Electromagnetic Field Theory - A Problem-Solving Approach .

CylindricalCartesianSphericalx r cosc/Sr sin 6 cos 4yz r sin4Sr sin 6 sin4 z ix Cos Oi-sin 4k 4 isin Ocos ki, cos 6 cos-sin Ois6 sin cos 6 io0 i, Cos 01i siniYrcos6 sin 0 sin 6i, Cosil41 C541S1 Viecos Oi,- sin OisSphericalCartesianCylindrical4iesin 6SrIx y7tan 1y/xz rcos6 sin Oi, cos 6iei4 cos (k ,, sin -sin ki,, Cos 4i, i, i,.i4 SphericalCartesianr0I4x y i, CosiOCylindricalIf,- z" z1 zcot-'cos i, -sin cos-1zx/ysinG cos kix sin cos6 sin ci, sini, Cos Oi,i. cos 6 cos oi, cos 6 sin oi,-sin Oi. Cos Oi,-sin Oi,i -sin 46i, Cos 0i, i4,i,Geometric relations between coordinates and unit vectors for Cartesian, cylirdrical, and spherical coordinate systems.

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MAXWELL'S EQUATIONSBoundary ConditionsDifferentialIntegralFaraday's Lawd CBE'-dl -B-dSVxE nx(E'-E') 0.dtJIatAmpere's Law with Maxwell's Displacement Current CorrectionH-dl Jf,-dSVXH Jf aDnX(H 2 -H 1 ) KfD'-dS t pfdVV - D pn - (D 2 -D 1 ) o-B-dS 0V-B 0 it-sD-dSGauss's LawConservation of ChargeJ,- dS sVdpfdV OV-J, at 0n - (J2-J) "a0Usual Linear Constitutive LawsD eEB HJf o-(E vX B) a-E' [Ohm's law for moving media with velocity v]PHYSICAL CONSTANTSConstantSymbolSpeed of light in vacuumElementary electron chargeElectron rest massceElectron charge to mass ratioValueunits8M,2.9979 x 10 3 x 1081.602 x 10 '99.11 x 10 3m/seccoulkge1.76 x 10"coul/kgI,kGkgjoule/*Knt-m2/(kg) 2m/(sec) 2M,g1.67 x 10-271.38 x 10-236.67 x 10-"9.807Permittivity of free space608.854 x 10 2 36rPermeability of free spacePlanck's constantA04r XImpedance of free spaceAvogadro's numberi1o1010*Proton rest massBoltzmann constantGravitation constantAcceleration of gravityfarad/mh6.6256 x 10-34henry/mjoule-sec4376.73- 120irohmsAr6.023 x 1023atoms/mole

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ELECTROMAGNETICFIELD THEORY:a problem solving approachMARKUS ZAHNMassachusetts Institute ofTechnologyKRIEGER PUBLISHING COMPANYMalabar, Florida

Original Edition 1979Reprint Edition 1987Reprint Edition 2003 w/correctionsPrinted and Published byKRIEGER PUBLISHING COMPANYKRIEGER DRIVEMALABAR, FLORIDA 32950Copyright 0 1979, John Wiley & Sons, Inc.Transferred to AuthorReprinted by ArrangementAll rights reserved. No part of this book may be reproduced in any form or by any means,electronic or mechanical, including information storage and retrieval systems without permissionin writing from the publisher.No liabilityis assumedwith respectto the use oftheinformation containedherein.Printed in the United States ofAmerica.FROM A DECLARATION OF PRINCIPLES JOINTLY ADOPTED BY A COMMITTEEOF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS:This publication is designed to provide accurate and authoritative information in regard tothe subject matter covered. It is sold with the understanding that the publisher is not engagedin rendering legal, accounting, or other professional service. If legal advice or other expertassistance is required, the services of a competent professional person should be sought.Library of Congress Cataloging-in-Publication DataZahn, Markus, 1946 Electromagnetic field theory: a problem solving approach/ Markus Zahn.-Reprint ed.w/corrections.p. cm.Originally published: New York : Wiley, c1979.Includes index.ISBN 1-57524-235-4 (alk. paper)1. Electromagnetic fields. 2. Electrodynamics. I. Title.QC665.E4Z32 2003530.14'1--dc2l20030474181098765432

to my parents

PrefaceVPREFACEElectromagnetic field theory is often the least popularcourse in the electrical engineering curriculum. Heavy reli ance on vector and integral calculus can obscure physicalphenomena so that the student becomes bogged down in themathematics and loses sight of the applications. This bookinstills problem solving confidence by teaching through theuse of a large number of worked examples. To keep the subjectexciting, many of these problems are based on physical pro cesses, devices, and models.This text is an introductory treatment on the junior level fora two-semester electrical engineering course starting from theCoulomb-Lorentz force law on a point charge. The theory isextended by the continuous superposition of solutions frompreviously developed simpler problems leading to the generalintegral and differential field laws. Often the same problem issolved by different methods so that the advantages and limita tions of each approach becomes clear. Sample problems andtheir solutions are presented for each new concept with greatemphasis placed on classical models of such physicalphenomena as polarization, conduction, and magnetization. Alarge variety of related problems that reinforce the textmaterial are included at the end of each chapter for exerciseand homework.It is expected that students have had elementary courses incalculus that allow them to easily differentiate and integratesimple functions. The text tries to keep the mathematicaldevelopment rigorous but simple by typically describingsystems with linear, constant coefficient differential anddifference equations.The text is essentially subdivided into three main subjectareas: (1) charges as the source of the electric field coupled topolarizable and conducting media with negligible magneticfield; (2) currents as the source of the magnetic field coupled tomagnetizable media with electromagnetic induction generat ing an electric field; and (3) electrodynamics where the electricand magnetic fields are of equal importance resulting in radi ating waves. Wherever possible, electrodynamic solutions areexamined in various limits to illustrate the appropriateness ofthe previously developed quasi-static circuit theory approxi mations.Many of my students and graduate teaching assistants havehelped in checking the text and exercise solutions and haveassisted in preparing some of the field plots.Markus Zahn

Notes to the Studentand InstructorA NOTE TO THE STUDENTIn this text I have tried to make it as simple as possible for aninterested student to learn the difficult subject of electromag netic field theory by presenting many worked examplesemphasizing physical processes, devices, and models. Theproblems at the back of each chapter are grouped by chaptersections and extend the text material. To avoid tedium, mostintegrals needed for problem solution are supplied as hints.The hints also often suggest the approach needed to obtain asolution easily. Answers to selected problems are listed at theback of this book.A NOTE TO THE INSTRUCTORAn Instructor's Manual with solutions to all exercise problemsat the end of chapters is available from the author for the costof reproduction and mailing. Please address requests on Univer sity or Company letterhead to:Prof. Markus ZahnMassachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer ScienceCambridge, MA 01239Vii

CONTENTSChapter 1-REVIEW OF VECTOR ANALYSIS1.1 COORDINATE SYSTEMS1.1.1 Rectangular(Cartesian)Coordinates1.1.2 CircularCylindricalCoordinates1.1.3 Spherical Coordinates1.2 VECTOR ALGEBRA1.2.1 Scalarsand Vectors1.2.2 Multiplicationof a Vector by a Scalar1.2.3 Addition and Subtraction1.2.4 The Dot (Scalar) Product1.2.5 The Cross (Vector) Product1.3 THE GRADIENT AND THE DELOPERATOR1.3.1 The Gradient1.3.2 CurvilinearCoordinates(a) Cylindrical(b) Spherical1.3.3 The Line Integral1.4 FLUX AND DIVERGENCE1.4.1 Flux1.4.2 Divergence1.4.3 CurvilinearCoordinates(a) Cylindrical Coordinates(b) SphericalCoordinates1.4.4 The Divergence Theorem1.5 THE CURL AND STOKES' THEOREM1.5.1 Curl1.5.2 The Curlfor CurvilinearCoordinates(a) CylindricalCoordinates(b) Spherical Coordinates1.5.3 Stokes' Theorem1.5.4 Some Useful Vector Relations(a) The Curl of the Gradient is ZeroIV x(Vf) O](b) The Divergence of the Curl is Zero[V - (V X A) 0PROBLEMSChapter 2-THE ELECTRIC FIELD2.1 ELECTRIC CHARGE2.1.1 Chargingby Contact2.1.2 ElectrostaticInduction2.1.3 Faraday's"Ice-Pail"Experiment2.2 THE COULOMB FORCE LAWBETWEEN STATIONARY CHARGES2.2.1 Coulomb's Lawix

xContents2.2.2 Units552.2.3 The Electric Field562.2.4 Superposition572.3 CHARGE DISTRIBUTIONS592.3.1 Line, Surface, and Volume Charge Dis tributions602.3.2 The Electric Field Due to a Charge Dis tribution632.3.3 Field Due to an Infinitely Long LineCharge642.3.4 Field Due to Infinite Sheets of SurfaceCharge65(a) Single Sheet65(b) ParallelSheets of Opposite Sign67(c) Uniformly Charged Volume682.3.5 Hoops of Line Charge69(a) Single Hoop69(b) Disk of Surface Charge69(c) Hollow Cylinder of Surface Charge71(d) Cylinder of Volume Charge722.4 GA USS'S LAW722.4.1 Propertiesof the Vector Distance Betweentwo Points rQp72(a) rQp72(b) Gradientof the Reciprocal Distance,V(1/rQp)73(c) Laplacianof the Reciprocal Distance732.4.2 Gauss's Law In Integral Form74(a) Point Charge Inside or Outside aClosed Volume74(b) ChargeDistributions752.4.3 SphericalSymmetry76(a) Surface Charge76(b) Volume ChargeDistribution792.4.4 CylindricalSymmetry80(a) Hollow Cylinder of Surface Charge80(b) Cylinderof Volume Charge822.4.5 Gauss'sLaw and the Divergence Theorem822.4.6 ElectricField DiscontinuityAcross a Sheetof Surface Charge832.5 THE ELECTRIC POTENTIAL842.5.1 Work Required to Move a Point Charge842.5.2 The ElectricField and Stokes' Theorem852.5.3 The Potentialand the Electric Field862.5.4 FiniteLength Line Charge88902.5.5 ChargedSpheres(a) Surface Charge90(b) Volume Charge91(c) Two Spheres92

Contentsxi2.5.6 Poisson'sand Laplace's EquationsTHE METHOD OF IMAGES WITHLINE CHARGES AND CYLINDERS2.6.1 Two ParallelLine Charges2.6.2 The Method of Images(a) GeneralProperties(b) Line Charge Near a ConductingPlane2.6.3 Line Chargeand Cylinder2.6.4 Two Wire Line(a) Image Charges(b) Force of Attraction(c) CapacitancePer Unit Length2.7 THE METHOD OF IMAGES WITHPOINT CHARGES AND SPHERES2.7.1 Point Chargeand a Grounded Sphere2.7.2 Point ChargeNear a GroundedPlane2.7.3 Sphere With Constant Charge2.7.4 Constant Voltage SpherePROBLEMS96979999100101Chapter 3-POLARIZATION AND CONDUCTION3.1 POLARIZATION3.1.1 The Electric Dipole3.1.2 PolarizationCharge3.1.3 The Displacement Field3.1.4 LinearDielectrics(a) Polarizability(b) The Local ElectricField3.1.5 Spontaneous Polarization(a) Ferro-electrics(b) Electrets3.2 CONDUCTION3.2.1 Conservationof Charge3.2.2 ChargedGas ConductionModels(a) Governing Equations(b) Drift-DiffusionConduction(c) Ohm's Law(d) Superconductors3.3 FIELD BOUNDARY CONDITIONS3.3.1 Tangential Component of E3.3.2 Normal Component of D3.3.3 Point ChargeAbove a DielectricBoundary3.3.4 Normal Componentof P and eoE3.3.5 Normal Component of J3.4 RESISTANCE3.4.1 Resistance Between Two Electrodes3.4.2 70932.693939696103103106109110110