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Electrodynamics: A Concise Introduction PDF

438 Pages·1996·12.177 MB·English
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Electrodynamics: A Concise Introduction Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo 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 the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Av 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 names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly 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,

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