆠ�ᨨ 矄矄 C 盐泐矄矄 ᨨ犨 � ��盐 ᨨᆠ├B A 矄� 矄ᨨ ᨨ� ᨨ ᨨ 矄盐 ᆠᨨ 矄�ᨨ 泐ᨨᆠ├ C ᨨᆠ├A矄� 矄ᨨ � �ᆠ 矄� ᨨ C ᨨ 犨� 矄鼀盐 � ᨨ鼀矄
ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 矄盐 ᆠ矄ᨨ ᨨᆠ├矄 �矄 ᨨᆠᆠᨨ 矄 ᨨ 矄� ��矄҈黨 犨ᨨ 矄⏌鼀ᔼ
鼀矄矄ᨨ ᨨᆠ├A�矄⧤�ᆠᆠ盐 鼀矄矄ᨨ ᨨᆠ├B �泐矄ᨨ � 矄� � ҈黨㜠�㝐├矄桌 D A
G
F E C 㡰 㡰 ᔼ ��盐矄 㡰 㡰ACDG GF � � ᆠ� 矄BC㦨EF㦨x AB㦨DE㦨y꾀矄鼀ᨨᆠ├ᨨ 鼀矄矄 M�矄矄 ᨨ鼀 ᨨ� Aᔼ矄� 矄 ᨨ犨 AC ᆠ 矄C犨� 矄 Vc 矄鼀 ᨨᆠ├ �矄矄 ᨨ鼀矄 ᨨ鼀 ᨨ� Dᔼ矄� 矄 ᨨ犨 DGF ᆠ 矄 ᨨ� F 犨� 矄 VF ᆠᨨ �ᆠ� 矄 ᨨ ├� �ᆠ �ᆠ�ᨨ 矄犨 ᨨᆠ├ ᨨ 矄盐 ᆠAC DGFm ҈黨桌 ᔴᆠ盐VC VF ҈ ␔ �矄 ᨨ犨 盐 鼀� �盐 ᨨ �矄鼀ᨨ鼀 盐鼀 ᆠ矄矄 � ᆠ矄矄 ᨨ矄 �矄 �ᨨ ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀� ᆠ�䁸 ҈黨䰘黨桌 꾀矄� �ᨨ ᨨᆠᆠᨨ 矄 k�矄ᆠ盐� ᨨ ⼄盐矄 ␔ �矄 ᨨᆠᆠᨨ 矄 ᨨ ᆠ 䁸 ҈黨䰘桌 꾀盐�矄 � ᨨ鼀� ᔼ � ᆠᨨ�矄 泐 鼀� 犨 ├� �ᆠ �泐ᨨ � �矄� ⼄盐 ᨨ矄矄 ᨨ 盐 ҈黨䰘桌 B
Topic-3 : Power
҈
S
ᆠᨨ矄ᨨ ҈㦨䰘␔�㦨҈鼀矄㡰㝐� 䰘 ᔼ桌 㡰 ҈䰘鼀矄҈ 㡰 ҈黨鼀矄҈ ҈ ᆠ㡰 ҈�鼀矄 㡰 鼀矄҈ 꾀�ᆠᨨ 鼀矄矄㝐�矄鼀ᨨ� �� ᆠ�ᆠᨨ � �盐矄 � 矄盐ᆠ 犨泐 �矄ᆠ �ᆠᆠ�ᨨ �鼀�矄 �� 泐 犨 �矄ᆠᨨ 矄 ᨨ犨 � ᨨ �ᆠ泐 ᨨᆠᆠ� �ᨨ ��矄 㝐� 졠 � 꾀�҈҈桌
S
҈ 㝐 㡰 㝐 ᆠ㡰 㝐 㡰 㝐 꾀ᆠᨨ 犨�� ␔�矄ᨨ � ᆠ� ᨨ �矄矄泐҈ 鼀ᨨ �矄 ᆠᨨ ҈⧤鼀 �矄ᆠᨨ 矄 �ᆠ�ᨨ
MCQs with One Correct Answer
1
꾀�ᆠ�矄鼀ᨨ� �盐 � �ᆠ�ᨨ 泐ᨨ ᨨ� ᨨ 盐 ᆠ�ᨨ ᨨ ᆠᨨ 矄 ᨨ犨 矄盐泐� � �泐 矄ᨨ盐ᆠ �矄ᆠ鼀 s㡰㜠�鼀 t㡰� 矄ᆠ�矄 鼀ᨨ�ᨨ ᨨ �ᆠ�矄 � 矄 犨 矄ᆠ 鼀�ᆠ泐 ᨨᨨ矄ᆠ㡰 㝐� 和 ᔼᔼ㡰桌 S
㡰
㡰
㡰 t
t
S
␔ ᨨ ᆠ␔ �鼀ᨨ� �盐 �ᨨ ᨨ 矄 ᨨ ҈鼀矄҈ ᆠ 矄ᨨ犨. ᔼ � 矄ᨨ犨
ᨨᆠ ᆠ㡰
㡰 t
t
꾀ᆠ�ᆠ鼀ᨨᨨ� 矄 ᨨ � �鼀 �鼀盐鼀 ᨨᨨ ᆠᆠ�泐ᨨ ├�ᔼ �ᆠ�ᨨ ᨨᆠᨨ ᨨ�矄 矄 ᨨ ᨨ 盐ᨨ �矄
犨 �鼀ᨨ� � ᨨ犨 �矄 v 盐ᨨ v �矄 㝐� 졠 � 꾀�黨҈桌 㡰 鼀矄҈ 㡰 �鼀矄҈ ᆠ㡰 ҈�鼀矄҈ 㡰 ҈鼀矄҈
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EBD_8341 P-38
�
䰘
Physics
꾀犨� 㜠ᨨ犨 � ᨨᆠᨨ 矄犨� �泐� ᨨᆠ�ᆠ �泐꾀矄矄盐鼀 � ᨨᆠᨨ 矄 � ᆠ�ᨨ
ᨨ 犨� �泐� ᆠ 泐�矄 矄� ᨨᆠ�ᆠ �泐 ᨨ犨� 矄 n ᆠ�ᆠᨨ犨ᨨ盐盐犨� 鼀ᨨ矄�├泐ᨨᨨ�ᨨ ᨨ 㝐� 졠 � 꾀��҈桌 㡰 n 㡰 n ᆠ㡰 n 㡰 n 꾀䰘├�鼀 矄�ᆠ泐� ᨨ � ᨨ鼀ᆠᨨ盐ᆠ � � ᨨ矄��ᨨ ᨨ├ 鼀 盐矄 矄 �ᨨ盐 犨� ᆠᨨ 矄 ᨨᆠFᨨ�矄 �鼀矄 ᆠ ᨨ ��泐 �矄矄泐�鼀 ᨨ 矄꾀 ᆠ��矄矄泐 ᨨ ҈鼀 鼀 �鼀盐鼀ᨨ犨 � 泐 鼀盐矄ᆠ矄 �矄 ├�㦨҈鼀矄㡰 㝐� 졠 � 꾀�҈桌 㡰 맔҈␔矄 矄 㡰 맔҈␔矄├ᨨ ᆠ㡰 맔҈␔矄 矄 㡰 맔҈␔矄├ᨨ 꾀 犨� 㜠ᨨ犨 � ᨨ ᆠᨨ 矄 犨� �泐 � ᨨ ᆠ�ᆠ �泐꾀矄矄盐鼀 � ᨨ ᆠᨨ 矄 � ᆠ�ᨨ ᨨ 犨� �泐� ᆠ 泐�矄 矄 � ᨨᆠ�ᆠ �泐 ᨨ犨� 矄 v ᆠ�ᆠᨨ犨 ᨨ盐盐犨�ᨨᨨ�ᨨ ᨨ 和桌 㡰 v 㡰 v ᆠ㡰 v 㡰 v 꾀�ᆠᨨ 鼀矄矄鼀�矄鼀ᨨ� �� ᆠ�ᆠ盐 ᨨ ᆠᨨ 矄 �盐矄r矄盐ᆠ �矄ᆠ �ᆠᆠ�ᨨ ac�矄泐� � 犨� �鼀t矄ac㦨krt犨 k�矄ᆠᨨ 矄 ᨨ犨 � ᨨ �ᆠ矄泐 ᨨᆠᆠ� �ᨨ ��矄 ҈黨黨㜠҈㝐├桌
㡰 mkrt p mk r t � mk r t 㡰 㡰 ᨨ ᆠ㡰 꾀 ᨨ 泐 �矄 鼀ᨨ ᨨ � 矄�� � 泐 鼀ᆠ � �� � ᆠᨨ 矄 ᨨ犨 �矄 ᆠ 鼀ᨨ 泐 ᨨ 泐� �鼀t�矄ᨨᨨ�ᨨ ᨨ ҈黨㜠㝐├矄桌 㡰 t ҈⏌ 㡰 t ⏌ ᆠ㡰 t ⏌ 㡰 t ᔼ 鼀ᆠ � �矄盐�ᆠ 犨� ᨨ� ҈黨桌 㡰 鼀ᆠ �ᆠ �ᨨ 鼀ᆠ � � ᆠ矄矄 㡰 鼀ᆠ �ᆠ �ᆠ� ᆠ泐ᨨ 鼀ᆠ � � ᆠ矄矄 ᆠ㡰 ᨨ �矄 鼀ᆠ �ᆠ � �ᆠ� ᆠ泐 � ᆠ矄 㡰 �矄 �ᆠ� ᆠ泐� ᆠ矄矄盐�矄鼀ᆠ �ᆠ � ᆠ矄矄
㡰
黨
҈
2
Integer Value Answer
҈҈ 꾀�ᆠᨨ 鼀矄矄├��矄鼀ᨨ� �� ᨨ �鼀 矄�ᨨ 盐 ᨨᆠ �矄ᆠᨨ 矄 ᨨ犨�␔ᨨ �ᆠ ᔼ � ��矄 � 鼀矄҈㡰ᨨ �ᆠ�矄 ᨨ 矄 � 鼀矄҈㡰 �矄�矄 꾀 ҈桌 3
Numeric Answer
҈ 꾀ᨨ 泐ᨨ 鼀矄矄├��矄 � 泐 �� �� � ᆠᨨ 矄 ᨨ犨 ᨨ ҈ ⏌矄 ᨨ 泐 矄矄 ᨨ鼀 矄 鼀ᨨ矄� 矄�� � 꾀 黨矄ᆠᨨ 矄 ᨨ 泐 矄 鼀ᨨ �矄 ᆠ � 鼀㡰힘힘힘힘힘힘 㝐� �和 ᔼᔼ㡰桌
Topic-4 : Collisions 1 ҈
MCQs with One Correct Answer
�ᆠ꾀ᨨ 鼀矄矄鼀҈鼀ᨨ� �犨� ᨨᆠ�泐 �$ + $ 㡰 鼀矄-҈ ᆠᨨ� 矄犨� ᨨ �ᆠᨨ 鼀矄矄鼀犨 �ᆠ �矄矄 r r � ��泐 V҈ V ᨨᆠ��矄ᨨ �ᆠ矄꾀 ᆠᨨ�矄�ᨨ 矄ᆠ�泐ᔼ 鼀҈㦨鼀 ᆠᨨ�矄�ᨨ
r r r V҈ = $� + $ 㡰 鼀矄 -҈ �犨 V҈ V �矄
㝐� 和 ᔼᔼ㡰桌 㡰 ҈�º 㡰 º ᆠ㡰 �º 㡰 ҈�º ᨨᆠ├矄ᨨ 鼀矄矄矄mmm m � � � ᨨ
�ᆠ�ᨨ 矄矄 ᨨᨨ꾀 ᨨ ᨨᆠ├ᨨ 鼀矄矄m鼀ᨨ� �犨� 矄 vᨨ � 矄鼀� 矄 ��盐㡰ᆠᨨ� 矄犨� 鼀矄矄 m� ᆠ泐� 矄�ᆠ鼀
꾀 矄盐矄⼄盐 ᆠᨨ�矄�ᨨ 矄 矄ᨨ ᆠ泐� 矄�ᆠ泐 �鼀 矄ᨨᆠ├ᨨ 鼀矄矄m矄矄鼀ᨨ� � ᨨ �泐ᨨ矄矄�矄p奠ᨨ ᨨ��� �泐盐ᨨ �p�矄ᆠᨨ矄ᨨ [4 Sep.2020 (I)] v
m
㡰 䰘䰘
m
m
㡰 黨
m
ᆠ㡰 䰘
m
㡰 䰘
�
꾀ᨨᆠ├ᨨ 鼀矄矄҈黨├��矄矄 �ᨨ ᨨ �� ҈鼀꾀盐ᨨ 鼀矄矄҈├�ᆠᨨ� 矄犨� ᨨᆠ├ 矄�ᆠ├矄ᨨ�ᔼ ᨨᆠ�泐ᨨ 盐�矄鼀⏌矄� ᨨ� ᨨ �ᆠ�ᨨ 盐矄 ᨨ ᆠᨨ�矄�ᨨ ├� �ᆠ �泐盐矄 ᨨ ᆠᨨ鼀� 矄泐矄鼀矄�├矄 ᨨᨨ �矄 ├g㦨҈鼀⏌矄꾀矄矄盐鼀 �矄 ᨨᨨ�ᨨ 鼀ᨨ�ᨨ
ᨨ矄矄矄ᨨ �泐 ᆠᨨ�矄�ᨨ �矄 ����桌 㝐� 和 ᔼᔼ㡰桌 㡰 㡰 ҈ ᆠ㡰 ҈黨 㡰 꾀�ᆠᨨ 鼀矄矄m犨� � ��ᨨᆠ�泐 u �ˆ ᆠᨨ� 矄 ᆠ泐矄�ᆠ泐犨� 鼀矄矄鼀矄ᔼ鼀ᨨ矄犨� ᨨᆠ�泐 v ˆ ᆠᨨ�矄�ᨨ v�矄�� 泐 [㝐� 2 Sep. 2020 (I)] u 㡰 v = u 㡰 v = u ҈ u ᆠ㡰 v = 㡰 v =
犨ᨨ �ᆠ矄 ᨨ ⼄盐 鼀矄矄 m 矄ᆠ� � �� æ �ˆ + ˆ ö ᨨᆠ��矄 u�ˆ u ç ÷ 泐ᆠᨨ� ᆠᨨ鼀泐 è ø � 矄�ᆠ泐 �泐ᨨ矄� ᨨᆠ矄矄�矄 㝐� 黨 ᔼ桌 ҈ ҈ 㡰 鼀盐 㡰 鼀盐
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P-39
Work, Energy and Power
鼀盐 㡰 鼀盐
犨ᨨ�ᆠ矄ᨨ 鼀矄矄矄㝐 㝐鼀ᨨ� �矄矄 ᨨ犨 犨� 矄 矄ᨨ ҈鼀⏌矄 �鼀⏌矄ᆠᨨ� 矄�ᆠ泐 ᨨ��� 꾀 ᆠᨨ�矄�ᨨ 泐鼀ᨨᨨ � � �ᆠ �ᆠ�ᨨ 矄犨� 矄 矄҈ 矄ᆠ�泐 盐矄 ᨨ ҈ 泐 㝐� ҈꾀�҈黨ᔼ桌
ᆠ㡰
䰘
㡰 �鼀⏌矄 鼀⏌矄 㡰 鼀⏌矄 鼀⏌矄 ᆠ㡰 �鼀⏌矄 鼀⏌矄 㡰 鼀⏌矄 ҈鼀⏌矄 꾀ᨨ 泐ᨨ 鼀矄矄鼀҈鼀ᨨ� �犨� 盐 ├ ᨨ犨 ᨨᆠ�泐ᨨ v҈ �ˆ 盐 �ᨨ矄ᆠᨨ� ᆠᨨ�矄�ᨨ 犨� ᨨ 泐ᨨ 鼀矄矄 m鼀ᨨ� �犨� ᨨᆠ�泐v �ˆ 꾀 ᆠᨨ�矄�ᨨ m҈ m 鼀ᨨ犨� ᨨᆠ��矄ᨨ v �ˆ v �ˆ 矄ᆠ�泐 ᔼ m㦨�鼀҈ v㦨�v҈ v҈�矄 㝐� 꾀�҈黨ᔼᔼ桌 v v 㡰 vv ᆠ㡰 v 㡰 v졔v 꾀 㜠�ᆠᨨ 鼀矄矄鼀矄盐 矄҈㜠 �鼀 矄�ᨨ 矄�ᆠ ᆠᨨ�矄�ᨨ 犨� 盐ᆠ盐矄矄ᨨ 盐 ├ ᨨ犨 鼀矄矄 ᔼ�矄 矄ᆠ �ᆠ泐ᆠ├犨 矄 ᨨ矄� � 奠 ᨨ �矄 � �� ├� �ᆠ �泐 鼀矄矄ᨨ 盐ᆠ盐矄�矄 㝐� ҈ ҈黨ᔼᔼ桌 㡰 鼀 㡰 �鼀 ᆠ㡰 ҈�鼀 㡰 鼀 꾀�ᆠ ᨨ 犨ᨨᨨ ᨨ 鼀矄矄 ├� �矄 ᨨ ᨨ鼀 ᨨᨨ ҈鼀 �� 盐� � �꾀 矄鼀�鼀盐 ᨨ 鼀矄矄├��矄 � �ᆠ泐盐犨 犨� ᨨᆠ�泐 ҈鼀矄҈ ᨨ鼀 �ᨨ盐 盐�矄鼀 �
犨ᨨᨨ 鼀 �鼀盐鼀 �� ᨨ 犨 �ᆠ ᆠᨨ鼀� 矄泐矄鼀 ᆠ 矄 ᨨ ᨨ ᨨ 盐� � � ᨨ � �ᨨ犨�矄 �㦨҈鼀矄㡰 㝐� ҈ ҈黨 ᔼ桌 㡰 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 ҈鼀
ᨨᆠ├꾀 ᔴ泐� �ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ 矄盐 ᆠ矄矄 ᨨ犨 � ��盐꾀 ⼄盐鼀矄矄矄 鼀犨 �ᔴ 矄鼀矄矄㝐ᨨᆠ├꾀�矄�� � �矄 v ᨨ犨 矄 盐 ᨨ 犨 �ᆠ � ᆠᨨ� 矄 犨� ᆠ泐 � 矄�ᆠ泐 ᆠᨨ鼀� 鼀矄矄ᆠᨨ� 矄犨� ᔴ矄ᨨ � ᆠ泐� 矄�ᆠ泐 ᨨ � ��├� �ᆠ �泐�矄 ᨨ矄� 犨 ᨨᨨᆠ矄矄␔ �矄盐ᨨ 㝐⏌鼀䁸
㡰 v
黨
҈
꾀 鼀 㡰 �
㡰
Main黨 ҈黨ᔼ桌 ᔴ 㝐
鼀 ᆠ㡰
㡰
҈҈ ᔼ ᆠᨨ� ᆠᨨ�矄�ᨨ �ᆠ犨� � ��矄 n 矄�├矄矄�ᨨ 泐�ᆠᨨ 矄鼀鼀矄矄ᔼ � ᨨ├� �ᆠ �泐�矄�奠� ᨨ��� ├� �ᆠ �泐 鼀� �盐 ᨨ �ᨨᆠ�泐犨 犨ᨨ�ᆠ矄 ᆠᨨ�矄�ᨨ �矄 㝐� ҈桌
n n n 㡰 n ᆠ㡰 㡰 ҈ 鼀矄矄ᨨ 泐 ᨨ� 鼀ᨨᆠ盐�矄맔҈䰘├�ᔼ ҈ 泐 ᨨ� 鼀ᨨᆠ盐矄矄�├矄ᆠᨨ � 犨ᨨ ᆠ鼀 � ᨨ � ᨨ ᨨ鼀 ᨨ盐 矄�ᆠ泐犨� 矄 ᨨ ҈鼀⏌矄 矄矄盐ᨨ
犨�矄 泐 㝐� ҈桌 㡰 �맔҈⏌鼀 㡰 䰘맔҈⏌鼀 ᆠ㡰 �맔҈⏌鼀 㡰 䰘맔҈⏌鼀 ҈ ᔼ�矄 ᨨ盐 � 盐ᨨ 矄盐 矄 矄�ᆠᆠᨨ� ᆠᨨ�矄�ᨨ 犨� 盐�盐鼀矄 ᆠ�ᨨ ᨨ矄矄ᨨ �矄 �泐 is p 錐犨 � ᨨ�矄矄�鼀�ᆠᨨ�矄�ᨨ 犨� ᆠᨨ 盐ᆠ盐矄 矄 ᆠ�ᨨ ᨨ矄矄ᨨ �泐�矄ᆠ 盐矄ᨨ ᆠ 矄ᆠ�泐 㝐� ҈桌 㡰
㡰 ( ×黨 ×) 㡰 ( × ×黨 ) ᆠ㡰 (0, 0) 㡰 (0, 1) ҈ 꾀 盐ᨨ 鼀ᨨ� � 犨� 矄 ἘἘ 鼀├矄 ᨨ
ᆠᨨ�矄�ᨨ 犨� 矄�ᨨ 泐 泐 ᨨ� ᨨ鼀� �ᨨ盐 矄
鼀� �鼀盐鼀├� �ᆠ �泐ᨨ 盐ᨨ ᨨ犨 �ᆠ � 矄�ᆠᆠᨨ�矄�ᨨ 犨�├ᆠ�矄 㝐� 졠 � 꾀�҈҈桌 㡰 㡰 ҈ ᆠ㡰 ҈҈ 㡰 ҈ ҈� 꾀�ᆠᨨ 鼀矄矄鼀鼀ᨨ� �� �ᆠ�ᨨ 犨� 矄 v�矄 �泐 ᨨ �ᆠᨨ 鼀矄矄m鼀ᨨ� �� y �ᆠ�ᨨ 犨� 矄 ᔼ ᆠᨨ�矄�ᨨ �矄 ᆠ泐� 矄�ᆠ ᆠ �ᨨ矄矄� �泐 盐� � ᆠᨨ�矄�ᨨ �矄 ᆠᨨ矄ᨨ 㝐� ҈�桌 㡰 �奠 㡰 奠 ᆠ㡰 奠 㡰 �奠 ҈ 鼀矄矄矄鼀鼀 鼀鼀ᨨ� �� 㜠泐 犨� 矄 盐盐 盐矄ᆠ�泐矄矄 ᨨ犨 � ��盐 鼀矄矄矄 ᆠᨨ� 矄鼀 ᨨ� 矄�ᆠ├ ᨨ� ᨨᆠ�泐ᨨ 矄盐� �鼀矄矄犨� 㝐� 졠 � 꾀�҈҈桌 泐 鼀盐 鼀盐
鼀盐
(
)
(
)
盐 ˆ 盐 ˆ � + ˆ � - ˆ 㡰 㡰 ҈ ҈ 盐 ˆ 盐 ˆ -� + ˆ -� - ˆ ᆠ㡰 㡰 ҈ ҈ ҈䰘 꾀
�矄�矄 ᨨ ᨨ ᨨ� ᨨ 矄鼀ᨨᨨ 矄盐 ᆠᔼ ᨨ盐 ᆠ矄 ᆠ├ ᨨ�矄 ᨨ��� ᨨ矄��ᨨ �� � 矄盐 ᆠ ᨨᆠ ᨨ 盐� � ᆠᨨ�矄�ᨨ �矄 ᨨᨨ�ᨨ ᨨ � ᨨ ᆠᨨ鼀矄矄�ᨨ ᨨ ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � 矄├ᆠ 矄 矄ᆠ�矄 ��ᨨ ᨨ �矄 ├� �ᆠ �泐 K 犨� �鼀 t 鼀ᨨ矄 ᨨ�泐䁸 ��盐ᨨ 泐�盐矄� ᨨᨨ 矄ᆠ 꾀 ҈桌
(
)
(
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WWW.JEEBOOKS.IN
EBD_8341 P-40
Physics K
K
㡰
2
㡰 t
t
K
K
ᆠ㡰
㡰 t
t
҈ 꾀�ᆠᨨ 鼀矄矄m �矄ᨨᆠ ᨨ鼀 �ᨨ盐 犨�
� ��矄 u �a犨� ᨨ� ᨨ 꾀 �� 矄ᨨ� ᨨ �矄ᆠᨨ泐�鼀├矄ᆠᨨ鼀泐� 矄�ᆠ ᆠᨨ�矄�ᨨ 犨� ᨨ � �ᆠ�ᆠ犨 �ᆠ 犨矄 ᨨ犨
�ᆠ泐盐犨 ᨨ鼀 �ᨨ盐 犨� 矄鼀� �� 矄 盐 � ᆠᨨ鼀ᨨ矄�矄泐矄鼀鼀├矄犨� ᨨ� ᨨ �鼀鼀 �泐 ᆠᨨ�矄�ᨨ �矄 꾀 ҈桌 p p p p -a 㡰 + a ᆠ㡰 㡰 꾀ᨨ 鼀矄矄├�矄矄ᨨ �ᆠᨨ矄ᨨ �� �鼀꾀 盐ᨨ 鼀矄矄҈├�� �犨� ᨨᆠ�泐鼀⏌矄� ᨨ� ᨨ �ᆠ�ᨨ �矄 ᆠ ᨨ 꾀 ᆠᨨ�矄�ᨨ 盐� 泐 �矄 �ᨨ盐 �矄 ᆠᨨ 鼀 盐 �矄 ᆠᨨ ҈鼀 ᨨ鼀 ᨨᨨᨨ ᨨ矄 ᨨᆠ�泐 ᨨ 盐�矄 ҈҈桌
㡰 ҈黨
㡰 �鼀⏌矄 㡰 � 鼀⏌矄 ᆠ㡰 鼀⏌矄 㡰 �鼀⏌矄
犨ᨨ 矄鼀 �ᆠ矄 ᨨ ⼄盐 鼀矄矄矄 矄 鼀ᨨ� � �
ᨨᨨ矄� �ᆠ�ᨨ 矄 ᨨ鼀ᨨ� A� ᨨ� ᨨ ᆠ�ᆠ盐 ᨨ� � � �ᨨᆠ��矄v v矄ᆠ�泐 矄矄 ᨨ犨 � ��盐犨 ᆠᨨ�矄�ᨨ 矄 �ᆠ矄 鼀ᨨ犨� ᆠᨨ 矄 矄 矄꾀 鼀├� � ᨨ犨鼀 泐矄�ᆠ ᆠᨨ�矄�ᨨ 矄ᨨ A 矄犨ᨨ�ᆠ矄犨� �� ᆠ ᨨ� A䁸 黨桌 v
A
v
Integer Value Answer
ᨨᆠ矄A, B C├� 矄�� � ᨨ �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ 矄 鼀矄矄矄m, 2m m, 矄ᆠ�泐 ᨨᆠA 鼀ᨨ矄ᨨ犨 矄B犨� 矄 黨鼀⏌矄 鼀├矄 矄�ᆠᆠᨨ�矄�ᨨ 犨� � B鼀├矄ᆠᨨ鼀泐� 矄�ᆠᆠᨨ�矄�ᨨ 犨� C.꾀 鼀ᨨ�ᨨ 矄ᨨᆠᆠ盐ᨨ 矄鼀矄�� � � � 矄 � 鼀⏌矄㡰ᨨ ᨨᆠᔴ 黨桌 m
m
m
꾀
ᔴ
꾀ᨨᨨ 鼀矄矄 m, 矄盐矄 泐矄� �ᨨ � l1, �矄 �� 鼀� �鼀盐鼀ᨨᆠ�泐⼄盐� ᨨᆠᨨ鼀 盐ᆠ�ᆠ�
�ᆠ 꾀 �� 矄ᨨ� �ᆠᨨ� 矄矄�ᆠ泐 犨� ᨨ ᨨᨨ 鼀矄矄m 矄盐矄 泐矄� �ᨨ � l2, 犨 �ᆠ �矄� ��泐矄ᨨ 矄� �矄鼀矄矄㜠矄矄 � 矄�ᔼ 矄ᆠᨨ ᨨ ᆠᨨ�矄�ᨨ ᆠ⼄盐�矄 鼀� �鼀盐鼀矄 ⼄盐� ᨨᆠᨨ鼀 盐ᆠ�ᆠ� l �ᆠ �ᨨ ҈ �矄 꾀 ҈桌 l 3
Numeric Answer
犨ᨨᨨ �矄ᨨ 矄鼀鼀矄矄鼀ᨨ� �犨� 矄鼀 矄 盐� � �ᆠ�ᨨ 矄� 泐 ᆠᨨ鼀泐� 矄�ᆠᆠᨨ�矄�ᨨ 鼀ᨨᨨ� 犨� � 矄 犨 �ᆠ �矄 ᨨ �� ��矄 �犨 � ��ᨨᆠ��矄ᨨ 犨ᨨᨨ �矄 �
㝐� 和 ᔼ㡰桌 �㡰�矄힘힘힘힘힘힘 � 꾀�ᆠᨨ 鼀矄矄m�矄鼀ᨨ� �ᨨ � x㜠 �矄犨� � �� ᨨᆠ�泐 u�ˆ ᔼᆠᨨ� 矄矄�ᆠ泐犨� �ᆠᨨ 鼀矄矄 ҈鼀矄 鼀ᨨ矄犨� �矄� ��├� �ᆠ �泐 矄 ��盐㡰ᔼ 矄� q҈ = n 矄� q 盐ᨨ n �矄 힘힘힘힘힘힘힘힘힘힘힘 [㝐� 2 Sep. 2020 (II)] 鼀 q҈ 鼀 ҈鼀 q u�$ ҈鼀 꾀ᨨ 泐Aᨨ 鼀矄矄m㦨҈├� 矄 � ��ᨨᆠ�泐ᨨ �ˆ 鼀矄҈ ᔼᆠᨨ� 矄矄�ᆠ泐犨� ᨨ ᨨ 泐Bᨨ 矄鼀 鼀矄矄 犨 �ᆠ 矄 � �� ᨨᆠ�泐ᨨ � ˆ 鼀矄҈ 꾀 r ᆠᨨ�矄�ᨨ A鼀ᨨ矄犨� ᨨᆠ�泐v = �ˆ + ˆ �泐 x ᨨ B ᆠᨨ�矄�ᨨ �矄犨� 矄 J 盐ᨨ x�矄 ҈ 힘힘힘힘힘힘힘 㝐� ᔼ桌
(
҈
㡰 㡰 ᆠ㡰 㡰 ҈
犨ᨨ�ᆠ矄ᨨ 鼀矄矄矄m҈ m� ᨨᆠ�鼀ᨨ�ᨨ
r r ᨨᆠ��矄 v҈ v 矄ᆠ�泐�鼀t㦨 泐 r r ᆠᨨ� �鼀 tᨨ �ᨨᆠ��矄ᆠᨨ鼀 v҈ Ἐ v Ἐ �鼀 t ᨨ 犨 � 矄� 鼀ᨨ� � � � 盐 ᨨ r r r r ҈和桌 ( m҈v҈ Ἐ+ mv Ἐ) - ( m҈v҈ + mv ) �矄 㡰 ᨨ 㡰 ( m҈ + m ) gtᨨ ҈ 㡰 m҈졔m㡰gtᨨ ᆠ㡰 ( m҈ + m ) gtᨨ
)
䰘 꾀�矄ᨨᆠ ᨨ鼀 �ᨨ盐 �ᨨ �ᨨ犨� ᨨ� ᨨ 矄盐 ᆠᔼᆠ 矄鼀 �鼀盐鼀 �� ᨨ ҈ m 盐 矄ᨨ �ᨨ盐 ᨨ �� � �ᨨ盐 ᨨ �矄�鼀�ᨨ矄矄 ᨨ �矄├� �ᆠ �泐ᔼ鼀鼀 �泐 ᨨ盐 ᆠ ᨨᆠ�泐ᨨ 鼀├矄 � ᨨ ᨨ犨� ᨨ� ᨨ 矄盐 ᆠ 鼀 �鼀盐鼀 �� � ᆠ 矄 ᨨ盐 ᆠ� metres�矄힘힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌
WWW.JEEBOOKS.IN
P-41
Work, Energy and Power
6
MCQs with One or More than One Correct Answer
꾀矄鼀�ᆠᨨ 鼀矄矄m鼀ᨨ� �� 矄� 泐 ᨨᨨ犨 矄�� 盐ᨨ � 盐 �矄盐 �ᨨ矄矄�ᆠ ᆠᨨ�矄�ᨨ 犨ᨨ 矄 盐 矄 ᨨ �ᆠ�ᨨ ��矄 ᆠᨨ矄 ᨨ 泐 矄盐 ᆠ犨 � ᨨ �矄 � 犨� 泐鼀ᨨ �矄ᨨ 矄矄 ᨨ犨 � ��盐 ␔ �矄 ᆠᨨ �矄ᨨ ᨨ鼀ᆠᨨ矄 �矄L㦨L �ᆠ矄 �矄v㦨v �矄ᨨ �矄鼀ᨨ � 犨 dL v 犨 dL�矄 泐ᨨ犨矄 V 矄盐ᆠ V >mw㡰 � ᆠ �� � ᨨ 矄� � �矄 矄 �� 泐 ᨨ�ᨨ 㝐� 黨 ᔼᔼ桌
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P-49
System of Particles and Rotational Motion
㡰
æ mw ö ç ÷ çè k ÷ø
mw 㡰 k
mw mw 㡰 k k 꾀�ᆠᨨ 鼀矄矄m�矄 � ᨨᨨ ᨨ �� 矄� � � � ᨨᆠ ᆠᨨ 矄 k 盐 矄ᆠ � l ᨨ �矄 � 矄泐矄鼀�矄�� �盐矄 w ᨨ盐 � ᨨ 矄� �矄盐ᆠ �ᨨ矄� ᆠ�ᆠ� ��泐 矄ᆠ 矄ᆠ � 矄� � �矄 㝐� ᔼ桌
ᆠ㡰
㡰
ml w
㡰
k - mw
ml w
ml w k + mw k + mw 꾀矄鼀ᨨᨨ 犨�ᨨ � p�矄 � ᨨᆠ�ᆠ ├�
�ᆠ 꾀 ᆠ 矄� 矄鼀ᨨᨨ 泐ᨨ 犨� ␔ ᆠ�ᆠ�矄ᨨ� �犨� �盐矄 wᨨ盐 �ᆠ �鼀꾀矄矄 ᨨ犨 � ��盐 �矄矄 犨� 矄ᆠᨨ ᆠ�ᆠ盐� �ᨨ矄��ᨨ 矄矄 ᨨ犨
盐ᨨ w�矄⼄盐ᨨ㝐� ҈꾀҈黨ᔼᔼ桌
ᆠ㡰 �
ml w k - wm
㡰
g g ᆠᨨ矄 q ᆠᨨ矄 q 㡰 l l g g 矄� q ᆠ㡰 㡰 矄� q l l ᔴᨨ ᆠ鼀� 盐�矄鼀 泐鼀� � �ᆠ鼀 矄ᨨ 矄 � ᨨ� �ᆠ泐� �ᆠ 盐鼀ᔼ 盐鼀ᨨ矄ᨨᨨ 矄 � � � 矄鼀� 矄盐ᆠ├ᨨ 犨ᨨ 盐鼀 ᨨ鼀� � �ᨨ � � � 矄 ᨨ矄 ᨨ├ᆠ 鼀 �鼀盐鼀ᨨ�ᨨ 矄 ᨨ 盐鼀� ᨨ盐�ᨨ 矄 鼀� 盐 鼀㡰ᨨ 矄盐ᨨ鼀� � ��矄ᆠᨨ矄ᨨ ├ �盐矄ᨨ 盐鼀ᨨ҈�鼀 �矄 ᨨ ᨨ� ᨨ 㡰 㝐� 졠 � 꾀�҈҈桌 㡰 䰘 㡰 ᆠ㡰 ҈ 㡰
犨ᨨ� �ᆠ �矄ᆠ矄ᨨ 矄鼀 �盐矄Rᨨ� �ᨨ盐 � 矄� ᨨᨨ矄� �ᆠ�ᨨ 矄犨� 矄鼀ᆠᨨ 矄 �盐矄 w �矄ᆠ矄� 矄鼀 ᨨ� ᨨ 꾀�鼀t㦨 ᨨ� 矄P Q ᆠ� �ᆠ ᨨ 矄 矄 ᨨ犨 � ��盐 �矄 犨 犨ᨨ ᨨ� 矄P Q�矄vrᔼ ᨨ �鼀�ᨨ T㡰ᨨ ᨨ�ᨨ ᨨ �矄ᆠ矄vr 矄 盐 ᆠ�ᨨ ᨨ �鼀�矄矄矄 泐 ҈桌
㡰
黨
㡰
㡰
g 㡰 g ⏌ r 㡰 r 㡰 �⏌r ᆠ㡰 g 㡰 ⏌ 꾀�ᆠ�矄鼀ᨨ� �犨� 盐 � ᨨ鼀矄 � ᆠ�ᆠ盐ᨨ� ᨨ �盐矄 � ᆠ ᨨᆠ� 矄泐ᨨᨨ�ᨨ ᨨ n ᨨ犨ᨨ ᔼ �ᨨ ᨨ ᨨ�ᨨ ᨨ �ᆠ�矄 㝐� ҈桌 ⏌ 㡰 µ ᨨ 泐 㡰 µ n ⏌+҈
㡰
䰘
ᆠ㡰 µ +҈㡰⏌ 㡰 µ ⏌ 꾀矄 盐 � ᨨ鼀ᨨ ᨨ 鼀矄矄㝐 � l �矄�ᨨ ᨨ 矄ᨨ �ᆠ ᨨ� �ᆠ 矄 ��盐㡰 �矄 ���� �ᆠ�ᨨ �ᨨ �矄 �ᆠ泐ᨨ �ᨨ 矄
�盐ᆠᆠ�ᨨ ᨨ ᨨ 犨 �鼀├矄 � 㝐� ҈䰘桌 q 犨� �ᆠ�矄
ᆠ㡰
㡰
҈ ᔴᨨ 矄� �矄ᆠᨨ� �� ᨨ� ᨨ 犨� ᆠᨨ 矄 �盐矄 w ᨨ盐�矄ᆠ 졠 �矄ᆠ 矄 矄 ��ᨨ ᨨ ᨨ 矄� ᨨ �鼀
盐 矄 ��ᨨ ᨨ ᨨ 矄� 矄矄 ᨨ犨 � ��盐 ␔ �矄ᆠ�矄� ᨨ� �ᨨ 矄矄 ᨨ犨 犨ᨨ矄 P Q矄�鼀盐 ᨨ盐矄泐ᨨᆠ �ᨨ犨 矄 R ᨨᆠ�泐ᨨ ᨨᆠ�ᨨ �矄� y㜠z �矄矄鼀 ᨨᨨ 矄犨� 矄ᆠᨨ �矄ᆠ꾀矄矄盐鼀 �㡰 泐 ᆠ├ᨨ �矄ᆠ ᨨ �矄ᆠ 矄ᆠᨨ鼀 ҈⏌ᨨ�ᨨ ��㡰 � ��矄矄矄 �矄ᆠ �盐矄 ���㡰w 鼀� 矄ᆠᨨ 矄 ᨨ盐� ᨨ盐 ҈桌
㡰 P 矄� 矄 ��ᨨ Q� 盐 矄 ��ᨨ 㡰 P 矄� 盐 矄 ��ᨨ Q� 矄 ��ᨨ ᆠ㡰 ᨨ P Q � 盐 矄 ��ᨨ 㡰 ᨨ P Q � 矄 ��ᨨ
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EBD_8341 P-50
Physics
4 ҈҈
Fill in the Blanks
和ᨨ�� S ᨨ矄� ᨨ� ᨨ 犨� ᆠᨨ 矄 �盐ᨨᆠ�泐ᨨ ҈ � ⏌矄ᆠᨨ 矄ᨨᨨ �� P 鼀ᨨ矄ᨨ � 犨 �矄 ᆠᨨ 鼀 ᨨᆠ�泐ᨨ 矄ᨨP犨 q 㦨� 矄 ��㡰�矄鼀⏌矄 ҈黨䰘㜠㝐├矄桌
ᨨ犨ᆠᨨ 矄� 矄鼀ᨨᨨ 矄ᨨᨨ � �鼀ᨨ �矄ᆠᨨ �盐矄 ᨨ� �ᆠᨨ盐 㜠ᆠᨨᆠ├犨�矄犨� ᆠᨨ 矄 �盐矄 w ᨨ盐�矄�ᆠ �矄 ᨨ盐� �矄ᆠ ␔矄矄�� ᆠᨨᨨ � 矄泐矄鼀犨� ᨨ��� ᆠ ᨨ �矄ᆠ 㜠 �矄ᨨ � 矄ᨨ 泐㜠 �矄 w
鼀 ⏌
�ᆠ盐ᨨ 矄ᨨ 㜠 �矄ᨨ � ᨨ�ᨨ �矄 ur 犨 㦨犨 ├ˆ 꾀矄鼀ᨨᆠ├ᨨ 鼀矄矄鼀�矄� 泐ᆠ � 矄ᨨ 8
Comprehension Based Questions
矄矄� 꾀 鼀ᨨ ᆠ �矄ᆠᆠ 犨� 矄ᆠᨨ � � 鼀ᨨ ᆠ�矄ᆠ ᨨ 㜠� � 鼀ᨨ ᆠ꾀 ᆠᨨᨨ � 矄泐矄鼀 � ᨨ ᆠ�ᆠ盐 �矄ᆠᨨ� �ᨨ盐 � �矄犨� ᆠᨨ 矄 �盐ᨨᆠ�泐w�矄 鼀ᨨ ᨨ 㜠 � � 鼀ᨨ ᆠ �ᨨ 矄 �犨 ᨨᆠ r ᨨ � ᆠ 泐�ᆠᨨ 鼀矄矄鼀鼀ᨨ� �ᨨ ᨨ� � r �矄ᆠ ᨨᆠ � � ᆠ 泐 �ᆠ� � � 鼀ᨨ ᆠ�矄 r r r ur ur r ur ᨨ 㦨 � 졔 鼀 ᨨ 犨 졔 鼀 犨 犨 r 犨 ᨨ �矄 ᨨᆠ�泐ᨨ �ᆠ� ᨨ� � 鼀ᨨ r ᆠ �矄 ᨨ矄��ᨨ ᆠᨨ ᨨ �ᆠ 犨� 矄ᆠᨨ ᆠ ᨨ �矄ᆠ
㡰 㡰
r
㡰
⏌ 㡰 ˆ� 㦨 �矄ᆠᨨ 矄� ᨨ鼀ᨨᨨ 泐ᨨ � 矄ᨨ ҈ �矄 ᆠᨨ ᨨᆠ├�鼀�矄 꾀 ҈桌
㡰
犨 㜠犨 졔
㡰
ᆠᨨ矄 犨
㡰
ᆠᨨ矄 犨 ҈ ᆠ�ᨨ ᨨ �矄ᆠᨨ ᨨᆠ├�矄꾀 ҈桌 ҈ 鼀犨 犨 㜠 㜠 犨 ˆ 졔 鼀�├ˆ 㡰 ҈ 鼀犨 犨 㜠 㜠犨 ˆ 졔 鼀�├ˆ 㡰 ᆠ㡰 㜠 鼀犨 ᆠᨨ矄犨 ˆ 㜠 鼀�├ˆ 㡰 鼀犨 矄� 矄犨 ˆ 㜠 鼀�├ˆ
ᆠ㡰
犨
졔 㜠 犨
㡰
㡰
㡰
㡰
Topic-3 : Torque, Couple and Angular Momentum 1 ҈
ᨨ盐O 矄犨� �矄矄矄盐ᨨ ᆠᨨ�矄�ᨨ 鼀├� �
MCQs with One Correct Answer
angle q ᨨ鼀ᨨ鼀 �泐ᆠᨨ鼀� �ᨨ矄ᔼ ᨨ 矄 鼀矄矄 M㦨 ├� � l㦨 ҈鼀 盐ᨨ q �矄 ᨨ �鼀泐 ├g㦨҈鼀⏌矄㡰 [Main Sep. 03, 2020 (I)]
ᨨ盐ᨨ� 鼀矄矄矄ᆠ ᨨ 鼀矄矄m � ᆠᨨ 矄 ᨨ 矄⼄盐ᨨ 矄� l 矄⼄盐�矄ᨨ� �犨� �盐 ⼄盐 ᆠ泐wᨨ盐 �矄矄矄� � ᨨ盐� ᨨ ᨨ ᆠᨨ 矄 ᨨ 矄⼄盐 ᨨ �矄 ��ᨨ 矄 矄 ᨨ犨 � ��盐 �盐鼀ᨨ鼀 盐鼀ᨨ 矄⼄盐 ᨨ盐 �矄 �矄�矄 㝐� 和 ᔼ㡰桌
O M, l q m
m
�矄
m v
㡰 ᆠ㡰 黨 㡰 鼀lw
㡰 鼀lw
ᆠ㡰 鼀lw 㡰 鼀lw 꾀ᨨᆠ├ᨨ 鼀矄矄m㦨҈├�矄� 矄犨� ᨨᆠ�泐v㦨鼀⏌矄ᨨ
�ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ ᆠᨨ� 矄犨� 盐 � ᨨ鼀 �ᆠᨨ 矄�ᆠ├矄ᨨ�矄矄 ᨨ犨 ᨨ �矄�ᨨ
㡰 �� 㡰 黨
A
�
�
䰘�
鼀
҈ B
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P-51
System of Particles and Rotational Motion
4.
Shown in the figure is rigid and uniform one meter long rod AB held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass 'm' and has another weight of mass 2 m hung at a distance of 75 cm from A. The tension in the string at A is : [Main Sep. 02, 2020 (I)] (a) 0.5 mg (b) 2 mg (c) 0.75 mg (d) 1 mg A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre 'O' perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is : [Main Sep. 02, 2020 (I)] F
O
æ R-aö (a) Mg 1 - ç ÷ è R ø
2
R
a 2
æ R ö (b) Mg ç ÷ -1 è R-aø
a a2 (d) Mg 1 - 2 R R Consider a uniform rod of mass M = 4m and length l pivoted about its centre. A mass m moving with velocity v making p angle q = to the rod’s long axis collides with one end of 4 the rod and sticks to it. The angular speed of the rod-mass system just after the collision is: [Main 8 Jan. 2020 I]
(c) Mg
5.
6.
7.
3 v (b) 7 2l
4v 3 2v (d) 7l 7 l A particle of mass m is moving along a trajectory given by x = x0 + a cos w1t y = y0 + b cos w2 t The torque, acting on the particle about the origin, at t = 0 is: [Main 10 Apr. 2019 I] 2$ 2$ (a) m ( - x0 b + y0 a ) w1 k (b) +my0 aw1 k 2 2 $ (c) zero (d) -m( x0 bw2 - y0 aw1 )k The time dependence of the position of a particle of mass r m = 2 is given by r (t ) = 2 t i$ - 3t 2 $j . Its angular momentum, with respect to the origin, at time t = 2 is : [Main 10 Apr. 2019 II] $ $ (a) 48 i + j (b) 36k$
(a)
(
(
)
(c) -34 k$ - i$ 8.
)
3v 7l
9.
(a) 4.0×10–6 Nm (b) 1.6×10–5 Nm (c) 7.9×10–6 Nm (d) 2.0×10–5 Nm A rectangular solid box of length 0.3 m is held horizontally, with one of its sides on the edge of a platform of height 5m. When released, it slips off the table in a very short time t = 0.01 s, remaining essentially horizontal. The angle by which it would rotate when it hits the ground will be (in radians) close to : [Main 8 Apr. 2019 II]
(a) 0.5 (b) 0.3 (c) 0.02 (d) 0.28 10. A particle of mass 20 g is released with an initial velocity 5 m/s along the curve from the point A, as shown in the figure. The point A is at height h from point B. The particle slides along the frictionless surface. When the particle reaches point B, its angular momentum about O will be : [Main 12 Jan. 2019 II] (Take g = 10 m/s2) O
(c)
(d) -48k$ A metal coin of mass 5 g and radius 1 cm is fixed to a thin stick AB of negligible mass as shown in the figure The system is initially at rest. The constant torque, that will make the system rotate about AB at 25 rotations per second in 5s, is close to : [Main 10 Apr. 2019 II]
a = 10 m A h = 10 m
B (a) 2 kg-m2/s
(b) 8 kg-m2/s
(c) 6 kg-m2/s
(d) 3 kg-m2/s uur uur 11. A slab is subjected to two forces F1 and F2 of same uur magnitude F as shown in the figure. Force F2 is in XYplane while force F1 acts along z-axis at the point r r 2i + 3 j . The moment of these forces about point O will
(
)
be :
[Main 11 Jan. 2019 I] Z F1 O
F2
y
30º 4m 6m x
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EBD_8341 P-52
Physics
㡰 ᆠ㡰 ҈
( $� - $ + k$ ) ( $� + $ k$ )
㡰 㡰
( $� - $ k$ ) ( $� + $ + k$ )
犨� 矄 矄 矄ᆠ�泐 鼀矄矄矄矄�ᆠ├ᨨ ᆠᨨ�矄�ᨨ �矄 ᆠ 矄ᆠ�泐 ᨨ鼀 ᆠ ᨨ ᔼ 矄矄ᨨ� �ᨨ盐 �矄ᆠ ᨨ 鼀矄矄矄矄盐ᨨ ᆠᨨ�矄�ᨨ �盐矄 ᨨ 犨� 㝐� 졠 � 꾀�҈�҈桌
鼀� �盐 ᨨ ᨨ⼄盐ᨨ �ᆠᨨ 鼀矄矄҈├��矄� 鼀ᨨ盐 ᨨ��� ᔼ ᨨᆠᆠ� �ᨨ ��矄҈ �矄 ᆠᨨ �ᆠ ᨨ鼀 ᨨ��� �矄�鼀 � 犨 ᨨᆠ ᨨ矄��ᨨ ᆠᨨ�矄 � � 矄㡰 㝐� ҈҈ ҈黨ᔼᔼ桌
p p p p 㡰 ᆠ㡰 㡰 ҈ ᨨ 鼀ᨨ㜠ᆠ ᨨᨨ ᆠ � � 鼀ᆠ � 矄矄矄 ᆠ�ᆠ盐 鼀ᨨ ᨨ �盐矄 �ᆠ泐 ᨨ犨 犨� ᨨ ᨨᆠ ᨨ矄 � 犨� ᆠᨨ 矄 �盐 矄 ᨨ盐 �矄 �矄 ᔼ ᨨᆠ �矄 �矄�盐 盐 � ᨨ鼀泐 ᨨ 鼀ᨨ � ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 鼀ᨨ ᨨᨨ �矄 m ᨨ⼄盐 � 泐 鼀ᆠ � ᨨ 鼀ᨨ�矄 㝐� ҈ ҈黨ᔼ桌 㡰 m ⏌ 㡰 m ⏌
⏌
v
㡰
μ m 꾀��� 鼀矄矄矄矄ᨨ ᨨ � l 矄犨ᨨ鼀矄矄矄ᆠ ᆠ 矄矄 ᨨ犨 � ��盐 ᨨ �矄�ᨨ ᨨ� ᨨ ᨨ� ᨨ �矄 矄 ��盐㡰␔ 矄 ᨨ鼀� �� ᨨ� ᨨ ᨨ矄��ᨨ �矄� 矄 ᨨ盐矄 �盐 ᆠᆠ�ᨨ 犨� 㝐� ҈ ҈黨ᔼᔼ桌 l l
ᆠ㡰 m ⏌ ҈
㝐ᨨ
� � � 䰘� 㡰 ᆠ㡰 㡰 ҈l l l l ҈� 꾀 㜠矄 ᨨᆠ鼀 ᨨ � ᨨ 矄ᨨ 盐 � ᨨ鼀鼀矄矄 矄�泐�矄矄盐矄 犨� 矄� �矄矄 ᨨ犨 � ��盐ᔼ 꾀㦨ᔴ �鼀 泐꾀犨� ᨨ犨 犨 �ᆠ�矄q 㝐� 黨 ҈黨ᔼ桌 㡰
㡰
q =
҈
㡰 q =
q =
졠
㡰 ᆠ㡰 㡰 � � � ҈䰘 꾀�ᆠᨨ 鼀矄矄m�矄鼀ᨨ� �ᨨ � 矄� ᨨ 矄⼄盐 ᨨ 矄� ἘaἘ犨� 盐 � ᨨ鼀矄 v� x㜠y 矄矄 ᨨ犨
� ��盐 㝐� ҈桌 泐
졠
ᔴ
꾀 �
␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 矄 ᨨ �盐 ur 鼀ᨨ鼀 盐鼀 ᨨ盐 ᨨ��� 䁸 ur é ù + ú ├$ 犨 �ᆠ�矄鼀ᨨ� � ᨨ鼀 㡰 = 鼀 ê ë û ᨨᔴ ur 鼀 ├$ 犨 �ᆠ�矄鼀ᨨ� � ᨨ鼀ᨨ꾀 㡰 㦨 ur 鼀 $ ├ 犨 �ᆠ�矄鼀ᨨ� � ᨨ鼀꾀ᨨ ᆠ㡰 㦨 ur é ù - ú ├$ 犨 �ᆠ�矄鼀ᨨ� � ᨨ鼀 㡰 = 鼀 ê ë û ᔴᨨ ҈ 꾀盐 � ᨨ鼀犨ᨨᨨ 矄�ᆠ├ᨨ 鼀矄矄҈├� � l矄矄 � � ᆠ� 鼀
ᨨ 矄鼀ᨨᨨ �ᆠ犨ᨨ �� l㡰矄盐ᆠ 矄鼀ᨨ�ᨨ ᨨ 矄�ᆠ├ 矄泐ᨨ 犨 ᆠ�ᨨ ᨨᆠᨨ 犨 ᨨ 矄�ᆠ├ �矄 �ᆠ盐ᨨ 矄�ᆠ├ 矄�ᆠ├鼀├矄 �ᨨ 犨� 犨 ᨨᨨ鼀ᨨ 矄�ᆠ├�矄ᨨ ᨨ盐� ᨨᨨ ᆠ�ᨨ ᨨ 犨 ᨨ 矄�ᆠ├ �矄 ⼄盐 �
鼀� �盐 ᨨ ᆠ�ᨨ ᨨ ᨨᨨᨨ 矄�ᆠ├ �ᨨ ⏌l �ᆠ�ᨨ ᨨᆠf ᨨᨨ鼀ᨨ 矄�ᆠ├ �㦨҈鼀矄㡰 꾀 ҈桌
҈
҈ ҈ 꾀 � 盐 � ᨨ鼀ᨨ � 鼀矄矄鼀�矄ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ 犨ᨨᨨ� 鼀矄矄矄鼀 鼀鼀ᨨ� ��
矄鼀 ᨨ� ᨨ ᨨ鼀ᨨᨨ矄�矄� 矄ᨨ ᆠ㡰
㡰
㡰
�㝐ᨨ
⏌
㡰
㡰 q =
҈ = = l ҈
㡰
҈ = = l ҈
҈ = 㡰 = = = l ҈ l ҈ 꾀�ᆠᨨ 鼀矄矄├��矄ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ 鼀ᨨ矄� ᆠ�ᆠ盐 ᨨ �盐矄鼀 �� ᨨ ᨨ鼀 �ᨨ盐 �矄鼀ᔼ �盐矄
ᆠ㡰
҈黨
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P-53
System of Particles and Rotational Motion
20.
21.
22.
23.
24.
25.
of the particle is 12 rad s–1, the magnitude of its angular momentum about a point on the ground right under the centre of the circle is : [Main Online April 11, 2015] (a) 14.4 kg m2s–1 (b) 8.64 kg m2s–1 2 –1 (c) 20.16 kg m s (d) 11.52 kg m2s–1 A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed w rad/s about the vertical. About the point of suspension: [Main 2014] (a) angular momentum is conserved. (b) angular momentum changes in magnitude but not in direction. (c) angular momentum changes in direction but not in magnitude. (d) angular momentum changes both in direction and magnitude. A ball of mass 160 g is th rown up at an an gle of 60° to the horizontal at a speed of 10 ms–1. The angular momentum of the ball at the highest point of the trajectory with respect to the point from which the ball is thrown is nearly (g = 10 ms–2) [Main Online April 19, 2014] (a) 1.73 kg m2/s (b) 3.0 kg m2/s 2 (c) 3.46 kg m /s (d) 6.0 kg m2/s A particle of mass 2 kg is moving such that at time t, its r position, in meter, is given by r (t ) = 5iˆ - 2t 2 ˆj . The angular momentum of the particle at t = 2s about the origin in kg m–2 s–1 is : [Main Online April 23, 2013] (a) - 80 k$ (b) (10iˆ - 16 ˆj ) (c) -40k$ (d) 40k$ A bullet of mass 10 g and speed 500 m/s is fired into a door and gets embedded exactly at the centre of the door. The door is 1.0 m wide and weighs 12 kg. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds [Main Online April 9, 2013] into it will be : (a) 6.25 rad/sec (b) 0.625 rad/sec (c) 3.35 rad/sec (d) 0.335 rad/sec A small mass m is attached to a z massless string whose other end is fixed at P as shown in the figure. The P mass is undergoing circular motion in the x-y plane with centre at O and constant angular speed w. If the m O angular momentum of the system, calculated about O and P are denoted w r r by LO and LP respectively, then r r (a) LO and LP do not vary with time r r (b) LO varies with time while LP remains constant r r (c) LO remains constant while LP varies with time r r (d) LO and LP both vary with time A particle is confined to rotate in a circular path decreasing linear speed, then which of the following is correct? [2005S]
r L (angular momentum) is conserved about the centre r (b) only direction of angular momentum L is conserved (c) It spirals towards the centre (d) its acceleration is towards the centre. 26. A block of mass m is at rest under the action of force F against a wall as shown in figure. Which of the following statement is incorrect? [2005S] a (a) f = mg [f friction force] (a)
a
(b) F = N [N normal force]
F
(c) F will not produce torque (d) N will not produce torque 27. A horizontal circular plate is rotating about a vertical axis passing through its centre with an angular velocity wo. A man sitting at the centre having two blocks in his hands stretches out his hands so that the moment of inertia of the system doubles. If the kinetic energy of the system is K initially, its final kinetic energy will be [2004S] (a) 2 K (b) K/ 2 (c) K (d) K/ 4 28. A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved? [2003S] (a) centre of the circle (b) on the circumference of the circle. (c) inside the circle (d) outside the circle. 29. Consider a body, shown in figure, consisting of two identical balls, each of mass M connected by a light rigid rod. If an impulse J=MV is imparted to the body at one of its ends, what would be its angular velocity? [2003S] L
(a) V/L (b) 2V/L M (c) V/3L
M
J = MV (d) V/4L 30. A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity w0. When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform w(t) will vary with time t as [2002S] w(t )
w(t )
(a)
w0
(b)
w0
t
t
w( t )
w( t )
(c) w0
(d) w0 t
t
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EBD_8341 P-54
҈
Physics
꾀 ⼄盐�� �ABC ᨨ鼀 ᨨ鼀盐 � ᨨ鼀犨� 矄犨ᨨ矄鼀� �ᆠ 矄� ��泐ᨨᆠ A � ��矄矄ᨨ� �ᨨ盐 �ᆠ �矄AO 矄矄 ᨨ鼀矄矄�鼀盐 ᨨ盐矄泐 ᨨ犨 ᨨ矄� ᨨ犨 ᨨ ᨨ �AB ᨨ ᨨ �AC矄 矄 ᨨ犨 �ᆠ� � �ᆠ�ᨨ ᆠ矄 ⼄盐 ��矄 ᆠᨨ 矄 矄 矄矄� ᨨ犨 和桌 㡰 �盐 ᨨᆠ�泐 ᨨ A �泐 ├� �ᆠ ᨨ �㡰 㡰 ᨨ �盐鼀ᨨ鼀 盐鼀 g ᨨ �泐 ᆠ㡰 �盐ᨨᆠ�泐 鼀ᨨ鼀 ᨨ � � ᨨ盐 �矄 ᨨ C B ᨨ�ᨨ
O 㡰 ᨨ �盐鼀ᨨ鼀 盐鼀 鼀ᨨ鼀 ᨨ � � ᨨ盐 �矄ᨨ ᨨ�ᨨ
꾀 �矄ᆠᨨ 鼀矄矄M �盐矄R�矄ᨨ� �犨� �盐 矄 wᨨ ᨨ� ᨨ 矄矄 ᨨ犨 � ��盐 鼀� �盐 ᨨ �盐鼀ᨨ鼀 盐鼀ᨨ �矄ᆠᨨ盐 ᨨ��� O�矄 ҈黨黨黨和㜠㝐├矄桌 Y 㡰 ҈⏌㡰MRw w 㡰 MRw M ᆠ㡰 ⏌㡰MR w X 㡰 MRw O A鼀矄矄M鼀ᨨ� �犨� ᆠᨨ 矄 ᨨᆠ�泐ᨨ X㜠 �矄ᔼ矄 �盐鼀ᨨ鼀 盐鼀犨� 矄ᆠᨨ ᨨ���
҈黨�㜠㝐├矄桌 㡰 �矄 ᨨ 㡰 鼀� 矄ᆠᨨ 矄 ᆠ㡰 �ᨨ矄ᨨ � ᆠ矄� � 㡰 �ᨨ矄ᨨ ᆠ矄� � 꾀 � ᆠ�ᆠ盐� �ᨨ 鼀矄矄�M �盐矄r�矄ᨨ� � ᨨ盐�矄 �矄犨� ᆠᨨ 矄 �盐 ᨨᆠ�泐w 犨ᨨ ᨨᆠ矄 ᆠ ᨨ 鼀矄矄 m ᆠ � 泐 ᨨ ᨨᨨ矄� 矄ᨨ �鼀ᨨ � � 犨 ᨨ犨 ᨨ矄犨� �盐ᨨᆠ�泐 ҈黨㜠҈㝐├桌 㡰
wM M + m㡰
㡰
w M m㡰 M + m㡰
ᆠ㡰
wM M + m㡰
㡰
w M + m㡰 M
2 �
Integer Value Answer
꾀 ᨨ� ᨨ ᆠ�ᆠ盐 ᨨ鼀ᨨ �盐矄�鼀 鼀矄矄�├��矄 ᨨᨨᨨ盐�矄 �矄 犨ᨨ 鼀矄矄矄矄矄� �ᨨ泐㜠�盐 矄ᆠ ᆠ泐� �矄ᨨ 鼀矄矄�├�ᆠ ᨨ ᨨ鼀 �矄 ᆠ�鼀 ᨨ鼀 ᆠ ᨨ �矄� 矄� 矄 ᨨ � �矄 �鼀 矄 ��盐㡰 쟼ᆠ �盐
矄�鼀盐 ᨨ盐矄泐 �矄 矄 ᨨ� ᨨ 泐 �ᆠ盐ᨨ �鼀� ᨨᨨ矄� �ᆠ�ᨨ 矄꾀 � � ᨨ鼀 矄 ᨨ� ᨨ 矄 ᨨ 黨 鼀矄҈犨� 矄ᆠᨨ �ᨨ盐 ᨨ�ᨨ 矄 ᨨ ᨨ鼀� 矄҈ 矄 ᨨ鼀�矄 꾀 ҈桌
꾀盐 � ᨨ鼀ᆠ�ᆠ盐 �矄ᆠᨨ 鼀矄矄�├� �盐矄鼀�矄 ᨨ� �犨� �盐ᨨᆠ�泐ᨨ ҈ 矄҈ ᨨ盐�矄 ᨨ犨 �矄犨 �ᆠ �矄�ᆠ 犨ᨨ盐 � ᨨ鼀ᆠ�ᆠ盐� �矄 ᆠ ᨨ 鼀矄矄�├� �盐矄鼀� 泐ᆠ 矄泐鼀鼀�ᆠ泐ᨨ �矄ᆠ� 矄盐ᆠ 鼀
泐 ᨨ盐ᆠ � � ᆠ ᨨ ᨨ � �矄 ᨨ �矄ᆠ ᨨ� ᨨ 꾀矄矄盐鼀 �ᆠ�ᨨ �矄� ᨨ盐� 矄盐ᆠ � �矄矄�ᨨ �矄ᆠ 矄泐矄鼀 ᨨ矄ᨨ盐 ᨨ��� �矄 犨 �盐ᨨᆠ�泐 꾀 ҈桌 � 矄҈㡰ᨨ 矄泐矄鼀�矄 䰘 꾀� 泐矄ᆠᨨ 矄�矄矄ᨨ 犨ᨨ矄矄꾀 鼀矄矄M矄㡰 鼀矄矄҈҈M矄㡰犨 Ms�矄 鼀矄矄ᨨ 矄盐 泐 矄 泐 �矄 ᆠ ᨨ� �ᨨ盐 �ᆠ ᨨ 鼀矄矄犨 �ᆠ �矄矄�ᨨ 泐 �ᨨᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀ᨨ � 泐矄ᨨ �盐鼀ᨨ鼀 盐鼀ᨨ 矄ᨨ盐 ᆠ ᨨ 鼀矄矄�矄 ҈桌 3
Numeric Answer
꾀 � ᨨ ᨨ 鼀矄矄黨├� � ҈鼀�矄矄盐矄 矄 ᨨ鼀 ᨨ 矄ᨨ � ᆠ 泐ᨨ矄ᆠ�� �ᆠ 꾀�ᆠᨨ 鼀ᨨ҈├�鼀ᨨ� �� 矄�� � 犨� ᨨᆠ�泐鼀⏌矄 �矄 ᨨ �矄ᨨᨨ鼀鼀ᨨ矄 ᨨ� 矄�ᆠ├矄ᨨ� 矄 ��盐㡰 �盐矄 � ⏌矄㡰ᨨ ᨨ �鼀鼀 �泐 ᆠᨨ�矄�ᨨ 犨� 힘힘힘힘힘힘힘힘힘힘힘힘힘힘 㝐� 和� ᔼᔼ㡰桌 黨 꾀矄ᨨ ᨨ ├�鼀矄矄�矄矄 � �ᨨ �鼀ᨨ ᆠ�ᆠ盐 ᨨ鼀 ᨨ 鼀矄矄 ├� ᨨ� � ᨨ盐 �矄 �矄 � ᨨ盐�ᨨ 矄 鼀� 盐 鼀㡰 矄ᨨ ᨨ犨 矄矄 鼀ᨨ� �ᨨ犨 矄 ᆠ ᨨ ᨨ鼀␔ 犨� ᨨ�ᨨ 矄 � 鼀㡰ᨨ ᨨ鼀犨 矄ᨨ ᆠ 矄�矄ᆠ 힘힘힘힘힘힘힘힘힘힘 [Main Sep. 03, 2020 (I)] 盐盐 � ᨨ鼀鼀矄ᆠ ᨨ� ᨨ 泐ᨨ 泐ᨨ盐 � � �矄犨� ᨨ ᆠ鼀 �� ᨨ 黨ᆠ鼀␔ 泐ᨨ盐鼀ᨨ鼀ᨨᨨ � �矄 矄ᨨ犨泐ᨨ犨 矄 ᆠ � ��泐ᨨ 泐 � �矄�矄 犨� 矄ᆠᨨ 矄ᆠ �� � � ᨨ矄 ᨨ 꾀 矄ᨨ鼀 �矄 ᆠ � �矄ᨨ矄 �� ᨨ 矄矄矄�� � �� � �矄ᨨ矄 �矄 ᆠxR ᨨ鼀 ᆠ �ᆠ鼀㡰ᨨ 矄ᆠ ᨨ 矄矄 矄�� � �� �矄 矄 ᆠ盐矄 ᨨ � ᆠ� �ᆠ�ᨨ ᨨᆠ矄ᨨ 犨ᨨ � �矄ᔼ ᆠᨨ �ᆠ� 矄ᨨ 矄�ᆠ 泐 鼀�ᆠ �ᆠ�ᨨ 犨 � �矄 矄ᆠ 矄ᆠ�泐 盐ᨨ xR � ᆠ鼀㡰�矄힘힘힘힘힘힘 꾀 桌 4
Fill in the Blanks
҈ 꾀矄ᨨ ᨨ 鼀矄矄m� ᨨ ᨨ 矄� ��矄犨 � ᨨ盐 � ᨨ� ᨨ ᆠ�ᆠ �ᆠ ᨨᆠ 盐ᨨ ��泐㡰 � ᨨ 矄� ��矄 盐ᆠ � 盐泐 ├� � �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄ᨨ ᨨ盐 ᆠ ᨨ ᆠ�ᆠᆠᨨ 矄 矄�ᨨ � 矄� � �矄 �� 泐 T 㦨 Arn 犨 A �矄 ᆠᨨ 矄 r �矄 � 矄 ᨨ盐矄 �盐矄ᨨ ᆠ�ᆠ n㦨 ҈黨黨㜠҈㝐├桌
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P-55
System of Particles and Rotational Motion
꾀ᆠ泐� ᨨ 鼀矄矄M �盐矄R�矄矄� �ᨨ ᨨ� ᨨ ᨨ鼀 犨 �ᆠ �矄ᨨ xy 㡰犨� �矄 �矄 � ᨨ � y㜠 �矄 ᨨᨨᨨ盐�矄 �矄 ᨨ鼀�矄�� 鼀ᨨ�ᨨ � x㜠 �ᆠ�ᨨ �� 泐x㦨A ᆠᨨ矄 w t㡰 �矄 ᨨ矄�� �犨 ᆠ泐� ᨨ鼀 鼀 �鼀盐鼀ᨨ⼄盐ᆠ� �ᨨ ᆠ泐� 盐� � �矄鼀ᨨ�ᨨ �矄 ҈黨㜠㝐├矄桌 꾀矄鼀ᨨᨨ 盐 � ᨨ鼀ᨨ ᨨ � L 鼀矄矄M 矄犨ᨨ � �ᆠ 矄ᨨ ����矄� ᆠ ᨨ 鼀矄矄m犨 �ᆠ ᆠ 矄� 泐ᨨ � ᨨ ᔼ ��泐 犨ᨨ 矄 ᆠ ᨨ ᨨ 矄泐矄鼀�矄ᨨ� �犨�
�盐ᨨᆠ�泐 w ᨨ盐 �矄 盐ᆠ盐ᨨ ᨨ 矄矄� � ᨨ盐� 鼀� ᨨ� ᨨ ᨨ 矄 ��盐㡰 ᨨ ᨨᆠ矄␔ 矄ᆠ 矄ᨨ ᨨ �盐ᨨᆠ�泐ᨨ 矄泐矄鼀�矄 ҈黨㜠㝐├矄桌
w
ᔴᨨ 矄� ᨨ 泐ᨨ 鼀矄矄҈├�矄 ᨨ��� �鼀 r t㦨꾀 ᨨᆠ F = at �ˆ + bˆ 㡰 �矄� ᨨ ᨨ 泐犨 a㦨҈矄҈ b㦨҈N ᨨ⼄盐ᆠ� �ᨨ ᨨ 泐 r ᨨ盐 ᨨ��� �鼀t 㦨 ҈ s �矄 t ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 㡰盐䁸 꾀 ҈桌 r ҈ 㡰 t = Nm r 㡰 ᨨ⼄盐 t �矄� �ᆠ�ᨨ ᨨ 盐 �ᆠᨨ졔 kˆ r ҈ ᆠ㡰 ᨨᆠ�泐ᨨ ᨨ 泐t㦨҈s�矄 v = �ˆ + ˆ 㡰ms ҈ ҈ 㡰 鼀� �盐 ᨨ �矄ᆠ鼀 ᨨ ᨨ 泐t㦨҈s�矄 m 䰘 ᨨ � �泐ᨨ �ᆠᨨ 鼀矄矄m �矄 ᆠ r ᨨ鼀 � ᨨ� O�矄�� 泐V r㡰㦨kr⏌犨 k�矄 ᨨ矄��ᆠᨨ 矄 ᨨ ᨨ� �鼀 矄�ᨨ 矄 �矄�ᆠ �矄鼀ᨨ� �� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄Rᨨ盐 ᨨ� O ᔼ v�矄 矄 ᨨ �ᆠ L�矄 鼀� �盐 ᨨ �矄 �盐鼀ᨨ鼀 盐鼀ᨨ盐O犨 �ᆠ ᨨ ᨨᨨ犨� � 矄鼀 矄�矄 㡰盐䁸 꾀 ҈桌 k R m
㡰 v㦨
㡰 v㦨
mk R 꾀犨 ᨨ �盐矄 鼀矄矄㝐�矄ᆠ ᨨᨨ鼀ᨨ � 矄ᨨ �� 矄矄 ᨨ犨 � ��盐꾀ᆠᨨ 矄 ᨨᆠ�矄ᆠᨨ � 盐ᨨ盐矄泐� ᨨ 矄盐 ᆠᨨ 犨 矄ᨨ �盐矄ᆠ�鼀矄 矄犨� ᨨ盐矄�� �ᔴᨨ 矄� ᨨ⼄盐tᨨ盐 �矄 ᨨ鼀ᨨ ᨨ 矄矄� � ᨨ盐� ᨨ� 턼 ␔ �ᆠ ᨨ ᨨᨨ犨� � 꾀 ҈䰘桌 ᨨ�ᨨ 矄�矄⏌ᆠᨨᆠ䁸
ᆠ㡰 L㦨 mk R
5 True / False 꾀 � 盐 � ᨨ鼀ᆠ�ᆠ盐 �矄ᆠᨨ 鼀矄矄M �盐矄R�矄 ᨨ� �� ᨨ� ᨨ ᨨ盐 �矄矄矄� � ᨨ盐� �矄ᆠ �ᆠ盐ᨨ�矄 犨� �盐 ᨨᆠ�泐 w 꾀 ᨨ �矄ᆠᨨ 矄鼀 �鼀 矄�ᨨ 矄盐ᨨ 鼀矄矄M⏌�矄ᆠ � 泐ᨨ �矄 �矄ᆠᆠᨨ �泐 �盐ᨨᆠ�泐ᨨ 矄泐矄鼀 ᨨ犨�矄 w ⏌ � ҈黨㜠㝐├矄桌 6
�
k R m
㡰 L㦨
和
MCQs with One or More than One Correct Answer
꾀ᨨ ᨨ 鼀矄矄m � L�ᨨ ᨨ ᨨ �矄 矄�矄 �� ��ᆠ泐꾀盐ᨨ 矄鼀鼀矄矄鼀ᨨ� � 矄 v矄�├矄 ᨨ ᨨ� ᨨ 泐 �矄 ᆠx ᨨ鼀�矄 �ᨨ �矄 鼀 � � ᆠᨨ鼀� 矄泐矄鼀 ᨨ犨ᨨ矄犨� �盐矄 wᨨ盐 �ᨨ
鼀 �鼀盐鼀 �盐矄 wM�矄ᆠ � ᨨx = xM.
꾀 桌
x
L
v
vx 㡰 w = L + x L ᆠ㡰 xM =
҈vx 㡰 w = L + ҈ x v 㡰 wM = L
턼
㡰 ᔼ ᨨᆠ�矄� ᨨ� � �泐
ᆠ矄矄ᆠᨨ � 盐ᨨ盐矄泐矄 犨 ᆠ�鼀矄 㡰 ᔼ ᨨᆠ�矄� ᨨ鼀ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t�矄ᆠᨨ 矄 ᆠ㡰 ᔼ ᨨᆠ�矄� ᨨ鼀ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t�矄 ᨨ 㡰 ᔼ ᨨᆠ�矄� � �泐ᨨ� 和 t ¹ 盐 犨 ᆠ�鼀矄 矄 黨 꾀��� 盐 � ᨨ鼀꾀ᨨ � �矄矄�� � ᨨ鼀�矄 �ᆠᨨ矄��ᨨ ᨨ �ᆠ�ᨨ 矄矄 ᨨᨨ 矄矄 ᨨ犨 � ��盐㡰 꾀矄ᨨ鼀� 矄 ᨨ �鼀 �鼀 泐 犨� �ᆠ�矄q␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄ᨨ盐�矄 꾀 ҈䰘桌 鼀ᨨ�ᨨ �矄⏌ᆠᨨᆠ䁸
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EBD_8341 P-56
Physics
꾀 q
졠
�
�҈
㡰 鼀� ᨨ� ᨨ 犨� �ᆠ泐 ᨨ犨 犨 㡰 ᆠᨨ泐ᨨ ᨨ� 꾀�矄ᨨ ᆠ㡰 ᔼ 矄 ᨨ盐矄ᨨ⼄盐ᨨ盐 ᨨ� � ᆠᨨ ᆠ犨� ᨨᨨ�矄ᨨᨨ�ᨨ ᨨ矄� q 㡰 ␔ 鼀├矄 �q犨� �ᆠ �矄ᆠ鼀 ᨨ �矄鼀� ᨨ� ᨨ鼀 � ��ᨨ矄��ᨨ
�矄ᨨᨨ�ᨨ ᨨ ҈ᆠᨨ矄q㡰
ᨨ矄��ᨨ ᆠᨨ r ᨨ �ᆠᨨ 鼀矄矄鼀�矄�� 泐 ᨨᨨ犨� �⼄盐�ᨨ
r 㡰㦨 a ˆ� + b ˆ 犨 a㦨҈⏌鼀矄b 㦨�鼀矄 鼀㦨҈├�꾀㦨҈矄 犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐ᨨ盐 �ᆠ䁸 꾀 ҈桌 r r ˆ ˆ 鼀矄҈ 㡰 ᨨᆠ�泐 �矄�� 泐 㦨 ҈� + ҈㡰 r 㡰 �盐鼀ᨨ鼀 盐鼀 犨� 矄ᆠᨨ ᨨ���
r �矄�� 泐 㦨�⏌㡰 ├ˆ 鼀矄 r r ᆠ㡰 ᨨᆠ �矄�� 泐 㦨 �ˆ + ˆ㡰 r 㡰 ᨨ⼄盐 t 犨� 矄ᆠᨨ ᨨ��� �矄�� 泐 r t 㦨 ⏌㡰 ├ˆ 鼀
犨ᨨ � ᆠ�ᆠ盐 �矄ᆠ矄ᨨ 鼀矄矄鼀 鼀 � � ��ᨨ 矄ᆠ�泐��� 泐 � 泐鼀矄矄矄矄
� 꾀� �ᨨ 鼀矄矄M �盐矄R�矄ᨨ� �犨� �盐 矄 wᨨ盐 � �ᆠ �矄矄矄� � ᨨ盐� �矄 M ᆠ O犨� 犨ᨨᨨ� 鼀矄矄矄ᆠ ᨨ 鼀矄矄 矄 O 矄鼀矄矄矄ᆠ 鼀ᨨ �泐ᨨ盐犨 矄ᨨ �犨ᨨ 鼀矄矄矄矄ᨨ 矄 � ᨨ � �矄矄 ᨨ犨 � ��盐꾀 矄ᨨ鼀� 矄 �盐矄 ᨨ 矄泐矄鼀�矄 w 黨 ᨨ ᨨ 鼀矄矄矄�矄 �矄 ᆠᨨ ᨨ鼀O꾀 �矄 � � 矄 �矄 ᆠᨨ ᨨ 鼀矄矄 ᨨ鼀O�矄 꾀 ҈�桌 w
졠
�
m҈
��� ᨨ ᨨ � l㦨 ᨨ盐� �ᆠ 矄 �矄 矄矄鼀泐�矄� ᨨ �鼀 矄盐 ᆠ 矄ᨨ� � 犨� ᨨ盐矄�� �ᨨ 矄盐 ᆠ矄ᨨ �盐矄 ᨨ盐 �矄ᨨ ᨨ �矄w �盐鼀ᨨ鼀 盐鼀ᨨ ur �矄矄鼀泐ᨨ盐 ᨨ� �졠�矄 矄 ��盐㡰␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌 鼀
鼀
l
m
w
l
졠 㡰 ᆠ ᨨ 鼀矄矄 ᨨ 矄矄鼀泐ᨨ矄 ᨨ盐 㜠 �矄犨� �盐矄 ᨨ w⏌� 㡰 鼀� �盐 ᨨ �盐鼀ᨨ鼀 盐鼀ᨨ ᆠ ᨨ 鼀矄矄ᨨ 矄矄鼀泐ᨨ盐 ᨨ� 졠�矄҈鼀w ᆠ㡰 鼀� �盐 ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄矄鼀泐ᨨ盐�矄ᆠ ᨨ 鼀矄矄�矄҈䰘鼀w⏌ ur 㡰 鼀� �盐 ᨨ 㜠ᆠᨨ鼀ᨨ ᨨ �矄��鼀w
҈ 㡰 ᆠ㡰 㡰 � � ᔼ ��盐 ᨨ 鼀矄矄m�矄矄 ᨨ犨 � ��� 矄 犨ᔼ�矄� 矄�ᆠ⼄盐���盐鼀鼀├� � �q犨� ᨨ� ᨨ ᨨᨨ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 犨 �矄m҈ 犨 ᨨᨨ �矄m ᨨ鼀ᆠ�ᨨ ᨨ 犨ᨨ �矄N҈ ᨨ ᨨᨨ�矄Nᔼ �矄 ᨨ盐ᨨ矄� 꾀 ҈桌
㡰
m҈ = m ¹ N q㦨
㡰
m҈ ¹ m
ᆠ㡰
m҈ ¹ m
mg ᔼ 矄盐 ᨨ ᨨᆠ矄ᆠ� �ᨨ 矄泐矄鼀 ᨨ �ᆠ矄�矄 ᨨ ᨨ鼀 � � 鼀ᨨ ᆠ
矄盐泐矄泐 黨桌 㡰 � 鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 ᨨ矄 ᨨᆠ ��
�鼀 㡰 ├� �ᆠ �泐ᨨ 矄泐矄鼀 ᨨ矄 ᨨ ᆠ � �
�鼀 ᆠ㡰 �盐鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 ᨨ矄 ᨨᆠ ��
�鼀 㡰 ᨨ � �泐ᨨ 矄泐矄鼀 ᨨ矄 ᨨᆠ ��
�鼀
㡰 �
mg mg = N҈ q = mg ¹ N = ҈ + m m ҈
㡰
m҈ = m ¹ N҈ q =
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P-57
System of Particles and Rotational Motion
��
ᨨ⼄盐 t ᨨ ᨨ 泐ᨨ盐�� ᨨ� �矄 ᨨ盐 ᨨ ⼄盐ᨨA맔L犨 A�矄ᆠᨨ 矄 ᆠᨨ L�矄 �盐鼀ᨨ鼀 盐鼀ᨨ ᨨ 泐ᨨ盐 ᨨ� ᨨ鼀 �矄� ᨨᨨ犨矄 ҈黨黨和㜠㝐├矄桌 dL �矄 �ᆠ盐ᨨL� 矄 矄ᨨ �鼀 dt 㡰 ᆠᨨ鼀ᨨ ᨨ L� �ᆠ�ᨨ ᨨ A ᨨ矄 ᨨ ᆠ �犨� �鼀 ᆠ㡰 鼀� �盐 ᨨ L ᨨ矄 ᨨᆠ �犨� �鼀 㡰 L ᨨ矄 ᨨᆠ �犨� �鼀 꾀�ᆠᨨ 鼀矄矄m�矄ᨨᆠ 犨� ᨨᆠ�泐v鼀├� � �ᨨ � 犨� ᨨ� ᨨ 鼀� �盐 ᨨ
�盐鼀ᨨ鼀 盐鼀ᨨ ᨨᆠ�ᨨ盐 ᨨ� ᨨ ᨨᆠ�ᨨ 犨 �ᆠ�矄�矄鼀 �鼀盐鼀 �� h�矄 ҈黨黨㜠㝐├矄桌 mv 㡰 ᨨ 㡰 g
㡰
�
ᆠ㡰
mv
㡰
g
m gh
10 Subjective Problems �䰘 꾀�ᆠ�矄ᨨᆠ �鼀t㦨 ᨨ鼀ᨨ� Pᨨ �ᨨ盐 犨� 矄 v �ᨨ � ᨨ ᨨ� ᨨ � 鼀� �盐 �ᆠ�ᨨ ᨨ �盐鼀ᨨ鼀 盐鼀 ᨨ �ᆠᨨ盐P�鼀t㦨v⏌g ҈黨㜠㝐├矄桌
Topic-4 : Moment of Inertia and Rotational K.E. 1 ҈
ᨨ盐 � 矄�矄I҈I㦨҈҈ 盐ᨨ a�矄 㝐� 和 ᔼᔼ㡰桌
MCQs with One Correct Answer
和 ᨨ犨 � ��盐�矄 ᨨᨨ犨�ᆠᆠ鼀ᆠᨨ ��矄ᨨ ᨨ㡰ᔼ �矄鼀矄矄�矄M �盐矄ᨨ �矄ᨨR �� H �矄鼀ᨨ鼀 ᨨ � �ᨨ盐�矄 �矄�矄 㝐� 和 ᔼ㡰桌
�
㡰 y
㡰
ᆠ㡰
㡰
O' O
ᆠ鼀
MR M R + H 㡰 㡰 㡰 MH MR ᆠ㡰 㡰
� 鼀矄矄 矄�泐ᨨ � ᨨ 꾀ᨨ � �矄 xö æ ᨨ鼀꾀ᨨ矄 l x 㡰 = l ç҈ + ÷ 犨 x�矄 �矄 ᆠ è ø ᨨ鼀꾀ᔼ 㝐�矄 鼀矄矄ᨨ ᨨ �矄鼀ᨨ鼀 ᨨ � � ᨨ盐 �矄矄矄� � ᨨ盐� 꾀 �ᆠ盐ᨨ ᨨ �矄 㝐� 和 ᔼᔼ㡰桌
5 䰘 㝐 㡰 㝐 ᆠ㡰 㝐 㡰 㝐 ҈ ҈ � 䰘 꾀犨 �矄ᨨ� � 泐犨� �盐矄 wᨨ 矄 鼀ᨨ鼀 ᨨ � �ᨨ 犨 �矄ᔼ 鼀ᨨ㜠 鼀 ᨨ � �ᨨ 矄 �矄 ����꾀 ᨨ 犨 ᨨ 鼀ᨨ鼀 ᨨ � �ᔼ� ��泐矄�矄矄盐 泐ᆠᨨ盐 ᨨ 矄鼀 矄 矄盐 ᆠ�ᨨ ᨨ矄矄 � ├� �ᆠ �泐ᨨ 矄泐矄鼀�矄 㝐� 和� ᔼ㡰桌
æ R L ö I = M çç + ÷÷ ᔼ 矄盐ᆠ ᆠ泐� �矄ᨨ鼀 ᨨ è ҈ ø �� 鼀矄矄ᨨ 鼀� �ᨨL/R ᨨ�ᨨ 鼀� �鼀盐鼀 ᨨ矄矄�I�矄 [Main Sep. 03, 2020 (I)]
㡰
� ҈ 㡰 ᆠ㡰 㡰 ᔴᨨ 矄� 犨ᨨ盐 � ᨨ鼀 �矄ᆠ矄ᨨ 矄鼀 �ᆠ├ 矄矄 � ��R҈㦨R R㦨aR鼀 ᨨ 矄鼀鼀� ᔼ �ᨨᨨ �鼀ᨨ鼀 矄ᨨ � �I҈ I矄ᆠ�泐
㡰
x ᆠ鼀 ᨨ盐 � ᨨ鼀ᆠ �盐矄 矄 ᨨ犨 � ��盐 �ᨨᨨ 鼀ᨨ鼀 矄ᨨ � �ᨨ盐 矄 �ᆠ盐ᨨ 矄 矄矄� � ᨨ盐� O ᆠ ᨨ 鼀矄矄㡰 OἘ ᆠᨨ ᨨ� 㡰�矄 㝐� 和 ᔼᔼ㡰桌 㡰 ⏌ 㡰 ҈⏌ ᆠ㡰 ҈⏌ 㡰 ҈⏌ 㝐ᨨ鼀 ᨨ � �ᨨ ᆠ泐� ᨨ 鼀矄矄M � L �盐矄 R ᨨ盐 �矄 矄矄� � ᨨ盐� �矄 ᆠ �ᆠ盐 ᨨ �矄 ᨨ ᆠ泐� �矄
㡰 ᆠ㡰 㡰
犨ᨨ盐 � ᨨ鼀ᆠ�ᆠ盐 �矄ᆠ矄ᨨ� �� 泐�
矄鼀 �ᆠ�ᨨ ᨨ盐 � ᆠᨨ鼀鼀ᨨ �矄 矄矄� � ᨨ盐� �ᆠ 矄 鼀ᨨ鼀 ᨨ � � �盐 ᨨᆠ�泐ᨨ �矄 �矄ᆠ ҈ ├�㜠鼀 ҈ 矄҈ 矄ᆠ�泐犨 � ᨨ矄 ᨨ 矄ᆠᨨ ᨨ ├�㜠鼀 � 矄҈矄ᆠ�泐꾀矄ᨨ鼀� 矄 泐�矄盐ᆠ├ ᨨ� 矄ᨨ� �矄矄� �矄泐矄鼀ᨨ盐 � ᆠᨨ鼀鼀ᨨ �矄犨� 矄ᨨ鼀 �盐矄 ├� �ᆠ �泐 ᨨ ᆠᨨ鼀� 矄泐矄鼀�矄 [Main Sep.02, 2020 (II)]
㡰 䰘
㡰
҈
㡰
ᆠ㡰
�
㡰
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EBD_8341 P-58
黨
Physics
矄ᨨ� 矄 矄ᆠ ᨨ 鼀矄矄m �鼀d 矄盐ᆠ├ᨨ� 矄盐ᆠ � 矄ᆠᨨ
ᆠ� � ᆠ 矄 ᨨ鼀 ⼄盐�� �ᨨ 矄� ᨨ � d �ᨨ ᔼ ᨨ 鼀ᨨ鼀 ᨨ � �ᔼᨨ 矄泐矄鼀ᨨ盐 �矄 ᔼ꾀 矄矄� � ᆠ ᨨ� ᨨ盐ᆠ ᨨ 泐ᨨ 矄 矄 ᔼ꾀 �ᆠ盐ᨨ ᨨ � ��矄 㝐� 黨 ᔼ桌 ҈ 㡰 ҈� 㡰 ҈ ᆠ㡰 ҈ ҈ 㡰 ҈� 꾀盐 � ᨨ鼀泐 �ᆠ├犨 犨� 鼀ᨨ鼀 ᨨ � �ᔼ �盐矄 �矄 ᨨ ᨨᨨ盐 �矄 ᆠ ᨨ 鼀矄矄 矄 ��㡰꾀 鼀矄矄矄矄矄� ��矄犨 ᨨ�矄�鼀 犨ᨨᨨᆠ├矄ᨨ 鼀矄矄矄m҈ m m҈>m㡰ᆠ ᨨ 矄ᨨ 矄� � 矄泐矄鼀�矄矄 ᨨ鼀矄 �盐矄 ᨨ 犨 犨 m҈ 矄ᆠ 矄泐 �矄 ᆠh�矄 㝐� 黨 ᔼᔼ桌
҈҈ �盐矄ᨨ �泐�ᨨ ᨨ 盐 � ᨨ鼀ᨨ ᨨ � lᨨ盐 l �矄矄矄� � ᨨ盐� ᨨ� 犨泐 ᨨ鼀 ᆠ ᨨ ᨨ �ᆠ盐ᨨ��矄㝐� 䰘 ᔼ桌 ҈ ҈ 䰘 l l l l 㡰 ᆠ㡰 㡰 ҈ 㝐矄矄盐 �ᨨ ᆠ�ᆠ盐 �矄ᆠᨨ �盐矄 矄ᨨ
�矄 ᆠr ᨨ鼀�矄ᆠ 矄s r㡰㦨A 졔Br 鼀ᨨ鼀 ᨨ � �ᨨ �矄ᆠᨨ盐 �矄 �ᆠ盐ᨨ 矄矄� � ᨨ盐� �矄ᆠ �矄 㝐� 䰘 ᔼᔼ桌
㡰
aB ö æ A 㡰 pa çè + ÷ø �
Bö æ aA + ÷ 㡰 pa çè �ø
Bö æ A aB ö æA ᆠ㡰 pa çè + ÷ø 㡰 pa çè + ÷ø � � ҈ 꾀ᆠ�ᆠ盐 �矄ᆠᨨ �盐矄b 矄 ᨨᨨ �盐矄a�矄ᆠ 矄 ��盐㡰ᔼ 鼀矄矄盐 �ᨨ �矄ᆠ�矄矄 æ s ö ç ÷ �盐矄ᨨ �泐�ᨨ ᨨ �矄ᆠᨨ盐�矄 �矄 è r ø 矄矄� � ᨨ盐� ᆠ �矄 㝐� ҈꾀҈黨ᔼ桌
҈⏌
㡰
a + b + ab
㡰
a+b
҈⏌
ᆠ㡰
a + b + ab
㡰
a+b
é m҈ - m 㡰 gh ù ú 㡰 ê ëê m҈ + m 㡰 + ҈ ûú é m҈ + m 㡰 gh ù ú 㡰 ê êë m҈ + m 㡰 + ҈ úû
҈ 犨ᨨᆠᨨ � �矄ᆠ矄 � �鼀ᨨ鼀 矄ᨨ � �ᔼ҈
҈⏌
é m҈ - m 㡰 ù ú ᆠ㡰 ê ëê m҈ + m 㡰 + ҈ ûú
gh
҈⏌
é ù m҈ + m ú gh 㡰 ê êë m҈ + m 㡰 + ҈ úû ҈ 꾀矄 矄 ᨨ犨 � ��盐 ᨨ ᨨ 鼀矄矄 m �矄 � 泐 鼀矄矄矄矄矄� �犨 ᨨ矄ᨨ ᨨ�ᨨ �矄犨ᨨ盐 ᨨ 泐 犨 �矄ᆠ㡰ᨨ �盐矄 鼀矄矄鼀␔ 矄 ᨨ鼀 矄 ᨨ矄矄 � ��ᆠ泐␔ � 矄ᆠᨨ �矄 ᆠᨨ �盐矄 ᨨ 犨 犨� 㝐� 䰘 ᔼ桌
ᔼ҈
w҈ ᨨ盐 �ᆠᨨ鼀鼀ᨨ �矄 泐ᨨ盐� � ᆠᨨ ᆠ犨� ᆠ ᨨ 泐ᨨ犨� ᆠᨨ鼀鼀ᨨ �盐 ᨨᆠ�泐ᔼ 쟼 쟼� � � ��ᨨ ��矄 쟼 쟼�㡰�矄 㝐� ҈꾀҈黨ᔼ桌
ᨨ� �犨� 矄ᆠ� �盐ᨨᆠ��矄w҈
ᔼ w ᔼ҈w҈ ᔼ w ᔼ҈w҈ 㡰 ҈ ҈ ᆠ㡰 㡰 - ҈ ҈ ҈ ҈� 꾀 � �矄ᆠᨨ 鼀矄矄㝐 �盐矄 矄鼀矄矄盐 � s r㡰㦨kr犨 r�矄 �矄 ᆠ ᨨ鼀�矄ᆠ ᔼ矄鼀ᨨ鼀 ᨨ � �ᨨ盐 �矄�ᨨ� � ᨨ盐� �矄ᆠ ᨨ 鼀矄矄 �ᆠ盐ᨨ�矄 �矄 㝐� ҈꾀҈黨ᔼ桌
㡰 -
㝐 㝐 㡰 㝐 㝐 ᆠ㡰 㡰 ҈ 꾀矄ᨨ� 矄 ᨨ 鼀矄矄㝐 �盐矄 �矄 �� � ᨨ犨ᨨ
㡰
㡰
҈ gh r
҈ gh ᆠ㡰 r
㡰 r
gh
r 㡰 gh
䰘M �矄 ᆠᨨ � ᨨ盐 � ᨨ鼀 �矄ᆠᨨ �盐矄 矄ᆠᨨ
盐 ⼄盐矄 �矄 矄鼀矄矄ᨨ
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P-59
System of Particles and Rotational Motion
part is converted into a uniform solid sphere. Let I1 be the moment of inertia of the new sphere about its axis. The ratio I1/I2 is given by : [Main 10 Apr. 2019 II] (a) 185 (b) 140 (c) 285 (d) 65 17. A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of q, where q is the angle by which it has rotated, is given as kqn. If its moment of inertia is I then the angular acceleration of the disc is: [Main 9 April 2019 I] k k k 2k q q (c) 2 I q q (a) (b) (d) 4I I I 18. A thin smooth rod of length L and mass M is rotating freely with angular speed w0 about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be: [Main 9 Apr. 2019 II] M w0 M w0 (a) (b) M +m M + 3m M w0 M w0 (c) (d) M + 6m M + 2m 19. A thin circular plate of mass M and radius R has its density varying as r(r) = r0 r with r0 as constant and r is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is I = a MR2. The value of the coefficient a is: [Main 8 April 2019 I] 3 (a) 1 2 (b) 3 5 (c) 8 5 (d) 2 20. Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm), about its axis be 1. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is: [Main 12 Jan. 2019 I] (a) 12 cm (b) 16 cm (c) 14 cm (d) 18 cm 21. The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is ‘I(x)’. Which one of the graphs represents the variation of I(x) with x correctly? [Main 12 Jan. 2019 II]
I(x)
I(x)
(a)
(b)
O
x
O
I(x)
I(x)
(c)
22.
x
triangle DEF is removed from ABC, the moment of inertia of the remaining figure about the same axis is I. Then : [Main 11 Jan. 2019 I] A
E
D
G B
23.
C
F
15 3 (b) I = I0 (a) I = I0 16 4 9 I (c) I = I0 (d) I = 0 16 4 a string is wound around a hollow cylinder of mass 5 kg and radius 0.5 m. If the string is now pulled with a horizontal force of 40 N, and the cylinder is rolling without slipping on a horizontal surface (see figure), then the angular acceleration of the cylinder will be (Neglect the mass and thickness of the string) [Main 11 Jan. 2019 II] 40 N
(a) 20 rad/s2 (b) 16 rad/s2 2 (c) 12 rad/s (d) 10 rad/s2 24. A circular disc D1 of mass M and radius R has two identical discs D2 and D3 of the same mass M and radius R attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis OO’, passing through the centre of D1, as shown in the figure, will be : [Main 11 Jan. 2019 II] O'
D2
O
D3
D1
(a) MR2
(b) 3MR2
(c)
4 MR 2 (d) 5
2 MR 2 3
m are connected at the two ends of a 2 massless rigid rod of length l. The rod is suspended by a thin wire of torsional constant k at the centre of mass of the rod-mass system (see figure). Because of torsional constant k, the restoring toruque is t= kq for angular displacement q. If the rod is rotated by q0 and released, the tension in it when it passes through its mean position will be: [Main 9 Jan. 2019 I]
25. Two masses m and
(d)
x x O O An equilateral triangle ABC is cut from a thin solid sheet of wood. (See figure) D, E and F are the mid-points of its sides as shown and G is the centre of the triangle. The moment of inertia of the triangle about an axis passing through G and perpendicular to the plane of the triangle is I0. If the smaller
(a)
3k q0 2 l
(b)
2k q0 2 l
(c)
k q0 2 l
(d)
k q0 2 2l
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EBD_8341 P-60
Physics
꾀ᨨ ᨨ � �ᆠ鼀�矄�ᨨ ᨨ ᔼ�矄�矄 矄盐ᆠ � 鼀├矄 �ᨨ ᨨ鼀 ᨨ� ᨨ 矄 矄 ᨨ犨 矄 ᨨ鼀 矄 ᔼ矄 �盐 矄 犨
� 矄矄矄 ᨨ盐� ᨨ� ᨨ � 矄҈㡰 犨� �㦨҈鼀矄㡰 㝐� 黨 ҈黨ᔼᔼ桌
�矄鼀 � � 盐 � ᨨ鼀 �矄ᆠ � �鼀矄矄㝐 �盐矄 矄矄 ᨨ犨 � ��盐 鼀ᨨ鼀 ᨨ � �ᨨ 鼀� � �ᨨ�ᨨ ᨨ �矄ᆠᨨ盐 �矄矄矄� � ᨨ盐� ᨨ� 졠 �ᆠ盐ᨨ ᨨ �矄ᆠ�矄 㝐� 졠 � 꾀�黨҈䰘桌 ҈黨 㝐 㡰 �
҈ 꾀ᆠ�ᆠ盐 ᨨᨨ �盐矄
䰘 㝐 ⏌ �҈ ᨨἘ 졠 ҈黨 㝐 ᆠ㡰 ⏌ �҈ ҈黨䰘 㝐 㡰 � ᨨ鼀 矄ᨨ� 矄 ᨨ 鼀矄矄 㝐 �盐矄 ᆠ盐 ᨨ 鼀 �鼀盐鼀ᨨ矄矄�ᨨ盐鼀�矄ᆠ盐㝐ᨨ鼀 ᨨ � �ᨨ ᆠ盐 ᨨ盐 �矄矄矄� � ᨨ盐� �矄ᆠ �ᆠ盐 ᨨᨨ ᨨ �矄 ᆠ矄�矄 㝐� ҈�桌 㝐 㝐 㡰 㡰 黨 p p 㝐 㝐 ᆠ㡰 㡰 p ҈ p ᔴᨨ 矄� � 盐 � ᨨ鼀 矄⼄盐 矄 鼀 ᨨ ��� 鼀�ᔼ �矄矄� �矄�a鼀矄矄m 鼀ᨨ鼀 ᨨ � �ᔼ ᨨ盐ᨨ ᨨ �矄 ��ᨨ 矄 㝐� 졠 � 꾀�҈҈�桌
㡰
㡰 ᆠ㡰 㡰 䰘 䰘 和 � �ᆠᆠ�ᆠ盐 �矄├矄ᆠ ᨨ 鼀矄矄㝐 �盐矄 犨 矄泐鼀鼀�ᆠ泐矄矄 ᨨ犨 鼀ᨨ鼀 ᨨ � �ᨨ �鼀 ᨨ盐 �矄 ᨨ鼀ᨨ 矄矄� � ᨨ盐� ᨨ� �矄 㝐� ҈桌 㡰
졠
҈黨 �� 䰘 ҈҈ 㝐 㡰 㝐 ᆠ㡰 㝐 㡰 㝐 ᨨ鼀盐 � ᨨ鼀ᆠ�ᆠ盐 �矄ᆠᨨ �盐矄 鼀矄矄黨㝐
㡰
�矄鼀ᨨ 矄矄 ᨨ犨 � ��盐
鼀ᨨ鼀 ᨨ � �ᨨ 鼀� � � �矄ᆠᨨ盐 �矄 �ᆠ盐ᨨ ᨨ �矄ᆠ 矄矄� � ᨨ盐� ᆠ ᨨ �矄ᆠ�矄 㝐� ҈桌 矄鼀 �矄ᆠᨨ �盐矄
㡰
䰘 㝐 ᆠ㡰҈㝐 㡰 㝐 黨 黨 꾀 � ᆠ�ᆠ盐 �矄├�矄� 泐 矄矄 ᨨ犨 � ��盐 �ᨨᨨ �矄鼀ᨨ鼀 ᨨ � �ᨨ盐 ¢ 矄 犨� 㝐� 졠 � 꾀�҈҈桌
㡰 㝐
黨
㡰
z
졠
x
z¢
y
㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈�
鼀ᨨ鼀 ᨨ � �ᨨ 盐 � ᨨ鼀ᆠ泐� ᨨ � l �盐矄 ᨨ盐�矄 �ᆠ盐�矄ᆠᨨ�矄ᔼ␔ �矄 �ᨨl⏌ 矄盐ᆠ 鼀ᨨ鼀 ᨨ � ��矄鼀� �鼀盐鼀䁸 㝐� ҈䰘桌 㡰 ҈ 㡰 ᆠ㡰 㡰
鼀 ҈
㡰
鼀 鼀
ᔼ=
Mm m M + m㡰 w R w R 㡰 M + m㡰 M Mm M + m㡰 M w R w R ᆠ㡰 㡰 M + m㡰 M + m㡰 � 꾀矄ᨨ� 矄 ᨨ 鼀矄矄M �盐矄R � �鼀ᨨ鼀 ᨨ � �ᔼᨨ盐�矄 �鼀�矄ᆠ矄� ᨨ矄ᨨ� �矄ᆠᨨ �盐矄r �ᆠ├ 矄矄t 鼀ᨨ鼀 ᨨ � �ᨨ �矄ᆠ ᨨ盐 �矄矄矄� � � �ᆠ盐ᨨ 鼀� 矄I R r 矄 㜠㝐҈桌 㡰
㡰
r=
ᆠ㡰
r=
R ҈� R ҈�
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r=
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r=
R ҈�
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R
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P-61
System of Particles and Rotational Motion
36. From a circular disc of radius R and mass 9M, a small disc of radius R/3 is removed from the disc. The m oment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is [2005S] (a) 4MR2 R 3
40 MR 2 (b) 9
R
(c) 10 MR 2 (d) 37.
2R/3
37 MR 2 9
IO and IP respectively. Both these axes are perpendicular to the plane of the lamina. The ratio IP / IO to the nearest integer is [2012] 42. Four solid spheres each of diameter 5 cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is N ´ 10–4 kg– m2, then N is. [2011] 3
One quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the center of the original disc. Its moment of inertia about the axis of rotation is [2001S]
Numeric Answer
43. ABC is a plane lamina of the shape of an equilateral triangle. D, E are mid points of AB, AC and G is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through G and perpendicular to the plane ABC is I0. If part ADE is removed, the moment of inertia of the NI 0 where N is an 16 integer. Value of N is _______. [Main Sep. 04, 2020 (I)]
remaining part about the same axis is (a)
38.
1 MR 2 (b) 2
A
1 MR 2 (d) (c) 2 MR 2 8 A thin wire of length L and uniform linear mass density r is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000S]
(a)
rL3 8p
(b)
2 3
(c)
39.
1 MR 2 4
5rL
16p
X
2
X' 90º
3
(d)
3rL
8p2 16p Let I be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle q with AB. The moment of inertia of the plate about the axis CD is then equal to [1998S - 2 Marks] (a) I (b) I sin2 q 2 (c) I cos q (d) I cos2 (q/2) 2
2
rL3
Integer Value Answer
40. The densities of two solid spheres A and B of the same r radii R vary with radial distance r as rA(r) = k æç ö÷ and R è ø 5 ærö rB(r) = k ç ÷ , respectively, where k is a constant. The è Rø moments of inertia of the individual spheres about axes passing through their centres are IA and IB, respectively. If IB n = , the value of n is [Adv. 2015] I A 10 41.
A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing though O and P is
E
D
G B
C
44. A circular disc of mass M and radius R is rotating about its axis with angular speed w1. If another stationary disc R having radius and same mass M is dropped co-axially 2 on to the rotating disc. Gradually both discs attain constant angular speed w2. The energy lost in the process is p奠 of the initial energy. Value of p is ___________. [Main Sep. 04, 2020 (I)] 45. An massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass m situated at its vertices. The moment of inertia of the system about the N ma 2 line EX perpendicular to EG in the plane of EFG is 20 where N is an integer. The value of N is _________. [Main Sep. 03, 2020 (II)] X
E
F
a
G
46. One end of a straight uniform 1 m long bar is pivoted on horizontal table. It is released from rest when it makes an angle 30° from the horizontal (see figure). Its angular speed when it hits the table is given as ns -1 , where n is an integer. The value of n is ________ . [Main 9 Jan. 2020 I]
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EBD_8341 P-62
䰘
Physics
4 Fill in the Blanks 꾀 矄泐鼀鼀�ᆠ 鼀� ᨨ 鼀矄矄 M ᆠᨨ 矄�矄矄 ᨨ A B 矄⼄盐 矄 犨� 矄鼀�ᆠ�ᆠ盐矄ᆠ�ᨨ ᨨ ᨨ �ᨨ 矄⼄盐 矄 矄 ᨨ犨 � �� 㜠҈ a
矄� ᨨ 矄⼄盐�矄 a 鼀ᨨ鼀 ᨨ � � O ᨨ 鼀� ᨨ盐
�矄 ᨨ盐� �矄ᆠ ᨨ 鼀矄矄 �ᆠ盐 ᨨ �矄҈Ma
鼀ᨨ鼀 ᨨ � �ᨨ 鼀� ᨨ盐 � AB � ᨨ 鼀� �矄 ҈黨黨䰘㜠㝐├矄桌
6
MCQs with One or More than One Correct Answer
鼀ᨨ鼀 ᨨ � �ᨨ � 矄⼄盐ABCD ��ᨨ 盐 � ᨨ鼀 �ᆠ├ 矄矄ᨨ盐 �矄矄矄� � ᨨ盐� ᆠ O �ᆠ盐ᨨ ᨨ �矄 ҈黨黨㜠㝐├矄桌 ҈ 㡰 I҈ + I 꾀 㡰 I + I 졠
ᆠ㡰
I҈ + I
㡰
I҈ + I + I + I
ᔴ
犨 I҈ I I I 矄ᆠ�泐 鼀ᨨ鼀 矄ᨨ � �ᨨ盐 �矄҈ 犨 �ᆠ � ᨨ
Topic-5 : Rolling Motion 1 ҈
MCQs with One Correct Answer
꾀盐 � ᨨ鼀矄 ᨨ 鼀矄矄��ᨨ矄犨� ᨨ盐矄�� �ᨨ ᨨ� ᨨ 矄盐 ᆠ犨� �矄ᆠ 鼀ᨨ� �矄 ᨨ �ᆠ鼀⏌矄ᔼ矄├� �ᆠ �泐�矄㝐� ᔼᔼ桌 㡰 䰘�맔҈ 㡰 䰘�맔҈ ᆠ㡰 �맔҈ 㡰 1.13 × 10 꾀矄鼀ᨨᨨ �鼀ᆠ鼀 矄 ᨨ �鼀҈ ᆠ鼀 矄 ��盐ᨨ犨ᨨ 㡰ᔼ�矄ᨨ ᨨ� ᨨ ᨨᨨ 鼀矄ᆠ�矄ᨨ矄��ᨨ ᨨ� ᨨ 泐ᨨ �矄 犨� ᨨ �ᨨ 矄ᆠᨨ ᨨᨨ 矄 ��盐ᨨ �� 㡰 矄ᆠ�矄 ᨨ犨盐矄 矄ᨨ犨泐ᨨ 矄ᨨ �鼀ᨨ矄犨� ᨨ盐矄�� �ᨨ ᨨ矄矄 ᨨ� �犨� ᨨ盐矄�� �꾀 ᨨ 矄鼀ᨨ �ᆠ鼀 ᨨ矄��ᨨ ᨨ 矄ᆠ犨�ᨨᨨ├�├ ��盐矄矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ矄ᆠ㡰 꾀 桌
꾀矄ᨨ� 矄 矄ᨨ� ᆠ泐� ᨨ � �ᆠ ��ᨨᆠ � ᆠ� 犨� 矄鼀� ᨨᆠ�泐 矄 ��盐㡰ᨨ ᨨ犨� ᨨ盐矄�� � ᨨ盐� ᨨ盐 犨ᨨᆠ�鼀鼀 �鼀盐鼀 �� 矄hsph hcylᨨ � ᆠ� �ᨨ 泐
hcyl
�矄��
㝐� 꾀҈黨ᔼᔼ桌
҈ 㡰 ҈� � �
ᨨᨨ犨� �ᨨ �矄鼀 ᨨᨨ盐 犨� ᨨ盐矄�� �㡰 矄鼀� ᆠ� ᨨ鼀 ᨨ� ᨨ �㡰� �ᨨ
㡰
hsph
㡰 ҈
ᆠ㡰
�盐矄 ��㡰矄ᨨ� ᆠ泐� ᨨ �盐矄
R ���㡰矄ᨨ�
R ᔼ � ᆠ ᆠ矄 矄 ᨨ ᆠ ᨨ 鼀矄矄 ᨨᨨ鼀ᨨ � ᆠ� �矄矄鼀 �ᨨᨨ 鼀 �鼀盐鼀 �� 矄 泐ᆠ�鼀�矄 㝐� 黨꾀�҈黨ᔼ桌 㡰 㡰 ҈҈�䰘 ᆠ㡰 ҈҈� 㡰 꾀 ᨨ鼀ᨨ� ᨨ盐矄矄ᨨ� ᆠ泐� �ᆠᨨᨨ �盐矄 鼀矄矄 㝐 �矄 盐 ᨨ ᆠ�ᆠ├ �ᆠ 泐 ᨨ� ᨨ ᨨᆠ 꾀矄矄盐鼀� � ᨨ� � 犨� ᨨ盐 矄�� � �盐 ᆠᆠ�ᨨ ᨨ ᆠ泐� �矄 㝐� ҈ ҈黨ᔼ桌
矄 ᨨ �盐矄
x㦨
x㦨�ᆠ鼀
㡰
x㦨䰘�ᆠ鼀
�
㡰
x㦨
ᆠ㡰
x㦨
x㦨�ᆠ鼀
x㦨
㡰
x㦨҈ᆠ鼀
㡰 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀 꾀ᨨ�矄鼀 泐ᨨ� � �ᨨ� 犨ᨨᆠᨨ 矄 � �ᆠ矄졠ᔼ�矄├ᨨ 犨ᨨ�矄꾀 ᔴ犨 �ᆠ ᆠ 矄泐鼀鼀�ᆠ泐 矄 ��盐㡰 犨� �矄 �矄 �ᆠ盐ᨨᔴ �矄ᆠ 졠 ᆠ ᨨ �
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P-63
System of Particles and Rotational Motion
sufficiently high so that the block does not slide before toppling, the minimum force required to topple the block is [2000S] (a) infinitesimal P F (b) mg/4 M L (c) mg/2
joining AB and Cd (see figure). It is given a light push so that it starts rolling with its centre O moving parallel to CD in the direction shown. As it moves, the roller will tend to: [Main 2016] B
D
(d)
O
2 A
7.
C
(a) go straight. (b) turn left and right alternately. (c) turn left. (d) turn right. A loop of radius r and mass m rotating with an angular velocity w0 is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero.What will be the velocity of the centre of the hoop when it ceases to slip ? [Masin 2013]
mg(1 - m) Integer Value Answer
12. Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v1 and v2, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v1 = 3 m/s then v2 in m/s is [Adv. 2015] (g = 10 m/s2) A
v1 = 3m/s 30m
rw0 rw0 rw0 (b) (c) (d) rw0 3 4 2 A small object of uniform density rolls up a curved surface with an initial velocity v. It reaches up to a maximum height 3v 2 of with respect to the initial position. The object is 4g [2007]
B
(a) 8.
v
9.
10.
11.
(a) ring (b) solid sphere (c) hollow sphere (d) disc A disc is rolling without slipping with angular velocity w. P and Q are two points equidistant from the centre C. The order of magnitude of velocity is [2004S] (a) vQ > vC > vP (b) vP > vC > vQ C P (c) vP = vC , vQ = vC/2 Q (d) vP < vC > vQ A cylinder rolls up an inclined plane, reaches some height, and then rolls down (without slipping throughout these motions). The directions of the frictional force acting on the cylinder are [2002S] (a) up the incline while ascending and down the incline descending (b) up the incline while ascending as well as descending (c) down the incline while ascending and up the incline while descending (d) down the incline while ascending as well as descending. A cubical block of side L rests on a rough horizontal surface with coefficient of friction m . A horizontal force F is applied on the block as shown. If the coefficient of friction is
C
v2 27m D
13. A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2N on the ring and rolls it without slipping with an acceleration of 0.3 m/s2. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is [2011] 3
Numeric Answer
14. Consider a uniform cubical box of side a on a rough floor that is to be moved by applying minimum possible force F at a point b above its centre of mass (see figure). If the coefficient of friction is m = 0.4, the maximum possible value b of 100 × for box not to topple before moving is a ________. [Main 7 Jan. 2020 II]
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EBD_8341 P-64
҈�
Physics
꾀� � �矄ᆠ� ��泐矄矄� 泐矄� ᨨᨨ � ᆠ� 犨 �ᆠ 鼀├矄 � 犨� ᨨ� ᨨ 泐矄ᨨᨨ犨� ᨨ盐矄�� � 矄鼀� 矄 ᨨ �鼀ᨨ � 矄 ᨨ矄 ᔼ �鼀 � ᆠ 犨 � ᆠ � � � ᨨ盐 �矄 㡰 ⏌ ҈ 矄 �� ᨨ ᨨᨨ � ᆠ� � 鼀矄�矄힘힘힘힘힘힘힘힘힘힘 ├g㦨҈鼀矄
w
w⏌ O
x
꾀 ҈桌 5 ҈
꾀� �ᨨ 鼀矄矄├� �盐矄҈鼀 矄ᨨ� ᆠ泐� ᨨ 鼀矄矄├� ᨨ 矄鼀 �盐矄�� 矄鼀 ├� �ᆠ �泐 矄 矄�鼀盐 ᨨ盐矄泐 ᨨ ᨨ� ᨨ 矄盐 ᆠ矄盐ᆠ 泐�� ᨨᨨ矄矄ᨨᨨ 矄 矄 ᨨ犨 矄犨犨 �ᆠ �矄 矄鼀 �矄 ᆠ ᨨ鼀 � � ᆠ泐� ᨨ� � �ᆠ�ᨨ � ᨨ ᆠ矄矄 �矄 ���� ᆠ泐� 犨�ᆠ 犨 �矄 ҈黨黨㜠㝐├矄桌 6
҈䰘
㡰 ᨨ� O 矄� ᨨᆠ�泐 w iˆ
True / False
҈黨
MCQs with One or More than One Correct Answer
꾀犨 ᨨ �盐矄 鼀矄矄㝐�矄ᆠ ᨨᨨ鼀ᨨ � 矄ᨨ �� 矄矄 ᨨ犨 � ��盐꾀ᆠᨨ 矄 ᨨᆠ�矄ᆠᨨ � 盐ᨨ盐矄泐� ᨨ 矄盐 ᆠᨨ 犨 矄ᨨ �盐矄ᆠ�鼀矄 矄犨� ᨨ盐矄�� �ᔴᨨ 矄� ᨨ⼄盐tᨨ盐 �矄 ᨨ鼀ᨨ ᨨ 矄矄� � ᨨ盐� ᨨ� 턼 ␔ �ᆠ ᨨ ᨨᨨ犨� � 꾀 ҈䰘桌 ᨨ�ᨨ 矄�矄⏌ᆠᨨᆠ䁸 和
턼
҈҈ Rwiˆ + Rwkˆ ҈ Rwkˆ ᆠ㡰 ᨨ� P 矄� ᨨᆠ�泐 Rwiˆ 㡰 ᨨ� P 矄� ᨨᆠ�泐 æ ö ˆ ˆ ҈ çè ÷ø Rwi + Rwk 꾀矄 �矄ᨨ� �犨� ᨨ盐矄�� �ᨨ � ᨨ� ᨨ 矄盐 ᆠᔼ ��盐꾀�矄 ᨨ� ᨨ ᆠᨨ ᆠ�矄 ᆠ ᨨ 矄 ᔴ�矄�矄ᨨ鼀ᨨ矄ᨨ� 黨桌 r r r r C 㡰 VC - VA = (VB - VC ) r r r r 㡰 VC - VB = VB - VA B r r r r ᆠ㡰 VC - VA = VB - VC r r r 㡰 VC - VA = VB A 꾀矄ᨨ� ᆠ泐� �矄ᨨ� � ᨨ犨 ᨨ盐� � ᆠ� ᨨ � ᆠ� �ᨨ q
㜠�㝐҈桌 㡰 �ᆠ�ᨨ ᨨᆠ�矄 �矄矄�� 㡰 �ᆠ�ᨨ ᨨᆠ�矄 ᆠ矄矄�泐ᆠ �� � ᆠ㡰 �ᆠ�ᨨ ᨨᆠ 犨� � ᨨ�ᨨ 盐 � 矄�ᨨ
㡰 �ᆠ�ᨨ ᨨᆠ�矄 盐ᆠ � q�矄 盐ᆠ
㡰 ᨨ� P 矄� ᨨᆠ�泐
9 㡰
҈
ᔼ ᨨᆠ�矄� ᨨ� � �泐
ᆠ矄矄ᆠᨨ � 盐ᨨ盐矄泐矄 犨 ᆠ�鼀矄 㡰 ᔼ ᨨᆠ�矄� ᨨ鼀ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t�矄ᆠᨨ 矄 ᆠ㡰 ᔼ ᨨᆠ�矄� ᨨ鼀ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t�矄 ᨨ 㡰 ᔼ ᨨᆠ�矄� � �泐ᨨ� 和
t ¹ 盐 犨 ᆠ�鼀矄 矄
��盐矄 ᨨ犨矄矄泐矄鼀ᆠᨨ 矄�矄� �ᨨ �㡰� �ᨨ ᨨ盐 �盐矄Rᨨ� �ᆠᨨᆠ├犨�矄犨� ᨨ盐矄�� �ᨨ ᨨ� ᨨ 矄盐 ᆠ犨� �盐矄 w ��㡰 �
�矄ᆠᨨ �盐矄 Rᨨ� � �㜠ᆠᨨᆠ├犨�矄犨� �盐矄 w⏌ � � �矄ᆠ矄 泐 �ᆠ�ᨨ 矄矄� �矄
ᨨ� Pᨨ �
�矄ᆠ�矄 �矄 ᆠR ᨨ鼀 ᨨ��� 犨 OP鼀├矄 �ᨨ 犨� ᨨ� ᨨ
犨� 矄ᆠ ᨨ ᨨ� ᨨ 矄盐 ᆠ ҈桌
Assertion and Reason Type Questions
҈ 和T꾀TE㝐ENT㜠҈ 犨ᨨᆠ泐� 矄ᨨ ᨨᨨ犨 鼀㡰 ᨨ 矄ᨨ� 犨ᨨᨨ 㡰犨� 矄鼀鼀矄矄 � �ᆠ �鼀 矄�ᨨ 矄 矄�鼀盐 ᨨ盐矄泐ᨨ犨 ᨨᨨ 犨� ᨨ盐 矄�� � ᨨ犨 � ᆠ� ᨨ鼀 矄鼀 �� ᨨᨨ犨ᆠ泐� 犨�ᆠ ᨨᨨ鼀ᨨ � ᆠ� �矄 和T꾀TE㝐ENT㜠泐 � ᆠ�ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐 ᨨ├� �ᆠ ��矄ᨨ ᨨ ᆠ泐� 矄 � �ᆠ犨 泐ᆠ ᨨᨨ鼀ᨨ � ᆠ� 桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐
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P-65
System of Particles and Rotational Motion
m
F
�
10 Subjective Problems
㡰 ᆠᆠ�ᨨ 矄ᨨ ├ ᆠ ᨨ M҈ 鼀矄矄ᨨ ᆠ泐� 㡰 鼀� �盐 矄 �ᆠ�ᨨ 矄ᨨ �ᆠ�ᨨ ᨨᆠ矄 ᆠᨨ ᆠ ᨨ� 矄
꾀鼀 盐矄 矄ᆠ泐� ᨨ 鼀矄矄m҈犨� ᨨ ├ᨨ 鼀矄矄m矄矄 ᨨ犨 � ��盐 � ᨨ矄�� � 泐ᆠᨨ ᆠ ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ � 泐 鼀 �矄F ҈黨黨黨㜠҈㝐├矄桌
Topic-6 : Miscellaneous (Mixed Concepts) Problems 1 ҈
ᆠᨨ 矄 �盐 ᆠᆠ�ᨨ a ᔼ ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨 ᨨ �矄 m ��泐�矄
�ᆠ �鼀 犨 �ᆠ 矄矄矄�� � �矄 和桌
MCQs with One Correct Answer
꾀 ᨨᨨᨨ �盐矄R�矄├ᨨ ᨨᨨ �盐矄r r < R㡰 鼀 ᨨ ├├ ᨨ� ᨨ 泐졠 ᨨ ├�矄
ᨨ犨� 矄ᨨ ��矄� 鼀├� � �q ᨨ鼀 ᨨ� ᨨ 矄矄 ᨨ犨 � ��盐ᨨ犨 鼀 �鼀盐鼀 盐ᨨ q矄ᨨ ᨨᨨ ᨨ矄 ᨨ矄ᨨ� � ᨨ犨
├矄�矄 �矄 ��盐�矄矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ 矄ᆠ㡰 꾀 桌 Plank
R
a
L
r
q
V
㡰 矄� q =
�
w
v® O
r t
r t
㡰
O
T
㡰
t
O
䰘
T
t
r t
r t ᆠ㡰
졠
t
㡰
O
t
T T 꾀ᨨ � ᨨ� ᨨ ᨨ 矄 犨 �ᆠ ᆠ 矄� ᨨ ��矄 � � ��泐ᆠ �矄 ᆠL ᨨ鼀ᨨ A ᨨ ᨨ ᨨ �矄矄� �盐鼀ᨨ�ᨨ ᨨ盐A犨�
҈ 㡰� � �矄�鼀 m ⏌ a 㡰 m ⏌ a ᆠ㡰 ma 꾀ᆠ盐�ᆠᨨᆠ├ᨨ 矄� �矄鼀ᨨ� �犨� ᨨᆠ�泐Vᨨ ᨨ� ᨨ 矄鼀ᨨᨨ 矄矄 ᨨ犨 � ��盐ᔼ �矄� � ᨨ� O �盐矄 ᨨ ᨨᆠ├ � �矄O�矄 ҈黨黨黨和㜠㝐├矄桌 a 㡰 V⏌ a㡰 㡰 V⏌ a㡰 M 㡰
r r 㡰 q = R R r r ᆠ㡰 矄� q = 㡰 ᆠᨨ矄 q = R R 꾀 � 盐 � ᨨ鼀ᨨ �ᨨ O�矄ᨨ� �� ᨨ� ᨨ 犨� ᆠᨨ 矄 �盐矄 w矄矄 ᨨ犨 � ��盐 꾀�鼀t㦨矄鼀� 矄ᆠ矄矄 ᨨ鼀O 鼀ᨨ矄犨� ᆠᨨ 矄 矄 v犨� 矄ᆠᨨ ᨨ ᨨ犨 矄 ᨨ ᔼᆠ 矄 ᨨ ᨨ t = T 矄ᨨ矄 �盐矄 ᨨ 矄泐矄鼀鼀� 矄w ᨨ盐� ᨨ盐 鼀� �盐 ᨨ ᨨ⼄盐 ( rt ) ᨨ盐 O 矄 盐 ᆠ�ᨨ ᨨ �鼀�矄矄矄 泐犨 �ᆠ ᨨ䁸 ҈桌 z
B
A
ᆠ㡰 V ⏌ a 㡰 㡰 ᨨ O 꾀矄鼀ᨨᨨ 矄 꾀�矄鼀ᨨ� �ᨨ �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 犨� �盐矄 w ᆠ ᨨ 鼀矄矄ᨨᆠ�泐u ᔼᆠᨨ� 矄矄�ᆠ泐 ᨨ 犨� � �ᆠ矄 矄�ᆠ �ᆠ�ᨨ 泐犨 꾀 ᆠᨨ�矄�ᨨ � �盐矄 矄w꾀 w矄ᆠ�泐
҈黨黨黨和㜠㝐├矄桌 㡰 w꾀w 㡰 w꾀㦨w ᆠ㡰 w꾀㦨w 㡰 w㦨w
犨ᨨᨨ� 鼀矄矄矄ᨨ ├� 䰘├� � 矄 ᨨ ᨨ ᨨ � ҈鼀 ᨨ ����鼀矄矄 ᨨ �矄 矄ᨨ� �ᨨ盐 �矄 �ᆠ盐ᨨ�矄 � 犨� 盐 � ᨨ鼀 �盐矄 ᨨ� ᨨ ᨨ ᨨ盐� 犨 �ᆠ �矄矄 ᨨ盐 矄矄� ᨨ 犨ᨨ├⼄盐� ᨨᨨ�ᨨ ᨨ ᨨ �矄鼀� �鼀盐鼀�矄ᨨᆠ �矄 ᆠ ᨨ ҈黨黨�和桌 㡰 m ᨨ鼀鼀矄矄ᨨ ├� 㡰 䰘m ᨨ鼀鼀矄矄ᨨ 䰘├� ᆠ㡰 黨m ᨨ鼀鼀矄矄ᨨ ├� 㡰 黨m ᨨ鼀鼀矄矄ᨨ 䰘├� 꾀ᆠ�矄鼀ᨨ� �� ᆠ�ᆠ盐 ᨨ� ᨨ ᆠ├ᨨ �盐矄҈鼀 犨� ᆠᨨ 矄 矄 ᨨ ҈ 鼀⏌矄꾀 盐盐鼀 ᨨ �矄 矄盐矄 ᨨ鼀 ᨨᨨ ᨨ ᆠ泐�� ��� ᨨ ᨨ � ҈鼀 �鼀 泐 ᨨ 犨� ᆠ├�矄 ҈黨黨㜠㝐├桌 㡰 ᨨ 㡰 ᆠ㡰 � 㡰 꾀 盐 ᨨ � L �矄 � ᆠᨨ鼀泐犨�
� ᆠᨨ鼀矄矄��⼄盐� ᨨ 鼀矄矄M ᆠᨨ矄 ᨨ 矄 盐�矄 ᨨ � ᨨ� ᨨ ᨨ盐 ᨨ ᨨ �矄 矄犨� 盐 � ᨨ鼀 �盐ᨨᆠ�泐 w ᨨᆠ 泐 �⼄盐� ᨨ �矄 ҈黨黨㜠㝐├矄桌
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EBD_8341 P-66
Physics
㡰 ᆠ㡰 2 黨
M w L 㡰 M w L 㡰 Integer Value Answer
ᆠ㡰
M w L
犨�
Mw L
F
졠 矄ᆠᨨ 泐� � ᨨᆠ矄 �盐矄 ᨨ �矄ᆠ� 矄҈�矄 꾀 ҈桌
҈
꾀盐 � ᨨ鼀ᆠ盐ᨨ 矄� a 鼀矄矄m矄矄ᨨ ᨨ盐� ᨨ� ᨨ 꾀 ᨨ� ᨨ ᨨᆠF�矄� ᨨ鼀ᨨ ᨨ ᨨ ᆠ矄ᨨ� �矄 �ᆠ泐ᨨ ᆠ ᨨ ᆠ �� a⏌ᨨ 矄 鼀� �鼀盐鼀 盐ᨨ F ᨨ犨 �ᆠ ᆠ盐�� 矄ᨨ�ᨨ盐 � �矄 꾀矄矄盐鼀 ᆠ盐 ᨨ矄 ᨨ矄� 㡰 ҈黨㜠㝐├矄桌 5
҈҈
True / False
꾀� �盐 ᨨ 盐 � ᨨ鼀 �ᆠ├ 矄矄 矄�泐�矄鼀 ᨨ ᨨ ᨨ盐 �矄 �ᆠ盐 ᨨ ᨨ 㡰 矄矄� � A B C ᨨ盐� A 㡰矄矄� � ᨨ盐� B泐 �ᆠ�ᨨ ᨨ F 矄鼀 ᨨᆠFC 鼀� ᨨ� ᨨ AB㡰矄矄 ᨨ犨 � ��盐 �盐ᆠᆠ�ᨨ � ᨨ ᆠ矄矄犨� 矄鼀 ҈黨�㜠㝐├矄桌 6
҈
Fill in the Blanks
MCQs with One or More than One Correct Answer
꾀 � 盐 � ᨨ鼀ᨨ ᨨ 鼀矄矄M � L�矄 �ᆠᨨ ᨨᨨ犨� � �ᆠ�ᨨ ᨨ �矄矄 ᨨ鼀 矄 矄ᨨ � 矄 泐ᨨ� � ᨨ盐 �矄 ᆠᨨ ᆠ㜠 ᨨ� 犨� ᨨᨨ 犨� ᨨ盐 矄�� � ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄⏌ᆠᨨᆠ犨 ᨨ 鼀├矄 � 犨� �ᆠ䁸 꾀 ҈黨桌 g�矄 ᆠᆠ�ᨨ 盐ᨨ��泐桌 g L
㡰
�盐矄 ᨨ ᨨ 犨�
㡰
�ᆠᆠᆠ�ᨨ ᨨ ᨨ 矄ᆠ ᨨ 鼀矄矄 g 犨�
Mg ҈
g L ҈ 犨ᨨ矄ᨨ� ᆠ泐� 矄P Qᨨ 矄鼀鼀矄矄 矄鼀 �盐矄 矄ᨨ� � ᨨ犨 � � ᆠ� ᨨ鼀 矄鼀 �� 矄鼀�鼀ᔴ泐� P 矄鼀ᨨ矄ᨨ �矄鼀矄矄 ᆠᨨ ᆠ �矄矄盐 ᆠ犨 �Q 矄鼀ᨨ矄ᨨ �矄鼀矄矄 ᆠᨨ ᆠ �矄 ␔ �ᆠ 矄鼀 矄㡰 �矄 㡰 ᆠᨨᆠ䁸 ҈桌
㡰
꾀盐 � ᨨ鼀ᆠ�ᆠ盐 �矄ᆠᨨ 鼀矄矄҈�├� �盐矄�鼀�矄 � ��泐矄ᨨ ᨨ� ᨨ �ᆠ�ᨨ 矄矄矄盐 ᆠ F ᨨᆠ矄ᨨ ⼄盐鼀� �盐 F㦨� X � 矄�鼀盐 ᨨ盐矄泐ᨨ � 矄� 矄ᨨ ⼄盐�� � O XYZ犨� �矄�ᆠ矄ᨨ �鼀 F Y Z ᨨ �矄ᆠ 矄 ��盐㡰
4
ᨨ鼀ᆠ�ᨨ ᨨᆠ ᨨ鼀 ᨨᨨᨨ ᨨ
�盐ᆠᆠ�ᨨ ᨨ ᨨ 犨�
㡰 ᨨ ᆠ泐� 矄P Qᆠ �ᨨ盐 矄鼀 �鼀 㡰 ᔴ泐� 矄P 矄�� ᆠᆠ�ᨨ ᆠ泐� Q ᆠ㡰 ᨨ ᆠ泐� 矄 ᆠ �ᨨ盐 犨� 矄鼀 矄�ᨨ ├� �ᆠ �泐 㡰 ᔴ泐� Qᆠ 矄 �ᨨ盐 犨� � �盐 矄 ҈ 꾀 � � �ᨨ 鼀矄矄├� �盐矄�鼀�矄ᨨ� �犨� ᨨ盐 ᨨ ᨨ� ᨨ 犨� ᨨᆠ�泐҈鼀⏌矄꾀矄鼀ᨨ 鼀矄矄҈├�鼀ᨨ� �犨� ᨨᆠ�泐鼀⏌矄� ᨨᨨ矄� �ᆠ�ᨨ �矄 � � �� ᨨ 䰘� 鼀 �ᨨ矄 �ᆠ泐盐犨� ᨨᆠ�泐҈鼀⏌矄ᔼ鼀鼀 �泐 ᆠᨨ�矄�ᨨ
҈҈桌
㡰 � � 矄盐ᨨ�ᨨ ᨨ盐�矄矄�ᨨ 泐ᔴ㝐 㡰 � �ᆠᨨ鼀矄ᨨᆠᨨ鼀矄ᨨ ᆠ㡰 �ᆠ�ᨨ 犨 � � �ᨨ盐 �矄ᨨ 㡰 �矄 ᨨ �ᆠ�ᨨ 犨 � � �ᨨ盐 ҈� 꾀盐 � ᨨ鼀ᨨ � a 鼀矄矄m�矄ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ 犨ᨨᨨ� 鼀矄矄矄m m鼀ᨨ� ��
矄鼀 ᨨ� ᨨ 犨� 矄 v v矄ᆠ�泐 矄�├ 矄矄 ᨨ犨 � ��桌 矄�ᆠ├ᨨ ᆠᨨ�矄�ᨨ ᨨ� � �盐ᨨᆠ�泐 ᨨ盐 ᆠ ᨨ 鼀矄矄㡰ᨨ �泐 ᆠ ᨨ 鼀矄矄ᨨᆠ�泐泐wE vc矄ᆠ�泐犨 ᆠᨨ�矄�ᨨ ҈黨黨҈㜠㝐├桌 㡰 vc 㦨 m v 㡰 w 㦨 a v �a ᔴ v ᆠ㡰 w = �a 㡰
mv E= �
a
a
v 鼀m
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P-67
System of Particles and Rotational Motion
7
Match the Following
҈ ᔼ ᔴᨨ盐鼀 㜠ᔼᨨ犨 ᨨ盐 � 矄ᨨ �ᆠ �� 矄 盐 ᆠ�ᨨ 矄ᨨ �鼀ᔼ 矄 盐 ᆠ�ᨨ 矄a b ᨨ矄��ᆠᨨ 矄 矄ᨨ ᨨ� �鼀 矄�ᨨ 矄 a ¹ bᔼ ᆠ ᆠ矄 ᨨᆠᆠ� �ᨨ �ᆠ�矄� ᨨᨨᆠᨨ 矄�ᔼ ᔴᨨ盐鼀 㜠ᔼᔼ � 泐矄�ᆠ⼄盐 ��矄 r r ᨨ �ᆠ鼀 �ᨨ �矄 � 鼀ᨨ鼀 盐鼀 �矄 �盐鼀ᨨ鼀 盐鼀ᨨ盐 ᨨ��� ⧤�矄 ├� �ᆠ �泐�矄 ᨨ � �泐 쟼�矄 ᨨ �泐 㝐ᆠ ᆠ � �矄㜠ᔼ犨� ᨨ矄⼄盐 ��矄� �矄㜠ᔼᔼ 犨 �ᆠ ᆠᨨ 矄 ᨨ 꾀 ҈桌
턼 和 㡰 㡰 ᆠ㡰 㡰
ᔴᨨ盐鼀 ᔼ r 㡰 = a �ˆ+ b ˆ r 㡰 = a ᆠᨨ矄 w �ˆ+ b 矄� w ˆ r 㡰 = a ᆠᨨ矄 w �ˆ+ 矄� w ˆ㡰 r b 㡰 = a �ˆ+ ˆ
ᔴᨨ盐鼀 ᔼᔼ r r ⧤
҈
� 쟼 ®҈�錐턼®�錐 ®�錐和®� ®҈�錐턼®�錐 ®�錐和®� ®錐턼®�錐 ®҈錐和®� ®҈�錐턼®�錐 ®�錐和®�
҈䰘 ᔴᨨ盐鼀 㜠ᔼᔼ矄 ᨨ犨矄 �矄泐矄鼀矄� 犨 �ᆠ 犨ᨨᨨᆠ矄 矄 Ӹ꾀矄ᨨ� ᆠ ᆠ矄ᨨ� �矄矄 ᨨ犨 ᔴᨨ盐鼀 㜠ᔼ ��矄矄ᨨ鼀矄鼀 矄ᨨ盐 ⏌ᨨӸ㝐ᆠ 矄矄鼀 矄ᨨ ᨨ�矄泐矄鼀 矄㡰 ᨨ鼀ᔴᨨ盐鼀 ᔼᔼ黨桌 ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 ᨨᆠ 泐ᨨ Ӹ 矄 㡰 Ӹ 鼀� �盐 Mg.
ᨨᆠ├Ӹᨨ 鼀矄矄M ᨨ � � ᆠ� 矄� 矄ᨨ �犨� ᆠᨨ 矄 ᨨᆠ�泐 㡰 ���ᨨ ᨨ � �泐ᨨ �矄ᆠᨨ � 盐ᨨ盐矄泐� ᆠ矄� �
⼄㡰
Ӹ
犨ᨨ� �鼀� 矄Ӹ ᆠ ᨨ 鼀矄矄M, ├� �ᆠ�ᨨ 矄矄�ᆠ 矄�ᆠ矄 矄ᨨ 泐ᆠ ᨨ Ӹ矄矄ᨨ 矄 �矄� � � ⼄盐���盐鼀 �矄 ᨨ鼀ᨨ矄ᨨ� ᨨ 矄 ᨨ ᆠᨨ鼀鼀ᨨ �矄ᨨ 犨ᨨ� �矄 犨 ᨨ矄泐矄鼀�矄� � �矄�ᨨ� �盐 犨� ᆠᨨ 矄 ᨨᆠ�泐 ᔴ㡰 㝐ᆠ �ᆠ �泐ᨨ 矄泐矄鼀 졔Ӹ�矄ᆠᨨ � 盐ᨨ盐矄泐 ᆠ矄� �
㡰 ᨨ⼄盐ᨨ 犨�� ᨨ Ӹᨨ盐 ᨨ� �矄 ᨨ
㡰 꾀盐泐Ӹᨨ 鼀矄矄m0 �矄 � ᨨ ᨨ盐� ᆠ鼀꾀ᨨᆠ├ᨨ 鼀矄矄㝐 �矄 ᨨ鼀矄� � �ᨨ矄 ᨨ 盐泐 �矄 � ᨨ� ᨨ 犨 ᨨ矄泐矄鼀�矄├� � �矄�ᨨ� � ᨨ犨 犨� ᆠᨨ 矄 ᨨᆠ�泐
Ӹ
矄㡰 Ӹ
꾀矄 Ӹᨨ 鼀矄矄M�矄盐� ᨨ 㜠�矄ᆠᨨ盐矄�⼄盐� ├� ᆠᨨ � 矄 矄 �矄矄 �鼀ᨨ矄 ᨨ犨 � �⼄盐� 㡰 꾀矄 Ӹᨨ 鼀矄矄M�矄 � �犨� �矄鼀� ᨨᆠ�泐 � �矄ᆠᨨ盐矄 �⼄盐� ├ � ᆠᨨ �
Ӹ
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EBD_8341 P-68
Physics
8
Comprehension Based Questions
Passage 1 One twirls a circular ring (of mass M and radius R) near the tip of one's finger as shown in Figure 1. In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is r. The finger rotates with an angular velocity w0. The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is m and the acceleration due to gravity is g. [Adv. 2017]
Now consider two similar systems as shown in the figure: Case (a) the disc with its face vertical and parallel to x-z plane; Case (b) the disc with its face making an angle of 45° with x-y plane and its horizontal diameter parallel to x-axis. In both the cases, the disc is welded at point P, and the systems are rotated with constant angular speed w about the z-axis. z
Q
w
y
P x
Case (a)
z
w
Q
45°
R x
r R Figure 1
Figure 2
18.
19.
The total kinetic energy of the ring is 1 Mw02 (R - r) 2 (a) (b) Mw02 R 2 2 3 Mw02 (R - r)2 (c) (d) Mw02 (R - r) 2 2 The minimum value of w0 below which the ring will drop down is (a)
g m(R - r)
(c)
3g 2m(R - r)
y
P
(b)
2g m(R - r)
(d)
g 2m(R - r)
Passage 2 The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded (rigidly fixed) horizontally at its rim to a massless, stick, as shown in the figure. When the disc-stick system is rotated about the origin on a horizontal frictionless plane with angular speed w, the motion at any instant can be taken as a combination of (i) a rotation of the centre of mass of the disc about the z-axis and (ii) a rotation of the disc through an instantaneous vertical axis passing through its centre of mass (as is seen from the changed orientation of points P and Q). Both these motions have the same angular speed w in this case [2012]
Case (b)
20. Which of the following statements about the instantaneous axis (passing through the centre of mass) is correct? (a) It is vertical for both the cases (a) and (b) (b) It is vertical for case (a); and is at 45° to the x-z plane and lies in the plane of the disc for case (b). (c) It is horizontal for case (a); and is at 45° to the x-z plane and is normal to the plane of the disc for case (b). (d) It is vertical for case (a); and is 45° to the x-z plane and is normal to the plane of the disc for case (b). 21. Which of the following statements regarding the angular speed about the instantaneous axis (passing through the centre of mass) is correct?
2w for both the cases
(a) It is
(b) It is w for case (a); and w/ 2 for case (b) (c) It is w for case (a); and 2w for case (b) (d) It is w for both the cases. Passage 3 A uniform thin cylindrical disk of mass M and radius R is attached to y two identical massless springs of spring constant k which are fixed 2d to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its centre. The axle is massless and both the d d springs and the axle are in R V0 horizontal plane. x The unstretched length of each spring is L. The disk is initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with r velocity V = V ˆi . The coefficient of friction is µ. [2008] 0
0
22. The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is (a) – kx (b) – 2kx (c) – 2kx/3 (d) – 4kx/3
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P-69
System of Particles and Rotational Motion
졠
ᆠ ᨨ 鼀矄矄ᨨ �矄├盐 �ᨨ矄矄�鼀 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ 犨� �盐 ⼄盐 ᆠ泐w⼄盐ᨨ ├ ├ ├ ├ 㡰 ᆠ㡰 㡰 㝐 㝐 㝐 㝐
鼀 �鼀盐鼀盐ᨨ ᨨ犨 �ᆠ �矄├犨�ᨨ犨� ᨨ盐 矄�� ��矄
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㡰
㡰
m�
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m�
㝐 㝐 �㝐 ᆠ㡰 m� 㡰 m� ├ ├ ├
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�ᨨx҈⏌x�矄 䰘桌 ҈ ҈ 㡰 㡰 ᆠ㡰 㡰
鼀
҈ �ᆠ矄AB Cᆠ ᨨ 鼀矄矄鼀ᆠᨨ
ᆠ ᨨ ᆠ ᨨ 泐 鼀矄矄矄矄��� ᨨ 矄ᨨ ᨨ鼀 ��� ⼄盐�� �盐ᨨ 泐ᨨ 矄� l �矄ᨨ 泐�矄ᆠ ᨨ ᨨ� ᨨ �ᆠ�ᨨ 矄矄 x㜠y 㡰 �矄 � � ᨨ � ᨨ� A矄ᨨ �ᆠ 鼀ᨨ犨� ᨨ盐 �ᆠ�ᨨ ᨨ盐 �ᆠ �矄 ᨨ盐� A 矄 ��盐㡰 ᨨ 泐�矄矄� ᨨ ᨨ�ᨨ 鼀ᨨ�ᨨ ᨨ ᨨ盐A犨� ᆠᨨ 矄 �盐 ᨨᆠ�泐w 㜠�㝐├矄桌 泐
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w
㡰
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10 Subjective Problems
黨
꾀 ᆠ �盐 ᨨ 鼀矄矄M �鼀 矄�ᨨ
a맔b�矄 � ᨨ� ᨨ ᨨ矄��ᨨ 泐矄�├� �
矄鼀矄 ᆠ ᨨ 鼀矄矄 m㡰 盐 � 矄ᆠᨨ
矄矄�├� �� 矄 ��ᨨ ᨨ
ᆠᨨ�矄�ᨨ ᨨ 矄犨� �矄矄�ᆠ␔ �矄v䁸 㜠㝐桌 � n㦨҈M 㦨├�m 㦨҈├�錐b㦨m錐a㦨҈鼀錐 g㦨҈鼀⏌矄㡰
犨ᨨ� �ᆠ 矄ᆠ ᨨ 鼀矄矄 M � L矄� �ᨨ ᨨ盐� ᨨ� ᨨ 矄盐 ᆠ矄矄 ᨨ犨
鼀 � ��盐꾀ᨨᆠ├ᨨ 鼀矄矄m q q �矄 ᨨ鼀P 꾀 ᔼ 矄泐矄鼀�矄� ⼄盐���盐鼀 � 鼀� �盐 �ᆠ�ᨨ ᨨ �ᆠ�ᨨ ᨨᆠA B �㜠㝐├矄桌 꾀 犨ᨨᨨ ᨨ� ᨨ 鼀矄矄 M � L �矄 � � 泐 �ᆠ�ᨨ 矄矄 �O꾀盐ᨨ 鼀矄矄m 矄�├矄犨� ᨨᆠ�泐 v 矄�ᆠ├矄 ᨨ � � �盐 ᨨᆠ�泐ᨨ 矄泐矄鼀 �鼀鼀 �泐 ᆠᨨ�矄�ᨨ ᨨ盐O�㜠㝐├矄桌
F
l ᔴ 㡰 � 鼀� �盐 ᨨ ᨨ� ᨨ ᨨᆠ 泐 � �ᨨ ᨨ 泐 㡰 꾀�鼀T犨 矄� BC�矄ᨨ x㜠 �矄 ᨨᆠF�矄� ᨨ Bᨨ �BC 矄矄 ᨨ犨 㡰졠�
x㜠ᆠᨨ鼀ᨨ y㜠ᆠᨨ鼀ᨨ ᨨ ᨨᆠ 泐 � �ᨨ ᨨ 泐�鼀鼀 �泐 �鼀T 犨ᨨ 泐 鼀�ᆠ 矄 ᨨ� A ᨨ� 黨ᨨᨨᆠ ᨨ 꾀鼀� 矄 턼 ᨨ 鼀矄矄├��矄 � � � AB ᨨ� � � B P 犨ᨨ 泐鼀�ᆠ 矄 � �矄 �ᆠ�ᨨ 矄矄 鼀ᨨ鼀 ᨨ � �ᨨ 鼀� 矄 ᨨ盐 �矄ᨨAB 矄矄� � ᨨ盐� �矄 ᆠ ᨨ 鼀矄矄 �矄 ҈ ├�㜠m 犨ᨨ 盐 ᨨ矄ᆠ矄P Q � ᨨ ᨨ ᆠ 鼀�ᆠ �矄 ᆠ�鼀 ᨨ鼀 � AB �矄 �矄 ᆠ�矄ᆠ ᨨ矄 矄ᨨ ᆠ�ᨨ 盐ᨨ � �矄ᨨ 鼀� 矄 �矄 ҈㜠҈㝐├矄桌 ᨨ 盐� � �鼀ᆠ ᔼ ��泐 鼀� 矄 �矄ᨨ ᨨ ᨨ矄ᆠ矄犨�
�盐ᨨᆠ�泐҈ ⏌矄 盐 矄ᆠ├ᔼ �鼀盐矄ᨨ
矄 盐ᨨᆠ ᨨ矄ᆠ�矄㜠矄 㡰 � ᨨᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄ᨨ 鼀� 矄 ᨨ鼀AB 㡰 꾀犨 �盐ᨨᆠ�泐 ᨨ矄 鼀� 矄 ᆠᨨ鼀 ᆠ├ �矄�鼀ᆠ䁸 ᆠ㡰 꾀 ᨨ犨鼀 泐�鼀ᆠ矄 ᨨ矄 鼀� 矄 ᆠᨨ鼀 ᨨ矄䁸
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EBD_8341 P-70
Physics
꾀 盐 � ᨨ鼀 ᆠ�ᆠ盐 A �矄ᆠ 矄 �盐矄R 鼀矄矄m꾀�ᆠ矄ᨨ R ᨨ 鼀矄矄m�矄 � ᨨ� Aᨨ �ᨨ C �矄ᆠ矄矄 ᨨ犨 �
R⏌ ��盐 �矄ᆠᆠ P Q ᨨ 泐ᨨ盐 � ᨨ� ᨨ ᆠ ᨨ PQ �矄 �矄 ᆠR⏌ ᨨ鼀 ᆠ Cᨨ �矄ᆠ
� AC�矄 �ᆠ盐ᨨPQ ᔼ ��泐 �矄ᆠ�矄 �ᆠ犨� ᨨ� A�矄 �� 矄ᨨ矄��ᨨ ᔼ�矄 ᨨ犨 ᨨ 矄ᨨ �矄矄 ᨨ� �ᨨ盐PQ � � 矄 ᨨ �ᆠ矄 ҈黨黨㜠㝐├矄桌 �ᆠ 矄�矄ᨨ犨矄ᨨ矄��ᨨ 꾀犨 �ᨨ 鼀矄矄鼀 � �盐ᆠᨨ矄矄㜠矄ᆠ�ᨨ AB㦨BC 㦨CA㦨R㡰�矄鼀ᨨ� �犨� ᆠᨨ 矄 ᨨᆠ�泐- v�ˆ ᨨ犨 矄 矄 ᨨ �盐矄R � ᨨ 矄鼀ᨨᨨ ᨨ� ᨨ 矄 矄 ᨨ犨 � ��盐 犨 �鼀├矄 矄�ᆠᆠᨨ�矄�ᨨ
犨� � 矄 盐 矄ᨨ � 矄鼀 犨� ᨨ盐 泐ᨨ�ᨨ �ᆠ �ᆠ�ᨨ 矄盐ᨨ矄 犨 �鼀� 矄� ᆠᨨ ᆠ犨� 矄 ᨨ泐 矄 ᨨ�鼀Dt 盐� �犨 �ᆠ 矄 矄ᆠᨨ 矄 ҈黨黨㜠㝐├矄桌 ᨨᆠFᨨ 犨 � A
y
R
v
x
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B
C
㡰 � ᨨᆠF 矄ᨨ ᨨ鼀 ᨨᆠN 泐 ᨨ 犨 � 盐� � �鼀D 㡰 h ᨨ �ᆠ盐 �矄 ᆠ犨 ᆠ ᨨ 鼀矄矄ᨨ 犨 � � ᨨ ᆠ�ᨨ ᨨ � 鼀� �盐 ᨨ ᨨ⼄盐 盐ᨨ ᨨ鼀 ᨨᆠNᨨ盐 ᆠ ᨨ 犨 � 盐� � � Dt
犨ᨨ � ᆠ�ᆠ盐 �矄├矄 ᨨ 鼀矄矄 ├� �盐矄҈ ᆠ鼀 ᆠ ᨨ� 泐��� 鼀矄矄矄矄 ᨨ ᨨ � ᆠ鼀
�矄ᨨ ᨨ �矄ᨨ � �ᆠ盐ᨨ 矄 ᨨ �矄├ ᨨ盐� �ᆠ 矄 �矄ᨨᆠ�矄├ᨨ 盐ᆠ├� 矄盐ᆠ 犨泐 �矄ᨨ ᨨᆠ�矄 ᨨ� ᨨ �ᆠ盐ᨨ �ᆠ�ᨨ ᨨ 鼀ᨨ�ᨨ ᨨ 盐ᆠ├ᔼ矄 �ᆠ�ᨨ 犨� ᨨᨨᨨ 盐ᆠ├�矄� ᨨ盐� 矄ᨨ ᨨᆠᆠ ᨨᨨ 盐ᆠ├犨� ᨨ盐矄�� �
├x �矄矄 �ᆠ�ᨨ ᨨ 鼀ᨨ�ᨨ ᨨ 盐ᆠ├ z �矄 矄 �ᆠ泐盐犨 矄 �ᆠ�ᨨ ᔼ 盐ᆠ├ 矄
ᆠᆠ�ᨨ ᨨ 黨鼀⏌矄 ᔴᆠ盐 ҈黨黨䰘㜠�㝐├矄桌 �㡰 ᨨᆠᨨ �ᆠ�ᨨ ᨨ ᆠ �矄├ ��㡰 鼀� �盐 �ᆠ�ᨨ ᨨ �ᆠ�ᨨ ᨨ⼄盐ᆠ� �ᨨ ᆠ �矄├ᨨ盐 ᆠ ᨨ 鼀矄矄 Oᨨ ᨨᆠ쟼 矄矄 ᨨ⼄盐� ᆠᨨ ᨨ鼀 � 鼀矄ᨨ 盐 �ᆠᨨ矄 �ˆ ˆ kˆ � xy z �ᆠ�ᨨ 矄
꾀ᆠ �盐��� � ᨨᆠ├ 矄ᨨ � ᨨ� ᨨ �꾀 矄ᨨ� ᨨ鼀ᨨ� ᨨ盐矄ᆠ泐� ᨨ �盐矄R�矄ᆠ ᨨ� ᨨ 泐 矄�矄 � ᨨ �矄盐ᆠ �矄ᨨ ᆠ泐� �ᨨ ᨨᆠ├� 矄鼀�ᆠ 矄 矄 ᨨ犨 � ��盐 ᨨ犨 �矄矄盐 �ᆠ� �ᆠ�ᨨ 矄 � 矄ᨨ 泐 矄鼀 �矄ᆠ鼀 ᆠ盐矄矄 ᆠ泐� ᨨ ᨨ ᨨ � 犨� ᨨ盐矄�� �鼀� ҈黨黨�㜠҈㝐├矄桌 㡰 �qᆠ ᨨ盐� 犨 �ᆠ ᆠ泐� ᨨ矄 ᨨ �矄ᆠᨨ ᆠ犨� � 㡰 矄 ᨨ ᆠ ᨨ 鼀矄矄ᨨ ᆠ泐� ᨨ � �ᆠᨨ ᆠ犨� � ᆠ㡰 �ᨨᨨ 矄�ᨨ ᨨ ᨨ�ᨨ ├� �ᆠ ��矄ᨨ ᆠ泐� 犨 �矄ᆠ ᨨ 鼀矄矄�矄�
ᨨ� ᨨ � 犨� � 䰘 犨ᨨ盐 � ᨨ鼀 � ᨨ 矄A Bᨨ � mᆠ ᨨ 鼀矄矄矄҈├� ├� 矄ᆠ�泐��� 泐ᨨ� ᨨ ᆠᨨ鼀� �ᨨ �矄�ᨨ �� P 꾀 矄矄 ᨨ犨 � ��和盐ᆠ �ᆠ 泐ᨨ ᨨ盐ᨨ� P � �ᆠ 꾀矄鼀 ᨨᆠᨨ 鼀矄矄�├�鼀ᨨ� � ᨨ� ᨨ 泐 �矄 ᨨ犨 ᨨ ᆠᨨ鼀� �ᨨ 矄�ᆠ├矄ᨨ� ␔ 矄 ᨨ盐 ᨨᆠ�泐ᨨ ᨨᆠ 矄ᨨ 矄泐矄鼀ᆠᨨ盐 盐矄�矄 ᨨ ᨨ� ᨨ ᨨ矄��ᨨ ҈黨黨㜠㝐├矄桌 꾀ᨨᆠ├Xᨨ 鼀矄矄�├��矄 泐ᨨ �鼀矄矄矄矄矄� � ᨨ �ᆠ�ᨨ 矄矄� ᆠ� ᨨ � ᆠ� �ᨨ ᨨ ᨨ� ᨨ 矄� � �矄 犨ᨨ盐 ᨨ 盐 � ᨨ鼀 矄ᨨ� ᆠ泐� �ᆠ 盐鼀Yᨨ 鼀矄矄├� ᨨ �盐矄m矄 矄 ᨨ犨 � ��盐 盐鼀�矄�� � �� �盐 ᨨᆠ�泐矄盐ᆠ ᨨᆠ├X矄矄鼀ᨨ� �盐 ҈黨黨㜠㝐├矄桌 Ӹ �㡰 � 矄�ᨨ � 矄� � 盐� � 鼀ᨨ�ᨨ ��㡰 꾀ᆠ� � 矄 ᨨ �鼀 鼀� �盐 ᨨ �盐 ᨨᆠ�泐 ᨨ Y �矄 ҈ s ҈ ᆠᆠ盐 �矄 ᆠ 泐X ᨨ鼀 � 矄 ᨨ �鼀盐 � �ᆠᨨ鼀矄ᨨ矄 黨 꾀 ᨨ鼀ᨨ� ᨨ盐矄ᨨ ABᨨ � L㦨҈鼀 鼀矄矄M�矄 �ᨨ ᆠ 졠� 矄盐ᆠ 犨泐 �ᆠ ᨨ 泐� �ᆠ ��㡰 ᨨ �矄� ��泐� ᨨ� ᨨ ᨨ矄��ᨨ 꾀 � 矄ᆠSᨨ 矄鼀鼀矄矄M 矄 �ᆠ泐犨� 矄 Vᨨ ᨨ� C鼀� 犨泐犨
ᨨ� 矄O Bᔼ鼀鼀 �泐 � � � 矄ᆠ 鼀ᨨ矄ᨨ犨 矄 B矄盐ᆠ ᨨ ᨨ矄犨� ᆠᨨ 矄 �盐ᨨᆠ�泐w ҈黨黨㜠㝐├矄桌 㡰 鼀� �盐ᨨᆠ�泐 w � 鼀矄ᨨ V L 㡰 ᔼ � 矄ᆠ ᆠ 矄 B 犨 ᨨ 矄 盐 ᨨ盐� �ᨨ 黨 鼀� V
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P-71
System of Particles and Rotational Motion
�矄 �� 矄ᨨᆠ� ᨨ矄 矄 ᆠᨨ � 盐ᨨᨨ ᨨ盐�矄ᆠ ᨨ 鼀矄矄䁸쟼 � ҈黨䰘㜠䰘㝐├矄桌
和 V
҈
O
L
C
L
L
꾀 � 盐 � ᨨ鼀�矄ᨨ �ᆠ�ᨨ 矄矄 ᨨ� ᨨ 矄盐 ᆠ �矄 ᨨ鼀ᨨ�
泐犨泐ᨨ 矄盐 ᆠᔼ矄鼀矄矄�矄҈
B
鼀
A
҈鼀⏌矄
꾀
├� � 鼀矄 犨ᨨ�ᆠ矄 鼀⏌矄 ᆠ ᨨ 鼀矄矄├�鼀ᨨ� �ᨨ 矄鼀 矄盐 ᆠ ᨨ犨 矄 � �ᆠ�ᨨ
�ᆠ盐ᨨ ᨨ 犨� ᨨᆠ�泐ᨨ ҈鼀⏌矄 ᨨ 犨� 鼀⏌矄矄矄 ᨨ犨 � �� �矄�ᆠ矄�├矄 ᨨ� A ᨨ ᨨ� Bᨨ� 矄A B �矄 ᆠᨨ �鼀 ᨨ鼀 ᆠ ᨨ �ᆠ矄矄�├ 矄鼀� 矄 ᨨ �鼀 矄�ᆠ├ ᨨ ᨨ ᆠᨨ�矄�ᨨ ᔴᆠ盐 ᨨ矄矄ᨨ ├� �ᆠ �泐ᨨ 矄泐矄鼀� ᨨᆠᨨ�矄�ᨨ ᨨᆠ矄矄 ҈黨黨㜠㝐├矄桌 꾀矄鼀矄 ᨨ矄 ᨨ犨 犨� ᨨ盐矄�� � ᨨ鼀 ᨨᨨ ᆠ├� �ᆠ ᆠ├ 矄 矄ᆠ�ᨨ
ᨨ� ᨨ ᨨ� ᨨ �矄҈鼀ᨨ �ᨨ盐 ᨨᨨ ᆠ├�矄鼀矄 ᨨ �ᨨ盐 � �矄 ᆠᨨ �ᨨ盐 犨� 矄ᆠᨨ ᨨ� B 犨 �ᆠ �矄�ᆠ泐ᨨ犨 ᨨ ᆠ├矄矄 ᨨ犨 � ��㡰犨 矄 矄盐� �
҈ 鼀
꾀�鼀矄矄M 矄鼀鼀矄矄m �犨ᨨ 矄ᨨ 矄� � 矄矄矄 ᨨ 矄鼀ᨨᨨ 盐 矄 l 矄 ᨨ犨 � ��盐 鼀矄矄鼀 鼀ᨨ矄ᨨ盐 ᆠ�ᆠ盐 犨 �ᆠ �矄� ᨨ� ᨨ 鼀
� ᨨ 矄� � ᨨ鼀 鼀矄矄 mᨨ ᨨᨨ 盐�矄l q�矄 � � �矄 � 鼀├矄犨� 㝐 �ᆠ ␔ 矄 ᨨ盐 ⼄盐 ᆠ泐 ᨨ ᨨ�ᨨ ᨨ 鼀矄矄 m 矄ᨨ 鼀矄矄 M 鼀� 矄矄�ᨨ 泐䁸 ҈黨䰘桌 꾀├�鼀矄矄 �� � ᨨ ᨨᨨ � l ᨨ矄ᆠ�矄� �ᆠ 犨� �盐鼀�盐 q0 ␔ �矄 矄�ᨨ � ᨨ犨 �鼀├矄 �q 犨� �ᆠ䁸ᔼ ├� �矄 � ᨨ ᨨ�矄 ├�犨 �矄 鼀 �鼀盐鼀鼀�盐 犨� 犨 �ᆠ 鼀矄矄 ᆠ ᨨ矄ᆠ�犨� ᨨ盐 ᨨ├� �䁸 ҈黨䰘桌
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EBD_8341 P-72
Physics
Answer Key Topic-1 : Centre of Mass, Centre of Gravity & Principle of Moments ҈
㡰
㡰
㡰
㡰
� 㡰
㡰
䰘 ᆠ㡰
҈҈
㡰
҈ 㡰
҈ ᆠ㡰
҈ 㡰
҈� 㡰
҈ 㡰
҈䰘 㡰
矄 ҈ ᆠ㡰
ᆠ㡰
㡰
ᆠ㡰
黨 㡰
҈ ᆠ㡰
æ d - x ö xw ҈ 㡰҈黨 ç ÷ w d è d ø
Topic-2 : Angular Displacement, Velocity and Aceleration ҈ ҈҈
㡰 㡰
ᆠ㡰 ҈ 㡰
ᆠ㡰 ҈ 㡰
㡰
� 㡰
ᆠ㡰
䰘 ᆠ㡰
㡰
黨 㡰
҈ ᆠ㡰
Topic-3 : Torque, Couple and Angular Momentum ҈ ҈҈ ҈ ҈
ᆠ㡰 㡰 ᆠ㡰 㡰
�ᆠ鼀㡰
҈ 㡰
��
ᆠ㡰 䰘 ᆠ㡰 ᆠ㡰 � 㡰
ᆠ㡰
҈
㡰 㡰 㡰 ᆠ㡰
҈
㡰 㡰 㡰 㡰
㡰 � 㡰 ҈� ᆠ㡰 � ᆠ㡰 � ҈ 㝐꾀 w 黨 ᆠ 㡰 ҈
ᆠ㡰 㡰 㡰 㡰
㡰 䰘 㡰 ҈䰘 㡰 䰘 㡰 䰘 Mw M + m � 㡰 �҈
҈
㡰 㡰 㡰 㡰
҈
㡰 㡰 㡰 㡰
黨 ҈黨 黨 黨
㡰 ҈ ᆠ㡰 㡰 ᆠ㡰 㡰 㡰 黨㡰
矄 � ᆠ 㡰 ᆠ㡰
� 㡰
� ᆠ 㡰 � 㡰
Topic-4 : Moment of Inertia and Rotational K.E. ҈ ҈҈ ҈ ҈ ҈
㡰 ᆠ㡰 㡰 㡰 㡰
҈
㡰 㡰 㡰 㡰 黨㡰
҈
㡰 ᆠ㡰 㡰 㡰 ҈҈㡰
҈
ᆠ㡰 㡰 㡰 ᆠ㡰 㡰
� ҈� � � �
㡰 㡰 ᆠ㡰 㡰 �㡰
҈
ᆠ㡰 㡰 㡰 㡰 ҈�㡰
䰘 ҈䰘 䰘 䰘 䰘
㡰 㡰 ҈ 㡰 㡰 㝐㡰
㡰 黨 ᆠ㡰 ҈黨 㡰 黨 㡰 黨 ᆠ㡰
㡰 ᆠ㡰 ᆠ㡰 㡰
҈
㡰 㡰 ᆠ㡰 㡰
Topic-5 : Rolling Motion ҈ ҈ ҈黨
㡰 㡰 ᆠ㡰
㡰 ҈҈ 㡰 㡰
ᆠ㡰 ҈ 㡰 ҈ ᆠ㡰
ᨨ 盐矄㡰 � ҈ ᆠ㡰 ҈ 䰘㡰 ᆠ 㡰 㡰
㡰 ҈� 㡰
ᆠ㡰 䰘 ҈ �㡰
ᆠ㡰 㡰 黨 ᆠ㡰 ҈䰘 䰘�㡰 ҈ 矄
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈ ҈҈
㡰
㡰
㡰
㡰
矄 ҈ ᆠ㡰 ҈ 㡰
҈ ᆠ㡰
҈䰘 꾀錐 ® ⼄矄 㡰 錐 ᔴ ® 㡰 ® ⼄ 㡰 ᆠ㡰 � 㡰 㡰 䰘 ᆠ㡰
� ᆠ㡰
ᆠ㡰
䰘 ᆠ㡰
㡰
㡰
҈ 㡰
黨 㡰
҈
鼀�
҈� ᆠ 㡰 ҈ 㡰 ҈ ᆠ㡰
҈黨 㡰
㡰 㡰
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P-73
Gravitation
Gravitation
7
Topic-1 : Kepler's Laws of Planetary Motion 1 ҈
㡰 � TA>TB MA>MB
ᔼ �盐鼀ᨨ鼀 盐鼀ᨨ ᨨ 鼀矄矄鼀鼀ᨨ� � ᨨ盐 和盐 � ᆠ�ᆠ盐ᨨ��矄ᨨ盐 ᆠ ᨨ 和盐 �矄ᨨᆠ�泐�矄 㝐� 黨 ҈黨ᔼ桌 㡰
ᆠ㡰
㡰 � TA>TB RA>RB
MCQs with One Correct Answer
鼀
㡰
鼀
㡰
鼀
ᆠ㡰
鼀
��盐矄 ᨨ犨矄��ᆠ ᆠ ᨨ ᨨ盐 ҈ ᨨ �矄 和 ��盐㡰␔� 矄 矄鼀�鼀ᨨ �矄 ᆠ 矄 矄鼀�鼀� ᨨ �矄ᔼ ҈�矄 �鼀├ ᨨ ᨨ�ᨨ ᨨ ᆠ ᨨ ├ ᨨᆠ 㝐� 졠 � 꾀�黨҈桌
�
ᆠ
和
҈㦨 ҈㦨
MCQs with One or More than One Correct Answer
ᔼ鼀�� �� ᨨ� �ᨨ盐 泐鼀矄矄� 矄� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄R犨� �ᨨ ᨨ ᨨ盐�ᨨ
Tᔼ ���ᨨ ᨨᆠᨨ ᆠ�ᨨ 犨 矄�矄ᨨᨨ�ᨨ ᨨR�⏌҈黨黨㜠㝐├桌 㡰 T�矄ᨨᨨ�ᨨ ᨨR 㡰 T�矄ᨨᨨ�ᨨ ᨨR䰘⏌ ᆠ㡰 T�矄ᨨᨨ�ᨨ ᨨR⏌ 㡰 T�矄ᨨᨨ�ᨨ ᨨR⏌䰘 10 Subjective Problems
㡰 ᆠ㡰
㡰 TA㦨TB ᔼ �矄 ᆠ犨 矄盐 犨 �矄 矄 盐 盐鼀ᨨ 泐矄� 泐犨ᨨ盐
҈黨黨㜠㝐├矄桌 㡰 � 㡰 ҈黨 ᆠ㡰 ҈� 㡰 䰘 6
矄盐 和矄盐ᆠ ᨨ � �ᆠ矄�矄
æ TA ö æ RA ö çè T ÷ø = çè R ÷ø B B
㡰 ҈㦨 㡰 ҈㦨
꾀� 泐矄矄泐矄鼀ᆠᨨ 矄�矄矄ᨨ 犨ᨨ矄矄A B犨 �ᆠ �鼀�ᨨ TA TB �盐矄RA RB 鼀矄矄 MA MB
㜠㝐҈桌
犨ᨨ矄�矄S҈ Sᨨᨨ盐 � ᆠᨨ ᆠ�ᆠ盐ᨨ�矄� 矄鼀矄 矄 ��ᨨ 矄ᨨ ᨨ盐�ᨨ
҈ ᨨ盐 ᨨ盐矄矄ᆠ�泐 �盐矄ᨨ ᨨ� ᨨ S҈�矄҈├鼀␔ S�矄ᆠᨨ矄矄ᨨS҈ � �㡰 矄 ᨨ S�ᨨS҈ ��㡰 �盐矄 ᨨ S矄ᆠ盐泐ᨨ矄 泐
矄ᨨ 盐� S҈ ҈黨㜠㝐├矄桌
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EBD_8341 P-74
Physics
Topic-2 : Newton's Universal Law of Gravitation 1 ҈
MCQs with One Correct Answer
꾀矄�� ᨨ ᨨ � 矄 ᨨ鼀x 㦨 aᨨx 㦨졔
���ᨨ ᨨᆠ� 矄ᨨ ᨨ� 鼀矄矄�鼀x 㦨 � 鼀矄矄盐 � � ᨨ ᨨ �矄꾀졔x�矄��
㝐� ҈ ҈黨ᔼ桌 泐 㡰
㡰
é æ ҈ ù ҈ö 鼀 ê 꾀 ç - ÷ - ú a + L a ø ë è û
㡰
é æ҈ ù ҈ ö 鼀 ê 꾀 ç ÷ - ú a a + L ø ë è û
ᆠ㡰
é æ ҈ ù ҈ö - ÷ + ú 鼀 ê 꾀 ç a + L a è ø ë û
㝐
é æ҈ ù ҈ ö 鼀 ê 꾀 ç ÷ + ú ë èa a+Lø û
├ 鼀 �矄 ᆠᨨ 鼀ᨨᨨ 矄盐 ᨨ鼀 ᨨ맔҈├鼀 ҈�맔҈├鼀矄ᆠ�泐
�鼀矄矄矄맔҈├� 맔҈├�矄ᆠ�泐
�盐矄ᨨ �矄├鼀D ҈ � ᆠ � ᨨᆠ矄 泐 鼀ᨨᨨ 矄 矄 ᨨ� 矄ᨨ D � ᆠ� ᨨᆠ 泐 矄盐 矄 矄ᨨ� 矄ᨨ
DF҈ �矄 DF 㝐� 졠 � 꾀�҈�҈桌 ᆠ㡰 ҈ 㡰
(
㡰
㝐
)
(
)
盐鼀ᆠᨨ矄矄ᨨ 㡰
㡰
㝐 ҈ 㝐 ҈+ ҈+ 㡰 ᨨ鼀矄 ᨨ 鼀矄矄㝐 �盐矄 矄鼀矄 ᨨ �盐矄 �矄ᆠ ᨨ盐矄盐ᆠ ᆠ�泐鼀 � ᨨ��� 矄 �矄犨 �矄 ᆠ � 泐 和 ��盐㡰 ᨨ ᆠᨨ ��盐�ᨨ � ��盐犨 �矄 ᆠ犨 ᆠ ᨨ ᨨ��� 矄 鼀ᨨ 矄 �矄 ���ᨨ ᨨᆠ犨 犨ᨨ矄 �矄 㝐� 졠 � 꾀�҈҈҈桌
ᆠ㡰
㡰
ᨨ盐�ᆠ矄ᆠ ᨨ 鼀矄矄㝐 ⼄盐� �矄 ᨨ鼀ᆠ ᨨ 鼀ᨨᨨ �ᆠ�ᆠᨨ �盐矄 盐 ᆠ�ᨨ ᨨ �鼀盐盐���ᨨ ᆠ�ᨨ 矄 ᨨ ᆠ �ᆠ�矄 㝐� ҈桌
㡰
҈ 㝐
㡰
�黨 㝐
҈ 㝐
ᆠ㡰 �
�
㡰
㝐
�
Topic-3 : Acceleration due to Gravity 1 ҈
矄� � ᆠᔼ 犨�� 矄 ᨨ盐 ᨨ矄鼀 h �矄 hR犨 R�矄 �盐矄ᨨ 㡰 㝐� �和 ᔼᔼ㡰桌
MCQs with One Correct Answer
盐ᨨ ᆠᆠ�ᨨ 盐ᨨ��泐�矄g҈ �� R R㦨 �盐矄ᨨ 㡰 ᨨ鼀 矄盐 ᆠᨨ ᔼ�矄�� ⼄盐ᨨg҈ dᨨ犨 矄盐㜠
h㦨
ædö ᆠᨨ �ᨨ ç ÷ ⼄盐矄 è Rø 㝐� �和 ᔼ㡰桌 � ҈ 䰘 㡰 ᆠ㡰 㡰 黨 黨 黨
ᆠᆠ�ᨨ 盐ᨨ��泐ᨨ 矄矄盐 ᆠ ᨨ矄�矄g �盐ᨨᆠ�泐ᨨ ᨨ盐 �矄矄矄� � ᨨ盐� ᨨ�矄w꾀 ᨨᆠ�矄犨�� ⼄盐ᨨ �� hᨨ ᨨ矄泐盐矄� �
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㡰
R w g
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R w g
R w R w ᆠ㡰 㡰 g g
�� ἘhἘ犨 �ᆠ 犨�� ᨨ ᨨ 泐犨� 矄鼀矄 矄鼀 ἘhἘ ᨨ鼀 矄盐 ᆠᨨ �矄 �盐矄ᨨ �矄R ᆠᨨ ᨨ�ᨨ
ᨨ �矄 �ᆠ 㡰 [Main 2 Sep. 2020 (II)]
㡰
� R-R
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ᆠ㡰
�R - R
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R
R - R
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P-75
Gravitation
�
꾀ᨨ 犨�� 矄҈黨ᨨ 矄� � ᆠ ᨨ ᨨ ᔼ矄犨�� ᆠᨨ ᨨ 矄鼀 ᆠ� ��矄矄 � ᨨ ⼄盐ᨨ�矄ᆠᨨ矄ᨨ ├g㦨҈鼀矄 ᨨ ᨨ �盐矄ᨨ 㦨├鼀㡰 㝐� 䰘 ᔼᔼ桌 㡰 ҈黨� 㡰 ҈黨 ᆠ㡰 ҈黨 㡰 ҈黨�
�ᨨᨨ 犨�� 矄ᨨ ᨨ 泐ᨨ 쟼 矄矄盐 ᆠᨨ ᨨ 矄盐 ᆠᨨ �矄 黨 鼀矄矄 ᨨ ҈ �矄 ᨨ ᨨ 쟼 ᔼ � �矄 �盐矄ᨨ 黨 쟼 犨 �矄 �盐矄ᨨ 䁸 ├ 矄 ᨨ 矄鼀鼀矄矄 矄�泐㡰㝐� ҈꾀�҈黨ᔼᔼ桌 R R R R 㡰 ᆠ㡰 㡰 黨
盐ᨨ ᆠᆠ�ᨨ 盐ᨨ��泐쟼 矄矄盐 ᆠ �矄黨鼀矄 �盐 ᨨ�矄矄盐 ᆠ犨 �ᆠ ᆠᆠ�ᨨ 盐ᨨ��泐 ᆠ矄矄ᨨ黨鼀矄�矄ᆠᨨ矄 ᨨ �盐矄ᨨ 㦨맔҈鼀㡰㝐� ҈꾀�҈黨ᔼ桌 㡰 맔҈鼀 㡰 맔҈鼀 ᆠ㡰 黨맔҈ 鼀 㡰 ҈맔҈鼀 和盐ᨨ矄 �盐ᨨᆠ�泐ᨨ ᨨ�ᨨ ᨨ �矄 � ᆠ矄 矄ᆠᨨ 矄⼄盐 ᆠ 㝐� 졠 � 꾀�҈҈桌 㡰 犨� ᨨᆠ �� 犨�� 泐犨 ᨨ 㡰 ␔�� ᨨ ᨨᆠ泐犨 ᨨ 犨� ᆠ矄 ᆠ㡰 ␔�� ᨨ ᨨᆠ泐犨 ᨨ 犨�� ᆠ矄 㡰 쟼 ᆠᨨ矄犨�� ᨨ ᨨᆠᨨ 犨� ᆠ矄
��ᨨ ᨨ ᆠᆠ�ᨨ 盐ᨨ��泐�犨� �矄 ᆠ ᨨ鼀 ᆠ ᨨ �矄 矄 矄 泐 㦨쟼 Ἐ矄 �盐矄㡰 㝐� ҈䰘桌 � �
黨
ᔼ 쟼 矄 ᨨᨨ�ᨨ 鼀ᨨ�ᨨ 犨�� ᨨ 矄ᨨ
ᨨ ⼄盐ᨨ�矄␔鼀� 矄 犨� 犨 �ᆠ 犨ᨨ盐 ᨨᨨᨨ盐�矄 �矄矄ᨨ 矄ᨨ
├鼀 �㦨҈鼀⏌矄
␔ �盐矄ᨨ 쟼 �矄 㝐� 졠 � 꾀�҈䰘桌
㡰 ҈҈맔҈ ⏌矄
㡰 맔҈ ⏌矄
ᆠ㡰 맔҈ ⏌矄
㡰 맔҈ ⏌矄
⼄盐ᨨ犨�犨��
҈ ´ ( �盐矄 ᨨ 쟼 ) 矄 ҈ 矄鼀鼀矄矄 矄�泐矄쟼 和ᆠ� �矄矄 ��犨ᨨ
҈ 꾀 ᨨ �盐矄 R =
㡰
䰘
㡰
㡰
졠
졠
� ᆠ㡰
졠
�
鼀矄矄ᨨ 鼀ᨨᨨ 㦨䰘맔҈├� �盐矄ᨨ 鼀ᨨᨨ 矄 ᨨ�㦨맔҈鼀㡰 㝐� 졠 � 꾀�҈桌 㡰 䰘맔҈�鼀⏌矄
㡰 䰘맔҈鼀⏌矄
ᆠ㡰 䰘맔҈鼀⏌矄 㡰 䰘맔҈鼀⏌矄 ҈ ᔼ �盐矄ᨨ 犨ᨨ矄 � ├泐ᨨ ᆠ �矄 鼀矄矄鼀� � � 矄鼀 ᆠᆠ�ᨨ 盐ᨨ��泐 ᨨ 矄矄盐 ᆠ犨ᨨ盐 ҈黨҈㜠㝐├矄桌 㡰 ᆠ矄 㡰 鼀� 盐 ᆠ � ᆠ㡰 � ᆠ矄 㡰 ᨨ 4
㡰
졠
R ᨨ � ᨨ犨犨�ᨨ 矄鼀 � � � 鼀矄矄 矄�泐҈├�鼀҈� ᨨ�ᔼ 犨��矄 ᨨᨨ盐ᆠ � � 泐犨 ᨨᆠ� ᨨᨨ 犨�泐矄ᨨ
ᨨ � ��� ᆠ�矄 ├ �盐矄ᨨ 쟼 㦨맔҈鼀 ᆠᆠ�ᨨ 盐ᨨ��泐ᨨ 쟼 �矄҈鼀矄㡰 꾀 ҈桌 㡰 黨 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈� ҈҈ ᆠ �� 盐ᨨ ᆠᆠ�ᨨ ᨨ ᨨ犨 矄 矄盐 犨 鼀ᨨᨨ ᆠᨨ鼀矄 ᨨ鼀 ᨨ矄��ᨨ ᨨ 矄ᨨ ᆠ�矄ᨨ ᨨ矄��ᨨ ᨨ ᨨ 矄� ᨨ � � 犨� 矄盐 �矄
Fill in the Blanks
҈ 盐鼀�ᆠ盐ᨨ �盐ᨨᆠ�泐ᨨ ᨨ�ᨨ ᨨ 矄 ᨨ盐 ⏌矄� ᨨ ᨨ鼀├ ᆠ�ᆠᆠ�ᨨ 盐ᨨ��泐⼄盐ᨨ ᨨ ҈黨㜠㝐├矄桌
Topic-4 : Gravitational Field, Potential and Potential Energy 1 ҈
MCQs with One Correct Answer
犨ᨨ 矄 鼀矄矄矄M ҈M � ��a a矄ᆠ�泐 矄�ᨨ 犨 ᆠ 矄 ᨨ 矄�矄҈a꾀ᨨ 泐ᨨ 鼀矄矄鼀�矄 � ᨨ鼀 矄盐 ᆠᨨ � ᨨ犨 矄 矄鼀 ᨨ � � ᨨ� � � �ᆠ 矄 ᨨ ᨨ 泐ᨨᨨ
ᆠ 矄盐 ᆠᨨ 矄鼀 鼀� �鼀盐鼀 �� � 矄 �矄 㝐� 和 ᔼᔼ㡰桌 㡰 ᆠ㡰
GM a
GM ma
㡰 㡰
GM a
�GM a
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EBD_8341 P-76
Physics
졠 x㜠 �矄 �矄 ᆠx ᨨ鼀 ᨨ��� ���ᨨ � 盐ᨨ鼀矄矄 �矄�盐�ᨨ �矄�� 泐 Ax
x + a 㡰⏌
A
x + a 㡰
҈
x + a 㡰
ᆠ㡰 A x + a 㡰 㡰 A x + a 㡰
鼀矄矄 矄�泐ᨨ ᨨ �盐矄R�矄犨�
㡰 r = R 㡰 r = R ҈ � R 㡰 r = ᆠ㡰 r = R 黨 ᔴᨨ 矄� 犨ᨨ矄ᨨ� 矄 矄ᨨ ��R҈㦨҈mR㦨m 鼀矄矄矄M҈ M矄ᆠ�泐 ���ᨨ � m҈ 盐ᨨ矄 ҈ 矄 ᨨ犨 盐ᨨ m �矄 㝐� ᔼ桌
��ᨨ � E
㝐 a
㡰 ҈҈
㝐 a
꾀矄�ᆠ�矄鼀ᨨ� �� ᆠ�ᆠ盐ᨨ�� ���ᨨ
ᆠ㡰 ⏌ �矄ᆠᨨ 矄 㡰 �矄ᆠᨨ 矄 꾀ᨨ 泐ᨨ 鼀矄矄鼀�矄鼀ᨨ� �� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄R ᨨ盐 ᨨ 鼀矄矄M꾀矄ᨨ鼀� 矄 �矄�矄� ᨨ 犨ᨨ⼄盐鼀矄矄矄 �矄鼀矄矄鼀ᨨ矄� ᆠ�ᆠ盐ᨨ� R ᨨ 鼀矄矄� ᆠ�ᆠ盐ᨨ�ᨨ
R � ᆠ犨 � � �� ᨨ ��矄�矄 㝐� 졠 � 꾀�҈�҈桌 �盐矄
GMm GMm GMm GMm 㡰 + ᆠ㡰 㡰 R R R R ᨨ鼀矄ᨨ� 矄 ᨨ 鼀矄矄㝐 �盐矄 矄 �ᆠ ᨨ�ᨨ ᨨ �盐矄 ⏌�矄鼀ᨨ 矄矄 ᨨ犨 � ��盐
├� ����ᨨ ᨨ �㦨㦨¥ ᨨ � ᆠ ᨨ ᆠ�泐 盐矄 ᨨ鼀 �矄 G = grav�tat�ᨨnal cᨨnstant㡰 㝐� ҈�桌
㡰
黨
-
҈
҈
҈
� �盐矄 R
҈ ҈ ҈ 㡰 ᆠ㡰 㡰 꾀矄ᨨ� 矄 ᨨ 鼀矄矄�㝐 �盐矄��矄矄盐ᨨ盐 泐 盐 � ᨨ鼀ᆠᨨ ᆠ �ᆠ矄 �ᆠ矄 ᨨ �ᆠ├ 矄矄 鼀矄矄㝐 ���ᨨ � �矄 ᆠ� ᨨ鼀 ᆠ 犨� 㝐� 黨꾀�҈黨ᔼ桌 㝐
㡰
㝐
ᆠ㡰
㝐
㡰
-㝐 -㝐 -㝐 -㝐 㡰 ᆠ㡰 㡰 ␔ �ᆠ ᨨ ᨨᨨ犨� �鼀ᨨ矄ᆠᨨ矄泐 �ᆠ矄 ᆠᨨᆠ ��ᨨ ᨨ ���ᨨ ᨨ � 㡰 盐ᨨ� ᨨ �盐矄 盐 � ᨨ鼀鼀矄矄 矄�泐䁸 ��盐矄 ᨨ 犨 ᨨ 矄ᆠ㡰㝐� 졠 � 꾀� ҈҈ ҈�桌
㡰
㡰
ᆠ㡰 ҈҈
ᨨ �盐矄
㡰
㝐 a
K ᔼ � 泐 r ᆠᨨᆠ�ᨨ 犨 �盐矄 ᨨ �ᆠ矄ᨨ� �矄�ᨨ 㝐� 꾀�҈黨ᔼᔼ桌 㡰 ⏌ �矄ᆠᨨ 矄 㡰 ⏌ �矄ᆠᨨ 矄
�
㡰 ҈҈
� ᨨ 盐ᆠ 泐鼀矄矄 矄�泐 r r 㡰 =
æ r ö �矄 ᆠr ᨨ鼀�矄ᆠ 矄 r r 㡰 = r çç҈ - ÷÷ è R ø ���ᨨ � �矄鼀 �鼀盐鼀㝐� 和 ᔼᔼ㡰桌
䰘
A
㡰 ҈
㝐 a
� x㜠 �ᆠ�ᨨ 鼀� �盐 ᨨ
���ᨨ ᨨ �ᨨ x㜠 �矄 �矄 ᆠx├� � �矄盐ᨨ ᨨ� � �泐�矄 [Main 4 Sep. 2020 (I)] 㡰
㡰 ҈�
҈
㝐
黨a a 黨a a ᨨ盐� �ᆠ�ᆠ矄ᨨ 鼀矄矄㝐ᨨᆠ ᆠᨨ 矄 ᨨ 矄⼄盐ᨨ 矄� �a␔ 矄 ᨨ盐 �矄 � ᆠ ᨨ 鼀 ᨨ矄 盐 � 盐 ᆠ ᨨ ᨨ 矄 ���ᨨ � � ᆠ�ᆠ盐ᨨ�ᆠ�ᆠ盐鼀矄ᆠ�� � 矄⼄盐䁸 㝐� 꾀�҈黨ᔼ桌
㡰 㡰
㡰
㡰 ᆠ㡰
㡰
졠
졠
㡰
졠
㡰
졠
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P-77
Gravitation
҈҈ ���ᨨ � � ��ᨨ �矄 �� 泐 ®
� = � ⏌ ├��ˆ + ҈ ⏌ ├�ˆ ᆠ �� ���ᨨ ᨨ � �泐ᨨ �ᆠᨨ 鼀矄矄҈├�犨 ��矄├
ᨨ鼀 ᨨ��� ᨨᨨ� 䰘鼀鼀㡰�矄 㝐� 졠 � 꾀�҈黨҈桌 㡰 䰘҈ 㡰 ҈ � ᆠ㡰 䰘҈ 㡰 ҈ 鼀 鼀 ҈ ҈ ҈
犨ᨨ 泐ᨨ �ᆠ 矄ᨨ 鼀矄矄矄鼀҈ 鼀矄 犨 泐 � � � �矄 ᆠ ᆠ盐矄ᨨ ���ᨨ ᨨᆠ 泐鼀ᨨᨨ犨 矄ᆠ ᨨ ᨨ � � ᨨ� � � �ᆠ 矄␔ �矄 �矄 犨
�矄�ᨨ �矄� 䁸㝐� 졠 � 꾀�҈҈桌 和 ᨨ 鼀҈�矄҈ ᨨ 鼀�矄㡰 㡰 ҈㦨 ҈ = 鼀
= 鼀҈ ( 鼀҈ + 鼀 ) ( 鼀҈ + 鼀 )
ᆠ㡰
҈ = 鼀҈
= 鼀 ( 鼀҈ + 鼀 ) ( 鼀҈ + 鼀 )
㡰
҈ = 鼀
= 鼀 鼀҈ 鼀
㡰
㝐矄矄ᨨ ᆠᨨ鼀 矄 㦨㝐
MCQs with One or More than One Correct Answer
҈䰘 鼀� �盐 矄ᨨ ���ᨨ � �矄 ᆠr҈ r ᨨ鼀 ᆠ ᨨ 盐 � ᨨ鼀矄 ᨨ �盐矄R 鼀矄矄mF҈ F矄ᆠ�泐 ҈黨黨㜠㝐├矄桌 F҈ r҈ 㡰 F = r � r҈ < R r < R 㡰
F҈ r = F r҈ � r҈ > R r > R
ᆠ㡰
F҈ r҈ = F r � r҈ > R r > R
F҈ r҈ = F r � r҈ < R r < R ҈ 꾀矄ᨨ� 矄 ᨨ 盐 � ᨨ鼀 矄�泐 �盐矄盐 �矄�矄 ᨨᆠ 犨� �矄ᆠ ᨨ��� Oᨨ ᆠᨨᨨ � 矄 犨ᨨ 矄 矄ᨨ ⼄盐 ��҈盐 �犨� �ᆠ 矄A 㡰 㡰矄ᆠ�泐├ ᨨ盐ᨨ 矄ᨨ� � � � 矄 �ᆠᆠ��矄矄矄 ᨨ犨 � �� y
҈黨黨㜠㝐├矄桌 㡰
㡰
҈� ᔼ g�矄 ᆠᆠ�ᨨ 盐ᨨ��泐ᨨ 矄矄盐 ᆠ �� � ᨨ � �泐ᨨ ᨨᆠᨨ 鼀矄矄m �矄 ᨨ鼀 矄盐 ᆠᨨ ᨨ �� ⼄盐ᨨ �盐矄 ᨨ �矄 ҈黨㜠҈㝐├桌 ҈ ҈ 㡰 mg R 㡰 mg R ᆠ㡰 mg R 㡰 mg R 3 Numeric Answer ҈ 꾀 矄ᨨ� �矄鼀ᨨ� � �ᆠ泐ᨨ犨 矄 ᆠ ᨨ ␔ �矄 ᆠᨨ ҈R R�矄 �盐矄ᨨ 㡰 ᨨ鼀 矄ᆠ � 矄矄 ᨨ ҈├鼀⏌矄 �ᆠ� � ᆠᨨ 矄鼀ᨨ矄 犨 犨� 矄 ᨨ 矄ᨨ� 犨 � �矄 矄盐 ᆠᨨ 矄ᆠᨨᆠ�泐 ᨨ鼀 �矄҈҈├鼀⏌矄㡰䁸� 泐ᨨ盐 矄犨ᨨ 矄� �� ├�ᨨ鼀⏌矄힘힘힘힘힘 㝐� ᔼᔼ桌
6
҈ ���ᨨ � 盐ᨨ Ἐ ᨨἘᨨ 盐 � ᨨ鼀 矄 ᨨ鼀犨 �ᆠ 矄矄 ᨨ犨 矄 Ἐ鼀ᨨ ᨨ盐Ἐ㡰泐 ᨨ ᨨ� ᨨᆠ 矄矄 ᨨ犨 犨ᨨ盐 泐㡰 㝐� 졠 � 꾀�黨҈桌 鼀ᨨ
ᆠ㡰
x
䰘 GM GM GM � GM 㡰 ᆠ㡰 㡰 x 䰘 x 黨 x x ҈ 꾀矄 �ᆠ泐矄泐鼀鼀�ᆠ���ᨨ 矄泐矄鼀ᨨ �ᆠ矄 㡰
ìr ᨨ £ î ᨨ >
矄鼀矄矄 矄�泐 r = í
犨 r�矄ᆠᨨ 矄 꾀矄鼀矄矄ᆠ 盐 �ᨨᆠ�ᆠ盐 鼀ᨨ�ᨨ 盐 � 盐 ᆠᨨ ���ᨨ � ᨨ �ᆠ矄ᔼ矄矄 矄 盐 ᆠ�ᨨ ᨨ �矄 ᆠ ¥㡰 ᨨ鼀 ᆠ ᨨ 矄泐矄鼀�矄矄 泐 桌
㡰
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A
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O
B
x
z
㡰 ���ᨨ ᨨᆠ 盐ᨨ �矄ᨨᆠ ᨨ��� �矄 ᨨ 㡰 ���ᨨ ᨨᆠ ᨨ� 㡰�矄 ᨨ ᆠ㡰 ���ᨨ ᨨ ��矄 矄鼀ᨨ� 矄ᨨ ᆠ�ᆠy졔z㦨 㡰 ���ᨨ ᨨ ��矄 矄鼀ᨨ� 矄ᨨ
ᆠ�ᆠy졔z㦨
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EBD_8341 P-78
Physics
Topic-5 : Motion of Satellites, Escape Speed and Orbital Velocity 1.
2.
3.
A satellite is in an elliptical orbit around a planet P. It is observed that the velocity of the satellite when it is farthest from the planet is 6 times less than that when it is closest to the planet. The ratio of distances between the satellite and the planet at closest and farthest points is : [Main 6 Sep. 2020 (I)] (a) 1 : 6 (b) 1 : 3 (c) 1 : 2 (d) 3 : 4 A body is moving in a low circular orbit about a planet of mass M and radius R. The radius of the orbit can be taken to be R itself. Then the ratio of the speed of this body in the orbit to the escape velocity from the planet is : [Main 4 Sep. 2020 (II)] 1 (a) (b) 2 2 (c) 1 (d) 2 A satellite is moving in a low nearly circular orbit around the earth. Its radius is roughly equal to that of the earth’s radius Re. By firing rockets attached to it, its speed is instantaneously increased in the direction of its motion so
7.
vA n = . vB 4 The value of n is :
(a) 4 8.
(a) 5.
T 2 µ R (b) T 2 µ R3 (c)
T2 µ
R3
(d) T µ R
A body A of mass m is moving in a circular orbit of radius R about a planet. Another body B of mass
6.
1
m collides with 2
r æ vö A with a velocity which is half çè ÷ø the instantaneous 2 r velocity v or A. The collision is completely inelastic. Then, the combined body: [Main 9 Jan. 2020 I] (a) continues to move in a circular orbit (b) Escapes from the Planet’s Gravitational field (c) Falls vertically downwards towards the planet (d) starts moving in an elliptical orbit around the planet The energy required to take a satellite to a height 'h' above Earth surface (radius of Eareth = 6.4 × 10 3 km) is E1 and kinetic energy required for the satellite to be in
(c) 2
(d) 3
A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it m so that subsequently the 10 satellite moves in a circular orbit. The kinetic energy of the rocket is (G is the gravitational constant; M is the mass of the earth): [Main 7 Jan. 2020 I]
3 times larger. Due to this the farthest 2 distance from the centre of the earth that the satellite reaches is R. Value of R is : [Main 3 Sep. 2020 (I)] (a) 4Re (b) 2.5Re (c) 3Re (d) 2Re
4.
(b) 1
[Main 9 Jan. 2020 II]
ejects a rocket of mass
that it become
K The mass density of a spherical galaxy varies as over r a large distance 'r' from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as : [Main 2 Sep. 2020 (I)]
a circular orbit at this height is E2. The value of h for which E1 and E2 are equal, is: [Main 9 Jan. 2019 II] (a) 1.6 × 103 km (b) 3.2 × 103 km (c) 6.4 × 103 km (d) 28 × 104 km Planet A has mass M and radius R. Planet B has half the mass and half the radius of Planet A. If the escape velocities from the Planets A and B are vA and vB, respectively, then
(a)
m æ 2 113 GM ö çu + ÷ 20 è 200 R ø
3m æ 5GM ö u+ (c) ç 8 è 6 R ÷ø
9.
2
æ 2 119 GM ö (b) 5m çè u ÷ 200 R ø mæ 2GM ö u(d) 20 çè 3 R ÷ø
2
A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet ? [Given : Mass of Planet = 8×10 22 kg, Radius of planet = 2×10 6 m, Gravitational constant G = 6.67×10–11Nm2/kg2] [Main 10 April 2019 II] (a) 9
(b) 17
(c) 13
(d) 11
10. A rocket has to be launched from earth in such a way that it never returns. If E is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that the earth’s volume is 64 times the volume of the moon. [Main 8 April 2019 II] (a)
E 64
(b)
E 32
(c)
E 4
(d)
E 16
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P-79
Gravitation
҈҈ 꾀矄�ᨨ 鼀矄矄㝐�矄� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄 ᨨ盐 ᆠ ᨨ 꾀鼀ᨨ�ᨨ 矄鼀鼀矄矄 � � ᨨ犨 矄 ᆠᨨ� 矄犨� 矄�ᆠᨨ鼀泐�
矄�ᆠ泐 矄 矄ᨨ 矄� 鼀ᨨ� 矄鼀盐矄 ᨨ ᆠᨨ�矄�ᨨ 矄盐矄⼄盐 鼀ᨨ�ᨨ
ᨨ ᆠᨨ鼀� ᨨ 泐犨�㝐� ҈ ҈黨ᔼ桌 㡰 矄盐ᆠ �矄ᆠᨨ� � �泐 㡰 ᔼ ��ᆠᨨ� ᆠ㡰 � 矄鼀ᆠ�ᆠ盐ᨨ�ᨨ �盐矄 㡰 � ᆠ�ᆠ盐ᨨ�ᨨ � �盐矄 ҈ 犨ᨨ矄�矄꾀 鼀矄矄矄鼀 鼀矄ᆠ�泐 꾀�矄� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄 �矄� ᆠ�ᆠ盐ᨨ� ᨨ �盐矄 ᨨ盐 �ᨨᨨ �├� �ᆠ ��矄 꾀⏌ �矄 㝐� ҈ ҈黨ᔼᔼ桌 㡰
҈
㡰 ҈
ᆠ㡰
҈
㡰
҈ 꾀矄��矄ᨨ� �� ᆠ�ᆠ盐ᨨ� �� ᨨ鼀 矄盐 ᆠ矄盐ᆠ 犨 �矄 �盐矄ᨨ 꾀矄矄盐鼀� � ᆠᨨ 矄鼀ᨨ矄 ᆠ �ᆠ 鼀� �鼀盐鼀 � ᆠ矄 � 矄 ⼄盐� 矄ᨨ 矄� ᆠᨨ盐 矄ᆠ ᨨ鼀 ���ᨨ � ᨨ �矄 㝐� ҈҈ ҈黨ᔼ桌 㡰 ᆠ㡰
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㡰 㡰
㡰
gR
(
㡰
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ᆠ㡰 㡰 � � ҈ 꾀 矄ᨨ 盐ᨨ 鼀矄矄鼀�矄犨ᨨ├� �ᨨ 矄�ᨨ�� � �矄 ᆠ ᨨ鼀 Ἐ矄矄盐 ᆠ �盐矄 ᨨ �矄 犨 ��矄鼀矄矄�矄㝐 ���ᨨ 盐 ᨨ 矄ᨨ 盐�矄 㝐� 졠 � 꾀�҈҈桌 㡰 ᨨ矄� ᆠ矄ᨨ 盐 矄犨�� 矄矄 㝐鼀
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10 Subjective Problems ҈ 꾀ᆠ泐ᆠ�ᆠᨨᆠ矄矄ABCA矄 ᨨ犨 � V㜠T ��鼀 ��㡰�矄 ᨨ鼀 犨� ᆠᨨ 矄 鼀矄矄ᨨ � �矄和 ᨨ犨 矄鼀ᨨᆠ矄矄ᨨ P㜠V ��鼀 ҈黨҈㜠㝐├矄桌
҈ ҈
ᔼ ��盐CA�矄ᨨ V㜠 �矄 BC�矄 ᨨ T㜠 �矄㡰
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EBD_8341 P-108
Physics
Topic-2 : Specific Heat Capacity and Thermodynamical Processes 1 1.
MCQs with One Correct Answer
Three different processes that can occur in an ideal monoatomic gas are shown in the P vs V diagram. The paths are lebelled as A ® B, A ® C and A ® D. The change in internal energies during these process are taken as EAB, EAC and EAD and the workdone as WAB, WAC and WAD. The correct relation between these parameters are : [Main 5 Sep. 2020 (I)]
D C
P A
2.
3.
4.
5.
6.
the gas molecule changes from t1 to t2. If
T2
T1
V (a) EAB = EAC < EAD, WAB > 0, WAC = 0, WAD < 0 (b) EAB = EAC = EAD, WAB > 0, WAC = 0, WAD > 0 (c) EAB < EAC < EAD, WAB > 0, WAC > WAD (d) EAB > EAC > EAD, WAB < WAC < WAD In an adiabatic process, the density of a diatomic gas becomes 32 times its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is : [Main 5 Sep. 2020 (II)] 1 (a) 32 (b) 326 (c) 128 (d) 32 A balloon filled with helium (32°C and 1.7 atm.) bursts. Immediately afterwards the expansion of helium can be considered as : [3 Sep. 2020 (I)] (a) irreversible isothermal (b) irreversible adiabatic (c) reversible adiabatic (d) reversible isotherm7al A thermodynamic cycle xyzx is shown on a V-T diagram.
(a) 2 7.
8.
9.
The P-V diagram that best describes this cycle is: (Diagrams are schematic and not to scale) [Main 8 Jan. 2020 I]
(a)
(b)
Cp Cv
= g for this
t2 gas then a good estimate for t is given by: 1 [Main 7 Jan. 2020 I]
T1>T2 B
A litre of dry air at STP expands adiabatically to a volume of 3 litres. If g = 1.40, the work done by air is: (31.4 = 4.6555) [Take air to be an ideal gas] [Main 7 Jan. 2020 I] (a) 60.7 J (b) 90.5 J (c) 100.8 J (d) 48 J Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between
(b)
1 2
(c)
æ 1ö çè ÷ø 2
g
(d)
æ 1ö çè ÷ø 2
g +1 2
A sample of an ideal gas is taken through the cyclic process abca as shown in the figure. The change in the internal energy of the gas along the path ca is – 180 J, The gas absorbs 250 J of heat along the path ab and 60 J along the path bc. The work down by the gas along the path abc is: [Main 12 Apr. 2019 I]
(a) 120 J (b) 130 J (c) 100 J (d) 140 J A cylinder with fixed capacity of 67.2 lit contains helium gas at STP. The amount of heat needed to raise the temperature of the gas by 20oC is : [Given that R = 8.31 J mol – 1 K – 1] [Main 10 Apr. 2019 I] (a) 350 J (b) 374 J (c) 748 J (d) 700 J One mole of an ideal gas passes through a process where é 1 æ V ö2 ù pressure and volume obey the relation P = P0 ê1 - ç 0 ÷ ú . êë 2 è V ø úû Here Po and Vo are constants. Calculate the charge in the temperature of the gas if its volume changes from Vo to 2Vo. [Main 10 Apr. 2019 II]
(a)
1 Po Vo 2 R
(b)
5 Po Vo 4 R
3 Po Vo 1 Po Vo (d) 4 R 4 R 10. Following figure shows two processes A and B for a gas. If DQA and DQB are the amount of heat absorbed by the
(c)
(c)
(d)
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P-109
Thermodynamics
矄泐矄鼀� 犨ᨨᆠ矄矄 D꾀 Dᆠ �矄 �
� ��矄矄ᆠ�泐 㝐� 黨꾀�҈黨ᔼ桌 㡰 D턼꾀D턼 D꾀D
҈ 졠 鼀ᨨᨨ � 鼀ᨨ ᨨᨨ鼀�ᆠ�矄�矄├ ᨨ � 꾀ᔴ꾀矄矄 ᨨ犨 � ��鼀 鼀 �鼀盐鼀 鼀盐� 泐 �矄ᨨ � ᔴ�矄��
泐 㝐� 졠 � 꾀�҈҈桌
㡰 D턼꾀>D턼 D꾀>D
҈
ᔴ
꾀
ᆠ㡰 D턼꾀>D턼 D꾀㦨D ҈҈
㡰 D턼꾀㦨D턼錐 D꾀㦨D 꾀 鼀泐� 矄盐 矄矄ᆠᨨ � 矄҈��ᨨ 犨 ᔴ
� ᨨ鼀 矄矄�矄盐鼀 ᨨ盐 ��ᆠ泐꾀 ᆠ�ᨨ ᨨ 犨盐 矄� ᨨ�ᆠ 矄ᨨ矄 ᔴ�矄 鼀矄矄ᨨ ᨨ 犨犨�ᆠᨨ矄 ᨨ ᨨ ᨨ� �ᨨ ᨨ 犨㦨҈맔҈├�҈ ᨨ 盐矄�ᨨ ᨨ 犨㦨맔҈�├�҈㡰 㝐� 꾀�҈黨ᔼ桌 㡰 ҈�� 㡰 � ᆠ㡰 ҈� 㡰 ��
�� ��鼀矄 ᨨ犨矄 ᨨ盐ᨨᆠ矄矄矄�.e.�矄ᨨᆠ ᨨ�ᆠ �矄ᨨ�ᆠ�矄ᨨ 鼀 ��ᆠ ᆠᨨᆠ矄矄�� 鼀 ᨨ ᨨᆠ矄矄矄� 矄鼀ᨨ �矄�� 泐 㝐� 꾀҈黨ᔼᔼ桌
㡰 � 㡰 � � � ᆠ㡰 㡰 ҈ ҈䰘 Ἐ Ἐ鼀ᨨ矄ᨨ � �矄盐 �ᨨ矄ᨨᆠ矄矄꾀®矄 矄 ᨨ犨 � ��盐 鼀 �鼀盐鼀鼀盐ᨨ �矄 盐� � ᨨᆠ矄矄犨� 㝐� ҈桌
꾀
黨 黨 黨 㡰 㡰 ᆠ㡰 㡰
҈ �ᨨᨨ 犨ᨨ├ ᨨ 泐 � 鼀ᨨ ᨨᨨ鼀�ᆠ�矄ᨨ 矄盐� ᨨ�� �矄ᨨ�ᆠᨨᆠ矄矄�矄 㝐� 졠 � 꾀�黨҈桌 ҈
㡰 ᆠ 㡰 ᆠ ᆠ㡰 ᆠ 㡰 ᆠ ᨨ �� ᆠ泐ᆠ�ᆠᨨᆠ矄矄ᔴ꾀矄矄 ᨨ犨 ᨨ�矄 犨ᨨ├ ᨨ �矄 㝐� ҈ ҈黨ᔼ桌
꾀
ᔴ
�
㡰
҈ ҈
҈
� 鼀㡰
㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 � 꾀��� �ᨨ鼀�ᆠ� �矄盐 �ᨨ矄 ��ᆠᨨᆠ矄矄 ᨨᨨ鼀鼀盐 �ᨨ 犨 鼀盐 ᨨ盐鼀 ᨨ �矄ᨨᆠ矄矄�矄 㦨ᆠᨨ 矄 �矄 㝐� ҈҈ ҈黨 ᔼ桌 � 㡰 ᆠ㡰 㡰 � � 鼀ᨨᨨ � 鼀ᨨ ᨨᨨ鼀�ᆠ�矄�矄 ᆠᨨ 矄 矄矄盐ᨨ ҈鼀 ᨨ鼀 ᔴᨨ黨 ᔴ␔ᨨ├ ᨨ 泐�矄�矄 ᆠᨨ矄ᨨ 矄ᆠᨨ 矄 㦨҈⏌鼀ᨨ㜠⧤㡰 㝐� ҈ ҈黨 ᔼᔼ桌 㡰 �҈ 㡰 黨҈ ᆠ㡰 ҈ 㡰 䰘
㡰
҈�
㡰 ᆠ㡰 㡰 � � ҈黨 꾀�矄�矄 ᆠᨨ矄 � ᆠ泐� 犨� 鼀ᨨ �ᆠ�ᨨ 矄矄 �矄ᨨ ᔼ矄� �� 鼀ᨨ 泐 鼀�ᆠ矄矄矄盐�㦨҈� ᨨ盐鼀�㦨҈鼀ᆠ �矄ᨨ � 矄 㦨 ҈⏌㡰맔҈� 㦨맔҈鼀� ��ᆠ⼄盐矄�㜠 矄�ᆠ ᨨᆠ矄矄 矄盐ᆠ � 㦨 ᆠᨨ 矄 ᔴᨨ 矄� ᨨ 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄 � �矄 矄泐矄鼀 ᨨ鼀 矄鼀� ��矄ᨨ 矄鼀 � 矄� 犨ᨨ 矄矄 �矄ᨨ�ᆠ 矄�ᨨ � ᨨᨨ犨 泐 �矄ᨨᆠ ᨨ�ᆠ �矄ᨨᨨ盐鼀�ᆠ㡰ᨨᆠ矄矄ᨨ盐鼀 鼀ᨨ盐 ᨨ 矄盐� ᨨ 矄泐矄鼀 � 犨ᨨ㜠矄 ᨨᆠ矄矄 �矄 ᨨ �鼀泐 꾀 ҈桌 㡰 ҈҈ 㡰 黨 ᆠ㡰 � 㡰 ҈ ᔴᨨ 矄� � �矄 ᆠᨨ � � �矄ᨨ ᆠᨨ矄 ᆠ 鼀꾀矄 �矄盐 �ᨨ矄 ��ᆠ 矄�ᨨ ��鼀ᨨ ᆠᨨ�矄�ᨨ 犨 鼀ᨨᆠ盐矄� ᆠ矄矄 as V⼄犨 �矄 ᨨ盐鼀ᨨ �矄 盐ᨨ ⼄
㡰
ᔴ ö æ �矄 ç g = 㝐� ҈�桌 ÷ ᔴ ø è g + � g - � g +҈ g -҈ 㡰 㡰 ᆠ㡰 㡰 ҈ ᔴᨨ 矄� 矄 �ᆠ矄 ᨨ �盐矄 鼀盐
ᆠ├ᨨ 泐 ��ᨨ � 矄� �ᆠ ᆠᨨ 矄� 矄
� �矄ᨨ ᨨᨨ 矄犨� � �泐盐 �ᨨ盐鼀 ҈æö 盐㦨 µ 矄矄盐 = ç ÷ ᔼ 矄 ᨨ犨 è ø 盐 �ᨨ矄 ��ᆠ 矄�ᨨ �ᨨ 犨
�矄 㝐� ҈�桌
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EBD_8341 P-110
҈ ҈ 㡰 µ ᆠ㡰 µ 㡰 µ 꾀 � �矄�ᨨ矄 ᨨ盐� 矄�ᆠ泐ᆠ®®ᆠ® 矄 㜠 ��鼀矄 ᨨ犨 ᨨ犨ᨨᆠ矄矄 ® ®ᆠ ��ᆠ 㡰
Physics
µ
ᆠ
ᆠᨨ矄ᨨ � �㜠 ��鼀 ᨨ ᨨᆠ矄矄�矄 ��盐矄矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ矄ᆠ㡰 㝐� 졠 � 꾀�҈҈�桌
㡰
ᆠ
㡰
ᆠ
� 꾀 � �矄 鼀ᨨ矄 �ᆠ 矄矄盐 �矄 ��ᆠ泐 ᆠᨨ鼀矄矄 矄ᨨ �矄 矄�泐ᆠᨨ鼀矄�鼀矄ᨨ �矄 � ��盐ᔼ � 矄矄盐ᨨ �矄�矄҈鼀ᨨ矄 矄 盐ᨨ �gᨨ �矄�矄 㝐� 졠 � 꾀�҈桌 㡰 ҈� 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ ��ᨨ �盐鼀�矄和 �矄 ��ᆠ泐ᆠᨨ鼀矄矄 ᨨ䰘� ├� � � ��鼀盐ᨨ ҈ 犨ᨨ├ ᨨ � ᨨᆠ矄矄�矄 㝐� ҈҈桌 ҈� 黨 黨 RT 㡰 RT 㡰 㡰 ᆠ㡰 RT RT ҈ ҈ ҈ ҈ 䰘 꾀 � �矄�矄� ��泐P҈V҈�矄 ᨨPV ᆠᨨ鼀矄矄 ��ᆠ泐ᨨ 矄鼀ᨨ盐鼀V҈ 矄矄盐 P ᔼ W �矄 犨ᨨ├ ᨨ 泐 �矄 �
ᆠᨨ鼀ᨨᆠ矄矄犨 �ᆠ ᨨ ᨨᨨ犨� ��矄盐 和桌 㡰 W>錐P>P҈ 㡰 W錐P>P҈ ᆠ㡰 W>錐PP҈ 㡰 W錐PP҈ PT ��鼀 ᨨ � �矄�矄矄 ᨨ犨 � ��盐 犨 AC�矄 ��ᆠᨨᆠ矄矄 � ᆠᨨ矄ᨨ � � PV ��鼀 和桌
A
P
ᆠ
C
B
ᆠ㡰
㡰
ᆠ
T A
졠 鼀ᨨᨨ �ᨨ鼀�ᆠ� �矄盐 �ᨨ矄ᆠ泐ᆠ�ᆠᨨᆠ矄矄 꾀ᔴ矄矄 ᨨ犨 � ��盐 ᨨᆠ矄矄ᔴ�矄 ��ᆠ 鼀盐矄꾀 ᔴ⧤⧤ ⧤ 矄ᆠ�泐ᔴ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㝐� ҈桌
A
P
P
㡰
C
B
㡰
V
V
A
A
P
P
ᆠ㡰
⧤
C
B
㡰
B C
V
꾀
⧤
├ ᔴ
B
C
V
黨 꾀 � �矄�矄├ ᨨ盐� ᆠ泐ᆠ A ® B ® C ® A 矄矄 ᨨ犨 � ��盐ᔼ 矄盐� ᨨ �矄 � ᆠ泐ᆠ�矄� 犨ᨨ├ ᨨ 泐 �矄� ᨨᆠ矄矄 和桌 C ® A �矄 C B 㡰� 鼀 㡰
㡰 ᆠ �� � �泐� 犨 ᨨᆠ泐ᆠ�ᆠᨨᆠ矄矄 �矄� 㡰 ᆠ �� � �泐� ᨨᆠ矄矄ᔴ꾀�矄䰘 ᆠ㡰 ᆠ �� � �泐� ᨨᆠ矄矄꾀�矄㜠� 㡰 ᆠ �� � �泐� ᨨᆠ矄矄ᔴ�矄� 盐� � ��ᆠᆠᨨ鼀矄矄�ᨨ ᨨ 犨ᨨ├�矄 ᨨ ᨨ
鼀ᨨ矄ᨨ �ᨨ鼀�ᆠ� �矄ᨨ 盐ᆠ�矄ᨨ盐鼀泐 �奠 ᆠ �� �矄鼀盐�矄 泐 㦨⧤҈鼀ᨨ҈㡰 㝐� 졠 � 꾀�҈҈҈桌 㡰 ⧤ 㡰 ⧤ ᆠ㡰 ⧤ 㡰 ҈⧤
㡰҈
҈
ᆠ㡰҈�
A
㡰 ⏌鼀㡰 ҈ P㜠V ᨨ矄 ᨨ 犨ᨨ�矄矄 盐� � ��ᆠ ᨨᆠ矄矄矄 矄 ᨨ犨 � ��盐 ᨨ矄 ҈ 矄 ᨨ盐 ᆠᨨ矄ᨨ 矄ᆠ�泐ᨨ ҈和桌 㡰 졠 㡰 졠 ᆠ㡰 꾀 㡰 졠
҈
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P-111
Thermodynamics
҈
和� � 犨� 矄鼀 � �� ᆠᨨ ��ᨨ 矄 � �矄 矄 ᨨ鼀ᨨ盐鼀V҈ᨨV� � 犨泐矄 犨ᨨ├ ᨨ 泐 �矄 �矄 W҈ � ᨨᆠ矄矄 �矄 盐泐 �矄ᨨ 鼀W� 盐泐�矄ᨨ�ᆠ W� 盐泐 ��ᆠ
和桌 㡰 W>W҈>W
y ᔼ矄ᨨ�ᆠ
㡰 W>W>W҈
W ᔼ矄ᨨ 鼀 ␔҈ 꾀 ��ᆠ ␔
P
ᆠ㡰 W҈>W>W 㡰 W҈>W>W
ᨨ
V҈
æ L҈ ö ç ÷ è L ø
⏌
㡰
L҈ L ⏌
æ L ö ÷ ᆠ㡰 㡰 ç è L҈ ø 꾀�� ⼄盐 �泐ᨨ � �矄�矄矄矄盐P 矄ᨨ盐 鼀盐T �矄ᨨ 鼀盐├鼀ᨨ 盐盐矄ᨨ �矄�矄 ҈黨黨和㜠㝐├矄桌 P 㡰 㡰 P P 㡰 P ᆠ㡰 꾀 � 鼀ᨨ ᨨ鼀�ᆠ�矄�矄├ ᨨ盐 ᆠ泐ᆠABCDA 矄矄 ᨨ犨 � PV ��鼀 矄 ��㡰 犨ᨨ├ ᨨ 盐� � ᆠ泐ᆠ�矄 ҈黨㜠҈㝐├桌 P 㡰 PV P, V P, 2V 㡰 PV B C L L҈
҈ PV 㡰 ᨨ
ᆠ㡰
2 �
x
꾀 鼀ᨨ ᨨ鼀�ᆠ � �矄 � ��泐 鼀盐 T҈ �矄 ᆠᨨ矄 � ᆠ泐� � 犨� �ᆠ�ᨨ 矄矄�矄ᨨ �矄�矄ᨨ犨 ᨨ ��ᆠ泐ᨨ鼀盐T 泐矄� � �矄ᨨ 矄盐 泐ᔼ L҈ L � ᨨ �矄ᆠᨨ盐鼀 ᨨ 矄�ᨨ 矄ᆠ�泐 T҈ T �矄�� 泐 和桌 㡰
V
꾀矄矄盐鼀� ��ᨨ �ᨨ鼀�ᆠ� �矄鼀 盐ᨨ ��� 鼀ᨨᆠ盐矄 ᆠ �� �矄� �泐 盐� � �矄 ᨨᆠ矄矄ᆠᨨ鼀矄ᨨ盐ᨨX├ 盐ᨨ Xᨨ 矄 � ��矄힘힘힘힘힘힘힘힘 [Main 2 Sep. 2020 (I)] 䰘 和� �鼀盐⧤ᨨ 鼀ᨨᨨ � �ᨨ鼀�ᆠ �矄 g㦨҈㡰�矄 �矄ᆠᨨ鼀矄矄 ��ᆠ泐 ᨨ鼀ᨨ盐鼀 ҈ ҈ᨨ㦨 ᔼ�矄 ᨨ犨 ᨨ �矄ᨨ�ᆠ泐ᨨ ҈ ᨨ盐鼀ᔼ ᨨᆠ矄矄矄 ⼄盐矄�㜠矄�ᆠ
� 鼀盐ᨨ �矄 � ⧤㡰�矄 ᨨ 矄 � �㡰힘힘힘힘힘힘 㝐� 黨 ᔼᔼ桌 ᔴᨨ 矄� ᨨ 鼀ᨨᨨ �盐鼀�矄 ᆠᨨ矄 � ᆠᨨ � � ��矄矄盐P҈ ᨨ盐鼀V҈ᔼ 矄�矄ᨨ 鼀泐 ᨨᨨ盐鼀V҈꾀 �矄 �矄 矄 ��ᆠ泐 �矄ᨨ盐鼀ᆠᨨ鼀矄V҈ 犨ᨨ├ ᨨ 泐 �矄 盐� ��矄ᨨ 鼀 ��ᆠ 矄�ᨨ ᨨᆠ矄矄矄 W�sᨨ = f W�sᨨ Wad�a矄ᆠ�泐ᔼ �ᨨ W ad�a f�矄힘힘힘힘힘힘힘힘 꾀 桌 黨 꾀矄 �ᆠ盐� 矄� 犨 矄 �盐矄R ├ 矄矄盐� 矄� 盐 犨矄矄盐ᨨ p 盐 ᨨ犨�矄ᆠᨨ鼀矄矄 �泐� �㜠 �ᆠ鼀
矄ᨨ �矄 �盐矄ᆠᨨ鼀矄 R a㡰 ᨨ a = R 鼀� �盐 ᨨ 犨ᨨ├ ᨨ � ᨨᆠ矄矄�矄�� 泐 犨 X�矄ᆠᨨ 矄 g = C p ⏌ Cv = ҈⏌ 盐 ᨨ X�矄힘힘힘힘힘힘힘힘 꾀 桌 4
꾀 � �矄犨� 矄矄盐Pᨨ盐鼀V 鼀盐T �矄 �矄ᨨ 鼀泐ᨨᨨ盐鼀V � 矄矄盐 P�ᔼ 矄鼀�矄�矄 ��ᆠ泐ᨨᨨ盐鼀 V � 矄矄盐�矄Pa �ᨨᨨ 矄ᆠ� �ᆠ 矄 Pa ᨨ �矄�矄҈䰘 �ᨨ P �矄 ҈黨黨㜠㝐├矄桌 ҈
5 A P, V
D
P,V V
Integer Value Answer
True / False
҈ ᆠ盐矄A B� ��盐矄 ᨨ犨 P㜠V� 矄 ᨨ
�矄ᨨ 鼀 ��ᆠᨨᆠ矄矄 ᨨ � �矄 �矄ᨨ 鼀ᨨᆠ矄矄�矄矄 泐 ᆠ盐A ҈黨�㜠㝐├矄桌
҈ ᨨ �矄� ��ᨨ盐鼀ᔼ � ��鼀盐ᨨ �矄�矄 T� �
꾀 �ᨨ鼀�ᆠ� �矄�矄ᆠᨨ鼀矄矄 ��ᆠ泐ᨨ
P
3
Numeric Answer
꾀 �� ├矄� �鼀ᨨᨨ � ᔴ ҈鼀 ᆠᨨ鼀矄矄矄� ��ᆠ泐ᨨ҈⏌҈ ᨨ ᨨ��� ᨨ盐鼀
A B
⧤� 㡰 � 鼀盐�矄 T� 盐ᨨ a�矄 ҈桌
Fill in the Blanks
6
V MCQs with One or More than One Correct Answer
꾀鼀� 盐ᨨ � �矄ᆠᨨ � � ��鼀ᨨ矄ᨨ 鼀ᨨ ᨨ鼀�ᆠ �矄 ҈鼀ᨨᨨ ��� �ᨨ鼀�ᆠ�矄�矄� ��泐矄矄盐
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EBD_8341 P-112
Physics
Pᨨ盐鼀V 鼀盐Tᔼ �矄鼀� 盐�矄 ��ᆠ泐ᆠᨨ鼀矄矄 ᨨ ᨨ盐鼀 V⏌ ᆠᨨᆠ矄鼀 矄㡰�矄⏌ � ҈㦨錐黨錐R�矄 �矄 ᆠᨨ 矄 㡰 꾀 ҈黨桌 㡰 犨ᨨ├␔ ᨨ 盐� � ᨨᆠ矄矄�矄҈RT 㡰 � 矄矄盐 ᨨ �矄 鼀� 盐 ᆠᨨ鼀矄矄�ᨨ �矄� 犨 黨P ҈P ᆠ㡰 �├� �ᆠ �泐ᨨ �矄鼀� 盐 ᆠᨨ鼀矄矄�ᨨ �矄� 犨 ҈RT ҈黨RT 㡰 꾀 ��ᆠᆠᨨ 矄 ᨨ �矄鼀� 盐�矄҈ 졠 鼀ᨨ ᨨ 鼀ᨨ ᨨ鼀�ᆠ � �矄 �ᨨ矄 ᨨ盐� 鼀ᨨ 泐 鼀�ᆠ ᆠ泐ᆠ 矄 矄 ᨨ犨 � ᨨ盐鼀 矄盐矄 鼀盐 VT㡰 ��鼀 ᆠᨨᆠ矄鼀 矄㡰�矄⏌ �矄 �矄ᆠᨨ 矄 꾀 ҈黨桌
V
V
҈
T⏌
T
T⏌
T
T
㡰 ␔ᨨ├ ᨨ � �矄 鼀ᨨ 泐 鼀�ᆠ ᆠ泐ᆠ
҈ RT 㡰 ᨨ 鼀ᨨ 泐 鼀�ᆠ ᆠ泐ᆠ ��矄 ᨨ 泐 �矄ᨨᆠ ᨨ�ᆠ ��ᆠᨨᆠ矄矄矄 ᆠ㡰 �ᨨᨨ 矄 盐� �ᨨᆠ矄矄矄҈® ҈ ® ® ® ® ҈㡰 �矄 W =
Q҈® � = ®�矄 Q ®
㡰 �ᨨᨨ 矄 盐� �ᨨᆠ矄矄矄 ҈ ® Q҈® ҈ = ®�矄 Q ®
졠 鼀ᨨᨨ � �矄� � ��矄꾀盐 �ᨨ矄ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄꾀ᔴ꾀矄矄 ᨨ犨 � ��盐ᔼ矄矄矄盐꾀�矄 Pᔴ ᨨᨨ矄 ᆠᨨᆠᨨ�ᨨ 矄㡰 ᨨ鼀 ᨨᨨ犨� � ҈桌 V V V
ᔴ
꾀 T
T 㡰 ᔼ ��矄꾀 矄鼀 㡰 ␔ᨨ├ ᨨ 泐 �矄� ᨨᆠ矄矄꾀�矄PVln P ᆠ㡰 矄矄盐ᔴ�矄 T 㡰 鼀盐ᔴ�矄
� ��盐矄 ᨨ犨矄 P㜠Vᨨᨨ � �矄├ ᨨ盐� ᆠ泐ᆠABCDA ABC �矄矄鼀�㜠ᆠ�ᆠ CDA�矄 ᨨ �矄 黨桌 P
A
B
D
҈
C ҈
V
㡰 ᨨᆠ矄矄 盐� � A®B�矄�矄ᨨ 鼀 㡰 ᨨ犨矄ᨨ盐ᨨ �矄 盐� � B®C®D ᆠ㡰 犨ᨨ├ ᨨ 盐� � A®B®C�矄 ᨨ 㡰 ᨨ矄��犨ᨨ├�矄 ᨨ 泐 �矄� ᆠ泐ᆠABCDA ᨨ � �矄 ҈黨黨㜠㝐├矄桌 㡰 ᆠ �� � �泐� ᆠᨨ 矄 矄矄盐 ᨨᆠ矄矄 ᨨ鼀 鼀盐 T҈ ᨨ T �矄⼄盐 ᨨnCv TT҈㡰犨 Cv�矄 鼀ᨨ矄ᆠ� �ᆠ ᆠᨨ 矄 ᨨ盐鼀 n 盐鼀ᨨ 鼀ᨨ矄ᨨ �矄 㡰 ᆠ �� � �泐ᨨ �矄 犨ᨨ├ ᨨ 泐 �矄⼄盐� 鼀� �盐 � ��ᆠ ᨨᆠ矄矄 ᆠ㡰 � �泐 ᨨ矄 ᨨᆠ �� �矄ᨨ 鼀 ᨨᆠ矄矄 㡰 ᨨ �矄 ᨨ鼀ᨨ � ��ᆠᨨᆠ矄矄 7
Match the Following
䰘 㝐ᆠ 鼀ᨨ 泐 鼀�ᆠ ᨨᆠ矄矄矄├� � ᆠ � 矄泐矄鼀犨� ᆠᨨᆠᆠᨨ ��ᨨ 矄ᔼ DQ�矄 矄盐� DW�矄 犨ᨨ├ ᨨ DU�矄ᆠ ��
� �泐ᨨ 矄泐矄鼀 㝐� 和 ᔼᔼ㡰桌 ᨨᆠ矄矄 ᔴᨨ ��ᨨ
ᔼ㡰 꾀 ��ᆠ 꾀㡰 DW㦨 ᔼᔼ㡰 ᔼ矄ᨨ 鼀 㡰 DQ㦨 ᔼᔼᔼ㡰 ᔼ矄ᨨᆠ ᨨ�ᆠ ᔴ㡰 DU ¹ DW ¹
DQ ¹ ᔼ㡰 ᔼ矄ᨨ�ᆠ 㡰 DU㦨 㡰 ᔼ㡰㜠 꾀㡰 ᔼᔼ㡰㜠 㡰 ᔼᔼᔼ㡰㜠 㡰 ᔼ㡰㜠 㡰 㡰 ᔼ㡰㜠 㡰 ᔼᔼ㡰㜠 꾀㡰 ᔼᔼᔼ㡰㜠 㡰 ᔼ㡰㜠 ᔴ㡰 ᆠ㡰 ᔼ㡰㜠 꾀㡰 ᔼᔼ㡰㜠 꾀㡰 ᔼᔼᔼ㡰㜠 㡰 ᔼ㡰㜠 ᔴ㡰 㡰 ᔼ㡰㜠 㡰 ᔼᔼ㡰㜠 㡰 ᔼᔼᔼ㡰㜠 꾀㡰 ᔼ㡰㜠 ᔴ㡰 ᔼ 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄ᨨ � 鼀ᨨ ᨨ鼀�ᆠ�矄 � � �矄�鼀 矄ᨨ 泐 �矄�矄 �� 泐
D犨 T�矄鼀盐ᨨ 矄泐矄鼀 D�矄 � � �矄�鼀ᆠ �� 鼀ᨨ 泐 鼀�ᆠ⼄盐 �泐Xᨨ 矄泐矄鼀 ᨨ 鼀ᨨ ᨨ 鼀ᨨ ᨨ鼀�ᆠ � �矄 æT ö æV ö X = ҈ ç ÷ + ҈ ç ÷ �矄�矄ᆠᨨ 矄 V T è Aø è VA ø �矄ᨨ盐鼀ᨨ �矄T꾀 V㜠ᆠᨨ 矄 矄
�矄㜠ᔼ ᨨ犨 ��矄 矄ᨨ鼀 ⼄盐 ��矄 � ᨨ � ᨨᆠ矄矄 �矄㜠ᔼᔼ��矄矄ᨨ鼀ᨨ矄矄�盐矄ᨨ 矄 ⼄盐 ��矄 꾀 ҈黨桌
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P-113
Thermodynamics
�矄㜠ᔼ ᔼ㡰
␔ᨨ├ ᨨ 泐
矄泐矄鼀� ᨨᆠ矄矄 ҈®® ᔼᔼ㡰 ᔴ �� � �泐� ᨨᆠ矄矄 ҈®® ᔼᔼᔼ㡰 矄ᨨ 泐 矄泐矄鼀�
ᨨᆠ矄矄҈®® ᔼ㡰 矄ᨨ 泐 矄泐矄鼀 � ᨨᆠ矄矄҈®
�矄㜠ᔼᔼ ҈ RT ⏌ ln 㡰
턼㡰
҈ RT
㡰
和㡰
҈ 㡰
졔l 㡰 � 㡰
ᔼ ᨨᆠ矄矄ᆠ� ᨨ盐ᨨ ᨨ 鼀ᨨᨨ 鼀ᨨ ᨨ鼀�ᆠ� ҈ �矄�矄矄 ᨨ犨 � ��盐㜠 ��鼀犨� PV㦨 RT ᆠᨨᆠ鼀ᆠ �矄 P P
P
҈ V
黨
V
V
㡰 ᔼ®턼ᔼᔼ® ᔼᔼᔼ®和ᔼ® 㡰 ᔼ®和ᔼᔼ® ᔼᔼᔼ®턼ᔼ®
ᆠ㡰 ᔼ®턼ᔼᔼ® ᔼᔼᔼ®ᔼ® 㡰 ᔼ®턼ᔼᔼ®和ᔼᔼᔼ® ᔼ® ᔼ 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄ᨨ � 鼀ᨨ ᨨ鼀�ᆠ�矄 � � �矄�鼀 矄ᨨ 泐 �矄�矄 �� 泐
D犨 T�矄鼀盐ᨨ 矄泐矄鼀 D�矄 � � �矄�鼀ᆠ �� 鼀ᨨ 泐 鼀�ᆠ⼄盐 �泐Xᨨ 矄泐矄鼀 ᨨ 鼀ᨨ ᨨ 鼀ᨨ ᨨ鼀�ᆠ � �矄 æT ö æV ö X = ҈ ç ÷ + ҈ ç ÷ R�矄�矄ᆠᨨ 矄 V è TA ø è VA ø �矄ᨨ盐鼀ᨨ �矄T꾀 V꾀ᆠᨨ 矄 矄
�矄㜠ᔼ ᨨ犨 ��矄 矄ᨨ鼀 ⼄盐 ��矄 � ᨨ � ᨨᆠ矄矄 �矄㜠ᔼᔼ��矄矄ᨨ鼀ᨨ矄矄�盐矄ᨨ 矄 ⼄盐 ��矄 꾀 ҈黨桌 �矄㜠ᔼ �矄㜠ᔼᔼ ҈ RT ln ᔼ㡰 ␔ᨨ├ ᨨ 泐 㡰 矄泐矄鼀� ᨨᆠ矄矄 ҈®® ҈ RT ᔼᔼ㡰 ᔴ �� � 턼㡰 �泐� ᨨᆠ矄矄 ҈®® ᔼᔼᔼ㡰 矄ᨨ 泐 㡰
矄泐矄鼀� ᨨᆠ矄矄 ҈®® ᔼ㡰 矄ᨨ 泐 和㡰 矄泐矄鼀� ᨨᆠ矄矄 ҈®
҈
졔l 㡰 � 㡰
ᔼ ᨨᆠ矄矄ᆠ� ᨨ盐ᨨ ᨨ 鼀ᨨᨨ 鼀ᨨ ᨨ鼀�ᆠ�
㡰
҈ �矄�矄矄矄 ᨨ犨 � TV㜠 ��鼀犨� 㦨
ᆠᨨᆠ鼀ᆠ �矄
㡰 ᔼ®ᔼᔼ® ᔼᔼᔼ®턼ᔼ®
㡰 ᔼ®和ᔼᔼ® ᔼᔼᔼ®턼ᔼ® ᆠ㡰 ᔼ®ᔼᔼ® ᔼᔼᔼ® ᔼ® 㡰 ᔼ®ᔼᔼ® ᔼᔼᔼ® ᔼ®和 � 졠 鼀ᨨ ᨨ 鼀ᨨ ᨨ鼀�ᆠ � �矄 盐 �ᨨ矄 ᨨ盐 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄矄矄矄 ᨨ犨 矄ᆠ 鼀�ᆠ泐� 㜠 ��鼀ᨨ犨꾀鼀ᨨ � 矄 ᨨ盐ᨨᆠ矄矄矄ᨨ �矄 �矄ᨨ�ᆠᨨ �矄�矄ᨨᆠ ᨨ�ᆠᨨ �矄�矄ᨨ 鼀 ᨨ �矄 ��ᆠ㝐ᆠ ᨨᆠ矄矄矄鼀 �ᨨ � �矄㜠ᔼ犨� ᆠᨨ矄ᨨ � �矄鼀 矄� �矄㜠ᔼᔼ 꾀 ҈桌 ᔼᔼ
ᔼᔼᔼ
ᔼ ᔼ
턼
和 㡰 㡰 ᆠ㡰 㡰
ᔼ和Tᔼ ᔼ ᨨᆠ矄矄ᔼ
ᔼ和Tᔼᔼ ␔ᨨ├ ᨨ 泐 �矄 �矄 ᨨ ᔼ ᨨᆠ矄矄ᔼᔼ 鼀盐ᨨ �矄 鼀� 矄盐 ᆠ � ᔼ ᨨᆠ矄矄ᔼᔼᔼ ᨨ �矄 ᆠ � 犨 �矄 �矄 矄盐ᨨ盐 � �矄 ᔼ ᨨᆠ矄矄ᔼ ␔ᨨ├ ᨨ 泐 �矄�矄 6P ®錐턼®錐 ®҈錐和® ®҈錐턼®錐 ®錐和® ®錐턼®錐 ®҈錐和® ®錐턼®錐 ®錐和®҈ ҈
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EBD_8341 P-114
Physics
ᔼ EᔴTᔼ졠N和턼Nᨨ�҈� �泐ᨨ�泐鼀ᆠ � � � ᨨ鼀�ᨨ �� � ᆠᨨ盐鼀 矄ᨨ ᨨᨨ犨� � 꾀 � �矄�矄盐 �ᨨ� �ᆠ泐ᆠ�ᆠ 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄� � 犨泐矄矄矄 ᨨ犨 � ᆠᨨ矄ᨨ � � ��鼀矄�
ᆠᨨ盐鼀 ᨨ ᔴᨨ 矄� ᨨ 泐 ᨨ鼀矄҈ᨨ矄␔ ᨨ矄 ᆠᨨ矄ᨨ � �犨ᨨ├ ᨨ ᨨ 矄泐矄鼀 ⼄盐�ᨨ 矄 ᨨ矄� 矄 ᨨ�ᨨ 矄矄盐矄 � 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄矄Ӹ�矄 �ᨨᨨ ᆠᆠ��矄 ᆠᨨ 矄 矄矄盐 ᆠᨨ 矄 ᨨ盐鼀 盐鼀ᨨ 鼀ᨨ矄� �矄�矄 꾀 ҈䰘桌 ᔴᨨ盐鼀 ҈ ᔼ㡰
␔҈® =
ᔴᨨ盐鼀
҈ - ҈҈ 㡰 g -҈
ᔼᔼ㡰 ␔҈® = - + ҈
�㡰
ᔴᨨ盐鼀
ᔼ矄ᨨ 鼀
��㡰 ᔼ矄ᨨᆠ ᨨ�ᆠ
㡰
턼㡰
҈
҈
ᔼᔼᔼ㡰 ␔҈® =
���㡰 ᔼ矄ᨨ�ᆠ
҈
㡰
҈ æ ö ᔼ㡰 ␔҈® = - ç ÷ è ҈ ø
�㡰
꾀 ��ᆠ
和㡰
�҈
�
�
�
␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄ᆠᨨᆠ泐矄 矄 鼀ᨨ 泐 鼀�ᆠᨨᆠ矄矄 �矄盐矄 矄ᆠᨨᆠ�ᨨ � 鼀� �ᨨ ᨨ 矄 ᨨ 矄ᨨ盐 � � �矄䁸 㡰 ᔼ㡰 ��㡰 턼㡰 㡰 ᔼ㡰 ��㡰 㡰 ᆠ㡰 ᔼᔼᔼ㡰 �㡰 㡰 㡰 ᔼ㡰 �㡰 턼㡰 ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � ᨨ�ᨨ 矄 �矄 ᆠᨨᆠ ᆠᨨ鼀� �ᨨ 䁸 㡰 ᔼ㡰 ��㡰 和㡰 㡰 ᔼᔼᔼ㡰 ��㡰 和㡰 ᆠ㡰 ᔼᔼ㡰 �㡰 㡰 㡰 ᔼᔼ㡰 �㡰 㡰 ␔ �ᆠ ᨨ ᨨᨨ犨� � ᨨ�ᨨ 矄 �矄 ᨨ 泐 ᆠᨨᆠ 矄 �ᨨ ᨨ ᨨᆠ矄矄� 犨 �ᆠ D㦨D턼D䁸 㡰 ᔼᔼ㡰 �㡰 㡰 㡰 ᔼᔼᔼ㡰 ���㡰 㡰 ᆠ㡰 ᔼᔼ㡰 ���㡰 和㡰 㡰 ᔼᔼ㡰 ���㡰 㡰 졠 鼀ᨨᨨ 鼀ᨨ ᨨ鼀�ᆠ� �矄�矄├ ᨨ �犨ᨨᆠ泐ᆠ�ᆠ ᨨᆠ矄矄矄쟼® ®®쟼 쟼® ®®쟼矄矄 ᨨ犨
� ��鼀 ᨨᆠ矄矄矄� ᨨ 盐泐�矄ᨨᆠ ᨨ�ᆠ �矄ᨨ�ᆠ�矄ᨨ 鼀ᨨ ��ᆠ 꾀 ҈桌
쟼
㝐ᆠ 矄� �矄ᔼ犨� 鼀� �盐 矄ᨨ 犨ᨨ├ ᨨ � �矄 ᔼᔼ 矄ᆠ ᆠᨨᆠ 矄犨 盐矄� � ᆠᨨ 矄�� ᨨ犨 �矄矄 �矄ᔼ �矄ᔼᔼ ®쟼 ҈ ҈ 턼 ® ® 和 ® ҈
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P-115
Thermodynamics
Codes: P (a) 4 (b) 4 (c) 3 (d) 1
Q 3 3 1 3
R 2 1 2 2
(a) Sketch the complete process on a p – V diagram. (b) Find the total work done by the gas, the total change in its internal energy and the final temperature of the gas. [Give your answer in terms of p1, V1, V2, Q and R] 59. One mole of an ideal monatomic gas is taken round the cyclic process ABCA as shown in Figure. Calculate [1998 - 8 Marks]
S 1 2 4 4
P
10 Subjective Problems 55.
56.
A cylinder of mass 1 kg is given heat of 20,000J at atmospheric pressure. If initially the temperature of cylinder is 20°C, find [2005 - 6 Marks] (a) final temperature of the cylinder. (b) work done by the cylinder. (c) change in internal energy of the cylinder (Given that specific heat of cylinder = 400 J kg–1 °C–1, coefficient of volume expansion = 9 × 10–5 °C–1, Atmospheric pressure = 105 N/m2 and Density of cylinder = 9000 kg/m3) A monoatomic ideal gas of two moles is taken through a cyclic process starting from A as shown in figure. The volume ratios are is 27oC.
V VB = 2 and D = 4. If the temperature TA at A VA VA
[2001-10 Marks]
V
VD
D
C
B
VB VA O
A TA
TB T
57.
Calculate, (a) the temperature of the gas at point B, (b) heat absorbed or released by the gas in each process, (c) the total work done by the gas during the complete cycle. Express your answer in terms of the gas constant R. Two moles of an ideal monatomic gas is taken through a cycle ABCA as shown in the P-T diagram. During the process AB, pressure and temperature of the gas vary such that PT = Constant. If T1 = 300 K, calculate [2000 - 10 Marks]
3P0
P0
B
A
C
V0
2V0
V
(a) the work done by the gas. (b) the heat rejected by the gas in the path CA and the heat absorbed by the gas in the path AB; (c) the net heat absorbed by the gas in the path BC; (d) the maximum temperature attained by the gas during the cycle. 60. One mole of a diatomic ideal gas (g = 1.4) is taken through a cyclic process starting from point A. The process A ® B is an adiabatic compression, B ® C is isobaric expansion, C ® D is an adiabatic expansion, and D ® A is isochoric. The volume ratios are VA / VB = 16 and VC / VB = 2 and the temperature at A is TA = 300 K.Calculate the temperature of the gas at the points B and D and find the efficiency of the cycle. [1997 - 5 Marks] 61. At 27°C two moles of an ideal monoatomic gas occupy a volume V. The gas expands adiabaticallly to a volume 2V. Calculate (i) the final temperature of the gas, (ii) change in its internal energy, and (iii) the work done by the gas during this process. [1996 - 5 Marks] 62. One mole of a mono atomic ideal gas is taken through the cycle shown in Fig: [1993 - 4+4+2 Marks] A ® B : adiabatic expansion
P
A B
B ® C : cooling at constant volume C ® D : adiabatic compression D
C
D ® A : heating at constant volume. V The pressure and temperature at A, B, etc. are denoted by
PA , TA , PB , TB etc., respectively. Given that TA = 1000 K,
58.
(a) the work done on the gas in the process AB and (b) the heat absorbed or released by the gas in each of the processes. Give answer in terms of the gas constant R. Two moles of an ideal monatomic gas, initially at pressure p1 and volume V1, undergo an adiabatic compression until its volume is V2. Then the gas is given heat Q at constant volume V2. [1999 - 10 Marks]
PB= (2/3)PA and PC = (1/ 3) PA , calculate the following quantities: (i) The work done by the gas in the process A ® B (ii) The heat lost by the gas in the process B ® C. (iii) Thetemperature TD. [Given : (2 / 3)2/5 = 0.85] 63. Two moles of helium gas undergo a cyclic process as shown in Fig. Assuming the gas to be ideal, calculate the following quantities in this process [1992 - 8 Marks]
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EBD_8341 P-116
Physics
鼀
҈鼀
꾀
ᔴ
⧤
⧤
㡰 ᆠ �� �泐 㡰 犨ᨨ├ ᨨ ᆠ㡰 ᆠ �� � �泐
鼀ᨨ矄ᨨ � �矄 æç C p = 䰘 Rö÷ 矄矄盐 PA è ø 鼀盐 TA �矄�矄ᨨ 鼀泐 ᨨ犨�ᆠ�矄 � ��ᨨ盐鼀ᔼ�矄 ᆠᨨ鼀矄矄 ᆠᨨ 矄 矄矄盐ᨨ �矄ᨨ��� ᨨ盐鼀 � 泐�矄�矄ᆠᨨ鼀矄矄 ᆠᨨ 矄 ᨨ盐鼀ᨨ�矄ᨨ��� 矄矄盐PA ҈黨黨҈㜠졔㝐├矄桌 㡰 和├ᆠ P 㜠 V P㜠 T ��鼀矄 ᨨ ᆠᨨ鼀 ᨨᆠ矄矄 㡰 ᔴᆠ盐 犨ᨨ├ ᨨ 泐 �矄 矄盐� ᨨ �矄 盐� � ᆠᨨ鼀ᨨᆠ矄矄
� 꾀 � �矄 � � � �� 矄矄盐 P ᨨ盐鼀 V 鼀盐T�矄ᨨ犨 ᨨ ��ᆠ泐盐 ��矄 T ᨨ盐鼀ᆠᨨ鼀矄�V犨 ��矄鼀盐 矄ᨨ ҈黨黨㜠䰘㝐├桌 �㡰 ᨨ犨鼀 泐 �矄ᨨ ᨨ鼀 ᨨ �矄鼀ᨨᆠ盐矄 䁸 ��㡰 졠� 犨ᨨ├ ᨨ 泐 �矄 盐� � 矄�ᨨ
矄 盐 ᆠ�ᨨ ᨨ � ��矄矄盐P ᨨ盐鼀V 꾀 � �矄 矄 矄ᆠ� �ᆠ ᆠᨨ 矄 矄矄盐 �R �矄�矄├� ᆠᨨ矄 矄矄ᨨ ᨨ盐鼀 CP = 鼀 鼀盐ᨨ ⧤ 矄矄盐ᨨ ҈맔 ҈⏌鼀꾀 鼀ᨨ盐 ᨨ 黨맔҈ᨨ盐矄ᨨ �泐 �矄矄盐� ᨨ �矄ᔴᆠ盐 � 鼀盐 矄矄盐ᨨ �矄 ҈黨䰘㜠䰘㝐├桌 䰘 ᔴᆠ盐 犨ᨨ├ ᨨ 犨 ᨨ 鼀ᨨᨨ ᆠ�矄�矄 ᆠᨨ鼀矄矄 ��ᆠ泐 � ��矄矄盐 ᨨ盐鼀 ᨨ �矄҈�⏌鼀 �矄矄ᆠ�泐 � ᨨ盐鼀ᨨ �矄�矄�㝐ᨨ矄ᆠ� �ᆠ ᨨ �矄 ᆠᨨ 矄 ᨨ盐鼀�矄 ⏌ ҈黨㜠㝐├矄桌
Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics 1 ҈
MCQs with One Correct Answer
꾀 �� ᨨ矄泐├� �鼀ᨨ ᨨ鼀�ᆠ� �矄 ᨨ盐� ᆠ泐ᆠ矄 ᨨ犨 � ��盐 ᆠ � �ᆠ� ᆠ泐ᨨ �� �矄ᆠᨨ矄ᨨ 㝐� 和 ᔼᔼ㡰桌 졠
ᔴ
犨ᨨ� ᔴ ᨨ �� 矄ᨨ� ᆠ矄ᆠ �� 盐泐ᨨ �� �矄盐矄 泐 ᨨ �� ᨨ ᨨ 盐ᆠ犨ᨨ├㡰犨 鼀盐矄 T҈ T 鼀盐ᨨ ᨨ矄ᨨ�ᨨ �矄 �� �矄T҈ 鼀盐ᨨ ᆠᨨ 矄ᨨ�ᨨ 矄ᆠᨨ �� �矄T �矄鼀盐ᨨ 矄� ├ᨨ �矄 �� 犨 �ᆠ �矄矄ᨨ 矄ᨨ盐ᆠ ᨨ 矄ᆠᨨ �� ᨨ犨�矄 T ᨨT҈ T� ᨨ �� 矄 ᨨ鼀⼄盐 鼀ᨨ盐 ᨨ 犨ᨨ├䁸 㝐� 䰘 ᔼᔼ桌
졠 졠
꾀 졠
졠
㡰 ҈黨 㡰 ҈ ᆠ㡰 㡰 꾀 �� �矄� ᨨ 犨� ᆠ �ᨨ ᨨ ҈黨҈� 졔҈� Q 盐� �ᨨ ᆠ泐ᆠᆠ �� �
�ᆠ� ᆠ泐ᨨ �奠 盐ᨨ Q�矄 [Main 2 Sep. 2020 (II)] 㡰 㡰 ᆠ㡰 黨 㡰 ҈ 꾀ᔴ ᨨ �� � � �ᆠ� ᆠ泐ᨨ �矄� �盐矄 ҈ 矄 ��ᨨᔼ 犨ᨨ├ ᨨ ᨨ ��ᨨ�矄҈ 鼀ᨨ盐 ᨨ 矄ᨨ ᨨ鼀 矄ᨨ�ᨨ犨 鼀盐�矄 㝐� ᔼᔼ桌 㡰 黨黨 㡰 ҈ ᆠ㡰 ҈ 㡰 黨
�
T҈T T҈ + T
㡰
T=
ᆠ㡰
T = T҈T
㡰 T =
T҈ + T
㡰 T㦨
꾀 ᔴ ᨨ �� 矄 �ᆠ� ᆠ泐 ᨨ ҈⏌ ␔ 鼀盐ᨨ 矄� ├�矄 盐ᆠ 泐ᨨᔴ�矄 �ᆠ� ᆠ泐 �矄 ᨨ盐 鼀盐矄ᨨ 矄ᨨ盐ᆠ 矄� ├ 矄ᆠ�泐 㝐� ҈꾀҈黨ᔼᔼ桌 㡰 ᨨᔴ҈ᨨᔴ 㡰 黨黨ᨨᔴ䰘ᨨᔴ ᆠ㡰 ҈ᨨᔴᨨᔴ 㡰 䰘ᨨᔴ黨黨ᨨᔴ
ᔴ ᨨ �� 矄ᨨ� 矄�矄犨 矄ᨨ盐ᆠ 鼀盐 ҈ 矄� ├ 鼀盐 矄 ��盐㡰 犨ᨨ ᨨ 矄ᨨ�矄鼀盐 矄矄 ᨨ犨 犨� ҈ > > > 㡰 �� 矄⼄盐泐 �ᆠ� � 㝐� ҈ ҈黨ᔼ桌
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P-117
Thermodynamics
A solid body of constant heat capacity 1 J/°C is being heated by keeping it in contact with reservoirs in two ways: [Main 2015] (i) Sequentially keeping in contact with 2 reservoirs such that each reservoir supplies same amount of heat. (ii) Sequentially keeping in contact with 8 reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature 100°C to final temperature 200°C. Entropy change of the body in the two cases respectively is : (a) ln2, 2ln2 (b) 2ln2, 8ln2 (c) ln2, 4ln2 (d) ln2, ln2 12. A Carnot engine absorbs 1000 J of heat energy from a reservoir at 127°C and rejects 600 J of heat energy during each cycle. The efficiency of engine and temperature of sink will be: [Main Online April 12, 2014] (a) 20奠 and – 43°C (b) 40奠 and – 33°C (c) 50奠 and – 20°C (d) 70奠 and – 10°C 13. p A B 2p 11.
7.
8.
9.
(a)
T2 = ( T1 T4 )
(b)
T2 = T12 T4
(
)
1/3
(c)
T2 = T1 T42
(
)
1/3
(d)
T2 = T13 T4
(
)
1/2
1/4
(
)
; T3 = T1 T42
(
)
; T3 = T12 T4
(
)
(
)
; T3 = T12 T4
; T3 = T1 T43
1/3
1/3
1/3
1/4
Two Carnot engines A and B are operated in series. The first one, A receives heat at T1 (= 600 K) and rejects to a reservoir at temperature T2. The second engine B receives heat rejected by the first engine and in turn, rejects to a heat reservoir at T3 (= 400 K). Calculate the temperature T2 if the work outputs of the two engines are equal: [Main 9 Jan. 2019 II] (a) 600 K (b) 400 K (c) 300 K (d) 500 K A Carnot's engine works as a refrigerator between 250 K and 300 K. It receives 500 cal heat from the reservoir at the lower temperature.The amount of work done in each cycle to operate the refrigerator is: [Main Online April 15, 2018] (a) 420 J (b) 2100 J (c) 772 J (d) 2520 J An engine operates by taking n moles of an ideal gas through the cycle ABCDA shown in figure. The thermal efficiency of the engine is : (Take Cv =1.5 R, where R is gas constant) [Main Online April 8, 2017] (a) 0.24 (b) 0.15
2P0
(d) 0.08
C
B
P P0
(c) 0.32
10.
0
D
A
V0
2V0
V
A Carnot freezer takes heat from water at 0°C inside it and rejects it to the room at a temperature of 27°C. The latent heat of ice is 336 × 103 J kg–1. If 5 kg of water at 0°C is converted into ice at 0°C by the freezer, then the energy consumed by the freezer is close to : [Main Online April 10, 2016] (a) 1.51 × 105 J (b) 1.68 × 106 J 7 (c) 1.71 × 10 J (d) 1.67 × 105 J
p0
D
C
v0 2v0 v The above p-v diagramrepresents the thermodynamic cycle of an engine, operating with an ideal monatomic gas. The amount of heat, extracted from the source in a single cycle is [Main 2013] æ 13 ö (a) p 0 v 0 (b) ç ÷ p0 v0 è2ø æ 11 ö (c) ç ÷ p0 v0 (d) 4p0v0 è2ø 14. Two identical containers A and B with frictionless pistons contain the same ideal gas at the same temperature and the same volume V. The mass of the gas in A is mA, and that in B is mB. The gas in each cylinder is now allowed to expand isothermally to the same final volume 2V. The changes in the pressure in A and B are found to be DP and 1.5 DP respectively. Then [1998S - 2 Marks] (a) 4mA = 9mB (b) 2mA = 3mB (c) 3mA = 2mB (d) 9mA = 4mB 3
Numeric Answer
15. If minimum possible work is done by a refrigerator in converting 100 grams of water at 0°C to ice, how much heat (in calories) is released to the surroundings at temperature 27°C (Latent heat of ice = 80 Cal/gram) to the nearest integer? [Main 3 Sep. 2020 (II)] 16. A Carnot engine operates between two reservoirs of temperatures 900 K and 300 K. The engine performs 1200 J of work per cycle. The heat energy (in J) delivered by the engine to the low temperature reservoir, in a cycle, is _______. [Main 7 Jan. 2020 I]
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EBD_8341 P-118
Physics
8
犨 �ᆠ 矄泐� ⼄盐���盐鼀ᔼ �矄ᆠᨨ ��ᨨ 䰘桌 �矄 ᆠᨨ �矄ᨨ ᨨ鼀 ᨨ�矄
Comprehension/Passage Based Questions
矄矄� 꾀 � 鼀泐ᆠᨨ 盐ᆠ� �ᆠ泐� 矄 �盐矄 �� ᆠ泐� �矄ᨨ �矄ᨨᨨ鼀 矄矄鼀 ᨨ�矄 ᨨ꾀�矄ᨨ ᨨ 鼀矄矄㝐�矄 �矄 ᆠ ᨨ鼀 ᨨ 矄盐 ᆠ矄矄 ᨨ犨 � ��盐 鼀ᨨ矄 �ᆠ矄矄盐�矄
㡰
æ ö÷ çç ÷ 㡰 ÷ çç ÷ ÷ 졔 㝐� è ø
æ 㜠 㝐� ÷ ö ÷ 㡰 ççç ÷ ÷ 㡰 çè ÷ ø
ᆠ㡰
æ ö çç 졔 㝐� ÷ ÷ 㡰 ÷ ç ÷ ÷ çè ø
æ ö÷ ÷ 㡰 㡰 ççç ÷ ÷ çè 㜠 㝐� ÷ ø
�矄ᨨ
�矄ᨨ �矄 ᨨ犨盐 ᨨ盐矄ᨨ犨泐 �矄 ᆠ ᨨ鼀 ᨨ 矄矄盐� ᆠ泐� 犨 �矄 ᨨ �矄ᨨ 犨� 䰘桌
҈䰘
㡰
㡰
҈黨 �矄ᨨ �矄├ ᆠᨨ鼀泐ᨨ盐ᨨ ᆠ泐� ᨨ ᨨ�矄矄 꾀犨 ├�矄ᨨ盐� ᨨ犨 ᆠ泐� 盐� ᨨ矄��ᨨ 矄ᨨ 犨矄盐 ᆠ�
├�矄 矄鼀矄 ᨨᨨ ᆠ泐� 矄 矄 ᨨ犨 � ��盐 矄�泐ᨨ 犨 �矄 r ᔼ
⼄盐���盐鼀 �� ᨨ 犨ᆠᨨ盐鼀 � ᆠ泐� 矄�矄 �矄 䰘桌
㝐� 㝐� 㜠 졔 㡰
ᨨ 矄矄盐� 矄� ᆠ泐� �矄 ᨨ盐� ᨨ盐 矄ᨨ犨盐� �ᨨᆠ矄矄 ␔ � �矄ᨨ �矄 �矄 ᆠ ᨨ鼀 ᨨ ᨨ ᨨ�矄矄 �矄ᨨ �矄 矄 ᨨᨨ矄��ᨨ
ᆠ㡰
҈
㡰 㡰 ᆠ㡰 㡰
r� 㡰졔 㡰졔㦨 r� 㡰 㡰㦨 r� 㡰졔 㡰㦨 r� 㡰 㡰졔㦨
Answer Key Topic-1 : First Law of Thermodynamics ҈ 㡰 㡰 㡰 黨 꾀㜠錐㜠錐ᔴ㜠⼄矄錐㜠
㡰
� 㡰
ᔴᨨ 矄 㡰 䰘 ᆠ㡰
ᆠ㡰
Topic-2 : Specific Heat Capacity and Thermodynamical Processes ҈ ҈҈ ҈ ҈ 黨
㡰 㡰 㡰 㡰 㡰 ᆠ㡰
҈ ҈ �
ᆠ㡰 㡰 㡰 㡰 盐㡰 ᆠ㡰
҈ �҈
㡰 㡰 㡰 㡰 㡰 㡰
҈ �
㡰 㡰 ᆠ㡰 㡰 ᆠ㡰 㡰
� ҈� � � �
㡰 㡰 㡰 㡰 㡰 㡰
҈ � �
ᨨ 盐矄㡰 䰘 㡰 ҈䰘 㡰 䰘 㡰 䰘 㡰 㡰
㡰 ᆠ㡰 ҈ 㡰 ҈҈㡰 ᆠ 㡰
ᆠ㡰 黨 㡰 ҈ 㡰 ҈黨 ᆠ㡰 㡰 黨 㡰 ҈䰘㡰 黨 �㡰 䰘 㡰
ᆠ㡰 㡰 㡰 㡰
Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics ҈ 㡰 ҈҈ 㡰
ᆠ㡰 ҈ 㡰
㡰 ҈ 㡰
㡰 ҈ ᆠ㡰
� 㡰 㡰 䰘 㡰 ҈� 䰘黨҈㡰 ҈ 㡰 ҈䰘 㡰
㡰 ҈ 㡰
黨 㡰 ҈黨 ᆠ㡰
҈ 㡰
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P-119
Kinetic Theory
Kinetic Theory
12
Topic-1 : Kinetic Theory of an Ideal Gas and Gas Laws 1 ҈
ᨨ盐ᨨ ᨨᨨ犨� �犨 �ᆠ ᨨ ᆠᨨᆠ泐矄 矄 㜠 ��鼀䁸 㝐� 졠 � 꾀�黨҈䰘桌
㡰
㡰
MCQs with One Correct Answer
꾀�ᆠᆠᨨ矄 ᆠ泐� �矄矄 � ᨨ犨ᨨ矄泐 �ᆠ�ᨨ 矄矄�矄ᨨ ᨨ 鼀矄矄鼀 ᨨ ���� �ᆠ├ 矄矄
�矄ᨨ �矄 ᨨ鼀ᨨᨨ � � ᨨ ᆠ泐�
� ᨨ ᆠ泐� ᨨ �矄ᨨ �矄l҈ ᨨ犨 �矄ᨨ �矄l矄盐ᆠ l҈>l쟼ᆠ ᨨ ᆠ泐� ᆠᨨ � 矄 鼀ᨨ矄ᨨ � �矄⼄盐鼀盐
ᔼ �矄ᨨ �矄矄�ᨨ 泐�矄鼀矄矄鼀犨��� 泐 �矄盐 �矄�矄ᆠᨨ 矄 ��矄 ᆠᆠ�ᨨ 盐ᨨ ��泐㡰 㝐� ҈ ҈黨ᔼᔼ桌 㡰
é l҈ - l ù ê ú
� ë l҈ I û
㡰
҈
҈ 㦨
�
҈
҈
ᆠ㡰
ᆠ㡰
é l҈ + l ù ê ú � ë l҈ I û
é l҈ - l ù
é ҈ ҈ ù ê ú 㡰 ê + ú � ë l҈ l û � ë l l҈ û
鼀盐ᨨ ᨨ ᨨᨨ鼀ᨨ ᨨ盐鼀鼀� ᆠ矄矄 ᨨ鼀҈䰘 ᔴᨨ䰘 ᔴ 盐ᨨ矄盐 矄 � 鼀ᨨ矄 �ᆠ矄矄盐 � ᨨᨨ鼀鼀� 矄҈맔҈�ᔼ � 盐鼀ᨨ 鼀ᨨᆠ盐矄� ᨨᨨ鼀 ᨨ � � � 犨� ҈䰘桌 㡰 �맔҈� 㡰 �맔҈� ᆠ㡰 ҈҈맔҈ 㡰 ҈맔҈ ᨨ 㜠 ��鼀�� ᨨ � �矄
҈
㡰
犨ᨨ ᨨ 㜠ᆠ�鼀ᨨ ᨨᨨ鼀�ᆠ� �矄矄 �ᨨ鼀�ᆠ 鼀矄矄矄� �ᨨ �ᨨᨨ ��矄矄盐矄 犨 ᆠᨨ矄 � 矄矄├ᆠᨨ 矄 鼀盐 �矄 �ᨨᨨ � 矄��矄�矄 꾀 ҈桌 㡰 ҈ 㡰 ҈ ᆠ㡰 黨 㡰 黨 꾀�矄 矄�├ � �矄� �矄 ҈桌 㡰 矄矄盐 鼀盐ᨨ �� 㡰 矄矄盐 鼀盐ᨨ ᨨ犨 ᆠ㡰 矄矄盐�矄 �� 鼀盐�矄ᨨ犨 㡰 矄矄盐�矄ᨨ犨 鼀盐�矄 �� ␔ �ᆠ ᨨ ᨨᨨ犨� �� 矄ᆠᨨᆠ泐矄 矄 dV ⏌ dP 犨� P ᨨ � �矄 V ᆠᨨ 矄 鼀盐䁸 和桌
��ᨨ ᨨ b = -
ᔴᨨ 矄
㡰
㡰
P
P
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EBD_8341 P-120
Physics
3 ᆠ㡰
䰘
Numeric Answer
ᔼ ��泐 �矄 ᨨ �ᨨ鼀�ᆠ鼀ᨨᆠ盐矄�矄 ᆠᨨ � � ᆠ泐� ᨨ ᨨ盐鼀҈矄矄盐҈ 鼀盐 �⧤꾀矄矄盐鼀� � �奠ᨨ 鼀ᨨᆠ盐矄� �矄矄ᨨᆠ� ᆠ盐矄� �ᆠ �� 盐鼀ᨨ 鼀ᨨ矄 矄矄盐ᨨ 矄盐� ��矄鼀盐⧤犨 ᆠᨨ � � ᨨ盐鼀҈�矄�� 泐 �ᨨ⏌҈�矄힘힘힘힘힘힘 㝐� 和 ᔼ㡰桌 ҈ ᆠ �� 鼀� �盐 ᨨ ᨨ盐鼀ᨨ � �矄 犨 矄鼀 ��ᨨ 矄矄盐 DP �矄 � ᆠᨨ 矄 鼀盐�矄 矄鼀矄 ᆠ �犨 鼀盐�矄 盐ᆠ 泐矄鼀⼄盐 �泐DT ᆠᨨ 矄 矄矄盐 � ��鼀盐 矄矄盐ᨨ �矄 黨
㡰
P P
犨ᨨ� �ᆠᆠᨨ � 矄꾀 犨� �ᆠ�ᨨ 矄矄�矄ᨨ 矄 ᆠᨨ � 矄鼀� �矄 矄鼀鼀盐 矄鼀ᨨ盐鼀 鼀矄矄ᨨ �矄� 꾀�矄鼀꾀 � �矄鼀 �矄� ᆠ ᆠ泐� �矄 ᨨ犨ᨨ犨 ᨨ �矄ᨨ 鼀泐ᨨ 矄鼀 � ᨨ盐鼀 ᆠ �矄� 矄矄盐� 꾀 ᨨ盐 ᨨD ҈�D矄ᆠ�泐
҈黨黨和㜠㝐├矄桌 㡰 鼀꾀㦨黨鼀 㡰 鼀꾀㦨鼀 ᆠ㡰 鼀꾀㦨鼀 㡰 黨鼀꾀㦨鼀
2 Integer Value Answer 꾀 鼀泐�矄ᨨ ᆠ泐� �ᆠᆠᨨ矄 矄矄ᨨ �� 鼀 �矄├�ᆠ泐ᔼ�矄 �� � ᨨ犨ᨨ⼄盐矄泐 � 鼀�ᆠ ᆠ 鼀ᆠᨨ 盐ᆠᨨ㡰 �ᆠ�ᨨ 矄矄��ᨨ
ᨨ 鼀矄矄├� 盐矄 ��ᨨ �矄 � ��泐 �矄 ᆠ ᨨ 鼀 ᨨ鼀 ᨨ矄矄 ᨨ犨 � 矄ᆠ 鼀�ᆠ ��盐ᨨ犨 쟼ᆠ ᨨ 犨ᨨ矄ᨨ 矄矄ᆠᨨ � 矄҈鼀ᨨᨨ
� �矄鼀盐⧤ ��ᨨ �矄 ᨨ犨矄 鼀ᨨ矄犨� ᨨ盐 泐�矄├� � ᨨ鼀ᨨ ᨨ 矄矄 ᨨ ᨨ ␔ ⼄盐���盐鼀 �矄 ᆠ �矄 ᆠᨨ ��ᨨ ᨨ鼀 ᨨ � 鼀㡰犨�힘힘힘힘힘힘힘 ├ ᆠᆠ�ᨨ 盐 ᨨ ��泐㦨҈ 鼀矄 盐 �矄�矄ᆠᨨ 矄 㦨鼀ᨨ҈⧤҈㡰
犨⧤ 鼀矄ᆠ�泐ᔼ DT = C DP
盐ᨨ C� ⧤⏌鼀㡰�矄힘힘힘힘힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌 4
Fill in the Blanks
҈҈ 盐� � �鼀 � �矄�矄 ᨨ盐 ᨨᨨ泐
��ᨨ 犨VP 㦨ᆠᨨ 矄 �矄�矄� ��泐 鼀盐T ᨨ盐鼀V␔ � 矄ᨨᨨ盐鼀 V 鼀盐ᆠᨨ鼀矄 ҈黨䰘㜠㝐├矄桌 5
True / False
҈ ᨨ盐鼀V矄盐矄鼀盐T� 矄 ᨨᆠ�
鼀ᨨ盐 ᨨ ᆠ�矄犨ᨨ矄矄盐p҈ p矄 矄 ᨨ犨 � ��ᔼ ᨨᨨ犨矄 ᨨ鼀 � 矄 p҈�矄� p ҈黨㜠㝐├矄桌 p҈ V
鼀
p
T
Topic-2 : Speed of Gas, Pressure and Kinetic Energy 1 ҈
MCQs with One Correct Answer
盐鼀ᨨ 鼀ᨨᆠ盐矄� ᨨ盐鼀ᨨ ᆠ鼀 ᨨ ᆠ 鼀ᨨ ᨨᨨ鼀�ᆠ�矄矄ᨨ鼀鼀盐T 矄矄盐 ᨨ ᆠ鼀ᨨ 鼀ᆠ盐泐�矄ᆠᨨ矄ᨨ䁸 � 鼀 ├� �ᆠ �泐 ᨨ 鼀ᨨᆠ盐 T㡰�矄맔҈҈�g㦨黨ᆠ鼀⏌矄 矄�泐 ᨨ 鼀ᆠ盐泐㦨҈�⏌ᆠ鼀㡰 㝐� 和� ᔼ㡰桌 㡰 맔҈҈ 㡰 맔҈҈ ᆠ㡰 �맔҈҈ 㡰 �맔҈҈ ᨨ �� �矄 ҈ 鼀 矄矄盐 鼀矄 矄 ᨨ 鼀ᨨᆠ盐矄�矄鼀⏌矄҈䰘 ᔴ꾀鼀矄矄盐 䰘 ᔴ 鼀矄矄 ᨨ 鼀ᨨᆠ盐矄犨� 㝐� 黨꾀�҈黨ᔼ桌 㡰 ҈鼀⏌矄 㡰 � 鼀⏌矄 ᆠ㡰 ҈ � 鼀⏌矄
㡰 鼀⏌矄
ᔼ ҈�矄鼀ᨨᆠ盐矄ᆠ ᨨ 鼀矄矄҈├�ᆠᨨ� 犨� 矄盐 ᆠ �ᆠ盐ᨨ�㡰矄�ᆠ泐矄ᆠᨨ ᨨ ҈鼀犨� 矄 ҈鼀⏌矄 矄矄盐 泐 �矄鼀ᨨᆠ盐矄犨�ᨨ ᨨ ᨨ 㝐� 꾀�҈黨ᔼ桌 㡰 ҈⏌鼀 㡰 ҈⏌鼀 ᆠ㡰 ҈⏌鼀 㡰 ҈҈⏌鼀
鼀盐犨 �ᆠ ᨨᨨ鼀 矄⼄盐ᨨᆠ�泐 ᨨ 泐 ᨨ� 鼀ᨨᆠ盐矄⼄盐矄 �矄ᆠᨨᆠ�泐 ᨨ鼀 �矄ᆠᨨ矄矄ᨨ 㝐� 꾀�҈黨ᔼᔼ桌 ᨨ 鼀
ᔴᨨ 矄 k㦨҈맔҈⏌⧤ 꾀ᨨ� ᨨ盐鼀꾀㦨맔҈⏌├� �盐矄ᨨ 쟼 맔҈鼀 ��ᨨ ᆠᆠ�ᨨ ᨨ 쟼 㦨҈鼀矄桌
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P-121
Kinetic Theory
5.
(a) 800 K (b) 3 × 105 K (c) 104 K (d) 650 K A mixture of 2 moles of helium gas (atomic mass = 4u), and 1 mole of argon gas (atomic mass = 40u) is kept at 300 K in a container. The ratio of their rms speeds
6.
é Vrms ( helium ) ù ê ú is close to : [Main 9 Jan. 2019 I] ë Vrms ( argon ) û (a) 3.16 (b) 0.32 (c) 0.45 (d) 2.24 N moles of a diatomic gas in a cylinder are at a temperature T. Heat is supplied to the cylinder such that the temperature remains constant but n moles of the diatomic gas get converted into monoatomic gas. What is the change in the total kinetic energy of the gas? [Main Online April 9, 2017] 1 nRT (b) 0 2 3 5 nRT nRT (d) (c) 2 2 In an ideal gas at temperature T, the average force that a molecule applies on the walls of a closed container depends on T as Tq. A good estimate for q is: [Main Online April 10, 2015] 1 1 (a) (b) 2 (c) 1 (d) 2 4 At room temperature a diatomic gas is found to have an r.m.s. speed of 1930 ms–1. The gas is: [Main Online April 12, 2014] (a) H2 (b) Cl2 (c) O2 (d) F2 In the isothermal expansion of 10g of gas from volume V to 2V the work done by the gas is 575J. What is the root mean square speed of the molecules of the gas at that temperature? [Main Online April 25, 2013] (a) 398m/s (b) 520m/s (c) 499m/s (d) 532m/s A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40 amu) is kept at 300 K in a container. The ratio of the rms speeds æ vrms (helium) ö [2012] çè v (argon) ÷ø is rms (a) 0.32 (b) 0.45 (c) 2.24 (d) 3.16 The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K is [1999S - 2 Marks]
(a)
7.
8.
9.
10.
11.
(a) 12.
13.
(2 / 7
(b)
(1 / 7)
(c) ( 3 ) / 5 (d) ( 6 ) / 5 A closed compartment containing gas is moving with some acceleration in horizontal direction. Neglect effect of gravity. Then the pressure in the compartment is [1999S - 2 Marks] (a) same everywhere (b) lower in the front side (c) lower in the rear side (d) lower in the upper side A vessel contains 1 mole of O2 gas (relative molar mass 32) at a temperature T. The pressure of the gas is P. An identical
vessel containing one mole of He gas (relative molar mass 4) at a temperature 2T has a pressure of [1997 - 1 Mark] (a) P/8 (b) P (c) 2P (d) 8P 14. The temperature of an ideal gas is increased from 120 K to 480 K. If at 120 K the root-mean-square velocity of the gas molecules is v, at 480 K it becomes [1996 - 2 Marks] (a) 4v (b) 2v (c) v/2 (d) v/4 15. Three closed vessels A, B and C are at the same temperature T and contain gases which obey the Maxwellian distribution of velocities. Vessel A contain only O2, B only N2 and C a mixture of equal quantities of O2 and N2. If the average speed of the O2 molecules in vessel A is v1 that of the N2 molecules in vessel B is v2, the average speed of the O2 molecules in vessel C is [1992 - 2 Marks] 1
v1 + v2 (b) v1 (c) (v1.v2 ) 2 (d) 2 where M is the mass of an oxygen molecule.
(a)
3
3kT M
Numeric Answer
16. Nitrogen gas is at 300°C temperature. The temperature (in K) at which the rms speed of a H2 molecule would be equal to the rms speed of a nitrogen molecule, is _____________. (Molar mass of N2 gas 28 g); [Main Sep. 05, 2020 (II)] 5
True / False
17. The root mean square (rms) speed of oxygen molecules (O2) at a certain temperature T (degree absolute) is V. If the temperature is doubled and oxygen gas dissociates into atomic oxygen, the rms speed remains unchanged. [1987 - 2 Marks] 18. The ratio of the velocity of sound in Hydrogen gas 7 5 ( g = ) to that in Helium gas ( g = ) at the same 5 3 21 temperature is . [1983 - 2 Marks] 5 19. Two different gases at the same temperature have equal root mean square velocities. [1982 - 2 Marks] 20. The root-mean square speeds of the molecules of different ideal gases, maintained at the same temperature are the same. [1981- 2 Marks] 6
MCQs with One or More than One Correct Answer
21. Let v, v rms and vp respectively denote the mean speed. root mean square speed, and most probable speed of the molecules in an ideal monatomic gas at absolute temperature T. The mass of a molecule is m. Then [1998S - 2 Marks] (a) no molecule can have a speed greater than
2 v rms
(b) no molecule can have a speed less than v p / 2 (c) v p < v < v rms 3 2 (d) the average kinetic energy of a molecule is mv p . 4
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EBD_8341 P-122
Physics
Topic-3 : Degree of Freedom, Specific Heat Capacity and Mean Free Path 1 ҈
㝐ᨨᆠ盐矄 ᨨ � �矄 ├ ᨨ犨 ᨨ 矄�ᨨ �矄ᨨ ᨨ鼀 犨ᨨᨨ�ᨨ �矄 ᨨ ᨨ鼀 �矄�矄鼀� � 鼀盐ᨨ
ᨨ� �泐ᨨ 鼀ᨨᨨ �矄�矄 æ Cp ö 盐ᨨ g ç = ÷ �� 矄ᆠ�泐泐 è Cv ø 㝐� 和 ᔼ㡰桌 � 㡰 㦨 g㦨 �
�
䰘 �
� 䰘 ᆠ㡰 㦨 g㦨 㡰 㦨� g㦨 � � ᔼ �盐�矄矄矄盐P 鼀盐T 鼀
�鼀 犨 矄盐ᆠᆠ矄矄�ᆠᨨ�矄�ᨨ 矄 ᨨ 鼀ᨨᆠ盐 �矄 犨� T�矄 㝐� 和 ᔼᔼ㡰桌
҈
҈ ᆠ㡰 㡰 T T T 㝐ᆠ ᔴ⏌ᔴ�ᨨ ᨨ� �矄矄犨� � 泐 ᨨ 鼀ᨨᆠ盐矄 㝐� 和 ᔼ㡰桌 ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀 㜠ᔼᔼ 㝐ᨨᆠ盐Ty ᔴ⏌ᔴ 꾀㡰 㝐ᨨ ᨨ鼀�ᆠ ᔼ㡰䰘⏌� 㡰 �ᨨ鼀�ᆠ��� 鼀ᨨᆠ盐矄 ᔼᔼ㡰黨⏌䰘 ᔴ㡰 �ᨨ鼀�ᆠ ᨨ 㜠��� 鼀ᨨᆠ盐矄 ᔼᔼᔼ㡰⏌ 㡰 �ᨨ鼀�ᆠ��� 鼀ᨨᆠ盐矄 ᔼ㡰�⏌ 㡰 꾀㡰㜠 ᔼ㡰 㡰㜠 ᔼᔼ㡰 ᔴ㡰㜠 ᔼ㡰 㡰㜠 ᔼᔼᔼ㡰 㡰 꾀㡰㜠 ᔼᔼᔼ㡰 㡰㜠 ᔼ㡰 ᔴ㡰㜠 ᔼᔼ㡰 㡰㜠 ᔼ㡰 ᆠ㡰 꾀㡰㜠 ᔼ㡰 㡰㜠 ᔼ㡰 ᔴ㡰㜠 ᔼᔼ㡰 㡰㜠 ᔼᔼᔼ㡰 㡰 꾀㡰㜠 ᔼᔼ㡰 㡰㜠 ᔼᔼᔼ㡰 ᔴ㡰㜠 ᔼ㡰 㡰㜠 ᔼ㡰 ᔴᨨ 矄� �矄ᨨ �ᨨ鼀�ᆠ鼀ᨨᆠ盐矄
鼀ᨨᆠ盐矄 矄矄盐鼀 ᨨ � �盐 鼀 ᨨ 鼀矄矄矄矄��� ᨨ 矄 犨 ᨨ矄 �ᆠ矄 ᨨᆠᆠ盐� 泐 ᨨ鼀矄 � �泐ᨨ 鼀ᨨᨨ �矄鼀盐T�矄 㝐� 和 ᔼ㡰桌
㡰 T
㡰 㦨� g㦨
㡰
꾀�矄鼀� 盐ᆠᨨ 矄�矄矄ᨨ 鼀ᨨ矄ᨨ ᨨ 泐� �鼀ᨨ矄ᨨ �ᨨ 鼀盐T꾀矄矄盐鼀� � �矄矄ᨨ� ᨨ 泐� ᨨ ᨨ��� ᨨ� �泐 � 盐 �矄ᨨ 㡰ᨨ 鼀� 盐�矄 㝐� 和 ᔼ㡰桌 㡰 ҈� 㡰 ҈ ᆠ㡰 㡰 ҈҈ 䰘 꾀 � �矄� ᆠᨨ矄 ᆠᨨ � �矄矄ᨨ犨泐 꾀矄�矄 鼀盐� ᆠ矄矄犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄 盐䁸 㝐� 和 ᔼᔼ㡰桌 ҈㡰 鼀 ᨨ 鼀ᨨᆠ盐矄 ᆠ矄矄 㡰 鼀 ᆠᨨ�矄�ᨨ �鼀 犨 鼀ᨨᆠ盐矄 ᆠ矄矄 㡰 鼀 鼀� 矄盐 ᆠ � 㡰 鼀 ᆠᨨ�矄�ᨨ �鼀鼀� 矄盐 ᆠ � 㡰 㡰 㡰 㡰 ҈㡰 㡰 ᆠ㡰 㡰 㡰 㡰 ҈㡰 㡰 ᔴᨨ 矄� 犨ᨨ� �ᨨ鼀�ᆠ �矄矄꾀 矄ᨨ鼀 鼀盐 㝐ᨨᆠ盐矄ᨨ �矄꾀��� 鼀矄矄 鼀 㝐ᨨᆠ盐矄ᨨ �矄 ��ᨨ m ��ᨨ 鼀ᨨ 鼀矄矄 �ᨨᨨ 矄ᆠ� �ᆠ 矄 ᔴ꾀 ᔴ 㡰 ᨨ �矄꾀 矄ᆠ�泐 �矄 㝐� 黨 ᔼ桌 㡰 䰘黨 㡰 �黨 ᆠ㡰 � 㡰 �䰘 黨
犨ᨨ�矄矄㜠�ᨨ ᨨ鼀�ᆠ �盐矄䰘 鼀ᨨ鼀�ᆠ犨�� 㡰 ᨨ ᨨ鼀�ᆠ �盐矄҈ 鼀ᨨ鼀�ᆠ犨�� ҈㡰 矄鼀 盐鼀 矄�泐 矄鼀 鼀盐 �ᨨᨨ �矄ᆠ�鼀 �鼀矄 �矄ᆠᨨ矄矄ᨨ 㝐� 黨 ᔼᔼ桌 㡰 䰘 㡰 ҈ ᆠ㡰 㡰 䰘 ҈ ᨨ �ᆠ矄 �ᨨᨨ 鼀 �鼀t �鼀犨 犨ᨨ矄盐ᆠᆠ矄矄�ᆠᨨ�矄�ᨨ 矄㡰 ᨨ 鼀ᨨᆠ盐矄 ᨨ � �矄矄 盐 ᆠ�ᨨ ᨨ 鼀盐 T㡰⼄盐��泐 �矄 矄矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ矄ᆠ㡰 㝐� ᔼ桌
MCQs with One Correct Answer
� 黨 RT 㡰 㡰 RT ᆠ㡰 RT 㡰 RT
ᨨ�矄 鼀盐ᨨ ᆠ� 鼀矄矄ᨨ �矄泐� ᔴ ᆠᨨ 矄 矄矄盐҈ᆠᨨ�矄ᨨ �矄⼄盐� ␔ 矄鼀鼀矄矄ᨨ �矄�矄ᆠᨨᨨ 泐҈ ᔴᆠᨨ 矄 ᨨ盐鼀ᆠᨨ�矄ᨨ �矄矄 ᨨ犨鼀 泐 �矄 ᨨ ᨨ鼀 ᨨ矄ᆠ 鼀ᨨᆠ盐ᨨ �矄�矄 矄矄盐鼀 �矄ᨨ� 㡰䁸 㝐� 和 ᔼᔼ㡰桌 㡰 � 㡰 ᆠ㡰 㡰 䰘
t
t
㡰
㡰 ҈ T
T
t
t
㡰
ᆠ㡰 ҈ T
T
҈҈ ᔴᨨ 矄� 鼀� 盐ᨨ n鼀ᨨ矄ᨨ �盐鼀�矄 n鼀ᨨ矄 ᨨ ᨨ 泐� �矄 鼀ᨨᆠ盐矄├ ᨨ��� 㡰矄 � �矄ᔼ矄CP⏌CV盐犨� 㝐� ᔼᔼ桌 㡰 ҈黨⏌҈ 㡰 䰘⏌� ᆠ㡰 ⏌䰘 㡰 ⏌҈�
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P-123
Kinetic Theory
҈
Cp �
犨ᨨ鼀ᨨ矄ᨨ � �矄犨� C = 鼀� 犨� V Cp 鼀ᨨ矄ᨨ ᨨ � �矄犨� C = 盐ᨨ V
Cp
҈
҈
҈�
҈
ᨨ 鼀� 盐�矄 㝐� 䰘 ᔼ桌 CV 㡰 ҈� 㡰 ҈� ᆠ㡰 ҈䰘 㡰 ҈
犨ᨨ鼀ᨨ矄 ᨨ �盐鼀 �矄 �矄 鼀� 犨� 鼀ᨨ矄 ᨨ 泐 ᨨ� 鼀ᨨᆠ盐矄 ├ ᨨ��� 㡰␔ �矄 鼀ᨨ 矄ᆠ� �ᆠ ᨨ 鼀� 盐ᆠᨨ 矄 ᨨ盐鼀䁸 㦨⏌鼀ᨨ⧤㡰 㝐� ҈꾀�҈黨ᔼ桌 㡰 ҈黨䰘⏌鼀ᨨ 㡰 ҈�䰘⏌鼀ᨨ⧤ ᆠ㡰 ҈䰘⏌鼀ᨨ⧤ 㡰 ҈⏌鼀ᨨ⧤ 꾀 �ᨨ鼀�ᆠ�矄犨� ��� 鼀ᨨᆠ盐矄 ᨨ矄҈ᨨ 犨ᨨ├ 犨 ᆠᨨ 矄 矄矄盐␔ 犨ᨨ盐 �泐矄ᨨ 泐 �矄� �矄ᨨᆠ矄矄䁸 㝐� ҈꾀�҈黨ᔼᔼ桌 㡰 � 㡰 � ᆠ㡰 㡰 꾀�맔҈鼀ᨨ盐鼀ᆠ泐� �矄 � 犨� ҈鼀ᨨᨨ 졠 �矄ᨨᨨ鼀鼀盐 ⧤㡰 鼀ᨨᆠ盐 �鼀 ᨨ 졠 �矄ᨨᨨ鼀 矄⼄盐矄 ᨨ盐 ᨨ
鼀 鼀⏌矄矄ᆠ�泐␔ �矄 �ᆠᨨ�矄�ᨨ
矄ᆠᨨ 㡰 ᨨ 졠鼀ᨨᆠ盐䁸 㝐� ҈꾀�҈黨ᔼ桌 㡰 ~҈҈ 㡰 ~҈҈҈ ᆠ㡰 ~҈҈ 㡰 ~҈҈ ␔ 턼 �矄 矄盐� ᨨ �ᨨ鼀�ᆠ �矄 ᨨ ��� 鼀ᨨᆠ盐矄ᆠᨨ 矄 ᨨ盐鼀�矄鼀盐� ᆠ矄矄 泐D ⼄盐� ᨨᨨ 盐ᆠ 矄鼀ᆠ ��
鼀盐ᆠᨨ 矄 矄矄盐�矄 㝐� ҈꾀�҈黨ᔼᔼ桌 � 䰘 Q Q Q 㡰 ᆠ㡰 㡰 Q �
矄ᆠ� �ᆠ 矄 ᔴ p ᔴv ᨨ �矄 ᨨ �ᨨ鼀�ᆠ 鼀ᨨᆠ盐矄꾀�� � 盐 �矄ᨨ 鼀ᨨ҈├҈㡰泐黨 矄ᆠ�泐꾀 ᨨ �矄ᨨ �ᨨ鼀�ᆠ鼀ᨨᆠ盐矄 矄 ᆠᨨ矄ᨨ � �盐矄 ҈ᔼ 泐 矄� �矄矄 㝐� 黨꾀�҈黨ᔼᔼ桌 㡰 꾀�矄��� 盐 矄��ᨨ 鼀ᨨ 㡰 꾀 矄��ᨨ 鼀ᨨ 盐 矄 ᨨ ᆠ㡰 꾀 矄ᨨ ��ᨨ 鼀ᨨ 矄犨ᨨ 㡰 ᨨ 꾀 ��ᨨ 鼀ᨨ ᆠ 꾀 � �矄ᨨᆠᆠ盐�矄ᨨ盐鼀ᨨ 鼀 矄矄盐ᨨ 맔҈ �泐ᨨ �矄 㝐� ҈ ҈黨ᔼ桌 㡰 黨맔҈ 㡰 맔҈ ᆠ㡰 ҈ 㡰 맔҈ 꾀 � �矄�矄 ᆠᨨ矄 � ᆠ泐� 矄矄盐ᨨ 鼀 鼀盐⧤ 鼀 �鼀犨 犨ᨨ 矄盐ᆠᆠ矄矄�ᆠᨨ�矄�ᨨ 矄�矄맔҈矄ᔼ 矄矄盐�矄 ᨨ盐 鼀盐 �矄� ᆠ矄 ᨨ� ⧤ 鼀 �鼀 犨 犨ᨨ矄盐ᆠᆠ矄矄�ᆠᨨ�矄�ᨨ 矄犨��ᆠᨨ矄ᨨ 㝐� ҈ ҈黨ᔼᔼ桌 㡰 맔҈䰘矄 㡰 맔҈矄 ᆠ㡰 �맔҈矄 㡰 맔҈矄
㡰 ҈䰘
҈
҈黨
꾀�矄鼀� 盐ᆠᨨ 矄�矄矄ᨨ 鼀ᨨ矄ᨨ ᨨ 泐� �鼀ᨨ矄 ᨨ �ᨨ 鼀盐 ᔴᨨ 矄� � �ᨨ 泐 矄�ᨨ ᨨ�ᨨ 鼀ᨨ 矄 ᨨ � �泐ᨨ 矄泐矄鼀�矄 㝐� ҈҈ ҈黨ᔼ桌 㡰 ҈�
㡰 ҈
ᆠ㡰
㡰
҈ 犨ᨨ├�ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ�矄�矄矄矄盐ᨨ 맔҈ ⏌鼀 矄�泐 ᨨ �矄 �矄 ├�⏌鼀 ␔ �矄 ᨨ ᨨ �泐ᨨ �矄 盐 ᨨ�矄 鼀 鼀ᨨ�ᨨ 䁸 㝐� ҈ ҈黨ᔼᔼ桌 㡰 ҈ 㡰 ҈� ᆠ㡰 ҈ 㡰 ҈ 꾀 ҈� � 鼀矄矄 ᨨ �ᨨ� �矄 �矄 ᆠᨨ矄 � 矄矄 鼀盐䰘 ᔴ꾀鼀ᨨ盐 ᨨ 矄 ᨨ �矄 矄ᨨ 鼀矄 ᨨᆠ�泐 ᨨ 鼀ᨨᆠ盐矄 �矄 ᨨ盐 �矄 ᨨ盐 ├ 㦨⏌⧤鼀ᨨ桌㝐� 黨 ҈黨 ᔼᔼ桌 㡰 黨├ 㡰 ├ ᆠ㡰 ҈├ 㡰 ҈├ 犨ᨨ鼀ᨨ矄ᨨ � 鼀ᨨ ᨨᨨ鼀�ᆠ�矄ᨨᆠᆠ盐�矄ᨨ盐鼀 䰘 ᔴ �矄 矄 ��ᆠ泐ᨨᨨ盐鼀 ᔴᆠ盐 ҈㡰 � 鼀盐ᨨ �矄 㡰ᆠ � � �矄� �泐 㝐� ҈桌 㡰 ҈㡰҈黨⧤ 㡰 䰘├ 㡰 ҈㡰҈黨�⧤ 㡰 䰘├ ᆠ㡰 ҈㡰҈黨⧤ 㡰 䰘├ 㡰 ҈㡰҈黨�⧤ 㡰 䰘├ 犨ᨨ鼀ᨨ矄 ᨨ �盐鼀 鼀� 犨� 犨� 鼀ᨨ矄 ᨨ ᔴ 泐 ᨨ� ᔼ = ᨨ 鼀� 盐 盐ᨨ
ᔴ �矄 㝐� 졠 � 꾀�҈҈桌 㡰 ⏌ 㡰 ᆠ㡰 ҈ 㡰 � ᔴ ᔴ 矄ᆠ� �ᆠ 矄 ᆠᨨ 矄 矄矄盐 ᆠᨨ 矄 ᨨ盐鼀矄ᆠ�泐ᔼ�矄ᨨ矄 ᔴᔴ㦨 ᨨ 泐 ᨨ� �矄 ᔴᔴ㦨 ᨨ �ᨨ� �矄
ᆠᨨᆠ�ᨨ 犨 �矄 㝐� ҈䰘桌 㡰 㦨҈ 㡰 㦨 ҈ b 㡰 㦨 ҈ 꾀 � �矄 矄鼀ᨨᆠ盐矄犨� � �矄ᨨ ᨨ鼀
�ᨨᨨ 矄ᆠ� �ᆠ 矄ᆠᨨ 矄 矄矄盐 ᔴ㡰 ᆠᨨ 矄 ᨨ盐鼀 ᔴ㡰�矄 㝐� 졠 � 꾀�҈䰘桌
ᆠ㡰 a =
䰘 䰘 � ᆠ㡰 㡰 � 䰘 꾀 � �矄盐 �ᨨ矄⼄盐矄�矄�ᆠ矄�ᨨᆠ矄矄 � 犨 �ᆠ �矄鼀ᨨ ᆠᆠ�泐ᔴ鼀� 矄ᆠᨨ 矄 ᔼ 盐� � �矄ᨨᆠ矄矄 �ᨨ ᨨ 矄矄盐 ᨨ盐鼀 �矄�� 泐 㦨ᆠᨨ 矄 �矄�� 泐 ᔴ ᔴ鼀ᨨ矄ᆠ� �ᆠ ᆠᨨ 矄 矄矄盐 ᆠᨨ 矄 ᨨ盐鼀矄ᆠ�泐㡰 㝐� ҈桌 ᔴ - ᔴ ᔴ - ᔴ 㡰 = 㡰 = ᔴ - ᔴ ᔴ - ᔴ
㡰
ᆠ㡰 =
㡰
ᔴ ᔴ
㡰 =
ᔴ ᔴ ᔴ ᔴ
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EBD_8341 P-124
黨
Physics
矄� � ⼄盐���ᨨ ᨨ �泐 矄ᆠ� �ᆠ � ├�҈⧤҈㡰ᨨ 盐鼀� �盐鼀ᨨᨨ鼀鼀盐ᆠ 矄�鼀 ᨨ ᨨ鼀�ᆠ犨�� ᨨ 盐鼀� �盐鼀㦨䰘㡰 㝐� 졠 � 꾀�҈҈ ҈�桌 㡰 ҈ 㡰 � ᆠ㡰 ҈� 㡰 黨� 㝐ᨨ ᆠ盐盐鼀盐鼀矄ᆠ ᆠ盐矄矄 ᨨ犨 ᨨ 矄矄盐ᨨ 맔҈҈�鼀ᨨᨨ鼀鼀盐 ⧤㡰
├� � 㦨⧤҈鼀ᨨ҈҈鼀㦨҈� 꾀ᨨ� ᨨ 㦨맔҈鼀ᨨ҈ 鼀 �矄 ᆠ犨 鼀ᨨᆠ盐矄 ᨨ �矄� ᆠ盐 矄矄犨�ᨨ ᨨ ᨨ 㝐� 졠 � 꾀�黨҈桌 㡰 m鼀 㡰 鼀鼀 ᆠ㡰 ᆠ鼀 㡰 鼀 ��盐矄 ᨨ犨矄 ��ᨨ � 鼀盐 D 㡰犨� 鼀ᨨ盐 ᨨ 矄盐� 턼㡰 � �矄ᨨ�ᆠ ᨨᆠ矄矄 ᆠᨨ矄ᨨ � �ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ 㝐㡰 �ᨨ鼀�ᆠ 㡰 ᨨ泐ᨨ鼀�ᆠ 㡰�矄 � ��矄ᨨ �矄矄 矄鼀 矄ᆠ矄 ᨨ 犨ᨨ 矄ᆠᨨ� ᆠ� ᔼ� ᨨ� � ��ᨨ �矄 ᨨ ᨨ鼀 � 矄 a b c 矄ᆠ�泐ᆠᨨ矄ᨨ ᨨ 㝐� 졠 � 꾀�黨҈桌 a
3
b
턼
c D
㡰 㝐 ҈
㡰 ҈ 㡰 ҈� ᆠ㡰 ҈� 㡰 䰘 � 꾀矄矄ᆠᨨ � 矄鼀� 盐ᨨ ᨨ 鼀ᨨᨨ ᨨ 泐� 犨ᨨ 鼀ᨨ矄 ᨨ �ᨨ� ⧤ �ᨨ ᨨ � ᨨ�ᨨ ├� �ᆠ �泐졠鼀ᨨᆠ盐ᨨ 鼀ᨨᆠ盐�矄 ҈黨黨和㜠㝐├矄㡰 㡰 ҈҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 矄 ᨨ 鼀ᨨ鼀 矄 ᨨ � � ᨨ 犨ᨨ 鼀ᨨᆠ盐矄 犨ᨨᆠ泐� 矄A B � 犨� �矄ᨨ 矄ᆠᨨ � ⼄盐 鼀ᨨ盐 矄ᨨ � �ᨨ鼀�ᆠ�矄⧤ �矄ᨨ ᨨ A �矄 ᨨ鼀ᨨ犨 � B�矄 � 矄鼀鼀ᨨ盐 ᨨ �矄�� ᨨ �矄� ᆠ ᆠ泐� ᔼ �矄�
鼀盐 ᨨ �矄 � A �矄 ⧤ �矄 �
鼀盐ᨨ �矄� B�矄 ҈黨黨和㜠㝐├矄桌 ᆠ㡰 �⧤ 㡰 ⧤ 㡰 ⧤ 㡰 ҈⧤ 䰘 ␔ � �ᨨ鼀�ᆠ�矄�矄 ᆠᨨ 矄 矄矄盐 ᆠ�ᨨ ᨨ �泐矄盐� 犨 �ᆠ � ᆠ矄矄 � �泐ᨨ �矄�矄 ҈黨黨㜠㝐├矄桌 � 㡰 㡰 ᆠ㡰 㡰 � � 䰘 䰘
㡰 㝐
ᆠ㡰 㝐 㡰 㝐 꾀�矄鼀� 盐ᆠᨨ 矄�矄矄ᨨ 鼀ᨨ矄ᨨ ᨨ 泐� 鼀ᨨ矄ᨨ �ᨨ 鼀盐 �ᆠ� ���ᨨ 鼀ᨨ 矄 ᨨ� �泐ᨨ 矄泐矄鼀�矄 ҈黨黨黨和㜠㝐├矄桌 ᆠ㡰 黨
㡰 ҈҈
㡰
㡰 ҈�
� 矄�ᨨ ├� �ᆠ �泐ᨨ 졠 � 鼀ᨨ鼀矄矄㡰鼀ᨨᆠ盐矄�ᆠ盐鼀盐�矄 矄�ᨨ ├� �ᆠ �泐ᨨ � 鼀ᨨ鼀矄矄㡰鼀ᨨᆠ盐矄� 矄鼀鼀盐�矄 ҈黨黨䰘㜠҈㝐├桌 㡰 ҈� 㡰 ᆠ㡰 㡰 䰘 ᨨ鼀 ᨨᨨ犨� �矄鼀 矄ᆠᨨ ᆠ � �� �矄 泐 �� 鼀盐 矄ᆠ ᆠᨨᆠᨨ 矄㡰҈黨黨�和桌 㡰 ᆠᨨ �ᆠ� ᨨ ᨨ盐鼀 矄�ᨨ ᆠᨨ 矄 矄矄盐�矄 矄鼀 ᨨ� �矄矄 㡰 � 矄�ᨨ ├� �ᆠ �泐鼀ᨨᆠ盐 ᨨ ᨨ 泐� �矄�矄├ ├� �ᨨ 鼀
ᆠᨨ 矄 ᆠ㡰 鼀 㜠 ᨨ 鼀ᨨᆠ盐矄 � ᆠ矄矄 犨� � ᆠ矄矄� 矄矄盐 㡰 ᔼ �矄ᨨ盐矄鼀� 盐 � 矄�ᨨ ├� �ᆠ �泐ᨨ 鼀ᨨᆠ盐矄ᨨ ᆠ ᆠᨨ鼀ᨨ �矄 � �ö æ ᔼ ᨨ 鼀ᨨᨨ 鼀ᨨ ᨨ鼀�ᆠ�矄 ç g = ÷ �矄鼀� 犨� ᨨ ø è 䰘ö æ 鼀ᨨᨨ �ᨨ鼀�ᆠ�矄 ç g = ÷ 盐ᨨ g ᨨ鼀� 盐 �ø è �矄 ҈黨㜠҈㝐├桌
Numeric Answer
꾀ᆠᨨ矄 矄矄ᆠᨨ � 矄҈鼀ᨨᨨ 鼀ᨨ ᨨ鼀�ᆠ� �矄⧤ᔼ �鼀ᨨᨨ 矄鼀�矄⧤�矄 ᨨ� � ⼄盐���盐鼀鼀盐 � ⧤㡰ᨨ �矄�
矄矄犨�ᆠᨨ矄ᨨ힘힘힘힘힘힘힘힘힘 㝐� 和 ᔼ㡰桌 4
Fill in the Blanks
黨 졠 鼀ᨨᨨ 鼀ᨨ ᨨ㜠ᨨ鼀�ᆠ� �矄�矄鼀� 犨� ᨨ 鼀ᨨᨨ �ᨨ鼀�ᆠ� �矄 鼀ᨨ矄ᆠ� �ᆠ ᨨ 鼀� 盐ᆠᨨ 矄 ᨨ盐鼀�矄 ҈黨㜠㝐├矄桌 5
True / False
꾀�� 鼀盐 矄ᆠ� �ᆠ ᨨ �矄ᆠᨨ 矄 矄矄盐�矄犨泐矄� �矄矄ᆠ� �ᆠ ᆠᨨ 矄 ᨨ盐鼀 ҈黨䰘㜠㝐├矄桌 6
MCQs with One or More than One Correct Answer
҈ 꾀ᆠᨨ � ᨨ � ᨨ盐鼀 矄鼀� 盐ᨨ ᨨ 鼀ᨨᨨ 泐 ᨨ� ᨨ 鼀ᨨ ᨨ �盐鼀 � ⼄盐���盐鼀 鼀盐T꾀矄矄盐鼀� � �矄矄� ᆠᨨᆠ 矄鼀 矄㡰�矄 㡰 꾀 ҈�桌 㡰 � �泐鼀ᨨᨨ �矄鼀� 盐�矄RT 㡰 �ᨨᨨ 矄 ᨨ 矄ᨨ盐 � �矄鼀� 盐ᨨ ᆠ㡰 㡰
� �盐鼀�矄�矄 ⏌ �
�ᨨᨨ 鼀矄矄 ᨨ �盐鼀ᨨ鼀矄ᨨ ᨨ 泐 ᨨ� 鼀ᨨᆠ盐矄�矄҈⏌
�ᨨᨨ 鼀矄矄 ᨨ �盐鼀ᨨ鼀矄ᨨ ᨨ ҈ 泐 ᨨ� 鼀ᨨᆠ盐矄�矄
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P-125
Kinetic Theory
42.
Cv and Cp denote the molar specific heat capacities of a gas at constant volume and constant pressure, respectively. Then [2009] (a) Cp – Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas (b) Cp + Cv is larger for a diatomic ideal gas than for a monatomic ideal gas (c) Cp / Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas (d) Cp . Cv is larger for a diatomic ideal gas than for a monoatomic ideal gas 9
43.
the product of its pressure and its volume. [2007] because Statement-2 : The molecules of a gas collide with each other and the velocities of the molecules change due to the collision. (a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1 (b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (c) Statement-1 is True, Statement-2 is False (d) Statement-2 is False, Statement-2 is True
Assertion and Reason Type Questions
Statement-1 : The total translational kinetic energy of all the molecules of a given mass of an ideal gas is 1.5 times
Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 1.
2. 3.
Two moles of ideal helium gas are in a rubber balloon at 30°C. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35°C. The amount of heat required in raising the temperature is nearly (take R = 8.31 J/mol.K) [2012] (a) 62 J (b) 104 J (c) 124 J (d) 208 J An ideal gas is expanding such that PT2 = constant. The coefficient of volume expansion of the gas is – [2008] (a) 1/T (b) 2/T (c) 3/T (d) 4/T When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied which increases the internal energy of the gas is [1990 - 2 Marks] 2 3 3 5 (a) (b) (c) (d) 5 5 7 7 3
4.
M1
MCQs with One Correct Answer
Numeric Answer
One mole of a monatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is 100 K and the universal gas constant R = 8.0 J mol –1 K –1, the decrease in its internal energy, in Joule, is__________.
M2
a
S
6 6.
MCQs with One or More than One Correct Answer
As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate (KMnO4) of different concentrations n1 and n 2 (n 1 > n 2 ) molecules per unit volume with Dn = (n1 - n2 ) F4 㡰 F 1> F2 > F 3 >F4 ᆠ㡰 F 1= F2 = F 3 㦨F4 㡰 F 1> F 3 ; F 2 F4
��ᨨ 犨 犨ᨨᆠᨨ ᆠ �ᆠ矄 矄ᨨ ��ἘἘ ἘἘ矄ᆠ�泐 矄 ��盐㡰 ᨨ盐鼀ᆠ � 矄�泐 꾀 㦨 犨 꾀�矄ᆠᨨ 矄 �矄 �矄 ᆠ ᨨ鼀 ᆠ 꾀 ᆠ ᨨ 矄 矄�矄ᨨ� ᆠ � 턼 盐ᨨ 꾀矄盐ᆠ ᆠ�ᆠ � � ��ᨨ
㝐� ҈桌 犨 矄 矄犨�ᆠᨨ 矄 �矄
㡰
҈ p҈ p ҈ p҈ p 㡰 pe x pe x ҈ p҈ p ҈ p҈ p ᆠ㡰 㡰 pe x pe x r ҈҈ ᔴᨨ 矄� ᆠ�ᆠ � E = E xˆ 犨 쟼�矄ᆠᨨ 矄
盐 ᨨ盐� 矄 矄矄 ᨨ犨 � ��盐㡰 盐ᨨ �矄 � �矄 ҈҈桌 㡰 Ea aa㡰
㡰
L x
D a
C
a y
O A
B
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P-161
Electric Charges and Fields
3 ҈�
Numeric Answer
ur 꾀 ᆠ�ᆠ � 쟼 = x�ˆ - y + ҈㡰 ˆ ⏌ᔴ 矄矄矄 ᨨ盐� ᨨ 矄 ᨨ犨 � ��盐 盐 ᨨ ᆠ�ᆠ � ᨨ盐� 矄盐 ᆠ矄꾀ᔴ ᔴ 鼀├ 矄f҈ f҈҈矄ᆠ�泐 � ᆠ犨 f҈f҈҈㡰�矄 �
鼀⏌ᔴ㡰힘힘힘힘힘힘힘 㝐� 黨 ᔼᔼ㡰桌 꾀 㡰 㡰
泐
҈
҈䰘
҈
㡰
㡰 ᆠ�ᆠ 盐 ᨨ盐� 矄 �矄 l ⏌Î 㡰 㜠ᆠᨨ鼀ᨨ ᨨ ᆠ�ᆠ � �矄 ᨨ ᨨ� 矄ᨨ 矄盐 ᆠᨨ 矄 ᆠ㡰 ᆠ�ᆠ 盐 ᨨ盐� 矄 �矄 l ⏌Î 㡰 ᆠ�ᆠ � �矄 ᨨ鼀ᨨ 矄盐 ᆠᨨ 矄 ᨨ� 矄 ҈黨 꾀ᨨ� ᆠ �졔턼�矄ᆠ 盐矄ᨨ盐矄� �鼀�� 泐 鼀�矄 �ᆠ矄盐 ᆠᨨ �盐矄 矄矄 ᨨ犨 � ��盐 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄⏌ᆠᨨᆠ䁸 턼 꾀 ҈䰘桌
ᔴ
㡰 쟼 㡰 㡰 㡰 㡰
꾀ᆠ�ᆠ盐 �矄ᆠᨨ �盐矄Rᆠ�矄矄盐 ᆠᆠ � 矄�泐 rö æ s r 㡰 = s ç҈ - ÷ 犨 s �矄ᆠᨨ 矄 r�矄 è Rø �矄 ᆠ ᨨ鼀 ᆠ ᨨ �矄ᆠ쟼ᆠ�ᆠ 盐 ᨨ盐� � 矄 �ᆠ 矄盐 ᆠ ᆠᨨ矄矄 ᆠ � �矄ᆠ ᆠᨨ鼀泐�矄f쟼ᆠ�ᆠ 盐 ᨨ盐� ᨨ 矄 �ᆠ R 矄盐 ᆠᨨ �盐矄 ᆠᨨ ᆠ �ᆠ犨� �矄ᆠ�矄f f
�ᨨ �矄힘힘힘힘힘힘힘힘힘 꾀 桌 f 6 MCQs with One or More than One Correct Answer 꾀ᆠ � 矄 ᨨ �盐矄 ᆠ�矄ᨨᆠ �턼�
f矄 盐 ᨨ ᆠ�ᆠ � ᨨ盐� ᆠᨨ矄 ᆠ泐� �ᆠ 矄盐 ᆠᨨ �� h �盐矄r 犨� �矄ᆠ 矄鼀矄 ᨨ 矄 ᆠ ᨨ ᆠ泐� �矄ᨨ� ᨨ
�矄ᨨ ᆠ泐� 犨 �ᆠ �矄⼄盐� �矄 ᨨ鼀�矄ᨨ ᨨᨨ鼀 矄盐 ᆠ矄␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄㡰 꾀 ҈黨桌 �矄⏌ᆠᨨᆠ䁸 e�矄 鼀���泐ᨨ 矄ᆠ桌 㡰 ᔼ h>R r 㦨R⏌� f㦨턼⏌�e 㡰 ᔼ h>R r>R f㦨턼⏌e ᆠ㡰 ᔼ hR⏌� r㦨R⏌� f㦨 㡰 ᔼ h>R r>R / 5 f㦨턼⏌�e 꾀 � � �泐ᨨ � � ᨨ 㜠ᆠᨨ 盐ᆠ� �犨��矄ᨨ 㜠 �矄 ᆠ�矄盐 � ᨨ鼀� ᆠ � 矄�泐lᔼ �ᆠ矄 � ᨨ 㜠ᆠᨨ 盐ᆠ� �矄 �ᆠ矄 ᨨ �盐矄 � 矄盐ᆠ 犨泐 ᆠ턼矄盐 矄 �҈ ᆠ 졠ᨨ 矄 �ᆠ矄 矄矄 ᨨ犨 � ��盐
鼀���泐 ᨨ 矄ᆠ �矄 Î ␔ �ᆠ ᨨ 꾀 ҈桌 ᨨᨨ犨� �矄鼀 矄�矄 㡰盐䁸
l
㡰
ᆠ�ᆠ 盐 矄矄� � ᨨ盐� ᆠ盐 矄盐 ᆠ
턼 æ ҈ ö ç҈ ÷ e è ø 㡰 ᨨ 盐 ᨨ盐� ᆠ盐 矄盐 ᆠ矄 턼 �矄 e ᆠ㡰 ᆠᨨ鼀ᨨ ᨨ ᆠ�ᆠ � ᨨ鼀ᨨ 矄盐 ᆠ�矄ᆠᨨ 矄 ᨨ 矄盐 ᆠ 㡰 ᆠ�ᆠ盐鼀 ᆠᨨ 矄盐 ᆠ�矄 ⼄盐�ᨨ �
ᨨ 鼀�矄 �矄 -
꾀 ᆠ盐�ᆠ ��ᨨ ᨨ 矄� a 矄 �矄ᆠ ᨨ��� ᔼ ᆠᨨ矄矄 � ᨨ� ᆠ �矄-q -a⏌㡰졔q 㡰 -q 졔a⏌ 㡰 ᔴ ᨨᨨ矄 ᆠᨨᆠ ᨨ�ᨨ 矄 矄㡰 ҈桌 z a q
q
q
x
㡰 ᆠ�ᆠ 盐 ᆠᨨ矄矄� � x 㦨 졔a⏌ �矄⼄盐ᨨ ᆠ�ᆠ 盐 ᆠᨨ矄矄� � x㦨-a⏌ 㡰 ᆠ�ᆠ 盐 ᆠᨨ矄矄� � y㦨졔a⏌�矄鼀ᨨ ᆠ�ᆠ 盐 ᆠᨨ矄矄� � y㦨-a⏌ q ᆠ㡰 ᆠ�ᆠ 盐 ᆠᨨ矄矄� � ���ᨨ �矄 e 0 㡰 ᆠ�ᆠ 盐 ᆠᨨ矄矄� � z 㦨졔a⏌�矄⼄盐 ᨨ ᆠ�ᆠ 盐 ᆠᨨ矄矄� � x㦨졔a⏌ ҈ 꾀 �矄ᨨ� ᆠ�泐�矄ᆠ 犨� � ᆠᆠᨨ 盐ᆠᨨ 꾀ᨨ矄��ᆠ �q�矄ᆠ ᆠ ᨨ ᆠ�泐
ᨨ� 矄A Bᨨ ᆠ�泐矄盐 ᆠ矄矄 ᨨ犨 �
��盐
҈黨黨黨和㜠㝐├矄桌
ᨨ 졠 ҈
턼
A
q
B
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EBD_8341 P-162
Physics
(a) electric field near A in the cavity = electric field near B in the cavity (b) charge density at A = charge density at B (c) potential at A = potential at B (d) total electric field flux through the surface of the cavity is q/e0
ur 22. The magnitude of electric field E in the annular region of a charged cylindrical capacitor. [1996 - 2 Marks] (a) is same throughout (b) is higher near the outer cylinder than near the inner cylinder (c) varies as 1/r, where r is the distance from axis (d) varies as 1/r2 where r is the distance from the axis.
Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 1.
2.
Dl 鼀ᨨ�ᨨ ᨨ P�矄҈黨黨和㜠㝐├矄桌 㡰 �ᨨ �ᆠ ᨨ盐矄ᨨ z矄�矄 泐� �z¥ 㡰 矄�鼀 鼀ᨨ �ᆠ ᨨ盐矄ᨨ z矄�矄 泐� �z£R ᆠ㡰 ᨨ �鼀泐矄�鼀 鼀ᨨ �ᆠᨨ� zR 㡰 矄盐ᆠ Pᆠᨨ矄矄矄O ᆠᨨ � 盐矄ᨨ鼀ᨨᨨ � ��z �矄ᨨ犨 矄 z = - ¥
Answer Key Topic-1 : Electric Charges and Coulomb's Law ҈ 㡰 ҈ ҈°
㡰
KQ L
㡰 ҈҈
㡰 矄 ҈
ᆠ㡰
� 㡰
㡰
盐 ҈ ᆠ㡰 ҈
㡰 ҈�
䰘 㡰
㡰
黨
҈ q 㡰 pÎ L
㡰
Topic-2 : Electric Field and Field Lines ҈ ҈҈ ҈ ҈ ҈
ᆠ㡰 㡰 㡰 矄 㡰
҈
ᆠ㡰 㡰 㡰
盐 㡰
҈
ᆠ㡰 㡰 ᆠ㡰
盐 㡰
҈
ᆠ㡰 � 㡰 㡰 㡰 ҈� 㡰 ҈ ᆠ㡰 ᆠ㡰 � 㡰 ᆠ㡰 ᆠ㡰 � 㡰 㡰 ᆠ㡰
䰘 ҈䰘 䰘 䰘
ᆠ㡰 㡰 ᆠ㡰 ᆠ㡰
҈
㡰 黨 ᆠ㡰 ҈黨 㡰 黨 㡰 黨
㡰 㡰 㡰 㡰
҈
ᆠ㡰 㡰 ⼄쟼꾀㡰 ᆠ㡰
Topic-3 : Electric Dipole, Electric Flux and Gauss's Law ҈ 㡰 㡰 ҈҈ ᆠ㡰 ҈ 㡰 ҈ ᆠ 㡰 ᆠ㡰
㡰 ҈ 㡰
㡰 ҈ 㡰
� 㡰 㡰 䰘 㡰 ᆠ㡰 ҈� 㡰 ҈ 㡰 ҈䰘 ᆠ㡰 ҈ 㡰
黨 ᆠ㡰 ҈ ᆠ㡰 ҈黨 㡰 ᆠ 㡰
Topic-4 : Miscellaneous (Mixed Concepts)Problems ҈ 㡰
㡰
㡰
҈㡰
� 矄
ᆠ㡰
䰘 㡰
ᆠ㡰
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EBD_8341 P-164
Physics
16
Electrostatic Potential and Capacitance
Topic-1 : Electrostatic Potential and Equipotential Surfaces 1 ҈
r
ᆠ �矄ᆠ ᨨ ᆠ�ᆠ盐鼀 ᆠᨨ ᆠ�ᆠᨨ �盐矄 R 犨� ᆠᨨ 矄 �盐 矄�ᨨ 犨
矄盐ᆠᆠ矄矄�ᆠ �矄꾀 ᆠ �矄҈�䰘黨 ᆠ � 졔q㡰ᆠ 犨 �҈ ᆠ � q㡰ᆠ
ᨨ �V ᆠ�ᆠ � E ᆠ ᨨ ᆠ�ᆠ矄ᆠ�泐 ├V㦨� � �泐㡰 㝐� 和� ᔼᔼ㡰桌 ҈q 錐 E = 㡰 V = pe R ҈ q 㡰 V = 錐 E = pe R ᆠ㡰 V㦨錐E㦨 ҈ q ҈ q 㡰 V = 錐E= pe R pe R
犨ᨨ�矄ᨨ ᆠᨨ 盐ᆠ� �矄 矄S҈ Sᨨ �盐矄 R ҈ R ҈mᔴ mᔴᆠ �矄矄ᆠ�泐 � �矄 ᆠ ᨨ鼀ᆠ ᨨ 泐 ᨨ犨ᆠᨨ
ᆠ 泐 ᆠᨨ 盐ᆠ� �犨�꾀ᨨ ��鼀 �矄�矄 ᨨ ᆠ �矄ᨨ
S҈ S矄ᆠ�泐 [Main Sep. 03, 2020 (I)] 㡰 �mᔴᨨ ᨨ 㡰 졔�mᔴ �mᔴ ᆠ㡰 mᔴ mᔴ 㡰 mᔴ mᔴ ᔴᨨ ᆠ �ᆠ鼀�ᆠ ᨨᨨ犨矄 矄ᨨ ��R R ᨨ ᆠ �矄Q҈ Q矄ᆠ�泐� 矄盐 ᆠᆠ � 矄��矄ᨨ ᆠᨨ ᆠ �ᆠ矄 矄⼄盐 ᨨ � � ᆠV R㡰V R㡰�矄 㝐� 和 ᔼᔼ㡰桌 㡰
MCQs with One Correct Answer
Q Q Q҈ 㡰 ᆠ㡰 pe R pe R ҈pe R
㡰
Q҈ pe R
꾀ᆠ �Q�矄 �矄�盐 ᨨ犨ᨨᆠᨨ ᆠ �ᆠᆠᨨ 盐ᆠ� � � 矄 �ᆠ矄 矄 ��r R R r㡰ᔼ 矄盐 ᆠᆠ � 矄��矄ᨨ 犨ᨨ矄 矄⼄盐 ᆠ�ᆠᨨ � ᆠᨨ鼀鼀ᨨ ᆠ �矄 㝐� 和 ᔼᔼ㡰桌
R
҈ R + r 㡰 ҈ R + r 㡰 Q Q 㡰 pe R + r 㡰 pe R + r 㡰 ҈ R + r 㡰 ҈ R + r 㡰Q Q ᆠ㡰 㡰 pe R + r 㡰 pe R + r 㡰 꾀盐 � ᨨ鼀泐ᆠ � � �ᨨ �盐矄 ᨨᆠ �⼄ �矄ᆠ � 泐㜠 ᆠ ᨨ��� 꾀ᨨ� ᆠ �⼄�矄 鼀ᨨ� �ᨨ犨 矄 � �ᨨ � 㜠 �矄 矄矄 㦨 鼀� �鼀盐鼀盐ᨨ 矄盐ᆠ �ᆠᨨ矄矄矄 ᨨ��� �矄 㝐� ҈꾀�҈黨ᔼ桌
㡰
�
҈⏌
㡰
æ ⼄ ö ç ÷ 鼀 è ҈� pe ø
ᆠ㡰
æ ⼄ ö ç ÷ 鼀 è ҈� pe ø
҈⏌
㡰
æ ҈ ⼄ ö ç ÷ 鼀 è � pe ø
㡰
æ ҈ ⼄ ö ç ÷ 鼀 è ҈� pe ø
҈⏌
䰘
҈⏌
꾀 矄ᨨ� ᆠᨨ 盐ᆠ� � 矄 � � ᆠ � 턼 �矄 矄盐ᨨ盐 泐 盐 ᆠ � ᆠᨨ 盐ᆠ� � ᨨᨨ犨矄 �ᆠ 矄 ᨨ � � ᆠ犨 矄盐 ᆠᨨ 矄ᨨ� 矄 ᨨ ᨨ盐矄盐 ᆠᨨ ᨨᨨ犨 矄 ᔼ 矄 �矄 ᨨ犨�� ᆠ �ᨨ 턼
犨ᨨ � � ᆠ犨 矄鼀犨ᨨ矄盐 ᆠ矄 �矄 㝐� 꾀�҈黨ᔼ桌 㡰 㡰 ᆠ㡰 㡰 r
ᆠ�ᆠ � � ��ᨨ �矄�� 泐 쟼 = ( 꾀x + ) �ˆ 犨 쟼�矄� ᔴ҈ x �矄 � 鼀矄 盐矄 ᨨ ᆠᨨ 矄 矄꾀㦨和ᔼ盐 � 㦨҈和ᔼ盐 �ᔼ ᨨ �x㦨҈�矄҈ x 㦨��矄
҈�矄 㝐� ҈黨ᔼᔼ桌 㡰 㡰 ᆠ㡰 ҈ 㡰 �
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P-165
Electrostatic Potential and Capacitance
�� � 矄 ᨨ犨矄��ᨨ 犨� �矄 ᆠ ᨨ鼀ᆠ 㡰 ᨨ 㝐� ҈҈ ҈黨ᔼ桌 졠
҈
졠
黨
㡰 쟼ᆠ�ᆠ � ᨨ 盐 � ᨨ鼀泐ᆠ � 矄 㡰 ᨨ �ᨨ 盐 � ᨨ鼀泐ᆠ � 矄 �ᆠ矄 ᆠ㡰 ᨨ �ᨨ 盐 � ᨨ鼀泐ᆠ � 矄 㡰 쟼ᆠ�ᆠ � ᨨ 盐 � ᨨ鼀泐ᆠ � 矄 �ᆠ矄
犨ᨨᆠ�ᆠ �ᨨ矄꾀犨� 矄ᆠ� �ᨨ鼀ᨨ鼀 矄 r r 꾀 = ⼄ �ˆ = ⼄ �ˆ ᆠ ᨨ x㜠 �矄 犨� 矄�ᨨ 矄矄 ᨨ犨 � ��盐
҈
�矄 ᆠ ᨨ鼀꾀犨 �ᆠ ᨨ ᨨ 鼀ᨨ 盐ᆠ 矄鼀ᨨ ��矄 㝐� ҈ ҈黨ᔼ桌 㡰 ᆠ㡰 㡰 㡰 +҈ -҈ +҈ -҈ 꾀 � 矄 �ᆠ � 矄盐� � 矄 ᨨ �盐矄 ᆠ�矄 盐 � ᨨ鼀泐 �矄�盐 ᆠ �矄盐ᆠ ᨨ ��矄 矄盐 ᆠ�矄꾀 ᨨ犨� 矄鼀apR a ҈㡰�矄 鼀 ᨨ 矄 犨� ᨨ盐 ᆠ� � 矄ᨨ 矄 ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� �矄鼀 矄�矄ᆠᨨᆠ䁸 꾀 ҈黨 㡰 鼀� �盐� ᨨ ᆠ�ᆠ � ᨨ� ᨨᆠ ᨨ
� 矄矄� � ᨨ盐� ᨨ 矄 矄ᆠ ᨨ
�矄 ᆠR ᨨ鼀 ᆠ ᨨ 矄 �ᆠ矄 aV 犨� 盐ᆠ 泐 R 㡰 �ᨨᨨ ᨨ � ᆠ ᨨ 矄 ᨨ ҈ ᨨ ᨨ� R ᨨ鼀 ᆠ ᨨ犨 矄 ҈- a ᨨ犨� ҈ - a ᆠ㡰 鼀� �盐 ᨨ ᆠ�ᆠ � ᆠ ᨨ aV R 㡰 ᨨ � ᆠ ᨨ 矄 �矄 盐ᆠ 泐 aV
ᆠᨨ ᆠ �ᆠ鼀矄 矄꾀 ᔴᨨ 矄ᆠ� �� ab c abc㡰 矄盐 ᆠᆠ � 矄��矄졔ss 졔s 矄ᆠ�泐 ᨨ �ᨨ 矄 �矄 ҈桌 s é - ù s é - ù 㡰 Î ê 졔ᆠ ú 㡰 Î ê 졔ᆠú úû úû êë êë ù ù s é -ᆠ s é - ᆠ ᆠ㡰 Î ê 졔 ú 㡰 Î ê ᆠ 졔 ú ëê ëê ûú ûú
�矄 盐 � ᨨ鼀 ᆠᨨ矄�ᆠ � � ��ᨨ ᨨ ��ᨨ盐矄ᨨ� 矄ᨨ 矄鼀矄 ᆠ � ��ᨨ �矄 ᨨ盐 ᨨ泐犨 � �鼀�矄�黨
矄 �矄 盐ᆠ 泐
҈҈
҈
҈
҈�
҈
҈䰘
ᨨ�黨␔ �矄 ᨨ �ᨨ� ᨨ 矄 犨 ᨨ矄 �盐矄ᆠᨨ鼀├矄 �ᨨ 犨� �ᆠ�ᨨ
ᨨ � 䁸 㝐� 졠 � 꾀�҈䰘桌 㡰 �黨� 㡰 �黨 ᆠ㡰 �黨 㡰 �黨
ᨨ � � ᨨ矄㡰ᨨ ᆠ � �矄�盐�ᨨ �矄�� 泐 㡰㦨� ᨨ £҈鼀 㡰㦨�҈ ᨨ ³ ҈鼀 㡰 ᨨ矄 ᨨ ᨨ 泐 ᔼ �矄 ᨨ � �矄 � 泐 ᆠᨨ 矄 ᆠ � 盐 � ᨨ盐鼀 r �
盐 �矄ᨨ e㡰犨 �ᆠ �矄矄 ᨨ ᆠ� ��ᨨ
ᆠ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㝐� 졠 � 꾀�黨҈桌 㡰 r㦨e� ���ᨨ
㡰 r㦨҈e ᨨ £҈鼀 㦨矄犨 ᆠ㡰 r㦨e ᨨ £҈鼀 㦨矄犨 㡰 r㦨e� ���ᨨ
꾀 盐 � ᨨ鼀泐 ᆠ � 矄ᨨ� 矄 ᨨ �盐矄 矄 ᨨ � 鼀矄盐 犨� 矄ᆠᨨ¥㡰ᨨ �矄矄盐 ᆠ ᨨ �矄矄 ⼄盐�ᨨ �矄盐 ᆠ矄犨� ᨨ �矄 � �盐矄 ҈ 矄ᆠ�泐
㝐� ҈�桌 㡰 ҈㦨 㡰 㡰 㦨 ᆠ㡰 ҈㦨 > 㡰 㡰 ҈¹ ҈㡰> 㡰 r -1 �矄矄� ��ᨨ ᨨ $ 꾀 ᆠ�ᆠ � E = (25i$ + 30j)NC 矄ᆠ ᔼ ᨨ � ᨨ��� �矄 ├ ᨨ ᨨ ᨨ � x㦨 m y㦨 m�矄 㝐� 졠 � 꾀�҈҈ ҈�桌 㡰 ҈҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ r 꾀矄矄盐鼀 ᆠ�ᆠ � 쟼 = ˆ� �矄矄� 矄ᆠ
ᨨ � � ᆠ 꾀 - 졠 犨 졠 �矄 ᨨ � ᨨ��� 꾀 ᨨ � 㦨鼀�矄㝐� ҈桌 㡰 ҈⏌ᔴ 㡰 ҈⏌ᔴ ᆠ㡰 ⏌ᔴ 㡰 ⏌ᔴ
犨ᨨ 矄鼀 ⼄盐 ᨨ� ᆠ �矄 ᨨ 鼀� �盐 q 矄盐矄 ᨨ鼀 ᆠᨨ鼀鼀ᨨ ᨨ� ᨨ ᆠ�� � 泐 � 矄盐� �鼀矄矄矄矄矄� �矄ᨨ ⼄盐 � 矄 泐ᆠᨨ鼀 ᨨ⼄盐���盐鼀犨� ᆠ 矄� �鼀├� � �q ᨨ鼀 �ᆠ ᔼ 鼀矄矄 ᨨ ᆠ ᆠ � �矄 m ᆠᨨ矄�ᆠᨨ � ᆠ ᨨ � ᨨ� � � 鼀犨� æ
ö ҈ = k÷ p Î è ø
ç
㝐� 졠 � 꾀�҈桌
㡰 k mg q
㡰
k mg q
ᆠ㡰 k mg ⏌ q 㡰 k mg ⏌ q ҈ 꾀ᨨ � ᨨᨨ犨ᆠᨨ 盐ᆠ� �ᆠ泐� �矄├ᆠᨨ �泐� 矄� ᨨ ᨨ � ᨨᨨ犨ᆠᨨ 盐ᆠ� �ᆠ泐� ᨨ � �盐矄 ᨨ ᆠ泐� 矄� ��泐ᆠ�ᆠ泐 盐䰘桌 㡰 꾀ᨨ � � ᆠ矄犨 犨ᨨᆠ泐� 矄 犨 ᆠ � 矄�泐�矄�� ᨨ �
ᆠ泐� 㡰 꾀ᨨ � � ᆠ矄犨 犨ᨨᆠ泐� 矄 犨 ᆠ � 矄�泐�矄�� ᨨ ᨨ盐ᆠ泐� ᆠ㡰 ᨨ ᨨ � � ᆠ 矄 犨 犨ᨨ ᆠ泐� 矄犨 盐 � ᨨ鼀� ᆠ ��矄├ᨨ � �矄ᨨ ᆠ泐� 矄 㡰 ᨨᨨ � � ᆠ矄犨 犨ᨨᆠ泐� 矄 犨 矄鼀ᆠ � 矄�泐�矄�� ᨨᨨ ᆠ泐� 矄
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EBD_8341 P-166
Integer Value Answer
� 꾀�ᆠᨨ 鼀矄矄҈kg ᆠ �҈C�矄� ��泐 矄꾀�鼀㦨 �ᆠᆠᨨ鼀矄盐 � 盐 ᆠ r ᨨ ᆠ�ᆠ � 쟼 㡰㦨쟼矄� w �ˆ犨 쟼㦨҈ ᔴ҈ w 㦨҈ 矄҈ᔴᨨ 矄� ᆠᨨ ᨨ 泐 ᆠ�ᆠ ᨨᆠᨨ �ᆠ 鼀 �鼀盐鼀矄 � 鼀矄҈� 泐 �ᆠ矄盐矄⼄盐 �鼀矄�矄 힘힘힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌 4 Fill in the Blanks ᆠ�ᆠᨨ �V 泐ᨨ� xyz � 鼀矄㡰�
矄ᆠ�矄�� 泐V㦨xᨨ矄 ᆠ�ᆠ � ᨨ� ҈鼀鼀㡰�矄⏌鼀 ҈黨黨㜠҈㝐├桌 䰘 ��盐矄 ᨨ犨矄� ᨨ ᆠᨨ 矄 ᨨ �� ��ᨨ � 犨 �ᆠ ᆠ�ᆠ � �矄矄 盐矄ᨨ ᨨ � 犨� � ᆠ├矄졠 ᨨ� 矄AB C 鼀� �盐 ᨨ ᆠ�ᆠ � �矄�矄 ᨨ� ҈黨㜠㝐├矄桌 꾀
҈
҈
㡰 ␔� S҈ᆠᨨ矄 V҈㦨҈�V㦨 㡰 ␔� Sᆠᨨ矄 V҈㦨V㦨� ᆠ㡰 ␔� S҈ Sᆠᨨ矄 V҈ = V = 㡰 ␔� S҈ Sᆠᨨ矄 V҈ = V V = V 꾀ᆠ �졔q�矄 � ᆠ ᨨ ᨨ� 矄x㦨xx㦨x x㦨�x x = ¥ ᨨ x �矄 ᆠ �q�矄 � ᆠ ᨨ ᨨ� 矄x㦨xx㦨xx㦨x x = ¥ x�矄ᨨ矄��ᆠᨨ 矄 ├ ᆠ�ᆠᨨ � ᨨ� 盐ᨨᆠ �Q �矄 ᆠr ᨨ鼀�ᨨQ⏌ per㡰 ᨨ � ᨨ��� 盐ᨨ ᨨ 矄泐矄鼀ᨨ ᆠ �矄�矄 ҈黨黨和㜠㝐├矄桌 q ln q 㡰 㡰 pe x ln ᆠ㡰 ¥ 㡰 pe x 꾀 矄ᨨ� ᆠᨨ 盐ᆠ� � 矄 � � ᆠ � Q �矄 矄盐ᨨ盐 泐 盐 ᆠ � ᆠᨨ ᆠ �ᆠ ᆠᨨ 盐ᆠ� � ᨨᨨ犨矄 �ᆠ矄 ᨨ � � ᆠ犨
矄盐 ᆠᨨ 矄ᨨ� 矄 ᨨ ᨨ盐矄盐 ᆠ ᨨ ᨨᨨ犨矄 Vᔼ 矄 �矄 ᨨ犨�� ᆠ � ᨨ Q 犨ᨨ � � ᆠ犨 矄鼀 ҈黨黨㜠㝐├矄桌 犨ᨨ矄盐 ᆠ矄�矄 㡰 V 㡰 V ᆠ㡰 V 㡰 V 꾀 ᨨᨨ犨鼀矄 ᨨ �盐矄�ᆠ鼀矄�矄ᆠ � 矄盐ᆠ ᨨ �ᨨ �矄矄盐 ᆠ�矄҈ᨨ矄 ᨨ � ᆠ ᨨ 矄 �矄 ҈黨㜠҈㝐├桌 㡰 ᨨ 㡰 ҈ᨨ矄 ᆠ㡰 矄鼀矄ᨨ� �ᆠ鼀矄犨泐 ᨨ鼀 矄盐 ᆠ 㡰 矄鼀矄ᨨ� �ᆠ鼀矄犨泐 ᨨ鼀 矄盐 ᆠ
犨ᨨ�ᆠ�ᆠ盐 �矄ᆠ矄矄 泐 �矄 ᆠᨨ ҈鼀 ᆠᨨ
ᆠ ᨨ 泐� 矄犨�ᆠ 矄矄 ᨨ犨 � ��盐ᔴ � ᨨ� ᨨ矄ᨨ 矄�泐黨├�鼀矄 ᨨ盐� � 泐 ᨨ ᆠ ᨨ ᨨ �矄ᆠ졠 ᆠ矄ᨨ鼀 ᨨ� ᨨ矄ᆠ �鼀� ᨨᆠ�泐 矄犨�ᆠ �矄ᆠᨨ矄 ᨨ 泐ᨨ�ᨨ ᆠᨨ矄矄 �矄ᆠ矄꾀矄矄盐 ᨨ� ᨨᨨ �盐矄맔҈䰘鼀矄ᨨ矄鼀ᨨ� ��ᆠ泐 ᨨ矄 犨 �矄ᆠ矄 盐鼀ᨨ ᆠᨨ 矄矄 � �矄 ᨨ� ᨨ �矄 힘힘힘힘힘힘힘힘 �ᆠ 盐ᨨ泐 ᆠ泐 ᨨᆠ ├ ᆠᆠ�ᨨ 盐ᨨ��泐㦨҈鼀矄 ᆠ �ᨨ
ᆠᨨ 㡰㦨҈맔҈҈黨ᔴ㡰 꾀 桌
2
�
꾀盐 � ᨨ鼀ᆠ�ᆠ � ᨨ� � �� ᨨ矄��x㜠 �ᆠ�ᨨ �矄矄 � ��ᨨ A ᨨ��� B ᨨ� ᨨ x㜠 �矄 x 㦨졔҈ᆠ鼀 C ᨨ� ᨨ y㜠 �矄y 㦨졔҈ᆠ鼀
ᨨ �矄 ᨨ� 矄AB C 矄�矄 泐 ҈和桌 㡰 VA VB 㡰 VA >VB ᆠ㡰 VA VC 㡰 VA >VC ᨨ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐犨 �ᆠ ᨨ ᨨᨨ犨� � 矄鼀 矄�矄盐䁸 ҈黨黨黨㜠㝐├矄桌
ᔴ
҈黨
Physics
6 MCQs with One or More than One Correct Answer 犨ᨨ ᨨ 㜠ᆠᨨ 盐ᆠ� � 矄 矄 ᨨ �� ҈ ᆠ泐� � 盐 � ᨨ鼀 ᨨ盐鼀 ᆠ � 矄��矄 졔r r 矄ᆠ�泐ᆠ 矄盐ᆠ 泐�泐ᨨ 矄矄 ᨨ犨 � ��盐꾀ᨨ� 矄� ᨨ� � ��ᨨ
꾀 ҈桌 r
–r
҈
㡰 ᆠᨨ矄�ᆠ � �矄 ᨨ 㡰 ᆠᨨ矄�ᆠᨨ ��矄ᆠᨨ 矄 ᆠ㡰 ᆠᨨ矄�ᆠ � �矄ᆠᨨ 矄 � 鼀� �盐 㡰 ᆠᨨ矄�ᆠ � 矄矄鼀 �ᆠ�ᨨ
黨 和� ᨨ� ᆠ �矄 ├ �ᆠ矄 ᨨ �盐 �ᨨ ᨨ 矄� ᆠ O矄矄 ᨨ犨 � ��盐�
҈ q 犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 pe 0 㡰ᆠᨨᆠ䁸 ҈桌
K =
졔q
E–q P
Sw�tch
A 졔q
S
T
O
҈鼀
D –q
R B졔q
– qC
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P-167
Electrostatic Potential and Capacitance
҈
㡰 ᆠ�ᆠ � O�矄⧤ᨨ �OD 㡰 ᨨ �O�矄 ᨨ ᆠ㡰 ᨨ �ᨨ� 矄ᨨ � PR�矄矄鼀 㡰 ᨨ �ᨨ� 矄ᨨ � ST�矄矄鼀 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄⏌ᆠᨨᆠ䁸҈҈桌 㡰 ᔼ ᆠ�ᆠ � 盐ᨨᨨ� ᆠ ��矄矄 � � 矄 ᨨ 盐矄矄犨犨�矄�� 㡰 盐矄矄 犨 ᆠ 盐矄 ᨨ ᆠᆠ盐 � �矄�盐�ᨨ ᨨ盐 ᆠ�ᆠ �ᨨ ᆠ㡰 ᔼ ᆠ�ᆠ � 犨 犨ᨨᨨ� ᆠ �矄�矄 ᨨ 矄ᨨ鼀犨 矄�� ᨨ 犨ᨨᆠ �矄�矄 矄鼀 㡰 犨ᨨ├ ᨨ 泐 ᨨᆠ � 鼀ᨨ� � 盐 � ᨨ矄�� ᆠ � ᨨ鼀 ᨨ� 꾀 ᨨ � 꾀 ᨨᨨ� ᨨ ��矄 - 꾀㡰 꾀矄 �ᆠ鼀矄 꾀ᨨ �盐矄 꾀 矄ᨨ� 鼀 矄 ᨨ �盐矄 꾀㡰├ ᆠ �矄 �� ᆠ ��졔턼ᨨ犨 泐ᆠᨨ
ᆠ 泐 � 鼀 犨�
҈҈桌 㡰 E �ns�de 㡰 턼꾀>턼 = A s A RB ᨨn surface ᨨn surface ᆠ㡰 = 㡰 E A < EB s B RA 꾀矄 �ᆠ矄泐鼀鼀�ᆠᆠ �矄泐矄鼀�矄ᆠ ᨨ��� � 쟼ᆠ�ᆠᨨ � 和㜠�㝐├矄桌 Q Q V= r £ R 㡰 V = r > R 㡰 pe R per V
㡰 㡰 ᆠ㡰 㡰
␔� � r㦨Rᨨ ᆠᨨ矄 ᆠ ��矄Q 쟼ᆠ�ᆠ � �矄 �矄ᆠᨨ � 盐 r㦨R ᔴ ��矄ᨨ 泐矄 r㦨R 쟼ᆠᨨ矄�ᆠ �泐�矄 ᨨ ᨨrR
9 Assertion and Reason Type Questions 和T꾀TE㝐ENT㜠҈ ᨨᆠ�ᆠ盐ᨨ矄矄 �矄 盐矄 矄 ᆠ ᨨᨨ �� ᆠ�ᆠᆠ�ᆠ盐�矄
和T꾀TE㝐ENT㜠 ᆠ�ᆠᨨ �ᨨ 矄 ᨨ �盐矄 犨� ᆠ �턼盐 � ᨨ鼀泐 �矄�盐 ᨨ 矄盐 ᆠ 턼 �矄�� 泐 pe
桌
㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐 10 Subjective Problems 꾀 ᆠᨨ 盐ᆠ� � �⼄盐� 盐 ᨨ �盐矄 �ᆠ├ 矄矄 t ta㡰�矄ᆠ � ᨨᨨ �Vᔼ 盐ᆠᨨ矄矄 ᨨ ᨨ � ᨨ �ᨨ ᨨ �㜠㝐├矄桌 � 犨ᨨ � ᆠ �矄Q Qᨨᆠ ᨨ� 矄 犨� ᆠᨨᨨ � 矄 a㡰 졔a㡰矄ᆠ�泐� x㜠y ҈黨黨҈㜠졔졔㝐├矄桌 㡰 和 ᨨ犨 ᨨ� 矄� x㜠y 犨 ᆠ�ᆠ ᨨ � 盐ᨨ 犨ᨨᆠ �矄�矄 ᨨ�ᨨ ᆠ�ᆠ � �矄 �盐矄 ᨨᆠ�ᨨ ᨨ �矄ᆠ 㡰 � 矄矄�ᨨ V x㡰� ᨨ� ᨨ x㜠 �矄 矄├ᆠ 盐 ᆠ�ᨨ V x㡰ᨨ 犨 ᨨx㜠 �矄 ᆠ㡰 ᔼ �ᆠᨨ ᆠ �졔q矄矄 ᨨ鼀矄 ᆠ ᨨ ᆠ�ᆠ 矄 ᨨ犨泐矄 ᨨ ⼄盐 ��盐鼀 �ᆠ 盐泐ᆠᨨ矄矄矄 ᆠ�ᆠ � �矄 矄 犨 � ᨨ矄矄ᨨ ᆠᨨ ᆠ �ᆠ矄 �ᆠ鼀�ᆠ矄 矄AB Cᨨ ��ab c abc㡰 矄盐 ᆠᆠ � 矄��矄 s s s 矄ᆠ�泐 ҈黨黨㜠䰘㝐├矄桌 �㡰 � ᨨ �ᨨ 矄 矄AB C ��㡰 ᔼ 矄 矄A C 矄鼀ᨨ �ᨨ�
�ᨨ 犨 ��ab c 䰘 꾀ᆠ ��Q�矄 �矄�盐 ᨨ犨ᨨᆠᨨ ᆠ �ᆠ ᨨᨨ犨 矄 矄 ᨨ �� �r �R >r㡰 矄盐ᆠ 矄盐 ᆠ 矄��矄 ⼄盐 � ᨨ � ᆠᨨ鼀鼀ᨨ
ᆠ ҈黨҈㜠㝐├矄桌
Topic-2 : Electric Potential Energy and Work Done in Carrying a Charge 1 ҈
MCQs with One Correct Answer
㡰 � ᆠ矄泐
꾀犨ᨨᨨ� ᆠ �矄q q � ᨨ x㜠 �矄 d d � x = - x = 矄ᆠ�泐ᔼ � ᨨ� ᆠ � �q�矄├ ᨨ鼀 ᨨ��� ᨨx = dᨨ � 矄鼀�ᆠ�ᆠ矄 矄 ᨨ犨 � ��盐 �泐ᨨ ᆠ �犨� [Main Sep.04, 2020 (I)] q
q pe d
㡰 � ᆠ矄泐
q pe d
q q 㡰 ᆠ矄泐 pe d pe d 泐 ᨨ� �ᨨ 矄� �泐�ᨨ � �盐鼀 ᨨ鼀 ᆠᆠ ᨨ鼀 矄 ᨨ盐� 矄鼀 ᨨ � � ᆠ �ᨨᨨ � 矄 矄ᨨ 泐 ᨨ� �盐鼀 �ᨨ 矄�矄ᆠᨨ矄ᨨ 㝐� 和 ᔼᔼ㡰桌 㡰 ҈ 㡰 ҈䰘 ᆠ㡰 ҈ 㡰 �䰘
ᆠ㡰 ᆠ矄泐
q
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EBD_8341 P-168
Physics
꾀矄泐矄鼀ᨨ ᆠ �矄ᆠ 矄矄 ᨨ犨 � ��盐
ᔼ >> ᨨ � �泐ᨨ 矄泐矄鼀�矄矄��
泐 㝐� 黨꾀�҈黨ᔼ桌 ҈ é - q - qQd ù ҈ é - q qQd ù + ê ú 㡰 㡰 ê ú p Î ë d D û p Î ë d D û ҈ é q qQd ù ҈ é q qQd ù + ú 㡰 - ú ê+ ê ᆠ㡰 p Î ë d p Î ë d D û D û
ᆠ �矄턼졔⼄ 졔⼄ᆠ �ᆠ矄ᨨ �� 㜠 ��矄ᨨ矄ᆠ矄� �矄矄 ᨨ犨 ᨨ犨 ᆠᨨ矄�ᆠ �泐ᨨ ᆠᨨ ��盐�ᨨ �矄 ᨨ� 盐 ᨨ 턼�矄 㝐� ҈҈ ҈黨ᔼ桌 턼
졔⼄
졔⼄
-⼄ - ⼄ ᆠ㡰 㡰 ⼄ ҈+ +҈ ᨨ盐 ⼄盐 ᨨ� ᆠ �矄턼ᆠ ᆠ � 泐 㡰 㡰 㡰 㡰 犨ᨨ├ ⼄盐� ᨨ 盐 � ᆠ � 턼 ᨨ��� ᨨ ᆠᨨᨨ � 矄泐矄鼀 犨� 㝐� ҈ ҈黨 ᔼᔼ桌 㡰 졔⼄
�
㡰
턼 pe
㡰
҈ ö æ ç҈ + ÷ ø è
턼 pe
㡰
턼 턼 㡰 pe pe 和鼀 ҈ᨨ犨ᨨ├�矄⼄盐� ᨨ ᨨ ᨨ鼀ᨨ矄 ᆠ �犨 泐犨ᨨᨨ� 矄ᨨ ⼄盐�ᨨ �矄盐 ᆠ 和鼀 쟼ᆠ�ᆠ� 矄ᨨ ᨨᆠ ⼄盐�ᨨ � 矄盐 ᆠ矄鼀盐盐泐 �ᆠ盐ᨨᆠ ᨨ 㝐� 졠 � 꾀��҈桌 㡰 和鼀 ҈�矄盐和鼀 �矄盐和鼀 �矄 ᆠᨨᆠ �ᨨ ᨨ 和鼀 ҈ 㡰 和鼀 ҈�矄盐和鼀 �矄盐和鼀 �矄
ᨨ ᆠᨨᆠ �ᨨ ᨨ 和鼀 ҈ ᆠ㡰 和鼀 ҈�矄盐和鼀 �矄 矄 㡰 和鼀 ҈�矄 矄和鼀 �矄盐 ᔴᨨ 矄� 矄泐矄鼀ᨨ ᆠ �矄q⏌ q⏌ q⏌ᆠ ᨨ� 矄 A, B C矄ᆠ�泐矄矄 ᨨ犨 � ��盐
├Oᨨ ᆠ ᨨ ᆠ�ᆠᨨ �盐矄 � CAB㦨 桌
ᆠ㡰
䰘
҈ ö æ ç҈ + ÷ �ø è
泐
ᔴ 꾀
졠
㡰 ᆠ�ᆠ � ᨨ� O�矄
q pe R
�ᆠ ᨨ �
�� 㜠 �矄 㡰 ᨨ � �泐ᨨ 矄泐矄鼀�矄 ᨨ ᆠ㡰 鼀� �盐 ᨨ ᨨᆠ犨 ᆠ �矄C B�矄
q �pe R
q ҈pe R ᨨ矄�� ��ᨨ� ᆠ �矄ᨨ ⼄盐鼀� �盐 aö æ -a ö æ ├ çè ÷ø ç ÷ 矄ᆠ�泐 è ø 犨ᨨ├ ᨨ 泐 ᆠ�ᆠ � 犨 ᨨ ᨨ矄��ᨨ� ᆠ ��矄鼀ᨨ ᨨ鼀 a㡰ᨨ a㡰�矄 䰘桌 㡰 ᨨ矄�� 㡰 �� ᆠ㡰 ᨨ 㡰 矄ᨨ ᆠᨨ
ᆠ� � � �� � ᨨ矄��ᨨ 矄 黨
犨ᨨ⼄盐ᨨ� ᆠ �矄 � x㦨a x㦨졔aᨨ
x㜠 �矄꾀 ᨨ ᨨ� ᆠ �Q�矄ᆠ ᨨ���
ᆠ �� ᆠ�ᆠᨨ � �泐ᨨ Q犨 � �矄 �矄ᆠ 泐 矄鼀 �矄 ᆠ x ᨨ � x㜠 �矄 �矄 ᨨ �鼀泐ᨨᨨ�ᨨ ᨨ 和桌 㡰 x 㡰 x ᆠ㡰 x 㡰 ҈⏌x ҈ ᆠ �矄Q졔q 졔qᆠ �ᆠ矄ᨨ �� 㜠 � �矄ᨨ矄ᆠ矄 � � 矄 矄 ᨨ犨 ᆠᨨ矄�ᆠ �泐ᨨ ᆠᨨ ��盐�ᨨ �矄 ᨨ� Q �矄 和桌 ⼄盐ᨨ -q 㡰 Q ҈+ -q 㡰 + +q +q ᆠ㡰 q a 㡰 졔q ҈҈ 犨ᨨᨨ� ᆠ �矄졔q q � dᨨ㡰 dᨨ㡰矄ᆠ�泐ᨨ ax㜠yᆠᨨᨨ � 矄泐矄鼀 ҈黨黨�和桌 㡰 ᆠ�ᆠ � Eᨨ� 矄ᨨ x㜠 �矄 矄 矄鼀 �ᆠ�ᨨ
㡰 쟼ᆠ�ᆠ � ᨨ� 矄ᨨ y㜠 �矄�矄ᨨ �x㜠 �矄 ᆠ㡰 ␔ᨨ├ 矄ᨨ ᨨ � � �� �矄ᆠ � ᨨ鼀 ¥ ᨨ ᨨ���
㡰 �ᨨ鼀ᨨ鼀 �矄qdᨨ � x㜠 �矄 ҈ 犨ᨨ � �ᆠ � � �矄 ᆠ ᨨ �盐矄 R 鼀矄
㡰 ᨨ �ᨨ� O�矄
ᆠᨨ �泐ᆠ �矄 ᆠR鼀矄ᔼ Q҈ ᆠᨨ盐ᨨ鼀 Q ᆠᨨ盐ᨨ鼀矄ᆠ�泐 ᆠ �矄盐 � ᨨ鼀泐 矄 ᨨ 犨ᨨ� �矄 犨ᨨ├ ᨨ � 鼀ᨨ� �ᆠ � q ᨨ鼀 ᆠ ᨨ ᨨ � �ᨨ ᨨ ᨨ �矄 ҈黨黨㜠㝐├矄桌 q Q҈ 㜠 Q 㡰 㜠҈㡰 㡰 ᨨ 㡰 pe R 㡰 ᆠ㡰
q Q҈ + Q 㡰 pe R㡰
㡰
q Q҈ + Q 㡰 + ҈㡰 pe R 㡰
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P-169
Electrostatic Potential and Capacitance
҈
꾀ᨨ� ᆠ �qᨨ 鼀矄矄m�矄矄盐矄 �ᆠ泐泐 r 矄� �ᨨ � l꾀ᨨ� �ᨨᨨ �ᨨ鼀ᨨ鼀 p �矄
ᨨ犨ᨨ盐� ᨨ犨 矄q ᨨ鼀� � �泐矄ᨨ ᆠ � 鼀ᨨ矄犨泐 � ⼄盐���盐鼀ᨨ矄��ᨨ ᨨ 矄泐矄鼀 � ᆠ盐 � � �ᆠ�ᨨ ᨨ �ᨨ �矄 �矄 ᆠ矄�矄矄 ᨨ犨 � ��盐ᨨ犨ᔼ 犨ᨨ├ ᨨ �
� �� � �ᨨᨨ �矄ᨨ矄��ᨨ �矄N 맔 mgh㡰犨 ��矄 ᆠᆠ�ᨨ 盐 ᨨ ��泐 盐 ᨨ N �矄 힘힘힘힘힘힘힘힘힘 ᨨ ᨨ ᆠᨨ ᨨᆠ矄├� � F �矄 矄鼀 ᨨ ᨨᆠ矄 矄� q 犨 F�矄 泐ᨨ ᨨ ᨨᆠ矄 q�矄 �犨
ᨨ 犨ᨨ ᨨᆠ矄㡰 꾀 桌
ᨨ� 鼀矄矄� ⼄盐���盐鼀
҈� ᔴ �矄졔q qᨨᆠ ᆠᨨ 矄ᨨ ᆠ盐ᨨ 矄� 矄矄 ᨨ犨� ��盐 � 犨ᨨ├ ᨨ ᨨ矄 ᆠ �矄ᨨ� � � �矄 ᆠ 㜠㝐├矄桌 –q +q
–q
+q –q
+q
҈ 꾀ᆠ�ᆠ盐� �ᨨ �盐矄R犨� 盐 � ᨨ鼀ᨨ矄��ᆠ � 矄�泐l盐 � � �矄ᨨᆠ � y㜠z 犨� �矄 ᆠ ᨨ��� O꾀�ᆠᨨ 鼀矄矄m ᨨ矄�� ᆠ �q�矄ᨨᆠ ᨨ鼀 ᨨ� P R 㡰 ᨨ
a
l l a
l矄�
q
h
p
ᨨ矄��x㜠 �矄 �ᆠ泐ᨨ犨 矄O犨� � ��矄 v � 矄鼀矄 ᨨ 㜠 ᨨ㡰盐 ᨨ 矄 矄盐ᆠ �ᆠ ᨨ矄 ᨨ盐 ᨨP ҈黨黨㜠㝐├矄桌 ҈䰘 犨ᨨ � ⼄盐ᨨ矄��ᆠ �矄ᆠ ᨨ 鼀� �盐 �맔 10�ᆠᨨ盐ᨨᆠ ᨨ� 矄A B矄 泐 �矄 ᆠ ᨨ 鼀 꾀 ⼄盐 ᨨᨨ矄� ᆠ � 鼀ᨨ矄 ᨨ犨 矄 鼀 ᨨ � � COD �ᆠ盐 �矄ᆠᨨᨨ � AB ҈黨�㜠㝐├矄桌 꾀
10 Subjective Problems ҈
+q –q
졔⼄
⼄
졠
꾀ᆠ �졔Q�矄 � ᨨ��� ᨨ ᆠᨨ㜠ᨨ � 矄泐矄鼀 uur 犨 �矄鼀ᆠ�ᆠ �ᨨᨨ �ᨨ鼀ᨨ鼀 p ᨨ� � � 犨泐 ᨨ鼀 ᆠ �ᨨ � x㜠 �矄�矄矄 ᨨ鼀 ᨨ� 犨泐 ᨨ鼀 ᨨ��� 㜠㝐├矄桌 㡰 ᔴᆠ盐 ⧤쟼ᨨ �ᨨ犨 �ᆠ 矄ᨨ ᨨ� d㡰 㡰 ᔴᆠ盐 ᨨᆠᨨ ᆠ �졔Q �矄鼀ᨨ鼀
ᔴ 졔⼄
鼀ᨨ� � ᆠ � 犨 � ᆠ 矄 ᨨ� C �矄 ᆠᨨ 鼀 ᨨ鼀O 矄├� �ᆠ �泐ᨨ ᨨ盐矄 ᔴᆠ盐 �矄 ᆠᨨ 矄ᨨ� D犨 �ᆠ
��ᆠ �犨�ᆠ ᨨ盐 � �ᨨ犨 矄C
Topic-3 : Capacitors, Grouping of Capacitors and Energy Stored in a Capacitor 1 ҈
MCQs with One Correct Answer
犨ᨨᆠᆠ�ᨨ矄ᨨ ᆠᆠ� ᆠ矄ᔴ ᔴᆠ � ᨨᨨ㜠 � � ᆠ矄 矄ᆠ�泐 矄
ᆠᨨ
ᆠ � � 矄盐ᆠ 鼀
ᨨ矄��鼀�㜠
ᨨ ᨨ �矄ᆠᨨ
ᆠ ᨨ ��鼀� ᨨ ᨨ
� �泐ᨨ �矄ᆠᨨ ��盐�ᨨ �矄 㝐� 和� ᔼ㡰桌 � 黨 㡰 ᔴ 㡰 ᔴ ᆠ㡰 ᨨ 㡰 ᔴ ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᆠ �ᨨ �m ᆠᆠ�ᨨ�矄 㝐� 和� ᔼᔼ㡰桌 m
m
꾀ᆠᆠ�ᨨ C �矄 盐泐ᆠ � 犨� ᨨ� V 꾀 �矄ᆠᨨ
ᆠ� � ᨨ�矄ᨨ盐ᆠ��矄ᆠᨨ
ᆠ � C 犨� ᨨ 盐 ᆠ � ᆠᆠ�ᨨᨨ ᆠᆠ� ᆠ �泐ᨨ矄矄� ᨨᆠ矄矄 ᆠ ��矄 �矄�盐 犨 犨ᨨᆠᆠ�ᨨ矄�矄 㝐� 和 ᔼᔼ㡰桌 ҈ ҈ ҈ ҈ CV 㡰 㡰 CV CV 㡰 CV ᆠ㡰 ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐 ᨨᆠ ��矄䰘�mᔴ ᨨ�ᆠᨨ矄矄 ᆠᆠ�ᨨ C �矄 ᆠ �ᨨ ᆠᆠ�ᨨC�矄 [Main Sep. 03, 2020 (I)]
C㦨m
� m
졔
O
㡰 ҈mᔴ 㡰 ҈黨mᔴ ᆠ㡰 ҈mᔴ 㡰 ��mᔴ
C
C҈㦨҈�m
㡰 �mᔴ
㡰 �黨mᔴ ᆠ㡰 ҈mᔴ
㡰 �mᔴ
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EBD_8341 P-170
�
䰘
黨
Physics
꾀҈m ᆠᆠ�ᨨ�矄 盐泐ᆠ � ᨨᨨ � � ᆠ ᨨ �꾀 鼀ᨨ� � 矄ᨨ盐ᆠᨨ���矄ᆠᨨ
ᆠ ᨨ 盐 ᆠ � ᆠᆠ�ᨨ� ᨨ犨 ᨨ � � ᆠᆠᨨ矄矄 鼀ᆠᨨ鼀矄 ᆠᆠ� ᆠᨨ 矄ᆠᨨ ᆠᆠ�ᨨ�矄 [Main Sep.02, 2020 (II)] 㡰 ҈m 㡰 ҈�m 㡰 m ᆠ㡰 m 쟼 ᆠ� ᆠᆠ� ᆠ ᨨ ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ ᆠᆠ�ᨨ矄C҈ C�矄҈m ␔ 矄ᆠᆠ�ᨨ矄 � �� 盐泐ᆠᨨ
ᆠ ᨨ ᨨ� 矄ᨨ盐ᆠᨨ ҈ �泐矄ᨨ � ᆠᆠ�ᨨC�矄�鼀矄 ᨨ C҈ᔼ 矄ᆠᆠ�ᨨ矄ᆠᨨ
ᆠ � 矄�矄 � ᆠ� ᆠᆠ� ᆠ犨� 㝐� ᔼ桌 㡰 m 㡰 m ᆠ㡰 ҈m 㡰 m 꾀ᆠᆠ�ᨨ 矄矄ᨨ 꾀矄 泐 �矄 ᆠ�d犨 鼀ᔼ�矄 � 犨� �ᆠ�ᆠ 犨 �ᆠ 矄 �ᆠ�ᆠᆠᨨ 矄 �矄矄k x㡰㦨K ҈졔 ax㡰犨 �x�矄 �矄 ᆠ鼀矄盐 ᨨ鼀ᨨ ᨨ 矄ᔼ ad㡰 ᨨᆠᆠ� ᆠᨨ 矄泐矄鼀�矄 矄�� 泐 矄矄�ᨨ 㝐� 䰘 ᔼ桌 AK Î æ ad ö 㡰 ç҈ + ÷ ø d è A Î K æ æ ad ö ö ç҈ + ç 㡰 ÷ ÷ d ç è ø ÷ è ø æ ö A Î K ç a d ÷ ҈+ ᆠ㡰 ÷ d ç è ø AK Î (҈ + ad ) 㡰 d
ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ� � ᆠᆠ�ᨨ矄 ᨨ ᆠᆠ� ᆠ ᔴ nᔴ �矄 ᆠᨨ
ᆠ ᨨ 泐ᨨ ᨨ� ␔ ᆠᆠ�ᨨ矄 盐泐 ᆠ � 泐�矄鼀ᨨ �ᆠ�ᆠ 鼀�ᨨ �ᆠ�ᆠᆠᨨ 矄 ⧤�矄ᆠ 犨 犨ᨨ 矄 ᨨ �矄 ᆠᆠ�ᨨ 犨 ᨨ � � ᆠᨨ ᆠᨨ鼀� 矄泐矄鼀�矄 㝐� 黨꾀�ᔼᔼ桌 n + ҈㡰 n 㡰 㡰 ᆠ㡰 㡰 ⧤ + n㡰 ⧤졔n ⧤졔n ᔼ �� ᆠ�ᆠ盐� ᆠ �ᨨ m ᆠᆠ�ᨨ犨� 㝐� ҈ 꾀� ҈黨 ᔼᔼ桌 㡰 �mᔴ
҈҈ 꾀ᆠᆠ�ᨨ犨� ᆠᆠ� ᆠ�m �矄ᆠ � ᨨ�mᔴᔼ 矄盐 ᨨ 盐ᆠ ᆠᆠ� ᆠᨨ ᨨ犨鼀盐ᆠ 犨ᨨ├�矄 ᨨ 䁸 㝐� 黨꾀�҈黨ᔼ桌 㡰 �맔҈ 㡰 䰘�맔҈ ᆠ㡰 ҈맔҈ 㡰 ��맔҈ ҈ 꾀ᆠᆠ�ᨨ 矄҈m ᆠᆠ� ᆠ졠 ᨨ �矄 犨ᨨ矄�矄�� 졔 mᔴᆠ � ᨨ 졔mᔴᆠ � ᨨ � � ᆠ ᨨ ᆠᨨ矄矄 ᆠᆠ�ᨨ�矄 㝐� 꾀�҈黨ᔼᔼ桌 㡰 㡰 ҈ ᆠ㡰 � 㡰 ҈ ᔼ ��盐矄 ᨨ犨 矄犨�ᆠ �和�矄盐 ᨨ鼀 ᨨ矄��ᨨ �꾀ᨨᨨ矄��ᨨ � �泐 �矄矄� � ᆠ�ᆠ盐�� 鼀矄ᨨ ᆠᆠ� ᆠ�ᔴ ᨨᆠ ��턼�矄 㝐� ҈ ҈黨 ᔼ桌 꾀
和 e
ᔴ
ᔴ
҈ 턼 턼 � 턼 턼 㡰 ᆠ㡰 㡰 ᔴ ᔴ ᔴ ᔴ ҈ 꾀ᆠᆠ�ᨨ犨� 矄ᨨ ҈鼀ᆠ 矄�ᨨ ᨨ ҈鼀ᔼ ᆠ�ᆠ � 犨 矄�矄҈⏌ᔴ 鼀� �盐 ᨨ ᆠ �ᨨ ᆠ �矄 ᔴ ҈ ├Î0㦨�맔҈ 㝐 㡰㝐� ҈ ҈黨ᔼᔼ桌 㡰 䰘�맔҈҈ᔴ 㡰 �맔҈҈ᔴ ҈ ᆠ㡰 �맔҈ ᔴ 㡰 黨�맔҈҈ᔴ ҈� ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ᔴ� ᆠ�ᆠᆠ� ᆠᨨ 犨 ᨨᆠ�ᆠ盐��矄ᨨ�Ҙ 꾀盐矄� ᆠ�ᆠ盐� � Ҙ 㝐� ҈ ҈黨ᔼᔼ桌 ᔴ 㡰
꾀
҈
㡰 黨mᔴ
ᆠ㡰 ҈mᔴ 㡰 mᔴ ҈
��盐 矄 ᨨ犨矄 ᆠ � ⼄㡰 矄盐矄 ᨨ� 㡰 � ᨨ 矄�矄 ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ�� ᆠᆠ�ᨨ矄
ᆠᆠ� ᆠ矄 㝐� ҈꾀�҈黨ᔼ桌
䰘 䰘 Ҙ 㡰 Ҙ ᆠ㡰 Ҙ 㡰 Ҙ ҈҈ � ҈ ҈ ᔼ ��盐矄 ᨨ犨 ᨨ犨 ᆠ �ᨨ ᨨ ҈m ᆠᆠ�ᨨ�矄mᔴ ᆠ �ᨨ �� ᨨ m ᆠᆠ�ᨨ�矄 㝐� ҈҈ ҈黨ᔼ桌
㡰
m ҈ m m
㡰 m ҈m ᆠ㡰 �m m
㡰 m m 㡰 m m
m
㡰 ҈ m ᔴ 㡰 졔҈m ᔴ ᆠ㡰 ҈m ᔴ 㡰 졔҈m ᔴ
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P-171
Electrostatic Potential and Capacitance
҈䰘
҈
꾀 ᆠᆠ�ᨨ � � ᆠᆠ� ᆠ҈ �矄 ᆠ � 泐 泐 ᨨ ᨨ � � ᆠ ᨨ ҈ 犨 �矄 矄 ᆠ �� � 泐 �矄 ᨨ犨 �矄ᆠᨨ
ᆠ ᨨᆠ� 矄ᨨ �ᆠ�ᆠᆠᨨ 矄 ��矄矄� 犨 矄 犨ᨨ├ ᨨ 泐 ᆠᆠ�ᨨ ᨨ 矄�矄 㝐� ҈ ҈黨 ᔼᔼ桌 㡰 黨 㡰 � ᆠ㡰 � 㡰 꾀 ᆠᆠ�ᨨ �矄 ᨨ ᆠ鼀 矄�ᨨ 鼀鼀 ��矄 � 犨� �ᆠ�ᆠ 鼀�矄ᨨ ⼄盐 �ᆠ├ 矄矄 矄 ��盐㡰犨� �ᆠ�ᆠ ᆠᨨ 矄 矄⧤҈㦨҈⧤㦨҈ ⧤㦨҈ 㡰 �ᆠ�ᆠ ᆠᨨ 矄 ᨨ 鼀� 犨 �ᆠ 犨 盐泐 � 矄 �
ᨨ ᆠᆠ�ᨨ ��矄 矄鼀 ᆠᆠ� ᆠ 犨ᨨ盐 㝐� ҈ ҈黨 ᔼ桌
æ ᔴ ö ç e = � ´҈-҈ ÷ ç 鼀 ÷ø è 㡰 ҈� 㡰 ҈ ᆠ㡰 � 㡰 꾀ᆠᆠ� ᆠᨨ m �矄⼄盐� � ᆠ�ᆠᆠ�ᆠ盐� ᆠᨨ矄矄ᨨ � � ᆠᨨ ҈├꾀� 盐鼀ᨨ ҈m ᆠᆠ�ᨨ矄 � 犨 �ᆠ ᆠ 犨� 矄 ᨨ � � ᆠᨨ ᨨ鼀ᨨ 鼀� �鼀盐鼀
盐鼀ᨨ ᆠᆠ�ᨨ矄⼄盐� ᨨᆠ � �矄�矄 㝐� ҈䰘桌 㡰 㡰 ᆠ㡰 㡰 ҈ 꾀ᆠᨨ鼀� �ᨨ ᨨ ᆠᆠ�ᨨ矄�矄矄 盐矄 矄 ᨨ犨 � ��盐 鼀� �盐 ᨨ ᆠ�ᆠ � 盐ᨨᨨ� ᆠ �턼 � �ᆠ �⼄盐ᨨ 矄盐鼀ᨨ ᆠ �矄 ᨨ m 黨m ᆠᆠ�ᨨ矄㡰ᨨ� �矄 ᆠ鼀 ᨨ鼀�犨ᨨ盐 ⼄盐 㝐� ҈桌 m
҈黨
m
㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 꾀ᆠᆠ�ᨨ�矄鼀 ᨨ 犨ᨨ矄⼄盐矄ᨨ 矄� � 矄 泐 �矄 ᆠ 㡰 ᨨ犨 � �盐ᨨ�ᨨ �矄 � 犨� �ᆠ�ᆠᨨ �ᆠ�ᆠ ᆠᨨ 矄 ⧤矄矄 ᨨ犨 � ��盐ᔴᆠ� ᆠᨨ �矄 ᆠᆠ�ᨨ�矄 㝐� 黨 ҈黨ᔼ桌
黨m m 졔
⧤
⧤ Î 㡰 ⧤ + ҈㡰
⧤ Î ᔼ ⧤ 㡰 ⧤ ҈㡰
⧤ Î ҈ ⧤ Î ᔼ ⧤ 㡰 ᆠ㡰 꾀ᆠᆠ�ᨨᨨ ᆠᆠ� ᆠ黨 �矄ᆠᨨ
ᆠ ᨨ泐ᨨ 鼀 ᔼ �ᆠ�ᆠ鼀�ᨨ �ᆠ�ᆠ � ᆠᨨ 矄 k 㦨 �矄 � 矄 犨 矄 鼀� �盐 ᨨ � 盐ᆠ ᆠ �犨� 㝐� ҈桌 㡰 ҈ ᔴ 㡰 ᔴ ᆠ㡰 ᔴ 㡰 黨 ᔴ ҈ ᔼ ᨨᨨ犨� �ᆠ�ᆠ盐� 矄犨�ᆠ 和�矄ᆠᨨ矄 㦨 ᆠ �ᨨ ᆠᆠ�ᨨᔴ҈矄 盐 ᆠ�ᨨ ᨨ �鼀犨� æ ᔴᔴ ö �� 泐 ç ᔴ⼄ = ҈ ÷ ᔴ҈ + ᔴ ø è 㝐� 졠 � 꾀�҈҈桌 㡰 ᔴ⼄쟼҈ ⏌ ᔴ⼄㡰桌 ᔴ ᔴ҈ 㡰 ᔴ҈쟼҈ ⏌ᔴ҈㡰桌
ᔴ
和
㡰 ᔴ⼄쟼 ⏌ ᔴ⼄㡰 쟼 꾀ᆠᆠ�ᨨ犨� ᆠ鼀 矄�ᨨ
犨 矄҈�ᆠ鼀�矄ᆠᨨ
ᆠ ᆠᨨ矄矄泐ᨨ 鼀 ᔼ ᨨᆠᨨ ᆠ�ᨨ 犨 矄�矄�맔 ҈ 盐ᨨ �矄ᨨ �鼀泐 㝐� 졠 � 꾀�҈�҈桌
҈
꾀
ᆠ㡰 ᔴ쟼҈ ⏌ ᔴ㡰桌
㡰 ⏌ᔴ 㡰 ⏌ᔴ ᆠ㡰 ⏌ᔴ 㡰 ⏌ᔴ � ��盐矄 ᨨ犨矄 犨ᨨ├ᨨ ᆠᆠ�ᨨ矄犨 盐鼀矄 � �ᆠ矄 ᆠᆠ� ᆠ矄 � 鼀�ᆠᨨ 盐 ᨨ ᆠᆠ� ᆠᔴ� ⼄盐� ᆠᆠ� ᆠ犨 ᨨ� 꾀 �矄ᨨ҈m �矄 㝐� 졠 � 꾀�҈҈桌
҈
�
҈ m 㡰 m ᆠ㡰 m 㡰 m 㡰
ᆠᆠ�ᨨ矄ᆠ ᨨ m ᨨᆠᨨ
ᆠ � 矄盐ᆠ 犨泐 ᆠ�ᆠᆠ� ᆠ�矄m �矄ᆠ ᨨ 泐ᆠᨨ
ᆠ� � 鼀 㝐� 졠 � 꾀�黨҈桌 㡰 � 矄�矄 㡰 � ᆠ㡰 犨ᨨ� ᨨ � 矄�矄 㡰 犨ᨨ� 矄�矄 ᨨ � ᔼ �� ᆠ�ᆠ盐� ᆠ � 턼 ᨨ Ҙ ᆠᆠ�ᨨ ᆠ �矄矄ᔴ�矄� ᨨ鼀҈Ҙ ᨨҘ 턼矄 盐 ᆠ�ᨨ ᨨ ἘᔴἘ�矄�� ᨨ泐泐 f�gures are drawn schemat�cally and are nᨨt tᨨ scale㡰 㝐� ҈�桌 ҈Ҙ
ᔴ
Ҙ
쟼
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EBD_8341 P-172
Physics ᔴ �
ᔴ �
턼
턼
㡰
㡰 ҈Ҙ
Ҙ
ᔴ
҈Ҙ
ᔴ
ᔴ �
ᔴ �
턼
턼
ᆠ㡰
㡰 ҈Ҙ
䰘
Ҙ
Ҙ
ᔴ
҈Ҙ
Ҙ
ᔴ
ᔼ ��盐 矄泐矄鼀 ᨨ ᨨ盐 ᆠᆠ�ᨨ矄 ᆠᨨ
ᆠ ᆠᨨ矄矄 ҈ 泐 �矄 矄 ᨨ犨 ᔴ � 犨� ᨨ犨 ᨨ鼀 矄犨�ᆠ 和 犨 � �矄ᆠᨨ矄 �矄 㝐� 졠 � 꾀�҈҈ ҈�桌 m m
和 m m
黨
҈ 㡰 �Ҙᔴ ᨨ鼀ᨨ 㡰 Ҙᔴ ᨨ鼀 ᨨ ᆠ㡰 ᨨ 㡰 �Ҙᔴ ᨨ鼀ᨨ 꾀ᆠᆠ�ᨨ�矄鼀 ᨨ 犨ᨨᆠ�ᆠ盐矄 矄 泐 �矄 ᆠ �鼀鼀 犨� �ᆠ�ᆠᨨ �ᆠ�ᆠᆠᨨ 矄 犨 鼀␔ ᆠ�ᆠ � � �ᆠ�ᆠ�矄 ´ ҈ 鼀 ᆠ � 矄�泐ᨨ ᨨ矄��犨�ᆠᨨ矄ᨨ 㝐� ҈桌 䰘 䰘 㡰 ´ ҈ ᔴ 鼀 㡰 ´҈ ᔴ 鼀 ᆠ㡰 ´҈ ᔴ 鼀 㡰 ´҈ ᔴ 鼀 꾀ᆠᆠ�ᨨ � �矄�ᨨ 犨 矄 꾀 鼀�犨� �ᆠ�ᆠᆠᨨ 矄 ⧤ 矄ᆠᆠ� ᆠᔴᨨ犨ᨨ 㜠 � ᨨ 鼀��矄 ᆠ 泐 ᨨ 鼀�犨� �ᆠ�ᆠᆠᨨ 矄 ⧤ 矄ᨨ ᆠ�泐 犨ᨨᆠᆠ�ᨨ矄ᨨ 犨� ҈ 꾀 �ᆠ�ᆠᆠᨨ 矄 ⧤ ᨨ 犨� 꾀 꾀 �ᆠ�ᆠᆠᨨ 矄 ⧤ᔼ ᆠᆠ� ᆠᨨ �矄 犨 ᔴ ᆠᆠ�ᨨ�矄ᔴ �矄 㝐� 졠 � 꾀��҈桌 ᔴ ҈ 㡰 ҈ 㡰 ᆠ㡰 㡰 ᔼ �� ᆠ�ᆠ盐�ᆠ �ᨨ 졔mC�矄�� ᨨ 盐 ᨨ mF ᆠᆠ�ᨨ � 矄 泐矄 ᆠ �ᨨ 盐ᨨ mFᆠᆠ�ᨨ�矄 ҈桌 㡰 졔mᔴ
҈ 꾀m ᆠᆠ�ᨨ�矄ᆠ � 矄矄 ᨨ犨 � ��盐 ᆠ �ᨨ �矄矄ᨨ �泐 �矄矄� 矄犨�ᆠ 和�矄盐 ᨨᨨ矄��ᨨ �矄 ҈҈桌 㡰 奠 㡰 奠 ᆠ㡰 䰘�奠 㡰 奠 꾀ᆠᆠ�ᨨC 犨� 矄ᨨ 盐 � 矄�ᨨ �矄 � 犨� �⼄盐� ᨨ �ᆠ�ᆠᆠᨨ 矄 K 㦨 ᨨ �⼄盐� �矄d⏌� ��泐和盐ᨨ矄 �⼄盐� ᆠ矄矄ᆠᨨ 矄 矄 v �鼀ᆠᨨ 矄 矄 盐 ᆠ�ᨨ ᨨ �鼀�矄 桌 e R 㡰 �d + vt ҈�d + 黨vt 㡰 e R C 㡰 d - dvt - 黨v t d R d⏌ e R ᆠ㡰 �d - vt ҈�d - 黨vt 㡰 e R 㡰 d - dvt - 黨v t 犨ᨨ� �ᆠᆠᆠ�ᨨ矄 矄鼀ᆠᆠ� ᆠC 졠 ᨨ 鼀�矄ᆠ � ᨨᨨ �V҈ ᨨ V
�� 矄 ᨨ ᆠᆠ�ᨨ矄 ᆠᨨ
ᆠ ᨨ� ␔ ᨨ矄�� 矄矄ᨨᆠᨨ
ᆠ ᆠ矄� �泐ᨨ ᆠᨨ鼀� 矄泐矄鼀�矄 和桌 ҈ ҈ C V҈ - V C V҈ + V 㡰 㡰 ҈ ҈ C V҈ - V C V҈ + V 㡰 ᆠ㡰 ᔴᨨ 矄� 矄�盐�ᨨ 矄 ᨨ犨 � ��盐 ᆠᆠ�ᨨ A 矄ᆠ �q ᨨ �犨 矄B �矄盐 ᆠ � ᆠ � � �ᨨ ᆠᆠ�ᨨB ᨨ ��鼀 矄犨�ᆠ �矄 ᆠᨨ矄 �矄 ҈和桌 ⼄
( (
) )
졔 졔 졔 졔 졔 졔
( (
) )
和
꾀
㡰 ᨨ 㡰 q⏌ ᆠ㡰 q 㡰 q � 꾀ᆠᆠ�ᨨᨨ A矄�ᨨ d ᆠᆠ� ᆠ C �矄 � 犨� � �ᆠ�ᆠ 鼀�矄 � � �ᆠ�ᆠ ᆠᨨ 矄 矄 k ҈ k k 矄 矄 ᨨ犨 ᔼ 矄� � �ᆠ�ᆠ鼀��矄ᨨ盐矄 ᨨ 矄鼀ᆠᆠ� ᆠC� �矄ᆠᆠ�ᨨ �矄 �ᆠ�ᆠ ᆠᨨ 矄 k�矄�� 泐 和桌 A⏌
A⏌
K҈
K
d⏌
d K
A
㡰 졔mᔴ ᆠ㡰 졔mᔴ 㡰 졔mᔴ
҈ ҈ ҈ 㡰 ҈ = ҈ + ҈ + ҈ 㡰 = + + K K K K K K҈ K K ҈ K҈ K ᆠ㡰 K = + K 㡰 K㦨K҈졔K졔K K҈ + K
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P-173
Electrostatic Potential and Capacitance
䰘
黨
犨ᨨ� �ᆠ鼀矄�� ᨨ矄��ᆠ �矄Q҈ Q Q҈㡰矄ᆠ�泐ᔼ 泐 ᨨ犨ᨨ盐� ᆠᨨ矄 ᨨ� ᨨ ᨨ鼀ᆠᆠ�ᨨ犨� ᆠᆠ� ᆠ C ᨨ � � ᆠ犨 鼀�矄 ҈黨黨黨㜠㝐├矄桌 㡰 Q҈졔Q㡰⏌ C㡰 㡰 Q҈졔Q㡰⏌C ᆠ㡰 Q҈Q㡰⏌C 㡰 Q҈Q㡰⏌ C㡰 꾀ᆠᆠ�ᨨᨨ ᆠᆠ� ᆠC�矄ᆠᨨ
ᆠ ᨨ 泐 �矄ᆠ � ᨨᨨ � � ᆠV꾀 ᨨ ᆠᆠ�ᨨ ᨨ ᆠᆠ� ᆠ C �矄 矄�鼀�泐 ᆠ � ᨨ ᨨ � � ᆠ V ᆠ �� � 泐�矄 ᨨ犨 �矄ᆠᨨ
ᆠ ᆠᆠ�ᨨ矄ᆠᨨ
ᆠ � ᨨᆠ ᨨ � 矄盐ᆠ 犨泐 ᨨ矄��鼀� ᨨ ᨨ �矄ᆠᨨ
ᆠ ᨨ ��鼀� ᨨ ᨨ � �泐ᨨ ᆠᨨ ��盐�ᨨ �矄 ҈黨黨�和桌 � 黨 CV 㡰 CV 㡰 ᨨ 㡰 CV ᆠ㡰 꾀 ᆠᆠ�ᨨ � �矄ᨨ S 矄�ᨨ 矄ᆠᆠ� ᆠC҈� �␔ 犨ᨨ �ᆠ�ᆠ矄ᨨ � ��鼀����矄 e҈ 㦨 e 㦨㡰� ᨨ 盐ᆠ 犨 犨ᨨ矄矄矄 ᨨ犨 �
C ��盐 ᆠᆠ� ᆠᆠᨨ鼀矄C �ᨨ C �矄 ҈ 꾀 ҈�桌
矄�矄 � 犨� �ᆠ�ᆠ泐矄ᨨ�矄 d �ᆠ�ᆠ 矄 ᆠ 犨� �ᆠ├ 矄矄 d = N æ mö ᆠᨨ 矄 ᨨ 鼀 泐�矄 K m = K ç҈ + ÷ ᨨ泐 è Nø K Î A ö æ � >҈㡰 ᆠᆠ� ᆠᔴ�矄 a ç ÷ 盐 è d ø ᨨ a犨�힘힘힘 Î �矄 鼀���泐ᨨ 矄ᆠ桌 꾀 ҈黨桌 ҈ 꾀�m ᆠᆠ�ᨨ�矄ᆠ � 盐泐泐矄盐泐ᔼ�矄 �矄ᆠᨨ
ᆠ ᨨ鼀 矄盐泐 �矄ᆠᨨ
ᆠ �
矄�矄ᨨ ᨨ 盐 ᆠ � �m ᆠᆠ�ᨨᔼ �泐 X ᆠ � 盐� � ᆠ � �矄�盐�ᨨ �矄 盐 ҈ ᨨ Xᨨ 矄� ��矄힘힘힘힘힘힘힘힘 [Main Sep.02, 2020 (I)] 꾀pFᆠᆠ�ᨨ�矄 盐泐ᆠ � 泐矄盐泐ᔼ �矄 �矄ᆠᨨ
ᆠ ᨨ鼀 矄盐泐 �矄ᆠᨨ
ᆠ ᨨ ᨨ 盐 ᆠ � pFᆠᆠ�ᨨ� ᆠᨨ矄�ᆠ �泐 �矄ᨨ矄� �矄ᨨᆠ矄矄泐 �鼀 ᆠ ��矄 �矄�盐 犨 鼀�矄 � 㡰 㝐� 䰘 ᔼᔼ桌 � �ᆠᆠᆠ�ᨨ矄C҈C C ᆠᆠ� ᆠ ᨨ ҈m ᆠ 泐盐 ᆠ � � ��泐 泐 ᆠᨨ
ᆠ � ᆠ�ᆠ盐�矄矄 ᨨ犨 � ��盐 C҈�矄
� ᆠᨨ鼀泐犨� �ᆠ�ᆠ 鼀� ᨨ � 鼀���泐er ᆠᆠᨨ鼀ᨨ� ᨨᆠ 鼀 㡰V㦨V �矄 矄犨�ᆠ S҈�矄ᆠᨨ矄 犨 � 矄犨�ᆠ S�矄├ ᨨ ␔ ᆠᆠ�ᨨC�矄 盐泐ᆠ � S҈�矄ᨨ S�矄ᆠᨨ矄 矄�鼀盐 ᨨ盐矄泐␔ ᆠᆠ�ᨨ矄 ᆠ ⼄盐���盐鼀 ᆠ �ᨨ C�矄 ᨨ盐 ᨨ�mC
盐ᨨ er㦨힘힘힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌
V
㡰 ⏌� 㡰 �⏌ ᆠ㡰 䰘⏌� 㡰 䰘⏌ 和 ᆠᆠ�ᨨ矄ᆠ ᨨ ᆠᆠ� ᆠ mF ᨨᆠᨨ
ᆠ � ᆠᨨ ��盐�ᨨ ᨨ ᨨ� ᆠ� ᆠᆠ� ᆠ ᨨ
S
C҈ S҈
æ ҈ ö çè ÷ø mF ␔ �ᆠ ᨨ ᆠᨨ鼀� �ᨨ 矄㡰矄 ᨨ犨 � ��盐犨� ҈҈ ᆠ � 矄� 矄盐䁸 ҈黨黨㜠㝐├矄桌 4 㡰
㡰
ᆠ㡰
㡰
2
Integer Value Answer
꾀ᆠᆠ�ᨨᨨ ᆠᆠ� ᆠC 矄矄ᆠ� �d 犨 犨ᨨ矄 � �A ��ᨨ 犨
C
C
Fill in the Blanks
犨ᨨᆠᆠ�ᨨ矄ᨨ ᆠᆠ� ᆠ矄C C ᆠᨨ
ᆠ � ᆠ � ᨨᨨ � � ᆠ V 泐�矄 �矄ᆠᨨ
ᆠ ��ᨨ 犨
矄ᨨ ᆠᆠ�ᨨC�矄ᆠᨨ鼀泐 � 犨� 鼀� ᨨ �ᆠ�ᆠᆠᨨ 矄 K ᨨ � � ᆠ矄ᆠᨨ矄矄 ᆠᆠ�ᨨ矄 ᨨ犨ᆠᨨ鼀矄 ҈黨㜠㝐├矄桌 � �� �ᆠᆠᆠ�ᨨ矄ᆠ ᨨ A � 矄盐ᆠ ᆠ 矄 �矄 ᆠd 矄ᆠᨨ
ᆠ ᨨ矄ᨨ盐ᆠᨨ 鼀 V矄矄 ᨨ犨 � ��盐 ҈黨㜠㝐├矄桌
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EBD_8341 P-174
Physics
ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐 犨ᨨ ᆠᆠ�ᨨ矄ᆠ ᨨ ᆠᆠ� ᆠC. 矄犨�ᆠ S҈ �矄 矄矄 �矄 ᨨ 盐泐ᆠ � ᆠᆠ�ᨨ C҈
矄 矄犨�ᆠ S �矄 矄矄 ᨨ ᆠ � ᆠᆠ�ᨨC. 꾀 矄ᨨ鼀�鼀S �矄矄 S �矄 矄矄 꾀 矄ᨨ鼀�鼀 꾀 ҈桌
– + ҈ �
6
MCQs with One or More than One Correct Answer
ᔼ ᆠ�ᆠ盐� 矄 ᨨ犨 � ��泐 �矄 ᨨ ᆠ � ᨨ
ᆠᆠ�ᨨ矄 ├泐矄和҈ 和ᨨ 盐矄ᨨ ᆠᆠ�ᨨ矄ᔴ҈㦨҈mFᔴ㦨mF C㦨C㦨mF 꾀 ҈黨桌
黨
䰘
␔ �ᆠ ᨨ 矄鼀 矄㡰�矄⏌ᆠᨨᆠ䁸 㡰 ᔼ ├泐 和 ҈ �矄 ├ ᆠᨨ矄 ᨨ ᨨ � �鼀 矄盐ᆠ ᆠᆠ�ᨨ矄 盐泐ᆠ � ᨨ� � ᆠ 犨 ᨨ� 矄 턼犨�҈ 㡰 ├泐和҈�矄├ᆠᨨ矄 ᨨᨨ ��鼀矄盐ᆠ ᆠᆠ�ᨨ矄 盐泐ᆠ � ᨨ犨├泐和�矄ᆠᨨ矄 �矄�鼀 � 矄 ᨨ盐矄ᆠ盐 ᆠᨨ矄矄矄�矄ᨨ 犨 ᨨ� 矄 턼㡰犨�꾀 ᨨ盐 ᨨ ᨨ҈矄 ᆠ�鼀ᆠ㡰 ᆠ㡰 꾀�鼀t㦨 ├泐和҈�矄ᆠᨨ矄 � 矄 ᨨ盐矄 ᆠ盐 � ᆠᨨ矄 ᆠ�ᆠ盐�犨��鼀꾀 㡰 ᔼ ├泐 和 ҈ �矄 ├ ᆠᨨ矄 ᨨ ᨨ � �鼀 矄盐ᆠ ᆠᆠ�ᨨ矄 盐泐ᆠ � ᨨ�ᆠᨨ矄矄 ᆠᆠ�ᨨ矄ᔴ҈犨� 꾀ᆠᆠ�ᨨ 矄 �ᆠ�ᆠ矄ᨨ �ᆠ�ᆠ ᆠᨨ 矄 K 犨 �矄矄 ᆠᨨ矄҈⏌ᨨ ᨨ �矄矄矄矄 ᨨ犨 � ��盐 ᨨᆠᆠ� ᆠ ᨨ ᆠᆠ�ᨨ�矄C犨 � ᨨ ᨨ�ᨨ 犨� �ᆠ�ᆠ � 犨 �矄C҈␔ ᆠᆠ�ᨨ�矄ᆠ � ᆠᨨ 泐 �ᆠ�ᆠ�矄ᆠ �Q҈ 矄 ᨨ �矄ᆠ �Q ᆠ�ᆠ � � �ᆠ�ᆠ �矄E҈ � ᨨ ᨨ�ᨨ �矄Eᔴ ᨨᨨ矄 ᆠᨨᆠ ᨨ�ᨨ ⏌ᨨ�ᨨ 矄�� ᨨ� � � ᆠ矄 꾀 ҈桌 E҈ 㡰 E = ҈ Q҈ E
Q҈ ᆠ㡰 Q = K
C + K 㡰 C = K ҈
�
�҈
҈
E҈ ҈ 㡰 E = K
�
Q
E
和
和҈
ᆠ �ᨨ ҈�矄 ᨨ �矄
ᔴ҈
和
ᔴ
ᔴ
㡰 ᆠ �ᨨ 盐ᨨ C҈ �矄CV 㡰 ᆠ �ᨨ 盐ᨨ C҈ �矄CV ᆠ㡰 ᆠ �ᨨ 盐ᨨ C �矄 㡰 ᆠ �ᨨ 盐ᨨ C �矄CV 꾀 �ᆠ�ᆠ矄ᨨ �ᆠ├ 矄矄d�矄� 矄 � ᆠᆠ�ᨨ犨 ᨨ矄 ���矄x㦨 ᨨ矄�� �矄x㦨d 矄�矄⼄盐� �矄 ᨨ鼀 矄 ᆠᆠ�ᨨ�矄�� 矄ᨨ鼀ᆠ �꾀矄ᨨ �ᨨ矄 ᨨ鼀ᨨd ҈黨黨和㜠㝐├矄桌 㡰 鼀� �盐 ᨨ ᆠ�ᆠ � 鼀� 矄 矄鼀 㡰 �ᆠ�ᨨ ᨨ ᆠ�ᆠ � 鼀� 矄 矄鼀 ᆠ㡰 ᆠ�ᆠᨨ �� ᆠ矄矄ᆠᨨ � 盐ᨨ盐矄泐 㡰 ᆠ�ᆠᨨ �� ᆠ矄矄 �矄 ᆠ矄矄 �� � ᆠ矄矄 꾀ᆠᆠ�ᨨᨨ A 矄�ᨨ
�矄ᆠ � ᨨᨨ � � ᆠV 泐 �矄 �矄ᆠᨨ
ᆠ 꾀矄ᨨ �ᆠ�ᆠᆠᨨ 矄 K �矄
� 矄 犨 矄ᨨ ᆠᆠ�ᨨ矄ᨨ矄ᨨ � 矄ᆠ 犨 矄 ᔼ Q E W ᨨ 矄ᆠ�泐 鼀� �盐 ᨨ ᆠ �ᨨ ᆠ ᆠ�ᆠ � 犨 矄 矄�矄� 矄 㡰 犨ᨨ├ ᨨ ᨨ 矄泐矄鼀� ⼄盐矄�ᨨ � ᨨᆠ矄矄 ᨨ � 矄� � 矄
҈黨黨҈㜠㝐├矄桌 e AV e KAV 㡰 Q㦨 㡰 Q㦨 d d V e AV é ҈ù ᆠ㡰 E㦨 㡰 W㦨 ҈- ú ê Kd d ë K û 꾀 ᆠᆠ�ᨨ �矄 ᆠ � ᆠ �� � 泐�矄 �矄ᆠᨨ
ᆠ ᔼ 矄ᨨ ᆠᆠ�ᨨ 鼀ᨨ 泐鼀 矄ᨨ � 矄盐� � 矄 ҈黨䰘㜠㝐├矄桌 㡰 ᆠ �ᨨ ᆠᆠ�ᨨ� ᆠ矄矄 㡰 ᨨ�ᆠᨨ矄矄 矄� ᆠ矄矄 ᆠ㡰 ᆠᆠ� ᆠ� ᆠ矄矄 㡰 ᆠᨨ矄�ᆠ �泐 矄ᨨ � ᆠᆠ�ᨨ � ᆠ矄矄 꾀�ᆠᆠ�ᨨ �矄ᆠᨨ
ᆠ ᨨ泐
⼄盐 ��矄ᆠ �ᨨ�ᆠ�ᆠ � �泐 矄矄ᨨᆠ� 犨� �矄ᆠᆠ�ᨨ�� 泐QVE U矄ᆠ�泐꾀 �ᆠ�ᆠ矄�矄 ᨨ犨� ᨨ 盐ᆠ ᨨ � 矄ᆠ 犨 矄 犨� 泐 矄� �
ᆠᨨ
ᆠ�ᨨ ᆠᨨ矄ᨨ � �⼄盐 ��矄 ᨨ犨�� 泐 QVE U ᨨ �ᨨ盐矄ᨨ 矄 ҈黨�㜠㝐├矄桌 㡰 Q > Q 㡰 V > V ᆠ㡰 E > E 㡰 U > U
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P-175
Electrostatic Potential and Capacitance
10 Subjective Problems �
犨ᨨᆠᆠ�ᨨ矄A B犨� ᆠᆠ��矄mF mF ᆠ � ᨨ ᨨ � � ᆠᨨ ҈ ҈ 矄ᆠ�泐 矄ᨨ ᆠᆠ�ᨨ矄ᆠᨨ
ᆠ 矄矄 ᨨ犨 � ��盐犨� ᨨ 犨� ᨨ鼀ᆠ ᆠᆠ�ᨨ 盐 ᨨ A �矄 ᨨ矄�� ᨨ B �矄
��꾀 盐 ᆠ � mFᆠᆠ�ᨨC犨� 犨�矄 矄ᨨ 矄ᨨᆠᨨ鼀 ᆠ�ᆠ盐�ᔴᆠ盐 ҈黨黨䰘㜠�㝐├矄桌
� 犨ᨨ ᆠᆠ�ᨨ矄 A B 矄鼀 矄�ᨨ d㦨�맔҈鼀犨 矄 ᨨ A B鼀 鼀矄ᆠ�泐꾀矄 ᨨ �ᆠ�ᆠ ᆠᨨ 矄 � 鼀���泐㡰 K 㦨 黨 矄 �鼀 矄�ᨨ 矄矄盐ᆠ �ᆠ ᆠ泐 � 矄ᆠ犨
矄ᨨ ᆠᆠ�ᨨB ҈黨黨㜠졔졔㝐├矄桌 꾀
҈҈ 㡰
�
��
�㡰 � ᆠ �ᨨ ᆠᆠ�ᨨ矄 ��㡰 鼀ᨨ盐 ᨨ ᆠᨨ矄�ᆠ �泐矄ᨨ � 矄泐矄鼀 ᨨ ᆠᨨ鼀�ᨨ ᨨ ᆠ�ᆠ盐�
ᆠᆠ� ᆠᨨ ᆠᆠ�ᨨ 犨� A 矄�ᨨ d�矄C 矄ᆠ犨
矄�矄 � 犨� 犨ᨨ犨 �矄 ᨨ �ᆠ�ᆠᆠᨨ 矄 矄 K҈ K 矄ᆠ�泐 � ᆠᆠ� ᆠ ᨨ ҈黨黨㜠㝐├矄桌 矄盐� �ᆠᆠ�ᨨ
犨ᨨ矄⼄盐鼀矄ᨨ 矄� ҈鼀├҈鼀 �├ᆠᆠ�ᨨ� �� 矄盐ᆠ 犨泐 ᨨ ᨨ � �矄�矄 �ᆠ盐ᨨ ᨨ�矄盐 ᆠ� ├ � 犨� � 矄盐� �ᨨ� 矄ᆠᨨ
ᆠ ᨨ 泐ᨨ 鼀 � 矄 ᨨ犨 �ᆠ泐 � ᨨ ᨨ�矄 ᨨ ҈鼀矄҈ᔴᆠ盐 ᆠ盐 犨 ᨨ鼀 泐 盐� � ᨨᆠ矄矄 �ᆠ�ᆠ ᆠᨨ 矄 ᨨ ᨨ�㦨҈҈e㦨�맔҈҈ᔴ҈鼀҈㡰 ҈黨黨㜠㝐├矄桌
㡰
꾀
ᆠ㡰
�㡰 �ᆠ�ᆠ矄�矄ᆠ � 矄� 꾀矄矄 ᨨ犨 � ��盐 㡰꾀�矄 ᆠ � ᨨᨨ � � ᆠᨨ ҈҈ ᔴᆠ盐 ᆠᆠ� ᆠᨨ A �泐矄ᨨ � � ��㡰 泐�矄 �矄ᆠᨨ
ᆠ �ᆠ�ᆠ 矄 �矄 鼀ᨨ ᨨ鼀 A � 犨ᨨ├ ᨨ 泐 � ᆠ泐� 鼀ᨨ� � 矄 ᨨ鼀A ���㡰 矄鼀 �ᆠ�ᆠ矄�矄 ᨨ犨ᆠ � 矄� B �� � �ᆠᨨ鼀泐 犨ᨨᆠᆠ�ᨨ矄A B
ᆠᨨ
ᆠ 矄矄 ᨨ犨 � ��盐 ᆠ㡰ᔴᆠ盐 �泐 矄ᨨ � 矄泐矄鼀 �䰘 ��盐 矄 ᨨ犨矄犨ᨨ � �ᆠ 和 ᆠᆠ�ᨨ矄 ᆠᨨ
ᆠ ᨨ 泐犨� 矄犨�ᆠ Sᆠᨨ矄 꾀 ᔴ ᔴ
矄犨�ᆠ �矄 ᨨ犨ᨨ 矄ᆠ犨 矄 ᨨ ᆠᆠ�ᨨ矄�矄 � 犨� �ᆠ�ᆠᨨ �ᆠ�ᆠᆠᨨ 矄 ᨨ�鼀���泐㡰 � �ᨨᨨ ᨨᆠᨨ矄�ᆠ �泐矄ᨨ �
ᨨ ᆠᆠ�ᨨ矄 ᨨ � ᨨ 盐ᆠ�ᨨ ᨨ �ᆠ�ᆠ ҈黨㜠㝐├矄桌
Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 ҈
犨ᨨ � �ᆠ 鼀 矄 � � 矄�ᨨ ᨨ ҈ᆠ鼀ᆠᨨ
ᆠ ᨨᔴᨨ�矄ᨨ盐ᆠ ᨨ ᨨ � � ᆠX꾀ᨨᨨ �矄矄 矄鼀� 犨泐 犨 犨ᨨ矄ᔼ�矄 ᨨ盐 ᨨ鼀ᨨ� ᨨ �ᆠ和 矄 X�矄 泐 ҈桌 㡰 ҈´ ҈�V 㡰 ҈´ ҈䰘V ᆠ㡰 ҈´ ҈黨V 㡰 ҈´ ҈҈V 6
㡰 ᆠ�ᆠ � A� ᆠ�泐㦨ᆠ�ᆠ � B � ᆠ�泐 㡰 ᆠ � 矄�泐A㦨ᆠ � 矄�泐B ᆠ㡰 ᨨ �A㦨ᨨ �B 㡰 ᨨᆠ�ᆠ � 盐 ᨨ盐� 矄盐 ᆠᨨ ᆠ�泐 �矄q⏌e
MCQs with One Correct Answer
MCQs with One or More than One Correct Answer
꾀 �矄ᨨ� ᆠ�泐 �矄 ᆠ 犨� � ᆠ ᆠᨨ 盐ᆠᨨ 꾀 ᨨ矄�� ᆠ � q �矄 ᆠ ᆠ ᨨ ᆠ�泐 ᨨ� 矄A Bᨨ ᆠ�泐矄盐 ᆠ矄矄 ᨨ犨 �
��盐
҈黨黨黨和㜠㝐├矄桌
A
q
B
7
Match the Following
和� ᨨ� ᆠ �矄 ᆠ ᨨ 矄鼀鼀� �盐 q, � � � 鼀
矄矄矄 ᨨ犨 � ᔴᨨ盐鼀 㜠ᔼᔼᔼ
ᆠ ᆠ矄ᨨ� 㝐 � 턼矄矄� � ᨨ盐� 㝐 矄 ᨨ犨 E ᆠ�ᆠ � V ᆠ�ᆠ ᨨ �㝐 ᨨ �� � �泐�矄 ᨨ㡰 盐ᨨ ��
ᆠ � �矄�盐�ᨨ 犨 ��矄矄ᨨ犨 犨 ᨨ矄泐矄鼀 �矄矄� ᨨᨨ�ᨨ 犨� ᆠᨨ 矄 �盐ᨨᆠ�泐ᨨ盐 � 턼 鼀� �ᆠ � 㝐 m 鼀� �ᆠ鼀ᨨ鼀 ᨨ 矄泐矄鼀� �矄ᆠᨨ ��ᨨ 꾀矄矄盐鼀 ᆠ ᨨ� �ᆠ �ᨨ ⼄盐� ᨨ矄 泐ᆠ盐 黨桌
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EBD_8341 P-176
Physics
ᔴᨨ盐鼀 㜠ᔼ
ᔴᨨ盐鼀 㜠ᔼᔼ 졔
꾀㡰
㡰
E㦨
턼
ᔴ �矄 ᆠᨨ 矄ᨨ �盐 �ᨨ 㝐�矄 ᆠ ᨨ �ᨨ 턼�矄 �ᆠ盐ᨨ ᨨ �ᨨ
졔
㝐
졔
㡰
V¹
졔
⼄㡰 졔
ᔴ㡰
㡰
B㦨
ᔴ �矄ᨨ � �ᆠ盐ᨨ턼⼄盐� 矄㝐�矄 鼀� 㜠ᨨ� 犨 犨ᨨ�
鼀ᨨ矄ᆠ �矄
턼
ᔴ �矄 ᆠ ᨨ 犨ᨨ ᆠᨨ � 矄盐� � � �矄 ⼄盐 � 矄㝐�矄 ᆠᨨ鼀鼀ᨨ ᆠ ᨨ � �矄턼�矄 �ᆠ盐 ᨨ ᨨ � �矄
㝐
졔
m ¹ 0
졔
㡰
졔 졔 㝐 턼
矄㡰
졔
㝐
ᔴ �矄ᆠ ᆠᨨ 矄ᨨ ᆠ �ᨨ 矄� 矄a a 鼀� ᨨ� 矄ᨨ ᨨ �矄� 矄㝐�矄 ᆠ ᨨ ᆠ �턼�矄ᨨ ᨨ �矄� 矄
턼
졔 졔
)
졔
졔
㝐
턼
Comprehension/Passage Based Questions 矄矄�㜠҈ ᔴᨨ 矄� 矄�鼀 ᔴᆠ�ᆠ盐�矄矄 ᨨ犨 � ��盐҈ ᨨᆠ矄矄҈ᔼ ᆠ�ᆠ盐� 矄犨�ᆠ 和�矄ᆠᨨ矄 㦨 ᆠᆠ�ᨨ�矄 盐泐ᆠ � ᨨᨨ� �ᆠ �� �ᆠᨨ � 盐矄 ᨨ�鼀 >> ᔴ㡰ᔼ ᨨᆠ矄矄矄ᨨ鼀 �矄矄��ᨨ 쟼㡰ᨨᆠᆠ盐矄 ᆠᨨ矄矄 矄�矄 ᆠ 鼀ᨨ盐 ᨨ �泐 � 泐矄ᨨ �
盐泐ᆠ � ᆠᆠ�ᨨ�矄쟼ᔴ ᨨᆠ矄矄ᔼ � ᨨᆠ矄矄 ᨨ��矄 �矄矄ᨨ 鼀� � ᨨᆠ �� ��鼀 >> ᔴ ᨨ��矄 8
犨� ᨨ盐 �矄ᆠ �� � ᆠᆠ�ᨨ ��
鼀� � ᨨ�鼀 >> ᔴ ᨨᆠ矄矄�矄 ᨨ 鼀ᨨ �鼀泐�矄� � ᨨ�ᨨ ᆠᆠ�ᨨ�矄ᆠ � ᨨ 矄鼀 � ᨨ�矄� ᨨᆠ矄矄҈
矄犨ᨨᨨᆠ矄矄矄 �ᆠ � ��盐
�矄 ᨨ
和
졔
ᔴ
��盐҈
ᔴ �矄ᆠ ᨨ 犨ᨨᆠᨨ � �ᆠ� 矄盐� �� �矄⼄盐 � 矄㝐�矄 鼀� 㜠ᨨ� 犨 ᆠ 矄ᨨ � �矄턼�矄 �ᆠ盐ᨨ � ᨨ� � � ᆠ 矄 ᆠᨨ ᨨ � �矄 㡰 쟼ᔴ㦨쟼 㡰 쟼ᔴ㦨쟼ᔼ ҈ ᆠ㡰 쟼ᔴ㦨 쟼 㡰 쟼ᔴ㦨쟼 � ᔼ ᨨᆠ矄矄ᨨ �泐 �矄矄� ᆠᨨ矄矄 矄�矄 ᆠ 쟼�矄 꾀 ҈䰘桌 æ҈ ҈ ö 㡰 쟼㦨 ᔴ 㡰 쟼㦨 ç ᔴ ÷ è ø ҈æ ҈ ö ᆠ㡰 쟼㦨 ç ᔴ ÷ 㡰 쟼㦨 ᔴ è ø 矄矄�㜠 ᔴᨨ 矄� ᆠ盐 ᆠ泐� �ᆠᆠ 鼀ᨨ �� � � ��� ᆠᨨ 盐ᆠ� �矄 矄 � 矄盐� �ᆠ盐 矄盐 ᆠ矄矄 ᨨ犨 � ��盐꾀 盐鼀 ᨨ 矄 �ᆠ矄 鼀 ᨨ �� 犨�� 矄ᨨ 鼀� ᆠᨨ 犨� ᆠᨨ 盐ᆠ� �鼀�ᆠ ᨨ ᨨᨨ鼀 矄 �盐矄 ᨨ犨 �� ᨨ�矄ᨨ盐ᆠ 㡰�矄ᆠᨨ
ᆠ ᆠᨨ矄矄 ᆠᨨ 盐ᆠ� �矄矄盐ᆠ ᨨᨨ鼀�矄 졔 ᨨ盐ᨨ �ᆠᨨ 盐ᆠ� �矄盐 ᆠ 矄犨��ᆠ � 犨�ᆠᨨ鼀⼄盐�ᨨ �犨� 泐� 矄犨� 盐泐ᆠᨨ� 犨� ᨨ犨 ᆠᨨ �ᆠ� ᨨ 矄�盐�ᨨ ᆠ ├ ᨨ ᨨ 盐ᨨ 矄ᨨ 盐ᨨ 鼀�ᨨ 矄 ᆠ�ᆠ � � ᆠ 鼀ᆠ ᆠᨨ 矄� ᨨ ᨨ ᆠᆠ�ᨨ꾀矄矄盐鼀 ᨨ ᆠᨨ�矄�ᨨ 矄犨 矄 � ᆠ�ᨨ 犨 鼀 �矄 ���� ᔼ� ᨨ��泐㡰 꾀
ᨨᆠ矄矄҈
⏌
ᨨᆠ矄矄
>> ᔴ
⏌
졔
��盐 ᔼ ᨨᆠ矄矄҈ �泐矄ᨨ � ᆠᆠ�ᨨ쟼ᔴ �矄矄� ᆠᨨ矄矄矄�矄 ᆠ쟼 泐 꾀 ҈䰘桌
␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� �矄鼀 矄�矄ᆠᨨᆠ䁸 꾀 ҈桌 㡰 矄犨�矄�ᆠ├ᨨ ᨨ 鼀� 㡰 矄 犨� ᨨ盐 ᆠ ᆠ├ ᨨ ᨨᨨ鼀 ᆠ泐� � 矄鼀ᆠ � 泐犨 盐犨�
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P-177
Electrostatic Potential and Capacitance
ᆠ㡰 矄 犨� ᨨ盐 ᆠ ᆠ├ ᨨ ᨨᨨ鼀 ᆠ泐� � ᨨᨨ矄�ᆠ � 泐犨 盐犨� 㡰 矄 犨� ᆠ盐 矄�鼀 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ
犨 犨ᨨ矄
�ᆠ盐 � 矄 泐矄��矄 泐 鼀鼀� ᆠ�ᆠ盐�犨� 꾀 ҈桌 㡰 ᨨ 㡰 ᨨᨨ�ᨨ ᨨ ᨨ �
䰘
ᆠ㡰 ᨨᨨ�ᨨ ᨨ ҈⏌ 㡰 ᨨᨨ�ᨨ ᨨ 10 Subjective Problems ᨨ盐ᨨ� ᆠ �矄졔mC҈mC҈mC 졔mC
䰘 䰘 鼀- 鼀 + 鼀 + 鼀 矄ᆠ�泐ᨨ 泐㜠 �矄꾀�ᆠᨨ 鼀矄矄맔҈├� ᆠ �졔҈mᔴ鼀ᨨ矄ᨨ � 㜠x �ᆠ�ᨨ ᔼ矄矄 x㦨졔 ¥ �矄V � 矄盐ᨨ V ᨨ犨 �ᆠ �ᆠ犨�ᆠᨨ矄矄 ᨨ��� � 矄ᨨ ├� �ᆠ �泐 ᨨ �ᆠ ᨨ��� 꾀矄矄盐鼀 矄ᆠ�矄��泐 ҈ 黨 � pe = 黨 ´ ҈ 鼀 ⏌ ᔴ 㜠҈㝐├矄桌 꾀 ᨨ 㜠ᆠᨨ 盐ᆠ� � �矄ᆠᨨ �盐矄 盐 � ᨨ鼀ᨨ矄�� 矄盐 ᆠᆠ � 矄�泐 s �矄ᆠ ᨨ �ᨨ盐 犨� �矄 �矄�ᆠ꾀�ᆠᨨ 鼀矄矄m ᨨ矄��ᆠ �q�矄 ᨨ ᨨ � �矄ᨨ �矄ᆠ ᨨ鼀 �� H犨� ᨨ� ��ᨨᆠ�泐 �ᆠ 矄 q m = Î g ⏌ s ҈黨黨黨㜠҈㝐├矄㡰 㡰 � 盐ᨨ H� �ᆠ盐矄ᆠ 矄 �矄ᆠ 㡰 和├ᆠ ᨨ � �泐ᨨ �ᆠ矄 盐 ᆠ�ᨨ
ᨨ �矄 �� � �矄⼄盐���盐鼀ᨨ矄��ᨨ 꾀 ᆠᨨ 盐ᆠ� � 矄 S҈ ᨨ �盐矄 �矄 ᆠ ᨨ
� 矄盐� � 꾀 ᨨ ᆠᨨ 盐ᆠ� �矄 Sᨨ �盐矄 R�矄鼀ᨨ盐 ᨨ � 矄盐� �矄 S�矄� ��泐盐 ᆠ �
� ᨨ� 矄 -
黨
҈
S҈�矄�� ᆠ �Qᨨ盐� � ᨨᆠᨨ ᆠ犨� S 鼀ᨨ S҈�矄ᆠ � 矄盐ᆠ ᆠ �ᨨ ��矄��
Q錐 ��矄�� ᨨ盐� � ᨨᆠᨨ ᆠ犨� S 鼀ᨨ
�矄ᨨᆠ 盐�矄 �鼀矄 ҈黨黨㜠㝐├矄桌 㡰 � ᆠᨨ矄�ᆠ �泐 ᨨ S 矄盐ᆠ ᆠᨨ ᆠ矄犨� S҈ 㡰 ␔ �矄 �鼀�� �盐ᨨ �矄 �泐矄 n ® ¥ 䁸 ҈҈ 㡰 꾀矄矄盐鼀 ᨨ 矄 ᨨ 盐 � ᨨ鼀 鼀矄矄 矄�泐ᔴᆠ盐 �矄 �泐�� ᨨ 盐ᆠᨨ 鼀矄矄 �盐矄ᨨ ᨨ�맔҈҈ ├�鼀꾀ᆠ �ᨨ Qᆠᨨ盐ᨨ鼀�矄盐 � ᨨ鼀泐 �矄�盐 ᨨ矄 �ᆠᨨ盐鼀ᨨ �盐矄R鼀矄졠�
矄矄�ᨨ ᨨ �泐ᨨ 矄泐矄鼀 㡰 ␔ 犨� ᆠᨨ矄ᨨ � � 矄矄�ᨨ ᨨ �泐 ᨨᆠᨨ鼀泐 �矄矄矄鼀 �� 矄 ���ᨨ 盐 鼀ᨨ �矄 �矄 ᆠᨨ 矄�盐 �ᆠ矄䁸 ᆠ㡰 ᔼ 矄鼀ᆠ �ᨨ Qᆠᨨ盐ᨨ鼀矄� 㡰ᨨ�矄 �� ᨨ矄 �ᆠᆠᨨ 盐ᆠᨨᨨ 矄鼀 �盐矄R犨 犨� �泐ᨨ 矄泐矄鼀䁸 ҈黨黨㜠҈㝐├矄桌 ҈ 꾀ᨨ� �ᆠᨨ 鼀矄矄M �矄ᆠ ᨨᨨ ᨨ 鼀矄矄矄矄��� ᨨ 㜠ᆠᨨ 盐ᆠ� � 턼졔⼄ ᨨ ᨨ � L 꾀 ᨨ q 쟼 ᨨ� �ᆠᨨ 矄鼀 턼㜠⼄ 鼀矄矄 �矄 ᆠ ᨨ ᨨ ᨨ ᨨ 犨ᨨ�ᆠ矄ᆠ泐ᆠ �矄졔q ⼄矄ᆠ�泐 �矄 �鼀 �矄 � ��ᨨ ᨨ 盐 � ᨨ鼀ᆠ�ᆠ � E 矄盐ᆠ ᨨ 鼀├矄矄鼀 � q 矄泐ᨨ ᨨ盐� �㡰犨� � �ᆠ�ᨨ �� � 矄矄�ᨨ ᨨ 鼀� �鼀盐鼀�鼀 ᨨ ᨨ ᨨᆠᨨ鼀ᨨ � ��矄矄 ҈黨黨㜠鼀├桌 ҈ �ᆠ矄ᆠ ᨨ 鼀矄矄҈�鼀 ᆠ泐� �ᆠ � q 矄盐矄 ᨨ鼀 ᆠᨨ鼀鼀ᨨ ᨨ� 泐 � 矄盐 鼀矄矄矄矄矄� �矄ᆠ ҈ᆠ鼀ᨨ �ᔼ �ᆠ矄�
⼄盐���盐鼀 ᨨᆠ ᆠᨨ 矄ᨨ ⼄盐� � �ᨨ 矄� � ᆠ鼀ᆠᆠ盐 ᆠ �qᨨ ᆠ �ᆠ ├g㦨҈鼀⏌矄㡰 ҈黨㜠�㝐├矄桌
Answer Key Topic-1 : Electrostatic Potential and equipotential surfaces ҈ ҈҈ ҈ ҈ ҈҈
ᆠ㡰 㡰 㡰 㡰 � ᆠ㡰 㡰 䰘 ᆠ㡰 㡰 黨 㡰 ҈ 㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ 㡰 ҈� 㡰 ҈ ᆠ㡰 ҈䰘 ᆠ㡰 ҈ 㡰 ҈黨 㡰 㡰 㡰 㡰 㡰 㡰 � 㡰 㡰 䰘 㡰 ᆠ 㡰 黨 ᆠ㡰 ᆠ 㡰 ҈ ᆠ 㡰 㡰 㡰 Topic-2 : Electric potential energy and work done in carrying a charge 㡰 ᆠ㡰 㡰 㡰 � 㡰 ᆠ㡰 䰘 ᆠ㡰 ᆠ㡰 黨 㡰 ҈ 㡰 㡰 ҈ 㡰 ҈ 㡰 Topic-3 : Capacitors, grouping of capacitors and energy stored in a capacitor
҈ ҈҈ ҈ ҈
㡰 㡰 㡰 ᆠ㡰
҈
㡰 㡰 ᆠ㡰 㡰
҈
҈㡰
㡰
㡰 黨 㡰
҈
㡰 㡰 㡰 㡰
㡰 � ᆠ㡰
㡰 㡰 ҈ 㡰 㡰 ҈�㡰 � 㡰 ⧤+ �҈ ᆠ 㡰 � 㡰 ҈
㡰 ᆠ㡰 㡰 ᆠ㡰
� ҈� � �
ᆠ㡰 㡰 㡰 㡰
㡰 㡰 㡰 ҈ ᆠ㡰 㡰 㡰 㡰 㡰 쟼꾀 -쟼 꾀 � 㡰 䰘 ҈䰘 䰘 䰘
黨 ҈黨 黨 黨
㡰 㡰 㡰 㡰
҈
㡰 㡰 㡰 㡰
䰘 ᆠ 㡰
Topic-4 : Miscellaneous (Mixex Concepts) Problems ҈
ᆠ㡰
ᆠ 㡰
꾀㜠 矄㡰㜠 矄㡰ᔴ㜠 ⼄㡰㜠 矄㡰
㡰
� ᆠ㡰
ᆠ㡰
䰘 㡰
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EBD_8341 P-178
Physics
17
Current Electricity
Topic-1 : Electric Current, Drift of Electrons, Ohm's Law, Resistance and Resistivity 1 ҈
MCQs with One Correct Answer
꾀ᆠ�ᆠ盐�ᨨ� 泐졠 鼀矄犨盐矄矄鼀鼀 ᨨ鼀 � 矄�矄ᨨᆠᨨ
ᆠ ᆠᨨᆠ泐ᨨ 矄�矄ᨨᔼ
ᆠ�ᆠ盐� 㝐� 和 ᔼᔼ㡰桌 㡰 鼀鼀�矄犨泐矄盐矄 � ᨨ鼀�矄矄�矄 㡰 ᨨ 鼀鼀 ᨨ鼀鼀盐矄ᆠᨨ
ᆠ � ᆠ㡰 鼀鼀�矄犨泐矄ᆠᨨ
ᆠ � 矄�矄 ᨨ鼀 � 㡰 ᨨ 鼀鼀 ᨨ鼀鼀盐矄ᆠᨨ
ᆠ � 矄�矄 ᔴᨨ 矄� ᨨ盐 ᆠᨨ 盐ᆠ� � 鼀�矄 ᆠᨨ 盐 �矄 鼀ᆠ盐泐 盐鼀� �盐鼀犨� 矄�矄��泐rCrTrM rA 矄ᆠ�泐 㝐� 和 ᔼ㡰桌 㡰 rC > r A > rT 㡰 r M > r A > rC ᆠ㡰 r A > rT > rC 㡰 r A > rM > rC
ᨨ� 泐졠 鼀矄 犨 矄盐 ᆠᨨ
ᆠ矄 ᨨ鼀 ᆠᨨ矄矄 泐矄矄 ᨨ犨 � ��盐 鼀矄盐 ᨨ� �矄ᨨ 矄 盐 ᆠ�ᨨ ᨨ ᆠ盐 ᨨᨨ犨� �� �矄ᨨ� 㝐� ҈꾀҈黨 ᔼ㡰桌
�
ᆠ㡰 ᨨ � � ᆠᆠᨨ矄矄 泐�矄҈�犨
�矄 矄ᆠ盐 ᨨ ҈鼀꾀 㡰 鼀 ᨨ 泐�矄҈� 盐ᨨ �矄҈�W 꾀 ᆠ盐 ᨨ �꾀 矄矄矄 ᨨ盐� ᆠᨨ ᆠᨨ 盐ᆠᨨ 矄�矄��泐㡰㦨҈䰘맔҈W鼀㡰ᨨ �盐矄ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ � 鼀鼀 � 鼀ᨨ��泐ᨨ ᆠ �矄� � � ᨨᆠ�泐 �矄҈҈맔҈鼀⏌矄 㝐� ҈꾀҈黨 ᔼ㡰桌 㡰 ҈鼀⏌矄 㡰 ҈�鼀⏌矄 ᆠ㡰 ҈鼀⏌矄 㡰 ҈鼀⏌矄 和ᆠ犨 犨ᨨᆠᨨ ᆠ �ᆠᆠᨨ 盐ᆠ� �矄 矄ᨨ �� a b b>a㡰�矄 � 犨� 鼀 �盐鼀ᨨ 矄�矄��泐r 矄�矄 ᆠ犨 犨ᨨ矄 矄犨� 㝐� ҈꾀҈黨 ᔼᔼ㡰桌 㡰
r æ҈ ҈ö ç - ÷ p è a b ø
r æ҈ ҈ö ç - ÷ p è a b ø
r æ҈ ҈ö r æ҈ ҈ö 㡰 ç + ÷ ç + ÷ p è a b ø p è a b ø ᔼ ᆠᨨ 盐ᆠᨨ� 盐鼀ᨨ ᆠᨨ 盐ᆠ�ᨨ ᆠᨨ 矄 盐 �ᨨ盐鼀�矄�맔҈ 鼀 鼀 �鼀�矄 �fs 鼀ᨨ矄ᆠᨨ 㡰�矄ᨨ �鼀矄�矄��泐�矄 鼀 㦨黨҈맔҈҈ ├�㡰 㝐� 黨꾀҈黨 ᔼᔼ㡰桌 㡰 ҈W鼀 㡰 ҈䰘W鼀 ᆠ㡰 ҈W鼀 㡰 ҈�W鼀 꾀W矄�矄ᨨ 矄ᆠ� ᆠᨨᨨᆠᨨ ᔼ ᨨ ᆠ矄 ᆠᨨᨨ泐� � ᆠᨨ 犨矄�矄 ᆠ犨� 㝐� 꾀�҈黨 ᔼ㡰桌
ᆠ㡰
䰘
㡰 ҈W
ᔼ ᨨ�矄鼀ᨨ矄 ᨨ� � 泐 ᆠᨨᆠ矄鼀 㡰 鼀 ᨨ 泐 �矄 ҈� �矄 � 矄�矄 ᆠ�矄҈�W 㡰 盐ᨨ 矄�矄 ᆠ �矄҈�W
㡰
㡰 W
ᆠ㡰 W
㡰
�W
ᆠ �ᨨ ᆠᆠ�ᨨ� ᆠ�ᆠ盐�矄 盐 ᆠ�ᨨ ᨨ �鼀�矄矄 ᨨ犨 � ��盐 㝐� ҈ ҈黨 ᔼᔼ㡰桌 � ⼄ Ҙᆠ㡰
矄㡰
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P-179
Current Electricity
␔ �矄 盐ᨨ ᆠ盐 㦨矄䁸 㡰 ᨨ 㡰 Ҙ꾀 ᆠ㡰 Ҙ꾀 㡰 ҈�Ҙ꾀 � 矄 ᨨ ᆠᨨ 矄犨 ҈�꾀ᨨ ᆠ盐 ᨨ犨矄� ᆠᨨ犨�ᨨ ᆠᨨ矄矄矄ᆠ�ᨨ �鼀鼀�矄vᔼ ᆠᨨ
矄�泐� ᆠᨨ�矄黨맔҈⏌鼀 盐ᨨ v� 鼀鼀⏌矄 ᆠᨨ矄ᨨ ├ᆠ �ᨨ ᆠᨨ ᨨ㦨҈맔҈҈黨ᔴ㡰 㝐� 黨 ҈黨 ᔼ㡰桌 㡰 㡰 ᆠ㡰 㡰 ҈ 꾀ᆠᨨ 矄�矄 ᆠ 矄 ᨨᨨ犨� �ᆠᨨᨨ盐ᆠᨨ ␔ �矄 盐 ᨨ 矄�矄 ᆠ䁸 㝐� 黨 ҈黨 ᔼᔼ㡰桌 黨
҈� 和盐ᨨ矄 � ᨨᆠ�泐n � 鼀�� 犨� � ᆠ�ᆠ � 쟼矄 n µ 쟼 ᔼ� ᨨ 犨�鼀 ᨨ 矄盐ᆠ 鼀��矄矄�� 泐 㝐� 졠 � 꾀�҈҈�桌
㡰
㡰 ᔼ
ᔼ
졠Ӹ ᨨ
㡰 �├W᠀�奠 㡰 �├W᠀�奠 ᆠ㡰 㝐W᠀�奠 㡰 㝐W᠀҈奠 ҈҈ 꾀 � � 鼀 矄 矄�矄 ᆠ ᨨ ҈W ᨨᨨ鼀 鼀盐␔ ��矄ᆠᨨ
ᆠ ᨨ矄盐泐ᨨ 矄 泐ᆠ盐 ᨨ 꾀矄矄矄� � 鼀盐�矄� ᔴ 鼀ᨨ ᨨᨨ鼀 鼀盐 ␔ �矄 鼀盐 ᆠᨨ �ᆠ� ᨨ 矄�矄 ᆠᨨ � �鼀 䁸 㝐� 졠 � 꾀�҈҈桌 㡰 ҈맔҈ ᔴ҈ 㡰 �맔҈ ᔴ҈ ᆠ㡰 맔҈ ᔴ҈ 㡰 �맔҈ ᔴ҈ ҈ 꾀盐 � ᨨ鼀犨�ᨨ � l �盐矄 矄矄�矄 ᆠᨨ ҈Wᔼ�矄ᆠ矄� ᨨ犨�ᨨ �盐矄 矄�矄 ᆠᨨ
犨犨�犨� 㝐� 졠 � 꾀�黨҈䰘桌 㡰 ҈W 㡰 W ᆠ㡰 W 㡰 ҈W ҈ 꾀 � � �� ᆠ �ᨨ 盐 � ᨨ鼀ᆠ�ᆠᆠ � 矄�泐l �矄ᨨ � �矄ᨨ ᆠ�ᆠ泐ᆠᨨ 盐ᆠ� �� � � ᆠ泐� �ᆠ矄 ᨨ �盐矄 꾀�鼀㦨 矄ᆠ� 矄� ᆠ泐� �矄 � 犨� 鼀�ᨨ 鼀���泐e ᆠ�ᆠᆠᨨ 盐ᆠ��泐s ᆠ�ᆠᆠᨨ 盐ᆠ�ᨨ � 鼀� ᨨᨨ犨矄졠 鼀Ἐ矄犨␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � � 矄 矄 矄ᆠ�矄 矄盐矄⼄盐 ��ᨨ ᨨ 鼀� �盐 ᨨ ᆠ盐 矄�泐 㡰 泐 ᨨ� � 鼀�䁸 꾀 ҈桌
ᆠ㡰
㡰
ᔼ ᔼ ҈ ᔴᨨ 矄� � 矄⼄盐矄 ᨨ 矄� L �ᆠ├ 矄矄t 鼀 ᨨ 鼀�ᨨ 矄�矄��泐r 矄�矄 ᆠ犨
犨ᨨ ᨨᨨ矄� ᆠ矄 矄 ᨨ犨 泐 矄 矄� ��盐�矄 ҈桌 㡰 �ᆠ泐ᨨᨨ�ᨨ ᨨL
㡰 �ᆠ泐ᨨᨨ�ᨨ ᨨt ᆠ㡰 � ᨨ L
t
㡰 � ᨨ t
L
҈䰘 ᨨ� 泐졠 鼀矄犨 矄盐 �矄ᨨ� 犨� 矄 矄�矄ᨨRT �� 矄�矄 ᆠR҈矄鼀矄�矄 ᆠR犨ᨨ � �ᆠ� ᨨ鼀矄G҈ G �ᨨ� 矄ᨨ盐ᆠV ᆠᨨᆠᆠ�ᆠ盐�ᨨᆠ泐ᨨ盐 �鼀 �矄 ҈桌 G҈ R
㡰
R
R҈
㡰
V
㡰
㡰
ᆠ㡰 㡰
㡰
㡰
㡰
㡰
㡰
㡰
҈ 쟼 矄矄 犨 �ᆠ ᨨ ᨨᨨ犨� � 矄 盐矄 ᆠ 盐矄 ᨨ 和桌 � 泐졠 鼀矄犨䁸
㡰 ᆠ㡰
㡰 㡰
㡰
҈ ␔ �ᨨ � � ᆠ�矄� ᆠᨨ矄矄犨�ᨨ � ҈鼀 � 矄 ᨨ ᆠᨨ 矄�矄�맔҈鼀矄҈ ᔼ ᆠᨨ 矄�泐 � 犨� �矄 맔 ҈ 鼀 矄�矄��泐ᨨ 鼀��矄ᆠᨨ矄ᨨ 㝐� ҈�桌 㡰 ҈맔҈W鼀 㡰 ҈맔҈�W鼀 ᆠ㡰 ҈맔҈W鼀 㡰 ҈맔҈䰘W鼀
㡰
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EBD_8341 P-180
Physics
outer radius is R2. If a voltage V0 is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage DV develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale) [Adv. 2020]
(c)
DV
(d) 19.
20.
A piece of copper and another of germanium are cooled from room temperature to 80° K. The resistance of [1988 - 1 Mark] (a) each of them increases (b) each of them decreases (c) copper increases and germanium decreases (d) copper decreases and germanium increases The temperature coefficient of resistance of a wire is 0.00125 per °C. At 300 K, its resistance is 1 ohm. This resistance of the wire will be 2 ohm at. [1980] (a) 1154 K (b) 1100 K (c) 1400 K (d) 1127 K 5
21.
True / False
The current –voltage graphs for a given metallic wire at two different temperatures T1 and T2 are shown in the figure. [1985 - 3 Marks]
T1 I
T2
V The temperature T2 is greater than T1. 22.
Electrons in a conductor have no motion in the absence of a potential difference across it. [1982 - 2 Marks] 6
23.
MCQs with One or More than One Correct Answer
Shown in the figure is a semicircular metallic strip that has thickness t and resistivity r. Its inner radius is R1 and
R2 R1
I
I
V0
v0 t æ R2 ö ln ç ÷ pr è R1 ø (b) the outer surface is at a higher voltage than the inner surface (c) the outer surface is at a lower voltage than the inner surface (d) DV µ I 2
(a) I =
9
Assertion and Reason Type Questions
24. Read the following statements carefully: [1993-2 Marks] Y : The resistivity of a semiconductor decreases with increase of temperature. Z : In a conducing solid, the rate of collisions between free electrons and ions increases with increase of temperature Select the correct statement(s) from the following; (a) Y is true but Z is false (b) Y is false but Z is true (c) Both Y and Z are true (d) Y is true and Z is the correct reason for Y 10 Subjective Problems 25. If a copper wire is stretched to make it 0.1奠 longer what is the percentage change in its resistance? [1978]
Topic-2 : Combination of Resistances 1 1.
MCQs with One Correct Answer
In the given circuit diagram, a wire is joining points B and D. The current in this wire is: [Main 9 Jan. 2020, (I)] (a) 0.4A
(b) 2A
(c) 4A
(d) zero
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P-181
Current Electricity
ᆠ盐 I҈ � A㡰 ᨨ犨� � ᨨ盐� ҈W矄�矄ᨨ� ᨨᨨ犨� �ᆠ�ᆠ盐��矄 㝐� 䰘 ᔼ㡰桌
㡰
CE
r҈ ҈ + 㡰
ᆠ㡰 ᔴ쟼 㡰
CE
r҈ + 㡰
㡰 㡰 � ᆠ㡰 㡰 � 꾀鼀犨�ᨨ 矄�矄 ᆠW�矄ᨨ � ᨨ鼀├盐 � ᨨ鼀 犨�ᨨ ᨨ盐�矄�ᨨ盐矄 � �矄 犨犨��矄 ᨨ犨 矄ᨨ� ᨨ鼀├ᆠ�ᆠᔼ 犨ᨨᨨ� 矄ᨨ
ᆠ�ᆠ鼀├ � ᆠ ⼄盐� 矄�矄 ᆠ犨 矄犨ᨨᨨ� 矄犨� 㝐� 黨꾀҈黨 ᔼᔼ㡰桌 ҈ � � 䰘 W W W W 㡰 ᆠ㡰 㡰 � ᔼ ��盐矄 ᨨ犨 犨 �矄 ᆠ盐 � 꾀鼀㡰 犨
ᨨ鼀 泐䁸Ӹᨨ盐�� 㝐� 꾀҈黨 ᔼᔼ㡰桌 ҈㦨҈�W 㦨҈W 㦨W 㦨�W �㦨�W 㦨W쟼㦨҈�
㡰
ᔼ ᨨᆠ�ᆠ盐� ᆠ盐 � ᆠ 矄�矄 ᆠ�矄 㝐� ҈䰘桌 㡰 �꾀 㡰 꾀 ᆠ㡰 ҈꾀 㡰 �꾀 黨
ᔼ 盐鼀� �盐鼀 꾀㡰ᨨ 矄⼄盐ᆠᨨ矄矄矄ᆠ�ᨨ 矄⼄盐 ᨨ�矄 � �矄 � 犨� �ᨨ 㡰矄矄 ᨨ犨 � ��盐
ᆠ�ᆠ矄�矄���矄ᨨ 꾀 䰘맔҈W鼀 ҈ 맔 ҈䰘 W 鼀 矄ᆠ�泐 ᆠ�ᆠ 矄�矄 ᆠ 犨 犨ᨨ ᆠ矄P Qᨨ ᆠᨨ鼀ᨨ矄��矄 꾀 ҈�桌 㡰
䰘� ҘW
㡰
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EBD_8341 P-182
҈
Physics
和� � �ᆠ矄�矄ᨨ矄ᆠᨨ
ᆠ 矄矄 ᨨ犨 � ��盐
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ᆠ�矄�矄 ᆠ犨 ᨨ� 矄 P Qᨨ ᆠ�ᆠᆠ�ᆠ盐�矄 ᨨ犨 � ��盐�矄 和桌 㡰
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Fill in the Blanks
҈䰘 ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ犨ᆠ 泐�矄� 矄
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├W
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҈ 꾀 ҈
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2 Integer Value Answer ҈� ᔼ ᨨᨨ犨� �ᆠ�ᆠ盐� ᆠ盐 ᨨ盐� 矄�矄ᨨ R 㦨W㡰�矄I鼀矄 盐ᨨ I�矄 꾀 ҈�桌
2W
9V
2W
8W
8W
4W
㦨W㡰 2W
㡰 㡰 ᆠ㡰 㡰
3
Numeric Answer
҈ 矄�矄ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ�矄ᨨ ᨨ 矄鼀 鼀 ҈盐 � � 矄�矄 ᆠᨨ W �W �矄ᆠᨨ
ᆠ ᨨ ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ矄�矄ᨨ矄 W W ᨨ� � ᆠᆠᨨ矄矄 泐ᨨ � 矄�矄 ᆠW�矄 ᨨ 盐ᨨ R � W㡰�矄 힘힘힘힘힘힘힘힘힘 㝐� ᔼᔼ㡰桌
2W
2W
W 矄�矄ᨨ�矄�꾀 W 矄�矄ᨨ�矄�꾀 W 矄�矄ᨨ�矄�꾀 W 矄�矄ᨨ�矄�꾀
10 Subjective Problems ҈ 꾀 � � � 犨ᨨ├ᨨ 矄�矄 ᆠ矄�矄ᆠᨨ 矄盐ᆠ 犨� ҈ᨨ 鼀 ᨨ 鼀矄�矄 ᆠ矄矄矄 ᨨ犨 � �� ҈黨䰘㜠䰘㝐├矄桌 꾀
҈W
W
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W
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Current Electricity
ᨨ泐犨 A B 矄 ����� 矄�矄 ᆠ �㡰 和 ᨨ犨 ᆠ�矄�矄 ᆠ犨 A B �矄ᨨ 鼀矄 ��㡰 ␔ �矄 ᆠ盐 矄矄矄 ᨨ盐� ᨨ 鼀 矄�矄 ᆠ 矄ᨨ 泐䁸
꾀矄�矄 ᆠ矄� ��鼀ᨨ犨� ᨨ 鼀矄 � ᆠ�矄�矄 ᆠ犨 ᨨ� 矄A B҈黨䰘黨桌
꾀
Topic-3 : Kirchhoff’s Laws, Cells, Thermo e.m.f. & Electrolysis MCQs with One Correct Answer
1 ҈
5W 10W
20 2W
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4W
ᔼ ��盐矄 ᨨ犨 ᆠ盐 � ҈泐�矄ᆠᨨ矄 ᨨ 㝐� 和 ᔼᔼ㡰桌 㡰 䰘҈꾀 ᨨ鼀ᨨ矄��ᨨ ��鼀� 㡰 꾀 ᨨ鼀ᨨ矄��ᨨ ��鼀� ᆠ㡰 ҈꾀 ᨨ鼀ᨨ矄��ᨨ ��鼀� 㡰 꾀 ᨨ鼀 ��ᨨᨨ矄��鼀� ᔼ ᆠ�ᆠ盐��� � ��盐ᆠ盐 矄� � ᆠ 矄 盐ᨨ ᨨ 矄�矄ᨨ矄 ᨨ犨 ᨨ �ᨨ� B犨� 矄ᆠᨨ ᨨ� A�矄㝐� 和� ᔼᔼ㡰桌
D
E ҈꾀
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F
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㡰 䰘 㡰 ᆠ㡰 䰘 㡰 꾀ᆠ ᨨ � 矄�矄 ᆠr �矄ᆠ盐 ᨨ盐�
矄�矄 ᆠ ᨨ犨 � 泐 ᆠᨨ 矄�矄 ᆠ犨�鼀 �鼀盐鼀犨 㝐� 꾀҈黨 ᔼᔼ㡰桌 㡰 㦨҈r 㡰 㦨҈r ᆠ㡰 㦨r 㡰 㦨r ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ � � ᆠ犨 꾀 �矄 㝐� ҈҈ ҈黨 ᔼᔼ㡰桌
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ᆠ ᆠᨨ矄矄 ҈W 矄�矄 ᆠ 矄 � 矄�矄 ᆠᨨ ᆠ ᆠ�矄 㝐� ҈꾀҈黨 ᔼ㡰桌
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㡰 ҈ 㡰 ᆠ㡰 㡰 ᔼ �� ᆠ�ᆠ盐� ᆠ矄 ᨨ� 矄�矄 ᆠ
ᆠ盐 矄 � 꾀鼀矄㡰 矄矄� � ᨨ盐� 矄�矄 ᆠ ҈ 矄ᆠ�泐 㝐� ҈ ҈黨 ᔼ㡰桌
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㡰 ҈ 㡰 ᆠ㡰 � 㡰 ҈ 黨 犨ᨨ�矄犨� 鼀 ҈ ҈ᆠᨨ
ᆠ �
ᆠᨨ矄矄 ᨨ 矄�矄ᨨ ᨨ ҈W � 矄�矄 ᆠ矄ᨨ 犨ᨨ�矄҈W W矄ᆠ�泐
ᨨ�ᆠᨨ矄矄 ᨨ �矄犨 㝐� ҈桌 㡰 ҈҈ ҈҈䰘 㡰 ҈҈� ҈҈ ᆠ㡰 ҈҈ ҈҈� 㡰 ҈҈䰘 ҈҈ ҈ ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᆠ盐 � ҈W矄�矄ᨨ�矄 㝐� ҈�桌
W ҈W
犨�ᨨ 矄鼀鼀� ᆠᨨ矄矄㜠矄ᆠ�ᨨ 盐ᨨ � L 鼀盐ᨨ 犨��矄�矄 泐 矄鼀鼀ᨨ盐 ҈和桌 DT� 矄鼀�鼀t 盐ᨨ N �矄 㡰 㡰 ᆠ㡰 㡰 黨 ҈ 꾀泐ᨨ � 矄�矄 ᆠW�矄ᆠᨨ
ᆠ ᨨ
犨ᨨ├ᨨ 矄�矄 ᆠ矄矄矄 ᨨ犨 ᔼ ᨨ 鼀 �鼀盐鼀 ᨨ犨ᆠ � ᨨ 犨ᨨ├ 盐ᨨ R� W 矄 ᨨ盐 ҈黨黨�和桌
E
R
R
W
㡰 2
R R
R
黨
㡰
ᆠ㡰
㡰 ҈
Integer Value Answer
҈� 犨ᨨ �矄 ᨨ � 鼀 矄 � � 矄�矄 ᆠ矄ᆠᨨ
ᆠ 矄矄 ᨨ犨 ᨨ�ᆠᨨ矄矄 꾀� ᨨ矄�矄 ҈҈桌
黨
W W
W
㡰 ҈꾀 ᨨ鼀턼ᨨ 㡰 ҈꾀 ᨨ鼀ᨨ턼 ᆠ㡰 ҈꾀 ᨨ鼀ᨨ턼 㡰 꾀 ҈҈ ᔼ ᆠ�ᆠ 犨ᨨ├ 矄 ᨨ犨 犨 ᨨ ᆠ盐 ᨨ犨矄 ᨨ盐� W 矄�矄ᨨ � 鼀 쟼 ᨨ � � ᆠ 犨 ᨨ� 矄꾀 犨� 㝐� 졠 � 꾀�҈҈҈�桌 W
R
R
B W
W
W W
Numeric Answer
҈
쟼
3
A
C
黨W
꾀
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ᨨ盐矄�矄 ᆠ矄WW黨W ҈҈W鼀├ 鼀矄 ᨨ ⼄盐 �ABCD꾀ᆠᨨ矄矄AC�矄泐ᨨ 鼀 � 矄�矄 ᆠ ���� ᨨ � � ᆠ ᆠᨨ矄矄BD� �矄힘힘힘힘힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌 ҈䰘 꾀 � ᆠᨨ 鼀 ҈�矄ᆠᨨ
ᆠ � ᆠ�ᆠ盐�矄 ᨨ犨 �
��盐쟼ᆠ 矄�矄 ᆠ�矄W ᨨ � � ᆠ �
㡰 ᆠᨨ矄矄 ᆠᆠ�ᨨ 犨 � �矄 盐泐ᆠ � �矄 힘힘힘힘힘힘힘힘힘힘 [Main Sep.02, 2020 (II)]
ᔼ
ᨨ犨鼀盐ᆠ �矄 ᆠ盐 ᔼ� ᆠ�ᆠ盐�� 矄 泐矄䁸 㡰 ҈꾀 㡰 䰘꾀 ᆠ㡰 �꾀 㡰 �꾀 ҈ 꾀犨�ᨨ � L � �ᆠᆠ矄ᨨ ����� 矄�矄 ᆠ矄ᆠᨨ
ᆠ � 矄�矄 盐ᨨ ᆠ盐 鼀盐ᨨ 犨��矄�矄 泐DT� �鼀t꾀
盐鼀N ᨨ 矄�鼀�ᆠ矄�矄 ᨨ犨ᆠᨨ
ᆠ � 矄�矄犨�
҈
ᔴ �
҈
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Current Electricity
5 ҈
㡰 㡰 ᆠ㡰 㡰
True / False
ᔼ ᆠᨨ泐�ᆠ矄ᨨ盐�ᨨ ᆠ�ᆠᆠ盐 �矄鼀� 泐 盐ᨨ 鼀ᨨ鼀 ᨨ ᆠᨨ 矄 ҈黨桌 6
10 Subjective Problems
MCQs with One or More than One Correct Answer
҈黨 犨ᨨ� �矄ᨨ 鼀 V҈ V 矄�矄 ᆠ矄 R҈R Rᆠᨨ
ᆠ 矄矄 ᨨ犨 � ��盐 ᆠ盐 � 矄�矄 ᆠR犨ᨨ盐 ᨨ� 꾀 ҈桌
R҈
V҈
ᆠ盐 ᨨ盐� 턼�矄 ᨨ ᔼ҈㦨꾀
ᨨ �和�矄矄矄 턼 ᔼ㦨꾀
҈ ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐쟼 ᆠ矄ᨨ 鼀 ҈ ҈ᨨ矄ᆠ�泐 �� 矄�矄 ᆠ矄 ҈ ҈ᨨ 鼀矄ᆠ�泐 ҈黨㜠㝐├矄桌 ᔴᆠ盐 �㡰 ᨨ � � ᆠ犨 ��㡰 ᨨ � � ᆠᆠᨨ矄矄 鼀� 矄ᨨ ᆠ ᆠ矄
R
E B
A
V
W
F
R 㡰 V҈㦨V R҈㦨R㦨R 㡰 V҈㦨V R҈㦨R㦨R ᆠ㡰 V҈㦨V R҈㦨R㦨R 㡰 V҈㦨V R҈㦨R㦨R ᨨ 矄�矄 ᆠ 犨ᨨ├矄 ᨨ犨 � ��盐ᆠ ᨨᨨ矄 ᆠᨨᆠᨨ�ᨨ 矄㡰 ҈㜠ᔼ桌
H C
D
G
ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��E҈㦨ᨨ矄E㦨ᨨ矄E㦨҈ ᨨ 㦨r҈㦨r㦨r㦨҈ᨨ 鼀 ҈黨҈㜠㝐├矄桌 r҈ 졔
A
R
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r r
E҈ 졔 E 졔 E
B D
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ᆠ ᨨ ᨨ� B � ᆠ盐 矄 ᨨ盐� E҈EE 矄�矄ᨨR
Topic-4 : Heating Effect of Current 1 ҈
MCQs with One Correct Answer
꾀 ᆠ�ᆠᨨ犨� � �ᨨ矄�矄 ᆠᨨ W �矄҈├␔ �ᆠ� ᆠ泐ᨨ 矄鼀�矄矄�ᨨ
� �矄ᨨ �鼀泐 㝐� 和� ᔼ㡰桌 㡰 䰘奠 㡰 黨҈奠 ᆠ㡰 �奠 㡰 黨奠 ᔼ 盐� � � ҈�盐矄ᨨ �␔҈�盐矄ᨨ ҈␔ ҈�矄鼀 矄ᨨ ҈␔ 矄ᨨ ҈├␔ ᨨ� ᨨ ᆠ�ᆠ鼀� �矄 鼀� �鼀盐鼀 盐矄ᆠᆠ�泐 盐㡰ᨨ 盐� � �犨� 㝐� 䰘 ᔼᔼ㡰桌 㡰 ҈꾀 㡰 �꾀 ᆠ㡰 ҈�꾀 㡰 꾀
矄�矄� 犨ᨨ├矄 ᨨ犨 ᨨ犨�矄ᆠᨨ
ᆠ ᨨᔴ 矄ᨨ盐ᆠᨨ ҈ ᨨ犨ᆠᨨ 矄盐鼀 泐 犨ᨨ├�矄 ␔ 盐ᨨ �矄 㝐� ҈꾀҈黨 ᔼ㡰桌
㡰 W 㡰 W ᆠ㡰 ҈W 㡰 ҈W 졠 ├�ᨨ 犨ᨨᔴ�矄 � ᆠ�ᆠ├ 犨 ᨨ矄 � �鼀 矄鼀 鼀盐� 㡰 矄�矄 ᆠᨨ W 鼀矄ᨨ�� 鼀� 矄�矄 ᔼ� ᨨ� � ᨨ矄矄 ᨨ鼀 ├�鼀├ ᨨ犨ᨨ ᨨ 盐泐�矄ᆠᨨ矄ᨨ 和ᆠ� �ᆠ ᨨ 犨㦨⏌ ├�ᨨᔴ㡰 ᨨ 犨 㦨├⏌├�桌 㝐� ҈꾀҈黨 ᔼᔼ㡰桌 㡰 ҈鼀� 盐矄 㡰 鼀� 盐矄 ᆠ㡰 鼀� 盐矄 㡰 鼀� 盐矄
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EBD_8341 P-186
5.
6.
7.
8.
Physics
Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers P1 and P2 respectively, then: [Main 12 Jan. 2019 (I)] (a) P1=16 W, P2=4 W (b) P1=16 W, P2=9 W (c) P1=9 W, P2=16 W (d) P1=4 W, P2=16 W Two equal resistances when connected in series to a battery, consume electric power of 60 W. If these resistance are now connected in parallel combination to the same battery, the electric power consumed will be : [Main 11 Jan. 2019 (I)] (a) 60 W (b) 240 W (c) 120 W (d) 30 W A current of 2 mA was passed through an unknown resistor which dissipated a power of 4.4 W. Dissipated power when an ideal power supply of 11 V is connected across it is: [Main 10 Jan. 2019 (II)] (a) 11 × 10–5 W (b) 11 × 10–3 W (c) 11 × 10–4 W (d) 11 × 105 W The figure shows three circuits I, II and III which are connected to a 3V battery. If the powers dissipated by the configurations I, II and III are P1, P2 and P3 respectively, then : [Main Online April 9, 2017]
12. Which of the four resistances P, Q, R and S generate the greatest amount of heat when a current flows from A to B? [Main Online April 23, 2013] P = 2W
Q = 4W
A
B
R = 1W
S = 2W
(a) Q (b) S (c) P (d) R 13. Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances R100, R60 and R40, respectively, the relation between these resistances is [2010] 1
(a)
R100
=
1 1 + R40 R60
(b) R100 = R40 + R60 1
1W
1W
1W 1W
1W
3V
1W
1W
1W
1W
3V 1W
1W 1W
(I)
1W
1W
1W
3V
(II)
(a) P1 > P2 > P3 (c) P2 > P1 > P3
(b) P1 > P3 > P2 (d) P3 > P2 > P1
9.
1 1 > R100 R60 R40 14. Figure shows three resistor configurations R1, R2 and R3 connected to 3V battery. If the power dissipated by the configuration R1, R2 and R3 is P1, P2 and P3, respectively, then – [2008]
(c) R100 > R60 > R40
1W
>
(d)
(III)
R
3V
3V
R
r
R1
In the circuit shown, the resistance r is a variable resistance. If for r = fR, the heat generation in r is maximum then the value of f is : [Main Online April 9, 2016] 1 1 3 (a) (b) 1 (c) (d) 2 4 4 10. In a large building, there are 15 bulbs of 40 W, 5 bulbs of 100 W, 5 fans of 80 W and 1 heater of 1 kW. The voltage of electric mains is 220 V. The minimum capacity of the main fuse of the building will be: [Main 2014] (a) 8 A (b) 10 A (c) 12 A (d) 14 A 11. Four bulbs B1, B2, B3 and B4 of 100 W each are connected to 220 V main as shown in the figure. [Main Online April 19, 2014]
(a)
P1 > P2 > P3
(b)
P1 > P3 > P2
R2
3V
(c) P2 > P1 > P3 (d) P3 > P2 > P1 R3
15. The three resistance of equal value are arranged in the different combinations shown below. Arrange them in increasing order of power dissipation. [2003S] i
220 V
B1
B2
B3
B4
i
Ammeter
The reading in an ideal ammeter will be: (a) 0.45 A (b) 0.90 A (c) 1.35 A
(d) 1.80 A
(I)
(II)
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ᆠ㡰 �盐矄ᨨ 犨��矄 ᨨ盐 㡰 � ᨨ 犨��矄 ᨨ盐 �
҈
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ᔼᔼᔼ㡰 ᔼ㡰 㡰 ᔼᔼᔼᔼᔼᔼᔼ 㡰 ᔼᔼᔼᔼᔼᔼᔼ ᆠ㡰 ᔼᔼᔼᔼᔼᔼᔼ 㡰 ᔼᔼᔼᔼᔼᔼᔼ 꾀҈ ␔盐B҈ 犨ᨨ ␔盐B B ᆠᨨ
ᆠ ᨨ�矄ᨨ盐ᆠ矄矄 ᨨ犨 � ��盐ᨨ犨W҈ W W ᨨ盐盐ᨨ犨矄ᨨ 盐矄B҈B B矄ᆠ�泐
和桌 B2
B1
B3
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҈䰘
㡰 W҈> W㦨 W 㡰 W҈>W>␔ ᆠ㡰 W҈ W㦨 W 㡰 W҈WW ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � �� ᨨ 盐ᆠ � �ᨨ 鼀 矄�矄ᨨ 盐ᨨ ᆠ盐 ᨨ犨� � ᨨ盐� ��矄҈ᆠᨨ�矄 矄ᆠᨨ ҈黨҈㜠㝐├矄桌 W 6W
5W
҈
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2
Integer Value Answer
҈黨 ␔ 犨ᨨ� �ᆠ�矄ᨨ � 矄�矄 ᆠ҈Wᆠ ᆠᨨ
ᆠ � 矄�矄ᆠᨨ矄矄矄�矄ᨨR ᨨ ᨨ 盐ᆠ � �矄J҈␔ 矄鼀�矄ᆠᨨ
ᆠ � ᆠᨨ矄矄 �矄Jᔼ J҈㦨�J 盐ᨨ R� W�矄 ҈桌 4
Fill in the Blanks
꾀 ᆠ�ᆠ盐 ᨨ�犨矄҈ᨨ矄�矄盐矄 �
ᆠ�ᆠ盐� � �ᨨ矄矄盐泐 矄�矄 ᆠR 鼀盐矄盐� 矄�矄犨� 盐矄ᨨ 盐 �矄 �犨�矄ᨨ 鼀 ҈黨䰘㜠㝐├矄桌 6 MCQs with One or More than One Correct Answer ҈ ᨨ ᆠ�ᆠ├�矄鼀 ᨨ 犨�ᨨ � L �鼀dᔼ├矄鼀� 盐矄ᨨ�矄 鼀盐ᨨ �├�犨泐⧤ �矄 �矄ᆠ 泐 犨 � �犨ᨨ犨�矄ᨨ 矄鼀鼀�ᆠ ᨨ � L �鼀d 犨泐 矄犨�矄ᆠᨨ
ᆠ �矄 �� � ᨨ�ᨨ 矄ᨨ犨鼀盐ᆠ �鼀� 鼀� 盐矄犨��├ ᨨ�矄 鼀盐ᨨ 矄鼀鼀ᨨ盐 ᨨ 犨泐 꾀 ҈桌 K䁸 㡰 � 犨�矄� 㡰 � 犨�矄� 矄�矄 ᆠ㡰 ҈� 犨�矄� 矄�矄 㡰 �� 犨�矄� 10 Subjective Problems 꾀 �矄 矄�� ᨨᨨ犨� ᨨ犨ᨨ ҈犨矄 � ҈ᨨ� ᔼ�矄ᆠᨨ
ᆠ � ᆠᨨ鼀� �ᨨ 矄犨� 矄�矄 ᆠᨨ ҈ᨨ 鼀矄 矄�矄 ᆠRᨨ҈ᨨ矄 鼀� 矄矄矄 ᨨ犨 � ��盐␔ 矄 ᨨ盐 盐ᨨ R矄ᨨ ᨨ矄犨� ᨨ犨ᨨ �犨矄 ҈黨䰘桌 ҈W R
҈
Topic-5 : Wheatstone Bridge and Different Measuring Instruments 1 ҈
MCQs with One Correct Answer
犨ᨨ矄�矄ᨨ矄W Wᆠᨨ
ᆠ � 矄�矄ᆠᨨ矄矄 泐 ᨨ � � ᆠ 鼀矄盐 泐 ᨨ鼀ᨨ ҈├Wᆠᨨ矄矄W矄�矄ᨨ�矄ᆠᨨ矄ᨨ 㝐� 和 ᔼᔼ㡰桌 㡰 㡰 ҈ ᆠ㡰 � 㡰 ҈黨� ␔ �ᆠ ᨨ ᨨᨨ犨� � 犨� 졠 ᨨ矄 犨 鼀盐�鼀 ᨨ� �� 矄�矄 ᆠ鼀矄盐� �鼀ᨨ 㡰ᨨ矄 ᆠᨨ
ᆠ ᆠᨨ矄矄ᆠᨨ鼀ᨨ 盐矄矄 䁸 㝐� 和 ᔼᔼ㡰桌
㡰 㝐盐�鼀矄 ᨨ犨矄 ⼄盐 ᆠ�ᨨ � ᨨ ᆠ矄矄 � ᨨ 矄� � ᨨ矄� ᆠ ᨨ矄
ᆠᨨ鼀ᨨ �矄矄�矄ᨨ 㡰 㝐盐�鼀 矄 ᨨ犨矄 졠 ᆠ�ᨨ � ᨨ ᆠ矄矄 � ᨨ 矄� � ᨨ矄� ᆠ ᨨ矄
ᆠᨨ鼀ᨨ �矄ᆠᆠ�ᨨ ᆠ㡰 㝐盐�鼀 矄 ᨨ犨矄 ᆠ�ᨨ ᆠᆠᨨ鼀 � 泐 矄矄 ᨨ �� ᨨ盐ᨨ ᆠᨨ
ᆠ 졠 ᆠ�ᨨ ᨨ
矄� � ᨨ矄� ᆠ ᨨ矄 ᆠᨨ鼀ᨨ �矄쟼 㡰 㝐盐�鼀 矄 ᨨ犨矄 졠 ᆠ�ᨨ � ᨨ ᆠ矄矄 � ᨨ 矄� � ᨨ矄� ᆠ ᨨ矄
ᆠᨨ鼀ᨨ �矄鼀犨�
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Physics
J
P
Q
r
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ᆠᨨ
ᆠ � 꾀犨 �ᆠ � ᨨ鼀矄 矄 ᨨ犨 � �� 矄 ᨨ犨矄 ᨨ ᆠ�ᨨ �矄 㝐� ҈ ҈黨 ᔼ㡰桌
꾀ᨨ �ᨨ鼀犨�PQᨨ ҈鼀 � �矄ᆠᨨ
ᆠ ᨨ 矄 ᆠE҈꾀 ᨨ ᆠEᨨ 鼀 ҈�矄ᆠᨨ
ᆠ 犨� 矄�矄 ᆠἘrἘ 矄犨�ᆠ S 矄矄 ᨨ犨 � ��盐㡰␔� 矄犨�ᆠ Sᨨ 盐ᨨ矄��ᨨ �矄ᨨ� �矄 ᆠᨨ 黨ᆠ鼀 ᨨ鼀Q ᨨ �� � � ᨨ �ᨨ鼀 犨��矄 [Main Sep.02, 2020 (II)] E҈
G
E S 㡰 ⏌ᆠ鼀 㡰 ҈⏌ᆠ鼀 ᆠ㡰 ⏌ᆠ鼀 㡰 ⏌ᆠ鼀
� ᨨ ᨨ �ᨨ鼀犨��矄҈ᆠ鼀 �ᆠ�矄 ᆠ盐 ᨨ 鼀꾀 ᨨ ᆠᨨ 鼀 � � 矄�矄 ᆠᨨ W 盐ᨨ� ᨨ ��矄 ᨨ盐 ᨨ ҈ᆠ鼀 矄�矄 ᆠᨨ 犨 ᨨ犨��矄 㝐� ᔼ㡰桌 㡰 W 㡰 ҈W ᆠ㡰 W 㡰 ҈W ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ盐㜠犨�ᨨ �ᨨ鼀�矄鼀 ᨨ ᆠ鼀ᨨ �犨�犨 �ᆠ 矄犨 꾀 矄�矄 ᆠ 盐 � � ᨨ ᨨ �ᨨ鼀 犨� �矄 r㦨҈W⏌ᆠ鼀ᔼ � ᨨ鼀�矄ᆠᨨ
ᆠ 矄矄 ᨨ犨
犨� ᨨᆠ├泐�ᆠ鼀 ᨨ鼀 꾀 ᆠ � �ᨨ ᨨ鼀犨� 㝐� 꾀҈黨 ᔼᔼ㡰桌
҈҈ ҈҈ ҈ � 㡰 ᆠ㡰 㡰 ҈ ҈
␔ 矄ᨨ � �矄 ᨨ犨 � �� �矄 ᆠ 犨 ᆠᨨ 矄�矄ᨨ盐矄 矄 ҈ 矄 ᆠᨨᨨ盐 ᆠᨨ 졠 � ᨨ犨 㡰 矄�矄ᨨ矄 W W矄ᆠ�泐 꾀矄矄盐鼀� � ᆠᨨᨨ盐 ᆠᨨ ᨨ ᆠᨨ 矄�矄ᨨ矄 ��矄 �ᆠᆠ盐盐矄 ᆠᨨᨨ盐ᆠᨨ ᨨ ᆠᨨ
矄�矄ᨨ盐矄 矄 犨ᨨ盐 㝐� ҈ ҈黨 ᔼᔼ㡰桌
㡰
҈
졔
㡰 � 㡰 䰘� ᆠ㡰 � 㡰 ᔼ ␔ 矄ᨨ � � 矄 ��㡰 矄�矄 ᆠ矄 턼 ᨨ �鼀泐⼄盐␔ 㦨W � ��矄 ᆠ 졠 � ᆠ �� � 턼 盐ᨨ ᨨ ᆠ�矄 �W 盐ᨨ �矄ᆠᨨ矄ᨨ㝐� ҈҈ ҈黨 ᔼ㡰桌 턼
꾀
⧤
ᔴ
㡰 ᨨ犨 盐ᨨ犨 㡰 ᨨ犨 盐ᆠ├ ᆠ㡰 ᨨ犨
㡰 泐ᆠ├ᨨ犨
黨 ᔼ ᨨ �ᨨ鼀 �鼀 ��矄 ᨨ盐 ᨨᆠ盐 矄矄矄 ᨨ盐� � ᨨ鼀犨 鼀� 矄ᨨ ᆠᆠᨨ
ᆠ ᆠᨨ矄矄�ᆠ鼀ᨨ ᨨ �ᨨ鼀 犨�ᔼ ᆠ�矄矄 盐 泐矄�矄 ᆠᨨ �W ᆠ �矄 ᨨ盐 犨 ᆠ�矄ᆠᨨ
ᆠ ᆠᨨ矄矄ᆠ鼀 ᨨ 犨� � � 矄�矄 ᆠᨨ ᆠ 㝐� ҈桌 㡰 ҈W 㡰 ҈�W ᆠ㡰 W 㡰 �W ҈ ᔼ 鼀� �矄矄 ᨨ犨 � ��盐��矄�� 矄�矄 ᆠӸ㦨҈�W ᆠ�矄ᨨ� �矄 ᆠ 黨� ᆠ鼀 ᨨ鼀 A 泐 ᨨᆠ├泐 㡰 꾀 � ᆠ �� � 矄�矄 ᆠ矄X Y 犨 ᆠᨨ� �矄 ᨨ盐 �矄 ᆠl ᨨ鼀 A␔ 盐矄ᨨ X l䁸 㝐� 졠 � 꾀�҈�҈桌
꾀
⧤҈
䰘
Ӹ
㡰 ҈�ᨨ 鼀 㡰 �ᨨ 鼀 ᆠ㡰 �ᨨ 鼀 㡰 �ᨨ 鼀 꾀ᨨ �ᨨ鼀犨�꾀 � � � 矄�矄 ᆠ ҈�矄ᨨ� ᨨᆠᨨ 鼀 e � 矄�矄 ᆠ 꾀ᆠᔴ � � 鼀 e⏌ � 矄�矄 ᆠ �矄
l҈
҈ l҈㡰
ᔴ
黨� 犨�
㝐쟼 쟼 和ᔴ꾀쟼
泐
㡰 ҈黨҈�W 黨�ᆠ鼀 ᆠ㡰 ҈黨҈�W �ᆠ鼀
⧤泐
㡰 ҈W �ᆠ鼀 㡰 ҈W 黨�ᆠ鼀
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҈҈ ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 矄䁸㝐� ҈䰘桌 㡰 꾀 ᨨ矄ᆠ 盐矄 矄ᨨ � �� 㡰 ⧤�ᆠ ᨨ Ἐ矄矄ᆠᨨ 犨矄 矄 �泐ᆠᨨ 矄�ᨨ
ᆠ㡰 ␔ 矄ᨨ � ��矄 鼀ᨨ矄矄 矄��犨 ᨨ盐矄�矄 ᆠ矄ᨨ 矄鼀ᨨ ᨨ 鼀� �盐 㡰 ᔼ ᆠ 犨 矄ᨨ � � � ᆠ � ᨨ鼀 ᆠ � 盐ᨨ� �矄 �矄盐 ҈ 꾀҈泐犨� � 矄�矄 ᆠ҈W ҈�泐 犨� � 矄�矄 ᆠW ᆠᨨ
ᆠ � ᨨ ᨨ鼀 矄 ��盐㡰 � �� ᨨ鼀犨� ᆠᨨ矄ᨨ 㝐� 졠 � 꾀�҈҈�桌 ҈ ҈W
�␔ 矄矄�矄 ᆠ矄� ᆠ � ᆠ ᨨ� 矄 � 矄 泐ᆠ鼀 �ᆠ� � 泐ᆠᨨᆠ�ᨨ 矄 䰘桌 盐 ├ ᨨ犨 矄�矄 ᆠ�矄 㡰 W 㡰W ᆠ㡰 �W 㡰W ҈ 꾀鼀ᨨ� �ᆠᨨ�� ᨨ鼀ᨨ 矄�矄 ᆠ҈W�矄盐矄 矄 鼀鼀 盐矄� � 矄�矄 ᆠ҈ W. 鼀 �鼀盐鼀 ᆠ�ᨨ ᆠ盐 � � ᨨ鼀�矄҈m꾀 � 鼀� �鼀盐鼀ᆠ盐 � ᆠ�ᆠ盐�矄ᨨ 鼀鼀矄 ᨨ犨矄 鼀 �鼀盐鼀 ᆠ�ᨨ
�和桌 㡰 ҈҈鼀꾀 㡰 ҈҈鼀꾀 ᆠ㡰 ҈҈鼀꾀 㡰 ҈҈鼀꾀 ҈䰘 ᔼ 矄 ᨨ犨 �鼀 ᨨ �鼀 ᨨ 鼀 � � � AC ᆠᨨ矄ᨨ � � ᨨ 盐 ᆠ�ᨨ ᨨ � ᨨ鼀�矄 犨 犨ᨨ盐 �矄盐� �盐矄ᨨ 犨�꾀�矄 ᨨ盐 䁸 和桌
҈� W
R
R҈ G
㡰 ҈� 㡰 � ᆠ㡰 ҈҈ 㡰 ҈҈黨 ҈ 盐� � �鼀 犨� 鼀� � � ᨨ鼀 矄 ᨨ犨矄 盐ᨨ� 犨 ᨨᆠ├泐�矄矄矄 ᆠ鼀 盐矄� �矄 矄�矄 ᆠᨨ 黨W矄矄 ᨨ犨 � ��盐
矄ᆠᨨ盐 ᨨ 矄ᆠ盐矄 � 鼀� ��矄 ҈鼀鼀 盐 ├ ᨨ犨 矄�矄 ᆠ�矄 꾀 ҈桌
黨W
A
㡰 x
x
㡰 x⏌
B
C
ᆠ㡰 x
҈ ᔼ ᆠ�ᆠ盐� P ¹ R � � ᨨ � ᨨ鼀�矄矄鼀犨� 矄犨�ᆠ Sᨨ ᨨᆠᨨ矄
҈黨黨黨㜠㝐├矄桌 㡰 IR㦨IG 㡰 IP㦨IG
㡰 x
Q
P S R
G
ᆠ㡰 IQ㦨IG 㡰 IQ㦨IR
㡰 ҈
ᆠ鼀 ± ҈� W
3 㡰 ҈� ± � W
ᆠ㡰 ± � W 㡰 ҈� ± W 꾀鼀� ��矄矄盐矄矄 ᨨ犨 ᨨ 鼀� 盐 ├ ᨨ犨
矄�矄 ᆠ � 盐矄� � 矄 ҈ ᨨ 鼀 矄�矄ᨨ � ᨨ鼀矄 ᨨ犨矄 盐ᨨ� 犨 � �㜠├泐�矄� ᆠ鼀 鼀├ 㜠ᆠᨨᆠ�ᨨ 矄 ҈ ᆠ鼀 ᆠ鼀 矄ᆠ�泐 ᨨ 矄꾀 鼀� 盐 ᨨ ��矄 ҈҈桌
㡰 ҈ᨨ 鼀 㡰 ҈ᨨ 鼀 ᆠ㡰 ҈ᨨ 鼀 㡰 ҈҈҈ᨨ 鼀 ҈� 꾀矄�矄 ᆠᨨ W�矄ᆠᨨ
ᆠ ᆠᨨ矄矄ᨨ �ᨨ 鼀㜠 � � � ᨨ 犨��矄҈ᆠ鼀㡰 盐 ├ ᨨ犨
矄�矄 ᆠ� W�矄ᆠᨨ
ᆠ ᆠᨨ矄矄 ᨨ
V
Numeric Answer
ᨨ盐矄�矄 ᆠ矄ᨨ ҈�W҈WW ҈W矄ᆠ�泐 � ᆠ泐ᆠ�ᆠ ᨨ ᨨ ᨨ鼀 ␔ 矄ᨨ 矄 犨ᨨ├ 矄�矄 ᆠ �矄 ᨨ ᆠᨨ
ᆠ � 犨� 矄�矄 ᆠᨨ ҈Wᨨ ᆠ 犨ᨨ├�矄힘힘힘힘힘W 㝐� ᔼ㡰桌 ᆠ� � � ᨨᆠ�矄�ᆠ鼀� ᨨ �ᨨ鼀 �鼀 ␔ 矄�矄 ᆠ ᨨ ҈ W �矄 ᆠᨨ
ᆠ � ᨨ ᆠ ᆠ� � � ᆠ �矄泐ᆠ鼀ᔼ � 矄�矄 ᆠᨨ ᆠ�矄 N W犨 �矄 � � 盐ᨨ �矄________ ҈ 㝐� 䰘 ᔼᔼ㡰桌 � ᔼ ᆠ ᆠᨨ ��ᨨ 盐矄ᨨ 矄�矄 ᆠ矄ᨨ ᨨ盐鼀矄ᨨ ␔ 矄ᨨ � �矄 ᨨ犨 � ��盐ᨨ犨 矄�矄 ᆠR 矄鼀盐ᆠᨨ �ᆠ� ᔴ҈ᔼ 鼀盐ᨨ R�矄� ᆠ矄 泐҈ ᔴ ᨨ� ᨨ 犨 S T犨�힘힘힘힘힘힘힘힘 ᨨ 꾀 桌
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EBD_8341 P-190
Physics
AB ᔴ � 犨 �ᆠ ᨨ ᨨ犨��� 鼀ᨨ矄 ᆠᆠ盐 � � 犨 泐䁸 �㜠㝐├矄桌
V
R҈㦨W
R㦨҈W
P
�
X R㦨W
S
R㦨�W
T
A Q
Assertion and Reason Type Questions
9
和T꾀TE㝐ENT㜠҈ᔼ 㝐� � �鼀 盐ᨨ� ᨨ 盐 ├ ᨨ犨 矄�矄 ᆠ�矄鼀矄盐 ᨨ犨 盐 ├ ᨨ犨
矄�矄 ᆠ�矄盐� 矄� ᆠᨨ矄盐鼀� � �� 鼀盐 盐ᨨ� ᆠ ᨨ� 矄鼀 ᨨ� 矄 ᨨ泐 ᆠ矄� � 盐ᨨ 矄 矄�矄 ᆠ 和T꾀TE㝐ENT㜠 矄�矄 ᆠᨨ 鼀� ᆠ矄矄犨� � ᆠ矄� 鼀盐 桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐
B
C
꾀 � 盐 � ᨨ鼀 犨� AB ᨨ � ҈鼀 盐 ├ ᨨ犨
矄�矄 ᆠX 矄�矄 ᆠᨨ ҈Wᆠᨨ
ᆠ 泐 �ᆠ├ᆠᨨ 盐ᆠ� �矄�矄矄矄 ᨨ犨 � ��盐꾀泐 � ᨨ鼀 犨� 矄� � �ᨨᆠ├泐ᆠᨨ
ᆠ ᨨ �㡰 矄ᨨ � ᔴᨨ
ᆠ�ᨨ 矄 ᨨ 鼀 ᨨ 鼀矄盐 盐 ├ ᨨ犨 矄�矄 ᆠX盐矄� � � ᆠ�ᨨ ␔ 矄ᨨ � �꾀 矄犨 ᨨᨨ犨� �⼄盐矄�ᨨ 矄 㜠�㝐├矄桌
҈W
X A
B
C
D
㡰 꾀 ᨨ矄�� �� 鼀� 矄 ᨨ � ᨨ鼀䁸 㡰 ᔴᨨ泐 ��盐� 泐ᨨ盐 矄犨ᨨᨨ├ 矄 ᨨ犨 泐 � ᨨ鼀 犨� ᨨᆠ├泐㡰 ᆠᨨ
ᆠ ᨨ�ᨨ� 矄
10 Subjective Problems 䰘
R㦨R҈ᨨRᨨR
R G
꾀 盐 ├ ᨨ犨 矄�矄 ᆠ X �矄 ᨨ 鼀� 盐矄� � 矄�矄 ᆠ矄R҈RᨨR �ᆠᨨ矄ᨨ � � 盐ᨨ� 矄
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈
㡰 㡰 ᆠ㡰 㡰
MCQs with One Correct Answer
1
꾀 ᆠ�ᆠ盐� �矄ᆠᨨ
ᆠ 矄矄 ᨨ犨 � ��盐犨� 矄犨�ᆠ Sᨨ ␔ 矄犨�ᆠ �矄ᆠᨨ矄 ᨨ鼀ᨨ盐 ᨨ ᆠ � ᨨ犨矄 ᨨ鼀YᨨX�矄 䰘桌 mF
m F
X
S W
W
Y
黨
㡰 㡰 �mᔴ ᆠ㡰 䰘mᔴ 㡰 ҈mᔴ ᔼ 矄 泐ᆠ盐 ᔼ�矄 ᨨ犨� � ᨨ盐� ᆠ泐� �ᆠ鼀 꾀ᔴᔴ ᨨᨨ矄 ᆠᨨᆠ�ᨨ 矄 � 桌 A
B
r
r
I
l⏌
C
�
V AB = VBC
ᨨ犨ᆠᨨ矄矄BC�矄�鼀矄 ᨨ犨ᆠᨨ矄矄AB ᔴ盐 矄��矄� AB BC⼄盐 쟼ᆠ�ᆠ � 盐ᨨᆠ盐 � 矄� AB BC ⼄盐 꾀 � �矄�矄 � � ᆠᨨ矄 ��� 鼀泐� 矄盐 ᆠᨨ � 꾀ᆠᨨ�ᨨ ҈W矄�矄ᨨᆠ泐� �ᆠ盐 ҈꾀 ᨨ �鼀� 盐矄矄盐�矄 ᨨ �矄 ᆠ �� � �泐ᨨ �矄�矄 �和桌 㡰 ҈├ 㡰 ├ ᆠ㡰 ├ 㡰 ├ 꾀mFᆠᆠ�ᨨ矄�矄 ᆠᨨ �MW�矄� 矄�矄犨� ҈ 泐 � �鼀 犨 �ᆠ ᨨ � � ᆠᆠᨨ矄矄 ᆠᆠ�ᨨ �矄 �鼀矄 ᨨ � � ᆠᆠᨨ矄矄 矄�矄ᨨ� n 㡰㦨黨桌�和桌 㡰 ҈矄 㡰 黨矄 ᆠ㡰 䰘矄 㡰 ҈矄 꾀ᆠᆠ�ᨨ�矄ᆠ � 盐矄� � 泐犨� 矄�矄 ᆠx� 矄�矄 矄 � 矄 ᨨ犨矄 ��ᨨ
ᨨ ᔼ犨� 矄ᆠᨨ�鼀ᔼ 矄�矄 ᆠ�矄ᆠ � ᨨ x 犨� 犨� 和桌
l⏌
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Current Electricity
㡰 P 㡰 Q
S R
҈n ᔼ
ᆠ㡰R
Q
P 㡰 S t 和 ᨨ犨 � ��盐�矄ᨨ矄졠 �ᆠᨨ ᔼ ᨨ ᨨᆠᆠ盐 盐ᨨ 矄�矄 ᆠ�矄 ᨨ盐 ᆠᨨ
ᆠ 和桌 犨
C B A
D
C'
䰘
R �
R ҈
Integer Value Answer
҈ 꾀� ᨨ鼀��矄 盐矄ᆠ ᆠ�ᨨ 犨� 꾀 ᆠ盐 泐ᆠᨨ
ᆠ� ��ᨨ黨黨W矄�矄 ᆠ�ᆠ ᆠᨨ � ᨨᨨ鼀ᨨ �ᔼ ᆠᨨ
ᆠ n W 矄�矄 ᆠ�ᆠᨨ鼀矄 鼀鼀ᨨ � ᨨ 黨 ҈�꾀 盐ᨨ n�矄 꾀 ҈桌 ҈҈ 꾀�鼀t 㦨泐ᨨ ҈�矄ᆠᨨ
ᆠ ᆠᨨ矄矄 ᨨ� 矄꾀 � �� ᆠ�ᆠ盐�ᔼ ᆠᆠ�ᨨ矄 ᨨᆠ �� ��泐犨 �鼀 � 矄ᆠᨨ 矄㡰 ᨨ矄 ᨨ�ᆠᨨ矄矄 鼀ᆠᨨ鼀䁸 ├l �㦨҈ l 㦨҈҈桌 ҈桌
R
R
m
㝐W
B'
㡰 BἘ CἘ 㡰 A D ᆠ㡰 C D 㡰 B D ᔼ �� ᆠ�ᆠ盐���矄 ᨨ矄 ᆠ盐 I �矄 � ᨨ 盐ᨨ 矄�矄 ᆠR 矄�矄 ᆠ盐矄鼀盐矄矄�矄 泐 ҈和桌
I
2
R
꾀 㝐W
6
m
MCQs with One or More than One Correct Answer
҈ ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ犨 ├泐�矄矄矄 �鼀㦨 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌 m
�├W
R
㡰 R҈RR�㦨RRR 㡰
㡰 R҈R㦨RR㦨R�R
黨
�W
1 1 1 1 + = + R 5 R6 R1 + R2 R3 + R 4
ᆠ㡰 R҈R㦨RR
꾀
꾀犨�ᨨ � L � �ᆠᆠ矄ᨨ ����� 矄�矄 ᆠ矄ᆠᨨ
ᆠ � 矄�矄 盐ᨨ ᆠ盐 鼀盐ᨨ 犨��矄�矄 泐DT� �鼀t꾀
盐鼀N ᨨ 矄�鼀�ᆠ矄�矄 ᨨ犨ᆠᨨ
ᆠ � 矄�矄犨� 犨�ᨨ 矄鼀鼀� ᆠᨨ矄矄㜠矄ᆠ�ᨨ 盐ᨨ � L 鼀盐ᨨ 犨��矄�矄 泐 矄鼀鼀ᨨ盐 DT� 矄鼀�鼀t 盐ᨨ N �矄 ҈和桌 㡰 㡰 ᆠ㡰 㡰 黨
ᆠᨨ矄�ᆠ � 盐ᨨᨨ� ᆠ � 矄ᨨ
҈ �矄 ᆠ r 矄 ᔼ �ᆠ 犨 �ᆠ ᨨ ᨨᨨ犨� � r ⼄盐 ��矄矄 ᨨ犨矄矄鼀 ᆠᨨ r ҈黨桌 㡰 ᔼ 矄�泐ᨨ �� ᨨ鼀ᨨ� 矄ᨨ盐ᆠ 㡰 쟼ᆠᨨ矄�ᆠᨨ � 盐ᨨᨨ� ᆠ � ᆠ㡰 쟼ᆠᨨ矄�ᆠᨨ � �矄 ᆠ ᨨ鼀 ᆠ ᨨ ᆠ � 鼀�ᆠ矄 � r �盐矄ᨨ 矄 㡰 ᨨ ᨨ 矄
졔 m
졔 ⧤泐
�
㡰
ᨨ鼀 �矄泐矄 � 矄 矄ᨨᨨ 矄 ├泐 �矄 矄矄 �矄泐矄졔� ᨨ ��鼀 㡰 ᨨ鼀犨� �矄泐�鼀㦨 矄ᆠᨨ 矄 ᆠ㡰 ᆠ盐 � 鼀鼀ᆠᨨ鼀矄҈⏌ᨨ � �� 盐 ҈矄ᆠᨨ 㡰 ᆠ盐 � 鼀鼀ᆠᨨ鼀矄 ᨨ ᨨ � �鼀 ҈ 꾀 � ᆠ 矄ᆠ 盐 矄 � �鼀 ᨨ 盐 �矄 �矄 ᨨ �� 鼀盐 泐 矄矄� � ᆠ�ᆠ ᆠ盐 ᨨ �鼀 鼀�矄ᆠ├ᨨ 泐 ��ᨨ �鼀 �矄ᨨ矄 ᨨ├盐 ᨨ鼀ᨨᆠ�ᨨ 矄 矄盐 �ᆠ� 泐ᨨ ��鼀ᨨ ᨨ�ᨨ 盐ᨨ ᨨ 盐 � ᨨ鼀 ᨨ�ᨨ ᨨ 盐 �矄 ᨨ鼀 �鼀 ᔼ 盐�矄 ᨨ犨 ᆠᨨ 矄 ᨨ� 犨 �ᆠ ᨨ ᨨᨨ犨� � 矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌 㡰 鼀盐 �矄�盐�ᨨ ᨨ �鼀 �矄 盐 � ᨨ鼀 㡰 矄�矄 ᆠᨨ矄鼀矄ᆠ�ᨨ 矄ᨨ �鼀 ᆠ矄矄犨� �鼀 ᆠ㡰 �鼀 鼀�矄 鼀ᨨ�� �� ᨨ ⼄盐 ᆠ�矄 ᨨ�├矄盐 㡰 �鼀 ᆠᨨ 矄盐鼀矄矄矄ᆠ�ᆠᨨ犨ᨨ犨 矄 ᨨ � ᨨ 盐
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҈
҈�
҈
Physics
␔ ᨨ � � ᆠ�矄� ᆠᨨ矄矄 ᆠ盐 矄矄� � ᨨ盐� ҈黨黨黨和㜠㝐├矄桌 㡰 � 矄盐ᨨ⧤�矄 ᨨ 㡰 矄鼀�ᆠᨨ 盐ᆠᨨ⧤�矄 ᨨ ᆠ㡰 鼀⧤�矄 � � 㡰 a p㜠n �ᨨ ⧤�矄 � �� ��矄矄�矄 ᔴᆠ�ᨨC҈ᨨ ᆠᆠ� ᆠ҈鼀�ᆠᨨ㜠 ᆠᆠ�ᨨ Cᨨ ᆠᆠ� ᆠ鼀�ᆠᨨ 矄泐ᆠ � 盐泐 泐 ᆠᨨ鼀鼀ᨨ 泐 犨ᨨ ᆠᆠ�ᨨ矄
矄泐ᨨ犨 ᨨ �矄ᆠ � ᨨ盐� ⼄盐矄�矄ᨨ矄 �鼀t㦨 ҈黨黨㜠㝐├矄桌 㡰 ᆠ盐 � ᆠ ᨨ 犨ᨨ �矄ᆠ �� �ᆠ�ᆠ盐�矄�矄 ᨨt㦨 㡰 ᆠ盐 矄� 犨ᨨ �矄ᆠ �� �ᆠ�ᆠ盐�矄t㦨 ⼄盐盐 ᨨ ᨨ ᆠ㡰 ᆠ盐 矄� 犨ᨨ �矄ᆠ �� �ᆠ�ᆠ盐�矄t㦨 盐 ⼄盐 㡰 ᔴᆠ�ᨨC҈ᨨ矄矄矄�奠ᨨ �矄� ��ᆠ �矄ᨨᨨ Cᨨ矄矄�奠ᨨ �矄� ��ᆠ � 7 Match the Following ᔴᨨ盐鼀 ᔼ��矄矄ᨨ鼀 �ᆠ矄 ᔴᨨ盐鼀 ᔼᔼ��矄矄ᨨ鼀 ᨨᆠ矄矄矄 ᨨ 犨 �ᆠ 盐 ᆠ�ᨨ � � ᨨ 矄 �ᆠ矄 㝐ᆠ �ᆠ矄� ᔴᨨ盐鼀 ᔼ犨� ᨨᆠ矄矄矄 � ᔴᨨ盐鼀 ᔼᔼ � �ᆠ 泐ᨨ盐 矄犨 泐 ├ � � ᨨ�盐矄� 맔鼀� �� � 졠 和
C
A
B
R
ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐 泐�矄 � ᨨ 犨� 鼀 V ᆠᆠ�ᨨ�矄� ��泐盐 ᆠ � 矄犨�ᆠ S�矄ᆠᨨ矄 �鼀t㦨 ҈黨黨㜠㝐├矄桌 㡰 � ᆠ �Qᨨ ᆠᆠ�ᨨ�鼀t 㡰 � ᆠ盐 � AB�鼀t␔ �矄�矄�鼀�� � 盐矄 t ® ¥ S
R
A
R
V
C
B
R
҈ 꾀 ᆠ�ᆠᆠ�ᆠ盐��矄矄 ᨨ犨 � ��ᔴᆠ盐 ᨨ � � ᆠᆠᨨ矄矄 矄�矄ᨨᨨ ᨨ 鼀矄犨�鼀矄盐 泐 ᨨ鼀 V ᨨ 矄�矄 ᆠ ᨨ 鼀 � 泐 泐� �⧤�ᆠ ᨨ 矄盐矄ᨨᨨ 犨�矄 ҈黨黨㜠�㝐├矄桌
䰘桌 꾀㡰 㡰 ᔴ㡰 㡰
ᔴᨨ盐鼀 ᔼᔼ 㡰 ��ᨨ ᨨ鼀 ᨨ ᨨ 泐 和鼀 �� ⼄㡰 쟼 �泐ᆠᨨ 矄�ᨨ
ᔼ ᆠ 矄ᆠ 鼀 㡰 㝐� � 쟼ᆠ�ᆠ 盐矄 矄㡰 鼀 矄�ᨨ ᨨ 矄ᨨ� 矄
V
ᔴᨨ盐鼀 ᔼ �鼀�ᆠ矄�
W
ᔼ �� ᆠ�ᆠ盐� 矄犨�ᆠ S�矄ᆠᨨ矄 �鼀t㦨 ᆠ � Q ᨨ ᆠᆠ�ᨨ 泐� 矄 t �矄 �� 泐 Q t㡰㦨Q ҈eat㡰 � 盐ᨨ Q a � 鼀矄ᨨ �� 鼀矄矄矄 ᨨ犨 � ᆠ�ᆠ盐��㜠㝐├矄桌 R҈
҈黨
R
҈黨㜠�㝐├矄桌 R҈ = R 㦨ᨨ 鼀矄
R = R 㦨ᨨ 鼀矄 C㦨 � m f � ᆠ盐 � �泐矄ᨨ � ᆠᆠ�ᨨ R҈
C
҈
҈
ᔼ �� ᆠ�ᆠ盐� E҈ = E = E 㦨ᨨ矄
S 졔
W
҈W
҈
10 Subjective Problems ҈䰘
҈W
҈W l
E҈
C
V
犨 ᆠ�ᆠ盐� ��鼀ᨨ� 泐졠 鼀矄犨犨� ᨨ 鼀� 矄�矄 ᆠᨨ ҈W 犨ᨨ� ᨨ鼀矄 ᨨ 矄�矄 ᆠ矄҈W ҈W 矄ᨨ盐ᆠᨨ 泐� � 鼀 和 ᨨ犨 ᆠᨨᆠᨨ矄��ᨨ 矄ᨨ ᨨ鼀 鼀鼀 㜠㝐├矄桌 ᨨ犨 泐�矄 ᨨ ᆠᨨ
ᆠ 矄ᨨ 矄 ᨨ犨
ᨨ矄犨� �├ᨨ � �� 䁸꾀矄ᨨ� �ᆠ ᨨ� 矄ᨨ盐犨 �ᆠ ᨨ盐盐ᆠ ├ 㜠㝐├矄桌
E
R
E
R
R
꾀ᨨ ᆠ�盐ᆠ盐�� 矄 泐矄ᨨ �犨� ᆠ盐 矄 ᨨ犨� �� ᆠ 矄 盐矄ᨨ 矄�矄 ᆠ矄ᆠ�矄 矄 ᨨ犨 � ��盐ᔴᆠ盐 �泐矄ᨨ � ᆠᆠ�ᨨC Ҙ 㡰 ҈黨㜠㝐├矄桌
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Current Electricity
� 和ᨨ 鼀矄犨 ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐ᨨ鼀 矄ᨨ矄 犨 ��矄ᆠᨨ
ᆠ ᆠᨨ矄矄ᨨ 鼀矄�矄 ᆠᔴᆠ盐 犨 矄鼀ᨨ鼀犨� 犨 ��矄ᆠᨨ
ᆠ ᆠᨨ矄矄 ᨨ 鼀矄�矄 ᆠ ҈黨桌
҈꾀 W
�W
W
꾀 ᔴ
҈W
m
W
W
҈W
꾀
W
W
W
҈꾀
ᔴᆠ盐 矄 泐矄ᆠ盐 � 㜠ᨨ 鼀矄�矄ᨨ 矄 ᨨ犨 � ᆠ�ᆠ盐�� ��盐 � 矄�矄 ᆠ ᨨ 泐�矄 ���� ᆠᆠ� ᆠ ᨨ ᆠᨨ 矄C�矄鼀�ᆠᨨ ҈黨㜠�㝐├矄桌
W W
꾀泐ᨨ 鼀 ᨨ矄 � 矄�矄 ᆠ҈ᨨ 鼀�矄 � �ᆠ � 犨� ᆠ盐 ᨨ �鼀矄 ҈黨桌 ᔼ 犨 �ᆠ�ᨨ 犨� ᆠ盐 ᨨ犨� 矄� 泐䁸 ␔ �矄 ᨨ � � ᆠ犨 犨ᨨ鼀� ᨨ 泐䁸 䰘 ᔼ ��鼀矄 ᨨ犨 � ᨨ � � ᆠ犨
ᨨ� 矄A B 犨 ᨨ� 矄B C� ҈黨䰘黨桌 矄 泐 矄 B
Ҙf Ҙf
W
C
҈Ҙf ҈Ҙf ҈Ҙf
.8 W
҈W
҈
W
C
A
V = 6 VOTS
Answer Key Topic-1 : Electric Current, Drift of Electrons, Ohm's Law, Resistance and Resistivity ҈ ᆠ㡰 㡰 㡰 㡰 ҈҈ ᆠ㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ 盐 矄 ᆠ 㡰 ᆠ㡰
� 㡰 ҈� ᆠ㡰
ᆠ㡰 ҈ ᆠ㡰
䰘 㡰 ҈䰘 ᆠ㡰
㡰 ҈ 㡰
黨 㡰 ҈黨 㡰
㡰
黨 㡰
҈ 㡰 㡰
Topic-2 : Combination of Resistances 㡰
ᆠ㡰
ᆠ㡰
ᆠ㡰
� 㡰
҈҈ 㡰
҈ ᆠ㡰
҈ 㡰
҈ ᆠ㡰
҈� ҈㡰
҈
㡰
䰘 㡰
҈ 㡰 ҈䰘 㡰
æRö ҈ ç ÷ èø
҈ 㡰
҈黨 㡰 㡰
Topic-3 : Kirchhoff's Laws, Cells, Thermo e.m.f. & Electrolysis ҈ ᆠ㡰 㡰 ҈҈ ᆠ㡰 ҈ 㡰 ᆠ 㡰
ᆠ㡰 ҈ 㡰
㡰 ҈ 㡰
� 㡰 ҈� �㡰
㡰 ҈ 㡰
䰘 㡰 ᆠ㡰 黨 㡰 ҈ 㡰 ҈䰘 㡰 ҈ 矄 ҈黨 㡰
Topic-4 : Heating Effect of Current ҈ 㡰 㡰 ҈҈ ᆠ㡰 ҈ 㡰 ҈ 㡰
㡰 ҈ 㡰
㡰 ҈ ᆠ㡰
� 㡰 ҈� 㡰
㡰 ҈ 㡰
䰘 㡰 ҈䰘 ᆠ㡰
ᆠ㡰 ҈ 㡰
黨 ᆠ㡰 ҈黨 㡰
҈ ᆠ㡰 W㡰
Topic-5 : Wheatstone Bridge and Defferent Measuring Instruments ҈ 㡰 㡰 ҈҈ 㡰 ҈ ᆠ㡰 ҈ 䰘㡰 㡰
㡰 ҈ ᆠ㡰
㡰 ҈ 㡰
� ᆠ㡰 ҈� 㡰
㡰 ҈ 㡰
䰘 ᆠ㡰 ҈䰘 㡰
㡰 ҈ 㡰
黨 㡰 ҈ 㡰 ҈黨 ҈㡰 ҈㡰
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈ ᆠ㡰 ҈҈ 㡰
㡰 㡰 ҈ ᆠ 㡰
㡰 � 㡰 㡰 䰘 ᆠ㡰 㡰 黨 㡰 ҈ �㡰 ҈ ᆠ 㡰 ҈ 㡰 ҈� 㡰 ҈ 꾀®矄錐®⼄錐ᔴ®⼄錐®⼄
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EBD_8341 P-194
Physics
Moving Charges and Magnetism
18
Topic-1 : Motion of Charged Particle in Magnetic Field 1 ҈
MCQs with One Correct Answer
꾀 ᆠᨨ �矄鼀ᨨ� �ᨨ �졔x �ᆠ�ᨨ 犨� ᨨᆠ�泐 ᨨ 맔҈鼀矄҈ᔼ 矄��ᨨ ᨨ 盐 � ᨨ鼀ᆠ�ᆠ � ᨨ ⏌ᆠ鼀ᨨ� � �ᨨ �졔y �ᆠ�ᨨ 鼀� �盐 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � 矄盐� �矄��ᨨ 矄盐ᆠ ᆠᨨ ├矄鼀ᨨ� �ᨨ � x �ᆠ�ᨨ 犨� 㝐� 和 ᔼ㡰桌 㡰 맔҈ ᨨ �졔z �ᆠ�ᨨ
㡰 �맔҈ ᨨ �z �ᆠ�ᨨ
ᆠ㡰 �맔҈ ᨨ �졔z �ᆠ�ᨨ
㡰 맔҈ ᨨ �z �ᆠ�ᨨ
꾀 �ᆠ ᨨ ᆠ � q 鼀矄矄 m �矄 鼀ᨨ� � 犨� ᨨᆠ�泐 v $� v ¹ 㡰 ᨨ犨 矄�矄ᆠ ᆠ �
�
㡰 ҈҈鼀
Ӹ㜠 �矄 ᆠ d ᔼ �矄 鼀� �ᆠ � ur B = B k$ 鼀� �鼀盐鼀盐ᨨ v ᨨ犨 �ᆠ �ᆠ 犨� ᨨ � 矄ᆠ �矄 㝐� 和 ᔼ㡰桌 㡰
qdB m
㡰
qdB qdB 㡰 m m 꾀ᆠ � �ᆠᆠ泐� �ᆠ �҈mᔴ�矄鼀ᨨ� �犨� ᨨᆠ�泐 �ˆ + ˆ + kˆ㡰 鼀矄҈ᔼ 鼀� �ᆠ �
ᆠ㡰 鼀 㡰 鼀 䰘
ᨨ ��ˆ + ˆ - kˆ㡰 ´҈- �矄矄� ��ᨨ 犨 �ᆠ �矄 鼀ᨨ� � ᨨᆠ ᨨ �ᆠ �矄 ur ur F ´҈-黨 ᆠᨨ F �矄 [㝐� Sep. 03, 2020 (I)] 㡰 - �ˆ + ˆ - 黨kˆ ᆠ㡰
- �ˆ + ˆ - 黨kˆ - �ˆ + ˆ - 黨kˆ
㡰
- �ˆ + ˆ - 黨kˆ
㡰
B
㡰 鼀
qdB m
ᆠ㡰
꾀鼀ᨨ ᨨᨨ 矄犨� 矄 맔҈�鼀矄҈ 矄盐 � ᨨ鼀 鼀� �ᆠ � ᨨ �ᨨ ᨨ 鼀� �ᆠ � �ᆠ ᨨ 矄盐� � �ᆠ ᨨ ᨨᨨ 矄�矄 ᆠᨨ矄ᨨ 㝐矄矄ᨨ ᨨᨨ 㦨҈䰘맔҈䰘├�ᆠ �ᨨ ᨨᨨ 㦨҈黨맔҈҈黨ᔴ㡰 [㝐� Sep.02, 2020 (I)] 㡰 ᆠ鼀 㡰 �ᆠ鼀 ᆠ㡰 ҈ᆠ鼀 㡰 ᆠ鼀
��盐矄 ᨨ犨矄��ᨨ ᨨ � �l犨� 盐 � ᨨ鼀 鼀� �ᆠ � ᨨ � � ᨨᨨ � � ��ᨨ
犨� ᨨᆠ�泐맔҈�鼀矄҈鼀├� � � 犨� � ᔼ ᨨᨨ ᆠᨨ鼀矄҈ᨨ盐�ᨨ 泐 �鼀� ᆠᨨ矄矄 ��ᨨ 矄 ᨨ犨 �l�矄ᆠᨨ矄ᨨ 鼀矄矄ᨨ ᨨᨨ 㦨 ҈䰘맔҈䰘├�ᆠ �ᨨ ᨨᨨ 㦨҈맔҈҈黨ᔴ㡰 [㝐� Sep.02, 2020 (II)]
l
꾀�ᆠ � � 矄鼀ᆠ �矄ᨨ ᆠᨨ 鼀ᨨ矄 � ᆠ�ᆠ盐 ᨨ �盐矄�ᆠ鼀盐 � 盐 ᆠᨨ 鼀� �ᆠ � ᨨ � ᔼ ᆠ�ᆠ � ᨨ ҈⏌鼀 鼀├矄�ᨨ鼀ᨨ� 矄�� 鼀矄矄ᨨ �ᆠ�矄 � ᆠ �ᨨ ᆠᨨ 㦨҈맔҈҈黨ᔴ㡰 㝐� ҈꾀�҈黨ᔼ桌 㡰 黨҈맔҈҈├� 㡰 ҈맔҈䰘├� ᆠ㡰 ҈맔҈҈黨├� 㡰 맔҈├� 꾀ᨨᨨ ᆠᨨ �盐鼀 盐ᆠ盐矄 矄鼀 �泐 泐� ᆠ�ᆠ盐ᨨ�矄� 盐ᨨ 鼀� �ᆠ � �ᆠ盐ᨨ �矄ᆠ� �� 㝐� ҈꾀�҈黨ᔼ桌 㡰 >㦨 㡰 㦨 ᆠ㡰 㡰 >>
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P-195
Moving Charges and Magnetism
9.
10.
11.
A proton and an α -particle (with their masses in the ratio of 1 : 4 and charges in the ratio 1 : 2) are accelerated from rest through a potential difference V. If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii rp : ra of the circular paths descrfibed by them will be: [Main 12 Jan 2019, I] (a) 1: 2 (b) 1: 2 (c) 1: 3 (d) 1: 3 In an experiment, electrons are accelerated, from rest, by applying a voltage of 500 V. Calculate the radius of the path if a magnetic field 100 mT is then applied. [Charge of the electron = 1.6 × 10–19 C Mass of the electron = 9.1 × 10–31 kg] [Main 11 Jan 2019, I] (a) 7.5 × 10–3 m (b) 7.5 × 10–2 m (c) 7.5 m (d) 7.5 ×10–4 m The region between y = 0 and y = d contains a magnetic r field B = Bzˆ . A particle of mass m and charge q enters the
mv region with a velocity vr = viˆ . if d = , the acceleration 2qB of the charged particle at the point of its emergence at the other side is : [Main 11 Jan 2019, II] qv B æ 1 ˆ 3 ˆö qv B æ 3 ˆ 1 ˆ ö j÷ i + j÷ ç iç (b) m è 2 2 ø m è2 2 ø qvB æ - ˆj + iˆ ö qvB æ iˆ + ˆj ö ç ÷ ç ÷ (c) (d) m è 2 ø m è 2 ø An electron, a proton and an alpha particle having the same kinetic energy are moving in circular orbits of radii re, rp, ra respectively in a uniform magnetic field B. The relation between re, rp, ra is : [Main 2018] (a) re > rp = ra (b) re < rp = ra (c) re < rp < ra (d) re < ra < rp A negative test charge is moving near a long straight wire carrying a current. The force acting on the test charge is parallel to the direction of the current. The motion of the charge is : [Main Online April 9, 2017] (a) away from the wire (b) towards the wire (c) parallel to the wire along the current (d) parallel to the wire opposite to the current Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed 'v' in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities s1 and s2 are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe [Main Online April 10, 2016] effects):
(a)
12.
13.
14.
15.
- Î0 vB Î vB , s2 = 0 2 2
(a)
s1 =
(b)
s1 =Î0 vB, s 2 = - Î0 vB
(c)
s1 =
Î0 v B - Î0 vB , s2 = 2 2
field occupies a region of space by width ‘d’. If a be the angle of deviation of proton from initial direction of motion (see figure), the value of sin a will be : [Main Online April 10, 2015]
B a d (a)
qV
Bd 2m
B qd 2 mV
(b)
B q q (d) Bd d 2mV 2mV 16. A particle of charge 16 × 10–16 C moving with velocity 10 ms–1 along x-axis enters a region where magnetic field ur of induction B is along the y-axis and an electric field of magnitude 104 Vm–1 is along the negative z-axis. If the charged particle continues moving along x-axis, the ur magnitude of B is : [Main Online April 23, 2013] (a) 16 × 103 Wb m–2 (b) 2 × 103 Wb m–2 3 –2 (c) 1 × 10 Wb m (d) 4 × 103 Wb m–2 ur 17. A magnetic field B = B0 Jˆ , exists in the region a < x < 2a, uv and B = – B0 ˆj , in the region 2a < x < 3a, where B0 is a positive constant. A positive point charge moving with a r velocity v = v0iˆ , where v0 is a positive constant, enters the magnetic field at x = a. The trajectory of the charge in this region can be like [2007]
(c)
B0 0
2a
a
x
3a
- B0 z
(a)
z
a
2a
3a
x
(b)
z
(c)
a
2a
x
3a
z
a
2a
3a
x
a
(d)
2a
3a
x
v
18. An electron travelling with a speed u along the positive x-axis enters into a region of magnetic field where B = –B kˆ B
s1 s2 (d) s1 = s 2 =Î0 vB A proton (mass m) accelerated by a potential difference V flies through a uniform transverse magnetic field B. The
0
(x > 0). It comes out of the region with speed v then [2004S] (a) v = u at y > 0 ×B y
(b) v = u at y < 0 (c) v > u at y > 0
e–
u
x®
(d) v > u at y < 0
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EBD_8341 P-196
҈黨
҈
�
ᨨ ᨨ矄��泐ᆠ � �ᆠ鼀ᨨ� �� x㜠y � ��泐ᨨ � x㜠 �矄 �矄矄盐 ᆠ �� �矄 盐ᨨ 矄 ᆠᨨ ᆠ�ᆠ ⏌ᨨ鼀� �ᆠ � 矄 泐ᨨ P ᆠ盐 �矄矄 ᨨ犨 � x㜠y �矄 ᨨ盐 ᨨ ᨨ 㜠ᆠ�ᆠ盐 ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � ᆠᨨ鼀� �ᨨ 矄�矄ᨨ矄矄�䁸 和桌 uur uur y 㡰 E = 錐 B = b�ˆ + ckˆ uur uur 㡰 E = a�ˆ錐 B = ckˆ + a�ˆ uur uur ᆠ㡰 E = 錐 B = cˆ + bkˆ uur uur 㡰 E = a�ˆ錐 B = ckˆ + bˆ 꾀�ᆠᨨ 鼀矄矄鼀 ᆠ �q鼀ᨨ矄犨� ᆠᨨ 矄 ᨨᆠ�泐ᨨ � ᨨ矄��x㜠 �ᆠ�ᨨ ᔼ 矄��ᨨ
ᆠᨨ � � �盐 � ᨨ鼀鼀� �ᆠ � B �ᆠ ᨨ �
�� 㜠 �ᆠ�ᨨ � � ᨨ鼀x㦨aᨨx㦨b 鼀� �鼀盐鼀盐ᨨ v⼄盐� 矄ᨨ �ᆠᆠ 盐矄 ��ᨨ x>b�矄 和桌 qbB q b - a 㡰 B qaB q b + a 㡰 B ᆠ㡰 㡰 㡰 㡰 m m m m
犨ᨨ�ᆠ矄A B ᨨ 鼀矄矄矄mA mB 矄ᆠ�泐 � � 矄鼀ᆠ �鼀ᨨ� �� 꾀 盐 � ᨨ鼀鼀� �ᆠ � �矄矄 �ᆠ盐ᨨ �矄
矄 矄ᨨ �ᆠ矄vA vB矄ᆠ�泐 ᆠᨨ�矄矄矄 ᨨ犨 � ��盐 ҈和桌 㡰 mAA mBB 꾀 㡰 mAA >mBB ᆠ㡰 mA mB A B 㡰 mA 㦨mB A 㦨B 꾀 �ᨨ � �矄ᆠᨨ � 矄ᨨ ᨨ矄�� ���ᨨ 矄 ᔼ ��矄矄盐ᆠ 矄�鼀盐 ᨨ盐矄泐ᨨ ᆠ�ᆠ � ᨨ � 졔 㜠 �ᆠ�ᨨ 鼀� �ᆠ � ᨨ � 졔z㜠 �ᆠ�ᨨ 和桌
㡰 ᨨ矄�� �ᨨ 矄 ᆠ ᨨ犨 矄 졔y㜠 �ᆠ�ᨨ
���ᨨ 矄ᨨ犨 矄㜠y �ᆠ�ᨨ
㡰 �ᨨ 矄 ᆠᨨ犨 矄졔y㜠 �ᆠ�ᨨ
ᆠ㡰 �ᨨ 矄 ᆠᨨ犨 矄y㜠 �ᆠ�ᨨ
㡰 ᨨ矄�� �ᨨ 矄 ᆠ ᨨ犨 矄 泐㜠 �ᆠ�ᨨ
���ᨨ 矄ᨨ犨 矄졔y㜠 �ᆠ�ᨨ 꾀�ᆠᨨ ᆠ �⼄ 鼀矄矄鼀鼀ᨨ矄� ᆠ�ᆠ盐ᨨ� ᨨ �盐矄犨� �盐矄 w �ᨨᨨ 鼀� �盐 ᨨ �矄鼀� �ᆠ鼀ᨨ鼀 ᨨ ᨨ �矄 �盐鼀ᨨ鼀 盐鼀 矄 ᨨ
和桌 㡰 w q 㡰 w q m ᆠ㡰 q m 㡰 w m 꾀ᆠ � �ᆠ�矄矄 ᨨ鼀矄� ��ᨨ ᨨ 矄 泐 盐 � ᨨ鼀ᆠ�ᆠ 鼀� �ᆠ � 矄犨 �ᆠ ᨨᆠ ᨨ �ᆠ犨�鼀ᨨ� ҈黨黨黨和㜠㝐├矄桌 㡰 矄�� � 㡰 ᆠ�ᆠ ᆠ㡰 � 㡰 ᆠ泐ᆠᨨ� 꾀泐�矄ᆠᨨ
ᆠ 犨 犨ᨨᨨ� 矄A Bᨨ ᆠ�ᆠ盐鼀 ᆠᨨ 盐 � ᨨ鼀ᆠᨨ 盐ᆠ� �� �ᨨ �盐矄 矄�矄 ᆠR졠 ᨨ ᆠ矄ABᨨ � �矄盐 矄
�q ᆠ 盐ᨨ 鼀� �ᆠ� 盐ᆠ�ᨨ
ᆠ 盐ᨨ ᆠ盐 � � ��矄 ҈黨黨�和桌 㡰 ᨨᨨ�ᨨ ᨨ ҈ q㡰 㡰 � 矄泐ᨨᨨ�ᨨ ᨨ ᆠ㡰 ᨨᨨ 泐� q㦨҈ 㡰 ᨨ ᨨ盐矄ᨨ q
Physics
犨ᨨ�ᆠ矄X Y � �⼄盐ᆠ �矄 � � ᆠᆠ ᨨ盐� 矄鼀ᨨ � � ᆠ ��ᨨ ᨨ 盐 � ᨨ鼀鼀� �ᆠ � 矄ᆠ�ᆠ�ᆠ盐 矄 ᨨ ��R҈ R矄ᆠ�泐 �ᨨᨨ 鼀矄矄ᨨ X ᨨ ᨨ Y�矄 ҈黨㜠㝐├矄桌 㡰 R҈ ⏌ R 㡰҈⏌ 㡰 R ⏌ R҈ ᆠ㡰 4
R҈ ⏌ R 㡰
㡰
R҈ ⏌ R
Fill in the Blanks
䰘 꾀 盐ᨨ ᨨᨨ ᆠᨨ �ᆠ ��ᨨ ᨨ ᆠᨨ 矄 鼀� �ᆠ � 犨� ⼄盐 ᨨᆠ��矄 鼀� �ᆠ � �矄ᨨ � � 犨 ᨨ鼀 ᨨ ᨨ ᆠ├矄ᨨ �ᆠ矄 � �� ᆠᨨ ᨨᨨ犨矄ᆠ├ �ᆠ ᨨᨨ犨矄ᆠ├ ҈黨㜠㝐├矄桌 C
B A
D
5
True / False
꾀 ᆠᨨ ᨨᨨ 鼀ᨨ� �犨� 矄鼀├� �ᆠ �泐ᨨ � 矄鼀 �ᆠ�ᨨ ␔ 泐矄矄 ᨨ盐� 盐 � ᨨ鼀鼀� �ᆠ � �ᆠ盐ᨨ �ᆠ�ᨨ ᨨ � 鼀ᨨ�ᨨ 泐 矄ᆠ� ᆠ�ᆠ盐 矄 ᨨ 矄鼀 �盐矄 ҈黨�㜠㝐├矄桌 黨 꾀ᆠ � �ᆠ 矄��ᨨ ᨨ 盐 � ᨨ鼀鼀� �ᆠ � �ᨨ � ᨨ 鼀� �ᆠ� ᨨ ᨨᆠ ᨨ �ᆠ�矄ᆠ�ᆠ ҈黨㜠㝐├矄桌 �矄 ᨨᆠ �� �泐ᨨ ᆠ � �ᆠ 鼀ᨨ� �� 鼀� �ᆠ � ᨨ盐� 鼀� �ᆠ ᨨᆠ�矄 ᆠ� �ᨨ � ҈黨㜠㝐├矄桌 6
MCQs with One or More than One Correct Answer
҈ 꾀盐 � ᨨ鼀鼀� �ᆠ � �矄矄� ��ᨨ 犨
㦨 㦨 ��ᨨ � ��盐㡰ᨨ� � � ᨨ鼀泐 � ᨨ ᨨ 꾀�ᆠ犨� ᆠ �졔턼 鼀ᨨ鼀 盐鼀 �ᆠ ᨨ � �矄 矄��ᨨ ᨨ鼀 ��ᨨ ҈ ᨨ� ҈ 泐 㦨 㡰 ␔ �ᆠ ᨨ ᨨᨨ犨� � ᨨ�ᨨ 矄㡰�矄⏌ᆠᨨᆠ䁸 꾀 ҈䰘桌 ��ᨨ ҈ 泐
졠 졔턼
҈
맔 맔 맔 맔 맔 맔
맔 泐㦨 㡰 맔 맔
��ᨨ 맔 맔 맔 맔 맔 맔 맔 맔 맔
��ᨨ 맔 맔 맔 맔 맔 맔
맔 맔 맔
⏌
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P-197
Moving Charges and Magnetism
�ᆠ犨� ��ᨨ ҈ 턼 �ᆠ 犨� ��ᨨ 㡰 ᨨ = ҈ 턼 ᨨ盐� ᨨ� ᨨ �矄 ᆠ㡰 ␔ �ᆠ 矄 ��ᨨ ҈ ᨨ盐� ᨨ �矄ᨨ矄矄� � ��ᨨ 鼀� �盐 ᨨ ᆠ �� �矄� 鼀ᨨ鼀 盐鼀犨 ᨨ� ҈ 矄ᨨ� ᨨ鼀泐 �矄�矄⏌ 㡰 ᨨ � �ᆠ矄ᨨ 矄鼀ᆠ �턼 矄鼀 ᨨᆠ�泐 �矄 ᆠ犨 ᨨ� ҈ ᨨ� ᨨ 泐 � ᨨ ��ᨨ ҈ �矄 � 矄泐 ᨨᨨ�ᨨ ᨨ 鼀矄矄ᨨ �ᆠ 꾀 ᆠᨨ 盐ᆠᨨ 矄 ᨨ犨 � ��盐㡰 ᆠ泐� � ᆠᨨ 矄 ᆠ盐 I�矄├� x㜠y � 盐 � ᨨ鼀鼀� �ᆠ � r B ᔼ F�矄 鼀� �盐 ᨨ ᨨ鼀� �ᆠ ᨨᆠᆠ� � ᨨ ᆠᨨ 盐ᆠᨨ ᆠᨨᆠ矄鼀 矄㡰�矄 㡰 꾀 ҈�桌 y R R
㡰 ᨨ >
p⏌
p⏌
x R R Lr L 㡰 ᔼ B �矄ᨨ � $z Fµ L졔R㡰 r 㡰 ᔼ B �矄ᨨ � xˆ F㦨 r ᆠ㡰 ᔼ B �矄ᨨ � yˆ Fµ L졔R㡰 r 㡰 ᔼ B �矄ᨨ � $z F㦨 꾀�ᆠᨨ 鼀矄矄M ᨨ矄��ᆠ �Q, 鼀ᨨ� �犨� r ᆠᨨ 矄 ᨨᆠ�泐 u҈ = �ˆ 鼀矄-҈ 矄��ᨨ ᨨ 盐 � ᨨ鼀 矄�ᆠ鼀� �ᆠ � ᨨ鼀ᨨ x㜠y ��ᨨ ᨨ 鼀� �ᆠ � 矄 ᨨ鼀x 㦨ᨨx 㦨L ᨨ盐矄 ᨨ y. 꾀 矄矄� � ᨨ盐� �矄��ᨨ �ᆠ鼀�矄 ᨨ ᨨ 矄� ҈ 鼀��矄ᆠᨨ 矄 犨� ᨨᆠ�泐 r u = �ˆ + ˆ 㡰 鼀矄 -҈ ᆠᨨᆠ矄鼀 矄㡰�矄 㡰 꾀 ҈桌 㡰 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � �矄 z �ᆠ�ᨨ
㡰 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � �矄졔z �ᆠ�ᨨ
ᆠ㡰 鼀� �盐 ᨨ 鼀� �ᆠ �
�pM 盐 �矄 Q
҈pM 盐 �矄 Q ᔴᨨ 矄� 鼀ᨨ�ᨨ ᨨ ᨨ矄��ᨨ� ᆠ �� ��ᨨ
犨 矄�鼀盐 ᨨ盐矄盐 � ᨨ鼀ᆠ�ᆠ 鼀� �ᆠ r r � 矄 E = E0 ˆj B = B0 ˆj . 꾀�鼀t㦨 �矄ᆠ � 矄 r ᨨᆠ�泐 v � � x㜠泐 鼀├� � �q 犨� x㜠 �矄␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄㡰�矄 㡰ᆠᨨᆠ ᨨ�鼀t>䁸 ҈桌 㡰 ᔼ q = ° ᆠ �鼀ᨨ矄� ᆠ�ᆠ盐 � x㜠z 㡰 ᔼ q = ° ᆠ �盐 �ᨨ矄 �ᆠ鼀ᨨ�ᨨ 犨� ᆠᨨ 矄 �ᆠ ᨨ � y㜠 �矄 ᆠ㡰 ᔼ q = ҈° ᆠ �盐 �ᨨ矄 �ᆠ鼀ᨨ�ᨨ 犨� �矄�ᆠ � ᆠ矄� �犨� �鼀ᨨ � y㜠 �矄 㡰 ᔼ q = 黨° ᆠ �盐 �ᨨ矄� 盐ᆠᆠ 鼀ᨨ�ᨨ ᨨ � y㜠 �矄
㡰 鼀� �盐 ᨨ 鼀� �ᆠ � �矄
� 꾀 ᆠᨨ ᨨᨨ 鼀ᨨ� �ᨨ 矄�� 矄犨� 矄鼀ᨨᆠ�泐 泐 矄鼀�� � ���ᨨ
ᨨ 盐 � ᨨ鼀鼀� �ᆠ � �ᆠ盐ᨨ ᨨᆠ�泐 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄⏌盐䁸 ҈҈桌 㡰 泐犨� ᆠᨨ鼀ᨨ盐ᨨ 鼀� �ᆠ � ��ᨨ 㡰 泐犨�ᆠᨨ鼀ᨨ盐� �ᨨ � 矄 ᆠ㡰 泐犨�ᆠᨨ鼀ᨨ盐 矄鼀�鼀 㡰 泐犨�ᆠᨨ鼀ᨨ盐 � �鼀矄 꾀�ᆠᨨ 鼀矄矄鼀 ᆠ �⼄鼀ᨨ� �犨� ᨨᆠ�泐 矄 ��ᨨ ᔼᔼ ᨨ鼀ᨨ ᨨ盐 泐矄矄 ᨨ犨 � ��盐 ��ᨨ ᔼᔼ 矄盐 � ᨨ鼀鼀� �ᆠ � 㜠 �ᆠ盐ᨨ ᨨ � ᨨ ��ᨨ
ᔼᔼ�矄 l ᔴ ᨨᨨ矄 ᆠᨨᆠᆠ ᨨ�ᆠ 矄㡰 桌 ��ᨨ ᔼ ��ᨨ ᔼᔼ ��ᨨ ᔼᔼᔼ 맔 맔맔맔맔맔 맔맔맔맔맔맔 맔맔맔맔맔맔 맔맔맔맔맔맔 맔맔맔맔맔맔 㡰 �ᆠ 矄 ��ᨨ ᔼᔼᔼ ᨨ 泐� �矄 ᨨᆠ�泐 ⼄l > 鼀 㡰 �ᆠ 矄 ��ᨨ ᔼᔼᔼ ᨨ 泐� �矄 ᨨᆠ�泐 ⼄l < 鼀 ᆠ㡰 � ᨨ �ᆠ� ��ᨨ ᔼᔼ�矄鼀 �鼀盐鼀 ⼄l 鼀 㡰 �鼀矄 � ��ᨨ ᔼᔼ�矄矄鼀 ᨨ 泐ᨨᆠ�泐矄 ᨨ �矄 �ᆠ盐 矄ᨨ ��ᨨ ᔼ 䰘 졔졔 졠졔졔 � � 矄鼀├� �ᆠ �泐矄矄 ᨨ盐� ��ᨨ � 犨 �ᆠ �矄盐 � ᨨ鼀鼀� �ᆠ � �ᆠ盐ᨨ �ᨨᆠ�泐 鼀矄矄矄ᨨ 졔졔 졠졔҈鼀盐鼀盐 ҈鼀盐矄ᆠ�泐
㡰 졔犨� ᆠ 鼀ᨨ矄 ҈黨黨㜠㝐├矄桌 㡰 졠졔犨� ᆠ 鼀ᨨ矄 ᆠ㡰 졔 졠졔犨� ᆠ ⼄盐泐 㡰 犨� ᆠ ⼄盐泐 꾀ᨨᨨ 鼀ᨨ� �犨� ᆠᨨ 矄 ᨨᆠ�泐矄矄矄 ᨨ盐� ��ᨨ ᨨ 矄ᆠ犨� ᨨ盐 泐ᆠ �� �矄ᨨᆠ�泐ᔼ E B矄 ᆠ�ᆠ 鼀� �ᆠ � 矄矄ᆠ�泐 �矄��ᨨ ᨨ 矄ᆠ鼀泐 ҈黨�㜠㝐├矄桌 㡰 E㦨B㦨 㡰 E㦨B ¹ ᆠ㡰 E ¹ B㦨 㡰 E ¹ B ¹
犨 ᨨᆠ�泐 =
8
Comprehension/Passage Based Questions
�ᆠ�ᨨ 矄 턼矄 ҈ ᨨ 㡰 泐ᨨ�泐鼀ᆠ � � � ᨨ鼀�ᨨ �� � ᆠᨨ盐鼀 矄ᨨ ᨨᨨ犨� � 꾀ᆠ � �ᆠ ᆠᨨ ᨨᨨᨨ 㡰�矄� ᨨ 盐ᆠ ᨨ���
®
㦨泐㦨 㦨㡰犨� �� � ��ᨨᆠ�泐 꾀盐 � ᨨ鼀 ®
®
ᆠ�ᆠ � 쟼 盐 � ᨨ鼀鼀� �ᆠ � �矄泐犨 ®
®
®
ᨨᆠ�泐 ᆠ�ᆠ � 쟼 鼀� �ᆠ � ��
� ᆠᨨ盐鼀 矄҈ 矄ᆠ�泐 ⼄盐 ��矄쟼 ᨨ矄��� 鼀� �盐
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EBD_8341 P-198
Physics
ᔼᔼ㡰
쟼ᆠᨨ 犨� 쟼 ® = ˆ 쟼ᆠᨨ 犨� ® 쟼 = 泐ˆ
�㡰
��㡰
®
ᔼᔼᔼ㡰 ᨨᨨ 犨� = ���㡰 ᔼ㡰 ᨨᨨ 犨� 쟼 ® = ˆ 黨
҈
�㡰
®
쟼 = 쟼 ˆ
ᔴᨨ盐鼀 㡰
®
쟼 = - 쟼 泐ˆ 턼㡰
®
쟼 = - 쟼 ˆ 㡰
®
쟼 = 쟼 ˆ
和㡰
®
= - ˆ
®
= ˆ
®
= 泐ˆ
®
= ˆ
ᔼ 犨 �ᆠ ᆠ矄犨� �ᆠ鼀ᨨ� 矄�� � 犨� ᆠᨨ 矄 ᨨᆠ�泐䁸 㡰 ᔼᔼᔼ㡰 ��㡰 㡰 㡰 ᔼ㡰 �㡰 和㡰 ᆠ㡰 ᔼᔼᔼ㡰 ���㡰 㡰 㡰 ᔼᔼ㡰 ���㡰 和㡰 ᔼ 犨 �ᆠ ᆠ矄犨� �ᆠ 矄ᆠ� �ᆠ 犨� �矄ᨨ � ᨨ矄�� �ᆠ�ᨨ 䁸 꾀㡰 ᔼ㡰 �㡰 和㡰 㡰 ᔼᔼ㡰 ��㡰 㡰 ᔴ㡰 ᔼᔼᔼ㡰 ���㡰 㡰 㡰 ᔼ㡰 ��㡰 㡰 ᔼ 犨 �ᆠ ᆠ矄犨ᨨ盐 �ᆠ鼀ᨨ� 矄�� � ᨨ � �� �ᆠ�ᨨ ᨨ 泐㜠 �矄 �鼀ᨨᨨ � 泐ˆ 㡰䁸 㡰 ᔼᔼ㡰 ���㡰 턼㡰 㡰 ᔼᔼᔼ㡰 ��㡰 㡰 ᆠ㡰 ᔼ㡰 ��㡰 和㡰 㡰 ᔼᔼᔼ㡰 ��㡰 㡰 10 Subjective Problems
꾀ᨨᨨ a㜠�ᆠ ᆠᆠ 犨� 矄鼀 ᨨ � � ᆠ 泐 � ��ᨨ ᨨ ᆠᨨ 矄 鼀� �ᆠ � B �ᆠ盐ᨨ ᨨᆠ�泐ᨨ �ᆠ矄 � �ᨨᨨ �盐矄ᨨ ᆠ盐盐ᨨ ᨨᨨ ᨨ �盐矄ᨨ ᆠ盐盐ᨨ a㜠�ᆠ 㜠㝐├矄桌
��ᨨ 犨 x㦨 x㦨L�矄 � 犨� 盐 � ᨨ鼀 矄 泐鼀� �ᆠ � B kˆ 꾀�ᆠᨨ 鼀矄矄鼀ᨨ矄�� ᆠ �⼄ ᨨᆠ�泐 n�ˆ 矄ᨨ �x㜠 �矄 矄 ��ᨨ ᨨ 鼀� �ᆠ � �ᆠ��泐 ᨨ盐� ᨨ盐 ⼄盐矄�ᨨ ҈黨黨黨㜠҈㝐├矄桌 㡰 � 盐ᨨ L� �ᆠ鼀�矄 ᨨ鼀 ��ᨨ ᨨ 鼀� �ᆠ � 犨� �矄 � ᨨᆠ�泐 � ᨨ�矄� ��ᨨᆠ�泐 㡰 � � ᨨᆠ�泐ᨨ �ᆠ �鼀 矄 泐�� 鼀� �ᆠ � � 鼀� �ᆠ �
ᨨ犨 矄盐ᨨ҈L
꾀 ᆠᨨ �盐 鼀�矄ᆠᨨ 矄 ᨨ �泐├� �� ᨨ矄��x㜠 �ᆠ�ᨨ ᆠᨨ 矄 ⼄盐� ᨨ � 矄ᨨS 犨 GS㦨҈鼀 � 和鼀├矄 �ᨨ 犨� 㜠 �矄 矄矄 ᨨ犨 � ��꾀盐 � ᨨ鼀
和
r ® 鼀� �ᆠ � B ᨨGS �矄矄 � B ᨨ GS �矄矄� ��ᨨ ᨨ盐矄� ᆠᨨ �盐 � 鼀� �鼀盐鼀盐ᨨ B ᨨ鼀├ ᆠᨨ 矄 �S ҈黨黨㜠䰘㝐├矄桌 � 꾀鼀ᨨ ᨨᨨ 矄犨� ᨨᆠ�泐맔 ҈�鼀⏌矄ᆠ 矄 盐 � ᨨ鼀鼀� �ᆠ � ᨨ 矄 �ᨨ ᨨ 鼀� �ᆠ � � �盐矄ᨨ �ᆠ ├ 泐 ᨨᨨ 鼀꾀矄ᨨ � �ᆠ ᨨ � 犨 �ᆠ �矄 �矄 ᆠ 泐ᨨᨨ � 鼀ᨨ 鼀� �ᆠ � 盐� �ᨨ �ᨨ ᨨ ᨨ�ᨨ 㡰 ҈黨㜠㝐├矄桌 꾀�ᆠᨨ 鼀矄矄m㦨҈맔҈䰘├� ᆠ �q㦨҈맔҈҈黨ᔴ 矄��ᨨ ᨨ 盐 � ᨨ鼀鼀� �ᆠ � ᨨ 矄 � ҈矄 ᨨ � �ᆠ�ᨨ 矄 ᨨ犨 � �� 矄 ᨨ �ᆠ�矄 ҈䰘鼀⏌矄 �㡰 鼀� �ᆠ � �矄 �ᆠ ᨨ � � 犨
ᨨ鼀ᨨ ᨨ �ᆠ矄 ��ᨨ ᨨ � ᨨ� F � �矄 ᆠEF �q ��㡰ᔼ �ᆠ�ᨨ ᨨ q � �矄ᨨ � ᨨ盐犨 ᨨ鼀 ᨨ ᨨ � �鼀 矄 泐 �ᆠ � 쟼 ��ᨨ ᨨ 鼀� �ᆠ � � ��E � ҈黨㜠㝐├矄㡰 䰘 꾀ᨨ � � ᆠᨨ ᨨ矄�矄� ᆠᨨ矄矄 矄ᨨ ᆠᨨ 矄 矄�ᨨ 犨
矄�矄鼀鼀꾀 ᆠᨨ ᨨᆠ �ᆠ泐 ᨨ 矄 犨� ᨨᆠ�泐 ᨨ 맔 ҈ 鼀⏌矄ᆠ 鼀ᨨ矄 盐 ᆠ 犨 矄 � 鼀� �盐 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � � ��ᨨ 犨 ᆠᨨ 矄矄 �ᆠ � ᆠ矄㡰 ᔴ �ᨨ ᆠᨨ 㦨҈맔҈҈黨ᆠᨨ盐ᨨ鼀㡰 ҈黨҈㜠㝐├矄桌 졠 和 졔
ᔼ㡰
ᔴᨨ盐鼀
졔
ᔴᨨ盐鼀 ҈
Topic-2 : Magnetic Field Lines, Biot-Savart's Law and Ampere's Circuital Law 1 ҈
®
MCQs with One Correct Answer
꾀ᆠ � �ᆠ�ᨨ� �ᨨ盐 � ᆠ�ᆠᆠ ᆠᨨ 矄� 㜠 ᨨᆠ盐 ᨨᨨ꾀�ᆠᨨ 鼀矄矄mᆠ泐� � ᆠ �q�矄鼀ᨨ� �� 犨�矄 v盐 � 盐㜠 ᆠᨨ 鼀� �ᆠ � ® 鼀� �ᆠ鼀ᨨ鼀 ᨨ �矄 B 鼀ᨨ� ��ᆠ 㝐� 和 ᔼᔼ㡰桌
㡰
mv B B
®
㡰
-
ᆠ㡰
-
B
pB ®
®
mv B
mv B
㡰
-
mv B B
WWW.JEEBOOKS.IN
P-199
Moving Charges and Magnetism
꾀犨�A � 矄 ᨨ ᆠᨨ ᆠ�ᆠᆠ泐� � ᆠ盐 ᨨ 꾀 � � �盐矄ᆠ鼀 ᨨ 犨�B 矄ᨨ � 矄 ᨨ ᆠᨨ ᆠ�ᆠᆠ泐� �ᆠ盐 ᨨ 꾀 � � �盐矄ᨨ ᆠ鼀ᆠ 矄矄 ᨨ犨 �
��盐 �ᨨᨨ 鼀� �ᆠ � 矄 盐ᨨ 犨�矄 A B ᆠᨨ鼀鼀ᨨ ᆠ O�矄[Main Sep.04, 2020 (I)] A O
B
黨 㡰 㡰 ᆠ㡰 � 㡰 � 㝐� �盐 ᨨ 鼀� �ᆠ � � 和ᔼ盐 �矄㡰 ᆠ ᨨ �ᨨ 矄 ᆠᨨ�ᨨ 矄� ҈ᆠ鼀�盐 矄 ᆠ泐� � m I �矄[Main Sep. 03, 2020 (I)] p 㡰 � 㡰 � ᆠ㡰 � 㡰 � 꾀泐ᨨ �犨�ABDMNDC�矄矄 ᨨ犨 � ��盐ᆠ泐� � ᆠ盐 IAB BC矄矄�� ᨨ � �� �꾀D犨� ᨨ鼀矄ᆠ�ᆠ盐盐 DMNDᨨ �盐矄R ABBC矄 � �ᨨᆠ�ᆠ盐盐 N D 㝐� �ᆠ � ᆠ ᨨ ᆠ�ᆠ�矄 㝐� ᔼᔼ桌 m I æ ҈ ö p+ 㡰 ÷ pR çè ø m I æ ҈ ö 㡰 pR çè p ÷ ø m I p + ҈㡰 ᆠ㡰 pR m I 㡰 R
犨ᨨ泐ᨨ �矄�� � 矄盐 犨�矄├黨 � ᨨ鼀ᆠ ᨨ � xy㜠 矄矄 ᨨ犨 � ��盐
ᆠ盐 I 꾀鼀㡰� 盐 �矄ᨨ
�
矄犨�矄ᆠ泐ᆠ盐 矄ᨨ ⼄盐鼀� �盐 ᔼ犨 ᨨ矄 �ᆠ�ᨨ 矄矄 ᨨ犨 � ��盐 鼀� �ᆠ � ᨨ� 犨� 㝐� ҈꾀�҈黨ᔼ桌 㡰 ᨨ
m I ( xˆ + yˆ ) pd
+m I m I ( zˆ ) ( xˆ + yˆ ) 㡰 pd pd � 鼀� �ᆠ � ᨨ� 盐ᨨ矄�� � 矄�鼀 꾀ᨨ � ᆠ鼀ᆠ泐� �ᆠ盐 ᨨ �꾀 和 ��盐㡰 mᨨ㦨p맔҈䰘㜠꾀㡰 㝐� ҈꾀�҈黨ᔼᔼ桌
ᆠ㡰
㡰
䰘
㡰 맔҈�
㡰 ҈�맔҈�
� ᆠ㡰 맔҈
㡰 �맔҈�
꾀矄矄 ᨨ犨 � ��盐犨ᨨ� � �泐ᨨ �� �ᆠ犨�矄 泐黨º ᆠ � 矄盐ᆠ 犨泐 矄�鼀 矄 턼㝐ᨨ � x㜠 �矄犨 �矄�鼀 矄和 턼ᨨ y㜠 �矄ᔼ 졠㦨졠턼㦨ᆠ鼀 鼀� �盐 ᨨ 鼀� �ᆠ � 졠�矄҈ 犨ᨨ 犨�矄ᆠ泐⼄盐ᆠ盐 矄 矄 ��盐㡰 鼀� �盐 ᨨ ᆠ盐 � ᆠ 犨� �ᆠ�ᨨ ᨨ 鼀� �ᆠ � 졠犨� Ҙ㦨p맔҈䰘꾀㡰 㝐� ҈ ҈黨ᔼ桌 和
y
졠턼
㝐
x
㡰 꾀 �ᆠ盐ᨨ盐ᨨ � 㡰 꾀 �ᆠ盐ᨨ盐ᨨ � ᆠ㡰 꾀 �ᆠ盐� ᨨ � 㡰 꾀 �ᆠ盐� ᨨ � 졠 ᨨ 犨ᨨ� �ᆠᆠᨨ 盐ᆠ� �犨�矄ᨨ � �矄 � ᨨ鼀ᨨ ᆠ�ᆠ盐ᨨᨨ ᨨ ᨨ � ᨨ ᆠ�ᆠ盐ᆠᨨ�ᨨ � �ᆠ盐 矄ᔼ 矄鼀ᆠ盐 �矄 矄矄 � ᨨ �ᨨ ᨨ 鼀� �ᆠ � ᆠ ᨨ ᨨᨨ ҈㡰ᨨ ᆠ ᨨ ᆠᨨ� ᔴ㡰�.e. 犨� 㝐� 黨 ҈黨ᔼᔼ桌 ᔴ ҈ ҈ 㡰 㡰 ᆠ㡰 㡰 黨
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�ᨨ ҈ �矄 㝐� ҈桌 B ҈ 㡰 㡰 ᆠ㡰 㡰 ҈ 꾀鼀 ᨨ ᆠᨨ� 矄�ᨨ ᨨᨨ矄ᆠ 犨� N盐 矄 �盐矄R 泐ᆠ ᆠᨨ �泐 �矄 ᆠR 矄鼀 ᆠ盐 I ᨨ犨矄 ᨨ盐� ᨨᨨ矄 � 矄鼀 �ᆠ�ᨨ 鼀� �盐 ᨨ 鼀� �ᆠ � P鼀� 犨泐 犨 ᆠ 矄A C�矄�� 泐 ᨨ ��盐㡰 㝐� 졠 � 꾀�҈�҈桌
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EBD_8341 P-200
Physics
m0 l m0 l 6 [ 3 - 1] 6 [ 3 + 1] (b) 4 pa 4 pa m0 l m0l 3 [ 3 - 1] 3 [2 - 3] (c) (d) 4 pa 4pa An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic r field, B as a function of the radial distance r from the axis is best represented by [2012]
(a)
2R
A
P
C
16.
R
(a)
11.
12.
15.
8 N m0 I 3/ 2
(c)
4 N m0 I 1/ 2
(d)
8 N m0 I
plgL tan q µ0
L q
plgL µ0 cos q plgL (d) 2sin q I I µ0 cos q Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field B1, at its centre when a current I passes through it. The ratio B1 : B2 is: [Main Online April 12, 2014] (a) 1 : 1 (b) 1 : 3 (c) 1 : 9 (d) 9 : 1 An electric current is flowing through a circular coil of radius R. The ratio of the magnetic field at the centre of the (c)
14.
3/ 2
(b)
5 R 5 R 5 R 51/ 2 R Two identical wires A and B, each of length 'l', carry the same current I. Wire A is bent into a circle of radius R and wire B is bent to form a square of side 'a'. If BA and BB are the values of magnetic field at the centres of the circle and B square respectively, then the ratio A is: [Main 2016] BB p2 p2 p2 p2 (a) (b) (c) (d) 8 16 8 2 16 2 Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle 'q' with the vertical. If wires have mass l per unit length then the value of I is : (g = gravitational acceleration) [Main 2015] pgL tan q (a) 2 µ0 (b)
13.
4 N m0 I
sin q
coil and that at a distance 2 2R from the centre of the coil and on its axis is : [Main Online April 9, 2013] (a) 2 2 (b) 27 (c) 36 (d) 8 A symmetric star shaped conducting wire loop is carrying a steady state current I as shown in the figure. The distance between the diametrically opposite vertices of the star is 4a. The magnitude of the magnetic field at the center of the loop is [Adv. 2017] I
4a
(a)
r B
(b)
(c)
r B
(d)
r B
r B
17. A loop carrying current I lies in the x-y plane as shown in the figure. The unit vector kˆ is coming out of the plane of the paper. The magnetic moment of the current loop is [2012] 2 ˆ (a) a I k æp ö 2 (b) çè + 1÷ø a I kˆ 2 æp ö 2 (c) - çè + 1÷ø a I kˆ 2 (d) (2p + 1)a2 I kˆ 18. A long insulated copper wire is closely wound as a spiral of ‘N’ turns. The spiral has inner radius ‘a’ and outer radius ‘b’. The spiral lies in the XY plane and a steady current ‘I’ flows through the wire. The Z-component of the magnetic field at the centre of the spiral is [2011] (a)
(b)
(c)
m0 NI æbö ln ç ÷ 2(b - a) èaø
æb+aö m0 NI ln ç ÷ 2(b - a) è b - a ø m0 NI æbö ln ç ÷ 2b èaø
m0 NI æb+aö ln ç ÷ 2b èb-aø 19. A long straight wire along the Z-axis carries a current I in r the negative Z-direction. The magnetic vector field B at a point having coordinates (x, y) in the Z = 0 plane is [2002S] m 0 I xiˆ + yjˆ m 0 I yiˆ - xjˆ (b) (a) 2p x 2 + y 2 2p x 2 + y 2
(d)
(
(c)
(
)
(
m 0 I xjˆ - yiˆ
(
2
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2
(
)
)
)
(d)
(
)
)
(
m 0 I xiˆ - yjˆ
(
2
2p x + y
2
)
)
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P-201
Moving Charges and Magnetism
꾀ᆠᨨ� � �N 盐 矄�矄犨ᨨ盐 �� 泐� ᨨ鼀ᨨ 矄�犨� �
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ᆠ盐 I 矄矄矄 ᨨ盐� ᆠᨨ� 鼀� �ᆠ � ᆠ �矄 ҈和桌 㡰
mᨨ NI b
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mᨨ NI a
mᨨ NI b mᨨ NI a
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҈
(
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2
Integer Value Answer
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鼀� �ᆠ � 盐ᨨ �矄ᆠ盐 ᨨ� M�矄H҈ ᨨ犨 ᨨ � � �泐ᨨ � 矄�� ᆠᨨ 盐ᆠᨨ QS �矄 ᆠᨨ
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4
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MCQs with One or More than One Correct Answer
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EBD_8341 P-202
29.
Physics
A current I flows along the length of an infinitely long, straight, thin-walled pipe. Then [1993-2 Marks] (a) the magnetic field at all points inside the pipe is the same, but not zero. (b) the magnetic field at any point inside the pipe is zero (c) the magnetic field is zero only on the axis of the pipe (d) the magnetic field is different at different points inside the pipe.
Y-axis such that OS = OR = 0.02 m. Find the magnitude and direction of the magnetic induction at the origin O. [1989 - 6 Marks] Y
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Ii Q Ii
10 Subjective Problems 30.
L
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A pair of stationary and infinitely long bent wires are placed in the XY plane as shown in fig. The wires carry currents of i = 10 amperes each as shown. The segments L and M are along the X-axis. The segments P and Q are parallel to the
P
R O
¥
M
S
Ii
X
Ii ¥
Topic-3 : Force and Torque on Current Carrying Conductor 1 1.
3.
MCQs with One Correct Answer
A square loop of side 2a, and carrying current I, is kept in XZ plane with its centre at origin. A long wire carrying the same current I is placed parallel to the z-axis and passing through the point (0, b, 0), (b >> a). The magnitude of the torque on the loop about z-axis is given by : [Main Sep. 05, 2020 (I)] (a)
m0 I 2a2 2 pb
(b)
m 0 I 2 a3
(a) T =
2pb2
2m 0 I 2 a 3 2m 0 I 2 a 2 (d) pb pb 2 A wire carrying current I is bent in the shape ABCDEFA as shown, where rectangle ABCDA and ADEFA are perpendicular to each other. If the sides of the rectangles are of lengths a and b, then the magnitude and direction of magnetic moment of the loop ABCDEFA is : [Main Sep. 02, 2020 (II)]
Z E
I
I
F
C Y
D O X
(a) (b) (c) (d)
A
a
B
æ ˆj kˆ ö abI , along ç + ÷ ç 2 2 ÷ø è æ ˆj kˆ ö 2abI , along çç + ÷ 2 ÷ø è 2 æ ˆj 2kˆ ö 2abI , along ç + ÷ ç 5 5 ÷ø è æ ˆj 2kˆ ö abI , along ç + ÷ ç 5 5 ÷ø è
b
2m IB
(b) T =
pm 2 IB
2pm pm (c) T = IB IB A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field Bo parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)
(c) T =
(c)
2.
A small circular loop of conducting wire has radius a and carries current I. It is placed in a uniform magnetic field B perpendicular to its plane such that when rotated slightly about its diameter and released, it starts performing simple harmonic motion of time period T. If the mass of the loop is m then : [Main 9 Jan 2020, II]
4.
Bo
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P-203
Moving Charges and Magnetism
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҈ 犨ᨨᆠᨨ �矄ᨨ ᨨ� 矄ᨨ � �盐矄ᆠ泐ᆠ盐 ᔼ�
uur 矄鼀 �ᆠ�ᨨ ҈ 鼀� �ᆠ ᨨᆠᨨ �
uur 矄ᨨ ᨨ� 盐ᨨ ᨨ盐ᨨ 鼀� �ᆠ ᨨᆠ ᨨ ᨨ盐矄ᨨ ᨨ� 盐ᨨ �
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EBD_8341 P-204
҈
Physics
犨ᨨ ᨨ � 矄�� 犨�矄 ᆠ泐� � 盐矄㡰 ᆠ盐 ᔼ҈ ᔼ├ �矄 ᆠ ᔼ ᨨᆠ � 犨 犨ᨨ犨�矄�矄├ 矄�ᨨ矄��犨 犨�矄ᆠ ᨨ � ��犨 犨�矄ᆠ ᆠ ᨨ � 矄 ᨨ犨� � ᆠᨨ � ᨨ
ᨨ 盐ᆠᔼ҈ᔼ犨ᨨ盐 㝐� 졠 � 꾀�҈҈҈�桌
졠
ᔼ҈ᔼ
㡰 㡰 ᆠ㡰 㡰
҈
I
㡰
ᔼ҈ᔼ
졠
졠
ᆠ㡰
i
㡰
҈ 꾀ᆠ �盐ᨨᨨᆠ泐� �ᆠ盐 ��矄矄�盐 ᨨ �矄�� 犨�矄盐ᆠ 犨��矄ᨨᨨ ᨨ 矄� 矄ᨨ ᨨᨨ �矄� ᨨ ᨨᨨᔼ 矄 泐ᆠ盐 I�矄矄�矄 � 犨�矄矄 ᨨ犨 � ��盐 ᨨᨨ犨� ҈黨�㜠㝐├矄桌
㡰
ᔼ҈ᔼ
2
ᔼ҈ᔼ
졠
矄�� ᆠ盐 ᆠ泐� �ᆠᨨ 盐ᆠᨨ矄 矄 ᨨ犨 � ��盐 ᨨᆠ � ᆠ 泐 鼀� ᆠᨨ 盐ᆠᨨᨨ � �ᆠ鼀�矄㝐� 졠 � 꾀�҈҈҈桌 ᔼ҈ 㦨꾀
ᔼ 㦨꾀
ᨨᨨ盐 �矄ᨨ 犨� 鼀ᨨ犨泐 ᨨ鼀 犨� 鼀ᨨᨨ犨 矄 犨� 鼀� 矄�ᨨ 泐 Integer Value Answer
҈䰘 犨ᨨ犨�矄� ᨨ �矄 ᆠX Aᨨ� ᆠ ��矄鼀ᨨ� �犨� 矄 u犨 犨�矄� 矄鼀 �矄 ᆠX҈ ᨨ鼀ᨨ ᨨ 犨�矄␔ 犨�矄ᆠ泐ᆠ盐 ᨨ 鼀� �盐 I � 矄鼀 �ᆠ�ᨨ �盐矄ᨨ ᆠ盐盐ᨨ ᨨ ᨨ� ᆠ ��矄R҈ᔼ ᆠᨨ 矄� ᆠ盐 矄I � 犨ᨨ 犨�矄 �ᆠ�ᨨ 矄ᨨᨨ矄�ᨨᆠ ᨨ �盐矄ᨨ R X ᆠ盐盐ᨨ �矄Rᔼ = 盐ᨨ ҈ �矄 R X҈ 꾀 ҈桌
�ᆠ鼀
ᆠ鼀
4
ᔼ㦨҈꾀
맔҈ᨨ犨 ��
㡰 㡰 맔҈ᨨ犨 �� ᆠ㡰 黨맔҈ᨨ犨 �� 㡰 ᨨ ҈
҈ 꾀犨�ABCDEF 犨� ᆠ 矄� ᨨ � L㡰 矄矄 ᨨ犨
� ��盐 ᆠ泐� �ᆠ盐 I�矄ᆠ � 盐 � ᨨ鼀 鼀� �ᆠ� 盐ᆠ�ᨨ Bᨨ ᨨ矄��y㜠 �ᆠ�ᨨ
ᨨᆠ � ᆠ 泐 犨��矄� �ᆠ�ᨨ ҈黨黨㜠㝐├矄桌 z D
꾀ᆠᨨ 盐ᆠ� �ᨨᨨᆠ泐� �ᆠ盐 I�矄ᆠ � 盐 � ᨨ鼀 鼀� �ᆠ � ᨨ� � � � ᨨ ᨨ 矄 矄 ᨨ犨 ᨨᨨ犨� ᆠ泐ᨨ 和桌
C
F E
㡰
y
ᔼ
ᆠ㡰 鼀ᨨᨨ犨 矄졔x㜠 �矄
B
㡰 鼀ᨨᨨ犨 矄㜠x㜠 �矄
犨ᨨ � ᨨ �犨�矄矄 泐 �矄 ᆠ�b ᆠ泐� �ᆠ盐 ��鼀ᆠ 鼀� �盐 ᨨ ᨨᆠ 盐 � � 泐ᨨ 犨�ᨨ ᨨ �矄 ҈黨㜠㝐├矄桌 㡰
m �
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㡰 ᆠᨨ ᆠ
҈�
Fill in the Blanks
㡰
m � p
ᆠ㡰
m � p
㡰
m � p
A
x
5
True / False
҈黨 ᨨ ᨨᆠᆠ矄ᨨ ᆠ �盐ᆠᨨ�ᆠ泐� �矄 泐 ᆠ盐 犨 矄盐矄 泐� 盐 � ᨨ鼀鼀� �ᆠ � ҈黨҈㜠㝐├矄桌
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P-205
Moving Charges and Magnetism Ӹ
10 Subjective Problems
҈
� � �泐ᨨ � � 犨�矄ᆠ ᆠ泐� �ᆠ盐 ��
矄鼀 �ᆠ�ᨨ � x㜠y ᨨ ��泐 矄ᆠ ᆠ 犨��矄ᨨ � y㜠 �矄犨 � ᨨ 犨ᨨᨨ �x㦨᠀d ҈黨黨䰘㜠�㝐├矄桌 �㡰 � ᨨᆠ盐矄ᨨ ᨨ� 矄 ᨨ犨 �ᆠ 鼀� �ᆠ � B�矄 ᨨ ��㡰 ᔼ ᆠ 犨��矄 �矄ᆠ ᨨ � Z㜠 �ᆠ�ᨨ
泐矄鼀鼀ᨨ盐 矄 矄 ᨨ犨 �犨� ᆠ盐矄�鼀 鼀ᨨ �ᆠ鼀ᨨ�ᨨ ᔼ � 矄�泐 ᨨ 犨�矄�矄l � ⼄盐 ᆠ泐ᨨ ᨨ矄ᆠ��ᨨ 꾀ᨨ � ᨨ� ᨨ 犨�AB犨 �ᆠ �矄 ᨨ鼀ᨨ� �ᆠ ᆠ�矄矄 泐ᆠ盐 ᨨ A�矄� ⼄盐���盐鼀 �� ᨨ ҈鼀ᨨ ᨨ ᨨ �犨�CD 犨 �ᆠ �矄 � � ᨨ� ᨨ ᆠ�矄矄 泐 ᆠ盐 ᨨ A矄矄 ᨨ犨 � ��盐和 ᨨ犨 犨 AB�矄 矄�� 泐 矄矄 � ᆠ盐矄矄�鼀 鼀ᨨ �ᆠ鼀ᨨ�ᨨ � �ᨨ ᨨ ᨨ矄ᆠ��ᨨ 矄 ҈黨黨㜠㝐├矄桌
꾀
ᔴ
꾀
ᔼ 犨�矄A Bᆠ ᆠ泐ᆠ盐 I鼀� ᨨ ᨨ ᨨ� 矄矄�ᨨ ᨨ ᨨᆠᆠ� � ᨨ 矄�鼀 OC␔ 犨� ᨨᆠᨨ OC� ᆠ盐 � 犨�B�矄矄 䁸 ҈黨黨㜠҈㝐├矄桌 犨ᨨ ᨨ � 矄�� 犨�矄 鼀矄 �ᆠ盐ᨨ ᨨ 矄 ��盐㡰 犨�Aᆠ�矄ᆠ盐 ᨨ 黨鼀矄 �ᆠ � ᨨ ᨨ 犨�Bᆠ�矄ᆠ盐 矄盐ᆠ 鼀� �ᆠ � ᨨ � 盐ᆠ�ᨨ ᨨ� P �矄 ᆠᨨ
ᔴ 꾀矄�� 矄�鼀 OC ᨨ � L鼀㡰ᨨ ᆠ�ᆠ盐� ᆠ泐� �ᆠ盐 I鼀�矄ᆠ ᨨ � 㜠 �矄 ��㡰
犨ᨨ� � 泐ᨨ �矄�� 犨�矄A Bᆠ � � ᨨ鼀z 㦨 - ¥ ᨨ +¥ � y㦨a鼀 y㦨졔a 鼀矄ᆠ�泐矄矄 ᨨ犨 � ��盐
҈黨䰘㜠䰘㝐├矄桌 A×
鼀
҈ 鼀 ᨨ鼀 犨�B�矄 ᨨ ҈҈ � �㡰 鼀� �盐 �ᆠ�ᨨ
ᨨ ᆠ盐 � B ��㡰 鼀� �盐 ᨨ 鼀� �ᆠ � ᨨ � 盐ᆠ�ᨨ ᨨ� S ���㡰 ᨨᆠ盐 � � ᨨ 犨�B
҈
ᔼ
졠
鼀
和 鼀 ҈
B
҈⏌ 鼀 ҈҈
Topic-4 : Galvanometer and its Conversion into Ammeter and Voltmeter 1 ҈
MCQs with One Correct Answer
꾀� ᨨ鼀ᨨ 矄�矄 ᆠ�矄ᆠᨨ � ᨨᨨ㜠 鼀ᨨ � ҈泐ᆠᨨ
ᆠ� �矄�矄 ᆠ ҈�
矄�矄犨� � ��ᨨ 矄�矄 ᆠ ҈� 矄�矄犨� �
��ᨨ 矄�矄 ᆠ 矄 ᨨ盐 ᆠᨨ
ᆠ � 矄㜠 �矄犨� ҈ᨨ� ᆠ矄 �ᨨ ᨨ鼀ᨨ 犨� 㝐� 和� ᔼ㡰桌 㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈졔 꾀� ᨨ鼀�矄盐矄 � ᨨᨨ泐 ᨨ ᆠ� � 盐 ᨨ� � ᆠ�ᆠ �鼀 矄ᔼ ᨨ 矄矄� �ᆠ盐 ᨨ 鼀꾀�ᨨ 盐ᆠ矄 ᆠ�ᨨ ᨨ �矄 ��盐ᨨ 鼀��矄ᆠᨨ矄 ᨨ 㝐� 和� ᔼᔼ㡰桌 㡰 꾀⏌ � 㡰 맔҈꾀⏌ � ᆠ㡰 꾀⏌ � 㡰 맔҈꾀⏌ � 꾀鼀ᨨ� �ᆠᨨ�� ᨨ鼀ᨨ犨矄 盐矄ᆠᆠ盐 ᨨ 10꾀꾀矄�矄矄�矄 ᆠᨨ 㝐W �矄⼄盐� ᨨᆠᨨ ᨨ� ᨨ鼀� ᨨᨨ鼀ᨨ ��
ᨨ 盐 ᨨ 矄 盐 矄�矄 ᆠ ⼄盐� ᨨ ᆠᨨ ᨨ� ᨨ鼀� ᨨ 鼀鼀ᨨ � ҈鼀꾀�矄 㝐� ҈꾀�҈黨ᔼ桌 㡰 �W 㡰 ҈W ᆠ㡰 W 㡰 ҈W
�
꾀鼀ᨨ� �ᆠᨨ�� ᨨ鼀 矄矄�矄 ᆠ�W � � �ᆠ矄 盐 ᆠ�ᨨ 鼀꾀ᆠ盐 꾀ᨨ鼀�矄 鼀 盐矄� � �矄� ᨨ鼀 �├W矄�矄 ᆠ 鼀 �鼀盐鼀 ᨨ� ᆠ 鼀矄盐 盐矄� � �矄 ᨨ鼀犨�ᆠᨨ矄ᨨ 㝐� 黨꾀�҈黨ᔼ桌 㡰 㡰 ҈� ᆠ㡰 㡰 ҈ 꾀鼀ᨨ� �ᆠᨨ�� ᨨ鼀 矄ᆠᨨ�犨� ҈䰘�盐 矄 ҈ᆠ鼀ᔼ盐矄矄ᨨ矄�ᨨ ᨨ ᨨ矄�ᨨ ᆠᨨ 矄 ҈㜠鼀⏌ ᆠᨨ��矄ᆠ � 鼀� �ᆠ � ᨨ�矄
ᆠᨨ� ᆠ矄泐҈ ᨨᆠ盐 ᨨ ҈鼀꾀 盐ᨨ �
矄㡰�矄ᨨ �鼀泐 㝐� 黨꾀�҈黨ᔼᔼ桌 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈҈ ᆠ㡰 ҈
� ᨨ鼀 ᆠ�ᨨ 犨 ├泐⧤҈�矄ᆠᨨ矄 盐 ⧤�矄ᨨ ⼄盐矄 θ 矄 ��盐㡰졠 ᆠᨨ矄� �⧤矄ᨨ 盐矄� � ᨨ�W ᆠ�ᨨ � � ᨨ鼀 θ ᆠᨨ鼀矄 矄�矄 ᆠᨨ � ᨨ鼀�矄 � �� 泐�ᆠ � 矄�矄 ᆠᨨ 泐桌 㝐� ҈ ҈黨ᔼ桌
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EBD_8341 P-206
Physics R2
K2 R1 = 220 W
G
K1
7.
8.
9.
10.
11.
12.
13.
(a) 5W (b) 22W (c) 25W (d) 12W A galvanometer, whose resistance is 50 ohm, has 25 divisions in it. When a current of 4 × 10–4 A passes through it, its needle (pointer) deflects by one division. To use this galvanometer as a voltmeter of range 2.5 V, it should be connected to a resistance of : [Main 12 Jan 2019, II] (a) 250 ohm (b) 200 ohm (c) 6200 ohm(d) 6250 ohm A galvanometer having a resistance of 20 W and 30 division on both sides has figure of merit 0.005 ampere/division. The resistance that should be connected in series such that it can be used as a voltmeter upto 15 volt, is: [Main 11 Jan 2019, II] (d) 125 W (a) 100 W (b) 120 W (c) 80 W A galvanometer having a coil resistance 100 W gives a full scale deflection when a current of 1 mA is passed through it. What is the value of the resistance which can convert this galvanometer into a voltmeter giving full scale deflection for a potential difference of 10 V? [Main 8 Jan 2019, II] (a) 10 kW (b) 8.9 kW (c) 7.9 kW (d) 9.9 kW In a circuit for finding the resistance of a galvanometer by half deflection method, a 6 V battery and a high resistance of 11kW are used. The figure of merit of the galvanometer 60mA/division. In the absence of shunt resistance, the galvanometer produces a deflection of q = 9 divisions when current flows in the circuit. The value of the shunt resistance that can cause the deflection of q/2 , is closest to [Main Online April 16, 2018] (a) 55W (b) 110W (c) 220W (d) 550W When a current of 5 mA is passed through a galvanometer having a coil of resistance 15 W , it shows full scale deflection. The value of the resistance to be put in series with the galvanometer to convert it into to voltmeter of range 0 - 10 V is [Main 2017] 3 (a) 2.535 ´ 10 W (b) 4.005 ´ 103 W (c) 1.985 ´ 103 W (d) 2.045 ´ 103 W A galvanometer having a coil resistance of 100 W gives a full scale deflection, when a currect of 1 mA is passed through it. The value of the resistance, which can convert this galvanometer into ammeter giving a full scale deflection for a current of 10 A, is : [Main 2016] (a) 0.1 W (b) 3W (c) 0.01W (d) 2W To know the resistance G of a galvanometer by half deflection method, a battery of emf VE and resistance R is used to deflect the galvanometer by angle q. If a shunt of resistance S is needed to get half deflection then G, R and S related by the equation: [Main Online April 9, 2016] (a) S (R + G) = RG (b) 2S (R + G) = RG (c) 2G = S (d) 2S = G
3
Numeric Answer
14. A moving coil galvanometer has 50 turns and each turn has an area 2 ×10–4 m2. The magnetic field produced by the magnet inside the galvanometer is 0.02 T. The torsional constant of the suspension wire is 10–4 N m rad–1. When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by 0.2 rad. The resistance of the coil of the galvanometer is 50 W. This galvanometer is to be converted into an ammeter capable of measuring current in the range 0 –1.0 A. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is __________. [Adv. 2018] 6
MCQs with One or More than One Correct Answer
15. Two identical moving coil galvanometers have 10 resistance and full scale deflection at 2mA current. One of them is converted into a voltmeter of 100mV full scale reading and the other into an Ammeter of 1mA full scale current using appropriate resistors. There are then used to measure the voltage and current in the Ohm’s law experiment with R – 1000W resistor by using an ideal cell. Which of the [Adv. 2019] following statement(s) is/are correct? (a) The resistance of the voltmeter will be 100k (b) The measured value of R will be 978 矄盐 � ᨨ鼀鼀� �ᆠ � B҈kˆ ��ᨨ y 矄 ᨨ 盐 � ᨨ鼀鼀� �ᆠ � B kˆ 꾀ᨨ矄��泐ᆠ � �ᆠ�矄ᨨᆠ ᨨ鼀 ᨨ��� ᨨ � ᨨ矄��y㜠 �矄犨� 矄 v㦨p 鼀矄 ҈t㦨矄矄 ᨨ犨 � ��盐�ᆠ��泐� �矄
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꾀ᨨ �ᆠ�ᆠ盐盐ᨨ � ҈鼀 �盐矄鼀ᆠ�矄 ᆠ盐 Iᨨ ��矄ᆠ盐 矄盐 ᆠ矄矄 ᨨ犨 꾀犨�㜠ᨨᨨᨨ 矄�矄 ᆠ�ᨨ 鼀 ᨨ �盐矄҈鼀�矄 ᆠ � 矄� 盐 犨� �矄 �矄 ᆠᨨ� ᆠ� � �犨� �矄ᨨ 盐 ᆠ盐 �矄矄 I㦨Iᆠᨨ矄 t㡰犨 I�矄 ᆠᨨ 矄 ᔼ 鼀� �ᆠ鼀ᨨ鼀 ᨨ ᨨᨨ�矄NmI矄� t㡰 �N�矄 ҈҈桌 4
Fill in the Blanks
꾀鼀�ᆠᨨᆠ├ᆠ泐� �ᆠ盐 ᔼ�矄矄盐ᆠ ᨨ盐 � ᨨ鼀 ur 鼀� �ᆠ� 盐ᆠ�ᨨ B 矄矄 ᨨ犨 � ��盐 B
Y
E
G F
H B
X I
A C
D Z
ur
鼀ᨨ� �ᆠ �矄 � ᆠ ᨨᆠ F �� 泐 犨 �ᆠ 矄盐矄� ᨨ犨� �ᨨ ᨨ �ᨨ ᆠ 꾀矄矄盐鼀 矄 ᨨ ᆠ�矄ᨨv ҈黨黨㜠㝐├矄桌
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EBD_8341 P-208
Physics
6 䰘
MCQs with One or More than One Correct Answer
犨ᨨ� � �泐ᨨ �矄�� 犨�矄�� xy㜠 ᨨ � � 矄x㦨 ᠀R 犨�ᨨᆠ x㦨 졔Rᆠ�矄 ᆠᨨ 矄 ᆠ盐 I҈ 犨�ᨨᆠ x㦨Rᆠ�矄 ᆠᨨ 矄 ᆠ盐 I 꾀 ᆠ�ᆠ盐 ᨨᨨ ᨨ �盐矄 R �矄 矄盐矄 犨� �矄ᆠ R 㡰 � ᨨ xy㜠 �矄ᨨᨨᆠ�矄ᆠᨨ 矄 ᆠ盐 I� ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 矄矄 ᨨ鼀ᨨ ᨨᨨ
ᆠ盐 � 犨��矄├ ᨨᨨ矄��� ��矄� + ˆ �ᆠ�ᨨ ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄� � � r 鼀� �ᆠ � Br �矄 㡰盐䁸 꾀 ҈桌 㡰 ᔼ I҈㦨I B ᆠ
ᨨ⼄盐ᨨ ᨨ ᨨ���
㡰 r 㡰 ᔼ I҈> I B ᆠ ⼄盐ᨨ ᨨ ᨨ��� 㡰 r ᆠ㡰 ᔼ I҈ I> B ᆠ ⼄盐ᨨ ᨨ ᨨ��� 㡰 㡰 ᔼ I҈㦨I z㜠ᆠᨨ鼀ᨨ ᨨ 鼀� �ᆠ � æ m I ö ᆠ ᨨ ᨨᨨ�矄 ç ÷ è R ø
犨ᨨ鼀�ᆠ� �矄A B� �ᆠ� 矄 矄� 盐 � � � 矄�矄���矄rA rB├ᨨ ᨨᨨ 犨ᨨ� �ᆠ 矄ᨨ ᨨ� 矄 矄 矄 ᨨ犨 � ��盐 ␔
ᆠ盐 I�矄矄犨�ᆠ ᨨ � ᨨ 矄ᨨ ᨨ� 矄� � �ᆠ 鼀
� �矄 A B 盐鼀 ᨨ �� 矄 hA hB 矄ᆠ�泐犨� hA>hB ᨨ矄矄��ᨨ 矄㡰犨
�矄�矄���矄 �鼀矄矄矄mA mB�矄 㡰 黨桌 A
㡰 rA>rB mA㦨mB ᆠ㡰 rA>rB mA>mB 黨
B
㡰 rArB mA㦨mB 㡰 rArB mAmB
7 Match the Following
犨ᨨ犨�矄ᆠ ᆠ泐� �矄 泐ᆠ盐 ᔼ矄 ᨨ犨 �
ᨨ盐ᆠᨨ ��盐�ᨨ 矄� ᔴᨨ盐鼀 ᔼ和ᨨ鼀ᨨ 矄盐� � ᆠ矄 矄ᆠ� � ᔴᨨ盐鼀 ᔼᔼ㝐ᆠ 矄鼀 矄 � ᔴᨨ盐鼀 ᔼ犨� 矄鼀 矄� ᆠᨨ盐鼀 ᔼᔼ � �ᆠ 泐ᨨ盐 矄犨 泐 ├ � � ᨨ� 盐矄 � 맔鼀� �� � 졠 和 䰘桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 ᨨ� �矄矄�盐 㡰 鼀� �ᆠ � 矄 㡰 鼀� 犨泐犨 盐ᨨ ᆠ盐 矄 犨�矄 � 犨�矄� 矄鼀 �ᆠ�ᨨ
㡰 ᨨ� �矄矄�盐 ⼄㡰 鼀� �ᆠ � 矄 㡰 鼀� 㜠ᨨ� ᨨ 盐ᨨ ᆠ盐 矄 � ᨨ� � � ᆠ 矄 � 犨�矄� ᨨᨨ ᨨ ᆠ�ᆠ盐 犨�矄 矄� �ᆠ�ᨨ 矄 犨 �ᆠ 矄鼀 ��
ᔴ㡰 ᨨ� �矄矄�盐 㡰 鼀� 㜠ᨨ� ᨨ � ᨨ� � � ᆠ 矄 ᨨ ᆠ�ᆠ盐犨�矄 犨 �ᆠ 矄鼀 ��
�矄 ᨨ鼀� �ᆠ �
㡰 ᨨ� �矄矄�盐 ᆠᨨ鼀鼀ᨨ ᆠ ᨨ 犨�矄
矄㡰 犨�矄ᆠ ᨨ
҈ ᔴᨨ盐鼀 ᔼ ��矄 ᆠ� 矄�盐�ᨨ 矄 � 犨 �ᆠ 矄�� 鼀�ᆠ犨�ᨨ 矄�矄 ᆠ �矄盐矄 ᔴᨨ盐鼀 ᔼᔼ��矄 矄ᨨ鼀矄盐� � ᆠ矄㝐ᆠ 矄鼀 矄� ᔴᨨ盐鼀 ᔼ 犨� 矄鼀 矄� ᔴᨨ盐鼀 ᔼᔼ � �ᆠ泐ᨨ盐 矄犨 泐 ├ � �ᨨ�盐矄� 맔鼀� ��
� 졠 和 䰘桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 꾀ᆠ � ᆠᆠ�ᨨ 㡰 꾀ᆠᨨ 矄 ᆠ盐 �矄ᆠᨨ
ᆠ ᨨ ᨨ犨矄 ᨨ盐� 犨� 矄ᨨ 犨� 㡰 犨��矄鼀ᨨ ⼄㡰 鼀 �泐�矄 �ᆠ盐ᨨ�矄 � � 犨� � 犨� ᆠᨨ 矄 ᨨᆠ�泐� 盐 � ᨨ鼀 鼀� �ᆠ � �ᆠ盐ᨨ ᨨ 鼀ᨨ�ᨨ
ᔴ㡰 犨��矄ᆠ � 㡰 꾀ᆠᨨ 矄 ᨨ � ᆠᨨ 矄 ᆠ�ᆠ � ᆠ ᨨ矄 � 矄 �ᆠ�ᨨ
犨 矄ᨨ ᨨ � � ᨨ 犨� 犨� 㡰 꾀泐ᨨ ᆠᨨ 矄 矄㡰 ᆠ �矄ᨨ ᆠᨨ 矄 鼀 �矄ᆠᨨ
ᆠ ᨨ 鼀� �盐 矄ᨨ 犨� 矄ᨨ 犨� ҈҈ 㝐ᆠ ᨨᨨ犨� �ᆠᨨ盐鼀 矄 㝐桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 �ᆠ�ᆠ� � 㡰 ᔴᨨ 矄 ᆠᨨ矄�ᆠ 盐 � ᨨ鼀泐ᆠ � � ᨨ盐ᨨ 矄泐矄鼀 㡰 �ᆠ�ᆠ� � ⼄㡰 㝐� �ᆠ � 矄 � 盐 � ᨨ鼀泐ᆠ � ᨨ� �犨� �盐 ᨨᆠ�泐w ᔴ㡰 ᔴᨨ 矄 ᆠ盐 �
㡰 쟼ᆠ�ᆠ � � 盐ᆠ 㡰 � �� 㡰 �㦨�ᆠᨨ矄w 矄㡰 㝐� �ᆠ �ᨨ鼀ᨨ鼀 8
Comprehension/Passage Based Questions
矄矄�㜠҈ ᔼ � ᆠ �盐鼀�ᆠ矄�ᆠᨨ 矄 ᆠ盐 I ᨨ犨矄 ᨨ � ᨨ矄�� 㜠 �ᆠ�ᨨ 矄矄 ᨨ犨 � ��盐 � 犨� �ᆠ├ 矄矄ᨨ 矄�lw d矄ᆠ�泐 su 꾀盐 � ᨨ鼀鼀� �ᆠ � B �矄� ᨨ 矄�ᨨ � ᨨ矄��泐㜠 �ᆠ�ᨨ 盐ᨨ �矄 ᆠ �ᆠ�矄 � ᆠ
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P-209
Moving Charges and Magnetism
ᆠ�ᨨ ᨨ � 㜠 �ᆠ�ᨨ �矄矄盐矄� ᆠᆠ盐鼀盐�ᨨ ᨨ ᆠ �ᆠ�矄ᨨ 矄盐 ᆠPQRS ᆠᨨ ⼄盐 ᨨᨨ矄� ᆠ �矄 ᨨ ᆠ ᨨᨨ矄� ᨨ PQRS 꾀 ᨨ � � ᆠ ᨨ � 㜠 �ᆠ�ᨨ �矄 盐矄 ᨨ ᔴ � ᆠᆠ盐鼀盐�ᨨ ᆠᨨ � 盐矄盐 � 鼀� �ᆠ ᨨᆠ�矄 ᆠ 泐 ᆠ�ᆠ ᨨᆠ ᆠ盐 �矄矄矄盐鼀 ᨨ盐 � ᨨ鼀泐 �矄�盐 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ 矄� ᆠ� 泐ᆠᨨ 矄
ᆠ㡰 ᆠ盐 � 犨�҈ 犨�� �ᆠ�ᨨ PQ SR矄ᆠ�泐 h » ҈a 㡰 ᆠ盐 � 犨�҈ 犨�� �ᆠ�ᨨ PQ RS矄ᆠ�泐 h » ҈a ҈� ᔴᨨ 矄� d>>a ᨨᨨ�矄ᨨ ᨨ盐�矄 �鼀 ᨨ 犨�矄泐 ᨨ鼀 ᨨ矄��ᨨ 矄 ᨨ犨 � ��盐 ᔼ ᆠ盐 矄 � 犨�矄 � ᨨᨨ矄� �ᆠ�ᨨ 矄 ᨨ⼄盐ᨨ ᨨᨨ�矄 犨ᨨ矄��ᨨ 犨� 矄矄盐鼀 � 盐ᨨ 犨�矄�矄ᆠᨨ 矄 ᨨ ᨨᨨ㡰 꾀 ҈桌 㡰
ᔴᨨ 矄� 犨ᨨ � 鼀�ᆠ矄�矄 ҈ 㡰ᨨ 矄鼀 鼀� � � 矄 矄鼀犨� 矄w҈ w �ᆠ├ 矄矄矄d҈ d矄ᆠ�泐 犨ᨨᨨ� 矄⧤ M矄泐鼀鼀�ᆠ泐ᨨᆠ ᨨ ᨨᨨ矄� ᆠ矄 ᨨ 㜠泐 矄 ��盐㡰V҈ V ᨨ � � ᆠ矄犨 K M� 矄�矄҈ 矄ᆠ�泐 ᨨ�� ᆠ盐 ᔼ ᨨ犨� � ᨨ盐� 鼀 � �� 鼀� �ᆠ � 矄 � ᆠᨨᆠ 矄鼀 矄㡰�矄 㡰 꾀 ҈�桌 㡰 ᔼ w҈㦨w d҈㦨d V㦨V҈ 㡰 ᔼ w҈㦨w d҈㦨 V㦨V҈ ᆠ㡰 ᔼ w҈㦨w d҈㦨d V㦨V҈ 㡰 ᔼ w҈㦨w d҈㦨d V㦨V҈ ҈ ᔴᨨ 矄� 犨ᨨ � 鼀�ᆠ矄�矄 ҈ 㡰 ᨨ 矄鼀 �鼀 矄�ᨨ 矄 � l犨� w �ᆠ├ 矄矄d㡰犨� ᆠ� 矄��矄n҈ n矄ᆠ�泐和�҈�矄ᆠ � 鼀� �ᆠ � B҈ 矄��矄ᆠ � 鼀� �ᆠ � Bᨨ ᨨ � ᨨ矄��泐㜠 �ᆠ�ᨨ 矄 V҈ V ᨨ � � ᆠ矄 ᨨ 犨 K M� 矄�矄҈ 矄ᆠ�泐꾀矄矄盐鼀� � ᆠ盐 I�矄 矄鼀 ᨨᨨ 矄�矄 ᆠᨨᆠᨨ�ᨨ 矄㡰�矄 㡰 꾀 ҈�桌 㡰 ᔼ B҈㦨B n҈㦨n V㦨V҈ 㡰 ᔼ B҈㦨B n҈㦨n V㦨V҈ ᆠ㡰 ᔼ B҈㦨B n҈㦨n V㦨�V҈ 㡰 ᔼ B҈㦨B n҈㦨n V㦨V҈ 矄矄�㜠
��盐矄 ᨨ犨矄ᆠ�ᆠ盐ᨨᨨᨨ �盐矄a犨� 犨ᨨᨨ � 犨�矄 盐鼀 ҈ 㡰� ᨨ �矄 ᆠ ᨨ ᆠ 犨� ᨨ鼀 ᆠ ᨨ ᨨᨨ�矄d ᨨᨨ 犨�ᆠ泐� � 矄鼀ᆠ盐 I ᆠ盐 � ᨨᨨ�矄�
ᆠᨨ盐 ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ � 矄 ᨨ鼀ᨨ ҈
Q
␔�҈ P ҈
d
d
S
␔� R
␔ d » a 盐犨�矄 ᨨᨨ盐ᆠ � � ᨨᨨ��矄 ᨨ盐 鼀� �ᆠ � ᨨ �矄ᨨ ᨨᨨ�矄 ᨨ �� hᨨ ᨨᨨᔼ ᆠ矄 꾀 ҈桌 㡰 ᆠ盐 � 犨�҈ 犨�� �ᆠ�ᨨ PQ RS矄ᆠ�泐 h » a 㡰 ᆠ盐 � 犨�҈ 犨�� �ᆠ�ᨨ PQ SR矄ᆠ�泐 h » a
ᆠ㡰
m I a d
㡰
m I a d
㡰
m I a d m I a d
矄矄�㜠 꾀 ᆠ ᆠᨨ盐 �矄鼀├� �盐矄ᨨ ᨨ犨 盐ᆠᨨ鼀� 矄 ᨨ鼀ᨨ� 矄泐 �� 矄 矄� 矄ᆠ 鼀� � 矄 � ᨨ鼀鼀� �ᆠ��ᨨ 㡰 矄� 矄 ᨨ ᨨ �盐� 犨泐 ᨨ ᨨ盐 ᨨ 矄�ᆠ├矄 ᔼ 矄 ᨨ 盐矄� � �� 矄 ᨨ ᨨ矄矄� 盐矄 泐鼀├盐矄ᨨ 鼀� �ᆠ � ᨨᆠ矄 鼀� � ᆠᨨ�矄 � � �盐� 犨泐犨 �ᆠ 矄 矄ᨨ �鼀� 矄ᆠ � � Ἐ矄 盐 ᆠ�� �矄 矄� 鼀ᨨᨨ �盐� 犨泐ᆠ �ᆠ ᆠ ᆠᨨ㜠 泐 鼀�ᆠ矄盐矄 矄�ᨨ ␔ ᆠ盐 矄矄矄� ᆠᨨ�矄ᨨ �盐� 犨泐泐�ᆠ鼀� �ᆠ � �矄矄盐犨 盐 ᆠ��ᨨ � �盐� 犨泐犨 �ᆠ 盐矄 矄 盐 � ᨨ � �盐� 犨泐 � �ᨨ ⼄盐�鼀
ᆠ├ᨨ �ᆠ�ᨨ �矄ᨨ 泐 鼀�ᆠ矄泐ᨨ犨矄 � ᨨ 鼀ᨨ泐 �� 矄 ҈ ��ᨨ ᨨ � �矄 盐ᨨ 㜠�㝐桌 㡰 㝐ᆠ �ᆠ ᨨᆠ 㡰 쟼ᆠᨨ矄�ᆠᆠ�ᨨ
ᆠ㡰 쟼ᆠᨨ矄�ᆠ盐矄�ᨨ 㡰 㝐� �ᆠ盐矄�ᨨ
҈䰘 �矄 �ᨨ 鼀�� 矄�矄 㜠�㝐桌 㡰 㝐ᨨ �ᆠ�ᨨ
㡰 矄矄ᨨ盐�ᨨ
ᆠ㡰 矄矄犨 㡰 �� � ��ᆠᨨ矄 ҈ ᨨᆠ犨 �ᆠ 鼀├矄鼀�鼀ᨨ 㜠�㝐桌 㡰 ��ᨨ � 㡰 㝐� �ᆠ � ᆠ㡰 盐ᆠ ᨨᆠ矄 㡰 꾀� � 10 Subjective Problems ҈黨 ᔼ 鼀ᨨ� �ᆠᨨ�� ᨨ鼀ᨨ⼄盐ᨨ ᆠᨨ�ᆠ 矄矄 矄 t = k� 犨 ��矄ᆠ盐 ᨨ盐� 犨� ├�矄ᆠᨨ 矄 ᆠ �盐ᆠᨨ�ᨨ � ᨨ鼀 � � 盐鼀ᨨ 盐 矄NA 鼀ᨨ鼀 ᨨ � �I�矄 ᆠ � 鼀� �ᆠ � B � �㜠㝐├矄桌 㡰 k� 鼀矄ᨨ �� 鼀矄NIA B 㡰 ᨨ矄�ᨨ ᆠᨨ 矄 ᨨ 矄� � � ᆠ盐 � ᨨ 盐ᆠ矄 ᆠ�ᨨ ᨨ p⏌� ᆠᨨ� ᆠ㡰 鼀 �鼀盐鼀 � ᨨ盐� 犨 �ᆠ ᆠᨨ� �矄 ᆠ � ᆠ � Q �矄 矄矄 ᨨ盐� ᆠᨨ� 鼀ᨨ矄 � 矄 ᨨ盐矄泐 �� ᨨ 鼀� � �
鼀ᆠ �ᆠᨨ矄ᆠ��ᨨ 矄㡰 꾀犨 ᨨ �盐矄R � �ᆠ �Q盐 � ᨨ鼀泐 �矄�盐 ᨨ �鼀 ᨨ 犨 �矄 ᨨᨨ ᨨ盐 �� ᨨ� ᨨ ᨨ ᨨ �矄矄盐矄 泐�� � 矄� 矄� �矄 鼀� �ᆠ � B�矄� 矄矄 ᨨ犨 �
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EBD_8341 P-210
Physics
��盐 � �� 矄�ᨨ 矄� 矄� �矄T ᔼ T ├� � 矄�ᨨ ᨨ 矄� �矄 � 鼀 �鼀盐鼀 �盐ᨨᆠ�泐w犨� 犨 �ᆠ 犨 ᆠ ᨨ 㜠㝐├矄桌 d
w0
㡰 꾀�ᆠᨨ ᆠ �q�矄矄 ᨨ��� 犨� ur uur ᨨᆠ�泐 v 㦨v �ˆ � � 矄 ᨨ盐矄 ᨨᆠ F ᨨ �ᆠ꾀矄矄盐鼀 矄ᆠ�矄��泐 㡰 ᔼ 盐 � ᨨ鼀鼀� �ᆠ � Bᨨ ˆ �矄� uur uur 鼀� ᨨᆠ F҈ F ᨨ 矄鼀�ᆠ�ᆠ矄 uur KLM KNM 盐ᨨ � ᨨᆠ F ᨨ ᨨᨨ 꾀犨�ᨨᨨᆠ泐� �ᆠ盐 ᔼ�矄ᆠ � x㜠y 矄矄 ᨨ犨 � �� ҈黨黨҈㜠졔㝐├矄桌
㝐 B
꾀ᆠ盐 ᨨ ҈꾀 ᨨ犨矄ᨨ盐 ᆠᨨ矄 � ᆠ�ᆠ盐�犨 �ᆠ �矄� ᨨ� ᨨ 矄矄 ᨨ犨 � ��盐 ᆠ�ᆠ盐� ᆠᨨ 矄�矄矄ᨨ �� � �ᆠ矄ᨨ ��r҈㦨鼀 r҈㦨҈鼀쟼ᆠ ᆠ矄盐 矄 矄鼀 � ᆠ ҈㜠҈㝐├矄桌 D r
C A
r҈
㡰 � 鼀� �ᆠ � ᨨ 盐ᆠ 泐 �矄ᆠ�ᆠ盐� ᆠ 㡰 꾀 � � �泐ᨨ �矄�� 犨�ᆠ泐� �ᆠ盐 ᨨ ҈꾀�矄矄矄� � ᨨ盐� ᆠ ᨨ ᨨᆠ�ᆠ盐� �ᆠ泐犨� �ᆠ�ᨨ ᨨ ᆠ盐 � �� ᨨ ᨨ ᆠ�ᆠ盐�␔ �矄 ᨨᆠᆠ� �ᨨ
犨� ᆠ 盐ᨨ ᆠ盐 � ᆠ�ᆠ盐�䁸 ␔ �矄 ᨨᆠᆠ� �ᨨ ᆠAC 矄�� 矄�鼀 CD 盐ᨨ ᆠ盐 ᆠ 䁸 꾀ᆠ�ᆠ盐ᨨᨨᨨ �盐矄R�矄 ᨨ � �鼀 �� 矄 矄矄 ᨨ犨 � ��盐졠 ᨨ 矄鼀�ᆠ�ᆠ矄 KNM㡰�矄� x㜠z ᨨ ᨨ KLM㡰� y㜠z 犨� �ᆠ 矄 ᨨ��� ᔴ盐 I�矄 ᨨ犨� � ᨨ盐� ᆠ ᨨ 矄鼀�ᆠ�ᆠ矄矄矄 ᨨ犨 � ��盐 㜠҈㝐├矄桌
҈
҈
졔턼
ᔼ
泐 졠
㡰 ᔼ �ᆠ犨� ᆠ �졔Q 鼀矄矄鼀�矄ᆠ uur ᆠ �� ᨨᆠ�泐 v ᨨ �NP 矄 ��盐㡰 � �矄� 矄 ᨨ盐矄ᆠᆠ�ᨨ 㡰 ᔼ uur 盐 � ᨨ鼀 鼀� �ᆠ � 盐ᆠ�ᨨ � B 㦨 B�ˆ �矄� � ᨨᆠ ᨨ⼄盐 ᆠ� �ᨨ ᨨᨨ 盐ᨨ �矄 � 犨ᨨᨨ �犨�矄ᆠ泐� �ᆠ盐 �鼀矄 ᔼ 鼀� 矄鼀 �ᆠ�ᨨ �ᆠ � ᨨ ᨨ 㡰 P Q矄ᆠ�泐矄盐ᆠ 泐 �ᆠ盐ᨨ ᨨ ᨨ� 矄P Q ᨨᆠ �矄 ᆠ ᨨ � 鼀矄 鼀矄 矄ᆠ�泐 ᨨ鼀ᆠᨨ� ᨨ� 矄 ��盐㡰 ҈黨黨㜠㝐├矄桌
�鼀
鼀 턼 - - - -졠 - - - - 졠 - - ® - -졠 �꾀 ᔼ꾀 �㡰 꾀 ᆠᨨ 鼀ᨨ� �犨� ᨨᆠ�泐ᨨ 맔҈�鼀⏌矄ᨨ � ᨨ矄�� 㜠 �ᆠ�ᨨ � ᆠ矄 ᨨᆠᨨ 鼀� �盐 맔҈ ᨨ� R � 盐ᨨ ᔼ ��㡰 � ᨨ矄��ᨨ 矄犨 �ᆠ � ᨨ � 犨�ᆠ泐� �ᆠ盐 ᨨ 鼀� �盐 �鼀矄鼀泐 ᆠ 矄ᨨ 鼀� �ᆠ� 盐ᆠ�ᨨ R�矄 ᨨ
泐
ᔼ
㝐
⧤
ᔼ
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P-211
Moving Charges and Magnetism
Answer Key Topic-1 : Motion of Charged Particle in Magnetic Field ҈ ҈҈ ҈ 黨
ᆠ㡰 㡰 ᆠ㡰 盐㡰 㡰
҈ 黨
ᆠ㡰 㡰 ᆠ㡰 㡰 㡰
҈ ҈
㡰 㡰 㡰 ᆠ㡰 㡰
҈
㡰 � ᆠ㡰 㡰 䰘 㡰 㡰 ҈� ᆠ㡰 ҈ 㡰 ҈䰘 㡰 ҈ 㡰 � ᆠ㡰 㡰 䰘 ᆠ㡰 ᆠ 㡰 㡰 � ᆠ 㡰
㡰 黨 㡰 ҈ ᨨ 盐矄㡰 㡰 ҈黨 㡰 㡰 矄㡰 矄㡰 ᆠ㡰 䰘 㡰
Topic-2 : Magnetic Field Lines, Biot-Savart's Law and Ampere's Circuital Law ҈ ҈҈
㡰 㡰
㡰 ҈ 㡰
ᆠ㡰 ҈ 㡰
㡰 ҈ 㡰
� 㡰 ҈� 㡰
㡰 ҈ 㡰
䰘 ᆠ㡰 ҈䰘 㡰
҈
㡰
ᆠ㡰
㡰
㡰
� �㡰
䰘㡰
䰘 =
㡰 ҈ 㡰
黨 ᆠ㡰 ҈黨 㡰
҈ 㡰 ᆠ㡰
m ᔼ é ҈ ҈ ù ê ú 㡰 黨 㡰 ë ҈ û
Topic-3 : Force and Torque on Current Carrying Conductor ҈ ҈҈ ҈黨
ᆠ㡰 㡰
盐
㡰
ᆠ㡰
㡰
� 㡰
ᆠ㡰
䰘 㡰
҈ 㡰
҈ 㡰
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҈� 㡰
҈ ᆠ㡰
҈䰘 㡰
㡰
黨 ᆠ㡰
҈ ᆠ㡰
҈ ᔼ ├ˆ ᨨ矄�� �ᆠ�ᨨ
Topic-4 : Galvanometer and its Conversion into Ammeter and Voltmeter ҈
㡰
㡰
҈҈
ᆠ㡰
҈ ᆠ㡰
ᨨ 盐矄㡰 ҈ 㡰
ᆠ㡰
� 㡰
㡰
䰘 㡰
ᆠ㡰
黨 㡰
҈ 㡰
҈ 㡰
҈䰘 㡰
҈ ��㡰 ҈� 㡰 ҈ ᆠ㡰 ҈䰘 ᆠ㡰 ҈ ᆠ㡰
Topic-5 : Miscellaneous (Mixed Concepts) Problems ҈ 黨 ҈҈ ҈
ᆠ㡰
㡰
㡰
㡰
꾀®⼄®ᔴ®⼄®⼄矄 꾀®®⼄矄ᔴ®⼄矄®⼄矄 㡰
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䰘 㡰 㡰
҈ 꾀®®⼄矄ᔴ®⼄矄®⼄矄 ҈ 㡰 ҈ ᆠ㡰 ҈ ᆠ㡰 ҈� 㡰
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EBD_8341 P-212
Physics
Magnetism and Matter
19
Topic-1 : Magnetism, Gaus’s Law, Magnetic Moment, Properties of Magnet ҈
꾀矄鼀鼀� ᆠ 犨� �矄 �矄 犨� � ᨨ � ᆠ矄 ᨨ⼄盐ᨨ ҈鼀 鼀� �鼀盐鼀 犨ᨨ├ ⼄盐� ᨨᨨ � ᨨ鼀 �矄 矄 ᨨ 盐 矄⼄盐���盐鼀ᨨ矄��ᨨ �矄[Main Sep.04, 2020 (I)] 㡰 맔҈ 㡰 黨맔҈ ᆠ㡰 䰘맔҈ 㡰 ҈҈䰘맔҈ 꾀ᆠ�ᆠ盐ᆠᨨ� 矄鼀ᨨ鼀 ᨨ � �├�鼀ᨨ盐 泐 �鼀 �矄ᆠ泐� �ᆠ盐 ᨨᨨ 盐ᆠ鼀� �ᆠ 鼀ᨨ鼀 ᨨ 꾀鼀 ᆠᨨ��矄├� ��泐� �ᆠ ᨨ矄��ᨨ � ᆠ ᨨ 泐 ᨨ盐 ᨨ� ᨨ �鼀␔ 盐 � ᨨ鼀鼀� �ᆠ � ᨨ �矄� ᨨ � �ᆠ�矄矄ᨨ� �ᨨ盐 �矄 ᨨ� ᨨ �鼀 �盐矄 ᆠᨨ�ᆠ⼄盐�矄 ᨨ� � 泐 犨� Main 和 ᔼᔼ㡰桌 㡰 ҈ 矄҈ 㡰 ҈p 矄҈ ᆠ㡰 p 矄҈ 㡰 矄҈
犨ᨨ鼀� �ᆠ �ᨨ矄 Ӹᆠ 矄�ᨨ d 犨� � 矄 �ᆠ盐ᨨᆠ ᨨ �ᨨ鼀ᨨ鼀 ᨨ Ӹ�矄犨�ᆠ ᨨ 꾀�ᆠᨨ ᆠ �q�矄矄矄� � ᨨ盐� �鼀� ᨨ� �q㦨� 犨� ᨨ� ᨨ � 矄矄 ᨨ犨 � ��盐␔ 犨ᨨ盐 鼀� �盐 ᨨ ᨨᆠᨨ �ᆠ � 矄 䁸 d �矄鼀盐ᆠ �
�鼀 矄�ᨨ 矄ᨨ �ᨨ㡰 Main 꾀�҈黨ᔼᔼ桌
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䰘
꾀 鼀� ᨨ ᨨ 鼀� �ᆠ 鼀ᨨ鼀 ҈ - �ˆ 꾀㜠鼀 �矄 ᆠ � �鼀泐� �鼀� �ᆠ � �ˆ ᆠᨨ矄w㡰犨 㦨 ҈ 矄 w 㦨 ҈� ⏌矄 犨ᨨ├ ᨨ ᨨ 矄� � �ᆠ�ᨨ ᨨ 鼀� �ᆠ鼀ᨨ鼀 㦨҈ 矄ᆠᨨ �矄 Main ҈ ҈黨ᔼ桌 㡰 ҈ 㡰 䰘 ᆠ㡰 㡰 ҈ 꾀鼀� �ᆠ �ᨨ�矄ᆠ 盐ᨨ 泐犨ᨨ鼀� �ᆠ � 矄 犨 �ᆠ � ᆠ� ᨨᆠ ᨨ �ᨨ 䰘� 졠 ᨨ � 矄 矄鼀� �盐 ᨨ ҈�鼀 �ᨨ� 矄 矄⼄盐���盐鼀 �ᨨ 犨� �矄 � 鼀� � 盐ᨨ ᨨ � � 鼀 㡰�矄ᆠᨨ矄ᨨ Main 졠 � 꾀�黨҈桌 㡰 ҈ 㡰 ҈҈ ᆠ㡰 㡰 ҈ 꾀�ᆠ鼀ᨨ �矄ᨨ ᨨ� 矄 �盐矄ᆠ鼀 �ᨨ 盐鼀 ᨨ 盐 矄ᔼᆠ�矄ᆠ盐 ᨨ ҈�꾀ᔼ ��矄⼄盐� ᨨ r 鼀� ᨨ 矄鼀矄� 鼀� � �ᨨ 㝐 鼀� �ᆠ uur 鼀ᨨ鼀 ⏌ᨨ盐鼀㡰 㝐 �矄 Main졠 � 꾀�҈҈�桌 㡰 p꾀鼀҈ 㡰 p꾀鼀҈ ҈ ᆠ㡰 꾀鼀 㡰 꾀鼀҈ 꾀ᆠ盐 ᆠ泐� �ᨨᨨ�矄ᆠ � 盐 � ᨨ鼀鼀� �ᆠ � � ᨨ盐 � ᨨ� �ᨨ 矄ᔼᔼᔼᔼᔼᔼᔼ � 鼀� ᆠ矄� �ᨨ ᨨ ᨨ �쟼 �泐和桌
ᔼ æ m ö M ´ qv ç ÷ 㡰 è p ø d 㡰 æm ö M æ m ö M ´ qv ´ qv ç ÷ çè ÷ø è ø p ᆠ㡰 㡰 p d d
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ᔼᔼᔼ
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P-213
Magnetism and Matter
6
ᔼ
㡰 ᔼ>ᔼᔼᔼ>ᔼᔼ>ᔼ 㡰 ᔼ>ᔼᔼ>ᔼᔼᔼ>ᔼ ᆠ㡰 ᔼ>ᔼ>ᔼᔼ>ᔼᔼᔼ 㡰 ᔼᔼᔼ>ᔼ>ᔼ>ᔼᔼ
鼀� �ᆠ � � 矄 盐ᨨ鼀� ᆠᨨᆠ泐 矄 ᨨ犨 �
和桌 N
N
MCQs with One or More than One Correct Answer
҈� 꾀ᨨ �ᆠ盐 ᆠ泐� �犨�ᆠ泐� �ᆠ盐 ᔼ҈矄盐ᆠ ᔼ҈�矄 ᨨ犨� �ᨨ盐 ᨨ鼀 ᨨ �矄ᆠ O꾀 矄 泐矄ᆠ盐 I�矄 ᨨ犨� �� ᨨᨨABCD 㜠�㝐҈桌 ᔴ ᔼ 졠
졠Ἐ
ᔼ҈
㡰
㡰 S
S
N
N
ᆠ㡰
㡰 S
黨
҈
꾀
S
꾀�ᆠᨨ ᆠ �⼄ 鼀矄矄鼀鼀ᨨ矄� ᆠ�ᆠ盐ᨨ� ᨨ �盐矄犨� �盐矄 w �ᨨᨨ 鼀� �盐 ᨨ �矄鼀� �ᆠ鼀ᨨ鼀 ᨨ ᨨ �矄 �盐鼀ᨨ鼀 盐鼀 矄 ᨨ
和桌 㡰 w q 㡰 w q m ᆠ㡰 q m 㡰 w m
犨ᨨ�ᆠ矄ᆠ ᨨ 鼀矄矄鼀 ᆠ �qᆠ ᨨ 犨ᨨ 矄ᨨ �� ��� ᨨ ᨨ � R ᨨ �矄 ᨨ ᆠᨨ 矄 �盐 矄 ᨨ盐 �ᆠ盐 �矄矄矄� � ᨨ盐� �矄ᆠ �ᨨᨨ 鼀� �盐 矄 ᨨ 鼀� �ᆠ鼀ᨨ鼀 ᨨ 矄泐矄鼀 �矄 �盐 鼀ᨨ鼀 盐鼀ᨨ盐 ᆠ ᨨ ᨨ �矄҈黨黨和㜠㝐├矄桌 q q q q 㡰 ᆠ㡰 㡰 m m m pm 꾀ᆠᨨ 盐ᆠ� �ᆠ�ᆠ盐ᨨᨨᨨ �盐矄rᆠ�矄ᆠᨨ 矄 r ᆠ盐 �ᔼ�矄ᆠ � 盐 � ᨨ鼀鼀� �ᆠ � B 矄盐ᆠ r B �矄 �ᆠ盐 ᨨ ᨨ ᨨᨨ 鼀� �ᆠ ᨨᆠᆠ� �ᨨ ᨨᨨ�矄 ҈黨㜠҈㝐├桌 㡰 �r B 㡰 p �r B ᆠ㡰 ᨨ 㡰 p �r B 꾀鼀� �ᆠ �矄├� ᨨ 盐 � ᨨ鼀鼀� �ᆠ � ᔼ � ᆠ矄 ҈黨㜠㝐├矄桌 㡰 ᨨᆠ ᨨ⼄盐 㡰 ᨨᆠ盐 ᨨᨨ⼄盐 ᆠ㡰 ᨨ⼄盐盐 ᨨ ᨨᆠ 㡰 � ᨨᆠ ᨨᨨ⼄盐
㡰
҈҈
҈
҈
҈
4 Fill in the Blanks ᔼ 泐 ᨨ� ᨨ鼀 ᆠᨨ 鼀ᨨ矄� ᨨ� ᨨ �盐矄 � 处 鼀├� � ҈҈ ᨨ盐�ᨨ 矄 矄ᆠᨨ 鼀� �ᆠ鼀ᨨ鼀 矄矄ᨨᆠ� 犨� ᨨ�鼀ᨨ�ᨨ ᨨ ᆠᨨ �矄 ҈黨㜠㝐├矄桌 꾀犨�ᨨ � L鼀ᆠ泐� �ᆠ盐 �鼀�矄 � ᨨ鼀ᨨ ᆠ�ᆠ 鼀� �盐 ᨨ �矄鼀� �ᆠ 鼀ᨨ鼀 �矄� MKS盐 �矄 ҈黨䰘㜠㝐├矄桌
㡰 ᨨᆠ�矄 ᨨ 㡰 ᨨ⼄盐�矄 ᨨ ᆠ㡰 矄矄 ᨨ鼀O ᨨᨨ犨�ᨨ� ᆠᨨᆠ├犨�矄ᨨ � OOἘ �矄 㡰 矄矄 ᨨ鼀O ᨨᨨ犨�ᨨ� �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ᨨ �OOἘ �矄 ҈ ᨨᨨ犨� � � � ᆠ 矄 㜠�㝐҈桌 㡰 � 盐ᆠ ᆠ�ᆠ � 㡰 鼀� ᨨ矄�ᆠ � ᆠ㡰 ���ᨨ � ᨨ 鼀矄矄矄 㡰 ᆠᨨ矄�ᆠ � 10 Subjective Problems ҈䰘 꾀盐 � ᨨ鼀ᆠᨨ 矄 鼀� �ᆠ � �矄 �ᆠ � ᨨ � ᨨ x �矄� xy㜠 PQRS �矄��� 矄⼄盐 犨� 鼀ᆠ泐� �矄 泐ᆠ盐 I犨� �矄ᆠ ᨨ��� O꾀�鼀t㦨 鼀�矄矄� ᨨ矄��ᨨ
矄矄 ᨨ犨 � ��盐犨� �矄矄� 矄ᨨ x y 矄쟼ᆠ 矄� ᨨ 鼀�矄ᨨ 鼀矄矄M � L 泐
和
졠
턼
㡰 ␔ �矄 ᨨ⼄盐t ᨨ盐Oᆠ� �ᨨ 鼀 盐 ᨨ 鼀� �ᆠ � 䁸 㡰 � �泐犨 �ᆠ 鼀ᨨ矄盐 ᆠ�ᨨ ᨨ �矄ᨨ⼄盐� 矄 ᨨ� ᨨ �鼀Dt �矄ᨨ盐 �矄ᨨ�ᨨ ᨨᆠᆠ盐矄 Dt�矄矄ᨨ矄 ᨨ 泐��ᨨ � ᨨ⼄盐 盐� � �矄� 鼀泐
�ᆠ 㡰� 鼀ᨨ鼀 ᨨ � �ᨨ 鼀 ᨨ盐 �矄 ᨨ盐� �矄ᆠ �ᆠ盐ᨨ�矄 �矄 ML ҈黨黨㜠㝐├矄桌 ҈ 꾀 ᆠᨨ � �ᨨ盐 矄 ᨨ 泐 ᨨ� ᨨ鼀 �矄 ᨨ� �� �ᆠᨨᆠ├㜠犨�矄 �ᆠ�ᨨ � ᆠ�ᆠ盐ᨨ�ᨨ �盐矄R ҈黨黨㜠�㝐├矄桌 �㡰 졠� 矄矄�ᨨ ᨨ ᨨ�鼀� �ᆠ �ᨨ 鼀ᨨ鼀 ᨨ ᆠᨨ
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EBD_8341 P-214
Physics
(ii) The ur atom is placed in a uniform magnetic induction B such that the plane-normal of the electron-orbit makes an angle of 30° with the magnetic induction. Find the torque experienced by the orbiting electron.
19. A bar magnet is placed with its north pole pointing north and its south pole pointing south. Draw a figure to show the location of neutral points. [1979] 20. A bar magnet with poles 25 cm apart and of strength 14.4 amp-m rests with centre on a frictionless pivot. It is held in equilibrium at an angle of 60° with respect to a uniform magnetic field of induction 0.25 Wb/m2, by applying a force F at right angles to its axis at a point 12 cm from pivot. Calculate F. What will happen if the force F is removed? [1978]
Topic-2 : The Earth Magnetism, Magnetic Materials and their Properties 1 1.
2.
3.
4.
MCQs with One Correct Answer
5.
An iron rod of volume 10–3 m3 and relative permeability 1000 is placed as core in a solenoid with 10 turns/cm. If a current of 0.5 A is passed through the solenoid, then the magnetic moment of the rod will be : [Main Sep. 05, 2020 (II)] (a) 50 × 102 Am2 (b) 5 × 102 Am2 (c) 500 × 102 Am2 (d) 0.5 × 102 Am2 A paramagnetic sample shows a net magnetisation of 6 A/ m when it is placed in an external magnetic field of 0.4 T at a temperature of 4 K. When the sample is placed in an external magnetic field of 0.3 T at a temperature of 24 K, then the magnetisation will be :[Main Sep. 04, 2020 (II)] (a) 1 A/m (b) 4 A/m (c) 2.25 A/m (d) 0.75 A/m A perfectly diamagnetic sphere has a small spherical cavity at its centre, which is filled with a paramagnetic substance. r The whole system is placed in a uniform magnetic field B. Then the field inside the paramagnetic substance is : [Main Sep. 03, 2020 (II)]
r (a) B P (b) zero r r (c) much large than | B | and parallel to B r r (d) much large than | B | but opposite to B Magnetic materials used for making permanent magnets (P) and magnets in a transformer (T) have different properties of the following, which property best matches for the type of magnet required? [Main Sep. 02, 2020 (I)] (a) T : Large retentivity, small coercivity (b) P : Small retentivity, large coercivity (c) T : Large retentivity, large coercivity (d) P : Large retentivity, large coercivity
The figure gives experimentally measured B vs. H variation in a ferromagnetic material. The retentivity, co-ercivity and saturation, respectively, of the material are: [Main 7 Jan. 2020 II] (a) 1.5 T, 50 A/m and 1.0 T (b) 1.5 T, 50 A/m and 1.0 T (c) 150 A/m, 1.0 T and 1.5 T (d) 1.0 T, 50 A/m and 1.5 T 6.
7.
A paramagnetic material has 1028 atoms/m3. Its magnetic susceptibility at temperature 350 K is 2.8 × 10–4 . Its susceptibility at 300 K is: [Main 12 Jan. 2019 II] (a) 3.267 × 10–4 (b) 3.672 × 10–4 (c) 3.726 × 10–4 (d) 2.672 × 10 –4 A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of 20 × 10–6 J/T when a magnetic intensity of 60 × 103 A/m is applied. Its magnetic susceptibility is: [Main 11 Jan. 2019 II] (a) 3.3 × 10–2 (b) 4.3 × 10 –2 (c) 2.3 × 10–2 (d) 3.3 × 10–4
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P-215
Magnetism and Matter
8.
9.
A bar magnet is demagnetized by inserthing it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The coercivity of the bar magnet is: [Main 9 Jan. 2019 I] (a) 285 A/m (b) 2600 A/m (c) 520 A/m (d) 1200 A/m The B-H curve for a ferromagnet is shown in the figure. The ferromagnet is placed inside a long solenoid with 1000 turns/cm.. The current that should be passed in the solenoid to demagnetise the ferromagnet completely is: [Main Online April 15, 2018]
12. The coercivity of a small magnet where the ferromagnet gets demagnetized is 3 ´ 103 Am -1. The current required to be passed in a solenoid of length 10 cm and number of turns 100, so that the magnet gets demagnetized when inside the solenoid, is: [Main 2014] (a) 30 mA (b) 60 mA (c) 3 A (d) 6 A 13. Three identical bars A, B and C are made of different magnetic materials. When kept in a uniform magnetic field, the field lines around them look as follows:
B(T)
A
C
B
2, 0
1, 0
H(A/m)
–200 –100 100 200 –1, 0 –2, 0
10.
(a) 2 mA (b) 1 mA (c) 40 µA (d) 20 µA Hysteresis loops for two magnetic materials A and B are given below : D
B
H (A)
11.
H (B)
These materials are used to make magnets for elecric generators, transformer core and electromagnet core. Then it is proper to use : [Main 2016] (a) A for transformers and B for electric generators. (b) B for electromagnets and transformers. (c) A for electric generators and trasformers. (d) A for electromagnets and B for electric generators. A fighter plane of length 20 m, wing span (distance from tip of one wing to the tip of the other wing) of 15m and height 5m is lying towards east over Delhi. Its speed is 240 ms–1. The earth's magnetic field over Delhi is 5 × 10–5T with the declination angle ~0° and dip of q such that sin q 2 = . If the voltage developed is VB between the lower and 3 upper side of the plane and VW between the tips of the wings then VB and VW are close to: [Main Online April 10, 2016] (a) VB = 40 mV; VW = 135 mV with left side of pilot at higher voltage (b) VB = 45 mV; VW = 120 mV with right side of pilot at higher voltage (c) VB = 40 mV; VW = 135 mV with right side of pilot at higher voltage (d) VB = 45 mV; VW = 120 mV with left side of pilot at higher voltage
Make the correspondence of these bars with their material being diamagnetic (D), ferromagnetic (F) and paramagnetic (P): [Main Online April 11, 2014] (a) A « D, B « P, C « F (b) A « F, B « D, C « P (c) A « P, B « F, C « D (d) A « F, B « P, C « D 14. The mid points of two small magnetic dipoles of length d in end-on positions, are separated by a distance x, (x > > d). The force between them is proportional to x–n where n is: [Main Online April 9, 2014] N
S
N
S
x
(a) 1 (b) 2 (c) 3 (d) 4 15. Two short bar magnets of length 1 cm each have magnetic moments 1.20 Am2 and 1.00 Am2 respectively. They are placed on a horizontal table parallel to each other with their N poles pointing towards the South. They have a common magnetic equator and are separated by a distance of 20.0 cm. The value of the resultand horizontal magnetic induction at the mid-point O of the line joining their centres is close to (Horizontal component of earth.s magnetic induction is 3.6× 10.5Wb/m2) [Main 2013] (a) 3.6 × 10.5 Wb/m2 (b) 2.56 × 10.4 Wb/m2 (c) 3.50 × 10.4 Wb/m2 (d) 5.80 × 10.4 Wb/m2 16. The earth’s magnetic field lines resemble that of a dipole at the centre of the earth. If the magnetic moment of this dipole is close to 8 × 1022 Am2, the value of earth’s magnetic field near the equator is close to (radius of the earth = 6.4 × 106 m) [Main Online April 25, 2013] (a) 0.6 Gauss (b) 1.2 Gauss (c) 1.8 Gauss (d) 0.32 Gauss
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EBD_8341 P-216
Physics
Topic-3 : Magnetic Equipment 1 ҈
MCQs with One Correct Answer
꾀� ��矄 盐 �ᨨ �ᔼᆠ ᨨ矄ᆠ�犨� ᨨ盐矄�� � ᨨ矄� � � �㡰� �矄 犨� �鼀�ᨨ T҈ ��㡰ᆠ├ ᨨ � �ᆠ�ᨨ �ᆠ盐ᨨ�矄 犨� �ᨨ T �ᨨ 㡰
T҈ 犨� 㝐� 和� ᔼᔼ㡰桌 T 㡰
꾀鼀� �ᆠᆠᨨ鼀矄矄 ᨨ矄ᆠ�矄�鼀矄鼀� 盐 ᆠ犨 ��矄�ᨨ �鼀矄鼀� 盐 犨 ��矄ᨨᔼ ҈ 矄ᆠ�泐 ᨨ 鼀� �ᆠ � 盐ᨨ 犨ᨨᆠ矄
�ᨨ҈⏌�矄矄�� 泐 㝐� ҈꾀�҈黨ᔼ桌 㡰 ҈ 㡰 䰘 ᆠ㡰 㡰 ᆠ㡰
㡰
꾀 ᨨᨨ 矄ᨨ� ᆠ泐� ᨨ 矄鼀鼀矄矄 �盐矄 鼀 ᨨ 鼀 鼀� �ᆠ 鼀� 犨� � 鼀� �ᆠ鼀ᨨ鼀 ᨨ �矄ᆠ� 矄盐 鼀� �ᆠ鼀ᨨ鼀 ᨨ ᨨᨨ�矄犨�ᆠᨨ 矄ᨨ� ᆠ泐�
泐ᆠ � 盐 � ᨨ鼀鼀� �ᆠ � � 矄盐ᆠ 鼀
�鼀� �ᆠ鼀ᨨ鼀 矄鼀├矄鼀 � 犨� � ᔼ ᨨ矄ᆠ��ᨨ �ᨨ 矄 ᨨ ᨨᨨ ᆠ泐� ᆠ矄ᆠ�泐 㝐� ҈ ҈黨 ᔼᔼ桌 㡰 㦨 ᆠ 㡰 㦨 ᆠ ᆠ㡰 㦨҈� ᆠ 㡰 㦨� ᆠ 꾀鼀� �ᆠ ᨨ 鼀� �ᆠ鼀ᨨ鼀 䰘맔҈꾀鼀 鼀ᨨ鼀 ᨨ � �䰘�맔҈├�鼀�矄 ᨨ鼀� �矄�鼀 鼀ᨨ �ᆠᨨ矄ᆠ��ᨨ 矄� 鼀� �ᆠ � ᨨ ҈ �鼀 ├ ᨨ҈ᆠᨨ鼀ᨨ矄ᆠ��ᨨ 矄�矄 㝐� ҈䰘桌 㡰 黨矄 㡰 䰘矄 ᆠ㡰 �矄 㡰 黨矄
Topic-4 : Miscellaneous (Mixed Concepts) Problems 8
� B҈犨 �ᆠ ᨨ ᨨᨨ犨� �� 矄矄 ᨨ犨矄 ᆠᨨᆠ��ᨨ ᨨ 犨� � 矄 � 矄䁸
Comprehension/Passage Based Questions
꾀和和꾀E 쟼ᆠ�ᆠ 矄�矄 ᆠ ᨨ ᆠ� 鼀�矄 ├ ᨨ犨 矄 矄盐ᆠᨨ 盐ᆠᨨ矄 ᆠ �矄 盐泐 ᨨ鼀 ᨨ ᨨ 盐 ᨨ ᨨ矄 �鼀盐�矄ᨨ犨 ᨨ犨ᆠ��ᆠ鼀盐 TC 㡰꾀 � 矄� �ᨨ泐ᨨ 矄盐ᆠᨨ 盐ᆠᨨ矄�矄 � ᆠ��ᆠ鼀盐ᆠᨨ鼀矄矄鼀 TC 㡰� 泐ᆠ � 鼀� �ᆠ � � ᆠ��ᆠ鼀盐TC B㡰�矄 盐 ᆠ�ᨨ
ᨨ 鼀� �ᆠ � 矄 � ᆠᨨ TC B㡰ᨨ �矄矄 ᨨ犨 � ��盐 ҈桌 TC B㡰
㡰
㡰
ᆠ㡰
㡰
TC 㡰
҈
졠 B ᔼ � 矄ᨨ犨 矄�矄 ᆠRᨨ 矄盐ᆠᨨ 盐ᆠᨨ �矄矄 ᨨ犨 矄 盐 ᆠ�ᨨ ᨨ �矄鼀盐T ᨨ犨ᨨ � 鼀� �ᆠ � 矄B҈ 矄ᨨ� � 㡰 B 矄 � 㡰ᔼ B�矄
꾀矄盐ᆠᨨ 盐ᆠᨨ 矄TC 㡰㦨҈⧤␔ 鼀� �ᆠ � ᨨ 䰘� 矄�矄� �矄TC ᆠ矄矄ᨨ䰘�⧤ ᨨ �矄鼀�ᨨ ᆠ � �泐矄泐 犨
㡰 B㦨� 矄TC 㡰㦨⧤ 㡰 B㦨� 矄䰘�⧤TC 㡰҈⧤ ᆠ㡰 B㦨҈ 矄䰘�⧤TC 㡰҈⧤ 㡰 B㦨҈ 矄TC 㡰㦨䰘⧤
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P-217
Magnetism and Matter
Answer Key Topic-1 : Magnetism, Gaus's Law, Magnetic Moment, Properties of Magnet ҈ ᆠ㡰
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҈҈ ᆠ㡰
҈ 㡰
㡰
ᆠ㡰
҈ ҈�맔҈꾀鼀㡰
� 㡰 ҈
ᆠ㡰
䰘 㡰
㡰
黨 ᆠ㡰
҈ 㡰
�L 㡰 ҈� ᆠ㡰 ҈ ᆠ 㡰 π
Topic-2 : The Earth Magnetism, Magnetic Materials and their Properties ҈ 㡰
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㡰
㡰
� 㡰
㡰
҈҈ 㡰
҈ ᆠ㡰
҈ 㡰
҈ 㡰
҈� 㡰
҈ 㡰
䰘 㡰
㡰
黨 㡰
҈ 㡰
Topic-3 : Magnetic Equipment ҈ 㡰
ᨨ 盐矄㡰 㡰
ᆠ㡰
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈ 㡰
㡰
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EBD_8341 P-218
Physics
Electromagnetic Induction
20
Topic-1 : Magnetic Flux, Faraday's and Lenz's Law MCQs with One Correct Answer
1 ҈
҈㡰
꾀矄鼀鼀� �矄鼀ᨨ ᨨ盐� ᆠᨨ�ᆠᨨ 矄 矄 ᨨ鼀ᨨ ᨨ ᨨ ␔ �ᆠ ᨨ ᨨᨨ犨� � 矄�矄ᨨ ᨨ矄�ᨨ 矄犨�矄 ᨨ � ᨨ鼀G ᆠ ᆠᨨ矄矄 ᆠᨨ�䁸 㝐� 和 ᔼ㡰桌
ᆠ
鼀�
㡰
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꾀 ��ᆠᨨᨨ � �矄�矄 ᆠRᨨ 矄鼀�鼀ᨨ �矄a 矄鼀�鼀� ᨨ �矄b�矄ᆠ � 鼀� �ᆠ � 矄矄 ᨨ犨
� ��盐ᔼ ᨨᨨ�矄ᨨ ᨨ盐 x㜠 �矄犨� �盐 ⼄盐 ᆠ泐w �ᨨ犨ᨨ矄矄� ᨨᨨ 盐ᨨᨨ盐 � ��矄 㝐� 和 ᔼ㡰桌
B
z
ᨨ矄��ᨨ 矄矄 ᨨ犨 矄ᆠ� ҈㡰 鼀� 矄 泐 㡰鼀� �矄ᆠᨨ鼀泐� 矄� 㡰鼀� 矄 � ҈㡰
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G
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b
x
㡰
a y
y ®
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҈㡰
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㡰 ®
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p a b B w R
㡰 ᨨ
ᆠ㡰
pabBw R
㡰
㡰
҈㡰 ᆠ㡰
®
®
㡰
x
㡰 ®
p a b B w R
꾀 盐 � ᨨ鼀 鼀� �ᆠ � B �矄矄 � �ᆠ�ᨨ
�ᆠ盐 ᨨ ᨨ 矄⼄盐 ᨨᨨ 鼀 ᨨ 鼀犨� 犨� 矄 �鼀ᨨ 鼀鼀 ᨨ � ᨨ ᆠ鼀 鼀� �ᆠ � ᆠ �矄犨� �鼀 矄 泐dB/dt㦨 矄҈ � 盐ᆠ ᆠ盐 �
ᨨᨨ �矄 ᆠᨨ矄 ᨨ 矄�矄��泐ᨨ 鼀 犨� �矄 ҈맔҈W 鼀㡰 㝐� 和 ᔼᔼ㡰桌 㡰 �꾀 㡰 꾀 㡰 ҈꾀 ᆠ㡰 꾀
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P-219
Electromagnetic Induction
�
䰘
꾀�鼀t㦨鼀� �ᆠ � ᨨ ҈盐矄矄�矄矄矄� � �ᆠ盐泐 ᨨ盐� � 泐 ᆠᨨ矄 ᨨᨨ矄 ᨨ犨 � ��盐ᔼ 鼀� �ᆠ � 盐ᆠ矄 � 泐ᨨ�盐矄矄� �矄 � 盐ᆠ 쟼㝐 � ᨨᨨ�矄 㝐� ᔼ桌
㡰 �m 㡰 m ᆠ㡰 m 㡰 m ᔴᨨ 矄� ᆠ�ᆠ盐ᆠᨨ�ᨨ 犨�ᆠ泐� �ᆠᨨ 矄 ᆠ盐 I ᨨ鼀� �鼀� �ᆠ �ᨨ 鼀� �ᆠ 盐 ᨨ盐�
� � � ᆠᨨ � 矄 ᆠ�ᆠ盐ᆠᨨ� ᆠ盐 � � ᆠ�ᆠ盐ᆠᨨ��矄�� 泐f� 鼀� �ᆠ 盐 ᨨ盐� ᨨ ᆠ�ᆠ盐ᆠᨨ��矄�� 泐f␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ �矄ᆠᨨᆠ䁸 㝐� 䰘 ᔼ桌 㡰 f�㦨f 㡰 f�>f ᆠ㡰 f�f 㡰 f�㦨f 꾀ᨨ �矄ᨨ ᨨ� ᨨ �盐矄Rᆠ�矄�鼀 t㡰㜠 ᆠ盐 I t㡰㦨It t㡰꾀� �ᨨ �盐矄R�矄ᆠ ᆠᨨ �泐
�矄鼀� 盐� � �鼀� £t£҈ � 盐ᆠ ᆠ盐 IR㡰 � 盐ᆠ EMF VR㡰� � � ᆠ �矄 㝐� 䰘 ᔼ桌 㡰 �ᆠ�ᨨ ᨨ IR鼀� 矄盐 ᆠ � VR�矄鼀 �鼀盐鼀 t㦨� 㡰 꾀t㦨� �ᆠ�ᨨ ᨨ IR 矄矄 VR�矄鼀 �鼀盐鼀 ᆠ㡰 �ᆠ�ᨨ ᨨ IR鼀� 矄盐 ᆠ � VR�矄 ᨨt㦨� 㡰 꾀t㦨� �ᆠ�ᨨ ᨨ IR矄矄 VR�矄 ᨨ 꾀 ᨨᨨᨨ 犨�ᨨ矄� 盐 � ᨨ鼀鼀� �ᆠ � ᔼ ��泐t㦨 ᨨ ᨨᨨ�矄 �ᆠ盐ᨨ 鼀� �ᆠ � ᔼ �ᨨ矄犨� �ᨨ ᨨ ҈矄ᨨ盐
�矄� �矄 鼀� �盐 ᨨ � 盐ᆠ 鼀 犨� 鼀 �鼀盐鼀 鼀� �鼀盐鼀矄ᆠ�泐㝐� 䰘 ᔼᔼ桌 㡰 �矄 䰘�矄 㡰 �矄 �矄 ᆠ㡰 �矄 䰘�矄 㡰 �矄 ҈矄 꾀 泐 ᨨ � 矄ᨨ ᨨ� ᨨ �盐矄 �矄 ᆠ泐� � ᆠ盐 ᔼ 㡰㦨├at k >㡰矄 盐 ᆠ�ᨨ ᨨ �鼀 t >㡰ᔴᨨ盐 ᆠᨨᆠ├犨�矄 ᆠ盐 �矄 ├ ᨨ ᨨ矄�� 꾀 ᆠ�ᆠ盐 ᆠᨨ 盐ᆠ� �ᆠᨨ�ᨨ �盐矄 �矄ᆠ � ⼄盐ᨨ� ᨨ 矄ᨨ ᨨ� ᆠᨨ ᆠ �ᆠ犨� 矄ᨨ ᨨ�
ᆠ盐 � 盐ᆠ � ᨨ盐ᆠᨨ��矄ᆠᨨᆠ泐 �ᆠ 矄 盐 ᆠ�ᨨ ᨨ �鼀泐 㝐� 黨꾀҈黨ᔼᔼ桌 㡰
矄 � 盐ᆠ 鼀 ᨨ ᆠᨨ� �矄 � ᨨ矄 ␔ ᆠ盐 � ��矄ᆠ � 盐 �� ᨨ鼀 ᨨ鼀҈꾀ᨨ� 꾀� ҈矄 ᆠ �� �泐ᨨ � 盐ᆠ ᆠ�矄 㝐� 黨 ҈黨ᔼᔼ桌 㡰 䰘 㡰 䰘� ᆠ㡰 � 㡰 䰘� ҈ 꾀ᆠᨨ 盐ᆠ� �ᆠ�ᆠ盐ᨨᨨ鼀 ᨨ � 犨� 矄 �맔҈鼀 矄�矄 ᆠ҈Wᔼ�矄ᆠ �ᆠ盐 ᨨ�鼀 鼀� �ᆠ � 㡰㦨 㡰矄� �p㡰
ᆠ � ᨨ犨� � ᨨ盐� ᨨᨨ 盐� �㦨 㝐� 黨 ҈黨ᔼ桌 矄 㦨҈鼀矄�矄ᆠᨨ矄ᨨ 㡰 ҈鼀ᔴ 㡰 䰘鼀ᔴ ᆠ㡰 ҈鼀ᔴ 㡰 鼀ᔴ ҈҈ ᔼ ᆠᨨ�ᨨ 矄�矄 ᆠ҈ W ᆠ盐 �矄� 盐ᆠ 泐ᆠ �� � 鼀� �ᆠ 盐 ᨨ盐� �矄矄 ᨨ犨 � ��盐 鼀� �盐 ᨨ ᆠ �� 盐 ᨨ盐� ᆠᨨ��矄 㝐� ҈䰘桌 黨
㡰 �␔ 㡰 䰘�␔ ᆠ㡰 ␔ 㡰 �␔ ҈ 꾀ᆠᨨ 盐ᆠ� �鼀ᆠ�ᆠ盐犨�ᨨᨨᨨ �盐矄�矄ᆠ �ᆠ盐ᨨ鼀� �ᆠ � 犨 �ᆠ �矄犨� �鼀矄 㦨
-
t
犨 tᆠᨨ 矄 矄�鼀㦨
ᔼ 矄�矄 ᆠᨨ ᨨᨨ�矄 � �
ᨨᨨ ᨨ ��鼀 ® ¥㡰 �矄錐 㝐� 졠 � 꾀�҈҈桌 㡰
p t
㡰
p t
p p 㡰 t t ҈ ␔ ᆠ盐 � ᆠᨨ�ᆠ �矄 ᨨ鼀�꾀ᨨ꾀� ҈矄 �ᨨ�ᨨ ��矄ᨨ 盐ᆠ 矄 㜠� 盐ᆠ ᆠ ᨨ ᆠᨨ��矄 㝐� 졠 � 꾀�҈҈�桌 㡰 㡰 䰘 ᆠ㡰 㡰 ҈䰘 ҈ ��盐矄 ᨨ犨矄ᆠ�ᆠ盐ᨨ �盐矄 犨 盐 � ᨨ鼀
ᆠ㡰
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鼀� �ᆠ � �矄 �ᨨ� �� ᨨ ᨨ � ᆠ矄� �� 鼀� �盐 ᆠᨨ 矄
㡰
ᆠ㡰
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ᔼ ᆠ矄 犨 �ᆠ ᨨ ᨨᨨ犨� � � 矄 犨
矄ᆠ 鼀�ᆠ泐ᆠᨨᆠ泐矄 ᨨ犨矄 ��ᨨ ᨨ � 盐ᆠ ᆠ�ᆠ � 쟼 㡰䁸 졠 � 꾀�҈黨҈桌
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EBD_8341 P-220
Physics
쟼
쟼
㡰
ᆠ㡰
쟼
쟼
ᆠ㡰
㡰
҈�
҈
꾀ᆠ�ᆠ盐ᨨᨨᨨ �盐矄ᆠ鼀�矄ᨨ鼀盐ᆠ ��� ᆠ�ᆠ盐ᨨᨨᨨ �盐矄ᆠ鼀 ᆠ ᨨ 矄鼀ᨨᨨ �矄ᨨ �矄ᨨ ��� ᨨᨨ �矄 ᆠ犨
�ᆠ 矄�矄҈�ᆠ鼀ᔼ ᆠ盐 ᨨ 꾀 ᨨ犨矄 ᨨ盐� 矄鼀ᨨᨨ 盐 � ├ 犨� ���ᨨᨨ�矄 㝐� ҈桌 ҈҈ ҈҈ 㡰 黨҈맔҈ 犨 㡰 맔҈ 犨 ᆠ㡰 맔҈҈҈犨 㡰 맔҈黨犨
��盐矄 ᨨ犨矄ᆠ� 犨�矄�鼀 矄ᨨ� ᨨ� ᨨ ᨨ鼀ᆠᨨ ᨨᨨ ᨨᨨ�矄ᆠ � �ᆠ盐 鼀� �ᆠ � � �ᆠ�ᨨ �ᨨ� �� ᨨ ᨨ ��盐 鼀� �盐 ᨨ � � ᆠ矄矄犨� �鼀I҈ I ᆠ盐 矄� 矄�鼀 矄b ᆠ 黨桌 d
c
a
҈䰘
b
㡰 I҈>I 㡰 I҈I ᆠ㡰 I҈�矄� �ᆠ�ᨨ ba I�矄� �ᆠ�ᨨ cd 㡰 I҈�矄� �ᆠ�ᨨ ab I�矄� �ᆠ�ᨨ dc 꾀矄鼀鼀� �矄� � 矄ᨨ犨泐� 矄 犨� ᆠᨨ 矄 ᨨᆠ�泐� 矄� 矄ᨨ ᨨ� 矄 矄 ᨨ犨 � ��盐␔ �ᆠ � 矄矄 矄 �ᨨ 矄 �犨 鼀 � 盐ᆠ 犨� �鼀 和桌
2
�鼀
㡰
�鼀
BR A
㡰
AB R
ᆠ㡰 ABR
㡰
B A R
Integer Value Answer
҈黨 꾀矄�矄 㜠ᔴᆠᨨ鼀� �ᨨ �矄ᆠᨨ
ᆠ ᨨ 꾀ᔴᨨ� ᨨ �盐 ⼄盐 ᆠ泐w 㦨� � ⏌矄ᔼ �鼀 ᆠᨨ 㜠ᔴ ᆠ�ᆠ盐� �矄 R ҈� �鼀 ᆠᨨ 矄 �
鼀��矄ᆠᨨ 㡰ᨨ ᆠ�ᆠ盐��矄 ҈҈桌 3
Numeric Answer
犨ᨨᆠᨨ ᆠ �ᆠᆠ�ᆠ盐ᆠᨨ�矄ᔴ҈ ᔴᆠ � Ӹ ᔴ҈ 矄�盐 矄 �盐矄ᨨ ҈ᆠ鼀ᔴ 矄 盐 矄 �盐矄ᆠ盐 ᆠ鼀ᔴᆠ�矄�鼀 㜠 ᆠ盐 ᔼ t㡰㦨 �tt졔㡰꾀␔ t�矄� 矄 x 盐ᨨ x�矄힘힘힘힘힘힘 㝐� 和� ᔼ㡰桌 ҈ 꾀ᆠ�ᆠ盐ᆠᨨ�ᨨ �盐矄҈ᆠ鼀�矄ᆠ � 盐 � ᨨ鼀鼀� �ᆠ � ᨨ 맔҈� 犨� �矄 �ᆠ盐ᨨ � � ��泐ᔼ�矄ᨨ ᆠᨨ 矄 �盐矄 ᨨ盐
�矄 ᨨ � �鼀 ᨨ ᆠᨨ� �ᆠ盐 ᨨ 鼀� �ᆠ � 矄ᨨ �盐 �ᨨ矄 ᨨ ᨨ�ᨨ � 矄
鼀 �鼀盐鼀盐ᨨ 쟼㝐 � 盐ᆠ � m㡰� ᆠᨨ�犨� ᆠᨨ矄ᨨ � �힘힘힘힘힘 㝐� 和 ᔼ㡰桌 ᔼ 盐ᨨ矄ᆠ 鼀ᆠ ᨨ├ 矄鼀 矄 ᨨ鼀㡰҈ᨨ 矄 ᨨ� �矄 ᨨ 盐ᆠ 犨 ᆠ ᨨ├ ᆠ盐 ᆠ �矄盐 � ᨨ鼀泐 ᨨ鼀�꾀ᨨ� 盐�ᨨ ᨨ � 鼀矄 矄 㜠� 盐ᆠ ᆠᨨ ᆠ ᨨ├ � 鼀㡰�矄矄�鼀 ᨨ힘힘힘힘힘힘 㝐� 黨 ᔼ桌
鼀 � 盐ᆠ � ᔴ҈ � 鼀㡰 � 矄 t㦨҈矄�矄
꾀ᨨᨨ꾀ᔴ쟼 꾀ᨨ 矄�� �矄 矄矄� ᆠᨨ ᨨ� 矄 A 㡰B{�㡰C ��㡰D �㡰E ��㡰 r F �㡰 鼀� �ᆠ � � �矄��ᨨ �矄 B 㦨 �ˆ 졔 kˆ 㡰
⼄盐 �泐ᨨ 盐 ᨨ盐� ᨨᨨABCDEFA � ␔㡰�矄 힘힘힘힘힘힘힘힘힘 . 㝐� 䰘 ᔼ桌 4
㡰
�鼀
҈ 꾀 � ᆠ�ᆠ盐� �ᨨ A�矄 �ᆠ盐ᨨ 盐 � ᨨ鼀鼀� �ᆠ � ᨨ � 盐ᆠ�ᨨ B. A 矄鼀ᆠ盐�矄鼀 � � � � ᨨ鼀�矄ᆠᨨ
ᆠ ᆠᨨ矄矄 矄矄盐ᆠ ᨨ矄�矄 ᆠᨨ ᆠ�ᆠ盐��矄R␔
� � �矄 矄盐 泐矄⼄盐 ᨨ ᨨ ᆠ � ᨨ犨� � ᨨ盐� � ᨨ鼀�矄 ҈黨黨�和桌 㡰
㡰
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㡰
Fill in the Blanks
ᔼ 矄�� ᆠᨨ 盐ᆠ� �犨�ᆠᨨ 矄 ᆠ盐 �矄 ᨨ犨� � ᨨ鼀 ᨨ�� 盐ᨨ矄ᨨ盐ᆠᨨ 鼀 ␔ 矄ᨨ盐ᆠ �矄矄犨�ᆠ ᨨ �ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 � 犨�犨� ҈黨黨㜠҈㝐├矄桌
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Electromagnetic Induction
5 �
꾀ᆠᨨ�ᨨ 鼀犨��矄├矄�ᨨ 泐� ᨨ 㜠盐 � ᨨ鼀 鼀� �ᆠ � 꾀 鼀 �矄� 盐ᆠ � ᆠᨨ� ҈黨㜠㝐├矄桌 꾀 鼀 ᆠ � 盐ᆠ 犨 犨ᨨ 矄ᨨ 矄�� ᆠᨨ犨�犨 ��矄鼀ᨨ ᨨ盐� 盐 � ᨨ鼀鼀� �ᆠ � ҈黨桌 6
䰘
ᆠ㡰 �矄矄盐 泐矄ᨨ �鼀T ᆠ盐
True / False
MCQs with One or More than One Correct Answer
꾀ᆠ盐 ᆠ泐� �� � �泐ᨨ �犨��矄├ᨨ � �鼀ᨨ ᆠ�ᆠ盐犨�ᨨᨨ犨� ᨨ盐ᨨ盐ᆠ � �� ᆠᨨᆠ矄鼀 矄㡰�矄 㡰 ҈桌 㡰 鼀 � 盐ᆠ � ᨨᨨ�矄 ᨨ� ᆠ盐 �矄ᆠᨨ 矄 㡰 鼀 � 盐ᆠ � ᨨᨨ�矄 � �� ᆠ盐 �矄ᆠᨨ 矄 ᆠ㡰 鼀 � 盐ᆠ � ᨨᨨ �矄 ᨨ � ᆠ盐 ᆠ矄矄矄 泐 㡰 鼀 � 盐ᆠ � ᨨᨨ�矄� � �� ᆠ盐 ᆠ矄矄矄 泐 10 Subjective Problems
꾀鼀AB ᆠ 矄� ᨨ
犨ᨨ �ᆠ├ 鼀�ᆠ �矄矄 泐 �矄 ᆠ l 꾀矄�矄 ᆠ R
� 盐ᆠ ᆠL ᆠᨨ
ᆠ ᨨ �矄矄矄 ᨨ犨 � ��盐꾀ᨨ �矄�� 犨� ᆠ泐� �ᆠᨨ 矄 ᆠ盐 I�矄ᆠ � ᨨ �矄 �ᆠ盐ᨨ 鼀矄矄 ᨨ犨 AB �矄 矄 �矄 ᆠx ᨨ鼀 ᨨ �犨�꾀t㦨 ��矄鼀 ᨨ矄� ᨨ �矄犨泐 ᨨ鼀 犨�꾀 矄犨 ᨨᨨ犨� �⼄盐矄�ᨨ 矄 㜠�㝐├矄桌 㡰 � �ᨨ 鼀ᨨ ��
d� df 犨 ��矄 dt dt
� 4
ᨨ盐� 矄�矄 ᆠR �矄 ᨨ盐 ᨨ 1 �鼀T � 盐ᨨ
L � 鼀矄ᨨ ᨨ �� ⼄盐 ��矄 R
黨 꾀ᆠ �盐 鼀ABCD鼀 ᨨ 盐 � ᨨ鼀 鼀 犨� 矄 矄�� ᆠᨨ
ᆠ�ᨨ 犨 E F 鼀 ᨨ 矄鼀犨�矄矄 ᨨ犨 � ��AEFD �矄矄⼄盐ᨨ 矄� ҈鼀 EB 㦨FC 㦨 �鼀 �ᆠ�ᆠ盐��矄ᆠ �
矄 �泐� ᆠ矄� �盐 � ᨨ鼀鼀� �ᆠ � �ᆠ � ᨨ ᨨ ᨨ鼀ᨨ�
ᨨ ᆠ �ᨨ 鼀� �ᆠ � �矄 ҈ ⏌矄 矄�矄 ᆠ盐 � � ᨨ 犨��矄 ҈W ⏌ 鼀 � 鼀� �盐 矄 �ᆠ�ᨨ 矄ᨨ ᆠ盐 矄� 矄�鼀 矄 AE, BE EF ҈黨黨㜠�㝐├矄桌 � �ᆠᆠᨨ矄 ᆠᨨ�矄AB C ᆠ 犨� � 矄ᨨᨨ ᨨ ᔴᨨ�矄A C ᆠ泐 ⼄盐ᆠ盐 矄矄矄 ᨨ犨 � ��ᔴᨨ�矄B C � �
ᨨ矄��ᨨ ᆠᨨ�A�矄鼀ᨨ ᨨ犨 矄B犨� 盐 � ᨨ鼀 鼀ᨨ�ᨨ ᔼ矄 泐� 盐ᆠ ᆠ盐 � B 䁸ᔼ ᨨ �� 矄ᨨ 矄ᔼ 泐矄鼀├ �ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 �
��鼀 ҈黨㜠㝐├矄桌
҈ 꾀ᆠ盐 ᨨ鼀A ᨨB �矄� ᆠ矄� �� 鼀� �盐 ␔ �矄 �ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 � 泐 � ᨨᨨ矄 矄 ᨨ犨 � ��盐䁸 ҈黨䰘黨桌
ᆠ盐 � ᆠ�ᆠ盐� f�矄 盐 ᨨ 鼀� �ᆠ � 盐 ᨨ ᨨ � 犨� ᨨ盐� ᆠ�ᆠ盐� 㡰 ᔼ�矄ᨨ矄 �鼀t㦨T 鼀AB �矄 �矄 ᆠᨨ x ᨨ鼀 ᨨ �犨� 矄�矄 ᆠ Rᆠ�矄ᆠ盐 �҈졠� 矄矄�ᨨ ᨨ ᆠ � 矄 ᨨ犨 ᨨ盐� 矄�矄 ᆠR ᨨ鼀t㦨 ᨨt㦨T
Topic-2 : Motional and Static EMI and Application of EMI 1 ҈
MCQs with One Correct Answer
꾀 � � �泐ᨨ �矄�� 犨�ᆠ泐� �ᆠ盐 Iᨨ 矄� ᨨ ᆠ �盐ᨨᨨ ᆠᨨ 盐ᆠᨨC犨� 矄� � � ᆠᨨ
ᆠᨨᨨᆠ � 矄鼀 矄矄 ᨨ犨 � ��盐 ᆠᨨ
ᆠᨨ 矄 � l 矄�矄 ᆠ R ᔼ
矄� 矄ᨨ �� 犨� ᨨᆠ�泐v 矄�矄 ᆠᨨ ᆠᨨ 盐ᆠᨨ 矄 � 盐ᆠ ᆠ ᨨ ᨨᨨ
���� � 盐ᆠ ᆠ盐 � ᨨᨨ矄 盐 ᆠ�ᨨ
ᨨ 矄�ᨨ r 犨 ᆠᨨ
ᆠᨨ 矄�� 犨��矄 㝐� 和� ᔼᔼ㡰桌
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Physics
ᨨ 矄� ᨨ ᨨ � ᆠᨨ 盐ᆠ� �犨�ᨨᨨ
C I
鼀� �ᆠ � ᨨ ҈ ᔼ 矄� �矄盐 ᨨ鼀 �矄 ⼄盐���盐鼀 ᨨ矄��ᨨ 矄 盐鼀 ᨨ ᨨ矄ᆠ��ᨨ 矄� ᨨ鼀矄 ᨨ�矄鼀�盐 ᆠ矄矄泐 ᆠᨨᨨ e�矄ᔼ 鼀矄矄ᨨ 矄��矄��鼀矄 �矄 矄�矄 ᆠ҈W � � ����犨�ᆠᨨ矄ᨨ 㝐� 꾀�҈黨ᔼ桌
R
v
l
r
m Ivl m Ivl m Ivl m Ivl 㡰 ᆠ㡰 㡰 p Rr p Rr p Rr p Rr 꾀�� �矄ᆠ鼀 ᨨ 盐鼀� �盐鼀 ᨨ 鼀� �ᆠ鼀�㡰�矄 ├ ᨨ� ᨨ 泐 �矄 ᨨᨨᨨ盐�矄 �矄 矄 矄 ᨨ犨 � ��盐꾀矄ᨨ �鼀� �矄 �ᆠ泐 ᨨ� ᨨ �矄ᆠ犨泐 ᨨ鼀�矄 �矄졠 ᨨ� � 鼀� ᨨ盐 �矄ᨨ �矄ᆠ �矄ᆠ犨� ��盐�矄 矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ矄ᆠ㡰 꾀 桌
㡰
䰘
和
㡰 ᨨ � �ᆠ�ᨨ ᨨᨨ矄� ᨨ �ᆠ�ᨨ ᨨ 鼀� 矄鼀ᨨ�ᨨ
㡰 ᨨ� 矄鼀 �ᆠ�ᨨ 矄 �ᆠ�ᨨ ᨨ 鼀� 矄 鼀ᨨ�ᨨ
ᆠ㡰 ᨨᨨ �矄鼀盐犨�鼀� 盐 ᆠ � 㡰 ᨨᨨ盐�矄鼀盐犨�矄ᨨ犨泐�矄
��盐矄 ᨨ犨矄矄⼄盐ᨨᨨᨨ 矄� �ᆠ鼀犨 �ᆠ �矄 ᆠᨨ
ᆠ ᨨ 犨ᨨ├ᨨ 矄�矄 ᆠ矄 犨 ᨨ矄盐 �矄鼀ᨨ� �ᨨ犨 矄�� 犨� ᆠᨨ 矄 矄 ᨨ ҈ᆠ鼀矄 ҈ 꾀矄ᨨ鼀� 矄 ᨨ �矄� 盐 � ᨨ鼀鼀� �ᆠ � ᨨ ҈ �ᆠ盐 ᨨ ᨨ ᨨᨨ ᔼ 矄�矄 ᆠᨨ �矄҈䰘! ᆠ盐 � ᨨᨨ � 矄 犨�ᆠᨨ矄ᨨ 㝐� ҈꾀҈黨ᔼ桌
�
㡰 ᆠ矄矄泐 ᆠᨨᨨ 黨 꾀 � 矄盐� � � ᨨ ᨨ � l 矄� ᆠ � x 矄�泐r x㡰㦨 r ᨨ � ᨨ �矄ᨨ ᨨ盐
l �矄矄矄� � ᨨ盐� ᨨ��� x㦨㡰 �ᆠ盐 ᨨ ᨨ ᔼ ᨨ 鼀├矄 ᨨ�ᨨ 矄矄ᆠᨨ
�鼀� 鼀� �ᆠ鼀ᨨ鼀 ᨨ ᨨ �矄 㝐� ҈ ҈黨 ᔼ桌 p p
r l ᆠ㡰
r l 㡰 r l 꾀ᆠᨨᨨ ᨨ 鼀矄矄鼀矄� 矄盐 ��泐ᨨ 犨ᨨ矄鼀ᨨᨨ �矄犨� 矄�ᨨ ҈ 矄 �ᨨ q 犨� ᨨ� ᨨ 꾀 ᨨᨨ鼀�矄ᨨ� 泐 矄�矄 ᆠ �矄盐 � ᨨ鼀鼀� �ᆠ � ᨨ鼀ᨨ ᨨ �矄矄矄 ᨨ犨 � ��盐 鼀� 矄 ᨨ ᆠᨨᨨ �矄 㝐� 졠 � 꾀�҈�҈桌
㡰 p r l
黨
㡰 Ҙ꾀 㡰 ҈䰘Ҙ꾀 ᆠ㡰 ҈�Ҙ꾀 㡰 ҈҈�Ҙ꾀
ᨨ 盐鼀 ᨨ 盐 矄 ᆠᨨ矄矄㜠矄ᆠ�ᨨ � 矄ᨨ ᨨ� �矄 � ᨨ犨�矄 � �矄� 泐 盐矄� � 矄�ᨨ 犨 犨� � �矄 � 盐ᆠ ᆠ ᨨ 矄ᨨ ᨨ� 犨�ᨨᨨ�ᨨ ᨨ 㝐� 黨꾀�҈黨ᔼ桌 㡰 㡰 ᆠ㡰 ҈⏌ 㡰 ҈⏌ 꾀 � 矄�҈ᆠ鼀ᨨ ��矄ᨨ 矄 犨�ᨨ ���� 矄�矄 ᆠ ��矄ᆠᨨ
ᆠ ᨨ矄� �ᨨ 矄� �ᆠᨨ 矄 �鼀҈ 矄 ��盐㡰 矄矄鼀泐�矄├� 盐 � ᨨ鼀
㡰 ҈ 㡰 � ᆠ㡰 � 㡰 ҈ 꾀҈鼀ᨨ � ᨨ� ᨨ 犨� 矄 ᨨ鼀ᨨ 쟼矄ᨨ 和ᨨ盐 ␔矄ᔼ�矄 � �犨� 矄 ᨨ �鼀矄҈�� �矄ᨨ ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ 矄鼀� �ᆠ � ᨨ 맔҈␔⏌鼀 盐ᨨ � 盐ᆠ 鼀 �
犨��矄 㝐� ҈ ҈黨ᔼᔼ桌 㡰 ҈�맔҈ 㡰 ҈҈맔҈ ᆠ㡰 �맔҈ 㡰 맔҈ 꾀ᆠᨨ犨��矄犨ᨨ盐 ᨨ 犨ᨨᨨ 鼀犨 ᨨ矄矄 �矄 ᨨ ⼄盐�� �ᔼ � �鼀 矄�ᨨ ᨨ ᆠ 矄� ᨨ 鼀�矄� ᆠ矄 泐 ᆠᨨᨨ ├� � 盐鼀ᨨ 盐 矄ᨨ ᆠᨨ�盐 � � ᨨ 鼀 矄鼀 矄 � 盐ᆠ ᆠᨨ ᆠᨨ� 㝐� ҈҈ ҈黨ᔼᔼ桌 㡰 ᆠ矄矄泐 ᆠᨨᨨ 黨 㡰 � ᆠ矄矄泐 ᆠᨨᨨ 䰘 ᆠ㡰 � ᆠ矄矄泐 ᆠᨨᨨ
㡰
®
B
l R
㡰 ᆠ㡰
鼀� ᆠᨨ矄 q
l 鼀� q l
q
㡰 㡰
鼀� 矄� q l 鼀� ᆠᨨ q l
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P-223
Electromagnetic Induction
꾀矄⼄盐 鼀ᨨ 矄� ҈ᆠ鼀 ᨨ �矄�� 犨� ᆠ泐� �ᆠ盐 ҈꾀� ᨨ 和� � ᨨ鼀ᆠᨨ矄ᨨ 犨� 鼀鼀ᨨ矄ᨨ犨 矄 �� 犨� ᆠᨨ 矄 矄 ᨨ ҈鼀矄҈ 矄 ��盐㡰꾀 桌
҈
ᔼ㦨҈꾀
҈ᆠ鼀
鼀 � 盐ᆠ �鼀 鼀ᨨ 鼀�矄 㦨҈ᆠ鼀 ᨨ鼀 犨��矄 졠 � 꾀�҈黨҈桌 ᆠ㡰 䰘�m 㡰 �m 㡰 m 㡰 ҈m 꾀鼀�ᆠᨨ ᨨ � �l�矄� ᨨ矄� �ᨨ � l 鼀 ᨨᨨ犨� �盐矄 wᨨ ᨨ� ᨨ 犨� ᨨ ᨨ 矄� � � ᔼ �矄�ᆠ 鼀� �ᆠ � �� ��ᨨ 鼀 � 盐ᆠ ᆠᨨ矄矄 矄ᨨ ᨨ �矄 ҈桌
҈҈
㡰
Bwl
㡰
Bwl ᆠ㡰
Bwl 㡰
�Bwl
犨ᨨ ᆠ�ᆠ盐 ᆠᨨ�矄 ᆠ � � 泐 ᨨ 矄�盐�ᨨ 矄矄 ᨨ犨 � ��盐 �鼀盐盐� 盐ᆠ ᆠ 犨� ҈和桌
҈
㡰
҈
㡰
ᆠ㡰
㡰 鼀 �鼀盐鼀� 矄�盐�ᨨ 㡰 㡰 鼀 �鼀盐鼀� 矄�盐�ᨨ 㡰 ᆠ㡰 鼀 �鼀盐鼀� 矄�盐�ᨨ ᆠ㡰 㡰 矄鼀� 矄�盐�ᨨ 矄 꾀鼀�ᆠ矄⼄盐ᨨᨨABCD �矄鼀ᨨ� �� �矄ᨨ犨 犨� ᨨᆠ�泐v � 盐 � ᨨ鼀鼀� �ᆠ � �ᆠ盐 ᨨ�矄 矄矄 ᨨ犨 � ��盐꾀 ᆠ�ᆠ � �矄 � 盐ᆠ ҈和桌 꾀
ᔴ
㡰 � AD盐 ᨨ� BC 㡰 � BC盐 ᨨ� AD ᆠ㡰 � � AD ᨨ� BC 㡰 � ᨨ AD BC
҈ 꾀 ᆠᨨ� ᨨ � 盐ᆠ ᆠ 鼀 矄�矄 ᆠ W �矄 ᆠᨨ
ᆠ ᨨ҈泐 ᆠ盐 � ᆠᨨ��矄҈꾀 ᨨ �鼀泐 �鼀 ҈黨黨黨和㜠㝐├矄桌 㡰 �矄 㡰 �矄 ᆠ㡰 �鼀矄 㡰 ҈鼀矄 ҈� 꾀 � 矄鼀�㜠ᆠ�ᆠ盐 ᆠᨨ 盐ᆠ� � � �ᨨ �盐矄 �矄 � �犨� �矄 �ᆠ� ᨨ� ᨨ 鼀� �ᆠ ur � 盐ᆠ�ᨨ B 꾀 ᨨ矄��ᨨ MNQ 矄 ᨨ � ��矄v ᨨ � � ᆠ ᨨ ᆠᨨ矄矄 � ��矄 㡰 ᨨ ҈黨黨㜠㝐├矄桌 㡰 BvpR ⏌ M �矄 �� ᨨ � ᆠ㡰 pRBv Q �矄 �� ᨨ � 㡰 RBv Q �矄 �� ᨨ � ҈ 꾀鼀ᨨ 鼀ᨨ矄ᆠᨨ 矄 ᨨᆠ�泐� �ᆠ�ᨨ
�ᆠ盐ᨨ�矄 � 꾀ᆠᨨ 矄 盐 � ᨨ鼀鼀� �ᆠ � �矄矄� 矄ᆠ� �ᆠ�ᨨ �ᆠ盐ᨨ ᨨ 矄犨矄�矄ᨨᆠ�泐和ᆠ ᆠᨨᆠ矄鼀 ᨨ鼀 ᨨᨨ犨� � ҈黨黨和㜠㝐├矄桌 㡰 �ᨨ �矄 矄鼀ᆠ�ᆠᨨ � 㡰 �矄 ᆠ�ᆠ � � ᨨ ᆠ㡰 ᆠ�ᆠᨨ ��矄 �� 矄 ᆠ ᨨ ᨨ ᆠ矄矄ᨨ犨 矄�矄 矄 㡰 ᆠ�ᆠᨨ ��矄ᨨ犨矄 ᆠ ᨨ ᨨ � ᆠ矄矄ᨨ犨 矄�矄 矄 ҈䰘 꾀矄鼀矄⼄盐ᨨᨨᨨ 犨�ᨨ 矄� l�矄ᆠ � 矄� � 矄⼄盐ᨨᨨ ᨨ 犨� ᨨ 矄� L >>l㡰 ᨨᨨ矄 ᆠᨨ㜠 �ᆠ 矄ᆠᨨ� ᆠ� 鼀盐盐� 盐ᆠ ᆠ ᨨ 矄泐矄鼀�矄ᨨᨨ�ᨨ ᨨ ҈黨黨和㜠㝐├矄桌 㡰 l⏌L 㡰 l⏌L ᆠ㡰 L⏌l 㡰 L⏌l ҈ 꾀ᆠᨨ 盐ᆠ� �矄⼄盐ᨨᨨᨨ 矄� L 矄�矄 ᆠR 鼀ᨨ矄� �矄 犨� 盐 � ᨨ鼀 ᨨᆠ�泐 v �ᆠ盐ᨨᨨ ᨨ �矄矄� 矄꾀 鼀� �ᆠ� 盐ᆠ�ᨨ Bᆠᨨ 矄 �
�鼀 矄ᆠᨨ� � � �ᆠ盐 � ᨨ ᨨ ᨨᨨ �矄矄泐犨
ᆠ盐 � 盐ᆠ � ᨨᨨ�矄 ҈黨黨㜠㝐├矄桌 㡰 BLv/R ᆠᨨᆠ├犨�矄 㡰 BLv/R �ᆠᨨᆠ├犨�矄 ᆠ㡰 BLv/R �ᆠᨨᆠ├犨�矄 㡰 ᨨ 4
Fill in the Blanks
҈黨 꾀 盐 � ᨨ鼀泐犨ᨨ盐 矄ᨨ ᨨ� ᆠᨨ� ᨨ 矄 � 盐ᆠ ᆠ ҈맔҈ 泐 矄�矄 ᆠᨨ 鼀�矄ᨨ├ 盐� ᨨ 犨ᨨ� �ᆠᆠᨨ�矄 矄� �ᆠᆠᨨ�矄 ᆠᨨ
ᆠ � ᆠᨨ矄矄҈�㜠ᨨ泐ᨨ ����矄�矄 ᆠ
�鼀ᆠᨨ 矄 ᨨ ᆠ盐 � ᆠ�ᆠ盐��矄 矄ᆠᨨ 矄 矄 泐矄ᆠ盐 ᨨ盐� 泐 �矄鼀矄 ҈黨黨㜠㝐├矄桌
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EBD_8341 P-224
Physics
5
True / False
꾀ᆠᨨ 盐ᆠ� �ᨨ AB 鼀ᨨ矄ᨨ 㜠 �矄 矄 ��㡰 � 盐 � ᨨ鼀 鼀� �ᆠ � ᨨ� � � � ᨨ矄�� 㜠 �ᆠ�ᨨ A ᨨ ᨨ �矄ᨨ矄��泐ᆠ � ҈黨䰘㜠㝐├矄桌
盐泐ᆠᨨ矄矄矄 ��ᨨ ᨨᆠ�ᨨ ᨨ �� �ᨨ ᨨᨨ 㡰ᔼ 㡰 㡰矄 ᨨᆠ�泐ᨨ ᨨᨨᆠ盐 � ᨨᨨ ᨨᆠᨨ ᨨᨨ矄ᆠ�泐矄 盐 ᆠ�ᨨ ᨨ ᔴᨨ盐 㜠ᆠᨨᆠ├犨�矄 ᆠ盐 �矄├ 矄ᨨ矄�� 꾀 ҈桌
y Ӹ
B
v
A
␔ �ᆠ ᨨ ᨨᨨ犨� �矄ᆠ 鼀�ᆠᨨ 矄㡰�矄 㡰ᆠᨨᆠ䁸 ᔼ� ᨨ��泐㡰
ᔼ 㡰
x 㡰
6 ҈
MCQs with One or More than One Correct Answer
꾀ᆠᨨ 盐ᆠ� �犨�ᨨ ᨨ�ᆠ矄 � ��泐y㦨x�矄鼀ᨨ� � ur 犨� ᨨᆠ�泐 V = V $� � ᨨ 盐 � ᨨ鼀鼀� �ᆠ �
㡰
㡰
æ æ y öb ö ur B = B ç҈ + ç ÷ ÷ k$ 矄矄 ᨨ犨 � ��盐ᔼ VBL b ç è Lø ÷ è ø
ᨨ矄��ᆠᨨ 矄 矄 Df �矄 ᨨ � � ᆠ ᨨ 犨 矄ᨨ 犨� ᆠᨨᆠ 矄鼀 矄㡰�矄⏌ 꾀 ҈黨桌
㡰
ᔼ 㡰
ᆠ㡰
㡰
y
꾀ᆠᨨ 盐ᆠ� �ᨨᨨ� 矄 ᨨ �� � �矄ᨨ矄ᆠ矄 � �ᨨ �� ҈ᆠ鼀�矄├矄盐ᆠ 黨 �矄 泐ᆠᨨ矄ᨨ � � �泐ᨨ �ᆠᨨ 盐ᆠ� �犨� 矄 ��盐㡰 犨��矄ᆠ�ᆠ泐� 矄盐 ᨨ鼀 ᨨᨨ 泐ᨨ 盐矄ᨨ � ��矄ᨨ 犨� ᆠ盐 � � �盐ᨨᨨ�矄� ᆠᨨ盐 ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ � ᆠ矄 ᆠᨨ 矄 ᨨ ҈꾀矄҈ ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌
B
L
V = V�
L
x
㡰 Df �矄ᨨᨨ�ᨨ ᨨ � ᨨ 犨�ᨨᆠ ᨨ 泐㜠 �矄 㡰 Df 鼀� 矄 矄鼀� ᨨ�ᆠ犨��矄ᆠ
泐矄�� 犨�y㦨x� ��泐ᨨ � L ҈ ᆠ㡰 Df = BV L ᨨb㦨 㡰 Df 㦨 BV L ᨨb㦨 꾀��� 犨�ᨨᨨᨨ 矄⼄盐矄 � �矄� ᨨ � 矄�矄 ᆠ �矄鼀ᨨ� �ᨨ � 㜠 �矄犨� ᆠᨨ 矄 ᨨᆠ�泐� ᨨ 꾀㦨 �� �ᨨ ᨨᨨ 矄��ᨨ ᨨ � 犨 �矄盐 � ᨨ鼀鼀� �ᆠ � � ᨨ ᨨ 矄矄 ᨨ犨 � ��盐 ᨨ矄盐 �ᆠ� 泐� ᨨᨨ
҈ᆠ鼀
黨
æ m ö 㡰 鼀� �盐 ᨨ � 盐ᆠ emf� 犨��矄 çè ÷ø ᨨ p 㡰 ᔼ ᨨᨨ�矄ᨨ ᆠᨨ 矄 �盐矄 ᨨ盐 æ m ö 犨� ��ᨨ emfᨨ çè ÷ø ᨨ�矄� 盐ᆠ �
p 犨� ᆠ㡰 � 盐ᆠ ᆠ盐 � 犨��矄� ᨨᨨ矄� �ᆠ�ᨨ
ᨨ ᆠ盐 ᨨ � 泐ᨨ 盐矄 㡰 �矄盐矄� ᨨᆠ犨 犨� ᨨᨨ SI 盐 �ᨨ � 盐ᆠ ᆠ 泐ᆠ 犨� 矄 ҈黨黨和㜠㝐├矄桌 㡰 犨⏌鼀 㡰 ᨨ㜠矄ᆠᨨ ⏌鼀 ᆠ㡰 ᨨ盐⏌ 鼀㡰 㡰 ᨨ 鼀㜠矄ᆠᨨ
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P-225
Electromagnetic Induction
�
犨ᨨ � ᆠᨨ�矄 矄 㜠� 盐ᆠ ᆠ矄 L҈ = 鼀 L㦨鼀 ᆠ盐 � ᨨ ᆠᨨ��矄� ᆠ矄 ᆠᨨ 矄 ᆠ盐 � 矄ᆠᨨ ᆠᨨ��矄矄ᨨ� ᆠ矄 矄鼀ᆠᨨ 矄 꾀ᆠ� � 矄 ᨨ �鼀 ᨨ犨 �� ᨨ 犨ᨨᆠᨨ�矄�矄 矄鼀꾀 �鼀 ᆠ盐 � 盐ᆠ ᨨ� �泐矄ᨨ � �矄ᆠᨨ� �҈V҈ W҈矄ᆠ�泐ᔴᨨ矄ᨨ � �盐矄 ᨨ 矄ᆠᨨ ᆠᨨ� 矄鼀 � 矄 � V W 矄ᆠ�泐 ҈黨黨㜠㝐├矄桌 �҈ ҈ 㡰 � =
㡰
�҈ = �
ᆠ㡰
W҈ ҈ = W 㡰
V҈ = V
꾀鼀ᨨ OA ᨨ 鼀矄矄�m � �r�矄├ᨨ� � 犨� ᆠᨨ 矄 �盐矄 w � �ᆠ ᨨ盐 ᨨ� ᨨ �矄 O A�矄 � ᨨ 矄� 犨� ᨨ盐 �ᆠ�ᨨ ᨨ � � ᆠᨨ 盐ᆠ� �ᆠ�ᆠ盐� � � 矄鼀 矄 ᨨ ᨨ�ᨨ 꾀盐 � ᨨ鼀 ᆠᨨ 矄 r 鼀� �ᆠ� 盐ᆠ�ᨨ B �矄� �ᆠ盐 � ᨨ ᨨ ᨨ�ᨨ 矄矄 ᨨ犨 � ��盐ᨨ犨꾀 � 盐ᆠᨨ 矄�矄 ᆠR ᆠᨨ
ᆠ ᨨ盐� 矄犨�ᆠ S 犨 ᨨ� O ᨨ� C ᨨ � �ᨨ ᨨ鼀 ᆠ�ᆠᆠ�ᆠ盐��ᆠ 矄�矄 ᆠᨨ � � ᨨ ᔼ ��泐 矄犨�ᆠ �矄ᨨ ҈黨黨�㜠҈㝐├矄桌
10 Subjective Problems
䰘
꾀鼀� �ᆠ � B㦨B y⏌a㡰 kˆ �矄� ᨨ � 졔z �ᆠ�ᨨ B ᨨ矄��ᆠᨨ 矄 矄꾀矄⼄盐ᨨᨨ EFGH ᨨ 矄� 鼀矄矄m 矄�矄 ᆠR� x y 矄矄 � �盐 � 盐 ᆠᨨ ��泐矄 ��盐㡰ᨨ �ᆠ�ᨨ 矄ᨨ x y 矄� ��盐 ҈黨黨黨㜠҈㝐├矄桌
� 㡰 � 盐ᆠ ᆠ盐 � ᨨᨨ � �ᆠ�矄 �ᆠ�ᨨ 㡰 ᨨᨨ ᨨᆠᆠ� �ᨨ ᨨᨨ � �ᆠ �矄 �ᆠ�ᨨ ᆠ㡰 矄矄�ᨨ ᨨ 矄 ᨨ ᨨᨨ v t㡰 �矄 鼀� 盐 꾀�ᨨ ᨨ� ᨨ ᆠᨨ 盐ᆠ� ��矄ᨨ ���� 矄�矄 ᆠ 矄 ᨨ ᨨ �矄 � ᨨ �矄 ᆠ犨 �矄�矄L꾀ᆠᨨ 盐ᆠ� �鼀矄矄矄矄ᨨ ᨨ 矄�矄 ᆠRᆠ 矄� ᨨ �矄 �ᆠ�ᨨ 矄矄泐 ᨨ �矄� ᨨ 鼀矄矄矄矄矄� � 犨 �ᆠ 矄矄矄ᨨ 盐泐 � ᨨ �ᨨ 꾀鼀矄矄m� ᨨ ᨨ ᨨ 矄� � �矄�ᆠ泐꾀ᆠᨨ 矄 鼀� �ᆠ � B �矄矄 �ᆠ盐ᨨ ᔼ 矄泐矄鼀�矄 矄 ᨨ鼀矄ᆠᆠ盐 ҈黨黨䰘㜠�㝐├矄桌
�㡰 鼀� ᨨᆠ�泐ᆠ � 泐 ᨨ ��㡰 ᆠᆠ�ᨨ ᨨ 鼀矄矄 � 矄 犨 ᨨᆠ�泐ᨨ ᨨ �矄 鼀� ᨨᆠ�泐
㡰 ␔ �矄 � 盐ᆠ 鼀 ᆠᨨ矄矄 鼀� 矄ᨨ 矄犨�ᆠ 䁸 㡰 矄犨�ᆠ S �矄ᆠᨨ矄 �鼀t㦨 �㡰 졠� 矄矄�ᨨ ᨨ ᆠ盐 矄 盐 ᆠ�ᨨ
ᨨ �鼀 ��㡰 ᔼ 矄 泐矄ᨨ� �鼀 ᆠ ᨨ ᨨ⼄盐⼄盐� ᨨ鼀� � ᆠᨨ 矄 �盐矄 �� ᨨ OA 犨矄ᨨ � ᨨ矄��㜠 �矄t㦨 R҈ 黨 犨ᨨ�ᆠ鼀�ᆠ�矄AB A C CD 矄 泐҈鼀 泐 L ᆠᨨ
ᆠ 犨ᨨ 矄 泐 矄�矄 ᆠ矄R҈ R 矄 矄 ᨨ犨 �
��盐꾀 ᨨ� ᨨ 鼀�ᆠL ᨨ 鼀矄矄├�矄� 矄犨� ᨨ盐 �ᆠ�ᨨ
R �ᆠ泐 ᨨ犨 �矄盐 D ᆠ�ᨨ ᨨ ��泐 �矄盐 � ᨨ鼀 B ᨨ� ᨨ 鼀� �ᆠ � ᨨ 矄 �ᆠ盐ᨨ ᨨ �矄ᔼ�矄ᨨ矄 犨 鼀� ᨨᆠ�泐 �矄 � ᨨ犨矄 �矄矄� � R ҈ R 䰘 ␔ ҈ 犨 矄ᆠ�泐 � 鼀� ᨨᆠ�泐ᨨ L 盐矄ᨨ R҈ R ҈黨黨㜠㝐├矄桌 和ᆠ�矄 �� 泐 � AD � ᨨ犨ᨨ��ᨨ 矄 ��ᨨ
ᔼ�矄 � ��ᨨ ᔼᔼ 矄盐 � ᨨ鼀鼀� �ᆠ � B �ᆠ � ᨨ ᨨ ACD �矄 矄鼀�ᆠ�ᆠ盐ᆠᨨ 盐ᆠ� �ᨨᨨᨨ �盐矄r犨� ᆠ O ᨨ ᨨᨨ� �� ᨨ ᨨᨨ�矄 ᨨ犨鼀 ᨨᨨ犨� ᆠᨨ 矄 �盐ᨨᆠ�泐 wᨨ盐 �矄矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ ᆠ�矄�矄 ᆠᨨ ᨨᨨ�矄R ҈黨�㜠㝐├矄桌
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EBD_8341 P-226
҈
Physics
�㡰 졠� 矄矄�ᨨ ᨨ 鼀� �盐 ᨨ � 盐ᆠ ᆠ盐 � ᨨᨨ ��㡰 和 ᨨ犨 �ᆠ�ᨨ ᨨ ᆠ盐 犨 ᨨᨨ�矄 � �� ᨨ ��ᨨ ᔼᔼ ���㡰 ᨨ� 犨 � 盐ᆠ 鼀 �鼀 ᨨ ᨨ�ᨨ ᨨ犨ᨨ�ᨨ 矄ᨨ ᨨ�ᨨ 꾀矄⼄盐鼀犨�ᨨᨨᨨ 矄� ҈ᆠ鼀矄 矄�矄 ᆠ҈ ᨨ 鼀�矄鼀ᨨ 犨� ᆠᨨ 矄 ᨨᆠ�泐v0 � 盐 � ᨨ鼀 鼀� �ᆠ � ᨨ � 盐ᆠ�ᨨ B㦨犨矄⏌鼀矄矄 ᨨ犨 �
��盐 鼀� �ᆠ � � 矄 �ᆠ盐ᨨ
ᨨ ᨨᨨ �ᆠ � ᨨ 㡰 ᨨᨨ �矄 ᆠᨨ
ᆠ ᨨ 犨ᨨ├ ᨨ 矄�矄ᨨ矄 ᆠ ᨨ 盐 ᨨ 鼀矄 矄�矄 ᆠ矄ᨨ 犨�矄OS PQ ����␔ 矄 ᨨ盐 矄 ᨨ ᨨᨨ矄ᨨ矄ᨨ 矄 泐ᆠ盐 ᨨ ҈鼀��鼀�
ᨨᨨ䁸� �ᆠ�ᨨ ᨨ ᆠ盐 � ᨨᨨ ҈黨㜠㝐├矄桌 犨ᨨ�矄ᨨ �犨泐ᆠ├� 矄盐 ᨨ鼀ᆠ ᨨ �ᨨ盐 ᆠᨨ
ᆠ ᨨ鼀��ᨨ鼀␔ �矄 � �ᨨ 鼀��ᨨ鼀犨 � 矄 矄 ᨨ ҈├鼀⏌ ᨨ盐ᨨ � ᆠ├�� �ᆠ ᆠᨨ鼀ᨨ ᨨ 矄鼀� �ᆠ � �矄맔҈犨⏌鼀 �矄矄 泐҈鼀䁸 ҈黨҈㜠㝐├矄桌
Topic-3 : Miscellaneous (Mixed Concepts) Problems 1 ҈
MCQs with One Correct Answer
꾀 � � �泐ᨨ �ᆠ泐� �矄├ᨨ 盐 � ᨨ鼀 鼀� �ᆠ � B �ᆠ ᨨ �ᨨ矄�� 㜠 �矄 �ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 矄矄 ᨨ鼀 㜠 �矄犨� �和桌 㡰 ᨨ 㡰 �ᆠᨨᆠ├犨�矄ᨨ 졔 �矄 ᆠ㡰 ᆠᨨᆠ├犨�矄ᨨ 졔 �矄 㡰 ᨨ � 鼀� �ᆠ � 꾀矄 ᨨ㜠ᆠ�ᆠ盐� ᆠᨨ��矄ᆠ � �鼀㜠泐� �鼀� �ᆠ � 쟼ᆠ�ᆠ ᨨ犨 �矄 �矄矄� 盐ᨨ ᆠ盐 � 盐ᆠ � ᆠᨨ� ᔼ 盐鼀 ᨨ 盐 矄 犨ᨨ ⼄盐 盐 犨� �盐矄 ᆠ�ᆠᨨ犨 �矄矄� 犨ᨨ盐 和桌 㡰 㡰 矄鼀 ᆠ㡰 ᨨ盐 㡰 ⼄盐 盐 꾀矄矄 ᨨ犨 � ��盐P Q 犨ᨨᆠᨨ �ᆠᨨ 盐ᆠ� � ᨨᨨ矄矄 泐矄ᨨ鼀 �矄 ᆠ␔ 矄犨�ᆠ S�矄 ᆠᨨ矄 ᆠᨨᆠ├犨�矄ᆠ盐 IP ᨨ犨矄� P 矄矄 泐E㡰 � 盐ᆠ ᆠ盐 IQ1 ᨨ犨矄� Q 矄犨�ᆠ 鼀� 矄 ᆠᨨ矄 ᨨᨨ ��鼀␔ S�矄ᨨ ᆠ盐 IQ ᨨ犨矄� Q �ᆠ�ᨨ IQ҈ IQ 矄矄 泐E㡰 和桌
�
㡰 矄ᆠ�泐ᆠᨨᆠ├犨�矄 �㜠ᆠᨨᆠ├犨�矄 㡰 ᨨ ᆠᨨᆠ├犨�矄 ᆠ㡰 ᨨ �㜠ᆠᨨᆠ├犨�矄 㡰 矄ᆠ�泐 �㜠ᆠᨨᆠ├犨�矄 ᆠᨨᆠ├犨�矄 꾀 ᆠᨨ� ᨨ 犨� � � � 盐ᆠ ᆠ 矄�矄 ᆠ 矄 ᆠᨨ 盐ᆠ� �� � ᆠ ᆠᨨ �泐犨� � � ᆠᨨ� �矄 ᆠᨨ
ᆠ ᨨ泐�鼀t㦨矄ᨨ �鼀㜠 ᆠ盐 ҈ t㡰矄矄 ᨨ犨� � ᨨ盐� ᆠᨨ�ᔼ I t㡰�矄 ᆠ盐 � 盐ᆠ � � � B t㡰�矄 鼀� �ᆠ � �矄ᨨ ᆠᨨ� 盐ᨨI҈ t㡰 矄 盐 ᆠ�ᨨ ᨨ �鼀 t>㡰 ᨨ 盐ᆠI t㡰B t㡰 和桌 㡰 � ᆠ矄矄犨� �鼀 㡰 ᆠ矄矄犨� �鼀 ᆠ㡰 ᨨ矄 ᨨ泐犨� �鼀 㡰 矄矄矄 ᨨ盐� 鼀 �鼀盐鼀 꾀盐 � ᨨ鼀盐�鼀㜠泐� �鼀� �ᆠ � B t㡰 �矄矄� ᆠ�ᆠ盐��ᨨ ᨨ �盐矄 �矄 �ᆠ � ᨨ ᨨ 矄矄 ᨨ犨 鼀� �盐 ᨨ � 盐ᆠ ᆠ�ᆠ � ᨨ� �矄 ᆠ ᨨ鼀 ᆠ ᨨ ᆠ�ᆠ盐 ��ᨨ
和桌 㡰 �矄 ᨨ 㡰 ᆠ矄矄矄҈⏌r ᆠ㡰 � ᆠ矄矄矄r 㡰 ᆠ矄矄矄҈⏌r
犨ᨨ� �ᆠᆠ�ᆠ盐ᨨᨨ矄ᨨ 鼀犨�泐� �ᨨ 犨� ᨨ盐ᨨ盐ᆠ � �ᆠ ᨨ ᨨᨨ㜠꾀ᆠ�矄ᆠ盐 犨 �ᆠ � ᆠ矄矄犨� �鼀ᔼ 矄ᨨ 矄 ᨨᨨ㜠 㡰 鼀� 矄矄�ᨨ 泐 ҈黨黨黨和㜠㝐├矄桌 㡰 �矄ᆠ 泐 ᨨᨨ㜠꾀 ᆠ㡰 �矄 泐 ᨨᨨ㜠꾀 㡰 ᨨ矄ᨨ盐�矄ᔴ㝐犨� ᔴ㝐 �
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P-227
Electromagnetic Induction
6 䰘
MCQs with One or More than One Correct Answer
꾀矄ᨨ盐ᆠᨨ ᆠᨨ 矄 ᨨ��矄ᆠᨨ
ᆠ ᨨ矄�矄 ᆠ 犨ᨨ� � 盐ᆠᨨ矄҈ ᨨ盐� 矄犨�ᆠ 和矄 矄 ᨨ犨 �矄 ᨨ鼀盐盐� 盐ᆠ ᆠ犨 犨ᨨ � 盐ᆠᨨ矄 矄犨�ᆠ 和�矄� ��泐ᨨ 꾀㦨 矄犨�ᆠ �矄ᆠᨨ矄 ᆠ盐 �� 矄ᨨ ᨨ犨␔ �ᆠ ᨨ ᨨᨨ犨� � ᨨ�ᨨ 矄�矄⏌ᆠᨨᆠ䁸 꾀 ҈䰘桌 黨
和 졔
҈
㡰 꾀 ᨨ � �鼀 ᆠ盐 ᨨ盐� ҈ 犨� ҈ +
㡰 꾀 ᨨ ��鼀 ᆠ盐 ᨨ盐� 犨� ҈ ҈ +
ᆠ㡰 �ᨨᨨ ᆠ盐 矄 ᨨ盐� ҈ �矄 � �鼀矄 >㡰 꾀ᆠ�ᆠ盐� 矄盐 ᆠᨨ犨�ᨨᨨ�矄犨�矄 ᨨ ᨨ鼀 犨ᨨᨨᨨ矄ᨨ 꾀 꾀矄矄 ᨨ犨 � ��盐꾀 ᨨ� ᨨ ᆠᨨ矄矄� � 犨�矄鼀� ᆠ�ᆠ泐� 矄盐 ᨨ鼀ᆠ ᨨ �ᨨᨨ�矄� ᨨ
㡰 꾀㦨 ᆠ盐 ᨨ盐� 矄�矄 ᆠ �矄
®
㡰꾀盐 � ᨨ鼀鼀� �ᆠ � ᨨ� 矄� ᨨ ᨨ 꾀㦨 ᨨᨨ矄矄ᨨ� �ᨨ盐 ᆠᨨ鼀鼀ᨨ �鼀矄 �矄犨� ᆠᨨ 矄 �盐ᨨᆠ�泐 w� 鼀� �ᆠ � ␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄�矄⏌ ᆠᨨᆠ䁸 꾀 ҈䰘桌
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔꾀 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 꾀 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 w
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
㡰 鼀 � 盐ᆠ � ᨨᨨ�矄ᨨᨨ�ᨨ ᨨ 矄盐鼀ᨨ 矄ᨨ 犨ᨨᨨᨨ矄
㡰 鼀�盐 ᨨ 鼀 �鼀盐鼀 鼀 � 盐ᆠ 盐ᨨ ᨨ ᨨᨨ矄�矄⼄盐ᨨ 鼀�盐 ᨨ 鼀 �鼀盐鼀 鼀 � 盐ᆠ � 矄鼀ᨨᨨᨨ ᆠ㡰 鼀 � 盐ᆠ 盐 ᨨ ᨨ ᨨᨨ矄 �矄 ᨨᨨ�ᨨ ᨨᆠᨨ矄w 㡰 ᨨ ᆠ �ᨨ 盐 �矄鼀 �鼀盐鼀犨 ᨨ ᨨᨨ矄�矄 �ᆠ盐 ᨨ ᨨ 꾀�鼀t 㦨鼀� 꾀 B D � ᆠ�ᆠ盐�矄 ᨨ犨 � A ��盐�矄ᆠᨨ
ᆠ ᨨB泐 ├泐 � � �V C㦨m ᆠ盐 I t㡰㦨Iᆠᨨ矄 w㡰 犨� I㦨҈꾀 w㦨� 矄҈矄矄 ᨨ犨� ��
R㦨҈W
䰘p w ├泐�矄矄犨�ᆠ ᨨ鼀B ᨨDᨨ犨ᨨ 犨 矄ᨨ 泐A Dᆠᨨ
ᆠ 꾀ᨨᆠ �Q ᨨ犨矄 ᨨ鼀 泐 ᨨᆠ � ᆠᆠ�ᨨ 盐泐ᔼ C 㦨m 㦨҈W 泐�矄 � 犨� 鼀 ᨨ � � � 泐 ᆠᨨᆠ 矄鼀 矄㡰 꾀 ҈桌 㡰 㝐� �盐 ᨨ 鼀 �鼀盐鼀ᆠ �ᨨ ᆠᆠ�ᨨ 䰘p ᨨ t = �矄҈맔҈ᔴ w 㡰 ᆠ盐 � ᨨ ᆠ�ᆠ盐�盐矄 ᨨ 䰘p t= �矄ᆠᨨᆠ├犨�矄 w ᆠ㡰 ᔼ鼀鼀 �泐 A�矄ᆠᨨ
ᆠ ᨨD ᆠ盐 � R �矄҈A 㡰 Q㦨맔҈ᔴ ҈ LC R 矄 泐矄�ᆠ⼄盐 ��矄� 盐ᆠ ᆠ ᆠᆠ� ᆠ 矄�矄 ᆠ矄ᆠ�泐 ᆠᨨ鼀� �ᨨ 矄㡰 犨 �ᆠ �鼀 矄�ᨨ 矄ᨨ ⼄盐 ᆠ泐 ҈黨㜠㝐├矄桌 㡰 ҈⏌RC 㡰 R/L ᆠ㡰 ҈⏌ LC 㡰 C/L �犨� � �� �ᆠ�ᨨ 矄 ᨨ犨 � ��盐꾀 t =
8
Comprehension/Passage Based Questions 矄矄�
꾀ᨨ� ᆠ �턼�矄鼀ᨨ� �� ᆠ�ᆠ盐ᨨ�ᨨ �盐矄 � 㜠泐 犨� �盐ᨨᆠ�泐w �矄ᆠ ᆠᨨ 矄� 矄 턼w ⼄盐� ᨨᨨᨨᆠ泐� �矄 泐ᆠ盐 꾀盐 � ᨨ鼀 p 鼀� �ᆠ � ᨨ � ᨨ矄�� 㜠 �矄�矄 ᨨ犨矄犨�ᆠ ᨨ 犨 �ᆠ � ᆠ矄矄ᆠᨨ 矄 ᨨ鼀ᨨ� ᨨ 矄ᆠᨨ 꾀矄矄盐鼀 �盐矄 ᨨ ᨨ� 鼀� 矄 ᆠᨨ 矄 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � � 盐ᆠ矄 鼀 � ᨨ� � 盐ᆠ 鼀 �矄 � 矄 犨ᨨ├ ᨨ 泐 � 盐ᆠ ᆠ�ᆠ � � 鼀ᨨ� �盐 �ᨨ矄��ᆠ �ᨨ盐 ᆠᨨ矄 ᨨᨨᔼ�矄 ├ ᨨ犨 ᨨ ᨨ�� �ᆠ � 鼀� �ᆠ �ᨨ鼀ᨨ鼀 �矄ᨨᨨ�ᨨ ᨨ �盐鼀ᨨ鼀 盐鼀犨� ᨨᨨ�ᨨ �泐 ᆠᨨ 矄 g 꾀 ҈桌
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EBD_8341 P-228
҈҈
Physics
鼀� �盐 ᨨ � 盐ᆠ ᆠ�ᆠ � � ᨨ� 泐� 矄 ᨨ �鼀 盐� � �鼀� ᨨ 鼀� �ᆠ � ᆠ ��矄 㡰 ᆠ㡰 㡰
ᆠ �� 鼀� �ᆠ �ᨨ鼀ᨨ鼀 矄矄ᨨᆠ� 犨� ᨨ� ᨨ �鼀� ᨨ 鼀� �ᆠ � ᆠ ��矄
㡰
҈
㡰 g턼 ᆠ㡰 g 9 ҈
턼
턼
㡰
-g
㡰
g 턼
Assertion and Reason Type Questions
和鼀 㜠҈꾀�ᆠ�ᨨ ᨨ 矄ᆠᨨ�ᨨ 犨�犨ᨨ盐 ᨨ� ᨨᨨ鼀 꾀
� �ᆠ盐 ᨨ犨矄� ᆠᨨ� ᨨ �ᨨ矄 ᨨ盐� ᆠᨨ 盐ᆠ� � � � 矄 矄 ᨨ犨 � ��盐 � �ᆠ ᨨ ᆠ� �� ᨨ ᆠᨨ� 和鼀 㜠ᔼ ᨨ矄�盐�ᨨ ᆠ盐 �矄� 盐ᆠ � � �犨 �ᆠ � ᆠ矄犨� ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ 鼀� �ᆠ � ᨨᨨ 盐ᆠ
� ᨨᆠ� 盐犨 �ᆠ�ᨨ 䰘桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐 10 Subjective Problems
҈
꾀ᨨ �矄ᨨ ᨨ� ᨨ �盐矄 盐鼀ᨨ 盐 矄盐 � � �矄 ᆠᨨ矄 泐ᆠ泐� �ᆠ 矄 ᨨ �盐矄 R �ᆠ├ 矄矄d dR㡰 � LA�ᆠ盐 � 㦨� 矄� w ᨨ犨矄 ᨨ盐� ᆠᨨ�ᔼ 矄�矄��泐ᨨ 鼀� ᨨ ᆠ泐� �ᆠ矄 �矄r � � 盐ᆠ ᆠ盐 � 矄 �㜠㝐├矄桌
҈�
꾀矄⼄盐ᨨᨨᨨ 矄� �a犨� ᆠᆠ�ᨨᨨ ᆠᆠ� ᆠC �矄ᨨᆠ 犨 犨ᨨᆠ盐 ᆠ泐� �ᨨ �犨�矄 矄矄 ᨨ犨 盐ᨨ I� 犨�矄�矄�� 矄I㦨I 矄� wt 㜠㝐├矄桌
㡰 ᔴᆠ盐鼀 �鼀盐鼀ᆠ盐 � 矄⼄盐ᨨᨨ 㡰 犨� 犨 ᆠ �矄ᨨ 盐ᨨ ᆠᆠ�ᨨ矄�鼀 ҈ 꾀ᆠ �盐ᨨᨨPQRS 鼀 ᨨ鼀盐 � ᨨ鼀犨� 矄 � a犨� b 鼀矄矄鼀ᔼ�矄 ᨨᨨᨨ盐 鼀PQ犨 �ᆠ 鼀� 矄 � � ᨨ � ᨨ� ᨨ � ├
矄 泐㜠 �矄 矄 ��盐㡰 ├ �ᆠ泐 盐犨 �ᆠ�ᨨ 矄 㜠 �矄 꾀 盐 � ᨨ鼀 鼀� �ᆠ � ur B = �ˆ + kˆ㡰 B �矄矄� ��ᨨ ᨨᨨ�矄 �
㜠泐 ᆠ盐 ᔼ�矄矄矄 ᨨ盐� � ᨨᨨ�矄 ᨨ犨矄 �矄 ᨨ盐 ᨨ矄泐� ᨨ� ᨨ ᨨ矄��ᨨ � ⼄盐���盐鼀 㜠�㝐├矄桌 z
P
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a x
S
R b 㡰 ␔ �矄 �ᆠ�ᨨ ᨨ ᆠ盐 I � PQ䁸 㡰 � 鼀� �ᆠ ᨨᆠᨨ 鼀RS ᆠ㡰 � 矄矄�ᨨ ᨨI� 鼀矄ᨨ Bab m ҈䰘 犨ᨨᨨ � ᨨ� ᨨ �矄 �矄 ᆠd ᆠ � �矄�矄 ᆠ l 盐 � � ᨨ� ᨨ 泐矄�矄 ᆠR꾀 ᆠ泐ᆠᨨ 盐ᆠ� �ᨨ MN ᨨ 鼀矄矄鼀�矄 ᨨ矄� ᨨ � �矄犨� ᨨ盐 �ᆠ�ᨨ
矄 ��盐㡰 �矄盐 � ᨨ鼀鼀� �ᆠ � ᨨ � 盐ᆠ�ᨨ
B ᨨ鼀ᨨ ᨨ �ᆠ � ᨨ 꾀� ᨨᆠF�矄� ᨨ ᨨ MN 矄盐ᆠ 矄 ᨨ 鼀ᨨ矄ᆠᨨ 矄 ᆠ盐 ᨨ犨矄 ᨨ盐� R ҈黨㜠㝐├矄桌
�㡰 � ᨨᆠ�泐ᨨ ᨨ � ᨨᆠF 矄 盐 ᆠ�ᨨ ᨨ �矄 ᆠx ᨨ ᨨ ᨨ鼀R ��㡰 ␔ ᆠ�ᨨ ᨨ 犨ᨨ├ ᨨ 矄ᆠᨨ 泐F �矄 ᆠᨨ � ᨨ 䁸
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P-229
Electromagnetic Induction
Answer Key Topic-1 : Magnetic Flux, Faraday's and Lenz's Law ҈ 㡰
㡰
㡰
㡰
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㡰
䰘 㡰
㡰
黨 㡰
҈҈ 㡰
҈ 㡰
҈ 㡰
҈ 㡰
҈� 㡰
҈ 㡰
҈䰘 ᆠ㡰
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҈黨 㡰
҈ ҈�㡰 ҈
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҈䰘�㡰 ᨨ��
䰘 ᆠ㡰
㡰
� 矄 盐
҈ ᨨ 盐矄㡰 �㡰
Topic-2 : Motional and Static EMI and Application of EMI ҈ 㡰 ҈҈ 㡰 盐
㡰
㡰
҈ 㡰 ҈ 㡰 ҈ 㡰 㡰
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҈ 㡰 ҈� 㡰 ҈ 㡰 㡰 ᆠ 㡰 � ᆠ㡰
䰘 ᆠ㡰
ᆠ㡰
҈䰘 㡰
҈ 㡰
黨 㡰
҈ 㡰
҈黨 맔҈矄҈꾀
Topic-3 : Miscellaneous (Mixed Concepts) Problems ҈ 㡰
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黨 ᆠ 㡰 ҈ ᆠ㡰
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EBD_8341 P-230
Physics
Alternating Current
21
Topic-1 : Alternating Current, Voltage and Power 1 ҈
MCQs with One Correct Answer
꾀 � �ᨨ�v t㡰㦨矄� ҈Àtᨨ�矄� ᨨ盐泐矄�矄�ᨨ ᨨ �W �鼀├ ᨨ ᆠ盐 ᨨ�矄 ᨨ鼀 ᨨ ├盐ᨨ ├ 盐�矄 㝐� 꾀�҈黨ᔼ桌 㡰 �鼀矄 㡰 鼀矄 ᆠ㡰 䰘鼀矄 㡰 鼀矄 꾀矄鼀ᆠ�ᆠ盐ᨨᨨᨨ 犨�ᨨ �盐矄a�矄ᨨᆠ ᆠ ᨨ 鼀盐ᆠ �ᆠ�ᆠ盐犨�ᨨᨨᨨ �盐矄b 犨ᨨᨨᨨ矄� 矄鼀 ᨨ盐ᨨᨨᨨ �盐矄 bᆠ�矄 � �ᆠ盐 ᔼ㦨ᔼᨨᆠᨨ矄 w㡰 鼀 � 盐ᆠ � 矄鼀�
ᨨᨨ�矄 泐 㝐� 졠 � 꾀�҈䰘桌
㡰
pm ᔼᨨ w 矄� w㡰 㡰 pm ᔼᨨ w ᆠᨨ矄 w㡰
ᆠ㡰 pm ᔼᨨ
w 矄� w㡰
㡰
pm ᔼ ᨨ w ᆠᨨ矄 w㡰
꾀矄� 盐矄ᨨ� ᨨ� 㡰㦨҈矄� �㡰�矄� ᆠᨨ矄矄 盐� 盐ᆠ ᆠᨨ 㦨 ᆠ盐 ᨨ盐� ᆠᨨ��矄 㝐� 졠 � 꾀�҈҈桌 㡰 ҈ᆠᨨ矄 �㡰 㡰 ҈ᆠᨨ矄 �㡰 ᆠ㡰 ҈矄� �㡰 㡰 ҈矄� �㡰
Topic-2 : AC Circuit, LCR Circuit, Quality and Power Factor 1 ҈
꾀 꾀ᔴᆠ�ᆠ盐� 矄R㦨҈WC㦨m L㦨鼀 ᆠᨨ
ᆠ � 矄�矄 ⼄盐�泐 ᆠᨨᨨ ᆠ�ᆠ盐��矄 㝐� 和 ᔼ㡰桌 㡰 㡰 � ᆠ㡰 㡰 꾀矄�矄L㜠Rᆠ�ᆠ盐��矄ᆠᨨ
ᆠ ᨨ泐ᨨ 鼀 Vᔼ ᆠ�ᆠ盐��矄矄犨�ᆠ ᨨ t㦨 �鼀犨 �ᆠ æ ҈ö �泐矄ᨨ � � 盐ᆠᨨ ᆠ 矄 ç ÷ �鼀矄ᨨ �矄 è nø 鼀 �鼀盐鼀盐�矄 㝐� 和 ᔼᔼ㡰桌 L æ n ö ÷ 㡰 R ç è n -҈ ø L æ n ö ÷ ᆠ㡰 R ç è n +҈ø
m � 矄�矄 �鼀� 犨 �ᆠ 矄�矄 ᆠ ᆠᆠ�泐⏌ ᔴ㡰犨�� 泐҈ ᔴ 矄矄盐鼀 ᨨᨨ矄矄 ᨨ ᨨ 矄盐ᨨ盐 � �矄㡰�矄ᆠᨨ矄ᨨ
MCQs with One Correct Answer
L æ n +҈ö ç ÷ 㡰 R è n -҈ ø
L æ n -҈ ö ç 㡰 ÷ R è n ø 꾀䰘� 鼀矄㡰矄ᨨ盐ᆠ�矄ᆠᨨ
ᆠ ᨨ矄�矄 ᆠᨨ ҈W � 盐ᆠ ᆠᨨ ҈ ᆠᆠ� ᆠᨨ ҈
㝐� 和 ᔼ㡰桌
�
㡰 ҈矄 㡰 �矄 ᆠ㡰 �矄 㡰 矄 꾀 � 盐ᆠ ᆠᆠᨨ� 矄ᆠ ᆠᨨ ҈W␔ 꾀ᔴ 矄�� ᨨ ⼄盐 ᆠ泐҈ �矄� ᨨ ᆠᨨ� � ᨨ� 矄 ᆠ盐 泐� 矄 㜠� 盐ᆠ ᆠᨨ ᆠᨨ��矄 㝐� 和 ᔼᔼ㡰桌 㡰 ҈҈맔҈
㡰 ҈҈맔҈҈
ᆠ㡰 ��맔҈�
㡰 䰘맔҈䰘
ᔴᨨ 矄� ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐ᔼ 矄犨�ᆠ 和�矄ᆠᨨ矄 㦨 鼀ᨨ盐 ᨨ ᆠ � 矄矄矄 L ᨨ盐� 泐犨 㦨 t = �矄 R 㝐� ҈꾀�҈黨ᔼᔼ桌
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P-231
Alternating Current
黨
҈㡰ᨨ矄 矄 � ᆠ犨 ᔼ҈ ᔼ�矄 㝐� ҈ ҈黨ᔼᔼ桌 㡰 㡰 ᆠ㡰 黨 㡰 ᔼ ᆠ�ᆠ盐�矄 ᨨ犨
㡰
ᆠ㡰
䰘EL R
EL
㡰
䰘EL
䰘 R
和
EL
㡰
R
和҈
䰘R
e
꾀 泐 � 盐ᆠᨨ ᆠᨨ� �矄 ᆠᨨ
ᆠ ᨨ ҈ ᨨ 鼀 矄�矄 ᆠ� 矄�矄矄矄 ᨨ犨 � ��盐 �鼀犨 �ᆠ ᨨ �矄矄��ᨨ ᨨ �泐 ᨨ盐矄 㡰 ᆠᨨ矄矄 矄�矄 ᆠ�矄⼄盐ᨨ 犨 �ᆠ 鼀� �ᆠ �泐�矄 矄ᨨ � � 盐ᆠᨨ�矄 㝐� 꾀�҈黨ᔼ桌
矄犨�ᆠ 和҈�矄ᆠᨨ矄 �鼀㦨 矄犨�ᆠ 和�矄 ├ᨨ 꾀矄ᨨ鼀�鼀 㡰 矄犨�ᆠ 和҈�矄ᨨ 和�矄ᆠᨨ矄 �ᨨ盐ᨨ ᆠ盐 ᔼ矄 盐 ᆠ�ᨨ
ᨨ �鼀��矄�� 泐 㝐� ҈҈ ҈黨ᔼᔼ桌 ᔼ
㡰
㡰 ln
䰘
ᔼ
҈ 㡰 ln ᆠ㡰 ln 㡰 ln ᔼ ��盐 矄 ᨨ犨 ᆠ�ᆠ盐� ᆠᨨ � 矄 犨ᨨ � �ᆠ 矄�矄ᨨ矄犨� 矄�矄 ᆠ 㦨�W � 盐ᆠ ᆠ犨� 㦨鼀꾀 � 泐ᨨ ҈��矄ᆠᨨ
ᆠ � ᆠ�ᆠ盐� ␔ 犨� ᆠ盐 ᨨ盐� 泐ᨨ � 矄犨�ᆠ �矄ᆠᨨ矄 䁸 㝐� ҈ ҈黨ᔼ桌
㡰 졠
졠
ᔼ
ᔼ
㡰
ᆠ㡰 졠
졠
和
҈ ᔼ ᔴᆠ�ᆠ盐� � 盐ᆠ ᆠ㦨鼀 ᆠᆠ� ᆠᔴ = 100 m ᔼ ᨨ� t㡰㦨҈矄� ҈t㡰�矄� ᨨ ᆠ�ᆠ盐� ᆠ盐 � ᆠ�ᆠ盐��矄�� 矄
㝐� 黨 ҈黨桌
㡰 ��꾀
㡰 �ᆠᨨ矄҈ ᆠ㡰 �ᆠᨨ矄҈
㡰 䰘�꾀
ᆠ㡰 꾀
㡰 꾀
㡰 ҈ᆠᨨ矄҈ 㡰 �矄� ҈
҈҈
ᔼ
҈
ᔴ
ᔼ҈
~
Ҙ 㦨W㦨 ҈ ҈㦨҈Wᔴ盐 � 㜠 ҈ �矄ᔼ҈ � ᔴ㜠 ��矄
ᔼ ᨨᆠ�ᆠ盐�ᔴ㦨
ᔼ ᨨ�ᨨ 꾀ᔴ矄ᨨ盐ᆠ�矄�� 泐㦨 矄�
꾀矄矄 ᨨ犨 � ��盐泐ᨨ 鼀 Î�矄ᆠᨨ
ᆠ ᨨ
� 盐ᆠᨨL 矄�矄 ᆠR� 矄�矄 矄犨�ᆠ �矄ᆠᨨ矄 t㦨 ᨨᆠ � ᨨ犨矄 ᨨ鼀 泐犨 t 㦨 t㦨tc tc�矄 �鼀ᆠᨨ 矄 ᨨ ᆠ�ᆠ盐�㡰�矄 㝐� ҈黨ᔼᔼ桌 Î L æ ҈ö ÎR 㡰 㡰 ç҈ - ÷ R è e ø eL ᆠ㡰
ÎL R
㡰
ÎR eL
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EBD_8341 P-232
҈
҈
҈
҈�
Physics
꾀ᔴ ᆠ�ᆠ盐� 矄�├ 鼀 鼀ᨨ �ᆠᨨ矄ᆠ�ᨨ ᔴᨨ鼀� � � 犨� 泐矄�ᆠ 矄� �㜠鼀矄矄 鼀 ᨨ矄ᆠ�ᨨ � � 鼀� � ᆠᨨ 矄 �b ᆠᨨᆠ ⼄盐� ᆠ犨ᨨ盐 㝐� 䰘 ҈黨ᔼ桌 㡰 L«mC«kR«b ҈ ҈ ҈ 㡰 L« C« R« b m k ᆠ㡰 L«kC«bR«m ҈ 㡰 L«mC« R«b k 꾀 鼀 ᨨ �矄� �鼀t㦨ᨨᆠ�ᆠ盐�ᆠᨨ � � � � 矄�矄҈鼀� 盐ᆠᨨ �W矄�矄ᨨ �ᨨᨨ ᆠ盐 矄�鼀t 㦨¥ t㦨矄�矄ᆠᨨ矄ᨨ
҈ 꾀 � 盐ᆠᨨ 㦨㡰 矄�矄ᨨ 㦨҈�├W㡰 ᆠᨨ
ᆠ � 矄�矄ᨨ泐ᨨ ҈�鼀 � ᆠ�ᆠ盐� 矄 ᨨ犨 ᨨ犨 ├泐⧤҈ 矄 ├ᆠᨨ矄 ᨨᨨ � �鼀 㦨⧤҈ �矄ᨨ ├泐⧤�矄 ᆠᨨ矄 矄�鼀盐 ᨨ盐矄泐꾀㦨鼀矄 ᆠ盐 � ᆠ�ᆠ盐�犨�
├e㦨䰘黨㡰
㡰 䰘鼀꾀 㡰 䰘鼀꾀 ᆠ㡰 ҈鼀꾀 㡰 䰘鼀꾀 ҈黨 꾀 ᔴ ᆠ�ᆠ盐��矄⼄盐� ᨨ 鼀 盐盐鼀ᔼ
ᔴ ᆠ�ᆠ盐� ᆠᆠ�ᨨ �矄 ᆠ � ᨨ 턼
ᆠᨨ
ᆠ ᨨ 矄矄 ᨨ犨 ᨨ犨
㝐� 䰘 ҈黨ᔼᔼ桌
㡰 ҈ 㡰 ҈҈� ᆠ㡰 ҈ 㡰 ᔼ ᆠᆠ�ᆠ盐� � 矄 ᨨ盐矄鼀 ᆠ盐 �� 泐 e㦨҈矄� t
pö æ �㦨矄� ç - ÷ ø è ᔼ ᨨ ᆠ泐ᆠᨨ ᆠ �ᨨ犨ᆠᨨ 矄盐鼀 泐 ᆠ�ᆠ盐� 犨矄矄ᆠ盐 矄ᆠ�泐 㝐� ҈桌 ҈ 㡰 �␔҈꾀 㡰 ␔҈꾀 � ␔ ᆠ㡰 㡰 �␔ ᨨ ᔴᆠ�ᆠ盐� � 犨� ᨨ�ᨨ 鼀�盐 v鼀 ⼄盐 ᆠ泐w㦨
҈
ᆠ盐 ��矄矄ᨨ ᆠ ᔴ ⼄盐�泐 ᆠᨨQ�矄�� 泐 㝐� ҈桌
ᔴ w w 㡰 ᆠ㡰 㡰 w wᔴ㡰 꾀矄� 盐矄ᨨ� ᨨ�ᨨ ├盐 �盐 ⼄盐 ᆠ泐⏌矄�矄� ᨨ矄�矄ᔴ ᆠ�ᆠ盐��
㦨�W㦨�鼀 ᔴ㦨҈m ᨨ�鼀 ᆠ 矄 � ᆠ 犨 ᨨ� ᆠᨨ矄矄 矄ᨨ盐ᆠ ᆠ盐 犨�矄ᆠ�泐 㝐� 졠 � 꾀�黨҈䰘桌
( � @ ҈� )
҈�桌
⧤
� 㡰 ҈W ҈ æç ö÷ 㡰 䰘W � è ø
æö � ᆠ㡰 ҈W ҈ ç ÷ 㡰 䰘W ҈ æç ö÷ èø è ø ҈䰘
꾀 ᆠ鼀⼄盐�矄 �ᆠᆠ盐 ᨨ ҈꾀ᨨ 盐 ᆠ�ᨨ ᔼ ��矄ᆠᨨ
ᆠ ᨨ 鼀矄㡰� 꾀ᔴ 矄盐泐 矄�矄� 盐ᆠᨨ ᨨ�ᨨ犨ᨨ├�矄ᆠᨨ矄ᨨ 㝐� ҈桌 㡰 㡰 � ᆠ㡰 㡰
⧤҈
҈�
ᔴ ᔼ 矄盐 ᨨ矄� 矄ᨨ 矄⼄盐ᨨ 鼀 �鼀盐鼀ᆠ � 턼 㝐 ᨨ ᆠᆠ�ᨨ犨� �鼀 㡰 ᨨ犨ᨨ � 盐矄҈ ҈>㡰ᨨ 犨 �ᆠ ᨨ ᨨᨨ犨� � 矄 矄 �矄� ᆠᨨᆠ泐䁸 plᨨts are schemat�c and nᨨt drawn tᨨ scale㡰 㝐� ҈�桌
(
)
҈
㡰
턼㝐
㡰
턼㝐
턼 ᨨᨨ ҈ 㡰
t
t
㡰
҈
҈�├W
ᆠ㡰
턼㝐
҈
턼 㡰 㝐 ҈
t
t
ᨨ ᔴ ᆠ�ᆠ盐�矄 ᨨ犨 ᆠ盐 �矄ᨨ矄 ᨨ � ᨨ�꾀 ��ᨨ ᆠᆠ�ᨨᔴ 犨 ᨨ� 犨� ᆠᆠ�ᨨᔴ矄 � ᆠ�ᆠ盐� 鼀├矄 ᨨ犨 ᆠᨨ ᨨ ᆠ�ᆠ盐� 盐 �泐 ᆠᆠ�ᨨ ᔴ 鼀盐矄 ᆠᨨ
ᆠ � 㝐� 졠 � 꾀�҈҈ ҈�桌
ᔴ
~ 㦨矄� w
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P-233
Alternating Current
C
㡰 矄�矄犨� ᔴ 矄鼀� �盐
(w2 LC –1)
㡰 矄�矄犨� ᔴ 矄 鼀� �盐
1 - w2 LC w2 L
ᆠ㡰 犨� ᔴ 矄鼀� �盐
҈
1 - w2 LC w2 L
C 㡰 犨� ᔴ 矄鼀� �盐 2 (w LC - 1) ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ� �ᔴ�矄├ᆠᨨ
ᆠ ᨨᨨ� �꾀� ᆠ盐 ᨨ犨� � ᨨ盐� ᆠ�ᆠ盐� ᆠᨨ鼀矄 ᆠᨨ 矄 꾀 犨 矄盐 泐 ᨨ� �ᔴ�矄 �矄ᆠᨨ
ᆠ ᨨ鼀ᨨ� �꾀 ᆠᨨ
ᆠ ᨨᨨ� � �鼀㦨 �ᨨᨨ ᨨ�ᆠᨨ矄矄矄�矄 ᆠ � 盐ᆠᨨ㦨⏌ 犨�⼄盐ᨨ 㝐� ҈桌 꾀
ᔴ
㡰
҈-
㡰 ҈
ᆠ㡰 ҈
㡰
҈-
␔ 鼀矄 ᨨ�矄 ᔴ 鼀矄盐 矄ᆠ�泐ᆠᨨ矄矄 � 盐ᆠᨨ ᆠᆠ�ᨨᔴ 矄�矄ᨨ � 矄�矄ᔴ ᆠ�ᆠ盐�ᆠᨨ
ᆠ ᨨ 꾀ᔴ矄ᨨ盐ᆠ ��矄 ᨨ盐 �ᨨᔴ 㦨҈ᔼ 鼀矄 ᨨ�ᨨ 꾀ᔴ矄ᨨ盐ᆠ矄�矄҈ �矄ᆠᨨ矄ᨨ 㝐� 졠 � 꾀�黨҈桌 㡰 � 㡰 䰘 ᆠ㡰 黨 㡰 ҈ ᔼ ᔴ ᆠ�ᆠ盐�矄矄 ᨨ犨 ᨨ犨ᨨ 矄犨�ᆠ 矄ᨨ
� ��泐ᨨ犨矄犨�ᆠ 和҈�矄ᆠᨨ矄 和├ᨨ ⼄�矄ᆠ � ᨨ ᆠᆠ�ᨨ t㦨 ᔴ�矄ᔴᆠ���鼀ᆠᨨ 矄 㡰 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 �矄ᆠᨨᆠ䁸㝐� ҈桌
꾀 矄�矄 ᆠ�ᆠ盐� �矄 ᆠᨨ
ᆠ ᨨ ᆠ 矄ᨨ盐ᆠ ᨨ ⼄盐 ᆠ泐w � 盐ᆠ�ᆠ ᆠ�矄⼄盐ᨨ 꾀 ᆠᆠ� ᆠᨨ ᆠᆠ��ᆠ ᆠ⼄盐ᨨ �矄 �
矄�矄犨� �ᨨᨨ 犨ᨨ犨 ᆠᨨᨨ ᨨ ᨨ �矄 㝐� 졠 � 꾀��҈桌 㡰
�
㡰
ᆠ㡰
㡰
�
� ᔼ 矄�矄㜠ᔴ㜠 ᆠ�ᆠ盐�ᔴ㦨҈҈҈ 㦨҈� 泐 㦨҈졠 鼀犨 ᆠᨨ 矄 ᔴᨨ�쟼�矄� ᨨ ᆠ�ᆠ盐� ᆠᆠ�ᨨᆠ⼄盐�矄ᆠ �҈ 黨ᔴ ᔴ矄ᨨ盐ᆠ�矄ᆠ 泐矄� 盐矄ᨨ� ᨨ�矄ᨨ盐ᆠ�
犨 �ᆠ ├ᨨ�쟼�矄⼄盐ᨨ ᆠᨨ 矄 ᔴ ᨨ� 쟼 꾀 矄ᨨ ᆠ ├ 盐 ᨨ ᆠ � ᆠ⼄盐� 泐 ᆠᆠ�ᨨ犨� 㝐� 졠 � 꾀�҈桌 㡰 ҈҈�ᔴ 㡰 ҈ᔴ ᆠ㡰 ҈҈ᔴ 㡰 ҈ᔴ 꾀 꾀ᔴ ᨨ� 矄ᨨ盐ᆠᨨ � �盐 ⼄盐 ᆠ泐w � 鼀�盐 V �矄 ᆠᨨ
ᆠ � 矄�矄 犨� ᆠᆠ� ᆠ C ᆠ�ᆠ 盐 ᨨ 矄�矄 ᆠ R � 盐ᆠ ᆠ ᨨ㡰␔ w�矄� ᆠ矄 ҈桌 㡰 盐�ᨨ犨矄 �鼀鼀 㡰 盐�ᨨ犨矄�� ᆠ㡰 ᨨ�鼀 ᆠᨨ ᆠ�ᆠ盐��矄盐 ᆠ � 㡰 ᨨ�鼀 ᆠᨨ ᆠ�ᆠ盐�� ᆠ矄矄 䰘 � �鼀ᆠᨨ 矄 � m矄㡰 ᨨ �� RC ᆠ�ᆠ盐�矄� �� ᨨ 矄ᆠ�泐 㜠㝐҈桌
҈
ᔴ҈
ᔴ
和҈ ᔴ
和
㡰 ␔ᨨ├ ᨨ 泐 泐 �矄 ᨨ �泐 �矄矄� � 矄�矄ᨨ 㡰 꾀㦨t⼄㦨ᔴ⏌ ᆠ㡰 At, t = 2t⼄㦨ᔴ ҈㡰 㡰 꾀㦨t⼄㦨ᔴ ҈҈㡰
R҈ = ҈W R = W C҈ = mF C = mF 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ 黨 黨 黨 黨 ␔ AC 矄ᨨ盐ᆠᨨ 鼀 e㦨E矄� ҈t㡰�矄ᆠᨨ
ᆠ ᆠᨨ矄矄ᆠ�ᆠ盐� 矄 � ᆠ犨 鼀 e ᆠ盐 �� ᆠ�ᆠ盐��矄ᨨ矄 ᨨ p 4 矄 矄 ᨨ犨 � ��鼀ᔼ ᆠ�ᆠ盐�ᆠᨨ 矄�矄矄ᨨ矄矄�泐ᨨ 泐 ᨨ R㜠CᨨR㜠LᨨL㜠C � 矄�矄 � �ᨨ 矄 �犨
和桌 犨ᨨ鼀 矄
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EBD_8341 P-234
Physics
㡰 R 㦨҈k W C 㦨҈ҘF
�
ᆠ�ᆠ盐�ᨨ �矄 矄鼀鼀� �盐 ᨨ ᆠᨨ鼀ᨨ� ᨨᆠ 鼀 㡰 ᨨ��矄矄� �ᆠ � ��盐 ᔼ� ᨨ 鼀盐盐 � 盐ᆠ ᆠ 犨 � 盐ᆠᨨ矄 ᆠ盐 I� 鼀� 犨�ᆠ 矄�矄 鼀 �鼀盐鼀鼀� �盐 I鼀 �鼀t㦨 t ␔ �ᆠ ᨨ 꾀 ҈桌 ᨨᨨ犨� �矄鼀 矄�矄 㡰盐䁸
㡰 R 㦨҈k W C 㦨҈ҘF ᆠ㡰 R 㦨҈k W L㦨҈H 㡰 R㦨҈k W L㦨҈H 2 黨
犨ᨨ� 盐ᆠᨨ矄҈ � 盐ᆠ ᆠ҈鼀� 矄�矄 ᆠ W㡰 � 盐ᆠ ᆠ鼀� 矄�矄 ᆠW㡰 矄�矄ᨨ 矄�矄 ᆠ҈W㡰ᆠᨨ
ᆠ � ᆠᨨ矄矄�泐 ᆠ�ᆠ盐��矄矄犨�ᆠ ᨨ �鼀㦨 �ᨨ ᨨ 鼀 �鼀盐鼀ᨨ 鼀� �鼀盐鼀 ᆠ盐 ᔼ鼀 ⏌ᔼ鼀� 㡰 犨 ᨨ鼀 泐�矄 꾀 ҈桌 3
꾀ᨨ ᆠᨨ鼀ᆠ�ᆠ盐��矄矄 ᨨ犨 � ��盐꾀矄ᨨ鼀 � 矄 盐ᨨ ᆠ盐 ᔼ�矄҈꾀 ��矄 ᆠ矄� � rateof 10꾀矄҈ 盐ᨨ ᨨ � � ᆠ턼 � ᨨ矄㡰 � 矄 �矄힘힘힘힘힘힘 㝐� 和 ᔼ㡰桌
V
㦨W
V R
㡰 I鼀 㦨
L L 㡰 t㦨 R R ᔼ ᆠ�ᆠ盐�矄 ᨨ犨 㦨҈mᔴ㦨҈m 㦨҈├W 泐 ᆠᨨ
ᆠ � 矄�矄犨� ᆠ矄ᨨ盐ᆠ㦨矄� w矄 矄 ᨨ犨 ␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄�矄⏌ᆠᨨᆠ䁸 꾀 ҈䰘桌
~ 矄� w 㡰
㝐� 和 ᔼᔼ㡰桌
� 盐ᆠᨨ矄ᨨ 犨ᨨLRᆠ�ᆠ盐�矄ᆠ ᨨᆠ ᨨ 矄 矄 ᨨ犨 � ��盐 盐矄 ᨨ 矄 㜠 � 盐ᆠ ᆠ ᨨ � 盐ᆠᨨ矄 矄�矄 ᆠ矄 鼀盐盐㜠 � 盐ᆠ ᆠ � ᨨ�矄矄ᆠ� � � ��
ᆠ�ᆠ盐�꾀 ᨨ 矄犨�ᆠ 矄ᆠᨨ矄 矄�鼀盐 ᨨ盐矄泐 ᨨ犨ᨨ├ ᨨ 泐 �矄�� 矄 � 盐ᆠ EMF� � 盐ᆠᨨ矄泐 �鼀 ᆠ盐 矄ᆠ � 矄 泐矄盐矄�矄힘힘힘힘힘힘힘힘鼀 꾀 桌
ᆠ盐 犨� � 矄 犨� ᨨ� � w㦨҈ 矄㜠҈ 㡰 ⼄盐 ᆠ泐犨 �ᆠ ᆠ盐 犨�� 矄 犨� ᨨ��矄� ᨨ ᆠ㡰 꾀 w ~ ᆠ盐 ᨨ犨� � ᨨ盐� ᆠ�ᆠ盐� ᆠᨨ鼀矄 泐 ᨨ 㡰 꾀 w >> ҈ 矄㜠҈ ᆠ�ᆠ盐� 矄 �├ ᆠᆠ�ᨨ � ᔼ �� ᆠ�ᆠ盐� 꾀ᔴ 矄ᨨ盐ᆠ 矄 w 㦨 ҈ ⏌矄 ᔴᨨ 矄� � � � 盐ᆠᨨ ᆠᆠ�ᨨᨨ� ᆠᨨᆠᆠ ᨨ�ᆠ 矄㡰�矄 㡰 ҈桌 ҈00 m
V 㦨
L 㦨鼀
M 㦨�鼀
L҈ 㦨҈鼀
V҈ 㦨�
҈00W
R 㦨҈W
0.5 H
I҈
V R
= ҈m ᔴ = ҈m = ҈├W
æ ö çè ÷ø m 盐ᨨ �矄힘힘힘힘힘힘 p
R҈ 㦨�W
S
ᆠ㡰 t㦨
҈ ᔼ 矄�矄 ᆠ�ᆠ盐�ᨨ犨ᨨ ␔�矄 �矄矄� ᨨ鼀 矄ᨨ盐ᆠᨨ �� ᨨ犨 ᆠᨨᨨ ᆠ�ᆠ盐��矄ᔼ
ᨨ ᨨ� � ᨨ犨 ᆠᨨᨨ盐 �泐ᆠᆠ�ᨨᨨ 盐ᔴ �矄 � 矄�矄ᨨ ├� � 盐ᔴ 矄
L
V
S҈
턼
R
I
㡰 I鼀 㦨
Numeric Answer
㦨�鼀 ᔼ
L
R
Integer Value Answer
I
~
50 W
0 V
㡰 ᆠ盐 ᨨ盐� ᆠ�ᆠ盐�I�矄꾀 㡰 ᆠ盐 ᨨ盐� ᆠ�ᆠ盐�I�矄 A
6
MCQs with One or More than One Correct Answer
ᔼ ��盐 ᨨ犨 矄犨�ᆠ 矄S҈ S ᆠᨨ矄 矄�鼀盐 ᨨ盐矄泐t㦨 ᆠ盐 矄矄ᨨ ᨨ犨�
ᆠ㡰 ᨨ�ᆠᨨ矄矄҈W 矄�矄ᨨ㦨 ҈ V 㡰 ᨨ�ᆠᨨ矄矄�W 矄�矄ᨨ㦨҈
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P-235
Alternating Current
10 Subjective Problems
ᔼ ��盐ᨨ ᆠ矄A B ᨨ ⼄盐鼀 � R ᨨ 犨 �ᆠ ᨨ � � ᆠᆠᨨ矄矄泐A犨� ᨨ ᨨ ��鼀 矄犨�ᆠ �矄ᆠᨨ矄 ᔼ 矄�矄 ᆠᨨ �矄A B r҈ r矄ᆠ�泐 r҈>r㡰 㜠㝐├矄桌
䰘
ᔼ 矄�矄 LR ᆠ�ᆠ盐� L 㦨 �鼀 R㦨҈҈W㡰� 鼀 矄ᨨ盐ᆠ V㦨V矄� w㡰ᨨ Vrms㦨 ⼄盐 ᆠ泐 � �矄 � � ᆠ盐 鼀�盐 � ᆠ�ᆠ盐� 矄 ᨨ ᆠ盐 犨� 矄ᆠᨨᨨ�犨ᆠ盐 㜠�鼀� ᨨ �� � p㦨⏌䰘㡰 㜠㝐├矄桌 꾀 � 盐ᆠᨨ ᨨ � 盐ᆠ ᆠL 㦨 鼀 矄�矄ᨨ矄 L E ᨨ 矄�矄 ᆠ矄 R ҈ R҈ 㦨W R㦨W ᆠᨨ
ᆠ ᨨ S R 泐ᨨ
鼀 E㦨҈矄矄 ᨨ犨 � ��盐 � 矄�矄 ᆠ ᨨ 泐�矄 ���� 矄犨�ᆠ S�矄ᆠᨨ矄 �鼀 t㦨 ␔ �矄 ᨨ � ᨨᆠᨨ矄矄L矄 盐 ᆠ�ᨨ ᨨ �鼀䁸꾀 矄 泐矄 �矄 ᆠ 矄犨�ᆠ �矄 ᨨ ␔ �矄 �ᆠ�ᨨ 鼀� �盐 ᨨ ᆠ盐 ᨨ盐� R҈矄 盐 ᆠ�ᨨ ᨨ �鼀䁸 ҈㜠�㝐├矄桌 黨 꾀矄ᨨ ᨨ� 矄 � 盐ᆠ ᆠᨨ ҈ 泐 矄�矄 ᆠ ᨨ ᨨ 鼀ᔼ�矄ᆠᨨ
ᆠ ᨨ҈ᨨ泐ᨨ犨ᨨ �犨� �├ ᨨ 鼀� �ᆠ �泐ᨨᆠ ҈⏌ᨨ �矄鼀 �鼀盐鼀 盐䁸 ҈黨黨㜠㝐├矄桌 꾀ᆠ�ᆠ盐�ᆠᨨ � � �犨ᨨᨨ矄��ᨨ 矄犨�ᆠ 和�矄矄 ᨨ犨 �
�� ҈黨黨҈㜠졔㝐├矄桌
㡰 矄犨�ᆠ S �矄� ᨨ矄��ᨨ �҈ � ᨨ � � ᆠVA VB ᨨ ᨨ 盐ᆠ�ᨨ ᨨ ᨨ盐 � R҈ 㡰 ᔼ ᨨ犨 矄犨�ᆠ S �矄盐� ᨨ矄��ᨨ t㦨 � �㡰矄 泐ᆠ盐 � R ��㡰 �鼀犨 ᆠ盐 � R�矄 矄 泐 盐꾀矄ᨨᆠᆠ盐 �泐矄ᨨ � � 盐ᆠᨨ L �鼀
Topic-3 : Transformers and LC Oscillations 1 ҈
ᨨ 㡰
MCQs with One Correct Answer
ᨨ �� � 盐ᨨ�犨 ᨨ鼀� t㡰 ᨨ盐盐 ᨨ�犨 ᨨ鼀ᨨ t㡰ᆠᨨ矄矄 ᆠᆠ�ᨨ�矄ᆠᨨᆠ泐 �ᆠ 泐 㝐� 和 ᔼ㡰桌
ᆠ㡰 �m矄 ᨨ 㡰
҈├W 졔� �m 矄 �m
҈ t
졠 㡰
㡰
矄
㡰
㡰
�m矄 ᨨ 㡰
҈m矄 ҈5m矄
�m矄
҈m矄 ҈5m矄
҈m矄 ҈5m矄
㡰 꾀 ᆠ㡰 �꾀
꾀 矄 ᨨ鼀ᆠᨨ 矄�矄� �ᨨ 盐 矄� �鼀泐 ҈�盐 矄� 矄ᆠᨨ 泐��矄ᨨ盐盐ᨨ犨ᨨ ├␔ ᔼ ᆠ盐 � 矄ᆠᨨ 泐ᆠᨨ��矄҈꾀 � 盐 ᨨ� ᆠ盐 � �鼀泐ᆠᨨ�
�m矄
ᨨ 㡰
҈m矄 ҈5m矄
㝐� ҈꾀�҈黨ᔼ桌 㡰 꾀 㡰 ҈꾀
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EBD_8341 P-236
3.
Physics
A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns. The output power is delivered at 230 V by the transformer. If the current in the primary of the transformer is 5A and its efficiency is 90奠, the output current would be: [Main 9 Jan. 2019 II] (a) 50 A (b) 45 A (c) 35 A (d) 25 A
A power transmission line feeds input power at 2300 V to a step down transformer with its primary windings having 4000 turns, giving the output power at 230 V. If the current in the primary of the transformer is 5 A, and its efficiency is 90奠, the output current would be: [Main Online April 16, 2018] (d) 25 A (a) 20 A (b) 40 A (c) 45 A
4.
Topic-4 : Miscellaneous (Mixed Concepts) Problems 6 1.
A series R– C circuit is connected to AC voltage source. Consider two cases; (A) when C is without a dielectric medium and (B) when C is filled with dielectric of constant 4. The current IR through the resistor and voltage VC across the capacitor are compared in the two cases. Which of the following is/are true? [2011] (a) 7
2.
MCQs with One or More than One Correct Answer
I RA > I RB
(b)
I RA < I RB
VCA > VCB
(c)
(d)
VCA < VCB
Match the Following
You are given many resistances, capacitors and inductors. These are connected to a variable DC voltage source (the first two circuits) or an AC voltage source of 50 Hz frequency (the next three circuits) in different ways as shown in Column II. When a current I (steady state for DC or rms for AC) flows through the circuit, the corresponding voltage V1and V2 , (indicated in circuits) are related as shown in Column I. Match the two [2010] Column I Column II V1
(A) I ¹ 0, V1 is proportional to I
(p)
V2 3m F
6mH
V V1
V2
6mH
(B) I ¹ 0, V2 > V1
2W
(q) V
(C) V1 = 0, V2 = V
V1
V2
6mH
2W
(r) V
V1
(D) I ¹ 0, V2 is proportional to I
(s)
V2
6mH
3m F
V1
V V 2
3m F
1kW
(t) V
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P-237
Alternating Current
Comprehension/Passage Based Questions Passage 1 In the given circuit the capacitor (C) may be charged through resistance R by a battery V by closing switch S1. Also when S1 is opened and S2 is closed the capacitor is connected in series with inductor (L). 8
5.
Given that the total charge stored in the LC circuit is Q0, for t ³ 0, the charge on the capacitor is [2006 – 5M, –2] t ö æp (a) Q = Q0 cos ç + ÷ è2 LC ø
(c) Q = - LC
At the start, the capacitor was uncharged. When switch S1 is closed and S2 is kept open, the time constant of this circuit is t. Which of the following is correct [2006 – 5M, –2] CV (a) after time interval t, charge on the capacitor is 2 (b) after time interval 2t, charge on the capacitor of CV (1 – e–2) (c) the work done by the voltage source will be half of the heat dissipated when the capacitor is fully charged (d) after time interval 2t, charge on the capacitor is CV (1 – e–1) When the capacitor gets charged completely, S1 is opened and S2 is closed. Then, [2006 – 5M, –2] (a) at t = 0, energy stored in the circuit is purely in the form of magnetic energy (b) at any time t > 0, current in the circuit is in the same direction (c) at t > 0, there is no exchange of energy between the inductor and capacitor (d) at any time t > 0, instantaneous current in the circuit
3.
4.
may be V
C L
d 2Q
t ö æp (b) Q = Q0 cos ç ÷ è2 LC ø
(d) Q = -
dt 2
1 d 2Q LC dt 2
Passage 2 A thermal power plant produces electric power of 600 kW at 4000 V, which is to be transported to a place 20 km away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method , a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with power factor unity. All the currents and voltages mentioned are rms values. [Adv. 2013] 6. If the direct transmission method with a cable of resistance 0.4 W km–1 is used, the power dissipation| (in 奠) during transmission is (a) 20 (b) 30 (c) 40 (d) 50 7. In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is 1 : 10. If the power to the consumers has to be supplied at 200 V, the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is (a) 200 : 1 (b) 150 : 1 (c) 100 : 1 (d) 50 : 1
Answer Key Topic-1 : Alternating Current, Voltage and Power 1.
(d)
2. (a)
3. (b) Topic-2 : AC Circuit, LCR Circuit, Quality and Power Factor
1. (a) 11. (a) 21. (c) 31.
(400)
39.
3.465 s
2. (a) 12. (d) 22. (c)
3. (d) 13. (a) 23. (c)
4. (a) 14. (b) 24. (d)
32. (55)
33. (b, d)
5. (b) 15. (a) 25. (d)
6. (c) 7. (d) 16. (b) 17. (b) 26. (b) 27. (b) 4 (r - r ) 34. (b, c) 35. (a,c) 36. 3 1 2
8. (Bonus) 9. 18. (b) 19. 28. (a) 29. p 37. 20 A, 38. 4
(b) (c) (8)
10. (a) 20. (c) 30. (33)
12e–5t V
Topic-3 : Transformers and LC Oscillation 1.
(a)
2. (c)
3. (b)
4. (c)
Topic-4 : Miscellaneous (Mixed Concepts)Problems 1.
(a,c)
2. A-r,s,t; B-q,r,s,t; C-p,q; D-q,r,s,t
3. (b)
4. (d)
5. (c)
6. (a)
7. (b)
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EBD_8341 P-238
Physics
Electromagnetic Waves
22
Topic-1 : Electromagnetic Waves, Conduction and Displacement Current 1 ҈
MCQs with One Correct Answer
ᨨ ᆠᨨ鼀� �ᆠ犨 鼀� �ᆠ � ᨨ� x �鼀t�矄 ®
x t 㡰 = ҈ ´ ҈-䰘 矄� � ´ ҈® x + ҈� ´ ҈҈҈t 㡰k$ 桌 ®
� 矄 ᨨ盐矄ᆠ�ᆠ � 쟼 ᆠᨨ矄ᨨ � �ᨨ �矄 矄 ᨨ �� ᆠ㦨맔҈鼀矄҈㡰㝐� 和 ᔼᔼ㡰桌 ® 㡰 쟼 x t 㡰 = - 矄� � ´ ҈ x + ҈� ´ ҈҈҈ t 㡰 $ 桌 鼀 ® ҈҈ 㡰 쟼 x t 㡰 = 矄� ҈ ´ ҈ x + � ´ ҈ t 㡰 $ 桌 鼀 ® ҈҈ $ ᆠ㡰 쟼 x t 㡰 = 矄� � ´ ҈ x + ҈� ´ ҈ t 㡰 k 桌 鼀 ® 㡰 쟼 x t 㡰 = 矄� ҈ ´ ҈ x + ҈� ´ ҈҈҈ t 㡰$�桌 鼀 꾀 ᆠᨨ �矄ᆠᨨ 矄� ᨨ鼀ᨨᨨ � y㜠 �矄犨� 矄 ᨨ ҈c c�矄 矄 ᨨ �� 㡰� 矄 ᆠᨨ ® ᆠᨨ鼀� �ᆠ 犨 犨 ᨨ矄 ᆠ�ᆠ � �矄 쟼 = $ 矄� ҈�맔҈䰘t�맔҈x㡰⏌鼀 鼀 �鼀盐鼀鼀� �ᆠ ᨨᆠ � ᆠ 泐 ᆠᨨ 犨� �� c㦨맔҈鼀矄҈ᆠᨨ ᆠ �㦨҈맔҈҈黨ᔴ㡰 㝐� 和� ᔼ㡰桌 ҈ 㡰 맔҈ 㡰 맔҈҈ ᆠ㡰 맔҈҈黨 㡰 ҈맔҈҈黨
ᆠ�ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄��
r 泐 E = E xˆ + yˆ 㡰 矄� kz - wt 㡰 ᔼ矄鼀� �ᆠ � 犨��� 泐㝐� 和 ᔼᔼ㡰桌 E 㡰 - xˆ + yˆ 㡰矄� kz - wt 㡰 c E 㡰 xˆ + yˆ 㡰矄� kz - wt 㡰 c
E xˆ - yˆ 㡰 矄� kz - wt 㡰 c E 㡰 xˆ - yˆ 㡰 ᆠᨨ矄 kz - wt 㡰 c
鼀� �ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄 ur ˆ B = ´ ҈- 矄� p y + ct 㡰桌�T
ᆠ㡰
犨 c㦨맔҈鼀矄҈�矄 矄 ᨨ ��
ᆠᨨ矄ᨨ � �ᆠ�ᆠ � �矄 㝐� 和 ᔼ㡰桌 ur 㡰 E = 黨矄� p y + ct 㡰桌kˆ ⏌鼀 ur 㡰 E = -҈- 矄� p y + ct 㡰桌kˆ ⏌鼀 ur ᆠ㡰 E = ´ ҈ 矄� p y + ct 㡰桌kˆ ⏌鼀 ur 㡰 E = -黨矄� p y + ct 㡰桌kˆ ⏌鼀 �
ᆠ�ᆠ � ᨨ ᆠᨨ鼀� �ᆠ 犨 ᨨ�� � ᨨ � x �ᆠ�ᨨ � ᆠ盐盐鼀 �矄 r r E = E ˆ ᆠᨨ矄 wt - kx 㡰 鼀� �ᆠ � B 鼀ᨨ鼀 t㦨�矄 㝐� 和 ᔼᔼ㡰桌 r r E 㡰 B = ᆠᨨ矄 kx㡰kˆ 㡰 B = E m e ᆠᨨ矄 kx㡰 ˆ m e
r r ᆠ㡰 B = E m e ᆠᨨ矄 kx㡰kˆ 㡰 B =
E m e
ᆠᨨ矄 kx㡰 ˆ
꾀 ᆠᨨ鼀� �ᆠ犨 矄 ⼄盐 ᆠ泐ᨨ 맔҈҈ �矄 �泐 矄�泐�矄҈맔҈⏌鼀� ᆠ盐盐鼀
鼀�盐 ᨨ 鼀� �ᆠ � ᨨ 犨�矄ᆠᨨ矄ᨨ
Nm ҈ = 黨 ´҈黨 矄 ᨨ �� 㦨맔҈鼀矄҈㡰 pe C 㝐� 和 ᔼ㡰桌 㡰 ҈�
㡰 ҈
ᆠ㡰 ҈
㡰 ҈黨
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P-239
Electromagnetic Waves
䰘
ᔼ ᆠᨨ鼀� �ᆠ犨 �ᆠ�ᨨ 矄ᨨ ᆠ�ᆠ � 鼀� �ᆠ � 矄 泐 kˆ �ˆ - ˆ 矄ᆠ�泐␔ �矄 盐 �ᆠᨨ ᨨ � �ᆠ�ᨨ ᨨ ᨨ��ᨨ ᨨ 犨 㝐� 和 ᔼᔼ㡰桌 㡰
҈ ˆ ˆ � + 㡰
㡰
҈
ˆ + kˆ㡰
҈ ˆ ˆ ҈ ˆ � + 㡰 � + ˆ 㡰 㡰 � � 꾀 ᆠᨨ鼀� �ᆠ犨�矄ᨨ�� �ᨨ � �ˆ + ˆ 犨� �矄ᨨ� �ᨨ ᨨ � �ᆠ�ᨨ
�ᆠ�ᨨ kˆ ᆠᨨᆠ ᨨ鼀ᨨ 鼀� �ᆠ � ᨨ 犨 犨ᨨ盐 �矄 ᨨ�ᆠᨨ 矄 㡰 㝐� 黨 ᔼᔼ㡰桌 æ �ˆ - ˆ �ˆ + ˆ ö ᆠᨨ矄 ç wt - k 㡰 è ÷ø ᆠ㡰
㡰
�矄 v kˆ ᨨᆠᆠ� �ᨨ � 盐ᨨ 犨�矄 㝐� 䰘 ᔼᔼ㡰桌 �ˆ + ˆ 㡰 ᨨ 㡰 ᨨ �ˆ + ˆ ᆠ㡰 �ᨨ 㡰 ᨨ kˆ ҈ 꾀 ᆠᨨ鼀� �ᆠ犨�矄矄 泐 ᆠ�ᆠ ur � E = E n$ 矄� w + 泐 - 㡰桌 ├� �盐 �ᆠᨨ矄�
x, y z �ᆠ�ᨨ 矄 ᨨ $� $ k$ �ᆠ�ᨨ ᨨ
҈
æ �ˆ + ˆ ö ᆠᨨ矄 ç wt + k è ÷ø
ˆ - �ˆ
æ �ˆ + ˆ ö ˆ ᆠ㡰 k ᆠᨨ矄 ç wt - k è ÷ø æ �ˆ + ˆ ö ᆠᨨ矄 ç wt - k è ÷ø 꾀 ᆠᨨ鼀� �ᆠ犨ᨨ ⼄盐 ᆠ泐� �矄 ᨨ�� � � ᆠ盐盐鼀 ᨨ � z㜠 �ᆠ�ᨨ 꾀 �ᆠ盐ᨨ� � 矄ᆠ �鼀 鼀� �ᆠ � �矄 r �� 泐 B = � ´ ҈- ˆ T ᆠᨨ矄ᨨ � � ᆠ�ᆠ r � E �矄 矄 ᨨ �� c㦨´鼀矄㡰 㝐� ᔼᔼ㡰桌 㡰 ҈´҈҈ �ˆ ⏌鼀 㡰 ҈´҈҈ �ˆ ⏌鼀
㡰
黨
҈
�ˆ + ˆ
҈
ᆠ㡰 ҈� �ˆ ⏌鼀 㡰 ҈� �ˆ ⏌鼀 ᔼ 鼀� �ᆠ � � ᆠᨨ鼀� �ᆠ犨�矄 ur �� 泐 B 㦨´҈矄� ´҈x졔´҈҈㡰 ˆ
犨 犨� 矄矄�ᨨ ᨨᆠ�ᆠ � 䁸 㝐� 䰘 ᔼ㡰桌 ur 㡰 E 㦨 矄� ҈´ x 졔´ ҈㡰 kˆ ⏌鼀㡰 ur 㡰 E 㦨 黨矄� ҈´ x 졔´ ҈㡰 kˆ ⏌鼀㡰 ur ᆠ㡰 E 㦨 ´ 矄� ´ x졔´ ҈㡰 kˆ ⏌鼀㡰 ur 㡰 E 㦨 ´ 矄� ´ x 졔´ ҈㡰 kˆ ⏌鼀㡰 ҈҈ ᆠ�ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄 �� 泐 �ˆ + ˆ r E 㦨 E ᆠᨨ矄 kz + wt 㡰 꾀 t 㦨 ᨨ矄��泐 ᆠ � �ᆠ �矄 ᨨ�
pö æ xyz㡰㦨 çè ÷ø ᔼ �矄� 矄 ᨨ盐矄ᨨᆠ�泐 t 㦨㡰 k
҈�
҈
ᨨᨨ��ᨨ s$ �矄 㝐� ҈꾀�҈黨 ᔼ㡰桌 $i - $j -k$ + $j 㡰 s$ = 㡰 s$ = � � $j - k$ $ + k$ ö æ 㡰 s$ = ᆠ㡰 s$ = ç ÷÷ ç � � è ø 꾀 ᆠᨨ鼀� �ᆠ 犨 � � ⼄盐 ᆠ泐 v㦨黨 ᨨ�矄ᨨ � ᨨ矄��z㜠 �ᆠ�ᨨ �
矄ᆠ ├盐ᨨ 쟼ᆠ�ᆠ � �矄⏌鼀 ␔ �ᆠ 鼀ᨨ � ᨨᨨ犨� ��矄 ᆠᆠ鼀� �ᆠ � ᆠᨨ鼀ᨨ � ᆠᨨ鼀� �ᆠ犨䁸 㝐� ҈꾀�҈黨 ᔼᔼ㡰桌 ur 㡰 B = ´ ҈䰘 矄� � ´ ҈ z + ҈� ´ ҈҈҈ t 㡰$� ur 㡰 B = ´ ҈-䰘 矄� � ´ ҈ z - ҈� ´ ҈҈҈ t 㡰$� ur ᆠ㡰 B = 矄� � ´ ҈ x + ҈� ´ ҈҈҈ t 㡰 k$ ur -䰘 ҈҈ $ 㡰 B = ´ ҈ 矄� ҈� ´ ҈ x + � ´ ҈ t 㡰
ᆠ�ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄��
ur 泐 쟼 = 쟼 $� ᆠᨨ矄 ├ 㡰 ᆠᨨ矄 w 㡰
ᆠᨨ矄ᨨ � �鼀� �ᆠ � �矄 �� 泐 㝐� ҈꾀�҈黨 ᔼ㡰桌 ur 쟼 $ 矄� ├ 㡰 矄� w 㡰 㡰 = ᔴ ur 쟼 $ 矄� ├ 㡰 ᆠᨨ矄 w 㡰 㡰 = ᔴ ur 쟼 $ ᆠᨨ矄 ├ 㡰 矄� w 㡰 ᆠ㡰 = ᔴ ur 쟼 ├$ 矄� ├ 㡰 ᆠᨨ矄 w 㡰 㡰 = ᔴ �� �矄 � ᆠ� ᨨ鼀泐ᨨ ᆠᨨ鼀泐矄ᨨ� � 矄盐 ᆠ犨� �泐 盐 ᨨ �␔ᆠ鼀ᔼ 矄盐 ᆠ 矄 ᨨ �ᆠ鼀 鼀ᨨ鼀 盐鼀 矄 ᨨ 矄盐 ᆠ � 鼀� �鼀 盐�ᨨ 犨� 㝐� ҈꾀�҈黨ᔼᔼ桌 㡰 맔҈矄 㡰 ҈맔҈矄 ᆠ㡰 �맔҈矄 㡰 �맔҈矄 꾀 ᆠᨨ鼀� �ᆠ犨ᨨ ⼄盐 ᆠ泐�㝐 矄 � 矄ᆠᨨ � ᨨ矄��x㜠 �ᆠ�ᨨ 꾀�ᆠ盐 r ᨨ� � 矄ᆠ �鼀 쟼 = ˆ ⏌ 鼀 r ᆠᨨ矄ᨨ � �鼀� �ᆠ � ᨨ� 犨� 㝐� 黨꾀�҈黨ᔼ桌 㡰 ҈黨맔҈ ├
ˆ ˆ 㡰 ҈맔҈ ├
ˆ ˆ ᆠ㡰 맔҈ ├
㡰 ҈黨맔҈ ├
WWW.JEEBOOKS.IN
EBD_8341 P-240
҈䰘
҈
҈黨
Physics
�␔⏌m �泐 矄�泐ᨨ 矄盐 �� �矄 ᨨ鼀泐� ᆠ� ᨨ 矄盐 ᆠᨨ 矄ᨨ 和ᨨ鼀ᨨ � ᆠ� �泐 �奠㡰�矄 ᆠ ᨨ鼀 矄盐 ᆠ 矄�矄 矄ᨨ ᨨᆠ ᨨ ҈鼀矄盐 ᆠ犨� ᆠᨨ矄ᨨ ᆠ㦨맔҈鼀⏌矄㡰 㝐� 黨꾀�҈黨ᔼᔼ桌 㡰 ҈�맔҈ 㡰 맔҈ ᆠ㡰 ҈맔҈ 㡰 �맔҈ 꾀 ᆠᨨ鼀� �ᆠ犨矄� 矄ᆠᨨ � the x㜠 �ᆠ�ᨨ ᆠ�ᆠ � ᆠᨨ鼀ᨨ ᨨ 犨 �ᆠ盐ᨨ� ᨨ 矄ᆠ �鼀�矄쟼㦨鼀 ҈ᨨ � y㜠 �ᆠ�ᨨ ᔼ矄ᆠᨨ矄ᨨ � �鼀� �ᆠ � ᆠᨨ鼀ᨨ 犨ᨨ盐 㝐� 꾀�҈黨ᔼ桌 㡰 맔҈ ᨨ �z㜠 �ᆠ�ᨨ
㡰 맔҈ ᨨ �x㜠 �ᆠ�ᨨ
ᆠ㡰 맔҈ ᨨ �z㜠 �ᆠ�ᨨ
㡰 맔҈ ᨨ �y㜠 �ᆠ�ᨨ
鼀� �ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄�� 泐 ur Wb = ҈ ´ ҈ ᆠᨨ矄 ´ ҈ 䰘 z + ´ ҈҈� t �ˆ + ˆ m
矄矄ᨨᆠ� ᆠ�ᆠ � 犨� 㝐� 꾀�҈黨ᔼᔼ桌
(
)(
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(
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V ur 㡰 쟼 㦨맔҈ᆠᨨ矄 맔҈䰘z맔҈҈�t㡰 �ˆ + ˆ m ur V 㡰 쟼 = ´ ҈ ᆠᨨ矄 ´ ҈ 䰘 z - ´ ҈҈� t㡰 - $j + $i㡰 m V ur ᆠ㡰 쟼 㦨맔҈ᆠᨨ矄 맔҈䰘z졔맔҈҈�t㡰 �ˆ + ˆ m V ur 㡰 쟼 㦨맔҈ᆠᨨ矄 맔҈䰘z졔맔҈҈�t㡰 �ˆ ˆ m
鼀 � 矄�泐ᨨ ��ᨨ ᨨ 矄盐 ᆠᨨ 和盐
�矄ᨨ盐҈␔⏌鼀 鼀矄盐ᨨ ᆠᨨ矄ᨨ � � 鼀� �ᆠ � �矄ᆠᨨ矄矄ᨨ 㝐� ҈ ҈黨ᔼᔼ桌 㡰 ҈
㡰҈
ᆠ㡰 ҈ 㡰 ҈
꾀 ᆠᨨ鼀� �ᆠ犨ᨨ � 矄�泐�␔鼀 矄� 鼀 �盐鼀ᨨ ᆠ�� � 犨� ᨨ盐 泐ᨨ矄矄 �ᨨ ᨨ 鼀� �盐 矄ᨨ ᆠ�ᆠ � 矄 �ᨨᨨ 鼀� �盐 矄ᨨ 鼀� �ᆠ � 矄ᨨ 犨 ᨨ � �� ᨨ 鼀 �盐鼀矄ᆠ�泐�� 泐 㝐� ҈҈ ҈黨 ᔼ桌 æ ҈ ҈ ö
㡰 ç 㡰 ÷ è ø ҈ ö æ æ ҈ ö ÷ ᆠ㡰 ç 㡰 ç ÷
ø è è
ø 꾀䰘鼀␔矄鼀 矄ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ ҈鼀鼀
鼀� �盐 ᨨ 鼀 �鼀盐鼀 ᆠ�ᆠ � � �矄 ᆠᨨ鼀� �ᆠ犨�矄�� 泐 � 鼀���泐ᨨ 矄ᆠÎ㦨黨맔҈҈和ᔼ盐 �矄和 ᨨ �� ᆠ㦨맔҈鼀⏌矄桌 㝐� ҈҈ ҈黨ᔼᔼ桌 㡰 ├⏌鼀 ᆠ㡰 䰘├⏌鼀 㡰 ҈├⏌鼀 㡰 ҈├⏌鼀 ᔼ 鼀� �ᆠ � ᨨ ᆠᨨ鼀� �ᆠ犨�矄 �� 泐 矄 ᨨ �� 㦨 맔 ҈ 鼀⏌矄㡰
(
(
҈
(
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)
é xö ù ҈� æ 㦨҈맔҈矄� ê p ´ ´ ҈ çè - ÷ø ú ᆠ û ë 鼀 �鼀盐鼀ᆠ�ᆠ � 矄矄ᨨᆠ� 犨� ��矄 㝐� ҈ ҈黨 ᔼ桌
㡰 맔҈⏌ᔴ 㡰 맔҈⏌ᔴ ᆠ㡰 맔҈ ⏌ᔴ 㡰 �҈⏌ᔴ ᆠ�ᆠ � ᨨ ᨨ� ᆠᨨ鼀� �ᆠ 犨� 矄ᆠ�鼀㦨�矄�� 泐 矄矄�ᨨ
ur 쟼 x y 㡰 = ҈ ˆ ᆠᨨ矄 xur+ z㡰桌
鼀� �ᆠ � x z t 㡰 �矄 �� 泐 ᆠ �矄 ᨨᆠ�泐ᨨ �� 㡰 㝐� ҈ ҈黨 ᔼᔼ桌 ҈ ˆ ˆ ᆠᨨ矄 [ x - z + ҈ᆠt 㡰 ] 㡰 ├ + �㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [ x + z - ҈ᆠt 㡰 ] ├ - �㡰 㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [ x + z - ҈ᆠt 㡰 ] ├ + �㡰 ᆠ㡰 ᆠ ҈ ˆ ˆ ᆠᨨ矄 [ x + z + ҈ᆠt 㡰 ] ├ - �㡰 㡰 ᆠ � 꾀 쟼㝐犨 ᨨ鼀� 矄鼀 �盐鼀 ᆠ�ᆠ � 矄 r é æ z öù E҈ = E҈ xˆ ᆠᨨ矄 ê pv ç - t ÷ú � � è c øû ë r E = E xˆ ᆠᨨ矄 [ k z - ct 㡰] � 鼀 �盐鼀 犨 犨
盐鼀k ⼄盐 ᆠ泐v ᨨ �盐矄� � 鼀 �盐鼀�矄 ᨨ 鼀� �ᆠᔼ Îr҈ Îr ᨨ� 鼀����矄ᨨ � 鼀 �盐鼀矄ᆠ�泐犨 �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄�矄ᆠᨨᆠ䁸 㝐� 黨 ҈黨ᔼ桌 Îr Îr ҈ ҈ = = 㡰 㡰 Îr Îr
ᆠ㡰
Îr Îr
҈
=
҈
㡰
Îr Îr
҈
=
҈
�泐矄矄ᨨᆠ� 犨� ᆠ�ᆠ � �矄 쟼㡰 犨� 鼀� �ᆠ � 矄�矄 㡰 ᨨ ᆠᨨ鼀� �ᆠ犨�
矄ᆠ 㝐� 黨 ҈黨 ᔼᔼ桌
㡰 쟼> ᆠ㡰 쟼 㡰 쟼㦨 䰘 꾀 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ 鼀 ᨨ �� 矄 ⼄盐 ᆠ泐 v= ´ ҈҈ �矄ᨨ�� �ᨨ � �ᆠ�ᨨ
p �ˆ + ˆ ᔼ�矄ᨨ� ᨨ � kˆ �ᆠ�ᨨ ᆠᆠ ᨨ鼀 ᨨ 鼀� �ᆠ � �矄 㝐� 졠 � 꾀�҈�҈桌 㡰 쟼 =
é æ �ˆ - ˆ ö r E æ �ˆ - ˆ ö ҈ ù 㡰 k ç ÷ ᆠᨨ矄 ê҈ ç ÷ r - ´ ҈ 㡰t ú C è ø êë úû è ø 㡰
é æ �ˆ + ˆ ö r E æ �ˆ - ˆ ö ҈ ù ç ÷ ᆠᨨ矄 ê҈ ç ÷ r - ´҈ 㡰t ú C è ø êë úû è ø
ᆠ㡰
é æ �ˆ + ˆ ö r E ˆ ҈ ù k ᆠᨨ矄 ê҈ ç ÷ r + ´҈ 㡰t ú C è ø ëê ûú
㡰
é æ �ˆ + ˆ ö r E �ˆ + ˆ + kˆ㡰 ҈ ù ᆠᨨ矄 ê҈ ç ÷ r + ´ ҈ 㡰t ú C è ø ëê ûú
WWW.JEEBOOKS.IN
P-241
Electromagnetic Waves
黨
ᆠ�ᆠ � ᆠᨨ鼀ᨨ ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ��ᨨ �矄�� 泐 ur 쟼 㦨쟼 $� ᆠᨨ矄├ ᆠᨨ矄w ur ᔼ矄鼀� �ᆠ � �矄 �� 泐 㝐� 졠 � 꾀�黨҈䰘桌 쟼ᨨ $ 쟼 矄� ├ ᆠᨨ矄w 㡰 - ᨨ $ 矄� ├ 矄� w 㡰 ᆠ ᆠ 쟼ᨨ $ 쟼ᨨ $ 矄� ├ 矄� w 㡰 ᆠ㡰 ᆠᨨ矄├ ᆠᨨ矄w ᆠ ᆠ ᔴᨨ 矄� ᆠᨨ鼀� �ᆠ犨ᨨ�� �� ᆠ盐盐鼀 ᔴ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㝐� 졠 � 꾀�҈҈桌 㡰 ᨨ ᆠᨨ鼀� �ᆠ 犨 ᨨ�� � � 졔泐 r ҈ �ᆠ�ᨨ ᆠ�ᆠ � �矄 쟼 = 쟼 泐 㡰 ˆ r ҈ 鼀� �ᆠ � �矄 = 㡰泐ˆ 㡰 ᨨ ᆠᨨ鼀� �ᆠ 犨 ᨨ�� � � 졔泐 r ҈ 쟼 泐 㡰泐ˆ �ᆠ�ᨨ ᆠ�ᆠ � �矄 쟼 = r ҈ 鼀� �ᆠ � �矄 = 泐 㡰 ˆ ᆠ㡰 ᨨ ᆠᨨ鼀� �ᆠ 犨 ᨨ�� � �
r ҈ 졔 �ᆠ�ᨨ ᆠ�ᆠ � �矄 쟼 = 쟼 泐 泐 㡰 ( 泐ˆ + ˆ ) 鼀� �ᆠ � �矄 r ҈ = 泐 泐 㡰 ( 泐ˆ + ˆ )
҈
㡰 ᨨ ᆠᨨ鼀� �ᆠ 犨 ᨨ�� � � 졔 r ҈ �ᆠ�ᨨ ᆠ�ᆠ � �矄 쟼 = 쟼 泐 㡰 ( 泐ˆ - ˆ )
r 鼀� �ᆠ � �矄 = ҈ 泐 㡰 ( 泐ˆ + ˆ ) ᨨ ᆠᨨ鼀� �ᆠ犨矄ᨨ�� �� 㜠 �ᆠ�ᨨ 犨 �ᆠ ᨨ ᨨ ᨨᨨ犨� �ᆠᨨ鼀� �ᨨ ��矄 ur ur ᆠᨨᆠ ᨨ矄矄� �ᆠ�ᨨ ᨨ 쟼 � 矄ᆠ�泐䁸 㝐� 졠 � 꾀�҈҈ ҈�桌 㡰 (2$i + 3$j) ($i + 2$j) 㡰 (-2$i - 3$j) (3$i - 2 $j) ᆠ㡰 (3$i + 4 $j) (4$i - 3$j) 㡰 ($i + 2$j) (2$i - $j) 盐� � ᨨ��ᨨ ᨨ ᆠᨨ鼀� �ᆠ犨矄� 鼀 �盐鼀 㝐� ҈桌 㡰 쟼ᆠ�ᆠ �泐 矄�泐�矄 ᨨ盐 ᨨ 鼀� �ᆠ �泐 矄�泐 㡰 쟼ᆠ�ᆠ �泐 矄�泐�矄 ᨨ 鼀� �ᆠ �泐 矄�泐 ᆠ㡰 쟼ᆠ�ᆠ �泐 矄�泐�矄⼄盐ᨨ 鼀� �ᆠ �泐 矄�泐 㡰 ᨨ ᆠ�ᆠ 鼀� �ᆠ �泐 矄��矄 ᨨ 꾀鼀鼀�矄鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ� �� 盐 � ᨨ鼀泐� �ᆠ�ᨨ 矄 鼀�矄奠 �ᆠ� � ᆠᨨ � �ᆠ�ᆠ ᨨ犨ᨨᆠᨨ鼀� �ᆠ犨矄 ᆠᨨ 矄盐鼀矄҈␔ᨨ ᨨ犨 鼀�盐 ᨨ ᆠ�ᆠ � 矄矄ᨨᆠ� 犨� ᆠᨨ鼀� �ᆠ ��ᨨ �矄 ᆠᨨ �鼀 ᨨ鼀 㝐� 졠 � 꾀�҈҈桌 鼀犨� 泐 㡰 ҈⏌鼀 㡰 ⏌鼀 ᆠ㡰 ⏌鼀 㡰 �⏌鼀
鼀� �ᆠ � � � �ᆠᨨ鼀� �ᆠ犨 矄├盐ᨨ
├盐ᨨ ᆠ�ᆠ � 矄 � �矄 㝐� ҈桌 㡰 ⏌鼀 㡰⏌鼀 ᆠ㡰 黨⏌鼀 㡰 ҈⏌鼀 꾀 ᆠᨨ鼀� �ᆠ犨� ᨨ 㜠鼀� �ᆠ �ᆠ�ᆠ ur ur 鼀 �盐鼀 �矄 �� 泐 E = E ´ ҈ -䰘 x - �t 㡰 犨� �矄 ᆠ� �� 鼀 �鼀� 矄ᆠᨨ 矄 �ᆠ�ᆠ ᆠᨨ 矄 ᨨ 鼀 �盐鼀�矄 㝐� 졠 � 꾀�҈桌 㡰 㡰� ᆠ㡰 㡰
Topic-2 : Electromagnetic Spectrum 1 ҈
㡰 㡰 ᆠ㡰 㡰
MCQs with One Correct Answer
ᔴ ᨨ矄矄 ᆠᨨᆠ ᨨ�ᨨ � � 犨 � 矄 ᨨ � 矄ᨨ ᆠᨨ鼀� �ᆠ犨矄ᆠ盐鼀 㝐� 和 ᔼ㡰桌 㡰 l �矄� < l 鼀�ᆠᨨ犨矄 < l �ᨨ犨矄 < l X -泐矄 㡰 l �ᨨ犨矄 > l 鼀�ᆠᨨ犨矄 > l �矄� > l x-泐矄 ᆠ㡰 l x- 泐矄 < l 鼀�ᆠᨨ犨矄 < l �ᨨ犨矄 < l �矄� 㡰 l �矄� > l x-泐矄 > l �ᨨ犨矄 > l 鼀�ᆠᨨ犨矄 꾀 � ᨨᨨ犨� � ᆠᨨ鼀� �ᆠ ��ᨨ 矄 ⼄盐 盐鼀� ᨨ ᨨ � ᆠ矄� � �泐㝐� ҈桌 꾀盐�� Ӹᨨ犨�� ᔴ㜠泐 �ᨨ犨 㡰 ᔴ꾀 㡰 꾀ᔴ ᆠ㡰 꾀ᔴ 㡰 꾀ᔴ 㝐�ᆠᨨ犨ᨨ ᆠ矄ᨨ � ᆠ�ᨨ 㝐� 졠 � 꾀�黨҈桌
�
��� �ᨨ�ᨨ �泐ᨨ犨鼀ᨨᆠ盐矄 ��� � 矄�ᨨ �泐ᨨ犨鼀ᨨᆠ盐矄 ��� ���ᨨ �泐ᨨ犨鼀ᨨᆠ盐矄 矄 � �ᆠᨨ 矄 ᨨ鼀ᨨ犨 ᨨ �� �泐 矄� 犨鼀ᨨᆠ盐 ᔼ 鼀�ᆠᨨ犨矄泐矄� �鼀鼀泐矄盐㜠�ᨨ �ᨨ 犨矄 �矄� 矄 ᨨ ᆠᨨ鼀� �ᆠ 矄ᆠ盐鼀 ᨨ 泐㝐ᔼ 犨 �ᆠ ᨨ ᨨᨨ犨� ��矄 �鼀 � 矄ᆠ � �ᨨ ᨨ 犨 � 䁸 㝐� 졠 � 꾀�҈黨҈桌 㡰 㝐ᔼ 㡰 㝐 ᔼ ᆠ㡰 ᔼ㝐 㡰 ᔼ㝐 ᨨᨨ 矄ᨨ ᆠᨨ鼀� �ᆠ ��ᨨ 矄 �泐 ҈҈├ᆠ ᨨ犨 �ᆠ ��ᨨ ᨨ ᆠᨨ鼀� �ᆠ矄ᆠ盐鼀 ᨨ矄�ᨨ �䁸 㝐� 졠 � 꾀�黨҈桌 㡰 㜠泐��ᨨ
㡰 �ᨨ��ᨨ
ᆠ㡰 ᔼ ��ᨨ
㡰 �矄���ᨨ
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EBD_8341 P-242
Physics
ᆠᨨᆠ鼀ᆠ 犨 �矄� ᆠᨨ盐鼀 ᔼ ᆠᨨ盐鼀 ᔼᔼ 㝐� 和� ᔼᔼ㡰桌 ᔼ ᔼᔼ ��ᨨ
W �h 꾀㡰 㝐�ᆠᨨ犨 �㡰 ҈鼀 㡰 鼀鼀泐矄 ��㡰 ҈҈�鼀 ᔴ㡰 꾀㝐 �ᨨ犨矄 ���㡰 ҈҈鼀 㡰 㜠泐矄 �㡰 ҈鼀 㡰 꾀㡰㜠 ��㡰 㡰㜠 �㡰 ᔴ㡰㜠 �㡰 㡰㜠 ���㡰 㡰 꾀㡰㜠 �㡰 㡰㜠 ���㡰 ᔴ㡰㜠 �㡰 㡰㜠 ��㡰 ᆠ㡰 꾀㡰㜠 ���㡰 㡰㜠 ��㡰 ᔴ㡰㜠 �㡰 㡰㜠 �㡰 㡰 꾀㡰㜠 �㡰 㡰㜠 ��㡰 ᔴ㡰㜠 �㡰 㡰㜠 ���㡰 � ᨨ犨� ᆠᨨ盐鼀 � 鼀ᨨ 矄ᨨ ᆠᨨ鼀鼀盐 �ᆠ�ᨨ 盐矄� � ├� 矄 ᨨ 犨矄�� � �� ᆠᨨ盐鼀 㝐� ҈꾀�҈黨ᔼ桌 꾀 졠�ᆠ � 矄ᨨ盐 ᔴᨨ鼀鼀盐 �ᆠ�ᨨ
턼 ᔼ �� ᔴ 和ᨨ 㝐�ᆠᨨ犨矄 㝐ᨨ� ᨨ 矄 和 �ᨨ ␔矄 ᨨ鼀 ᨨ�ᨨ 矄�� ᨨ犨 � 鼀ᨨ矄ᨨ� 鼀ᆠ 犨 �矄� �� ᆠᨨ盐鼀 㡰 꾀턼和ᔴ 㡰 꾀和턼ᔴ ᆠ㡰 꾀턼和ᔴ 㡰 꾀 ᔴ和턼 㝐ᆠ �矄㜠ᔼ 쟼ᆠᨨ鼀� �ᆠ犨泐㡰犨� �矄㜠ᔼᔼ ᔼ矄矄矄ᨨᆠ��ᨨ ⏌�ᆠ�ᨨ 㡰 矄ᆠ ᆠᨨᆠᨨ�ᨨ
ᨨ鼀 ᆠ ᨨ�ᆠ矄�� ᨨ犨 �矄矄 㝐� ҈桌
䰘
�矄҈ ᔼ 犨矄 �ᨨ犨矄 㜠泐矄
�矄 ҈ �㡰 ᨨ鼀盐矄ᆠ盐矄�
��㡰 ᨨᨨ ᆠ矄� � ���㡰 ᨨ ᆠ ᆠ盐ᨨ ᨨ 矄 �ᨨ泐矄 �㡰 꾀矄ᨨ 泐 ᨨ ᨨ 泐ᨨ 鼀ᨨ矄 ҈ 㡰 �㡰 ���㡰 ��㡰 �㡰 㡰 �㡰 ��㡰 �㡰 ���㡰 ᆠ㡰 ���㡰 ��㡰 �㡰 �㡰 㡰 �㡰 ��㡰 ���㡰 �㡰 㝐ᆠ �矄㜠ᔼ ᨨ鼀 ᨨ 矄矄ᨨᆠ� 犨� ᆠᨨ鼀� �ᆠ ��ᨨ 㡰 犨� �矄㜠ᔼᔼ ᨨ
黨
ᆠᨨ鼀� �ᆠ矄ᆠ盐鼀㡰 矄ᆠ ᆠᨨᆠᆠᨨ ᨨ鼀 ᆠ ᨨ�ᆠ矄�� ᨨ犨 �矄�矄矄 㝐� 졠 � 꾀�҈҈҈桌
Match the Following
7
ᔼ ᔼᔼ
�矄ᔼ ᨨ 盐 ᨨ 矄ᨨ �盐 鼀
�矄ᔼᔼ 꾀 㡰 �矄� ��ᨨ
␔ � ᆠᨨ 矄 ᨨ � � ᨨ 鼀 盐矄矄ᨨ ᆠ� 犨 � �矄ᨨᨨ �ᆠ � �ᨨ �� �矄ᆠ
㡰 㝐 �ᆠ ᨨ犨
ᔴ 㡰 和 ᨨ �ᨨ 犨
ᔼᔼᔼ ␔ � 鼀 � 泐 ᨨ 鼀�ᆠ 泐 ᨨ � � � 矄 矄 ᆠ
ᔼ ␔ � ᨨ � �ᨨ 㡰 㜠泐矄 �矄� � ᨨ 鼀犨ᨨᆠᨨ矄 � 泐 矄 � 泐 ᨨ �
㡰 ᔼ㡰㜠 꾀㡰 ᔼᔼ㡰㜠 㡰 ᔼᔼᔼ㡰㜠 㡰 ᔼ㡰㜠 ᔴ㡰 㡰 ᔼ㡰㜠 꾀㡰 ᔼᔼ㡰㜠 㡰 ᔼᔼᔼ㡰㜠 ᔴ㡰 ᔼ㡰㜠 ᔴ㡰 ᆠ㡰 ᔼ㡰㜠 㡰 ᔼᔼ㡰㜠 ᔴ㡰 ᔼᔼᔼ㡰㜠 꾀㡰 ᔼ㡰㜠 㡰 㡰 ᔼ㡰㜠 㡰 ᔼᔼ㡰㜠 꾀㡰 ᔼᔼᔼ㡰㜠 㡰 ᔼ㡰㜠 꾀㡰 ҈ 㝐ᆠ �矄 ᔼ ␔ � � ᨨ ᆠᨨ鼀� �ᆠ 矄ᆠ盐鼀㡰 犨� �矄 ᔼᔼ 㝐 ᨨ ᨨ ᨨ 盐ᆠ�ᨨ ᨨ 矄 犨矄㡰 矄ᆠ ᆠᨨᆠᨨ�ᨨ ᨨ鼀 ᨨ�ᨨ 矄��
ᨨ犨 �矄矄 㝐� 졠 � 꾀�黨҈桌 �矄ᔼ
�矄 ᔼᔼ
҈㡰 䰘 鼀ᨨ �㡰 ��ᨨ ᨨ ᨨ鼀矄 ҈鼀鼀 鼀ᨨᆠ盐矄 㡰 ҈ 鼀ᨨ 鼀
��㡰 ᔼ
矄 ᆠᨨ 矄 � ᨨ鼀矄鼀ᨨ� � ᨨ鼀ᨨ �泐 ᨨ ᨨ犨
㡰 ҈ 鼀 ���㡰 �ᨨᆠ� ᆠ泐ᨨ 盐ᆠ盐矄 㡰 ҈鼀鼀ᨨ ҈鼀
㡰 㡰 ᆠ㡰 㡰
�㡰 㝐� ᨨ
҈㡰㜠 �㡰 㡰㜠 ���㡰 㡰㜠 ��㡰 㡰㜠 �㡰 ҈㡰㜠 ���㡰 㡰㜠 �㡰 㡰㜠 �㡰 㡰㜠 ��㡰 ҈㡰㜠 ��㡰 㡰㜠 ���㡰 㡰㜠 �㡰 㡰㜠 �㡰 ҈㡰㜠 �㡰 㡰㜠 ��㡰 㡰㜠 ���㡰 㡰㜠 �㡰
Answer Key Topic-1 : Electromagnetic Waves, Conduction and Displacement Current ҈ ҈҈ ҈ ҈
㡰 ᆠ㡰 ᆠ㡰 ᆠ㡰
҈
ᆠ㡰 ᆠ㡰 㡰 㡰
҈
㡰 㡰 㡰 㡰
҈
㡰 㡰 㡰 㡰
� ᆠ㡰 ҈� ᆠ㡰 � ᆠ㡰
㡰 ҈ 㡰 㡰
䰘 㡰 ҈䰘 㡰 䰘 ᆠ㡰
㡰 ҈ 㡰 ᆠ㡰
黨 㡰 ҈黨 ᆠ㡰 黨 㡰
҈ 㡰 㡰 㡰
㡰
黨 㡰
҈ 㡰
Topic-2 : Electromagnetic Spectrum ҈
㡰
ᆠ㡰
ᆠ㡰
ᆠ㡰
� 㡰
㡰
䰘 ᆠ㡰
WWW.JEEBOOKS.IN
P-243
Ray Optics and Optical Instruments
Ray Optics and Optical Instruments
23
Topic-1 : Plane Mirror, Spherical Mirror and Reflection of Light 1
MCQs with One Correct Answer
҈ 졠ᆠ
� ᆠ鼀㡰
҈
҈
꾀矄 �ᆠ鼀�ᨨ�矄ᨨ� 矄矄 ᨨ犨 � ��盐 ᨨ鼀 ᨨᨨ犨�矄矄矄 ᔼ ᨨᆠ�矄ᨨ矄��ᨨ � ᨨ ᨨ 鼀�ᨨ犨 犨� 盐 鼀� � �ᆠ�ᨨ ᨨ �鼀�ᨨ ᨨᆠ䁸 ��盐 犨 矄矄ᆠ 鼀�ᆠ ᨨ ᨨ矄ᆠ㡰 [Main Sep.02, 2020 (I)] 㡰 ᔼ 鼀� � � 㡰 쟼ᆠ�盐 鼀� � � ᆠ㡰 쟼ᆠ�盐 盐 鼀� � � 㡰 ᔼ 盐 鼀� � � 꾀ᆠᨨ ᆠ鼀�ᨨ ᨨ ᆠ�犨� � 矄 ᨨᆠ � ᨨ 鼀
�矄 ᆠ犨 �ᆠ 泐ᨨ盐 ᨨ 鼀�ᨨ ᨨ鼀泐ᨨ盐 ᆠ � ᨨ ᨨ矄泐ᨨ盐�鼀�盐�� 犨� 鼀� � �ᆠ�ᨨ ᨨ ��矄 㝐� 黨꾀�҈黨ᔼ桌 㡰 鼀 㡰 ҈鼀 ᆠ㡰 鼀 㡰 ҈鼀 꾀ᨨ� 矄ᨨ盐ᆠᨨ �� 和�矄ᆠ �矄 ᆠ� ᨨ ᨨ ᆠ ᨨ 鼀�ᨨᨨ 犨� 犨 �ᆠ �矄 �� � �ᆠ泐ᨨ 犨꾀鼀 犨├矄� ᨨ ᨨ 鼀�ᨨ ᨨ �� ᨨ 鼀�ᨨ �矄 ᆠ矄矄 ᨨ犨
ᨨ犨 �矄 ᆠᨨ犨 �ᆠ 鼀 ᆠ 矄 �鼀� ᨨ �� 矄ᨨ盐ᆠ� 鼀�ᨨ�矄㝐� ҈ ҈黨ᔼ桌 㡰 㡰 和 ᆠ㡰 d 㡰
犨ᨨ 鼀�ᨨ矄� ᆠ� ᨨᆠ ᨨ 矄盐ᆠ 泐ᨨ �� � ᆠ� ᨨ �矄鼀�ᨨ 㝐҈㡰 ᨨ 矄ᆠᨨ 鼀�ᨨ 㝐㡰�矄 � 泐 ᆠ ᨨ鼀 矄ᆠᨨ 鼀�ᨨ 㝐㡰ᨨ �矄鼀�ᨨ 㝐҈㡰 � 犨 犨ᨨ鼀�ᨨ矄犨� 㝐� 黨 ҈黨ᔼᔼ桌 㡰 � 㡰 ᆠ㡰 䰘� 㡰 黨 꾀 ᨨᆠ�矄� 盐泐鼀ᨨ� �犨泐 ᨨ鼀 ᨨᆠᨨ� ᨨ ᆠᨨ ᆠ 鼀�ᨨ ᨨ � �矄 ᨨ 鼀�ᨨ � �ᆠ 矄 �ᨨ ᨨ 鼀� �盐 ᨨ � 鼀� � �ᆠ�ᨨ m㡰矄盐矄 �矄 ᆠᨨ ᨨᆠ ᨨ鼀 鼀�ᨨ x㡰�矄ᆠᨨᆠ泐�� 泐 矄 犨 矄ᆠ 鼀�ᆠ泐 ᨨᨨ矄ᆠ㡰 㝐� ҈黨ᔼᔼ桌
㡰
㡰
ᆠ㡰
㡰
꾀�ᆠ�矄ᨨ矄ᆠ�� �ᨨ 㜠 �矄犨� 鼀�盐 ᆠ鼀ᨨ盐 ᨨ� x㦨҈ᆠ鼀犨� ⼄盐 ᆠ泐w꾀ᆠᨨ ᆠ 鼀�ᨨᨨ ᨨᆠ � �ᆠ鼀�矄ᆠ ᨨ��� 矄 ��盐㡰 ᔼ � 泐 ᆠᨨᆠ矄鼀 矄 㝐� 졠 � 꾀�҈�҈桌 꾀㡰 �鼀� ᆠ盐矄�ᨨ �ᆠ鼀ᨨ�ᨨ
㡰 �鼀� ᆠ盐矄 ᨨ 㜠�ᨨ �ᆠ鼀ᨨ�ᨨ
ᔴ㡰 盐 � �ᨨ� 矄ᨨ �鼀�矄泐鼀鼀�ᆠ犨 �鼀�ᨨ ᨨ� x㦨҈ᆠ鼀 㡰 �矄 ᆠ 犨 盐 � � ᨨ� 矄 ᨨ ҈ x㦨҈ᆠ鼀 ᨨ矄ᆠ��ᨨ ᨨ �鼀��矄 ҈ x㦨 㡰 㡰 㡰 㡰 㡰 ᔴ㡰 ᆠ㡰 꾀㡰 ᔴ㡰 㡰 㡰 꾀㡰 㡰
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EBD_8341 P-244
Physics
Ӹᨨ盐矄├ ᨨ 矄�� 矄 � �鼀�ᨨ矄矄盐鼀� � 矄ᨨ ├矄�҈ᆠ鼀 ᨨ鼀 �矄 ᆠ �犨矄 鼀� � � �鼀�ᨨ ᆠ ᆠᨨ矄矄 ᆠᨨ鼀 ᨨ �矄 ᆠᨨ �ᆠ鼀 �盐矄ᨨ ᆠ盐盐ᨨ 鼀�ᨨ犨ᨨ盐 㝐� 졠 � 꾀�҈҈�桌 㡰 ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ᆠ鼀 ҈ 꾀泐ᨨ �� � �� �ᆠ�ᨨ �ˆ + ˆ 㡰 �矄� ᆠ� ᨨ 鼀�ᨨ꾀 ᆠ�ᨨ �矄ᨨ � �ᆠ�ᨨ
҈ ˆ � - ˆ 㡰 �ᨨ � ᆠ� ᆠ�矄 꾀 ҈桌 㡰 ᆠ㡰 㡰 � 㡰 䰘� 黨 ᔼ �鼀 ᨨ 鼀� ᨨᆠ � f 㡰ᨨ ᆠᨨ ᆠ鼀�ᨨ泐 盐㜠鼀 ᨨ 矄盐 ᆠ矄 ᨨᆠ� 꾀ᨨ � ᆠ� �矄 �矄 ᆠ ᨨ鼀 ᨨP 矄盐 ᨨᨨ├矄 � �矄� �鼀� ᨨ鼀 �矄 ᆠ├� � �矄⏌ 泐� � 犨� PA␔
矄盐 矄 � 矄 �矄⏌ 泐ᨨ犨 矄 �鼀�矄 ᨨ �� ᨨ ᨨᆠ� 䰘桌 㡰 xf 㡰 fxf ᆠ㡰 x㦨f 㡰 x>f ҈ 꾀ᨨ� 矄ᨨ盐ᆠᨨ �� B�矄ᆠ B �矄 ᆠL� ᨨ ᨨ ᆠ ᨨ d 鼀�ᨨᨨ 犨� ἘdἘ 盐 ��ᆠ泐ᨨ 犨꾀鼀 犨├矄� ᨨ ᨨ 鼀�ᨨ L ᨨ �� ᨨ 鼀�ᨨ 2L �矄 ᆠL ᨨ鼀�矄矄 ᨨ犨 � �� �矄 �矄 ᆠᨨ犨 �ᆠ ᆠ 矄 �鼀�ᨨ �� 矄ᨨ盐ᆠ� 鼀�ᨨ�矄 和桌 㡰 d⏌ 㡰 d ᆠ㡰 d 㡰 d ҈҈ 꾀矄 ᨨ� ᨨᆠᨨ � �矄ᨨ � �矄ᨨ ᆠᨨ ᆠ鼀�ᨨᨨ ᨨᆠ � �矄 盐 ᨨ鼀 ᨨᨨ 鼀�ᨨ 矄� ᨨ �鼀��矄ᨨ �鼀泐 ⼄盐ᨨ ҈黨㜠㝐├桌 䰘
㡰
æu - f ö bç è f ÷ø
ᆠ㡰
æu- f ö bç è f ÷ø
2 ҈
æ f ö bç è u - f ø÷
㡰
Integer Value Answer
ᔼ鼀�ᨨ ᨨᆠᨨᆠ � �ᆠᨨ 鼀�ᨨᨨ �盐矄 ᨨ ᆠ盐盐 鼀 ᨨ � �矄ᨨ�ᆠ �矄�矄ᨨ矄 ᨨ � � 鼀ᨨ ᨨ鼀 鼀 ᨨ 鼀� 矄ᆠᨨ 矄␔ �矄 矄 䰘 ᨨ ᨨᆠ� ├鼀 ᨨ盐䁸 ҈桌 3
҈
æ f ö bç è u - f ø÷
҈⏌
㡰
҈⏌
Numeric Answer
␔ ᨨᆠ�矄├ �矄 ᆠᨨ ᆠ鼀 ᨨ鼀ᆠᨨ ᆠ 鼀�ᨨ �鼀��矄 ᨨ鼀 �矄 ᆠᨨ ҈ᆠ鼀 ᨨ鼀 鼀�ᨨ ᔼ ᨨᆠ �矄 鼀ᨨ 犨� 矄 ᨨ 黨 ᆠ鼀矄҈ 矄 � ᆠ鼀矄҈㡰犨� 犨 �ᆠ �鼀�鼀ᨨ矄 � 矄 �矄힘힘힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌
4
Fill in the Blanks
f ҈ 꾀 � ᨨ ᨨ � �矄ᆠ ᨨ � ᨨ�ᆠ �矄ᨨ ᆠᨨ ᆠ鼀�ᨨᨨ ᨨᆠ � 矄盐ᆠ �矄�鼀�犨 �ᆠ �矄 ᨨ � 盐矄 ᨨ盐ᆠ 矄 ᨨ 鼀� � �ᆠ�ᨨ �矄 ҈黨黨҈㜠҈㝐├桌
6
MCQs with One or More than One Correct Answer
҈� 꾀犨��矄 � 矄 ᨨ �� � � � �矄 ᆠ � ᨨ ᨨ ᆠᨨ ᆠ鼀�ᨨᨨ ᨨᆠ � f矄 矄 ᨨ犨 � ��盐␔ �ᆠ ᨨ ��盐矄矄 ᨨ犨 � ᨨ盐ᨨ�ᨨ 矄⼄盐��泐矄 矄㡰 矄 ᨨ �鼀�ᨨ 犨�䁸 矄 ��盐矄 ᨨᨨ矄ᆠ㡰 꾀 ҈桌
ᨨ
� f
f
¥
㡰
a
ᨨ a >�
㡰
¥
ᆠ㡰
a
ᨨ a � 㡰
҈ 꾀矄盐 ᨨ鼀 �鼀 ᨨ 鼀� �ᨨ ᨨ ᨨᆠ � ᨨ ᆠᨨ ᆠ鼀�ᨨ泐u㜠v 鼀 ᨨ 盐矄� �
ᨨ�ᆠ ᆠ ᨨ � ҈�鼀 ᨨᆠ � ᨨ 鼀�ᨨ盐矄 �矄ᆠ鼀 鼀 �鼀盐鼀ᨨ� ᨨᆠ�ᨨ ᨨ �鼀�ᆠ ᆠ鼀 �矄矄ᨨ u, v 㡰盐矄ᆠᨨ 泐 矄盐 � ᆠ鼀㡰 �㡰 㡰 㡰 㡰 䰘黨㡰 矄 矄㡰 ᆠ
ᨨᆠᨨ鼀 ᨨ鼀 �鼀 �矄 㡰� ᆠᨨᆠ泐 ᆠᨨ �矄 㡰 黨桌 㡰 �㡰 㡰 㡰 ᆠ㡰 㡰 㡰 䰘黨㡰 9
Assertion and Reason Type Questions
҈䰘 和T꾀TE㝐ENT㜠҈ ᨨ鼀盐ᆠᨨ
ᆠ� �uv f ᨨ 矄 �ᆠ鼀�ᨨ�矄� ᨨ鼀�ᨨ矄犨 ᨨ矄矄� 矄泐 矄鼀ᆠᨨ鼀 ᨨ � ��ᨨ ᆠ盐盐 和T꾀TE㝐ENT㜠犨矄ᨨ ᆠ�ᨨ 矄�ᆠ泐� ᨨ 矄盐 ᆠ矄盐 ᨨ ᨨ�矄 �ᆠ矄盐 ᆠ矄 䰘桌 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐
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P-245
Ray Optics and Optical Instruments
Topic-2 : Refraction of Light at Plane Surface and Total Internal Reflection 1 ҈
MCQs with One Correct Answer
꾀矄矄ᨨ �矄 � 犨� �⼄盐� ᨨ ᆠ� � 盐 犨� ᨨ �⼄盐� ᨨ ᆠ� � �⼄盐� 矄 �鼀鼀�矄ᆠ� ᨨ �
矄盐 ᆠᨨ ᨨᨨ鼀ᨨ 矄矄 犨� 㝐� 黨 ᔼ桌 h h 㡰 㡰 + ҈㡰
h
h ᆠ㡰 㡰
�矄矄鼀矄ᨨ盐ᆠᨨ �� 矄ᨨ鼀 ᨨ犨 矄盐 ᆠᨨ 犨 ᆠ�� 㦨 㡰� ├ᨨ � ᆠᨨ矄矄 矄ᆠ�ᨨ 矄盐 ᆠ �ᆠ� � 泐 ᆠ�ᨨ
ᨨ鼀 ᨨᨨ鼀 矄ᨨ�ᨨ 泐犨ᆠ �ᨨ �� 鼀�矄ᨨ盐ᨨ 矄盐 ᆠ�矄 泐㡰 矄 ᆠ 矄盐 ᆠᨨ 矄 �ᆠᆠᨨ �� h �盐矄ᨨ ᆠ盐盐r�矄prh桌㝐� 黨 ᔼᔼ桌 㡰 ҈奠 㡰 奠 ᆠ㡰 ҈䰘奠 㡰 �奠
ᆠ��ᆠ �ᨨ 鼀 �盐鼀 ᨨ矄ᆠ� �ᆠ犨 � � 鼀 �盐鼀 矄 � 鼀���泐 �
鼀��泐 ᨨ �矄犨 � 犨� 㝐� ᔼ桌 㡰 ҈� 㡰 ᆠ㡰 � 㡰 꾀 矄 ᆠ盐 ᨨ 矄� 鼀 ᨨ 鼀� ᨨ ᆠ�� m�矄�鼀鼀矄 � �⼄盐� ᨨ ᆠ� � m҈ m҈ m㡰꾀泐�矄� ᆠ� ᨨ ᆠ꾀
�q 矄 ᨨ犨 � ��盐㡰 ᨨ� ᆠ�ᨨ ├矄 ᆠᨨ� 쟼ᨨ ᆠᔴ
䰘
㡰
+
ᆠ㡰
+
�
㡰
q < 矄� -҈
ᆠ㡰
q < 矄� -҈
m҈ m m m҈
-҈
㝐� ҈꾀҈黨ᔼᔼ桌 㡰
q > 矄� -҈
㡰
q > 矄� -҈
m m҈
-҈
m҈ m
꾀泐ᨨ �� 꾀졠� ᆠ盐盐鼀�矄� ᆠ� ᨨ �矄矄矄 �ᨨ ᆠ �ᨨᨨ �졠矄矄 ᨨ犨
� ��盐 ᨨ�ᆠ � ᨨ �� 泐 ᨨ鼀꾀ᨨ �矄 㝐� ҈꾀҈黨ᔼ桌
+
㡰 졔 ᔼ ��盐 ᨨ�ᆠ ��矄l㦨鼀ᨨ � 矄 �鼀 ᨨ d㦨m鼀ᔼ 泐ᨨ �� �矄� ᆠ� ᨨ ᨨ ᨨ � �q҈㦨 盐鼀ᨨ ᆠ�ᨨ 矄�鼀├矄 ᨨ鼀�� � ᨨ鼀 ᨨ �矄ᆠᨨ矄ᨨ ᆠ�� ᨨ ��矄҈҈ 矄� 㦨㡰 㝐� 꾀�҈黨ᔼ桌
㡰 �� 㡰 ᆠ㡰 � 㡰 �䰘 꾀 �� 犨 �矄 � ᆠ� ᨨ鼀泐 ᨨ �矄矄 矄 ᨨ ᆠ�� ҈�ᔼ 奠ᨨ �� �矄 ᆠ 鼀�盐 ᨨ ᆠ�ᆠ � ᨨ � ᆠ� �� �矄⏌ 鼀 鼀�盐 ᨨ ᆠ�ᆠ � ᨨ 犨 ᨨᨨ�� �� �矄矄鼀 �盐鼀犨� 㝐� ҈ ҈黨ᔼ桌 㡰 ⏌鼀 㡰 ҈⏌鼀 ᆠ㡰 ⏌鼀 㡰 ⏌鼀 ᆠ�� ᨨ 矄鼀 �盐鼀犨� 矄ᆠᨨ 鼀 �盐鼀 ҈ �矄ᆠ��ᆠ �qᔴ꾀
�ᨨ � ᆠ� ᆠ꾀犨 �� �矄� � ᨨ鼀 矄 鼀 �盐鼀ᨨ鼀 �盐鼀ᨨ �� �矄 ᆠ 矄�矄 ᆠ �犨 ᆠ ᆠ 泐矄�矄黨 꾀 �꾀�矄�� 泐 㝐� 졠 � 꾀�҈䰘桌 㡰
q鼀盐矄矄�矄 泐
㡰
҈ -҈
ᆠᨨ矄 矄� qᔴ 㡰
㡰
҈ -҈
矄� qᔴ 㡰
ᆠᨨ矄҈ 矄� q
黨
ᆠ㡰 㡰 ҈ 矄� qᔴ㡰 ᔴ㡰 꾀 � ᨨᨨ├� � 盐 ᨨ盐� 犨 矄矄 ᨨ盐矄� 犨ᨨ ᆠᨨ � � ᆠ�ᆠ盐 ᨨ� ᨨ ᆠ�� ᨨ 犨�矄 �矄泐矄҈�ᆠ鼀ᨨ犨 矄盐 ᆠᨨ 犨 �盐矄ᨨ ᆠ�ᆠ�矄 㝐� 졠 � 꾀�黨҈桌 㡰
҈� ´ ´ � ᆠ鼀
㡰 ҈� ´ 䰘 ᆠ鼀
ᆠ㡰
҈� ´ 䰘 ᆠ鼀
㡰
҈� ´ 䰘
ᆠ鼀
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EBD_8341 P-246
10.
Physics
A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is [Adv. 2014]
Liquid
Block
11.
S (a) 1. 21 (b) 1. 30 (c) 1. 36 (d) 1. 42 A light ray falls on a square glass slab as shown in the diagram. The index of refraction of the glass, if total internal reflection is to occur at the vertical face, is equal to : [Main Online April 23, 2013]
Air r Water
r
Glass
i
(a) (4/3)sini (b) 1/sini (c) 4/3 (d) 1 16. Two plane mirrors A and B are aligned parallel to each other, as shown in the figure. A light ray is incident at an angle 30° at a point just inside one end of A. The plane of incidence coincides with the plane of the figure. The maximum number of times the ray undergoes reflections (including the first one) before it emerges out is [2002S] B
2 3m
0.2m 30°
45° Incident ray
A
(a) 28 (b) 30 (c) 32 (d) 34 17. An observer can see through a pin-hole the top end of a thin rod of height h, placed as shown in the figure. The beaker height is 3h and its radius h. When the beaker is filled with a liquid up to a height 2h, he can see the lower end of the rod. Then the refractive index of the liquid is [2002S] 5 3 3 ( 2 + 1) (b) (c) (d) 2 2 2 2 A ball is dropped from a height of 20 m above the surface of water in a lake. The refractive index of water is 4.3. A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is 12.8 m above the water surface, the fish sees the speed of ball as [Take g = 10 m/s2.] [2009] (a) 9 m/s (b) 12 m/s (c) 16 m/s (d) 21.33 m/s A ray of light traveling in water is incident on its surface open to air. The angle of incidence is q, which is less than the critical angle. Then there will be [2007] (a) only a reflected ray and no refracted ray (b) only a refracted ray and no reflected ray (c) a reflected ray and a refracted ray and the angle between them would be less than 180° – 2q (d) a reflected ray and a refracted ray and the angle between them would be greater than 180° – 2q A point object is placed at the centre of a glass sphere of radius 6 cm and refractive index 1.5. The distance of virtual image from the surface is [2004S] (a) 6 cm (b) 4 cm (c) 12 cm (d) 9 cm A ray of light is incident at the glass-water interface at an angle i, it emerges finally parallel to the surface of water, then the value of m g would be [2003S]
(a)
12.
13.
14.
15.
3h h
2h
5 3 5 3 (b) (c) (d) 2 2 2 2 18. A ray of light passes through four transparent media with refractive indices m1, m2, m3 and m4 as shown in the figure. The surfaces of all media are parallel. If the emergent ray CD is parallel to the incident ray AB, we must have [2001S] (a)
m1
m2
D
m3
m4
C B
A
(a) m1 = m2 (c) m3 = m4
(b) m2 = m3 (d) m4 = m1
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P-247
Ray Optics and Optical Instruments
҈黨
꾀ᆠ �盐�矄矄矄ABCDᨨ ᆠ�� n҈�矄 �鼀鼀矄 � 犨ᨨ ᆠ�� n n҈>n㡰꾀泐ᨨ �� �矄� ᆠ� 矄盐 ᆠABᨨ 矄矄矄 ᨨ犨
鼀 �鼀盐鼀盐ᨨ �ᨨ � ᆠ� ᆠa鼀 矄盐ᆠ 泐ᆠᨨ鼀矄ᨨ盐ᨨ 泐 ᨨ鼀 ᨨ 矄盐 ᆠCD�矄 �� 泐 和桌 꾀
㡰
ᔴ
én æ æ n ööù 矄� -҈ ê ҈ ᆠᨨ矄 ç 矄� -҈ ç ÷ ÷ ú è n҈ ø ø úû è êë n
é æ æ ҈ öö ù 㡰 矄� -҈ ên҈ ᆠᨨ矄 ç 矄� -҈ çè n ÷ø ÷ ú è êë ø úû ᆠ㡰
æn ö 矄� -҈ ç ҈ ÷ è n ø
鼀 鼀D
鼀҈ 鼀҈㡰D
æn ö 矄� -҈ ç ÷ è n҈ ø
㡰
꾀 ��� �鼀ᨨ �� ᨨ鼀ᨨ� 矄ᨨ盐ᆠS � � �� ᆠ � a 矄矄泐鼀鼀�ᆠ泐ᨨ �矄矄矄矄 矄 ᨨ犨 �矄ᨨ � ᆠ� ᆠᨨ 犨ᨨ 鼀泐矄 ⼄盐ᔼ �ᆠ├ 矄矄ᨨ �矄矄矄�矄t ᆠ�� n �� ᆠ �ᨨ 鼀� 鼀�矄 和桌 S i
a
i
m
㡰 ᨨ ᆠ㡰 ҈
æ ҈ö 矄� -҈ ç ÷ è nø
t
㡰
a
㡰
æ ҈ö 矄� -҈ ç ÷ è nø
꾀泐ᨨ �� ᨨ鼀 矄鼀 �盐鼀矄�├鼀 �盐鼀 � ᨨ � ᆠ� ᆠ � 矄 ��㡰 ᆠ ᆠ 泐矄鼀├ �ᨨ 黨 犨� ᆠ ᨨ �矄ᨨ ᆠ�ᨨ ᆠ�ᨨ r r ᆠ��ᆠ ��矄 ҈黨㜠҈㝐├桌
i
Integer Value Answer
2
꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� �矄� �� 鼀 �盐鼀ᨨ ᆠ� � 㦨҈ᔼ 矄矄ᆠ├ᨨ �矄矄泐矄 ᨨ鼀 ᨨᨨ鼀矄� �q㦨 � ᆠ矄ᨨ �矄矄 泐矄ᨨᆠ ᨨ ᆠ�� �ᆠ矄ᨨ � �矄矄泐矄鼀ᨨ ᨨᨨ �ᆠ泐 ᆠ矄� �矄 鼀㦨
鼀D 犨 鼀�矄 ᆠ�� ᨨ 鼀 矄 D 㦨҈ 矄 ��盐㡰 泐�矄 ᆠ ᨨ盐ᨨ � ᆠ犨 鼀҈㡰 鼀 矄矄 ᨨ鼀 �� 矄� ᨨ 矄ᆠ├␔ �矄 盐ᨨ 鼀䁸꾀 ҈䰘桌
҈
a鼀
␔ 泐ᨨ �� 矄�矄矄矄 ᨨ鼀� ҈黨桌 㡰 �矄犨 � ᆠ矄矄 㡰 �矄犨 � � ᆠ矄矄 ᆠ㡰 ᔼ矄 ⼄盐 ᆠ泐 ᆠ矄矄 㡰 � �矄犨 � ᨨ�矄 ⼄盐 ᆠ泐ᆠ �矄
r
҈
q
␔ 犨� ᆠ�� 㦨 㡰� ├�矄҈ᆠ鼀 䰘 졠�ᨨ ᆠ�� �矄ᨨ 犨鼀├� �ᆠᨨ 矄盐 ᆠᨨ �盐矄ᨨ ᆠ盐盐� 㦨ᆠ鼀矄矄 ᨨ犨 ᔴᨨ 矄� ᨨ�ᨨᆠ矄 � 矄꾀 ᨨᆠ�和�矄ᆠ ᆠ鼀ᨨ 犨矄盐 ᆠ ᨨᆠ�ᨨ ᨨ �矄�鼀��矄� ᆠ鼀ᨨ ᨨᨨ鼀ᨨ ├ � �矄 ҈҈桌
� 꾀��矄矄矄 m = � ⏌ 㡰 ᨨ �ᆠ├ 矄矄ᆠ鼀�矄ᆠ ᨨᨨ� 矄ᨨ盐ᆠᨨ �� ᨨ 矄盐 ᆠᔼ�矄矄
�� 鼀�矄ᨨ盐ᨨ ᨨ矄盐 ᆠᨨ 矄 ᨨ鼀 ᆠ�ᆠ盐ᨨ �盐矄 ᆠ鼀␔ �矄 盐ᨨ 䁸҈桌 3
rἘ 㡰
sin –҈ tan r 㡰
㡰
sin –҈ tan i 㡰
ᆠ㡰
sin –҈ tan rἘ㡰
㡰
҈ 矄� � 㡰
D
D
D
Numeric Answer
꾀 ᨨ矄ᆠ 矄 ᨨ盐� 矄鼀 ᨨᨨ 矄� ᨨ �盐矄҈�ᆠ鼀㡰ᨨ� �� ᨨ ҈�ᆠ鼀 ᨨ鼀 ᨨᨨ鼀 矄 ��盐㡰 ᨨ�矄 �� ᨨ �ᆠ鼀␔
�矄 � 犨� �⼄盐� 盐ᨨ �� ᨨ ᆠ鼀 矄鼀 ᨨ矄ᆠ 矄 � ᨨᨨ鼀ᨨ ᔼ ᆠ�� ᨨ �⼄盐� �矄N⏌҈犨 N�矄 � � 盐ᨨ N�矄힘힘힘힘힘힘힘힘힘힘힘 [Main Sep. 03, 2020 (I)]
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EBD_8341 P-248
Physics
�ᆠ鼀 ҈�ᆠ鼀
҈�ᆠ鼀
䰘
黨
҈
꾀�� 泐 矄矄ᨨ� �矄矄矄 ᨨ ᆠ�� m = � ᨨ � ᆠ� ᆠ 泐 �矄 ᨨ ᆠ ᆠ 矄盐 ᆠᨨ 矄
� � �矄㡰犨 ᆠ ᆠ 泐矄 �矄矄盐 ᆠ�矄힘힘힘힘힘힘힘힘힘힘힘 [Main Sep.02, 2020 (II)] 꾀 矄盐ᆠ盐ᨨ � L 犨� W�矄鼀 ᨨ 犨ᨨ � ᨨ�ᆠ鼀 �ᨨ ᆠ�� �ᆠ矄n҈㦨҈� n 㦨҈矄矄 ᨨ犨 � ��盐ᔼ L>>W泐 � � ᨨ鼀 꾀犨�鼀� ᨨ鼀 ᔴᨨ 泐� ᨨ� ᆠ�ᨨ ᆠᨨ ��ᨨ �矄鼀� 矄� 矄盐ᆠ盐 ᨨL㦨黨m � � ᆠ� �q�矄� 鼀 �鼀盐鼀�鼀├
泐泐ᨨ � ᔴ�矄t맔҈㜠黨s犨 t�矄힘힘힘힘힘 꾀 ҈黨桌
4 Fill in the Blanks 꾀�� ᨨ 犨 � 处� � 矄鼀 �盐鼀犨� ᆠ�� ҈�ᔼ 矄� 鼀 �盐鼀�矄 ⼄盐 ᆠ泐�矄 �矄犨 � �矄处 ҈黨黨䰘㜠㝐├矄桌 꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ鼀ᨨ �� ᨨ 犨 � 处�
ᆠ盐盐鼀 矄鼀 �盐鼀ᨨ ᆠ�� ҈�ᔼ 鼀 �盐鼀�矄犨 � �矄�矄 ⼄盐 ᆠ泐�矄 ҈黨�㜠㝐├矄桌 꾀�� 犨ᨨ ⼄盐 ᆠ泐�맔҈҈ 矄鼀 �盐鼀ᨨ ᆠ�� ҈�ᔼ 鼀 �盐鼀 ᨨᆠ�泐ᨨ �� 犨�矄 �矄犨 � �矄҈黨㜠㝐├矄桌 6 MCQs with One or More than One Correct Answer 꾀 矄 矄ᨨ �ᆠ├ 矄矄 矄 ᆠ�� 㡰 � ᆠ矄矄犨� �矄 �ᆠ �矄 ᆠ� 矄� 矄鼀矄盐 ᨨ鼀 ᨨ 矄�矄ᆠ 犨
犨ᨨ鼀 �犨� 盐 � ᨨ鼀 ᆠ�� �ᆠ矄 ҈ > ҈㡰 矄矄 ᨨ犨 � ��盐꾀泐ᨨ �� �矄� ᆠ� 犨� � q� ᨨ鼀鼀 �盐鼀҈ 鼀�矄� 鼀 �盐鼀犨� ᆠ�ᨨ
꾀 ҈桌 �q犨� �矄ᆠ鼀 l
␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 㡰 矄� q�㦨 矄� q 㡰 ҈矄� q�㦨 ҈㡰矄� q ᆠ㡰 l�矄� ᨨ 㡰 l�矄 ᨨ 㡰 꾀泐ᨨ �� � �� 矄 鼀 �盐鼀 矄ᨨ 矄盐 ᆠ矄� � 鼀 �盐鼀 ᨨ鼀� �ᨨ � ᆠ� ᆠ ᨨ � 泐盐 �ᨨ矄ᨨ� ᆠ�ᨨ ᔼ n �矄 ᆠ�� ᨨ 鼀 �盐鼀犨� 矄ᆠᨨ�矄ᆠ ᨨ矄矄�盐 矄㡰ᨨ ᨨ鼀 ᨨᨨ犨� �҈黨黨㜠㝐├矄桌 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈� 㡰 ҈ Comprehension/Passage Based Questions 矄矄� �� �盐� ᆠ� ᨨ�ᆠ �ᆠ 盐 矄ᨨᨨ 泐 ᆠᨨ 矄� � � 矄盐ᆠ盐 ᆠᨨ鼀�矄� � ᨨ � 矄ᨨ� �矄矄 ᆠ泐� ᨨ ᆠ�� n҈矄盐ᨨ盐 泐鼀 �盐鼀ᨨ ᨨ犨 ᆠ� � n �� �盐� ᆠ � 矄盐ᆠ盐 ├矄 ᆠ 盐 ᨨ 矄盐ᆠᆠ矄矄� ᨨ � ᆠ�ᨨ 矄 � ᆠᨨ 鼀 �n҈ n矄矄 ᨨ犨
� ��盐꾀泐矄犨� �ᨨ � ᆠ� ᆠ�矄矄 �ᆠ盐盐�mᆠᨨ � � 鼀 �盐鼀ᨨ ᆠ�� n҈ 盐鼀�ᆠ盐 꾀㡰ᨨ 矄盐ᆠ盐�矄 � 矄s�n�鼀 8
ᨨ犨ᨨ矄盐ᆠ盐 鼀泐S҈犨� n҈ = � ⏌ n㦨⏌ S犨� n҈㦨⏌� n㦨䰘⏌� ├� � ᆠ� � ᨨ 犨ᨨ⏌ ᨨ �ᨨ҈ ᆠᨨᆠ ᨨ�ᨨ 矄㡰�矄 㡰 꾀 ҈�桌 㡰 NAᨨ S҈�鼀鼀矄 � 犨�矄 矄鼀矄 ᨨ S ҈ �鼀鼀矄 � �⼄盐� ᨨ ᆠ�� ҈� 㡰 NAᨨ S҈�鼀鼀矄 � �⼄盐� ᨨ ᆠ�� �矄 ҈� 矄鼀矄 ᨨ S�鼀鼀矄 � 犨 ᆠ㡰 NA ᨨ S҈ ᆠ � � �矄 矄鼀 矄 ᨨ S �鼀鼀矄 � �⼄盐� ᨨ ᆠ�� ҈� 㡰 NAᨨ S҈ᆠ � ��矄 矄鼀矄 ᨨ Sᆠ � 犨 � ᔼ 犨ᨨ矄盐ᆠ盐ᨨ 矄鼀ᆠᨨ矄矄㜠矄ᆠ�ᨨ 盐 �
盐鼀�ᆠ盐矄NA҈ NA NA2NA1㡰ᨨ� ᨨ ��盐 � 泐 盐鼀�ᆠ盐ᨨ ᆠᨨ鼀� 矄盐ᆠ盐�矄 꾀 ҈�桌 㡰
NA҈ NA NA҈ + NA
ᆠ㡰 NA ҈
㡰 NA҈졔NA 㡰 NA
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P-249
Ray Optics and Optical Instruments
10 Subjective Problems
꾀⼄盐ᆠ泐� ᨨ �盐矄R ᆠ�� ҈��矄 ᆠ ᨨ 꾀ᨨ� ᨨᆠ�矄├ �矄 ᆠᨨ mR ᨨ鼀� � 盐ᨨ m ᨨ犨 �ᆠ 泐 ᨨ鼀P犨� 鼀�ᨨ 矄矄 ᨨ犨 � ��盐 ҈黨黨黨㜠�㝐├矄桌
黨 꾀鼀ᨨ �� � �� 犨 ᆠ�� 㦨⏌㡰�矄 ᆠ 泐矄 �ᆠ�盐ᨨ �盐矄鼀鼀 矄�盐 � 犨꾀矄矄盐鼀� � �� 泐矄ᨨ � ҈黨㜠㝐├矄桌 �㡰 � ᨨ矄��ᨨ ᨨ �鼀� 盐ᨨ ᆠ�ᨨ �矄矄盐 ᆠ ᨨ矄��ᨨ ᨨ � �鼀� ��㡰 犨泐 ��鼀矄 ᨨ犨� � ᨨ矄��ᨨ 矄ᨨ ᨨ �鼀�矄 꾀�� �矄鼀�矄ᨨ鼀 泐矄ᆠ� �ᨨ鼀� �矄A B B £ A㡰 矄矄 ᨨ犨 � ��盐ᔼ�矄 矄� 泐ᨨ �� � ᆠ� ᨨ ᆠ꾀鼀�矄 ᨨ � ᆠ� �ᆠ�ᨨ 犨ᨨ � ᆠ�ᨨ 矄
䰘
鼀
xy �矄 ᨨ盐 泐犨 犨ᨨ 矄 鼀 �㝐 �盐鼀҈犨� z ³ 矄 ᆠ�� 2 鼀 �盐鼀犨� z £ 矄 ᆠ�� 3 꾀 泐 ᨨ �� � 鼀 �盐鼀 ҈ �� 泐 ᆠᨨ A = � + ҈ ├ �矄 � ᆠ� ᨨ ᨨ 矄�ᨨ � 盐 � ᆠᨨ � �ᆠ�ᨨ ᨨ ᆠ 泐� 鼀 �盐鼀 ҈黨黨黨㜠҈㝐├矄桌 �� �矄� ᆠ� � a ᨨ ᨨ ᨨ 矄 ᆠ泐� �ᆠᨨ ᨨ ᆠ�� m鼀� 矄盐ᨨ m矄ᨨ �� � � ᨨ ᨨ矄
ᨨ鼀� ᨨ鼀 ᆠ盐 矄盐 ᆠᨨ ᨨ �矄ᆠ�ᨨ 盐ᨨ a ҈黨黨㜠㝐├矄桌
꾀
ᔴ �㡰
␔ 矄 ᨨ盐 鼀� �鼀盐鼀 ᆠ�� n ᨨ �矄ᨨᨨ矄矄�䁸 ��㡰 ᨨ n = � �矄�ᨨ矄矄�ᨨᆠ � �矄犨� � B⼄盐ᨨ �矄䁸 ҈黨䰘㜠䰘㝐├矄桌
m a
Topic-3 : Refraction at Curved Surface Lenses and Power of Lens
꾀ᨨ� �├ᨨᆠ�矄ᆠ �矄 ᆠᨨ ҈鼀� ᨨ ᨨ ᆠᨨ 矄ᨨ ᨨᆠ � �鼀꾀 鼀�ᨨ�矄ᆠ �矄 ᆠᨨ 鼀 � 矄 ᨨ矄��ᨨ 盐ᨨ � �鼀� ᨨ鼀 泐 矄泐矄鼀�矄㝐� 和 ᔼ㡰桌 㡰 鼀 ᨨ鼀 鼀�ᨨ 㡰 ҈鼀 ᨨ鼀 鼀�ᨨ�盐 ᆠ㡰 ҈鼀 ᨨ鼀 鼀�ᨨ 㡰 鼀 ᨨ鼀 鼀�ᨨ�盐 꾀 ᨨ盐ᆠᨨ 矄 矄ᨨ犨P 矄鼀 ��ᨨ ᆠ盐㜠 盐Rᨨ ᨨ 矄盐 ᆠ矄 �盐矄ᨨ ᆠ盐盐ᨨ 矄盐 ᆠᨨ ᨨ㜠ᆠᨨ 矄鼀 ᨨ 矄鼀鼀� 犨� ᨨ犨҈�P�矄 㝐� 和 ᔼᔼ㡰桌 R R R 㡰 R 㡰 ᆠ㡰 㡰
ᨨᆠᨨ ᆠ 矄ᨨ ᨨᆠ � f �ᨨ 犨
ᨨᆠ �鼀� �矄 ᆠ矄u v矄ᆠ�泐 ᨨ鼀�矄 ᨨᆠ 矄矄 泐 u㦨v�矄 ᆠ� 㡰 㝐� 和� ᔼ㡰桌 v f
盐 㦨 v
҈
MCQs with One Correct Answer
㡰
f v f
盐
盐 㦨 v
1
㡰
f
盐
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EBD_8341 P-250
= u
v f
v
Physics
(c)
u
f v
v f
= u
8. (d)
f 4.
5.
(b) 9
(c) 5
(d)
17
One plano-convex and one plano-concave lens of same radius of curvature ‘R’ but of different materials are joined side by side as shown in the figure. If the refractive index of the material of 1 is m1 and that of 2 is m2, then the focal length of the combination is : [Main 10 Apr. 2019 I]
(a)
R m1 - m2
(b)
2R 2(m1 - m2 )
(d)
2R m1 - m2
R 2 - (m1 - m2 ) The graph shows how the magnification m produced by a thin lens varies with image distance v. What is the focal length of the lens used ? [Main 10 Apr. 2019 II]
(c) 6.
u
A thin lens made of glass (refractive index = 1.5) of focal length f = 16 cm is immersed in a liquid of refractive index 1.42. If its focal length in liquid is fl ,then the ratio fl /f is [Main 7 Jan. 2020 II] closest to the integer: (a) 1
4 3 (b) (c) (d) 3 2 3 2 An upright object is placed at a distance of 40 cm in front of a convergent lens of focal length 20 cm. A convergent mirror of focal length 10 cm is placed at a distance of 60 cm on the other side of the lens. The position and size of the final image will be : [Main 8 April 2019 I] (a) 20 cm from the convergent mirror, same size as the object (b) 40 cm from the convergent mirror, same size as the object (c) 40 cm from the convergent lens, twice the size of the object (d) 20 cm from the convergent mirror, twice the size of the object A convex lens (of focal length 20 cm) and a concave mirror, having their principal axes along the same lines, are kept 80 cm apart from each other. The concave mirror is to the right of the convex lens. When an object is kept at a distance of 30 cm to the left of the convex lens, its image remains at the same position even if the concave mirror is removed. The maximum distance of the object for which this concave mirror, by itself would produce a virtual image [Main 8 Apr. 2019 would be : II] (a) 30 cm (b) 25 cm (c) 10 cm (d) 20 cm What is the position and nature of image formed by lens combination shown in figure? (f1, f2 are focal lengths) [Main 12 Jan. 2019 I]
(a)
9.
10.
2 cm A
20 cm
B
f1 = + 5 cm
f2 = –5 cm
(a) 70 cm from point B at left; virtual (b) 40 cm from point B at right; real 20 cm from point B at right, real 3 (d) 70 cm from point B at right; real 11. Formation of real image using a biconvex lens is shown below : [Main 12 Jan. 2019 II]
(c)
a b b2 b2c (b) (c) (d) c c ac a A thin convex lens L (refractive index = 1.5) is placed on a plane mirror M. When a pin is placed at A, such that OA = 18 cm, its real inverted image is formed at A itself, as shown in figure. When a liquid of refractive index µi is put between the lens and the mirror, the pin has to be moved to A’, such that OA’ = 27 cm, to get its inverted real image at A’ itself. The value of µi will be:[Main 9 Apr. 2019 II]
(a)
7.
2f 2f
f
screen
f
If the whole set up is immersed in water without disturbing the object and the screen positions, what will one observe on the screen ? (a) Image disappears (b) Magnified image (c) Erect real image (d) No change
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P-251
Ray Optics and Optical Instruments
҈
҈
҈
҈�
҈
꾀 ᨨᆠ�矄 �矄 ᆠᨨ 鼀 ᨨ鼀ᆠᨨ 矄ᨨ ᨨᆠ � 鼀 矄 ᨨ鼀矄 �鼀�ᨨ ᨨᆠᔼ ᨨᆠ鼀ᨨ矄犨泐 ᨨ鼀 矄矄 ᨨ �鼀⏌矄 矄 �ᆠ�ᨨ ᨨ �鼀�犨� 㝐� ҈҈ ҈黨ᔼ桌 㡰 맔҈鼀⏌矄犨泐 ᨨ鼀 矄 㡰 黨맔҈鼀⏌矄犨泐 ᨨ鼀 矄 ᆠ㡰 맔҈鼀⏌矄ᨨ犨 矄 矄 㡰 ҈҈맔҈鼀⏌矄ᨨ犨 矄 矄 꾀 ᨨ ᆠᨨ 矄 ᨨ ᆠ� � m҈ ᨨᆠ � f҈�矄├� ᆠᨨ ᆠ犨� ᨨ ᨨᆠᨨ ᆠ 矄 ᨨ ᆠ� � m ᨨᆠ � f ᔼ �盐矄 ᨨ ᆠ盐盐 ᨨ � 矄 �ᆠ ᆠ矄 �矄 ᆠ f҈ 㦨 f m҈ m 矄 㝐� ҈ ҈黨 ᔼ桌 㡰 m҈졔m㦨 㡰 m҈m㦨҈ ᆠ㡰 mm҈㦨҈ 㡰 mm҈㦨҈
泐ᆠ � 矄矄� � ᆠ� �矄盐 ᆠ
�盐矄ᨨ ᆠ盐盐 ᨨ �矄 矄盐 ᆠ�矄⼄盐 ᨨ ᨨ ᆠᨨ 䰘鼀鼀㡰 �矄矄盐 ᆠ矄矄犨ᨨ鼀 �ᨨ ᆠ�� �ᆠ矄҈ ҈ᔴᆠ盐 �矄 ᆠ ᨨ鼀 ᆠ� �矄盐 ᆠ犨 �ᆠ 鼀ᨨ �� 㝐� ҈ ҈黨 ᔼᔼ桌 犨�ᆠᨨ鼀ᨨ ᨨᆠ盐矄 㡰 ҈ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ҈ᆠ鼀 꾀ᆠᨨ 矄�矄盐҈ᆠ鼀 ᨨ鼀 �� 矄ᨨ盐ᆠ � 鼀├矄矄 �鼀�ᨨ 矄ᆠ ├҈ᆠ鼀 ᨨ鼀 矄ᨨ犨�矄矄ᨨᆠ├ ᆠ�� ҈�㡰ᨨ ҈�ᆠ鼀 �ᆠ├ 矄矄�矄ᆠ � ᆠᨨ ᆠ犨� �� 矄ᨨ盐ᆠ ᨨ � 矄 �鼀� �� 矄ᆠ鼀 �矄 矄 � 泐 �矄 ᆠ �矄 㝐� 黨 ҈黨ᔼ桌 㡰 ҈҈ᆠ鼀犨泐 ᨨ鼀 矄 㡰 ᆠ㡰 ��ᆠ鼀ᨨ犨 矄 矄 㡰 ��ᆠ鼀犨泐 ᨨ鼀 矄 꾀 ᨨᆠᨨ 矄ᆠᨨ鼀矄 ᨨ�ᆠ矄泐矄鼀ᨨ ᆠ鼀 ᨨᆠ � 犨 �矄 矄盐 ᆠ �矄 矄� �盐鼀� ᨨ鼀 ᨨ�� 矄矄 ᨨ犨 � ��㜠꾀ᔼ 矄鼀 矄 �矄 � 矄 矄� ᨨ ᆠ盐 矄盐 ᆠ �盐鼀� ᨨ鼀ᨨ 矄� 矄� ���ᆠ矄�├ ᨨ�ᆠ 矄泐矄鼀ᨨ ᨨᆠ � ҈ᆠ鼀 ᆠ�� ᨨ 鼀�ᨨ 矄� 㝐� 졠 � 꾀�҈�҈桌
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㡰 ҈� 㡰 ҈�� ᆠ㡰 ҈䰘� 㡰 ҈�҈ ᔼ �鼀 ᆠᨨ 矄ᨨ ᨨᆠ � ҈�ᆠ鼀�矄 ᆠ ᆠᨨ �泐ᨨ ᨨ�ᆠ ᆠ � ᨨ ᨨ ᆠᨨ 鼀�ᨨ �矄 ᆠᨨ �ᆠ鼀 ᨨ鼀�ᔼ�矄 ᨨ盐
ᨨᆠ �矄�鼀�ᆠᨨ� ᆠ� � ᨨᆠ�矄ᆠ �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 矄 ᨨᆠ � ᨨ 㝐� 졠 � 꾀�黨҈䰘桌 ᆠᨨ 鼀�ᨨ�矄 㡰 䰘�ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 �ᆠ鼀 㡰 �ᆠ鼀 꾀 鼀�矄 �ᆠ�矄矄ᨨ 泐ᨨ �盐矄҈ᆠ鼀 ᆠ� � ҈��矄矄� ᨨ �矄ᆠ盐 矄盐 ᆠ꾀矄鼀�盐 �矄ᆠ鼀ᨨ犨 矄盐 ᆠ� 矄� �ᨨ � �矄 ᨨ矄��ᨨ ᨨ �鼀�ᨨ �盐鼀 泐 鼀�ᨨ �矄矄 㝐� 졠 � 꾀�҈҈桌
҈ᆠ鼀 ᆠ鼀 졠
和� 㡰 ҈ᆠ鼀ᨨ犨 矄盐 ᆠ 㡰 ᆠ鼀ᨨ犨 矄盐 ᆠ ᆠ㡰 ҈ᆠ鼀ᨨ犨 矄盐 ᆠ 㡰 ᆠ鼀ᨨ犨 矄盐 ᆠ ҈黨 꾀ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ鼀ᆠᨨ ᆠ 矄ᨨ ᨨᆠ � ҈ᆠ鼀 鼀�ᨨ � 矄 矄 ᨨ犨 ᨨ ᨨᆠ├ �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 ᆠᨨ 矄 � �鼀� ᨨ鼀 泐 ᆠᨨ鼀� �ᨨ �矄 �鼀� �矄 ᆠᨨ 㝐� 졠 � 꾀�黨҈桌
ᨨᆠ � ᨨᆠ � 㦨ᆠ鼀㦨҈ᆠ鼀 ᆠ鼀 ᆠ鼀 䰘ᆠ鼀 㡰 ᆠ鼀 ᨨ鼀 ᆠᨨ 矄 㡰 ᆠ鼀 ᨨ鼀 ᆠᨨ ᆠ 矄 ᆠ㡰 䰘ᆠ鼀 ᨨ鼀 ᆠᨨ 矄 㡰 䰘ᆠ鼀 ᨨ鼀 ᆠᨨ ᆠ 矄 꾀 � ᆠᨨ 矄ᨨ ᨨᆠ � � �矄盐ᨨ 鼀�ᨨ矄矄 ᨨ犨 � ��盐␔ ᨨᆠ�矄├ �矄 ᆠ� ᨨ鼀 矄㜠鼀�ᨨᆠᨨ鼀� �ᨨ �矄�鼀� a �矄 ᨨ鼀 �矄 ᆠ � ᨨ ᨨ ᆠᨨ鼀� �ᨨ 3 㝐� 졠 � 꾀�҈҈ ҈�桌
盐ᨨ ��矄
㡰
㡰
3 2
ᆠ㡰
㡰
ö æ ҈ 꾀 � ᆠᨨ 矄鼀 ᨨ鼀ᆠᨨ犨 �矄矄 ç m = ÷ 矄 ø è ᨨᆠ � ␔ ��矄鼀矄盐 � 犨ᨨ � �⼄盐� 矄 � � � ᆠ�� �ᆠ矄 � 矄 ᨨᆠ � 矄 ҈ 矄ᆠ�泐 ᆠᨨᆠ �ᨨ 犨 ᨨᆠ � 矄�矄 㝐� ҈桌 㡰 ҈㦨 㡰 ҈> ᆠᨨ鼀矄 �� ᆠ㡰 > ҈ᆠᨨ鼀矄 �� 㡰 ҈ ᨨ ᆠᨨ鼀 ��
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EBD_8341 P-252
Physics
ᆠ�� ᨨ 鼀�ᨨ ᆠᨨ ᆠ 矄�矄mᔼ �矄�鼀鼀矄 � 鼀 �盐鼀ᨨ ᆠ�� m҈꾀 鼀ᨨ �� �矄� ᆠ� ᨨ 矄 ᨨ 鼀� 泐矄犨 m҈>m�矄 㝐� 졠 � 꾀�҈҈桌 m m҈
m҈
㡰 ҈鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀 꾀矄鼀ᨨᆠ�矄ᆠ �ᆠ鼀ᨨ ᨨ � ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ鼀꾀ᆠᨨ 矄 �ᆠ鼀�ᨨᨨ �盐矄ᨨ ᆠ盐盐҈ᆠ鼀�矄ᆠ ᨨ �� ᨨ 矄 �矄 ᆠᨨ �ᆠ鼀 鼀�ᨨ�矄� 矄盐ᆠ �矄ᨨ 鼀�ᨨ�矄 �q㦨 ᨨ �矄ᨨ 矄矄矄 ᨨ犨 � ��盐 㦨ᆠ鼀
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m
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ᆠ㡰
-�㡰
ᔼ ᨨ��� ᨨ ᆠᨨᨨ � 矄泐矄鼀�矄├ ᨨ ᆠ ᨨ 矄 ᆠᨨᨨ � 矄 � ᆠ鼀㡰 ᨨ ᨨ� 泐㡰犨 �ᆠ �鼀��矄 ᨨ鼀 꾀 ҈桌
㡰
m҈
㦨 ҈ ᆠ 鼀
� � 㡰
㡰 㡰
� 㜠 � �㡰 ҈� ⏌ � 㡰
䰘 꾀�㜠ᆠᨨ 矄�矄 ᨨ鼀 犨� 犨ᨨ � ᨨ㜠ᆠᨨ 矄矄矄矄 ᨨ犨 � ��盐 ᆠ�� nᨨ �矄 矄�矄҈� ᨨ 矄ᆠᨨ 矄�矄҈ᨨ ᆠ盐 矄盐 ᆠᨨ 矄鼀 �盐矄ᨨ ᆠ盐盐R㦨҈ ᆠ鼀 ᨨ �矄�㜠ᆠᨨ 矄 ᨨ ᨨᆠ �矄 ᆠᨨ ᆠ鼀 �鼀� �矄 ᆠ犨� ҈桌
m m҈
m҈
㡰
�
�鼀ᨨ ᨨ㜠ᆠᨨ 矄�矄ᆠ鼀 �ᆠ├ 矄矄 ᆠ �矄鼀鼀ᔼ 矄 ᨨ �� � 鼀�ᨨ 矄�矄 맔҈鼀⏌矄 ᨨᆠ � ᨨ 矄�矄 㝐� ҈桌 㡰 ҈�ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ҈ᆠ鼀
�鼀�ᨨ �盐鼀� 矄⼄盐�矄ᨨ� ᨨ 矄ᆠ
犨� ᨨ ᆠᨨ �� � 矄 �矄 ᆠᨨ 矄⼄盐 ᨨ鼀 矄�矄ᆠ鼀 ᨨ �鼀��矄黨 �鼀矄 ᨨ 矄⼄盐 ᨨᆠ � ᨨ 矄�矄 㝐� 졠 � 꾀�҈桌 㡰 ᆠ鼀 㡰 䰘ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ᆠ鼀
�鼀�ᨨ ᨨᆠ ᨨ鼀 泐 ᨨ㜠ᆠᨨ 矄 �矄 ᆠᨨ 鼀 � 矄�矄�矄ᨨ 㜠 � 矄� ᨨ ᨨᆠ 犨 � ᨨ �� � 矄� 矄 �矄 �鼀矄 犨 � � 矄ᆠ �盐矄ᨨ ᆠ盐 矄盐 ᆠᨨ 矄�矄 꾀 ҈桌
黨
㡰 㜠ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ҈�ᆠ鼀 㡰 ҈ᆠ鼀 泐矄ᨨ �� ᨨ鼀和盐 矄ᨨ �ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ�ᆠ盐�鼀�ᨨ 和盐 ᨨ �盐矄�矄 ᨨ鼀 ᨨ ᨨᆠ ᨨ 矄
䰘桌 㡰 꾀ᨨ �鼀��矄p �矄 �ᆠ泐ᨨᨨ�ᨨ ᨨ f 㡰 꾀ᨨ �鼀��矄p �矄 �ᆠ泐ᨨᨨ�ᨨ ᨨf ᆠ㡰 ᔼ 矄�泐ᨨ �鼀�� ᆠ矄矄� �矄� ᆠ矄 㡰 ᔼ ᨨ犨 ᨨ 矄�矄ᆠᨨ 犨� ᆠ├ 犨�ᆠᨨ鼀 꾀�ᆠᨨ 矄ᨨ ᨨᆠ � ҈�ᆠ鼀�矄� ᨨ ᨨ 鼀�ᨨ �矄 ᆠ犨 矄 鼀�ᨨ�矄҈ ᆠ鼀꾀矄鼀ᨨᆠ�矄├ �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 矄 � �鼀��矄 ҈桌 㡰 �盐 �矄 ᆠᨨ ҈ᆠ鼀 ᨨ鼀 鼀�ᨨ 㡰 �矄 ᆠᨨ ҈ᆠ鼀 ᨨ鼀 鼀�ᨨ ᆠ㡰 �盐 �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 鼀�ᨨ 㡰 �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 鼀�ᨨ
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� 矄 ᨨ犨矄 �ᨨ 矄 � 犨 ᨨᆠ �矄 ᆠ �鼀� �矄 ᆠ ᨨ ⼄盐�ᆠᨨ 矄 ᨨᆠ � ᨨ 矄�矄 㜠㝐桌桌 ҈ ᆠ鼀
҈
盐ᆠ鼀҈
҈
҈
黨졔黨㡰
㡰 �᠀�ᆠ鼀 㡰 �᠀҈ᆠ鼀 ᆠ㡰 �᠀�ᆠ鼀 㡰 �᠀҈ᆠ鼀 ᨨᆠ � ᨨ ᨨ㜠ᆠᨨ 矄�矄҈�ᆠ鼀꾀矄鼀 ᨨᆠ�矄ᆠ 꾀矄矄 ᨨ犨 � ��盐 矄盐 ᆠ�矄矄� �鼀�犨� ᨨ鼀㜠㝐҈桌
꾀 �鼀� �矄 �鼀�ᨨ ᨨᆠ�矄ᨨᨨ� ᨨ 矄ᆠ ҈鼀 ᨨ鼀� �矄ᆠ ᆠ � 泐ᨨ�泐 ᆠ� � ҈黨黨�和桌 㡰 ᆠᨨ ᆠ鼀�ᨨᨨ 矄盐� ᨨᆠ � 㡰 ᆠᨨ 鼀�ᨨᨨ 矄盐� ᨨᆠ � ᆠ㡰 ᆠᨨ 矄ᨨ ᨨᆠ � 矄矄 �鼀 㡰 ᆠᨨ ᆠ 矄ᨨ 矄盐� ᨨᆠ � 䰘 和 �ᆠ�ᨨ � � 矄ᆠ 盐ᆠ 泐 㡰 盐矄� �鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ҈黨黨㜠҈㝐├桌 㡰 盐矄� � ᨨ盐ᆠᨨ鼀� �ᨨ
ᆠ㡰 盐矄� �ᆠ�ᆠ盐
盐鼀├ᨨ 矄 㡰 � ᆠ矄� � 矄� ᨨ 矄 ᔴᨨ 矄� ᆠᨨ ᆠ鼀�ᨨ ᆠᨨ 矄 ᆠ� � 㦨҈�㡰ᨨ ᨨᆠ � ҈ᆠ鼀ᆠ 矄 泐 �矄 ᆠᨨ �ᆠ鼀� � ᆠ�� 㦨҈㡰矄矄 ᨨ犨 �
��盐꾀 ᨨᆠ�矄ᆠ �矄 ᆠᨨ ҈�ᆠ鼀 ᨨ鼀 鼀�ᨨᔼ矄ᆠ�鼀� ᨨ鼀 泐 �矄ᆠᨨ鼀� �ᨨ 矄 鼀� � �ᆠ�ᨨ M҈␔ 矄㜠盐�矄├� 鼀 �盐鼀ᨨ 䰘 ᆠ�� 鼀� � �ᆠ�ᨨ ᆠᨨ鼀矄M 鼀� �盐
꾀
M �矄 M҈
ᆠ鼀
㡰 ᆠ鼀ᨨ ᨨ 矄 㡰 ҈ᆠ鼀ᨨ ᨨ 矄 ᆠ㡰 ᆠ鼀ᨨ �� ᨨ 矄 㡰 ᆠ鼀ᨨ ᨨ 矄 꾀 ᆠᨨ 矄 �矄 � ᆠᨨ ᆠ 犨� ᆠᨨ ᆠ 矄 鼀� �盐 ᨨ �ᨨᨨ � ᨨᆠ � �矄⏌ � ⼄盐� ᨨᆠ � �矄ᆠ鼀␔ �� �� 盐 ᨨᆠ � 矄䁸 �和桌 㡰 ҈�҈ 㡰 ҈҈� ᆠ㡰 䰘�� 㡰 䰘�� 矄� ᨨ �鼀�ᨨ ᨨᆠ犨 �ᆠ �矄� � �泐矄 ᨨ鼀 泐ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ鼀�矄ᆠ鼀ᔼ ᆠᨨ ᆠ 矄ᨨ ᨨᆠ � ᆠ鼀�矄ᆠ 犨 ᆠᨨ 矄 �鼀� �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 ᆠᨨ 矄ᆠᆠ盐 犨矄� ᨨ �鼀�和桌 㡰 ҈�ᆠ鼀 㡰 �ᆠ鼀 ᆠ㡰 ҈�ᆠ鼀 㡰 ᆠ鼀 꾀 ᨨᨨ犨 ᨨ盐 ᆠᨨ ᆠ 矄 �矄 鼀 ᨨ 泐 �
矄 鼀�ᔼᆠ � 犨� �ᨨ� ᨨ 犨ᨨ �⼄盐� 矄 L҈ ᨨ L � � ᆠ� � �ᆠ矄 m҈ m 矄ᆠ�泐 m>m҈>҈㡰 矄犨� �� 鼀ᨨ �� � ��矄 � 犨� 和桌 㡰 � ᆠ � � 㡰 � �鼀鼀矄 � L҈ ᆠ㡰 L҈ �鼀鼀矄 � L 㡰 L �鼀鼀矄 � L҈ � 꾀 ᆠᨨ ᆠ 矄 ᨨ �矄矄 ᆠ� � ҈� 矄 ᨨ 矄盐 ᆠ矄ᨨ 矄鼀 �盐矄ᨨ ᆠ盐盐R졠 �鼀鼀矄�ᨨ � 鼀 �盐鼀ᨨ ᆠ�� ҈䰘��犨� 矄 ҈黨黨黨和㜠㝐├矄桌 㡰 ᆠᨨ � 矄ᨨ ᨨᆠ � �R 㡰 ᆠᨨ � 矄ᨨ ᨨᆠ � R ᆠ㡰 �� 矄ᨨ ᨨᆠ � �R 㡰 �� 矄ᨨ ᨨᆠ � R
҈�ᆠ鼀
꾀 ҈�桌 �ᆠ鼀
2
Integer Value Answer
黨 ᨨᆠ � ᨨ � �ᆠᨨ 矄�矄ᆠ鼀␔
ᨨᆠ�矄鼀ᨨ ᨨ鼀 �矄 ᆠᨨ �ᆠ鼀� ᨨ ᨨ �ᨨ� ᆠ鼀 鼀� � �ᆠ�ᨨ ᨨ �矄�鼀�ᆠ �矄 ᨨ鼀鼀�ᨨ m� 鼀� �ᨨ �矄 ҈桌 m� 3
Numeric Answer
�矄 ᆠ犨 ᨨᆠ 矄ᆠ �矄҈ᆠ鼀꾀 矄ᆠ ᨨ 盐ᆠ�鼀�ᨨ ᨨᆠᨨ 矄ᆠ ᨨ 犨ᨨ � ᨨ矄��ᨨ 矄犨 矄ᆠ ᨨᆠ
�矄 ᆠ犨 矄犨ᨨᨨ矄��ᨨ 矄�矄ᆠ鼀ᔼ æ N ö ᨨ犨ᨨ 矄�矄ᆠᨨ矄ᨨ ç D 犨 N�矄 � � è ҈ ÷ø 盐ᨨ N�矄힘힘힘힘힘힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌 ҈ 꾀ᨨ� ᨨᆠ� ��矄� ᨨ ᨨ ᆠ盐 矄盐 ᆠᨨ planᨨ㜠cᨨnvex 矄 �盐矄ᨨ ᆠ盐盐ᨨ ᆠ盐 矄盐 ᆠ�矄ᆠ鼀 ᆠ�� ᨨ 矄鼀� �矄 ҈� ᨨᆠ � ᨨ 矄 � ᆠ鼀㡰 �矄힘힘힘힘힘힘힘힘힘힘 㝐� ᔼ桌 4
Fill in the Blanks
犨ᨨ � 矄矄犨 � ᆠᨨ ᆠᨨ 盐ᆠᆠᨨ鼀� �ᨨ
ᨨ ᨨ犨졔҈ �ᨨ矄␔ 泐�鼀 ᨨ犨 盐ᆠ矄 ᨨ졔 �ᨨ矄 ᨨᆠ � ᨨ 矄矄m m ҈黨黨䰘㜠㝐├矄桌
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�
Physics
꾀 � 矄ᨨ ᆠ�� ҈� 矄 ᨨᆠ � ᨨ ҈�ᆠ鼀 � �␔ 矄�矄ᆠ � 鼀 �盐鼀ᨨ ᆠ�� �矄 ᨨᆠ � 犨�ᆠᨨ鼀ᆠ鼀 ҈黨䰘㜠㝐├矄桌 꾀ᆠᨨ 矄Aᨨ ᨨᆠ � ᆠ鼀 ᆠᨨ ᆠ 矄 Bᨨ ᨨᆠ � �ᆠ鼀├ᨨ � 矄鼀 �矄犨� �矄 ᆠd犨 鼀ᔼ 鼀ᨨ �� � � ᨨ A矄B矄鼀 d�矄⼄盐ᨨᆠ鼀 ҈黨�㜠㝐├矄桌
5 True / False 꾀鼀ᨨ 犨 ��� ᨨ ᆠᨨ鼀� �ᨨ ᨨ ᆠᨨ ᆠ ᆠᨨ 矄 ᨨ ᨨ 矄鼀鼀�
� ᨨᆠ � 矄҈�ᆠ鼀 ᆠ鼀矄ᆠ�泐 ᨨ 鼀 犨 � � 犨 ��� 졠 ᨨ 矄� ᨨ 矄 矄泐矄鼀 ᨨ 矄矄 ᆠᨨᨨ盐 矄 犨� �ᨨ ᆠᨨᨨ盐 ᨨ盐 � ҈黨㜠㝐├矄桌 꾀ᆠᨨ 矄ᨨ ᨨᆠ � ҈鼀 ᆠᨨ ᆠ 矄 ᨨ ᨨᆠ � �鼀├䰘�鼀꾀 鼀ᨨ �� �矄矄矄矄 ᨨ盐� ᆠᨨ 矄
ᨨ盐� ᆠᨨ ᆠ 矄 ᆠᨨ鼀矄 ᨨ ᨨᆠ盐矄 � 鼀 犨泐 ᨨ鼀 ᆠᨨ ᆠ 矄 ҈黨㜠㝐├矄桌 6
䰘
MCQs with One or More than One Correct Answer
꾀├ᨨ �盐矄r�矄 � 犨� 犨 ᆠ�� 㡰盐 ᨨ �� H矄矄 ᨨ犨 � ��盐ᨨ ├�矄 ├ᨨ ᨨ� ᨨ ᨨ� �犨� �盐矄 w �矄 鼀├矄 犨矄盐 ᆠᆠ盐 矄ᨨ � ᆠ� �� ᨨ 犨 ᆠ ᆠ�ᆠ盐鼀 ᆠᨨ ├�矄 h h = H h = r 㡰 矄矄 ᨨ犨 � ��盐ᨨ
�� ├ �矄矄盐 ᆠᨨᨨ �鼀泐矄 �ᆠ犨� �盐矄ᨨ ᆠ盐盐R␔ �ᆠ ᨨ ᨨᨨ犨� ��矄⏌ᆠᨨᆠ䁸 g�矄 ᆠᆠ�ᨨ 盐ᨨ��泐㡰 w
h
H
㡰 R =
h + r h
r h ᆠ㡰 꾀 ᨨ ᨨᨨ鼀 ᨨ ├ �矄ᆠᨨ矄
꾀 ᨨᆠᨨ 矄�矄鼀 ᨨ 鼀�ᨨ ᆠ� � ␔ 矄鼀ᨨᆠ�矄ᆠ ᆠ鼀犨泐� ᨨ ᨨ ᆠ盐 矄盐 ᆠᨨ 矄 �鼀�ᨨ ᨨ盐 矄� ᨨ ᨨᆠ�矄ᨨ 盐ᆠ 盐ᨨ ᆠ�ᨨ ᨨ鼀 ᆠᨨ 矄盐 ᆠᨨ 矄 ᨨ � �鼀��矄ᨨ矄 �矄 ᆠᨨ ҈ᆠ鼀犨泐 ᨨ鼀 矄␔ �ᆠ ᨨ 꾀 ҈桌 ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 㡰 ᆠ�� ᨨ 矄�矄� 㡰 �盐矄ᨨ ᆠ盐盐ᨨ ᆠᨨ 矄盐 ᆠ�矄�ᆠ鼀 ᆠ㡰 � �鼀��矄ᆠ 㡰 ᨨᆠ � ᨨ 矄�矄ᆠ鼀 黨 꾀矄 �ᆠ矄盐 ᆠᨨ �盐矄ᨨ ᆠ盐盐 矄矄� ᆠ�� ҈㡰 ᨨ鼀�矄矄 ᆠ�� ҈�㡰 ᆠ ᨨ ᆠ盐盐�矄� �矄矄꾀ᨨ� ᨨᆠᆠ � ��矄 ᨨ盐 ᨨ �鼀�Q� �矄矄 � PQ ᆠ盐矄 矄盐 ᆠ ᨨ� O PO 㦨 OQ ҈黨黨㜠㝐├矄桌 �矄 ᆠPO�矄⼄盐ᨨ 㡰 �R 㡰 R ᆠ㡰 R 㡰 ҈�R � 꾀�鼀�ᨨ �矄 ᨨᆠ�矄 ᨨ鼀 泐 ᨨ㜠ᆠᨨ 矄ᨨ �矄� ᆠ� �矄和 �ᆠ�ᨨ
㡰 �矄矄 ҈黨黨㜠㝐├矄桌 㡰 �矄矄鼀� ᆠ盐 矄盐 ᆠᨨ 矄 ᆠ矄 ᨨᆠ ᆠ㡰 �矄矄鼀� 矄盐 ᆠᨨ 矄 ᆠ矄 ᨨᆠ 㡰 �矄 矄鼀犨 �ᆠ 矄� ᨨ 矄 ᆠ矄 ᨨᆠ �҈ ␔ �ᆠ ᨨ ᨨᨨ犨� � ᨨ鼀 矄㡰�盐 ᆠ�鼀� ᨨᨨ矄��ᨨ 矄ᨨ ᨨᆠ䁸 ҈黨黨㜠㝐├矄桌 㡰 ᔴᨨ 矄 㡰 ᔴᨨ ᆠ 矄 ᆠ㡰 ᔴᨨ 鼀�ᨨ 㡰 ᔴᨨ ᆠ鼀�ᨨ � 꾀ᆠᨨ �� � 矄�矄盐矄 ᨨ ᨨ鼀 �鼀�ᨨ 矄ᆠ ␔ 盐 ᨨ 矄�矄ᆠᨨ 泐 ᨨ⼄盐 矄ᆠ
҈黨㜠㝐├矄桌 㡰 �鼀�犨� �矄 㡰 ᆠᨨ鼀�鼀�犨� ᨨ鼀 ᆠ㡰 � 矄�泐ᨨ �鼀�犨�� ᆠ矄 㡰 � 矄�泐ᨨ �鼀�犨� ᆠ矄 � 꾀ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ鼀�矄� ᆠᨨ ᆠ犨� ᆠᨨ ᆠ 矄 ᨨ ᨨᆠ � � ᆠ鼀 ᨨ犨 ᨨ ᆠᨨ鼀� �ᨨ �矄 ҈黨㜠㝐├矄桌 㡰 ҈� �ᨨ矄 㡰 � �ᨨ矄 ᆠ㡰 졔� �ᨨ矄 㡰 졔䰘 �ᨨ矄 � ᨨ盐ᆠᨨ鼀� �ᨨ 矄ᨨ 犨ᨨ � 矄矄�� � �矄㜠ᔼ
�盐矄ᨨ ᆠ盐盐ᨨ ᆠ盐 矄盐 ᆠ矄�矄r ᆠ� � ᨨ 矄矄 �矄 ҈� 㝐ᆠ 矄 ᆠᨨ鼀� �ᨨ 矄� �矄㜠ᔼ犨� � ᨨᆠ � � �矄㜠ᔼᔼ 矄ᆠ ᆠᨨᆠ 矄犨盐矄� � ᆠᨨ �� ᨨ犨 �矄矄 꾀 ҈桌 �矄㜠ᔼ �矄㜠ᔼᔼ
㡰 R =
҈
r
턼
r
-҈
H æ w H ö ç҈ + ÷ ᨨ çè g ÷ø 㡰 꾀 ᨨ ᨨᨨ鼀ᨨ ├�矄ᆠᨨ矄ᨨ H æ w H ç҈ + çè g
ö ÷ ÷ ø
-҈
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ᔴᨨ盐鼀 ᔼ
��
r
和
r
꾀㡰
ᔴᨨ 矄 㡰 㜠҈턼㜠 㜠和㜠 㡰 㜠턼㜠 㜠和㜠҈ ᆠ㡰 㜠턼㜠҈ 㜠和㜠 㡰 㜠턼㜠҈ 㜠和㜠
犨ᨨ 矄 鼀 �ᨨ ᆠ�� �ᆠ矄 m1 m 矄ᨨ� 矄 矄 矄 鼀� ᨨ ᆠ� � m 犨 鼀矄矄 ᨨ犨 � ��盐矄� ᔴᨨ盐鼀 ᔼᔼ꾀 꾀 泐矄� � 矄鼀 ��矄矄ᨨ矄 ᨨ犨 � ��盐矄ᔼ
ᔴᨨ盐鼀 ᔼ � �ᨨ 矄 �矄犨 m҈ m m �� 㝐ᆠ 鼀ᨨ 泐 ��鼀矄矄 ᨨ犨 � ᔴᨨ盐鼀 ᔼᔼ ҈桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ
꾀㡰 m҈ < m
㡰
㡰 m҈ > m
⼄㡰 m
ᔴ㡰 m = m
㡰
m
m҈
m
m҈
m
m
㡰
ᔴ㡰
㡰
ᔴᨨ盐鼀 ᔼᔼ
和
㡰 �鼀�
和
⼄㡰 �盐�鼀�
和
㡰 鼀� � � �鼀�
和
矄㡰 �鼀�� � �泐
10 Subjective Problems �䰘 꾀 ᨨᆠ�矄 鼀ᨨ� �犨� ᨨᆠ�泐҈鼀⏌矄ᨨ犨 矄 ᆠᨨ 矄ᨨ ᨨᆠ � 鼀 � 鼀� �盐 ᨨ ᨨ 矄�ᨨ ᨨ �鼀� ᨨ鼀 矄犨 ᨨᆠ �矄 �矄 ᆠᨨ 鼀 ᨨ鼀 矄꾀矄ᨨᆠᆠ盐 鼀� �盐 ᨨ ᨨ ᆠ � ᨨ 㜠㝐├矄桌 鼀� � �ᆠ�ᨨ � 꾀 � ᨨ㜠ᆠᨨ 矄ᨨ ᨨᆠ � f �矄矄�� ᨨ犨ᨨ 矄ᨨ ᨨ 矄�矄矄 � ᨨ � ᨨ�ᆠ �矄
矄�ᨨ 犨 ᨨᆠ �鼀� 矄�矄҈鼀
鼀� � �ᆠ�ᨨ ᨨ �鼀� ᨨ鼀 泐ᨨ ᨨ 㜠 矄矄�矄 � ᨨᆠ㜠 � ᨨ 矄 矄�ᨨ
犨 犨ᨨ 矄犨 泐 ��鼀 ᨨ�鼀� ᨨ鼀�ᨨ ҈黨黨㜠�㝐├矄桌
m҈
m
졠
㡰 m > m
矄㡰 m
㡰 m
�
m
҈鼀
m҈
m
m҈
꾀 ᨨ�ᆠᆠᨨ鼀ᨨ ᨨᆠ和ᆠ ᨨ ��矄 ᨨ�ᆠ �矄�� � ᔴᨨ盐鼀 ᔼ �矄 ᆠ犨 ᨨᆠ ᆠᨨ鼀ᨨ ᆠ � ᨨ�矄 ᨨ �鼀�矄�� � ᔴᨨ盐鼀 ᔼᔼ㝐ᆠ ᨨ�矄 ᨨ �鼀�矄 ᨨ鼀ᔴᨨ盐鼀 ᔼᔼ犨� ᨨ�ᆠᨨ鼀ᨨ㜠
矄�� � ᔴᨨ盐鼀 ᔼᔼ �ᆠ泐ᨨ盐 矄犨泐 ├ 㜠 � � ᨨ�盐矄ᨨ 맔鼀� �� � 졠 和 桌
�黨 꾀 �鼀�Y�矄 ᨨ鼀 ᨨ ᨨ� ᨨᆠX泐 矄犨 ᨨ矄 ᨨ�ᆠ �矄�矄AB矄矄 ᨨ犨 � ��盐犨泐 ��鼀ᨨ ᨨᆠ 矄 �矄 ᨨᆠ盐矄ᔼ �鼀�Ӹᨨ ᨨᆠX �矄 ᨨ鼀 泐ᆠᨨ ᆠ鼀�ᨨ � � 矄鼀 �矄矄 AB㡰� 矄 ᨨ 矄 犨 ᨨ 泐 ��鼀ᨨᨨᆠ 鼀�ᨨ �矄 ᨨᆠ盐矄␔� ᨨ犨 矄矄ᨨ ᆠᨨ 矄盐ᆠ�ᨨ
҈黨黨㜠㝐├矄桌 ᨨ 泐 ��鼀矄
꾀
Ӹ
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EBD_8341 P-256
60.
Physics
In Fig., S is a monochromatic point source emitting light of wavelength l = 500nm. A thin lens of circular shape and focal length 0.10 m is cut into two identical halves L1 and L2 by a plane passing through a diameter. The two halves are placed symmetrically about the central axis SO with a gap of 0.5 mm. The distance along the axis from S to L1 and L2 is 0.15 m while that from L1 and L2 to O is 1.30 m. The [1993 - 5+1 Marks] screen at O is normal to SO. L1
A
0.5mm
S
O
62. A plano convex lens has a thickness of 4 cm . When placed on a horizontal table, with the curved surface in contact with it, the apparent depth of the bottom most point of the lens is found to be 3 cm. If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the centre of the plane face is found to be 25/8 cm. Find the [1984- 6 Marks] focal length of the lens. 63. The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed on a horizontal surface. [1981- 6 Marks]
Screen
L2 0.15m
1.30m
(i)
61.
If the third intensity maximum occurs at the point A on the screen, find the distance OA. (ii) If the gap between L1 and L2 is reduced from its original value of 0.5mm, will the distance OA increase, decrease, or remain the same? Monochromatic light is incident on a plane interface AB between two media of refractive indices n1 and n2 (n2 > n1) at an angle of incidence q as shown in fig. The angle q is infinitesimally greater than the critical angle for the two media so that total internal reflection takes place. Now if a transparent slab DEFG of uniform thickness and of refractive index n3 is introduced on the interface (as shown in the figure), show that for any value of n3 all light will ultimately be reflected back again into medium II. Consider [1986 - 6 Marks] separately the cases (i)
n3 < n1 and
E
MEDIUM I
(n 3 )
G
F
B
A q
Where should a pin be placed on the optic axis such that its image is formed at the same place? (ii) If the concave part is filled with water of refractive index 4/3, find the distance through which the pin should be moved so that the image of the pin again coincides with the pin. 64. What is the relation between the refractive indices m1 and m2, if the behaviour of light rays is as shown in the figure? [1979] m1
m
m1
m2
m m2
n3 > n1 .
(ii) MEDIUM I (n1 )
D
(i)
65. A pin is placed 10 cm in front of a convex lens of focal length 20 cm, made a material of refractive index 1.5. The surface of the lens farther away from the pin is silvered and has a radius of curvature are of 22 cm. Determine the position of the final image. Is the image real as virtual? [1978]
MEDIUM II
(n 2 )
Topic-4 : Prism and Dispersion of Light 1 1.
MCQs with One Correct Answer
The variation of refractive index of a crown glass thin prism with wavelength of the incident light is shown. Which of the following graphs is the correct one, if D m is the angle of minimum deviation ? [Main 11 Jan. 2019, I]
Dm
(a)
1.535 1.530
400 500 600 700
n2
l (nm)
1.525 1.520 1.515 1.510
400
500
600
700
l (nm)
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P-257
Ray Optics and Optical Instruments 鼀
� 矄 ᨨ犨矄 �ᨨ 矄 � 犨 ᨨᆠ �矄 ᆠ �鼀� �矄 ᆠ ᨨ ⼄盐�ᆠᨨ 矄 ᨨᆠ � ᨨ 矄�矄 㜠㝐҈㡰 ҈ ᆠ鼀
㡰
l 鼀㡰
� 䰘
鼀 ҈
盐ᆠ鼀҈
ᆠ㡰 l 鼀㡰
� 䰘
䰘
鼀
㡰
�
黨졔黨㡰
҈
㡰 �᠀�ᆠ鼀 㡰 �᠀҈ᆠ鼀 ᆠ㡰 �᠀�ᆠ鼀 㡰 �᠀҈ᆠ鼀 㝐ᨨ ᨨᆠ ᨨ鼀�ᆠ�� �矄� ᆠ� ᨨ �矄矄�矄鼀ᨨ � 꾀ᔼ ᆠ�� ᨨ 鼀�ᨨ �矄鼀�矄Ҙ 泐 � ᆠ� � q ᨨ ᆠ 꾀 犨ᨨ盐 � 矄鼀� ᨨ盐� ᆠ꾀ᔴᨨ �矄鼀ᨨ� 㝐� ҈�桌 꾀
l 鼀㡰
� 䰘
q
꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� �矄� ᆠ� ᆠ� �ᨨ
⼄盐�� �盐�矄鼀 矄盐 矄鼀� �鼀盐鼀 ��ᨨ ᔼ ᆠ�� ᨨ 鼀�ᨨ �矄鼀�矄 �ᨨ � ᆠ� ᆠ�矄 㝐� ҈҈ ҈黨ᔼᔼ桌 㡰 黨 㡰 ᆠ㡰 㡰 � 꾀泐ᨨ �� �矄� ᆠ� �ᨨ ᨨ ᨨ ᆠᨨ �矄鼀ᨨ � 鼀� 泐ᨨ �� 鼀├矄
�ᨨ 犨� � ᆠ� 泐 �鼀 泐 鼀� 泐犨� 矄ᆠᨨ ᆠᨨ �矄鼀犨� 㝐� 졠 � 꾀�҈҈桌 㡰 㡰 黨 ᆠ㡰 㡰 � ᔼ �鼀 ᨨ 鼀� �ᨨ ᨨ ᆠ�� ᨨ �矄矄ᨨ �矄鼀泐�dᨨ�犨矄 ᨨ盐 泐� ᆠ� �� 矄盐 矄 ��ᨨ ᨨ �鼀�矄 �䰘黨 ᔼ ᆠ矄犨 �ᆠ ᨨ ᨨᨨ犨� ��矄ᆠᨨ矄矄 ᨨ 鼀 �鼀盐鼀ᨨ矄矄�盐ᨨ ᆠ�� 䁸 㝐� ҈桌 㡰 ҈䰘 㡰 ҈ ᆠ㡰 ҈� 㡰 ҈ 꾀鼀ᨨ �� �矄� ᆠ� ᨨ鼀� �aᨨ
矄� 턼ᨨ �� � � �盐�矄鼀ᨨ ᆠ� � 㦨 �� 盐 �ᨨ矄ᨨ� ᆠ�ᨨ �
�矄鼀 ᆠ 犨 a 矄鼀� �鼀盐鼀盐ᨨ � �qᨨ �矄鼀�矄 꾀 ҈桌
ᔴ
㡰
é æ æ ҈ öù q > ᆠᨨ矄 -҈ êҘ矄� ç 꾀 + 矄� -҈ ç ÷ ú è Ҙ ø ûú è ëê
㡰
é æ æ ҈ öù q < ᆠᨨ矄-҈ êҘ矄� ç 꾀 + 矄� -҈ ç ÷ ú è Ҙ ø ûú è ëê
ᆠ㡰
é æ æ ҈ öù q > 矄� -҈ êҘ矄� ç 꾀 - 矄� -҈ ç ÷ ú è Ҙ ø ûú è ëê
㡰
é æ æ ҈ öù q < 矄� -҈ êҘ矄� ç 꾀 - 矄� -҈ ç ÷ ú è Ҙ ø ûú è ëê
� 犨 �ᨨ ��ᨨ d㡰 �ᨨ � ᆠ� ᆠ �㡰 ᨨ� �盐�矄鼀�矄矄 泐 㝐� ҈桌 d
d
㡰
㡰 ᨨ
q
ᨨ
�
a
d
d
ᆠ㡰
=
㡰 ҈�
턼 㡰 �
ᆠ㡰
㡰 ᨨ
㡰 �
�
ᨨ
�
�
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EBD_8341 P-258
9.
Physics
A beam of light consisting of red, green and blue colours is incident on a right-angled prism on face AB. The refractive indices of the material for the above red, green and blue colours are 1.39, 1.44 and 1.47 respectively. A person looking on surface AC of the prism will see [Main Online May 26, 2012] A
(a) are parallel to each other (b) are diverging (c) make an angle 2 [sin–1 (0.72) – 30°] with each other (d) make an angle 2 sin –1 (0.72) with each other 14. A glass prism of refractive index 1.5 is immersed in water (refractive index 4/3). A light beam incident normally on the face AB is totally reflected to reach on the face BC if [1981- 2 Marks] (a) (b)
45°
C (a) no light (b) green and blue colours (c) red and green colours (d) red colour only Two beams of red and violet colours are made to pass separately through a prism (angle of the prism is 60°). In the position of minimum deviation, the angle of refraction [2008] will be (a) 30° for both the colours (b) greater for the violet colour (c) greater for the red colour (d) equal but not 30° for both the colours An equilateral prism is placed on a horizontal surface. A ray PQ is incident onto it. For minimum deviation [2004S]
B
10.
11.
R
Q
sin q >
8 9
A
2 8 < sin q < 3 9
2 3 (d) None of these
(c)
2
B q
sin q
m
턼
e ®g
m>m҈ m>m
e ®h
m҈㦨m
和
e ®�
mm҈ m m>m
ᆠ㡰 꾀鼀� �鼀盐鼀 ��ᨨ � ᆠ� ��҈ ᆠ� � � ҈ �矄 ᆠ� �矄盐 ᆠ 泐҈㦨 �҈⏌㡰 㡰 ᨨ �矄�矄鼀 鼀� 泐 矄ᆠᨨ 矄盐 ᆠ 犨� � �ᨨ 矄盐 ᆠ犨 �ᨨ � ᆠ� ᆠ �矄矄盐 ᆠ�矄
é ù 꾀 �҈ = 矄� -҈ ê矄� 꾀 ᆠᨨ矄 - ҈ - ᆠᨨ矄 꾀 ú ë û
҈
꾀 � �矄鼀҈犨� � 鼀 ᨨ鼀�矄矄ᨨ ᆠ�� ҈��矄ᆠᨨ鼀� 犨� ᨨ � �矄鼀 P鼀 ᨨ鼀�矄矄ᨨ ᆠ�� ҈䰘ᨨᨨ 盐ᆠ �矄矄�ᨨ 犨� ᨨ盐 ��ᨨ �ᨨ �矄鼀P�矄 ҈黨黨㜠㝐├矄桌 㡰 � 㡰 ᆠ㡰 㡰 꾀鼀ᨨ �� ᆠᨨ 矄�矄� �ᨨ � 盐ᆠᨨᨨ盐矄 �矄� ᆠ� ᨨ �� � �矄鼀 �� ᆠ� � �ᆠ矄ᨨ 鼀�ᨨ �矄鼀 ᨨ ᨨ �
盐犨 � 矄҈黨҈ ҈䰘矄ᆠ�泐
�矄鼀犨� ҈黨黨㜠㝐├桌
�矄ᔼᔼ
�
m҈
�
m
m
ᔴᨨ 矄
턼
和
㡰
҈
㡰 ҈
ᆠ㡰
҈
㡰
҈
10 Subjective Problems 犨ᨨ� �ᆠ�矄鼀矄ᨨ ᆠ�� ├矄 矄 ᨨ犨 � ��盐꾀�� 泐矄�├矄 �矄�矄鼀 ᆠAB � �㜠㝐├矄桌
� 㡰 矄ᨨ ᆠᨨᨨ盐 ᨨ鼀 � 盐ᆠᨨᨨ盐矄 㡰 矄ᨨ 盐ᆠᨨᨨ盐 ᨨ鼀 �
ᆠᨨᨨ盐矄 ᆠ㡰 矄 ᆠᨨᨨ盐矄 ᨨ鼀ᨨ ᨨ 㡰 ᨨ矄 �泐 泐ᆠᨨᨨ盐 ᨨ鼀 ᨨ 犨ᨨᆠᨨᨨ盐矄
�
꾀
ᔴ
쟼
㡰 �ᨨ � ᆠ� ᆠ矄ᨨ 鼀� 泐 ᨨ鼀 �矄�矄鼀 矄鼀� �鼀盐鼀 ��ᨨ 㡰 ᨨ盐� 犨 � �矄鼀DCE矄 ᨨ盐 ᨨ ᨨ盐 C 矄ᨨ � 鼀� 泐 矄ᨨ 矄 鼀� �鼀盐鼀 ��ᨨ
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EBD_8341 P-260
�
Physics
和 ᨨ犨 � ��盐�矄�矄鼀ᨨ �º ᆠ� � m p = ᆠ꾀ᔴᨨ �矄鼀�矄ᆠᨨ 犨� � �鼀ᨨ ᆠ�� m 㦨꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ 犨 � l㦨�� 鼀 ᨨ ᆠAB �ᨨ � ᆠ� ᆠᨨ º 㜠㝐├矄桌 꾀
º
㡰 ᔴᆠ盐 犨 � l ᨨ犨 �ᆠ 泐矄� ᆠ� 泐 �ᨨ � ᆠBC矄矄 ᨨ盐� 犨� ᨨ盐 � � � ᆠ 㡰 ᨨ�� ᨨ 犨 � l � �ᨨ � ᆠ� ᆠ �ᨨ ᆠAC矄盐ᆠ ��ᨨ ᨨ 盐ᆠ 泐 ᆠᨨ鼀� �ᨨ ᨨ �矄鼀矄�矄鼀� �鼀盐鼀 䰘 꾀�� � �矄鼀 � 黨 � 㡰ᨨ ᆠ�� 矄 ᨨ ᆠ�� n҈ n҈n㡰ᆠ鼀 ᨨ�矄 ��ᨨ ᆠ 矄矄鼀泐�矄� �꾀泐�矄� ᆠ� ᨨ AB
º
꾀 m =
ᔴ
m 㦨
ᔴᆠ盐 㡰 �ᨨ 鼀� ᆠ 㡰 鼀� �鼀盐鼀盐ᨨ �ᆠ├ 矄矄t 矄ᨨ � 矄�泐ᨨ 鼀� 泐�矄鼀 �鼀盐鼀 꾀�矄鼀ᨨ ᆠ�� n҈ ᨨ �矄鼀ᨨ ᆠ� � n 矄盐ᆠ├ ᨨ� 犨� ᨨ盐 �矄 矄 ᨨ犨 �
��盐 �矄ᨨ �矄鼀矄矄矄 ᨨ犨 n҈ n ᨨ l 犨 � ᨨ �� ᆠᆠᨨ � � ᨨ
n҈ = ҈ +
҈ ´ ҈
l 犨 l �矄� nm ᔴ
n = ҈� +
҈ ´ ҈
l ҈黨黨㜠㝐├矄桌
䰘
҈
�㡰 ᔴᆠ盐 �ᨨ � ᆠ� ᆠAB ᨨ犨 �ᆠ 泐矄�├矄 ��ᨨ ᆠ ᆠ��ᆠ � ��㡰 꾀矄矄盐鼀� �n㦨҈�ᆠᆠ盐 �ᨨ � ᆠ� ᆠ AB ᨨ犨 �ᆠ ᆠ 泐矄矄矄 ᨨ盐� ��ᨨ ᆠ盐 � ҈黨黨㜠㝐├矄桌 꾀泐ᨨ �� �矄� ᆠ� �ᨨ ᨨ ᨨ ᆠᨨ �矄鼀犨 �ᆠ 矄 �ᨨ 泐鼀�� �ᨨ盐ᨨ �矄鼀鼀├矄 �ᨨ 犨� � ᆠ� 泐和 ᨨ犨 鼀� 泐�矄 �ᆠ盐ᨨ ᆠ ᨨ盐� 犨 �ᆠ �鼀�矄 ᆠᆠ盐 ᆠ�� ᨨ 鼀�ᨨ �矄鼀 ҈黨䰘桌
҈
꾀
Topic-5 : Optical Instruments 1 ҈
MCQs with One Correct Answer
鼀� � 泐� �ᨨ犨ᨨ 矄ᆠᨨ犨� 盐 � ᆠ鼀�矄�␔ �矄 ᨨᆠ � ᨨ �矄泐�ᆠ䁸 㝐� ᔼ桌 㡰 ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 ҈ᆠ鼀 ᔼ 犨 鼀� � �ᆠ�ᨨ ᨨ 䰘� ᨨ鼀 ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨᨨ 盐 � ҈�鼀鼀 ᨨᆠ�ᨨ ᨨᆠ � �鼀鼀 ᨨᆠ � ᨨ 泐㜠�ᆠ矄 ᨨ盐 ᆠᨨ矄ᨨ 㝐� 䰘 ᔼ桌 㡰 鼀鼀 㡰 ҈鼀鼀 ᆠ㡰 鼀鼀 㡰 鼀鼀 꾀 ᨨ矄ᨨᨨ├矄 �矄 ᨨ �� ҈鼀犨� 矄ᆠᨨᨨ 鼀� � 泐� �ᨨ犨ᨨ ᨨ ᨨ矄 矄 㝐� ҈桌 㡰 �鼀矄 㡰 �鼀矄 ᆠ㡰 ҈�鼀矄 㡰 ҈�鼀矄
ᨨ 鼀� ᆠ� � ᨨ �矄矄 矄 盐矄� � � � 鼀�ᆠᨨ矄ᆠᨨ 鼀� �鼀盐鼀 盐鼀 ᨨ � �矄
�
䰘
⼄盐� 㝐� 졠 � 꾀�҈҈桌 㡰 犨ᨨ 㡰 ᨨ盐 ᆠ㡰 㡰 � 꾀矄ᆠᨨ 矄 ᨨᆠ� 矄ᨨ ᨨᆠ � ҈�ᆠ鼀 泐�ᆠᨨ ᨨᆠ � �ᆠ鼀ᔼ �鼀ᨨ犨 �矄 ᆠᨨ ҈├鼀�矄ᨨ矄 ᨨ盐� �矄矄ᆠᨨ�
ᨨ鼀矄� � � ᨨ鼀 泐 �鼀�ᨨ ᨨ犨 �矄q q�矄ᆠᨨ矄ᨨ 㝐� 졠 � 꾀�҈҈�桌 㡰 㡰 ҈� ᆠ㡰 㡰 ҈ ᔼ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ ᨨᆠ � ᨨ ᨨᆠ� 矄�矄҈ᆠ鼀 ᨨᆠ � ᨨ 泐�ᆠ�矄ᆠ鼀␔
ᨨᆠ�矄├҈�ᆠ鼀� ᨨ ᨨ ᨨᆠ� � �鼀��矄 ᨨ鼀 � � �泐㝐� � 泐� �ᨨ犨ᨨ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ矄 ᨨ盐 㝐� 졠 � 꾀�҈҈҈桌 㡰 㡰 ҈ ᆠ㡰 㡰 ҈� ᔼ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ � 鼀 ��鼀��矄 和桌 㡰 �盐ᆠ 鼀� � � 㡰 ᆠ 鼀� � � ᆠ㡰 � 鼀� � � 㡰 �盐ᆠ 盐ᆠ
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P-261
Ray Optics and Optical Instruments
ᨨᆠ � 矄ᨨ ᨨᆠ� 泐�ᆠᨨ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨᆠ鼀 ᆠ鼀矄ᆠ�泐
�矄 ᆠ犨 ᨨᆠ� 泐�ᆠ�矄 ҈�ᆠ鼀 � �鼀� ᨨ鼀 泐 泐�ᆠ�矄 � � �泐 犨ᨨ 矄矄 � �矄 ᆠ� ᆠ鼀ᨨ ᨨᆠ �鼀�ᨨ 盐ᆠ 泐 ᨨᆠ�鼀矄盐 ᨨ鼀 ᨨᆠ� 矄矄ᆠ�泐 ҈黨黨�和桌 㡰 ҈ 㡰 ҈� ᆠ㡰 ҈ 㡰 3
黨
҈
Numeric Answer
꾀ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨᆠᨨ 矄�矄矄ᨨ ᨨᆠ� 矄ᨨ ᨨᆠ � ҈ᆠ鼀 泐�ᆠᨨ ᨨᆠ � �ᆠ鼀 犨� 矄�ᨨ ᨨ ҈ᆠ鼀
�矄 ᆠ犨 ᨨᆠ ᨨᆠ� 矄 n 犨 �ᆠ 矄� ᨨ 泐�矄鼀� �鼀盐鼀�矄 ᆠ鼀
盐ᨨ n �矄힘힘힘힘힘힘 㝐� 和� ᔼ㡰桌 ᔼ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ 鼀� � � �盐�鼀��矄 ᨨ鼀 �矄 ᆠᨨ �ᆠ鼀 ᨨ鼀 泐㜠�ᆠ ᨨᆠ � ᨨ �矄ᨨᆠ� 矄�矄҈ᆠ鼀ᔼ 鼀� � �ᆠ�ᨨ �矄 ҈ 盐 � ᨨ 鼀�ᆠᨨ矄ᆠᨨ�矄ᆠ鼀
ᨨᆠ � ᨨ 泐㜠�ᆠ 矄 � ᆠ鼀㡰 �矄 [Main Sep.04, 2020 (I)] 힘힘힘힘힘힘힘힘힘힘
4
Fill in the Blanks
҈҈ 矄ᨨ� �ᨨ犨ᨨ ᆠᨨ 鼀�ᆠᨨ矄ᆠᨨ�矄 �� ᨨ ᨨ�ᆠ鼀�ᆠᨨ矄ᆠᨨᆠ盐矄 犨 � ᨨ ᆠᨨ 矄�矄 犨 � ᨨ �矄� �� ҈黨黨㜠҈㝐├桌 6
MCQs with One or More than One Correct Answer
҈ 꾀 �矄 ᨨ矄 泐 矄ᨨ ᨨ鼀�ᆠ ᆠ� � 矄ᆠᨨ � � ᨨᆠ�ᨨ ᨨᆠ � ҈鼀
泐�ᆠᨨ ᨨᆠ � ᆠ鼀 ҈黨黨㜠㝐├矄桌 㡰 �矄 ᆠ犨 ᨨᆠ� 泐�ᆠ �矄҈鼀 㡰 �盐鼀� � �ᆠ�ᨨ ᨨ �矄 ᆠ㡰 �鼀�ᨨ �矄� 㡰 ᨨᆠ��矄� 泐�ᆠ ҈ 꾀 矄ᨨ ᨨ鼀�ᆠ矄ᆠᨨ 矄 �盐鼀� � �ᆠ�ᨨ ᨨ 鼀� �盐 � ᨨ �矄 ᨨᆠ矄 矄�ᨨ 犨
ᨨᆠ� 泐�ᆠ�矄ᆠ鼀 � �鼀� �矄 ᨨ鼀 � � �泐 ᨨᆠ � fᨨ ᨨᆠ� ᨨᆠ � fᨨ 泐�ᆠ ҈黨黨㜠㝐├矄桌 㡰 f㦨�ᆠ鼀 fe㦨黨ᆠ鼀 㡰 f㦨�ᆠ鼀 fe㦨҈ᆠ鼀 ᆠ㡰 f㦨䰘ᆠ鼀 fe㦨�ᆠ鼀 㡰 f㦨ᆠ鼀 fe㦨ᆠ鼀
Topic-6 : Miscellaneous (Mixed Concepts) Problems 1 ҈
��ᨨ ᔼ
MCQs with One Correct Answer
꾀�� 泐� �� �矄矄鼀 �盐鼀�矄� ᆠ� ᨨ �矄矄㜠 �� ᆠ �ᨨ � ᆠ� ᆠq ᆠ 㡰 矄鼀� 㡰� 矄��矄ᨨ 矄 盐 ᆠ�ᨨ ᨨ q ᨨ ᆠᨨᆠ矄├ᆠ �矄 ҈҈桌
��ᨨ ᔼᔼ
ᔼ 矄�泐
鼀
㡰 ᆠ鼀
ᔼ 矄�泐
ᔼ 矄�泐
㡰 矄� ҈ ⏌㡰 㡰 矄� ҈ ҈⏌㡰 ҈ ᆠ㡰 矄� ҈⏌㡰 㡰 矄� ҈ ҈⏌㡰 꾀ᆠᨨ � �矄 � 犨� 犨 m㦨҈㡰盐ᨨ �� ᨨ �ᆠ鼀꾀ᆠᨨ ᆠ鼀�ᨨ�矄ᆠ ҈�ᆠ鼀ᨨ 犨 �鼀�ᨨ ᨨᆠᆠ ᨨᨨ鼀�矄 ᨨ鼀 �ᆠ鼀ᨨ犨 犨 ᨨᆠ � ᨨ 鼀�ᨨ�矄 �和桌 㡰 ҈�ᆠ鼀
㡰
㡰҈奠 ᔼ 矄�泐
ᆠ㡰
��ᨨ ᔼ
鼀
㡰
��ᨨ ᔼᔼᔼ
҈�ᆠ鼀
黨 q
꾀�� 鼀�矄� � ᨨ鼀 ��ᨨ ᔼᨨᔼ ��盐㡰
ᆠ�� � ��ᨨ 矄ᔼᔼᔼᔼᔼᔼ ᔼ
矄ᆠ�泐 �ᨨ � ᆠ� ᆠq ᨨ犨 �ᆠ 鼀盐矄鼀�矄矄矄 � ���ᨨ ᔼ�矄 桌
ᆠ㡰 ҈҈ᆠ鼀
m㦨҈
�ᆠ鼀
�ᆠ鼀 㡰 ҈ᆠ鼀 졠
ᔼ
WWW.JEEBOOKS.IN
EBD_8341 P-262
�
䰘
Physics
꾀ᨨ� ᨨᆠ�矄ᆠ ᆠ ᨨ �矄矄矄 ᨨ �盐矄ᆠ鼀 ᆠ�� ҈� �矄 ᆠᨨ �盐 和桌 �鼀� ᨨ鼀 矄盐 ᆠ�矄 㡰 ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 ҈ᆠ鼀 㡰 黨ᆠ鼀 꾀矄ᨨ盐ᆠ鼀�矄矄ᨨ盐 ᨨ ⼄盐 ᆠ泐 � 矄� 犨
⼄盐 ᆠ泐 � � 犨� ⼄盐 ᨨ ᨨᆠ�泐 ᨨ 矄ᨨ盐 � 犨 㦨 ҈� 鼀⏌矄 ᨨᆠ�泐ᨨ 矄ᨨ盐 � � 㦨鼀⏌矄㡰 和桌 㡰 㡰 ҈ ᆠ㡰 㡰 꾀鼀ᨨ 犨 ��� �矄� ᆠ� ᨨ �矄矄�� ᆠ ᨨ鼀�矄矄ᨨ�矄盐ᆠ � �� 盐矄矄盐 矄ᨨ � ᆠ�ᨨ ᆠᨨᨨ矄ᨨ �� 犨 �ᆠ 犨�ᆠᨨ鼀 ᨨ盐ᨨ� 和桌 㡰 �ᨨᔼ ��ᨨ盐 㡰 꾀ᆠᨨᨨ矄 ᆠ�
ᆠ㡰 Ӹᨨ犨졠 � 㡰 ␔ ��� ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � 矄 �ᆠ 矄矄 ᨨ矄 ᨨ �� �矄矄�ᨨ 䁸 ��ᨨ ᆠ盐盐ᨨ 矄盐 ᆠ矄 和桌 ᨨ 矄矄矄�� � ��鼀矄
㡰
҈
ᆠ㡰
2
҈
㡰
㡰
꾀
ᔴ
= ᔴ
=
졠
҈�ᆠ鼀
ᆠ鼀
5 True / False ҈҈ � 矄�泐ᨨ �� �矄 ᆠ�r ᨨ鼀 �矄ᨨ ᨨ �ᆠ泐� �ᆠ矄ᨨ盐ᆠ�矄� 矄泐ᨨᨨ�ᨨ ᨨ�r ҈黨҈㜠㝐├矄桌 ҈ 矄� �矄盐 矄 �� � 矄├泐 �泐�矄 ҈黨桌 6
MCQs with One or More than One Correct Answer
҈ �矄矄ᆠ泐� 矄ᨨ ⼄盐 �� 㦨ᆠ鼀 矄鼀 ᆠ�� n㦨҈�ᆠ ᨨ ᨨ� ᨨ 矄盐 ᆠ 矄矄 ᨨ犨 � ��盐ᔴ泐� ᔼ 矄 ᨨᆠ泐� ᔼᔼ 矄ᆠᨨ ᨨ ᆠ泐� ᔼᔼᔼ 矄ᆠᨨ ᆠᨨ ��ᨨ ᆠ盐盐ᨨ 犨ᨨᆠ盐 ᨨ矄矄鼀 㦨 鼀㡰ᔼ H҈H H 矄ᨨ ᨨ� X ᨨ ᨨᨨ鼀 ᨨ ᆠ泐� 矄 矄ᆠ�泐 꾀 ҈黨桌 ᆠᨨᆠ矄鼀 矄㡰�矄⏌
Integer Value Answer
和盐 �� ᨨ � 矄�泐҈├␔鼀�矄� ᆠ� ᨨ鼀泐ᨨ � ᆠᨨ 矄ᨨ ᨨᆠ � ᆠ鼀ᔼ� ᨨ �泐 ᨨ矄矄 ᨨ �� 盐 ᨨ 矄 矄矄盐鼀 矄 盐矄� �矄鼀盐ᆠ 矄鼀 �矄 ᨨᆠ � �� 矄�泐ᨨ �� � ├␔鼀 �矄 ᆠᆠ鼀 ᨨ鼀 矄ᨨ ᨨ 矄� �矄힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌 4
黨
҈ = ҈
Fill in the Blanks
ᔼ e m 矄ᆠ�泐 ᆠ�ᆠ鼀���泐 鼀� �ᆠ鼀��泐ᨨ 矄ᆠ e m ᆠᨨ矄ᨨ � � ⼄盐 ��矄 � 鼀 �盐鼀 � ᨨ ᆠ�ᨨ ᨨ 鼀 �盐鼀� 鼀矄ᨨ ᨨ鼀矄 �矄 ҈黨黨㜠҈㝐├桌 꾀矄ᨨ 鼀�ᨨ ᆠ�� 矄 ᨨ犨 � �� 矄 ᆠ盐 矄盐 ᆠAPBᨨ �盐矄ᨨ ᆠ盐盐҈ᆠ鼀 矄盐 ᆠCD졠 ᨨ APB�矄� ᨨ �� ᨨ CD�矄犨犨� ᆠ�� �ᆠ矄矄�� � ��盐 꾀 ᨨᆠO�矄ᆠ �矄 ᆠᨨ ҈�ᆠ鼀 ᨨ鼀 ᨨ P矄矄 ᨨ犨 �矄 ᆠᨨ � �鼀�ᨨ O ᨨ鼀P矄 �犨 ᨨ鼀 �矄 ҈黨黨҈㜠㝐├矄桌
㡰 ᆠ鼀 㦨҈㡰黨ᆠ鼀 㡰 >҈ ᆠ㡰 >҈ 㡰 > ҈ 꾀 � ᆠᨨ 矄�矄鼀 ᨨ 犨ᨨ鼀�矄犨� ᆠ� � �ᆠ矄n҈ n矄矄 ᨨ犨 � ��盐 �盐矄ᨨ ᆠ盐盐 ᨨ �� 矄 �ᆠ矄盐 ᆠ⼄盐f�矄 ᨨᆠ � ᨨ 矄犨 n҈㦨n㦨n ᨨᆠ � �矄f졔 Df犨 n҈㦨n n㦨n졔Dn꾀矄矄盐鼀� �Dn n҈㡰 ҈n ᆠᨨᆠ矄鼀 矄㡰�矄⏌꾀 ҈黨桌
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P-263
Ray Optics and Optical Instruments
㡰 ᔼ
� ᆠ 矄 �矄ᆠ 泐�ᆠ 矄�ᨨ D 矄 矄 ᨨ犨 � �� ҈黨黨㜠㝐├矄桌
Df Dn > < f n
Df Dn 㡰 �ᨨ 犨 鼀� 矄 f n 盐 ᆠ � � ᨨ ᆠᨨ 矄盐 ᆠ矄ᆠ 泐ᆠᨨ ᆠ矄盐 ᆠᨨ 矄鼀 �盐矄ᨨ ᆠ盐盐 ᆠ㡰 For n㦨҈�Dn㦨҈ f㦨ᆠ鼀 盐ᨨ Df 犨�ᆠ鼀 ᨨ盐 ᨨ ᨨ ᆠ�鼀ᆠ㡰 Df Dn < f n
犨ᨨ� �ᆠ�矄矄ᨨ 矄S҈ S ᆠ�� 㦨҈�㡰 ᨨ ᆠᨨ ᨨ �盐矄ᨨ ᆠ盐盐҈ᆠ鼀 泐 ᆠ 犨� ᆠ盐 矄盐 ᆠ矄 �矄 ᆠ d 矄 矄 ᨨ犨 � ��盐犨� � 矄 矄 ᨨ犨 泐 矄 � 㡰�� ␔ ᨨ� 矄ᨨ盐ᆠᨨ �� P�矄ᆠ � 矄� ᨨ S҈ᨨ �矄 �矄 �矄 ᆠᨨ �ᆠ鼀 ᨨ鼀 ᆠ盐 ᆠ �� 泐矄鼀 � � ᨨ鼀� ᨨ盐 ᨨᨨ �矄� 矄� S �矄 ᆠ �矄 꾀 ҈�桌
㡰 ҈�
S҈
�ᆠ鼀
҈
S
P d
㡰 ᆠ鼀 㡰 䰘ᆠ鼀 ᆠ㡰 ᆠ鼀 㡰 黨ᆠ鼀 꾀泐졠ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� �矄� ᆠ� ᨨ ᆠ 꾀ᨨ �矄鼀꾀ᔴ � ᆠ� �ᨨ 矄 ��盐㡰ᔼ ᆠ�� ᨨ 鼀�ᨨ �矄鼀�矄 犨 �ᆠ ᨨ ᨨᨨ犨� ��矄 㡰ᆠᨨᆠ䁸҈桌
҈䰘
҈
㡰 泐�矄ᨨ泐� 泐 ᆠ ᆠᔴ 㡰 泐ᆠᨨ鼀矄ᨨ盐 ᨨ盐� ᆠ꾀 ᆠ㡰 �犨 � ᆠ� 泐 鼀� 泐�矄黨 㡰 �犨 � ᆠ� 泐 鼀� 泐�矄҈ 꾀ᆠᨨ ᆠ鼀�ᨨ�矄ᆠ ᨨ ᨨ� ᨨ 犨� �矄 �矄 �ᆠ �ᆠ泐盐犨 矄O ᨨᨨ 鼀�ᨨ ᔴ�矄ᆠ ᨨ ᆠ盐盐꾀ᨨ� ᨨᆠ�矄ᆠ Cᔼ 矄�鼀�矄ᨨᨨᆠ Cᔼ 鼀�ᨨ�矄
ᨨ犨 � 犨� 犨 �鼀�犨�҈黨黨㜠㝐├矄桌 㡰 犨�鼀� C 㡰 ᨨᆠ ᨨ� 犨 C ¥ ᆠ㡰 �盐 ᨨᆠ ᨨ� 犨 C O 㡰 ᨨᆠ ᨨ� 犨 C O
犨ᨨ � ᆠᨨ 矄矄ᨨ ᨨᆠ � 矄 f҈ f 矄 泐 ᨨ� ᨨ �矄 ᆠ 犨 df҈ f㡰
泐 D
ᨨ
├� � ᨨ��� ᨨ ᆠᨨᨨ � 矄O ᆠ ᨨ �矄 矄 泐ᆠᨨᨨ � 矄ᨨ ᨨᆠᨨ� ᨨ �矄 矄 矄泐矄鼀 ᨨ鼀ᨨ 泐矄ᆠᨨ鼀� � ᨨ鼀 �� 泐 f҈ f y= D f҈ + f
㡰
x=
㡰
f f + d㡰 D x= ҈ y= f҈ + f - d f҈ + f
ᆠ㡰
x=
㡰
f f + d f҈ - d 㡰 y = x= ҈ f҈ + f - d
f҈ f + d f҈ - d 㡰 D f҈ - d 㡰 y= f҈ + f - d f҈ + f - d
Comprehension/Passage Based Questions
8
꾀和和꾀E 㝐ᨨ矄鼀�矄 ᆠ�� n>҈和ᨨ犨 �� 泐 ᨨ鼀� 矄 盐泐ᨨᆠᆠ盐� �鼀�
sin q҈ n ��矄盐 矄ᨨᨨ = 泐和 矄犨 sin q n҈ ᆠ 泐 矄 ᨨ犨 矄 ᨨ鼀 盐 � 鼀�矄ᨨ 矄鼀矄� ᨨ ᨨ鼀矄 � ᆠ� 泐 꾀ᆠᆠᨨ � �ᨨᆠᨨ鼀� �矄鼀 ᆠ�� ᨨ 鼀 �盐鼀 �矄 �� 泐 �ᨨ
㦨c/ v = ± er mr 犨 c�矄 矄 ᨨ ᆠᨨ鼀� �ᆠ 犨矄� ᆠ盐盐鼀v�矄矄 � 鼀 �盐鼀er mr �鼀���泐 鼀��泐ᨨ 鼀 �盐鼀 矄ᆠ�泐 ᔼ ᨨ鼀鼀�矄ᨨ e r mr ᨨ矄���鼀泐� � ᨨ矄��n ᨨ 鼀 �盐鼀 ␔ ᨨ e r mr
�� ᨨ 鼀盐矄ᆠ ᨨᨨ矄 ��ᨨᨨᨨ n 和盐ᆠ
�� ᆠ�� 鼀�矄ᆠ ᨨ犨� �ᆠ�泐 ᆠ 鼀㜠鼀�矄 泐 �� 矄�� � �ᆠ 泐 � ᨨ�ᆠ �ᨨ犨� ᨨ盐�ᨨ� � 泐 泐矄�ᆠ犨矄和� ᆠn�矄 ���矄盐矄� ᆠ � � �ᆠ�ᨨ ᨨ ᨨ��ᨨ ᨨ ᆠ �� ᨨ犨矄�鼀�ᨨ ᨨ鼀鼀�矄 ⼄盐 ᆠ泐ᨨ �� 鼀� 矄盐 ᆠ � 盐ᨨ ᆠ�ᨨ � 鼀㜠鼀�矄 ҈桌 ҈黨 ᨨ �� � ᆠ� ᨨ鼀 � ᨨ 鼀㜠鼀� ᨨ�泐 ��鼀�矄
q1
㡰
Air
㡰
Meta㜠material q2
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EBD_8341 P-264
Physics
q҈
q1
ᆠ㡰
Air
㡰
Meta㜠material
Air
Meta㜠material q
q2
ᔴ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㡰 矄 ᨨ �� � 鼀㜠鼀��矄v㦨cn 㡰 矄 ᨨ �� � 鼀㜠鼀��矄v㦨
c
n ᆠ㡰 矄 ᨨ �� � 鼀㜠鼀��矄v㦨c 㡰 犨 � ᨨ �� � 鼀㜠鼀� lm㡰 �矄�� 泐lm㦨l�n犨 l��矄犨 � ᨨ �� � �
10 Subjective Problems ҈
꾀 泐 �矄 � ᆠ� ᨨ 鼀 �盐鼀 ᆠᨨ 矄�矄� � ᨨ 犨ᨨ ᨨ盐 �矄ᨨ ᨨ ᆠ盐 矄矄 ᨨ犨 � ��盐 矄盐 ᆠ鼀├矄 �º犨� ᨨ� ᨨ ᆠ盐 矄盐 ᆠ 矄 �盐矄 ᨨ ᆠ盐盐 鼀 ᆠ�� �ᆠ矄ᨨ 鼀 �盐鼀 �矄 �ᨨ 鼀 矄 ᨨ犨 � ��盐ᔼ ᆠ�ᨨ ᨨ 矄盐 ᆠ矄 泐鼀矄� ᆠ� �矄P � OP㜠㝐├矄桌
㦨Ö
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졠
㦨҈
㦨҈�҈
º
� ᨨᆠ � ᨨ 矄矄 ᨨ犨 � ��盐 ��ᨨ ᆠ盐盐ᨨ ᨨ 矄盐 ᆠ矄⼄盐ᨨR 㜠㝐├矄桌 m҈ < m < m m҈
m
m
ᆠ�� �ᆠ矄ᨨ ᆠᨨ犨 �矄矄 ᨨ盐 �� 矄҈�҈ ҈黨矄ᆠ�泐 ᨨ矄ᨨ � �矄矄 ҈䰘䰘 ҈䰘矄ᆠ�泐꾀 �矄ᨨ矄ᆠ矄�矄鼀ᨨ � ᨨ�矄鼀 ᨨ ᆠᨨ犨 �矄矄꾀鼀ᨨ 犨 ��� �矄� ᆠ� 矄鼀 �ᨨ �矄�矄鼀 ᨨ � �矄矄�矄ᨨ矄ᆠ矄 �矄鼀�矄ᆠᨨ鼀� 犨� ᆠᨨ犨 �矄矄�矄鼀矄盐ᆠ �矄 ᨨ ��ᨨ ᨨ � ᆠ� �� 㡰鼀� �ᨨ � �矄矄�矄鼀 㡰ᔴᆠ盐 �矄矄�ᨨ ᨨ ᆠᨨ鼀� 矄泐矄鼀 ҈㜠�㝐├矄桌 � 꾀泐ᨨ �� � �� ��矄� ᆠ� � � � � � ᆠ� � @ 黨° 㡰ᨨ ᨨ �ᆠ �盐矄ᨨ 矄 鼀 �盐鼀ᨨ �ᆠ├ 矄矄㦨҈鼀 矄 ��盐ᨨ犨㡰
ᨨ� ᨨ 泐 � ᆠ� ᆠ�矄 ᨨ���
A 㡰 鼀 �盐鼀 ҈ 泐҈ 㡰 꾀ᔼ 矄�� ᨨ ᆠ�ᨨ n y㡰 ��
= ҈鼀 泐㡰 泐 n y 㡰 = ky ⏌ + ҈桌҈⏌
꾀 � �ᆠᨨ 矄ᨨ ᆠ�� ⏌�矄ᆠ ᨨ ᨨ� ᨨ 鼀�ᨨ矄矄 ᨨ犨 � ��盐 矄ᆠ 犨 矄 鼀�ᨨ�矄 � 犨� 犨ᨨ ᆠ�� ⏌ᔼ�矄 ᨨ盐 犨 ᨨ� ᨨᆠ�矄 ᆠ ҈�ᆠ鼀ᨨ 矄ᨨ �矄� ᆠ� �矄 ᨨᆠ ᆠᨨ� ᆠ� 矄犨� �矄ᨨ犨 �鼀�졠 � �犨� ᨨ �⼄盐� ᨨᆠ �鼀��� ᆠᨨ� ᆠ� �矄 ᆠ �ᆠ鼀 ᨨ鼀 矄ᔴᆠ盐 ᆠ�� ᨨ ҈㜠�㝐├矄桌 �⼄盐�
꾀 㡰
꾀ᔼ
犨 ├㦨҈ 鼀㡰 - ⏌
ᆠ�� ᨨ ��矄҈ ҈黨黨�㜠҈㝐├矄桌 㡰 졠� �ᨨ 犨 矄ᨨᨨ ᆠᨨ泐 ᨨ 泐� B xy㡰� 鼀 �盐鼀 � ᆠ� � ᨨ� 㡰 졠� ⼄盐�ᨨ ᨨ ᆠᨨ泐y x㡰ᨨ 泐�
鼀 �盐鼀 ᆠ㡰 鼀� ᆠᨨᨨ � 矄 x҈ y҈ 㡰 ᨨ ᨨ� P 犨 泐� 矄ᆠ矄 盐矄盐 ᆠᨨ 矄㜠 �ᨨ盐 泐 㡰 ᔼ �ᆠ ᨨ 泐矄盐矄⼄盐 泐 犨ᨨ 鼀矄 ᨨ �� P Q 矄�ᨨ 㡰 ᆠᨨ � � � ��ᨨ 矄ᨨ 犨 � 矄处 �处 犨 �ᆠ 鼀盐盐泐 ᆠᨨ � ᆠ 犨 � 矄泐㡰� ᆠ� ᨨ鼀泐ᨨ �矄鼀矄矄 ᨨ犨 �
�� ᆠ�� ᨨ �矄鼀矄 盐 ᆠ�ᨨ ᨨ 犨 � �矄�� 泐 �ᨨ m l㡰 㦨҈졔 l 犨 l �矄� 处 b�矄ᨨ矄��ᆠᨨ 矄 盐ᨨ b �矄矄盐ᆠ ᆠᨨ ��ᨨ ᨨᨨ ᆠ�ᨨ ᨨ ᆠ AC�矄盐矄矄�矄 � ᨨᨨ 犨 � �矄 ᨨ矄�矄 � ᨨ ᨨ ҈黨黨҈㜠졔졔㝐├矄桌 꾀 q
㝐쟼ᔼ㝐
矄� q =
黨
턼
ᔴ
㡰 � 盐ᨨ b 㡰 � ��ᨨ ᨨ 鼀矄 矄鼀� ᨨ盐� ᆠAC ᆠ㡰 꾀ᆠᨨ � 矄�矄盐矄 ᨨ� � 矄 矄鼀� 鼀矄 � ᨨ ᨨᆠ盐矄 ᔼ � 矄��矄 ᨨ 矄鼀�矄矄�ᨨ
ᨨ鼀 ᆠAC҈ I矄ᆠ�泐 � 矄盐 � 矄�泐 ᨨᆠ盐矄
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P-265
Ray Optics and Optical Instruments
和ᆠ 和�矄�盐鼀� 泐犨ᨨᨨ� 矄ᨨ盐ᆠ矄A B 꾀 ᨨ 矄ᨨ盐ᆠC矄 矄鼀ᨨ �� ᨨ犨 矄 ᨨ� Pᨨ 矄ᆠ 矄 ��盐㡰� AP�矄 ᨨ鼀ᨨ 矄ᆠ � 矄APBP CP� ᨨ �矄 ᆠAPBP CP鼀҈�鼀 ҈�鼀 矄ᆠ�泐 � C ᨨ犨矄ᨨ 矄ᨨ盐ᆠ矄A B黨犨矄 ҈ 犨矄 矄ᆠ�泐 鼀 ᨨ鼀 ᔴ �矄 ᨨ 60° P � 矄�泐 犨矄⏌鼀 A× 60° ᔴᆠ盐 � 矄�泐 Pᨨ 矄ᆠ ҈黨㜠�㝐├矄桌
䰘
B
꾀 ᨨᆠ�矄ᆠ ҈ᆠ鼀� ᨨ ᨨ ᆠᨨ ᆠ鼀�ᨨᨨ �盐矄ᨨ ᆠ盐盐҈ᆠ鼀꾀�矄矄矄ᨨ �ᆠ├ 矄矄ᆠ鼀 ᆠ�� ҈��矄 ᆠ ᆠᨨ矄ᨨ 鼀�ᨨ�
矄ᆠ犨 ᨨᆠ 鼀�ᨨ � ᨨ矄��ᨨ ᨨ � �鼀� ᨨ鼀 Ӹᨨ盐鼀泐 ├ �矄 ᆠᨨ 矄盐 ᆠᨨ 矄 ᨨ鼀 鼀�ᨨᨨ҈ᆠ鼀 ҈黨桌 黨 꾀ᆠ �盐ᨨᆠ├ᨨ �矄矄�矄ᆠ ᨨ � � 泐� �ᨨ ᨨ� ᨨ 矄盐 ᆠ � 鼀� �鼀盐鼀盐ᨨ ᆠ�� ᨨ �矄矄 ᨨ犨 �ᆠ 矄ᨨ � ᨨ�矄� ᨨ鼀 泐ᨨ �ᆠ ᆠ矄ᨨ ҈黨䰘黨桌 ᨨᆠ├
S
Answer Key Topic-1 : Plane Mirror, Spherical Mirror and Reflection of Light ҈ 㡰 ҈҈ 㡰
ᆠ㡰 ҈ 㡰
ᆠ㡰 ҈ ҈㡰
㡰 � ᆠ㡰 ҈ ҈�㡰 ҈� 㡰
ᆠ㡰 䰘 ᆠ㡰 ҈ ᆠ 㡰 ҈䰘 ᆠ㡰
㡰
黨 㡰
҈ 㡰
Topic-2 : Refraction of Light at Plane Surface and Total Internal Reflection ҈ ҈҈ ҈
㡰 ᆠ㡰 㡰 ҈ ᆠ㡰 㡰 㡰 处�맔҈҈ 㡰
҈ ҈
㡰 ᆠ㡰 � 㡰 ᆠ㡰 ҈ 㡰 ҈� 㡰 ҈ 㡰 㡰 � 㡰 맔҈鼀⏌矄맔҈䰘鼀㡰 ᆠ 㡰
㡰 䰘 ᆠ㡰 㡰 ҈䰘 㡰 ҈�㡰 䰘 黨㡰 ᆠ 㡰 ᆠ㡰
҈ �
㡰 㡰 �㡰 㡰
黨 㡰 ҈ ᆠ㡰 ҈黨 㡰 㡰 黨 处㡰
Topic-3 : Refraction at Curved Surface Lenses and Power of Lens ҈ ҈҈ ҈ ҈ ҈ � �
㡰 㡰 㡰 㡰 ҈ 㡰 ҈ 㡰 ҈ 㡰 㡰 ᆠ㡰 㡰 㡰 㡰 㡰 ҈�鼀�鼀㡰 㡰 �҈ ᆠ㡰 � 㡰 � 꾀㜠⼄矄錐㜠⼄錐ᔴ㜠⼄矄錐㜠⼄矄
㡰 � 㡰 ҈� 㡰 � 㡰 � ᆠ鼀㡰 㡰 �
㡰 㡰 ҈ ᆠ㡰 㡰 ҈�ᆠ鼀㡰 � 㡰 ��
㡰 䰘 㡰 ᨨ盐 矄㡰 黨 㡰 ҈䰘 㡰 ҈ 㡰 ҈黨 ᆠ㡰 䰘 㡰 㡰 黨 ᆠ㡰 䰘 ᆠ㡰 䰘㡰 黨
盐錐 矄錐 䰘 㡰 꾀㜠錐㜠⼄矄錐ᔴ㜠㜠⼄矄
ᆠ㡰 ҈ 㡰 㡰 㡰 㡰 黨
㡰 㡰 ᆠ㡰 䰘҈黨㡰 㡰
Topic-4 : Prism and Dispersion of Light ҈ 㡰 ҈҈ 㡰 ҈ ᆠ㡰
ᆠ㡰 ҈ ᆠ㡰 㡰
ᆠ㡰 ҈ ᆠ㡰 㡰
ᆠ㡰 ҈ 㡰
� 㡰 ҈� 㡰
ᆠ㡰 ҈ ҈�㡰
䰘 ᆠ㡰 ҈䰘 ҈� 㡰
ᆠ㡰 ҈ 㡰
黨 㡰 ҈ 㡰 ҈黨 盐錐 ᆠ 㡰
㡰
黨 �㡰 ҈ 㡰
Topic-5 : Optical Instruments ҈ 㡰 㡰 㡰 ҈҈ 矄鼀㡰҈ ᆠ 㡰 ҈ 㡰
ᆠ㡰
� ᆠ㡰
㡰
䰘 ᆠ㡰
Topic-6 : Miscellaneous (Mixed Concepts) Problems
҈ ᆠ㡰
㡰
҈ ᆠ鼀㡰 ҈҈ 盐錐 㡰
ᆠ㡰 ҈ 盐錐
㡰
� ᆠ㡰
ᆠ㡰
҈ 㡰 ҈ ᆠ㡰҈� 㡰
䰘 ᆠ㡰
҈㡰
҈ ᆠ㡰҈䰘 㡰
黨
me 㡰 me
҈ ᆠ㡰
҈黨 ᆠ㡰
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EBD_8341 P-266
Physics
Wave Optics
24
Topic-1 : Wavefront, Interference of Light, Coherent and Incoherent Sources 1 ҈
MCQs with One Correct Answer
ᔼ ��盐 ᨨ犨 턼 犨ᨨ ⼄盐泐 � 矄 ᆠᨨ 矄ᨨ盐ᆠ矄鼀�� � ��ᨨ ᨨ 犨 � 鼀
矄�ᨨ 犨 턼�矄�鼀 矄ᨨ �矄 ᨨ ᨨ 턼泐黨º꾀 ᔴ �矄� ᆠ ᨨ� 矄ᨨ ᨨ矄�ᨨ ᆠ ⼄盐� �矄 ᨨ鼀 鼀� ᨨ� ᨨ 턼 � 矄��矄ᨨ ��ᨨ 꾀ᔴ犨�� �ᨨ 㝐� 和 ᔼ㡰桌
p æ L L҈ ö p æ L҈ L ö 㡰 ç - ÷ ç - ÷ l è n҈ n ø l è n҈ n ø p p n L҈ - n҈ L 㡰 n҈ L҈ - n L 㡰 㡰 ᆠ㡰 l l 꾀鼀ᨨ �� 矄�├矄�ᆠᨨ 矄 �矄矄 � �ᆠᨨ矄矄矄ᆠ�ᨨ 矄矄 ᨨ犨 � ��盐ᨨ犨ᔴᨨᆠ 矄 ᨨ 鼀� 犨 ᨨ 犨� ��盐矄 矄ᆠ 鼀�ᆠ ᨨ 犨 ᨨ矄ᆠ㡰 꾀 桌
㡰
Glass
L�ght
턼
ᔴ
㡰 �
鼀
S ҈�鼀
鼀
L
d
S҈
꾀�
꾀�
꾀
㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈
犨ᨨᆠᨨ 矄ᨨ盐ᆠ矄ᨨ 矄ᨨ盐 S҈ Sᨨ 盐ᆠ矄ᨨ盐 犨矄ᨨ 矄鼀犨 � l㦨҈鼀� 矄S҈ S ᆠ ҈�鼀 矄 ��㡰꾀�矄 ᨨᆠ L �ᆠ泐� ᨨ ᨨ S � 矄 � 矄�泐�矄鼀� �鼀盐鼀 犨 �矄鼀犨泐 ᨨ鼀S �矄 鼀ᨨ矄犨泐 ᨨ鼀S҈├� � �矄 �矄 ᆠ ᨨ鼀S � ᆠ 鼀 �鼀盐鼀ᨨ � 矄�泐�矄ᨨ矄 犨 �矄 �矄 �矄 ᆠd ᨨ鼀S҈ d�矄 㝐� 和� ᔼᔼ㡰桌
㡰 ҈鼀 㡰 �鼀 ᆠ㡰 鼀 㡰 鼀
犨ᨨ�� 犨矄 � � 矄鼀犨 � l� ᆠ盐盐鼀 � 矄� ��泐 �矄犨矄 L҈ ᨨ盐� 鼀 �盐鼀ᨨ ᆠ�� n҈犨 � 矄ᆠᨨ 犨矄 ᨨ � L ᨨ盐� 鼀 �盐鼀 ᨨ ᆠ�� n꾀 �矄 矄 � ᆠ犨
犨ᨨ犨矄�矄 㝐� 和 ᔼᔼ㡰桌
䰘
㡰
ᆠ㡰
㡰
ᔼ � ᆠ �鼀 �ᨨᨨ 鼀�盐 矄ᨨ a҈ ҈ ᆠᨨ 犨矄�矄 a = �ᨨᨨ 鼀 �鼀盐鼀 鼀� �鼀盐鼀� 矄��矄ᨨ � �矄犨� 㝐� 꾀�҈黨ᔼ桌 㡰 㡰 ҈ ᆠ㡰 㡰 黨
犨ᨨᆠᨨ 矄ᨨ盐ᆠ矄ᨨ 盐ᆠ犨矄ᨨ � � 矄��矄 犨 �ᆠ � 꾀 � ᆠ �ᨨᨨ 鼀 �鼀盐鼀 � 矄�泐ᨨ 鼀� �鼀盐鼀� 矄�泐�矄҈ � 矄�泐ᨨ 犨矄� �ᨨ 㝐� 黨 ҈黨ᔼ桌 㡰 ҈黨 㡰 �黨 ᆠ㡰 ҈ 㡰 � 졠 ᨨ矄盐鼀鼀 �� ᆠ�� ᨨ ��矄矄鼀矄
�ᨨ盐 � ᆠ矄矄犨� �� ᨨ鼀 �ᨨ盐 ␔ �� 鼀�矄 �ᆠ ᨨ� ᨨ 泐 盐泐� 矄Ἐ � ᆠ� 矄盐矄ᨨᆠᨨ ᆠ盐 矄�矄 �� 鼀 㝐� ҈�桌 㡰 矄 ᨨ犨 犨 矄 㡰 矄 盐犨 矄 ᆠ㡰 ᆠᨨ鼀矄 ᨨ犨 㡰 �ᨨ矄 ᨨ� ᨨ 泐犨� ᨨ盐 泐 ᆠ�ᨨ
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P-267
Wave Optics
ᔼ ᆠ �矄 ᨨ矄 � 盐 ᨨ 矄盐�鼀ᨨ矄��ᨨ ᨨ 犨ᨨ泐矄ᆠᨨ鼀� �ᨨ盐 ᨨ鼀矄ᨨ盐ᆠ�和 矄矄 ᨨ犨 � ��盐 盐ᨨ � ᨨ犨 �ᆠ 鼀 �鼀�矄 ᨨ� ��矄 㝐� 졠 � 꾀�҈҈桌 �矄 ᆠ ᆠ� �矄盐 ᆠ㡰 ҈
和
㡰 = ᆠ㡰
黨
=
l -҈ ( - ҈) l
(
-
)
㡰
=
( - ҈) l
㡰 =
(
)
-҈
( - ҈) l
-҈ ᔼ ᆠ ��鼀 CP矄 矄犨 ᨨ AO BP ᆠᨨ矄ᨨ � �犨ᨨ泐矄 � ᆠᨨ ��ᨨ
ᨨ q ᨨᆠᨨ 矄盐ᆠ�� ᆠ犨 泐BP ᆠ 泐OP 和桌 턼
졠
㡰 ᆠᨨ矄 q 㦨 l ⏌d ᔴ
㡰 ᆠᨨ矄 q 㦨 l ⏌d
ᆠ㡰 矄ᆠ q ᆠᨨ矄 q 㦨 l ⏌d
꾀
㡰 矄ᆠ q ᆠᨨ矄 q 㦨 l ⏌d
҈
҈҈
҈
犨ᨨ鼀矄ᨨ �� � �� 矄��矄I I � ᨨ ᨨ 盐ᆠ � � ᨨ 矄ᆠ 矄 � ᆠ 犨 鼀矄�矄p⏌ᨨ� A pᨨ� B
� ᆠ犨 矄盐 � 矄��矄 A B �矄 ҈和桌 㡰 I 㡰 I ᆠ㡰 �I 㡰 䰘I 꾀 � 矄�ᆠ�矄ᆠ盐ᨨ盐ᨨ �矄矄ᆠ泐� ᨨ � ᨨ�矄 �矄 矄�ᆠ�矄ᆠ ᨨ �矄矄 矄矄 ᨨ犨 � ��盐
ᨨ矄 � ᆠ � �矄 ᨨ鼀 �矄ᆠᨨ鼀� �ᨨ
矄 ҈黨黨黨和㜠㝐├矄桌 㡰 矄�� 㡰 ᆠ�ᆠ盐 ᆠ㡰 ⼄盐泐矄ᆠ 㡰 � � � � 矄ᆠ� � 犨 �ᆠ � ᆠ矄矄 矄 犨 �ᨨ ᨨ盐犨 矄
犨ᨨᆠᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� 鼀矄ᨨ � 矄��矄I I矄盐ᨨ矄 鼀 �鼀盐鼀 鼀� �鼀盐鼀ᨨ矄矄� � 矄��矄� 矄盐� �鼀 ҈黨㜠҈㝐├桌 㡰 �I I 㡰 �I I ᆠ㡰 黨I I 㡰 黨I I
6
MCQs with One or More than One Correct Answer
҈ 犨ᨨᆠᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠᨨ� 矄ᨨ盐ᆠ矄和҈ 和ᨨ 犨 � l㦨 鼀ᆠ 矄泐鼀鼀�ᆠ泐ᨨ � 矄� ᨨ ᆠ ᨨ ᆠ�ᆠ矄矄 ᨨ犨 矄ᨨ盐ᆠ矄 矄 泐 �矄 ᆠ 㦨҈鼀鼀 �矄 �鼀 ᨨ 盐ᆠ矄� ᆠ � �矄�矄�矄 �� ├ 矄ᨨ矄ᨨ ᆠ�ᆠ盐鼀 ᆠᨨ ᆠ�ᆠ �盐矄�ᨨ 犨 犨ᨨᆠᨨ 矄ᆠ盐��� 矄ᨨ矄 �矄Dq␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨ�ᨨ 矄�矄⏌ᆠᨨᆠ䁸 꾀 ҈䰘桌 ҈ 㡰 꾀 ├ 矄ᨨ 犨� ᨨ鼀 ᨨ� Dq 㡰 꾀 ᨨ ᨨ � �犨�鼀 �鼀盐鼀 ᆠ㡰 ᨨ 盐鼀 ᨨ 和҈ 和 � �矄ᨨ 盐ᆠ 犨
҈ � �矄 ⼄盐 �矄ᆠᨨ矄ᨨ 㡰 �盐 矄�ᨨ 犨 犨ᨨᆠᨨ 矄ᆠ盐� �� 矄ᨨ矄 ᆠ矄矄 矄 犨 鼀ᨨ ᨨ鼀 ҈ ᨨ ᨨ � �矄⼄盐 10 Subjective Problems ҈ 꾀ᨨ� 矄ᨨ盐ᆠS鼀�� ��� ᨨ 犨 � 鼀�矄 ᆠ 泐矄鼀 �� ᨨ ᆠ� � 矄盐 ᆠAB 矄 ��盐㡰 � 矄�泐ᨨ ᆠ �� �矄奠ᨨ � ᆠ� � 矄�泐ᔼ ᆠ � �矄 ᨨ矄 ᨨ 矄ᆠ ᆠ ᨨ ᆠ� � 矄盐 ᆠ泐� �矄 ᆠD ᨨ鼀�㜠�㝐├矄桌 㡰 ␔ �矄 矄 ᨨ 和ᆠ
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矄ᆠ 䁸 㡰 ᔴᆠ盐 �ᨨ ᨨ 鼀� �鼀盐鼀 ᨨ 鼀 �鼀盐鼀� 矄��矄�
� ᆠ � �矄 ᨨ鼀 ᨨ� P 和 꾀 矄 ᨨ犨 � ��盐㡰 ᆠ㡰 ᔼ � 矄�泐ᨨ� Pᆠᨨ矄ᨨ 矄ᨨ鼀 �鼀盐鼀 ᆠᆠ盐 鼀� �鼀盐鼀 �矄 ᆠ ᨨ盐� 犨 �ᆠ ᆠ� �矄盐 ᆠAB矄 ᨨ盐 矄 � 矄ᨨ � 矄�泐P�� ᆠᨨ鼀矄鼀 �鼀盐鼀 ҈� 꾀 ᨨ犨鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ鼀ᨨ �� ᨨ 矄矄 � 矄�泐ᔼ �矄 � ᆠ� ᨨ �矄矄 矄 矄 ᨨ犨 � ��盐꾀 ᨨ � �ᆠ�矄矄 ᔼ �矄├ᆠᨨ矄ᨨ �矄ᨨ ᨨ �쟼ᆠ �矄矄 ᆠ矄�ᆠ ᨨ �� � ᆠ� ᨨ � 矄鼀�矄 鼀� � � � �ᨨᨨ 鼀� �鼀盐鼀 矄矄 鼀 �鼀盐鼀� 矄��矄� � ᆠ ᨨ鼀 泐 犨ᨨ鼀矄ᨨ� ᨨ ᆠ�ᨨ ᆠ ҈黨黨㜠䰘㝐├桌
Topic-2 : Young's Double Slit Experiment 1 ҈
MCQs with One Correct Answer
ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 �� ᨨ � 鼀�矄盐矄 ᨨ ᨨ 盐ᆠ � ᆠ ␔ �矄 ᆠ
犨 矄�矄�矄�鼀鼀 �盐犨� � �㡰 ᨨ � �矄 ᨨ鼀 ᨨ �矄 ᆠ矄ᆠ �矄ᆠᨨ矄ᨨ 㡰 ҈䰘
㡰 �䰘
[Main Sep. 03, 2020 (I)] ᆠ㡰 ҈䰘 㡰 䰘
WWW.JEEBOOKS.IN
EBD_8341 P-268
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䰘
ᔼ ᆠ � �矄 ᨨ矄 ᨨ 矄ᆠ 泐 �盐鼀� � �犨ᨨ � 矄�矄҈鼀鼀犨� �� 矄ᨨ盐ᆠ l 㦨 鼀㡰 �矄 ᆠ犨 矄ᆠ 矄�矄 �矄҈ᆠ鼀ᔼ �� � ��矄ᨨ矄 ᨨ 矄ᆠ �矄 ᆠᨨ ҈䰘鼀鼀 ᨨ鼀 ᆠ �� � � � ᆠ犨 犨矄 犨 �ᆠ ᆠ � � �矄ᨨ� ᨨ鼀 矄�矄�矄ᆠᨨ矄ᨨ [Main Sep.02, 2020 (I)] 㡰 ҈䰘m鼀 㡰 䰘 鼀 ᆠ㡰 鼀 㡰 �m鼀 ᔼ Ӹᨨ盐 �Ἐ矄 ᨨ盐矄� �鼀 ҈ � �矄ᨨ矄 � ᆠ� 矄�鼀 ᨨ 矄ᆠ 犨 �� ᨨ 犨 � 䰘 鼀�矄盐矄 ᔼ 犨 � ᨨ �� �矄ᆠ � ᨨ 鼀 盐鼀 ᨨ � �矄ᨨ矄 � 矄鼀 矄�鼀 ᨨ 矄ᆠ 犨ᨨ盐 [Main Sep.02, 2020 (II)] 㡰 㡰 ᆠ㡰 ҈ 㡰 ᔼ ᨨ盐㜠矄� �鼀 ᆠ� ᨨ� ᨨ 矄ᆠ
� ᆠ犨 犨ᨨ� � �犨矄�矄 ҈ ᨨ 犨 � �ᨨᨨ � 矄�泐ᨨ �� ᨨ� ᨨ ᆠ ᨨ �� � ��矄 㝐� ᔼᔼ桌 㡰 � 㡰 䰘 ᆠ㡰 � 㡰 䰘 ᔼ Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 矄�ᨨ
犨 矄�矄�矄҈�鼀鼀ᔼ �鼀 矄ᨨ盐ᆠ ᨨ �� ᨨ 犨 � �黨 鼀�矄盐矄 � ᆠ �矄 ᨨ矄 ᨨ 矄ᆠ ├ ҈� 鼀 犨泐 矄�ᨨ 犨 矄盐ᆠᆠ矄矄��� � �矄ᨨ 矄ᆠ �矄 㝐� 䰘 ᔼᔼ桌 㡰 黨鼀鼀 㡰 黨鼀鼀 ᆠ㡰 �黨鼀鼀 㡰 黨鼀鼀 ᔼ ᨨ盐矄� �鼀 犨 � �鼀ᨨ �ᆠ├ 矄矄t � � ᆠ�� Ҙ�矄� ᨨ 盐ᆠ � ᨨ ᨨ ᨨ ᨨ 矄�矄 鼀 �鼀盐鼀 ᆠ ᨨ � �
矄 � 矄 泐 ᨨ � � 犨� 盐 ᨨ t �矄 » �矄 犨 � ᨨ �� 盐矄 㡰 㝐� ҈꾀�ᔼ桌 l l l l 㡰 㡰 ᆠ㡰 㡰 m - ҈㡰 m - ҈㡰 m - ҈㡰 m - ҈㡰
��盐矄 ᨨ犨矄Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄盐 ᔼ �矄 ᨨ矄 犨 � 矄 矄 ᨨ �ᆠ├ 矄矄 ᆠ�� �矄盐� ᨨ ᨨ ᨨ ᨨ 矄�矄 ᆠ 鼀 �鼀盐鼀�矄矄 � 泐 �矄 ᆠ ⼄盐ᨨ � �犨� 矄ᔼ 犨 � ᨨ �� 盐矄 �矄 »犨� 㝐� 黨꾀�҈黨ᔼ桌
nDl Dl nDl Dl 㡰 㡰 ᆠ㡰 a (m 힘 ҈) a ( m 힘 ҈) a (m 힘 ҈) a (m 힘 ҈) ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 � ᆠ ᆠ� ᨨ� ᨨ 矄ᆠ 犨 犨ᨨ� � �犨矄 ҈ �矄 ᨨ 犨 � �ᨨᨨ � 矄�泐 �矄 ᨨ� ᨨ ᆠ ᨨ �� � ��矄ᆠᨨ矄ᨨ 㝐� ҈҈ ҈黨ᔼ桌 㡰 䰘 㡰 � ᆠ㡰 黨 㡰 ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 犨� 矄�矄�ᨨ
҈ ҈鼀鼀ᨨ ᨨ矄矄�� � � �
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Physics
泐 盐矄� � �� ᨨ 犨 � l҈ ␔ �� ᨨ 犨 � l �矄 盐矄 �� � � �矄 矄 矄鼀 �� 矄鼀矄盐� l҈ l � �矄� � 鼀ᨨ䰘 鼀㡰 �盐矄 㝐� ҈ ҈黨 ᔼ桌 㡰 � 鼀� 鼀 㡰 鼀�� 鼀 㡰 鼀� 鼀 ᆠ㡰 鼀� 鼀 ҈ ᔴᨨ 矄� Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄矄 ᨨ犨 �
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� - 㡰 � - 㡰 � - 㡰 � - 㡰 ҈҈ ᔼ Ӹᨨ盐 �Ἐ矄 ᨨ盐矄� �鼀 矄�矄ᆠ 鼀鼀 �� ᨨ 犨 � l 㦨 � 鼀 �矄 � ᆠ� ᨨ 矄�矄 ᨨ 盐鼀ᨨ �� � �矄 ᨨ矄 � �盐 � £q£ �矄 㝐� 黨 ҈黨ᔼᔼ桌 㡰 㡰 ᆠ㡰 ҈ 㡰 ҈ ҈ ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄�矄�ᨨ d �矄鼀鼀 矄ᆠ �矄 ᆠD�矄҈鼀꾀鼀 ᨨ �� ᨨ 犨 � 鼀�矄� ᆠ� ᨨ 矄�矄 �a矄矄 ᨨ犨 � ��盐졠 矄ᆠ ᨨ� O�矄 ⼄盐� �矄 ᨨ鼀 矄�矄 �矄 ᆠ졠�矄҈҈鼀鼀 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄⏌ᆠᨨᆠ䁸 꾀 ҈黨桌
㡰 ᨨa㦨 犨�ᆠᨨ 矄盐ᆠ�� ᆠ ᨨ� P 㡰 � �矄ᆠ� � 矄ᨨ a ᆠ㡰 ᨨ a㦨 � 犨� 矄盐ᆠ� p � ᆠᨨ� P 㡰 ᨨ a 㦨 � 犨� 矄盐ᆠ� p � ᆠᨨ� O ҈ ᔼ Ӹᨨ盐 �Ἐ矄 ᨨ盐矄� �鼀 矄�矄矄 泐 �鼀鼀 矄ᆠ �矄ᆠ ҈�ᆠ鼀犨泐꾀鼀ᨨ �� ᆠᨨ 矄�矄� �ᨨ 犨ᨨ犨 � 矄� 鼀 � 鼀 �矄盐矄 ᨨᨨ� � ᆠ � �矄ᨨ 矄ᆠ 矄 �矄 ᆠ ᨨ鼀 ᆠᨨ鼀鼀ᨨ ᆠ 鼀 �鼀盐鼀ᨨ ᨨ� 犨 �� � �矄 盐ᨨᨨ 犨 � 矄 ᆠᨨ� ᆠ� �矄 㝐� ҈䰘桌 㡰 黨䰘�鼀鼀 㡰 ҈�鼀鼀 ᆠ㡰 ҈�鼀鼀 㡰 䰘鼀鼀
WWW.JEEBOOKS.IN
P-269
Wave Optics
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ᔼ Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 犨� �� ᨨ 犨 � l 矄�ᨨ ᨨ 矄�矄�矄 �矄 ᆠᨨ 矄ᆠ �矄矄盐ᆠ >> >>lᔼ � �犨� �矄b �矄 ᆠ ᨨ鼀ᨨ� ᨨ 鼀 �鼀盐鼀� 矄�泐ᨨ ᨨ� 犨 � 矄�泐 矄ᨨ ᨨ 鼀 �鼀盐鼀� 矄�泐 ᨨ � 矄� �矄 㝐� 졠 � 꾀�҈҈ ҈�桌 b b b b 㡰 㡰 ᆠ㡰 㡰 ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 �矄 ᆠ犨
犨ᨨ� �ᆠ矄�矄�矄҈�鼀矄� 矄�犨�
盐鼀ᨨ � 矄�泐鼀 �鼀ᨨ矄 犨� � ᆠ 鼀 �鼀盐鼀ᨨ 矄� �矄� � ᆠ�ᨨ �矄 㝐� 졠 � 꾀�҈黨҈桌 㡰 㡰 ᆠ㡰 ҈ 㡰
犨ᨨᆠᨨ ᨨ� 矄ᨨ盐ᆠ矄和҈ 和矄 泐 矄鼀 �矄 ᆠἘ Ἐ矄矄 ᨨ犨 � �矄ᨨ� ᨨ 矄ᆠ 犨� 㝐� ҈桌 和҈ 和
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� l l�矄犨 � ᨨ �� 盐矄 㡰 ᆠ� � m 㦨 ҈� �矄 � 矄 犨 ᨨ ᨨ 矄�矄 矄ᆠ � Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 꾀ᨨ� ᨨ 矄ᆠ ⼄盐� �矄 ᨨ鼀 矄�矄 �ᨨᨨ � 矄��矄 ᨨ � ᨨ 盐ᆠ�ᨨ ᨨ �矄矄�矄 㝐� 졠 � 꾀��҈桌 ᆠ㡰 ҈ 㡰 㡰 ҈ 㡰 ҈ ᔼ Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 盐矄� � 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ 犨 � l, � ᆠ � 鼀矄ᨨ � �n㡰ᆠᨨ矄ᨨ � �ᨨ 泐ᨨ� � � ├� 矄�泐�矄 꾀 ҈桌 l l l l 㡰 n + ҈㡰 㡰 n +҈㡰 ᆠ㡰 n + ҈㡰 㡰 n +҈㡰 ҈ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 �矄 ᆠ� ᨨ盐 泐盐矄� � � 盐�� ᨨ ᆠᨨᨨ�鼀 � � 犨� 矄ᆠᨨ GR B矄ᆠ�泐 ҈桌 㡰 G>B>R 㡰 B>G>R ᆠ㡰 R>B>G 㡰 R>G>B ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 � 矄�泐ᨨ� �矄 ҈⏌ 㡰ᨨ 鼀 �鼀盐鼀� 矄�泐꾀 �盐ᨨ矄��ᨨ ᨨ �矄ᨨ� �矄 �和桌 㡰 矄� ҈ l⏌ 㡰 㡰 矄� ҈ l⏌ 㡰 ᆠ㡰 矄� ҈ l⏌ 㡰 㡰 矄� ҈ l⏌ 㡰 㝐ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ 犨 � 鼀 � 鼀 � ᆠ� 矄�鼀盐 ᨨ盐矄泐 ᨨ鼀泐ᨨ ᨨ盐矄�矄 盐矄犨 ᨨ矄矄�矄矄�ᨨ �矄҈鼀鼀 矄ᆠ
�矄 ᆠ�矄҈鼀�矄 ᆠ犨 矄ᨨ ᨨ ├ 矄矄 犨� 和桌 㡰 鼀鼀 㡰 �鼀鼀 ᆠ㡰 ҈鼀鼀 㡰 鼀鼀 ᔼ � ᨨ盐㜠矄� �鼀 犨 �矄矄㜠 ᆠ�� ҈�㡰ᨨ �ᆠ├ 矄矄�矄� ᨨ 盐ᆠ �
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ᨨ ᨨ ᨨ � � �鼀矄 犨㜠 � l 㡰 � 矄�泐 ᨨ矄��ᨨ 犨 ᆠ 鼀 �鼀盐鼀 ᨨᆠᆠ盐 �ᨨ盐矄泐鼀� 矄盐 ᆠ � 鼀� �鼀盐鼀 �ᆠ├ 矄矄ᨨ �矄矄㜠�矄 和桌 㡰 l 㡰 l ⏌ ᆠ㡰 l ⏌ 㡰 l ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 ҈ � �矄ᨨ矄 ᨨ ᨨ鼀 � ᆠ� 矄�鼀 ᨨ 矄ᆠ 犨 �� ᨨ 犨 � 鼀�矄盐矄 ᔼ 犨 � ᨨ �� �矄 ᆠ � ᨨ 鼀 盐鼀ᨨ � �矄ᨨ矄 � 矄鼀矄�鼀 ᨨ 矄ᆠ �矄�� 泐 ҈和桌 㡰 ҈ 㡰 ҈ ᆠ㡰 㡰 ᔼ ᨨ盐矄� �鼀 � 矄 ᨨ ├� �矄�矄ᨨ ⼄盐 犨� 矄ᨨ 矄��矄鼀 犨�ᆠ矄犨� 矄 ᨨ �
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⏌ 和
盐矄ᨨ q ᔼ Ӹᨨ盐 �矄 ᨨ盐㜠矄� �鼀 矄�ᨨ 犨
矄�矄�矄 �矄 ᆠ犨 矄�矄 矄ᆠ �矄 ᨨ盐 � �犨� �矄҈黨҈㜠㝐├矄桌 㡰 盐 ᆠ � 㡰 ᆠ㡰 ᨨ盐 㡰 ⼄盐 盐 2
Integer Value Answer
䰘 꾀Ӹᨨ盐 �Ἐ矄 ᨨ盐矄�� ᆠ �鼀 犨� 矄�矄 S҈ S�矄�鼀鼀矄 � 犨 ᆠ�� 㦨 㡰矄 矄 ᨨ犨 � ��盐 ᨨ矄��ᨨ 矄ᨨ 鼀 �鼀盐鼀ᨨ
矄盐 ᆠ ᨨ 犨 S �� 泐 x 㦨 pml d 꾀� x d 犨 l �矄 犨 � ᨨ �� � � d ᆠ�� 㦨҈㡰d�矄 S ␔ 矄�ᨨ 犨
矄�矄 m �矄
� � 盐ᨨ p�矄 꾀 ҈�桌 ҈
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EBD_8341 P-270
Physics
Numeric Answer
3
黨
꾀泐ᨨ盐 �矄 ᨨ盐㜠矄� �鼀 �矄 ᨨ鼀 盐矄� � 鼀ᨨ ᨨᆠᨨ鼀�ᆠ�� ᨨ 犨 � l �
矄�泐ᨨ �� ᨨ� ᨨ 矄ᆠ 犨 � ᆠ�矄 l�矄⧤盐 �矄 � 矄�泐ᨨ �� ᨨ� 犨 l
⧤ � ᆠ�矄 �矄�� 泐 犨 �矄 � 㜠 ҈ � 盐ᨨ n�矄힘힘힘힘힘힘 㝐� 和 ᔼᔼ㡰桌 ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 ҈� � �矄ᨨ矄 ᨨ 矄鼀ᨨ�ᨨ ᨨ 矄ᆠ 犨 �� ᨨ 犨 � � 鼀�矄盐矄 � �矄ᨨ矄 ᨨ 矄鼀 矄ᆠ�ᨨ ᨨ 矄ᆠ 犨 ᨨ �� 矄ᨨ盐ᆠ ᨨ 犨 � l�矄 盐矄 盐 ᨨ l �矄 � 鼀㡰 힘힘힘힘힘힘 㝐� 黨 ᔼᔼ桌 Fill in the Blanks
4
ᔼ Ӹᨨ盐 �矄 ᨨ盐㜠矄� �鼀 犨ᨨ矄�矄ᆠ 矄 ᆠᨨ 矄ᨨ盐ᆠ矄ᨨ ⼄盐鼀�盐 �A ᨨ 犨 � � l ᔼ ᨨ �鼀 犨� 矄鼀矄㜠盐 犨ᨨ 矄�矄矄ᨨ盐ᆠ矄ᨨ ⼄盐鼀�盐 �A 犨 � � l 盐� ᆠᨨ �ᨨᨨ � 矄�泐ᨨ �� 鼀� ᨨ� ᨨ 矄ᆠ � �矄ᆠ矄ᨨ � 矄ᆠᨨ ᆠ矄�矄 ҈黨㜠㝐├矄桌 True / False
5 ҈
ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 ᨨ鼀 犨� 矄ᨨ盐ᆠ ᨨ 犨 ��� ᨨ 泐ᆠ├ 犨 � � �矄ᨨ矄 ҈黨䰘㜠㝐├矄桌
犨ᨨ矄�矄 � Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 �盐鼀� 泐犨ᨨ � 矄ᨨ �盐鼀鼀矄鼀�� ��� ᨨ 矄鼀 犨 � ᨨ � ᆠ 犨� ᨨ矄 ᨨ 矄ᆠ ҈黨㜠㝐├矄桌
MCQs with One or More than One Correct Answer
6
졠 和҈
和
和ᆠ
␔ �ᆠᨨ 盐ᆠ� � Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄盐 ᆠ 犨ᨨ矄�矄犨� �ᨨ⼄盐�
㜠泐 ᆠᨨ � � �犨ᨨ矄鼀 ᨨ矄 ᆠ矄犨ᨨ ᆠᨨ ᨨ� 矄ᨨ盐ᆠ矄 和҈和㡰鼀�� ��� ᨨ 犨 � 鼀 矄盐 鼀�矄├ 泐ᆠ 矄ᆠ ᨨ 㜠 ᨨ >㡰 �矄 ᆠ㦨鼀 ᨨ鼀 鼀� 㜠ᨨ� ᨨ 和҈和矄矄 ᨨ犨 矄ᆠ 鼀�ᆠ泐� ��盐
�矄 ᆠ犨 矄ᨨ盐ᆠ矄 㦨 鼀鼀 ᨨ��� 졠�矄 � 矄ᆠ�ᨨ ᨨ 矄ᆠ � ᨨ� � �和҈和␔ �ᆠ ᨨ ᨨᨨ犨� ��矄 㡰盐ᨨ � 矄�泐 ᨨ 矄ᆠ 䁸 꾀 ҈桌
㡰 和�� �� ├ 矄ᨨ 㜠 �矄 㡰 ��ᨨ 泐ᆠᨨ矄ᨨ ᨨ� 졠犨� ├ ᆠ㡰 泐ᨨ�ᆠ �� ├ 矄 犨� ᨨᆠ� 矄泐鼀鼀�ᆠ泐ᆠ ᨨ盐졠� 㜠 �ᆠ�ᨨ
㡰 和鼀�ᆠ�ᆠ盐�� ├ 矄ᆠ ᨨ� 꾀�� 矄ᨨ盐ᆠ犨 �ᆠ 鼀�矄犨ᨨ犨 � l҈㦨 鼀 l㦨 鼀�矄盐矄 � Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 ᔼ ᆠᨨ � �犨� 矄 ᨨl҈ lb҈ b 盐鼀ᨨ � �矄 ᨨ 鼀犨� � �矄 ᆠyᨨ ᨨ 矄� ᨨ ᆠ 鼀 �鼀盐鼀m҈ m矄ᆠ�泐 꾀 ҈桌
㡰 b>b҈ 㡰 m҈>m ᆠ㡰 ᨨ鼀 ᆠ 鼀 �鼀盐鼀 鼀 �鼀盐鼀ᨨ lᨨ矄 犨� � 鼀� �鼀盐鼀ᨨ l҈ 㡰 �盐 矄�ᨨ ᨨ � �矄 ᨨ l҈�矄 � l � ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄�ᨨ 㜠 犨 犨ᨨ矄�矄�矄 犨 � ᨨ �� �矄 l � 矄�泐ᨨ �� � �ᨨ 矄�҈�矄 ᨨ盐�鼀矄 � 矄�泐ᨨ �� � � ᨨ 矄� ᔴ ᨨᨨ矄 ᆠᨨᆠ ᆠ ᨨ�ᆠ 矄㡰 桌 㡰 ᔼ 㦨l 矄ᆠ 犨�ᆠᨨ � ᨨ 泐ᨨ 鼀 �鼀盐鼀 㡰 ᔼ l l矄ᨨ 鼀ᨨ鼀 �鼀盐鼀 矄� 矄 ᆠ 鼀 �鼀盐鼀㡰犨�ᨨ矄 ᨨ 矄ᆠ
ᆠ㡰 ᔼ � 矄�泐ᨨ �� � �ᨨ 矄�҈�矄 盐ᆠ 矄ᨨ �ᆠᨨ鼀矄⼄盐ᨨ ᨨ 矄� � 矄��矄 ᨨ ᨨ矄 ├ �� � �矄犨�� ᆠ矄 㡰 ᔼ � 矄�泐ᨨ �� � �ᨨ 矄��矄� ᆠ矄 矄ᨨ �ᆠᨨ鼀矄⼄盐ᨨ ᨨ 矄�҈ � 矄��矄 ᨨ ᨨ矄 ├ �� � �矄犨�� ᆠ矄 ␔ ��� �矄盐矄 ᨨ�盐鼀� 犨ᨨ矄�矄� Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 矄�ᨨ 犨 矄�矄�矄 b 矄ᆠ �矄 �矄 ᆠd >b㡰 ᨨ鼀 矄�矄꾀 ᨨ� ᨨ 矄ᆠ �ᆠ泐� ᨨ ᨨ ᨨ ᨨ 矄�矄 ᆠ� 犨 � 矄鼀�矄矄� �和ᨨ鼀ᨨ 矄鼀�矄矄� � ҈黨㜠㝐├矄桌 犨 � 矄 b b b b ᆠ㡰 l = 㡰 l = 㡰 l = d d d d 䰘 ᔼ Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 � ᆠ �矄 ᨨ盐 ᨨ � 矄�泐�ᨨ犨 �� ├ � �矄矄黨 �矄�鼀�矄 ҈黨㜠㝐├矄桌 㡰 � 矄��矄 矄ᆠ 盐ᨨ 犨ᨨ矄�矄� 盐 �矄 盐 �矄矄ᆠ�泐 㡰 � 矄��矄 矄ᆠ 盐ᨨ 犨ᨨ矄�矄 盐 �矄 ҈盐 �矄矄ᆠ�泐 ᆠ㡰 鼀�盐 �ᨨ�矄 㡰 鼀�盐 �ᨨ�矄
㡰 l =
7
泐
Match the Following
ᔴᨨ盐鼀 㜠ᔼ 矄 ᨨ犨矄 ᨨ盐 矄�盐�ᨨ 矄 ᨨ 矄 Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 犨� 矄ᆠ ᆠ 犨泐 ᨨ鼀 矄�矄和҈ 和ᔼ ᆠ ᨨ 矄ᆠ矄矄和҈㦨和 和҈҈㜠和҈㦨 l/4 和҈ 和㦨 l⏌ 犨 l �矄 犨 � ᨨ �� 盐矄 ᔼ ᆠ矄矄ᔴ
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P-271
Wave Optics
矄 矄 ᨨ ᆠ�� m �ᆠ├ 矄矄t�矄 矄 ᨨ 矄�和 �ᆠ├ 矄矄矄ᨨ 矄 矄 � � � ᆠ矄矄 矄 � ᆠ犨 �� 犨矄ᆠ � �ᨨ� ᨨ 矄ᆠ ᨨ鼀 犨ᨨ矄�矄 �矄 ᨨ 泐d 㡰 � 矄�泐泐I 㡰㝐ᆠ ᆠ 矄�盐�ᨨ �� � ᔴᨨ盐鼀 㜠ᔼ 犨� 矄鼀 矄㡰 �
黨桌 ᔴᨨ盐鼀 㜠ᔼᔼ� ᨨ 矄�盐�ᨨ ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀 㜠ᔼᔼ ҈
和
꾀㡰 和҈
㡰 d 㡰㦨 㡰 m ҈㡰t㦨l⏌
泐
҈
和
和҈
⼄㡰 d ҈㡰㦨
ᔴ㡰 m ҈㡰t㦨l⏌ 㡰 I ҈㡰㦨 和
㡰 m ҈㡰t㦨3l⏌ 和
҈
҈
矄㡰 I 㡰>I ҈㡰 I 㡰>I ҈㡰
10 Subjective Problems
和ᆠ
和҈
黨
㦨҈鼀鼀
㦨҈鼀
҈
和
㡰
㡰 � ᨨᆠ�ᨨ ᨨ ᆠ 鼀 �鼀盐鼀 �� � � 犨� ᨨ � ᆠ㡰ᨨ 泐 �矄 㡰 � �� � 矄�泐 ᨨ� O � ᨨ 鼀 �鼀盐鼀 � �� 矄�泐 ᆠ㡰 ᨨ犨� 鼀�� �矄ᆠ 泐犨 ��� ᨨ � ᨨ䰘 鼀 � 犨 � 矄ᨨ �� ᨨ鼀鼀 �鼀 ᆠ泐ᨨ� O 꾀犨 � 矄� �矄ᨨ鼀 ᨨ ��
鼀 �盐鼀ᨨ ᆠ�� ⏌ᔼ� ᨨ �矄矄�ᨨ 桌 ҈ 꾀ᆠᨨ 鼀ᨨ 鼀�ᆠᨨ犨矄ᨨ 犨 � l 㦨 � 鼀鼀 矄 ᨨ Ӹᨨ盐 �矄 ᨨ盐 矄� 盐矄 矄�ᨨ 犨 矄�矄�矄҈鼀鼀 � 矄�泐ᨨ 鼀�ᆠᨨ犨矄�矄鼀矄盐 ᨨ 矄ᆠ ᆠ ᨨ ᨨ 矄�矄 �矄 ᆠᨨ ҈鼀 ᨨ鼀�矄矄 ᨨ犨 �
��
ᔼ YDSE�� ᆠᨨ � � �犨ᨨ犨 � 矄� 鼀 䰘 鼀盐矄 � 鼀� �鼀盐鼀 �矄 ᆠ犨 鼀 �鼀 ᨨ 犨ᨨ犨 � 矄ᆠᨨ� ᆠ� � D/d㦨҈犨 D�矄 �矄 ᆠ犨 矄�矄 矄ᆠ d�矄 �矄 ᆠ犨 矄�矄 㜠㝐├矄桌
Ӹᨨ盐 �矄 ᨨ盐矄� �鼀 �矄 ᨨ � 鼀 �盐鼀ᨨ ᆠ�� ⏌꾀�� ᨨ 鼀犨 � �矄 � � ᨨ 矄�矄 � ��鼀鼀矄�ᨨ ᨨ犨矄�S�矄 ᆠᨨ 泐 � �矄矄矄 ᨨ �ᆠ├ 矄矄҈m鼀 ᆠ�� ҈� � ᆠ �矄ᨨ矄 ᨨ 矄ᆠ ᆠ ҈�鼀 ᨨ鼀 矄�矄矄矄 ᨨ犨 � ��盐 ҈黨黨黨㜠҈㝐├矄桌 Ӹ
和҈ 졠
和ã 和
㡰 ᔼ � ᆠ� 鼀 矄 ᨨ鼀泐ᨨ ᨨ盐矄� 盐矄 � 泐㜠ᆠᨨᨨ � 矄 ᨨ � ᆠ鼀� �鼀ᨨ 矄ᆠ 㡰 ᔼ � ᆠ� 鼀鼀├矄 �ᨨ 犨� x �矄 矄� ᨨ ᨨ犨矄 ᨨ犨 � ��盐㡰 � y㜠ᆠᨨᨨ � ᨨ �矄鼀� �鼀ᨨ � 矄� ᨨ ҈黨黨㜠㝐├矄桌 ᆠ 鼀 �鼀盐鼀 ᔼ Ӹᨨ盐 �矄 �鼀 盐矄��矄ᆠᨨ 泐 �
�矄矄ᨨ ᆠ�� ҈犨 � ᨨ犨矄��矄 ᆠᨨ 泐 ᨨ �矄矄 � � 矄鼀 �ᆠ├ 矄矄 矄 �矄ᨨ 盐 � � ᆠ�� ҈䰘ᔼ ᆠ �矄ᨨ矄 盐矄� ��� ᨨ 犨 � �处ᔼ�矄 ᨨ盐 ᨨ� Pᨨ 矄ᆠ 犨 ᆠ 鼀 �鼀盐鼀 㦨㡰 矄 ᨨ �矄矄矄犨� 矄
ᨨ犨 矄⏌ ᨨ��� � 矄�泐ᔼ�矄 盐 ᨨ矄 犨 盐矄 ᨨ � 鼀 �鼀盐鼀��矄ᨨ犨 ᨨ� P犨 � 矄� 鼀� �鼀盐鼀�矄ᨨPᔴᆠ盐 �ᆠ├ 矄矄ᨨ �矄矄 꾀矄ᨨ�ᨨ ᨨ �� 泐 �矄矄鼀泐 �ᆠ 㡰 ҈黨黨䰘㜠�㝐├矄桌 꾀 ᨨ盐㜠矄�盐矄�矄�鼀鼀矄 � �⼄盐� ᨨ ᆠ� � ҈ᔼ 矄矄�矄�ᨨ ᨨ ҈鼀鼀 �矄 ᆠ 犨 ᨨ 矄�矄 矄ᆠ �矄҈鼀 矄�矄 �盐鼀� 泐 鼀 ᨨ �� 犨 ᨨ矄 犨 � � ��矄处 ҈黨黨㜠㝐├矄桌 �㡰 ᔴᆠ盐 � �㜠犨� ��㡰 졠 ᨨ 矄�矄ᨨ 盐矄�矄ᆠᨨ 泐 �
�矄矄矄 ᨨ ᆠ�� ҈� � 矄鼀矄 �ᆠ├ 矄矄ᨨ 矄 ᨨ� � ᆠ 鼀� �鼀盐鼀 ᨨ �矄 ᔼ 鼀ᨨ � � Ӹᨨ盐 �矄 ᨨ盐 矄� �鼀 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ 盐 � ᨨ鼀 鼀 ᨨ �� ᨨ 犨 � 处 � 矄�泐 ҈⏌ p 㡰␔鼀�矄� ᆠ�
ᨨ鼀泐ᨨ 犨ᨨᆠ�ᆠ盐盐矄꾀 ᨨ ��҈鼀 鼀矄ᆠ�泐꾀 ᆠ泐 矄 �鼀ᨨ �ᆠ├ 矄矄 处 ᆠ� � ҈� ᨨ 犨 � ᨨ 处�矄ᆠ � ᨨ ᨨ 盐A矄
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EBD_8341 P-272
Physics
fig. Calculate the power (in watts) received at the focal spot F of the lens. The lens is symmetrically placed with respect to the apertures. Assume that 10奠 of the power received by each aperture goes in the original direction and is brought to the focal spot. [1989 - 8 Mark]
A
F
B
45. A beam of light consisting of two wavelengths, 6500Å and 5200Å, is used obtain interference fringes in a Young’s double slit experiment : [1985 - 6 Marks] (i) Find the distance of the third bright fringe on the screen from the central maximum for wavelength 6500Å. (ii) What is the least distance from the central maximum where the bright fringes due to both the wavelengths coincide? The distance between the slits is 2 mm and the distance between the plane of the slits and the screen is 120 cm.
Topic-3 : Diffraction, Polarisation of Light and Resolving Power 1 1.
2.
3.
4.
5.
MCQs with One Correct Answer
A beam of plane polarised light of large cross-sectional area and uniform intensity of 3.3 Wm–2 falls normally on a polariser (cross sectional area 3 × 10–4 m2) which rotates about its axis with an angular speed of 31.4 rad/s. The energy of light passing through the polariser per revolution, is close to : [Main Sep. 04, 2020 (I)] –5 (a) 1.0 × 10 J (b) 1.0 × 10–4 J –4 (c) 1.5 × 10 J (d) 5.0 × 10–4 J The aperture diameter of telescope is 5m. The separation between the moon and the earth is 4 × 105 km. With light of wavelength of 5500 Å, the minimum separation between objects on the surface of moon, so that they are just resolved, is close to: [Main 9 Jan. 2020 I] (a) 60 m (b) 20 m (c) 200 m (d) 600 m A polarizer - analyser set is adjusted such that the intensity of light coming out of the analyser is just 10奠 of the original intensity. Assuming that the polarizer analyser set does not absorb any light, the angle by which the analyser need to be rotated further to reduce the output intensity to be zero, is: [Main 7 Jan. 2020 I] (a) 71.6° (b) 18.4° (c) 90° (d) 45° The value of numerical aperature of the objective lens of a microscope is 1.25. If light of wavelength 5000 Å is used, the minimum separation between two points, to be seen as distinct, will be : [Main 12 April 2019 I] (a) 0.24 µm (b) 0.38 µm (c) 0.12 µm (d) 0.48 µm A system of three polarizers P1, P2, P3 is set up such that the pass axis of P3 is crossed with respect to that of P1. The pass axis of P2 is inclined at 60o to the pass axis of P3. When a beam of unpolarized light of intensity Io is incident on P1, the intensity of light transmitted by the three polarizers is I. The ratio (Io/I) equals (nearly) : [Main 12 April 2019 II]
6.
7.
8.
9.
(a) 5.33 (b) 16.00 (c) 10.67 (d) 1.80 Diameter of the objective lens of a telescope is 250 cm. For light of wavelength 600 nm. Coming from a distant object, the limit of resolution of the telescope is close to: [Main 9 April 2019 II] (a) 1.5 ´ 10 -7 rad(b) 2.0 ´ 10-7 rad (c) 3.0 ´ 10 -7 rad (d) 4.5 ´ 10-7 rad Calculate the limit of resolution of a telescope objective having a diameter of 200 cm, if it has to detect light of wavelength 500 nm coming from a star. [Main 8 April 2019 II] (a) 305 × 10–9 radian (b) 610 × 10–9 radian (c) 152.5 × 10–9 radian (d) 457.5 × 10–9 radian In a double-slit experiment, green light (5303Å) falls on a double slit having a separation of 19.44 µm and a width of 4.05 µm. The number of bright fringes between the first and the second diffraction minima is : [Main 11 Jan 2019 II] (a) 10 (b) 05 (c) 04 (d) 09 Consider a tank made of glass (refractive index 1.5) with a thick bottom. It is filled with a liquid of refractive index m. A student finds that, irrespective of what the incident angle i (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of m is: [Main 9 Jan. 2019 I]
(a)
5 3
(b)
3 5
(c)
5 3
(d)
4 3
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P-273
Wave Optics
҈
҈҈
҈
҈
�盐犨� ᨨ ᆠ 鼀 �鼀盐鼀� 矄� �矄� � ᆠ�ᨨ �矄 犨� ᨨ 矄��矄҈m鼀 矄� �矄 �盐鼀� 泐鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ 犨矄 ᔼ ᨨ 矄�ᨨ 矄鼀犨� �矄鼀 �Ӹᨨ盐 �Ἐ矄 � �矄 ᆠ ᨨ矄 ᨨ 矄ᆠ ᆠ �矄 ᆠ�ᆠ鼀 ᨨ鼀 矄�矄ᔼ ᨨ矄 � �犨� �矄҈ᆠ鼀犨 �矄 矄�矄�ᨨ �矄 ᆠ䁸 � �矄 ᆠ犨 ᆠ 矄ᨨ ᆠ 矄�㡰㝐� ҈桌 㡰 �m鼀 㡰 �m鼀 ᆠ㡰 䰘�m鼀 㡰 ҈m鼀 ᨨ� �� ᨨ � 矄�泐ᔼ矄矄矄 ᨨ盐� � ᨨ� 꾀꾀 ᨨ � �ᆠᨨ� �矄ᆠ � ᔼ 꾀 � 矄�泐ᨨ �� 泐ᨨ �矄 ᨨ盐 ᨨ ᨨ犨 ᨨ � �ᆠᨨ� ᔴ�矄ᆠ 犨 꾀
� 矄�泐泐ᨨ �矄 ᨨ犨 ᨨ盐 ᨨ ᔼ � 犨 ᨨ� 꾀 ᔴ�矄 㝐� ҈桌 㡰 㡰 ᆠ㡰 � 㡰 �� ᨨ 犨 � �� 鼀 矄 ᨨ鼀泐ᨨ 矄�ᨨ 犨� 맔҈�ᆠ鼀 �盐ᨨ矄��ᨨ ᨨ 矄ᆠᨨ 鼀� �鼀 ᨨ鼀 ᆠ 鼀 �鼀盐鼀犨� � � 矄㡰 㝐� 졠 � 꾀�҈�҈桌 p p p p 㡰 㡰 ᆠ㡰 㡰 ҈ ᨨ� �� ᨨ � 矄�泐ᔼ�矄� ᆠ� ᨨ 矄泐矄鼀ᨨ 犨ᨨᨨ� 矄꾀 ᨨᨨ犨 泐 � 矄�泐ᨨ 鼀� �� �矄ᔼ⏌ᔼ � ᨨ� ᔴ�矄ᆠ 犨 꾀 � 矄�泐ᨨ 鼀� �� �矄 盐ᆠ ᨨᔼ⏌ � 犨 ᨨ� 矄꾀 ᔴ�矄q
㝐� 졠 � 꾀�҈҈桌 ҈⏌
æö 㡰 ᆠᨨ矄 q = ç ÷ èø
҈⏌
҈
҈�
҈
҈䰘
҈
҈黨
҈⏌
æ҈ö 㡰 ᆠᨨ矄 q = ç ÷ èø
҈⏌
æ҈ö æö ᆠ㡰 ᆠᨨ矄 q = ç ÷ 㡰 ᆠᨨ矄 q = ç ÷ èø èø 꾀 ᨨ矄�矄鼀ᨨ� �犨� 矄 ᨨ �� ᨨ犨 矄 矄�ᨨ 泐鼀�ᆠᨨ犨矄ᨨ盐ᆠ鼀�� �犨矄 ⼄盐 ᆠ泐 ҈ ␔ �矄 ⼄盐 ᆠ泐ᨨ 鼀�ᆠᨨ犨鼀矄盐 泐 ᨨ矄䁸 矄 ᨨ �� 㦨맔҈鼀矄҈㡰㝐� ҈䰘桌 㡰 ҈䰘 㡰 ҈� ᆠ㡰 ҈҈ 㡰 ҈҈ 꾀 矄� � 矄� ᨨ 犨� b �矄 �盐鼀� 泐 ᆠᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ 犨 � lᔼ 矄ᆠᨨ ᨨ盐 鼀� �鼀� � ᆠ�ᨨ �矄 ᆠ҈鼀 ᨨ鼀 矄�ᆠ鼀 ᆠ鼀矄ᆠ�泐 ᨨ鼀 ᆠ 鼀 �鼀盐鼀犨 �矄 犨� ᨨ ᆠ 鼀 �鼀盐鼀䁸 � �矄 ᆠ犨 �矄鼀� �鼀盐鼀ᨨ � 矄� ᨨ ᆠ 鼀 �鼀盐鼀㡰 㝐� 졠 � 꾀�҈䰘桌 㡰 ҈�ᆠ鼀 㡰 ᆠ鼀 ᆠ㡰 �ᆠ鼀 㡰 ᆠ鼀
ᨨ ᨨ � ᨨᆠ鼀ᨨ � 矄 ᨨᨨ �盐矄ᔼ�矄矄矄盐鼀 犨 ᨨ�矄�盐鼀� 泐 鼀ᨨ �� ᨨ 犨 � l 矄 ᨨ 矄ᨨ ᨨ� ᨨ ᨨᨨ矄�犨ᨨ ᆠ鼀㡰�矄 矄盐鼀 ᨨ �矄 �ᨨ鼀�ᆠ 矄 矄 盐 ᨨ � ᆠ�ᨨ 矄ᨨ犨ᨨ盐 �矄鼀� �鼀盐鼀矄� (say b鼀� 㡰犨 㝐� ҈桌 㡰 = l 鼀� = l l 鼀� = l 㡰 =
æ l ö l 鼀� = ç ÷ ç ÷ è ø æ l ö ÷ 㡰 = l 鼀� = çç ÷ è ø
犨ᨨ矄矄҈�� 泐矄犨泐 ᨨ鼀 泐 矄 ᨨ盐� 矄ᆠᨨᨨ ᨨᆠ� �鼀ᆠ鼀 犨 � ᨨ �� �矄 鼀 ᨨ矄 矄矄盐矄矄ᨨ 泐 矄ᆠᨨ 鼀� �鼀盐鼀 �矄 ᆠ 犨 鼀 矄 ᨨ盐 ҈�� 泐㦨黨맔҈҈�鼀㡰ᨨ ᨨ ᨨ 㝐� 졠 � 꾀�҈҈桌 ҈ 㡰 ҈ ├鼀 㡰 ҈ ├鼀 ᆠ㡰 ҈҈҈├鼀 㡰 ҈├鼀 꾀矄矄盐鼀� � 盐鼀 盐�ᨨ �盐矄ᨨ �ᆠ鼀 ᆠᨨ鼀 ᨨ �犨� � �矄 ᆠ ᨨ � ᆠ鼀 鼀� �鼀盐鼀 矄�ᨨ 犨 犨ᨨᨨᆠ矄 盐鼀 泐ᆠ 矄ᨨ � 鼀犨 � �矄 㝐� ҈�桌 㡰 ҈Ҙ鼀 㡰 Ҙ鼀 ᆠ㡰 ҈Ҙ鼀 㡰 Ҙ鼀 ᨨ� �� ᨨ � 矄�泐ᔼ�矄� ᆠ� ᨨ 矄盐 ᆠᨨ ᨨᆠ├ᨨ �矄矄犨矄矄 �ᔼ ᆠ矄犨 �ᆠ ᨨ ᨨ ᨨᨨ犨� � 矄鼀 矄 �矄 盐 䁸 㝐� 졠 � 꾀�҈҈҈�桌 㡰 ᆠ �� �矄ᆠᨨ鼀泐ᨨ� 犨� � 矄�泐 I0 矄矄 2 㡰 矄鼀� �� �矄ᆠᨨ鼀泐ᨨ� 犨� � 矄�泐 I0 矄矄 2 ᆠ㡰 矄鼀� �� �矄�泐ᨨ� 犨� � 矄�泐 I0 2 㡰 ᆠ �� �矄 �泐 ᨨ� 犨� � 矄�泐 I0 2
犨ᨨ鼀矄꾀 ᨨ ᨨ� �� 犨� 鼀盐盐泐 �ᆠ盐 矄ᨨ ᨨ� �ᨨ 矄 ᨨ盐� ᨨᨨ� ᨨ鼀 ᨨ矄��ᨨ 犨 鼀꾀 矄鼀 �鼀盐鼀 � 矄�泐 鼀 矄 ᨨ� 矄�泐㡰ᨨ�ᨨ ᨨ ᨨᨨ� ᨨ盐� ° 鼀├矄 犨ᨨ鼀矄⼄盐泐 �� ᔼ � ��� 矄��矄ᨨ 犨ᨨ鼀矄ᔼ꾀 ᔼ ᔼ矄ᆠ�泐 꾀 ⼄盐矄 㝐� ҈桌 ᔼ ҈ 㡰 㡰 ᆠ㡰 ҈ 㡰 꾀泐ᨨ �� �矄� ᆠ� ᨨ鼀 矄ᨨ鼀 �盐鼀
ᆠ��ᆠ � ᨨ ᨨ � ᆠ�ᨨ �矄 q�ᔴ 犨矄矄 �ᨨ � ᆠ� ᆠ�矄q�矄盐ᆠ 矄� q�ᔴ⏌ 矄� q�㦨h㦨҈ � ᆠ�� ᨨ 犨ᨨ 鼀 ��矄 㝐� 졠 � 꾀�҈黨҈桌 㡰 㡰 ᆠ㡰 㡰 黨 ᔼ �鼀 ᨨ 矄� �矄� � ᆠ�ᨨ �矄 鼀� �鼀盐鼀 ᨨ �� ᆠᨨ� ᆠ� 矄犨� �矄鼀 �鼀盐鼀ᨨ 矄ᨨ鼀 ᨨ 犨 � ᔼ 犨 � ᨨ �� �矄处 犨 � ᨨ �矄鼀 �鼀盐鼀犨� 㝐� 졠 � 꾀�҈҈桌 㡰 处 㡰 处 ᆠ㡰 ��处 㡰 处 ᆠ㡰 =
҈
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EBD_8341 P-274
23.
24.
Physics
Abeam of unpolarised light of intensity I0 is passed through a polaroidAand then through another polaroid B which is oriented so that its principal plane makes an angle of 45° relative to that of A. The intensity of the emergent light is [Main 2013] (a) I0 (b) I0/2 (c) I0/4 (d) I0/8 A person lives in a high-rise building on the bank of a river 50 m wide. Across the river is a well lit tower of height 40 m. When the person, who is at a height of 10 m, looks through a polarizer at an appropriate angle at light of the tower reflecting from the river surface, he notes that intensity of light coming from distance X from his building is the least and this corresponds to the light coming from light bulbs at height 'Y' on the tower. The values of X and Y are 4 ) 3 [Main Online April 9, 2013]
respectively close to (refractive index of water ;
L
40 m
10 m
Y
X
50 m
25.
(a) that the central maximum is narrower (b) more number of fringes (c) less number of fringes (d) no diffraction pattern 26. Consider Fraunhoffer diffraction pattern obtained with a single slit illuminated at normal incidence. At the angular position of the first diffraction minimum the phase difference (in radians) between the wavelets from the opposite edges of the slit is [1995S] (b) p / 2 (c) 2 p (a) p / 4 (d) p 27. A beam of light of wave length 600 nm from a distance source falls on a single slit 1 mm wide and a resulting diffraction pattern is observed on a screen 2m away. The distance between the first dark fringes on either side of [1994 - 1 Mark] central bright fringe is (a) 1.2 cm (b) 1.2 mm (c) 2.4 cm (d) 2.4 mm 28. A parallel monochromatic beam of light is incident normally on a narrow slit. A diffraction pattern is formed on a screen placed perpendicular to the direction of the incident beam. At the first minimum of the diffraction pattern, the phase difference between the rays coming from the two edges of the slit is [1998 - 2 Marks] (a) 0 (b) p/2 (c) p (d) 2p
(a) 25 m, 10 m (b) 13 m, 27 m (c) 22 m, 13 m (d) 17 m, 20 m Yellow light is used in a single slit diffraction experiment with slit width of 0.6 mm. If yellow light is replaced by X– rays, then the observed pattern will reveal, [1999S - 2 Marks]
3
Numeric Answer
29. Orange light of wavelength 6000 × 10–10 m illuminates a single slit of width 0.6 × 10–4 m. The maximum possible number of diffraction minima produced on both sides of the central maximum is _____________. [Main Sep. 04, 2020 (II)]
Topic-4 : Miscellaneous (Mixed Concepts) Problems 4 1.
A point source emits sound equally in all directions in a non-absorbing medium. Two points P and Q are at a distance of 9 meters and 25 meters respectively from the source. The ratio of amplitudes of the waves at P and Q is ............ [1989 - 2 Marks] 7
2.
8
Fill in the Blanks
Match the Following
A simple telescope used to view distant objects has eyepiece and objective lens of focal lengths fe and fo, respectively. Then [2006 - 6M] Column I Column II (A) Intensity of light received (p) Radius of aperture by lens (B) Angular magnification (q) Dispersion of lens (C) Length of telescope (r) Focal length of objective lens and eyepiece lens (D) Sharpness of image (s) Spherical aberration
Comprehension/Passage Based Questions
Passage The figure shows a surface XY separating two transparent media, medium-1 and medium-2. The line ab and cd represent waveforms of a light wave travelling in medium-1 and incident on XY. The lines ef and gh represent wavefronts of the light wave in medium-2 after refraction. [2007] b d
X
a c
medium-1 Y f h medium-2
e g
3.
Light travels as a (a) parallel beam in each medium (b) convergent beam in each medium (c) divergent beam in each medium (d) divergent beam in one medium and convergent beam in the other medium.
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P-275
Wave Optics
矄矄ᨨ �� 犨cde fᆠf f f 矄ᆠ�泐ᔼ�矄�� fᆠ¹f 㡰 fᆠᆠ
ᨨ⼄盐ᨨf 㡰 f ᆠ ⼄盐ᨨf ᆠ㡰 f f 㡰�矄⼄盐ᨨ fᆠf㡰 㡰 f fᆠ㡰�矄 ᨨ⼄盐ᨨ f f㡰 和 ᨨ �� �矄 㡰 矄鼀� 鼀 �盐鼀㜠҈ 鼀 �盐鼀㜠 㡰 �� 鼀 �盐鼀㜠҈ � 鼀 �盐鼀㜠 ᆠ㡰 �� 鼀 �盐鼀㜠 � 鼀 �盐鼀㜠҈ 㡰 � b d
�
㡰 꾀ᆠᨨ 矄ᨨ ᨨᆠ � ҈�ᆠ鼀 ᆠ ᆠ 鼀�ᨨᨨ ᨨᆠ � ᆠ鼀├犨� �ᨨ�ᆠ 矄PQ RS 盐 矄 � �ᆠ �ᆠ�ᨨ 泐ᆠ鼀矄矄 ᨨ犨 �矄 ᆠ犨
矄 鼀�ᨨ�矄ᆠ鼀꾀 盐�� ᨨᆠABᨨ �� ҈ᆠ鼀�矄ᆠ ᨨ ᨨ�ᆠ �矄PQᨨ 矄 �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 矄ᔼ AἘ B Ἐ �矄 �鼀� ᆠ�ᨨ ᨨ鼀 矄 ᆠ�ᨨ
ᨨ鼀 鼀�ᨨ � �矄 ᆠᨨ AἘ B Ἐ ᨨ鼀 ᨨᨨ 鼀�ᨨ ᨨ� �矄鼀� � �ᆠ�ᨨ 꾀矄ᨨ ᨨᆠ ᨨ矄��ᨨ ᨨ A Ἐ B Ἐ 犨� 矄ᆠ ᨨ ᨨ�ᆠ �矄RS 㜠㝐├矄桌
䰘
10 Subjective Problems 꾀矄矄ABCDᨨ ҈ᆠ鼀犨� 矄犨ᨨ矄鼀矄�矄S҈ S矄 犨� � �ᆠ�矄矄矄ᨨ ⼄盐 �ᆠ├ 矄矄
�矄 ᆠ犨 矄�矄�矄鼀鼀POQ�矄 � �ᆠ盐ᨨ AB 矄矄� � ᨨ盐� O 鼀� ᨨ� ᨨ S҈ S꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� 矄ᨨ盐ᆠ�矄 ├ 和ᆠ鼀ᨨ犨P 鼀 ᨨ鼀 矄矄 ᨨ �盐鼀� 矄�矄矄矄 ᨨ犨 � ��盐ᨨ犨ᔴᆠ盐 ᨨ矄��ᨨ ᨨ ᆠ �� � �ᨨ ᨨ 犨 CD犨� 矄ᆠᨨ � OQᨨ犨�⼄盐� �矄ᨨ盐 � ᨨ 矄矄 � 盐ᨨOQ ᆠ �� � � �矄 ᨨ盐 ᨨQᔴᆠ盐 ᆠ�� ᨨ �⼄盐� ҈㜠�㝐├矄桌
A
㡰 꾀�矄矄ᨨ ᆠ�� ҈��矄ᆠᨨ 犨� � 泐ᨨ �ᆠ├ 矄矄 ᆠ�� ҈�� ᨨ 犨 � l� �� ��矄� ᆠ� ᨨ鼀泐 ᨨ 泐ᔼ�矄泐 ᆠ 盐 ᨨ犨矄盐 ᆠᨨ 泐 犨ᨨ ᆠ 泐矄 � ␔� ᆠᨨ ��ᨨ ᨨ �ᆠᨨ 矄盐ᆠ� � ᆠᔼ l㦨 鼀ᨨ� 矄盐ᨨ ᨨ犨 �ᆠ 泐矄� ᆠᨨ 矄盐ᆠ�泐 㜠㝐├矄桌
D
S҈ P
Q
O S
ᆠ鼀 鼀
S
҈ᆠ鼀 B
C
Answer Key Topic-1 : Wavefront, Interference of Light, Coherent and Incoherent Sources ҈ ҈҈
㡰 㡰
㡰 ҈ ᆠ㡰
ᆠ㡰 ҈ ᆠ㡰
㡰
� ᆠ㡰
㡰
䰘 㡰
ᆠ㡰
黨 㡰
҈ 㡰
Topic-2 : Young's Double Slit Experiment ҈ ҈҈ ҈ ҈
㡰 㡰 㡰 ҈ ᆠ㡰 ҈ 㡰 㡰 꾀和쟼 쟼 꾀㜠矄錐㜠⼄錐ᔴ㜠錐㜠矄㡰
㡰 㡰 㡰 㡰
҈
㡰 � ᆠ㡰 ᆠ㡰 ҈� ᆠ㡰 㡰 � ᆠ㡰 ᆠ㡰
҈ �
ᆠ㡰 䰘 ᨨ 盐矄㡰 㡰 ҈䰘 ᆠ㡰 ҈ 㡰 䰘 㡰 㡰 ᆠ㡰 䰘
㡰 㡰 黨㡰 㡰
黨 㡰 ҈ 㡰 ҈黨 㡰 ᆠ㡰 黨 䰘�㡰
Topic-3 : Diffraction, Polarisation of Light and Resolving Power ҈ ҈҈ ҈
㡰 ᆠ㡰 ᆠ㡰
㡰 ҈ 㡰 㡰
㡰 ҈ 㡰 ᆠ㡰
㡰 ҈ 㡰 㡰
� ᆠ㡰 ҈� 㡰 � 㡰
ᆠ㡰 ҈ 㡰 ᆠ㡰
䰘 㡰 ҈䰘 㡰 䰘 㡰
㡰 ҈ 㡰 㡰
黨 㡰 ҈ 㡰 ҈黨 㡰 㡰 黨 ҈黨㡰
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈ 䰘
� 黨 㡰
꾀㡰® 㡰錐 㡰® 㡰錐 ᔴ㡰® 㡰錐 㡰® 㡰 ⼄㡰 㡰錐
㡰
ᆠ㡰
� 㡰
㡰
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EBD_8341 P-276
Physics
Dual Nature of Radiation and Matter
25
Topic-1 : Matter Waves, Cathode and Positive Rays 1 ҈
MCQs with One Correct Answer
꾀 ᆠᨨ ᨨ盐泐�ᨨ � �盐鼀�ᨨ 졔졔㡰 ᨨᨨ � � 矄鼀├� �ᆠ �泐 �ᨨ
犨 �矄ᆠ� 㜠ᨨ��犨 � 矄ll졔졔 l�矄 㝐� 和 ᔼ㡰桌 㡰 l> l졔졔>l 㡰 l< l졔졔㦨l ᆠ㡰 l> l>l졔졔 㡰 l< ll졔졔 꾀矄矄盐鼀� � �ᨨ� 鼀ᨨᆠ盐 �矄 鼀ᨨ� � 犨� 鼀矄ᨨᆠ�泐 ⧤ 㜠ᨨ�� 犨 � ᨨ
�ᨨ � 鼀ᨨᆠ盐�矄ᆠᨨ矄ᨨ � �ᨨ� 鼀ᨨᆠ盐犨�� 맔҈├�ᨨ 鼀
ᆠᨨ 矄 ҈맔҈⏌⧤ ᆠ├ᆠᨨ 矄 맔҈矄㡰 㝐� 和 ᔼᔼ㡰桌 㡰 处 㡰 处 ᆠ㡰 处 㡰 处
�
䰘
m 鼀ᨨ� �ᨨ � x㜠 �矄犨� ᨨᆠ�泐vᆠᨨ� 矄矄�ᆠ泐犨� ᨨ �ᆠB矄
�ᆠAᨨ 鼀矄矄 mA =
m ᔼ ᨨ �ᆠ矄鼀ᨨᨨ � x㜠 �矄 ᆠᨨ�矄�ᨨ ᆠ � Dl � 㜠ᨨ�� 犨 � ᨨ �ᆠA� 鼀矄ᨨ �矄 㜠ᨨ��犨 � l㡰 ᨨᆠᨨ�矄�ᨨ �矄 [㝐� Sep.04, 2020 (I)]
� �鼀矄矄 mB =
㡰
Dl = l
㡰
㡰
l ҈+
l
㡰 ҈+
㡰
e E t m v l
e E t e E t + m v m v 꾀�ᆠ � �矄 ᨨ鼀 盐 ᨨ ᆠᨨ鼀泐� 矄�ᆠ ᆠᨨ�矄�ᨨ ᨨ �ᆠ矄 �x �y � � 㜠ᨨ�� 犨 � 矄 �gx �g泐矄ᆠ�泐 ᔼ x y 犨 鼀ᨨ� � � ᨨᨨ矄� �ᆠ�ᨨ 矄 㜠ᨨ�� 犨 � ᨨ ��矄 㝐� 黨꾀҈黨 ᔼᔼ㡰桌 ҈+
e E t m v l
ᆠ㡰
� Dl = l
ᆠ㡰 Dl = l 㡰 Dl = l 꾀�ᆠ�矄鼀ᨨ� ���鼀矄矄 矄矄 ᆠᨨ �ᨨᨨ 㜠ᨨ��犨 � ᨨ �ᆠᨨ ᨨ ᆠᨨ �矄҈䰘맔҈ 鼀矄矄ᨨ �ᆠ�矄 ᆠᨨ矄ᨨ [㝐� Sep.02, 2020 (II)] 㡰 맔҈䰘├� 㡰 黨҈맔҈҈├� ᆠ㡰 ҈맔҈├� 㡰 黨䰘맔҈├�
꾀�ᆠ鼀ᨨ� �犨� ├� �ᆠ �泐쟼 矄 ᨨ�� 犨 � lᔼ �泐D쟼�矄 ᨨ�矄 �泐 犨 � l ᆠᨨ鼀 盐ᨨ D쟼�矄 㝐� 黨 ᔼ㡰桌 㡰 쟼 㡰 쟼 ᆠ㡰 쟼 㡰 쟼 꾀 ᆠᨨ ᨨ 鼀矄矄鼀 鼀� �盐 ᨨ ᆠ �e� ��泐 矄�矄ᆠᆠ 泐ᆠᨨ 矄 ᆠ�ᆠ � 쟼 ᨨ ᆠ �ᨨ 㜠ᨨ��犨 � ᨨ �矄ᆠᨨ �鼀t �� ᨨ� ���矄�ᆠ ᆠ矄�矄 㝐� 黨 ᔼᔼ㡰桌 h e 쟼 㡰 㡰 e쟼 t h -h h ᆠ㡰 㡰 e 쟼 e 쟼 r 꾀 ᆠᨨ 鼀矄矄鼀㡰犨� � ��ᨨᆠ�泐 v = v�ˆ + v ˆ �矄 r � ᆠ�ᆠ � E = - E kˆ ᔼ l �矄� �� 㜠ᨨ�� 犨 � ᨨ ᆠᨨ �矄 㜠ᨨ��犨 � �鼀 t�矄�� 泐 㝐� ᔼᔼ㡰桌
g xg y g xg y 㡰 g + g 㡰 g - g ᆠ㡰 g x - g y x y x y
㡰 g x + g y
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Dual Nature of Radiation and Matter
黨
꾀�ᆠ꾀ᨨ 鼀矄矄�鼀 ᆠ ��⼄�矄ᆠᆠ 泐 ᨨ � � ᆠᨨ �꾀 ᨨ �ᆠᨨ 鼀矄矄�鼀 ᆠ ��⼄ �矄 ᆠᆠ 泐 ᨨ � � ᆠ ᨨ
l � �ᨨᨨ 㜠ᨨ��犨 � A �矄 lB 㝐� ҈ ҈黨 ᔼ㡰桌 ҈
㡰 ҈ 㡰 䰘 ᆠ㡰 ҈҈ 㡰 䰘 ᔼ ᨨ��犨 � ᨨ ᆠᨨ �矄⼄盐ᨨ҈ �鼀矄 犨 � ᨨ ᨨᨨ ᨨ ⼄盐 ᆠ泐맔҈҈ 矄 ᨨ ᆠᨨ �矄⼄盐ᨨ 和 ᨨ �� 㦨맔҈鼀⏌矄㡰 ᆠ├矄ᆠᨨ 矄 㦨맔҈矄 㝐矄矄ᨨ ᆠᨨ 㦨黨҈맔҈҈├�㡰㝐� ҈҈ ҈黨 ᔼ㡰桌 㡰 ҈҈맔҈鼀⏌矄 㡰
҈҈
ᆠ㡰 ҈맔҈鼀⏌矄 㡰 ҈�맔҈鼀⏌矄
犨ᨨᆠᨨ 矄 鼀ᨨ� � 犨� ᨨ 㜠��矄�ᆠ 矄 矄 �ᆠ盐ᨨᆠ ᨨ ᔼ ᆠᨨ矄ᨨ � � ᨨ�� 犨 � 矄l҈ l � ᨨ��犨 � �
鼀ᨨ ᆠᆠ ᨨ �ᆠ ᨨ 鼀矄矄�矄 㝐� 졠 � 꾀�҈�҈桌 㡰 lᔴ㝐㦨l҈㦨l
㡰
҈ ҈ ҈ = + l҈ l҈ l
l҈l
æ l + l ö lCM = ç ҈ ÷ è ø 꾀�ᆠ꾀ᨨ 鼀矄矄鼀 � ��ᨨᆠ�泐ᆠᨨ� 矄犨� m �ᆠᨨ 鼀矄矄 犨 �ᆠ �矄矄 ᆠᨨ�矄�ᨨ �矄 ᨨ 矄�ᆠ �ᨨ ᨨ 㜠ᨨ�� ᆠ㡰
҈
҈䰘맔҈鼀⏌矄
lCM =
l҈ + l
㡰
犨 � 矄 l꾀 ᨨ lB ᆠᨨ�矄�ᨨ �矄㝐� ҈䰘桌 㡰
lA = lB
㡰
lA ҈ = lB
lA lA ҈ = 㡰 l = lB B 꾀鼀ᨨ ᆠᨨ 矄� �� x㜠 �ᆠ�ᨨ 矄 ᨨ 矄�ᨨ 犨� 矄 ��盐㡰ᔼ 矄矄� � 矄�
ᆠᨨ ᆠ⼄盐�矄鼀ᨨ鼀 盐鼀泐� 泐㜠 �ᆠ�ᨨ
ᨨ鼀ᨨ�泐ᨨ ᆠᨨ 矄矄矄� � ᨨ盐� 矄� �矄 ᆠ├矄ᆠᨨ 矄 㡰 㝐� 졠 � 꾀�҈҈�桌
ᆠ㡰
҈
Ӹ
㡰 泐 > 㡰 泐 ᆠ㡰 泐 ; 㡰 泐 >> ҈ ᨨ犨 �ᆠ ᨨ ᨨᨨ犨� ��ᆠ矄犨��鼀ᨨ矄 � �ᆠ盐 ᨨ �鼀 泐� 泐 㜠ᨨ���ᨨ 矄 �䁸 㝐� 졠 � 꾀�黨҈桌 㡰 ᆠᨨ
㡰 ᨨᨨ
ᆠ㡰 a㜠�ᆠ 㡰 盐矄�ᆠ ҈� 쟼ᆠᨨ 矄ᆠᆠ ᨨ盐� ᨨ � � ᆠ ᨨᨨ 矄ᆠᆠ ᨨ盐� ᨨ � � ᆠ 㜠ᨨ��犨 � 矄le lp ᨨᆠᨨ 矄
l ᨨᨨ 矄矄ᆠ�泐 �ᨨᨨ e �矄�� 泐 lp �� me�矄鼀矄矄ᨨ ᆠᨨ mp�矄鼀矄矄ᨨ ᨨᨨ 㡰 㝐� 졠 � 꾀�҈桌 㡰
mp le = lp me
㡰
me le = lp mp
ᆠ㡰
le ҈ me = l p mp
㡰
mp le = lp me
҈ 꾀ᨨᨨ 矄├� �ᆠ �泐E㦨҈├犨 �ᆠ �矄⼄盐ᨨ ᨨ ᨨᨨ 犨 � ᨨ ᨨᨨ �矄l ᨨ ᨨᨨ �矄l҈ �ᨨ ᨨ l⏌l҈�矄ᨨᨨ�ᨨ ᨨ 和桌 ᆠ㡰 E҈ 㡰 E ҈⏌ 㡰 E 㡰 E ҈⏌ ҈䰘 꾀�ᆠᨨ 鼀矄矄M矄 ᆠ泐矄� ᨨ犨ᨨ�ᆠ矄ᨨ 鼀矄矄矄m҈ m � � ᨨ 㜠 ᨨᨨᆠ��矄 �ᨨ ᨨ ᨨ��犨 � 矄ᨨ �ᆠ矄l҈⏌l�矄 ҈黨黨黨和㜠㝐├矄桌 㡰 m҈⏌m 㡰 m⏌m҈ ᆠ㡰 ҈
㡰
m ⏌ m҈
2 Integer Value Answer ҈ 꾀 a㜠�ᆠ ᨨᨨ ᆠᆠ ᨨ鼀矄泐 ᨨ � � ᆠᨨ ҈꾀 �矄 � ᨨ�� lp 犨 � 矄la l矄ᆠ�泐 �ᨨ ᨨ la 矄� ��矄 ҈桌 10 Subjective Problems ҈黨 ᨨ � �泐ᨨ �ᆠᨨ 鼀矄矄m�矄�� 泐 ì E 錐 £ x £ ҈ü V x㡰 = í ý x >҈ þ î 錐 l҈ l 㜠ᨨ��犨 � 矄ᨨ �ᆠ 犨 £ x £ ҈ x>҈矄ᆠ�泐ᔼ ᨨ �泐 ᨨ �ᆠ�矄E � l҈⏌l �㜠㝐├矄桌
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EBD_8341 P-278
Physics
Topic-2 : Photon, Photoelectric Effect X-rays and Davisson-Germer Experiment 1 ҈
MCQs with One Correct Answer
� ��盐矄 ᨨ犨矄 犨 ᨨ� 矄� ᨨᨨᆠ�ᆠ ᆠ �鼀 ᨨᆠ� 鼀 鼀� �鼀盐鼀 �泐 ᨨᆠ�ᨨ ᨨ ᆠᨨ ᨨ鼀�矄矄盐 ᆠ�矄 ᆠ├矄ᆠᨨ 矄 h㦨맔҈矄㡰 [Main Sep.04, 2020 (I)]
矄ᨨ 㡰
Ӹ
ᔴ
㡰
䰘
��㡰 꾀 �
f ҈ 㡰
㡰 䰘 㡰 �黨 ᆠ㡰 ҈黨 㡰 ҈ ᔼ ᨨᨨᆠ�ᆠ ᆠ �鼀 � ᨨ 矄ᨨ� � ᨨ �V矄盐矄ᆠ�ᨨᆠᨨ 犨 � ᨨ� �矄 矄 ᨨ犨 � ��盐꾀矄 � 矄�泐ᨨ � ᆠ� ��ᨨ
�矄� ᆠ矄 Main 和 ᔼᔼ㡰桌 V
q
҈⏌l
㡰 和�� � 矄 � 矄ᨨ�� 㡰 和ᨨᨨ 矄�� � �鼀ᨨ矄 ᆠ㡰 和�� � 矄 � 矄ᨨ 㡰 ᨨ矄 ᨨᆠ � ␔ 犨 � ᨨ ��ᨨ � �ᨨ 鼀�矄 ᆠ � ᨨ鼀� 鼀ᨨ 鼀 鼀 �鼀盐鼀├� �ᆠ �泐 ᨨ ᨨᨨᆠᨨ 矄ᆠᨨ鼀矄 �鼀矄� 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀�矄ᆠᨨ矄ᨨ [Main Sep. 03, 2020 (I)] 㡰 ҈ 㡰 ҈ ᆠ㡰 � 㡰 ҈
犨ᨨ矄ᨨ盐ᆠ矄ᨨ �� 鼀�㜠泐矄ᨨ 犨 � ҈ 鼀 �矄��� ᨨ 犨 � � 鼀矄ᆠ�泐ᨨ 矄ᨨ盐ᆠ矄鼀��� ᨨ 矄鼀ᨨ犨␔ �ᨨᨨ 盐鼀 矄�泐ᨨ ᨨᨨ 矄ᨨ 㜠泐矄ᨨ 盐鼀 矄�泐 ᨨ ᨨᨨ 矄 ᨨ �矄� �� ᨨ ��
犨 � 矄�矄 Main 和 ᔼᔼ㡰桌 㡰
�
҈
㡰 ҈҈ 㡰 ᆠ㡰 ҈ 㡰 ҈ ␔ ᨨᨨ ᨨ �泐矄�├矄 矄盐 ᆠᨨ 鼀 A ᆠ ᨨᨨᆠᨨ 矄 鼀 �鼀盐鼀 ├� �ᆠ �泐TA 㜠ᨨ��犨 � lA 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᨨᨨᆠᨨ 矄� ᨨ鼀 ᨨ 鼀B泐 ᨨᨨ ᨨ �泐��矄 T B 㦨 T A ҈�㡰 ᔼ 㜠ᨨ��犨 � ᨨ 矄 ᨨᨨᆠᨨ 矄lB㦨lA 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀 �矄 Main ᔼ桌 㡰 㡰 ᆠ㡰 ҈� 㡰
矄ᨨ� �ᨨ � � ᨨ㡰矄 盐 ᆠ�ᨨ ᨨ ⼄盐 ᆠ泐 v㡰 ᨨ矄ᨨ �盐鼀鼀��矄矄 ᨨ犨 � ��盐 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 矄ᨨ �盐鼀 ᨨ鼀 ᨨ � ��盐犨� � ᆠ├矄ᆠᨨ 矄 h㡰㦨맔҈矄ᆠᨨ
ᆠ �e㦨҈맔҈҈黨ᔴ㡰 Main ҈꾀҈黨ᔼ桌
҈ 㡰 � �
ᆠ㡰
҈ �
㡰 �
��ᨨ 犨� 犨 � �҈处 矄ᨨ 鼀矄盐 ᆠ ᨨᨨ 盐ᆠ ᨨᨨᆠᨨ 矄 ᆠᨨ 矄鼀 ᨨ 盐 � ᨨ鼀鼀� �ᆠ � ᨨ 맔҈ ᔼ �盐矄ᨨ �矄ᆠ�ᆠ盐 ᨨᨨ犨 泐 ᆠᨨ 矄�矄҈鼀鼀 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀�矄ᆠᨨ矄ᨨ Main 黨 ᔼ㡰桌
黨
㡰 ҈ 㡰 ҈ ᆠ㡰 ҈黨� 㡰 ҈ 꾀鼀␔矄ᨨ矄犨 � ᨨ � 鼀
盐鼀ᨨ ᨨᨨ 矄 犨�鼀� 矄ᆠᨨ �矄 � ᆠ├矄ᆠᨨ 矄 㦨맔҈矄矄 ᨨ �� c㦨맔҈鼀⏌矄桌 Main ҈꾀҈黨 ᔼᔼ㡰桌 ҈� ҈ 㡰 �맔҈ 㡰 ҈�맔҈ ᆠ㡰 맔҈҈ 㡰 ҈맔҈҈
ᆠ�ᆠ � ᨨ �� 犨�矄�� 矄 r æ p x ö N - p ´ ´ ҈҈ t ÷ x$ E = ҈ ᆠᨨ矄 ç -䰘 è � ´҈ ø C
�矄�� 矄ᨨ 鼀ᨨ 犨ᨨ├ 盐 ᆠ�ᨨ 矄ᨨ� �ᨨ �ᨨ ᨨᨨ㜠ᆠᨨ 矄�矄
҈䰘� � 쟼 � 㡰㦨 ( 㝐� 黨꾀�҈黨 ᔼ㡰桌 l � 处 ) 㡰 㡰 䰘 ᆠ㡰 㡰 ҈ ␔ ᆠ� ᨨᨨ矄 矄�矄�矄盐 ᆠ�矄�盐鼀� 犨� 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ ⼄盐 ᆠ泐v 矄ᨨ� �ᨨ � ᨨ ᨨᨨ ᆠ盐 �矄 ⏌ ␔ 矄盐 ᆠ �矄 �盐鼀� 泐鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ ⼄盐 ᆠ泐v⏌ 矄ᨨᨨ� �ᨨ ��矄 矄 ᨨ ⼄盐 ᆠ泐 ᨨ ᨨᨨᆠ�ᆠ鼀�矄矄�ᨨ �矄 Main ҈ ҈黨ᔼᔼ桌 �v v 㡰 㡰
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P-279
Dual Nature of Radiation and Matter
v ᔼ ᨨᨨᆠ�ᆠ �鼀 犨 � ᨨ �� � ᆠ� ᨨ 鼀�矄ᆠ � ᨨ鼀 鼀ᨨ 鼀 ᆠ矄�
矄ᨨ� �ᨨ ��矄ᆠᨨ矄ᨨ Main ҈҈ ҈黨ᔼᔼ桌 æ ᆠ ö çè = ҈ 鼀㜠÷ø 㡰 � 㡰 ҈� ᆠ㡰 ҈ 㡰 꾀 鼀 ᨨ ҈ 맔 ҈ 鼀 �矄 �盐鼀� 泐 ��ᨨ ᨨ � 矄�泐҈鼀␔⏌鼀 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀�矄� �泐ᨨ � ᆠ� ᨨᨨ 矄�矄҈ ᨨ 泐҈奠ᨨ �ᨨ 盐ᆠ矄 ᨨᨨᆠᨨ 矄 盐鼀 ᨨ 鼀� ᨨᨨᆠᨨ 矄矄ᆠᨨ �鼀 �鼀盐鼀 �泐矄ᆠ�泐犨� ҈㦨҈맔҈҈黨桌 㝐� ҈ ҈黨 ᔼᔼ㡰桌 㡰 ҈҈ ҈ 㡰 ҈҈ � ᆠ㡰 ҈҈҈ � 㡰 ҈҈ � 和盐 ᆠᨨ ᆠ� 鼀�矄 �矄�盐鼀� 犨� �� ᨨ 犨 � l҈㦨� 鼀 泐�� ᨨ 犨 � l㦨� 鼀ᔼ�矄 ᨨ盐 鼀 �鼀盐鼀矄 ᨨ ᨨᨨᆠᨨ 矄� 犨ᨨᆠ矄矄 � 泐 ᆠᨨᨨ 㡰
犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀 � 㡰�矄ᆠᨨ矄ᨨ ҈ 쟼 �泐ᨨ ᨨᨨ 㦨 l � 鼀 㡰 ( ) Main 黨 ҈黨ᔼ桌 㡰 ҈ 㡰 � ᆠ㡰 � 㡰 ҈
鼀� �ᆠ � 矄矄ᨨᆠ� 犨� �� 犨�矄��
ᨨ��� 泐 㦨矄� ҈맔҈䰘㡰ᆠ졔矄� 맔҈䰘㡰ᆠ桌 ᔼ �矄�� 矄ᨨ 矄� � �犨ᨨ├ 盐 ᆠ�ᨨ
ᨨ 䰘犨 犨� 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᨨᨨᆠᨨ 矄䁸 Main 黨 ҈黨 ᔼᔼ桌 ᆠ㦨맔҈鼀矄҈ 㦨맔҈㜠矄㡰 㡰 㡰 ҈� ᆠ㡰 � 㡰 䰘䰘 꾀 ᆠᨨ 鼀�矄ᆠᆠ 泐ᨨ � � ᆠᨨ �鼀�ᆠ�ᨨᨨ 盐ᆠ㜠泐矄ᔼᨨ 盐ᆠ矄ᆠᨨ � 盐ᨨ盐矄 矄犨矄ᆠ ᆠ�矄�ᆠ㜠泐矄ᔼ l鼀� �矄 矄鼀矄ᨨ矄矄� 犨 � ᨨ 㜠泐� 矄ᆠ盐鼀 ��ᨨ ᨨ ᨨ�l鼀�
Main ҈䰘桌 犨� ᨨ��矄ᆠᨨᆠ泐矄 �
ᆠ㡰 v
҈҈
҈
҈
҈
҈�
㡰
ᆠ㡰 ҈
㡰
㡰
㡰
鼀 �鼀盐鼀ᨨᆠ�泐ᨨ ᨨᨨᆠᨨ 矄鼀� ᨨ鼀 矄盐 ᆠ�矄犨 �� ᨨ ⼄盐 ᆠ泐 矄ᨨ 鼀 矄盐 ᆠᔼ � ᆠ� ⼄盐 ᆠ泐�矄� ᆠ矄 ᨨ 鼀 �鼀盐鼀ᨨᆠ�泐ᨨ ᆠ ᨨᨨᆠᨨ 矄犨� Main 졠 � 꾀�҈䰘桌
㡰 㡰 矄矄 ᆠ㡰 鼀ᨨ 㡰 ⼄盐ᨨ ҈䰘 꾀 ᨨᨨᆠ�ᆠ 鼀� � � 犨ᨨ├ 盐 ᆠ�ᨨ f �矄 æ ᆠ ö �盐鼀� 犨� �� ᨨ 犨 � l ç l < f ÷ ø è 矄矄 ᨨᨨᆠᨨ 矄 ᨨ��犨 � l 꾀 ᆠ �� 犨 � ᨨ � ᆠ� �� 泐Dl 矄盐�
ᆠ �Dl � l �ᨨDl ⏌Dl�矄ᨨᨨ�ᨨ ᨨ 꾀 ҈䰘桌
㡰 l ⏌l ᆠ㡰
l ⏌ l
㡰
l ⏌ l
㡰
l ⏌ l
҈ ��ᨨ ᨨ 犨 � l�矄� ᆠ� ᨨ ᨨᨨᆠ 矄矄鼀� ᆠᨨ 矄矄 ᔼ 犨 � �矄 l ᆠ � ᨨ 矄 ᨨ 矄矄鼀� ᆠᨨ
犨� Main ҈桌 ҈
㡰
æ ö = ç ÷ èø
҈
㡰
æ ö = ç ÷ èø
҈
҈黨
҈
æ ö æ ö ᆠ㡰 > ç ÷ 㡰 < ç ÷ è ø èø 꾀 ᨨᨨᆠ�ᆠ 矄盐 ᆠ �矄 �盐鼀� 矄盐ᆠᆠ矄矄�泐 泐 l 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ �� ᨨ 犨 � 矄 l ᔼ 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ 鼀� ᨨᨨᆠᨨ 矄�
矄ᆠᨨ ᆠ矄�矄�鼀矄 � �矄ᆠ矄 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 矄盐 ᆠ�矄 Main 졠 � 꾀�҈҈桌
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ᆠ l
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ᆠ l
ᆠ㡰
ᆠ l
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ᆠ l
ᔼ �矄ᨨ�ᆠ �鼀 ᨨ 鼀� ᆠ├Ἐ矄ᆠᨨ 矄 鼀 矄盐 ᆠ 犨矄 � � 犨� �� ᨨ � 犨 � 矄 鼀� ᨨᨨᆠᨨ ��矄 犨 鼀矄盐 泐泐� �矄ᨨ� �ᨨ � ᨨ 犨 � l㡰 ᨨ � ᆠ� �� ᆠᨨ矄ᨨ � �矄ᨨ� �ᨨ � 㡰�� ᨨ犨 l m鼀㡰 ᨨ㡰 �
҈
� ᆠ㦨맔҈鼀矄҈ 㦨҈맔҈҈黨ᔴ ᆠ├Ἐ矄 ᆠᨨ 矄 � 盐 �矄ᨨ 矄㡰 ᨨ盐 ᨨ鼀矄盐ᆠ �鼀 �矄 꾀 ҈桌 㡰 맔҈ 㡰 맔҈ ᆠ㡰 맔҈ 㡰 맔҈ ҈ 㝐ᆠ �矄㜠ᔼ 盐 鼀 쟼 �鼀 㡰犨� �矄㜠ᔼᔼ �矄 ᆠᨨ ᆠ盐矄�ᨨ 㡰 矄ᆠ ᆠᨨᆠᨨ�ᨨ ᨨ鼀 ᆠ ᨨ�ᆠ矄 �� ᨨ犨 �矄 Main ҈�桌
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EBD_8341 P-280
Physics
�矄㜠ᔼᔼ �矄㜠ᔼ 꾀 ᆠ├㜠 �㡰�ᆠ 盐ᨨ �� 쟼 �鼀 ᨨᨨ㜠ᆠ�ᆠ ��㡰�矄ᆠ �泐矄ᨨ ᨨ鼀 �鼀 ᔴ�矄ᨨ 㜠鼀 ���㡰␔ 盐ᨨ ᆠᨨ
�鼀 �㡰和盐ᆠ盐ᨨ ᨨ鼀
�
㡰 꾀㡰㜠 ��㡰錐 㡰㜠 �㡰錐 ᔴ㡰㜠 ���㡰 㡰 꾀㡰㜠 �㡰錐 㡰㜠 ���㡰錐 ᔴ㡰㜠 ��㡰 ᆠ㡰 꾀㡰㜠 �㡰錐 㡰㜠 �㡰錐 ᔴ㡰㜠 ���㡰 㡰 꾀㡰㜠 ��㡰錐 㡰㜠 �㡰錐 ᔴ㡰㜠 ���㡰 꾀鼀ᨨ �� 矄犨ᨨ犨 � 矄ᨨ 黨䰘处 ҈处 犨� ᨨ� 矄�泐ᨨ 맔҈␔鼀⼄盐泐 �矄�盐 鼀ᨨ � 犨ᨨ犨 � 矄 鼀 矄 ᨨ鼀泐ᨨ
ᨨ ҈ᆠ鼀ᨨ ᆠ 鼀�ᆠ矄盐 ᆠᨨ 犨ᨨ├ 盐 ᆠ�ᨨ
꾀矄矄盐鼀 �矄 ᨨᨨ矄矄ᨨ �� 泐 ᆠ�ᨨ
ᆠ ᆠ ᨨᨨ ᆠ矄 ᨨ ᆠᨨ
盐鼀ᨨ ᨨᨨᆠᨨ 矄� � 矄�矄ᨨ �鼀泐 Main 졠 � 꾀�҈҈桌 㡰 맔҈҈҈ 㡰 黨맔҈҈҈ ᆠ㡰 ҈҈맔҈҈҈ 㡰 ҈�맔҈҈҈ 꾀鼀矄盐 ᆠ�矄�盐鼀� 泐�� ᨨ 犨ᨨ � 犨 � 矄 鼀 ҈ 鼀 鼀 �鼀盐鼀矄 矄ᨨ ᨨᨨᆠᨨ 矄ᆠᨨ矄ᨨ � � ᨨ 矄 犨 � 矄 u҈ u矄ᆠ�泐ᔼ �ᨨu҈u㦨҈ hc㦨҈
鼀 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀�矄 泐 꾀 ҈桌 㡰 䰘 㡰 ᆠ㡰 㡰 � ᔼ lᔴ盐�矄 犨 � ᨨ ⧤a㜠泐� ᨨ ᆠᨨ ᨨ鼀�ᆠ
盐鼀黨㡰 l㝐ᨨ�矄 犨 � ᨨ ⧤a㜠泐 � ᨨ 鼀ᨨ泐 盐鼀 ᨨ鼀�ᆠ 盐鼀㡰 �ᨨ lᔴ盐⏌l㝐ᨨ�矄ᆠᨨ矄ᨨ 꾀 ҈桌 㡰 ҈黨黨 㡰 ҈ ᆠ㡰 � 㡰
ᨨ ᨨ�ᨨ ᨨᨨᆠ�矄├ � 犨 � lᨨ �� � �ᨨ ᆠ ᨨ �矄� 盐泐ᆠ � ᆠ盐 ᔼᨨ ᨨᨨᆠ�矄矄 ᨨᨨ犨矄Main ҈桌
ᔼ
㝐꾀 㝐
҈⏌l
ᨨᨨ├� � � 矄泐ᨨ盐ᆠ 鼀ᨨ矄ᨨ�泐矄泐 㡰 ␔ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀�矄� ᨨ 鼀꾀 㡰 ᨨ�� ᨨ ᆠ� 犨 � � �ᨨ ᨨ 鼀 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᆠᨨ 矄鼀� ᨨ鼀꾀 犨�� ᨨ矄鼀� ᨨ鼀 ᆠ㡰 ␔ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀꾀�矄� ᨨ 鼀 㡰 和盐 矄 �矄 ᨨᆠᨨᆠ 䰘 꾀盐矄ᨨ �� ᨨ 盐�ᨨ ҈ 矄�矄矄ᨨ ᆠᨨ鼀泐 泐矄鼀ᨨᆠ� ��泐矄ᨨ犨ᨨ 盐矄�矄 鼀␔ 矄 ᨨ �� �矄 맔҈ 鼀矄҈ � 鼀ᨨ鼀 盐鼀ᨨ ᨨᆠ�矄 꾀 ҈桌 㡰 맔҈҈䰘├�鼀矄҈ ᆠ㡰 맔҈҈䰘├�鼀矄҈ 㡰 ҈맔҈҈䰘├�鼀矄҈ 㡰 黨맔҈҈䰘├�鼀矄҈ ᨨᨨᆠ�ᆠ ᆠ �鼀 矄 ᨨ鼀 盐矄� � � 鼀矄⼄ � �犨ᨨ├ 盐 ᆠ�ᨨ 矄 fp 㦨fq㦨� fr㦨矄ᆠ�泐꾀�� 鼀ᆠᨨ � � �犨 � 矄ᨨ �� 鼀� 鼀 �
鼀犨� ⼄盐� 矄��矄�盐鼀� 矄ᆠ ᨨ 矄
ᆠᨨᆠI㜠V� ᨨ �鼀 �矄 ├hc = ҈ 鼀桌 Main 黨桌 ᔼ
ᔼ ⼄ 㡰
㡰
졠
ᆠ㡰
㡰
졠 l l ᔼ �鼀 ᨨ ᨨᨨᆠ�ᆠ ᆠ矄盐 ᨨ矄 矄ᨨ� �ᨨ ��� 矄ᆠ�ᨨᆠᨨ 犨 � l ᨨ � ᆠ� �� ᨨ犨ᨨ � 鼀矄꾀 矄 Main 졠 � 꾀��҈桌 矄 ᨨ犨 � ��盐 졠
l
ᔼ
ᔼ
ᔼ
ᆠ㡰
⼄ 㡰
⼄
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l
ᔼ
졠
⼄
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黨 ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� �矄鼀 矄�矄␔ 졠� ᆠᨨ ᨨ 㜠泐矄� ᨨ鼀㜠泐盐䁸 桌 㡰 ␔ � ᨨ ᆠ ᆠ�矄�ᆠ㜠泐矄 ᆠ矄矄犨
ᨨ鼀�ᆠ 盐鼀ᨨ �� ᆠ矄矄 㡰 ᔴ盐㜠ᨨ 犨 � ᨨ ᆠᨨ � 盐ᨨ盐矄 㜠泐矄 矄ᨨ ᨨ鼀�ᆠ 盐鼀ᨨ � ᆠ㡰 ᔼ 矄�泐ᨨ ᆠ ᆠ�矄�ᆠ㜠泐矄 矄ᨨ ᆠ�ᆠᨨ犨�� ᨨ 㜠泐盐 㡰 ᔴ盐㜠ᨨ 犨 � ᨨ ᆠᨨ � 盐ᨨ盐矄㜠泐矄 矄 ᨨ �泐ᨨ ᆠᨨ 矄� 㜠泐盐 쟼ᆠᨨ 矄犨� 㜠ᨨ��犨 � l ᨨ � � 㜠泐盐 ᆠ盐㜠ᨨ 犨 � ᨨ 鼀� 㜠泐矄�矄 䰘桌
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P-281
Dual Nature of Radiation and Matter
㡰
l =
mcl h m c l
㡰
l =
h mc
㡰 l = l h ҈ Ka 犨 � 鼀� 泐 ᨨ鼀 ᨨ ᨨ鼀�ᆠ 盐鼀 Z㦨҈҈�矄l � ᨨ鼀�ᆠ 盐鼀 ᨨ ᨨ鼀 鼀�矄 Ka ��ᨨ 犨� 犨 � l. �和桌 㡰 Z㦨 㡰 Z㦨 ᆠ㡰 Z㦨҈҈ 㡰 Z㦨 ᔼ ᨨᨨᆠ�ᆠ �鼀 ᨨ ᨨ ��矄ᨨ �� 矄ᆠ盐 和桌 ᆠ㡰
l =
ᔼ ᔴ
꾀
㡰 A B犨� � � 矄��矄犨 �B C 犨� � ⼄盐 ᆠ�矄 㡰 B C犨� � � 矄��矄犨 �A C 犨� � ⼄盐 ᆠ�矄 ᆠ㡰 A B犨� � � 矄��矄犨 �A C 犨� ⼄盐 ⼄盐 ᆠ�矄 㡰 B C犨� ⼄盐� 矄��矄犨 �A B犨� 矄鼀 ⼄盐 ᆠ�矄
ᨨ � � ᆠ� ᨨ 㜠泐盐�矄�├ ᆠ盐 ᨨ盐� ��矄mA 盐鼀ᨨ ᆠᨨ 矄矄�├� � �矄ᆠᨨ �矄 和桌 㡰 ´ ҈҈ 㡰 � ´ ҈ ᆠ㡰 ҈´ ҈҈䰘 㡰 ´ ҈҈�
� 矄�泐ᨨ 㜠泐矄 ᨨ鼀ᔴᨨᨨ� �盐�矄ᨨ �� 矄犨 � l矄矄 ᨨ犨 � ��盐 鼀� �鼀盐鼀 犨 � ᨨ盐 �矄lC 犨 � ᨨ Ka� �矄lK꾀矄 ᆠᆠ� �ᨨ��矄� ᆠ矄 ҈和桌 㡰 lK㜠lC� ᆠ矄矄
ᔼ
㡰 lK㜠lC ᆠ矄矄 ᆠ㡰 lK� ᆠ矄矄 㡰 lK ᆠ矄矄 �
lᔴ
l⧤
l
쟼ᆠᨨ 矄犨� �泐├� ᆠ� ᨨ 盐 �矄 � ᨨ 㜠泐盐⧤㜠矄 ᆠᨨ 矄ᨨ 盐 �矄 䰘�├ �泐㜠泐矄鼀� 泐 盐ᆠᨨ � ᨨ 泐 和桌 㡰 ᆠᨨ � 盐ᨨ盐矄㜠泐矄ᆠ盐鼀 鼀矄矄 盐 �㡰犨� 鼀� �鼀盐鼀犨 � ᨨ ҈��处 㡰 ᆠᨨ � 盐ᨨ盐矄㜠泐矄ᆠ盐鼀 鼀矄矄 盐 �㡰犨� 犨 � 矄 ᆠ㡰 ᆠ ᆠ�矄�ᆠ㜠泐矄ᆠ盐鼀ᨨ 盐 �矄
㡰 ᆠᨨ � 盐ᨨ盐矄㜠泐矄ᆠ盐鼀 鼀矄矄 盐 �㡰犨� 鼀� �鼀盐鼀 犨 � ᨨ ҈��处 ᆠ ᆠ�矄�ᆠ㜠泐矄ᆠ盐鼀ᨨ 盐 �矄
犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 矄盐矄 ᆠ�矄 ᨨ �矄 犨 � ᨨ �� ᆠ ᆠ盐矄 ᨨᨨᆠᨨ 鼀�矄矄�ᨨ
ᨨ鼀 �矄矄盐矄 ᆠ�矄ᨨ �鼀泐 ҈黨黨和㜠㝐├矄桌 㡰 � 鼀 㡰 鼀 ᆠ㡰 ҈ 鼀 㡰 鼀 䰘 㜠泐矄ᨨ 盐ᆠ � 㜠泐盐ᨨ� ���
ᆠᆠ� �ᨨ� 犨 � ᨨ ᆠᨨ � 盐ᨨ盐矄 㜠泐矄 矄盐矄 ᨨ鼀 ҈黨黨和㜠㝐├矄桌 㡰 ᨨ ¥ 㡰 l 鼀� ᨨ ¥ 犨 l 鼀� > ᆠ㡰 ᨨ l 鼀 犨 l 鼀 < ¥ 㡰 l 鼀� ᨨ l 鼀 犨 < l 鼀� < l 鼀 < ¥ 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᨨᨨᆠᨨ 矄鼀� ᨨ鼀矄盐 ᆠ犨 ᨨᨨ 矄ᨨ �泐 ᨨ ��矄 矄ᨨ� �ᨨ �� ᨨ�矄 ҈黨黨䰘㜠҈㝐├桌 㡰 㡰 ᆠ㡰 㡰 ҈ 黨 꾀 �泐ᨨ �矄⼄盐� ᨨ鼀ᨨᨨ ᨨ ᆠᨨ 矄 ᨨ鼀 盐 �盐鼀ᨨ鼀 �泐� 㡰 ⼄盐� ᨨ鼀ᨨᨨ ᆠᨨ 矄 ᨨ鼀 盐 �盐鼀 ᨨ鼀�矄 ҈黨黨�和桌 㡰 㡰 黨 ᆠ㡰 �҈ 㡰 䰘黨 㜠泐鼀ᆠᨨ鼀� � ᨨ鼀 㜠泐盐犨� 㡰 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ҈黨�㜠㝐├矄桌 㡰 � �犨 � 矄矄鼀 ᆠ� 鼀 �鼀盐鼀 犨 � ᆠ㡰 � �犨 � 矄� ᆠ� 鼀� �鼀盐鼀 犨 � 㡰 � �犨 � 矄泐� �犨 鼀� �鼀盐鼀 鼀 �鼀盐鼀犨 � ҈ 矄 ᨨ矄犨 � ᨨ 㜠泐矄鼀� ᨨ鼀 㜠泐 盐 矄 ᨨ
҈黨㜠㝐├矄桌 㡰 ᆠ盐 � 盐 㡰 ᨨ�� ᨨ 盐 ᆠ㡰 盐ᨨ �矄� 盐 㡰 ᨨ鼀�ᆠ 盐鼀ᨨ �鼀� 2
Integer Value Answer
꾀 ᆠ泐 ᆠ� �鼀�ᨨ ᨨ 鼀矄矄㝐鼀ᨨ盐 ᨨ 矄� � ᆠᨨ 矄�盐矄 矄� �㜠鼀矄矄 矄泐矄鼀 ᨨ �盐 ⼄盐 ᆠ泐矄盐ᆠ 㦨҈m㜠犨� h 矄 ᆠ├矄ᆠᨨ 矄 N ᨨᨨ 矄ᨨ 犨 � l㦨p맔҈㜠鼀矄�├ 鼀�ᨨ 矄�鼀盐 ᨨ盐矄泐 ᨨ鼀� ᆠ� ᆠ矄盐ᆠ 鼀�ᨨ �矄 �矄ᆠ 泐҈m鼀ᔼ 盐ᨨ N�矄x맔҈҈
盐ᨨ x�矄힘힘힘힘힘힘 ᔴᨨ 矄� 矄� �矄鼀矄矄矄矄桌 꾀 ҈黨桌
犨ᨨ├ 盐 ᆠ�ᨨ 矄ᨨ 和� 和ᨨ �盐鼀 矄ᆠ�泐 �ᨨᨨ 矄ᨨᨨ 矄ᨨ� � ᨨ �矄盐矄 ⼄盐 ᆠ泐ᨨ ᨨ和�ᨨ ᨨ 和ᨨ �盐鼀 �矄 꾀 ҈㜠ᔼ桌
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EBD_8341 P-282
Physics
꾀矄�矄 ᨨ �盐矄҈ᆠ鼀 犨ᨨ├ 盐 ᆠ�ᨨ 䰘�矄 矄盐矄 ᨨ鼀 � 矄盐� � � 矄ᆠ ᔼ�矄 盐 ᆠᨨ � 盐ᨨ盐矄�盐鼀� �ᨨ ᨨ 鼀犨 � �� 꾀矄 ᨨᨨᆠᨨ 矄鼀� 矄 �矄ᆠ � ᆠ⼄盐�矄 ᨨ � 鼀 �鼀盐鼀 盐鼀 ᨨ ᨨᨨᆠᨨ 矄鼀� ᨨ鼀 矄 �矄꾀´ ҈ 犨 ҈꾀҈㡰 盐ᨨ � �矄 ҈҈桌 3
�
䰘
Numeric Answer
꾀鼀 ᨨ ᆠᨨ 矄 ᨨ �泐쟼 矄ᆠ矄 ᨨ鼀 � � �ᨨ鼀�ᆠ矄ᆠ� �ᨨ ҈处 �矄鼀 �鼀盐鼀� 矄�泐 ᨨᆠᆠ盐矄q㦨º 쟼 � 㡰�矄힘힘힘힘힘힘 ├ᆠᨨ 矄 h㦨맔҈矄҈㦨҈맔҈҈黨 ᆠᨨ 鼀矄矄m㦨黨҈맔҈҈├�㡰㝐� 和� ᔼ㡰桌
矄盐 ᆠ ᨨ 鼀 �矄 �盐鼀� 泐犨� ᨨᨨ 矄ᨨ ��矄E҈㦨 E㦨�矄ᆠ�泐
�ᨨᨨ 鼀 �鼀盐鼀矄 矄ᨨ ᨨᨨᆠᨨ 矄鼀� � 犨ᨨᆠ矄矄�矄 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀� 㡰 �矄 힘힘힘힘힘힘힘힘힘힘힘힘힘힘힘 㝐� 和� ᔼᔼ㡰桌 ␔ ��ᨨ ᨨ 犨 � l�矄盐矄 ᨨ�盐鼀� 鼀�ᆠ矄盐 ᆠ 矄ᨨ� �ᨨ ��矄V␔ 矄鼀矄盐 ᆠ�矄�盐鼀� 犨� ��ᨨ ᨨ 犨 �
V l 矄ᨨ� �ᨨ ��矄 ᔼ 矄 ᨨ 犨 � ᨨ 鼀�ᆠ 矄盐 ᆠ �矄 nl 盐 ᨨ n 犨� 힘힘힘힘힘힘힘힘힘 [Main Sep.02, 2020 (I)] 꾀鼀ᨨ ᆠᨨ鼀� �ᆠ ��ᨨ ᨨ � 矄�泐맔 ҈�␔⏌ᆠ鼀 �矄ᆠᨨ鼀�矄 ᨨ 犨 � l㦨҈ 鼀ᔼ 矄 ᨨ鼀泐ᨨ 鼀 犨ᨨ├ 盐 ᆠ�ᨨ j 㦨㡰ᨨ 矄盐 ᆠ ᨨ ҈ᆠ鼀ᔼ ᨨ � ҈ ᨨᨨ 矄ᆠ矄 ᆠᨨ ᨨ
盐鼀ᨨ ᆠᨨ 矄ᆠ � ҈矄�矄҈x hc㦨҈ 鼀 㦨҈맔҈҈黨㡰 x�矄힘힘힘힘힘힘 Main 䰘 ᔼ桌 黨 ᔼ ᨨᨨᆠ�ᆠ �鼀 鼀 ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� 犨� ᨨ犨ᨨ W�矄� ᆠ� ᨨ ᆠ泐矄ᨨ� �ᆠ ᨨ ᨨ 犨ᨨ├ 盐 ᆠ�ᨨ � ⼄盐 ᆠ泐ᨨ �� �矄盐矄ᨨ 矄 ᨨ ⼄盐 ᆠ泐 矄ᨨ ᨨᨨᆠᨨ 矄 鼀� 犨� ���� ├� �ᆠ �泐꾀矄矄盐鼀 ᨨᨨᆠᨨ 鼀�矄矄�ᨨ
�ᆠ� ᆠ泐�矄҈奠꾀ᨨ � � ᆠᨨ �V�矄� 犨 ᆠ ᨨ ᨨ 꾀 鼀� ᆠᨨ 矄 � ᆠ� ᨨ鼀泐ᨨ ᨨ 矄ᨨ ᨨ � ᆠ矄 ᨨᆠ 㦨 맔҈ 盐 ᨨ �鼀ᆠ ᨨ ᆠᨨ 矄 盐 ᨨ n �矄 힘힘힘힘힘힘힘힘힘힘㝐矄矄ᨨ ᆠᨨ m㦨黨맔҈҈kg ҈㦨҈맔҈҈黨J 꾀 ҈桌
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Fill in the Blanks
犨 � ᨨ ⧤a㜠泐矄ᨨ 盐ᆠ 泐 㜠泐盐 �矄䰘处 ᨨ鼀�ᆠ 盐鼀ᨨ ᨨ 鼀�ᨨ 盐�矄 ҈黨黨㜠㝐├矄桌 꾀ᨨ � � ᆠᨨ ├�矄� ᆠᨨ矄矄 㜠泐 盐 鼀� �鼀盐鼀 犨 � ᨨ 㜠泐矄 � �矄处 ҈黨黨㜠㝐├矄桌
� ᔼ 㜠泐盐ᆠᨨ 矄ᆠᆠ ᨨ盐� ᨨ � � ᆠᨨ ҈�ᨨ矄矄�├ᆠᨨ� 矄 ᨨ 鼀� 㜠泐� 矄� 盐�矄鼀⏌矄 ҈黨黨㜠҈㝐├桌 � 犨 � ᨨ ᆠ ᆠ�矄�ᆠ㜠泐 ⧤ a � 鼀� 泐 泐 ᨨ� �├ 鼀 �矄 处 犨 � ᨨ ⧤ b � 鼀� 泐 矄鼀鼀 犨� ҈黨黨㜠㝐├矄桌 � ␔ 盐鼀ᨨ ᆠᨨ 矄矄�├� � ᨨ ᨨ
㜠泐盐�矄� ᆠ矄 ᨨ 鼀� 㜠泐矄 � ᆠ矄矄犨 �犨 矄 矄ᨨ ᆠᨨ 矄矄�├� � ᨨ � ᆠ矄 ᆠ盐㜠ᨨ 犨 � ᨨ 鼀� 㜠泐矄 ҈黨㜠㝐├矄桌 �� 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᆠᨨ 矄鼀� � ᨨᨨᆠ�ᆠ ᆠ�矄� 泐 ᨨ ᨨ � ᆠ� ��ᨨ ҈黨㜠㝐├矄桌 � ᨨᨨ 盐ᆠᆠ ᆠ�矄�ᆠ㜠泐矄盐矄� � 盐 �矄 � � 㜠泐� ᨨ ᆠᆠ� �ᨨ�矄 ᨨ盐 � 힘힘힘힘힘힘힘힘힘힘힘ᨨ矄 �泐ᨨ ᆠ ᆠ�矄�ᆠ ��ᨨ �矄힘힘힘힘힘 � � � �泐ᨨ �
鼀ᨨ矄ᆠᨨ � 盐 �矄
҈黨㜠㝐├矄桌 �矄├㡰 5
True / False
�䰘 ᔼ ᨨᨨᆠ�ᆠ鼀�矄矄�ᨨ ᨨᆠ矄矄 鼀 �鼀盐鼀 �泐 ᨨ ᨨᨨ㜠ᆠᨨ 矄� ᆠ矄矄犨� � ᆠ矄� �� 矄�泐 ᨨ � ᆠ� �� ҈黨㜠㝐├矄桌 � ├� �ᆠ �泐 ᨨ ᨨᨨᆠᨨ 矄 鼀� 泐 ᨨᨨ矄 矄�� 矄盐 ᆠ 矄 ᨨ � 矄�泐ᨨ � ᆠ� ��ᨨ ҈黨҈㜠㝐├矄桌 6
MCQs with One or More than One Correct Answer
�黨 ᔼ 㜠泐盐ᆠᨨ 矄鼀� ᨨ鼀 �鼀 ᆠ ᨨ 㡰 ᆠ泐� �ᆠ盐 I �� ᨨ 㡰 �矄 ᆠd ᨨ鼀 ᆠ ᨨ ��矄├ᨨ �V ��
ᆠ ᨨ 矄盐� � � 鼀�矄矄�ᨨ ᨨ ᆠᨨ � 盐ᨨ盐矄 ᆠ ᆠ�矄�ᆠ㜠泐矄ᔼ �鼀 ᆠ盐 I�矄 ᆠ矄 I ᨨ ᨨ � � ᆠV�矄� ᆠ矄 ᨨV d 矄�ᨨ �矄 ᆠd�矄 盐ᆠ ᨨ
㡰 ᆠ盐㜠ᨨ 犨 � 犨� 盐ᆠᨨ 犨 � 矄ᨨ ᆠ ᆠ�矄�ᆠ㜠泐矄犨�鼀�
矄鼀 㡰 ᆠ盐㜠ᨨ 犨 � 矄犨矄 犨 � 矄ᨨ ᆠ ᆠ�矄�ᆠ㜠泐矄犨�鼀� 矄鼀 ᆠ㡰 ᆠ盐㜠ᨨ 犨 � 犨� 盐ᆠᨨ � 矄��矄ᨨ 㜠泐矄犨� ᆠ矄 㡰 ᆠ盐㜠ᨨ 犨 � 犨�ᆠᨨ鼀犨ᨨ�鼀矄� � 矄�泐ᨨ 㜠泐矄犨� ᆠ矄 �� ᨨ 犨 � l 矄ᨨ ᆠ ᨨ � 矄� ᆠ盐盐鼀盐矄矄 ᨨ犨 � ��盐 犨ᨨ├ 盐 ᆠ�ᨨ
ᨨ ᆠ ᨨ 矄盐 ᆠ�矄f ᨨ �矄犨�鼀矄 ᨨ ᆠᨨ 盐ᆠ� �鼀�├ �矄 ᆠ ᨨ鼀 ᆠ ᨨ 꾀 ᨨ � � ᆠ �矄 鼀� � 犨 ᆠᨨ 矄ᔼ 鼀� �鼀盐鼀 ᨨ��犨 � ᨨ ᆠᨨ 矄矄矄� � ᨨ盐� ᨨ �矄l犨 �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸 꾀 ҈桌
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P-283
Dual Nature of Radiation and Matter ��
쟼ᆠᨨ 矄
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㡰 l ᆠ矄矄犨� � ᆠ矄� f l 㡰 l�矄ᨨ �鼀泐 � �矄 ᨨ盐 ᆠ㡰 ᨨ � ᨨ � � ᆠ >> f⏌㡰 l �矄 ᨨ �鼀泐 � �矄鼀 ᨨ盐�鼀矄 㡰 l� ᆠ矄矄 矄鼀矄l ᨨl ᆠ⏌f ᨨ ᨨᨨ㜠ᆠ�ᆠ ᆠ犨� � ᆠ� ᨨᨨ 犨 � l 矄ᨨ� � ᨨ � �矄 V ᔼ � 泐 ᆠᨨᆠ ��ᨨ 矄㡰ᨨ V犨� l ҈⏌l 꾀 ҈�桌 V
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� 犨 矄ᨨ� �ᨨ � V㡰 ç ÷ èlø �矄矄 ᨨ犨 � ��盐 f҈ f f 犨ᨨ├ 盐 ᆠ�ᨨ 矄 犨 �ᆠ ᨨ ᨨᨨ犨� ��矄⏌ᆠᨨᆠ 㜠�㝐҈桌 㝐҈ f҈
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f҈ f f = ҈ f҈ f f = ҈ hc ᆠ㡰 q µ e 㡰 盐�ᨨ�� ᆠ 盐矄 ᨨ鼀� ᨨᨨᆠᨨ 矄 ᨨ鼀鼀 鼀ᨨ 泐 ␔ ᨨᨨ 矄ᨨ �泐�矄�├ 矄盐 ᆠᨨ 鼀 A ᆠ ᨨᨨᆠᨨ 矄 鼀 �鼀盐鼀├� �ᆠ �泐 TA ᨨ��犨 � lA 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᨨᨨᆠᨨ 矄� ᨨ鼀 ᨨ 鼀B泐 ᨨᨨ 矄ᨨ �泐䰘�矄 TB = TA - ҈�㡰 ᔼ ᨨ��犨 � ᨨ 矄 ᨨᨨᆠᨨ 矄�矄 l B = l A
҈黨黨㜠㝐├矄桌 㡰 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ A�矄� 㡰 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ B�矄 ᆠ㡰 TA㦨 㡰 TB㦨䰘� 㡰 㡰
㝐
␔ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠᨨ� 矄ᨨ盐ᆠᨨ �� �矄 �矄 ᆠ ᨨ 鼀 ᨨ鼀 ᨨᨨᆠ�ᆠᆠ ᆠ盐ᨨ ᨨ� 矄盐�ᨨ ᆠ盐 矄ᆠ�泐 ҈鼀꾀 ᔼ 矄鼀 矄ᨨ盐ᆠ �矄 ᆠ 鼀 犨泐 ᨨ鼀 ᨨᨨᆠ�ᆠᆠ
҈黨黨㜠㝐├矄桌 㡰 矄ᨨ� �ᨨ �犨�ᨨ 㡰 矄ᨨ� �ᨨ �犨�ᨨ ᆠ㡰 矄盐�ᨨ ᆠ盐 犨�鼀꾀 㡰 矄盐�ᨨ ᆠ盐 犨�鼀꾀 � ᨨ � � ᆠ� ᨨ 㜠泐盐�矄� ᆠ矄 꾀矄矄盐� 鼀� ��ᨨ
҈黨㜠㝐├矄桌 㡰 � 矄�泐� ᆠ矄矄 㡰 鼀� �鼀盐鼀犨 � � ᆠ矄矄 ᆠ㡰 � 矄�泐鼀� 盐 ᆠ � 㡰 鼀� �鼀盐鼀犨 � ᆠ矄矄 ᨨᨨᆠ�ᆠ ᆠ 矄盐ᨨ矄 ⼄盐 盐鼀 盐ᨨ �� ᆠ盐矄 ҈黨䰘㜠㝐├矄桌 㡰 �矄鼀� �鼀盐鼀 ⼄盐 ᆠ泐ᨨ �� ᨨ犨犨 �ᆠ ᨨ ᨨᨨᆠᨨ 矄鼀� 㡰 鼀 �鼀盐鼀 ├� �ᆠ �泐ᨨ ᨨᨨ ᆠᨨ 矄 矄ᨨ 泐ᨨ ⼄盐 ᆠ泐ᨨ �� ᨨᨨ
�矄� 矄�泐 ᆠ㡰 犨 鼀矄盐 ᆠ�矄 � 泐�盐鼀� ᨨᨨᆠᨨ 矄 矄盐 ᆠ�鼀鼀 �泐 㡰 ᆠ�ᆠᆠ �ᨨ ᨨᨨᆠᨨ 矄�矄⼄盐 � 䰘 矄 ᨨ 犨 � ᨨ ᨨᨨᆠ�ᆠ鼀�矄矄�ᨨ ᨨ鼀 鼀��矄�处 ᨨᨨᆠᨨ 矄犨�鼀� 犨
�矄鼀��矄�盐鼀� 犨� 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ��ᨨ
ᨨ鼀 ҈黨㜠㝐├矄桌 㡰 �犨� 鼀 㡰 ҈㜠犨� 㜠 鼀 ᆠ㡰 �犨盐�ᨨ鼀 㡰 ҈㜠犨盐�ᨨ鼀 9
Assertion and Reason Type Questions
和T꾀TE㝐ENT㜠҈ 䰘桌 ᔼ ᆠᆠ� �ᨨ �� 㜠泐盐�矄� ᆠ矄 犨 � 矄ᨨ ᆠ ᆠ�矄�ᆠ㜠泐矄 ᨨ ᨨᆠ � 和T꾀TE㝐ENT㜠 ␔ ᆠᨨ 鼀矄�├矄 �� 㜠泐盐 ᨨ ├� �ᆠ �泐�矄ᆠᨨ � ᨨ㜠泐 �泐 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ 㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 盐錐和鼀 㜠 �矄졠 ᆠᨨᆠ �ᨨ ᨨ和鼀 㜠҈ ᆠ㡰 和鼀 㜠҈�矄 盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄 盐 10 Subjective Problems 黨 ᔼ ᨨᨨᆠ�ᆠ �鼀 矄盐 ᨨᨨ 矄ᨨ �泐� 矄ᨨ ᆠ ᨨ � �犨ᨨ├ 盐 ᆠ�ᨨ 㡰 ᔼ 矄盐�ᨨ ᆠ盐 �矄�A㦨mA ᨨ� 矄�泐҈�␔⏌鼀 ᨨ� 犨 ᨨ ᨨ � ᆠ盐 㡰꾀矄ᨨ 犨� ᨨ� 矄�泐ᨨ � ᆠ� ��ᨨ
맔҈�␔⏌鼀 㜠㝐├矄桌 䰘 ⼄盐 ᆠ泐 ᨨ ᨨᨨ 鼀� 盐 ᨨ 矄��ᨨ ᨨ ᆠᨨ ᨨ ᆠ� 鼀 ᨨ鼀LᨨK矄 �矄 ᨨ盐 ᨨ 맔҈҈ 矄� �㝐ᨨ矄泐矄犨 � ᨨ鼀�ᆠ
盐鼀ᨨ 鼀 �� 泐 �矄ᆠᨨ 矄 R㦨҈҈맔҈䰘鼀҈ 㜠㝐├矄桌 䰘҈ 犨ᨨ鼀�ᆠ矄A Bᆠ ᨨ �맔҈鼀 ᆠ ᨨᆠ ᨨ 矄�ᨨ ᨨ ҈ᆠ鼀B ᆠ�矄ᨨ矄��ᆠ �ᨨ 䰘맔҈҈C꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ 鼀ᨨ �� 犨� ᨨᨨ 矄ᨨ �泐�ᆠ 矄矄
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EBD_8341 P-284
Physics
� �ᨨ At㦨矄ᨨ ҈҈ ᨨᨨ 矄 ᨨ � 矄⼄盐鼀矄ᆠᨨ 꾀矄矄盐鼀 ᨨ ᨨᨨᆠᨨ
�矄鼀� ᨨ 泐҈ � ᆠ� ᨨᨨ 矄꾀矄ᨨ 矄矄盐鼀 鼀� ᨨᨨᆠᨨ 矄ᆠᨨᆠ 泐B 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ A鼀� 矄ᆠᨨ 矄 盐鼀� 㜠�㝐├矄桌 㡰 盐鼀ᨨ ᨨᨨᆠᨨ 矄鼀� 盐ᨨt㦨҈矄 㡰 鼀� �盐 ᨨ ᆠ�ᆠ � 犨 矄 A Bt㦨҈矄 ᆠ㡰 ├� �ᆠ �泐ᨨ 鼀ᨨ矄 ��ᆠ ᨨᨨᆠᨨ
鼀� t㦨҈矄犨 �ᆠ 矄B �ᆠ �鼀├ 泐 ᨨᨨᆠᨨ ᨨᆠ B
├e㦨�맔҈҈ᔴ⏌㜠鼀 ᨨᨨᆠᨨ 矄鼀� 犨 鼀 ��ᨨ �矄� ᆠ� ᨨ 矄盐 ᆠᨨ 犨ᨨ├ 盐 ᆠ�ᨨ ҈黨 矄 ᨨᨨᆠᨨ 矄 矄矄 ᨨ盐� ��ᨨ ᆠᨨ � � �a㜠�ᆠ矄꾀鼀 �鼀盐鼀
䰘
�泐ᆠᨨ ᆠᨨ鼀� 矄犨� a�ᆠᨨ ᨨ鼀 졔�ᨨ 鼀�� �矄� � ᨨᨨ � �矄ᨨᆠ矄矄졔�ᨨ 矄 盐矄 ᨨ鼀 � � ᨨ盐 ᆠ� 矄 � ��矄� ᨨ ᨨᨨ 矄泐� �� ᨨ � �├泐 ᨨ 鼀� 盐� � ᆠᨨ鼀� �ᨨ ├ 㦨҈맔҈҈�矄桌 ҈黨黨黨㜠�㝐├矄桌 䰘 꾀矄矄盐鼀 ᨨ�� 犨 矄矄ᨨᆠ� 犨�
ᆠᨨ ᆠ ᨨ鼀矄 � �犨犨 ᨨ鼀矄 � � ᨨ �鼀 矄�ᨨ 泐犨� ᨨ 矄ᆠ ᨨ ᨨ鼀�ᆠ矄�矄ᔼ�矄 ᨨ盐 ᨨ 矄盐ᆠ 矄 � �犨 �矄 ᨨ鼀 � �矄 ᆠ 犨 ᨨ鼀矄ᨨ 泐 �矄处꾀矄�鼀�矄 � �犨�矄�� ᨨ鼀 � �矄 � ᆠ矄 ᨨ�处盐 ᨨ ᨨ 泐� 鼀 �盐ᨨ � �泐ᨨ ᆠᨨ 矄� ᆠᨨ ᨨ矄 矄盐ᨨ ᨨ犨 �ᆠ 矄 � �犨ᨨ 泐 ҈黨黨䰘㜠�㝐├矄桌 矄ᆠ� ᨨᆠ ᨨ鼀
Topic-3 : Miscellaneous (Mixed Concepts) Problems 2 Integer Value Answer 꾀ᨨᨨ �矄 � ᨨ鼀泐 犨泐ᨨ犨 矄 盐ᆠ盐矄 犨� ᆠ �Q㦨҈e犨 e�矄 ᆠᨨ �ᆠᆠ � ᔼ鼀├矄ᆠᨨ矄矄ᨨᆠ ᨨ ҈ 鼀ᨨ 盐ᆠ盐矄 ᨨ��犨 � � 盐 �矄ᨨ 鼀㡰ᨨ ᨨᨨ �矄 矄�矄 ├ ᨨᨨ 鼀矄矄mp㦨 �⏌㡰´ ҈䰘├�錐h⏌e㦨 ҈ ´ ҈҈�矄⏌ᔴ錐 㦨黨맔҈黨鼀⏌ 錐҈ 鼀㦨҈҈�鼀҈㜠ᔼ桌 pe
҈
6
MCQs with One or More than One Correct Answer
�鼀 ᨨ �� 盐 矄矄盐 ᆠ鼀鼀 �鼀 ᆠ ᆠᨨ 矄� 矄ᆠ├ᨨ 泐鼀盐 �⧤鼀�� � ��ᨨ �├ᨨ� 矄ᨨ盐ᆠ犨 �犨 ᨨ鼀 꾀 �� �� 盐�矄ᨨ矄 ᨨ鼀 �矄 ᆠ
ᨨ ҈鼀꾀矄矄盐鼀 盐�ᨨ 泐矄ᨨ ᨨ矄ᨨ ᆠ�ᆠ盐犨� �盐矄鼀鼀
├和 㜠ᨨ 鼀
ᆠᨨ 矄 㦨�䰘맔҈␔鼀⧤ ␔� 矄 �矄ᆠ鼀 ᆠᨨ 矄 㦨黨맔҈鼀㜠⧤ ᆠ├矄 ᆠᨨ 矄 㦨맔҈矄矄 ᨨ �� � ᆠ盐盐鼀 㦨맔҈鼀矄҈㡰 㡰 ᨨ犨 � 泐 �鼀 �矄� �␔ ᨨ�␔ 㡰 � ᨨ犨 � �� ᨨᨨ 泐ᨨ ᨨ矄 �矄� �҈�맔҈␔ᨨ�맔҈␔ ᆠ㡰 犨 � ᆠᨨ矄ᨨ � � ᨨ 鼀 �鼀盐鼀 � 矄�泐ᨨ �� �矄҈҈ 鼀 㡰 ├� � �犨 � ᨨ 鼀� ��ᨨ ᨨ ҈䰘 鼀 ᨨ 盐鼀ᨨ ᨨᨨ 矄 � � 矄ᆠᨨ � ᨨᨨ 泐ᨨ ᨨ矄 �矄� � 䰘�맔҈҈҈ᨨ�맔҈҈҈
Answer Key Topic-1 : Matter Waves, Cathode and Positive Rays ҈ ҈҈
ᆠ㡰 ᆠ㡰
㡰 ҈ 㡰
㡰 ҈ 㡰
㡰 ҈ 㡰
� ᆠ㡰 ҈� 㡰
㡰 ҈ 㡰
䰘 ᆠ㡰 ҈䰘 ᆠ㡰
㡰 ҈ 㡰
黨 ᆠ㡰
҈ 㡰
Topic-2 : Photon, Photoelectric Effect X-rays and Davisson-Germer Experiment ҈ ҈҈ ҈ ҈ ҈ �҈ � ҈
㡰 ᆠ㡰 ҈ 㡰 㡰 㡰 处㡰 � 맔҈㡰 㡰 � 䰘㡰
㡰 㡰 㡰 � 㡰 㡰 䰘 㡰 ᆠ㡰 ҈ 㡰 ҈ 㡰 ҈� ᆠ㡰 ҈ ᆠ㡰 ҈䰘 㡰 ҈ 㡰 㡰 㡰 � 㡰 㡰 䰘 㡰 㡰 㡰 㡰 � 㡰 ᆠ㡰 䰘 㡰 ҈㡰 ҈㡰 䰘㡰 � �㡰 㡰 䰘 黨㡰 맔҈鼀⏌矄㡰 � 䰘处㡰 � ᆠ矄矄㡰 �� �䰘 矄 � 矄 �黨 ᆠ㡰 ᆠ㡰 ҈ ᆠ㡰 ᆠ 㡰 ᆠ㡰 䰘 ᆠ 㡰 㡰 Topic-3 : Miscellaneous (Mixed Concepts) Problems ᆠ 㡰
㡰 黨 ᆠ㡰҈ ᨨ 盐矄㡰 ᆠ㡰 ҈黨 㡰 㡰 㡰 黨 㡰 㡰 㡰 黨 㡰 ᆠ㡰 ҈҈㡰 黨 㡰 � ҈㡰 ⼄盐 ᆠ泐㡰 ᆠ㡰 ᆠ㡰
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P-285
Atoms
Atoms
26
Topic-1 : Atomic Structure and Rutherford's Nuclear Model 1 ҈
MCQs with One Correct Answer
� 犨 �ᆠ �ᆠ矄 矄盐矄ᨨ 盐 ᨨ �ᨨ ᨨ� �鼀 犨� 㝐� ᔼ桌 a㜠�ᆠ矄�矄 q和ᆠ� � � Y盐鼀ᨨ 矄ᆠ a㜠�ᆠ矄 ᆠ ᨨ矄矄ᆠ 鼀�ᆠ ᨨᨨ矄ᆠ㡰
㡰
ᆠ㡰
㡰
㡰
꾀 �ᆠᨨ �泐�㝐�矄矄ᆠ ᨨ盐� ҈ 泐 � 盐 �盐鼀 盐ᆠ盐矄 �矄 ᆠᨨ ᆠᨨ矄矄 ᨨᆠ �矄ᨨ ᨨ ᨨ 㝐� ҈黨҈㜠㝐├矄桌 㡰 ҈处 㡰 ҈҈ᆠ鼀 ᆠ㡰 ҈҈ᆠ鼀 㡰 ҈҈�ᆠ鼀
Topic-2 : Bohr's Model and the Spectra of the Hydrogen Atom 1 ҈
MCQs with One Correct Answer
ᔼ 泐 ᨨ� ᨨ鼀 ᆠᨨ 鼀├矄 矄��ᨨ ᨨ鼀 n졔҈㡰 ᨨ n ᔼ n>>҈ ⼄盐 ᆠ泐ᨨ ��ᨨ 鼀� �矄ᨨᨨ�ᨨ ᨨ [㝐� Sep.02, 2020 (II)] ҈ ҈ ҈ ҈ 㡰 ᆠ㡰 㡰 n n n n
�泐⼄盐� ᨨ�ᨨ �矄 泐 ᨨ� �├�ᨨ � �矄 �ᨨ盐 矄�矄黨 泐 �矄␔ �矄 犨 � ᨨ ��ᨨ 鼀� 犨 ᆠᨨ � �矄�ᨨ 盐鼀矄 ᨨ鼀 矄ᆠᨨ ᆠ� 矄ᨨ �ᨨ盐 矄䁸 㝐� 黨 ᔼᔼ桌 㡰 鼀 㡰 ҈҈ 鼀 ᆠ㡰 � 鼀 㡰 鼀
�鼀�ᨨ ᨨ ᨨ盐�ᨨ ᨨ ᆠᨨ � �矄�ᨨ盐 矄ᨨ�� 泐 ᨨ� ᨨ鼀�矄҈´ ҈҈ 矄
㡰
�
⼄盐 ᆠ泐ᨨ ᨨ盐�ᨨ ᨨ ᆠᨨ � �矄 �矄 ᆠ� 矄 � s҈㡰�矄 㝐� 䰘 ᔼ桌 㡰 ҈´ ҈҈ 㡰 䰘´҈҈ ᆠ㡰 ´ ҈҈� 㡰 �´ ҈҈
ᆠᨨ � 泐 ᨨ� ᨨ鼀 �矄盐鼀矄 ᨨ鼀 � ᆠ� 矄ᨨ 矄ᆠᨨ ᆠ� 矄 矄盐矄⼄盐 泐 ᨨ �矄 ᆠ� 矄 �ᨨ ᨨ 矄ᆠ� 犨 � 矄l҈⏌ lᨨ ᨨᨨ 矄鼀� � �矄ᨨᆠ矄矄 �矄 㝐� ҈꾀�҈黨ᔼᔼ桌 㡰 ⏌䰘 㡰 䰘⏌� ᆠ㡰 䰘⏌� 㡰 黨⏌䰘 ᔼ �졔졔ᆠᨨ � �矄ᨨ ᨨ��矄 ᆠ� ᨨ泐 ��ᨨ ᨨ 犨 � l␔ �ᨨ �矄 ᆠ� ᨨ �ᨨ盐 矄� ᨨ矄矄�犨泐矄 � ᆠ盐 � �� 鼀 � 鼀�矄矄�ᨨ 矄㡰ᨨᨨ 矄� 矄ᆠ� 矄ᨨ矄 ␔ �矄 盐ᨨ l䁸 㝐� ҈꾀�҈黨ᔼᔼ桌 � h㦨맔҈矄錐c㦨맔҈鼀矄҈㡰 㡰 ҈҈ 鼀 㡰 黨 鼀 ᆠ㡰 ҈ 鼀 㡰 ҈ 鼀
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EBD_8341 P-286
6.
7.
8.
9.
10.
Physics
Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be; [Main 9 April 2019 I] (a) 889.2 nm (b) 488.9 nm (c) 642.7 nm (d) 388.9 nm Radiation coming from transitions n = 2 to n = 1 of hydrogen atoms fall on He+ ions in n = 1 and n = 2 states. The possible transition of helium ions as they absorb energy from the radiation is : [Main 8 April 2019 I] (a) n = 2 ® n = 3 (b) n = 1 ® n = 4 (c) n = 2 ® n = 5 (d) n = 2 ® n = 4 A hydrogen atom, initially in the ground state is excited by absorbing a photon of wavelength 980Å. The radius of the atom in the excited state, in terms of Bohr radius a0, will be: [Main 11 Jan 2019 I] (a) 25a0 (b) 9a0 (c) 16a0 (d) 4a0 In a hydrogen like atom, when an electron jumps from the M-shell to the L-shell, the wavelength of emitted radiation is l. If an electron jumps from N-shell to the Lshell, the wavelength of emitted radiation will be: [Main 11 Jan 2019 II] (a)
27 l 20
(b)
16 l 25
(c)
25 l 16
(d)
20 l 27
An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let ln, lg be the de Broglie wavelength of the electron in the nth state and the ground state respectively. Let Ln be the wavelength of the emitted photon in the transition from the nth state to the ground state. For large n, (A, B are constants)[Main 2018] B (a) L n » A+ 2 (b) Ln » A + Bln ln (c) L 2n » A + B l 2n
11.
12.
(d) L 2n » l
If the series limit frequency of the Lyman series is v1, then the series limit frequency of the P-fund series is : [Main 2018] (a) 25 nL (b) 16 nL (c) nL/16 (d) nL/25 The energy required to remove the electron from a singly ionized Helium atom is 2.2 times the energy required to remove an electron from Helium atom. The total energy required to ionize the Helium atom completely is: [Main Online April 15, 2018] (a) 20 eV
13.
(b) 79 eV
(c) 109 eV
(d) 34 eV
Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths r = l1/l2, is given by [Main 2017]
(a) r =
3 4
(b) r =
1 3
(c) r =
4 3
(d) r =
2 3
14. According to Bohr’s theory, the time averaged magnetic field at the centre (i.e. nucleus) of a hydrogen atom due to the motion of electrons in the n th orbit is proportional to : (n = principal quantum number) [Main Online April 8, 2017] (a) n –4 (b) n –5 (c) n –3 (d) n –2 15. A hydrogen atom makes a transition from n = 2 to n = 1 and emits a photon. This photon strikes a doubly ionized lithium atom (z = 3) in excited state and completely removes the orbiting electron. The least quantum number for the excited state of the ion for the process is : [Main Online April 9, 2016] (a) 2 (b) 4 (c) 5 (d) 3 16. As an electron makes a transition from an excited state to the ground state of a hydrogen - like atom/ion : [Main 2015] (a) kinetic energy decreases, potential energy increases but total energy remains same (b) kinetic energy and total energy decrease but potential energy increases (c) its kinetic energy increases but potential energy and total energy decrease (d) kinetic energy, potential energy and total energy decrease 17. If one were to apply Bohr model to a particle of mass ‘m’ and charge ‘q’ moving in a plane under the influence of a magnetic field ‘B’, the energy of the charged particle in the nth level will be : [Main Online April 10, 2015]
æ hqB ö (a) n çè ÷ 2pm ø
æ hqB ö (b) n çè ÷ 8pm ø
hqB ö (c) n æç è 4pm ÷ø
æ hqB ö (d) n ç è pm ÷ø
18. The radiation corresponding to 3 ® 2 transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of 3 × 10–4 T. If the radius of the largest circular path followed by these electrons is 10.0 mm, the work function of the metal is close to: [Main 2014] (a) 1.8 eV (b) 1.1 eV (c) 0.8 eV (d) 1.6 eV
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P-287
Atoms
҈黨
泐 ᨨ� ҈҈㡰盐�盐鼀 ҈㡰矄� �泐�ᨨ �矄 �盐鼀
(
)
+
ᨨ盐泐�ᨨ �矄 � �盐鼀 ( �
)
++
ᨨ ᆠᨨ ᨨ盐 盐ᆠ盐矄 ᔴᨨ 矄� ᆠᨨ
矄��ᨨ ᨨ鼀 㦨ᨨ 㦨҈ᔼ 犨 � 矄ᨨ 鼀� ��ᨨ l҈ l l l 矄ᆠ�泐
ᨨ �鼀泐犨 �ᆠ ᨨ ᨨ ᨨᨨ犨� ��矄ᆠᨨᆠ䁸 㝐� ҈桌 㡰 l҈ = l = l = l 㡰 l҈ = l = l = l ᆠ㡰 l҈ = l = l = 黨l
҈
㡰 l҈ = l = l = l
� � � �泐ᨨ ᆠᨨ � 泐 ᨨ� ᨨ鼀�矄 ҈ �泐⼄盐� ᨨ鼀ᨨ ᆠᨨ ᨨ鼀 �矄 ᆠ� 矄ᨨ �졔졔�矄 㝐� 졠 � 꾀�黨҈桌 㡰 ҈ 㡰 ᆠ㡰 ҈ 㡰 ᔼ 泐 ᨨ� �├ᨨ鼀ᆠᨨ 鼀├ 矄��ᨨ ᨨ鼀
�泐 犨� ⼄盐 盐鼀 盐鼀 ᨨ ᨨ 犨� ⼄盐 盐鼀 盐鼀 ҈㡰ᔼ >>҈ ⼄盐 ᆠ泐ᨨ ��ᨨ
鼀� �矄ᨨᨨ�ᨨ ᨨ 㝐� ҈桌 ҈ ҈ 㡰 㡰
҈ ҈ ᆠ㡰 㡰
ᔼ ᨨ 矄鼀ᨨ ᨨ 泐 ᨨ� 㜠�├ ᨨ鼀 ᨨᆠ 犨 盐ᆠ盐矄 ᆠᨨ �矄 鼀ᨨ � � 矄 e æ ҈ b ö F= ç + ÷ 犨 b�矄ᆠᨨ 矄 ᨨ �矄ᨨ鼀 pe è r r ø �盐矄ᨨ n ᨨ� � 鼀矄 ᨨ ᨨ �盐矄 æ e h ç a = ç mp e è
ö ÷ �矄 ÷ ø
㝐� 졠 � 꾀�҈桌
㡰 rn㦨anb
㡰 rn㦨an졔b
ᆠ㡰 rn㦨anb
㡰 rn㦨an졔b
犨 � ᨨ �矄矄ᆠ� � 鼀矄�矄 ᨨ 泐 ᨨ� ᨨ鼀�矄�҈꾀 犨 � ᨨ 矄ᆠᨨ 矄ᆠ� � 鼀矄�矄ᨨ 矄� �泐㜠�ᨨ � �盐鼀 ᨨ鼀�矄 ҈҈桌 㡰҈҈�꾀 㡰 ҈꾀 ᆠ㡰꾀 㡰 䰘꾀 �矄犨 � � 盐�ᨨ��ᨨ ᨨ 泐 ᨨ� 矄ᆠ盐鼀�矄҈ 鼀 矄鼀矄犨 � �
� ��ᨨ ᨨ 泐 ᨨ� 矄ᆠ盐鼀 ᨨ 矄 � �㡰�矄 䰘桌 㡰 鼀 㡰 鼀 ᆠ㡰 ҈ 鼀 㡰 ҈ 鼀 � 꾀 ᨨᨨ ᆠᨨ� 矄 犨� 矄�ᨨ 泐 泐 ᨨ� ᨨ鼀 �
�ᨨ盐 矄� 矄�ᆠ泐쟼 �泐ᨨ ᆠᨨ� � � ᨨᨨ
�矄҈꾀 �鼀� ᨨ ᨨ ᨨ 鼀�ᆠᨨ矄ᆠᨨ
ᨨ ᨨᨨ ᆠᨨ� 矄 犨� 矄鼀 泐 ᨨ� ᨨ鼀 � 矄�ᆠ泐犨� �泐ᨨ ҈�␔ 犨�ᨨ矄 泐 ᆠᨨ䁸 �和桌 㡰 졠 ᨨᨨ ᨨ �泐҈ ᆠᨨ ᨨ �泐҈ 㡰 ᨨᨨ ᨨ �泐ᨨ ҈ ᆠ㡰 ᨨᨨ ᨨ �泐҈ 㡰 졠 ᨨᨨ ᨨ �泐҈ ᨨ ᨨᨨ ᨨ ҈ ᔼ ᨨ鼀҈Fm�䰘 ᨨᨨ犨矄 ᨨ 鼀ᨨ �盐矄 ᨨ ҈Fm�䰘�矄n�鼀矄 ᨨ �盐矄 � n 和桌 㡰 ҈ 㡰 ᆠ㡰 㡰 ҈⏌ 䰘 ᆠ�ᆠᨨ �犨 ᨨᨨ ᆠᨨ �矄 r 犨 �矄ᆠᨨ 矄 꾀矄矄盐鼀� � r ᨨ 矄鼀ᨨ ᨨ�ᆠ犨���ᨨ ᨨ rn犨� n n� � � ᆠ�⼄盐 盐鼀 盐鼀䁸 和桌 㡰 rn µ n 㡰 rn µ ҈⏌n 㡰 rn µ ҈⏌n ᆠ㡰 rn µ n 졔졔 꾀泐 ᨨ� ᨨ鼀 � �ᨨ ᨨ � 矄ᆠᨨ
�� 泐V㦨Vln
ᆠ� 矄ᔼ l l � �矄ᆠ�ᆠᨨ �ᆠ �盐鼀ᨨ鼀 E E� �矄ᆠ� ��矄
和桌 㡰
l > l � E>E�
㡰
l 㦨 l � EE�
ᆠ㡰
l 㦨 l � E>E�
㡰 l l � EE� 黨 矄��ᨨ ᨨ鼀 矄n 㦨ᨨn 㦨� 泐 ᨨ� 㜠 �├ᨨ鼀矄盐矄� 盐�ᨨ ��ᨨ ᔼ ��ᨨ
҈和桌 犨�ᨨ� � 矄��ᨨ
㡰 ®҈ 㡰 ® ᆠ㡰 ® 㡰 �® ᆠᨨ � 泐 ᨨ� ᨨ鼀鼀├矄 矄��ᨨ ᨨ鼀 ᆠ� 矄ᨨ �ᨨ盐 矄␔ �ᆠ ᨨ ᨨᨨ犨� � 矄鼀 矄�矄盐䁸 和桌 㡰 ᔼ矄├� �ᆠ �泐� ᆠ矄矄 �矄ᨨ � ᨨ ��矄 ᆠ矄矄 㡰 ᔼ矄 ├� �ᆠ �泐 ᆠ矄矄 ᨨ � �泐 � ᆠ矄矄 �矄ᨨ �泐鼀� 矄 矄鼀 ᆠ㡰 ᔼ矄├� �ᆠ ᨨ ��矄 ᆠ矄 �矄ᨨ � �泐� ᆠ矄矄 㡰 ᔼ矄├� �ᆠᨨ � ᨨ ��矄 ᆠ矄 ҈ ᔼ鼀�� ᨨ鼀鼀 盐ᨨ ᨨᨨ 泐ᨨ �ᆠ �ᆠᨨ ᨨ盐 鼀矄矄ᨨ ᆠᨨ 盐 � � 矄鼀ᆠ �矄 ᆠᨨ 꾀泐 ᨨ ᨨ鼀鼀ᨨ ᆠᨨ 矄� ᨨ矄矄� 矄��ᨨ 矄ᨨ �矄 泐ᨨ �ᆠ �ᆠᨨ �矄 ᆠ� ᨨ �矄犨 � ᨨᨨ 犨�鼀� 矄犨 � l �� �
鼀矄ᨨ 泐 �ᆠᨨ 矄 R ᨨ 泐 ᨨ� ᨨ鼀㡰 ⼄盐ᨨ 和桌 㡰 黨⏌ �R㡰 㡰 ⏌ �R㡰 ᆠ㡰 ҈⏌ �R㡰 㡰 ⏌R
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EBD_8341 P-288
Physics
ᔼ 泐 ᨨ� 矄ᆠ盐鼀 犨 � ᨨ a� �矄�
鼀犨 矄� 矄ᆠ盐鼀ᨨ �矄 � 泐a� 犨 � �矄䰘 鼀쟼矄�鼀 矄 ᨨ � 泐犨� 矄ᆠᨨ �矄 ҈黨黨黨和㜠㝐├矄桌 㡰 맔҈鼀⏌矄 㡰 맔҈䰘鼀⏌矄 ᆠ㡰 맔҈鼀⏌矄 㡰 맔҈�鼀⏌矄 꾀矄ᨨ 鼀ᨨ 鼀� �鼀盐鼀 �泐 � 㡰⼄盐� ᨨ 鼀ᨨ ᆠᨨ ᨨ鼀 �ᨨ盐 矄 ᨨ ᨨ盐泐 �ᨨ � �ᨨ鼀 Z 㦨㡰�矄 ҈黨黨䰘㜠҈㝐├桌 㡰 ҈�҈ 㡰 ҈ ᆠ㡰 㡰 ҈ ᔴᨨ 矄� 矄ᆠ� 矄盐� � ᨨ鼀 矄��ᨨ
n = ® n = ҈ � ᨨ鼀矄 �ᨨ 矄�� ᨨ犨 矄 ᨨ矄犨 � �矄ᨨ 盐ᆠ 泐 ҈黨㜠҈㝐├桌 㡰 泐 ᨨ� ᨨ鼀 㡰 盐�盐鼀ᨨ鼀 ᆠ㡰 和� �泐�ᨨ � �盐鼀 㡰 ᨨ盐泐�ᨨ �矄 � �盐鼀 2
�
Integer Value Answer
ᔴᨨ 矄� 泐 ᨨ� 㜠�├�ᨨ � ᨨ鼀犨� ᨨ鼀�ᆠ 盐鼀 犨� 矄� �ᆠᨨ ᔼ 鼀�矄矄�ᨨ 矄ᆠ盐鼀ᨨ �矄 ᨨ鼀 ᨨᨨ 鼀� � 㦨ᨨ 㦨҈ 矄��ᨨ
矄 �泐䰘 �� ᨨᨨ 鼀� �
㦨ᨨ 㦨 矄��ᨨ �ᨨ � �ᨨ �泐ᨨ 泐 ᨨ� ᨨ鼀�矄҈ 盐ᨨ Z�矄힘힘힘힘힘힘힘힘힘힘 꾀 ҈桌 꾀 ᆠᨨ � 泐 ᨨ� ᨨ鼀盐 �ᨨ矄 矄��ᨨ
ᨨ鼀 ᨨ�犨� ⼄盐 盐鼀 盐鼀 �ᨨ ᨨ 犨� ⼄盐 盐鼀 盐鼀 � 矄ᆠ�泐 � ��
� 㦨� 矄鼀矄ᨨ矄矄� �矄 꾀 ҈䰘桌 䰘 꾀 泐 ᨨ� ᨨ鼀� �矄�ᨨ盐 矄�矄� � 泐�� ᨨ 犨 � 黨䰘处 ├� � ᆠ⏌㦨҈䰘맔҈鼀 �ᨨ盐 矄 �泐ᨨ 泐 ᨨ� ᨨ鼀矄҈
盐鼀ᨨ � 矄矄 � 鼀�矄矄�ᨨ 矄ᆠ盐鼀�矄 꾀 ҈桌 꾀 ᆠᨨ �矄 ᆠ� 矄 ᨨ �졔 �ᨨ 矄 �盐 鼀ᨨ鼀 盐鼀 ⏌p ᨨ��犨 � ᨨ ᆠᨨ
� �矄矄�矄p 犨 �矄 ᨨ �盐矄㡰 盐ᨨ p�矄 꾀 ҈�桌 黨 ᔴᨨ 矄� 泐 ᨨ� ᨨ鼀犨� �矄ᆠᨨ � n ᨨ� 꾀 ᆠᨨ鼀� �ᆠ ��ᨨ ᨨ 犨 � 黨 鼀�矄盐矄 ᨨ �ᨨ � ᨨ鼀 ᔼ ├� �ᆠ �泐ᨨ ᆠ ᆠᨨ �矄҈ 盐ᨨ n�矄 hc㦨҈nm㡰 꾀 ҈�桌
� ᨨ � ��矄ᨨ ᆠᨨ ᔼ
3
Numeric Answer
꾀�ᆠᨨ 鼀矄矄㝐⏌ᆠᆠᨨ� 矄犨� 泐 ᨨ�
ᨨ鼀矄和ᨨᨨ ᆠᨨ�矄�ᨨ �ᆠᆠᨨ鼀矄ᨨ 矄 ᨨ鼀ᆠᨨ�矄 �ᨨ矄ᨨ�矄 �矄 ᆠ� 矄
� ��├� �ᆠ �泐ᨨ �ᆠ � 㡰�矄 盐ᨨ �矄 㝐� 和� ᔼ㡰桌
� 鼀矄矄ᨨ 泐 ᨨ� ᨨ鼀ᨨ҈⏌ᆠ㡰 ҈ ᔼ � 矄ᆠᨨ 泐 ᨨ� ᨨ鼀 � ᆠ犨
�矄 矄 ᨨ矄 犨 � 矄 ᨨ 泐鼀
矄�矄 �矄 处 ᆠᨨ矄ᨨ � � � ᆠ ᨨ 矄ᆠ 矄�矄� 处�矄힘힘힘힘힘힘힘힘힘힘 [㝐� Sep.04, 2020 (I)] �矄鼀鼀ᨨ 鼀矄�矄ᨨ 泐 ᨨ� ᨨ鼀 矄 犨 � ᨨ �҈ 处 犨 � ᨨ 矄ᆠᨨ 鼀鼀ᨨ 鼀矄�矄 � 鼀㡰�矄힘힘힘힘힘힘 㝐� ᔼᔼ桌 4
Fill in the Blanks
ᔼ ᨨ 鼀ᨨ ᨨ 泐 ᨨ� ᨨ鼀 �ᨨᨨ ├� �ᆠ �泐ᨨ ᨨ �泐ᨨ ᆠᨨ � ⼄盐 盐鼀矄n�矄 ҈黨黨㜠҈㝐├桌 ᨨ �盐矄ᨨ � ᆠᨨ ᨨ ᨨ矄 ᨨᨨ盐矄ᨨ鼀 ᨨ鼀�ᆠ 盐鼀㦨҈�㡰ᆠ� �矄 ᨨ � 矄��ᆠᨨ � �ᆠ�ᆠᆠᨨ 矄 㦨҈㡰�矄处҈黨黨҈㜠҈㝐├桌 6
MCQs with One or More than One Correct Answer
� 꾀�ᆠᨨ 鼀矄矄m鼀ᨨ矄� ᆠ�ᆠ盐ᨨ�矄犨� ᨨ � �泐V r㡰㦨Fr犨 F�矄ᨨ矄��ᆠᨨ 矄 r�矄�矄 �矄 ᆠ ᨨ鼀 ᨨ��� ᔼ矄 ��矄ᆠᆠ盐 盐矄� � ᨨ 鼀ᨨ ᔼ �盐矄ᨨ �ᆠ矄 ᨨ� �矄 ᨨ 泐R �矄矄 �泐 ᨨ 泐v E矄ᆠ�泐 ᨨ n ᨨ� h�矄 ᆠ├矄ᆠᨨ 矄 㡰 꾀 桌 㡰 R µ n҈⏌ v µ n ⏌ 㡰 R µ n ⏌ v µ n҈⏌ æ n h F ö ᆠ㡰 E = ç ÷ è p m ø
҈⏌
҈⏌
æ n h F ö E = 㡰 ç ÷ è p m ø 꾀 泐 ᨨ� ᨨ鼀 矄ᨨ� � ᨨᨨ ᨨ 犨 � l�矄 ᆠ� ᨨ鼀 矄n㦨҈ᨨ 矄 n 㦨ᔼ鼀鼀 �泐 ᆠᨨ 盐鼀矄ᨨn 㦨m 矄 泐 鼀�� � ᨨᨨ ᨨ 犨 � l ᆠ �� 鼀ᨨ鼀 盐鼀ᨨ ᨨ鼀 盐ᨨ 矄ᨨ�ᨨ ҈ 鼀�矄矄�ᨨ Dpa Dpe矄ᆠ�泐ᔼ l a ⏌ le = � 犨 �ᆠ ᨨ ᨨ�ᨨ 矄㡰�矄⏌ᆠᨨᆠ䁸 矄hc㦨҈ 鼀錐҈ 鼀㦨҈黨鼀h c ᆠ├矄 ᆠᨨ 矄 矄 ᨨ �� 矄ᆠ�泐桌꾀 ҈黨桌 ҈ 㡰 Dpa ⏌ Dpe = 㡰 �ᨨᨨ ├� �ᆠ �泐ᨨ ᆠᨨ � 矄 ҈ n㦨m ᨨ 矄n㦨҈�矄 ᆠ㡰 m㦨 㡰 l㦨҈ 鼀
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P-289
Atoms
�� 泐 ᆠ� 矄矄 ᨨ 泐 ᨨ� �├ᨨ鼀矄 矄ᨨᆠ 泐 � 矄矄㡰 犨� 盐ᆠ ᆠ � � 泐 �� ᆠ�⼄盐 盐鼀 盐鼀 犨 >>҈␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰盐䁸꾀 ҈桌 㡰 � ᆠ � � �� ᨨ 犨ᨨ ᆠᨨ 矄ᆠ盐� ᨨ�矄 ᨨ矄 ᨨ ᨨ 㡰 � ᆠ � � �� ᨨ 犨ᨨ ᆠᨨ 矄ᆠ盐� ᨨ�矄�矄矄҈⏌
ᆠ㡰 �ᆠ � � �泐ᨨ 犨ᨨᆠᨨ 矄ᆠ盐� ᨨ�矄�矄矄҈⏌ 㡰 � ᆠ �� �盐 鼀ᨨ鼀 ᨨ 犨ᨨ ᆠᨨ 矄ᆠ盐�ᨨ�矄�矄矄҈⏌
�盐矄ᨨ ᨨ�ᨨ ᆠᨨ � 泐 ᨨ� 㜠�├ ᨨ鼀�矄�犨 �矄 ᨨ �盐矄ᔼ矄ᨨ� �盐 䰘
ᔼ�矄�� �矄 ᆠ├ᆠᨨ 矄 p �矄 泐 �ᆠᨨ 矄 ᨨ矄矄�犨 � 矄㡰犨
ᨨ鼀 㜠 ᆠ�矄�矄 㡰 꾀 ҈桌
鼀ᨨ鼀 盐鼀�矄
㡰
黨
㡰
ᔴ㡰 㝐ᨨ矄泐Ἐ矄 犨 㡰 泐 ᨨ� 矄ᆠ盐鼀 㡰 ᔴ �ᨨ ᨨᨨ �泐 矄㡰 b㜠 ᆠ泐 � ᨨ├� �ᆠ �泐ᨨ ᆠᨨ 矄 8
矄矄�㜠҈
├泐 盐ᨨ ᨨ 矄 ᨨ泐ᨨ 矄ᆠ盐鼀ᨨ 泐 ᨨ� ᨨ鼀 �矄 ⼄盐 � �ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 犨
ᆠᨨ �矄ᨨ� �ᨨ盐 ᨨᨨ ␔犨� �矄ᨨ � ᨨ�ᨨ 鼀ᨨ�ᨨ ᨨ � ⼄盐 � ᨨ�ᨨ �泐ᨨ �ᨨ鼀�ᆠ鼀ᨨᆠ盐矄矄盐鼀� ��ᨨ��� 盐ᨨ � �矄ᨨ 矄⼄盐 � �ᨨ ᆠᨨ ��ᨨ ҈桌 � 꾀 �ᨨ鼀�ᆠ鼀ᨨᆠ盐 矄鼀ᨨ鼀 ᨨ � �I泐ᨨ 矄 ⼄盐 � �ᨨ ᆠᨨ ��ᨨ �矄ᨨ�ᨨ �泐� 㦨�矄 ᨨᨨ犨 㡰�矄 㡰
҈ æ h ö ç ÷ n è p I ø
ᆠ㡰
æ h ö nç ÷ è p I ø
黨 ҈
黨 㡰 �
ᆠᨨ � 泐 ᨨ� ᨨ鼀 鼀├矄 矄��ᨨ
ᆠ㡰 黨
n҈ ® n 犨 n҈ n � ᆠ� ⼄盐 盐鼀
盐鼀矄ᨨ 犨ᨨ矄矄꾀矄矄盐鼀 ᨨ 鼀ᨨ ᨨ � �鼀�ᨨ ᨨ ᆠᨨ � � ��矄�矄 �� �鼀矄 � � 矄 ᨨ矄矄�盐矄ᨨ n҈ n ҈黨黨和㜠㝐├矄桌 㡰 n҈ = n = 㡰 n҈ = n = �
ᆠ㡰 n҈ = n = ҈ 㡰 n҈ = n = ᔼ ᨨ 矄鼀ᨨ ᨨ 泐 ᨨ� ᨨ鼀҈黨㜠㝐├矄桌 㡰 �盐矄ᨨ ᨨ��矄ᨨᨨ�ᨨ ᨨn 㡰 ᨨ �泐 ᨨ ᆠᨨ � ᨨ� �矄 � 矄泐ᨨᨨ�ᨨ ᨨn ᆠ㡰 �盐鼀ᨨ鼀 盐鼀ᨨ ᆠᨨ � ᨨ��矄 h � �鼀盐�ᨨ p 㡰 鼀� �盐 ᨨ ᨨ � �泐ᨨ ᆠᨨ �
泐ᨨ��矄� �矄⧤쟼 7
�҈
Match the Following
和ᨨ鼀犨矄⏌ᨨᆠ矄矄矄�� � ᔴᨨ盐鼀 ᔼ 㝐ᆠ 矄犨� 泐矄�ᆠ ᨨ鼀 �� � ᔴᨨ盐鼀 ᔼᔼ � �ᆠ 泐ᨨ盐 矄犨 泐 ├ � � ᨨ� 盐矄� 맔鼀� �� � 졠 和 䰘桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 矄��ᨨ 犨 犨ᨨ 㡰 ᔴ ᆠ�矄�ᆠ㜠泐矄 ᨨ鼀�ᆠ �泐矄 㡰 쟼ᆠᨨ 鼀�矄矄�ᨨ ᨨ鼀 ⼄㡰 ᨨᨨᆠ�ᆠ ᆠ 鼀�
Comprehension/Passage Based Questions
㡰
҈ æ h ö n çè p I ÷ø
㡰
æ h ö n ç ÷ è p I ø
� ᔼ�矄 ᨨ盐 ᆠ��ᨨ ⼄盐 ᆠ泐 ᨨ鼀�ᨨ盐 ᨨ �矄 ᆠ� 矄ᨨ ᨨ�ᨨ ᨨ ᔴ졠鼀ᨨᆠ盐�矄 ´ ҈҈҈ 鼀ᨨ鼀 ᨨ � � ᨨ ᔴ졠 p 鼀ᨨᆠ盐ᨨ盐�矄ᆠ ᨨ 鼀矄矄�矄ᆠᨨ矄ᨨ
ᆠᨨ矄ᨨ
├ = p ´ ҈ 矄㡰 㡰
䰘 ´ ҈ ├� 鼀
㡰 ҈䰘 ´ ҈ ├� 鼀
ᆠ㡰 䰘 ´ ҈䰘 ├� 鼀 㡰 ҈҈䰘 ´ ҈ 䰘 ├� 鼀 � ᔼ ᔴ졠鼀ᨨᆠ盐 �矄 ᆠ犨 ᔴ 鼀矄矄㦨҈鼀盐㡰 � 졠 鼀矄矄㦨҈鼀盐㡰犨 ҈鼀盐 = ´ ҈ 䰘 ├� �矄 ᆠᨨ矄ᨨ
㡰
´ ҈ ҈ 鼀
㡰 ҈黨 ´ ҈ ҈ 鼀
ᆠ㡰
҈ ´ ҈ ҈ 鼀
㡰
´ ҈ ҈҈ 鼀
矄矄�㜠 ␔ �ᆠ�矄矄�ᆠ ᨨ鼀ᨨᨨ �x 㜠 �矄犨
x 㦨 x 㦨a, 犨 a �矄ᨨ ᨨ鼀 �鼀 矄�ᨨ �矄 �泐 ᆠ ├ᨨ 泐ᆠ� 矄ᆠ� �ᆠ盐矄 ᨨ犨 ��矄ᨨ �ᆠ鼀ᨨ� �� 矄盐ᆠ 矄�ᆠ ��ᨨ ᆠᨨ矄ᨨ ᨨ ᨨ鼀�ᨨ ᨨ 矄 � �犨矄犨� ᨨ 矄�矄 矄x = x = a . 犨 � ᨨ �矄矄 � �犨�矄 ᨨ � 鼀ᨨ鼀 盐鼀p ᨨ �ᆠᆠᆠᨨ � �ᨨ ᨨ�� �ᨨ �泐ᨨ �ᆠᨨ 鼀矄矄m �矄 ᨨ�矄
p . 盐矄 �泐ᨨ �ᆠ m ᆠ ᨨ 泐⼄盐 盐鼀 盐鼀�n ├� �盐矄҈
� 鼀ᨨ鼀 盐鼀矄 E =
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EBD_8341 P-290
㦨҈ᆠ �ᨨ盐 矄㡰ᆠᨨ矄ᨨ � �ᨨ 盐鼀ᨨ ᨨᨨ矄� 矄 � �犨 矄 鼀ᨨ 矄ᆠ� ᨨᨨ 矄犨 ᨨᨨ犨� � ⼄盐矄�ᨨ 矄 ᨨ�ᆠ鼀ᨨ� �� � x = ᨨx = a. ├ h 㦨맔҈J s e㦨҈맔҈҈黨C �� ᨨ犨 �泐 ᨨ �ᆠ ᨨ�ᆠ盐盐ᨨ n �矄ᨨᨨ�ᨨ ᨨ 黨桌 ᆠ㡰 a҈ 㡰 a 㡰 a 㡰 a⏌ � ᔼ 鼀矄矄ᨨ �ᆠ�矄m = ҈맔҈├� a㦨
鼀 �泐ᨨ �ᆠ� �矄�ᨨ盐 矄�矄ᆠᨨ矄矄 ᨨ 黨桌 㡰 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀 �䰘 矄 ᨨ �ᆠ ᆠ ├ �矄ᆠ盐矄�矄 ᨨᨨ�ᨨ ᨨ 黨桌 㡰 n⏌ 㡰 n҈ ᆠ㡰 n҈⏌ 㡰 n 矄矄�㜠 ᔼ 鼀� 盐ᨨ 㜠졔�矄 졔�矄矄� �泐�ᨨ � ᨨ鼀㡰 ᨨ鼀矄 졔�ᨨ 矄 ᆠ� ᨨ �矄ᆠ� �矄 ᆠ� 矄矄 和盐矄⼄盐 泐 ᨨ鼀矄 矄 � ᨨ ᆠ��ᨨ
�泐ᨨ졔�ᨨ 矄 泐ᆠᨨ�矄�ᨨ 矄㡰꾀矄矄盐鼀 ᨨ 鼀ᨨ 桌 ᨨ ᨨ鼀�矄 ᆠ泐� � ⼄盐 盐鼀 盐鼀 ᨨ 矄 � 泐ᨨ盐 �
졔�ᨨ 矄�矄 㡰 㡰 ᆠ㡰 㡰 � �黨 犨 � ᨨ �� 鼀� � �矄���ᨨ 泐 졔�ᨨ 矄 ᆠᨨ�矄�ᨨ 矄犨� ᨨ鼀矄�矄 㡰 �맔҈䰘鼀 㡰 �맔҈䰘鼀 䰘 ᆠ㡰 맔҈ 鼀 㡰 맔҈䰘鼀 �ᨨᨨ ├� �ᆠ �泐ᨨ 㦨ᆠᨨ ᨨ ᨨ鼀ᨨ ᨨ 졔�ᨨ �矄 㡰 ҈⏌ 㡰 ҈⏌ ᆠ㡰 ҈ 㡰 10 Subjective Problems ҈
ᔼ 泐 ᨨ� 㜠�├ᨨ鼀 z㦨҈҈㡰 � ᨨ 泐鼀 矄�矄 矄犨 � l 㜠ᨨ��Ἐ矄 犨 � ᨨ ᆠᨨ � ᨨ鼀犨 �ᆠ �ᨨ��� �矄矄ᨨl � 盐ᨨ n䁸 㜠㝐桌
ᨨᨨ 矄 ᨨ鼀 鼀矄�矄� 泐 ᨨ� 矄ᆠ盐鼀 � �犨 � 犨 � 鼀ᨨ䰘 鼀� ᆠ� ᨨ 鼀矄盐 ᆠᨨ 犨ᨨ├ 盐 ᆠ�ᨨ � 鼀 �鼀盐鼀 ├� �ᆠ �泐ᨨ ᆠ ᆠᨨ � hc㦨҈ 鼀㡰 㜠㝐├矄桌 꾀 泐 ᨨ� 㜠�├ᨨ鼀 矄ᆠ� 泐 ᨨ 鼀ᨨ 㡰�矄 ᨨ矄 ᨨ鼀�矄� 犨 � 矄ᨨ��� � � ᨨ鼀 ᨨ矄矄� 矄��ᨨ 矄 犨 �ᨨ盐 ᨨ 矄 矄 矄 ��矄 犨 㜠� 㜠� � ᆠ盐 � �ᨨ 矄盐矄㡰 㜠�㝐├矄桌 㡰 � ᨨ鼀�ᆠ 盐鼀ᨨ ᨨ鼀 㡰 ᔴᆠ盐 矄鼀矄犨 � 鼀� � 矄 矄��ᨨ 矄 ├hc㦨҈㜠 鼀�ᨨ盐 矄 �泐ᨨ 泐 ᨨ�
ᨨ鼀㦨҈㡰
Physics
꾀 泐 ᨨ� 㜠�├ᨨ鼀ᨨ ᨨ鼀�ᆠ 盐鼀Z�矄� ᆠ� 矄ᨨ ⼄盐 盐鼀 盐鼀nᔼᆠ 鼀�鼀 �鼀盐鼀 �泐 ᨨᨨ ᨨ ᔼ �鼀├矄 矄��ᨨ ᨨ⼄盐 盐鼀矄 n ᨨᨨ ᨨ �泐�矄鼀� � nZ �ᨨ盐 矄 �泐 � 㡰 ᨨ �矄ᨨ鼀꾀矄ᨨᆠᆠ盐 鼀� �鼀盐鼀 �泐 � 㡰 ᆠ 鼀� 泐 �矄 ᨨ鼀 盐� � 㜠 ᆠ��ᨨ ᨨ盐 矄 �泐 ᨨ 泐 ᨨ� ᨨ鼀�矄҈ 㜠㝐├矄桌 � 꾀 ᆠᨨ � 泐 ᨨ� 㜠�├ᨨ鼀�矄� ᆠ� 矄 ᔼ 矄ᨨ �泐ᨨ ᔴᆠ盐 �㡰 ├� �ᆠ �泐 ��㡰 ᨨ��犨 � ᨨ ᆠᨨ ҈黨黨㜠㝐├矄桌 꾀 泐 ᨨ� �├ᨨ鼀 ᨨ鼀�ᆠ 盐鼀Z㡰�矄� �� ᆠ� 矄ᨨ ⼄盐 盐鼀 盐鼀n ᆠ� ᨨ鼀ᆠ
鼀├ 矄��ᨨ ᨨ �矄 ᆠ� 矄泐矄盐ᆠᆠ矄矄�泐 鼀�� � 犨ᨨ ᨨᨨ 矄 ᨨ �泐 ҈ ҈䰘 矄ᆠ�泐꾀 泐 ᨨ鼀 ᨨ鼀 矄鼀 ᆠ� 矄ᆠ 鼀├ 矄��ᨨ ᨨ 矄ᆠᨨ ᆠ� 矄泐 矄盐ᆠᆠ矄矄�泐鼀�� �犨ᨨ ᨨᨨ 矄ᨨ ��矄� �黨�矄ᆠ�泐 ҈黨黨㜠㝐├矄桌 鼀� 盐矄ᨨ n Z ᔼᨨ � �ᨨ �泐ᨨ 㜠 ᨨ鼀㦨҈㡰 䰘 꾀�ᆠᨨ ᆠ �⼄盐ᨨ ᨨ ᆠᨨ e 鼀矄矄�鼀矄 鼀矄矄ᨨ ᆠᨨ ᆠ 鼀盐㜠 鼀矄ᨨ 㡰 鼀ᨨ矄 � ᆠ�ᆠ盐 ᨨ� ᨨ盐 盐ᆠ盐矄 ᨨ ᆠ �졔e ├ 鼀矄矄ᨨ 盐ᆠ盐矄ᨨ� � �㡰 꾀矄矄盐鼀� � ᨨ 鼀ᨨ ᨨ ᨨ鼀�矄�ᆠ ᨨ �矄矄泐矄鼀 ҈黨㜠㝐├矄桌 �㡰 � 矄矄�ᨨ ᨨ �盐矄ᨨ ᨨ ᨨ� ��㡰 � 盐ᨨ n ᨨ犨 �ᆠ �盐矄ᨨ ᨨ��矄 ᨨ �鼀泐 矄鼀矄 ᨨ �矄ᨨ ᨨ� ᨨ 泐 ᨨ� ᨨ鼀 ���㡰 � 犨 � ᨨ ��ᨨ 鼀� 犨
鼀盐㜠鼀矄ᨨ 盐鼀矄 ᨨ鼀 � ᨨ�ᨨ �矄 ᨨ� 꾀 ᨨ盐 �ᨨ �矄 � �盐鼀 ᨨ鼀 �矄 泐 ᨨ� 㜠�├犨� ᨨ鼀�ᆠ 盐鼀 ҈黨�㜠㝐├矄桌 �㡰 � 犨 � ᨨ ��ᨨ ⼄盐� ᨨ ᆠ� ᆠᨨ � �졔졔 ᨨ鼀 �矄ᨨ � ᨨ ᨨ� ᔼᨨ �矄�ᨨ �泐ᨨ 泐 ᨨ� ᨨ鼀 ⼄盐矄҈㡰 ��㡰 ᨨ犨 鼀 泐 矄ᆠ � 矄 ᨨ矄 � 鼀�矄矄�ᨨ 矄ᆠ盐鼀ᨨ ᨨ ᆠ� 矄泐矄鼀䁸 黨 �ᨨ � �ᨨ �泐ᨨ 泐 ᨨ� �├ᨨ ᨨ鼀�矄 泐 �矄 �㡰␔ �矄 犨 � ᨨ ��ᨨ
鼀� 犨 ᆠᨨ 盐鼀矄 ᨨ鼀 �矄 ᆠ� 矄ᨨ �ᨨ盐 矄䁸 ��㡰␔ �矄 �盐矄ᨨ �矄ᨨ� ᨨ �矄ᨨ鼀䁸 ҈黨㜠㝐├矄桌 䰘 泐 ᨨ� ᨨ鼀� �矄�ᨨ盐 矄�矄 ᆠ� 泐鼀 矄ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ��ᨨ ᨨ 犨 � 黨䰘�处ᨨ犨鼀 泐 � � 矄 ᨨ矄矄� � 矄盐� � 矄ᆠ盐鼀 䁸 ᔴᆠ盐 ᨨ �矄犨 � 鼀ᨨ �矄 鼀Ӹᨨ盐鼀泐 矄矄盐鼀 �ᨨ � �ᨨ �泐 ᨨ 泐 ᨨ� ᨨ鼀矄҈
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P-291
Atoms
Topic-3 : Miscellaneous (Mixed Concepts) Problems 10 Subjective Problems �� ᨨ鼀 �矄ᆠ �盐ᆠᨨ � � � 泐 ᨨ� ᨨ鼀矄 矄ᨨ 矄盐 ᆠᨨ �ᆠᨨ 矄ᨨ �盐鼀 ├� �ᆠ �泐 ᨨ 矄矄 ᨨᨨᆠᨨ 矄鼀� ᨨ鼀矄ᨨ �盐鼀�矄䰘
犨ᨨ├ 盐 ᆠ�ᨨ ᨨ矄ᨨ �盐鼀�矄҈ � ҈黨黨㜠҈㝐├矄桌 㡰 �泐ᨨ ᨨᨨ 矄ᆠ盐矄� � ᨨᨨᆠ�ᆠ 鼀�矄矄�ᨨ 㡰 ⼄盐 盐鼀 盐鼀矄ᨨ 犨ᨨ矄� ᨨ �
鼀�矄矄�ᨨ ᨨ 矄 ᨨᨨ 矄 ᆠ㡰 ᆠ �� �盐鼀ᨨ鼀 盐鼀ᨨ ᆠᨨ
� 泐 ᨨ� ᨨ鼀� ᨨ 矄��ᨨ 㡰 ᆠᨨ�矄 ᨨ 鼀�� �ᨨ鼀矄矄盐鼀� ��ᨨ 矄 ᨨ 矄��ᨨ ᔼᨨ � �ᨨ ᨨ �ᨨ 泐 ᨨ� �矄҈㡰 쟼ᆠᨨ 矄� 泐 ᨨ� �├ᨨ鼀 Z㦨㡰鼀├ 矄��ᨨ 矄 ᨨ鼀 � ᨨ ᨨ盐 ᨨ� ᨨ鼀 ᨨ盐 ᨨ � ᨨ� 矄盐� � ��ᨨ 矄� ᆠ� ᨨ鼀泐 ᨨ 鼀 ᆠ ᨨᨨᆠᨨ 矄 矄ᨨ� � ᨨ � ᨨ ᨨᨨᆠᨨ 矄 ᆠ 泐 矄 ᨨ 犨 � �矄黨�ᨨ矄ᔴᆠ盐 犨ᨨ├ 盐 ᆠ�ᨨ ᨨ 鼀 矄ᨨ� � ᨨ � ᨨ ᨨᨨᆠᨨ 矄ᆠ 泐 ᨨ �犨 �
҈
泐 �ᆠᨨ 矄 㦨҈黨맔 ҈ 䰘 鼀 -҈ 㡰҈黨黨㜠䰘㝐├矄桌 꾀�矄ᨨ � �ᆠ 泐 ᨨ� 㜠�├ᨨ鼀矄 矄矄ᨨ鼀ᨨ鼀矄�
ᨨ犨矄 �ᨨ盐 㡰 �泐A 矄ᨨ鼀ᨨ鼀矄� �ᆠ盐盐 ᆠ� 㡰 �泐B ᨨ
ᨨ鼀矄� 泐ᨨ �泐 ᨨ鼀矄ᨨ �矄 鼀├ 矄��ᨨ ᨨ �� �泐 泐矄ᨨ� � 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ�� ᨨ ᨨᨨ �泐䰘和盐矄⼄盐 泐 ᨨ鼀矄 鼀� ��ᨨ ᨨ ᨨ 泐矄� � ᨨᨨ
��矄和ᨨ鼀ᨨ 鼀� ᨨᨨ 矄 �泐䰘 矄ᨨ鼀 �泐鼀ᨨ 矄ᨨ鼀 矄矄 䰘 ҈黨黨㜠㝐├矄桌 �㡰 � � ᆠ�⼄盐 盐鼀 盐鼀ᨨ � ��泐 ᆠ� B ��㡰 � �ᨨ � �ᨨ �泐 ᨨ �矄ᨨ鼀矄 ���㡰 � 鼀 �鼀盐鼀 鼀� �鼀盐鼀 ��矄ᨨ 鼀� ᨨᨨ 矄 꾀矄� �ᆠᨨ ᨨ�矄ᨨ盐 矄�ᨨ 泐 盐ᆠ盐矄ᨨ ᆠ �졔Ze␔ Z�矄ᆠᨨ 矄 e�矄 鼀� �盐 ᨨ ᆠᨨ �ᆠᆠ �ᔼ⼄盐�矄䰘ᨨ ᆠ� ᆠᨨ ᨨ鼀 矄ᆠᨨ ᨨ ᨨ�ᨨ � ᨨ ᨨ� � ҈黨҈㜠҈㝐├矄桌 �㡰 盐ᨨ Z ��㡰 �泐⼄盐� ᨨ ᆠ� ᆠᨨ ᨨ鼀 � ᨨ ᨨ盐 ᨨ ᨨ� ���㡰 犨 � ᨨ ᆠᨨ鼀� �ᆠ ��ᨨ
⼄盐� ᨨ鼀ᨨ ᆠᨨ ᨨ鼀 �矄ᨨ ᨨ�ᨨ� � �泐 �㡰 ├� �ᆠ �泐ᨨ � �泐 �盐 鼀ᨨ鼀 盐鼀ᨨ ᆠᨨ � �矄ᨨ ᨨ� 㡰 �盐矄ᨨ �矄ᨨ ᨨ� �ᨨ � �ᨨ �泐ᨨ 泐 ᨨ� ᨨ鼀㦨҈ᨨ �盐矄㦨�맔҈҈҈鼀ᨨᆠ�泐ᨨ �� 㦨맔҈鼀⏌矄ᆠ ᆠ├矄ᆠᨨ 矄 㦨맔҈ᨨ盐矄㜠矄ᆠ㡰
Answer Key Topic-1 : Atomic Structure and Rutherford's Nuclear Model ҈
ᆠ㡰
ᆠ㡰 Topic-2 : Bohr's Model and the Spectra of the Hydrogen Atom
҈ ҈҈ ҈ ҈ ҈ � �䰘
㡰 㡰 ҈ 㡰 ᆠ㡰 ҈��҈㡰 ᆠ 㡰�҈ 㡰 �
㡰 㡰 ᆠ㡰 㡰
㡰 㡰 � ҈ 㡰 ҈ 㡰 ҈� 㡰 㡰 � 㡰 㡰 � 㡰 ҈㡰 꾀®錐®⼄矄錐ᔴ®錐®⼄ ᆠ㡰 �黨 ᆠ㡰 㡰
㡰 㡰 㡰 㡰 ҈㡰
҈ � �
㡰 ᆠ㡰 㡰 �㡰 ᆠ㡰 㡰
䰘 ҈䰘 䰘 䰘 �
㡰 ᆠ㡰 㡰 㡰 ᆠ㡰 㡰
҈ 䰘 �
ᆠ㡰 黨 㡰 ҈黨 㡰 黨 㡰 黨 㡰 ᆠ㡰 ��
㡰 ᆠ㡰 㡰 㡰 ᆠ㡰 㡰
҈ 黨 �
㡰 㡰 㡰 �҈㡰 㡰 㡰
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EBD_8341 P-292
Physics
27
Nuclei
Topic-1 : Composition and Size of the Nuclei ҈
1 MCQs with One Correct Answer
�盐矄 R ᨨ 盐ᆠ盐矄 ᨨ 鼀矄矄 盐鼀 A ᆠ 矄�鼀 泐 ᨨ鼀盐R㦨 ҈맔҈҈�㡰A҈⏌鼀ᔼ ᨨᨨ犨矄 鼀矄矄 矄�泐 ᨨ 盐ᆠ盐矄 �矄 ᨨ ᨨ ᨨ M ᨨ @ M 盐 ; ҈䰘 ´ ҈ -䰘 ├�㡰 㝐� 和 ᔼᔼ㡰桌 㡰 ҈├�鼀 㡰 ҈҈├�鼀 ᆠ㡰 ҈├�鼀 㡰 ҈҈䰘├�鼀
�ᨨᨨ 鼀矄矄 矄��矄ᨨ 盐ᆠ�ᨨ ᔴ ҈졠�矄 ᆠᨨ矄ᨨ 㝐� 꾀�҈黨 ᔼᔼ㡰桌 㡰 ҈ 㡰 ҈ ᆠ㡰 � 㡰 꾀 盐 矄 泐 盐ᆠ盐矄矄├矄� ᨨ犨ᨨ 盐ᆠ� 犨 �ᆠ 鼀ᨨ犨泐犨� ᨨᆠ��矄� �ᨨᨨ 䰘 �ᨨᨨ ��ᨨ 盐ᆠ� 矄矄盐鼀 ᨨ矄 �ᆠ㡰�矄 㝐� 졠 � 꾀�҈�҈桌 㡰 䰘 㡰 ᆠ㡰 㡰 黨
�
ᨨ盐 �盐鼀 盐ᆠ盐矄 ᨨ犨 ᨨ矄�矄鼀矄矄泐犨� ᨨ盐鼀䁸 和桌 㡰 m µ V 㡰 m µ ҈⏌V ᆠ㡰 m µ V 㡰 m µ V 졠 ᨨ 鼀� �盐 ᨨ 矄�泐ᨨ 盐 �盐鼀 盐ᆠ盐矄 �矄 ҈黨黨黨和㜠㝐├矄桌 mp㦨҈䰘맔҈䰘├�桌 㡰 ҈├�⏌鼀 㡰 ҈҈䰘├�⏌鼀 ᆠ㡰 ҈҈├�⏌鼀 㡰 ҈҈҈├�⏌鼀
5 True / False
ᨨ ᨨ 鼀� �盐 ᨨ 矄�泐ᨨ 盐ᆠ鼀�矄 ҈├�鼀 ҈黨黨㜠㝐├矄桌
6 䰘
MCQs with One or More than One Correct Answer
鼀矄矄 盐鼀ᨨ 盐ᆠ盐矄�矄 ҈黨㜠㝐├矄桌 㡰 犨泐矄矄矄 �矄ᨨ鼀�ᆠ 盐鼀 㡰 犨泐矄鼀ᨨ �矄ᨨ鼀�ᆠ 盐鼀 ᆠ㡰 矄ᨨ鼀�鼀矄⼄盐ᨨ�矄ᨨ鼀�ᆠ 盐鼀 㡰 矄ᨨ鼀�鼀矄 鼀ᨨ 矄ᨨ鼀�鼀矄 ⼄盐 ᨨ �矄 ᨨ鼀�ᆠ 盐鼀
Topic-2 : Mass-Energy Equivalence and Nuclear Reactions 1 ҈
º 盐ᨨ ␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨᆠ矄矄�矄ᨨ犨 泐 鼀ᨨ鼀 盐鼀 �泐ᆠᨨ 矄�ᨨ 䁸 㝐� 和 ᔼᔼ㡰桌 㡰 졔 ® 盐�盐鼀ᨨ鼀 ᆠᨨ ᨨ盐 ᨨ 盐ᆠ盐矄㡰
MCQs with One Correct Answer
Ӹᨨ盐�� 鼀矄矄ᨨ 䰘 � 㦨䰘҈盐 㝐矄矄ᨨ 㦨盐 㝐矄矄ᨨ ҈҈ 㦨҈䰘黨盐 ␔ � ᨨ 䰘 � �矄 ᆠᨨ � ᨨ 泐 ᨨᨨ
ᆠ盐 �泐� � ├␔ 㡰�矄 㝐矄矄ᨨ 盐ᆠᨨ 㦨҈⏌ᆠ桌 㝐� 和 ᔼ㡰桌 㡰 �맔҈� 㡰 맔҈ ᆠ㡰 맔҈� 㡰 ҈맔҈ � 鼀矄矄矄ᨨ �ᨨ盐矄ᨨ鼀�ᆠ�ᆠ矄鼀㦨҈䰘盐 鼀 㦨҈䰘盐鼀㦨�盐 鼀v 㦨鼀 㦨҈҈盐犨 º ᨨᨨ º 盐ᨨ º ᆠᨨ v º � 盐� ᨨ
㡰 ® 졔졔졔 v ᆠ㡰 졔® 졔g 㡰 졔졔®g
� � � � �泐 盐ᆠᨨ ᨨ ҈ � 和 㝐矄矄ᨨ ᨨᨨ mp㦨҈䰘鼀矄矄ᨨ 盐ᨨ mn㦨҈䰘 鼀矄矄ᨨ � 盐ᆠ盐矄m和 㦨҈҈黨黨҈黨黨 ├҈㦨黨҈㝐㡰 㝐� 和 ᔼᔼ㡰桌 㡰 䰘�㝐 㡰 黨㝐 ᆠ㡰 㝐 㡰 �㝐
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�
䰘
ᔼ ᆠᨨ├�ᨨ 黨� 盐�矄 盐泐盐矄 盐� 泐矄
�泐矄 �矄矄�ᨨ �矄㝐� 꾀ᨨ� ᨨ 盐鼀㦨맔҈├�ᨨ鼀ᨨ ҈ 㦨҈맔҈҈黨 ᨨ犨ᨨ盐盐ᨨ ᆠᨨ�矄ᆠᨨ矄ᨨ [㝐� Sep.02, 2020 (I)] 㡰 �㝐␔ 㡰 㝐␔ ᆠ㡰 ҈�㝐␔ 㡰 �㝐␔ ᔴᨨ 矄� 盐ᆠ �矄矄�ᨨ
Ne®졔ᔴ҈ � � � � �泐⏌ 盐ᆠᨨ ᨨ ᔴ҈矄ᆠ�泐㝐䰘䰘㝐 䰘㝐 � � 泐 ᆠᨨᆠ 矄鼀 㝐� ҈ ҈黨ᔼᔼ桌 㡰 �泐ᨨ ҈㝐犨�矄盐� 㡰 㝐 �泐犨�矄 ᆠ㡰 �泐ᨨ 㝐犨�矄 㡰 �泐ᨨ ҈҈黨㝐 矄ᨨ矄盐� ᔼ鼀�� ᆠᨨᆠᨨ 矄�� 鼀矄矄� ᨨ �泐 �ᨨ矄ᨨ犨ᨨ ҈黨犨 鼀矄矄 ᨨ 盐 ᆠᨨ 矄盐鼀 ᨨ盐 � ᆠᨨ 犨� ᨨᆠ�泐ᨨ �� ᆠ�矄맔҈鼀⏌矄㡰 㝐� 졠 � 꾀�黨҈䰘桌 㡰 黨�鼀 㡰 �鼀 ᆠ㡰 맔҈�鼀 㡰 맔҈��鼀 ␔ �盐鼀�矄ᨨ鼀 犨� 盐ᨨ 矄�盐 �ᨨ矄 �矄矄�ᨨ �矄矄�ᨨ ᆠ�ᨨ ᆠ 犨� 矄 + n҈ ® � ҈҈ + ⧤ 黨 + x + 턼 �泐㡰 犨 �ᆠ矄 鼀 xᨨ 盐ᆠ �泐턼 �矄矄 ␔ �矄 鼀ᨨ �ᆠx䁸 㝐� 졠 � 꾀�黨҈桌 㡰 ᆠᨨ
㡰 a㜠�ᆠ ᆠ㡰 盐ᨨ
㡰 盐� ᨨ ᔼ ᨨ�ᨨ 矄�� ᨨ犨E ᨨ 矄鼀矄矄 �泐 ᨨ 盐ᆠ盐矄 n 盐ᨨ ᆠᨨᆠᨨ�ᨨ �矄 㝐� 䰘桌 æ ö æ҈䰘 ö æ 黨䰘 ö U > Eç I + E ç Y ÷ + E n 㡰 㡰 E ç è 黨 ÷ø è � ÷ø è 黨 ø æ ö æ҈䰘 ö æ 黨䰘 ö 㡰 E ç U < Eç I + E ç Y ÷ + E n 㡰 è 黨 ÷ø è � ÷ø è 黨 ø æ ö æ҈ ö æ 黨 ö U÷ < Eç Ba ÷ + E ç Kr ÷ + E n㡰 ᆠ㡰 E ç è 黨 ø è � ø è ø æ ö æ ҈ ö æ 黨 ö U = Eç Ba ÷ + E ç Kr ÷ + E n㡰 㡰 E ç è 黨 ÷ø è � ø è ø ᔼ 矄ᆠ ᆠᨨ 盐ᆠ�ᆠᨨ鼀泐� ᨨᨨ 泐�
盐ᆠ� �泐矄 ᨨ 泐� 盐ᆠ��矄㝐矄矄ᨨ 盐ᆠ盐矄�矄鼀盐 鼀矄矄ᨨ 졠 泐� 盐ᆠ盐矄�矄 �和桌 ҈�黨黨黨鼀盐桌 㡰 䰘㝐 㡰 �҈㝐 ᆠ㡰҈㝐 㡰黨㝐 � � � �泐 盐ᆠᨨ 矄鼀矄矄 盐鼀 ᆠ盐 ᨨ
盐ᆠ��矄矄 ᨨ犨 � ��盐W, X, Y Z ᨨ盐 盐ᆠ� � �ᆠ ᨨ ᆠ盐 ᨨᆠ矄矄 犨ᨨ盐 矄 �泐�矄 ҈黨黨黨和㜠㝐├矄桌
黨
黨
҈
�
Binding Energy/nucleon in Mev
Y 8.5 8.0 7.5
X W
5.0
Z 0
90 30 60 Mass number of nuclei
120
҈҈
㡰 W ® X + Z 㡰 Y ® Z ᆠ㡰 W ® Y 㡰 X ® Y + Z 矄 盐ᨨ 矄ᆠ 矄�泐矄ᨨ犨 ᨨ犨 泐 㡰 盐矄ᨨ 矄 � � � ҈黨黨㜠҈㝐├桌 㡰 矄矄� � 鼀 ᨨ盐� 犨 ᆠ㡰 矄�ᆠᆠᨨ�矄�ᨨ 矄犨� 泐 盐ᆠ� 㡰 泐� �矄ᨨ �ᆠ�ᆠ � 4
Fill in the Blanks
҈ ᔴᨨ 矄� ᨨᨨ犨� �ᆠ�ᨨ ҈ + ҈ = + 턼 㝐矄矄ᨨ 盐�盐鼀ᨨ鼀㦨҈҈u 㝐矄矄ᨨ �盐鼀ᨨ鼀㦨u
�矄�矄 盐ᆠᆠ�ᨨ � 犨 �ᆠ �泐Q 矄 �矄㝐 ҈黨黨㜠㝐├矄桌 ҈ ᔼ 盐ᆠ ᨨᆠ矄矄 ᔴ҈҈ ¾¾ ® � ҈҈ + b + + 矄 矄 ᨨ ҈黨黨㜠҈㝐├桌 ҈ � � � ��矄 盐ᆠᨨ ᨨ 盐ᨨ ҈㡰 �盐鼀 㡰҈҈㝐 䰘㝐矄ᆠ�泐 �泐矄 犨 犨ᨨ 盐ᨨ 矄 盐矄ᨨ ᨨ鼀 �盐鼀
盐ᆠ盐矄 㡰�矄 ҈黨㜠㝐├矄桌 6 MCQs with One or More than One Correct Answer ҈� 꾀矄矄盐鼀 盐ᆠ� � � �泐 盐ᆠᨨ ⏌꾀㡰 矄盐矄鼀矄矄 盐鼀 꾀㡰�矄矄矄 ᨨ犨 � ��盐矄 �矄 ᨨᨨᆠ ᨨᨨ矄 ᆠᨨᆠᆠ ᨨ�ᆠ 矄㡰�� ᨨ犨 桌 ⏌꾀
҈
꾀
㡰 盐矄�ᨨ ᨨ 犨ᨨ 盐ᆠ�犨� 鼀矄矄 盐鼀矄泐� �� �ᨨ ҈꾀�犨�矄 �泐 㡰 盐矄�ᨨ ᨨ 犨ᨨ 盐ᆠ�犨� 鼀矄矄 盐鼀矄泐� �� �ᨨ �҈꾀҈犨�矄 �泐 ᆠ㡰 �矄矄�ᨨ ᨨ 盐ᆠ盐矄泐� �� 鼀矄矄 �ᨨ ҈꾀 犨�矄 �泐犨 ᨨ├ � ᨨ犨ᨨ⼄盐 �鼀 矄 㡰 �矄矄�ᨨ ᨨ 盐ᆠ盐矄泐� �� 鼀矄矄 �ᨨ 꾀 犨�矄 �泐犨 ᨨ├ � ᨨ犨ᨨ⼄盐 �鼀 矄 ҈ mp 鼀矄矄ᨨ ᨨᨨ mn 鼀矄矄ᨨ 盐ᨨ M҈ 鼀矄矄ᨨ ҈ 盐ᆠ盐矄 M 鼀矄矄ᨨ ᔴ
盐ᆠ盐矄
҈黨黨和㜠㝐├矄桌 㡰 M = M ҈ 㡰 M > M҈ ᆠ㡰 M < M҈ 㡰 M҈ < ҈ mn + m p 㡰 ҈䰘 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄㡰�矄 㡰ᆠᨨᆠ䁸 ҈黨黨㜠㝐├矄桌 㡰 矄鼀矄矄ᨨ 矄 盐ᆠ盐矄�矄矄矄 矄盐鼀 ᨨ 矄鼀矄矄矄ᨨ �矄矄 盐ᆠᨨ 矄 㡰 矄鼀矄矄ᨨ 矄 盐ᆠ盐矄�矄� 矄盐鼀ᨨ 矄鼀矄矄矄ᨨ �矄矄 盐ᆠᨨ 矄 ᆠ㡰 ᔼ 盐ᆠ �矄矄�ᨨ �泐�矄矄 泐 盐矄� �犨ᨨ
盐ᆠ�ᨨ 鼀 �盐鼀鼀矄矄 ᨨ �鼀泐҈鼀盐㡰 㡰 ᔼ 盐ᆠ �矄矄�ᨨ �泐�矄矄 泐 �鼀 �ᨨ
ᨨ 泐 泐 盐ᆠ盐矄
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҈
҈黨
Physics
盐� � 盐ᆠ 盐矄�ᨨ ᆠ�ᨨ
҈黨䰘㜠㝐├矄桌 㡰 泐 盐ᆠ盐矄├矄� ᨨ犨ᨨ �鼀 矄泐�矄 㡰 �� 盐ᆠ盐矄ᨨ鼀 泐 鼀 盐ᨨ 矄├矄盐 ᆠ㡰 泐 盐ᆠ盐矄 ᨨ鼀 泐 鼀 盐ᨨ 矄 ├矄盐 㡰 犨ᨨ�� 盐ᆠ�ᆠᨨ鼀� ᨨ�� � 盐ᆠ盐矄 ᨨ矄矄�泐ᨨ ᨨ 盐ᆠ矄 ᨨ鼀 ᨨᨨ犨� �⼄盐�ᨨ 矄�ᆠ├ᨨ盐 ᨨ矄矄� 盐ᆠ 盐矄�ᨨ ᆠ�ᨨ 矄 ҈黨㜠㝐├矄桌 㡰 ᔴ҈ + ҈҈ ® ᔴ҈ + 㝐 㡰 ᔴ҈ + ҈҈ ® 䰘 ҈ + 㝐 ᆠ㡰 㡰
҈ ҈ ҈� 䰘 + ҈ ® 졠 + 䰘㝐 � ҈ + n ® � ҈ + 和 黨 黨
10 Subjective Problems ᔼ 盐ᆠᆠ�ᨨ �盐 �ᨨ矄 �矄矄�ᨨ �� � 㝐ᨨ �泐 ᆠᨨ 矄҈奠 �ᆠ� ᆠ泐 ᨨ 盐ᆠ矄 ҈㝐␔ᨨ犨ᔼ ᆠᨨ�矄ᨨ 盐 ᆠ�ᨨ ᨨ҈泐矄 � ᨨ鼀矄矄ᨨ 盐 �盐鼀⼄盐� ҈㜠�㝐├矄桌 ҈ ᔼ�矄ᨨᨨ矄 ᨨ盐矄 盐ᆠ 盐矄�ᨨ ᆠ�ᨨ
҈ +҈
҈黨黨㜠㝐├矄桌 ® � 盐ᆠᆠᨨᨨ 㝐␔� �ᔼ �泐 ᨨ鼀 ᨨᆠ�ᨨ �矄盐矄 犨� �ᆠ �ᆠ� ᆠ泐�
ᆠᨨ ᨨ犨鼀 泐�鼀矄ᨨ 盐�盐鼀 盐犨� 泐 鼀矄矄矄ᨨ ҈ ҈҈ᨨ鼀�ᆠ 鼀矄矄盐 �矄 ᨨ鼀�ᆠ鼀矄矄盐 �矄矄ᆠ�泐㡰
+ n҈
+ n҈ + g + 㝐
Topic-3 : Radioactivity 1 ҈
MCQs with One Correct Answer
꾀ᆠ���矄ᨨ �ᨨᆠ�矄盐矄 ᆠ矄꾀 ᔴ 矄 泐 ᆠ盐矄꾀 ᔴ� ��盐
� 㜠�矄 ҈ 꾀㡰 ҈ 㡰 ҈ ᔴ㡰 � �ᨨ
[Main Sep. 05, 2020 (I)] ᔼ
䰘
꾀 ᔴ
泐矄㡰 㡰 ҈҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ 꾀 �ᨨᆠ� 盐ᆠ盐矄 ᆠ泐矄泐犨ᨨ � ᨨᆠ矄矄矄
� ᨨ �矄ᨨᆠ矄矄�矄҈矄 ᨨ 矄ᆠᨨ �矄҈矄 ᆠ� � ᨨ 盐ᆠ盐矄�矄 ᆠᨨ矄ᨨ 㝐� 和� ᔼᔼ㡰桌 㡰 黨矄ᆠ 㡰 矄ᆠ ᆠ㡰 ��矄ᆠ 㡰 ҈矄ᆠ ᔼ �ᨨᆠ� 鼀� ᆠ�ᨨ ᨨ ᆠ� 鼀� 鼀� � � �鼀 t �矄 黨⏌҈ ᆠ�ᨨ 犨矄 鼀� � � t⏌�矄 [Main Sep. 03, 2020 (I)] �
҈
䰘 㡰 ᆠ㡰 㡰 � �
ᆠ��泐ᨨ �ᨨᆠ�矄鼀 矄 ᨨ鼀䰘矄҈ᨨ� 矄҈� 鼀� 盐矄ᔼ矄 � �矄ᆠᨨ矄ᨨ䰘 ᔼᔼ桌 㡰 䰘鼀� 㡰 鼀� ᆠ㡰 鼀� 㡰 �鼀�
犨ᨨ �ᨨᆠ� 矄盐矄 ᆠ矄 꾀 ᆠ泐 ᆠᨨ 矄 矄�l l矄ᆠ�泐꾀t㦨矄鼀 矄 矄鼀 盐鼀ᨨ 犨ᨨ 盐ᆠ� �鼀├ ᨨ
㡰
�
æ ҈ö �ᨨᨨ 盐鼀ᨨ 盐ᆠ�ᨨᆠᨨ鼀 ç ÷ 犨� è eø 㝐� ҈꾀�҈黨ᔼᔼ桌
㡰 1/2l 㡰 ҈⏌l ᆠ㡰 ҈⏌l 㡰 ⏌l ᔼ �ᨨᆠ� ᆠ泐ᆠ � � �� 盐ᆠ盐矄�矄 黨 꾀 a㜠�ᆠ矄 b㜠�ᆠ矄犨 �ᆠ 鼀� ᔼ 盐ᆠ盐矄�矄 꾀 꾀 �� 泐 㝐� ҈ ҈黨ᔼᔼ桌 㡰 꾀㦨錐㦨 㡰 꾀㦨錐㦨 ᆠ㡰 꾀㦨錐㦨 㡰 꾀㦨錐㦨҈ 矄� � 盐ᆠᆠᨨ盐 ᆠᨨ盐 ᨨ 鼀� �ᆠ矄 ᨨ鼀 �ᨨᆠ�矄ᨨ盐ᆠ�矄鼀矄盐 꾀㦨�犨矄҈ ᆠᨨ盐 矄矄ᆠᨨ 㦨矄ᆠᨨ 矄�犨矄҈ᆠᨨ盐 矄 矄ᆠᨨ ᆠᨨ盐 ᨨ矄 矄ᆠᨨ盐 矄 矄ᆠᨨ 㦨 矄ᆠᨨ 矄 �矄 ᆠᨨ矄 ᨨ 㝐� ҈ ҈黨ᔼ桌 㡰 㡰 ҈� ᆠ㡰 㡰 꾀 �� � 矄 矄泐㦨 犨ᨨ �ᨨᆠ�矄盐矄 ᆠ矄꾀
ᨨ �ᆠ���矄 꾀 �鼀�矄 ᆠ泐矄犨� �鼀矄ᔼ 㜠� ᨨ 꾀�矄l 㜠� ᨨ �矄 㝐� 黨 ҈黨ᔼᔼ桌
⼄盐ᆠ���矄 �ᨨ
l l ᆠ㡰 㡰 l ᔴᨨ � �ᆠ��泐 黨 꾀矄ᨨ盐�ᨨ ᆠᨨ � � �ᆠ�ᆠᨨ䰘 ᨨ Ҙᔴ� ᆠ泐ᆠᨨ 矄 l�矄� ᆠ � �鼀Ἐ矄 ᨨ 泐ᔼ ҈ᆠ鼀ᨨ ᨨᨨ �矄 犨 ᨨ鼀 �鼀Ἐ矄ᨨ 泐 ҈ 矄ᨨ � ᆠ�ᨨ ᆠ��泐 ᨨ盐 犨矄 ᆠ泐矄 鼀� 盐␔ �矄 ᨨ盐鼀ᨨ ᨨᨨ �矄 ᨨ犨� ��
ᨨ 泐䁸 ҈ᔴ�㦨䰘맔҈҈ ᆠ泐矄ᆠᨨ 㦨҈ 矄l㦨㡰 㝐� 졠 � 꾀�҈�҈桌 㡰 �矄 㡰 䰘�矄 ᆠ㡰 �矄 㡰 ��矄 ҈ 꾀 �ᨨᆠ� 盐ᆠ盐矄꾀犨� � ᆠ泐矄� ᨨ
盐ᆠ盐矄꾀㦨 �矄 ᨨ 盐ᆠ盐矄꾀矄ᨨ鼀�鼀 �ᨨᨨ 盐鼀ᨨ ᨨ ᨨ 꾀�矄 �矄��
泐 㝐� ҈䰘桌 㡰 l
㡰
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T 㡰 t = ᨨ� ҈㡰 ᨨ� ᨨ�҈ 㡰 t = ᆠ㡰 t = T ᨨ�҈ ᨨ� 㜠�矄 ᨨ 犨ᨨ �ᨨᆠ�鼀 矄꾀 鼀� 盐矄 鼀� 盐矄矄ᆠ�泐ᔼ ��泐 矄鼀矄 ⼄盐 盐鼀ᨨ 盐ᆠ�꾀 鼀� 盐矄 �ᨨᨨ ᆠ泐 盐鼀ᨨ 꾀 盐ᆠ�犨�㝐� ҈桌 㡰 ҈ 㡰 � ᆠ㡰 ҈҈ 㡰 ҈ b 盐鼀ᨨ b�ᆠ矄鼀� 泐҈�鼀 ᨨ �ᨨᆠ� 盐ᆠ� � 㦨҈� 矄㡰� 䰘� ᨨ盐矄 b �矄ᆠᨨ矄ᨨ 꾀ᨨ� ᨨ 盐鼀㦨맔҈⏌�鼀ᨨ㡰 㝐� 졠 � 꾀�҈҈҈�桌 㡰 맔҈҈ 㡰 䰘�맔҈҈ ᆠ㡰 ҈�맔҈ 㡰 ҈䰘�맔҈ 꾀�ᆠᨨ 犨ᨨᨨ ᨨ鼀ᆠ 泐ᆠ盐矄 ᨨ犨矄 ᆠ泐矄 鼀� 盐 꾀 犨ᨨᨨ �ᆠ ᨨ 矄鼀 矄� ᆠ � 鼀盐矄盐鼀 ᨨ� ᨨ鼀ᆠ盐鼀 泐泐矄ᆠ├㡰矄 ᨨ犨矄 ᆠ泐矄鼀� 盐ᔼ � ᨨ ᔴ҈�矄�䰘泐矄
� ᨨ 犨ᨨᨨ �ᆠ ᆠ � 鼀盐矄盐鼀 �矄 ᨨ �鼀泐 㝐� 졠 � 꾀�҈黨҈桌 㡰 ҈黨泐矄 㡰 ҈黨泐矄 ᆠ㡰 ҈黨黨泐矄 㡰 黨黨泐矄 꾀 �ᨨᆠ� 盐ᆠ�犨� ᆠ泐ᆠᨨ 矄 �⏌矄�矄� � ᨨ 盐ᆠ ᆠᨨ 矄 ᨨ ҈ 盐ᆠ�⏌矄ᔼ 㦨 犨 ᨨ 盐ᆠ� �鼀犨 � 盐ᆠ��矄 㝐� 졠 � 꾀�҈҈҈桌
㡰 t = T ᨨ� ҈㡰
҈҈
҈
҈
҈
㡰 l æç ö÷ 矄 è ø æ ö 㡰 l ç ÷ 矄 è ø
㡰 ҈矄 ᆠ㡰 l 矄 ҈�
҈
҈䰘
҈ 䰘 �矄 �ᨨᆠ�矄盐矄 ᆠ � � � ᨨ 泐矄
� ᨨ��泐 盐ᆠ盐矄盐 �ᨨ矄 ᆠ泐 犨ᨨ �矄 㜠㝐҈桌 ҈
ᆠ㡰
A
X z + g ¾¾ ® X Z -҈ + a + b
㡰
A
X z + ҈nᨨ ¾¾ ®
ᆠ㡰
A
㡰
A
A-
㡰
҈
和桌
X Z - + c
A
X z ¾¾ ® XZ + f A
X z + e-҈ ¾¾ ® X Z -҈ + g ҈�
㜠� ᨨ 꾀�矄҈ m s �鼀├ ᨨ �ᨨᆠ��泐ᨨ 矄鼀ᨨ ҈�꾀ᨨ ᆠ泐ᨨ҈⏌҈ ᨨ �矄 � ��盐�矄 和桌 㡰 m s 㡰 m s ᆠ㡰 m s 㡰 m s 꾀 �ᨨᆠ� 矄鼀 ᆠᨨ 矄�矄矄 ᨨ 犨ᨨ �矄� ᆠ 矄ᆠ�矄 � �⼄盐 盐鼀ᨨ ᨨ鼀矄� ��泐 鼀 � �鼀 ᨨ ᨨ 矄ᆠ�矄�矄t ᨨ ᨨ �矄�t ᆠ泐 ᨨ 盐ᆠ矄� ᨨ ᆠ矄矄矄꾀ᨨ�矄鼀 ᨨ ᨨ 盐鼀ᨨ �ᨨᆠ� 盐ᆠ�矄 盐 ᆠ�ᨨ ᨨ �鼀 ␔ �ᆠ ᨨ ᨨᨨ犨� � ��盐矄矄矄 ᨨ鼀ᨨ �矄ᨨ䁸 ҈和桌
㡰
盐ᨨ 犨�� l ҈㦨桌 꾀 ҈黨桌 㡰 黨 㡰 ᆠ㡰 ҈҈� 㡰 꾀 �ᨨᆠ�矄鼀和҈ � � ᆠ��泐�Ҙᔴ� 矄犨�ᆠ 盐鼀ᨨ 盐ᆠ�矄 ᨨ 矄鼀和犨 �ᆠ 矄
ᆠ��泐ᨨ ҈Ҙᔴ� �矄ᨨ 和҈ 和ᆠ 桌 㡰 泐矄 �泐矄矄ᆠ�泐 㡰 泐矄 ҈泐矄矄ᆠ�泐 ᆠ㡰 ҈泐矄ᆠ 㡰 �泐矄ᆠ
㡰
҈
A
㡰
ᔼ �ᨨᆠ�矄鼀 ҈黨 K 盐ᆠ� � ᆠ泐� ᨨ 矄 Ca 盐ᆠ�犨� ᆠ泐ᆠᨨ 矄 �맔҈҈ 泐ᨨ� ᨨ矄 ҈ Ar 盐ᆠ�犨� ᆠ泐ᆠᨨ 矄 �맔 ҈҈泐� � �矄矄鼀 矄 Ca ҈ 盐ᆠ�ᨨ 盐ᆠ 泐 盐ᆠ�ᨨ 泐 Ar ҈黨 K ᔼ �鼀맔҈黨泐矄� �ᨨᨨ 矄盐鼀ᨨ 矄 Ca Ca 盐ᆠ�ᨨ �ᨨᆠ� ҈黨 K 盐ᆠ��矄黨黨
㡰 ҈
҈ 꾀 泐矄ᨨ �ᨨᆠ�矄盐矄 ᆠ矄 ᨨ犨矄 ᆠ��泐 ᨨ 矄҈ 泐矄�矄ᆠ��泐ᆠᨨ鼀矄 矄 ␔ 犨矄�矄� ��ᆠ��泐䁸 和桌 㡰 矄 㡰 矄 㡰 ҈ 矄 㡰 矄 ҈黨 꾀 盐ᆠ盐矄犨� 鼀矄矄 盐鼀� ��泐矄鼀�矄
a 㜠�ᆠ ᔼ Q 盐 ᨨ ᆠ�ᨨ �矄 �� 㝐 ᆠᆠ盐 ├� �ᆠ �泐ᨨ a 㜠�ᆠ 和桌 㡰 㝐 㡰 �㝐 ᆠ㡰 �㝐 㡰 �㝐 ␔ �ᆠ ᨨ ᨨᨨ犨� �ᨨᆠ矄矄矄矄 矄 g 㜠 ᆠ泐䁸
㡰 t
t
t
ᆠ㡰
㡰 t
ᆠᨨ 鼀� � ��ᨨ ᨨ��� 矄 ᨨ鼀 㡰 �
ᨨ�矄ᨨ ᨨ鼀矄 ҈和桌 㡰 ᆠᨨ 矄 �矄� �� 盐ᆠ� ᆠ㡰 ᆠ泐ᨨ 盐ᨨ � 盐ᆠ盐矄 㡰 ᨨᨨ 矄ᆠ� � ᨨ鼀 盐ᆠ盐矄 犨ᨨ �ᨨᆠ�鼀�矄X҈ X ᆠ泐ᆠᨨ 矄 矄 ҈l l 矄ᆠ�泐ᔼ � ��泐 泐 矄鼀
盐鼀ᨨ 盐ᆠ� �ᨨᨨ 盐鼀ᨨ 盐ᆠ�ᨨ X҈ᨨ ᨨ X犨�҈⏌e �鼀 和桌 ҈ ҈ ҈҈ ҈ 㡰 ᆠ㡰 㡰 㡰 ҈l ҈҈l ҈l 黨l � ␔ �ᆠ ᨨ ᨨᨨ犨� ��矄ᆠᨨᆠ矄鼀 䁸 ҈黨黨黨和㜠㝐├矄桌 㡰 泐矄矄鼀矄ᆠ ᨨ 泐矄 㡰 鼀鼀泐矄 �� �泐 盐ᨨ 矄 ᆠ㡰 꾀 �ᆠ矄矄� �泐�ᨨ �矄 �盐鼀ᨨ鼀矄 㡰 ᨨᨨ 矄 盐ᨨ 矄 ᆠ泐 矄鼀鼀矄矄
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EBD_8341 P-296
26.
27.
28.
Physics
A radioactive material decays by simultaneous emission of two particles with respective half-lives 1620 and 810 years. The time, in years, after which one-fourth of the [1995S] material remains is (a) 1080 (b) 2430 (c) 3240 (d) 4860 Consider a particles, b particles and g - rays, each having an energy of 0.5 MeV. In increasing order of penetrating powers, the radiations are: [1994 - 1 Mark] (a) a, b, g (b) a, g , b (c) b, g, a (d) g, b, a The equation
35. To determine the half life of a radioactive element, a student dN (t ) dN (t ) is the rate versus t. Here dt dt of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is [2010]
plots a graph of ln
4 11 H + ¾¾ ® 42 He 2 + + 2e – + 26 MeV represents [1983 - 1 Mark]
29.
30.
(a) b -decay (b) g-decay (c) fusion (d) fission Beta rays emitted by a radioactive material are (a) electromagnetic radiations [1983 - 1 Mark] (b) the electrons orbiting around the nucleus (c) charged particles emitted by the nucleus (d) neutral particles The half life of radioactive Radon is 3.8 days. The time at 1 th of the radon sample will remain the end of which 20 undecayed is (given log10 e = 0.4343 ) [1981- 2 Marks] (a) 3.8 days (b) 16.5 days(c) 33 days (d) 76 days. Integer Value Answer is an isotope of Iodine that B decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with 131I is injected into the blood of a person. The activity of the amount of 131I injected was 2.4 × 105 Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from person's body, and gives an activity of 115 Bq. The total volume of blood in the person¡¦s body, in liters is approximately (you may use ex » 1 + x for |x| < < 1 and ln2 » 0.7). [Adv. 2017] For a radioactive material, its activity A and rate of change dN dA and R = – of its activity R are defined as A = , dt dt where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life t) and Q (mean life 2t) have the same activity at t = 0. Their rates of change of activities at t = 2t R n are RP and RQ, respectively. If P = , then the value of RQ e n is [Adv.2015] A freshly prepared sample of a radioisotope of half-life 1386 s has activity 103 disintegrations per second. Given that ln2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is [Adv. 2013-I] The activity of a freshly prepared radioactive sample is 1010 disintegrations per second, whose mean life is 109 s. The mass of an atom of this radioisotope is 10–25 kg. The mass (in mg) of the radioactive sample is [2011] 2
31.
32.
33.
34.
131I
4
Fill in the Blanks
( )
36. When Boron nucleus 105 B is bombarded by neutrons, a -particles are emitted. The resulting nucleus is of the element ........ and has the mass number ...... [1986 - 2 Marks] 37. In the Uranium radioactive series the initial nucleus is
238
92
U
206
and the final nucleus is 82 Pb . When the Uranium nucleus decays to lead, the number of a-particles emitted is ......and the number of b-particles emitted is ..... [1985 - 2 Marks] 38. The radioactive decay rate of a radioactive element is found to be 103 disintegration/second at a certain time. If the half life of the element is one second , the decay rate after one second is _______ and after three seconds is _______. [1983 - 2 Marks] 6
MCQs with One or More than One Correct Answer
39. In a radioactive decay chain,
232 90Th nucleus
decays to
212 82 Pb nucleus. Let Na and Nb be the number of a and b particles, respectively, emitted in this decay process.
Which of the following statements is (are) true? [Adv. 2018] (a) Na = 5 (b) Na = 6 (c) Nb = 2 (d) Nb = 4 40. The half-life period of a radioactive element X is same as the mean-life time of another radioactive element Y. Initially both of them have the same number of atoms. Then [1999S - 3 Marks] (a) X and Y have the same decay rate initially (b) X and Y decay at the same rate always (c) Y will decay at a faster rate than X (d) X will decay at a faster rate than Y 41. The half-life of 131I is 8 days. Given a sample of 131I at time t = 0, we can assert that [1998S - 2 Marks] (a) no nucleus will decay before t = 4 days (b) no nucleus will decay before t = 8 days (c) all nuclei will decay before t = 16 days (d) a given nucleus may decay at any time after t = 0
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P-297
Nuclei
ᆠ泐ᆠᨨ 矄 ᨨ �ᨨᆠ�矄鼀 �矄 l 㜠� 鼀 㜠� ᨨ 矄鼀矄ᆠ�泐��
泐 ҈黨黨㜠㝐├矄桌 㡰 ҈⏌ l ln㡰⏌ l 㡰 ln㡰⏌ l ҈⏌ l ᆠ㡰 l ln㡰 ҈⏌ l 㡰 l ⏌ ln㡰 ҈⏌ l 꾀 矄 泐 �ᨨᆠ�矄ᨨ盐ᆠᨨ � 鼀�矄 ��ᨨ ᨨ � 矄�泐犨 �ᆠ �矄�鼀矄 鼀�矄矄�矄 鼀� �鼀盐鼀�鼀 犨 �ᆠ �犨ᨨ盐 ᨨ矄矄�ᨨ 犨ᨨ├矄 泐犨� �矄矄ᨨ盐ᆠ�矄 ҈黨㜠㝐├矄桌 㡰 㡰 ҈ ᆠ㡰 㡰 ҈ 盐� � �� ᆠ泐 ҈黨䰘㜠㝐├矄桌 㡰 ᨨ鼀�ᆠᆠᨨ �矄ᆠ 㡰 ᆠᨨ 犨 �ᆠ �矄 泐矄 犨� � 盐ᆠ盐矄 �矄ᆠ ᆠ㡰 盐ᨨ � 盐ᆠ盐矄 ᆠ泐矄鼀�� � ᆠᨨ
㡰 ᨨ � � � �泐 ᨨ 盐ᆠ盐矄 �矄 ᆠᨨ � ᨨ ᆠᨨ
7
�
Match the Following
㝐ᆠ 盐ᆠᨨᆠ矄矄矄�� � ᆠᨨ盐鼀 ᔼ犨� ᨨ�ᨨ�ᨨ 矄㡰� ᆠᨨ盐鼀 ᔼᔼ 꾀 ҈�桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 盐ᆠ 盐矄�ᨨ
㡰 꾀矄ᨨ�ᨨ ᨨ 鼀 㡰 �矄矄�ᨨ � 盐ᆠ ᆠᨨ ᔴ㡰 b㜠 ᆠ泐
㡰 쟼 �泐ᨨ 盐ᆠ�ᨨ � 矄矄 � 泐 ᨨ� ᆠᨨ 矄�ᨨ
ᨨ �盐鼀 㡰 g㜠泐鼀�矄矄�ᨨ
矄㡰 泐犨 㡰 盐� ᨨ鼀�矄矄�ᨨ
㝐ᆠ �矄ᔼᨨ 盐ᆠᨨᆠ矄矄矄犨� �矄ᔼᔼᆠᨨ � � � 盐ᆠ盐矄 ᨨ ᨨ ᨨ 盐ᆠ矄ᨨ ᆠ ᨨᆠ矄矄 矄ᆠ ᆠᨨᆠ 矄犨盐矄� � ᆠᨨ 矄��
ᨨ犨 �矄矄 꾀 ҈㜠ᔼᔼ桌 �矄ᔼ �矄ᔼᔼ
䰘
⼄㡰
�
盐ᨨ 矄泐 黨 ᔴᨨ 盐ᆠ盐矄 䰘
꾀 ᆠ泐
҈
턼 b졔 ᆠ泐
�矄矄�ᨨ
҈� ҈� 졠 ® 䰘 졠 + ҈ 黨 ® 黨 + ҈� ҈ � ® + 黨 ҈ 黨 盐 ® �䰘 +
和 ᨨᨨ 鼀�矄矄�ᨨ
ᔴᨨ 矄 턼 和 㡰 ҈ 㡰 ҈ ᆠ㡰 ҈ 㡰 ҈ � ᨨ犨ᆠ� 鼀ᆠ � �泐⼄盐矄�ᨨ 矄犨 犨ᨨᆠᨨ盐鼀 矄 ᆠ � ��鼀矄㡰�� ᔼ鼀鼀 �泐 ᆠᨨ盐鼀 矄 鼀ᆠ � ��� �矄�� 犨 ᆠ �鼀 ᨨ ᔴᨨ盐鼀 ᔼ 矄 ᨨ鼀ᆠ 犨� �鼀矄 ᨨ ᔴᨨ盐鼀 ᔼᔼ泐 ᆠ�ᆠ� � ᆠᨨᆠ鼀ᆠ 矄㡰ᨨ �鼀ᨨ ᆠᨨ盐鼀 ᔼᆠ 鼀ᆠ 犨� 鼀ᨨ ᨨ �鼀ᨨ ᆠᨨ盐鼀 ᔼᔼ꾀 �鼀矄ᨨ ᆠᨨ盐鼀 ᔼᔼ鼀盐矄鼀ᆠ 㝐ᆠ ᨨᨨ犨� � 㜠㝐桌
ᔴᨨ盐鼀 ᔼ 꾀㡰 盐ᆠ 盐矄�ᨨ
㡰 盐ᆠ �矄矄�ᨨ
ᔴ㡰 b㜠 ᆠ泐 㡰 쟼 ᨨ 鼀�ᆠ 盐ᆠ ᆠ�ᨨ
8
ᔴᨨ盐鼀 ᔼᔼ 㡰 ᔴᨨ 矄矄ᨨ鼀鼀� ᨨ �泐 ⼄㡰 泐ᨨ矄矄� ᨨ
盐ᆠ�犨� ᨨ犨ᨨ鼀�ᆠ
盐鼀 㡰 泐ᨨ矄矄� ᨨ
盐ᆠ�犨� �� ᨨ鼀�ᆠ
盐鼀 矄㡰 쟼矄矄 �泐ᨨᆠ 矄泐 犨├ 盐ᆠ ᨨᆠ矄
Comprehension/Passage Based Questions
矄矄�㜠҈
鼀矄矄ᨨ 盐ᆠ盐矄 꾀 �矄矄矄 矄盐鼀ᨨ 鼀矄矄矄ᨨ 꾀㜠㡰 盐鼀ᨨ 盐ᨨ 矄 盐鼀ᨨ ᨨᨨ 矄� 盐ᆠ盐矄
�泐⼄盐� ᨨ ᆠᨨ矄ᨨ � �鼀矄矄 � ᆠ�矄 ├ ᨨ犨 矄 � � � �泐ᨨ 盐ᆠ盐矄꾀 泐 盐ᆠ盐矄 ᨨ 鼀矄矄㝐ᆠ ├� ᨨ犨ᨨ�� 盐ᆠ�ᨨ 鼀矄矄矄鼀҈ 鼀 ᨨ 泐� 鼀҈졔鼀㡰㝐꾀矄ᨨ犨ᨨ�� 盐ᆠ�ᨨ 鼀矄矄矄鼀 鼀 ᆠ 盐 �ᨨᆠᨨ鼀 盐矄�ᨨ ᨨ鼀 泐 盐ᆠ盐矄ᨨ 鼀矄矄 㝐Ἐᨨ 泐� 鼀졔鼀㡰>㝐Ἐ 鼀矄矄矄ᨨ 矄ᨨ鼀 盐ᨨ鼀矄 �� � ᨨ犨 ҈ ҈
҈䰘�盐
҈
҈҈盐
҈
҈�盐
盐
�
҈�҈盐
䰘 �
䰘҈盐
䰘
黨黨��盐
和
҈黨҈䰘黨盐
҈�
҈�҈黨҈黨盐
�黨䰘��盐
黨 �
黨盐
҈ ᨨ
黨黨䰘盐
҈盐㦨黨㝐⏌ᆠ㡰 꾀 ҈桌 ├� �ᆠ �泐 � ├㡰ᨨ �ᆠ犨 ҈
盐ᆠ盐矄 ᨨ 矄盐 �ᨨ矄 ᆠ泐�矄 㡰 �҈黨 㡰 � ᆠ㡰 �䰘䰘 㡰 �҈ 黨 ᆠᨨᆠ矄鼀 �矄
㡰 盐ᆠ盐矄 � ᆠ 鼀� �ᆠ ҈ 㡰 盐ᆠ盐矄 ᨨ ᆠ 鼀�ᨨᨨ
ᆠ㡰 盐ᨨ �ᆠᆠ 盐 �ᨨᆠᨨ鼀 盐矄�ᨨ
㡰 盐ᆠ� 䰘 和 ᆠ 盐 �ᨨᆠᨨ鼀 盐矄�ᨨ
10 Subjective Problems � 꾀 �ᨨᆠ� 矄鼀 ᨨ ᆠ泐矄 ᨨ ᨨ盐� ᨨᆠ矄矄 ᨨ犨 �ᆠ 㜠� �矄�맔҈黨泐矄 � �ᨨᨨ 盐鼀 ᨨ 盐ᆠ� ᨨ ᨨ �鼀 ᨨ ҈�맔҈黨泐矄� 㡰҈⏌㦨҈ 㜠㝐├矄桌 �҈ 꾀 �ᨨᆠ�矄鼀鼀�矄nb㜠�ᆠ矄� 矄ᆠᔼ 矄ᆠ�鼀�矄䰘�nb㜠�ᆠ犨 �矄 鼀 � ᨨ 矄鼀䁸 㜠㝐├矄桌
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EBD_8341 P-298
�
�
�
Physics
꾀 �ᨨᆠ� 盐ᆠ盐矄 X ᆠ泐矄 ᨨ 盐ᆠ盐矄 Y 犨� ᆠ泐ᆠᨨ 矄 lx㦨҈矄㜠҈Y 盐 ᆠ泐矄ᨨ矄
盐ᆠ盐矄Z犨� ᆠ泐ᆠᨨ 矄 lY㦨҈⏌矄㜠҈ᔼ ��泐 ᨨ 泐X 盐ᆠ� � 盐鼀�矄Nᨨ㦨҈和 盐 ⼄盐�ᨨ 矄 ᨨ ᨨ盐�ᨨ 矄ᨨ XY Z
ᨨ盐�ᨨ ᨨ Y 盐ᆠ盐矄矄 盐 ᆠ�ᨨ ᨨ �鼀�矄��
泐NY t㡰㦨 NᨨlX⏌ lX㜠lY㡰㡰{ 㜠lYt㡰㜠 㜠lxt㡰晤 � �鼀 犨 �ᆠ NY �矄 鼀 �鼀盐鼀 鼀� ᨨ盐�ᨨ 矄X Z � 矄 ҈㜠�㝐├矄桌 盐ᆠ�ᨨ �ᨨᆠ�鼀 A� �ᨨ 盐ᆠ ᆠᨨ 矄 a 鼀 矄 ᆠ泐ᆠᨨ 矄 l꾀�鼀 t㦨 N 盐ᆠ�ᨨ 鼀 ҈黨黨㜠㝐├矄桌 㡰 ᔴᆠ盐 盐鼀Nᨨ 盐ᆠ�ᨨ A�鼀t 㡰 ᔼ a㦨Nlᆠᆠ盐 盐鼀ᨨ 盐ᆠ�ᨨ A ᨨ 㜠� ᨨ A 矄ᨨ �鼀�� �盐ᨨ N矄 t ®¥ 꾀�� � 矄 �奠盐 ᆠ泐 �ᨨ㜠ᆠ�
盐ᆠ� � 矄鼀 꾀 ҈ 矄ᆠᨨ 矄 盐鼀 ᨨ 盐 ᆠ泐 盐ᆠ� 盐ᆠ矄ᨨ҈�奠ᔴᆠ盐 �㡰鼀 㜠 � ᨨ 盐ᆠ� ��㡰 �鼀� 犨 �ᆠ 盐鼀ᨨ 盐 ᆠ泐 盐ᆠ� 犨� 盐 盐ᆠ ᨨ �奠 ᨨ 盐ᆠ 盐鼀 ҈黨黨㜠㝐├矄桌
�� 꾀矄鼀⼄盐 �泐ᨨ 矄ᨨ盐�ᨨ ᆠᨨ � � � �ᨨ 盐ᆠ� � 㦨҈� ᨨ盐㡰ᨨ ᆠ��泐҈鼀�ᆠᨨᆠ盐��矄� ᆠ � ᨨ ᨨᨨ ᨨ 矄ᨨ 꾀矄鼀ᨨ ᨨᨨ ᨨ ᨨ盐鼀҈ᆠ鼀 ├ � ᨨ盐矄 ᨨ犨 ᆠ��泐ᨨ 黨 �矄� ��ᨨ 矄 鼀� 盐鼀� ᨨᨨ盐鼀ᨨ ᨨᨨ � ᨨ 泐ᨨ 矄ᨨ 꾀矄矄盐鼀 �ᨨᆠ�矄ᨨ盐�ᨨ 鼀� 矄 盐 � ᨨ鼀泐� ᨨᨨ ᨨ 矄ᨨ ҈ᆠ盐�㦨䰘맔҈҈ �矄� ��ᨨ 矄矄ᆠᨨ 㡰 ҈黨黨㜠㝐├矄桌 � 꾀 盐ᆠ盐矄 X � ��泐 矄 盐 �ᨨ矄 ᆠ泐 ᆠᆠᨨ � �ᨨ ⼄盐�ᨨ A 黨 X
® Z Y +a
҈黨黨҈㜠졔㝐├矄桌
㡰 � 盐矄ᨨ A Z� ᨨᨨᆠ矄矄 㡰 �ᆠᨨ 盐ᆠ � ᨨᨨᆠ矄矄�矄 ᨨ盐 ᨨ鼀ᨨ� ᆠ�ᆠ盐ᆠ├ᨨ �盐矄҈҈鼀� 盐 � ᨨ鼀鼀� �ᆠ � ᨨ 矄 � �泐 ᔼ
㝐㡰矄 盐� � ᨨᆠ矄矄 � � � �泐ᨨ 盐ᆠ盐矄X
( ) ҈
� m Y㡰㦨u錐m n 㦨҈黨u
( )
( )
m 㦨u錐m ҈҈ 㦨҈u
Topic-4 : Miscellaneous (Mixed Concepts) Problems 1 ҈
꾀 ᆠᆠ� � 盐ᆠᨨᨨ泐矄盐 � ᨨ矄��ᨨ
ᨨ ᆠ� 鼀ᨨ盐 ᨨ �ᨨᆠ�鼀�ᨨ 㜠� ҈ 泐矄� 矄� ᨨᨨ泐 矄矄 ��ᨨ
犨矄�鼀矄鼀ᨨ 鼀�矄矄�⼄盐� ᨨ 矄 ᨨ�ᨨ ᨨ ᨨᨨ泐␔ �矄 鼀� �鼀盐鼀
盐鼀 ᨨ 泐矄 犨 �ᆠ ᨨᨨ泐ᆠ ᆠᨨ 矄� 矄 ᨨ盐矄䁸 꾀 ҈桌 㡰 㡰 黨 ᆠ㡰 ҈ 㡰 ҈
ᆠᨨ矄�ᆠ �泐ᨨ ᨨᨨ 矄盐 � ᨨ鼀泐 �矄�盐 ᨨ盐� ᨨ盐矄 �ᆠ 盐ᆠ盐矄ᨨ �盐矄 �矄�� 泐 쟼㦨
�矄 盐泐 盐 ᨨ ᆠᨨ矄�ᆠ �泐 �盐矄 ᨨ � ᨨ 盐ᆠ��矄 ҈ 鼀㦨҈҈�鼀㡰 꾀 ҈桌 㡰 � 鼀 㡰 鼀 ᆠ㡰 鼀 㡰 鼀 盐ᆠ盐矄 矄ᨨ� � �泐 ᆠ泐矄� ᨨ犨ᨨ a㜠�ᆠ矄 盐 ├ ᨨ犨 盐ᆠ盐矄 盐 ├ ᨨ犨
盐ᆠ盐矄�矄 ҈黨黨黨和㜠㝐├矄桌 㡰 �ᨨ� 㡰 ᆠᨨ ᆠ㡰 ᨨᨨ
㡰 ᨨ 泐�
�矄ᨨᨨ ҈ � � � 鼀矄矄 ҈҈ 盐 盐 �ᨨ矄
MCQs with One Correct Answer
҈ b ᆠ泐ᨨ ҈ ᔴ ᔴ 矄 ᆠ� 矄ᨨ 盐ᆠ盐矄
( ҈ ᔴã) ҈㝐ᨨ�矄�ᨨ盐 矄ᔼ ҈� 쟼 ᆠ泐矄
㜠 ҈㡰 � πε
ᨨ ҈ ᔴ ã 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ b�ᆠ�
҈�
鼀矄盐 鼀矄矄矄ᨨ 盐ᨨ ҈҈ ҈� 䰘 졠
҈�盐҈䰘�盐҈�҈黨盐 ҈��盐 ҈� 矄ᆠ�泐� ��ᨨ ᨨ ҈� 䰘 졠
盐ᆠ�矄鼀҈盐㦨黨҈�㝐⏌ᆠ ᆠ�矄 矄 ᨨ �� 㡰 ⏌ pe㡰㦨҈㝐 鼀꾀矄矄盐鼀� � ҈� � ᆠ犨 � � � ��矄ᨨ ҈� 䰘 졠
�
盐 �矄ᨨ 㝐�矄 ҈盐㦨黨҈�㝐⏌ᆠ犨 ᆠ�矄 矄 ᨨ �� � ᆠ盐盐鼀㡰 꾀 ҈桌 꾀 盐ᆠ ᨨ犨 矄盐泐� �ᆠ�ᆠ ᨨ犨 ᨨ ��盐矄矄 �ᨨᆠ�鼀�ᨨ � T泐矄矄 盐 鼀ᨨ盐 ᨨ 盐 ��
� ��矄矄盐ᆠ ᨨ ᨨ犨 ⼄盐�鼀 ᨨ ���矄 ҈�奠 ᨨ ᆠ�ᆠᨨ犨� ᨨ鼀 �鼀ᔼ �矄ᨨ鼀 ᨨᨨ犨 矄ᨨ �� ᨨ鼀 �鼀盐鼀�ᨨ ᨨ nT泐矄 盐ᨨ n�矄 꾀 ҈�桌
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P-299
Nuclei
Fill in the Blanks
4
꾀ᨨ鼀矄 � � 矄鼀盐 � ᆠ �矄ᨨᨨ矄 ҈黨㜠㝐├矄桌
MCQs with One or More than One Correct Answer
6 䰘
҈ 꾀 �矄矄�ᨨ ᆠ�ᨨ �矄�� 泐 黨 ® � +
黨 和
+ x + y
犨 x y犨ᨨ�ᆠ矄ᔴᨨ 矄� � � 黨 ᨨ 矄 ├� �ᆠ ��矄ᨨ ᨨ 盐ᆠ矄 ᨨ 泐 KK和K 㝐㡰 ⧤泐 㝐㡰矄ᆠ�泐 黨 ҈ � � � ��矄 盐ᆠᨨ ᨨ 黨 � 和
䰘�㝐�㝐 �㝐矄ᆠ�泐ᔴᨨ 矄� � � � ᆠᨨ 矄�ᨨ 犨矄 ᆠᨨᆠᨨ�ᨨ 矄㡰�矄 㡰 꾀 ҈�桌 㡰 x㦨ny㦨n⧤和㦨҈黨㝐⧤㦨㝐 㡰 x㦨py㦨⧤和㦨҈黨㝐⧤㦨㝐 ᆠ㡰 x 㦨py㦨n⧤和㦨҈黨㝐⧤㦨㝐 㡰 x㦨ny㦨n⧤和㦨㝐⧤㦨҈黨㝐 꾀 矄 � ��泐 矄 ҈ 盐ᨨ 矄 ᔼ ᨨ 盐ᆠ矄 �泐 � ᨨᆠ矄矄矄 ҈ +҈ ®҈ + p ҈
+҈ ® + n ᔼ �ᨨ犨 �
泐 矄�矄҈ ҈W 盐ᨨ 矄盐泐ᨨ 矄�矄 盐矄 � �鼀ᨨ ᨨ ᨨ ҈黨黨㜠㝐├矄桌 㡰 ҈s
㡰 ҈s
ᆠ㡰 ҈҈ s 㡰 ҈҈ s
鼀矄矄矄ᨨ 盐ᆠ�矄 ᨨᨨ犨矄 M 㡰 = ҈ 鼀盐錐
M p㡰㦨҈䰘鼀盐錐M n㡰㦨҈鼀盐錐 M 㡰 = ҈鼀盐 7 黨
Match the Following
ᨨ盐 泐矄�ᆠ⼄盐 ��矄�矄 � ᔴᨨ盐鼀 ᔼ �盐矄 �矄 � ᔴᨨ盐鼀 ᔼᔼ� ᨨ鼀ᨨ ҈黨䰘㜠㝐├矄桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 㡰 鼀 �泐ᨨ 㡰 �鼀ᨨᆠ盐矄ᨨᨨ鼀 鼀 㡰 � � � �泐ᨨ 㡰 泐 盐ᆠ�
盐ᆠᨨ
ᆠ㡰 㜠泐 ᨨᨨ �泐 �㡰 ҈├ 㡰 ᨨᨨ �泐ᨨ 㡰 䰘㝐 �矄���
ᆠᨨᆠ鼀ᆠ � �ᨨ ᔴᨨ盐鼀 矄ᔼ ᔼᔼ�矄�� 泐 㡰 a e, b h, c g, d f 㡰 a e, b g, c f, d h ᆠ㡰 a f, b e, c g, d h 㡰 a f, b h, c e, d g
҈ ᔴᨨ盐鼀 㜠ᔼᔼ��矄ᆠ� 矄泐矄鼀矄盐 �ᨨ� �ᨨᆠ矄矄 ᔴᨨ盐鼀 㜠ᔼ矄盐��矄矄ᆠ �矄� 矄ᨨ鼀ᨨ 鼀矄 ᨨ 矄泐矄鼀㝐ᆠ 矄鼀 矄� ᔴᨨ盐鼀 㜠ᔼ ᨨ ᨨ�ᨨᆠ矄矄 矄㡰 ᨨ鼀ᔴᨨ盐鼀 㜠ᔼᔼ 黨桌 ᔴᨨ盐鼀 㜠ᔼ ᔴᨨ盐鼀 㜠ᔼᔼ 꾀㡰 �泐ᨨ 㡰 System 꾀ᆠᆠ�ᨨ� ��泐 矄泐矄鼀�矄� ᆠ矄 盐 ᆠ � Prᨨcess ᔼ�矄ᆠᨨ
ᆠ ᨨ泐 㡰 㝐ᆠ �ᆠ �泐 ⼄㡰 System 꾀�矄�
�矄ᨨ� ᨨ ��ᆠᆠᨨ � � 矄泐矄鼀犨 �ᆠ �矄 犨� ��ᆠ�矄ᨨ
ᆠᨨ � ᨨ �泐 Prᨨcess �矄�矄 ᨨ ᨨ鼀鼀ᨨ�ᨨ ᨨ ᆠᨨ鼀矄矄 泐盐矄 � � �矄矄 �矄ᨨ
ᔴ㡰 ᔼ �泐 㡰 System꾀�矄� ��� ᨨ 矄泐矄鼀�矄 ᆠᨨ � ᆠᨨ � ᨨ�矄 Prᨨcess �矄�矄 鼀ᆠ �ᆠ �泐 ᆠᨨᨨ 盐ᨨᆠᨨ 鼀ᨨ矄 矄盐ᨨ盐 � �� 㡰 㝐矄矄ᨨ 矄泐矄鼀 矄㡰 System꾀 泐 盐ᆠ盐矄 �矄 ᆠ矄 � ��泐矄 Prᨨcess 盐ᆠ盐矄 �矄矄�ᨨ 矄� ᨨ犨ᨨ �鼀 矄 ᨨ 泐⼄盐鼀矄矄矄 矄ᨨ鼀 盐ᨨ 矄 鼀� 㡰 System 꾀矄�矄�犨�ᨨᨨ Prᨨcess ᨨᨨ�矄 ᆠ � �鼀泐� � 鼀� �ᆠ � �ᆠ盐ᨨ�矄 ҈҈ ᔼ ᨨᨨ犨� �ᆠᨨ盐鼀 ᔼ�矄矄矄ᨨ鼀 泐矄�ᆠ⼄盐 ��矄 ᆠᨨ盐鼀 ᔼᔼ ��矄 ᨨ �鼀 �泐 盐矄 矄矄ᨨᆠ� 犨� 矄ᨨ鼀ᨨ 鼀ᔴ ᨨᨨ矄 ᨨ�盐 ᨨ �泐 ᨨ鼀ᆠᨨ盐鼀 ᔼᔼ ᨨᆠ ᨨ 泐矄�ᆠ⼄盐 ��矄 � ᆠᨨ盐鼀 ᔼ 犨� ᆠᨨ矄ᨨ � �⼄ᆠ �� 矄 盐鼀 꾀㡰 㡰 ᔴ㡰 㡰ᆠᨨ 泐矄�ᆠ ⼄盐 �泐� 矄犨ᨨᨨ├ᔼ 泐ᨨ盐 矄犨 矄⼄盐 ᆠ ᨨ ᆠᨨ盐鼀 ᔼ矄 ᨨ盐 鼀� � ҈黨黨䰘㜠㝐├矄桌 ᔴᨨ盐鼀 ᔼ ᔴᨨ盐鼀 ᔼᔼ 꾀㡰 쟼 �泐ᨨ 鼀 㡰 �
盐ᨨ 矄 㡰 쟼 �泐ᨨ 㜠泐矄 ⼄㡰 � ᔴ㡰 � � � �泐 㡰
盐ᆠᨨ
㡰 ᨨᨨᆠ�ᆠ 矄㡰 矄 ᨨ ᨨ 鼀 㡰 ҈├ 盐㡰 㝐
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EBD_8341 P-300
Physics
8
Comprehension/Passage Based Questions
矄矄�㜠҈
b 㜠 ᆠ泐ᨨᆠ矄矄 �矄ᆠᨨ ᨨ盐 ҈黨�矄矄�ᆠ泐 ᆠ泐ᨨ 盐ᨨ n㡰ᔼ ᨨᨨ泐ᨨᨨ p㡰
ᆠᨨ e㡰ᨨ矄 矄 ᆠ泐ᨨ 盐ᆠ矄ᨨ 盐ᨨ
ᨨ ᆠᨨ 矄� � � ᆠ泐ᨨ 盐ᨨ 矄 犨ᨨ㜠ᨨ 泐 ᆠ泐ᨨᆠ矄矄 � 犨矄 �ᆠ ᨨ�ᆠ泐 ├� �ᆠ �泐ᨨ ᆠᨨ 矄 ᨨ盐 ᆠᨨ 矄 盐 �鼀 泐 �犨矄ᨨ矄 ᆠᨨ ├� �ᆠ �泐 矄ᆠᨨ � 盐ᨨ盐矄 矄ᆠ盐鼀 ᔴᨨ 矄� � � 㜠ᨨ 泐 ᆠ泐 ᨨᆠ矄矄 � n ® p + e - + n e ᨨ盐 ҈黨 盐� � ᨨ矄
ᆠᨨ �泐矄ᆠ盐鼀꾀矄矄盐鼀� � �㜠 盐� ᨨ ne 㡰 ᨨ 鼀矄矄矄矄 ᨨ矄矄矄矄� � ���� �泐 盐ᨨ
ᨨ矄鼀ᨨ鼀 盐鼀 �泐ᆠᨨ 矄�ᨨ � ᆠ�矄 � ᨨ鼀 �矄ᆠᆠ盐�ᨨ 鼀 �鼀盐鼀├� �ᆠ �泐ᨨ ᆠᨨ �矄맔҈ ├� �ᆠ �泐ᆠ� 泐 ᨨᨨ �矄ᨨ 泐 ᆠᨨ� �泐 ҈ ᔼ �㜠 盐� ᨨ 鼀矄矄ᨨ ⏌ᆠ 犨 ᆠ�矄 矄 ᨨ �� 㡰� 矄 ᨨ ᨨ鼀矄矄犨 矄 ᨨ盐 �ᨨ ├� �ᆠ �泐Kᨨ ᆠᨨ 䁸 ҈桌 6
㡰 0 £ K £ 0.8 ´ ҈0 eV 6 㡰 .0 eV £ K £ 0.8 ´ ҈0 eV 6 ᆠ㡰 .0 eV £ K < 0.8 ´ ҈0 eV 6
㡰 0 £ K < 0.8 ´ ҈0 eV ҈
␔ �矄 鼀 �鼀盐鼀 �泐ᨨ �㜠 盐� ᨨ䁸 ҈桌 㡰 ᨨ 㡰
㝐盐ᆠ 矄矄 맔҈
ᆠ㡰 泐맔҈ 㡰 㝐盐ᆠ � 맔҈ 矄矄�㜠 和ᆠ� �矄矄犨ᨨ├� � ᨨ ᨨ 盐ᆠ 盐矄�ᨨ ᆠᨨ 盐ᆠ�ᨨ 泐 泐 ᨨ� ҈ H ├ ᨨ犨 矄 盐ᨨ ᨨ 泐ᆠ ᨨ盐� ᨨ 矄ᆠ � ᨨ 盐矄�ᨨ ᆠᨨ 㜠ᆠ�ᨨ �矄 ҈ + ҈ ® + n + �泐ᔼ ᆠᨨᨨ 盐矄�ᨨ ᆠᨨ �矄ᨨ 泐 泐 ᨨ� �矄 盐泐�ᨨ � � ᨨ 盐ᨨ 盐ᆠ� ᆠᨨ 矄 �矄ᆠᨨᆠ�ᨨ ᨨ ҈ H 盐ᆠ� ᆠᨨ 矄�矄├ ᨨ犨 矄矄鼀 盐ᆠ�鼀ᨨ ᨨ鼀泐� ᆠᨨᆠᨨ ᨨᆠᆠ矄�ᨨ 泐ᆠᨨ鼀ᆠᨨ矄 ᨨ盐� ᨨ 盐ᆠ 盐矄�ᨨ ᨨ├ᆠ矄盐泐 鼀盐矄� ᆠᨨ ᆠᨨᨨᨨ �� ᨨ鼀�犨ᆠ 盐矄 ᨨᆠᨨ � 矄鼀和ᆠ�ᆠ �⼄盐矄盐矄 犨 �ᆠ ᆠᨨ � 矄鼀 ᨨ�鼀t0 ᨨ �ᆠ矄 泐犨泐 ᨨ鼀 ᆠᨨᔼ n �矄 矄�泐 盐鼀⏌ᨨ盐鼀㡰ᨨ 盐ᨨ 矄 ᨨ 盐ᆠnt0 �矄ᆠ
犨矄ᨨ 盐鼀 ᔼ ᨨ ᨨ ᆠ�� ᆠᨨ �矄 鼀 矄盐ᆠᆠ矄矄 盐� 犨矄ᨨ 盐鼀�矄� �맔҈ ҈矄⏌ᆠ鼀 ᔼ鼀泐 盐ᨨ盐矄 ᨨᨨ犨� �ᨨ 鼀
ᆠᨨ 矄 k = 맔҈�⏌⧤錐
e 㦨҈맔҈黨鼀 pe
黨桌
҈ ᔼ ᆠᨨ ᨨ 盐ᆠ 盐矄�ᨨ ᆠᨨ �矄 ᆠᨨ鼀矄 矄鼀ᆠ盐矄ᨨ 㡰 矄ᨨ � 盐ᆠ ᨨᆠᆠ� �犨 盐ᨨ 矄 㡰 ᆠᨨ盐ᨨ鼀 ᨨᆠᆠ� �犨 盐ᨨ 矄 ᆠ㡰 ᆠᨨ盐ᨨ鼀 ᨨᆠᆠ� �犨 盐ᨨ 㜠ᆠᨨ �矄 㡰 �� 鼀盐鼀� � � 矄� ᆠᨨᆠᨨ ҈� 꾀矄矄盐鼀 犨ᨨ 盐ᨨ 盐ᆠ� � ᆠᨨᨨ 盐矄�ᨨ
ᆠᨨ鼀盐T鼀ᨨ� �ᨨ犨 矄ᆠ ᨨ ᆠ 犨� ├� �ᆠ �泐 ҈� kT, 犨 矄�ᨨ
犨 鼀�矄� ᨨ盐� ᨨ �ᆠᆠᨨ盐ᨨ鼀ᨨ � �泐꾀矄ᨨ �ᆠ 泐� ᆠ�ᨨ ᨨ鼀ᨨ �ᆠ矄 � ᆠᨨ 鼀� �鼀盐鼀鼀盐T⼄盐� ᨨ 鼀 ᨨᆠ 矄�ᨨ ᨨ 맔҈҈�鼀�矄� � 㡰 ҈맔҈黨⧤T맔҈黨 ⧤ 㡰 맔҈黨⧤T맔҈黨 ⧤ ᆠ㡰 맔҈黨⧤T맔҈黨 ⧤ 㡰 맔҈黨⧤T�맔҈黨 ⧤ ҈ 矄盐矄ᨨ ᆠᆠ盐�ᨨ 矄 ᨨ ᨨ盐 � 矄�� 矄ᨨ 盐矄�ᨨ
ᆠᨨ盐矄� �㜠ᆠ�ᨨ �� ᨨ犨␔ �ᆠ ᨨ 矄 �矄鼀ᨨ矄ᨨ鼀�矄� �矄 ᨨ 犨矄ᨨ ᆠ��ᨨ 䁸 㡰 盐ᨨ 矄�泐㦨맔҈҈ᆠ鼀 ᆠᨨ � 鼀 �鼀㦨�맔҈矄 㡰 盐ᨨ 矄�泐㦨맔҈҈ᆠ鼀 ᆠᨨ � 鼀 �鼀㦨黨맔҈҈矄 ᆠ㡰 盐ᨨ 矄�泐㦨맔҈ᆠ鼀 ᆠᨨ � 鼀 �鼀㦨҈맔҈҈҈矄 㡰 盐ᨨ 矄�泐㦨҈맔҈ᆠ鼀 ᆠᨨ � 鼀 �鼀㦨맔҈҈矄 10 Subjective Problems ҈䰘 �� 泐 ��ᆠᆠᨨ 矄ᨨ鼀 ᨨ �ᨨ 鼀 ᆠᨨ � � � 盐ᨨ 矄 �ᨨᨨ ��ᨨ
盐ᆠ盐矄ᨨ ᨨ �盐鼀 盐ᆠ盐矄�矄 ҈㡰҈⏌ � 㡰ᨨ鼀�ᆠ
盐鼀ᨨ 盐ᆠ盐矄 㡰 ⼄盐 ᆠ泐ᨨ Ka� ᨨ 㜠泐ᨨ 盐ᆠ R㦨҈҈맔҈䰘鼀҈ c㦨맔҈鼀⏌矄㡰 �㜠㝐├矄桌 ҈ 꾀 盐ᆠ盐矄矄盐 �ᨨ矄 ᆠ泐鼀�� � a�ᆠ ᨨ 㜠ᨨ��犨 � l㦨�䰘´҈㜠҈�鼀ᔼ 鼀矄矄ᨨ 盐� 盐ᆠ盐矄�矄҈鼀盐 ᨨ a �ᆠ矄�矄鼀盐 鼀� ᨨ├� �ᆠ �泐 � � 矄 ᆠᨨ� 鼀矄矄ᨨ
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P-301
Nuclei ҈
鼀 ᔴ盐�盐鼀 248 96 ᔴ鼀 矄 鼀 � ᨨ ҈
҈黨
矄ᆠᨨ 矄 ᔼ矄 �鼀泐 ᆠ泐 鼀ᨨ 矄 矄ᨨ ᨨ盐矄 �矄矄�ᨨ a㜠 ᆠ泐 ᨨ鼀犨� ᨨ��泐ᨨ 奠 犨� ᨨ��泐ᨨ 黨奠쟼ᆠ �矄矄�ᨨ 矄矄 㝐ᨨ �泐 鼀矄矄矄� ᨨ � a㜠 ᆠ泐矄
和
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꾀鼀ᨨ ᨨᆠ ᨨ鼀�ᆠᨨ� 矄ᨨ盐ᆠ �� �犨 � 处犨� ᨨ犨犨 盐Aᨨ �鼀҈鼀 � 矄ᆠ SC ᆠ 矄 矄 ᨨ犨 � �� 꾀 ᨨᨨ鼀�矄矄� ᆠᨨDᨨ 矄盐 ᆠ�ᆠ鼀�矄ᆠ ᆠ ᨨ 矄ᆠ �ᆠ� ᆠ泐ᨨ ᆠᨨ ᨨ ᨨᨨᆠᨨ � �ᨨ � ᆠ� ᨨᨨ �矄 黨 ҈黨黨҈㜠졔졔㝐├矄桌
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㡰 ᔴᆠ盐 ᨨᨨ 盐 ᆠ ᨨ 矄ᆠ
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Answer Key Topic-1 : Composition and Size of the Nuclei ҈
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黨 ᆠ㡰 ҈ ᆠ㡰 ҈䰘 㡰 ҈ 㡰
Topic-3 : Radioactivity ҈ ҈҈ ҈ ҈ 黨
ᆠ㡰 㡰 㡰 �㡰 ᆠ㡰 ᆠ㡰
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Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈ 黨 ҈�
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EBD_8341 P-302
Physics
Semiconductor Electronics: Materials, Devices and Simple Circuits
28
Topic-1 : Solid Semiconductors and P-N Junction Diode 1 ҈
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P-303
Semiconductor Electronics : Materials, Devices and Simple Circuits
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EBD_8341 P-304
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Physics
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P-305
Semiconductor Electronics : Materials, Devices and Simple Circuits
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黨 ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐ᆠᨨ � 矄犨ᨨ �ᨨ 矄ᆠ 犨� ᨨ犨 矄�矄 ᆠᨨ �ᨨ 鼀矄 犨� � � � ᆠ├犨 矄�矄 ᆠᔼ 泐ᨨ��矄 ᆠ盐 ᨨ盐� ҈ᨨ 鼀矄�矄 ᆠ � 鼀矄㡰�矄 ҈黨黨䰘㜠҈㝐├桌
50 W
ᆠ盐 矄ᔼᔼ ᔼ矄ᆠ�泐 㡰 ҈�鼀꾀�鼀꾀҈鼀꾀 㡰 ҈�鼀꾀䰘�鼀꾀䰘�鼀꾀 ᆠ㡰 ҈�鼀꾀�鼀꾀䰘�鼀꾀 㡰 ҈�鼀꾀䰘�鼀꾀�鼀꾀
ᔼ㜠ᆠ ᆠ�矄�ᆠᨨ 쟼�矄 㝐� ҈桌
㡰
ᔼ
㡰
졠
Ӹ
ᔼ
Ӹ 㡰
ᆠ㡰 졠
䰘
졔
t
1
ᔼ
V A
㡰 �鼀꾀 㡰 ҈鼀꾀 ᆠ㡰 鼀꾀 㡰 �鼀꾀 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 矄�矄 ᨨ盐䁸 ҈黨黨䰘㜠҈㝐├桌 㡰 矄�矄 ᆠᨨ � � 矄�ᆠ矄鼀�ᆠᨨ 盐ᆠᨨ矄 ᆠ矄 犨� � ᆠ矄ᨨ 鼀盐 㡰 ᨨ� �盐和�犨� � �鼀盐��矄��p㜠泐 矄鼀�ᆠᨨ 盐ᆠᨨ矄 ᆠ㡰 鼀ᨨ�泐ᆠ�矄� n㜠泐矄鼀�ᆠᨨ 盐ᆠᨨ矄 ᨨ矄 㡰 꾀p㜠n盐 ᆠ�ᨨ ᆠ ᆠ矄矄鼀�ᆠᨨ 盐ᆠᨨ �ᨨ
out-put +
졠
҈
0
��盐矄 ᨨ犨矄ᆠ�ᆠ盐�� 犨 �ᆠ � �ᆠ �ᨨ 矄 盐矄 쟼ᆠ �ᨨ 矄 ᨨ犨 矄�矄 ᆠᨨ W � � � ᆠ├犨 矄�矄 ᆠ 矄�矄ᨨ矄 ҈㦨 㦨 㦨�W.泐 ᨨ��矄 ᆠ盐 ᨨ盐� �矄 㝐� 졠 � 꾀�҈桌
҈
㡰 ᨨ 㡰 ᆠ㡰 㡰 꾀 盐㜠犨ᆠ� �ᆠ�ᆠ盐�ᨨ �犨� ᨨ盐㜠盐�矄矄 ᨨ犨
� ��盐 ᆠᨨ �盐�ᨨ 矄㡰 ᨨ鼀 �ᨨ ҈�矄 㡰 ҈黨黨㜠㝐├矄桌 V
졠
100 W
6V
B
C D
0 t 㡰 ᔴ out-put 㡰 꾀ᔴ ᆠ㡰 㡰 꾀ᔴ ҈ ᨨ��泐ᨨ ᆠᨨ 矄ᨨ ᨨ盐 � ᆠᨨ 盐ᆠ�ᨨ
ᨨ � � 矄�ᆠ矄鼀�ᆠᨨ 盐ᆠᨨ � �鼀盐 ҈黨黨�和桌 㡰 � ᆠ矄矄 ᨨ �泐犨� � ᆠ矄� � � 㡰 ᆠ矄矄 ᨨ �泐犨� � ᆠ矄� � � ᆠ㡰 ᆠ矄矄犨� � ᆠ矄� �鼀盐 㡰 �矄� ᨨ 鼀盐 � 3
Numeric Answer
ᨨ �ᨨ 矄盐矄 � ᆠ�ᆠ盐�矄 ᨨ犨 矄矄盐鼀 ᨨ � ���� 矄�矄 ᆠ 犨 矄 ᨨ犨 �矄 盐�� ᨨ �� ᆠ �ᨨ �矄䰘 ᨨ � 盐 ᨨ�矄 矄 ᨨ犨 � ��盐 ᨨ� �
ᨨ矄㡰ᨨ� 꾀�矄힘힘힘힘힘힘힘힘 㝐� 黨 ᔼ桌
ᆠ�ᆠ盐�矄 ᨨ犨 ᨨ犨�矄犨ᨨ├� �矄 ᆠ�盐 ᨨ�矄ᨨ盐ᆠ␔ ҈�矄盐矄 矄� 盐 ᨨ犨 �矄矄� � 鼀␔㡰� ᆠ �ᨨ �矄錐 ᆠᨨ 矄� � �ᨨ �ᨨ 矄� �ᆠ㡰힘힘힘힘힘 㝐� 黨 ᔼᔼ桌
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EBD_8341 P-306
Physics
҈
ᔼ
6 4 �
䰘
ᔼ �矄 㜠 盐 ᆠ�ᨨ ᨨ犨ᨨ ᨨ矄�矄 ᨨ鼀 n��ᨨ ᨨ p��ᨨ ҈黨黨㜠҈㝐├桌 ᨨ �� ᆠ�ᆠ盐�矄 ᨨ犨 � ��ᨨᆠ矄 盐犨ᆠ� � ᆠ� 盐矄 ᨨ盐 ᆠᨨ
ᆠ ᆠᨨ矄矄 ᆠ ᨨ盐盐 犨ᨨ盐 ᆠᨨ矄矄 ҈黨黨҈㜠҈㝐├桌
꾀
ᔴ
�矄� �ᨨ p㜠n盐 ᆠ�ᨨ ᨨ 矄 �� 矄�矄 ᆠᨨ ᆠ盐 ᨨ犨ᆠᨨ矄矄 盐 ᆠ�ᨨ �矄� ��矄ᨨ� 泐ᆠᨨ
ᆠ� � p㜠矄� ᨨ 鼀� ᨨ 泐 ҈黨黨㜠㝐├矄 ᔼ ᨨ犨 �矄 �鼀 ᨨ p㜠n盐 ᆠ�ᨨ ᆠ� � p �矄ᆠᨨ
ᆠ ᨨ 鼀� ᨨ 泐 �ᆠ�ᨨ ᨨ ᆠ盐 �矄 ᨨ鼀ᨨ�
ᆠ� � ҈黨㜠㝐├矄桌 5
Fill in the Blanks
True / False
ᨨ �ᨨ ��ᨨ ᨨ �矄 ᨨ ᆠ盐 Ia犨� ᨨ ᨨ�Va犨ᨨ � ᆠ ᨨ 鼀盐矄T҈ T �矄矄 ᨨ犨 � ��盐 鼀盐 T �矄 � T҈ ҈黨㜠㝐├矄桌
MCQs with One or More than One Correct Answer
黨 ᔼ p㜠n盐 ᆠ�ᨨ �ᨨ ᨨᆠᨨ
ᆠ ᨨ 泐ᆠ�ᆠ盐� ҈黨黨和㜠㝐├矄桌 㡰 ᨨ ��矄 矄鼀泐犨 㡰 p㜠泐矄� �矄 �� ᨨ � n㜠 泐 矄� ᆠ㡰 �矄 ᆠ�ᆠ � 盐 ᆠ�ᨨ �ᆠ ᨨ鼀 n㜠泐矄� ᨨ p㜠泐矄� 㡰 �矄 ᆠ�ᆠ � 盐 ᆠ�ᨨ �ᆠ ᨨ鼀 p㜠泐矄� ᨨ n㜠泐矄� ᨨ矄ᆠ �ᆠ�矄�
҈黨黨㜠㝐├矄桌 㡰 � � 矄�ᆠ矄鼀�ᆠᨨ 盐ᆠᨨ矄 㡰 �ᨨ �ᆠ矄ᨨ� 矄 ᆠ㡰 p㜠泐 矄鼀�ᆠᨨ 盐ᆠᨨ矄 㡰 鼀矄 ҈ 犨ᨨ� �ᆠp㜠n盐 ᆠ�ᨨ 矄鼀泐ᆠᨨ
ᆠ � 矄�矄 犨� 泐� 犨泐矄 �� ᨨ � ᨨ矄ᆠᨨ矄矄 犨ᨨpn盐 ᆠ�ᨨ 矄⼄盐� ҈黨黨㜠㝐├矄桌 �鼀盐�泐ᨨ鼀矄犨� 犨 �ᆠ 盐矄��ᆠᨨ 矄 ᨨ盐 ᨨ ᨨ鼀├ p㜠泐矄鼀�ᆠᨨ 盐ᆠᨨ ᨨ矄ᨨ ҈黨㜠㝐├矄桌 㡰 ᨨ矄 ᨨ盐矄 㡰 ᨨᨨ
ᆠ㡰 �鼀ᨨ 泐 㡰 盐鼀� �盐鼀 和ᆠ ᆠᨨᆠ矄鼀 ᨨ鼀 ᨨᨨ犨� � 㡰 꾀 �ᨨ ᆠ 盐矄 矄ᆠ� �҈黨㜠㝐├矄桌 㡰 꾀�ᨨ ᆠ
ᨨ盐矄 矄ᆠ� � ᆠ㡰 ᆠ盐 � �ᨨ �矄犨泐矄ᨨᨨ�ᨨ ᨨ � ᨨ� 㡰 � ᨨ�ᨨ ᨨ ᔼᆠ ᆠ�矄�ᆠᨨ �ᨨ �矄盐矄 ᨨ鼀� �ᆠ�ᨨ 犨� ᨨ盐 �矄ᨨ�ᨨ
Topic-2 : Junction Transistor 1 ҈
MCQs with One Correct Answer
矄 ᆠ ᆠ�矄�ᆠᆠ盐ᨨ 矄�矄ᨨ � � � 盐 ᨨ盐盐矄�矄 ᆠ҈W ҈├W矄ᆠ�泐 �矄矄 ᨨ犨 � ��盐 ᨨ� ᨨ犨�� 矄ᆠ�泐 㝐� ҈꾀҈黨ᔼ桌
㡰 �맔҈�맔҈ 㡰 �맔҈�맔҈ � ᆠ㡰 �맔҈ �맔҈ 㡰 �맔҈�맔҈ 꾀 矄�矄ᨨᨨ矄矄ᆠᨨ鼀鼀ᨨ 鼀�鼀� � 犨� ᨨ犨�� ᨨ � 盐ᆠ�ᆠ盐�矄�矄 ᆠ�矄 ҈W ᨨ盐盐ᨨ 矄�矄 ᆠ�矄҈├W ᆠᨨ鼀鼀ᨨ
鼀�ᆠ盐 �� b�矄 㝐� ҈꾀҈黨ᔼ桌 㡰 ҈ 㡰 ᆠ㡰 맔҈ 㡰 ҈ 꾀 矄�矄ᨨ�矄盐矄 � ᆠᨨ鼀鼀ᨨ 鼀�ᆠᨨ ��盐�ᨨ
矄 鼀� �犨� ҈├!ᨨ 矄�矄 ᆠ和�� ᨨ� ᨨ ҈鼀�矄� ᆠᨨ矄矄 矄㜠鼀� �矄ᨨ 盐ᆠ矄 鼀꾀ᆠ �� ᆠᨨᆠᨨᆠ盐 ҈�꾀ᆠ ��
矄ᆠ盐 ᨨ 鼀� � � 盐矄�矄 ᆠ 㝐� 黨꾀�҈黨ᔼ桌 ᨨ��� 㡰 ├W҈� 㡰 䰘├W ᆠ㡰 䰘├W 㡰 ├W
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P-307
Semiconductor Electronics : Materials, Devices and Simple Circuits
4.
A common emitter amplifier circuit, built using an npn transistor, is shown in the figure. Its dc current gain is 250, RC = 1 k &! and VCC = 10V. What is the minimum base current for VCE to reach saturation ? [Main 8 Apr. 2019 II]
(a)
1 -1 (W ) ri 0.01
(b)
1 -1 (W ) ri 0.01
0
0
(c)
0.6 VBE (V)
1 -1 (W ) ri
1
(d)
0.6 1 VBE (V)
1 -1 (W ) ri 0.1
10
(b) 100 mA (d) 10 mA
0
IC C
RB B
5.
6.
v1 ~
VBB
RC
E
In the figure, given that VBB supply can vary from 0 to 5.0 V, VCC = 5 V, bdc = 200, RB = 100 kW, RC = 1 KW and VBE = 1.0 V. The minimum base current and the input voltage at which the transistor will go to saturation, will be, respectively : [Main 12 Jan. 2019 II] (a) 25 µA and 3.5 V (b) 20 µA and 3.5 V (c) 25 µA and 2.8 V (d) 20 µA and 2.8 V In a common emitter configuration with suitable bias, it is given than RL is the load resistance and RBE is small signal dynamic resistance (input side). Then, voltage gain, current gain and power gain are given, respectively, by: (b is current gain, IB, IC, IE are respectively base, collector and emitter currents:) [Main Online April 15, 2018] (a)
b
RL DI E 2 RL , ,b RBE DI B RBE
R
DI
R
2 L 2 L C (c) b R , DI , b R BE E BE
7.
8.
DI
R
R
2 C L L (b) b R , DI , b R BE B BE
R
DI
R
2 L L C (d) b R , DI , b R BE B BE
In a common emitter amplifier circuit using an n-p-n transistor, the phase difference between the input and the output voltages will be : [Main Online April 2, 2017] (a) 135° (b) 180° (c) 45° (d) 90° For a common emitter configuration, if a and b have their usual meanings, the incorrect relationship between a and b is : [Main 2016] b (a) a= 1 +b
(b) a=
b2 1 +b2
1 1 = +1 (d) All of these a b A realistic graph depicting the variation of the reciprocal of input resistance in an input characteristics measurement in a common emitter transistor configuration is : [Main Online April 10, 2016]
(c)
9.
~ v0
VCC
IE
0.6 VBE (V)
1
0
0.6 VBE (V)
1
10. An n-p-n transistor has three leads A, B and C. Connecting B and C by moist fingers, A to the positive lead of an ammeter, and C to the negative lead of the ammeter, one finds large deflection. Then, A, B and C refer respectively to: [Main Online April 9, 2014] (a) Emitter, base and collector (b) Base, emitter and collector (c) Base, collector and emitter (d) Collector, emitter and base. 11. For a given plate voltage, the plate current in a triode valve is maximum when the potential of [1985 - 2 Marks] (a) the grid is positive and plate is negative (b) the grid is zero and plate is positive (c) the grid is negative and plate is positive (d) the grid is positive and plate is positive 12. The plate resistance of a triode is 3 × 103 ohms and its mutual conductance is 1.5 × 10 –3 amp/volt. The amplification factor of the triode is [1981- 2 - 3 Marks] (a) 5 × 10–5 (b) 4.5 (c) 45 (d) 2 × 10 5 3
Numeric Answer
13. The output characteristics of a transistor is shown in the figure. When VCE is 10 V and IC = 4.0 mA, then value of bac is ______. [Main Sep. 06, 2020 (II)] (IB ) 60 mA 50 mA 40 mA 30 mA 20 mA 10 mA
8
(IC) in mA
(a) 40 mA (c) 7 mA
6 4 2 2
6
4
6 8 10 12 14 (VCE) in volts
MCQs with One or More than One Correct Answer
14. A transistor is used in the common emitter mode as an amplifier. Then [1998S - 2 Marks] (a) the base-emitter junction is forward-biased (b) the base-emitter junction is reverse-biased (c) the input signal is connected in series with the voltage applied to bias the base-emitter junction (d) the input signal is connected in series with the voltage applied to bias the base-collector junction
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EBD_8341 P-308
҈�
Physics
� 矄� ��ᨨ ᨨ ᨨ盐� 矄꾀�� ᨨ�ᨨ ҈ ᨨ ᨨ ᆠ盐 ᔼ � 鼀��鼀矄㡰�矄�� � 鼀矄 ᨨ ᨨ� � ᨨ矄㡰泐 ��ᆠ�ᨨ ᔼ㦨҈�䰘� ᨨ�� ᨨ�ᨨ ᨨ矄 ᆠ盐 ᨨ ᨨ� ᨨ ᨨ矄�矄�鼀��鼀鼀� 矄�矄 ᆠ rp㡰 矄ᆠᨨ 盐ᆠ ᆠ gm㡰 鼀� �ᆠ�ᨨ ᆠᨨ Ҙ㡰 ᨨ �ᨨ ҈黨䰘㜠䰘㝐├矄桌
ᔼ n㜠p㜠n 矄�矄ᨨᆠ�ᆠ盐� ᆠᨨᆠᨨᆠ盐 �矄҈ 鼀꾀ᔼ 黨奠ᨨ ᆠᨨ 矄鼀� ᆠ ᆠᨨᆠᨨ 㡰 鼀�ᆠ盐 犨�黨鼀꾀 ҈黨黨㜠㝐├矄桌 㡰 矄ᆠ盐 犨�҈鼀꾀 ᆠ㡰 鼀�ᆠ盐 犨�҈҈鼀꾀 㡰 矄ᆠ盐 犨�҈鼀꾀 10 Subjective Problems
҈
꾀�ᨨ 矄ᆠ ᆠ�矄�ᆠ矄� ᨨ鼀ᨨ
Topic-3 : Digital Electronics and Logic Gates 1 ҈
MCQs with One Correct Answer
ᔼ � 泐 ᆠᨨᆠ ᨨ盐盐 矄�� Ӹ � ��
ᆠᨨ鼀� �ᨨ ᨨ �矄 矄矄 ᨨ犨 㡰 ᨨ �� � 盐矄꾀 㝐� 和 ᔼ㡰桌 Ӹ 꾀
A
҈
҈�
Z
B
꾀
�
ᔼ ᨨᨨ犨� � ����ᆠ�ᆠ盐�犨 犨� ᨨ盐盐 �Z犨 � 盐 A, B㡰 ҈㡰 㡰 ҈҈㡰 ҈㡰 [Main Sep.02, 2020 (II)]
㡰 ҈ 㡰 ҈҈҈ ᆠ㡰҈҈҈ 㡰 ҈ ᨨᨨ �ᨨ ᨨ盐盐矄�㜠Y ᨨ ᨨᨨ犨� � ᆠ�ᆠ盐��矄 㝐� ᔼ桌
㡰
�
㡰
㡰 A + B 㡰 A졔B ᆠ㡰AB 㡰 AB ᔼ �� ᆠ�ᆠ盐�盐ᨨ Y�矄 㝐� ᔼᔼ桌
ᆠ㡰 �
҈ ҈�
㡰 ᔼ � 泐 ᨨ�ᨨ ᨨ鼀 泐 ᆠ�ᆠ盐��� ᨨ犨 㝐� 和 ᔼᔼ㡰桌
꾀
㡰 㡰 ᨨ��矄犨 ҈ ᆠ㡰 犨� ᨨ ᆠ盐 㡰 ҈ ␔ �ᆠ ᨨ ᨨᨨ犨� ���矄矄�ᨨ�ᨨ 䁸 㝐� 䰘 ᔼ桌 㡰
㡰
ᆠ㡰
㡰
ᔴ
㡰 꾀 㡰 졠
ᆠ㡰 꾀
㡰 졠
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P-309
Semiconductor Electronics : Materials, Devices and Simple Circuits
䰘
盐 ᨨ ᆠ�ᆠ盐��� � ���矄 㝐� 黨꾀�҈黨ᔼ桌
҈҈
盐 ᨨ �� ᆠ�ᆠ盐�犨� 㝐� 졠 � 꾀�҈�҈桌 x
z é꾀 Ӹ ù é꾀 Ӹ ù é꾀 Ӹ ù ê ҈ú ê ҈ú ê ҈ú ê ú ê ú ê ú ê ҈ ҈ú ê ҈ ú ê ҈ ҈ú ê ú ê ú ê 㡰 ҈ ҈ 㡰 ҈ ᆠ㡰 ҈ ú 㡰 ê ú ê ú ê ú êë ҈ ҈ ҈ úû êë ҈ ҈ úû êë ҈ ҈ úû
黨
é꾀 Ӹ ù ê ú ê ú ê ҈ ú ê ú ê҈ ҈ú êë ҈ ҈ ҈ úû
y
㡰
ᨨ��ᆠ�⼄盐� ᨨ �� ᨨ��ᆠᆠ�ᆠ盐��矄 㝐� 黨꾀҈黨ᔼᔼ桌
㡰 꾀 㡰 졠 ᆠ㡰 졠 㡰 꾀
ᨨ盐盐ᨨ �� ᨨ��ᆠᆠ� ᆠ盐��矄 㝐� ҈ ҈黨ᔼ桌 꾀
҈ ҈
泐 ҈ ҈
҈ ҈ ҈
㡰
҈ ҈
泐 ҈ ҈
ᆠ㡰 ҈ ҈ ҈
泐 ҈ ҈
҈ ҈ 㡰 ҈ ҈
҈ ҈
泐 ҈ ҈
҈ ҈ ҈
҈ ᔼ ᆠ � 盐矄ᨨ� �矄�矄ᨨ盐盐 矄 ᨨᨨ犨� ��鼀� ��矄 㝐� ҈桌 ᆠ 㡰 졠 㡰 꾀 ᆠ㡰 졠
㡰 꾀 ҈ ᨨ� ᨨ盐盐ᨨ ҈ ᨨ鼀 ᆠ�ᆠ盐�矄 ᨨ犨 � ��盐 � 盐鼀盐矄 㝐� 졠 � 꾀�҈҈桌
Ӹ
Ӹ
㡰 꾀 + 꾀 㡰 꾀 + 꾀 ᆠ㡰 꾀 㡰 꾀 ҈ ᨨ � ᨨ盐盐 �҈ ᨨ �� ᨨ��ᆠ � ᆠ�ᆠ盐� � 盐 盐矄 鼀盐矄 㝐� ҈ ҈黨ᔼ桌
㡰 㦨Ӹ㦨҈ ᆠ㡰 㦨҈Ӹ㦨
ᆠ
㡰 㦨㦨ᆠ㦨҈ 㡰 㦨҈㦨ᆠ㦨 ᆠ㡰 㦨҈㦨ᆠ㦨҈ 㡰 㦨㦨҈ᆠ㦨 ҈ ᔼ � 泐 � 鼀ᆠ 꾀Ӹ� ᆠ├ᨨᆠ ᆠ├ 㝐� 졠 � 꾀�黨҈桌
㡰 㦨҈Ӹ㦨҈ 㡰 㦨Ӹ㦨
꾀
꾀
㡰 㡰 ᆠ㡰 㡰
꾀 꾀㦨҈㦨҈Ӹ㦨҈㡰 졠 꾀㦨҈㦨҈Ӹ㦨㡰 졠 꾀㦨҈㦨҈Ӹ㦨҈㡰 졠 꾀㦨㦨Ӹ㦨㡰
Ӹ
Answer Key Topic-1 : Solid Semiconductors and P-N Junction Diode ҈ ҈҈ ҈ ҈ 䰘 ҈
㡰 ᆠ㡰 㡰 ҈ 㡰 ҈ ᆠ㡰 㡰 㡰 ҈㡰 졔 㡰p㜠矄� n㜠矄� 㡰 㡰 㡰
ᆠ㡰 㡰 㡰 㡰
盐 㡰
҈ 黨
ᆠ㡰 � ᆠ㡰 ҈� 㡰 � 矄㡰 ᆠ㡰
㡰 㡰 㡰
҈ �
ᆠ㡰 䰘 ᆠ㡰 ᆠ㡰 ҈䰘 㡰 㡰 䰘 㡰 錐꾀ᔴ㡰
҈
㡰 黨 㡰 ҈ 㡰 㡰 ҈黨 ᆠ㡰 㡰 ᆠ㡰 黨 㡰 㡰 矄 ��㡰
ᆠ㡰
Topic-2 : Junction Transistor ҈ ᨨ 盐矄㡰 㡰 ҈҈ 㡰 ҈ 㡰
㡰 ҈ ҈�㡰
㡰 � 㡰 ҈ ᆠ㡰 ҈� ᆠ㡰
㡰
䰘 㡰
㡰
黨 ᆠ㡰
҈ ᆠ㡰
㡰
黨 ᆠ㡰
҈ ᆠ㡰
Topic-3 : Digital Electronics and Logic Gates ҈ 㡰 ҈҈ ᆠ㡰
ᆠ㡰 ҈ 㡰
㡰 ҈ ᆠ㡰
㡰 ҈ 㡰
� 㡰
㡰
䰘 ᆠ㡰
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EBD_8341 P-310
Physics
Communication Systems
29
Topic-1 : Communication Systems 1 ҈
MCQs with One Correct Answer
�
꾀 鼀�盐 鼀ᨨ 盐 犨 �矄 矄 泐 矄矄�ᨨ vm = � ҈ + ᆠᨨ矄 t 㡰 矄� ҈҈´ ҈ t 㡰 ᨨ矄
鼀� �鼀盐鼀 鼀 �鼀盐鼀鼀�盐 矄ᨨ 鼀�盐 鼀ᨨ 盐 犨矄ᆠ�泐 [Main Sep.02, 2020 (I)] � � 㡰 㡰 � ᆠ㡰 � ᔼ 鼀�盐 鼀ᨨ 盐ᨨᆠ�ᆠ盐� ᆠ�犨�矄��
泐ᔴ t㡰㦨矄� pt㡰犨 �鼀ᨨ 盐� �矄�� �矄��
泐m t㡰㦨矄� pt㡰 盐矄ᨨ 鼀ᨨ 盐�ᨨ � ᨨ犨矄� ⼄盐 ᆠ泐 㝐� ҈꾀�҈黨 ᔼᔼ㡰桌 㡰 � ҈├ 㡰 ҈├ ᆠ㡰 黨├ 㡰 � 黨├ 꾀 鼀矄矄� 矄�� ᨨ ⼄盐 ᆠ泐 ҈ 㝐 ├ ᨨ�҈�矄盐矄 ᨨ ᆠ盐鼀�盐 鼀ᨨ 盐�ᨨ ᨨ
ᆠ�犨ᨨ ⼄盐 ᆠ泐 ├ᨨ� 鼀ᨨ 盐�ᨨ � � ᆠ犨 犨ᨨ 矄� ⼄盐 ᆠ�矄 㝐� ҈꾀�҈黨 ᔼᔼ㡰桌 㡰 錐҈맔҈ 㡰 錐맔҈ ᆠ㡰 �錐맔҈ 㡰 �錐҈맔҈ 꾀矄�� 꾀ᆠᨨ矄w�矄 矄鼀� 盐矄� �v矄� wĀ矄ᆠ� 犨 ᆠᨨᆠ鼀�盐 鼀ᨨ 盐 꾀㝐㡰矄�� �矄 㝐� 黨꾀�҈黨 ᔼ㡰桌
㡰
㡰 v矄� wĀ t + 꾀 矄� ( w w ) t + 꾀 矄� ( w + w ) t 뾼 㡰 v矄� w ҈졔҈꾀矄� w㡰桌 ᆠ㡰 v矄� w졔꾀ᆠᨨ矄w 㡰 v졔꾀㡰ᆠᨨ矄w矄� wĀ
䰘
泐矄�ᆠ矄� 矄ᨨ 矄鼀� ᆠ�
� ᆠᨨ鼀鼀盐 �ᆠ�ᨨ 矄泐矄鼀㝐� 黨꾀�҈黨 ᔼᔼ㡰桌 㡰 � ᨨ ᨨ ᆠ� 鼀ᨨ 盐�ᨨ ⼄盐 ᆠ泐 㡰 � 矄泐ᨨᨨ�ᨨ ᨨᆠ� ⼄盐 ᆠ泐 ᆠ㡰 � 矄泐ᨨᨨ�ᨨ ᨨ鼀ᨨ 盐�ᨨ ⼄盐 ᆠ泐 㡰 ᨨᨨ�ᨨ ᨨᆠ� ⼄盐 ᆠ泐
犨 � ᨨ ᆠ�犨矄� 鼀ᨨ ᨨ�ᆠ �ᆠᨨ鼀鼀盐 �ᆠ�ᨨ 犨ᨨ├�矄ᆠᨨ矄ᨨ 㝐� 꾀�҈黨 ᔼ㡰桌 㡰 鼀 㡰 ҈� 鼀 ᆠ㡰 鼀 㡰 黨 鼀 ᔼ � ᨨ 矄�� �ᨨᆠᨨ鼀鼀盐 �ᆠ�ᨨ �矄 ᆠᨨ ᨨ盐 � ├鼀 �矄├犨 矄鼀�� � ᆠ�� �
矄ᔼ �� ᨨ ᆠ�� �
�矄䰘鼀 鼀� �鼀盐鼀 �� ᨨ 矄鼀�� �
矄 ᨨ盐 㝐� 꾀�҈黨 ᔼᔼ㡰桌 �盐矄ᨨ 쟼 㦨맔҈鼀㡰 㡰 鼀 㡰 �҈鼀 ᆠ㡰 鼀 㡰 鼀 꾀҈ᆠ�犨�矄鼀 ᨨ泐犨 ҈ 泐鼀ᨨ 盐� �矄�� ␔ �矄 鼀ᨨ 盐�ᨨ � 䁸
㝐� ҈ ҈黨 ᔼ㡰桌 ᆠ㡰 㡰 㡰 㡰 � 黨
ᨨ ᨨ盐 ᆠᨨ� � �ᨨ 矄鼀��ᨨ ᨨ犨 �矄 �� 矄 ᨨ盐 鼀盐�� 泐 㝐� ҈ ҈黨 ᔼᔼ㡰桌 ҈ 㡰 㡰 㡰 ᆠ㡰 ҈ 꾀 鼀�盐 鼀ᨨ 盐 矄�� �矄�� 泐 㡰㦨҈҈졔 ᆠᨨ矄 맔҈㡰桌矄� ��맔҈�㡰�矄� 矄ᆠᨨ 矄
矄� ⼄盐 ᆠ�矄 � ├ 㡰� p㦨⏌䰘桌 㝐� ҈҈ ҈黨 ᔼᔼ㡰桌 㡰 ҈䰘� ҈䰘҈� 㡰 ҈䰘� ҈䰘҈� ᆠ㡰 黨� �䰘� 㡰 黨� �䰘�
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P-311
Communication Systems
҈҈
꾀 鼀�盐 鼀ᨨ 盐 矄�� �矄ᨨ ᨨ犨 ҈
㡰
m矄 ҈m矄 ␔ �ᆠ ᨨ ᨨ ᨨᨨ犨� � 矄 矄ᆠ�矄 ᨨ 矄�� 䁸 㝐� ҈҈ ҈黨 ᔼᔼ㡰桌 㡰 黨졔矄� �p맔҈�㡰㡰矄� p맔҈㡰 㡰
҈
҈졔黨矄� p맔҈㡰㡰矄� �p맔҈�㡰
ᆠ㡰 黨졔矄� p맔҈㡰㡰矄� �p맔҈�㡰 㡰 黨졔矄� p맔҈㡰㡰矄� �p맔҈�㡰 꾀 矄鼀�矄矄�ᨨ ᨨ犨 矄 �� ᨨ ҈鼀 �� ᨨ ᆠ�� �
�矄 鼀 ␔ �矄 鼀 �鼀盐鼀 �矄 ᆠ盐ᨨ犨 �ᆠ 矄�� 矄ᆠ ᨨ ᆠ矄 ᨨ鼀 �矄ᨨ犨� 졠和 � ᨨ 和�� 㡰鼀ᨨ 䁸 � �盐矄ᨨ 㦨맔҈鼀㡰 㝐� ҈ ҈黨 ᔼ㡰桌 㡰 �├鼀
҈
҈
҈�
҈
҈䰘
㡰 ├鼀
ᆠ㡰 ├鼀
㡰 ├鼀
鼀ᨨ 盐�ᨨ ⼄盐 ᆠ泐ᨨ 꾀㝐 �ᨨ矄�ᨨ �矄� ├ 犨 �ᆠ �矄҈奠ᨨ ᆠ�犨ᔼ ᨨ 꾀㝐矄㜠 �ᨨ ᨨᆠ 矄泐ᨨ盐 ᨨ�ᆠ 矄犨 ᨨ ᆠ矄 ⼄盐 ᆠ泐 犨�泐ᨨ盐ᨨ䁸 㝐� ҈ ҈黨 ᔼ㡰桌 㡰 䰘�├ 㡰 黨├ ᆠ㡰 �├ 㡰 ├ ᔼ ᆠᨨ鼀鼀盐 �ᆠ�ᨨ 矄泐矄鼀ᨨ� �犨 �
鼀ᨨ 泐ᨨ ᆠ ᨨ 矄ᨨ盐ᆠ ⼄盐 ᆠ泐�矄�矄 矄�� 犨� 盐鼀ᨨ ᆠ
矄ᆠᆠᨨ鼀ᨨ ᨨ 矄鼀�� � 矄�� 矄ᨨ 犨� 㝐 ├ ᨨᆠ�泐ᨨ �� ᆠ㦨맔҈鼀⏌矄 㦨맔҈㜠矄㡰 㝐� 黨 ҈黨 ᔼᔼ㡰桌 㡰 䰘�맔҈ 㡰 맔҈ ᆠ㡰 �맔҈� 㡰 䰘맔҈� 꾀 ᨨ �ᆠᆠᨨ鼀鼀盐 �ᆠ�ᨨ 矄�ᆠ�矄犨ᨨ├� �ᆠ� ⼄盐 ᆠ泐 ᨨ ҈ 졠 泐҈奠 ᨨ � �矄 盐�� ᨨ 矄鼀�矄矄�ᨨ ᨨ犨鼀 泐 ᨨ �ᆠ ᆠ
矄ᆠ 矄鼀� 矄�鼀盐 ᨨ盐矄泐� ᆠ ᆠ
⼄盐�矄 犨� ᨨ �├ 䁸 㝐� ҈桌 㡰 맔҈ 㡰 맔҈ ᆠ㡰 맔҈� 㡰 맔҈ 꾀ᆠ�犨ᨨ ├ᨨ�҈�矄盐矄 ᨨ 矄鼀�� � 鼀矄矄�矄�� ├ᨨ�ᨨ 鼀ᨨ 盐� �矄�� �� ᨨᆠ �鼀ᨨ 盐�ᨨ � ᨨ 奠犨� 㝐� ҈桌 㡰 ҈҈ 㡰 䰘 ᆠ㡰 㡰
盐鼀ᨨ 鼀�盐 鼀ᨨ 盐 ᨨ ᆠ矄矄�ᨨ 矄 ᆠ ᆠᆠᨨ鼀鼀ᨨ � ├ 犨� ᨨ �� 矄鼀ᨨ 盐� � ⼄盐 ᆠ泐҈�├ 犨� 㝐� 졠 � 꾀�҈�҈桌 㡰 㡰 ҈ ᆠ㡰 㡰 ҈�
҈ ᆠ� ⼄盐 ᆠ泐ᨨ 矄鼀��矄ᨨ� 泐 ├ᆠ�ᆠ盐�ᨨ ᆠᨨ�ᨨ � 盐ᆠ ᆠ黨m ᆠᆠ� ᆠ ᨨ � ᔼ�矄鼀ᨨ 盐 泐 盐 �ᨨ矄�� ᨨ ҈├ ⼄盐 ᆠ泐 �ᨨᆠᆠ盐� 泐 矄� 矄�矄 㝐� 졠 � 꾀�҈�҈桌 㡰 ҈├ ├ 㡰 ├ 䰘�├ ᆠ㡰 ├ ├ 㡰 ҈├ ҈黨├ ҈黨 ᔼ 鼀�盐 鼀ᨨ 盐�ᨨ 矄� 盐矄ᨨ� ᆠ� ⼄盐 ᆠ泐盐矄 �矄 ᨨ 泐 ωc 矄�� ⼄盐 ᆠ泐�矄 ᨨ 泐
ωm 犨� Dωm 㡰ᨨ 矄�� �矄矄盐ᆠ Dω m ωc ␔ �ᆠ ᨨ ᨨᨨ犨� � ⼄盐 ᆠ�矄�矄 ᨨᆠᨨ � � 鼀ᨨ 盐 犨䁸 㝐� ҈䰘桌 㡰 ω m + ω c 㡰 ω c - ω m ᆠ㡰 ω m 㡰 ωc 꾀 矄�� �矄ᨨ 矄鼀� ᨨ盐� 犨ᨨ 犨㜠 � l盐矄� ��
� ҈ᨨ 㜠
ᆠ�ᨨ犨 � 犨��� 㜠 矄ᆠ�泐矄 ⧤�矄ᆠᨨ 矄 ᨨ ᨨᨨ�ᨨ �泐㡰 㝐� 졠 � 꾀�黨҈䰘桌
l æ҈ö 㦨 ⧤ ç ÷ èlø
æ҈ö 㡰 l, 㦨⧤ ç ÷ èlø
㡰
l ҈ ᆠ㡰 㦨 ⧤ æç ö÷ ҈ èlø
l æ ҈ ö 㡰 㦨 ⧤ ç ÷ � èlø
҈
҈ 꾀矄�� ᨨ ⼄盐 ᆠ泐├ ├ᨨ�ᨨ �ᨨ�矄 盐矄 ᨨ鼀ᨨ 盐ᆠ�犨ᨨ ⼄盐 ᆠ泐҈㝐 ├ᨨ��ᨨ矄ᔴ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㝐� 졠 � 꾀�҈䰘桌 㡰 㝐ᨨ 盐�ᨨ � 㦨�矄� ⼄盐 ᆠ泐 矄 ҈├ ҈├ 㡰 㝐ᨨ 盐�ᨨ � 㦨�矄� ⼄盐 ᆠ泐 矄 ҈├ ҈├ ᆠ㡰 㝐ᨨ 盐�ᨨ � 㦨矄� ⼄盐 ᆠ泐 矄 ҈҈├ ҈├ 㡰 㝐ᨨ 盐�ᨨ � 㦨矄� ⼄盐 ᆠ泐 矄 ҈├ ҈҈├ ᔴ ᨨᨨ矄 ᆠᨨᆠ矄鼀 㝐� ҈桌 㡰 ᔼ ⼄盐 ᆠ泐鼀ᨨ 盐�ᨨ 鼀�盐 ᨨ �� ⼄盐 ᆠ泐ᆠ�犨�矄鼀 ᨨ泐� ᨨᨨ�ᨨ
ᨨ 鼀�盐 ᨨ 盐 �ᨨ矄�� 㡰 ᔼ ⼄盐 ᆠ泐鼀ᨨ 盐�ᨨ 鼀�盐 ᨨ �� ⼄盐 ᆠ泐ᆠ�犨�矄鼀 ᨨ泐� ᨨᨨ�ᨨ
ᨨ ⼄盐 ᆠ泐ᨨ 盐 �ᨨ矄�� ᆠ㡰 ᔼ 鼀�盐 鼀ᨨ 盐�ᨨ 鼀�盐 ᨨ �� ⼄盐 ᆠ泐ᆠ�犨�矄鼀 ᨨ泐� ᨨᨨ�ᨨ
ᨨ 鼀�盐 ᨨ 盐 �ᨨ矄�� 㡰 ᔼ 鼀�盐 鼀ᨨ 盐�ᨨ ⼄盐 ᆠ泐ᨨ �� ⼄盐 ᆠ泐ᆠ�犨�矄鼀 ᨨ泐� ᨨᨨ�ᨨ
ᨨ 鼀�盐 ᨨ 盐 �ᨨ矄��
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EBD_8341 P-312
�
䰘
黨
Physics
꾀鼀ᨨ 盐 矄�� ᔴ鼀 㡰 矄 ᨨ鼀ᔴ鼀 㡰㦨矄� p 졔҈ ᆠᨨ矄pᆠᨨ矄p㡰 ᆠ� ⼄盐 ᆠ泐 ᆠ 鼀ᨨ 盐� � ⼄盐 ᆠ泐 鼀矄矄� ⼄盐 ᆠ泐㡰 w 鼀ᨨ 盐�ᨨ � � m 矄ᆠ�泐�� 泐 㝐� 졠 � 꾀�҈҈桌 ҈ 㡰 ᆠ㦨 錐 犨㦨� 錐m 㦨 㡰 ᆠ㦨҈� 錐 犨㦨� 錐m 㦨 ҈ ᆠ㡰 ᆠ㦨҈� 錐 犨㦨 錐m 㦨 ҈ 㡰 ᆠ㦨 錐 犨㦨 錐m 㦨 꾀 盐 �ᨨ矄�� ᆠᨨ 矄�矄矄ᨨ 犨ᨨ �矄� ᆠ 矄ᨨ盐 矄 ᨨ 盐鼀 矄ᆠ 矄�� � ⼄盐 ᆠ泐 ᨨ ᨨ 䰘 犨 � ᨨ �矄 �� ⼄盐 ᆠ泐鼀盐矄�ᆠ矄�� � ⼄盐 ᆠ泐 ᨨ ҈ ᨨ҈� �ᨨᨨ 꾀㝐矄�� 犨� ⼄盐� ᨨ矄 ᨨ 矄�� 矄 ᨨ� ᨨ 꾀㝐 矄�� 犨� ⼄盐� ᨨ矄 盐矄 盐鼀 矄ᆠ �矄 㝐� 졠 � 꾀�黨҈桌 㡰 㡰 � ᆠ㡰 㡰 꾀矄�� ᨨ �├ ⼄盐 ᆠ泐�矄鼀�盐 鼀ᨨ 盐 ᨨ ᆠ�犨ᨨ ⼄盐 ᆠ泐㝐 ⼄盐 ᆠ�矄ᨨ 矄盐 矄�� �矄⏌ 㝐� ҈�桌 㡰 �├ ├ ҈黨黨�├ 㡰 ├ ҈黨黨�├ ᆠ㡰 㝐 ᨨ 泐 㡰 �├ ҈黨黨�├ ᨨ � � �ᨨ 矄鼀�矄矄�ᨨ �矄ᨨ矄矄�犨 �ᨨ 犨矄 ᆠ ᨨ鼀 �ᨨ ᨨ矄 ᨨ �矄ᨨ
⼄盐 ᆠ泐ᨨ �ᨨ犨矄鼀盐矄� � 㝐� 졠 � 꾀�҈黨҈桌 㡰 㜠҈�㝐 㡰 㜠�㝐 ᆠ㡰 ҈㜠㝐 㡰 ҈�㜠҈�├ ᨨ矄├泐犨ᨨ��ᨨ �ᨨ犨矄鼀盐矄 ⼄盐 ᆠ泐 �� 犨 㝐� 졠 � 꾀�҈҈桌 㡰 ҈㝐 ᨨ㝐 㡰 �㝐 ᨨ�㝐 ᆠ㡰 �㝐 ᨨ㝐 㡰 �㝐 ᨨ�㝐 꾀 矄鼀�� �
ᨨᨨ ᨨ犨 矄 �� 鼀 �� ᨨ ᆠ�� �
�矄�鼀␔ �矄 鼀 �鼀盐鼀 �矄 ᆠ 犨 鼀 ᨨ 矄�矄 ᆠᨨ泐 ᆠᨨ鼀鼀盐 �ᆠ�ᨨ � � ᨨ 矄�� 졠和㡰鼀ᨨ 䁸 㝐� 졠 � 꾀�黨҈桌 㡰 ��├鼀 㡰 ��├鼀 ᆠ㡰 ��├鼀 㡰 ��├鼀 꾀 �ᨨ ᆠᨨ�矄盐矄 ᨨ ᆠ 鼀�盐 鼀ᨨ 盐 犨ᨨ 奠鼀ᨨ 盐�ᨨ 泐盐矄� �ᆠᨨ 矄ᨨ ᆠᆠ�泐 ��ᆠᨨ � 犨� ᨨ 矄�矄 ᆠ҈├�ᨨ
ᨨ 鼀 � 鼀 �鼀盐鼀鼀ᨨ 盐 ⼄盐 ᆠ泐犨 �ᆠ ᆠᨨ盐 ᆠ 泐� 㝐� ҈桌 和��
҈
ᔴ
㡰 ҈㝐 㡰 ҈├ ᆠ㡰 �҈㝐 㡰 �҈├ ␔ �ᆠ ᨨ ᨨᨨ犨� � 鼀ᨨ 盐 矄�� 矄 矄
ᨨ�矄㜠ᨨ ᆠ䁸 㝐� 졠 � 꾀��҈桌 㡰 ᨨ �㜠犨 㡰 和 ᨨ㜠犨 ᆠ㡰 㝐 �盐鼀㜠犨 㡰 꾀鼀�盐 㜠鼀ᨨ 盐 ␔ �ᆠ ᨨ ᨨᨨ犨� �矄鼀 �矄졠 ᆠᨨᆠ䁸 㝐� 졠 � 꾀�҈桌 㡰 ᨨ盐 犨 矄�� 矄 鼀ᨨ 矄 矄├泐 犨矄�� 矄 㡰 ᆠ��ᆠ ⼄盐 ᆠ泐ᨨ �ᨨ ᨨ矄 �ᆠ泐�矄 �� 矄 ⼄盐 ᆠ泐 犨� ᆠ ᆠ├泐 泐犨 ��矄�ᆠ泐� ᆠ� ᆠ㡰 쟼ᆠᨨ鼀� �ᆠ 犨矄 ᨨ ⼄盐 ᆠ�矄 ��
ᨨ盐㝐 ᆠ
ᨨ �ᨨ ᨨ矄 㡰 和├泐犨矄�� 矄� ᨨ ᆠ矄 ⼄盐 ᆠ泐 � 矄ᨨ � �� � 泐�鼀
�矄⼄盐矄�ᨨ 矄和鼀 㜠҈ 和鼀 㜠졠 ᨨ盐ᆠ ᨨ�ᆠ矄�� 和鼀 矄ᆠ ᨨᨨ矄 ᨨ 矄 矄ᆠ�矄 犨ᨨ和鼀 矄 和鼀 㜠҈和 ᨨ犨 矄鼀�矄矄�ᨨ �矄ᆠ � 盐ᨨ ᨨ � ᆠ�ᨨ ᨨ 㜠鼀 犨 ᨨ鼀
ᨨ� �� � �ᨨ ᨨ矄 和鼀 㜠 ᆠ�� ᨨ 矄鼀�矄� ᨨ ⼄盐 ᆠ泐ᨨ 㜠鼀犨矄 㝐� 졠 � 꾀�҈桌 㡰 和鼀 㜠҈�矄盐和鼀 㜠�矄 矄 㡰 和鼀 㜠҈�矄 矄和鼀 㜠�矄盐 ᆠ㡰 和鼀 㜠҈ �矄 盐 和鼀 㜠 �矄 盐 盐 和鼀 㜠 �矄 ᨨ ᆠᨨᆠ �ᨨ ᨨ 矄鼀 㜠҈ 㡰 和鼀 㜠҈ �矄 盐 和鼀 㜠 �矄 盐 和鼀 㜠�矄 ᆠᨨᆠ �ᨨ ᨨ 和鼀 㜠҈ ᔼ ᆠ�犨ᆠ 㡰㦨꾀矄� wᆠ�矄鼀�盐 鼀ᨨ 盐 泐 鼀ᨨ 盐ᨨ矄�� 鼀 㡰㦨꾀矄� w鼀 ⼄盐�ᨨ ᨨ 鼀ᨨ 盐 矄�� ᔴ鼀 㡰桌 �矄鼀ᨨ 盐�ᨨ � 矄ᆠ�泐 㝐� 졠 � 꾀�黨҈桌 㡰 Cm t㡰㦨A ҈졔矄� wmt㡰矄� wct 㡰 Cm t㡰㦨A ҈졔矄� wmt㡰矄� wmt ҈ ᆠ㡰 Cm t㡰㦨A ҈졔矄� wmt㡰矄� wct ҈ 㡰 Cm t㡰㦨A ҈졔矄� wct㡰矄� wmt
Answer Key Topic-1 : Communication Systems ҈ ҈҈ ҈ ҈
㡰 ᆠ㡰 㡰 ᆠ㡰
҈
㡰 㡰 ᆠ㡰 㡰
҈
ᆠ㡰 㡰 㡰 ᆠ㡰
㡰 ҈ ᆠ㡰 ᆠ㡰
� 㡰 ҈� ᆠ㡰 � 㡰
㡰 ҈ 㡰 㡰
䰘 ᆠ㡰 ҈䰘 㡰 䰘 㡰
ᆠ㡰 ҈ ᆠ㡰 㡰
黨 ᆠ㡰 ҈黨 ᆠ㡰 黨 㡰
҈ ᆠ㡰 㡰 㡰
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EBD_8341 PS-2
Physics ᆠ㡰 쟼 �泐 E µ AaT b P c �㡰 ᨨ E = kAaT b P c 犨 k �矄 �鼀 矄�ᨨ 矄矄 ᆠᨨ 矄 a, b c ᨨ 矄 �鼀 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀 P = M ҈ L҈T -҈ �鼀 矄�ᨨ ᨨ A㦨L �鼀 矄�ᨨ ᨨ �鼀T㦨T ҈ 盐� � 矄盐� ⼄盐�ᨨ �㡰犨�
桌㦨a桌b桌 㝐 㦨 㝐҈ b桌 ҈
M ҈LT - = M c La + cT b - c
泐 ᆠᨨ鼀�矄ᨨ
c㦨҈ a 졔 c㦨 b c㦨
҈
c㦨҈a㦨҈⏌b㦨҈
\ E = A҈⏌ T -҈ P҈
m m = = m c e e m
䰘
㡰
mc ® 㝐 꾀 맔 ҈ 㝐 꾀 �鼀 矄�ᨨ 矄 ᨨ 矄�矄 ᆠ 㡰 㦨�Ӹ
ÞY µ
�㡰
Z
A T 桌 Q Q = = W MLT - 桌
㦨 㝐 ҈ 꾀桌 F Z =B= IL 㦨 㝐 꾀҈桌
QF㦨ILB桌
M -҈ L-T A 桌
é e ù ê ú= ëê m úû
㡰
Q F=
҈
é e =ê m e êë m e
ù ú 㦨e CLT ҈桌맔e 桌 0 0 úû é ù ҈ = Cú êQ m e êë úû
pe r
AT 桌 MLT - 桌 ´ L 桌
= A M -҈ L-T 桌
é e ù \ ê ú = LT -҈ 桌 ´ A M -҈L-T 桌 = M -҈L-T A 桌 ëê m úû b㡰 꾀矄 犨 ├ ᨨ犨
élù êë úû = [ ] [ ᆠ] = [ 꾀 ] é l ù é ù \ê ú= ê = é 꾀 ҈ ù ë ᆠ û ë 꾀 ûú ë û ҈҈
҈�
矄� � �㡰㡰
e
q
Þ e 桌 =
\ V µ h I ҈G ҈C � ҈
æ - ö ç ÷ ç a├ ÷ ø abè
҈䰘
é ù
\ ê a├ ú 㦨 㝐
Þ
Þa桌 㦨 㝐҈
҈
�矄 �泐 矄�泐 ᨨ 鼀� �ᆠ m
é B ù 쟼 �泐 ᨨᆠ ´ �矄ᆠ鼀 = ê ú= �矄ᆠ鼀 㡰 ëê m ûú Vᨨlume é MLT ù ҈ = ú = ML T ê êë L úû
b㡰 ├ � �矄盐 �ᨨ � lp㦨҈҈黨맔҈� 鼀
lp =
úû
= 㝐 [ a ]㝐 -
㡰 ⼄盐 �泐 �
҈ � = - \ µ ᔴ�
Nᨨ 㡰 和ᨨ� � ᨨ � V 㡰 µ h x I y G Z C r -҈ h 㦨 ᆠ├矄ᆠᨨ 矄 = éë ML T ùû I㦨ᆠ盐 㦨A桌 G 㦨 ��ᨨ ᆠᨨ 矄 㦨 M҈LT桌 c㦨矄 ᨨ �� 㦨LT ҈桌 V 㦨 ᨨ �㦨 ML T A҈桌 \ ML T A ҈桌㦨ML T ҈桌 A桌泐M ҈LT桌 LT ҈桌r Mx z錐L2x졔z졔r錐Tx2zr錐Ay ᔴᨨ鼀� � �鼀 矄�ᨨ ᨨ 㝐 꾀犨� 泐㦨҈ 㦨 㦨҈㦨�
Þ
b㡰 ᨨᆠᨨ � ᆠ�ᨨ 犨 犨ᨨᨨ鼀矄 = 和� ᆠ ᨨ �鼀矄 �鼀 矄�ᨨ 矄矄
êë
㡰 Ӹ桌㦨桌 桌꾀桌ᆠ 㝐҈ 桌 㦨 ҈桌 㝐 桌 桌ᆠ 㝐҈ 桌 㦨 㝐졔졔ᆠ ᆠ桌 ᔴᨨ鼀� �ᨨ犨ᨨ 矄� ᨨ 矄�鼀�鼀矄犨� 㦨҈졔졔ᆠ㦨҈ᆠ㦨 矄ᨨ� �ᨨ⼄盐�ᨨ 矄犨� 㦨㦨҈ᆠ㦨 矄ᨨӸ桌㦨 꾀桌㦨꾀 桌 b㡰 �鼀 矄�ᨨ ᨨ h桌㦨ML T҈桌 ᔴ桌 㦨 LT ҈桌 G桌 㦨 M҈L T桌 ᆠ �鼀 矄�ᨨ ᨨ é hC � ù é MLT -҈ ù × é L�T -� ù û ë û ê ú㦨 ë é M -҈LT - ù ê G ú ë û ë û 㦨 ML T桌 㦨 �泐 ᆠ㡰 µ 泐 ᔴ �鼀 矄�ᨨ 矄 ᨨ 㦨 㝐҈ 桌 㦨㝐 ҈桌 ᔴ㦨 ҈桌 桌㦨 㝐 ҈ 桌 㝐 ҈桌泐 ҈桌 㝐 ҈桌 㦨 㝐 졔泐 졔泐졔 泐 桌 泐ᆠᨨ鼀� � ᨨ犨矄 ᨨ 㝐 ᨨ 矄� 矄 졔泐㦨Þ 㦨泐 졔泐졔 㦨Þ� 졔 㦨 �㡰 泐 㦨҈ Þ 졔 㦨҈ ��㡰 和ᨨ� �⼄ 矄 �㡰 ��㡰
= 泐=
- -҈
MT A 桌 Ӹ 㦨 㝐 꾀桌 黨
҈
X
X = ᔴᆠ� ᆠ 㦨
Y=
æ ҈ ö ç = c÷ ç m e ÷ è ø
Þ b桌㦨㝐
hG c
㡰 鼀矄矄 矄㝐 µ ᔴ泐 㝐҈ 㦨 Ἐ㡰 ҈ ҈㡰泐 㝐҈ ҈㡰 㝐҈ 㦨 㝐 泐 졔 졔 泐
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PS-3
Physical World, Units and Measurements 㦨҈ 泐졔 㦨 泐㦨
҈黨
Q
泐 㦨 졔҈㦨 㦨҈
㝐 㦨 ҈ ᔴ ҈桌 b㡰 ␔ ├ ᨨ犨 矄�矄��泐
㦨
꾀
=
ᔼ 턼 㦨 ᔼ㡰 꾀
桌ᔼ桌 ␔ ␔ Q = = é 㝐 ù ⼄ � ê ú ´ 桌 ᔼ桌 桌 ëê ûú
ᆠ㡰 Ҙ 犨� e m c h 矄 ᨨᨨ犨矄 m 㦨 keambcchd MLTA桌 㦨 AT桌a M桌b LT҈桌c MLT҈桌d 㦨 Mb 졔 d Lc 졔 d Ta c d Aa桌 졠 ᆠᨨ鼀� � ᨨ 矄� 矄 犨 � a 㦨 �㡰 b 졔 d 㦨 ҈ ��㡰 c 졔 2d 㦨 ҈ ���㡰 a c d 㦨 �㡰 泐 ⼄盐�ᨨ �㡰 ��㡰 ���㡰 �㡰 犨 � a 㦨 b 㦨 c 㦨 ҈ d 㦨 ҈
é h ù ú ë ce û
u 㦨
�
㦨
é 㝐 ҈꾀҈ ù ë û
�鼀 矄�ᨨ 泐 f E ¹ E ᆠ�ᆠ � 㡰
�ᨨ鼀ᨨ鼀 p㦨ᆠ � q㡰맔 �矄 ᆠ d㡰 ᆠ㡰 ᨨ犨ᨨ ᨨ �矄 盐鼀ᨨᆠᨨ 矄
az = M ° L°T °桌 kq
kq z MLT - q -҈ 桌q桌 é = MLT - ù \ a㦨 ë û L桌 ᨨ盐 = MLT - q-҈ 桌 z㦨L桌 꾀矄 k = ├�
a a Þ b㦨 꾀 �鼀 矄�ᨨ 泐P㦨 b P MLT - = M LT 桌 \ b桌㦨 ML-҈T - F é -҈ - ù êQ P = A = ML T 桌ú ë û Dq e A = 㡰 ᔴᆠ� ᆠC = Dt L DV Dq㡰 L DV = L \ X = e L Dt A DV 㡰 Dt \
p é 㝐 -҈- 꾀 ù é 㝐 -҈ - ù é 㝐҈ ù ë ûë ûë û é 꾀 ù ë û 㦨 p é㝐 -҈-҈+ -+ - 꾀 ù ë û é 꾀 ù ҈ ë û 㦨 㦨 ù p é p 㝐 꾀 ë û
턼 㝐 - = 㦨 鼀 㝐 ᔴ � = 㝐 -҈- 꾀 ᔴᆠ� ᆠᔴ㦨 ur ur 㡰 쟼ᆠ�ᆠ 盐 f E = E S
ᨨ �鼀 矄�ᨨ 泐
e a hc
pe 鼀
⼄҈⼄
㦨㝐҈ 꾀桌 㡰 꾀 �盐鼀ᨨ鼀 盐鼀㦨鼀맔맔㦨㝐 ҈
\ 䰘
= é 㝐 ҈꾀҈ ù ë û ҈ e = é 㝐 꾀 ù ë û -҈ - ù é 鼀 = é 㝐҈ ù = 㝐
ë û ë û
҈ ⼄҈⼄ pe
㦨
b㡰 �鼀 矄�ᨨ ᨨ鼀盐ᨨ
ᨨ犨
b㡰 꾀矄犨 ├ ᨨ犨 =
p ᔴ 꾀 桌 ᆠ e = = 鼀 㝐 - 桌 桌
㡰 盐 � �u 犨� e a h c 矄 ᨨᨨ犨矄 u桌 㦨 e桌a a桌b h桌c C桌d 矄� � �鼀 矄�ᨨ 鼀 ᨨ M҈LT졔A졔桌 㦨 A҈T҈桌aL桌bML2T҈桌cLT҈桌d M҈LT졔A졔桌 㦨 Mc Lb졔c졔d Tacd Aa桌 a 㦨 b 㦨 ҈ c 㦨 ҈ d 㦨 ҈ \
㝐 - 꾀 - 桌 �鼀 矄�ᨨ 矄 ᨨ m 㦨 �鼀 矄�ᨨ 矄 ᨨ Î 㝐 -҈- 꾀 桌
Þ e =
-
\ m 桌 = ê ҈
㦨 㝐 꾀 桌 㦨 桌
㦨 㝐㜠 ҈㜠 桌ᔼ 桌 㦨 㝐㜠 ҈㜠 ᔼ 桌
矄鼀 盐� � 矄泐矄鼀矄 ᨨ 盐 �矄 ᆠ㡰 �鼀 矄�ᨨ 矄 ᨨ m 㦨 㝐 꾀桌 �鼀 矄�ᨨ 矄ᨨ Î 㦨 㝐҈ 꾀桌 �鼀 矄�ᨨ 矄 ᨨ 㦨 㝐 꾀桌 \
҈ ᔴᨨ 盐ᆠ��泐㦨 矄�矄��泐 㦨 꾀
㦨
҈ �矄 �鼀 矄�ᨨ 矄矄 盐矄 ᆠᨨ鼀� �ᨨ 犨ᨨ盐 p pe 鼀
a㦨
Dq 㡰 L DV Dq = = I ᆠ盐 㡰 DV L Dt Dt æ ҈ö ç ÷ e E 矄 矄 �泐 矄�泐 � �泐 è ø
=
黨
ᆠ㡰
盐 �ᨨ盐鼀
- é҈ ù [ 쟼 �泐] ML T \ ê e E ú = = = ML-҈T - 桌 ᨨ盐鼀 ] ë û [ L
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EBD_8341 PS-4
Physics 㡰
d µ r x S y f z = K r x S y f z 犨 ⧤ �矄
bᆠ㡰꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ �鼀 矄�ᨨ 矄ᨨ �盐鼀ᨨ鼀 盐鼀 mvr桌㦨M LT ᨨ 鼀矄矄m桌㦨M LT
�鼀 矄�ᨨ 矄矄 ᆠᨨ 矄
MLT ҈ 㦨MLT ÞT㦨 L
M L҈T = M x L-x M yT - yT - z 쟼⼄盐� � �鼀 矄�ᨨ 矄ᨨ 矄� 矄
mvr M LT = = L-҈ r L ҈ ҈ mv 㡰 E = mv = = L- m E L- L- P= = = = L- t T L - E L F= = = L- x L
㝐ᨨ鼀 盐鼀
p㦨mv㦨
M L҈T = M x + y L-xT - y - z
\x졔y㦨x㦨҈
쟼 �泐
҈ ҈ \ x = - y㦨
ᨨ犨
\n㦨
҈
ᔴᨨ 盐ᆠ��泐 s =
J � /A � t = = E F/q FA
ᨨᆠ 䰘
= é M ҈LT A ù ë û
MLT é a ù = P 桌 Þ a桌㦨PV 桌㦨 ú L ëV û
ê
s=
é q ù - ê ú = Fr 桌 = U ´ r 桌 = ML T 桌 e ëê ûú
L 㦨ML� T
kBT桌㦨U桌 㦨 MLT 桌 �鼀 矄�ᨨ ᨨ l㦨L桌
J � A � t = = = M -҈L-T A 桌 E F q Fa
㡰
\
Y桌㦨
M -҈ L-T Q 桌 MT -҈Q -҈ 桌
= U ´ r 桌 ´
æ ö X �� ÷ çQ Y = Z è ø
= M - L-T Q 桌 㡰
e k BT 㡰
和㦨
nq
矄� �E㦨hv Þ ᆠ├矄ᆠᨨ 矄
b㡰� ᨨ矄��ᨨ 㦨 a桌 和 ҈㦨b桌 꾀ᆠᆠ�ᨨ 㦨桌 � 鼀ᨨ鼀 盐鼀㝐 ҈㦨⼄桌 ᨨᆠ 㝐 㦨 桌
ᆠ㡰
㡰
和 b = =
�鼀 a-b
桌 =
� 鼀ᨨ鼀 盐鼀 �鼀
⼄ 桌
a-b
桌
Þ 㦨⼄졔b - a Þ 㦨⼄졔b - b - 㡰 Þ 㦨⼄b + Þ 졔⼄㦨b ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ
k B T 㡰
=
n q e 㡰
҈ ҈ = = L桌 - n桌r 桌 L 桌 L桌
和㦨
æ q ö ҈ ҈ ´ ç ÷ ç e ÷ k BT 㡰 n è ø
和 =
҈ L- 桌
҈
= L桌 ᆠ㡰 矄� � C =
U 桌 n桌U ´ r 桌
= L 桌
æ q ö ҈ ҈ ´ ç ÷´ ç e ÷ k B T 㡰 n҈ è ø
= U ´ r 桌 ´ U 桌 ´
=
=
҈
a-b + 㦨b \ a 졔㦨2b ᆠᨨ�ᨨ 㡰�矄ᆠᨨᆠ ᨨᆠ =
= U ´ r 桌 U 桌
ᨨ矄��ᨨ
a 桌 = �鼀 = = a-b 和 b 桌 꾀ᆠᆠ�ᨨ =
҈ ´ n = n桌 ´ r 桌 U 桌
= L- 桌 L桌 = L-҈ 桌
é - ù E PS ë ML T û é -҈ ù = = ML T h㦨 = ë û ҈ v éT -҈ ù ë û T
�
æ q ö ҈ ´n ç ÷´ ç e ÷ k BT 㡰 è ø
和㦨
X桌㦨C桌㦨M ҈ L T Q桌 Z桌㦨B桌㦨MT ҈ Q ҈ 桌
-
b 㡰 �鼀 矄�ᨨ 矄ᨨ n桌㦨L桌q桌㦨AT桌
҈ L-҈ 桌 ҈
me
҈ ҈
n
桌
= r 桌 ´
n
桌
= L桌 R =
m V mI㦨eV Þ =
m e
æm ö = R = ç ÷ e I è e ø m I ҈ = = QV㦨RI㡰 㡰 eI㦨mV Þ e V R 㡰
҈ ҈
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PS-5
Physical World, Units and Measurements ᆠ㡰
�鼀 矄�ᨨ 泐� ᆠᨨᆠ I㦨eC V
\
҈ V = =R \ eC I
㡰 \
҈ e
㝐 - = = 㝐-҈ -҈ p ´ -҈ 쟼 㝐 - = 㝐- -҈ ᆠ├ᆠᨨ 矄 h㦨 = v
-҈ l
鼀 ᆠᨨ 盐ᆠ��泐 ⧤ᆠᨨ 盐ᆠ��泐 㦨 꾀D
㝐 - ´ = 㦨 㝐
⧤҈
´ ´ ⧤
ᔴᨨ �ᆠ� ᨨ �矄ᆠᨨ矄�泐h桌㦨
҈ m e
=R
�鼀 矄�ᨨ 泐 ᆠᨨᆠ mC I㦨eV
m V m R = = = ´ e IC C e
҈ 㦨Ҙ ҈ m e
�鼀 矄�ᨨ 泐� ᆠᨨᆠ 黨 ᆠ 㡰 꾀矄 ᆠ├矄ᆠᨨ 矄 h矄 ᨨ �� c ���ᨨ ᆠᨨ 矄 G盐矄 矄矄�ᆠ盐 �矄 ᨨ � L 㝐矄矄M矄ᨨ Lµhx cy Gz �㡰 㝐µhp cq Gr ��㡰 �鼀 矄�ᨨ 矄ᨨ h桌㦨M L T҈桌c桌㦨L T҈桌 G桌㦨M҈ L T桌 矄� � � ᆠ� ᨨ ᨨ鼀ᨨ� �泐 ᨨ �鼀 矄�ᨨ 矄 ᨨ⼄ �㡰
M L T 桌 = M x L x T x 桌 Ly T y 桌 M z L z T z 桌
M L T㦨M xz㡰L x졔y졔z㡰T x y z㡰 졠 ᆠᨨ鼀� �ᨨ犨矄 ᨨ鼀 ᨨ 矄� 矄犨� xz㦨x졔y졔z㦨҈xyz㦨 졠 矄ᨨ� � 矄⼄ 矄犨�
҈ ҈ y㦨 z㦨 hG L = K 錐K�矄ᆠᨨ 矄 \ c
�
ᔼ 矄鼀犨泐矄ᨨ� �⼄ ��㡰犨�
bᆠ㡰矄� �F㦨
C =
҈
m e pe 㡰 r Q 桌 I T = ᆠe桌㦨 㦨M҈L T I F 桌 r 桌 MLT - L ҈ ҈ LT - 桌 = Ҙ桌㦨 -҈ - e 桌C 桌 M L T I 桌 = MLT - I - 桌
҈
bᆠ㡰 泐 ᨨ 矄 盐鼀 㦨ᔴᨨ �ᆠ� ᨨ �ᆠ�ᨨ 㦨MLT桌 � �鼀 矄�ᨨ 矄矄 ᔴ盐��矄 盐 �ᨨ �ᨨᆠ��泐 盐鼀ᨨ ᨨ鼀矄 ᆠ泐� � 矄ᆠᨨ 㡰 ⼄盐 ᆠ泐矄ᨨ 矄 盐 �矄ᆠᨨ �.e.MLT҈桌
ᆠ㡰
ᨨ 鼀
ᆠᨨ 矄 K 桌 = u B
\⧤桌 = 㝐 - ⧤ -҈
q
҈ m e
\ C =
҈ m e
\ m = e-҈L-T QC㦨LT҈桌 F A 㦨 M҈LT҈Q҈桌 �㡰㝐� �ᆠ 盐 f = BA = qv F ��㡰 㝐ᨨ 盐盐矄 ᨨ ��� �泐㦨 㦨 ML҈T桌 A
꾀矄⼄盐矄�ᨨ TµPadbEc ᨨT桌㦨P桌ad桌bE桌c ᨨ MLT҈桌 㦨 ML ҈T桌a ML桌b MLT桌c ᨨ MLT҈㦨Ma 졔 b졔 c Lab 졔 c Ta c \ a졔b졔c㦨 �㡰 ab졔c㦨 ��㡰 ac㦨҈ ���㡰 和ᨨ� �⼄ 矄 �㡰 ��㡰 ���㡰犨� a㦨�⏌b㦨҈⏌c㦨҈⏌
Topic-3 : Errors in Measurements & Experimental Physics ҈
b㡰 � ᨨᨨ ��矄�ᨨ ᨨ ᆠ�ᆠ盐矄ᆠᨨ 矄ᆠ犨�盐�㦨� �ᆠ 㦨�鼀鼀 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐�
�ᆠ ᨨᨨ ��矄�ᨨ ᨨ ᆠ�ᆠ盐矄ᆠ � = 鼀鼀 = ҈ ´ ҈� 鼀㦨҈m鼀 �
=
W Q 㦨 ��ᨨ ᨨ �㦨 m m
�鼀 矄�ᨨ 泐矄鼀 MLT桌 㡰 ᨨ⼄盐t㦨F맔r맔矄� q 犨ᨨ├ ᨨ W㦨F맔d맔ᆠᨨ矄q �鼀 矄�ᨨ 泐t㦨W㦨MLT 桌 �鼀 矄�ᨨ 泐�� 泐㦨 犨 � 㦨L桌
\쟼㦨ᔴ㦨 ҈ 㡰 ␔├ ᨨ犨
C=
䰘
hc 錐K'�矄ᆠᨨ 矄 G Q҈Q
҈ q = 錐C桌㦨M҈LTQ桌 u u ᔼ 盐ᆠ ᆠL㦨 錐 L桌 㦨MLQ桌 � RA L R = \r桌㦨MLT҈Q桌 矄�矄��泐r㦨 l t E ᆠ㡰 矄� � C = 犨 ᔴ㦨矄 ᨨ �� B
ᔴᆠ� ᆠC㦨
x㦨
M = KἘ
꾀 �盐㝐ᨨ鼀 盐鼀㦨mvr㦨 MLT҈桌 L桌㦨Q⏌mL桌㦨LT桌
ᨨ⼄盐t㦨F 맔rt桌㦨MLT桌
҈ æ ö çQ u = K BT ÷ è ø
꾀 盐 ᨨ ᨨ ᨨ �矄 ᨨ矄�� 㡰 꾀� �鼀d㦨��䰘�鼀鼀 ��ᨨ ᨨ Dd㦨䰘黨�鼀鼀 꾀矄 鼀矄盐 盐ᨨ犨ᨨ ���矄 ᆠ�鼀 ᨨ 矄犨矄 ᨨ盐 � 犨ᨨ ���矄 ᆠ�鼀
\ d = �� ± 䰘㡰 鼀鼀 a b ⏌ 㡰 � Z = cd
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EBD_8341 PS-6
Physics ᆠ �ᨨ� Z,
҈
DZ Da Db ҈ Dc Dd = + + + Z a b c d ҈ = ´ + ´ ҈� + ´ + ´ � = ҈�奠
҈҈
㦨
��矄�ᨨ
和ᨨᆠᨨᆠ鼀矄盐鼀 �矄鼀矄盐鼀 ᨨ � �鼀盐�ᨨ ᔴ ᆠ㡰 ᔴᨨ �ᆠ�矄㦨҈㝐和҈和
䰘 ��矄�ᨨ ᨨ � 矄ᆠ ᆠᨨ� ᆠ� 矄 犨� ��矄�ᨨ ᨨ 鼀� 矄ᆠ ᨨᨨ �矄ᆠ�矄泐� ��� 矄� ᨨ ᨨᨨ 鼀� 矄ᆠ ᨨᨨ㦨䰘맔҈㦨䰘鼀鼀㦨䰘ᆠ鼀 � ᨨ ᆠ泐� 㦨鼀矄盐 盐 ᨨᨨ 㦨 ҈졔맔҈㡰䰘㦨䰘ᆠ鼀 㡰 ␔ 矄ᆠ犨 ᨨ 矄ᆠ犨㜠�盐� �矄 �� 矄� ᨨ�ᨨ 矄 � 鼀ᨨ矄泐鼀鼀ᨨ 鼀� 矄ᆠ
=
҈
䰘
D҈㦨A҈졔B҈졔C҈㦨졔䰘졔�㦨
҈
҈ ҈�
D㦨A졔B졔C㦨�졔҈黨҈졔҈㦨䰘 ᨨ ᨨ ᨨ�ᨨ 鼀ᆠ 矄 ᆠ㡰 � � ᨨ 矄�鼀 盐盐鼀l㦨�ᆠ鼀
�鼀ᨨ ᨨ矄ᆠ��ᨨ T㦨�s
�鼀 �ᨨ ᨨ 盐盐鼀
l g
p l p l Þ g = g T Dg Dl DT = + Þ ᆠ�ᨨ ᨨ� �㦨 g l T Dg æ ҈ ö ҈ æ ö Þ =ç ÷ + çè ÷ø = g è � ø � Dg \ ᆠ � ᨨ � g = ´ ҈ = 奠 g M M ᨨ 盐矄㡰 d = = = Ml - V l ҈ æ ҈ö Dd DM Dl + ç = + = è ҈ ÷ø 㦨҈├�⏌鼀 ҈ d M l
㦨�맔҈㦨�맔҈鼀鼀 졔ᨨ㦨맔�맔҈ 鼀鼀 㦨҈�맔҈鼀鼀㦨҈�鼀鼀 � �㦨㝐和 졔ᔴ和 졔ᨨ㡰 㦨��鼀鼀졔 맔�맔҈㡰҈� 㦨��졔҈�㦨�䰘�鼀鼀 ᆠ㡰㝐 �鼀盐鼀ᨨ㦨҈�奠졔 ҈奠㡰㦨�奠 㡰 㝐 �鼀盐鼀ᆠ �ᨨ� 꾀
㦨҈�졔졔҈�졔҈�㦨�奠 ҈
Þ T =
黨
�ᆠ ᨨ ᨨ ��矄�ᨨ
= 奠 ᨨ � 㡰 + 奠 ᨨ � 턼㡰 ҈ + 奠 ᨨ � 㡰 + ҈ 奠 ᨨ � 和㡰 ҈ = ´ � + ´ ҈ + ´ + ҈ ´ ҈�
D㦨A졔B졔C㦨�졔҈黨҈졔҈黨�㦨҈
T = p
҈-
Þ㦨 ᆠ㡰 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐� ᔴ㦨
D㦨A졔B졔C㦨졔҈졔㦨䰘
�ᆠ 盐鼀ᨨ ��矄�ᨨ ᨨ ᆠ�ᆠ盐矄ᆠ
� ´ ҈ - =
�ᆠ =
ᨨ 盐矄㡰
҈ æ ҈ö ´ ҈ + ç ÷ ´ ҈ è ø ��
㦨҈졔䰘㦨奠 b㡰 矄ᆠᨨ盐 ᨨ 鼀� 矄ᆠᨨ 矄ᆠ犨�盐�㦨҈鼀鼀 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐�
=
= � 鼀鼀 �ᆠ � 鼀鼀 = \ 矄 ᆠᨨ盐 ᔴ = � CSD ҈ 鼀鼀 = ҈ mm = ҈ᆠ鼀 = ҈ \
l l ᨨ g = p g T
Dg DR DT ´ ҈ = ´ ҈ + ´ ҈ g Q T
�ᆠ ҈ = = ᆠ鼀 ᔴ�ᆠ盐矄ᆠ ��矄�ᨨ �
黨ö æ = ç҈ - ÷ ´ ҈ = ҈ 鼀鼀㦨҈ᆠ鼀 ҈ è ø
T = p
㡰 �ᆠ├ 矄矄㦨㝐和 � �졔ᔴ�ᆠ盐和ᆠ � � ᔴ㡰 ᔴ =
�
㡰 A㦨䰘맔�黨㦨䰘ᆠ鼀
矄盐矄 ᨨ盐 矄�� � �ᆠ ��盐矄 矄ᨨ A㦨䰘ᆠ鼀 㡰 ␔
㡰 矄ᆠᨨ盐 㦨
㦨
盐ᨨ ҈ᨨ 鼀� 矄ᆠ 盐鼀ᨨ 矄ᨨ �矄ᆠ
� ᆠ鼀 㦨�맔҈ᆠ鼀 �맔҈
� �㦨맔�ᆠ鼀졔맔�맔҈ᆠ鼀 㦨 졔҈�㡰ᆠ鼀㦨҈�ᆠ鼀 �ᆠ├ 矄矄ᨨ 犨�㡰 ҈䰘
㡰 和盐 ᆠ 矄�ᨨ T =
rhg ´ ҈
�ᨨ� 矄盐 ᆠ 矄�ᨨ
DT Dr Dh = + + Q�҈ᆠᨨ 矄 㡰 T r h ᆠ �ᨨ
҈ ´
DT æ ҈ ´ ҈ ҈ ´ ҈ö =ç + ÷ ҈ T è ҈� ´ ҈ ҈� ´ ҈ ø
㦨 졔黨㡰 㦨 ҈黨㡰㦨҈黨奠@ ҈�奠
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PS-7
Physical World, Units and Measurements ҈
D ҈ D D Dᆠ D = + + + ᆠ ҈ = ´ + ´ ҈ + ´ + ´ � 㦨奠 ҈黨
d 㦨 � 졔 맔 ҈ 졔 㦨 �黨 ᆠ鼀
ᆠ㡰 � 㦨҈⏌ ᆠ 㝐 �鼀盐鼀�ᨨ
b㡰 \
ᨨ 犨㦨L㦨鼀
�鼀├ 泐矄ᨨ ᨨᆠ ᨨᨨ鼀 ᨨ 犨
t҈㦨
҈ ö L æ ç 盐矄� �L = gt ÷ è ø g
㝐 �鼀 㦨 �
쟼ᨨ㦨᠀҈ 泐 ᨨ � 盐 ᆠ�ᨨ 㡰 㦨 mA㡰 ´
L v
ᨨ�鼀 ├ � ᨨᆠ矄矄�矄 �� 泐
㦨҈졔 㦨 \
d 㦨
L L + g v
䰘
\
҈ dL
D ҈ + D + D + D +҈+ + = = ҈�
ᆠ㡰
D =
꾀矄 矄ᨨ盐�ᨨ ᨨ 鼀矄盐� �ᆠᨨᆠ├�矄҈� ᨨ 鼀
�鼀矄 ᨨ盐 黨᠀҈� b㡰 ᨨ ᔴ҈ � ᆠ� ᔴ㦨҈㝐和҈和㦨҈鼀鼀黨鼀鼀㦨҈鼀鼀㦨҈ᆠ鼀 Q҈和㦨黨㝐和㦨黨鼀鼀桌 � �㦨㝐和 졔ᔴ맔和 㦨졔 ҈㡰맔䰘㦨䰘ᆠ鼀 ᨨ ᔴ � ᆠ� ᔴ㦨҈鼀鼀҈҈鼀鼀 Q҈和҈҈㝐和㦨҈҈鼀鼀桌 ᔴ㦨҈鼀鼀㦨҈ᆠ鼀 � �㦨졔 ҈䰘㡰맔҈㦨ᆠ鼀 㡰 꾀矄�㦨 p
L
T Dg DL DT 和ᨨ ´҈ = ´҈ + ´҈ g L T ҈ ҈ ´҈ + ´ ´҈ 㦨䰘 ; 奠 㦨 黨 ҈ 㦨 ҈ ᆠ鼀 b㡰 矄 ᆠᨨ盐 㦨 ҈ d҈ 㦨 � 졔 맔 ҈ 졔 㦨 ҈ ᆠ鼀 d 㦨 � 졔 맔 ҈ 졔 㦨 �䰘 ᆠ鼀
Þ
҈҈ � - D =
\
D = -� ᨨ
b㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ
Dt ҈ = t 黨 Dl ҈ l㦨 ᠀҈㡰ᨨ = l Dg 奠=䁸 g
t 㦨 黨᠀҈㡰ᨨ
� 㡰 ᔴ = 㦨҈鼀鼀 � ᨨᨨ㦨�맔҈㦨�鼀鼀 ��㡰 � �㦨 �졔�맔҈㡰졔�㦨鼀鼀
҈
b㡰 㝐矄盐 � ᨨ ᨨ 㦨�ᆠ鼀 ᨨ �和ᆠ犨� ҈㝐� 和ᆠ��矄�ᨨ 㦨҈鼀鼀 黨㝐� 和ᆠ��矄�ᨨ 㦨҈ �和ᆠ��矄�ᨨ 矄ᆠᨨ盐 㦨҈㝐和҈和㦨҈鼀鼀 㡰 ᔼ ᨨ鼀
µl
҈ dL dL + g L v
dT dL ´ ҈ = ´ ҈ ᆠ�ᨨ ᨨ L ҈ L ҈ = ´ ´ ҈ ; ҈奠 ҈
҈ ´ ҈㡰 㦨鼀꾀
㦨├l ᨨ犨��矄�� 쟼㦨҈҈ᨨ ᨨl҈㦨ᆠ鼀 㦨�ᨨ ᨨl㦨ᆠ鼀 ᨨ� � �ᨨ ᨨ鼀D ҈҈㦨⧤ �D㡰㦨⧤
졠 矄盐矄�盐� � �� 盐矄犨� d 㦨
ᆠ盐 ᨨ��ᨨ ᨨ �ᨨ �矄
I = e҈ V ⏌T - ҈㡰 鼀꾀 �� 㡰 ␔ I = �mA e҈ V ⏌T = mA ҈ 꾀矄ᨨ dI = e҈ V ⏌ T 㡰 ´ T
�鼀├ 泐�鼀ᆠ矄ᨨ盐 ᨨᆠ 矄ᨨ
t =
㡰
҈ + �䰘 + �黨 㦨 �黨 ᆠ鼀
꾀矄 犨 ├ ᨨ犨
l p 2l Þ g = g t Dg D l D t æ ö æ ҈ + ´ ҈ ö ᨨ = ± ç + ÷ 㦨 ç ÷ 㦨䰘 黨 ø è l g t ø è t = p
\ 黨
Dg 奠 = 䰘奠 g
b㡰 ᔼ 鼀矄盐鼀 �鼀ᨨ ᆠ泐� 㦨㝐和 졔 和 㡰맔 ᔴ㡰ᔴ㦨 ҈㝐和҈和㡰
æ � ö ÷ = ҈ ᆠ鼀 è � ø
㦨 �҈� - �҈㡰 - ç
\ 㡰
㦨�҈졔맔҈㦨�҈ᆠ鼀
鼀 �鼀盐鼀ᨨ矄矄�ᨨ� Y 盐ᨨl d
DY Dl Dd = + Y l d
矄ᆠᨨ盐 㦨
㦨
�ᆠ ᨨ ᨨ ��矄�ᨨ ᨨ ᆠ�ᆠ盐 矄ᆠ
� 鼀鼀 㦨 � 鼀鼀 ҈
Dd㦨Dl㦨�鼀鼀 쟼ᨨᆠᨨ �盐�ᨨ ᨨ l 㦨
Dl � 鼀鼀 ҈ 㦨 㦨 � 鼀鼀 � l
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EBD_8341 PS-8
Physics 쟼ᨨᆠᨨ �盐�ᨨ ᨨ d㦨
҈
ᆠ㡰
҈ Dd 맔 � 鼀鼀 㦨 㦨 � 鼀鼀 � d
�
ᆠ㡰
�ᆠ
ᨨᨨ ��矄�ᨨ 矄ᨨ ᆠ�ᆠ盐矄ᆠ � = ҈鼀鼀 㦨 �
矄ᆠᨨ盐 =
�鼀r㦨㝐和 졔 ᔴ和 㡰맔 ᔴ㡰
ᨨᨨ㦨�맔LC.㦨�맔҈鼀鼀㦨�鼀鼀 �鼀ᨨ 㦨 � �ᨨ 鼀� 矄ᆠ桌졔 � �ᨨ ᆠ�ᆠ盐 矄ᆠ맔LC桌ᨨᨨ 㦨�맔졔�맔҈�㦨҈鼀鼀
� 㦨䰘鼀鼀 � ҈ Dr ҈ Dr = ᨨ ´ ҈ = 䰘 䰘 r r m m ᨨ犨 矄�泐 d = = V æ r ö pç ÷ èø Dd ´ ҈ ᆠ �ᨨ� 矄�泐 d ì Dm æ Dr ö ü + ç 㦨 í ÷ ý ´ ҈ è r øþ î m Dm æ Dr ö ´ ҈ + ´ ç 㦨 ÷ ´ ҈ m è r ø ҈ = ҈҈奠 㦨奠졔맔 䰘
�鼀r㦨�鼀鼀졔맔
\
䰘
\
㡰
黨
㡰
ᨨ盐鼀ᨨ ᆠ盐 V㦨 l 㦨 ҈맔҈ 鼀㡰 㦨҈䰘맔҈鼀 ÞV㦨҈䰘맔҈鼀 꾀矄 � 矄 犨ᨨ矄�� � �ᆠ ��盐矄矄ᨨ ᨨ盐鼀 矄 矄ᨨ 犨ᨨ 矄�� � �ᆠ ��盐矄 E t㡰㦨Aeµt㦨Aet
\
ᨨ�E㦨ᨨ�At
졠 � �� �犨�
dE dA dt = ´ t E A t 꾀矄 ᨨ矄 犨泐矄 盐
= ´ ҈
) (
⏌鼀
)
㦨�졔�㦨҈҈�
꾀 ᆠᆠ盐泐├ ᨨ犨 \ᆠ �ᨨ� Ӹᨨ盐 �矄鼀ᨨ 盐盐矄
DY DD Dl ᨨ犨 = + l Y D Q 盐ᨨ mg L ᆠ桌 ҈ � + 㦨 ´ 㦨맔�졔� Þ DY㦨Y맔҈҈�㦨맔҈҈҈맔҈҈�㦨�맔҈҈҈ = ´ ҈҈҈ ⏌鼀 � �ᨨ Ӹᨨ盐 �矄鼀ᨨ 盐盐矄㦨 Y졔DY㡰 \ = ( ± ) ´ ҈҈҈ ⏌鼀
dE æ dA ö æ dt ö ´ ҈ = ç ´ ҈ ÷ + t ç ´ ҈ ÷ E A t è ø è ø dE Þ 맔҈㦨맔҈�奠졔맔�맔҈�奠㦨奠 E FL 㡰 Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 Y = A´l \
㦨҈鼀鼀鼀鼀㦨鼀鼀
(
盐� � 盐矄ᨨ DllDT T犨�쟼҈㦨҈�奠 矄鼀ᨨ �쟼҈쟼 쟼
҈҈
ᆠ �ᨨ� 矄�泐
Dr ´ ҈ = ´҈ = 奠 r ҈
l l l Þ T = p Þ g = p g g T Dg ´ ҈ ᆠ �ᨨ� g㦨 g Dl DT ´ ҈ + ´҈ = l T
F A mgL ´҈ ´ 黨 ´ = = l L pD l p ´ ҈- ´ ´҈-
�㡰
Dr = r ҈
҈ ҈ ´ ҈鼀鼀 㦨鼀鼀 ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠ㦨
Y=
m m m 㦨 = l p r v Al
ᨨ鼀 ⼄盐矄�ᨨ 盐� � 盐矄ᨨ Dl 㦨ᆠ鼀 l = ᆠ鼀錐 Dr㦨�ᆠ鼀錐r㦨�ᆠ鼀 m㦨�錐Dm㦨�� ⼄ �㡰犨�
␔├ ᨨ犨 矄ᆠᨨ盐 ᔴ㦨҈㝐和҈和 b㡰 �鼀�ᨨ ᨨ 矄�鼀 盐盐鼀�矄�� 泐
b㡰
矄�泐 r =
Dr Dm Dr Dl = + + l r m r
T = p
㡰
├� �ᨨ� � � ᨨᨨ矄
㡰 ��矄�ᨨ 矄ᨨ �矄ᆠ 㦨҈ ��矄�ᨨ 矄ᨨ 鼀� 矄ᆠ \҈ ��矄�ᨨ ᨨ �矄ᆠ 㦨
Dg Dl DT = + g T l
Dl DT矄 盐鼀ᨨ � �矄├ 鼀 �鼀盐鼀 � ᨨ�ᨨ 㡰 ᨨ 鼀矄盐鼀 ᨨ g�矄鼀ᨨ矄ᆠᆠ盐
ᆠᆠᨨ �盐�ᨨ ᨨ 鼀 �鼀盐鼀ᨨ矄矄�ᨨ� 鼀矄盐鼀 ᨨ y 盐ᨨl d�矄 矄鼀 矄ᆠᨨ盐 ᨨ 矄ᆠ犨�盐�
�ᆠ � = = = ҈鼀鼀 㦨 Dr ��矄�ᨨ 矄ᨨ ᆠ�ᆠ盐矄ᆠ �
㡰 �ᨨ� g
҈
҈ ´ ҈ -� DY DL ´ ҈ 㦨奠 ´ ҈ = ´ ҈ 㦨 Y l � ´ ҈ -�
҈�㡰 矄�泐 r =
M M Þ r = MD - = p V æ Dö pç ÷ è ø
æ Dr ö Dm æ DD ö \奠ç ÷ = + ç = + ´ ҈� = ҈�奠 è D ÷ø è rø m
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PS-9
Physical World, Units and Measurements ᔴᨨ �ᆠ�㦨҈㝐和҈和
æ Dr ö ҈� æ x ö 奠ç ÷ = 奠=ç 奠 è ҈ ÷ø è r ø ҈
æ è
㦨 ç ҈ -
\ x = ҈�
꾀矄黨㝐和㦨҈和
҈㡰
犨ᨨᆠᆠ�ᨨ矄ᔴ҈ ᔴ ᆠᨨ
ᆠ � 矄�矄
\
ᔴᔴ ҈ ҈ ҈ = + Þ ᔴ⼄ = ҈ ᔴ ⼄ ᔴ҈ ᔴ ᔴ҈ + ᔴ
ᔴ ⼄ =
�
㦨 ҈MSD -
Dᔴ Dᔴ Dᔴ ҈ ҈ ҈ = + Þ - = - ҈ - ᔴ⼄ ᔴ҈ ᔴ ᔴ⼄ ᔴ҈ ᔴ
Q҈㝐和㦨҈鼀鼀㡰 � �ᨨ 矄� l㦨㝐和 졔和 ᔴ㡰㦨҈鼀鼀졔҈맔҈鼀鼀 㦨҈҈鼀鼀㦨҈҈ᆠ鼀
é ù Dᔴ҈ Dᔴ Þ Dᔴ = ᔴ⼄ ê + ú ᔴ úû êë ᔴ҈ é ҈ ҈� ù Þ Dᔴ = ´ + ê � � ë ´ ´ úû
ᨨ犨 矄�泐㦨
黨 + ҈� ´ = � � 쟼 �泐矄ᨨ � ᆠᆠ�ᨨ = ҈ ´
҈ ᔴ Þ D Þ ´ ҈ =
和� ᆠ
\ Y㦨
␔├ ᨨ犨
�㡰 䰘
=
D
D 䰘䰘 ´ ҈ = ´҈ = ҈奠 = ҈黨奠
㦨
҈ ´ ҈ ´ ҈ æ � ´ ҈ pç ç è
҈ � ´ ҈- 㡰 ´ ҈� ´ ҈- 㡰
Fl AY
ö ҈҈ ÷÷ ´ ´ ҈ ø
; 鼀鼀
㦨맔҈҈҈⏌
鼀 \㝐 �鼀盐鼀ᨨ矄矄�ᨨ� 鼀矄盐鼀 ᨨ Y DY㦨 黨㡰Y 㦨黨맔맔҈҈҈ 㦨҈黨맔҈҈⏌鼀 ᆠ鼀 㡰
҈鼀鼀 = ҈鼀鼀 ҈
ᔴ盐 矄盐 ᆠ㦨 prl = p
\D 㦨䰘䰘 ᆠᆠ �ᨨ� 鼀矄盐鼀 ᨨ ᨨᆠ �
㡰␔├ ᨨ犨 Dl㦨
� ´ ´ ҈҈ ´ ҈-
�鼀㦨MSR졔CSR맔 ᔴ㡰 㦨҈鼀鼀졔䰘맔 ҈㡰鼀鼀㦨҈䰘鼀鼀
( ) ( ) ( )
X F L ´ 㦨 L X pD
W㦨�錐D㦨�ᆠ鼀㦨�맔҈鼀錐 X㦨҈�ᆠ鼀㦨҈�맔҈鼀錐 L㦨҈҈ᆠ鼀㦨҈҈맔҈鼀
矄ᆠᨨ盐 ᔴ㦨
和盐矄�盐� � 盐矄� ⼄ �㡰
�
F A
æ DY ö æ DD ö DX DL = ç + + çè ÷ è D ÷ø X Y ø 鼀 L æ ҈ö æ ҈ö æ ҈ ö 㦨 ç + + = 黨 è � ÷ø çè ҈� ÷ø çè ҈҈ ÷ø
æ҈ ҈ö 盐±D盐㦨 䰘��㡰± ç + ÷ 㦨 ±�㡰ᆠ鼀 è ø æ҈ ҈ö ±D㦨 ҈�䰘�㡰± ç + ÷ 㦨 ±�㡰ᆠ鼀 è ø
+
ᨨ盐 � �ᨨ ᨨ矄�� � �ᆠ ��盐矄㡰 ҈黨×҈҈N⏌鼀㡰
㝐 �鼀盐鼀ᨨ� Y
㦨�졔㦨҈奠
盐矄 ᆠ �ᨨ� ᆠᆠ盐�ᨨ ᨨ �泐矄ᨨ �矄҈奠 ҈黨㡰
�
m 䰘 g 鼀矄矄 = = = �⏌ᆠ鼀 ᨨ盐鼀 l ҈҈㡰
Ӹᨨ盐 �矄鼀ᨨ 盐盐矄Y =
D Dᔴ D ´ ҈ = ´ ҈ + ´ ҈ ᔴ⼄ ´� ´҈ + ´ ´҈ �
҈ ҈ - = 盐 D D盐 D \+ + = 盐 ҈ ҈ ҈ ҈ ҈ ҈ = ᨨ犨 - = Þ 盐 - ҈ ҈ Þ = \ = ᆠ鼀
� 鼀├� � ᨨ � 矄ᆠ 犨�ᆠᨨ� ᆠ� 犨� 鼀�
矄ᆠᆠ盐矄矄ᆠᨨ盐 �矄҈鼀鼀 �ᆠ鼀㡰 ᔴ㦨҈MSD ҈VSD
黨 MSD Q黨MSD 㦨҈VSD㡰 ҈ 黨ö æ æ ҈ ö ҈ MSD 㦨 ç ´ ҈÷ 鼀鼀 㦨҈鼀鼀 㦨 ç҈ - ÷ MSD 㦨 è ҈ ø è ҈ ø ҈
´ = � �
=
黨 ö ÷ = ҈ 鼀鼀 ҈ ø
㦨
D l 㦨pDl
맔҈䰘맔�鼀鼀㦨�䰘ᆠ鼀 䰘
㦨ᆠ鼀 ᨨ盐 � �ᨨ ᨨ犨ᨨ矄�� � �ᆠ ��盐矄㡰 b 㡰
鼀 =
� + � + �䰘 + � + �黨 = �� » � 和
쟼ᨨ� � � 鼀 ҈錐 鼀 錐 鼀 錐 鼀 錐
鼀 �
㝐 ᨨ =
+ + ҈ + + = ҈ » 和
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EBD_8341 PS-10
Physics
\
奠ᨨ� 鼀矄盐鼀 ᨨ �
ᨨ�鼀 㦨 矄ᆠᨨ 矄 =
ᨨᨨ ᨨ矄ᆠ��ᨨ 矄 DT DT ҈ ҈ \ = Þ = Þ DT = �矄ᆠ T
�鼀�ᨨ T =
D
´ ҈ = ´ ҈ = �䰘奠
� 奠ᨨ� 鼀矄盐鼀 ᨨ �
D� D
æ D + D ö ´҈ = ´҈ + ç ÷ ´҈ �
è - ø æ ҈ +҈ ö 㦨 ( �䰘 ) + ç ÷ ´҈ » ҈҈奠 è - ҈ ø
黨
ᨨ犨 ᨨ鼀矄盐� �ᨨ� �g
\ � = p
D ҈ ´ ҈ = ´ ҈ = ҈奠 ҈
\ᆠ �ᨨ� �g
bᆠ㡰 �ᆠ�矄
҈ᆠ鼀 㦨҈�ᆠ鼀
�和㦨㝐和 \҈和 =
�
´ ҈� 㦨҈ᆠ鼀 �
ᔴ㦨҈㝐和҈和 㦨҈�ᆠ鼀҈ᆠ鼀㦨�ᆠ鼀 和ᆠ犨�盐� ᔼ �ᆠ ᨨ 矄ᆠ犨�盐��矄犨�ᆠ ᔴ ᨨ �ᆠ�矄 �ᆠ 㦨맔ᔴᨨ �ᆠ�㦨맔�㦨�ᆠ鼀 ᔴᨨ 矄ᆠ犨�盐�
�ᆠ
ᨨ ᨨ ᨨ ��矄�ᨨ 矄 ᨨ ᆠ�ᆠ盐 矄ᆠ � ᆠ鼀 㦨�ᆠ鼀㦨�鼀鼀 㦨 ҈
�
ᨨ犨� 矄ᆠᨨ盐 ᨨ � 矄ᆠᨨ 矄ᆠ犨�盐��矄 犨�ᆠ 矄ᆠᨨ盐 ᨨ �ᆠ�矄
ᔴᨨ � 矄ᆠᨨ 矄ᆠ犨�盐�㦨맔�㦨�ᆠ鼀
�ᆠ 㦨맔�㦨҈ᆠ鼀
҈ \ᔴᨨ 矄ᆠ犨�盐�㦨 ᆠ鼀㦨҈ᆠ鼀㦨҈鼀鼀 ҈ l 㡰 Q 矄� q㦨 l \ 㦨 ᆠᨨ矄ᆠ q �㡰 㡰 l = [ - ᆠᨨ矄ᆠ q ᆠᨨ q] \ q l \ 㡰㦨 ᆠᨨ矄ᆠqᆠᨨq q ��㡰
Dg Dl DT ´ ҈ = ´ ҈ + ´ ҈ g l T Dg æ ҈ ö \ ´҈ = + ç ÷ ´҈ 㦨�奠 g è ø Dr D ҈ - a㡰 D ҈ + a㡰 æ ҈- a ö = + b㡰 r = ç ÷ Þ r ҈ - a㡰 ҈ + a㡰 è ҈+ a ø Dr Da Da Da ҈ + a + ҈ - a 㡰 = + = Þ r ҈- a ҈+ a ҈ - a 㡰 ҈ + a 㡰 Da ҈ - a 㡰 D a \ Dr = = ҈ - a 㡰 ҈ + a 㡰 ҈ + a 㡰 ҈ + a 㡰 ᆠ㡰 矄� � N = N e -lt Þ N = N - l t
҈ dN = -t N dl DN Dl = = = Nt ´ ҈
Þ
�
n졔҈㡰和㦨n㝐和 \
҈和㦨
n ö æ = ç҈ è n + ҈÷ø 㝐和 ҈ a 㦨 㝐和㦨 盐 �矄Q ҈㝐和㦨盐 �矄桌 n +҈ n +҈
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈
ᆠ㡰
҈ 㦨 ҈桌
ᨨᨨ ᆠᨨ 矄盐 �ᨨ盐鼀㦨桌鼀���泐ᨨ
꾀矄 q � ᆠ矄矄 ᨨ鼀 ᨨ 黨 ᆠᨨ q ᆠ矄矄 ᨨ
꾀 桌 矄ᆠ Îᨨ 㦨 㝐҈ 꾀
㡰 ᆠ矄矄
Dt DT ҈ = = t T
ᔴ � e㦨ᔼ㦨꾀 桌 �盐 ⼄盐 ᆠ泐 w㦨
㡰 㦨ᆠᨨq q
\
n 㝐和 n +҈
矄ᆠᨨ盐 ᔴ㡰㦨҈㝐和҈和
�� � � ��㡰泐 �㡰犨�
�.e. ᆠ�ᨨ ᨨ� d ᆠ矄矄 ᆠ㡰 � ᨨ 矄� �ᨨ 矄�鼀 盐盐鼀�矄 ᆠ泐鼀 �� 㡰 ᨨ ᨨ� � Dl㦨 盐 ᨨ矄矄���泐ᨨ ᨨ� 鼀矄盐� ��鼀�矄҈矄� 矄矄 矄 ᆠᨨ盐 ᨨ 矄ᨨ 犨ᆠ �矄 ҈矄
� ��ᨨ 犨� 矄ᆠ ᨨ l
㦨
�҈
l
奠ᨨ� 鼀矄盐鼀 ᨨ r
҈㝐和㦨
�
Þ T = p l g g
盐矄� � T = p
ᨨ犨矄盐矄�盐� � �鼀 矄�ᨨ 矄ᨨ ᨨ⼄盐 ��矄犨�
Ne 㦨 m Îᨨ
L- AT M M -҈ L- AT
= T - = T -҈ 㦨w
Ne c m Îᨨ = pn = p Þ l = pc l m Îᨨ Ne
b㡰
wp =
㦨 ´
҈ - ´҈-҈҈ ´ ´ ҈ 㦨 鼀 䰘 ´ ҈䰘 ´ ҈ ´ ҈ -҈黨 㡰
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PS-11
Motion in a Straight Line
Motion in a Straight Line
2
Topic-1 : Distance, Displacement & Uniform Motion ҈
b㡰 � v㦨 b x ᨨ
dx = b x҈⏌ dt
ᨨ
x
t
-҈⏌ ò x dx = ò bdt
ᨨ꾀 �� � 矄 ᨨ �鼀 ᨨ 泐 �矄 犨ᨨ � ᨨ矄��ᨨ 矄 A B � �� ᨨ矄��ᨨ
�鼀 � 犨 �ᆠ �矄 ᨨ ᨨ矄矄�
Topic-2 : Non-uniform Motion
x҈⏌ bt 㦨t ᨨ x㦨 ҈⏌ � �� �犨�鼀犨�
҈
㡰
ᨨ
dx b ´ t = dt
�
�鼀
vu㦨aS 盐 a =
�矄 ᆠ
®
C
Þv㦨 ҈
R O �矄ᆠ鼀
B
쟼
和
æFö \ v = u - ç ÷ S èmø
҈ ҈ 鼀⏌矄 = 䰘鼀⏌矄 Þ v =
b㡰 x㦨at졔btct
dx d = at + bt + ct 㡰 dt dt 㦨a졔btct
ᨨᆠ�泐 v =
꾀ᆠᆠ�ᨨ
® vf
Dv ␔
F m
é � ´ ҈- ù Þ v = ҈㡰 - 㡰 ê ú - ëê ´҈ ûú ҈
p 㡰 �矄ᆠ鼀 㦨 矄 ᨨ矄 �矄 ᆠ
pr 㦨 pR A b㡰 꾀� ᆠᆠ�ᨨ
uur ur uur ur r r v f - v� v f + -v� 㡰 D v a= = = t t t
҈ æ ҈ ö ҈ � � 䰘 鼀 + = çè + ҈÷ø ´ + ´ ´ = b㡰 ᨨ鼀 � ⼄盐�ᨨ ᨨ 鼀ᨨ�ᨨ
=
r ҈ = ´ = 鼀⏌矄 Q r㦨҈鼀錐t㦨҈矄㡰 t ҈
㦨 ACB =
� 矄㡰
҈ ҈ = ҈ � SD = - = �矄 ᆠ ᆠᨨ 泐 ᨨ 泐㦨 ᨨ v㜠t � 㦨 OABS㡰졔 SCD㡰
�矄ᆠ鼀 b㡰 꾀�ᨨᆠ�泐㦨 �鼀
㦨AB㦨AOB㦨 R �矄 ᆠ 㦨 �
t
OS = +
b t ᨨ v㦨 b㡰 矄� ᨨ�ᨨ ᆠ㡰ᨨ矄��ᨨ 㜠 �矄 ᆠ �鼀 ᨨ�ᨨ 㡰 ᨨᆠ�泐㜠 ᨨ矄��ᨨ ᆠᨨ矄ᨨ � � ᨨ ᨨᆠ�泐㜠�鼀� ᨨ�ᨨ 㡰 �矄 �矄 ᆠ㜠�鼀� �矄 矄 ��
ᨨ犨 ᆠ �矄 ᆠ㜠�鼀� ᨨ�ᨨ 㡰 �矄 � ᆠᨨᆠ
㦨
t㦨t㡰
A B v 鼀⏌矄㡰 S D O ҈� C
æb ö ÷ è c ø
ᨨ㦨bc맔t \ t = ç
vf㦨졔�鼀⏌矄
ö æ æb ö æ b ö = a+b c ç ÷ ÷ ç ÷ c ø è è c ø è c ø
v㦨 a + b ç
㜠v� 㦨졔�鼀⏌矄
r Dv = v f + v� = � + � = � 鼀⏌矄 r r Dv � ҈ = = a = 鼀⏌矄 � ᨨ 㜠犨矄 �ᆠ�ᨨ Dt ҈
dv d a 졔 bt 㜠 ct 㡰 㦨 dt dt
㡰 ᨨ ᆠᨨ 矄 ᆠᆠ�ᨨ �矄 矄�� � ® ᨨ㜠 �矄ᨨ a - t ® ® ᔼ ᆠ� 矄�� � ᨨ v - t ᨨᨨ x - t
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EBD_8341 PS-12
�
Physics
ᆠ㡰 �鼀├ 泐꾀ᨨᆠ � �矄 � �ᨨ� �矄 \ �鼀├ 泐ᨨᆠ � �矄 � �ᨨ� 㦨졔
ᨨἘtἘᨨ鼀� �鼀盐鼀
x
盐㦨
҈é ҈ x ê- + v êë d + x
꾀 㦨҈
҈ ҈ = 꾀 = ҈ = + 㡰 Þ ҈ = ( + ) Þ Þᨨ㦨
(
)
҈
(
҈ +
҈
)
㦨 ҈ +
ᨨ = ҈
ᆠ㡰 矄� �⼄盐�ᨨ a㦨
v u ҈ S㦨ut졔 at t
�矄 ᆠ 泐ᆠ� ҈�矄ᆠ㦨
҈ �㡰 ҈�㡰 ҈�
䰘� 鼀 �矄 ᆠ 泐矄ᆠᨨᨨ� ҈�矄ᆠᨨ 矄㦨맔҈�㦨� Q �矄 ᆠ㦨矄 맔�鼀㡰 � ᆠ 犨 �矄 ᆠ 泐ᆠ 矄ᆠᨨᨨ � ҈�矄ᆠ�䰘�㦨҈҈�鼀 ᆠᆠᆠ 矄 矄ᆠᨨᨨ � �鼀錐
䰘
QM - x t҈ = �㡰 v
�鼀├ ᨨ �矄 ᆠἘRP Ἐ� �
t =
d + x v
��㡰
ᨨ�鼀矄 ᨨ鼀ᨨ ᆠ ᨨ鼀QᨨP
t = t҈ + t =
QM - x d + x + v v
턼
㝐
d = ᨨ x = - ҈ ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ᨨᆠ �矄 鼀ᨨ� � 犨� ᆠᨨ 矄 �� ᆠᆠ�ᨨ � 㦨 ᆠᨨ 矄 ᔴ㡰
黨
├ = - ᔴ + ├ = + ᔴ ᔴ ᆠ � 㡰 矄 矄 ᆠᨨᆠ泐 b㡰 �矄 ᆠ ᨨ � � � �矄ᆠ鼀 s㡰 㦨t Q s µ t �� 㡰 泐 ᨨ盐 � ��ᨨ ᨨ �矄ᆠ鼀 犨 � ᆠᆠ�ᨨ
ds dt dv d t = = t = = t a = dt dt dt dt a㦨tᨨ a µ t ᆠ � 㡰 �矄 ᆠᨨᆠ ҈ 㡰 ⼄盐�ᨨ ᨨ �� v㜠x � �矄 v v㦨 x + v �㡰 x
V=
v dv =- dx x
㦨
䰘� + � t ҈�㡰 = t 䰘�졔�䰘�㦨 Þ ҈�㦨䰘� Þ 㦨�矄ᆠ 㡰 ᆠ盐 ᨨ �� 犨泐 �矄 ᆠἘxἘ ᨨ鼀 ᨨ� 㝐和ᨨ 㝐㦨x 꾀 � 矄 ᨨ ᆠ� � �矄v �鼀├ 泐 ᆠ ᨨᆠᨨ �矄 ᆠ QR㦨 QM x ᨨ �� 犨泐
= -ᔴ 㦨ᔴ
= ( ) =
҈ =
盐� � �矄盐ᨨ � ⼄盐�ᨨ �㡰
ù ú= úû
d
㦨 졔㡰 꾀㦨 Þ 㦨҈ 졔㡰㦨 ҈㡰 �㡰
dt = dx
\a㦨 v Þ
ù v é v dv v = ´ v = ê x + v ú ᨨ鼀 �㡰 dx x x ë x û
v v a㦨 x x x
��㡰
졠 ᆠᨨ鼀� � ⼄盐�ᨨ ��㡰 犨� ⼄盐�ᨨ ᨨ 矄�� � y㦨mx졔c
v
犨�m㦨 = + x
� q 㦨졔� q �矄ᆠ盐
v 꾀矄ᨨc㦨 -
x � y㜠� ᆠ �矄 �� ᆠ� 㡰ᆠᨨᆠ泐 �ᆠ矄 ᆠᨨ矄ᨨ � �a㜠x � ҈҈ b㡰 ᔴ � � ᨨᆠ�泐㦨 盐 ᆠᆠ�ᨨ 㜠�鼀 � ҈ 㦨 ´ ҈ ´ ҈҈ = �� 鼀 ⏌ 矄 和� ᆠ � �� ᨨᆠ�泐�矄 ᨨ � ᨨᆠ�泐� 鼀 �鼀盐鼀 ᨨᆠ�泐�矄��鼀⏌矄
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PS-13
Motion in a Straight Line ҈ 㡰 A
鼀矄 ҈ a㦨鼀矄 x B
鼀矄 ҈
鼀
턼
鼀
ᨨA
u҈ = 鼀矄 -҈ a҈ = -鼀矄 - s҈㦨xt҈㦨t ҈ a҈t҈ x㦨t t
s҈ = u҈t҈ +
\
�㡰
ᨨB u㦨鼀矄҈a㦨鼀矄s㦨xt㦨t
҈ s㦨ut졔 x㦨t 졔t 꾀 � �⼄ �㡰 ��㡰
҈
t㦨
��㡰
㦨矄 �
u
㡰
A
B t � �矄 ᆠ 㦨꾀 ᨨ 矄 㜠�鼀 � O
҈ ´ � ´ = 鼀 ҈ 㡰 �矄 ᆠ�矄犨� �鼀矄x㦨at졔bt졔c =
Þ x
和 ҈
ᆠ �� ᨨᆠ�泐 = = 鼀⏌矄 �鼀 ҈ ᨨ QS�矄 �ᨨ 㦨鼀⏌矄 ᨨ OPa ᆠᆠ�ᨨ 㡰>鼀⏌矄 ᨨ PS ᆠᆠ�ᨨ 㡰鼀⏌矄 ᨨ ORᆠᆠ�ᨨ a 鼀⏌矄 ᨨ RS �ᨨ a >鼀⏌矄 ᆠa³矄ᨨ鼀ᨨ� ᨨᨨ� 矄� �矄 ҈ �㡰t҈ �鼀├ 泐 ᆠᨨ� 鼀 �鼀盐鼀 ᨨᆠ�泐vm 犨 � � �矄 ᆠᆠ�ᨨ 矄� �㦨u졔at v vm = + at҈ ᨨt҈ = m �㡰 a 和� ᆠ ᨨ �鼀 矄 �矄 t ᆠ ᆠᨨ矄 ᨨ �鼀t㦨 tt҈㡰ᨨᆠᨨ鼀泐矄a㦨b v㦨 矄� v㦨u졔at v = vm - b t - t҈ 㡰ᨨt҈ = t + m ��㡰 b 矄� � �㡰� ��㡰犨� =
\
㦨� \
졠 ᨨ OQᆠᆠ�ᨨ a㡰
vm v æ ҈ ҈ö = t - m ᨨt = vm ç + ÷ a b èa bø t ab ᨨ vm = a + b㡰
dx dx at + b㡰 dx = at + b Þ x = at + b Þ = dt dt x dt
���㡰
Þx
d x æ dx ö +ç ÷ =a dt è dt ø
æ dx ö æ at + b ö a -ç ÷ a -ç ÷ d x è dt ø = è x ø Þ = x x dt
=
ax - ( at + b )
2
=
ac - b
x x Þ µ ᆠ 㦨 ҈� ᆠ 㡰 꾀t㦨 t㦨҈ᨨ 泐�矄矄 ᔼ ��泐 a �矄 ᨨ矄�� 矄ᨨ ᨨ 泐 ᆠ⼄盐�矄 矄ᨨ鼀 ᨨᆠ�泐 a 矄 ᨨ盐 �� 矄ᨨ ᨨ 泐 ᆠᨨ鼀矄ᨨ矄 ᆠa ᆠ
ᨨ鼀� ᨨ矄�� ᨨ�鼀 � � £ t £ ҈
ᨨ盐 泐�矄 �ᆠ � ᨨᨨ犨� � v㜠t �
ᨨ�鼀ᨨ ᨨ盐 泐㦨҈矄ᆠ
ᨨ �矄ᆠ鼀 㦨҈鼀㦨꾀盐 v㜠t㡰� v鼀 =
��㡰 ᨨ �矄 ᆠ 㦨 ᨨ DABC ҈ ҈ 㦨 ´ 矄 ´ �盐 㦨 ´ t ´ v鼀
s ´҈ = = 鼀⏌矄 t ҈
҈ æ ab ö ab ҈ t 㦨 ç ´t ´ è a + bø÷ a+b Topic-3 : Relative Velocity in One Dimension 㦨
҈
㡰 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ � A B 盐
� � ᨨ
ᆠ├矄 � ᨨᨨ矄� �ᆠ�ᨨ
҈├鼀⏌
├鼀⏌ A
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EBD_8341 PS-14
Physics 꾀矄 ᨨᆠ�泐ᆠᨨ鼀ᨨ ᨨ A Bᨨ � 矄鼀� OA
VA = ├鼀⏌ 㦨҈鼀⏌矄
䰘├鼀⏌
VB = -䰘 ├鼀⏌ 㦨鼀⏌矄
VMA㦨҈├鼀⏌ 㦨�鼀⏌矄 V鼀 B㦨V鼀 A졔VA, B 㦨V鼀 A졔VA VB㦨�졔҈ 㡰 㦨�졔㦨黨�鼀⏌矄 㡰 턼 �矄 ᆠ턼㦨맔 ᨨ �矄 ᆠ㦨矄 맔 �鼀㡰 �矄 ᆠ턼 㦨
ᆠᨨ矄 ° =
\
t0 =
vA ҈ = = 鼀矄-҈ ᆠᨨ矄°
�矄ᆠ鼀 � � = = = �矄 ᨨᆠ�泐 vB 矄� ° ´ ҈
æ ö ç ÷ 쟼ᆠ 矄ᨨ KLMN鼀ᨨ矄犨� 盐 � ᨨ鼀矄 è ø v矄盐ᆠ K犨泐矄 鼀ᨨ �ᆠ泐ᨨ犨 矄LL �ᆠ泐 ᨨ犨 矄 M M �ᆠ泐ᨨ犨 矄 N N �ᆠ泐ᨨ犨 矄 K ᨨ 矄�矄ᨨ 矄泐鼀鼀泐犨ᆠ 矄泐 KLM N犨�鼀 ᆠ ᨨ 矄⼄盐ἘOἘ 꾀 泐� 矄 ᨨᆠ�泐ᆠᨨ鼀ᨨ ᨨ � KO 㦨 v ᆠᨨ矄 �
\
҈ ³矄ᆠᨨ³ 矄ᆠᨨ 和 � � 矄ᆠᨨ 犨� ᨨ盐 犨├� � 矄 �矄 矄ᨨ 矄 矄 犨├� � ᨨ 泐�矄
和ᨨ 矄ᨨ 矄 矄 �矄
҈ ҈ ҈� ³矄ᆠᨨ³ + = ҈ 矄ᆠᨨ
和ᨨ �鼀ᨨ�ᨨ盐 矄ᆠᨨ = �㡰
҈ 㦨 矄ᆠᨨ �
꾀㦨҈
꾀
º
矄�
�
º º
ᆠ鼀
ᆠᨨ矄
졠
�ᨨᆠ�泐ᨨ B犨� 矄ᆠᨨA�矄 �ᆠ盐 ᨨ � ᨨ 鼀ᨨ�ᨨ ᨨ A�OA
黨
㝐
O
d
⧤
�
�矄 ᆠ
�鼀├ ᨨ鼀ἘOἘ = ᨨᆠ�泐
ᨨ t = 䰘
v
�矄 ᆠKO㦨dᆠᨨ矄� 㦨
҈ ³矄ᆠᨨ³ 矄ᆠᨨ ␔├� � 犨� 矄ᆠᨨ �ᨨ� � 矄
�
vB =
盐矄
= ҈ 矄ᆠᨨ ᆠ㡰
Þ
㦨
鼀⏌矄ᆠ
鼀 � 盐ᔴ㦨盐㦨ᔴ㦨鼀⏌矄㦨鼀⏌矄 ᆠ�ᆠᆠ�ᨨ ᔴ 㦨 鼀⏌矄ᆠ ҈ ᨨ犨犨├ ᨨ犨 矄 = 盐 + ҈ = ´ Q 盐 = ᆠ ᆠ犨�ᆠᆠ 盐犨� 盐矄 �鼀
vBᆠᨨ矄 㦨vA㦨 ҈
졠矄㡰
鼀⏌矄ᆠ ᔴ
ᆠ㡰
턼 턼
҈ ´ = Þ =
\
B
d
v
=
d v
矄㡰꾀矄 ᨨ � �矄 鼀ᨨ� � � �ᆠ�ᨨ ᨨ 矄 ᨨ�ᨨ ᨨ � ᨨᨨ矄� �ᆠ�ᨨ 和ᨨ 矄矄盐 犨� � b㡰 ᨨ鼀ᨨ� �ᨨ矄 ҈
泐ᨨᆠ矄 ᨨ鼀ᨨ � ᨨᨨ矄� �ᆠ�ᨨ �矄 �矄 �矄 ᆠ盐矄 �盐 矄 ᨨ 泐 ᨨᆠ犨ᨨ矄�矄�꾀矄 ᨨᆠ 鼀ᨨ矄 犨泐 �盐 ᨨᆠ�泐 ᆠ矄矄 ᨨ �矄 矄 矄鼀矄 ᨨ q 矄矄 �矄 ᨨᆠ 鼀ᨨ矄 q҈ 鼀� 矄 矄�ᨨ 泐
Q q > q҈ \w > w҈ ᨨ鼀 ᆠᨨ ᆠ ᨨ � ᨨᆠ�泐 r r r ҈ 㦨 ҈ 犨 �矄 ᨨᨨ泐 鼀 㡰 㡰 ᨨ Bᨨ � ᨨ泐㜠A�ᨨᆠ�泐 ᨨ 犨AvBA 矄 ᨨ盐 ᨨ �OA和ᨨvBA 犨�鼀├
�� 犨� 졔 㡰X㜠 �矄 B v BG 㡰 q㦨� vTG q ´ � = = ° � \ f㦨 ҈� 犨� 졔X㜠 �矄 � A ҈� ᔼ DOMA f q 㦨� q 㦨� Þ ÐOAM㦨� O
M
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PS-15
Motion in a Straight Line \ ÐOAB㦨҈� 꾀矄ᨨÐBOA㦨 � 㦨҈� 矄� �矄� 犨� DOBA
ᨨ
�
vT 矄� ҈�° - ҈㡰 ´ 䰘҈ = ; 鼀⏌矄 矄� ҈�° � ᆠ矄 ᨨ 犨�ᨨ盐 vB㦨鼀⏌矄
ᨨ犨v㦨u졔at Þ
҈ s = ut + at
Þ -h =
gh t -
҈ ҈ gt Þ gt - gh t - h =
gh ± gh + ´ ᨨ t =
´
g ´h
ædö
䰘
矄� � S = ut +
꾀矄⼄盐�ᨨ �矄⼄盐 �ᆠ矄ᨨv㜠h� 犨�ᨨ v
Þ 㦨 t h=d
d ҈
ᔼ ��泐ᨨᆠ�泐 �矄 ᨨ犨 犨 矄 㡰 ᆠᨨ�矄�ᨨ �矄矄�矄 �ᆠ�ᨨ 犨� 矄矄鼀� �盐 � ᨨᆠ�泐 �矄 盐犨 矄 졔㡰 ᨨ �鼀t㦨ᆠᨨ矄ᨨ 矄ᨨ ᨨ� ᨨ � 犨 h㦨d �鼀ᆠᨨ�矄�ᨨ ├矄ᆠ 㡰 ᨨ ᨨ 泐 ᨨ犨 �ᆠ泐 盐犨 矄 ᆠᆠ�ᨨ
鼀� 矄 ᆠᨨ 矄 a 㦨 g㡰 ᨨᆠ�泐 泐�鼀 t �矄 �� 泐V㦨 u gt 盐� � �矄 ᨨᆠ�泐 ᆠ矄矄 � 泐 盐� � ᨨᆠ�泐 � ᆠ矄矄 � 泐 �ᆠ�ᨨ �矄 ᨨᨨ矄� ᨨ ᆠ ᨨ ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 ᨨᆠ�泐 矄盐矄 �鼀 b㡰 泐҈㦨҈�錐泐㦨� ᨨ 泐҈㦨鼀㦨矄 \ 泐泐҈㦨 ᨨ矄
㦨gt
Þt=
҈ ® V� ᆠ矄矄 ᨨ犨 犨 矄 h ® ᨨᆠ�泐ᆠ �矄�矄 �ᆠ�ᨨ
® V ᆠ矄矄盐犨 矄
҈ ҈ gt Þ S = ´ t + gt Q S = ҈m桌
g
… �㡰
æ ҈ö ҈ sᨨ 泐矄 �矄 ᆠᨨ ҈黨m矄ᨨ� ç t ÷ è ø �矄 ᆠ 㦨 ҈
ᔼ 矄
ᆠᨨ�矄�ᨨ ├矄 ᆠ
v æ vö çè ÷ø = ᨨv㦨 v Ἐ Þ v Ἐ = vἘ 꾀矄 �ᆠ�ᨨ �矄 矄 矄 �矄 ᆠ矄 矄ᨨ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 矄 ᆠᨨ ��ᨨ 矄 㡰 ├矄 �鼀ᨨᆠ �ᨨ盐 \
ᨨ t =
v = u ± gh
nu = ᨨ鼀⼄盐矄�ᨨ 㡰 g g Þ gH㦨n n㡰u 㡰 ᨨ 盐 � ᨨ鼀泐 ᆠᆠ ᨨ ᆠ�ᨨ 鼀ᨨ�ᨨ
v = u ± gh ⼄盐�ᨨ �矄 ⼄盐 �ᆠ ᆠ v h � 犨� ᨨ ᔼ �� ᨨᆠ�泐 ᨨ犨 犨 矄 矄ᨨ �� ᆠᨨ�矄�ᨨ � 矄矄 �矄 �ᆠ�ᨨ 犨� 矄鼀 盐 ᆠ ᨨᆠ�泐 �矄 盐犨 矄 盐 v Ἐ = g ´ ç ÷ = gd 錐 è ø
g
h gh h ҈ + 㡰 Þ t = ҈ + 㡰 ᨨ t = g g g ᆠ㡰 ᨨ 盐 � ᨨ鼀泐ᆠᆠ ⏌ ᆠᆠ 鼀ᨨ�ᨨ
u g
u + u + gH
Þt =
gh
ᨨ犨ᆠ├犨�矄鼀ᨨ� �盐 ��泐 ἘtἘ �鼀 ├ 泐 ᨨᨨ ᆠ├ ᨨ ᆠ �ᨨ盐
u + gh = -u + gt
�鼀├ ᨨᆠ �� 矄 ᨨ� �矄 t =
ᆠ㡰 ᨨ 盐犨 鼀ᨨ�ᨨ ᨨ �ᆠᨨ
v = u + gh Þ v = + gh Þ v =
盐
Topic-4 : Motion Under Gravity
ᆠ㡰
v㦨 u + gh
vB =
҈ � 和 ᨨ ᆠ � � �ᨨ盐
泐泐҈㦨
vB vT = 矄� ҈�° 矄� ҈�°
҈
>矄
ᨨ犨
҈ æ ҈ö æ ҈ö g ç t ÷ = ҈ \ g ç t ÷ = ҈´ è ø è ø
҈´ æ ҈ö Þ çt ÷ = g è ø \
҈ ҈ = ҈ ´ 㡰 g
Þ
g = ҈ 黨 㡰
Þ
ᨨ鼀⼄ �㡰
g =
\�㦨m⏌s
T盐㡰꾀矄 矄ᆠ├ᨨᨨ� ᨨ ᨨᆠ�ᨨ 矄ᨨs㦨 \ vu㦨맔g맔s 㦨 \ v㦨u
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EBD_8341 PS-16
Physics
Motion in a Plane
3
Topic-1 : Vectors ҈
㡰 矄� � R㦨A҈졔A졔A҈Aᆠᨨ矄q �㦨졔�졔맔맔�ᆠᨨ矄q ᨨ ᆠᨨ矄q㦨 ®ö æ ® çè A҈ + A ÷ø
�ˆ
®ö æ ® ç A҈ - A ÷ 㦨A 맔A ҈ ҈ è ø
҈
㦨A҈졔黨A҈Aᆠᨨ矄qA҈Aᆠᨨ矄q A 㦨A҈A졔�A㜠҈Aᆠᨨ矄q 㦨맔맔�졔�맔맔� 㡰㦨҈҈�
ᆠ㡰 ᨨ鼀 ��盐 r ˆ ˆ r = � + ├ Þ = ˆ + ├ˆ
䰘
( )
ö r r æ ö æ \ = ç ˆ + ├ˆ ÷ ç ˆ� + ├ˆ ÷ = ˆ ˆ� è ø è ø r r 㡰 鼀� �盐 ᨨ 犨ᨨᆠᨨ矄 꾀 㦨 r r 꾀 + = + + ᆠᨨ矄 q r r 꾀 = + éëᆠᨨ矄 (҈° q ) ùû
ᨨ
Þ
r r
꾀
(҈+ ҈ ᆠᨨ矄 q)
Þ
(҈+ ᆠᨨ矄 q ) = (҈ ᆠᨨ矄 q)
\
r \ 㝐� �盐 ᨨ ᆠᨨ ᆠᨨ ᔴ =
�
+ ᆠᨨ矄 wt = - ᆠᨨ矄 wt ҈ + ᆠᨨ矄 wt = ҈ - ᆠᨨ矄 wt 㡰 p ҈ \ ᆠᨨ矄 wt = ᆠᨨ矄 wt = ᨨ wt = p p ´t= \ t = 矄ᆠᨨ 矄
Topic-2 : Motion in a Plane with Constant Acceleration
盐矄� � ᆠᨨ鼀ᨨ ᨨ �� ᨨ ᨨ鼀 犨 �
r
r r r r r P 졔 Q R = P Þ P+Q = P P P졔Q졔PQᆠᨨ矄q㦨P Þ Q졔Pᆠᨨ矄q 㦨 q a Q Q Þ ᆠᨨ矄 q = �㡰 P P 矄� q a = = ¥ Q ᆠᨨ矄 q + Q = 㡰 Q + P ᆠᨨ矄 q Þ a㦨黨 r r r r 㡰 � A + B = A - B
Þ
æ ҈ ö q = ᆠᨨ矄 ҈ ç ÷ r r r r r è +҈ ø 㡰 ᔼ ᔴ = �ˆ + ˆ 꾀ᔴ = 꾀 졔㦨҈ �㡰 rr r r ᔴ = 꾀 㦨҈ ��㡰 和ᨨ� �⼄盐�ᨨ �㡰 ��㡰犨� ҈ = =
㝐� �盐 ᨨ ᨨ⼄盐 t = ҈黨� 黨㡰�
\
+ + ᆠᨨ矄 q = + ᆠᨨ矄 q
(҈+ ҈ + ᆠᨨ矄 q)
kˆ
\ a �ˆ + a ᆠᨨ矄 wt �ˆ + a 矄� wt ˆ = a �ˆ - a ᆠᨨ矄 wt �ˆ - a 矄� wt ˆ Þ ҈ + ᆠᨨ矄 wt 㡰�ˆ + 矄� w t ˆ = ҈ - ᆠᨨ矄 w t 㡰�ˆ - 矄� w t ˆ
㦨 + ᆠᨨ矄 q ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ
r r 꾀 + =
ˆ
t = ҈ - = 䰘�ˆ - ҈҈ ˆ + �kˆ
졔 A㡰 A҈㡰ᆠᨨ矄q A҈㡰 A㡰ᆠᨨ矄qA맔A
҈黨�㡰 r � F = �ˆ + ˆ + kˆ㡰 r 꾀 r = �ˆ + ˆ - kˆ㡰 - �ˆ + ˆ + kˆ㡰桌 = �ˆ + ˆ - kˆ r r
ᨨ⼄盐 t = r ´ F = �ˆ + ˆ - kˆ㡰 ´ �ˆ + ˆ + kˆ㡰
҈
㡰 ᨨ鼀 ��盐⏌ ��ᨨ ᨨ鼀泐 h҈ h = �° \ h҈ = d d h҈ h҈ + h 꾀 = ° d + d A
�
d
Þ h҈ + h 㡰 ´ = d
҈ + = 黨 黨
r r r S-P r r r 㡰 P + bR = S Þ R = b r r r 꾀矄ᨨ R = Q - P r r r r r r S-P r r = Q - P Þ S - P = bQ - bP \ b r r r \ S = bQ + ҈ - b㡰 P
Þ h҈ + h 㡰 =
� 黨
d
Þ d + h =
\ h = d r 㡰 � u = � ˆ 鼀⏌矄 r 꾀ᆠᆠ�ᨨ a = ҈�ˆ + ˆ � ᆠᨨᨨ � y㡰� �鼀t ҈ S x = ux t + ax t
B d C
d
Q ux = 桌
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PS-17
Motion in a Plane
Þ = +
�
҈ ´ ҈ ´ t Þ t = 矄
҈ S y = u y ´ t + ay t ҈ y = � ´ + ´ ´ = ҈ 鼀 ҈ ᆠ㡰 矄� � S = ut + at ҈ y = u y t + a y t ᨨ � y 꾀 �矄㡰 ҈ ҈ Þ = ´ t + 㡰t Þ ´ ´ t = Þt㦨s ҈ S x = u xt + a xt 꾀ᨨ �x 꾀 �矄㡰 ҈ Þ x = ´ + ´ ´ = 㡰 � ᨨ矄��ᨨ ᆠᨨ r r = ᆠᨨ矄 wt�ˆ + 矄� wt ˆ r r dr = w 矄� wt�ˆ + ᆠᨨ矄 wt ˆ 㡰 ᨨᆠ�泐 v = dt 꾀ᆠᆠ�ᨨ r r dv a= = - w ᆠᨨ矄 wt�ˆ + 矄� wt ˆ 㡰 dt r r a = -w r r \ a �矄 �ᨨ rr r r r r 꾀矄ᨨ v r = \v ^ r
盐矄 �ᆠ�矄 ᨨ鼀� � 盐 � ᨨ鼀 ᆠ�ᆠ盐 鼀ᨨ�ᨨ ® 㡰 r = ҈�t �ˆ + - t 㡰 ˆ ®
v =
®
d r = t�ˆ - tˆ dt
®
®
꾀ᆠᆠ�ᨨ a = d v = �ˆ - ˆ
dt
䰘
\ a = + = � 鼀⏌矄 r r ҈r b㡰 꾀矄 和 = 盐 + r ˆ + ҈ �ˆ+ ˆ㡰 和 = ��ˆ + 㡰 = ҈�ˆ + ˆ + �ˆ+ ˆ r r - � = ҈�ˆ + ҈ˆ r r r 矄 矄 = ᆠ � � ᨨ矄��ᨨ = - � 桌 r = �ˆ + ˆ r = ᆠ㡰 ᨨ鼀 �� ⼄盐�ᨨ r = ⧤ 泐�ˆ + ˆ
(
)
泐 = ├泐 = ├
泐 = = 泐 Þ 泐 泐㦨 ᨨ犨 泐 ᔼ �� � ᨨ 矄� 泐㦨 졔ᆠ �㡰 ᨨ��ᆠ�A XA㦨t졔t졔҈ r VA = t 㡰�ˆ r aA = �ˆ
ᨨ�ᆠ�B YB㦨�t
r VB = t ˆ r aB = -tˆ
꾀t㦨҈矄ᆠ
r r VA = t 㡰�ˆ = �ˆ vB = ˆ r r r \ V B ⏌ A = v A + vB = �ˆ ˆ \和 ᨨ B犨A v = +
= + �䰘 = �
黨
\ v㦨� 鼀⏌矄㡰 bᆠ 㡰꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ ⼄盐�ᨨ
泐=
泐=
꾀㦨
= 泐 = ü ý ��
盐 =҈ þ
㦨
㦨҈鼀⏌矄
泐= 泐 ҈ = × = Þ 泐㦨 ᨨ犨 � ᆠ�犨�鼀
泐
泐 =
=
æ ö +ç ÷ è ø
× +
泐 = + 㡰ᔼ 㦨҈ �ᆠ�矄ᨨ��� 㦨泐㦨㡰 泐㦨 泐㦨҈㦨҈鼀⏌矄 㡰 㦨 泐㦨 졔 Þ泐㦨 ᔼ 㦨 㦨ᆠᨨ 矄 㦨҈ Þ泐㦨ᔼ㦨҈ ᆠ㡰 꾀㦨 㦨泐㦨 矄 㦨҈錐泐㦨 Þ 㦨҈ 㡰 㦨�鼀�矄 㦨҈矄 泐㦨 졔 Þ泐㦨 Þ泐㦨 ᔼ 㦨Þ 㦨ᆠᨨ 矄 � ��泐 㦨҈㡰Þ泐㦨҈㦨҈ 꾀㦨҈矄ᆠ 泐㦨졔泐맔㦨҈맔҈㦨҈ 泐 q = = q® �犨� �矄㡰 \ q =
泐
҈ = = ҈ Þ q = �° ҈
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EBD_8341 PS-18
҈
Physics x㦨aᆠᨨ矄 Þ ᆠᨨ矄 㡰㦨
bᆠ㡰
x a
y b 和⼄盐� � � �⼄ �㡰 ��㡰犨� y㦨b矄� Þ 矄� 㡰㦨
x
+
�㡰
R 㦨vxT =
��㡰
y
=҈ a b ᆠ ᨨ �ᆠ �矄 �矄 ᨨ鼀 �� ⼄盐�ᨨ 矄 犨ᆠ �
Þ R =
u h ´ g
u g
ᆠ㡰 � y㦨x黨x 졠 ᆠᨨ鼀� �犨�
y = x q -
gx
u ᆠᨨ矄 q
␔ q㦨ᨨᆠᨨ矄q㦨
҈
d x dx = v x = - ap 矄� pt 錐 = ax = - ap ᆠᨨ矄 pt dt dt dy = v y 㦨 ᆠᨨ矄 pt dt
dy dt
= a y = -bp 矄� pt
꾀�鼀t㦨
p p ᨨpt㦨
p = 㡰 졠 泐 vx ay ᨨ 犨 ᆠ 矄泐 ᨨᆠ�泐 �矄 ᨨ �
�� x㜠 �矄 ᆠᆠ�ᨨ ᨨ � �� y㜠 �矄 p ᨨᆠ�泐 ᆠᆠ�ᨨ ᨨ �ᆠ ᆠt㦨 p ᨨ鼀 ᨨᆠ ᨨ 꾀t㦨 tᨨ矄��ᨨ ᨨ �ᆠ r r t 㡰 = x�$ + y $ = a ᆠᨨ矄 pt $� + b 矄� pt ˆ a x v y ᆠᨨ鼀 ᨨ ᆠ盐矄 ᆠᨨ矄
=黨 = 黨 ᨨ u ҈ ⏌ �㡰 u ᆠᨨ矄 q \ u㦨�⏌鼀⏌矄 㡰 R犨�矄鼀 ᨨq 黨 q.
�鼀ᨨ �� 矄 u 矄� q t҈㦨 g u 矄� 黨° - q㡰 u ᆠᨨ矄 q = t㦨 g g
æ u 矄� q ö R æ u 矄� q ö æ u ᆠᨨ矄 q ö ÷= ᨨ犨t҈t㦨 ç 㦨 g ç g g è g ÷ø çè g ÷ø è ø b㡰 ᨨ 矄鼀 � � ᨨ ᨨᆠ�ᨨ 矄 q 黨 q.和ᨨ h҈㦨
u 矄� q g
h㦨
u 矄� 黨° - q㡰 u ᆠᨨ矄 q = g g
ᆠᆠ�ᨨ ᨨ �ᆠ
u 矄� q 꾀矄ᨨR㦨 g
ᨨ ᆠᆠ�ᨨ ᨨ �ᆠ �矄 犨泐矄 �ᆠ ᨨ犨 矄 ᨨ��� p �ᆠ�矄 꾀t㦨�ᆠ�矄 a㡰 t㦨 p b㡰 ᨨ �矄 ᆠ ᆠᨨ �矄 ᨨ ᨨ盐 ᨨ ��ᆠ ᨨ a
h҈h㦨
r a t 㡰 = ax $� + a y $ = - p a ᆠᨨ矄 pt $� + b 矄� pt $ 桌 r 㦨p x�$ + y $ 桌 = - p r t 㡰
Topic-3 : Projectile Motion ҈
g
㡰 矄� � � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ ᨨ� ᨨ 鼀ᨨ�ᨨ 犨 mvx㦨mu졔muᆠᨨ矄
u ᨨ �ᆠ 鼀ᨨ�ᨨ
vx =
h ҈ gT Þ T = g R �矄 ᨨ� ᨨ �矄 ᆠ 泐 ᨨ 泐 ҈ R = v xT + 㡰 T 㡰 ᨨ ᨨ� ᨨ 鼀ᨨ�ᨨ 㡰
�
҈
u 矄� q u ᆠᨨ矄 q 맔 g g
u u 矄� q ᆠᨨ矄 q㡰 R 㦨 ҈ g ҈ ᨨR㦨҈h҈h 㡰 졠 � ᆠ� �鼀ᨨ �� 㡰�矄�� 泐 u 矄� q T= g ᆠᨨ矄 a 和盐矄�盐� � 盐矄 犨 � 㡰 矄� ҈�°㡰 矄� ҈�° T= = g ᆠᨨ矄 ° ҈ ᆠᨨ矄° ҈ �矄 ᆠ和 = ᆠᨨ矄҈�°㡰T - g 矄� ° T 㡰 㦨
�
h = +
泐
鼀⏌矄 q 㦨҈�
�矄�
� �ᆠᨨ矄
a 㦨
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PS-19
Motion in a Plane 矄� ҈�° æ҈ ö ҈矄� ҈�° = ᆠᨨ矄҈�°㡰 - ç ´ ҈矄� °÷ ø ҈ᆠᨨ矄 ° ҈ ҈ᆠᨨ矄 ° è
黨
҈ - ҈ ; ҈黨�鼀 ; ᆠ鼀 b㡰 ҈鼀⏌矄
q
� ᨨ
҈
)
泐㦨 � ҈ 鼀 ⏌ 矄 = ҈ �
ᔴ �ᆠᆠ�ᨨ 㦨
� +҈ + 䰘� ҈ �㡰
҈ᆠᨨ矄 q ᨨ鼀 ��盐 盐矄� � �㡰㡰
=
䰘
=
҈ � = Þq= ҈�° �
(
҈ ҈ᆠᨨ矄҈�
҈҈
) = 鼀
㡰 꾀矄犨├ ᨨ犨 � =
盐 矄� q �
꾀㦨p \꾀µ ᨨ꾀µ 盐
�ᆠᨨᆠ�泐 � ��㡰�㦨uy t졔 Þ uy맔졔
\
q =
҈ gt
҈ ҈㡰맔
ᨨ �㦨uy ᨨ uy㦨
uy ux
=
� 䰘 =
Þ 꾀 �q㦨 ҈
䰘
q
u⏌a t҈
u҈
q
u⏌a t
u
q
u⏌a m
t u
q tn un
ᨨ �矄ᆠ鼀 꾀� ᨨᆠ�泐 = ᨨ�鼀├
u 矄� q
�鼀ᨨ �� T = g
\
鼀
ᨨ 盐 = ҈ = ҈ 鼀矄 -҈ �� ᨨ 犨꾀 ҈ � 졠꾀㡰 㦨 졠꾀 q 盐 ᆠᨨ矄 q ҈ ҈ ´ ҈ = - ´ = 鼀 ҈ ҈ ҈ ´ ҈ ´ ´ (4) u҈ u u ᨨ� ᨨ ᨨᆠ�泐ᨨ ᨨᆠ�矄 t҈ t t �鼀├ 矄矄 ᨨ犨 �
��盐 u
꾀҈ 盐҈ é ҈ ù ҈ = = = 꾀 盐 êë úû ҈ ᆠ㡰 ᨨ鼀 ⼄盐矄�ᨨ ᨨ� ᨨ ᨨᆠ�泐 � ��㡰 ux = = 鼀⏌矄
꾀
盐 矄� q � ´ � � ´ ´ ҈ \ 盐 = = = ҈ 矄� q æ ҈ ö ç ÷ è ø
꾀 㦨҈矄ᆠ ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐x㦨� 鼀⏌矄 �ᆠ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐
q=
犨
㝐 �鼀盐鼀 �� 㦨
v泐㦨҈ᆠᨨ矄 㦨 � 鼀⏌矄
+ 泐
� 鼀
꾀矄 �矄 ᨨᆠ � � ᨨ ᨨ� ᨨ ᨨ �㦨 ҈ = � 鼀 ᨨ ҈㦨Þ = Q �㦨鼀졔鼀㦨҈鼀㡰
ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 x㦨҈ᆠᨨ矄 㦨�鼀⏌矄 �ᆠ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐
Þ =
b㡰 졠
� ҈ - � 㡰
(
… ��㡰
�
�ᆠᨨ矄q
… �㡰
҈ y = t - ҈t 㡰 ᨨ鼀 �㡰 ��㡰y㦨x�x
=
r
b㡰 ᨨ鼀⼄盐�ᨨ v = �ˆ + ˆ Þx㦨t
䰘 = �鼀 ⏌ 矄
ᨨ�� 盐ᨨ q盐ᨨ T犨�ᆠ �犨� 盐 ᨨ u \ ᨨ�鼀├ 㦨t҈졔t졔t졔 t t t ta = t҈ + ҈ + ҈ + = ҈ = ҈ ҈ a a a - ҈㡰 ҈a
ᨨ �矄ᆠ鼀 㦨盐҈҈졔盐졔盐 졔
u҈ t҈ u҈ t҈ ut u t a + + = ҈ ҈ = ҈҈ ҈ a a a a ҈ - a - ҈㡰 a ᨨ �矄ᆠ鼀 \ 꾀� ᨨᆠ�泐 = ᨨ�鼀 u҈t҈a a - ҈㡰 u҈a ´ = = a +҈ t҈a a - ҈㡰 = u҈t҈ +
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EBD_8341 PS-20
Physics 犨 a㦨҈�鼀⏌矄 ᨨ uᆠᨨ矄 q 㡰t㦨졔䰘�t ��㡰 ���㡰 ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 ᨨ 矄ᨨ 㦨 �ᆠ ᆠᨨ鼀ᨨ 矄 ᨨᆠ�泐 ᆠᨨ �矄 � ᆠ� � 犨� ᨨ� ᨨ \ uᆠᨨ矄 q 㡰㦨gt u矄� q 㡰 ���㡰 �� 矄� �矄 犨� gt u 矄� q ᆠ盐矄 矄ᨨ �矄� �矄 ᨨ犨 犨 鼀ᨨ�ᨨ \gt>u矄� q ᔼ 盐犨 鼀ᨨ�ᨨ 盐矄� q > gt㡰 㝐盐�泐� �⼄盐�ᨨ ���㡰犨� t uᆠᨨ矄 q 㡰t졔 u矄� q 㡰t㦨҈ t �㡰 ᨨ犨 �㡰 ��㡰 �㡰��矄�t�㦨ᨨt㦨҈矄 和盐矄�盐� �t㦨҈矄� �㡰 ��㡰犨� u矄� q 㦨�鼀⏌矄ᨨuy㦨�鼀⏌矄 uᆠᨨ矄 q 㦨䰘�鼀⏌矄 r ᨨ ux㦨䰘�鼀⏌矄 \ u = u x $� + u y $
꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ � ᨨᆠ�泐㦨 ҈ u҈a ᨨ ҈ = �㡰 a +҈
u҈ ´ t҈ = u҈ ��㡰 t҈ ᨨ鼀⼄盐�ᨨ �㡰 ��㡰 a㦨a졔ᨨaa㦨 = ᨨ a㦨ᨨ a = ҈ �㡰 u㦨҈鼀⏌矄q㦨 盐泐 盐㦨҈鼀⏌矄 盐矄� q 㦨� 鼀⏌矄
V҈ =
r
盐 盐ᆠᨨ矄q 㦨�鼀⏌矄 \ ux㦨uᆠᨨ矄q㦨҈맔ᆠᨨ矄 㦨�鼀⏌矄 uy㦨u矄� q㦨҈맔矄� 㦨 � 鼀⏌矄 졠
´� = = 矄 g ҈ a ᆠᆠ�ᨨ ᨨ � ҈ ҈҈� = u xt - at ҈ Þ ҈҈� = � ´ - ´ a ´ 㡰 a Þ = � - ҈҈� = � - ҈҈� = 䰘� \ a = 䰘� ´ = � 鼀⏌矄 T盐㡰 쟼㦨쟼졔⧤쟼㦨ᔴᨨ 矄 꾀 ᨨ⧤쟼�矄鼀� �鼀盐鼀 쟼�矄 鼀 �鼀盐鼀和� ᆠ ⧤쟼�矄鼀� �鼀盐鼀矄ᨨ矄 �矄矄ᨨ鼀� �鼀盐鼀 ἘtἘ �鼀 犨 �ᆠ 矄ᨨ �矄 ᨨᆠ r q � 犨 �ᆠ ᨨᆠ�泐ᆠᨨ u 鼀├矄 犨� ᨨ� ᨨ \
҈ ҈
҈�
t=
u y
y
v x 㦨u ᆠᨨ矄 q
u u矄� q
q
졠 u ᆠᨨ矄 q
gtu 矄� q 㦨vy 桌 ҈�鼀
� v
x
꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ 犨 ᨨᨨ犨� � ᆠᨨ ��ᨨ 矄 �㡰 �ᆠ �矄ᆠ鼀 ᨨ 矄ᨨ �矄҈�鼀 ҈ \ ҈�㦨 u矄� q 㡰t gt 犨 �㦨҈鼀⏌矄 �㡰 ᨨ u矄� q 㡰t㦨҈�졔� t ��㡰 ᨨ� ᨨ �矄ᆠ鼀 ᨨ 矄ᨨ 㦨졔 �矄ᆠ鼀 ᨨ ᨨᆠA ҈ \ uᆠᨨ矄 q 㡰t㦨졔 at
ᨨ u = 䰘� $� + � $ 㡰 鼀⏌矄 꾀 �鼀 犨 �ᆠ 矄ᨨ �矄 ᨨᆠ t 㦨 ҈矄 㡰 � ᨨᆠ�泐ᨨ �ᆠ 犨� 矄ᆠ ᨨ ᨨ �矄uᨨ犨 �ᨨᆠ�泐� x y㜠 �ᆠ�ᨨ
ux uy 矄ᆠ�泐 和� ᆠ �矄 ᨨ ᨨᆠ�泐ᨨ ᨨ � y㜠 �ᆠ�ᨨ ᨨ �矄 �矄 �ᆠ ᨨᆠ�泐 ᨨ �ᆠ 犨� 矄ᆠ ᨨ �ᨨ盐 矄ᨨ
├� ��鼀矄犨ᨨ Y㜠 �ᆠ�ᨨ 鼀ᨨ�ᨨ
uy㦨졔u矄� a ay㦨gᆠᨨ矄q sy㦨 ᆠ��泐�矄├ � �鼀 �ᆠᆠᨨ鼀矄 ᆠ├ᨨ ᨨ 㡰 ty㦨t ҈ ҈ sy㦨 u y t + a y t Þ 㦨 u矄� a㡰t g ᆠᨨ矄 q ´ t u 矄� a Þ t㦨ᨨt㦨 g ᆠᨨ矄 q
├� ��鼀矄犨ᨨ X㜠 �ᆠ�ᨨ 鼀ᨨ�ᨨ
ux㦨졔uᆠᨨ矄a ; ax㦨tx㦨tsx㦨sx
҈
sx㦨 u xt + a xt Þ sx㦨uᆠᨨ矄a맔
u 矄� a u 矄� a = g ᆠᨨ矄 q g ᆠᨨ矄 q
æ u 矄� a ö b㡰 �矄 ᆠ 泐 ᨨ � �鼀 ç ÷ è g ᆠᨨ矄 q ø 矄 ᨨ盐 ⼄盐ᨨ �ᨨ �ᆠ ᨨ ᨨ矄 ᨨ �ᨨ盐 ᨨ 矄 ᨨ� ᨨ �矄ᆠ鼀 ᨨ ᨨ 矄 ᨨ ᨨ �鼀ᨨ ᨨᆠ�ᨨ ᨨ �ᆠ U ᨨ 鼀ᨨ�ᨨ ᨨ ᨨ 犨� 矄ᆠᨨ�ᨨ盐 ux㦨U 錐 ax㦨g矄� q 錐t y㦨
u 矄� a -u 矄� a 錐 sx㦨 g ᆠᨨ矄 q g ᆠᨨ矄 q
҈ sx㦨 u xt + axt
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PS-21
Motion in a Plane
æ u 矄� a ö ҈ -u 矄� a æ u 矄� a ö = -U ç - g 矄� q ç g ᆠᨨ矄 q è g ᆠᨨ矄 q ø÷ è g ᆠᨨ矄 q ø÷
ᨨ鼀矄矄鼀hᨨ犨 ᨨ鼀C ᨨ ᨨ� ᨨ 鼀ᨨ�ᨨ QM㦨҈ᆠᨨ矄 桌t QM㦨�t
ᨨ�ᆠ鼀ᨨ�ᨨ vy㦨҈矄� 㦨� 錐ay㦨g sy㦨y졔҈錐ty 㦨t
u ᆠᨨ矄 a + q㡰 ᆠᨨ矄 q
� 盐矄ᆠᨨ� � �ᨨ� P 泐 ᨨᨨ犨 � ACP ABP. t҈㦨 �鼀├ 泐 �矄盐� ᆠ � �P t㦨 �鼀├ 泐矄ᆠᨨ 盐� ᆠ � �P y u B ᨨ ABP �R㦨 uᆠᨨ矄 㡰t u �R㦨 u҈㡰 A C ᨨ ACP h 和ᨨ� �⼄ �㡰犨�U㦨
҈
\
ut = ut҈
ᨨ t㦨t҈ �㡰 �� ᨨ ABP
ᨨ犨vy㦨� 졔gt ҈ sy 㡰㦨uy t졔 ay t ҈ Þ y졔҈㦨 � t + gt ҈ ᨨ犨盐� �盐ᨨ y = gt � ⼄ �㡰 ҈ ҈ 矄ᆠ gt + ҈ = � t + gt Þ t㦨 �矄 ᆠ 犨 ᨨ ᨨ犨矄
P
҈鼀
㡰
ᨨ ᨨ
-� ´ t
\
㦨҈䰘鼀 ��㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 盐� � ᆠᨨ�矄�ᨨ ᨨ 鼀矄矄矄 M㡰 � ᨨ� ᨨ �ᆠ�ᨨ
ᨨᆠ�泐 ᨨ ᆠᨨ鼀� 鼀矄矄 ᨨ � ᨨ� ᨨ 㦨 v x m맔҈m҈ᆠᨨ矄 㦨m맔 vx Þ ҈m҈m㦨m맔vx Þ vx㦨 和� ᆠ ᨨ� ᨨ 鼀ᨨ鼀 盐鼀ᆠᨨ鼀矄ᨨ盐ᨨ ᨨ ᆠᨨ鼀� �ᨨ ᨨ 鼀矄矄矄犨� ᨨ�ᆠ泐 ᨨ犨 犨 矄 E \ 쟼㦨㝐㦨҈t㦨҈맔 㦨҈҈�䰘鼀
-u ´ t ҈ ҈ + gt = gt҈
҈t҈ ҈t҈ = -҈�t + ҈t = ҈҈ ᨨ - t + t = t҈ +
- ´ t҈ 㡰 + t҈ 㡰 = t҈ ᨨ -t҈ + ´ 㡰t҈ = t҈ ᨨ t҈ = t҈ ᨨ t҈㦨҈矄ᆠ \ t㦨t҈㦨矄ᆠ ᨨ犨R㦨ut҈ ᨨ ACP
��㡰 ���㡰
u 矄� q 黨㡰 矄� ´ �° = 㦨 � 鼀 g 黨 �矄 ᆠ 犨 ᨨ� 矄 ᨨ ᨨ犨 ᨨ �ᆠ矄 �矄 矄ᨨ �鼀 �矄鼀 矄 �ᆠ矄矄 ᨨ盐 ᆠᨨ� 鼀� 犨泐 犨 A B � 鼀 �鼀盐鼀 �� H 㦨
�㡰
u 矄� q 黨㡰 矄� �°㡰 㦨 㦨҈�鼀 g ´ 黨
ᨨ
ᨨ
R = � ´҈ = � 鼀
҈
҈ gt҈ ᨨ ACP ҈ = ´҈ ´ ҈㡰 = � 鼀
҈䰘
u ᆠᨨ矄 �° = 黨
y㜠ᆠᨨᨨ � ᨨ P㦨҈h㦨҈�㦨�鼀 㡰
�鼀� 犨 �� �矄㦨 tt҈㡰 㦨 ҈㡰㦨҈矄ᆠᨨ
҈ y㦨 gt ��㡰 ᨨᆠ�泐 � �ᆠ �ᆠ�ᨨ
vy㦨uy졔ay t㦨gt ���㡰
鼀⏌矄
꾀泐� �ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ� ᨨ ᆠᨨ�矄�ᨨ
m҈ u ᆠᨨ矄 �°㡰 - m u ᆠᨨ矄 �°㡰 = mv - m҈ u ᆠᨨ矄 �°㡰
��㡰 ᔴᨨᨨ � 矄ᨨ ᨨ� P㦨 Ry㡰 = � 鼀�鼀㡰 �㡰 t �鼀├ ᨨᆠᨨ�矄�ᨨ
� � ᨨ� ᨨ ᆠᨨ�矄�ᨨ ᨨᆠ矄 ᨨ犨 ᨨ鼀꾀 ᔴ 盐 ᨨ � ᨨ�ᆠ 矄 ᨨ鼀矄矄mhᨨ犨 hᨨ� ᨨ y ᨨ鼀A ᨨ ᨨ� ᨨ 鼀ᨨ�ᨨ
㝐㦨VA맔t㦨҈t �㡰 ᨨ �ᆠ 鼀ᨨ�ᨨ
uy㦨錐sy㦨y錐ay㦨g錐ty㦨t \
� R 㦨
쟼ᆠ �ᆠ 矄 矄鼀 矄 ᨨ 黨 鼀⏌矄 ᨨ� ᨨ ᨨᆠ�泐 ᨨ ᆠ �ᆠ �矄 矄鼀 �
꾀�� h =
\ �㡰
�㡰
BD㦨PM졔MQ㦨҈t졔�t㦨҈�t㦨҈�맔
҈ h = - u 矄� °㡰t + gt ҈ h = gt҈ �� ᨨ ACP
㡰
x
R
�㡰
맔黨맔
҈ ҈ ҈ ´ 黨 ´ + V 㦨 ´ 黨 ´
\ V㦨 �ᆠQ �矄 矄 ᆠᨨ�矄�ᨨ 犨� �ᆠ泐 ᨨ犨 犨 矄 鼀� 犨泐
�鼀├ 㦨
H ´ ҈� 㦨 㦨�矄 g 黨
Topic-4 : Relative Velocity in Two Dimensions & Uniform Circular Motion ҈
㡰 R㦨҈鼀
w=
p p = = ҈� ⏌矄 T
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EBD_8341 PS-22
Physics
꾀ᆠᆠ�ᨨ ᨨ � ᨨ ᆠᨨᆠ├ 矄ᆠᨨ Ἐ矄
a = w R = ҈�㡰 ҈㡰 = ҈҈ = ҈҈ ´ ҈- 鼀⏌矄
㡰
ᆠ � ᆠᆠ�ᨨ �矄 ᨨ ᨨ ᨨ ҈
㡰 �� 矄�盐�ᨨ �矄 矄 ᨨ犨 � ��鼀 vr ᨨᆠ�泐ᨨ � ᨨ
҈
q
vr
vr
p - q㡰
vr
�
- ҈
vᆠ 㦨 b 졔҈㡰v
vᆠ 㦨v
v
ᔴ � � ᨨᆠ�泐
D = ҈ + + ҈ ᆠᨨ矄 ( p q )
␔ ᆠ�矄鼀ᨨ� �犨� 矄 v v ° = r �㡰 v ␔ ᆠ�矄 鼀ᨨ� �犨� 矄 ҈ + b㡰v
vr b + ҈㡰v �� � � �㡰泐 ��㡰犨� �° =
㦨矄�
ᆠ㡰 矄� q =
��㡰 䰘
b㡰
u ҈ = = v
q 盐
ˆ ᨨ 㡰
B rBA
҈
㦨҈鼀⏌矄 ᆠ㡰 和 V 㦨 ᆠᨨ 矄 ᨨ鼀 ⼄盐矄�ᨨ 㡰 ᔴ � ᆠᆠ�ᨨ
V r ra 㦨 ᆠᨨ 矄 ᆠ � ᆠ㡰 ᆠᨨᆠ泐 矄ᆠ�矄 �ᨨ 犨
ᆠᆠ�ᨨ �盐矄 r b㡰 和 ᨨ矄 ᨨ盐 ᆠᨨ矄ᨨ 矄 ᨨ vb �ᆠ盐 ᨨ � ᨨ犨 ® vr d d \t= = vb ® - vr vbr ® vb d v ᨨ t =
ᨨ
r r �ˆ + ҈� ˆ 㡰 -�ˆ - � ˆ 㡰 rBA )( · vBA ) ( t鼀� �鼀盐鼀 = = ҈ ҈㡰 ( vrBA )
҈䰘 \t= = = 矄 ҈ ҈ ´ ҈ ҈ ҈ p p = ᆠ㡰 ᨨ鼀q㦨w㦨w w 和ᨨ ᨨ ᆠᨨ鼀 ⼄盐 ᆠ�ᆠ
d ҈├鼀 = ҈ vb ҈
҈ =
҈ � - vr
Þ vr = ├鼀⏌ 黨
҈䰘
㡰 �鼀├ ᨨᆠᨨ矄矄 �t㦨
vsᆠᨨ矄q
vs w ҈ 꾀
� ᨨᆠ�泐
( )
w
d vs ᆠᨨ矄 q
d ␔
q
꾀 = w ҈ �ˆ - w ( � )
= w ( ҈ ) �
r
br
�ˆ 쟼矄㡰
A r ˆ vA = � + � ˆ ├鼀⏌ r vB = -҈�ˆ㡰 ├鼀⏌ rBA = �ˆ + ҈� ˆ 㡰 ├鼀 r r r vBA = vB - v A = -҈�ˆ - �ˆ - ��ˆ = �ˆ - � ˆ
�
r
(Q ҈ = ) =
a 㦨
ᨨq㦨 犨� 矄ᆠ ᨨ ᨨ犨 㦨黨 졔 㦨҈
q
㦨 맔҈㡰맔矄� 㡰㦨맔҈맔
v = b + ҈㡰v Þ b = - ҈ = 䰘
҈
vr
쟼 和
ᨨ�鼀ᨨ鼀� �鼀盐鼀ᆠᨨ矄q㦨鼀 �鼀盐鼀�q㦨 ᆠ矄犨�鼀鼀矄 ᨨ盐 矄犨�鼀 盐 ᨨ
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PS-23
Laws of Motion
Laws of Motion
4
Topic-1 : Ist, IInd & IIIrd Laws of Motion ҈
㡰 �
r F = k v y �ˆ + vx ˆ 㡰 \ Fx = kv y �ˆ Fy = kv x ˆ mdvx dv k = kv y Þ x = v y dt dt m 和�鼀�泐
dv y
dv y
dt
=
\ = �
꾀 㦨㦨鼀盐 h =
k v m x
䰘
dp d mv㡰 æ dm ö F= = = vç �㡰 ÷ dt dt è dt ø dM t 㡰 = bv t 㡰 ␔ �� ��㡰 dt
盐矄 ᨨ 矄� æ dm ö F = -v ç ÷ = -v bv 㡰 = -bv 矄� � �㡰 ��㡰桌 dt è ø -bv Þ F = M t 㡰 a = -bv Þ a = M t 㡰
H
g졔kv 㡰㦨a ᆠᆠ�ᨨ 㡰
u
r F = mkv - mg Q mg mkvᆠ
mV mV = 鼀 L nr 㡰 L nr V 鼀 㡰 ᨨ鼀 ⼄盐矄�ᨨ 㝐矄矄ᨨ ᨨ 泐m㦨�├� ᨨᆠ�泐t㦨 u㦨 �ˆ - ˆ㡰 鼀⏌矄 ᨨᆠ�泐t㦨҈矄 v㦨졔 ˆ 鼀⏌矄 ᨨᆠF㦨䁸
꾀ᆠᆠ�ᨨ a㦨
ᨨᨨ矄�ᨨ ᆠ ᨨ 㡰
r r F a= = -kv + g 桌 m dv ö æ dv Þ v× = -kv + g 桌 çèQ a = v ÷ø dh dh h v × dv ҈ Þò = ò dh éëkv + g ùû = - h Þ u k u kv + g ҈ é ku + g ù ê Þ ú= h k ë g û 㡰 ᆠᆠ�ᨨ
㦨㦨 �졔g㡰 �鼀⼄盐� ᨨ�矄ᨨ�矄 � 㦨㡰矄ᨨ - = ò ò ᨨ鼀 㦨桌 � + g
é ù ├ = ├ Þ = ├ [ ] ê ú Þ = Þ = ò ò êë úû ├ 㡰 㦨 㡰 ᨨ鼀 㦨 㡰 鼀 = ò ᔼ �� � ᨨ 矄� 矄 ò 鼀 ҈ \ µ ᔼ = 鼀 b㡰 和ᆠ 犨 矄盐ᨨ矄 ᨨ 鼀ᨨ�ᨨ ᨨ 矄 �矄 꾀� ᨨᆠ � ᆠ 泐 ᆠ 矄盐ᨨ
F=
黨
u P = f
盐犨 �ᆠ�ᨨ �矄 ᨨ矄�� ᨨ犨 犨 �ᆠ�ᨨ �矄
�� b㡰 ᨨ鼀 犨ᨨ 矄 矄ᆠᨨ 犨 = = ├ ᔼ �� � ᨨ 矄� 矄 犨 �
b㡰 ᨨ鼀 犨ᨨ Ἐ矄 矄ᆠᨨ 犨
v㦨
㡰 㦨盐 � ᨨ㦨 u - gh 㝐ᨨ鼀 盐鼀㦨鼀 = m u - gh
v = x Þ ò v y dv y = ò v x dvx dvx v y v y = v x + C v y - v x = ᆠᨨ 矄 k r r v ´ a = v x �ˆ + v y ˆ 㡰 ´ v y �ˆ + vx ˆ 㡰 m k k = vx kˆ - v y kˆ㡰 = vx - v y 㡰 kˆ = ᆠᨨ 矄 m m
㡰
æ g ö -҈ ç ç � ÷÷ g� è ø
҈
҈
v - u ˆ - �ˆ - ˆ 㡰 -�ˆ + ˆ 㦨 鼀⏌矄 㦨 ҈ � t
ᨨᆠF㦨ma -�ˆ + ˆ 㡰 㦨 � ´ = -�ˆ + ˆ 㡰 N � ur 㡰 � 鼀ᨨ鼀 盐鼀 p t 㡰 = A �$ ᆠᨨ矄 kt 㡰 - $ 矄� kt 㡰桌
ur ur d p = Ak -$� 矄� kt 㡰 - $ ᆠᨨ矄 kt 㡰桌 꾀 ᨨᆠ F = dt ur ur r r F P = 盐 F p = Fp ᆠᨨ矄 q \ ᆠᨨ矄 q 㦨Þ q 㦨黨 ᆠ �犨 ᨨᆠ鼀ᨨ鼀 盐鼀q㦨黨 ҈҈ �㡰 꾀 盐 F t � ��矄 �鼀盐矄 �鼀 ᨨ ᨨ 泐
鼀� �盐 ᨨ ᨨ�鼀盐矄ᨨ ᨨᆠᨨ ᨨ 泐 ᨨ鼀 t㦨Ҙ矄ᨨt㦨҈Ҙ矄 㦨 BCDFEB㡰 㦨ᨨ BCFEB졔CDFC
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EBD_8341 PS-24
Physics F � 㡰
C
A
B
҈
҈
҈
҈�
D
F
E
҈ ҈ ҈ ҈ T
҈ ҈ - - 㦨 + 㡰 ´ ´ ҈ 졔 ´ ҈ ´ ´ ҈ ᆠ�ᨨ
㦨҈졔㦨�矄 矄 ᨨ犨 鼀ᨨ�ᨨ ᨨ ᨨᆠ├ �矄 盐 ᨨ 盐矄 �矄矄 ᨨ犨
ᆠ├犨 ᨨ 盐ᨨ矄盐ᨨ盐 � �� 盐矄 � � ᆠ├犨 矄 ᨨᆠ├ 盐矄 �矄矄 ᨨ犨
ᆠ├犨 矄 �矄 ᆠ�ᨨ ᨨᆠ├ 鼀ᨨ� � ᨨ犨 �矄 ᆠ�ᨨ 쟼 盐矄�矄矄
�矄 ᨨ鼀 ᨨ ├矄ᆠ� 꾀ᆠ�ᨨ
矄 ᆠᨨ � 矄 犨 b 㡰쟼 �矄 ᆠᆠ 鼀 ᆠᆠ
ᨨ � � 鼀 쟼 �矄 ᨨ� � ᨨ盐 矄盐 �矄 ᨨ� � ᨨ盐 �矄 ᨨ犨 �矄 b㡰 ᔼ �矄 矄� ᨨ盐 泐ᨨᆠ ᨨ 盐矄 � ᨨ �ᨨ盐 �矄�矄ᆠ盐矄 ᨨ鼀ᆠ�ᨨ � ᆠ矄 ᨨ 盐� ��矄矄矄矄ᆠᨨ鼀 泐盐矄 � � 㦨m 㡰 ᨨ �ᆠ�ᨨ ᨨᆠ�矄矄鼀� ᆠ矄ᨨ 盐� �
鼀� �盐 ᨨ �ᆠ�ᨨ ᨨᆠ 矄 ᨨ 盐ᨨ 犨ᨨ 矄盐 ᆠ矄 � ᆠᨨ ᆠ 盐 �矄 ᨨ ᆠ盐ᆠ �ᨨ ᨨ 矄鼀 㜠҈ b㡰 ᔴᨨ ᆠ 盐 ᨨ盐犨� ᨨ盐 �矄ᨨ �� � �矄 矄 ᨨ鼀 ᆠ盐矄 ᨨ � � 犨ᨨ � ��矄 犨ᨨ Ἐ矄 �矄犨ᨨ 鼀ᨨ�ᨨ ᨨ 泐 ᆠ�ᨨ �矄 ⼄盐 ᨨᨨ矄� ᆠ�ᨨ
�矄�矄犨ᨨ Ἐ矄 � 犨ᨨ 鼀ᨨ�ᨨ Topic-2 : Motion of Connected Bodies, Pulley & Equilibrium of Forces
҈
�
= ҈ 鼀矄 - + ҈ +
矄�ᨨ 鼀� ᨨ� 㝐矄矄ᨨ ᨨ ö æ
= ç 㝐 + ÷ è ø 㦨 ҈졔㡰맔҈㦨҈黨 =
\ 㦨҈
҈
T a T ma
T
a
T矄� q T矄� q
q Tᆠᨨ矄q
x
x
m
�㡰
矄ᨨ F㦨T矄� q Þ T㦨 盐� � �矄盐ᨨ � ⼄ �㡰
F 矄� q
æ F ö ᆠᨨ矄 q ÷ è 矄� q ø m
a㦨 ç
䰘
졠 �� � � �㡰泐 ��㡰犨�
b㡰 ᨨ鼀 ��盐 ᆠᆠ�ᨨ
ᨨ 鼀矄矄m�矄 盐ᨨ ᨨᆠT ᆠᨨ矄 q Tᆠᨨ矄 q 㦨ma T ᆠᨨ矄 q Þ a㦨 m
��㡰
T ᆠᨨ矄 �° ҈ = Þ F㦨҈ T 矄� �° F 㡰 ᨨᆠ ᨨ ᨨ 㦨 ᨨᆠ ᨨ ᨨ 졔 �ᆠ�ᨨ ᨨᆠ Þ F㦨 ҈m졔M㡰g졔f 犨 m㦨 鼀矄矄ᨨ 矄ᨨ M㦨鼀矄矄ᨨ ᨨ f㦨 �ᆠ�ᨨ ᨨᆠ Þ F㦨 ҈맔졔黨㡰맔黨졔Þ F㦨N Þ P㦨FV㦨맔㦨W 㡰 꾀⼄盐���盐鼀 ᨨ � 鼀� ҈ = � 㦨 ᨨ �
F
\
㡰 ᨨ鼀 ᨨ 泐 ��鼀 Tᆠᨨ矄� 㦨҈N �㡰 T矄� � 㦨F
b㡰 ᨨ鼀 ��盐 꾀ᆠᆠ�ᨨ a㦨 Ra … �㡰 a mgT = ma … ��㡰 ᨨ鼀⼄盐�ᨨ �㡰 ��㡰 æ aö
T맔R㦨mRa㦨mR ç ÷ è Rø ᨨ T㦨ma
Þ mgma㦨ma 鼀 g Þ a= 鼀� 㡰 꾀ᆠᆠ�ᨨ ᨨ 盐ᆠ � 盐犨 �ᆠ�ᨨ
= 㝐҈ + 㝐 + 㝐矄矄ᨨ 鼀ᨨ
é F F x a - x ù êQ q ú = m q = m x ê ú a - x ë û ᆠ㡰 ᨨ 矄� � �矄 ᆠ盐 矄�ᨨ T 矄 ᨨ ᨨ ᨨ 鼀矄矄矄 m m ᨨ T㦨mg ␔ 矄� ��矄ᆠ盐 鼀矄矄m�矄 泐 � �ᨨ 泐 �矄 ᆠᆠ�ᨨ 㦨 ᆠᆠ�ᨨ 盐ᨨ��泐㦨 g ᨨ鼀矄矄m盐矄 矄� ��矄ᆠ盐 鼀� T鼀� 矄 mgᆠ盐矄ᨨ 矄�ᨨ
ᨨ 矄� � 鼀 鼀 g \ mgmg㦨m맔a Þ = a 鼀� 鼀� 㡰 꾀⼄盐���盐鼀T㦨Mg ᨨ 盐泐 T
mg
T T
Mg
T㦨㝐�
F
F҈㦨 m졔M㡰g
矄盐 ᨨᆠ ᨨ 盐泐
F҈㦨 m졔M㡰g
F㦨 F҈ + T 㦨 m + M 㡰 + M 桌 g
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PS-25
Laws of Motion 黨
ᆠ㡰 矄�ᨨ � ᨨ 矄� �矄 犨� 矄鼀 盐 ᨨ 矄泐鼀鼀泐
T ᆠᨨ矄q T ᆠᨨ矄q
ᨨ鼀 ��盐
㦨鼀�
q q
T T T 矄� q A
T T 矄� q
B
T C
Ö mg mg
mg
ᨨ⼄盐���盐鼀
mg = T ᆠᨨ矄 q + T ᆠᨨ矄 q = T ᆠᨨ矄 q \
҈
mg = mg 㡰 ᆠᨨ矄 q ҈ Þ q = �° \ ᆠᨨ矄 q = 矄 ᨨ鼀 矄 ᨨ犨 � ᆠ矄 㡰 ᨨ鼀矄矄m Tmg㦨 ma �㡰 ᨨ鼀矄矄m ᔴ矄 㡰
a
mgT㦨ma ��㡰 ᨨ鼀 �㡰 ��㡰 a㦨g⏌ ���㡰 ᔴ矄 b㡰 Tmg㦨maἘ Þ mgmg㦨maἘ Q T㦨mg桌 \ aἘ㦨g �㡰 ᆠ ᨨ鼀⼄ ���㡰 �㡰aaἘ ҈҈
㡰 꾀ᆠᆠᨨ � � ᨨ ᆠᨨ 矄� �ᨨ
ᨨ鼀 ��盐
m
T a mg
҈
a T
T
U Q
҈鼀
T= mg
m
a' mg
q
q �㡰 x졔z㦨 l � �� � O M ⼄盐�ᨨ �㡰犨ᨨt dz dl = l 졔z Þ zvM㦨 lU dt dt zö æ U U l = Þ vM㦨 U = çèQ ᆠᨨ矄 q = ÷ø l z z ⏌ l ᆠᨨ矄 q l㦨㝐矄矄ᨨ 盐 � � ᨨ 犨�├�⏌鼀 P U
mg
a҈ kx
T
x ҈ æ ҈㝐� ö 㝐� = ç ÷= è ├ ø ├ � ├ 㝐� � ´ = ᨨ鼀⼄ �㡰҈㦨 䰘 ҈㝐 x 䰘 x �ᆠ�矄鼀 ᨨ矄��ᨨ �矄ᨨᆠ�泐㦨꾀 꾀w 꾀 㦨 㝐� ├ x ├ = 㦨 ҈㝐 ├ ҈㝐 ҈ b㡰 ᨨ鼀 ��盐AN㦨x 㦨ᆠᨨ 矄 矄盐泐A dz B � 㡰NO㦨z ᨨᆠ�泐ᨨ 鼀矄矄m㦨 꾀矄ᨨ dt dl A x N B =U OA㦨 l dt Z ᨨ鼀DANO l ᨨ x㦨
F㦨mg
T
l
҈鼀
T
T + Þ졔㦨҈ T T a Þ 졔㦨҈졔҈ a Mg Þ ҈㦨҈ M Þ 졠�ᨨ 㡰�矄 ᆠᨨᆠ �x 矄�ᨨ ᨨ 矄� �ᆠ� � 矄 ├ 㦨㝐҈ 㝐� 㦨㝐 㝐� 㦨㝐 ҈ 졠 矄ᨨ� � 犨 � ├
㝐 � ├x ҈㦨 �㡰 䰘 ҈㝐 㝐� ├ ├ \ w 㦨 \ w㦨 ҈㝐 ҈㝐 㝐� ├x + ��㡰 㦨 䰘 䰘 ᨨ҈㦨 㝐 �鼀盐鼀 矄�ᨨ ᨨ 矄� �㡰犨 ᨨ鼀 �㡰
� ├x 㦨 䰘 ҈㝐 ├x 㝐�
\ �㦨 \ x㦨
㝐 ├ 㝐 ҈㝐� \ x㦨x㦨 㝐 ├
m҈
m҈g 졔 l l�
҈㦨
鼀҈ 㦨҈黨├� 鼀 㦨黨├�
�㡰 矄�ᨨ T鼀� ᨨ� ᨨ ᨨ犨犨� l㦨 㜠 � 㦨�鼀 \ T 鼀҈졔ll㡰�㦨 鼀҈졔ll㡰a T㦨 鼀҈졔ll㡰 a졔g㡰 㦨҈黨졔 맔�㡰桌 졔黨㡰㦨맔҈㦨 ��㡰 矄�ᨨ T Ἐ鼀� 㜠ᨨ� ᨨ 盐犨� \T Ἐ㦨鼀҈졔 t맔l㡰졔m桌a졔mg졔l맔l�졔m҈g桌 ᨨT Ἐ㦨m҈졔 l맔l㡰졔m桌 a졔g㡰 㦨҈黨졔 맔҈㡰졔黨桌졔黨桌㦨�맔҈㦨�
Topic-3 : Friction ҈
㡰 ᨨ ᆠ� � mg 矄� q = f = mmg ᆠᨨ矄 q
Þ q = m = Rᆠᨨ矄q q h
= 䰘� f = mgᆠᨨ矄q
R h q mg矄� q
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EBD_8341 PS-26
Physics
Rᆠᨨ矄q
3 + 50 㦨졔
R
q
�
q
h
�
æ ö R h = R - R ᆠᨨ矄 q = R - R ç ÷ = è �ø � R \ h = = 鼀 Q �盐矄R㦨҈鼀桌 � ᆠ㡰 꾀N㦨��졔矄� ҈ 㦨�졔맔 㦨 f 꾀ᆠᆠ�ᨨ a҈㦨
N ᆠᨨ矄 ��
F- f ᆠᨨ矄 ° - ҘN = m �
矄�
é ù - ´ ú ê ´ ú 㦨҈鼀⏌矄 㦨 ê f � ê ú êë úû
矄�
N
ᆠᨨ矄
䰘
\㦨҈㦨 b㡰 � 鼀҈㦨�├�錐鼀㦨҈├�錐m㦨҈� ᨨ鼀҈鼀҈� 㦨鼀҈ Þ� 㦨�맔 鼀 ҈� 鼀졔҈㡰�
鼀 㦨 ҈졔鼀㡰 m 鼀졔鼀㡰� 鼀졔鼀㡰� ᨨ矄a㦨 ᨨ�㦨҈� 鼀졔҈㡰҈ 鼀 + ҈㡰 Þ �=
鼀҈ 鼀҈�㦨�
҈ = 鼀 + ҈ \鼀㦨├�錐 ᆠᨨ矄 ᨨᨨ�ᨨ 㡰 b㡰 ᆠᨨ �ᆠ� 矄 ᨨ ├� �ᆠ �ᆠ�ᨨ 犨 ᨨᆠ � ᆠ� ҈ ҈ö æ m = q ç҈ - ÷ Þ m = ҈ - Q q㦨� 㡰 è ø
b㡰 m 鼀� �鼀盐鼀ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ
m鼀�ᆠᨨ矄q
��
N㦨��矄�
҈ 㦨
㦨�맔
m�矄� q
F - f é ᆠᨨ矄 ° - ´ ù =ê a㦨 ú 㦨҈鼀⏌矄 m � ë û ᨨ犨aa҈㦨҈҈㦨鼀⏌矄 b㡰 ├� � 꾀졔㡰矄矄泐矄鼀 Fm M졔m㡰g㦨 M졔m㡰a F m M + m㡰 g Þa㦨 M + m㡰
�
꾀 ⼄盐���盐鼀 鼀矄矄 ᨨ矄 ᨨ 鼀ᨨ 矄ᨨ 鼀�矄� q㦨m鼀�ᆠᨨ矄q
\ μ 鼀� = θ
F - 㡰 ´҈ æ F - ö =ç ÷ �㡰 è ø 盐a鼀 㦨m�㦨맔҈㦨
㡰
f҈
҈
꾀矄矄盐鼀� � ᨨ ᨨᆠ├矄 矄�ᨨ 泐 㦨 f҈㦨 f㦨҈졔㦨҈
F - \ = Þ 㦨҈ ҈ ├�
꾀
a=
µ 㦨
µ鼀�ᆠᨨ矄q
㡰
黨
ᔴᨨ 矄� � � 犨ᨨ ᨨᆠ├矄 矄 ᨨ 矄泐矄鼀 盐 ᨨ ⼄盐���盐鼀f㦨҈ 㡰 f 㦨 Ҙ M 졔 m㡰 g
f Ҙ M + m㡰 g = = Ҙg M +m M + m㡰 㦨 � 맔 ҈ 㦨 � 鼀矄
҈
a=
졔鼀�矄� �q
f
f҈
�
鼀�矄� � 㦨
҈
㦨 �
Q鼀 = ҈├� � = 黨 鼀 矄 - 桌 m鼀�ᆠᨨ矄q㦨맔鼀�맔
҈
졔鼀�矄� q 㦨졔m鼀�ᆠᨨ矄q
= ´ �
ᔼ �� 鼀ᨨ鼀 盐鼀 �V V = = M + m㡰 ҈�
m㦨�� n
M㦨҈├�
v u 㦨 as u 㦨 as u 㦨 as
æ �v ö ç ÷ = ´ � ´ è ҈� ø
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PS-27
Laws of Motion 和ᨨ� � 犨 � v = ҈ 졠ᆠ � � ᨨ鼀 �� H
V = ҈
҈
mg
x
Þ
҈
mgᆠᨨ矄q
P
mg矄� q
q
mgᆠᨨ矄q
f mg
m q
y
x㦨᠀҈
҈
x ҈ = m 㡰 ᨨᆠᨨ �ᆠ�ᨨ ᨨ� ҈ �ᆠ�ᨨ = 㝐 矄� q ҈ 㦨 㝐 矄� 黨° q㦨黨 桌 㝐 㦨 ᆠ㡰 㝐� �鼀盐鼀 ᨨᆠᨨ A 㦨 �ᆠ�ᨨ ᨨᆠ 犨 矄盐 ᆠ矄 㦨҈N 꾀
N
q
q
ᨨ犨 y =
҈҈
q
mg矄� q
dy x = q㦨m 㦨 ᨨ鼀⼄盐矄�ᨨ 㡰 dx Q ᔴᨨ �ᆠ� ᨨ �ᆠ�ᨨ m㦨� � =
f
P
gH
҈ = ´҈ ´ H ҈ H㦨鼀㦨├鼀 㡰 꾀�鼀�� �⼄盐���盐鼀 m㦨 q
\
N
├� �├�
和鼀ᨨᨨ
␔ P㦨mg 矄� qf㦨 ␔ ᨨᆠP�矄� ᆠ矄 盐 ᨨᆠ├ 矄 ᆠ泐 ᨨ 鼀ᨨ 盐犨 矄 ᨨ � � ᆠ� ᆠ �ᆠ�ᨨ ᨨᆠ ᆠ矄 ᨨ犨 犨 矄 ᨨ � � ᆠ� ⼄盐���盐鼀P㦨f졔mg 矄� q Þ f = P mg 矄� q ᨨ犨矄P� ᆠ矄矄f� ᆠ矄矄� 泐犨P ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 矄�盐�ᨨ b㡰 㝐 �鼀盐鼀 � ᨨ ᨨ 矄� ᨨᆠ├ � ᨨ f � ᆠ� �ᨨ 㦨 �ᨨ ᨨ矄 졠 ҈� ᆠ q
� -҈ m = -҈ = ° ᨨ ᨨᆠ├ ᨨᨨ ᆠᨨ ��ᨨ ᨨ ᨨᆠ├ 矄 矄 ᨨ犨 � ��盐
㝐
鼀
ᆠ鼀
҈
m=
犨
q
� ᆠ鼀 PM ҈ ⏌ = = = OM ҈� ⏌ 䰘� ᆠ鼀 和ᨨq ᨨ鼀 �矄犨ᆠ ᆠᨨ ᆠ盐 ᨨᆠ├犨� ᨨ矄矄 �ᨨ � ᆠ� �ᨨ ᔴ泐 ᨨᆠ├犨� 鼀� 矄ᨨ 盐ᨨᆠ� �q
�犨�ᨨ ҈䰘 㡰 和� ᆠ ᨨᆠ├�矄 ᨨ鼀ᨨ� � ᨨ犨 ᨨ 鼀 �鼀盐鼀 N ᨨᆠ F � ᨨ Fᆠᨨ矄 㦨f㦨ҘN ᨨ� ᨨ �ᆠ�ᨨ 㡰 F ᆠᨨ矄ᨨ ᨨ�ᆠ⼄盐���盐鼀 ᨨ mg ᨨ ᨨᆠ├ f N㦨鼀�졔F矄� F \ Fᆠᨨ矄 㦨m㦨Ҙ F矄� ᨨ F矄� 졔mg桌 ᔼ D POM q =
҈
҈ ҈
ᨨ 鼀 �鼀盐鼀 ᆠᆠ�ᨨ
҈ = 鼀 ⏌ 矄 a鼀 㦨 ├� ᆠ鼀 �鼀盐鼀 ᨨᆠ 鼀 㦨ᨨ鼀矄矄맔a鼀 㦨黨맔㦨䰘 b㡰 㡰 ᆠᨨᆠ泐 ��ᆠ矄 ᆠᆠ�ᨨ 㜠�鼀 � ᨨ ᨨᆠ├ 㡰 ␔ ᨨ 泐 矄 鼀 �鼀盐鼀 矄
´ ´ ҈ m mg � = = Þ F㦨 㦨 ᆠᨨ矄 ° - m 矄� ° ҈ ҈ ҈
҈
m = = �° ҈�
\ 㦨鼀 㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ q > m 矄ᨨ ᨨᆠ├ 矄 ᆠ泐 ᨨ 鼀ᨨ ᨨ犨 � ᆠ� ᨨᆠ P �矄 � 盐犨 矄 ᨨ � � ᆠ� ᨨ ├ ᨨᆠ├ 矄�ᨨ 泐 ⼄盐���盐鼀P졔f 㦨mg矄� q Þf㦨mg 矄� q - P ᨨ犨矄P� ᆠ矄矄f ᆠ矄矄 � 泐犨� 矄ᆠᨨP
҈黨
-
´
㡰 犨ᨨ ᨨᆠ矄 ᆠ� � ᨨ � 矄ᆠ mg N
犨ᨨ ᆠᨨ鼀ᨨ 矄 ᨨ 鼀� mgᆠᨨ矄 a ᆠ矄 N m㦨҈⏌ a mg矄� a�矄 ᆠ f N 泐 �ᆠ�ᨨ ᨨᆠ \ N㦨mgᆠᨨ矄a mgᆠᨨ矄a a f㦨mg矄� a mg mg矄� a 盐f㦨ҘN㦨Ҙmgᆠᨨ矄a ҈ \ Ҙmgᆠᨨ矄a㦨mg矄� a Þ ᆠᨨa㦨 Þ ᆠᨨa㦨 Ҙ m 㦨� b㡰 �鼀�� � �ᆠ�ᨨ ᨨᆠ fl㦨Ҙ矄N㦨�맔�㦨�N f � ᨨ�ᆠ⼄盐���盐鼀 � ᨨ ᨨᆠ├ � �ᆠ�ᨨ ᨨᆠf㦨鼀�㦨黨N
鼀�㦨 ҈맔黨 㦨黨㦨␔
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EBD_8341 PS-28
Physics
㡰 ᨨᆠ├�矄 矄 ᨨ⼄盐���盐鼀 �ᆠ�ᨨ ᨨᆠ
N
和盐矄�盐� �g㦨҈鼀⏌矄㡰 Þ � + � m = � - � m㡰 Þ � + � m = - m Þ � m = ҈�
f
f㦨鼀�矄� q㦨鼀�矄�
= ´ 黨 ´ ҈
mg矄�
҈ = 黨
mg
= = � ҈ I 和ᨨ = ҈ ҈ Þm=
mgᆠᨨ矄
�㡰 ᨨᆠ├ 鼀ᨨ� � 盐犨 矄�ᨨ 泐
ᨨᆠ├
盐矄
鼀� 矄
f㦨mN F҈ q mgᆠᨨ矄q mg矄� q f mg
� F mg矄� q
�
ᨨ盐犨 鼀ᨨ� �ᨨ ᨨᆠ├盐矄 � � ᨨᆠF҈㦨mg矄� q졔 f \ F҈㦨mg矄� q졔mmgᆠᨨ矄q㦨mg 矄� q졔mᆠᨨ矄q㡰
ᨨᆠ⼄盐� ᨨ盐矄 � ᨨ鼀 矄� � � ᨨ犨 ᨨ ᨨᆠ├ 盐矄 鼀� 矄 矄�ᨨ 泐 F㦨mg矄� qm㦨mg 矄� qmᆠᨨ矄q㡰 � F҈㦨F \ 矄� q졔mᆠᨨ矄q㦨 矄� q mᆠᨨ矄q㡰 \ ҈졔m㦨 ҈m㡰 Q 矄� q㦨ᆠᨨ矄q桌 \ m㦨 Þ m㦨� \ 㦨҈m㦨҈맔�㦨� �N㡰 �ᆠ�ᨨ ᨨᆠ�矄矄ᨨ 矄�ᨨ鼀ᨨ 㦨鼀� ᨨᆠ├ᨨ 鼀矄矄҈├�犨� ᆠᆠ�ᨨ
ᨨ �鼀⏌矄 f
ᨨ �ᆠ�ᨨ ᨨᆠ f㦨m맔a㦨҈맔�㦨�N 鼀� 㡰 꾀ᆠᆠ�ᨨ ᨨ ᨨᆠ├ 犨 �鼀ᨨ� � 盐 � ᆠ�
a҈ = g 矄� q + mg ᆠᨨ矄 q Þ a҈ = g 矄� ° + mg ᆠᨨ矄°
g mg + 矄� � v - u = a s 㡰 =
�㡰
Þ v - = a҈ s㡰
Þ
v
v a҈
Q q 㦨ᨨ㡰
Q 盐㦨㡰
��㡰
꾀ᆠᆠ�ᨨ 犨 � 鼀ᨨ� � ᨨ犨 � ᆠ� a = g 矄� q - mg ᆠᨨ矄 q Þ a = g 矄� ° - mg ᆠᨨ矄 °
g m g ���㡰 矄� ��� v - u = as ᨨ ᨨ犨 犨 鼀ᨨ�ᨨ
Þ a =
v æv ö Þ ç ÷ = a s 㡰 Þ s = a è ø
ᔼ ᆠ矄ᨨ P矄 矄 ᨨ犨 � ��盐 泐� 鼀 � ᨨ矄��ᨨ ᨨ� ᨨ ᨨᆠ�泐 �矄 � ᆠ矄矄 盐 ᨨ N 矄� qᆠ 矄 鼀 盐O ᆠ矄矄 ᆠ�犨泐矄鼀� 矄� v和ᨨtPtQ 䰘 㡰
- a҈ s 㡰 =
Þs=
矄 �ᆠ�ᨨ ᨨᆠ ᨨᨨ矄矄 � 鼀ᨨ�ᨨ ᨨ 矄盐 ᆠ ᨨ ᆠᨨ ᆠ 꾀矄 盐矄 矄 矄盐 ᆠ � ᆠ├犨 �ᆠ�ᨨ 矄ᨨ �ᆠ�ᨨ ᨨᆠ 泐 矄盐 ᆠ ᨨ 矄ᨨ �矄 �
�ᆠ�ᨨ ᨨ �矄 鼀ᨨ�ᨨ ᆠ㡰 �ᨨ盐矄 ᨨᆠ矄 ᆠ� � ᨨ ᨨᆠ├ 矄 矄 ᨨ犨 � 턼 ��盐 cos q ␔ q㦨�º矄� q㦨ᆠᨨ矄q q ҈
ᨨᆠ├ 犨� 鼀�
矄� q q 矄�ᨨ 泐 �ᆠ�ᨨ 矄� q ᨨᆠ�矄 ᨨ ᆠᨨ矄 q ҈ ␔ q>�º矄� q>ᆠᨨ矄q
ᨨ �ᆠ�ᨨ ᨨᆠᆠ矄 q ᔼ쟼 ᨨ犨 矄 Q ␔ q� ᆠᨨ矄q>矄� q
ᨨ �ᆠ�ᨨ ᨨᆠ ᆠ矄 ᨨ犨 矄 P 㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ A ᨨ� ᨨ 矄 矄 ᨨ Q q ᨨ 鼀矄矄矄�矄 矄鼀 A B v 꾀矄 ᨨ ᨨᆠ �矄 ᆠ� � � Nᆠᨨ矄q N ᨨ� ᨨ �ᆠ�ᨨ q N矄� q ᨨᆠ�泐 ᨨ Q 鼀� 矄 P 矄鼀� ᨨ� ᨨ O mg
�㡰
쟼⼄盐� �⼄盐�ᨨ ��㡰 �㡰
æg m ö g mg v v = ç = Þ a҈ = a Þ + ÷ ç ÷ø a҈ a è
ᨨᆠ├犨� ᨨ矄�ᨨ犨�矄� m҈졔m㡰g 矄� q£mmg ᆠᨨ矄q
q £
mm g ( m҈ + m ) g
mm Þ q £ m҈ + m
f㦨mN
N q m҈ 졔m㡰g ᆠᨨ矄q
m҈ 졔m㡰矄� q
´ ҈ m҈졔 m㡰g Þ q £ £ q ҈+ � Þ q£�q £҈҈� �ᔼ �q҈҈� �ᆠ�ᨨ ᨨᆠ�矄矄矄
mN㦨mm� 㦨맔맔g㦨� �矄⼄盐ᨨ m҈졔m㡰g矄� q ᨨᆠ├矄 犨� ᨨ 矄� ᨨ � ᆠ� �ᆠ�ᨨ �矄 矄�ᆠ 꾀q>҈҈� ᨨ �矄矄鼀ᨨ� �ᨨ � ᆠ� �ᆠ�ᨨ �矄├� �ᆠ ⼄盐ᨨmmgᆠᨨ矄 q ᨨ鼀ᆠ�ᨨ N҈㦨mma矄� q N㦨mg 꾀泐� � 矄盐 ᨨ ᨨᆠ ma 矄ᨨ� � � F 㦨mar maᆠᨨ矄 q f҈졔f㡰㦨mar maᆠᨨ矄 q ҘN҈ҘN㦨mar
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PS-29
Laws of Motion maᆠᨨ矄 q Ҙma矄� q Ҙmg㦨mar Þ ar㦨aᆠᨨ矄 q Ҙa矄� q Ҙg
- ´ � ´ - ´ ҈ = ҈ 鼀 ⏌ 矄 � � � � 꾀ᆠᆠ�ᨨ ᨨ ᨨᆠ├ ᨨ犨 mN = f K mg 矄� q - m k mg ᆠᨨ矄 q a㦨 m
矄ᆠᨨ ��ᨨ 矄犨�矄�矄 � 犨 T҈�N矄泐T㦨҈ f҈㦨f㦨҈N
ᨨ ᆠᨨ ��ᨨ f҈ 㦨 f 犨� ᨨ 矄�矄 �
盐矄 m҈ m ᨨ ᆠ Ἐ 鼀� 矄�ᨨ 泐 ᔼ 矄ᆠᨨ ᆠ矄犨 m҈ m ᨨ 鼀ᨨ f㦨 f㡰鼀 㦨҈�N \ f҈㦨f㦨N 和� ᆠ f҈ f҈㡰鼀 �矄 ᨨ � 鼀ᨨ�ᨨ 犨 m҈ M �.e. 鼀矄矄矄 鼀ᨨ 犨� 矄鼀 ᆠᆠ�ᨨ 矄泐 ἘaἘ ᨨ 泐 ��鼀矄 ⼄盐�ᨨ 矄ᨨ 鼀ᨨ�ᨨ 矄 ᨨᨨ犨矄
㦨 � ´
\
aA㦨g矄� q Ҙk,Agᆠᨨ矄 q
q
mg矄� q
mgᆠᨨ矄q
㦨�矄� � Ҙᆠᨨ矄�
N = mg ᆠᨨ矄 q
mg
æ ҈ ö æ ҈ ö = ҈ ç - 㡰 ҈㡰 ç = è ÷ø è ÷ø 꾀 aB㦨g矄� q Ҙk,Bgᆠᨨ矄 q 㦨�矄� � m├�ᆠᨨ矄� æ ҈ ö æ ҈ ö = � 鼀⏌矄 = ҈ ç - 㡰 ҈㡰 ç ÷ è ø è ÷ø aAB�矄�ᆠᆠ�ᨨ ᨨ A犨B
aAB㦨aA aB
� �矄 ᆠ犨 A BL ҈ L㦨 a AB t 㡰 L L = = ᨨ t㦨 a AB a A - aB - �
(
) (
)
Þt㦨ᨨt㦨s �矄 ᆠ鼀ᨨ 泐A 盐� � �鼀�矄 �� 泐 ҈ ҈ SA㦨 a t 㦨 ´ � ´ = 鼀
҈
和�鼀�泐 ᨨB 㦨 䰘 鼀 ᆠᨨ ᨨᆠ├矄犨�ᆠᨨ鼀� � A 矄 �矄 ᆠ 鼀 ᨨ犨 㡰 ᨨ 泐 ��鼀ᨨ 鼀矄矄 M T T
T
m mg �㡰 mg ᆠᨨ矄 q + m 矄� q � �� � ᨨ ⼄盐�ᨨ 犨 q 犨 � dF m mg = - 矄� q + m ᆠᨨ矄 q桌 = d q ᆠᨨ矄 q + m 矄� q 㡰
f҈
F
T
㡰 鼀 �鼀盐鼀 盐ᨨ ᨨᆠ ᨨ �ᆠ�ᨨ 犨 鼀҈ 㝐 f҈㡰鼀 㦨 㡰 㡰 ҈㡰㦨N �㡰
鼀 �鼀盐鼀 盐 ᨨ ᨨᆠ ᨨ �ᆠ�ᨨ 犨 鼀 㝐 f㡰鼀 㦨 㡰 �㡰 ҈㡰㦨҈�N ��㡰 ᨨᆠ矄ᨨ m҈ m� ᨨ� ᨨ �ᆠ�ᨨ 矄 ᨨᨨ犨矄
m҈
f҈
f
m
F㦨
\ q 㦨 ҈Ҙ
�矄�矄 � ᨨ鼀� �鼀盐鼀 ᨨᆠ
ᨨ � 鼀� �鼀盐鼀 ᨨᆠ 矄盐矄�盐� � 矄 盐矄 �
⼄盐�ᨨ �㡰 Ҙ ҈ 矄� q 㦨 ᆠᨨ矄 q 㦨 Ҙ +҈ Ҙ +҈
N
T
T
M
F f҈㦨 f㦨҈� ᨨ鼀矄矄m ҈T㦨a ���㡰 ᨨ鼀矄矄m T҈�㦨�a �㡰 ᨨ 鼀矄矄 㝐F㦨�a 㡰 꾀 � �⼄ ���㡰 �㡰犨� ᆠᆠ�ᨨ a㦨 鼀 ⏌ 矄 � �´ ᨨ鼀⼄ �㡰 ҈�㦨 Þ 㦨҈ � ᨨ鼀⼄ 㡰 㦨 � ´ Þ 㦨 � ᨨᆠ � ᨨ鼀ᨨ ᨨ 泐 � q ᨨ ᨨ� ᨨ F矄� q
ᨨ 泐犨� 鼀ᨨ犨 F N Fᆠᨨ矄 q 㦨ҘN q ᨨ鼀 ��盐 ᨨ鼀ᆠ�ᨨ ҘN Fᆠᨨ矄q N㦨mgF矄� q ᆠᨨ矄q㦨Ҙ 鼀� 矄� q㡰 Þ
N҈ Mg
m
m҈
a
F㦨
Ҙ mg ҈ Ҙ + ҈
T
+
ᨨ 泐 犨ᨨ ᨨ矄矄����矄 ᔴ矄ᔼ쟼� ᨨ m҈ m犨�鼀� 矄�ᨨ 泐 犨 �ᨨ盐 㡰 ᔴ矄ᔼᔼᨨ m҈ m犨�鼀ᨨ 犨�ᨨ盐 㡰 �矄ᆠ矄 �矄 ᨨ矄矄� 犨
ᨨ T f҈㡰鼀 ᨨ TN T f㡰鼀 ᨨ T҈�N
m
+1
f҈㦨
a T
Þ F㦨
Öm
黨
m Ҙ + ҈
Ҙ mg Ҙ + ҈㡰
Ҙ +҈
=
´m
q
1
Ҙ mg
Ҙ + ҈
Þ F㦨mg矄� q 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ 鼀矄矄 M҈ 鼀ᨨ矄 ᨨ犨 犨 矄 犨� 盐 � ᨨ鼀 ᨨᆠ�泐� ᆠᆠ�ᨨ ᨨ 矄泐矄鼀 �矄 ᨨ 졠 盐� � ᨨᆠ ᨨ 矄泐矄鼀�矄 ᨨ
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EBD_8341 PS-30
Physics
ᨨ⼄盐���盐鼀 㡰 M҈g㦨Mg矄� 䰘 졔mMgᆠᨨ矄䰘 졔mMg ᨨ M҈㦨M矄� 䰘 졔mMᆠᨨ矄䰘 졔mM
F
æ ö æ ö = 㡰 ç ÷ + �㡰 㡰 ç ÷ + �㡰 㡰 = ├� è �ø è �ø 㡰 和� ᆠM�矄 鼀ᨨ� �犨� 盐 � ᨨ鼀ᨨᆠ�泐
㦨m҈mg㦨 �㡰맔맔黨㦨黨 T 矄�ᨨ � ᨨ ᨨ� P a ᆠᆠ�ᨨ ᨨ 盐ᆠ � ᨨ F 㝐矄矄 盐 � � ᨨ ᨨ�矄 m = L A P B F Ll l L ᨨ AP錐 T㦨m L㡰a �㡰 ᨨ 錐 FT㦨m⏌a ��㡰 é T ù F - T = ml ê 矄� �⼄ �㡰桌 ú ë m L - l 㡰 û Tl é l ù ÞT ê + l ú = F錐 L-l L l ë û é L ù æ L -lö Tê = F ÞT = Fç è L ÷ø ë L - l úû l ᨨ T = F é҈ - ù ê Lú ë û 꾀矄ᆠ盐矄 ᨨ ᨨ矄� ᨨ犨
矄 ᆠ 泐 矄 a 矄鼀 ᆠᆠ�ᨨ 鼀҈ 矄盐 ᨨ ᨨᆠ㡰 ᔴᨨ 矄� ᨨ a ᆠ盐矄 ᨨ � � ᆠ� 鼀҈ �
졔鼀҈ ᆠᨨ矄a㦨m҈g矄� a® �㡰
졔鼀�矄� b㦨mgᆠᨨ矄b® ��㡰 鼀 ᆠᨨ矄b 鼀 쟼⼄ �㡰쟼⼄ ��㡰 b 鼀 鼀҈ᆠᨨ矄a졔鼀ᆠᨨ矄b㡰f b 㦨 鼀҈矄� a졔鼀矄� b㡰� Þ
�
T
f = m m A + mB 㡰 g
f҈ = m mA + mB + mC 㡰 g 꾀矄C�矄鼀ᨨ� �犨� ᆠᨨ 矄 矄 F㦨f҈졔f졔T �㡰 ᨨ鼀矄矄B�矄 mm A g = f m m A + mB 㡰 g = f
B
T
꾀矄B�矄鼀ᨨ� �犨� ᆠᨨ 矄 矄 f졔 f㦨T ��㡰 和盐ᆠ� �⼄ ��㡰 ᨨ鼀 �㡰 F f졔 f㡰㦨f҈졔f졔TT㦨f҈졔f Þ F 㦨f҈졔f졔f㦨Ҙ mA졔mB졔 mC㡰g졔 Ҙ mA졔mB㡰g졔ҘmAg F㦨Ҙ mA졔mB졔mC㡰g � 鼀꾀㦨├�鼀㦨├�㝐㦨�├� m㦨�㡰 㦨�맔졔맔졔�桌맔黨㦨䰘҈�N ᆠ ᨨᆠ ᆠ矄矄泐ᨨ � 㦨 䰘҈�N
Topic-4 : Circular Motion & Banking of Road ҈
F -T =
鼀҈矄� a + m 矄� b㡰 f = g 鼀҈ ᆠᨨ矄 a + 鼀 ᆠᨨ矄 b㡰
C
mv = mrw r w㦨pn㦨p맔�㦨䰘p ⏌矄ᆠ �盐矄r㦨҈�ᆠ鼀㦨҈�맔҈鼀 ᔴᨨ �ᆠ� ᨨ �ᆠ�ᨨ Ҙ㦨䁸 m rw 㡰 mmg = Qv㦨rw㡰 r Ҙmg㦨mrw
㡰 矄� � mmg =
ö æ ҈� ´ ҈- ´ ç 䰘 ´ ÷ è rw 䰘ø = Þ m= g ҈
O
�矄ᆠ
鼀҈�矄� a
鼀�矄� b
鼀�
é m 矄� a + m 矄� b ù ᨨ鼀⼄ �㡰 = 鼀҈ g 矄� a㡰 - 鼀҈ᆠᨨ矄 a㡰 ê ҈ úg ë m҈ ᆠᨨ矄 a + m ᆠᨨ矄 b û é m҈ ᆠᨨ矄 a 矄� a + m҈m ᆠᨨ矄 b 矄� a - m҈ ù ê ú ᆠᨨ矄 a 矄� a - m҈m ᆠᨨ矄 a 矄� b ú ᨨ = g ê ê ú 鼀҈ᆠᨨ矄 a + 鼀 ᆠᨨ矄 b㡰 ê ú êë úû m m [ ᆠᨨ矄 b 矄� a - 矄� b ᆠᨨ矄 a ] g ᨨ T = ҈ 鼀҈ᆠᨨ矄 a + 鼀 ᆠᨨ矄 b 㡰 ␔ ᨨᆠF�矄� ᨨ ᨨᆠ├C犨�鼀ᨨᨨ犨 矄 ᨨᆠ├ἘἘ犨�鼀ᨨᨨ犨 矄�� 盐ᨨᆠ�ᨨ ᨨ Cᨨ B犨 �ᨨᆠ├A犨泐矄鼀� 矄矄 =
ᨨ鼀矄矄C �矄
҈�ᆠ鼀
=
҈� ´ ҈- ´ = ҈
㡰 � q㦨� 㦨鼀�㦨҈鼀⏌矄 鼀 �㡰
矄� q =
ᆠᨨ矄 q = 鼀� ��㡰 ᨨ鼀⼄盐�ᨨ �㡰 ��㡰犨
q =
q
�
㦨� Qq㦨� ᆠ 矄 ᨨ 盐盐鼀 � �矄 ᆠ�ᆠ盐
= � = ´ ҈ 㦨鼀⏌矄 ᆠ㡰 㡰 ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ 矄�ᨨ ᨨ� 矄
ᆠ矄矄泐 ᆠ � ᨨᆠ \ 矄� q =
T´
鼀 㦨鼀 w
q L
R R = mRw Q矄� q㦨 桌 L L
T = = \w= � ´ � mL
=
Tᆠᨨ矄q T q
T矄� q R
mg
҈ = ⏌矄 = � �
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PS-31
Laws of Motion �
㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 矄 犨� 犨 �ᆠ ᨨᆠ├ 矄 ᆠ├�矄 矄鼀� ᆠ├矄 ᨨᆠ├�矄矄ᨨ 矄鼀 �� 矄ᨨ ᨨ鼀犨ᨨ ᆠᨨ 矄�ᨨ
ᨨ �泐矄 ᨨ ᨨᆠ├ �� 矄ᨨ� 犨�矄鼀 � ᨨ盐ᆠ矄矄 ᨨᆠ�泐 �� 矄 ᨨ� v
bᆠ㡰꾀ᨨ � �盐矄 ᨨᆠ㦨 ᆠ � ᨨᆠ
æ Mv ö ç ÷ ç l ÷ è ø 꾀 ᨨ � � ᨨᆠ 㦨鼀矄 鼀ᨨ�ᨨ ᨨ 盐盐鼀�矄 ᨨ ᆠ�ᆠ盐 鼀ᨨ�ᨨ
\ 졔鼀�㡰ᨨ� 矄 ᆠ � ᨨᆠ
mv 犨 l l�矄 � ᨨ 盐盐鼀 v�矄 ᨨᆠ�泐ᨨ ᨨ ᨨ⼄盐���盐鼀ᨨ ᨨTmgᆠᨨ矄 =
\ mg ᆠᨨ矄 q - N =
mv R
黨
q Mg Mgᆠᨨ矄 q
g
gᆠᨨ矄q
꾀 mat㦨mg 矄� q Þat㦨g矄� q �盐矄ᨨ ᆠ�ᆠ㦨 ᔼ DOCPOC㦨CP㦨R \ ÐCOP㦨ÐCPO㦨 Þ ÐOCP㦨 \ DOCP�矄 ⼄盐�� � Þ OP㦨R
C
R R
ᨨ
mgᆠᨨ矄
A h
N v q
mg矄� q
mg
P mg
O
R R = =x
ᨨᆠ矄 ᆠ� � 矄 ᨨ犨 � ��盐 �㡰 ᨨ鼀 ᨨ � � ᨨᆠ � � � �ᆠ�ᨨ
㦨Fᆠᨨ矄 졔mgᆠᨨ矄 㦨kx졔mg桌ᆠᨨ矄 \
쟼 矄�ᨨ ᨨ 矄� �㦨 R -
N B
N = mg ᆠᨨ矄 q - mg + mg ᆠᨨ矄 q N = mg ᆠᨨ矄 q - mg N = mg ( ᆠᨨ矄 q - ) ᔴ泐犨 ᆠᨨ矄 q > N�矄ᨨ矄��ᆠ矄 �泐ᨨ盐犨 矄 � ᆠᨨ矄 q < Nᆠ矄 �泐� 犨 矄
N F㦨kx
M
mg
mv Þ N = mg ᆠᨨ矄 q �㡰 R mg ᆠᨨ矄q 꾀 泐 �泐ᆠᨨ 矄�ᨨ q ҈ mv = mg [ R - R ᆠᨨ矄 q ] v = g (҈ - ᆠᨨ矄 q ) \ R v 盐� � �矄盐ᨨ � ⼄ �㡰 R N = mg ᆠᨨ矄 q - m [ g - g ᆠᨨ矄 q] Þ Þ Þ
Mg矄� q
q g矄� q
T
꾀矄 �矄鼀ᨨ� �꾀ᨨ�� ᆠ�ᆠ盐
q
l
ᆠᆠ泐T>mgᆠᨨ矄 㡰
T
mv T㦨mgᆠᨨ矄 졔 l
mv �矄 犨泐矄 ᨨ矄�� ⼄盐 �泐 l 䰘
Mv TMg ᆠᨨ矄 q = l
鼀 ᨨ ᨨ 泐
mv N + mg = R mv N= - mg ᨨ R �盐矄ᨨ ᆠ盐盐㡰� �矄ᆠ矄�矄鼀� �鼀盐鼀 ᆠ ᨨ鼀ᆠ�ᨨ 犨�鼀 �鼀盐鼀� �矄ᆠ矄 矄 �盐鼀�盐 ᨨ 盐盐鼀�矄 �� 犨 �矄 �盐 �矄ᆠ鼀 �矄
\
l
v
졔鼀�
q
F矄� ᨨ
ᨨ
F 졠
ᨨ
鼀�矄� ᨨ ᨨ
Fᆠᨨ矄ᨨ 鼀� 鼀�ᆠᨨ矄ᨨ
� �
� mg � \Ft㦨mat Þ at㦨 � 꾀矄ᨨ犨 � � �矄 盐矄 矄 N졔F矄� 㦨mg矄� mg ö ҈ mg æ Þ N㦨 mgF㡰矄� 㦨 ç mg ÷´ = è ø Ft
㦨
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EBD_8341 PS-32
Physics
Work, Energy & Power
5
ᔼ �� � ᨨ 矄� 矄
Topic-1 : Work Done by Constant & Variable Force ҈
W =
㡰 ��
矄�盐�ᨨ ᆠ F 犨 � �ᆠ泐矄 B A 矄 ᨨ犨 � ��盐 ␔ᨨ├ ᨨ 㦨 꾀 盐 F㜠x � E ҈ 㦨 꾀 ᨨ ᆠ � ABCD 졔 꾀 ᨨ C F D �盐鼀 BCFE ҈�鼀 鼀 ҈ W = ´ ҈�㡰 + ҈ + 㡰 ´ ҈� = + �
䰘
ᆠ㡰 ᔼ AC㦨l ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ BC㦨l AB 㦨l ᨨ盐� 犨ᨨ├ ᨨ 泐 m ᨨᆠ矄 �矄 ᨨ q W �ᆠ�ᨨ 졔Wmg㦨 A
和鼀ᨨᨨ
mg l 㡰矄� q - mmg ᆠᨨ矄 q l 㡰 =
\k =
r uur
ᆠ㡰 ␔ᨨ├ ᨨ W = ò F × ds
(
)(
= - x�$ + yˆ × d ´ �$ + dyˆ
)
黨
҈
҈ ҈ Þ W = ò xdx + ò ydy = æç + ö÷ + = ҈J è ø ҈
�
鼀� 鼀� Þ㦨 㡰 鼀� 㦨鼀 㦨 㦨 ᨨ鼀ᆠ�ᨨ
ᨨ犨犨ᨨ├ ᨨ 泐 ᨨ鼀ᆠ�ᨨ �ᨨ
r r æ 鼀� ö æ ҈ ö ᨨᆠ├� �鼀 ␔㦨和㦨 çè ÷ø çè �⏌ ÷ø 鼀� ᨨ ␔ 㦨 矄 㡰 ᨨ鼀 ⼄盐矄�ᨨ = 矄 = ᨨ 矄 = Þ 和 = = 鼀´ 鼀 ҈ ␔ᨨ├ ᨨ = ´ = 鼀 ᆠ㡰 ␔ᨨ├ ᨨ � 矄ᆠ � � 盐㜠 泐 �矄 ᆠ dx �矄 dW㦨F dx㦨 ax졔bx㡰dx
㡰 �盐矄ᨨ ᆠ�ᆠ盐 㦨
⼄盐�ᨨ ᨨ ᆠ�ᆠ�矄 졔泐㦨 � ᨨᆠ
泐
l矄� q
Þ mmg ᆠᨨ矄 ql = mgl 矄� q Þ m = q = k q
aL bL +
é �ˆ r 泐ˆ ù = ⧤ ê ⏌ + ⏌ ú 㡰 㡰 ûú ëê 㡰
B
C
L
ò axdx + ò bx dx =
é ù r �ˆ 泐ˆ + = ⧤ê ú ⏌ + 泐 㡰 ⏌ ûú ëê + 泐 㡰
Þ W = ��
L
矄 泐㡰
ˆ r ⧤ 泐 = éë �ˆ + 泐ˆùû 㡰 �ˆ
ᨨᆠ ᆠ矄 �泐ᨨ盐犨 矄 矄 矄 ᨨ犨 � ��盐 �矄ᆠ鼀 �矄 � � ᨨ ᆠ�ᆠ盐 � 犨 ᨨᆠ 犨 �ᆠ ᆠ矄 �泐 ᨨ盐犨 矄 �矄ᆠ鼀 犨 �ᆠ �矄 � � ᨨ ᆠ�ᆠ盐 �矄黨 \ ␔ᨨ├ ᨨ ␔㦨 和ᆠᨨ矄q㦨 b㡰 ᔼ ᆠᨨ 矄� � 犨ᨨ├ ᨨ ᨨ矄 ᨨ ᨨ �.e., � ���ᨨ � �矄 ᆠᨨ 矄� � \ W҈㦨W㦨W r r r ᆠ㡰 dW㦨 F dS F = - K y�ˆ + xˆ㡰 ��
r dS = dx �ˆ + dy ˆ + dz kˆ dW = K y �ˆ + x ˆ 㡰 dx �ˆ + dy ˆ + dz kˆ㡰
㦨K ydx졔xdy㡰㦨K d xy㡰桌 \
a a
a a
ò dW 㦨 K ò d xy㡰 a a
҈
ᨨ W = K xy 桌 ᨨ W㦨Ka ᆠ ᨨ 犨ᨨ├ ᨨ 泐 ᨨᆠ ᨨ �ᆠ W㦨Ka 㡰 犨ᨨ├ ᨨ � � �� � 鼀矄矄盐犨�⼄盐 ᨨ ᆠ � � ᨨ � �泐ᨨ 鼀矄矄 h = ⏌ 㝐g 㝐矄矄㦨 醨 ᔴ㦨ᔴ ᨨ 鼀矄矄
C
�.e. W 㦨 ᔴ � � ᨨ � �泐 M L MgL ´g´ = 㦨mgh㦨 ҈ ҈҈ 㡰 ␔ 矄泐矄鼀 �矄 矄 T mg = ma �㡰 Mg T = Ma ��㡰 ᨨ鼀⼄ �㡰 ��㡰 M - m㡰 g = g ⏌ a= M +m T㦨mg⏌
T a
m 㦨├� T mg M a Mg
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PS-33
Work, Energy & Power ᨨbᨨᆠ├鼀=├� u㦨a㦨g⏌ t㦨҈s㦨䁸 ҈ ҈ g s㦨ut졔 at 㦨졔 ´ ´ ҈ = g ⏌ m㦨䰘├�㡰 \ ␔ᨨ├ ᨨ 泐 矄� � ᨨ m ´ ´ ҈ ´҈ mg g ´ ´҈ = = T sᆠᨨ矄 㦨 ´ ҈ 䰘�㡰 r � ᨨᆠ F 㦨 ay�ˆ + axˆ㡰 a㦨҈鼀҈ r r ␔├ ᨨ犨 ␔ = × = a泐 $� + a $ × $� + 泐 $ \ ␔㦨a泐 졔a 泐 Wᨨ├ ᨨ ᨨ鼀꾀® 泐㦨矄泐㦨҈
(
҈
ò
)(
҈ ´ = Þ = 鼀⏌矄 ᔼ �� ᆠᨨ鼀矄矄�ᨨ ᨨ 矄� � ҈�맔҈ 㦨Þ » 盐矄� � 犨ᨨ├ �泐 ᨨ鼀 ␔�졔␔矄㦨D⧤쟼
҈
ò
�
ᨨ鼀®ᔴ 㦨矄 㦨҈ �
\␔ =
ò
҈
�
a 泐 =
ò a 泐 = a (-�) = -a ҈
ᨨ鼀ᔴ® 泐㦨矄泐㦨� �
\␔ =
a
ò a ´ � = -
҈
ᨨ鼀®E 㦨矄 㦨�
\␔ =
a
ò a ´ � 泐 = -
�
ᨨ鼀E® 泐㦨矄泐㦨 \␔� = ò a ´ ´ = ᨨ鼀 ®꾀 㦨矄 㦨
ò
\␔ = a ´ = \ ᨨ犨ᨨ├ ᨨ ␔㦨␔ 졔␔ 졔␔ 졔␔ 졔␔ 졔␔ ҈
a a a - ( -҈) = 㦨䰘� 㦨aa - =
�
Topic-2 : Energy ҈
䰘
ᆠ㡰 ␔├ ᨨ犨 盐 㜠 � ��矄 犨ᨨ├ ᨨ 泐 ᨨ 泐
҈ ´ + 㡰 ´ - 㡰 + ´ 㦨�졔㦨� 矄� � 犨ᨨ├ �泐 ᨨ鼀 D ⧤쟼㦨犨ᨨ├ ᨨ \D ⧤쟼㦨� nl l l = ᆠ㡰 l҈ + l = l l҈ = nl \ l҈ = n +҈ n +҈ ҈ 꾀矄k µ l k҈ l ⏌ n + ҈㡰 ҈ \ = = k nl 㡰 ⏌ n + ҈㡰 n b㡰 ᨨᆠ�泐ᨨ ҈├�ᨨᆠ├盐矄 ᨨ�ᆠᨨ� 矄犨� ├�
\W=
ᨨᆠ├ = � = 鼀⏌矄
├�
҈ ҈ ´ ҈� ´ ҈ - 㡰 = - ´ ´ 矄ᨨ� � » ᆠ鼀 㡰 ␔㦨盐 盐� æ mg L ö MgL = -ç´ = è n n ÷ø n ᆠ㡰 ␔├ ᨨ犨 盐 㜠 � ��矄 犨ᨨ├ ᨨ 泐 ᨨ 泐 ҈ \ W = ´ + 㡰 ´ - 㡰 + ´ 㦨�졔㦨� 矄� � 犨ᨨ├ �泐 ᨨ鼀 D ⧤쟼㦨犨ᨨ├ ᨨ \D ⧤쟼㦨� 㡰 㝐 �鼀盐鼀矄 �矄鼀 ᨨ矄��ᨨ ᨨ⼄盐���盐鼀 꾀 ⼄盐���盐鼀 ᨨ矄��ᨨ
㦨├ Þ = ├ ᨨ鼀 犨ᨨ├㜠 �泐 ᨨ鼀 ␔ + ␔矄 = D⧤쟼 ҈ ҈ 㡰 ├ = 鼀 - ҈ æ ö ҈ æ ö ç ÷ - ├ ç ÷ = 鼀 Þ ҈ = ҈ 鼀 è ├ø è ├ø ⧤ ᨨ 鼀 = 鼀├ ⧤ ¶盐 ˆ = ˆ ᆠ㡰 = ¶ 和� ᆠ�ᆠ�矄鼀ᨨ� �� ᆠ�ᆠ盐 鼀 ⧤ ⧤ = = Þ 鼀 = ҈ ⧤ \ ⧤쟼 = 鼀 =
ᨨ �泐㦨쟼졔⧤쟼 ⧤ ⧤ ⧤ = - + = ᨨ Q 쟼 = - �� 㡰 㡰 Vf �矄 � 矄 ᨨ ᨨ 泐 ᨨ鼀 ⼄盐矄�ᨨ 矄 V ҈ ҈ mV f = mV = � m⏌ s Þ Vf = dV æ dV ö F = mç 㦨kV ÷ = -kV \ ҈㡰 dt è dt ø Þ ´ +
)
\␔҈ = a泐 = a = a
矄� � � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 盐矄 ᨨ 盐矄 ᆠᨨ�矄�ᨨ 犨 �
�
ò
dV
҈ V
҈
= -҈ K ò dt
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EBD_8341 PS-34
黨
҈
Physics
҈ ҈ - = ҈ K ҈㡰 ᨨ K㦨҈kgm҈ � ҈ 㡰 ⧤쟼㡰Ἐ㦨�奠ᨨ ⧤쟼 �� ҈ � ҈ 鼀 Ἐ = ´ 鼀 Þ Ἐ = ҈ ҈ ᔴᨨ �ᆠ� ᨨ 矄�盐�ᨨ = ᨨ犨 ᨨ �矄 ᆠ 泐 ᨨᆠ �矄 ҈ö æ æ ҈ + ö ç ҈+ ÷ = ç ÷ = ç ÷ = ç ҈ - ÷ ҈ è ø çç ҈ - ÷÷ è ø dV ᆠ㡰 矄� � 㦨鼀㦨 m dt dV t = ҈ Q m㦨҈├��� 桌 dt
ar ar � + = ar ҈ b㡰 꾀泐� � 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
m҈u҈ 졔 mu 㦨 m҈v҈ 졔 mv ҈u졔m 㡰㦨҈ 㡰졔m 㡰 ҈u㦨 m 㦨
҈ ҈ ҈u = m 㡰 和ᨨ� � 犨 � u 㦨
҈ ҈ æ xö ҈ kx = K ç ÷ + ҈㡰 èø kx = 黨 Þ ´ ҈ kx = 黨 Þ ҈ \ Kx = 0.6 (ᨨ � �� �泐 ᨨ 矄� �)
҈
ét ù ҈ ò dV = ò t dt V = êë úû = 鼀矄 Qt㦨҈矄ᆠ�� 桌 ᨨ鼀 犨ᨨ├㜠 �泐 ᨨ鼀 v
(
҈҈
ᨨ �泐 㦨 ⧤쟼 졔 쟼
҈�
盐 - ҈㡰 ᨨ = 盐 - =
ᔼ � � ᨨᆠ � 泐ᆠ ├
盐矄� � 犨ᨨ├ �泐 ᨨ鼀
)
҈ ҈ ␔㦨D⧤쟼㦨 m V - u 㦨 ´ ҈ ´ 黨 = � 㡰 ␔ᨨ├ ᨨ 泐 �ᆠ�ᨨ 턼 㦨Ҙ鼀�x
\
҈ 㦨 ÞPQ㦨m PQ ␔ᨨ├ ᨨ 泐 �ᆠ�ᨨ 턼㦨Ҙ鼀�맔ᔴᨨ矄 맔
҈ ҈ ҈ ҈ ( - ҈) 㦨 鼀盐 鼀㦨 鼀盐 鼀盐
ù é ҈ ҈ - ( - ҈) ú 㦨 鼀盐 ê ê ú
ë û ҈ æ - ҈ ö = 鼀盐 ç ÷ è ø 盐鼀ᨨ ├矄⼄盐� ᨨ矄ᨨ 盐
ᨨ �矄 ᆠ 泐 盐 ᨨ ᆠᨨ鼀� � ᨨ 矄㦨 矄� � 犨ᨨ├㜠 �泐 ᨨ鼀 ��
´ 㦨 Ҙ鼀� 和� ᆠ犨ᨨ├ ᨨ 泐 �ᆠ�ᨨ ᨨ 矄PQ QR⼄盐 Ҙ鼀�x㦨 盐鼀�
㦨Ҙ鼀�맔
Þ x㦨 @�m 矄� �犨ᨨ├ �泐 ᨨ鼀鼀�矄� 맔㦨 Ҙ鼀�졔Ҙ鼀�x
҈
Þ 㦨 Ҙ Þ Ҙ㦨0.29 ␔ 鼀� ´ ҈ ҈ ´ 黨 ´ ҈ ´ ҈ = = b㡰 㦨 � 盐 � 盐 � 盐 黨 ᔼ 盐㦨 㦨黨맔҈ 黨 ´ ҈ 㦨҈黨맔҈├� 盐矄 㦨 ´ ҈䰘 b㡰 꾀矄 犨 ├ ᨨ犨 㦨 r
U =
ò ar
dr =
ar
�㡰
mv 꾀矄 = ar r mv 㦨 mar ҈ ᨨ鼀 ⧤쟼㡰㦨 ar ��㡰
ᨨ �泐 㦨 ᨨ � �泐 졔 ├� �ᆠ �泐 ᨨ犨 ᨨ鼀 ⼄ �㡰 ��㡰
盐㦨
ᔼ � �矄� 㦨
҈
㡰 盐 � ��ᨨᆠ�泐ᨨ 盐ᨨ 鼀矄矄 鼀 꾀 矄矄� � ᨨ盐� ├ ᨨ 犨� �矄 ᨨᆠ�泐 ᆠ矄矄 ᨨ
( ) =
҈ 鼀盐 -
é҈ ( - ҈) ù ҈ = 鼀盐 \ =
ú ᨨ ( ) = ê 鼀盐
û - ҈ ë
҈
㡰 � ├ 꾀 = ⏌ 鼀 ├ = ⏌ 鼀 犨 ᆠᨨ鼀� �ᨨ ᨨ 矄� �矄 �矄 ᆠᨨ鼀矄矄 泐 ᨨᆠ 和� �꾀�矄ᆠᨨ鼀矄矄 泐 ᨨᆠᨨ鼀矄矄�ᨨ
� 矄� �
= 䰘� - 㡰 ᆠ鼀 = ´ = 䰘� - 㡰 和ᨨ� �犨� 㦨�ᆠ鼀 = 䰘� - � = 䰘�ᆠ鼀 ҈ ├ 㡰 쟼꾀 꾀 꾀 ´ �㡰 = = = ҈ 쟼 ├ 㡰 ´ 䰘�㡰
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PS-35
Work, Energy & Power ҈䰘
ᆠ㡰 꾀盐 F t � ��矄 � � 鼀ᨨ鼀 盐鼀 ᨨ ᨨ 泐
Ux㦨x㦨 x㦨
F t㡰
꾀ᨨ DAOB㦨
҈ ´ ´ = 矄 A N
\ \
OA CD �s = 졠 C t s OB CB CD D = Þ CD㦨 ҈� é҈ ù 꾀ᨨ Dᔴ = - ê ´ ҈� ´ ú = -҈� 矄 ë û
� � 鼀ᨨ鼀 盐鼀㦨҈�㦨�矄
p �㡰 = = � m ´ ҈ ᆠ㡰 矄 犨 ᨨᆠ├ B �矄 �矄ᆠ ᨨ犨 矄 犨 ҈ ᨨ 泐矄� �S҈�矄 ᆠᨨ鼀矄矄 S �矄 � �矄 盐 矄 矄 ᨨ ᨨ S �矄 ҈
ᨨ �泐矄ᨨ � 矄泐矄鼀 㦨 k҈ x ␔ ᨨᆠ├ �矄 矄 � 犨� ᆠᨨ鼀 ᆠ├ ᨨ ⼄盐���盐鼀 ᨨ矄��ᨨ �� 鼀ᨨ鼀 盐鼀 ᨨ矄 ᨨᨨ ᨨ ⼄盐���盐鼀 ᨨ矄��ᨨ 鼀ᨨ ᨨ犨 矄 犨 꾀矄 �矄 矄 矄� � S҈ᆠᨨ鼀矄 ᨨ�矄 盐 � S �矄 ᆠᨨ鼀矄矄 쟼 矄ᨨ � 矄� � S҈ �矄 矄ᨨ 矄 쟼ᨨ 矄� �S犨 ᨨᆠ├Bᆠ 矄�矄 鼀ᨨ矄��ᨨ ᆠᨨ鼀矄矄� � S 泐 y ᔼ �矄 ᆠ盐矄 ᨨ �ᆠ�ᨨ 泐犨 和ᨨ �泐 �矄 ᆠᨨ 矄 ҈ ҈ ҈ ҈ \ k҈x㦨 ky Þ 맔 kx 㦨 맔ky y ҈ Þ x㦨y\ = x ҈黨 b㡰 ᨨ ᆠᨨ 矄� ᨨᆠ矄 DU㦨 W \
⧤� �ᆠ �泐ᨨ ᨨᆠ├ =
x
x
DU㦨 - ò Fdx ᨨDU㦨 - ò k xdx
Þ U x㡰U 㡰㦨 -
Q U 㡰㦨㡰
kx -U x Þ x㦨 k ᔼ矄 矄 ᨨᨨ犨x �矄 矄泐鼀鼀�ᆠ b㡰 x 鼀 �鼀盐鼀 矄�ᨨ ᨨ 矄� � 鼀ᆠ �ᆠ �泐 �矄 ᆠᨨ 矄 矄ᨨ ᆠ矄 � ���ᨨ ᨨ � �泐ᨨ 矄� �鼀矄矄矄泐矄鼀 Mgx㡰 \
kx
U x㡰㦨 -
æ҈ è
ö ø
㦨 �� � 矄� � 矄�ᆠᨨ � �泐 ç kx ÷
҈ Mg kx Þ x = k du 㡰 ᨨ鼀F㦨 dx dU x㡰㦨Fdx
Mgx =
҈
x
Ux㦨 -ò - kx + ax 㡰dx \
x
kx ax Ux㦨 - ò Fdx =
k a ␔ ᨨ � �泐 ᨨ犨�ᆠᨨ盐ᨨ 犨 �ᆠ ᨨ �矄 ᨨ��� 꾀矄ᨨ犨 犨盐x 㦨� �� 盐 ᆠ�ᨨ dU 犨�F㦨盐F㦨 dx dU = � 矄ᨨᨨ � 矄 ᨨ盐 Þ 꾀x㦨錐 dx ᨨ
矄 ᆠ ᆠ�矄�ᆠ矄 矄 泐 � 㡰 b㡰 ␔ 矄� � �矄 ᆠ盐 � ᨨ �ᆠ矄 � ᨨ 꾀 Ux㦨 �� ᨨx>
꾀矄ᨨ Þ
k 錐 a
L ᨨ��� � ᨨ 矄� �L 矄� �ᆠᨨ 矄 �矄K �� 㡰 ᨨ矄� �K맔L㦨ᆠᨨ 矄 K Ἐ 矄� � ᆠᨨ矄 ᨨ ᨨ � �ᆠ ᨨ 矄� � L \ K맔L㦨 맔K Ἐ Þ K Ἐ㦨 K p ᆠ㡰 ⧤쟼㦨 錐⧤쟼㦨⧤� �ᆠ �泐錐㦨鼀ᨨ鼀 盐鼀桌 m p҈ p = ⧤쟼҈㦨⧤쟼 \ m҈ m ᨨ � �ᆠ ᨨ 矄� � 㦨
\ �㡰
p҈
pl m҈ ml ҈ ҈ = = = Þ p m m ␔ᨨ├ ᨨ 㦨ᔼ ᆠ矄� 쟼졔�� � ⧤쟼 F맔d㦨mgh졔�� � ⧤쟼 ҈맔�㦨҈맔҈맔졔�� � ⧤쟼 � � ⧤쟼㦨�㦨҈
n㦨�
p
=
\ \ � 㡰 ᨨ矄矄� ⧤쟼ᨨ ᨨᆠ├㦨�� � 쟼ᨨ 矄� � 졔 犨ᨨ├ ᨨ �� 矄 �ᆠ�ᨨ
҈ ҈ mmgx + kx mv 㦨 kx + mmgx Þv㦨 m ´ ҈ ´ ҈ ´ ҈ ´ + ´ ´ v㦨 ҈ \v㦨 㦨 \㦨 ҈ ҈ ҈�㡰 ᨨ鼀犨ᨨ├ �泐 ᨨ鼀
W = F × s = DKE =
҈ mv
V = gh
\F ×s = F ´
҈ ҈� = ´ ´ ´ ҈ ´ ҈ ҈
\ F = ҈�
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EBD_8341 PS-36
䰘
Physics
҈㡰 ⧤� �ᆠ �泐㦨 ᆠ � � ᨨ � �泐ᨨ �ᆠ ⧤쟼㦨鼀�D � m㦨҈├� Dh㦨 ҈㦨҈㦨҈m \KE㦨҈맔҈맔҈㦨҈J 㡰 꾀ᨨ� Ӹ ᆠ � ᨨᆠᨨ� 泐 ᆠᨨ鼀ᨨ ᨨ 犨�� 鼀�
\ 鼀� 矄� ° =
\ =
҈ ҈ 鼀 = 鼀� + 鼀
鼀�矄�
\ 㦨 �
�㡰
\ ᨨ鼀⼄ �㡰
\
t
Þ
v
L
B L
u
A
盐� � 盐ᨨ v ᨨ鼀⼄ �㡰
r D v 㦨 u - g l㡰 ᆠ 㡰 꾀矄 �ᆠ �矄 ᆠ 盐ᨨ 泐 ᨨᆠ ᨨ ᆠᨨ 矄 鼀� �盐 �矄 �ᆠ盐 ᨨ ᨨᆠ�泐 ᨨ �ᆠ矄ᨨ��矄ᆠ矄ᨨ 盐 � ᨨ鼀ᆠ�ᆠ盐鼀ᨨ�ᨨ
ᨨᆠ�矄 ᆠᨨ 矄 � 鼀� �盐 �矄 矄 ᨨ犨 矄 �矄
ᨨᆠ �� � ᆠ├� �ᆠ �泐犨�鼀� ᆠᨨ 矄 v
mV gt =
F
v
F F
v
v
ᨨᆠ�泐 ᆠᆠ�ᨨ ᆠ �矄 ᆠᨨ � 盐ᨨ盐矄泐 盐 ᨨᆠ � � �ᆠ�ᨨ 꾀®⼄錐®⼄矄錐ᔴ®⼄矄錐® ᨨ꾀 dU d é U æ æ x ö ù -U ê ç҈ - ç ÷ ú = = x - a㡰 x x + a㡰 㦨 dx dx ê ç è a ø ú a è ë û 㦨 x㦨x㦨ax㦨a u㦨 x㦨ax㦨a U
dV dé g ù g =m ê ´ t ú 㦨m = gm 㦨ᆠᨨ 矄 ᨨ犨 F㦨ma㦨 m dt dt ë m û m b㡰 ᨨ鼀 ��盐�� � ⼄盐矄�ᨨ ᨨ � �泐ᨨ ᨨ� 꾀㦨鼀� 꾀 ᨨ � �泐ᨨ ᨨ� 㦨 ᨨ � �泐ᨨ ᨨ� ᔴ㦨鼀� ᔴ
ᨨ �泐ᨨ� 꾀쟼꾀㦨⧤꾀졔鼀� 꾀
ᨨ �泐ᨨ� 쟼㦨⧤
ᨨ �泐ᨨ� ᔴ쟼ᔴ㦨⧤ᔴ졔鼀� ᔴ 꾀ᆠᆠᨨ � � ᨨ 犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐 쟼꾀㦨쟼㦨쟼ᔴ �㡰 쟼꾀㦨쟼Þ쟼ᔴ>⧤ᔴ ��㡰 쟼꾀㦨쟼ᔴ ⧤꾀졔鼀� 꾀㦨⧤졔鼀� ᔴ ⧤ - ⧤꾀 ᨨ 꾀 - ᔴ = ᔴ ���㡰 鼀�
U U0
g 矄 g g ´ t �µ 錐V = = Þ 矄= 鼀 鼀 m 和ᨨ V �矄 ᨨᨨ�ᨨ ᨨ � �矄 ᆠ ᆠ
ᨨ ᨨᨨ�ᨨ ᨨ ἘἘ
\V =
�㡰
� = - � ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ 꾀ᨨ� ᨨ ᆠ�ᆠ盐 ᨨ� 矄矄鼀 �� 盐 矄矄 和ᨨ ᨨᆠ�泐鼀ᨨ ᨨ� 泐 鼀 和ᨨ⼄盐� ᆠ � ᨨᆠ = �矄鼀 �鼀盐鼀ᨨ� 矄 d æ҈ dk ö ҈ b 㡰G 㦨gt k = mV \ ç mV ÷ 㦨gt dt dt è ø m dV dV ´ V 㦨gt \ mV 㦨gt Þ dt dt V
҈ ҈ mv + mgL = mu 졔
r r 㦨 v - u = v + u
� ��㡰
m ò VdV = g ò tdt
ᨨ
Þ v㦨 u - gL �㡰 r ᔴ � � ᨨᆠ�泐 D v
鼀
ᨨ犨 泐 �泐 ᆠᨨ 矄�ᨨ 犨 ᨨᨨ鼀 ᨨ� ᨨ� Ӹ 泐
黨
҈
Þ 꾀> ᔴ錐⧤ᔴ⧤꾀 ᨨ鼀⼄ �㡰 ��㡰 �㡰犨� 꾀> ᔴ \ ⧤>⧤ᔴ 㡰 ᨨ鼀 � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐 KE㡰B졔 PE㡰B㦨 KE㡰A졔 PE㡰A
a
졔a
x
x U
U
졔a
ᨨ Fx =
x
a
졔a
x
dU æ xö = -U ç ÷ dx èaø
dU e- x ⏌ a = U x x - a 㡰 x + a 㡰 dx a U dU ᨨ Fx = = - x - a㡰 x + a㡰桌 dx a
ᨨᔴ Fx =
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PS-37
Work, Energy & Power b㡰
N mgᆠᨨ矄 q犨�ᨨ� ⼄盐� ᆠ � ᨨᆠ
mv R mv N㦨mgᆠᨨ矄q졔 R
Nmgᆠᨨ矄q㦨
҈ ҈´ ҈㡰 + 㦨䰘� � b㡰 ␔ᨨ├ ᨨ �� 矄 矄� �ᆠ�ᨨ 㦨 ᔼ �� 鼀ᆠ �ᆠ 鼀 �泐 � 鼀ᆠ �ᆠ �泐 턼 ҈ 鼀�矄� ҈�㦨mgh 鼀 鼀� ҈ 鼀�ᆠᨨ矄 Þ ҈�㦨mg R矄� 鼀 Q h㦨R矄� 㡰 ҈ ҈ Þ ҈�㦨҈맔҈맔맔 맔҈맔v \ v㦨҈鼀⏌矄 ᆠ㡰 ᔼ �矄 ᆠ矄 鼀ᆠ �ᆠ �泐�矄 ᆠᨨ鼀泐 ᆠᨨ � ᨨ ᆠ盐矄 ᨨ �ᆠ�ᨨ �.e. ᆠ矄 �
҈ 鼀ᆠ �ᆠ �泐㦨 mv ␔ � �矄 矄ᆠᨨ ᆠ矄 ᨨ 鼀ᆠ �ᆠ �泐 �矄 ᆠᨨ � ᨨ 盐 ᨨ �ᆠ�ᨨ 盐 ᨨ ᨨ 鼀ᆠ �ᆠ �泐 �矄 � � ᨨ鼀 ᨨ ᨨ � 㦨҈맔҈맔
�泐 ᨨ ᨨᆠ├ �.e. ᆠ矄 � 鼀ᆠ �ᆠ �泐
҈ mv mgh
ᨨ 矄鼀 ҈ �矄 ᆠᨨᆠ 和鼀 �矄 犨ᨨ � ᆠᨨ �ᆠ� ᨨ �ᆠ�ᨨ 犨
ᨨᆠ├ 矄盐 ᆠ ᨨ矄 ᨨ ᨨ �ᨨ � ᆠ� �ᨨ 䰘 ᨨ� 鼀矄矄 � 犨犨� 矄 v 㦨
ᨨᆠ�泐ᨨ 鼀矄矄m �矄� 矄 㦨vᆠᨨ矄q㦨
v � 盐 M犨� �矄 ᆠᨨ ҈鼀犨 �m犨��矄盐泐
҈鼀 � ҈㡰鼀 ᨨ鼀 �泐 鼀 ᆠᨨ 矄 �ᨨ 졠 㝐 q ᆠ矄� 쟼 ᨨ M㦨� ᆠ矄 ҈鼀 h vᆠᨨ矄q � 쟼 ᨨ m 졔 �鼀 v � ᆠ矄� ⧤쟼 q ᨨ ᨨ h' �҈㡰鼀 v 㦨v ᆠᨨ矄q ᨨᆠ├矄 鼀 ҈ ҈ Mgh㦨mgh'졔 mv + m ᆠᨨ矄 q㡰 ҈ ᨨ 㡰 黨㡰 ҈㡰㦨 �㡰 黨㡰 � - ҈㡰 + ´ ´ v æ v ö ҈ 졔 ´ � ´ ç ÷ è �ø 和ᨨ� �犨�v㦨黨鼀⏌矄
⧤쟼ᨨ ᨨᆠ├㦨犨ᨨ├�� 矄 �ᆠ�ᨨ 졔쟼ᨨ 矄� � ҈ ҈ mv = m k mg BD + x 㡰 + kx ҈ ҈ mv 㦨Ҙkmg ҈졔x㡰졔 kx ҈ ҈ 맔�맔㦨맔�맔黨 ҈졔x㡰졔 맔x ҈졔x졔x㦨� \ x졔x҈҈㦨 ҈ = ҈ 졠 矄ᨨ� �犨�x㦨 ҈
矄� � �矄 ᆠᨨ鼀矄矄 泐҈ 鼀 矄ᨨ� � ᨨᆠY㦨kx㦨맔҈㦨N �ᆠ�ᨨ ᨨᆠY㦨Ҙsmg㦨맔�맔黨㦨҈䰘N 和� ᆠ �ᆠ�ᨨ ᨨᆠ>矄ᨨ� � ᨨᆠ ᨨ 泐犨�矄ᨨ �.e. ᆠᨨ鼀矄矄� � 矄� � 泐x \ ᨨ �矄 ᆠ 㦨AB졔BD졔x㦨졔҈졔҈㦨鼀
A
B 鼀
D
C
҈鼀
x
Y
黨
ᨨᆠ ᆠ� � v N ᨨ A B �矄 ᨨ和� ᆠ T ᨨᆠ├矄 꾀 B 鼀ᨨ� � 犨� ᆠᨨ 矄 C T f ᨨᆠ�泐 mBg 矄�ᨨ ᨨ T v 矄� � x A
犨� ᨨ �ᆠ�ᨨ
mg ᨨᆠ 犨 ᨨᆠ├ A C Q f 㦨 ҘN �矄 ᨨ ᨨ鼀 ᆠ�ᨨ ᨨ A ᆠ盐矄 A �矄 ᨨ 盐矄 � � C 矄ᨨ �ᆠ�ᨨ ᨨᆠ 犨
ᨨᆠ├꾀 ᔴ f㦨m㦨 꾀泐� �F 㦨maᨨ A mAgT㦨mA맔 \ T㦨mAg �㡰 꾀泐� �F 㦨maᨨ B Tf㦨mB맔 \ T㦨f㦨ҘN 㦨ҘmBg ��㡰 ᨨ鼀 �㡰 ��㡰 A
mA = = ҈ ├� m ᨨᆠᆠ� �ᨨ 矄� ��矄 矄�ᨨ ⼄盐ᨨ 矄ᨨ� � ᨨᆠ ma g T ҈黨 \ T㦨kx Þ x㦨 \ x㦨 㦨 k k k 쟼 �泐矄ᨨ � 矄� � ҈ ҈ ҈黨 ҈黨 ҈黨 ´ ҈黨 = 黨 U = kx = ´ k ´ ´ 㦨 k k ´ ҈黨 ҘmBg㦨mAg Þ mB㦨
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EBD_8341 PS-38
Physics
⧤쟼C㦨ᨨ矄矄� 쟼犨ᨨ├ ᨨ 泐 �ᆠ�ᨨ
ᔼ ᨨ ᆠ矄矄 犨ᨨ├ ᨨ 泐 �ᆠ�ᨨ 㦨 mmgx
ᆠ =
A
҈ mvc = mg y - mmgx = gy - mgx
ᨨ
q(
\ vc㦨 g y - mx 㡰 B ᨨ犨⧤쟼F㦨ᨨ矄矄� 쟼犨ᨨ├ ᨨ 泐 �ᆠ�ᨨ
҈ D mv F = mgy - mmgx a( G M ҈ vF = gy - mgx Þ \
C
�.e. vC㦨vF㦨 gy - mgx
E
F
N
҈
ᨨ 犨 �矄 ᨨ犨 盐 ���ᨨ 盐 ᨨ �矄 ᆠ� � ᨨ 犨 �ᆠ �矄 矄ᨨ 矄� ᨨ ᆠ � � 鼀ᨨ鼀 盐鼀
ᨨ 矄� � 矄� �ᆠᨨ 矄 㡰 k맔 � 㡰l 㦨 ᔴᨨ 矄 l ᔼ � �矄 鼀 ᨨ � �.e. 矄� � ᆠᨨ 矄 ᆠᨨ鼀矄 �鼀矄 �.e ⧤ 矄� �⧤� �ᆠ �泐 ⧤쟼㡰 鼀ᨨ鼀 盐鼀�ᨨ
⧤쟼㦨
҈ p ᨨ⼄盐盐ᨨ p,⧤쟼µ 鼀矄矄 m
� 䰘
\ P = mv
dv v dt
dv Þ P dt = mv dv dt t
ᔼ �� � ᨨ 矄� 矄
黨
ò
�矄 ᆠ s = v dt =
ᨨ
\
ᆠ㡰
w E ᨨ犨 P = = = ᆠᨨ 矄 t t
t
P ҈⏌ t dt = m
ò
Þ
V㦨krt
\
v = ᆠᨨ 矄 k 㡰 \ t
҈ mv = ᆠᨨ 矄 t
v㦨kt҈⏌
ᨨ ds = kt҈⏌ dt 泐� �� � 犨 �
P t ⏌ × m ⏌
P ⏌ × t Þ 矄 µ ⏌ 黨m 和ᨨ� 㡰�矄 ᆠᨨᆠ b㡰 ᨨ ᨨᆠ ⼄盐� ᨨ� 鼀 �鼀盐鼀 ᨨ ᆠᆠ�泐 �� 矄 �ᆠ�ᨨ ᨨᆠ㦨N Fᨨ =Mg졔 �ᆠ�ᨨ
㦨맔҈졔 㦨졔㦨N 矄� �ᨨ犨P㦨F맔v ×䰘㦨맔v Þv㦨҈鼀⏌矄»҈黨m/s ᆠ矄 ᨨ ᨨ 盐ᨨ �矄ᆠᨨ矄ᨨ҈黨鼀矄҈
Þs=
d d æ dm ö ÷ = v r´ ᨨ盐鼀㡰 = vr ᨨ盐鼀㡰 dt dt è dt ø 㦨vr맔 Av㡰㦨Arv ᨨ犨㦨 ᨨᆠ맔ᨨᆠ�泐㦨Arv맔v㦨Arv ᆠᨨ犨 Pµv b㡰 ᆠ � ᆠᆠ�ᨨ ac㦨 krt ᆠ㡰 F㦨 v ç
ᨨ鼀 犨ᨨ├㜠 �泐 ᨨ鼀 ᨨ 犨ᨨ├ ᨨ 㦨 ᆠ � �
├� �ᆠ �泐
æ P ö ҈⏌ ҈ ÷÷ t = mv Þ v = çç è m ø t
\和 ᨨ ᆠ犨 �鼀ᨨ� � ᨨ犨 �㦨҈�鼀⏌矄 㡰 b㡰
v
ò P dt = m ò v dv
q ᨨ矄 ҈ 犨ᆠ q = ҈
犨 æ 犨 犨 ö = ç - ÷ è ҈ ø v Þ v = ҈� 鼀⏌矄
v = k rt r dv 㦨kr at㦨 dt ᨨ犨 P㦨Ftv㦨matv \ ᨨ犨㦨mkrt
b㡰 ␔ ├ ᨨ犨 ᨨ犨P = Fv 盐 F = mav = m
㦨 㦨 ᆠ 㦨 㦨 ᨨ犨㦨鼀㦨鼀 㦨㝐 ᆠ㡰 ␔ � 鼀ᨨ� � ᨨ犨 � ᨨ犨
= =
Topic-3 : Power ҈
=
wö æ P = ç w 矄� q + ÷҈ è ø 犨 æ犨 犨ö = ç + ÷҈ = è ҈ ø
b(
vF㦨 g y - mx 㡰
b㡰 ᔴ � ᆠᆠ�ᨨ ᆠ 㦨
Þ s㦨 ҈
kt ⏌ + C Þ sµt⏌
�.e.�矄 ᆠ鼀ᨨ Sµt⏌ b㡰 ᔼ 鼀ᆠ � �矄盐�ᆠ 犨� ᨨ� �ᆠ�ᨨ �矄 盐ᆠ 졠盐盐 犨ᨨ├ 㝐ᆠ �ᆠ �ᆠ� ᆠ泐㦨 ᔼ 盐 �泐 盐 ᨨ 矄矄 �ᆠ�ᨨ ᨨ盐盐 犨ᨨ├ 犨� � ᆠ矄 盐矄 鼀ᆠ �ᆠ �ᆠ� ᆠ泐� ᆠ矄矄 㝐ᆠ �ᆠ � 㝐꾀 㦨
҈҈
ds = kt҈⏌ dt
ᨨ ᨨ
�㡰 矄� � 犨ᨨ├ �泐 ᨨ鼀 D ⧤쟼 㦨 ␔ 㦨 맔 ҈ 鼀 = ´ \ =
= 鼀
´ � ´ � = �鼀矄 -҈
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PS-39
Work, Energy & Power ҈
� ᨨᆠ�泐vἘ盐矄� �鼀ᨨ鼀 盐鼀ᆠᨨ 矄�ᨨ
҈㡰� 㝐矄矄ᨨ ᨨ 泐m㦨├� ᨨ犨 � 泐 �� P㦨҈⏌矄
�鼀t㦨黨矄ᆠᨨ 矄 ᨨ犨P㦨Fv
Q F = ma桌 dv ö æ çèQ a = ÷ø dt
Þ P = mav Þm
v ҈ ҈ ᨨ犨� �� �泐 E� = mv v ҈ ҈ mv � �泐 E f = ´ ҈m ´ æç ö÷ = ҈ è ҈ ø mv = ҈ mv Ἐ Þ v Ἐ =
dv v=P dt
҈ ҈ v mv - m ҈ ҈ é ҈ù ҈ é҈� ù Þ mv ê҈ - ú Þ mv ê ú ë ҈ û ë҈ û
쟼 �泐ᨨ矄矄 E� - E f =
P Þ v dv = dt m ᔼ �� � ᨨ 矄� 矄 犨 � v
Þ ò v dv =
Þ
dx = dt x
Þ ò dx =
t
P dt m ò
Þ
v Pt æ Pt ö = Þv= ç è m ÷ø m
҈⏌
ᨨ �泐ᨨ矄矄 �矄 P奠 ᨨ ᨨ��� �泐 쟼 �泐ᨨ矄矄 ´ ҈ 졠��� �泐 ҈ é҈� ù mv ê ú ë҈ û ´ ҈ = 黨䰘�奠 = ҈ mv \ 奠P =
dx ö æ çèQ v = ÷ dt ø
P ҈⏌ t m t
P ҈⏌ t dt m ò
P t ⏌ P ⏌ = ´ t \ �矄 ᆠ x = m ⏌ m ´ ҈ ⏌ Þx= ´ ´ 黨 = ´ 䰘 = ҈
Topic-4 : Collisions ҈
㡰 ᨨ ᆠᨨ�矄�ᨨ ᨨᆠ�泐ᨨ �ᆠA u҈ = �ˆ + ˆ 㡰 鼀⏌矄 ᨨᆠ�泐ᨨ �ᆠBu㦨 꾀 ᆠᨨ�矄�ᨨ ᨨᆠ�泐ᨨ �ᆠA v҈ = �ˆ + ˆ㡰 ᨨᆠ�泐ᨨ �ᆠBv㦨 矄� � � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀
r r r r m҈u҈ + m u = m҈v҈ + m v r Þ m �ˆ + ˆ 㡰 + m ´ = m �ˆ + ˆ 㡰 + m ´ v r Þ �ˆ + ˆ = �ˆ + ˆ + v r r Þ v = - ҈㡰�ˆ - - ҈㡰 ˆ Þ v҈ = �ˆ + ˆ r r ᨨ �犨 v҈ v r r v ×v - ҈㡰 ҈ - 㡰 ҈ - ᆠᨨ矄 q = ҈r r = = v҈v ´ - ҈㡰 Þ q = ҈�° r r 꾀 �犨 v҈ v �矄҈�
ᆠ盐ᨨ P�矄ᆠᨨ矄ᨨ黨 b㡰 � 㝐矄矄ᨨ ᨨᆠ├m҈㦨҈黨├� 㝐矄矄ᨨ 盐m㦨҈├� ᨨᆠ�泐ᨨ 盐v㦨鼀⏌矄 v ᨨᆠ�泐 ᨨ ᆠᨨ鼀� 矄泐矄鼀 ᔼ �矄
� 矄�ᆠ ᆠᨨ�矄�ᨨ 矄� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀
m҈ ´ + m ´ v = m҈ + m 㡰v Þ ҈ ´ = ҈ + ҈黨㡰 ´ v Þ v = ҈ 鼀⏌矄 矄� � 犨ᨨ├ �泐 ᨨ鼀 ␔ᨨ├ ᨨ 㦨ᔴ � � ⧤� �ᆠ �泐 K ⧤� �ᆠ �泐ᨨ ᆠᨨ鼀� 矄泐矄鼀 鼀҈졔鼀㡰� ҈ = ⧤ - 鼀҈졔鼀㡰 ҈ Þ ´ g ´҈ = K - ´ ´ ҈ Þ K = ҈ ᆠ㡰 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ur mu�ˆ + = mvˆ + mv Ἐ ur u v v Ἐ = �ˆ - ˆ
b㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ᆠᨨ�矄�ᨨ 矄 ᆠ泐 � 矄�ᆠ矄ᨨ � ᆠᨨ�矄�ᨨ ᨨᆠ├矄鼀ᨨ� � ᨨ� m
ᨨ鼀 ├� �ᆠ �泐 ᆠᨨ 矄�ᨨ æ æ u ö æ v ö ö ҈ ҈ ҈ mu = mv + m㡰 ç ç ÷ + ç ÷ ÷ çè ø è ø ÷ è ø ᨨ mu = mv +
\v = m
m v'
ᨨ ᆠᨨ�矄�ᨨ
v = - ҈㡰$� - - ҈㡰 $
m
m
x
�
v
m
m u
v҈ = $� + $
v
y
m
꾀 ᆠᨨ�矄�ᨨ
mu mv +
u
m
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EBD_8341 PS-40
�
Physics
b㡰
盐⏌
黨
泐
ᆠ㡰
├� ҈鼀
鼀
盐
鼀
盐⏌
҈鼀⏌矄
졠
├�
x㜠 �ᆠ�ᨨ
mu u mu + = mvἘx Þ VxἘ = mu u Ἐ = mv y Þ v Ἐy = 泐㜠 �ᆠ�ᨨ + ù é ҈ ҈ æ uö æ uö K E� = m u + m ê ç ÷ + ç ÷ ú è ø è ø ú êë û
�鼀├ ᨨ �ᆠ矄 ᨨ ᆠᨨ�
҈ = =҈矄ᆠ ҈ 和 ᨨ 犨ᨨᨨ 盐矄 ᨨ ᆠᨨ�矄�ᨨ 㦨 � 㦨 ҈ 鼀⏌矄 矄 ᨨ 盐 盐矄 ᨨ ᆠᨨ�矄�ᨨ 㦨 � 㦨҈҈㦨黨鼀⏌矄 ҈ 和㦨҈맔҈ 맔҈맔҈㦨黨�鼀 ᨨ犨 盐矄� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 盐矄 ᨨ ᆠᨨ�矄�ᨨ
㡰 ҈㡰졔 㡰 黨㡰㦨 �㡰v Þ҈�㦨�v \v = 鼀⏌矄 㝐 �� ᆠ 泐ᨨ 泐 㦨
҈ mu mu mu + = ҈ ҈ K 쟼 f = ( m) vxἘ + ( m) v Ἐy éæ u ö æ u ö ù � ҈ = m êçè ÷ø + çè ÷ø ú = mu êë úû \ᨨ矄矄� KE㦨KEfKE� =
( )
( )
㦨 =
ᨨ ├� ҈鼀⏌矄
æ � ö mu = mu ç - ÷ = è ø
㡰 꾀泐ᆠᨨ ᆠ�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀� Ӹ �ᆠ�ᨨ ᨨ 矄泐矄鼀
㝐 ҈ᆠᨨ矄 㡰졔㝐 �ᆠᨨ矄� 㡰㦨㝐 ҈ᆠᨨ矄 㡰 졔 㝐 ᆠᨨ矄� 㡰
� +� =
҈ +
├�
҈
҈㡰
꾀矄ᨨ 㝐 �矄� � 㡰㝐 ҈矄� 㡰㦨㝐҈矄� 㝐矄� �
� - � = ҈ -
㡰 和ᨨ� � ⼄盐�ᨨ ҈㡰 㡰 + ҈ ҈ = � + ҈ - � Þ ҈ = �鼀⏌矄
(
)
㦨鼀⏌矄 䰘
b㡰 m҈v҈졔mv㦨m҈v졔mv҈ ᨨ
m҈v҈졔 �m҈㡰v㦨m҈�v҈졔 �m҈㡰v
졠 矄ᨨ� �v҈㦨vv 㡰 矄� � ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀 mv㦨鼀鼀҈
҈ ҈ 鼀҈ = ´ 鼀 Þ ҈ =
ᆠᨨ�矄�ᨨ �矄 矄�ᆠ 和ᨨ
a
҈҈
҈
㝐
\ 㝐㦨鼀
Þ 鼀 = 鼀㝐
黨鼀⏌矄
�├�
\�� ᨨᨨ犨㦨鼀 ᆠ㡰 ⧤� �ᆠ �泐ᨨ ᨨᆠ├꾀 ҈ ├҈ = 鼀 \ ᨨ鼀� ᆠ� ᨨ � 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
鼀 鼀 = ( 鼀 + 㝐 ) Þ = 鼀 + 㝐 � 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ ᨨ � �� ├� �ᆠ �泐 �矄 ᨨ矄 � 犨 ᨨ ᨨᆠ矄矄 ҈ 鼀 ├� = \ = Þ ├ ҈ æ 鼀 ö ( 鼀 + 㝐 ) çè 鼀 + 㝐 ÷ø 鼀 + 㝐 㝐 Þ =\ = 鼀 鼀 꾀 ᔴᨨ�矄�ᨨ
b㡰 ᨨᔴᨨ�矄�ᨨ
鼀
꾀 ᆠᨨ�矄�ᨨ
a
꾀
㝐
҈ ҈ 㝐 = ´ 鼀 \ 㝐㦨鼀矄矄ᨨ 盐ᆠ盐矄桌 鼀 Þ = ´ 鼀 = 鼀㝐 ´ 鼀 ´ 㝐
Þ҈ 鼀 = 鼀㝐
´ = � 鼀 ´ ҈
鼀
Þ
҈ 鼀
鼀
和�ᨨ 泐
҈ ҈ æ҈ ö 鼀҈ + 鼀 = ç 鼀 ÷ è ø Þ ҈ + = �㡰 ᨨ鼀 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
鼀㦨鼀 ҈졔㡰 ��㡰 和⼄盐� � ᨨ 矄� 矄 ҈졔㡰㦨 Þ҈졔졔҈㦨
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PS-41
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҈ = - ҈ - 㡰 = ҈ + - ҈ =
⧤쟼ᨨ矄�矄盐矄 ᨨ盐鼀 ᨨ鼀҈矄ᨨ�ᨨ ᨨ� D⧤쟼㦨҈ 㝐� �鼀盐鼀⧤쟼ᨨ 盐ᨨ ᨨ� 矄�ᆠᆠᨨ�矄�ᨨ
҈ 鼀 㦨 ҈ 㦨
+
和ᨨ� � 犨 � � ᨨᆠ�泐犨 犨ᨨ�ᆠ矄
҈ - = ҈
Ӹ 㦨鼀
㡰 ᔴ �� 鼀ᨨ鼀 盐鼀
^ ˆ ˆ ˆ ˆ + + �� D = ^ ˆ = ᔼ 鼀ᨨᆠ盐 D = ^ � � � ^� ˆ ^ Þ ᔼ犨 = 矄矄盐
Q = ᨨᨨ �ᆠ矄㡰 㦨 = 꾀 꾀 ´ ´ ҈ -䰘 ´ ҈ ´҈ = 㦨�맔҈⏌鼀 ´ ҈- ҈ 㡰 ᨨ ᆠᨨ�矄�ᨨ ᨨ 盐ᨨ 犨� 盐�盐鼀 ҈ 鼀 鼀 鼀 鼀 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀 鼀졔㦨鼀҈졔鼀 �㡰 ҈㦨 ��㡰 Q ᔴᨨ�矄�ᨨ �矄矄�ᆠ㦨҈ ᨨ鼀⼄ �㡰 ⼄ ��㡰 ҈ =
҈�
㡰
鼀
ᔼ �� 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀
^�
�㦨 鼀 㡰 ´ 鼀 㡰 桌 㦨 鼀 ´ � 鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀㦨鼀 泐 犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀
鼀v = 鼀 Þ
҈
v = ᆠᨨ鼀�
ᨨ矄矄 � �泐 ҈ ҈ ҈ D쟼 = 鼀҈҈ + 鼀 - 鼀҈ + 鼀 㡰ᆠᨨ鼀� � D쟼 = 鼀 - 鼀 = 鼀 㦨����奠 ᆠ �ᨨ矄矄� �泐 盐� � ᆠᨨ�矄�ᨨ ; �奠 㡰 ᨨ鼀 犨ᨨ ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀犨├ ᨨ犨 鼀҈盐҈졔鼀盐졔㦨鼀҈҈졔鼀졔 � 鼀҈㦨鼀鼀㦨鼀 鼀㦨鼀 盐҈㦨盐盐㦨盐 盐㦨盐
Ӹ 鼀盐
鼀҈
鼀҈ 졔 鼀 㡰
矄� 鼀盐
�
�ᆠᨨ矄 㡰矄� 鼀盐
ˆ 졔鼀맔盐 -ˆ� ᆠᨨ矄 ° - ˆ矄� °㡰 鼀맔盐 �㡰 ® 졔鼀맔盐 -ˆ� ᆠᨨ矄 ° + ˆ矄� °㡰 㦨 鼀졔鼀졔鼀㡰
҈ ҈ æ ҈҈ ö 鼀 - 鼀 ç ÷ è ҈ ø = » ᆠ = ҈ ҈黨 鼀
�ˆ
® æ ˆö æ ö �ˆ ÷ - 鼀盐 + 鼀盐 ç ˆ ÷ = 鼀 ÷ ç ÷ è ø è ø
Þ 鼀盐�ˆ - 鼀盐 - 鼀盐 ç ç
®
ˆ 鼀盐�ˆ - 鼀盐�ˆ 鼀盐 = 鼀 ® ҈ ˆ 鼀盐 = 鼀 Þ - 鼀盐�ˆ ® 盐 ˆ - � - ˆ Þ = ҈ ҈䰘 b㡰 꾀矄
�矄 �矄 ᨨ 矄ᨨ� ��ᨨᆠ�泐u㦨 ҈ ҈ ҈ K.E. 㦨 mv 㦨 m [ u + at ] 㦨 m [ + gt ] ҈ \ K E = mgt \ K E µ t �㡰 Þ
҈ 㦨 ҈ 졔 ҈㡰
® 㡰 ᨨ 㡰 㡰 ® 꾀 ҈ ҈ æö ҈ 鼀 鼀㡰 ᨨ矄矄� ⧤쟼㦨 ç ÷ = 鼀 è ø 㦨
ᨨᆠ�泐犨 泐矄�ᆠ├ 㦨
ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ
҈҈ ҈
㡰 ᨨ� 矄�ᆠᆠᨨ�矄�ᨨ Ἐ 㦨
�
v
ᨨ犨 ᨨ ᆠᨨ�矄�ᨨ ᨨ 盐ᨨ 犨� ᆠᨨ 盐ᆠ盐矄 ҈ 鼀 鼀 ҈鼀 ҈鼀 꾀泐� � ᔴᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀 mv졔㦨mv1졔҈mv2 ���㡰 㦨҈ �㡰 ᨨ鼀⼄ ���㡰 ⼄ �㡰
҈
�
®
҈ ҈ 鼀 - 鼀҈ = = 黨 = ҈ 黨 鼀
҈ = -
鼀 v
(
)
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EBD_8341 PS-42
Physics
�.e. �ᨨ 犨 k t �矄 ᨨ�ᆠ �矄 ├� �ᆠ �泐犨� � ᆠ矄 矄 ⼄ �㡰 꾀矄 矄 ᨨ盐ᆠ 矄 �ᨨ盐 � 矄矄 ᨨ鼀� � ᨨ矄矄 �矄 ⧤쟼犨 ᨨ鼀�ᨨ �矄鼀 �鼀盐鼀⧤쟼㦨꾀矄 鼀ᨨ矄盐�ᨨ矄矄⧤쟼 �� ���ᨨ ᨨ � �泐� 矄鼀�鼀� 矄ᆠ ᆠ�矄�ᆠ矄 矄 �盐矄 泐kt � 矄 ᨨ犨 � 㡰
ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 犨 ᆠᨨ�矄�ᨨ �ᆠ矄 鼀ᨨ 犨� ᆠᨨ 矄 矄 꾀 �矄 ᆠᨨ�矄�ᨨ ᨨ �ᆠ � � 矄 v 犨� ᨨ 맔 ҈ 㦨 犨 � ᨨ �ᆠ � � 矄 v 犨� ᨨ҈ ᆠ �矄ᆠᨨ�矄�ᨨ ├矄ᆠ꾀 �矄 ᆠᨨ�矄�ᨨ 泐犨� ᆠ � �ᨨᆠ��矄矄 ᆠᨨ�矄�ᨨ
�矄矄�ᆠ �ᆠ矄 ⼄盐鼀矄矄矄꾀�� 矄ᆠᨨ ᆠᨨ�矄�ᨨ ├矄ᆠᔴ
盐 矄� a g 盐矄� �盐㦨� é u 矄� a ù v҈ - u = - g 㡰 ê ú ëê g ûú
҈ 㡰
Þ Þ
泐
a 鼀
҈
盐
鼀盐ᨨᆠᨨ矄a
鼀
꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 �
Ӹ㜠 �ᆠ�ᨨ
鼀矄� q㦨鼀҈㦨鼀盐ᆠᨨ矄a �㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 �
㜠 �ᆠ�ᨨ
鼀ᆠᨨ矄q㦨鼀盐 ᆠᨨ矄a ��㡰 �� � � �㡰 ��㡰犨� p q㦨҈\q㦨� 㦨 㡰 Fᨨr vert�cal mᨨt�ᨨn ᨨf bullet ᨨr ball 盐㦨 㦨�鼀㦨䁸�㦨҈鼀⏌矄 鼀㦨҈├�
�鼀
ᔴ ᆠᨨ�矄�ᨨ
r
� 㦨�鼀⏌矄 ҈ ᆠ� ��ᨨᆠ�泐ᨨ 盐㦨�鼀⏌矄
r
r
r
Dp㦨F Dt㦨 ( m҈v҈ Ἐ+ mv Ἐ) - ( m҈v҈ + m v ) 㦨 m҈졔m㡰g맔t 㡰 ᆠ 盐矄 ᆠᨨ�矄�ᨨ 犨� Aᨨᆠ�泐ᨨ B m - m A 㡰u B m A u A + m ´ 黨 vB = B + = = 鼀⏌矄 mB + mA m A + mB m + m \
m
m
m
A
B
C
ᆠᨨ�矄�ᨨ 犨 B C �矄 ᆠᨨ鼀泐� 矄�ᆠ \ mB vB㦨 mB졔mc㡰v
´ m = 鼀⏌矄 m + m �㡰 盐 ᨨ 矄�ᆠ ᨨ ᆠᨨ�矄�ᨨ ᨨ ⼄盐 鼀矄矄 m ᨨ ᨨᨨᆠ�泐 �� 矄ᨨ� ᨨ ᨨ� ᨨ矄� �l҈�矄 ᆠ⼄盐� 泐 ᨨ � ᨨ 矄� � l l \ �l҈ = ��l Þ ҈ =� l ҈㡰 m v=
v
盐
҈ ҈ h = ut + gt Þ � = ´ ҈ ´ t Þ 㦨҈矄ᆠ ᨨhᨨ� ᨨ 鼀ᨨ�ᨨ ᨨ b 㦨Þ 㦨맔҈Þ㦨鼀⏌矄 ᨨhᨨ� ᨨ 鼀ᨨ�ᨨ ᨨ b盐 盐㦨盐맔Þ ҈㦨盐맔҈Þ盐㦨҈鼀⏌矄 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 盐� � ᆠᨨ�矄�ᨨ 鼀㦨鼀盐졔㝐 ҈㦨҈맔҈졔맔 \㦨
҈
v ᨨ犨矄矄 ᨨ犨 � ��盐 犨ᨨᆠᨨ�矄�ᨨ 矄 泐犨���
ᆠ ᨨ� 꾀 㡰 F 㦨 m҈졔m㡰g Dt㦨t Dp F 㦨 Dt
Þ
㝐㦨├�
鼀 ҈鼀
v
v
鼀
盐 矄� a = �
盐
҈
҈
v
҈矄ᆠᨨ�矄�ᨨ
q 鼀ᆠᨨ矄q
҈黨
v
v҈ = u ҈ - 矄� a㡰 = u ᆠᨨ矄 a ҈㦨盐ᆠᨨ矄a 鼀矄� q
꾀
v
�� h㦨
q q
鼀 v⏌
v
m
�
㝐ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ ᨨ � x �ᆠ�ᨨ v ҈ mv ᆠᨨ矄 q = m Þ ᆠᨨ矄 q = ᨨ q = ° ᆠ �犨 � ��ᨨᆠ��矄ᨨ 犨ᨨᨨ �矄 = q + q = ° + ° = ҈° ҈㡰 ᨨ鼀 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ � �ᆠ盐 �ᆠ�ᨨ
ᨨ � ��鼀ᨨ�ᨨ mu҈ 矄� q҈ = ҈mv҈ 矄� q �㡰 ᔼ�矄�� �泐ᨨ m 盐ᆠ 泐 ᔼ u҈ᨨᆠ�泐 ᨨ m ᆠᨨ�矄�ᨨ
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- = = ␔ �矄 �矄 ᆠ 泐 �矄ᨨ � �鼀
鼀� 盐矄矄�� � �ᆠ矄 ᆠ矄� �犨� �鼀
u æ ҈ ö ҈ ҈ Þ u҈ = çè mu ÷ø = mu҈ ᔼ v ҈ ᨨᆠ�泐 ᨨ 鼀矄矄 ҈ 鼀 ᆠᨨ�矄�ᨨ
҈ ҈ u u ´ ҈ m ´ v҈ = m Þ v҈ = ᨨ鼀⼄盐�ᨨ �㡰犨
\
矄� q҈ = ҈ 矄� q
ᨨᆠᨨ�矄�ᨨ
v
v
v
ᨨ
=
= 犨 = ⏌ ҈ \ ⧤쟼 ⏌ = 鼀 㡰 ⧤쟼 ⏌ ҈ \ = \ ⧤쟼 = 鼀 ⧤쟼 黨 b 㡰 犨 =
졔
盐
盐
ᨨᆠᨨ�矄�ᨨ
҈
盐矄 ᆠᨨ�矄�ᨨ
v -v v + v ҈= ҈ ҈= v +u uv \v҈㦨u졔v \v㦨uv \Dv҈㦨u졔v Dv㦨uv dp҈ = rA u + v 㡰 u + v 㡰 ᨨ犨 F҈ = dt dp F = = rA 盐 - 㡰 盐 - 㡰 dt \ F҈㦨rA u졔v㡰 F㦨rA uv㡰 DF�矄 ᨨᆠ 盐ᨨ �鼀ᨨᆠ盐矄 ᨨ
꾀 ᆠᨨ�矄�ᨨ
v졔v
=-ò
\ l
= - l
҈㡰 ᨨ矄�ᆠᆠᨨ�矄�ᨨ KE�㦨KEf
҈ ҈ ҈ ҈ m ´ � + ´ m ´ 黨 = m ´ + mvB = + VB Þ VB = ҈ ҈ ҈ KEB = mvB = ´ ҈ ´ = ҈J = J ҈ \x㦨҈ 䰘 㡰㝐 �鼀盐鼀 �� æ҈ö u ç ÷ u 矄� q èø 㦨 Þ҈㦨 g g \ u㦨� ᨨ �� � �ᨨ盐 �ᨨ矄矄 ᨨ �矄├� �ᆠ �泐 ҈ \ ⧤쟼� ��㦨 mu 㦨鼀� ҈ ⧤쟼 � 㦨 鼀�㡰㦨҈鼀� ҈ mv 㦨҈鼀� \㦨� \ 꾀 ᨨ盐 ᆠ 鼀 �鼀盐鼀 �� ᆠ 矄 æ҈ö g ´ ç ÷ v 矄� q èø \ ¢㦨 㦨 㦨鼀 g g 㡰
ò
\
�矄ᨨ
␔ 矄鼀 �ᆠ 鼀ᨨ� � 犨� ᨨᆠ�泐 v 盐 �ᨨ矄 矄�ᆠ ᆠᨨ�矄�ᨨ 犨� 泐 鼀ᨨ �矄ᨨ 鼀ᨨ� �犨� ᨨᆠ�泐v�ᆠ⼄盐�矄 犨ᨨᆠ�泐v 졔v和ᨨ � ᆠ矄� ᨨᆠ�泐 泐ᆠᨨ�矄�ᨨ �矄v
�鼀�ᨨ ᨨ ᆠᨨ�矄�ᨨ 犨 �矄ᨨ �矄 �矄 ᆠ � ᨨ鼀 ᆠᨨ矄 �矄 ᆠ
= = 矄 ␔ v�矄 矄 ᨨ �ᆠ �鼀 \ ⼄盐 ᆠ泐ᨨ犨 �ᆠ �ᆠ矄�├矄 �矄ᨨ
㦨
ᨨ ᆠ � ᨨ 矄 ᨨ �ᆠ = 㦨 ⼄盐 ᆠ泐㡰맔 \ =
DF P= = ruv ҈ DF = F҈ - F = rAuv \ A
ᨨᆠFnet㦨FDF㦨ma \FrAuv㦨ma ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ 犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 m҈u҈졔mu㦨m҈v҈졔mv ҈맔u҈졔�맔㦨҈ 㡰졔� v㡰 Þu҈㦨졔�v �㡰
ᆠᨨ �ᆠ� ᨨ 矄�盐��ᨨ
v - v҈ v - -㡰 Þ ҈= u u҈ - u ҈ Þu҈㦨v졔 ��㡰 ᨨ鼀⼄ �㡰 ��㡰u҈㦨鼀⏌矄 v㦨҈鼀⏌矄 ᆠᨨ鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀㦨├�鼀⏌矄 ⧤쟼ᆠ鼀㦨䰘� e=
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EBD_8341 PS-44
Physics
҈ 㡰 ᨨ鼀犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀
� ��� 鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 p�ˆ - p�ˆ = \ � � 鼀ᨨ鼀 盐鼀矄 ᨨ盐 矄ᨨ ᨨ�.e.p҈¢졔p¢㦨 졠�ᨨ p҈¢졔p¢ = a҈ + a 㡰�ˆ + b҈ + b 㡰 ˆ + c҈kˆ 㦨 � 鼀ᨨ鼀 盐鼀 ᔼ�矄�� ҈҈ᆠ҈ ᆠ ᨨ 㜠 ᨨ盐矄 ᔼ ҈㦨 㦨 꾀矄ᨨ� ҈㦨泐 㦨泐
ˆ� ˆ ᆠᨨ鼀ᨨ 矄ᆠᨨ鼀 ᨨ盐 � 鼀 � � ├ˆ ᆠᨨ鼀ᨨ �矄 ᨨ 㜠 ᨨ �矄 ��矄 � � � 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 犨 �ᆠ �ᨨ矄 ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 矄ᨨ �矄�矄 犨ᨨ �ᨨ�ᨨ 졠�ᨨ p҈¢졔p¢㦨 a҈ + a 㡰�ˆ + b҈ ˆ ¹ᆠ盐矄b҈¹ ᨨᨨ犨� � 矄鼀 矄ᨨ � � 矄 ᨨ ᨨ�ᨨ �矄 矄ᨨ 犨ᨨ � b 㡰 盐矄 ᆠᨨ�矄�ᨨ ᨨ C 犨� A C 矄ᨨ矄 Aᆠ⼄盐�矄ᨨᆠ�泐vᆠ盐矄ᨨ 㜠ᨨ 矄�ᆠᆠᨨ�矄�ᨨ
犨 � �ᆠ 鼀矄矄矄
␔ A 矄矄 鼀ᨨ� � ᨨ犨 矄 �� 矄� � 矄盐 ᆠᨨ鼀矄矄�ᨨ 盐 ᨨ 犨 �ᆠ B 矄ᨨ 矄矄 鼀ᨨ� � ᨨ犨 矄 �� ᆠᨨ鼀矄矄�ᨨ ᨨ 矄� �ᆠᨨ � 盐矄� �矄 � ᨨᆠ�泐 犨 A B ␔ �矄 � ᨨᆠ�泐ᆠᨨ鼀矄 ᨨᨨ A B 鼀ᨨ犨� 矄鼀 ᨨᆠ�泐 vἘ 矄� � �矄 � 矄 ᨨ 鼀 �鼀盐鼀 ᆠᨨ鼀矄矄�ᨨ 矄泐 x 㦨 ᔴ
Ἐ
Ἐ
꾀
㝐 ᆠᨨ鼀矄矄�ᨨ
ᨨ鼀 � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 v mv㦨mvἘ졔mvἘ Þ vἘ㦨 \ ⧤쟼ᨨ AB矄泐矄鼀鼀 �鼀盐鼀ᆠᨨ鼀矄矄�ᨨ
҈ ҈ mv æQ v Ἐ = v ö mv Ἐ + mv Ἐ = mv Ἐ = çè ÷ ø 꾀泐� � �泐ᆠᨨ 矄�ᨨ
҈ ҈ ҈ ҈ mv = mv Ἐ + mv Ἐ + Kx ҈ ҈ ҈ Þ mv = mv + Kx m ҈ ҈ Þ x㦨 v Kx = mv \ K 㡰 v¢ 矄 ᨨ ᨨ �ᆠ ᨨ 矄 ᆠᨨ�矄�ᨨ 꾀矄 ᨨ �ᆠᆠ矄 �矄 矄 ᨨ �矄�ᆠ盐矄 ᨨ矄�ᨨ 矄 ᨨ盐 v ᆠᨨ矄 q v
v矄� q q
h
D
ᔴ 꾀
꾀
A
㦨
ᔴ
꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 �� 矄 ᨨ� m m m vᆠᨨ矄 q 㡰㦨 ´ v¢ - ´ v ᆠᨨ矄 q \ v¢㦨vᆠᨨ矄q ᆠ 㡰ᔼ � 矄�ᆠᆠᨨ�矄�ᨨ ᨨ 泐鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 鼀泐鼀� ᆠᨨ 矄 㡰 �矄 � ᆠᨨᆠ ᆠ盐矄 鼀ᨨ鼀 盐鼀 ᨨ ᆠ �矄 � 鼀� �盐 矄 犨矄 �ᆠ�ᨨ 㡰 �矄 � ᆠᨨᆠ ᆠ盐矄 ᨨ ᆠᨨ�矄�ᨨ 矄ᨨ鼀 鼀ᆠ �ᆠ �泐�矄 ᆠᨨ � ᨨ 矄ᨨ盐 �泐 ᆠ㡰 �矄 ᆠᨨᆠ ᆠ盐矄 ᨨ 졔 矄泐矄鼀 �鼀ᆠ ᨨᆠ�矄 � ᨨᆠ 㡰 �矄 ᆠᨨᆠ ᨨ �泐ᨨ �矄 ᆠᨨ 矄 � b㡰 꾀矄 � ᆠ� �矄 �ᆠ�ᨨ 矄矄
⧤쟼㦨쟼A ҈ mv = mgh Þ v = gh
졔 m
B
vᆠᨨ矄 v矄� = �
鼀
쟼
ᔼ DADB 㦨 \ h㦨鼀
C
鼀
\v㦨 � = 鼀 ⏌ 矄
�矄 �矄 ᨨᆠ�泐ᨨ ᨨᆠ├盐矄 ᨨᆠᨨ�矄�ᨨ
�矄ᨨᆠ�泐鼀├矄 �ᨨ 犨� �ᆠ 꾀矄ᨨ � �� � � � BEC ÐEBC㦨
ᨨ 鼀├矄 �ᨨ 犨� 矄ᆠᨨ � ᆠ� BC ᆠᨨ鼀ᨨ ᨨ ᨨ �BC�矄v ᆠᨨ矄 ᔼ�矄�� ᆠᨨ�矄�ᨨ B�矄 ᆠ泐� 矄�ᆠ ᨨ �鼀ᆠ ᨨᆠ矄 ᆠ ᨨ鼀 ᨨ 矄盐ᆠ �ᆠᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐ᆠᨨ鼀矄 ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 ᨨ � � ᆠ� BC鼀� 矄盐 ᆠ � �矄 ⼄盐ᨨv ᆠᨨ矄
҈ = = � 鼀 ⏌ 矄 ҈ BE BE Þ = Þ BE = 鼀 b㡰 ᔼ D BCE = CE ᨨ鼀 鼀ᆠ �ᆠ �泐ᆠᨨ 矄�ᨨ � ᆠ� 㝐ᆠ �ᆠ �泐B㦨 鼀ᆠ �ᆠ �泐 C ҈ ҈ M � + M ´ ҈ ´ = Mvc 㦨 ´
( )
\vc㦨 ҈� 鼀 ⏌ 矄 䰘 ᆠ㡰 ᨨᆠ�泐 ᨨ ᨨᆠ├ ᨨ � BC 盐矄 ᨨ ᆠᨨ�矄�ᨨ �矄ᆠᨨ矄 �鼀ᆠ ᨨᆠ矄ᆠ �ᆠ盐 ᨨ 矄盐 ᆠ 矄ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 ᨨ � � ᆠ� 鼀� 矄 盐 ᆠ � �졔㦨 vc
vᆠᨨ矄q
vᆠᨨ矄q
v' m
m
vᆠᨨ矄q
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PS-45
Work, Energy & Power
v矄� v
B
vᆠᨨ 矄
C
盐矄 ᆠᨨ�矄�ᨨ
꾀矄ᨨ 矄� ᆠ ᆠᨨ�矄�ᨨ �矄 矄�ᆠ �ᆠ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 v矄� 㡰 ᨨᆠᨨ�矄�ᨨ ᆠ �矄� �ᆠ�ᨨ 鼀� �盐 鼀� � � 矄鼀矄矄 ᨨ犨 � ��盐 和ᨨ ᆠ �盐 ᆠᨨ鼀ᨨ 矄 ᨨ ᨨᆠ�泐 ᆠᨨ�矄�ᨨ
矄矄 ᨨ犨 � ��盐 �矄鼀 矄 � ᨨᆠ�泐 ᨨ ᨨᆠ├矄 ᨨ盐 ᨨ� ᨨ 鼀├� � �º犨� BC ᨨ �ᆠ ᆠᨨ鼀ᨨ ᨨ � ᨨᆠ�泐 ᨨ ᨨᆠ├ �矄 ᨨ
㡰 㝐 �鼀盐鼀 �泐ᨨ矄矄㦨
鼀 鼀 + 㝐㡰 é ҈ ù = 鼀 ú êQ ⧤쟼 = 鼀 ëê ûú
é 㝐 ù ҈ ì 㝐 ü = 鼀 í ý ê ú 鼀 ë 鼀 + 㝐㡰 û î鼀 + 㝐 þ 和鼀 ᔼᔼ �矄 ᆠ矄 ᨨ ᆠ泐� 矄�ᆠᆠᨨ�矄�ᨨ 泐ᆠᨨ鼀� � ⼄盐�ᨨ �� � 矄鼀 ᔼ犨� ᨨ ⼄盐�ᨨ 犨 �\ æ 㝐 ö æ 鼀 ö =ç � 矄 ᨨ ÷ ç ÷ è 鼀 + 㝐ø è㝐+鼀ø ᆠ 矄鼀 ᔼ �矄 犨ᨨ � 矄鼀 ᔼᔼ�矄 ᆠᨨᆠ 㡰 和鼀 ҈ ᨨ 矄�ᆠᆠᨨ�矄�ᨨ ᆠᨨ �ᆠ� ᨨ 矄�盐�ᨨ 㦨 ҈ v - v҈ Þ vv҈㦨u҈u e㦨 u҈ - u Þ � ᨨᆠ�泐 ᆠᨨ�矄�ᨨ �矄 ⼄盐 ᨨ � ᨨᆠ�泐 ᨨᆠᨨ�矄�ᨨ 盐� 矄鼀 �矄 �矄 �� 和鼀 � 鼀ᨨ鼀 盐鼀鼀� 矄ᆠᨨ 矄 �
矄�ᆠ ᆠᨨ�矄�ᨨ �矄 矄鼀 �矄 盐 ᨨ鼀PᨨQ � u㦨҈鼀⏌矄 m㦨�錐s㦨鼀 é mg 矄� q + f ù é mg 矄� q + mmg ᆠᨨ矄 q ù = -ê a㦨 - ê ú úû m m ë û ë =
黨
R
M uA
鼀
Q
N m mg矄� q
q mgᆠᨨ矄q P mg f q
㦨g矄� q 졔Ҙgᆠᨨ矄 q 桌㦨g矄� q 졔Ҙᆠᨨ矄 q 桌 㦨҈�졔�맔黨黨桌㦨黨黨鼀⏌矄 矄� �vu㦨as Þ v㦨҈졔 黨黨㡰맔Þ v㦨鼀⏌矄
ᆠᨨᆠ�泐ᨨ 鼀矄矄 m盐矄 ᨨᆠᨨ�矄�ᨨ 㦨鼀⏌矄
ᨨᆠ�泐ᨨ 鼀矄矄M盐矄 ᨨᆠᨨ�矄�ᨨ 㦨鼀⏌矄 �� 㡰 v҈ ᨨᆠ�泐ᨨ 鼀矄矄 m ᆠᨨ�矄�ᨨ v ᨨᆠ�泐 ᨨ 鼀矄矄 M ᆠᨨ�矄�ᨨ ᨨ y ᨨ 鼀矄矄 M 鼀ᨨ� � ᨨ鼀QᨨR ᆠᨨ鼀� �ᨨ矄꾀 ᆠᨨ�矄�ᨨ u㦨v v㦨 a㦨黨黨鼀⏌矄 s㦨�
vu㦨as Þ 㡰 v 㦨 黨黨㡰맔� Þ v㦨҈䰘鼀⏌矄 ᨨ yᨨ 鼀矄矄m鼀ᨨ� � ᨨ鼀QᨨP ᆠᨨ�矄�ᨨ
h S㦨 h㦨矄� q 㦨矄� 㦨맔� q P u㦨v҈ v㦨졔҈鼀⏌矄 ⧤쟼졔쟼㡰� ��㦨 ⧤쟼졔쟼㡰 � 졔␔ �ᆠ�ᨨ
҈ ҈ mv҈ + mgh = mv + + m mgs ҈ ҈ v҈ + ҈ ´ ´ �㡰 = ҈㡰 + � ´ ҈ ´ v҈㦨�鼀⏌矄 \ ᔴᨨ �ᆠ� ᨨ 矄�盐�ᨨ
矄� q =
e㦨
Q h
� ᨨᆠ�泐 ᨨ 矄�ᨨ
� ᨨᆠ�泐 ᨨ ᨨᆠ
-� - ҈䰘 = - 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ ᆠᨨ�矄�ᨨ mv졔M맔㦨m맔v҈졔m맔v 맔졔M맔㦨맔 �㡰졔M ҈䰘㡰 = ҈� ├� \ M㦨 ҈䰘 ҈ ␔ 矄ᨨ ᆠ 矄 ᨨ� B ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐� 졔Z �ᆠ�ᨨ vᆠᨨ矄 q㡰ᆠᨨ鼀矄 ᨨ 盐ᨨ ���ᨨ ᨨᆠ � Z �ᆠ�ᨨ 㦨
Z
졔Z
졔Y v
vᆠᨨ矄q q v矄� q A vx 㦨m t 㦨
L
B
v' Y
vx 㦨鼀⏌矄
X t=t
矄ᨨ 矄 犨ᨨ ᨨᆠ��矄 B vx� 졔X �ᆠ�ᨨ 鼀⏌矄㡰 v矄� q � 졔Y �ᆠ�ᨨ 矄� 㦨鼀⏌矄㡰 矄盐 ᨨᆠ�泐ᨨ 矄ᨨ vἘ㦨 v x 㡰 + v 矄� q㡰 㦨 + = �鼀 ⏌ 矄 �㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 B ᨨ ᆠᨨ�矄�ᨨ 犨� 鼀矄矄 ᨨ ⼄盐 鼀� �盐 和� ᆠ ᆠᨨ�矄�ᨨ �矄ᆠᨨ鼀泐� 矄�ᆠ 犨ᨨ鼀矄矄矄 犨�矄�ᆠ├ᨨ� v�矄 ᨨᆠ�泐ᨨ 犨ᨨ鼀矄矄矄盐矄 ᆠᨨ�矄�ᨨ 桌 m맔�㦨m맔v \ v㦨�鼀⏌矄
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EBD_8341 PS-46
Physics
��㡰 ␔ 矄� � �矄 盐 �ᨨ� � ᆠ�ᆠ盐 鼀ᨨ�ᨨ
Ҙgd 㦨 - Ҙgd �
�� 矄�� 矄 ᨨ犨矄 鼀矄矄 m ᨨ盐 矄 ᔼ ᨨᨨ犨矄 ᨨᆠ� 鼀ᨨ�ᨨ �矄 矄 ᨨ�ᆠ uy㦨sy㦨day㦨gty㦨䁸 ᨨ �ᆠ 鼀ᨨ�ᨨ = - ´ �
mv 泐�泐ᨨ矄��ᨨ Tmgᆠᨨ矄a㦨 l 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ T㦨犨 a㦨黨 mv Þ v㦨 l �.e.� ᨨ� ᨨ ᨨ矄��ᨨ v㦨 꾀泐� � �泐 ᆠᨨ 矄�ᨨ ᨨ鼀 B ᨨ C a \
㦨
҈ mv = mg l Þ l=
d ҈ ҈ at Þ d㦨 gt Þ t㦨 g
ᨨ� ᨨ �矄 ᆠ 泐 鼀矄矄 m 盐� � �矄 �鼀t ᨨ鼀 犨 矄
S㦨 ut +
C
T
v �㡰 = 㦨҈鼀 g ´ 黨
mg 矄� a
a
mg ᆠᨨ矄a
mg
d = Ҙd 㦨 d µ g ᔼ ��泐犨 ᨨᨨ 盐盐鼀�矄꾀�矄쟼㦨mgl ␔ ᨨ矄 ᨨ鼀꾀 矄�├矄ᨨ 犨 쟼 ᆠ �矄 ᨨ ⧤쟼 犨 � �矄 ᨨ矄��ᨨ �ᔴ �盐鼀�盐 �矄 ᔼ DOCM x㦨 Ҙgd ´
B
졠ᆠᨨ 鼀矄矄mᆠᨨ� 矄犨� ᨨᆠ├Bᨨ 鼀矄矄m 和� ᆠ ᆠᨨ�矄�ᨨ �矄 矄�ᆠ � 盐 泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 mv㦨 鼀㡰u졔mv¢ 犨 u�矄 ᨨᆠ�泐ᨨ 鼀矄矄鼀 ᆠᨨ�矄�ᨨ v¢�矄 ᨨᆠ�泐 ᨨ 鼀矄矄 m Þ v¢㦨vu �㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ ├� �ᆠ �泐
҈ ҈ ҈ mv = 鼀㡰 u + mv Ἐ m Þ v㦨u졔vἘ ��㡰 ᨨ鼀⼄ �㡰 ��㡰
v
҈ mvB Þ vB㦨 gl n ᆠᨨ�矄�ᨨ 矄 �盐 鼀�盐 �矄 犨
ᨨ�� 鼀ᨨ矄ᨨ犨 矄 犨 ᨨ鼀C ᨨᆠ�泐 vἘB ᨨ ᆠᨨ�矄�ᨨ
mg l =
m
A
꾀矄ᨨ
l OM Þ OM㦨 l
ᨨᆠ�泐 ᨨ ᨨ B
ᆠᨨ矄 㦨
B
m
d
l ҈ = mv ἘB Þ vἘB㦨 gl
�矄 鼀 矄 ᨨᆠ�泐 ᨨ ᨨ 矄 ᨨ盐 盐ᆠ
v v㦨u졔 vu㡰 Þ u㦨 � ᨨᆠ├ B 矄矄 鼀ᨨ� � 盐 ᨨᆠ├ A 鼀� 矄 矄 꾀矄 �矄 ᨨ �ᆠ�ᨨ 犨 A B ᨨᨨᆠ├Aᨨᨨᨨᆠ├B矄 ᨨ盐 鼀ᨨ �矄 ᆠd �ᨨ ᨨ 盐ᆠ � B 盐 ᨨ �ᆠ�ᨨ ᨨᆠ 犨 B F㦨mN Þ m㡰a㦨m mg㡰Þ a㦨҈�Ҙg ᨨ 鼀ᨨ�ᨨ ᨨ B u㦨
mg
ᨨ鼀 gl ᨨ
� ᨨᆠ�泐 nᆠᨨ�矄�ᨨ 矄 �矄 \
n
m҈㦨mm㦨mu҈㦨 � Ҙgd u㦨 Þ
҈ æ ö = ᨨ� nᨨ� ç Þ n㦨҈ ÷ è �ø
v=
m҈ - m 㡰 u҈ m u + m҈ + m m҈ + m
v҈㦨
m - m㡰� Ҙgd + m + m
n
g l Þ æç ö÷ = ҈ è �ø
├� � ᨨ� ᨨ ᨨ 矄� 矄 犨 �
v҈㦨
gl
æ ö çè ÷ø ´ g l = �
v v㦨s㦨 a㦨҈�Ҙg �
� µgd ᨨ 矄�ᆠ ᆠᨨ�矄�ᨨ 犨 犨ᨨ ᨨ �矄
en g l 㡰 =
gl
犨 e �矄 ᆠᨨ �ᆠ� ᨨ 矄�盐�ᨨ
æ v ö ᨨ犨vu㦨asÞ 㡰 ç ÷ = -҈�Ҙg 㡰 d è �ø Þ
gl 盐ᨨᆠᨨ�矄�ᨨ 矄犨� 犨矄
ᆠ 盐鼀ᨨ ᆠᨨ�矄�ᨨ 矄㦨 ᨨ� 鼀矄矄 � 犨犨� 矄 v
ᨨᆠ�泐ᨨ 鼀矄矄m �矄� 矄 㦨vᆠᨨ矄q㦨
v � 盐 M犨� �矄 ᆠᨨ ҈鼀犨 �m犨��矄盐泐 � ҈㡰鼀 ᨨ鼀 �泐ᆠᨨ 矄�ᨨ ᆠ矄� 쟼ᨨ M㦨� ᆠ矄 � 쟼ᨨ m졔� ᆠ矄� ⧤쟼ᨨ ᨨ ᨨᆠ├矄
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Work, Energy & Power 꾀泐� � 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
҈鼀 鼀 졠 ҈鼀
㝐
mv㦨mv¢졔mv¢\
v¢㦨
v
��㡰
ᨨ鼀⼄ �㡰 ��㡰
q
h
v - ´
vᆠᨨ矄q �鼀
v q v㦨v ᆠᨨ矄q
䰘
h' �҈㡰鼀 鼀
mv v k 矄� �ᆠᨨ 矄 㡰 = x Þk㦨 黨 m x
꾀 �� 矄ᨨ� P ᨨᆠ�泐ᨨ 盐㦨uᆠᨨ矄q ᔼ DAQR R �㡰 ᨨ鼀 ��盐
QR � 矄� 㦨 QR㦨 ҈ ⏌
҈ ҈ Mgh㦨mgh'졔 mv + m ᆠᨨ矄 q㡰 ҈ ᨨ 㡰 黨㡰 ҈㡰㦨 �㡰 黨㡰 � - ҈㡰 + ´ ´ v
졠
Y ҈⏌
P
uᆠᨨ矄q
u
R ,H 㡰
�
æ v ö ҈ 졔 ´ � ´ ç ÷ è �ø 和ᨨ� �犨�v㦨黨鼀⏌矄 ᔼ ��ᨨ矄��ᨨ ᨨ ᔴ㝐 m҈x҈ + m x ҈ ´ + ҈ ´ 黨 x҈㦨 m + m 㦨黨鼀 㦨 ҈ ҈�㦨 ᨨ 㡰 ᔼ �� ᨨᆠ�泐ᨨ ᔴ㝐
q O
Mu ᆠᨨ矄 q u ᆠᨨ矄 q = M ᨨ鼀 �泐ᆠᨨ 矄�ᨨ � ᆠ�KEᨨ 盐㜠鼀矄矄 矄泐矄鼀P㦨PEᨨ 盐㜠鼀矄矄 矄泐矄鼀R v㦨
m҈u҈ + mu ҈ ´ 黨 + ҈ ´ 㦨�鼀矄҈ m҈ + m 㦨 u҈ 꾀ᆠᆠ�ᨨ ᨨ ᔴ㝐 ac㦨黨鼀矄 ᔴ㝐 �矄ᆠ鼀 ᨨ ᔴ㝐 �矄 sc㦨黨鼀 꾀泐� � 黨鼀矄҈ ҈ S㦨ut졔 at ҈g 졠 ᨨ��� 㡰 黨㦨�t黨t t�t҈㦨 � ± � + � ± � t㦨 㦨 㦨�矄 ᆠᨨ�矄�ᨨ 犨 C A�矄 矄�ᆠ �鼀矄矄矄 ⼄盐 矄ᨨ 泐 犨� ᆠ � � ᨨᆠ��矄 A 犨矄 � ��泐矄 ᨨ 矄盐ᨨ ᆠᨨ�矄�ᨨ 犨� C犨�ᆠᨨ鼀ᨨ矄 A犨�� ��泐矄鼀ᨨ� �犨� ᨨᆠ�泐v 盐 矄� ᆠ A �矄 ᆠᨨ
ᆠ ᨨ B 犨� 矄� � 矄� � 犨� � ᆠᨨ鼀矄矄 v¢ ᆠᨨ鼀鼀ᨨ ᨨᆠ�泐ᨨ A B m v
ᨨ
A m
B m
㦨
C
A m
҈ M 㡰v = M 㡰 gh
k x m
黨 ´ � ´ ´ ҈ Þ q 㦨䰘 � ´ � Qu㦨�鼀⏌矄�� 㡰 ��㡰 �ᆠ ᆠᨨ鼀ᨨ ᆠᨨ矄 q 㦨
R u 矄� q � ´ �矄� ´ 䰘o = = = ҈ 鼀 g ´ 黨 ᨨ� ᨨ ᆠᨨ鼀ᨨ H㦨
u 矄� q � ´ � ´ 矄� 䰘°㡰 = = 鼀 g ´ 黨
mV A
C
G
vἘ
҈ ҈ ҈ ҈ mv = mv Ἐ + mv Ἐ + kx 犨 x�矄 ᆠᨨ鼀矄矄�ᨨ � 矄� �t㦨t
v = v Ἐ +
҈ u ᆠᨨ矄 q æ ҈ � ö M 㡰 = Mg ´ ç + ÷ è ø ҈
vἘ B m
꾀t㦨t ᨨᆠ��矄ᨨ A Bᆠᨨ鼀矄鼀 꾀泐� � �泐ᆠᨨ 矄�ᨨ 錐
\
�㡰
X
ᨨ犨 ᨨ鼀ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 �� 矄 ᨨ� 㜠 M uᆠᨨ矄q㡰졔M맔㦨 M졔M㡰v
uc㦨
C
Q
A
B B
C
ᨨᆠᨨ�矄�ᨨ
mV
꾀 ᆠᨨ�矄�ᨨ
ᨨ ᆠᨨ�矄�ᨨ 泐 矄泐鼀鼀泐 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 �矄 ᨨ 꾀 ᆠᨨ�矄�ᨨ 鼀ᨨ鼀 盐鼀�矄矄ᨨ ᨨ矄 ᨨ矄 ᨨᆠ �矄ᆠ� �ᨨ 矄泐矄鼀꾀ᆠᨨ鼀矄ᨨ矄矄ᨨ ᔴ矄 ᨨ盐 ⼄盐 ᨨᨨ矄�鼀ᨨ鼀 矄ᨨᨨᆠ�泐ᨨ C矄 ᨨ盐 矄鼀 � 鼀� �盐 �.e. V 盐 � ᨨᨨ矄� �ᆠ�ᨨ ᨨ ᨨᆠ�泐ᨨ B
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EBD_8341 PS-48
黨
Physics
꾀泐� �ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 ᨨ 盐 Amv㦨mv҈졔m҈v �㡰 ᨨ v㦨v҈졔҈맔v
㦨
m㦨├�
v҈
v
Topic-5 : Miscellaneous (Mixed Concept) Problems
v
҈ v
ᨨ
꾀�� 泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ ᆠᨨ�矄�ᨨ B mv҈㦨 m졔m㡰v ᨨ v҈㦨 黨졔㡰v㦨v
v҈ ᨨ鼀⼄ �㡰 ��㡰 Þ v㦨
t
\s㦨 v ò e
v = Þ v㦨 v҈ Þ v҈
t ⏌ t
R
奠ᨨ矄矄� ᨨᆠ�泐
v v q = wt = t R R $ vp㦨 ( v 矄� q� + v ᆠᨨ矄 qˆ )
pf㦨 mV + M v
ᨨ
Mv ҈ Mv Þa㦨
mV mV 犨 �ᆠ ��矄 �ᆠ�ᨨ ᨨ 鼀ᨨ鼀 盐鼀
y r pf a
æ è
vp㦨 ç v 矄�
x
éæ v ö $ æ v ö ù 㦨 m êç v 矄� t ÷ � + ç v ᆠᨨ矄 t v҈ ÷ ˆ ú ø û R ø è R ëè 㡰 ᨨ ᆠ�ᆠ盐 鼀ᨨ�ᨨ ᨨ ᆠ矄矄泐 ᆠ � ᨨᆠ �矄 ᨨ� 泐 mg ᆠᨨ矄 q N㡰 mgᆠᨨ矄 q N㦨
Mvˆ
⧤쟼 �
p f m + M 㡰 ⧤쟼 f ҈ 㦨҈ 㦨 ҈ ҈ ⧤쟼� mV + Mv
㦨҈
m V +M v
m + M 㡰 mV + Mv 㡰
v $ v ö t� + v ᆠᨨ矄 tˆ ÷ R R ø
vm㦨 v҈ ˆ \ � 鼀ᨨ鼀 盐鼀ᨨ �ᆠ犨鼀 矄 盐 ᆠ�ᨨ
ᨨ �鼀 Lpm㦨m vpvm㡰
a㦨
mV�ˆ
x
t㦨
w㦨
� ᨨ 泐㡰
��㡰
ds 㦨vet⏌t Þds㦨ve⏌tdt dt
dt 㦨vt ҈e҈㡰㦨�맔맔㦨鼀
v - v҈ = � v
⧤쟼 � ⧤쟼 f
҈ J = 㦨�鼀矄҈ m
꾀 �盐矄 ᨨ �ᆠᨨ盐ᆠ ᨨ ᆠ�ᆠ盐 y v v q
҈ v - v҈ 맔҈㦨 ´ ҈ = �奠 v �㡰꾀ᆠᆠᨨ � � ᨨᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 r r p f 㦨 p� 㦨 mV�ˆ + Mvˆ 㝐� �盐 ᨨ 鼀ᨨ鼀 盐鼀ᨨ \
㦨
v҈ v ҈ = Þ ҈ - ҈ = ҈ - = = � v v
\
㦨鼀Þ
꾀矄ᨨ 㡰㦨ve⏌t \
��㡰
v҈ v㦨v҈졔
�
㡰 � 鼀矄矄m㦨├�錐�鼀盐矄㦨҈矄 t㡰㦨 vet⏌t錐t㦨矄 e҈㦨䰘 ᔼ鼀盐矄㦨ᔴ �� � 鼀ᨨ鼀 盐鼀
m㦨黨├�
m ҈㦨҈├�
Þ
m + M 㡰 mV + Mv 㡰
DK E mM v + V 㡰 㦨 K E� m + M 㡰 mV + Mv 㡰
B
A
mV + mMv + mMV + M v m V M v
mv �㡰 dö æ R + çè ÷ ø
B C A D
N㦨 ᨨ鼀ᆠ�ᨨ �q 꾀ᆠᆠᨨ � � ᨨ �泐 ᆠᨨ 矄�ᨨ
R
mgᆠᨨ矄q
q
NA
d⏌
q
V
mg矄� q
mg
dö ҈ æ mv = mg ç R + ÷ ҈ - ᆠᨨ矄 q㡰 è ø dö æ Þ v㦨� ç R + ÷ ҈ᆠᨨ矄q㡰 ø è
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PS-49
Work, Energy & Power 盐� � �矄盐ᨨ v� ⼄ �㡰 N㦨mg ᆠᨨ矄 q 㡰 㡰 犨�ᨨ矄ᆠᨨ ᆠ犨� �
矄 犨
N㦨
æ ö è ø 꾀 �矄 � 鼀├矄 ᆠᨨ ᆠ 犨� ᨨ盐 矄 ᨨ鼀 ᆠ�ᨨ 矄矄 ᆠ� � ᨨ犨 矄 ᆠ
ᨨ
ᆠᨨ矄q㦨ᨨq㦨ᆠᨨ矄҈ ç ÷
和ᨨ ᨨ
æ ö è ø
q £ᆠᨨ矄҈ ç ÷
\ ux㦨҈ᆠᨨ矄 졔 � 㦨 �� \ 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 ᨨ ᆠ
ᨨ 㜠ᨨ泐 矄泐矄鼀 � ᨨ� ᨨ �ᆠ�ᨨ mux졔M맔㦨 m졔M㡰vx
mu x 犨 m 㦨 鼀矄矄 ᨨ ᆠ
ᨨ
m+M M㦨鼀矄矄ᨨ ᨨ泐vx㦨ᨨᆠ�泐ᨨ ᆠ
ᨨ 㜠 ᨨ泐 矄泐矄鼀㡰
\
vx㦨
҈´ �� = �� 鼀矄 ҈ ҈+ 黨 ᨨ� ᨨ �矄 ᆠ ᆠᨨ 泐 ᆠ
\
NB㦨 NA㦨mg ᆠᨨ矄q㡰
ᨨ� ᨨ ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐 ᨨ ᆠ
ᨨ 鼀� 矄 矄鼀
æ ö ᨨ q ÷ è ø NA㦨 NB㦨mg ᆠᨨ矄q㡰
NAᆠᨨ矄q NBᆠᨨ矄q� 矄矄 ᨨᨨ犨矄 ³ᆠᨨ矄҈ ç
NB �鼀�
NA
vx 㦨
P 㦨 ҈ 맔 � = 鼀 Q 和ᆠᨨ 犨矄 ᨨᆠ ҈ 矄ᆠᨨ 㡰 和� ᆠ 矄ᆠᨨ 矄ᨨ 矄盐ᆠ├ ᨨ泐 \ � �鼀 ҈ 矄ᆠᨨ 矄 ᨨ泐 ᆠᨨ矄 �矄 ᆠ ᨨ
鼀 ᨨ ᨨ泐 ҈ 矄ᆠᨨ 矄錐
u = � 鼀⏌矄 v㦨䁸 t㦨҈矄 m
mg
s㦨 鼀 鼀�
҈
⏌
ᆠᨨ矄q
졔҈
҈
⏌ 졔҈
s㦨 ut +
ᆠᨨ矄q
ᔴᨨ 矄� �ᆠ 鼀ᨨ�ᨨ ᨨ ᆠ
ᨨ uy㦨졔҈矄� �ᆠᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐㡰 sy㦨҈鼀 �ᆠ �矄ᆠ鼀 㡰 ay㦨҈鼀⏌矄
A
m 㦨黨├� B
u 㦨 T
t 㦨t �鼀ᨨ �矄�鼀ᆠᨨ ᆠ��㡰 ҈
sy㦨uy t + a y t \҈㦨�t� t
Þ \ \
t ҈t㦨
-҈㡰 ± ҈ - ҈㡰 -㡰 㦨҈ᨨᨨ� 桌 t㦨 t㦨҈矄ᆠ
҈ at
҈ 㦨 �� 맔҈졔 맔a맔҈
\
a㦨 ⏌ ҈ 鼀矄 \v㦨u졔at㦨 � 鼀⏌矄
æ ö v㦨u졔at㦨 �� çç ÷÷ ´҈ è ҈ ø
ux
C
ux
Þ
졔 P uy u ᨨ 힘
҈m
M
v㦨 � 鼀矄 ҈
ᨨ � � ᨨᆠ�泐vfᨨ ᆠ�� 矄ᆠᨨ �鼀ᆠ 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 � ᨨ� ᨨ �ᆠ�ᨨ
mux졔 M졔m㡰vx㦨 M졔m㡰vf \
҈맔 �� 졔 黨졔҈㡰 � 㦨 黨졔㡰vf vf 㦨҈�䰘�鼀⏌矄
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EBD_8341 PS-50
Physics
System of Particles and Rotational Motion
6
㝐矄矄 ᨨ 鼀� � �
Topic-1 : Centre of Mass, Centre of Gravity & Principle of Moments ҈
pR r p ҈㡰 r ᔴ ᨨ 鼀矄矄ᨨ 鼀� � � M r + M r X ᔴ졠㝐 = ҈ ҈ M҈ + M M 鼀� � �㡰 =
b㡰 �
æxö ÷ èLø
� 鼀矄矄 矄�泐 r x 㡰 = a + b ç
X CM =
ò xdm ò dm
é ù é ù êë pR rúû + êë p ҈㡰 r㡰 úû R ҈桌 Þ R 㡰 = pR r + p ҈㡰 r㡰
L
ò dm = ò r x㡰dx Lé
æxö = ò êa + b ç ÷ èLø ê ë
Þ
ù
bL ú dx = aL + úû L Læ æ aL bL ö bx ö xdm = ax + dx = + ç ÷ ç ÷ ò òç ç ÷ø L ÷ø è è æ aL bL ö + çç ÷ ÷ø è \ X ᔴ㝐 = bL aL + L æ a + b ö Þ X ᔴ㝐 = ç ÷ è a + b ø b㡰
R ҈㡰
R ҈㡰
= R
Þ
R ҈㡰 R ҈㡰 R + R + ҈㡰
= R
Þ R졔R졔҈㡰 R㡰㦨҈
㡰
Xcm =
X cm
m҈ x҈ + m x + m x m҈ + m + m ҈ ´ + ҈� ´ + � ´ ҈� ´ = = = 黨cm ҈ + ҈� + � �
m y + m y + m y Ycm = ҈ ҈ m҈ + m + m ҈ ´ + ҈� ´ + � ´ � ´ Ycm = = = cm ҈ + ҈� + � � ᆠᆠ ᨨ 鼀矄矄ᨨ 矄泐矄鼀 �矄ᨨ� 黨㡰 ᨨ�� 鼀� x y m҈㦨҈C҈㦨 ҈��㡰 m㦨C㦨 �҈�㡰 m x + m x ҈� + ҈� = = 䰘� X cm = ҈ ҈ m҈ + m
m҈ y҈ + m y � + � = = ҈䰘� m҈ + m \ ᔴᨨᨨ � ᨨ ᆠ ᨨ 鼀矄矄 ᨨ � 矄 鼀� 䰘�҈䰘�㡰 㡰 㝐矄矄 ᨨ 矄 㦨 ᨨ盐鼀 ᨨ 矄 矄�泐 ᨨ 矄
�
=
p r
㝐矄矄 ᨨ ᆠ�泐 M ᆠ�泐 =
p ҈㡰 r
m㡰aˆ + m ´ a�ˆ + ma -�ˆ㡰 + m ´ a - ˆ 㡰 m + m + m + m a�ˆ - aˆ a ˆ ˆ = = � - 㡰 ҈ � 㡰 ␔� 矄ᆠ ᨨᨨ� q ᔴ㝐 ᨨ ᆠ盐㜠ᨨ ᨨ�ᨨ
æa bö ç ÷ è ø MX - mx 矄� � xᔴ㝐 = M -m M a M ´- ´ =- a = M ҈ M =
Ycm =
㡰 꾀ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 aᆠ鼀㡰 �矄 �� 泐 r r m a + m a + r \ aᆠ鼀 = ҈ ҈ m҈ + m +
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PS-51
System of Particles and Rotational Motion
b ҈ 和ᨨᔴ㝐 ᆠᨨᨨ � 矄 ᨨ yCM = -
҈
a a �a - = ҈ ҈ b b �b = y = ҈ ҈ ᆠ㡰 ᨨᨨ 盐ᆠ鼀 �鼀盐鼀鼀ᨨ鼀 ᨨ ᨨᆠ� ᨨ ᆠ�ᨨ
ᨨ ᨨᆠ鼀盐矄 �ᆠ盐ᨨ� 꾀
ᆠ㡰 ᔴ ᨨ 鼀矄矄 xᆠ鼀
x =
䰘
꾀 q
黨
҈҈
h D
B
O
C
h
C
h
h'
B
P
B D = h' - h = OB - OP = OC - OP ҈ æ ö = h - h = h - ҈ = h -҈ ç æ pö ÷ ç ᆠᨨ矄 çè ÷ø ÷ è ø n
(
)
ᆠ ᨨ 泐盐ᨨ n D = h
҈ æ ö -҈ ç æ pö ÷ ç ᆠᨨ矄 çè ÷ø ÷ è ø n
꾀
ydy
h
y
�㡰
a
dy h
r
R
ᔴ
h
é y ù ê ú ë û ycm㦨 㦨 h 㦨 h h ҈ b㡰 ᔴ ᨨ 鼀矄矄 ᨨ ᨨ �矄 �� 泐 ò y dy
ò ax +
L
bx 㡰 dx L bx
ò a + L 㡰dx
A O
h
ò r 㦨
ò ydm ò dm
҈ ҈ pR hr R h ᨨ ᆠᨨ 犨├ ᨨ犨 r y y = \r㦨 R R n n
xcm =
B
D D
dyr
L
n㦨㡰 OD 㦨h 矄ᨨO�矄 ᆠ ᨨ� OA h Ἐ = = Q OA = OB = h Ἐ㡰 \ OD h ҈ Þ h'㦨h æ ҈ ö æ ҈ ö D = h Ἐ- h = h - h = h ç -҈ = h -҈ ç p ÷ è ҈ ⏌ ÷ø çè ᆠᨨ矄 ÷ ø ᨨ犨ᆠᨨ 矄� 矄⼄盐 n㦨㡰
h
㦨
h'
O
C
ò y pr
ᔴ
泐
㡰 矄�泐ᨨ ᆠᨨ 㦨 r
h
O
㡰
q
҈ \ q = = ᆠ㡰 꾀ᆠᆠᨨ � �ᨨ� ᆠ�ᨨ 鼀ᨨ鼀 矄犨 矄泐矄鼀�矄 矄 ᨨ ᆠ �㜠ᆠᨨᆠ├犨�矄 鼀ᨨ鼀 �矄 ⼄盐 ᨨ ᆠᨨᆠ├犨�矄 鼀ᨨ鼀 �ᨨ 맔ᨨ 鼀㦨 ᨨ맔 ᨨ鼀 ␔ �鼀� 犨�� �矄 ᆠ ᨨ 鼀 矄 � 矄 ᨨ 矄� ᆠ 鼀 ᆠᨨ鼀矄 矄 ᨨ �� 鼀 b㡰 ᔴᨨ 矄� 矄�鼀矄 ᨨ泐�ᨨ � � � A A C
꾀
Bc y ҈ + \ = = ҈䰘 AB x
鼀
æ x ö҈ rx 㡰 ç ÷ + ry y ⏌ èø r x + y 㡰
҈ y y + = x x
Þ
ᔴ ᨨ 鼀矄矄ycm㦨
鼀
x =
A
aL bL L æ a + b ö + ç ÷ = è ø = bL b aL + a+ a b + 䰘L = ᨨ犨 b ҈ a+ 졠 矄ᨨ� � 犨 � b 㦨 a ҈ ᆠ㡰 ҈ 㡰 ҈ 㡰 犨� � �� � ��盐 �矄 鼀 盐 ᨨ � ᨨ �矄 � ᆠ�ᆠ矄 犨ᨨ 矄�� � ᨨ 盐 � ᨨ鼀 鼀矄矄 �矄�盐�ᨨ ᨨ犨ᆠ 矄矄盐鼀 矄泐矄鼀ᨨ鼀 盐 ᨨ � ᨨ� 鼀矄矄矄 犨 鼀矄矄 ᨨ ᆠ �矄 ᆠᨨ 矄� �矄 �ᨨ鼀�ᆠ ᆠ Y a a㡰 m
m
Y
m 㡰
a㡰 m
a a㡰 m
a a㡰 m
a a㡰
X
m
㡰
X
m a㡰
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EBD_8341 PS-52
҈�
҈
Physics
泐㜠ᆠᨨᨨ � ᨨ ᆠ ᨨ 鼀矄矄 �矄 m y + m y + m y + m y + m� y� ycm = ҈ ҈ m҈ + m + m + m + m� m ´ + m ´ + m ´ a + m ´ a + m - a 㡰 \ ycm = m + m + m + m + m ma a = = ҈ m ҈ 㡰 㝐盐盐 ᨨᆠᨨ ᆠ�ᨨ 犨� 犨 �ᆠ 犨ᨨ�ᆠ矄꾀 鼀ᨨ ᨨ犨 矄 ᆠ ᨨ �矄 � ᨨᆠ ᨨ ᨨᆠ矄 ᆠ� � ᨨ 矄泐矄鼀 ␔├ ᨨ犨F 㦨 Maᆠ鼀 aᆠ鼀㦨 Þ vᆠ鼀㦨ᆠᨨ 矄 和� ᆠ� ��泐vᆠ鼀㦨 \ � ᆠ鼀㦨 㡰 ᔴ ᨨ 鼀矄矄ᨨ 矄ᨨ� 鼀�矄 ᨨ �盐矄R�矄
R ᨨ ᆠ ᨨ 矄� ᨨ 鼀�矄 R ´ \ hᆠ鼀 = = = ᆠ鼀 㡰 s 鼀矄矄 矄�泐ᨨ ᆠ�ᆠ盐 �矄ᆠ 졠��� 鼀矄矄ᨨ �矄ᆠ m = pa s �矄 ᆠ
҈䰘
a s æ a ö æ p - ҈ö 鼀� � �鼀矄矄 m Ἐ = ç pa - ÷ s = a ç s è ÷ø è ø Y 鼀ᨨ 鼀矄矄 m =
a a
X ҈
犨 ᨨ矄��ᨨ ᨨ ᆠ ᨨ 鼀矄矄
a a ´ m x - mx X CM = = m - m a pa a -a ⏌ -a -a = = = = æ p - ҈ ö a p - ҈㡰 p - çè ÷ ø \ x = RA㦨ᨨ鼀ᆠ�ᨨ A RB㦨ᨨ鼀ᆠ�ᨨ B ᨨ ᨨ ᨨ� ⼄盐���盐鼀� ᨨ� ᨨ ᨨ矄��ᨨ RB 鼀ᨨ鼀 ᨨ ᨨᆠ矄 R A x d x 㡰 ᨨ盐 A ᨨ ᨨ盐 B 矄 ᨨ盐 ᨨ C B \ 㝐ᨨ鼀 ᨨ ᨨᆠ矄 ᨨ盐 A A pa ´ -
҈
W 㦨犨�� ᨨ ᨨ 㡰 xW RB = d
xW㦨RBd ᨨ
꾀�� 鼀ᨨ鼀 ᨨ ᨨᆠ矄 ᨨ盐 B,
æ d - xö RA = ç W ��㡰 è d ÷ø xW æ d - xö \ ᨨ鼀ᆠ�ᨨ ᨨ A = ç W ᨨ B = è d ÷ø d ҈黨 矄 �矄 ᨨ ᨨᆠ�矄ᆠ� �ᨨ 犨ᨨ�ᆠ 矄泐矄鼀 和ᨨ ac.m 㦨 ᆠVc.m㦨ᔴᨨ 矄 bᆠ㡰 ᨨᆠ├ ᨨ 鼀矄矄 㝐 鼀ᨨ矄 泐 �矄 ᆠ ᨨ犨 矄 Mx㦨m Rx㡰 mR Þ x= ᨨ犨 矄 M +m mR \ x= M +m ᔼ �矄 ᨨᆠ�泐ᨨ 鼀矄矄Ἐ鼀Ἐ矄�矄 ᨨᆠ├ �矄 ᨨᆠ�泐 ᨨ ᨨᆠ├ � 矄 ᆠᆠᨨ � � ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 鼀㦨㝐 泐 �泐 ᆠᨨ 矄�ᨨ
҈ ҈ 鼀� 㦨 mv + MV d x㡰W = RAd
ᨨ
m gR gR 㦨 m m M ҈+ ҈+ M M ҈ ᆠ㡰 ␔ � � 矄ᨨ 盐 ᆠ泐ᆠ�矄 � 鼀ᨨ�ᨨ 犨 矄鼀ᨨ� �ᆠ�ᨨ 矄盐ᆠ ᆠ ᨨ 鼀矄矄 ᨨ 犨 鼀ᨨ ᨨ犨 ᨨ� � �ᆠ�ᨨ 犨�ᆠ� ᨨᨨ矄� �ᆠ�ᨨ ᨨ � 鼀ᨨ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄ᨨ ᨨ 泐犨� 矄ᆠᨨ�ᨨ盐 ᆠ ᨨ� � �ᆠ�ᨨ 犨�ᆠ� ᆠ├犨 �ᆠ�ᨨ � ᨨ 犨 矄 盐� � � � 矄� � � �ᆠ�ᨨ 犨� ᆠ � ᨨ犨 �ᆠ�ᨨ ᨨ 犨 㡰 �矄 ᨨ ᨨ⼄盐ᨨ ᨨ 泐ᨨ盐�矄ᆠ ᨨ 鼀矄矄矄ᨨ ᨨ �矄ᆠᨨ 矄 ᨨᨨᆠ�泐ᨨ ᆠ ᨨ 鼀矄矄ᨨ 鼀� ᆠᨨ 矄 ᨨᆠᆠ� � ᨨ ᨨ 泐鼀盐矄 ᨨ
� 鼀ᨨ鼀 盐鼀ᨨ �矄ᨨ 矄泐矄鼀鼀� 矄ᆠᨨ 矄 � � ᨨ 矄 矄 矄 � �� 矄 ᨨ � ᨨ 矄 矄 矄 � 矄 和ᨨ� �犨�㦨
Y R
4m O
LἘ㜠� R
L
m (L,O) (L',O) R
X
L'+�R
␔ 矄鼀矄 Mᆠ �矄�矄ᨨ矄��ᨨ ᨨᨨ 鼀 ᨨ矄��ᨨ �矄 ᨨ ᨨᆠ � ᨨ� ᨨ �ᆠ�ᨨ ᨨ 㜠ᆠᨨᨨ � ᨨ ᆠ鼀犨� ᨨᆠ � xᆠ鼀桌� ��㦨xᆠ鼀桌 � M҈ x҈ + M x M҈ x Ἐ҈ + M x Ἐ Þ = M҈ + M M҈ + M
m ´ L + m ´ � R + L 㡰 m ´ L Ἐ+ m ´ L Ἐ- � R 㡰 = m + m m + m Þ �L졔�R㦨�LἘ�R Þ �L졔҈R㦨�LἘ Þ L졔R㦨LἘ Þ
W
�㡰
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PS-53
System of Particles and Rotational Motion
꾀矄 � �� 盐 ᆠ ᨨ 鼀矄矄 ᨨ 犨ᨨ 矄 矄 矄yᆠᨨ㜠ᨨ � ᨨ� �矄� ��矄 �矄 � 矄矄ᨨ y㜠ᆠᨨᨨ � ᨨ ᆠ鼀ᨨ 犨ᨨ矄 矄泐矄鼀犨�鼀� ᨨ ᆠ ᆠᨨᨨ � ᨨ ᆠ ᨨ 鼀矄矄ᨨ ���矄 L졔R㡰 �㡰 ᨨ 矄 矄 �ᆠ ᨨ � 泐 泐 �矄 ␔ � ᨨ� � ᨨᆠ ᆠ� � ᨨ ᨨ �矄 犨��
ᨨ鼀 ᆠ�ᨨ 矄 ᨨᆠ矄 �ᆠ ᨨᆠ矄 ᆠ
ᨨ ᆠ ᨨ� ᨨ 鼀ᨨ�ᨨ ᆠ ᆠ ᨨ 鼀矄矄 鼀ᨨ矄 �ᆠ泐 ᨨ犨 犨 矄 盐矄 ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᨨ �矄 矄�� � ��㡰 ᔴᨨ 矄� �泐 ᨨ� P ᨨ ᨨ ᨨᆠ xy㡰 �矄 ᆠr ᨨ鼀 B ᨨ 鼀├矄
�q 犨� ᨨ� ᨨ �矄 ᨨ矄��ᨨ y ᔼ DBNP 矄� q = �㡰 A Y r x + BN x + r ᆠᨨ矄 q ᆠᨨ矄 q = = C L⏌ L⏌ P xy㡰 x y r Þ ᆠᨨ矄 q = q ��㡰 L O N B X -r x Q 矄� q + ᆠᨨ矄 q = ҈
y
�
m
r
+
㦨 㦨
s p R - s p r - ҈㡰 ´ 䰘
㡰 - ҈㡰
=-
҈ ´ ҈ ´ 䰘 = -黨 ᆠ鼀 䰘 ´ 黨
�.e.ᆠ ᨨ 鼀矄矄�矄 �矄 ᆠᨨ 黨ᆠ鼀 ᨨ鼀 ᨨ���
ᨨ犨 矄
Topic-2 : Angular Displacement, Velocity and Aceleration ҈
dy = q = Cx dx 꾀P q = Ca
㡰 y = Cx Þ
y
w
mw aᆠᨨ矄q N
R r
R r v O 졔Y
ᔴ矄 �㡰
졔X
r
v'
a x
2
ᔴ矄 ��㡰
ᔴ矄 ��㡰 ᆠ ᨨ 鼀矄矄 ᨨ 矄泐矄鼀 � x㜠ᆠᨨᨨ � M ´ x + m ´ x = x xἘC㦨 M +m xC㦨xἘC Q 㦨� x㜠 �ᆠ�ᨨ 㡰 mx҈ m R - r 㡰 = \ x㦨 M +m M +m m R - r 㡰 �.e. ᨨᆠ├鼀ᨨ x = 犨
M +m
ᆠ泐� ᆠ 矄 ᨨᨨ鼀 ᨨ� b ᨨ ᆠ├ 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 P�㦨Pf 㦨MVmv MV \ v㦨 m 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ �泐 ҈ ҈ Þ mg Rr㡰㦨 MV + mv
MV 桌 m
s ´ p R 㡰 - s ´ p r ´ 䰘
졔Y x 㦨 Rr㡰
M
Q v =
m Þ mg Rr㡰㦨MV졔 M V m é Mù ém + M ù mg Rr㡰㦨MV ê҈ + ú = MV ê ú m ë û ë m û g R - r 㡰 m g R - r 㡰 Þ = V Þ V㦨m M m + M 㡰 M m + M 㡰 s 鼀矄矄盐 �ᨨ 盐 � ᨨ鼀 \ 㝐矄矄ᨨ 犨 ᨨ �矄ᆠ㦨s맔p R 㝐矄矄ᨨ ᨨ�ᨨ 鼀ᨨ 㦨 s맔p r R R㦨ᆠ鼀錐r㦨҈ᆠ鼀錐OP㦨䰘ᆠ鼀 ᨨ矄��ᨨ ᨨ ᆠ ᨨ 鼀矄矄 r m҈ x҈ - m x x㦨 O P m҈ - m
1
r
M V
x
=҈ æL ö çè - r ÷ø ᆠ矄 ᨨ ᨨ �矄 ᨨ� �矄 �矄 ᔴ矄 �㡰 ᆠ ᨨ 鼀矄矄 ᨨ 矄泐矄鼀 ᨨ 犨ᨨ ᨨ �矄 �
x㜠ᆠᨨᨨ � M ´ + mx҈ mx҈ = xC 㦨 M +m M +m
\
Þ mg Rr㡰㦨MV졔 m
q P a, b㡰 mg
q
mg矄� q
mw a x
mgᆠᨨ矄q
ᨨ 矄 泐 ᆠ�ᆠ盐 鼀ᨨ�ᨨ
mw a ᆠᨨ矄 q = mg 矄� q
Þw=
g q a
\w =
g ´ aC = gC a
ᆠ㡰 rdr w r = rgdh R
h
Þ w ò rdr = g ò dh
w
dh dr
w R = gh � R㦨�ᆠ鼀㡰 Þ
\h =
w R �w = g g
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EBD_8341 PS-54
Physics
ᆠ㡰 ᨨ 泐 ��鼀� 鼀 ᨨ �矄ᆠ kx
® mw l + x 㡰 ¬¾¾ m ¾¾
㦨 � ᨨᨨ�
ᨨ�ᨨ 矄 ᨨ 盐鼀
\ mw l + x 㡰 = kx
Þ x=
ml w
k mw
x mw = k l
b㡰 꾀 ᨨ � ᨨ矄��ᨨ x㡰 mv F � = = mr w r
\ kx = m l + x㡰w Q r = l + x 㡰 ├ 㦨鼀lw졔鼀 w
mlw k mw b㡰 N矄� q㦨鼀w r⏌㡰 \x=
�
�㡰
矄� q =
r⏌ ҈ = Þ q = ° r
Nᆠᨨ矄q㦨鼀�
w r ᨨ q㦨 g w r w r ҈ ᨨ 㦨 ᨨ 㦨 g g g \
ᆠ㡰
w㦨
r
犨� ᨨ 矄 Q �矄 ᨨ��� �矄ᨨᆠ�泐犨� 泐 ᨨ �턼 矄ᨨ�犨� � 盐 矄 ҈҈
҈
ᨨᆠ㦨
Þ w µ +҈
p w
�鼀�ᨨ µ ᆠ㡰 ᨨ⼄盐 � q
=
p p
和
҈
鼀 㡰
+҈
턼
ᨨᆠ�泐 v㡰ᨨ 矄ᨨ 㦨 dx⏌dt �盐矄 w㡰ᨨ 矄ᨨ�� 㦨 w,a
ᨨ鼀DSOP
x Þ h dx d f = h 矄ᆠ f ᨨ dt dt v㦨矄ᆠ� 맔҈ v㦨맔맔҈㦨鼀⏌矄
f㦨 Q Q
\
ᆠ鼀
턼
+҈
l t = Mg 矄� q 꾀矄ᨨ t㦨la l \ l a = Mg 矄� q
Ml l a = Mg 矄� q 矄� q la =g Þ
矄�
�
턼
҈
�鼀 �ᨨ =
䰘
��㡰
鼀w = ᨨᆠ µ Þ w µ
� ҈ 㦨 㦨 ҈�
鼀 �鼀盐鼀ᨨ�ᨨ 矄 ᨨ 盐鼀� ᨨ盐�ᨨ 矄 鼀� 盐 ҈ ω 鼀㡰 㦨 㦨 䰘 p ҈� T 黨 㡰 꾀t㦨 t = t = T �ᨨᆠ�泐犨� ᨨ T T 꾀t = t = �ᨨᆠ�泐犨�鼀 �鼀盐鼀 � 鼀� �盐 ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 vr 矄盐矄 t � w ҈ ҈ ᆠ㡰 ᨨ ᨨ ᨨ�ᨨ ᨨ �矄ᆠ R Q p ҈ p O t= ´ = w w x㜠ᆠᨨᨨ � ᨨ P㦨 wRT P pR > R 矄� �° = t㦨
ᨨᆠ 盐 矄 �ᆠ㜠 P 矄ᨨ ᨨ� ᨨ �> 턼 R矄� � ; 䰘 졠 盐�矄 ��矄矄矄 ⏌ 矄ᨨ� ω㦨
ᨨk >>mw
Þ
㡰 ᨨ 盐矄 ᆠᨨ鼀 ᨨ�ᨨ
é Ml ù êQ I rᨨd = ú û ë g 矄� q a= l
\ \ \ ҈ 㡰
x㦨h f v㦨 h矄ᆠf㡰w Q q + f = 黨° 桌
ᨨᆠᨨ ᨨᆠ├ᨨ �矄ᨨ㦨鼀w㦨鼀㦨鼀
\
df dt
ò = ᨨ
ò
w Þ = w -
=
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PS-55
System of Particles and Rotational Motion
\
= ò w
졠 矄ᨨ� � 犨�
ᨨ
\
=
w 㦨
w - w 졔 =
w ( + -w )
w k㡰
®
� 㦨鼀 Ù ᨨ w
y 㡰
�㡰
w é + -w ùû ë
O ¬
� = 鼀
\ \
q
´ ҈ = ⏌矄 � ´҈ � m = ҈├� ᨨ犨 盐矄� � �泐 ᆠᨨ 矄�ᨨ ᆠᨨ�矄�ᨨ
҈ l I w = mg ҈ - ᆠᨨ矄 q㡰 + mgl ҈ - ᆠᨨ矄 q㡰 ҈ æ � ö 黨v Þ ç ml ÷ = mgl ҈ - ᆠᨨ矄 q 㡰 ø �l è Þ mv = mgl ҈ - ᆠᨨ矄 q㡰 �´ 䰘 ´ = ҈ - ᆠᨨ矄 q Þ ҈ = ᆠᨨ矄 q ҈ ´ ҈ � ᨨ ᆠᨨ矄 q = \q ; ° � 㡰 ᨨ⼄盐t ᨨ盐B�矄 ᨨ⼄盐���盐鼀 \ TA ´ ҈ - mg ´ � - mg ´ � =
Þ TA ´ ҈ = ҈mg
vr
Þ TA = ҈ mg 矄�ᨨ � 矄� � A㡰
Ù w w - -w ) w (
TA
Ù 鼀 w w � = - -w ) (
�ᆠ鼀 �ᆠ鼀
®
®
ᆠ�ᨨ =
TB B
A
Ù Ù 鼀 w w - -w ) + 鼀� ⧤ (
Topic-3 : Torque, Couple and Angular Momentum ҈
x®
w -w 㦨 㦨 éë w - w ùû ��㡰 ᨨ鼀 �㡰 ��㡰
\
®
M㦨├�
ᨨ w =
® Ù Ù Ù æ ö æ Ù Ùö ᨨ = � + 鼀 ç ᨨ � ÷ ´ w ├+ 鼀 ç w ├´ � ÷ ´ w ├ è ø è ø Ù ® Ù Ù \ 鼀w � = � + 鼀 ᨨ wæç - ö÷ + 鼀w � è ø
盐 㦨
w
� v Þ mvl = ml w Þ w = �l
®
҈ b㡰
㡰 矄� � ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀
æ ml ö w mvl = ç ml + ÷ø è
ᨨ
+ -
ò
mg mg
F
㡰
ᆠ㡰 꾀 �盐鼀ᨨ鼀 盐鼀 L = I w R Ra 꾀 �矄
l
Mg
m
x
N f
a
ᨨ矄盐 F ´ R ³ Mg ´ x l⏌
m
m
l⏌
x=
R - R - a㡰 ᨨ鼀 ��盐
F鼀�
Mg æ R-aö = ´ R - R - a㡰 = Mg ҈ - ç ÷ R è R ø
æ l ö I = m 㡰 + m ç ´ + m l 㡰 è ÷ø
�
ᆠ㡰
ml + ml = ml 꾀 �盐鼀ᨨ鼀 盐鼀 L = I w = ml w =
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EBD_8341 PS-56
Physics
꾀ᨨ盐 ᨨ� 졠 �盐 鼀ᨨ鼀 盐鼀 � ��㦨 �
Þ
mV
´
҈ é mL mL ù =ê + ú´w ë ҈ û
V V = 䰘L 䰘 L b㡰 � 㦨 졔ᆠᨨ矄w1 泐㦨泐졔矄� w2 \w=
=
泐 = 泐㦨w2ᆠᨨ矄 w2㡰 泐 = 㦨w21ᆠᨨ矄 w1㡰 㦨泐㦨w22矄� w2㡰 꾀㦨 㦨 졔泐㦨泐 2 㦨w 1 ,泐 㦨 r r r r r ᨨ犨 t = ´ = 鼀 ´ 㡰 Þ 㦨w1矄� w1㡰
䰘
㦨 + 㡰 �ˆ+ 泐 ˆ桌 ´ 鼀 - w҈ ˆ�㡰 = + 鼀泐 w҈ ├ˆ r 㡰 ␔ �� r = t�ˆ - t ˆ
r r t = 㡰 = �ˆ - ҈ ˆ r r dr v = = �ˆ - tˆ ᨨᆠ�泐 dt r v t = 㡰 = �ˆ - ҈ ˆ r r r L = mvr 矄� qnˆ = m r ´ v 㡰 = �ˆ - ҈ ˆ 㡰 ´ �ˆ - ҈ ˆ 㡰 = -kˆ
㡰 꾀 �盐 ᆠᆠ�ᨨ w - w � ´ p - a= = 㦨҈p ⏌矄 t � t㦨ᔼa
æ�
ö æ�ö mR ÷ a » ç ÷ � ´҈- 㡰 ҈- 㡰҈p èø ø � 㦨맔҈ 鼀 黨 㡰 ҈ ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ 犨ᨨ├㜠 �泐 ᨨ鼀 Þ t㦨 çè
鼀� 㦨
҈ ҈ 鼀 鼀 꾀
� 㦨 꾀 맔҈맔҈㦨 �
ÞvB㦨҈�鼀⏌矄 꾀 �盐 鼀ᨨ鼀 盐鼀 ᨨ盐 O LO㦨 mvr 㦨 맔҈맔 LO㦨 ├�鼀⏌矄
r ˆ ҈҈ 㡰 � ҈ = -ˆ�㡰 + - 㡰 r ҈ = �ˆ + ˆ
ᨨ⼄盐 盐 ᨨ ҈ ᨨᆠ æ r r r ˆ ö ˆ t ҈ = ҈ ´ ҈ = ˆ ´ çç -ˆ�㡰 + - 㡰 ÷÷ = ├㡰 è ø
ᨨ⼄盐 盐 ᨨ ᨨᆠ r ˆ ´ ├ˆ = �ˆ + - ˆ㡰 t = �ˆ + 㡰 r r r ˆ t = t ҈ + t = �ˆ + -ˆ㡰 + ├㡰 ˆ = �ˆ - ˆ + ├㡰 r r r ҈ 㡰 ᨨ⼄盐ᨨ盐 ᨨ��� 㦨 t = ´ 㦨 矄� q Þ �㦨҈맔�矄� q ҈ p 矄� q= � = Þ q= ҈ 㡰 ᔴᨨ 矄� 矄�ᨨ �盐矄 �ᆠ├ 矄矄
ᨨ⼄盐 盐 ᨨ �ᆠ�ᨨ ᨨ �矄 矄� ᨨ⼄盐㦨 å ᨨ⼄盐 ᨨ � �
ò t =
ò
m p p
Ҙ · Ҙ t= ҈ 㡰 꾀泐� � ᨨ⼄盐 ⼄盐�ᨨ ᨨ盐 ᨨ� t 㦨ᔼa㦨㝐 l㡰졔�㝐l桌a Þ �㝐�l 㝐�l㦨㝐 l㡰졔�㝐l桌a Þ t=
Þ 㝐�l 㦨 ҈㝐�l㡰a
� ҈l ҈� 㡰 � ᨨ �矄 ᨨ 盐 � ᨨ鼀 鼀矄矄 矄�泐 꾀㦨ᔴ 鼀矄矄ᨨ ᨨ ᨨ �矄鼀 ᆠ� � ᨨ⼄盐 ᨨ盐 � � ᨨ� 鼀� ᔴ҈㡰㦨鼀� ᔴ㡰 \ a 㦨
æL ö æL ö mg ç 矄� q ÷ = mg ç ᆠᨨ矄 q - L 矄� q ÷ è ø è ø mgL Þ mgL 矄� q = ᆠᨨ矄 q 矄� q ҈ ҈ Þ 㦨 ᨨ q㦨 ᆠᨨ矄 q ҈ 㡰 ҈䰘 㡰 ␔├ ᨨ犨 㦨mvr^ 泐
꾀
⏌
졠
ᔴ
ᔼ ᨨ ᨨ ᆠ矄矄 �ᆠ盐
æ R ö + a÷ �矄 ᆠ ^ �矄 ç è ø
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PS-57
System of Particles and Rotational Motion ҈
㡰 泐�ᆠ⼄盐���盐鼀
ᆠᨨ矄
졔矄� 㦨҈� g Þ㦨 … �㡰 泐 ᨨ� ᨨ ⼄盐���盐鼀 㦨ᆠᨨ矄
҈ N= 㦨
ᨨ⼄盐 ᨨ盐꾀
矄�
x
h
l⏌ ҈�
҈�맔꾀㦨맔 꾀 f l g l ´ x \ = x … �㡰 ҈�맔 ᆠᨨ矄 ° =
h x ᨨ鼀⼄ �㡰 ��㡰
\ x=
盐ᆠᨨ矄 㦨
҈黨
�
h … ��㡰 ᆠᨨ矄 °
l h h = \ = ᆠᨨ矄 ° l ҈ 㡰 꾀 �盐 鼀ᨨ鼀 盐鼀 L 㦨 鼀 矄� 黨 㦨 맔 맔 ҈ 맔 ҈ 맔 ҈ 꾀矄V 㦨 rw 和� 黨 㦨 ҈桌 和ᨨ L 㦨 ҈ ├�鼀⏌矄
ᨨ 鼀
㡰 ᨨᆠ├�矄� ⼄盐���盐鼀 ᨨ 矄�ᨨ ⼄盐���盐鼀 SFx㦨 Þ F㦨N SFy㦨 Þ f㦨mg
y
f a
鼀
x
҈鼀
a C
b
졠
b㡰 P ᆠ㡰 ᨨᨨᆠ�ᨨ 矄 ᨨ 鼀 �盐 鼀ᨨ鼀 盐鼀 ᨨ P mass mᨨ盐O �.e., Lᨨ�矄mrw �矄 �ᆠ ᨨ犨 졔 z �ᆠ�ᨨ �盐 鼀ᨨ鼀 盐鼀ᨨ 鼀矄矄mᨨ盐P l � Lp �矄 mvl �矄 �ᆠ ᨨ �� ᨨᆠ�ᨨ ᨨ m 矄 矄 ᨨ犨 � ��盐 ᨨ O v � ᨨᆠ�ᨨ ᨨ m r �ᆠ�ᨨ ᨨ P 鼀� 矄 m r O ᆠ �� � b㡰 꾀矄�ᆠ�矄ᨨ� �� ᆠ�ᆠ盐 � 矄 �矄 ᆠ矄� � L �矄 ᨨ ᆠᨨ 矄 � 鼀� �盐 和� ᆠ �ᆠ 矄 犨ᨨ ᆠᆠ�ᨨ 矄 ac at ᨨ
ᆠᆠ�ᨨ �矄ur ᨨ ᨨ犨 矄 ᆠ
�ᆠ�ᨨ ᨨ L 鼀� 矄 矄鼀 犨 矄 ᆠ矄矄
F
N
ᆠ㡰 ᨨ⼄盐犨ᨨ├� �ᨨ ᨨᨨ 鼀矄矄鼀�矄t㦨mg맔l矄�
q �ᆠ�ᨨ ᨨ ᨨ ᨨ�ᨨ ᨨ �ᆠ㡰
mg
l 鼀
҈
q
ᨨᨨ�ᨨ ⼄盐���盐鼀 Stc㦨
ᨨ⼄盐ᆠ 泐 �ᆠ�ᨨ ᨨᆠ f 㡰ᨨ盐 C㦨 f맔 a �
ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
�矄 ᨨ ᨨ⼄盐 犨 �ᆠ 矄 ᨨ盐 ᆠᨨ盐 �矄 ᨨ⼄盐
ᨨ鼀 ᆠ�ᨨ N ᨨ ᨨᆠ├ 犨� ᆠ ᨨ⼄盐 N맔b� �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 和盐ᆠ f맔a㦨N맔b ᆠ N 犨� ᨨ 盐ᆠ ᨨ⼄盐
l 鼀�
r 꾀矄 t �矄 �ᆠ盐 ᨨ L �ᆠ�ᨨ ᨨ L ᆠ �矄 盐 鼀� �盐 鼀� 矄 矄鼀 ᆠ㡰 � 鼀㦨҈├� q 㦨 㦨҈鼀⏌矄 ® ® 꾀 �盐鼀ᨨ鼀 盐鼀 = ´ 鼀 㦨鼀ᆠᨨ矄q é 矄� q ù 矄� q = ᆠᨨ矄 q ê ú 㦨 � ûú � ëê
҈2 ´ 矄� °´ ᆠᨨ矄 ° ´҈ 㦨├�鼀⏌矄 㡰 꾀 �盐鼀ᨨ鼀 盐鼀㦨鼀 맔㡰 æ ö 㦨 ├� ç ´ ÷ = ├� ´ �� - ˆ㡰 è ø 㦨├� ├ˆ 㡰㦨├�맔맔鼀矄҈ ├ˆ 㦨 ├ˆ
䰘
㦨
黨
b㡰 KE㡰ᨨ�ᨨ 㦨
L I
L㦨ᆠᨨ 矄 \ KE㡰ᨨ�ᨨ 맔 I 㦨 ᆠᨨ 矄 ᆠ 犨 I �矄 ᨨ盐 KEᨨ�ᨨ ᆠᨨ鼀矄 㡰 ᨨᆠᆠ� �ᨨ �ᆠ盐 �ᨨ� �盐 � ᨨ鼀 ᆠ�ᆠ盐 鼀ᨨ�ᨨ �矄 ᆠ � ᨨᆠ ᆠᨨ犨 矄 ᆠ ᨨ ᆠ�ᆠ ᨨ⼄盐 盐ᨨ �矄 ᨨᆠᨨ盐 ᆠ �矄 ᨨ ur ᨨ �盐鼀ᨨ鼀 盐鼀 L ᨨ盐ᆠ �矄ᆠᨨ 矄 㡰 꾀泐� �ᆠ �� �盐鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 㦨 �盐�鼀盐矄�� ᨨ 矄泐矄鼀ᨨ盐 ᆠ ᨨ 鼀矄矄 꾀 �盐 鼀ᨨ鼀 盐鼀㡰f 꾀 �盐 鼀ᨨ鼀 盐鼀㡰� L 㦨Mv맔 㦨IC w �㡰 w �矄 �盐 ᨨᆠ�泐ᨨ ᨨ 㝐ᨨ鼀 ᨨ ᔼ � ᨨ 矄泐矄鼀 ᨨ盐 �矄 �矄 ᨨ ᨨ�ᨨ
ᆠ ᨨ 鼀矄矄 ᨨ 矄泐矄鼀桌
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EBD_8341 PS-58
Physics
æ Lö
æ Lö
IC㦨 M ç ÷ + M ç ÷ = è ø è ø
ML ML =
ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 鼀㡰㦨Iw ҈ 맔�맔黨맔�㦨 맔�맔 �㡰맔w \w㦨 矄҈ 㡰 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 I҈w҈㦨ᔼw
� 㡰
V MVL æ ML ö =ç ÷ w Þ w㦨 L è ø b㡰 �矄 ᨨ ᨨ⼄盐 C �盐 鼀ᨨ鼀 盐鼀 鼀� 矄 ᆠᨨ 矄 꾀矄 鼀ᨨ鼀 ᨨ � � � ��泐 犨 ᨨᨨ�矄 鼀ᨨ矄 B ᨨ鼀 꾀 ᨨ ᆠ矄矄
� ᆠ矄矄 犨 ᨨᨨ�矄 鼀ᨨ矄 ᨨ鼀 ᨨ ᔴ 矄ᨨ w 犨� � ᆠ矄 A � ��泐 ᆠ矄矄和ᨨ ��ᨨ ᨨ w�矄 ᨨ � ҈ b㡰 盐ᨨ� ᆠ矄� �ᆠ盐 �矄 ᆠᨨ 犨� �矄 ᨨ ᨨ�ᨨ MI ᨨ盐 �矄 ᨨ ᨨ�ᨨ �矄
ᨨ ᆠᨨ 矄 和ᨨ I � ᆠ矄矄
�矄 ᨨ ᨨ⼄盐�矄 ᆠ� � dL \t 㦨 Þ L㦨ᆠᨨ 矄 Þ Iw㦨ᆠᨨ 矄 dt 盐 I � ᆠ矄矄 矄ᨨw ᆠ矄矄 ᆠ㡰 ␔ �矄ᆠ ᨨ矄 � 矄 犨ᨨ 泐矄 ᨨ 鼀ᨨ�ᨨ � 矄�ᨨ ᨨ�ᨨ ᨨ 犨ᨨ 泐矄 ᨨ �盐鼀ᨨ鼀 盐鼀 ᨨ �盐 鼀ᨨ鼀 盐鼀�矄 ᆠᨨ 矄盐鼀 ᨨ 矄 犨ᨨ y ur ur ur L = LT + L R LT㦨 �盐鼀ᨨ鼀 盐鼀 w 盐 ᨨ 矄�ᨨ 鼀ᨨ�ᨨ R LR㦨 �盐鼀ᨨ鼀 盐鼀 v 盐 ᨨ ᨨ�ᨨ 鼀ᨨ�ᨨ
C ᨨ盐 C.M L㦨MV맔R졔Iᆠ鼀w x O \
҈ 㝐 ´ ω҈ I҈ω҈ \ ω = = ᔼ ì҈ ü í 㝐 + 鼀 桌ý î þ � M㦨�├�R㦨mw҈㦨҈ᨨ ⏌矄 m㦨�├� r㦨m
҈ ´ � ´ ´ ´ ҈
=
㦨 㦨 ⏌ 矄 ҈ ´ � ´ ´ + ´ � ´ ´ 桌 졔҈ 䰘 㡰 ᆠ ᨨ 鼀矄矄 ᨨ � 泐矄矄泐矄鼀 ᨨ��� ἘOἘ�矄 ᆠ犨 AO㦨x OB㦨 dx㡰 졠 꾀 x d㜠x ᆠ鼀 ҈҈Ms Ms ᨨ���
d M s - x㡰 + ҈҈M s d - x 㡰 = M s + ҈҈M s �d ᨨ� 泐矄矄泐矄鼀 �盐矄 wᨨ盐 ᆠ ᨨ 鼀矄矄Ἐ졠Ἐ�矄矄鼀 ᨨᨨ 矄矄
Þ㦨Ms 㡰졔҈҈Ms d x㡰 Þ x =
Iᆠ鼀㦨M.Iᨨ盐ᆠ ᨨ 鼀矄矄C V㦨Rw
҈ 㦨M Rw㡰R졔 MR w 㦨 MR w
\
Y
b㡰 꾀 �盐 鼀ᨨ鼀 盐鼀
ur ur L = r^ ´ p
v
M
w
꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 㝐 y ᆠᨨ 矄 ᆠ �矄 �盐鼀ᨨ鼀 盐鼀犨ᨨ��� 鼀� 矄 ᆠᨨ 矄 ᆠ㡰 和� ᆠ 犨ᨨ ᨨᆠ矄 ᆠ � 泐ᨨ � � ᨨ ᨨ ᨨ⼄盐�矄ᆠ� �ᨨ 矄泐矄鼀 ᆠ �盐 鼀ᨨ鼀 盐鼀 鼀� 矄 ᆠᨨ 矄 w1 w2
m
r
m
I㦨Mr I㦨Mr 졔mr \ I҈w҈㦨Iw Þ Mr맔w҈㦨 Mr졔mr㡰w (Q w҈ = w ) w㦨
Mw M + m
㦨
㡰
M, L㡰
X
O
\
ædö ҈҈ M s ç ÷ ´ w è ø
y
L㦨Mv맔y
LTᨨtal = LB
æ �d ö ædö M s ç ÷ w + ҈҈ M s ç ÷ ´ w è ø è ø
m
v
ᨨᆠᨨ�矄�ᨨ 꾀 ᆠᨨ�矄�ᨨ
矄� �� ᆠ�ᨨ ᆠᨨ 矄�ᨨ ᨨ �盐鼀ᨨ鼀 盐鼀犨 r r L� = L f Þ mvL = I w
æ ML ö Þ mvL = ç + mL ÷ w è ø æ 黨 ´ ҈ ö Þ ҈ ´ ´ ҈ = ç + ҈ ´ ҈ ÷ w è ø
æ ҈ö Þ= ç + ÷wÞ= w è ҈ ҈ ø ҈ Þ w = ⏌矄ᆠ
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PS-59
System of Particles and Rotational Motion 黨
I҈w҈ = I w
I҈ = I M
␔ m�矄 鼀矄矄ᨨ 矄ᨨ w�矄 �盐ᨨᆠ�泐 � 矄 ᨨ �鼀t L 꾀矄ᨨL㦨Iw Þ w = ҈ 盐� � �矄盐ᨨ w�矄⼄ �㡰
黨㡰 M㦨├�m㦨├� 矄� � ᆠᨨ 矄�ᨨ ᨨ �盐鼀ᨨ鼀 盐鼀L� = Lf
M
æ M R ö + I m 㡰 = ç + mR ÷ ç ÷ è ø
m
҈ I = M R w҈ = � 鼀 æ M R ö � \w = çç + mR ÷÷ ´ è ø M R �R + ҈㡰 = 黨 鼀 = ´ ҈ R � ᆠ鼀㡰
҈
鼀�
dx dx = - Aw 矄� wt Þ = - Aw ᆠᨨ矄 wt dt dt \㝐� �盐 ᨨ 鼀 �鼀盐鼀ᆠᆠ�ᨨ ᨨ ᨨ鼀 \ 㝐 ᆠᆠ�ᨨ 㦨Aw ␔ ᨨ鼀 鼀ᨨ矄 ᨨ⼄盐 ᆠ矄 ᨨ ᆠ泐� ᆠ泐� ᨨ矄 ᨨ盐 �矄 �矄 f 꾀ᆠᆠ�ᨨ ᨨ ᆠ泐� a҈ = m
ᨨ⼄盐 t 㦨 f R 㦨 Ia fR fR = Þ a = I MR ⏌
҈ᆠ鼀
ᔼ ��泐
㝐� 㝐� = 黨 黨
ᨨ⼄盐ᨨ盐 ᆠ tᆠ 㦨 � �㝐� \ ҈ �㡰㦨 㡰 = 㝐� = 黨 黨 �㦨 졔 㦨㝐�
=
҈ 㦨 m⧤
҈ 㦨m和
҈ 㦨
҈ 㦨
⧤
⧤
f f ᨨ Ra 㦨 㦨a㡰 MR M \ ᨨᨨ鼀 � ᆠᆠ�ᨨ f f f + = a鼀 㦨 a҈ + a 㦨 M M M f MAw ᨨ Aw 㦨 Þ f㦨 M
盐矄鼀 �鼀盐鼀ᨨ⼄盐
⧤ 㦨 Þ 㦨 ҈ 㦨� = 㡰 Þ = � ᨨ犨 㦨 �矄 ᆠ犨 �� � �矄ᨨ矄 � � 矄矄 鼀ᨨ� � ᨨ⼄盐 ᨨ盐 ᆠ ᆠ 㦨 桌 㦨 㡰 㦨 ⧤ Þ 㦨 ҈ 㦨 㡰 = �
҈
҈ = � ᆠ鼀 �
mv 㦨mrw r
㦨
MAw R ҈ 㦨 MARw �矄 ᨨ ᨨᆠ ᆠ ᨨᨨ⼄盐�矄� �盐 鼀ᨨ鼀 盐鼀 鼀� 矄 ᆠᨨ 矄 �.e.錐 l҈㦨 l \ I҈w҈㦨Iw ML ´ w M w Iw ҈ \ w㦨 ҈ ҈ = 㦨 M + m I ML æ Lö + m ´ ç ÷ è ø ҈ 矄 ur I҈w҈㦨Iw Q L = ᆠᨨ 矄 㡰 t鼀 㦨
f맔R㦨
҈ I҈㦨 MR
҈ ҈ M æ + ҈ö � MR + R =ç MR = MR ÷ è ø w҈㦨w ; w㦨䁸 ҈ MR ´ w I҈w҈ =+ = w= w w㦨 � I ´� � MR
I㦨
泐� 矄 ᨨ �鼀 t �盐矄 ᨨ ᨨ� ᨨ 矄盐 ᆠ㦨r T㦨
a
ᨨ
㦨 㦨 ⧤ ᔼ 㦨 �矄 ᆠᨨ � � ᨨ鼀ᆠ 犨 �� � � 矄矄 鼀ᨨ� � t 㦨㡰ᨨ盐ᆠ Þ 㦨 㡰
\ =
m
æ L ö = ´ r Þ T = ç ÷ r - è mø mr 㡰 mL
Þ
黨ᆠ鼀
T
æ L ö = A ᆠᨨ 矄 ÷ çè ø m ᔴᨨ 矄� � � 鼀ᨨ�ᨨ ᨨ ᨨ鼀 x㦨Aᆠᨨ矄wt �� 㡰
ᆠ鼀
�
I
r
ᨨ = Ar - \n 㦨
T㦨
mrL
w
�㡰
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EBD_8341 PS-60
�
Physics
ᆠ 㡰 ᨨ鼀 �盐 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ ᨨ盐 �ᨨ ᨨ�
䰘
꾀泐� � F =
�ᨨ ᨨ�
æ 鼀 ö 鼀 = ç + 鼀 ÷ w ç ÷ è ø
bᆠ㡰� ᨨ � �泐 㡰㦨⧤⏌
ᨨr㦨RF㦨kR
dv d é kr ù = ê ú = kr dr dr êë ûú
mv Q �ᆠ�矄鼀ᨨ� �� ᆠ�ᆠ盐ᨨ�㡰 R mv k = kR Þ v = ´R ᨨ R m
꾀矄ᨨ F =
꾀矄 ᆠᨨ鼀� 矄泐矄鼀 ᨨ矄 犨� �盐 矄 w ᨨ盐 �ᨨ桌 鼀 鼀 \w = = 鼀 + + 鼀 w= + ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ w = ᨨ鼀 �鼀盐鼀 �盐ᨨᆠ�泐
盐
æ k ö R÷ R = kmR ÷ m è ø
꾀 �盐 鼀ᨨ鼀 盐鼀 L = mvR = mçç
ᆠ㡰 ᔼ ᨨᆠ 㡰�矄 � � � t 鼀� 矄ᆠᨨ 矄 t 㦨 맔 ᔼ ᨨᆠ�矄� ᨨ鼀ᨨ 矄 犨 矄ᆠ�鼀矄 �ᆠ盐 �矄 ᆠ ᨨ ᨨᆠ ᨨ鼀 턼 犨� �ᨨ ᨨ ᆠ �� � � ��泐 �ᆠ盐 �矄 턼㝐 � ᆠᨨ鼀矄 턼㝐Ἐ
ö æ ç + ÷ = ÷ çè ø
Þ
x
+ = Þ x =
㝐¢
和ᨨ ᨨ�ᨨ ᆠ㡰 �矄 ᆠᨨᆠ
턼
\ 鼀 =
w鼀 =
+
=
=
æ ö + ç ÷ è ø ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ ur r ᆠ㡰 � F = at�ˆ + bˆ ᨨ F = t�ˆ + ˆ r md v \ = t �ˆ + ˆ dt r \ dv = t dt �ˆ + dt ˆ \
v
r
黨
Q a = 1NS -҈ b = 1N 㡰
ᔼ ᨨᆠ 㡰 �矄 � ᨨ鼀 ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t �矄 ᨨ ᔼ ᨨᆠ 㡰�矄� � �泐ᨨ� 和
t = F ´ R 犨 ᆠ�鼀矄 ᆠ 㡰 ᨨᆠᆠ� �ᨨ ᆠᨨ鼀 㦨\ 㦨 ᨨ ᨨᆠ ᆠ� �� �ᆠ �ᆠ�ᨨ 犨�鼀ᨨ 鼀� ᨨ� ᨨᆠᨨ鼀 ᨨ �ᆠ泐 ᨨ犨 犨 矄 Ӹ 㡰
꾀
Q m = ҈桌
t
t
ò dv = ò t dt �ˆ + ò dt ˆ
黨q
r t v = �ˆ + tˆ ҈ r ҈ -҈ 꾀 t = ҈s v = �ˆ + ˆ = �ˆ + ˆ 㡰鼀矄 r r dr t ˆ ˆ = � + t \ drr = t dt �ˆ + t dt ˆ 꾀矄ᨨ v 㦨 dt ᨨ
r
r
㝐
t t
t
r
t
t
ò d r = ò dt �ˆ + ò t dt ˆ \ r = �ˆ + ˆ
r ҈ ҈ 꾀 t = ҈ r = �ˆ + ˆ
r \ r =
r r r æ҈ˆ ҈ t=r ´F =ç � + è
r ҈ ᨨ t = - kˆ
\
҈ ҈ ҈ + =
ˆ ö÷ ´ �ˆ + ˆ 㡰 t㦨҈矄㡰 ø
r ҈ t = 鼀
泐㡰
q
⏌
L 矄� q
q
⏌ 鼀�
L ᆠᨨ矄 q
␔ 鼀├矄 �q �� ᨨ �矄 ᆠᨨ鼀 L 㦨 ᆠᨨ矄 q \ �矄ᆠ鼀 ᨨ �矄鼀� ᨨ� ᨨ鼀 � ��ᨨ矄��ᨨ
L L L - ᆠᨨ矄 q 㦨 ҈ - ᆠᨨ矄 q㡰 ᔼ 矄 ᨨ盐矄 ᨨ⼄盐 ᨨ盐 ᨨ� ᨨ ᆠᨨ ᆠ L t = mg ´ 矄� q
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PS-61
System of Particles and Rotational Motion ᨨ犨 x =
�
é ma m a㡰 ù ҈䰘 LCM = I CM w = ê + ma w úw = ëê ûú 㝐� �盐 ᨨ �盐鼀ᨨ鼀 盐鼀 ᨨ ᔴ㝐 ᨨ 矄泐矄鼀ᨨ盐 ᨨ� O.
L 矄� q y = L 矄� 黨° - q㡰 = L ᆠᨨ矄 q
x y æ x ö æ y ö \ç ÷ + ç ÷ = ҈ ᨨ + = ҈ è L ø è Lø L L
盐矄 ᨨ 꾀�矄 �矄 ᨨᨨ b㡰 r � r = at �ˆ + bt ˆ
黨w 黨 ҈鼀w ҈ 鼀w ´ = = � � � � 㝐� �盐 ᨨ 㜠ᆠᨨ鼀ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 ᨨ盐 ᨨ� Lz = L'ᆠᨨ矄q Lᔴ㝐 矄� q ¢ = �鼀 ´
r ҈ r = t �ˆ + �t ˆm r r dr = ҈t �ˆ + ҈tˆ ms ҈ \ v= dt r r dv a = = t�ˆ + ҈ ˆms - dt r ҈ 꾀=҈矄 rt =҈ = �ˆ + � ˆm 錐 r r - vt =҈ = ҈�ˆ + ҈ ˆ ms ҈ at =҈ = �ˆ + ҈ ˆ ms r pt =҈ = �ˆ + ˆ kgms ҈ ˆ kˆ �ˆ r r r L = r ´ p = ҈ � = kˆ é ҈ - �ù = - � kˆ ├�鼀矄 -҈ êë úû
҈ mwa ҈䰘 ҈ ´ - ma w ´ � � � � ҈黨 ҈䰘 æ ö = ma w ç è � ҈ ÷ø � 㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ �盐 鼀盐鼀 盐鼀 ᨨ盐 �矄 =
MR맔w㦨MR맔 和ᨨ� � 犨 � Þ �
҈ ҈ r r F = ma = �ˆ + ˆ 㡰 N ˆ kˆ �ˆ
�҈
P
Þ �
l
ᆠᨨ矄 q =
a
q
O
w
l l �
҈ l +a
=
a
a + a
=
�
éQ l = ë
a �� ùû
OA㦨 l 㡰 + a㡰 = 黨a + a = ҈a ᆠᨨ鼀 矄泐矄鼀 ᨨ矄 ᨨ盐 z㜠 �矄 犨� ᆠᨨ 矄 �盐 ᨨᆠ�泐 w¢
w' p a㡰 w = Þ w' = � w p ҈a㡰 㝐� �盐 ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 ᨨ盐 �矄 ᆠ ᨨ 鼀矄矄
\
㡰 �
r
A
҈ q =
å
f҈㦨 m҈҈
N҈
m҈
f㦨 m
鼀� r Fext = r d p矄泐矄鼀
r 㦨 Þ p矄泐矄鼀 㦨ᆠᨨ 矄 dt � � 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 ᨨ矄 ᨨ ᆠ � �
�鼀 盐ᨨ� ᨨᆠ矄ᆠ� �� 矄泐矄鼀 ├� �ᆠ ᨨ � �泐鼀泐ᆠ � 犨� �鼀 꾀矄ᨨ ᨨ ᨨᆠ 鼀泐 ᆠ ᨨ⼄盐 盐矄 ᨨ⼄盐犨�ᆠ � �盐鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀�
�鼀 �� bᆠ㡰 ur uur æ r dL ö r ur ur dL ur ur = A´ L � t = A ´ L Þ çQ t = dt ÷ è ø dt uur dL ᨨ鼀 ᆠᨨ矄矄㜠ᨨ 盐ᆠ 盐 �矄 �ᆠ盐 ᨨ ᨨ dt ur ur A L r ur ᨨ鼀 ᨨᨨ 盐ᆠ盐 L L = L � �� � 犨� 矄ᆠ ᨨ �鼀
å Fext =
a
黨a �
ᆠ 㡰␔ m҈¹ m¹ ᨨ� ᨨ ⼄盐���盐鼀҈㦨m �ᆠ⼄盐���盐鼀鼀�㦨졔m1҈ 和ᨨ� � ᨨ ⼄盐�ᨨ 矄 犨 �
l 鼀� ´ ᆠᨨ矄 q = ҈ ´ l 矄� q
m m
R �
鼀� N m mg ҈ + m҈m q ␔ m҈㦨 錐 ᨨ⼄盐ᨨ盐
m 㡰 + m l 㡰 l = m + m �
w'
㦨
=
r r r ҈ é҈ ù - ˆ t=r ´F = � = kˆ ê - ҈ú = k 鼀 ë û ҈ ᆠ㡰ᨨ矄��ᨨ ᨨ ᔴ㝐 ᨨ �矄ᨨ ᨨ xᔴ㝐 =
w M 黨 R w M w 졔 맔 맔 졔 r 맔 黨 黨 黨 �
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EBD_8341 PS-62
Physics
uur uur uur ur dL ur dL ur dL dL dL = L L× + L× = L Þ L × dt dt dt dt dt ur dL L^ 和� ᆠ dt uur r dL dL = \ L × = Þ dt dt Þ L 㦨 ᆠᨨ 矄
盐矄 r 鼀� �盐 ᨨ L 犨泐矄 鼀� 矄 ᆠᨨ 矄 꾀矄 A �矄 ᆠᨨ 矄 ᆠᨨ � �矄 犨泐矄 �ᆠ盐 r ᨨ t r r 꾀矄ᨨ L �矄 �ᆠ盐ᨨ A r r rr Q L ^ A \ L A = r ᆠ � ᆠ ᆠᨨ ᆠ盐 ᆠᨨ鼀ᨨ ᨨ L ᨨ � r A �矄 ᨨ�ᆠᨨ 矄 � b 㡰꾀 �盐鼀ᨨ鼀 盐鼀 L = r ^ ´ P 꾀 �盐 鼀ᨨ鼀 盐鼀 ᨨ盐 ᨨ� O mv ´h L㦨 �㡰 vᆠᨨ矄� h
v
꾀 �盐鼀ᨨ鼀 盐鼀L㦨mVy x Vx y桌 ᨨ
év æv v ö v ù ö æ v ´ ç - v ÷ - ç - ÷ ú L㦨m ê ø è g g ø ú êë g è û
é v v v v ù - - + ú L㦨m ê g g g ûú ëê g - mv mv é ҈ ҈ ù 㦨 ú g êë û g ur r ur 꾀�� L = r ´ p ᆠ �ᆠ�ᨨ ᨨ L �矄 �ᆠ盐 ᨨ ᨨ 鼀ᨨ�ᨨ �矄 �ᆠ 犨泐 ᨨ鼀 L㦨
Topic-4 : Moment of Inertia and Rotational K.E. ҈
㡰 ᨨᨨ犨�ᆠ㜠ᆠ鼀ᆠᨨ ᆠ 矄矄盐鼀矄矄矄 ᨨ �矄ᆠ矄 � � � �盐矄 矄ᨨ
I = ò dI = ò dm r 㡰
�㡰
R
v Ö
h
�
r
H
O
꾀矄ᨨh㦨
v 矄� q v = g g
Q q = �°桌
q
��㡰
ᨨ鼀⼄ �㡰 ��㡰 L㦨
m
ᨨ鼀 ��鼀
gh 㡰 h = m gh
r R R = q = ᨨ r = h h H H
v mv ´ = 꾀矄ᨨ L㦨 g g mv
�䰘
㝐矄矄 ᨨ 鼀 dm = r pr 㡰dh ᨨ鼀⼄ �㡰 ��㡰 ���㡰
v 꾀 �鼀 t 㦨 x y㡰 ᆠᨨᨨ � 矄 ᨨ
꾀ᨨ 鼀 dA = prdl = pr
g �ᆠ vxvy ᆠᨨ鼀ᨨ 矄ᨨ ᨨᆠ��矄 �鼀 ᨨᆠ�泐� ᨨ� ᨨ �ᆠ�ᨨ
v
o
vx㦨 v ᆠᨨ矄 � =
Ö
v
I = ò dI =
v
x
Ö
꾀泐� �v㦨u졔� �ᆠ �ᆠ�ᨨ
H
H
H
æR ö ò dm r 㡰 = ò r pr 㡰dh çè H × h÷ø
æ æR
ö ö ÷ dh ø
=
v v ´ = ᔼ ᨨ� ᨨ �ᆠ�ᨨ x = vx ´ t 㦨 g g ҈ 2 ᔼ �ᆠ �ᆠ�ᨨ 泐� �S㦨ut졔 at
和ᨨ� �犨� I =
)
v
v v v ҈ v ´ - g = - y㦨 g g g g v
ò r çè p çè H × h÷ø
æv ö v v y = v 矄� �o - g ç ÷ 㦨 v - g ´ v 㦨 - v è gø g
(
dh ᆠᨨ矄 q
R R + H ᆠᨨ矄 q
r = h q 㡰
졠
���㡰
Mh dh
㝐矄矄 ᨨ 鼀 dm =
泐
�
��㡰
MR dm
b㡰 A
B x
dx 㝐矄矄 ᨨ 矄鼀鼀 ᨨ ᨨ
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PS-63
System of Particles and Rotational Motion
y
dm = l × dx 㝐ᨨ鼀 ᨨ � �ᨨ 矄鼀鼀 xö æ dI = dm × x = l ç҈ + ÷ × x dx è Lø 㝐ᨨ鼀 ᨨ � �ᨨ ᆠᨨ鼀ᨨ ᆠ ᨨ� 泐 � ��ᨨ
҈ ᆠ鼀
O
= l
L
䰘l L ҈ 㝐矄矄 ᨨ � ᨨ ÞI =
L
L
æ M = ò l dx = ò l ç҈ + è
IO Ἐ = IO + M �㡰
éL L x x + = l ê + ú L û ë
ù
�㡰
l L xö ÷ø dx = L
M \ l = L 䰘 æ M ö 䰘 ML \I = ç ÷L ÞI = ҈ è L ø ҈ 㡰 泐 �盐 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ Lc = L f w wI + I ´ = I w Ἐ Þ w Ἐ = ҈ w KE 㡰� = I w ҈ KE 㡰 f = I + I 㡰w Ἐ
DKE =
dx
OἘ dmR dI = + dmx n= - L ⏌ dmR M Þ I = ò dI = ò + ò dx ´ x L n= L ⏌
I I
҈ ҈ Iw - Iw = Iw
Iw DKE = = \ ᆠ�ᨨ ᨨ矄矄� ⧤쟼 = ҈ KEl� Iw ᆠ㡰 p 矄�泐ᨨ �矄ᆠ矄 t�矄 �ᆠ├ 矄矄 ᨨ �矄ᆠ矄 㝐ᨨ鼀 ᨨ � � ᨨ �矄ᆠ �矄 �� 泐
MR r pR 㡰t 桌R = 꾀矄r t矄鼀㡰 I µ R
I=
�
I æ R ö ҈ = Þ = a Þ a = I҈ çè R҈ ÷ø ҈ b㡰 㝐ᨨ鼀 ᨨ � � ᨨ ᆠ �盐 矄 ᨨ盐 �矄 矄矄� � ᨨ盐� O M M a + b 㡰 = 㡰 + 㡰 桌 IO = ҈ ҈
MR ML + ҈ M V ML MV ML ÞI = ´ + ÞI= + pL ҈ pL ҈ dI mV M ´ L =+ = ҈ dL pL Þ V = pL Þ pR L = pL L \ = R b㡰 ᔼ �� �盐鼀ᨨ鼀 盐鼀 = I҈w҈ + I w w �盐 矄 ᨨ ᆠᨨ鼀� 矄泐矄鼀 � �盐鼀ᨨ鼀 盐鼀 = I҈w + I w 꾀ᆠᆠᨨ � � ᨨ ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 I҈ + I 㡰w = I҈w҈ + I w � I =
I w ҈ æ wö ´ I 㡰 ´ ç ÷ = è ø
M + 㡰 IO ҈ ҈ = = M IO Ἐ + 㡰 + M �㡰 ҈ ᆠ㡰 ᆠ泐� ᨨ 鼀矄矄 m � L �盐矄 Rᨨ犨├鼀 泐 �矄ᆠᨨ �盐矄R �ᆠ├ 矄矄dx �矄 ᆠᨨ x ᨨ鼀 �矄 OOἘ 鼀ᨨ鼀 ᨨ � � ᨨ盐OOἘ ᨨ �矄 鼀 O
R
=
ᆠ鼀
x ᆠ鼀 ᨨ鼀 �矄 ᨨ鼀鼀ᨨ鼀 ᨨ � �ᨨ盐O Ἐ
æ x ö I = l ò ç x + ÷ dx Lø è L
O'
䰘
I҈w҈ + I w ҈ ´ ҈ + ´ � = = I҈ + I ҈ + � ᨨ�ᨨ ├� �ᆠ �泐 Þw=
҈ ҈ ҈ æ ö I҈w + I w = ҈ + 㡰 ´ ç ÷ è ø Þ Kf =
Kf =
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EBD_8341 PS-64
Physics
㡰 㝐ᨨ鼀 ᨨ � �
æd ö m ç ÷ + m ( AO ) � èø d AO =
I҈ =
㝐ᨨ鼀 ᨨ � � ᨨ盐 �졠 é æ d ö æ d ö ù I = I҈ = ê m ç ÷ + m ç ÷ ú êë � è ø è ø úû
҈ Md ҈ é æ d ö ù æ d ö 꾀 I A = ê M ç ÷ + Md ú + M ç ÷ êë � è ø úû � è ø Md Þ IA = ҈ ҈ Md IO ҈ ҈ \ = = IA Md ҈ 㡰 矄� � � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐 Þ I =
黨
҈ ҈ m҈ + m 㡰v + I w ҈ ҈ m҈ m 㡰 gh = m҈ + m 㡰 wR 㡰 + I w Q v = wR 㡰
m҈ m 㡰 gh = Þ
Þ
w é m҈ + m 㡰 R + I ù û ë m҈ m 㡰 gh
m҈ m 㡰 gh =
Þ w=
m҈ + m 㡰 R + I
҈ 㡰 ␔ ᨨ ᆠᨨ �矄 ᆠ �
҈ ҈ mv + I w ҈ ҈ mr = 鼀 w 㡰 + ´ ´ w Q v = wr nᨨ sl�pp�ng 㡰 Þ mgh = mw r gh ҈ gh Þw= = r r ҈҈ ᆠ㡰 㝐ᨨ鼀 � � ᨨ ᨨ 矄矄� � ᨨ盐� ᨨ� l 犨泐 ᨨ鼀 ᆠ ᨨ ᨨ I=Ig졔鼀l æ I ö 䰘 MI MI Þ I= + M ´ç ÷ = ç ҈ ÷ ҈ è ø 矄� � mgh =
䰘MI 矄� � I = MK = K㦨 �盐矄ᨨ �泐�ᨨ 㡰
Þ K=
䰘 I
҈ 㡰 � 鼀矄矄盐 �ᨨ ᆠ�ᆠ盐 �矄ᆠs㦨A졔Br 꾀ᨨ � �㦨pdr 㝐矄矄ᨨ � �dm㦨sprdr 㝐ᨨ鼀 ᨨ � �
I = ò dmr = ò s prdrr a é Aa Ba� ù Þ I = p ò A + Br 㡰r dr = p ê + ú � úû êë é A Ba ù Þ I = pa ê + ú ë � û b
҈ ᆠ㡰 I㦨 ò dm㡰r
a b
æ s ö ps b r a 㦨 ò ç r ´ pr dr ÷ r 㦨 ø aè
ps b - a 㡰 㝐矄矄 ᨨ �矄ᆠ 㦨
b
s m㦨 ò r ´ pr dr 㦨ps ba㡰 a
�盐矄 ᨨ �泐�ᨨ k㦨
I m
ps ⏌ 㡰 b - a 㡰 a + b + ab 㦨 㦨 ps b - a 㡰 ҈ 㡰 꾀矄 ᨨ ᨨ⼄盐�矄ᆠ� � 矄ᨨ �盐鼀ᨨ鼀 盐鼀 矄 ᨨ盐 ᆠᨨ 矄 ᔼ҈졔ᔼ㡰w㦨ᔼ҈w҈졔ᔼwwᆠ㦨ᆠᨨ鼀鼀ᨨ �盐ᨨᆠ�泐ᨨ 矄泐矄鼀 犨 �矄ᆠ矄 � ᆠᨨ ᆠ桌 ᔼw ᔼ҈w҈ + ҈ ҈ æ � ´ öw wᆠ = ç ÷ ҈ ᔼ ø ᔼ҈ + ҈ è �w҈ wᆠ = ҈ ҈ ҈ 쟼 - 쟼� = ( ᔼ҈ + ᔼ ) wᆠ - ᔼ҈w҈ - ᔼ w �w҈ 盐ᔼ㦨ᔼ҈⏌ wᆠ㦨 �w҈⏌ ␔ � ᔼ w 쟼 쟼� = ҈ ҈ ҈� b㡰 꾀矄 ᨨ鼀 ⼄盐矄�ᨨ 矄�泐 s㡰㦨 ├
㝐矄矄ᨨ �矄ᆠ 㝐 =
ò ├
Þ├=
㡰 p = p├
p├ =
㝐
p \㝐ᨨ鼀 ᨨ � � ᨨ盐 �矄 ᨨ �矄ᆠ
�㡰
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PS-65
System of Particles and Rotational Motion
= ò = ò ( 鼀 ) = ò s 꾀 = ò ├ 㡰 p 㡰
=
æ 㝐 ö p´ ç ´ ÷ p ├ è p ø = ò p├ � = = = 㝐 盐� �盐ᨨ ├ ᨨ鼀⼄ �㡰桌 ҈ b㡰
é ù pr R ú R = MR � êë � û
b㡰
҈
㡰 꾀ᆠᆠᨨ � � ᨨ 矄 ᨨ鼀
ᔼ=
鼀 + 鼀 �
ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 ᔼ矄 㡰 鼀矄矄ᨨ �� �㦨㝐 和� ᨨ �� �㦨l 㝐ᨨ鼀 ᨨ � �ᨨ �� �㦨鼀
㝐 㝐矄矄ᨨ 矄鼀� �㦨 � ᨨ 矄鼀� �㦨
҈ æ 䰘M ö ö ҈ æ 䰘 I҈ = ç = ç ´ ÷ MR = 鼀 ÷ R㡰 è ҈ ø è ø ö æ MR æ M ö R æM ö Þ = = I ç ÷ I = ç ÷ r ç ÷ � è ø çè ÷ø �è ø ҈ ù é ê pr r = pR ´ r ú ê ú ëêÞ r = R ⏌ ûú I҈ ҈ ´ = = ҈ I
㝐ᨨ鼀 ᨨ � �ᨨ 鼀ᨨ � �㦨
㝐 æö ç ÷ ᔼ \ 鼀ᨨ = è ø 㝐 㡰 ᔼᨨ��� ᔼ ᔼ 鼀ᨨ = ҈
æ M w ö \w f = ç ÷ è M + m ø ҈黨 ᆠ㡰 ├� �ᆠ�ᆠ盐� �ᨨ �盐矄r �ᆠ├ 矄矄dr矄 鼀矄矄 鼀 矄ᨨᨨ 鼀矄矄 R
M = ò r r ´ prdr =
R
pr R
IC = ò r r ´ prdr ´ r =
㦨 a
ᨨ鼀 犨ᨨ 矄 矄ᆠᨨ 犨 졔 㦨鼀 a㡰
├� � ᨨ⼄盐ᨨ盐 犨 � 맔 맔 㦨ᔼa 맔 맔 㦨鼀 a 㦨鼀 a 和ᨨ� �⼄盐�ᨨ �㡰 ��㡰 a= = ҈ ⏌ 矄 鼀
pr R � �
矄� � �矄 ᨨ鼀 � æ ҈ ҈ ö ҈ pr R \ I = I C + MR = pr R � ç + ÷ = ҈� è �ø
æ ML ö æ Lö ö + m ç ÷ ÷ w f ÷ w = ç ÷ ç ҈ è ø ÷ø ø è
㝐æö çè ÷ø
ᔼ ҈�ᔼ 和ᨨᔼ㦨ᔼ - = ҈ ҈ b㡰
졠
҈ I w = kQ æ k ö ᨨ w = çç ÷÷ Q è I ø dw K æ dQ ö æ k ö = ᨨ a = ÷w ç ÷=ç p I è dt ø çè I ÷ø æ k öæ k ö =ç q k q ÷ç ç I ÷ç ÷÷ = I I è øè ø ҈ ᆠ㡰 I�w�㦨Ifwf ҈䰘 㡰
æ ML ᨨ çç ҈ è
l
�㡰
��㡰
b㡰 㝐ᨨ鼀 ᨨ � �ᨨ �矄ᆠ҈ᨨ盐졠졠¢㦨ᔼ҈㦨
㝐
㝐졠ᔼᨨ ᨨ盐졠졠¢
҈ æ 㝐 ö 㝐 + 㝐 ÷ + 㝐 = 㦨ᔼ㦨 ç è ø 㝐졠ᔼᨨ ᨨ盐졠졠¢ ҈ æ 㝐 ö 㝐 + 㝐 ÷ + 㝐 = 㦨ᔼ㦨 ç è ø 矄ᨨ矄盐 㝐졠ᔼᨨ盐졠졠¢�矄ᔼ㦨ᔼ҈졔ᔼ졔ᔼ Þᔼ =
æ 㝐 ö 㝐 + ç + 㝐 ÷ è ø
=
㝐 㝐 + + 㝐 㦨㝐
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EBD_8341 PS-66
Physics
ᨨ 鼀矄矄 ᨨ �矄ᆠ 㦨s맔pR㦨黨M
鼀矄矄 ᨨ ᆠ�ᆠ盐 �矄ᆠᆠ盐
├ � ᆠ㡰 꾀矄犨├ ᨨ犨 w = ᔼ
w=
é êëQ ᔼ
├ 鼀l
ᨨ =
҈ ù 鼀l ú û
矄�ᨨ 犨 � 矄矄矄 ᨨ盐� 鼀 ᨨ矄��ᨨ
= 鼀w q
l ├ l = 鼀 q = ├q 鼀l l
҈ æ Rö 㦨 ´ M ´ ç ÷ è ø
é҈ æ Rö æ R ö ù ú 㦨 ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è ø ú êë û 矄� � �ᆠ盐 �矄 ᨨ鼀桌 \ ᨨMIᨨ 矄泐矄鼀ᨨ盐 �矄矄矄� � ᨨ盐� O �ᆠ盐ᨨ ᨨ �矄ᆠ�矄 I㦨I҈졔I
ᔼ ��ᨨ矄��ᨨ
ᨨ
é҈ ҈ æ Rö æ R ö ù 㦨 黨MR - ê ´ M ´ ç ÷ + M ´ ç ÷ ú è ø è ø ú êë û
� ᨨ矄��ᨨ
泐 ᨨ犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐 쟼ᨨ ᨨ 㦨 ᨨ�ᨨ ⧤쟼 鼀�
l ҈ 和� q 㦨 ᔼw2
l ҈ ҈ 鼀l 2 ҈ 鼀l l w ω Þ 鼀� ´ = 和� 㦨 ᨨ ᆠᨨ鼀 � ᨨ ᨨ
ᨨ 矄 ҈ 㡰 矄� � 矄 ᨨ鼀鼀ᨨ鼀 ᨨ � �ᨨ盐�O ᔼᨨ㦨ᔼᆠ鼀졔鼀 w = � ( l ) =
=
䰘㝐 ��㝐 + 㝐 ´ 㡰 㡰 =
꾀�� 鼀ᨨ鼀 ᨨ � � ᨨ盐 ᨨ� PIp㦨Iᨨ졔md
��㝐 ҈҈ 㝐 + 䰘㝐 㡰 = 㡰 s 鼀矄矄盐 � =
R/ O'
R ⏌ O
黨MR 黨MR 黨 - ҈㡰MR = MR 㦨 ҈ 黨 ᆠ㡰 꾀矄犨├ ᨨ犨鼀ᨨ鼀 ᨨ � �ᨨ �矄ᆠᨨ盐 �矄 矄矄� � ᨨ盐� ᔴ �ᆠ盐 ᨨ�矄 㦨
Þ 鼀�
䰘
\ MI I㡰ᨨ ᆠ盐ᨨ盐ᨨ�ᨨ ᨨ盐 �矄 矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ �矄ᆠ
㡰 㦨�ᆠ鼀
p R =M 黨 盐矄 ᆠᨨ 矄� ᨨ 矄泐矄鼀 矄 ᆠᨨ鼀 �矄ᆠ ᨨ 鼀矄矄黨M ��鼀矄矄 M矄盐�鼀ᨨ矄 ᨨ � 㝐ᨨ鼀 ᨨ � � I ҈ 㡰 ᨨ ᆠᨨ鼀 �矄ᆠ ҈ 㦨 黨MR ᨨ盐 �矄 矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ �矄ᆠ MI ᨨ ᆠ盐 ᨨ盐 ᨨ�ᨨ ᨨ盐 �矄 矄矄� � ᨨ盐� OἘ �ᆠ盐 ᨨ ᨨ �矄ᆠ
æ Rö
㦨 s ´ p ç ÷ 㦨 s ´ è ø
鼀 㝐ᨨ鼀 ᨨ � � ᨨ �矄ᆠ ᨨ盐 � � �矄 �ᆠ盐 ᨨ�矄 ᨨ犨 ᔼ =
ᔼ Ἐ =
鼀
\
鼀 ᔼ ᔼ Ἐ =
졠
⼄
⼄
鼀 =҈
ᆠ㡰 꾀矄 犨 ├ ᨨ犨 鼀ᨨ鼀 ᨨ � � ᨨ 矄ᨨ� ᆠ泐� ᨨ盐 �矄 犨 �ᆠ �矄 �ᆠ盐 �矄ᆠᨨ
mR ml + ҈ mé l ù I = êR + ú ë û m é V l ù 㦨 ê + ú ë pl û I=
l Þ
dl m é -V l ù = + = dl êë pl úû
pl l l pl Þ = ᨨ = pR l = R R
V l = pl
ÞV =
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PS-67
System of Particles and Rotational Motion ҈
b㡰 㝐ᨨ鼀 ᨨ ᔼ � ᨨ ᆠᨨ鼀 �矄ᆠ ᨨ盐 Ἐ졠Ἐ ᨨ� 㝐 ᔼᨨ = �盐矄ᨨ 鼀ᨨ �矄ᆠ㦨 ⏌ \ 㝐矄矄 ᨨ 鼀ᨨ �矄ᆠ 㦨 㝐⏌҈ 꾀矄㝐µ 桌 㝐ᔼᨨ 鼀ᨨ �矄ᆠᨨ盐�矄ᨨ犨 �矄 졠Ἐ㡰
�
b㡰 A' M
A
B'
㝐 㝐 æ ö ҈黨 㝐 + ç ÷ = �҈ �҈ ҈ è ø 㝐ᔼᨨ 鼀� � � �矄ᆠ 䰘 㝐 ҈黨 㝐 㝐 = �҈ �҈ R 㡰 a = pR M ᨨ犨 = M¢ a pR = = p æ ö R÷ ç è ø M M¢= p ᔼ鼀� � � =
R MR = Mr Þ r㦨 ҈� � 㡰 s 鼀矄矄盐 �ᨨ �矄ᆠ \ ᨨ 鼀矄矄 ᨨ �矄ᆠ \
R/
㦨s맔pR㦨黨M 㝐矄矄 ᨨ ᆠ�ᆠ盐 �矄ᆠ ᆠ盐
æ Rö 㦨 s ´ p ç ÷ è ø 㦨 s ´
M ¢a
M æ ö R÷ ´ç p è ø = MR = 黨 p 㡰 ᨨ � 盐 � ᨨ鼀 矄⼄盐 矄
ma I҈㦨 I㦨 I 㦨 ҈
㝐w 㝐 + 鼀 和ᨨ� �犨�ᨨ矄矄� ⧤쟼 㝐鼀 w 㦨 㝐 + 鼀㡰
R ⏌
O
p R =M 黨
҈ æ Rö 㦨 ´ M ´ ç ÷ è ø 矄� � �ᆠ盐 �矄 ᨨ鼀 \ MI I㡰ᨨ ᆠ盐ᨨ盐ᨨ�ᨨ ᨨ盐 �矄 矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ �矄ᆠ é҈ æ Rö æ R ö ù ú 㦨 ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è ø ú êë û \ ᨨMIᨨ 矄泐矄鼀ᨨ盐 �矄矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ �矄ᆠ I㦨I҈I
I҈ I
҈ ᆠ㡰 ⧤� �ᆠ �泐 ᨨ�ᨨ 㡰⧤ 㦨 ᔼw ҈ ⧤� �ᆠ �泐 矄�ᨨ 㡰⧤ 㦨 㝐 㝐ᔼ � ��㡰ᔼ� �㦨㝐 錐w� ��㦨w 㝐ᔼ 犨㡰ᔼ¢ 矄泐矄鼀㡰㦨 㝐 + 鼀 w¢ 矄泐矄鼀㡰 㦨
O'
҈ 㝐ᨨ鼀 ᨨ � � I҈㡰ᨨ ᆠᨨ鼀 �矄ᆠ㦨 黨 MR ᨨ盐 �矄 矄矄� � ᨨ盐� O �ᆠ盐 ᨨ ᨨ �矄ᆠ MI ᨨ ᆠ盐 ᨨ盐 ᨨ�ᨨ ᨨ盐 �矄 矄矄� � ᨨ盐� OἘ �ᆠ盐 ᨨ ᨨ �矄ᆠ
㝐ᨨ鼀 ᨨ � � ᨨ ᆠ盐 ᨨ盐 �� �矄
Y'
ᨨ矄ᨨ� 矄 鼀ᨨ鼀 ᨨ � � ᨨ盐 �鼀 IAB㦨 MR = I � ᨨ 矄ᨨ� �矄ᆠ ҈ IA'BἘ㦨IYYἘ졔Mr㦨 Mr + Mr = Mr IAB㦨IA'BἘ �� 㡰
㝐ᔼ ᨨ 鼀ᨨ �矄ᆠᨨ盐 졠 ᔼ鼀ᨨ �矄ᆠ㦨ᔼᆠ鼀졔鼀
I=
M
r
B
R
҈ 㝐æ ö 㝐 = çè ÷ø = ҈ �҈
=
Y
I
㦨 w㡰
䰘
é҈ ҈ æ Rö æ R ö ù ú 㦨 黨 MR - ê ´ M ´ ç ÷ + M ´ ç ÷ è ø è ø ú êë û 黨MR 黨MR 黨 - ҈㡰MR = MR ᨨ I㦨 㦨 ҈ 㡰 㝐矄矄ᨨ �矄ᆠ㦨㝐 ҈ ᔼ �矄ᆠ㦨 MR \ 㝐ᨨ鼀 ᨨ � � ᨨ ᨨ ⼄盐ᨨ �矄ᆠ ҈ é҈ ù ҈ 㦨 ê M 㡰 R ú = MR ë û
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EBD_8341 PS-68
Physics
IXXἘ㦨
MR + MR = MR
鼀矄矄M㦨Lr �盐矄R㦨 \ 黨
ᨨ鼀⼄ 矄 �㡰 ��㡰
㡰 꾀ᨨ盐 �鼀ᨨ ᆠ�ᆠ盐 ᨨᨨ � �㡰 ҈ I㦨 MR 矄� � �矄 ᨨ鼀 㝐ᨨ鼀 ᨨ � �ᨨ ᨨᨨᨨ盐 XXἘ �矄
L 錐 p
Lr æ Lö = ÷ è p ø p
IXXἘ㦨 Lr㡰 ç
㡰 ᨨ鼀矄泐鼀鼀泐IAB㦨IAἘBἘ ICD㦨ICἘDἘ AἘBἘ ^ AB CἘDἘ ^ CD A' ᨨ鼀 ᨨ鼀 ᨨ C'
D
�ᆠ盐 矄 Izz㦨IAB졔IAἘBἘ㦨ICD졔ICἘDἘ
q
A
Þ IAB㦨ICD \
B
O
IAB㦨ICD 㦨I
D'
C
B'
㡰 ᨨ 矄ᨨ� 矄 A 矄�泐
R
æ rö rA r 㡰 = k ç ÷ è Rø
dx
æ rö R
s
x
꾀 PB r㡰㦨K ç ÷ è ø
ᔴᨨ 矄� 矄 �ᆠ矄 ᨨ �盐矄 x �ᆠ├ 矄矄 dx 㝐矄矄ᨨ 矄 dm㦨 矄�泐맔 ᨨ盐鼀
æ xö = ç k ÷ p x dx 㡰 è Rø 和ᨨ鼀ᨨ鼀 ᨨ � �ᨨ 矄 ᨨ盐�矄 �鼀
dI =
dm㡰 x
æ xö ÷ p x dx 㡰 x Rø
㦨 ç k è
æ p k ö � =ç x dx è R ÷ø R
\
㝐ᨨ鼀 ᨨ � �ᨨ 矄 A I A = ò dI
pk � pk é x x dx = = ê ú ò R R ëê úû
I = kml
G
鼀ᨨ鼀 ᨨ � �ᨨ DEF㦨I҈ᨨ盐G
D I ml æ m öæ l ö ᨨ I҈ = ÷ç ÷ µ ҈ ҈ è øè ø m⏌ B I ADE = I BDF = I EFC = I I �I \ I + I҈ = I Þ I + = I Þ I = ҈ ҈
和ᨨ I҈ µ ç
�㡰
R
æ pk ö é x҈ ù I B = � ò x dx = ç � ÷ ê ú è R ø ëê ҈ ûú R
IB =
A
I µ ml
B
æ pk ö � \ IA = ç R è ҈ ÷ø 和�鼀�泐 ᨨ矄 B
\
㦨 é é ù ù 㦨 ê MR + Ma ú + ê MR ú ë� û ë� û 꾀 Ӹ é ù =M 㡰 + a ú = ´ MR + Ma ê� R � ë û é æ ù �ö - 㦨 � ê ´ ç ÷ + ´ ú ´҈ ç ê � è ÷ø ú ë û 㦨 � [ + ҈] ´ ҈ - = 黨 ´ ҈- 㦨N맔҈├�m \ 㦨黨 ҈҈㡰鼀矄矄ᨨ � �盐鼀� 㦨m � ᨨ 矄� 㦨l 鼀ᨨ鼀 ᨨ � �ᨨ 鼀� ᨨ盐 �矄矄矄� � ᨨ盐� ᆠ ᨨ� G �ᆠ盐 ᨨ
ùR
R
pk
I B ҈ n = = = \n= I A ҈ ҈ ҈ 㡰 s 矄盐 ᆠ鼀矄矄 矄�泐 㝐ᨨ鼀 ᨨ � � ᨨ 鼀� ᨨ盐 矄 矄矄� � ᨨ盐� Ἐ졠Ἐ ҈ IO㦨 sp R 㡰 桌 ´ R㡰 é҈ ù êë s p R 㡰 + s p R 㡰 ´ R úû ҈ 㦨 psR 㝐ᨨ鼀 ᨨ � �ᨨ 鼀� ᨨ盐 矄矄矄� � ᨨ盐� ἘἘ IP 㦨 p s 졔sp 㡰맔 㡰 é҈ æ ö ù ê s pR 㡰 R + s pR 㡰 ç R㡰 + R ÷ ú è ø ûú ëê 㦨 p s ��s p 㦨҈�p s I P ҈�psR 䰘 = = » \ ҈ IO psR ҈ 黨㡰 ᨨ盐矄 矄꾀ᔴ m㦨m㦨mᔴ㦨m㦨�├� 㝐ᨨ鼀 ᨨ � � ᨨ ᔴ 矄泐矄鼀 ᨨ盐 ��ᨨ ᨨ 矄⼄盐 ᔼӸ 㦨ᔼ҈졔ᔼ졔ᔼᔴ졔ᔼ㦨ᔼ꾀졔ᔼ ᆠ鼀
R
黨
pk � R
��㡰
A
l⏌ G m⏌
F
E
C
ᆠ鼀ᨨ鼀 ᨨ � �ᨨ DECB� 鼀ᨨ ADE æ � I ö æ I ö ҈҈I NI = I + I҈ = ç ÷ + ç ÷ = = ҈ è ҈ ø è ҈ ø ҈
ᨨ盐ᨨ N㦨҈҈
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PS-69
System of Particles and Rotational Motion
҈ MR
(20) 꾀矄犨├ ᨨ犨鼀ᨨ鼀 ᨨ � � �矄ᆠ I �矄ᆠ = w҈
w M
I҈ R҈㦨R
Iᨨ㦨I҈졔I㦨I졔I ᨨ Iᨨ㦨I㦨2I \I㦨I I҈㦨I㦨I㦨I
ᨨ Iᨨ ᆠ ᨨ� 泐 � � 泐犨ᨨ � Iᨨ㦨I҈졔I㦨I҈졔I 㦨I҈졔I㦨I졔I 㦨I졔I㦨I졔I
w M
R҈ M㡰 R M㡰
I R㦨R/2
矄� � �盐 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
I҈w҈ + I w = I҈ + I 㡰 ´ w f æ MR MR ö + çç ÷wf ÷ø è ҈ ҈ æ MR ᔼ ��⧤쟼 K � = I w = çç è MR ´w+ =
҈
ö MR w ÷÷ w = ø
MR w MR w � = ´ ҈ = 奠 = P奠 MR w X F rF ᆠ盐ᨨ P㦨 �㡰 a 㝐ᨨ鼀 ᨨ � � ᨨ 矄泐矄鼀 ᨨ盐 �矄 XE I = I E + I F + IG E
a
rG Þ I = m rE 㡰 + m rF 㡰 + m rG 㡰 � � æaö ma Þ I = m ´ + m ç ÷ + ma = ma = èø \ N = �
a
G
҈�㡰 � ᨨ 㦨҈鼀
�q㦨
҈ Iw l ҈ æ ml ö Þ 鼀� l ´ ҈ = ҈ æç ml ö÷ w Þ 鼀�㡰 矄� ° = ç ÷w çè ÷ø çè ÷ø
DPE = DKE ᨨ mgh =
Þ w = ҈� ⏌矄
O
D
C
㡰 KE ᨨ 矄 㦨 矄�ᨨ KE 졔 ᨨ�ᨨ KE
҈ ҈ mv + I w ␔ I㦨 鼀ᨨ鼀 ᨨ � � w 㦨 꾀 �盐 ᨨᆠ�泐 ᨨ ᨨ�ᨨ
m 㦨 鼀矄矄 ᨨ 矄 v 㦨 � ᨨᆠ�泐ᨨ ᆠ ᨨ 鼀矄矄 ᨨ 矄 Q 㝐ᨨ鼀 ᨨ � �ᨨ 矄 I = mR � ҈ ҈ \ K E = mv + ´ mR ´ w � v ö ҈ ҈ ævö æ Þ K E = mv + ´ mR ´ ç ÷ çQw = ÷ Rø � è Rø è ҈æ öæ v ö Þ KE = ç mR + mR ÷ç ÷ è� øè R ø ҈ 䰘 v 䰘 ҈ � Þ KE = mR ´ ´ = ´ ´ � R ҈ ҈ � Þ KE = ´ ҈ ᨨ盐 Þ⧤쟼㦨䰘�맔҈ᨨ盐 b㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ᨨ �矄 鼀ᨨ � ᆠ鼀 � ᆠ 㦨�ᆠ鼀 㦨҈ᆠ鼀 ᨨ = ҈ 㦨�ᆠ鼀 �盐矄ᨨ ᨨ = ᨨ ᨨ 矄�� � �ᨨ盐 ᆠ 㦨w \ ᨨᆠ�泐ᨨ 矄ᆠ㦨 ᆠ 졔w㡰 \ �矄 ᆠ鼀ᨨ 泐矄ᆠ㦨 ᆠ 졔w㡰
꾀矄矄盐鼀� � 矄泐鼀鼀�ᆠ 鼀� ᨨ � xy Ix 㦨 Iy 泐矄泐鼀鼀泐 Ix졔Iy㦨Iz �ᆠ盐㜠 �矄 ᨨ鼀㡰
ö æ ö æ = ç ᆠ + ᆠ ÷ çQ w = ÷ è ø è ø
Iz B = Ma A Ix㦨Iy㦨 a a졔a=2a Q Iz㦨҈Ma �� 㡰 a ᨨ犨 ᨨ鼀 ᨨ 㜠 矄 X ᨨ鼀 O IAB㦨Ix졔M a㡰 㦨Ma졔Ma 㦨Ma bᆠ㡰 和� ᆠ ABCD �矄 矄⼄盐 鼀� ᆠ 泐 矄泐鼀鼀泐I҈㦨I I㦨I ᨨ鼀 �ᆠ盐 矄 ᨨ鼀 㝐ᨨ鼀 ᨨ � � ᨨ盐 �矄 �ᆠ盐 ᨨ 矄⼄盐 矄矄� � ᨨ鼀ᆠ O
=
Y
B
=
҈ æ MR MR ö ҈ MR w + ç ÷ w = ÷ø � � çè ᆠ � ᨨ矄矄 � ├� �ᆠ �泐 奠 ᨨ矄矄
䰘
҈
Topic-5 : Rolling Motion Þ wf = w �
� ⧤쟼 K f =
�
A
ᆠ × = ´ � = 䰘� ᆠ鼀 꾀
x 㦨
ᨨ
ᨨ �ᆠ鼀
꾀
҈ᆠ鼀
ᆠ㡰 ᨨ 矄 ҈ ҈ mv 졔ᔼw㦨 mgh 䰘v ҈ ҈ æ öv ᨨ mv 졔 ç mR ÷ = mgh ᨨh㦨 ҈ g è� øR
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EBD_8341 PS-70
Physics
ᨨ ᆠ泐�
䰘
҈ ҈ æ mR ö mv + ç ÷ = mgh Ἐ çè ÷ø h 䰘v ⏌ ҈ g ҈ = = \ h Ἐ v ⏌ g ҈�
ᨨ
҈ ҈ mvcm + I cm w I cm ö ҈æ ҈ ҈ æv ö = mvcm + I cm ç cm ÷ = ç m + ÷ vcm R è R ø è ø mR ö ҈æ ᨨ� �鼀� = çç m + ÷÷ vcm R ø è v \ h = cm g mR ö ҈æ ÷ vcm ᨨ矄ᨨ� ᆠ泐� mgh = çç m + R ÷ø è v \ h = cm g ҈æ mR ö m = + ç ÷ vcm ᨨ矄 mgh ç � R ÷ø è 䰘v \ h = cm ҈ g
䰘 Þ ҈� ҈ ҈
㡰
㦨鼀
mR a ��㡰 ᨨ盐ᨨ� � 㦨a ���㡰 ᨨ鼀 ҈㡰 㡰 㡰 mRa F = maR F = mRa F a= mR ᆠ㡰 꾀矄矄 ᨨ犨 � ��鼀 ᨨ鼀ᆠ�ᨨ ᨨ 꾀 ᨨ ᨨ犨�矄 � ᨨ犨 矄졠
�矄犨� ᨨ � �ᨨ 矄泐矄鼀ᨨ ᆠᨨ 矄ᨨ盐
졠
꾀
ᔴ
a
ᨨ
ᨨ
黨
ᆠ㡰
҈
ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ盐 泐 � ᨨ� ᨨ 矄盐 ᆠ mrw㦨mrw w r [Q v = r w ] Þ w㦨w⏌ Þ v = 㡰 꾀泐� � �泐 ᆠᨨ 矄�ᨨ � ᆠ� ᨨ
㦨ᔼa㦨
�㡰
mR F 㝐矄矄㦨m a = fR 졠 aù é mR a = fR êQ a = ú f Rû ë R ma = f ��㡰 盐 �矄 盐 � ⼄盐�ᨨ �㡰 ma ma ma 㦨 F ᨨ F = 㡰 꾀矄犨├ ᨨ犨 鼀� 矄� q 꾀ᆠᆠ�ᨨ = ᔼ 鼀+ 㝐 ᆠ � 矄� qᆠ 㝐 ᆠ � 矄� qᆠ ᨨᆠ泐� ᆠ = 㦨 ҈ 㝐ᆠ 㝐ᆠ 㝐ᆠ + 㝐 + ᆠ ᨨ ᆠ㦨 � 矄� qᆠ ᨨ 矄 㝐 � 矄� q矄 㦨 㝐矄 � 矄� q矄 矄 = 矄 ᔼ 㝐 㝐矄 + 㝐 矄 + 矄 � � 矄 = � 矄� q矄 ᨨ 䰘 �� ᆠ㦨矄 � � 矄� qᆠ 䰘 ҈� � � = = � 矄� qᆠ = � 矄� q矄 \ 矄� q矄 ҈ 䰘 �
�㡰
ᆠ㡰 ᨨ鼀 ��盐 ma 㦨 F f 꾀 ᨨ⼄盐 t 㦨 Ia
ᨨ 盐矄㡰 mgh =
�ᨨᨨ �� 矄 ҈ �
v hἘ = g
æ v ö ҈ ҈ mv + I w = mg ç ÷ è g ø ҈ ҈ v mv + I = mv Qv㦨 w桌 \ R ҈ v ҈ ҈ ҈ I = mv - mv = mv Þ I㦨 mR \ R
�矄�矄 鼀ᨨ鼀 ᨨ � �ᨨ �矄ᆠ ᆠ ᨨᆠ�矄 �矄ᆠ ҈҈ b㡰 ᔼ 盐 ᨨ� � ᨨ� ᨨ ᆠᨨ ᆠ �矄 � 矄 ᨨ盐矄 ᆠ ᨨ ᨨ�ᨨ ᨨ �ᆠ矄 ᨨ �矄ᆠ졠 泐� �v㦨rw ␔ � w�矄矄鼀 ᨨ �ᆠ矄 vµr rQ rC rP \ rQ rC rP
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PS-71
System of Particles and Rotational Motion b㡰 ᔴ泐� ᨨ 鼀ᨨ� � ᨨ �ᆠ�ᨨ 矄矄 矄盐 ᆠ ᔼ
ᨨ ᆠ矄矄 ᨨ 盐 ᨨ ᨨ犨 ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� �矄 � 矄� q 꾀矄ᨨ ᨨ ᨨ⼄盐 ᨨ盐 ᆠ ᨨ ᆠ泐� �矄 ᆠ� �ᨨ ᆠ泐� 矄� ᆠ 犨 矄矄盐鼀 矄盐 ᆠ ᨨ �ᆠ�ᨨ 矄矄 ᨨᆠ矄 ᆠ� �ᨨ ᆠ泐� �矄鼀� 犨 �ᆠ 矄矄 ᨨ盐� ᆠ ᨨ F ᆠ泐� ᨨ g s�nq ᨨ� ᨨ N 鼀ᨨ鼀 ᨨ ᨨ� ᨨ ᆠᨨ ᆠ ᆠᨨ ᆠ � ᨨ ᆠ矄矄 �矄 g s�nq � ᨨ犨 犨 �ᆠ�ᨨ
ᆠ �ᆠ�ᨨ mg s�n q ᨨᆠ 犨泐矄 ᆠ矄 � mg q 盐犨 �ᆠ�ᨨ ҈ ᆠ㡰 盐 ᨨ � F ᨨᆠ ᨨ鼀 ᆠ�ᨨ 矄 � 矄 L N G ᨨ ᨨ ᆠᨨ 꾀 �矄 矄�盐�ᨨ ᆠ盐�ᆠ ᨨᆠ├ L⏌ 矄矄ᨨ� � ᨨ盐ᨨ� O O 꾀ᨨ盐 ᨨ� 졠 ᨨ⼄盐 mg L mg F맔L㦨mg맔 Þ F㦨 ᆠ鼀� �鼀盐鼀 ᨨᆠ⼄盐� ᨨᨨ ᨨᆠ├ mg F= ҈ 䰘㡰 K.ETᨨtalᨨ ᨨ� � �矄ᆠK.Etrans 졔K.Erᨨt ҈
҈ ҈ ҈ ҈ æ ҈ mR ö æç v ç ÷ 㦨 鼀 졔 ᔼw 㦨 mv 졔 ø çè R è
ö ÷ ÷ ø
mv ᨨ 矄盐 ᆠ AB k.E�졔 ᨨ矄矄 � ���ᨨ ᨨ � �泐㦨 K.Ef m 㡰졔mg 㡰㦨 m VB �㡰 ᨨ 矄盐 ᆠ CD m v㡰졔mg 䰘㡰㦨 m VD ��㡰 Q VB㦨VD\ ᨨ鼀⼄ �㡰 ��㡰 m 㡰졔mg맔㦨 m v㡰졔mg맔䰘 \V㦨䰘m/s ҈� 㡰
矄�ᆠ├ �矄 㡰 ᨨᆠ矄ᨨ ᨨ� ᨨ ᆠᨨ ᆠ 졠 ᨨ � � 犨� �ᨨ盐 矄 ᨨ矄� 盐 �ᆠ�ᨨ ᨨᆠ f ᨨ矄 ᨨ ᨨ犨 �矄 ᆠ矄 ᨨ⼄盐 ᨨ盐 ἘᆠἘ 犨 �ᆠ 矄矄ᨨ� � � �犨 � � 矄�ᆠ├ �ᆠ�ᨨ
ᨨᆠ f҈ 矄ᨨᆠ矄 Ff㦨ma f㦨맔 \ f㦨҈ C 꾀泐� � t 㦨Iaᨨ盐 C R R f҈ a ff҈㡰R㦨Ia㦨 I f R 졠 Q a㦨Ra桌 Q I㦨MR桌 Þ ҈m맔桌맔�㦨맔 �㡰맔 � P \ m㦨㦨 㦨 \ P㦨 ҈ ҈ K.ETᨨtal㦨
҈ �㡰 ᨨ ᨨ ᨨ矄� F㦨mmg㦨鼀� ᨨ ᨨ ᨨ� �
a æa ö F ç + b ÷ £ mg è ø a æa ö Þ mg ç + b ÷ £ mg Þ a졔b£�a è ø b Þ £ a �b£䰘�a盐 �矄�矄 ᨨᨨ矄矄� 꾀矄 鼀 �鼀盐鼀盐ᨨ bᆠ ⼄盐ᨨ�a ҈b Þ = � a ҈䰘 䰘�㡰
�鼀├ ᨨᆠ �ᨨ盐 ᨨ 矄ᆠ
t=
҈ h æ K ö ç҈ + ÷ 矄� q g çè R ÷ø h K Q =҈ g R
ᨨ� � t҈ =
҈ h (҈ + ҈) = 矄� ° g
ᨨ �矄ᆠ t =
҈ h æ ҈ ö h K ҈ + = = ҈ Q ç ÷ 矄� ° g è ø g R
� t҈ - t = \
- ҈
h - h = g g ҈
Þ h - h = -
\ h » 䰘� 鼀 �� ᨨ ᨨᨨ � ᆠ� 㡰 ҈ 矄 ᨨ ⧤� �ᆠ �泐ᨨ � � E҈ 㦨 K.E.㡰 ᨨ�ᨨ 졔 K.E.㡰 矄�ᨨ ҈ ҈ 㦨 I w + mvc ҈ ҈ 㦨 ´ mr w + m r w 㡰 Q I㦨mr vc㦨rw) Þ h ҈䰘㡰㦨҈䰘҈�Þ h =
E҈ = mr w ᨨ ├� �ᆠ �泐ᨨ ᆠ泐� E 㦨 K.E㡰 ᨨ�ᨨ 졔 K.E㡰 矄�ᨨ
҈ ҈ I Ἐ w Ἐ + Mv Ἐc ҈æ҈ ö ҈ 㦨 ç Mr ÷ w Ἐ + M r w Ἐ㡰 è ø E㦨 Mr w Ἐ 쟼⼄盐� �E҈㦨E mrw㦨 Mr w Ἐ m wἘ = = ´ = ҈ Þ wἘ㦨w Þ M w ᆠᨨ � � ᆠ泐� 犨�ᆠ 矄鼀�鼀
㦨
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EBD_8341 PS-72
҈黨
Physics
ᆠ㡰 ᔼ ᨨᆠ 㡰�矄 � � � t 鼀� 矄ᆠᨨ 矄 t 㦨 맔 ᔼ ᨨᆠ�矄� ᨨ鼀 ᨨ 矄 犨 矄 ᆠ�鼀矄 �ᆠ盐 �矄 ᆠ 㝐¢ ᨨ ᨨᆠ ᨨ鼀 턼 犨� �ᨨ 㝐 ᨨ ᆠ �� � � ��泐 �ᆠ盐 �矄 턼㝐 턼 � ᆠᨨ鼀矄 턼㝐Ἐ ᔼ ᨨᆠ 㡰�矄�
ᨨ鼀 ᨨ ᆠ�ᆠ盐鼀 ᆠ ᨨ� t �矄 ᨨ ᔼ ᨨᆠ 㡰�矄� � �泐ᨨ� 和
t = F ´ R 犨 ᆠ�鼀矄 r b㡰 ᨨᆠ�泐ᆠ Ἐ졠Ἐ \ vo = Rw iˆ R w ᆠᨨ矄º kˆ
R w R w 矄� º -�ˆ 㡰
졠
҈
º
ur Rw Rw VP = Rw�ˆ 矄� °�ˆ + ᆠᨨ矄 °kˆ ur Rw ˆ ù Rw ˆ é �ú+ k \ VP 㦨 êRw �ˆ û ë ur ҈҈ R kˆ ᨨ VP = Rw �ˆ + ur w bᆠ㡰 VA = V C = V ur ur VB = V ur ur VC = V ur ur VB = V \ VC -VB ur ur = VB - VA
ur
R w �ˆ
ur
ur
ur
F
m f҈
f
mgᆠᨨ矄q
ᆠ �ᆠ�ᨨ ᨨᆠ � 矄 ᨨ�ᨨ 盐 � 矄 矄�ᨨ 鼀ᨨ�ᨨ 꾀泐� � F 㦨 ma ᨨ � �ᆠ�ᨨ ᨨ � ᆠ� mg矄� q f㦨mac 犨 ac 㦨 ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� \ f㦨mg矄� q mac … �㡰 g 矄� q g 矄� q = = g 矄� q 盐ac 㦨 Ic mR ⏌ ҈+ ҈+ mR mR mg 矄� q f㦨 ᔴ泐� q �矄 盐ᆠ �ᆠ�ᨨ ᨨᆠ�矄 盐ᆠ
a
m҈
a҈
和� ᆠ �矄 ᨨ 矄�� � 泐犨 \ a҈㦨a
VA =
V C - V A = V B - V C N ᆠ 㡰 矄� � � �ᆠ�ᨨ
ᆠ矄 � ᨨᨨ矄� �ᆠ�ᨨ ᨨ 鼀� 矄� q ᨨ ᨨᨨ矄 � 鼀ᨨ�ᨨ q ᆠ盐矄ᨨ �ᆠ�ᨨ ᨨᆠ
q 矄� g mg m ᆠ泐� ᨨ矄 q
㡰 ᔼ ᆠ矄ᨨ 盐ᨨ� �ᨨ � ᆠ� ᆠᆠ�ᨨ
ᨨ ᨨ 泐 � 矄� q = ᔼ ҈+ 㝐 ᔼ 㝐 ᨨ ᨨᨨ犨 ᆠ泐� = =҈ 㝐 㝐 ҈ 㝐 ᔼ ҈ ᨨ矄ᨨ� ᆠ泐� = = 㝐 㝐 ᆠ ᆠᆠ�ᨨ ᨨ 矄ᨨ� ᆠ泐� �矄 鼀ᨨ ᨨᨨ犨 ᆠ泐� ᨨ 矄ᨨ� ᆠ泐� 犨� ᆠ ᨨᨨ鼀 ᨨ � ᆠ� �矄 ᔼ ᆠ矄ᨨ ᨨ� � 犨� ᨨ犨ᨨ├ ᨨ 泐 �ᆠ�ᨨ
ᨨ ᨨ 鼀ᆠ �ᆠ �泐 鼀� 矄 ᆠᨨ 矄 \ ⧤쟼㡰矄ᨨ� 㦨 ⧤쟼㡰 ᨨᨨ犨㦨 ᆠ矄� 쟼㦨鼀� 鼀 �矄 ᨨᆠ ᔼ矄 ᨨ� ᨨ ᆠᨨ鼀ᨨ F 盐矄 矄 ├ ᆠ ᨨ� ᨨ ᆠᨨ ᆠ ᨨ ├ 犨� ᆠ泐� 犨� ᨨ 鼀ᨨ ᨨ犨 矄 �� �ᆠ�ᨨ ᨨᆠf҈犨�ᆠᨨ犨 矄 ᨨ ├ f҈ 㦨 �ᆠ�ᨨ 犨 ├ ᆠ泐� f 㦨 �ᆠ�ᨨ 犨 ᆠ泐� �ᨨ盐 a҈㦨ᆠᆠ�ᨨ ᨨ ├ a 㦨 ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� a㦨 �盐 ᆠᆠ�ᨨ ᨨ ᆠ泐� ᨨ盐 �矄 ᔴ㝐 f҈
F - f҈ a҈㦨 m f҈ - f a㦨 m҈ f҈ - f 㡰 R f҈ - f 㡰 R a= = ҈ I m R ҈ f҈ - f 㡰 a= m҈R f҈ - f 㡰 a㦨Ra = m҈ 㡰 和ᨨ� �⼄矄 �㡰ᨨ 㡰犨�
f
�㡰 ��㡰 ���㡰
�㡰 㡰
F m҈ + m 꾀 꾀ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� F a = m҈ + m m҈F 㡰 �ᆠ�ᨨ 犨 ├ ᆠ泐� f҈ = m҈ + m m҈F �ᆠ�ᨨ 犨 ᆠ泐� �ᨨ盐 f = m҈ + m 和� ᆠ ⼄盐 ��矄 ᨨ矄�� 泐 ᆠᨨᆠ泐 矄 ᨨ犨 � ��盐矄 꾀ᆠᆠ�ᨨ ᨨ ├ a҈ =
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PS-73
System of Particles and Rotational Motion �
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈
㡰 鼀 �鼀盐鼀盐ᨨ ⼄�⼄鼀 ᨨᨨ�矄ᨨ盐 ᨨᨨ 㦨 ᨨᆠ矄 鼀� ҈㡰鼀盐矄矄矄 ᨨ盐� ᆠᨨ ᆠ ᨨ� ἘἘ 졠턼 = ᨨ矄� q鼀 = \ ᆠᨨ矄 黨 q鼀 㡰 = 졠 졠 q 鼀 턼
ᨨᨨ 鼀�
҈
鼀�
q
b㡰 꾀 �盐鼀ᨨ鼀 盐鼀 ᨨ L 㦨 Iw ᨨ盐 �矄 ᨨ ᨨ 㡰 㝐ᨨ鼀 ᨨ � � ᨨ ᨨ 㜠 � 矄ᆠ 矄泐矄鼀 I 㦨 Iᨨ 졔 mx 㦨 Iᨨ 졔 mvt m 㦨鼀矄矄ᨨ � 矄ᆠ \ L㦨 Iᨨ 졔mvt㡰w
q鼀
w
(
x㦨v t
dL = mv t w㡰 ᨨ t µ t dt � � �矄 矄�� � 矄矄� � ᨨ盐� ᨨ��� 꾀 �鼀TL 㦨ᆠᨨ 矄 dL = \ t ᨨ dt �犨 � 矄ᆠ矄ᨨ矄鼀ᨨ� �L ᨨ矄 ᨨᆠ � ᨨ T ᆠᨨ鼀矄 ᆠᨨ 矄 㡰 ␔ 犨 ��� � �盐 ᆠᆠ�ᨨ a㡰 ᨨ ᨨ 矄 � 矄 ᨨ盐矄 ᆠᆠ�ᨨ ᆠ� 矄 㦨 La 矄 ᆠ泐 ᨨ 鼀ᨨ 犨泐 ᨨ鼀 ᆠ 盐 盐 ᨨ �ᆠ�ᨨ 犨 ᨨ �矄 ᨨ矄 ᨨ ᔼ � 矄 ᨨ盐矄 �盐 ᨨᆠ�泐�矄 w
ᆠ矄矄泐 �ᆠ�ᨨ ᨨᆠ �矄 ᨨ� 泐 �ᆠ�ᨨ ᨨᆠ mLw㦨Ҙ ma㡰ÞmLw㦨ҘmLa Þ w㦨Ҙa
\
at㦨Ҙa Þ t㦨 a
㡰
a
v
Ҙ a
mv T 矄� q 㦨 �矄 ᨨ 盐ᆠ� � r
M
r w
q
T q
Tᆠᨨ矄q
T矄� q
mg
ᆠ矄矄泐ᆠ � ᨨᆠ ᨨ ᆠ�ᆠ盐 鼀ᨨ�ᨨ
L
)
Tᆠᨨ矄 q 㦨mg
v ҈ ´ ҈ = = ҈ ᨨ q 㦨� rg ҈ ´ ҈ 㡰 ᨨᆠ ᆠ� � ᨨ 鼀矄矄 ᨨ �⼄盐� dm ᨨ � dx �矄 ᆠx ᨨ鼀 �矄ᨨ ᨨ�ᨨ O æ mv ö ÷ dF㦨 dm㡰xw çQ F = r ø F è \
C
M
q 㦨
x
O
O
a a ᨨ r = 泐�ᨨ鼀泐 ᨨ⼄盐ᨨ盐 ᨨ� O �矄 ᨨ ᆠ �盐 鼀ᨨ鼀 盐鼀 L㡰ᨨ盐 O �矄 ᆠᨨ 矄 ᨨL�㦨Lf æ aö MV ç ÷ = I w = Icm + Mr 㡰w è ø ìï Ma æ a ö üï v + M ç ÷ ý w = Ma w 㦨 w㦨 í a è ø þï îï r㦨
䰘
a
矄� �w㦨w졔at Þ w㦨at
dW = dx ҈x d é w \ x + ҈䰘 - ҈黨 x ù úû dx êë 䰘├� Þ x҈黨㦨 ҈黨 = 黨 鼀 \ x㦨 ᆠ㡰 ᨨ�矄盐 �ᨨ� �ᆠ�ᆠ盐鼀ᨨ�ᨨ 矄 ᆠ�矄 鼀ᨨ� �� ᆠ�ᆠ盐 ᨨ� ᨨ ᆠ├犨� ᆠᨨ 矄 矄
ᨨ �矄 盐 � 盐 ᆠ ᨨ 犨ᨨ ᨨᆠ矄 �㡰 矄�ᨨ T � ᨨ v ��㡰 ␔�� ᨨ ᨨ mg ᨨ犨ᨨ├ ᨨ ᨨ鼀� �鼀盐鼀
ᨨ犨 t =
ᆠ㡰 꾀矄 � �矄 ᨨ ᆠᨨ�矄�ᨨ 犨 犨ᨨ � �ᆠ 矄 矄 泐 犨� ᆠ � � � ᨨᆠ��矄 和� ᆠ 矄 矄矄鼀ᨨᨨ 犨� ᨨ �ᆠ�ᨨ ᨨᨨ⼄盐㡰 ᨨ 犨� ᨨ 矄 ᨨ �盐鼀ᨨ鼀 盐鼀
盐矄 A ᆠᨨ�矄�ᨨ 犨� 鼀� 犨� �矄 � �� �盐 鼀ᨨ鼀 盐鼀 � w A = w wB = ᆠ㡰 ᨨᨨ� 鼀矄矄I㦨mr 㝐ᔼ ᨨ 矄泐矄鼀ᨨ盐 �矄 ᨨ ᨨ�ᨨ O I㦨I҈졔I㦨x졔䰘 ҈x㡰 㦨x졔䰘 ҈黨졔xx㡰 㦨x졔҈䰘҈黨x
犨ᨨ├ ᨨ � ᨨ� � ᨨ �矄 ᆠᨨ � ᨨ �矄 ᨨ�ᨨ ├� �ᆠ �泐 ├� ҈ \ W㦨 I w x ҈ = + ҈䰘 ҈黨 桌 w x x 졠
M dx dF dx ´ xw L 졠 M 犨 �矄� 鼀矄矄 矄�泐ᨨ �⼄盐� L \ ᨨᆠ ᆠ� � ᨨ �矄 ᨨ 犨 ᨨ �⼄盐� � 盐 LM M L w x dx = w ò x dx F㦨 ò L L \
dF㦨
L
ù MLw M é x ù M é L w ê ú = w ê - ú = 㦨 L ëê úû L ëê ûú
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EBD_8341 PS-74
Physics
҈ù é êF ´ r ´ ú = I a û ë ҈ ҈ ´ � ´ � ´ 㦨 ´ ҈� ´ � ´ � ´ a Þa㦨 矄҈ w㦨w졔atÞw㦨졔맔҈㦨 矄҈ ҈ ␔ ᆠ盐 �� 矄 ᨨ � ᨨ盐 � ᨨ鼀 ᆠ�ᨨ N 犨� 矄矄 ᨨ盐� � D ᨨ盐 犨 �ᆠ ᨨ�ᨨ ├矄ᆠ 鼀ᨨ鼀 矄 盐ᨨN �ᆠ�ᨨ ᨨᆠf犨� ᨨ
├� � 鼀ᨨ鼀 ᨨ ᨨᆠ B C 鼀�ᨨ盐 D F 黨
҈
㡰
a a = mg ´ \ F㦨 mg a⏌ A f D a mg ᆠ鼀� �鼀盐鼀盐ᨨ F㦨 mg t 矄ᨨ⼄盐t㦨Ia \a㦨 I t 㦨 F ´ r ^ ᨨ⼄盐 �矄 矄鼀 � ᨨ ᆠ矄矄 盐 矄� ᆠ I 犨� � 盐 ᨨ � 鼀矄矄 �矄�盐�ᨨ ᨨ盐 �矄矄ᨨa犨� � b ᆠ㡰 ᨨ �矄 矄 ᨨ鼀 矄 矄ᨨ � 矄 泐 ᨨ� � ᨨ盐 �矄 ᆠᨨ ᆠ ᨨ� 犨� ᨨᨨ 犨� ᨨ盐 矄�� �
㡰 꾀矄 犨 ├ ᨨ犨 ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� ᨨ� � ᨨ犨 � ᆠ�
g 矄� q I ҈+ MR ᔼ ᆠ矄 ᨨ P 鼀矄矄 �矄 ᆠᨨ ᆠ 㜠 犨泐 ᨨ鼀 �矄 和ᨨIP>IQ \ aPaQ Þ vv턼Þww턼 ҈
҈
ac
ᆠ㡰 �盐 �鼀盐矄 ᆠ 泐 �ᆠ�ᨨ ᨨᆠ 犨 � � 矄ᨨ ᆠ矄 �盐 矄 w ᨨ � � ᨨ盐 졠 òt
F맔
҈҈
C
ac =
㦨㜠矄 ᨨ 泐㦨҈矄
ᆠ
҈�
꾀 ᆠᨨ�矄�ᨨ w ᆠ矄矄盐 � � 鼀� 矄ᨨ� � �
�ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ᆠ �ᆠ�ᨨ 犨 � � �ᨨ盐 ᨨ� ᨨ ᆠᨨ ᆠ �矄 ᨨ ᆠ 㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀� P�㦨 Pf m v㡰졔m v㡰졔m맔㦨 m졔m졔m㡰vc Þ vc㦨 + -
m
a
v a
a
v m
⏌ ᆠᨨ矄
ar
at
arᆠᨨ矄 졔at ᆠᨨ矄
� � ├� �ᆠ �泐㦨 ᨨ矄矄 � ᨨ � �泐
҈ l Iw 㦨鼀� ҈ᆠᨨ矄 㡰 l ml � \ w 㦨鼀� Þ w㦨 l ᨨ犨t㦨ᔼa ҈ l � \ 鼀�맔 矄� 㦨 鼀la Þ a㦨 l l � 盐 㦨 a㦨 � l l � ´ 㦨 꾀矄ᨨ 㦨w 㦨 l ᨨ �ᆠ 鼀ᨨ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄 鼀�㦨鼀 ᆠᨨ矄 졔ᆠᨨ矄 㡰 \
é � ҈ � ù ´ ú 鼀�㦨鼀 ê ´ + û ë
\
㝐� 㦨 ҈
҈
ᨨ犨 泐� � ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ盐 ᆠ ᨨ 鼀矄矄 � L�㦨 Lf mv맔a졔m v㡰맔a㦨Iw ҈ I㦨 m㡰맔 a㡰졔m맔a졔m맔a㦨ma ҈ v \ mv ( a ) + m ( v ) ´ a = ma ´ w Þ w = �a ҈ I w \ 꾀 ᆠᨨ�矄�ᨨ �泐E㦨 v mv ҈ = 㦨 ´ ma ´ � �a r dr r r 㡰 r t 㡰 = at�ˆ + btˆ ® v 㦨 dt 㦨 a�ˆ + bˆ 㦨 ᆠᨨ 矄 r a 㦨 \ r r 盐 P = mv 㦨 ᆠᨨ 矄
҈ 鼀 㦨 ᆠᨨ 矄 r æ ¶U ˆ ¶U ˆ ö F = ç �+ ÷ 㦨 ¶y ø è ¶x
⧤㦨
\ 㦨ᆠᨨ 矄 꾀矄ᨨ쟼㦨⧤졔㦨ᔴᨨ 矄 r \
dL r r r = t= r´F = dt
Q ar �矄 ᆠᨨ 矄 㡰
r
\ L 㦨 ᆠᨨ 矄
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PS-75
System of Particles and Rotational Motion
r r dr r ˆ ˆ v = 턼 r t 㡰 = a ᆠᨨ矄 wt � + b 矄� wt ® dt = -aw 矄� wt 㡰 ˆ� +bw ᆠᨨ矄 wt 㡰 ˆ r r dv r a= = -w a ᆠᨨ矄 wt �ˆ + b 矄� wt ˆ桌 = -w r \ dt r r r r r 꾀矄ᨨ t = r ´ F = \ 㦨ᆠᨨ 矄 㡰 \ r F ҈҈㡰
⼄㡰 � 鼀ᨨ矄 盐 X 矄ᨨ鼀ᨨ矄 盐 ᨨ ���ᨨ �泐ᨨ X�矄 ᆠᨨ � 盐ᨨ盐矄泐 � ᆠ矄� � T ᨨ 犨�� ᨨ Y ᨨ盐 P 矄 �ᆠ盐 �矄 ᆠ ᨨ � ᨨ ᆠ�ᨨ ᨨ ᨨᆠ ᨨ鼀 ᨨ� P�矄 ᨨ Ӹ ᨨᆠ 泐Xᨨ Y 㝐� 㦨 Mg + Mg = mg 犨 Mg�矄犨 ᨨ Y Mg�矄 ᨨᆠᨨ Y 盐ᨨZ. 㡰 Mg
r r uur mw r DU = - ò F dr = + ò mwrdr = \Uµr
꾀矄ᨨ = a ᆠᨨ矄 wt + b 矄� wt \ r㦨f t㡰 ᨨᆠ�矄ᆠ \쟼㦨 r dr r r ˆ ˆ r t 㡰 = a t� + 矄� w t ® v 㦨 dt ˆ 矄�
t � ᆠᨨ矄 t ˆ 桌 a w w + w w 㦨 \ 㦨aw㦨ᆠᨨ 矄 r r 㦨 dv 㦨awᆠᨨ矄wt ˆ 졔矄� wt ˆ 桌 � a dt r a 㦨 w rr \ r r r t 㦨 r ´ F 㦨 \ 㦨ᆠᨨ 矄 ᨨᆠ�矄ᆠ � 盐 ᆠ ⧤矄ᨨᆠᨨ 矄 r dr b r 和 r t 㡰 = a ᆠᨨ矄 w t�ˆ + t ˆ ® vr 㦨 㦨 at �ˆ + bt ˆ \㦨f t㡰 dt r a 㦨 bˆ 㦨ᆠᨨ 矄 r r F 㦨 ma 㦨 ᆠᨨ 矄
鼀 �
\ ᨨᆠᨨ Y�矄 ᨨ� Mg = B + Fv B졔Fv 泐Xᨨ Y
���ᨨ ᨨ � �泐 ᨨ X �矄 ᆠᨨ � ᨨ盐矄泐� ᆠ矄� � ᆠ盐矄矄 Y 鼀ᨨ矄 ᨨ犨 ᆠ ᨨ 鼀矄矄 ᨨ X 鼀ᨨ矄 盐
҈
N f
҈黨 q
q
mgᆠᨨ矄q mg
㦨 q + ҈ 㡰 mg ᆠᨨ矄 q = 矄ᆠ qmg ᆠᨨ矄 q = mg 㦨犨�� ᨨ Y 꾀�� 盐 ᨨ 矄 ᆠ ᨨ �ᆠ�ᨨ ᨨᆠ 犨 Y X 鼀ᆠ �ᆠ �泐 ᨨ 矄泐矄鼀 X + Y 㡰 ᆠ矄矄 ᆠᨨ � ᨨ盐矄泐矄 Y 矄� 矄 ᨨ犨
B
Fv
Mg
鼀ᆠ �ᆠ �泐ᨨ 矄泐矄鼀 X졔Y㡰�矄ᆠᨨ � ᨨ盐矄泐 ᆠ矄� � ᨨ ᨨᆠᨨ鼀 �矄ᆠᨨ盐矄 ᨨᆠ矄
f + N = m N + N = m + ҈N
mg矄� q
Mg
R = Mg 㡰 + m + M 㡰 g 桌 ¹ Mg
鼀ᆠ �ᆠ �泐ᨨ 矄泐矄鼀 X 졔 Y㡰�矄ᆠᨨ � ᨨ盐矄泐 ᆠ矄� �矄 矄泐矄鼀 �矄 ᆠᨨ鼀� � ᨨ犨 �矄 ᨨ � �泐�矄 ᆠ矄� � ├� �ᆠ �泐鼀� � � 矄鼀
ᨨ⼄盐ᨨ 犨�� ᨨ Yᨨ盐 P ¹ 矄㡰 ᨨᆠ ᨨ Y 泐 X �矄 㦨 犨 ᨨ �⼄盐� �矄ᆠ 犨 �ᆠ ᆠ
ᨨ⼄盐ᨨMg 矄 矄�泐ᨨ Y> 矄�泐ᨨ X Y�矄矄� ├� �㡰
���ᨨ ᨨ � �泐 ᨨ X � ᆠ矄矄 ᆠᨨ � ᨨ盐矄泐ᆠ盐矄矄 Y鼀ᨨ矄 ᨨ犨 ᆠ ᨨ 鼀矄矄 ᨨ X 鼀ᨨ矄 盐 㡰 和 Y�矄鼀ᨨ� �犨� 鼀� ᨨᆠ�泐
t
v
R
ᔼ �矄ᆠ矄 ᨨᆠ 泐Xᨨ Y㦨 ᨨᆠ 泐 Yᨨ X ᨨᆠᨨ X 盐ᨨY�矄
r r mbt D㦨 ò F × dr 㦨 m ò bˆ × a�ˆ + b tˆ 㡰dt 㦨
҈ ҈ ⧤㦨 mv = m a + b t 㡰 ҈ 꾀矄ᨨ쟼㦨⧤졔㦨 ma 㦨ᆠᨨ 矄 ҈䰘 꾀 ® 㡰 錐 ® ⼄矄㡰 錐 ᔴ ® 㡰 ® ⼄ 㡰 꾀矄 ᨨᆠ�泐�矄 ᆠᨨ 矄 f = mg 矄� q \ 盐 f = mN = mmg ᆠᨨ矄 q \ mmg ᆠᨨ矄 q = mg 矄� q Þ m = q
ᨨᆠ 泐Xᨨ Y�矄 矄盐 ᨨ f N
mg
Ӹ
҈
æ R - rö ÷ R ø
ᨨ ├� �ᆠ �泐ᨨ � � 㦨 ⧤� �ᆠᨨ�ᨨ 졔 ├� �ᆠ �泐 矄�ᨨ 㡰 ҈ K Etᨨtal = MR 㡰w = M w R - r 㡰 㡰 mM w鼀� R - r 㡰 = Mg g \ w鼀� = m R - r 㡰 㡰 꾀 �矄 ᨨ ᨨ�ᨨ �矄 ᨨ 㜠 �矄 ᆠ ᨨ ᨨ ᆠ矄矄 � 矄 ᨨ盐矄 �矄 矄矄� � ᨨ盐� �矄 �ᆠ 㡰 ᨨ��� ᨨ 泐w �矄 矄鼀 ᨨ 泐 ᨨ� ᨨ ᨨ 泐 㡰 kxf㦨Ma a㦨Ra a kx - f é fR ù \ = Rê ú a ҈ M ê MR ú ë û 2kx é fR ù = Rê ú ҈ f ê MR ú ë û ᆠ㡰 w R - r 㡰 = wR \ w = wᨨ çè
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EBD_8341 PS-76
Physics
和ᨨ� � �矄 ⼄盐�ᨨ 犨 � ├ ├ \ = ├ - = ├ =
�矄 �矄 ᨨᨨ矄� ᨨ �矄ᆠ鼀 kx f = kx \ Fnet = �ᆠ ᨨ犨 矄 ⼄盐���盐鼀
æ kx ö ÷x è ø k F æ k ö x = -w x Þ w = \ a = net = - ç è M ÷ø M M ᆠ㡰 㝐ᆠ �ᆠ �泐�矄ᆠᨨ 矄 � ᆠ矄ᨨ 盐ᨨ� � 鼀ᨨ�ᨨ
҈ é҈ ù ҈æ҈ ö æ v ö Mv + MR \ ç ÷ ç ÷ = ê kx 鼀 ú è R øè ø ë û M x鼀 = v \ k kx鼀 k M F鼀 = mMg = v = k M v = mg \ k � b㡰 ᨨ矄矄 � ├� �ᆠ �泐㦨 KE㡰� �� KE㡰 � 㡰 Fnet = ҈ - ç
é҈ é҈ æ ö ù ҈ ù 㦨 ê I w 㡰 + I 㡰 w ú - ê I + I 㡰 ç w ÷ ú è ø ú ë û êë û
䰘
I w 㦨 I w - I w = 㡰 ␔ �矄ᆠB �矄ᨨ盐� � ᆠᨨ ᆠ 犨� �矄ᆠ A w ᆠᨨ鼀鼀ᨨ ᨨᆠ�泐 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ 犨ᨨ �矄ᆠ 矄泐矄鼀 I w㡰졔I w㡰㦨 I졔I㡰wἘ Þ wἘ㦨 w
ᨨ⼄盐 ᨨ �矄ᆠ A I ´ w - I ´ w DLA L f - L� - I w = = = tA㦨 t t t t ��矄�� � �ᆠ矄 ᨨ⼄盐ᆠ矄 �盐 �ᨨ ᆠ㡰 ᨨ �矄ᆠ A 2w ҈ ҈ kx҈ = I w 㡰 Þ kx҈ = I w A ᨨ �矄ᆠ B ҈ ҈ kx = ´ I w Þ kx = I w w x k x҈ I w B Þ ҈ = = \ x kx Iw �矄 � ᨨ� ᨨ ᨨ矄��ᨨ
\ ᨨ⼄盐 ᆠ� � ᨨ �矄 ᨨ m b⏌
F
a
b⏌
mἘ
Mg
b⏌
b b =F´ �㡰 n 盐鼀ᨨ 矄ᆠ ᨨ 鼀矄矄 m矄�├� �犨� ᨨᆠ�泐v ᨨ 矄 ᨨ�ᨨ ᨨ b dp F㦨 n 맔A㦨n맔 mv㡰맔a맔 Q A㦨㡰 dt 盐� � �矄盐ᨨ � ⼄ �㡰 b b b Mg맔 㦨n맔 mv㡰맔a맔 맔 ´ Þ 맔҈㦨҈맔맔҈맔v맔҈맔 \ v㦨҈鼀⏌矄 Y 黨 和� ᆠ 矄泐矄鼀�矄� ⼄盐���盐鼀 P Mg맔
\
X
SFy㦨
mg
Mg졔mg졔MgNN 㦨
Þ Mg졔mg졔Mg㦨N졔N
N
Mg f
A
N
Mg f
B
m2
mg
M + m㡰 g Þ N㦨 �㡰 �ᆠ�ᨨ ᨨᆠ꾀 ᔴᆠ盐� �ᨨ⼄盐矄 ᨨ盐 P Q P Mg맔PQ졔f맔PM㦨N맔OP O q Þ Mg맔
L ᆠᨨ矄 q + f ´ L 矄� q = NL ᆠᨨ矄 q
MgL NL ᆠᨨ矄 q ᆠᨨ矄 q Þ f㦨 L 矄� q
N A
l⏌ q f
Mg M
Mg ᆠᨨ q éæ N + m ö Mg ù ᆠᨨ q Þ f㦨 ê ç ÷ø g è úû ë gù é Þ f㦨 ê( M + m) ú ᆠᨨ q û ë 꾀盐ᨨ 鼀矄矄m矄�├矄 犨ᨨᨨ ᨨ�ᨨ 鼀矄矄 M � L 矄�ᆠ├矄ᨨ� ur ur r r dL Þ t ´ dt = d L
ᨨ⼄盐 t = dt r r ␔ �盐 �鼀盐矄 t ´ dt 㡰 �矄 ᨨ �盐 鼀ᨨ鼀 盐鼀 �矄 ᆠᨨ 矄 ᔼ �矄 ᆠ矄 ᨨ 犨ᨨᨨ
ᨨ�㜠盐矄泐矄鼀 �盐�鼀盐矄ᨨ盐O�矄ᆠᨨ 矄 \ �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀桌� �� 㦨 �盐 鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀桌 � Þ mv맔L㦨I맔w �㡰 犨 I�矄 鼀ᨨ鼀 ᨨ � �ᨨ 犨ᨨᨨ ᨨ�㜠盐 矄泐矄鼀 ᆠᨨ�矄�ᨨ ᨨ盐 O I㦨I犨ᨨᨨ ᨨ�졔I盐 ҈ 㦨 ML + ML ��㡰 ᨨ鼀⼄ �㡰 ��㡰 mv mv ´ L w㦨 = M + m㡰 L é҈ ù êë ML + mL úû = N ᆠᨨ q -
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PS-77
System of Particles and Rotational Motion ҈
㡰 ᨨ� Ἐ졠Ἐ ᨨ ᆠ ᨨ 鼀矄矄 ᔴ㝐㡰 ᨨ 矄泐矄鼀 \
l
OA㦨 r =
+ w� = + -҈㡰 ���㡰 I AB ҈ + x ᨨ鼀⼄ ��㡰 ���㡰 é ù - ҈ + ҈ú Q w� = ҈ᨨ ⏌矄㡰 㦨맔x ê ҈ + x ë û é ù ҈㦨�x ê ú ë҈ + x û Þ ҈졔x㦨�x졔桌㦨҈�x \ x҈�x҈㦨 和ᨨ� �犨�x㦨҈ᨨ 盐x㦨鼀wfᆠᨨ鼀矄ᨨ盐ᨨ �� �ᨨ ⏌矄㡰 犨 �ᆠ �矄 ᨨ ᆠᆠ ᨨ 㡰 x㦨 �矄 ᆠᨨ ᔴ㝐 ᨨ鼀� 꾀㦨҈鼀 㡰 和盐矄�盐� �x㦨҈鼀� ⼄ ��㡰犨�wf㦨҈ᨨ ⏌矄 �盐 ᨨᆠ�泐犨� 犨 �ᆠ 矄 ᆠᨨ鼀矄 ᆠ├ �矄�鼀ᆠ�矄҈ ⏌矄 ᆠ㡰 꾀矄 矄 盐 矄 犨� 矄鼀 �盐 ᨨᆠ�泐ᨨ ҈ ⏌矄 矄 犨� ᆠᨨ鼀ᨨ矄
\
泐 �ᨨ鼀泐㡰 A m㡰 y x
l
l
O CM F
l
r
l
B m㡰
C m㡰
F㦨ᆠ � ᨨᆠᨨ F㦨 m㡰rw
æ ҈ ö w F = m㡰 ç è ÷ø ᨨ F = ml w 㡰 a �盐 ᆠᆠ�ᨨ ᨨ 矄泐矄鼀 ᨨ盐 ᨨ� A æ ö F 㡰 çç l÷ ÷ø tA F è a= = = IA ml ml ᨨ犨ᆠᆠ�ᨨ ᨨ ᔴ㝐ᨨ �x㜠 �矄�矄
ᨨ
æ l ö æ F ö ax = r a = ç ÷ ÷ç è ø çè ml ÷ø
F ᨨ ax = m Fx ᨨᆠ� 泐 � �ᨨ � X㜠 �矄 \ Fx 졔 F = m㡰ax F ö ᨨ Fx + F = m㡰 æç ᨨ Fx + F = F ÷ è m ø F Fy ᨨᆠ� 泐 � �ᨨ �Y㜠 �矄 \ Fy 㦨 ᆠ � ᨨᆠ ᨨ
Fx =
ᨨ Fy = ml w x �矄 ᆠᨨ ᆠ ᨨ 鼀矄矄 ᨨ鼀� AB \ 㝐ᔼᨨ 鼀� 矄 ᨨ盐AB IAB㦨Iᆠ鼀졔mx ᨨ鼀 矄 ᨨ鼀㡰 IAB㦨҈졔x �㡰
wf㦨
ᔼ ��泐 �矄ᆠ �矄 �ᆠ犨� ᨨ� A�矄 �� 矄 ᨨ矄��ᨨ ᔼ 矄矄 ᨨ� �犨 ᨨ犨 ᨨ ᆠ �矄 ᨨ � �泐 ᆠ �矄 ᨨ ᨨ�ᨨ ├� �ᆠ �泐
A m R C
꾀泐� � �泐 ᆠᨨ 矄�ᨨ
턼
P
ᆠ鼀
B m㦨├�
IAB wfw�㡰㦨맔�㦨
R
ᆠ ᨨ쟼
R ö R ö ҈ æ æ + mg ç R + = mgR + I w ÷ ÷ è è ø ø ␔ I㦨 MI ᨨ �矄ᆠ㜠鼀矄矄矄泐矄鼀ᨨ盐PQ ҈ ҈ R R ҈ + mg = mgR + I w Þ mgR㦨 I w mg맔 mgR I I㡰PQ㦨 I �矄ᆠ㡰PQ졔 I鼀矄矄㡰PQ
Þ w㦨
�㡰
é mR æ Rö ù æ �R ö ê + M ú + m 㦨 çè ÷ø çè ÷ø ûú ëê
҈ MR 桌
mR + ҈ + �桌 ҈�mR = ҈ 盐� � �矄盐ᨨ I� ⼄ �㡰 ᔼ㦨
mgR ´
҈ g �R ҈�mR
� 矄 ᨨ �ᆠ 矄 � ᆠ 矄 �矄 ᨨ犨矄 ᨨ矄��ᨨ
w㦨
ᆠ盐矄 ᨨ �鼀盐矄 �盐 ᨨᆠ�泐 ᨨ 鼀� 矄 犨� ᆠ � 泐�鼀ᆠ ᔼ鼀盐矄㦨ᆠ �� � 鼀ᨨ鼀 盐鼀 㦨 Vf V�㡰 㦨맔x wfw�㡰 ��㡰 꾀矄ᨨᆠ �� �盐鼀ᨨ鼀 盐鼀㦨鼀ᨨ鼀 ᨨ �鼀盐矄 \ IABwfIABw�㦨ᔼ鼀盐矄맔 �矄 ᆠ
ᔴἘ
mg ç R +
Q MIᨨ �矄ᆠᨨ盐 �鼀㦨 x
Q
R⏌
v
쟼 �� �泐㦨 � �泐
A
鼀 �
R/4
P
=
҈ g �R Rö æ ´ = � gR \v㦨 ÷ �R è ø 㡰 ᆠᨨ�矄�ᨨ �矄 矄�ᆠ 矄 �矄 � ᆠ 犨 � 犨� 盐 ᆠ├ 犨� 矄鼀 ᨨᆠ�泐 � 鼀� �盐 㡰 v㦨w ç R +
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EBD_8341 PS-78
Physics Z
Y
꾀矄ᨨ
X
\
F
t fR f f = = = = ҈ f I mR ⏌ mR ´ ҈ a㦨҈f ��㡰 m ├� ├� z a=
ᨨᆠ矄ᨨ 矄� ᨨᆠ � ᨨᆠ�泐ᨨ 犨 � � X㜠 �ᆠ�ᨨ �矄Fx Fx맔Dt㦨mv mv㡰 ᔼ鼀盐矄㡰㦨 ᔴ �� 鼀ᨨ鼀 盐鼀㡰
҈鼀 O y
mv mv mv Þ F㦨 Þ Fᆠᨨ矄 㦨 Dt Dt Dt Q Fx㦨Fᆠᨨ矄 Fy㦨F矄� ᔼ ᆠᨨ 鼀矄 ur F 㦨 Fx $� + Fy - k$ 㡰 = F ᆠᨨ矄°�$ + F 矄� ° -k$ 㡰
\
Þ
f
ᨨ
$ ҈ � + F ´ - k$ 㡰
mv F + mg = + mg Dt
�
mv Dt
ᨨ ᨨ ᨨ
� ᨨ ᨨ 㦨ᆠ鼀㦨鼀
\
a
ᨨ f =
(
) ( )
r t҈ = -҈ ˆ - ҈kˆ ´ �ˆ N - m
r t҈ = kˆ - ˆ r t҈ = kˆ - ˆ 㡰 N - m
�㡰
ᨨ
(
)
t = - ˆ - kˆ 㜠鼀
r t = 㡰 + 㡰 = ´ = � 㜠鼀 r r t҈ = t = � 㜠鼀
�盐矄ᨨ ᆠ �矄ᆠ㦨R㦨҈ᆠ鼀㦨҈鼀
a㦨� ᆠᆠ�ᨨ ᨨ ᆠ ᨨ 鼀矄矄ᨨ �矄ᆠ a 㦨 �盐 ᆠᆠ�ᨨ ᨨ盐 �矄 ᆠ ᨨ 鼀矄矄 f f a = = �㡰 m
ᨨ f =
r r r 和�鼀�泐 t = r ´ f
� 㝐矄矄ᨨ ᆠ �矄ᆠm㦨├�
�㡰
f =a
㦨�㜠鼀
�矄ᆠ㜠ᨨ 矄泐矄鼀ᆠ ᨨᨨ 盐ᆠ├犨� ᨨ盐矄�� � 矄 �ᆠ�ᨨ ᨨ �矄ᆠ㜠ᨨ 矄泐矄鼀 犨� ᨨᨨ ᨨ 鼀ᨨ� �盐ᆠ├�矄�
�矄ᨨ �矄ᆠ㜠ᨨ ᨨᆠ�矄 ᨨ� ᨨ �ᆠ盐 ᨨ �ᆠ�ᨨ ᨨ 鼀ᨨ�ᨨ ᨨ 盐ᆠ├z㜠 �矄�矄 �ᆠ泐 盐犨 �ᆠ�ᨨ
fö ÷ 졔 ҈㡰 ҈f 㡰㦨 ᨨ鼀 �㡰㡰 ø
r t҈ = 㡰 + 㡰 = ´ = ҈҈ ´
´h
꾀ᆠᆠ�ᨨ ᨨ 盐ᆠ├ᨨ �x㜠 �矄㦨 黨鼀⏌矄
æ çè
��㡰 �ᆠ�ᨨ ᨨ⼄盐 t = r ´ f r 꾀 P r҈ = ҈ ˆ - ҈kˆ r 꾀 Q r = ҈ ˆ - ҈kˆ r r r \ t҈ = r҈ ´ f
æ mv ö + mg ÷ k$ N㦨 ç è D t ø
�矄�矄 ᨨ鼀 ᨨᆠ 泐 ᨨ 犨 � 盐� � �鼀D b㡰 ᨨ⼄盐矄 ᨨ 犨 � ᨨ盐 ᆠ ᨨ 鼀矄矄 ᨨ 犨 � F맔h ᨨ⼄盐 盐ᨨN졔mg맔㦨 h�矄 �ᆠ盐 �矄 ᆠ犨 ᆠ ᨨ 鼀矄矄ᨨ 犨 � � ᨨ ᆠ�ᨨ ᨨ
ᨨ⼄盐 盐ᨨN㦨F맔h㦨
f
´ 黨 = N
�矄 ᨨᆠᨨ �ᆠ�ᨨ f�矄 ᆠ� �� ᨨ矄��x㜠 �ᆠ�ᨨ �ˆ �矄 盐 �ᆠᨨᨨ � x㜠 �矄 r \ ᔼ ᆠᨨ ᨨ鼀 f = �ˆ㡰 犨ᨨ
���㡰 ᨨ
쟼⼄盐���盐鼀ᨨ ᨨᆠ� Z㜠 �ᆠ�ᨨ ᆠ� �ᨨ 犨 �㡰 Fy졔mg㦨N
\
Q
和� ᆠ �矄 ᨨ 矄�� � 犨 �矄ᆠ 盐ᆠ├ ᆠᆠ�ᨨ ᨨ P㦨 ᆠᆠ�ᨨ ᨨ Q a졔Ra㦨a
ur F mv $� - k$ 㡰 F 㦨 $� - k$ 㡰 = Dt
Þ N㦨
P
Fx㦨
㦨F맔
҈鼀
㡰 ᆠ泐� ᨨ �盐矄 鼀矄矄ἘmἘᨨ矄ᨨ �ᨨ ᆠ �盐 ᨨᆠ├犨� ᨨ盐矄�� � ᨨ�ᨨ ᨨ ᆠ泐� �矄 ᨨ盐 ᨨ� ᨨ ᆠᨨ ᆠ 矄ᨨ �泐 ᨨ ᆠ泐� �矄 ᆠᨨ 矄 ᨨ��� ᨨ矄��ᨨ ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� O␔ �ᨨ� � ᨨ犨 ᨨ � ᆠ泐� 矄盐ᆠ ᨨ矄��ᨨ �矄ᆠ ᨨ 鼀矄矄�矄ᨨ矄��ᨨ OἘ ÐNPO q 꾀矄 ᆠ泐� �矄ᨨ� � ᆠ鼀ᨨ矄
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PS-79
System of Particles and Rotational Motion � ᆠ�ᆠ盐 ᆠ � ᨨᆠ ⼄盐� ᨨ ᆠ�ᆠ盐 鼀ᨨ�ᨨ N
҈ ҈ ҈ æ g ö mgR 㦨 ´ mR ´ ç ÷ + m v Ἐc 㡰 R 䰘 è ø
O
mvc mgᆠᨨ矄 q N㦨 R
ᆠᨨ ��ᨨ ᨨ ᆠ泐� � � � N㦨
R
h M Rh
w q os
gc
m
q
OἘ
q R mg P
mg矄� q
vc mvc Þ ᆠᨨ矄 q 㦨 �㡰 Rg R 꾀泐� � �泐ᆠᨨ 矄�ᨨ ᨨ鼀 O ᨨ OἘ ᨨ矄矄 ᨨ ᨨ � �泐ᨨ ᆠ泐� 㦨�� � 矄�ᨨ KE졔�� � ᨨ�ᨨ KE
\
vc gR ⏌ 䰘 = R R mgR mgR Þ mgR 㦨 㦨 矄�ᨨ KE 䰘 䰘 ҈ mgR 꾀矄ᨨᨨ�ᨨ KE㦨 Iw = 䰘
矄�ᨨ K E = \ ᨨ�ᨨ K E w㦨
Q
vc
mgᆠᨨ矄 q 㦨
҈ ҈ mgh㦨 mvc + I w ��㡰 ␔ I�矄 鼀ᨨ鼀 ᨨ � �ᨨ ᆠ泐� ᨨ盐OἘ ҈ I = mR , w �矄 �盐 矄 Vc �矄 ᨨᆠ�泐 ᨨ ᆠ ᨨ 鼀矄矄
dL 䰘 和� ᆠt㦨 t㦨 dt
P
\ L �矄 ᆠᨨ 矄
꾀 �盐鼀ᨨ鼀 盐鼀 ᨨᆠᨨ�矄�ᨨ 㦨mv맔 l l 꾀 �盐鼀ᨨ鼀 盐鼀 ᆠᨨ�矄�ᨨ 㦨Iw I㦨Im졔I꾀졔I
vc R 盐� �盐ᨨ I w� ⼄ ��㡰
m 㦨�├�㡰
é ì æ l ö æ l ö ï ìï æ l ö æ l ö üï ù ï ê m l 㡰 + 㦨 í mA ç ÷ + ç ÷ ý + í mB ç ÷ + ç + l÷ ý ú è ø ï ï è ҈ ø è ø ï ú ҈ êë îï è ø þ î þû ü
mgh 㦨
é æ l l ö æ l 黨l ö ù + ÷ + mB ç + 㦨 ê ml + m A ç ÷ú ø úû êë è ҈ ø è ҈
é ë
���㡰
gr ҈ - ᆠᨨ矄 q㡰 Rg
䰘 b㡰 和 ᨨ ᔴ㝐ᨨ ᆠ泐� ᨨ� �ᆠᨨ ᆠ犨� �
gR gR æ ö gR ᨨ鼀 ���㡰桌 Þ vc㦨 ç҈ - ÷ = 䰘 è 䰘ø 䰘 ᆠ㡰 ᨨ ᆠ ᨨ 鼀矄矄 ᨨ ᆠ泐� ᆠ 矄 ᨨ� ᨨ � ᨨ ��矄 ᆠᨨ ᆠ犨� � q = ᆠᨨ矄 -҈ = ��҈�° 䰘 ᆠ ᨨ�ᨨ K.E 犨 �ᆠ ᆠ泐� �� 矄 �鼀ᨨ � �ᆠᨨ ᆠ犨� �鼀� 矄 矄鼀� �矄 盐 鼀ᨨ�ᨨ ᆠ泐� �� 矄 矄�ᨨ K.E 꾀�� 泐� � �泐 ᆠᨨ 矄�ᨨ ᨨ鼀 O ᨨ ᨨ� 犨 ᆠ ᨨ 鼀矄矄�矄 � ᨨ� ᨨ � 犨� � ҈ ҈ mgR㦨 I w + m v Ἐc 㡰
䰘 ù m l = 黨 ├� m B úû
Iw㦨mv맔 l
mv ´ l � ´ v ´ ´ = 㦨䰘v 黨 I 꾀泐� �ᆠᨨ 矄�ᨨ ᨨ 鼀ᆠ �ᆠ �泐 ᆠᨨ�矄�ᨨ ᨨ矄矄ᨨ KE㦨�� � PE Þ w㦨
҈ æ lö æ l ö I w = mg l㡰 + m A ç ÷ g + mB g ç ÷ è ø è ø
Þ ᆠᨨ矄 q 㦨ᆠᨨ矄 q Þ ᆠᨨ矄 q 㦨
vc =
҈
㦨 ê ml + m A l +
ᨨ鼀⼄ �㡰 ���㡰犨�
ᆠᨨ矄 q =
mB B
v
꾀矄ᨨ ᨨᨨ� �vc㦨wR Þ w㦨
v ҈ ҈ ҈ mvc + ´ mR ´ c R ҈ ҈ gh Þ gh㦨 vc + vc = vc Þ vc = R-h ᔼ DO¢MP ᆠᨨ矄 q = R Þ h㦨R ҈ᆠᨨ矄 q 㡰 g R ҈ - ᆠᨨ矄 q㡰 \ vc =
mA A
҈ ´ 黨 ´ 䰘 v㡰 ҈ ù é 㦨 ê � ´ + ҈ ´ + ´ ú ´ 黨 ´ û ë Y Þ v㦨鼀⏌矄 R T w �㡰 꾀ᆠᆠᨨ � � ᨨ M ⼄盐矄�ᨨ 盐鼀 �矄 m T �� � �� �盐 X ᨨᆠ�泐 矄盐ᆠ q ᨨᆠ├ X 矄矄 鼀ᨨ� � 盐 mg矄� q mg mg ᆠᨨ矄q Þ
q=30 o
꾀矄 �鼀矄矄矄 ᨨᆠ├ X �矄矄 盐 ᨨᆠ�泐 ᨨ ᨨᆠ├ ᆠ矄矄a � �ᨨ ᨨ ᨨᆠ├X. \ mg 矄� q - T = ma �㡰
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EBD_8341 PS-80
Physics
� �ᨨ ᨨ ᨨᆠ├ a �盐 ᆠᆠ�ᨨ ᨨ 盐鼀 a㡰 矄
a R 犨 R�矄 �盐矄ᨨ 盐鼀
� �ᨨ⼄盐ᨨ 盐鼀�矄 盐ᨨ 矄�ᨨ T� 矄� � t㦨T맔R㦨Ia Q t㦨Ia㡰 a㦨 a Þ a =
\
҈ T맔R㦨 MR a
҈ é ù êëQ I = MR úû
a T ҈ MR Þ a㦨 R M 和盐矄�盐� � �矄 盐ᨨ � ⼄ �㡰
Þ TR㦨
mg矄� q T㦨m맔
T æ m ö Þ mg矄� q 㦨 ç҈ + ÷T è Mø M
mg 矄� q㡰 ´ M � ´ 黨 ´ 矄� ° ´ = 㦨҈ M + m + ´ � ��㡰 �矄 ᆠ 泐x
\
T㦨
T ´ ҈ = = ҈ m ⏌ s M 盐矄� �V㦨RwÞ V 㦨맔҈㦨鼀⏌矄 Q R㦨m w㦨҈ᨨ ⏌矄�� 㡰 \ �矄 ᆠ 泐 ᨨᆠ├ x
a=
´ v = a ´ ҈ \ �矄 ᆠ 泐x㦨҈鼀 㡰 꾀 �盐鼀ᨨ鼀 盐鼀ᨨ 矄泐矄鼀 ᨨ 졔� 矄ᆠ㡰ᨨ盐 ᆠ ᨨ 鼀矄矄Ἐ졠Ἐ ᨨᆠᨨ�矄�ᨨ 㦨 �盐鼀ᨨ鼀 盐鼀 ᨨ 矄泐矄鼀 ᆠᨨ�矄�ᨨ L M Mv맔 = I w v ␔ I �矄 鼀ᨨ鼀 ᨨ � � ᨨ 矄泐矄鼀 盐矄 ᆠᨨ�矄�ᨨ w A B O �矄 �盐 ᨨᆠ�泐 L⏌ L⏌ 盐矄 ᆠᨨ�矄�ᨨ as㦨盐Þ s =
黨
Þ
Þ
Mv
ù L é æ Lö ҈ = ê M ç ÷ + ML ú w êë è ø ҈ úû
L ML é ҈ ҈ ù ML é + ù Mv ´ = + úw= ê ë û êë ҈ úû
҈ v 䰘 ML ´ ´ w Þ w㦨 䰘 L ҈ b㡰 꾀 ᆠᨨ�矄�ᨨ �矄 鼀矄矄Mᨨ � 矄ᆠ 犨 �ᆠ ᆠ矄 ᨨ⼄盐 � ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 犨 �ᆠ 矄 ᨨ ᆠ �盐 ᆠᆠ�ᨨ � ᨨ 盐 矄鼀 �矄 ᆠᨨ鼀 矄 泐 鼀ᨨ鼀 ᨨ � 矄ᆠ ᨨ犨 矄 B 盐ᨨ犨 �ᆠ 鼀ᨨ鼀 ᨨ � �Iᨨ 矄泐矄鼀� ᆠ矄矄 泐 � 矄 ᨨ �鼀 t � 矄ᆠ �矄 ᆠ x 㦨
ᨨ鼀 ᆠ ᨨ ᨨ ᨨ 矄 盐 ᨨ盐� � q 㦨wt㡰犨�矄ᨨ��� ᨨ矄��ᨨ
dL d A = I w㡰 㦨 dt dt M
dI = w dt 㦨 w
A
q =w t
O x
d é҈ ù ML + Mx ú ê dt ë҈ û
B N B
dx Mg dt
�矄 ᨨ⼄盐 �矄 ᆠ 泐 ᨨ⼄盐 盐 ᨨ 犨�� ᨨ � 矄ᆠ dx æ g ö ᆠᨨ矄 wt dt \ Mwx 㦨Mg xᆠᨨ矄q㡰 Þ dx㦨 ç è w ÷ø dt 졠 � ��ᨨ ├� ��鼀�矄 L⏌ g p ⏌ w ò L ⏌ dx = w ò ᆠᨨ矄 wt dt L 犨
x = wt = L⏌ g p ⏌ w [ x] L ⏌ = [ 矄� wt ] w p L 犨
x㦨 wt = g é p æ L Lö ù Þ ç - ÷= 矄� - 矄� ú è ø w êë û 㦨Mwx
Þ
L g g = Þ w㦨 L w
盐 w㦨
҈ v ҈ v = Þ 䰘 L 䰘 L
g 䰘 gL Þ v㦨 L ҈
䰘 ´ ҈ ´ ҈ 㦨�鼀矄҈ ҈ ᨨ ᆠᨨ�矄�ᨨ ⧤� �ᆠ �泐ᨨ 矄泐矄鼀 ҈ ҈ ҈ ⧤쟼� 㦨 m҈v҈ + m v + MV ҈ ҈ �㡰 㦨 ´ ҈ + ´ + = � Þ v㦨
M 㦨҈├� ҈鼀⏌矄
A
m҈ 㦨├�㡰
x = �鼀 C x = �鼀
m
鼀⏌矄 㦨├�㡰
B
꾀泐� � 犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 盐� � ᆠᨨ�矄�ᨨ
m҈맔v҈졔m맔v㦨 M졔m҈졔m㡰Vc 犨 Vc �矄 ᨨᆠ�泐 ᨨ ᆠ ᨨ 鼀矄矄 ᨨ �ᆠ矄 矄�ᆠ├ ᨨ � ᆠᨨ�矄�ᨨ
맔҈졔맔㦨 ҈졔졔㡰Vc Þ Vc㦨鼀⏌矄
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PS-81
System of Particles and Rotational Motion \
⧤쟼 矄 ᆠᨨ�矄�ᨨ
҈ 㦨 M + m҈ + m 㡰Vc = � ��㡰 꾀泐� � ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ 犨ᨨ �ᆠ 矄泐矄鼀 ᨨ盐 ᆠ ᨨ TExternal㦨 \ L㦨ᆠᨨ 矄 L � 㦨m҈v҈맔xmvx 㦨맔҈맔�맔맔� 㦨㦨҈├�鼀矄҈ ᔼ ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 㡰 Lf㦨Iw
\
҈
҈ ҈ ⧤쟼 ᨨ㦨 I w = ´ ´ = ҈ �㡰 K.E.f 㦨 矄�ᨨ KE졔 ᨨ�ᨨ KE 㦨�졔҈㦨䰘 D KE㦨KE� � KEf 㦨�䰘㦨䰘 PED㦨PEQ졔 KE㡰Trans졔 KE㡰Rᨨt ᨨ鼀 �泐ᆠᨨ 矄�ᨨ � ᆠ� ҈ ҈ Þ mg 㡰㦨mg ҈㡰졔 mv + I w ҈ ҈ æ ö v mv + ç mr ÷ ør è� vö æ çèQ I = mr w = ÷ø � r
䰘v ᨨ g ´ ҈ = Þv㦨鼀⏌矄 ҈ 꾀 ᨨ� Q ᨨ D ᨨ 泐ᨨ�ᆠ ҈ Q 鼀⏌矄 鼀 h㦨 gt Þ҈㦨黨 t y ҈鼀 ҈ �矄ᆠ = \ ty㦨 B C A 黨 �矄 ᆠ矄BC㦨V맔ty㦨맔�㦨鼀 盐� � �矄 �� 矄 ᨨᆠ� 矄 ᆠᨨ � 盐矄 ᨨ ᨨ ᆠ盐矄 ᨨ ᆠᨨ 矄�ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 矄鼀鼀矄矄m鼀ᨨ矄ᨨ盐 ᆠ�ᆠ盐 ᨨ �盐矄r 矄� �鼀├矄 �q犨� �ᆠ T 矄�ᨨ
Tᆠᨨ矄 q 㦨mg �㡰 T矄� q 㦨mrw ��㡰 l \
q 㦨
q
r w g
ᨨ鼀 ��盐矄� q 㦨
Þ r㦨 l 矄� q
T
r l
r
q
T ᆠᨨ矄q
T矄� q mg
q 㦨 l 矄� q
Þ w =
w g
qg ᨨ w = l 矄� q
g l ᆠᨨ矄 q
g ҈ ���㡰 Q w = pn p l ᆠᨨ矄 q ᨨMᨨ鼀� 矄�ᨨ 泐T㦨Mg Tᆠᨨ矄q㦨mg \ Mg ᆠᨨ矄 q 㦨 mg m Þ ᆠᨨ矄 q 㦨 M 盐� � �矄盐ᨨ ᆠᨨ矄q�矄⼄ ���㡰犨� Þ n㦨
é M l ù + m҈ x + m x ú w 㦨 ê ëê ҈ ûú é ҈㡰 㡰 ù + ´ �㡰 + 㡰 �㡰 ú w 㦨 ê ҈ êë úû 㦨w w㦨҈ Þ w㦨 ⏌矄 ���㡰
mg ( - ҈) =
\
n㦨
҈ p
gM l m
O
OM ᔼ DOPMᆠᨨ矄 q = l Þ OM㦨 l ᆠᨨ矄 q OM Ἐ ᔼ DOPἘMἘ ᆠᨨ矄 q = l Þ OMἘ㦨 l ᆠᨨ矄q
q0
q
T
㝐 㝐Ἐ
P P'
ᆠ mg矄� q ᨨ쟼
q mg 鼀�ᆠᨨ矄 q
OMἘOM 㦨 l ᆠᨨ矄 q - ᆠᨨ矄 q 㡰 ᨨ矄矄 � ᨨ � �泐㦨 � � ├� �ᆠ �泐 꾀ᆠ��泐P ᨨ PἘ㡰 ҈ Þ mg l ᆠᨨ矄 q - ᆠᨨ矄 q 㡰 = mv Þ v㦨 g l ᆠᨨ矄 q - ᆠᨨ矄 q 㡰 �㡰 Tmgᆠᨨ矄 q 㦨
mv l
mv + mg ᆠᨨ矄 q \ T㦨 ��㡰 l ᨨ鼀� �㡰 ��㡰 m T㦨 ´ g l ᆠᨨ矄 q - ᆠᨨ矄 q 㡰 + mg ᆠᨨ矄 q l \ T㦨mgᆠᨨ矄 q mgᆠᨨ矄 q Þ T㦨mg ᆠᨨ矄 q ᆠᨨ矄 q 㡰 ᨨ鼀 ⼄盐�ᨨ ��㡰� �矄 ᆠ 矄�ᨨ �矄 鼀 �鼀盐鼀 犨 ᆠᨨ矄 q 㦨҈� q 㦨 ᆠT鼀 㦨
mv + mg l
ᨨ鼀⼄ �㡰 v㦨 g l ҈ - ᆠᨨ矄 q 㡰 ᨨ鼀 ���㡰 �㡰 m T鼀 㦨 g l ҈ - ᆠᨨ矄 q 㡰桌 + mg l \ T鼀 㦨mgmg ᆠᨨ矄 q 㦨맔맔 ᆠᨨ矄 q
���㡰 �㡰
҈ Þ ᆠᨨ矄 q 㦨 Þ ᆠᨨ矄 q 㦨 Þ q = °
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EBD_8341 PS-82
Physics
Gravitation
7
v
Topic-1 : Kepler’s Laws of Planetary Motion ҈
꾀 꾀 ҈ q ҈ ҈ = 꾀㦨 q Þ 㦨 w 꾀矄ᨨ㦨鼀㦨鼀 w 꾀 ҈ \ = 鼀 ᆠ㡰 ᨨ �矄ᆠ 㦨 � ᨨ ᆠ 졔 和ᆠ和㡰 꾀ᨨ 和ᆠ和㦨 졔 꾀ᨨ �矄졔꾀ᨨ � �㡰 ᆠ
= 꾀ᨨ 和 ᆠ和㦨 꾀 ᨨ 和ᆠ和 ⏌ 㦨 㦨 ⏌ 꾀ᨨ 和 ᆠ和
҈
\
v҈㦨
҈
RA RB
v - v҈ ҈ ´ ҈ ´ � ⏌҈ 㦨 㦨맔҈ ⏌矄 R - R҈ ´ ҈ ´ ҈
㡰 � l㦨 꾀졔 㡰
├� �矄鼀鼀 鼀ᨨ � �矄 ᆠ ᨨ鼀 㦨 㦨 矄ᨨ 鼀㦨l 鼀㦨 꾀졔 㡰
R
mV 㦨 mRw ���ᨨ ᨨᆠᨨ ᆠ�ᨨ
R mRw㦨GMmR�⏌
+ 鼀 꾀 + 㡰
鼀
鼀 鼀
Þ = ò
R = ҈ T R
R T = ҈ Þ R = R҈ ´ T R T҈
R
pR p㡰 ´҈ 㡰 = = p´ ҈ ├鼀⏌ T
=
MA
T҈
T҈
R҈
Topic-2 : Newton’s Universal Law of Gravitation
Þ mR ´ p = GMm Þ TµR䰘⏌ T R� ⏌ �㡰 꾀ᆠᆠᨨ � � ᨨ⧤Ἐ矄 � 犨 ᨨ 泐鼀ᨨ�ᨨ
µ
\
㦨
⏌
S҈ P
ᆠ 矄 ᨨ 矄� S犨 S҈ 㦨vv҈㦨p맔҈p맔҈㦨p맔҈├鼀⏌ ��㡰 꾀 �盐矄 ᨨ S犨 S҈
⏌ æ R ö é R ⏌ù = ҈黨 泐矄 ÷ 㦨� ê Þ T㦨T҈ ç ú ë R û è R҈ ø b㡰 ᆠ � ᨨᆠ�矄ᨨ� 泐ᔴ � ᨨᆠ
㦨
S
v҈
pR҈ p´҈ = = p´ ҈ ├鼀⏌ T҈ ҈ v =
T҈㦨� 泐矄錐T㦨䁸錐 R҈㦨R R㦨
�
´
� 矄 ᨨ 矄� S
和
҈ 㦨 ᨨ 㦨 ҈ 㡰 ᔼ ᆠ矄 ᨨ � 泐 矄 矄泐矄鼀 ���ᨨ ᨨᆠ ᨨ ᆠ�ᨨ
犨 矄矄 犨� ᨨ� MB
ᆠ矄矄泐 ᆠ � ᨨᆠ矄 和ᨨ �盐 ᨨᆠ�泐 w ᨨ ᨨ 矄矄 �矄 矄鼀 ᨨ�鼀�ᨨ T p 㦨 鼀� 矄 矄鼀 w b㡰 꾀ᆠᆠᨨ � �ᨨ⧤Ἐ矄犨T µR
( )
R = ҈
= ´ ҈҈ ҈ Þ R㦨맔҈├鼀 � 矄 ᨨ 矄� S҈
ᆠ㡰 꾀 ᨨᆠ�泐錐
㦨
Þ
+ é æ҈ ù ҈ ö é 꾀 ù = 鼀 ê 꾀 ç = 鼀 ê - + ú ÷ + ú ë û ë è + ø û 㡰 꾀矄 犨├ ᨨ犨 ��ᨨ ᨨᆠᨨ ᆠ�ᨨ 㝐鼀 = 㝐 鼀 㝐 㝐 矄 ҈ = = ҈
D ҈ =
㝐 鼀 ҈
D҈ D =
㝐 㝐 矄
D
æ 鼀 ö æ ö æ D҈ ö D ҈ 鼀D҈ = =ç ÷ ÷ç ÷ç D ҈ 㝐矄 D è 㝐矄 ø è ҈ ø è D ø 矄� �D҈㦨D㦨 錐鼀㦨맔҈ ├�錐 㝐矄㦨맔҈├� ҈㦨맔҈├鼀 㦨҈�맔҈├鼀
D ҈ æ ´ ҈ ö æ ҈� ´ ҈ ö =ç ÷ç ÷ ´҈ @ D è ´ ҈ ø è ´ ҈ ø
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PS-83
Gravitation
F ᆠᨨ矄 �° + F ¢ =
㡰
␔ F = F ¢ =
GM
R㡰 GM
R ´ GM
Þ
Ἐ
㝐
Mv ᨨ鼀 ��盐㡰 R
R + h㡰 G 㦨 ���ᨨ ᆠᨨ 矄 M㦨鼀矄矄ᨨ
ᆠᆠ�ᨨ 盐 ᨨ ��泐 h �矄
ᨨ
+
GM
=
Mv R
GM æ ç҈ R è � g㦨g' gἘ =
㝐
㝐
R 㡰 R GM é ҈ ҈ ù + ú = Mv R êë û
Þ
\
GM æ + ö ҈ GM ҈ + 㡰 ç ÷= R çè ÷ø R 㡰 ᨨ盐鼀ᨨ 鼀ᨨ 矄
æRö æ ҈ö ÷ = pR ç ÷ èø è ø
V鼀ᨨ㦨 p ç
æ҈ö æ䰘ö èø èø
ᨨ 鼀矄矄 ᨨ 矄 ᆠ 鼀� � � 矄 ҈ 䰘 矄ᆠ�泐 㝐 㝐
ᨨ ���ᨨ ᨨᆠ 犨 矄 犨ᨨ 矄 䰘M ҈ GM m G ´ M ҈ M 䰘 GM ; F㦨 㦨 = r ´ 黨 R R 㡰
Topic-3 : Acceleration due to Gravity
�
g d =
GM R - d 㡰
R Rö æ çè R + ÷ø GM GM R - d 㡰 R - d 㡰 = Þ = 黨 R R æ R ö çè ÷ø Þ R = 黨 R - 黨d Þ � R = 黨 d d � \ = R 黨 b㡰 盐ᨨ g⼄盐ᨨ g A = g × - Rw 盐ᨨ g �� h ᨨ ᨨ
㡰 ␔�� ᨨw㦨mg㦨҈黨N Þ m㦨҈黨├� ␔�� ⼄盐ᨨw㦨mg㦨m gwR㡰 é ù æ p ö p ö = ҈黨 ê҈ ç ´ ´ ҈ ú æçQ w = ÷ ÷ è ø ´ è T ø êë úû 㦨҈黨҈桌㦨҈黨�N We mge 黨 ge 黨 㡰 W = mg = ᨨ g = p p p
GM ⏌ R
b㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ gh = g d = g҈
GM
GM æ hö ç҈ - ÷ R è R ø
- R ± R + ҈㡰 R - R + �R � - ҈㡰 = R =
V鼀� 㦨 pR - pR ç ÷ 㦨 pR ç ÷
gh =
=
Þh=
ᨨ盐鼀ᨨ 矄 鼀� � �㡰
҈
GM R + h㡰
hö ÷ Rø
\ R = R + h㡰 R - h㡰 = R + h + hR 㡰 R - h㡰 Þ R = R + h R + hR - R h - h - h R Þ h + h R - R 㡰 - R h = Þ h + h R - R h = Þ h + hR - R =
\ v=
GM
g=
㝐
ᆠ㡰 ᆠᆠ�ᨨ 盐ᨨ��泐 �� h �矄 ��
泐
ᨨ
h㦨 R/ d
G M ⏌ 黨㡰 ⏌
\ Rp= R㜠d㡰
æ h ö gB = g × ç҈ - ÷ Rø è 꾀矄 ᨨᆠ �矄 犨�� ⼄盐泐 ⼄盐ᨨ ᨨ矄 � 鼀 矄g �矄 矄鼀 矄ᆠ矄 g A = gB
R w æ h ö gh Þh= Þ g - Rw = g ç҈ - ÷ Þ Rw = g Rø è R
R p
㡰 �
꾀ᆠᆠ�ᨨ 盐ᨨ ��泐 �� ᨨ鼀 矄 矄盐 ᆠ �矄
æ ö = ç҈ ÷ è ø
҈
=
黨
R
æ ö � = � ç҈ + ÷ è ø
䰘
=
(
-
æ ö Þ 黨 = 黨 ç҈ + ÷ è ø
-
矄 桌
)
-҈
㦨맔҈├鼀㦨맔҈鼀 㡰 ␔� ᨨ�ᨨ ᨨ ᨨ�盐 ᆠᆠ�ᨨ 盐ᨨ ��泐泐矄�Ἐ㦨�w ᆠᨨ矄f ␔ f�矄�盐 犨� ᨨᆠ �� ��泐 ᨨ矄矄f 㦨黨 꾀 ᨨ ᨨ� 矄 矄 w � ᆠ矄矄 �Ἐ 犨� ᆠ矄矄 ᆠ 犨�� ␔ 㦨 鼀� ᆠ矄矄 b㡰 ��ᨨ ᨨ ᆠᆠ�ᨨ 盐 ᨨ ��泐 g 犨� �矄 ᆠ Ἐd Ἐ ᨨ鼀 ᆠ ᨨ
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EBD_8341 PS-84
Physics
ᔼ d < R g = ᔼ d㦨Rgs㦨
Gm R Gm
҈
R Gm
҈
Þ
Q
g㦨
҈a x㡰 ҈a
GMm G ҈M 㡰m GMm G ҈M 㡰m + KE = a a a a é ҈ ҈ ҈ ҈ ù KE = GMm ê + ë a a a a úû
-
é҈ + - - ҈ ù Þ KE = GMm ê úû a ë ҈ 黨GM é � ù Þ mv = GMm ê ú Þ v = a a ë û
Rp � Rp
�GM a 㡰 � ��ᨨ � Ax V¥ = EG = x + a 㡰⏌ Vx x r r dV = ò ò EG × d x Þv=
¥
V¥
x
Ax
Þ Vx - V¥ = - ò
x + a 㡰⏌
¥
dx
A A -= x + a 㡰҈⏌ x + a 㡰҈⏌ 㡰 㝐矄矄ᨨ 矄鼀鼀 ᨨ ᨨ �盐矄x �ᆠ├ 矄矄 dx. æ x ö dm = r ´ px dx = r çç ҈ - ÷÷ ´ px dx è R ø 㝐矄矄 ᨨ \ Vx =
R
g¢㦨
GM
黨黨 R㡰 R �盐矄 ᨨ 矄 � ├矄 泐҈奠
gἘ æ R ö =ç ÷ Þ gἘ>g g è 黨黨 R ø ҈ ҈×҈ ⏌矄㡰 矄� � g¢㦨 gRwᆠᨨ矄 f \
꾀 ⼄盐ᨨ
f㦨
\
g¢㦨 gRw
g¢㦨 \ w㦨
a
A ᨨ� 犨 ���ᨨ � ᨨ ᨨ 矄 ᆠ ᆠ ᆠ ᨨ � ᨨ GM G ҈ M 㡰 = x ҈a - x 㡰 ҈ Þ = Þ x = ҈a - x Þ x = a �㡰 x ҈a - x㡰 矄� � ᆠᨨ 矄�ᨨ ᨨ �泐 犨
GMm
GM
A
a x
㡰 ᆠ㡰 ᨨ鼀mg㦨
҈M
M
�㝐 ᆠᆠ�ᨨ 盐ᨨ��泐ᨨ� 㝐犨 �ᆠ Rp �矄 鼀� ᨨ� ᨨ 犨� �矄 ᨨ ҈ é R p ⏌҈ ù - g pM = g p ê҈ ú = ҈[҈ - ҈] = 黨 鼀矄 R p úû êë \ ᨨᆠ㦨鼀矄矄ᨨ 犨�맔gpM㦨҈맔黨㦨҈ ҈҈
㡰
҈
�.e.g µ d d 黨 ᆠ㡰 ␔├ ᨨ犨�Ἐ㦨�w ᆠᨨ矄q � = � - w � � Ἐ = � � w = � ҈ w= = ´ ´ ҈ ҈ = ´ ҈ - ⏌矄 = ´ ´ ҈ R ´҈ = ´҈� m ҈ b㡰 � R p = e = ҈ ҈
鼀矄矄 ᨨ 犨� Rp 㦨҈맔҈맔҈�㦨҈├� 꾀ᆠᆠ�ᨨ 盐ᨨ��泐 ҈ M 矄盐 ᆠᨨ g p = prGRP 錐 ge㦨 prGRE g p Rp ҈ ҈ = ҈ 鼀矄 \ Þ g p = = = ҈ ge RE ҈ ᔼ d>R�㦨
Topic-4 : Gravitational Field, Potential and Potential Energy
�.e.g µ d 矄�� � 㡰
d
g 㦨҈맔҈ ⏌矄 R
r
æ x ö M = pr çç x - ÷÷ dx R ø è
ò
r dx
x
�
r r - �R ��ᨨ � Þ M = pr
E=
GM
r
=
æ r r� ´ pr ç ç - �R r è G
ö ÷÷ ø
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PS-85
Gravitation
æ r r ö Þ E = pGr ç - ÷ ç �R ÷ è ø dE E�矄鼀 �鼀盐鼀犨 = dr æ ҈ r ö dE Þ = pGr ç - ÷ = Þ r = � R ç �R ÷ dr è ø b㡰 ��ᨨ � 矄盐 ᆠ Gm E= r Gm Gm \ E҈ = ҈ E = r҈ r ᨨ鼀 ��鼀 �� � ⼄盐矄�ᨨ E҈ = E r҈㦨҈鼀R㦨m�� 㡰
\
�
Ef㦨
黨
矄
=
Y
GM a
D
A
r
Mv 㦨 矄盐 ᨨᆠ r ᨨ犨 矄 ᆠ \
҈ö Mv GM æ = + ÷ ç è ø æ a ö a çè ÷ø
O
-㝐 æ ҈҈ ç çè
ù
ú úû
和ᨨ� 矄
ö 㝐 ÷ = -҈҈ ÷ ø
ᔴ�泐
㝐 㝐 =ᆠ�泐 = 和ᨨ ᨨ � ᆠ ᨨ ᆠ�泐 = 矄 - ᆠ�泐
B
҈ ö æ çè҈ + ÷ ø
Þ v=
GM a
GM ҈ ö æ çè҈ + ÷ = ҈҈ a ø
C
r dV 㡰m GMm = òa 㡰 F = r r R
)
泐 é ù = - ê ò ᔼ + ò ᔼ 泐 泐 ú 㦨 é ᔼ + ᔼ 泐 ù 泐 û ë êë úû 㦨 éë � ( 䰘 - ) + ҈ ( - - ) ùû ᔼ=-
a
GM a
҈҈㝐 æ 㝐 ö -㝐 -ç÷= è ø GM R r 㡰 ҈ ᆠ㡰 꾀矄 V = R ᆠ㡰鼀ᨨ矄ᆠᨨ矄泐 �ᆠ矄 ᆠᨨᆠ ��ᨨ ᨨ v r㡰 ҈҈ 㡰 ��ᨨ � ᔼ = ��ˆ + ҈ˆ ⏌├� =-
(
a
�
Þ v =
R
-㝐 é æ ö = ê - ç ÷ èø êë
盐 ᨨ ᆠ�泐 ᨨ � ᨨ� P
AC a a = = 矄盐 ᨨᆠ ᨨ ᨨ 泐 GM ˆ GM ˆ GM ᆠᨨ矄 �°�ˆ + 矄� �° ˆ 㡰 B㦨 � + + a a a 㡰
a
GMm æ ҈ ö GMm ç + ÷ 㦨 R è ø R 㡰 盐 ᨨ ᆠᨨ鼀 矄ᨨ� 矄 ᨨ � ᨨ� P Ef E�㦨
㡰 +
GMm ⏌ GMm ⏌ GMm GMm 㦨 æRö æ R ö R R ç ÷ ç ÷ èø è ø
GMm GMm =R R \ � ᆠ 犨 � �� � �泐
Q r=
GM
GMm R
㦨
æm ö ҈ Þ ç ҈ ÷ = è m ø GM GM G M 㡰 ᆠ㡰 E g = = + a 㡰 a 㡰 a b㡰 AC = a
ᆠ㡰 ᔼ �����ᨨ ᨨ � �泐 E�㦨 � ���ᨨ ᨨ � �泐
E҈ æ r ö æ m҈ ö æ ö æ m҈ ö =ç ÷ ç ÷ Þ =ç ÷ ç ÷ è ҈ ø è m ø E è r҈ ø è m ø
Þ F =
䰘
k pr dr æ ҈ö pkGm = - pkGm ç ÷ = = mG ò è ø r R r r 矄� � 犨ᨨ 矄 矄ᆠᨨ 犨 犨 mv pkGm = R R ᨨv㦨ᔴ ᆠᨨ 矄㡰 T pR pR
�鼀�ᨨ T = = ᨨ㦨 㦨ᆠᨨ 矄 R v C
X
҈
㦨 - éë � + ( - ) ùû = ҈ ⏌ ├� � ᆠ �� ���ᨨ ᨨ � ҈ ⏌├� ᆠ ᆠ � � ���ᨨ ᨨ � �泐҈ b㡰 ␔ ᆠ ᨨᨨ矄 ᆠ ᨨ� � � �泐 犨 ᨨ �泐ᨨ 矄泐矄鼀 �矄 ᨨ 和ᨨ � �� �泐ᨨ 矄泐矄鼀 㦨
҈ ҈ 鼀 ҈鼀 鼀 ҈ ҈ + 鼀 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ �泐 ᔼ �� �泐㦨 � �泐 ҈ ҈ 鼀 ҈ 鼀 \ = 鼀 ҈ ҈ + 鼀
� �泐㦨
ᨨ
鼀 ҈ 鼀 ҈ ҈ 鼀 ҈ ҈ + 鼀 ҈ = ҈㡰
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EBD_8341 PS-86
Physics
泐 ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀 鼀 鼀 鼀҈҈ + 鼀 = ᨨ ҈ = - Þ = ҈ ҈ 鼀҈ 鼀 盐� � 盐 ᨨ � ⼄盐�ᨨ ҈㡰 犨�
GMm ù é êëQ ��ᨨ ᨨ � �泐 = r úû
Þ
æ 鼀 ö 鼀҈鼀 + 鼀 çç - ҈ ҈ ÷÷ = è 鼀 ø 鼀҈鼀 ҈ + 鼀҈ ҈ 鼀҈鼀 = 鼀
鼀҈҈
҈ =
鼀 鼀҈ + 鼀 㡰
= 鼀
҈䰘
鼀҈ + 鼀 㡰
㝐矄矄 ᨨ ᨨ ᨨ 矄 㝐 䰘 㝐Ἐ = 㝐= 㝐
ᨨ���ᨨ � 盐ᨨ ᨨᨨ 矄
F҈ =
GM r҈
F㦨
҈�
F ᨨ ҈ = F r
GM
r r҈ GM R
´r
F҈㦨
´ r҈ F㦨
GM
Y
=
҈ r
A 㜠㡰
æ æ ö ö çQ M = ç pR ÷ r ÷ è ø ø è
æ ö mv é GM ù = mê M㦨 ç pR ÷ r ú r è ø ëR û
X
Z
���ᨨ � 盐ᨨ鼀矄矄矄꾀 �졠�矄 ⼄盐 ᨨᨨ矄�꾀矄 ᨨ 矄ᆠ�ᆠ矄�矄Ӹ㜠^ ᨨ㜠 �矄矄ᨨᨨ � 泐ᨨ� ᨨ 矄犨ᨨᆠ�ᆠ矄犨� ⼄盐 盐ᨨ鼀矄矄M 鼀矄矄矄M m
Þ v µ r
� v r� �矄矄�� � 矄矄� � ᨨ盐� ᨨ��� 㡰 ᔴ � � ᨨ � �泐 B \ DU㦨UfU�
Topic-5 : Motion of Satellites, Escape Speed and Orbital Velocity
R
é ҈ ҈ ù GMm A 㦨GMm ê 㦨 ë R R úû R GM R ÞGM㦨gR Q g㦨 R O gR m mgR = \ DU㦨 R ҈ ҈㡰 矄� � 犨 ᨨ ᆠᨨ 矄�ᨨ ᨨ �泐
ᨨ �泐 �� ҈ 㦨 ᨨ �泐 GM E m ҈ GM E m ҈ + mV = + mV ҈ R R
B 㡰
O
ᨨ \
GM
r
´ r R R F҈ r҈ = \ F r ҈ ᆠ 㡰 ���ᨨ � 쟼㡰 � 矄�泐 ᨨ� O � ᆠ ᨨ 矄ᨨ� 矄 �矄 ᨨ ᨨᆠᆠ� �ᨨ 矄 鼀矄矄mᆠ O F㦨mE㦨m맔㦨 ᨨ犨 y졔 z 㦨 矄 矄 ⼄盐�ᨨ ᨨ ᆠ�ᆠ犨� ᆠ 㡰 �盐矄 盐 �矄 ᨨ ᆠ�ᆠ�矄 �ᆠ盐 ᨨ x㜠 �矄 \
mv é GM ù 犨 M �矄 ᨨ 鼀矄矄 ᨨ = m´ê ú r ë r û 矄 �ᆠ 矄泐矄鼀 \ v µ
GM
ᨨrR ���ᨨ � F㦨
䰘 㝐 ҈ ᆠ㡰 ᨨ r ³ R =
Q 㦨҈├鼀⏌矄�� 桌
b㡰 ᨨr>R ���ᨨ � F㦨
\
鼀҈ + 鼀 㡰 ᆠ㡰 鼀矄矄ᨨ 矄鼀矄 犨 �ᆠ 矄ᨨ鼀ᨨ 㡰�矄鼀 �盐矄 = ᨨ鼀 ��盐㡰 㝐 㝐 鼀 Þ鼀= = æ ö p p ç ÷ èø
㝐 Ἐ
҈ ö V V 黨 æ ҈ = Þ V = V + gR çè ÷+ ҈ ø �
黨 Þ V = V + gR » ҈ km ⏌ s �
和�鼀�泐 = - 鼀҈
҈
GM E R
҈
㡰 泐 �盐 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
mr鼀� v鼀 = mr鼀 v鼀�
v鼀�
r鼀�
r鼀
v鼀 � v鼀� =
v鼀
\
r鼀� v鼀� ҈ = = r鼀 v鼀
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PS-87
Gravitation
㡰 졠�矄 ᨨ ᨨ 泐犨 �ᨨ矄泐ᆠᨨ矄 ᨨ 矄盐 ᆠ ᨨ
GM �㡰 R G 㦨 ���ᨨ ᆠᨨ 矄 쟼矄ᆠ 矄 ᨨ鼀 矄盐 ᆠ ᨨ
�
V =
GM R �� � � �㡰泐 ��㡰犨 Ve =
GM ҈ R = GM R
V = Ve
��㡰
ᆠ㡰
hv =
Q p K G ᆠᨨ 矄 矄 \ T µ R r r 㡰 ᨨ鼀犨ᨨ ᆠᨨ 矄�ᨨ ᨨ 鼀ᨨ鼀 盐鼀 p� = p f m҈u҈졔mu㦨MVf mv ö æ çè mv + ÷ ø �v Þ vf = = m ᔴ泐vfv�\ 犨���ᆠ b㡰 ⧤쟼ᨨ 矄��矄 ᨨ 矄盐 ᆠ �� ᨨ鼀 �泐ᆠᨨ 矄�ᨨ
矄盐 ᆠ졔쟼҈㦨
V
㝐 鼀 㝐 鼀 + 쟼҈ = ( + )
æ ҈ 㝐鼀 ҈ ö ´ Þ 쟼҈ = 㝐 鼀 ç ÷ Þ 쟼҈ = ( + ) è + ø ��ᨨ ᆠ�ᨨ
Re
㦨鼀ᆠ㦨
鼀 =
R鼀
쟼 =
V鼀�
GM Re
졠� ᨨᆠ�泐 V =
ᨨ鼀 �泐ᆠᨨ 矄�ᨨ
GMm ҈ æ ö GMm ҈ + mç V÷ = + mV鼀�
Re R鼀 è ø ᨨ鼀 �盐 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ
䰘
҈㡰
VRe = V鼀� R鼀 㡰 和ᨨ� �⼄盐�ᨨ ҈㡰 㡰犨� R鼀 㦨Re 㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 鼀矄矄 矄�泐ᨨ 矄 �ᆠ k � 泐�矄矄 r 㝐矄矄 M = ò rdV
ÞM =
r = R
ò
R
k pr dr Þ M = pk ò r dr r
pkR = pkR GMm = mw R = FC 㡰 FG = R
ᨨ M =
M R
pkR G = w R Þ w = pKG æQ w = p ö Þ ç ÷ T ø è R R p p R pR pR \T = = = Þ T = w KG KG pKG
m
㝐 鼀 鼀 = ( + ) ( + )
㝐 鼀
( + )
㝐 鼀 鼀 = ( + ) 쟼҈㦨쟼 ᔴ泐 = ҈ Þ = = ├鼀
GM A RA 犨 MA RA 鼀矄矄 �盐矄ᨨ A 꾀ᆠᆠᨨ � � ᨨ�� ᨨ鼀 M R M B = A RB = A GM A M G A RA n \ VA = = =҈ \ VB = GM VB RA A RA Þ n㦨 M M R u b㡰 R Þ R v m ҈ GMm ҈ GMm mu + = mv + R R ҈ GMm Þ m v u 㡰 = R 㡰 쟼矄ᆠᨨᆠ�泐ᨨ A�矄 VA =
Þ = = 盐 v =
GM R
㝐
\ vrad =
�㡰 m´v = ҈ v æ mö çè ÷ø ҈
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EBD_8341 PS-88
Physics
쟼ᆠ� �ᨨᆠ├ ᨨ 鼀矄矄
⧤� �ᆠ �泐ᨨ 矄�
m ҈
GM m GM 黨m ´ = ´ vt Þ Vt = ҈ ҈ R ҈ R ⧤� �ᆠ �泐 ᨨ ᨨᆠ├ ҈M VT + Vr ⧤쟼 rᨨcket = ҈ ҈ m æ GM GM ö 㡰҈ + ҈ = ´ ´ ç u ÷ ҈ è R R ø vT m æ GM ҈ GM ö M M ´ ҈ ç u + 㦨 ÷ è R ø R ҈ R ҈҈黨 GM ö æ 黨M = �m ç u ÷ ҈ è R ø 㡰 �鼀 �ᨨ ᨨ ᨨ盐�ᨨ ᨨ 矄� pr T= v GM r r v= \ T = p r = p r GM GM 和盐矄�盐� � 盐矄 犨 �
҈
㦨 鼀
\
(
黨
)
㡰 ´ ҈҈
T = p
䰘 ´ ҈ -҈҈ ´ ´ ҈
㦨䰘҈矄
;҈䰘 Þ ҈҈ᨨ盐�ᨨ 矄 ҈ 㡰 쟼矄ᆠ ᨨᆠ�泐 vc =
GrV = R
GM = R
矄ᆠ
GS ´ pR = R
prGR
prGRm R � pR = ´ pRm ᨨ Rm = Ἐ ᨨ 鼀ᨨᨨ vc =
Ἐ \ ve =
҈҈
v æ Rö prG ç ÷ = c è ø
҈ mve E v v E = = Ἐe = e = ҈ ᨨE 㦨 v ҈ EἘ ҈ æ ö e mv Ἐe vc çè ÷ø r b㡰 鼀�ˆ + 鼀ˆ = 鼀 r Þ = ˆ� + ˆ
҈ 㝐 = ´ b㡰 졠�ᨨᆠ�泐㦨 ҈
� + 㡰 » �
쟼矄ᆠ ᨨᆠ�泐 㡰 �矄
=
� + 㡰 » �
㝐
Q 桌
D = - = - ҈㡰 �
vR
҈ b㡰 꾀 �� ᨨ鼀ᆠ ᨨ ᨨ� ᨨᆠ�泐
㝐 r 泐 � ᆠ� ᨨ �泐 ᆠᨨ 矄�ᨨ
æ 㝐鼀 ö ⧤쟼ᨨ �鼀졔 çè ÷ =+ ø Q 꾀� � �泐쟼㦨⧤쟼㦨㡰 v=
㝐鼀 æ 㝐 ö 㦨ç 鼀㦨鼀 ᨨ ⧤쟼ᨨ �鼀㦨 ÷ø è ҈� 㡰 㝐�矄鼀矄矄ᨨ 矄鼀�矄鼀矄矄ᨨ 鼀ᨨ� 泐 �泐ᆠᨨ �ᨨ 犨 ¥. 㝐鼀 㝐鼀 ҈ + + 鼀 矄 = + 㝐 ´ 䰘 ´ ҈ ҈҈ ´ ´ ҈҈ = \ 㦨 ҈҈҈ ; ´҈� 鼀 ⏌ 矄 ҈ b㡰 꾀泐� � �泐ᆠᨨ 矄�ᨨ
GM e m GM e m ´ ´҈� ҈ mVs = Re � ´҈ Re
Vs GM e é ´҈� ù = ê҈ + ú Re ë � ´҈ û ö Gme æ GM e ö æ Vs = ҈ ç = ҈҈ ├鼀 ⏌ 矄 ÷ ççQ Ve = ÷ ÷ Re è Re ø è ø \ Vs㦨 ҈ ´ ҈҈ » 鼀 矄 ҈䰘 b㡰 ᨨ ᨨ�ᨨᆠ�泐�矄 � 쟼矄ᆠ ᨨᆠ�泐㦨 � \ 鼀� �鼀盐鼀� ᆠ矄� �矄ᨨ�ᨨᆠ�泐 ҈
㦨 � � 㦨 � ҈㡰 ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ 盐 �矄 犨 ᨨ ��ᨨ ��ᨨ ᨨᆠ 㦨
⧤� �ᆠ �泐ᨨ 矄�꾀
꾀㦨 鼀 꾀 꾀
㡰 ᨨ 矄� ᨨ�� � ᆠᨨ矄 ᨨ ᨨ� ᨨᆠ�泐�矄 �� 泐
=
r æ ö æ ö Þ = ç ÷ +ç ÷ = è ø è ø
҈
҈
㝐 鼀´
꾀 Þ 㦨 㝐 㦨҈ 鼀 ´
꾀矄ᨨ 盐
㝐鼀 졔 㡰
mv GM GM m = ҈黨 㡰 F = m ᨨ r Lr Lr GMm é p ù æ p ö mr ç ÷ = Qv = r w w = T úû Lr êë èT ø Þ T µr
쟼
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PS-89
Gravitation
㡰 꾀矄犨├ ᨨ犨
-GMm ��ᨨ ᨨ � �泐㦨 r ᨨ�ᨨᆠ�泐v㦨 㝐 ⏌ + ҈ 㝐鼀 ҈ 㝐 㝐鼀 쟼 = 鼀 = 鼀 㝐鼀 æ ҈ ö - 㝐鼀 = ç - ҈÷ = è ø -㝐鼀 +⧤ 쟼� = E�㦨Ef
҈ 맔⧤쟼ᨨ 矄ᆠ ҈ GMm ҈ ᨨ mv 㦨 ´ R GMm GMm GMm =\ R R R+h ҈ ҈ Þ R 졔 h = 2R ᨨh = R =Þ R+h R æ ö � ç
ᨨ鼀ᆠ �ᆠ �泐鼀矄矄 ma ç 㝐҈ 졔 㝐 㡰 ÷÷ è ø 鼀� 犨泐ᨨ� 犨 ᆠ 矄 ᨨ 鼀ᨨᨨ
⧤쟼ᨨ� 㦨
�㝐鼀 b㡰 �ᨨ 犨 矄ᆠ 矄 V e
ᨨ鼀� �鼀盐鼀⼄盐� �泐 ⧤ =
҈
ᨨ�ᨨᆠ�泐Ve㦨 V ├� �ᆠ �泐 �鼀ᨨ ᆠ�ᨨ
ì é GM ҈M GM ҈m GM m ù é GM ҈M ù ü - 㦨 í ê úý d d ⏌ d ⏌ úû êë d ûþ îë
҈ ҈ mVe = m V 㡰 = mV 㡰 h �� ᨨ 犨 �ᆠ 盐 �矄矄 犨� �� ᆠᆠ�ᨨ 盐ᨨ��泐�矄 矄
KE =
æ è
g҈㦨g ç҈ +
hö ÷ Rø
Þ
g hö æ 㦨� ç҈ + ÷ è Rø
Gm M҈ + M 㡰 㦨� � � �泐㡰 d ⏌ ⧤� �ᆠ �泐⼄盐� ᨨ矄ᆠ 鼀矄矄 �mᨨ � � �泐 ҈ Gm \ M҈ + M 㡰 mV e = d 犨 Ve�矄 ᨨᆠ�泐犨� 犨 �ᆠ 鼀矄矄m�矄 ᨨᆠ G M҈ + M 㡰 Þ Ve㦨 d 㦨
Þ
h㦨R
GM 㦨 v N �� 㡰 �㡰 R ᨨ犨 泐� � ᆠᨨ 矄�ᨨ ᨨ �泐 � ᆠ� ᨨ矄矄ᨨ ├� �ᆠ �泐㦨�� � ���ᨨ ᨨ � �泐 GMm æ GMm ö ҈ ç mv 㦨 \ R R ÷ø è
␔├ ᨨ犨矄ᆠ矄 ve㦨
GM R ᔴᨨ鼀� �⼄ �㡰 ��㡰犨� N㦨 \
\
vp ve
gp
=
g
´
Rp R
gp
�㡰
ve
=
\ vp =
´ ´ = = ҈҈ ҈҈ ´ ҈҈
´ ve = ´҈҈├鼀⏌矄 = ├鼀⏌矄 ҈҈ ҈҈
h㡰꾀ᆠᆠᨨ � �ᨨ⧤Ἐ矄犨 T µ R \
T҈
R҈ = T R
R҈ = R + R = 䰘 R R㦨�R졔R㦨�R T҈㦨 ᨨ盐矄 \
gR
� 㦨 ��㡰 ge ҈҈ g p r p Rp 꾀矄ᨨ g = p G rR \ = ´ r ge R r é ù R p = �� 㡰ú êQ \ = ´ ҈҈ R ë r û Rp \ = ���㡰 R ᨨ鼀 �㡰 ��㡰 ���㡰 vp
��㡰
v㦨
㡰 ␔ ├ ᨨ犨 矄ᆠ 矄 v =
㡰 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ 鼀ᆠ �ᆠ �泐 GMm ҈ æ GMm ö mv + ç ÷= - R+h R ( ) è ø
䰘
´
T
=
䰘 ´ 䰘 ´ 䰘 ´ R � ´ � ´ � ´ R
Þ T㦨
矄 犨 � �矄 ᨨ ᨨ ⼄盐ᨨ� ᨨ矄�ᨨ 泐矄��矄盐 ᆠ ᨨ ⼄盐ᨨ� b 㡰 矄 P Q 矄鼀 盐 � ᨨ鼀 矄�泐�r 矄盐 ᆠ 矄 A A 矄ᆠ�泐 鼀矄矄ᨨ P, MPm
éAù
m = r´ pr = r´ p ê ú ë p û
é Aù
鼀矄矄ᨨ MQ㦨 r´ p ê ú ë p û
⏌
⏌
=鼀
\ 鼀矄矄ᨨ R㦨鼀졔鼀㦨黨鼀 ᔼ �盐矄ᨨ P㦨r
�盐矄ᨨ Q㦨r ⏌ ⏌ ù é æ Aö æ Aö = ç êQ rQ = ç ú ÷ ÷ è p ø è p ø êë úû ҈⏌ �盐矄ᨨ R㦨黨 r
éQ M R = M P + M Q ù ê ú êë rR = r + r 㡰 = 黨 r úû
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EBD_8341 PS-90
Physics
꾀矄 犨 ├ ᨨ犨 矄ᆠ ᨨᆠ�泐 ᨨ鼀 Ve㦨
vQ㦨
vR 㦨 黨
MB㦨҈M
GM P GM = \ vP㦨 Rp R GM Q
G 黨 鼀㡰 ҈⏌
黨
r
⏌
T҈ æ R҈ ö ҈ =ç ÷ 㦨 T è R ø 㡰 꾀 矄ᨨ � ᨨ�� � 矄ᆠ 矄�ᨨ ᨨ � ᆠ矄犨�� 矄矄矄 �矄� 矄ᨨ
ᨨᆠ ᆠ� � ᨨ 矄ᨨ �矄 盐��矄 � ᨨ� � �
ᆠ矄矄泐 ᆠ � ᨨᆠ 꾀泐� � 鼀ᆠ �ᆠ �泐 ᆠᨨ 矄�ᨨ
ᨨ �泐A㦨 ᨨ �泐 B ⧤쟼㡰A 졔 쟼㡰A 㦨 쟼㡰B µ \
҈ GM é -GMm ïì æ 黨黨 R ö ïü ù GMm m´ +ê R - ç í ýú = ÷ è ø ҈ ï ú R R+h êë R ïî þû
B
㝐 �鼀盐鼀 �� � 泐 ᨨ 泐 ᨨ鼀 矄盐 ᆠᨨ 㡰
ve A R
R ҈ 黨黨R ҈
ᨨᆠᨨ ᆠ�ᨨ �矄 ᨨ �矄 ᆠx ᨨ鼀 ��� 矄
GM S m
㦨 -
=
Þ
҈ M
M
=
Þ x㦨a x ҈a - x㡰 x ҈ a - x㡰 v ᨨ ᨨᆠ�泐犨� 犨 �ᆠ ᨨ 泐ᨨ 鼀矄矄 m �矄 � 矄ᨨ � ᆠᨨ矄矄矄 P ᨨ � OP �泐 �矄 ᆠᨨ 矄
�泐ᨨ 矄泐矄鼀 ᨨ 矄 鼀矄矄矄㡰 � ��泐 㦨 � �泐犨 m�矄P GM B M S GM B m GM S m ҈ + mv ҈a a a
和ᨨ� � ᨨ ⼄盐�ᨨ 犨 �h㦨黨黨�R
和鼀 矄
x
GM B m
= 黨҈⏌ v P
ᆠ㡰 ᨨᆠ ᨨ 矄� �矄 犨泐矄 ᨨ犨 矄 犨 �ᆠ ᆠ矄 矄�犨� ���ᨨ ᨨᆠF ᨨ ᆠᆠ�ᨨ ᨨ 矄�S�矄犨泐矄 �ᆠ ᨨ犨 矄ᆠ ᨨ
ᨨ⼄盐ᨨ �矄���ᨨ ᨨᆠFᨨ盐ᆠ ᨨ �矄 ᨨ ᨨ �盐 鼀ᨨ鼀 盐鼀 ᨨ �
鼀� �盐 �ᆠ�ᨨ 㡰 ᨨ S ᨨ盐 ᆠ ᨨ �矄 ᆠᨨ 矄 ᨨ盐� ᨨ盐 꾀矄 ᨨᆠ F �矄 ᆠᨨ 矄� � 盐 ᨨ 鼀ᆠ �ᆠ �泐ᨨ 矄�鼀� 矄 ᆠᨨ 矄 和 ᨨ S �矄 鼀 �鼀盐鼀 犨 � �矄 矄 ᨨ 鼀� �鼀盐鼀 犨 � �矄 矄 b㡰 Q 졠� ᨨᆠ�泐
Þ
a O'
҈a
V҈ R GM ҈ ᨨVµ = \ 㦨 R ҈ V R҈ R L҈ m҈v҈ R҈ ´ ´҈ ҈ 㦨 㦨 㦨 L m҈v R ҈´҈´ ҈ GMm ⧤� �ᆠ �泐K㦨 R k҈ m҈ R҈ ´ ´ \ 㦨 㦨 㦨 ҈´ ҈ ҈ k m R ᨨ鼀⧤Ἐ矄犨ᨨ 泐鼀ᨨ�ᨨ
P
ᨨᆠᨨ 鼀矄矄m 盐ᨨ���矄㦨 ᨨᆠᨨ 鼀矄矄m 盐ᨨ矄鼀矄
V㦨
҈
O a ��� 矄
G 鼀㡰 = v P r
=
RQ
Gm r
MS=M
GM B M S GM B m GM S m ҈a a a
Q M B = ҈M 錐 M S = M 桌 �GM a �㡰 ᆠ � ᨨᆠ �矄 ᨨ� 泐 ���ᨨ 盐 v mv GMm A m = \ R + h㡰 R + h㡰 \ v㦨
h
B
Þ
GM v㦨 R+h
R
M
E
҈ ҈ GM ve = R ҈ æ GM ö GM Þ R졔h㦨R\ h㦨R㦨├鼀 \ ç ÷= è R ø R+h ��㡰 矄 犨� 犨 �ᆠ 矄� �矄 矄盐 ᆠ ᨨ ᔼ 矄� �矄 矄ᨨ �矄 ├� �ᆠ �泐�矄 ᨨ␔ � 矄 泐ᨨ 쟼 �矄 ᨨ � �泐 ᆠ矄矄 ᆠᨨ 矄 � ᨨ ├� �ᆠ �泐 \ 쟼㡰A 쟼㡰B㦨⧤쟼 -GMm æ -GMm ö ҈ -ç = mv Þ è R ÷ø R � v㦨
GM = gR = 黨 ´ ´ ҈ R 㦨䰘黨鼀⏌矄㦨䰘黨├鼀⏌矄 � � ��ᨨᆠ�泐 矄 ᨨ盐 �� ᨨᆠ �ᆠ � ᆠ � ᨨᆠ �矄 ᨨ� 泐 矄盐 ���ᨨ ᆠ�ᨨ ᨨ 犨ᨨ 鼀矄矄矄 Þ v㦨
m
FR㦨 F + F + F ᆠᨨ矄 ° 㦨 F = G \
mv G = r a
m´m a
O
m
a⏌ Ö
F
FR
m
F
a
m
v
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PS-91
Gravitation
æ mv ö = ᆠ � ᨨᆠ÷ ç è r ø
㝐矄矄ᨨ � �dM㦨spxdx㦨
�盐矄ᨨ ᆠ�ᆠ盐 r㦨
Gmr
a a a 㦨
R
K dr Þ Gm
K pr mG
r r 㡰 K r K = = ᨨ m m pr m G pr mG 㡰 㝐矄矄 盐 � ᨨ 矄 M M 鼀矄矄 = = s㦨 p R㡰 - R㡰 㡰 䰘 pR 盐矄 ᆠᨨ 矄� � � ᨨ �盐矄 x �ᆠ├ 矄矄 dx 矄 矄 ᨨ犨 � ��盐 \
r=
䰘pR
R
ò
R
xdx ҈ R + x
R GM p é GM ҈ R + x ù = - �㡰 ëê ú û 䰘R 䰘 pR R ␔ᨨ├ ᨨ � 鼀ᨨ� �盐 �鼀矄矄 ᨨ鼀ᨨ¥㦨V¥VP
æ - GM ö GM - �㡰 ÷ = - �㡰 䰘R è 䰘R ø 䰘㡰 ᨨᨨ� 鼀矄矄 �矄 ᆠr㦨l
ᨨ WP¥㦨 ç
GMm l 㡰 㝐
+
Gm l
㦨ma
鼀
鼀 l
r㦨l ᨨᨨ� 鼀矄矄 �矄 ᆠr㦨l
GMm l 㡰 \
K dM = dr Gm
pr dr r =
R㡰 + x 㡰
GM p
VP㦨
mv GMm GMm GMm ҈ = Þ mv = ÞK= r r r r Kr K \ M = Gm � �� � ᨨ m ⼄盐�ᨨ 犨ἘrἘ犨� dr r dM dM K = dr Gm M
\
=-
和ᨨ� � 犨 �
b㡰 ��ᨨ 盐ᨨ 鼀矄矄ἘM Ἐ矄 � 矄 ᨨ �盐矄Ἐr Ἐᨨ� ⼄盐� ᆠ � ᨨᆠᨨ �ᆠ ᨨ 鼀矄矄�mᨨ盐� ᆠ�ᆠ盐
ᨨ
GdM
-
R
a pr p a ⏌ = = p
㦨 Gm v Gm a
҈
ò
VP㦨
G ma
Topic-6 : Miscellaneous (Mixed Concepts) Problems
䰘pR
ᨨ � ᨨ� P 盐 ᨨ矄
Gm v = = Þ v = a a ´ a
�鼀�ᨨ ᨨ ᆠ�ᆠ盐鼀ᨨ�ᨨ
pMxdx
-
Gm l
GMm l㡰
+
㦨ma
Gmm l
=
GMm l㡰
Gmm l
䰘M é҈ ҈ ù Þ m㦨 \ K㦨䰘 ú ë 黨 ҈ û b㡰 ᨨ鼀 ᆠᨨ 矄�ᨨ ᨨ 鼀ᆠ �ᆠ �泐 -GMm GMm ҈ + mv 㦨졔 L L \
m㦨 M ê
ᨨ
҈ GMm mv 㦨 L
GM GM = L L
V=
\
㝐
ᔴ
鼀
㝐
P R dx R
R x
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EBD_8341 PS-92
Physics
Mechanical Properties of Solids
8
Topic-1 : Hook's Law & Young's Modulus ҈
-A
ᆠ㡰 � U =
r
+
B
�
r҈
ᨨ⼄盐���盐鼀
p ´ ´ ҈- 㡰 ᆠ㡰 Dl҈㦨Dl ᨨ
dU = - A -r -䰘 㡰㡰 + B -҈ r -҈ 㡰 = dr A ҈ B A ҈ Þ = 䰘 - ҈ Þ = ҈ B r r r F=
æ B ö \ 和�ᨨ 犨 鼀ᨨᆠ盐矄 r = ç è A ÷ø
Fl҈
Fl pr y
҈⏌
和矄矄҈ =
- A A - A + = B B B FL F FL = ᆠ㡰 La DT㡰㦨 \ a㦨 AYT pr YT AY ᔴᨨ �ᆠ� ᨨ ᨨ盐鼀 矄矄�ᨨ
F g㦨a㦨 pr YT ᨨ 盐矄㡰
和矄矄 æ Dl ö çè ÷ø L
s 矄矄矄
ᨨ ᨨ ��ᨨ Dl =
é҈ ҈ù Dl = s ê + ú ë Y҈ Y û
sL҈ sL + Y҈ Y
QL҈㦨L㦨҈鼀桌
æ YY ö s = Dl ç ҈ ÷ è Y҈ + Y ø
N æ ҈ ´ ö = ´ ҈- ´ ç ´ ҈黨 = ´ ҈ è ҈ ÷ø m ᆠ㡰 � �盐矄ᨨ 犨�r㦨鼀鼀 㝐矄矄ᨨ ᨨ 鼀㦨├� F mg 和矄矄 = = A p r 㡰
R ´䰘
=
҈�
´
҈� ᨨᆠ҈ = ҈ ҈ 㡰
和矄矄 =
҈� ᨨᆠ = 㡰
x ᆠ �� � = ᨨ��� � x �矄 �矄ᆠ鼀 ᨨ 矄ᆠᨨ ᨨᆠ├ 和� =
ᨨ矄鼀鼀�
ᨨ
䰘
҈㡰
矄矄矄҈ 矄矄矄 = 矄� ҈ 矄�
҈�
҈�
Ӹᨨ盐 �Ἐ矄 鼀ᨨ 盐盐矄 Y =
ᆠ �� � ҈ � ´ ҈- = ᨨ��� � ҈ ҈ ᨨ 矄ᆠᨨ ᨨᆠ├
A B + ÷=B ⏌ A B ⏌ A ø
和
ᨨ
和� ҈ =
҈⏌ ö
矄矄
= ҈ ´ ҈ ⏌鼀
\R㦨҈䰘�鼀鼀 b㡰 ᨨ矄鼀鼀� �ᨨᨨ 矄矄矄ᨨ矄� �矄矄鼀 ᨨ �矄 ᆠ盐
=
=
pr҈ y҈
ᨨ � �泐
æ æ B ö U çr = ç ÷ è Aø è
´ ҈p
=
=
㡰 x
� ´ ҈ - ҈ 和ᨨ� �犨�x㦨�ᆠ鼀 ᆠ㡰 ᔴᨨ 矄� 矄鼀鼀 ᨨ �盐矄
졔 꾀 ⼄盐���盐鼀 ᆠ �� � ᨨ 犨�
㦨
҈
ò = ò
㝐�
é ù + ú 泐 pê ë û
├� ��鼀� ᨨ鼀ᨨ
é ù 㝐� ê ú 㝐� ê ҈ ú 㝐� D = ´ = ú p泐 ê é ù ê ú p 泐 + úû êë êë úû
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PS-93
Mechanical Properties of Solids
⼄盐���盐鼀 � ᨨ 犨�㦨졔D
= +
æ ҈ 㝐� ö = ç҈ + ÷ p Ӹ è pӸ ø 㝐�
҈
㡰 � lc㦨 鼀錐 ls 㦨 ҈ 鼀錐 Yc 㦨 ҈ 맔 ҈҈҈ ⏌鼀 Ys㦨맔҈҈҈⏌鼀
DL = α DT L 꾀矄 � �矄 ᆠᨨ 矄 㡰 㦨맔҈҈҈맔҈҈맔҈�맔҈㦨맔҈ æ 鼀㝐 ö b㡰 矄�ᨨ � 犨� = ç ÷� è鼀+㝐ø ᨨᆠ ⏌ 矄�ᨨ
鼀㝐 � = 和矄矄㦨 꾀 꾀 鼀 + 㝐㡰 鼀 ´ 鼀㡰� = 㝐㦨鼀�� 㡰 꾀 鼀 + 鼀㡰
҈鼀
鼀
Ts
Tsᆠᨨ矄
和� 㦨
黨
҈
鼀 鼀� �= = 鼀꾀 꾀 ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ l矄 泐 Dl矄 = 矄 = 矄 = ᆠ =䁸 l 泐 Dl 꾀矄 泐 =
Dl 矄 =
Dl =
鼀�l 矄 p矄 泐矄
p 泐
҈
Þ
Dlc æ Tc ö æ lc = ç ÷´ ç Dls è Ts ø è ls
Tc ҈ = Ts
Þ
Yc ҈´ ҈҈҈ ҈ = = Ys ´҈҈҈
Fl ADl As㦨Ac
Dl㦨
Fl AY
ö æ Ys ö æ ҈ ö æ ö æ ö ÷ ´ ç ÷ 㦨 ç ÷÷ ´ ç ÷ = ÷ ´ çç ø è Yc ø è ø è ҈ ø è ҈ ø
├� �矄矄矄s㦨맔҈䰘鼀 꾀 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ 㦨 ҈0 ᆠ鼀
s=
T ml w = A A
(
)
´ ҈䰘 A ml w £ ´ ҈䰘 Þ w £ A ml
F A Dl l
F A
lc = ҈ ls
Ts = Tc
㦨㝐lw
⏌ p 㡰 ⏌ p Ӹ= = Dl҈ ⏌ Dl ⏌ 㡰 矄� �Y㦨
\
Þ
㝐 �鼀盐鼀 �盐矄 w㦨䁸
Dl 矄 p 泐 = 矄 矄 = \ Dl 㝐�l ᔴ p 泐
҈
Ts Tc =
㝐矄矄ᨨ ᨨ 泐鼀㦨҈├�
Q = 㝐�桌
ᆠ㡰 矄� �Y㦨
Þ
҈� 㡰 � ␔� � l㦨鼀
㝐�l 矄
҈҈
Tcᆠᨨ矄
꾀⼄盐���盐鼀Tsᆠᨨ矄 㦨Tcᆠᨨ矄
\
Q 矄 = 㝐 + 㝐㡰�桌
㝐�l
Tc
҈
ᨨ鼀Y㦨
l l Þ Dl = 꾀Dl 꾀泐
矄矄矄 㡰 Ӹᨨ盐 �Ἐ矄鼀ᨨ 盐盐矄Ӹ㦨 矄�
和矄矄㦨Ӹ´矄�
和矄矄 � 矄犨� 㦨꾀� 矄矄盐 矄矄盐㦨矄矄矄㦨Ӹ맔矄�
( ´ ҈ )(҈ ) = ҈ Þw £ 䰘
Dl \ = Dl҈
´ ҈ Dl F l = = - - l A Dl ҈ ´ ҈
= ´ ҈҈҈ ⏌ 鼀 T⏌A T ´l T l 㡰 矄� �Y㦨 Þ Dl = = ´ Dl ⏌ l A´Y Y A æ T ö l \ Dl µ çQ �矄 ᆠᨨ 矄 ÷ è Y ø A
l �矄 �矄 � 㡰 ᆠ�矄 矄�ᨨ 꾀
Þw
-
鼀 㦨 ⏌矄 ҈ ´ ҈ b㡰 ᨨ鼀� 鼀 �鼀盐鼀矄矄矄 ᆠ 犨� 矄 ᨨ├� � �矄 � Q
和�
㝐 矄� 㡰 㝐 矄� 턼㡰
㝐 �鼀盐鼀 矄矄矄ᨨ 턼
和矄矄
㝐 矄矄矄 ᨨ
矄� ᨨ P�矄鼀ᨨ Q ᨨP�矄鼀ᨨ 盐ᆠ�
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EBD_8341 PS-94
Physics
矄矄矄 和ᨨ�� 矄� 矄矄矄�矄鼀ᨨ ᨨQ 矄�
\ Ӹ턼>Ӹ
Q Ӹ =
矄� �⼄ �㡰 ��㡰 \
Topic-2 : Bulk & Rigidity Modulus and Work Done in Stretching a Wire ҈
㡰 ᔼ ᨨᆠFᆠ矄ᨨ � � ᨨ 犨�ᨨ ᆠᨨ矄矄㜠 矄ᆠ�ᨨ A �泐矄ᨨ � 盐 � ᨨ盐鼀 ᨨ 犨� �矄 �� 泐
�
ᨨ鼀 ᨨᆠ 㦨 㦨 꾀 p 㡰 꾀 和矄矄 㦨 맔矄�
㡰
和矄矄 㦨
㦨
F X ö ҈ F F æ = ´ ´ çèQ 矄矄矄 = 矄� 㦨 ÷ø A AY A AY ҈ F ҈ F ´ ҈ ҈ F ´ ҈ = = AY pd 㡰 Y pd Y ᔼ u҈ u 矄��矄 ᨨ 犨ᨨ犨�矄
dr mg = ᆠ�ᨨ ᆠ鼀 � �盐矄㡰 r Ka
p 㦨 p 㡰 p
҈ 쟼 �泐 矄�泐㦨 矄矄矄 맔 矄�
=
dr mg = r Ka
d҈ d u҈ æ d ö ҈ = ( ) Þ ҈ = ҈ =ç ÷ Þ d d u è d҈ ø 㡰 ␔ ᆠ盐�矄 矄ᆠ 盐 ᨨ � l 矄ᨨ �泐� �㦨D├쟼Þ
p 㡰
p ᨨᆠ ᨨ 盐矄 ᆠᨨ├ 㦨 m 㦨 mp 㦨 pm㡰D ᆠ㡰 和 � � 矄� �矄 ᆠ ᨨ � 矄� 矄盐 ᆠ ᨨ 盐 ᆠ �矄├ᨨ ᨨᆠ矄 ᨨ �矄├ ᨨ � ᆠ�ᆠ盐鼀 ᆠ ᨨ �矄├ ᨨ ᨨᆠ ᨨ �矄 ᆠ ᨨ 泐
mv L ҈ æ YA ö ҈ ç ÷ DI 㡰 = mv Þ y = è Lø D Dl 㡰
鼀㦨├� 㦨鼀矄҈ 㦨鼀 꾀㦨 p 㡰⏌ 㡰 㦨맔҈鼀 Dl 㦨鼀 y=
盐矄 矄矄盐鼀 矄 � � 矄矄矄 ᨨ � 矄� 矄盐 ᆠᨨ �矄├ �矄 盐 � ᨨ鼀
=
´ ´ ´
ᆠ㡰
Þ
ᨨ盐鼀ᨨ 矄 V =
æ҈ è
矄盐 ᆠ
ö ø
㦨 黨䰘 ´ ҈ ; ´ ҈
�㡰
pR
҈ æ YA ö x çè L ÷ø ␔ᨨ├ ᨨ W㦨 犨 K㦨
dV dr = V r
꾀
- - 㦨 � ´҈ ´ ç ´ ҈ ÷ ´ ´҈ ´ p
ᆠ�ᨨ ᆠ �� ᨨ盐鼀
矄盐 ᆠ
ò
æö ÷ èø
mg æ dV ö aç ÷ èV ø
dV mg = V Ka
s 鼀 꾀 = s 鼀
㦨 ò s 鼀 꾀 = s鼀 p ç
㦨맔҈⏌鼀
ᨨ盐鼀�ᆠ 矄矄矄 盐├ 鼀ᨨ 盐盐矄 ⧤ = ᨨ盐鼀�ᆠ 矄�
K=
ò
鼀 =
矄盐 ᆠ
p ´ ´ ҈ ´ 和ᨨᨨ �矄҈
ò
��㡰
\
YA L
҈ Kx
x㦨 矄�ᨨ � 犨�
҈ æ YA ö ÷x è L ø
W㦨 ç
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PS-95
Mechanical Properties of Fluids
Mechanical Properties of Fluids
9
Topic-1 : Pressure, Density, Pascal's Law and Archimedes' Principle ҈
= A ҈ 㡰r°c g
㡰 ᔼ ⼄盐���盐鼀mg = Fe FB = V r g 鼀矄矄㦨 ᨨ盐鼀맔 矄�泐
ᆠ鼀
p R - r 㡰r g = pRr w g r 䰘 � � 矄�泐 = rw
ᔴ
\
r 黨 ҈ r r = Þ ҈- = Þ = R R R 䰘 ҈⏌ r æ ö r Þ = ç ÷ Þ҈- = è ø R 䰘 R
Þ ҈-
r
҈黨 Þ = ҈= 䰘 䰘 R
�
R
pr R ´ = pR rl �
x҈ x + rSx 㡰 g ×
U f = rSx f 㡰 g ×
xf
´
泐ᨨ盐鼀ᆠᨨ 矄�ᨨ
x҈
xf xf
Sx҈ + Sx = S x f 㡰 x +x xf = ҈
x
=
ù rSg é x҈ x - x҈ x ú = rSg x҈ - x 㡰 ê + êë úû
\ rl =
r �
㡰 P҈ P P 矄矄盐 ᨨ� 矄 M N O 矄ᆠ�泐 M P҈ 㦨 r F҈ N P㦨�rg F 㦨҈� g P O 矄矄盐�矄�� 泐P㦨rgh ᨨ犨P҈㦨 Qh㦨㡰 P 㦨rg �㡰 P 㦨rg ҈�㡰 㦨҈�rg
P҈ + P 㡰 A P + P 㡰 ᨨᆠ ᨨ ᨨ犨 F = A F �rg � ҈ \ ҈ = = = F rg
ᨨᆠᨨ 盐 F҈ =
␔ �矄 ᨨ ᨨ矄矄 � ᨨ � �泐 ᨨᆠᆠ盐 � 犨 ᆠᨨ鼀矄鼀
DU = U � - U f éæ x x ö ù DU = rSg êç ҈ + ÷ - x f ú ø êëè úû éx x æ x҈ + x ö ù ҈ Sg = r + ê ú ç ÷ è ø úû êë
ᆠ㡰 ᨨ 鼀� �鼀盐鼀 矄�泐 ᨨ �⼄盐� 矄ᨨ� 矄 矄 ᨨ ᨨ ᆠᨨ鼀泐�鼀鼀矄 㡰 � �⼄盐� mg㦨FB 矄ᨨV�鼀鼀矄 㦨Vᨨ㡰
é r r� ù Þ pr ê ú = pRrl êë �R úû
㡰 ᔼ �� ᨨ � �泐
� ᨨ � �泐
ᔴ
pR r l é ù æ r ö êr r 㡰 = r ç ҈ ÷ < r £ R �� ú êë úû è R ø R æ r ö Þ ò r p ç ҈ ÷ r dr = pR r l R ø è
ᆠ㡰 � �盐矄ᨨ �盐㦨҈ᆠ鼀 犨 ᆠᆠ�ᨨ ᨨ 盐a㦨黨ᆠ鼀⏌矄
U҈ = rSx҈ 㡰 g ×
r°c = = ҈҈ r°c 䰘黨
҈ᆠ鼀
ᨨ ò rdV =
\ r = 黨 R = R 黨
盐ᨨ泐 r犨 㦨҈�ᆠ鼀 p p ᨨ盐鼀 V = r = ´ ҈㡰 = ҈黨 ᆠ鼀 a F盐ᨨ泐 F盐ᨨ泐 - mg = ma Þ m = mg g 졔a V rw g 㡰 V rw ҈黨㡰 ´ ҈ ҈黨 \m = = = = = ҈�� a 黨 g+a ҈҈ ҈+ ҈+ g 黨
҈ᆠ鼀
␔ ᆠ泐� �矄 ᨨ� �� 犨 ᔴ 盐矄 = A h h҈ 㡰r°c g = A ҈ ҈㡰r°c g
é æ r ö ù 䰘 Þ ê҈ - ç ÷ ú rw = rw è Rø ú ëê û
ᆠ㡰 ␔ ᆠ泐� �矄 ᨨ� �� 犨 ᔴ 盐矄 = A h h҈ 㡰r°c g
䰘
b㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ ҈졔 㦨黨맔졔҈ Þ ҈졔 㦨 矄矄盐� 犨ᨨ鼀矄ᨨ 盐㜠盐�矄⼄盐 \ 졔r├� ҈㡰졔r犨� ҈҈㡰㦨졔r犨� r├ 㦨 矄�泐ᨨ ├ᨨ矄 r犨 㦨 矄�泐ᨨ 犨桌
҈㡰
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EBD_8341 PS-96
Physics
Þ r├� ҈㡰졔r犨� ҈r犨�맔 ҈㡰㦨r犨� Þ 맔҈맔҈졔҈맔҈맔 ҈҈맔҈맔҈㦨҈맔 ҈맔 Þ ҈ ҈ 㡰㦨 \ ҈ 㦨 㡰 ᨨ鼀⼄ 矄 ҈㡰 㡰 ҈㦨� 㦨
\
ᆠ 矄�ᨨ ᨨ 矄� �犨 � �矄 � ⼄盐���盐鼀 �矄
= ҈
҈ � � = =
㡰 V҈㦨 ᨨ鼀�ᨨ盐鼀ᨨ ᆠ泐� �ᆠ矄 V㦨ᨨ� 矄� ᨨ盐鼀ᨨ ᆠ泐� �ᆠ矄 y㦨 ᆠ�ᨨ ᨨ Vᨨ盐鼀 � 犨� 犨
ᨨ犨�� 㦨 盐矄 ᔴᨨ ��ᨨ ᨨ ᨨ�ᨨ 㡰
҈
æ V҈ + V ö ÷ ҈㡰g è ø
\
V҈rc g졔 yV㡰 ҈㡰g㦨 ç
ᨨ
V y㦨�졔 �rc㡰 ҈
҈鼀
h҈
㝐� æ 꾀s ö ç҈ ÷ ├ è 㝐 ø
V
和ᨨ� rc� y>� 㡰 盐├鼀ᨨ 盐盐矄
h
㦨
´ ҈㡰 ´ g = M ᆠ盐 g ҈
㡰 ᨨ鼀 꾀ᆠ �鼀 矄� ᆠ� 盐矄㦨 犨 ᨨ 盐� �矄ᆠ Fᨨᨨ鼀Fᨨ㦨Vrg Þ Fᨨᨨ鼀㦨Fᨨ졔Vrg
æ V ö Mg = ç ÷ rwg è � ø æ M ö rw rw ᨨ ç ÷ = ᨨ r = � èV ø � ␔ ᨨᆠ├ ᨨ矄 盐泐� 犨 ᨨ�
a ᆠᨨ矄a
矄� a 矄� a
҈䰘
矄� a
꾀
\ R ᆠᨨ矄 a + R 矄� a 㡰d g
= R ᆠᨨ矄 a - R 矄� a 㡰d҈ g d҈ ᆠᨨ矄 a + 矄� a ҈ + a = = Þ d ᆠᨨ矄 a - 矄� a ҈ - a
҈
꾀� = 㝐�
Q鼀矄矄㦨 矄�泐맔ᨨ盐鼀桌
Þ ├ = 㝐� - s 꾀�
Þ =
㝐� -
s꾀� = 㝐� æ҈ - 꾀s ö ç ÷ ├ è 㝐 ø ├
㡰 ␔�� ᨨ ᆠ泐� 㦨 盐 盐矄 盐 ᨨ 盐 �⼄盐� 졔盐 盐矄 盐 ᨨᨨ犨�⼄盐�
b㡰 矄矄盐 � �鼀 ᔼ B㦨矄矄盐� �鼀ᔼᔼA hr҈g㦨hrg Þ r҈㦨r ᆠ 矄ᆠ� �ᆠ ��泐 ᨨ �⼄盐� ᔼᔼ㦨sp��泐ᨨ �⼄盐� ᔼ㦨҈҈ 㡰
h P҈ ᔼᔼ A
h P ᔼ B
æ ö ç- ÷ è ø æ Dl ö Þ = - Þ ç ÷ = è l ø
꾀矄犨├ ᨨ犨盐├鼀ᨨ 盐矄 =
ᆠ㡰 ᨨ鼀 ��盐 ├ + = 㝐�
├ + s
V
æA ö æ Aö æ ö æ Aö æ L ö D ç ´ L ´ g ÷ = d ç ÷ ç L ÷ g + d ç ÷ ç ÷ ´ g è � ø è ø è� ø è �ø è ø �d \ D㦨
d҈
h
㦨rgpRh졔V桌
V æV ö rV 㡰 g = ç ÷ rᨨ� g + rwg ᨨrᨨ�㦨 rw = rw èø � ᆠ㡰 矄矄盐� ᆠ꾀鼀盐矄矄鼀 ᨨ鼀ᨨ 矄� 矄ᨨ � ⼄盐���盐鼀 d
P҈
㦨P҈맔A졔Vrg 㦨 hrg㡰맔 pR㡰졔Vrg
҈
a
҈҈
҈�
ᆠ㡰
Mg = Fb҈ + Fb
҈
㡰 ␔ ᆠᨨ� 矄 ᨨ鼀 ᨨ ᨨ ᨨᆠ├ � ᨨ犨 ᨨᆠ├ 鼀ᨨ矄 盐犨 矄 ᆠ盐矄 犨�� ᨨ ᨨ� � ᨨ 泐 ᆠᨨ鼀矄矄矄 ᆠl ᆠ矄矄꾀矄ᨨ�犨� �矄ᆠᨨ盐鼀 ᨨ 犨 ⼄盐ᨨ �矄 ᨨ犨 ᨨ盐鼀 犨 矄 犨 � �矄 ᨨ ᨨᆠ├ � �矄ᆠ矄 鼀ᨨ ᨨ盐鼀 ᨨ ᨨ犨 ᨨ盐鼀 ᆠ盐矄 矄�泐ᨨ ᆠᨨ� �矄� 矄�泐ᨨ 犨 ᆠh犨�矄ᨨ ᆠ矄
�㡰
鼀 鼀矄矄矄 ᨨ盐 ᆠ
��㡰 �㡰맔 ҈㡰맔g㦨 Mᆠ盐졔m㡰g 和盐ᆠ� �⼄盐�ᨨ �㡰 ᨨ鼀⼄盐�ᨨ ��㡰犨� Þ 鼀�㦨 �㡰맔� ҈㡰㦨҈�맔䰘맔҈ � Þ 鼀㦨䰘�├� 黨
҈ b㡰 ␔ ᨨ 泐 ᨨ矄 犨�� ᨨ ᨨ 泐㦨盐 盐矄
\ �㡰 ´
҈҈� ´ ҈� - ҈҈ ´ ҈� 㡰 D = 㦨҈��맔҈� D ⏌ ҈
⧤
㝐�
æ - ö l æ r� ö l Dl = ç ÷ =ç ÷ è ø è ø =
҈ ´ ҈ ´ � ´ ҈ 黨
䰘 ´ ҈
҈ ´ = ´ ҈ -
\ Dl 㦨鼀鼀 ; 鼀鼀
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PS-97
Mechanical Properties of Fluids ҈黨
K㦨
-DP DV ⏌ V
Mg DV Mg \ = A V AK V f - V� 㡰 Mg V� - V f Mg Þ Þ = = V� AK V� AK pR - p R - d R 㡰 Mg Þ = AK pR dR Mg R - R - R dR桌 Mg = Þ = Þ R AK AK R 矄 ␔ ᨨᆠ├ᨨ �ᆠ鼀矄 矄 ᨨ犨�盐�鼀泐矄� ├�
犨␔ 矄 ᨨ矄� ├矄 �犨� �矄ᆠ犨⼄盐ᨨ �矄 ᨨ犨 ᨨ盐鼀 盐 犨 矄 ᨨ 犨矄 鼀 � �ᆠ � �矄ᆠ 鼀ᨨ ᨨ盐鼀 ᨨ 犨 �矄 ᨨ犨 ᨨ盐鼀 ᆠ盐矄 d >d犨 DP 㦨
犨
Q ᨨ盐鼀㦨 ҈
㦨
GMr R P
\
k WFS R r
pRrg pRrg ᨨ x = Þkx = k �
ᆠ㡰 d 矄�泐ᨨ 鼀�ᨨ 矄 ␔�� ᨨ 矄 㦨 盐矄 盐ᨨ�졔 盐矄 盐ᨨᨨ� Vdg㦨
V V d � g + dᨨ� ´ g
Þ d㦨
d� + dᨨ�
=
҈ + = 䰘� ⏌ ᆠ鼀
bᆠ㡰 ␔ ᨨᆠ├ᨨ 鼀矄矄m�矄 A � 矄� �⼄盐� 盐 盐矄FT犨�ᆠ ᨨ 鼀矄矄 犨 �ᆠ 犨� ᆠ矄 m � �ᨨ A�犨� 矄矄 ├� 꾀ᆠᆠᨨ � �ᨨ犨ᨨ Ἐ矄 � 犨ᨨᆠ F F 泐 ᆠ�ᨨ �矄 ⼄盐 B ᨨᨨ矄� ᆠ�ᨨ 和ᨨ泐ᆠ�ᨨ ⼄盐 ᨨᆠ犨� ᨨ �⼄盐� ᨨ ├犨 �ᆠ 犨�� ᆠ矄 � �� B� 犨� 鼀ᨨ �├� 㡰 犨 ᨨ矄泐矄鼀 矄 泐盐 ��泐 矄ᨨg㦨 꾀ᆠᆠᨨ � � ᨨ꾀ᆠ �鼀 矄 � ᆠ� 盐矄 㦨 犨�� ᨨ 盐� �矄ᆠ 㦨 鼀矄矄ᨨ 盐� �矄ᆠ 㡰맔g \ 盐矄 㦨 ᆠ㡰 ᔴᨨ 矄� ⼄盐���盐鼀ᨨ 犨ᨨᨨ ᆠ泐� �ᆠᨨᆠ├ ᨨᆠ矄ᆠ� �� ᨨ犨 犨 �ᆠ�ᨨ T
䰘
r p㡰dr
GM r
r
ò rdr
R R
졔
W 㦨 p r 㡰 ´ h ´
= p ´ r
GM r é ù R -r \ P㦨 û R ë é 黨R ù êR ú 䰘R ҈ úû P = ⏌ 㡰 êë = = ҈ = \ P = ⏌ 㡰 é R ù �R êR ú 黨 黨 ûú ëê é 黨R ù êR ú � ûú ҈ P = ⏌ �㡰 ëê = = P = ⏌ �㡰 é R ù ҈ êR ú � ûú ëê
pR rg pRrg
㡰 ᆠᨨ鼀矄泐矄鼀�矄矄矄 ᨨ犨 � ��盐 x ᨨ ��ᨨ ᨨ 矄� � 꾀⼄盐���盐鼀 ᨨ盐 矄 W졔 FS㦨FB
r
R r
T
P҈ ´ p r 㡰
�㡰 ␔�� ᨨ 犨ᨨᨨ ᆠ泐�
ò dp =
Þ kx㦨
鼀矄矄 ᆠᨨ 犨犨� 矄�泐
矄 ␔ 鼀盐 �矄 � ᆠ矄 矄�泐 ᨨ 鼀ᆠ盐泐 r㡰 ᆠ矄矄矄rt㦨r ҈gt㡰 ᆠ ᨨ 鼀ᆠ盐泐�
ᨨ鼀盐 㡰� ᆠ矄矄 Q hr 㦨 ᆠᨨ 矄 矄
ᨨ 犨犨�鼀� 矄鼀� ᨨ ␔ 鼀 � ├矄 矄ᨨ鼀 犨 ᨨ鼀 ᨨ �矄 犨�� � ᆠ矄矄 ᨨ ᨨ犨�矄� ├ 盐 꾀 �矄矄� ├� � ᨨ ᨨ 犨� �矄ᆠ 矄鼀 ᨨ盐鼀 ᨨ 犨 � ᨨ 矄 盐 ├泐鼀 bᆠ㡰盐矄ᆠᨨ 矄� 鼀 鼀矄矄 鼀矄 ᨨ犨 � 矄 ᨨ�ᨨ ᨨ ��盐 Pp P졔dP㡰pr
FB
pR rg졔kx㦨 pR r㡰g
r ´g
hr g
P҈
P
W
P
��㡰 ᨨᆠ 盐 ᨨ矄矄盐 P҈㡰 ᆠ 泐 �⼄盐� ᨨ �� h ҈ ᨨ 犨ᨨᨨ ᨨᆠ├ F҈ 㦨P҈맔p r㡰㦨P졔h҈rg桌맔p r㡰 ���㡰 ᨨᆠ ᆠ� �ᨨ 盐犨 �ᆠ�ᨨ 盐 ᨨ 矄矄盐 P ᨨ鼀ᨨ犨 犨ᨨᨨ ᨨᆠ├ 鼀ᨨ矄 �ᆠ矄矄盐 F = P ´ p é r 㡰 - r ù + P ´ p r 㡰
ë û = [ P + h҈ + h 㡰rg ] ´ p ´ r + P pr
ᔼ 矄�盐�ᨨ ҈ �ᨨ �矄� � ᨨᆠ├F = F҈졔W
[ P + h҈ + h㡰rg ] ´ ( p ´ r ) + pr P = [ P + h҈rg ] ´ pr +
黨
�h p ´ r hrg ᨨ h҈ =
b㡰 꾀�� ᆠᨨ 矄� � �⼄盐���盐鼀ᨨ 犨ᨨᨨ ᨨᆠ├
ᨨ ᨨ犨 犨 ᨨᆠ㦨 ᨨ ᨨᆠ 盐犨 矄 ␔ ᨨ ᨨᆠ├졔 ᨨᆠ 盐ᨨ鼀ᨨ矄 �ᆠ矄矄盐㦨 ᨨᆠ 盐ᨨ 矄矄盐 ᨨ �⼄盐� 졔 ᨨᆠ 盐 ᨨ鼀ᨨ矄 �ᆠ 矄矄盐
r p ҈r 㡰 ´ g + P p ´ ҈ r 㦨hrg졔P桌p ҈㡰r桌 졔P맔r
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EBD_8341 PS-98
Physics
犨 h㦨 �� ᨨ 犨� 矄�盐�ᨨ ᨨ犨 �ᆠ ᨨᆠ├ 鼀� 矄 � �矄 ᨨ��� ᨨ矄��ᨨ
ᆠ犨 矄ᨨ 矄盐� 犨 ᨨ 犨 矄
\ h㦨 h 黨
҈
g + l d 犨ᨨᨨ g = l d 犨 g
㡰 ᔼ 矄�盐�ᨨ 犨 �� hᨨ 犨�矄 盐 ᆠ矄 盐犨 ᨨᆠ ᆠ� � ᨨ 犨ᨨᨨ ᨨᆠ├ ᆠ矄矄 ᨨ ᨨᆠ ᨨ犨 犨 鼀� 矄 矄鼀 �矄 � ᆠ 犨� ᆠᨨ鼀 矄 泐 ᨨ鼀 ᆠ�ᨨ 泐 ├犨ᨨ 犨ᨨᨨ ᨨᆠ├ ᆠ ᨨᆠ├ ᨨ矄 ᨨ鼀ᨨ矄 盐 鼀� 矄�矄ᨨ��� ᨨ矄��ᨨ 㡰 꾀矄 泐 ᨨ矄�ᆠ ᨨᆠ 泐�⼄盐� 꾀ᨨ ᆠ泐� ᨨ鼀 矄� 矄 泐ᆠ ᆠ矄ᨨ盐 盐�矄 ᨨ b㡰 ᔼ ⼄盐���盐鼀 盐ᨨ泐 ᨨᆠ㦨 犨�� ᨨ ᨨ 泐 Þ hArAAg졔hBrB Ag㦨 hA 졔 h 졔 hB㡰ArC g 犨 rC 㦨 矄�泐 ᨨ ᆠ泐� 㡰
æ hAr A + hB rB ö ÷ - hA + hB 㡰 rC è ø
矄� ᨨ �ᨨ ᆠ盐 l ␔ 鼀矄矄m㦨��矄ᨨ ᆠ盐ᨨ 犨ᨨᨨ
Þ l d 犨ᨨᨨ = l d 犨 - �㡰 ␔ 鼀矄矄m㦨��矄鼀ᨨ
l d 犨ᨨᨨ = l - 㡰 l d 犨 ��㡰 ᨨ鼀⼄ �㡰 ��㡰
l d 犨 - = l - 㡰 l d 犨 Q d 犨 = ҈ 㡰
l - = l l - 㡰 Þ l = ҈ ᆠ鼀
\
h㦨 ç
�
泐 �矄 ᆠx ᆠᨨ盐鼀 ᨨ 鼀ᆠ盐泐 �矄ᆠ
和盐矄�盐� �盐ᨨ hAhBrArB re 矄ᨨ� �犨�h 㦨�ᆠ鼀 ᆠ㡰 ᆠᆠ�ᨨ a㦨
A
F盐ᨨ泐 - Mg
p1
M
mg
҈鼀
q
�鼀
├� �鼀ᨨ鼀 ᨨ盐ᨨ� O
Þ p㦨
�㡰
꾀矄ᨨFT㦨犨ᨨ 盐� �矄ᆠ 㦨 l - x 㡰 A桌 ´rw g ��㡰 꾀 m 㦨 l A㡰 �rw ���㡰 ␔ A�矄 ᨨ ᆠᨨ矄矄矄ᆠ�ᨨ ᨨ ᨨ ᨨ鼀⼄ �㡰 ��㡰 ���㡰
Vl Vs㦨ᨨ盐鼀ᨨ �⼄盐� 矄ᨨ 矄ᆠ�泐 dl ds 㦨 矄�泐 ᨨ �⼄盐� 矄ᨨ 矄ᆠ�泐 ␔ 矄ᨨ 矄 犨 � ᨨ 犨�� ᨨ 矄ᨨ 矄 犨 ᆠ 泐 盐ᨨ泐 ᨨᆠ
� ´ 䰘d � ´ g � - x
��㡰
Þ p҈졔҈d�맔g㦨p
���㡰
ᨨ鼀⼄ �㡰 ��㡰 ���㡰
� ´ 䰘 ´ d � g
� + x
� � = \ q 㦨� ҈ - x 䰘䰘
和� ᆠds> dl \Vs Vl
�㡰
p҈맔A졔҈맔A맔d�g㦨p맔A
l æ l - xö 矄� q 㦨 l - x㡰 A桌rw g ´ ç 矄� q è ÷ø
Vsds㦨 Vl dl
� ´ 䰘d � ´ g � + x
M�矄 鼀� 㜠ᨨ� ᨨ 盐AB. 꾀 ⼄盐���盐鼀 p҈맔A졔mg㦨p맔A
l = ҈ 鼀 \ ҈x㡰㦨�Þx㦨黨鼀 ᨨ鼀 ��鼀
ᆠᨨ矄 q =
B
ᨨ�矄 � ᆠᨨ盐鼀 턼 p 맔�맔A㦨p맔 �x㡰맔A
O
l A㡰 �rw g ´
45 cm
B
Þ p҈㦨
鼀�
l æ l - xö 矄� q mg ´ 矄� q = FT ç è ÷ø
Q
꾀 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ 盐 x �矄 ᆠ ᆠᨨ盐鼀 ᨨ 鼀ᆠ盐泐 �矄ᆠ � 盐�矄 �ᆠ泐 꾀泐� � P҈V҈ 㦨 PV ᨨ�矄 � ᆠᨨ盐鼀 꾀 p맔�맔A㦨p҈맔 �졔x㡰맔A
x
q
M
Q'
p2
45 cm
P
10 cm 10 cm
ᔴᨨ 矄� ⼄盐���盐鼀ᨨ ├ � � ᨨ� �졠 ᨨ⼄盐���盐鼀F 㦨 t 㦨
F҈
P'
M
é hAr A + rB h + hB 㡰 - h + hA + hB 㡰 rC ù 㦨 ê ú g rC h + hA + hC 㡰 ë û g 㦨 盐犨 矄
x
A
Þ
+ ҈ d � ´ g =
� ´ 䰘 � ´ 䰘 + ҈ = � + x � - x
� ´ 䰘 ´ d � ´ g � - \ 㦨黨�ᆠ鼀
Topic-2 : Fluid Flow, Reynold's Number and Bernoulli's Principle ҈
㡰 矄� � ᨨ盐�Ἐ矄 ⼄盐�ᨨ
҈ ҈ P҈ + ru҈ + rgh҈ = P + ru + rgh
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PS-99
Mechanical Properties of Fluids
ᨨ ᨨ� ᨨ �h҈㦨 h㦨 ├� �
҈҈ 㦨
P P҈㦨PP㦨 犨�
p � 㦨 p
҈ P ҈ P ҈ ҈ Þ P + ru = + rV Þ + ru = rV P Þ V = u + r b㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ AaA㦨 ᆠ鼀 BaB㦨ᆠ鼀 ᨨᆠ�泐ᨨ �⼄盐� ᨨ犨A㦨VA B㦨VB 矄� �⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐aAVA㦨aBVB VA㦨VB Þ VA㦨VB ᨨ犨盐矄� � ᨨ盐�矄⼄盐�ᨨ
҈ ҈ ҈ PA + rVA = PB + rVB Þ PA - PB = r VB - VA
(
� 㦨
和盐矄�盐� � 盐ᨨ � ⼄盐�ᨨ �㡰
� 졔 � 㦨 � ᨨ = é ù ê + ú ë û 䰘
㡰 �ᆠ鼀
x
�
( DP ) ´ = ( 䰘 ) ´ 鼀⏌矄 = ҈䰘 ´ ҈ ᆠ鼀⏌矄
҈� ҈� ᨨ盐鼀 ᨨ犨턼㦨aB맔vB 㦨맔҈맔VB㦨䰘ᆠ鼀⏌矄»䰘ᆠ鼀⏌矄 㡰 ᨨ鼀 ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐 A҈v҈㦨 Av v҈ v ᨨᆠ��矄犨ᨨ 矄 ᨨ � 꾀҈ 꾀 ᨨ � 犨ᨨ 矄
v҈ A p 㡰 黨 = = = ҈ v A҈ p 㡰 b㡰 矄� � ᨨ盐�矄⼄盐�ᨨ
Þ
\ - ҈ = � ҈㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐�ᨨ ᨨ ᆠᨨ � 盐泐
҈ P + v҈ - v 㡰 + rgh = P
Þ v = v҈ + gh Þ v = v҈ + gh
꾀҈҈ = 꾀 꾀҈ 鼀鼀 = 꾀 ҈ 鼀鼀
쟼⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐 꾀҈҈ 㦨 꾀
ᨨ鼀⼄盐�ᨨ 㡰
æ ҈� ö ҈ᆠ鼀㡰 ҈鼀⏌矄㡰㦨 꾀㡰 ç ҈㡰 + ´҈ ´ ҈ ÷ è ø ҈ 맔҈㦨꾀 맔
ᨨ =
҈ ҈
éQ � = ҈鼀 ⏌ 矄 = ҈ ᆠ鼀 ⏌ 矄 ù ê ú êë = � ᆠ鼀 úû ҈ 和ᨨ� �犨� ҈ =
䰘 = p = ( p´ ´҈ ) ´ � 䰘 ´҈ 䰘 Þ � = Þ � = p p
ᨨ 犨 �ᆠ 犨 ᨨ犨矄 ᨨ盐�
䰘 ´ 䰘 䰘 ´ 䰘 Qp =҈ ) Þ = » 鼀 ( ´ ´҈ ´ ´
� ᨨ ᆠ ᨨ ᨨ � � ᨨ鼀 ᨨ 犨� ├�矄ᆠᨨ矄ᨨ鼀 㡰 꾀ᆠᆠᨨ � � ᨨ ᨨ盐�Ἐ矄 � ᆠ�
҈ ҈ r ҈ 졔 r � 㦨 r
҈ 졔 � 㦨 � 졔 � 㦨
꾀҈ = = 꾀 ҈ ҈
æ ö ç ҈ ÷ - ( ҈ ) = ´҈ ´ � è ҈ ø
b㡰 ᨨ盐鼀�ᆠ ᨨ犨
Þ � =
㡰
盐� � �矄盐ᨨ � ⼄盐�ᨨ ҈㡰
㦨
� ᨨ �ᆠᨨ盐鼀 ᨨ鼀ᆠ盐泐� 盐�矄 P졔x㦨P Þ P㦨 䰘x㡰 Þ 맔A맔䰘㦨 䰘x㡰맔꾀맔 �x㡰 \ x㦨
盐矄 � ᨨ �ᆠᨨ盐鼀
㦨�㦨҈ᆠ鼀 㡰 꾀ᆠᆠᨨ � � ᨨ ᨨ盐�矄 ᨨ鼀
҈ ҈ ҈ + r҈ = + r
҈- \ 꾀 㦨 㦨�맔҈�鼀
�
�x㡰
)
P
ᆠ鼀
æ ҈ V ö V Þ DP = ҈ çVB - B ÷ Þ DP = � ´ B ç ÷ø è
Þ VB =
҈
盐㦨 꾀҈҈ = 꾀 = 黨
´ ҈ = 䰘�ᆠᆠ ⏌ 矄
㡰 ᨨ鼀 ᨨ盐�Ἐ矄 ᨨ鼀ᨨ� ҈ h
҈
�㡰
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҈ rV҈
㦨졔졔
҈ rV
P졔rgh㦨P҈
�㡰
꾀 ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐A҈V҈㦨AV ᨨ
V㦨
A҈V҈ A
҈
҈҈
҈
L p
҈�
҈
\ P㦨�Pa ᆠ㡰 �矄ᨨ �矄盐矄 矄 v҈㦨�鼀⏌矄 �ᆠᨨ鼀矄ᨨ盐ᨨ ᨨ 犨� 矄 v ᨨ鼀 � ᆠ� ᨨ ᆠᨨ � 盐�泐 a҈v҈㦨av Þ
pr҈ v҈ = pr v Þ r҈ v҈ = r v
Þ
( ) ´ � = (҈) ´ v
\
v㦨鼀鼀矄҈㦨鼀矄҈
㡰 PX - PY =
҈ ra va
X
Y Z
\
҈ ҈ rl vl = ra va Þ vl =
\
ᨨ盐鼀 ᨨ犨 µ
ᨨ鼀 ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐
和 = ´ 和 - 和 㡰 = ´ 和 ҈䰘
��㡰
ᨨ鼀⼄矄 �㡰 ��㡰
- =
҈
ra ´ va rl
ra rl
㡰 ᨨ盐鼀 ᨨ犨 Q㡰 ᨨ � ᆠᨨ鼀矄矄� 盐� �
矄 泐 ᨨ犨 鼀� 矄 ᆠᨨ 矄 ᨨ鼀 ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐 av㦨ᆠᨨ 矄 \
� 鼀鼀
҈
v A㦨�ᆠ鼀
A҈㦨҈ᆠ鼀
盐 PZ㦨PX
҈ æ ҈ 䰘 ö ҈ 䰘 r ç - ÷ = r = r \ 㦨黨 ø 黨 è黨 㡰 ᔼ ��泐矄矄盐 ᨨ � ᆠᨨ盐鼀
P҈ = ҈� N ⏌ m ᨨ犨 P҈ 㦨 ҈� 鼀 V҈ = � - H 㡰 A ᨨ盐鼀 V҈ = � - H 㡰 A 犨 A�矄 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ H 矄矄 � 泐 ᨨ盐鼀ᨨ �ᆠᨨ盐鼀 ᨨ 犨V㦨 �㡰꾀㦨꾀ᔼ P �矄 矄矄盐 ᨨ �
ᔼ ��泐
҈
v҈㦨҈鼀矄
和 和
P
҈ PZ - PY = rl vl
�㡰
Þ=
P҈㦨Pa
҈ = P + ´҈ ´
҈ ҈ + r = + r ҈ r - 㡰
\ � �� ᨨ 犨 盐ᨨ ᨨ � �ᨨ ᨨ� �ᆠ㦨 㦨鼀鼀 꾀ᆠᆠᨨ � �ᨨ ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐 A҈v҈㦨Av Þ ҈맔҈㦨�맔v Þ v㦨鼀⏌矄 ᨨ犨盐矄� � ᨨ盐�Ἐ矄 ᨨ鼀
҈ Þ + ´҈ ´҈
ᆠ㡰 � v҈㦨҈鼀矄҈ A҈㦨҈鼀 v㦨ᨨᆠ�泐ᨨ 犨矄鼀҈�鼀ᨨ犨 A㦨䁸 ᨨᆠᆠ盐� �v 盐矄� � vu㦨as u㦨҈鼀⏌矄錐s㦨҈�鼀a㦨g㦨҈鼀⏌矄 v㦨v㦨䁸 v҈㦨맔҈맔҈� Þ v㦨v㦨鼀⏌矄 泐⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐v҈A҈㦨 vA v A ҈ ´ ҈ - = � ´ ҈ -� m \ A㦨 ҈ ҈ = v 黨㡰 꾀泐� � ᨨ盐�Ἐ矄 ⼄盐�ᨨ
- =
� 泐
҈ ҈ P҈ + rv҈ = P + rv
gy 㡰 ´ L = g ´ y 㡰 pR R㦨
P = 黨 ´ ҈ N ⏌ m
\ ҈� ´ � - H 㡰 A = 黨 ´ ҈ 㡰 ´ A Þ 㦨鼀鼀
㡰 꾀矄犨├ ᨨ犨ᨨᆠ�泐ᨨ 盐 V㦨 gh ᨨ鼀⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐A҈V҈㦨AV
Þ
\
V = A
PV ҈ ҈ = PV
��㡰
Q h = - �� = 䰘� 鼀 㡰
L㦨pR\
P + ҈ ´ ҈ ´
꾀矄矄盐鼀� � 鼀盐鼀� 矄 ᆠᨨ 矄 ᆠᆠᨨ � �ᨨᨨ泐矄犨
ᨨ鼀⼄ �㡰 ��㡰 gh ´ ҈ ´ 䰘� v㦨 ᨨ v㦨 㦨�鼀⏌矄 ҈ - ҈㡰 æ aö ҈- ç ÷ è Aø
҈
P
= ҈� ҈
\
鼀鼀
졔rgh졔
Q㦨a맔v 㦨ᆠᨨ 矄 ᨨaµ
҈ v
犨 a㦨ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ v㦨ᨨᆠ�泐 Þ ᔼ v ᆠ矄矄a� ᆠ矄矄 �ᆠ㜠矄 ␔ 矄鼀 ᨨ 犨 鼀ᨨ矄 盐 �矄 矄 v㡰 ᆠ矄矄 ᨨ �a � ᆠ矄矄 � 犨 矄 矄 ᨨ盐 矄 ᨨ盐 � ␔ 矄鼀 ᨨ 犨 ᨨ鼀 ᨨ矄 � 鼀ᨨ矄 ᨨ犨 �矄 矄 � ᆠ矄矄 ᨨ ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᆠ矄矄 � A҈㦨p맔 맔҈鼀㡰 A㦨p맔 ҈맔҈鼀㡰 v҈㦨�鼀⏌矄
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PS-101
Mechanical Properties of Fluids v҈
A
A҈
A ´L´D´g H H � P㦨P졔 dg + ´ d ´ g + A �d ù é æ H L ö + ÷ dg êQ D = Þ P㦨P졔 ç úû è ø ë b㡰 �㡰 꾀泐� � ᨨ盐�Ἐ矄 ᨨ鼀 éH ù æH ö P졔 ê ´ d ´ g + ç - h÷ d ´ g ú è ø ë û
v
h
x ᨨ鼀⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐A҈v҈㦨Av \
v㦨
H - h㡰 ҈ g 㦨P졔 d 㡰v Þ v㦨 ��㡰 ᨨ�ᆠ鼀ᨨ�ᨨ ᨨ �⼄盐� � � ᨨ鼀 ᨨ�鼀├ ᨨ ᆠ �⼄盐� ᨨ ᨨᨨ鼀 uy 㦨Sy㦨hay 㦨gty㦨t
A҈ p ´ ´ ҈ - 㡰 ´ � v҈ 㦨 㦨鼀⏌矄 A p ´ ҈ ´ ҈- 㡰
ᨨ鼀 h㦨
h ҈ gt Þ t㦨 g
S㦨ut졔
ᆠ ᨨ� ᨨ �
h ´ ҈� = ´ = 鼀 g ҈
x 㦨v맔t㦨v ҈黨
� r㦨҈├�⏌鼀h҈㦨鼀h㦨�鼀 A҈㦨맔҈鼀A㦨맔҈鼀v҈㦨҈鼀⏌矄 ᨨ鼀 ⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐
H H 㜠h
A҈v҈ = � 鼀⏌矄 A҈v҈㦨Av \v㦨 A
v
ᆠ ᨨ쟼
꾀泐� � ᨨ盐�Ἐ矄 ᨨ鼀 P Q p҈p㡰㦨rg hh҈㡰
x = v × t 㦨 H - h㡰
dx 㦨 dh dx ҈ = Hh㡰h桌҈⏌{Hh晤 dh ҈ H Hh㡰桌҈⏌Hh桌㦨 Þ h㦨 ᆠx犨�鼀 �鼀盐鼀h㦨
҈ ҈ ù é҈ 꾀 r v - v҈ 㡰 = ´ ҈ ê - ҈ú = - ´ ҈ ⏌鼀 ë û 㦨䰘�맔҈⏌鼀 \ ␔ᨨ├ ᨨ ⏌ᨨ泐矄矄盐 ҈㦨黨맔҈䰘�맔҈⏌鼀㦨黨�맔҈⏌鼀 㡰 �㡰 和� ᆠ ᆠ泐� �矄� ⼄盐���盐鼀� �⼄盐� \ ␔�� ᨨ ᆠ泐� 㦨盐 盐矄 盐ᨨ盐�⼄盐� 졔盐 盐矄 盐ᨨᨨ犨�⼄盐�
d L
A⏌� F D
d
鼀 �鼀盐鼀盐ᨨ x
é æ H ö ù H xm㦨 êH - ç è ÷ø úû ë
H⏌
㦨
H⏌ A
҈ H H H H ´ = =
Topic-3 : Viscosity and Terminal Velocity
mg
L
Pᨨ h x
g h ´ 㦨 H - h 㡰 h g ���㡰 ᨨ � � � 盐ᨨ h ᨨ犨 �ᆠ x�矄鼀 �鼀盐鼀
҈ r v - v҈ 㡰
ᨨ犨 犨ᨨ├ ᨨ ⏌ᨨ 泐��泐 ᨨᆠ矄 㦨rg hh҈㡰㦨҈맔黨맔㦨黨맔҈⏌鼀
h ҈ ҈ at Þ h 㦨 gt Þ t 㦨 g Pᨨ
҈
㡰 矄� �vu㦨gh
Þ v - = gh Þ v = gh A A L A L ´ L ´ D ´ g = ´ ´ d ´ g + ´ ´d ´g � � �
d d �d + = Þ D㦨 ��㡰 ᔴᨨ 矄� � � �ᆠ ⼄盐���盐鼀 ᨨ 犨ᨨ �⼄盐� 矄 ᆠ泐�
ᨨ 矄矄盐 ᨨᨨ鼀 ᨨ ᆠ泐� 㦨 꾀鼀ᨨ矄 �ᆠ 矄矄盐졔矄矄盐 盐ᨨ�⼄盐� ᨨ 矄�泐d졔矄矄盐 盐ᨨ �⼄盐� ᨨ 矄�泐d졔矄矄盐 盐ᨨᆠ泐� ␔�� ⏌꾀桌
鼀� ᨨᆠ�泐
VT =
r r - s 㡰 g h 黨
꾀 � � ᨨ盐� h ᨨᆠ�泐矄 ᨨ盐 ⼄盐ᨨ鼀� ᨨᆠ�泐
\ gh =
r r - s 㡰 g 黨 h
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EBD_8341 PS-102
Physics
Þh=
r g r - s㡰 ҈h
㡰 䰘맔
�� � �⼄盐�ᨨ ��㡰泐 �㡰
r g r - s㡰 ҈ h
Þ gh =
҈ R = Þ R҈ = R R҈
Þ h a r
pr 㦨 pr
ᨨr㦨
鼀� ᨨᆠ�泐 vµr
ᨨ盐鼀 V =
R
v҈ r҈ \ v = r
æ r ö æ R⏌ö ҈ ᨨ v㦨 ç ÷ v҈ = ç ÷ v҈ 㦨 黨 è R ø è r҈ ø
ᨨ
v҈ v 㦨黨
= = 䰘 鼀 䰘 ´ ҈ ´ ҈� ´ ҈-
r rs㡰g 黨h r҈ s - r҈ 㡰 g VP 黨h҈ r s - r҈ 㡰 h = = ҈ ´ VQ r s - r 㡰 g r s - r 㡰 h҈ 黨h
�
- 桌 ´ = �㡰 - ҈桌 ҈
r r s҈ 㡰 � 黨h҈
Q
r҈s㦨s҈r㡰
r1
҈ s1
r2
� s2
҈
ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ 矄矄盐� 矄� ҈矄 矄ᨨ盐
DP҈ = P҈ - P = ҈ =
T R҈
꾀 DP = P - P = =
�㡰
T R
��㡰
R=
T d r㡰 g
b㡰 꾀矄 犨 ├ ᨨ犨
㡰 ␔ 盐�矄 ᆠ 盐ᨨ泐 ᨨᆠ㦨 ᨨᆠ 盐ᨨ矄盐 ᆠ 矄�ᨨ
q
꾀�� VPVQ�.e. ��矄VP VQᨨᨨ矄�ᨨᆠ ᨨ
Topic-4 : Surface Tension, Surface Energy and Capillarity
R d r㡰 g Þ
ᆠᨨ矄 q = r�
� = 䰘�
␔ r� ᆠᨨ矄 q� ᆠᨨ矄҈�° ҈ = ҈ = = ᆠᨨ矄 q␔ ᆠᨨ矄 ° r␔ æ ö q �
ᆠᨨ矄 æ � ö r ␔ æ � ö =ç ÷ç ÷ç ÷ ␔ çè ␔ ÷ø çè r� ÷ø çè ᆠᨨ矄 q ␔ ÷ø ҈ ҈ = 䰘� ´ ´ = = ҈ �
r s r҈ 㡰 � 黨h
\
㡰 ᨨ ᨨ矄ᨨ� ⼄盐���盐鼀盐犨 ᨨᆠᨨ ᨨ㦨 ᨨ犨 犨 ᨨᆠ ᨨ ᨨ
Þ T=
h VP = ҈ h VQ
pR dg pR rg Þ T pR㡰 = pR d r㡰 g
␔ �矄 �盐矄ᨨ 矄 ᨨ犨 犨 鼀� ᨨᆠ�泐
VQ =
T pR =
㡰和� ᆠ矄� ��矄盐 r҈s҈ rs ᨨ ᨨ� � 犨�� ᨨ 矄泐矄鼀 㦨 盐 盐矄 r҈졔r㡰Vg㦨 s҈졔s㡰Vg 犨 鼀� ᨨᆠ�泐
VP =
T ᆠᨨ矄 q rgr
´ � ´
-҈
VT㦨
=
V҈ R҈ R = = = ҈ V R R
\
b㡰 � 꾀 �ᨨ ᆠᨨ ᆠq㦨 和盐 ᆠ 矄�ᨨ T㦨�鼀҈ �盐矄ᨨ ᆠ�泐盐r㦨҈�鼀鼀㦨҈�맔҈鼀 矄�泐ᨨ 鼀 泐 �ᨨ � r㦨䰘├�鼀 ᔴ�泐�矄 h =
b㡰 꾀ᆠᆠᨨ � � ᨨ ᨨ�ᆠ�矄 ᨨ鼀 ᨨᆠ�泐ᨨ V 㦨 gh = ´ 黨 ´ � @ 黨 鼀矄 㡰 꾀矄 犨 ├ ᨨ犨 鼀� ᨨᆠ�泐
pR
rq
T맔dl
ᨨᆠ 盐ᨨ ᆠ矄矄矄矄盐㦨 盐 盐矄 꾀ᆠᆠ矄矄矄矄盐� �盐㦨
T p R rw g p r 㡰 = R T
Þ r =
R rw g rw g Þ r = R T T
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PS-103
Mechanical Properties of Fluids
㡰 꾀ᆠᆠ�ᨨ 盐 ᨨ ��泐ᆠ �矄 犨�
꾀ᆠᆠᨨ � �ᨨ� �矄⼄盐�ᨨ PV㦨nRT
ö æ �¢ = � ç҈ - ÷ è ø
PAVA㦨nARTB Þ
p 㡰㦨nARTA �㡰 PBVB = nB RTB Þ ҈ ´ p 㡰 = n B RTB … ��㡰
矄矄盐㦨r� ᆠ�ᨨ 䰘
æ ö �矄 ç҈ - ÷ 泐 è ø
㡰
C
�� � �⼄ ��㡰泐 �㡰
R
a
q 졔a b
M
҈ ´ p 㡰 n = B nA ҈ ´ p 㡰 nB \ = nA
q
P
҈
S ö æ 꾀矄ᨨ ç P ÷ + hrg = P R ø è
Þ
hrg 㦨
S R
T= ҈҈
S S ᆠᨨ矄 ( q + a ) = Rrg brg ҈ pR = k ´ pr \ R = K r 㡰
é R ù ҈⏌ \D㦨 ps ê k ´ ⏌ - R ú = pSR éëk - ҈ùû ë k û
҈ ´ ҈- 㡰 éë҈a ⏌ - ҈ùû p
\ ҈ = ҈ a ⏌ - ҈ �ᆠ� �
a = \ a = T 㡰 矄矄盐PA㦨P�졔 RA Q 쟼 ᆠ矄矄矄矄盐 盐ᨨ矄盐 ᆠ 矄�ᨨ T㡰� 矄ᨨ盐
҈㦨҈a⏌Þ
҈
ᆠ鼀
ᆠ鼀
T ´ = + Þ PA = ҈ N ⏌ m RA T ´ = + 和�鼀�泐 PB㦨졔 㦨҈⏌鼀 RB
\ 㦨 ҈ 鼀鼀㦨䰘҈ bᆠ㡰 ᨨᆠ矄ᔼ
=
�㦨鼀
PA = +
⏌
ᆠᨨ矄 q҈ ´ 䰘� ´ ᆠᨨ矄 ° = = 䰘�ᆠ鼀 r � ´ ҈- ´ ҈ ´ ҈
ᨨᆠ矄ᔼᔼ
B ҈ v
犨ᨨ 鼀҈ 䰘㡰꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ 犨ᨨ �鼀矄 �矄ᆠᨨ �ᆠ├ 矄矄Ἐ Ἐ � �矄鼀�ᆠ�ᆠ盐 �矄 㦨 ⏌ 矄矄盐 ᨨᨨ鼀ᨨ �矄ᆠ㦨矄矄盐 盐 ᨨ 矄盐 ᆠ 矄�ᨨ
҈ =
㦨T⏌r㡰
A
rhrg ҈� ´ ҈-� ´ ҈� ´ ҈- ´ 黨 ´ ҈ = = ҈҈ 鼀��
é ҈ ҈ ù ҈ ù
é \r� = ê + = ú = ê + ⏌ úû ë ë ҈ û
´ 䰘 ҈ ´҈ Þ = Þ = = = r� r� ҈ ´ ҈ ҈ ´҈
a ⏌ - ҈ùû Q K㦨҈a桌 \D㦨 pSR éë҈
黨
ᆠᨨ矄 q = ᆠᨨ矄° = ҈桌
æ ҈ ҈ ö r� = ç + ÷ è ҈ ø ҈ ҈ > \ ҈ 㦨 ⏌
DU = Sk ´ pr - pR 桌
\҈㦨 p ´
T ᆠᨨ矄 q rgr
和盐 ᆠ 矄�ᨨ
h=
[Q TA = TB ]
҈҈㡰 � �盐矄 ᨨ ᆠ�泐盐 r㦨҈�ᆠ鼀㦨҈�맔҈�鼀鼀 h㦨҈�ᆠ鼀㦨҈�맔҈鼀鼀 矄� � h =
R �盐矄ᨨ 鼀 �矄ᆠ盐矄 ᨨ鼀 犨� ᆠᨨ 矄 �q P 鼀ᨨ矄 �ᆠ矄矄盐 aö b æ b ᔼ Dᔴ㝐 ᆠᨨ矄 ç q + ÷ = Þ R㦨 aö ø R æ è çq+ ÷ ø è
\
҈맔
҈
ᆠᨨ矄 q ´ 䰘� ´ ᆠᨨ矄° = = 䰘�ᆠ鼀 r� ´ ҈- ´ ҈ ´ ҈
ᆠᨨᆠ�ᨨ � �� ᨨ 犨ᆠᨨ盐鼀 �矄 � 盐 盐ᨨ犨�� ᨨ 犨ᆠᨨ � � 鼀 �矄ᆠ盐矄犨� � ᨨ ᨨ ᆠ矄矄 ᔼ ᆠ矄 ᔼᔼ� ᆠ�泐ᨨ� �矄�ᆠ鼀 ᨨ 犨矄盐 ᆠ 犨 � ᆠ�泐 犨� ᨨ ᆠ � ᆠ␔ 犨� ᆠ ᨨ 泐�䰘�ᆠ鼀 s ᆠᨨ矄 q ᆠ㡰 꾀矄犨├ ᨨ犨 h = rrgeff ҈ 꾀矄�r� ᆠ矄矄h ᆠ矄矄hµ r ᨨ 鼀矄 鼀� � � ᆠᨨ 矄 桌
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EBD_8341 PS-104
҈
Physics
꾀矄ᨨhµs 盐 � � �矄�ᨨ� �盐犨� ᆠᆠ�ᨨ �a g 㦨g졔a꾀矄g � ᆠ矄矄�h ᆠ矄矄 꾀矄ᨨhµ ᆠᨨ矄q ᨨhµq ᆠ㡰 �ᆠ ᨨᆠ 盐ᨨ矄盐 ᆠ 矄�ᨨ
q FV 㦨 T ᆠᨨ矄q㡰맔pr r
҈黨
\
q
pr T ærö = T ç ÷ ´ pr = è Rø R
Þ ҈
Þ Þ
ᨨ ⼄盐���盐鼀 ᨨ 犨� � �ᆠ �ᆠ�ᨨ ᨨ鼀 ᨨ 泐 ��鼀ᨨ 犨�矄 ᨨ犨 � ��盐 T ᆠᨨ矄q \ Tlᆠᨨ矄q㦨llg ᨨ Tᆠᨨ矄q㦨lg
y a
\
T
q q
T矄� q
lag T ´ y 㦨lg ᨨ T㦨 a y
㡰 \
ᨨ R㦨҈맔҈ 鼀 b㡰 和盐 ᆠ �泐㦨 맔p
犨 �ya
҈
҈
pr T pR rg = R æ r T ö � ´҈- 㡰 ´ ҈҈ = ´ R = ç 㦨҈맔҈҈ ç rg ÷÷ ҈ ´ ҈ è ø
ᆠᨨ矄q㦨
T rv
Topic-5 : Miscellaneous Problems
´ ҈ ´ ҈ - 㡰 = 䰘 ´ ҈ - 㦨 ҈҈ ´ ´ 䰘 ҈䰘
T T ´ A Þ R㦨 R rv
R
㡰 ␔ ᨨ�矄ᨨ盐ᨨ ᆠ ᨨ鼀 ᨨ ␔�� 㦨�ᆠ ᨨᆠ 盐ᨨ矄盐 ᆠ 矄�ᨨ mg㦨FV㡰 \
rAv㦨
�盐矄犨 �ᆠ 盐矄矄 ᨨ鼀 � � 㦨
rù é êQ ᆠᨨ矄 q = R ú ë û ҈�
盐犨�矄 ᨨ鼀 � �犨 ᨨᆠ 盐ᨨ� 㦨 rAV � 盐 ⼄盐矄 矄盐 ᆠ 矄�ᨨ ᨨᆠ T⏌R � 矄� 盐
҈ dr = ᆠᨨ 矄 r dt pR d é m ù 㦨ᆠᨨ 矄 m dt ê ú ê pR ú ë û
�
d - R 㡰 㦨ᆠᨨ 矄 dt dR dR µR R - R - 㡰 㦨ᆠᨨ 矄 \ dt dt R
b㡰 ␔├ ᨨ犨 ᆠ矄矄矄矄盐� 矄ᨨ盐�矄� 矄泐 ᨨᨨ�ᨨ ᨨ�矄 �盐矄 �.e. P 㦨
T
T矄� q w㦨lg
␔ 盐�矄 ᨨ 盐矄� � ᨨ盐�Ἐ矄 ᨨ鼀
҈ ҈ P + P + rv҈ + rgH = rv + P ҈ Þ P졔rgH㦨 r v - v҈ 㡰
T r
矄ᨨ 盐 ҈ 矄 矄鼀 �盐矄矄ᆠᨨ鼀 ᨨ 矄ᨨ 盐 �� 㡰
ᨨ ᆠ矄矄矄矄盐҈ ҈ �矄鼀ᨨ ᆠ � ᨨ犨矄 ᨨ鼀 ҈ ᨨ ᨨ盐鼀 ᨨ 矄ᨨ 盐 ҈ ᆠ矄矄 ᆠ㡰 ᔼ ᨨᆠᆠ矄ᨨ 犨ᨨ犨 矄�� 犨 矄矄ᆠᨨ � � � 犨�矄�� ᆠᨨ 矄 ᆠᆠ�ᨨ aᨨ犨 矄 �� 꾀矄
ᆠ�ᨨ 犨 矄 ᨨᆠ ᨨ 犨 ᨨ 矄矄 ᨨ犨 矄 �� 꾀矄 ᆠ�ᨨ 犨 盐矄 矄 犨 ᨨ犨 矄 꾀ᆠᆠᨨ � �ᨨ犨ᨨ Ἐ矄 犨ᨨ 鼀ᨨ�ᨨ 矄ᨨᨨ 犨�矄 矄矄 ᨨ犨 � ��盐
ᨨ鼀⼄盐�ᨨ ᨨ ᆠᨨ � 盐�泐A҈v҈㦨 Av v҈㦨
A v A҈
P
v㦨v桌
҈ éê æ A ö ùú v v r \ P졔rgH㦨 ç ÷ ê è A҈ ø úû ë ҈ é æ A ö ù ê v ҈ r ç ÷ ú P졔rgH㦨 ê è A҈ ø ú ë û
P졔rgH㦨DP 꾀ᆠᆠᨨ � � ᨨ ᨨ�矄矄盐�Ἐ矄 ⼄盐�ᨨ
p DP 㡰a p DP㡰a Q = Þh㦨 Ql hl
é æ A öù ҈ p P + rgH 㡰 a p = ´ rv ê҈ - ç ÷ ú ´ a Ql Ql ê è A҈ ø ú ë û p ҈ é d ù é A d ù h= ´ rv ê҈ - ú ´ a êQ = ú Ql ëê D ûú ëê A҈ D ûú
\h㦨
黨
a
ma
矄盐
mg
P�矄�ᆠᨨ 犨
矄盐 ᆠᨨ 犨矄 ᨨ盐 矄�矄 黨 ᨨF矄盐 㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ��ᨨ ᨨ 矄�泐ᨨ � 犨� �� ᨨ鼀 �ᨨ盐
r 㡰 = r-
꾀 㦨҈鼀鼀矄矄鼀㦨├� \ 鼀�㦨�r ҈㡰 Þ 맔�㦨r҈� 꾀 㦨҈�鼀鼀矄矄鼀㦨 㡰├� \ 鼀�㦨�r ҈�㡰 Þ 㡰�㦨r�
r�
鼀�
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PS-105
Mechanical Properties of Fluids
ö - ⏌ - r æ Þ ç҈ = = ÷= r҈ è ø - ҈ ⏌ � � ´ Þ ҈=҈ \ = = �
҈ -
-
=
�
\
v'
㦨
V'd'mg
Vdmg
ᔼ 矄ᆠᨨ ᆠ矄犨 鼀盐�矄� ᆠ矄 泐DT v'd'Hgg = V'd'm g
黨
dἘ vἘ = m 㦨 ᆠ�ᨨ ᨨ ᨨ盐鼀 ᨨ 鼀 矄盐鼀� �
Þ V Ἐ d ἘHg 鼀ᆠ盐泐㦨 x' 矄泐㡰 \
x Ἐ d Ἐm ´ d Hg d Ἐm ´ d Ἐ Hg ҈ + g DT 㡰 ҈ + g DT 㡰 = = = x d Ἐ Hg ´ dm d Ἐ Hg ´ d Ἐm ҈ + g҈DT 㡰 ҈ + g҈DT 㡰 DT㡰҈
㦨 ҈졔gDT㡰 ҈졔g҈ 㦨 ҈졔gDT㡰 ҈g҈DT㡰 㦨 ҈졔 g g҈㡰DT g҈g DT㡰
xἘ - ҈ = ( g - g҈ ) DT Þ x
㡰 和� ᆠ 矄�泐ᨨ d㡰 矄�泐ᨨ �⼄盐� dL㡰 盐 盐矄> 犨�� \ 矄 犨� �ᨨ � 矄� �⼄盐� \
�ᨨ 㦨
ᨨ
a㦨
Vd L g - Vdg Vd
ᨨ
t㦨
v gt҈ = a a
t㦨
gt҈ dt҈ = d L - d 㡰 æd -d ö ç L ÷g d è ø
v = gh
ᨨ
t㦨t҈졔
ᨨ
㦨
d d 㦨 h h 1 2
��㡰
´ dt҈ d L - d 㡰
é
ᨨ t㦨t҈ ê҈ +
ë
ù d ú d L - d 㡰 û
t҈dL dL - d
æd -d ö 㡰 和� ᆠ �ᨨ 㦨 ç L ÷ g �矄 ᨨ ᨨᨨ㜠�ᨨ è
d
ø
ᨨ �矄ᆠ鼀 鼀ᨨ�ᨨ ᨨ �矄 ᨨ 矄�鼀 鼀ᨨ �ᆠ ᆠ㡰 ᔼ d㦨 dL �ᨨ = �ᨨ 㦨和� ᆠ 矄�├矄 犨矄盐 ᆠ犨� 矄ᨨ鼀ᨨᆠ�泐�犨�ᆠᨨ � 盐 犨� 矄鼀ᨨᆠ�泐v㦨gt҈⏌ � ᨨ犨 犨 �ᆠ�ᨨ
dx
x A
�㡰
\ �鼀ᨨ �矄ᨨ�⼄盐� 矄盐 ᆠ㦨t 㡰 Q t㦨 �鼀├ 泐 ᨨᆠᨨ鼀ᆠ├ᨨ ᨨ矄��ᨨ
ᨨ鼀犨 �ᆠ �犨矄矄 ᨨ t㦨t҈졔t矄盐矄�盐t ᨨ鼀 ��㡰
ᔼ ᆠ矄矄 d = h҈h = ҈ 鼀 ᔼ geff㦨 ᨨ犨├矄ᨨ盐矄 犨� ᨨ矄矄盐 � ᆠ d w 盐矄 ᆠᨨ 矄� 矄鼀鼀 鼀矄矄 dm �矄 ᆠ x ᨨ鼀 A 矄 矄 ᨨ犨 � ��盐ᔴ � ᨨᆠ ⼄盐� ᨨ 鼀矄矄dmᨨᨨ㦨 H dm㡰xw \ ᨨ ᆠ � ᨨᆠ ⼄盐� ᨨ 鼀矄矄ᨨ � Lᨨ ᨨ
æ dL - d ö ÷g è d ø
ᨨ a㦨 ç
Q ᨨᆠ�泐ᨨ 盐矄 ᨨ� ᆠᨨ� 矄 犨� �⼄盐� V㦨gt҈⏌桌
h҈ g
㦨 h҈h geff> g geff㦨g geffg
盐 盐矄 - 犨�� 鼀矄矄
ᆠᨨ鼀ᨨ矄 � 矄� �⼄盐� � �鼀 t \
ᆠ㡰 ᨨ� ᨨ �矄 ᆠ d㦨v맔t
ᔼ
xἘ- x = ( g - g҈ ) DT x
u æ dv ö ÷ ᨨ 㦨h꾀 h dx è ø u æ dv ö 犨 㦨ᨨᆠ�泐� � ç ÷ h è dx ø ҈ µh錐 µ꾀錐 µm µ h
gh ´
( )
pd ´ H ´ r´ g
ᆠ 㡰 㦨h꾀 ç
㦨
ö pd ´ dx÷ ´ xw ø
w L pd pd w ´ L \ ´ H ´r´ g = r´ ´ ᨨH㦨 g
T +D T
T
ò çè r ´
�矄 ᆠ � ᨨᆠ �矄 ᨨ� 泐 犨�� ᨨ �⼄盐� ᨨ �� H ␔�� ᨨ �⼄盐� ᨨ �� H
V' v
Læ
ᨨᆠ � ᨨᆠ 㦨
pd ´ dx
L pd ´ w ò x dx L pd ´ w ´ 㦨 r ´
v'd'Hgg
V
ò
dm㡰 xw 犨 dm㦨 r ´
㦨 r ´
d v = m = x 矄泐㡰 V d Hg vdHgg
䰘
㦨
ᆠᨨ ��ᨨ ᨨ ᨨ�ᨨ vdHgg㦨Vdmg ᆠ�ᨨ ᨨ ᨨ盐鼀ᨨ 鼀矄盐鼀� � 鼀ᆠ盐泐 㦨
L
L
dm
B
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EBD_8341 PS-106
Physics
Thermal Properties of Matter
10
Topic-1 : Thermometer & Thermal Expansion ҈
㡰 Ӹᨨ盐 �矄鼀ᨨ 盐盐矄Ӹ㦨
Dl Dl Þ = aD \Ӹ = 꾀 ( aD ) lD
l ᆠ㡰 꾀矄犨├ ᨨ犨盐├鼀ᨨ 盐盐矄
㡰 L'҈ L' � 矄 ᨨ 犨� 犨
鼀盐�矄 ᆠ � 泐 DT °ᔴ 꾀T ᔴ
Leq = L҈ + L
⏌ 꾀 矄矄矄 㦨 꾀 Dl ⏌ l) ( 矄�
矄� � ᆠᨨ �ᆠ� ᨨ � 矄�ᨨ
�
a=
꾀 T + D°ᔴ L' = L' + L' eq
҈
\ Leq ҈ + a ⼄ DT 㡰 = L҈ ҈ + a҈DT 㡰 + L ҈ + a DT 㡰 Þ L҈ + L 㡰 ҈ + a ⼄ DT 㡰 = L҈ + L + L҈a҈DT + La DT
a҈ L҈ + a L L҈ + L 㡰 ᔴ � � � ᨨ 鼀 犨� Dl㡰 犨 �矄 鼀盐�矄 ᆠ � 泐DT �矄 �� 泐 Dl = l aDT a㦨ᔴᨨ �ᆠ� ᨨ � 矄�ᨨ
Dl㦨奠DT㦨҈ºᔴ Dl -� \a = = Þ a = ´ ҈ l DT ҈ ´ ҈ ᨨ盐鼀ᆠᨨ �ᆠ� ᨨ 矄�ᨨ g = a = ´ ҈-�
䰘
M Qr = V DV -� - ´ ҈ 㦨gDT = ´ ҈ ´ ҈ ´ ҈㡰 = ´ ҈ V ᨨ盐鼀 � ᆠ矄 泐奠 ᨨ 矄�泐 ᆠ矄 泐 奠 ᨨ 盐矄㡰D鼀㦨Dᨨ A㦨pr㦨p ҈㡰㦨p맔҈ FL L aDT㦨 AY F ´ ᨨ 맔҈�맔㦨 p ´ ҈ - 㡰 ´ ҈҈҈ f \ F㦨pN\ m = 㦨p㦨├� g 㡰 ᔴ �� � � ᨨ ᨨ 矄矄鼀� Dl҈ = D l
Þ a ⼄ =
la҈D q҈ = la Dq a҈ Dq = a Dq҈ q = ҈ q㦨 ᔴ
é a҈ ù êQ = ú ë a û
Þ
DV P = V K g= 3a
V㦨V ҈졔gDt㡰
[Q L Ἐ = L ҈ + a DT 㡰]
DP æ -DV ö çè ÷ V ø
K=
DV P P P = gDt \ = gDt Þ Dt = = V K gK aK ᆠ㡰盐ᨨ 鼀 ᆠ �� � Dl㡰㦨laD �㡰 ᨨ鼀矄矄矄 Ӹᨨ盐 �矄鼀ᨨ 盐盐矄 Ӹ㡰 = ᨨ ��盐 � 矄�
꾀 Dl Ӹ= Þ = Dl l l 꾀Ӹ
D =
l 꾀Ӹ
ᨨ鼀⼄ �㡰
҈ ᔴ
l = l a D
꾀Ӹ
㦨꾀Ӹa D
㡰 꾀 矄 泐 矄 ᨨ犨 �矄 矄鼀 ᨨ鼀 P ᨨ Q ᨨ鼀QᨨS
㝐 ⧤
턼
⧤
和
ᔴ
⧤꾀 ( - ) ⧤꾀 ( - ҈ ) = Þ 㦨҈ ᔴ l l
鼀盐 � � \
҈ - ҈ = \ 㦨҈ ҈
ᨨᆠ �鼀盐ᆠᨨ矄矄㜠矄ᆠ�ᨨ 㝐 �矄 � ᨨ鼀�矄 D 㦨҈ 쟼 矄�ᨨ � 矄鼀鼀 � � �矄 l㦨 a D 㦨 a (҈ ) ҈
\ ò dl = ҈a ò
҈ \Dl㦨҈맔҈맔҈�맔 㦨䰘맔҈�鼀
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PS-107
Thermal Properties of Matter 黨
� �ᨨ 泐矄ᆠ - 㦨 ᆠᨨ 矄 ᨨ 矄ᆠ矄
ᆠ㡰
҈
Þ
mg ⏌ A mg l = Dl ⏌ l ADl 꾀矄ᨨ Dl = l a DT ᨨ鼀 �㡰 ��㡰
҈҈
l + l҈ a a + a s l҈ as = = Þ as l҈ l҈ + l a a + a s 㡰 ᨨ⼄盐��b�盐鼀犨h 鼀盐°ᔴ �� 㡰 盐矄 㦨 ␔ ᨨ ᨨ 泐 \ K҈Vdg㦨 Vd҈g Þ K҈㦨
d҈ d
�㡰 K҈Vdg
d҈
KVd ' 'g d'҈
V
Vd g
d
V'
҈
㡰
㡰
ᨨ⼄盐��b�盐鼀犨h 鼀盐� ᆠ矄矄ᨨ° �� b㡰 ␔ 鼀盐 �矄 � ᆠ矄 矄�泐 犨� ᆠ矄 \ d҈Ἐ㦨d҈ ҈졔gFe맔㡰 dἘ㦨d ҈졔gHg맔㡰 꾀�� 盐 盐矄 㦨 ␔ ᨨ ᨨ 泐 \ KVἘdἘg 㦨 VἘd҈Ἐg \
҈
é ù d d҈ ú= K ê êë҈ + g Hg ´ úû ҈ + g Fe ´
é ҈ + g Fe ´ ù ú= \ K ê êë҈ + g Hg ´ úû ᆠ㡰 W҈㦨mgVda g
W㦨mg V'd'a g㦨mgV ҈졔�gb㡰
é ҈ + � g b ù 㦨mgV dag ê ú ë҈ + � g a û � gbga \ ҈
҈졔�gb҈졔�gaᨨ
\ m =
da g ҈ + � g a 㡰
҈ + � g b W҈ᨨW҈W 㡰 꾀矄 ᆠᨨ 矄 ᨨ盐鼀 �矄 鼀ᨨ鼀 犨ᨨ├矄 ᨨ
ᔴ Ἐ矄犨
��㡰
YA a DT ҈҈҈ ´ p ҈- 㡰 ´ ҈-� ´҈ = =p» g ҈
㡰 ᨨ盐鼀ᆠᆠ�泐ᨨ ├V㦨�ᆠᆠ
Vb = V + V g ├ DT ␔ ├�矄�泐 � 犨� Vmᨨ盐鼀ᨨ 鼀ᆠ盐泐
Vb҈ = Vm + Vm g m DT � ᨨ盐鼀 V - Vm 㡰 = Vb - Vm҈ 㡰
Þ V g ├ = Vm g M ᨨ Vm =
\Vm =
V g ├ gM
� ´ ´ ҈-
= ᆠᆠ ҈� ´ ҈- ᨨ盐鼀V㦨Ibh
DV Dl Db Dh = + + V l b h g 㦨 ᆠᨨ �ᆠ� ᨨ ᨨ盐鼀 矄�ᨨ 㡰 Þg㦨�맔҈�졔�맔҈졔�맔҈ 㦨맔҈⏌ C \ 盐ᨨ C㦨 ҈䰘 b 㡰L ᨨ��� � ᨨ 矄� ᔴᨨ㜠 �ᆠ� ᨨ � 矄�ᨨ ᨨ 矄矄 �矄 �
ᨨ ᆠᨨ� a B > a C 矄矄 \ g=
LB㦨L ҈졔aBDT㡰 R졔d 㡰q 㦨L ҈졔aBDT 㡰 꾀�� LC㦨L ҈졔aCDT 㡰㦨Rq
\
҈ + g Fe ´ K d҈ Þ ҈ = K ҈ + g Hg ´ d
�㡰
mg l mg = Al a DT A a DT
҈ 㡰
V'd' g
d'
҈
Y=
҈�
Þ
F Dl ⏌ A l
\ Y =
- 䰘 °Ӹ - -҈㡰 䰘 泐 + ҈ = Þ = 䰘 - 䰘 -� - -҈㡰 ҈ ҈҈ \ Ӹ㦨 Ӹ ҈ ᆠ㡰 ␔ � 矄 ᨨ ᆠ ᨨ � ᆠ矄矄 泐 矄鼀 鼀ᨨ盐 \ Dl a = Dl s Þ l҈a a t = l a s t a l aa l Þ +҈ = a +҈ = l҈ as l҈ a s
㡰 ␔├ ᨨ犨 Y =
ᨨ
(R + d )q Rq
ᔴᨨ
҈ + a B DT = ҈ + aC DT
q
R+d = (҈ + a B DT ) (҈ - a C DT ) R 泐� ᨨ鼀� 矄�ᨨ 桌
d = ҈ + ( a B - aC ) DT - a B aC DT 㡰 R d d = ( a B - a C ) DT ᨨ R = ᨨ R ( a B - a C ) DT ҈ ҈ R µ \ Rµ a B - aC DT ҈ ᔼ ��泐 鼀盐 T ᨨ泐 ᨨᆠ FB㦨鼀�ᨨ Axrl g = ALrb g ᨨ
҈+
\ xr l = Lrb
�㡰
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EBD_8341 PS-108
Physics 꾀
F'B
r
o
rl
Þ
ᨨ A Ἐ xr Ἐl g = ALrb g
r Ἐl = rl ҈ - gDT 㡰
Þ
\
A ҈졔a DT㡰x rl ҈ - gDT 㡰 g 㦨ALrbg
Þ
xrl ҈ + a DT 㡰 ҈ - gDT 㡰 = Lrb
Þ
xrl ҈ + a DT 㡰 ҈ - gDT 㡰 = xrl
ᨨ鼀⼄ �㡰桌
҈
ᨨ
�ᔴ
Þ K҈ 㡰 = K �㡰 = K 㡰
ᨨ
黨�ᔴ 矄
h҈㦨�ᆠ鼀h㦨�҈ᆠ鼀h㦨黨ᆠ鼀 矄矄盐 B㦨 矄矄盐C P㦨d黨� ᔴgh҈d� ᔴgh㦨P졔d� ᔴghd黨� ᔴgh
ᨨ
K҈ K K Þ K҈ K K = ҈ ҈� = = ҈ ҈� Þ K҈ K = 㡰 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ ᨨ ᨨ �矄 ├� �ᆠ �泐 �矄 ᆠᨨ � ᨨ � 犨ᨨᨨ Þ
ᨨ
ᔴ
҈ - 䰘㡰 K҈ A 䰘 - 㡰 K A - 㡰 K A = = l l l
Þ K҈ ҈ - 䰘㡰 = K 䰘 - 㡰 = K - 㡰
ᔴ
㡰 꾀矄 ᨨ 矄� �ᆠ矄ᨨ 泐 矄鼀 � l㡰 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ A㡰 泐ᆠᨨ
ᆠ � 矄�矄 和ᨨ ᆠ盐 犨�矄鼀 ᨨᨨ 矄
Þ
꾀
b W - W҈ 㡰 W - W҈ + W - W 㡰 t - t҈ 㡰 W - W 㡰
æ DQ ö æ DQ ö æ DQ ö ᆠ盐 = ç ÷ =ç ÷ =ç ÷ t t D D è ø AB è ø BC è Dt øCD
҈
gl =
Topic-2 : Calorimetry and Heat Transfer
Þ ҈졔a DTg DT㦨҈ Þ g㦨a ᨨg�㦨µ矄 矄�泐ᨨ �⼄盐� ᆠ矄矄矄鼀盐�矄矄ᆠᆠᨨ � � ᨨᨨ� d㦨dt ҈졔gt㡰犨 g㦨ᆠᨨ㜠 �ᆠ� ᨨ ᆠ盐�ᆠ 矄�ᨨ ᨨ �⼄盐�
ᨨ
W - W҈ ҈ + g l DT W - W҈ ҈ + g l t - t҈ 㡰 Þ = = W - W ҈ + bDT W - W ҈ + b t - t҈ 㡰
Þ WW҈㡰҈졔b tt҈㡰㦨 WW㡰҈ + g l t - t҈ 㡰桌
ᨨ犨AἘ㦨A ҈졔aDT㡰
矄
V ҈ + b DT 㡰 ´ d l ´ g 㡰 ҈ + g l DT
W - W҈ Vd l g ҈ + g l D T 㡰 = W - W V ҈ + bDT 㡰 d l g
\ FB Ἐ = mg
�ᔴ ᨨ 黨�ᔴ
�㡰
�� � � �㡰泐 㡰犨�
꾀 鼀盐 T 졔 DT ᨨ盐鼀 ᨨ ᆠ盐 � ᆠ矄矄 盐 矄�泐 ᨨ �⼄盐� ᆠ矄矄 矄ᨨ 盐ᨨ 犨 �ᆠ ᆠ盐�矄�鼀鼀矄 � �⼄盐� 鼀� 矄矄鼀
҈黨
dl = dl Ἐ ҈ + g l DT 㡰
WW㦨
mg
鼀�
���㡰
ᨨ鼀⼄ ��㡰 ���㡰 �㡰
x
x
꾀矄ᨨV Ἐ㦨V ҈졔b DT㡰
҈ ҈ mv ´ = ms DT
d黨� ᔴ h҈졔h㡰㦨d� ᔴ h졔h㡰
d黨�°C h + h d ⏌ (҈ + 黨�g ) h + h = = ᨨ d �°C h҈ + h d ⏌ (҈ + �) h҈ + h
҈ + �g �҈ + 黨 ҈ ҈ = = = ҈ + 黨�g � + 黨 ҈҈ ҈҈ ᨨ ҈졔黨�g㦨҈҈졔�맔҈҈g ҈ ´ ´ ҈ ᨨ 黨�g �黨g = ҈ ᨨ g = 黨黨҈ g ´ ҈ \ ᔴᨨ �ᆠ� ᨨ � 矄�ᨨ a㦨 = \ a㦨䰘맔҈� ᔴ gl ᆠᨨ �ᆠ� ᨨ ᨨ盐鼀 矄�ᨨ ᨨ ᆠ盐 犨�� 犨�� 㦨 盐 盐矄
ᨨ
WW҈㦨 ´ l ´ �
�㡰
WW㦨 Ἐ´ Ἐl ´ �
��㡰
v ҈ ´ ҈ = = 䰘�°ᔴ ´ s ´ ´ ´ ҈ 㡰 �ᆠ鼀矄 盐ᨨ犨 鼀ᨨ盐 ᨨ �ᆠ鼀矄㦨m�ᆠ
Þ DT =
mw sw Dq = m�ᆠ L�ᆠ =
\ m�ᆠ =
´ ´ �
´ ҈�
mw sw Dq L�ᆠ
= ҈䰘 ├�㦨҈䰘�
㡰 �� 泐犨 = mw Cw T鼀� - Tw 㡰 = ´҈´ ҈ - �㡰 ├ 泐矄鼀㦨mL矄鼀졔m Cw Ts T鼀� 㡰 㦨鼀맔�졔鼀 ҈㡰맔 ҈҈㡰㦨鼀맔�졔鼀 ҈㡰맔 黨㡰 ᨨ鼀 � ᆠ�ᨨ ᆠᨨ�鼀 ᨨ矄 㦨 ��
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PS-109
Thermal Properties of Matter
\ 㡰 ҈ - �㡰 = m ´ � + m ҈㡰 黨㡰 쟼 �泐 盐
Þ ҈ = m 黨㡰 Þ m » �
㡰 矄� � � ᆠ� ᨨ ᆠᨨ�鼀泐
Þ M �㡰졔M맔҈맔 ҈㡰㦨맔졔맔҈맔 ÞM㦨ÞM㦨g
㦨�� 鼀맔矄Dq 㦨鼀҈矄矄矄 Dq1 졔鼀矄犨졔Dq2㡰 Þ҈黨맔和맔 ҈҈�㡰㦨҈맔黨맔 ҈�㡰
㡰 M҈C�ᆠ맔 ҈㡰졔M҈L = MCw �㡰
和ᨨ� �犨�졔맔맔 ҈�㡰
ᨨM҈맔ᔴ�ce 㦨�㡰맔҈졔M҈L = M맔҈맔�
和㦨黨҈├�҈├҈
�M Þ = -� M҈
҈ 㡰 q d
d
䰘
㡰 H҈㦨H q
æ q - q ö æ q - q҈ ö æ q + 黨q ö = kA ç ᨨ q = ç ҈ ᨨ k 㡰 A ç ÷ ÷ è d ø è d ø è ҈ ÷ø 㡰 쟼 ᆠ� 鼀 ᆠᨨ 盐ᆠ��泐ᨨ 矄泐矄鼀 ⧤ ⼄ = =
= 黨
k
k q҈
⧤҈꾀҈ + ⧤ 꾀 꾀҈ + 꾀
⧤҈ p 㡰 + ⧤ p 㡰
⧤
=
⧤҈ + ⧤
㡰 鼀�鼀ᨨ �ᆠ�矄 ᨨ鼀� ᆠ�ᨨ ᆠᨨ�鼀 �� 泐�ᆠ㡰 㦨 ᨨ矄 泐犨㡰 \맔҈맔鼀졔 鼀㡰맔㦨�맔맔 䰘鼀㦨졔 鼀㦨҈
҈
ᆠ㡰 ᨨ矄矄㦨�� 㦨鼀和Dq 和ᨨ鼀꾀和꾀Dq꾀㦨鼀和Dq Þ ҈맔和꾀맔 ҈黨㡰㦨�맔和맔 黨䰘�㡰
和 ᨨ犨҈맔和꾀맔 ҈q㡰㦨�맔和맔 q�㡰
和꾀㦨҈�和Þ和꾀㦨
҈ 꾀
鼀ᨨ
鼀ᨨ 矄ᨨ盐ᆠ 矄ᨨ� ҈鼀 ҈҈ 㡰 ҈ ⧤ ҈ ⧤ æ 턼 ö ├꾀D
çè ÷ø 㦨 l
⏌
⏌
⏌
⏌ 턼
⏌
졠
҈ ´ � 㦨� ᔴ 맔 㦨 � ҈ 㡰 꾀ᆠᆠᨨ � � ᨨ � ᆠ� ᨨ ᆠᨨ�鼀泐 턼�� 㦨턼盐矄 맔和맔 ҈�㡰㦨҈�맔҈맔 䰘㡰졔�맔 䰘㡰 맔和맔҈҈㦨҈�맔҈졔�맔҈ 和ᆠ� �ᆠ ᨨ 盐鼀� �盐鼀 D 턼 =
҈ ´ � ´ 䰘 = ⏌├�㜠 ᔴ ´ ҈҈ ҈� 㡰 꾀ᆠᆠᨨ � � ᨨ � ᆠ� ᨨ ᆠᨨ�鼀泐 ᨨ矄㦨 ��
҈맔҈ 䰘�㡰㦨҈맔҈맔�졔҈䰘맔҈맔� ҈T䰘�㦨�졔䰘�㦨҈ÞT䰘�㦨҈ T㦨� C
和=
҈
b㡰 ᆠᨨᨨ㦨 \ ᆠᨨᨨ㡰㦨
æ ö 맔 çè ÷ø 맔 ҈q㡰㦨 q�㡰 q㦨q҈ 㦨�qÞq㦨 ᔴ
⏌
ᔼ 矄 泐矄 鼀盐 � ᆠ 犨 턼
⧤҈꾀҈
p 㡰
D 꾀 ҈ ҈ ´ � = = 꾀 �
꾀
⧤҈p + ⧤ p 㡰 - p 桌
p
( ҈)( 黨)
= 黨 ␔⏌鼀 ҈ ҈ ᆠ㡰 矄ᆠ� �ᆠ ᨨ 盐 ├ ᨨ犨 鼀 �矄꾀ᆠᆠᨨ � � ᨨ � ᆠ�ᨨ ᆠᨨ�鼀泐 ᨨ矄
㦨
M�ᆠLf졔m�ᆠ 㡰Cw㦨m矄鼀Lv졔 m矄鼀 ҈㡰Cw
҈ æ 턼 ö ├D
çè ÷ø 㦨 꾀 l
҈䰘
mc D T V rc DT = t t
҈ ´ ҈ ´ ´҈ ´ ´ ´
\ᆠᨨᨨ㦨䰘␔ b㡰 꾀矄 Pt 㦨 mCDT 和ᨨ P맔 ҈맔 㦨 mC ҈ �㡰 P 맔 ��맔 㦨 鼀 ��㡰 �� � � ⼄盐�ᨨ �㡰 泐 ��㡰 犨 �
҈ C ´҈ = �� L \ L 㦨 �� ᆠ⏌�
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EBD_8341 PS-110
҈
Physics
ᆠ㡰 ᨨ ᨨ犨 �矄�� 泐
҈
p D r = p - - D 㡰 桌
⧤꾀 q҈ - q 㡰 l ␔ ⧤ 㦨 ᆠᨨ �ᆠ� ᨨ 鼀 ᆠᨨ 盐ᆠ��泐 l㦨 � ᨨ ᨨ 꾀 㦨 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ ᨨ
턼㦨
҈ ᔴ
和
Þ r D = - + D - D 桌
Þ r D = D
T rL 㡰 꾀矄 矄盐ᨨ盐 � � �矄 � �ᆠ 矄矄 �矄 � �ᆠ �鼀├ ᨨᆠᨨᨨᨨ 犨 �⼄盐� ᨨ鼀 ᔴᨨ� ᔴ㡰 �矄 矄鼀鼀� 盐矄 ᨨ
矄矄 ᔴ
ᔴ
æ 턼 ö æ 턼 ö =ç ÷ çè ÷ø 犨 è ø �⼄盐�
ᔼ 盐 ᆠ�ᨨ 鼀盐�矄
턼ᔴᨨ㦨턼矄矄졔턼和
鼀犨 ᔴ 犨 + ␔㡰D 鼀l ᔴl + ␔㡰 D
= ␔ 㦨 犨 ⼄盐� ᨨ 矄矄㡰 ᨨ 鼀 犨 ᔴ 犨 = 鼀 l ᔴ l
黨 ´ ҈ - T 㡰
ᨨ
´ ´ T - 㡰 ҈ ´ ´ T - 㡰 졔 ҈ ҈ Þ 㦨 졔 Þ 㦨 ᔴ
㦨
҈黨
\和ᆠ� �ᆠ ᨨ �⼄盐� ᔴl㦨
黨 ´ ´ = ᆠ⏌矄 ᆠ㡰 ᔼ �� ᨨ鼀 � 鼀盐 ᨨ 矄 \
Qᔴᨨ =
= ( - ) =
鼀盐 ᨨ 矄盐ᨨ盐 � � 矄盐 㦨 ᔼ ��鼀盐ᨨ 矄 � ��㦨 和ᆠ� �ᆠ ᨨ 鼀� ᨨ 矄 �矄 矄
ᆠ = a 盐 �鼀矄矄 a㦨ᆠᨨ 矄 㡰 꾀泐� � ᨨ鼀盐
(
s꾀 =
- 矄盐 㝐ᆠ
)
s p é Þ = ) - ( ) ùú ( ê ë û 㝐a ( ) Þ =
㝐a 䰘 s p ´
和ᨨ� � 犨 �
�鼀├ ᨨ 矄 ᨨᆠᨨᨨ ᨨ犨 鼀盐
㝐a
æ ҈ ö ç ÷ ҈p s è ø 㡰 T � 矄 泐 矄 鼀盐 ᨨ ᆠ├ ᨨ 泐 ᔼ 矄 泐 矄 쟼 �泐ᨨ矄 㦨 쟼 �泐�� s T T㡰맔pR㦨I pR㡰 =
D �矄泐矄鼀桌
ÞR=
ᔴᨨ
㡰 ␔ �盐矄 �矄 ᆠ矄 泐D
\ �䰘 ´ ҈ - éêT - ( ) ùú ´ = 黨҈ ë û \ T㦨⧤
㦨
鼀␔ᔴ␔ 鼀l
� ´ ҈ = � ├ᆠ ⏌ ├� ҈
㡰 쟼⼄盐� 鼀 矄�矄 ᆠ � ᆠᨨ ��盐�ᨨ 㜠ᔼ
L L L + = KA KA KA 쟼⼄盐� 鼀 矄�矄 ᆠ � ᆠᨨ ��盐�ᨨ 㜠ᔼᔼ R = R҈ + R =
R l KA KA ᨨ ҈ ҈ ҈ RP = = s = + = + � KA RP R҈ R l l � 鼀矄�矄 ᆠ� ᆠᨨ ��盐�ᨨ ᔼᔼRP�矄��鼀矄 矄矄 鼀 矄�矄 ᆠ� ᆠᨨ ��盐�ᨨ ᔼ Rs ts 黨 = s = s � � ᆠ㡰 꾀矄 矄 ᨨ犨 � ��盐 ҈⏌矄 �� 泐 犨 ᨨ�矄 �矄 鼀盐 㦨 ҈҈㡰㦨⏌矄 l ᨨ犨 ⼄盐� ᨨ�矄 鼀盐ᨨ 犨 ᨨ鼀䰘 ᔴᨨ䰘䰘 ᔴ Q㦨mcDt㦨맔맔� ᆠ�鼀⼄盐� ᨨ�� 턼 ҈⏌矄 鼀ᨨ盐 ᨨ \
�t P = t s Þ t P =
´ ´ � Q = 㦨�矄㦨鼀� 矄 � 㡰 ҈ ᔴᨨ� �矄 鼀ᨨ盐 ᨨ ⼄盐� ᨨ �矄 鼀盐ᨨ ҈�鼀ᨨ 犨 ᨨ鼀҈� ᔴᨨ҈�� ᔴ䰘 鼀鼀ᨨ � 㡰 ␔鼀� �ᨨ �矄矄ᨨ 盐 盐ᨨ �鼀 �矄�鼀�泐 盐 ᨨ ��ᨨ 꾀 鼀 �盐鼀 �矄 ⼄盐� ᨨ ᆠᨨ ᆠ�ᨨ
ᨨᆠ矄矄 꾀矄 盐 �矄 鼀ᨨ矄 ᆠ盐 ᨨ鼀 �鼀 �矄 矄鼀� ᨨ盐� ��ᨨ t㦨
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PS-111
Thermal Properties of Matter 䰘
㡰 꾀ᆠᆠᨨ � � ᨨ ␔� Ἐ矄 �矄ᆠ鼀 犨
҈
lm맔T㦨ᆠᨨ 矄 ᨨ鼀 � l m < l m < l m҈
҈
T1 T2
T
\ T>T>T҈
鼀盐ᨨ 和盐 T�矄 �� ᨨ 犨 � � ᆠT犨 �ᆠ � 盐 �矄 � 盐 �矄 �鼀 㜠T҈ 㡰 ᔼ ᆠᨨ鼀� �ᨨ ᨨ ᨨ 矄 KP㦨K҈졔K㦨K졔K㦨K
ᨨ
꾀
҈ ᔴ K
ᨨ
ᔴ
l ⧤
꾀
ᔼ 矄�矄 ᆠᨨ鼀� �ᨨ
ᨨ
æ -dT ö æ dT ö ᆠ㡰 ᨨ鼀 � ç > è dt ÷ø x çè dt ÷ø y
æ -dT ö ÷ µ 鼀�矄矄��泐 㡰 dt ø
ᨨ ᆠᨨᨨ� � ç è
KK K K - K = KS = ҈ 㦨 K K҈ + K
҈ ᔴ ⧤
b㡰 矄 犨 �├�ᨨ 犨 ᔴ 矄ᨨ ᔴ 鼀盐 㦨mCw DT㦨�맔҈맔㦨҈├ᆠ ⼄盐� 泐├��ᆠ ᔴᨨᆠᨨ � ᨨ├�ᨨ �ᆠ ᔴ 㦨mC�ᆠ DT㦨맔�맔㦨├ᆠ ᆠ鼀� � �҈㦨├ᆠᨨ 犨�鼀鼀├��ᆠ ᔴᨨ犨 ᔴ Q㦨mL Þ 㦨m맔 Þ m㦨҈├�
ᨨ 鼀ᨨ盐 ᨨ 犨 ᔴ 㦨�├�졔҈├�㦨├� 꾀 鼀ᨨ盐 ᨨ �ᆠ ᔴ㦨҈㦨҈├�
\ Ex>Ey 꾀矄ᨨ ax > ay 矄 �ᨨᨨ 矄ᨨ矄 �ᨨᨨ 鼀�矄 ᨨ
꾀
ᔴ
l
l
K A ҈㡰 KA ҈㡰 q = l l ҈ q KA ҈㡰 = ´ = \ q҈ l K A ҈㡰 ᆠ㡰 ᨨ鼀�Kᨨᨨ�� �鼀盐T� ᆠ矄矄� 泐 矄 Q 㦨 mc DT ᆠ T t � 犨� 矄�� � � ᆠ� ᨨ�鼀 �矄 盐� �ᨨ�� � Q㦨mL
鼀盐 鼀� 矄 ᆠᨨ 矄 � ᨨ�� � �矄 ᆠᨨ鼀 � 犨�矄�� � 矄ᨨ�鼀 �矄 꾀 鼀盐� ᆠ矄矄� 泐T㜠t� 犨� 矄�� � � ᆠ� ᨨ�鼀 �矄 b㡰 ᨨ鼀犨� Ἐ矄 �矄ᆠ鼀 犨 l鼀T㦨ᔴᨨ 矄 ᆠᨨ 矄 \ T= lm q҈㦨
黨
C C C ü ì T = T = í\ TA = ý -䰘 B -䰘 C ´ ҈ ´ ҈ � ´ ҈ -䰘 þ î
y x
҈
㡰 꾀ᆠᆠᨨ � �ᨨ⧤�ᆠ ᨨ Ἐ矄犨�ᨨᨨ 矄ᨨ矄�ᨨᨨ 鼀�矄 ᆠᨨ矄 꾀 �� 鼀盐 � 盐 ᆠ㡰矄� ᆠ�矄ᨨ矄鼀ᨨ �泐 � 鼀�矄 鼀ᨨ ��ᨨ 矄 矄 犨 ᆠ �矄 �� 矄 � �矄 ├矄 ᨨ 泐� ��泐
b㡰 q°ᔴ 鼀盐ᨨ 盐 ᆠ�ᨨ BQ�矄 ᨨ犨� �矄ᆠᨨ ᨨ鼀A黨 CᨨBq C ᨨ
ᆠᆠᨨ盐 ᨨ 鼀盐 � ᆠ \
Q㦨
KA 黨 - q 㡰 l
꾀 矄鼀 ᨨCᨨB.
�㡰
Q=
KA 黨 - q㡰 l A
꾀�� ᨨ鼀 矄 Ἐ矄 犨 Q㦨sAT\ QA㦨s.p 맔҈㡰맔 QB
㦨s.p 맔҈㡰맔
҈ ´ ҈ -
Q D
C
ᨨᔴ
� ´ ҈ -
QC㦨s.p 맔҈㡰맔 ᆠQB>QC>QA
C
黨ᨨᔴ
C
턼
ᨨᔴ
qᔴ ᨨ
턼
� ´ ҈
B
C 黨ᨨᔴ
-
\
ᨨ犨� � 矄ᆠᨨ ᨨ鼀Bᨨ D
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EBD_8341 PS-112
Physics
KA q - 㡰 ��㡰 l �� � �⼄ ��㡰泐 �㡰 q 㦨 Þ q = ° 黨 - q ᆠ鼀盐ᨨ 盐 ᆠ�ᨨ q㦨 ᔴ b㡰 꾀ᆠᆠᨨ � �ᨨ␔� Ἐ矄 �矄ᆠ鼀 犨lT㦨ᆠᨨ 矄 ᨨ鼀� l҈ll \ T҈>T>T 㡰 졠®꾀 鼀盐ᨨ �ᆠᆠ �矄 ᨨ鼀҈ ᔴᨨ ᔴ 꾀®�ᆠ ᔴ鼀矄� ᨨ犨 ᔴ ®ᔴ犨 ᔴᆠ �矄� ᨨ犨҈ ᔴ ᔴ®犨҈ ᔴᆠ �矄� ᨨ矄鼀҈ ᔴ
黨
Q㦨
�
҈⏌ ⏌ù æ ҈ö DT esAT esT é æ m ö k êp ç ú = = 㦨 çè ÷ø m Dt mc mc ê è prø÷ úû ë
ᨨ ᨨ �� 犨ᨨ ᨨ �矄
ᨨ
DT҈ ⏌ Dt҈ æ m ö = DT ⏌ Dt çè m҈ ÷ø
鼀
꾀 ᨨ ᔼᆠ
ᔴ ҈ᨨ ␔
䰘
lmT㦨맔҈nmK
U E
´ ҈ nmK = ҈ nm K
U҈
U l鼀
黨黨 �㡰
黨黨黨 ҈黨黨 ҈㡰 ҈�㡰
l
T - Tc
E ⏌ t ҈㡰 㡰 = = = � �㡰 ҈㡰 ᨨ
E 㦨�맔㦨҈␔ t
A 2a
a B
a
C
TC
2T
=
K2 R
K1
2R l
E = sT ´ A = sT ´ pr t ᨨ �㡰
T
Tc - T Þ T - Tc = Tc - T a a Tc Þ T = Tc + ҈㡰 Þ = T +҈ ҈ ᆠ㡰 K 㦨 鼀 ᆠᨨ 盐ᆠ��泐ᨨ 矄泐矄鼀
ᨨ 矄 ᨨ 矄ᆠᨨ ᨨ盐� ᆠᨨ鼀� 矄泐矄鼀 㦨 矄 矄ᆠᨨ ᨨ鼀 鼀� 犨� 鼀 ᆠᨨ 盐ᆠ��泐K҈ 졔 矄 矄ᆠᨨ ᨨ鼀 鼀� 犨� 鼀ᆠᨨ 盐ᆠ��泐K
ᔼ ᨨᨨ犨矄 �泐 � 犨 黨黨nmᨨ�nm 犨�矄矄 鼀� 犨 黨黨黨nmᨨ҈nm �U҈UᨨU>U҈ 㡰 ᨨ鼀 和 Ἐ矄 犨 �泐 � 矄ᆠᨨ 泐 ᆠ├ ᨨ 泐
E = � 犨 T㦨�Kr㦨҈鼀 t \ �㦨ps �㡰 ҈㡰 E 犨 T㦨҈Kr㦨鼀 = 䁸 t E \ = p s ҈㡰 㡰 t �� � �⼄ ��㡰泐 �㡰犨�
DQ k A DT = Dt l
æ DT ö ÷ = ç è a øCB
ᨨ 矄ᆠ盐鼀 lm㦨
҈⏌
æ DT ö ᨨ矄� 矄AC CB ç ÷ è a ø AC
Þ
犨 � ├
æ ҈ö = ç ÷ è ø
b㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 鼀盐 B = T > 鼀盐A㦨T矄ᨨ ᨨ犨 ᨨ鼀BᨨAAᨨC C ᨨ B ᨨ 矄 泐矄ᆠᨨ ��ᨨ D턼 D �矄 矄鼀
矄盐�
㡰 꾀ᆠᆠᨨ � � ᨨ ␔� Ἐ矄 �矄ᆠ鼀 犨
҈⏌
꾀泐� � ᆠᨨ 盐ᆠ�ᨨ ᨨ鼀盐
҈ᨨ 和鼀
ᨨ ␔
졠 ᨨ ҈ ᔼᆠ
㡰 꾀ᆠᆠᨨ � � ᨨ和 Ἐ矄 犨 DQ㦨esATDt 꾀矄ᨨDQ㦨mcDT ᨨ mcDT㦨esATDt
KADT K҈ A҈DT K A DT = + l l l Kp R㡰 㦨K҈pR 졔 Kp R㡰R桌
K ҈ + K 㡰 ᨨ鼀� ᆠ�ᨨ ᆠᨨ�鼀泐 ᨨ
K 㦨
ᨨ矄泐矄鼀㦨�� 泐 犨졔ᆠᨨ�鼀㡰 mL졔m맔c맔 ҈㡰㦨҈҈맔c맔 ҈�㡰 m�졔҈맔桌㦨҈҈맔҈맔� m㦨҈├�
��㡰
b㡰
Q Q nCv DT ҈ = = Þ Q = ҈ g Q҈ nC p DT g
Þ
Q =
䰘 = � ᆠ ҈
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PS-113
Thermal Properties of Matter
㡰 ᨨ鼀鼀矄㦨 㝐㦨
=
鼀矄
黨
RT m
´ ҈ ´ 黨 ´ ҈ = � ҈黨 ´ ҈黨
b㡰 鼀⼄盐���盐鼀 q A - q B = °C � 㡰 KA㦨KB � 㡰
KA q + l A q ᔴ 㦨 KA + l
\
q C 㦨
qᔴ
q꾀 KB qB l KB l
q
KA
KB
l
l
q A + q B q A + q A - q A - ҈㡰 = =
\ q A - qC = ҈°C ᆠ 鼀盐 � ᆠ ᆠᨨ矄矄 泐 A㦨҈ C 黨㡰 꾀ᆠᆠᨨ � �ᨨ矄 Ἐ矄犨 P µ T ᨨ P = PT \ ᨨ� P = ᨨ� P + ᨨ� T \ ᨨ�
P = ᨨ� T P
P = ᨨ� 䰘 = ҈ … �㡰 꾀 㦨䰘 ᔴ㦨䰘⧤ ᨨ� P 꾀 㦨䰘䰘 ᔴ㦨⧤ r ᨨ�e = ᨨ� = ᨨ� 䰘 ´ 㡰 r = éëᨨ� 䰘 + ᨨ� ùû = ᨨ� 䰘 + = ҈ + = 黨 䰘 㡰 ᨨ鼀 �㡰 和 㜠ᨨ 鼀
犨 P 㦨 sAT ��㡰 ␔� Ἐ矄 �矄ᆠ鼀 犨㦨lm맔T㦨ᆠᨨ 矄
PA AA = PB AB
L ҈� = = 䰘°C C � �҈ 㡰 ᨨ ᨨ犨犨�矄鼀
\
ᨨ ᨨ犨 犨 R =
A l = A´ B TB AB l A ҈ é ´ ù ê ú
é R P l A é AA =ê ´ ú =ê A´ Bú = l B ë AB PA û ë ҈ êë RB PA úû l \ A 㦨 lB 黨㡰 꾀ᆠᆠᨨ � � ᨨ矄 㜠ᨨ 鼀
犨
p ( r ) L҈ L K A = \ ҈ = = = [Q L҈ = L = L ] K҈ A҈ K A K A҈ pr
û
�
ᨨ ᨨ �泐 � 泐 꾀 ᨨ ᨨ �泐 � 泐
æT ö æ r ö = ç ҈ ÷ ´ç ҈ ÷ 㦨 sT pr 㡰 è T ø è r ø æ l m 㦨 çç l è m҈
2
æ ҈� ö æ ö 㦨黨 㦨 ç è � ÷ø çè ҈ ÷ø
和ᨨᨨ犨ᆠ� 泐鼀 㦨 ҈␔⏌鼀㡰 鼀㡰㦨␔ 㝐矄矄ᨨ �ᆠ㦨�㦨├� ⼄盐� ᨨ鼀�ᆠ㦨鼀 㦨 㡰 맔҈�㡰㦨黨맔҈ ᔼ t�矄 �鼀├ ᨨ �ᆠᨨ鼀 Eù é \ 㡰t㦨黨맔҈ êQ P = t ú ë û
ù ö æ r҈ ö é T҈ l m 泐 ␔� Ἐ矄犨 ú ÷ ç ÷ êQ = ÷ è r ø ê T l m úû ҈ ë ø
L KA
҈ ù
҈ PB ù
sT҈ pr҈ 㡰
dQ 鼀 � ᆠ ҈ = = ( T T҈ ) 鼀矄�矄 ᆠ R dt
҈ = ᨨR҈㦨R R҈ R
TA
\
҈ ҈ ´ ´ ҈� ´�+ ҈ ҈ \鼀= = � Þ = 鼀 ´ ҈� + ҈� � 䰘°ᔴ㡰C 矄ᆠ� �ᆠ ᆠᆠ�泐ᨨ �⼄盐� ᨨ ᨨ盐�矄�ᨨ ᨨ鼀� ᆠ�ᨨ ᆠᨨ�鼀泐 ᨨ矄㦨 ��
mCSCD 㦨mCDT졔mL ᨨmCSC ҈҈㡰㦨�ᔴ 㡰졔� �㡰 ␔ mC㦨鼀矄矄ᨨ ᆠᨨ�鼀 SC㦨矄 ᨨ ᆠᨨ�鼀 꾀�� 犨 � �⼄盐� �矄 ᨨ盐 ⼄盐���盐鼀 鼀盐�矄� ᔴ mCSC �㡰㦨ᔴ �㡰 ��㡰 ᨨ鼀⼄ �㡰 ��㡰 ҈ᔴ㦨�ᔴ졔� Þ = 鼀 ´
ᆠ �矄 �矄 �
㡰 꾀矄 �矄 ᨨᨨ ᆠ �� �矄ᨨᆠ矄矄和ᨨ 矄盐� 㦨 盐矄 � ᆠᨨ � � 鼀 �鼀矄 ᨨ �ᆠ ᨨ鼀� ᔴᨨ ᔴ졔盐矄 � ᆠᨨ � �҈�鼀ᨨ �ᆠ ᔴᨨ犨 ᔴ ᨨ㦨mcD q졔ml
ᨨ �
t㦨
黨 ´ ҈ 矄 = 矄 º ��鼀�
t 鼀盐 ᨨ � ᆠ 矄 泐 矄 �矄 ᆠ
矄 矄ᆠᨨ ᨨ盐� A Q҈㦨K҈A ҈t㡰
矄 矄ᆠᨨ ᨨ盐� B Q㦨KA t㡰 꾀矄 泐矄K҈A ҈t㡰㦨KA t㡰 Þ ҈t㡰㦨 t㡰 Þ t㦨t Þ t㦨 ᔴ
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EBD_8341 PS-114
Physics
�
ᔼ ├矄 t 矄ᆠᨨ 矄 ᨨ 矄盐矄 ᆠ ᨨ 矄ᨨ� � 泐 �� 㡰
ᨨᨨ 矄 � t矄ᆠᨨ 矄㦨P맔t 㦨mL 盐矄�ᨨ
P ´t \ L 盐矄�ᨨ 㦨 m
��
ᨨ 矄 ᨨ
P=
P=
鼀盐 � 鼀 矄�矄 ᆠ
t ⏌ pR K
t=
pKTR P 쟼 �泐 � 矄ᆠᨨ dT sTA㦨mc dt Þ
r ´ p r cdT dt 㦨 = sT A s T ´ pr mcdT
҈
ᨨ ᆠ �
l -t 㡰 ù é êQ 鼀 矄�矄 ᆠ㦨 KA ú ë û
T
pKTR P �.e. �ᆠ├ 矄矄 t 矄 ᨨ盐 Þ
�黨
⧤
졠 ® 꾀 鼀� �矄 � 矄ᨨ� 矄 ᨨ 泐 鼀盐 �矄 � ᆠ矄� � ᨨ鼀꾀® 鼀��矄泐矄ᨨ� 泐 �⼄盐� 和� ᆠP�矄ᨨ� 犨 A B ᨨ 鼀� �矄 泐矄ᨨ� 泐�⼄盐� 矄㡰 쟼 �泐 � 矄ᆠᨨ 泐 �矄 矄 E҈㦨esT A㦨es 㡰맔p맔҈҈맔҈҈ 㦨҈맔p맔҈҈맔es 쟼 �泐 � 矄ᆠᨨ 泐 矄ᆠᨨ 矄 E㦨es맔 㡰맔p맔맔 㦨҈p맔҈҈맔es \ E҈ = E ᆠ 㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ 턼 ᨨ犨矄ᨨ 泐 ᨨ鼀 ᨨ�� ᨨ盐� ᨨᆠ├矄 ᆠ ᨨ犨 ᨨ盐� 矄꾀 쟼 矄鼀 꾀
턼꾀
rrc dT s T ᔼ �� � ᨨ 矄� 矄
ò dt =
�䰘
\
�
꾀
rrc rrc é ҈ ù = s ò T s êë T úû
( ) (
)
\
sT 맔pR㦨҈맔pr ᨨ
ᔴ
ᨨ
T =
ᨨ
T =
⧤
�⧤
턼ᔴ
턼
턼쟼 쟼
RA =
L ҈ L = RB = = K L㡰W KW K LW 㡰 KW
RC =
L ҈ L = RD = = K LW 㡰 KW �K LW 㡰 �KW
҈ L = K LW 㡰 KW ᨨ犨DT㦨QR 和� ᆠ 矄�矄 ᆠ ᨨ ᨨ犨 �矄 矄 ᨨ 矄 E 鼀盐 � ᆠᆠᨨ矄矄 E �矄 矄鼀矄 꾀矄ᨨ DTC DTC QC = = = KW DTC 㡰 RC ҈⏌ KW DTB DTC KW DTC 㡰 QB = = = Q DTB㦨DTC桌 RB ⏌ KW RE =
҈ ´ r sR
DTC �KW DTc 㡰 DTD = = Q DTD = DTC 桌 RD ⏌ �KW KW DTC 㡰 � KW DTC 㡰 QB + QD = + KW DTC 㡰 = = KW DTC 㡰 = QC
҈ ´ ҈� ´ ҈҈҈㡰
QD =
�䰘 ´ ҈ 㡰 䰘 ´ ҈ 㡰
҈ ´ � ´ ҈
ᨨ T㦨�⧤ �䰘 ´ 黨 ´ ҈
⼄盐� ᨨ҈ � ᨨ �ᆠ ᔴ ᨨᆠ � � ᨨ 犨 ᔴ㦨mL㦨҈맔맔㦨
矄 泐� ᨨ 犨 � ᔴ ᨨ ᆠ ��矄 鼀盐ᨨ ᔴ㦨mcDT㦨맔맔�㦨҈� ᆠᆠᨨ鼀�ᆠ犨� ᨨ鼀矄ᨨ � 鼀盐 ᨨ 鼀� 盐犨� ᔴ
l KA
犨� ᨨ 矄矄 ␔
和
T =
쟼 ⧤
␔├ ᨨ犨 鼀矄�矄 ᆠ 㦨
E
쟼 �泐 = sT ´ pR �鼀 쟼 �泐 ᆠ� ᨨ 盐 ��鼀㦨҈맔pr
턼
턼꾀 㦨턼 졔턼ᔴ 졔턼 㦨턼쟼
҈
҈ dT
rrc é ҈ ҈ ù t㦨 ê ú 黨s êë ҈㡰 㡰 úû 䰘rrc 䰘rrc » = ҈䰘҈rrc t㦨 䰘 ´ ҈ 㡰 s 䰘 ´ ҈ �䰘 ´ ҈ - 㡰 ᆠ �鼀 ⼄盐� ᨨ 鼀盐 ᨨ 矄 ᨨ ᨨᨨ҈├㦨҈䰘҈rrc 꾀ᆠᆠᨨ � � ᨨ和 Ἐ矄 犨 쟼 �泐 � 盐 � �鼀 R 盐 �㦨sT r \
⧤
ᔴ
é ù êëQ m = r ´ pr A = pr úû
Þ dt 㦨
t
⧤
�.e. QC㦨QB졔QD ᆠ 㡰 和� ᆠ 鼀盐ᨨ ᆠ├ᨨ 泐 ᆠ 鼀鼀� 矄 ᆠᨨ 矄 矄ᨨ �泐鼀� 㦨 �泐矄ᨨ 泐ᆠ├ ᨨ 泐
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PS-115
Thermal Properties of Matter
b㡰 쟼 �泐鼀� 矄ᆠᨨ 泐ᨨ 泐A B矄鼀
\㦨eAs TA A㦨eBs TB A ҈⏌
æ eA ö ´ T 㦨҈黨⧤ \TB㦨 ç A è e ÷ø
҈䰘 = 黨� ├�\㝐矄矄ᨨ ᆠᨨ � 㦨黨�├� ҈ ᨨ矄 泐矄鼀 㦨 �� 泐犨 msL 盐矄㦨mwc DT ᨨ m =
Þ ms 㦨
B
꾀ᆠᆠᨨ � � ᨨ␔� Ἐ矄 �矄ᆠ鼀 犨 l m µ
҈ T
䰘
lm㡰ATA 㦨 lm㡰BTB
Þ
l m 㡰 A TB � = = l m 㡰 B TA ҈黨
�㡰
和� ᆠ鼀盐ᨨ A�矄鼀ᨨ ᨨ lm㡰A�矄矄矄 \ lm㡰B lm㡰A㦨҈맔҈鼀 �� 㡰 ��㡰 和ᨨ� �⼄ �㡰 ��㡰犨� lB㦨҈�맔҈鼀
�
ᨨ矄泐矄鼀҈ ᔴᨨᆠ �ᨨ҈ ᔴ犨 Q҈㦨 mL㦨�맔맔҈㦨҈҈ ᨨ矄泐҈ ᔴ犨ᨨᆠ �ᨨ ᔴ犨 Q㦨MCDT㦨�맔맔҈㦨҈ ⼄盐� 泐� ├� ᨨ �ᆠ ᨨᆠ � �矄 鼀盐 ᨨ鼀�K ᨨ䰘K Q㦨m맔C�ᆠ맔DT㦨�맔҈맔㦨҈黨 ⼄盐� 泐�├��ᆠ䰘⧤ᨨᆠᨨ � ᨨ� ├�犨䰘K Q㦨mL 盐矄�ᨨ 㦨�맔맔҈㦨҈�҈ ᨨ鼀 ᨨ � �矄 ᆠ Q҈ 졔 Q > Q 盐 Q҈졔QQ졔Q矄ᨨ犨 ᨨ�ᆠ犨� ᨨ鼀
ᨨ � 鼀盐犨�䰘⧤ᨨ ᔴ m 鼀矄矄 ᨨ ᆠᨨ � 꾀ᆠᆠᨨ � � ᨨ� ᆠ� ᨨ ᆠᨨ�鼀泐 ᨨ矄 㦨 ��
ᨨ矄泐ᆠᨨ � 㦨msdTᨨdQ㦨m A졔BT㡰dT Q
ᨨ
ò dQ = mò� ( A + BT ) dT
ᨨ
é BT ù Q = m ê AT + ú úû ëê �
ᨨ
é ´ ҈ Q = m ê҈T + ëê
mwcDT ҈ ´ ´ = 㦨҈├� L fus � ´ ҈ ´
�泐鼀� 矄ᆠᨨ 泐ᨨ 矄 矄 犨� 矄鼀矄 鼀盐 矄盐 ᆠ 矄鼀 ␔├ ᨨ犨 Q = mcDT
dQ mcdT = dt dt 和� ᆠQ�矄矄鼀 c�矄矄ᨨ矄鼀 ᨨ ᨨ ᆠᨨ㡰
ᨨ
dQ ҈ µ dt m 㝐矄矄 ᨨ ᨨᨨ犨 矄 �矄 矄矄 矄ᨨ ᨨᨨ犨 矄 犨� ᆠᨨᨨ 矄 � �奠 ᨨ �矄 矄ᨨ 泐 ᨨ矄ᆠ
ᨨ䰘�奠 �矄盐矄 � 鼀� �ᨨ ᔼ ��鼀 㦨䰘 ᔴ 㝐� �ᨨ� ᨨ 㦨䰘 ᔴ\ D 㦨䰘䰘㦨 ᔴ \
䰘�㡰맔
҈ Mv 㦨McDT졔ML
҈ 䰘�㡰맔 v 㦨 맔졔㡰맔 v㦨҈黨鼀⏌矄 黨 �㡰 ᔼ ��ᨨ 㜠꾀 �矄 矄ᨨ 泐 鼀� ᆠᨨ 矄 鼀盐 �矄 盐矄 � 矄 ᆠ �� � ᨨ鼀 矄ᨨ� ᨨ�⼄盐� ᔼ ��ᨨ ᔴ �矄 矄ᨨ 泐 鼀� ᆠᨨ 矄 鼀盐 �矄盐矄 � 矄ᆠ �� � ᨨ鼀�⼄盐� ᨨ�矄 ��㡰 ᨨ ᨨ�矄�ᨨ 㦨 ᨨ 盐矄�ᨨ 㡰 ���㡰 矄ᨨ DE � �ᆠ矄 鼀盐 ᨨ 矄ᨨ� �� 矄 ᨨ �矄 �㡰 矄ᨨ ᨨ 졠꾀 > 矄ᨨ ᨨ ᔴ �矄 � �ᆠ矄 �矄 �矄� 矄ᆠ� �ᆠ 䰘 � 鼀ᆠᨨ 盐ᆠ��泐 KX 㦨黨ᆠ⏌矄ᆠ㜠ᆠ鼀㜠 ᔴ KY㦨ᆠ⏌矄ᆠ㜠ᆠ鼀㜠 ᔴ ᔴ 桌 ᔴ
ù Tú ûú �
é ) ´ ( �) ù ( ê ú ᨨ Q = m ҈ ( � ) + ҈ êë úû ᨨ Q㦨m҈桌ᆠᨨ� ᨨQ㦨҈mᆠᨨ� �㡰 �� 泐�ᆠ� 鼀� �㦨mL \Q҈㦨҈맔㦨ᆠ �� 泐犨ᨨ ᨨ�ᆠ㦨 msDT \Q㦨҈맔҈맔䰘㦨䰘ᆠ \ ᨨ �� 㦨졔䰘 Q҈졔Q㦨҈䰘ᆠ ��㡰 \ᨨ矄㦨�� ᨨ҈m㦨҈䰘
꾀 ᨨᔴ桌
Ӹ
桌 Ӹ
쟼 ҈ᨨᔴ桌
Ӹ 桌
ᨨ鼀 ��盐
KY A - TB 㡰 K X A TB - ҈㡰 KY A TB - ҈㡰 = + l l l 和ᨨ� � ᨨ⼄盐�ᨨ 犨�TB㦨 ᔴ ᨨ 犨� ᨨ犨 ᨨ盐� ᔴ 꾀矄C�矄ᨨ� 鼀� ᨨ 쟼 ᨨ鼀盐 TB + TE °C + ҈°C = = °C 和�鼀�泐鼀盐�矄矄ᨨ ᔴ
C =
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EBD_8341 PS-116
Physics
Topic-3 : Newton’s Law of Cooling ҈
T҈ - T éT + T ù = K ê ҈ - T ú t ë û
犨 q �矄 鼀盐 ᨨ 矄盐ᨨ盐 � �
҈㦨� ᔴ 㦨 ᔴ ᨨ㦨 ᔴ㦨和㦨�鼀� 盐矄
ᨨ犨 ᨨ犨ᆠᨨᨨ矄 ᨨ鼀 ᔴᨨ� ᔴ� ҈鼀� 盐矄
� - æ � + ö = Kç - ÷ �㝐�
è ø
- � é + � ù = -⧤ ê - q ú ҈ ë û
�㡰
- T æ + T ö = Kç - ÷ � è ø �� � �⼄ ��㡰泐 �㡰犨�
� - é � + ù = -⧤ ê - q ú ҈ ë û
��㡰
�� - q ҈ = - q ҈ �� - q = - q
= °ᔴ
- ҈q = - q q =
æ dT ö ÷ = esAT dt ø
æ dT ö çè - ÷ø dt B
q = ҈°ᔴ �
dT es ´ A ´ T dT ҈ = Þµ dt r ´ Vᨨl ´ S dt rS
æ dT ö çè - ÷ø dt A
T Þ - �T = T �
b㡰 ᨨ ᨨ矄矄㦨 mS çè
-
��㡰
�� � �⼄盐�ᨨ 矄 �㡰泐 ��㡰犨�
- T + T - T = = ҈ � + - �
\T =
�㡰
꾀�� �ᆠᨨᨨ矄 ᨨ鼀� ᔴᨨ ᔴ� ҈鼀� 盐矄
T 鼀盐 ᨨ 矄 � 鼀� 盐矄
Þ - T =
泐 犨ᨨ 矄 犨 ᨨ ᆠᨨᨨ� �
q҈ - q é q + q ù = -⧤ ê ҈ - q ú ë û
b㡰 ᨨ鼀 犨ᨨ Ἐ矄 犨 ᨨ ᆠᨨᨨ� �
Þ
b㡰
ᆠ㡰 꾀ᆠᆠᨨ � �ᨨ犨ᨨ Ἐ矄犨ᨨ ᆠᨨ� � 鼀盐 �ᨨ矄 ᨨ ᆠ矄� � 犨� �鼀 ᨨ 㜠� 泐 Topic-4 : Miscellaneous (Mixed Concepts) Problems
҈
r S ҈ = B´ B = ´ r A S A ´ ҈
㡰 ᨨ犨
P=
dQ d dT = mc 㡰T = H 㡰 dt dt dt d éT ҈ + bt҈ 㡰ù û dt ë
æ dT ö æ dT ö Þ ç- ÷ > ç- ÷ è dt ø A è dt ø B
= H 㡰
和ᨨ꾀ᆠᨨᨨ矄 ᨨ犨 矄 㡰 꾀ᆠᆠᨨ � � ᨨ犨ᨨ 矄 犨 ᨨ ᆠᨨᨨ� �
P = H 㡰T =
æ q҈ - q ö æ q҈ + q ö - q ÷ çè ÷ø = K çè ø t æ - � ö æ + � ö - �÷ çè ÷ = K çè ø ҈ ø æ � - q ö æ � + q ö - �÷ ÷ = K çè ø ҈ ø
çè
�� � �⼄ �㡰泐 ��㡰
҈ = Þ q = � ᔴ @ ᔴ � - q㡰 q
\
H 㡰 =
�㡰
盐 t҈ =
��㡰
\
t =
bt - 犨 H㦨 ᆠᆠ�泐
Pt Tb
�㡰
T t 㡰 - T bT
[T t 㡰 - T ] bT
��㡰
ᨨ鼀⼄ �㡰 ��㡰
P [T t 㡰 - T ] P [T t 㡰 - T ] = Tb b T bT
H 㡰 =
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PS-117
Thermal Properties of Matter
b㡰 ᔼ �矄 ᨨᆠ�矄ᆠ 盐ᨨ矄矄盐 � 矄ᆠᨨ ᨨᆠ�矄 盐 ᨨ矄盐 ᆠ 矄�ᨨ T
턼 = s × D
꾀 ᨨ犨 ᨨ鼀⼄ �㡰
P0 2R
FT h/2 P0 졔
h rg 2
鼀矄
= s꾀 - 㡰
Þ
s꾀 + D 㡰 - 桌 = 鼀矄
=
hrg ö éæ ù
ᨨᆠ�矄 ê ç P + ÷ø ´ R ´ h㡰ú - RT è ë û \ ᨨᆠ㦨PRh졔RrghRT 㡰 ␔ 矄 �ᆠᨨ 泐�矄├� 矄� ᆠ泐ᆠ├ ᨨ 泐 ᨨ � 泐 ᨨ 泐�矄 ⼄盐 ᨨ ᨨ ᆠ├ ᨨ 泐 ᔼ �矄⼄盐矄�ᨨ �� ᨨ�ᨨ 矄犨ᨨ �矄 ᨨ盐 ᨨ�ᨨ 矄 ᆠᨨ � e� ᆠᨨ s ᆠ㡰 ᨨ �� 鼀矄矄 ᨨ �矄 ᆠᨨ 矄 矄矄盐
ᨨ
턼 = s × D
꾀 和� ᆠ ᨨ ᨨ矄矄 ᨨ 㦨 ᆠ� 矄ᆠᨨ 䰘㦨 ⧤⏌꾀㡰 鼀矄㡰D
⧤맔鼀矄㦨맔҈桌
DV ҈ ҈ = d ᨨ dT = ҈ = ᨨ T T V DT
⼄盐�ᨨ 矄 矄 ᆠ �盐 泐ᨨ ᨨ ᨨ鼀xy㦨c �ᆠ 泐� c㡰
�
黨㡰 ᔴ
l
҈ ᔴ
䰘
ᨨh ᨨ犨 ᨨ鼀ᨨ
dQ҈ dm҈ KA = Lf = lx dt dt ᨨh ᨨ犨 ᨨ鼀ᨨ
�㡰
é æ D ö ù 턼 = s + D 㡰 - 㡰 = s êç҈ + ÷ - ҈ú 꾀
ø êè ú ë û
éæ D ö ù = s êç҈ + ÷ - ҈ú
ø úû êëè
é ù ҈ ö æ ê AT ҈ s + - ҈ú t ç 㦨 ÷ êëè T ø úû
é ù
㦨 sAT ê T ú ´ t = ´҈´ ´҈ = ҈J ë û
P=
쟼 �泐 � = sAT - sAT �鼀
dp
Þ
ᆠ㡰 쟼 �泐 � 泐 ᨨ 泐㦨 sA T - T 㡰 ᨨᆠ├ᨨ 泐㦨҈桌 㦨 sA + ҈㡰 - T 桌t
�㡰
dm҈ dm ù dQ dm KA é = = = Lvap ��㡰 ê � ë dt dt úû dt ҈ x - lx dt �� � �⼄ �㡰泐 ��㡰 矄ᨨ� �犨�l㦨黨 㡰 ᨨ ᨨ矄矄 ᨨ
턼 = s 꾀 - 㡰
䰘 ´ 꾀 䰘 ´ � ´҈ - � � = = = ⧤ ´ 鼀矄 ҈ - ´ \ D 㦨 Þ D =
҈ l
ù s꾀 éêæ D ö ú
´ ç ҈ + ҈ ÷ 鼀矄
ø êè ú ë û
= ⧤D 錐 æ ö s꾀 ᔴᨨ 矄 ᨨ 犨ᨨ Ἐ矄 犨 ᨨ ᆠᨨᨨ� � ÷ ç⧤ = ç ÷ 鼀矄 è ø ⧤ Þ s꾀 = 鼀矄㡰 꾀 ᨨ鼀⼄ �㡰
ᨨ
ᔴ
Q 턼㦨鼀矄D 桌
s꾀 =
× D
鼀矄
V V + DV V \ = 㦨 ᔴᨨ 矄 T T + DT T VT졔VDT㦨VT졔TDV ᨨ VDT㦨TDV
꾀
��㡰
\ dT = sA T 㡰 \ dp = sA T 㡰 dT
\ DP = sAT 矄 盐鼀 ᨨ 泐�矄 ᨨᆠ├ᨨ 泐和ᨨᨨ�ᨨ 㡰 㡰 � ᆠᨨᆠ 쟼 �泐 � µ꾀 犨 꾀 �矄 矄盐 ᆠ ᨨ ᨨ 泐 ᆠ ᨨ�ᨨ ᆠ㡰 �矄 ᆠᨨᆠ
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EBD_8341 PS-118
Physics
Thermodynamics
11
ᨨᆠᨨ 矄� �泐�矄 㦨 kx kx \ P = P҈ + 꾀㦨ᨨ ᆠᨨ矄矄矄ᆠ�ᨨ ᨨ �矄ᨨ 㡰 ��㡰 A ␔ V㦨V҈T㦨T҈
Topic-1 : First Law of Thermodynamics ҈
㡰 D鼀� 矄矄鼀 ᨨᨨ 矄꾀ᔴ 꾀 D턼꾀ᔴ㦨D␔꾀ᔴ졔D꾀ᔴ Þ 㦨졔D꾀ᔴ Þ 꾀ᔴ㦨 \ D꾀㦨D꾀ᔴ㦨 D턼꾀㦨D꾀졔D␔꾀 㦨҈졔㦨 㡰 ᨨ盐鼀ᨨ 犨 ᨨ矄 ᨨᆠ � ᨨ犨ᨨ├�矄 ᨨ ᨨ
ᨨ泐 矄泐矄鼀 ␔㦨㡰 꾀ᆠᆠᨨ � � ᨨ �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄
\
盐� � �矄盐ᨨ P� ⼄ ��㡰犨� PA kx P҈ = P҈ + Þ kx = ҈ A V V҈ V҈ ҈ V҈ x= = = A A A 쟼 �泐矄ᨨ � 矄� � PV ҈ ҈ = kx = ( kx )( x ) = ҈ ҈ 㡰 ᔴ � � � �泐
턼 㦨 졔 ␔
�
䰘
ᨨ ᔼ矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄 턼 㦨 D㦨mᆠ 㦨맔҈맔㦨҈䰘├ 㡰 꾀矄 犨├ ᨨ犨 ᔼ矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄㡰 D턼 = D盐 + D犨 Þ D턼 = D盐 + D ᨨ ҈� = D盐 + ҈ (҈ - ) 㦨 D盐 - ҈ \ D盐 = ҈� + ҈ = �
盐矄 � �泐ᨨ �矄 � ᆠ矄矄 泐� 㡰 ᨨ鼀 �矄犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 dQ㦨dU졔dW dW㦨 �� 㡰 \ dQ㦨dU ᨨ犨矄� ᆠdQ �� 㡰 �, dQ �矄 �� 矄ᨨ dU ᆠ矄矄 ᔼ �泐ἘUἘ ᆠ矄犨 鼀盐T ᆠ矄矄 㡰 꾀泐� � �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 ᨨ � Q�af㦨DU�af졔W�af �㦨DU�af졔 \DU�af㦨 ᨨ犨 Q�bf 㦨 DU�bf 졔 W�b 졔 Wbf 㦨졔�졔҈ Q�b 졔 Qbf㦨� �㡰 꾀矄ᨨ Q�b = DU �b + W�b \Q�b㦨҈졔�㦨҈� ��㡰 Qbf Q�b g Q�b = = = ᨨ鼀⼄ �㡰 ��㡰 Q�b Q�b ҈� ᔴᨨ 矄 㡰ᔼ �矄 矄�ᨨ ᨨ犨ᨨ├�矄 ᨨ ᆠ盐矄 �矄 矄 ᆠᨨ 矄 � ᆠ盐盐鼀\DW㦨 꾀矄ᨨQ㦨矄 ᨨᆠ矄矄�矄 ��ᆠ ᆠ ᨨ鼀 �矄犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄DU㦨 �.e., 鼀盐 鼀� 矄 ᆠᨨ 矄 bᆠ㡰 矄� � �矄 ᆠᨨ鼀矄矄 泐x ᨨ � � �矄 P҈ PV Ἐ ҈ἘT҈
x
P҈
P ἘV ἘT kx
kx
x + x
㡰 꾀矄 �矄 �矄 � 鼀ᨨ ᨨᨨ鼀�ᆠ PV PV ҈ ҈ = \ T҈ T
PV ҈ ҈ = P ´ ( V҈ ) Þ P = P ҈ T҈ T҈
f æ ö PV PV ( ҈ ҈ ) = ç P҈ ´ V҈ PV ҈ ҈ ÷ 㦨P҈V҈ ø è ᆠ㡰 꾀�� 犨 V㦨V҈ T㦨T҈
ᨨ鼀⼄ �㡰 P ( V ) PV ҈ ҈ = ҈ Þ P = P҈ T҈ T҈ V V V x= ҈ = ҈ A A ᨨ鼀⼄ ��㡰 PA kx P҈ = P҈ + Þ kx = ҈ A ␔ᨨ├ ᨨ 泐�矄 㦨 ␔ᨨ├ ᨨ 泐�矄 ᨨ 鼀ᨨ矄 졔 쟼 �泐矄ᨨ � 矄� � ҈ ҈ æ P A ö æ V ö Wg = P҈ Ax + kx = P҈ ( V҈ ) + ç ҈ ÷ ç ҈ ÷ è øè A ø ҈ 䰘 PV = PV ҈ ҈ + PV ҈ ҈= ҈ ҈ 㡰 DQ㦨Wg졔DU 䰘 = PV ҈ ҈ + ( PV PV ҈ ҈) 䰘 æ ö = PV ҈ ҈ + ç P҈ ´ V҈ PV ҈ ҈÷ ø è bᆠ㡰
�矄 ᆠ矄� ᨨ盐鼀 盐� �鼀� �ᨨ �ᆠ矄 䰘 ⧤ ᨨ �� 犨ᨨ├ �矄 ᨨ 泐�ᆠ㜠犨 矄泐矄鼀ᨨ 鼀ᨨ矄 ᨨᨨ矄��犨ᨨ├�矄 ᨨ ᨨ �ᆠ㜠犨 矄泐矄鼀 泐 鼀ᨨ矄 ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ 和ᆠᨨ 泐 �矄 矄ᨨ 盐� � 鼀� � � dQ �矄 ᨨ矄��㡰 矄 犨 矄 犨ᨨ├ ᨨ 泐�ᆠ㜠犨 矄泐矄鼀 �矄 �� dW �矄 ��㡰 ᨨ ᨨ鼀 �矄 犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄dU㦨dQdW ᆠ �� � �泐ᨨ �ᆠ㜠犨矄泐矄鼀dU 犨� ᨨ矄��ᨨ � �泐犨� � ᆠ矄 DU =
�㡰
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PS-119
Thermodynamics 黨
҈
꾀㜠錐㜠錐ᔴ㜠⼄矄錐㜠 꾀㡰 ᨨᆠ矄矄꾀®
�矄 �矄 �矄ᨨ�ᆠᨨᆠ矄矄 P 㦨 ᆠᨨ 矄 ᨨ盐鼀 㡰 ᨨ �矄 ᆠ矄矄 ᨨ 犨ᨨ├ �矄 ᨨ ᨨ �矄 ␔㦨 㡰㦨 꾀矄ᨨ V ᆠ矄矄 矄ᨨ 鼀盐 ᆠ矄矄 \ ᔼ �泐 ᆠ矄矄 ᨨ鼀 Q㦨 U졔 W矄 U ␔ ᆠ矄矄 矄ᨨ Q ᆠ矄矄 鼀 矄 �矄 ᨨ矄 㡰ᨨᆠ矄矄®ᔴ
�矄�矄 �矄ᨨᆠ ᨨ�ᆠᨨᆠ矄矄V㦨ᆠᨨ 矄 矄矄盐 ᆠ矄矄 P µ T 矄ᨨ 鼀盐 矄ᨨ ᆠ矄矄 W㦨錐DU㦨 ��矄ᨨDQ �� ᆠ �矄 ᨨ矄 ᔴ㡰ᨨᆠ矄矄ᔴ®
�矄 �矄 �矄ᨨ�ᆠ 矄矄盐 P 㦨 ᆠᨨ 矄 V � ᆠ矄矄 VµT矄ᨨT� ᆠ矄矄 ᆠD␔D D턼졔矄ᨨ �� 泐 �矄 㡰ᨨᆠ矄矄®꾀 꾀泐� �㦨
黨 PV ᨨ 黨㡰㦨҈ \ 㦨 R 黨 PV ᨨ꾀 㡰㦨҈ 꾀\ 꾀㦨 R � ᨨᆠ矄矄�矄�矄ᨨ 鼀\ D㦨 ᨨ犨D턼㦨D졔␔\ D턼㦨␔ 꾀矄 ᨨ盐鼀 ᆠ矄 � �矄 ᨨᆠ矄矄 矄ᨨ W �� � 犨ᨨ鼀 ᨨ ᨨ �矄 DQ �� ᆠ �矄ᨨ矄 꾀㡰㜠 ⼄㡰 꾀矄 � �矄 矄 � ᆠ盐盐鼀 ᆠᨨ � �矄� 矄盐 ᨨW㦨Q㦨 ᆠᆠᨨ � � ᨨ �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 D㦨턼졔␔ Þ D㦨 ᆠ �矄 ᨨᆠ � � 鼀盐ᨨ �矄 ᨨ T �矄 ᆠᨨ 矄 㡰㜠 㡰� P µ
PV PV� ᆠ矄和ᨨT nR � ᆠ矄矄 ᨨ盐鼀 � ᆠ矄矄 矄ᨨ W � ᆠ矄矄 ᨨ鼀Q㦨DU졔WQ� ᆠ矄矄 ᆠ �矄 �� 矄 ҈҈ ᨨ鼀 �� PV � ᔼ ᨨᆠ矄矄 J ® K ᨨ盐鼀 V 㦨ᆠᨨ 矄 p�矄 ᆠ矄� � PµT ᨨT矄 ᨨ盐 矄ᨨ ᆠ矄 \W㦨Pdv㦨DV㦨DQ ��㡰 ᨨ鼀 DQ㦨DU졔DW ᔼ ᨨᆠ矄矄K®L p㦨ᆠᨨ 矄 㦨nCvDTV�矄� ᆠ矄� �和ᨨ 鼀盐 矄 ᨨ盐 矄ᨨ� ᆠ矄 \DW㦨Pdv>DU㦨nCvDT > Q㦨mCDT> ᔼ ᨨᆠ矄矄L®M V㦨ᆠᨨ 矄 P� ᆠ矄矄ᨨT� ᆠ矄矄 \W㦨DU> Q> ᔼ ᨨᆠ矄矄 M®J V�矄 ᆠ矄� �\DW T�矄矄ᨨ ᆠ矄� �\ DU DQ ҈ ᆠᨨ矄ᨨ � �㜠� ᨨ 꾀 �� 㜠 � �矄 矄 ᨨ犨 � ��盐 ᨨᆠ矄矄꾀®꾀矄�� � � V㜠T � � �ᆠ矄VµT\ 矄矄盐�矄 ᆠᨨ 矄 ᨨ V ᔴ T � ᆠ矄� � ᨨᆠ矄矄®ᔴᨨ盐鼀�矄ᆠᨨ 矄 和� ᆠ 鼀盐 �矄 ᆠ矄� � 和ᨨ 矄矄盐犨� 矄ᨨ ᆠ矄 ᨨᆠ矄矄 ᔴ ® 꾀 鼀盐 �矄 ᆠᨨ 矄 盐 ᨨ盐鼀 I ᆠ矄矄 ᆠ 矄矄盐 犨� � ᆠ矄 矄 P µ V 㡰㜠 ⼄矄㡰 ᨨ鼀PV㦨nRT ÞT㦨
Topic-2 : Specific Heat Capacity and Thermodynamical Processes ҈
҈
ᨨ 㦨ᆠᨨ 矄 V ᨨ nRTV 㦨 ᆠᨨ 矄 \맔 㦨ᆠᨨ 矄 꾀矄 �矄 矄�矄ᨨ盐鼀� ᆠ矄矄矄ᨨ鼀盐 ᆠ矄矄 ␔├ ᨨ犨 턼㦨 ᔴD �㡰 ᨨ ᨨ泐ᨨ�ᆠ ᨨᆠ矄矄 ᔴ㦨ᔴ졔 㦨ᆠᨨ 矄 ҈㜠
㦨ᔴᨨ 矄 \ 㦨 \ᔴ㦨ᔴ졔 㦨ᔴ 㦨 㦨 ҈㜠 턼㦨 ᔴD 㦨 맔 맔D �D �矄 �� Q�矄 ��矄ᨨ �矄ᨨ矄 꾀矄 ᨨ盐鼀 � ᆠ矄矄 矄ᨨ 鼀盐 ᆠ矄矄 ��
҈ P µ ⏌ Þ PV ⏌ 㦨 ᆠᨨ 矄 V \n㦨 - \ᔴ㦨ᔴ졔 㦨 졔 㦨 㦨 -҈ ҈ æ - ö \턼㦨nCVDT㦨 ç ÷ Δ è ø 꾀矄D �矄 �� 턼 �矄 ᨨ矄�� 和ᨨ �矄 �� 矄
b㡰 鼀盐ᆠ �DT�矄矄鼀 ᨨ ᨨᆠ矄矄矄 A ® B錐 A ® C A ® D DU = nCv DT 㦨矄鼀
E AB = E AC = E AD
␔ᨨ├ ᨨ W = P ´ DV AB ® ᨨ盐鼀�矄� ᆠ矄� � Þ WAB > AD ® ᨨ盐鼀�矄 ᆠ矄� � Þ WAD < AC ® ᨨ盐鼀�矄 ᆠᨨ 矄 Þ WAC = ᆠ㡰 ᔼ ��ᆠ ᨨᆠ矄矄 PV g = ᆠᨨ 矄 g æ mö æmö \ P ç ÷ = ᆠᨨ 矄 çèQ V = r ø÷ èrø 꾀矄 鼀矄矄 �矄 ᆠᨨ 矄
\ P µ rg
ᔼ P� Pf � �� � 矄矄盐ᨨ �矄 r� r f � �� � 矄�泐ᨨ �矄
g Pf æ r f ö 䰘⏌� =ç ÷ = 㡰 P� è r� ø nP Þ � = � 㡰䰘 ⏌� = 䰘 P�
Þ n = 䰘 = ҈
b㡰 盐矄� � ᨨ �盐鼀 ᨨᨨ �矄 �矄� � �矄 ᨨᆠ矄矄 DQ = 矄ᨨ ��ᆠ 㡰 ᨨ鼀 ᆠᨨ矄ᨨ � �㜠 � �� � ⼄盐矄�ᨨ ᨨᆠ矄矄 xy ® ᔼ矄ᨨ�ᆠ 矄�ᨨ ᨨᆠ矄矄 yz ® ᔼ矄ᨨᆠ ᨨ�ᆠ 矄矄盐 ᆠ矄矄㡰
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EBD_8341 PS-120
Physics
ᨨᆠ矄矄 zx ® ᔼ矄ᨨ 鼀 ᆠᨨ鼀矄矄�ᨨ
ᨨ ᆠᨨ矄ᨨ � � PV� �矄 矄 矄 ᨨ犨 � ��盐
҈
҈҈
�
b㡰 � V҈㦨҈�P҈㦨҈鼀 V㦨�g㦨҈ g g 矄� �PVr㦨 ᆠᨨ 矄 Þ PV ҈ ҈ = PV ҈
æ҈ö Þ P = P҈ ´ ç ÷ è ø
҈ atm ��� PV PV \␔ᨨ├ ᨨ W = ҈ ҈ g ҈ ҈ æ ö ´ ÷҈҈� ´ ҈� ´ ҈ ç҈´҈ ��� ø è = 㦨黨҈J
=
\ = ҈�
҈
g ҈
Þ T µ V ҈+ 盐矄� � �㡰
Þ T
҈+g µV
҈+g
҈+g æ V ö Þ =ç ÷ = 㡰 T� è V ø b㡰 DUᆠ㦨 DUᆠ㡰㦨 ҈㡰㦨҈ Q㦨�졔㦨҈ ᨨ犨Q㦨DU졔W ᨨ ҈㦨҈졔W ᨨ W㦨҈ ᆠ㡰 꾀矄 ᨨᆠ矄矄 �矄 �矄ᨨᆠ ᨨ�ᆠ 矄ᨨ 䰘 턼 = ᆠ DT = ´ ´ = 黨 = 黨 ´ ҈ ; 䰘 b㡰 ␔ ��
Tf
黨
҈
V
�㡰 T 쟼⼄盐�ᨨ ᨨ ��ᆠ ᨨᆠ矄矄 �矄 TVg҈ 㦨 ᆠᨨ 矄 ҈ Þ T µ g-҈ V
҈
ᔴᨨ矄矄盐ᨨ W㦨黨�J ᨨ 盐矄㡰␔├ ᨨ犨 �ᨨ �鼀
Tµ
䰘
҈
ᆠ㡰 ᔼ �泐 矄ᨨ 泐ᨨ � �� � 矄 和ᨨD꾀㦨D 꾀矄ᨨD턼㦨D졔␔ 꾀矄␔꾀>␔Þ D턼꾀 >D턼 b㡰 和盐ᨨ矄 鼀ᨨ盐 ᨨ 犨 ᨨ 㝐 �鼀
҈�㝐㡰�鼀犨ᆠᨨ � ᨨ�ᆠ 矄ᨨ ᆠᨨ 矄盐鼀 � ᨨᨨ�ᨨ 㦨矄 � 盐矄�ᨨ
㝐맔㦨 ҈�㝐㡰맔矄 㝐맔҈맔҈㦨 ҈�㝐㡰맔맔҈� Þ㝐㦨� 㡰 ® ᔼ矄ᨨ矄�ᆠ ® ᔼ矄ᨨ 鼀 ᆠ ® 꾀 ��ᆠ ® ᔼ矄ᨨᆠ ᨨ�ᆠ b㡰 ᨨ 犨ᨨ├ ᨨ 泐 �矄 盐� � ᆠ泐ᆠ �矄 ⼄盐 ᨨᨨ � �꾀ᔴ ҈ \ D␔ = ´ ´ � = ҈ b㡰 쟼⼄盐�ᨨ ᨨ ��ᆠ ᆠ � �矄
g-1 㦨 ᆠᨨ 矄 䰘 䰘 盐 g = 犨� g - ҈ = - ҈ � �
�
b㡰 ␔ᨨ├ ᨨ
҈ ␔㦨D㦨 D 㦨 ´ ҈´ 䰘 ; 黨҈ 㡰 쟼⼄盐�ᨨ ᨨ ᔴ = - - 㡰 盐矄� �㦨
鼀盐 =
=
-
+ ҈´ Q 㦨҈鼀ᨨ�� 㡰
é ù ê� ú êë úû
� = Þ�= Þ =
é � � ù �
= ê� ´ ´ ú = ë ҈ û ҈䰘 ᆠ㡰 ⼄盐�ᨨ ᨨ � �矄
é ҈ æ V ö ù P = P ê҈ - ç ÷ ú êë è V ø úû
ᨨ
é
҈ù
P
␔ ҈㦨 Þ ҈㦨 ê҈ - ú = ë û ␔ 㦨 é ҈ æ ҈ ö ù æ 䰘 P ö Þ 㦨 ê҈ - ç ÷ ú = ç ÷ ë è øû è ø PV P V é PV ù DT = T - T҈ = ҈ ҈ - êQ T = nR nR ë nR úû æ ҈ ö æ ҈ ö æ PV 䰘 PV ö DT = ç ҈ ҈ - PV 㡰 = ç ÷ PV ÷ç nR ÷ø è ø è nR ø è �P V �P V = = Qn㦨҈㡰 nR R
ᆠ ᨨ
q ᨨ
ᨨ
q ᨨ ᨨ
- + 졔㦨 盐 㦨
\㦨 㦨
ᨨ
矄ᨨ㦨
- ᆠ㦨桌 �㡰 ��㡰
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PS-121
Thermodynamics
졔㦨 \ 졔 㦨 ���㡰
ᨨ鼀盐ᨨ鼀 �鼀盐鼀 㦨 � �� �⼄ ���㡰泐�犨�
졔 㡰㦨
\ 㦨
- 㦨 㦨
P 㦨 ᨨ鼀 �㡰桌 \ P= 黨 \ 鼀 㦨 ᨨ鼀 ���㡰桌 㡰 쟼 �ᆠ� ᆠ泐ᨨ �� �矄 �� 泐 ᔴ 犨 㦨 ҈㜠 㦨 㦨 㦨 h㦨 � 턼 ᔴ ᔴ � Qᔴᔴ㦨 㡰 � ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ �矄 ᔴ 㦨 ᆠ㡰 Q PV r = ᆠᨨ 矄 � �㦨ᆠᨨ 矄 Þ�⏌㦨ᆠᨨ 矄 \ g = ᨨ鼀 �㡰 ��㡰
҈
҈黨
�
ᨨ 犨ᨨ 矄 ᨨᆠ矄矄
(
)
)
W = p� V f V� = ҈� 䰘 ´ ҈ = 䰘 ´ ҈
(
)
҈ æ҈ f ö Pf V f PV çè ´ ҈ ҈ ÷ø � � = g ҈ 黨 D U = ´ ҈ = ´ ҈ ᨨ鼀 �矄犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 QW㦨DU 䰘 黨 ´ ҈ = � \ Q = 䰘 ´ ҈ ´ ҈ = ҈ 㡰 t = æ ö
2p ç ÷ èø 㝐 tµ
꾀矄 g҈㦨⧤ 和ᨨtµg졔҈⏌ g +҈
ᨨ ⼄ 㦨 DU =
҈æU ö 㡰 꾀矄 P = ç ÷ èV ø 盐 = ⧤
҈ 和ᨨ = ⧤
盐 ҈ = ⧤ 꾀矄㦨盐 桌 ᨨ p = ᆠᨨ 矄 ҈
ᨨ µ b㡰 ᔼ VT � 㜠ᨨᆠ矄矄 ᔼ矄ᨨ�ᆠ 鼀盐 � ᆠ矄矄 ᆠ ᨨᆠ矄矄 꾀 ��ᆠ 矄矄盐 ᆠ矄矄 ᆠ ᨨᆠ矄矄 ᔼ矄ᨨ�ᆠ ᨨ盐鼀 ᆠ矄矄 ᨨᆠ矄矄 꾀 ��ᆠ 矄矄盐 � ᆠ矄矄
ᨨ ᨨᆠ矄矄矄 ᆠᨨᆠ泐 矄 � P㜠V ��鼀 㡰 㡰 ᔼ ᆠ泐ᆠ�ᆠᨨᆠ矄矄ᆠ �� ᨨ� �泐�矄 ᨨ Dᆠ泐ᆠ�ᆠ㦨 �R DT Dᔴ㦨 nCv DT = ҈ ´ ␔ ᔴ 㦨 鼀ᨨ 矄ᆠ� �ᆠ ᆠᨨ 矄 ᨨ盐鼀 ᨨᔴD 㦨⧤ \ Dᔴ㦨� ᆠ㡰 � 犨ᨨ├ ᨨ W㦨 ᨨᨨ 鼀ᨨ矄ᨨ �矄m㦨 ᨨ �ᨨ鼀�ᆠ�矄g㦨҈ ␔ᨨ├ ᨨ 盐� � ��ᆠ ᆠ �
mR T҈ - T 㡰 g -҈ ´ DT 㡰 ´ DT 㡰 Þ 㦨 㦨 ҈ - ҈ ´ 㦨⧤ Þ DT㦨 ´ � b㡰 ᨨ盐鼀ᨨ �矄 鼀 = 矄� � g = ᆠᨨ 矄 W㦨
��ᆠ
(
҈
��
�
҈
Ἐ æ Ἐö = =ç ÷ Ἐ è ø
g
ᨨ҈㦨 㡰g 䰘 \ g = = ҈ � 㡰 ᔼ ��泐 ᨨ盐鼀 ҈ 㦨 �l 鼀盐 㦨 T҈ � g = ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ �矄㡰 �l ҈
盐鼀ᨨ 鼀ᨨ矄ᨨ �矄 n = = l � 泐 ��ᆠᆠᨨ鼀矄矄�ᨨ 㡰㦨䰘l ᨨ ��ᆠ ᨨᆠ矄矄 T҈V҈g-҈ = TV g-҈
æ V҈ ö \ T = T҈ ç ÷ è V ø
g -҈
�
-҈
æ � ö ⏌ = T҈ ç ÷ 㦨 T҈ 㡰 = T҈ è 䰘 ø
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EBD_8341 PS-122
䰘
Physics
␔ᨨ├ ᨨ � ��ᆠ ᨨᆠ矄矄 ҈ R T T҈ ] nR DT [ ҈ 黨 = = RT҈ W= � g -҈ é ù êë ҈úû b㡰 ᔼ �矄ᨨᆠ矄矄WAB�矄졔矄DV�矄ᨨ矄��� 矄ᆠᨨ ᨨᆠ矄矄 WBC�矄矄 DV�矄 盐 ᆠ盐 ᨨ 矄ᆠᨨ ᨨᆠ矄矄 �矄 鼀ᨨ \ ␔ᨨ├ 矄ᨨP>P҈ C
P
꾀 ��ᆠ
P1
A
ᔼ矄ᨨ 鼀 B
P2
黨
�
b㡰 ᨨ鼀 P T � 犨 � AB ᨨ �矄ᨨ 鼀 ᨨᆠ矄矄AC�矄 ��ᆠᨨᆠ矄矄�� 꾀矄ᨨ ᨨ 矄�ᨨ
ᨨᆠ矄矄 矄ᨨᨨ ��ᆠ ᆠ盐 �矄 鼀ᨨ ᨨ犨 ᆠ 矄泐 盐 PV � ᨨ�矄ᨨ 鼀ᨨᆠ矄矄 �矄 鼀ᨨ ��ᆠᨨᆠ矄矄 ᨨ矄鼀� ᆠ矄� ᨨ盐鼀㡰 ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 ᆠᨨ矄ᨨ � � � 㡰 ᨨ ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄錐 Qᆠ泐ᆠ�ᆠ㦨WAB졔WBC졔WCA㦨҈졔졔WCA㦨� \ WCA㦨� b㡰 ᨨ ��ᆠ ᨨᆠ矄矄 PV g 㦨 ᆠᨨ 矄 꾀矄ᨨ ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄�├ �盐鼀g㦨҈䰘 ᨨ �ᨨ鼀�ᆠ�矄�├ᨨ 泐� g㦨҈
g-҈
æV ö = TV Þ T҈ = T ç ÷ è V҈ ø V҈ T҈ = T� T = aT� g㦨҈ ᨨ �ᨨ鼀�ᆠ�矄㡰 V = ҈ -҈ aT é҈ù é҈ù \ T� = aT� ê ú = aT = � \㦨 �ê �ú ë û ë û 㡰 ᨨ ��ᆠ ᨨᆠ矄矄 TV g -҈ = ᆠᨨ 矄 ᨨ T҈V҈g -҈ = TVg -҈ 䰘 V T҈ = °ᔴ + 䰘 = 黨 ⧤ V = ҈ g = � ҈
⏌�
䰘
\ T҈V҈g ҈ = TVg ҈ ҈ -҈
æ ö çV ÷ Þ T = 㡰 ´ ç ҈ ÷ Þ T㦨맔 ҈㡰 çç V҈ ÷÷ è ҈ ø ᔼ �矄⼄盐�ᨨ PV㦨 nRT
nRT P Þ V 㦨 kT 矄� ᆠ 矄矄盐 �矄 ᆠᨨ 矄 ᨨ �矄ᨨ�ᆠ ᨨᆠ矄矄㡰 和ᨨ 盐� � �矄ᨨ�ᆠ ᨨᆠ矄矄 V㦨kT �㡰 㦨kTf ��㡰 �� � � �㡰泐 ��㡰 ҈ T = Tf
V
⏌
dPV dV ᨨ �矄ᨨ 鼀 ᨨᆠ矄矄錐 PV 㦨 ᆠᨨ 矄 \ PdV졔VdP㦨 dP =K \ P㦨 盐├鼀ᨨ 盐盐矄㡰 dV ⏌ V
b㡰 盐├鼀ᨨ 盐盐矄K㦨
盐� � ᨨᆠ矄矄 ᆠ � � � �泐 � DU = NCV DT = � ´ ´ ´ ; ´ ҈ 㦨 X ├ \ X = ҈҈㡰 ᨨ ��ᆠ ᨨᆠ矄矄
g҈ 㦨 ᆠᨨ 矄
\ V=
和� ᆠg �ᨨ鼀�ᆠg鼀ᨨ ᨨᨨ鼀�ᆠ \P �ᨨ鼀�ᆠ>P鼀ᨨ ᨨᨨ鼀�ᆠ ᆠ� ҈ �矄 ᨨ �ᨨ鼀�ᆠ�.e. ᨨ 泐� � �矄 ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�.e. ᨨ �盐鼀 㡰 ␔ᨨ├ ᨨ �矄 ⼄盐 ᨨ 盐 ᆠ盐 ᨨ PV ��鼀 ᆠ ᨨ鼀 ��鼀w>w҈>w 㡰 ᨨ鼀TVg҈ 㦨 ᆠᨨ 矄 � ᨨ鼀ᨨ ᨨ鼀�ᆠ�矄 g㦨 ᆠTV⏌ 㦨 ᆠᨨ 矄 ⏌ ⏌ Q VµL㡰 \ T҈ L҈ = T L
æL ö T ᆠ ҈ = ç ÷ T è L҈ ø
g -҈
æ ҈ö Þ 黨 = T ç ÷ Þ T = 黨 ҈㡰 ⏌ � ; 䰘 ⧤ è ҈ ø DT = 䰘 - 黨 = ⧤
҈
g-҈
æV ö T҈ V҈ 㡰 g -҈ = T ç ҈ ÷ è ҈ ø
P
҈
㡰 犨ᨨ├ ᨨ 盐� � ᆠ泐ᆠ㦨 ᆠᨨ矄 � PVᆠ盐 㡰 ᨨ ��ᆠ ᨨᆠ矄矄 泐� � TVg҈ 㦨 ᆠᨨ 矄
T҈V҈g-҈
V1
Tf㦨T㦨맔맔 ҈㡰㦨҈҈K ҈䰘䰘ᨨ ҈䰘㡰 � ҈ ᨨ鼀ᨨ ᨨ鼀�ᆠ�矄 g = ҈⏌ ᔼ ��ᆠ ᨨᆠ矄矄 P҈҈g 㦨g
Þ
҈
҈
҈
҈ ҈㡰�⏌ = ҈ 㡰�⏌
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PS-123
Thermodynamics
Þ =
�⏌
҈ æ ҈ ö çè ÷ø
=
҈ ҈
ᆠ㡰 PV� ᨨ �� VT� �矄�� ᨨ犨
҈҈ - ҈ ҈ 㡰 ҈҈ - ҈ = 꾀 ␔ � = � g -҈ -҈ ⏌ 㡰 黨 = ҈ ҈ = ҈҈ ⏌ ᔼ �矄ᨨ 鼀 ᨨᆠ矄矄
㡰 ҈
㡰
黨
҈
⧤쟼 = ᔴ鼀� ´
ᨨ鼀T҈V҈g҈㦨TVg҈ ᨨT㦨T҈ 㡰⏌�㦨T
\ ⧤쟼 = ᔴ鼀� ´ =
㡰 ␔ᨨ├ ᨨ 盐� � ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄 ҈®®®҈㡰 W
㦨 ᆠᨨ矄 � ᨨᨨ =
= Q 㦨 \
= \ ␔ᨨ├ ᨨ W = [ 矄 = ҈] ҈
㡰 ᨨᆠ矄矄҈®�矄�矄ᨨ�ᆠ ᨨᆠ矄矄 ® �矄 �矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄®�矄�矄ᨨ�ᆠ ᨨᆠ矄矄 ® ҈�矄 �矄ᨨᆠ ᨨ�ᆠ ᆠ ᨨ ��ᆠ ᨨᆠ矄矄 �矄 �鼀ᨨ ᆠ㡰 D턼҈® = ᔴ D = ᔴ ( - ) = ᔴ
D턼 ® = D = ᔴ D = ᔴ ᔴ � D턼҈® \ = = D턼 ® ᔴ
㡰 D턼® = ᔴ
ᔴ
D턼҈® \ = =
҈ D턼®
ᔴ
b㡰 ᨨᆠ矄矄꾀ᨨ TA㦨TB\UA㦨UB 꾀矄ᨨPV㦨맔V
WAB = nRT ᨨ� e
�
V
V
P ␔ᨨ├ ᨨ Þ PB =
҈
æ ö ҈ \ = ç ÷ = ( ) è ø 㦨 㡰҈㦨㦨黨 ␔ᨨ├ ᨨ 盐� � ᨨᆠ矄矄 - ҈ ҈ 黨 ´ ⏌ 㡰 - ҈ ␔= = = ҈- g ҈ - ҈ 盐㦨 矄 㦨�졔҈㦨㡰 ҈ ( ) \ ␔= = -҈ \␔㦨҈RT
�⧤쟼ᨨ �矄鼀� 盐
㡰
㡰
æ ö ␔�矄ᨨ㦨m ᨨ�҈ ç ÷ è ҈ ø æ ö Þ ␔�矄ᨨ㦨҈҈ ç ҈ ÷ = ҈҈ ᔼ è ҈ ø ␔ ᔼ ҈ \ �矄ᨨ = ҈ ҈ = = 黨 ␔ � 黨 ҈҈ ҈ 和ᨨ = =҈䰘䰘䰘 »҈䰘 黨 �㡰 ᨨ ��ᆠ 矄�ᨨ
P P PVg㦨Pa맔 V㡰g Þ Pa㦨 g = ҈䰘 ᨨ �矄ᨨ 鼀 矄�ᨨ
P P맔V㦨P�맔V Þ P�㦨 Pa P = ´ = \ 㦨 P� ҈䰘 P ҈䰘 T盐㡰 矄ᨨ ᨨ P㜠V ᆠ盐 �矄 鼀ᨨ ᨨ ��ᆠ ᨨᆠ矄矄 ᨨ �矄ᨨ 鼀 ᨨᆠ矄矄 ᨨ鼀 � � �矄 ᆠ 矄ᨨ ᨨB�矄� 矄ᨨ ᨨA. 和ᨨ �矄ᨨ 鼀 ᨨᆠ矄矄 �矄 矄 泐 ᆠ盐 A b 㡰 꾀 ��ᆠ ᆠᨨ 矄 ᨨ �矄 鼀� 盐 � 䰘
҈ᔴ҈ + ᔴ � ´ + ҈ ´ g鼀 = = = ҈
҈ᔴ ҈ + ᔴ � ´ + ҈ ´ � ᨨ ��ᆠᨨᆠ矄矄 Pv g 㦨 ᔴᨨ 矄
V
V V
ᔴ
꾀 T
T
㦨 PV ᨨ� Q P0V0 = nRT0 桌
ᨨᆠ矄矄 ᔴ �矄 ᨨ ᆠ ᨨ ᨨ 盐 �鼀 ᆠ
鼀 C� � �P T b 㡰 㡰 ᨨᆠ矄矄AB�矄 ᨨ�矄ᨨ 鼀ᔼ ᆠ矄ᨨ �矄ᨨ 鼀 ᨨᆠ矄矄 犨� ᆠ �盐 泐ᨨ� 㜠 ��鼀 㡰 ᔼ ᨨᆠ矄矄®ᔴ®D�矄 �� ᆠ矄矄 ᨨ盐鼀矄ᨨ ᆠ矄矄 ᨨ␔ �矄 �� ᨨ鼀 �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ 턼 �矄 �� � �矄 ᨨ矄矄
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EBD_8341 PS-124
Physics ᔼ 턼҈®㦨 ᔴD ҈®㦨 ᔴ �㡰
ᆠ㡰 ␔ 꾀 > ␔ ᔴ ᨨ 犨ᨨ├ ᨨ 盐� � 꾀®®ᔴ�矄ᨨ矄�� 㡰 ␔ᨨ├ ᨨ �矄ᆠᨨᆠ├犨�矄ᆠ泐ᆠ� ��鼀�矄ᨨ矄�� bᆠ 㡰 㡰 ᨨ 鼀ᨨᆠ矄矄矄ᆠ ��矄� �泐�
ᆠᨨ 矄 矄矄盐ᨨᆠ矄矄
� é 㡰 ù = ҈´ ê Q㦨 桌 ë úû � � æ ö � = = ç ÷ = è ø \ ᔼ
DU = nCv DT 犨 DT = (T - T҈ )
b㡰 꾀ᆠᆠᨨ � � ᨨ �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄
DQ = DU + DW
䰘
ᔼ ��ᆠᨨᆠ矄矄 DQ = ᨨ = DU + DW ᨨ DU = DW ᆠ㡰 ᔼ �矄ᨨ 鼀ᨨᆠ矄矄 DT = \ DU㦨 Q DU㦨nCvDT 㡰 ᔼ ��ᆠᨨᆠ矄矄 DQ = 㡰 ᔼ㡰 꾀 �b�ᆠᨨᆠ矄矄 ᨨ ᆠ �ᨨ ├矄 ᆠ 犨� 矄盐ᨨ盐 � �矄
黨
ò P × dV =
ᆠ 犨ᨨ├ ᨨ �矄 ᨨ ᔼ㡰 ᔼ �矄ᨨ�ᆠ ᨨᆠ矄矄 矄矄盐 鼀� 矄 ᆠᨨ 矄 W = P × DV ¹ f f DU = nR DT = P DV 桌 ¹
\ DQ = nC p DT ¹
҈ Q = �� 桌 \ ᔼ®턼 ᔼᔼ D҈®®㦨D҈®졔D® 㦨 ᔴD ҈®졔 ᔴD ® = ҈´ - � 㡰҈® + ҈´ - � 桌® é ù ´ - ´ ú 㦨 - 桌 + ê ë û ҈ 㦨㦨맔 RT 㦨RT ᔼᔼ® ᔼᔼᔼ 턼҈®®㦨턼҈®졔턼® 㦨 ᔴD ҈®졔 ᔴD ® � é ù = + ҈ ´ ê ´ - 㡰 ú ë û
= ´ = ᔼᔼᔼ和 =
l + = l [Q D҈® = ] ᔼ ᆠ㡰 ᨨᆠ矄矄 ᔼ�矄 ��ᆠ ᨨD턼 㦨 ᨨᆠ矄矄ᔼᔼ�矄�矄ᨨ�ᆠP㦨 ᆠᨨ 矄 ᨨ␔㦨 ҈㡰 㦨 㡰㦨 ᨨᆠ矄矄 ᔼᔼᔼ�矄 �矄ᨨᆠ ᨨ�ᆠV㦨 ᆠᨨ 矄 ᨨ ␔㦨 ҈㡰㦨 ᨨᆠ矄矄ᔼ�矄�矄ᨨ 鼀鼀盐T㦨ᆠᨨ 矄 \D盐㦨 㡰 ᆠἘ矄 ᆠᨨᆠ�ᨨ ᨨ 矄 ᨨ 矄ᨨ盐 � � �矄 �矄 ᨨ ��ᆠ ᨨᆠ矄矄 PVᆠ盐� ��ᆠᨨᆠ矄矄�矄矄 �矄ᨨ 鼀 b㡰 ␔ᨨ├ ᨨ � �矄ᨨᆠ ᨨ�ᆠᨨᆠ矄矄 �矄 ᨨ ᨨ 犨 �ᆠ 犨 ��ᆠ� � 㜠� � s㡰 㡰 D㦨D턼D W㦨PDV㦨P VV҈㡰㦨PV졔PV҈ 犨 �ᆠ �矄 ᨨ鼀盐 ᨨ �矄ᨨ�ᆠᨨᆠ矄矄 矄ᨨ� P㡰 㡰 ␔쟼㦨 㡰㦨҈ ␔㦨 㡰㦨 Pf Vf PV � � P 㡰 - 㡰 ␔ 㡰 ��ᆠ = 㦨 = � ҈ g ҈ æVf ö æV f ö ␔ 㡰�矄ᨨ 鼀 = nRT ç ç V ÷÷ = PV çç V ÷÷ è � ø è � ø
=
㡰 ᔼ ␔҈®®㦨␔҈®졔␔® ␔®㦨DV㦨桌 㦨맔졔졠
= =
éQ = ҈ ␔ ® �矄ᨨᆠ ᨨ�ᆠ ù ê ú ë ᨨ ᆠ � � ᨨ盐鼀 D = û
\ ᔼ ᔼᔼ D҈®®㦨D҈®졔D® 㦨 졠 + ᔴ D = D [ ᨨᆠ矄矄҈ ® �矄ᨨ 鼀�ᆠD 㦨]
ö æ = ҈´ ç - ÷ = ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄 㦨桌 è ø \ᔼᔼ ᔼᔼᔼ 턼҈®®㦨␔҈®®졔D҈®® ᨨ鼀 �矄 犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 = l + = [ l + ] \ᔼᔼᔼ
ᔼ 턼҈®㦨␔҈®졔D҈®
ᔼᔼ㡰 ᔼ矄ᨨh鼀 ᨨᆠ矄矄 鼀盐 鼀� 矄 ᆠᨨ 矄 f \ DT = Þ DU = nR DT Þ DU = ᨨᆠ �� � �泐DU㦨桌 ᔼᔼᔼ㡰 ᔼ矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄 ᨨ盐鼀 鼀� 矄 ᆠᨨ 矄
Vf
+ Pdv V�
= ҈ l + ᨨ RT = 㦨nRT
Þ DQ =
DV = Þ W =
ᆠ㡰 ᔼ ␔҈®® = ␔҈® + ␔-
�
�҈
� �
�
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Thermodynamics
㦨ᨨ��㦨҈ᨨ� 㡰 �矄 ᆠᨨᆠ ᨨ�ᨨ
㡰 ᨨ鼀DQ㦨msD
DQ DT = = = �°C ms ҈ ´ æ ҈ ö � 㡰 DT = V gDT = çè ÷ 黨 ´ ҈ ( �) 黨 ø 㦨�맔҈䰘鼀 \W㦨pDV㦨 ҈㡰� �맔҈䰘㡰㦨� 㦨҈ 㡰ᨨ�
��
(
)
ᆠ㡰 DU = DQ W = ( �) J = ҈黨黨黨黨黨� J �
VB
ò
V 㦨 ᆠᨨ 矄 ᔼ矄ᨨ�ᆠ ᨨᆠ矄矄㡰 T VA VB \ = TA TB é VB ù V = ú Þ TB㦨 B ´ TA 㦨맔㦨K êQ VA ë VA û 㡰 矄ᨨ ᨨ 矄 ᔼ ᨨᆠ矄矄 Aᨨ B �矄 �矄ᨨ�ᆠᨨᆠ矄矄㡰 � \ Q㦨nCpDT㦨맔 R ´ 㦨҈�R � é ù êQ C p = R ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ �矄 ú ë û �矄 矄ᨨ ᔼ ᨨᆠ矄矄 Bᨨ C �矄 �矄ᨨ 鼀 ᨨᆠ矄矄㡰 \ ᔼ �泐ᆠ �dU㦨 Q DT㦨 \ ᨨ鼀 �矄犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄dQ㦨dW Vf \ Q㦨맔nRTᨨ�҈ V� 㦨맔맔R맔맔ᨨ�҈ 㦨䰘맔ᨨ�҈맔R㦨҈R 矄ᨨ 㡰 ᔼ ᨨᆠ矄矄 C ᨨ D �矄 �矄ᨨᆠ ᨨ�ᆠᨨᆠ矄矄㡰 dW㦨Q dV㦨 æ ö \ Q㦨nCvDT = n ç R ÷ ( TA - TB ) è ø R 㦨맔 맔 㡰㦨黨R 矄 㡰 ᔼ ᨨᆠ矄矄 D ᨨ A �矄ᨨ 鼀 ᨨᆠ矄矄㡰 Vf \ Q㦨맔nRTᨨ�҈ V� æVf ö 㦨맔R맔맔ᨨ� ç ç V ÷÷ 㦨҈R è � ø �矄 矄 ᆠ㡰 ᨨ 犨ᨨ├ ᨨ W㦨QA®B졔QB®C졔QC®D졔QD®A 㦨 ҈�R졔҈R㡰 黨R졔҈R㡰㦨R 㡰 盐鼀ᨨ 鼀ᨨ矄n㦨T҈㦨⧤ 盐� � ᨨᆠ矄矄꾀® pT㦨 ᆠᨨ 矄 ᨨ pV㦨 ᆠᨨ 矄 㦨 K 矄泐㡰 \
p=
K V
VB
K V
dV
㦨 K VB VA 桌 㦨 KVB KV A 桌 㦨
( P V )V B B
B
( P V )V A A
A桌
㦨PBVBPAVA桌 㦨nRTBnRTA桌㦨nR T҈T҈桌 㦨 㡰 㡰 R㡰桌㦨҈R \ ␔ᨨ├ ᨨ ᨨ �矄 � ᨨᆠ矄矄 AB �矄 ҈R 꾀 鼀hᨨ
㡰 ᔼ �� V㜠T� AB�矄矄�� � \
�䰘
ò
\ WA® B = p = VA VA
ᔴ
P҈
҈
҈
T҈
꾀
T҈
pV㦨nRT \ pdV 졔 Vdp = nRdT
( nRT )
dp = nRdT �㡰 P ᨨ鼀 �� ᆠᨨ ��ᨨ
pT 㦨 ᆠᨨ 矄 pdT졔Tdp㦨 ��㡰 ᨨ鼀쟼⼄矄 �㡰 ��㡰犨� pdV㦨nRdT ᨨ pdV 졔
\ WA® B =
ò
pdV = nR
TB
òT
dT = nR TB TA 㡰
A
㦨nR T҈T҈㡰㦨 㡰 㡰 R㡰 㡰 ᨨ WA® B = ҈ R 㡰 ᨨ 㦨ᔴᨨ 矄㝐ᨨ ᆠᆠ�泐 R R R R C= + = + � ҈ g ҈ ҈ x ҈ ҈ 㦨ᆠᨨ 矄 ᨨ҈⏌㦨ᆠᨨ 矄 ҈ \x = Þ C = �R 턼꾀®㦨 ᔴD 㦨 � 㡰 㡰㦨҈ ᨨᆠ矄矄 ᔴᨨᆠ矄矄 �矄 �矄ᨨ�ᆠ æ� ö 턼®ᔴ㦨 ᔴD 㦨 㡰 çè R ÷ø ᔴ 㡰 æ� ö 㦨 çè R ÷ø ҈ ҈㡰㦨 � 㡰 㡰㦨҈� �矄 矄ᨨ ᨨᆠ矄矄 ᔴ 꾀ᨨᆠ矄矄 �矄 �矄ᨨ 鼀
æ PC ö è PA ÷ø
D 㦨\D盐㦨和ᨨ턼ᔴ®꾀㦨␔ᔴ®꾀㦨 ᔴ ç
æ P҈ ö 㦨 㡰 㡰 㡰 㡰㦨҈ 맔黨 㦨 ҈㡰 ç è P҈ ÷ø 턼ᔴ®꾀㦨҈ 矄ᨨ 㡰
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EBD_8341 PS-126
�
Physics
n㦨 ᨨᨨ 鼀ᨨ矄㦨 꾀㡰 ᆠᨨ鼀ᨨᆠ矄矄ᨨ 㜠 ��鼀�矄矄 ᨨ犨 ᨨ犨
ᔴ � �ᆠᨨ 矄 ᨨ盐鼀 꾀 ��ᆠ ᆠᨨ鼀矄矄�ᨨ
҈
꾀
҈
㡰 �㡰 ᨨ犨ᨨ├ ᨨ W㦨WAB졔WBC㦨
҈
ᔼ矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄㡰
PV ҈ ҈ - PV 㡰 + g - ҈㡰 Q WBC㦨PDV㦨P맔㦨桌 g
æV ö 꾀ᆠᆠᨨ � �ᨨᨨ�矄矄ᨨ Ἐ矄 犨P㦨 P҈ ç ҈ ÷ è V ø g ù æV ö ҈ éê P҈V҈ - P҈ ç ҈ ÷ V ú \ W㦨 g -҈ ê ú è V ø ë û g -҈ é V҈ æ V҈ ö ù æ ҈ ê �ö P҈V҈ - P҈ V ç ÷ ú çQ g = ÷ 㦨 ø g -҈ ê V è V ø ú è ë û ⏌ é æ V ö ⏌ ù æ V҈ ö ù éê ҈ ú P V P V P V W㦨 㦨 ҈ ҈ ê҈ - ç ÷ ú ÷ ҈ ҈ ҈ ҈ç V ê V ê ú è ø úû è ø ë ë û ��㡰 ᨨᆠ �� � �泐DU㦨DUAB 졔DUBC 㦨QW é æ V ö ⏌ ù P V 㦨Q ҈ ҈ ê҈ - ç ҈ ÷ ú ê è V ø ú ë û ᆠᆠᨨ � � ᨨ �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄桌 é B ® C Q = DU BC + ù ê A ® B Q = DU + W ú AB ë û ���㡰 ᨨᨨᆠ矄矄BCDUBC㦨nCvDT㦨Q Q WBC㦨桌 R = R ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄Cv㦨 g -҈ Q R DT ÞDT㦨 \ DUBC㦨Q㦨맔 R 꾀ᆠᆠᨨ � � ᨨᨨ�矄矄�ᨨ Ἐ矄 犨 g -҈
ᨨ ᨨᆠ矄矄 AB TAVB
æV ö ᨨ TB㦨TA ç ҈ ÷ è V ø
�黨
g-҈
g -҈
= TBVB
PV æ V ö = ҈ ҈ç ҈ ÷ nR è V ø
g-҈
P g ҈-g PV � ⏌ V- ⏌ \ TB㦨 ҈ V҈ V = ҈ ҈ R R � ⏌ - ⏌ PV V Q ᆠTC㦨TB졔DT㦨 ҈ ҈ + R R 盐鼀ᨨ 鼀ᨨ n㦨҈ ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄 Cp㦨
�R R Cv =
㡰 ␔ᨨ├ ᨨ 泐 �矄 w㦨 꾀 ᨨ DABC ҈ ҈ W㦨 맔矄맔 �� 㦨 V맔P㦨PV 㡰 ᆠ 泐 �矄 � CA � �矄ᨨ�ᆠ ᨨᆠ矄矄 DQCA㦨nCpDT㦨҈맔 �R⏌㡰 TA TC㡰
PV PV TA = I ´R I´R � R é PV PV ù � = - PV DQCA 㦨 ê ú ë R R û 矄ᨨ 泐 �矄 � AB �.e �矄ᨨᆠ ᨨ�ᆠ ᨨᆠ矄矄 DQAB㦨nCvDT㦨҈맔 R⏌㡰 TBTA㡰 R é PV PV ù = PV 㦨 êë ҈ ´ R ҈ ´ R úû ᆠ㡰 꾀矄DU 㦨 � ᆠ泐ᆠ�ᆠᨨᆠ矄矄 ᆠ ᨨ鼀 �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄DQ㦨DU졔DW \ DQ㦨D␔ DQAB 졔DQCA졔DQBC㦨DW PV PV DQBC㦨PV = é P ù 㡰 쟼⼄盐�ᨨ ᨨ� BCP㦨 ê ú V졔�P ë V û (Q y = mx + c ᨨ P = mV + C )
TC㦨
RT ᨨᨨ 鼀ᨨ桌 V P \ RT㦨 V + �PV V P㦨
ᨨ鼀 �鼀盐鼀錐
�㡰
P dT = - 맔V졔�P㦨錐 V dV
�V ��㡰 ᆠ ᨨ鼀⼄盐�ᨨ �㡰 ��㡰
\ V㦨
- P æ �V ö æ �V ö ´ç + � P ç ÷ RT鼀 㦨 è ÷ø è ø V � �PV � PV + = 㦨PV맔 ҈ � PV \ T鼀 㦨 R V VA = ҈ C = � VB VB TA㦨KTB㦨䁸TD㦨䁸h㦨䁸 n㦨҈ ᨨ �ᨨ鼀�ᆠ�矄g㦨҈ ᨨ ��ᆠᨨᆠ矄矄 A®B g -҈ = TBVB g-҈ æV ö ᨨ TB = ç A ÷ TA è VB ø
ᔴ
g -҈ TAVA
꾀
= ҈㡰
⏌�
´ = 黨黨 K
\ ᨨ �矄ᨨ�ᆠ ᨨᆠ矄矄 B ® C 矄矄盐 㦨 ᆠᨨ 矄 VB VC æV ö = ᨨTC㦨TB ç C ÷ 㦨黨黨桌㦨҈҈K TB TC è VB ø
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PS-127
Thermodynamics ᨨ ��ᆠᨨᆠ矄矄C®D VA V V = Q A = ҈ C = 錐 \ VC VB VB g -҈ TCVC
҈
= TDVDg -҈
ᨨᨨ 鼀ᨨ矄n㦨҈R㦨҈ 鼀ᨨ҈k҈TA㦨҈⧤
g -҈ ҈҈ é҈ù é VC ù = 㦨䰘黨҈K \ TD㦨TC ê ú 㦨҈҈ êë úû 㡰҈⏌ � ë VD û ᨨB®Cᨨᆠ矄矄矄ᨨ Q҈㦨nCp TC TB㡰 gR 䰘 ⏌ �㡰 R TC - TB 㡰 = ҈ ҈҈ - 黨黨㡰 㦨 n g -҈ ⏌ �㡰 䰘R ´ 黨黨 @ ҈ R 㦨 ᨨD®Aᨨᆠ矄矄 矄 R T - TA 㡰 Q㦨nCv TDTA㡰㦨n g -҈ D �R R ´ 黨҈ 䰘黨҈ - 㡰 = 㦨 ҈ ⏌ �㡰 Q ᨨ �矄 ᆠ � � ��ᆠ ᨨᆠ矄矄矄㡰 nR T - TA 㡰 ᨨ犨WAB㦨 g -҈ B �R R ´ 黨 黨 - 㡰 = 㦨 ⏌ �㡰 WBC㦨nR TCTB㡰㦨҈맔R ҈҈黨黨㡰㦨黨黨R nR R TC - TD 㡰 = + ҈҈ - 䰘黨҈㡰 WCD㦨 g -҈ ⏌ �㡰 �R ´ ҈䰘 㦨 �R �R W 㦨黨黨R졔 ҈䰘黨㡰㦨黨黨R졔 맔҈ 㦨黨黨R졔҈�R㦨҈黨�R ҈黨� R \ 쟼 �ᆠ� ᆠ泐㦨҈맔 W ⏌Q҈㡰㦨҈맔 㦨҈奠 ҈ R � �㡰 T�㦨䰘졔䰘㦨K錐g㦨 ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄㡰 和� ᆠ �矄 矄 ��ᆠ泐 矄ᨨ 盐矄� � ⏌�
T�V�g -҈ = T f V fg -҈
�㡰 WAB ��ᆠ ᨨᆠ矄矄㡰
nRTA - TB 桌 g -҈ g ҈g ᨨ鼀T P 㦨ᆠᨨ 矄
WAB㦨
g ҈-g
TA PA
g ҈-g
= TB PB
æP ö Þ ç A÷ è PB ø
҈-g
g-҈
g
æT ö = ç A ÷ �㡰 è TB ø ҈-� ⏌
éP ù g é ù �⏌ Þ TB㦨TA ê A ú 㦨�⧤ = ҈ ê ú ë û ë PB û ҈ ´ ҈҈ - �桌 \ WAB㦨 㦨҈䰘 �⏌ -҈ ��㡰 Hᨨ矄� hᨨᆠ矄矄 ® ᔴ�矄ᨨᆠ ᨨ�ᆠᨨᆠ矄矄 QBC㦨nCvDT㦨nCv TBTC㡰 P P
ᨨ � TC犨盐矄 B = C ᨨ盐鼀V㦨ᆠᨨ 矄 㡰 TB TC ҈ é ù ê PC PA ҈ ú PC ҈ ´ TB = ´ � = � K êQ = = ú Þ TC㦨 PB ê PA PA ú ë û \ Q㦨҈맔 맔҈��桌㦨�黨 ���㡰 ᔴ® 꾀® ��ᆠᨨᆠ矄矄矄 g
æ T ö҈ g P \ PC҈ gTCg = p҈D gTDg Þ C = ç D ÷ PD è TC ø
�㡰
PA҈ g TAg = p҈B g TBg g
æ T ö҈ g P Þ A =ç B÷ ��㡰 PB è TA ø 㝐盐�泐� �쟼⼄矄 �㡰 ��㡰犨� g
g -҈
�⏌ -҈ æV ö éV ù = ҈黨 K Þ Tf㦨T� ç � ÷ 㦨 ê ú çV f ÷ ë V û è ø ��㡰 ␔ᨨ├ ᨨ �矄 ��ᆠ ᨨᆠ矄矄 -nR T - T҈ 㡰 - ´ ҈ ҈黨 - 㡰 = W㦨 g -҈ �⏌ -҈ + ҈ ´ ҈҈҈ ´ = + 䰘䰘 㦨 꾀ᆠᆠᨨ � � ᨨ �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄 DQ㦨DU졔DW 盐DQ㦨 矄 ᨨᆠ矄矄 �矄 ��ᆠ㡰 \ DU㦨DW㦨䰘䰘 ���㡰 ␔ᨨ├ ᨨ 泐 �矄 W㦨䰘䰘 꾀 � TA㦨҈⧤ ҈ PB㦨 PA 錐PC㦨 PA
CP � = g㦨 ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄㡰 CV
PC PA æ TDTB ö҈ g =ç ÷ PD PB è TC TA ø ᨨᆠ矄矄矄®ᔴ ®꾀�矄ᨨᆠ ᨨ�ᆠ V 㦨 ᆠᨨ 矄 㡰 PC TC P T = A = A
ᨨ PB TB PD TD 㝐盐�泐� � 矄 犨ᨨ ⼄盐�ᨨ 矄 犨 � PC PA TC TA = �㡰 PB PB TB PB ᨨ鼀쟼⼄矄 ���㡰 �㡰犨 g
g
æ TC TA ö҈ g æ TC TA ö æ TD TB ö҈ g æ TC TA ö =ç =ç ç ÷ ÷ ᨨ ç ÷ ÷ è TB TD ø è TD TB ø è TB TD ø è TC TA ø TC TA =҈ Þ TD TB
ᔴ
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EBD_8341 PS-128
Physics
TC TA ( �)(҈) = ᨨ TD = TB � ᨨ TD㦨�⧤ ᨨ ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄 ��
ᨨᆠ矄矄 㜠ᔴ QBC = nCP TC TB㡰
҈ ö æ䰘 ö æT = ( ) ç R÷ ç A TA ÷ = RTA è øè ø 㦨��RTA ᨨᆠ矄矄ᔴ㜠꾀㦨ᆠᨨ 矄 \WCA㦨 ᨨQCA㦨DUCA㦨nCV TATC㡰
鼀
҈鼀
꾀
ᔴ
⧤
⧤
Wᨨ├ ᨨ ᨨᨨᆠ矄矄AᨨB ᔼ矄ᨨb�ᆠᨨᆠ矄矄㡰 WAB㦨nR TBTA㡰 㦨nR맔҈㦨맔맔㦨҈ Wᨨ├ ᨨ ᨨᨨᆠ矄矄CᨨD ᔼ矄ᨨb�ᆠᨨᆠ矄矄㡰 WCD㦨 WAB 㦨҈J 㦨҈nR㦨맔㦨҈ Wᨨ├ ᨨ ᨨᨨᆠ矄矄BᨨC ᔼ矄ᨨh鼀ᨨᆠ矄矄㡰
PB WBC㦨nRTᨨ�҈ PC 㦨nR맔ᨨ�҈ 㦨䰘䰘nR ҈ 㦨��맔㦨҈ Wᨨ├ ᨨ ᨨᨨᆠ矄矄DᨨA ᔼ矄ᨨh鼀ᨨᆠ矄矄㡰 WDA㦨nRTᨨ�҈
PD ҈ 㦨nR맔ᨨ�҈ PA
㦨䰘黨nR 㦨҈�맔㦨�黨�
ᨨ犨ᨨ├ ᨨ W㦨WAB졔WBC졔WCD졔WDA 㦨҈҈� 㡰 ᨨ鼀 �矄犨ᨨ 鼀ᨨ 泐 鼀�ᆠ矄DU㦨QW ᨨᆠᨨ鼀ᆠ泐ᆠDU㦨 \ Q㦨␔㦨҈҈� 㡰 犨ᨨ├ ᨨ W㦨҈҈� ᆠ㡰 DU㦨和� ᆠ ᨨᆠ矄矄�矄ᆠ泐ᆠ�ᆠ�DT㦨 㡰 p㜠V p㜠T ��鼀矄 矄 ᨨ犨 ᨨ犨 P PA PA
P PA
A
C VA
( pV ) A
B
PA
VA
A C
TA
B
VA
TA 㡰 ᨨᆠ矄矄 A㜠B T 㦨 ᆠᨨ 矄 ҈ \ P µ V �矄 ᨨ盐 矄ᨨ p 犨�ᆠᨨ鼀 V nRTA RTA 꾀矄ᨨ VA = = PA PA DU AB =
( pV )C =
Þ\ TC =
æ V ö \ Q AB = WAB = nRTA ᔼ ç A ÷ è VA ø 㦨RTA� 㡰㦨RTA
�
T ö æ� öæ 㦨 㡰 ç R÷ ç TA A ÷ = 䰘�RTA è øè ø ᔼ ᆠ泐ᆠ�ᆠ ᨨᆠ矄矄 DU㦨 \Qnet㦨QAB졔QBC졔QCA㦨�RTA �㡰 ᨨ ��ᆠᨨᆠ矄矄 g ҈ 㦨 ᆠᨨ 矄 T TV g -҈ = � V 㡰 g -҈ Þ 㦨 �㡰g ҈
├� �ᨨ�ᨨ ᨨ 矄� 矄ᨨ�㦨 g҈㡰ᨨ�� \ g㦨҈ 盐g㦨҈졔
Þ ҈㦨҈졔 f f
= � �ᨨ ᨨ鼀㡰 ��㡰 ᨨ ��ᆠ ᨨᆠ矄矄 ᨨ犨 盐矄� � PVr 㦨 ᆠᨨ 矄 Þ f㦨 g
g
PV ҈ ҈ = PV
P ҈҈ �㡰 ␔ᨨ├ ᨨ ᨨ ��ᆠ ᨨᆠ矄矄 æ P ö PV - ç � V 㡰 è ҈҈ ÷ø PV P V ҈ ҈ = 㦨҈�PV W㦨 g -҈ ҈ - ҈ 꾀泐� �� �矄 ⼄盐�ᨨ PV 㦨 nRT Þ P㦨
P
҈
=
PV ҈ ´ ҈6 ´ ҈ 㦨�鼀ᨨ矄 = = RT ´ �R R � Cp 㦨 \ Cv = 黨맔҈ᨨ �泐�矄矄盐� ᆠᨨ 矄 ᨨ盐鼀 ᆠ ᨨ鼀 Q㦨nCvDT Q 黨 ´ ҈ = \ DT㦨 㦨䰘�⧤ nCv � ´ ´ \ � 鼀盐㦨졔䰘�㦨䰘�⧤ ᨨ犨 ᨨ鼀泐盐矄矄ᆠἘ矄犨 P҈ P P҈T = Þ P㦨 T҈ T T҈ \ n㦨
҈ ´ ҈ ´ 䰘� 㦨맔҈鼀 䰘 ᔼ ��ᆠ ᨨᆠ矄矄 犨ᨨ├ ᨨ ҈ P V - PV 桌 ␔㦨 ҈- g ҈ ҈ P҈㦨҈�⏌鼀V҈㦨 l 㦨맔҈鼀 =
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PS-129
Thermodynamics g
æV ö P㦨P҈ ç ҈ ÷ V㦨 l 㦨맔҈ è V ø � � Cv㦨 R \Cp㦨 R Q C p - Cv = R桌 g=
\
Cp Cv
= ҈䰘 ҈䰘
éù 㦨҈�맔 㡰҈䰘㦨맔҈�⏌鼀 ë û ҈ \ W㦨 맔҈�맔맔҈҈�맔맔҈桌 ҈ - ҈䰘 \ P㦨҈� ê ú
҈ � 㦨黨䰘҈ ҈�桌㦨 䰘 -䰘 �� 矄�� � �ᆠ矄 �矄 �矄 ᆠᨨ鼀矄矄 W㦨
Topic-3 : Carnot Engine, Refrigerators and Second Law of Thermodynamics ҈
P
҈㡰
B
C
P
P
D
A
V
V
V
ᨨ鼀 ��盐 ␔ᨨ├ W = PV �� Q� = WAB + WBC = n × CV DTAB + nCP DTBC R n� R TB - TA 㡰 + TC - TB 㡰 =n
R �R ö æ CP = çèQ Cv = ÷ ø
� = PBVB - PAV A 㡰 + PCVC - PBVB 㡰 � = ´ PV - PV 桌 + PV - PV 桌 ҈� ҈ PV = PV + PV = PV W = = 쟼 �ᆠ� ᆠ泐 h = ҈ Q� ҈ PV h奠 =
» ҈黨 ҈
W ␔ᨨ├ ᨨ = 矄ᨨ SQ Q҈ + Q + Q + Q = � = Q҈ + Q Q҈㦨҈黨҈�Q㦨 Q㦨҈� ҈黨҈� - + ҈� + Q \ = � ҈黨҈� + ҈�
ᆠ㡰 쟼 �ᆠ� ᆠ泐 h =
�
Þ ҈黨҈� - + ҈� + Q = ҈ Þ Q = ҈ - Þ Q = -Q = -黨 Þ Q = 黨 㡰 ᨨ ᆠ ᨨ ��ᨨ Q҈ Q 쟼 �ᆠ� ᆠ泐 = Q҈ ␔ Q҈ 㦨 ᨨ矄 ᨨ鼀矄ᨨᨨ盐 � � Q 㦨 矄ᨨ ᨨ鼀矄ᨨ� ᨨ犨 鼀盐 Q Q w 꾀矄ᨨ ҈ = Q҈ Q҈ ҈ 犨 Þ = ҈ 턼҈ Þ Q҈㦨w맔҈㦨҈J 和ᨨ턼҈Q㦨w Þ Q㦨Q҈w Þ҈҈㦨Q㦨黨J b㡰 QH㦨 ├ 泐 �矄 �� QL㦨 ᆠ 泐 �矄 �� Q2 㦨 ᆠ 泐矄ᆠᨨ �� ␔ᨨ├ ᨨ 泐 ҈矄 �� 㦨犨ᨨ├ ᨨ 泐 �� W㦨QHQL㦨QLQ ÞQL㦨QH졔Q q q = H + qL qL 鼀盐ᨨ ᆠᨨ 矄ᨨ�ᨨ �矄 ��
� ᆠ ᨨ �� QH T҈ Q T = L = QL T Q T T T Þ = ҈ + 盐矄� � �㡰 T T T +T Þ T = T҈ + T Þ T = ҈ T b㡰 矄� �n㦨 ҈ T҈ T ҈ n㦨 㦨 ҈ T҈ T - T = ҈ T҈ 졠 矄ᨨ� � 犨 � T҈㦨黨黨 ᔴ T㦨䰘 ᔴ b㡰 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ h҈㦨 h㦨 h
\҈ = ҈ = ҈
҈
Q �� 矄 ⼄盐泐 �ᆠ� 桌
Þ = =
҈ Þ = ҈
�㡰
=
��㡰
ᨨ鼀 �㡰 ��㡰
= ҈ 㡰
҈
҈
㦨 ҈ 㡰
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EBD_8341 PS-130
䰘
Physics
҈ 犨 꾀 =
턼҈
␔ = h =
턼 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ␔꾀㦨␔
㡰 h꾀 =
҈
턼 - ҈ \ ҈= ҈´ = 턼 ҈ -
\ =
+
ᨨ h =
+ 㦨�⧤ 㡰 � 鼀盐 ᨨ ᆠᨨ ᨨ 泐 㦨 � ⧤ 鼀盐ᨨ ᨨᨨ 泐錐 ҈㦨 ⧤ ᆠ� Q㦨�ᆠ犨ᨨ├ ᨨ ␔㦨䁸 =
쟼 �ᆠ� ᆠ泐㦨 ҈
黨
҈
㡰 D㦨鼀㦨�맔맔҈㦨턼矄� ├
턼矄� ├
㦨 矄� ├ 턼矄ᨨ盐ᆠ 矄ᨨ盐ᆠ
҈ - ´ ҈奠 ҈
h = 奠 ᨨ犨 ᨨ ᆠ ᨨ ᆠ泐ᆠ
´ ҈ 㦨⧤ 㦨䰘
=
҈
\ = -°ᔴ b㡰 �矄 ᆠ ᨨ鼀 矄ᨨ盐ᆠ� 꾀 꾀�矄
DQ =
æ PV ö � æ PV ö R + R çè R ÷ø çè R ÷ø
æ ҈ö � PV + PV = ç ÷ PV è ø ҈ ᆠ㡰 ᔴᨨ � 꾀 ᔴᨨ � 㝐矄矄ᨨ �矄㦨mA 㝐矄矄ᨨ �矄㦨mB ᔴ �� 矄矄盐㦨 DP ᔴ �� 矄矄盐㦨҈�DP Þ
PAV㦨
mA RT M
PἘA 㡰㦨
矄ᨨ盐ᆠ ´ 턼矄� ├
矄� ├
쟼 �泐 ᆠᨨ 矄盐鼀 泐
Þ PAPἘA㦨 Þ D㦨
� 矄ᨨ盐ᆠ 㦨 䰘 ᔴ 졔 䰘 㦨 ⧤
PBP'B 㦨
æ
ö - ҈÷ = ҈䰘 ´ ҈� ␔ᨨ盐盐㦨 � ´ ´҈ ç è 䰘 ø ҈҈ 㡰 ᨨ泐 ᆠ � ᨨ ᨨ 泐 � 犨ᨨ ᆠ矄矄 �矄 矄鼀 矄 ᨨ泐 �矄 矄 盐 ᆠ�ᨨ
PBV㦨
mA RT M
æ 矄ᨨ盐ᆠ ö - ҈÷ \ 犨 ᨨ盐盐 㦨 턼矄ᨨ盐ᆠ 㜠 턼矄� ├ = 턼矄� ├ ç è 矄� ├ ø
矄� ├ 㦨 ᔴ 졔 䰘 㦨 䰘 ├
턼 = 턼҈ ҈
= ҈
T W W � = = Þ ҈ T҈ Q + W Q + W
Q � ´ J 㦨 = ␔㦨 � � b㡰 ␔ᨨ├㜠 ᨨ ␔㡰㦨 꾀ᆠᆠᨨ � � ᨨ � ᆠ� ᨨ ᆠᨨ�鼀泐 �� 㦨턼꾀㦨턼ᔴ 㦨 ᔴ 꾀졔 ᔴ ᔴ � = - 꾀 㡰 + ᔴ - 㡰 � = - 㡰 + - 㡰 ҈ = ␔
鼀 �ᆠ� ᆠ泐ᨨ �� h㡰 = = = ҈� 턼 �� ҈
\ 턼矄ᨨ盐ᆠ =
b㡰 � 턼҈㦨҈ 턼㦨
҈㦨҈䰘 ᔴ㦨⧤
㦨䁸 h㦨䁸 쟼 �ᆠ� ᆠ泐ᨨ ᆠ ᨨ �� ␔ h= ´҈奠 턼҈ 턼 - 턼҈ ´҈奠 ᨨ h = 턼҈
P'B V㡰㦨
mB RT M
m A RT m A RT MV M V 㡰
m A RT MV
�㡰
mB RT mB RT MV M V 㡰
mB RT MV �� � �⼄ ��㡰泐 �㡰 ҈�D㦨
mB RT M
��㡰
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PS-131
Thermodynamics
҈�
m ҈�DP mB Þ = B Þ mA㦨mB = mA DP MA 䰘黨҈㡰 � 矄ᨨ Q㦨mL㦨맔҈㦨ᔴ
鼀盐ᨨ �ᆠT㦨䰘⧤
鼀盐 ᨨ 矄盐ᨨ盐 � � T҈㦨䰘졔䰘㦨⧤ 쟼 �ᆠ� ᆠ泐㦨
҈
Þ p㦨
w Q҈ - Q T҈ - T - 䰘 = = 㦨 Q Q T 䰘
T W = T҈ Q҈
҈ Þ ҈ = 黨 Q҈ Þ
҈䰘
҈ = Þ Q҈ = ҈ Q҈
ᨨ �泐 � 泐 �� ᨨ ᨨ犨 鼀盐矄ᨨ�턼㦨턼҈␔㦨҈҈㦨J 㡰꾀矄 ᆠ泐� �矄ᨨ �矄ᨨᨨ鼀 矄矄鼀 ᨨ �矄 ᨨ ᆠ 矄矄盐 � 矄� ᆠ泐� �矄 鼀ᨨ矄 �ᆠ 矄矄盐 ᨨ盐� ᨨ盐 矄ᨨ犨 盐� � ᨨᆠ矄矄
+ P
P ´ L p é P ´ pR ù P = ´ L = ê ú ´ L é Mg ù êë P pR - Mg úû P ê ú pR û ë
P맔 맔p 㡰㦨px맔pRÞ x =
Q҈ - 䰘 = Þ Q҈ = 䰘黨҈ ᔴ 䰘 㡰 � 錐T҈㦨黨KT㦨KW㦨҈J 矄� � ҈
- Mg
pR 0 和� ᆠ ᆠ泐� �矄ᨨ 鼀泐ᆠᨨ 盐ᆠ� � \鼀盐 T 㦨 ᆠᨨ 矄 꾀泐� �P҈V҈㦨 PV
Þ
҈
㡰 x �矄 ᆠᨨ �矄ᨨ ᨨ鼀 ᨨ 和 꾀⼄盐���盐鼀 Mg㦨 Pp㡰pR
҈黨
ᆠ㡰 꾀⼄盐���盐鼀p㦨 P Þ p㦨P졔 LH㡰r�
�㡰
꾀矄ᨨP맔 pRL㡰㦨pp LH㡰桌
(Q PV ҈ ҈ = PV )
L P L - H 和盐矄�盐� � �矄 盐 ᨨ E ᨨ鼀 �㡰 ��㡰 P L P = P + L - H 㡰 r g P L - H Þ LP㦨P LH㡰졔 LH㡰r g Þ r� LH㡰졔P LH㡰LP㦨 Þ p㦨
P
p H
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EBD_8341 PS-132
Physics
Kinetic Theory
12
Topic-1 : Kinetic Theory of an Ideal Gas and Gas Laws ҈
䰘
㡰 ᔴ泐 ᨨ鼀 ��盐 꾀㦨 ҈꾀졔鼀�
× 꾀
ᨨ 꾀l 㦨
l҈
l
Þ
PAV㦨
× 꾀 + 鼀� 꾀l҈
҈꾀 꾀
鼀�
æ ҈ ҈ö
ç ÷ 㦨鼀� è l l҈ ø
\鼀㦨 � ç l × l ÷ è ҈ ø b㡰 � 鼀盐T�㦨҈䰘졔䰘㦨黨K
鼀盐Tf 㦨䰘졔䰘㦨K 꾀鼀ᨨ矄 �ᆠ矄矄盐P㦨҈맔҈�Pa ᨨ盐鼀ᨨ ᨨᨨ鼀V㦨m � ᆠ� 盐鼀ᨨ 鼀ᨨᆠ盐矄nfn�㦨䁸 矄� �� �矄⼄盐�ᨨ PV㦨nRT N㡰 㦨꾀ᨨ� ᨨἘ矄 盐鼀
n=
\
nf n�㦨
�
PV æ ҈ ҈ ö - ÷ ç R è T f T� ø
҈´ ҈� ´ ҈ ö æ ҈ ´ ´ ҈ ç 㦨 ÷ ҈ è 黨 ø � 㦨�맔҈ ᆠ㡰 ᨨ鼀㜠�
µ
PV 㡰 RT
Þ
҈ 㦨ᆠᨨ 矄 矄矄盐�矄� ᆠ矄� � ᨨ鼀ᨨ҈
矄ᨨᨨ�ᨨ ᆠ㡰 矄 矄 ᆠᨨᆠ 㜠 � 㡰 ҈㝐҈㦨҈ 㝐㦨
㝐 \ ҈´ ҈ = ҈ 㝐
҈ ҈ = ´ = \ 黨 㡰 꾀�矄 矄矄 � �矄ᨨ犨矄矄盐 �� 鼀盐 㡰 꾀ᆠᆠᨨ � �ᨨᨨ泐Ἐ矄犨 PV 㦨 ᆠᨨ 矄 \ PdV 졔 VdP 㦨 Þ
mA RT M
ᔴᨨ � 㝐矄矄ᨨ �矄㦨mB ᔴ �� 矄矄盐㦨҈�DP PBV㦨
mB RT M
mA mB RT RT P'B V㡰㦨 M M mA RT mA RT Þ PAPἘA㦨 MV M V 㡰 m A RT Þ D㦨 �㡰 MV mB RT mB RT PBP'B 㦨 MV M V 㡰 ҈�D㦨 ��㡰 mB RT MV �� � �⼄ ��㡰泐 �㡰 m ҈�DP mB Þ = B Þ mA㦨mB = mA MA DP ᨨ 盐矄㡰⏌ ᔼ ��泐 ��ᨨ �矄 �矄 ᆠᨨ 鼀 ᨨ鼀 ᨨ
鼀盐 㦨 ⧤ ҈ 鼀 㦨꾀 㦨
꾀㦨҈ 맔 ᨨᨨ 㡰 �矄ᨨ 矄 � 矄 泐 맔 鼀 � 鼀盐�矄
꾀 㡰㦨҈
҈
\ 꾀 = = ҈꾀 = - ҈ - 㡰 ҈ + 㡰 PἘA 㡰㦨
æ l҈ l ö
ᆠ㡰 ᔴᨨ � 꾀 㝐矄矄ᨨ �矄㦨mA ᔴ �� 矄矄盐㦨DP
PdV æ ҈ ö æ dV ö æ ҈ ö Þ b맔P㦨҈ = -V 錐 b = - ç ÷ ç = è V ø è dP ÷ø çè P ÷ø dP
ᆠ� 犨 b P犨�ᆠ �盐 泐ᨨ
� 泐 꾀 - ҈꾀 = 鼀� = =
Þ
æ ҈ ҈ ö ç ÷ è ҈ - + ø
æ ö = ҈ çè ҈ - ÷ø
ᔴ맔졔鼀� 㦨ᔴ
Þ ´ ´ + = ´
҈ ҈ æ 黨 ö æ ö Þç + ÷ = ç ÷ ҈ è ø è ҈ ø
ö æ ö æ æ ö
Þ ç + = Þç + ÷ = ÷ ø çè ҈ - ÷ø ø ҈ è è ҈ ö æ ö æ Þ ç + ÷ç = ø è ҈ - ÷ø è 黨 + 㡰 㡰 Þ =҈ ҈ - 㡰 Þ ҈ 졔 㦨 Þ � 졔҈ 㦨 \ 㦨҈䰘» ᆠ �矄 ᆠ ᨨ鼀ᨨ犨 ᆠ ⼄盐���盐鼀㦨졔㦨鼀
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PS-133
Kinetic Theory 黨
�㡰矄� �� �矄⼄盐�ᨨ PV = nRT
Þ PV ҈ ҈ = nR ´ �
P V҈ =
�n R ´
Q T҈ = � ⧤桌
�㡰
Q T = ⧤桌
��㡰
�� � �⼄ �㡰泐 ��㡰
´ � P҈ P ҈ = Þ ҈ = P � P � ´ ҈
\
P = � P҈
RT kT = ҈҈ ´ ҈ ᨨ = ҈҈ ´ ҈ M m 꾀矄 鼀盐 �矄 ᆠᨨ 矄 㡰
DP V �㡰 P ᔼ 矄ᆠᨨ ᆠ矄盐矄� �� �矄⼄盐�ᨨ ��
�
P DV = -nR DT nR DT P 쟼⼄盐� � �㡰 ��㡰犨� DV = -
��㡰
ᔴᨨ鼀� � ᨨ⼄盐�ᨨ 犨� DT = C DP 犨
� 鼀ᨨ ᨨᨨ鼀�ᆠ 㡰 = + - 㡰
҈ � = +
ᨨ犨ᆠ �� ᨨ├� �ᆠ �泐ᨨ �矄
T 㡰 PV㦨nRT ᔼ �矄⼄盐�ᨨ 㡰
nRT V
� VP㦨ᆠᨨ 矄 ᨨ鼀 �㡰 ��㡰
T = ᆠᨨ 矄 ᨨ T µ v V Q V҈㦨V T҈㦨T V V T҈ T = T Þ T㦨T҈ = T \ = V҈ V V҈ V ҈ 矄㡰 ᨨ�ᆠ盐鼀盐TV µ P ᨨ盐鼀�矄� ᨨ矄矄盐P҈ \ p҈p
�㡰 ��㡰
D = 턼 =
\
҈
Topic-2 : Speed of Gas, Pressure and Kinetic Energy ҈
kT = ´ ҈ -҈ P㦨ᆠ鼀ᨨ �V㦨ᆠ鼀 PV PrgV ´ ҈ ´ 黨 ´ N= = ; ´ ҈҈ KT KT ´ ҈ -҈ RT ᆠ㡰 Vrms = M v҈ T҈ 䰘 + ҈䰘㡰 = = = = = � � v T 䰘 + 䰘㡰 �
䰘
� � v҈ = ´ = ҈ � 鼀⏌矄
҈
ᆠ㡰 矄矄盐 P 㦨
҈ mN Vrms V
mN 㡰T V ᔼ �矄 鼀矄矄 鼀盐 ᆠᨨ 矄
P µ V鼀矄㡰 µ T 和ᨨ ᨨᆠ µ V鼀矄㡰 µ T � 盐 ᨨ q 㦨 ҈ ᨨ
P=
㡰 Q ᔴ =
㝐
´ ҈ ´ 㝐 ´ ҈ ´ » ´ ҈ - ├� 㝐= ҈黨 ´ ҈黨
�矄�矄
(҈黨 ) =
㡰 � ⧤쟼鼀 㦨
\ v =
鼀矄 ( ) 㝐 꾀 鼀矄 ( 꾀 ) 㦨 㝐 㦨 㦨҈ 㡰 쟼 �泐矄矄ᨨᆠ� 犨� 鼀ᨨ矄ᨨ �ᨨ鼀�ᆠ�矄
=
ᨨ �泐犨 鼀ᨨ矄 ᨨ �ᨨ鼀�ᆠ �矄 ᆠᨨ � ᨨ
DT ⧤ V = = = ҈� ⧤⏌鼀 nR DP 鼀
Þ P㦨
\ T㦨҈ ⧤
� � =
쟼 �泐矄矄ᨨᆠ� 犨� 鼀ᨨ矄ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ�矄
DP nR DT V =V Þ DT = DP nR P P
҈҈
´ ҈ ´ ҈- T
= ҈҈ ´ ҈ ´ ҈- ҈鼀矄 㝐 㡰 矄� � 鼀矄 㦨 㝐҈ ᨨ
DV = -
C=
҈ ´ ´ ҈ ´ ҈ = N ⏌ m ҈ ´҈ ᆠ㡰 v鼀矄㦨ve =
҈�㡰ᔼ �矄ᆠ矄 ᨨ鼀� �矄⼄盐�ᨨ
PV = nRT
P D V + V DP =
ᨨ盐 矄㡰 ᨨ ᆠ �ᨨ 鼀ᨨ鼀 盐鼀 盐� �ᆠᨨ�矄�ᨨ
mv mv 㡰 mv N 㦨 = Dt Dt N ´ mv 㡰 矄ᨨ 矄矄盐 P = Dt ´ A
r 鼀矄矄ᨨ �矄
黨
ᆠ㡰
鼀矄 =
҈
㡰
矄� �V鼀矄㦨
v鼀矄 �盐鼀㡰 v鼀矄 �ᨨ 㡰
=
RT ҈ Þ Vrms µ M M
M �ᨨ
M �盐鼀
㦨
= ҈ » ҈
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EBD_8341 PS-134
҈҈
ᆠ㡰
Physics ᨨᆠ�泐ᨨ 矄ᨨ盐 � �矄 v =
҈
҈
҈
=
g 㝐 g 㝐
gp = p
VP 꾀 �├� �ᆠ �泐ᨨ �矄ᨨ盐矄鼀ᨨᆠ盐矄 ҈ ҈ æ ö E 㦨 mvrms = m ç v p ÷ = mv p è ø
䰘 ⏌�´ = � ⏌ ´ �
=
b㡰 ␔ ᆠᨨ矄 �矄 �矄 ᆠᆠ � ᨨ矄�� 㜠 �ᆠ�ᨨ 矄矄盐 ᨨ �矄 ᆠ矄矄 ᨨ � ᨨ矄�� 㜠 �矄 ᨨᨨ犨矄 ⼄盐�ᨨ
DP㦨ra dx 和ᨨ鼀ᨨ矄矄盐ᨨ 矄� 矄矄矄矄盐ᨨ ᨨ 矄�
ᆠ㡰 ᨨ鼀㦨 Þ㦨 ᨨ µ
Q 矄鼀㡰
ᨨ� �矄 ᨨ盐 矄矄盐矄ᨨᆠᨨ鼀矄犨ᨨ�鼀矄� ҈ Vrms 㡰҈ V T = = ҈ Þ b㡰 Vrms 㡰 T Vrms 㡰
Þ
ᨨ鼀 矄 矄矄�ᨨ 矄犨ᆠ ᆠᨨ ᆠ盐 vp v vrms
gRT M
҈ = Þ V鼀矄㡰㦨V Vrms 㡰
꾀矄ᨨ鼀矄㦨
Topic-3 : Degree of Freedom, Specific Heat Capacity, and Mean Free Path ҈
ᆠ㡰 ᨨ �ᨨ ᨨ鼀f㦨졔㦨� nfRT � RT
ᨨ �泐 U = = Cp 䰘 꾀 g = = ҈+ = ҈+ = � � Cv f
b㡰 㝐 l =
V
҈�
b㡰 꾀ᆠᆠᨨ � �ᨨ㝐 犨Ἐ矄 �矄�盐�ᨨ ᨨ 矄 �矄
҈
ᨨ 鼀ᨨᆠ盐矄ᨨ � �矄µ \ ᨨᆠ�泐ᨨ ᨨ 泐� 鼀ᨨᆠ盐矄犨�矄鼀� 꾀矄犨 矄ᔴ矄 鼀盐ᨨ A C矄鼀 ҈㡰 ᨨᨨ鼀鼀 矄⼄盐矄 �矄�� 泐
RT M M㦨㝐ᨨ鼀矄矄ᨨ �矄鼀ᨨᆠ盐 T㦨鼀盐ᨨ �矄鼀ᨨᆠ盐 vrms =
盐 v鼀矄 =
␔ �� v = v
\ ҈䰘
҈
RT 㝐
=
RT M
Þ
҈
=
�䰘 Þ T = ҈ ⧤
矄㡰␔ ├ ᨨ犨 鼀矄矄 R T 㡰 RT v㦨 \ vἘ㦨v vἘ㦨 M M ⏌ g ҈RT CH 㡰҈ M҈ g҈ M = = ´ 矄㡰 C M҈ g H e 㡰 g RT M
䰘⏌� 䰘 ´ = ´ ´ = �⏌ � � � 矄㡰 ᨨ �ᆠ盐 鼀盐 ᨨᨨ 鼀 矄⼄盐 矄 ҈ Vrmsµ M �Vrms犨� � 盐矄 ᨨ � �矄矄
㝐 �鼀矄 矄ᨨ 鼀盐 鼀ᨨ鼀矄矄 ᆠ鼀矄 矄 犨� � ᨨ � � �矄矄 ᆠ 㡰 ␔├ ᨨ犨
矄㡰 㝐和矄 鼀矄 =
v=
RT 錐 vrms = pM
kT R
\t =
l ҈ Þtµ T kT m
ᆠ㡰 꾀矄 犨 ├ ᨨ犨 Cp g= = ҈ + 犨 f㦨 �ᨨ ᨨ鼀 Cv f 꾀㡰 㝐ᨨ ᨨ鼀�ᆠf㦨
� = 㡰 �ᨨ鼀�ᆠ��� 鼀ᨨᆠ盐矄f㦨� 䰘 \ g = ҈+ = � � ᔴ㡰 �ᨨ鼀�ᆠ ᨨ 㜠��� 鼀ᨨᆠ盐矄f㦨䰘 黨 \ g = ҈+ = 䰘 䰘 㡰 �ᨨ鼀�ᆠ��� 鼀ᨨᆠ盐矄f㦨 \ g = ҈+ = 㡰 �ᨨ ᨨ鼀f㦨졔㦨 ᨨ�ᨨ鼀�ᆠ ᨨ 㜠� 鼀ᨨᆠ盐 ᔼ �泐ᨨ 鼀ᨨᨨ �矄鼀盐T \ g = ҈+
T
㦨
҈黨
҈
pnd 犨 d㦨 �鼀ᨨ 鼀ᨨᆠ盐 n㦨 盐鼀ᨨ 鼀ᨨᆠ盐矄盐 �ᨨ盐鼀 l 盐鼀 �鼀ᨨ ᆠᨨ�矄�ᨨ t = v鼀矄
RT RT vp 㦨 M M
f nRT = RT = RT b㡰 Cp Cv 矄ᆠ� �ᆠ ᆠᆠ�泐 ᨨ �矄 ᆠᨨ 矄 矄矄盐 ᨨ盐鼀 꾀 ᆠᨨ 矄 矄矄盐 ⼄盐�
U =
�
DQ҈ = nC p DT Þ ҈ = nC p × �
�㡰
꾀 ᆠᨨ 矄 ᨨ盐鼀 ⼄盐�
DQ = nCv DT Þ = nCv ×҈
��㡰
�� � � �㡰泐 ��㡰犨� Cp ҈ C p � = × Þ = Cv ҈ Cv
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PS-135
Kinetic Theory g=
Cp
Cv
=
=҈+ f
ᆠ � 犨 t ⏌矄
f㦨 �ᨨ ᨨ鼀㡰
Þ f =
䰘
҈҈
㡰 ᨨ �泐ᨨ �矄鼀� 盐 f҈n҈ RT҈ f n RT + E鼀� = � � = ´ RT + ´ RT = ҈�RT 㡰 꾀矄犨├ ᨨ犨鼀
� R CP҈ = R � 䰘 CP = R ᨨᨨ 泐� CV = R � 䰘 n R + n R N҈CP҈ + N C P = ҈黨 nR ´ g= Þ g = � ҈nR㡰 N҈CV҈ + N CV n R + n R æC ö ҈黨 \ç P ÷ = C è V øm�xture ҈
҈ N æ ö ç ÷ pd èV ø N㦨 ᨨᨨ 鼀ᨨᆠ盐 V㦨ᨨ盐鼀ᨨ ᆠᨨ � d㦨 �鼀ᨨ 鼀ᨨᆠ盐 盐 PV = nRT = nNKT
҈
N P ҈ KT = =n Þ l= V KT pd P ᨨᆠᨨ 矄 ᨨ盐鼀 ᆠᆠᨨ 矄 盐鼀 矄�泐n ᨨ �矄 P 鼀ᨨᆠ盐矄 �矄 ᆠᨨ 矄 T 和ᨨ鼀 鼀� 矄矄鼀 꾀矄鼀盐� ᆠ矄矄 ᨨᨨ ᆠᨨ�矄�ᨨ � ᆠ矄矄矄ᨨ �ᨨ
�鼀 ᆠ矄矄 㡰 和ᆠ� �ᆠ ᨨ �矄 ᆠᨨ 矄 ᨨ盐鼀 ҈ Cv = fR錐 f = �ᨨ ᨨ鼀 ᨨ�矄꾀 �ᨨ鼀�ᆠ㡰 㦨� 矄�ᨨ 졔ᨨ�ᨨ 㡰
黨
ᨨ�矄 �ᨨ鼀�ᆠ㡰� ��ᨨ ᨨ 矄�ᨨ 졔ᨨ�ᨨ 㡰 ��ᨨ �ᨨ ᨨ鼀 � CvA R � 䰘 B \ Cv = R ᆠ = = CvB 䰘 R 䰘 ᨨ 盐矄㡰㝐 ᨨ �矄鼀ᨨᆠ盐�矄�� 泐
҈
҈
҈�
p d n n㦨 盐鼀 ᨨ ᆠᨨ�矄�ᨨ 矄 盐 � ᨨ盐鼀 d㦨 �鼀ᨨ 鼀ᨨᆠ盐 ᔼ �矄 ᨨ 鼀ᨨᆠ盐�矄
\ tµ
҈ pnd v
=
҈ pnd M҈
M RT
d M \ t҈ = ´ t M d҈ d
æ ççQ v = è
҈
æ ҈ ö ´ç ÷ = ҈黨 ҈ è 䰘 ø
ᆠ㡰 �ᨨ �鼀 t㡰 µ v µ T
\tµ
҈ T
RT M
ö ÷÷ ø
n҈Cv҈ 桌 + n Cv 桌 n҈ + n
R �R ù é ê ´ + ´ ú 㦨 ê ú + ê ú ë û 㦨҈ 㦨҈맔㦨҈䰘⏌鼀ᨨ├ � Cv ҈ ҈ = = b㡰 F㦨 㦨 C p r 䰘 ⏌ �㡰 䰘 䰘 䰘 ´ ҈ � W = ҈- = ᨨ Q㦨 W = 㦨� 䰘 䰘 Q ᆠ㡰 㦨�맔҈鼀 㦨҈鼀ᨨᨨ 졠
㦨⧤ 鼀矄㦨鼀⏌矄 ҈ \ l= p
҈
҈ = = p t l
´ ´ ´ ҈
p ´ ҈ -҈ ´ 黨 � ´ ҈ -
ᆠᨨ矄矄盐� �� ᨨ�ᨨ �矄㦨҈҈ ᆠ㡰 꾀鼀ᨨ盐 ᨨ ⼄盐� 턼㡰 ᨨ �矄 鼀盐 ᆠᨨ 矄 ᨨ盐鼀 턼㦨 ᔴD
�㡰 꾀鼀ᨨ盐 ᨨ ⼄盐� 턼҈㡰 ᆠᨨ 矄 矄矄盐 턼҈㦨 ᔴD
��㡰 =
=
ᆠ㡰 Cv桌鼀� 㦨
꾀��鼀
l 㝐 �鼀 t = v Þ t=
҈
n҈Cv + nCv
ᨨ
҈
l=
n҈C p + n C p
� n҈ n n +n + = Þ + = ҈ Þ � g m ҈ g ҈ ҈ g ҈ g m ҈ ҈ ҈ � ҈䰘 黨 ´ � Þ gm = = ҈ Þ g 鼀 ҈= Þ + = ҈ ҈ ҈ gm ҈
� R
v
㡰 矄� �g鼀� 盐㦨
҈
Þ
\C A =
�矄 矄�� � 犨 �ᆠ �矄 T ᆠᨨᆠ泐 �ᆠ 泐� 矄 ᨨ犨 � ᨨ�ᨨ ᆠ㡰 㡰 �盐鼀�矄鼀ᨨ ᨨᨨ鼀�ᆠ�矄 졠 泐� �矄 �ᨨ鼀�ᆠ�矄 ᨨ �盐鼀 CV҈ =
l=
҈
�� � �⼄盐�ᨨ ��㡰泐 �㡰犨�
鼀 ҈ Þ t µ 矄
æ 䰘ö Þ Q҈ = Q㡰 ç ÷ è �ø
\
Q҈ C p = Q Cv
C p 䰘ö æ çQ g = C = � ÷ è ø v
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EBD_8341 PS-136
҈䰘
Physics
b㡰 g A = g B =
҈
CP 黨 = = ҈ < ҈ �ᨨ鼀�ᆠ㡰 Cv
g 鼀� 盐
҈ = = ҈ > ҈ ҈ 䰘
b㡰 矄� � t=
\ µ
҈ p �
ᨨᨨ 鼀ᨨᆠ盐矄 ù é êë\ = úû ᨨ盐鼀
\盐 =
R R ᨨ �ᨨ� M㦨㡰Cp Cv㦨 b = a = ҈ ᨨ a㦨҈b \ b
� 㡰 + ´ �
ᔴ ᔴ
æ ö 䰘 = ç҈ + ÷ = ᔴ è � ø � 䰘
㡰 ᨨ ᨨ泐ᨨ�ᆠ ᨨᆠ矄矄
\ ᔴ - ᔴ = ҈-
҈-
\ ҈ - = \ ҈=
ᔴ - ᔴ ᔴ - ᔴ ᔴ = ᔴ +
ᨨᨨ 㦨҈ ᆠ㡰 矄 Q = nC D 矄�矄� ᆠᨨ矄 矄矄
ᨨ ᨨ盐 鼀矄矄 鼀盐 矄 ᨨ盐 �鼀矄� Ἐ
\ =
㦨 矄鼀矄㦨 RT ⏌ M
҈� � ´ R ´ ´ ( T - T )
=
䰘 é CP ù êë\ CV = γ d�atᨨm�c = � C p - Cv = R úû
=
㡰 ⏌
㦨҈黨 ⧤ � 鼀盐㡰 ᔴ �� � �泐 D =
æ öæ � ö = ç ÷ç ÷ -҈҈҈㡰 = -䰘 ├ è øè ø
D
ᔴ - ᔴ ᔴ - ᔴ
Q ᔴ - ᔴ = 㡰
㡰 矄� � ⼄盐���ᨨ ᨨ �泐 犨
C=
� ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄 g =
ᔴ - ᔴ - ᔴ - ᔴ - ᔴ + ᔴ = ᔴ - ᔴ ᔴ - ᔴ
KT = mCT
ᨨ 턼㦨҈㦨҈├ ᆠ㡰 ᔼ ��ᆠ ᨨᆠ矄矄
g҈ 㦨ᔴᨨ 矄 ᨨ ҈҈g҈ 㦨 g҈
㡰⏌㦨 㡰⏌ Þ
æ ö = g = ç҈ + ÷ è ø
犨 �矄 �ᨨ ᨨ鼀
æ ö = = = ( ) ├ ÷ è ø 꾀 æ鼀ö æ ö 㦨 ç ÷ = ç ÷ 㦨 ´ ´҈ ´ = ҈� ´҈ è r ø èø
㡰 �ᨨ ᨨ 矄ᆠ� �ᆠ 矄 ᆠᨨ 矄 矄矄盐 ᔴ 㡰 ᆠᨨ 矄 ᨨ盐鼀 ᔴ㡰
ᔴ
㦨҈�
ᆠ㡰 鼀 �泐ᨨ 鼀ᨨᆠ盐
\턼㦨
R M
ᨨ 泐 ᨨ� M㦨㡰Cp Cv 㦨 a =
㦨 ç
㡰 꾀矄犨├ ᨨ犨CpCv㦨R犨 Cp Cv鼀ᨨ矄ᆠ� �ᆠ ᆠᆠ��矄 ᨨ Cp Cv㦨
㡰 = ҈ ҈ +
ö = ÷÷ ø鼀�
� 䰘 g ҈ = g = 矄ᨨ� �犨� 㦨 � �
ᔴᨨ 矄� � � 矄�ᨨ ᨨ�ᨨ 鼀ᨨ 矄 �矄 ᨨ ᨨ鼀 ҈㦨� 㦨
g g -҈
g -҈
盐� � 盐ᨨ ҈㦨 㦨 ç
҈ � = ´ » ´҈ ᨨ ´҈
҈
ö ÷÷ ø 鼀�
҈g҈ + g -҈ = ҈
҈ + g҈ - ҈
æ ᔴ çᔴ è
= = ´ ҈ 㡰 㡰 = ´ ´ ҈
ᔴᨨ 矄� � ��矄�矄鼀ᨨ ᨨᨨ鼀�ᆠ� 㦨 쟼 �泐쟼㦨黨맔҈
æ ᔴ = çç è ᔴ
矄꾀 矄鼀ᨨ �㜠 �矄ᨨ ᨨ鼀 㡰 쟼 �泐ᨨ �矄쟼
=
҈黨
b㡰 矄� � ᨨ鼀盐
黨
´҈ ´҈ ´ ´҈ 䰘 ´҈
\ ᔴ 㦨 黨� ⏌├�⧤ b㡰 b㡰 졠 ��� �矄鼀鼀ᨨ盐 ᨨ ᆠᨨ 矄 矄矄盐 �矄 ᨨᆠ �� 鼀盐 ᨨ鼀ᨨ ᨨ � ᨨ泐ᨨ鼀�ᆠ
æ ᨨᨨ 鼀ᨨᆠ盐矄 ö D턼㡰 = mᔴ D ç m = è 꾀ᨨ� ᨨ Ἐ矄 ᨨ ÷ø ᨨ
D µ
҈
ᨨᨨ 鼀ᨨᆠ盐矄
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PS-137
Kinetic Theory ҈
҈ U㦨 nfRT 犨 f㦨 盐鼀ᨨ �矄ᨨ ᨨ鼀
䰘
㡰
䰘㡰犨ᨨ├ ᨨ ᨨ �矄 矄盐� ᨨ �矄 ᨨ T � ⼄盐���盐鼀鼀盐ᨨ �矄� 矄矄
ᨨ� �泐ᨨ �矄矄鼀� 矄鼀
fO = � fAr㦨 æ� ö æ ö U = U O + UAr = ç RT ÷ + ç RT ÷ è ø è ø
\
\ ᆠ㡰
ᨨ� �泐U㦨҈҈
꾀� 矄�ᨨ ├� �ᆠ �泐ᨨ � �矄鼀ᨨᆠ盐
�矄
├ 犨 �ᆠ 矄 ᨨ 鼀盐 ᨨ 泐 ᨨ �
鼀盐 �矄 矄鼀 矄�ᨨ ├� �ᆠ �泐ᨨ 졠 ᨨ 犨�⼄盐 㡰 ᨨ � �矄㦨
ᔴᨨ �ᆠ� ᨨ ᨨ盐鼀 矄�ᨨ
æ D ö = ᔴᨨ 矄 ç ÷ = è D ø 꾀� 矄�ᨨ ⧤쟼 ᨨ졠㦨
� u҈ + u = u Ἐ҈ + u Ἐ ᨨ n҈Cv DT҈ + n Cv DT = n҈ + n 㡰CvT
Þ ҈㡰Cv 㡰 + �㡰Cv 㡰 = ҈�㡰CvT \T = 黨
├
҈
=
n҈Cv + nCv ҈
n҈ + n
�
Cp Cv
=
R = ҈� R
T盐㡰 ᔴ泐Cp>CvCpCv㦨R Þ CP = CV 졔 R bᆠ㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ ᨨᨨ 鼀ᨨᨨ 泐 ᨨ� 㦨 ᨨᨨ 鼀ᨨᨨ �盐鼀㦨҈
ᨨ� �泐盐
Vs 㡰m�x = Vs 㡰 He
㡰 꾀ᆠᆠᨨ � �ᨨ 犨ᨨ ⼄盐���ᨨ ᨨ �泐 � ├� �ᆠ �泐矄矄ᨨᆠ� 犨� ᆠ �ᨨ ᨨ鼀鼀ᨨᆠ盐 �矄
=
҈ ├ ᔼ �矄 ᆠ矄 졠 ᨨ 犨ᨨ �矄 ᨨ
g m�x M He ´ M m�x g He
ᨨ�ᨨ ├� �ᆠ �泐 矄� ᆠ 鼀盐�矄矄ᨨ矄鼀
´ KT �.e.҈҈ �ᨨᨨ �ᨨ�ᨨ ├� �ᆠ �泐 ´ KT
㡰 �矄ᨨ A�矄 ᨨ鼀ᨨ ᨨ 犨�矄盐� ᆠᨨ 矄 矄矄盐 \ DQA㦨nCpDTA �㡰 �矄ᨨ B�矄 � ᨨ 犨�矄盐� ᆠᨨ 矄 ᨨ盐鼀 \ DQB㦨nCvDTB ��㡰 盐 DQA㦨DQB �� 㡰 \
nCpDTA㦨nCvDTB
DTB 㦨g DTA㡰 㦨 ҈㡰 ⧤㡰 \ DTB㦨⧤
æ Cp ö ÷ DTA è Cv ø
\DTB㦨 ç
g㦨҈ �ᨨ鼀�ᆠ㡰桌
� R + ҈´ R = R ҈+҈
҈ ҈ n҈ + n n n = + = ҈ + Þ 䰘 g 鼀� - ҈ � g 鼀� x ҈ g҈ - ҈ g ҈ -҈ -҈ � ҈ � = + 㦨 Þ g 鼀� - ҈ = \ g 鼀� = g 鼀� - ҈
æ ö æ �ö ҈㡰 ç R÷ + ҈㡰 ç ÷ R è ø è ø = = R ҈+ ҈
Cp ᨨ 鼀� 盐㡰㦨Cv 졔 R 㦨 R
\ g 鼀� 盐 =
n҈ + n
҈´
f҈ f � nRT + nRT Þ u = RT + RT = RT d u RT \꾀�� �泐鼀ᨨ㦨 㦨 㦨RT n gRT ␔├ ᨨ犨 V矄ᨨ盐 㦨 M
g҈ =
Cv ᨨ 鼀� 盐㡰
n҈Cv҈ + n CV
=
� 鼀 矄�矄�矄鼀ᨨ ᨨᨨ鼀�ᆠᨨ Cr = R ҈ 䰘 � g = 鼀 矄�矄�矄 �ᨨ鼀�ᆠᨨ Cv = R �
b㡰
= 䰘 ⧤
㝐ᨨ矄ᆠ� �ᆠ ᨨ 鼀� 盐ᆠᨨ 矄 ᨨ盐鼀
Cv =
�矄ᨨ ᨨ鼀 ᨨ 矄�ᨨ 鼀ᨨ�ᨨ 㡰 ᨨ犨 ᆠ矄� 矄矄盐 � ᆠ矄矄 ᨨ盐鼀 ᔼ� ᆠ矄矄鼀 ᨨ 鼀ᨨᆠ盐矄꾀矄ᨨ�⧤쟼 ᨨ矄 ᨨ ᨨ �矄矄ᨨ鼀ᨨᆠ盐矄ᨨ ᆠ ᆠᨨ鼀ᨨ ᨨ 鼀� 盐ᨨ �矄矄 矄鼀� 矄�ᨨ �泐
҈ � DU nCv DT Cv ҈ = = = = = Q p nC p DT C p g 䰘 ⏌ � 䰘
㡰 � �泐ᨨ 鼀ᨨ矄ᨨ � �矄
␔├ ᨨ犨 Vrms㦨
Vrms 㡰 He = rms 㡰 H
\ V
´ = = � � ´ ҈ ´ + ҈´ é ù = ú êëQ M m�x = û
RT M
M H M HE
=
҈ =
b 㡰 ␔├ ᨨ犨 ᨨ�矄矄CpCv㦨R
� � R 錐 C p = R 錐 g = ҈� \ C p Cv = C p + Cv = � 䰘 䰘 ᨨ �ᨨ鼀�ᆠ�矄 Cv = R 錐 C p = R 錐 g = � � + = C C \ C p Cv = p v
ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄 Cv =
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EBD_8341 PS-138
Physics
b㡰 ᨨ 矄�ᨨ ├� �ᆠ �泐ᨨ 鼀ᨨ矄 ᨨ �矄 㦨
҈�
æ ҈ ö ÷和 꾀 ø è 꾀
㦨D꾀㦨 ç
Q PV㦨 㡰
㦨 ҈ 㡰├ 和㦨D ├ 和
Ӹ矄 鼀ᨨᆠ盐矄ᨨ �矄ᆠᨨ� 犨� ᆠ ᨨ ᨨᆠ��矄 ᨨ 鼀ᨨᆠ盐矄ᆠ � 盐ᨨᆠᨨ�矄�ᨨ
æ ö = ⧤ ᨨ 鼀
ᆠᨨ 矄 㡰 ÷ çQ è 꾀 ø
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈
㦨
ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ
㡰 �矄矄盐� ᆠᨨ 矄 矄矄盐�.e. ᨨᆠ矄矄 �矄 �矄ᨨ�ᆠ \Q㦨n CpDt
=
� é� ù êë R úû ´ Dt 㦨맔 맔҈맔�㦨 � æ ö çQ C p = R ᨨ 鼀ᨨ ᨨ㜠ᨨ鼀�ᆠ �矄 ÷ è ø
ᆠ㡰
҈和
æ nRT ö \ç T = ᆠᨨ 矄 ᨨT V ҈ = ᆠᨨ 矄 è V ÷ø \PV㦨nRT㡰 � �� � ⼄盐�ᨨ 犨�
ᨨ 盐鼀ᨨ 鼀ᨨᆠ盐矄⏌矄ᆠ
T T T dT dV = ᨨ dT = dV V V V
= ҈和 =
�㡰
ᨨ鼀 ⼄盐�ᨨ
dV㦨VgdT
g㦨ᆠᨨ �ᆠ� ᨨ ᨨ盐鼀 矄�ᨨ ᨨ �矄㦨
dV V
dV = V T ҈ � DU nCv DT Cv ҈ = = = = = Q p nC p DT C p g 䰘 ⏌ � 䰘
ᨨ鼀⼄ �㡰g㦨 㡰
黨J㡰
䰘
ᨨ ��ᆠ ᨨᆠ矄矄 TVg҈ 㦨 ᆠᨨ 矄
æVf ö T Þ � =ç ÷ T f è V� ø
g ҈
æ V ö T Þ � = ç �÷ T f è V� ø � ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄 g = T� T \T f = = � � ҈
(
g ҈
)
ᔴ �� � �泐Du㦨nCVDT
æT ö æ ö = ҈ ´ R ç � T� ÷ = ´ ç ÷ ´ ҈ = 黨 J è ø ø è �
鼀ᨨ矄ᨨ犨� 盐矄ᨨᨨ矄��ᨨ 犨� ⼄盐矄矄盐 ᨨ � 矄� 矄 ᨨ҈㦨
鼀
҈
鼀
㦨 㦨 㦨 ҈ 㝐 㝐҈ ° 㝐҈ ´ = ҈黨° \ a㦨 = = = + 䰘㡰 ҈ 㝐 䰘
҈㦨 Þ
bᆠ㡰 ᨨᆠ㦨矄矄盐맔꾀
҈ =
҈
= 꾀 꾀
D ҈ 㡰 和 = D D
D ├ 和 æ D ö æ ├ ö =ç ÷和 ÷ç b l è l øè b ø
졠�ᨨ ᆠ㡰 �矄 ᆠᨨᆠ 꾀矄 D 犨� ᆠ矄 犨� �鼀 矄ᨨ ᨨ 鼀ᨨᆠ盐矄 �� � 矄 ᨨ盐� 盐 ᆠ矄矄犨� �鼀 ᆠ ᨨ�ᨨ 㡰 �矄 � ᆠᨨᆠ bᆠ 㡰 㡰 꾀矄犨├ ᨨ犨Q㦨mcDT
Þ
� T�㦨҈├Vf㦨V�
ᨨT�V�g҈㦨 TfVf g҈
和
ᨨᆠ ᆠ㦨 矄矄盐맔㦨ᨨ 盐鼀ᨨ 鼀ᨨᆠ盐矄맔 b D ├ 和㦨l ҈和bb㦨�矄ᆠᨨ盐矄 ᨨᆠ �� 㡰桌 Þ ҈bl 㦨D ├
和ᨨ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ
pT㦨ᆠᨨ 矄 �� 㡰
D ├ 和 b
dQ dT dQ = mc µ C �ᨨ 矄ᨨ�ᨨ µᔴ ᨨ dt dT dt
ᔼ � ᨨ ҈⧤ ᨨ鼀 � ᔴ � ᆠ矄矄 犨� 鼀盐 盐 ᨨ � 泐 ᨨ 犨 �ᆠ �矄 矄ᨨ �矄 犨� 鼀盐 盐 ᨨ � 泐 㡰 꾀矄 盐ᨨ ᔴ�矄�� 鼀盐 �㜠 �⧤ 矄ᨨ � � ᆠ矄� � 鼀盐 ᨨ鼀㜠 ҈⧤�矄矄矄 ⼄盐� ᨨ� ᆠ矄� � 鼀盐 ᨨ鼀㜠�⧤ ᆠ㡰 ᨨ鼀 � 盐ᨨ ᔴ ᨨ矄 ᨨᆠ �� 鼀盐 �㜠�⧤ ᨨ �矄 ᨨᆠ ��
ᨨ 矄ᨨ�ᨨ � �矄 � 㡰 꾀矄 盐ᨨ ᔴ � ᆠ矄矄 ᨨ鼀 㜠⧤ ᨨ 矄ᨨ�ᨨ � ᆠ矄矄� �㜠⧤ b 㡰
㡰
ᨨ
ᔼ矄ᨨ 鼀 ᨨᆠ矄矄
꾀
�� �㡰
ᨨ
�� ��㡰
␔ᨨ├ ᨨ 㦨꾀盐 PVᆠ盐 V㜠 �矄 \W҈>W 㡰 ᨨ 矄盐 泐㜠 ��鼀 ᔼ �� ᨨᆠ矄矄AB�矄矄�� � ᔼ 矄 ��矄ᨨ ᨨ矄�� � ᆠ
⼄盐�ᨨ ᨨ � �矄P㦨a졔b �㡰
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PS-139
Kinetic Theory ␔ a bᨨ矄��ᆠᨨ 矄 矄
Q PV = nRT nRT nRT \ = aV + b \P = V V aV bV ᨨT㦨 + nR nR
ᨨ ��㡰
㦨 - ´ ���㡰
�矄矄 矄ᨨ� 鼀矄ᨨ T V \ ABᆠᨨ鼀矄ᨨ ᨨ ᆠ㡰 ᨨP㜠T ��鼀�鼀� � �㡰 ��㡰
҈҈
҈
nRT \ P = a + b ᨨ P Pb = anRT P
dV
d T dV V㦨
黨
=
҈-
é P + H rl g ù \ T = T ê ú ë P + H - y 㡰rl g û
㡰
- ´
a a += nR nR
�矄 ��ᔼ鼀 矄T 矄矄ᨨ鼀鼀 �鼀盐鼀盐
҈
b �矄 盐ᨨ 鼀 �鼀ᨨ 鼀盐 a
T P + H - y 㡰rl g 桌 � nR ´ V= P + H - y 㡰rl g 桌 P + H rl g 桌� nRT =
\ 盐ᨨ泐 ᆠ泐 ᨨᆠ
DW҈ + DU҈ = DQ҈ DW + DU = DQ
DQ҈ + DQ =
g 桌�
[ P + H rl g ] T҈ T 꾀ᆠᨨ 矄 矄矄盐VµTᨨ = V҈ V T T \ ҈ = \V㦨A맔h㡰 Ah҈ Ah T h ´҈ = 鼀 Þ h㦨 ҈ = T҈ P + H - y 㡰rl
҈
\ T㦨��⧤
nRT rl g
= V rl g =
㡰 꾀矄 矄矄盐ᨨ �矄矄� ᨨ ᆠᨨ鼀鼀 矄�矄 矄鼀 \ nC p҈ ( 䰘 - T ) = nC p ( T - )
P + H - y 㡰rl g 桌� [ P + H rl g ] �
䰘 R ( 䰘 - T ) = R (T - ) Þ ҈ - T = 䰘T - Þ ҈T㦨黨 \ T㦨黨⧤
��T㦨䰘T Þ ҈T㦨 꾀泐� � �矄 犨 ᨨ 鼀ᨨ 泐 鼀�ᆠ矄
é P + H - y 㡰rl g ù � T = T ê ú ë P + H rl g û b㡰 Q PV㦨nRT nRT nRT Þ V = = P P + H - y 㡰rl g ␔ P�矄矄矄盐ᨨ 盐 �泐ᨨᆠ�ᨨ �矄 泐 ᨨ鼀 ᨨᨨ鼀 和盐矄�盐� � 盐ᨨ P T ᨨ鼀ᨨ犨�
꾀矄ᨨ PAVA = PBVB RTA = RTB ᨨr TA = TB ᔼ �ᨨ� � ᨨ鼀꾀ᨨ 鼀盐ᨨ �矄 �矄� ᆠ矄矄ᨨ 鼀 �鼀盐鼀V㦨b⏌a
鼀盐 ᆠ矄矄 矄ᨨ ᨨᨨ��� 盐 㡰 T � 鼀盐ᨨ �矄矄犨 ⼄盐���盐鼀 �矄ᆠ � ᨨ矄泐鼀ᨨ ᨨ鼀�ᆠ�矄 ᆠᨨ 矄 ᨨ盐鼀 㦨 �� 泐 �ᨨ鼀�ᆠ�矄ᆠᨨ 矄 矄矄盐
� 䰘 R ( 䰘 - T ) = R (T - )
� �⏌
é P + H rl g ù � = T ê ú ë P + H - y 㡰rl g û
�㡰
\ nCv҈ ( 䰘 - T ) = nC p (T - )
҈
ᨨᨨ犨ᨨ├ ᨨ 㦨҈ 㡰 ᨨᆠ矄 ᆠ� �矄� 矄 盐ᨨ泐 ᆠ泐 ᨨᆠ 犨 �ᆠ �矄 ᨨᆠ 盐ᨨ矄矄盐ᨨ �⼄盐� �㡰 ᨨᆠ ᨨ ��泐 �ᆠ泐 ᨨ犨 犨 矄㡰 ��㡰�矄ᆠᨨ盐矄 ᨨᆠ �ᆠ泐 ᨨ犨 犨 矄㡰 b㡰 꾀矄 盐 ᨨ矄 ᨨ ᆠ � 泐 犨� �⼄盐� 和ᨨ ᨨᆠ矄矄 �矄 ��ᆠ 꾀泐� � �ᨨ T g P҈ g = ᆠᨨ 矄
éP ù g T = T҈ ê ҈ ú ë P û T҈㦨T P҈ = P + H rl g T㦨TP㦨P � P = P + H - y 㡰rl g g =
ᨨ鼀 eq ( ��� ) T =
d T
R �R ´ ҈䰘� + ´ ´҈� 㦨��R졔�R㦨҈R
҈-g
Pb P + ᨨT㦨 �㡰 a nR a nR
�矄矄 矄ᨨ� 鼀矄ᨨ T P \ ABᆠᨨ鼀矄ᨨ ᨨ 㡰 ��ᨨ ᨨ ᨨ �AB
aV bV + nR nR dT aV b \ = + dV nR nR b dT = V = ␔ dV a
- ( DW҈ + DW ) = DU҈ + DU
㦨 nCv҈ ( �� - 䰘 ) + n Cv ( �� - )
�
矄�矄ᆠᨨ鼀矄矄 犨� ᨨ盐 ᆠ �和ᨨᨨᆠ矄矄�矄 ��ᆠ ᨨ鼀T҈V҈r҈ 㦨 TVg҈
æ V ö \T '҈㦨T ç ÷ è V҈ ø
g-҈
æö = ç ÷ èø
⏌�
㦨⧤
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EBD_8341 PS-140
҈�
Physics
␔ ᆠᨨ � �矄 矄ᨨ ᨨᆠ�泐 ᆠ矄矄 泐 v
ᆠ㡰
҈ ᆠ├� �ᆠ �泐 ᆠ矄矄泐 MV
dv ҈ = k T ҈⏌ dT kdT ᨨ dv = 犨 dT = ҈K = ⧤ T dv kdT ҈ ҈ æ dT ö = ´ = ç ÷ ᨨ v T k T è T ø ҈ dT dv \ ´ ҈ = ´ ҈ v T ҈ æ ҈ ö ҈ ᨨ奠ᆠ �� v㦨 ´ ç ÷ ´҈ = = ҈䰘奠 è ø
ᨨ
ᆠ矄� ├� �ᆠ �泐㦨ᔼ ᆠ矄� � �泐ᨨ �矄
҈ mv 㦨 nCVDT
mv ҈ m æ ö mv = ç R ÷ DT \ DT = M è ø R ҈
T鼀盐ᨨ �矄
�矄 ᆠ 泐 ᨨ鼀ᨨ �盐鼀�
҈ 矄��鼀 �
㡰
ᨨ ��ᆠᆠ �PV㦨ᆠᨨ 矄
犨 犨ᨨ矄盐ᆠᆠ矄矄�ᆠᨨ�矄�ᨨ 矄 㦨l㦨m Q l㦨҈鼀㡰
\ V ( dP ) + P gV g ҈ ( dV ) =
�矄 ᆠ = = ҈ 鼀⏌矄 鼀矄㦨 �鼀 ҈⏌ �
ᨨ 1m
㡰
Q ᔴᨨ鼀矄矄���泐 ( C ) =
҈ \ DC = C C҈ gP ҈ ҈ ҈æ ҈ ҈ö ᨨ DC = = ç ÷ gP gP҈ g è P P҈ ø
├
- ´ ҈ ⧤쟼㦨 ├ = ´ ҈ ´ ҈
(
)
㦨҈맔҈҈
ᆠ㡰
ᨨ鼀� �矄⼄盐�ᨨ 㦨nRT㦨
꾀�� P҈ =
鼀
㝐
ᨨ҈㦨
ᆠ鼀矄矄ᨨ �盐鼀�矄� ᨨ 鼀㦨҈� 㡰 盐鼀ᨨ 鼀ᨨ矄ᨨ �矄B㦨n
盐鼀ᨨ 鼀ᨨ矄ᨨ �矄꾀㦨҈
nRT ҈黨 Q U = g ҈ m = ҈ \ Um = U A + U B
( nA + nB ) RT g m ҈
=
ᨨ
ᨨ㦨nᨨn㦨
gRT M n A M A + nB M B ᨨ鼀� 盐M㦨 n A + nB (҈ ´ ) + ( ´ ) = g 鼀ᨨ ҈ M㦨 ҈+ ҈黨 ҈ ´ ´ ᨨ v = ҈鼀⏌矄 \v = ´ ҈ ´ ҈
㡰
和 ᨨ 矄ᨨ盐 v 㦨
V
(҈ + ) ´ ҈ ´
ᨨ P҈ =
䰘䰘黨 V
��㡰
���㡰
҈æ ҈ ҈ö ҈ é҈ ù ÷= ҈ú ç g çè P҈ ´ �g P҈ ÷ø gP҈ êë �g û ҈V é ҈ ù ҈ ᨨ DC = ҈黨 ´ 䰘䰘黨 êë �҈黨⏌҈ úû (҈ ´ 黨) V (҈ ´ ҈) V ҈V = 㦨 [ ҈ ҈] = ҈黨 ´ 䰘䰘黨 ҈黨 ´ 䰘䰘黨 ҈黨 ´ ҈ \ DC =
n A RT nB RT + g A ҈ g B ҈
�n ᨨ ҈ + ҈n = 黨 + ҈�n = +
( n A + nB ) RT
盐� �盐ᨨ P҈P� ⼄ �㡰
n ҈+ n ҈ = + ҈黨 � 䰘 æ ö æ ö æ ᨨ ç ҈ ÷ ç ҈ ÷ ç ҈ ö÷ è ҈ ø è ø è � ø
҈ (҈ + n )
�㡰
V g g æ �V ö P æ V҈ ö = ç ÷ = ç ҈ ÷ = �g 꾀�� P҈ è V ø è V҈ ø ҈黨 ö g æ �� ÷ ᨨ P = P҈ ´ � ç 犨 g = è ø ҈
PVM ҈ ´ ҈ ´ = Þ 鼀㦨҈�鼀 \ m = RT � ⏌ ´ ҈
\
҈ 盐├鼀ᨨ 盐盐矄 ( E )
\C =
꾀�├� �ᆠ �泐ᨨ ᨨ鼀ᨨ 鼀ᨨ ᨨᨨ鼀�ᆠ �盐鼀
�矄⧤쟼㦨 \
gP dP dPV = ᨨ = gP dV V dV
ᨨ盐├鼀ᨨ 盐盐矄㦨쟼㦨g
´ � ⏌ ´
Þ ҈㦨 \ 㦨҈⧤ 㡰 鼀矄㦨 㝐 ´ ҈-
҈䰘
ᨨᆠ�泐ᨨ 矄ᨨ盐 v µ T \ v㦨 ᆠᨨ 矄 k㡰 T҈⏌
҈
㦨䰘V㦨䰘맔҈�V
ᨨ矄矄盐 泐 鼀� 盐P㦨҈�鼀
鼀盐T㦨⧤錐ᨨ盐鼀㦨鼀 x�鼀ᨨ \鼀矄矄ᨨ 꾀㦨 x㡰g
x 錐 - x 盐鼀ᨨ �鼀鼀ᨨ矄ᨨ 꾀�ᨨ n㦨
盐鼀ᨨ �鼀鼀ᨨ矄ᨨ ᨨ n҈㦨
盐 ᆠᆠᨨ � � ᨨ ᨨ Ἐ矄 犨 ᨨ �矄矄盐
n҈RT n RT + V V x RT x 㡰 RT + ҈�㦨 V V
P㦨p҈졔p㦨 ᨨ
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PS-141
Kinetic Theory
҈� ´ ´ Þ 㦨x졔 Þ x㦨䰘g ҈ ´
҈黨
\
nR T - T 㡰 g -҈ ´ - 㡰 㦨黨맔҈� 㡰҈졔 � -҈
㦨P҈ VV҈㡰졔
Pf㦨P鼀졔
Þ Pf㦨҈�졔
´ ҈ -
ᔼ 㝐ᨨ ᨨᨨ鼀�ᆠ 矄
㦨맔҈�⏌鼀 꾀鼀矄矄盐㦨҈�⏌鼀
҈㦨⧤
҈
k 㦨⏌鼀
� ᨨ盐鼀 Vf 㦨V҈졔xA 㦨맔҈졔҈맔맔҈㦨맔҈鼀
PV 꾀泐� � ҈ � =
Pf V f
T�
ᨨ
Tf㦨
Tf
Þ Tf 㦨
�ᆠ鼀
PV � �
´ ҈� ´ ´ ҈- ´ ҈� ´ ´ ҈ -
= ⧤ 꾀
҈
㡰
\
\
P҈맔맔҈㦨맔맔 P҈㦨黨맔҈�鼀 ᨨᨨᆠ矄矄 ҈ ® �矄ᨨ�ᆠ㡰
V҈ V ᨨ = Þ T 㦨⧤ = T҈ T ᨨᨨᆠ矄矄® �b�ᆠ㡰 ҈
é T ù g -҈ é ù � -҈ V㦨V ê ú 㦨҈҈l = ê ú ë û ë T û � Q g㦨 ᨨ鼀ᨨ ᨨᨨ鼀�ᆠ�矄桌
ᨨ犨 PV㦨nRT
҈㦨
꾀
� ´ � ҈�
鼀ᨨ矄 矄 ᨨ鼀 �� 鼀盐矄� ᨨᨨ犨鼀盐 矄� 盐 V
V
n
n ᔼ ��泐
TVg -҈ = TVg -҈ \
ᔴ
졠
맔 㦨
��㡰 ᨨ � � ᨨ盐鼀 矄矄盐ᨨ �矄 꾀泐� � P҈V҈ 㦨 nRT҈
҈
�ᆠ鼀
ᔴ
P2
� ´ � ´ ҈� é + � ù � ´ � ´ Þ 맔 ê úû = ҈� ë + Q 㦨 ҈ 桌 Þ 㦨䰘�ᆠ鼀
l
盐� � �矄盐ᨨ P҈� px㦨P҈x҈
㡰
l
P1
㦨҈졔�矄� 㦨҈졔� 和盐矄�盐� � �矄 盐� ���㡰 ҈맔 ҈㦨҈졔�桌맔 ᨨ ҈맔㦨҈졔�桌맔�
x1
ᆠ鼀 㦨
鼀� 矄� ° � 矄� ° = = 꾀 꾀 꾀 꾀 ´ �㡰 ´ � 矄� ° 㦨�矄� ᆠ鼀ᨨ � 㦨 꾀
҈҈㡰
x
鼀 �ᆠ
㦨
ᨨᆠ矄矄ᨨ PV ��鼀�矄矄矄 ᨨ犨 ᨨ犨 ҈㦨
x
x 2㦨
矄矄盐 盐ᨨ鼀ᆠ盐泐ᆠᨨ盐鼀
҈ Q㦨 P D V + nCv D T + k x PV ҈ ҈ ҈ Cv DT + kx 㦨҈�맔҈桌졔 RT҈ � - ҈ ҈ ´ ´҈ 㦨졔 ´ ´ ´ � 졔 ´ 맔҈㦨䰘 ´ �㡰
㦨黨졔䰘䰘㦨҈� 꾀 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ 盐 和� ᆠ 鼀盐 �矄 矄鼀 泐� � ᨨ泐Ἐ矄 犨 � PV 㦨 ᆠᨨ 矄 � 犨ᨨ矄� 矄ᨨ 鼀ᆠ盐ᆠᨨ盐鼀
맔 맔꾀㡰㦨맔 맔꾀㡰 맔 맔꾀㡰㦨҈맔 ҈맔꾀㡰 \ ҈맔 ҈맔꾀㡰㦨맔 맔꾀㡰 ᨨ ҈ ҈㦨 �㡰 犨 㦨҈졔矄矄盐 盐ᨨ鼀ᆠ盐泐ᆠᨨ盐鼀
Pf V f T�
矄盐� 泐 �矄盐矄 ᨨ 矄�ᨨ ᨨ �矄 � ᆠ矄� � �矄 鼀盐 矄ᨨ� � ᨨ � �泐� 矄� �
nRT ´ ´ = 㦨맔҈� ⏌鼀 V ҈҈ ´ ҈-
Q T㦨T҈��
���㡰 ␔ᨨ├ ᨨ 泐 �矄 W㦨W҈졔W
\ 㝐矄矄ᨨ ᨨ 㦨䰘g 鼀矄矄ᨨ 꾀�ᨨ 㦨䰘㦨黨g � 矄矄盐
kx A ´ ҈
P㦨
Q 꾀泐� � PV 㦨 nRT ᨨ 盐 䰘맔㦨 맔䰘 Ἐ맔㦨 졔 㡰 맔䰘 �� � �犨� Ἐ + = 䰘
n졔x 졠ᔴ
n㜠x � 泐
졠ᔴ
ᔼ ��泐 � 泐 �㡰
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EBD_8341 PS-142
Physics
ᨨ�� 盐 䰘맔㦨 맔䰘 Ἐ맔㦨 㡰 맔� 꾀�� �� � �犨�
é ù P ´ V - PV ú êë 䰘 û æ黨 ö ҈� 㦨 - ç - ҈÷ PV = - ´ RT 㦨҈䰘�RT è ø 㦨 -
ᔼ ��泐 � 泐
Ἐ - � = ´ 䰘 䰘
��㡰
ᨨ鼀⼄ �㡰 ��㡰
+ - � = ´
䰘 Þ 㦨
㦨
䰘 Ἐ ´ 䰘 = 䰘� ᆠ鼀 � \Ἐ㦨 = ҈+ 䰘 �㡰
�㡰
Þ
´ Vg
V æ ö =ç ÷ V è ø
�⏌
\
n㦨
҈� ´ ´ ҈ = ҈黨䰘 ´ ҈䰘 ´ ´
҈-g
ᨨ 㦨
= 㝐 g - ҈㡰
´ ҈ 㦨�鼀矄҈ ´ ҈҈ - ҈㡰 ҈
� ҈㦨㦨鼀鼀�錐䁸 ҈㦨 錐㦨錐 ҈㦨 錐 㦨 ҈ 㦨黨黨
҈҈ ҈ ´ 黨黨
= = 䰘黨 鼀鼀�
҈
\㦨 =
҈
҈- �⏌
æ P ö =ç ÷ è P ø
҈ 㝐
R ҈ Mv = Cv 盐 Cv 㦨 g -҈ ҈ R \ Mv = g -҈
��㡰
꾀�� 盐矄� �PT⼄盐�ᨨ ᨨ�� ᆠ 鼀�P҈gTg㦨ᆠᨨ 矄
æ T҈ ö æ P ö ç ÷ =ç ÷ T è ø è P҈ ø
´ ҈ ´ = 㦨鼀⏌矄 㝐 ⏌҈
\
V = V 䰘 䰘 ´ V ´ ´ T ᨨ鼀⼄ �㡰 ��㡰T҈㦨 V ´ 䰘 䰘 T = ҈黨T ᨨ T҈㦨 ҈ V҈㦨VV㦨V
g
v㦨
㦨nCvD 㦨҈맔Cv맔҈㦨Cv
V Þ V㦨 䰘
盐 V҈졔V㦨V \
҈� 鼀
⧤쟼ᨨ ҈�鼀鼀ᨨᨨ ᨨ 泐�
⏌ �
�㡰
��㡰 ⧤� �ᆠ �泐 ᨨ 鼀ᨨ�ᨨ ᨨ ᨨ 泐� 鼀ᨨᆠ盐矄 犨� ᆠᨨ � ᨨ �泐
ᨨ �� ᆠ 鼀 ��ᆠᆠᨨ鼀矄矄�ᨨ ᨨᆠᆠ盐矄 \P Vg = P ´
\ n 㦨
ᨨᨨ 鼀 矄⼄盐 ᨨᆠ�泐
PV P ´ = ´ V҈ T ´ T҈ V҈T ´ V
D Þ D = ´ D = ҈� ´ ҈ = ҈� D
맔鼀㦨҈�
ᨨ ᆠ 鼀
Þ T҈㦨
ᨨᆠ 㦨맔꾀㦨҈�맔҈㦨҈�
ᨨ犨鼀ᨨ鼀 盐鼀ᆠ �矄ᆠᨨ D㡰㦨 맔鼀 ��㡰 ␔ 㦨 盐鼀ᨨ ᆠᨨ�矄�ᨨ 矄矄ᆠᨨ 矄⼄盐鼀 ᨨ鼀 �㡰 ��㡰
���㡰
和盐矄�盐� � �矄盐ᨨ � ⼄ �㡰
�㡰
Þ
æ T ö ç ÷ è T ø
��㡰
␔ᨨ├ ᨨ 泐 �矄� �� ᆠ 鼀 ��ᆠᨨᆠ矄矄㡰
W㦨
҈ PV - PV 㡰 ҈- g
ᨨ
T㦨�T
\ ᨨ矄矄盐� 㦨䰘黨졔�㦨҈䰘鼀鼀� ᨨ鼀㦨
␔ 矄鼀 n µ V 꾀矄ᨨ盐鼀矄矄鼀�҈ᆠᆠᨨ ᆠ 泐 ᨨ� ᨨ 泐� 和ᨨ ᨨ 矄鼀矄犨� ⼄盐 盐鼀ᨨ 鼀ᨨᆠ盐矄
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PS-143
Oscillations
Oscillations
13
Topic-1 : Displacement, Phase, Velocity and Acceleration in S.H.M. ҈
㦨w 矄� w泐㦨w ᆠᨨ矄w 㦨w
㡰
҈
Þ = + 泐 +
( w 矄� w ) + ( w ᆠᨨ矄 w )
Þ=
+ ( w )
= w
ᆠ㡰 ᨨ鼀 犨ᨨ鼀盐盐泐 �ᆠ盐和㝐矄 � ⼄盐�ᨨ ᨨ �矄矄ᨨ盐矄 ��盐
x A
+
y B
-
㦨
ᔴ泐 A ¹ B ᆠ �矄 b㡰 㝐 �鼀盐鼀 ᨨᆠ�泐� 和㝐 鼀 㦨 w 㝐 �鼀盐鼀ᆠᆠ�ᨨ � 和㝐꾀鼀 㦨w 犨 w鼀 �鼀盐鼀鼀�盐 �盐 ⼄盐 ᆠ泐 �
꾀 鼀 = ҈ 鼀
b㡰 ␔├ ᨨ犨 㦨 w 꾀 -
æ 꾀 ö ᔼ ��泐 㦨 w 꾀 - ç ÷ è ø
졠 �� � �犨�
꾀 ᨨᆠ�泐ᨨ 㡰
p⏌
꾀҈ 㦨꾀
黨
㡰
�鼀㦨㡰
꾀
p
쟼⼄盐���盐鼀ᨨ� 꾀
p 犨� 矄ᆠᨨx҈ t㡰矄矄 ᨨ犨 � ��盐
꾀 㦨꾀
é 꾀 ù 䰘꾀 꾀 黨 ê 꾀 - 黨 ú 㦨꾀Ἐ \꾀Ἐ㦨 黨 ëê ûú �
ᨨ = - 犨 �矄ᨨ矄��ᆠᨨ 矄 b㡰 犨ᨨ矄� 盐矄ᨨ� �矄ᆠ鼀 矄x҈ t㡰㦨A矄� wt x t㡰㦨
f㦨
␔ 꾀Ἐ㦨 � 鼀�盐 � 㦨
æ 꾀 ö 꾀 - ç ÷ è ø
ᆠ ᆠᨨ泐 ᨨ �ᆠ 犨� ᨨᨨ├ �├ 矄 ᆠ㡰 㡰 ᔼ � 和㝐 矄ᨨ� � ᨨᆠᆠ� �ᨨ �ᆠ矄 ᨨ盐 犨泐矄 ᨨᨨ�ᨨ ᨨ �矄ᆠ鼀 ᨨ �ᆠ �ᆠ ᨨ犨 矄 ⼄盐���盐鼀 ᨨ矄��ᨨ � µ
졔f㡰� �矄 鼀矄矄ᨨᆠᨨ鼀矄 �矄�矄ᨨ矄矄�犨 鼀�盐 ᨨ � B 㦨 A �矄 � � 矄 � ᆠ ᨨ
æ 꾀 ö � 泐㦨 w 꾀Ἐ - ç ÷ è ø
㦨 ҈
p 錐 x t㡰 㦨 a y t㡰 㦨 w
ö æ p ÷ 鼀�盐 꾀ᆠ 犨� 矄 � ᆠ ø è p ᨨ ᔼ�矄�� 矄� 盐矄ᨨ� �矄ᆠ鼀 x t㡰㦨B 矄� wt
ᆠ㡰 꾀 t 㦨 x t㡰 㦨 錐 y t㡰 㦨 x t㡰 �矄 矄� 盐矄ᨨ� 盐 ᆠ�ᨨ
A矄� ç wt +
㝐 �鼀盐鼀ᆠᆠ�ᨨ 꾀鼀 㦨w㦨 � 鼀⏌矄
æ 꾀 ö 꾀Ἐ - ç ÷ è ø
q p 㦨
㦨 p p
꾀 t 㦨 䰘
�w㦨҈矄҈ �矄ᆠ鼀 �矄�� 泐 㦨矄� w졔p⏌㡰 㦨 㦨� �㦨矄� p⏌ �㦨矄� � Þ㦨 �
p
ᔼ 泐ᆠᨨ矄矄ᆠ ᨨ �鼀
xy ᆠᨨ矄 d = 矄� d AB
x㦨A矄� at졔d㡰 y㦨B矄� bt졔r㡰
꾀 �ᆠᨨ ᨨ鼀q㦨 㦨
졠
㦨꾀 㡰 ⼄盐�ᨨ ᨨ 和㝐x㦨a矄� wt
Þ
p æ p ö æ p ö x = a 矄� ç ÷ ´ t = ҈矄� ç ÷ t = 矄� t è ø èT ø Q a㦨҈ᆠ鼀T㦨s ᨨ鼀� 㡰
꾀 ᨨ ᨨᆠ�泐 v =
dx d é æ p ö ù p æ pö = ê矄� ç ÷ t ú = ᆠᨨ矄 ç ÷ t è ø dt dt ë è ø û
\ 꾀ᆠᆠ�ᨨ a㦨
dx dt
æ pö æ pö = - ç ÷ 矄� ç ÷ t è ø è ø
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EBD_8341 PS-144
Physics
꾀 t =
a㦨
= ҈
dx dt
p -p p æ pö = - ç ÷ 矄� ´ = 矄�
è ø ҈
Þ
ᔼ 和㝐Y㦨A矄� w졔j㡰 y҈㦨҈矄� pt졔p⏌㡰
��㡰
Þ A = x Þ x =
é҈ ù ᆠᨨ矄p ú 㦨҈矄� pt졔p⏌㡰 \y㦨�맔 ê 矄� p + ë û A҈
�ᨨᨨ 鼀�盐 矄 = = ҈҈㦨҈҈ A
bᆠ㡰 �� ⼄盐�ᨨ
x㦨A矄� wt졔Bᆠᨨ矄wt졔C矄� wtᆠᨨ矄wt �� � ⼄盐�ᨨ ᨨ和㝐 矄� ᆠᨨ矄� 盐 ᆠ�ᨨ 矄 矄 ᨨ盐 � ᨨ犨
A B C 矄� wt 㡰 + ᆠᨨ矄 wt 㡰 + 矄� wt ᆠᨨ矄 wt 㡰 A B C 㦨 ҈ - ᆠᨨ矄 wt 桌 + ҈ + ᆠᨨ矄 wt 桌 + 矄� w t 桌 C 㡰 ᨨA㦨 B㦨x㦨 矄� wt 㡰
ᨨ ⼄盐�ᨨ 矄 矄 和㝐 㡰 ᔼ A㦨B C㦨B x㦨B졔B矄� wt
�矄�矄 ⼄盐�ᨨ ᨨ 和㝐 ᆠ㡰 A㦨BC㦨B錐 \ x㦨Bᆠᨨ矄wt졔B矄� wt
犨ᨨ 和㝐矄 矄盐ᨨ矄 ᨨ �� ᨨ 和㝐 ⼄盐�ᨨ 㡰 A㦨BC㦨\ x㦨A
�矄 ⼄盐�ᨨ ᨨ矄 ᨨ矄 矄 和㝐 ᆠ㡰 꾀泐� � 矄盐ᨨ矄��ᨨ � ᆠ� y 㦨y҈졔y졔y 㦨a矄� wt졔a矄� wt졔� 㡰졔a矄� wt졔黨 㡰 㦨a矄� wt졔矄� wt졔黨 桌졔a矄� wt졔� 㡰 㦨a矄� wt졔� 㡰ᆠᨨ矄� 졔a矄� wt졔� 㡰
Þ
Topic-2 : Energy in Simple Harmonic Motion ҈
ᆠ㡰
ᨨ鼀� ⼄盐�ᨨ ᨨ 和㝐
X = A ᆠᨨ矄 wt ҈㡰
꾀
T �ᆠ �矄 鼀 ᨨ矄��ᨨ
\ 꾀ᆠᆠ�ᨨ 㦨 ᨨᆠ㦨
= A ᆠᨨ矄 wt
Þt=
T
㡰 ⧤� �ᆠ �泐 ├ =
҈ 鼀w 꾀 ᆠᨨ矄 w
҈ 鼀w 꾀 矄� w ├ p ҈ = ᆠᨨ w = ᆠᨨ ҈㡰 = 黨 ҈ ᆠ㡰 ᨨ � �泐 㡰 㦨 kx ҈ ҈ ⧤� �ᆠ �泐 ⧤㡰㦨 kA - kx 꾀ᆠᆠᨨ � �ᨨ ⼄盐矄�ᨨ 㦨├
҈ ҈ ҈ \ kx = kA - kx A Þ 㦨꾀ᨨ x = ±
b㡰 ᨨ�ᆠ ᆠ盐� �和㝐 꾀鼀 ᨨ矄��ᨨ 錐t㦨wt㦨y㦨V㦨Vmax㦨aw
҈ mwa T p 꾀 鼀ᨨ矄��ᨨ t㦨 w㦨 y㦨AV㦨Vm�n㦨
\
K.E. 㦨KEmax㦨
\
K.E.㦨KEm�n㦨
⧤� �ᆠ �泐� SHMKE㦨
҈ ù mA w ú û
E矄盐 㦨 + 㡰 E矄� �
\
ᨨ � �泐 =
ᔴ泐 矄盐 鼀ᨨ�ᨨ �矄和㝐 ᨨ 鼀�盐 A㦨 + ҈ 㡰a � � 矄泐� ᨨ泐黨 �ᨨ �矄
E矄盐 æ A ö = ç ÷ = + ҈㡰 = + 㡰 è aø E矄� �
A
+A
\x㦨A犨 �ᆠ �矄 ᨨᨨ矄矄� \҈, ᆠᨨᆠ
㦨 + ҈ 㡰a矄� wt졔� 㡰 㦨A矄� wt졔� 㡰
é ë
҈ ҈ k A - x 㡰 = kx
A㦨鼀�盐 ᨨ 和㝐 x㦨 �矄ᆠ鼀 ᨨ鼀鼀 ᨨ矄��ᨨ
�㡰
쟼 �泐E� 和㝐µ 鼀�盐 㡰 Q E = ê
T �ᆠ �矄 鼀 ᨨ矄��ᨨ 矄ᨨ ᨨᆠ�泐 �矄
鼀 �鼀盐鼀 꾀ᆠᆠ�ᨨ 㦨 㡰 ␔ ⧤쟼㦨쟼
\ x㦨
҈
꾀 t =
㡰
- p ᆠ鼀 ⏌ s
y㦨�矄� pt졔 � ᆠᨨ矄pt
҈҈
㡰 꾀 T �ᆠ�� 鼀 ᨨ矄��ᨨ 矄ᨨ ᆠᆠ�ᨨ �矄 鼀 �鼀盐鼀
s ᆠᆠ�ᨨ
㦨 �
҈ mw a y 㡰
҈ mwaᆠᨨ矄wt
ᆠ� 㡰ᆠᨨᆠ泐 �ᆠ矄├� �ᆠ �泐�鼀� b㡰 和� ᆠ 矄泐矄鼀 �矄矄�矄 �矄 �泐 � 盐泐 ᆠ 鼀�盐 犨�矄ᨨ ᆠ矄矄犨� �鼀 ᆠᆠᨨ � �ᨨ 㦨⏌鼀 �㡰 Q 쟼 �泐ᨨ ��ᨨ ᨨᨨ ᨨ �矄� ��盐 쟼㡰矄 쟼µÞµ 쟼
=
҈ - Þ = = 鼀 ҈ ҈
ᨨ鼀 ⼄ �㡰
= - ҈
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PS-145
Oscillations
҈ = - ҈ ᨨ = ҈ = \㦨�矄ᆠᨨ 矄 ҈ ҈ ҈ 㡰 ⧤쟼㦨 k A - d 㡰 쟼㦨 kd
é êë
㡰 � U x㡰㦨K ҈ - e - x
ᆠ㡰
҈ - ᆠᨨ矄 wt ö y æ ҈ - ᆠᨨ矄 wt 㡰 çQ 矄� wt = ÷ø è y -y Þ y - = ᆠᨨ矄 wt Þ y = A ᆠᨨ矄 wt y \ 꾀鼀�盐 = Þ y=
꾀鼀 ᨨ矄��ᨨ d㦨꾀 鼀ᨨ矄��ᨨ 矄d㦨 꾀
䰘
҈ k ҈ YA = p m p mL y = y 矄� wt f =
ù \F㦨 - dU = kxe - x úû dx
ᆠᨨ矄ᨨ � � u x � ᨨ �� u x㡰 㦨K ҈ ex 桌 犨 �ᆠ �矄 ᨨ �泐� ᆠ矄� �� ᨨ u 犨� x �矄矄矄 ᨨ犨 � ��盐 U(x)
꾀 �盐 ᨨᆠ�泐㦨 w
ky k g = mg Þ = m y 꾀矄ᨨ 矄� � ᆠᨨ 矄 k = m w㡰 k g g ҈ g Þ w = = Þw= = m y y y
ᨨ⼄盐���盐鼀 ᨨ 鼀矄矄
C
K
B
A x x=0 ᨨ鼀 � ��矄ᆠ ᨨ � �泐�矄鼀� �鼀盐鼀 x㦨 ᆠx㦨�矄 矄ᨨ 矄⼄盐���盐鼀ᨨ犨� 犨 �矄ᆠ �ᆠ ᨨ鼀x㦨 ᨨ �矄ᆠ鼀 矄 �ᆠ 矄ᨨ�� ᨨ矄��ᨨ x㦨犨� ᨨᆠF㦨
黨
kx ex
ᨨ矄鼀盐矄ᨨ x犨 Fµx ᆠ ᨨ 矄鼀 �矄ᆠ鼀 ᨨ鼀 x 㦨 鼀ᨨ�ᨨ �矄 矄�鼀 鼀ᨨ �ᆠ ᆠ㡰 盐� � ᨨ ᆠᨨ鼀 ᨨ矄ᆠ��ᨨ ├� �ᆠ �泐 ᨨ �ᆠ ᆠ盐� �和㝐犨�ᆠᨨ鼀鼀 �鼀盐鼀犨�ᆠ
ᨨ ⼄盐 ᆠ泐犨� 犨 �ᆠ �矄├� �ᆠ �泐ᨨ矄ᆠ�矄 犨�f
鼀� � � �泐㦨
Þ AἘ =
ᆠ㡰
A
T = p
l g
␔ �鼀鼀矄 ᨨ �矄ᆠᨨ盐矄�⼄盐�
g ö ҈� g æ amt = ç g - ÷ = ҈ ø ҈ è l ᨨ犨 T ¢ = p = p
bᆠ㡰 쟼 �泐ᨨ ᨨ矄ᆠ�ᨨ
҈ ҈ 㦨 kA 㦨 ´ ´ ҈ ´ ҈㡰 = ҈ �
㝐 ᨨ矄��ᨨ
҈ æ ҈ ö ҈ ç kA ÷ø = kA Ἐ è
A' 㦨犨 �矄 ᆠᨨ ᨨᆠ├ ᨨ鼀鼀 ᨨ矄��ᨨ
҈ ҈ mw a = m ´ p v a ҈ � � ´ ´ a \ a = 鼀 Þ 黨㦨 ´ ´ p ´ p p
쟼 鼀 ᨨ矄��ᨨ
ᆠ㡰
l t = -⧤ ᆠᨨ矄 q
k=
YA L
Dl ù l úû
=
҈�
T
=
l
q
b㡰 꾀 矄�ᆠ 犨� ᆠ 矄 矄� � �矄 矄� � ᆠᨨ 矄
é F êQ Y = A ë
l
҈� g ҈
ᨨ⼄盐 盐ᨨ矄� � ᨨᆠ
x㦨A x㦨 ⧤쟼㦨 ⧤쟼㦨҈ E㦨҈ E㦨҈ 쟼㦨҈ 쟼㦨 Topic-3 : Time Period, Frequency, Simple Pendulum and Spring Pendulum ҈
҈ kA
꾀⼄盐���盐鼀ᨨ矄��ᨨ ᨨ 鼀矄矄ᨨ ᨨᆠ├├矄ᨨ 矄ᨨ �矄 ᨨ � �泐ᆠᨨ鼀矄
� � m㦨├�w㦨 ⧤쟼㦨� 쟼㦨 p
ᨨ鼀ᆠ �ᆠ �泐E㦨҈
҈ kx
꾀 鼀 ᨨ矄��ᨨ ᨨ � �泐 =
꾀矄ᨨ 쟼 㦨
҈
ᨨ � �泐ᨨ 矄� � =
x㦨 �矄 ᆠᨨ ᨨᆠ├ ᨨ鼀鼀 ᨨ矄��ᨨ k㦨矄� �ᆠᨨ 矄
ᆠ
a㦨䁸
쟼㦨⧤쟼졔쟼㦨 �졔㡰㦨黨
㡰
q
⧤
⼄盐 ᆠ泐ᨨ ᨨ矄ᆠ��ᨨ
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EBD_8341 PS-146
Physics
æ ⧤l ö Þ t=ç ÷ q = -ᔴq è ø
é ⧤l ù ê ᔴ = ú úû êë
\ f2 = f1 Þ
㡰
⧤l = ҈ ⧤ 㝐l p 㝐 ҈
҈ ᔴ ҈ = p ᔼ p
=
�ᨨ ҈
� y㦨 � éë矄� p㡰 + ᆠᨨ矄 p㡰 ùû
㡰 꾀ᆠᆠ�ᨨ 盐ᨨ��泐�㦨
㝐 æ = ç � 㝐 çè
㝐矄矄ᨨ ᨨ ᨨ鼀ᨨ 矄�㦨
҈҈
㡰
⼄盐���盐 㦨鼀�
҈
鼀
҈
盐矄� �㦨w
f =
҈ C p ҈
҈ C ML ҈ p
p =
├ 鼀
鼀㦨҈├�
C M Mö æ L ç + ÷ ø è
꾀矄 ⼄盐 ᆠ泐 盐ᆠ矄 泐奠
=
҈ ├´ ҈ = p �
ᆠ㡰
�鼀ᨨ矄⏌�� 泐 =
├� ҈ µ Dq ´ 矄ᆠᨨ
꾀矄犨 ├ ᨨ犨�鼀 �ᨨ = p
ᆠ㡰
l �
␔ ��ᨨ 鼀矄矄㝐�矄
p r� 鼀
l + Dl g
TM = p
␔;䰘黨� 矄҈
=
Þ
҈ ├ = ҈ p 鼀
q㦨q q㦨҈Þ q 㦨� ᔴ
f҈ 㦨
⼄盐 ᆠ泐ᨨ 矄� � 㡰 =
҈ a q㡰 ´ �㡰 ҈ = a q 㡰 ´ ��㡰 q 졠 �� � �犨� = q
( p r� ) 㦨
ᆠ㡰
ᆠ㡰
҈ =
鼀� 쟼 ᨨ泐 ᨨᆠ㦨r꾀 � 졔㦨鼀�졔鼀 \㦨鼀㦨r꾀 �㦨 pr�㡰
黨
)
´҈- ├�
ᔼ ᨨᆠ├ᨨ 鼀矄矄鼀㦨҈├��矄ᆠ ├㦨p ᨨ犨� �ᆠ矄� �矄ᆠ � 犨� 鼀矄矄鼀㦨 ├� ᆠ ├⼄㦨├ ├
ᨨ 盐矄㡰
Þw=
´ ҈
和ᨨ� �犨�矄� �ᆠᨨ 矄 ⧤㦨䰘҈⏌鼀
㝐
(
҈
҈ ├ ´ ´ ҈ = ҈҈ p ҈ ´ ҈-
Þ = 矄
҈ ├ = ҈҈ p 鼀
犨 鼀㦨鼀矄矄ᨨ ᨨ ᨨ鼀
ö æ҈ö ҈ ÷÷ = ç ÷ = èø ø
҈ � 꾀矄ᨨ µ Þ = = � �
�
鼀 㦨 䰘 㝐
b㡰 꾀矄犨├ ᨨ犨 ⼄盐 ᆠ泐� 和㝐 㦨
pö æ Þy㦨 ҈矄� ç p + ÷ ø è \꾀鼀�盐 㦨҈ᆠ鼀 p p 㦨 㦨 矄
�鼀�ᨨ 㦨 w p 䰘
���㡰
和ᨨ� �⼄盐�ᨨ 矄 �㡰 ��㡰 ���㡰
Þ和ᨨ ⼄盐 ᆠ泐ᨨ 矄盐� �ᨨ矄ᆠ��ᨨ 矄
f = f҈
㝐 =
�㡰
l + Dl l
⏌
⏌
鼀
鼀
��㡰 㝐
l + Dl æ 㝐 ö = ÷ è ø l
Þç
㝐� æ 㝐 ö = ҈+ ᨨ ç è ÷ø 꾀Ӹ \
ù 꾀 ҈ éæ 㝐 ö = êç - ҈ú ÷ Ӹ êè ø úû 㝐� ë
㝐�l ù é êëQ Dl = 꾀Ӹ úû
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PS-147
Oscillations ҈
㡰 ᔼ 矄�鼀 鼀ᨨ �ᆠ鼀ᨨ�ᨨ 矄� � ᨨ鼀 矄 꾀t㦨x㦨A x 㦨 꾀ᆠᨨ矄w �㡰 ␔ t㦨 t x㦨Aa ␔ 㦨 t x㦨Aa ᨨ鼀 ⼄盐�ᨨ �㡰 ��㡰 Aa㦨Aᆠᨨ矄w t Aa㦨Aᆠᨨ矄w t ���㡰 꾀矄ᆠᨨ矄w t 㦨ᆠᨨ矄 w t ҈ �㡰 ᨨ鼀⼄盐�ᨨ ��㡰 ���㡰 �㡰
æ P xg ö Mg - ç g ÷ A = F矄ᨨ� � ç x - x㡰 ÷ è ø g æ ö x ÷ = F矄ᨨ� � x - x » x 桌 P A ç҈ g ç x - x㡰 ÷ è ø gP Ax F=- x
\ ⼄盐 ᆠ泐 犨� 犨 �ᆠ �矄ᨨ ᆠ盐矄 和㝐
A - a æ A-a ö = ç ÷ -҈ A è A ø A - a A + a - Aa - A = Þ A A Þ
AaA㦨A졔aAa
Þ
a ҈ a㦨aAÞ A㦨aÞ = A
҈
҈�
ᆠᨨ矄 wt =
A- a ҈ Þ ᆠᨨ矄 wt = ᨨ A
Þ 㦨 t b㡰 ᨨ鼀� ��矄 ᆠ 犨
㦨҈鼀 㦨矄 꾀矄 犨 ├ ᨨ犨
\ ҈
p p t= T
+ ҈㡰 ҈ =
+ ҈㡰 ´ p
æ ö ҈ æ ö ÷ ´ =ç ÷ è 䰘 ø è 䰘 ø �= = 黨䰘鼀 ⏌ 矄 黨
҈黨
ᨨ
鼀 p 鼀 = p ᨨ w ⧤ ⧤
҈ = �
ᆠ㡰
҈ [Q 鼀 = ҈├�錐 w = � ⏌ 矄 ] ⧤
� 鼀矄 -҈
ᆠ�矄ᨨᆠ 犨� 矄 V㦨 � 鼀矄҈ 㡰 ᔼ 矄� �ᨨ 矄� �ᆠᨨ 矄 k҈�矄ᆠᨨ鼀矄矄 泐x҈ ᨨ 矄� �ᆠᨨ 矄 k �矄ᆠᨨ鼀矄矄 泐x
x҈ = ҈
… ��㡰
ᆠ㡰 ᨨ鼀 ��盐ᆠᨨ鼀矄矄�ᨨ ᨨ 矄� �x㦨
ì î
�矄ᨨ
L q
L ü qý L þ
꾀矄ᨨᨨ⼄盐㦨Ia㦨
kL
æ kL ö ML a = -ç ÷q ç ÷ ҈ è ø
L⏌
x q
P Ax g = PA x - x 㡰 g
… �㡰
k A k + k҈
ᨨ⼄盐㦨 í k 㡰
㝐� 㦨 꾀 … ҈㡰 �矄ᨨ �矄 �矄ᆠ 泐 �矄 ᆠ x
x - x 㡰g
\ V =
和ᨨ� �⼄矄 �㡰 ��㡰犨�
PV g = PV g
P=
V 矄� �º g
kx k҈ x҈ = k x Þ x = ҈ ҈ k
Mg = P A
g P x
҈=
V sin q g
x҈ + x = A
ᨨ ⧤㦨�鼀҈
鼀 �鼀盐鼀 ᨨᆠ 泐 矄泐矄鼀ᨨ ᨨᨨ 㦨㝐�졔⧤ 졔鼀� 㦨맔҈졔�맔҈맔҈졔҈맔҈㦨졔҈졔҈㦨 ҈䰘
\
l = 㡰 ´ p � �
Þ 㦨҈ \ 졔҈㦨҈졔҈㦨 㡰 �鼀ᨨ �� ᨨ ᨨᆠ� T㦨
ᆠ㡰 㝐矄矄ᨨ ���ᨨ 泐㝐㦨 ├� 㝐矄矄ᨨ 矄鼀ᨨ 泐鼀㦨҈ ├� 和鼀鼀矄矄 鼀㦨҈├�㡰 ᆠ盐矄和㝐ᨨ �盐 ⼄盐 ᆠ泐w 㦨� ⏌矄 꾀鼀�盐 㦨҈ᆠ鼀㦨҈맔҈ 꾀矄 犨 ├ ᨨ犨
= p
҈ = p � �
꾀矄 l҈ < l ᨨ ҈ < 㦨 泐矄矄犨� �� �ᨨ� 和� ᆠ ��鼀�ᨨ 矄 � 矄犨� �� � 犨� ᨨ � 盐 �矄ᨨ 犨泐矄 졠 泐 犨 盐鼀ᨨ ᆠᨨ鼀 ᨨ矄ᆠ��ᨨ 矄 � 泐 � � 犨ᨨ 盐盐鼀矄犨��� �� ᨨ矄犨� �ᨨ� ᨨ � � 盐盐鼀 ᆠᨨ鼀 ᨨ矄ᆠ��ᨨ 矄 ᨨ � 盐盐鼀ᆠᨨ鼀 졔҈㡰ᨨ矄ᆠ��ᨨ ᨨ盐 �矄ᨨ 矄犨� �� �
p
= p Þ � = �
㦨 ´ ç
gP A MV
҈ �鼀�ᨨ ᨨ 犨ᨨ 盐盐鼀矄
㡰
҈ = p
ᨨ犨Aa㦨꾀ᆠᨨ矄wt Þ
gP A ҈ x M 㦨 p
҈ p
f =
ᔴ泐� ᆠᨨ � � � � �矄
҈ æ ö çQ I = ML ÷ è ø
q
L⏌
x
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EBD_8341 PS-148
\ \
b㡰 \ ᨨ
Physics
҈ k w = f㦨 p p M
dy d y = v = kt = a = k dt dt ay = 鼀 ⏌ 矄 Q k㦨҈鼀⏌矄�� 㡰
l l ᨨT҈㦨 p g g
T҈
\
T
=
g + ay g
\
k桌㦨
Þ
k = m+m
T҈ T
=
҈ = ҈ �
\
è m ø
ᨨ
m
æ kA ö kA = mç = è m ÷ø
Ӹ g g 矄� a 㦨ay g 矄� a g 矄� a 矄� a g ᆠᨨ矄 a 矄� a 㦨ax
g 䰘
a ax = g 矄� a ᆠᨨ矄a 㦨aᆠᨨ矄a ay = g g 矄� a = g ҈ 矄� a㡰 㦨 g ᆠᨨ矄 a
ᆠ�ᆠᆠ�ᨨ 盐ᨨ��泐ᆠ� �ᨨ ᨨ
㦨
g 矄� a ᆠᨨ矄 a + g ᆠᨨ矄 a
㦨gᆠᨨ矄a 矄� a + ᆠᨨ矄 a \
T㦨p
L geff
㦨p
L g ᆠᨨ矄 a
kk Ἐ k +kἘ �㡰
Y, AL
k҈㦨
k
F YA 㦨 l L
YA L
k
m
m
\
ᨨ
k҈㦨
\
T㦨 p
�㡰
YA L
��㡰
��㡰
m k + k҈ 㡰 m 㦨 p keq kk҈
YA ö æ mçk + ÷ m YA + kL㡰 L ø è 㦨 p 㦨 p YA YAk k´ L x 㡰 A㦨L q㦨 L 㝐ᨨ 盐盐矄ᨨ ��� �泐h㦨F⏌Aq \ 矄ᨨ� � ᨨᆠ F㦨hAq㦨hLx ᨨ ᆠᆠ�ᨨ a㦨
keq㦨
ᨨᆠ 㦨k �
ᨨᆠ YA 㦨 � L
ax + a y
Þ
F L Y㦨 A l
㡰 꾀矄矄 ᨨ犨 � ��盐 ᆠᆠ�ᨨ ᨨ犨 a 㦨 gs�na �矄 矄盐 ᨨ ᆠᆠ�ᨨ � 泐 ᨨ矄 � ᆠᆠ 鼀
= ML-҈T -
҈
ᨨ 犨�
k m
�矄 ᆠᆠ�ᨨ ᨨ ᨨ犨 ᨨᆠ├ �矄 ᨨ� 泐 �ᆠ�ᨨ
\ 㝐 �鼀盐鼀 ᨨᆠᨨ �ᆠ�ᨨ
L
ᨨᆠᆠᨨ 矄 ᨨ 矄� �㦨k
geff㦨
MLT -
҈ ҈ ҈ = + keq k k Ἐ
a鼀 = Aw = A æ k ö = KA ç ÷
x 桌
=
b㡰 盐矄ᆠᨨ 矄� 犨�矄ᨨ矄矄� �ᨨ 矄� �ᆠᨨ 矄 k' ᆠ矄ᆠᨨ鼀矄 ᨨ 犨ᨨ矄� �ᆠ � 矄�矄
⼄盐� 矄� � ᆠᨨ 矄
㝐 �鼀盐鼀ᆠᆠ�ᨨ ᨨ 矄泐矄鼀� 和㝐
f鼀 = ma鼀
U 桌
a
㡰 ᆠᆠ�ᨨ �矄矄鼀 ᨨ ᨨ P Q 矄 ᨨᆠ├ Q ᨨ矄ᆠ�矄 盐 ᨨ矄 ᨨ 矄� P Q㡰 矄泐矄鼀 ᨨ矄ᆠ�矄 犨� �盐 ⼄盐 ᆠ泐w 矄� ��矄矄ᆠ 泐A 꾀 �盐 ⼄盐 ᆠ泐ᨨ 矄泐矄鼀
w=
� U x㡰㦨kx
ᨨTµ
l g + ay
T㦨 p
㡰
ᨨ犨 �鼀 �ᨨ 鼀泐 ᨨ T µ 鼀矄矄㡰 x 鼀�盐 㡰y k㡰z \ MLT桌㦨M桌xL桌yML҈T桌z 㦨Mx졔zLyz Tz桌 쟼⼄盐� � ᨨ犨矄犨� z㦨҈ᨨz㦨҈⏌ yz㦨ᨨy㦨z㦨҈⏌ ᆠTµ 鼀�盐 㡰҈⏌ µa҈⏌
� y㦨kt
␔├ ᨨ犨 T㦨 p
�
æ k ö ÷q = - w q èMø
a㦨 - ç
x
q
F hL =x M M
和� ᆠaµx和ᨨᨨ矄ᆠ��ᨨ 矄和㝐 ᨨ�鼀�ᨨ ᨨ 犨 �ᆠ �矄�� 泐
㦨gᆠᨨ矄a ᨨ
T㦨p
�矄ᆠ鼀 x 㦨 p ᆠᆠ�ᨨ
a
T㦨p
M hL
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PS-149
Oscillations
b㡰 ᔼ x�矄 �矄ᆠ鼀 矄ᨨ� � ᨨᆠ㦨 盐 盐矄졔 矄� � ᨨᆠ \
M w x = [ rAg + k ] x
é rAg + k ù Þ w㦨 ê ú ë M û
黨
҈⏌
҈ é rAg + k ù Þ v㦨 p êë M úû
҈⏌
ᔴ泐쟼҈>쟼
㡰 � 鼀 �鼀盐鼀ᨨᆠ�泐v҈㦨v ᨨa҈w҈㦨aw Q V鼀 �鼀盐鼀 㦨 aw Þ
\
æ ö æ 㝐 ö çQ = ç ÷ ҈ ᨨ鼀 ⼄ �㡰 ÷ 㝐 鼀 + è ø è ø 㝐 ҈ ҈æ 㝐 ö 맔 㦨 ç ÷ ҈ 㝐 + 鼀㡰 è㝐+鼀ø
a҈맔
p p = a ´ T҈ T
a҈ T҈ = = a T
m p k҈ m p k
ᔼ 矄 ᨨ盐矄 矄 ᨨ ᆠᨨ鼀� 鼀矄矄矄 㦨
=
k k҈
盐
�盐
ᔴ
��㡰
盐
꾀
�盐
쟼⼄盐���盐鼀ᨨ矄��ᨨ
ᨨᆠ�泐 ᨨ �ᆠ ᆠᨨ鼀矄 �u �鼀 t 矄� � ⼄盐�ᨨ 㦨鼀 ᆠᨨ矄wt �u㦨uᆠᨨ矄wt
├�ᨨᆠ├犨� ᨨ鼀和㝐 ᆠᨨ�矄�ᨨ 矄ᨨ矄� �盐 矄ᨨ �矄 盐 矄ᆠ ᨨ矄��ᨨ ᨨ �矄 �鼀
T m =p = p 㦨҈矄 㦨 p k
ᨨ �矄�鼀㦨t㦨
҈
\
㝐 꾀㦨 꾀 꾀 Þ 㝐+鼀 ҈
㦨
p
鼀 p + ├
㦨
䰘π
鼀 ├
A = A҈
҈ 꾀矄ᨨ 쟼҈㦨 㝐҈
҈ ҈ 㝐졔鼀㡰㦨 㝐졔 鼀㡰
T
鼀 p 鼀 \t㦨 ├ ├
鼀 é҈ ҈ ҈ ù 鼀 π 鼀 + =π 졔 졔 ├ ├ ├ êë úû
�鼀 ├ ᨨ �ᆠ ᨨ 矄矄 ᨨ盐� ⼄盐���盐鼀 ᨨ矄��ᨨ 矄ᆠᨨ �鼀
⧤ 㝐+鼀
M M +m
p 鼀 ├
�鼀├ ᨨ 鼀 �鼀盐鼀ᆠᨨ鼀矄矄�ᨨ
㦨t꾀졔t꾀졔t꾀ᔴ
㝐 ҈㦨 㝐졔鼀㡰 �㡰 㝐 꾀҈ 맔 w҈㡰 㦨 㝐졔鼀㡰 꾀맔w㡰
⧤ 㦨 㝐졔鼀㡰꾀 꾀맔 㝐
t=
�鼀 ├ 泐 �ᆠ ᨨ矄矄 ᨨ盐� ⼄盐���盐鼀
b 㡰 ᔴ矄 �㡰 꾀泐� � � ᆠ� ᨨ ᆠᨨ 矄�ᨨ ᨨ � 鼀ᨨ鼀 盐鼀
㝐꾀҈맔
Þ
�鼀�ᨨ = p
´ ҈ 㦨黨㦨黨鼀
\
p p = ´T t
\
�矄 ᆠ ᨨ ⼄盐� 矄�ᨨ 犨 ᨨᆠ├矄
쟼㦨
V = ҈ Þ㦨�u �u
�.e. �ᆠ ᨨ盐 矄 犨� 矄鼀 矄 ᨨ �ᆠ 犨�盐 ᨨ �矄 ⼄盐���盐鼀 ᨨ矄��ᨨ 犨� 矄 u
v㦨 鼀⏌矄v҈㦨 鼀⏌矄
҈
e=
v҈ + v
Þ v졔v҈㦨 ᨨ鼀⼄矄 �㡰 ��㡰
㦨v҈t㦨
鼀+㝐 ⧤
꾀矄ᨨ꾀㦨꾀҈ 쟼㦨 쟼҈ ᔼ �矄 ᆠ矄 矄ᨨ � 矄 ᨨ盐矄 矄 ᨨ ᨨ ᆠᨨ鼀� 鼀矄矄矄 ᆠ矄矄 㡰 �ᆠ ᆠᨨ� 矄 矄�ᆠ泐 犨� ��� 犨 \
ᨨ鼀 鼀ᨨ鼀 盐鼀 ᆠᨨ 矄�ᨨ � ᆠ� m҈u҈졔mu㦨m҈v҈졔mv 졔҈맔㦨҈v҈졔v �㡰 ᔴᨨ�矄�ᨨ �矄矄�ᆠ ᆠe㦨҈
t㦨
㝐҈ ҈ 㝐+鼀
ᔴ矄 ��㡰 犨�鼀�ᨨ 㦨
黨㡰 ᨨᆠ��矄ᨨ ҈├� ├�ᨨᆠ├矄盐矄 ᆠᨨ�矄�ᨨ v҈ v 矄ᆠ�泐 盐矄 ᆠᨨ�矄�ᨨ
v v҈ ҈├� ├�
e㦨҈㦨
鼀+㝐 ⧤
犨�鼀�ᨨ 㦨
éπ
㦨 ê
ë
鼀ù ú졔π ├û
鼀 æ ö � 鼀 鼀 㦨 π ç 졔 ҈÷ 㦨 π ├ è ø ├ ├
和㝐a �矄 ᆠyᨨ 鼀 ᨨ矄��ᨨ ᨨᆠ�泐ᨨ ᨨᆠ├ v = w a - y 꾀 ᆠ � � ᨨ鼀 矄� � ᨨ犨 犨 ᆠᆠ�ᨨ ᨨ ᨨᆠ├㦨g \ �� � 泐 ᨨᆠ├㦨h
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EBD_8341 PS-150
Physics
(
)
w a - y v ᨨ h = y + \ h= y+ g g dh ᨨhᨨ鼀 �鼀盐鼀 = y = y ã dy dh w \ = ҈+ ( - y ã) ᨨ = ҈ - w y ã dy g g
p 㦨 q
w y ã g = ҈ ᨨ y ã = g w 和� ᆠ aw > g ( �� )
g
ᨨ盐鼀ᨨ ᨨ 㦨LS ␔�� ᨨ ᨨ 㦨LSd҈g 盐矄ᆠ� �ᨨ ᨨ 㦨LSdg 和� ᆠd>d҈ �� 㡰 \ ᨨᆠ ᆠ� � ᆠ ᨨ 鼀矄矄 ᨨ ᨨ � ᨨ矄��ᨨ
LSdgLSd҈g㡰
ᨨ⼄盐ᨨ盐 �矄 ᨨᆠᨨ盐P t㦨F맔r҈㦨 LSdgLSd҈g㡰맔PN
L \ t㦨LSg dd҈㡰맔 矄� q 犨 q �矄矄鼀矄� q » q \
t㦨
L Sg d - d҈ 㡰q
ᨨ ᨨ
LSd 2g
\ q q
L/2
M
\
LSd 1g P N
=
d2
��㡰
L Sg d - d҈ 㡰
Þ Iw 㦨
L Sg d - d҈ 㡰
��㡰
鼀ᨨ鼀 ᨨ � �Iᨨ ᨨ ᨨ盐P I㦨
҈ ҈ ML = LSd҈L
҈ a p q ҈ k ⼄盐 ᆠ泐ᨨ ᆠ = p m ⼄盐 ᆠ泐ᨨ ᨨ矄ᆠ��ᨨ f =
҈ ´ ҈ ҈ = 矄ᆠ -҈ ��㡰 p p ҈ v鼀 ᨨᆠ�泐 鼀 ᨨ矄��ᨨ
L L Sg Sd҈ = d - d҈ 㡰 d - d҈ 㡰 g Sg d - d҈ 㡰 Þ w㦨 Þ w㦨 d҈ L LSd҈ w ´
I a = - kR q 犨 a㦨 �盐ᆠᆠ�ᨨ
æ k ö mR㡰a㦨kRq ᨨ a = - ç ÷ q è mø
㦨黨�맔҈
盐� � �矄盐ᨨ I� ⼄ ��㡰
졠 㝐
ᆠ �盐 ⼄盐 ᆠ泐ᨨ ᨨ矄ᆠ��ᨨ
g d - d҈ 㡰 w㦨 d҈ L �
턼
\ ᨨ矄矄� 矄�ᆠᨨ � �泐㦨�� � ├� �ᆠ �泐 é҈ æ pö ù é҈ ù ê K ç R ÷ ú = ´ ê mv鼀 ú ø úû ë û êë è K Rp ´ = p 鼀⏌矄 㦨鼀⏌矄 \ v鼀 = m é҈ ù ���㡰 ᨨ �泐㦨 ê mv鼀 ú 㦨 ⧤쟼鼀 ᨨ矄��ᨨ 㡰 ë û ҈ é ù 㦨 ê ´ ҈ p㡰 ú ë û
�㡰
和� ᆠt µ q ᆠ鼀ᨨ�ᨨ �矄矄�鼀 鼀ᨨ �ᆠ 졠 ᆠᨨ鼀� ��犨� t㦨C q犨� C㦨
q
ᔴᨨ鼀矄矄�ᨨ 㦨Rq㦨ᨨ ��ᨨ 㦨x \ ᨨᆠ 泐ᆠ 矄� �ᨨ ᆠ 㦨kx \ ᨨ ᨨᆠᨨ ᆠ 盐ᨨ犨ᨨ矄� �矄㦨kx \ 矄ᨨ� �ᨨ⼄盐ᨨ盐ᨨ��� O㦨 kx㡰R \ t㦨k Rq㡰R犨 q㦨꾀 �盐 �矄ᆠ鼀 ᨨ t㦨kRq 和� ᆠᨨ⼄盐 t㡰�矄ᨨᨨ�ᨨ ᨨqᆠ ᆠ盐矄 �盐 和㝐ᨨ盐 ᆠ O 꾀�� t㦨kRq
\ a > y ã y ã ᨨ鼀鼀 ᨨ矄��ᨨ a w g ᆠ yã = w
q R
ᨨ
\ a >
꾀
�㡰 � 盐 � ᨨ 矄� �l㦨p㦨pR 和� �ᆠᨨ 矄 k㦨҈⏌鼀 㝐矄矄ᨨ ᆠ ᨨᆠ├꾀 m㦨҈├� �盐矄ᨨ ᆠ�ᆠ 㦨鼀 꾀矄ᨨ 矄 �矄ᆠ 泐 � q = p ⏌ � 犨� 矄ᆠᨨ �鼀PQᨨ ᆠ�ᆠ 矄 ᨨ鼀矄ᔼ 矄盐矄 � ᨨ ᆠᨨ鼀矄矄�ᨨ ᨨ 矄� � � 盐 矄�鼀
⼄盐ᨨ ��ᨨ ᨨ 矄� �� ᨨ犨矄�鼀 �xPB QA ᨨx� ��盐
ᨨ
q -q �
�
턼 ⧤
鼀�
鼀҈�
꾀 ⼄盐���盐鼀
æ ᨨ⼄盐 ⼄ 盐 ᨨ ö æ ᨨ⼄盐 盐 ᨨ ö ç ÷= ç ÷ è �⼄盐� ᨨ 矄�泐 r ø è �⼄盐� ᨨ 矄�泐҈� r ø mg맔QM㦨m҈g맔PN
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PS-151
Oscillations mg R矄� � 졔 q 㡰㦨m҈gR矄� � q 㡰
\
VrgR矄� � 졔 q 㡰㦨҈�VrgR矄� � q 㡰
Þ
矄� � + q㡰 = ҈� 矄� � - q㡰
Þ
矄� �° ᆠᨨ矄 q + ᆠᨨ矄 �° 矄� q = 矄� �° ᆠᨨ矄 q - ᆠᨨ矄 �° 矄� q
꾀�� �鼀├ ᨨ ᨨ鼀ᔴ ᨨ �㡰
和ᨨ� �犨�
q =
t㦨
\ ᨨ �鼀 ᨨ ᨨ矄ᆠ��ᨨ
㦨t҈졔t졔t졔t 㦨䰘�졔�졔҈�졔�㦨 꾀ᨨ �鼀泐㡰
҈ �
㡰 盐矄 ᨨ犨 �矄ᆠ �⼄盐� 矄 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
ᨨ � ᆠ�ᆠ盐鼀 ᆠᨨ 盐 ᨨ盐� �a \ᨨ⼄盐 t㦨mgR矄� � 졔 q 졔a㡰m҈� 矄� � q a㡰 㦨VrgR矄� � 졔 q 졔a㡰҈�VrgR � q a㡰 㦨 VrgR 矄� q 졔 � 㡰 ᆠᨨ矄 a 졔 VrgR ᆠᨨ矄 � 졔 q 㡰 矄� a ҈� VrgR矄� � q㡰ᆠᨨ矄a 졔҈�rgRᆠᨨ矄 � q㡰矄� a 矄� �⼄ �㡰犨�
m m
1
m
2
2
x҈ 矄�ᨨ � ⼄盐���盐鼀犨 ᨨ m҈ m 矄盐矄 \
m҈ + m 㡰 g k
m҈졔m㡰g㦨kx҈Þx҈㦨
t = V rgR éëᆠᨨ矄 ( �° + q ) 矄� a + ҈�ᆠᨨ矄 ( �° - q ) 矄� aùû
x 矄�ᨨ 犨 ᨨ 泐m�矄
t㦨VrgRᆠᨨ矄 � 졔q㡰졔҈�ᆠᨨ矄 � q㡰桌矄� a 犨 a�矄矄鼀 �� 㡰 \ 矄� a » a \ t㦨VrgRᆠᨨ矄 � 졔q㡰졔҈�ᆠᨨ矄 � q㡰桌a 和� ᆠt a ᨨᨨ�ᨨ 鼀ᨨ�ᨨ �矄 矄�鼀 鼀ᨨ �ᆠ 㝐ᨨ鼀 ᨨ � �ᨨ盐 O I㦨m҈R졔mR㦨VrR졔҈�VrR㦨�VrR
\
ᨨ犨 ᨨ鼀
ᨨ
T㦨 p
A㦨x҈x㦨
w㦨 黨
g
ᆠ鼀
k҈ 꾀
ᔴ
\
t҈㦨
m
=
��㡰
M M = Þ Þ M㦨҈├� 黨 M + M +
ᆠ㡰 �鼀ᨨ 鼀�盐 �矄㦨s 矄� �A㦨Aekt
A = Ae e - k ´ A = Ae e- kt ҈
T m =p =p = 䰘� 矄ᆠ k
�矄 ᆠ = � 㦨 ᨨᆠ�泐 ҈
m =p = ҈� 矄ᆠ k҈ ҈
M + k
Topic-4 : Damped, Forced Oscillations and Resonance
ᨨ犨�鼀├ ᨨ ᨨ鼀ᨨᔴ
t㦨 p
�㡰
�� � �⼄ �㡰泐 ��㡰
ᨨ犨 ᨨᆠ├ 犨� ᆠᨨ鼀矄矄 矄� �
矄� ��� � 矄�矄 ᨨ鼀 � �鼀├ ᔴᨨ꾀
M k M = p k
㦨 p
҈
k m
␔ 鼀矄矄�矄� ᆠ矄 泐├�
k
v
m҈ g k
꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ � 鼀矄矄�矄� ᆠ矄 泐├� �ᨨ � ᆠ矄矄 泐҈ 矄 �鼀 �ᨨ 矄� �
ᨨ
T m 㦨 犨 T㦨p k
ᨨ �矄 ᨨᆠ矄矄 �鼀 ├
��㡰
T㦨 p
҈ö æ çQ q = ÷ �ø è
҈R
m g k
꾀 �盐 ⼄盐 ᆠ泐犨 m҈�矄鼀ᨨ ᨨ 泐m�矄
ᆠᨨ矄 � + q㡰 + ҈�ᆠᨨ矄 � - q㡰桌V rgR T㦨p
ᨨ x҈㦨
�㡰
ᨨ鼀 ⼄矄 �㡰 ��㡰 鼀�盐 ᨨ ᨨ矄ᆠ��ᨨ
和� ᆠ꾀�矄矄鼀ᨨᨨ 和ᨨ ᨨᆠᆠ�ᨨ ᨨ �ᨨ �
�矄 ��ᨨ 矄ᨨ 鼀ᨨ�ᨨ ᨨ鼀 ᔴ ᨨ ᨨ ᔴ �矄 盐 � ᨨ鼀 犨� 矄 ҈鼀⏌矄
鼀矄矄 犨� 矄�├ �� 矄� � ᆠᨨ鼀矄矄 � ⧤쟼 ᨨ 鼀矄矄 犨� ᆠᨨ � ᨨ 쟼 ᨨ 矄� �꾀�� 矄� � 犨�盐 ᨨ�矄 ᨨ鼀矄� 泐ᆠᨨ � ��矄쟼ᨨ⧤쟼 ᨨ ᨨᆠ├
t㦨
kx㦨mg
I C
V r ´ �R 㡰
㦨p
䰘
= � ҈
�㡰 ��㡰
�� � � �㡰泐 ��㡰 矄ᨨ� �犨� t㦨矄 b㡰꾀鼀�盐 ᨨ ��ᨨ �鼀 㦨 �矄 �� 泐 꾀㦨꾀 ҈맔㦨҈맔꾀㦨꾀 矄ᨨ㦨� 꾀 =
꾀
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EBD_8341 PS-152
Physics ҈
҈ Þ = ҈
㦨҈ ;䰘矄 b㡰 ᔼ �矄ᆠᨨ�矄�ᨨ 鼀盐鼀ᨨ鼀 盐鼀犨��鼀 ᨨ矄泐矄鼀 � 矄ᆠᨨ ᆠᨨ�矄�ᨨ 犨 鼀ᨨ鼀 盐鼀 ᨨ 㝐 졔 鼀㡰 �矄 � ᨨᨨ矄� �ᆠ�ᨨ 鼀盐鼀ᨨ鼀 盐鼀ᨨ �ᆠ犨�鼀├�矄鼀ᨨ鼀 盐鼀 ᨨ 졠 ҈ ᆠᨨ�矄�ᨨ 鼀 ® 㝐 + ҈ 錐 㝐 + ҈鼀 ®
E҈㦨
鼀盐 盐 = 鼀盐㦨 㝐졔҈ 鼀㡰 Þ = 㝐 + ҈鼀 ҈� 㦨w꾀 Þ
盐 = ҈�
�
҈ 䰘� ҈ = ҈� ҈
Q 꾀 = 꾀
鼀
E㦨
-
�㡰
-
… ��㡰
꾀㦨䰘黨꾀 \a㦨䰘黨 㡰 r r ᆠ㡰 矄盐 ᨨ 矄矄 a t � a rᆠᨨ鼀ᨨ ᨨ
ᔼ
r
㡰 ᔼ 和㝐 ᨨᆠ�泐ᨨ �ᆠ矄 �ᨨ矄 ᨨ ᆠ矄� � ᨨ鼀 鼀 �鼀盐鼀盐ᨨ ᨨ矄 �ᆠ矄 ᨨ鼀鼀 ᨨ矄��ᨨ
ᨨ 鼀 ᨨ矄��ᨨ
ᨨ � �鼀 ├ ᨨ ᨨ 泐 ᨨ �ᨨ ᨨ鼀 O ᨨ A⏌�矄T҈ ᨨ�ᨨ ᨨ鼀A⏌ᨨA�矄T ᨨ�ᨨ盐矄泐T҈T
FL æ YA ö ÞF = ç ÷l 㡰 꾀矄犨├ ᨨ犨 Y = è Lø Al
a a 㦨n 㦨n ᨨ鼀⼄ 㡰 b R
w҈ w҈ E҈ 㦨 ´ n 㦨 w E w ҈҈
\
꾀 �盐 ⼄盐 ᆠ泐w㦨
n㦨
K YA ᨨ w = m ml
n ´҈ ´ 黨 ´҈ ҈´҈ 黨
E҈ E 㦨 w҈ w
㡰 꾀泐� � ᨨ⼄盐 t = I a
æl ö 矄� q÷ + Mg l 矄� q㡰 = I A a A è ø
ᨨᆠ矄꾀 mg ç
l/
YA æ YA ö Kl = ç ÷ l Þ K㦨 L è Lø
ᨨ ҈ =
�㡰
㡰
꾀矄ᨨ 㦨Kl
\
���㡰
ᨨ鼀⼄ �㡰
ᨨ
x R
w 㦨n w҈
黨
Ew
E҈ w҈ a 㦨 E w R
ᆠᆠ�ᨨ �矄 矄 泐 a
p
ᨨ鼀⼄ 矄 �㡰 ���㡰
҈�㡰 鼀
ᨨ鼀⼄ 矄 �㡰 ��㡰
䰘
҈ mw R
a w 㦨 b w҈
… �㡰
꾀 ҈鼀ᨨ矄ᆠᨨ
꾀 = 꾀
��㡰
p鼀 㦨mv鼀 㦨mwR R㦨mwR Þ mw㦨҈ ᨨ鼀⼄ 矄 ��㡰 �㡰
犨 꾀 㦨鼀 �鼀盐鼀鼀�盐 㡰 꾀ᆠᆠᨨ � �ᨨ ⼄盐矄�ᨨ 矄 � 矄ᆠᨨ
黨꾀 = 꾀 鼀
x
a
ᨨ 矄ᆠᨨ 鼀ᨨ �ᆠ ᨨ矄ᆠ�ᨨ 㝐矄矄㦨 m 꾀 �盐 ⼄盐 ᆠ泐㦨w 꾀鼀�盐 㦨 R 㝐 �鼀盐鼀鼀ᨨ鼀 盐鼀p鼀 㦨R
ᨨ �泐㦨E
b㡰 ᆠ � � �鼀 �ᨨ ᆠᨨ鼀 ᨨ 盐 鼀 ᨨ矄ᆠ�ᨨ � ᆠ矄矄 泐 奠 ᆠ㡰
b
�㡰
҈ a 㦨 b mw҈
⧤ ´꾀 㝐 - ҈鼀
-
҈ mw҈ a
E҈w҈
p鼀 㦨mv鼀 㦨m aw҈ Þb㦨m aw҈
盐� �盐ᨨ 㝐鼀盐 ⧤犨�鼀�盐
꾀=
p
b 㡰 ᨨ �矄 鼀ᨨ �ᆠ ᨨ矄ᆠ�ᨨ 㝐矄矄㦨 m 꾀 �盐 ⼄盐 ᆠ泐㦨w҈ 꾀鼀�盐 㦨 a
ᨨ �泐㦨E҈ 㝐 �鼀盐鼀鼀ᨨ鼀 盐鼀 p鼀 㦨b
�㡰
q
l/矄� q l/
mg
l矄� q
-䰘
Q Y㦨黨맔҈黨�� 㡰 Mg
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PS-153
Oscillations
æl ö ᨨᆠ矄 mg ç 矄� q÷ + Mg l 矄� q㡰 = I B a B ��㡰 è ø ᨨ鼀⼄ �㡰 ��㡰 IA aA㦨IB aB
҈
҈
҈䰘
҈ at ᨨᆠᨨ 矄 ᨨ矄��ᆠᆠ�ᨨ ᔼ �矄ᆠ矄犨
�ᨨ ᨨ矄� � 犨 s矄盐矄t矄矄 ᨨ犨
泐� 矄㡰 㡰�矄盐 ᨨ盐ᆠ盐矄� a�矄 v�矄ᨨ矄�� R S㦨S 졔vt� 㡰
҈
u 矄� q Þ R ¥ u g
㡰 T㦨 2π ҈
l g
E҈ = E
b㡰 ␔ ᨨ矄��ᨨ ᨨ 鼀矄矄�矄ᨨ 鼀 � ᨨ矄��矄� ᨨ鼀ᨨ矄ᨨ� 㡰 鼀ᨨ鼀 盐鼀�矄 ᨨ꾀矄 鼀矄矄鼀ᨨ矄ᨨ犨 矄 鼀 ᨨ矄��ᨨ 鼀ᨨ鼀 盐鼀� ᆠ矄矄 � �� �ᆠ�ᨨ 꾀矄 鼀矄矄 �矄 ᨨ矄�ᆠ�� � � 犨 �矄 鼀�盐 犨� �ᨨ ᨨ
ᆠ矄� � 鼀�盐 犨� ᆠ矄犨� �鼀 ᆠ㡰 �ᆠ 犨� ᨨ ᨨ鼀 ᨨ矄ᆠ��ᨨ 矄 � �泐£ ᨨ쟼>ᔼ E㦨V ᨨ � �泐犨� ᆠᨨ鼀ᆠᨨ 矄 矄 �ᆠ � � �� ᔼ �矄ᆠ矄矄ᨨ �ᆠ 犨� ᨨ ᨨ矄ᆠ� \ E>V \ VE> b㡰 ᨨ � �泐V㦨µx��
T ¥ l
҈黨
㡰
F=
Q 꾀矄V x㡰㦨ᆠᨨ 矄 ᨨx>X \ F㦨 ᆠa㦨 ᨨx>X 꾀ᨨ� 鼀矄矄 m �矄矄盐矄 ᨨ 鼀矄矄矄矄 犨� ᨨ � L �� 㡰 ᨨ鼀 �� 㡰 盐ᨨ⼄盐���盐鼀 T㦨mg �㡰
l
꾀®錐®⼄錐ᔴ®錐®⼄ 꾀㡰 ᨨ 矄�鼀 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ v 㦨 w a - x
졠
ᆠᨨ鼀� ��犨� v㦨 c҈ c - x �矄⼄盐�ᨨ �矄和㝐犨� 犨 㦨c҈ a㦨c 㡰 v㦨kx 犨 x �矄 ᨨ矄��錐 v�矄 矄 x ᆠ矄矄 v ᆠ矄矄
ᨨ ├� �ᆠ �泐犨� ᆠ矄 ␔ x 㦨 v 㦨
ᨨ ᨨᆠ ᨨ矄 ᨨᆠ ��矄 �ᆠ�ᨨ ␔ x�矄 ��v�矄ᨨ矄��盐矄x ᆠ矄矄� 鼀� �盐 v矄ᨨ ᆠ矄矄 ᨨ├� �ᆠ �泐 ᆠ矄矄␔ x㦨 v㦨 ᨨ ᨨᆠ ᨨ矄 ᨨᆠ ��矄 �ᆠ�ᨨ ᔴ㡰 ␔ a㦨 矄� � 矄�ᨨ x k x㦨mg ␔ ᨨ 矄矄 �ᨨ� � 盐犨 矄 犨� ᆠᨨ 矄 ᆠᆠ�ᨨ 矄矄 泐 ᨨ矄� ᨨ ᨨᆠ�矄 矄 \ ma졔mg㦨kxἘ Þ ma㦨k xἘx㡰 和� ᆠa�矄ᆠᨨ 矄 㡰 㡰 ᨨᆠ �矄 ᨨᆠ 犨� 矄 �矄
ᔼ犨� ᨨ ᨨᆠ � �ᆠ�ᨨ ├矄ᨨ 鼀ᨨ� � 犨� ᆠ矄� � 矄 㡰 ␔ �矄 ᨨ犨 盐犨 矄 ᨨ� ᨨ ᨨ犨 졠㡰 � 鼀ᨨ鼀 盐鼀 �矄 � 盐犨 矄 �ᆠ�ᨨ 矄 鼀 �鼀盐鼀 盐 㡰 ᨨ矄��ᨨ �矄 ᨨ꾀矄 �鼀 矄矄矄 鼀ᨨ矄盐犨 矄 �矄鼀ᨨ鼀 盐鼀 �ᨨ矄ᨨ ᆠ矄� � ᨨ矄��ᨨ ᆠᨨ鼀矄 ᨨ矄�� 鼀ᨨ鼀 盐鼀 ᆠᨨ鼀矄 ᨨ ᨨ鼀ᨨ矄 ᨨ� 꾀矄 �鼀 � ᆠ矄矄 矄矄 鼀ᨨ� � ᨨ犨 犨�
� ᆠ矄� �� 鼀ᨨ鼀 盐鼀� ᨨ犨 犨 �ᆠ�ᨨ ��㡰 ᆠ 矄 ᆠ├ᨨ�矄ᨨ��� ᨨ矄��ᨨ
M -
ML T -
L =T
-dV x㡰 dx
T⏌A L l⏌ L YAl Þ T㦨 ��㡰 L
ᨨ鼀
Y㦨
鼀
l
x
ᨨ鼀⼄ �㡰 ��㡰 mg㦨
YAl L
Ἐ
鼀�
���㡰
�� 㡰
�� 㡰
鼀� �� ᆠ㡰
ᨨ鼀 �� ᆠ㡰 矄ᨨ� � ᨨᆠ
é YA l + x 㡰 YA l ù L L úû ë
㦨T' mg桌㦨 ê
�鼀矄
GM e 矄ᆠ矄 Ve㦨 ᨨ ᨨᆠ犨� Re
= ML-T - 桌
҈ m ҈ = A a L
x
=
MLT -
ᨨ犨
T Þ
ᨨ � �泐
a=
u
҈
\
\
ᔴ㡰 R㦨
ᔼ 和㝐鼀ᆠ �ᆠ �泐E µ 鼀�盐 㡰 쟼҈ µ 㡰 쟼 µ
ᆠ㡰
\
和� ᆠ I A > I B \ a A < a B ᆠ wAwB 꾀®錐 ®⼄錐 矄錐 ᔴ ®矄錐 ®⼄ 꾀㡰 ᨨ � �泐 ├ 鼀� �鼀盐鼀 鼀 ᨨ矄��ᨨ 鼀 �鼀盐鼀 鼀 ᨨ矄��ᨨ ᔼ ᆠ矄 ᨨ 和㝐 犨 � ᨨ ᨨ ᨨ � �泐 矄盐矄 �矄ᆠ鼀 � 㡰和㦨盐 ᨨ㦨 ᨨ犨 �矄�� � 矄矄� � ᨨ盐� ᨨ��� S㦨ut졔
҈�
㦨
ᨨ鼀 ���㡰桌
-YAx L
졠 ᆠᨨ鼀� � �矄⼄盐�ᨨ 犨� F 㦨mwx mw㦨
YA YA p = Þ w㦨 Þ L mL T
ᨨ ⼄盐 ᆠ泐 f =
YA mL
҈ ҈ YA = T p mL
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EBD_8341 PS-154
Physics
Waves
14
Topic-1 : Basic of Mechanical Waves Progressive and Stationary Waves ҈
҈ æ gRT ö 錐 C鼀矄㦨 ç C鼀矄µ ÷ è M ø m � é ù ê g mᨨnᨨatᨨm�c = R T ᆠᨨ 矄 ú ë û C鼀矄҈ æ m ö \ = ç C è m ÷ø
㡰 꾀矄 犨 ├ ᨨ犨
w ´ rV V é Bù w êQ K = V = ú rû V ë DP ҈ ҈ 鼀= 鼀鼀 » 鼀鼀 Þ S = » ҈ rV w ҈ ´ ´ ҈ b㡰 � �矄 ᆠ犨 ᨨ ᆠ矄 ᨨ ᨨ盐� 㦨҈�鼀 l = n҈ + ҈㡰 �矄 ᆠ犨 犨ᨨᆠ矄矄㦨�鼀 = n l ҈� n҈ + ҈㡰 = Þ n = ҈n҈ + � � n n҈ n� � ᔼ n҈ = ҈ n = � \l = ҈ n҈ = n = ҈� \l = ҈⏌ n҈ = 䰘 n = � \l = ҈⏌ � ҈ ҈ ҈ ᆠᨨ矄矄�犨 � 矄 鼀 ҈ � b㡰 꾀t㦨x㦨y㦨 f㦨p 㡰 矄� �b㦨҈ 矄矄盐 鼀�盐 DP = aKB = S KB = S ´
�
䰘
æ I ö ᨨ҈㦨҈ ᨨ�҈ ç -҈ ÷ �㡰 è ҈ ø P = 꾀矄ᨨI㦨 ��㡰 pr pr 졠 矄ᨨ� �ᨨ⼄盐�ᨨ 矄犨� r㦨ᆠ鼀 㡰 ᔴᨨ鼀� � �� ⼄盐�ᨨ
泐㦨҈矄� �졔 㡰犨� 矄 ⼄盐�ᨨ 泐㦨矄� w├ 㡰 Þ 犨�矄鼀ᨨ� �ᨨ � 㜠 �矄犨� 矄 w � = � 鼀⏌矄ᆠ = Þ= ⧤ 㡰 � 鼀�盐 㦨҈ᆠ鼀 犨ᨨᆠ�泐㦨 맔鼀 �鼀盐鼀 �ᆠᨨᆠ�泐 � wl = w p p ᨨ l㦨㦨맔҈㦨ᆠ鼀 㡰 �ᆠ ᨨᆠ�泐 vp �矄 ᨨ �矄ᆠ鼀 ᨨ �ᆠ ᨨ鼀 鼀 ᨨ矄��ᨨ 矄 泐 v p = pv A y ævö v p = p ç ÷ A y èlø
p ˆ p 鼀⏌矄 ´ ҈ ( ҈) ( �) = � � 和� ᆠ 犨�矄矄� 盐矄ᨨ� 鼀ᨨ� �� ᨨ矄�� 㜠 �矄 ᨨ� 犨� 鼀ᨨᨨ泐㜠 �矄 ᨨᨨ�ᨨ 矄 ᆠ㡰 㡰盐 ᨨ盐꾀矄 犨鼀ᨨ矄 ᨨ犨 � ᨨ矄��㜠 �ᆠ�ᨨ ᨨ� 矄 ᨨ盐 鼀ᨨ盐犨 矄�� ᨨ矄��Ӹ㜠 �ᆠ�ᨨ
㦨
b㡰 和 ᨨ 矄ᨨ盐 � �矄
鼀矄
黨
҈
҈
dy = - Aw ᆠᨨ矄 kx - w t 㡰 \ v鼀 㦨Aw 㡰 v 㦨 dt 㝐 �鼀盐鼀�ᆠ ᨨᆠ�泐ᨨ �ᆠ� 和㝐v鼀 㦨 Aw xù é b㡰 y㦨y矄� p ê f t - ú lû ë \ ᨨ
dy é x öù æ = y ᆠᨨ矄 p ç f t - ÷ ú ´ pf dt êë l øû è é dy ù = y ´ pf êë dt úû 鼀
� 㝐 �鼀盐鼀 �ᆠᨨᆠ�泐㦨c㦨 f맔T㡰 \
y맔pf㦨 f맔l㡰
l㦨
p y
҈
҈҈
� y =
҈
꾀t㦨犨�y㦨҈犨 x㦨 ҈ 꾀�� y = ҈ + ( x - ҈) 和ᨨt㦨y㦨҈犨 x㦨҈
犨盐矄 矄 �矄 ᆠᨨ ҈鼀� 矄ᆠ ҈ -҈ \ ᨨᆠ�泐ᨨ 犨盐矄 v㦨 = � 鼀矄 �ᆠᨨᆠ�泐鼀�盐 㦨鼀 㦨Aw㦨Apn �ᆠᆠᆠ�ᨨ 鼀�盐 㦨a鼀 㦨Aw 㦨pnA v ҈ T 㡰 ␔ � ᨨ 盐矄 l = = ᨨ µ
f f m ␔ 㦨 矄�ᨨ ᨨ 矄� � ҈> \ l҈>l
ᨨᆠ��矄ᨨ 犨ᨨ盐矄矄ᆠ
ᨨ矄鼀鼀� 㜠ᨨ� 矄 ᨨᆠ�泐 � � ᆠᨨ ⼄盐 �泐 矄 �ᆠ�ᨨ
҈
(҈ + x)
T 矄ᨨ矄 泐ᨨ矄��ᨨ 犨�矄鼀 ᨨᨨ 盐矄矄 m ᨨ�鼀├ 泐ᨨ 盐矄矄犨�矄鼀� 꾀졠㦨 졠꾀 ᆠ 㡰 ᨨ 犨� 矄�泐� �泐ᆠᨨ矄矄� �盐 � 盐 � �鼀�矄ᆠᨨ 矄 ᨨ� 矄 盐 ᨨ 矄 �ᆠ 犨 � 矄�泐 �矄 ᆠ ᨨ鼀 ᨨ� 矄ᨨ盐ᆠ ҈ Iµ r 盐 ᨨ� 矄�y ᨨ 矄 �ᆠ 犨ᨨ 矄 �ᆠ 矄盐 ᆠᆠ 矄ᨨ盐ᆠ鼀� 矄ᆠᨨ 矄 �鼀矄 㦨
҈
ᨨ� 矄ᨨ盐ᆠIµ
҈ 矄 �ᆠ犨�矄 ᨨᨨ 盐ᆠ 泐 � r
矄ᨨ盐ᆠ
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PS-155
Waves ҈� ҈
bᆠ 㡰ᔼ 犨鼀ᨨ�ᨨ y = a ( kx - wt ) yᆠ 矄 ᆠ�ᆠ 鼀� �ᆠ � 矄 � ᆠᨨ鼀� �ᆠ 犨矄 �矄ᆠ鼀 矄矄盐 � 矄ᨨ盐 犨矄 bᆠ 㡰ᔴᨨ鼀� � �� ⼄盐�ᨨ y xt㡰 㦨 = + �㡰 + � � ù é ҈ ê + ú + � û ë ␔� ⼄盐�ᨨ ᨨ 鼀ᨨ� �盐矄 y㦨f x졔t㡰 � -҈ � -҈ 鼀矄 㦨 鼀矄 = 泐 和ᨨ 犨犨� �矄 ᆠ ҈鼀 ᨨ �鼀� 矄ᆠ
҈䰘
҈
( x + �t ) + � = = ҈ 鼀 졠 � \㝐 �鼀盐鼀 �矄ᆠ鼀 �矄҈鼀
矄 ᨨ 盐矄x㦨 t㦨�矄矄矄 ᨨ犨 � ��盐 � �矄 矄泐鼀鼀�ᆠ ᆠ㡰 ᨨ 矄矄 矄� 盐矄ᨨ � 犨 � � ᨨ 矄� � 鼀 �鼀盐鼀ᨨᆠ�泐v鼀 㦨aw
\l =
҈
҈ -
=
= v
v҈ = v
T҈ v Þ ´= T v
(Q V µ T ) ´ ҈ T
´ ҈ = �҈� ´ ҈ N Þ T㦨�҈�맔҈N 㡰 � l㦨ᆠ鼀m㦨�A㦨҈鼀鼀v㦨黨鼀⏌矄 Y㦨 ҈맔҈҈҈鼀 Þ T =
�
¶ y
f ( x t ) = f ( x + l t ) ���㡰 bᆠ 㡰� y㦨҈矄� t졔x㡰 ᔴᨨ鼀� � �� ⼄盐�ᨨ 犨� 矄 犨⼄盐�ᨨ
y㦨a矄� wt졔k x㡰 꾀鼀�盐 a㦨҈鼀錐k㦨鼀҈ 꾀 w㦨 ⏌矄Þpf㦨 \ f㦨 p p p = l = 㦨p 鼀 k w 和 ᨨ 犨㦨 = 㦨鼀⏌矄 k
f x � Tx T 黨 = = \ x = = � f z Tz Tz b㡰 ᨨᆠ�泐 ᨨ 矄矄 犨 � 矄ᆠ 犨� �矄 �� 泐
\
�㡰 ¶t ¶x
犨 鼀ᨨ�ᨨ �矄 ᆠ ᆠ� 泐 犨ᨨ ᆠᨨ ��ᨨ 矄 f ( x t ) = f ( x t + T ) ��㡰
҈ Tx ҈ Tz f z = l m l m
T m ␔ T㦨 矄�ᨨ � 犨� m㦨 � 矄�泐ᨨ 犨�
҈ ҈�p ҈ ´ 㦨҈�鼀⏌矄 Þ l㦨 㦨鼀 㦨 p � �
¶ y
T҈ ├�
v=
p p = ҈p = ҈�p l l
ᨨ
l㦨ᆠ鼀
g l҈ = l҈ = ҈ ᆠ鼀 T҈㦨� ᨨᨨ鼀㡰 g ҈ T 㡰 矄� � f = l m 鼀矄矄 犨 T㦨 矄�ᨨ m = � fx =
҈ p
ᆠ㡰 ᨨ 犨 鼀ᨨ�ᨨ � � ⼄盐�ᨨ �矄 ¶ y æ w ö ¶ y é wù = ç ᆠᨨ 矄 ÷ êQ v = k ú ¶t k ¶ x è ø ë û
T T l = l҈ M T҈
V = l
҈鼀├�
=
p ´ ҈ ҈ = p ´ ҈ - 鼀 꾀 l㦨 = n ҈ ⏌ p bᆠ㡰� ⼄盐�ᨨ y㦨A矄� ҈px졔҈�pt졔p⏌㡰 ᔴᨨ鼀� � �矄 ⼄盐�ᨨ 犨� 矄 ⼄盐�ᨨ ᨨ 犨 � �� X �ᆠ�ᨨ
T
T㦨� ᨨ㡰
p é p ù é p ù t+ x + fú y㦨A矄� ê t + x㡰 + f㡰 ú Þ y㦨A矄� ê l l l ë û ë û
҈黨
ᆠ㡰 矄� � V = f l
n=
҈ = = ҈鼀⏌矄 ҈ ҈ aw㦨҈ Þ ҈맔2pn㦨҈Q w㦨pv桌
Þ n㦨
҈҈ ´ ҈ ´ ҈ = = ҈� �
V҈ V V = Þ l = l҈ l҈ l V҈ 꾀�� 盐矄� �
� 鼀 �鼀盐鼀ᨨᆠ�泐㦨 \
㡰 l҈㦨҈䰘ᆠ鼀 l㦨�ᆠ鼀V㦨鼀⏌矄f㦨䁸 l = l - l҈ 㡰 = ´ � - ҈䰘㡰 = ҈� ᆠ鼀 ᨨ犨 ᨨ鼀 v = f l Þ = l ´ ҈� ´ ҈-
㦨
꾀 x = t = y =
Topic-2 : Vibration of String and Organ Pipe
矄� � v =
T mv ´l ÞT = m I
T DL ⏌ L A Tl mv ´ l ´ ҈ ´ 黨 DL = = = = ´ ҈ m YA l Ӹ꾀㡰 ҈ ´ ҈҈҈ ´ ҈ 㦨鼀鼀
꾀�� ᨨ鼀 Y =
b㡰 ␔矄 㦨
m
犨 ᆠ�矄矄㦨
\ =
㝐� m
和�鼀�泐犨 ᆠ�矄鼀ᨨ� �犨� ᆠᆠ�ᨨ
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EBD_8341 PS-156
Physics
(
㝐 � 졔 � 㦨
)
鼀 ´ ´� ��㡰 l ᨨ鼀 ҈㡰 㡰 = � l
҈
m
Þ � =
矄�ᨨ =
� 졔 �
� 졔 æ � ö 㦨҈졔 Þ � 졔 㦨 � 졔 � 맔 çè ҈졔 ÷ø 㦨 � � 㦨� 㦨 犨 �ᆠ �矄ᆠᨨ矄矄ᨨ�⏌�桌 䰘
b㡰
-҈⏌ = �
黨
҈
҈
꾀 l = Þ l = l 㦨҈鼀 Q 㦨ᆠ鼀㦨鼀 �� 㡰 矄� �㦨 l
l⏌
҈�
꾀
Þ f =
҈
=
犨 m =
T mù m = ú lû m
Tl T Y Dl Þ = ADl A l
҈ l
y Dl lr
�㡰
盐� � 盐ᨨ l
Dl r y� ⼄ �㡰犨� l
҈ ´ 䰘 ᨨ f » ҈䰘 㡰 ᨨ � ᨨ 犨�㦨҈҈ᆠ鼀
҈ 㦨҈ ҈ � 矄ᨨ 矄 f =
҈ Qr㦨 r҈
꾀矄 µ
҈
҈ ҈ ҈ = ҈ ҈
\
҈㦨
\
æ ҈ ö ҈ = ç ´ ҈҈÷ = 䰘 ᆠ鼀 è ҈ + + ø
�㡰
鼀 鼀矄矄 ᨨ 矄� � = � ᨨ 矄� �
é êQ v = ë
Dl 㦨҈ l r 㦨䰘䰘맔҈ ├�⏌鼀 �� 㡰 y㦨맔҈҈҈⏌鼀 �� 㡰
㡰 ␔├ ᨨ犨 ᨨᆠ�泐� 矄� ��矄 �� 泐
m
T Ar
l 㦨҈�鼀
l p r 꾀矄 l矄鼀 ᨨᨨ 犨�矄
҈㦨 ҈ ҈ =
v ҈ T ҈ = = l l m l
꾀矄ᨨ Y =
=
Þ
l
盐 鼀 ⼄盐 ᆠ泐� ᆠ矄 㡰�矄 = = f '㦨 l ⏌ 㡰 l ᆠ㡰 � ᨨ �㦨�ᆠ鼀㦨�鼀 ⼄盐 ᆠ泐ᨨ ᨨ矄ᆠ��ᨨ 矄 ᨨ �ᆠᨨ盐鼀 � ᆠᨨ矄 ᨨ� � �矄 �� 泐 n - ҈㡰u n - ҈㡰u f = £ ҈� Þ f = L L - ҈㡰 ´ £ ҈� Þn҈҈�» Þ � ´ b㡰 盐 鼀 ⼄盐 ᆠ泐
f =
�� ´ ҈ Þ = = l ҈ 㦨맔҈ ; �⧤ b㡰 ҈㦨
®和鼀 ®和鼀 l®和鼀 ⼄盐 ᆠ泐ᨨ ��ᨨ
҈ = r҈ r
㡰
盐 鼀 ⼄盐 ᆠ泐� ᆠ矄 㡰�矄 =
҈
Ӹ 黨䰘 ´ ҈҈ = r 䰘 ´ ҈ 㦨��맔҈鼀⏌矄ᆠ 和� ᆠ ᨨ �矄 ᆠ鼀 鼀� 盐 鼀 犨 矄 �矄 矄 ᨨᨨ犨
l 㡰
( - ҈) =
=
( - ҈) ҈ l 犨 ҈㦨 盐 鼀 ⼄盐 ᆠ泐ᨨ ��ᨨ 㡰 ᆠ㦨 ҈㡰맔҈� Þ 㦨䰘҈»䰘 \ 盐鼀ᨨ ᨨᨨ 矄㦨 ᨨᨨ 鼀ᨨ ᨨ ��ᨨ 㡰҈ 㦨䰘҈㦨 㡰 ᔼ 矄ᨨ� 矄 ᨨᆠ�泐ᨨ 犨
l = = � ҈
f
b㡰 l
l VA u r = B = B = = A Þ l = l Þ P = ҈ A B lB VB u A rA q 㡰 ᔼ ᆠᨨ矄 � ��ᨨ � 鼀ᨨ ⼄盐 ᆠ泐 ᨨ
=
҈҈
= �´ \ =
= 鼀⏌矄ᆠ
= ᆠ㡰 꾀矄 鼀盐矄 ᨨ ᨨ 矄 ᨨ� ᨨ 犨�A B矄ᨨ
��ᨨ
Þ
꾀矄ᨨ f҈ =
l
\ ò -҈⏌ - � ò
l = ᨨl = m
ᨨᆠ�泐v㦨 f l = ´
l
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PS-157
Waves æ ö = ç ´ ҈҈÷ = ᆠ鼀 è ҈黨 ø
҈䰘
҈
҈
æ ö = ç ´ ҈҈÷ = ҈ ᆠ鼀 è ҈黨 ø ᆠ � �矄矄 ᨨ盐 ᆠ 䰘ᆠ鼀 䰘졔㦨黨 ᆠ鼀 ᨨ鼀ᨨ b㡰 ᔴᨨ 矄� � � ᆠᨨᆠ�ᨨ e 㦨 D 犨 㦨 �鼀桌 é v ù f = n ê l + e ú ᨨ �矄矄ᨨ ᆠn㦨҈ )û ë ( v v \f㦨 ( l + D ) Þ l = f D æ ´ ҈ ö ᨨ l = ç ÷ ´ = ҈�ᆠ鼀 è ´ �҈ ø 㡰 ⼄盐 ᆠ泐 ᨨ 鼀ᨨ �ᆠ ᨨ 矄� � 㦨 盐 鼀 ⼄盐 ᆠ泐ᨨ 盐ᆠ � �
㡰 ⼄盐 ᆠ泐ᨨ 鼀ᨨ �ᆠ ᨨ ᨨ � v v f҈㦨 = l l ⼄盐 ᆠ泐ᨨ 鼀ᨨ �ᆠ ᨨ ᆠᨨ矄 �
v nv = l l n�矄ᨨ 盐鼀 ᨨ鼀⼄ �㡰 ��㡰
��㡰
f㦨
f㦨
�㡰
n f҈ ᨨ �矄矄ᨨ ᆠn㦨� � f҈ ᔼ 矄ᨨ ᆠ盐
\f㦨
ᆠ㡰
l҈ 졔
l 졔
�鼀
ᨨ �矄矄ᨨ ᆠ
l l҈ + e = 盐 㦨nl \ 㦨n l + e㡰
鼀
æv ö v ç ҈ ÷ = è l҈ ø Al é ҈ Tù v \ ´ê ú= ë l҈ m û l
҈黨
҈ � \ = Þ m㦨├�鼀҈ � m ´ ᆠ 鼀矄矄ᨨ 矄� �m҈㦨ml҈ 㦨맔�├�㦨҈� 㡰 ᨨ 鼀� 矄 ᨨ 矄ᨨ盐 盐矄� �矄ᨨ ᆠᆠᨨ盐鼀 ᨨ �矄 ᨨ 盐 � � ᨨ├├� �ᆠ 꾀矄矄 ᨨ犨 � ��盐 � �矄ᨨ 盐 � � ᨨ├├� �ᆠ
㡰
⼄盐 ᆠ泐ᨨ �矄 鼀ᨨ �ᆠ� 꾀㦨f҈㦨
⼄盐 ᆠ泐ᨨ 矄ᆠᨨ 鼀ᨨ �ᆠ� ᔴ㦨f
㦨n l҈ 졔e㡰
\
l + e =
l
v l҈ + e = n
Þ l + e =
�㡰
Þ
��㡰
和盐ᆠ� � �㡰 ��㡰
= n l - l҈ 㡰 \ D㦨n Dl + Dl҈ 㡰 㦨맔�҈맔 ҈졔҈㡰ᆠ鼀⏌矄㦨ᆠ鼀⏌矄 ᆠ鼀 �鼀盐鼀ᨨ矄矄�ᨨ� 矄 Dv㦨ᆠ鼀⏌矄 b㡰 ⼄盐 ᆠ泐ᨨ �矄 ᨨᨨ � ᆠᨨ矄 ᨨ� �
v P l҈ r҈ ⼄盐 ᆠ泐ᨨ �矄 ᨨᨨ � ᨨ ᨨ� �
n㦨
҈ P l r v㦨vἘ nἘ㦨
҈ T l m
v P ҈ = l҈ r҈ l
҈ T l m
f҈㦨f �� 㡰
҈ T҈ ҈ T = l m l m ᨨ ҈㦨 �㡰 쟼⼄盐� � ᨨ⼄盐矄 盐 ᨨ T ҈ T ᨨ盐 O ᨨ ᨨ�ᨨ ⼄盐���盐鼀 꾀 \T҈x㦨T Lx㡰 ᨨ 矄�ᨨ ⼄盐���盐鼀 T҈졔T㦨鼀�
҈ ᨨ鼀� �㡰 ���㡰 mg mg 졠 T҈㦨 㦨 � � 㜠 ᨨ犨 ᨨ鼀⼄ ��㡰 \
mg L mg ´x= ( L x) Þ x = L x \ x = � � �
ᨨ矄ᆠᨨ 矄ᨨ ᆠ
P r
r l҈ ҈ r
\
l =
㡰
盐 鼀 ⼄盐 ᆠ泐f =
P T l m
� 黨g Mg = Þ M㦨�├� l Ҙ l Ҙ 꾀矄 ⼄ 盐 ᆠ泐 �矄 ᆠᨨ矄ᨨ 矄 ᨨ � 鼀ᨨ �ᆠ P㦨� P㦨矄ᆠ�泐 ҈ æ T ö ҈ æ T ö 㡰 n҈㦨 ç ÷ n㦨 l ç ÷ l è pr rø è pr rø 㦨
�
n㦨
鼀
\
v ҈ = l l
T l 犨 㦨 � ᨨ 矄� �桌 m
҈ n҈ = ´ =҈ n
é ù 鼀矄矄 r ´ A ´ � = r Aú êQ m = � = � ë û
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EBD_8341 PS-158
Physics
㡰 ᨨbᨨh ᨨ ᨨ� � 盐 鼀 ⼄盐 ᆠ泐
ᨨ ᨨ ᨨ� � l
c �㡰 l ᨨᨨ ᆠᨨ矄 ᨨ� � ᨨ � 鼀ᨨ �ᆠ ⼄盐 ᆠ泐 æ cö n㦨 çè ÷ø ��㡰 l ���㡰 � nn҈㦨҈ ᨨ鼀⼄ �㡰 ��㡰 n ⏌ = = Þ v҈ = v �㡰 n҈ ҈⏌ 꾀�� 矄ᨨ� �⼄ ���㡰 �㡰犨�n҈㦨 ᆠ㡰 ᔴᨨ鼀� � �� ⼄盐�ᨨ y x t㡰 㦨 ᆠᨨ矄
n2㦨
n҈ =
䰘
pö æ çè �pt + ÷ø ᆠᨨ矄 ҈px㡰 y xt㡰㦨Aᆠᨨ矄 wt졔p⏌㡰ᆠᨨ矄k x ᔼ k x㦨p⏌ ᨨ ᨨᆠᆠ盐矄錐 \ ҈p x㦨p⏌Þ x㦨�鼀 ᔼ k x㦨p � ᨨ ᨨᆠᆠ盐矄 Þ ҈px㦨pÞ x㦨҈鼀 꾀矄ᨨ 矄 ᨨ 犨 w �p = � 鼀⏌矄 l㦨p⏌k㦨p⏌҈p㦨鼀 v㦨 = k ҈p 㡰 In a�rT㦨鼀�㦨rVg T ҈ rVg = \ f㦨 �㡰 m l m 犨 ᨨᆠ�矄 �鼀鼀矄 � 犨 T㦨 mg盐 盐矄 V Vg 㦨Vrg rw g = r - rw 㡰 Vg r - rw 㡰 ҈ ҈ Vgr r - rw 㡰 \ f Ἐ㦨 㦨 l m l m r r - rw fἘ æ r - rw ö = f Ἐ = f ç f r è r ø÷
l
Q
҈
n1㦨 ᨨ �矄 鼀ᨨ �ᆠ l҈ l1
l ҈ = Þ ҈ = l҈ l l ᨨ盐ᆠᨨ矄
n҈ = n \
\ l = l
lἘ = l 盐 鼀 鼀ᨨ 㡰 c c \ n㦨 = = �҈ l l nἘ㦨
\ l Ἐ = l �� 㡰
c c æ cö = = ç ÷ 㦨맔�҈㦨҈ è l ø l Ἐ l
ᨨ ᆠᨨ矄 ᨨ� � æ vö f = n ç ÷ 犨 n = ҈� è ҈ø \ l =
nv f
ᨨn㦨҈ l҈ =
(҈)( )
´ ҈ ᆠ鼀 = ҈� ᆠ鼀 ´ ᨨn㦨 l = l҈ = 黨䰘� ᆠ鼀
ᨨn㦨� l� = �l҈ = ҈�� ᆠ鼀 �㡰 �矄 ᆠ 犨 犨ᨨ 矄盐ᆠᆠ矄矄� ᨨ 矄
� ´ T ⏌m v l = = D㦨 = f f �
҈⏌
黨
2
l = l 盐 鼀 鼀ᨨ 㡰 ᨨ ᨨ 盐
҈⏌
é r - ҈ ù 㦨 ê ú ë r û ᆠ㡰 和�ᨨ 泐犨 �矄 ᨨ 盐ᆠ 犨 犨ᨨ 犨矄 �
ᨨᨨ矄� �ᆠ�ᨨ ᨨ犨y㦨aᆠᨨ矄 k xwt㡰aᆠᨨ矄 k x졔wt㡰 \ y㦨a矄� kx矄� wt�矄⼄盐�ᨨ ᨨ 矄�ᨨ 泐犨犨 �ᆠ ��矄 ᨨ x㦨 y㦨aᆠᨨ矄 kx wt㡰ᆠ
ᨨ矄 � �ᆠ�ᨨ 矄 ᨨᨨᨨ矄� l ᆠ㡰 ␔ 盐犨矄ᨨ ᨨ 矄 = l v v \ f㦨 = l l lἘ l 犨 盐�矄 � � 犨 = Þ lἘ㦨 l v v = f \fἘ㦨 = l Ἐ l ᆠ㡰 ᨨ ᆠᨨ矄 ᨨ� �
㡰
ᨨ � 鼀ᨨ �ᆠ l
҈ ´ ҈
=
҈ = �ᆠ鼀
�㡰 � �矄泐ᨨ 犨� s = 黨 ´ ҈ - ├�ᆠ鼀 Ӹᨨ盐 �Ἐ矄鼀ᨨ 盐盐矄ᨨ 犨�Y㦨黨맔҈҈鼀 和� 㦨黨맔҈ 和矄矄 T ⏌ A Y= = 和� 和�
T \ = Y ´ 和� 㦨黨맔҈黨맔黨맔҈ A 꾀矄ᨨ鼀矄矄ᨨ 犨� m = Al s m 㝐矄矄盐 � � m = = As J 盐 鼀 ⼄盐 ᆠ泐� 矄� �
f = =
҈ T ҈ T = l m l sA ҈ 黨 ´ ҈黨 ´ 黨 ´ ҈ - ´҈ 黨 ´ ҈
҈ ҈ 黨 ´ ҈黨 - - = ´ 䰘 = � ᨨ㡰 l҈㦨 � �� � ᨨ � l㦨 犨 � ᨨ � 㦨和 ᨨ 盐 � � ᨨ├ =
ᔼ ᆠᨨ矄 ᨨ� � =
l҈
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PS-159
Waves æ ␔ 盐 � � ᨨ├�矄鼀ᨨ Ἐ = = ç è -
Þ
æ ö ç ÷= l҈ è - ø l
Þ
w 㦨҈鼀矄҈ ├ ᆠ ᨨ�ᨨ 矄 㡰 ᆠ㡰 㡰 矄�矄 泐 ᨨ ᆠᨨ ��ᨨ 矄 bᆠ㡰 泐㦨҈矄� 㡰桌ᆠᨨ矄 㡰� 桌
ö ÷= ø l
和 ᨨ 犨 v =
- l = l҈
㝐
-
-
l l -l -҈ = -҈ Þ ҈ = l҈ l҈ ᆠ �ᆠ �⼄盐� � � ᨨ � l - l҈ - ´ ҈ = ´ ҈ = -�奠 l҈ ᆠ矄鼀矄盐ᨨ ᆠ �ᆠ �⼄盐� � � ᨨ ��矄�奠 ᔼ ��盐 㡰ᨨ 矄ᨨ 盐 � � Þ
䰘
l = l Þ l㦨 l c c \f㦨 = l l ᔼ ��盐 㡰犨 ᨨ 盐 �ᆠ泐 � �
l 犨 lἘ l = Þ l Ἐ = l c c = f \ fἘ㦨 = (a) l Ἐ l ҈ T 盐 鼀 ⼄盐 ᆠ泐f㦨 ᨨfµ T l m
l/2
(b)
f µ 犨�� ᨨ 鼀矄矄 ( w ) f Ἐ µ 犨盐 盐矄 ( F )
u ( x y ) = x = L y = L [Q 矄� p = 矄� p = ]
fἘ wF w F = ᨨ f Ἐ = f f w w 和盐矄�盐� � 盐矄犨� \
f Ἐ = 黨
( �䰘) g - ( 䰘�) (҈ ) g ( �䰘) g
졠�ᨨ 㡰 u ( x y ) = y = y = L [Q 矄� = 矄� p = ] u ( x y ) ¹ x = x = L [Q ᆠᨨ矄 = ҈ ᆠᨨ矄 p = ҈]
= Hz
和 ᨨ矄 �矄 ᆠ ᨨ鼀 犨 犨 �ᆠ � �ᆠ矄 � = ҈�鼀 鼀 �鼀盐鼀鼀�盐 ᨨ ��ᨨ 㦨l⏌㦨 c = � 鼀 Q c㦨vl \l㦨 = v bᆠ㡰꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ � ᨨ �ᆠᨨ盐鼀 �矄 � 泐 ᆠ �� � ᨨ 犨 � 矄ᨨ ᆠ 盐 l 矄ᨨ n졔҈㡰 㦨�䰘졔e �㡰 l n졔㡰 㦨黨졔e ��㡰 �� � �⼄ �㡰泐 ��㡰 ᔼ n㦨҈ Þ \
l ⏌ �䰘 + e = �l ⏌ 黨 + e
e㦨҈
p p = ҈鼀 㦨\ l = l l ҈ = � 鼀 � ᨨ 矄� �l㦨� ´ = � ´
鼀� ᨨ� 㝐 �矄 � ᨨ 矄 鼀 �鼀盐鼀 �矄ᆠ鼀 㦨҈鼀 ω⏌├
盐 鼀 ⼄盐 ᆠ泐v㦨 = l l = = ´ � ´ bᆠ㡰 和 � � 犨矄 ᨨ 盐ᆠ 泐 犨ᨨ� �ᆠ 犨矄 矄盐ᨨ矄� � 犨 � � � � ᨨᨨ矄� �ᆠ�ᨨ bᆠ㡰 �矄ᨨ y ᆠ鼀 矄ᨨ �矄 �矄ᆠ鼀 矄 ᨨ � x 泐 矄 犨� Q L L㡰 R L㡰 � �� 盐泐 ᨨ �矄 졠�ᨨ 㡰 u ( x y ) = x =L y =L P x u ( x y ) ¹ x = y = 㡰 L㡰 졠�ᨨ 㡰 u ( x y ) = x = y = [Q 矄� = ] u ( x y ) = x = L y = L [Q 矄� p = ] 졠�ᨨ ᆠ㡰 u ( x y ) = x = y = [Q 矄� = ] ᨨ鼀 �� ⼄盐�ᨨ ├㦨
�
æ vö b 㡰 ᨨᆠᨨ矄 ᨨ� �v㦨n ç ÷ 犨 㦨҈�䰘 è l ø v v �v \ n㦨 㦨 l l l ҈
ᆠ㡰 ⼄盐 ᆠ泐 n = ᨨ �矄 鼀ᨨ ᨨ ��ᨨ
l 鼀 ᨨ�nᨨ鼀 �鼀盐鼀�l矄 ᨨ盐 鼀� �鼀盐鼀 和� �㜠҈ =
҈
m
和� �㜠 =
҈
= m
和� �㜠 =
҈
= m
和� �㜠 =
҈
= m
\맔黨졔e㦨�맔�䰘졔�e
l 㦨�䰘졔黨㦨�҈
\e㦨黨ᆠ鼀 Þ l㦨ᆠ鼀㦨鼀
꾀矄ᨨ矄 ᨨ 矄ᨨ盐 㦨vl㦨�맔鼀矄҈㦨鼀矄҈ ҈ ᆠ 㡰
矄 ᨨ盐 �矄ᆠ鼀 ᨨ x㦨 �矄ᆠ鼀 � ᨨ x㦨鼀 ᨨy㦨x㦨 y = ± A x㦨鼀
䰘
b㡰꾀矄 n =
n l 鼀
\ = l 鼀
m 和� �㜠҈ = l
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EBD_8341 PS-160
Physics ҈ ҈ dE㦨 dm㡰 A w = µdx 㡰 a 矄� k x 㡰 pn㡰 㦨pҘna矄� kxdx Q 㦨nl T ⏌Ҙ éQ = T ⏌ m ù \ n㦨 Þ n㦨 = ë û l l l
和� �㜠 =
æö ç ÷ ( m ) èø
( )
=
æ�ö ç ÷ ( m ) è ø 和� �㜠 = = ҈ �
\
和� �㜠 =
æ䰘ö ç ÷ ( m ) èø
(҈)
\
= ҈
l
E㦨 ò dE = ò p Ҙ
꾀㜠錐 㜠矄錐ᔴ㜠⼄矄錐 㜠⼄ 꾀㡰�ᆠᨨ矄 ᨨ ␔矄ᨨ 盐ᆠ ᨨ ��盐 �
= L
�҈
lf
\ l 㦨 㡰 �ᨨ bᨨh 矄 犨矄ᨨ 盐ᆠ ᨨ ��盐 �
L
lf
�
= L
\ l 㦨
lf
黨
vA = ᨨr ᨨ vB
= L
lf
\ l 㦨 㡰 和ᆠh 犨ᆠ鼀 bᨨh 矄 鼀� ᨨ� ␔矄ᨨ 盐ᆠ 矄矄 � 盐
ᨨ
l 㦨
æ � ⏌ ö æ ҈ ö ÷ç ÷ 㦨 çè 䰘 ⏌ �ø è 黨 ø
lf
黨 aw = Þ w㦨 ⏌矄 �㡰 aw 和盐矄�盐� � 盐ᨨ w� ⼄ �㡰犨� = ҈鼀 a㦨 㡰 p p w = = = = ᨨ犨k㦨 �㡰 ⏌ n l ᨨ犨盐� � 盐ᨨ a w k犨�犨 ᨨ鼀 y㦨҈矄� t᠀ x᠀f桌
矄� � �矄 � 盐 鼀 鼀ᨨ ᨨ l =
p p p = 㦨 l l l 꾀鼀�盐 ᨨ ��ᨨ �矄 ᆠx ᨨ鼀x㦨A㦨a矄� kx 㝐ᆠ �ᆠ �泐xᨨ � dx�矄 ᨨl㦨 l 和� ᆠk㦨
gA gB
g A RTA lB g B RTB MB
=
(Q l A = lB )
MB M g æ ҈ ö = ( 矄TA = TB ) Þ A = A ç ÷ MA MB gB è 黨 ø
� 䰘ö æ çè g A = g B = ÷ø �
M A æ � ö æ ҈ ö =ç ÷ç ÷ = M B è ҈ ø è 黨 ø ҈黨 㡰 �ᨨᨨ 盐 鼀 ⼄盐 ᆠ泐� � A � � B, 犨 �ᆠ �矄 ᨨ犨ᆠᨨ矄 ᨨ 矄
矄 犨 ᨨ鼀ᨨ 矄矄 鼀ᨨ �ᆠ �矄盐 ᆠ y㦨a矄� wt᠀kx᠀f㡰 � 鼀 㦨aw㦨鼀⏌矄 �㡰 A鼀 㦨aw 㦨黨鼀⏌矄 ��㡰 ᨨᆠ�泐ᨨ 犨v㦨鼀⏌矄 ���㡰 �� � �⼄ ��㡰泐 �㡰
\
f A v A ⏌ l A v A = = f B vB ⏌ lB vB g A RTA MA g A MB = 㦨 g B RTB gB MA MB
�
p Ta l ᔼ 盐 鼀 鼀ᨨ v l l + r 㡰 = = Þ v㦨f l + r 㡰 f r㦨 㜠ᆠᨨᆠ�ᨨ � 盐 r 㦨 �盐矄ᨨ 盐㡰㦨�ᆠ鼀 l 㦨 � ᨨ 盐㡰㦨҈ᆠ鼀 f 㦨 ⼄盐 ᆠ泐ᨨ 盐 � � ᨨ├㡰㦨 \ᨨᆠ�泐ᨨ 矄ᨨ盐 � �v = 鼀⏌矄 㡰 ⼄盐 ᆠ泐ᨨ 矄ᆠᨨ 鼀ᨨ �ᆠ� �Aᨨ ᨨ 矄 㦨 ⼄盐 ᆠ泐ᨨ � 鼀ᨨ �ᆠ� �Bᆠᨨ矄 ᨨ
æ vA ö æ vB ö \ ç l ÷ = ç l ÷ è Aø è Bø
ᆠ㡰 和ᆠh 犨�ᆠ鼀 bᨨh 矄 ␔矄ᨨ 盐ᆠ 矄矄� 盐
lf
T ⏌Ҙ æ p x ö a 矄� ç ÷ dx è l ø l
ᨨE 㦨
L
lf
T ⏌ Ҙ ìæ p ö ü a 矄� íç ÷ xý dx l îè l ø þ
ᨨ �泐ᨨ 矄� �
dE㦨pҘ
( 矄l A = lB ) (Q TA = TB )
M B ҈黨 � 䰘 和盐矄�盐� � M = g A = g B = A
fA � ҈黨 = ´ = ҈ fB 和 ᨨ 矄ᨨ盐 v㦨鼀⏌矄 l � ᨨ � ᆠᨨ盐鼀 ᆠᨨ矄ᨨ � � ᨨ 盐 鼀 ⼄盐 ᆠ泐
v v = ҈� Þ l 㦨 = = 鼀 l ҈�㡰 ҈�㡰 \
l
�
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PS-161
鼀
Waves
鼀
鼀
鼀
҈鼀
鼀
鼀
҈鼀
l
矄矄盐
ᨨ 㡰 �矄ᆠ鼀 � ᨨ x㦨 矄矄盐��ᨨ
�矄 ᨨ
㡰 꾀 泐 ᨨ矄��ᨨ x 矄矄盐 DP㦨DPᆠᨨ矄kxᆠᨨ矄wt
ᔼ ᆠᨨ矄 � ᨨ 泐 ᨨ 鼀ᨨ �ᆠ矄 ᨨ� ᨨ犨 l҈ l l l ᆠ � 矄 ᆠᨨ矄ᨨ � � ᨨ 鼀ᨨ �ᆠ� 鼀ᨨ �ᆠ䰘 鼀ᨨ �ᆠᆠ
鼀�盐 A㦨DPᆠᨨ矄kx㦨DPᆠᨨ矄
҈� ҈� L = 鼀 鼀� ᨨ� 㡰 㦨 ´ ҈ é p ҈� ù DP ´ ú= 꾀鼀�盐 㦨DPᆠᨨ矄 ê ⏌ 㡰 û ë ᆠ㡰 꾀ᨨ ᨨ �矄矄盐�矄犨泐矄矄鼀�⼄盐ᨨ
æ v ö 䰘ç ÷ 㦨҈� Þ l 㦨鼀錐 è l ø ��
\
æ - dH ö a gH 㦨 꾀 ç è dt ÷ø 和ᨨᨨ ᨨ 犨� �
æ - dH ö a gH çè ÷= dt ø A 和盐矄�盐� � 盐矄犨� ´ ҈ ´ H
- dH 㦨 ҈҈҈맔҈㡰 H dt 犨 �矄 犨ᨨ 矄ᨨ ᆠ矄 犨 矄 ᨨ鼀 鼀ᨨ鼀
Þ
�
\
dH
Þ
ò
Þ
- 桌 = - ҈҈ ´ ҈- 㡰t
ᨨ 㡰
ᨨ 矄ᆠᨨ ᨨᨨ ᨨ ᆠᨨ矄 �
H
= -҈҈҈ ´ ҈ - dt
dH H
t
= - ҈҈҈ ´ ҈- 㡰ò dt
t » s
æ vö æ ö ᨨ = � ç è L÷ø è L ÷ø � ´ ҈� Þ L㦨 Þ = 鼀 ´ ҈
⼄盐 ᆠ泐 f = P ç
鼀 矄矄盐 Q DP = P鼀 = P鼀� = P 㡰 꾀 ᆠᨨ矄 ᨨ � 㝐 �鼀盐鼀矄矄盐鼀 㦨P졔DP 㝐� �鼀盐鼀矄矄盐鼀� 㦨PDP px ᆠᨨ矄 ( 黨pt ) �㡰 y㦨矄� ҈� æ p xö 鼀�盐 A㦨矄� ç ÷ è ҈� ø At x㦨�ᆠ鼀 꾀鼀�盐 ᨨ 鼀 �鼀盐鼀 �矄ᆠ鼀 æ p ´ �ö A㦨 矄� ç 㦨맔㦨ᆠ鼀 è ҈� ÷ø ��㡰 ᨨ 矄 ᨨ矄��ᨨ 犨 A㦨
æ px ö x㦨҈�
矄� ç ÷ = = 矄� np \ è ҈� ø 犨 n㦨҈ x㦨҈�ᆠ鼀ᆠ鼀ᆠ鼀 ���㡰 꾀x㦨䰘�ᆠ鼀t㦨�ᆠ鼀ᨨᆠ�泐ᨨ �ᆠ dy æ p xö = 矄� ç v㦨 黨p矄� 黨pt㡰桌 è ҈� ÷ø dt æ p ´ 䰘� ö v㦨 矄� ç 黨p矄� 黨p맔�㡰桌 è ҈� ÷ø æ pö 㦨 矄� ç ÷ - 黨 p 矄� p㡰桌 㦨 è ø æ p xö �㡰 y㦨 矄� ç ÷ ᆠᨨ矄黨 p t 桌 è ҈� ø é ù æ p xö 㦨 ê 矄� ç ᆠᨨ矄 黨 p t 㡰ú è ҈� ÷ø ë û é æ p xö p xöù æ 㦨 ê矄� ç 黨 p t + ÷ø - 矄� çè 黨 p t ÷ è ҈� ҈� ø úû ë p xö p xö æ æ 㦨 矄� ç 黨 p t + ÷ - 矄� çè 黨 p t ÷ è ҈� ø ҈� ø 㦨y҈졔y ᆠ ᆠᨨ鼀ᨨ 矄 犨矄 p xö y҈㦨矄� æç 黨 p t + ÷ è ҈� ø
ᨨᆠ�泐ᨨ 盐 v㦨 gH ᔴᨨ � 盐�泐⼄盐�ᨨ ҈ ��矄
- dH ҈ ´ ҈ - = dt ҈ ´ ҈-
p x l
ᨨ x =
æ v ö ç ÷ 㦨҈� Þ l҈ 㦨҈鼀錐 è l҈ ø æ v ö �ç ÷ 㦨҈� Þ l 㦨鼀 è l ø
æ v ö 黨ç ÷ 㦨҈� Þ l 㦨鼀 è l ø ᨨ �� 矄 ᨨ 犨 㡰 鼀 ҈㡰 鼀 㡰鼀 㡰鼀 ᆠ �� 矄ᨨ 犨鼀鼀҈鼀 鼀 A ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ 矄 ᨨ � ᨨ 矄ᆠ�泐
A㦨p 맔҈㡰㦨҈맔҈鼀 a㦨p ҈㡰㦨҈맔҈鼀
矄矄盐 � ᨨ 㡰 ᨨ �矄ᆠ鼀 ᨨ 㝐ᔼ졠ᔼ x㦨졠 矄矄盐��ᨨ
x=x �矄鼀 ᨨ鼀�
�
p xö æ y㦨矄� ç 黨 p t ÷ è ҈� ø
盐ᨨ ᨨ 矄 � ᨨ 盐 l 㦨 ᆠ鼀 㦨 鼀 盐 鼀 ⼄盐 ᆠ泐 f㦨 ᨨᆠ�泐ᨨ 矄ᨨ盐 � �v㦨鼀⏌矄 v㦨 \ 㦨 l + D㡰 + ´ D㡰
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EBD_8341 PS-162
Physics
졔D㦨� Þ D㦨 Þ D㦨 ´ ҈ ᆠ鼀㦨ᆠ鼀 �鼀ᨨ 盐㡰
盐ᆠᨨ矄 ᨨ
졠矄 O D q
= ⼄盐 ᆠ泐v㦨 » ҈ l + D㡰 + ´ 㡰
Topic-3 : Beats, Interference and Superposition of Waves ҈
㡰 ᆠ㡰 ⼄盐 ᆠ泐 㦨 � ᆠ � ⼄盐 ᆠ�矄ᨨ 犨ᨨ犨矄 㦨҈҈黨㦨 㡰 㡰 ᨨ ⼄盐 ᆠ�矄ᨨ 盐 � � ᨨ├n᠀� ҈ ⼄盐 ᆠ泐ᨨ 矄ᨨ ᨨ鼀 犨� n µ l n + � ҈ \ = Þ 黨� n + �㡰 = ҈ n - �㡰 n - � 黨� ᨨ 黨�n졔䰘�㦨҈n�ᨨ�n㦨黨䰘�
黨䰘� = ҈黨� � 㡰 ␔� � ᆠ矄� 矄�ᨨ ⼄盐 ᆠ泐ᨨ 盐ᆠ 泐矄� � � ᆠ矄矄꾀矄 矄⏌矄ᆠ ᆠ矄矄 ᨨ ⼄盐 ᆠ泐ᨨ 盐 � � ᨨ├ ævö æ ö f = ç ÷ + = ç ÷ + = è l ø è ´ 䰘� ø ᨨ n =
�
b㡰 ᨨ矄� � ⼄盐 ᆠ泐 f =
和ᨨ盐ᆠ V
ᔼ ��泐q犨�矄矄矄ᨨᆠᨨ矄q鼀ᨨ \vᨨ矄 鼀ᨨ �犨� ᆠ矄
æ v + vc ö æ v + vc ö f = ç ÷ f҈ = ç v - v ÷ f è v ø c ø è é � + vc ù ҈ � + vc Þ = ê = ú Þ v � ҈҈ � - vc cû ë
Þ vc = � ├鼀⏌
҈ F l m
�鼀 �ᨨ T = l
� ´ ҈ - 㦨҈矄ᆠ ҈
�鼀⼄盐� ᨨᆠᨨ 矄盐ᆠ�� ᆠ⼄盐ᨨ �鼀 �ᨨ ᨨ 犨盐矄 ᆠD㦨҈矄 ætö b㡰 � ⼄盐�ᨨ y㦨ᆠᨨ矄 çè ÷ø 矄� ҈t㡰
\ T 㦨 ´
䰘
= + + ´ ´ ´ ᆠᨨ矄
A҈ + A + A҈ A ᆠᨨ矄 f
p = ҈ + 黨 + = �
Topic-4 : Musical Sound and Doppler’s Effect ҈
t æ ö 㦨 ç ᆠᨨ矄 矄� ҈t ÷ è ø ᨨy㦨ᆠᨨ矄t졔҈桌矄� ҈t ҈㡰 ᨨy㦨ᆠᨨ矄t矄� ҈t졔矄� ҈t 㡰 ᨨy㦨矄� ҈҈t졔矄� 黨黨黨t졔矄� ҈t 㡰 ᆠ ᨨ �� �ᨨ �ᆠ鼀ᨨ�ᨨ � 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ 矄 矄盐ᨨ矄 �㡰 矄 � ᆠ犨 犨ᨨ犨矄 f㦨 p 矄盐 鼀�盐 A =
b㡰 ⼄盐 ᆠ泐 泐 ᨨ矄
vᨨ矄
æ ö v矄ᨨ盐 =ç ÷ v v q ᆠᨨ矄 è 矄ᨨ盐 ø
㡰 ᨨ鼀 ᨨἘ矄 ᆠᨨ 矄ᨨ盐 ⼄盐 ᆠ泐 犨 fw = ×f �㡰 - v v㦨矄 ᨨ 盐矄 f㦨ᆠ盐 ⼄盐 ᆠ泐ᨨ 矄ᨨ盐ᆠ ⼄盐 ᆠ泐 ᆠ�ᨨ ᨨ鼀犨 f'㡰 �矄
+ v + v × fw = ×f - v + v Þ 黨 = × - v ´ 䰘 Þv= » � 鼀⏌矄 㦨 黨҈├鼀⏌矄 黨҈ ᆠ㡰 ᨨ鼀ᨨ矄 ᆠ ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 f҈㡰犨
矄ᨨ盐ᆠ�矄 ᨨᆠ � � c f҈ = f cv c 㦨 ᨨᆠ�泐 ᨨ 矄ᨨ盐 v㦨 ᨨᆠ�泐ᨨ 矄ᨨ盐ᆠ ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 f㡰犨 矄ᨨ盐ᆠ�矄ᆠ � � c f = f c+v ⼄盐 ᆠ泐㦨f҈f ҈ ù é ҈ Þ = f҈ f = f c ê ë c v c + v úû v = f c é v ù c ê҈ ú ëê c ûú ᨨ c>>v c c � ҈ Þ v= = = = 鼀⏌矄 f ҈ f
f Ἐ=
m F ҈ - 鼀矄矄 = � ´ ҈ - F㦨҈ m = 㦨 �
\
æ v ö 㡰 f҈ ⼄盐 ᆠ泐 泐犨 f҈ = ç ÷ f è v - vc ø v㦨ᨨᆠ�泐ᨨ 矄ᨨ盐 vc㦨ᨨᆠ�泐ᨨ ᔴ f㦨ᆠ盐 ⼄盐 ᆠ泐ᨨ ᆠ ᨨ
f ⼄盐 ᆠ泐 泐 � ᆠ�ᨨ ᨨ鼀犨
�
ᆠ㡰 f҈㦨 f
v - v v + v f㦨 f v v
盐 ⼄盐 ᆠ泐
v ff҈㦨 f ´ v
S҈
L
u
S
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Waves u \ u㦨�鼀⏌矄 b㡰 ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 矄ᨨ盐ᆠ 㡰 㦨 � ␔ ᨨ矄�矄鼀ᨨ� �犨泐 ᨨ鼀 矄ᨨ盐ᆠ æ n - n Ἐ ö 꾀 ⼄盐 ᆠ泐 ҈ = = ç �㡰 ÷ ç n ÷ è ø 꾀 犨 ᨨ矄�矄 鼀ᨨ� �ᨨ犨 矄 矄ᨨ盐ᆠ æ n - n³ ö = � = ç ÷ ��㡰 ç n ÷ è ø ᨨ鼀 ⼄盐�ᨨ �㡰 æ - Ἐ ö = � ç ÷ ç ÷ è ø 㦨҈鼀⏌矄 ᨨ鼀⼄盐�ᨨ ��㡰 æ ἘἘ ö � = � ç҈ + ÷ \ 㦨҈鼀⏌矄 ç ÷ø è v - v 㡰 f '㦨 f v + vs ᨨ
䰘
é v ù f = f ê Hz ú= f´ + v v sû ë
҈㦨 ´
҈
҈
�矄 ᔴ矄҈␔ 矄ᨨ盐ᆠ �矄鼀ᨨ� �ᨨ犨 矄矄�ᨨ 泐�矄
- \f㦨� + b㡰 꾀ᆠᆠᨨ � �ᨨᨨ矄 ᆠ犨 矄ᨨ盐ᆠ�矄鼀ᨨ� �盐 ᨨ矄 矄 ᨨ = f
é V ù 㦨 ê ú Þ 㦨 ëV Vs û
é ù ê ú ë û
é ù 㦨 ê ú ë ҈䰘 û
҈
҈䰘 ҈黨 = 㦨 ᨨ ҈ = ҈ 㡰 ⼄盐 ᆠ泐 ᨨ 矄ᨨ盐 ᨨ 盐ᆠ 泐ᨨ 盐
\
黨
æ v ö ´ f = ç ÷ = = Hz è l ø ´ � � � m⏌s = ҈ 黨 꾀矄 矄ᨨ盐ᆠ�矄鼀ᨨ� �ᨨ犨 矄 ᨨ矄 ᨨᆠᆠᨨ � � ᨨ ᨨἘ矄 ᆠ \ ⼄盐 ᆠ泐 ᆠ 泐ᨨ矄 é � ù é v + v ù ê 黨 + ú Ἐ㦨 ê ú f 㦨 ê ú ë v û ê ú ë û 黨黨� ´ ᨨ Ἐ㦨��� ; 㦨 黨 ´ 㡰 ␔ ├ ᨨ犨 ⼄盐 ᆠ泐 æ - ö Ἐ= ç ÷ ᨨ鼀 ᨨἘ矄 ᆠ è - 矄 ø 犨 㦨矄㦨鼀⏌矄ᨨᆠ�泐ᨨ ᨨ矄 矄ᨨ盐ᆠ 和 ᨨ 矄ᨨ盐 㦨鼀⏌矄 ᨨᆠ�泐ᨨ ᨨ矄㦨 ҈ ´
҈
졔 � 㦨 㜠 Q ⼄盐 ᆠ泐ᨨ 犨 �矄 㡰㦨� é v ù Hz 㡰 f҈ = f ê ú= f´ ë v - vs û
æ f ö æ ö - ҈ ÷ ´ ҈ ; ҈奠 ç - ҈ ÷ ´ ҈ = ç f è ø è ҈ ø 㡰 � ⼄盐 ᆠ泐 ᨨ 矄ᨨ盐 ᨨ 盐ᆠ 泐 矄� f㦨 和 ᨨ ᨨ矄u㦨鼀⏌矄 ᨨᆠ�泐ᨨ 矄ᨨ盐 v㦨鼀⏌矄 ᨨᨨ 矄 矄ᆠᨨ 㦨䁸 ᨨᨨ 犨矄ᆠ� 泐 ᨨ矄矄ᆠᨨ 㦨졔l \ ᨨᨨ 矄⏌矄ᆠ æ æ ö Vö ´ 㦨 - ç - ÷ = 㦨 çQ l = ÷ f ø l è lø l è ´ ´ = ҈ 㦨 㦨� ᆠ㡰 꾀 鼀⏌矄 ᔴ
҈�
҈
æ ö æ ö ⼄盐 ᆠ泐 h Ἐ = h ç ÷ = � ç ÷ 㦨� è 矄 ø è ø ᔴ矄␔ 矄ᨨ盐ᆠ�矄鼀ᨨ� �犨泐 ᨨ鼀 矄�ᨨ 泐�矄 æ ö æ ö h³ = h ç ÷ = � ç ÷ 㦨黨 è ø è + 矄 ø ᔼ ᆠ矄҈ 盐鼀ᨨ 矄 �矄 � ᆠ矄 盐鼀ᨨ 矄 �矄҈ ᨨ ⼄盐 ᆠ泐ᨨ 矄ᨨ盐ᆠ㦨�҈ 㡰 꾀 ⼄盐 ᆠ泐ᨨ 矄� 泐 ᆠ � f¢ é v + vᨨ ù 㦨 f ê ú ë v - vs û 㦨鼀⏌矄 �� 㡰 � vᨨ㦨v矄㦨 ´ 㦨҈鼀⏌矄 ҈ é + ҈ ù ᨨ f ¢ = ê ú = ´ ҈ » � ├ ë - ҈ û b㡰 f҈㦨 ⼄盐 ᆠ泐 ᨨ ᨨ�ᆠ ᆠ 泐鼀ᨨᨨᆠ泐ᆠ�矄 f 㦨 ⼄盐 ᆠ泐 ᨨ 矄� 泐鼀ᨨᨨᆠ泐ᆠ�矄 - v + v ´ ҈䰘錐 f㦨 ´ ҈�錐 f҈㦨 - Q f҈f㦨\ v㦨鼀⏌矄
æ v + vö b㡰 矄� � ᨨ鼀盐f' = f ç A è v ø÷ v A + v �� VB + V v = \ B = = � � v V v A
҈䰘
҈㡰꾀 ⼄盐 ᆠ泐 泐ᨨ矄 盐 ᨨ矄ᨨ盐ᆠ和
é + ù é + ҈ ù v҈ = v ê ú = ҈ ê ú = ҈ ´ = ҈ ë û ë û
鼀矄҈ 䰘 和
ᆠᨨ矄䰘
\ Ἐ㦨
҈҈
҈
҈鼀⏌矄
� ҈ᆠᨨ矄�
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EBD_8341 PS-164
Physics
꾀 ⼄盐 ᆠ泐 泐ᨨ矄 盐 ᨨ 矄ᨨ盐ᆠ 和
æ v + v҈ ö f҈ = f ç è v v҈ ÷ø
é + ù é + ҈ᆠᨨ矄 �° ù v = v ê ú = ҈ ê ë - ᆠᨨ矄 䰘° úû ë - 矄 û
҈鼀
꾀 ⼄盐 ᆠ泐 �졠 盐ᨨ矄ᨨ盐ᆠB
A
q 鼀矄
q
q
v é ù nB = n ê úû + ҈ᆠᨨ矄 q v ë \
҈ᆠᨨ矄 q
é
v
ù
é
v
ù
⼄盐 ᆠ泐vb = n êë v - ᆠᨨ矄 q úû - n êë v + ᆠᨨ矄 q úû
҈ ҈ ù é = n vê ë v - ᆠᨨ矄 q v + ᆠᨨ矄 q úû ҈ ҈ é ù = ҈ ´ ê � �ú + ú ê - ´ ҈ ҈ û ë ҈ ҈ é ù = ҈ ´ ´ ҈ ê ë ´ ҈ - ҈ ´ ҈ + � úû ҈ ù é ҈ = ҈ ´ ´ ҈ ê » � ë 黨� úû ҈黨
㡰
- l æ ö Þ = ç ÷= l҈ l҈ è - ø l l l - l҈ -
-
Þ -҈ = -҈ Þ = l҈ l҈
ᆠ �ᆠ �⼄盐� � � ᨨ �
- l - l҈ ´҈ = ´҈ = -�奠 l҈
ᆠ
ᨨ 㦨�鼀矄㜠҈
和ᨨ盐ᆠ 㦨黨
⼄盐 ᆠ泐ᨨ 矄ᨨ盐 ᆠ ᨨ鼀 犨
é + ù é + � ù ; ú Þ vἘ㦨 ê ë - � úû ë - s û
nἘ㦨n ê
矄盐 ᨨ 鼀�盐 꾀
p p é ù = I ê矄� O + 矄� + 矄�
+ 矄� p ú ë û é ù = I ê + ú = I ë û
\ IR㦨AR㦨I \ n 㦨 䰘㡰 �矄泐 ᆠ 犨� 矄 ᨨ矄 犨 �ᆠ �矄 ᨨᆠ � � 矄ᨨ盐ᆠ �犨� 矄矄ᨨ盐ᆠ犨 �ᆠ �矄鼀ᨨ� �� �ᆠ�ᨨ ᨨ 矄ᨨ盐 ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 ᆠ 泐 ᆠᔴ҈
矄㦨�鼀矄҈ �鼀�ᨨ 盐矄 ᨨ鼀 泐 犨 矄矄矄ᨨ盐ᆠ㡰
é v ù é ´ ù 黨 ´ = u ê c ú = 黨 ê ú = = ë û ë v vc û
҈
ᆠ矄鼀矄盐ᨨ ᆠ �ᆠ �⼄盐� � � ᨨ ��矄�奠
ᨨ矄 矄ᨨ盐ᆠ鼀ᨨ� �ᨨ犨 矄ᆠ ᨨ �鼀� ᨨ 矄ᨨ盐ᆠ矄矄矄矄ᨨ盐ᆠᨨ ᆠ 矄ᨨ盐 � 矄 矄 ᨨ矄㡰
vc㦨鼀矄҈
é v + vc v + vc ù é v + vc v + vc ù = u ê ú = u ê ú v vc v vc ë û ë û
㡰
ö ÷= l ø
Þ
⼄盐 ᆠ泐ᨨ ᆠ 矄ᨨ盐 矄ᨨ矄 矄ᨨ盐ᆠ
æ
è
æ v + vc ö v ÷ø
l҈
␔ 盐 � � ᨨ├�矄鼀ᨨ Ἐ = ç
u҈㦨 u çè
\ ⼄盐 ᆠ泐㦨vv
ᨨ㡰 l҈㦨 � �� � ᨨ � l㦨 犨 � ᨨ � 㦨和 ᨨ 盐 � � ᨨ├ ᔼ ᆠᨨ矄 ᨨ� � =
⼄盐 ᆠ泐 ᨨ矄 ᆠ
æ v ö æ v + vc ö u㦨 u҈ ç ÷ = v ç ÷ è v vc ø è v vc ø
ᔴ
( v҈ v ) æ ҈ ö \ç f= f è ҈ ÷ø v v ҈ ´ ҈ = = ҈黨 鼀矄 ҈ = 䰘├鼀 ҈ ᨨ ( v҈ v ) =
҈鼀矄 ҈ B
�鼀
�鼀
vv҈㡰㦨 vv㡰»v 矄v҈ v泐泐矄矄 v
҈鼀 和�ᨨ 泐㡰
҈
矄
æ v + v ö f = f ç ÷ è v v ø é v + v҈ v + v ù æ ҈ ö ÷ø f = f ê ú \ f҈ f = èç ҈ ë v v҈ v v û v ( v҈ v ) æ ҈ ö f ÷ø f = ᨨ çè ҈ ( v v҈ )( v v )
\ ⼄盐 ᆠ泐v 㦨҈҈҈䰘҈㦨҈� b » ҈ �㡰 꾀 ⼄盐 ᆠ泐 졠 盐 ᨨ 졠矄⏌�矄 和�ᨨ 泐 矄ᨨ盐ᆠA
v é ù nA = n ê ᆠᨨ矄q ë v - ᆠᨨ矄 q úû
҈
꾀 ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 ᆠ 泐 ᆠᔴ
é + ҈ ´ ù é ù \ v = ҈ ê = ҈ ê ú ú = ҈҈䰘 ë - ´ û ë û
҈
ᔴ҈
\ ⼄盐 ᆠ泐ᨨ 矄 㦨 nἘn㦨㦨 矄� 矄ᨨ盐 ᆠ 矄 ᨨ矄 � � ᆠ ᨨ鼀 矄�ᨨ 泐 矄盐 ᆠ 鼀 �盐鼀 �矄 矄ᨨ 졠矄 ҈ 矄�ᨨ 泐 �鼀�ᨨ 矄ᨨ盐ᆠ犨� 和 和 ᆠᨨ鼀 矄ᨨ盐ᆠᨨ ᆠ 矄ᨨ盐
盐矄� ᨨ ᆠ矄矄ᨨ 矄ᨨ盐 ᆠᨨ鼀� � �ᆠ泐 ᨨ鼀 矄ᨨ盐ᆠ ᨨ ᆠᨨ鼀� � ᆠ�ᨨ 犨� 矄鼀 ⼄盐 ᆠ泐 ␔ ᆠ ᨨ矄犨�
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Waves �
矄 � 矄�泐 ᨨ ᨨ盐 矄矄 ᨨ 矄ᨨ盐
I㦨
҈ r V w A
æ v + vm ö f ἘἘ = ç ÷f è v vb ø
꾀矄ᨨ � 矄�泐�矄 矄 �矄 ᆠ ᨨ鼀 ᨨ� 矄ᨨ盐ᆠ 矄
҈
䰘
ᆠ ⼄盐 ᆠ泐 泐鼀ᨨᨨ�矄
Iµ r 꾀矄 ᨨ ᨨ 鼀矄ᆠ �� �� ᆠ矄ᨨ ᆠ �� ᨨ ᆠ 泐矄ᨨ � 矄�泐犨�鼀� 矄鼀 꾀® ⼄錐® 錐ᔴ® 꾀㡰 �ᆠ ⼄ ⼄盐 ᆠ泐 㡰 ⼄盐�泐 犨 ᨨ鼀 ᔴ㡰 ᨨ盐 矄矄 � 矄�泐 泐 ᨨἘ矄 ᨨ鼀盐
æ v + vm ö æ v + vm ö fb = f ἘἘ f Ἐ = ç ÷ f ç ÷f è v + vb ø è v vb ø ᨨ fb =
vb ( v + vm ) v vb
é + 鼀 ± ù ú ë + 鼀 ± 矄 û
和�� ᆠᨨ �ᨨ ᨨVm�矄矄 ᨨᨨ犨矄 ᔼ 鼀 �盐鼀�矄鼀ᨨ� � ᨨ鼀sᨨO 졔 �ᆠ矄 和�鼀�泐v vsᨨ矄��� 矄 �ᆠ ᨨ鼀SᨨO �ᆠ矄 㡰
ᆠ 犨
鼀
nἘ㦨n ê
B = r
ᨨᆠ�泐ᨨ 矄ᨨ盐 � 犨v㦨
´ ҈黨
졠 黨
和
犨 �矄 犨 �ᆠ �矄 鼀�� � 矄ᨨ盐 �矄 � � ᨨ � ᆠ�ᆠ
ᨨ矄�矄� �矄 ᆠ ᨨ鼀 犨 �矄�졠 矄㦨w㦨鼀⏌矄
҈
꾀
㦨҈�鼀⏌矄
҈鼀⏌矄
㦨 ҈�鼀
鼀⏌矄
S 矄鼀�㡰
� ␔
졠
O ᆠ�㡰
w㦨 㜠҈
ᨨ㦨
⼄盐 ᆠ泐ᨨ 矄ᨨ盐 � 犨 v㦨
v犨 ҈� = = ҈� T犨 ҈� ´҈
㦨w 矄 㦨鼀⏌矄
� r㦨҈�鼀 w㦨 矄҈ 和 ᨨ 矄ᨨ盐ᆠvs㦨rw 㦨҈�맔㦨鼀矄҈ ␔ 矄ᨨ盐ᆠ�矄 ᨨ矄��ᨨ A ⼄盐 ᆠ泐 泐 ᨨ矄犨�鼀 �鼀盐鼀
⼄盐 ᆠ泐�矄� ᨨ 鼀 �盐鼀\v犨㦨v� 鼀㦨졔鼀⏌矄錐㦨錐矄 㦨҈鼀⏌矄
é v + vm v ù ú ë v + vm vs û
꾀泐� � ᨨἘ矄 ᨨ鼀盐 vἘ 㦨 v ê \
(
)
ù = é ù = êë - úû ú ë - s û
é
nἘ㦨 n ê
� é ҈� + - ù \nἘ㦨҈䰘맔҈� nἘ㦨 ҈ ´ ҈ ê ë҈� + - ҈ úû
␔ 矄ᨨ盐ᆠ�矄 ᨨ矄��ᨨ B ⼄盐 ᆠ泐 泐 ᨨ矄犨�鼀� �鼀盐鼀
b㡰 ᔼ �ᨨᆠ�泐ᨨ 矄ᨨ盐
g RT 㦨 㦨 M
é ù ù ú = ê + ú = ë û ë + 矄 û é
(҈) ´ ( ҈) ´ 黨 = 鼀⏌矄
nἘἘ㦨 n ê
´ ҈
ᆠ � ᨨ ⼄盐 ᆠ�矄 泐 ᨨ矄 㦨 ᨨ
vm㦨�鼀矄㜠҈ S
O ҈鼀⏌矄
= ҈ ⏌ 矄
ᆠ�㡰
æ v v m v ö ÷ è v vm vs ø
꾀泐� � ᨨἘ矄 ᨨ鼀盐 vἘ㦨 v ç \
(
�
nἘ㦨 ҈ ´ ҈
�盐 ⼄盐 ᆠ泐ᨨ ᆠᨨw㦨pn = p ´
- � - ù 㦨҈맔҈ ) éêë - � - ҈ úû
� p
�盐 ⼄盐 ᆠ泐ᨨ ᆠᨨ 矄ᨨ盐ᆠᨨ 矄ᨨ盐 ⼄盐 A w㦨҈ ⏌矄 ҈
�
鼀ᨨᨨ�矄 矄 ⼄盐 ᆠ泐矄 ᆠ�矄犨ᨨ � ⼄盐 ᆠ�矄 ᨨ �ᆠ泐 ᨨ鼀 矄ᨨ盐 矄ᨨ盐ᆠ ᨨ f' ᨨ ᆠ ᨨ鼀 犨 꾀 ⼄盐 ᆠ泐犨 �ᆠ矄ᨨ盐 ᨨ �矄 泐 鼀ᨨᨨ�矄
æ v + vm ö f Ἐ= ç ÷f è v + vb ø 꾀�� ⼄盐 ᆠ泐犨 ᨨ �ᆠ ᆠ 矄ᨨ盐 泐 鼀ᨨᨨ�矄
鼀
A A
B
鼀 C
D
A
Þ ␔ ᆠᨨ�矄C鼀ᨨ� �ᨨ犨 矄D 矄ᨨ盐ᆠ�矄 A҈鼀ᨨ� � 犨 矄ᔼ�矄� �矄矄�盐�ᨨ ⼄盐 ᆠ泐 �矄鼀� �鼀盐鼀
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EBD_8341 PS-166
Physics
é - ù - 㡰 = �䰘 n鼀� 㦨n ê ú = ´ + + 㡰 矄 ë û Q㦨꾀w㦨맔҈㦨鼀⏌矄 矄㦨 w㦨맔҈㦨鼀⏌矄㡰 꾀�� 犨 ᆠᨨ�矄C鼀ᨨ� �ᨨ犨 矄B 矄ᨨ盐ᆠ�矄 A鼀ᨨ� ��� 犨 ᔼ�矄� �矄矄�盐�ᨨ ⼄盐 ᆠ泐 �矄鼀 �鼀盐鼀
v㦨�鼀⏌矄
é + ù + 㡰 = 䰘 ú = ´ - 㡰 ë 矄û
矄
犨
é c - ù nἘ㦨 n ê ú ë c - s û
矄
s졔� 泐 �ᆠ ᨨ鼀矄ᨨ盐ᆠᨨ ᨨ矄 � 泐 �ᆠ ᨨ鼀ᨨ矄ᨨ矄ᨨ盐ᆠ
é - -�㡰 ù ú = ë - � û
nἘ㦨� ê
҈ 졠矄 ᨨ ᨨ├ ҈ 鼀ᨨ� � ᨨ犨 矄 ᨨ矄
\ ⼄盐 ᆠ泐㦨 vἘv㦨�㦨
v f Ἐ= f ( v vs ) 犨 vs㦨ᨨᆠ�泐ᨨ ᨨ├
Topic-5 : Miscellaneous (Mixed Concepts) Problems
ᨨ ᨨ├ 鼀ᨨ� �犨泐 ᨨ鼀 ᨨ矄
f ἘἘ =
v f v + vs 犨 v㦨ᨨᆠ�泐ᨨ 矄ᨨ盐
҈
= l
n =
g
㝐 ´ ҈
-
´
҈ Þv㦨vl㦨n맔l l
v 㡰맔맔l㦨䰘鼀⏌矄ᨨ�鼀⏌矄 \l㦨�졔�桌 ᨨ鼀ᨨ ᨨ鼀�ᆠ�矄g㦨҈䰘
vf vf é ( v + vs ) ( v vs ) ù ᨨ = vf ê ú v vs v + vs êë ( v vs )( v + vs ) úû vf ´ vs vf ´ vs ᨨ = ᨨ = 矄vsv v v v s v v ´ v = = = ҈�鼀⏌矄 ᨨ vs = ᨨ s vf f ´
ᨨ =
\ =
g
-
㝐 ´ ҈
= ҈ g ´
҈ 㝐
҈ ҈ = 㝐 㝐 䰘 ᨨNᨨ 㝐㦨\ = ´ 㦨鼀矄҈ ҈ ҈䰘 㦨鼀矄҈ ᨨ꾀�ᨨ 㝐㦨 \㦨 ´ 㦨 ҈䰘 ´
\和 ᨨ 盐 � � ᨨ├㦨҈�鼀⏌矄
ᆠ鼀㦨l
ᨨ �ᨨ鼀�ᆠ�矄g=҈
l⏌
l҈㦨�ᆠ鼀 l
㡰 Þ
\ ⼄盐 ᆠ泐㦨f' f''㦨
ᔼ 和ᨨ盐ᆠ㡰
和 졠矄㡰
n鼀 㦨 n ê ҈
v㦨�鼀⏌矄
= ҈
҈ ҈ = �黨 ´ 㝐 㝐 黨 㦨҈䰘鼀矄҈ ҈ \ = �黨 ´ 㦨�鼀矄҈ �
ᨨ졠 y� M㦨\ = �黨 ´ 泐 ᆠ矄� � 矄�ᨨ � 矄� � ⼄盐 ᆠ泐�矄 ᆠ矄� � \ �矄ᨨᨨ ⼄盐 ᆠ泐ᨨ 矄� � 盐 鼀 ⼄盐 ᆠ泐ᨨ ᆠᨨ矄 � 㦨
æ v҈ ö æ v ö é v ù ÷ç ÷ = ᨨ v҈ = l҈ ê + ú \ ç l û è l҈ ø è l ø ë 和盐矄�盐� � 盐犨
ù é = �m ⏌ s v҈ = � ê + ë ´ úû ᨨ犨 v҈ =
T \ T = mv҈ m
æ m ö æ � ´҈ ö ÷ �㡰 = 䰘 ᨨT = ç ÷ v҈ = ç ç � ÷ èl ø è ø
ᨨN�ᨨ� M㦨
b㡰 x ᆠᨨᆠ�ᨨ
l ᨨ l = l҈ + x㡰 l l + x 㡰 = ᨨ l = l + x㡰 = = \ n҈ 㦨 \n㦨 l҈ l҈ + x 㡰 l l + x 㡰 l҈ + x =
҈鼀㦨l҈
l҈ +x
�鼀㦨l
l +x
ᔼ 矄ᨨ盐 ᆠ 矄 ᨨ矄 � � ᆠ ᨨ鼀 矄�ᨨ 泐矄盐 ᆠ 鼀 �盐鼀�矄矄ᨨ矄�ᨨ 泐 �鼀� ᨨ 矄ᨨ盐ᆠ ᔼ㡰� ᆠ� �矄盐 ᆠ犨�ᆠᨨ鼀 矄ᨨ盐ᆠ ᨨ ᆠ 矄ᨨ盐
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PS-167
Waves � n҈㦨n \
�
= ᨨ l҈ + x 㡰 l + x㡰
\ x㦨�鼀
v dn æ -҈v ö Vc 矄� a = - c 矄� a ´ ç ÷= dt v è x 矄ᆠ a ø x 矄ᆠ a vc 矄� a dn = vc 矄� a 꾀a㦨黨 \ dt = ҈ ᆠᨨ a 㡰 矄ᆠ a
ᆠ㡰 꾀矄ᨨ w㦨pn Þ w µn E �鼀 =҈꾀鼀�盐 㦨A n҈㦨n E �鼀 =꾀鼀�盐 㦨A n㦨n
E A n = = \ E㦨E҈ E҈ A n҈
dn = max dt
b㡰 矄 ᨨ ᆠ 盐矄�矄ᆠ鼀⏌矄 � ��泐犨ᨨ盐矄矄 ᆠ鼀 鼀ᨨ� �ᨨ犨 矄ᆠ ᨨ
꾀 犨ᨨ 矄ᆠᨨ 矄 盐矄矄 犨� ᨨ ᆠ ᨨ 꾀ᆠᆠᨨ � � ᨨ 矄盐ᨨ矄��ᨨ � ᆠ� 矄� � 犨� ᨨ 泐 �矄ᨨ�ᨨ 犨� 矄�� ᆠ 犨� ᨨ쟼 ᨨ �泐犨�ᨨ 泐├� �ᆠ
黨
ᨨᆠ�泐ᨨ 矄ᨨ盐 泐 矄ᆠ 矄� �㦨
矄ᨨ ᆠ �矄矄ᨨ鼀犨 矄 ᨨ
+e=
和 ᨨ 矄ᨨ盐 犨矄 � 犨 �矄 � � � 矄ᨨ 犨 矄 鼀 �盐鼀 ᆠ 矄ᨨ盐 犨 犨� ᆠ � ᨨ犨 犨泐 ᨨ鼀 ᨨ鼀 bᆠ㡰
é v + vc ᆠᨨ矄 q ù é v + vc ᆠᨨ矄 q ù n P = V V鼀 㡰 ê ú = ҈҈ - ҈҈ êë úû v v ë û n Q = ( vN vM ) = ҈҈ - ҈҈ = é v + vc ᆠᨨ矄 q ù é v - vc ᆠᨨ矄 q ù n R = ( vN vM ) ê ú = ҈҈ ҈҈㡰 êë úû v v ë û \ n P + nR = nQ vᆠᆠᨨ矄q q vᆠ 꾀 a Vx ᆠ ҈҈ ҈҈ ҈ 턼 㝐
ᔼ � 犨 ᆠ�矄矄矄� � ᨨ盐� 꾀
é v + vc ᆠᨨ矄 a ù n = ê úû v ë \
… �㡰
dn é v 矄� a ù d n = - ê c �矄鼀 犨 矄� a㦨҈ da ë v úû d a
�a㦨黨 턼㡰
d n vc da = 矄� a㡰 … ��㡰 dt v dt ҈ dx ҈ da =- 꾀矄ᨨ a = \ 矄ᆠ a dt x x dt
ᨨ鼀⼄ �㡰
҈
ᨨ 犨 � ᨨ 矄ᨨ盐 � � 盐ᨨ ᆠᨨᆠ�ᨨ 㡰
TἘ ҈�x T = \ Ἐ㦨v 㦨v 㦨҈ T x Ἐ TἘ
䰘
b 㡰 ␔ 矄ᨨ盐 盐矄�矄 ᆠ ᨨ鼀 ᨨ ᨨ � 矄 ᆠ �矄泐҈ 꾀 �� 矄矄盐盐矄�ᆠᨨ鼀矄矄�ᨨ �矄 ᆠ 矄ᨨ犨矄矄盐盐矄� ᆠ�ᨨ ␔ 矄ᨨ盐 盐矄�矄 ᆠ ᨨ盐� ��� ᨨ盐 泐 ᆠᨨ矄 ᨨ �㡰 ᨨ 矄ᆠ �ᨨᆠᆠ盐矄矄ᨨ �� 矄矄盐盐矄 �矄 ᆠ 矄 �� 矄矄盐盐矄 e 㡰
� ᨨ �ᆠᨨ盐鼀 �矄
T m
ᨨ鼀ᨨᨨ├Ἐ矄犨 矄�ᨨ � 矄� �T µ 矄�ᨨ x
… ���㡰
ᨨ鼀⼄ ��㡰 ���㡰
쟼 �泐 EµAn犨 A㦨鼀�盐 n㦨 ⼄盐 ᆠ泐
㡰
da -҈v = dt x 矄ᆠ a
҈
҈҈
҈
l
Þ
=
l -e
ᆠ矄ᆠᨨ 矄ᨨ ᆠ � ᨨ �ᆠᨨ盐鼀 �矄鼀ᨨ矄 ᆠᨨ鼀 ᨨ �矄矄ᨨ ᆠᨨ犨ᨨ � � ᨨ �ᆠᨨ盐鼀 鼀ᨨ �矄 矄ᨨ�ᨨ ᨨ �泐 矄矄 �矄 � 矄�泐 ᨨ 矄ᨨ盐 b㡰 矄 ᨨ 矄ᨨ盐 矄ᨨ 鼀ᨨ ᆠᨨ ᨨ矄 和꾀㦨졔㦨鼀⏌矄 和㦨㦨҈鼀⏌矄 㡰 �矄 ᨨ�鼀ᨨ�ᨨ 犨 矄ᨨ盐ᆠ ᨨ矄 ᨨ 矄矄 �矄� � 꾀和� ᆠ 矄矄 �矄� � A 鼀ᨨ� �犨� ᨨᆠ�泐ᨨ 鼀⏌矄 ᨨ �矄�盐�ᨨ ᨨ 矄ᨨ盐 � 矄�泐ᨨ 犨 �矄 泐 矄矄 �矄� � A�矄盐 � ᨨ鼀 㡰 ᨨ 矄矄 �矄 � � 矄ᨨ盐ᆠ �矄 ᨨᆠ � � 犨� ᨨᆠ�泐 鼀⏌矄 ᨨ矄 �矄 ᆠ � � 犨� ᨨᆠ�泐 鼀⏌矄
é v - v ù ҈ é - ù n Ἐ = n҈ ê ú = ê - ú = ´ v v ë û ë sû é v - v ù ҈ n ἘἘ = n ê ú = ҈҈ ´ ë v - vs û \
n ἘἘ- n Ἐ 㦨 ҈҈㡰맔
҈ ҈ = ´ = ҈
҈ 㡰 ҈ ᆠ㡰 ҈� 㡰 �� ⼄盐�ᨨ 矄 y҈ 㦨 A ᆠᨨ矄 � px ҈ pt㡰 y 㦨 A ᆠᨨ矄 px 黨 pt㡰 矄 矄 犨ᨨ ᨨ�矄矄� 犨 � �� 矄鼀 �ᆠ�ᨨ ᨨ �x㜠 �矄犨� 矄�� � ᆠ � ⼄盐 ᆠ泐 ᔴᨨ鼀� ��犨� 矄 ⼄盐�ᨨ
y㦨Aᆠᨨ矄 kxwt㡰犨� w҈㦨҈p Þ pf҈㦨҈p Þ f҈㦨� K҈㦨�p Þ
p = �p Þ l҈㦨鼀 l҈
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EBD_8341 PS-168
Physics
␔ᨨᆠ�泐㦨l҈f҈㦨鼀⏌矄꾀 �泐盐矄v㦨 w㦨黨p Þ pf㦨黨p Þ f㦨
ᨨ ⼄盐 ᆠ泐㦨f҈f㦨
w 桌 K
p = p Þ l㦨 K㦨p Þ l ␔ᨨᆠ�泐㦨 맔㦨鼀⏌矄
v㦨
ᨨ
é
ì ax + bt 㡰 + ax - bt 㡰 ü ý î þ ë ì ax + bt 㡰 - ax - bt 㡰 ü ù 矄� í ý ú Aᆠᨨ矄 ax졔bt㡰桌 î þû Þ y㦨҈A矄� ax矄� bt졔Aᆠᨨ矄 ax졔bt㡰
= A ê - 矄� í
�鼀� 犨 犨ᨨ矄盐ᆠᆠ矄矄�鼀 �鼀矄�矄p맔҈矄ᆠ ��㡰 ᆠᨨ ᆠ ᆠ � 矄盐 � 矄�泐> A \ 矄盐 鼀�盐 > A
犨 ҈A矄� ax矄� bt㡰�矄 ⼄盐�ᨨ ᨨ 矄 � �犨 Aᆠᨨ矄 ax졔bt㡰�矄 ⼄盐�ᨨ ᨨ � �犨
犨�矄� �� x �ᆠ�ᨨ 꾀 � ᨨ 矄 ᨨ� 矄犨 鼀�盐 �矄鼀 �鼀盐鼀
A ᆠᨨ矄 p n҈ - n 㡰 t ³ A ҈ ᨨ ᆠᨨ矄 p n҈ - n 㡰 t ³ ᨨ ҈ é w - w 㡰t ù ᆠᨨ矄 ê ҈ ú³ ë û ᨨ
é w҈ - w 㡰t ù
ᆠᨨ�矄� 犨 盐矄ᨨ ᆠᨨ矄 ê ú �矄 ë û ҈ 犨
\
p p w҈ - w 㡰t �矄犨
dy 㦨Ab矄� ax졔bt㡰Ab矄� ax bt㡰 dt
㦨Ab矄� ax졔bt㡰졔矄� axbt㡰桌 㦨Ab矄� axᆠᨨ矄bt졔ᆠᨨ矄ax矄� bt 졔矄� axᆠᨨ矄btᆠᨨ矄ax矄� bt桌 v㦨Ab҈矄� axᆠᨨ矄bt졔ᆠᨨ矄ax矄� bt桌
鼀 �鼀盐鼀 ᨨᆠ�泐 犨� ᨨᆠᆠ盐 犨 矄� ax 㦨 ҈ ᆠᨨ矄bt㦨҈盐 矄ᆠᨨ ��ᨨ ᆠᨨ矄ax㦨 矄� bt㦨 \ v鼀 㦨҈Ab 꾀矄ᨨv鼀� 㦨 㡰 y㦨Aᆠᨨ矄 ax졔bt㡰桌Aᆠᨨ矄 axbt㡰桌 㦨Aᆠᨨ矄 ax졔bt㡰졔Aᆠᨨ矄 ax졔bt㡰桌 Aᆠᨨ矄 axbt㡰桌 㦨Aᆠᨨ矄 ax졔bt㡰Aᆠᨨ矄 ax bt㡰桌 졔Aᆠᨨ矄 ax졔bt㡰桌
w҈ - w 㡰 np t = np ᨨ t㦨 w҈ - w
p p np n - ҈㡰p 矄 = ᨨ w҈ - w w҈ - w w҈ - w ҈
b p
b㡰 犨�矄 ᆠ 泐 ᨨ矄ᆠ�犨�矄盐 矄 � ᆠᨨ p � 矄�泐ᨨ ᆠ 犨�矄�鼀矄 ᨨ � ᆠ� 犨 ᔼ 矄�泐ᨨ ᨨ��� 犨Iµ꾀 ᔼ 矄�泐ᨨ ᆠ 犨I Ἐ㦨I Þ I ἘµAἘ Þ IµAἘ Þ AµAἘ Þ 꾀ἘµA 和ᨨ ⼄盐�ᨨ ᨨ ᆠ 犨 y㦨Aᆠᨨ矄 ax bt졔p㡰㦨Aᆠᨨ矄 axbt㡰 ᆠ㡰 矄盐 犨 ⼄盐�ᨨ
y㦨y�졔y 㦨Aᆠᨨ矄 ax졔bt㡰졔Aᆠᨨ矄 axbt㡰桌 �ᆠ ᨨᆠ�泐
㦨A矄� p n҈졔n㡰tᆠᨨ矄p n҈n㡰t 犨 鼀�盐 AἘ㦨Aᆠᨨ矄p n҈n㡰t ᨨ犨 � 矄�泐 µ 꾀鼀�盐 㡰 Þ IµAἘ IµAᆠᨨ矄p n҈n㡰t ᔼ 矄�泐犨�鼀 �鼀盐鼀 犨
ᆠᨨ矄p n҈n㡰t㦨҈ ᨨ ᆠᨨ矄p v҈v㡰t㦨҈ ᨨ p n҈n㡰t㦨np
�鼀� 犨 犨ᨨ矄盐ᆠᆠ矄矄�鼀 �鼀
㡰 ᔴᨨ鼀 �� ⼄盐�ᨨ y҈㦨Aᆠᨨ矄 ax졔bt㡰犨� 矄 ⼄盐�ᨨ ᨨ ᨨ�矄矄�犨
\ ⼄盐 ᆠ泐ᨨ � ᆠ� 犨 n㦨
pn҈ + pn 㡰t pn҈ + pn 㡰t ᆠᨨ矄
\
p p p p - 㦨 㦨 ´ ҈ 矄 w҈ - w w҈ - w 㡰 w҈ - w 㡰
꾀矄ᨨpn㦨b
\ᔼ ҈矄ᆠᨨ y҈졔y㦨x㦨 ᨨ黨�鼀矄 犨ᨨ �ᨨ 犨矄 矄 泐 ⼄盐�ᨨ 矄 y҈㦨A矄� pn҈t y㦨A矄� pnt 和� ᆠA҈㦨A㦨A ᆠᨨ�矄x㦨
⼄盐�ᨨ ᨨ 矄盐 犨 ᆠᆠᨨ � � ᨨ 矄盐ᨨ矄��ᨨ
� ᆠ� y㦨y҈졔y㦨A矄� pn҈t졔A矄� pnt 㦨A矄� pn҈t졔矄� pnt桌
Þ
�鼀�㦨t҈t
p p æ p ö = a Þ l㦨 x + pn t÷ Þ l a è l ø
n + ҈ 㦨ᆠᨨ矄 n졔҈㡰p 黨pt Þ t㦨 ҈黨 ҈ ҈黨҈ = 黨� 犨 t㦨n㦨 - 犨 t㦨҈n㦨
㦨A맔矄�
\
y㦨Aᆠᨨ矄 ç
꾀x㦨 y҈졔y㦨 Aᆠᨨ矄҈pt㡰졔 Aᆠᨨ矄黨pt㡰㦨 Þ ᆠᨨ矄҈pt㦨ᆠᨨ矄黨pt㦨ᆠᨨ矄 黨pt㡰
҈
t҈㦨
㦨 ҈䰘
p p t㦨 w҈ - w w҈ - w 㡰
\
é
n
pù
� 矄� ax㦨҈㦨矄� ê np + -҈㡰 úû ë
é ë
ᨨ ax = ê np + ( ҈) ҈
n
pù úû
ᨨ
é
x㦨 ê n +
êë
-҈㡰n ù p ú úû a
�㡰 ⼄盐 ᆠ泐ᨨ 犨 �矄 泐ᨨ矄ᨨ �
é
v + vw ù é ҈ + ù = �黨黨 ú 㦨� ê v v v + ë҈ + - úû ë w sû
nἘ㦨n ê
��㡰 ᆠ ᨨ ᨨ鼀 ��矄 泐 �B犨 �ᆠ �矄 �矄 ᆠx ᨨ鼀 �矄ᨨ盐 ᨨ 盐ᆠ 泐� �矄 ᆠ҈
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PS-169
Waves ├鼀 ᨨ鼀 �
�鼀├ 泐 �ᨨᆠ ᨨ鼀AᨨB
k x - y㡰 n + ҈ = p n + ҈㡰 p 犨 㦨졔҈졔ᆠ xy㡰㦨 k
Þ
x 꾀
҈├鼀
҈- x t҈㦨
ᨨ �
�㡰
�鼀├ 泐 ᆠ ᨨᨨᆠ ᨨ鼀 � t㦨tAH졔tHB
҈ x + t㦨 ҈ + 㡰 ҈ - 㡰
��㡰
n㦨
é v - vw 㡰 + vs ù ú ë v - vw 㡰 - v û é ҈ - 㡰 + ù 㦨 � ê ú ë ҈ - 㡰 - û
҈
nἘἘ㦨n ê
҈黨
Þ k x㦨
Al r ù T é ê 犨 m = 鼀矄矄 盐 � � = l = Ar ú M ë û YAaDq Y aDq = = Ar r
v=
㦨҈
\v =
x - y㡰 é k x + y 㡰 - wt ù ᆠᨨ矄 ê úû ë
矄盐 � 矄�泐犨� ᨨ 犨
k x - y㡰 k x - y㡰 = Þ ᆠᨨ矄 =
黨 ´ ҈
҈ 鼀ᨨ �ᆠ
㦨䰘鼀⏌矄
鼀ᨨ �ᆠ
n + ҈ n + ҈㡰 p p Þ x㦨 犨 n㦨҈ k
Þ z㦨Aᆠᨨ矄
҈ ´ ҈҈҈ ´ ҈䰘 ´ ҈ -� ´
꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ � �鼀矄 A B 矄 � ᨨ ��ᨨ 矄ᨨ 矄鼀 ⼄盐 ᆠ泐� �鼀 C�矄 ᨨ 和ᨨ A B � ᨨ 矄 ᨨ鼀
n + ҈㡰 p
��㡰 矄矄犨z҈� �� 졔x㜠 �矄 z� � � 졔y㜠 �矄 z҈㦨Aᆠᨨ矄 k xwt㡰 z㦨Aᆠᨨ矄 k ywt㡰 ᔴᨨ鼀� ᨨᨨ 盐ᆠ犨� �� �ᆠ�ᨨ 鼀├� �
�ᨨ � 犨� ᨨ矄��x ᨨ矄��y 矄
矄盐 犨z㦨z҈졔z z㦨Aᆠᨨ矄 k xwt㡰졔Aᆠᨨ矄 kywt㡰
A ᆠᨨ矄
҈ ´ ҈҈҈ ´ ҈ - ´ ҈҈ ´ ҈ -� ´ 㦨҈҈ ´҈ ҈ F A Dl Dl㦨l a Dq 㡰 矄� �y㦨 l \ F = YAaDq 和 ᨨ 矄矄 犨
�㡰 ␔ 犨ᨨ � �ᆠ犨矄 � � � ᨨᨨ矄� �ᆠ�ᨨ
矄盐�鼀ᨨ矄 犨 � 矄 � � 犨矄 ᆠ ᨨᨨ犨� �犨ᨨ⼄盐�ᨨ 矄 犨�ᨨ 盐ᆠ矄 � �犨 z҈㦨Aᆠᨨ矄 k xwt㡰 z㦨Aᆠᨨ矄 k x졔wt㡰
矄盐 犨�矄�� 泐z㦨z҈졔z Þ z㦨Aᆠᨨ矄 k xwt㡰졔Aᆠᨨ矄 k x졔wt㡰 㦨Aᆠᨨ矄k xᆠᨨ矄wt
矄盐 � 矄�泐犨� ᨨ 犨
Aᆠᨨ矄k x㦨
Þ ᆠᨨ矄k x㦨ᆠᨨ矄
҈ T ҈ YA a Dq = l m l m
㦨
҈- x ҈ x = + ҈ + ҈ -
Þ x㦨黨�├鼀
⼄盐 ᆠ泐ᨨ ᆠ ᨨ矄 泐 �
Y㦨
⼄盐 ᆠ泐ᨨ 盐 鼀 鼀ᨨ ᨨ ��ᨨ
犨 tAH 㦨 �鼀 ├ 泐矄ᨨ盐 ᨨ鼀 A ᨨ � 犨� ᨨᆠ�泐 ҈졔㡰 tHB㦨�鼀├ 泐矄ᨨ盐 ᨨ鼀 �ᨨB犨� ᨨᆠ�泐҈ ᨨ鼀⼄ �㡰 ��㡰 t҈㦨t Þ
矄矄矄 꾀
\T㦨YAa Dq = = 矄� Dl l a꾀Dq Dl = a Dq 㡰 (Q Dl = la Dq Þ l 矄� �
졠 ᨨ ᔴ
꾀 � ᨨ 꾀 �鼀
v҈ v ´ p҈ = ´ p Þ l l
꾀 � ᨨ �鼀
p҈ v҈ = = ᨨ҈҈p҈㦨p p v ҈҈
� � 鼀ᨨ �ᆠ� AC�矄⼄盐ᨨ҈҈ 鼀ᨨ �ᆠ� CB ᨨ犨 盐 鼀 ⼄盐 ᆠ泐� AC 㦨
v҈ ҈҈ = = �� l ´ �
盐 鼀 ⼄盐 ᆠ泐� CB 㦨
v = = �� l ´ �
\ ⼄盐 ᆠ泐� AC㦨맔��㦨҈� ⼄盐 ᆠ泐� CB㦨҈҈맔҈�㦨҈�
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EBD_8341 PS-170
Physics
Electric Charges and Fields
15
ÞF 㦨Fᆠᨨ矄q
Topic-1 : Electric Charges and Coulomb's Law
Fnet =
҈
æ y + a ö ç ÷ è ø q æ ö kq ç ÷ y Fnet = è ø ⏌ y + a 㡰
㡰 졔턼
⼄
⏌
=
ᨨᆠ 盐ᨨᆠ �⼄
졔턼
⏌
=
ᨨᆠ 盐ᨨᆠ �졔턼
⧤턼턼
⧤턼⼄
æ ö ç ÷ èø
Þ
ᨨ ⼄盐���盐鼀
r r + =
b㡰
Þ
├턼턼
├턼⼄
+
㦨 ( ⏌ ) 꾀 턼 = ò r = ò ⏌ ( p )
æ ç ⏌ = p꾀 ò = p 꾀 ç ç è 턼 㦨 p꾀 ҈ ⏌㡰
⏌
\⼄ = -
㡰
q
a
a
Þ F矄� q
q
x
y + a
Qya㡰
和ᨨFµy
㡰 q l
⼄ = 鼀� p Î ´ 鼀�
矄� q ⼄ ⏌ 矄ᨨ q » 矄� = l ⼄ ⼄ l Þ = ᆠ l = p Î 鼀� p Î ´ 鼀�
q
ᆠᨨ矄q
⼄
鼀�
҈⏌
䰘
æ ⼄l ö ÷÷ \ = çç è p Î 鼀� ø ҈⏌
ᨨ µl
ᆠ �㦨 㡰 ␔ ᨨ矄�� ᨨ� ᆠ � �矄 ᆠ ᨨ盐矄� ᆠᨨ 盐ᆠ� �矄 + �鼀 ᨨ ᆠ �├矄ᆠᨨ
+ + 矄盐 ᆠ矄矄 ᨨ犨 � ��盐盐 +
ᆠ � ᨨ 矄 �矄 ᨨ 矄 ᨨ 졔⼄ + ᆠ � 矄 ᨨ 矄 b㡰 ᔴ � q 矄ᨨ �� ᨨ⼄盐���盐鼀 ҈ ҈ ҈ 졔턼 ҈ ⼄ l
졔턼 l
ᔴᨨ 矄� � �⼄盐���盐鼀ᨨ ᆠ �q졔҈ F҈㦨F҈
KQ ´ Q l㡰 黨
=
KQ ( - q ) l
\q㦨 -
Q
æ ҈ q ö çç ÷ ÷ è p Î L ø ᨨᆠᨨ ⼄㡰 盐ᨨᆠ �犨� �ᆠ ᆠ ᨨ盐泐 ᨨᆠᨨ ⼄㡰 盐ᨨ ᆠ �ᨨ 꾀和�鼀�泐 ᨨᆠᨨ ⼄ 盐 ᨨᆠ �쟼犨��ᆠ ᆠ ᨨ盐 盐ᨨ ᆠ �ᨨ 和ᨨ ᨨᆠ 犨� ᆠ盐矄 ᨨ ᆠ � ⼄ ᨨ ᔴ ᨨ 泐 �
F矄� q F ᆠᨨ矄q
y
q =
ö ÷ æ ö ⏌ ҈) ÷ = p꾀 ç ÷ ( è ø ÷ ø
F
a
×
鼀� = ᆠᨨ矄 q
ᆠ㡰 ᔼ 矄� ᆠ�泐 ᆠ � �矄 ᨨ \ Q҈ 㦨 s1 㦨
�矄 ᨨ ᆠᨨ ᨨ� ᆠ �矄졔QQ � 盐ᆠ ᆠ � ᨨ �
矄盐 ᆠ ᨨ ᨨ盐 矄盐 ᆠ \ Q 㦨 s2 㦨 y
kq y
ᔼ ⼄盐���盐鼀 = 矄� q
턼
æ ö ç ÷ ҈ = ᨨ� ç ÷ ç҈ 턼 ÷ p꾀 ø è 㡰 和 矄꾀 ᆠ泐⼄盐ᆠ �矄泐Ἐ⼄Ἐ ├⼄⼄ \ ᨨᆠ犨 鼀 = ⼄ ␔ 꾀 ᔴᨨ盐ᆠ ᆠ �ᨨ ᨨ ⼄ 꾀 = ⼄ ᔴ =
犨 ᔴᨨ盐ᆠ ᆠ �ᨨ ⼄ +⼄ ⼄ ⼄ = = ᨨ犨 ᨨᆠ犨 ᆠ �⼄꾀 ⼄ ⼄ ⼄ ├⼄ 꾀 ⼄ ├ ´ ´ ├⼄ Ἐ = = = =
F �
æqö kq ç ÷ èø
F=
҈ q 犨ᨨ ᆠ�� 盐 p Î L
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PS-171
Electric Charges and Fields
҈
҈°
Y”
KQ
҈҈
FR㦨Fᆠᨨ矄 q
㡰ᨨ 泐ᆠᨨ矄�ᆠ ᨨᆠᨨ 盐矄�ᨨ �矄ᆠ� � L 犨 �ᆠ 犨�├ 犨ᨨ矄矄 矄ᨨ矄矄� �犨
犨ᨨ矄� �矄犨�҈
矄�ᨨ � ᆠ 矄� �犨�⼄盐ᨨ ᆠᨨ矄�ᆠ ᨨᆠ ᨨ 盐矄�ᨨ
졠 ҈ 턼 턼 턼҈㦨턼㦨턼 㦨
㦨맔
㦨
\
҈ pe
(R
Qx
)
⏌
+x
=
҈Q ´ R ù⏌
pe é R + ( R) ë û
=
QR
pe � � R
҈�
ᨨᆠ㦨qE
-QR
҈
҈
�.e. ᨨᆠ�矄 ᨨᨨᨨ�ᨨ ᨨ �矄ᆠ鼀 pe � �R ᆠᆠ ��ᆠ犨� ᨨ ᆠ盐矄�鼀 鼀ᨨ �ᆠ鼀ᨨ�ᨨ T盐
鼀�ᆠ矄 犨 �ᆠ �矄 ��泐ᆠ � �� 矄ᆠᨨ 矄 ᆠ�矄鼀矄矄� ᆠ矄矄
鼀�ᆠ矄 犨 �ᆠ �矄ᨨ矄��泐ᆠ � ᨨ矄矄ᆠᨨ 矄 ᆠ�矄鼀矄矄 ᆠ矄矄 ᆠ㡰 ᆠ�ᆠ ᨨᆠ ᨨ 泐 矄 �矄 q 矄矄盐泐ᔴᨨ盐ᨨ鼀犨 ᨨᆠ 盐 ᨨ ᨨ
矄 ᨨ ᨨ 鼀�
ᆠᨨ矄 q 矄鼀 q ᔼ ⼄盐���盐鼀
矄�
q 졠
ᆠᨨ矄q㦨鼀� 鼀� 矄� q㦨 \ q = �㡰 鼀� 꾀矄 ᨨᆠ犨 犨ᨨᆠ �ᨨ �矄ᆠᆠᨨ � �ᨨᆠᨨ盐ᨨ鼀Ἐ矄犨 ᨨ矄 ᨨ 盐ᨨ 鼀 �盐鼀 ᆠ ᨨᆠ 犨 鼀 鼀� 矄鼀 ᆠ �鼀鼀矄 � �ᆠ�ᆠ鼀 �盐鼀 꾀矄�� ᨨᆠ �� �q
e
æ r ö Ἐ ᆠᨨ矄 q= 鼀� ç ҈ - ÷ è r矄 ø \ q = e r ��㡰 鼀� ҈ - 㡰 r矄 ᨨ犨⼄盐� �⼄ �㡰 ��㡰 r ҈ ҈ ҈- = Þ҈= ҈ r矄 e r矄 \ r矄㦨├�⏌鼀 ᆠᨨ�ᨨ ᆠ㡰�矄ᆠᨨᆠ
Ἐ 矄� q=
꾀 矄�ᨨ Ἐ =
҈
x + a 㡰
㜠q
´
F矄� q
㡰 F
x
x + a
k Q q x
a㡰 턼
q q
ᆠᨨ矄q F 㜠q
x + a 㡰 ⏌
㜠a㡰
F 矄� q
Q´Q Q ҈ ҈ T㦨 ´ = ´ p e L㡰 p e L 矄 쟼ᆠ�ᆠ � ᨨ � �矄ᨨ � � E=
kQq
= e ҈
ᆠ ᨨ�ᨨ 㡰 �矄 � ᆠᨨᆠ 㡰 盐矄ᆠᨨ 矄� ᨨ矄��ᆠ �Q �矄 ᆠx ᨨ鼀 ᨨ��� ᨨᆠ犨 Q qF졠 矄ᨨ� � �矄 ᨨᆠF矄� q ᆠ ᆠ矄ᨨ盐 矄盐 ᨨᆠ
҈
和� ᆠFR�矄 ᨨᨨᨨ�ᨨ ᨨx 鼀ᨨ�ᨨ �矄 ᨨ矄�鼀 鼀ᨨ �ᆠ
ᆠ �Q犨�ᆠᆠ� ᨨ��� �� ᨨᆠ�泐꾀 ᨨ��� ᨨᆠ�矄 ᨨ盐 盐ᨨ鼀ᨨ鼀 盐鼀�犨�ᆠᨨ矄矄 ᨨ���
鼀ᨨᨨ犨 矄 꾀矄� ᆠᨨ鼀矄 ᨨ �� x㜠 �矄 ᨨᆠ�矄 �� ᨨ犨 矄 ᨨ��� ᆠᆠ �턼犨� ᆠ盐ᨨ矄ᆠ�ᨨ泐盐 ᨨ矄�鼀 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ 㡰 ᔼ Q҈QQ Qᨨ矄�� ᨨᆠ犨� ᨨ �졔y㜠 �ᆠ�ᨨ 矄ᆠᨨ鼀ᨨ 矄ᨨ ᨨᆠ矄ᨨ �x㜠 �矄ᆠ ᆠ ᨨ盐ᆠ ᨨ ᔼ Q҈Qᨨ矄�� QQ �� ᨨᆠ犨�ᆠ ᨨ � 졔 x㜠 �ᆠ�ᨨ 矄 ᆠᨨ鼀ᨨ 矄 ᨨ ᨨ矄 ᨨ � y㜠 �矄ᆠ ᆠᨨ盐ᆠ ᨨ ᔼ Q҈ Q ᨨ矄�� Q Q �� ᆠ� ᨨᆠ犨� ᨨ鼀� 盐矄� ᨨᆠ ᨨᆠ犨�ᨨ � y �ᆠ�ᨨ ᔼ Q ҈ Q ᨨ矄�� Q Q �� ᆠᨨ鼀ᨨ 矄 ᨨ ᨨᆠ矄 ᨨ �y㜠 �矄ᆠ ᆠᨨ盐ᆠ ᨨ 和ᨨ ᨨᆠᨨ ᆠ �qᨨ � x㜠 �矄 쟼ᆠ �ᆠ犨� � ⼄盐���盐鼀 盐 ᆠ ᨨ ᨨᆠ 矄�ᨨ ᨨ 矄� �T犨�� � �矄盐 ᆠᨨ矄�ᆠ ᨨᆠFᨨ 犨ᨨᨨ ᆠ �矄 ᨨᆠF 鼀� �ᆠ盐
矄ᨨ� � T � �ᆠ�ᨨ ᨨ mg F 泐� � ᆠᨨ ��ᨨ ᨨ ⼄盐���盐鼀 Tᆠᨨ矄 q 㦨mg錐 T矄� q 㦨F F \ q 㦨 �㡰 mg F㦨 FCA + FCB + FCA FCB ᆠᨨ矄 a
\
+ FCA + FCA ´ F㦨 FCA
F㦨 FCA = ´
kq
҈
��㡰 ur ur a㦨 桌 F CB = F CA T鼀├ � q 犨� �ᆠ 㡰 - ҈�㡰 㦨҈䰘鼀 OC㦨 \ OM㦨黨黨黨䰘 OC ҈䰘 = \ q 㦨 ���㡰 OM 黨黨黨䰘 ᨨ鼀⼄ �㡰 ��㡰 ���㡰
CA㡰
҈䰘 ´ 黨 ´ ҈黨 ´ ⼄ = 黨黨黨䰘 㡰 ´ ҈- ´ 黨
\q㦨҈맔҈黨ᔴ
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EBD_8341 PS-172
� 㝐矄矄ᨨ �ᆠm㦨黨맔҈├� ᔴ �ᨨ �ᆠq㦨҈ᔴ Q + + r㦨҈鼀錐 + + Q㦨҈�ᔴ + r +
ᆠ�ᆠ � 盐盐 � ᨨ鼀泐 + + ᆠ � � �ᨨ �盐矄rᨨ� + �矄 x ᨨ鼀�矄ᆠ ᨨ �矄 �矄 x
Qx
E㦨 k
r + x 㡰 ⏌ \ ᨨᆠᨨ ��ᆠ � qF㦨qE
\ ᨨ
- kQq
´x ᨨ r + x 㡰 ⏌ Qq A㦨k ´x m r + x 㡰 ⏌
F㦨
ᨨ xr A = -
kQq
+
+ + + +
҈䰘
Physics
+
+ + + +
ᨨᆠᨨ ᆠ �q├A 盐ᨨᆠ �矄├B C F҈㦨Fᆠᨨ矄
æ q ö F҈㦨 ´ ç 黨 ´҈黨 ÷ ç a ÷ø è
ᨨᆠᨨ q 盐ᨨᆠ � q㡰├D q ö 黨 æ ´ ç 黨 ´ ҈黨 ´ ÷ a ⏌ 㡰 çè a ÷ø ᔴ泐 犨ᨨ ᨨᆠ矄 ᨨ⼄盐꾀矄ᨨ矄F>F҈ ᆠ �矄 犨�鼀ᨨᨨ犨 矄 ᆠ
q
F㦨黨맔҈黨
Fixed ring
-k Q q
mA㦨
r + x 㡰 ⏌
w㦨
\
p = T
kQq mr
´x
kQq mr
B
kQq mr
é 㦨 ê K ëê
é 黨 ´ ҈- ´ ҈ ù T㦨맔҈ ê 黨 -� - ú ëê 黨 ´ ҈ ´ ҈ ´ ҈ ûú 㦨҈桌҈⏌㦨҈桌㦨矄 ᔼ ᆠ �矄q҈ q矄�鼀�� 盐 ᆠ�ᆠ ᨨᆠ qq F = ҈ 犨�盐矄� �矄ᨨᨨ矄�ᨨ 矄�ᨨ ᨨᆠ T' l 꾀
⼄҈
P
a 㡰
҈黨
q a
+K
q
C
é q ù q´q ù + êK ´ ú ú 黨 a ⏌ 㡰 ûú a ûú ëê
Topic-2 : Electric Field and Field Lines ҈
ᆠ㡰 쟼ᆠ�ᆠ � 盐ᆠ �Q E = 쟼ᆠ�ᆠ � 盐ᆠ �Q҈ E҈ = Q҈ A
mg
ᨨ⼄盐���盐鼀ᨨ P mg ᆠᨨ矄 㦨 TF㡰ᆠᨨ矄a �㡰 N҈㦨mg ᆠᨨ矄 졔 TF㡰矄� a ��㡰 和�鼀�泐 ᨨ ⼄盐���盐鼀ᨨ Q mg 矄� 㦨 TF㡰矄� a ���㡰 N㦨mg ᆠᨨ矄 졔 㡰ᆠᨨ矄a �㡰 �㡰�� � �⼄盐�ᨨ ���㡰泐 �㡰犨� 㦨 a �a 㦨 㡰 ᨨ犨 ��㡰 ᨨ鼀⼄盐�ᨨ ���㡰TF㦨mg ᨨ 㦨 졔mg㦨 q҈q⏌l㡰졔mg �㡰 ���㡰 ᨨ鼀⼄盐�ᨨ ��㡰N҈㦨mg ᆠᨨ矄 졔 㡰矄� N҈㦨mg ᆠᨨ矄 졔mg矄� \N҈㦨 mg ��㡰 ᨨ鼀⼄盐�ᨨ �㡰N㦨mg ᆠᨨ矄 졔mg ᆠᨨ矄 㦨mg ���㡰 꾀�� � ᆠ ᨨ �矄 ᆠ盐 T ᆠᨨ鼀矄 ᨨ ᆠ ᨨ鼀 ⼄盐�ᨨ �㡰 q҈q㦨mgl � 㡰
�矄矄 ᨨ犨矄 q҈ q鼀盐矄 盐 �├ᆠ �矄 ᨨ 矄ᨨ 鼀� 矄�ᨨ 泐 �㡰 ᆠ �矄ᆠ ᨨ 盐qᆠ ᆠᨨ 矄 ᨨ ⼄盐�� � ᨨ盐 ᆠ �Q�矄ᆠ ᆠ ᨨ � �矄矄 ᨨ犨 � ��盐
kQ x
kQ҈ x҈
x҈
ᔴ mg
F
Aq
a
㡰
q q q 㦨K맔 + K 㦨 + 㡰 K a a a
�矄�矄 鼀ᨨ盐 ᨨ 犨ᨨ├ ᨨ鼀ᨨ ᆠ �矄ᨨ� � �泐
Q ⼄
黨 a
F
qQ 黨 q q =Q 맔 K ´ 㦨 ´ K ´ Þ 黨 a a
ᆠ �턼矄 ᨨ盐 �� ᨨ � �泐ᨨ 矄泐矄鼀
mr ÞT㦨p kQq
҈
q
F ᆠᨨ矄 a
��㡰 ᨨ ᆠ �矄 ᨨ 鼀� 矄�ᨨ 泐
҈⏌
҈
a D Q
a
´x
ᨨ w㦨
=
A q
r ᆠ 鼀ᨨ�ᨨ �矄矄�鼀 鼀ᨨ �ᆠ� 盐 ᔴᨨ鼀� � ᨨ⼄盐�ᨨ 犨� A = -w x \
q
q E
x
q E
ᨨ鼀 ��盐 E x q = = ҈ Þ E҈ x
q
黨q
O
B
Q
E҈
kQ x = ҈ kQ҈ x ´ x҈ Q x x Q x Q x Þ ҈ = ҈ Þ = ᨨ ҈ = ҈ x Q҈ x҈ Q x Q҈ x ᆠ㡰 ᨨ 矄 �ᆠ 矄 ҈ Q E= � r ³ R 㡰 pe r 㦨 � rR㡰 ᨨᆠᨨ ᆠ �� ᆠ� � F㦨qE ᨨrR㡰 \F = ҈ Qq F= ᨨr R㡰 pe r x
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PS-173
Electric Charges and Fields
ᆠ㡰 ᆠ�ᆠ � ᨨ 盐ᆠ 盐ᨨ盐 � ᨨ鼀泐ᆠ � � � � �矄盐 � ᨨ鼀和ᨨᨨ�ᨨ 㡰 㡰犨ᨨ � 꾀 졔 ᆠ � 矄�泐s졔 �矄 ��� � 鼀� �盐 矄ᨨ �矄 � ᨨ �Y �ᆠ�ᨨ 犨���� � ᨨ ᆠ � 矄�泐s � X �ᆠ�ᨨ ᆠᨨ�ᨨ ᆠ㡰�矄ᆠᨨᆠ s E҈ ER ER s E
E A ҈ ´ � æ 黨 ö 黨 ҈ = =ç ÷= ´ = ´ ҈䰘 è ҈䰘 ø ҈䰘 EB ᆠ㡰
쟼ᆠ�ᆠ � E = E ҈ - x 㡰
b㡰 矄� � vu㦨aS �㡰 u㦨s㦨x 꾀矄ᨨ Fᆠ�ᆠ 㦨 ma
\ ᨨᆠ F = qE = qE ҈ - x 㡰
ÞqE㦨ma Þ a =
E E E ҈
E
E
ᆠ㡰 �
꾀矄ᨨ F = ma = mv
dv dx
dv ö æ çèQ a = v ÷ø dx
v = 黨
qE ҈ - x 㡰dx Þ v dv = m ᔼ �� �ᨨ 矄� 矄犨� v
qE ҈ - x 㡰 dx 黨x ö v qE æ x Þ = ç ÷= ò m m çè ÷ø x
ax =
ᆠᆠ�ᨨ ᨨ �ᆠ�矄ᆠᨨ 矄 � �� ᨨᆠ�泐�矄 ᨨ ᨨ �矄 矄�� � ay = g
泐
쟼
졔s 졔s
r s ᆠᨨ矄 ° xˆ 矄� ° yˆ 㡰 E = e
s æ ҈ ö yˆ ÷ çç xˆ e è ÷ø 쟼ᆠ�ᆠ � � ��ᨨ 矄 ᨨ犨 � ��盐 㡰
E m
r r r æ s é ҈ ö ù EP = E҈ + E = ê xˆ + çç҈ ÷ yˆ ú e ëê ÷ø úû è
a=
æ E ö çè ÷ +g mø
҈
Rö æ b㡰 쟼ᆠ�ᆠ � 꾀 ç R Ἐ = ÷ ø è q E A ds = e
A 3R
r s ( R ⏌ ) æ sR ö =ç ÷ Þ E A = e è e ø æ Rö e × p ç ÷ èø 쟼ᆠ�ᆠ � 矄�B
r Þ EA =
=
r F = qE�ˆ + mgˆ
æ Rö r´ p ç ÷ èø
쟼҈
㡰
ᨨ鼀 ��盐 r s E҈ = yˆ e
Þx=
qE x m
a 㡰 ᨨᆠ ᆠ� � ᨨ
�ᆠ
)
qE qE Þ a= m m 和盐矄�盐� � 盐矄� �㡰犨�
dv \ mv = qE ҈ - x 㡰 dx
Þ ò v dv =
r r E 鼀盐矄 �ᨨ p \ Eˆ �矄ᨨ �ˆ + ˆ - kˆ
(
졔s
졔s
E
�
r r
和� ᆠ r × p =
䰘
҈
r ҈䰘 æ sR ö sR sR Þ EB = Þ EB = ç ÷ e �e � è e ø
æRö k ´ r´ pR k ´ r´ p ç ÷ r èø EB = R æ Rö ç ÷ è ø
r sR æ ҈ ö ( s ) p æ R ö Þ EB = -ç ÷ ç ÷ e è pe ø æ R ö è ø ç ÷ è ø
⏌
r xˆ ù s éæ ö ᨨ EP = êç҈ ÷ yˆ ú e ëêçè ÷ø ûú ᆠ㡰 ᔼ �ᆠ�ᨨ
Fx㦨qE Þ ma x㦨qE E q Þ ax = m ᨨ 矄 ᨨ ᨨ盐
v + vx = v 㡰 Þ v x = v = a x t Þ v = + ®
®
®
®
®
qEt Þ t= m
v m E q
҈҈ 㡰 E = E + E 㡰 + E + E 㡰 ҈ ᨨE㦨Eᆠᨨ矄aEᆠᨨ矄b –q d q d d q d –q
E3 a D
E1
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㦨
Physics kq
D + d 㡰
D
´
D + d
kqD
=
⏌
D + d 㡰 ᨨ
Eµ
D
-
-
kq
D + d 㡰
D
´
D + d 㡰
kqD
D + d 㡰 桌⏌
ᆠ㡰 � � 矄 ᨨ��� �ᆠ盐 ᨨ鼀 ᨨ矄�� ᆠ � 鼀� �ᆠ盐 ��ᆠ � 盐 �矄矄泐矄鼀 ᆠ 矄 ᆠ�ᆠ �ᨨ ҈ 㡰 쟼ᆠ�ᆠ � 盐 ᨨᆠᨨ鼀 �矄ᆠ R 㦨 a㡰 �矄 ᆠ x ᨨ �矄 �矄 ҈�
é ù x s ê҈ ú E҈ = e êë R + x úû s é h ù é x 㦨h ù ҈ = e êë a úû ëê R = a ûú E҈ =
҈
µ
D D ҈ 㡰 꾀⼄盐���盐鼀矄盐 ᨨᆠᨨ ᨨ鼀盐矄 ᨨ矄ᨨ Tᆠᨨ矄q㦨mg �㡰 T矄� q㦨qE ��㡰 Y 和ᨨ� � �㡰 ��㡰犨� q =
qE mg
-
� ´ ҈ ´
҈ = q = - ´ ҈ ´ ҈
҈
T
X q mg q㦨�맔҈ ᔴ 쟼㦨⏌鼀鼀㦨맔҈├�桌
ᆠ c =
rö
҈ Q · 錐 pÎ R Q⏌ ҈ Q ҈ E = E = · · pÎ R 錐 pÎ R
҈
쟼Ἐ =
\쟼Ἐ =
é R R ù é R ù = pr ê - ú = pr ê ú úû êë êë ҈ úû
=
pr R
æ pr R ö Epr = ç ÷ è Î ø \
쟼ᆠ�ᆠ � ᨨ盐矄� E =
s Î
r R ҈ Î r
ö ÷ ÷ ø
æ ç҈ ç + è
ö ÷ ÷ ø
s æ - ö ç ÷÷ = 쟼 Î çè ø ҈
ö æ çè 쟼 - 쟼÷ø ´ ҈ ҈ ҈ 奠 ; 䰘䰘奠 = = 쟼 ҈ 㡰 Wh h ᆠ�ᆠ � �矄 ᨨ
ᔼ ⼄盐���盐鼀 ᨨᆠ 盐ᨨᆠ�ᆠ � 㦨犨�� qE㦨mg Þ qE㦨
æ ç҈ ç + è
\奠 盐ᆠ�ᨨ � 盐ᨨ ᆠ�ᆠ �
҈黨
s Î
ᨨ鼀⼄盐矄�ᨨ 㦨 �盐矄ᨨ �矄ᆠ㡰
ò çè҈ - R ÷ø r dr = pr ò r dr - rR dr R
s �� 㡰 Î
쟼ᆠ�ᆠ � ᨨ � �矄 泐 �矄 ᆠ ᨨ鼀 ᆠ ᨨ �矄ᆠ
éé ù R é ù R ù ù é ê ê r ú - ê r ú ú = pr ê R - R ú = pr êê ú ê R ú ú êë R úû ë ë û ë û û
q=
ᔴ泐E>E҈>E 㡰 쟼ᆠ�ᆠ � � 矄�泐 ᆠ ᨨ �矄ᆠ
쟼=
Qdv㦨p 㡰
s ae
E҈ =
҈䰘 ᆠ㡰
ò
Ræ
a ᨨ
h ù s é hù s é ҈ ҈ 㦨 sh e êë a úû e êë a úû e a
q�n = dq = rdv
= pr
a
s æ hö ç҈ ÷ e è a ø 쟼ᆠ�ᆠ � 盐 ᨨ �� �矄ᆠ E 㦨 E҈ E
)
rö æ = r ç҈ - ÷ pr dr è Rø
ù ú úû
E =
)
(
é h ê҈ êë a + h
和�鼀�泐 ᆠ�ᆠ � 盐 ᨨ �矄ᆠ R 㦨 a㡰
q
æ҈ö Þ -҈ ç ÷ èø b㡰 쟼ᆠ�ᆠ � ᨨ �矄ᨨ � �ᨨ �盐矄 �矄 ᆠ ᨨ鼀 ᆠ ├턼 쟼= ⏌ + 쟼 = ᔴᨨ ��ᨨ ᨨ鼀 �鼀盐鼀 ᆠ�ᆠ � é ù d ê kQh ú Þ ú =0 dh ê ⏌ ê R +h ú ë û 泐盐矄� � ᆠᨨ ᆠᨨ 鼀 �鼀 鼀� �鼀犨� = rö æ 㡰 ᔴ � 矄�泐 r = r ç ҈ - ÷ è Rø dq㦨rdv
(
҈
q qE
s e
pR rg pR rg \ q = E
�㡰
Wh h ᆠ�ᆠ � �矄 矄犨�ᆠh ᨨ ␔�� ᨨ ᨨ㦨 �矄ᆠᨨ盐矄 ᨨᆠᨨ ᨨ 鼀�㦨 phRvt
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PS-175
Electric Charges and Fields
pR rg = phRvt \ R =
黨hvt rg
é s s $ s s ù $ + + ú - k 㡰 㦨 - e k ë e e e û
㦨 ê
��㡰
é 黨hvt ù rg ´ ᨨ鼀⼄ �㡰 ��㡰 q = p ê ë rg úû E
ᆠ �矄 ᨨ 矄盐 ᆠ矄 ᨨ 鼀�ᆠ 矄 矄
矄 ᨨ犨 � ��盐 ᆠ � 矄�泐 σ㦨 ⼄盐矄�ᨨ
Q҈ pR =
=
졔
Q҈ + Q p R 㡰
p R 㡰
졔
Q҈ + Q Þ Q = Q҈
졔
졔
Q҈+ Q㡰 졔
졔
–Q҈
졔
졔 Q҈
R
졔
졔
졔
졔
졔
Q҈+ Q+ Q
졔
졔
�
졔
졔
졔
R
졔
R
졔
졔
Q҈ + Q + Q = Q҈ 黨
졔
졔
䰘
졔 졔
b㡰 ᆠ � 盐 � ᨨ盐鼀 s O ᆠ ᨨ 盐 � ᨨ鼀矄 ᆠ � 矄ᨨ� 矄 盐矄ᆠᨨ 矄� 盐 � ᨨ鼀泐ᆠ � 矄 ᨨ �� ᆠ � 矄�泐s � ��矄ᆠ OἘ꾀矄ᨨ OOἘ⼄盐ᨨa
r҈
ur s uur E = r Ἐ \ ᨨ ᆠ�ᆠ � e ur ur ur s E = E҈ + E = a e ᆠᆠ�ᆠ � 犨� � �盐犨 �ᆠ 犨�盐 � ᨨ鼀 ᆠ㡰 �ᆠ�ᨨ ᨨ ᆠ�ᆠ � 矄ᆠᆠᨨ � �ᨨ ᆠ � ᨨ 矄 矄�ᨨ � ��z㜠 �矄 E1 E3 Z=a
E2
2 -2s
ᨨᆠ�ᆠ � ᨨ� P
ur ur ur ur E = E҈ + E + E 㦨E҈ - k$ 㡰 + E - k$ 㡰 + E - k$ 㡰
3 -s
E
+ R
E E
S
\E㦨
rR pR r ҈ pR r ⏌ rR = 㦨 e ´ ´ e 黨 ҈ pe R pe R
\k㦨 㡰 矄 �ᆠ矄 ᨨ �盐矄x �ᆠ├ 矄矄dx ᨨ盐鼀ᨨ �矄矄 㦨px dx㡰ᔴ � ᆠᨨ矄 � �矄矄 dq = px 㡰 dx ´ kx a ᨨ dq = pkx + a dx ᨨr=R
ᨨᆠ � ᆠᨨ矄 � 矄 ᨨ �盐矄R R
Q=
\
ò
E҈ =
pk x + a dx = pk
dx x
R + a + a
쟼ᆠ�ᆠ � r㦨R
҈ pkR + a ҈ pk ҈+ a = R pe + a 㡰 R pe + a
ᨨr=R⏌
ᨨᆠ � ᆠᨨ矄 � 矄 ᨨ �盐矄R⏌ R⏌
Z = –a
O
E
Q ҈ pe ( R )
lᆠ泐� 㦨 pR r
꾀矄ᨨ ᆠ�ᆠ � 盐ᨨ矄鼀矄
P
pe ( R )
졠¢
Z = 3a
g
졠
ur s ur E҈ = r҈ e
1 -s
r
+T
pR r æ Rö 턼矄 㦨 p ç ÷ r = è ø
盐矄 ᆠᨨ 矄� �泐 ᨨ� P � 矄鼀矄 ᆠ�ᆠ � 盐ᨨᆠ �ᨨ ��矄 盐矄� �盐矄矄Ἐ矄 ᨨ鼀
-
r҈
E U
㡰 ᆠ�ᆠ� 矄ᨨ ᨨᆠᆠ
ᨨ 鼀�ᆠ矄 矄 쟼 㦨 � 矄� 矄ᨨ� 鼀�ᆠ 矄 꾀矄ᨨ ᆠ�ᆠ � 矄 ᨨ ᨨᆠ ᨨ 鼀�ᆠ矄盐 ᆠ ᨨ鼀泐 ᆠ㡰 쟼ᆠ�ᆠ � 泐犨 � 矄� 鼀�ᆠ ᨨ�ᨨ � ᆠᨨ 盐ᆠᨨ ᨨ 矄 �矄 ᨨ 쟼ᆠ�ᆠ � � 矄犨泐矄 ᨨ鼀ᨨ矄盐 ᆠ ᆠ㡰 쟼ᆠ�ᆠ� 矄ᨨ ᨨᆠ ᨨ ᨨ ᨨ鼀ᆠᨨ矄 ᨨᨨ矄 㡰 鼀� �盐 ᨨ ᆠ�ᆠ � ᨨ� P犨 �ᆠ �矄 �矄 ᆠR ᨨ鼀 �矄ᨨ ᆠ泐� 쟼㦨쟼ᨨ쟼ᆠ�泐 㦨
ÞQ҈Q㦨Q ÞQ҈Q҈㦨�Q҈㦨Q \ Q҈ Q Q = ҈ � ҈
E
ur ur E졔 E 㦨 E
졔
Q҈+ Q㡰
졔
Q҈ + Q + Q
\ Q҈ =
ᆠ � 矄 졔
r
和�鼀�泐 쟼ᆠ�ᆠ � 盐 ᨨ Q ᨨ O�矄ᆠ ᆠ 泐ᆠ�ᆠ � 盐ᨨT O
ᆠ�ᆠ � 盐ᨨRᨨ O�矄 � 矄鼀 �ᆠ�ᨨ 矄 ᨨ U ᨨ O ᆠ ᆠ�ᆠ �
ᨨq㦨䰘맔҈҈黨ᔴ b㡰
Kq
쟼ᆠ�ᆠ � 盐ᨨPᨨ O�矄ᆠ ᆠ 泐ᆠ�ᆠ � 盐ᨨS ᨨ O P+Q
é 黨 ´ ҈ ´ ҈-� ´ ´ ҈- ù 黨 ´ 黨 ´ 䰘 = ´pê ú ´ ´ 黨 ´ 黨 ҈p ´ ҈� ë û
ur
쟼ᆠ�ᆠ � E =
ᆠ㡰
QἘ =
\
ò
pk x + a dx =
pk R ⏌ 㡰+ a + a
ᆠ�ᆠ � r = R⏌
E =
҈ pk R ⏌ 㡰+ a ҈ pk æ R ö = ç ÷ pe + a R ⏌ 㡰 pe + a è ø
҈+ a
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EBD_8341 PS-176
Physics
҈ E ҈ ҈+ a ҈ pk æ R ö ҈ ҈ pk ҈+ a = ´ R çè ÷ø pe + a 㡰 pe + a
Q E = \
盐 矄� q =
�鼀ᨨ �� ҈ = 泐
=
Þ ҈졔a㦨\a㦨 黨
⼄E꾀㡰和� ᆠᆠᨨ矄�ᆠ � �矄ᆠᨨ 矄� � \ WPQ 졔 WQR 졔 WRS㡰졔WSP㦨 ᨨ WPS졔WSP㦨ᨨ WPS㦨WSP \
(
=
)
҈
(
�鼀
E
B � � A
E
KP R
P O
⧤
⧤
� �ᆠ�ᨨ ᨨᨨ矄� ᨨ �ᨨ ᆠ�ᆠ � 矄 ᨨ盐 ᆠ ᆠᨨ盐 �矄 � �EB㦨 ®
r - ⧤
ᆠ�ᆠ � 꾀
q
E
ᨨ ᆠ�ᆠ �ᨨ ᆠ�ᆠ � 盐ᨨ �ᨨ�矄
®
⼄ 㦨鼀 㦨⼄쟼\泐 = 쟼 泐 㦨 맔҈҈ 鼀 é ⼄ ù ҈ ᔴ├� -҈ ҈ �� 㡰 ú Q = êë 鼀 û
쟼
´҈҈
�ᆠ ᨨ鼀졠ᨨ꾀㡰矄ᨨ� 꾀�矄ᨨ �� ᨨ ᆠ�ᆠ �ᨨ ᆠ�ᆠ � 쟼矄 ᨨ盐 矄ᨨ� �ᆠ�ᨨ
ᨨ 졠ᨨ꾀 ᨨ犨ᆠᨨ 矄� � �ᨨ� 犨 �ᆠ �矄ᨨ� ᨨ ⼄盐�ᨨ��
� 㦨�鼀
盐 = ҈ ´҈ 鼀矄 -҈
ᨨ犨ᆠᨨ 矄� � �ᨨ� 꾀 ᆠ�ᆠ � 盐ᨨ �ᨨ
盐 쟼 =
Þq㦨 ᨨ҈Þ q㦨 ᨨ �ᆠ �矄 �� q㦨 ᨨq㦨
)
ᆠ�ᆠ �矄 ⼄盐�ᨨ � 꾀矄ᨨ KP/R ᆠ�ᆠ � �矄 ᨨ鼀ᨨ矄盐ᆠ E ⼄ � �矄 �ᆠ�ᨨ ᨨ ᆠ�ᆠ � �矄 � � ᨨ 鼀� �盐 ᨨ ᆠ�ᆠ � 矄 ᨨ盐 ᨨᨨ� 矄ᨨ ᆠ�ᆠ
Þ 矄� q =
´ ҈ ´҈ ´
P ˆ ˆ � +
ᔼ ��矄 ᨨ ᆠ � �ᆠ 盐 = ҈ ´ ҈ 鼀⏌矄
盐 矄� q ´ ҈҈ 矄� q Þ�= � 㦨 泐 ´ ҈҈
´ ҈҈
b 㡰 � �ᨨ鼀ᨨ鼀 ᨨ ᆠ�ᆠ �ᨨ P =
\ WPS㦨WSP㦨qEa \␔ᨨ├ ᨨ 泐 � ᨨ �ᨨᆠ矄矄PQRS㦨qEa 矄 ᨨ�ᆠᨨ鼀ᨨ� ᆠ�ᆠ盐鼀ᨨ�ᨨ 犨 ᆠ � ᨨᆠ犨 �ᆠ �矄 ᨨ� T盐
ᆠ�ᆠ � ᨨ 盐ᆠ 犨 ᆠᆠ�ᨨ �矄 盐 � ᨨ鼀 ᨨᆠ ᆠ� � ᨨ ᆠ � �ᆠ ᆠ �
ᆠ�ᆠ � �矄�� 泐F㦨qE ᆠ ᨨᆠ矄 ᨨ 犨ᨨ ᨨᨨ 矄 � �ᆠ � � ⼄盐 ᆠ � 盐 � ᨨ鼀ᆠ�ᆠ � T盐 ␔ �� �泐X㜠泐鼀 矄 �犨�├ ᨨᆠ├ᨨ盐 ᆠᨨ 矄 ᨨ鼀 矄鼀鼀 盐ᨨ ᨨᨨᆠ�ᆠ ᆠ 鼀├� �� ᨨ矄��泐ᆠ � ᆠᨨ矄�ᆠ ᨨᆠ� �ᆠ�ᨨ ᨨ ᆠ�ᆠ � ᆠ矄 ᨨ 犨� ᆠ � �ᆠ�ᨨ ᨨ ᆠ�ᆠ � r -҈ bᆠ㡰� ᆠ�ᆠ � 쟼 泐 = - ᔴ
҈
� m矄 ᨨ q= ° 㡰
ur
WSP = qEa
� m矄 ᨨ q = ° 㡰
盐 矄� q =
�鼀ᨨ �� = 泐
r uur ␔ᨨ├ ᨨ 㦨 qE SP (\ F = qE ) r r WSP = qE�ˆ ( rP rS ) (Q E ҈҈´ �矄 )
WSP qE�ˆ a�ˆ + bˆ
´ ҈ ´ ҈ ´
쟼꾀 =
r ⧤
�㡰
R ®
®
®
®
+ 쟼 㦨 - 쟼 + 쟼 = - 쟼 ҈ ´ 꾀�� ᨨ鼀⼄ �㡰쟼㦨 pÎ R 㡰 \ �
㦨 p Î 쟼
҈⏌
é ù \ 㦨 ê ú 쟼 pÎ û ë
㡰 盐矄ᆠᨨ 矄� ᨨ� 㝐� 矄� ᆠ�泐犨 ᆠ�ᆠ � 矄ᨨᆠᆠ盐 꾀矄矄盐鼀 ᆠ�泐ᨨᆠᨨ � 矄�鼀�ᆠ � �矄�盐�ᨨ ᨨ ᨨ矄�� ��ᆠ � 矄 矄ᨨ 矄 쟼ᆠ�ᆠ � M 盐ᨨ盐 � ᨨ鼀泐 �矄�盐 ᆠ �ᨨ 犨 ᨨ 矄 ᨨ �盐矄R҈
r r r E҈ = r e
쟼ᆠ�ᆠ � 㝐 盐 ᨨ
�� ᆠ � �矄�盐�ᨨ � ᆠ�泐
ur r uuur E = MP e
㝐
졠
R҈
\ ᨨᆠ�ᆠ � 㝐� 矄� ᆠ�泐
ur ur ur r r r uuur E = E҈ + E = r+ MP e e
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PS-177
Electric Charges and Fields
ur r r r r r r uuur r a r éQ r + MP = a ù E= r+ ë û e e ur r r \ E= a e ur ᆠ� 矄� ᆠ�泐 E �矄 盐 � ᨨ鼀 ᨨ �矄 鼀� �盐 r �ᆠ�ᨨ ᨨ a
(
\
)
r
ᆠ㡰
҈
쟼ᆠ�ᆠ � 쟼҈ 盐ᨨ矄鼀矄 �꾀
҈ E҈ = pe
꾀
Q
E҈㦨 pÎ r
졔Q 졔 졔 졔
r
졔졔 졔졔 졔졔
r
E㦨
҈
l pÎ r
҈ 턼 s = Þ 턼 = ps pÎ Î
㡰
쟼ᆠ�ᆠ � E 盐ᨨ���矄 �B
p R r R r R ҈ ( ) E = = pe e ( R ) E҈ = E
쟼 r㡰㦨쟼 r㡰
҈ l s l = Þ = pÎ Î sp ᆠ㡰
= Q
쟼҈ ⏌㡰㦨
҈ 턼 p Î
´ l ҈ ҈ l ´ = p Î p Î E҈ r㡰㦨E r㡰
\ Q㦨lr
黨
é ҈ l ù ҈ l ú= ë pÎ û pÎ
꾀 E r⏌㡰㦨 ê
҈ 㡰 E r⏌㡰㦨 p Î s s = E r⏌㡰㦨 Î Î 䰘
p R r ҈ ( ) E = pe ( �R ) E҈졔E㦨
҈ pe \
pR r҈ p ( R ) r ҈ + = pe R ( ) ( � R )
r҈ = r �
r҈ R r R = Î ´ Î
Þ
r҈ = r
㡰 ᔴ泐 ᨨ鼀 ��盐ᆠ�ᆠ � � 矄ᨨ�� � � ᨨ鼀Q҈ 鼀� � �ᨨ Q \ Q҈�矄ᨨ矄�� Q �矄 �� 盐鼀ᨨ ᆠ�ᆠ � � 矄ᨨ��� � � ᨨ鼀 Q҈�矄 鼀ᨨ
鼀� � � Q \Q҈>Q 和� ᆠ � � �矄 ᆠᨨ �� ᨨ Q ᆠ�ᆠ � �矄 ᨨ
ᆠ�ᆠ � ᆠ 泐Q�ᆠ盐ᨨ� 犨�ᆠ ᆠᨨ盐 ᆠ�ᆠ � ᆠ 泐Q҈盐 � � �矄 ᆠᨨ ᨨ Q҈ ᆠ�ᆠ � �矄 ᨨ 㜠 ᨨ ᆠ㡰
ᆠ�ᆠ � � 矄� 矄
҈ Qq r pe R
E
ᨨEµr졠盐矄� 矄 Rr¥㡰E㦨 ᨨEµ
҈ Q pe r
A
E
҈
B r
r
O r=R ᆠE� ᆠ矄矄 ᨨrR ᆠ矄矄 ᨨR¥
b 㡰 쟼ᆠ�ᆠ � E҈ 盐ᨨ矄鼀矄 ἘAἘQ
쟼ᆠ�ᆠ � E 盐ᨨ���矄 ἘBἘQ
\
rR㡰E㦨
l l ҈ l = = ⏌ p Î pÎ l = ´ Î p
pR r҈ ҈ E҈ = pe ( R )
졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 ® ® 쟼 쟼҈ 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 r
s E㦨 2Î
쟼҈ r㡰㦨쟼 r㡰
\
졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 졔 r҈
턼
E҈ r㡰㦨E r㡰㦨E r㡰 � 㡰 㡰
r҈ ´ pR r pR r҈ R ҈ ´ ҈ = 㦨 p Î Î R㡰 ´
b㡰
ᨨᨨ� ᆠ �쟼㦨
쟼 µ
҈ d
kp
d
�
ᨨ �ᨨ
a
k l r
҈
쟼㦨 �쟼 µ d d ᨨ � � �ᨨ �� ᆠ � 쟼㦨
kQ
k l ҈ �쟼 µ d d
k l r
졔l l ᨨ犨ᨨ� � �犨�矄ᆠ泐� �盐 � ᨨ鼀� ᆠ � 矄�泐
쟼㦨
l
l k l k l l k l ´ = ᆠᨨ矄 a 㦨 r d + l d +l d +l
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Physics
ᨨ E µ
҈ d
Q l s ) e
rr = 쟼
盐 ᨨ盐� ᆠ矄㦨
␔ᨨ├ ᨨ 泐ᆠ�ᆠ � W㦨qv㦨 ( q E ) a ᆠᨨ矄 �o
\
W㦨
q s҈ - s 㡰 a e
Topic-3 : Electric Dipole, Electric Flux and Gauss's Law ҈
b㡰 v 矄 ᨨ �ᨨ 矄� � �泐ᆠᨨ 矄�ᨨ
K� + U � = K f + U f Þ -
k × p҈ r
҈ ҈ p ᆠᨨ矄 ҈°㡰 = mv + mv +
qE md
㦨쟼ᆠᨨ矄q ␔ q㦨 �犨 �ᨨ �ᆠ盐ᨨ � 桌 㦨 ҈黨㡰 ҈㡰ᆠᨨ矄� 㦨䰘䰘맔҈ 㦨䰘맔҈䰘 b㡰 ␔ ᆠ盐�矄ᨨ 矄� ᨨ� ᆠ �턼�矄 ᆠ ᨨ ᆠ盐 ᨨᆠ�ᆠ 盐 盐 ᨨ �矄 ᆠ � 犨� 矄矄 泐 ᨨ盐� 矄� ᆠ矄 ᨨ ᆠ盐 和ᨨ ᆠ�ᆠ 盐 ᨨ盐� ᨨ ᆠ犨�⼄盐ᨨ ҈⏌ ᨨ ᨨᆠ�ᆠ 盐 盐ᨨ �矄ᆠ �
\ᆠ�ᆠ � E㦨E҈E
=
b㡰
㦨qE맔
�
ᔼ ᨨ盐鼀ᆠ � 矄�泐�矄ᆠᨨ 矄 쟼µ
㦨
Q ҈ Q ÞE= r³R pe r e
b㡰 t㦨PE矄� q ᨨ Ia㦨PE q㡰
a=
ö æ 4π \ ⼄㦨 çè - ÷ø ᨨ㦨 ò π ò
S E㦨
q҈l e
E맔pr㦨
ᔴ �� 矄 鼀 ⼄㦨맔
®
Ñò E d A = ᨨ
和盐 ᆠᆠ � 矄�泐 矄ᨨ 泐 盐ᨨ Q꾀矄ᨨ
®
ᆠ㡰
㡰 b㡰
p ҈ a p Î ma
\ 䰘
턼 Îᨨ
盐 ᨨ盐� ҈ ᆠ㦨
턼 Îᨨ
㡰 T㦨쟼矄� q ᨨ⼄盐 � ᆠ 泐 �ᨨ� ᆠ�ᆠ r r r � T = P ´ E
r p 㦨pᆠᨨ矄q �ˆ 졔p 矄� q ˆ r r E҈ = E� r r r T҈ = p ´ E҈ 㦨 pᆠᨨ矄q �ˆ 졔p矄� q ˆ 㡰맔E �ˆ 㡰 t kˆ 㦨pE矄� q kˆ 㡰 �㡰 r ˆ E = E҈ r T = p ᆠᨨ矄 q�ˆ + p 矄� qˆ 㡰 ´ E҈ ˆ ��㡰 tkˆ = pE҈ ᆠᨨ矄 qkˆ
ᨨ鼀⼄ 矄 �㡰 ��㡰 pE矄� q㦨 pEᆠᨨ矄q q 㦨
\
q㦨
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PS-179
Electric Charges and Fields
黨
(c) 盐 � ├ 犨� ᆠᨨ矄 矄盐 ᆠ矄和҈和和和
҈
҈ ⼄㡰 ᨨ矄盐 ᆠ和҈ f҈ = e ҈ ҈ ⼄ + ⼄ + ⼄ - ⼄㡰 = ⼄ ᨨ矄盐 ᆠ和 f = e e ҈ ҈ ⼄ + ⼄㡰 = ⼄㡰 ᨨ矄盐 ᆠ和 f = e e ҈ ҈ ⼄ - ⼄ - ⼄㡰 = ⼄㡰 ᨨ矄盐 ᆠ和 f = e e ᆠ f҈ 㦨 f 㦨 f 㦨 f � ᆠ�ᆠ 盐 �矄 矄鼀 ᨨ 矄盐 ᆠ矄 ⧤� 鼀� ᆠ�ᆠ � 盐ᨨᆠ �ᨨ盐矄� 和 和㡰 盐矄矄� 矄盐 ᆠᆠᨨ �盐矄 ᨨ 盐 ᨨ盐� 矄盐 ᆠᆠ盐矄矄鼀 泐� 矄 盐ᨨ ᆠ � 矄盐 ᆠ 矄� ᆠ㡰 꾀泐� �盐矄矄矄 犨 r uur Q Q + pAr - pAa Ñò S E × ds = Î \ E맔pr㦨 Î
dr dV Q㦨rpr r=
盐矄矄�鼀 矄盐 ᆠ
r
A Q㦨 ò pr dr 㦨pAra桌 r a
E㦨
ù ҈ é 턼 - p꾀 + p꾀 ú ê p Î ëê ûú
ᨨEᨨ� ᨨ �r QpAa 㦨 \A㦨
҈
҈
ᆠ㡰
҈҈
҈ ҈ p Î ur ᆠ㡰 � E = E xˆ
졔⼄
� ᆠ � 졔졠 q
⏌ ⏌
Nm C f҈f҈҈㦨
⼄
f҈҈ =
ur A 㦨a´ a = a
E a 㡰 ´
҈
҈
㡰 盐矄ᆠᨨ 矄� � �鼀 ᨨ �盐矄 �ᆠ├ 矄矄 和盐 ᆠᆠ � 矄�泐ᨨ �矄ᆠᨨ �盐矄
ö æ s 㡰 = s ç҈ - ÷ è ø
ö æ s 矄㡰 = s ç ҈ - ÷ è ø
ᔴ �ᨨ �矄ᆠ鼀
㡰 쟼 q 꾀 㡰
2
泐
f =
ö æ ⼄ = s ç҈ - ÷ p è ø
ö æ ⼄ = s ç҈ - ÷ p è ø ᨨ犨 ᨨ鼀盐矄矄Ἐ矄 ᨨ鼀쟼ᆠ�ᆠ 盐 ᨨ盐� �矄 �ᆠ 矄盐 ᆠ ᆠᨨ矄矄 ᆠ � �矄ᆠᆠᨨ鼀泐
q㦨�
㡰
q
\ f҈҈ = é ´ �ˆ y + ҈㡰 ˆ ù �ˆ = ҈ x ë û
ur
㡰
ᆠ泐�
r uur ᨨ矄盐 ᆠBCGF fn = ò E dA
Q 꾀 �犨 쟼 꾀턼㦨� 㡰
҈ a⏌ = a ⏌
\ f҈ = ò E A ᆠᨨ矄 黨° =
� ᆠ�ᆠ � E ᆠ矄 ᨨ � 졔 x �ᆠ�ᨨ �矄 ᆠᨨ 矄
ᨨ ᆠ�ᆠ 盐 ᨨ盐� 矄 ᨨ�ᨨ 犨 ᨨ矄
㦨Ea
䰘C
\쟼ᆠ�ᆠ 盐 矄矄� � ᨨ盐� 꾀 ᔴ ᆠ泐� �ᆠ矄盐 ᆠ� ᨨ lL �㦨 ᆠ 㦨 e ҈� 㡰 r r 盐 ᨨ ᆠ�ᆠ � E ᨨ盐� 泐 A �矄 � 矄 f = ò E A ᆠᨨ矄 q q㦨 �犨 ᆠ�ᆠ � ᆠᨨᨨ 矄盐 ᆠ ᨨ矄盐 ᆠABCD꾀 �q㦨黨
r r f = E A = EA ᆠᨨ矄 q= E a 㡰 ᆠᨨ矄 � 㦨
ᨨ鼀 ��盐 q㦨
( ␔ L = � ᨨ ᆠ泐� )
=
⼄
C
\ q㦨 쟼ᆠ�ᆠ 盐 ᨨ盐� ᆠᨨ鼀ᆠ泐� 泐 盐矄矄矄 ᨨ鼀 q �n lL 㦨 fᆠ泐� 㦨 e e
졔⼄
҈
X C
㡰 盐 ᨨ盐� 盐矄矄� 矄盐 ᆠ�矄 盐ᨨ ᆠ �矄 � 矄� 盐矄矄� 矄盐 ᆠ盐 ᆠ�ᆠ � ᨨ 盐矄矄�
矄盐 ᆠ�矄 ᆠᨨ矄盐鼀ᨨ ᆠ�ᆠ � 矄 盐ᨨ ᆠ �矄 �.e.⼄҈⼄҈ ⼄
҈ 㡰
Q
pa ᨨᆠ ᨨ � ᆠ�ᨨ
Z
q -C f = �n = e e
턼
Y
㡰 ᨨ鼀 ��盐 ᨨ ᆠ � ᆠᨨ矄 � ᆠ盐�ᆠ 矄盐 ᆠq� 㦨C졔C䰘C㦨 C 꾀ᆠᆠᨨ � � ᨨ 盐矄矄Ἐ矄 ᨨ鼀 ᆠ�ᆠ 盐 ᨨ盐� ᆠ盐
ò ⼄ = e
æ
ö
ò s çè҈ - ÷ø p
e
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EBD_8341 PS-180
Physics
쟼ᆠ�ᆠ 盐 ᨨ盐� ᨨ 矄 �ᆠ矄盐 ᆠᨨ �盐矄 ⏌
҈黨
⏌
ö æ s ç ҈ - ÷ p ø ò ⼄ = è f= e e
ò
҈䰘
æ ö s p ò ç - ÷ ç ÷ f ø è = = \ = = ⏌ æ f � ö s p ò ç - ÷ ç ÷ ´ ø è bᆠ㡰 R 㡰 ᨨh>R r㦨
Q e
矄ᨨ� � 矄盐 泐 矄盐 ᆠ P
R � = = 䰘° R � é ù Q q�n = Q [҈ - ᆠᨨ矄䰘°] = Q ê҈ - ú = ë �û �
R R �
ᨨ鼀盐矄矄Ἐ矄 ᨨ鼀Q㦨
㡰
҈ ö æ = p (҈ ᆠᨨ矄 q ) = p (҈ ᆠᨨ矄 �° ) = p ç҈ ÷ è ø \ 盐 矄矄� � ᨨ盐� ᆠ盐 矄盐 ᆠ
q�
e
Q \ f = � Î ᨨh>R r>R
졔Q h
R
f=
R
q�n Q = Î Î
r
ᆠ㡰 ᨨh
R
㡰
R r㦨 R � � q�n = Î
q
R
R � = = Þ q = �° 矄� q = R �
é ù Q q�n = Q [҈ - ᆠᨨ矄 q ] = Q ê҈ - ú = � ë �û \ f =
q �矄ᆠᆠᨨ � �ᨨ盐矄矄Ἐ矄 Î
ᨨ鼀 졠�ᨨ 㡰 㡰 ᨨ ᆠ盐盐 犨 �ᆠ �矄 � ᨨ� 矄A Bᆠ盐矄ᆠ�泐�矄��ᆠ ᆠ㡰 쟼ᆠ�ᆠ � �矄 ᆠ r ᨨ鼀 �矄ᨨ ᆠ泐� + + l + + E㦨 + + l pe
҈ \Eµ l㦨ᆠ �盐 � � 㡰
Q � Î
b㡰꾀ᆠᆠᨨ � �ᨨ盐矄矄Ἐ矄犨 쟼ᆠ�ᆠ 盐 f
҈ ö æ p ç҈ ÷ ҈ ö QἘ è Q æ ø ==҈÷ e p e çè ø ᆠ 㡰 盐ᨨ矄泐鼀鼀泐 ᆠ�ᆠ 盐 矄矄� � ᨨ盐� a x =+ a a z 㦨 + �矄矄鼀 x㦨 꾀ᆠᆠᨨ � � ᨨ 盐矄矄Ἐ矄 ᨨ鼀 ᆠ�ᆠ 盐 q�
ᆠ�ᆠ 盐 ᆠᨨ矄矄� � ᨨ盐� 泐 ᆠᨨ矄 矄盐 ᆠ f 㦨 e - q + q - q q = 㦨 e e ᆠ 㡰
ᨨ �ᨨ ᨨ� 矄 � ᨨ ᆠᨨ 盐ᆠᨨ �矄矄鼀 盐矄 ᨨ �A㦨ᨨ �B
ᨨᆠ�ᆠ 盐 ᨨ盐� ᆠ�泐 =
ᨨh>R r> R � R �
҈
҈
f=
R R � �
� q
矄� q =
q
졔턼
ᆠᨨ鼀ᨨ ᨨ ᆠ�ᆠ � 㜠 �ᆠ盐ᨨ 쟼 쟼ᆠᨨ矄q 矄盐 ᆠ�矄Eᆠᨨ矄q E矄犨矄qᆠ �矄 ᨨ � ᨨ� ᨨ 矄盐 ᆠ
ᨨ 盐 ᨨ盐� ᆠ盐 矄盐 ᆠ矄 ᨨ盐 矄矄
�
㡰 ᆠ�ᆠ盐鼀㜠 ᆠ ᨨ 矄盐 ᆠ �矄
⼄盐�ᨨ �ᆠ盐矄 ᆠ�ᆠ盐鼀 ᆠ �矄 ⼄盐� �矄 ᨨ鼀졔턼
-
+ r
+
+
+
-
Topic-4 : Miscellaneous Problems z
l
҈ ҈ 㦨 ⼄ 㦨 l맔 矄� 桌 Î � Î
꾀
l R 㦨 Î 꾀㦨 矄� ᨨ꾀ᔴ㦨 矄� 꾀矄ᨨᆠ�ᆠ � �矄 �ᆠ盐ᨨ 犨� ᨨ �矄 㜠ᆠᨨ鼀 ᨨ 犨� ᨨ
ᔴ
҈
㡰
⼄盐 ᆠ泐ᨨ 和㝐 ᨨ鼀 泐犨ᨨᨨ ᨨᆠ├
쟼
҈
n㦨
҈ p
k m
졔턼
�矄 盐 ᨨ ⼄盐 ᆠ泐 犨� 鼀� 盐 ᆠ � ᆠ盐矄 犨
ᆠ�ᆠ � �矄矄犨�ᆠ ᨨ 盐ᨨ ├ 鼀�矄 ᨨ ᆠ QE 鼀 ᨨ矄��ᨨ ᨨ 和㝐矄 � 矄 ᨨ犨 矄�� 泐l㦨 k 盐ᨨ ᨨᆠᆠ� �F㦨qE
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PS-181
Electric Charges and Fields
㡰 ᆠᨨ矄�ᆠ ᨨᆠ 盐 � ᆠᨨ矄�ᆠ 矄矄盐 ᨨ� ᨨ 矄盐 ᆠ ᨨ 盐 � ᨨ鼀泐 ᆠ �
s ҈ e E = Î s ´ p \ ᨨᆠᨨ 鼀�矄 �ᆠ矄 F㦨 Î s R ᨨ F µ e 矄 =
é kq ҈ ù ᆠᨨ = + ê ´ ú = ûú a ëê a 和盐 ᆠ 矄�ᨨ ᨨᆠF矄㦨ga ᔼ ⼄盐���盐鼀
q
a
q
æ q ö a k = ç ÷ \ ç g ÷ è ø ҈㡰 y
├⼄
l Fnet㦨 pe x�fµx
├⼄
´ ᆠᨨ 矄 㦨 ga
l x \Fnet㦨 pe d x ␔ xd
´ ᆠᨨ 矄
a
├⼄
҈⏌
x
l éd + x d + xù \Fnet㦨 pe êë d - x úû
㡰 ᆠᨨ矄�ᆠ ᨨᆠᨨ ᨨ ᆠ � 盐ᨨ鼀� � � ᆠ �矄
kq
\ N㦨
䰘
p
q x
㡰 l 졔 x0
⧤ d 鼀 ᔼ �ᨨ 矄 ⼄盐���盐鼀 ⧤ ⼄㦨⧤ 졠��� ⼄盐 ᆠ泐 =
�㡰
l + 㡰 ␔ �矄ᆠ ᨨ犨 矄�� 泐 � =
⧤ l + + 㡰
× ⼄ ⧤ 졔 㡰
鼀 =
x ù ⧤⼄ é ҈+ ú ⧤ x졔x㡰 ê x û l l + x 㡰 ë
⧤⼄ é x ù = ҈ú ⧤ x졔x㡰 ê x û l + l + x 㡰 ë
=
⧤⼄ l + x 㡰
⧤ =
é x ù x + ҈ú ⧤x⧤x = - ⧤ ê l + x l + x ë û
-⧤ 鼀 ҈ ⧤ ҈ ⧤ 犨 ⼄盐 ᆠ泐 Ἐ = = = p 鼀 p 鼀 \ d 㦨p㦨҈ 矄 쟼ᆠᨨ矄�ᆠ ᨨᆠ�矄ᆠᨨ 矄�� 盐 ᆠ犨ᨨ├ ᨨ �矄 � 꾀矄ἘlἘ�矄 ��鼀㦨⧤ Þ =
�
ᆠ㡰 ᨨᆠᨨ ᆠ �q犨 ��矄�� 矄鼀 �矄ᆠ鼀 x l l Fnet㦨F҈F 㦨 l l ҈ ҈ 졔q F pe d x pe d + x F҈
x dx d
d � ᆠ �҈
l
� ᆠ �
l
�矄ᆠ鼀 ᨨ q ᆠ �졔q犨� ᆠ盐和㝐 F ᔼ ᆠ矄ᨨ ᆠ � q㡰F> F҈ F ҈ �.e. ᆠ � q �矄 盐 矄 ᨨ ᆠ � q ᆠᨨ � 盐矄 ᨨ 鼀ᨨ� �ᆠ�ᨨ ᨨ �矄 � ᆠ �҈ � ᆠ � �矄ᆠ鼀 v 㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 盐 � 盐 ᆠᨨ ᆠᨨ盐ᨨ鼀 ⼄ � ᨨ ᆠ �졔턼ᆠ �q�矄 鼀ᨨ� � ᨨ盐 � � ��ᆠ 졔Q ᨨ� �矄 矄�盐�ᨨ �矄 矄 ᨨ犨
� ��盐犨 �ᆠ �矄矄�鼀�ᨨ ᨨ� �ᨨ盐 矄盐 �矄 ᆠᨨ q ᨨ鼀 졔 Q �矄 ᆠ �� � ᨨ ᨨᆠ犨 ᆠ �矄犨�ᆠ � 矄 ᨨ ᆠ �q犨��犨 ᆠ ��矄 ᨨ졔Q矄ᆠᨨ鼀 ᨨ犨 ��矄 ᆠ �盐ᨨᆠ�泐 ᨨ ᆠ �q�矄 ᨨᆠᨨ 矄 �ᆠ�ᨨ ᨨ ᨨᆠ�泐ᆠ �矄 ᆠᨨ � 盐ᨨ盐矄泐 ᨨ� 鼀ᨨ鼀 盐鼀�矄 ᨨᆠᨨ 矄 �盐鼀ᨨ鼀 盐鼀ᨨ ᆠ � q㡰ᨨ盐Q�矄ᆠᨨ 矄 ᆠ盐矄 ᨨ⼄盐ᨨ盐ᆠ �졔Q�矄 ᨨ y ᆠ㡰 Q ᆠ �ᨨ
� � ��ᆠ � x Q q �矄矄 ᨨ鼀ᨨ� P 㡰 ᆠ�ᆠ � P E 盐ᨨ ᆠ � � �犨� q O z ᨨ � ᨨ矄�� z㜠 �矄 �矄 P 㡰 鼀� �盐 犨� R Qz ҈ E㦨 pe R + z 㡰⏌
ᨨ ᨨᆠᨨ ᆠ � P犨�ᨨ犨 矄ᆠ 矄 矄 ᨨ犨 �矄鼀� �盐 �矄 ҈ Qq Fe㦨qE㦨 ҈㡰 pe R + Z 㡰⏌ 和�鼀�泐 犨 �ᆠᨨ矄矄矄 ᨨ��� ᨨᆠ�矄�� ᨨ犨 矄 ᆠ O
盐矄 鼀ᨨ�ᨨ ᨨ �ᆠ �矄 �ᨨ �ᆠ ᨨ 盐矄 ᨨ Z 泐� �犨 ¥ 和ᆠᨨ 泐� ZR R졔 Z 㡰⏌ » R
҈ Qq Z ᨨ鼀⼄盐�ᨨ ҈桌 pe R �.e. 矄ᨨ� � ᨨᆠFe µ Z ᆠ 鼀ᨨ�ᨨ ᨨ �ᆠ 犨� 矄�鼀 鼀ᨨ �ᆠ �� 矄�� �鼀�矄 ᨨᆠ �矄 ᨨ犨 矄 �矄鼀 ᨨ矄��ᨨ 㡰 Fe㦨
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EBD_8341 PS-182
Physics
Electrostatic Potential and Capacitance
16
ᔴ �ᨨ �
矄 Q҈ = s pr ᔴ �ᨨ ᨨ盐矄 Q = s pR
Topic-1 : Electrostatic Potential and Equipotential Surface ҈
\ ᨨᆠ � Q = s p r + R 㡰
KQ ᆠ㡰 ᨨ � ᆠ VC = QQ = \VC =
Þs=
ᨨ � ᆠᨨ鼀鼀ᨨ ᆠ
æ ҈ ö KQ҈ KQ ç 犨 K = ÷ + pe ø r R è K s pr K spR + = K sp r + R 㡰 = r R KQp r + R㡰 ҈ r + R 㡰Q = = pe r + R 㡰 p r + R 㡰 VC =
Eᆠ�ᆠ � ᨨ 盐ᆠ 泐ᆠ ᆠ � ᆠ
矄盐 ᆠ�ᆠ � 犨� EC 㦨 矄� ᆠ ⼄盐ᆠ�ᆠ � ᆠᨨ矄ᆠ� �⼄盐 �矄ᨨ �矄盐 �矄⼄盐ᨨ ᨨ E
䰘
E
䰘
䰘
E
�
䰘
㡰 ᨨᆠ � Q҈ + Q = Q Ἐ҈ + Q Ἐ
鼀� �鼀盐鼀 ᨨᆠ�泐 㡰 矄 ᨨ盐 盐矄矄盐 �ᆠ� ᨨ ᆠ ᨨ� ᆠ � ᆠ ᨨ
é ⧤⼄ ⧤⼄ ù ҈ 鼀 = ⼄ ᔴ - 桌 = ⼄ ê ë � úû ⧤⼄ ҈ ⼄ = = ҈�鼀 ҈� pe 鼀
KQ Ἐ҈ KQ Ἐ = 㦨҈㦨黨Ҙᔴ R R Q Ἐ҈ = Q Ἐ Þ Q Ἐ + Q Ἐ = 黨mC \ Q Ἐ҈ = mC Q Ἐ = mC
kQ kQ VA = ҈ + R R
R R
æ
kQ҈ kQ + R R
ç ├㦨
è
ᨨ � � ᆠ
DV = VA - VB =
҈
㡰 ␔ �� 犨ᨨ鼀�ᆠ ᨨᨨ犨矄 矄ᨨ ��R R � �ᆠ �矄Q҈ Q矄ᆠ�泐 ᨨ �ᨨ 矄盐 ᆠᨨ �
Q Q҈ 矄 A㡰
VB =
B
ᨨ犨犨 ᆠ �턼�矄 �� ᨨ ᨨᨨ犨 矄 쟼ᆠ�ᆠ � 犨
矄盐 ᆠ鼀� 盐 ᆠ �
҈ ö ÷ pe ø
졔턼 턼
졔턼
쟼
졔턼 턼
턼
r KQ E= r
矄 � � 矄� ᨨᨨ犨 矄 �ᆠ 矄 㦨 \ᨨ � � ᆠ犨 鼀鼀� 盐 ᆠ �
s
ò
r r
� Edr = V 䰘
C r
㡰 ␔ ᆠ �턼�矄ᨨ �
矄ᨨ� ᆠᨨ 盐ᆠ� �矄 쟼ᆠ�ᆠ � 犨 矄 �ᆠ矄盐 ᆠ
r KQ r r E = Sᨨò Edr = V g�ven r
A
㡰 s 矄盐 ᆠᆠ � 矄�泐ᨨ 矄 矄
R
ö æ ⼄ ç ÷ 鼀 è ҈� ´ pe ø
Þ =
Q kQ҈ × = × ҈ ҈p Î R R
s
+
l = + = �
\ V҈ = V =
ᨨ �ᨨ 矄盐 ᆠᨨ ᨨ盐 矄 B㡰
⧤⼄
㦨 㦨
E
= ҈mC - mC = 黨mC
犨ᨨ�矄ᨨ ᆠᨨ 盐ᆠ� �矄 矄S҈ S ᨨ犨ᆠᨨ
ᆠ 泐 ᆠᨨ 盐ᆠ� �犨�
ᆠ㡰 ᨨ � 泐ᨨ� ᨨ ᆠ � � �
=
䰘 E
Q
p r + R 㡰
uur
ᆠ㡰 � E = ( Ax + B ) �ˆ ᨨ E㦨x졔҈
ò
矄� �V = Edx 犨 ҈
VV҈㦨
ò ( x + ҈) dx
-�
㦨҈ᨨ V҈V㦨҈
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PS-183
Electrostatic Potential and Capacitance 黨
b㡰 쟼ᆠ�ᆠᨨ ��矄ᆠᨨ 矄 � 矄� ᆠ � 矄 �ᆠ矄 㡰 �矄 ᆠ� ᨨ鼀ᨨ� ᨨ �ᨨ矄ᨨ 盐ᆠ 矄鼀 ᨨ �
⼄
⼄
⼄ ⼄ \ = ( + )
( )
Þ
= + Þ =
҈
҈
ᨨ �矄 ᆠ ᨨ鼀꾀犨 �ᆠ ᨨ ᨨ 鼀 ᨨ 盐ᆠ 矄鼀 ᨨ �
= ҈
҈
+ =
҈
b㡰 Q ᨨᆠ � ᨨ 矄盐 ᆠ ᨨ 矄 �ᆠ 矄 矄盐 ᆠᆠ � 矄�泐�矄
҈
Q 꾀 ᨨ�矄 ᨨ犨ᆠ盐 pR
ᨨ a p 㡰 ᆠ �ᨨ �矄 ᨨ矄泐q
⧤⼄ 꾀 ⧤⼄ ⧤⼄ᔴ + + ᆠ
=
҈ p Î
=
s Î
s ᔴ s s 꾀
é s p s p s pᆠ ù + ê ú ᆠ ûú ëê
ᆠ
é - ù + ᆠú ê úû ëê
ᆠ㡰 ᨨ �� � �矄�� 泐 D㦨쟼 㦨쟼 鼀 㡰 D㦨쟼 ᆠᨨ矄q㦨맔ᆠᨨ矄㦨 ᆠ鼀 �鼀盐鼀ᨨ �ᨨ� ᨨ 矄 㦨�黨
- = ҈ - S = = ҈
b㡰 S҈ =
ᆠᨨ 矄 쟼㡰
\ 矄ᨨ盐ᆠ �矄 � � �泐� ᨨ ᆠᨨ 盐ᆠ� � �ᆠ├ᨨ �ᆠ├ 矄矄鼀
q Q = \q㦨a턼 Q a pR 㡰 pR
r×꾀 µ e r 㦨 ҈ ᨨ £ ҈鼀 dv 㡰 꾀矄 犨 ├ ᨨ犨 쟼 㦨 dx \҈ ×҈꾀 =
ᆠ�ᆠ ᨨ � 盐 ᨨ盐 � ᨨ鼀泐ᆠ � � 矄 �ᆠ 矄 r>R㡰 V =
=
KQ R ҈
ᨨ � ᨨ� 㦨 鼀 泐 㦨 鼀 �矄 �� 泐 V
R⏌ P
ò dV =
M A R
B R ҈�
ᨨ犨 ᨨ �
KQ K aQ KQ 꾀ᨨ� A VA = = ҈ - a 㡰 R R R VP ҈ - a = \ V A ҈ - a
҈
ò
�dx + dy 㡰
ᨨ 矄ᨨ� � 犨 � 㦨 ҈҈ ᨨ ᆠ㡰 ᨨ � � ᆠ 犨 泐犨ᨨ ᨨ� 矄 � ᆠ�ᆠ � �矄�� r uur泐 㦨 E × dx VA
KQ K aQ KQ 꾀ᆠ VP = = ҈ - a 㡰 R R R = V ҈ - a 㡰 = V - aV
ò
dV = - ò x dx
VO
V A - VO = -҈ x 桌 = - ⏌ ᔴ
ᆠ㡰
졠
ᆠᨨ矄q q ᔴ 矄� q ⼄ x q
⼄
Eᆠ�ᆠ �
KQ R㡰 KQ K aQ㡰 KQ = - aV ( EB ) � = R 㡰 R 㡰 R K aQ㡰 aV ( EP )� �� = ( EP ) � = R = R ⧤턼 b㡰 ᨨ � ᨨ盐矄� 矄 ᨨ盐矄� =
( EB )� �� =
҈҈
犨 �矄 �矄 ᆠᨨ ᨨ� ᨨ鼀 ᆠ ᨨ 矄 ᨨ �� 矄� 矄 � 矄� 犨 � �矄 �盐矄ᨨ 矄
⧤턼 =
鼀� ᔼ ⼄盐���盐鼀 = 矄� q 鼀�㦨 ᆠᨨ矄 q
q =
⼄ = = 鼀� p Î ´ 鼀� \
⼄ p Î q 鼀�
쟼ᆠ�ᆠᨨ � ᆠ ᨨ �
= ҈䰘
├⼄ ├⼄ + = ├鼀� ⏌ q ⏌ ⏌
㡰 꾀ᆠᆠᨨ � � ᨨ 盐矄矄Ἐ矄 ᨨ鼀 ᆠ�ᆠ � 犨 犨ᨨ
l �矄ᆠ�ᆠ � 犨�ᨨ 盐ᆠᨨ � peurr r � ᆠ矄dV㦨 Ed r ␔ ᆠ � 矄�泐�矄�� ᨨ
ᆠ泐� 矄 E =
�
ᆠ泐� ᨨ � ᨨ �矄 矄盐 ᆠ�矄 �
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EBD_8341 PS-184
҈
Physics
ᨨ鼀 ᨨ ᨨ盐 ᆠ泐� �矄 �矄 ᆠ盐矄 ᨨ � ᆠ矄矄犨� �矄 ᆠ ᨨᆠ � ᆠᨨ 盐ᆠ� �ᆠ泐� 犨
ᨨ� ᨨ ᆠᨨ 矄� �ᨨ �矄 ᨨ盐矄� ᆠ泐� 盐犨 ᆠ � 矄�泐�矄�� ᨨ ᨨ盐ᆠ泐� �犨� ᆠ � �矄 ᨨ � 泐 矄鼀 鼀ᨨ盐 矄 ᨨ �
ᆠ泐� ᆠ ᨨᨨ � � ᆠ犨�ᨨ 盐ᆠ 犨
ᆠ泐� 矄 b㡰 꾀矄犨鼀ᨨᨨ � �ᆠ�ᨨ ᨨ ᆠ�ᆠ � ᨨ � ᆠ矄矄 \ VA>VB VA㦨VCᨨ ⼄盐�ᨨ �矄盐 ᆠ
dv dt qEdt q 矄� ҈t �ˆ dt Þ 㦨 㦨 m m Q 쟼㦨矄� ҈ �ˆ �� 㡰
㡰
v
\ ò dv =
泐
҈㡰
҈㡰
\
㡰 ᨨ � � ᆠ ᆠᨨ矄矄 ᆠᆠ�ᨨ 矄 鼀� 矄 盐 ᆠ � 犨 矄犨�ᆠ S�矄ᆠᨨ矄 ␔� ᆠᨨ矄� �ᨨ 矄犨�ᆠ S S ҈ �� ᆠ � ᨨ C 犨� ᆠ ᨨ矄�� ᆠ �ᨨ C 泐鼀� � � � ��ᆠ �ᨨ C҈
��ᆠ �ᨨ C҈犨�ᆠ ᨨ矄��ᆠ �ᨨ C҈ᨨ 矄 ᨨ ᆠ �犨�├ᆠ ᨨ ᆠᨨ矄矄C҈ C 犨� 矄ᆠ�泐 㡰 ᨨ � ᨨ��� 犨��� 泐
�
㦨맔҈⏌鼀 = ҈
p⏌ w
҈ ҈
´
ᆠᨨ矄҈t 桌p ⏌ w ҈
é ë
p ù ᆠᨨ矄 ú 㦨҈҈桌㦨鼀矄҈ ҈ û
d x 㡰 -dV = - x Þ Ex㦨 dx dx
ᆠ�ᆠ � ᨨ� ҈㡰 x㦨҈㡰 Ex 㦨⏌鼀
鼀� �盐 ᨨ ᆠ�ᆠ � �矄�矄ᨨ� Bᆠ盐矄B ⼄盐�ᨨ � 矄盐 ᆠ矄 ᆠᨨ矄矄 ᆠ 㡰 쟼ᆠᨨ矄�ᆠ � P �矄 P
r ᔴ҈
ur ur ur r uuuur ( r) uuuur E p = E҈ + E = C҈ P + C P e e r uuuur uuuur r uuuur uuuur r uuuuur = C҈ P C P = C҈ P + PC = C҈C e e e bᆠ㡰uuur uuur uuur E D uuur uuur E A uuur = EB = EC = = E E = EF = K
(
\
)
(
졔q F 졔q A
㦨҈鼀
E q
(
)
҈ ҈ 졔 ⧤졔⧤㡰맔 㦨⧤
q D
O
鼀� q
졔q B
C
EB 졔 EE
꾀矄⼄㦨 鼀 = p r \ 맔҈맔҈҈黨맔맔҈
p 맔҈䰘㡰맔黨맔҈ ´҈ -䰘 㡰 ´ 黨 ´҈ Þ = p´ \ 㦨 ´ ҈ ´ ҈ ´ ҈ -҈黨
)
EO㦨EA졔ED 졔 EF졔 EC㡰ᆠᨨ矄º졔 EB 졔 EC㡰ᆠᨨ矄º
㦨⧤졔⧤졔 ⧤졔⧤㡰맔
和犨�ᆠ
–r ᔴ
҈
쟼⼄
=
㦨
ᆠ鼀 �鼀盐鼀 矄 � 泐 �ᆠ � ᆠ�ᆠᨨ �V 㦨 x ᨨ矄
ᨨᆠ�ᆠᨨ �ᆠ �矄 ᨨ 泐ᨨ �x㜠 �矄
␔ 鼀� ᨨᆠ�泐�矄ᆠ � ⼄쟼㦨鼀�
㦨
T p ù é 矄� ҈t dt ê 鼀 矄 �矄 = w´ úû ë
㦨 êᆠᨨ矄҈ ´
␔├ ᨨ犨Ex㦨
䰘
㡰 ᨨ �� 矄� 矄 �矄⼄盐ᨨ ᨨ �ᨨ �矄 矄盐 ᆠ␔ 犨 Qᆠ �ᨨ 矄盐 ᆠ ᨨ �ᨨ 矄盐 ᆠᆠ �矄泐 矄鼀鼀ᨨ盐 矄 � 矄� ᨨ ᨨ � � ᆠ鼀� 矄 矄鼀 v㡰 b㡰 ᆠ�ᆠᨨ � 矄盐 ᆠᨨ ᨨᨨ犨ᨨᆠᨨ 盐ᆠ� � 矄 �矄 矄鼀 矄 ᨨ � ᆠ ᨨ 矄 泐 ᨨ� � 矄� 矄 ᆠ ᆠ�ᆠ ᨨ � 矄盐 ᆠ ᨨ 矄 㦨 ᨨ � ᆠ ᨨ 矄 㦨҈ 㡰 ᨨ ᨨ ᆠᨨ 矄矄 � ᨨ� ᨨ 쟼ᆠ�ᆠ � 쟼 =
ò
æQ m = ҈ kg q = ҈C E = ҈NC ҈ ö ç ÷ ç and w = ҈ 矄҈ ��
÷ è ø
ù ҈ ҈ ҈ q é҈ + + ú V㦨 ê pe ë x x x x û q 㡰 V㦨 pe x ҈
p⏌w
쟼
҈黨
q m
q é ᆠᨨ矄҈t ù \ 㦨 ê m ë ҈ úû
ᔴ
㡰 꾀
Q 㦨鼀\⼄쟼㦨 m
º O
º
EA
ED
EF 졔 EC
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PS-185
Electrostatic Potential and Capacitance
( )
쟼ᆠ�ᆠ ᨨ � O
\ pR = pa t Þ R = a t
҈ VO㦨 q졔q졔qqqq桌㦨 p e L
黨
҈ QA ҈ QB \ = =V p Î RA p Î RB Q RA>RB\ QA>AB
sA s EB = B Î Î E A s A RB = = < ҈ \ 쟼 쟼 꾀 EB s B RA V
졔졔졔졔 졔 졔졔 졔
졔
졔졔졔졔졔졔졔 졔
R
R
r
졔졔졔졔졔 졔졔 졔 졔
ᨨ �ᨨ �⼄盐� 盐 V =
㡰 Pᨨ� � X㜠Y 犨� ᆠᨨᨨ � 矄 x, y㡰 犨 �ᆠ ᨨ � 盐 ᨨ ᆠ �矄 Q ᨨᆠ a㡰 졔Q ᨨᆠ 졔㡰 ᨨ Y
P x y㡰
y
y A 㜠Q X' 㜠a㡰 O x a 졔x㡰
Y'
K Q 㡰
V 㦨 a �a 黨ax a
B 졔Q C X a㡰 �a㡰 a㜠x㡰
=
K +Q㡰 a - x㡰 + y
K -Q㡰 K +Q㡰 ҈ + 犨 k㦨 x + a x - a㡰 pe ù é ҈ \ V x㡰㦨KQ ê ᨨx>a ë x - a x + a úû V x㡰㦨
졔졔졔졔 졔 졔졔 졔
Q Q = C pe R
ᨨ � M
㡰 ᨨrR ᨨ ��矄 E ᆠᨨ 矄 ᆠ�ᆠ� 矄�泐 �矄 ᨨ 졠�ᨨ盐矄泐 ᆠᨨ矄�ᆠ �泐�矄 ᨨ ᨨr R R r b㡰 �矄盐矄 矄 ᆠ ᨨᨨ �� ᆠ�ᆠ ᆠ�ᆠ盐�矄 ᨨ ᆠ�ᆠ 盐ᨨ矄矄 ᨨ 矄 �ᆠ ᆠᆠ�ᨨ ᆠᆠ� ᆠ ᔴ 㦨 peR ᆠ�ᆠ ᨨ � V =
҈
�矄�矄 ⼄盐�ᨨ ᨨ ᆠ�ᆠ犨� ᆠ �a㡰 �盐矄a b㡰 ᨨxa 盐矄 ᆠᨨ 矄� ᨨ� �泐㡰 M �矄 ᆠ x ᨨ鼀 ᨨ��� ᨨ x㜠 �矄
RB RA
㡰 E A =
æ aö =Vç ÷ è t ø
Þ a - x㡰 + y = a + x㡰 + y Þ x�a㡰졔 y㡰㦨 a㡰
A
b 㡰
�� V㜠r� �矄ᨨ ᆠ � ᆠᨨ 盐ᆠ� �矄 ᨨ �盐矄 㡰 犨 ᨨᆠ � Q 犨� ᆠᨨ矄 � 矄 ᨨ �鼀R 㡰 쟼ᆠ�ᆠ � E 㦨 ᨨ� 矄� 矄 ᆠ ᆠ�ᆠ � �矄 �矄ᆠᨨ � 盐 r㦨R ᆠ㡰 ᔴ �矄 � E ᆠᨨ � 盐ᨨ盐矄泐矄 ᨨr>R 졠�ᨨ ᆠ㡰 �矄 � ᆠᨨᆠ
(a t )
҈
a + x㡰 + y
QA
҈
æ Va ö Kç ÷ è kø
Kq V҈ = = R
\
R R pÎ 꾀V pR A Q s = B´ A= B´ ᆠ㡰 꾀矄 A = 㦨 Q sB B R QB R p Î RBV A
ᨨ � ᨨ ᨨ
ᨨ �ᨨ� 矄ᨨ � �矄矄鼀 ᨨᨨ � 和 PR �矄 �ᆠ盐 �矄ᆠᨨ ⼄盐ᨨ�� 㡰 ᨨ ᆠ�ᆠ �ᨨ矄ABFE BC ᨨ ᆠ�ᆠᨨ �犨� ᨨ 泐ᨨ� ᨨ PR ᆠ 㡰 㡰 盐矄矄Ἐ矄犨 �矄� ᨨ 泐犨 쟼 µ 㡰 盐矄矄Ἐ矄犨ᆠ
ᨨ盐矄 ᨨᆠᆠ盐 � �矄�盐�ᨨ
ᨨ盐 ᆠ�ᆠ �ᨨ ᆠ㡰 �矄 ᆠᨨᆠ 矄 犨 犨ᨨ ᨨ� ᆠ �矄 犨 犨� � ᨨ� 犨 ᆠ�ᆠ � 盐ᨨ 犨ᨨᨨ� ᆠ �矄ᆠ ᆠ ᨨ盐ᆠ ᨨ ᔼ 犨ᨨᨨ� ᆠ �矄ᨨ ᨨᨨ矄�矄�� 犨ᨨ � 矄ᨨ � �ᆠ�ᨨ ᆠ 泐ᆠ
ᨨ ᨨ 㡰 WAB㦨q DV㡰㦨҈ VBVA㡰 ᨨWAB㦨VBVA bᆠ 㡰 㡰 쟼ᆠ�ᆠ � � 矄� 矄 �ᆠ鼀�ᆠ矄 犨� ᆠ �ᨨ
矄盐 ᆠ�矄犨泐矄 ᨨ �.e.EA� 矄� 㦨 㡰 ␔ 矄 矄ᆠᨨ
ᆠ 犨� � 鼀犨�
ᆠ�ᆠᨨ �矄犨�⼄盐�.e.VA㦨VB㦨V
pRB
҈
和�鼀�泐 ᨨxa
ù é ҈ ë a - x a + x úû
V x㡰㦨KQ ê
和� ᆠᆠ�ᆠᨨ ᨨᨨ �ᆠ盐矄 x㜠 �矄 a㡰 黨a㡰 ᆠV x㡰㦨x㦨ax㦨黨a 掸 ᨨ鼀 ᨨ 矄矄�ᨨ 矄 V x㡰®¥x®a V x㡰®¥x®a 掸 V x㡰®矄x® ±¥ 掸
V x㡰�矄
҈ � � x
盐 ᆠ�ᨨ x㡰ᨨ 犨 ᨨx �矄矄矄 ᨨ犨 ᨨ犨 V
a a
a
黨a
x
kq Va Þq= a K
␔ q 㦨ᆠ �ᨨ �⼄盐� 盐 ᆠᨨ矄� � �盐矄 ᨨ ᨨ ᆠᨨ鼀矄 R 盐 ᨨ盐鼀 鼀� 矄 ᆠᨨ 矄
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EBD_8341 PS-186
Physics \ ᨨᨨ � ᆠᨨ鼀鼀ᨨ ᆠ �졠
ᆠ㡰 ᨨ鼀 �泐ᆠᨨ 矄�ᨨ � ᆠ� v 矄 ᨨ �ᆠ ⧤쟼졔쟼㡰ᆠ 㦨 ⧤쟼졔 쟼㡰ᆠ�ᆠ盐鼀 ᆠ
V㦨V�졔V㦨
é Qq Qq ù ҈ é Qq Qq ù = mv + K ê 졔K ê ú a a ë û ë a ҈a úû ҈ KQq \ v㦨 mv = a �
KQq = ma
҈ æ Qq ö ç ÷ pe è ma ø
҈
KQR
R +r
㦨
kq A kqB kqC + + a b c
q
kq A kqB kqC s + + ᨨ VB㦨 b b c e
é a ù ê - b + cú êë b úû
矄�� 矄 ᨨ犨矄 �泐ᨨ ᆠ ��矄 ᆠ矄� � ᆠ㡰 꾀ᆠᆠᨨ � �ᨨ犨ᨨ├ �泐 ᨨ鼀�� � ├� �ᆠ �泐�矄 ⼄盐ᨨ犨ᨨ├ ᨨ � �矄ᆠ鼀 ᨨ ᆠ �
\
vµ
m
v = = ҈ v ҈ é ù ê ҈ - q 㡰Q ú q - q㡰 qQ ê ú 㡰 U = + + dö æ dö ú p Î ê d æ çè D + ÷ø çè D - ÷ø ú êë û é ù q qQd ҈ = - ú ᔼ� ᨨ� � d êp Î ëê d D ûú
b㡰 ᆠᨨ矄�ᆠ �泐 ᨨ 矄泐矄鼀
pR
qR㦨Qrqr
æ q = ᆠ �ᨨ �
矄 ö \ q㦨 çè Q q 㦨ᆠ �ᨨ ᨨ盐矄 ÷ø R +r
é ⼄ 턼⼄ 턼⼄ ù = ⧤ê + + = úú ëê û
Qr
ᨨ �O 盐ᨨ�
矄
턼
é ҈ ù Þ ⼄ = -턼 ê҈ + ú ë û
æ ö q = K ç Qr ÷ ç r è R + r ÷ø r Qr
V�㦨 K
҈
\
qö æ çè 和盐 ᆠ ᆠ � 矄�泐s = ÷ø A
V�㦨 K
҈ mv = q DV
㝐矄矄ᨨ 泐 ᨨ� �ᨨ 鼀 㦨҈ 㝐矄矄ᨨ �盐鼀�ᨨ 鼀 㦨
Q-q
ᨨ qR㦨 Qq㡰r ᨨ
Þ턼=
R + r
ᨨ � O 盐 ᨨ ᨨ盐 矄 ù é Q - q㡰 = K êQ - Qr ú V㦨 K R R + r ûú ëê R
-⼄ 졔҈
졠㡰 턼
�
ù é KQR K ( QR ) K ëQR + Qr - Qr û = 㦨 㦨 R + r R R + r 㡰 R R + r
(
d
DV㦨ᨨ � � ᆠ犨 犨ᨨᨨ矄��ᨨ 矄ᨨ ᆠ �q ᨨ矄鼀q DV.
ᨨ acbc졔c㦨ab졔cᨨc㦨a졔 b � 矄盐 ᆠᆠ � 矄��矄⼄盐
pr
d⏌
q æ k q k q ö Du = q ç =÷ è d ⏌ d ⏌ ø pe d
kq A kq B kqC + + c c c ù é s a -b +c ᨨVC㦨 ê ú e ëê c ûú 㡰 Q VA = VC �� 㡰 s s é a - b + c ù a - b + c 㡰 = ê ú c e e ëê ûú
=
R + r
X
q
O
d⏌
꾀 VC㦨
q
KQ R + r 㡰
q
s a - b + c桌 e
和�鼀�泐VB㦨
\
+
Y
҈ pa ҈ pb 㡰 ҈ pc 㡰 ´s´ s + ´s 㦨 a b c pe pe pe
R +r
㡰 ᔴ �� ᨨ � �泐 Du = q V f - V� 㡰 ᨨ �ᨨ q�矄矄鼀矄� �� � ᨨ� ᨨ
k s p a 㡰 k ( -s ) pb 㡰 k s pc 㡰 + + 㦨 a b c
㦨
Topic-2 : Electric Potential Energy and Work Done in Carrying a Charge
ᨨ � 泐矄 �矄 盐ᆠ �矄 矄 꾀 ᔴ ᨨ �ᨨ 矄 A VA 㦨
KQr
) (
)
졔⼄
졔⼄
턼 졔㡰
b㡰 졠㡰 턼
턼 㡰
ᨨ � ᨨ���
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PS-187
Electrostatic Potential and Capacitance
⧤턼 ⧤턼 ⧤턼 ⧤턼 + + + ҈ ö æ ᨨ �¥㦨㦨⧤턼 ç ҈+ ÷ �ø è
㦨
q
� �泐
\ ␔ᨨ├ ⼄盐� ᨨ 盐 � ᆠ � 턼 ᨨ��� ␔ 㦨 턼
ᆠ㡰 犨ᨨ├ ᨨ � 鼀ᨨ� �ᆠ �ᨨ � ⼄盐�ᨨ � 矄盐 ᆠ �矄 犨泐矄 ᨨ
�ᆠ�ᨨ ᨨ ᆠ�ᆠ � �矄 �ᆠ盐ᨨ ⼄盐�ᨨ � 矄盐 ᆠᨨ� 矄
䰘
ᆠ㡰 㡰 ᆠ�ᆠ � 盐ᨨᆠ �
æ q ö çè ÷ ø
꾀矄ᨨ
º
⼄
҈
҈҈
ᔴ 꾀 㦨矄� ºÞᔴ㦨 㦨 꾀
é( ⼄ ⏌ )( ⏌ ) ù é ( ⼄⏌)( -⼄ ⏌ ) ù é( ⼄⏌)( -⼄ ⏌ ) ù ú+⧤ê ú +⧤ê ú ë û ë û ë û ├⼄ é ҈ ù -㦨 ú ¹ 黨 êë û ê
ᆠ㡰 㝐� �盐 ᨨ ᨨᆠ犨 ᔴ
(
)
҈
é +q ⏌ ù é +q ⏌ ù é -q ⏌ ù 졔⧤ ê 졔⧤ ê 㦨 ú ú ë û ë û ë úû
)
)
)
Qq q Qq + + = a a a
ᨨ
Q + q + Q = \ Q㦨 -
q +҈
=-
q
+
b㡰 㡰 ᨨdxd ᆠ�ᆠ � �矄ᨨ �졔x㜠 �矄 ᨨᨨ ᨨ� 矄쟼�矄ᨨ � ��x㜠 �矄 泐 쟼 ᆠ쟼犨� ᨨ 矄鼀 �ᆠ�ᨨ
졔q㡰 ᨨ � �x㜠 �矄 쟼 㡰 쟼ᆠ�ᆠ � Pᨨ� ᨨ
쟼 q㡰 泐㜠 �矄 �矄 ᨨ x㜠 �矄 ᔼ �矄 ᨨ ᨨ �x㜠 �矄�矄 ᆠ㡰 쟼ᆠ�ᆠ ᨨ � q 졔q x ᨨ��� 㦨 ᨨ d㡰 㡰 d㡰 ᨨ犨ᨨ├ 矄ᨨ ᨨ � � �� �矄ᆠ � ᨨ鼀� � �泐ᨨ ᨨ��� 㡰 �ᨨ鼀ᨨ鼀 �矄 �ᆠ ᨨ鼀qᆠ �ᨨ졔qᆠ � ᨨ � �� �ᆠ�ᨨ ᨨ x㜠 �矄 \�ᨨ鼀ᨨ鼀 㦨qdᨨ �x㜠 �矄 b㡰 犨ᨨ├ ᨨ � 鼀ᨨ� �ᆠ �q ᨨ鼀ᆠ ᨨ ᨨ � � ᨨᨨ 㦨ᆠ �� ᨨ � �泐 Q҈
㡰 ᨨ �졠㦨⧤ ê
(
\
⼄ ҈ ( ⼄ ⏌ )( ⼄ ⏌ ) 㦨 4π Îᨨ �π Îᨨ
ax � 泐
b㡰 和� ᆠ ᆠᨨ矄�ᆠ �泐㦨
졠
㡰 ᨨ � �泐ᨨ 矄泐矄鼀 ⧤
㦨
Qq pe
(
졠 犨�� ᆠ ᆠ ᆠ�ᆠ � O 盐ᨨᆠ �
q ҈ q ⏌ 㦨 4π Îᨨ R π Îᨨ 꾀ᔴ - ⼄ ᔴ ᔼ D꾀ᔴ 㦨矄� º 꾀 º 꾀 꾀 ⼄ Þ꾀ᔴ㦨 㦨
x a졔x
(
q q 꾀 ᆠ �
쟼=
q
Qqa ҈ ù+ ҈ q = é ҈ êë a + x + a - x úû pe a pe a - x é ù Qq ê ҈ a ú U - Uf = \ � pe ê a a - x ú ëê ûú é ù Qq ê a - x - a ú -Qqx = = pe ê a a - x ú pe a ëê ûú 犨 x ҈桌 犨ᆠ �㦨KCV 犨 ᨨ � �泐 㦨
V d
҈ KCV
E㦨
矄鼀ᆠᆠ� ᆠᨨ ᨨ ᨨ�ᨨ
V d
�㡰 ᨨ盐 ᆠ � ᆠᆠ�ᨨ 矄 ᔴ �ᨨ ᆠᆠ�ᨨA qA㦨맔҈맔҈㦨맔҈ᔴ ᔴ �ᨨ ᆠᆠ�ᨨ B qB㦨맔҈맔҈㦨맔҈ᔴ ␔ 盐 ᆠ � ᆠᆠ�ᨨ 矄 ᆠ �ᨨ ᆠᆠ�ᨨAC B q҈ q q 矄ᆠ�泐 泐ᆠ � ᆠᨨ 矄�ᨨ
҈ ҈ ҈ = + dC dC҈ dC
犨 dC҈㦨 ᆠᆠ� ᆠᨨ ᆠᆠ�ᨨ犨� �ᆠ�ᆠK҈ dC 㦨ᆠᆠ� ᆠ ᨨ ᆠᆠ�ᨨ 犨� �ᆠ�ᆠ K + + +
A
m q҈
҈
+ + +
K҈ bdx 㡰 e dἘ d é xù d'㦨dx = d ê҈ - ú l ë lû
dC҈㦨
B
m q ҈
\
dC҈㦨
K҈ A e ( dx) K҈b ( dx) e K҈bl ( dx) e = d l - x 㡰 d l - x 㡰 é xù d ê҈ - ú ë lû
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PS-197
Electrostatic Potential and Capacitance
和�鼀�泐dC㦨
K e bdx 㡰 K e bdx = xd d -dἘ d -d + l
㦨
㦨
K e bldx K e Adx = xd xd \
҈ d l - x 㡰 xd = + dC K҈ Ae ( dx ) K Ae ( dx )
Þ
K҈K A e dx = dC K ld + d K҈ - K 㡰 x
l
dx
ò K ld + d K҈ - K 㡰 x
CAἘ㦨
l
é ᨨ� K ld + d K҈ - K 㡰 x桌 ù ú d K҈ - K 㡰 ë û
�
Ae ´ � ´ ҈ -҈ = 㦨맔҈黨F d � ´ ҈-
q㦨CAV㦨맔҈黨맔҈҈㦨맔҈䰘C \
K҈ K Ae K ᨨ� ҈ d K҈ - K 㡰 K
W㦨
ù ´ ҈-䰘 㡰 é ҈ ҈ ê -黨 -黨 ú ´ ҈ û ë ´ ҈
ᨨW㦨맔҈�
e e r AB d
矄泐矄鼀 矄矄 ᨨ犨 � ��盐矄 �矄 ⼄盐� ᨨ 犨ᨨ ᆠᆠ�ᨨ矄� ᨨ 犨� � ᨨ 犨� ᨨ�犨� �ᆠ�ᆠ C㦨C҈졔C
���㡰 ᔴᆠ� ᆠᨨ B㦨
k e x ´ ҈㡰 e ҈ - x 㡰 ´ ҈㡰 + 㦨 d d
㦨
e kx + ҈ - x桌 C㦨 d
ᔴ �ᨨ AqA㦨맔҈䰘 C�矄 �矄�盐 � ᨨ犨ᨨ矄 \ q҈졔q㦨맔҈䰘C 꾀矄ᨨ ᨨ � � ᆠᆠᨨ矄矄A㦨 ᆠᨨ矄矄 B
�㡰
x㦨矄� ᨨ �ᆠ�ᆠ㦨҈m d㦨҈m
和ᨨ I =
\
A ᨨ犨ᆠᆠ� ᆠ CA㦨C҈졔C
´ ҈-䰘 㦨҈맔҈䰘C ҈
ᨨ �泐矄ᨨ 㦨
㦨
+
+ - A⏌ A⏌ -
´ ҈-黨 CA q 㦨q q㦨 CB ҈ ´ ҈-黨
Þ q҈㦨맔҈䰘C
A ᆠᆠ�ᨨ矄� ᨨ � � �ᆠ�ᆠ 矄 A ᨨ ᨨ �矄犨� ᨨ盐 �ᆠ�ᆠ矄 +
㦨҈맔҈黨F
q졔q㦨맔҈䰘
Þ q㦨
�㡰 ᆠᆠ�ᨨA犨� �ᆠ�ᆠ矄ᆠ ᆠᨨ 矄� 矄犨ᨨ
҈҈
� ´ ҈-
Þ q҈㦨
� ´ ҈ ҈ ´ � [҈҈ ҈] ´ ҈ = � ´ ҈ 黨 A ҈
+ + +
� ´ ҈-҈ ´ 黨 ´
q҈ q = C Ἐ A CB
e V \ q = CV = ëé҈ + x ( K ҈)ûù ҈ dq e V dx = [ K ҈] \ᔴ盐 I = dt ҈ dt
�䰘
҈ ҈ C V 㦨 맔맔҈黨맔 ҈҈㡰 A
q q W㦨 C A Ἐ C A
㦨K҈KAe ê
ᨨC㦨
҈ + 黨桌 㦨맔҈黨F
㦨҈҈맔҈� ��㡰 ␔ �├� �ᨨ盐 �ᆠ�ᆠ ᆠ �ᨨ ᆠᆠ�ᨨ 鼀� 矄 矄鼀 盐 ᆠᆠ� ᆠ ᨨ 꾀犨� � ᆠ � ᆠ �泐犨�ᆠ � �矄ᆠ �� �泐�矄⼄盐ᨨ犨ᨨ├ ᨨ
ò K ld + d K҈ - K 㡰 x
㦨K҈KAe
´ � ´ ҈-
UA㦨
K҈ K Ae dx
l
´ � ´ ҈-҈
쟼 �泐矄ᨨ � �矄ᆠᆠ�ᨨ犨 ᆠᨨ
ᆠ ᨨV㦨҈҈ᨨ 泐
ᆠᆠᆠ� ᆠ ᨨ 犨 ᨨᆠᆠ�ᨨ C㦨
A ⏌ 㡰 e A ⏌ 㡰 e er A e + = ҈ + e r 桌 d d d
´ ´ ҈-҈ ´ ´ ҈-黨
+
q҈ q + C Ἐ A CB
҈ ´ ҈ ´ ҈-҈ ´ ҈ ´ ҈-
㦨맔҈�졔黨맔҈�㦨҈҈맔҈� �
ᨨ � � ᆠᆠᨨ矄矄ᆠ ᆠᆠ�ᨨ�矄V矄ᨨ ᆠᆠ�ᨨ矄 �
ᨨ �泐㦨 �泐� A졔 �泐� B
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EBD_8341 PS-198
㦨
Physics
҈ ҈ CV + CV 㦨CV
�㡰
␔ 矄犨�ᆠ ᨨ �ᆠ�ᆠ�矄� 矄 犨 矄 ᨨ ᆠᆠ�ᨨ矄 犨 ᆠᆠ� ᆠ �矄 ᔴ 盐 ᨨ � � ᆠᆠᨨ矄矄��矄V 쟼 �泐� A㦨
҈ C 㡰V = CV
쟼 �泐� B㦨
CV ҈ q ҈ CV 㡰 = ´ 㦨 KC C
ᆠ �ᨨ ᆠᆠ�ᨨB鼀� 矄矄鼀犨 矄犨�ᆠ �矄ᨨ 㡰 ᨨ �泐㦨 �泐� A졔 �泐� B \
ᨨ �泐㦨
҈ � CV + CV = CV
��㡰
ᨨ �泐 � ��泐 CV = = �
ᨨ �泐 � 泐 � CV
\ \
V=
쟼ᆠ�ᆠ � 盐ᨨ �
鼀ᨨ矄ᨨ矄�� ��ᆠ �矄
é
qù ú ᨨ犨 矄 ë r û
M E҈ = ê k
é q ù E = ê k ᨨ犨 矄 �� 쟼ᆠ�ᆠ � 盐 ᨨ ú ëê r 㡰 ûú ᨨ盐鼀ᨨ矄ᨨ矄�� ��ᆠ �矄 M
ᆠ㡰 꾀ᆠᆠᨨ � �ᨨ ⼄盐矄�ᨨ ᨨᨨ �矄矄 矄鼀� 㜠 犨泐犨 犨ᨨ矄 �矄 ᨨ盐 ᨨ 鼀ᨨ � 矄ᨨ
ᨨᆠ �矄 � ᨨ鼀 �ᆠ 犨ᨨ ᨨᆠ矄ᆠ� �ᨨ qE 졔 ᨨᨨ 盐矄 矄矄 q 㦨� 矄 ᨨ犨 � ��盐 qE㦨mg
æV ö q ç ÷ = mg èdø
졔 졔 졔 㝐 턼
쟼ᆠ�ᆠ � 盐 ᨨ ᨨ矄�� �� ᆠ �矄M
Topic-4 : Miscellaneous (Mixed Concepts) Problems ҈
ᆠ�ᆠᨨ � 盐ᨨ ᆠ �矄M �矄 ᨨ �㦨㝐 ␔ 矄泐矄鼀 ᨨ ᆠ �矄 �矄 ᨨ ᨨ盐 � PM ᆠ盐 犨� ᨨ ᆠ鼀� �ᆠ � B㦨M. ␔ B㦨 m㦨 ⼄㡰 㝐 �矄 鼀� 㜠ᨨ� 犨 犨ᨨ �
鼀ᨨ矄 ᆠ �矄
v
mg
mgd ҈䰘 ´ ҈-䰘 ´ ҈ ´҈ - = ҈-黨 㦨 -҈黨 q ҈ ´ ҈
é r r r r q ù E = ê k ᨨ犨 矄 � E҈ + E + E = d ú ëê r 㡰 ûú
ᆠ�ᆠᨨ � 盐ᨨ ᆠ �矄M
é +q -q q q q q ù + - + = ë r r r r r r úû
V =kê
ᆠ盐 盐ᨨ �
鼀ᨨ矄ᨨ矄�� ��ᆠ �矄�矄 ᨨ 和�鼀�泐 ᆠ盐 盐 ᨨ ᨨ ᆠ �矄 � �矄 �矄 ᨨ ᨨB㦨\m㦨 㡰
졔
ᆠ�矄 泐҈맔҈黨
ᨨᆠ�ᆠ 盐 ᨨ盐� ᆠ�泐 =
ᆠ 㡰 ᨨ �ᨨ ᨨ� 矄�ᨨ ᆠᨨ 盐ᆠᨨ�矄矄鼀
盐矄ᨨ �A㦨ᨨ �B
q �矄ᆠᆠᨨ � �ᨨ盐矄矄Ἐ矄 Î
ᨨ鼀 졠�ᨨ 㡰 㡰 ᨨ ᆠ盐盐 犨 �ᆠ �矄 � ᨨ� 矄A Bᆠ盐矄ᆠ�泐�矄��ᆠ 㡰 턼�矄 �ᆠ盐ᨨ ᨨ �ᨨ
졔
졔
㝐 졔
턼
ᔴ泐ᆠ�ᆠ � ᆠ Mᨨ �ᨨ 盐ᨨ ᆠ �矄 ᆠᨨ 矄ᨨ �盐 �ᨨ �矄 ᨨ
쟼
쟼 쟼졔 ҈ ҈ 쟼졔 ҈ 쟼졔 E+ = E- 쟼
� E㦨M 泐矄泐鼀鼀泐㡰
턼
㝐
졔
졔
ᆠ�ᆠ � 盐ᨨ ��ᆠ �矄쟼�
㦨和�鼀�泐 ᆠ�ᆠ � 盐ᨨᨨ矄��ᆠ �矄쟼ᨨ盐㦨 \쟼㦨
ᆠ�ᆠᨨ � 盐ᨨ ��ᆠ �矄M�矄 � ᨨ鼀 ᆠ�ᆠᨨ � 盐ᨨ ᨨ矄��ᆠ �矄 M \¹㝐 ␔ 矄泐矄鼀 ᨨ ᆠ �矄 ᨨ 犨 � ᆠ盐 I҈ 盐 ᨨ
��ᆠ �矄 ᨨ ᆠ盐 I 盐 ᨨ ᨨ矄�� ᆠ �矄
鼀� �盐 ᨨ B M 盐 ᨨ ᆠ盐 矄 �矄 � \B¹ m ¹ ᆠ�ᆠ � 盐ᨨ � ᆠ �矄ᆠ ᆠᨨ盐� �矄 \쟼㦨 矄㡰
a
졔
a
M
P
졔
Q
ᆠ�ᆠ � M 盐ᨨ ᆠ �矄�矄 ᨨᆠ盐矄
ᨨ �M 盐ᨨ ᆠ �矄
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PS-199
Electrostatic Potential and Capacitance
é +q q æ q öù V = kê + - ¹ ç �a ÷ ú a⏌ a⏌ ê ú ç ÷ è ø úû ëê
㡰
졔
졔
㝐
҈ ���㡰
Q 턼µ 桌
ᨨ鼀⼄ �㡰 ��㡰 ���㡰
␔ 犨 ᨨ 矄泐矄鼀 �矄 矄 � ᨨ ᨨ�ᨨ 犨� ᆠᨨ 矄 �盐ᨨᆠ�泐ᨨ盐 � PQ犨 � ᨨᨨ矄� 犨 �ᆠ ᆠ盐 �矄 ᨨ犨� �
鼀� �ᆠ � 盐 ᨨ 矄 ᆠ盐 矄 ᨨ 盐ᆠ 矄盐 鼀� �ᆠ � M犨 �ᆠ �矄 ᨨ⼄盐ᨨ ᨨ ᨨ 鼀� �ᆠ �ᨨ鼀ᨨ鼀 犨�ᨨ 盐ᆠ \B¹ m¹
졔
µ
\
ᔼ µ
㦨ᔼ µ ҈
P 泐ᨨ� �矄 ᆠx ᨨ鼀 ᨨ��� 꾀矄矄 ᨨ犨 �
��盐 犨ᨨ ᨨᆠ矄 ᨨ 盐矄�ᨨ ᆠ� � 盐 ᨨ 犨ᨨ ᆠ �矄ᨨ 졔鼀ᔴ ᨨᆠ�矄Fᆠᨨ矄aᨨ犨 矄�� 和�鼀�泐 犨ᨨ ᨨᆠ矄ᨨ ᆠ�ᨨ 盐ᨨ犨ᨨᆠ �矄ᨨ ҈鼀ᔴ ᨨᆠ 盐ᨨ 矄 ᨨᆠ�矄FἘᆠᨨ矄bᨨ犨 矄
턼
쟼¹矄 犨� ᆠ�ᆠ � 盐ᨨ �鼀 ᨨ ᆠ �矄Mᨨ犨 矄 �� 矄� 쟼ᆠ�ᆠᨨ �V㦨M 盐ᨨ矄泐鼀鼀�ᆠ �鼀 ᨨ ᨨ矄�� ��ᆠ �矄 ᆠ盐 �矄 ᨨ 犨 矄泐矄鼀ᨨ ᆠ �矄 ᨨ矄 ᨨ盐 PQ 盐ᨨ矄泐鼀鼀�ᆠ �鼀 ᨨ ᆠ �矄\B㦨 m㦨 㡰 ᔼ ᨨᆠ矄矄҈犨ᨨ├ ᨨ 泐泐犨㦨⼄맔㦨ᔴᨨ맔ᨨ㦨 ᔴᨨ
҈ 쟼 �泐矄ᨨ � 泐쟼ᔴ㦨 ᔴ
K ´ ´ ҈ - ´ ҈ ´ ҈-
�矄矄� ᆠᨨ矄矄 矄�矄 ᆠ 쟼㦨␔쟼ᔴ㦨ᔴ �
V ö ù V ö ҈ éæ Vᨨ ö æ V æ - ÷ + ç ᨨ - ᨨ ÷ + çVᨨ - ᨨ ÷ ú C êç ø è è ëè ø ø û
䰘
҈ CVᨨ
㡰 꾀�ᆠ盐 ᔼ µ 턼µ
턼
x +
´
䰘
x
x +
Þ x㦨᠀
�
� m ᨨᆠ�矄盐矄�ᨨ犨 矄졔vex㜠 �矄 ᨨx
´ 黨 ´ ҈黨 ´ ´ ҈ - � 䰘 +
é ë
-
҈ù
´ 黨 ´ ҈黨 ´ ҈ - � +
㦨맔黨맔҈黨맔҈ ê - ú = 䰘 ´ ҈
û
⧤� �ᆠ �泐�矄 ⼄盐� ᨨ ᨨᆠᨨ鼀 ᨨᆠ ᨨ 盐矄�ᨨ
�㡰 ��㡰
҈ F qE ö ҈ 턼쟼 ҈ æ 턼 ´ ö æ 㦨 ç 㦨 ÷ ÷ ´ çèQ a = = m mø 鼀 è 鼀 ø ᨨ鼀和㦨盐졔
æ ö ç x + ÷ è ø
x
´
� m ᨨᆠ�矄 ᆠ�ᨨ犨 矄 �� x㜠 �矄 � 쟼ᆠ�ᆠᨨ �ᨨ ᨨ盐ᆠ �矄x㦨 V㦨
ᆠ㡰 꾀 ᆠᨨ� � � ᨨ 犨� �� �� ᆠ � � 泐 ᨨ ᨨ盐 ᆠᆠ├ᨨ ᨨᨨ鼀 盐 ᨨ ᨨ ᆠ盐 矄�鼀 鼀ᨨ �ᆠ 鼀ᨨ�ᨨ
矄 ᨨᆠᨨ �µx
K ´ ҈ ´ ҈- ´ ҈ ´ ҈-
꾀x>
\ ᨨ �泐 �矄矄� ᆠᨨ矄矄 矄�矄 ᆠ
ᨨEd㦨
㦨
҈ ᔴ � 㡰
҈ 㦨 ᔴ � 㡰
ED =
æ 䰘 ö ç x + ÷ è ø
҈ ҈ ᔴ = ᔴ
\쟼ᔴ㦨쟼 ᆠ㡰 � � �� � ᨨ�� ᆠ 矄ᨨ ᨨᆠ矄矄
쟼 �泐 �矄矄� 㦨␔泐 D 㦨 ᔴ � 㡰
ᨨ ᨨᆠ ᨨᆠᨨ鼀 ᨨ Fᆠᨨ矄a㦨F'ᆠᨨ矄b
ᨨ鼀µᨨx㦨
�
犨ᨨ├ ᨨ � �矄ᨨᆠ矄矄W㦨q V㡰 \ W㦨҈맔҈맔䰘맔҈㦨䰘맔҈ 泐 �泐ᆠᨨ 矄�ᨨ
҈ mV 㦨䰘 맔҈
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EBD_8341 PS-200
Þ
Physics
҈ ´ ´ ҈- V = 䰘 ´ ҈- Þ V㦨鼀⏌矄
K.E. at the ᨨr�g�n ᨨ � ᨨ���
Vx 㦨㦨
҈
´ 黨 ´ ҈黨 ´ ´ ҈ - 䰘
-
´ 黨 ´ ҈黨 ´ ҈-
㦨맔҈ 꾀�� 泐 �泐 ᆠᨨ 矄�ᨨ
ë
r
黨
졠 � ᆠᨨ ᆠ
é R
sa e
ᔼ
Sn = ᨨ
a
qs a + x - x㡰 e
\ U 鼀� = 鼀� ᨨ鼀 ⼄盐�ᨨ ��㡰 � 犨
U x㡰 xᨨ�矄矄矄 ᨨ犨
ᨨ犨
҈
QR é æ R ö ê҈ ç ÷ r êë è R + r ø
nù
ú úû
qn qn = C ( pe R )
��㡰
U( A) = mgH + mg é a + H - H ù ëê ûú
H㦨
) 犨 r �矄ᆠᨨ鼀鼀ᨨ �ᨨ
é QR ìï æ R ö n üï ù ҈ ҈ ᨨ U n = ´ ´ê í҈ ç ÷ ýú pe R ê r ï è R + r ø ï ú î þû ë 㡰 �鼀�� �盐ᨨ �泐矄 n ® ¥ 盐矄ᆠᆠ盐qn犨 n 矄ᨨ ¥ a ᨨ S ¥ = 犨 r҈ = ᆠᨨ鼀鼀ᨨ �ᨨ ҈ - r҈ QR é QR ҈ ù ᨨ q¥ = \ q¥ = R ú R+r ê r ê҈ ú ë R+r û q¥ ҈ æ QR ö \ U¥ = =ç ´ ÷ C è r ø ´ ( pe ) ´ ( R )
b㡰 ᨨᨨ � �泐ᨨ �ᆠ �� H U x㡰㦨mgx졔qV x㡰
(҈ r҈ )
qn =
Un =
mgH㦨mg a - { a + H 㡰 - H 晤桌
= mg é a + H - H ù êë úû dU ᨨ⼄盐���盐鼀 = dH a
(
a ҈ r҈n
쟼ᆠᨨ矄�ᆠ �泐ᨨ S n 矄盐ᆠ ᆠᨨ ᆠ矄
㦨mgx졔mg a + x 㡰 - x桌
û
n é R æ R ö æ R ö ù +ç + + qn = Q ê ÷ ç ÷ ú R+r è R+r ø è R + r ø ûú ëê
盐� � �矄盐� ⼄盐�ᨨ �㡰
㦨mgx졔
æ R ö ù
ë
s mgH㦨q a { a + H 㡰 - H 晤桌 �㡰 e e g = Þ sq = mg ᨨ鼀 �� �ᨨ m s e
H㦨
졠 ᆠᨨᆠ
␔ �ᆠ �矄 ᨨ ᨨ � �矄 ᨨ �矄 �泐 �矄 ᆠᨨ 矄 ᨨ矄矄ᨨ ���ᨨ ᨨ � �泐㦨�� � ᆠ�ᆠ ᨨ � �泐 mgH㦨qDV㦨qV 㡰V H㡰桌
ᨨ
æ R ö
q㦨Q ê R + r + çè R + r ÷ø + çè R + r ÷ø ú ê ú
s a + H - H 桌 e
=
æ R ö ᨨ rq = RQ Rq + RQ ç è R + r ÷ø éæ R ö æ R ö ù ᨨq㦨 Q ê ç ÷ +ç ÷ ú êë è R + r ø è R + r ø úû
꾀x㦨H
꾀 V O㡰㦨
kq҈ ҈ æ R ö 犨 k = ᨨ q҈ = Q ç ÷ R pe è R+r ø
kq r R ᨨ rq = RQ Rq + Rq҈
s a + x - x桌 e
V H㡰㦨
=
k éëQ ( q q҈ ) ùû
⧤쟼㦨҈맔҈䰘맔҈맔҈桌 㦨҈맔҈맔맔҈㦨맔҈ 㡰 � a 㦨 �盐矄 ᨨ �矄ᆠ s 㦨 矄盐 ᆠ ᆠ � 矄�泐 q⏌m㦨eg⏌s ᨨ � 盐 ᨨ ᆠ � �矄ᆠ 泐 � ᨨ� 矄�盐 �矄 ᆠx ᨨ鼀O V x㡰㦨
�矄 矄 ⼄盐���盐鼀 ᨨ矄��ᨨ
졠 矄ᆠᨨ ᆠᨨ ᆠ 矄�鼀�泐
û
㡰 S҈ᆠᨨ � ᆠ �Q ᨨᆠᨨ ᆠ犨� S 졠 ᆠᨨ ᆠᆠ �q҈矄 � ᨨ鼀S҈ᨨS
k (Q q҈ )
V - Vx = ù ⧤쟼㦨q é x= � ê ú \
a
ᆠ H =
���㡰
ᨨ ҈҈
U¥ =
Q R
r ´ ´ pe R
ᨨ U ¥ =
Q R ( pe ) r
㡰 꾀ᆠ � �矄 盐 � ᨨ鼀泐 �矄�盐 ᨨ ᨨ 㜠ᆠᨨ 盐ᆠ� � 矄 �ᆠᨨ盐鼀ᨨ �盐矄R盐矄ᆠᨨ 矄� 矄 ᨨ �ᆠ├ 矄矄 dx �矄 ᆠx ᨨ鼀 ᆠ ᨨ 矄 ᨨ盐鼀ᨨ 矄 㦨
é ù p ê x + dx 㡰 - p x ú ë û
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PS-201
Electrostatic Potential and Capacitance
ᆠ㡰 ᔼ 矄鼀ᆠ � ᨨ Q ᆠᨨ ᨨ鼀 �矄 �� ᨨ 矄 �ᆠ ᆠᨨ 盐ᆠᨨ ᨨ 矄鼀 �盐矄 C = pe R �泐
㦨 p éë x + dx㡰 - x ùû ù é æ dx ö p ê ҈ + x 㦨 çè ÷ø - ҈ú x ëê ûú 㦨
E= ҈
é dx ù p x ê҈ + - ҈ú x ë û
=
҈
r ᆠ �盐 �ᨨ盐鼀ᨨ 矄 \ ᔴ �ᨨ 矄 㦨dq㦨pxrdx �㡰 ᨨ � 矄盐 ᆠᨨ 矄 ᨨ �盐矄x
҈ 㦨 ´ pe
r ´ px x
㝐� �鼀盐鼀 �鼀 ᨨ �ᨨ ᨨ �� ᨨ � �ᆠ�ᨨ ᨨ ᆠ�ᆠ � t㦨
dx px ´ 㦨pxdx x
҈ Q ҈ Q = pe R C
T p ML p ML = = qE qE
쟼ᆠ �ᆠ犨�� ⼄盐���盐鼀盐 ᆠᨨ ᨨᆠ 矄�ᨨ ᨨ 矄� � T 犨�� � � 矄盐 ᆠᨨ矄�ᆠ ᨨᆠ F ᨨ 犨ᨨᨨ ᆠ �矄 ᨨᆠF 鼀� �ᆠ盐
qù é êëQ V = k r úû
\ ᨨ � 矄盐 ᆠ ᨨ 矄 ᨨ �盐矄
x㦨
rx e
␔ᨨ├ ᨨ � � �� � ᆠ �dqᨨ 矄 ᨨ �盐矄 xdW
rx rx 㦨 맔dq Þ dW㦨 맔p xrdx e e
\ ␔ᨨ├ ᨨ � ᆠᆠ盐鼀盐� � ᆠ � Q ᨨ 矄 �ᆠ ᨨ盐鼀ᨨ �盐矄R R pr
pr x dx = W㦨 ò e e
R
é x� ù pr R� ê ú = e � ëê � ûú
p æ Q ö R� Q = 㦨 ç ÷ e è ⏌ pR ø � pe R
矄ᨨ� � T � �ᆠ�ᨨ ᨨ mg F 泐� � ᆠᨨ ��ᨨ ᨨ ⼄盐���盐鼀
F \ q 㦨 mg
Q KQ ҈ = 犨 K㦨 � ´ pe 㡰 R �R pe ҈ ᆠ� � ᨨ⧤泐G Q泐 M pe
\ g㦨
E㦨
GM �R
\
gR
Þ G㦨 M R
E =
F㦨 FCA =
´
҈
kq
��㡰
CA㡰 ur ur F CB = F CA a㦨 桌
T鼀├ � q 犨� �ᆠ OC㦨 \
㡰 - ҈�㡰 㦨҈䰘鼀
OM㦨黨黨黨䰘
\ q 㦨
gR M ҈ 㦨 gMR = ´ ҈ ´ � ´ ҈ � M R � � = ҈� ´҈
\
F㦨 FCA + FCA + FCA ´
GM
�㡰
F㦨 FCA + FCB + FCA FCB ᆠᨨ矄 a
�矄犨ᨨ├ ᨨ �矄 ⼄盐ᨨ �泐矄ᨨ � 矄泐矄鼀 b㡰 ᆠᨨ矄ᨨ � � �泐 ᨨᆠᨨ鼀泐 �矄矄矄鼀 �� 矄 ���ᨨ 盐 E㦨
T矄� q 㦨F
Tᆠᨨ矄 q 㦨mg錐
OC ҈䰘 = OM 黨黨黨䰘
���㡰
ᨨ鼀⼄ �㡰 ��㡰 ���㡰
҈䰘 ´ 黨 ´ ҈黨 ´ ⼄ = 黨黨黨䰘 㡰 ´ ҈- ´ 黨 \q㦨҈맔҈黨ᔴ
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EBD_8341 PS-202
Physics
Current Electricity
17
Topic-1 : Electric Current, Drift of Electrons, Ohm's Law, Resistance and Resistivity ҈
҈�
黨 ´ ҈ ´҈ ´ ҈-҈黨 ´ � ´ ҈ - ҈
ᆠ㡰 꾀鼀鼀 ᔼ 矄�矄 ᆠᨨ
ᆠ�ᨨ 矄鼀ᆠ盐 ᨨ犨矄 ᨨ盐� ᆠᨨ鼀ᨨ 矄 ᔼ �鼀矄 鼀矄盐� � ᆠ盐 ᨨ犨� � ᨨ盐� ᆠ�ᆠ盐� ᆠ��矄ᆠᨨ
ᆠ � 矄�矄 ᨨ鼀꾀ᨨ鼀鼀矄盐矄ᨨ�ᆠ �犨 犨ᨨ ᨨ� 矄� ᆠ�ᆠ盐�和ᨨ犨 ᨨᆠ ᨨ鼀� 犨� ᆠ�ᆠ盐� ᆠᨨ鼀ᨨ b㡰 rM = 黨 ´ ҈-
r A = � ´ ҈- rC = ҈䰘 ´ ҈- rT = �� ´ ҈- \rM > rT > r A > rC
㡰 㡰
쟼
҈҈
Þ m=
쟼 쟼꾀 ᔼ r㡰 = Þ 쟼= ᔼ 꾀
꾀 = = 쟼 ᔼr
(
҈҈ ´҈ ´ p ´ � ´҈
)
҈
dR =
� ´ ҈䰘 ´҈
px
䰘
dx px
b
Þ
R=
r é -҈ ù p êë x úû a
æ r ö æ҈ ҈ö R = ç ÷ ç - ÷ è p ø è a b ø m ᆠ㡰 r = ne t 黨҈ ´ ҈-҈ = � ´҈ ´ ҈ ´҈-҈黨 㡰 ´ � ´ ҈ -҈� 㦨҈W㜠鼀 㡰 盐鼀�矄矄矄ᨨᆠ� 犨� ᆠᨨᨨ盐 �矄ᆠᨨᨨ盐�矄 ᆠ 泐 � Q ᔴᨨᨨ盐ᆠᨨ ��盐 ᨨ� �矄� \犨矄�矄 ᆠ㦨�W 㡰 ᔴ泐 ᨨ鼀 � ᔴ盐 ᔼ㦨
黨
㡰
� ҈㦨҈WἘ㦨⏌ 㦨䁸
ᨨ a = ´ ҈
r
-
°ᔴ - ҈
rl Q꾀맔 � 㦨ᨨ盐鼀 꾀
꾀
⼄ = = 矄 和� ᆠ⼄�矄ᆠᨨ 矄 桌
矄� �ᔼ㦨 꾀
\� 矄 =
҈
꾀
µ
҈
ᆠ㡰
쟼ᆠ�ᆠ � �矄 ᆠr ᨨ鼀� ᆠ �
ᨨ µ
҈
Q꾀㦨p
꾀 = ҈ Þ = ҈ ´ ҈ = ҈ W 矄�矄 ᆠᨨ 犨犨� ҈
b
㡰
ᆠ =
r㡰 dx㡰
ò
a
҈ ҈ ´ �
和� ᆠr ®ᆠᨨ 矄 ®ᆠᨨ 矄
R = dR
ò dR = rò
a=
矄�矄��泐ᨨ 犨� =
鼀 m = ҈ 矄 㡰
= = ҈҈W ᔼ
㦨҈ �� 㡰鼀盐ᆠᨨ �ᆠ� ᨨ 矄�矄 ᆠa㦨䁸 盐矄� � 㦨 ҈졔a㡰 ҈҈㦨҈ ҈졔a�㡰
�
矄�矄 ᆠ 鼀盐 � ᆠ矄矄 泐� ᔴ �
ᆠ㡰
=
[ ␔ = � ᨨᆠ�泐 ]
矄�矄��泐 ( r ) =
㡰 ᔴᨨᨨ盐ᆠᨨ ᨨᆠᨨ 矄�矄ᨨ 졠Ӹ盐␔ ҈�䰘黨 矄�矄 ᆠ 㦨꾀맔ᔴ᠀ 矄꾀 �矄犨ᨨ矄�� � �ᆠ ��盐矄ᨨ 矄�矄 ᆠ ᔴ� �ᆠ矄 ᆠ�鼀鼀盐��ᨨ 盐鼀ᨨ ᨨ矄 ᨨᨨ犨꾀 �矄ᨨ ᆠᨨ 㦨᠀�奠 和�㦨᠀҈奠ᨨᆠᨨᨨ盐㦨᠀奠
= � ´ ҈ ± �奠 = �├W ± �奠
␔ �㦨V㦨e㦨҈�ᨨ ᔴ � 鼀ᨨ��泐
m㡰 =
㦨鼀鼀矄҈
҈
dV l = per dr l dV = dr per E=
l㦨� ᆠ � 矄�泐ᨨ 犨�㡰
ᔴ盐 ᨨ盐� 鼀 矄
l dr lsl dV per I= = = dr ҈ dr e ´ s prl Q = r
l ҈ dr 㡰 \ = r = 꾀 prl s prl
�矄 ᆠ盐 �矄 �泐 ᨨ盐犨 矄
d lsl \ ( ll ) = Þ dt e ( s ⏌ e )t
l
ò
l
t
dl æ sö = ç ÷ dt è eø l
ò
Þ l = l e lsl ls I \ J= = = prl perl per
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PS-203
Current Electricity
æ l s ö ( s ⏌ e )t e Þ J = J e ( s ⏌ e ) t per ÷ø l b㡰 V = IR = neAvd 㡰r A \ r㦨
҈
ᨨ J = ç è
҈
和ᨨᨨ � 㦨
�
´ ҈
´ ҈ ´ ҈
-҈黨
´ � ´ ҈- ´ ҈
ᆠ㡰
� 㦨 neAVd Vd µ
E � 㡰
ᨨ � µ E � µ E � µ V ᆠ� ᆠ㡰ᆠᨨᆠ泐 ��ᆠ矄 V㜠I� ᨨ犨�鼀 ᨨ 矄盐ᆠ 泐 ᨨ 鼀� ҈
\ R=
҈䰘
L
ᆠ R�矄� ᨨ L � 矄泐ᨨᨨ�ᨨ ᨨ t ᆠ㡰 꾀ᨨ鼀�矄鼀 泐ᆠᨨ
ᆠ� � �� 矄�矄 ᆠR҈ �
矄�矄犨� � ᨨ鼀G҈
R҈
꾀 鼀鼀�矄鼀 泐ᆠᨨ
ᆠ� �ᨨ犨矄�矄 ᆠR� 犨� � ᨨ鼀G
ò
G
҈
pr æ ö l ç ÷ è ҈ ø
҈黨
I V
҈ s
ᔼ ᆠᨨ 盐ᆠ� �矄ᨨ� ᆠᨨ�矄�ᨨ 矄ᆠᨨ鼀鼀ᨨ ⼄盐 犨� � ᆠ矄ᨨ 鼀盐
R
҈
æ ö l ç ÷ è ҈ ø Þ ᔼ= = pr
ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ ᆠ㡰 ᆠᨨ 盐ᆠ��泐 s㡰 ᨨ 矄鼀�ᆠᨨ 盐ᆠᨨ � ᆠ矄矄 犨� � ᆠ矄 � 鼀盐 � 矄�矄��泐 f㡰 ᆠ矄矄 犨� � ᆠ矄� 鼀盐矄P㦨
ᨨ鼀 �矄ᆠᨨ
ᆠ � 犨� 矄 矄�矄ᨨ RT �
ᆠ�ᆠ盐� 꾀鼀鼀�矄ᆠᨨ
ᆠ � 矄�矄犨� 矄矄�矄ᨨRT� ᆠ�ᆠ盐� 꾀� ᨨ�矄ᨨ盐ᆠ V�矄ᆠᨨ
ᆠ � 矄�矄犨� 矄 矄�矄ᨨRT � ᆠ�ᆠ盐� 㡰 ᔼ ᆠ�ᆠ ᆠ�ᆠ盐� 鼀鼀 �矄 ᆠᨨ
ᆠ � 矄�矄 犨� 矄�矄 ᆠ ᨨ鼀 犨� 矄�矄 ᆠ ᔼ ᨨ 鼀Ἐ矄犨犨ᆠ ᆠ├V㦨IR 泐泐� � 矄�矄 ᆠᨨ ᆠ�ᆠ盐� 㡰 ᔴᨨ �矄 鼀犨 矄 鼀 �盐鼀�矄 和鼀�㜠ᆠᨨ 盐ᆠᨨ 矄�矄 ᆠᨨ 鼀 ᆠ矄矄 矄鼀�㜠ᆠᨨ 盐ᆠᨨ� ᆠ矄矄犨� ᆠ矄� 鼀盐 㡰 矄�矄 ᆠ � ᆠ矄矄 犨� 鼀盐 矄 졔 ҈졔µ㡰 R҈㦨R ҈졔a t҈㡰Þ҈㦨 ҈졔҈�맔䰘桌 R㦨R ҈졔a t㡰Þ㦨R҈졔҈�맔 t桌 和ᨨ� �犨� T㦨� ᔴ㦨� ᔴ졔䰘 ᔴ㦨҈҈䰘⧤
T2
҈ æ ö Þ = l ç ÷ pr è ҈ ø rp
r r 鼀 鼀 = ⼄쟼 Þ 쟼 = ⼄ R҈ r 鼀ᔼ 쟼=
꾀 ⼄ I ᔼ ö æ Q � ᨨᆠ�泐 = çè ÷
꾀 ø r uur D = ò 쟼 ᨨ D µ \D µ ᔼ
t
G҈
҈ 矄ᨨᨨ �
ᆠᨨ�ᨨ 㡰�矄ᆠᨨᆠ 꾀 ᨨᆠ�ᆠ盐鼀ᨨ�ᨨ ᨨ ᆠᨨ D ᨨ矄 ᔼ
矄盐 ᆠ �� ᨨ � 矄ᨨ ᆠ�ᆠ � ᨨ矄 �泐 ᨨ盐犨 R 和ᨨ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ DV
L
rL r = L´t t
҈ =
矄�矄 ᆠ =
l ᆠ㡰 ␔├ ᨨ犨矄�矄 ᆠ R = r a l㦨L a㦨L맔t
T1 I
和ᨨ㡰T 和ᨨ㡰T҈\ 矄�矄 ᆠ㡰T> 矄�矄 ᆠ㡰T ҈ V ᨨ 鼀�ᆠ 犨� 矄�矄 ᆠ � ᆠ矄矄 犨� 鼀盐 \T>T҈ 矄盐ᨨ 鼀 �泐ᆠᨨ 矄� ᆠᨨ 盐ᆠᨨ 鼀ᨨᆠ��矄 ᆠᨨ 矄 鼀ᨨ�ᨨ � ᨨ鼀 �ᆠ�ᨨ 矄 � 矄 ᆠᨨ ᨨ � � ᆠ ᆠ 㡰 矄�矄 ᆠ ᨨ 鼀 泐矄�矄
\ò
㦨҈맔҈� W鼀 ҈�
I ҈ = V R
Þ 矄�矄 ᆠ R =
㦨ᨨ � � ᆠ 㦨 � ᨨ 犨�
㦨 ᨨᨨ ᆠᨨ 矄盐 �ᨨ盐鼀ᨨ ᆠᨨ 盐ᆠᨨ 㦨 ᨨᨨ ᆠᨨ 矄 ᆠ� � 盐ᨨ ᨨ鼀矄 犨�矄�矄��泐
r=
T盐 ᨨ鼀 �� ᨨ�ᆠ盐 �
�
l A V 꾀 V = Al Þ A = (V = ᨨ盐鼀 ᨨ 犨� ) l rl rl \R = = V ⏌l V r Vᆠᨨ 矄 \R µ l DR ´ ҈ = \ ᆠ � ᆠ � � 矄�矄 ᆠ R 奠ᆠ �� l㡰㦨 ҈奠㡰㦨奠 꾀矄犨 ├ ᨨ犨矄�矄 ᆠ R = r
Topic-2 : Combination of Resistances ҈
b㡰
ᨨ鼀ᆠ�ᆠ盐� ��鼀
҈ ҈ ҈ = + Þ R҈ = R҈ ҈ � ҈ ҈ ҈ = + Þ R = R �
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EBD_8341 PS-204
Physics
꾀
ᔴ
䰘
R = R҈ + R =
�=
ᆠ㡰 ᆠ㡰
+ = W � �
´ ҈ � = W ҈
⼄ 㦨
ᔴ
ᔴ
ᔴ
㡰 ᔼ 矄 泐矄 ᨨ犨 ᨨ ᆠ盐 ᨨ盐� ᆠᆠ�ᨨ 犨� ᨨ ᔴ盐 ᨨ盐� ᆠ�ᆠ盐�
E r + r
Vc =
R㦨맔㦨҈W R⼄㦨
ᔴ
꾀
ᨨ � � ᆠ ᨨ盐� ᆠᆠ�ᨨ
␔ � ᆠᨨ鼀矄 ᨨ盐 �矄 矄�矄 ᆠ ᆠᨨ鼀矄
R Ҙ l 㡰
�=
� � � v = = ҈ A \ I BC = - = = A � � � R
ᔴ
黨
æ E Q = E - �r 㦨 E - ç C è r + r
ö r ÷ r \ Q = CE r + r ø
b㡰 ᨨ � � ᆠ� ᆠ ᨨᨨ�矄 ᨨ \ᨨᆠ盐 犨� ᨨ犨ᨨᆠ盐 � ᆠ 矄�矄 ᆠ�矄ᨨ b㡰
꾀矄矄�矄 ᆠᨨ 犨�R㦨
òA l
rFe ´ lFe ҈䰘 ´ � ´҈ � = = ´҈ RFe㦨 AFe ´ ҈
ᆠ㡰 RR R�� 矄�矄矄ᨨ �⼄盐� R㦨졔�졔�㦨�W
�矄 �矄 犨� R 矄ᨨ 矄�矄 ᆠᨨ ᆠ�ᆠ盐�
r Al ´ l Al 䰘 ´ ҈ ´ � ´ ҈ 䰘 ´ � = = 맔҈� RAl㦨 AAl 黨 㡰 ´ ҈ �
㦨맔҈ 꾀矄ᨨ � � ᆠᆠᨨ矄矄ᨨ 矄�矄ᨨ矄�矄 矄鼀矄ᨨ 泐 � ᆠᨨ鼀� �ᨨ
\ RPQ㦨 ҈
矄�矄 ᆠ犨 턼
� ´ æ ö = � 턼㦨 P ç + ÷ = è ø � ҈҈ +
҈ æ ҈ ´ � ö W ÷ + ҈� + = è ҈ + � ø e ҈� 黨 = = A 和ᨨ � = R⼄ (҈ ⏌ )
R⼄ = ç
�
b㡰
RFe ´ RAl ҈� ´҈ ´ ´ ҈ ҈䰘� = = mW RFe + RAl ҈ ´҈
矄�矄 ᆠ犨 턼
´ 턼 = P + 㡰 = = + ҈҈
㡰
矄�矄 ᆠ犨
矄�矄 ᆠ µ l 矄ᨨ 矄�矄 ᆠ ᨨ ᆠ 矄� ᨨ ⼄盐� � �㦨W 矄�矄 ᆠ ⼄ 犨 泐犨ᨨ �ᆠ矄
҈ ҈ ҈ 㦨 졔 Þ ⼄ = W ⼄ ҈
㡰
矄�矄㦨 ҈졔 졔졔
҈ ҈ ҈ ҈ = + + + ҈
´ æ ö = = P ç + ÷ = è ø + ҈҈ ҈҈
ᆠ��矄ᆠ 턼�矄鼀 �鼀盐鼀 㡰 ᆠ盐 � W矄�矄ᨨ㦨 ᨨᆠ盐矄��矄 ᨨᨨ 泐ᆠᨨ矄 ᨨᨨ ҈
�W
҈W
W
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PS-205
Current Electricity � 矄�矄 ᆠᨨ ᆠ 矄�矄ᨨἘ Ἐ
ᆠ㡰
RPQ ҈
҈
㡰
� R RQR = R RPR = R = ҈҈ ҈҈ ҈҈
\ 턼�矄鼀 �鼀盐鼀 㡰 ᆠ�ᆠ盐� ᨨ ᨨ犨 �矄 POQ 矄泐鼀鼀�ᆠ 矄 矄 ᆠ 犨 矄ᨨ � � ᆠ ᆠ 矄�矄 ᆠ �矄� ᆠ� R
R
R
R
r
r
R
R
R r
矄�矄 ᆠ ᨨ W �矄 � 犨� R � ᆠ� 矄�矄 ᆠ
r
P
Q R
R
P
2R
Q
҈ ҈ ҈ = + R Ἐ R R RἘ = + R
ᨨ ⼄盐� ᆠ�ᆠ盐��矄 犨 矄 ᨨᨨ犨矄
r + r + R ҈ ҈ ҈ + + 㦨 Rr RPQ R R r Rr ᨨ RPQ㦨 R+r
҈
\
҈
ᆠ㡰
=
꾀矄ᨨ 㦨 ᔼ
R Ἐ = ҈� =
V 㦨 㦨҈꾀鼀 ᔴ盐 � ᆠ�ᆠ盐�I = R ҈㡰
҈䰘
ᔼ ��盐 㡰 ᨨᆠ盐 ᨨ犨矄 ᨨ盐� W矄�矄 ᆠ
W
W
W
W
W
�
҈W ҈W
ᆠ 犨 矄ᨨ � � ҈ ҈ ҈ = + ⼄
W
\ ⼄ =
´ = W +
I㦨
҈
æ Rö ç ÷ 꾀 矄�矄 ᆠ矄 R R R� 矄 èø
矄 ᨨ犨 � ��盐
R ҈ ҈ ҈ ҈ = + + = ᨨRAB㦨 R AB R R R R R
ᆠ
R
A
´ ҈ 黨 = W \R⼄㦨
B
R
W W
W
ᆠ 犨 矄ᨨ � �
҈黨
㡰 I
2k W
҈W
R
W
㡰
ᨨ ᆠ盐 ᨨ盐� 矄�矄ᨨR㦨W
V � � = = = ҈A 黨 � R +
R҈
V
҈W
k W
I=
矄� � �㡰
= � 꾀 ҈ W
꾀�� ⼄盐� 矄�矄 ᆠᨨ ᆠ 犨 矄ᨨ � � �� 㡰 ᨨᆠ盐 ᨨ盐� ҈W矄�矄 ᆠ
�
R + R
ᆠᨨ鼀 � � V㦨EIR 㦨 �㡰 �꾀㡰 W㡰 㦨��㦨
㡰
ᔼ
´ R Ἐ+ �
Þ졔R㦨R Þ R㦨W 㡰� ᆠ�ᆠ盐��矄ᆠ ᨨ � ᨨ 鼀 ᨨ ᆠ�ᆠ盐�㦨 �㡰㦨V ᔼ 矄�矄 ᆠᨨ ᆠ ᆠr㦨W 矄�矄 ᆠ� ᆠ�ᆠ盐�㦨 W㡰㦨҈W \ ᔴ盐 ᨨ犨� � ᨨ盐� ᆠ�ᆠ盐�
҈W W
ᔼ
Þ ҈ =
Þ R Ἐ+ � = Þ R㦨҈�
ᔼ �� ᆠ�ᆠ盐�
黨 ҈ ҈ ҈ = + = ÞR⼄㦨W R⼄ ´
҈�
�㡰
Rp =
I RL 㦨҈�kW
҈
R҈
V Rp
R ´ RL ´ ҈� 黨 = = kW R + RL + ҈� 䰘�
\Rᨨ㦨R҈졔Rp㦨졔 () I =
黨 㦨kW 䰘�
V = 鼀꾀 = 䰘� 鼀꾀 R
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EBD_8341 PS-206
Physics
I㦨IR
҈
\
æ RL ö æ ҈� ö IR = ç ÷I = ç ÷ ´ 䰘� = ҈� mA è ҈� + ø è RL + R ø
Þ R 졔졔R㦨R졔R Þ RR㦨 Þ RR졔R㦨 Þ R R㡰졔҈ R㡰㦨 Þ R졔҈桌R桌㦨 Þ R㦨W
\IR 㦨IR IR 㦨䰘�҈�㦨鼀꾀
㡰
҈
ᨨ � � ᆠ ᆠᨨ矄矄 ᨨ ��㡰
VRL = ( IRL ) RL = ´҈� = 黨V ᆠ㡰
\
҈
㡰
␔ R҈ R� ᆠ �
R҈P =
kW 䰘
6W
6W
3W
6W
6W
3W 3W
3W
6W
꾀
W
W
6W
3W
W
W
D
3W
3W 2W
Topic-3 : Kirchhoff ’s Laws, Cells, Thermo e.m.f. & Electrolysis ҈
ᆠ㡰
�W
�
A
B
D
W
҈W W ᨨ¥
E �҈
҈
W W
� �
C
W
W
F
矄� �⧤�ᆠ ᨨ Ἐ矄 ᨨᨨ犨 � ᨨᨨ ABCD
´R R+
C ҈W
6W
҈W
W W W ᨨ � � ᆠ犨 A B VAVB㦨黨҈ 졔㡰㦨V㦨�҈ \�҈㦨�꾀 \�㦨҈�҈㦨�꾀 和�鼀�泐ᨨ � � ᆠ犨 ᔴ VCVD㦨 VAVB㡰� 졔㡰 㦨�㦨 �㡰㦨V㦨� \�㦨�꾀 \ �㦨��㦨��㦨�꾀 �㡰 ᔼ ᆠ矄 ᨨ � � � ᨨ 矄�矄 ᆠ矄 ᆠ� 矄�矄 ᆠ 鼀� 矄 矄鼀 犨 ᨨ � �ᆠ�鼀 �矄 ᨨ 鼀ᨨ ᨨ鼀� 쟼 ᆠ�矄�矄 ᆠ犨 ᨨ� 矄C D R
ᆠ�ᆠ盐�ᆠ 犨 矄矄 ᨨ犨
쟼 ᆠ�矄�矄 ᆠ犨 A B
A ҈W
3W
6W
3W
�҈ 졔 �
�
�҈
Req㦨҈졔
3W
6W
ᆠ矄�矄 ᆠ犨 A BRAB㦨W
҈ = ᨨ ᨨ犨 黨
ᔴ
�
黨
3W
3W
黨 = = ҈꾀 㦨ᆠ盐 ᨨ盐� W矄�矄ᨨ 黨 W
3W
Þ
6W
3W
3W
Þ
3W 3W
3W
V ᨨPµV犨� ᆠ矄泐 ᆠᨨᨨ 黨 R
҈꾀
B
6W
3W
6W
3W
6W
㡰 矄�矄 ᆠᨨ ᆠ�ᆠ盐�㦨黨W \ᔴ盐 犨 ᨨ鼀 泐
Þ
3W
҈
6W
3W
� ᨨ犨 ᨨ � ᆠᨨ鼀矄
�㦨
3W 3W
3W
æ ⏌䰘 ö = ç = V è + 䰘 ÷ø
�矄矄� 㦨
㦨꾀
= ҈�꾀 꾀
3W 3W
3W
ᨨ犨 ᆠᨨ矄矄RL Ἐ VRL
�CD 㦨�EF 㦨
6W
R RL ´ ҈� = = kW R + RL � 䰘
RἘᨨ㦨졔R ἘP㦨졔
RAB㦨҈W졔҈W㦨W\IAB㦨
�CD 㦨�EF 矄矄�矄 ᆠ矄RCD 㦨REF
�ᨨᨨ ᨨ犨矄 �矄矄� � R҈ R
PR҈ ( I R ) R҈ ( 䰘�) ´ � = = = PR ҈� ´ ( ) I R (R )
´R 㦨 R+
҈졔
-�� - ҈ �҈ + � 㡰 - � + = Þ - ҈�҈ - ҈䰘� + =
�㡰
矄� �⧤�ᆠ ᨨ Ἐ矄ᨨᨨ犨 � ᨨᨨBEFC
Þ -҈ + �҈ + ҈ �҈ + � 㡰 = Þ ҈�҈ + ҈� + ҈ =
��㡰
㝐盐�泐� �⼄盐�ᨨ �㡰泐҈ 犨
҈�҈ + ҈䰘� = 㡰 ´ ҈ Þ ҈ �҈ - ҈䰘 � =
���㡰
㝐盐�泐� �⼄盐�ᨨ ��㡰泐҈䰘犨
҈�҈ + ҈� = ҈㡰 ´ ҈䰘 Þ �҈ - ҈䰘� = ҈䰘
�㡰
졠 矄ᨨ� �⼄盐�ᨨ 矄 ���㡰 �㡰犨� A
B
-҈�҈ = Þ �҈ = -
1W C
R
2W D
F
= -҈䰘 ҈
�҈ �矄 �� � 鼀 矄 ᆠ盐 ᨨ犨矄 ᨨ鼀 ᨨ矄�� ᨨ �� 鼀�
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PS-207
Current Electricity
㡰
D
E ҈꾀
�
B
ᨨ犨 � ᨨ
W
҈꾀㦨�҈ �
�㦨꾀
A
æ e ö P㦨� R㦨 ç ÷ R è R+r ø dP = ᨨ鼀 �鼀盐鼀 dR
F
C 꾀 ҈ 盐矄矄矄盐鼀 ᨨ �A㦨VA㦨 矄� �⧤�ᆠ ᨨ Ἐ矄盐 ᆠ�ᨨ 盐C犨�
�҈ + � = � ҈꾀 + � = 꾀 Þ � = A
ᨨ
ᨨ犨盐矄� �⧤�ᆠ ᨨ Ἐ矄ᨨᨨ犨ᨨ �ACDB
VA + ҈ + � 㡰 - = VB Þ VA + ҈ + � ҈㡰 - = VB Þ VB - VA = - = ҈ ᨨ
䰘
ᆠ㡰 ⼄盐� ᆠ�ᆠ盐�ᆠ 犨 矄 W
A
�
� ҈ W
C
꾀
쟼҈ 쟼 쟼 ҈ + + + + ҈ ҈ ҈ ҈ = ᔴ = = ҈ ҈ ҈ ҈ ҈ ҈ = = + + + + ҈ ҈ ҈ ҈
ᆠ㡰
ᔴ盐 矄矄� � ᨨ盐� 矄�矄 ᆠ ҈
W
W
d éæ e ö ù êç ÷ Rú = dR êè R + r ø ú ë û
ᨨ R㦨r b㡰 � 쟼҈㦨҈쟼㦨쟼㦨҈㦨҈W 㦨҈W 㦨҈W
W W
æ e ö �=ç ÷ èR+rø
㡰
҈ = = �꾀 �㦨 ҈ ҈ ҈W b㡰
҈ W 和 ҈ 턼 ҈W
�҈ =
ᨨ�ᆠᨨ矄矄AC㦨 矄�矄 ᆠRAC㦨졔㦨W
�҈ =
b㡰
黨
V = = ҈ 꾀鼀 RAC +
ᨨ �� ᆠ�ᆠ盐�
矄� �⧤�ᆠ ᨨ 矄犨P犨�
- ҈ - ҈ - + + = ҈ ҈ ᨨ �턼㦨 㦨
�=
҈ ҈ Þ + + = + + ҈ ҈ ҈ ҈ ҈ + � + ҈ + ҈ æ ҈ ö 䰘 Þ = Þ ç ÷ = ҈ è ҈ ø 䰘 ´҈ 䰘 Þ= = = ҈҈� ᨨ ҈ ´
+
ᨨ犨ᨨ�ᆠᨨ矄矄꾀
�´ =
´ = +
Þ黨㦨졔
҈ Þ= W �
㡰
҈
㡰 ᨨ鼀 ⧤ 졔ᔼ҈졔҈ ᔼ�ᔼ㡰㦨
꾀泐� � ᆠᨨ鼀� �ᨨ ᨨ �矄
E҈ E E + + ҈+҈ ҈ +҈ ҈ ҈ ҈ + + ҈ +҈ ҈+҈ + + = �´ ҈ ҈ ҈ + + ҈ = 㦨ᨨ
W 黨 ҈W
W
q
W
㦨ᔼ҈졔ᔼ҈ᔼ 錐ᔼ҈ᔼ㦨 ҈㡰 黨졔ᔼ ᔼ҈ᔼ㡰졔ᔼ㦨 ᔼ҈졔ᔼ㦨黨 㡰 졠 矄ᨨ� � ҈㡰 㡰 ᔼ҈㦨҈꾀
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EBD_8341 PS-208
Physics
�ᆠ�ᨨ 턼 ᨨ矄� ᆠᔼ҈>ᔼ ᆠ㡰 꾀矄 ᨨ ᆠ盐 ᨨ犨矄 ᨨ盐� 鼀 EB
㦨 쟼 㦨 㦨 꾀㦨� 和ᨨ ᨨ � � ᆠ 犨 ᨨ� 矄 A D VA VD 㦨 �
҈҈
҈
b㡰
ᔼ
�m
W
ᔼ
ᔼ҈
R
R
黨W
҈
ᔼ 矄 泐矄 ᆠᆠ�ᨨ �矄 盐泐ᆠ � ᆠ ᨨᆠ盐 犨� ᨨ犨 ᨨ盐� � 泐 矄�鼀泐 � � ᆠ�ᆠ盐�
ᨨ鼀 �鼀盐鼀ᨨ犨R 㦨R� R㦨W \ R㦨W �㡰 � ᆠ盐 ᨨ犨� � � ᆠ�ᆠ盐� 꾀泐� � ⧤�ᆠ ᨨ 矄 犨⧤� ᨨᨨᔴ쟼 ᔴ �� 졔㦨 Þ�㦨 \�㦨҈꾀 v ᨨ犨 ᨨ盐ᨨᨨ犨 ҈W ᔴ �
꾀졔҈맔҈㦨
\
҈
҈
盐矄ᆠ盐 ᔼ = ҈
b㡰
W �҈ A
æ V 㡰 V ö t÷ ´ t çQ H = R R ø è
҈҈W
黨V ´ t Q 盐Q㦨mcDT㡰 R
�㡰
ᔴ盐 ᨨ盐� AB �҈ =
= +
ᔴ盐 ᨨ盐� AD � =
҈ = 黨 + ҈҈ �
矄� �⧤� 꾀ᨨᨨ
NV 㡰 㦨 ´ t (Q R µ l ) R
VB + �҈ 㡰 - � 黨㡰 = VD ҈ Þ VB - VD = 黨㡰 - 㡰 Þ VB - VD = ҈ - ҈ = � ҈
꾀矄ᨨHἘ㦨m'cDT㦨 m㡰CDT
N V ´t R
��㡰
҈䰘
㡰 W
�� � �⼄ �㡰泐 ��㡰
A
C
mcDT 黨V ´ t ⏌ R ҈ 黨´ = = ᨨ mc DT N V t ⏌ R N
�
ᨨ N 㦨҈맔\N㦨 b㡰 ⼄盐� ᆠ�ᆠ盐�矄矄 ᨨ犨 � ��盐
҈
C
� 黨W
RἘ 矄�矄 ᆠᨨ 犨�ᨨ ᨨ盐 � �l \ ᨨ犨 �泐矄 � 㜠矄ᆠᨨ 矄'
鼀ᆠD 㦨
D
B
W W
�҈
W
�
F
E �
W
�
҈
R
R
쟼
D
꾀矄ᨨH㦨mcD 㦨
�
W
R 矄�矄 ᆠᨨ 犨�
쟼 �泐矄 � t矄ᆠᨨ H㦨
W
v
B
㡰
= 㦨 ᨨᔼ㦨 㦨䰘꾀 + 黨 ҈
B
꾀㦨�
ᆠ 矄盐 ᨨ � � ᆠ ᆠᨨ矄矄 矄�矄 ᆠ矄 犨�
A
ᨨ꾀맔҈㦨
黨W
R 4W E
\꾀㦨�
W
R
҈ ҈ ҈ R 㡰 R㡰 = + = R Þ Req = Req R R R㡰 + R㡰
҈�
R
E 4W
�
�
� ҈
R
R
꾀矄ᆠᆠ�ᨨ�矄 盐泐ᆠ � ᨨᆠ盐 犨� ᨨ犨 ᨨ盐� �
R E
W
R
R R
R
W
R
R
R
҈꾀
R
A
҈꾀 E W ᆠ ␔ 矄ᨨ � �
ᆠ�ᆠ盐� 矄 矄 ᆠ ␔ 矄ᨨ � � ᆠ ᨨ ᆠ盐 犨� ᨨ犨ᆠᨨ矄矄 W 矄�矄 ᆠ
W W
E 꾀
꾀
҈
B W
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PS-209
Current Electricity ␔ ᆠ盐 �矄�盐�ᨨ 矄矄 ᨨ犨 � ��盐
҈V VP = V ᨨ VP = V
æ ´ ö + 쟼⼄盐� 矄�矄 ᆠ ⼄㦨 ç è + ÷ø
VA VQ = I ´ W = (҈A)( W)
҈V VQ = V
҈ ҈ ´ = = 꾀鼀 ҈ +
VQ = ҈V V = V
�҈㦨꾀 �= ҈꾀 VAEB㦨҈맔졔맔㦨 ҈ 矄ᔼ ᆠᨨ泐�ᆠ 矄ᨨ盐�ᨨ ᆠ�ᆠ ᆠ盐 �矄 鼀� 泐 盐ᨨ 鼀ᨨ鼀 ᨨ �ᨨ 矄 ᨨᆠᨨ 矄 ҈黨 b 㡰 ᨨᆠ盐 �矄 ᨨ犨� � ᨨ盐� 矄�矄 ᆠR 꾀泐� � ⧤ � ᨨᨨ
\ � =
҈ + �㡰 ҈ +
V VS = V Þ VS = V
\VsVQ ҈
졔 II҈㡰졔҈졔҈맔I졔I㦨
Þ I҈�I㦨҈ A
1V
�
R
ᔼ
W
҈
矄�矄 ᆠᨨ 盐鼀
I
R⼄
I҈㦨⏌҈꾀 I㦨 �㡰
҈V \ I҈ = = A W æ ҈ ö I = ç ´ = A è + ҈ ÷ø I = I҈ I = ҈A
ᨨ � � ᆠᆠᨨ矄矄A P
VA VP = I ´ W = ( A)( W)
ᨨ � � ᆠ犨 B D
é� ù - ú ´ = VD \VB BD㦨 ҈ ҈ ҈ ë û
VB졔 ê
W
( W )(҈W ) = W RR = ҈ = R҈ + R W + ҈W
��㡰
� 꾀 ҈
ᆠᨨ矄矄G 㦨 -
ᆠᨨ矄矄 H㦨҈졔҈맔
҈W
쟼⼄盐� 矄�矄 ᆠᨨ ᆠ�ᆠ盐�
ᔼ҈
ᔼ㜠ᔼ҈
Þ I҈I㦨҈ 和ᨨ� �⼄ �㡰 ��㡰犨�
��㡰
R҈ = W + W + W = W
I҈
3W
҈ H
黨 - ҈ ҈ ´= = ҈ ҈ ҈
Q ᆠ�矄� �矄ᆠ �� �鼀ᨨ 桌
矄�矄 ᆠᨨ ᨨ犨鼀
R = W + W + W = ҈W
1W
D
W
1W
C G ᔼ҈ 꾀泐� �⧤�ᆠ ᨨ Ἐ矄 犨⧤� ᨨᨨ BCDB 犨 � II҈㡰졔҈졔I҈졔I҈㦨
F
W
B
E
F
V
A \ ҈ ҈졔҈ 㦨҈ ҈졔 ҈ Þ ҈ 㦨 ҈ ᔼ ҈㦨 ҈㦨 㦨 ᔼ ҈㦨 ҈㦨 㦨 ᔼ ҈㦨 㦨 ҈ ᔼ ҈㦨 㦨 ҈㦨 bᆠ 㡰 矄�矄 ᆠᨨ 鼀PQ STᆠᨨ鼀矄� ᆠ�矄PQ S T 矄鼀 ᨨ � ⼄盐� ᆠ�ᆠ盐� �矄 矄 矄 ᨨ犨 � ��盐 W 和 W W I
I҈
ᔼ
2W
E
B
҈ ҈ + = ҈ ҈ +
턼
�㡰
R
҈ - � ҈ = \ � = ��㡰 ҈
꾀泐� �⧤�ᆠ ᨨ Ἐ矄 犨 ⧤ � ᨨᨨ BDAB
R҈
V҈
꾀泐� �⧤� ᨨᨨᔴ쟼
ᨨ鼀⼄ �㡰 ��㡰
VP VS = ( A)( W ) = V
D �
C
҈ - � ҈ + - � =
ᨨ � � ᆠᆠᨨ矄矄P S
2W
ᆠ盐 � =
ᨨ � � ᆠᆠᨨ矄矄A Q
I҈
I
҈
�㡰
҈黨 = ҈ ҈ Q ᆠ�矄� ᆠ �� �鼀ᨨ 桌
和� ᆠ ᨨᆠ盐 �矄├ ᨨ鼀 泐
E҈ E E + + r҈ r r SE ⏌ r = \ VAB㦨E⼄㦨 ҈ ҈ ҈ ҈ S + + r r҈ r r
( ҈ + ҈ + ҈҈ ) = V 㦨 ҈ ( ҈ ) + ( ҈҈ ) + ( ҈҈ ) ᔼ 矄�矄 ᆠ ᨨ ⼄盐� 泐㦨
�҈ =
҈ W
VB V A + E҈ + = = ҈A r҈ ҈
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EBD_8341 PS-210
Physics
� =
VB V A + E + = = r ҈
� =
VB VA + E + ҈ = = ҈A r ҈
b㡰
�
㡰
��㡰 ᔼ r �矄 矄 ᨨ㜠ᆠ�ᆠ盐� 矄�矄 ᆠ ᨨ �矄 ᆠ �
ᨨ r҈ Q P I҈ E҈ A
R U
E
I I
I҈졔I졔I
T
r
E
R
҈ = � ҈ =
B
S
꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 � PQRUP 矄� � ᨨ鼀 P 鼀ᨨ� � ᆠᨨᆠ├犨�矄 I҈r҈ E҈졔E㦨ᨨI҈졔㦨 \I҈㦨҈꾀 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 � URSTU 矄� � ᨨ鼀 U 鼀ᨨ� � ᆠᨨᆠ├犨�矄 E졔EIr㦨ᨨ졔҈I㦨\I㦨҈꾀 矄�� ᨨ I � �ᆠ矄 �ᆠ�ᨨ ᨨ ᆠ盐 �
ᆠ UTSR �矄 ᨨᨨ矄�ᨨ 矄矄盐鼀 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 � AURBA 矄� � ᨨ鼀 A 鼀ᨨ� � ᆠᨨᆠ├犨�矄 I҈졔I졔I㡰RE㦨ᨨ
䰘
黨 ҈
æ ҈ ö ´ Pᨨ矄矄 = I R = ç è ÷ø
P ᨨ犨P㦨VI Þ I = V ҈� ´҈�� + \ I 鼀� = = ҈黨 A » A
b㡰
R ´ R R ´҈ R +R+ + R 㦨R졔R졔R졔R㦨R R + R R + ҈ R
V ҈ 矄� �P㦨 R Þ㦨 ⼄ R R㦨
҈ 㦨W ´
҈ Þµ
ᆠ㡰 �泐 犨� 鼀 �鼀盐鼀 犨 矄�矄 ᆠ 犨� 鼀� �鼀盐鼀 ᆠ㡰 ᨨ ᨨ犨 ᆠᨨ 矄盐鼀 泐 ᆠ�ᆠ � ᆠ矄 � 盐� � �ᨨ㦨�␔ ␔㦨 ᨨ맔 鼀 �㦨V맔IÞ�㦨IÞ I =
� = ҈҈ »҈꾀
㝐� �鼀盐鼀 ᆠᆠ�泐ᨨ 鼀� 盐矄㡰 ҈҈
ᆠ㡰
ᔴ盐 � ᆠ 盐㦨
ᨨ犨 ҈ = �꾀 㦨 ᨨ�
ᔴ盐 ᨨ盐� 鼀鼀㦨�맔㦨҈�꾀
㦨 W 턼㦨W ҈
b㡰
ᔼ҈ 꾀 ᔼ
쟼⼄盐� 矄�矄 ᆠ
R⼄㦨
\
ᆠ㡰 ᨨ鼀 �� ᆠ�ᆠ盐� 矄�矄 ᆠ矄 ᔼ㦨҈W ᔼᔼ㦨҈⏌W ᔼᔼᔼ㦨⏌W ᔼ�矄ᆠ > ҈> ᆠ҈
Þ
҈ ´ ҈ = 黨奠 \쟼 �ᆠ� ᆠ泐 = ҈ + Pᨨ矄矄
e ⏌㡰
҈҈ ҈҈ ´ ´҈ =҈҈´҈ � ␔ 㦨 ҈҈ ҈҈
꾀矄ᨨ犨 㡰 =
㡰 ᨨ犨P 㦨҈�맔�졔҈�맔҈졔҈�맔҈졔맔҈ 㦨҈�맔҈��졔␔
e e = ⼄
ᨨ'㦨㦨␔ Q㦨␔㡰 㡰 ᨨ犨㦨ᔼ 㦨맔҈맔 Þ 㦨҈҈맔҈W ␔ 矄盐泐ᨨ ҈҈�矄ᆠᨨ
ᆠ ᨨ犨㦨
R = W V =
㦨҈␔ �
和�鼀�泐㦨� 㦨 ␔ b㡰 ␔ 犨ᨨ矄�矄 ᆠ矄ᆠᨨ
ᆠ � 矄�矄 ⼄㦨
犨 ᨨ犨 ¢ =
b㡰 � ᨨ犨P㦨҈├␔㦨҈␔
P ҈ = V
æ ö ç � + ҈ ÷ è ø
´
ᔼ ᆠᨨ ��ᨨ ⼄ 㦨 ⏌
Topic-4 : Heating Effect of Current
ᔴ盐 I =
ᨨ犨ᆠᨨ 矄盐鼀 =
҈졔I҈㡰R 㦨\I㦨꾀
ᆠᆠ盐 ᨨ盐� R㦨I҈졔I졔I㦨꾀
҈
矄ᨨ ҈ = = � ҈ ᔴ盐 ᨨ犨 � = ҈ + 꾀矄 =
㦨҈W 和㦨W ҈㦨졔턼㦨W졔W㦨W 㦨 졔和㦨҈W졔W㦨W ᔼ҈ ҈㦨ᔼ
ᔼ҈ =
ᔼ ᔼ = ᔼ = ҈
ᨨ ᔼ = ᔼ҈ ᨨ犨㦨ᔼ
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PS-211
Current Electricity
æ R ö ÷ 㦨䰘IR ø
ᔼ ᨨ턼 턼 = ᔼ҈ 턼 = ´ = ᔼ
ᨨ犨 �矄矄��ᨨ P㦨I çè
ᨨ和 和 = ᔼ 和 = ᔼ × = ᔼ
R
\ 矄鼀ᨨ盐 ᨨ � 泐和 ҈
㡰 ␔├ ᨨ犨ᨨ犨 P =
ᔼ �矄 �� ᨨ犨矄 ᨨ 盐矄 � ᨨ ҈␔>␔>␔
\
R҈
ᔴ矄 �㡰
1W 1W
㡰 \
2W
2W
1W
V V \ R㦨 P R V V R = = R 錐 R҈㦨 ҈
꾀矄犨├ ᨨ犨P㦨
R
V҈
V V㦨 VR R ҈졔R
V҈㦨 VR ҈ R ҈졔R
�矄�矄ᆠ矄ᨨ ᆠ ␔ 矄ᨨ � �R҈㦨҈W ᔴ矄 ��㡰
R
1W
�
1W
�㡰 W㦨 V 㦨 R3 R
2W 1W
1W
㦨�
҈
҈ ҈ ҈ = + + Þ R = = ҈ � ҈ R
W҈㦨
V҈
=
R҈
ᔴ矄 ���㡰
1W
W =
1W
2W
2W
ᨨ
1W
1W
㦨
盐矄R㦨W 和� ᆠRR҈R\P>P҈> P 꾀矄犨├ ᨨ犨P㦨IR ᔼᆠᨨ 矄 \P µ R
I
R R R ᨨ犨 �矄矄��ᨨ P҈ 㦨 I R㡰 㦨 I R R
ᨨ
�㡰 V
R ҈䰘
V R
´
R
�㡰 �㡰 R �㡰 R R҈ R R҈ + R 㡰 R҈ + R 㡰 �㡰
´
�㡰V V é ҈ ҈ù êë҈ + úû V ´ ҈
҈ ´ ҈ ´ ´ ҈ ´ ҈ ´ ´ ҈ ´ ҈ ´ ҈ ´ ҈ ´ ҈
㦨�҈�
ᨨW҈WW ᆠ㡰 I҈맔�㦨I 졔㡰
I I
V V R = R R҈ + R 㡰
҈ù é ҈ êë҈ + úû V WWW҈
R
I
R҈ + R 㡰
W㦨
WWW҈
㦨
I
V R҈
WWW҈㦨
1W
㡰
2W
1W
҈�
R҈
1W
1W
1W
æ ö R÷ 㦨҈� IR ø
\ᔼᔼᔼᔼᔼᔼᔼ
1W 1W
1W
҈ R ҈
1W
R⏌
ᨨ犨 �矄矄��ᨨ P㦨 I çè
꾀 V�矄ᆠᨨ 矄 � ᆠ矄矄 \ P µ
1W
I
R
R
V ␔ ├ ᨨ犨ᨨ犨 P = R
҈ ᆠ㡰
R I
҈ ҈ > R R
>
R/3
R ᨨ犨 �矄矄��ᨨ P㦨 I ( R ⏌ ) 㦨I R
҈ \ ᨨᆠᨨ 矄 �ᆠ盐鼀盐 P µ R
҈
I
R
I
V R
W
W
I
I
R
I҈
�W
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EBD_8341 PS-212
Physics
I҈ \I҈㦨IÞ I =
� � �W 矄�矄 ᆠ矄ᆠᨨ I҈ ´ � = � � W 矄�矄 ᆠ矄ᆠᨨ I҈ ´
Þ
҈
�矄犨ᨨ盐 ᨨ矄矄�ᨨ 泐犨 R㦨W�矄� 矄�矄犨� 盐矄�矄 ᆠ R盐㦨W㡰ᆠ盐矄� ᆠ矄ᨨ 矄�矄 ᆠ矄 犨�矄 ⼄盐 ᨨ ҈ᆠ ҈
҈ ҈ = � Þ H� W = = ᆠ⏌矄 H� W �
犨 = r
\ � � W矄�矄ᨨ㦨ᆠᨨ�矄⏌矄 b㡰 ᨨ 盐ᆠ 矄ᆠᨨ
H=
V R = R ö
ò
é rl ù 㦨 ´ = ú ë p û
ᨨ矄�矄 ᆠ㦨 ҈졔 㦨 ê
æ ö rl \ H = V ç pr ÷ l = è rl ø A pr
\ =
æ pV r r H =ç ᨨ H µ ÷ l l r è ø
ᨨ
㡰
´ ⏌
ᨨ矄�矄 ᆠ 㦨
ᔴ矄ᆠᨨ
ᆠ � 矄�矄 R
R
\ =
r
E
r
æ E ö J҈ = I R = ç R è r + R ÷ø R
R
矄�矄 ᆠᨨ ᆠᨨ�R 㦨
\
E
r
ᔴ盐 � ᆠ�ᆠ盐�㦨 I 㦨
r
A
æ E ö J = I R = ç ´R r÷ çR+ ÷ è ø E 㡰
r + R 㡰 Þ
R = �
r + R 㡰
=
rö æ çè R + ÷ø
R
҈
\
R盐㦨
㡰 ᨨ鼀ᨨ 矄�矄 ᆠ҈├W�矄ᨨ 矄�矄 ᆠ ᨨ W和ᨨ �⼄盐� 矄�矄 ᆠ�矄
Q = ҈W桌
V ҈ ´ ҈ = = W P �
҈ ҈ ҈ ҈ ҈ = + = + R Ἐ ҈ k W W ҈
R
盐
V ᨨ犨P㦨 R
R
V Ἐ = I ´ R Þ �㦨�맔R \R㦨�W
Þ R졔�㡰㦨҈� 졔R㡰\R㦨W W W㡰␔ ├ ᨨ犨
Topic-5 : Wheatstone Bridge and Defferent Measuring Instruments
Þ R + �桌 = � + R桌
V 䰘� = = 䰘� A R ҈
҈ ᨨ鼀 졠 鼀Ἐ矄犨 矄�矄 ᆠ ᆠᨨ矄矄矄�矄ᨨ R
r R + 㡰 �
B ҈W
� 㡰
E
V ҈ ´ ҈ = 㦨҈W P ҈
�矄 ᆠ盐 �� 矄 � ᨨ 犨ᨨ 矄 I҈ ᆠ盐 矄矄矄 ᨨ盐� \V'㦨I҈ Þ�㦨I҈맔҈\ I҈㦨�A
盐矄ᆠ盐 ᨨ盐� 矄�矄 ᆠἘRἘI㦨䰘��㦨�꾀
I r
\҈㦨�
���㡰
\ V'㦨 ´ Ἐ = ҈ ´ � = � = � 和� ᆠ ᨨ� ᨨ ᆠᨨ矄矄 �矄 �V ᆠ ᨨ� ᨨᆠᨨ矄矄҈W矄�矄ᨨ�矄 ҈�㡰㦨䰘�V
ᔴ矄ᆠᨨ
ᆠ �
E
´ ⏌
ᨨ� ᆠᨨ矄矄 ᆠᨨ� ᨨ � � ᨨ矄 犨� ᨨ犨 P Ἐ㦨�␔
E r
E
҈ ⏌ = ҈ = = ҈ + ҈
ᨨ鼀⼄ �㡰 ���㡰㦨�鼀�
I
E
��㡰
ᨨ鼀⼄ �㡰 ��㡰㦨鼀� ␔ 矄�矄 ᆠᆠᨨ
ᆠ �
盐矄 ᨨ H㡰�矄 ᨨ盐 � ᨨ � l㡰 �盐矄 r㡰 ᨨ盐 ҈黨
l rl = 꾀 p
␔ 矄�矄 ᆠ矄ᆠᨨ
ᆠ � 矄�矄
æ ö = ´ �㡰 ÷ è ø
b 㡰 ᨨ 盐ᆠ = ç
Þ
҈ ҈ + � ҈ = = Þ RἘ = W R Ἐ ҈ ҈
矄� �졠 鼀Ἐ矄犨ᆠ盐 � ᆠ�ᆠ盐�
I=
ᨨ� = 矄�矄 ᆠ ҈ +
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PS-213
Current Electricity ᨨ � � ᆠ 鼀矄盐 泐ᨨ鼀
҈� ҈ ÞV = 㦨҈黨�ᨨ V = IR Ἐ = ´ ҈ 䰘䰘 +
b㡰 㝐盐�鼀 矄 ᨨ犨矄 ᆠ�ᨨ � ᨨ ᆠ矄矄 � ᨨ 矄� � ᨨ矄� ᆠ ᨨ矄 ᆠᨨ鼀ᨨ �矄ᆠᆠ�ᨨ
㡰 ᨨ �� � x =
黨
҈
b㡰
ᨨ ᆠ 鼀� �
Ӹ ÞӸ㦨黨�㦨맔 ҈黨�㡰 = 黨� ҈ - 黨�㡰 ҈� ´ 黨� ᨨ = = ҈ W �
ᨨ � ᨨ �
҈ = ᨨ⏌ᆠ鼀 ҈ - 黨
矄� � ᨨ鼀盐 � 矄�矄 ᆠ
æ l -l ö � - ö r = ç ҈ ÷ s = æç ÷ ´ � = ҈�W l è ø è ø
ᨨ � ᨨ㦨҈ ᆠ� � � ᨨ鼀㦨҈黨
\x =
b㡰
҈҈
␔ Ӹ � ᆠ � l҈ ҈ l҈㡰 犨� 矄ᨨ � ᆠ �矄ᨨl㦨�ᆠ鼀 㡰 �矄 ᨨ ᆠ � � 盐ᨨ� � ᆠ � ᨨ鼀 ᆠ � � ᆠ 犨 矄ᨨ � �
㡰
矄�矄 ᆠᨨ 犨 ᨨ犨� ᨨ � � � ᨨ ᨨ �ᨨ鼀 犨�
졠 ᆠ� �ᆠᨨ ��ᨨ
dV I ´ R é ´ R ù = = ê ú mv ⏌ m dl l ë l AB û æ dV ö ´ R =ç lAP = ´ ҈mV ÷ d l ҈ è AB ø
꾀 ᆠ �
Ἐ AB Ἐ = V AP
R҈ R = R R
ÞVAP㦨� mV 꾀矄ᨨVAP㦨�V ᨨ ᆠᨨ� P㡰
\R = �
V AP � ´ ҈
=
� = ҈W � ´ ҈
矄�矄 ᆠ ᨨ ᨨ �ᨨ鼀 犨� 㦨҈맔㦨W ᔴ盐 � 犨� ᆠ㡰
�=
҈ V = = A RT + � + �䰘 + ҈
҈ ᨨ犨V㦨�R 꾀㦨 맔 ҈맔�㡰㦨� 䰘
㡰 ᆠ㡰 � 꾀犨 �ᆠ � ᨨ鼀矄 ᨨ犨矄 盐 ᆠ�ᨨ ᆠ盐
�=
e æ ö e = ᨨ � ç ҈ ÷ = ҈ + ҈ è ø
e é e é ù e ù e ҈ Þ ·҈ ú = Þ ê ·҈ ú = ᨨ 㦨 ҈ êë ҈ û ë û
졠 ᆠ� �ᆠᨨ ��ᨨ
㡰 � ᆠᨨᨨ盐 ᆠᨨ ᨨ 矄�矄 ᆠ ҈㦨졠 � ᨨ犨
\ ҈㦨맔҈㦨 盐矄� � ᆠ 犨 矄ᨨ � � � ᆠ�
҈ = Þ =
\ 㦨҈�ᆠᨨᨨ盐ᆠᨨ ᨨ ᨨ犨 盐 ᨨ犨
҈
R҈ R = R R
ᆠ㡰 꾀矄 犨ᨨ ᆠ矄 ᨨᨨ矄 ᆠ ᨨ ᆠ ᆠ� 鼀 � ᆠᨨ矄 ᆠ�ᆠ盐� �矄 ҈� ҈ 㦨 � 矄�矄 ᆠ �矄 ҈ 졔 㦨 ҈ W ᆠ盐矄 � ᆠᨨ矄 ᆠ�ᆠ盐� � 矄�矄 ᆠ ᨨ 犨ᨨ ᆠ矄 � 矄�矄 ᔴ盐 � ᆠ�ᆠ盐� I 㦨
ᆠ� 鼀 � A = ᨨ 矄�矄 ᆠ ҈
ᨨ � � ᆠ ᆠᨨ矄矄 ᨨ鼀 犨� 矄鼀 矄 鼀� ᨨ� ᨨ � ᆠ 和� ᆠ ᆠ盐 �矄 犨 ᨨ鼀 ᆠ ᨨ ҈� \ V҈ 㦨 E҈ Ir҈ 㦨 ҈� ҈ ᆠ㡰
� ´ = ҈҈ ҈
ᔼ ᆠ矄ᨨ ᆠ 鼀� �
R X = � X㦨黨Wl㦨ᆠ鼀 l ҈ - l 黨 ´ Xl = = W \ R = ҈ - l DR Dl D (҈ - l ) = + \ ҈ - l R l DR ҈ ҈ = + \ DR = �W Þ
ᨨ R = ( ± � ) W
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EBD_8341 PS-214
҈
Physics
b㡰
҈W
R҈ A
x R
G C
R R
B
ᔼ �盐矄ᨨ 犨� �矄 ᨨ盐 矄�矄 ᆠ ᨨ AC 犨�
R ᆠ � 矄ᨨ 矄�矄 ᆠᨨ CB 犨�ᆠ �盐矄� ᆠ ҈ ᨨ矄
꾀
�ᆠ鼀
ᆠ鼀
P R = 꾀泐� � ᆠ 犨 矄ᨨ � � ᆠᨨ ��ᨨ Q S X ҈ = l҈ l l҈ 㦨�졔 ᆠᨨᆠ�ᨨ 㦨�졔҈㦨�ᆠ鼀
R
ᨨᆠ �矄ᨨ ҈
l 㦨졔 ᆠᨨᆠ�ᨨ 㦨졔㦨�ᆠ鼀 ᨨ ҈�
X ҈ � = \ X = = ҈W � � �
P R = 꾀泐� � ᆠ 犨 矄ᨨ � �ᆠᨨ ��ᨨ Q S R㦨W
l
҈㜠l
G
҈黨
S
R
꾀矄␔ 矄ᨨ � � ᆠ ᆠᨨ ��ᨨ
҈ R Ἐ Þ�졔�R㦨҈R ҈� ҈ R Ἐ Þ �= = + ҈ R Ἐ ҈ + R Ἐ ҈ � \ RἘ = = ҈W �
␔ 矄�矄 ᆠ矄 � ᆠ � ᨨᆠ├泐 矄 � 矄 ᆠ鼀
X = ÞX l X㦨 l 졔 l + - l æ ö ᨨ X㦨 l X졔㡰졔㦨 çè ÷ X + 㡰 + X + ø ᨨ X㦨 㦨W
\
\
E ҈ = k �㡰 r + ҈
G
r + ҈ � = Þ�r졔�㦨� ҈ � N Þ r = W = W ÞN㦨҈ � ҈ Þ
S
矄� �IgG㦨 IIg㡰S
æ G ö = ҈ ´ ҈- æ ҈ + ҈ö +҈ çè ÷ è S ÷ø ҈ ø
I㦨ᔼ� ç ҈䰘
��㡰
�� � � �㡰泐 ��㡰犨� I㜠Ig
ᨨ 㡰
҈㡰 ␔ ├ ᨨ犨
E µ l 犨 l�矄 ᆠ� � � \쟼㦨k �㡰 �㡰 ␔ ᆠ� � � ᆠ �矄泐ᆠ鼀
Ig 㦨҈맔҈A錐 G㦨҈W; S㦨҈W I
P S = Q R
矄�矄 ᆠ �矄 ᆠᨨ
ᆠ � 犨� 矄�矄 ᆠ 和 ᨨ ҈W
ᨨ ҈RR l 㦨 l X ᨨ l 㦨 l Xᨨ l 㦨 X +
Ig
B
҈㡰
B
㡰
G
㜠l
l졔
R X = l ҈ - l
҈
Q
P
㦨W
X
ᨨ� C ᨨ矄 ᨨᆠ � 㡰 和� ᆠ ᨨ � �ᨨᆠᨨ矄� � 矄犨�ᆠ �s ᨨ矄 ᨨ ᨨ矄 ᨨᆠ � � �ᨨ � ᨨ鼀�鼀 矄 � ᨨ ᆠ矄矄 �矄 ᨨᆠ盐 矄矄� � ᨨ盐� S ᔼ �矄 ᆠ矄 ᨨ ᆠ 犨 矄ᨨ � � 盐矄 ᨨ �A �矄⼄盐ᨨᨨ �B
\ IP㦨IQ IR㦨IG
␔ � ��矄 ᆠ x>W
㡰
AC 矄 ᨨ盐 矄ᨨ ᨨᆠ � 盐ᨨ� ᨨ A CB
I㦨҈맔҈맔҈҈㦨҈҈鼀꾀 ᔼ ᆠ矄ᨨ 鼀� � 盐ᨨ�
R҈ R AC AC = = = R R CB ҈ CB
҈
䰘㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ 矄�矄 ᆠ 矄 鼀盐 ᆠᨨ �ᆠ� a 㦨 ᔴ҈ 꾀 鼀盐�矄� ᆠ矄 泐҈ ᔴ �D 㦨҈ ᔴ Ἐ㦨 ҈졔aD 㡰㦨 ҈졔aD 㡰 \ Ἐ㦨҈W
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PS-215
Current Electricity
R҈㦨W
ᔼ
ᔼ
R㦨҈W
QV = IR 㡰
黨㦨I 졔㡰ÞI㦨҈A
m
m
和
�
� � = ᔼ = = ҈ + 䰘 +
W
W
� � - � ´ 㦨҈黨҈䰘㦨䰘 䰘
ᆠᨨ� ᨨ 犨 和 㦨䰘ᨨ 㡰 ␔ 鼀盐 ᨨ 鼀 � ᆠ矄矄錐 �矄 矄�矄 ᆠ � ᆠ矄矄Rt㦨R ҈졔µt㡰 ├ ᨨ犨 矄�矄 ᆠ㡰 和 矄�矄 ᆠ㡰 X S
黨 ᨨ � � ᆠᆠᨨ矄矄W矄�矄 ᆠ 㦨V ᨨ � � ᆠᆠᨨ矄矄W矄�矄 ᆠ 㦨 \ ᆠᨨ矄矄ҘFᆠᆠ�ᨨ㦨V ᆠᨨ矄矄ҘFᆠᆠ�ᨨ㦨V \ ᔴ �ᨨ ҘFᆠᆠ�ᨨQ҈㦨맔㦨黨ҘC ᔴ �ᨨ ҘFᆠᆠ�ᨨQ㦨맔㦨ҘC \ ᔴ � ᨨ犨矄 ᨨ鼀yᨨx㦨黨㦨䰘ҘC b㡰 꾀矄ᨨ 犨�ᨨ 矄鼀鼀�\ᔼ꾀㦨ᔼᔴ \
和 㦨҈ᔼ҈�ᔼ 㦨 ҈ ´
졔턼 턼
R㦨�W
RἘ = ҈W ᔼ = ҈
졔턼҈ 턼҈
r
꾀
ᔴ
ᔼ
N
l⏌
l⏌
r l
҈
l
㡰
J G
\V AB =
ᨨ鼀� � 盐ᨨ� ᨨ ᆠ ᆠᨨ ��ᨨ
X S X l = = ᨨ l ҈ - l S ҈ l ᨨ 盐ᨨ� ἘἘ矄鼀ᨨ� ᨨ 盐 ᆠ �
l҈ ᨨ鼀�
҈ - l �
\
� ᆠ矄� � 矄 矄�矄 ᆠἘsἘ矄 ᨨ盐 矄ᨨ� ᆠ矄 ᨨ矄��ᨨ 㜠ᨨ 盐ᨨ� ��矄鼀ᨨ矄ᆠᆠ盐 � �盐ᨨ� �矄鼀ᨨ矄 ᆠ ᨨ 矄�矄 ᆠ犨�犨 矄ᨨ � � �矄鼀ᨨ矄ᆠᆠ盐犨 ᨨ盐矄�矄 ᆠ矄ᨨ ᨨ盐鼀矄 ᨨ 矄鼀ᨨ ᨨ 鼀� �盐
ᆠ㡰
P X = Q R
㡰 ᨨ ᨨ ᨨ矄�� �� 鼀� 矄 ᨨ � ᨨ鼀␔ �矄 ᨨ ᆠ盐 ᨨ� ᨨ � ᨨ鼀�矄 ᨨ 㡰 C X A D Ý ҈W ᆠ㡰Q � ��矄 ᆠ
R AJ r ҈ W = = \ X 㦨W RJB r X
犨 r�矄 矄�矄 ᆠ盐 � �
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈
㡰
X 矄 ᨨ盐 矄ᨨ鼀� 盐 ᆠ � ᆠ� X�矄 S
G
VBC
ᔼ ᆠ矄ᨨ ᆠ 犨 矄ᨨ � �ᆠᨨ ��ᨨ
VAB I AB R AB R = = AB = VBC I BC RBC RBC
ᆠ㡰 ␔ 矄犨�ᆠ 和�矄ᨨ ᨨᆠ �Q҈졔Q㦨CV҈졔 CV㦨맔졔 맔㡰㦨␔ 矄犨�ᆠ �矄ᆠᨨ矄 矄 泐矄�矄ᆠ ᆠ盐 I ᆠᨨ鼀� � ᨨ鼀 泐�矄
㡰
PBC I RBC = = PAB I RAB
r
l p ´ r 桌 ҈ = l r pr 桌
l p ´ r 桌 ҈ = l r pr 桌
PAB㦨PBC
I J AB p ´ r ҈ = = 錐 \JBC㦨JAB I J BC p´r éV AB ù E AB êë l ⏌ úû ҈ = = 錐 \쟼ᔴ㦨EAB E BC é VBC ù ê l⏌ ú ë û
b㡰 ᆠ �� � �泐㦨 矄盐� � 㦨DU = Q = I2Rt 㦨҈맔҈맔҈맔�맔㦨㦨├ 㡰 꾀 泐 � 矄 ᨨ �鼀 t 盐� � ᆠ �� � ᨨᆠ矄矄 矄� ᆠ盐 � ᆠ�ᆠ盐�
V -t ⏌ RC e \ ᨨ � � ᆠᆠᨨ矄矄矄�矄ᨨ R é V -t ⏌ RC ù VR = ê e ú´R ëR û
I㦨
= Ve - t ⏌ RC
\
Q
ᨨ � � ᆠᆠᨨ矄矄 C Vc㦨VVet/RC 㦨V ҈et/RC㡰
Vc = VR
�㡰 ��㡰
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EBD_8341 PS-216
\
Physics
(
)
\ �鼀ᆠᨨ 矄 t 㦨RC㦨矄ᆠ ᨨ �ᆠᨨ矄矄 m ᆠᆠ�ᨨ -t ù 泐�鼀� é
V ҈ - e -t ⏌ RC = V e - t ⏌ RC
Þ ҈et/RC 㦨et/RC Þ ҈㦨et/RC
V = V êë҈ - e t úû
├� �ᨨ�ᨨ ᨨ 矄� 矄
æ t ö ᨨ�e҈㦨ᨨ�e졔 ç è RC ÷ø t Þ 㦨맔ᨨ�҈ RC Þ t㦨맔ᨨ�҈桌맔�맔҈맔맔҈ᨨt㦨҈矄 �
b㡰
é
Þ = ҈ ëê҈ - e Þt㦨矄 Þ e
҈
ᔴᨨ鼀� �犨� y㦨mx 졔C
҈ æE ö ᔼ ᆠ㦨ln ç ÷ 矄ᨨ㦨 è Rø RC
䰘
和ᨨ犨 x �矄 ᆠ � ᨨ x 矄ᨨ � ᆠ矄矄 ᆠ盐 ᆠᨨ鼀矄矄矄 ᆠ 犨� �矄Q b㡰 ᨨ矄ᨨ �ᆠ ᨨ �矄矄ᨨ犨ᨨ├矄 ᨨ � ᆠ� ᨨ ᆠ 犨 矄ᨨ � � 盐矄 ᨨ鼀矄盐 盐 ├ ᨨ犨 矄�矄 ᆠ ᨨ 矄�矄 ᆠ犨� ᨨ矄�矄 ᆠᨨ 犨 ᨨ � ᨨ ᨨ ᔼ矄 ᨨ盐 ᆠᨨ
ᆠ 犨 A D ᆠ㡰 꾀矄 ᆠ盐 ᔼ�矄� ᨨ R � ᨨᨨ犨矄 矄�矄 ᆠ R ҈ R R R R 鼀盐矄 ᨨ鼀 ᆠ ␔ 矄ᨨ � �\R҈R 㦨RR b㡰 R 矄�矄 ᆠᨨ 犨� 쟼 �泐矄 � t矄ᆠᨨ H㦨 꾀矄ᨨH㦨mcD 㦨
҈ �㡰
æ I - Ig ç Ig è
ᨨ鼀 ç
҈
⼄ 㦨
ᔴ ҈ -
҈ 錐V µ 錐V µ r r r
-
ᔴ
ᔴ
�҈ =
㡰
- ᔴ - ᔴ
㡰
� =
Ҙ 㦨 ᔴ
㦨�KW
ᔴ㦨Ҙ
- ᔴ҈ é ᔴ ù 㦨 ê҈ - úû 㦨 ë ᔴ
�ᨨ
҈ V ᨨr P µ (Q V = ᆠᨨ 矄 ) R R
꾀矄 ᨨ�ᨨ �矄 ᨨ 㜠盐 � ᨨ鼀 ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ 犨� � � ᆠᨨ矄矄㜠矄ᆠ�ᨨ ᨨ 鼀盐 �矄�盐�ᨨ 犨� ᨨ 㜠盐 � ᨨ鼀 �鼀 犨�├ ᨨ� 犨 鼀盐�矄鼀 �鼀盐鼀
ö V -R ÷S = ÷ I g ø
㡰 ⼄盐� ᆠ�ᆠ盐�ᨨ �� ᆠ�ᆠ盐��矄矄矄 ᨨ犨 ᨨ犨 R㦨҈㝐WC㦨m
-
盐 꾀 ᨨ � �鼀 ᨨᆠ盐 ᨨ犨矄 矄� ᆠ ᨨ ᆠᆠ�ᨨ ᨨ鼀 ᨨ ᨨ ᨨ犨 ᆠ盐 ᨨ ᨨ犨 ᆠ 㡰 ␔� 盐矄ᨨ �鼀 ᨨ�ᨨ � ᨨ �鼀 犨�ᆠᨨ鼀 �
泐 ᆠ矄� � ᨨ ᆠᨨ矄矄㜠矄ᆠ�ᨨ
� ᆠ矄� � 矄�矄 ᆠ ᨨ �鼀 犨�ᆠᨨ 矄盐鼀 矄矄ᨨ犨ᨨ犨 矄 ᨨ � 꾀矄 ᨨ犨 P =
҈
黨 ҈� - n =� ´ = - 黨黨 \ n = 黨 黨 ҈҈
ᔴ ҈ -
éQ � = � - t ù ᨨ ëê ûú ҈ �꾀 ҈矄ᆠ盐 � 鼀鼀ᆠᨨ鼀矄 ᨨ � �� e
N 㦨҈맔\N㦨
ᔼ 矄�泐ᨨ �� Iµ
⼄҈ 㦨
�꾀 �鼀 㦨 矄ᨨ鼀犨� �矄泐 � �졠
mcDT 黨V ´ t ⏌ R ҈ 黨´ = = ᨨ mc DT N V t ⏌ R N
㡰
bᆠ 㡰 꾀 㦨 � 矄 矄ᨨᨨ 矄 ├泐 �矄 矄矄 ᔴᆠ�ᨨ矄 ᆠ 矄 矄 ᨨ ᆠ�ᆠ盐� ᨨ鼀 �矄泐 � �� 꾀㦨µ� ᨨ ��鼀ᨨ ├泐矄矄ᔴᆠ�ᨨᆠ矄矄 ᨨ ᆠ�ᆠ盐� ᨨ ᆠ盐 ᨨ犨矄 ᨨ盐� ᨨ鼀 Q 泐 �� 矄�矄 ᆠ ᨨ ᨨ鼀㡰 矄ᨨ � �矄泐졔�
꾀t㦨҈矄ᆠ�㦨
�� � �⼄ �㡰泐 ��㡰
Þ t = ( l � l ) = �
\ D 㦨�맔 졔
�㡰
��㡰
҈
�
�
꾀矄ᨨHἘ㦨m'cDT㦨 m㡰CDT
黨
t = l l �
�├W
NV 㡰 ´ t (Q R µ l ) R
N V ´t 鼀ᆠD 㦨 R
= =
�
ᨨ
t
V 㡰 æ V ö t÷ ´ t çQ H = ç R R ÷ø è
黨V ´ t Q 盐Q㦨mcDT㡰 R
t ù û ú
�҈
RἘ 矄�矄 ᆠᨨ 犨�ᨨ ᨨ盐 � �l \ ᨨ犨 �泐矄 � 㜠矄ᆠᨨ 矄' 㦨
-
├� �ᨨ�ᨨ ᨨ 矄� 矄
ᔴ盐 � RC ᆠ�ᆠ盐� I㦨Iet/RC
æ -t ö t + lnI ᨨ lnI㦨lnI ᨨlnI㦨 ç è RC ÷ø RC æE ö æ -t ö + ln ç ÷ lnI 㦨 ç è RC ÷ø è Rø
m
҈㝐W
æ è
␔ �鼀 鼀盐�矄 �� ç l n µ ҈
҈ö ÷ �鼀�矄�� Tø
ᨨ ᨨ犨犨 � ᨨ �� ᨨ ⼄盐 ᆠ�矄 b 㡰 ᆠ ᆠ盐 矄矄� �㜠 ᨨ盐� � 矄盐ᨨ 矄鼀�ᆠᨨ 盐ᆠᨨ졠⧤�矄 ᨨ犨 ᨨ � � ᆠ�矄 꾀⧤ � 矄盐ᨨ ᨨ矄 ᨨ鼀� 泐ᆠ盐 ᨨ ᨨ犨 ᨨ盐� � 꾀⧤矄鼀�ᆠᨨ 盐ᆠᨨ 矄矄 � 矄盐ᨨ
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PS-217
Current Electricity
C
҈�
ᔼ 矄�矄� �K矄鼀 � �ᆠ盐 ᨨ犨矄 ᨨ盐� p㜠n 盐 ᆠ�ᨨ �ᨨ ᔼ ᆠ矄ᨨ 鼀 ᆠ盐 ᨨ犨� �犨�泐㜠泐 �� ᆠ盐矄 鼀ᆠᨨ鼀矄矄盐ᆠᨨ 盐ᆠᨨ⧤ b 㡰ᔼ ᔴᆠ�ᆠ盐� �矄ᆠ �� �ᆠ盐 I 㦨 Iet/RC C҈
ᔼ ��ᆠ盐 I㦨
҈黨
C
V R
꾀矄V R ᨨᨨ ᆠᆠ�ᨨ矄 矄鼀 矄ᨨ� �� �矄ᆠ �� � ᆠ盐 犨�矄鼀盐 ᨨ 㜠 ᨨ 꾀矄ᨨq 㦨 qet/RC
҈䰘
\�҈㦨
M
e 졔 t/RC 㦨Þ t㦨RCᨨ�e\tµC
\
t҈ C҈ ҈ = 㦨 = � t C
V 㦨�҈R 졔
q q 㦨qet/RC ᨨ t҈ 㦨�t
ᆠᆠᆠ�ᨨC҈ᨨ矄矄矄�奠ᨨ �矄� ��ᆠ �矄ᨨᨨ
Cᨨ矄矄矄�奠ᨨ �矄� ��ᆠ � 꾀®s �鼀�ᆠ矄��矄矄 ᨨ 鼀 矄�ᨨ ᨨ 矄ᨨ� 矄�l㦨l ҈졔µt㡰 ®q 和鼀 �� �矄矄 ᨨ �泐ᆠᨨ 矄�ᨨ ᔴ®p, q ᔼ ᆠ 矄ᆠ 鼀�矄矄 ᨨ �泐ᆠᨨ 矄�ᨨ �ᆠ�ᆠ �泐 � ᨨ�� �泐 ��ᨨ ᨨ鼀 ᨨᨨ 泐 ®q, r 쟼ᆠ�ᆠ 盐矄 �矄 矄 ᨨ 鼀� � ᨨ 盐矄 鼀� 犨 �ᆠ �
盐 矄ᨨ � � ᆠᨨ ᆠ盐 � Q㦨Q҈eat桌 Q㦨鼀 �鼀盐鼀ᆠ �
Q + �R C
T
B
ᨨ
a㦨
��㡰
쟼�鼀� � �� ᨨ鼀⼄盐�ᨨ �㡰 ��㡰犨� V㦨�҈R졔 ᨨ
�҈㦨
Q ᨨ C
�҈R㦨V
҈ æ Q ö çV ÷ ᨨ R è C ø
Q C
dQ ҈ æ Q ö = çè V ÷ dt R Cø Q t dt dQ ò æ Q ö = ò R çè V ÷ C ø
dQ dt ᨨ = Q ö R æ çè V ÷ Cø CV ᨨQ㦨 ҈et⏌RC㡰 ù dQ d é CV ҈ - e-t ⏌RC ú = ê 㡰 �҈㦨 dt ë dt û V -t ⏌ RC CV e = ´ ´ e -t ⏌ RC = R RC ᨨ
(
҈ ҈ = t c C q
ᨨ犨 鼀 �鼀盐鼀ᆠ � Q㦨CV桌犨 V㦨鼀 ᨨ � � ᆠᆠᨨ矄矄C
é
ù V ´ R ú ë R҈ + R û
)
ᨨ鼀 ⼄盐�ᨨ �㡰
= C ê
V+
V -t ⏌RC e \ ᔴ盐 ᨨ盐� AB R
ç
V + �҈R �㦨 = R
\ a =
V -t ⏌ RC e V - e -t ⏌ RC �㦨��҈㦨 R R V V -t ⏌ RC V e Þ �㦨 ᨨ �㦨 矄t® ¥ R R R
æ
ö V ÷ �矄 矄 泐矄 ᆠ盐 ᨨ盐� R è R҈ + R ø é R҈R ù 꾀 t c = C Req = C ê ú ë R҈ + R û (Q R҈ R � ᆠᨨ鼀� �ᨨ )
҈
R
dQ dt
㡰 꾀泐� �⧤ � ᨨᨨ MNABM V㦨 ��҈㡰R 졔�R ᨨ V 㦨�R�҈R �㡰 和�鼀�泐 泐� � ⧤ � ᨨᨨ MNSTM
␔ q㦨
҈
ᨨ ᨨ矄 ᨨ �├ ᨨ � B A �� 泐 矄 ᨨ盐 ᆠᨨ
ᆠ R ᆠᨨ矄矄A B졠盐盐ᆠ ├ ᆠᨨ矄矄 鼀� 矄A C ᨨB C V 꾀ᆠᆠᨨ � �ᨨ ⼄盐矄�ᨨ ᆠᆠ�ᨨ �矄� ��泐 盐 ᆠ � 矄犨�ᆠ s�矄ᆠᨨ矄 �鼀矄t㦨 泐�鼀tᆠ �ᨨ ᆠᆠ�ᨨ R A �҈ S C Qᆠ盐 矄矄矄 ᨨ犨 N � �㜠�҈㦨� � ��和� ᆠᆠ �Q 犨�� ᆠ矄 Q+ 犨� �鼀 ἘtἘ R C V
R + R ҈ = ҈ tc CR҈R
҈
ᨨ� 泐ᨨ 鼀Ἐ矄犨 ᆠ�ᆠ盐� ��鼀�矄矄矄 ᨨ犨 ᨨ犨 꾀 鼀鼀�矄ᨨ犨矄�矄 ᆠ� ᨨ鼀犨 �ᆠ �矄ᆠᨨ
ᆠ ᨨ 犨� 矄�矄 ᆠ ᨨ鼀�矄 �� 矄�矄 ᆠ � ᨨ鼀犨 �ᆠ �矄ᆠᨨ
ᆠ � 矄�矄犨� 矄�矄 ᆠ 꾀鼀鼀
ᨨ鼀
҈ W
G
G҈
�ᔴᨨ�
V+
҈ W
҈W
␔ᆠ 犨 ᆠ�ᆠ盐�矄 ᨨᨨ犨矄 �矄�矄 ᆠ 犨 矄ᨨ � � E
쟼⼄盐� 矄�矄 ᆠ 犨
G D
´ = W RGD 㦨 + RGE REB = RGD RDB
G
æ ᆠᨨ ��ᨨ ᨨ ᆠ ö ç ÷ è 犨 矄ᨨ � � ø
҈W
W I⏌
I⏌
҈W
M W V W
D
B
҈W
҈
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EBD_8341 PS-218
Physics
E⼄盐� 矄�矄 ᆠᆠᨨ矄矄G B RGB 㦨
҈ ҈ ´ V = ҈� W \ ᔴ盐 I㦨 = = + RGB ҈� ҈�
ᔴ盐
ᔼ 盐 �� 矄 � ᨨ 犨ᨨ ⼄盐 矄 㝐
ᆠ ᨨ � � ᆠ ᆠᨨ矄矄 ᨨ鼀 㦨ᔼ = �m 㦨C E
�
= ҈� 꾀 ҈ + Q P d �矄 矄鼀 \ I҈ = I Þ I = I҈ + I = I҈ + I҈ = ҈� \ I҈ = ´ ҈� = 黨 A �
ᆠ盐 ᨨ盐� 泐㦨
҈ I = ´ = ´ 㦨䰘 ҈�
� ᆠᨨ矄矄 W矄�矄 ᆠ㦨V \Pdᆠᨨ矄矄 W矄�矄 ᆠ㦨V㦨V 꾀矄 ᨨ鼀 �矄 � 犨� W 矄�矄 ᆠ � ᆠᨨ鼀� 矄�矄 ᆠ
R ҈=W
RἘ =
E҈=
G �
R 㦨W 㦨E
D E 㦨
�
A
�҈ �+�҈
R' 㦨 W矄 ᨨ � � ᆠᨨ V �矄 ⼄盐泐 矄 犨 W W矄�矄 ᆠ
�҈ B F R㦨W �
R㦨W
�
\ =
C
�㦨 -
R ἘἘ =
҈ A �
I=
㦨V㦨IR'' =
ᆠᆠ盐 � R㦨�҈졔�㦨҈䰘㦨҈�A 꾀ᆠᨨ矄矄 ᆠᆠ�ᨨ ᨨ � � ᆠ VE맔㦨VG\ VEVG 㦨V
㦨
꾀泐� �⧤�ᆠ ᨨ Ἐ矄 �矄犨盐 ᆠ�ᨨ M Pᆠ盐 �҈㦨 A �㦨 ҈꾀矄ᆠ�泐
鼀 A
҈ CV
҈ 맔�맔҈맔 㡰 㦨 ҈맔҈� ҈A W 4V W M �҈ �W
V V A \ ᨨ鼀 � � = = R W
㦨ᨨ � � ᆠᆠᨨ矄矄 W 矄�矄 ᆠ
æ ҈ö �
�҈㦨҈� ç - ÷ 㦨҈䰘A è ø
쟼 �泐矄ᨨ � ᆠᆠ�ᨨ㦨
R ´ ҈ = = W + R 㡰 + ҈㡰
\ ᨨ矄�矄 ᆠ� ᆠ�盐�㦨졔㦨W \ᔴ盐 � ᆠ�ᆠ盐�
和盐矄�盐� � �矄盐� ⼄ �㡰犨�
\
䰘
q q - + ҈ = Þ q㦨҈�ҘC m F
҈ ҈ CV = 맔맔҈맔맔㦨맔҈
矄�矄 ᆠᨨ W W�
mF
҈mF
mF
N
҈mF
mF
҈mF
҈mF ҈mF
C mF ҈W ҈W P � W O W W ҈A
´ = ҈ W Req㡰AB 㦨 +
´ = �V
ᔼ 矄� 泐 ᆠ盐 ᨨ犨矄 ᨨ鼀 ᨨ矄�� 鼀� ᨨ ��鼀� ᨨ 泐 盐� � ᆠ �� � ᨨ � � ᆠ 犨 犨ᨨ 鼀� 矄 ᨨ 泐 V㦨E 졔Ir 㦨졔�맔҈㦨� V 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 KVL � ᨨᨨ AQBRC
҈mF
W
҈W
҈W W
҈
mF
꾀 矄 泐矄矄 ᨨᆠ盐 ᨨ犨矄 ᨨ盐� ᆠᆠ�ᨨ 㝐ᨨ� � ᨨᨨᨨ �MNO ᨨP \ VM �맔�҈҈맔I҈맔�㦨VP \ VM VP㦨�҈졔�㦨맔졔맔҈㦨V \ V㦨맔졔맔҈㦨V 쟼 �泐矄ᨨ � ᆠᆠ�ᨨ 㦨
R Þ R㦨҈W + R 㡰
� 矄�矄 ᆠ ᨨ ᨨ�鼀 㦨 ҈W␔ ᨨ鼀�矄ᆠᨨ
ᆠ ᆠᨨ矄矄 W矄�矄 ᆠ �ᆠᨨ鼀� 矄�矄 ᆠ
꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 � ᨨᨨ ABFGA �҈졔�㡰㦨 �㡰 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 � ᨨᨨ BCDEFB �맔졔�졔 �졔�҈㡰㦨 ��㡰 盐� � 盐ᨨ �҈졔�㡰㦨� ⼄ ��㡰 ��졔�졔㦨 \
R R㦨矄�矄 ᆠᨨ ᨨ鼀㡰 + R 㡰
Q 졔q 㜠q
҈ mF
B
졔q 㜠q
A
҈ D
\
҈mF
R C
E
ᨨ � � ᆠ 犨 AB V㦨
q ҈� = �V VAB㡰 㦨 C
\ ᨨ � � ᆠ 犨 BC VBC㦨҈�㦨䰘�V
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PS-219
Moving Charges and Magnetism
Moving Charges and Magnetism
18
Topic-1 : Motion of Charged Particle in Magnetic Field ҈
�
r
m㦨鼀矄矄ᨨ ᨨᨨ
q㦨ᆠ �ᨨ ᨨᨨ
B㦨鼀� �ᆠ � � �矄 ᆠ � ᨨ ᨨ盐�ᨨ p㦨T v ᆠᨨ矄q㡰 v㦨 ᨨᆠ�泐ᨨ ᨨᨨ 㡰 \ � ᨨ ��ᨨ l = ҈ ´ v ᆠᨨ矄 q㡰T
ᆠ㡰 E = ˆ ⏌ᆠ鼀 = ´ ҈ ⏌鼀
r V = ´ ҈ �ˆ E E =
y x
⏌ᆠ鼀 = ´ ҈ ⏌鼀 e V $ V = ´ ҈ �
pm qB pmv ´ ҈ ´ ҈䰘 ´ ҈ -䰘 ´ ´ ҈� Þl= = qB ҈ ´ ҈ -҈黨 ´ Þ l = 鼀
E = VB
E ´ ҈ = = � ´ ҈- T V ´ ҈ ᆠ鼀� �ᆠ � B㦨�맔҈ ᨨ �졔z �ᆠ�ᨨ ᆠ㡰 ᔼ 盐 � ᨨ鼀鼀� �ᆠ � �ᆠ鼀ᨨ矄� ᆠ�ᆠ盐 � �盐矄ᨨ ᆠ�ᆠ盐 �矄'r'�ᆠ犨� ᨨ � 矄ᆠ \ B=
鼀 �㡰 ⼄ �矄 矄�� �
⼄쟼㦨⼄ 쟼 쟼㦨 Þ = ��㡰 ᨨ鼀⼄盐�ᨨ �㡰 ��㡰
é mv ù mv = qvB ú êQ qB r ë û ᆠ鼀� �鼀盐鼀盐ᨨ v ᨨ犨 �ᆠ �ᆠ犨� ᨨ � 矄ᆠ qB d v= m 㡰 � q = ҈mC = ҈ ´ ҈- C錐 r V = �ˆ + ˆ + kˆ㡰 鼀⏌矄 r B = ��ˆ + ˆ - kˆ㡰 맔҈ 桌
䰘
r r r F = q V ´ B㡰 = ҈ - ´ ҈ -
= -�ˆ + ˆ - 黨kˆ㡰 ´ ҈-黨
� - r \ F = -�ˆ + ˆ - 黨kˆ㡰
pm qB
pm \ �ᆠ = V ᆠᨨ矄 q㡰 qB p æ ҈䰘 ´҈-䰘 � = ´ ҈ ᆠᨨ矄 ° 㡰 ç çè ҈黨 ´҈-҈黨
´ ( �) ´ � ´҈ ҈
b㡰 꾀矄鼀㦨⼄Þ =
҈ 꾀矄 鼀 = ⧤쟼 Þ 鼀 = 鼀 ⧤쟼
鼀 = ⼄
鼀⧤쟼 ⼄
Þ 鼀 = 鼀 ⧤쟼桌 ᨨ ᨨᨨ ᆠᨨ a㜠�ᆠ 鼀㦨鼀 鼀>>鼀 꾀矄ᨨ㦨⼄ ⼄㦨⼄ \ 꾀矄⧤쟼ᨨ �ᆠ矄�矄矄鼀 鼀 ⼄ \㦨> a
kˆ
�ˆ ˆ
㡰 �ᆠ = v ᆠᨨ矄 q㡰T T =
҈黨
⼄ ҈ ´ ҈ = 쟼 \鼀㦨맔҈├� 鼀=
r=
㡰 꾀矄 �ᆠ�矄 鼀ᨨ� �ᨨ �ᆠ�ᆠ盐
\ =
d
pm qB
Þ l = ҈ ´ v ᆠᨨ矄 ° ´
z
r B 鼀盐矄� 졔z �矄 r r r qE + qV ´ B =
ᆠ㡰 �鼀 �ᨨ ᨨ ᨨ ᨨ盐�ᨨ ᨨ ᨨᨨ T =
㡰 �盐矄ᨨ ᆠ�ᆠ盐 犨� =
鼀 ⼄
鼀⧤쟼 Q = 鼀 = 鼀⧤쟼 㡰 ⼄ Q⧤쟼㦨⼄D Þ =
鼀⼄D 鼀 Þµ ⼄ ⼄ ҈ \ = a
\ = ö ÷÷ 㦨ᆠ鼀 ø
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EBD_8341 PS-220
黨
Physics
㡰 �盐矄ᨨ 㡰�矄�� 泐 =
= =
(
鼀├ 鼀
鼀 ⼄
Q = 鼀 = 鼀├
҈
)
꾀 犨 ├ ᨨ犨 Þ R 㦨
Q ├ = 㡰 ´ 黨҈ ´ ҈ -҈
=
鼀 =
=
黨҈ ´ ҈ -҈ ҈ = ´ ҈ - -҈ ҈
鼀 ⼄
= 㡰
⏌
鼀
ᔴ
⼄ é - � - ù ê ú 鼀 ë û �矄 ᨨ�ᨨ �矄 ᨨ �� � ᨨ ᨨ盐 ᆠ ᨨ�ᆠ矄 b㡰 꾀矄犨├ ᨨ犨 �盐矄ᨨ ᆠ�ᆠ盐 � 鼀� �ᆠ � =
҈҈
=
⧤鼀 ⼄
ᨨ ᆠᨨ = ᨨ ᨨᨨ = ᨨa�ᆠ a =
҈
҈
⧤鼀
�㡰
⧤鼀 ⧤鼀 ⼄a
��㡰 =
⧤鼀
=
\ 㦨a Q 鼀鼀㡰 b㡰 ᨨᆠ �矄 ᨨ �ᆠ�ᨨ
ᨨ ᆠ盐 � 鼀� �ᆠ � r ᆠ = ⼄ ´ 㡰 ᔼ 꾀ᆠᆠᨨ � � ᨨ 鼀� �Ἐ矄 盐
犨 �ᆠ�ᨨ ᨨ 鼀ᨨ�ᨨ ᨨ ᆠ ��矄 ᨨ犨 矄 犨� b㡰 턼 㦨⼄쟼 㦨⼄ \쟼㦨 꾀 盐矄矄Ἐ矄犨 � 쟼ᆠᨨ矄�ᆠ矄 쟼 㦨
s 㦨 Þ s 㦨 e e σ҈㦨 σ 쟼㦨
R a d
dqB mv
쟼 ҈ = = ҈ 犨 ⏌ 鼀 ҈ ҈ 㡰 矄 ᆠᨨ ᨨ鼀 ᨨ B v � ᨨ鼀盐 ur r ur F = q v ´ B㡰 ᨨ � � 矄 ᨨ盐矄 �ᆠ�ᨨ ᨨ ᨨᆠ x㦨a x㦨a ᔼ ��ᨨ axa ᨨᆠ鼀盐矄� �ᆠ�ᨨ �ˆ ´ ˆ �.e. 졔z㜠 �ᆠ�ᨨ 矄ᨨ �ᆠ泐盐犨 꾀 � ��ᨨ
axa� z㜠 �ᆠ�ᨨ �ᆠ泐 ᨨ犨 犨 ҈䰘 b㡰 㝐� �ᆠ ᨨᆠ ᨨ ᆠ � �ᆠ ᨨ矄 ᨨ ᆠ � �矄 矄 \ u㦨v
ᨨᆠ ᆠ� � ᨨ
X B y ᆠᨨ 犨� �ᆠ盐 ᨨ �ᆠ�ᨨ ᨨ ᨨᆠ�泐� e u x ᆠᨨ 鼀� 矄 �
鼀� �ᆠ � 和ᨨ ᆠᨨ 犨� ᨨᨨ犨 矄�� ᨨ犨 ᆠᆠᨨ ᆠᨨ鼀矄ᨨ盐 犨� 矄 v㦨u y r ҈ b㡰 ␔ E 㦨 ᨨ ᆠ � �ᆠ泐ᨨ P犨� � ᆠᨨ�ᨨ 㡰 ᆠ㡰� ᆠᨨᆠ ᨨᆠ�泐 P�矄 � X㜠 �ᆠ�ᨨ �� 㡰 r v = k�$ 꾀 ᨨ矄��泐ᆠ � �ᆠ�矄 ᆠ � x y ᨨ犨 y �ᆠ�ᨨ �矄 ᨨ 㜠ᆠ�ᆠ盐 ur r ur ᨨ犨 F = q v ´ B㡰 ur Þ F = q k�$ ´ ckˆ + a�$㡰桌 ᨨᨨ�ᨨ 㡰 Þ=
㡰
a
q éQ qV = ҈ mv ù ê ú û mV ë ҈� ᆠ㡰 和� ᆠ �ᆠ �矄 鼀ᨨ� � 盐 ᆠ 和ᨨ ⼄ 쟼 = ⼄
= 䰘� ´ ҈ 졠N和㡰 꾀矄矄盐鼀� � �ᆠ 矄 ᨨ鼀 㡰
=
mv 㦨 qvB R
矄� a 㦨 Bd
-
҈
mv qB
\ 矄� a =
�㡰
҈ ´ ҈ -҈黨 ҈ ´ ҈-
㡰 ᨨ鼀 ��盐 矄� a 㦨 d⏌R
s e
⧤鼀
���㡰
ˆ $ ´ $� 桌 㦨kcq - $ 㡰 㦨 q kc�$ ´ kˆ + ka�
Ä
҈黨
ur
和� ᆠ � ᨨ�ᨨ 㡰 ᆠ�ᆠ � �矄 矄ᨨ 矄 E = a�$ ᨨ�犨�矄ᨨ ᨨᆠ� 졔X �ᆠ�ᨨ
矄盐 ᨨ 犨ᨨ ᨨᆠ矄 犨� ᨨ 㜠ᆠ�ᆠ盐 졠 泐ᨨ�ᨨ 㡰 �矄 ᨨ ᨨ ᨨ��ᆠ b㡰 ␔ ᆠ � �ᆠ �矄 鼀ᨨ� � � 盐 � ᨨ鼀 鼀� �ᆠ � ᨨ鼀 ᨨ�矄鼀ᨨ�ᨨ
qvB㦨
q x=a
mv R X v
x=b
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PS-221
Moving Charges and Magnetism ␔� ᨨ 鼀� �ᆠ � ��ᨨ ba㡰R錐犨 ἘRἘ�矄 �矄 �盐矄 ᨨ ᆠ盐盐 � 矄� 鼀� �ᆠ �
mv b - a 㡰 qB ³ ba㡰 Þ v鼀� 㦨 qB m b㡰 ␔ ᆠ � �ᆠ �矄 鼀ᨨ� � ᨨ鼀 ᨨ 鼀� �ᆠ � ᨨᆠ ᆠ矄 ᨨ � 犨 �ᆠ 矄 矄 ᆠ � ᨨᆠ 鼀ᨨ矄 �ᆠ� ᆠ�ᆠ盐鼀ᨨ�ᨨ \ R㦨
㡰 㝐� �ᆠ � ᆠ 盐 ᨨᆠ盐 � ᆠAB m I҈ q B҈㦨 犨 矄㡰 p r C
I
p㜠q q
A
mv mv Þ r㦨 \ F㦨qvB㦨 qB r ᔼ q B矄鼀 ᨨᨨ rµmv
B
I҈ I
B
E
㝐� �ᆠ � ᆠ 盐 ᨨ ᆠ盐 � ᆠ ACB
B
和� ᆠrA>rB Þ ҈
mAvA mBvB
r
ᆠ㡰 � ᆠ�ᆠ � ᨨ � 졔x �ᆠ�ᨨ E = E �ˆ r 鼀� �ᆠ � ᨨ �졔z �ᆠ�ᨨ B = Bkˆ ᨨᆠ�泐 ᨨ �ᨨ �矄 �ᆠ 犨� ᨨ � �ᆠ�ᨨ
r 鼀� 泐 qE r ᨨᨨᆠ�泐 v = AqE�ˆ 犨 A �矄 ᨨ矄�� ᆠᨨ 矄 A E B ᨨ矄�� ᆠᨨ 矄 矄 ᔴ � ᨨ �ᨨ 矄 q㡰 鼀泐 ᨨ矄�� ᨨ �� r r r 㝐� �ᆠ ᨨᆠ F = q v ´ B 㡰 r \ F = q AqE�ˆ㡰 ´ Bkˆ㡰桌 㦨q AEB �ˆ ´ kˆ 㡰 㦨q AEB - ˆ 㡰 㦨 ᠀q㡰AFBᨨ � ��y㜠 �ᆠ�ᨨ 꾀矄 鼀� �ᆠ ᨨᆠ�矄 ᨨ � ��y㜠 �矄 ᆠ�ᨨ 矄 犨 ᨨ矄��ᨨ �� 犨� ᆠ ᨨ犨 矄 �� y㜠 �矄 ᆠ㡰 �盐 鼀ᨨ鼀 盐鼀 ᨨ �ᆠ L㦨mvr㦨mrw犨 w㦨pn Qv㦨rw桌
q w wq 錐 盐 �㦨 㦨q 맔n㦨 t p p wq qr w ´ pr Þ M㦨 㝐� �ᆠ鼀ᨨ鼀 M㦨�A㦨 p
\ ⼄盐 ᆠ泐n㦨
\
Ҙ I ( p - q) ᨨ犨 犨 矄㡰 p r \ 鼀� �ᆠ � ᆠ m I҈ q m I p - q㡰 B㦨 p r p p r E E EA é lù QR =r ú = = 꾀矄ᨨI҈㦨 R rl ⏌ A rr q ëê Aû B㦨
A
M q wqr = = L mr w m
r r 㡰 � ᨨ鼀 ᆠ�ᆠ � E 鼀� �ᆠ � B ᨨᆠ ᨨ FE㦨qE ᨨᆠ 盐 ᨨᆠ�ᆠ � 㡰 FB㦨evB矄� q 㦨qvB矄� 㦨 ᨨᆠ 盐 ᨨ鼀� �ᆠ � 㡰 E 졔q
B 和� ᆠ ᨨᆠ 盐 ᨨ鼀� �ᆠ � �矄 ᨨ 和ᨨ ᨨᆠ 盐 ᨨ ᆠ�ᆠ � 犨� 鼀├ ᆠ � �ᆠ ᆠᆠ ᨨ ᆠᨨ�矄 � 矄�� �
҈
I㦨
҈
E E EA = = R rl ⏌ A rr p - q㡰
m é EA q EA p - q㡰 ù ´ ´ ê ú= p ë r r q r rr p - q㡰 r û � ᆠ㡰 꾀 � � 鼀� �ᆠ � 鼀� �ᆠ ᨨᆠ ᆠ矄 ᨨ ᆠ � �ᆠ 犨 �ᆠ 鼀ᨨ矄 ᆠ � �ᆠ� ᆠ�ᆠ盐 \ B㦨
mv m K mv 㦨qvB Þ R㦨 ᨨ R = qB qB R KqB⼄盐 \ Rµm \
m҈ R҈ = m R 盐 � ᔴ �矄 ᨨ ᆠ �矄矄 �ᆠ 盐ᨨ 꾀ᆠᆠᨨ � �ᨨ 鼀� �Ἐ矄 盐 ᨨᆠᨨ ᆠᨨ 矄 ��ᆠ �㡰犨�ᨨ犨 矄�� �.e.ᆠ├D 꾀矄ᨨ泐 矄鼀盐犨 � ᨨᆠᨨ ᨨᨨ a㜠�ᆠ�.e.졔 㡰泐ᆠ � �ᆠ�矄ᨨ犨 矄 ᨨ犨 矄� ᆠ 鼀� �ᆠ ᨨᆠ犨� 矄 ᆠ � ᨨᆠ m \ mv = qvB ᨨ µ q r ҈ ᨨᨨᨨ rµ = ҈ 錐 ᨨa㜠�ᆠ r µ = ҈ �.e. (r㡰a> r㡰p \ a㜠�ᆠ犨�├ B 䰘 矄 ⧤� �ᆠ �泐⧤ᨨᨨ 㦨⧤ᆠᨨ 㦨⧤ ᆠ
mv = qvB r
Þ
r㦨
mK mv Þ r㦨 qB qB
Qmv㦨P㦨 mk 桌 \ rµ m ᨨᆠᨨ 矄 qK B桌 \ rµ m 和� ᆠ 鼀矄矄 ᨨ ᨨᨨ > 鼀矄矄 ᨨ ᆠᨨ ᨨ �盐矄 ᨨ ᨨᨨ 犨�鼀ᨨ
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EBD_8341 PS-222
Physics
矄
ᨨ �ᆠ�矄 ᆠ�ᆠ犨 � 矄 ᨨ鼀 ᨨ 鼀� �ᆠ � ᨨᆠ�泐 ᆠᨨ鼀ᨨ v 犨� 矄ᨨ 矄�� 鼀ᨨ� � ᆠ � �ᆠ� ᆠ�ᆠ
B v v㦨v矄� q
q
)
v҈㦨v ᆠᨨ矄q
ᆠ㡰ᔴ �� 鼀ᨨ鼀 盐鼀㦨 p
黨
ᨨᆠ�泐ᆠᨨ鼀ᨨ v҈ 犨� 矄ᨨ 矄� � 鼀ᨨ� � ᆠ � �ᆠ � ᨨ� ᨨ �ᆠ�ᨨ ᨨ ᆠ � �ᆠ犨�� �ᆠ T盐
鼀� �ᆠ ᨨᆠ ᆠ� � ᨨ ᆠ � �ᆠ �矄 � �ᆠ�ᨨ �ᆠ盐 ᨨ �ᆠ�ᨨ ᨨ ᨨᆠ�泐 ᆠ � ᆠ
ᨨ ᆠ � 矄 ᨨ ᆠ � �ᆠ
æ ҈ ö 鼀 ÷ ᨨ矄 ᨨᆠ � è ø 꾀矄犨ᨨ├ ᨨ 泐 �矄 ᨨᆠ�矄 ᨨ矄ᨨ �泐ᨨ 鼀ᨨ� � ᆠ � �ᆠ矄 鼀� 矄 盐 ᆠ � b㡰 㡰 ᨨ ᆠ �졔턼ᨨ盐 ��ᨨ ҈
ᨨ ├� �ᆠ �泐 ç =
ù p é mv = QB ê �盐矄r = R ú = QvB Þ ë û R R ⏌ 㡰 p \ B = QR p
ᨨ ᨨB³ �ᆠ犨�㜠 ��ᨨ ҈ QR p ҈QR mv æ p ö = Qv ç ÷ r è ҈QR ø
mv mv = qvB \ r = \ rµm qB r �.e. �矄 ᆠ �矄 �ᆠ泐ᨨᨨ�ᨨ ᨨ 鼀矄矄 ҈ bᆠ㡰 㝐� �ᆠ ᨨᆠᆠ� �ᨨ ᆠ盐 ᆠ泐� �犨�ᆠ �
盐 � ᨨ鼀鼀� �ᆠ � r r r F = I l ´ B 㡰 r l 㦨 �矄ᆠ鼀 ᨨ 犨�㦨 L졔R㡰 xˆ r r \ F = I L + R 㡰 xˆ ´ B 㡰 r ᔼ B = Bxˆ
r F = I L + R㡰 xˆ ´ xˆ 㡰 B㦨 㡰 盐
泐 㝐
㡰␔ B =
\ r =
҈ R
盐矄ἘᔴἘ�矄 ᨨ ᆠ ᨨ ᆠ�ᆠ盐 ᨨ �盐矄
҈R
Ӹ
r ᔼ B = Byˆ
r F = I L + R 㡰 xˆ ´ yˆ 㡰 B 㦨IB L졔R㡰 zˆ ᨨ Fµ L졔R㡰 r ᔼ B = Bzˆ
r F = I L + R 㡰 xˆ ´ zˆ㡰 B = IB L + R 㡰 yˆ ᨨ Fµ L졔R㡰 ᆠ㡰 꾀ᆠᆠᨨ � � ᨨ 鼀� �矄 盐 鼀� �ᆠ � 矄 ᨨ盐 � �ᆠ�ᨨ vy = ᨨ鼀 ��盐 q = vx 泐
졠
ᔴ
�R
졠
R ⏌
҈ 졔턼
\ q =
꾀矄ᨨ CP = CO + OP
æ � R ö æ R ö = ç ÷ ÷ +ç è ø è ø
҈ R
盐矄 �ᆠ 犨� ��ᨨ ᨨ盐� ᨨ� ҈ ᨨ 㜠 �矄 \ᔴ㦨
p
맔 맔 맔 맔 턼㝐맔졔 맔 맔
q
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 盐� = 盐 = �ˆ 맔 맔 맔 맔
盐 㦨
맔 맔 泐 㦨 맔 q 맔 맔 = 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 맔 맔 맔 맔 맔
ᆠ 矄 맔�鼀 = �盐矄 �盐矄 㝐 ù é p ´҈ ´ ҈- = êQ �盐矄 = 턼 úû M ´ ⏌ QB ë
꾀 �ᨨ 泐 �ᆠ =
\ B =
�pM Q
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PS-223
Moving Charges and Magnetism
ᆠ 㡰 ᔼ q 㦨 º ᆠ � �ᆠ �矄 ᨨᆠ ᨨ � x㜠 r �矄 盐ᨨ鼀� �ᆠ � B ᆠ � �ᆠ 犨� ᨨ鼀ᨨ� ᆠ�ᆠ盐 � y㜠 z 盐 盐 ᨨ ᨨᆠᨨ
掸
y
r ᆠ�ᆠ � E �ᆠ犨� 鼀ᨨ� �ᆠ 犨� � ᆠ矄� � �ᆠ ᆠ ᨨ�ᨨ 矄 꾀㡰 㡰 犨ᨨ �
x
ᔼ q 㦨 ҈º 犨 ᆠ 矄ᨨ ᨨᆠ�泐� ᨨ 犨ᨨ ᆠ �盐 ᆠᨨ鼀ᨨ 矄졠 ᨨ �x㜠 �矄 vᆠᨨ矄҈º㡰 ᨨ ᨨ �y㜠 �矄 v矄� ҈º㡰盐ᨨvᆠᨨ矄҈º �ᆠ犨�鼀ᨨ�
ᆠ�ᆠ盐 盐 ᨨ v 矄� ҈º 盐矄 ᨨᆠ 盐 ᨨ ᆠ�ᆠ � �ᆠ犨�盐 �ᨨ �ᆠ鼀ᨨ�ᨨ 犨� �矄 �ᆠ � ᆠ矄� �
⼄l 鼀 Þ= 鼀 ⼄ 掸 �鼀 ├ 泐 �ᆠ ᨨ 鼀ᨨ � ��ᨨ ᔼᔼ ᨨᆠᨨ鼀� �ᆠ├� ��ᨨ ᔼ
p鼀 = = 犨 �ᆠ �矄 � ᨨ v�.e. �鼀 矄 �矄 ⼄ 矄鼀� ��ᨨ 㜠ᔼᔼ�矄矄鼀 ᨨ 泐ᨨᆠ�泐 ᆠ㡰 ␔ ᆠ � �ᆠ矄 鼀� �ᆠ � ᨨᆠᆠ矄ᨨ �ᆠ犨 �ᆠ 犨�ᆠ矄ᆠ � ᨨᆠ \
B
E
×
×
×
×
×
×
×
×
×
×
×
×2q
×
×
×
×
×
×
×
×
×
2q
q q q
B
鼀 꾀矄ᨨr㦨 ᨨµ鼀 ⼄ Q鼀鼀\ p鼀 㦨 ᨨ µ鼀 ⼄
\ 㦨 㦨 ᨨ � ᆠ 㡰 꾀矄 ^ ��ᨨ ᔼ ��ᨨ ᔼᔼ 盐 � ᨨ鼀矄ᨨ ᨨ ᆠ � 맔맔맔맔 �ᆠ � ��ᨨ 㜠ᔼᔼ �矄 ᆠ�ᆠ盐 맔맔맔맔 �盐矄ᨨ ᆠ�ᆠ盐 맔맔맔맔 鼀 r = 맔맔맔맔 r ⼄ 掸 ᨨ �ᆠ ᨨ 맔맔맔맔 ��ᨨ ᔼᔼᔼr> l 矄 ᨨ犨
맔맔맔맔 泐 �矄 � 㡰 ⼄l 鼀 > l Þ> 鼀 ⼄
맔맔맔맔 맔맔맔 맔맔맔맔 l
��ᨨ ᔼᔼᔼ
\ rµ
m q
r㦨
mv qB
⧤� �ᆠ �泐�k �B矄鼀桌
҈ ҈ 錐r +µ 錐 r ++ µ ҈ He ҈ O Þ r + µ ҈錐 r + µ 錐 rO+ + µ H He
䰘
×
Þ
mk qB
H+
×
mv r
ᨨ r㦨
r
A
×
l=
qvB㦨
ᔼ q 㦨 黨º ᆠ � �矄 鼀ᨨ� � ᨨ � 鼀� �ᆠ �
ᨨ ᨨᆠ 盐ᨨ鼀� �ᆠ � �矄 ᨨ盐 ᨨᆠ 盐ᨨᆠ�ᆠ � 犨�ᆠᆠ �ᆠᨨ �y㜠 �矄 b 㡰
泐 � ᨨ ᆠᨨ ᨨᨨ � 鼀� �ᆠ � 鼀├矄 矄鼀 �犨� 꾀矄矄 ᨨ犨
ᨨᨨ 犨� ᆠᨨ鼀ᨨ盐 � � � 矄 q ×
ᨨ鼀 �鼀盐鼀h �h� ��ᨨ ᔼᔼ㦨l
黨
µ
ᆠ졔 졠졔졔犨� ᆠ ⼄盐泐 졔犨� ᆠ 鼀ᨨ矄 矄� ᆠ�矄 �盐矄 �矄矄鼀矄 b 㡰
�矄 ᨨ ᆠ � � ᨨᆠ�泐 ᔼ ᆠ ᨨ矄矄� 犨
掸 쟼ᆠ�ᆠ 鼀� �ᆠ � 矄矄 �.e.쟼㦨㦨 和ᨨ 鼀ᨨ ᨨ 掸 졠犨 ᆠ�ᆠ 鼀� �ᆠ � 矄矄 盐 ᨨᆠ 盐ᨨᆠ�ᆠ � �矄⼄盐 ᨨᨨ矄�ᨨ 鼀� �ᆠ ᨨᆠ�.e. 졔 鼀㦨 �.e.E¹B¹㡰 掸 졠犨 쟼㦨B¹ᨨ� F㦨qvB矄� q㦨 矄� q 㦨�.e. q 㦨Þv Bᨨ �㜠 㡰 ᨨ �ᆠ ᨨ 鼀ᨨ � 矄�� � 犨� ᆠᨨ 矄 ᨨᆠ�泐 r r Fe + FB =
r r FE = -eE = -e - E xˆ 㡰 = eE xˆ r r éE ù FB = q v ´ B㡰 = -e ê yˆ ´ B zˆ ú ë B û r FB = -eE xˆ
㡰
S㦨
E xˆ B
FB B = B zˆ
E = E zˆ Ӹ
ᨨᆠ 盐 ᨨ 鼀� �ᆠ � 㡰 犨� ᨨ�
ᆠ矄矄泐ᆠ � ᨨᆠ ᨨ ᆠ�ᆠ盐 鼀ᨨ�ᨨ 犨 �ᆠ 犨�
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EBD_8341 PS-224
Physics
� 㜠Ӹ ᨨᆠ 盐ᨨᆠ�ᆠ � 쟼㡰 쟼犨� ᆠᆠ ᨨᨨ � 㜠 �ᆠ�ᨨ 盐矄 犨� �ᆠ 犨� � ᆠ矄� � �ᆠ
mv ᨨ鼀⼄ �㡰 ��㡰㦨 qB
b㡰 �ᆠ犨�鼀ᨨ� 矄�� � ᨨ �㜠 yˆ
㡰 ᨨ犨 鼀� �ᆠ � 矄 盐ᨨ҈҈
- v �ˆ E = - E ˆ
㦨
v �ˆ
X ҈ mv ⏌qB
B = B ˆ
Ӹ
ᔼ ᆠ�ᆠ � 犨� 泐 ᨨᆠ ᨨ Ӹ �矄 泐 ᆠᆠ� � ᆠ � ᨨ �Ӹ �矄 꾀 鼀� �ᆠ ᨨᆠ 㦨⼄矄� q 㦨 q㦨҈ ҈
mv mv 㦨qvBÞr㦨 �㡰 qB r ᨨ � �泐㦨 ├� �ᆠ �泐
҈ qv mv ᨨ 㦨 ��㡰 m ᨨ鼀⼄ �㡰 ��㡰 qV㦨
r㦨
m qV ´ qB m
ᨨ rµ
ᨨ r㦨
qVm qB
m q
T맔v҈㦨GS Þ
҈ qa ҈ ´ 㦨 㦨 ma qp ҈ rq ᆠ � �ᆠ �졠 鼀�ᨨ盐 ᨨ鼀 ��ᨨ ᨨ 鼀� �ᆠ � ᨨ� P ᨨᆠ�泐 r ᆠᨨ v 鼀├矄 � 犨� x㜠 �矄 �ᆠ盐 ᨨᆠ�ᆠ盐 P� 矄ᆠ矄 ��y㜠 �矄ᨨ� A \
rp
㡰
B
㦨
和ᨨ 犨��ᨨ ᨨ 鼀� �ᆠ � ҈R >R 㦨
盐矄 ⼄盐� � ᨨᆠ�泐㦨 -v $� ᨨ鼀 ��盐 ᆠ � �ᆠ ᆠᨨ矄 ᆠ�ᆠ �
��ᨨ ᨨ 鼀� �ᆠ � 和� ᆠ �鼀�ᨨ ᨨᆠᨨ鼀ᨨ盐�ᨨ T㦨pm⏌qB
ᨨ �鼀├ 泐 �ᆠᨨᆠᨨ矄矄 ��ᨨ
T ᨨ 鼀� �ᆠ � �.e. ᆠ�ᆠ㦨 㦨pm⏌qB 㡰 ᨨᆠ�泐 ᨨ ᆠᨨ 鼀├矄 � 犨� 鼀� �ᆠ � v҈ᆠᨨ鼀ᨨ ᨨ ᨨᆠ�泐鼀ᨨ ᆠ � �ᆠ � �ᆠ�ᨨ ᨨ 鼀� �ᆠ � 犨 矄 v ᆠᨨ鼀ᨨ ᨨ ᨨ� � ᆠ � �ᆠ � ᆠ�ᆠ盐 鼀ᨨ�ᨨ ᨨᨨ �矄 �ᆠ �ᆠ矄 犨� ��矄犨� 鼀� �鼀盐鼀盐ᨨ � �ᆠ ᨨ �
mp
맔 A 맔 M 맔 맔 맔 O 맔 맔 맔 맔 맔 맔 X=
´
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔 맔
맔 맔 vᨨ 맔 P 맔 맔 B = B kˆ 맔 X 맔 맔 맔 맔 맔 맔 X=L
\AO㦨AP㦨R㦨 �盐矄ᨨ ᆠ�ᆠ盐
mv mv = B qv ÞR㦨 �㡰 qB R L R = L ��㡰 ᔼ DAPM 㦨矄� ÞR矄� 㦨L Þ R
p m 맔vᆠᨨ矄 㦨҈ qB
B㦨
p mv ᆠᨨ矄 ° q ´ ҈
盐
҈ E mv 㦨E Þ v㦨 m
S
r 㦨҈鼀 v҈㦨v ᆠᨨ矄
pm E ´ ´ ᆠᨨ矄 ° \ B㦨 q ´ ҈ m 㦨
B
v G
v㦨 v 矄� ᨨ
p ´ ҈ ´ mE ´ ᆠᨨ矄 ° 㦨 q ´ ҈ ҈ ´ ҈-҈黨 ´ ҈
㦨 ´ 黨҈ ´ ҈ -҈ ´ ´ ҈ ´ ҈ ´ ҈ -҈黨 ´
҈
҈黨
맔҈맔҈㦨䰘䰘맔҈ ҈-҈黨 ᨨ B㦨䰘䰘맔҈
꾀 鼀 ᨨ ᨨᨨ 矄 矄 盐 � ᨨ鼀 鼀� �ᆠ � 㦨
�q㦨 犨� ᨨᆠ�泐㦨맔҈�鼀⏌矄 㦨
҈ᨨ� 矄 ᨨ� ᨨ 鼀ᨨ�ᨨ ᆠ�ᆠ盐鼀ᨨ�ᨨ
ᨨ 鼀 ᨨ ᨨᨨ 矄 矄ᨨ ᨨᨨ 矄 ᨨᨨ犨 �ᆠ
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PS-225
Moving Charges and Magnetism
mv = qv B r mv ҈䰘 ´ ҈-䰘 ´ ´ ҈� ´ = 㦨҈鼀 r 㦨 qB ҈ ´ ҈-҈黨 ´ ´ �ᆠ ᨨ � 㦨v҈맔T
B
\
犨 T㦨
p r p r = v v 矄� q
qvB㦨ma Þa㦨
㦨
҈ ´ ҈
-҈黨
qvB m
q ´
´ ´
O
-䰘
V V2= ¾ Ö
´ ´
x
x
x x
x
x x
E
x x
v x x x E
�
´
B㦨
⏌ ´ ҈-
´ ҈
=
´ ҈
-
´ ´ ҈
㦨҈矄
Topic-2 : Magnetic Field Lines, Biot-Savart’sLaw and Ampere’s Circuital Law
´ ´
ᨨ
V ¾ E´ V 1=Ö
㦨҈҈�鼀⏌矄
x x
�ᆠ�ᨨ ᨨ ᨨᆠ 盐ᨨᆠ�ᆠ � ⼄쟼㡰�矄ᨨᨨ矄� ᨨ鼀� �ᆠ � ⼄ 맔㡰 쟼�矄 � ᨨ矄�� �ᆠ�ᨨ
\맔矄 ᨨ盐 � ��x㜠 �ᆠ�ᨨ ᨨ矄 ᨨ盐 � 㡰z㜠 �ᆠ�ᨨ ^ ᨨ� 犨 矄
´
b
G
´ ҈ ´ ҈
҈ ´ ҈
\
F ´ ´
x -
� V㦨㦨맔҈鼀⏌矄 㦨鼀鼀 㦨맔҈鼀
´
䰘
x x
+ x
p r v 矄� q 㦨prᆠᨨq㦨맔҈맔҈맔ᆠᨨ 㦨鼀 m㦨҈맔҈䰘├�q 㦨҈맔҈҈黨ᔴ B㦨҈
v㦨҈䰘鼀⏌矄 F㦨qv B矄� a ᆠ� �ᨨ犨 矄O泐 鼀� �Ἐ矄 盐㡰 Þ F㦨qvB Q a㦨黨 桌 \ �ᆠ ᨨ � 㦨 v ᆠᨨ矄 q ´
�
+ x x
҈
㡰
v
´
ÐOEF㦨� Q OEᆠ矄 �盐矄㡰 泐矄泐鼀鼀泐Ð OFE 㦨� \ ÐEOF㦨黨 泐 �ᨨ鼀泐㡰
�矄 �矄 ᆠ � ᆠᆠ�ᨨ
졔
҈҈ v = ҈҈� Þ r㦨 ҈� = ҈鼀 r ҈ ҈ \쟼 㦨rᆠᨨ矄� 㦨맔 ҈ ´ 㦨҈҈鼀 꾀 �q㦨� ᔼ 鼀� �ᆠ � �矄� ᨨ盐犨 �ᆠ�ᨨ �ᆠ 矄� 矄鼀犨泐E ᆠᆠᨨ � �ᨨ 鼀� �Ἐ矄 盐 �ᆠ犨� 盐 ᨨ犨 矄 ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
ᆠᨨ⏌ ᨨ ᆠ�ᆠ矄矄 ᨨ犨 � ��盐 \ ac㦨
졔q
� ᨨ ᆠ�ᆠ盐 l = pr
q qv = T pr 㝐� �ᆠ鼀ᨨ鼀 M㦨ᔴ盐 맔꾀 qv ´ pr = � ´ pr = pr ҈ M = q×v×r ᔴ盐 � =
졠
�
�盐矄ᨨ ᆠ�ᆠ盐 � 鼀� �ᆠ � r =
E
mv qB
mv mv ҈ qv ´ ÞM = qB B r r �ᆠ�ᨨ ᨨ M �矄 ᨨᨨ矄� ᨨ B ᨨ r r - mv B M = B 泐鼀盐�泐� � ᨨ 盐鼀ᨨ ᨨ鼀� ᨨ 泐B㡰 \M =
O 졠
� �졠
F
æ p ö é p r ù \ �鼀⼄盐� 㦨 T = ç ÷ 㦨 ´ ê è w ø ë v úû
㦨䰘҈맔҈矄 꾀矄 ᆠᨨ 矄矄矄 盐 � \ ᨨᆠ 盐ᨨᆠ�ᆠ � 㦨 ᨨᆠ 盐ᨨ鼀� �ᆠ � eE = evB \ B 㦨
E V ⏌d = v v
Vù é êëQ E = d úû
㡰 � IA㦨꾀RA 㦨ᆠ鼀 q A = p -
p p =
p �p = m I q 矄� �鼀� �ᆠ � B = 4pR IB㦨꾀RB㦨ᆠ鼀 q B = p -
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EBD_8341 PS-226
Physics
B A I A q A RB = ´ = BB I B qB RA
p ´ = �p � ´ ´ ´
Ҙ � 矄� a + 矄� b㡰 p r
b㡰 B㦨
r㦨 � - 㦨ᆠ鼀 a㦨b㦨䰘
� ´ 矄� 䰘° 㦨҈�맔҈�
ᆠ㡰 ᔼ ᆠ盐 � ᆠ 犨� �ᆠ � 犨 矄㡰 㝐� �ᆠ � �졠 盐ᨨ 턼㝐犨� ᨨ �㦨和졔턼 m � m � \鼀� �ᆠ � = + p p m � ´ ҈ -䰘 ´ � ᨨ ҈ - = + p ´ ҈ - \ � 㦨 꾀 �ᆠ�ᨨ ᨨ 鼀� �ᆠ � �矄 �ᆠ盐 � ᨨ 㡰 \ B㦨 ҈
ᆠ㡰
䰘
a
ᔼ a
㝐� �ᆠ � 盐ᨨ ᨨ 矄� ᨨ �ᨨ
m I 矄� ° + 矄� °㡰 B= a p
m I æ ҈ ҈ ö m I çè + ÷ø = a ap ᨨ犨鼀� �ᆠ � 盐ᨨᨨ �ᨨ ᆠᨨ� m I B = ´ ap 꾀�� 鼀� �ᆠ � ᆠ ᨨ �ᨨ 矄 ᆠᨨ� ᨨ �盐 矄
ᨨ B =
m I
҈�m I ´ ҈ ´ p
㦨p
= �
m I p
㦨맔p
ᨨᨨ㦨
\ 黨
m� m � m � m � = = ᆠᨨ�㦨 æ ö ç ÷ è ø ҈
= ᔴ
ᆠ㡰 㝐� �ᆠ � ᆠ ᨨ ᨨᨨ ҈ =
(
㡰
鼀 = ᔼp m ᔼ =
(
m I m I m I = [ 矄� 黨° 矄� �° ] + + 矄� � 졔矄� 黨 桌 pR R pR mᔼ æ ҈ ö m ᔼ mᔼ æ ҈ ö ҈ + + ҈+ ç ÷ ç ÷ è ø è p p ø
e uuur m ᔼ æ ҈ ö e = ç p + ÷ p è ø
�
®
®
®
㡰 B = B + B ҈
=
( )
µ æ � º ˆ � º ˆ ö ç k + -k ÷ = ø d p è d
\ ҈ = ҈
)
= ᔼp = 鼀҈
)
B㦨B҈졔B졔B졔B
=
m ᔼ
�ᨨ鼀ᨨ鼀 ᨨ ᆠ�ᆠ盐ᨨᨨ�矄鼀㦨ᔼ꾀 鼀҈㦨ᔼ꾀㦨ᔼp { 㦨 �盐矄ᨨ ᨨᨨ晤 ᔼ 鼀ᨨ鼀 �矄 ᨨ盐 ├� � ᆠ盐 ᆠᨨ 矄 㡰 㜠 ᆠᨨ鼀矄
ᔴᨨ�
㦨Þ㦨
҈ é ù êëQ a = ҈ = ҈ 鼀 úû
ap
ᨨᨨ
ÞB=
B = � ´ ´
-䰘
(
m ᔼ m ᔼ
=
)
b㡰 ᨨ� P�矄矄�盐 鼀� 㜠ᨨ� ᨨ � ᨨ� � � ᆠ 矄ᨨ ᆠ�ᆠ盐犨�矄犨 �ᆠ 矄鼀 �� 㡰
r
鼀� �ᆠ � 矄 B 㡰 盐 ᨨ ᆠ盐 矄 � 犨�矄� 矄鼀 �ᆠ�ᨨ 㝐� �盐 ᨨ 鼀� �ᆠ � ᨨ� P
ì ü m NIR m NI m NIR B㦨 ï ïï 㦨 ⏌ 㦨 ⏌ ï ⏌ � R � í æ R ö ý ïç R + ï ÷ ø þï îï è
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PS-227
Moving Charges and Magnetism ҈� ҈҈
㡰 ᨨ鼀 ��盐
b㡰 ᔴ矄 㡰 ᔼ D졠꾀ᔴᆠᨨ矄 㦨
Ҙ ᔼ Ҙ ᔼ ´ p Qp 㦨l㡰 꾀㦨 ´ p = p p l ⏌ p Ҙ ᔼ ´ p㡰 㦨 p l � ᔴ矄 b㡰
꾀
ᔼ
㦨 ´
҈
矄� q㦨
҈ =a 㝐� �ᆠ � Ἐ졠Ἐ 盐ᨨ鼀 꾀 㦨
ᆠᨨ矄q
m I é ҈ ù m I ҈ - ú= ´ - ҈㡰 ê p a ë û pa \ 㝐� �ᆠ � ᆠ 盐 ᨨᆠᨨ鼀 ᨨᨨ
ll㡰�
é FB m I ´ I ù m ᔼ ´ ᔼl = êQ p l 矄� q úû p 2 矄� q ë l
m I é m I ҈ ù ´ - ҈㡰 ú ´҈ = ´ - ҈㡰 㦨 ê p a ë p a û ҈
B=
m ᔼ
҈ =
m ᔼ 꾀 p ᆠ 㦨 �盐矄ᨨ ᨨᨨ m ᔼ m ´ ´ ( ᔼ ⏌ ) ´ p ( ⏌ ) = 㦨 p ( ⏌ )
҈ = ҈ ҈ b㡰 � �盐矄㦨 �矄 ᆠ = ⏌
= 黨㡰⏌ = 䰘
B=0
R £ r < R
µ0
R
é R ù êr úJ r ûú ëê
B
⏌
r
R
æ aö
é
pù
N㦨҈A㦨a졔p ç ÷ 㦨a ê҈ + ú ë û èø ᨨ鼀 和ᆠ犨 犨 �ᆠ�ᨨ ᨨ m �矄 ᨨ盐犨 ᨨ � 졔 㜠 �ᆠ�ᨨ \
æ 㡰 ö = çç҈ + ÷ ÷ø è
r P
R
R r ᆠ � d㡰 ᆠᨨᆠ泐 �ᆠ矄 B 矄盐矄 r � r r ҈䰘 b㡰 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᆠ盐 ᆠ泐� �ᨨᨨ M = NIA
҈ m ᔼ ⏌ = m ᔼ ⏌
ö ÷÷ ø
R
µ i ᨨ r>R B = 0 pr ҈ �.e. B µ r
ᨨ犨 ᨨᆠᨨ� =
ᆠ æ = çç҈ + �矄 è
㡰 ᨨ r < ᨨ
m ᔼ b㡰 ᨨᨨᨨ = 犨 �矄 �盐矄ᨨ ᨨᨨ
\
m I [ 矄� ° - 矄� °] p a
=
pl�
ᨨ ᔼ = 矄� q 盐 ᆠᨨ矄 q ҈
\졠ᔴ㦨 a ´
矄� q
矄� q
矄� q
a
a㦨桌
p B Þ A = BB 㡰 盐矄 ᆠᨨ 矄� ἘlἘ � ᨨ ᆠ盐 ᆠ泐� � 犨� 꾀⼄盐���盐鼀 ᆠᨨ矄q㦨l�l
졠
Ҙ ᔼ Ҙ ᔼ m ᔼ ´ ´ = ´ = p l ⏌ p l pl
q
ᔴ
⏌
Ҙ ᔼ 㦨맔 矄� � 졔矄� � 桌 p ⏌
OC OC 㦨 OA a
҈ ҈黨
r a é pù M = Ia ê҈ + ú kˆ ë û
a a
a 㡰 㡰 犨� ᆠ�矄 ᆠ盐 I � �� z㜠 �ᆠ�ᨨ ␔ ᨨᆠᨨ 矄� 鼀� �ᆠ ᆠᨨ � ur B xy㡰� z㦨
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EBD_8341 PS-228
Physics
ur
ur ҈ B= $� + k$ 㡰 ᆠ㡰 㝐� �ᆠ � M 盐ᨨPQ QR
㝐� �ᆠ � B �矄 �ᆠ盐ᨨOP ur \ B = B 矄� q�ˆ - B ᆠᨨ矄 qˆ
y x ᆠᨨ矄 q = B = r r ur m I B= y�ˆ - xˆ pr
矄� q = \
ᨨ
(
(
)
)
ur m I y�ˆ - xˆ B= p x + y
(
m I x 졔y 㦨r pr y
P(x, y) r O맔
)
m I ҈ é m I ù += ê ú ë pR û pR Q㝐� �ᆠ � B㦨 泐ᨨ� ᨨ ᆠ盐 ᆠ泐� � 矄�� ᆠᨨ 盐ᆠᨨ桌 H҈㦨
q
q y
x
®
M
B
x
R ᨨ µ P
ᆠ㡰 盐矄ᆠᨨ 矄� 鼀 ᨨ �ᆠ├ 矄矄dxᨨ 犨� � �矄 ᆠx ᨨ鼀 ᆠ O
I⏌ I
Q
ᨨ µ S
I⏌
dx
R
ᨨµ
ᨨ犨犨 犨�QS�矄ᨨ� H㦨 㝐� �ᆠ � M 盐ᨨPQ㡰졔 鼀� �ᆠ � M 盐ᨨQR㡰졔 㝐� �ᆠ � M 盐ᨨQS㡰
x
a O
҈ é m I ù ҈ é m I ⏌ ù m I ++ ê = 㦨 ê ú ë pR û ë pR ûú pR m I m I H҈ 㦨 㦨 \ pR pR H
b
N b-a N \盐鼀ᨨ 盐 矄� �ᆠ├ 矄矄dx㦨 b-a 㝐� �ᆠ � 盐ᨨ �矄矄鼀鼀 O m NI dx dB㦨 b - a㡰 x b m NI dx m NI = B㦨 ò a b-a x b - a 㡰 b dx m NI b òa x = b - a 㡰ᨨ�e x桌 a 盐鼀ᨨ 盐 矄盐 � � 㦨
m NI b
\ B㦨 b - a 㡰 a ҈ 㡰 ᨨᨨADEFA� y㜠z ᨨᨨABCDA� xy Z 泐ᆠ ᨨᨨ矄� � ᨨᨨ矄 犨 Y E � � ᨨ ᨨᨨ 犨 ᨨ ├ ᆠ盐 ᨨ鼀 I AᨨD � ᨨ ᨨ I ᨨ鼀 D ᨨ A 쟼 ᆠ�泐 C D �矄 ᨨᆠ盐 � 꾀 F ᆠ 矄 犨ᨨ ᆠ ᆠ X ᨨ盐 ᆠᨨ ᆠ ᨨ I 矄 矄ᆠ� �鼀� �ᆠ � ᨨ� P �矄 A B ᆠᨨ 矄�
ᨨ� 㡰�矄� X㜠Z
鼀� �ᆠ � 盐ᨨᆠ盐 � ABCDA犨�� 졔 Z㜠 �ᆠ�ᨨ 盐 ᨨ 矄泐鼀鼀泐 y㜠ᆠᨨ鼀ᨨ 矄 x㜠ᆠᨨ鼀ᨨ 矄 犨� ᆠ ᆠ ᨨ盐 ᆠ ᨨ 和�鼀�泐 鼀� �ᆠ � 盐ᨨᆠ盐 � ADEFA犨� � x㜠 �ᆠ�ᨨ \ �ᆠ�ᨨ ᨨ 矄盐 鼀� �ᆠ � P aa㡰
b㡰 ᆠ盐 ᨨ犨� �ᨨ盐ᨨ ᨨ 矄ᨨ鼀� �ᆠ � ᨨ� 矄ᨨ �� ᨨ 犨�犨�盐犨 矄 ᨨ 犨� ᨨ犨 犨 矄 盐矄ᆠᨨ 矄� ᆠ� ᨨ� 矄 ᨨ� ᔴ 鼀� ᨨ� b犨 A B㡰 鼀� �ᆠ � ur C 盐 ᨨ A BCA 㡰 �矄 � 盐犨 �ᆠ�ᨨ 盐 鼀� �ᆠ � C 盐ᨨB�矄� ᨨ犨 犨 �ᆠ�ᨨ � �矄 ᨨ ᨨ� E㝐� �ᆠ � 盐ᨨA�矄盐犨 鼀� �ᆠ
ur
ur
� 盐ᨨ�矄 ᨨ犨 犨 盐 B 쟼A < B 쟼B \鼀� �ᆠ � �矄� ᨨ犨 犨 �ᆠ�ᨨ ur ur ᨨ� B DA > B DB � 盐犨 矄 和�鼀�泐 ᨨ ᨨ� 矄 ᆠ ᆠᨨ 矄� BDA
BGB
BCA BEA
FB
FB
A D BDB
BFA
C
E
B
BGA G
BCB
BEB
ᆠ � 㡰 矄�矄 �矄 矄 ᆠᨨ ��ᨨ 㡰 㝐� �ᆠ � ᆠ 盐 ᨨᆠ盐 � ᆠAB m I҈ q B҈㦨 犨 矄㡰 p r C I
p㜠q q
A
B
I҈ I
E
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PS-229
Moving Charges and Magnetism 㝐� �ᆠ � ᆠ 盐 ᨨ ᆠ盐 � ᆠ ACB
\
Ҙ I ( p - q) B㦨 ᨨ犨 犨 矄㡰 p r \ 鼀� �ᆠ � ᆠ m I҈ q m I p - q㡰 B㦨 p r p p r E E EA é lù QR =r ú = = 꾀矄ᨨI҈㦨 ê R҈ rl҈ ⏌ A rr q ë Aû I㦨 \ B㦨 �
꾀矄ᨨ ᆠᨨ矄 q҈ =
B=
E E EA = = r ⏌ r l R r p - q㡰 A
m é EA q EA p - q㡰 ù ´ ´ ê ú= p ë r r q r rr p - q㡰 r û
… �㡰 … ��㡰
m I ém I ù m I ´ [ + ] = p ´ x ´ ҈ = 䰘 êë px úû p x
é m I ù ú 犨�├㦨䰘 ë p x û
䰘
m I c æ ö p ç a ÷ è ø
鼀� �ᆠ � ᆠ C 盐 ᨨ ᆠ盐 ᆠ泐� � 犨�矄PQ RS�矄 ᨨ 㝐� �ᆠ � 盐ᨨᆠ盐 � 矄鼀�㜠ᆠ�ᆠ盐ᆠQAR ҈ é m I ù ҈㦨 ê R ú ^ ᨨᨨ盐犨 矄桌 ë ҈û ᔼ B ᔼ A
æ æ pa ö ö I t J pa = 㡰 Ic J ç = ç ÷÷ è øø è m é pa J pa J ù 㦨 ê ú ҈ û pa ë �maJ ᨨ 㦨 ҈ N ᔴᨨ鼀� ��犨� maJ ҈ ␔�N㦨� 䰘㡰
鼀� �ᆠ � 盐ᨨ犨�矄 턼㦨졠 泐犨� 턼 犨�ᨨ 盐ᆠ鼀� �ᆠ � ᨨ鼀ᨨ� 㝐^턼 Q a m é It I c ù pa êë úû
q҈
… 㡰
ᔴᨨ鼀� ��犨� 㦨 k ê
ᆠ盐 I 㦨 ÞI㦨꾀 A 㝐� �ᆠ � 鼀ᨨ� �ᆠ�泐 ᔴ㡰 㦨ᨨᆠ�泐
㦨
a ҈ x = = x x
� x - a � x - ҈ x = = = ᆠᨨ矄 q = x x
ᨨ ᨨ鼀⼄ �㡰 㡰 �㡰 ��㡰
�㡰 ᔴ盐 矄�泐J㦨
m I t pa
PM = 黨 x - x = �䰘 x = x
Q P S R 㝐� �ᆠ � 盐ᨨᆠ盐 � 矄鼀�㜠ᆠ�ᆠ盐 ᆠSBP
҈ é Ҙ I ù 㦨 ê ^ ᨨ � 犨 矄桌 ú ë R û \ 㝐� �ᆠ � 㦨҈ ҈ é m I ù ҈ é m I ù 㦨 ê ú- ê ú ë R҈ û ë R û m I é ҈ ҈ ù ê - ú ë R҈ R û �ᆠ ᨨ犨 矄 �ᆠ盐ᨨ ᨨ 㡰 ᔼ ��ᨨ 鼀� �ᆠ � �矄 盐 ᨨ ᆠ盐 � 矄ᨨ ᨨ� ᨨ 㦨
M
�x a㡰
I x
q P x 㝐� �ᆠ � � 盐ᨨ犨� 턼 m I ᆠᨨ矄 q҈ + ᆠᨨ矄 q 㡰 B= p PM ᔼ D턼㝐黨x 㦨PM졔a ᔼ D 㝐 ҈x㦨PM졔 �x a㡰 Þ 䰘 x = � x - ҈ xa Þ ҈xa㦨҈x Þ a㦨҈x ᨨ鼀⼄ ��㡰 �㡰
黨 x = PM + ҈ x㡰
… ���㡰
ᔼ ��ᨨ r> 鼀� �ᆠ � �矄矄 盐ᨨ ᆠ盐 � ᆠ泐� ᨨ ��ᨨ r 鼀� �ᆠ � �矄 � ᨨ � ᆠᨨ鼀鼀ᨨ �矄 ᨨ � �ᨨ ᆠ�ᆠᨨ �盐矄r b㡰 鼀� �ᆠ � 泐ᨨ� � 矄� ᨨᨨ �矄 ᨨᔴᨨ 矄� 泐 ᨨ� � 矄� � 犨 � P 盐矄 ᆠᨨ 矄� ᆠ�ᆠ盐 ᨨᨨ 盐矄� � 꾀鼀Ἐ矄ᆠ�ᆠ盐�犨
… �㡰
Ñò B × d l = Ҙ I
R … �㡰 … ��㡰
黨
r r
和� ᆠᆠ盐 � 矄� ur ᨨᨨ �矄 ᨨ \ 㝐� �ᆠ � B 㦨
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EBD_8341 PS-230
Physics ¥
Þ pR IBq =
� Q
¥
M
L R �
P
O
S
�
Qm㦨IA 鼀ᨨ鼀 ᨨ � �ᨨ ᆠ�ᆠ盐ᨨᨨ I =
¥
�
Þ pRIBq = ᨨ¥
\ 㝐� �ᆠ � 盐ᨨᆠ盐 ᆠ泐� �ᆠᨨ 盐ᆠᨨP ᨨ� O
Ҙ � ^ ᨨᨨ盐犨 矄桌 p OR㡰 㝐� �ᆠ � 盐ᨨᆠ盐 ᆠ泐� �ᆠᨨ 盐ᆠᨨQ ᨨ� O Ҙ � B㦨 ^ ᨨᨨ盐犨 矄桌 p OS 㡰 㝐� �ᆠ � 盐 ᨨ ᆠ盐 ᆠ泐� � ᆠᨨ 盐ᆠᨨ矄 L MO�矄 ᨨ \ 矄盐 鼀� �ᆠ � O B 㦨B҈졔B 꾀矄҈ � 矄鼀 �ᆠ�ᨨ 㡰 m � m � m é ҈ ҈ ù + = �ê + B㦨 p OR p OS p ë OR OS úû
mR a
B҈㦨
mR 㡰
mR wq
Þ w=
pIB Þ m
Þ T=
pm IB
p = T
pIB m
r
r
b㡰 ᨨ⼄盐 � ᆠ 泐ᆠ�ᆠ盐ᨨᨨt㦨 㝐 ´ 㦨㝐 矄� qq㦨p⏌桌
r
犨 鼀� �ᆠ鼀ᨨ鼀 㝐 㦨 �꾀 㦨 �p 㦨 �盐矄 ᨨ ᆠ�ᆠ盐ᆠᨨ�桌 鼀� �ᆠ � 㦨 \ t 㦨㝐矄� q 㦨�p 㦨 ᨨᨨ 盐 矄桌
D Þ D 㦨 鼀� �盐 ᨨ �盐㡰 D 鼀ᨨ鼀 盐鼀�� 㦨tD㦨�p D㦨p턼 Q�D㦨턼桌
꾀矄ᨨᨨ⼄盐 t =
҈ ù é ҈ + úû 㦨҈ ë
㦨҈䰘맔҈맔 ê
Topic-3 : Force and Torque on Current Carrying Conductor ҈
ᆠ㡰 ᨨ⼄盐 ᨨ ᨨᨨ
�
㡰
t = M ´ B = MB 矄� q = MB 矄� 黨° 㝐� �ᆠ � B =
m I pd
꾀矄 ᨨᆠᨨ � 犨�ᔴ�矄 ᨨ
æm I ö \ t = I҈ a㡰 ç ÷ 矄� 黨° è pd ø
m I҈ I m I a ´ a = pd pd b㡰 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᨨᨨ ABCD M҈㦨ᨨ ᨨᨨ맔ᆠ盐 r ab㦨ᨨ ᆠ �㡰 M҈ = abI 㡰 ˆ 㡰 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᨨᨨ DEFA r M = abI 㡰 �$㡰 =
鼀� �ᆠ鼀ᨨ鼀 r r r r M = M ҈ + M Þ M = abI $� + $ 㡰
䰘
r Ҙ ᔼ Ҙ ᔼ Þ = ҈ + = p p - 㡰 Ҙ ᔼ҈ Ҙ ᔼ = p p - 㡰 ᔼ҈ ᔼ҈ 㦨ᔼ ᔼ = ҈ ᔼ҈ - ᔼ
犨ᨨᆠ矄矄鼀泐ᨨ矄矄�� ᔼ҈>ᔼᨨᔼ>ᔼ҈ m æ �� � �҈� ö m �� ´a = ҈ ᆠ㡰 F = p çè a a ÷ø p r r r b㡰 t = m ´ m㦨ᔼ꾀桌 㦨ᔼ꾀맔矄� 黨ᨨ 꾀㦨p桌 Þ t 㦨ᔼp
æ ˆ kˆ ö abI ç + ÷ ø è ᆠ㡰 ᨨ⼄盐ᨨ ᆠ�ᆠ盐ᨨᨨt㦨MB矄� q 犨 M㦨鼀� �ᆠ鼀ᨨ鼀 B㦨鼀� �ᆠ � ᨨ犨盐矄� �t 㦨Ia \ t 㦨MB矄� q㦨Ia r Þ M =
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PS-231
Moving Charges and Magnetism
㡰 㝐� �ᆠ 鼀ᨨ鼀 m = ᔼ꾀 =
҈
⼄ p 㡰 p
⼄w ҈ p 㡰 = ⼄ w p ᆠ㡰 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᆠ盐 ᆠ泐� �ᆠ �盐ᨨᨨ ᨨ 꾀�矄�� 泐㝐㦨ᔼ꾀 鼀� �ᆠ鼀ᨨ鼀 ᨨ ᆠ盐 ᆠ泐� � ᆠᨨ� �矄 ᆠᨨ �矄 �ᆠ�ᨨ �矄 �� 泐 �� 盐鼀 盐 ᨨ r ᆠ �盐 ᨨᨨ ᆠ 盐 ᨨᆠ盐 � ᨨᨨ r 㝐 犨泐矄 ᨨ m =
黨
ᆠ㡰 ᨨᨨ 犨� 鼀ᨨ ᨨ犨 矄 犨� AB ᨨ ᆠ �盐ᨨᨨ犨��ᆠ ᨨ ᨨ �矄�� 犨�矄 ᆠ盐 矄 � 矄鼀 �ᆠ�ᨨ 犨 矄 CD 犨� ᆠ盐矄 ᆠ盐 矄 � ᨨᨨ矄� ҈ �ᆠ�ᨨ 盐矄� ᆠ �矄 ᨨᆠ F µ 犨 r�矄 �矄 ᆠ r 犨 犨�矄 ᨨ 犨� ᆠ� ᨨᆠ ᨨ ᆠ �盐 ᨨᨨ ᨨᆠ ᨨ BC �矄 ⼄盐 ᨨᨨ矄�ᨨ ᨨ AD矄ᨨ 矄 ᨨᆠ矄犨�ᆠ ᆠᆠ ᨨ I
B
C
I
A 㝐 Ä ᔼ 犨 矄
㝐
D
҈
ᆠ ᆠ㡰 ᆠᨨ矄ᨨ 矄 ᨨ矄 ⼄盐���盐鼀 r r ᆠ㡰 ҈ = = ᆠ盐矄 ᨨ ᆠ�ᨨ ᆠ�ᨨ �
҈҈
㡰 ᨨ矄⼄盐���盐鼀 㝐
mv mv 㦨qvBÞ R = R qB ҈ ᨨ R µ mqv 矄鼀桌 B R B \ ҈ = R B҈
ᨨ盐 矄⼄盐���盐鼀 㝐 㡰 㡰 I҈ I 㦨 ᨨ矄�� ᆠ㡰 F 㦨 �� I҈ I㦨 �� 㡰 F 㦨 ᨨ矄�� ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ 矄犨 㡰 I҈ 㦨꾀 I 㦨҈꾀 I 㦨꾀
é ҈ ù Ҙ ҈ ´ I ê + ú X X X p R҈ ҈û ë ҈ = ᨨ R Ҙ é ҈ ù ҈ ´ I ê ú p ë X҈ X - X҈ û X é ù X - X҈ + X҈ X = 㦨 ê � X = ú X - X҈ - X҈ X - X҈ ë ҈ û
e 졠盐犨 矄
҈
҈
r
r
҈䰘
r
r
ᆠ鼀
\
�ᆠ鼀
҈
҈�
턼
㡰
꾀矄ᨨ �� 錐 � ᨨ 犨�턼 㦨�ᆠ鼀㦨�鼀 ᨨᆠᨨ 犨�턼 盐ᨨ犨� ´ ´ ҈ ´ � FQR㦨҈䰘 ´ � � 㦨맔҈ ᨨ犨 矄 㡰 ᨨᆠᨨ 犨�턼 盐ᨨ犨� ´ ´ ҈ FQP㦨҈䰘맔 ´ � 㦨�맔҈� ᨨ犨 矄�� 㡰 ᆠF 㦨 FQP FQR 㦨�맔҈�맔҈� 㦨맔҈ᨨ犨 矄�� b㡰 ᔼ 盐 � ᨨ鼀鼀� �ᆠ � ᨨᆠᨨ ᆠ盐 ᆠ泐� � ᨨᨨ�矄 ᨨ ᆠᨨᨨᆠ
ᨨ鼀ᨨ 鼀� �Ἐ矄 盐 犨 � ᨨᆠ�矄ᆠ� � � �泐ᨨ盐犨 �ᆠ�ᨨ
ᨨ盐� ᨨ盐 ᆠ�ᆠ盐鼀 ᆠ ᨨ ᆠᨨ 盐ᆠ� � ᨨᨨ b㡰 꾀矄 犨├ ᨨ犨 ᨨᆠ盐 � � ⏌ ҈ 犨 犨ᨨ犨�矄 ᆠ泐� �ᆠ盐 矄 I҈ I矄 泐 �矄 ᆠr � �
P m I I m � 㦨 ҈ = QI҈㦨I㦨� r㦨b桌 l pr pb
b
X X҈
R҈ = = = X R - - X҈
҈
ᨨᆠᨨ ᨨᨨEDCBE犨� ᨨ꾀矄ᨨ ᨨᆠ 盐 ᨨ 矄�鼀 FE BA 犨� ᨨ ᨨᆠ 盐 ᨨ 矄�鼀 EB ur F = I L�$ ´ B $ 桌 㦨 ILBk$ � ᨨ矄��z �ᆠ�ᨨ
҈黨
T盐 꾀 ᆠ盐 ᆠ泐� � ᆠᨨ� �矄 鼀� �ᆠ �ᨨ 鼀� �ᆠ ᨨᆠ ᨨ 鼀� �ᆠ �ᨨ ᆠ � 盐 � ᨨ鼀 鼀� �ᆠ � �矄 ᨨ �㡰㝐� �ᆠ � 盐ᨨ � � �泐ᨨ �ᆠ盐 ᆠ泐� � 犨� �矄 ᆠ�矄�� 泐
B㦨
m æ � ö ç ÷ p è r ø Y
C
� A
B
P O Z d
x
X
d
ᔼ ᆠ矄ᨨ � �ᆠ犨�矄 矄盐 � ᆠ ᨨ ᨨ 泐 � ᨨ� P �矄 犨 犨ᨨ 犨�矄 ᨨ 犨�矄
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EBD_8341 PS-232
Physics
� B 盐ᨨ 犨�矄犨�� 矄鼀 �ᆠ�ᨨ 矄ᨨ矄盐 B ᆠ
ᨨ ᨨ ᆠ � ᨨ� P �矄 �矄 ᆠ ᨨ鼀 ᆠ 犨�矄矄 ᨨ犨 � ��盐 ur ur ur ur B P = B PA + B PB + B PC ur 犨 B PA 㦨鼀� �ᆠ � P 盐ᨨA ur B PB 㦨㝐� �ᆠ � P 盐ᨨB ur B PC 㦨㝐� �ᆠ � P 盐ᨨ C �ᆠ�ᨨ ᨨ B�矄�� 泐�� 鼀盐
ur m ҈ ҈ ù $ é ҈ B P = � ê + - k 㡰 p ë d + x x d - x úû ur ᨨ B P = ҈ ҈ ҈ + 㦨 d+x x dx d 和ᨨ� �犨�x㦨 ± ��㡰 ᨨᆠ盐 � � 犨 犨ᨨᆠ盐 ᆠ泐� � 犨�矄 m �҈� 㦨f 矄泐㡰 p r 꾀ᆠ� � ᆠ盐 矄 � 矄鼀 �ᆠ�ᨨ d
d B
A r
X
Z
f
҈
␔ 犨�꾀�矄矄�� 泐 矄矄 ᨨᆠᨨ 盐矄�ᨨ 盐 ᨨᆠ盐 ᔼ �ᨨᨨ矄�ᨨ 犨�ᔴ犨�盐矄 �盐犨 矄 꾀 鼀泐� ᔼ ᨨᆠᨨ ᆠ�ᨨ �矄矄 ᔴ 犨� ᆠ�ᨨ 꾀 ��ᨨ ᨨ 꾀犨� ᨨᨨ矄矄� ␔ ᨨ �矄 矄矄 泐 �矄 ᆠx ᨨᆠᆠ� � ᨨ 盐犨�� ᆠ矄矄 矄矄矄ᨨ� � ᨨᆠ Fmag I҈㦨A
A' r 㦨҈鼀
mg
C
I㦨A
mg m I҈I 㦨 r p r
㝐矄矄盐 � � 㦨
zù m � � F z é = f ᆠᨨ矄 q = ҈ ´ ê 矄 ᆠᨨ矄 q = ú r p r l r ë û ᨨ
F m � z = × l p d + z 㡰
矄I҈㦨I r㦨d졔z桌
l w㦨
m � Þ f㦨 pl pd
ᨨ鼀⼄ �㡰 ��㡰
m I҈I m I҈I = ´ w Þ p r p rg Þ
g r
w㦨
p g 㦨
r
҈ r = p = 矄 g 黨 鼀� �ᆠ � 盐 ᨨ ᆠ盐 矄� A BPB҈ B 矄ᆠ�泐 � �ᆠ�ᨨ 矄 �ᆠ盐ᨨ꾀 矄 矄 ᨨ犨 � ��盐 Ð BPO㦨ÐAPO㦨 q ur ur m I B҈ = = B Z p a + x \ T㦨p
Y B a
xq O
㜠a A
P
B 矄� q
B
q
B҈矄� q
C
q
Bᆠᨨ矄q B҈ᆠᨨ矄q
X
B҈
졠 矄ᨨ� �B҈ B犨� 矄� q ᆠᨨ鼀ᨨ 矄 ᆠ ᆠ ᨨ盐 ᆠᨨ矄 q ᆠᨨ鼀ᨨ 矄 盐 \ B㦨B҈ᆠᨨ矄 q
Ҙ� m � ´ Þ w㦨 p d pd l
� Þ p f㦨 d
��㡰
x 졔
F m �.e. = - w z l l 㦨l wz犨�
F㦨mwz
m I҈ I p rg
Öa
m æ � ö F = - ç ÷ z 矄d>>z F�矄ᨨᨨ矄�ᨨz桌 �㡰 l p è d ø
ᆠᨨ鼀ᨨ 矄 ᨨ ᨨᆠᨨ � x㜠 �矄 犨� ᆠ ᆠ Fµz 鼀ᨨ�ᨨ �矄矄�鼀 鼀ᨨ �ᆠ ᔴᨨ鼀� �⼄ �㡰犨� 矄 ⼄盐�ᨨ ᨨ 和㝐
ᨨ
D
m I҈I mg x 㦨 r p r 和� ᆠFµx �ᆠ ᨨ⼄盐���盐鼀ᨨ矄��ᨨ
ᨨ 鼀ᨨ�ᨨ �矄 矄�鼀 鼀ᨨ �ᆠ m I҈ I \ 㦨 鼀矄矄盐 � � 㡰w Qa㦨lw㡰 �㡰 p r
B
ᔼ 犨�B�矄 �矄ᆠ ᨨ �Z㜠 �矄泐矄鼀 �矄 ᆠ F Z 矄ᨨ� � ᨨᆠ盐 � � ᨨ 犨�B 盐 l ᨨ犨�矄A C犨�
B'
矄ᨨ� � ᨨᆠ⏌ � F㦨
꾀⼄盐���盐鼀
Z
x
m I҈ I m I҈ I 矄ᨨ� � ᨨᆠ⏌ � 㦨 p r - x p r m ҈ ù m I I x é ҈ - ú 㦨 ҈ I҈I ê 㦨 p ë r - x r û p r r - x㡰 ␔ x�矄矄鼀�.e.xr r㦨x»r
f
q q
B
A
m pl
㦨
m p
I
a +x
´
x
a +x
=
m Ix p a + x 㡰 ᨨ犨 矄 Y �ᆠ�ᨨ 㡰
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PS-233
Moving Charges and Magnetism 盐矄 ᆠᨨ 矄� 矄鼀ᨨ�ᨨ ᨨ 犨�OC P ᨨ � dx ᆠ盐矄 鼀� �ᆠ � ᨨ 盐ᆠ � ᨨ� 矄 ᨨ 졠ᔴ犨� � 㡰 矄鼀鼀ᨨ盐 ᨨ ᨨᆠᆠ� � ᨨ 矄鼀 ᨨ�ᨨ
黨 黨 ´ ҈҈ I B ´ ҈҈ = 㦨꾀 Þ IB㦨 ҈ ��㡰 �鼀 矄�ᨨ 矄 �� 矄 ᨨ犨矄 SA졔SB㦨AB\ ÐASB㦨黨 泐泐 �ᨨᨨ盐矄 ᨨ鼀㡰 㝐� �ᆠ � 盐ᨨAS m I A m ´ 黨 × = ´ BSA㦨 �ᆠ Sᨨ B㡰 p rSA p ҈ 㝐� �ᆠ � 盐ᨨB S m I B m ´ × = BSB㦨 �ᆠ Sᨨ A㡰 p rSB p ҈ 和� ᆠBSA BSB鼀盐盐泐 �ᆠ盐 \ 鼀� �ᆠ � Þ
ur ur ur dF = I dx ´ B㡰 \dF㦨I dx B矄� 黨
m Ix Þ dF 㦨I dx맔 ´ p a + x 㡰
xdx m Þ dF㦨 I p a + x 㡰 \
ᨨ ᨨᆠF㦨
L m xdx ´ Iò p a + x 㡰
L
m é҈ ù ´ I ê ᨨ� e a + x 㡰 ú 㦨 p ë û
é m a + L ù ´ I êᨨ�e ú p a ûú ê ë ur
�ᆠ�ᨨ ᨨ F �矄ᨨ犨 矄Z �ᆠ�ᨨ 矄 鼀� �矄矄 盐 ␔ �ᆠ�ᨨ ᨨ ᆠ盐 � B �矄 矄 鼀� �ᆠ � �矄 ᨨ � ᆠ盐 ᆠ ᨨᆠ F㦨IlB矄� q q㦨҈ �矄 ᨨ 犨ᨨ犨�矄 �ᆠ盐ᨨ ᨨ 犨�꾀ᆠ�矄ᆠ盐 �ᆠ � ᨨ ᨨ
ᆠ盐 � 犨�B矄 ᨨ盐 � 盐犨 �ᆠ�ᨨ 矄ᨨ矄 ᨨᆠ ᆠ 鼀� �ᆠ � 盐ᨨA P 泐��
盐鼀 盐㡰
m æ 黨 ö æ ö B 㦨 BSA + BSB = ç ÷ +ç ÷ p è ø è ø 㦨҈䰘맔҈㦨҈맔҈
���㡰 ᨨᆠ盐 � � ᨨ 犨�B
Þ F 㦨
m I A I B p rAB ҈-䰘 ´ ´ 黨 ´ 㦨 㦨맔҈䰘⏌鼀
㦨
Topic-4 : Galvanometer and its Conversion into Ammeter and Voltmeter ҈
A ´
㡰 ᨨ鼀 ᨨ 矄�矄 ᆠ 㡰 ᆠᨨ � ᨨ ᨨ鼀ᨨ �㜠҈ R҈ �g
V = ҈ = �g G + R҈ 㡰
鼀 ҈
鼀
R҈
10 m 11
S
҈鼀
�㡰
ᨨ� ᆠ矄 �ᨨ ᨨ鼀㜠
= �g R҈ + R + G㡰
M BA
P
BB N
�㡰 鼀� �ᆠ � P 盐ᨨᆠ盐 � 犨�A
m I A m ´ 黨 = ´ �ᆠ�ᨨ PᨨM㡰 �㡰 ҈ ö p rAP p æ çè + ÷ø ҈҈
鼀� �ᆠ � P 盐ᨨᆠ盐 � 犨�B BA㦨
I m m ´ 黨 ´ = ´ B ҈ ö p æ ҈ ö p æ çè + ÷ø çè ÷ø ҈҈ ҈҈
��㡰
�� � �⼄ �㡰泐 ��㡰 G + R҈ ҈ Þ = G + R҈ + R
B
m I B ´ BB㦨 p æ ҈ ö çè ÷ø ҈҈ ᨨ鼀⼄ �㡰 ��㡰
R
Þ G + R҈ + R = G + R҈ \ R = G + R҈ 㡰 � ᔴ盐 矄矄� � ᨨ盐� � ᨨ鼀 I 㦨 鼀꾀 ᆠ�ᨨ q㦨 ��盐 ᨨ 鼀� ᨨ � ᨨ鼀 I ´҈- = = ´҈- 꾀⏌ � q ᨨ 盐矄㡰㦨�� 졔㡰 Þ�㦨҈ 맔҈졔 㡰 㦨҈黨�맔҈W ᆠ㡰 㦨�� 졔 㡰㦨맔҈ �졔�㡰㦨 㡰 Cq㦨NB�A矄� 黨 æ p ö = ҈䰘�B ҈- 㡰 ´ ҈- ᨨ ҈- ç è ҈ ÷ø \B㦨҈
=
��㡰 �
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EBD_8341 PS-234
Physics
b㡰 ␔ ├泐⧤҈ �矄 ᆠᨨ矄 ├泐⧤ �矄 ᨨ
쟼 �� = = ᔴq + � ␔ ᨨ ├泐矄 ᆠᨨ矄
�㡰
ᔴq 쟼 � æ ö ´ = ç � � ÷ � + �㡰 � ç + ÷ + � è �ø ᔴq �쟼 Þ = ��㡰 � � + ҈҈ � 쟼 = ᔴq �㡰 + � �� � � �㡰泐 ��㡰犨� � � + ҈҈ Þ =� ҈҈ + � � Þ��졔� �㦨� �졔҈҈ �㦨 �㦨W b㡰 ᨨ鼀 矄� ��矄�ᨨ 矄ᔼ�㦨맔҈맔�㦨҈꾀 � � �� =
䰘
�W
黨
҈
҈ ´ ҈ ´ 䰘 ´ ҈- 和= = ҈҈ W æ ö -ç ÷ è ø ҈҈ ᆠ㡰 � ᔴ盐 ᨨ盐� � ᨨ鼀 �g㦨�맔҈A ᨨ鼀矄�矄 ᆠ G 㦨 ҈�W 矄�矄 ᆠ Rᨨ盐� 矄�矄 犨� � ᨨ鼀 ᨨ ᆠᨨ � � ᨨ ᨨ鼀 V㦨�g R졔G㡰 ҈㦨�맔҈ R졔҈�㡰 \ R㦨҈�㦨҈黨�㦨҈黨�맔҈W ҈ ᆠ㡰 ᔼ�㦨 ᔼᔼ�㡰矄 \҈맔҈㦨 ҈҈㡰맔和 \和»҈W ҈ 㡰 꾀ᆠᆠᨨ � �ᨨ졠 鼀Ἐ矄犨 ᔼ = ᔼ� 㦨 졔 犨 ᔼ�㜠 ᨨ鼀ᆠ盐 㜠ᨨ ᨨ鼀矄�矄 ᆠ ᔼ ҈҈ ´ ҈ ´
㦨ᔼ�
�
␔ 矄 盐 ᨨ 矄�矄 ᆠ 和 �矄 ᆠᨨ
ᆠ ᨨ 和 ᨨ鼀 㦨 졔 和 \ ᔼ㦨 和 졔 졔 和 ᔼ
㦨ᔼ� 졔 㡰 �㦨 �졔 㡰҈\ 㦨W ᆠ㡰 ᆠ�ᨨ ᆠ盐 㦨ᔼ�鼀 㦨 ├㦨�맔 ␔ 㦨盐鼀ᨨ ��矄�ᨨ 矄㦨 ├㦨�鼀⏌ ��矄�ᨨ
㦨҈�맔҈㦨҈� 㦨ᔼ�졔 桌 ҈�㦨҈�졔 桌 ҈㦨졔 㦨W 㡰 � 矄�矄 ᆠᨨ � ᨨ鼀 G㦨҈W ᔴ盐 ��㦨҈mA 꾀 � ᨨ鼀 ᆠ ᆠᨨ � ᨨ ᨨ鼀 泐 ᆠᨨ
ᆠ� � � 矄�矄 ᆠ � 矄�矄 犨� �
ᨨ矄�矄 ᆠᨨ ᆠᨨ鼀� �ᨨ 㦨 G졔 R 꾀ᆠᆠᨨ � �ᨨ졠 鼀矄犨V㦨�� G졔R㡰 \҈㦨҈맔҈ ҈졔R㡰 Þ҈҈㦨黨黨W㦨 Þ 㦨黨黨├W b㡰 ��盐ᨨ 鼀�ᨨ � ᨨ鼀�矄 ᆠᨨᆠ⼄盐� ᨨᨨ 盐ᆠ ᆠ�ᨨ ᨨ ᨨ ��矄�ᨨ � � ᨨ鼀 ᔼ � ��盐ᨨ 鼀� = q e ҈ ᔼ= = ⧤W + 黨 ҈ e 和 ҈ e和 = ´ Þ = + 和 和+ 和 + 㡰 + 和 和 +和
҈ ᔼ 和= + 㡰ᔼ e ´
和 쟼⼄盐 ᨨ � � ᆠ �矄 �� 泐
ᔼἘ� 㦨 ᔼ 㜠 ᔼἘ� 㡰和 ᔼ Ἐ� + 和㡰 = ᔼ和
쟼
쟼
=
Þ + 和 + 和 = 和 + 㡰 Þ = 和 + 㡰 - 和 + 㡰 \ = 和 + 㡰
ᔼ⏌
҈
ᔼ�
ᔼ和 +和 和 Þ = ´ + 㡰 + 和 +和 +和 ҈ 和 Þ = + 㡰 + 和㡰 + 和 Þ + 和㡰 + 和 = 和 + 㡰 Þ
��㡰 ҈ q㦨 㦨�
� 㦨 ᔴ㦨҈鼀
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PS-235
Moving Charges and Magnetism 꾀㦨맔҈鼀桌 ␔├ ᨨ犨ᔴq㦨꾀ᔼ�
҈䰘
Cq ҈ ´ 㦨҈꾀 㦨 NBA � ´ ´҈ ´
ᨨᆠᨨ � ᨨ鼀ᨨ鼀鼀矄 盐 矄�矄盐矄 �
ᨨ � ᨨ鼀 ᔼ�맔㦨 ᔼᔼ�㡰和 \ ᔼ�㦨
҈ ´ � � \ 和㦨 㦨 㦨 㦨��W I Ig ҈ ҈ 黨 ҈� b 㡰 㦨 ҈W錐 ᔼ� 㦨 맔 ҈꾀 㦨 ҈ 鼀 㦨҈ᔼ㦨҈꾀 矄� �㦨ᔼ� 졔 㡰 㦨 矄�矄 ᆠᆠᨨ
ᆠ � 矄�矄 犨� � ᨨ鼀桌 Þ҈㦨맔҈ \ 㦨�맔҈ W 矄�矄 ᆠᨨ ᨨ鼀㡰 꾀矄ᨨᔼ�㦨 ᔼᔼ�㡰矄 맔҈맔҈㦨 ҈맔҈㡰矄 \和㦨맔҈W 矄�矄 ᆠ ᨨ 鼀鼀㡰 I gG
bᆠ㡰 ᨨᆠᨨ 鼀鼀� ᨨᨨ鼀 �� 矄�矄 ᆠ �矄 ᆠᨨ
ᆠ � 矄�矄 ᨨ犨 矄�矄 ᆠ�矄 ᆠᨨ
ᆠ �
ᨨᆠᨨ � ᨨ鼀� ᨨ 鼀鼀 ᨨV㦨Ig G졔R㡰㦨�맔҈�҈졔桌㦨҈
æG ö é҈ ù + ҈÷ = � ´ ҈-� ê + ҈ú 㦨�鼀꾀 èS ø ë ҈ û
ᨨI㦨Ig ç ҈
θ NBA ᆠ㡰 和 矄���泐ᨨ 鼀ᨨ� �ᆠᨨ�� ᨨ鼀㦨 㦨 ᔼ C θ ᔼ � ᆠ矄矄泐盐矄� ��ᨨ ᆠᨨ �.e.錐矄 矄���泐� ᆠ矄矄 ᔼ 和ᨨ �ᨨ ᆠ 矄�泐鼀� �矄 鼀� �
Topic-5 : Miscellaneous (Mixed Concepts) Problems ҈
ᆠ㡰 盐矄 ᆠᨨ 矄� 鼀 � dl 矄盐 � �
� dq ᆠ ᨨ ᆠ�ᆠ �졠 FB 鼀� �ᆠ ᨨᆠ ᆠ� � ᨨ �矄 �
Tᆠᨨ矄 q
和 ҈ ´ 꾀 = = » W + 和 ҈ +
ᔼ=
T
쟼 쟼 = � ´ ҈ + 黨҈ �҈
=
q
쟼 黨黨 = ´ = 黨黨W ᔼ 黨҈ 쟼 ⏌ 黨҈
T矄� q T矄� q
q q
I
쟼 � ´ ҈ 쟼 ´ = ´ 黨黨 黨҈ �҈ 黨҈
\ 鼀矄盐 =
Tᆠᨨ矄 q
ᔼ
꾀
쟼
é ᆠ dl 㡰 ù êQ � q㡰 = ú �盐矄 R û ë
꾀⼄盐���盐鼀 T 矄� ç
\ E㦨黨黨黨鼀 和� ᆠ�A㦨҈꾀
҈
I
L æ q ö q ÷ = BI p è ø
E æ ҈ ö ´ç ÷ ´ � ´҈ 黨 è �҈ ø
\ 㝐 �鼀盐鼀 � �ᨨ R㦨
R
Lù é æ Lö = BI ç ÷ d q êQ pR = L Þ R = p ú è p ø ë û T 矄�ᨨ � 犨�ᆠ� �ᨨ �ᨨ 矄ᨨ 鼀 �
ᔼ ᨨ鼀 矄 ᨨ犨矄 盐 矄ᆠ ᆠ�ᨨ
҈㦨
T
\ FB 㦨BI dl㡰 盐犨 矄㡰 㦨BI Rdq㡰
҈W
q
Þ T
黨黨黨 ´҈ ҈ ´ ҈
㦨黨黨黨W
ᆠ㡰 � ᨨ ᨨ鼀 V㦨Ig Req졔G㡰 \ 㝐 �鼀盐鼀 ᨨ� ᆠ ᨨ� � ⼄盐� 矄�矄 ᆠ ᨨ ᆠᨨ鼀ᨨ 矄 �矄 鼀 �鼀盐鼀 �.e. 犨 ᆠᨨ鼀ᨨ 矄 ᆠᨨ
ᆠ � 矄�矄
�ᨨ 鼀鼀
G ö æ I㦨 I g ç ҈ + ÷ è S eq ø \ 㝐 �鼀盐鼀ᆠ盐 �ᆠ ᨨ� � ⼄盐� 矄 盐 矄�矄 ᆠ�矄鼀� �鼀盐鼀�.e.犨 ᆠᨨ鼀ᨨ 矄 ᆠᨨ
ᆠ �
dq L = BI dq p
é dq ù êëQ 泐㜠泐矄鼀 úû
BIL p 㡰 鼀� �ᆠ� 矄ᨨ ᨨᆠᆠ 盐ᨨᆠ盐 犨� � 矄盐ᆠ 犨泐 ᨨ x y 矄 � 矄 犨� �ᆠ盐 矄� 矄犨�� ᆠ�ᆠ盐ᨨᨨ矄 盐鼀 ᨨ � 矄 鼀ᨨ� � ᨨ犨 犨 矄 � x y 犨� 矄鼀 �
盐鼀ᨨ ᆠᨨ鼀� �盐犨 矄ᨨ x y ᨨ ᨨ鼀� �ᆠ 盐 ᨨ盐� xy 犨� ᨨ 㡰 鼀� �ᆠ � 盐 ᨨ ᨨ 犨�矄 犨� ᨨ犨 犨 矄 矄 ᨨ犨 ᨨ犨 � ��盐 和� ᆠ � ur r 犨 v B �矄҈ ur r ur
ᨨ 鼀� �ᆠ ᨨᆠ F m = q v ´ B㡰 = \ T =
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EBD_8341 PS-236
Physics Z
v
Y
I
㜠I
X ␔�矄� X㜠Y ᨨᆠ�泐� Z㜠 �ᆠ�ᨨ
B
d
㡰 꾀�矄 ᨨ �x㜠 �矄vx㦨
y mv mv r҈㦨 r㦨 qB҈ qB
d҈ + d t҈ + t
v r҈
Q҈㦨
�
B
\
䰘
r҈㦨r
r B҈
r B
r
x
p r LR
⏌ 䰘㡰 ҈㡰 㦨�黨ᨨ; ҈㡰 �㡰
R )
Z
BR B
B҈
I҈ 졠
P
Z
I Y
æ m pr ö dI ᨨ M㦨 ç ÷× �㡰 è LR ø dt � I㦨Iᆠᨨ矄 t㡰
N㦨
(
X
æ m pr ö dI df = ç ÷× è L ø dt dt
æ p r ö ÷ m I 矄� t 㡰 M㦨 ç è LR ø
P
r m I ⼄盐ᨨ B = - kˆ㡰 R
æ m I ö f㦨BA㦨 ç ÷ pr 㡰 è L ø
dI 㦨I矄� t㡰 dt
I
R Y ᆠ㡰 ᔼ I҈ I> � 鼀� �ᆠ � ᨨ���
犨� � Z �ᆠ�ᨨ 鼀� �ᆠ � r 盐 ᨨ I ᨨ��� 犨�矄ᨨ� Z �ᆠ�ᨨ ᨨ B ᨨ���
ᆠ
ᨨ ᨨ 㡰 ᔼ I҈㦨I 矄盐 ᨨ 鼀� �ᆠ � P�矄ᨨ �졔X �ᆠ�ᨨ ᨨ 鼀� �ᆠ � �矄 ᨨ 泐 盐 ᨨ ᆠ盐 ᔼ 犨 �ᆠ �矄 � Z �ᆠ�ᨨ �矄
Iù m I é Q n� = ú ê ë L û L 盐 矄矄� � ᨨ盐� ᆠ�ᆠ盐 ᆠᨨ�
\
I 졠
\ 㦨
æ m pr ö dI e ᔼ 盐ᆠ ᆠ盐 �㦨 = ç ÷× R è LR ø dt 㝐� �ᆠ鼀ᨨ鼀 M㦨�A㦨�pr
�ᆠ�ᨨ ᨨ ᨨ犨ᨨ ᆠᨨ 矄�矄ᨨᨨ矄�ᨨ ᨨ ᆠ盐 盐ᨨ �矄 ᨨᆠ ᨨ犨� �ᨨ ᨨ �ᨨ ᆠ 꾀ᔴ 矄 ᆠᨨ ᆠᨨᆠ �矄 ᆠ b 㡰 㡰 ᔼ I҈㦨I 鼀� �ᆠ � 矄 盐ᨨI҈ Iᨨ��� Ἐ졠Ἐ犨�ᆠ ᆠᨨ盐ᆠ ᨨ 盐 鼀�㜠
�ᆠ � Ἐ졠Ἐ 盐ᨨ � �犨�矄 ᨨ ᆠ
ᨨ ᨨ ᨨ��� 㡰 ᔼ I҈> I 鼀� �ᆠ � 盐ᨨᨨ ᆠ盐 犨� � 졔 Z �ᆠ�ᨨ 㜠盐 鼀� �ᆠ � 盐ᨨᆠ盐 ᔼ犨�� Z �ᆠ�ᨨ � �矄鼀�㜠
�盐 �矄 ⼄盐 ᨨ ᆠᨨ鼀� 鼀� �盐 矄 ᨨ I҈ I ᆠ ᨨ ᨨ��� X I҈ R
pm
�鼀矄 泐ᆠ � �ᆠ� B҈t҈㦨 qB҈ pm
�鼀矄 泐ᆠ � �ᆠ� Bt㦨 qB
ᨨ �矄 ᆠ ᨨ � x㜠 �矄 d҈졔d㦨r҈졔r㦨 r҈졔r㡰㦨 �r㡰㦨҈r ҈ r mv qB 꾀�矄 㦨 �t 㦨 qB ´ pm 㦨鼀矄҈ Qv㦨p鼀矄҈㡰 㡰 ᔴᨨ 矄� ᆠ�ᆠ盐盐矄ᨨ �矄ᨨ ᨨ� 犨�矄 ᆠᨨ矄泐犨ᨨ盐 㝐� �ᆠ � � 矄� 矄ᨨ ᨨ� 㦨mn�
ᔼ 盐ᆠ 鼀 e㦨
ur r ur
ᨨᆠ � ᆠ 泐鼀ᨨ� � ᆠ � F = q v ´ B㡰 㦨 - e㡰 - v�$ ´ B $ 㡰 㦨evB k$
黨
I
df ᨨ� �ᆠ� �矄� 盐ᆠ 鼀 dt 犨�矄鼀盐ᆠ盐 矄犨� � � hA>hg æ v ö ᆠVA>VB矄 çç h = g ÷÷ è ø ᔼ rA>rB IAIBᔼ �矄ᆠ矄�� ᆠᨨ ��ᨨ ᆠ 盐 � � mAmB ᔼ rArB IA>IBᔼ �矄ᆠ矄�� ᆠᨨ ��ᨨ ᆠ 盐 � � mA£mB 꾀⼄ 盐ᨨ盐犨�� ᨨ犨 犨 �ᆠ�ᨨ 盐ᨨ ᨨ犨犨�� 盐犨 �ᆠ�ᨨ ᆠ q �矄 ᆠᨨᆠ 꾀矄P�矄 鼀� ᨨ� 犨ᨨ鼀� �ᆠ � 矄ᆠ ᆠᨨ盐 ᆠ ᨨ ᨨ r �矄 ᆠᨨᆠ 盐ᨨᆠ盐 � ᨨᨨA�矄ᨨ � �� ᨨ犨 矄 �� 盐 ᨨ ᆠ盐 � ᨨᨨ B �矄 矄ᨨ ᨨ � � � ᨨ犨 矄 �� b 㡰ᔼ 盐ᆠ 鼀 e =
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PS-237
Moving Charges and Magnetism
҈
҈҈
҈
ᆠ 盐ᨨ ᆠ盐 矄 � 犨�矄 � 矄鼀 �ᆠ�ᨨ ᔴ⼄
鼀� �ᆠ � 盐ᨨᆠ盐 � ᨨᨨAP�矄⼄盐 ᨨᨨ矄�ᨨ 鼀� �ᆠ � 盐ᨨᆠ盐 � ᨨᨨBP ⼄矄 �P 盐ᨨᆠ盐 � �
ᨨᨨ�矄 �ᆠ盐ᨨ ᨨ �ᆠ �ᆠ泐盐犨 矄 �P 盐ᨨᆠ盐 � ᨨ盐ᨨᨨ�矄 �ᆠ盐ᨨ ᨨ �ᆠ �ᆠ泐 ᨨ犨 犨 꾀矄 ᆠ盐 � ᨨᨨ矄� �ᆠ�ᨨ 犨�矄 ᆠ ᨨ 盐 ᨨᆠᨨ ᆠ 犨��矄 ᨨ 꾀⼄
鼀 �泐�矄 � � 犨� 犨 ᆠ � ᆠᆠ�ᨨ �矄 ᆠᨨ
ᆠ ᨨ 矄 ᨨ 犨� � ᆠ盐 ᆠ矄� �� 鼀� �盐 犨� �鼀㡰矄矄矄 ᨨ盐� 犨� 矄 ᨨ犨 矄 矄�矄ᨨ㡰 ᨨ � � ᆠ ᆠᨨ矄矄 犨� 矄ᨨ ᆠ矄矄 犨� �鼀 ᆠ � ᨨ
ᆠᆠ�ᨨ 矄ᨨ ᆠ矄矄 犨� �鼀 矄
犨� �矄 鼀ᨨ �ᆠ盐 ᨨ�矄 � l㡰 犨� ᆠᨨ 矄 ᨨᆠ�泐 v㡰� 盐 � ᨨ鼀鼀� �ᆠ � 㡰^ ᨨ ᨨ 鼀ᨨ�ᨨ e 㦨Blv ␔ B l v ᆠᨨ 矄 �矄 ᆠᨨ 矄 꾀 ᆠᨨ 矄 ᨨ � � ᆠ ᨨ矄 ᆠᨨ矄矄 矄 ᨨ 犨� ᆠ �矄 ᨨ ᆠᨨ 矄 鼀� �盐 矄 ᨨ 犨� ᔴ矄 ␔ 犨� �矄 ᆠ 鼀 ᆠᨨ 矄 ᆠ�ᆠ � 矄 �ᆠ�ᨨ ᨨ � � ᨨ 犨� ᆠᨨ 矄 鼀ᨨ 盐 � 盐 ᆠ ᨨ ᆠ�ᆠ � ᨨᨨ矄� ᨨ �ᆠ�ᨨ ᨨ ᆠ�ᆠ � �矄 鼀ᨨ鼀 ᨨ e ᆠᨨ � 盐矄 � ᆠ�ᆠ � � 矄� 犨� �矄 ᨨ ᔴ �矄 ᨨ ᆠᨨ 矄 鼀� �盐 矄 ᨨ 犨� ⼄ 和� ᆠ emf ᨨ 泐 E R ᆠᨨ 矄 ᆠᨨ 矄 ᆠ盐 ᨨ犨矄 � 犨�盐ᨨ � � ᆠᨨ ᆠ盐 鼀 �泐 �矄 � � 犨� 꾀矄ᨨ ᆠᨨ 矄 ᨨ � � ᆠ ᨨ矄犨 矄ᨨ 犨� 꾀錐⼄矄錐ᔴ⼄矄錐⼄矄 꾀㡰 ᔴ �ᨨ �ᆠ�ᆠ� � 犨�ᆠᆠᨨ矄�ᆠ � 犨 �ᆠ �矄 �鼀 � 㡰 ᨨ� �ᆠ ��矄�├ᆠ盐 �矄犨�ᆠ 鼀� �ᆠ � 鼀� �ᆠ鼀ᨨ鼀 ᔴ㡰 ᔴᨨ 矄 ᆠ盐 � � � 矄ᨨ ᆠ � �矄 ᨨ 犨� ᨨ �鼀 � ᆠ�ᆠ � ᆠ盐 ᨨ 盐ᆠ矄 鼀� �ᆠ � 鼀� �ᆠ 鼀ᨨ鼀 㡰 � 㦨 � ᆠᨨ矄 wt錐꾀 ᆠ �� � 鼀� �ᆠ � 犨� ᨨ 盐ᆠ �矄 犨� ᆠ � 盐ᆠ ᆠ�ᆠ � 꾀矄ᨨ ᆠ �� � 鼀� �ᆠ 鼀ᨨ鼀 犨� ᨨ 盐ᆠ 㡰␔ 鼀� �ᆠ ᨨᆠ ᆠ矄 ᆠ�ᆠ ᨨᆠ FB㦨FE Þ q vd B㦨q E \ vd B㦨
V w
Q V㦨E맔w桌
é I ù ú´B ë newd û
\ V㦨wvdB㦨 w ê \ V =
I ´B ned
ᨨ V µ
҈ d
Þ
V҈d҈㦨Vd
I I ù é êvd = neA = newd ú ë û
ᔼ d҈㦨dV㦨V҈ � d҈㦨dV㦨V҈ ҈
ᆠ㡰 Q㦨
I ´B ned
Vn V n B Þ ҈҈= n B҈ B ᔼ B҈㦨B n҈㦨nÞV㦨V҈ vfB҈㦨B n҈㦨nÞV㦨�V҈ ҈ ᆠ㡰 B҈矄� q B矄� qᆠ ᆠ ᆠ ᨨ \ V µ
B B矄� q
P
q q 黨ºq㡰 B
B҈矄� q B҈
q B q B҈ᆠᨨ矄졔ᆠᨨ矄
a +h
h q ␔�҈ a a ␔� ᨨ ᨨ鼀� �ᆠ � �P 㝐� �ᆠ � 盐 ᨨ ᆠ盐 ᆠ泐� � ᆠ�ᆠ盐 ᨨᨨ 㦨 㝐� �ᆠ � 盐ᨨ矄�� 犨�矄 B㦨B҈ᆠᨨ矄q졔Bᆠᨨ矄q㦨B҈ᆠᨨ矄q
é m I ê 㦨 ⏌ êë p a + h a + h
m Ia
(
҈�
)
ù a ú´ úû a + h
和ᨨ� � 犨 � h»҈a
ᆠ盐 �矄 ᨨ鼀PᨨQ RᨨS� 犨�҈ 犨� 矄ᆠ�泐 b㡰 ␔ ├ ᨨ犨 ᨨ⼄盐 r r r t = M ´ B 㦨MB矄� q
(
)
m Iù é 㦨 I ´ pa ´ ê ´ ú 矄� ° pd û ë
҈ ҈䰘 ҈ ҈黨
m I a \ t㦨 d 㡰 ��ᨨ ᨨ � �矄 盐ᨨ鼀� �ᆠ盐矄�ᨨ
鼀� �矄 ᆠᨨ�矄 盐
� � ᨨ � ᆠ├ � 鼀� 矄 ᨨ � Ἐ矄 盐 ᆠ�� 㡰 �� � ��ᆠᨨ矄 b㡰 㝐��矄 ��ᨨ ᨨ 鼀� �ᆠ��ᨨ 鼀� �ᆠ ᨨᆠ犨�盐 鼀�� 矄 㡰 ᨨ⼄盐ᆠ� �ᨨ ᆠ �盐ᆠᨨ�ᆠ � 盐 � ᨨ鼀 鼀� �ᆠ � r uur ur t = M ´ B Þ t = MB 矄� q M㦨N � A q 㦨黨 ᨨ鼀ᨨ� �ᆠᨨ�� ᨨ鼀㡰 \ t㦨N � A B矄� 黨 㦨N � A B t㦨k � �� 㡰 \ k �㦨N�AB Þ k㦨NAB 㡰 ᨨ矄�ᨨ ᆠᨨ 矄 ᔴ 㦨
t N�AB = q q
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EBD_8341 PS-238
Physics ᨨ鼀⼄ ��㡰 ���㡰
� �㦨� q = p ⏌ N � A B \ C㦨 p ᆠ㡰
T҈㦨
�㡰
ò tdt 㦨 ò N�AB dt 㦨NAB ò � dt Qt㦨N�AB㡰
㦨NABQ
ò t dt = I w NABQ I 盐矄 泐 犨 ᨨ �泐 ᆠᨨ 矄�ᨨ ᨨ � � ᨨ ᨨ�ᨨ ᨨ�ᨨ ├� �ᆠ �泐 ᨨ ᆠᨨ� 㦨 ᨨ � �泐 鼀 �鼀盐鼀 ᆠ�ᨨ
҈ ҈ I w = C q 鼀 l ҈ æ NIAB ö ҈ æ NABQ ö Iç ÷ q鼀 ÷ 㦨 ç è p ø è I ø
pBNA I ␔ 犨 �矄 ᨨ ᨨ� � ␔ ᨨ 犨 㦨 矄�ᨨ � 矄� � mg㦨T
\ T㦨
mg
�㡰 d
T҈
T w
P
O
B
␔ 犨 �矄 ᨨ� � 鼀� �ᆠ � �矄 � 鼀� �ᆠ鼀ᨨ鼀
Q Q 맔pr㦨 w맔pR T p ᨨ犨 矄�ᨨ 矄� 矄� �矄 犨�ᆠᨨ鼀盐 ⼄盐 矄�ᨨ 矄� 矄� �矄 T҈ T ᨨ 矄�ᨨ ⼄盐��b�盐鼀 T҈졔T㦨鼀� ��㡰
ᨨ⼄盐 ᆠ� � ᨨ � � ᨨ盐 ᆠ ᨨ � � r uur ur t= M ´B t㦨M맔B맔矄� 黨 M㦨�A㦨
Q QwBR w ´ pR ´ B = p ᨨ ᨨ�ᨨ ⼄盐���盐鼀 ᨨ⼄盐 ᨨ盐 ᆠ ᨨ � � 矄 ᨨ盐 ᨨ t㦨
QwBR d
dT
BQR ҈ 㝐� �ᆠ � ᨨ 盐ᆠ 泐 ᆠ�ᆠ盐� �矄 ᆠ B㦨B�
ᆠ矄졔Bᨨ盐ᆠ矄 矄 矄�� ᨨ ᆠ�ᆠ盐� 犨� ᨨ 盐ᆠ ᨨ 鼀� �ᆠ � ᆠ m� m� + \ B㦨 r҈ r � �㦨҈꾀r҈㦨鼀 r㦨҈鼀桌
҈ ö m æ ҈ ´ ҈ ´ ç + ÷ è ҈ ø \ B㦨 �맔҈�㡰T ^ ᨨ ᨨ盐犨 矄桌 ᨨ鼀�� 盐鼀盐㡰 㡰 ᨨᆠ ᆠ� � ᨨ ᆠ盐 ᆠ泐� � ᆠᨨ 盐ᆠᨨ ᆠ � 鼀� �ᆠ � ur r ur F = I l ´ B㡰 = I lB 矄� q ᨨ ᨨᆠᆠ� �ᨨ h犨�hᆠ ᔼ h�矄ᆠ矄q㦨҈ \ F㦨IlB矄� ҈ 㦨 졠 ᆠAC 盐ᨨᆠ盐 hᆠ ur m I B ᨨ ACB㦨 p r҈
�ᆠ�ᨨ ᨨ �矄 鼀� �ᆠ � ᨨ 泐矄鼀矄�鼀 ᨨ AC犨� � � \ q = ҈° ÞF㦨 졠 矄�鼀 CD m I 㝐� �ᆠ � �矄 ᆠx 盐ᨨᆠ 犨�B㦨 px
r㦨҈鼀 D
dx
D
O
r҈㦨鼀
x
C
B
ᨨᆠᨨ 矄鼀矄�鼀 dx �矄 r ᨨ鼀O dF㦨I dxB
m I �m I dx 㦨 p x p x 졠 � �� �
㦨҈맔dx맔
d d QwBR \ T҈맔 - T ´ = Þ T҈T㦨
mg ù é êëQ T = úû
Qw鼀 BR T = T + d
\ w鼀 㦨
T
ᨨ B㦨
\ q鼀 㦨 Q
盐 鼀 �鼀盐鼀 矄�ᨨ ᨨ├� � 矄�ᨨ 㦨 \
\ ᔼw㦨NABQ Þ w㦨
mg QwBR + d
���㡰
\ F㦨
�m I p
r
òr҈
dx x
\ F㦨
� m I [ ᨨ� e x ] rr҈ p
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PS-239
Moving Charges and Magnetism
� m I �m ´ ҈ r æ ҈ ö ᨨ� e = ᨨ� e ç è ÷ø p p r҈ 㦨҈맔҈ � 犨 矄㡰 泐 鼀� � 盐㡰 ur 㡰 㝐� �ᆠ � B㡰 ᨨ��� 㦨鼀� �ᆠ � 盐ᨨ 矄鼀�ᆠ�ᆠKLM졔鼀� �ᆠ � 盐ᨨᨨ 矄鼀�ᆠ�ᆠKNM ur m I m I \ B = -�$㡰 + $ 㡰 R R Q⧤㝐�矄� y㜠z ⧤㝐� x㜠z 桌 ur m I m I m I ᨨ B = - $� + $ = -$� + $ 㡰 R R R \ 㝐� �ᆠ ᨨᆠ ᆠ� � ᨨ �ᆠ ur r ur m Iü ì F = q v ´ B㡰 = q í - v �$㡰 ´ -$� + $ 㡰 ´ ý R þ î \ F㦨
-m qv I $ k ᨨ F㦨 R ur ur ur 㡰 F KLM = F KNM = F KM ur F KM = BI R㡰$� = BIR�$
ur
ur
M
㡰 I
y v x
N �㡰 ᔴ盐 I ᨨ盐� ᆠ MN 矄�� 犨� NM 矄盐鼀� �ᆠ � ᆠ P
ᆠ � �ᆠ �矄 ᆠ 盐ᨨ 泐 鼀� �ᆠ ᨨᆠ 盐 ᨨ 矄 � 矄 鼀� �ᆠ ᨨᆠ ᨨ 盐ᆠ矄
� 矄 ᨨ盐矄 ᆠᆠ�ᨨ � ᆠ � �ᆠ � 盐ᨨᆠ MN ҈ m I B҈㦨 ᨨ盐犨 矄 a ҈ m I m I ᨨ B҈㦨 ᨨ盐犨 矄㦨 ᨨ盐犨 矄 a a
҈ m I ˆ r k B҈ 㦨 a � 盐ᨨ矄�� 犨�NM m I 矄� 졔矄� 㡰犨 r㦨aᆠᨨ矄 B㦨 pr m I ᨨ B㦨 矄� 㡰 p a ᆠᨨ矄° m I ´ � 犨 矄㡰 ᨨ B㦨 pa 䰘 m I ˆ r k ᨨ B 㦨 a r r r \ Bnet 㦨 B҈ + B ᨨ
҈҈ m I ˆ r k Bnet 㦨 a r v 㦨vᆠᨨ矄 �ˆ 졔v矄� ˆ
v ˆ r v v 㦨 �ˆ + \ 㝐� �ᆠ ᨨᆠ r ҈҈m IQv ˆ r r �ˆ㡰 \ Fm 㦨 Q v ´ Bnet 㡰 㦨 a r Fm r \ 꾀ᆠᆠ�ᨨ 鼀矄矄 a 㦨 m m IQv ҈҈ r ( ˆ �ˆ ) ᨨ a 㦨 ma ��㡰 ᨨᆠ ᨨ⼄盐 ᨨ ᨨᨨ r ᔼ 盐 � ᨨ鼀鼀� �ᆠ � B = B�ˆ ᨨᆠᨨ ᆠ盐 ᨨᨨ�矄 ᨨ r r
ᨨ⼄盐ᨨ ᨨᨨ㦨 M ´ B r M = IA㡰 kˆ ᨨ
pa a 矄� ҈ ᨨ꾀㦨҈a r r \ M 㦨 ҈Ia㡰 kˆ B = B�ˆ r r r r \ t 㦨 M ´ B ᨨ t 㦨 ҈IaB㡰 kˆ ´ �ˆ 㡰 r ᨨ t 㦨 ҈Ia B 㡰 ˆ �㡰 㝐� �ᆠ � 盐ᨨᆠ盐 � 犨�PR m I p m ´ � Ҙ ´ = ´ 㦨 ҈㦨 p p rPR p � � ᨨ ᨨ犨 犨 矄桌 ᨨ A㦨
ᨨ F҈ = F = BIR�$ r r r r \ ᨨ ᨨᆠᨨ ᨨᨨ F = F҈ + F Þ F 㦨BIR �ˆ
a v 졔Q r a P
ᨨ
�꾀 맔
ᔼ꾀 맔 턼
� v㦨҈ 鼀⏌矄 졔
鼀 㦨�鼀
和�鼀�泐 鼀� �ᆠ � 盐ᨨᆠ盐 � 犨�QR Ҙ ´ I Ҙ I � ᨨ ᨨ犨 犨 矄桌 = 㦨 ´ p p
ᨨ鼀� �ᆠ � R 盐ᨨP Q桌 Ҙ Ҙ Ҙ + I = ҈ + I 㡰 B 㦨B҈졔B㦨 p p p � ᨨ ᨨ犨 犨 矄桌 ᨨᆠ � ᆠ 泐 ᆠᨨ
F 㦨qvB矄� q Ҙ 㦨pvB矄� 黨 㦨҈맔҈҈黨맔맔҈�맔 ҈졔I㡰 p \ 맔҈㦨҈맔҈҈黨맔맔҈�맔҈䰘 ҈졔I㡰 ᨨ I㦨꾀 ��㡰 ᨨ � Rᨨ ᨨ鼀� �ᆠ � 盐ᨨ � 犨�ᆠ泐� �ᆠ盐 ᨨ �꾀矄 ᨨ盐 B㦨҈䰘 I졔҈㡰㦨�맔҈䰘
r �矄 ᆠᨨ �矄犨� ᨨ鼀 m ´ � ´ ᨨ r㦨҈鼀 \ �맔҈䰘㦨 p r �.e. � ᨨ �� ҈鼀 ᨨ鼀 �ᆠ�ᨨ ᨨ ᆠ盐 ᨨ盐犨 矄 � 犨 矄 矄ᆠ�泐
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EBD_8341 PS-240
Physics
Magnetism and Matter
19
Topic-1 : Magnetism, Gaus's Law, Magnetic Moment, Properties of Magnet ҈
ᆠ㡰 q = ° t㦨҈㜠鼀B㦨
ᨨ⼄盐ᨨ 鼀� t = MB 矄� q ҈ = M ´ ´ 矄� ° ҈ Þ ҈ = M ´ ´ Þ M = 꾀㜠鼀 ᨨ矄��ᨨ ᨨ 矄⼄盐���盐鼀 q㦨 㡰 ᨨ矄��ᨨ ᨨ 盐 矄⼄盐���盐鼀 q㦨҈ 㡰 㝐� �鼀盐鼀犨ᨨ├⼄盐� ᨨᨨ鼀� ᨨ鼀矄 ᨨ盐 矄⼄盐���盐鼀 DU = U f - U � = -MB ᆠᨨ矄҈° - - MB ᆠᨨ矄°㡰 W = MB = ´ ´ \ W = 䰘 ´ ҈- 㡰 � 㝐ᨨ鼀 ᨨ � �ᨨ ᆠ�ᆠ盐ᆠᨨ�I㦨├�鼀 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᆠ�ᆠ盐ᆠᨨ�M㦨꾀鼀 ᨨ�ᨨ ├� �ᆠ �泐ᨨ ᆠ�ᆠ盐ᆠᨨ� ҈ ⧤쟼 = I w w㦨 �盐矄 ᨨ ᆠᨨ� ᨨ � �泐ᨨ 鼀� 㦨MBᆠᨨ矄f ᨨ鼀 �泐ᆠᨨ 矄�ᨨ
҈ I w = U � - U f = - MB ᆠᨨ矄 ° - - MB 㡰 MB ҈ Þ = Iw ´ ҈ Þ = 㡰w Þ ҈ = w Þ w = ҈
b㡰 B҈ =
�
䰘
黨
µ K p ( d ⏌ )
҈
B =
꾀 �盐鼀ᨨ鼀 盐鼀L㦨Iw㦨 mR㡰w
µ M p ( d ⏌ ) µ M p ( d ⏌ )
B = 㦨҈ \ q = B҈ µ M p ( d ⏌ ) ᨨ q㦨�º
uur
矄盐 � �矄�º ᨨ鼀҈ �犨 B uur v ᨨ矄ᨨ ᨨᆠᨨ �ᆠ�矄 ᨨ ᆠ㡰 ␔ᨨ├ ᨨ ␔㦨鼀· 㦨맔҈맔҈ᆠᨨ矄 ҈�㡰㦨 b㡰 ␔├ ᨨ犨 鼀� �ᆠ �ᨨ鼀ᨨ鼀 㝐 = �꾀 ᆠᨨ矄 θ � 㝐 µ ᆠᨨ矄θ ␔ 犨ᨨ鼀� �ᆠ � 矄� ᆠ� �ᨨ 䰘� ⼄盐���盐鼀犨� 矄ᨨ ҈ ᆠᨨ矄 θ 㦨 ᆠᨨ矄 䰘� 㜠 㡰 㦨 ᆠᨨ矄 � 㦨 ҈� 㦨 \ » ҈҈ r N�A ᆠ㡰 M 鼀�鼀ᨨ鼀 ⏌ᨨ盐鼀㡰㦨 Al N� �㡰҈� 㦨 㦨 㦨꾀鼀 꾀鼀҈ l � ´ ҈ uur ur 㡰 ᨨ � �泐 U㦨 - M B 㦨MBᆠᨨ矄q ᔼ ᆠ矄ᔼ q 㦨҈ U㦨졔MB ᔼ ᆠ矄ᔼᔼ q 㦨黨 U㦨 ᔼ ᆠ矄ᔼᔼᔼ q 㦨ᆠ盐U 㦨졔 矄矄 졔MB㡰 ᔼ ᆠ矄ᔼ q 㦨ᨨ盐矄U㦨 \ ᔼ>ᔼᔼᔼ>ᔼᔼ>ᔼ 㡰 㝐� �ᆠ� 矄ᨨ ᨨᆠ ᨨ鼀ᆠᨨ矄 ᨨᨨ矄 ᔼ 矄� 鼀� 矄 �ᆠ ᨨ鼀矄ᨨ盐 ᨨ ᨨ ᨨ ᨨ盐矄� 鼀� ᨨ鼀 ᨨ ᨨ矄ᨨ盐 ᆠ㡰 �盐鼀ᨨ鼀 盐鼀ᨨ �ᆠ L㦨mvr㦨mrw犨 w㦨pn Qv㦨rw桌 q wq w \ ⼄盐 ᆠ泐n㦨 錐 盐 �㦨 㦨q 맔n㦨 t p p wq qr w ´ pr Þ M㦨 㝐� �ᆠ鼀ᨨ鼀 M㦨�A㦨 p M q wqr \ = = L mr w m qw æ wö q 㡰 ᔴ盐 �㦨 㦨 ⼄盐 ᆠ泐㡰 ᆠ �㡰㦨 çè ÷ø q㡰 = p p t 㝐� �ᆠ鼀ᨨ鼀 æ qw ö w M㦨 �㡰 A㡰㦨 çè ÷ø pR 㡰 = qwR 㡰 p
M qw R q R R = = L mR 㡰 w m (q,m) (q,m) ᆠ㡰 ᔼ 盐 � ᨨ鼀鼀� �ᆠ � 鼀� �ᆠ ᨨᆠᨨ ᆠ盐 ᆠ泐� �ᨨᨨ�矄 ᨨ 鼀� �ᆠ � �矄 �ᆠ盐ᨨ ᨨ 盐矄ᆠᨨ 矄� 犨ᨨ �鼀�ᆠ泐ᨨᨨ矄� 鼀 矄泐 鼀� �Ἐ矄 盐ᨨ 鼀 AB ᨨᆠ dF 㦨 I ( d l ) B 矄� 黨° = I ( d l ) B 犨 矄㡰
\
҈҈
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PS-241
Magnetism and Matter X
X
X
X
dF X
X
B A
X
X
X
X
X
ᔴX
l X
X
X
l
D
X
X
X
dF
X
X
X
X
X
X
ᨨᆠᨨ ADFAD㦨IdlB҈矄� 黨 IdlB҈ B҈ D
d
ᔼ X
҈
҈
҈
҈�
X
X
X
졠 鼀 CD ᨨᆠ dF 㦨I dl㡰B �� ᨨ ᨨ 桌
矄犨ᨨ ᨨᆠ矄犨�ᆠ ᆠᨨ盐 和�鼀�泐 ᨨᆠ矄ᆠ� �ᨨ �鼀�ᆠ泐ᨨᨨ矄� 鼀 矄犨�ᆠ ᆠᨨ盐� � ᆠ ᨨᆠᆠ� � ᨨ ᨨᨨ犨� ᨨ ur B 㡰 ᨨᆠᨨ ᨨ ᨨ㦨 mB҈ ur
ᨨᆠᨨ 矄ᨨ盐 ᨨ㦨 mB N 和� ᆠ 犨ᨨᨨ矄ᨨ 鼀� �ᆠ
犨�ᆠ 盐ᨨ 泐 S � ᨨᆠ矄� � �ᆠ�ᨨ 矄 ᨨ 鼀� �ᆠ � ᆠ矄 ᨨᆠ矄犨矄ᨨ⼄盐 q ne � 㦨 = 㦨ve㦨҈҈맔҈맔҈҈黨㦨҈맔҈꾀 t t 㝐� �ᆠ鼀ᨨ鼀 M㦨�맔A㦨�맔pr 㦨҈맔҈맔҈맔�맔҈҈맔�맔҈҈ 㦨҈�맔҈꾀鼀 ␔�ᨨ � L�矄 � ᨨ鼀ᨨ ᆠ�ᆠ \ �鼀ᨨ ᆠ�ᆠ㦨 � ᨨ 犨� L Þ pr 㦨L Þ r㦨 p pL L \ 꾀ᨨ ᆠ�ᆠ㦨pr㦨 = p p 㝐� �ᆠ鼀ᨨ鼀 ᨨ ᨨᨨ� 犨 �ᆠ ᆠ盐 � ᨨ犨矄 �L ᨨ M㦨�A㦨 p ᆠ㡰 ᨨᆠᨨ AB 鼀� �ᆠ � 盐ᨨᆠ盐 I҈�矄ᨨ � AB \ F㦨IdlB矄� qAB 㦨I d l ´ B ´ 矄� °㡰 = C B
I
O
O' I҈ A
D
ᨨᆠᨨ CD和�鼀�泐 鼀� �ᆠ � 盐ᨨᆠ盐 I҈�矄 ᨨ �DCᆠ盐矄q㦨҈ ᨨ ᨨᆠᨨ DC�矄 ᨨ ᔴ㦨IdlB矄� q㦨Idl矄� ҈ 㦨 ᨨᆠᨨ BC ᔴᨨ 矄� 矄鼀鼀 dl B҈
I
C
B
FBC㦨IdlB҈矄� 黨 Þ dF㦨IdlB҈ 泐 鼀� �Ἐ矄 盐 �ᆠ�ᨨ ᨨ �矄 ᨨᆠ�矄 �ᆠ盐ᨨ ᨨ �ᆠ ᨨ盐犨 矄
I A 泐 鼀� �Ἐ矄 盐 �ᆠ�ᨨ ᨨ �矄 ᨨᆠ�矄 �ᆠ盐ᨨ ᨨ �ᆠ � 犨 矄和� ᆠ ᆠ盐 鼀 矄ᨨᆠ 矄泐鼀鼀�ᆠᨨᆠ盐 I҈ ᨨ ᨨᆠᨨ BC犨�ᆠ ᆠᨨ盐 ᆠᨨ ᨨᆠᨨ AD ᆠ ᨨᆠᨨ ᨨᨨABCDF 㦨
ᨨᆠᨨ BC AD犨�ᆠᨨ⼄盐ᨨ ABCD� ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ᨨ盐OOἘ �矄矄矄 泐 ᨨ矄O ҈ ᆠ 㡰 졠 泐� 盐ᆠ ᆠ�ᆠ � 鼀� ᨨ矄�ᆠ � ᨨ鼀 ᆠᨨ矄 ᨨᨨ矄ᨨ � � 矄 ur ҈䰘 㡰 꾀矄 鼀� �ᆠ � B �矄 �ᆠ �ᨨ � ᨨ x㜠 �矄� xy ᆠᨨ鼀ᨨ 矄ᨨ 鼀� �ᆠ � Bx㦨Bᆠᨨ矄� 㦨 B ⏌ By㦨B矄� � 㦨 B ⏌ Iᨨ
S
Y
R B
t
A
ᨨ
�
P
X
Q
和ᨨ� ᆠᨨ ᨨ鼀 ur uur æ B ö $æ B ö B = �$ ç + ç M = �A㡰kˆ = I L k$ ÷ ÷ è ø è ø
ᨨᨨ⼄盐ᆠ� �ᨨ ᨨᨨ r uur ur æ B $ B �+ t = M ´ B = I L k$ ´ ç è
$ ö = I L B ( ˆ - �ˆ) ÷ø
I L B �矄 �ᆠ ᨨ �� QS ᨨ鼀QᨨS
㡰 ᨨ⼄盐�矄� ˆ �ˆ �ᆠ�ᨨ �.e.ᨨ턼和 ᆠ ᨨᨨ犨�ᨨᨨ盐 �矄矄矄� � ᨨ盐� 턼 和꾀ᆠᆠᨨ � � ᨨ ᨨ鼀ᨨ �ᆠ盐 矄鼀ᨨ鼀 ᨨ � �ᨨ 鼀ᨨ盐 QS ҈ ҈ æ ö IQS㦨 I z = çè ML ÷ø = ML 꾀矄ᨨᨨ⼄盐t㦨Ia
t I L B ´ I B = = I M ML a�矄ᆠᨨ 矄 ᨨ ᨨ鼀
\ a㦨
҈ q = w t + a t w㦨 ҈ ҈ æ I B ö Dt 㡰 \ q = a t = çè M ÷ø
ᨨ
q=
IB Dt 㡰 M
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EBD_8341 PS-242
҈
Physics
�㡰㝐� �ᆠ鼀ᨨ鼀 M㦨IA 犨 I ᆠ盐 盐 ᨨᨨ� 鼀ᨨ�ᨨ ᨨ ᆠᨨ A㦨ᨨ ᨨᨨ鼀 泐ᆠᨨ M㦨IA㦨
w e㜠
R
␔
҈黨
I ´ = ´ Þ I = = 䰘� 꾀⏌鼀 b㡰 ␔ 鼀� �ᆠ � �矄 � ᨨ �鼀� �ᆠ 矄盐矄 ᆠ�ᨨ 盐ᆠ矄鼀� �ᆠ � � ᨨᨨ矄� �ᆠ�ᨨ
矄ᨨ 鼀� �ᆠ � � 矄� ᆠ�泐ᨨ 矄 犨� ᨨ 和ᨨ � � 矄� 鼀� �ᆠ 矄盐矄 ᆠ ├ � 矄� ᆠ�泐�矄 ᨨ 㡰 鼀 鼀� 矄 P㡰鼀 ᨨ 鼀�矄犨� � ��泐 �ᆠᨨᆠ��泐 矄 ᨨ鼀ᆠᨨ矄 T㡰 鼀 ᨨ 鼀�矄犨� ᨨ犨 ��泐 ᨨ犨ᆠᨨᆠ��泐 Þ
�
㡰
㡰 꾀ᆠᆠᨨ � �ᨨᔴ盐�犨 ᨨ鼀� �ᆠ矄盐矄 ᆠ
和
Þ
҈ 㦨�䰘꾀㜠鼀 � � l㦨�ᆠ鼀㡰
m㦨
㦨pm맔B맔矄� 㦨҈맔�맔
c㦨 䰘
´ � ´҈ 㦨맔҈
㡰 㝐� �ᆠ矄盐矄ᆠ���泐 c=
ᔼ
犨 ᔼ =
Topic-2 : The Earth Magnetism, Magnetic Materials and their Properties b㡰 � ᨨ盐鼀ᨨ �ᨨ ᨨ V = ҈- 鼀 �鼀��泐 m r = ҈ 盐鼀ᨨ 盐 矄盐 � � n㦨҈ 㝐� �ᆠ鼀ᨨ鼀 ᨨ �ᨨ ᆠᨨ矄ᨨ ᨨ�
ᔴ
´ ҈ 㦨 c �
ᨨ⼄盐 盐ᨨ ᨨᆠ㦨F맔҈
ᨨ⼄盐���盐鼀F맔҈㦨҈맔�맔 Þ F㦨�黨 ᔼ ᨨᆠ �矄鼀ᨨ ᨨ⼄盐 盐ᨨ鼀� �ᆠ � 犨�鼀ᨨ 鼀� ᔼ犨�矄ᨨ矄ᆠ�� �ᨨ盐 鼀
r r ᨨ矄��ᨨ 犨 �犨 pm B �矄 ᨨ
c
҈
c µ Þ ҈ 㦨
c ᔴ҈ ᔴ
ᨨ⼄盐 盐ᨨ鼀� �ᆠ �
҈
҈ ´ � ´ ҈ - = 黨黨� » � ҈ - 㡰 ᨨ鼀� �ᆠ鼀�꾀ᆠᆠᨨ � �ᨨᆠ盐�矄犨 ҈ cµ T ᨨ犨ᨨ鼀盐矄T҈ T c҈T҈ = cT I 盐 c = B I I \ ҈ T҈ = T B҈ B
쟼
P㦨m맔l㦨҈
V N Þ M = mr - ҈㡰 ´ �V l l
Þ M = 黨黨黨 ´
nh nh Þ Rw㦨 QV㦨Rw㡰 pm p e nh nhe eh = = \M㦨 ´ Q n㦨҈ ᨨ�ᨨ盐 矄㡰 pm pm pm �ᆠ�ᨨ ᨨ 鼀� �ᆠ鼀ᨨ鼀 盐鼀m �ᆠ盐ᨨ ᨨ ᨨ� B nˆ ��㡰 ␔├ ᨨ犨 ᨨ⼄盐 t ᨨ r r r Þt㦨MB矄� q t = M ´B 犨 q�矄 �犨 M B q㦨 he heB e ´ B 矄� ° = \ t㦨MB矄� q㦨 pm pm �ᆠ�ᨨ ᨨ ᨨ⼄盐ᆠ ᨨ盐 泐�� 盐鼀盐
�ᆠ�ᨨ ᨨ ᨨ⼄盐�矄 �ᆠ盐ᨨᨨ 㝐
Þ mRw㦨
Þ M = mr - ҈㡰 ´ N�
eö e ew æ ´ pR Þ M㦨 ´ pR çQ҈ = ÷ è sø p T
҈ \ M㦨 ewR 盐ᆠᆠᨨ � �ᨨᨨ Ἐ矄 ᨨ泐 mvx㦨nh⏌p
M = m r - ҈㡰 ´ N�A
㝐� �ᆠ 鼀ᨨ鼀 ´҈ 㦨⏌鼀 = ᨨ盐鼀 ҈
҈ = ´҈ = ´҈ ´҈ ö æ b㡰 ᔴᨨᆠ��泐 = m = m � çè = ÷ø l
ᨨ犨 c =
ᨨ㦨
҈ � 㦨 맔�㦨꾀⏌鼀 l
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PS-243
Magnetism and Matter
黨
҈ ҈҈ ҈
҈ ҈
҈�
b㡰 � 盐鼀ᨨ 盐 矄
㦨҈盐 矄⏌ᆠ鼀㦨҈맔҈盐 矄⏌鼀 ᔴᨨᆠ��泐ᨨ ᨨ鼀� 㦨҈꾀⏌鼀 ᔴ盐 ᨨ 鼀� �矄 ᨨ鼀� ᔼ㦨䁸 矄� � 㦨 ᔼ ᨨ 100 = 10�맔ᔼ ҈ \ ᔼ = � = ҈ 鼀꾀 ҈ b㡰 꾀桌�矄 ᨨ鼀�盐矄 ᨨ鼀├� �鼀 鼀� 矄 �� ᆠᨨᆠ��泐㡰 桌�矄 ᨨ鼀├� �ᆠᨨ鼀� 矄 矄 ᨨ鼀矄 㡰 㦨l㦨맔�맔҈�ᆠᨨ矄 q㡰맔�㦨䰘鼀 泐�� 盐 ᆠ �鼀ᨨ矄ᨨ ᨨ �ᨨ ᆠ㡰 㝐� �ᆠ � � 矄ᨨ ᨨ� B㦨mn� B = n� ␔ n㦨 盐鼀ᨨ 盐 矄盐 � � 㡰 Þ m B N� ҈� = Þ ´ ҈ = Þ m L ҈ ´ ҈- Þ �㦨꾀 b㡰 �鼀� �ᆠ鼀�矄 � 鼀� �ᆠ � B矄 矄 �鼀� �ᆠ鼀�矄 㡰 ᔼ 鼀� �ᆠ �ᨨ ҈ ᨨᆠ µ ᔼ �� ⼄盐矄�ᨨ ᨨᆠµ
ᆠ 㦨 b㡰 � M҈㦨҈Am M㦨҈Am錐 r =
cm = ҈鼀
㦨҈졔졔
Bnet =
҈
҈ 졠
和
-䰘
҈ ҈ + ҈㡰 ҈㡰
TµI \
ᨨ
T҈
҈ ⏌ 鼀 㝐 ᔴ泐 㦨 ᆠ ᆠ㡰 � 㝐� �ᆠ鼀ᨨ鼀 M㦨䰘맔҈Am 㝐� �ᆠ � 㦨҈T 㝐ᨨ鼀 ᨨ � �I㦨䰘�맔҈⧤�鼀
矄� � T = p
I MB
䰘� ´ ҈-
p ´ ҈ s 䰘 ´ ҈ ´ ҈ ҈
�鼀├ ᨨ҈ᆠᨨ鼀ᨨ矄ᆠ��ᨨ 矄 t㦨҈T㦨p맔҈ 㦨� » �矄 = p
-
=
Topic-4 : Miscellaneous Problems ҈
T
B æ ö ´ çè ÷ø = B҈ B҈ 黨 \ B = = 0.46 㡰 矄� ��鼀⏌ᨨ矄ᆠ��ᨨ �ᨨ
ᆠ㦨p
X
B ᆠᨨ矄 �° B ´ æ ö = çè ÷ø = ҈� B҈ ᆠᨨ矄 ° ´ B҈ ´
鼀
㦨p ( 㝐)
+ ´ ҈ -� = � ´ ҈ - 犨⏌鼀
꾀 �矄ᨨ ᨨ�ᨨ
Bx҈
ᔼ 㝐 ␔ 㝐㦨鼀� �ᆠ鼀ᨨ鼀 ᔼ鼀ᨨ鼀 ᨨ � � 㦨 鼀� �ᆠ �
和
MR = MR
Bx
㦨 p
㡰 � 㝐㦨맔҈꾀鼀 㦨 㦨맔҈鼀 m 㝐 쟼 矄鼀� �ᆠ � 㦨 p p ´ ҈ -䰘 ´ ´ ҈ = ´ @ 盐矄矄 p ´ ҈ 㡰
I = MR +
T
=
㡰 I҈ I 鼀ᨨ鼀 ᨨ � �� �矄 矄ᆠᨨ ᆠ矄 矄ᆠ�泐 I҈ = MR
I҈ = I
Topic-3 : Magnetic Equipment ҈
T҈
I mgd
MR = MR I ␔ T㦨 p MB x
T҈ = T
ᨨ 盐矄㡰 \
和
r
r
m M҈ + M 㡰 = + BH p r
�鼀�ᨨ T = p
㡰 ᔼ�矄ᆠ ᨨ鼀 �� � 犨� � ᆠ矄ᨨ 鼀� �盐 ᨨ 鼀� �ᆠ � B㡰 ᆠ��ᆠ 鼀盐TC B㡰 ᆠ矄矄 ᔼ B > B҈ ᨨ 鼀盐 犨 �ᆠ 矄�矄 ᆠᆠᨨ鼀矄 ᨨ矄 ᨨ盐 矄矄 b㡰 � 㦨TC㦨҈⧤ 㦨䰘�TTC㦨䰘�⧤ ␔� � ᆠ矄� TC ᆠ矄矄 \꾀㦨�TTC矄 ᨨ盐 犨 䰘�⧤ ҈⧤
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EBD_8341 PS-244
Physics
Electromagnetic Induction
20
Topic-1 : Magnetic Flux, Faraday's and Lenz's Law ҈
b㡰 ᔴ矄 㡰␔ 鼀� �矄 � �犨� ᆠᨨ 矄 矄 盐 f㡰 犨� ᆠ � 鼀 �矄 � 盐ᆠ 矄ᨨ � ᨨ鼀 犨� ᆠ � ᨨ矄�� �ᆠ�ᨨ ᔴ矄 b㡰␔ 鼀� �矄ᆠᨨ鼀泐� 矄� 盐 f㡰犨�
ᨨ ᆠ � 矄ᨨ � ᨨ鼀 犨� 矄 ᨨ犨 盐 ᆠ�ᨨ ᔴ矄 ᆠ㡰␔ 鼀� �矄鼀├� �ᨨ ��� 盐 f㡰 犨� ᆠ � 鼀 �矄 � 盐ᆠ � ᨨᨨ矄� �ᆠ�ᨨ
矄ᨨ � ᨨ鼀 犨� ᆠ � �� �ᆠ�ᨨ � 矄 �ᆠ�ᨨ 㡰 꾀矄犨├ ᨨ犨鼀 e = NABw ᆠᨨ矄 wt N㦨҈ 꾀� ᨨ犨
�
㡰 꾀矄 鼀� �ᆠ � � 矄 ᨨ鼀 ᆠᨨ矄ᨨᨨ ᆠ泐 鼀� �ᆠ � � ᆠ� �鼀� �ᆠ 盐 ᨨ盐� �
��ᨨ f�㡰 鼀盐矄 矄矄� � ᨨ盐� ᨨ盐 ��ᨨ 和� ᆠ 盐 � 犨ᨨ ��ᨨ 矄 � ᨨᨨ矄� ��ᨨ \ f�㦨f
㡰 꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ I t㡰㦨It ҈t㡰 \ I㦨ᔼIt f=B.A f㦨 m nI㡰맔 p 㡰
Q B = m ᔼ and A = pR 㡰 f VR = dt VR = m npR I I t 㡰
A B w æ ҈ö A B w ᆠᨨ矄 wt e > 㦨 < > 㦨 çè ÷ø R R R
ᨨ�ᨨ犨ᨨ矄矄� ᨨᨨ 盐ᨨᨨ盐 � �
=
㦨
҈ s b㡰 ␔ �� �鼀�ᨨ T㦨҈矄 p p = \ 꾀 �盐 ᨨᆠ�泐 w = ҈ � 㝐� �ᆠ 盐 f 㡰㦨BAᆠᨨ矄wt d f = BAw 矄� wt = BAw 矄� ( wt ) 쟼鼀 � 盐ᆠ E = dt p ᔼ 盐ᆠ 鼀 e �矄鼀 �鼀盐鼀犨 wt㦨 p Þ t = = � 矄 p � ᨨ� 盐ᆠ 鼀 ᨨ鼀� �鼀盐鼀� ᨨ p wt = p Þ t = = � s p � \ᔼ 盐ᆠ 鼀 �矄 ᨨ㦨�矄 㡰 Q㦨BA 㦨 mn�㡰A 㦨m n kt eat㡰A d dQ -a t e= = -m nAk te 㡰 dt dt Þ VR = t =
䰘
= -m nAkt -҈㡰e- at + e -at ´҈桌
꾀A㦨ᨨ X ᨨ D㦨 ҈맔맔꾀ᨨ � �㡰ᆠ鼀
҈ æ ö = ç ´ ´ ´ ÷ ᆠ鼀 㦨�맔҈鼀 è ø dB \ e� 盐ᆠ = A = � ´ ҈ ´ ҈ = � ´ ҈ V = �mV dt
黨
= - m nAk e -at ҈ - t 㡰桌 e - m nAk -at �= = e ҈ - t 㡰桌 R R 꾀t㦨�Þ b㡰 꾀ᆠᆠᨨ � � ᨨ 泐矄 犨 ᨨ ᆠᨨ鼀� �ᆠ - f � 盐ᆠ�ᨨ = � � ҈� ´ = � Þ ´ = � ᨨ = ҈
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PS-245
Electromagnetic Induction ᔴ � � �泐ᨨ � 盐ᆠ ᆠ ҈ ҈ � D쟼 = �҈ � = ´ ´ � ҈ 㡰 � = ´ �� = 䰘� ᨨ 盐矄桌 ᆠ � 턼 = Df = ҈ 꾀 - 㡰 = ҈ ´ � ´ ҈ - � ҈ ҈
(
҈
҈
)
dq dt dq ҈ d f BA ҈ = \ Þ ò dq = ò d f \ q = dt R dt R R ҈黨 㡰 �鼀ᆠᨨ 矄 㦨 ᔴ ᔼ鼀 ᆠ㦨
҈ � ´҈- 㡰 - 㡰 㦨҈맔҈ ҈ ᨨ ᨨ�ᨨ 鼀ᆠ 矄 和ᨨ� 矄 ᨨ盐 ᨨ 盐矄 ҈҈ 㡰 꾀ᆠᆠᨨ � � ᨨ 泐Ἐ矄 犨 ᨨ ᆠᨨ鼀� �ᆠ df � 盐ᆠ�ᨨ e = dt 꾀矄ᨨe 㦨�R df Þ ò d f = R ò �dt \ �R = dt 㝐� �盐 ᨨ ᆠ �� 盐 f㡰㦨R맔盐 ᆠ盐 矄 �鼀 � ҈ ҈ ᨨdf 㦨 ҈ ´ ´ ´ ҈ 㦨�␔ ҈ b㡰 쟼ᆠ�ᆠ 盐 �矄�� 泐 f㦨꾀
æ ҈ ö ÷ wC ø
\ R ҈� 㦨 R + ç è
\ ᔴ㦨
턼 㦨 㜠 ⏌ t 㡰
�㡰 ᨨᆠᨨ�C҈ᨨᨨ 盐 矄N҈㦨� �盐矄r㦨҈ᆠ鼀 ᨨᆠᨨ�Cᨨᨨ 盐 矄N㦨 �盐矄R㦨ᆠ鼀
d f æ m N ö d� æ m N҈ N pr ö =ç pr N҈ = ç ÷ ÷ ҈t - 㡰 dt è r ø dt è R ø 꾀t㦨҈矄
æ m N N pr ö m N҈ N pr e=ç ҈ = ÷ R R è ø
=
p㡰҈ -䰘 ´ ҈ - ´ � ´ - p ҈
= ´ p ´ ҈-䰘 ´ ҈ ´ ҈ ´ ҈- - = ´ ҈ ᨨ㦨鼀㦨
҈
Þ x = �
҈�㡰 B㦨맔҈� R㦨҈ᆠ鼀㦨҈鼀
p p = T r r 盐 矄 盐 ᆠ�ᨨ ᨨ �鼀 f = B × A = AB ᆠᨨ矄 wt 㡰
㡰 꾀矄犨├ ᨨ犨㝐� �ᆠ 盐 f = B A
w=
m 㡰 ´ ҈ - 㡰
҈䰘
= ´ ҈ 鼀矄 \ ᔴ㦨鼀矄 w �
e=
㡰 꾀ᆠᆠᨨ � � ᨨ 泐矄 犨 ᨨ ᆠᨨ鼀� �ᆠ � 盐ᆠ�ᨨ Ld� ᔼ 盐ᆠ 鼀 e 㦨 dt æ � ö � ´ ҈ � � = L ç = 㦨 ҈䰘 ÷ Þ L = è ҈矄ᆠ ø 㡰 ᔼ 矄� 矄 � �矄 � 泐�쟼µ犨� ҈ �矄 ᆠ ᨨ盐矄� �矄 ᆠᆠᨨ � � ᨨ 쟼 µ ᆠ ��ᨨ �矄 矄 ᨨ犨 泐ᆠ盐 㡰
´ p ´ ҈ - 㡰 㡰 + ҈�㡰 桌 졠 矄ᨨ� � 㦨黨҈맔҈҈҈㦨黨맔҈҈҈犨 c I ҈ 㡰 鼀� �ᆠ � �矄 a I � ᆠ矄� � � ᨨ犨 犨 ҈ �ᆠ�ᨨ ᨨ ᆠᆠᨨ � � ᨨ 矄犨 ᆠ盐 I҈犨� ᨨ犨� �ᆠ�ᨨ ab I � �ᆠ�ᨨ dc
dI = ҈t - 㡰 dt æ m IN ö f矄鼀 = BA = ç ÷ pr 㡰 è R ø ᔼ 盐ᆠ 鼀 � 矄鼀ᆠᨨ�
f p - ⏌ t = t ¥ p e = ò = t
҈�
I = �t - t + 㡰 Þ
ᔼ 盐ᆠ 쟼鼀 e =
҈
æ ҈ ö R + ç è wC ÷ø
� 㦨 R ҈�
=
҈
e ҈ df Þ �= R R dt
盐 � =
p æ ö ç 矄� - ÷ è ø
f = p - ⏌ t
b㡰 ᔴ盐 � 盐ᆠ � =
-d f = ABw 矄� wt 㡰 dt 㝐 盐ᨨ 쟼鼀 㦨ABw㦨pRBw p = ҈ ´ ҈ ´ ҈ ´ ´ ҈-� ´ 쟼鼀 � 盐ᆠ e =
d b
ᆠ㡰 ᨨ�泐ᨨ 鼀 犨�ᨨᨨ矄�犨 鼀� 矄 矄 ᆠᨨ� 졠 泐� ᆠ㡰 矄 ᨨ犨矄 矄 ᆠ ᆠ�矄�ᆠ
= ҈� ´ ҈- = ҈� m
҈㡰 � dI㦨�㦨�꾀 dt㦨�鼀矄 ᔼ 盐ᆠ ᨨ� E�nd㦨҈ 和 㜠� 盐ᆠ ᆠ㦨 䁸
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EBD_8341 PS-246
Physics
L � 㡰 Df Þ ҈ = Dt � ´҈- Þ㦨҈㦨҈鼀 ҈䰘�㡰
Dq =
矄� � E� =
Df L�҈ R R
��㡰 x = x
Df = f f f� =
ò
x = x
µ I µ I l ldx = ᔼ ( ) p x p
和ᨨ ᨨ鼀쟼⼄ ��㡰ᆠ � ᨨ犨 ᨨ盐� 矄�矄 ᆠ盐ᨨ �鼀t㦨 犨 ᆠ盐 �҈�矄
Dq =
҈ é µ I l ù ᔼ ( ) L�҈ ú ê R ë p û
� 㦨 �et⏌tL
L � � = ҈ � = �҈ t = ( T T ) = T t L = R 和盐矄�盐� � 矄 盐矄 � 㦨 �et⏌tL L T � = � e T ⏌ L ⏌ R㡰 Þ t L = = R In
盐 ᨨ盐� ᨨᨨ꾀ᔴ쟼 꾀
r r ˆ ˆ + �├㡰 ˆ f = 꾀 = �ˆ + ├㡰 ��
�
Þf= 맔�㡰졔 맔�㡰㦨҈䰘�犨 ␔ 矄ᨨ盐ᆠ�矄 矄犨�ᆠ ᨨ ᆠ盐 ᨨ �� ᆠ矄矄 ᨨ ᨨ � 盐ᆠ ᆠ盐 ᨨᨨ矄矄 ᆠ盐矄 ᨨᆠ �矄 Ἐ矄犨 ᨨ � 盐ᆠ ᆠ盐 犨� ᨨ鼀 ᨨ�� 矄 ᆠᨨ� ᨨ 鼀 犨��矄 ├ 矄�ᨨ 泐� ᨨ 㜠 盐 � ᨨ鼀鼀� �ᆠ � 꾀 ᨨ� 盐ᆠ 鼀 ᨨ ᨨ�
ᆠᨨ� 鼀� �ᆠ 盐 ᨨ盐� ᆠᨨ�鼀盐矄 ᆠ � 盐� �矄 ᆠ矄鼀� �ᆠ 盐 � ├ 犨� ᆠᨨ��矄 ᨨ ᆠ �� � T盐犨 犨��矄� 鼀ᨨ�ᨨ ᆠᨨ 矄
ᨨᆠ�泐� �ᆠ�ᨨ ᨨ 鼀ᨨ�ᨨ 꾀ᆠ � �ᆠ鼀ᨨ� �� 鼀� �ᆠ � � ᆠ矄
r r ur ᨨᆠ = ⼄ n ´ 㡰
䰘
矄ᨨᆠ ᆠᨨ � ᆠ矄 ᨨᆠ ᨨ ᆠᨨ 矄 犨� 鼀ᨨ ᨨ犨 矄 ᨨ ᆠ� � 鼀 犨 犨ᨨ 矄 ᨨ 矄�� ᆠᨨ 犨� ᆠ㡰 ᔼ ᆠ盐 �矄 ᆠᨨ 矄 鼀 � 盐ᆠ � ᨨᨨ ᨨ쟼鼀 犨�� 盐ᆠ � I ᆠ�ᆠ盐犨�ᨨᨨ犨 盐 ᨨ盐� � ᆠ �矄 犨� �鼀 ® Df B e=Dt ® 犨 ᆠ盐 �矄 ᆠᨨ 矄 B 盐 ᆠ �� � ᨨ盐� � Circular 犨� ᨨ loop 꾀矄ᨨ � ᆠ盐 ᆠ矄矄 矄 泐 鼀 � 盐ᆠ � ᨨᨨ �矄 ᨨ ␔ ᆠ盐 �矄 ᆠ矄� � 矄 泐 ᆠ �� 盐 ᆠ矄� �� 犨 矄㡰ᨨ �� ᨨ 犨��矄⼄盐ᨨ ᆠ �� 盐 ᆠ矄� �ᨨ盐犨 矄㡰 ᨨ ᨨ 犨�矄盐ᆠ Df ᨨ盐� ᆠ�ᆠ盐 ᨨᨨ �矄 ᨨ d� df 㡰 ᨨ � �ᨨ 鼀ᨨ � � dt dt 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 矄ᆠᨨ 犨 df d� �R L = dt dt df d� = �R + L �㡰 ᨨ dt dt b㡰 쟼 矄矄�ᨨ ᨨ ᆠ � ᨨ盐� R ᨨ鼀t㦨ᨨ t㦨T
d f = �Rdt + Ld� ( ᨨ鼀 ⼄ ( � ) ) ᔼ �� � 犨 �
Df = R Dq + L�҈
黨
�ᆠ�ᆠ盐� �矄 ᆠ � 矄 �泐� ᆠ矄� � 盐 � ᨨ鼀 鼀� �ᆠ � �ᆠ � ᨨ ᨨ ᨨ鼀 ᨨ� � 盐ᆠ 鼀 犨�ᨨ 盐ᆠ � AEFD矄犨 矄� ᆠ�ᆠ盐�EBCF矄盐ᆠ ᆠ盐 ᨨ犨� �� ᨨᨨᆠ矄鼀� �ᆠ� 矄ᨨ ᨨᆠ� 盐犨 �ᆠ�ᨨ
ᨨ ᨨ 㡰
ᨨ ᆠ盐 矄 ᨨ盐 ᨨ犨 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ � ᨨ ᨨᨨ矄 ᔼ 盐ᆠ 鼀 � ᨨᨨ AEFD
df d dB = - BA = - A = -҈ ´ ҈ = -҈ dt dt dt ᔼ 盐ᆠ 鼀 � ᨨᨨ EBCF df d dB = - BA Ἐ = - A Ἐ e㦨 㦨�맔҈㦨� dt dt dt 꾀泐� �盐 ᆠ�ᨨ 犨 F犨�ᆠ盐 � ᆠ FE �҈ �㡰 꾀泐� �⧤�ᆠ ᨨ 矄 犨 � ᨨᨨ EADFE ҈맔�҈҈맔�҈졔҈҈맔�҈҈ �҈�㡰㦨 Þ �҈�㦨҈ �㡰 꾀泐� �⧤�ᆠ ᨨ Ἐ矄 犨 � ᨨᨨ EBCFE 졔��҈�졔҈�졔��҈ �҈�㡰㦨 �҈졔�㦨� ��㡰 和ᨨ� �⼄ �㡰 ��㡰 䰘 A ҈҈�҈㦨� Þ �҈㦨�⏌҈҈ = A 꾀矄ᨨ҈҈�㦨 Þ �㦨⏌҈҈A = 䰘 \ᔴ盐 � 矄�鼀 AE = �҈ = A 錐 BE = � = A ҈ A EF = �҈ - � 㡰 = e= -
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PS-247
Electromagnetic Induction
҈
Ӹ矄 ␔ ᆠᨨ� A 鼀ᨨ矄 ᨨ犨 矄 B 盐鼀 ᨨ 鼀� �ᆠ � 矄 ᨨ ᨨᆠ 矄矄� � ᨨ盐� B ᆠ �矄
ᨨ � 盐ᆠ 鼀 ᆠ � 盐ᆠ ᆠ盐 �矄 ᨨ 盐ᆠ � B
�ᆠ�ᨨ ᨨ ᆠ盐 � B犨�矄盐ᆠ 矄 ᨨ ᨨᨨ矄 � ᆠ � � B ᨨ犨�� ᨨᨨ矄� �ᆠ�ᨨ ᨨ A
�ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 � ᨨᨨ 矄 矄 ᨨ犨
ᨨ犨 꾀矄 ᆠ盐 � ᆠ矄矄 盐鼀 ᨨ 鼀� �ᆠ � 矄 ᨨ ᨨᆠ 矄矄� � ᨨ盐� ᨨᨨ � ᆠ矄矄 � ᨨ盐犨 �ᆠ�ᨨ ᨨ ᨨᨨ矄 �矄 ᆠ � ᆠ盐 犨� ᨨ犨 � ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 矄 Ἐ矄犨
F = -kx - �IB = -kx -
B I ´v R 꾀ᨨ 矄矄�ᨨ 矄 ᨨ犨矄 � �矄 ᆠ矄 ᨨ 鼀 ᨨ矄ᆠ��ᨨ 矄ᨨ �矄 鼀�盐 ᆠ �� 泐
F = -kx -
Þ A = A e
Þ t=
m I Þ f = l dr pr
䰘
V r
�
m N A L ᆠ㡰 ᨨᆠ ᨨ 矄� 犨 � �矄 矄ᆠ ᨨ矄��ᨨ x ᨨ鼀鼀 ᨨ矄��ᨨ �矄
҈ 桌
ᨨ犨�� ᆠ矄 ᨨ� 和ᨨ矄 � 盐ᆠ ᆠ犨�ᆠᨨ鼀�鼀矄 ᆠ㡰
㝐� �ᆠ鼀ᨨ鼀 㝐㦨ᔼ꾀 턼㦨r
dQ w p 㝐㦨 ᔼ맔꾀 ᔼ㦨
w r x p x dx 㦨 p l
e � ´҈- = ; ҈䰘0 m 꾀 R
㡰 ᔼ 盐ᆠ ᆠ 㦨
µµ矄㦨⏌Þµ
´ + ҈䰘 㦨 + ҈䰘 ; W +
ᔴ盐 �㦨
m ; 矄 k 矄ᨨ ⼄盐� 盐鼀 ᨨ ᨨ矄ᆠ��ᨨ 矄 ҈ N= = � 㡰 ᔼ 盐ᆠ 鼀 e㦨vl 㦨맔҈맔�맔҈㦨҈�맔҈ ᆠ㡰 꾀矄ᨨ � ᨨ 犨�犨�鼀� ᆠᨨ 矄 㦨 㡰 㦨ᨨ盐 矄㡰 � ᨨ 犨� � �㦨 㡰
æ ö 㦨m ç ÷ è ø
dr
e m Ivl = × R p Rr b㡰 盐 ᨨ 鼀ᨨ�ᨨ ᨨ 鼀� ᨨ �矄ᆠ 鼀ᨨ矄 ᨨ盐� 鼀� �ᆠ 盐 盐ᨨ犨 �ᆠ 쟼㝐 �矄 � � 泐 ᆠ盐 矄 � 盐ᆠ
矄ᆠ盐 矄矄盐ᆠ �ᨨᨨ矄矄 �鼀ᨨ�ᨨ
矄ᨨ �矄ᆠ犨�ᨨ� 矄鼀 �ᆠ�ᨨ 矄 �ᆠ�ᨨ ᨨ 鼀� Ἐ矄 鼀ᨨ�ᨨ
�矄 盐矄 �矄 ᆠ 꾀�ᨨἘ矄 �矄├ ᆠ 犨矄 �矄ᆠᨨ � ҈ 泐꾀�ᨨ b㡰 ᔼ 盐ᆠ 鼀 e㦨Bvl㦨҈맔҈맔�㦨�맔҈ 쟼⼄盐� 矄�矄 ᆠ R㦨
´ � ´ ҈- ´ ҈ ҈´ ҈
㦨 �鼀ᨨ 犨�㡰 矄 � 盐ᆠ ᆠ 㦨 m 꾀l
ᔼ 盐ᆠᆠ盐 � ᨨᨨ � =
=
A 桌 e
�鼀�ᨨ T = p
m I pr 㝐� �ᆠ 盐 ᨨ矄鼀 �矄ᆠ鼀 dr, f = B × A = Bldr Q A = l dr B.A㦨BAᆠᨨ矄 桌
I
m
矄 ⼄盐矄�ᨨ A =
æ B I ö çç R ÷÷ è ø � m㦨�맔҈├� 㦨҈
l㦨҈鼀 㦨҈W ├㦨�
B=
d f m Il dr = × dt pr dt m Ivl × Þ e = p r
bt m
bt
㡰 㝐� �ᆠ � �矄 ᆠr ᨨ鼀 犨�
쟼鼀 e =
-
A Þ = A e m e
Topic-2 : Motional and Static EMI and Application of EMI ҈
BIv ´ IB R
黨
l
r Þ㝐㦨 l n p ò x dx
p 㦨 nrl b㡰 ᨨ鼀 泐矄 犨 ᨨ ᆠᨨ鼀� �ᆠ� 盐ᆠ�ᨨ d f d BA㡰 d Bll 㡰 Bdl ´ l = = BVl e= = = dt lt dt dt
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EBD_8341 PS-248
Physics ҈
B l V æ BV ö l B 㡰 = 꾀矄ᨨ F = �lB = ç ÷ è R ø R 꾀⼄盐���盐鼀
®
B
l
BlV mgR 矄� q mg 矄� q = ÞV = R R B l ҈
q
b㡰 ᔼ �� ⼄盐矄�ᨨ ᔴ盐 ᨨ犨� � ᨨ盐� 犨�ᔼ㦨҈꾀
Speed of the frame, v = 10 ms҈ 和� ᨨ 矄⼄盐ᨨᨨl㦨҈ᆠ鼀 �矄 ᆠ ᨨ 矄⼄盐 鼀 ᨨ鼀 ᆠ盐 ᆠ泐� � 犨�矄 㦨҈ᆠ鼀 ␔ ᨨ � 鼀 � 盐ᆠ 㦨䁸 꾀ᆠᆠᨨ � � ᨨ�ᨨ㜠和矄 犨
=
҈҈
m ᔼ 矄� q p´҈ 㦨 p p
-䰘
´
҈´ ҈
҈䰘
(҈ )
µ I 矄� a + 矄� b㡰 pR 꾀泐� � ᨨ ᨨ鼀盐 ᨨAB ᨨ � � � 鼀� �ᆠ � ᆠ O ᨨ ᨨᨨ µ I 矄� �° + 矄� �°㡰 㦨 µ I B= p L ⏌ 㡰 pL I I ᨨ B µ ᨨB = K ( K = ᆠᨨ 矄 ) L L
㦨҈
ᔼ 盐ᆠ 鼀 㦨 㦨҈맔҈҈맔҈㦨҈m 㡰 � 盐ᆠ 鼀 w l l
l
l㡰 l㡰 桌 �Bl w = l ҈ 㡰 ␔ ᆠ盐 ᨨ犨矄� 泐ᨨ ᆠᨨ�矄 盐 � ├ 犨� ᨨ ᆠᨨ��矄鼀 �鼀盐鼀犨 矄盐 ᆠᨨᆠ� 盐 �矄鼀 �鼀盐鼀ᔴ泐 盐 � ├��矄鼀 �鼀盐鼀� ᆠ矄 㡰 盐ᨨ 矄� �鼀 ᨨ 犨ᨨᆠ�ᆠ盐ᆠᨨ�矄 e=
ò wx㡰 Bdx = Bw
B A 꾀 鼀ᨨ�ᨨ 鼀 㦨 �矄 � 盐ᆠ � ᨨ 졠 犨 ᆠ 矄泐 ᨨ � � ᆠ �矄 � 盐ᆠ l v 犨 犨ᨨ 矄ᨨ ᨨ 꾀 犨� 꾀 �� ᨨ � ᨨ犨 B ᨨ � 盐 ᨨ �矄 ᨨ � � ᆠ �矄 ᆠ�ᆠ � � ᨨ b㡰 ␔├ ᨨ犨鼀� �ᆠ � 盐ᨨᆠ盐 ᨨ犨� �� 犨�ᨨ � � � 鼀� �ᆠ
B=
-҈
-҈
b㡰 ᔼ �矄ᆠ矄 �矄 ᆠ�ᆠ � � ᨨ
㝐� �ᆠ 盐 � ├ 犨� KI l 矄鼀 ᨨᨨ f = B A = L
꾀 f = ME
犨 M 㦨 㝐盐盐 � 盐ᆠ ᆠ KI l l \M µ \ MI = L L
df �矄 ᨨ dt ᆠ� 盐ᆠ 鼀 ᨨ ᆠ盐 � ᨨᨨ �矄 ᨨ ҈黨 ᆠᨨ� �矄 ᨨ├ � ᨨ 犨ᨨ� �ᆠ ᆠᨨ�矄 ᆠᨨ
ᆠ �
҈
㡰 和� ᆠ ᨨ ᆠ �ᨨ 鼀� �ᆠ 盐
L ⏌ ´ L ⏌ L ҈ ´ ҈ = = 㦨�맔҈ L⏌+ L⏌ R⏌´R⏌ R = = = ҈� W \ Req㦨 R ⏌ + R ⏌ Leq � ´ ҈ -
�鼀ᆠᨨ 矄 t㦨 㦨맔҈矄 = R q ҈� E ҈� = 和 泐矄ᆠ盐 ᨨ盐� 泐 I㦨 㦨҈꾀 Req ҈� T盐꾀ᆠᆠᨨ � �ᨨ 鼀� �Ἐ矄 盐犨 ᆠᨨ 盐ᆠ�ᨨ ᨨ 꾀 B 鼀ᨨ矄ᨨ 㜠 �矄� 盐 � ᨨ鼀 Fm 鼀� �ᆠ � ᨨ� � � � e v ᨨ矄�� z㜠 �ᆠ�ᨨ ᆠᨨ 矄 犨� � ᆠ ᨨᆠ ᨨ犨 矄 ᆠ 꾀 犨� 졔 졔 A ᆠᨨ鼀 ᨨ矄�� 盐 ᨨ �ᆠ� ᆠ泐 ᨨ ᆠᨨ 矄 꾀 \ Leq㦨
҈
҈
҈�
㡰 쟼ᆠ�ᆠ � 犨� A B � 盐ᆠ 矄ABCD 鼀ᨨ矄�
ᨨ AD BC 鼀�ᆠ 矄⼄盐 ᨨᨨ 鼀ᨨ矄 v � �矄ᨨ犨 犨� ᨨᆠ�泐 v� 盐 � ᨨ鼀鼀� �ᆠ � C �ᆠ盐ᨨ ᨨ D 矄⼄盐ᨨᨨAD BC �ᆠ盐 ᨨ ᨨᆠ�泐矄 犨 矄 �ᆠ盐 ᨨ � � 㡰 矄� �ᆠ盐 㜠�鼀⼄盐�ᨨ � LR ᆠ�ᆠ盐� I㦨I ҈et/t㡰 V L 盐 I = t = R R - ù V ҈ é \ I = ҈ - e - Rt ⏌ L 㡰 = ҈ - e -t ⏌ ´҈ ú 㦨҈ �� 㡰 ê R ë û \ t㦨黨䰘맔҈矄 »҈鼀矄 㡰 ᔼ 盐ᆠ 鼀 ᨨ 盐ᆠ ᆠᨨ矄矄 MNQ 犨� 矄鼀 矄 � 盐ᆠ 鼀 ᨨ 盐ᆠ � 矄�� 犨�MQ \ eMNQ =eMQ㦨 Bvl 㦨Bv맔R犨� 턼 �� ᨨ �
҈
é æ y ö f ù ur b 㡰 � B = B ê҈ + ç ÷ ú Kˆ V = V�ˆ êë è L ø úû ␔ ᨨ犨 ᆠᨨ 矄� � � � 矄鼀 � ᨨ 犨� 泐 �矄 ᆠ 泐 ᨨ鼀 ᨨ���
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PS-249
Electromagnetic Induction 쟼鼀 � 盐ᆠ ᆠᨨ矄矄 �
泐 Df = ( 泐 )
é æ 泐 ö ù Df = ê҈ + ç ÷ ú 泐 è ø ú ëê û
泐
泐
\ ᔼ 盐ᆠ 鼀 ᆠᨨ矄矄 ᆠᨨ鼀 ᨨᆠ�ᨨ
é b æ 泐ö ù é ҈ ù Df = ò ê҈ + ç ÷ ú 泐 = ê҈ + è ø β + ҈ úû ë úû ë ê ᨨ β = Df = ᔴ泐DfµL
BV L ᨨ矄�� 犨�ᨨ � ᆠ ᨨ �泐㦨
盐ᨨ Df 犨�鼀� 矄鼀矄�矄 ᨨᆠ�ᨨ ᨨ 泐㜠 �矄�矄矄鼀L矄 ᨨ �ᨨ盐矄 ᨨb㦨Df㦨
b㡰
�㦨
e 㦨 �㡰ᔴᨨ盐 ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
R 犨 � � � ᨨ 犨 ᆠᨨ鼀泐 � 矄� ᆠᨨᆠ├犨�矄犨 � �� �桌
㦨�㦨
��㡰 ᨨ犨 犨 � � � �� � ᨨ犨 ᆠᨨ鼀泐� 矄� 桌
\ - 鼀
=
\
ᨨ - 矄ᆠᨨ e ᨨ 泐 = 鼀 d� ⏌ dt 㡰 ҈ ᆠ㡰 쟼 �泐 U = L� ᨨ盐 U ᨨ 泐 = \ L= 鼀㡰 � ҈ 㡰 U = L� = � Rt \ L㦨 ᨨ 泐㦨ᨨ 鼀㜠矄ᆠᨨ � ᆠ㡰 \ L = -
鼀
d�҈ dt
= ᆠᨨ 矄 = m say㡰
���㡰 \ 㦨 鼀 ᆠ矄矄 ᨨ鼀 㦨ᨨ 㦨鼀� 矄 ᆠᨨ 矄 ᨨ 㦨ᨨ 㦨�� ᆠ矄矄 ᨨ鼀 㦨ᨨ 㦨 ᆠ� 㡰�矄ᆠᨨᆠ桌 ᨨ鼀 �㡰 ���㡰
é ù ê ú �㦨 ë 鼀 û � ᆠ矄矄 ᨨ鼀 㦨ᨨ 㦨�ᆠᨨ鼀矄 ᨨ ᨨ鼀 㦨ᨨ 㦨�ᆠ �矄 �ᆠ�ᨨ ᆠ矄矄 ᨨ鼀 㦨ᨨ 㦨桌 ᆠ � 㡰 �矄 ᆠᨨᆠ 㡰
盐 矄矄� � ᨨ盐� � �盐 犨� � � ᆠ盐 ᨨ犨矄 ᨨ盐� � � � �泐ᨨ � ᆠᨨ 盐ᆠ� � 犨� 꾀 ҈ m � x df㦨 ò px ´ pdx dx ҈ᆠ鼀 x x m� f= = M� ҈p � � m \ M = ҈p
æ d� ö ÷ dt ø
㡰 ᔼ 盐ᆠ emf e = - L ç è
꾀ᆠᆠᨨ � � ᨨ ⼄盐矄�ᨨ ᨨ ᆠ � ᨨ ᆠ盐
ò = - 鼀 ò Þ㦨
m d� m = ´ ҈ = V p dt ҈p
犨� ᨨ � 盐ᆠ 鼀 � 犨� ᆠ盐矄 �矄 ᨨᆠ � � 鼀� �ᆠ 盐 盐 ᨨ ᨨ�ᨨ ᨨ ᨨᨨ 꾀矄 ᆠ盐 � � �盐 犨� �矄 ᆠ矄� � � 盐ᆠ ᆠ盐 � 꾀�矄� 矄鼀 �ᆠ�ᨨ 矄 ᆠ盐 � 泐ᨨ 盐矄 ᨨ � �盐 犨� ᨨ ᨨᆠ 犨�盐矄� bᆠ 㡰 犨 f 㡰 ᔼ 盐ᆠ ᆠ L = ᨨ 泐 = � 鼀 ᔼ 盐ᆠ 鼀 � 犨�e㦨 M
æ dI ö -҈ çè dt = ҈ AS �� ÷ø
꾀 ᆠᆠᨨ � � ᨨ 泐Ἐ矄 犨
d� ᔼ 盐ᆠ 鼀 V҈ = - L҈ ҈ = - ´ ҈ - ´ m dt
V L ´ ҈ ҈ = = = V҈ L҈ ´ ҈ 和� ᆠᨨ犨 �� ᨨ 犨ᨨᆠᨨ�矄 ⼄盐 \ V҈�҈㦨V� � V ҈ Þ ҈ = = � V҈ ҈ 쟼 �泐 W = L�
\
�㡰
҈ L� W ҈ = = ´´ = \ W҈ ҈ L � ҈҈ æ yö � b = B ç ÷ kˆ 矄� ᨨ 矄⼄盐ᨨᨨ㦨 a è aø 和盐ᨨ矄t㦨y㦨 t㦨ty㦨y ur ur 㡰 ᨨ鼀� �ᆠ 盐 f㦨 B A
B y a = B ya a df dy = - B a = - B av t 㡰 鼀 e = dt dt
\
f=
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EBD_8341 PS-250
Physics 盐ᨨ �矄ᆠ盐 ᨨ � � B犨� � ᆠ ᨨᆠ
e B av t 㡰 = [ 꾀 �ᆠᨨᆠ├犨�矄] R R 꾀矄ᨨᨨ�ᨨ矄 ᨨ犨 鼀� �ᆠ 盐 � ├ 犨� �� ᆠ矄矄 ᆠ � 盐ᆠ ᆠ盐 ᨨ犨矄 � 矄盐ᆠ �ᆠ�ᨨ 矄ᨨ 矄 ᨨ 盐ᆠ 鼀� �ᆠ 盐 � ├ 犨� � ᨨ� 盐ᆠ ᆠ盐 ᨨ犨矄 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 㡰 쟼ᆠ 矄� 犨� � ᆠ ᨨᆠ矄矄 ᨨ犨 � ��盐 꾀 ᆠ盐 ᆠ泐� �矄�鼀 � 鼀� �ᆠ � � ᆠ矄 ᨨᆠ㡰 ur r ur B y ö æ F ҈ = � l ´ B 㡰 = � ç - a�ˆ ´ kˆ÷ = B y �ˆ ´ ˆ 㡰錐 è a ø ur B y + a 㡰 æ ö F = � ç + a�ˆ ´ kˆ÷ = �B y + a 㡰 ˆ è ø a ur ur F = - F ᆠ 犨� X F҈ ᆠ ᆠ ᨨ盐 ᆠ ᨨ y ᨨᆠ \�㦨
ur ur ur ur ur F = F҈ + F + F + F
和ᨨ ᨨᆠ � 矄泐矄鼀 T F = 맔 a�T㦨F
mgR B L vT = � vT = 錐 mR B L ��㡰␔ ᨨᆠ�泐ᨨ ᨨ �矄 ᨨ �矄鼀� ᨨᆠ�泐 v mgR v= T = B L 和盐矄�盐� � �矄 盐 ᨨ ᨨᆠ�泐� ⼄ ��㡰
F
g-
F Y
ᆠ㡰 ᨨᆠ ᨨ ᨨᨨ
ur é B a v t 㡰 ù ˆ F = mg ˆ + F = ê mg - ú 錐 R ëê ûú \
B a v t 㡰 dv m = mg - dt R
꾀ᆠᆠ�ᨨ a = g -
dv
v
ᔼ �� ��犨� ò
B a v t 㡰 g- mR
v 㡰t é B a v t 㡰 ù ᨨ� ê g - ú mR ûú ëê
- B a mR
=
t
ò dt
w
v
=
v t 㡰 =
)
X
t
B a v t 㡰錐 mgR
(
q
O B
A dx
L
- Ba t mR ù mgR é ҈ e ê ú B a ëê ûú ␔ 鼀� ᨨᆠ�泐�矄 � v t㡰 ᨨ矄 ᨨ ᨨ t mgR \ v t㡰㦨 B a �㡰 v ᨨᆠ�泐ᨨ ᨨ 泐�鼀t \ᔼ 盐ᆠ 鼀 e㦨BvL 矄ᨨ� 盐ᆠ ᆠ盐 � ᨨ ᔼ 盐ᆠ 鼀 BvL I= = R R
\
䰘
)
x
R
( Bat ) mR
(
㡰 �㡰 盐矄ᆠᨨ 矄� 矄鼀鼀 ᨨ � dxᨨ 鼀 ᨨ OA �矄 ᆠx ᨨ鼀 ᨨ��� 和鼀鼀ᨨ盐 ᨨ 鼀 � 盐ᆠ � �矄矄鼀 � 盐ᨨ盐 � ᨨ鼀鼀� �ᆠ � de㦨B dx㡰v … �㡰 犨 v�矄 ᨨᆠ�泐ᨨ 矄鼀 � dx v 㦨xw … ��㡰 Y
S
B a v t 㡰 =e ҈ ᨨ mgR - B a t mR
B L ҈ mgR g ҈ ´ = g- g= mR B L
=t
é B a v t 㡰 ù êg ú mR ú = - B a t ᨨ ᨨ� ê g mR ê ú ê ú ᨨ ë û
ᨨ ҈ - e
Q ᨨ �矄鼀矄矄矄矄桌
T B Lv = g … ��㡰 m mR 和ᨨᨨ 犨�ᆠ⼄盐�鼀� ᨨᆠ�泐犨 �矄ᆠᆠ�ᨨ
�矄 ᨨ�
F
… �㡰
Þ mg T㦨maÞa㦨 g -
y졔a
B a v t 㡰 ˆ = -�B aˆ = - R � 盐犨 �ᆠ�ᨨ
B L v ᨨᨨ矄� ᨨ �矄 鼀ᨨ�ᨨ 㡰 R
F = BIL =
\ ᨨ 鼀 ᆠᨨ矄矄 犨 ᨨ鼀�ᆠᨨ OA r
é x ù Br w e = ò Bxw dx = Bw ê ú = êë úû 㡰 �㡰 ᨨ ��鼀 ᆠ ᆠᨨ 矄盐ᆠ 矄 ᨨᨨ犨矄 和犨�ᆠ s�矄ᆠᨨ矄 �鼀t㦨 ᆠ��矄ᆠ矄ᨨ �ᨨ犨 ᨨ ᆠ盐 � L㜠Rᆠ�ᆠ盐� ᨨᆠ盐 泐�鼀ἘtἘ I 㦨I ҈et⏌t㡰 I S e B wr I = = O R R R A L t = R æ Rö -ç ÷ t L Bwr \ I = ҈ - e è L ø 桌 R r
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PS-251
Electromagnetic Induction ��㡰 ᔼ 矄 泐 矄
=-
æ Rö B wr Q t 矄�盐 - çè L ÷ø t ® 桌 I= e R 和� ᆠ ᨨ �矄ᨨ� �� �ᆠ 犨ᨨ├ 矄ᨨ ᨨ ᨨ ├ � ᆠᨨ 矄 �盐 矄 꾀 ᆠᨨ 矄 �盐 矄 ᨨ⼄盐 㦨 t㦨t ᨨ犨 ᨨ矄矄 盐ᨨᆠ盐 I
d é æ҈ öù B ç r q÷ ú = - ҈ Br d q = - ҈ Br w [Q q = wt ] øû dt êë è dt
\ ᔼ㦨
E ҈ Br w ҈ Br w =Þ I = R R R
æ Br w ö P= I R= ç ÷ R è R ø ᔼ ᨨ⼄盐⼄盐� ᨨ �矄 ᨨ犨 �矄 t1
r⏌
q
r ᆠᨨ矄q
t 㦨
mg
B r w R
ᨨ⼄盐 ᨨ 犨�� 鼀�㡰 ᨨ盐 ᆠ r t = mg ´ ᆠᨨ矄 q
ᨨ ᨨ⼄盐 t㦨t҈졔t ᔴᨨᆠ├犨�矄㡰
P = t҈w Þ t҈ =
B r w mgr + ᆠᨨ矄 wt R
⼄盐� ᨨ⼄盐 犨� ᨨ 矄鼀 鼀� �盐 �
�ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ᨨ⼄盐 犨� ᨨ �盐矄 ᨨ ᨨ 犨�鼀� � ᆠᨨ 矄 黨 �ᆠ�ᨨ ᨨ � 盐ᆠ ᆠ盐 R҈ ᆠ ᨨ盐 犨� ᨨ A C �҈ 矄犨 ᨨ � � ᆠ ᆠᨨ矄矄 B �҈+�)l ᆠᨨ鼀� �ᨨ 矄 鼀� 矄 e 矄鼀 꾀矄ᨨP҈㦨e�҈㦨䰘␔ E L �҈+� F P㦨e�㦨҈␔ mg � ҈䰘 ҈䰘 � � … ��㡰 Þ �҈ = \ ҈ = D B ҈ � ҈ R
ᨨ� ᨨ 鼀�ᆠL鼀ᨨ矄犨� 鼀� ᨨᆠ�泐 � ᨨᆠᨨ �矄 ᨨ \ B �҈졔�㡰㦨mg mg ´ 黨 黨 Þ �҈졔�㦨 鼀 ���㡰 = = Bl ´ ҈ ҈� ᨨ鼀⼄ ��㡰 ���㡰 ҈䰘 黨 � + � = ҈ ҈� ҈黨 䰘 Þ �㦨꾀 Þ �҈ = 꾀 Þ e = = V ҈� ҈黨 ⏌҈� ᔼ 盐ᆠ 鼀 ᆠᨨ矄矄 L 盐 ᨨ 鼀ᨨ鼀 ᨨ L � 鼀� �ᆠ � e = = ҈鼀⏌矄 e㦨BvTL Þ vT㦨 BL ´ ҈ 꾀矄ᨨ ᨨ鼀⼄ �㡰 e e = 䰘W R = = R҈ = = = W �҈ ҈黨 ⏌҈� � �㡰 ᔼ 盐ᆠ 鼀
��㡰 ␔ ᨨᨨ�矄 � �� 鼀� �ᆠ � ��
��ᨨ ᔼᔼ鼀� �ᆠ� 矄ᨨ ᨨᆠ矄矄� � ᨨ盐� ᨨᨨ �矄� ᆠ矄� �� ᨨ犨 犨 �ᆠ�ᨨ ᨨ矄 Ἐ矄犨ᆠ盐 犨� ᨨ犨� ᨨᨨ� 矄盐ᆠ �ᆠ�ᨨ
犨 �ᆠ 犨�ᨨᨨ矄 ᆠ � ᆠ ᆠ盐 犨� ᨨ犨 � �㜠ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ���㡰 犨 � 盐ᆠ 鼀 �ᨨ ᨨ ᨨ�ᨨ ᨨ �矄 ᨨ�ᨨ t㦨 T⏌㡰 犨 ᨨᨨ 矄 � ᆠ盐 �矄 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 㝐� �盐 ᨨ ᆠ盐 鼀� 矄ᆠᨨ 矄 I = Bwr ⏌ R ᨨ ᨨ�ᨨ 犨 ᨨᨨ ᆠᨨ鼀矄 ᨨ盐 ᨨ � ᆠ盐 ᨨ 矄鼀鼀� �盐 �矄 矄 盐 ᆠᨨᆠ├犨�矄 꾀 �ᆠᨨᆠ├犨�矄 ᆠ盐 �矄 矄盐ᨨ矄 ᨨ ᨨ矄�� I㜠t � ᨨ 犨ᨨ�ᨨ 矄 ᨨ ᨨ�ᨨ 矄 矄 ᨨ犨 ᨨ犨 I Bw r R
t=
E=-
df d = - B ´ A㡰 dt dt
T⏌ T T⏌ T
҈
t
犨ᨨ├ 矄 �├ ᆠ 犨 矄ᨨ � � 和ᨨ ᨨ ᆠ盐 ᨨ犨矄 ᨨ盐� AC
\쟼⼄盐� 矄�矄 ᆠᨨ ᆠ�ᆠ盐�R㦨졔҈㦨W
鼀 � 盐ᆠ e㦨BVl IR e㦨IR㦨BVlÞV㦨 Bl ҈ ´ ҈ ´ ᨨ V = = 鼀⏌矄 ´ ҈
ᆠᨨ 矄 � ᨨ�ᨨ MN ᨨ ᨨ ᨨᆠ�泐 v � �� �ᆠ�ᨨ 꾀泐� � 鼀� �矄 盐犨 � ᨨᆠᨨ ᆠᨨ 犨�ᨨ犨 矄 N ᆠM犨�졔 N犨� ��ᔴ盐 犨� ᨨ犨� ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
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EBD_8341 PS-252
Physics
B㦨맔҈wb⏌m l㦨҈鼀 I҈(t)
180 km/h=50m/s
B (t) I t㡰B t㡰
1m ᔼ2 (t)
�鼀
鼀 ᨨ 盐ᨨ 鼀ᨨ�ᨨ 鼀 㦨 l 㦨�맔맔҈맔҈㦨҈㦨҈鼀
� �ᨨ 鼀��ᨨ鼀�矄҈mv
�
Topic-3 : Miscellaneous Problems ҈
Þ E pr 㡰 = pa
㡰 꾀矄 ᆠ泐� �矄 ├ ᨨ 盐 � ᨨ鼀 鼀� �ᆠ � 矄ᨨ ᨨ ᆠ � � 鼀� �ᆠ 盐 ᆠ � 盐ᆠ ᆠ盐 犨� ᨨ
҈ a dB ÞEµ r r dt ᆠ鼀� �盐 ᨨ � 盐ᆠ ᆠ�ᆠ � �矄 ᆠ
l 桌 A
ᨨ P㦨
pr æ dB ö N r N A ç ÷ Þ P µ è dt ø l rl N 㡰 r ⏌ 㡰 l
P ҈ = \ᨨ犨鼀� 矄 矄鼀 rἘ ҈ 㡰 ␔ 矄犨�ᆠ S�矄ᆠᨨ矄 鼀� �ᆠ � �矄矄㜠盐�
矄ᆠᨨ盐 P � � 矄 � � Q � ᆠ矄矄 � �ᆠ�ᨨ ᨨ鼀�� ᨨ 꾀ᆠᆠᨨ � �ᨨ Ἐ矄犨 IQ҈ 犨� ᨨ犨 矄ᨨ 矄 ᨨ ᨨᨨ矄 ᆠ盐矄 ᨨ ᆠ � ᨨ犨 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ 矄 矄 泐E 졠ᨨ矄� �矄 ᆠ矄犨 S �矄 ᨨ IQ 犨�ᆠᨨᆠ├犨�矄 b㡰 꾀ᆠᨨ 盐ᆠ� �� ��矄ᆠ ᆠᨨ �泐犨� � ᆠᨨ� ᆠᨨ� 矄� 盐ᆠ ᆠ矄犨矄矄�矄 ᆠ 鼀� �ᆠ � ᆠ ᨨ ᆠᨨ� B t㡰㦨ҘnI҈ 꾀矄 ᆠ盐 � ᆠ矄矄B 犨�矄ᨨ� ᆠ矄犨� �鼀� � ᆠ 矄 鼀 �鼀盐鼀 盐 犨 ᆠ盐 ᆠᨨ鼀矄 矄 泐㡰 ᔼ 盐ᆠ 鼀 � � � df d ur ur d e= = - B A㡰 = - A µ nI҈㡰 dt dt dt \ᔼ 盐ᆠ ᆠ盐 � � � e µ nA dI҈ I t 㡰 = = R R dt dI҈ ᆠ矄矄犨� �鼀 ᆠI矄ᨨ ᆠ矄矄犨� dt �鼀桌 ␔ I҈㦨Imax ҈et⏌t㡰
� 矄 矄 ᨨᨨ犨矄
䰘
\
҈ r ᆠ㡰 ᨨᨨ㜠꾀ᆠ�矄 ᆠ盐 꾀 ᆠ盐 � ᨨᨨ A � ᆠ矄矄 犨� �鼀 鼀� �ᆠ � 矄 ᨨ ᨨᆠ �
ᨨᨨB 矄ᨨ� ᆠ矄矄矄ᨨᨨA �矄ᆠ ᨨᨨB �矄 � 盐ᆠ矄 鼀 � B � 矄盐ᆠ �ᆠ�ᨨ ᆠ盐 ᨨ犨矄 ᨨᨨ矄�� B 矄ᆠᨨ鼀 ᨨA ᆠ ᨨᨨB �矄 泐ᨨᨨ A b ᆠ㡰 꾀 㦨 � 盐ᆠᨨ矄 L҈ L 犨� ᨨ � � � 矄�矄 ᆠ ᆠ ᆠ盐 ᨨ盐� ᆠ�ᆠ盐� �矄 ᨨ 꾀 ᨨ ��鼀 ᆠ盐 ᨨ盐� 矄�矄ᨨ�矄ᆠᨨ 矄 ᔼ犨� �� � ᨨ犨ᨨ矄҈ 犨 �ᆠ � \ᔼ҈҈㦨ᔼᔼ㦨ᔼ҈졔ᔼ桌 r ᨨ鼀 ᆠ ᨨ ᆠ�ᆠ盐 ��ᨨ ᆠ矄矄 矄
␔ 盐鼀 ᨨ 盐 矄 ⼄盐 盐 犨� �盐矄 ᨨ犨 r Ἐµ
dB fᨨr r ³ a dt
Þ E=
b㡰 ᨨ犨 P = E = pr çæ d f ö÷ = pr é d NBA㡰 ù úû R rl è dt ø rl êë dt E㦨� 盐ᆠ 鼀 R = r
�鼀
ᆠ 矄 盐 ᆠ�ᨨ ᨨ �鼀 >㡰 ᨨ 盐ᆠ I t㡰 t㡰 ᆠ矄矄 犨� �鼀 ur uur d f d ur ur d dB b㡰 Ñ ò Ed l = dt = dt B A㡰 = dt BA ᆠᨨ矄 °㡰 = A dt
I҈ =
V é L҈ ù R êë L҈ + L úû 꾀矄ᨨ �ᨨᨨ ᆠ盐 矄 ᨨ盐� ҈ �矄 � �鼀矄꾀㦨 I » b 㡰 鼀� �ᆠ 盐 ᨨ盐� ᨨᨨ矄�鼀t f㦨BA ᆠᨨ矄wtBAᆠᨨ矄wt 㦨B AA㡰ᆠᨨ矄wt㦨BAᆠᨨ矄wt df = BwA 矄� wt \ dt
V é L ù R êë L҈ + L úû I =
df �矄鼀 �鼀盐鼀犨 f㦨wt㦨p⏌ dt
鼀 � 盐ᆠ � 矄鼀ᨨᨨ 和ᨨ
d ( BA ᆠᨨ矄 wt ) = BwA 矄� wt dt \꾀鼀�盐 ᨨ 鼀 �鼀盐鼀 鼀 � 盐ᆠ � ᨨ ᨨᨨ矄 㦨꾀鼀�盐 ᨨ 鼀 �盐鼀鼀 � 盐ᆠ � 矄鼀ᨨᨨᨨ dQ =I = ᆠ 㡰 ᨨ鼀 �鼀盐鼀ᆠ �ᨨ ᆠᆠ�ᨨ dt I㦨Iᆠᨨ矄wt㦨ᆠᨨ矄�t
e矄鼀 =
黨
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PS-253
Electromagnetic Induction I
ᨨ鼀⼄ �㡰 ��㡰
҈
D㝐㦨 -g 鼀 ç
p w
æ 턼 ö - g턼 = ÷ è 鼀 ø
䰘p w
p w
t
2p w
҈ ҈ D
B A
�
C㦨m F
R㦨҈W
҈
䰘p p
� = ᆠ �犨�鼀 �鼀盐鼀 w w p ⏌ w
턼¢ =
ò
p ⏌ w
é 矄� �t ù ᆠᨨ矄 �t 㦨 ê ú ë � û
҈ p ö ҈ æ 矄� ç � ´ ᔴ 㦨 ÷= � è ´ � ø � ҈ c = ´ ҈ C �Qmax㦨 �
\
ᨨ鼀 � � �矄 ᆠ 盐矄 ᨨ t = ᆠ盐 �矄 � �ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ ᔼ鼀鼀 �泐 A�矄 ᆠᨨ
ᆠ ᨨD 꾀 t = 䰘p w
æ
䰘p
dfù é êQ e = dt ú ë û
ö
pµ na �w ᆠᨨ矄 wt e = pRp 㦨 矄�矄 ᆠ Ld
҈ ҈ ´ = -҈ - ᔴ = � ᨨ犨泐� �⧤
҈
҈҈
ò \
҈
- � ´ ҈ = Þ�㦨҈꾀
´ ҈ -
鼀 �鼀盐鼀ᆠ �ᨨ CQ㦨CV 㦨맔҈맔�㦨҈ᔴ
ᨨ ᨨᆠ � ᨨ犨 ᨨ鼀 泐㦨맔҈ᔴ ҈ bᆠ㡰 R⏌L ҈⏌ LC �鼀 矄�ᨨ 矄 ᨨ RC ⼄盐 ᆠ泐�M L T҈桌 b㡰
® ®
쟼 㦨
L
pl A r ´ pR pRr \ 矄�矄 ᆠ R = ᨨ 矄�矄 ᆠ㦨 L´d Ld \ᔼ 盐ᆠ ᆠ盐 � 矄 ᔼ
҈ -
d
矄�矄 ᆠᨨ 矄 R =
ò ᆠᨨ矄 �t = � 矄� çè � ´ ´ � ÷ø
� +
df = pµ na � wcᨨ wt dt
ᨨ e = pµ n a � w ᆠᨨ矄 wt
䰘p w
䰘p ᆠ � ᨨ 盐 ᨨ ᆠᆠ�ᨨ w
҈
㡰 矄�� 矄 ᨨ犨矄 ᆠ � �矄 ᨨᨨ矄� ᨨ �ᆠ�ᨨ ᨨ 鼀� �ᆠ � 㡰 �矄 ᆠᨨᆠ ᨨ�ᨨ 㡰 ᨨ� ᨨ ᆠᨨ鼀ᨨ F I ᨨ 鼀� �ᆠ � I � ᆠ矄 犨� � 盐ᆠ ᆠ盐 ᨨ 盐ᆠ � ᆠᨨ 盐ᆠ� � � � 犨 �ᆠ BH ᨨ 盐ᆠ矄 � ᨨᆠ �
盐犨 �ᆠ�ᨨ �矄 �矄 ᔴᨨ 盐ᆠ� �� � ᆠᆠᨨ ᆠ 犨� 鼀� �Ἐ矄 盐 꾀 ᆠ泐� �ᆠ 矄 ᨨ � �ᆠ├ 矄矄 �盐矄 矄盐ᨨ盐 矄 ᆠᨨ � 矄ᨨ ᨨ� ᨨ �盐矄 a ᆠᨨ� ᨨ 矄ᨨ ᨨ� ᆠ�矄 �ᆠ盐 �㦨 �矄� wt 졠盐矄� 矄ᨨ ᨨ� 鼀� �ᆠ � �矄 ᨨ 㝐� �ᆠ � � 矄� 矄ᨨ ᨨ� 㦨Ҙn�矄� wt \㝐� �ᆠ 盐 � ├ 犨� 矄ᨨ ᨨ� f㦨BA R ᨨf㦨 Ҙn�矄� wt㡰 pa㡰 a
- f = - p 㡰 = -p
\ᔼ 盐ᆠ ᆠ盐 㦨 ҈�
µ na Ld � w ᆠᨨ矄 wt rR
㡰 盐矄 ᆠᨨ 矄� 矄鼀 矄� ᨨ �ᆠ├ 矄矄 dx �矄 ᆠx ᨨ鼀 犨�꾀矄 矄 ᨨ犨 � ��盐 鼀� �ᆠ � �矄矄� 盐ᨨᨨ 犨�矄 B㦨BA졔BB �ᆠ盐 ᨨ ᨨ �ᆠ 盐犨 矄㡰 B㦨
m I m m I é ҈ I ҈ ù + + 㦨 ê p x p a - x 㡰 p ë x a - x úû
= - p dx
쟼맔p 㦨p
- \ 쟼㦨 b㡰 � 㝐㦨g \ 㝐㦨g 鼀w \ 㝐㦨g 鼀 Dw㡰 턼맔 盐Dw㦨 鼀
I
x
҈㡰 㡰
和鼀鼀ᨨ盐 ᨨ 鼀� �ᆠ 盐 矄矄� � ᨨ盐� 矄�ᨨ �ᆠ├ 矄矄 dx
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EBD_8341 PS-254
Physics
ᨨ⼄盐 ᨨ盐 PQ 泐 �矄 ᨨᆠ r r ur $ $ $ t RS = r ´ F 㦨 � a㡰 ´ k - � 㡰 IbB 㦨 I abB $ 㡰
m Ia ´ a dx p x a - x 㡰
ᨨ 盐 ᨨ盐� 矄⼄盐 ᨨᨨ df㦨B맔adx㦨
f㦨 ò
a a
��㡰
ᨨ⼄盐ᨨ盐PQ 盐 ᨨ犨�� ᨨ 犨�PQRS
m I ´ a m Ia dx = x a - x㡰 p p
m a 㡰 I矄� wt㡰 p m aI w df = 㡰 ᆠᨨ矄 wt
鼀 ᨨ 盐ᆠ e㦨 dt p ᔴ �矄ᨨ � ᆠᆠ�ᨨ m aI w q㦨C맔e㦨C맔 㡰ᆠᨨ矄wt �㡰 p \ ᔴ盐 � ᨨᨨ ᨨf 㦨
dq C ´ m aI w = 㡰矄� wt �㦨 dt p
æ aö ���㡰 㡰㝐� �ᆠ ᨨᆠᨨ 矄 ᨨ鼀⼄ �㡰FRS㦨ᔼ ékˆ �ˆ ù ë û ᆠ㡰 ᨨ 犨�ᨨᨨᨨ ᨨ� ᨨ t㦨mg ç ÷ è ø
a mg Þ I㦨 �㡰 bB 꾀矄 ᆠ盐矄 ᨨ⼄盐 盐 ᨨ鼀� ᆠ盐 � ᨨᨨ矄� �ᆠ�ᨨ 矄 ᨨᆠ盐 �矄 ᨨ鼀 Pᨨ Q ҈䰘 �㡰 꾀矄矄 ᨨ犨 � ��盐� ᨨᆠF�矄� ᨨ ᨨ MN矄盐ᆠ 矄 ᨨ 鼀ᨨ矄� 盐 � ᨨ鼀鼀� �ᆠ � 꾀 ᆠᨨ 矄 ᆠ盐 ᨨ犨矄 ᨨ盐� R ᔴᨨ 矄� ᨨᨨ MPQN. MN �矄 ᆠx ᨨ鼀PQ. � ᨨ �矄� ᨨᨨ㦨x IabB㦨mg
m aI w C 㡰 p 㡰 ᨨ鼀⼄ �㡰 � 犨 ᆠ � �鼀 q t㡰 �矄 矄 矄 ᨨ犨 ᨨ犨 \ �鼀 㦨
q
q
p/w p/2w
2p/w
t
3p/2w
q
ᔴ �ᨨ 盐ᨨ ᆠᆠ�ᨨ q㦨 qᆠᨨ矄 wt
C ´ m aI w 㡰 p ҈ 㡰 盐矄ᆠᨨ 矄� ᆠ盐 � ᆠᨨᆠ├犨�矄 �ᆠ�ᨨ
� ᨨᨨPQRS �PᨨQ� 犨�PQ ᨨᆠᨨ 犨�QR ur r ur F = I l ´ B㡰 㦨 I a�$㡰 ´ �$ + k$ 㡰 B 桌 q㦨
QR
㦨 I B a�$ ´ �$ + a�$ ´ k$ 桌 㦨 I B졔a $ 㡰桌㦨 -aB I $ Z
\ 矄�矄 ᆠᨨ �矄� ᨨᨨ㦨xl \ ᨨ矄�矄 ᆠᨨ ᨨᨨ R 㦨R졔lx V ᨨᆠ�泐ᨨ ᨨ �矄� 矄 盐ᨨ 鼀ᨨ�ᨨ ᨨ ᨨ 鼀 � 盐ᆠ e㦨Bvd \ ᔼ 盐ᆠ ᆠ盐 I㡰 =
e Bvd = Rnet R + lx
R + lx 㡰 I �㡰 Bd 꾀 盐 ᨨ � 盐ᆠ ᆠ盐 I 犨� 犨� � ᆠ ᨨᆠ FM 㦨 BId ᨨᨨ矄� ᨨ �矄 鼀ᨨ�ᨨ ⼄盐�ᨨ ᨨ 鼀ᨨ�ᨨ ᨨ 犨� F FM = ma Þ F = FM 졔 ma 盐矄 FM = BId ᨨ鼀⼄盐�ᨨ �㡰 和ᨨ ᨨ ᆠᨨ 矄 I
v=
dv lI dx lIv lI = = = R + l x 㡰 dt Bd dt Bd Bd 㡰 lmI \ F = BId + R + lx 㡰 Bd 㡰 ��㡰 꾀矄 犨ᨨ├ ᨨ 泐 ᨨᆠ F 矄ᆠᨨ é ù é R + lx ù lmI dW P Fv BId R + l x 㡰 ú ê Iú = = = + ê dt û Bd 㡰 ëê ûú ë Bd
a=
B
>
k 졔 �
>
>
P
k
Q
Y
X S
R
ᨨᆠᨨ 犨�PS ur r ur F PS = I l ´ B 㡰 = I a -$� 㡰 ´ $� + k$ 㡰 B 桌 㦨 aB I $ � ᨨᆠᨨ QR �矄 ⼄盐 ᨨᨨ矄� ᨨ ᨨ PS ᆠ ᆠ ᨨ ᨨᆠ ᨨ RS ur r ur F RS = I l ´ B 㡰 = I b - $ 㡰 ´ $� + k$ 㡰 B 桌 㦨 I bB k$ - �$桌
�㡰
é
� P = ê I R + lx 㡰 +
êë
lmI
B d
ù R + lx 㡰 ú úû
ᨨ 盐ᆠ 矄ᆠᨨ
H = I R + lx 㡰 \
f =
H é lmI R + lx㡰 ù = ҈+ ú P êë B d û
-҈
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PS-255
Alternating Current
Alternating Current
21
Topic-1 : Alternating Current, Voltage and Power ҈
㡰 꾀矄V t㡰㦨矄� ҈p
Topic-2 : AC Circuit, LCR Circuit, Quality and Power Factor ҈
矄ᨨI t㡰㦨 矄� ҈p � �I㦨Im㦨矄� ҈p㡰 ᨨI㦨Im t҈ =
Þ
Im
Im ҈ p = Im 矄� ҈ pt 㡰 Þ = ҈ pt Þ t = s
҈ ҈ ҈ = = s = 鼀矄 㡰 ᨨ 犨ᨨᆠᨨ ᆠ �ᆠ ᆠ�ᆠ盐 ᆠᨨ� \ t⼄ =
㝐盐盐 ᔼ 盐ᆠ ᆠ㝐 =
m p҈
҈㦨㦨҈
m p �㡰 �� ᔼ㦨ᔼᆠᨨ矄w ��㡰 ᨨ犨 ᆠᆠᨨ � � ᨨ 泐Ἐ矄 矄ᆠᨨ 犨 � 盐ᆠ 鼀
ᆠ㝐 =
ᔼ ᨨ鼀⼄ ��㡰 = -㝐
-m p = ᔼ ᆠᨨ矄 w 㡰
=
҈ L ҈ ´ ҈ - = R C ҈ ´ ҈- ҈ ´ ҈ = = = ҈ ҈ 㡰 ᨨ � �泐 矄ᨨ � � 盐ᆠᨨ ҈ U = LI 盐� � �ᨨ犨 ᨨ ᆠ盐 �㦨I鼀 ҈eRt/L㡰 I U ᨨUᨨ 鼀 錐� 矄ᨨ 鼀 n n I 鼀 \ = I 鼀 ҈ - e - Rt ⏌ L 㡰 n ҈ n -҈ Þ e - Rt ⏌ L = ҈ = n n æ ö L æ n ö Rt n -҈ Þ - = çç Þ t = çç ÷÷ ÷ L R è n - ҈ ÷ø è n ø 㡰 R㦨҈XL = Lw㦨҈맔䰘�맔p 㦨�W, ҈ ҈ XC = = -� = ҈W C w ҈ ´ p ´ 䰘� Q=
p ҈ ҈ 矄ᆠ ´ = ҈ p
ᨨ I =
㡰 턼盐�泐 ᆠᨨ
ᔼ鼀 ᆠZ㦨 R + X L - X C 㡰
㦨 ҈ + � - ҈㡰 㦨䰘䰘 ; �
æV ö H = �鼀矄 Rt = ç 鼀矄 ÷ RT è Z ø
m p ᔼ 矄� w w 㡰
pm ᔼ = w 矄� w b㡰 ᔼ 盐� 盐ᆠ�ᆠ�ᆠ盐�ᆠ盐 犨泐矄�矄 � p 鼀 泐 ᔼ ( ) = 矄� w pö æ ᔼ = ᔼ 矄� ç w - ÷ è ø ᨨ犨�� 㡰㦨҈矄� �㡰 쟼 ҈ ᔼ = = [Q = ] w � ´ pö æ ᔼ = ҈矄� ç � - ÷ ø è ᔼ = -҈ᆠᨨ矄 ( � )
҈W
҈
҈m
㦨 ms㡰Dt
⏌䰘� ´ ´ ҈t = 㡰 ´ ҈ � � QV鼀矄 = D㦨҈ ᔴ \ �鼀t㦨҈矄 㡰 � ᆠ ᆠᨨ � 盐ᆠ ᆠᆠᨨ�Z 㦨 ҈W ⼄盐 ᆠ泐ᨨ 꾀ᔴ矄�� v㦨҈ 矄 �f㦨� X f = L = �° = ҈ Þ X L = R R
Þ
ᆠ ᆠ Z = ҈ =
Þ ҈ =
X L + R
R + R
Þ R = ҈ Þ R = � \ X L = �
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EBD_8341 PS-256
Physics
Q X L = wL㡰
Þ Lw = �
ÞL=
� p ´ ҈
=
Q w = pv㡰
� 鼀㦨҈҈맔҈ p e -t ⏌c 㡰 b㡰 ␔ �㦨� ҈et⏌c㡰㦨 ҈ - e R =
�
҈҈
t
ᔴ �q㦨 ò �dt
e
t
EL 㡰dt 㦨 E t 㦨 E ´ L ⏌ R 㡰 㦨 Re R e 䰘 R æ d� ö d� � = ᆠ㡰 � R = çè t ÷ø � Þ dt dt t 㦨 R ò ҈ - e
䰘
-t ⏌t
e e ´ ҈� = = = 꾀 � æ ´ ö çè ÷ + ø ᨨ 盐矄㡰 ᔴᆠ�� ᆠ ᆠ
Xc 㦨
҈ ´ ҈ 㦨 㦨 wᔴ ҈ ´ ´ ҈
ᔴ
XL㦨wL㦨 ҈ ´ R҈㦨҈
黨
ᆠ盐 犨� �ᨨ犨 ᨨ �鼀 㦨 ᨨ 㦨 ᆠ泐ᨨ ᆠ盐 ├矄 ᆠ 㡰 � ᔼ 盐ᆠ ᆠL㦨鼀 ᔴᆠ� ᆠC㦨҈mF ᔼ鼀 ᆠZ㦨XCXL
҈ wL wC
ÎL
= kx bv dt md x dx Þ + b + kx = �㡰 dt dt q Ld� �R = ᔼ ᔴ ᆠ�ᆠ盐� C dt d dq q L + R + = ��㡰 dt C dt
q㦨 Þ q㦨 ҈ 㡰 q㦨 和ᨨ 矄 � ᆠ ᆠᨨ鼀矄 � ᨨ盐黨 졔 㦨҈� ᔼ �矄⧤W 矄 � ᆠᆠᨨ鼀矄ᨨ盐ᨨ졔㦨黨 b㡰 ᔼ
Þ Z=
\q =
md x
q2
҈
tc Îé t + tC e t ⏌ tc ù ë û R t Îé ù Þ q = êtC + C tC ú Þ q = Î L Rë e R û
)
R e ҈ 㡰 ᔼ 鼀 鼀ᨨ �ᆠ ᨨ矄ᆠ��ᨨ
q1
q㦨 ҈
(
E ҈ e t ⏌ tc dt R
Þq=
㦨 ҈ ҈
ò �dt
Þ ò dq = ò
�
҈
q҈㦨 㦨 q҈�矄ᆠᨨ矄ᨨ黨 ᨨ㜠 ᆠ�ᆠ盐�
Tc
e
V 矄� wt + p ⏌ 㡰 ᔴ盐 � = Z ҈ Þ �= ᆠᨨ矄 wt 㦨�ᆠᨨ矄 ҈㡰 ҈黨 㡰 ᨨ 矄�矄 ᆠᨨ
ᆠ�ᨨ ᨨ 矄�矄ᨨ � 盐ᆠᨨ �鼀 ��ᨨ ᨨ ᆠ盐 �矄 I = I ҈ e t ⏌Tc 㡰 L TC = R q=
L é ù = ú Þ㦨t l 㦨l ê 矄t = = R ҈ ë û 㡰 ᨨ � �鼀 矄犨�ᆠ �矄 ᆠᨨ矄 � 盐ᆠᨨ 犨� � 矄ᨨ ⼄盐� ��鼀犨�矄ᨨ犨 ᔼ=
҈ ҈ ´ ´ ҈ 㦨҈黨W ҈ ´ ҈ ´ ҈
҈ æ ö X L = wL÷ çèQ X c = ø wC
҈
ᔴᨨ鼀� �⼄盐�ᨨ 矄 �㡰 ��㡰 ҈ L « m C « R « b k 㡰 ᆠ盐 ᔼ㡰� 矄�矄ᆠ�ᆠ盐��矄�� 泐 tR æ ö V I = ç҈ e L ÷ ÷ Rç è ø 꾀t㦨¥ ¥ ö æ ç L I¥ = I e ⏌R ÷ = �㡰 ÷ � ç è ø 꾀 t㦨矄
´ � ö æ = ҈ e 㡰 çè҈ e ÷ø ҈ ´ ҈ �� � � �㡰泐 ��㡰犨�
��㡰
I¥ ҈ = I ҈ e ҈ b㡰 꾀矄犨├ ᨨ犨�ᨨ犨�㦨鼀矄ᔼ鼀矄ᆠᨨ矄q Þ
æ öæ ᔼ ö æ ҈ ö æ ö = ç ÷ ç ÷ ᆠᨨ矄 q = ç ÷ç ÷ ᆠᨨ矄 �° Q q 㦨� 㡰 è øè ø è øè ø
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PS-257
Alternating Current � =
҈
犨
␔矄矄ᆠ盐 ᔼ㦨 ᔼ鼀矄 矄� q ᔼ 矄� �° = ҈꾀 = 矄� q = w w = ҈� 㡰 턼盐�泐 ᆠᨨ 턼 = Dw ҈ b㡰 � 㦨ᨨw㦨 㦨�W㦨�鼀ᔴ㦨҈Ҙ 㦨w㦨맔�맔҈㦨W ҈ ҈ ᔴ = = = ҈ W wᔴ ´ ҈ ´ ҈ -
ᨨ�鼀 ᆠᨨ ᆠ�ᆠ盐�
҈䰘
= + - ᔴ 㡰 = � + 黨㡰 = 䰘 W 矄 � ᆠ 犨 ᨨ� ᆠ盐 - ᔴ f = 黨 f = » ҈ Þ f = �° � b㡰 �㦨
e
R +
e
R +w L
҈㦨
҈
=
X L
҈�´҈
q d� - �R - L = c dt
졔
�
�
졔 ⼄ ᆠ
dq q dq Ld q = � =- Þ + R+ dt c dt dt
R dq q + = L dt Lc dt ᨨ鼀 鼀 鼀ᨨ �ᆠ ᨨ矄ᆠ�ᨨ 鼀�盐 �矄 ��
dt 泐꾀 = Aᨨ e m ᨨ盐 � � ⼄盐�ᨨ
d x dt
+
+
b dx k + x= m dt m -
Rt
-
Rt
= Qᨨe L Q鼀 = Q ᨨe L Þ Q 鼀 ᆠ 鼀� �犨� 矄 ᨨ 矄矄矄 � 盐ᆠ ᆠ
w L
ᔴ
VR = -҈ VC
ᆠ㡰 � ᔴ 㦨҈ 㦨҈ 㦨䁸 꾀矄犨├ ᨨ犨
= + ( - ᔴ )
和ᨨ� �犨� ; 黨 ᆠ㡰 ᔴ � ᨨ ᆠᆠ�ᨨ 泐 �鼀 �矄 ��
by q㦨ᔴ ҈et⏌t㡰 t㦨t q㦨ᔴ ҈e㡰 㡰 ᨨ犨 ᆠᨨ ᨨ 㡰 㦨
R
= ҈
҈ w LC
ᆠ㡰 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 ᨨ ᨨ� � ᆠᨨ矄 꾀 ᨨᨨVRVC㦨 Þ
I t㡰㦨҈ e = 䰘mA ᆠ㡰 ᨨ鼀⧤ 泐�鼀t
d q
҈
+ � ¥㡰
ᆠ ᨨ�ᨨ ᆠ㡰 �矄 ᆠᨨᆠ 矄犨
R + p v L
I t㡰㦨҈ e L ⏌ R = ҈ e L
҈黨
CἘ=
㦨桌 Q 㦨 = ᔼ ҈
t L I t㡰㦨ᔼ 㡰ᔼ ¥㡰桌 e ⏌ R t
é ù ҈ R + ê wL ú Ἐ㡰 w C + C ë û 졠 矄ᨨ� � 犨 � ҈ wL = w C + C Ἐ㡰
+ p �㡰 졠 矄ᨨ� �犨 � 㦨� ҈� ´ ҈ = ҈A b㡰 I 㡰 = ҈� ´ ҈ I ¥㡰㦨
R
ᆠᨨ矄 f =
e
=
ᆠ㡰 ᨨ犨 ᆠᨨ
=
=
� + + 㡰 ᨨ犨 ᆠᨨ 犨㡰 = 㦨 㦨 + - ᔴ 㡰 + - 㡰 犨ᨨ犨 ᆠᨨ = � \ 㦨 졠 ᨨ犨 ᆠᨨ � � 㡰 V Vrms b㡰 I rms = rms = 2 Z æ ҈ ö R + ç ÷ è wC ø 꾀矄w� ᆠ矄矄Irms ᨨ盐� 盐� ᆠ矄矄 ᆠ 盐 �ᨨ犨矄 �� 䰘 b㡰 �鼀ᆠᨨ 矄 ᨨ R C ᆠ�ᆠ盐�t㦨Req Ceq �㡰 R҈R� 矄�矄 C҈C� t҈㦨 졔҈㡰 졔㡰㦨҈Ҙ矄 ��㡰 R҈R� C҈C� 矄�矄 æ ´҈ ö æ ´ ö t = ç ÷ç ÷ = Ҙ矄 è +҈ø è + ø 黨 ���㡰 R҈R� C҈C� æ ´҈ ö t = ç ÷ ´ + 㡰 = Ҙ矄 è +҈ ø
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EBD_8341 PS-258
黨
Physics
㡰 ᨨ鼀e㦨E矄� ҈t㡰\ w㦨҈ ᨨ鼀 �� � ᆠ盐 矄 鼀 ᆠ �矄�矄 RC ᆠ�ᆠ盐� X f = C R p \ XC㦨R f㦨� 㦨 ҈ = R ÞRCw㦨҈ wC ҈ -҈ s \R㦨҈kWC㦨҈mF ᨨ RC㦨 ҈ 㡰 꾀㦨ᔼ鼀� 㦨
� ҈
꾀㦨¥ � � ҈ = = ᔼ鼀 㦨 Req ⏌
��㡰 � 㝐盐盐� 盐ᆠ ᆠ㝐㦨�鼀 ҈㦨҈鼀҈㦨�㦨鼀㦨
҈ � = = ҈꾀錐 ᔼ = = = 꾀 ҈ � ҈ 꾀 ᨨ 矄犨�ᆠ 矄 ᆠᨨ矄 矄�鼀盐 ᨨ盐矄泐 ᨨ 犨ᨨ├ ᨨ 泐 �矄 �� 矄 � 盐ᆠ 쟼㝐 㦨 � ᆠ矄� 鼀� �ᆠ �泐
ᔼ҈ =
҈ ҈ ҈ᔼ҈ + ᔼ + 㝐ᔼ҈ᔼ ҈ ҈ = ´ ҈ ´҈- 㡰҈ + ´ ´҈- 㡰 ´ 졔 �맔҈㡰맔҈맔 㦨 �졔졔҈㡰맔҈ \ ␔㦨��鼀 b 㡰 ᔼ졔ᔼ㦨ᔼ҈ \ ᔼ㦨ᔼ҈ᔼ \ ␔ = D =
҈鼀
W
鼀
W
҈W
҈ ҈ ҈ ù é ҈ ê R = + + ҈ = ҈ ú � ë eq û I 鼀 ҈ ҈ = ´ = \ I 鼀�
� 㡰 L㦨�鼀㦨�맔҈錐I㦨҈꾀R㦨W
dl - + RI = VQ dt Þ VP - VQ = � ´ ҈ - ´ ҈ + - ҈ ´ 㦨�졔㦨 㡰 � ᨨ犨P㦨␔ᨨ�V = � P = Vm × I 鼀矄 × ᆠᨨ矄 f Þ = � ´ I 鼀矄 ´ Þ I 鼀矄 = 꾀 矄� � P = I 鼀矄 R I 鼀矄 㡰 × R = P Þ ´ R = Þ R = ҈W ᨨ犨 ᆠᨨ �矄
\ I =
Þ I=
- Rt ù - Rt ù é é Vê V ҈ - e L ú - ê҈ - e L ú Rê ú Rê ú ë û ë û
- Rt ù é - Rt Vê L e - e L ú ú Rê ë û L
R
VP - L
҈
ᆠᨨ矄 f = Þ =
҈ + X L
Þ XL =
æ ҈ ö -҈ + ç Þ X L = 䰘�W è ÷ø
␔ ᨨ犨 ᆠᨨ �矄 盐 �泐 ҈ = 䰘� X C = X L = 䰘� Þ wC ҈ ҈ F ÞC= = 䰘� ´ p ´ � 䰘�p æ ҈ ҈ö = ´ ÷ mF = mF ç p è � p ø N㦨
I҈
V
V
I
dI = dt
ᨨᔼ鼀
- Rt - Rt ù é V ê -R L æ - R ö L ú \ -ç = e ÷e R ëê L è L ø ûú - Rt
L ҈ æ Rö \ t= Þ ç ÷ t = R è L ø
�矄�矄 �鼀犨 ᔼ�矄鼀 �鼀盐鼀 盐� � �矄盐ᨨ �鼀� ⼄ �㡰 \ e L =
æ ҈ ö Þ ҈ + X L = ç è ÷ø
L
R
I
R
R + X L ҈
�㡰
-R æ L ö öù é - R æ L ç l ÷ V ê L çè R l ÷ø L è R øú e e 盐 ᔼ鼀 ú Rê êë úû V é҈ ҈ù V Þ I 鼀 = ê - ú = R ë û R bᆠ㡰 ⼄盐 ᆠ泐犨 �ᆠ ᆠ盐 �矄� 矄犨� ᨨ� �矄 矄ᨨ ᆠ ⼄盐 ᆠ泐
=
wr =
҈
=
҈
= ҈ 矄 ҈ LC ҈ ´ ҈ - 㡰҈⏌
�矄 ⼄盐 ᆠ泐 �矄 � ᨨ ' ' -
꾀 w » ᆠ盐 � =
V R + w L -
҈ 㡰 wC
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PS-259
Alternating Current
�
� ᆠ盐 ᨨ盐� ᆠ�ᆠ盐� 泐ᆠᨨ鼀矄 ᨨ If w>>wXL>XC矄ᨨᆠ�ᆠ盐� 矄�├ � 盐ᆠᨨ ᆠ㡰ᔼ鼀 ᆠᆠᨨ矄矄 AB, RC ᨨ ᆠ�ᆠ盐�
䰘
Z ҈҈ ҈ R p = \f㦨 矄 ᆠᨨ矄f㦨 = Z ҈҈ ᔼ L㜠R ᆠ�ᆠ盐� ᨨ� 矄 ᆠ盐 ᔼ� 矄 ᨨ盐矄 㦨
V = Z҈ ҈ 矄鼀 泐f҈桌 ҈ ҈ R = = 犨 ᆠᨨ矄f҈㦨 Þ q㦨�º Z҈ ҈ I҈㦨
A
(
҈00 W
I҈ 0.5 H
50 W
I O
0V
ᔼ鼀 ᆠᆠᨨ矄矄 CDLR ᨨ ᆠ�ᆠ盐� 㦨 X L + R = wL㡰 + R 㦨 � ´҈㡰 + �㡰 = � W
I
5º
0V
I
R � ҈ = = Þf = �º Z � ᔴ盐 I ᨨ鼀 ᆠ�ᆠ盐�
犨 ᆠᨨ矄 f =
I =
R æR ö ҈ + R÷ ´ 졔҈졔㦨 졔r҈졔r è ø
Req㦨 ç
R
A
B
R
r҈
r
R R
R
L
R
C
S
e+e e e = Þ I㦨 I 㦨 Req Req R ⏌ 㡰 + r҈ + r 矄�矄 ᆠἘ Ἐ ᨨ犨 �ᆠ ᨨ � � ᆠᆠᨨ矄矄 泐 A�矄 ᨨ e r҈ eI r҈㦨 Þ e㦨 R ⏌ 㡰 + r҈ + r Þ r҈㦨r졔R⏌ᨨR㦨 r҈ - r 㡰
A
é d êE dt ê R ë
C
I҈
L
E R҈
R
G
F
dI df d = - LI 桌 = - L V㦨 dt dt dt 㦨 - L
I
B I
R - tù E éê ᨨ I = ҈ - e L ú R ê ú ë û 꾀矄ᨨ犨├ ᨨ犨 鼀 V㡰 ᨨ 盐ᆠ ᆠᨨ矄矄 � 盐ᆠᨨ
+
㦨ᔼ҈졔ᔼ ; 꾀 꾀 ҈ � 꾀 ᨨ ��鼀ᆠᆠ�ᨨ犨� 盐泐ᆠ � ᆠ ᨨ ᆠ盐 犨� ᨨ犨 ᨨ盐� ᆠᆠ�ᨨ ᆠ盐 犨� ᨨ犨 ᨨ鼀 � 盐ᆠᨨ 盐矄 矄�矄 ᆠ ᆠᨨ矄矄 ᆠᆠ�ᨨ ᆠᨨ鼀矄 � � � ᆠᨨ矄矄 � 盐ᆠᨨ ᆠᨨ鼀矄 ᨨ \
dI - I R = dt 12V
5º
矄鼀 泐f桌
\
t
� R҈㦨R㦨WE㦨҈V L㦨鼀
ᆠ�ᆠ盐� �� �矄 R㜠L ᆠ�ᆠ盐� � 犨 �ᆠ ᆠ盐 �ᨨ犨矄 矄 矄ᨨᨨ 矄 ᆠ盐 �矄 矄犨�ᆠ ᨨ 꾀泐� �⧤�ᆠ ᨨ 矄 犨� ᨨᨨ꾀ᔴ 꾀 犨�矄� � ᨨ鼀鼀ᨨ� �ᆠᨨᆠ├犨�矄
E-L
I҈
V = I㦨 Z �
I= I 矄� w t- p ⏌㡰
D
~
V = V 矄� w t
V
I
\
)
矄� wt p ᆠ盐 �鼀� 矄 矄 ᨨ犨 ᨨ犨
B
C
V = A \ I㦨 rms
V㦨V鼀矄
㦨 ҈㡰 + ҈㡰 = ҈
҈00 µ
V鼀矄㦨 n㦨� L㦨�鼀R㦨҈҈W
ᔼ鼀 ᆠZ㦨 wL㡰 + R = ҈҈ W V 꾀矄ᨨᆠ盐 鼀�盐 I㦨 Z
æ ҈ ö ҈㦨 X c + R҈ 㦨 ç ÷ + R҈ è wC ø
\
�
D
- R ö ù æ t ç҈ - e L ÷ ú ç ÷ú è øû
R t - L V㦨E e
��矄�� 矄 ᨨ犨矄 ᨨᨨ矄��ᨨ
ᨨ �ᨨ犨 ᨨ ᆠ盐 -
t
\ V㦨 ҈ e ´҈ - 㦨҈ e� ␔ 矄犨�ᆠ �矄 ᨨ ᆠ盐 ᨨ犨矄 ᨨ盐� ᆠ�ᆠ盐�ᔴ ᨨ 泐� �ᆠ�ᨨ 矄 矄 ᨨ犨 � ��盐 B C 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 I æ L dI ö = I R҈졔 R㡰 ç è dt ÷ø L R R R + dI æ ҈ ö \ dt = -ç ҈ è L ÷ø I \ 졠 � �� � R I dI R + R 㡰 t òI I = - ҈ L ò dt \
-
R҈ + R 㡰t L
F
D
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I㦨I e
EBD_8341 PS-260
Physics
R + R ҈ = L
+
��㡰 �ᨨ犨 ᨨ ᆠ盐 � LRᆠ�ᆠ盐�
= ҈
´ ҈-
R - tù I㦨I é êë҈ - e L úû
E ҈ = = 꾀 R҈ + R I㦨e҈tAᆠᨨᆠ├犨�矄
I㦨 \ 黨
꾀矄 犨├ ᨨ犨 �泐 U =
R é - tù I = I ê҈ - e L ú ê ú ë û -R R t - t ҈ ҈ Þ = ҈ - e L Þ e L =
├� � ᨨ� ᨨ ᨨ 矄� 矄
҈ L� � U µ �
҈ ᨨ �矄 鼀 �鼀盐鼀 盐 犨 ᆠ盐 �矄 ᆠ ᨨ �矄鼀 �鼀盐鼀盐 ᔼ L㜠R ᆠ�ᆠ盐� ⼄盐�ᨨ ᨨ ᆠ盐 �ᨨ犨 �㦨� ҈e⏌tL㡰 �㦨㝐 �鼀盐鼀盐ᨨ ᆠ盐 ҈H = �s tL㦨 �鼀ᆠᨨ 矄 㦨L⏌R = W \ �㦨�⏌㦨� ҈e⏌�㡰 ҈ ҈ ᨨ = ҈ e t ⏌� Þ e t ⏌� = æ ҈ö ᨨ⏌�㦨ᔼ çè ÷ø Þ t ⏌ � = ᔼ ( ) = 黨 \ t = ( �)( 黨) ᨨr t = �s 㡰 �㡰 ␔ 矄犨�ᆠ 和�矄� ᨨ矄��ᨨ Ἐ҈Ἐ ᔼ 矄 泐矄 ᆠ盐 ᨨ盐� ᆠᆠ�ᨨ �矄 ᨨ 犨� ᆠ
R
F
R҈
G I
1W
A
C
҈V D E I
V
R W
矄� �⧤�ᆠ ᨨ 矄犨� ᨨᨨ꾀ᔴ꾀 졔 I맔҈졔I҈맔㦨 Þ I҈I㦨黨 꾀泐� �⧤�ᆠ ᨨ 矄犨� ᨨᨨ쟼 I҈졔҈ I҈졔I㡰㦨 Þ I҈졔I㦨
҈ ᨨ鼀⼄ �㡰 ��㡰I҈㦨 A
\ ᨨ鼀 ��㡰I㦨҈A ᨨ � � ᆠ 犨 A B VA졔 ҈㡰맔㦨VB Þ VAVB㦨�V ᨨ ᨨ 盐ᆠ�ᨨ ᨨ � R҈
æ ҈ö PR҈ = I҈ R҈ = ç ÷ ´ = �W è ø b㡰 �㡰␔ 矄犨�ᆠ �矄 盐 � ᨨ矄��ᨨ
ᆠ�ᆠ盐� �矄 矄 矄 ᨨ犨
ᨨ犨 ␔ 矄 泐 矄 ᆠ盐 �矄 ᆠ
� 盐ᆠᨨ泐矄 ᨨᨨ � ᆠ�ᆠ盐� E㦨I R졔R㡰
Þ
I=
E = = 꾀 R + R �
R - t L
= ᨨ� e
҈
R L 黨 ´ ҈ ´҈- t = 黨 Þ t = 黨 = L R + 㡰 犨 R㦨R졔R \t㦨҈맔҈矄 쟼 �泐矄ᨨ 泐 � 盐ᆠᨨ
Þ
E=
҈ ҈ æ ö LI = ´ ҈ ´ ҈- ´ ç ÷ = � ´ ҈ - è ø
Topic-3 : Transformers and LC Oscillations
E҈
W
ᨨ� e
m
E W
I҈졔I
I I = 矄 泐 盐
␔
B
҈
㡰 ␔ �矄 盐矄 �矄 � ᨨ � ᆠᨨ矄矄 ᆠᆠ�ᨨ ҈ ö æ V t 㡰 = V� ç҈ - e RC ÷ è ø
꾀 t = �ms = � ´ ҈- s �㡰
��㡰
҈m
�
㡰
�´҈- æ ö -黨 ҈ V t 㡰 = � ç ҈ - e ´҈ ´҈ ÷ = � ҈ - e -� 㡰 = V ç ÷ è ø
␔ ᨨ盐矄 �矄 � ᆠᆠ�ᨨ 犨� �矄ᆠ � ᨨ犨V� 㦨鼀 矄 �矄ᆠ �� � ҈
V t 㡰 = e RC = e -� = ҈҈ ᨨ犨 ᨨ �m矄
V t 㡰 =
҈ RC � - 䰘黨e
꾀 �m矄�� V t 㡰 = 䰘黨 ᨨ » ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄
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PS-261
Alternating Current
ᆠ㡰 ᨨ犨ᨨ盐盐 ᔼ㡰㦨├␔
\ =
├␔ = ᨨ矄 (҈꾀 )
b㡰 ␔ ᆠ �� � �矄 ᆠᨨ鼀 ᨨ � � ᆠ 犨 ᆠᆠ�ᨨ矄犨�V ᆠ �矄ᨨ � �矄ᆠ矄犨�鼀 �鼀盐鼀 \Q㦨CV
é
\ ᔼ 盐 ᨨ� ᨨ 矄㜠 ᨨ犨 矄 ᨨ鼀
҈ ҈ = =
V
� 盐㦨맔ᨨ盐盐㦨맔 㦨 꾀矄ᨨ
R
S
ᔼ҈ =
L
´ ᔼ矄 Þ ᔼ矄 = 黨 ´ � = �꾀 ´ � 졠盐盐 ᆠ盐 㦨 �꾀 ᆠ㡰 � 㦨矄㦨ᔼ㦨�꾀 㦨黨奠㦨黨 矄 Þ 矄 = 黨 쟼 �ᆠ� ᆠ泐 㦨黨 =
㡰 ᔼ 矄 ᨨ盐矄 ᆠ � ᨨ 矄 泐�鼀 t 盐� � �矄ᆠ �� � Q� 矄㦨Qᆠᨨ矄wt \ ᔼ 矄 ᨨ盐矄 ᆠ盐
Þ 黨 =
S҈
C
ᔼ҈ = ᔼ ҈
҈ ´ ҈ = �꾀 ᔼ b㡰 쟼 �ᆠ� ᆠ泐 h= ᨨ盐 = 矄 矄 � ᔼ \
- t ù
␔ t㦨tQ㦨CV êë҈ = e t úû 㦨ᔴ҈桌
I� 矄 㦨
dQ = Q w 矄� wt dt 힘 C 졔
I
S
矄ᔼ矄㦨黨맔ᔼ Q㦨ᔼ㡰
ᔼ矄 =
黨 ´ ´ � = �꾀 \
Topic-4 : Miscellaneous Problems
҈
æ ҈ ö ᆠ㡰ᔼ RC㜠ᆠ�ᆠ盐��鼀 ᆠ㦨 R + ç è WC ÷ø
Q㦨CV w㦨
ᆠᆠ� ᆠ � ᆠ矄 �矄 ᨨ盐 �鼀矄 ᆠᆠ� ᆠ�
ᆠ矄 꾀 \ᔼ鼀 ᆠ� ᆠ矄�矄矄矄 ᨨ ᆠ矄꾀 꾀㡰 ᨨ犨ᔼ㦨
VRA
�
VCA
Q ᔼ �矄 � ᨨ � � ᆠᆠᆠ矄矄 矄�矄 㜠ᔴ ᆠ�ᆠ盐� V =
VR
+ VC
桌
꾀㜠矄錐㜠⼄矄錐ᔴ㜠⼄錐㜠⼄矄 ᨨ ᔴ ᆠ�ᆠ盐� � 矄 泐矄 ᆠ盐 I ᨨ盐� ᆠᆠ�ᨨ ᆠ㡰 �矄 ᨨ ᔼ ᆠ矄 ᨨ 㜠ᔴ ᆠ�ᆠ盐� ᨨ � � ᆠ 㡰ᆠᨨ矄矄 � 盐ᆠᨨ 㡰�矄 ᨨ ᆠᨨ矄矄 ᆠᆠ�ᨨ㦨� ᨨ � � ᆠᔼ ᆠ矄ᨨ 㜠 ᆠ�ᆠ盐� 㦨 㡰 ᆠᨨ矄矄 � 盐ᆠᨨ 㡰 㦨 ᆠᨨ矄矄 㡰 㦨 � ᨨ� ᨨ 꾀ᔴ ᆠ�ᆠ盐� � 矄 泐矄 I鼀矄ᆠ盐 ᨨ犨矄 ᨨ盐� ᆠᆠ�ᨨ ᆠ㡰 � 盐ᆠᨨ 㡰 㡰 矄�矄ᨨ 㡰 ᨨ � � ᆠᆠᨨ矄矄 矄�矄ᨨ� 盐ᆠᨨ ᆠᆠ�ᨨ I 꾀 ᨨ ᆠ �� � ᆠ盐 ᨨ � � ᆠ ᆠᨨ矄矄 㡰� 盐ᆠᨨ 㡰ᆠᆠ�ᨨ ᆠ㡰ᨨ矄�矄ᨨ 㡰µᔴ盐 ᔼ㡰
LC
I鼀 㦨CV맔
Þ Q㦨LC
> VCB
҈
҈ C 㦨V L LC ᆠ㡰 꾀泐� � ⧤�ᆠ ᨨ Ἐ矄 犨 Q dI Q dI = L -L = Þ C dt C dt \
V \I RA < I RB Z < VRB Þ
L
I鼀 㦨Qw
d æ dQ ö d Q çè ÷ø = - LC dt dt dt
㡰 和 盐 矄 ᨨ鼀
矄 矄 矄 ҈ Þ = = ҈
\ 矄㦨 和 ᨨ犨 矄 ᨨ鼀 䰘
b㡰 ᨨ犨㦨맔ᔼ
矄
=
矄
㦨
= ҈
´ ҈ = 㦨҈�꾀 꾀
ᨨ矄�矄 ᆠ㦨맔㦨W \ ᨨ犨 �矄矄� 矄 㦨 ᔼ 㦨 ҈�㡰 맔 㦨҈␔㦨҈├␔ ҈ ´ ҈ = 奠 \ 奠ᨨ矄矄㦨 Þ
ᔼ㦨
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EBD_8341 Ps-262
Physics
Electromagnetic Waves
22
Topic-1 : Electromagnetic Waves, Conduction and Displacement Current ҈
㡰 �ᨨ 犨 ᆠ�ᆠ � E 鼀� �ᆠ � Bᨨ
ᆠᨨ鼀� �ᆠ犨�矄�� 泐
E c㦨和 ᨨ �� 㡰 B Þ E = B ´ c = ҈ ´ ҈-䰘 ´ ´ ҈ = c=
꾀矄 犨�矄 ᨨ�� � ᨨ � x㜠 �ᆠ�ᨨ 鼀� �ᆠ � �矄 ᨨ � z㜠 �ᆠ�ᨨ
r Eˆ ´ Bˆ 㡰 Cˆ \ E 矄 ᨨ盐 ᨨ �y㜠 �ᆠ�ᨨ r r 和ᨨᆠ�ᆠ � E = E 矄� E × x t 㡰
V = - 矄� � ´ ҈ x + ҈� ´ ҈ t 㡰 ˆ 桌 m
҈҈
ᆠ㡰 ᔼ ᆠᨨ鼀� �ᆠ犨
䰘
E =C B
qV E C
盐 �ᆠᨨ Cˆ �矄 �ᆠ盐ᨨᨨ Eˆ Bˆ
ur
E C
� V㦨҈ᔴ E㦨㡰 ҈ ´ ҈ -҈黨 ´ ҈ ´ ´ ҈ ´ = = ´ ҈ -҈黨 ´ ҈ r 㡰 E = E xˆ + yˆ 㡰 矄� kz - wt 㡰 �ᆠ�ᨨ ᨨ ᨨ��ᨨ ᨨ 鼀犨 = + kˆ
�ᆠ�ᨨ ᨨ ᆠᨨ鼀� �ᆠ 犨 �矄 �ᆠ盐 ᨨ ᨨ ᆠ�ᆠ 鼀� �ᆠ �
\ kˆ = Eˆ ´ Bˆ æ �ˆ + ˆ ö ˆ -�ˆ + ˆ Þ kˆ = ç ÷ ´ B Þ Bˆ = è ø r E \B = - xˆ + yˆ 㡰矄� kz - wt 㡰 c ˆ 㡰 � B = ´ ҈- 矄� p y + ct 㡰桌�T - \ B = ´ ҈ E = CB Þ E = ´ ҈ ´ ´ ҈ - = 黨 ⏌鼀
黨
҈
E = c Þ E = Bc B ÞE㦨�맔҈ 맔맔҈㦨҈�N⏌C r Þ E = ҈��ˆ Vr ⏌ m b㡰 � B = ´ ҈ - 矄� ҈ ´ ҈ x + ´ ҈҈ 㡰 矄� � E = B ´ C = ´ ҈ ´ ´ ҈ = 黨 V⏌m \ 쟼ᆠ�ᆠ � r E = 黨矄� ҈ ´ ҈ x + ´ ҈҈ t 㡰 ├ˆ V ⏌m
҈҈
ᆠ㡰 꾀t㦨z㦨
\
E ´ B 㡰 C Bˆ = �ˆ Cˆ = - ˆ \ Eˆ = - kˆ \ E = E 矄� p y + ct 㡰桌 - kˆ㡰 ⏌鼀 ᨨ E = -黨矄� p y + ct 㡰桌kˆ ⏌鼀
ᔼ 矄ᆠ �矄矄 c =
E c
m 㦨 矄ᨨ盐 鼀��泐 e 㦨 矄ᨨ盐 鼀���泐
\ B =
E E = = E m e c ҈⏌ m e
r E E E = �ˆ + ˆ 㡰 ᆠᨨ矄 p 桌 = �ˆ + ˆ 㡰
ᨨᆠ 盐ᨨ ᆠ�ᆠ � 犨�� �ᆠ�ᨨ
҈ m e
p k
r r FE = qE
ᆠ㡰 �ᨨ 犨 ᆠ�ᆠ � 鼀� �ᆠ � ᨨ
ᆠᨨ鼀� �ᆠ犨� ᆠ盐盐鼀�矄 B =
kˆ �ˆ + ˆ �ˆ + ˆ Þ Cˆ = ҈ =
㡰 �ᆠ�ᨨ ᨨ ᨨ�矄�ᨨ 㦨 Eˆ = kˆ $ $ µ´B µ = �+ �ᆠ�ᨨ ᨨ ᨨ��ᨨ 㦨 E ˆ- r r � 盐 EB = \ Bˆ = 㡰 꾀鼀�盐 ᨨ ᆠ�ᆠ � E㡰 㝐� �ᆠ � B㡰ᨨ
ᆠᨨ鼀� �ᆠ犨 泐 �ᨨ
�ᆠ�ᨨ ᨨ 犨 ᨨ��ᨨ
�
ur
� E = k$ B = $� - $ �ˆ ˆ ҈ \ Cˆ = Eˆ ´ Bˆ = ҈ -҈
�ˆ + ˆ �ᆠᨨ� �ᆠ�ᨨ ᨨ ᆠ�ᆠ � Eˆ =
\ B = ´ p ´ ҈ -䰘 ´ ҈ ´ ҈ - = ҈ ´ ҈ -黨 㦨҈
㡰 쟼ᆠᨨ鼀� �ᆠ 犨 犨� ᨨ� �ᆠ盐 ᨨ �ᆠ�ᨨ ᨨ 쟼ᆠ�ᆠ 㝐� �ᆠ � 矄
Cˆ = Eˆ ´ Bˆ
\㝐 �鼀盐鼀盐ᨨ 鼀� �ᆠ � B =
F鼀 = qVB鼀 矄� 黨° =
꾀矄 ᆠᨨ鼀� �ᆠ犨�矄ᨨ�� �ᨨ �x �ᆠ�ᨨ ᆠ�ᆠ � �矄ᨨ �y �ᆠ�ᨨ \ Erˆ ´ Bˆ Cˆ Cˆ 㦨 �ᆠ�ᨨ ᨨ ᨨ��ᨨ ᨨ 犨㡰 \ B 矄 ᨨ盐 � kˆ �ᆠ�ᨨ \ 㦨 쟼 m e ᆠᨨ矄 w├ 㡰 ├ˆ 꾀㦨 㦨 쟼 m e ᆠᨨ矄 ├ 㡰 ├ˆ ҈ B b㡰 쟼 �泐 矄�泐 = Þ B = ´ m ´ 쟼 �泐 矄�泐 m ҈ m = = p ´ ҈ -䰘 C e
҈
- �ˆ + ˆ 㡰
ᨨᆠ 盐 ᨨ鼀� �ᆠ � �矄 � �ᆠ�ᨨ
r r r r r q v ´ B㡰 v k ᨨ � �矄 ᨨ E r r r �ˆ + ˆ Þ F = FE + FB �矄 �ᨨ ˆ + kˆ ˆ + kˆ ᆠ㡰 Sˆ = = � +
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PS-263
Electromagnetic Waves ҈
E = C ´ ҈ 㦨맔҈ 㦨맔҈䰘
w pf p ´ 黨 ´ ҈黨 = K㦨 = 㦨� v v ´ ҈
҈ ��㡰 ├ Î m �� � �⼄盐�ᨨ �㡰泐 ��㡰犨� ᔴ \ = ├=
҈ ҈ Î 쟼ᔴ 㦨 � 矄�泐㦨 Î ├쟼 ᔼ 鼀 �盐鼀 =
b㡰 B㦨
®
ᨨ B = B 矄� kz - wt 㡰
㦨맔҈䰘 矄� � ´ ҈ z - ҈� ´ ҈҈҈ t 㡰� ҈
ᔴᨨ矄ᨨ � �泐
҈�
\ 쟼 ᔴ = ├쟼
쟼 쟼 㡰 = ᔴ Þ = ᔴ r � 쟼 = 쟼 ᆠᨨ矄 ├ 㡰 ᆠᨨ矄 w 㡰 �ˆ r 쟼 쟼 = éëᆠᨨ矄 ( ├ w ) ˆ� ᆠᨨ矄 ( ├ + w ) �ˆùû r = éëᆠᨨ矄 ( ├ w ) ˆ ᆠᨨ矄 ( ├ + w ) ˆùû r = ´ 矄� ├ 矄� w r æ쟼 ö = ç 矄� ├ 矄� w ÷ ˆ è ᔴ ø I ᆠ㡰 矄矄盐 P = C F I IA Dp I Þ = Þ F = Þ Dp = ADt = A C C Dt C � ´ �㡰 ´ ҈ ´ ҈ ´ ´ 㜠矄 = ´ ҈
Þ
҈䰘
҈
r 쟼 ˆ = ᔴˆ = = ҈ ´ ҈- 쟼ˆ ´ = ᔴ ´ ҈ ˆ = �ˆ [Q EM 犨矄ᨨ �졔 㡰x㜠 �ᆠ�ᨨ 桌 ˆ ´ r ˆ \ ˆ = ├ˆ ᨨ = ҈ ´ ҈ ├
IA b㡰 F = ҈ + r 㡰 C ҈ + �㡰 ´ � ´ ҈ ; ´ ҈- = ´ ҈ 㡰 �ᨨ 犨 鼀�盐 矄ᨨ ᆠ�ᆠ 鼀� �ᆠ � � 矄ᆠ �矄�� 泐
�
uur uur
uur
uur
ᨨ �ᆠ�ᨨ ᨨ E ® ( - �ˆ + ˆ )
҈
· ᔴ Þ = ᔼҘ Ҙ ᔴ ᔼҘ ҈ p ´ ҈-䰘 ´ = ; 맔҈
Þ 鼀矄 = ᔴ ´ ҈ ␔ �ᆠ �矄ᆠᨨ矄矄ᨨ҈ ᆠ㡰 矄 ᨨ ᆠᨨ鼀� �ᆠ犨� 矄ᆠ�矄�� 泐 ҈ ᔴ= �㡰 m Î
㡰 ᔼ =
(
)(
)
ᆠ㡰 ᨨᆠ�泐ᨨ 쟼㝐犨 �矄�� 泐 v =
�ˆ + ├ˆ = ᆠˆ �
҈ mÎ
w =C k C ᨨᆠ�泐� 鼀 �盐鼀㦨 m҈㦨m㦨҈矄鼀 �盐鼀�矄 ᨨ 㜠鼀� �ᆠ ҈ Îr҈ ҈ Îr҈ C \ = = = Þ ҈ C æ ö Îr ç ÷ Îr è ø
ᨨ - K Þ E ´ �ˆ + ˆ 㡰
b㡰 矄� � ᨨ鼀盐쟼㦨맔ᔴ 㦨҈맔҈맔맔҈ 㦨맔҈ ⏌ᔴ 犨 矄矄盐鼀 㦨҈맔҈�矄� 矄 㡰盐 �矄 r b㡰 쟼 = ҈ˆᆠᨨ矄 éë �ˆ + ├ˆ �ˆ + ├ˆ ùû
ᨨᆠ�泐� �㦨
꾀矄ᨨ S Þ E ´ B
uur
҈ ´ 黨 ´ ҈ ҈ ´ 쟼 ´ ´ ҈
( ├ˆ �ˆ ) ᆠᨨ矄 ( + ҈ᆠ )
\ 鼀� �ᆠ � ᆠᨨ鼀ᨨ 犨�ᨨ �z �ᆠ�ᨨ ᆠ㡰 쟼㦨ᆠ㦨맔҈맔҈맔҈ 㦨맔҈⏌鼀
uur
=
犨矄 � �ᆠ�ᨨ ᨨ �ᆠ �ˆ + ├ˆ ᔴˆ ´ 쟼ˆ = � r 쟼 ҈ = = ᔴ ᔴ r ҈ æ �ˆ + ├ˆ ö æ �ˆ + ├ˆ ö \ ç = ÷÷ çç ᔴ ÷÷ ᔴ çè � ø è ø r ҈ ᨨ鼀� �ᆠ � ( ) = ᔴ
B =
҈黨
҈ ´҈
r r 㦨 ҈ ˆᆠᨨ矄 éë ⧤ ùû 쟼 ҈ ˆ㡰 r ˆ �ᆠ�ᨨ ᨨ \ ⧤ = �ˆ + ⧤錐
㦨�맔҈ 㜠矄 b㡰 꾀矄 r犨 ├ ᨨ犨
E = = ´ ҈ - T c ´ ҈ ᨨ��ᨨ �ᆠ�ᨨ = Eˆ ´ Bˆ �ˆ = ˆ ´ Bˆ Þ Bˆ = kˆ
䰘 ´ ҈
Þ 쟼 = ´҈ ├ ⏌ 鼀 =҈├ ⏌ 鼀
҈
쟼 ├ 쟼 = = Þ =
쟼 ᔴ
쟼 矄�鼀�泐 ᔴ ҈ = Þ = m m
㡰 쟼㝐 犨 � 矄�泐 ᨨ犨 ҈ = e 쟼ᆠ Þ ᔼ = 꾀 犨 쟼㦨鼀 �鼀盐鼀ᆠ�ᆠ � 桌 Þ
㡰 꾀� �泐 矄�泐ᨨ 鼀� �ᆠ �
盐 =
m
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EBD_8341 Ps-264
Physics
꾀� �泐 矄�泐 ᨨ ᆠ�ᆠ � e 쟼 盐쟼 = ҈ ᨨ犨쟼㦨ᔴ ᔴ㦨 m Î e ҈ e ´ = = 盐 盐쟼 㦨 ´ ᔴ = ´ m e m \ 盐쟼㦨盐 和� ᆠ �泐 矄�泐ᨨ ᆠ�ᆠ 鼀� �ᆠ � �矄矄鼀 矄ᨨ �泐矄矄ᨨᆠ� 犨� ⼄盐ᨨ盐鼀犨�⼄盐� 盐쟼㦨盐 䰘
ˆ ´ ˆ 矄 ᨨ盐 �� �ᆠ�ᨨ ᨨ 犨ᨨ��ᨨ
ᆠ㡰 쟼 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) æ ˆ ˆö ˆ ˆ P�´Þ⧤ ˆ ´ � + = - - �㡰 = � + P � + Þ ⧤´ ç ÷ è ø 졠�ᨨ 㡰 ᨨ�ᨨ 㡰 ᨨ�ᨨ 㡰 ᨨ矄 ᨨ 矄�矄 泐 ˆ� + ˆ ᆠ㡰 � 쟼ᆠ�ᆠ � ᆠᨨ鼀ᨨ ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ ��ᨨ
ˆ 矄 ᨨ盐 ᨨ � ␔ᨨ��ᨨ ᆠᨨ ⧤
ᨨ 쟼 =
҈
쟼ᆠ矄矄
�㡰
├ 矄� ├ 矄� w w
�
␔矄ᨨ├ ᨨ犨 쟼 w = =ᆠ ├ 㝐� �ᆠ � ᆠᨨ
r 쟼 ˆ 矄� ├ 矄� w = ᆠ
㡰 ␔� 㜠 �ᆠ�ᨨ 鼀 矄쟼 矄 ᨨ盐 盐 ᆠ�ᨨ ᨨ
҈
uur ur uur ur b㡰 꾀矄 犨 ├ ᨨ犨 E B = Q E ^ B 桌 uur ur E ´ B 矄 ᨨ盐 ᨨ � �ᆠ�ᨨ
꾀矄 $� $ 㡰 ´ $� $ 㡰 = � k$ ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ 矄犨 ҈ ᆠ㡰 E㦨CB C = m e ҈ 쟼ᆠ�ᆠ �泐 矄�泐㦨 e E = m E ҈ Bᨨ = mB 㝐� �ᆠ �泐 矄�泐㦨 m
盐矄mE㦨mB 쟼 �泐�矄⼄盐泐 �� 犨 ᆠ�ᆠ 鼀� �ᆠ � b㡰 ␔ � ᨨ 鼀ᨨ ᨨᆠ ᨨ鼀�ᆠ� �� 㦨��맔҈�ᆠ鼀 ҈ ´ ( ⏌҈ ) ᨨ犨 ␔鼀 - ᔼ 矄�泐ᔼ㦨 㦨 㦨 꾀 ҈ p p ( � ) ᨨ犨 ᨨ �矄� 矄�泐 ᔼ㡰ᨨ �矄ᨨᆠ�ᆠ � ᨨ ᨨ鼀� �ᆠ � ᨨ ᔼ ҈ = e 쟼 ᔴ
ᆠ㡰
g㜠泐矄㜠泐矄盐㜠泐矄 �矄�泐矄 ᔼ 泐矄 �ᨨ ᔼӸ졠 㝐�ᆠᨨ犨矄 犨矄
�ᨨ犨泐ᨨ犨 �� 盐�� 㜠泐矄 ᔼ ᆠ矄� �ᨨ ᨨ �泐㡰 ᆠ㡰 㝐�ᆠᨨ犨ᨨ ᆠ矄ᨨ � ᆠ�ᨨ ��� ���ᨨ �泐ᨨ犨 鼀ᨨᆠ盐矄 ᆠ㡰 鼀鼀 泐矄 㜠泐矄 �ᨨ �矄� 泐矄 ᔼ 泐矄 㝐�ᆠᨨ犨矄 �ᨨ 犨矄 㡰 쟼 = Þ l =
) ) ) ) 泐- ^ 泐+
b㡰 ᨨ 泐 �鼀 ᨨ � 矄ᨨ 쟼㝐 犨�
ᆠ矄� �ᨨ ᨨ 犨 � �矄矄 ᨨᨨ犨矄
l �ᨨ犨矄 > l 鼀�ᆠᨨ犨矄 > l �矄� > l 㜠泐矄
= +쟼 ├ 矄� ├ ò ᆠᨨ矄 w
黨
)
Topic-2 : Electromagnetic Spectrum
㝐� �ᆠ � �矄�� 泐
= +쟼
æ ö p÷ ´ç è ҈ ø 㦨 ´ 㦨 䰘 㦨 æ ö ҈ � ´ ´ ҈ ç ÷ è p´ 黨 ´ ҈黨 ø \ 쟼㦨⏌鼀 b㡰 ᨨ鼀 ⼄盐矄�ᨨ B㦨nT㦨맔҈黨T 鼀矄҈㡰 Qᨨᆠ�泐ᨨ �� � ᆠ盐盐鼀C㦨맔҈ r r r E = B ´ C r r r E = B × C = ´҈ -黨 ´ ´҈ 㦨⏌鼀 b㡰
(
r 쟼㡰 = 쟼ˆ� ᆠᨨ矄├ ᆠᨨ矄 w
쟼 ␔├ ᨨ犨 = 쟼 = -쟼 ├ 矄� ├ ᆠᨨ矄 w = 㦨졔쟼├矄� ├ ᆠᨨ矄w ᔼ �� �⼄ �㡰犨
ᔼ e ᔴ
䰘
黨 ҈
ᆠ ᆠ Þ l = l 쟼 ´҈- ´ ´҈
҈҈´҈ ´҈ ´҈-҈黨
ᔼ ᆠ矄� �ᨨ ᨨ ⼄盐 ᆠ泐
㜠泐矄 盐㜠泐矄 �矄� ᔼ 㦨҈处 犨 � �ᨨ �矄���ᨨ �矄处ᨨ䰘处 㡰 쟼 �泐矄⼄盐 ᆠᨨ ��ᨨ 矄�矄
Eg 㜠 泐矄 > E㜠 泐矄 > E鼀�ᆠᨨ犨 > E꾀㝐 �ᨨ犨矄 \ l g 㜠 泐矄 < l 㜠 泐矄 < l 鼀�ᆠᨨ犨 < l 꾀㝐 �ᨨ犨矄
ᨨ鼀 ᨨ矄⼄盐 ᆠ犨 㡰 㝐�ᆠᨨ犨 ® ҈- 鼀 �㡰 㡰 鼀鼀 泐矄 ® ҈-҈� 鼀 ��㡰 ᆠ㡰 꾀㝐 �ᨨ 犨 ® ҈ 鼀 �㡰 㡰 㜠 泐矄 ® ҈-҈ 鼀 ���㡰 ᆠ㡰 졠�ᆠ �ᔴᨨ鼀鼀盐 �ᆠ�ᨨ ᔼ �� �ᨨ␔矄 和ᨨ 矄ᨨ盐 㝐ᨨ� ᨨ 矄 㝐�ᆠᨨ犨矄 㡰 ҈㡰 ᔼ 泐矄盐矄 ᨨ鼀盐矄ᆠ盐矄� ᆠ盐矄 矄 泐矄 㡰 �ᨨ犨矄盐矄 ᨨᨨ ᆠ矄� �ᆠ盐矄 矄犨矄 泐ᨨ �犨 � �� � ᨨ鼀 犨ᆠ �鼀矄ᨨ 犨 盐 ├�ᨨ鼀矄 㡰 㜠泐矄 盐矄 ᨨ ᆠ ᆠ盐 ᨨ ᨨ 矄 ᆠ盐矄 泐 �� � � ᨨ犨 盐 泐 ᆠ Ἐ ᨨ盐� 矄鼀 �盐鼀�├ ᨨ 矄 㡰 �ᨨ 泐矄 矄ᨨ 泐ᨨ ᨨ ᨨ 鼀ᨨ矄 㡰 ␔ � 鼀� 泐ᨨ鼀�ᆠ 泐 ᨨ� � � 矄矄ᆠ 㜠 ᨨ 矄 ᨨ �ᨨ犨ᨨ ᆠᨨ鼀� �ᆠ矄ᆠ盐鼀 ᨨ盐 ᨨ 矄ᨨ �盐鼀 㜠�矄� ��ᨨ 㡰 ��ᨨ ᨨ ᨨ鼀矄 鼀ᨨᆠ盐矄䰘 鼀ᨨ҈鼀鼀 �ᨨᆠ� ᆠ泐ᨨ 盐ᆠ盐矄҈ 鼀 㝐� ᨨ ҈鼀鼀ᨨ҈鼀
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Ray Optics and Optical Instruments
Ray Optics and Optical Instruments
23
Topic-1 : Plane Mirror, Spherical Mirror and Reflection of Light ҈
㡰 졠ᆠ�矄ᆠ 泐ᨨ �盐矄ᨨ ᆠ盐盐 R㡰ᨨ ᆠᨨ ᆠ 鼀�ᨨ ᆠ �鼀� ᨨ鼀 �矄 � �鼀� �矄 ᨨ 盐 鼀� � �
ᔼ鼀� =
h C h�
O
F ᔼ鼀�
P
䰘
h� < h
␔ ᨨᆠ�矄ᆠ鼀 ´ 盐 �´ ᔼ鼀� ҈ = ᆠ鼀 = =盐 - -� ␔ ᨨᆠ�矄҈ᆠ鼀 ᆠ㡰
´ 盐 � ´ ҈ = =ᆠ鼀 盐 - ҈ - � 䰘
҈ 和�ᨨ = ҈ - = ᆠ鼀 = 䰘 ҈ 和ᨨ꾀ᔴ ᆠᨨᆠ矄鼀 矄 ᆠ㡰 ᔴᨨ 鼀ᨨᨨ �矄 盐矄 矄 矄 � � 鼀�ᨨ 졠
v ᆠ㡰 +� = - Þ v = -�u u ҈ ҈ ҈ ҈ ҈ ҈ + = 矄� � + = ᨨ v u f -�u u
҈ᆠ鼀
\ 盐㦨鼀 ᨨ鼀 ⼄盐矄�ᨨ v 㦨 ҈� ᆠ鼀 u 㦨 ҈ ᆠ鼀 �盐矄 ᨨ ᆠ盐盐 R 㦨 f 㦨 䁸 ҈ ҈ ҈ 矄� � 鼀�ᨨ ᨨ鼀盐 + = v u f
和
ᆠ㡰
ᨨ �矄 ᆠ =
+ =
b㡰 M1
q
ᨨ鼀 ��盐 �㦨
黨 O
q
q 턼
M2
�犨 犨ᨨ鼀�ᨨ矄q 泐턼 P 鼀�ᨨ㝐҈ 矄 P 鼀�ᨨ㝐 \ 㝐҈ 矄㦨Ð졠 턼㦨Ð㝐҈졠㝐㦨 q 和�鼀�泐 Ð㝐턼㦨 Ð졠턼 㦨Ð㝐졠㝐҈㦨 q
�
҈⏌
=
҈
ᔼ D졠 턼q㦨҈ Þq㦨
ᆠ㡰
矄� �鼀�ᨨ ᨨ鼀盐鼀� � �ᆠ�ᨨ �矄�� 泐
҈ f = u f ҈ u f 꾀 ᨨᆠ盐矄 鼀� � �ᆠ�ᨨ �矄 ¥ 꾀 u㦨f 鼀� � �ᆠ�ᨨ �矄҈ ᆠ � 㡰 ᆠᨨᆠ泐 �ᆠ矄 �m 矄盐矄 �矄 ᆠ ᨨ ᨨᆠ�x�
b㡰 꾀矄矄 ᨨ犨 � ��盐犨 ᨨᆠ O㡰�矄ᆠ 犨
F C �鼀� I㡰�矄 ᨨ鼀 泐ᨨ Cᔼ�矄� �矄ᆠᨨ ��ᨨ 犨 矄盐 矄 � 矄 �矄泐矄ᨨ犨 矄 �鼀�矄ᨨ �� ᨨ ᨨᆠ� 졠ᆠ졠�矄犨 ᨨᆠ盐矄 f 㡰 ᆠ ᨨ ᆠ盐盐 f㡰f xf 㝐ᨨ鼀 ᨨ犨 矄
F
҈ 㦨
\
m=
҈ �
⏌ \ �㦨 꾀 �ᨨ � ᆠ� ᆠ㡰
q
҈ ҈ ҈ + = Þ f 㦨 ᆠ鼀 ҈� -҈㡰 f
ᨨ �盐矄 ᨨ ᆠ盐盐 R 㦨 f 㦨 ᆠ鼀 㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ 泐 ᨨ �� �矄 � � �
҈ ᨨ鼀 � + 㡰 �ᆠ�ᨨ � ᆠ� ᔼ ᆠ� � ᨨ 鼀�ᨨ 泐
q
҈�ᆠ鼀
I
҈
C
P
O
V
㡰 泐 ��鼀�矄矄 ᨨᨨ犨矄 ᨨ鼀 ��盐WX㦨PQ㦨d
d ZS㦨RS 㦨
Q
PW㦨PS
\
d VW㦨RS㦨 㦨d
P
q
R q S
d
W
Z 턼
X
T L
U
和�鼀�泐 XY㦨d \ VY 㦨VW졔WX졔XY㦨d졔d졔d㦨d
L
Y
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EBD_8341 P-266
҈҈
㡰 \
Physics ᨨ鼀鼀�ᨨ ᨨ鼀盐
҈ ҈ ҈ dv du ᨨ鼀鼀�ᨨ ᨨ鼀盐 + = \ = v u f v u æ f ö dv - v æ f ö = = -ç Þdv㦨 çè - f ÷ø dx ÷ du u èu- f ø
æ f ö ᔼ鼀�矄� 㦨 ç 맔b è u - f ø÷ 㡰 矄� �鼀�ᨨ ᨨ鼀盐 ᨨ �矄ᨨ矄��ᨨ
� æ Rö ᆠ鼀 f = +҈ᆠ鼀 ç = ÷ u҈ = 䁸 v҈ = è ø ҈ ҈ ҈ ҈ ҈ + = + = v҈ u҈ f � u҈ ҈ \u҈㦨�鼀 矄� �鼀�ᨨ ᨨ鼀盐 ᨨh矄ᆠᨨ ᨨ矄��ᨨ
\
҈
u㦨䁸v㦨
҈
҈
҈
҈ ҈ ҈ + = v u f u v盐矄ᨨ ᨨ�ᨨ 矄 㡰 㡰鼀ᆠ 犨� 鼀�ᨨ ᨨ鼀盐 ␔ 矄ᨨ�ᨨ ᆠ㡰 㡰 ᨨ ᨨ鼀ᆠ 犨� 鼀�ᨨ ᨨ鼀盐 ᨨ 㡰
꾀泐� � 鼀�ᨨ ᨨ鼀盐
Þ
� f㦨҈ᆠ鼀 䰘
҈ ҈ ҈ 䰘 ҈ ҈ ҈ ҈ 䰘 + = Þ + = Þ = v u f � u ҈ u ҈ � u㦨�鼀 u҈ - u � ҈ ´ = ├鼀 -҈ 和 ᨨ ᨨᆠ㦨 㦨 �鼀 � ҈㡰 �矄 ᆠᨨ ᨨᆠu㦨ᆠ鼀 �矄 ᆠᨨ �鼀�v㦨҈ᆠ鼀 -v -҈㡰 ҈ = = 㝐� � �ᆠ�ᨨ m = u - ҈ 和 ᨨ �鼀�㦨m맔矄 ᨨ ᨨᆠ = ´ 黨 㦨҈ᆠ鼀矄҈ 黨 和� ᆠ �鼀� ᨨ鼀 �矄 ᨨ � ᨨᆠ �矄 犨 ᨨᆠ盐矄 ᆠ ᨨ ᆠ盐盐 ᔴ 矄�盐�ᨨ �矄 矄 矄 ᨨ犨 � ��盐和� ᆠ �鼀�ᨨ B�矄 ᨨ鼀 BἘ�矄
A
҈䰘
f⏌
҈ ҈ ҈ Þv㦨䰘 + = v -䰘 -
矄ᨨ盐ᨨ�矄黨䰘㦨犨 �ᆠ �矄� ᆠ鼀 ᆠ㡰 ᨨ鼀盐 ᆠᨨ
ᆠ� �u v f ᨨ 矄 �ᆠ 鼀�ᨨ
҈
㡰 꾀
B�矄矄�盐 ᆠ ᨨ ᆠ盐盐 �矄 �矄 ᆠ ᨨ f ᨨ鼀 ᨨ
\
҈ ҈ ҈ ҈ ҈ ҈ + = Þ + = -� f v - f u v f ҈ ҈ \v㦨�f Þ =- + v f �f ᔼ鼀� � ᨨ ᨨ 㦨� f f㦨�f
�f v = 䁸
ᨨ鼀 ᨨ鼀盐
t҈ t h h h h + = + = = m҈ m �
ᆠ㡰
ᆠ�� m =
矄� q = ҈矄� 黨° 䰘 Þ 矄� q㦨 錐 ᆠᨨ矄 q = 和ᨨ� �W㦨p ҈ᆠᨨ矄q㡰 = p ҈ 䰘 ⏌ 㡰 ᆠ�ᨨ ᨨ �泐 矄鼀�
B¢
p ҈ ᆠᨨ矄 q㡰 ҈ 䰘 ⏌ = = ҈䰘 p ᆠ �ᨨ �� 鼀�矄ᨨ盐ᨨ 矄盐 ᆠ 㦨҈䰘맔҈㦨҈䰘奠 b㡰 ᨨ鼀 ⼄盐矄�ᨨ � 鼀���泐 =
B
A
D=
v � f fù é \ 㝐� � �ᆠ�ᨨ M = = = ҈� êG�ven u = ú û u f ë 㡰 �矄 ᆠᨨ ᨨ� 꾀 ᨨ鼀 鼀�ᨨ�矄f⏌
A¢
m҈ =
m =
f �f PA = f - =
盐矄 � �鼀�ᨨ A ᨨᨨ� A u = -
҈�
v=-
҈ ҈ ҈ + = �矄 � ᨨ 泐 ᨨ 鼀�ᨨ矄 ᨨ 矄鼀盐矄 犨 u v f 矄� ᨨ 盐 �矄 泐矄鼀矄 ᆠᨨ鼀 ᨨ �盐矄 ᨨ ᆠ盐盐ᨨ 鼀�ᨨ 犨矄 ᨨ 鼀�ᨨ � ᨨ 矄犨 矄 � 矄 �ᆠ 矄盐 ᆠ矄 犨矄ᨨ ᆠ�ᨨ � 犨 �� �矄 ᆠ Topic-2 : Refraction of Light at Plane Surface and Total Internal Reflection
B
\
҈ ҈ ҈ ҈ ҈ - + - = - = + = = v f u - ´ ´
´ = -䰘䰘 盐 盐ᨨ v㦨 矄ᨨ盐ᨨ�矄䰘䰘㦨䰘ᆠ鼀 犨 �ᆠ �矄� ᆠ鼀 ᨨ犨ᨨ � � � ᨨ 䰘黨㡰犨 u㦨䰘
Þ
A'
B' P
҈ ҈ ҈ ҈ ҈ ҈ + = Þ = - = \ v㦨f v f ⏌ - f v f f f ᔼ鼀�A¢B¢ᨨ � AB矄 ᨨ盐 ᔼ� ᆠ� �矄ᔼ鼀�ᨨ F犨� ᨨ鼀 � � �泐 꾀矄ᨨ �� 泐 ᨨ鼀 � � �泐ᨨᨨ犨 矄� � �泐矄鼀矄ᨨ � ᆠ� �矄 ᨨ 鼀�ᨨ ᆠ 㡰 � f㦨ᆠ鼀
F
er =
e = Þ e = e e
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P-267
Ray Optics and Optical Instruments m � 鼀��泐 m r = m = Þ m = m
矄� 꾀 矄� 黨° - 꾀㡰 矄� 꾀 矄� q ᔴ = ᆠᨨ矄 꾀
ᨨ鼀⼄ �㡰 矄� q ᔴ =
\ me = m e m e v ҈ æ = = çQ c = c çè me
ö ÷ m e ÷ø ҈
矄� q ᔴ = 꾀 黨
´ =
n = mr er =
�
ᆠ㡰
҈ ҈ ҈ Þ 矄� qc = = n 矄� qc ᔴ��ᆠ �qᆠ㦨
Þ 矄� ᔴ =
æ ö -҈ ç Ҙ ÷ ᨨq鼀 㦨 矄� ç - ҈ ÷ Ҙ ҈ è ø æ Ҙ ö -҈ ᨨT҈Rq 矄� ç - ҈ ÷ ç Ҙ҈ ÷ è ø 㡰 ᨨ鼀 �� ��盐 꾀矄矄� ᨨ㦨m 矄� ᨨ
҈
=
䰘
҈҈
졠
= ´ ҈�ᆠ鼀 䰘 䰘
n ҈҈�鼀鼀 q
P
҈҈�⏌
n㦨 ᆠ�� ᨨ �⼄盐� ҈ + ҈҈�⏌㡰 ҈҈� ⏌ n = 䰘 ´ \n㦨҈ \ æ ҈҈� ö (҈) + ç ÷ è ø 㡰 꾀 ᨨ� 꾀泐和 矄 犨
矄� �° ҈ Þ 矄� = … �㡰 矄� m 꾀 ᨨ� ᨨ ᨨ� ᆠ�ᨨ m=
-
´҈ ´ ᆠᨨ q
矄� �҈ =
= �䰘䰘� » �䰘
´ ᆠ㡰 꾀矄奠 ᨨ �� �矄 ᆠ 矄ᨨ ᨨ 泐 ҈ 㦨 黨奠㡰 ᨨ �� ᆠᨨ鼀矄 ᆠ�ᨨ 矄ᨨ 黨 黨 ᆠ = ᔼ Þ ⧤ 꾀 = ⧤҈꾀� ҈ ҈ 黨 ҈ 黨 ´ ´ 㡰 ҈꾀� Þ 꾀 = Þ 꾀 = ҈ ҈ 꾀 ´ 㡰 = ҈ 㡰
m 矄� � �㡰 = m 矄� Q Ð� = 꾀 Ð = 黨° - 꾀㡰 m ␔矄ᨨ├ ᨨ犨 矄� qᔴ = m
ᨨ鼀和 Ἐ矄犨
ᆠ㡰
矄� q㦨
㡰 矄� �和 矄犨ᨨ ᆠ�ᨨ ҈맔矄� 㦨҈҈矄� q Þ 矄� q = = 黨 » � ҈҈ Þq㦨 x㦨Ҙ鼀맔ᆠᨨq
´ ҈
䰘 㦨黨 ᨨ =
=
꾀泐� � 和 矄 犨 ᨨ� P n҈矄� q㦨n矄� 黨
= +
\盐鼀ᨨ ᆠ�ᨨ 矄 =
ᨨ
҈鼀鼀 n҈ q
졠�ᆠ � 㦨 꾀졠 졔 m졠
= + 㡰
+ ҈ 㦨黨 졔黨
�⼄盐�
= ᆠᨨ矄 ᨨ Þ 꾀졠 = 꾀졠 = ᆠᨨ矄 ᨨ Þ 졠 = 졠
҈ = m
Þ
=
矄� ᨨ
矄� 黨° =m 矄� ᔴ
-҈ 矄� �矄� q鼀 㦨 Ҙ҈ Ҙ - Ҙ҈ 㦨 Ҙ҈
Þm=
ᔴ
Ҙ
矄� ᨨ
㦨҈�ᆠ鼀 㦨䁸
꾀 n = \
㡰 � Ҙ㦨
ᨨ꾀㦨 ҈ 矄� qᔴ㡰
ᔼ ᆠ� 泐 矄 꾀 꾀 Ҙ㡰 黨 Ҙ 㡰
҈ m
�
꾀�
ᨨ鼀 ��盐 �҈ = 黨° - \ 矄� 黨° - 㡰 = Þ ᆠᨨ矄 =
҈ m
�
҈ m
… ��㡰
ᨨ犨 ᆠᨨ矄 = ҈ - 矄� = ҈ -
ᨨ鼀⼄矄 ��㡰 ���㡰
꾀 �҈ 黨 m
҈ = m
҈ m
=
m - ҈
… ���㡰
m
m - ҈ m
和⼄盐� �ᨨ 矄� 矄 矄ᨨ� �犨� m =
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EBD_8341 P-268
҈
Physics
ᆠ㡰 � �PᨨQ 꾀泐� �v u㦨 gh
v - = ´ ҈ ´ 䰘 Þ v = ҈ 鼀⏌矄鼀 ᨨᆠ�泐ᨨ 矄 ᆠ� 泐 �矄
턼
矄� �° � = 矄� � 꾀泐� � 和 Ἐ矄 犨 A 矄� � Ҙ ҈ Ҙ = = 矄� r҈ Ҙ҈ ҈ Ҙ=
҈ v
´ ҈ = ҈ 鼀⏌矄
v Ἐ = wm ´ v =
ᨨ鼀⼄ �㡰 ��㡰
u 㦨 g 㦨҈鼀⏌矄 h 㦨䰘鼀
㡰
꾀�
҈鼀
m҈
m
m
␔
泐
҈ q
泐
B
㡰 �矄 ᆠᨨ �盐�鼀� ᨨ鼀矄盐 ᆠ㦨ᆠ鼀
泐矄ᆠᨨ鼀� � ᨨ鼀 ᨨ� ᨨᆠ ᨨ �矄矄㜠�� ᆠ
ᨨ鼀泐 ᆠ矄矄盐 � ᨨ� 犨ᆠ ᨨ ᆠ 泐矄ᆠ├犨 矄 �鼀�犨� ᨨ鼀 ᆠ ᨨ 泐 b㡰 ᨨ鼀 矄 Ἐ矄犨 Ҙ� 矄� � = m a�r 矄� 黨°
b㡰
㝐 �鼀盐鼀 盐鼀ᨨ ᆠ�ᨨ =
犨 x㦨 =
x ҈黨
l㦨 x
m =
\ ҈䰘
30° 30°
㝐 �鼀盐鼀 盐鼀ᨨ ᆠ�ᨨ 㦨
b㡰 ᨨ �鼀�ᨨ ᨨ� P ᨨ ᨨ PQᨨ矄 泐 ᨨ矄�矄 ᨨ盐 ᨨ鼀 ᨨ� Q ᔼ DQNS NS㦨QS㦨h \ ÐNQS㦨� Q \ r㦨� ᨨ犨� DQMA h ÐMQA㦨� \ MA㦨QA㦨h P 矄� r 矄� �° ҈ m= = 矄� � 矄� � ᔼ DPMB PM㦨h졔h㦨�h \
矄� � =
h
�h
=
҈ �
⏌
P qC n ҈
Q a
C
n n҈矄� qC㦨nᨨ 矄� qC = n҈ 꾀泐� � 和 Ἐ矄 犨 Q n矄� a㦨n҈ᆠᨨ矄qC
r
2h
Þ
�
S 2h
ᨨ
B h
n
黨 q C
B
M
A
D
꾀泐� �和 Ἐ矄犨 P ᆠ꾀ᨨ 矄
㦨 N
�
���㡰
m 㦨҈ \ m҈㦨m m҈ ᔼ 鼀� 泐�矄 ᨨ � ᆠ� 泐 � �
盐鼀ᨨ � ᆠ矄 ᆠ�� ᨨ �矄 矄鼀 �盐鼀�矄犨泐矄矄鼀 � ᆠ� 泐�矄鼀 �盐鼀㜠҈졠꾀ᔴ鼀� 泐� 鼀 �盐鼀㜠 \m҈㦨m 㡰 ᨨ鼀 ��鼀 泐 犨� ᆠᨨ鼀 ᨨ盐 ᨨ鼀 CD � � 矄盐 矄 ᨨ� ᆠ�ᨨ 矄盐 ᆠ AD � � 矄�├矄 矄盐 ᆠADᆠ��ᆠ �qC �鼀�� �ᆠ矄㡰 A
鼀
��㡰
\
҈ 矄� �
mg =
m =
矄� r m = m 矄� � 㝐盐�泐� � ⼄ �㡰 ��㡰 ���㡰 m m m 矄� � 矄� r҈ 矄� r ´ ´ = ´ ´ m҈ m m 矄� r҈ 矄� r 矄� �
ᆠ
ᔼ
ᆠ�
泐
泐
҈
矄� r҈ m = 矄� r m 꾀�� 泐� �和 Ἐ矄 犨 C
\
� C
꾀泐� � 和 Ἐ矄 犨 B
҈�
r2
B
O
␔
A
҈
泐
P 꾀�
O q q
ᆠ
A r҈
�҈
ᆠ㡰 � ᆠ� 泐�矄 泐 ᆠ 泐 ᆠ ÐMOB㦨҈ q >Ð졠 �犨 ᆠ ᆠ 泐 M
m D r
r҈
�矄
҈
�㡰
ìï æ n ö üï n҈ ᆠᨨ矄 í矄� -҈ ç ÷ ý n è n҈ ø þï ïî é æ n ö ïü ù n ïì a = 矄� -҈ ê ҈ ᆠᨨ矄 í矄� -҈ ç ÷ ý ú è n҈ ø þï ûú îï ëê n 矄� a =
b㡰
n҈ n
�㡰
a⏌
a ⏌ ᔴ
졠Ἐ
��㡰
졠
꾀 a⏌
a⏌
쟼
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P-269
Ray Optics and Optical Instruments
҈
� ᆠ� 鼀� 泐졠꾀ᔴ졠쟼 ᨨ �矄矄矄 ᨨ �� ᆠ �鼀� 矄 矄鼀 㡰 꾀泐� � 和 Ἐ矄 犨
矄� r Ἐ 矄� 黨 - r 㡰 = Q � 㦨 r錐 r' 졔 r㦨黨 桌 矄� � 矄� r ᆠᨨ矄 r ҈ m= \ 矄� r æ 矄� r ö æ҈ö ᔴ��ᆠ � qc = 矄� -҈ ç ÷ 㦨 矄� -҈ ç ÷ è ҈ ø è ᆠᨨ矄 r ø \qc㦨矄� ҈ r㡰 㡰 ⼄盐 ᆠy 㡰 ᨨ矄 ᨨᆠ �犨� 鼀 �盐鼀 矄矄 矄 �矄 ᨨ�ᆠ泐 矄 鼀 �盐鼀 ᨨᆠ�泐 ᨨ �� ᆠ矄矄 ᨨ 犨 � � �矄矄 ᆠ矄矄 㡰 n矄� q㦨 nmDn㡰矄� 黨 Þ n맔矄� 㦨nm맔҈桌矄� 黨 \ ҈맔矄� 㦨҈m맔҈\m㦨 㡰 ᨨ ᆠᨨ 矄 �ᆠ ᆠ� � 矄盐 ᆠ � �㜠ᨨ� � ᆠ 䰘 u㦨ᆠ鼀v㦨䁸u҈㦨҈m㦨 R㦨ᆠ鼀 -m҈ m m - m҈ + 㦨 u v R ᆠ鼀 ҈
\
m=
䰘 -҈ -҈ 䰘⏌ + = -㡰 v
䰘
Q
r' P
҈�
҈�㡰 ᨨ鼀 ��盐 矄� � =
҈� +
C m=
҈� ҈� +
= ҈ ´ 矄� �° =
r
ᆠ鼀
҈矄� ° =
҈
矄� r Þ 矄� r = Þ r = °
ᨨ鼀�ᨨ鼀泐CP = CQ
\ r Ἐ = °
꾀�� 泐矄 Ἐ矄 犨 Q
矄� r Ἐ = ҈矄� e Þ = 矄� e Þ e = °
ᨨ鼀 �ᨨ鼀泐
r Ἐ+ q + e = ҈° 꾀矄 �矄�矄ᨨ 矄�� � 㡰 Þ ° + q + ° = ҈° Þ q = 黨°
�㡰 ἘᔴἘ ᆠ��ᆠ � ҈�矄� ᔴ㦨҈矄� 黨 Þ
矄� ᔴ =
҈ ҈� B C
q
A
鼀 �盐鼀
꾀 꾀 꾀 ҈ \ = 꾀 ҈�
盐矄� ᔴ =
\
矄� r = 矄� �°
꾀 =
҈� 꾀 ҈
ᔼ 犨 ᆠ꾀 泐 黨 ᨨ � 泐�� 㦨
҈� ´ 黨 = ҈鼀 ҈
҈
㝐 �鼀盐鼀�鼀├ 泐泐ᨨ �
= ᔴ⏌ ҈ \ = = � ´ ҈ 矄 = � ´ ҈ -黨 矄 ´ ҈ ⏌ ҈�
� ҈�ᆠ鼀 P ҈�ᆠ鼀
ᆠ鼀
n D
黨鼀
=
҈�ᆠ鼀
�
꾀泐和 Ἐ矄 犨 P
ᨨ鼀和 Ἐ矄犨 m ´ 矄� � = ҈ ´ 矄� r
Þm´
e q S r' R ᆠ 泐
r
ᨨ�
\ v㦨҈ᆠ鼀 V¢ ҈ᆠ鼀
�矄�鼀�犨� ᨨ 矄ᨨᆠ 犨 ᨨ 犨㜠ᨨ�� ᆠ I 䰘 u㦨҈ᆠ鼀v㦨v¢m҈㦨 m㦨 R㦨¥ -䰘 + 㦨 +҈ V ¢ \ V¢ 㦨҈ᆠ鼀
ᨨ �矄 ᆠᨨ �鼀� ᨨ鼀 ᨨᨨ鼀ᨨ ├ 㦨҈҈㦨ᆠ鼀 㡰 ᔼ ��盐C㦨ᆠ��ᆠ � R 졠㝐 R = ᔼ D졠㝐 C = 㝐 ᔴ ᔴ ҈ ᔴᔴ Q 矄� C = = \ C = m � 㝐 졠 R R = \ R = ´ = ᆠ鼀 \
黨㡰 ᔼ ��盐QR�矄 ᆠ 泐 QS�矄 ᆠ 泐CQ �矄 ᨨ鼀
t㦨ᆠ 鼀
�
҈ = ҈� ´ ҈ - = N \m = ҈� ҈ ҈҈� ᆠ盐ᨨ N ; ҈�
�ᆠ鼀
黨
ᔴᨨ鼀� ��犨� 맔҈黨矄犨�㦨� ⼄盐 ᆠ泐 ᨨ矄 ᨨᆠ �犨� 鼀 �盐鼀
´ ҈ c = = � ´ ҈҈ l ´ ҈ -҈ l 处 ␔ � � 鼀 �盐鼀l1 = = = 处 Ҙ ҈� f =
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EBD_8341 P-270
æ ҈ ö ÷ ᨨS ç � ns = ҈� ø è
V V ´ ҈ f鼀 �盐鼀 = med�um = a�r = = � ´ ҈҈ l med�um l a�r ´ ҈-䰘 c ᨨᆠ�泐ᨨ �� � 鼀 �盐鼀 V = m
NA =
v ´ ҈ = ´ ҈-䰘 m ␔ � l = = f � ´ ҈҈ ᆠ 㡰 ᨨ鼀 和 Ἐ矄犨 n҈ 矄� q� = n 矄� q f Q ҈ � ᆠ矄
꾀�
桌 l 矄ᨨ ᆠ�� ᨨ 矄 矄n z㡰盐 ᨨ ᨨ nᔼ ᆠqf 矄ᨨ n ᆠ盐矄 �矄ᆠ鼀 l㡰 �矄ᨨ矄矄� 盐 m ᨨ 矄 �ᨨ � ᆠ� ᆠq� ᆠ 㡰 ᨨ 泐盐 �ᨨ矄ᨨ� ᆠ�ᨨ
n \ n҈
æ n ö ᆠᨨ矄C㦨 ҈ - ç ÷ è n҈ ø
҈
n n҈
ns
� 黨 = NA㦨 ҈
� 黨 黨 = NA㦨 ҈ ҈
҈� � 黨 NA㦨 ҈
҈�
ᨨ 㦨҈�m졔҈
҈�m㦨
ᨨ m㦨
Y
NA =
X
O 6Ö 3�
黨 � �
M¢
��㡰
黨 ҈� ҈� = NA = � � �
҈�
>
X
8Ö 3 O �+ 6 Ö3 -10K M
꾀㦨Ö � + Ö 10k ��b㡰
M
҈� �
ᨨS � n矄=
Y
Z
M¢ Ö3 +8 � 3 6Ö 8Ö3
ᨨS � 犨㡰
=
ᨨS � �㡰
҈� � = R ҈� m + ҈㡰 R
>
è
ö ÷ ҈� ø
ᨨ
>
æ
҈ ҈� ҈ - ҈� = ¥ - ҈�mR + R㡰 -R
x㜠y �矄 ᨨ盐 泐犨 鼀 �盐鼀㜠҈ 鼀 �盐鼀㜠
䰘
n҈ n
ᨨs҈ ç � n矄 =
ᨨ
ᨨ
ᨨS҈ � 犨㡰
NA =
R
ᨨ v҈㦨 ҈�mR㡰
ᨨ ᆠ盐 矄盐 ᆠ
ᔴᨨ
ᨨS҈ � �㡰
=
҈� ҈ ҈� - ҈ = ¥ v҈ - mR
\
m҈ m m҈ - m 犨 u㦨v҈졔Rv㦨¥ = v u r
n҈ n҈ NA㦨矄� �m㦨 n ᆠᨨ矄 C = n s s NA =
m 㦨҈�
P ҈�mR v҈
ᨨ犨 ᨨ鼀⼄ �㡰 盐鼀�ᆠ 盐
\
R
N
�m
꾀矄ᨨ矄� C㦨
m m҈ m - m҈ = v u r
�矄 ᆠ�ᨨ ᨨᆠᆠ盐矄ᨨ 矄盐 ᆠ ᨨ犨 �ᆠ r㦨¥
n҈
ᔴ 黨 ᔴ
>
n矄
m㦨҈� mR
n
ᔴ � �
꾀�
P
꾀 �ᨨ � ᆠ� ᆠ�>ᆠ��ᆠ � q ᆠ ҈ ҈ ҈ ᨨ 矄� �° > ᨨ > ᨨ n > ᨨ n > ҈҈ n n ᆠ㡰 矄� �和 矄犨P錐 ns矄� �m㦨n҈矄� 黨 ᔴ㡰 �㡰 n矄㦨 ᆠ�� ᨨ 矄盐ᨨ盐 � � 턼
n҈ n
>
҈ ns
NANA҈ \ 盐鼀�ᆠ盐ᨨ ᆠᨨ鼀� 矄盐ᆠ盐�矄⼄盐ᨨ 矄鼀盐ᨨ 犨ᨨ 盐鼀�ᆠ盐矄 꾀泐 ᨨ鼀盐 ᨨ ᆠ�ᨨ 矄 �ᆠ 矄盐 ᆠ 犨 �� 矄 ᨨ鼀� m҈㦨҈㡰ᨨ鼀 �盐鼀 m㡰
´ ҈ = ´ ҈ 鼀 ⏌ 矄 錐 ҈�
盐鼀�ᆠ盐 NA =
>
㡰
黨 黨 = � � ҈
>
�
҈� ҈
>
=
l med�um =
>
҈
l a�r = = 处 m ҈� ⼄盐 ᆠ泐 ᨨ矄 ᨨᆠ �犨� 鼀 �盐鼀
\
>
Physics
㡰
黨 = � �
��盐 㡰矄 ᨨ犨矄ᆠᨨOM Ἐ㦨 � + $
ur
��盐 㡰矄 ᨨ犨矄 ᆠᨨ A = $� + $ - ҈ k$
�ᆠ盐ᨨ� MOMἘ�矄Z㜠 �矄犨 �ᆠ 矄盐 �ᆠᨨ ᨨ k$
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uuur
uuur
ᨨ鼀盐ᨨ ᆠ�ᨨ 矄 �ᆠ矄盐 ᆠ ᨨ AB 矄 ��鼀 ᨨ犨
꾀 �犨 ᆠᨨ MP OP ᆠ ᨨ盐 泐 ᨨᨨ 盐ᆠ
uuur uuur MP OP 㦨 MP㡰 OP㡰ᆠᨨ矄�
(
Ҙ Ҙ҈ Ҙ҈ - Ҙ - ⏌ ҈ ҈ - ⏌ + = Þ + = u v R -¥ v \ v㦨鼀鼀
�矄 �矄 ᨨ矄��ᨨ ᨨ �鼀� 盐 ᨨ ᆠ�ᨨ �矄 矄盐 ᆠ�鼀鼀 ᨨ �矄矄盐 ᆠ �矄�鼀�犨� 矄�盐ᨨᆠ ᨨ ᆠ�ᨨ 矄ᆠᨨ 矄盐 ᆠ u㦨㦨҈鼀鼀 꾀�� 盐矄� � ᨨ鼀盐ᨨ ᆠ�ᨨ 矄 �ᆠ矄盐 ᆠ ᨨ CD -
)
$� + $ - ҈ k$ - k$ 㡰
= ᆠᨨ矄 � 㡰 + 㡰 + -҈㡰 + -҈㡰 Þ �㦨 �ᆠᨨ� �ᆠ�ᨨ ᨨ MOMἘ ᨨ鼀 �� 㡰 $� + $ $ $ $ n$ = n = � + � � 㡰 + 㡰 桌҈⏌
ᨨ � �ᨨ ᆠ�ᨨ 犨盐矄和 Ἐ矄犨
矄� � 矄� ° = Þ r 㦨� 矄� r 矄� r
=
-҈ Ҙ҈ Ҙ +Ҙ - Ҙ҈ ҈ ⏌ - + = - + = u Ἐ vἘ R ҈ v Ἐ - \ vἘ㦨�鼀鼀 � � �鼀��矄 ᨨ鼀 �鼀鼀ᨨ ᨨ 矄ᆠᨨ 矄盐 ᆠ ��㡰 泐 ��鼀矄 ᨨ犨� � ᨨ矄��ᨨ 矄 ᨨ ᨨ �鼀�矄
ᔼ ᆠ� 泐 A x㜠y
� M
^r
M'
O
矄� r㡰 n^
^ ᆠᨨ矄r㡰 k C
A
ᆠ 泐
V'
I'
ᨨ犨 r$ = 矄� r 㡰 n$ - ᆠᨨ矄 r 㡰 k$
é ù 㦨 矄� � 㡰 ê $� + $ ú - ᆠᨨ矄 �°㡰 k$ ë� � û ҈ �$ + $ - �k$ 桌 \ r㦨 �
�ᨨ � ᆠ� ᆠ ᆠ盐 矄盐 ᆠᨨ ᆠ泐� �ᆠᨨ �矄�� 泐 黨 b㡰
�� � � ᨨ ᨨ矄 ᨨ鼀� ᨨ鼀 ᆠ盐 矄盐 ᆠ ᨨ ᨨ � � 黨 b㡰�矄� ᆠ��ᆠ � ᔴ
I
V
A
b
鼀鼀
�鼀鼀
�㡰 x�矄 � ᆠ� � ᨨ ᆠ�ᨨ ᆠAC ᨨ �矄鼀 ᨨᨨ� ᆠ�ᨨ x>�ᔴ ᆠ��ᆠ �㡰
A
x
y
x
黨 x
y
y
B
黨 y
M
C
N
y � ᆠ� �ᨨ 泐ᨨ ᆠCB ᨨ �矄鼀 ᨨ ᨨ � ᆠ�ᨨ
y>�C \ x졔y>�C 盐 x㦨ÐA y㦨ÐB ᨨ鼀 �ᨨ鼀泐㡰 \ x졔y㦨黨 Þ 黨>�C Þ �C� \ 㝐� �鼀盐鼀盐ᨨ ᆠ�� nᨨ 鼀 �盐鼀 ᨨ �矄 ᨨ ᨨ矄矄�
҈ 犨 C�矄 ᆠ��ᆠ � 矄� C C㦨黨 b ҈ ҈ Þ Ҙ 矄� 黨° - b㡰 Þ Ҙ ᆠᨨ矄b ҈ 꾀矄�鼀�� �ᆠ矄Ҙ㦨 �㡰 ᆠᨨ矄b 꾀泐� � 和 Ἐ矄 犨 A 矄� a 矄� a Ҙ㦨 Þ 矄� b㦨 ��㡰 矄� b Ҙ ᔼ a㦨p⏌矄�鼀�� �ᆠ矄ᨨ ᨨ ᨨ 矄� a 矄� p ⏌ ҈ \ 矄� b㦨 㦨 Þ Ҙ㦨 ���㡰 m 矄� b Ҙ ᨨ鼀⼄ �㡰 ���㡰矄� b㦨ᆠᨨ矄b ᨨ b㦨� ҈ ҈ = \ Ҙ㦨 ᨨ Ҙ㦨 ᆠᨨ矄 �° ҈⏌ \ 矄盐ᨨ ᆠ�� ᨨ ᨨ ᨨ�� � � � Ҙ
黨
D
␔꾀 쟼
黨 b
ᨨ ᨨ� �� ᨨ鼀 ᆠ盐 矄盐 ᆠ�矄 �㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ 鼀ᨨ �� �矄� � � 犨泐矄 �ᆠ盐ᨨ �盐矄 鼀 ᔼ ��泐 ᨨᆠ �矄 � 矄 鼀 �盐鼀 u 㦨 ¥ 盐矄� �
鼀鼀 꾀ᔼ 쟼
B
B a
鼀鼀
n㦨 ��㡰
҈ ҈ = = 矄� �C 矄� �°
ᨨ n㦨
�
҈ ҈ = = Þ �CἘ㦨䰘 Ҙ �⏌ � y㦨 � 㡰\x㦨 x>�CἘ 盐 y�CἘ \ ᨨ� ᆠ�ᨨ 犨�├ᆠᨨ ᆠAC盐 ᨨᨨ
CB 矄� �CἘ㦨
Topic-3 : Refraction at Curved Surface Lenses and Power of Lens ҈
㡰 ᨨᆠ � ᨨ ᆠᨨ 矄f㦨�鼀 졠ᆠ�矄f矄ᨨ�鼀� I҈㡰犨�矄ᨨf ᔼ鼀�ᨨ I҈�I犨�҈鼀 � 鼀�ᨨ
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EBD_8341 P-272
Physics
ᨨ犨I犨�ᨨᆠ ᨨ 矄 ҈鼀
҈鼀
졠ᆠ
�
҈鼀
҈鼀
I
I҈ 㝐�ᨨ
f㦨�鼀
\ u = -҈㡰 + -҈㡰 + -҈㡰 = - 鼀 ҈ ҈ ҈ - = v u f ҈ ҈ ҈ ҈ ҈ = + = + ᨨ v = = 鼀 v f u +� - � ᆠ �矄 ᆠᨨ �鼀� ᨨ鼀鼀�ᨨ 㦨졔㦨鼀 㡰 � 盐矄� � 矄鼀├Ἐ矄 ᨨ鼀盐
矄� � 矄 ᨨ鼀盐
æ ҈ ҈ ҈ö = k - ҈㡰 ç - ÷ f è R҈ R ø R҈㦨R㦨R ᨨ ᨨ盐ᆠᨨ 矄㡰 ҈ ҈ ö æ҈ \ = m - ҈㡰 ç ÷ è f R -R ø
҈ Þ P = = m - ҈㡰 �㡰 f R ᨨ ᨨᆠᨨ 矄 R҈ = R Ἐ R = ¥ 矄� � 矄鼀├Ἐ矄 ᨨ鼀盐�� 犨 æ ҈ ҈ö ҈� P = ( m - ҈) ç - ÷ ��㡰 è RἘ ¥ø
k k ҈ ҈ = k҈ + �� � � ᨨ 矄� 矄泐㡰 Þ - k҈ u v v u ҈ ҈ ҈ ᔴᨨ鼀� �犨� 矄 ᨨ鼀盐 v - u = f 犨� Þ
k҈ = 䰘
m -҈ Þ P= RἘ ᨨ鼀 �㡰 ��㡰 R RἘ = Þ RἘ = R 㡰 ᨨ鼀 矄 ᨨ鼀盐
҈ b ҈ \ f= = f ├㦨҈ c 矄ᨨᨨ m - v �
҈ = f҈ fl f҈㦨҈ᆠ鼀ᨨf҈㦨黨ᆠ鼀 ҈ 和ᨨ 黨 = f ᨨfl㦨҈ᆠ鼀 l 㡰
҈ ҈ æö æö = m - ҈㡰 ç ÷ ᨨ = ҈� - ҈㡰 ç ÷ fl ҈ èRø èRø \R㦨҈ᆠ鼀 犨 �⼄盐� �矄盐 犨
҈ ҈ = + ᨨ 䰘 ⏌ 㡰 = ҈ + f f fl f ᨨf㦨�ᆠ鼀 ҈ ҈ æ ҈ ö = m҈ - ҈㡰 ´ = m҈ - ҈㡰 ´ ç ᨨ犨 ÷ � R è -҈ ø ҈ \m҈ = + ҈ = ´ ᨨ盐 矄㡰 v҈ = ( ) = cm 矄� �
uf ҈ ҈ ҈ - = Þv= v u f u+ f
㡰 ᨨᆠ � ᨨ ᨨ㜠ᆠᨨ 矄㜠 ҈ ҈ ö m҈ ҈ æ҈ Þ ҈ = = ( m҈ ҈) ç ÷= ( m҈ ҈) ҈ è ¥ ø ᨨᆠ � ᨨ ᨨ㜠ᆠᨨ ᆠ 矄㜠 ҈ ҈ ö m ҈ æ ҈ Þ = ÷= = ( m ҈) ç ( m ҈) è ¥ ø ᨨ ᆠᨨ鼀� �ᨨ ᨨ 犨ᨨ 矄㜠 ҈ ҈ ҈ m ҈ m ҈ m m ҈ Þ = = + = ҈ ⼄ ⼄ ҈ = m ҈ m 㡰 ᨨ鼀 ⼄盐�ᨨ ᨨ � 鼀㦨├҈졔├ Qy㦨mx졔c㡰 v vö æ Þ = k҈v + k çQ m = ÷ u uø è
ᔴ矄㜠ᔼᔼ v㦨u Þ f + u = f Þ u = ᔴ矄㜠ᔼᔼᔼ u = ¥ v = f ᆠ ᆠᨨᆠ u 矄盐矄 v � 矄�矄 �矄 �矄 ᆠᨨ ��ᨨ �矄 㡰 b㡰 矄� � 矄鼀├矄 ᨨ鼀盐
盐㦨㦨ᆠ鼀
öé ҈ ҈ æ mg ҈ ù =ç ҈ ÷÷ ê ú f çè ma R R û øë ҈ mg ma ᆠ� � ᨨ �矄矄 � 矄ᆠ�泐
æ ҈ ҈ ҈ ö Þ = ҈� ҈㡰 ç ÷ f è R҈ R ø ␔ �鼀鼀矄 � �⼄盐�
�㡰
\v =
黨
´҈ = cm ( ҈ )
\ᔼ鼀�ᆠ矄ᆠ├ᨨᨨᆠ�矄 矄�鼀� ᨨ鼀 泐 矄�矄 ᆠ ᨨ ᆠ盐盐ᨨ 鼀�ᨨ ᆠ㡰 ᨨ 矄
öæ ҈ ҈ æ mg ҈ ö = çç ҈ ÷÷ ç ÷ fl è ml R R è ҈ ø ø ml㦨 ᆠ�� ᨨ �⼄盐� 桌
҈ æ ҈� öæ ҈ ҈ ÷ ç =ç fl è ҈ ø è R҈ �� � � �㡰 泐 ��㡰 fl ҈� ҈㡰҈ Þ = = f Þ
҈ ö ÷ R ø
��㡰
҈ ҈ = »黨 ҈ ҈
҈ ҈ ҈ - = ᨨ v u f \ v㦨졔ᆠ鼀
҈ ҈ ҈ = v -
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҈
҈҈
҈
꾀ᆠᆠᨨ � �ᨨ ᆠᨨ ��ᨨ �鼀� ᨨ鼀 泐 矄矄 ᨨ盐 ᆠ ᨨ ᆠ盐盐ᨨ 鼀�ᨨ 矄ᨨ f㦨ᨨf㦨҈ᆠ鼀 ҈ ҈ ҈ 㡰 泐 矄矄 ᨨ鼀盐 - = 盐 ᨨ �矄 矄盐҈㦨桌 ҈ ҈ ҈ = Þ ᔼ = ҈ - � ᔼ鼀� ᨨ鼀 泐 �矄 矄犨� 矄 ᨨᆠ ᨨ矄ᆠᨨ 矄 ҈ -= 矄ᨨ 盐 = ҈ ҈ ҈ = Þ = 䰘 ᆠ鼀 ҈ -� 㡰 꾀ᆠᆠᨨ � �ᨨ 矄鼀├Ἐ矄 ᨨ鼀盐
҈�
矄� � 矄 ᨨ鼀盐 ҈ ҈ ҈ ҈ ҈ ҈ - = Þ = Þ = �ᆠ鼀 盐 ҈ -҈ æ ҈ö 和 � 盐ᨨ矄 = ç҈ - ÷ � �ᆠ�ᨨ ᨨ � ᆠ� 泐 è mø
æ ö ᨨ = ҈� ç ҈ - ÷ = � è ø ᨨ犨盐㦨黨�
æ ҈ ҈ ö ҈ 㦨 m ҈㡰 ç ÷ è ҈ ø
꾀�� 盐矄� � 矄 ᨨ鼀盐矄
ᨨᆠ � ᨨ 矄犨�ᆠ � 盐ᨨᆠ �� ᆠ�� m 和ᨨ �鼀� 犨� ᨨ鼀 犨 ᨨ矄��ᨨ ᆠ �鼀� �矄矄 㡰 泐 矄 ᨨ鼀盐 ҈ ҈ ҈ ҈ ҈ ҈ = - = 錐 -㡰 盐
Þ
҈ ҈黨䰘 ҈ ҈ ҈ = 錐 = = 錐 ҈黨䰘 㝐� � �ᆠ�ᨨ ᨨ 矄 鼀㡰�矄�� 泐
҈
҈ f 鼀�ᨨ㦨¥㡰 f
졔
�� � �
҈ ҈ ҈ ҈ = [Q盐 = ¥] ¥ 䰘 \㦨䰘鼀鼀㦨䰘ᆠ鼀;҈ᆠ鼀
҈ æ m ҈ö = ç ÷ è R ø
ᔴ矄㜠
҈ ҈ö æ ҈ = (Ҙ - ҈) ç - ÷ è - ¥ ø
Þ
㦨
Qᨨ犨P㦨
æ ҈ ҈ ҈ ö ҈ ҈ ö ҈ æ҈ = (Ҙ҈ - ҈) ç = (Ҙ - ҈) ç ÷ 錐 ÷= ¥ è ø è ҈ ø ҈ 和�鼀�泐 ᨨ ᨨ㜠ᆠᨨ ᆠ 矄
m҈㦨1 m㦨҈
졔
㦨҈졔 Þ
҈
b㡰 ᔴ矄㜠҈
҈ æ m ҈ö =ç ÷ f҈ è R ø
æ ´ ҈ ö ᔼ⏌ = ç �㡰 è ҈黨䰘 ´ ÷ø 㦨҈҈맔҈鼀⏌矄ᨨ犨 矄 矄 b㡰 ᨨ鼀 矄鼀├矄 ᨨ鼀盐
҈ ҈ 泐 犨� ҈ (Ҙ҈ - ҈) ( Ҙ - ҈) = ᨨm҈Ҙ㦨҈ m m҈ m m҈ 㦨䰘鼀鼀 㡰 盐矄� � = 盐
҈ ҈ ҈ = -黨� �
҈ ҈ 黨 = - = � ҈黨 黨� 黨� = ҈��ᆠ鼀 ᨨ = 黨
盐矄 矄ᆠ �矄矄 � 泐 �矄 ᆠ 㦨҈��҈㦨��ᆠ鼀犨泐 ᨨ鼀 矄
æ ö ç ÷ æ ö è ҈黨䰘 ø 鼀= ç ÷ = è 盐ø ᨨᆠ�泐ᨨ �鼀�犨ᨨ 矄�矄�� 泐 ᔼ⏌㦨鼀졠⏌ �ᆠ�ᨨ ᨨ ᨨᆠ�泐ᨨ �鼀��矄矄鼀矄 ᨨ ᨨᆠ 졠⏌㦨�鼀⏌矄
҈
㡰
҈䰘
R ҈ æ m ҈ ö f㦨҈ᆠ鼀 f = =ç ÷ f҈ è R ø ҈ æ m ҈ö = ç 㦨҈졔2Þ ÷+ ҈ è ø R ҈ ҈ = + Þ = ҈ - = ҈ ᨨ ҈ R R ҈ ᨨ = ᆠ鼀 黨 ҈ � æ m ҈ ö = ç ᨨ ÷ 黨 Þ m҈㦨 è ø 黨 � ҈ \ m㦨 ҈ + = 㦨҈�� 黨 黨 㡰 � ᨨᆠ � ᨨ 矄 㡰㦨҈�ᆠ鼀 ᨨᆠ�矄ᆠ �矄 ᆠ 盐㡰㦨ᆠ鼀 泐 矄 ᨨ鼀盐 ҈ ҈ ҈ = - 盐 ҈ ҈ ҈ ҈ ҈ = + = 盐 ҈� ҈ - = ᨨ㦨ᆠ鼀
㦨҈�ᆠ鼀
㦨�ᆠ鼀
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҈
Physics
�鼀�ᔼ�矄 ᨨ鼀 ᆠ鼀ᨨ �� ᨨ 矄 � 犨� �
泐矄 ᨨ鼀 �鼀� ᔼ㡰 ᨨ鼀 盐 矄ᨨ ᆠᨨ 鼀�ᨨ ᨨ鼀矄 ᨨ �鼀� �矄�鼀�矄 ᨨ盐 ᨨ鼀 � 矄盐ᆠ 犨泐 �ᆠᨨ� ᆠ� 犨� ᨨᆠ 矄鼀ᨨ� 盐ᨨ ᆠ�ᨨ
├矄ᆠ 泐ᆠᨨ 鼀�ᨨ �矄 ᆠ犨 矄 鼀�ᨨ犨� 㦨�鼀� �矄 ᆠ 㡰 �盐矄ᨨ ᆠ盐盐ᨨ ᆠᨨ 鼀�ᨨ �㦨 ᨨ 㦨� �� = = 䰘� ᆠ鼀 ᆠᨨ 鼀�ᨨ㡰 b㡰 � �盐矄ᨨ 鼀�矄 �ᆠ�矄矄 㦨҈ᆠ鼀
҈ 㦨�ᆠ鼀 盐㦨 ҈㡰㦨ᆠ鼀 泐盐矄� �鼀�ᨨ ᨨ鼀盐 ҈ ҈ ҈ ҈ ҈ ҈ + = Þ + = Þ㦨ᆠ鼀 盐 - -� m ҈ = ᆠ鼀 ᨨ犨 꾀 �� ha = hr ҈ = ´ ҈� m 矄盐 ᆠ ҈ ҈ ҈ = 㡰 Ἐ矄 ᨨ鼀盐�矄�� 泐 盐 \ ᨨᆠ � 㦨
҈黨
æ ҈ 1 ö ҈ = ( 鼀 m l - ҈) ç ÷ 鼀 è ҈ ø ᔼ �矄 ᨨᆠ � ᨨ 矄� �
æ ҈ 1 ö ҈ 鼀 ( ml - ҈) = ( ml - ҈) ç ÷ Þ = è ҈ ø ( 鼀 ml - ҈) ᔼ m҈> m 鼀 ᨨᨨ矄� 矄�� 矄 盐 ᨨ 矄ᆠ �矄�ᆠᨨ 矄 ��矄 �� 泐矄 ᆠᨨ ᆠ 矄ᆠᨨ �矄 �� 泐矄 盐矄�� ᨨ�ᨨ 㡰�矄ᆠᨨᆠ
҈ ҈ ҈ ҈ ҈ = Þ = -҈ �盐ᨨᆠ҈ᆠ鼀 � 鼀�ᨨ ᆠ�鼀�҈ᆠ鼀� ᨨ ᨨ 鼀�ᨨᨨᆠ鼀 ᨨ鼀ᆠᨨ 矄 㡰 ␔ ᨨᆠ �矄 ├ �矄 ᆠ �a ᨨ鼀 � ᆠᨨ 矄
泐 矄 ᨨ鼀盐
҈ ҈ ҈ = v u f
҈ ҈ ҈ = V a 㡰 f
\ =
ᨨᆠ�泐 ᨨ �� � ᆠ盐盐鼀 ᨨᆠ�泐 ᨨ �� � 鼀 �盐鼀
졔 鼀鼀㡰㦨 Þ 졔 鼀鼀㡰졔 鼀鼀㡰㦨 Þ »҈�ᆠ鼀 ҈ æ öæ ҈ ö = ç ҈ ÷ ç ÷ Þ 㦨ᆠ鼀 è ø è ҈� ø 㡰 ᔼ 矄� ᨨ ᨨᆠ矄⼄盐㦨l 矄� ᨨ �鼀�矄⼄盐㦨l¢
㦨ᆠ鼀 鼀鼀
l Ἐ lἘ =黨 = ᨨ l l �鼀� � �ᆠ�ᨨ 鼀㦨 盐㦨ᆠ鼀 㦨맔㦨҈ᆠ鼀 㦨䁸 ҈ ҈ ҈ ᨨ鼀 ᨨ鼀盐 - = 盐 ҈ ҈ ҈ ҈ ҈ ҈ ҈+ = + = ᨨ = ҈ - ҈ ҈ \ 㦨ᆠ鼀 l� ҈ = = = ҈� ᆠ㡰 Ҙ = l鼀 �盐鼀 ᨨ鼀⼄盐矄�ᨨ
和�鼀�泐 ᨨᆠᨨ ᆠ 矄
ᆠ㡰
\ =
҈ ҈ ҈ ҈ ҈ = + Þ =
ᨨᆠᨨ 矄
㡰 ᔼ 矄ᨨ ᆠ�� m�矄�鼀鼀矄 � 鼀 �盐鼀ᨨ ᆠ�� m҈ �矄 ᨨᆠ � � 鼀 �盐鼀�矄�� 泐
O I҈ v
�
҈ ҈ ҈ a = �㡰 v f a 㝐�ᨨ ᨨ鼀矄 �鼀� ⼄盐 �矄 ᆠ ᨨ鼀 鼀�ᨨ
꾀矄ᨨ 鼀 =
ᨨ
҈ = - Qv㦨m㡰 盐
\ 盐 = - ᆠ鼀
ᨨ ᨨ㜠ᆠᨨ 矄 I҈
v
v
҈ Ҙ - ҈㡰 ҈ ҈ ҈� - ҈ ҈ æ ҈ ö ҈ ҈ ҈ = = - ᨨ = -ç = ÷= + è - ø 盐
I
\ 㦨鼀
ᨨ犨 �� ᨨ鼀 矄 ᨨ鼀盐 I
I
⏌ v
҈
ᆠ㡰 Þ \
Fᨨr cᨨnvex lens
҈ ҈ ҈ = -� 㦨䰘�ᆠ鼀
ᨨ m L҈ =
f҈ = R
� ᨨ m L = f = -� R Þ f㦨 㡰
Qv㦨�ᆠ鼀f㦨ᆠ鼀㡰
� ᆠ� �矄ᨨ 鼀�ᨨ ᔼ¢
҈ ҈ ҈ ҈ ҈ = + = ᨨ ᨨ鼀 ⼄ �㡰桌 a V f a f a f ᆠ a 㦨 f b㡰 泐 矄鼀├Ἐ矄 ᨨ鼀盐 ᨨᆠᨨ 矄 öæ ö ҈ æ m =ç - ҈÷ ç ÷ f è mL øè R ø
҈ ҈ ҈ - = 盐
�
�ᆠ鼀 ᔼ
� 꾀 � ᆠ� �矄ᨨ 矄
㝐
㝐�ᨨ
�鼀� ᨨ鼀 泐ᆠᨨ 矄ᆠ矄矄 ᨨᆠ ᨨ鼀�ᨨ
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Ray Optics and Optical Instruments ҈ ҈ ҈ ҈ ҈ ҈ + = \ + = \㦨�ᆠ鼀 盐 � �
�鼀�ᔼ犨ᨨ盐 ᨨ鼀 矄矄 ᨨ犨 鼀�ᨨ
矄�� 盐 鼀�ᨨ�矄� 泐 ᨨ �鼀� 犨�� 泐 犨� ᨨ鼀 꾀 㝐꾀㦨�ᆠᨨ矄 㦨�ᆠ鼀 x㜠ᆠᨨᨨ � ᨨ �鼀�㡰 ᔼ¢꾀㦨�矄� 㦨 � ᆠ鼀 y㜠ᆠᨨᨨ � ᨨ �鼀�㡰 ᆠ ᆠᨨᨨ � ᨨ ᨨ� 犨 �ᆠ �鼀� �矄 ᨨ鼀 � � 㡰 b㡰 ᨨᆠ � f ҈㡰 ᨨ ᨨ㜠ᆠᨨ 矄 犨� n㦨҈�盐矄� � 矄㜠鼀├ ᨨ鼀盐
䰘
҈ ҈ ҈ ҈ ҈ ҈ � - = + = Þ = \ v㦨ᆠ鼀 v ҈ ҈� v ҈� ҈ ҈�
ᨨ � �鼀��矄 ᨨ鼀 �矄 ᆠᨨ ҈ᆠ鼀 ᨨ鼀 鼀�ᨨ ᆠ㡰 ᨨ鼀 � u㦨҈ᆠ鼀錐v㦨҈ᆠ鼀Du㦨Dv㦨҈ ҈ ҈ ҈ ҈ ҈ ᨨ鼀 矄 ᨨ鼀盐 = - 㦨 ҈ -҈ f v u \ f 㦨�ᆠ鼀 Þ
Fᨨr m�nᨨr
҈ vᆠ鼀
é҈ ҈ ҈ ù é҈ ҈ù ҈ = n҈ - ҈㡰 ê ú = ҈� - ҈㡰 ê - ú = f҈ R R ë҈ ¥ û û ë ҈
ᨨᆠ � f㡰ᨨ ᨨ㜠ᆠᨨ 矄犨� n㦨҈
҈
é҈ ҈ ҈ ù ҈ ù ҈ é҈ = n - ҈㡰 ê ú = ҈ - ҈㡰 ê ú= f ë ¥ -҈ û 䰘 ë R҈ R û ᨨᆠ � ᨨ ᆠᨨ鼀� �ᨨ
҈ ҈ ҈ ҈ = + = F f҈ f ᨨ犨泐� � 矄 ᨨ鼀盐 ᨨ ᆠᨨ鼀� �ᨨ ᨨ 矄
\ b㡰
ᨨ鼀 ��盐� DABC b =
Þ
҈ ҈ ҈ = V -
Þ AB㦨AC b Þ r㦨f b \ r㦨 \
� Ҙ㦨ᆠ鼀桌
AB AC
f b
꾀ᨨ �鼀� ᨨ鼀 泐矄盐 㦨pr 㦨 p
b f µ f
和盐
泐矄 ᨨ鼀 ᨨᆠᆠ � � �泐
ᨨᆠ
ᔼ鼀� ᨨ鼀 ᨨ ᨨᆠ
b
a
r
C
B
黨
I
҈ ҈ ҈ Þ v㦨҈ᆠ鼀 = v - -҈� ⏌ ��矄�� � �ᆠ矄 �鼀��矄҈ᆠ鼀� ᨨ ᨨ ᨨ 鼀�ᨨ 㡰 ᨨᆠ � ᨨ ᆠᨨ ᆠ 矄f㦨 - f҈ f ҈㦨 ᨨᆠ � ᨨ ᆠᨨ 矄 ҈ ҈ ҈ ҈ ҈ Þ = = f҈ f҈ f f҈ f
f҈㦨҈ᆠ鼀 f㦨 -
´ ҈ 㦨҈�ᆠ鼀
b㡰 �鼀� ᨨ鼀 泐ᆠᨨ 矄I҈ᆠ矄矄�盐ᨨᆠ ᨨ ᆠᨨ ᆠ 矄ᔴᨨ ᆠ 矄 ᨨ鼀矄 �鼀�ᨨ I҈I 和� ᨨ �鼀� I҈㦨ᆠ鼀 I҈
I
I҈
ᆠ鼀 ᆠ鼀
ᨨ鼀 �鼀盐鼀ᨨ� 㡰
M
L
O
黨졔黨㡰
҈ ҈ Df = + Þ Df㦨�맔҈맔맔҈㦨� � ҈㡰 ҈㡰 \ ᨨᆠ � f᠀Df㦨 �᠀�㡰ᆠ鼀 b㡰 ᨨᆠ � f ᨨ ⼄盐� 鼀�ᨨ 犨 矄 �矄 矄� ҈ ҈ ҈ ҈� = + = + Þ = ᆠ鼀 f f҈ f m ҈� ¥ 和� ᆠ 矄 ᨨ矄�� 盐 ᆠᨨ鼀� �ᨨ 矄 矄 ᆠᨨ �� �鼀�ᨨ ҈� u㦨ᆠ鼀 f = - ᆠ鼀 v㦨䁸 ҈ ҈ ҈ 꾀ᆠᆠᨨ � �ᨨ鼀�ᨨ ᨨ鼀盐 + = v u f
\
�ᆠᨨ 矄
b㡰 ᨨᆠ � ᨨ �ᆠᨨ 矄L�矄҈�ᆠ鼀꾀矄鼀ᨨᆠ �矄ᆠ �矄 ᆠᨨ ᆠ鼀 ᨨ鼀 矄� �矄 ᆠᨨ ᨨ �鼀�矄 ᨨ盐 ᨨ鼀ᆠ鼀 ᨨ鼀 矄I҈
҈
Þ
f
A
Þ
҈
҈ ҈ ҈ - = V U F V㦨ᆠ鼀
uᆠ鼀҈
� �� � 矄 ᨨ鼀盐 ҈ ҈ ҈ Df Dv Du = - Þ = + f v u f v u
҈ᆠ鼀
ᆠ鼀
ᆠ鼀
ᆠ鼀
�鼀� I҈ᆠ矄 矄�盐ᨨᆠ ᨨ 鼀�ᨨ 鼀�ᨨ ᨨ鼀矄 �鼀�I �矄 ᆠᨨ ᆠ鼀� ᨨ ᨨ �
�鼀�Iᆠ矄矄 ᨨᆠ ᨨ 矄 u㦨졔҈ᆠ鼀f㦨졔҈�ᆠ鼀 ҈ ҈ ҈ 꾀泐� � 矄 ᨨ鼀盐 - = v u f
I
ᨨᆠᨨ ᆠ 矄 ᨨ ᨨ
҈ ҈ ҈ - = v u f
҈ ҈ ҈ ҈ ҈ ҈ ҈ - == ᨨ = - + = � v v v㦨�ᆠ鼀㦨�矄 ᆠᨨ I ᨨ鼀ᆠᨨ ᆠ 矄
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\
Physics 㝐� � �ᆠ�ᨨ m㦨
v 矄� ᨨ �鼀� � = = u 矄� ᨨ ᨨᆠ
\
矄� ᨨ �鼀� = ҈� \ 和� ᨨ �鼀� 盐ᨨᆠᨨ ᆠ 矄㦨�ᆠ鼀 㡰 矄鼀├ ᨨ鼀盐 ᨨ
�
҈ ҈� + ҈ � ҈ = = = ҈� ҈� v v㦨ᆠ鼀
\
꾀 鼀� � �ᆠ�ᨨ m҈㦨 -
҈ æ Ҙ ö æ ҈ ҈ ö = -҈ f èç Ҙ҈ ø÷ èç R҈ R ø÷ m ҈ m m ᔼ m > m҈ ᆠᨨ ᆠ 矄 鼀� � 矄 �矄
盐 ᨨ 犨�矄 盐 ᨨ 矄 犨� 矄 和ᨨ 矄矄 ᨨ盐 � 犨� L �鼀鼀矄� L҈ 㡰 ᨨᆠᨨ ᆠ 矄矄矄 ᨨ犨 � ��盐� �矄ᆠ矄 R҈㦨R R㦨졔R
鼀 㦨҈ᆠ鼀
l 㦨졔҈ᆠ鼀
҈�ᆠ鼀
ᆠ鼀 �ᆠ鼀
gm
mm
- v =㦨 -҈� u
ᨨ犨 ᨨ ᆠ�ᨨ ᨨ鼀 矄u㦨 �㡰㦨ᆠ鼀
҈ ҈ ҈ ҈ ҈ ҈ - = Þ = v u f҈ v - ҈
R2
R1
㝐쟼ᔼ㝐 mm
҈ ᨨ鼀 矄鼀├ ᨨ鼀盐 = f gҘ m gҘ= mҘ
=
(
)
m g Ҙ -҈
҈ ҈ ҈ ҈ = = \ v㦨ᆠ鼀 v ҈
æ ҈ ҈ ö çè R - R ÷ø ҈
v = 㦨҈ u - 㝐� � �ᆠ�ᨨ ᨨ 盐ᆠ 泐 ᆠᨨ鼀� �ᨨ M҈㦨m҈맔m㦨 㡰맔 ҈㡰㦨 꾀�� 犨 矄泐矄鼀�矄├� 鼀 �盐鼀ᨨ ᆠ�� 䰘⏌
�矄 ᨨᆠ � ᨨ鼀�ᨨ� �矄ᆠ矄 㝐� � �ᆠ�ᨨ m㦨
҈� ҈䰘�
҈ æ ҈� � ´ ö æ ҈ ҈ö =ç - ҈÷ ç - - ÷ = + f è ҈䰘� ø è R R ø ҈䰘� R ᨨ f 㦨졔�R ᆠ 矄 矄 矄 ᆠᨨ � 矄 ᨨ ᨨᆠ � f 㦨� ᆠ㡰 꾀ᆠᨨ 鼀�ᨨ ᆠᨨ ᆠ 矄犨泐矄ᨨ 盐ᆠ�盐 �鼀�␔ �ᆠ ᆠ
ᨨ├ ᨨ 矄ᆠ
ᨨᨨ�ᨨ 㡰 㡰 ᨨᆠᨨᆠ �鼀� ᨨ鼀 泐ᆠᨨ 矄�矄 �鼀� �矄 犨 ᨨᆠ�矄ᆠ 泐ᨨ f u㦨f 졔x \
ᨨ 矄
҈ æ ҈ ҈ö ҈ ö æ / ö æ ҈ =ç - ҈÷ ç - ÷ Þ ¢ = ç - ÷ 䰘 è R҈ R ø fl fl è 䰘 / ø è R҈ R ø 꾀矄ᨨ 犨 矄犨矄� � ҈ ö ҈ ö ҈ æ ҈ æ ҈ ҈ ҈ fl 㦨 ҈�҈㡰 çè R҈ R ÷ø 㦨 ҈ \ çè R҈ R ÷ø 㦨 � 矄� � �矄矄盐� ⼄ �㡰犨�
䰘
҈ fl¢
=
� ҈ ´ \ f ¢l㦨 ᆠ鼀 䰘 �
҈ ҈ ҈ 꾀�� 盐矄� � 矄 ᨨ鼀盐 - = v u fl¢ ҈ ҈ ҈ ҈ ҈ = Þ = = \v㦨҈ᆠ鼀 v - � v � ҈
f + x㡰 £҈ f +x
\ f졔x㡰 £ f졔x �矄�矄盐 ᨨf�鼀 ᆠ㡰 和 �ᆠ�ᨨ ᨨᆠᆠ盐矄 盐ᨨ � ��泐ᨨ 矄ᨨ ᆠᨨ �鼀�� 泐矄ᨨ 矄鼀犨 � ᨨ ᨨᆠ盐矄矄� ᆠᨨ �矄 �泐矄� 鼀�� �矄泐矄 ᨨᆠ盐矄矄 � ᆠ矄ᨨ �矄ᨨ 矄 ᨨ �鼀�矄ᨨ ᨨ鼀 �矄盐 �矄�ᨨ ᆠ 盐ᆠ 泐 盐矄� �ᆠ�ᆠ盐
盐鼀矄├ᨨ 矄 䰘㡰 꾀泐� � 鼀�ᨨ ᨨ鼀盐 ҈ ҈ ҈ ҈ ҈ ҈ ҈ ҈ + = ᨨ = = + v u f v f u ҈ ҈�
҈ö æm öæ ҈ = ç l - ҈÷ ç - ÷ è m s ø è R҈ R ø
¢
f f + x 㡰 £҈ f +x
é f ù f + x ê҈ + £҈ Þ f x úû + ë
fl¢
҈
҈ ҈ ҈ ҈ ҈ ҈ 矄� � - = Þ = v u f v - f + x 㡰 f ҈ ҈ ҈ f +x- f f + x㡰 = + = = Þ v f f + x f f + x 㡰 f f + x 㡰 盐u졔v㦨҈ �� 㡰 f + x㡰 +
҈
v ҈ = 㦨䰘 u - 㝐� � �ᆠ�ᨨ ᨨ 盐ᆠ 泐 ᆠᨨ鼀� �ᨨ M㦨m҈맔m¢㦨 㡰맔 䰘㡰㦨҈
㝐� � �ᆠ�ᨨ m¢㦨
\ 黨
҈ Ἐ M 㦨 㦨䰘 M҈
㡰 꾀泐� � 和 矄 犨 A 矄� 㦨n矄� r � �� �犨�n犨� 㦨矄� r졔nᆠᨨ矄 r맔
dr dn
�㡰
��㡰
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Ray Optics and Optical Instruments 盐矄� �
꾀
r
r r
æ ҈ ҈ ҈ ö = m - ҈㡰 ç ÷ f R R è ҈ ø
矄㜠鼀├Ἐ矄 ᨨ鼀盐桌
æ ҈ ҈ ҈ ö = ҈� - ҈㡰 ç ÷ ҈� R R è ҈ ø
q
ᔴ
�㡰
æ ҈ ҈ ҈ ö = ҈҈� - ҈㡰 ç ��㡰 ÷ fἘ è R҈ R ø �� � �⼄ �㡰泐 ��㡰 fἘ ҈� - ҈ � = = = ҈� ҈҈� - ҈ ҈� \ f Ἐ㦨ᆠ鼀 ᨨ鼀 ��鼀 � �矄 ᆠ ᨨᆠ盐矄 ᨨ ᨨ 矄矄 矄 ᨨ盐 ᆠᨨ� ᆠ� \d = 20 5㦨҈�ᆠ鼀
꾀�� 泐� �和 矄犨 B 矄� q㦨n矄� r㡰 � �� �犨�n犨�
q é ù 㦨矄� r㡰졔nᆠᨨ矄 r㡰 ê ú ë û
ᆠᨨ矄q \
���㡰
é r ù q ᆠᨨ矄q 㦨矄� r㡰 nᆠᨨ矄 r㡰 ê n úû
ë ᨨ鼀 ��㡰桌 q ҈ = 矄� r㡰졔ᆠᨨ矄 r㡰 r桌 ᆠᨨ矄 q
\
ᨨ鼀⼄ �㡰 ᨨ n =
犨�r㦨
ᨨ鼀⼄ ���㡰 ᨨ n =
r㦨 犨�q㦨
�㡰
鼀
和盐矄�盐� � 盐矄ᨨ r q� ⼄ �㡰犨�
q ҈ æ ҈ ҈ö = + ᆠᨨ矄 ° 矄� 졔ᆠᨨ矄 桌㦨 çè ÷ø 㦨 䰘҈黨㡰 � �矄 ᆠ犨 ᨨᆠ 矄ᆠ D㦨҈ᆠ鼀 �矄 ᆠ犨 犨ᨨᨨ矄��ᨨ ᨨ 矄d㦨ᆠ鼀 ᨨᆠ � ᨨ 矄 f =
ᆠ鼀
�
T盐錐
҈ ҈ ҈ - + ҈ ҈ ҈ ҈ + = Þ = = + ҈� F F f҈ f \F㦨ᆠ鼀 ᆠᆠᨨ鼀� �ᨨ ᨨ 矄矄 矄矄ᆠᨨ ᆠ 矄ᨨ ᨨᆠ � ᆠ鼀 和� ᆠFvFr�ᨨᆠᨨᨨ盐�矄 � 鼀ᨨ ᆠᨨᨨ盐犨
犨 ��� 矄矄矄 ᨨ盐� 矄 ᨨ�矄鼀 \ 졠 矄矄 ᆠᨨᨨ盐 犨� �ᨨ ᆠᨨᨨ盐 ᨨ盐 � 矄錐
D - d ҈ - ҈ + 㡰 ҈ - 㡰 = = = ҈ ᆠ鼀 D ҈㡰 ҈㡰
ᨨ犨 P =
N ҈ ҈ = = ҈ ҈ f
\ N = 䰘҈黨 ҈
㡰 � m㦨҈�錐Rᆠ盐 㦨ᆠ鼀 矄� � 矄㜠鼀├ ᨨ鼀盐
�ᆠ鼀
�鼀� ᨨ鼀 泐 ᆠᨨ 矄 ᨨᆠ盐矄ᨨ ᆠᨨ ᆠ 矄 ᨨᔼ犨�ᆠ矄�盐ᨨᆠ ᨨᆠᨨ ᆠ 矄 �鼀�犨� ᨨ鼀 � � �泐 ᆠ ᆠ�ᨨ ᨨ鼀ᨨ 矄矄泐矄犨�ᆠᨨ鼀ᨨ ᨨ�ᆠ �矄
æ ҈ ҈ ҈ ö = m ҈㡰 ç ÷ f R R è ҈ ø ᨨ ᨨ㜠ᆠᨨ 矄
R҈ ® ¥ R = R R = = ᆠ鼀 m ҈ ҈� ҈ ␔ 犨ᨨ 矄矄� ᆠᨨ ᆠP㦨P҈졔P㦨҈ ␔ 犨ᨨ 矄矄矄 泐 �矄 ᆠ 㦨�鼀㡰P㦨P҈ 졔P �㡰P҈P㦨 ᨨ鼀 矄 犨ᨨ 矄矄�ᨨ 矄 犨 � P҈P㦨҈鼀 \f =
P҈ - P = P҈ + P 㡰 - P҈P
\
= ҈-҈ 㡰 - ҈-҈ 㡰 = 鼀 -҈ P҈㦨鼀҈ P㦨鼀҈
\
f҈ =
m gm
=
gm mm
҈ ҈ ҈ = 鼀 = ҈�鼀 f = = 鼀 = �鼀 P҈ P
=
҈� = ҈҈�
ᔼ �盐 ᨨᆠ ᨨᆠᨨ ᆠ 矄
䰘�鼀
䰘
㡰 ᨨ鼀 ��盐� D졠꾀 㦨 㡰졔 㦨 졔 졔 Þ 㦨 졔
\ =
+
ᆠᨨ�ᨨ 㡰�矄ᆠᨨᆠ ᨨ犨 ᨨ鼀⼄盐�ᨨ ᨨ 矄盐 ᆠ
�鼀
졠
–
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Physics
泐 = 泐 + =
u㦨xv㦨졔xR㦨졔RҘ҈㦨҈Ҙ㦨҈�
w �
-҈ ҈� ҈� - ҈ + = \ x㦨�R 㦨PO -x x R
w �
�
ᨨ犨盐矄� � ᨨ鼀盐 ᨨ ᆠ�ᨨ ᆠ盐 矄盐 ᆠ �҈
m m҈ m - m҈ - = 盐 ҈҈ = - - 㡰
Þ
҈ ҈
�
b㡰 和 �ᆠ�ᨨ �矄 矄鼀 犨 ᆠ盐 矄盐 ᆠ �矄 ᆠ� � ᨨᆠ ᆠ盐矄 ᨨ ��ᨨ �矄 矄 犨
ᆠ盐 矄盐 ᆠ矄 bᆠ㡰ᔼ鼀� ᨨ鼀 泐ᆠᨨ ᆠ 矄 ᆠᨨ 鼀�ᨨ�矄�盐 ᆠᔴᨨ ᆠ 矄 ᆠᨨ 鼀�ᨨ ��� �� 盐
ᨨ ᆠ ⏌ ᆠ 泐矄 ᨨ ᨨ 鼀
矄泐矄ᨨ 盐ᆠ ᆠ├犨 矄ᨨ鼀├ 鼀鼀 b 㡰 �鼀� ᨨ鼀 犨�ᆠᨨ鼀ᆠ盐矄�� 泐矄 ᨨ鼀 矄ᨨ ᨨᆠ 犨�矄�├ᨨ ᨨ犨
Q 桌
A
҈ = ҈ é w ù Þ =ê҈ ú ëê � ûú
Þ
é w ù \ = ê҈ + ú � úû êë
-҈
꾀 㡰
ᆠ鼀
�
�
ᔼ
ᆠ鼀
f㦨ᆠ鼀
Q
v v m㦨 Þ㦨 Þv㦨ᆠ鼀 u -
҈ ҈ ҈ ҈ ҈ + = = Þ + = Þ R = ᆠ鼀 v u f R ҈ - R
O u
C v
r
PO㦨OQ
é҈ ҈ ҈ ù = ( m - ҈) ê - ú f ë R҈ R û
҈ \ f㦨r r ҈ = r r \Feq.㦨r
҈ ҈ ҈ ҈ = 㦨 + ÞFeq.㦨r r r r Feq ��
Q �鼀�㡰
Feq =
ᨨ 和㡰ᆠᨨ鼀� �ᨨ ᨨ ᨨ ᨨ盐ᆠᨨ ᨨ ᨨ㜠ᆠᨨ ᆠ 矄
ᆠ�� ᨨ 矄㡰 㡰 矄� � ᨨ鼀盐 ᨨ矄 �ᆠ ᆠ� �矄盐 ᆠ
- µ҈ µ µ - µ҈ + = u v R
é҈ ҈ ҈ ù = ( m - ҈) ê - ú f ë R҈ R û
҈ æ ҈ ҈ö - ÷ Þf㦨r 㦨 ҈�҈㡰 çè f -r ¥ ø ҈ ҈ ҈ ҈ ҈ + 㦨 + = ÞFeq.㦨r f f -r -r Feq
҈ æ nl ҈ ö öæ ҈ = ç - ҈÷ ç ÷ f è ns R R ø øè ҈ ҈ æ nl öæ ҈ ö \ = ç - ҈÷ ç ÷ \ l㦨� è ҈ ø è ø
P
ᨨ ᨨ盐ᆠᨨ 矄 㡰
ᨨ 㡰 ᨨ㜠ᆠᨨ ᆠ 矄
�鼀� ᨨ鼀 泐ᆠᨨ 矄� �矄 � ᆠ �盐 泐 矄鼀├ ᨨ鼀盐
矄矄
b㡰
é ҈ ҈ ù � = (҈� - ҈) ê - ú = = r ë ¥ -r û ҈ ҈ ҈ ҈ ҈ = + = + = Feq f҈ f r r
ᨨ ᆠ�ᨨ 泐� �鼀�ᨨ ᨨ鼀盐
m2
҈ ҈ ҈ ҈ ҈ = + = + f f҈ f -� 㦨҈�
ᨨ 턼㡰 ᨨ㜠ᆠᨨ 矄
\
m1
P=
\
҈ ҈ ҈ ҈ ҈ ҈ - = Þ = v u f v - f ҈ ҈ ҈ ҈ ҈ ҈ = Þ = + - f f
꾀�
盐矄� ᆠ 盐 �� 泐矄ᆠ盐ᨨ � 矄�泐犨� 盐ᆠ 㡰 ᨨ犨ᨨ 矄ᆠᨨ鼀� �ᨨ
æ҈ ҈ö éù ҈ Þ ҈� - ҈㡰 ç - ÷ = (҈� - ҈) ê ú = Þ f = r èr rø ër û r ҈ ҈ ҈ ҈ ҈ = + = + = Feq f҈ f r r r
n
黨
F
A¢
ᆠ ᨨ�ᨨ 㡰 �矄 ᆠᨨᆠ 㡰 ᨨ 矄 y� � 矄 ᨨ鼀盐
졠
B¢
O B
꾀㜠錐㜠⼄矄錐ᔴ㜠㜠⼄矄 ␔ 泐ᨨ �� 矄 ᨨ鼀鼀 �盐鼀ᨨ 矄鼀 �盐鼀 � 矄 ᨨ犨 矄 ᨨ鼀 �矄 ᨨᨨ矄� 犨 泐 ᨨ �� 矄 ᨨ鼀 矄ᨨ� 矄犨泐 ᨨ鼀 ᨨ鼀 㡰 꾀矄 m҈ m 泐 ᨨ �� � � 矄 犨� ᨨ犨 矄 ᨨ鼀
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�
黨
ᨨ鼀⼄ ���㡰 �㡰
㡰 꾀矄m҈>m 泐ᨨ �� � � 矄犨� 犨泐 ᨨ鼀 ᨨ鼀 ᔴ㡰 꾀矄 m 㦨 m 泐 ᨨ �� ᆠᨨ鼀� � ᨨ盐 ᨨ鼀 矄 犨� ᨨ盐 泐 ��ᨨ 㡰 꾀矄m>m 泐ᨨ �� ᆠᨨ鼀� �ᨨ盐ᨨ 矄 �矄 犨泐 ᨨ鼀 ᨨ鼀 꾀㜠⼄矄錐㜠⼄錐ᔴ㜠⼄矄錐㜠⼄矄 ᔼ鼀� ᨨ鼀 泐ᆠᨨ ᆠ鼀�ᨨ 꾀㡰ᆠᨨ 矄 ᔴ㡰 ᆠᨨ ᆠᨨ㜠 ᆠᨨ 矄 㡰 ᔴ �盐鼀� � � �鼀� �矄 꾀 �鼀��矄� � �泐 犨 ᨨᆠ�矄 ᨨᆠ盐矄 ᔼ鼀� ᨨ鼀 泐ᆠᨨ 鼀�ᨨ 㡰�矄犨泐矄�盐 � f㦨鼀u 㦨鼀
҈ ҈ ҈ Þ v㦨҈鼀 = v - ҈ ҈ ҈ ᨨ犨 � �� � 矄 ᨨ鼀盐 - = 犨�鼀t v u f
Þ \
d㦨鼀
҈
Þ
�黨
du ҈ dv ҈ du = ҈ 鼀⏌矄 + = �� dt dt dt v u æ dv ö ҈㡰 ´ ҈ 㦨黨鼀⏌矄 çè ÷ø = dt 㡰
-
ᨨ鼀� �盐 ᨨ ᨨ ᆠ �ᨨ ᨨ矄��ᨨ ᨨ �鼀� 犨 矄㡰㦨黨鼀⏌矄
҈ ҈ ҈ + = f ҈ - d + d ҈ ҈ ҈ + = ҈ - + f
Y҈
\
f 㦨鼀
M
X
F A
B Y
� 㝐� �盐 ᨨ ᨨ ᆠ � ᨨ 鼀� � �ᆠ�ᨨ
Y
��㡰 和矄ᨨ ᆠᨨ 矄盐ᆠ�ᨨ ᨨ y ��鼀 ᨨᆠᨨ ᆠ鼀�ᨨ ҈㡰 犨� ӸӸἘ �ᆠ盐ᨨ꾀矄盐ᆠ Ӹ㦨ӸἘ
㦨�矄҈
�矄 矄 ҈㡰 ᨨ鼀矄�鼀�ᨨ 졠꾀졠¢꾀¢ 矄ᆠᨨ 矄 㡰 ᨨ鼀矄�鼀�ᨨ 졠꾀졠¢꾀¢ ud 꾀
҈
�㡰 和� ᆠY�矄ᨨ犨ᨨ ᨨ�ᆠ �矄AB Xᨨ ᨨ �鼀��矄 � 矄矄 ᨨ盐 ᆠᨨ 矄 和矄ᨨ ᆠᨨ 矄盐ᆠ�ᨨ ᨨ 泐 ��鼀 ᨨ 矄 ҈㡰 ᨨ� XY �矄矄 矄 泐ᨨ��� � � ᨨ鼀 矄ᨨ盐ᆠ X 鼀� � �鼀�Y和� ᆠ 泐�矄盐 � 矄矄� � ᨨ盐� 矄 ᨨO�矄 ᨨ�ᆠᆠ ᨨ 矄犨 Y҈OY �ᆠ盐ᨨAB 㡰 犨泐 ᨨ鼀XᨨAB XMᔼ矄�├矄Y҈OY Mᨨ� MYᔼᆠ盐矄AB F �矄�矄 ᨨᆠ盐矄ᨨ ᆠᨨ 矄
udv vdu v dm = dt dt 鼀� � �ᆠ�ᨨ m㦨 \ u dt u - 㡰 黨㡰 - ҈㡰 ҈㡰 -҈ = -�矄 㦨 㡰
�
҈
和盐矄�盐� � �矄盐� ⼄ �㡰 v + u = f
矄� � 矄 ᨨ鼀盐
\
҈ ҈ ҈ ҈ + = + v + d u - d v u ҈ ҈ ҈ ҈ + = + ҈ - d + d + d - d ҈ - d + d
Y'
N M
v 졔d
X F
A O
҈
C
B
Y
졠
졠¢ 꾀¢¢
d u
ᔼ鼀� v
҈鼀
꾀¢
u졔v㦨҈鼀
鼀� � �ᆠ�ᨨ ᨨ 矄 ҈㡰�矄 �� 㡰
�㡰
\
��㡰
v + d 㦨 u - d
㡰 ᨨ� YἘX � �ᆠ盐矄 � ABᨨ� O
ᨨ� O�矄 ᨨ ᨨ 鼀�ᨨ 㡰 ᨨ� X Yᔼᆠ盐矄AB ᨨ� C �矄ᨨ� C�矄 ᆠ ᨨ ᆠ盐盐ᨨ 鼀�ᨨ 㡰 犨XMᨨ � ᆠ� �矄 ᨨ� MᨨY � ᆠ盐ADF ᨨF�矄 ᨨᆠ盐矄ᨨ ᆠᨨ ᆠ鼀�ᨨ �㡰 犨ᨨ� �ᆠ 矄ᨨ ᆠᨨ 矄҈ 犨� ᨨ鼀 犨ᨨ 矄�鼀�矄 S҈ Sᨨ 矄ᨨ盐ᆠ S ⼄盐 �矄 ᆠ 矄 �鼀�矄 S҈ S 犨� 矄 犨ᨨ ᆠᨨ 矄ᨨ盐ᆠ矄 u L҈
v S҈
ᨨ鼀⼄ �㡰 ��㡰
u = + d v㦨҈d
S
꾀泐� � 矄 ᨨ鼀盐 ᨨ 矄 ҈㡰
҈ ҈ ҈ + = v + d u - d f
���㡰
L ҈�鼀
꾀�� 泐� � 矄 ᨨ鼀盐 ᨨ 矄 㡰
҈ ҈ ҈ + = v u f
O
�㡰
A
d
O҈
O
鼀 S
D ҈鼀
ᨨ 矄L҈ L u㦨҈�鼀v㦨䁸f 㦨졔҈鼀
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҈ ҈ ҈ ҈ ҈ ҈ ҈ ҈ Þ = + = - = + v u f v f u ҈ -҈� \ v㦨鼀
Q
v � 鼀� � �ᆠ�ᨨ m㦨 u 㦨 \ m㦨 -҈�
\ 犨ᨨ�鼀�矄S҈ Sᨨ S犨� ᨨ鼀 鼀 ᨨ鼀 矄 S҈犨��鼀鼀ᨨ�矄ᨨ�ᆠ �矄矄m㦨 S犨��鼀鼀ᨨ犨�矄ᨨ�ᆠ �矄矄m㦨 \ d 㦨S҈S㦨҈�鼀鼀㦨҈�맔҈鼀 D㦨҈㦨҈鼀 l 㦨� 鼀㦨�맔҈黨鼀㦨�맔҈䰘鼀
Dl d -䰘 ҈㡰 � ´ ҈ 㡰
\
� �犨� b㦨
ᨨ
b㦨
҈� ´ ҈
-
矄� r>
\ 泐 PR 犨� ᨨ泐 � ᆠ ᨨ � RQ 졠
ᆠ � �Q 泐犨� ᆠ � 鼀 ᔼᔼ ᆠ 泐犨� 盐�鼀泐 ᆠ ᆠ├� 鼀 �盐鼀ᔼᔼ 矄盐 ᆠ�矄 ᆠ� �矄盐 ᆠ\R㦨¥
O
Ҙ Ҙ Ҙ -Ҙ Ҙ Ҙ - ҈ + = ҈ Þ- ҈ + = u v R - -
Ҙ = Ҙ҈
҈ 맔҈鼀
æ҈ è
Ҙ Ҙ Ҙ -Ҙ - ҈+ = ҈ u v R
- ö
÷ 㦨҈鼀 ø
\ OA㦨҈鼀 ��㡰 ��ᨨ ᨨ OA 犨� �犨 L҈ L ᔼ �犨 L҈ L�矄 盐ᆠ d犨� ᆠ矄 Q ҈
b㦨
Dl \ b犨�� ᆠ矄 d
\ �矄 ᆠOA犨�� ᆠ矄 泐ᨨ �� �矄 矄 ᨨ鼀鼀 �盐鼀ᔼᔼᨨ鼀 �盐鼀ᔼᔼᔼ� ᨨ鼀 矄ᨨ鼀 �盐鼀 ᔴ矄 �㡰Wh nn҈ �.e. nn҈n
æ n҈ ö æ n ö n n҈ ᨨ 矄� ҈ ç n ÷ 矄� ҈ ç n ÷ < è ø è ø n n 졠�ᨨ盐矄泐nn �q�矄� ᆠ��ᆠ � ⼄盐� ᨨ 泐 矄矄� � ᨨ鼀 鼀 �盐鼀 ᔼᔼ ᨨ 鼀 �盐鼀 ᔼᔼᔼ
ᨨᨨ� ᆠ�ᨨ 犨�矄ᨨ├ᆠ犨 泐 矄�├矄 犨� 矄鼀 � � ᆠ ᨨ 鼀 �盐鼀 ᔼᔼ 鼀 �盐鼀 ᔼᔼᔼ ᔴ矄 ��㡰nn҈b盐nn
泐犨�� ᆠ � 鼀 �盐鼀ᔼᔼᔼ矄 �q犨� ᨨ犨 矄矄 ᆠ��ᆠ �⼄盐� ᨨ鼀 �盐鼀ᔼᔼ 鼀 �盐鼀ᔼᔼᔼ � 㝐 ᔼ
R
Þ -
҈ Ҙ ⏌ Ҙ҈ Ҙ ⏌ Ҙ҈ 㡰 - ҈ + = u v R
Þ
-
҈ ⏌ ⏌ -҈ + = Þ 㦨�ᆠ鼀 - -� ⏌ R
v
u
I O
ᨨ犨 泐� � Ἐ矄 鼀├ ᨨ鼀盐
æ ҈ ҈ ҈ö ҈ ҈ö = m - ҈㡰 ç - ÷ = æç - ҈öæ - ÷ \f 㦨䰘�ᆠ鼀 ÷ç f R R è ҈ è øè � µ ø ø
ᨨ
㝐 ᔼᔼᔼ
�㡰
꾀�� ᆠ�ᨨ ├矄ᆠ ᨨ鼀矄 �ᆠ矄盐 ᆠ
和� ᆠᨨ� A�矄 � 鼀 �鼀 OA㦨b㦨맔 ç ´҈
v I
u
\ ᨨ b㦨
n҈ n Þ 矄� r>矄� �'' Þ r>�''
�㡰
矄� ᆠᨨ ᆠᨨ㜠ᆠᨨ 矄 矄�├鼀�ᨨ
犨 ᨨ矄 ᨨᆠ �
҈ ҈ = + f f҈ f
㦨ᆠ鼀 f㦨 ᨨᆠ � ᨨ ᆠᨨ ᆠ鼀�ᨨ㦨 㦨҈ᆠ鼀
f҈㦨 ᨨᆠ � ᨨ ᆠᨨ ᆠ矄盐 ᆠ㦨
r
r R҈㦨ᆠ鼀
P
G q
㝐 ᔼᔼ
F R 㦨ᆠ鼀
꾀泐� � 和 Ἐ矄 犨 P
n 矄� q n 矄� q = \ 矄� r㦨 n 矄� r n
Q n>n\ r >q ␔ ᆠ 泐PR鼀矄 ᨨ盐 泐DE��矄� � ᨨ鼀 矄鼀 �盐鼀ᨨ鼀 �盐鼀 ᆠ 泐犨� ᨨ泐� 泐 ᆠ DE� �矄 �ᨨ � ᆠ� ᆠ�矄鼀ᨨ ᆠ��ᆠ � ᨨ鼀 ᔼᔼᔼ ᔼ 矄� �ἘἘ㦨
n҈ n
\
҈ ҈ = + = - Þf 㦨䰘�ᆠ鼀 f - -҈
P� ᆠ �矄 ᆠxᨨ ᨨ�ᆠ �矄矄盐ᆠ �矄 �鼀��矄 ᨨ鼀 矄鼀ᆠ�u㦨v㦨x 矄� �鼀�ᨨ ᨨ鼀盐
҈ ҈ ҈ ҈ ҈ ҈ = + Þ = + f v u -䰘� - x - x
\
x㦨҈�ᆠ鼀
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␔ ᆠᨨ ᆠ�矄 � 犨� 犨ᨨ m㦨
ᨨ犨 ᨨ 矄�├� �犨� ᆠᨨ ᆠ 矄盐 ᆠ 泐 �矄 �矄 ᆠ ᨨ鼀 矄盐 ᆠ 和ᨨ y �矄 ᆠ ᨨ � 矄盐 ᆠ犨� ᨨ鼀�矄�鼀� �矄 ᆠ 㦨mh㡰 ᨨ鼀�
æ 꾀 + d鼀 ö æ 䰘 + d 鼀 ö 矄� ç ÷ 矄� ç ÷ è ø= è ø 盐 m = 䰘 矄� 꾀 ⏌ 矄�
d鼀 ö � æ 矄�
䰘 = ° + çè ÷ø
y
m m҈ m - m҈ = v u R ҈� ⏌ ҈� - ⏌ = - - x - ҈ ҈ = ᨨ x㦨҈ᆠ鼀 x
矄� �
ᨨ
�
m 鼀 ᆠ
� � �矄 m 鼀 �矄矄矄 = ҈䰘
盐d鼀犨�矄矄 矄ᨨ
\ Dx㦨x҈x㦨҈�ᆠ鼀҈ᆠ鼀㦨҈҈ᆠ鼀 ᨨ鼀 ��盐 �㡰 �矄 ᨨ ᆠ�ᨨ ᨨҘ҈㦨Ҙ ᨨ鼀 ��盐 ��㡰 ᆠᨨ 矄 矄矄 ��� � 矄
ᨨ ҘҘ\m҈m ᨨᆠ � ᨨ ⼄盐� 鼀�ᨨ
�
m
矄� ҈ êm 矄� 꾀 矄�
Mw ⏌ = = Mg ⏌ 黨
矄� ᔴ㦨
꾀泐� � 矄 矄 犨 Ӹ ҈맔矄� q㦨 矄� 㦨 矄� 䰘� ᔴ㡰 ᨨ鼀⼄ �㡰
�㡰
��㡰 䰘� Ӹ 黨黨 ᔴ q 黨 ᔴ R× ᔼ =
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Ray Optics and Optical Instruments ᨨq㦨
\ r҈ = A 矄�
= 矄� 䰘�° ᔴ㡰 \ ᔴ㦨� ᨨ鼀⼄ ��㡰 = ҈䰘
矄� �° =
m=
A
҈
ᨨ
ᨨ
ᨨ
矄� ° = 矄� r Þ r㦨�
҈
r 㦨�ᨨ
B
C
dm = �҈ - A d m = A \ �҈㦨꾀㦨e 㦨꾀 \
�҈
턼 ҈
e
A
� r҈ = ҈
ᨨ �� ᨨ犨 �ᆠ m҈҈ 犨� ᆠ � ᆠ꾀犨 矄 B 盐 � �� 犨� 矄盐 ᨨ � ᆠ�ᨨ 㡰 ® ␔ 泐 矄 ᨨ鼀 ᆠ �盐ᨨᆠ├ᨨ �矄鼀 �ᨨ � ᆠ� ᆠ> �ᨨ ᆠ�ᨨ 矄ᨨm>m҈
泐 鼀ᨨ矄犨泐 ᨨ鼀 ᨨ鼀犨 �鼀�矄ᨨ盐 ᨨ ᆠ �盐ᨨᆠ├ ᨨm>m ® �꾀矄 �矄 ᨨ ��ᨨ ᨨ 泐矄�鼀�矄ᨨ盐ᨨ �矄鼀\m㦨m҈ ® 꾀矄 泐鼀�矄ᨨ盐ᨨ �矄鼀�鼀ᨨ矄犨泐 ᨨ鼀
ᨨ鼀\mm҈꾀 泐鼀ᨨ矄犨泐 ᨨ鼀 ᨨ鼀矄 � 鼀�矄 ᨨ盐 ᨨ ᆠ �盐 ᨨᆠ├ \m>m ® � 꾀 �矄鼀矄盐 ᆠᨨ� ᆠ�ᨨ 矄├
ᆠ \ᔴ��ᆠ �� >ᔴÞ矄� � >矄� ᔴ Þ
ᔴ
m > \ m > m ҈ m҈
҈
ᨨ ᨨ � ᆠ�ᨨ ᨨ � ᆠ 꾀
҈
Ҙ
҈ = = 矄� �㦨矄� C҈ 㦨 ҈ = Ҙ Ҙ
�ᨨ 犨 m A
m=
҈ ҈ = = ҈҈ 矄� C 矄� �°
�=45o
ᆠᨨ ��ᨨ �矄 ᨨ鼀� �盐鼀 ��ᨨ ᔼ �矄ᆠ矄 泐� 矄� �矄鼀ᆠᨨ鼀矄ᨨ矄 ᨨ �鼀 泐 泐� 矄� �矄鼀犨� 矄ᨨ �矄鼀�矄 T盐錐 ᨨ �� ᨨ矄� 鼀� ᨨ盐� 犨 �ᆠ �� 矄矄矄 矄 ᨨ盐 ᆠ�� � ҈ 和� ᆠ �矄鼀 �矄 ᨨᨨ犨犨 � ᨨ 矄ᆠ盐鼀 矄 ᨨ ᆠ�ᨨ
ᨨ �矄矄�ᨨ ᨨᆠᆠ盐 �ᆠ├ 矄矄 ᨨ �矄矄 矄矄 ᨨ盐� 犨 �ᆠ �矄鼀�矄鼀 ᆠ �ᆠ ᆠ 㡰 ᨨ鼀� �鼀盐鼀 ��ᨨ
꾀 �҈㦨
A
҈� - ҈㡰 ° = ° ҈䰘 - ҈㡰
㡰 ᆠᨨᨨ盐 鼀 ᨨ �� ᨨ犨 �ᆠ � > c 犨� � ᨨ � ᆠ�ᨨ
� ᨨ �ᨨ � ᆠ� ᆠᨨ � ᆠᨨᨨ盐犨�矄盐 ᨨ� ᆠ�ᨨ
ᨨ犨 �ᆠ m>҈҈
+ fm = �° Þ fm = °
҈㦨㦨 矄泐㡰㦨
A =
Ҙ҈҈㡰A҈㦨 Ҙ҈㡰Aᨨ
m=
(Ð꾀㦨 ᨨ ⼄盐�� �㡰
҈黨
é -҈ ҈ ù êm 矄� A - 矄� m 㡰 ú ë û
矄� �҈ -҈ \ �҈ = 矄�
矄� r҈
ᨨ 鼀� �鼀盐鼀 ��ᨨ ᨨ盐� �矄ᨨ矄ᆠ矄 �矄鼀 � ÐB㦨Ðᔴ턼ᔴ ᆠ㡰 꾀 �ᨨ ��ᨨ ᨨ �矄�矄鼀P҈ d҈㦨 Ҙ҈҈㡰A҈ 꾀 �ᨨ ��ᨨ ᨨ 矄ᆠᨨ �矄鼀 P d㦨 Ҙ҈㡰A ᨨ �矄矄�ᨨ 犨� ᨨ盐 ��ᨨ d 㦨 \
æ A + dm ö æ + m ö 矄� ç ÷ 矄� ç è ÷ø è ø \ = m= 矄� A 矄�
Þ
҈ö çm÷ è ø
꾀泐� � 和 Ἐ矄 犨 ἘἘ
\ 㦨҈� 꾀 ᆠ꾀ᨨ �矄鼀泐 矄
ᨨ鼀泐盐矄� �和 Ἐ矄犨 ᨨ ᆠ�ᨨ AC Ҙ矄� �㦨 ҈㡰矄� r
꾀 �ᨨ ��ᨨ ᆠAC 㦨� 㦨҈�
-҈ æ
矄� �҈ 矄� A 矄� A ⏌ ᆠᨨ矄 A ⏌ = = = ᆠᨨ矄 A ⏌ 矄� r҈ 矄� A ⏌ 矄� A ⏌
␔ 鼀� 泐 � �ᨨ 矄盐 ᆠ
矄� 黨° ҈ -҈ æ ҈ ö m= = Þ r = 矄� ç ÷ 矄� r 矄� r èmø 꾀
҈
Þ � 㦨ᔴ҈㦨�
m҈ = B
A � � C
m = m =
D
ᨨ ᨨ � ᆠ�ᨨ ᨨ � ᆠ ᔴ
�҈
盐r҈졔r㦨A\r҈㦨Ar
턼
m ҈ Þ�㦨C㦨 矄� � = 矄� C = = = m m
ᔴ
ᆠ 鼀� �鼀盐鼀 �ᨨ � ᆠ� ᆠ矄盐ᆠ ᨨ� ᆠ�ᨨ ᨨᆠᆠ盐矄ᨨ ᨨ 矄盐 ᆠ矄�矄
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�
Physics ᨨ犨��矄�� r㦨qc \r҈㦨Ar㦨 � qc㡰 꾀泐� � 和 Ἐ矄 犨 ᆠAB
ᨨ ᆠ�ᨨ �矄 矄盐 ᆠ � ᨨᆠ ᆠ � � �泐 �鼀� �矄 ᆠ�ᨨ 犨� ᨨ鼀 �矄 ᆠv҈
m m҈ m - m҈ = +R v҈ - ¥
�㡰
矄� �҈ n㦨 矄� r҈
ᨨ ᆠ�ᨨ 矄ᆠᨨ 矄盐 ᆠ
m m m - m = +R v v҈
��㡰
m
m҈
v
㦨矄� ҈
ᨨ ᨨᆠ � ᨨ �� 矄矄泐矄鼀㦨
Ҙ R Ҙ - Ҙ҈
ᨨ
l
æ 黨 ´ ҈
l㦨 ç
è
= ҈� +
î
l
犨 l�矄� 鼀㡰
㦨 鼀
ᨨ \
D C 䰘 n2 n1 r 㦨
B
A
矄� � n҈㦨 Þ 矄� �㦨n҈矄� r 矄� r é ҈ ´ ҈ ù ҈ + ҈ ê ú´ Þ 矄� �㦨 㡰 ûú ëê
䰘
Ql㦨 鼀㡰
矄� �㦨 è ø
\
꾀 �ᨨ � ᆠ� ᆠ�㦨矄� ҈ ç ÷ è ø
�㡰
꾀 ᆠAC
æ ö
æ ҈ö n ᔴ��ᆠ �qc㦨矄� ҈ ç n ÷ ᨨ 矄� qc㦨 ҈ n è ø
矄� �҈ 矄� r҈
矄� �҈ 矄� �° æ ҈ ö 矄� �҈㦨 ҈�㡰 ç ÷ 矄� �҈㦨黨� è ø
҈�㦨
\ �҈㦨矄� ҈ 黨�㡰»䰘 � ⼄盐� �ᨨ � ᆠ� ᆠ�㦨䰘
矄�盐�ᨨ ᆠ 矄 ᨨ犨 矄� ��盐 A � 㦨 A㦨 d㦨 e㦨䁸 꾀 A졔d㦨�졔e Þ e㦨A졔d� 㦨 졔 㦨 � ᆠ 鼀� 泐 �矄 r
ᨨ鼀ᨨ ᆠ ᨨ鼀犨 �ᆠ �鼀�矄 B ␔ e㦨r¢㦨 r¢졔r㦨A㦨r㦨 ᨨ鼀和 Ἐ矄犨 ᆠ�� ᨨ �矄鼀
Ҙ=
æ ö
ᨨ
n
ü n - n҈ - n҈ 㡰 ý þ
ᨨ犨泐� �和 Ἐ矄犨 ᆠAB犨 n㦨
E
C
��㡰 n҈ ¹ n ᆠ盐矄 n҈ n �矄�� 㡰 矄ᨨ ᨨ 泐ᨨ 矄矄 盐 � ᨨ盐� AC r㦨 泐^ᨨ ᆠAC㡰 ᨨ r҈㦨Ar㦨� 㦨�
A ° r㦨 = 㦨
�
�
ì ü æ n҈ n҈ ö÷ ï ï n ç ҈ í ý n n ÷ø ï ïî çè þ
ì ҈
㡰 ᨨ�� ᨨ 犨 � l㦨 鼀 ᆠᨨ鼀� �ᨨ ᨨ �矄鼀ᆠ矄矄矄� ��矄鼀矄 �├ �矄ᨨ矄ᆠ矄� �ABE 矄矄 ᨨ犨 � ��盐꾀 鼀� �鼀盐鼀 ��ᨨ 泐� 矄� �矄鼀 犨�ᨨ 矄 \
n
�҈㦨矄� ҈ í
҈ ´ ҈-
ö ҈⏌
÷ � ø
n҈
�矄�矄 ⼄盐� �ᨨ � ᆠ� ᆠ �҈㡰 ᆠAB ᨨ犨 �ᆠ 泐矄�├矄 ��ᨨ ᆠACᆠ��ᆠ ��矄
㡰 泐矄 � ᆠ� 泐 � ᨨ � ᆠ ᔴ 矄矄 ᨨ盐� 犨� ᨨ盐 � � �矄矄盐 ᆠ� n҈㦨n
҈ ´ ҈-
r
ü ҈ - 矄� qc - 矄� qc 㡰 ý î þ
␔ � ᆠ� 泐矄�鼀� ᨨ鼀 ᨨᆠ盐矄
҈졔
r҈
ì n
m R Ҙ Ҙ - Ҙ҈ Þ v㦨 m - m = ҈ v R
�҈
㦨矄� ҈ í
v҈
꾀 � �⼄ �㡰 ��㡰
�
ᨨ 矄� �҈㦨n矄� r҈ B \ �҈㦨矄� ҈ n矄� r҈㡰 和盐矄�盐� �盐ᨨ r҈犨� �҈㦨矄� ҈{n矄� � qc㡰晤 㦨矄� ҈{n 矄� � ᆠᨨ矄qcᆠᨨ矄� 矄� qc㡰晤
I҈
I
m
A
d rἘ
e C
矄� � 矄� ° ⏌ = = = = ҈䰘 矄� r 矄� ° ҈⏌
Topic-5 : Optical Instruments ҈
㡰 ᨨ 矄ᆠᨨ
盐 � 㡰㦨fᨨ졔fe㦨 鼀� � �ᆠ�ᨨ m㡰 =
fᨨ = � Þ f = � fe fe
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Ray Optics and Optical Instruments \ fᨨ㦨�ᆠ鼀 f㦨҈ᆠ鼀 ᆠ ᨨᆠ � ᨨ 泐㜠�ᆠf㦨҈ᆠ鼀
㡰 꾀ᆠᆠᨨ � �⼄盐矄�ᨨ M㦨䰘� L=҈�鼀鼀f㦨�鼀鼀 fe㦨䁸 矄� �鼀� � �ᆠ�ᨨ M ;
Þ 䰘� =
�
� �矄 ᆠ犨 ᨨᆠ� 泐�ᆠ㦨҈�ᆠ鼀 \ �矄 ᆠᨨ �鼀� ᨨ鼀 泐 ᨨᆠ� 㦨҈�㦨҈ᆠ鼀 盐 ᨨᆠ �矄 ᆠ ᨨ鼀 ᨨᆠ� ᨨ ᨨᆠ� 矄
L æ Dö ç҈ + ÷ f è fe ø
҈� æ � ö ç҈ + ÷ f e ø QD㦨�ᆠ鼀㦨�鼀鼀㡰 � è
Þ ҈� = ҈ +
� � = ҈䰘 » mm Þ fe = fe ҈҈�
b㡰 꾀矄ᆠᨨ鼀� � �矄泐鼀├� � ᨨᆠ� �ᆠᨨ矄 ᆠ㡰 � �ᨨ 犨� ᨨ盐 矄 � �犨ᨨ 犨� 矄 � � 犨� 矄犨 盐矄 㝐� �鼀盐鼀 � �矄⼄盐� ᨨ 鼀� ᆠ�� ᨨ �矄矄矄盐矄� �� �鼀�ᆠᨨ矄ᆠᨨ ᆠ㡰 㝐� � 泐� � ᨨ犨 ᨨ 矄ᆠᨨ
MP =
Þ 黨
҈ ҈ ҈ - = v u f ҈ ҈ ҈ ҈ ҈ ҈ � = Þ = Þ u = - ᆠ鼀 v u f � u ҈
�
ᨨ ҈
a=
� ҈ = ҈
\
b 㦨 q 㦨 㝐 맔 a = ´
꾀矄ᨨ 鼀� � �ᆠ�ᨨ m҈ =
҈ ҈ ҈ x = Þ v҈ = v҈ - x ҈ x -҈
҈ v҈ = u҈ x -҈
ᨨ 矄ᨨ泐㜠�ᆠ �矄�矄ᆠ� �矄ᨨᆠ
꾀¢
꾀 꾀¢¢ 졠
x ö æ \ u = - - v҈ 㡰 = - ç ÷ v㦨�ᆠ鼀 x -҈ ø è
쟼
¢ ¢¢
ᨨᆠ �鼀� ᨨ鼀 泐҈矄 矄 �鼀� ᨨ鼀 泐 矄
쟼泐㜠�ᆠ
ᆠ鼀
ᨨ �矄 矄ᨨ ᨨᆠ� =
Q D㦨� ᆠ鼀矄 �矄 ᆠᨨ �矄� ᆠ�矄�ᨨ 㡰㦨 ᆠ 鼀� � 泐� � ᨨ犨 ᨨ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ �矄 ᆠ㡰 � 鼀 � �鼀� � ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨ �矄 � 鼀� � � ᨨ
O I҈ I
v҈
矄� � 矄 ᨨ鼀盐
� v D ´ ´ 㦨 ҈� u f e
I
졠ᆠ�
㝐� � �ᆠ�ᨨ � � �泐
䰘
�ᆠ鼀 I҈
x
҈ ҈ ҈ = ҈ v -҈�㡰
= �ᆠ鼀
㡰 꾀ᆠᆠᨨ � �ᨨ⼄盐矄�ᨨ � �鼀��v㦨�ᆠ鼀 f㦨҈ᆠ鼀鼀� � �ᆠ�ᨨ m㦨m҈m㦨҈
O
҈ ҈ ҈ Þ v㦨ᆠ鼀 = v ҈ ҈�
M ¥ 㦨 -
\N =
f㦨҈ᆠ鼀
㡰 � f㦨҈ᆠ鼀錐fe㦨ᆠ鼀 u㦨҈�ᆠ鼀錐M¥㦨䁸
Þ
� N =
fe
҈ = = ҈�
҈° ; ° ᨨ b = ҈� ´ p
҈ ҈ ҈ ᨨ鼀 Þ = f v u
�㡰 � � ᨨ ᆠᨨ鼀ᨨ盐 鼀�ᆠᨨ矄ᆠᨨL㦨҈ᆠ鼀 ᨨᆠ � ᨨ ᨨᆠ� f 㦨 ҈ ᆠ鼀 ᨨ 泐㜠�ᆠ fe = �ᆠ鼀 u㦨fe㦨�ᆠ鼀 � �鼀� ᨨ鼀 � � �泐 ¥ 㡰ve㦨 ¥ v㦨҈�㦨� 矄� � 矄 ᨨ鼀盐
b �矄盐 泐�鼀�泐�ᆠ㡰 a �矄盐 泐ᨨᆠᨨ ᨨᆠ�㡰 fe
-҈ ҈ ҈ ҈ ҈ ҈ + = + = ᨨ uᨨ ҈ u v f -҈ ҈ ҈ � ҈ = - = \ uᨨ = - = - ᆠ鼀 uᨨ ҈ ҈ �
-
f ᨨ ҈� = 㦨 꾀矄ᨨ MP =
-҈ ҈ ҈ + = Þ u㦨ᆠ鼀 ue µ [Q � �鼀� ᨨ鼀 泐 泐�ᆠµ桌
Þ
㡰 fᨨ㦨ᆠ鼀 fe㦨ᆠ鼀 矄� � 矄 ᨨ鼀盐 ᨨ泐�ᆠ
꾀 �盐鼀� � �ᆠ�ᨨ m A =
� D = u u
ᨨ鼀� � �ᆠ�ᨨ m = m҈mA = ҈
� æ ҈ öæ ö ç ÷ç ÷ = ҈ è x - ҈ ø ç - x ÷ x -҈ ø è � Þ = ҈ Þ ҈ = x - ҈㡰 - x x - ҈㡰 - x Þ 䰘 x = ҈ Þ x =
҈ 䰘
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EBD_8341 P-286
Physics
æ ç Þ u = - ç çç è
ᨨ犨 ᆠ�ᨨ 犨�ᨨᆠᆠ盐ᆠᨨ ᆠ鼀�ᨨ ᨨ �矄IἘ 矄 矄 ᨨᆠ�矄 ᆠᨨ ᨨᆠ ᨨ鼀鼀�ᨨ u㦨 ҈�졔�㡰㦨ᆠ鼀
҈ ö 䰘 ÷ = -҈黨 ÷ ҈ ÷ � -҈÷ 䰘 ø
é ë
꾀�� 盐矄� � 矄 ᨨ鼀盐 ᨨ泐㜠�ᆠ
҈ ҈ ҈ � ´ ҈黨 = Þ fe = » ᆠ鼀 ҈黨 fe -� ҈ � ҈҈
҈
҈ ҈ ҈ ҈ ҈ ҈ = + Þ = + f v u f - -
矄ᨨ� �ᨨ犨 ᨨ ᆠᨨ 鼀�ᆠᨨ矄ᆠᨨ �矄 ��
ᨨ ᨨ�ᆠ鼀�ᆠᨨ矄ᆠᨨᆠ盐矄 犨 � ᨨ ᆠᨨ 矄 �矄矄鼀 犨 � ᨨ �矄��� bᆠ 㡰ᔼ 矄ᨨ ᨨ鼀�ᆠ矄ᆠᨨ �矄 ᆠ犨 ᨨᆠ� 矄 泐�ᆠ 矄 L㦨f졔 f㦨҈졔㦨҈鼀
f ᨨᆠ� f 泐 �ᆠ
=
-҈ = -
�鼀�矄 泐 矄ᨨ ᨨ鼀�ᆠ矄ᆠᨨ�矄� 꾀矄ᨨ ᨨᆠ��矄� 泐�ᆠ 㡰 ᔼ 矄ᨨ ᨨ鼀�ᆠ矄ᆠᨨ犨 � �鼀��矄 ᨨ鼀 � � �泐 꾀 �盐鼀� � �ᆠ�ᨨ
\
�
fᨨ = � Þ f ᨨ = � f e 㝐㦨 fe
V
䰘
҈ ᔼ n
ᔼᔼᔼ n
ᔼᔼ n
æ ҈ö
G
Y
O
qC
R
q
a a
b
҈㡰 � ��ᆠᨨ 泐�� � ᆠᨨ 泐�� ᆠ鼀 ᨨ鼀 ��盐DAFB DCFD矄�鼀� 꾀
҈ d = = D ҈
ᔼ
ᔴ
ᆠ鼀
ᆠ鼀
꾀矄 �矄 ᨨ �泐ᨨ矄矄\ID㦨Id \꾀�� 矄�泐ᨨ �� �矄 ᆠᆠ鼀 I 㦨
b
é ù � = �ᆠ鼀 êQ 꾀 = ú m ҈ ë û
æ ҈ ҈ ҈ö = m - ҈㡰 ç - ÷ = f è R҈ R ø
҈ \ �ᨨᨨ = = ҈ D
ᔼ n
ᆠ㡰 �鼀�I Ἐ ᨨ �矄 ᆠ�ᨨ �犨 泐ᆠᨨ鼀矄 ᨨ盐ᨨ �⼄盐� 㡰�矄 ᨨ
=
B
d
矄� a㦨 矄� b㦨 矄� 黨º
ᨨ q㦨矄� ҈ ç ÷ è ø
I
꾀� �� 㡰
ᆠ 泐ᨨ犨 ᨨ � ᆠᨨᨨ盐矄 ᨨ �� ᨨ 犨 �ᆠ qC>� ᆠ� ᆠ �犨�ᆠᨨ鼀ᨨ盐ᨨ� ᆠ㡰 和� ᆠ ᨨ 矄盐 ᆠ矄 矄鼀 �盐矄 ᨨ ᆠ盐盐 R҈㦨R㦨Rᨨ 矄鼀矄� ᨨ �矄矄�ᨨ 犨�ᨨᆠᆠ盐
\
�ᨨ ᆠ�ᨨ � ��ᨨ ᔼ鼀盐矄黨 矄 鼀 盐矄鼀�矄矄矄 � � ��ᨨ ᔼ 矄� q㦨
҈ ҈ Ҙ µ Ҙ l
ᨨ ᨨ �矄矄�鼀
ᆠ㡰 ␔ �� �矄� ᆠ� ᨨ �矄矄㜠 � ᆠ� � ᨨ鼀 �矄矄 �矄矄 ᆠ��ᆠ �矄鼀ᨨ �� 犨� ᆠ 鼀ᨨ矄 犨� 矄鼀� ␔ �� �矄� ᆠ� � ᆠ��ᆠ ���矄 ᆠᨨ鼀 ᆠ ᨨ� ᆠ�ᨨ 㡰矄ᨨ奠 矄鼀�矄矄�ᨨ
҈奠 ᆠ�ᨨ
矄 ᆠ ᆠ�矄�ᆠ矄 �ᆠ � ᨨ�ᨨ ᆠ㡰 b㡰 ᨨ ᆠ�ᨨ � ᆠ矄 ᨨ鼀 和 Ἐ矄犨
\
矄� qC =
矄矄
L㦨 fᨨ + f e = Þ � f e + f e = ᨨ fe㦨ᆠ鼀 fᨨ㦨�fe㦨�맔㦨ᆠ鼀
矄� q =
ᆠ㡰
l � ᆠ矄矄
Topic-6 : Miscellaneous (Mixed Concepts) Problems ҈
\ f 㦨҈҈ᆠ鼀 㡰 �矄 ᆠᨨ �盐�鼀� ᨨ鼀矄盐 ᆠ㦨ᆠ鼀
泐矄ᆠᨨ鼀� � ᨨ鼀 ᨨ� ᨨᆠ ᨨ �矄矄㜠�� ᆠ
ᨨ鼀泐 ᆠ矄矄盐 � ᨨ� 犨ᆠ ᨨ ᆠ 泐矄ᆠ├犨 矄 �鼀�犨� ᨨ鼀 ᆠ ᨨ 泐 ᆠ㡰 ⼄盐 ᆠ泐 ᨨ矄 ᨨᆠ �犨� ᆠ �ᨨ 鼀 �盐鼀 \ ⼄盐 ᆠ泐ᨨ 矄ᨨ盐 � 犨 㦨 ⼄盐 ᆠ泐 � � 㦨
\ 矄� qCµ l ᨨ �� 盐ᨨ l ᆠ��ᆠ �qC矄ᨨ� ᆠ矄矄
꾀 矄�ᨨ 犨 ᨨᆠ�矄 泐㜠�ᆠ \
� �矄 ᨨ �鼀� ҈
矄� �鼀�ᨨ ᨨ鼀盐
҈ l used
꾀 �盐鼀� � �ᆠ�ᨨ 㦨
҈
␔
矄ᨨ� �ᨨ犨ᨨ 鼀�ᆠᨨ矄ᆠᨨ
µ
� ù û
v = - ê҈� + ҈ ú
黨
D d
I 㦨҈맔҈㦨҈├␔鼀
ᨨᆠ�泐ᨨ �� � ᆠ盐盐鼀 c = ᨨᆠ�泐ᨨ �� � 鼀 �盐鼀 v =
ᆠ�� m㦨
҈ Ҙ e ҈
Ҙe
c ҈⏌ Ҙe Ҙe = = v ҈⏌ Ҙe Ҙe
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ᨨ ᆠ�ᨨ APB
ᨨ ᆠ盐盐 ᨨn㦨҈�Dn㦨҈ f 㦨ᆠ鼀 ⼄ �㡰
Ҙ Ҙ Ҙ - Ҙ - + ҈= ҈ u v R - ҈ ҈ - + = Þ v㦨ᆠ鼀 -҈� v -҈
Þ ҈҈
҈ - Df =Þ Df = - ᆠ鼀 ҈� - ҈㡰 ҈�
ᨨ �鼀�ᨨ O犨� ᨨ鼀 ᆠ鼀ᨨ �� ᨨ P T盐錐
I=
n n n n҈ ҈+ =
p P ҈ = ᨨ I µ A prl r
u
T盐錐
҈
= = ᆠ鼀 ҈� ҈� - ҈ - ҈ + = ᔴ矄㜠ᔼᔼ 盐 ҈� ҈ ҈ - ҈� \ + = \㦨ᆠ鼀 - - - ҈� ҈ ҈ - ҈� + = ᔴ矄㜠ᔼᔼᔼ \ - - ҈ ҈ ҈ \ =\㦨҈黨�ᆠ鼀 - ҈ R = n - ҈㡰 Þ f = bᆠ㡰 f R n - ҈㡰 ҈ n - ҈㡰 n + Dn - ҈㡰 n - ҈㡰 + Dn = + = f +Df R R R R \ f +Df = n - ҈㡰 + Dn f +Df R [ n - ҈㡰 ] \ = ´ f n - ҈㡰 + Dn R n ҈㡰 [ ] = [ n - ҈㡰 + Dn] \
D f n - - n + - Dn = f [ n - ҈㡰 + Dn] -Dn -Dn = = n - + Dn n - ҈㡰
Df Dn > f n
v R ҈ ҈� ҈� ҈ + = d �㡰 µ ҈ ҈ ҈ Þd�㦨\d㦨䰘ᆠ鼀 = d - �
bᆠ㡰 㡰 꾀泐� � 和 矄 犨 n҈矄� �㦨n矄� r Þ n҈矄� 㦨n矄� q
҈ =
ᨨ鼀⼄盐�ᨨ �㡰� 꾀矄ᨨ
\ ҈
b 㡰꾀矄犨├ ᨨ犨m㦨 ᔴ矄㜠ᔼ
҈� ҈ ҈ - ҈� n҈ n n n҈ + = + = Þ � V ҈ u v R
Þv㦨�ᆠ鼀 꾀�� ᨨ ᆠ�ᨨ � 和
�矄�矄 盐ᨨ鼀ᨨ矄 �ᆠ ᆠ�ᨨ �� ᆠᨨ鼀� � ᨨ鼀矄盐
矄 ᨨ犨 矄 ᨨ鼀꾀矄 犨 �ᨨ �� � 鼀ᨨ矄 矄�泐 泐 ᆠ矄矄 ᨨ 矄盐 矄 �� ҈
D f 㦨ᆠ鼀 ᨨ ᆠ�ᨨ � 和҈
ᔼ 矄�泐ᨨ � 矄ᨨ盐ᆠᨨ ᨨ犨 P㡰 �矄 ᆠ r ᨨ鼀 矄ᨨ盐ᆠ
҈
ᨨ b㡰
Þ
矄� ° = 矄� q Þ q 㦨 㦨r
ᔼ ⼄盐 � ᔴ턼 졠
㦨q ҈� � �
턼 �
䰘�
a㝐
꾀
졔 黨 졔 㡰졔҈� 졔Ð턼ᔴ㦨 Þ Ð턼ᔴ = �° Þ � 㦨�
ᆠ��ᆠ � ᨨ�矄鼀㜠��ᨨ 鼀 � C㦨 矄�
-҈ æ
҈ ö çè ÷ø 犨 �ᆠ �矄矄矄 �
ᨨᨨ� ᆠ�ᨨ ├矄ᆠ ᆠᔴ ᔼ D턼㝐Ð턼㝐㦨҈ � 졔䰘� 㡰㦨
ᨨ �ᨨ � ᆠ� ᆠᨨ 泐턼㝐ᨨ 꾀�矄
�矄 � �矄 矄矄 ᆠ��ᆠ � ᆠ 泐 鼀�矄ᨨ盐ᨨ ᆠ꾀 ᆠ㡰 ᨨ鼀 ��盐 �犨 � ᆠ� 泐졠 鼀� 泐㝐 �矄黨 㡰 泐 ��鼀�矄矄 ᨨ犨 � ��盐 ᨨ �鼀� 犨� 犨 C O␔ 鼀�ᨨ�矄 � 犨� 犨 㡰
҈䰘
�㡰 Q Dn n҈㡰桌
ᨨ鼀P 犨� 矄矄 ᨨ盐� C
C
C
ᔼ
Dn Df <
> n f
�ᨨ 犨
ᔴ
ᔼ鼀� P
҈
Df Dn 鼀� 矄 盐 ᆠ � � f n
O
O
ᆠ㡰 �鼀�I҈ᨨ 泐矄 ᨨ鼀 泐 矄L҈犨�ᆠ矄 �盐ᨨᆠ ᨨ矄ᆠᨨ 矄L
ᆠᨨ 矄盐 ᆠᆠ 泐ᆠᨨ ᆠ矄盐 ᆠᨨ 矄鼀 �盐矄
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Physics L
�ᆠ├ 矄矄␔ 泐 ᨨ鼀矄盐 ᆠACᨨ �矄鼀 矄 �鼀 鼀盐� ᆠ�ᨨ 矄ᨨᆠᆠ盐� �鼀
� 矄�泐ᨨ 鼀� 泐犨� 鼀 �鼀盐鼀 � 矄鼀� 犨矄 盐 �ᨨ ᆠᨨ 矄盐ᆠ� � ᆠ \ ᨨ鼀� �鼀盐鼀 �ᆠ├ 矄矄Dx㦨l Þ D x㦨mt㦨l 犨 t㦨 �ᆠ├ 矄矄ᨨ �鼀㡰
L҈
O
I
I҈
\ d
f҈
꾀泐� � 矄 ᨨ鼀盐 ᨨ 矄 L
Þ \
f f҈ - d 㡰 ҈ ҈ ҈ = Þ v = f v f҈ - d f + f҈ - d
ᨨ� ᨨ �矄 ᆠᨨ �鼀�I ᨨ鼀O
x=d+
f f҈ - d 㡰 f f + d f҈ - d 㡰 = ҈ f + f҈ - d f҈ + f - d
t㦨
l = ҈� 鼀 Ҙ
㡰 ᆠ盐矄S�矄ᨨ� 矄ᨨ盐ᆠ矄ᨨ� ᆠ � �矄犨� ᆠ�ᆠ盐 㡰 ᔼ � 矄�泐ᨨ �� � �ᨨ P �ᆠ泐 ᨨ鼀S I҈㦨I
� 矄�泐ᨨ �� � �P ᆠ�ᨨ ᨨ鼀 鼀�ᨨ ABI㦨奠ᨨ I㦨 I
I 鼀� æ I - I ö ҈ æ ö =ç ÷ø = ÷ = çè ҈ ҈ I 鼀 è I + I ø
ᨨ � y㜠ᆠᨨᨨ � 犨盐矄鼀� � �ᆠ�ᨨ ᨨ鼀盐 ᨨ 矄 L
ᆠ㡰 ᨨ鼀 �鼀盐鼀� 矄�泐 P � ᆠ㦨 nl ᔼ AB�矄 矄 � 泐 �矄 ᆠ x�犨�ᆠ盐矄 ��ᨨ � ᆠᨨ x
f f҈ - d 㡰 v f + f -d f = m= = ҈ u f҈ - d f҈ + f - d
h D ´ f 꾀矄ᨨ m = Þ h = D f҈ + f - d \
҈黨
Df D f҈ - d 㡰 f҈ + f - d 㦨 f҈ + f - d
꾀
和
ᆠ㡰 n҈㦨 ᆠ�� ᨨ � n㦨 ᆠ�� ᨨ 鼀鼀�犨 �ᆠ �矄 ��
ᨨ鼀� �鼀盐鼀盐ᨨ xn㦨҈
矄� q҈ n ᨨ鼀和 Ἐ矄犨 =
\
Q
y㜠ᆠᨨᨨ � y 㦨Dh
= D-
和ᆠ
矄� q
n҈
n ��\ q�矄矄ᨨ �� ᆠ� ᆠ㡰�矄ᆠᨨᆠ
C b㡰 和 ᨨ �� � 鼀 �盐鼀V㦨 n
x㦨lÞ x㦨
l = = 鼀
�㡰 � D҈㦨鼀D㦨҈ᆠ鼀y҈㦨ᆠ鼀 O�矄 鼀� ᨨ� ᨨ 犨ᨨ矄�矄 S҈ S S҈S㦨d㦨鼀鼀
n ᨨ鼀㜠鼀�㦨n
\ ҈
㡰
A
C 和 ᨨ �� � 鼀㜠鼀�V㦨 n 꾀泐� �和 Ἐ矄 犨 矄盐 ᆠ AB m҈矄� �㦨m矄� r
Þ
Ҙ�矄� 㦨Ҙp矄� r҈ Þ
Q \
r҈졔r㦨A r㦨A r҈㦨
S1 P
= 矄� r҈ Þ r҈㦨
ᆠ鼀
ᨨ
\ Q
N' M
Q
S2 S
a =
A
ᨨ
a
y҈
y
O q
B 鼀 D҈
盐 ÐQMN㦨 Þ ÐAMQ㦨
C R
D ҈ᆠ鼀 D
y҈ ҈ = = D҈ �
矄� a㦨
҈
=
҈ ҈ » » a �҈ ҈�
� ᆠDX҈㦨SS҈SS
ᨨ ᨨ
ᨨ
N
æ ҈ö DX҈ = d 矄� a = ( 鼀鼀 ) ç ÷ 㦨҈鼀鼀 �㡰 è �ø
R ᨨ矄 ᨨ矄��ᨨ ᨨ ᆠ �� � � � ᆠ㦨 B
C
ᨨ ᆠ 泐� 矄� �矄鼀 �矄 ᨨ ᆠ 黨 ᆠ �ᨨ 鼀� ᆠe㦨 㡰 㝐盐� ᆠ�ᨨ 矄 ᨨᆠᆠ盐 � �鼀 ᨨ 鼀� �鼀盐鼀
ᨨ犨 DX = S R - S҈R ᨨ DX = d 矄� q ᨨᆠ �� � �
��㡰
DX - DX҈ = ᨨ d 矄� q - DX҈ =
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Ray Optics and Optical Instruments ᨨ
d 矄� q = DX ҈ = ҈ 鼀鼀
ᨨ
㡰矄� q㦨҈ ᨨ 矄� q =
\
q =
ᨨ
҈ y = � D
҈
犨 k㦨҈ 鼀㡰⏌ ᆠ�� ᨨ ��矄҈
泐矄 ᨨ鼀�ᨨ 矄鼀 �盐鼀ᔼᆠᨨ � 盐ᨨ盐矄泐 矄 ᨨ犨 矄 ᨨ鼀 和ᨨB�矄 q꾀 �ᨨ � ᆠ� ᆠ㦨�B
҈ ҈ = �
\
҈ ҈ » = 矄� q \ q = y 黨 � D D ҈ ᨨ y = = = ᆠ鼀 � �
=
ᨨ
\ ᨨ矄��ᨨ ᨨ ᆠ �� � ��矄 ᆠ鼀ᨨᨨ� Qᨨ
矄� CD ��㡰
A
ᨨ
㡰 쟼⼄盐�ᨨ ᨨᆠᨨ泐y x㡰ᨨ 泐� 鼀 �盐鼀 꾀泐� � 和 Ἐ矄犨 ᨨ� 矄 A B mA矄� �A㡰㦨mB矄� �B㡰 mA㦨҈ ᨨ鼀 �㡰ᆠ盐矄y㦨 �A㦨黨 ᆠ盐矄ᨨ � � �� ᆠ� ᆠ �B㦨� mB㦨ky⏌졔҈桌 犨 k㦨҈鼀⏌ ᨨ mB㦨y⏌졔҈桌 ᨨ鼀 ���㡰 �㡰犨� ҈㡰 ҈㡰㦨y⏌졔҈桌矄� �B㡰
C
S҈ P
졠
S
S B
y҈
Q
D t ҈鼀鼀
矄� �㦨
ᨨ ᨨ
ᆠᨨ�㦨 y ᆠᨨ�㦨y⏌
㡰 � ᆠ 犨 犨ᨨ 泐矄 泐㜠҈ ᆠ ᨨ鼀 盐矄盐 ᆠAB 泐㜠 ᆠ ᨨ鼀ᨨ犨矄盐 ᆠ CD
ᨨ
dy 㦨y⏌ ᨨ y⏌dy㦨dx dx
l D x㦨amm맔t졔 l �矄
ᨨ
ᨨ
m҈㦨҈ ᨨ m㦨҈҈
҈
A t
amm맔t졔
B am
C
� ᆠ矄 泐҈矄盐 ᆠ�ᨨ ᨨ鼀 矄 鼀 �盐鼀ᨨ 矄盐 ᆠAB ᨨ ᆠᨨ 矄盐ᆠ� � ᆠ � ᆠ㦨nl犨 n�矄҈
\
m㦨҈
D
mm
=
y⏌
x
dy = ò dx ᨨ y҈⏌㦨xÞ y㦨 x⏌㡰
q
M
l҈
q q
d l
q=�
Q
N r �
d
ᨨ鼀 �� 矄矄�ᨨ ᨨ
= 黨 鼀 ´ ҈
b
㡰 㡰 꾀泐ᨨ �� � � � � �矄 � ᆠ� � � � � � ᆠ� � @ 黨 㡰 ᨨ 矄 ᨨ� ᨨ � ᆠ� ᆠ�矄 ᨨ��� A 㡰 y P x҈ y҈㡰 꾀�
B x y㡰
� q A 㡰 ᆠ�� ᨨ 鼀 �盐鼀㦨m m y㡰㦨ky⏌졔҈桌҈⏌
�
am 㦨҈� g
l ҈ö æ a 㦨nl Þ m m t = ç n - ÷ l è ø
t㦨҈鼀
�
l t㦨 a
⏌
\ 쟼⼄盐�ᨨ ᨨ ᆠᨨ泐ᨨ 泐� 鼀 �盐鼀�矄y㦨 x⏌㡰 㡰 ᨨ ᔼ ᨨ ├ ᆠ q 矄 ᨨ盐 � C ᨨ 矄鼀盐矄 ᨨ C 盐矄ᨨ m矄 ᨨ盐 �� ᨨ� ᨨ 犨ᨨ 矄 盐ᨨ l矄 ᨨ盐 矄鼀
ᨨ矄盐ᨨ �ᆠ├ 矄矄tn㦨҈ \
y
�㡰
҈
y ⏌ + ҈㡰
y ⏌
ò
���㡰
y⏌ + ҈㡰
҈
ᨨ
�⼄盐� ᨨ ᆠ�� m �矄 ᨨ盐 � ᨨ 矄矄 ᆠ �� � ��矄 ᨨ盐 Q� � ᆠ㦨 \ m҈㡰t㦨DX҈ ᨨ m҈㡰 ҈㡰㦨҈
dy 㦨 q 꾀ᆠᆠᨨ � �ᨨ�ᨨ鼀泐�졔q㦨黨 dx p æ dy ö q㦨 � \ 矄ᨨ ç ÷ 㦨 q è dx ø æp ö dy dy 㦨 ç - � ÷ ᨨ 㦨ᆠᨨ� ��㡰 dx dx è ø
㝐 �盐鼀 꾀�
Ҙ l㡰㦨҈졔 l
ᨨ� ᆠ�ᨨ 犨��� 泐l҈㦨处
矄� q = �� 㡰 Þ q = �҈°
҈ ҈ = = ҈� 矄� q b
\
Ҙ㦨
\
Ҙ㦨҈졔
\ 㡰
b㦨맔҈҈鼀 꾀泐� �和 Ἐ矄犨 N ᨨ犨 � l 㦨�处
´ ҈ -҈ 㡰
= ҈�
矄� r ´ ҈ -҈ 㦨҈ 犨 Ҙ㦨҈졔 矄� � � ´ ҈-҈ 㡰 矄� r Þ r㦨 Þ ҈㦨
Ҙ㦨
x �㡰
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EBD_8341 P-290
Physics
ᨨ鼀 ��盐��矄ᆠ ��ᨨ d㦨r� 㦨 �҈ 㦨䰘 ᆠ㡰 � 矄��矄 ᨨ 矄鼀� 鼀矄 I I 矄ᆠ�泐
ᨨ ᨨᆠ �矄 ᆠ u㦨OA¢㦨OAAA¢㦨҈҈㦨ᆠ鼀
ᨨᆠA¢矄�盐ᨨᆠ ᨨᆠᨨ ᆠ鼀�ᨨ
҈ ҈ ҈ + = v u f ´ � uf = = ᆠ鼀 = 䰘ᆠ鼀 v= u - f - �
ᨨ犨 ᨨ鼀鼀�ᨨ ᨨ鼀盐
꾀 r �
�
r
㝐
ᆠ 泐矄�� 矄矄 ᨨ盐� �矄矄矄 矄盐 � � 盐 矄 � ᨨ ҈ᆠ鼀 �鼀�矄 ᨨ盐 ᨨ鼀 B� 矄 ᆠᨨ �矄矄矄盐 盐ᨨ�矄矄 ᆠ �鼀��矄 ᨨ鼀 BἘ
턼
� ᆠDx㦨 m MB㡰 - AN 㦨
矄� r AB 矄� � 㡰 - AB 矄� q 㦨 矄� �
O
B
矄 � ᆠ犨 泐矄P Q�矄 ᨨ ᆠ鼀 �鼀盐鼀 � 矄�泐犨� ᨨ� ᨨ鼀 � � ᆠ \ 矄盐 ᔼ 矄�泐
I 鼀 = 䰘
(
I҈ + I
) =(
I + I
)
㦨 I 㡰 = 黨ᔼ
ᔼ 矄�泐ᨨ� IP㦨IA 졔 IB 졔 IC
IA =
ᔼ盐鼀� �ᨨ ᨨ犨㡰 ´ ᆠᨨ矄 q
pr 黨 ´ ᆠᨨ矄 ° ҈ = = p 犨 ⏌ 鼀 A 黨␔ p´
和�鼀�泐
IB =
҈ ´ ᆠᨨ矄 ° p ´ ҈�㡰
黨
C ␔
A
졠
P
B ҈␔
Ip =
黨C
泐矄 ᨨ��� � � ᨨ鼀 ᨨ� ᨨᆠA矄盐 矄 ᆠ�ᨨ
盐ᨨ�矄矄矄 ᨨ矄�├� � ᆠᨨ ᆠ鼀�ᨨ ᨨ 鼀�ᨨ 泐矄ᆠᨨ鼀� � ᨨ鼀AἘ ᆠ鼀
҈ᆠ鼀
A
҈ᆠ鼀
æ è
D
꾀泐� � 和 Ἐ矄 犨 M
矄� 黨° Þ 矄� 黨° - C 㡰 ҈ ҈ 꾀矄ᨨ Ҙ = 矄� C
҈ Ҙ
=
҈ ᆠᨨ矄 C
�㡰 ��㡰
␔ C�矄 ᆠ��ᆠ �
\ A¢
҈m
M
ᨨ鼀 �㡰 ��㡰
O
m
q
҈ Ҙ=
҈ ҈ + + ҈ = ҈黨␔ ⏌ 鼀 p p
꾀 矄 � AA Ἐ = t ç҈ -
C �ᆠ 矄�
C
졠
IC㦨ᆠᨨ矄 㦨҈ \
ᨨ�鼀� �矄 ᆠOB㦨OB 졔 BBἘ 㦨䰘졔҈㦨䰘䰘ᆠ鼀 ᨨ� � �� ᆠ� 泐BD ᨨ犨 �ᆠ �»黨 �ᨨ ᆠ�ᨨ 黨 C㡰 �矄 鼀 �鼀盐鼀 ᨨ �矄 C �矄矄 C �矄 � ᆠ��ᆠ �
B
҈ 犨 ⏌ 鼀 = p
B'
҈ Ҙ
=
҈ ҈ = ÞC㦨� ᆠᨨ矄 C 矄� C
҈ = = ҈҈ 矄� �°
ᨨ 鼀� �鼀盐鼀盐ᨨ m�矄矄㦨҈҈ ᨨ犨 �ᆠ 矄 ᨨ � ᨨ�矄� ᨨ鼀 泐ᨨ �ᆠ ᆠ矄ᨨ ᨨᆠ├
҈ö ҈ ö æ ÷ = ç҈ ÷ 㦨 ҈ ᆠ鼀 Ҙø ҈� è ø
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PS-291
Wave Optics
Wave Optics
24
␔ �矄 鼀ᨨ ᨨ鼀L �矄鼀 �鼀犨�� � ᆠ�矄� �鼀盐�ᨨ 犨 � ᨨ 鼀 n㦨҈ ᨨ �矄鼀 �鼀㡰 Dx = nl = ҈l
Topic-1 : Wavefront, Interference of Light, Coherent and Incoherent Sources ҈
b㡰 ᨨ 꾀㡰
xP
P
Q
xQ
A
d
xP - xQ = d + �㡰 - d - �㡰 = � 鼀 矄 � ᆠDf 盐ᨨ � ᆠ
D = ct = c ´
p p p Dx 㡰 = �㡰 = = l 꾀 A, Q �矄 ᨨ P 泐 矄 犨 鼀� 泐 Q ᆠ 矄 ᨨ犨鼀� 泐P p p \ ᨨ 矄 � ᆠA - = 盐ᨨP� � ᨨ Q泐黨 㡰
ᨨ鼀⼄盐�ᨨ �㡰桌
c c = md ��㡰 Q m = 㡰 v v
�矄 �矄 ᆠ�矄 ⼄盐 �矄 ᆠ� ᆠ盐盐鼀 �矄ᆠ ᨨ�ᆠ 졠�ᆠ ᨨ �矄 泐 犨 �ᆠ 矄 L ҈ ᨨ盐� 鼀 �盐鼀ᨨ ᆠ�� n҈㦨 n҈ L҈ 졠�ᆠ ᨨ 矄ᆠᨨ 泐犨 �ᆠ 矄 L ᨨ盐�
I ᆠᨨ矄 㡰 = I
鼀 �盐鼀ᨨ ᆠ�� n = n L � ᆠ = n҈ L҈ - n L ᨨ犨 矄 � ᆠ
� ᆠ xQ - xP = �鼀 矄 � ᆠDf 盐ᨨ � ᆠ
=
d v
= d
I A = I҈ + I + I҈ I ᆠᨨ矄 Df = I + I + I ᨨ C
\ Dx = l = S҈ L Ἐ- S L Þ ҈ = d - Þ d = 鼀 ᆠ㡰 �矄 ᆠ 矄 泐�� � 鼀 �盐鼀 ᨨ ᆠ� � 鼀� �鼀t�矄�� 泐 d = vt �㡰 犨 v�矄ᨨᆠ�泐ᨨ �� � 鼀 �盐鼀 �矄 ᆠ矄 泐�� � ᆠ盐盐鼀� �矄�鼀
p p ´ � ᆠ = ´ n҈L҈ - n L 㡰 l l 㡰 ᔴ泐 鼀� ᨨ �矄矄 �矄 ��� � 盐 ᨨ犨 ᆠᨨ �� � 矄ᨨ ᆠᨨᆠ 矄 ᨨ 鼀� 犨 ᨨ �矄矄矄 ᨨ犨 � ��盐 =
p p p Dx 㡰 = �㡰 = l
p p
ᨨ 矄 � ᆠC = + = p
和 ᨨ 鼀� 犨 ᨨ
I = I҈ + I + I҈ I ᆠᨨ矄 Df㡰 ᔴᨨ �� �
= I + I + I
I ᆠᨨ矄 p) = ᨨB, � ᆠ xP - xQ = 矄 � ᆠ Df =
p
I B = I + I + I I ᆠᨨ矄
��� �
盐ᨨP� � ᨨ Q泐黨 㡰
p = I
ᨨ� 矄��矄ᨨ ��ᨨ A, B C犨�� �ᨨ
ᔴᨨ �� �
I A I B IC = I I = ҈ 㡰 ᔼ ��泐 S L = 鼀
�
ᆠ㡰 � 鼀�盐 �ᨨᨨ 犨矄 �矄
矄ᨨ
� æö S҈ L = + ç ÷ = = � 鼀 èø
和 d
æ a ö +҈ ça ÷ æ + ҈ö æ ö = = 㦨 = ç ҈ ÷ = ç ç ÷ ÷ a è ø è ҈ø ҈ ç ҈÷ çè a ÷ø ҈
l
L'
N/c
� ᆠ Dx = S҈ L - S L = � 鼀 㦨
/c N
ᔼ鼀 æ a 졔 a҈ ö ᔼ 鼀� çè a a҈ ÷ø
a҈ = a ҈
L
b㡰
꾀矄 犨 ├ ᨨ犨
ᔼ鼀 æ 꾀҈ + 꾀 ö =ç ÷ ᔼ 鼀� è 꾀҈ - 꾀 ø
ᔼ҈ 꾀҈ ᔼ = 꾀
和҈
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EBD_8341 PS-292
Physics ᨨ 㡰 ᨨ�ᨨ
Dx㦨d矄� qÞdDx㦨dᆠᨨ矄dq
꾀 ᔼ 鼀 꾀 + 꾀 = ҈ Þ 鼀 = Þ ҈ = ᔼ 鼀�
꾀҈ - 꾀 ҈ 꾀 鼀�
矄� �ᆠᨨ鼀ᨨ ᨨ �� ᨨ
꾀҈ � ᔼ � æ�ö = Þ ҈ =ç ÷ = 꾀 ᔼ è ø 黨 䰘
黨
҈
␔ Ҙ� ᆠ矄矄 ᆠ矄矄
b㡰 �� 矄 盐犨 矄㡰 ᆠ ␔ ᆠ㡰 b㡰 ᔼ DOPM
O
R
⼄盐�㜠ᆠᨨ 矄ᨨ m҈㦨
��㡰
ᨨ㜠ᆠᨨ ᆠ犨 矄ᨨ m㦨
C
d
\
F㦨 R O
q
I P
l l d ᆠᨨ矄 q d = CO + OP + = + + ᆠᨨ矄 q ᆠᨨ矄 q
҈�ᆠ鼀 R҈㦨졔R
l l d ᆠᨨ矄 q + ҈㡰 + = d ᆠᨨ矄 q + ᆠᨨ矄 q ᨨᆠᨨ 矄盐ᆠ�� ᆠP � ᆠ㦨nl d ᆠᨨ矄 q +
҈
R㦨R
l n - ҈㡰 l = nl Þ ᆠᨨ矄 q = d
ᨨn㦨҈ ᆠᨨ矄 q =
҈
l d
꾀矄犨├ ᨨ犨I㦨I҈졔I + I҈ I ᆠᨨ矄 f ␔ 矄 � ᆠ�矄p⏌ Ip⏌㦨I졔I ÞIp⏌㦨�I 꾀�� 犨 矄 � ᆠ�矄p b㡰
҈�㦨 R Þ R㦨҈ᆠ鼀 ᨨ犨m ᆠ�� ᨨ �⼄盐� 犨 矄 鼀�ᨨ �鼀�ᆠᨨ� ᆠ� 矄犨� ᨨᆠ犨 ��矄ᆠ � ᆠ鼀\F¢㦨�ᆠ鼀 ᔼ �矄ᆠ矄 \
\ Ip⏌Ip㦨�II㦨I 㡰 ᨨᆠ盐矄ᨨ ⼄盐 � ᆠ� 矄盐
� �ᨨ �矄 ᨨ ᆠ泐� ᆠ 矄�� � ᆠ � �矄 犨� ᨨ矄 ᆠ㡰 I҈㦨I I㦨I
҈ ö ҈ö ҈ æ҈ æ ҈ - ÷ ÷ø 졔 m҈㡰 çè 㦨 m҈҈㡰 çè ø R R ¥ R F¢ Þ
\ I 鼀 = I҈ + I 㡰 = I + I 㡰 = I 㡰 = 黨 I
bᆠ㡰
� ᆠ 泐ᨨ� ᨨ ᆠ�ᆠ盐鼀 ᆠDx㦨d P҈ 矄� q d㦨҈鼀鼀 Dx㦨d 矄� q 꾀P҈ Dx㦨d矄� q Qq㦨 㡰 꾀P Dx㦨d矄� q Qq㦨黨 㡰
d矄� q S҈
҈ æ ö m - ҈㡰 㦨 çè - ҈÷ø ´ R R F¢
҈ ҈ ҈ m - ҈㡰 - m -m = ᨨ 㦨 㦨 \ m㦨҈ F¢ R R R � ҈ 꾀矄矄 ᨨ犨 � ��盐 � ᆠ犨�犨 �I㦨 I҈ ҈ I㦨I矄ᆠ ᆠ矄�奠 矄鼀�矄䰘�奠ᨨ �� ᔼ ᆠ ├矄ᆠ犨 泐矄꾀 Þ
I 鼀� = I҈ - I 㡰 = I - I 㡰 = I ҈
R = ¥
꾀ᆠᨨ 矄ᆠ ᨨ 鼀�ᨨ 矄�├ᆠᨨ ᆠ 鼀�ᨨ �鼀� �矄 ᨨ鼀 ᨨᆠ �矄 � ᨨᆠ �矄 ᆠ ᆠ ᨨ ᆠ盐盐ᨨ ᆠᨨ ᆠ鼀�ᨨ 꾀 ᆠ�ᨨ ᨨ盐� 矄 泐矄 ᨨ 鼀�ᨨ
ᨨ鼀泐 ᆠ � ᨨ ᨨ鼀�鼀� ᨨᆠ�矄 \ ᨨᆠ � ᨨ 矄泐矄鼀 F㡰㦨҈�ᆠ鼀
I p = I + I + I I ᆠᨨ矄 p 㦨 I
҈҈
R҈ 㦨R
=
\
㦨҈ ҈ö æ ҈ - ÷ 졔 m҈㡰 çè -R ¥ø
æ ö æ ö æ -҈ö ҈ ҈ = 㦨 çè - ҈÷ø + çè - ҈÷ø çè ÷ø 㦨 R R R R R
q q
d ᆠᨨ矄 q OC = ᆠᨨ矄 q � ᆠ犨
犨ᨨ泐矄ᆠ � �P
㦨҈�
�㡰
҈ ҈ ҈ ҈ ö æ҈ = + ç ÷ F f҈ f 㦨 m҈҈㡰 è R - R ø
M
Q
d ᆠᨨ矄 q ᔼ DCOP OP =
҈ Rl q\Rdqµ d ᆠᨨ矄 q ᆠᨨ矄 q 꾀矄犨鼀ᨨ ᨨ鼀P҈ᨨPq� ᆠ矄矄ᆠᨨ矄q ᆠ矄矄 ␔ 矄ᆠ 犨 矄 鼀�ᨨ �矄 � 犨� 犨 矄泐矄鼀 ᨨ 犨ᨨ 矄矄
R㦨dᆠᨨ矄qRdqÞRdq㦨
q
S
P
d
Dx ҈ 鼀鼀 = 盐鼀ᨨ 鼀 �鼀n㦨 㦨 l 鼀 꾀P Dx㦨l \� � � �桌 꾀P 鼀 �鼀�矄 ᨨ鼀
҈�
I 鼀 = I 鼀�
( (
) I )
I҈ + I I҈ -
矄矄
矄矄
ᔼ
䰘�I
�I
꾀
� I + ҈ I 桌
黨 = 㦨 ҈ � I - ҈ I 桌
҈䰘�I ҈I
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PS-293
Wave Optics
Topic-2 : Young's Double Slit Experiment ҈
l � ´ ҈-黨 = = ҈ 㦨�䰘 d � ´ ҈- 㡰 � ᆠ DP = d 矄� q = d q
d㦨 �矄 ᆠ犨 矄�矄㦨҈鼀鼀㦨҈鼀鼀 D㦨 �矄 ᆠ犨 矄�矄 矄ᆠ 㦨҈ᆠ鼀㦨҈鼀 y 㦨 �矄 ᆠ 犨 ᆠ �� � � ᨨ矄 � � 㦨҈䰘鼀鼀
\ DP =
҈
ᨨ�� � � � ᆠ鼀盐矄� �鼀盐�ᨨ l \�㦨 l1 =鼀l2 \l1 =� ᨨ 㦨㡰 l2 = 500 ᨨ鼀㦨�㡰 㡰 ҈㦨 㦨 � ᨨ �矄 鼀� �鼀 D =
҈҈
㡰 n҈ � �矄�矄�犨� �� ᨨ 犨 � l҈ n 犨� �� ᨨ 犨 � l
Þ
n l҈ 䰘 = Þ n = ´ ҈ = n҈ l
㡰
� � ᆠ Dx =
\
䰘
҈
㦨
´ ҈ - � ´ ҈
=
p ҈+ ҈ = � =
҈ + ᆠᨨ矄
-黨
´
�
)
҈ =
ᆠ㡰
� ᆠDx㦨d矄� a졔d矄� q㦨da졔
yd
犨 a q矄鼀桌
l
ᔼ æ pö æ pö ᔼ = ᔼ ᆠᨨ矄 ç ÷ ᨨ = ᆠᨨ矄 ç ÷ è ø è ø ᔼ
(
l
ᆠᨨ ᨨᨨ 鼀 �鼀矄ᨨ矄 � �盐 � £ q £ 㦨졔҈졔㦨҈
nlD ö æ çèQ b = ÷ d ø
æpö I I æpö ç ÷ Þ = ᆠᨨ矄 ç ÷ = ᆠᨨ矄 ç ÷ Þ I = � I èø èø ᆠ㡰 � �矄 ᆠ犨 矄ᆠ 矄�矄 D㦨҈�鼀 和�ᨨ 犨 矄�矄 㦨҈� 鼀鼀 ␔ � ᨨ 矄ᨨ盐ᆠᨨ �� l㦨�黨 鼀 � �㜠犨� D ҈� ´ �黨 ´ ҈黨 m w= l = d ҈� ´ ҈ 㦨�黨맔҈鼀鼀㦨�黨鼀鼀»�黨鼀鼀 ᆠ㡰 � D㦨b l D m ҈㡰t Dl = ᨨ \ t㦨 m - ҈㡰 d d ᨨ 盐矄㡰和 � 㦨 b �� 㡰 D m - ҈㡰 ù m - ҈㡰t n lD é = \ D êQ 和 � 㦨 úû a ë a a nl ᨨ t = m - ҈㡰 l b㡰 � � ᆠ D = 矄 � ᆠ Df㡰�矄�� 泐 p ( p) l = p Df = D 㡰 Df = l l ᨨ 犨ᨨ 矄ᨨ盐ᆠ矄 � � 矄矄
l Þ =
㡰 ᨨἘ Ἐ 盐鼀ᨨ 鼀 �鼀矄 矄� q㦨 l ´҈矄� 㦨 ´�´҈黨
Y P
S҈
p p l p Dx = ´ = 矄 � ᆠ矄Df = l l æ Df ö I = I ᆠᨨ矄 ç ÷ è ø
�
l
\D 㦨 ҈㦨 � =
dy ҈- ´ ҈䰘 鼀鼀 = = ҈䰘 m鼀 D ҈鼀
n Dl n Dl b= ҈ ҈ = d d
҈ 鼀鼀㦨� 鼀
㡰
b㡰 � ␔ � ᨨ �� l㦨� 鼀 �矄 ᆠ犨 矄�矄d㦨�鼀鼀 꾀 �盐犨� ᨨ � � ᨨ鼀
q=
� ᆠ㦨 矄� q » q 㦨҈맔
黨
d s�n
a
aq d S
㡰 㦨
y O
q �n ds D
ᨨa㦨 � ᆠDx㦨
yd
´҈҈ 㦨맔҈鼀鼀 ҈
Dx ´ ҈ - ҈҈ = 㦨 l ´ ҈- ҈҈ l \ Dx㦨 l Þ Dx㦨 n҈㡰 ᨨ犨
ᆠ� ᆠP�矄 矄盐ᆠ� 㡰
� �犨� b㦨
ᆠ㡰
ᨨa㦨
l �矄� ᨨ µ d
� ᨨ� 㡰 p
é ҈҈´ ҈ yù é Dx㦨 d ê a + ú 㦨맔҈ ê ҈ + ҈ êë Dû ë
- ù
ú 鼀㦨黨 鼀 úû
Dx 黨 ҈ = l 㦨 l ᆠ 矄盐ᆠ�� ᆠ 㡰 ᨨa㦨 � ᨨ� 졠㡰 p Dx㦨da㦨맔҈맔 ҈ 㦨맔҈黨鼀㦨 鼀 Dx ᨨ犨 㦨҈ Þ Dx㦨҈l l ᆠᆠᨨ 矄盐ᆠ�� ᆠ졠 ᨨ犨
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Physics
㡰 ᨨᆠᨨ鼀鼀ᨨ 鼀 �鼀n҈l҈㦨nl
n҈ l � ´ ҈ -黨 = = = n l҈ � ´ ҈-黨 �
Þ
ᨨ鼀� �鼀盐鼀tn㦨҈錐\t㦨
ᨨl҈
y= y㦨 ҈
n҈l҈D l҈㦨� 鼀 d ´ � ´ ҈ -黨 ´ ҈�
� ´ ҈ -
盐 Df =
p Dx l
ᨨ
y㦨䰘鼀鼀 Df = 矄ᨨ Dx =
p
l
ᆠ㡰 㡰 ᆠ㡰
ᔼ犨�ᆠᨨ ᆠ �ᆠᆠ�ᆠ矄
҈
b㡰
ᔼ 矄�泐 ᔼ 㦨 ᔼᨨ ᆠᨨ矄
�
ᔼ ᔼ f p ᔼ = ᨨ \ ᨨ = ᔼᨨ ᆠᨨ矄 \ f = n + ҈㡰 p � f = p p \ 꾀 � ᆠ æ lö l n + ҈㡰 l Dx㦨 çè ÷ø f \ Dx㦨 l p 㡰
␔├ ᨨ犨 � �犨� b =
Q ᆠ㡰
lR>l>lB \ b R>b G>b B ᔼ㦨ᔼ鼀 ᆠᨨ矄
㦨 ᔼ 鼀 ᆠᨨ矄
p 矄� q l
p d 矄� q 㦨 l
҈
Ds㦨
㡰
d⏌ d⏌
d p = ᨨq㦨黨
\
\
æ p ö d=ç ÷ ҈�㡰 ҈㡰 = p ᨨ è ø
q q �
ᔼ 鼀 æ p 矄� q ö = ᔼ鼀 ᆠᨨ矄 ç ÷ è l ø p 矄� q p p 矄� q ҈ æ l ö = Þ = \ q = 矄� -҈ ç ÷ ᨨ ᆠᨨ矄 l l è ø 㡰 꾀 ᨨ ᨨ ├ 矄矄鼀� �鼀犨�ᨨᆠᆠ盐 ᨨᨨ 犨 � 矄� ᆠ� 矄�鼀盐 ᨨ盐矄泐 ᨨ鼀泐 n + ҈㡰 m + ҈㡰 l҈ = l Þ n졔҈㡰l҈㦨 m졔҈㡰l \ n + ҈㡰 � 䰘 = = ᨨ҈n㦨҈m졔 ᨨ m + ҈㡰 � 泐� 矄ᆠ�ᨨ ᨨm㦨n㦨 ᨨm㦨䰘n㦨҈ �矄 ᆠ 犨 鼀 犨� �矄 ᆠ犨 矄盐ᆠ ᨨ� 矄 Dl҈ ì n + ҈㡰 - n҈ + ҈㡰 ü � Ds = í ý d î þ 盐 n㦨҈n҈㦨 Þ
æ pd 矄� q ö I q㡰 = I ᆠᨨ矄 ç ÷ è l ø y
S҈ q D S
æ p ö æ҈ö p d=ç ÷ ҈�㡰 ç ÷ = è ø èø
A
d矄
I鼀 µ a졔 a 㡰㦨 �a㡰
v ´ ҈ l= = = m ҈ h d㦨҈�鼀 ᨨ q㦨
B
a
I鼀� µ a a㡰㦨 㦨҈䰘a㡰 ᔴ泐 � 矄��矄ᨨ ᨨ 鼀 �鼀 鼀� �鼀� ᆠ矄 ᆠ㡰 ␔├ ᨨ犨 � 矄�泐ᨨ ��
\
lD d
d
k=
d pd q I q㡰 = I ᆠᨨ矄 犨 d= l
犨 ᔼᨨ �矄 ├� 矄�泐
҈黨
҈l҈ D k l D = d d
҈ ´ = ҈ ᆠ 盐鼀ᨨ � �矄ᨨ矄 � 矄鼀矄�鼀 ᨨ 矄ᆠ 㦨҈ 㡰 ␔ 矄�矄ᨨ ⼄盐犨� I鼀 µ a졔a㡰 㦨a㡰 I鼀� µ aa㡰 㦨㡰 ␔ ᨨ 矄�Ἐ矄犨� �矄犨�ᆠ ᨨ ᨨ
\
b
҈� ҈ ҈䰘
� �犨� w㦨
I҈ W҈ a W a = = = Þ b = Þ I W b W b
lD dy l = b ��㡰 = �㡰 d D 㝐盐�泐� � ⼄盐�ᨨ �㡰 ��㡰 犨 �
y=
b㡰 Þ
æ Df ö ÷ è ø
I 㦨 I ᆠᨨ矄 ç
ᆠ㡰
nl l = 㦨l m - ҈㡰 ҈� - ҈㡰
ᨨq㦨 d㦨ᨨ \
ᔼ q㡰㦨ᔼ
㡰
b =
䰘
㡰 \
æ pö I I q㡰 = I ᆠᨨ矄 ç ÷ = ᨨ è ø d p = I q㡰 =
d =
lD l D㡰 lD = = b b Ἐ = d d ⏌ d
� �犨� �矄⼄盐 盐 ᨨ鼀 �鼀 ᆠ㦨ml SAS҈A㦨ml S҈ x + d
d x
O
A
d
҈ ´ ´ ҈ -黨 㡰 é ҈ - 䰘 ù ê ú 㦨鼀鼀 ҈ ´ ҈ - ë û � ᆠ㦨 Ҙ҈㡰t 㦨nl錐
S
\
é n - ҈㡰 d + x + d + x ù d - x 㦨ml ëê ûú
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Wave Optics
\
n҈㡰 d + x 㡰 㦨ml
\
æ ö ç ҈÷ d + x 㦨ml \ è ø
d + x 㦨ml
\ d졔x㦨黨ml \ x㦨黨mld ᔴᨨ鼀� � �矄 ⼄盐�ᨨ 犨� �� ⼄盐�ᨨ
x㦨pmld犨� p㦨黨 \ p㦨 黨㡰 ᔼ 泐ᨨ盐 �Ἐ矄 ᨨ盐矄� �鼀 � 矄�泐ᨨ� �矄�� 泐
I = I ᆠᨨ矄
꾀矄 �矄犨 ᨨ 盐鼀 ᆠᨨ ��ᨨ ᨨ鼀� �鼀�矄矄�矄 �
ᨨ�졠犨� ├�鼀� �鼀�矄 ᨨ鼀 �졠 꾀矄ᨨ 矄 矄ᆠ �矄 �ᆠ盐 ᨨ ᆠᨨ � � � 矄�矄和҈和 ᨨ � �矄ᨨ� 犨�矄鼀�㜠ᆠ�ᆠ盐 ᨨ 泐ᨨ ᨨ 矄ᆠ �矄�㡰
f
bᆠ㡰 Q
´
犨 f㦨 矄 � ᆠ
꾀 �盐矄�ᨨ �
矄� �⼄盐�ᨨ �㡰
æf ö æ p ö ᆠᨨ矄 ç ҈ ÷ ᆠᨨ矄 ç ÷ è ø è ø = æf ö æ pö ᆠᨨ矄 ç ÷ ᆠᨨ矄 ç ÷ è ø è ø K ҈ K 黨 K Þ = = Þ I = = I ҈ \ n = 黨 I҈ = I
Þ
Þ ҈�l҈㦨҈l Þ l㦨҈�l҈҈�l҈㦨҈�맔� 鼀 Þ l㦨䰘� 鼀 ᨨ ᆠᨨ 矄ᨨ盐ᆠ矄 ᨨᆠᨨ 矄盐ᆠ� � ᆠ
鼀�盐 鼀� ᨨ� 㦨A졔A㦨A Imax µ (2A)2 Þ Imax㦨I㦨ᔼ҈ 㦨 ᔼ 矄�泐 盐 ᨨ ᨨ 矄� ᨨ � ᆠᨨ 矄ᨨ盐ᆠ矄 � 矄�泐 盐 ᨨ鼀泐 ᨨ � ᆠ㡰
ᨨ ᨨ� 矄�泐I㦨I 졔I 㦨I ��㡰 ᨨ鼀⼄矄 �㡰 ��㡰
l 㦨 犨� 㦨҈
d 矄
�
和ᆠ
l ù é êëQ = 矄� q úû
l < l 矄� q
鼀 b
맔㦨�맔
p 맔 � ᆠ l ᨨ � ᆠl 矄 � ᆠ f҈ = p 矄� � 矄 � ᆠ f =
l>l҈\b >b ҈
l
盐鼀ᨨ � �矄� �� 犨� mµ
�㡰
p l ᨨ � ᆠ 矄 � ᆠ f =
␔├ ᨨ犨 � �犨� b =
(
ᔼ҈ + ᔼ
)
ᔼ鼀� 㦨
(
ᔼ҈ - ᔼ
)
ᔼ ��泐ᔼ҈㦨ᔼ ᔼ㦨ᔼ \ ᔼ鼀 㦨黨ᔼ ᔼ鼀� 㦨ᔼ ␔ ᔼ҈㦨ᔼ㦨ᔼ ᔼ鼀 㦨ᔼ ᔼ鼀� 㦨 � ␔ � 矄��矄 ᆠᨨ鼀 ⼄盐 ᔼ鼀� 盐ᆠ矄 ᨨ ᨨ ᆠ㡰 ᨨ矄犨 � 矄犨�鼀�矄矄� � ᨨ犨 �ᆠ � ᆠ Dx㦨 n҈㡰 l⏌
lD y = n - ҈㡰 d ld 㦨 n - ҈㡰 b Q d㦨b D㦨d㡰 b ld = n - ҈㡰 Qy㦨b⏌㡰 \ b
P
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Þ
䰘
Physics
l=
b 犨 n㦨҈ n - ҈㡰d
l=
b b d d
I b 㡰 鼀 = I 鼀�
\
( (
n҈㦨҈n㦨 �㦨� r㦨䁸 矄和 Ἐ矄犨B n҈矄� �㦨n矄� r \
矄� r㦨
҈ ´ 矄� �°
=
҈ Þ r㦨
\ 꾀 � 鼀 泐 �矄 矄盐 ᆠ 犨� ᆠ 泐 BC �矄 ᆠ ᆠ 泐Bᨨ ᨨ� ᨨ
I҈ = I
B �
A҈ = 꾀鼀�盐 �ᨨ A
黨
) =黨 ҈ I )
I҈ + I I҈ -
I P㡰>I P҈㡰 I P㡰>I P҈㡰
I҈ = I
C
O
A n҈ = ҈
=
n =
n㦨҈�҈
꾀㜠矄錐㜠⼄錐ᔴ㜠錐㜠矄 ᨨ � ᆠl⏌ 矄 � ᆠ�矄p⏌ ᨨ � ᆠl⏌ 矄 � ᆠ�矄p⏌
ᨨ犨 ᨨ ᆠ�ᨨ 矄 �ᆠ矄盐 ᆠ
S҈ P - S P = \ d P㡰㦨
� ᆠ ᨨP҈ P犨� ᨨ ᨨ � 矄��矄 �矄鼀 �鼀盐鼀 ᔼ 矄�泐ᆠᨨ � 盐ᨨ盐矄泐 ᆠ矄矄 ᨨ鼀 P ᨨ犨 矄 P
Þ
\
I P 㡰 > I P҈ 㡰
㡰 꾀 P ҈ � ᆠ �矄 ᨨ ᆠ P҈ �矄 �� 矄 ᆠ � � dP҈㦨
dP =
n n҈ n - n҈ - = Qu㦨¥㡰 v u R ҈�҈ ҈�҈ - ҈҈ = \ OP㦨�鼀 OP
㡰
ᆠ 鼀 �鼀�矄ᨨ� �矄 ᆠxᨨ犨졠
\
Dx҈㦨 S҈PSP㦨
æ mg
Dx㦨 ç
è mm
l l dP҈ = dP = ҈
xd D
ö - ҈÷ 졔 盐ᨨ�矄矄矄 㡰 ø
ᔴ㡰 d P 㡰 = -l ⏌ 錐 d P҈ 㡰 = -l ⏌ d P 㡰 = -l ⏌
� ᆠ �矄 ᨨ \ Dx҈㦨Dx
I P 㡰 = I҈ + I + I҈ I ᆠᨨ矄 p㡰
\
= I҈ + I - I҈ I = I + I - I =
I P҈ 㡰 = I҈ + I + I҈ I ᆠᨨ矄 -p ⏌ 㡰 = I҈ + I = I + I = I
æ pö I P 㡰 = I҈ + I + I҈ I ᆠᨨ矄 ç - ÷ è ø = I҈ + I +
\
I҈ I = I + I + I = I
I P 㡰 > I P҈ 㡰 >I P㡰
ö xd æ m g =ç - ҈÷ t D è mm ø
æ mg
\
矄 � ᆠ
è mm
=
(
)
= I҈ + I - I҈ I = - I ᔴ泐I P҈㡰㦨
ö - ҈÷ t ø
QDx҈㦨㡰
ö p p æ m g Dx = - ҈÷ t ç l l è mm ø
´ ҈ æ ҈� ö - ҈÷ ҈ ´ ҈- 㡰 㦨 -䰘 çè ⏌ ø ´ ҈ f I \ 㦨ᆠᨨ矄 㡰㦨䰘� I
Q
I㦨Iᆠᨨ矄
ᨨ
I 㦨
ᆠ㡰
ᨨ鼀 �鼀盐鼀 � 矄�y 졠
I P҈ 㡰 = I҈ + I + I҈ I ᆠᨨ矄 -p㡰 I P 㡰 = I҈ + I + I҈ I ᆠᨨ矄 [ -�p ⏌ ]
d⏌ S
ᨨ졠 � ᆠ㦨 ç
æ -p ö I P 㡰 = I҈ + I + I҈ I ᆠᨨ矄 ç è ÷ø
I =
O x P
㡰
f㦨
= I҈ + I = I + I = I
S҈ d⏌
t æ mg ö D - ҈÷ t ´ Þ x 㦨ç d è mm ø - æ ҈� ö ҈ ´ ҈ 㡰 ҈�㡰 - ҈÷ ´ =ç 㦨맔҈鼀 è⏌ ø � ´ ҈ -
㡰 dP = l ⏌ 錐 dP҈ = -l ⏌ 錐 dP = -�l ⏌҈
= I҈ + I - I҈ I = I + I - I
P
I æ mg
꾀�� � ᆠ㦨 ç
è mm
ö - ҈÷ t ø
ᨨ鼀 �鼀 � ᆠ㦨nl
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Wave Optics \
æ mg
ö - ҈÷ t ø
nl㦨 ç
è mm
P S҈ Na
- æ mg ö t æ ҈� ö ҈ ´ ҈ = ҈ ҈ ç ÷ Þ l㦨 ç ÷ n è mm ø n è ⏌ ø
҈
҈ ´ ҈ 鼀 ҈ 㦨
鼀 n n
� ᆠ犨 泐҈ ᆠ � �P㦨NS҈졔S҈M
盐� �n㦨҈ᨨ � 犨 � � �ᨨ 맔҈鼀ᨨ䰘맔҈鼀犨� l㦨�맔҈䰘鼀 맔҈䰘鼀 㡰 ᔼ � ᆠ� 鼀 矄 ᨨ鼀泐 � ᆠ Dx㡰 ᨨ鼀 泐 矄� � ᨨ鼀 S҈ S ᆠ � �ᨨ� P Dx㦨SPS҈P㦨 d 矄� q
Dx㦨d矄� a졔d矄� q
S҈ Na
P
҈ y
d
Q S
q q
ᨨᆠ 鼀 �鼀 � ᆠ矄 ᨨ盐 ᨨ \ Dx҈㦨ᨨDx㦨
l 犨
\ \
d 矄� a㦨d 矄� q µ㦨q㦨
Q
y㦨
n㦨҈
d 矄� q 㦨 n҈㡰
l
n - ҈㡰l n - ҈㡰� n - ҈ = = d ´ ҈ 꾀矄ᨨ -҈ £ 矄� q £ ҈ \ᨨ矄矄�盐矄ᨨ 鼀᠀҈ ± ᨨ鼀DPOQ ᨨ矄��ᨨ ᨨ 鼀� �鼀
D 矄� q
҈ - 矄� q
㦨
矄� q ҈ - 矄� q
҈ - 矄� q
㦨 �鼀
d 矄� q =
\
矄� q =
\
y㦨
l
l + d 矄� a
� ҈ ҈ l + 矄� a = + 矄� ° = + = d ´҈
矄� q ҈ - 矄� q
㦨҈҈鼀
Fᨨr f�rst m�n�ma ᨨn the ᨨther s�de
҈ ᨨm㦨졔҈ 矄� q = y㦨 m㦨҈ 矄� q = - y = -҈҈ 鼀 m㦨졔 矄� q = \y㦨졔҈҈鼀 ҈ m㦨 矄� q = - \y㦨鼀
d 矄� q㦨
矄� q㦨 \
l ᨨ
l � ҈ = ᨨ 矄� q㦨 d ´ ҈
y㦨
矄� q ҈ - 矄� q
㦨鼀
ᆠ y㜠ᆠᨨᨨ � 矄 ᨨ �矄 鼀� �鼀 ᨨ � 矄� ᨨ ᆠ 鼀 �鼀盐鼀鼀 ҈҈鼀
㡰 ᔼ � ᆠ� 鼀鼀├矄 �ᨨ 犨� x㜠 �矄 � ᆠ犨 泐҈ ᆠ � � P㦨SMNS҈ Dx҈㦨 d 矄� q - d 矄� a
Þ
QD㦨҈鼀㡰
ᨨ y㜠ᆠᨨᨨ � 矄ᨨ � ᆠ鼀� �鼀ᨨ 矄ᆠ 矄 ᨨᨨ犨矄
\
矄� q
Fᨨr f�rst m�n�ma錐 d 矄� q - d 矄� a =
矄� q =
y㦨D q =
y
O
M
꾀 � ᆠ ᨨ鼀� �鼀盐鼀� 矄�泐㦨 n҈㡰
Þ
M
P
D 㦨҈鼀㡰
\
d q Qq a S
S҈
d矄 � q
҈
q d q aQ M S
-
㦨
y
矄 � ᆠf㦨
p p Dx = �l + D 㡰 l l
␔ Dl⏌
æ fö
␔├ ᨨ犨 I f㡰㦨I鼀 ᆠᨨ矄 ç ÷ è ø
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EBD_8341 PS-298
Physics
ᔼ 矄�泐 ᨨ 㦨
I 鼀
㦨҈졔맔҈졔 ҈ - ´ ´ ҈ - ᆠᨨ矄 �
f f p I 鼀 = I 鼀 ᆠᨨ矄 Þ = ° =
\
t 㦨黨맔҈ 鼀
�㡰
a mm
\
=
la Þ lm㦨 lm
la
l D ´ ҈ ´ ҈ D l 㦨 a = - a d ҈ ´ ҈ m Ҙd
yn =
Dt é g m ù - ҈ú 㦨 ê d ë mm û
矄 � 矄 ᨨ 矄盐ᆠ 鼀� �鼀 矄 � 矄 ᨨ �矄 �
b
t m
S҈
҈
b⏌
㝐� �鼀
´ ҈
= ҈� ´ ҈ - 鼀
㡰 � ᔼ 矄�泐I 㦨␔鼀 꾀 A㦨맔҈鼀 꾀 �盐矄 w㦨҈ ⏌矄
҈ ᨨ盐�ᨨ
㡰 ´ ҈- 㡰 \꾀� �泐 = ; � ´ ҈-
㡰
Dt é g m - m m ù b ê ú= d ë mm û
Þ 㦨
´ � ´ ҈-҈ ´ ҈ ´ ҈-
Q < ᆠᨨ矄 q > =
㝐 �鼀
D
㦨҈҈䰘맔҈鼀
꾀� �泐 = I A < ᆠᨨ矄 q >
d
\
´ ҈-
Topic-3 : Diffraction, Polarisation of Light and Resolving Power
졠
S
´ � ´ ҈-҈ ´ ҈ ´ ҈-
矄 �矄 ᆠ ᨨ鼀 ᆠ 鼀 �鼀盐鼀 盐矄ᆠᨨ矄ᨨ 矄ᨨ �� � �ᨨ l҈㦨�处ᨨ � �� � �ᨨ l㦨�处 ᔼ矄 �矄 ᆠ ᨨ鼀 ᆠ 鼀 �鼀
㦨맔҈鼀 ��㡰 �t �ᆠ├ 矄矄ᨨ �矄矄 ᆠ犨 �ᆠ ᆠᨨ矄ᨨ ᨨ 矄�矄 矄 � ᨨ � �矄犨 ᨨ 矄��矄ᆠᨨ 犨� �
�矄矄 矄
矄 � � �ᨨ � �矄㦨
y㦨
nl҈ D ml D n � = = = \ d d m � �
-҈
� �犨� 㦨
n lD ᨨ �� � �n㦨 d
��㡰 �� � �ᨨ 犨 � �处ᆠᨨ� ᆠ� 犨� 鼀 �� � � ᨨ 犨 � �处 �矄 ᆠ 犨� 矄鼀 ᨨ鼀 ᆠ ��
a mҘ
␔ � ᨨ �� � �� �⼄盐� 㡰 \
�㡰 꾀矄 犨 ├ ᨨ犨 �矄 ᆠ ᨨ �� � � ᨨ鼀 ᆠ 鼀 �鼀
yn =
l p p = �l + D 㡰 Þ Dx�l졔 㦨t l
Þ
p 㦨䰘맔҈␔
b dm µ ´ ҈- ´ ҈- ´ ҈ 㦨 g Ҙ - m µ㡰 ´ D ҈� - ҈㡰 ´ ҈
㦨҈�䰘�맔҈䰘鼀㦨҈�䰘�맔҈鼀 ᨨ犨 矄鼀� 泐 A P҈㦨҈奠ᨨ I prA 㡰 㦨 ê҈奠 ᨨ ç
和鼀矄 �盐 矄�ᨨ 犨 犨ᨨ �矄 ᨨᆠ矄 鼀ᨨᨨ
和�鼀�泐 ᨨ犨 矄鼀� 泐 B
q = ҈
é
æ ҈ ö ù ´ p ҈㡰 㦨҈␔ è p ÷ø úû
ë
é
æ ҈ ö ù ´ p ´ 㡰 㦨맔҈␔ è p ÷ø úû
㦨 盐 �鼀 ᨨ 矄ᆠᨨ �矄 ᆠ졠҈졠㦨 q㡰d 㝐� �鼀盐鼀矄�ᨨ 犨 ᨨᆠ矄ᨨ 矄盐 ᆠᨨ 鼀ᨨᨨ
P㦨 ê҈奠 ᨨ ç
ë
Df 矄 � ᆠ� ᨨ 盐ᆠ 泐 �鼀 \ 㦨 \
Df 㦨
lö æ = ç ҈ ÷ d è aø
p � ᆠ� ᨨ 盐ᆠ 泐 �鼀㡰 l
p p p ҈ ´ Ҙ - ҈㡰 t = -҈ ҈�҈桌맔맔҈ 㦨 l ´ ҈ ᨨ犨ᆠ� ᨨᆠ矄ᨨF
P 㦨P҈졔P졔 P҈ P ᆠᨨ矄 D f
l a
=
҈㡰 �� ´ ҈ -҈ 㡰 ´ ´ ҈� ´ ҈ �
㦨�맔҈鼀㦨�鼀»鼀 b㡰 꾀ᆠᆠᨨ � � ᨨ⼄盐矄�ᨨ � 矄�泐ᨨ �� ᆠᨨ鼀� �ᨨ盐 ᨨ 泐矄�矄盐矄 ҈奠ᨨ ᨨ��� � 矄�泐 I㡰 矄� �I㦨Iᆠᨨ矄q
WWW.JEEBOOKS.IN
PS-299
Wave Optics
I ҈ = I ᆠᨨ矄 q Þ = ᆠᨨ矄 q ҈ ҈ ҈ Þ ᆠᨨ矄 q = = ҈ Þ q » 䰘҈° ҈
Þ
ᨨ � 泐犨 �ᆠ 泐矄 ᨨ ᨨ 盐 ᨨ 盐ᆠ ᨨ盐盐 � 矄�泐 ᨨ ᨨ f=黨 q 㦨黨 䰘҈ 㦨҈
�
l´ 錐 � �犨� b = Ἐ
҈- =
� ´ ҈- ҈ - ´ � ´҈ - ´ = Ἐ ´ Ἐ
ᨨ矄� 矄�ᨨ �矄 ᆠ 㦨�m鼀 ᆠ㡰 꾀 �矄 ᨨ 矄鼀�矄矄�ᨨ ᨨ 꾀
҈҈
ᨨ�矄꾀
҈l Ҙ矄� q
㡰 x㦨
㦨
ᨨ l =
҈ ´ � 㦨Ҙ鼀 ´ ҈�
ᔼ
æ I ö ᆠᨨ矄 ᆠᨨ矄 ÷ø
ᆠ㡰 I㦨 çè
꾀 � ᨨ 盐ᆠ� �ᨨ�矄ᔴ 犨 꾀
ᔼ
I ҈ ´ ´ 㦨
҈l ҈ ´ ´ ҈-黨 = d � ´ ҈-
㦨맔҈䰘 䰘 黨
㡰 q =
b㡰 b㡰 꾀ᆠᆠᨨ � � ᨨ 犨矄矄 犨 ᆠ� � ᨨ 鼀� m㡰�矄⼄盐ᨨ � ᨨ ᨨ�矄� � �
҈
҈
҈� ҈
Þ μ 졔 (҈�) ( μ 맔҈�)
҈
- ҈㡰 l l ´ �� ´ ҈ -黨 = = ´ ´ ҈ -䰘 p q ;
㡰 ᨨ�矄꾀 矄鼀 �� 鼀 ᨨ 矄鼀�矄矄�ᨨ
�矄 矄矄矄盐鼀ᨨ�矄ᆠ�矄� ᨨ 盐ᆠ q �
ᨨ
ᆠᨨ矄 q =
æ ö çè ÷ø
҈
c+ 錐 V 㦨 � 矄 ᨨ ᨨᆠ c+
f㦨҈
�鼀� �鼀盐鼀盐ᨨ m 矄 ᨨ盐 �
l 㡰 꾀 �盐犨� ᨨ ᆠ 鼀 �鼀㦨
Þ ᆠᨨ矄 q =
㡰 矄 ��矄�ᆠ ᨨἘ矄 ᆠ 矄 ᨨᆠ�泐ᨨ ᨨ矄 �矄
ᨨ矄鼀矄ᆠᨨ鼀 ᨨ��
f = f
�
҈ ᨨ q = �°
҈ ҈ ᆠᨨ矄 q ´ ᆠᨨ矄 q =
ᨨ μ 졔 (҈� )