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Maxwell's Equations Maxwell's Equations Paul G. Huray ♦IEEE IEEE PRESS ©WILEY A John Wiley & Sons, Inc., Publication Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com . 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QC670.H87 2010 530.l4'l-dc22 2009019340 Printed in the United States of America 10 9 8 7 6 5 4 3 21 This book is dedicated to the author's lifelong partner Susan Lyons Huray Contents Acknowledgments Introduction 1. Foundations of Maxwell's Equations 1.1 Historical Overview 1 1.2 Role of Electromagnetic Field Theory 2 1.3 Electromagnetic Field Quantities 3 1.4 Units and Universal Constants 6 1.5 Precision of Measured Quantities 12 1.6 Introduction to Complex Variables 12 1.7 Phasor Notation 15 1.8 Quaternions 19 1.9 Original Form of Maxwell's Equations 20 Endnotes 22 2. Vector Analysis 23 Introduction 23 2.1 Addition and Subtraction 23 2.2 Multiplication 26 2.3 Triple Products 29 2.4 Coordinate Systems 30 2.5 Coordinate Transformations 35 2.6 Vector Differentiation 39 2.7 Divergence Theorem 46 2.8 Stokes's Theorem 50 2.9 Laplacian of a Vector Field 51 Endnotes 55 3. Static Electric Fields 56 Introduction 56 3.1 Properties of Electrostatic Fields 56 3.2 Gauss's Law 58 3.3 Conservation Law 71 vii viïi Contents 3.4 Electric Potential 72 3.5 Electric Field for a System of Charges 76 3.6 Electric Potential for a System of Charges 77 3.7 Electric Field for a Continuous Distribution 79 3.8 Conductor in a Static Electric Field 80 3.9 Capacitance 84 3.10 Dielectrics 87 3.11 Electric Flux Density 90 3.12 Dielectric Boundary Conditions 91 3.13 Electrostatic Energy 93 3.14 Electrostatic Field in a Dielectric 98 Endnotes 103 4. Solution of Electrostatic Problems 104 Introduction 104 4.1 Poisson's and Laplace's Equations 104 4.2 Solutions to Poisson's and Laplace's Equations 106 4.3 Green's Functions 110 4.4 Uniqueness of the Electrostatic Solution 114 4.5 Method of Images 116 5. Steady Electric Currents 129 Introduction 129 5.1 Current Density and Ohm's Law 129 5.2 Relation to Circuit Parameters 134 5.3 Superconductivity 136 5.4 Free Electron Gas Theory 139 5.5 Band Theory 145 5.6 Equation of Continuity 150 5.7 Microscopic View of Ohm's Law 151 5.8 Power Dissipation and Joule's Law 155 5.9 Boundary Condition for Current Density 155 5.10 Resistance/Capacitance Calculations 157 Endnote 158 6. Static Magnetic Fields 159 Introduction 159 6.1 Magnetic Force 160 6.2 Magnetostatics in Free Space 161 6.3 Magnetic Vector Potential 165 6.4 The Biot-Savart Law 167 6.5 Historical Conclusions 170 6.6 Atomic Magnetism 171 Contents ix 6.7 Magnetization 182 6.8 Equivalent Surface Current Density 184 6.9 Equivalent Magnetic Monopole Charge Density 184 6.10 Magnetic Field Intensity and Permeability 187 6.11 Ferromagnetism 188 6.12 Boundary Conditions for Magnetic Fields 193 6.13 Inductance and Inductors 194 6.14 Torque and Energy 198 Endnotes 202 7. Time-Varying Fields 203 Introduction 203 7.1 Faraday's Law of Induction 204 7.2 E&M Equations before Maxwell 205 7.3 Maxwell's Displacement Current 206 7.4 Integral Form of Maxwell's Equations 208 7.5 Magnetic Vector Potential 209 7.6 Solution of the Time-Dependent Inhomogeneous Potential Wave Equations 213 7.7 Electric and Magnetic Field Equations for Source-Free Problems 214 7.8 Solutions for the Homogeneous Wave Equation 215 7.9 Particular Solution for the Inhomogeneous Wave Equation 221 7.10 Time-Harmonic Fields 227 7.11 Electromagnetic Spectrum 228 7.12 Electromagnetic Boundary Conditions 228 7.13 Particular Solution for the Wave Equation with Inhomogeneous Boundary Conditions 236 7.14 Memristors 238 7.15 Electric Vector Potential 241 Endnotes 246 Appendix A: Measurement Errors 247 Appendix B: Graphics and Conformai Mapping 260 Appendix C: Vectors, Matrices, Orthogonal Functions 274 Bibliography 285 Index 287