Physics: A Student Companion [Paperback ed.] 1904842682, 9781904842682

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ys1c 11

A Student Companion Lowry A Kirkby

Magdalen College, University of Oxford, Oxford, UK Currently at Department of Biophysics, University of California, Berkeley, USA With a foreword by Frank Close Professor of Theoretical Physics, University of Oxford, Oxford, UK

Scion

Contents xv

Foreword Preface

XVll

Acknowledgements

xvm

How to use this book

Part I

Newtonian Mechanics and Special Relativity

Introduction

Chapter 1

linear motion

XIX

1 2

3

1.1

Newton's Laws of Motion 1.1.1 Newton's First and Second Laws 1.1.2 Newton's T hird Law

3 3 4

1.2

Force and momentum 1.2.1 Impulse 1.2.2 Conservation of momentum

4 4 5

1.3

Force and energy 1.3.1 Work done by a force 1.3.2 Power

6 6 7

1.4

Mechanical energy Kinetic energy (KE) 1.4.1 1.4.2 Potential energy (PE) 1.4.3 Conservation of energy 1.4.4 Dissipated energy

8 8 8 9 9

1.5

Conservative forces

Chapter 2

Practical applications of linear motion

10

15

2.1

Two-body collisions 2.1.1 Elastic and inelastic collisions Elastic collisions in two dimensions 2.1.2

15 15 17

2.2

Equations of motion 2.2.1 Constant force and projectiles 2.2.2 Variable mass problems

17

19 19

V

Part I

Newtonian Mechanics and Special Relativity

14

1.5 Conservative forces

Hooke's Law approximation (Derivation 1.4):

potential energy V(x) plus kinetic energy T(x) such that the total energy is conserved:

Restoring force for a small displacement from equilibrium, Ax:

Ea,b,c

F(Ax) = -kAx

(1.24)

where the spring constant is: (1.25) (Ax= x x 0, where x 0 is the equilibrium position).

Bound and unbound motion The motion of an object is classified as bound motion if it is restricted to a particular region in space, and unbound if it is not. For example, consider the poten­ tial energy function shown in Fig. 1.6. Let the total mechanical energy of the object be Ea, Eb, or Ee for Figs. 1.6a, b, and c, respectively. This is divided into

V(x) + T(x)

The distribution of energy between these two forms changes as a function of x. If the potential energy func­ tion does not exceed the total energy of the object, as illustrated in Fig. 1.6a, then there are no restrictions on the body's displacement (it can take any value of x), thus the motion is unbound. If however the total energy falls below V(x) for a given range of x, as illustrated in Figs. 1.6b and c, then the object cannot have dis­ placements that correspond to these regions. Regions with E < V(x) are called classically forbidden regions.2 In these cases, the body is confined to regions where E > V(x), thus its motion is bound. A body at Xtow or xhigh (refer to Figs. 1.6b and c) has zero kinetic energy. Therefore, at these positions, the body is stationary and all its energy is stored as potential energy. 2 In quantum mechanics, it is possible for a particle to enter into the classically forbidden region by a process known as quantum tunneling ( see Section 17.2.2).

Chapter 2: Practical applications of linear motion

system described above are illustrated in Fig. 2.9. In one normal mode, when Wf = klm, the two masses oscillate in the same direction (in phase): X1 = X2; in the other normal mode, when wi = 3klm, the two masses oscillate in opposite directions (in anti-phase): X1 = X2. Since the equations of motion are linear, then the general solu­ tion is a linear superposition of normal modes: x1 = A cos cqt + Bsin cqt + C cos m2t + D sin li½_t x2 = Acoscqt+Bsincqt-Ccosli½_t-Dsinli½_t

The method described above can be used to solve various configurations of coupled oscillators, for example, two coupled pendula or two masses on a vibrating string.

N coupled oscillators Normal modes exist for any linear system of springs and masses, with any number of components. For a system of N coupled oscillators, there are N normal modes; in three dimensions, there are 3N normal modes. The nor­ mal modes of masses on a vibrating string in the limit N oo describes the propagation of a transverse wave on the string. Similarly, the propagation of waves through matter are equivalent to the vibrations of N OO coupled atoms in the solid. ➔

The constants A, B, C, and D are fixed by initial con­ ditions (two initial positions plus two initial velocities).

33

➔

Chapter 12:

Electromagnetic waves

Derivation 12.6: Derivation of the Brewster angle (Eq. 12.54)

cos0t -

n2

n,

cos0I

• Starting point: the Fresnel equation for r11 is:

=0

'II

sin Be cos Be -sin 0i cos 0i = 0 I . I . -sm20c --sm2 0i = 0

r11 = 0: :. n, cos et

sin(Be - 0i)cos(0c +Bi)= 0 For this to be true:

nin1.

or et +0i = 2

:. Using the latter in Snell's Law (let 0i n, sin{% = n2 sin et

. (TC )

2 = n2 cos(%

:. tan(%=

n2

n,

n 2 sin 0,) to eliminate

• Use the trigonometric identities:

TC

.

- n2 cos 0; = 0

• Use Snell's Law (n 1 sin 0i

either et - Bi = O

:. n1 sm (% = n2 sm

n, cos et - n2 cos ei n 1 cos Be + n2 cos 0i

• There is no reflected parallel component of E when

2

2

=

-(%

O n):

I . sm 0cos 0 = -sm 20 2 and sin201 -sin202 = sin(01 - 02)cos( 01 + 02) to manipulate the expression for zero reflection. • For zero reflection, we find that either the refractive indices are equal (0i O, ), or the sum of incident and transmitted angles is 0i + 0, n 12. • Use the latter condition in Snell's Law to eliminate 0,. The angle of incidence at which there is zero reflection is the Brewster angle, 08 .

193

ar

Quantum Physics

The Schrodinger equation

Chapter 17:

Therefore, a state is defined by the three integers n, l, and p, which represent quantization along each direction. Energy degeneracy

For a cube of equal sides, say a, the wavefunction's energy levels are:

ti 2 t?-

Entp

--2

2ma

(n 2 +l 2 +p 2)

The ground state is defined by the integers (n, !, p) = (1, 1, 1), and the ground-state energy is therefore: E111

=

3n 2 t? 2 ma 2

The first excited state, however, has three different permutations (n, !, p) = (2, 1, 1), (1, 2, 1) or (1, 1, 2).

259

These each correspond to a different wavefunction­ and thus to a different state-but they all have the same energy: E211

=

E121

=

E112

=

6n2 t?

- 2

2ma

States that correspond to the same energy level, but have different wavefunctions, are called degenerate states. Since there are three states that correspond to the first energy level in this case, this level is said to have a three-fold degeneracy. Degeneracy is a prop­ erty of symmetric systems. For example, the poten­ tial box has cubic symmetry. Atoms possess spherical symmetry, which also gives rise to degenerate states, as we shall see in the case of the hydrogen atom (see Chapter 20).

Chapter 24:

'Derivation 24.14:

Derivation of the Stefan-Boltzmann Law (Eq. 24.57)

f

uT = u(v)dv 0

= =

8ffh = v3 dv f h vlk T 3 c oe B -1 8ffh

15c 3

(k8 ) 4 Tff

h

Statistical mechanics

• Starting point: the total energy density, uT, is found by integrating Planck's Radiation Law (Eq. 24.53) over all photon frequencies, v: 8ffh = v3 UT = -3 f h vlk T d V c oe B -1 • To integrate, use the standard integral: 4 3 =

f0 ca: -1 dx= l�(�)

• From kinetic theory, we found that the particle flux, fm-2 s-1], of particles with average velocity c is: 1 :. j = CUT 4 = 2,?kri y4 15h 3 c 2 =