Physics Principles

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Defense of th e opment of the anti-aircraft g uided missile, the Nike. See page 446. Courtesy Jr"E 1\l aga=in P. Wes tern Electric Compan y, Incorporated.

STANLEY S. BALLARD, M.A., Ph.D. Professor and Chairman, Department of Physics, Tufts College

EDGAR P. SLACK, S.B., M.S. Professor and Chairman, Undergraduate Physics Polytechnic Institute of Brooklyn Director, Training School, National Board of Fire Underwriters

ERICH HAUSMANN, E.E. Sc.D. Dean Emeritus, Polytechnic Institute of Brooklyn Consultant, Nussbaumer, Clarke g Velzy, Inc.

Physies Prineiples

D. Van Nostrand Company, Jue. TORONTO

NEW YORK

LONDON

NEW YORK

D. VAN NOSTRAND COMPANY, INC. 250 Fourth Avenue, New York 3 TORONTO

D. VAN NOSTRAND COMPANY (Canada), Ltd. 25 Hollinger Road, Toronto LONDON

MACMILLAN & COMPANY, Ltd. St. Martin's Street, London, W.C. 2 Copyright 1954, by D. Van Nostrand Company, Inc. Published simultaneously in Canada by D. Van Nostrand Company (Canada), Ltd.

All rights in this book are reserved. Without written authorization from D. Van Nostrand Company, Inc., 250 FourtJi Ave., New York 3, N. Y., it may not be reproduced in any form in whole or in part (except for quotations in critical articles or reviews), nor may it be used for dramatic, motion-talking-picture, radio, television, or any other similar purpose.

Library of Congress Catalog Card No. 54-6280 Printed in the United States of America

PREFACE This is an introductory textbook on the principles of Physics, written especially for students who are majoring in one of the physical sciences or in engineering. It is designed both for the 3- or 4-semester courses offered in some schools and for the 2-semester course in others. For the shorter courses certain sections can be omitted without disturbing the continuity of treatment; a suggested list of sections for omission is given in Appendix III. It is assumed that the students who use this book will be taking calculus concurrently, or will already be familiar with its elements. Calculus is introduced in Chapter 3, and is used in later chapters in those places where its powerful methods are most helpful. The treatment is based principally on three systems of units: the British gravitational system because of its extensive use in engineering, the cgs system because of its wide acceptance in scientific work, and the mks system because of its growing use in electrical theory. The relationships between the mks units and the older electrical and magnetic units are given so that the student can make the transition to the older units when he encounters them in the literature. The authors have striven for that hard-to-define property called "teachability," which despite its elusive nature is so very real both to instructor and to student. Introductions of new concepts and definitions of units are clearly stated, and many completely solved problems are given, especially in the earlier chapters. The many figures are designed to illustrate the basic concepts, devices, and circuits in the most straightforward manner; line drawings are chiefly used because of their simplicity. More than 800 problems are included at the ends of the chapters to give the student ample practice in applying physical principles. Answers to the odd-numbered problems are given at the back of the book. The selection of topics and the order of their presentation were carefully considered. For instance, the concept of mass is def erred until definitely needed in Chapter 4, rotation is taken up before work and energy so that kinetic energy of translation and rotation can be covered together, and the transfer of heat precedes a treatment of the effects of heat to attain a more logical development. Such important fields as electronics and thermionics, high-frequency circuits, and semiconductors are given treatments appropriate lll

lV

PREFACE

to their theoretical and practical importance. In the division on light, emphasis is given to physical optics and to up-to-date treatments of visual phenomena, photometry, color specification, and polarization-fields sometimes neglected in introductory courses. Topics in so-called modern physics are introduced throughout the text wherever they follow naturally from the "classical" treatment. Additionally, two chapters at the end of the book cover briefly the major phenomena in the fields of atomic and nuclear physics. It is urged that this important material be included in all basic courses, so that students of science and engineering may be introduced to atomic and sub-atomic physics early in their college careers. The authors believe that the book reflects the great strides taken in the physical sciences in recent years and that it provides a good background of basic physics for the engineers and scientists of the future. The entire text has been taught for two years in preprint form, and has been improved greatly thereby. Sincere appreciation is expressed to our many colleagues who have offered suggestions for the betterment of the book, especially to Professors Kathryn A. McCarthy and Charles R. lvfingins of Tufts College for their criticism of large portions of the manuscript and preprint and for other assistance. Thanks are expressed to Professor Charles A. Hachemeister of the Polytechnic Institute of Brooklyn for a critical review of Chapter 24 and to Professor Leo E. A. Saidla, also of the Polytechnic, for the editorial reading of the entire manuscript. February, 1954.

STANLEY

8.

EDGAR

BALLARD

P.

SLACK

ERICH HAUSMANN

CONTE.NTS MECHANICS CHAPTER

1 2 3 4 5 6

7 8 9 10

PAGI (QdT2), with the net result that the entropy increases. 14-9. The Reciprocating Steam Engine. The reciprocating steam engine, invented by Watt, utilizes the expansion of steam for producing mechanical work. The simple engine, represented in Fig. 7, has a cylinder with ports A and B, a piston, and a slide valve. Depending on the position of this ;Steam from boiler Slide valve f, Exhaust Steam chest ~~~~~~=J::::::_::Connecting rod

Crank Frn. 14-7.

Eccentric

The reciprocating steam engine.

valve, the ports lead either to the steam chest or to an exhaust pipe entering from the side. The piston is joined to the cross-head, and thence, through the connecting rod and crank, to the main driving shaft of the engine. Upon this shaft is mounted the flywheel and an eccentric that controls the slide valve. Steam enters the steam chest under high pressure from a boiler and, at the instant shown, is admitted through port A to one end of the cylinder; the other end is connected simultaneously through port B to the low-pressure exhaust. The piston at this position is subjected to an unbalanced or "effective" pressure and is forced to the right (forward stroke); the valve closes port A at a suitable point and the stroke is completed by the expansion of the steam enclosed in the cylinder. The slide valve next interchanges the port connections, joining port A to the exhaust and port B to the steam chest. With this position of the valve, the steam drives the piston in the opposite direction (return stroke). This cycle is repeated over and over, and the reciprocating motion of the piston is converted to rotary motion of the driving shaft and flywheel. A diagram showing the relation between the pressure and volume in an

§ 14-10

INTERN AL COMBUSTION ENGINES

249

engine cylinder throughout a cycle can be obtained by an indicator. This device, which is piped to the cylinder like a pressure gage, has a tracing pencil which moves up and down as the pressure changes. The pencil rests against a card which follows the forward and backward motion of the piston to a reduced scale. The vertical motion of the tracing point combined with the ----->-j I I horizontal motion of the card results in I I I . ,I I a diagram of the form shown in Fig. 8. MN,1 The student should correlate the different portions of the diagram with the Volume various steps of the actual cycle of the engine, and should also note the simi- Frn. 14-8. A steam-engine indicator card. larity between this diagram and that of the Rankine vapor cycle. As before, the area of the indicator diagram represents that portion of the energy supplied during a cycle that is available for mechanical work. In the ordinary steam engine the same ports are used alternately to admit the highInlet--..... yExhaust valve "'I. valve temperature steam and to exhaust this steam at a lower temperature after expansion. The repeated heating and cooling .0f the ports is a wasteful process and is avoided in a later development known as the unijlow engine. In this engine the steam is adSpark mitted at the ends of the cylinder and is plug exhausted at the center, the piston serving as an exhaust valve by covering and uncovering the exhaust ports at the proper times. 14-10. Internal Combustion Engines. The internal combustion engine differs from the steam engine in that the fuel is burned directly in the cylinder. This type of engine, as used for automobile propulsion, usually has six or eight cylinders of the form represented in Fig. 9. The watercooled cylinder, fitted with mechanically Frn. 14-9. An internal-combustion operated valves, encloses a piston that is engine. connected by a piston rod to the crank shaft. The fuel used is gasoline, which is atomized and mixed with air in a carburetor so as to form an explosive mixture. The mixture is ignited at the spark gap by means of an electrical discharge. I

250

HEAT

Chap. 14

The complete cycle consists of four strokes. The figure shows the piston starting downward on the first stroke; the inlet valve is open, and a charge of fresh fuel is drawn in through it from the carburetor. When the piston has reached the bottom of its stroke and starts back, the inlet valve closes and the piston compresses the charge in the upper part of the cylinder. As the piston reaches the end of its upward stroke, the compressed charge is ignited and the resulting explosion drives the piston downward during the next, or working, stroke. On the return upward stroke of the piston, the exhaust valve opens and the piston forces the burned gases out through the exhaust pipe, leaving the cylinder ready for the beginning of a new cycle. At each explosion, the heat of combustion, § 12-15, of the gasoline consumed is liberated and part of this energy is converted into mechanical work. The Diesel engine eliminates spark plugs and uses fuel oil. Air is drawn into the cylinder and highly compressed, thus raising it to a high temperature. A charge of fuel is then sprayed into the cylinder under high pressure and ignites spontaneously as it mixes with the hot compressed air. Burning takes place without explosion, and the fuel supply is regulated so that the pressure remains almost constant during combustion. 14-11. Engine Horsepower and Efficiency. It will be recalled that of the heat energy supplied each cycle to an engine, the portion that becomes available for mechanical work is represented by the area of its indicator diagram. This fact makes it easy to calculate the horsepower of an engine from such a diagram. Since the area of any figure is the product of its average height and its base, for an indicator diagram such as in Fig. 8 the area is the average of all the ordinates MM', NN', etc., multiplied by the base b. The average ordinate is known as the mean e.ffective pressure-that is, the average difference in pressure on opposite sides of the piston; this value is denoted by p. The base represents the volume swept out by the piston during one stroke; that is, b = LA, where L is the length of the stroke and A the area of the piston. When p is expressed in pounds per square inch, L in feet, and A in square inches, the product pLA gives the work in foot· pounds for each working stroke. If there are N working strokes per minute, the output of the engine becomes pLAN ft· lb per min. Since this value is obtained from the indicator diagram, the output in horsepower is known as the indicated horsepower (ihp), and has the value

. pLAN (14.10) ihp = 33 ooo· ' The actual engine may be regarded as a combination of a heat engine and a mechanical device. First, as a heat engine, the input is the heat absorbed from the source, which, per unit time, can be expressed in horsepower, and the output is the indicated horsepower. The ratio of the latter to the former is known as the indicated thermal efficiency, or

*1-l-12

THE STEAM TURBINE

. Indicated t h erma1 effi c1ency

251

indicated horsepower = horsepower . · of heat supply'

naturally this will be less than the efficiency of an ideal engine operating over the same temperature range, given by Eq. 14.7. Second, as a mechanical device, § 6-12, the input is the indicated horsepower and the output is the horsepower as measured by a brake (bhp). Hence, . . Mechamcal efficiency

brake horsepower m d'1cat ed h orsepower ·

= .

The overall efficiency is the product of the indicated thermal efficiency and the· mechanical efficiency. For the purpose of assigning horsepower ratings to automobile engines, standard conditions have been adopted that are equivalent to the following: a mean effective pressure of 67.2 lb per in.2, and an average piston speed of 1000 ft per min. In the automobile engine the piston travels 4 L for each working stroke; consequently, 4 LN = 1000. From these values, the rated horsepower per cylinder is found by Eq. 14.10 to be h

_ pLAN _ 67.2 X (1000/4) X (,rrd2 /4) _ !.!!.._ P - 33,000 33,000 - 2.5'

where d is the piston diameter in inches. cylinders, the rating is

For an automobile engine having C

a2c

hp=-· 2.5

14-12. The Steam Turbine. The steam turbine makes use of the kinetic energy of a steam jet rather than the expansion of a vapor as in a reciprocating engine. High-velocity jets are formed by Fixed blades passing tp.e steam through a set of fixed nozzles; t these jets impinge against a series of curved vanes or blades evenly spaced around the rim of a rotary disk, and set the disk into rapid motion. The energy of the steam jets cannot be absorbed by a single row of blades without causing excessive speed. In the turbine construction shown in Fig. 10, the steam issues from the fixed nozzles and after passing one row of rotating blades strikes a corresponding Rotating row of fixed blades and is redirected against a blades second row of rotating blades mounted on the Fm. 14-10. Arrangement of same rotor. blades in a turbine. The theoretical output developed by a turbine is equal to the reduction in kinetic energy of the steam in passing through the machine. If W lb of steam moving at a speed of v1 ft per sec are supplied

\

J

252

HEAT

Chap. 14

to the blades int min, and if this steam is discharged at a speed of v2 ft per sec, then the reduction of energy in foot·pounds is given by Eq. 6.10 as

Wvi2

Wv2 2

2g

2g

This value represents the theoretical energy output of the turbine; the corresponding power output in foot· pounds per minute is consequently

w

p = 2 gt (v12 - V22), and can be converted to horsepower by dividing by 33,000. 14-13. The Jet Engine. The turbo-jet is a gas turbine system used in airplanes, in which a powerful jet of combustion products is expelled backward from the engine, and the forward reaction to this jet is the source of propulsion. The air for combustion is rammed into the front of the engine by the forward motion of the plane; it is compressed by a compressor and discharged into the combustion chamber. Here fuel oil is sprayed into the air and ignited, raising its temperature to perhaps 1400°F and increasing its volume at constant pressure. From the combustion chamber the gases are delivered to a turbine, which in turn drives the compressor mentioned above. The gases leave the turbine at considerable pressure and high speed and are discharged at the back of the plane through the exhaust nozzle. Here the gases expand adiabatically, and their internal energy is largely transformed into the kinetic energy of the final jet. If a mass m of air passes through the engine in time t, and if its velocity relative to the plane is v0 at the front of the engine and v1 at the mouth of the exhaust nozzle, then the thrust given to the plane is basically F

=

(m v.tft) -

(m

Vo/t),

although this result may be influenced by pressure differences in different parts of the engine. 14-14. Refrigeration. The manufacture of ice, the cooling of rooms, and the preservation of food in cold-storage spaces are processes that require apparatus for the production of low temperatures. In order to cool the articles under refrigeration, heat must be removed from them and given off to surroundings which are at higher temperatures. · The evaporation of a 1iquid and the expansion of a gas or vapor are known to be processes in which heat is absorbed. These actions can be illustrated in the making of carbon-dioxide snow, by allowing some liquid carbon dioxide at the pressure of its saturated vapor to escape from the containing cylinder through a small opening into the atmosphere. The resulting evaporation and expansion take sufficient heat from the issuing stream to cause the C02 to solidify as snow; at atmospheric pressure its temperature is -78°C. In the commercial manufacture of ice, mentioned in § 13-::5, anhydrous ammonia circulates continuously around a closed system, such as that rep-

REFRIGERATION

§ 14-14

253

resented in Fig. 11. The vapor is compressed in a compressor cylinder and passes through the coils of a condenser. In these coils, which are cooled by water, the vapor liquefies and the liquid flows to an expansion valve at the freezing tank. Here the liquid vaporizes into an evaporator or brine coil and is then drawn into the compressor to repeat the cycle. The evaporation and expansion which occur at the expansion valve absorb heat from the brine within the freezing tank and lower its temperature. Cans filled with water are placed in the brine tank and their contents are frozen. This system can be regarded as a reversed heat engine, operating as described in § 14-5. The working substance (ammonia vapor) absorbs heat from the refrigerator (freezing tank) and gives it off to the higher-temperature source (condenser), work in driving the compressor being done on the vapor during the process. Expansion valve

Frn. 14-11.

The compression system of refrigeration.

The electric refrigerator for household use operates as a compression system, essentially like the ice machine of Fig. 11, but with certain modifications to adapt it to domestic purposes. Sulfur dioxide or ethyl chloride is commonly used as the refrigerant; the evaporator coils are located in the food compartment, and the condenser coils, which are air cooled, are outside. The compressor is driven by an electric motor, which is controlled by a thermostatic switch to start and stop as needed in order to keep the temperature at the desired value. PROBLEMS 1. How high would a 1-ton weight have to be raised in order that its potential energy, if transformed into heat, would be sufficient to convert 1 lb of ice at 32°F to steam at 212°F? 2. Water flowing through a pipe at the rate of 10 gal/hr is heated by a 1200-watt immersion heater. Assume that all the heat generated is transmitted to the water, and compute the temperature rise of the stream. 3. Some gas expands adiabatically, doing 400 ft· lb of work. It then expands isothermally, doing an additional 900 ft·lb of work. How much heat was supplied externally to the gas? 4. Heat is supplied to 1 kg of hydrogen at constant pressure, raising its temperature from 20 to 70°0. Compute (a) the quantity of heat supplied, (b) the work done by the gas in expanding, and (c) the increase of internal energy of the gas. See Table I of Chap. 13.

254

HEAT

Chap. 14

5. Refer to the expansion described in Prob. 1 of Chap. 13, and compute (a) the work done by the expanding gas, and (b) the quantity of heat supplied externally to it during the process. 6. What mass of air will do 5 X 104 joules of work in expanding from 14 to 6 newtons/m2 absolute pressure, if its temperature is maintained constant at 200°0? 7. Air under standard conditions is to be compressed to a pressure of 100 psia (pounds per square inch, absolute), without change of temperature. Compute the amount of work needed per pound of air. 8. Refer to Prob. 9 of Chap. 9 and compute the amount of work expended in compressing the air. Assume no change in temperature. 9. A quantity of gas expands adiabatically, doing 1000 joules of work; it then expands further isothermally while it is supplied with 200 cal of heat. Determine, for the entire expansion, (a) how much external work was done by the gas, and (b) how much its internal energy was changed. 10. Some gas which initially occupies 20 ft 3 at a gage pressure of 100 lb/in.2 expands isothermally to a volume of 30 ft 3 and then expands further at constant pressure to a final volume of 50 ft3. Compute the work done by the gas during the entire expansion. 11. Compute the efficiency of an ideal engine that delivers an output of 200 ft· lb for each British thermal unit of heat absorbed from the source. 12. An ideal engine at the beginning of its isothermal expansion contains 1 liter of air at an absolute pressure of 10 atmospheres and a temperature of 230°C. The air expands isothermally to a volume of 3 liters, and then expands adiabatically until its temperature is lowered to 100°C. At this temperature the isothermal compression takes place, followed by an adiabatic compression to complete the cycle. Determine the efficiency of the engine operating under these conditions. 13. Compute the amount of work done by the air during the isothermal expansion in the cycle described in Prob. 12, and also the quantity of heat absorbed per cycle from the source. 14. Calculate the mass of air used in the ideal engine described in Prob. 12. 15. Compute the amount of work done during the adiabatic expansion (or compression) in the cycle described in Prob. 12. 16. A steam engine receives steam at 212°F and exhausts into the atmosphere at fi0°F; it delivers 24 hp and operates at 120 cycles per min. For an ideal engine operating under these conditions, compute (a) the efficiency, and (b) the quantity of heat absorbed each cycle from the source. 17. An ideal engine exhausts into a refrigerator at 40°C. (a) What is the temperature of the source if the engine operates at an efficiency of 20 per cent? (b) To what temperature must the source be raised in order to increase the efficiency to 30 per cent? 18. In a pressure-volume diagram of a Carnot cycle the areas, converted to foot· pounds, are as follows: under the isothermal expansion curve, 23,000; under each adiabatic curve, 15,880; under the isothermal compression curve, 17,920. Determine (a) the quantity of heat supplied by the source per cycle, and (b) the efficiency of the engme. 19. Compute the change of entropy of the air in Prob. 7. 20. What is the change of entropy of the hydrogen in Prob. 4? 21. Refer to Prob. 15 of Chap. 12 and calculate the total change of entropy of the substance. 22. Construct a temperature-entropy diagram for the cycle described in Prob. 12. 23. The area of an indicator diagram as measured by a planimeter is 2.11 in. 2 Each inch of height corresponds to a pressure of 80 lb/in.2 and each inch of length corresponds to a volume of 0.25 ft 3 • Compute the work in foot· pounds represented by the diagram. 24. A reciprocating steam engine ]urn a piston diameter of JO in. ::md fl :citroke of

PROBLEMS

255

12 in. Steam is admitted to the cylinder in such a way as to produce a mean effective pressure of 110 lb/in. 2 (a) How much work does the engine perform during each stroke? (b) What horsepower does it develop at a speed of 240 rev/min? 25. A 4-cycle (that is, 4 strokes per cycle) tractor engine has 4 cylinders with 3!-in. bore and 4i in. stroke. If the mean effective pressure is 90 lb/in. 2, what horsepower does the engine develop when it operates at 1400 rev/min? 26. A Diei'iel engine operating at 780 rev/min uses 0.625 lb of fuel in 10 min while developing 58 lb· ft of torque. What is the fuel consumption of the engine in pounds per horsepower· hour? 27. The unit of refrigeration called a "ton" is a rate of heat absorption sufficient to produce 1 ton of ice per 24-hr day from water at its freezing point. What is a ton of refrigeration expressed in (a) British thermal units per minute, and (b) horsepower?

ELECT-RICITY & MAGNETISM 15 ELECTRO ST A TICS Electricity and Magnetism is that part of Physics which deals with electric charges at rest and in motion, with magnetism, and with the forces exerted by electric and magnetic fields. The subject has three broad subdivisions: Electrostatics-concerned with the production of electric charges and the action of electric fields upon them; Electric Currents-concerned with the chemical, heating, and magnetic effects due to charges in motion; and Electromagnetic Induction-concerned with the generation of electromotive force and the behavior of alternating currents. The principles of electricity and magnetism underlie the operation of electrical machinery, such as generators, motors, and transformers; also the various systems of electrical communication, such as the telephone, radio, and television. Several systems of units can be used to express electrical quantities; in this book consistent use is made of the meter-kilogram-second system (abbreviated mks system), but occasional reference is made to the older systems so that the student can coordinate his references to published material expressed in electrostatic and electromagnetic units. The mks system incorporates the practical units used in electrical engineering and eliminates the powers of 10 so often present in equations utilizing other systems. Two forms of the mks system have developed, one called rationalized and the other unrationalized. As the rationalized form has been employed widely in advanced texts, it was chosen for inclusion here. A comparison of the mks units with those of other systems is given in the Appendix, and conversion factors are given there also. Although most students of physical science have some familiarity with electric currents and with certain practical applications of electricity, they need to learn the basic concepts of electricity and magnetism and the way 257

258

ELECTRICITJ' & MAGNETISM

Chap. 15

these concepts are related to one another. As a logical approach to the subject, it is desirable to begin with the fundamental principles of electrostatics. 15-1. Electric Charges. The earliest electrical experiment ever recorded is probably that attributed to the Greek philosopher Thales (c 624-c 546 B.c.), who observed that a piece of amber when rubbed with cloth was able to attract light objects placed near it. Nowadays, the act of bringing about a Yery close contact between the amber and the cloth is said to give the amber a charge of electricity, and the attraction is called an electric or electrostatic attraction. The term electricity is derived from elektron, the Greek word for amber. Many substances can be charged in the above manner. A glass rod will become charged when it is rubbed with a silk cloth. A hard-rubber rod stroked with fur becomes highly charged; it can exert sufficient force upon a meter stick to turn it horizontally about a pivot at its midpoint.. The forces due to electric charges can be demonstrated best by using very light objects, because the effects produced can then be observed readily. Two pith balls, each suspended by a thread and hung a few centimeters apart, will serve very well. When each of them is touched with a charged glass rod the balls fly apart and remain separated; when both are touched with a eharged hard-rubber rod they also repel each other. But if one pith hall that has been touched with the glass rod is brought near one that has been touched with the rubber rod, they will attract each other. From these tests it is evident that there is a difference between the electricity on the glass and that on the rubber. The charge on the glass is called positive and that on the rubber negative, as originally named by the versatile American, Benjamin Franklin (1706-1790). Furthermore, the tests show a fundamental fact of great importance-namely, that like charges of electricity repel each other, whereas unlike charges attract each other. The presence of an electric charge on a body can be detected by an electroscope, the construction of which is indicated in Fig. 1. Two leaves, ordinarily of aluminum or gold foil, hang side by side from a metal rod which passes through an insulating bushing and terminates in a metal knob outside of the case. If the knob is touched with the charged body, the leaves will acquire electricity of the same sign and will repel each other, as represented in the figure. The larger the quantity of charge, the farther the Leaves leaves will stand apart. Thus, a calibrated electroscope can be used to measure amounts of charge. Frn. lS-1. A 15-2. Structure of Atoms. A more detailed considersimple form ation of electric charges requires an insight into the composiof electrotion of matter itself. The atoms of matter are extremely scope. small, and the atoms of one element differ from those of another. Despite their minuteness, it is known that atoms are made 11p of still smaller particles, some of which have electric charges. These parti-

§ 1D--2

S1'RUC'l'UR1l OF ATOMS

259

cles are called electrons, protons, and neutrons. The electron has a tiny lmt definite charge, which is negative. The proton has 1836 times as much mass as the electron but has the same amount of charge; however, the charge is of opposite sign-that is, positive. The neutron has about the same mass as the proton and possesses no charge. The actual structure of the atom is subject to conjecture, but for many purposes it may be pictured as consisting of a nucleus composed of protons and neutrons, around ,vhich the electrons whirl in much the same way that the planets move around the sun. When an atom is in an electrically neutral or uncharged condition, the number of its electrons is equal to the number of protons in its nucleus. This planetary picture of the atom is useful in clarifying many phenomena in Physics and Chemistry. The simplest atom is that of hydrogen; it is pictured as having a single proton for the nucleus and a single electron whirling around it; the necessary centripetal force on the electron is provided by its attraction to the oppositely charged nucleus. Next in order of simplicity is the helium atom, composed of a nucleus and two planetary electrons; the nucleus is regarded as a stable combination of two protons and two neutrons. The more complex atoms have more and more protons and neutrons in the nucleus, together with a corresponding increase of planetary electrons. The electrons are thought to take positions in so-called shells around the nucleus, their arrangement under normal conditions being as follows: The hydrogen (H) atom has its one electron in the first shell, and the helium (He) atom has both its electrons in that shell. This innermost shell accommodates only two electrons. The lithium (Li) atom, with a total of three planetary electrons, has one of them in the second shell; beryllium (Be) has two in the second shell, boron (B) three, carbon (C) four, nitrogen (N) five, oxygen (0) six, fluorine (F) seven, and neon (Ne) eight; the last number fills the second shell. The sodium (Na) atom has a total of eleven planetary electrons, of which two fill the first shell, eight fill the second shell, and the remaining one is in a third shell. Other elements follow that have additional electrons in the third shell. By a continuation of this process to include the more complex atoms, the entire Periodic Table of the elements can be constructed (see Appendix). Certain elements are quite inactive chemically, from which it is concluded that their atomic structures are inherently stable. Perhaps because of compactness or symmetry, such stability is associated with electron shells that are completely filled. Helium with its first shell complete, neon with the first and second shells complete, and other elements similarly located in the table (argon, krypton, xenon, and radon) are chemically inert. The ability of atoms to combine and form molecules is determined by the planetary electrons, and the tendency in combining is apparently to form arrangements in which the electron shells are completely filled. A lithium or a sodium atom, with one electron in its outer shell, is in a condition which

260

ELECTRICITY & MAGNETISM

Chap. 15

favors losing this electron, while a fluorine or a chlorine atom, with one electron less than is needed to complete its outer shell, is.in a condition ,vhich favors gaining one. When sodium and chlorine are allowed to mingle, each sodium (Na) atom joins a chlorine (Cl) atom to form a molecule of sodium chloride (NaCl), in which process the loosely held electron of the sodium atom is transferred to the chlorine atom, thus leaving both atoms with completely filled shells. The measure of the ability of atoms to form molecules by combining in this manner is known as valence; for example, sodium, which has one electron more than is needed to fill its outer shell, is said to have a valence number of + 1, and chlorine, which has one electron less than is needed to fill its outer shell, is said to have a valence number of -1. The charge on the nucleus of an atom is determined by the number of protons in it, and this number is called the atomic number; it also represents the number of planetary electrons in the neutral atom. The total number of protons and neutrons in the nucleus is called the mass number. The atoms of any one element are not all alike-some are heavier than others. Elements containing atoms that have the same atomic number but different mass numbers are called isotopes; they differ in the number of neutrons in the atomic nucleus. The average weight of the atoms of any one element, as it occurs in nature, is nearly always the same. Instead of stating such quantities in weight units, the value for oxygen is taken as 16.000, and the values for the other elements are assigned proportionately. The number thus obtained for any element is called its atomic weight. Atomic quantities are much too small to be measured directly, but results of great precision have been obtained from indirect measurements. For example, the hydrogen atom, which is composed of a proton and an electron, has a mass of 1.673 X 10-24 gm. Of this amount, the electron forms only a small part; its mass when at rest is 9.107 X 10-28 gm. The dimensions of the hydrogen atom, with its components assumed to be spherical, are of the following order of magnitude: radius of nucleus and radius of electron, each about 2 X 10-13 cm; least radius of electronic orbit, about 5 X 10-9 cm. A better appreciation of the relative proportions of these quantities can be obtained by imagining the atom to be magnified until the electronic orbit is as large as that of the earth about the sun. The electron would then be represented by a sphere about the size of the earth itself and would rotate around a nucleus of approximately equal size. 15-3. Production of Electric Charge. The process of charging a body by rubbing it with another material may be viewed as a stripping of electrons from some of the atoms at the contacting surface. Atoms of some elements release electrons with comparative ease, and others acquire them readily. A neutral or uncharged body contains equal amounts of positive and negative electricity; when electrons are added it becomes negatively charged, and when electrons are removed it becomes positively charged. Thus a hardrubber rod, when brought into intimate contact with fur, gains electrons and

§ 1,.~:)-4

CHARGING BY INDUCTION

261

becomes negative, whereas the fur loses these electrons and becomes positive to an equal extent. A glass rod rubbed with silk loses electrons and becomes positive, while the silk gains these electrons and so becomes negative. Examples illustrating the production of electric charge are familiar to everyone. The effect may be observed in dry weather by passing a rubber comb through the hair or by shuffling the feet on a woolen carpet. A leather belt traveling on iron pulleys may acquire sufficient electricity to produce a spark to a person's finger held near it. The paper in a printing press usually manifests a charge when it is separated from the rollers, and means are provided to dissipate· this charge. It is possible to charge a body from another that is already charged, simply by bringing the two into contact. Thus, a metal sphere gains negative electricity if it is touched with a negatively charged rod, Fig. 2. It appears that some electrons leave the rod at the point of contact by virtue of their mutual repulsion and attach themselves to the sphere, making it negative also. Again, if the sphere is touched with a positively charged rod it becomes positive, because electrons are attracted away from it to the rod at the point of contact. It is assumed that the sphere referred to is supported in such a way that the electricity acquired ,vill not leak Frn. 15-2. Chargaway. This can be done by suspending the sphere with ing a metal sphere a dry silk string or by supporting it on props of mica or by contact. glass. Evidently materials like these do not transfer or conduct electricity to any appreciable extent, and they are called insulators or dielectrics. If the sphere had been suspended by a metallic wire or mounted upon a metal support, practically the entire charge might escape to the earth. It can be concluded that metals are good conductors of electricity. Many substances are neither good insulators nor good conductors but may be classed in an intermediate group as fair electrical conductors; for example, the human body, a piece of damp wood, and the earth. 16-4. Charging by Induction. There is a method of charging a conductor from a charged object which does not require bringing the two into contact; the process is called induction. In order to charge a conductor by induction, it is necessary (1) to bring the charged object close to, but not in contact with, the conductor to be charged; (2) to connect the conductor to ground; (3) to break the ground connection; and (4) to remove the initially charged object. The conductor will then have acquired a charge which will be found to be opposite in sign to the original one. This process can be explained by reference to Fig. 3, vvherein the conductor is represented as a brass tube with rounded ends mounted on a glass stem for insulation, and the charged object as a negative rod. The mechanism of the process is as follows: (1) When the charged rod is brought near the tube, it will repel some of the electrons of the tube; this adion makes the distant end

262

ELEC'l'RICITY & MAGNETISM

Chap. 15

of the tube negative and leaves the adjacent end positive. A state of equilibrium will soon be reached in which any other electrons repelled by the rod are prevented from moving to the distant end of the tube by the repulsion of the negative electricity already accumulated there. (2) When the tube is connected to ground, a path is provided for some of the electrons to escape, +

+

+

d6 (1)

FIG. 15-3.

(2)

(3)

(4)

Procedure in charging a metal tube by induction.

and there will be a flow of electrons through the ground connection to the earth. (3) When the ground connection is broken, the tube is again isolated and will have a positive charge, since it has lost some electrons. Finally, (-!) the removal of the inducing object allows the charges on the tube to distribute themselves in a normal manner, and as a result the tube becomes positive over its entire surface. 15-5. Forces between Charged Bodies. It has been shown that charges of like sign repel and those of unlike sign attract each other, but nothing has been said thus far about the magnitude of the forces of repulsion or attraction. The first quantitative measurements of these forces were made by the French physicist Charles A. Coulomb (1736-1806), using a torsion balance. This instrument, sketched in Fig. 4, has a stationary sphere A and a suspended element consisting of two spheres B and C connected by a slender rod, all within a glass enclosure. When spheres A and B carry charges of the same sign, the repulsion between them twists the supporting wire S until the torque balances that set up in the wire. Coulomb and later investigators showed that the force between two charged bodies, whether of repulsion or attraction, depends upon the distance between the two bodies, their shapes, and the Fm. 15-4. The toramount an_ dt dt . express10n, . L -- N dd; F 1 om th1s ~. .

On the assumption that the flux grows

uniformly with the current, and that a value of flux if> will be reached when the current is I, the inductance of the coil becomes Nif>

L=-, I

(21.3)

where the product of the number of turns N and the number of flux loops if> linked with them is spoken of as flux-linkages. It follows that a circuit which has an inductance of 1 henry will have one flux-linkage per ampere of current in the circuit. Equation 21.3 can be applied to a ring solenoid in which a coil of N turns is wound upon a toroidal core of length l and cross-sectional area A. The magnetic flux set up in this core by a current of I amperes in the coil is = µHA, where µ is the permeability of the core and H is the magnetic intensity within it.

This intensity is given in § 19-1 as H = ~I, and conse-

§ 21-4

367

GROWTH AND DECAY OF CURRENT

quently the inductance can be expressed as

L =NAµ NI= N 2 Aµ I l showing that the inductance of a solenoid varies directly with the square of the number of turns, and depends upon the cross-section, length, and permeability of the core. When the dimensions of the core are expressed in meters, the inductance will be in henrys. The expression indicates that the unit of permeability may be expressed in simpler units than before; the permeability unit is the henry per meter. In summary, the different units in which permeability can be expressed are as follows: weber = newton = Q_enry. µ = ampere· turn· meter ampere 2 meter

21-4. Growth and Decay of Current in Inductive Circuits. The current in a circuit containing inductance and resistance does not instantly assume its ultimate value when connected to a unidirectional source of emf, because of the counter emf of self-induction. Although the entire current growth usually occurs in a fraction of a second, the current rises rapidly at first and then builds up more and more slowly as it approaches its final value. - A solenoid of 10 ohms resistance and 2 henrys inductance is being connected across 120-volt direct-current supply mains. Determine the momentary values of the current when it is increasing at 50 amp/sec and at 10 amp/sec. At an instant when the rate of current growth is 50 amp/sec, the counter emf has a value given by Eq. 21.2 of e = L ~: = 2 X 50 = 100 volts, and hence the current in the solenoid is (120 - 100)/10 = 2 amp. When the rate of current growth has fallen to 10 amp per sec, the counter emf is momentarily 2 X 10 = 20 volts, and the current value is (120 - 20)/10 = 10 amp. The current will have the final value of V /R = 120/10 = 12 amp.

The foregoing illustration also indicates that Ohm's Law in its simple form applies only to steady currents. A broader statement of the law includes the counter emf of self-induction, and the current at any instant can be expressed as E- Ldi i

dt =----,

(21.4)

R

where E is the emf impressed upon the circuit. At the instant that an inductive circuit is connected to an electrical source, all of the emf causes the current to grow, for then the current i is zero and E = L ~;; in contrast, whei1 the current has reached its final value I, then~!

=

0 and E

=

RI.

368

ELECTRICITY & MAGNETISM

Chap. 21

To solve the differential equation above, the variables are separated, and the resulting equation is multiplied through by - R and integrated. Thus,

dt = L

r

and

• dt

=

-

di = __!!_ - R di , E - Ri -RE - Ri

L s-Rdi R E - Ri ;

t = LR In (E - Ri)

hence

+ constant.

The value of the constant of integration must be such that i = 0 when t = 0, and therefore the constant is =

L ln E - Ri_

R

E

~

ln E.

Then -t

=

~

ln (E - Ri) -

~ lnE

·when transposed to exponentials, the result is

E - Ri _LRt =e ' E where e is the base of natural logarithms and has the value 2.7183. Consequently, the current in amperes at an instant t seconds after an emf of E volts

.. ...

-----------

c:

Cl)

:::>

u

Time

Fru. 21-3.

Graph of current growth and decay in an inductive circuit.

is impressed upon a circuit having a resistance of R ohms and an inductance of L henrys will be i =

! ( e-P). 1 -

A similar method of analysis shows that when the applied emf is suddenly withdrawn and replaced by a short-circuit, the current does not fall to zero instantly. The current values during this period are given by that part of the foregoing equation which is subtracted from E/R, that is,

E _lli i = - e L· R In these expressions the ratio of the inductance L to the resistance R determines the rate of current growth and decay in the circuit. For this reason the ratio L/R is called the time constant of the circuit. If the time is chosen so that t = L/R, then the exponent of e becomes -] , and the latter equa-

ENERGJ' OF A MAGNE'PIC FIELD

§ 21-5

369

tion indicates that the current falls to 1/e = 0.368 of its steady value in this time. Figure 3 shows a graph of current in an inductive circuit from the instant it is connected to a constant source of supply until the current subsequently falls to zero after the source is replaced by a short-circuit. At instant t1 a constant emf is applied to the circuit, and at instant t2 the source is replaced by a short-circuit. 21-5. Energy of a Magnetic Field. In order to produce current in an inductive circuit, work must be done against the emf of self-induction so that the magnetic field can be created around the circuit. When the current has the , momentary value of i and the emf impressed upon the circuit is E

= L ~; + Ri

(§ 21-4), the energy supplied in a time dt is

Ei dt = Li di

+ Ri

2

dt,

which shows that part of the energy is converted into heat (Ri 2dt) and the rest is used to build the magnetic field. Consequently, while the current is reaching its final value I, the energy expended in the magnetic field, as found by integration, is W

=

I

t=t

i e dt =

fi=IiL d"..!:. dt = [L -JI 2

_i

=0

or

df

i=O

W

=

!

2

L/2•

,

0

(21.5)

The2nergy of the magnetic field, W, will be in joules if the inductance of the circuit is in henrys and the current through it is in amperes. Once established, the field requires no further energy to maintain itself; all energy supplied to the circuit subsequently is dissipated in heating. Equation 21.5 is analogous to the expressions for the kinetic energy of a moving mass, § 6-5, and affords an energy concept of the unit of inductance. A circuit of one henry inductance, carrying a current of one ampere, would have ! joule of energy stored in its magnetic field. The energy per unit volume of the magnetic field can be expressed in terms of magnetic quantities rather than in quantities of the associated electric circuit. Let it be assumed that the energy is stored in a toroidal core of length land cross-sectional area A, and that the core material has a permeability µ when the magnetic flux is . Then, since the inductance of a ring solenoid is L

=

i' the energy per unit volume of the magnetic field is:

Energy density =

1. 2

LJ2

lA

Nl

= 2 lA ·

But the number of ampere· turns NI of the winding equals the product of the l

flux and the reluctance µA of the core, and the flux per unit area

/ A

370

ELECTRICI1T & MAGNETISM

equals the flux density B. field is:

Chap. 21

Hence, the energy per unit volume of the mag,netic Energy density

=

B2 2 µ'

(21.6)

and the unit is the joule per cubic meter. CAPACITANCE

21-6. Capacitance of an Isolated Conductor. The study of capacitance is based on the principles of electrostatics, developed in Chap. 15. There, consideration is given to electric charges in space, to electric fields near them, and to the potentials at points within such fields. It is pointed out, for example, that a charged sphere isolated in space has an equi-potential surface, and that the electric lines of force around it are arranged radially, exactly as though the charge were concentrated at a point. The potential at any distance r from a point charge Q in free space is given by Eq. 15.6 as V = k

Q, r

where k has the numerical value 9 X 109 •

This

would be the potential also of the surface of a sphere of radius r which carries the same charge. Basically, the equation shows that the potential V is directly proportional to the amount of charge Q on the sphere. This conclusion is found to be true for an isolated conductor of any shape. The reverse of this conclusion is also true--that is, the charge on an isolated conductor is proportional to its potential; hence Q o: V, or (21.7)

Q = CV,

where C is a proportionality constant determined by the size and shape of the conductor and is called the capacitance of the conductor. In this expression, when Q is in coulombs and V in volts, the capacitance unit is the coulomb per volt; it is called the farad, derived from the name of Faraday. For an isolated conducting sphere of radius r, the foregoing equation gives the capacitance as

Q

c= v= 9x

Q 109 Q = 9

r

x

109 '

(21.8)

r

which shows that the capacitance of the sphere is directly proportional to its radius. When another conductor is brought nearby, the capacitance of the sphere is increased. 21-7. Dielectrics. In the earlier consideration of electric fields near and between charged bodies, the medium was restricted theoretically to free space, but practically includes air also. Attention is now given to the presence in electric fields of insulating substances-so-called dielectrics. The molecules of such substances are normally uncharged, that is, the positive component particles neutralize the negative ones. But in an electric field, these charges

DIELECTRICS

§ 21-7

371

are displaced and the molecules become polarized. Thus, when a sheet of dielectric is placed in a uniform electric field between parallel plates, the entire sheet becomes polarized and one surface becomes positive and the other negative. The effect of dielectric polarization is shown in Fig. 4. In part I the electric field between parallel plates in a vacuum is directed toward the right from the positive to the negative plate. In part II the space between the plates is almost filled by a dielectric sheet, and charges of opposite sign are induced on its adjacent surfaces. The field set up within the dielectric by these bound surface charges is directed toward the left and is opposite in direction to the initial field, causing the field in the dielectric sheet to be less intense than in a vacuum. This effect is allowed for through the concept of permittivity. Plate s

Dielectric .. -

--+

+-.

+ , ... + .. -t-----,--

--

1. . -

+--

.. -

+ ..

-

ll Fm. 21-4.

Uniform field between parallel plates--I, in vacuum; II, with dielectric.

The electric field produced at a point near a charged sphere depends upon the amount of charge on the sphere and the distance of the point from it. If the sphere is located in a vacuum and undisturbed by neighboring charges, the field intensity at a distance r is 8 = k Q_ = - 1~ Q , r2 4 7rf:o r 2

as in § 15-6, where Eo is the permittivity of free space. If similarly located in a medium of permittivity E, the field intensity can be expressed as 8

=

_l_Q_ 4 7rE r 2

The ratio of the permittivity E of the medium to the permittivity Eo of free space may be called the relative permittivity of the medium and defined by E

E,.

= -.

(21.9)

Eo

The relative permittivity Er is a pure number and has a characteristic value for every imbi-tanee. Th1rn, for a medium having a relative permittivity of 2,

372

ELECTRICITY & MAGNETISM

Chap. 21

the permittivity e is twice that of vacuum, and hence the intensity of the electric field at the reference point above, distant r from a charge Q, is half as much as it would be in free space. The relative permittivity of vacuum is unity, and for air at standard temperature and pressure is 1.000586. The displacement of the charges that occurs in the molecules of a dielectric when polarized in an electric field corresponds to a so-called displacement current. It persists only as long as these charges are shifting-that is, while the electric field across the dielectric is changing. 21-8. The Capacitor. A capacitor or condenser consists essentially of two conductors separated by some insulating medium; the conductors are often called the plates and the insulation the dielectric. Capacitors are used chiefly to reduce arcing at contact points, to neutralize the effects of inductance, to tune radio circuits, and to obtain pulses of current for various purposes. In an automobile ignition circuit the condenser is ordinarily composed of two long strips of tinfoil separated by waxed paper and rolled into cylindrical form. The type commonly used in radio reception consists of two sets of aluminum plates separated by air, one set being mounted on a shaft so it can be rotated for changing the overlap or effective area of the plates. A capacitor can be charged by connecting its plates to the terminals of a battery or other source of direct current. This will cause electrons to leave one plate and flow to the other through the source and associated circuit, until a state of equilibrium is reached in which the plates have a potential difference equal to the emf of the battery. In this process the plates will receive equal charges, one positive and the other negative. If a battery of higher emf were used, the charges would be correspondingly larger; consequently, the ratio of the charge on either plate to the potential difference between the plates would remain the same. This ratio of the charge on one of the plates to the potential difference between them is the capacitance of the condenser; it is given by Eq. 21.7 as

It follows from this expression that a condenser has a capacitance of one farad when a charge of one coulomb of electricity produces a potential difference of one volt across its plates. The farad is an enormously large unit of capacitance, and, for convenience, smaller units, the microfarad (abbreviated µf) and the micro-microfarad (µµf), are generally used. One farad = 106 µf = 1012 µµf. Oceanic cables have large amounts of capacitance. The conductor at the center serves as one plate of a capacitor; the sheathing, together with sea water, forms the other plate; and the gutta percha or other insulating layers between them serve as the dielectric. Such a cable 2000 miles long has a capacitance of about 0.001 farad. 21-9. The Parallel-plate Condenser. The capacitance of a parallel-plate condenser is determined entirely by the dimensions and properties of the di-

THE PARALLEL-PLATE CONDENSER

§ 21-9

373

electric and is not affected, for instance, by the materials used for the plates, provided only that these are electrical conductors. This will be proved for a capacitor, Fig. 5, which consists of conducting plates separated by a dielectric layer of thickness s, effective area A, and permittivity e. It is assumed that the + plates are large compared with the thickness of the dielectric. When the plates are charged and the potential difference between them is V, the electric field in the dielectric is uniform and has the value given by Eq. 15.8 FIG. 21-5. Dimensions of a parallelas 8 = V /s. The polarization of the diplate condenser. electric produced by this field is represented by the number of lines of electric flux '¥ that extend from one plate to the other per unit of area; this is the same as the electric flux density

Since by § 15-7 the number of flux lines is numerically equal to the charge Q on either plate, and the flux density is D = E8, it follows that D = Q/A and D = eV /s. Hence

v -Q = e-· A

s

Finally, the capacitance of the condenser, being the ratio of the charge on one of the plates to the difference of potential between them, is

c

Q= eA .

= -

V

8

If the dimensions of the condenser are given in meters, the capacitance will be in farads. The ratio of the permittivity e of a dielectric to the permittivity Eo of empty space,- called the relative permittivity (Er), is also known as the dielectric constant of the dielectric; the dielectric constant of a material is defined as the ratio of the capacitance of a condenser with that material as dielectric to its capacitance when the dielectric is a vacuum. When e is replaced by e0 Er, the expression for the capacitance of a condenser becomes

C=EoErA,

(21.10)

8

where

eo

=

1 4 1r 9 X 109 numerically.

The result also affords a simpler unit for permittivity than used previously. Since the dielectric constant is a pure number, the permittivity unit is the

374

ELECTRICITY & MAGNETISM

Chap. 21

farad per meter. In summary, the different ways in which the unit of permittivity can be expressed are as follows: €

=

coulomb coulomb2 farad = = --· 2 volt· meter newton· meter meter

Some typical values of dielectric constant are given in Table I. TABLE

I.

DIELEC'l'RIC CONSTANTS

Glass, crown ....... Glass, flint ......... India rubber ....... Mica .............. Paper, dry ......... Paraffin wax ....... Teflon ............. Water (pure) .......

. . . .

5 7 2.1 5.7 2 2 2.0

.

. . .

to to to to to to to

7 10

2.3

7 2.5 2.3 2.05

81

Find the capacitance of a condenser formed of 21 square metal plates measuring 10 cm along each edge, separated by sheets of mica 0.01 cm thick and having a dielectric constant of 6. Alternate plates are connected to one terminal of the capacitor and the remaining plates are connected to the other terminal. The 20 dielectric sheets may be regarded as equivalent to a single one having an area of 20 X 10 X 10 = 2000 cm2 = 0.2 m2 • The dielectric thickness is 10-4 m. Hence . given . b y E q. 21 .10 as 7T' X 6 XX 0. 2 X _4 th e capac1'tance of t h e con d enser 1s 9 109 4 10 = 0.106 X 10-6 farads = 0.106 µf.

By the use of a ballistic galvanometer, § 18-8, the capacitance of a condenser can be compared experimentally with that of a standard capacitor. The capacitors are charged separately from the same battery and each is discharged in turn through the instrument, the maximum throw being noted for each. With constant potential difference the deflections are proportional to the charges; these, in turn, are proportional to the capacitances, and therefore the capacitances are in the same proportion as the deflections. 21-10. Energy of a Charged Capacitor. When a condenser is connected across a battery, it charges very quickly, and the current falls from an initially high value to zero within a fraction of a second. Its value at any instant during this brief period is i

. - cJ!J. dt'

dq being the small amount of charge that is transferred in the time interval dt, when this interval is vanishingly small. Since the capacitance C of the condenser is constant, Eq. 21.7 shows that this current can be written as

• = C dv "

dt'

(21.11)

§ 21-11

CAPACITORS IN PARALLEL AND IN SERIES

375

proving that the momentary value of the charging current is proportional to the time rate of change of potential difference, dv/dt, across the capacitor. The amount of energy supplied to a capacitor in charging it can be determined by evaluating the work dW done in transferring a small charge dq from one plate to the other when the potential difference between them is v; this is found to be dW = v dq according to Eq. 15.,5. If a series of amounts like this, or its equivalent v i dt, is summed up during the entire charging period t in which a potential difference V is being established across the condenser, the result is given by t=t

W =

( vi dt = Jt=O

or

w

f

i•=V

2

vC dv dt = [Cv V=O dt 2

= ~

cv

J, V

0

(21.12)

2•

When the capacitance C of the condenser is expressed in farads and the potential difference V is in volts, the energy W stored in the condenser will be given in joules. Since the energy of a charged condenser resides in its electric field, it is of interest to express the foregoing result in terms of field quantities rather than circuit quantities. On the assumption that the energy is stored uniformly in a dielectric of area A, thickness s, and permittivity E, the energy per unit volume of the field is: Energy density

1.

CE 2

Q V -, A s

= ~ = .!. -

As

2

where the fraction Q/ A is the same as the density D of the electric lines of force, and the potential gradient V / s is the same as the electric field intensity 8. Finally, since D and 8 are related by D = E8, it follows that in an electric field the

n2

Energy density = ~ '

(21.13)

and is in joules per cubic meter when D is expressed in coulombs per square meter. 21-11. Capacitors in Parallel and in Series. Condensers ~re often connected in parallel in a circuit in order to increase the capacitance of that circuit. Let G\, C2 , C:i, · · · be the capacitances of several capacitors that are F1u. 21-6. Capacicom1ected in parallel, as in Fig. 6, to form a condenser of tors connected m equivalent capacitance C. The potential differences parallel. across the condensers will be represented by V1, V2, V3, · · · , and the charges on them by Q1, Q2, Q3, · · · . The corresponding potential difference and charge for the equivalent capacitor will be taken as V and Q, rrn;pectivcly.

376

ELECTRICITY & MAGNETISM

Chap. 21

For the parallel connection, each condenser has the same potential difference as the source, or

V = V1 = V2 = V3 = · · · , and the total charge is distributed among them, or hence, by division, Q

= Qi +

+

Q2

V

Q3 • • •

= Qi +

Q2

Vi

V2

V

+

Q3

+

V3

But C = ~' Ci = ~:, and so on, from Eq. 21.7; and consequently,

C

=

Ci

+

C2

+

C3

+ ··· ,

(21.14)

which shows that the combined capacitance of several condensers connected in parallel is equal to the sum of the individual capacitances. Condensers are sometimes connected in series in order to lessen the potential difference across each of them; the result of this arrangement is to reduce the capacitance. In the series connection of Fig. 7 the same momentary flow of electrons occurs in all of the condensers; this gives each an equal charge, or Fm. 21-7. Capacitors connected in series.

Q

= Qi = Q2 = Q3 = ' ' ' ,

and the applied potential difference is divided among the individual con

v

_

' .... .... /

E

::>

v

...:E w

II

Ill

Fw. 22-12.

Curves of emf, current, and power plotted with respect to time for: I, capacitive circuit; II, resistive circuit; and III, inductive circuit ,vith resistance.

tive, their product will have a positive value. The pm,ver curve is similar to that shown in part I but is entirely above the axis, and the area under it represents energy that is dissipated as heat. For circuits in which the current is neither in phase nor 90° out of phase with the emf, the power curve will have positive lobes that are larger than the negative lobes, as in part III; the difference between their areas represents the amount of energy expended. The corresponding vector diagrams for these typical circuits are given respectively as parts I, II, and III of Fig. 13, in which E and I represent the effective values of emf and current. The average power supplied to a circuit

.,,,

lcoso£E

I

8 .... ,

---

I

E

II Fw. 22-13.

Ill

Vector diagrams corresponding to the sine curves of Fig. 12 for alternatingcurrent circuits.

392

ELECTRICITY & MAGNETISM

Chap. 22

is found much more readily from such vector diagrams than from the areas of the power curves indicated in the preceding figure. It is necessary only to compute the component of the current which is in phase with the emf and to multiply it by that emf in order to determine the power. In part I the current I has no component in phase with E, and hence the power supplied is zero; in part II the current I is in phase with E and the power is P = E I; in the general case of part III the component of current I which is in phase with E is I cos e, and consequently the power is

P =EI cos fJ.

(22.11)

Naturally this equation applies to the preceding circuits for which e is 90° and 0° respectively. Since cos e is a factor by which the product EI must be multiplied in order to give power, it is known as the power factor of the circuit. Hence, p Power factor = cos e = EI, (22.12) so that the power factor is defined either as the cosine of the angle of lag or lead, or as the ratio of the watts to the volt· amperes. Reference to Fig. 10 shows that the power factor of a circuit can be expressed also as the ratio of its resistance R to its impedance Z. For a circuit containing resistance only, the phase angle is 0, so that cos e = 1; hence the power factor is unity, and the power is P = EI, as for direct-current circuits. A highly inductive circuit has a low power factor, as the current lags the emf considerably; a highly capacitive circuit also has a low power factor, as the current leads the emf by a large angle. Low pmver factor is a disadvantage, because on the ordinary constant-potential circuit it necessitates a relatively large current in order to supply a given amount of power, and the large current causes more waste of energy in heat. The rating of electrical machines is limited chiefly by their ability to dissipate the heat produced in them; for this reason alternating-current machinery is rated in kilovolt· amperes (abbreviated kva) instead of kilowatts. An inductive load takes 8 amp when connected to 2200-volt, 60-cycle supply mains. The power factor has the low value of 0.75, and improvement is sought by connecting a 2-µf capacitor in parallel with the load. Find the resulting power factor of the combination. To help in visualizing the problem, the student is advised to construct a vector diagram and to use the applied emf E as a basis of reference. The current in the inductive load lags the applied emf by the angle cos-1 0.75 = 41.4°. It has a component of 8 cos 41.4 ° = 6.00 amp in phase with E, and a component of 8 sin 41.4 ° = 5.29 amp, which lags Eby 90°. The capacitor has a reactance of 1/ (2 1r X 60 X 2 X 10-6) = 1327 ohms and takes a current of 2200/1327 = 1.66 amp, which leads E by 90°. The resultant current has two components. One of these is 6.00 amp in phase with E. The other is found by subtracting the capacitor current from the lagging component of the inductive load current; it equals 5.29 - 1.66 = 3.63 amp and lags Eby 90°.

§ 22-10

THE ALTERNAT/NG-CURRENT GENERATOR

The resultant current thus lags Eby the angle tan-1 (3.63/6.00) power factor of the combination is cos 31.2° = 0.855.

393

= 31.2°; hence the

22-10. The Alternating-current Generator. The stationary and rotating members of an alternator are called respectively the stator and the rotor; generally the stator is the armature and the rotor is the field structure. Alternators must provide an emf that has not only the desired magnitude but also the desired frequency-a characteristic that is controlled by the speed of the rotor and the number of poles. High-speed alternators usually have two poles, while those driven at slow speeds have many pairs of poles. The rotational speed at which an alternator gives its rated frequency is called the synchronous speed; its value is given by Eq. 22.1 as n

=

!f

rev per sec,

where f is the frequency in cycles per second, and P is the number of field poles. The armature is designed to have a sufficient number of coils or conductors to develop the desired emf at the synchronous speed. The value of the emf generated in each conductor is obtained from Eq. 20.3 in terms of the magnetic flux density B, the length l of the conductor, and the linear speed v of cutting the flux. This emf has the maximum value of Em = Blv and is assumed to be sinusoidal; consequently, the effective value of the generated emf per conductor is Blv E -_ V2,

(22.13)

where Bis in webers per square meter, l in meters, and v in meters per second. The simple form of alternator shown in Fig. 1 generates a single alternating emf; it is called a single-phase machine to distinguish it from the more usual type of alternator that generates two or three alternating emfs at the same time and which is called a polyphase alternator. + These emfs have the same magnitude and have definite phase relations with each other. The two-phase alternator consists of two inde!! pendent armature windings with a definite angular g separation; in a bipolar tield they are at right angles 'i to each other. When such a machine is driven, an w alternating emf is generated in each winding; and, because of the relative positions of the windings, the emfs are go 0 apart in phase. These emfs are Fm. 22-14. Emf curves plotted with respect to time in Fig. 14. The ma- of a two-phase alternachine has four terminals in order that it can supply tor. a ~ur-wire circuit; the potential difference across one pair of wires is displaced go 0 from that across the other pair, as indicated. Sometimes one wire is common to the two phases. The three-phase alternator consists of three like windings symmetrically

394

ELECTRICITY & MAGNETISM

Chap. 22

placed on the armature so as to produce three equal emfs that are 120° apart in phase, as shown in Fig. 15. The windings are connected in Y or in Li, § 17-15. With either connection, the total power developed in a three-phase alternator, + when the load is the same on all three phases, is given by

P

=

V3 El cos e,

(22.14)

where E represents the emf acting across any pair of lines, I is the line current, and () is the phase angle between E and I. 22-11. The Induction Motor. The most Fm. 22-15. Emf curves of a threewidely used alternating-current motor is phase alternator. the polyphase indnction motor. It consists essentially of a stationary field structure and a rotating element resembling a squirrel cage. The rotor is usually formed of heavy copper bars welded to end rings and has no connection with the supply circuit. Part I of Fig. 16 shows the arrangement of a two-phase induction motor. The field windings on alternate poles are connected to one phase, and the windings on the others are connected to the other phase of the supply circuit. The directions of the currents through the motor during their positive half-cycles are indicated by the arrows.

... c:

...~ a... ~-+---+--+-_g

Phase A

0

V) I I

I I

i

Phase B I Frn. 22-16.

II

Diagram of a two-phase induction motor and its current and flux relations.

The motor currents are shown graphically by curves IA and le in part II of the figure, and the direction- of the magnetic flux which they set up through the rotor is represented by the arrows below the curves. At the instant indicated by 0, I A = 0 and le has its maximum negative value; hence only the horizontal poles (phase B) will be energized and the current through their windings will produce a fl~x that is horizontally directed toward the right. After l- cycle, le = 0 and IA has its maximum positive value, at point 1 on

§ 22-12

TRANSFORMERS

395

the curves; now only the vertical poles (phase A) will be energized and the flux will be directed upward. At the instant corresponding to 2 on the curves, IA = 0 again and IB is positive, directing the flux horizontally toward the left, and so on. In this manner, the same effect is produced upon the rotor as if the flux through it were rotating mechanically. The result is to induce currents in the rotor bars, and these currents, reacting with the flux, develop a torque which makes the rotor follow the rotating field. The rotor does not rotate as fast as the field; if it did, no magnetic flux would be cut by the rotor bars and no current would be set up in them. The difference between the actual rotor speed and the synchronous speed of the rotating field is called the slip of the motor. As the load on the motor is increased, the machine slows down a little, thereby increasing the slip and causing larger currents to be induced in the rotor; these larger currents set up a greater torque, which enables the motor to drive the increased load. The three-phase induction motor is similar in construction and performance to the two-phase machine described, but it has three sets of windings for connection to a three-phase supply circuit; it is the type most frequently used. The polyphase induction motor operates as a constant-speed machine like the direct-current shunt motor. The power supplied to an induction motor can be computed from Eq. 22.14. 22-12. Transformers. In transmitting electrical power over long distances it is of advantage to use large potential differences, because a given amount of power can be transmitted with a correspondingly small current. This results in reducing the energy wasted in heating the transmission lines and permits the use of relatively Core small line wires. Transformers are used to change the potential difference from one value to another; in To supply Load power transmission they step it up mains to a high value at the alternator end of a line and then step it down at Primary Secondary the other end to a value suitable for the apparatus in the consumer's Fm. 22-17. Connections of a transformer. premises. The transformer consists essentially of two coils of wire, entirely separate electrically but wound upon the same core of laminated iron, as represented in Fig. 17. The primary winding is connected to the power supply mains, and the se~ondary winding is connected to the load circuit. The alternations of the primary current set up an alternating flux in the core, and the continual building up and collapsing of this flux induces an emf in the secondary coil. The value of this emf depends upon the number of turns Ns of the secondary

~f webers per second through its turns, in accordance with Eq. 20.2, which gives e. = Ng ~f volts.

coil and upon the rate of change of flux

396

ELECTRICITY & MAGNETISM

Chap. 22

The variations of flux which produce the secondary emf also affect the primary coil of NP turns and induce in it an emf ep

=

NP~~ volts.

This

emf, by Lenz's Law, opposes the impressed potential difference, in somewhat the same manner as the counter emf of a motor, § 20-7. This explains the fact that a transformer under no load takes very little current from the supply circuit, for the emf induced in the primary coil is very nearly equal to the potential difference of the supply; for practical purposes the two may be considered equal. The foregoing expressions can be combined and changed to effective values, yielding (22.15)

which shows that the emfs in the transformer coils are directly proportional to the numbers of turns on their windings. Upon connecting a load across the secondary winding, the emf induced in that coil will set up a current. This current will oppose the magnetizing effect of the primary current and will reduce the flux in the transformer core slightly. As a result, the counter emf induced in the primary winding is lessened, and more current is taken from the supply mains. In this manner the input to a transformer automatically accommodates itself to the output. The efficiency of a transformer may be expressed as the ratio of the output volt· amperes to the input volt· amperes, since the transformer itself has little effect upon the power factor of the circuit. Hence Es f s , Effi ciency (22.16) = -.- - . Eplp The efficiency of transformers is very high, as might be expected from the absence of moving parts, and usual values range well over 95 per cent. If the efficiency were taken as 100 per cent, the input and output volt· amperes would be the same, and it follows that the ratio of primary and secondary currents would be inversely proportional to the numbers of turns on the respective windings. 22-13. Alternating-current Measurements. The instruments most commonly used in alternating-current circuits are those which measure current, potential difference, and power. The d' Arsonval type of instrument described in §§ 18-8 and 9 is not suitable for use with alternating current, for the coil tends to follow the rapid reversals of the current and the pointer merely oscillates near the zero mark of the scale. The iron-vane ammeter makes use of the fact that iron is attracted by a coil carrying current regardless of the direction of the current, and hence will be attracted if the current is alternating. Current in a stationary coil attracts a soft-iron vane, which, as it moves, swings a pointer over a scale. The iron-vane voltmeter differs from the ammeter in that it has a high resistance in series with the coil to permit direct connection across the supply mains.

§ 22-14

THE SYNCHRO

397

The dynamometer type of ammeter has two current coils in series, one of which is stationary. The other coil carries a pointer and is pivoted at right angles to the field of the first, being normally held in this zero position by a coiled spring. An alternating current through the instrument reverses simultaneously in both coils, and the resulting torque on the moving coil is, therefore, always in the same direction. When a high resistance is connected in series with the coils,· this type of instrumen'.t can be used as a voltmeter. The power delivered to a load connected in an alternating-current circuit cannot be measured with a voltmeter and an ammeter; for this purpose it is customary to use a wattmeter. This instrument, described in § 18-10, operates with alternating as well as with direct currents. The torque producing the deflection is proportional to the instantaneous values of potential difference e and current i, and hence is proportional to the product of e and i. With sinusoidal waves of emf and current, and the latter lagging by the angle (), the instantaneous values are e = Em sin wt and i = Im sin (wt - ()); hence the torque is proportional to e i = Em Im sin wt X sin (wt - e).

The deflection of the instrument is proportional to the average product of ei, which, following the procedure in § 22-3 by integrating over a half-cycle, is found to be (ei)av = Em2Im

COS()=

EI

COS()=

P,

and shows that the deflection is proportional to the power delivered. The power factor of a motor or other load circuit can be determined by measuring the power with a wattmeter, the current with an ammeter, and Wattmeter Current coil

Ammeter

Potential coil

Load

High resistance

FIG. 22-18.

Connections for measuring the power factor of

a,

load.

the potential difference with a voltmeter, connected as shown in Fig. 18. The power factor is obtained by dividing the watts by the volt· ampere proquct. 22-14. The Synchro. The field of instrumentation concerns itself not only with .instruments for measuring physical quantities, such as those considered

398

ELECTRICITY & MAGNETISM

,Chap. 22

in the preceding section, but also devices for controlling industrial processes. A typical example of these devices is the Selsyn, a self-synchronizing arrangement for controlling angular position at a distance; it is commonly called a synchro. The synchro is a combination of two or more identical units, each of which has a stator and a rotor as in an induction motor. The connection plan for two units is indicated in Fig. 19. The stators are wound for three phases, and the terminals A, B, and Care connected to each other in proper order. The rotors R 1 and R2 are single phase and connected to a common supply of alternating current in order to develop magnetic fields that are synchronous.

c

8

B

120V Frn. 22-19.

c l20V

Scheme of Synchro as position controller. The rotors are supplied from a common supply of alternating current.

When the rotor R1 of the sending synchro has assumed a definite position, determined by some machine element to which it is coupled, emfs will be induced in the stator windings by transformer action. These set up currents in the three-phase circuit that extends to the stator of the receiving synchro and produce a torque that will cause its rotor R2 to turn. This movement continues until the rotor has the same position as that of the sending synchro. When this position is reached, the emfs developed in the two stators will be equal, and there will be no current in the connecting lines. When the machine element moves and turns rotor R1 through some angle, then rotor R2 will turn through the same angle, just as though both were mounted upon a common shaft that might be a few miles long. Commercially available synchros have an accuracy of about one minute of arc. PROBLEMS 1. A 24-pole alternating-current generator delivers a sinusoidal curren\ having a maximum value of 250 amp. What will be the instantaneous current value after the machine has turned through -,h rev from the position at which the current has its maximum value? 2. What is the period of an alternating current that has a frequency of 25 cycles/sec? If this current were passed through the coil of an electromagnet, how many times per second would the electromagnet tend to attract its armature? 3. The effective value of a sinusoidal alternating emf is 120 volts. Over what emf range do its instantaneous values extend?

PROBLEMS

399

4. The effective emf and current values of the alternating-current supply to a capacitive circuit are respectively 120 volts and 10 amp, and once each cycle their instantaneous values are 120 volts and -10 amp at the same moment. What is the phase relation of the emf and current waves? 5. It is desired to have a resistor develop heat at the rate of 70 cal/sec when connected across 120-volt alternating-current mains. What should its resistance be? 6. Two alternating-current generators operating at the same frequency are connected in series. One generates 120 volts and the other 50 volts. What is the resultant emf of the combination if the smaller emf (a) is in phase agreement with the larger, (b) is in phase opposition to the larger, and (c) lags the larger by 90°? 7. A circuit that has an inductance of 0.1 henry carries a 60-cycle current. What is the co}mter emf of self-induction in that circuit at an instant when the current has (a) it::,jmaximum value of 5 amp, and (b) half its maximum value? 8. An inductive coil of negligible resistance draws a current of 3.5 amp when connected across 120-volt, 25-cycle supply mains. What will be the current in the coil when it is connected across 240-volt, 60-cycle mains? 9. A coil of 0.15 henry inductance and 15 ohms resistance is connected across a 220-volt, 60-cycle line. Find (a) the reactance of the coil, (b) its impedance, (c) the current in it, and (d) the angle by which this current will lag the applied potential difference. 10. A 2-µf condenser is used in a telephone set. Find its capacitive reactance (a) for the "ringing current" of 16i, cycles/sec, and (b) for the "speech current" averaging 1000 cycles/sec. 11. A 60-cycle potential difference of 120 volts is impressed across a condenser of 20-µf capacitance. Compute the reactance of the condenser and the current in the circuit. 12. What should be the capacitance of a condenser so that it will have a capacitive reactance of 5 ohms at (a) 500 cycles/sec, (b) 500 kilocycles/sec, (c) 500 megacycles/sec? 13. At a certain frequency a series circuit has a capacitive reactance of 30 ohms, a resistance of 80 ohms, and an inductive reactance of 90 ohms. What is the current in the circuit when an alternating emf of the same frequency and of 240 volts effective value is impressed on the circuit? 14. What is the impedance at 1 megacycle/sec of a series circuit containing a 1-µµf condenser and a I-megohm resistor? What will be the result when the frequency is 10 kilocycles per sec? 16. A series circuit has a resistance of 25 ohms, an inductance of 0.01 henry, and a capacitance of 2 µf. Calculate the impedance of the circuit at 1000 cycles/sec. 16. Determine the natural frequency of the circuit of Prob. 15, and also the current that the circuit will take at that frequency with an impressed emf of 90 volts. 17. It is desired to vary the natural frequency of an oscillating circuit from 200 to 1000 kilocycles/sec. The circuit has an inductor of 0.5 millihenry and a variable condenser. What range in capacitance should the condenser have? 18. What inductance connected in series with a 10-µf condenser will produce resonance at a frequency of 25 cycles/sec? 19. Across 60-cycle supply mains are connected in series a condenser and an inductive coil which have reactances respectively of 75 and 100 ohms. What should be the capacitance of a series-connected condenser which, when added to the circuit, would establish resonance? 20. In the solved problem of § 22-8, suppose the frequency of the applied emf to be 240 cycles/sec. Determine the current taken by the parallel circuit and the total impedance.

+

400

ELECTRICITY & MAGNETISM

Chap. 22

21. Find the power developed in a coil having a resistance of 150 ohms and an inductance of 0.50 henry when connected across a 220-volt, 60-cycle supply line. 22. Find the power factor of each branch circuit and of the entire circuit for the solved problem of § 22-8, the frequency being 500 ~ycles/sec. 23. A fluorescent lamp unit takes 0.75 amp when connected across 115-volt alternating-current mains and draws 53 watts. When equipped with a condenser for power factor improvement, the unit takes 0.50 amp and draws the same power. Determine the power factor of the unit with and without the condenser. 24. The alternators at the hydroelectric power station at Bonneville Dam have 96 poles and develop 13,800 volts at 60 cycles/sec. At what speed do these waterwheel generators revolve? 25. A three-phase balanced inductive load of 80 per cent power factor is supplied with 60-cycle power over a three-wire circuit. The potential difference between the wires is 220 volts at the load, and the current in each wire is 15 amp. What power is supplied to the circuit? What is the angle between the current and emf for any phase? 26. A three-phase induction motor operating on 440-volt supply mains has a power factor (lagging) of 0.82 and an efficiency of 86 per cent when delivering 30 hp. What is the line current? 27. The alternators mentioned in Prob. 24 are three-phase machines and most of them are rated at 60,000 kva. How much current will a generator of this rating supply to the lines at full load? How much power can each supply with rated current when the currents are 15° out of phase with the generated emfs? 28. A 10-kva transformer, having 800 primary and 80 secondary turns on its windings, is connected to 2200-volt service mains and delivers full-load currents to a nonreactive load. Determine the currents in the windings. Neglect losses. 29. A coil of wire and a capacitor are connected in series to an alternator that generates 120 volts at 1000 cyles/sec. At that frequency the coil has a resistance of 40 ohms and a reactance of 60 ohms, and the capacitor has a reactance of 90 ohms. The impedance of the alternator itself may be neglected. (a) What should be the reading of a wattmeter when connected to this circuit to measure the power delivered by the alternator? (b) What should be its reading when the coil and capacitor are connected in parallel?

23 ELECTRONICS Of the various divisions of physical science, perhaps no division of this vast subject has developed faster and brought about more industrial applications than that dealing with electronic behavior. This field of Electronics includes Thermoelectricity, which deals with contact potentials and their depJ,idence upon temperature, Thermionics, which deals with the emission of electrons from heated cathodes, and the new branch dealing with Semiconductors. The basic concepts and important applications of these branches are presented in this chapter. Other topics that come within the scope of Electronics, such as photoelectric and photovoltaic action, radio and microwaves, television and radar, are considered in other chapters. THERMOELECTRICITY

23-1. Thermoelectric Effects. When two different metals are placed in contact with each other they assume slightly different potentials. This phenomenon, discovered by Volta, is now explained as a transfer of certain of the free electrons present in all metals across the boundary between the two; it is found that more electrons pass in one direction than in the other, and consequently one metal becomes negative and the other positive. The difference of potential between the metals, spoken of as contact potential difference, depends in amount upon the metals used and is also influenced by the temperature of the junction. This contact potential difference provides an explanation for three thermoelectric effects which have been known for years. The most important of these states that an emf will be produced in a circuit formed of two wires of different metals when their junctions are at different temperatures. Another states that an unequal heating of the junctions of a circuit formed of two different metals will result when a current traverses the circuit. The third states that an emf will be developed in a circuit formed of a single metal when there is a temperature gradient along its length. 23-2. The Seebeck Effect. The first of the thermoelectric effects was discovered by the German physicist Thomas J. Seebeck (1770-1831). This effect is the production of an emf in a thermocouple formed of two different metals when one of their junctions is at a higher temperature than the other. 401

402

ELECTRICITY & MAGNETISM

Chap. 23

A circuit formed by joining an iron and a copper wire is shown in Fig. 1; a galvanometer is included in order to indicate the amount of emf produced. It will be assumed that the low-temperature junction is kept at 0°C while the temperature of the other junction is raised. When the latter is, say, at room temperature, the thermal emf developed sets up a current which, for the particular circuit shown, is directed clockwise. As the second junction is heated more and more, the thermal emf increases to a maximum value, then diminishes to zero, reverses in direction, and again increases in magnitude. The values of emf developed in such a copper-iron thermocouple are plotted in Fig. 2; the curve is approximately a parabola. The thermal emf for any specific couple between any two given temperatures is always the same, provided the metals are of the same purity and crystalline state. As implied, the emf is affected by any structural changes that may be produced by mechanical and heat treatments.

+ 1.5

Cl)

+l.O

]

Galvanometer

I

E +o.5

....E UJ

I

v

,,.... ........

''

,, ',,..

I

I\ \

O

-0.5

i

'

''\ I\

-1.0 0

200

400

\ 600

Temperature, °C

Fw. 23-1.

Thermoelectric circuit of two metals.

Jcrn. 23-2.

Thermal emf diagram of a copper-iron couple.

The temperature at which the thermal emf of a couple has a maximum (or minimum) value is called the neutral point, and the temperature at which this emf reverses is called the inversion point. Each pair of metals has its own characteristic neutral and inversion temperatures; for the copper-iron couple these temperatures are shown in the figure to be approximately 205 and 480°C, respectively. The slope of the thermal emf curve at any point is called the thermoelectric power of the thermocouple at that temperature-an unfortunate use of the term "power." For the copper-iron couple the thermoelectric power is 13.7 microvolts per degree at 0°C, and 3.7 microvolts per degree at 150°C. The value of the emf produced by a couple formed of any two metals can be computed from a knowledge of the thermoelectric powers of these materials with respect to some metal chosen as a standard, such as lead or platinum. For this purpose graphs of thermoelectric power against temperature are useful. Graphs for a few metals against lead are shown in Fig. 3; they

THE SEEBECK EFFECT

§ 23-2

403

are straight lines over the temperature range shown. An example will clarify the procedure. Compute the emf developed by an iron-copper couple over the range from Oto 150°C. Reference to Fig. 3 shows that the thermoelectric power for iron falls from a value of 16.5 microvolts per degree at 0° to 8.1 at 150°C, and that the value for copper rises from a value of 2.8 microvolts per degree at 0°C to 4.4 at 150°C. Hence the total emf generated by an iron-lead couple over this temperature range is the average of 16.5 and 8.1 microvolts per C 0 multiplied by 150°, or 12.3 X 150 = 1845 microvolts; similarly the total emf of a copper-lead couple over the same range is 3.6 X 150 = 540 microvolts. Consequently, the emf of an iron-copper couple would be 1845 - 540 = 1305 microvolts; also see Fig. 2. The same result is obtained by computing the area between the graphs for iron and copper in Fig. 3 over the temperature range from O to 150°C and observing that each square represents 5 microvolts. Over this range iron is positive with respect to copper.

20

r,...

..._

~-

-

l

Antimony

--

i--

-

'--

.E o ......

f

- ~- --

I I I

fro,.,

})

. . . . . 10

...

~-- --

..._ ,.._

r-..

'--

-- - -

-Copper -...... ,.._

i.-

-

,_.

- --

J Use scale I lat righf .- -cadmium

-

.......

:::,

50

r-.

c 0

!'"---

>,.

0

Plati~um

'.E-10 Cl)

Bismuth

1E ...

-30

E :.:: c

so]

f Use scale

u

Cl)

5

i--

1

Nickel

-,--.r--.1o-.

100

-- - 200

100 § ~

- --300

Temperature, °C

Frn. 23-3.

Thermoelectric powers of several metals ·with respect to lead (Computed from data given in the International Critical Tables).

Measurements of radiant heat energy may be made with a multiplicity of couples formed of two metals and placed close together; such a compact group is called a thermopile. This device is often constructed of short lengths of bismuth and antimony connected in series alternately, and put together in zigzag fashion so that alternate junctions can be exposed to the radiation while the intermediate ones are shielded from it. The hot-wire ammeter, which utilizes a thermal junction in contact with a hot wire, is often used for measuring small alternating currents. The heat

404

ELECTRICITY & MAGNETISM

Chap. 23

evolved in the wire by the current which is being measured causes the thermocouple to generate an emf, thereby producing a deflection in a calibrated instrument connected across the couple. Increased sensitivity can be obtained by placing the wire and couple in a vacuum. Since the emf of a thermocouple depends in a known manner upon the temperatures of the junctions, the thermocouple can be used as a pyrometer, for the measurement of temperature. Such an instrument is commonly used to measure the temperatures of furnaces and of molten metals, care being exercised to protect the thermal junction from furnace gases and from direct contact with the molten substances. 23-3. The Peltier Effect. The French physicist Jean C. A. Peltier (17851845) discovered that in a circuit formed of two different metals, the maintenance of current from an outside source of emf caused one of the junctions to be heated and the other cooled. As compared with the Seebeck Effect, the hot and cold junctions are interchanged for the same direction of current. In the copper-iron couple of Fig. 1, when a current traverses the circuit at ordinary temperatures in a clockwise direction, the left junction will become heated and the right junction cooled. The extent to which the junctions are heated or cooled by a given current depends solely upon the metals used. For a copper-iron junction at room temperature, about 4 calories are developed per hour per ampere of current. The heating or cooling due to the Peltier Effect is directly proportional to the current which is passed through the junction. If a current of I amp is maintained fort sec, and if the contact potential difference in volts caused by electron diffusion at the junction is represented by the symbol II, then the heating or cooling is given in joules by W = IIIt. This effect is small and is often masked by the heating of both metals brought about by the Joule Effect, which for a junction of resistance R ohms is given by Eq. 17.6 as W = Rl2t. The combination of both effects at a junction can be expressed as P

=

RI2 ± III;

where Pis the power in watts; the plus sign is used for Peltier heating, the negative sign for cooling. 23-4. The Thomson Effect. A thermodynamic analysis of the foregoing effects prompted William Thomson to predict that an emf must exist between different parts of the same metal if they are at different temperatures. He demonstrated that, if a uniform metal bar is heated at the middle and a current is sent through it from end to end from an external source, the heat is conducted unequally along the two halves. In a copper bar, Fig. 4, the region A where current is directed from a colder to a hotter part will be cooler than it would be if there were no current, and the region B where current is directed from a hotter to a colder part will be warmer; thus B is warmer than A. The same is true for cadmium, silver, and zinc, but the effect is reversed in iron and nickel, to mention but a few metals. This effect and the emf involved are named after Thomson. Lead shows no appreciable Thomson

§ 23-5

THERMIONIC EMISSION

405

Effect, and this accounts for its frequent use as a reference metal in Thermoelectricity (see Fig. 3). The evolution or absorption of heat at the junctions of a thermoelectric circuit demonstrates that there must be a difference of potential at places where two dissimilar metals are in contact. If both junctions are at the same temperature, the thermal emfs at the two junctions are in opposite directions and annul each other; but if there is a temperature difference, these two emfs do not balance, and their resultant, together with the Thomson emfs, establishes a current in the circuit.

FIG. 23-4.

Experiment to show the Thomson Effect.

FIG. 23-5. Two-element

electron tube circuit.

THERMIONICS

23-5. Thermionic Emission. A discovery of far-reaching importance was made by Edison in 1883. He experimented with an evacuated tube containinf a heated filament and a separate electrode, and found that a current would be set up between them if the electrode were positive with respect to the filament, but not if it were negative. In the light of present knowledge this current is explained by the emission of electrons from the filament into the space surrounding it, and the attraction of these emitted electrons to the nearby electrode. The escape of electrons from the surface of a metal is comparable in many respects to the escape of molecules from a liquid during the process of evaporation. The situation is a little different, however, because an electron about to leave the metal induces a positive charge on the surface behind it and hence is attracted backward toward the metal. This action gives rise to the idea of a potential barrier, somewhat like surface tension, which must be overcome before the electron can escape. When the metal is heated, the electrons in it are given more kinetic energy, which assists them in passing the potential barrier at the surface. For each metal, there is a definite minimum amount of energy needed to release an electron from the surface; this energy is known as the work function of the metal. The thermionic emission of electrons by a hot body is the operating principle of the modern electron tube. The simplest of these is the two-element tube, or diode. This device consists of an evacuated bulb or tube with a filament somewhat like that of an incandescent lamp, and a separate metal plate, as shown in Fig. 5. When the filament is heated to incandescence by battery A,

406

ELECTRICITY & MAGNETISM

Chap. 23

it will emit electrons. These will be attracted to the plate when it is maintained positive by battery B, as indicated, and the galvanometer G will show a deflection. If the plate were made negative by reversing battery B, the electrons evaporated from the filament would be repelled by the plate and, since no electrons would be emitted from the cold plate, the galvanometer would not show a deflection. Hence the electrons can flow only from filament to plate, or, what corresponds to the same thing, the conventional direction of current, § 16-1, can be only from plate to filament. Consequently, this electron tube acts as a rectifier, the plate being the anode and the heated filament the cathode. The number of electrons emitted from the filament in a unit of time depends upon the substance of which it is made and upon its temperature. The rate of electron emission is generally expressed as the current per unit of surface area of the hot body; at absolute temperature T this current density is expressed by b

J = AT2 e-T

'

as given by the English physicist Owen W. Richardson (1879-1945).

(23.1) Herein

J is expressed in amperes per square centimeter, Tis in degrees K, e is the

base of natural logarithms, and A and b are constants. The value of A is 60.2 for the materials used as emitters. The constant b incorporates the work function of the material; its values found experimentally for three metals are: molybdenum 50,900, thorium 38,900, and tungsten 52,400; dimensionally these values of b are in degrees K. The current from plate to filament of an electron tube, as expressed by the foregoing equation, is the steady value that results when the plate potential is high enough to sweep all the electrons from the region around the filament as fast as they are liberated; this is the so-called saturation current. At lower potential differences between the two electrodes of the tube the current will be less, because some of the evaporated electrons are driven back into the filament by the negative charge that builds up in the space near the plate. The accumulation of electrons in this region of the tube is called the space charge. In most types of electron tubes it is not expedient to measure the saturation current, because its value is so large as to change the emitting conditions or damage the tube. When there is a copious supply of electrons in a diode and the potential difference across its electrodes is much below the value needed to produce the saturation current, then the current is limited by the space charge; it can be expressed in terms of the applied potential difference E by means of the generalized expression (23.2) where K and n are constants determined by the geometry of the tube. For a vacuum diode with plane parallel electrodes, the value of n is l That this

§ 23-6

407

RECTIFIER TUBES

figure is appropriate can be appreciated from the facts that the speed of the electrons reaching the anode is proportional to the square root of the potential difference between the electrodes, § 18-12, and that the number of electrons that are within the interelectrode space to neutralize the field around the anode is proportional to the potential difference. 23-6. Rectifier Tubes. The two-element tube permits current to be established through it in one direction only, and hence can serve as a rectifier of alternating currents. The relation between the current in such a tube and the potential difference between its filament and plate is indicated graphically in Fig. 6. The curves show the operating characteristics of a vacuum diode and a gas diode, the latter being used for rectifying the larger amounts of current.

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DC load circuit FIG. 23-7.

Full-wave tube rectifier circuit.

A rectifier that eliminates every other half-cycle of the alternating-current wave is called a half-wave rectifier, and one which utilizes both half-cycles is called a full-wave rectifier. The circuit of the full-wave rectifier is shown in Fig. 7; it comprises an iron-core transformer T with three windings, and a double-anode tube. The primary winding joins to the source of alternating yurrent, the secondary winding 1-2 connects with the two anodes P 1 and P 2, lnd the third winding supplies current to heat the filament F. ]\!Jid-taps 3 and 4 on the latter windings lead to the load circuit. At an instant when terminal 1 of the lower winding is positive, the current is directed from P 1 to F, to 4, through the load, to 3 and back to 1. When terminal 2 is positive, the path of the current will be from P2 to F, to 4, through the load, to 3 and back to 2. Consequently, the load current will be unidirectional. Since this current would pulsate between zero and a maximum value, a filter is used to make the current steady, § 24-2. High alternating potentials can be rectified similarly; a tube for this purpose bears the trade name of Kenotron. In the gas-filled diode the gas atoms are bombarded by the electrons emitted from the cathode and become ionized when electrons are knocked out of them. These gas ions are positively charged and tend to neutralize the space charge about the cathode; hence the emission of electrons from this electrode is in-

408

ELECTRICITY & MAGNETISM

Chap. 23

creased. The Tungar rectifier and the mercury-vapor rectifier are examples. The Tungar rectifier tube contains argon or other inert gas at low pressure, a cathode of tungsten coiled into a closely wound spiral, and an anode of graphite having a relatively large area. Tungar rectifiers are available for half-wave as well as for full-wave rectification, and are often used for charging small storage batteries from alternating-current service mains. The mercury-arc rectifier contains an electrode of mercury and one of graphite, within an evacuated container of glass or steel. To start the action, an auxiliary electrode is touched for a moment to the mercury pool and then withdrawn, striking an arc. Thereupon mercury vapor is formed which becomes ionized by electrons proceeding from the mercury surface, and a current is established between the electrodes. When the rectifier is connected to alternating-current mains, there will be current only during those intervals when the mercury electrode is negative; this fact accounts for the rectifying action of the device. Two anodes are used for full-wave rectification from a singlephase supply; more are used when converting from polyphase systems. Steeltank rectifiers, designed for currents up to several thousand amperes, are used in supplying power for traction systems. Other types of rectifiers, utilizing the properties of semiconductors, are considered in § 23-16. 23-7. Three-element Electron Tubes. The idea of adding another electrode to the two-element electron tube, in order to control conveniently the number of electrons passing from the filament to the plate, came from the American inventor Lee DeForest. The introduction of this so-called grid electrode between the filament and the plate makes the electron tube more versatile by enabling it to serve a number of functions, especially in telephone and radio circuits. The three-element tube and its circuits then appear as in Fig. 8. Input The effect of the grid is like that of a shutter which, opening and closing, controls the flow of electrons going through it from the filament to the plate. This control is accomplished by changing the potential of Frn. 23-8. Three-element electron the grid. When the grid is positively charged, it attracts electrons and increases tube circuit. their flow from the filament to the plate, for most of them pass through the relatively wide spaces between the grid wires. When the grid is negatively charged it repels the electrons and they cannot go to the plate. Consequently, when the grid G is made alternately positive and negative by joining the input terminals to a source of alternating potential, the electron flow from F to P is increased and decreased accordingly, thereby varying the direct current in the plate circuit. The grid potential might change thousands or millions of times per second and the plate current

§ 23-7

409

THREE-ELEMENT ELECTRON TUBES

would change accordingly. Actually, the grid is not made positive with respect to the filament, but only more or less negative. This is done by inserting a so-called C battery, as shown, to "bias" the grid negatively; when so biased there will be no current in the grid circuit. Thus, the grid serves as a gate-valve to control the plate current while taking practically no power itself. The cathode of the tube is often a thin metal sleeve coated with thorium or other material having a low work function; a heating coil of tungsten wire is mounted within but separated from the sleeve. This construction makes it possible to heat the cathode with alternating current without introducing disturbing effects.

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Rectifiers utilizing different semiconductors. current.

Silicon Ill Arrows indicate the direction of

The copper-oxide rectifier has an electrode of copper on which a layer of the oxide is produced by heat treatment. Contact with the exposed surface of the oxide layer is usually made in small rectifiers by evaporating a layer of silver or gold on the oxide, in intermediate units by clamping a lead washer against the oxide face, and in large rectifiers by reducing the top surface of the oxide to copper and then nickel plating it. The rectifier is pictured sectionally in part I of Fig. 21; the barrier layer is located between the copper and the oxide. Electrons pass easily from the copper to the oxide, and hence the direction of the conventional current is from oxide to copper. Small rectifiers are used in communication circuits, and the larger ones as power rectifiers. Copper-oxide and selenium rectifiers form the basis of the photovoltaic or barrier-layer type of photoelectric cell, §§ 27-8 and 34-2. The seleni'um rectifier is made by depositing a thin coating of metallic selenium on steel or aluminum and, after heat treatment, applying an alloy having a low melting point. The barrier layer is located between the selenium and the alloy, as indicated in part II of Fig. 21. Electrons pass easily from

424

ELECTRICITY & MAGNE'l'ISM

Chap. 23

the alloy to the selenium and hence the "forward" current direction is from selenium to alloy. A selenium rectifier does not require as many seriesconnected units as the copper oxide rectifier for a given potential difference and hence has the advantage of smaller size and weight in power applications. Tiny selenium rectifiers are used in telephone switching for contact protection; in this application two rectifiers are connected in series but in opposed relation to suppress sparking across a pair of contacts. The silicon rectifier is a development of the early crystal detector with the "catwhisker" contact. The schematic arrangement is indicated in part III of the figure, the barrier layer being at the contact point. The silicon is in the form of a wafer with one terminal soldered to it and the other touching it; the latter is the catwhisker usually formed of tungsten, platinum, or phosphor-bronze wire. The forward direction of current is from silicon to catwhisker. To give an idea of size, the silicon wafer for a rectifier that operates at microwave frequencies up to 25,000 megacycles per sec is 0.05 in. square and 0.01 in. thick. A modification of the silicon rectifier which is suitable for high potentials makes use of silicon-carbide-that is, carborundum. The granules are mixed with a binder and are fired to form ceramic-like rods or disks, contact with which is made by plating opposite faces. Within these forms the individual crystals constitute a series-parallel grouping of rectifier units connected with random polarity, and for that reason form a symmetrical varistor. These rectifiers are used in protective circuits, such as lightning arresters, and are known under the trade name of Thyrite. The germanium rectifier, which resembles the silicon rectifier in construction, is the most widely used point-contact varistor; because it serves the same purpose as a diode tube, it is generally called a germanium diode. Its forward current direction is usually from catwhisker to germanium. Most of these diodes can withstand reverse potentials of 75 volts, and some twice that. Germanium diodes are widely used in communication circuits and in industrial electronics. A possible competitor of the germanium and silicon rectifiers is a recently developed and inexpensive semiconducting compound formed of aluminum and antimony. E c 23-17. Transistors. The aim to utilize semiconductors as amplifiers was realized IZ:'Z:Z?;Z~ n-type R by John Bardeen and Walter H. Brattain Input 8 at the Bell Telephone Laboratories through the invention in 1948 of the trans~ - -- - - . . i istor. This device is similar to the triode FIG. 23-22. Input and output cirin action, and can be used as an oscillator cuits of a transistor. as well as an amplifier. A transistor consisting of a tiny block of n-type germanium with three electrodes is shown schematically in Fig. 22. Two of the contacts are rectify-

'l'RANSIS'l'ORS

§ 23-17

425

ing points, such as those described in the preceding section; they are the emitter E, connected to the input circuit, and the collector C, connected to the output circuit R. The third electrode is the base B and is common to both circuits. When the emitter is biased to establish current in the forward direction, as shown, and the collector is close by, the input current from Eis found to control the output current to C in such a way as to produce amplification. It will be recalled that there are n-type and p-type varistors, and that conduction may take place by means of electrons or by "holes" that are equivalent to positive charges; the action of the transistor depicted can be explained by considering the emitter to inject holes into n-type germanium (which otherwise would contain only electrons) and the collector to draw them over to it, thus augmenting the output current many-fold. ,i()

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Comparison of transistor and triode characteristics. ton-type transistor.

60

Signs on symbols apply

The performance of a transistor is compared with that of a triode in Fig. 23, which shows in part I a few of the collector circuit characteristic curves of a particular transistor, and in part II a few of the plate circuit characteristic curves of a particular triode. While the families of curves are similar in appearance, the difference in the variables should be emphasized, for the transistor curves show the effect of one current upon another, whereas the triode curves show the effect of one potential upon another. Like the triode, the transistor has several operating factors; for example, its amplification factor measures the effect which the emitter current ie has upon the collector current i 0 and is defined by a

=

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42G

ELECTRICITY & 1v!A.GNETISM

Chap. ~3

This factor can be likened to the amplification factor µ of a triode, for the application of positive current bias to the emitter shifts the collector circuit curve of the transistor to the right just as negative potential bias shifts the plate circuit characteristic of the triode to the right. Another form of transistor has been developed by interposing a thin p-type layer serving as base electrode between two sections of n-type germanium which form the emitter and collector. Such units show better operating characteristics, have exceedingly small power consumption, possess mechanical strength, and give high amplification. PROBLEMS 1. Compute the emf that is generated by a copper-nickel thermocouple when one of the junctions is at 0°0 and the other is at 300°C. 2. What emf is produced by an antimony-bismuth thermocouple when the temperatures at the junctions are maintained at 20°0 and 175°0? 3. The thermoelectric powers of aluminum and silver, both against lead, are expressed by the following equations over the temperature range under consideration: Silver E = 2.1 + 0.15 t, Aluminum E = -0.8 + 0.004 t where Eis in microvolts per C degree and tin degrees C. Find the emf produced by an aluminum-silver couple when the temperature of one junction is 0°C and the other is (a) 100°C, and (b) 200°0. • 4. A thermopile being designed for use in fire detection is to have the same number of antimony-bismuth junctions on both faces. It is desired to generate a total emf of at least 0.10 volt when the exposed face of the thermopile reaches 160°C, the other face remaining at the room temperature of 15°C. What is the smallest number of junctions that may be used on the exposed face? 5. A thermopile has 64 antimony-bismuth junctions on each face. When radiation falls upon one face while the other is at the room temperature of 20°0, the thermopile generates an emf of 0.18 volt. 'What is the temperature of the thermopile face that receives the radiation? 6. A demonstration magnet for operation by a thermocouple formed of copper and nickel is able to exert a large attractive force when one junction is heated by a Bunsen burner. If the winding consists of 8 massive turns and has a resistance of 50 microhms (including the thermocouple), and if the junction temperatures are 450 and 60°0, how much magnetomotive force is developed by the winding? Extrapolation of the curves of Fig. 3 gives the thermoelectric powers at 450° for copper as 7 .5 and for nickel as -32.5 microvolts per 0°. 7. Compute the rate of electron emission in amperes per square centimeter from a tungsten filament operating at 2300°0. 8. A thoriated-tungsten cathode is operated at a temperature of 1900°K and produces an electron emission of 0.70 amp/cm2 • What value do these data give for bin Richardson's equation? 9. The constant bin Richardson's equation is found by theoretical analysis to equal the ratio of the work function of the metal to the gas constant per molecule. This gas constant is obtained by dividing the universal gas constant by Avogadro's Number. Calculate the work function for molybdenum; express the result in electron· volts. 10. The potential difference applied to a diode has a steady value of 250 volts and the current through the tube is 0.10 amp. (a) What is the value of the tube constant Kin Eq. 23.2? (b) How many electrons arrive each second at the anode? (c) With what velocity do the electrons impinge upon that electrode?

PROBLEMS

427

11. Determine the amplification factor of a Type 6.T5 tube having the characteristics shown in the accompanying figure. Choose the operating current to be 7.0 milliamp. Type 6J5 12. A triode, for which the amplification E1=6.3 volts factor is 70 and the transconductance 1600 micromhos, is connected to a load circuit of a. 0.1 megohm resistance. What power is supE plied to the load when an emf of 1 volt rms 6 f-------1----+1----1-----.~ value is applied to the input circuit? "i§ 13. For a triode that has an amplification c factor of 10, the plate current is found to ~ 4>-----, where the reciprocal of 1' is replaced by the frequency f. These expressions are applicable to all wave motions. The speed v of a wave is determined completely by the properties of the transmitting medium, and the frequency f of the source may be assumed to remain constant; it follows that, if a wave from this source enters a new medium wherein it moves slower or faster, there will be a corresponding change in wavelength to satisfy the equation. 25-3. The Mechanism of Wave Propagation. The process by which a wave advances will be explained by referenre to a mechanical model, Fig. 2,

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'\fochanical wave model illustrating a transverse wave moving toward the right.

452

WA VE MOTION & SOUND

Chap. 25

consisting of a long coiled spring with small masses A, B, C, which represent the vibrating particles of the medium. The spring is fastened at the distant end, and particle A is given harmonic motion along the path 0-1-2-0. During the time that the end particle A is moving up to position 1, particle B is pulled upward by the spring tension, and proceeds in that direction, as shown in part I of the figure. Because of its inertia, B continues to move upward when A reverses its direction. As A moves to 2, a downward force acts on B which soon arrests its upward motion and causes it to move down. The same behavior is repeated at the lower end of the path, and it follows that B will have the same kind of motion as A but will reach a corresponding point of the path a little later than A will. As a consequence of the inertia of the weights and the elasticity of the spring, similar motion will be imparted successively to all parts of the spring, and the weights will reach their maxiA B

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A longitudinal wave and its transverse representation.

mum displacements in the sequence A, B, C, and so on-in other words, a wave is set up which advances toward the right. At the instant represented in part II of the figure, A has been given a complete vibration and the wave has advanced to J; the wave along the spring has a crest at G and a trough at C. The same mechanical model will also serve to illustrate the motion of a longitudinal wave, as represented in Fig. 3. Motion of A toward the right compresses the spring, and motion toward the left extends it. The spring acts on B and gives it the same kind of motion as A, except for a slight lag in phase. Similarly, B produces a corresponding motion of C, and so on. The result is a series of condensations in which the weights are close together, separated by rarefactions in which the weights are farther apart. Both configurations move to the right and constitute the advancing wave. At the instant shown in the figure the wave in the spring has advanced to J. The wave form of a longitudinal wave is not apparent but can be made so by laying off the displacements of the particles at right angles to the direction in which they actually occur. Such a construction is shown at the bottom of the figure. In this diagram the normal rest positions of the particles are indicated by corresponding lower-case letters, and the displacements along the axis are shown turned counterclockwise through 90 degrees by means of arcs centered at these points.

THE MECHANISM OF WAVE PROPAGATION

§ 25-3

453

From the behavior of the mechanical wave model, some general inferences can be made. \Vave motion is evidently not due to bodily transfer of the medium through which the wave advances. It is caused by vibrations of individual particles over short ranges about normal rest positions, all the particles having the same kind of motion, but with a progressive change of phase along the direction of propagation. Moreover, mechanical wave motion requires that the transmitting medium possess both inertia and elasticity; for electromagnetic waves these properties are replaced by their electrical equivalents, inductance and capacitance.

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Fm. 25-4.

Motion of air particles transmitting a sound wave. The wave is moving toward the right, and the arrows represent the velocities of the particles.

The progress of a sound wave in air is represented in Fig. 4. Line 1 shows the positions and velocities of the vibrating particles at a particular instant, line 2 shows them t of a period later, line 3 shows them t of a period later still, and so on. It will be observed that each particle moves back and forth with harmonic motion, that there is a progressive change of phase in considering one particle after another, that condensations and rarefactions are formed, and that these configurations advance in a definite direction-toward the right in this example. In the condensations the particles move in the same direction as the wave, while in the rarefactions they move the other way. Water waves are of interest in that the particles which transmit the disturbance move longitudinally as well as transversely. The particles follow elliptical or circular paths; the wave is neither longitudinal nor transverse, but a combination of the two. A typical construction for deep water waves

454

WA VE MOTION & SOUND

Chap. 25

is shown in Fig. 5; the progressive change of phase in the motion of the particles and the resulting wave shape are indicated. 25-4. Energy Transmission by Waves. Whenever waves pass through a medium, energy is transmitted through it in the direction of propagation. This fact can be illustrated by the mechanical model of Figs. 2 and 3, which show waves progressing to the right. For either type of wave, each section of the spring exerts a force on the weights at its ends, and the weights at the right move in the direction of this force, but those at the left are constrained to move in opposition to it. Each spring section thus does work on the weight ahead of it and has work done upon it by the weight behind it, § 6-1, and each weight, in turn, performs a similar action on the adjoining spring sections. Hence, a continuous transfer of energy takes place in the direction of wave travel. In the model described, if no energy were wasted as the wave advances, the amplitude of vibration would be the same for all of the weights.

Frn. 25-5.

The displacement of water particles in a surface wave.

On the other hand, in a wave that spreads out as it advances, such as a circular wave on the surface of water, the amplitude of vibration diminishes as the wave progresses, since the energy is spread out over a larger and larger surface. The energy of a wave may be transformed in various ways; for example, that of a sound wave may be converted into mechanical energy in setting the ear drum into vibration. When a wave encounters a medium of a different character, some of its energy will be reflected back into the initial medium, and the rest will be transmitted into the second medium; also, as the wave advances, part of its energy will be absorbed. For a light wave that impinges upon a sheet of glass, most of the energy is transmitted to the region beyond the glass, part is returned by reflection at the surfaces, and a small portion is absorbed within the glass itself. When a light wave strikes black velvet, practically no light is reflected from it nor transmitted through it; the velvet absorbs the energy and its temperature is raised very slightly. 25-5. The Wave Equation. It has been repeatedly pointed out that wave motion is associated with vibrations of the particles in the transmitting medium. To investigate how these two motions are related, it is convenient to imagine a source vibrating harmonically and producing a sinusoidal wave. If the source vibrates with frequency f and amplitude r, its displacement at any instant is given by the expression Ys = r sin 2 1rft,

(25.2)

where tis the time reckoned from the instant when the vibrating source passes

§ 2!i-G

WA VE IN A STRETCHED CORD

455

through the midposition of its path in a positive direction. This motion is imparted to the particles of the surrounding medium, and as the disturbance moves forward it will reach these particles one after another and give each the same type of motion as the source. If the disturbance moves with a speed v, then to travel a distance x will require a time x/v, and consequently the displacement of a particle in the medium at a distance x from the source will be (25.3)

y P = r sin 2 1rf ( t - : }

Although this expression gives the displacement of the particle in the medium, tis still reckoned from the time the vibrating source passes the midposition of its path. It is clear that the displacement of the source at a given instant will be duplicated by the vibrating particle after a time interval x/v; thus, if t is given· a particular value in Eq. 25.2 and is increased by x/v in Eq. 25.3 the values of y. and YP will be identical. The wave equation, 25.3, shows how the displacements of the vibrating particles vary with respect to their location and also with respect to time. By selecting a particular value for t and pl