Electrodynamics: A Concise Introduction

438 Pages·1996·12.177 MB·English

by  James Blake Westgard (auth.)

Electrodynamics:
A Concise Introduction
Springer
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J aITles Blake Westgard
Electrodynamics:
A Concise
Introduction
With 121 Figures
Springer
James Blake Westgard
Department of Physics
Indiana State University
Terre Haute, IN 47809
USA
Cover illustration: The cover illustration shows three radiation patterns from
an accelerating charge. At small velocities, as compared to the velocity of light,
the observed distribution is proportional to the square of the component of the
acceleration orthogonal to the line of sight of the observer. At higher velocities,
the distribution is thrown more toward the direction of the velocity, and the
diagram illustrates linear breaking-radiation and synchrotron radiation dis
tributions. The diagram was created with the symbolic mathematics program,
Mathematica.
Library of Congress Cataloging-in-Publication Data
Westgard, James B.
Electrodynamics: a concise introduction/James B. Westgard.
p. cm.
Includes bibliographical references and index.
ISBN-13:978-1-4612-7514-5 e-ISBN-13:978-1-4612-2356-6
DOl: 10.1007/978-1-4612-2356-6
1. Electrodynamics. I. Title.
QC631.W47 1995
530.1'41-dc20 95-37687
Printed on acid-free paper.
© 1997 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 1st edition 1997
All rights reserved. This work may not be translated or copied in whole or in part without
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enue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or
scholarly analysis. Use in connection with any form of information storage and retrieval.
electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that such
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be used freely by anyone.
Production managed by Francine McNeill; manufacturing supervised by Jeffrey Taub.
Typeset in TEX from the author's Microsoft Word files.
9 8 7 6 5 4 3 2 1
ISBN-13:978-1-4612-7514-5 Springer-Verlag New York Berlin Heidelberg SPIN 10425511
Preface
This textbook is intended for advanced undergraduates or beginning
graduates. It is based on the notes from courses I have taught at Indiana
State University from 1967 to the present. The preparation needed is
an introductory calculus-based course in physics and its prerequisite
calculus courses. Courses in vector analysis and differential equations
are useful but not required, since the text introduces these topics.
In writing this book, I tried to keep my own experience as a stu
dent in mind and to write the kind of book I liked to read. That
goal determined the choice of topics, their order, and the method of
presentation. The organization of the book is intended to encourage
independent study. Accordingly, I have made every effort to keep the
material self-contained, to develop the mathematics as it is needed,
and to present new material by building incrementally on preceding
material. In organizing the text, I have taken care to give explicit cross
references, to show the intermediate steps in calculations, and to give
many examples. Provided they are within the mathematical scope of
this book, I have preferred elegant mathematical treatments over more
ad hoc ones, not only for aesthetic reasons, but because they are often
more profound and indicate connections to other branches of physics.
I have emphasized physical understanding by presenting mechanical
models.
This book is organized somewhat differently from the traditional
textbook at this level. The first three chapters (about 100 pages) present
the experimental foundation of the Maxwell field equations in three
dimensional notation, while detailed applications are reserved for later
chapters. This organization presents an immediate overview of the sub
ject, treats electricity and magnetism on an equal footing, and allows
exploitation of many symmetries between the two-even at this level of
discussion. This organization offers flexibility to instructors, who may
choose to emphasize certain applications over others, with the assur
ance that students have the necessary foundation. These chapters also
serve as a review for well-prepared students, who may proceed more
quickly to later sections.
vi PREFACE
Chapter 4 introduces special relativity and the concept of a local
Lorentz transformation, which is then used to formulate the Maxwell
field equations in four dimensions. In this treatment, the logical
connection between relativity and electrodynamics is very clear and
immediate, and the local transformation concept lays a foundation for
general relativity and quantum field theory. After expressing electro
dynamics in the four-dimensional form, subsequent calculations are
made in either three- or four-dimensional form, depending on the ap
plication. Being able to choose is a great advantage, since a calculation
in four-dimensional notation can be much easier than the same calcula
tion done in three-dimensional notation. The remaining chapters deal
with steady-state fields, radiation in dielectric media, charged particle
motion, radiation by moving charges, and classical electron theory.
I have introduced a few innovations into the content of this book.
In Chapter 7 the Lorentz equations of motion of a particle in an elec
tromagnetic field are reformulated as a relativistic fluid model. In this
formulation a conserved vector field Gk is introduced, and the equa
tions of motion become integral equations with infinite series solutions.
These solutions generate corresponding diagrams that are analogous
to the Feynman diagrams in quantum electrodynamics. I have also in
corporated two recent advances in computer software by including the
numerical solution of differential equations, and by using a symbolic
mathematics program, Mathematica. Many of the illustrations in the
text were generated by Mathematica calculations, and short program
listings (notebooks) on special topics are provided in the appendixes.
If they have Mathematica available, students may use these notebooks
in the exercises.
I am deeply grateful to my own teachers, and to my students who
have contributed to this work by their enthusiasm and comments. I
therefore dedicate this book to them, as a way of passing the torch
from one generation to the next.
Terre Haute, Indiana James Blake Westgard
Contents
Preface v
CHAPTER 1 Introduction to Electrodynamics 1
1-1 A Brief History of Electromagnetism 1
1-2 Vectors 14
1-3 Vector Calculus 18
1-4 Curvilinear Coordinate Systems 26
1-5 Integration of Vectors 37
1-6 Delta Functions 46
1-7 Selected Bibliography 48
1-8 Problems 49
App. 1-1 Vector Operators 50
App. 1-2 Curvilinear Coordinates 59
CHAPTER 2 Experimental Foundation 71
2-1 Fields 72
2-2 Coulomb's Law 73
2-3 Ampere's Laws for the Magnetic Field 78
2-4 Faraday's Induction Law 84
2-5 Maxwell's Equations 87
2-6 A Mechanical Model of the Electromagnetic Field 90
2-7 The Michelson-Morley Experiment 95
2-8 Systems of Units 98
2-9 Selected Bibliography 102
2-10 Problems 102
App.2-1 Field Plotter in 2-D 105
CHAPTER 3 Dielectric and Magnetic Materials and Boundary Conditions 111
3-1 Dielectric Materials 112
3-2 Currents 117
3-3 Magnetic Materials 121
3-4 Boundary Conditions 126
3-5 Some Mathematical Aspects of Boundary Conditions 130
viii CONTENTS
3-6 Selected Bibliography 138
3-7 Problems 138
CHAPTER 4 Electromagnetic Equations 141
4-1 Tensor Notation 142
4-2 Integral Theorems 149
4-3 Relativity: A New Kinematics 152
4-4 The Electromagnetic Field 164
4-5 Electromagnetic Potentials and Gauge Conditions 173
4-6 Lorentz Transformed Fields 175
4-7 The Lagrangian Method 178
4-8 Selected Bibliography 181
4-9 Problems 182
App. 4-1 Lorentz Transformation of the Field Tensor 184
CHAPTER 5 Electromagnetic Fields in Steady States 187
5-1 Steady-State Equations 187
5-2 Multipole Expansion 189
5-3 Laplace's Equation: Separation of Variables 196
5-4 The Cauchy-Riemann Equations and Conformal
Mapping 208
5-5 Numerical Solutions by Finite-Element Analysis 214
5-6 Magnetic Fields 217
5-7 Selected Bibliography 219
5-8 Problems 220
App. 5-1 Relaxation Solution of Laplace's Equation 222
App. 5-2 Gram-Schmidt Orthogonalization 224
App. 5-3 Schwartz Transformations 226
App. 5-4 Fourier Series 231
App. 5-5 Laplace Equation 240
CHAPTER 6 Radiation and Optics in Dielectric Media 247
6-1 Wave Equation in Uniform Media 247
6-2 Spherical Waves 252
6-3 Radiation in Conductive and Dispersive Media 261
6-4 Refraction and Reflection at a Dielectric Boundary 267
6-5 Momentum and Energy 274
6-6 Huygens' Principle and Diffraction 282
6-7 Selected Bibliography 292
6-8 Problems 293
App.6-1 Bessel and Legendre Functions 294
App. 6-2 Multipole Radiation Patterns 300
App. 6-3 Single Slit Diffraction by Helmholtz Integral 305
App. 6-4 The Electromagnetic Stress Tensor 306
CONTENTS ix
CHAPTER 7 Particle Motion in Electromagnetic Fields 311
7-1 Uniform Fields 311
7-2 Numerical Solutions 314
7-3 An Example: Particle Optics 320
7-4 Velocity Field Model of Single Particle Kinematics 324
7-5 Cross Sections 337
7-6 Selected Bibliography 341
7-7 Problems 342
App.7-1 Finite-Element Solution of a Differential Equation Using
a Spreadsheet 342
App.7-2 Taylor Series Solutions of Differential Equations 345
CHAPTER 8 Radiation by Moving Charges 349
8-1 Multipole Expansion 350
8-2 A Physical Model 358
8-3 Frequency Analysis of Radiation 360
8-4 Calculating with Delta Functions 368
8-5 Radiation by Charged Particles 371
8-6 Selected Bibliography 388
8-7 Problems 389
App.8-1 Radiation by a Fast Charged Particle 390
App. 8-2 Tensor Potentials 396
CHAPTER 9 Beyond the Classical Theory 401
9-1 Radiation Reaction 402
9-2 Classical Models of the Electron 405
9-3 Quantization 414
9-4 Unification with Weak Interactions 424
9-5 Selected Bibliography 429
Index 431
1
CHAPTER
Introduction to
Electrodynamics
The history of science is science itself; the history
of the individual. the individual.
Johann Wolfgang von Goethe
Mineralogy and Geology
Mathematics, rightly viewed, possesses not only
truth, but supreme beauty-a beauty cold and
austere, like that of sculpture.
Bertrand Russell
The Study of Mathematics (1902)
1-1 A
BRIEF HISTORY OF ELECTROMAGNETISM
The history of electromagnetism is intertwined with the revolutions in
astronomy and mechanics, and with the history of science and intel
lectual history in general. It is convenient to distinguish three periods
in the history ~hich we might call early, classical, and modern, each
initiated by certain clusters of critical, distinguishing discoveries. Cu
riously, these clusters occurred around the centennial years 1600, 1800,
and 1900.
The Early Period: 16~1800. Of the tumultuous social and in
tellectual changes of the late Renaissance, none had a more profound
effect on Western thought than the scientific revolution of the seven
teenth century, which overturned traditional beliefs about the nature
of the physical universe and promoted development of the scientific
method. In explorations of unknown parts of the world, in religious
beliefs, in political structures, in the arts, and in the physical sciences,