Classical Electromagnetism in a Nutshell

703 Pages·2012·23.95 MB·English

by  Anupam Garg

Copyright© 2012 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton,
New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street,
Woodstock, Oxfordshire OX20 lTW

press.princeton.edu

Ali Rights Reserved

Library ofCongress Cataloging-in-Publication Data

Garg, Anupam Kumar, 1956-
Classical electromagnetism in a nutshell / Anupam Garg.
p.cm.
Includes bibliographical references and index.
ISBN-13: 978-0-691-13018-7 (cloth: alk. paper)
ISBN-10: 0-691-13018-3 (cloth: alk. paper) l. Electromagnetism-Textbooks. l. Title.
QC760.G37 2012
537- dc23
2011037153
British Library Cataloging-in-Publication Data is available

This book has been composed in Scala

Printed on acid-free paper. oo

Typeset by SR Nova Pvt Ud, Bangalore, India
Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To my parents, and to Neelja, Gaurav, and Aljun

Contents

Preface
XV

List of symbols
xxi

Suggestions for using this book
xxxi

ITJ lntroduction
1
1 The field concept
1
2 The equations of electrodynamics
2

3 A lightspeed survey of electromagnetic phenomena
7
4 SI versus Gaussian
10
[I] Review of mathematical concepts
18
5 Vector algebra
18

6 Deriva ti ves of vector fields
25

7 Integration of vector fields
30
8 The theorems of Stokes and Gauss
32

9 Fourier transforms, delta functions, and distributions
37
10 Rotational transformations of vectors and tensors
45
11 Orthogonal curvilinear coordinates
51

[TI Electrostatics in vacuum
55
12 Coulomb's law
55
13 The electrostatic potential
57
14 Electrostatic energy
58
15 Differential form of Coulomb's law
63
16 Uniqueness theorem of electrostatics
65

viii
Contents

17 Solving Poisson's equation: a few examples
68
18 Energy in the electric field
71
19 The multipole expansion
73
20 Charge distributions in external fields
80
0 Magnetostatics in vacuum
82
21 Sources of magnetic field
82
22 The law of Biot and Savart
89
23 Differential equations of magnetostatics; Ampere's law
93
24 The vector potential
101
25 Gauge invariance
105
26 \1 · B and \1 x B for a point dipole
108
27 Magnetic multipoles
112

ITJ lnduced electromagnetic fields
114
28 Induction
114
29 Energy in the magnetic field-Feynman's argument
117
30 Energy in the magnetic field-standard argument
120
31 Inductance
121
32 The Ampere-Maxwell law
125
33 Potentials for time-dependent fields
128
0 Symmetries and conservation laws
132
34 Discrete symmetries of the laws of electromagnetism
132
35 Energy flow and the Poynting vector
137
36 Momentum conservation
140
37 Angular momentum conservation*
144
38 Relativity at low speeds
148
39 Electromagnetic mass*
150
0 Electromagnetic waves
152

40 The wave equation for E and B
152
41 Plane electromagnetic waves
154
42 Monochromatic plane waves and polarization
156
43 Nonplane monochromatic waves; geometrical optics*
160
44 Electromagnetic fields in a laser beam*
165
45 Partially polarized (quasimonochromatic) light*
168
46 Oscillator representation of electromagnetic waves
171
47 Angular momentum of the free electromagnetic field*
174
w
1 nterference phenomena
178
48 Interference and diffraction
178

49 Fresnel diffraction
50 Fraunhofer diffraction
51 Partially coherent light
52 The Hanbury-Brown and Twiss effect; intensity interferometry*
53 The Pancharatnam phase*

0 The electromagnetic field of moving charges
54 Green's function for the wave equation

55 Fields of a uniformly moving charge
56 Potentials of an arbitrarily moving charge-the Lienard-Wiechert

solutions

57 Electromagnetic fields of an arbitrarily moving charge
58 Radiation from accelerated charges: qualitative discussion

ITQJ Radiation from localized sources

59 General frequency-domain formulas for fields
60 Far-zone fields
61 Power radiated
62 The long-wavelength electric dipole approximation
63 Higher multipoles*
64 Antennas
65 Near-zone fields
66 Angular momentum radiated*
67 Radiation reaction

ITIJ Motion of charges and moments in externa! fields
68 The Lorentz force law
69 Motion in a static uniform electric field
70 Motion in a static uniform magnetic field
71 Motion in crossed E and B fields; E< B
72 Motion in a time-dependent magnetic field; the betatron
73 Motion in a quasiuniform static magnetic field-guiding center

drift*

74 Motion in a slowly varying magnetic field-the first adiabatic
invariant*

75 The classical gyromagnetic ratio and Larmor's theorem
76 Precession of moments in time-dependent magnetic fields*

@] Action formulation of electromagnetism

77 Charged particle in given field
78 The free field
79 The interacting system of fields and charges
80 Gauge invariance and charge conservation

Contents 1 ix

182

186

187

191

195

200

200

204

207

210
214

217

217

219

223

227

229

233

237

239

241

245

245

246

248

251

255

257

261

264

268

273

273

276

279

283

x 1 Contents

02] Electromagnetic fields in material media

81 Macroscopic fields
82 The macroscopic charge density and the polarization
83 The macroscopic current density and the magnetization
84 Constitutive relations
85 Energy conservation

~
Electrostatics around conductors
86 Electric fields inside conductors, and at conductor surfaces
87 Theorems for electrostatic fields

285

286

289

293

297

300

302

303

306
88 Electrostatic energy with conductors; capacitance
308
89 The method of images
313
90 Separation of variables and expansions in basis sets
320
91 The variational method*
329
92 The relaxation method
334
93 Microscopic electrostatic field at metal surfaces; work function and
contact potential*
339

~
Electrostatics of dielectrics
344
94 The dielectric constant
344
95 Boundary value problems for linear isotropic dielectrics
347
96 Depolarization
350
97 Thermodynamic potentials for dielectrics
354
98 Force on small dielectric bodies
360
99 Models of the dielectric constant
361

~
Magnetostatics in matter
370
100 Magnetic permeability and susceptibility
370
101 Thermodynamic relations for magnetic materials
371
102 Diamagnetism
375

103 Paramagnetism
378
104 The exchange interaction; ferromagnetism
378
105 Free energy of ferromagnets
382
106 Ferromagnetic domain walls*
391
107 Hysteresis in ferromagnets
394
108 Demagnetization
397
109 Superconductors*
399

[22] Ohm's law, emf, and electrical circuits
404

110 Ohm's law
405

111 Electric fields around current-carrying conductors-a solvable
example*
407

112 van der Pauw's method*
113 The Van de Graaff generator
114 The thermopile
115 The battery
116 Lumped circuits
117 The telegrapher's equation*
118 The ac generator

@_] Frequency-dependent response of materia Is

119 The frequency-dependent conductivity
120 The dielectric function and electric propensity
121 General properties of the ac conductivity*
122 Electromagnetic energy in material media*
123 Drude-Lorentz model of the dielectric response
124 Frequency dependence of the magnetic response*

[2I] Quasistatic phenomena in conductors

125 Quasistatic fields

Contents

126 Variable magnetic field: eddy currents and the skin effect in aplanar

1 xi

409

412

413

414

417

422

424

427

427

429

431

435

437

441

443

443

geometry
445
127 Variable magnetic field: eddy currents and the skin effect in finite
bodies*
450
128 Variable electric field, electrostatic regime
455
129 Variable electric field, skin-effect regime
457
130 Eddy currents in thin sheets, Maxwell's receding image construction,

and maglev*
131 Motion of extended conductors in magnetic fields*
132 The dynamo*

l 201 Electromagnetic waves in insulators

133 General properties of EM waves in media
134 Wave propagation velocities
135 Reftection and refraction ata ftat interface (general case)
136 More reftection and refraction (both media transparent and
nonmagnetic)
137 Reftection from a nonmagnetic opaque medium*
lm Electromagnetic waves in and near conductors

138 Plasma oscillations
139 Dispersion of plasma waves*
140 Transverse EM waves in conductors
141 Reftection oflight from a metal

459

465

467

470

470

472

475

479

483

487

487

488

490

492

xii
Contents

142 Surface plasmons*
143 Waveguides
144 Resonant cavities

l 22 I Scattering of electromagnetic radiation

145 Scattering terminology
146 Scattering by free electrons
147 Scattering by bound electrons
148 Scattering by small particles
149 Scattering by dilute gases, and why the sky is blue
150 Raman scattering
151 Scattering by liquids and dense gases*

~
Formalism of special relativity
152 Review ofbasic concepts
153 Four-vectors
154 Velocity, momentum, and acceleration four-vectors
155 Four-tensors
156 Vector fields and their derivatives in space-time
157 Integration of vector fields*
158 Accelerated observers*

l 24 I Special relativity and electromagnetism

159 Four-current and charge conservation
160 The four-potential
161 The electromagnetic field tensor
162 Covariant form of the laws of electromagnetism
163 The stress-energy tensor
164 Energy-momentum conservation in special relativity
165 Angular momentum and spin*
166 Observer-dependent properties oflight
167 Motion of charge in an electromagnetic plane wave*
168 Thomas precession*

~
Radiation from relativistic sources
169 Total power radiated
170 Angular distribution of power
171 Synchrotron radiation-qualitative discussion
172 Full spectral, angular, and polarization distribution of synchrotron
radiation*

173 Spectral distribution of synchrotron radiation*
174 Angular distribution and polarization of synchrotron radiation*
175 Undulators and wigglers*

493

496

502

505

505

506

508

510

512

515

516

524

524

532

537
540

543
544

548

553

553

556

556

559

561

564

565

567

572

576

581

581
584

588

589
592

595

597

Appendix A: Spherical harmon,ics

Appendix B: Bessel ftmctiom

Appendix C: Time averages of bilinear quantities in, electrodynamics

Appendix D: Caustics

Appendix E: Airy functions

Appendix F: Power spectrum of a ran,dom fun,ction,

Appendix G: Motion, in, the earth 's magn,etic field-the Stormer problem

Appendix H: Altemative proof of Maxwell's recedin,g image con,struction

Bibliography

fodex

Contents 1 xiii

605

617

625

627

633

637

643

651

655

659

[ Preface [

In the United States, a graduate course in classical electromagnetism is often seen as
a vehicle for learning mathematical methods of physics. Students mostly perceive the
subject as hand-to-hand combat with the v symbol, with endless nights spent gradding,
divvying, and curling. Instructors ha ve a more benign view, but many of them too regard
the subject as a way of teaching the classical repertoire of Green functions, Sturm-
Liouville theory, special functions, and boundary value problems. Perhaps this viewpoint
is traditional. In the 1907 preface to the first edition of his classic text Electricity and
Magnetism, James Jeans writes:

Maxwell's treatise was written for the fully equipped mathematician: the present book is

written more especially for the student, and for the physicist of more limited mathematical

attainments.

The emphasis on mathematics continues throughout the preface: words derived from
mathematics appear ten times in the single page, while derivatives of physics appear just
five times.
To me, this point of view is both a puzzle and a pity, for while the mathematics of
electrodynamics is indeed very beautiful, the physics is much more so. And so, while this
book <loes not shy away from mathematical technique, the emphasis is very much on the
physics, and I have striven to provide physical context and motivation for every topic in
the book. Naturally, greater mastery of the mathematics will lead to greater enjoyment of
the physics. I try to model good and efficient methods of calculation, but this is never an
end in itself, and Ido not hesitate to use the physicist's quick and dirty methods when ap-
propriate. I urge readers, students in particular, to approach the subject in the same spirit.
This book started out many years ago as lecture notes for students in a graduate course.
Over the years, to keep my own enthusiasm from waning, I varied the topics in the latter
part of the course. A few others that I never got to teach are also included in the book.

xvi 1 Preface

As a result, there is more material here than can be covered in a single full-year course. I
hope that all instructors will find something to interest them among the more advanced
topics and applications, and that the book will serve as a reference for students after the
course is complete.
The sequence of topics is as follows. After a very brief survey [1] and a longish math
review [2], 1 we proceed to electrostatics (3] and magnetostatics [4] in vacuum, Faraday's
law of induction [5], and electromagnetic waves in vacuum [7]. Chapter 6 is on the
symmetries of the laws of electromagnetism, and it is here that the concepts of field
energy and momentum are introduced. These chapters are part of the core of the subject
and belong in all courses.

Chapter 8 deals with interference and diffraction. This material is somewhat nonstan-
dard in an electromagnetism course and may be skipped without loss of continuity. On
the other hand, the phenomena of interference and diffraction are the most striking
manifestations of the wave aspects of light, and the related topics of coherence and
intensity interferometry (the Hanbury-Brown and Twiss effect) are among the more
subtle aspects of light that were understood in the twentieth century. They also connect
to our understanding of quantum mechanics, and similar ideas are active research topics
even today in other areas of physics. A section on the Pancharatnam phase and how the
polarization oflight affects interference rounds out the chapter.

Next we come to radiation from accelerated charges [9, 10]. Although the field of a

moving charge is found in full generality [9], only nonrelativistic sources are considered
for the present [10], and relativistic sources are treated later [25]. While this is also core
material, the more advanced sections, such as those on higher multipoles, radiation of
angular momentum, and radiation reaction, can be omitted in a first reading.

The motion of charges and magnetic moments in external fields is studied next

[11]. In addition to charges in uniform and static fields, and the betatron, I have
included a moderately complete account of AlfVen's guiding center method for charges
in inhomogeneous magnetic fields. This method is indispensable for the study of
plasmas. I also discuss Larmor precession of magnetic moments, in both static and
time-dependent magnetic fields. The concept of adiabatic invariants is developed both
for charges and moments in magnetic fields. Again, subsequent chapters do not depend
on this development in an essential way and many sections in this chapter may be skipped
in a first pass.

Chapter 12 puts electromagnetism within the framework of the action principle. It is
thus an essential launching point for the study of quantum electrodynamics, and even
within the classical theory it is a deep unifying principle. For the study of electromagnetic
phenomena, however, it is inessential, and less formally minded readers may choose to
skip it.

Chapters 3 through 12 constitute a fairly complete treatment (except for relativity, for
which see later) of the basic laws and phenomena of electromagnetic fields in vacuum.
With these fundamentals over, we turn to electromagnetic fields in matter [13-21 ]. This

1 Chapter numbers are in square brackets.

Preface 1 xvii

separation of the study of fields in vacuum and matter is a little nonstandard, 2 and it
is more common to study the electrostatics of conductors (14] and dielectrics (15] along
with electrostatics in vacuum, and electromagnetic waves in matter (20, 21] along with
those in vacuum. I believe that no good can come of this fusion. It appears to me to
be motivated by the incidental similarity of the mathematics required. However, the
interesting physical questions are quite different in the two situations, and the basic
electromagnetic equations in vacuum and in macroscopic materials have rather different
meaning and ontological status. Thus, studying the propagation of light in glass and
vacuum as common instances of the solution of the wave equation obscures the physical
content of the dielectric constant, or the Kramers-Kronig relations, or the interplay
between electromagnetic energy, mechanical work, and heat dissipation in a medium.
A mathematically oriented approach also misses out on the beautiful physics involved in
modeling constitutive relations. With specific reference to electrostatics, most students
have studied it with conductors and in vacuum as one topic as undergraduates, so they
already know that the field is strong near sharp points and edges and other similar
facts, and there is little compelling reason to follow the same path again. Nevertheless,
instructors who wish to stress the solution of Laplace's equation through methods such
as separation of variables and orthogonal functions, can cover chapter 14 earlier.
In more detail, the treatment of electromagnetism in matter is organized as follows.
A short survey chapter [13] introduces the concepts of spatially averaged or macroscopic
fields, magnetization and polarization fields, and the vital role of constitutive relations.
It is particularly stressed that the latter are largely phenomenological, but no pejorative
connotation is attached to this word. It is also stressed that the similarity of form of the
macroscopic or material Maxwell equations with those in vacuum is deceptive and sweeps
important physics under the rug. This chapter is very important, and readers are urged
not to skip any of it.
Chapter 14 covers electrostatics with conductors. My treatment is quite traditional, and
it is with sorne ambivalence that so much detail is given. The prevalence of numerical
methods has rather reduced the importance of the classical mathematical techniques,
but they provide invaluable insight, and even the numerical solver would be unwise to
ignore them. I have also included a section on the work function and contact potentials,
if for no other reason than to show the reader that there is subtle physics in this
otherwise mathematical territory. The coverage of electrostatics with dielectrics (15] is
also traditional. In both chapters, there are several advanced sections that may be skipped
without loss of continuity.
Magnetostatics in matter comes next (16]. The central point of physics here is to
understand the constitutional differences (constitutive relations, if you will) between
para-, dia-, and ferromagnets. Much of the chapter is devoted to ferromagnets, and
relevant phenomena, such as domain walls, hysteresis, and demagnetization, ancient
topics that are astonishingly current for the design of hard disks and other computer
elements. The dangers of uncritical use of equations such as B = µ,H are dwelt on. I have

2 The same approach is taken in the Landau and Lifshitz Course of Theoretical Physics, vols. 2 and 8.

xviii 1 Preface

tried hard to obey Paul Muzikar's exhortation to explain the difference between B and H
carefully, and I am sure I will hear from him if I have let him clown! The chapter also
includes a short section on superconductors and the Meissner effect.

The next chapter is on Ohm's law, emf, and electrical circuits [17]. These tapies are

of great practica! importance, but they are often regarded as belonging to the realm of
engineering. My focus is on understanding of the meaning of emf, and of the physics of
the lumped circuit approximation. I also discuss the distribution of current in extended
conductors, and van der Pauw's method of measuring resistivities.

With this, we turn to dynamic phenomena, beginning with the general issue of
frequency-dependent response or constitutive relations [18]. This chapter covers the
Drude and Drude-Lorentz models for the frequency-dependent conductivity and dielectric
functions, which are of great pedagogical value because they fix several key physical con-
cepts in the student's mind. I also discuss Kramers-Kronig relations and electromagnetic
energy in material media. N one of this material should be skipped.

Quasistatic phenomena [19], such as the skin effect, eddy currents, and maglev, come

next, followed by electromagnetic waves in insulators [20] and conductors [21 ]. The former
chapter covers dispersion, and refiection and refraction at interfaces, while the latter
includes plasma oscillations, ultraviolet transparency and metallic refiection, waveguides,
and resonant cavities.
This ends the discussion of macroscopic electromagnetism. Completeness would

demand that I include tapies such as spin waves and Walker modes in ferromagnets,
light in anisotropic media, sorne magnetohydrodynamics, and nonlinear optics. But life
is short, and the interested reader must seek these elsewhere.
Scattering of light [22] is a vast subject, and the selection of subtopics refiects my

personal tastes. I do not discuss scattering of charged particles, collision radiation, virtual
quanta, or the energy loss formula, as there are excellent treatments of these tapies by
other authors, and my own knowledge of them is limited.
The formalism of special relativity [23], its role in electromagnetism [24], and radiation
from relativistic particles [25] bring up the rear, but they could in fact be covered any
time after chapter 12. I have chosen not to present electromagnetism as an outgrowth of
relativity as many other authors do, partly because that is ahistorical, but mainly because
I have found that students find it hard to relate the formal relativistic underpinnings
to actual electromagnetic phenomena, especially in matter, and the machinery of four-
vectors and four-tensors seems unconnected to concepts such as the multipole expansion
and mutual inductance. This machinery also requires more sophistication to master, so
I have found it helpful to delay introducing it. Nevertheless, relativity is used to provide
insight throughout the book, the first-order transformation of E and B fields is found
early on, and much of the treatment of radiation and charged particle motion is fully
relati vistically correct.
Severa! appendixes provide mathematical reference material (Bessel and Airy

functions, the Wiener-Khinchine theorem, etc.), which need not be covered explicitly.
Appendix A on spherical harmonics is an exception, as these are used throughout the
book, and an easy familiarity with them is a must. I have also included two appendixes

Preface 1 xix

on caustics (the rainbow and the teacup nephroid) and the motion of charged particles in
the earth's magnetic field as specialized applications of matter in the main text.
Several colleagues have asked me if my book would include numerical methods. It
does, but in a very limited way. My experience has been that good use of numerical
methods requires so much attention to basic calculus and linear algebra that it reduces
the time students can devote to physical concepts. Further, solving problems numerically
requires much trial and error, making the exercise closer to experimental than theoretical
physics, and thus ill suited to a traditional lecture-based course. Secondly, the peripheral
aspects of computer use-the operating system, the precise computer language, the
graphics package-are so varied and riddled with minutiae that they often end up
dominating the learning of the numerical methods themselves. I do not advocate the
use of canned black-box packages, as the student learns neither much physics nor much
numerical analysis from them.
The book includes over three hundred exercises. These appear in each section rather
than at the ends of chapters, because in my view the immediacy of the problems makes
them more relevant. Sorne exercises are quite simple and designed to develop technical
skills, while others may require significant extension of concepts in the text. The latter are
often accompanied by short solutions or hints. I have tried to make sure that all exercises
have a point and to explain that point, often by adding explicit commentary, and that none
are just make-work for the sake of having a large number. A few exercises are purely
mathematical or formal, but these are generally used to develop results used elsewhere in
the book.

The book does not require in-depth understanding of quantum mechanics or thermo-

dynamics, and though quantal and thermodynamic concepts are used or mentioned in a
matter-of-fact way when needed, they are not critical to the central development, and an
acquaintance with them at the undergraduate level will suffice.

I have benefited from encouragement and discussions with so many colleagues and
students over the years that I am sure I cannot remember them all. I am indebted,
among others, to Greg Anderson, Mike Bedzyk, Carol Braun, Paul Cadden-Zimansky,
Pulak Dutta, Shyamsunder Erramilli, André de Gouvéa, Nick Giordano, C. Jayaprakash,
John Ketterson, Mike Peskin, Mohit Randeria, Wayne Saslow, Surendra Singh, Mike
Stone, and Tony Zee. I am materially indebted, in addition, to David Mermin and Saul
Teukolsky for lectures at Cornell University, which I relied on in writing appendix A
on spherical harmonics and chapters 23 and 24 on relativity. I am likewise indebted to
Jens Koch for his invaluable help with sorne of the figures. I am particularly indebted to
Onuttam Narayan and Paul Sievert for reading large sections ofthe text, and their valuable
feedback and criticisms. None of these people are, of course, responsible for any errors or
omissions in the book.
I also wish to thank Karen Fortgang and Ingrid Gnerlich at Princeton University Press
for their gentle prodding and guidance through the production process.

I owe a different kind of debt to the authors of the texts from which I learned this
subject. These include the delightful Electricity and Magnetism by Purcell; The Feynman
Lectures (all three volumes) by Feynman, Leighton, and Sands; Classical Electrodynamics