Electromagnetic Field Theory - A Problem-Solving Approach ...

1 Cartesian Cylindrical Spherical x = r cosc/ S r sin 6 cos 4 = r sin4 S r sin 6 sin4 z = z = rcos6 y ix= Cos Oi-sin4k 4 = sin Ocos ki,+cos 6 cos 4ie -sin Ois iY =sin 0 i,+ Cos 01i = sin 0 sin 6i, + Cos 6 sin ie = S il41+C541 +cos 6 io i1 = cos Oi,-sin Ois Cylindrical Cartesian Spherical =~V Ix+ y7 Sr sin 6 tan~1y/x = z = rcos6 = cos (k ,, +sin i, = sin Oi,+cos 6ie i4 . = -sin ki,,+Cos 4i, = i4 = i, = cos i, -sin iO Spherical Cartesian Cylindrical If,-+z"r I4x +y +z _1 z -1 z0 = Cos = cos = cot-' x/y i, = sinG cos kix+sin 6 sin ci, = sin i, +Cos Oi, +cos i. i, = cos 6 cos oi,+cos 6 sin oi, = Cos Oi,-sin Oi, -sin Oi.)

2 I = -sin 46i, +Cos 0i, = i4, Geometric relations between coordinates and unit vectors for Cartesian, cylir drical, and spherical coordinate systems. CartesianCoordinates(x, y, z) Vf = Ofi.+Ofi,+ Ofi. ax ay Oz+-A=,++-iV-aA,, aA, aA, 2 ax ay az aA aA)VxA ay_ a (aA,(-_)+i=i. (LAI )+ _ 8A\ -(ay az ) az ) ay 2fV2f+!L+f+aOx2 j Z CylindricalCoordinates (r, 4, z) Of. 1 Of. OfVf= r+ i,+ iz +A -A= Ia(rA,.)+ -. r Or r ao az VxA=i -+ixaz Or + OA r a4 0/f a Of\ 1a2f a2f V'f= r + +42 r Or On) r O SphericalCoordinates (r, 0, 4,) Vf=i,.+ afi+Or r aO r sin 0 a4 V A = (r2A,)+ 1 (sin OA.) 1 oA* r2 ar r sin 0 aI r sin 0 a4 VxA=i1 a(sin OAs) aA.

3 'r sineL 80 a4 I rIAM, a(rA) 1 [a(rA#) OA,. r sin{ 0r] rL Or aeJ 2fV f = r + r s n sin0 ) +Of I a rf ar' Or. r--s i n aG ai r sin 04, 0 MAXWELL'S EQUATIONS Integral Differential Boundary Conditions Faraday's Law d C BE'-dl=--B-dS VxE=-nx(E'-E')= at Ampere's Law with Maxwell's Displacement Current Correction H-dl= Jf,-dS VXH=Jf+aD nX(H 2-H 1 )=Kf + D-dS~it-s Gauss's Law V -D=p n -(D2-D 1)= o-D'-dS=t pfdV B-dS=0 V-B=0 Conservation of Charge J,-dS+ pfdV=O V-J,+ =0 n -(J2-J)+" 0 sVd at a Usual Linear Constitutive Laws D=eE B= H Jf= o-(E+vX B)= a-E' [Ohm's law for moving media with velocity v] PHYSICAL CONSTANTS Constant Symbol Value units Speed of light in vacuum c x 108= 3 x 108 m/sec Elementary electron charge e x 10~'9 coul Electron rest mass M, x 10 3 kg Electron charge to mass ratio e x 10" coul/kgM, Proton rest mass I, x 10-27 kg Boltzmann constant k x 10-23 joule/*K Gravitation constant G x 10-" nt-m2/(kg) 2 Acceleration of gravity g m/(sec)}

4 2 10 * Permittivity of free space 60 x 10~2~36r farad/m Permeability of free space A0 4r X 10 henry/m Planck's constant h x 10-34 joule-sec Impedance of free space i1o 4 ohms Avogadro's number Ar x 1023 atoms/mole I Electromagnetic Field Theory : a problem solving Approach MARKUS ZAHN Massachusetts Institute of Technology KRIEGER PUBLISHING COMPANY Malabar, Florida Original Edition 1979 Reprint Edition 1987 Reprint Edition 2003 w/corrections Printed and Published by KRIEGER PUBLISHING COMPANY KRIEGER DRIVE MALABAR, FLORIDA 32950 Copyright 0 1979, John Wiley & Sons, Inc. Transferred to Author Reprinted by Arrangement All rights reserved.

5 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 permission in writing from the publisher. No liabilityisassumedwith respectto the use oftheinformation containedherein. Printed in the United States ofAmerica. FROM A DECLARATION OF PRINCIPLES JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS: This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold with the understanding that the publisher is not engaged in rendering legal, accounting, or other professional service.

6 If legal advice or other expert assistance is required, the services of a competent professional person should be sought. Library of Congress Cataloging-in-Publication Data Zahn, Markus, 1946 Electromagnetic Field Theory : a problem solving Approach / Markus 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. 2003 '1--dc2l 2003047418 10 9 8 7 6 5 4 3 2 to my parents V Preface PREFACE Electromagnetic Field Theory is often the least popular course in the electrical engineering curriculum.

7 Heavy reli ance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked examples. To keep the subject exciting, many of these problems are based on physical pro cesses, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering course starting from the Coulomb-Lorentz force law on a point charge. The Theory is extended by the continuous superposition of solutions from previously developed simpler problems leading to the general integral and differential Field laws.

8 Often the same problem is solved by different methods so that the advantages and limita tions of each Approach becomes clear. Sample problems and their solutions are presented for each new concept with great emphasis placed on classical models of such physical phenomena as polarization, conduction, and magnetization. A large variety of related problems that reinforce the text material are included at the end of each chapter for exercise and homework. It is expected that students have had elementary courses in calculus that allow them to easily differentiate and integrate simple functions. The text tries to keep the mathematical development rigorous but simple by typically describing systems with linear, constant coefficient differential and difference equations.

9 The text is essentially subdivided into three main subject areas: (1) charges as the source of the electric Field coupled to polarizable and conducting media with negligible magnetic Field ; (2) currents as the source of the magnetic Field coupled to magnetizable media with Electromagnetic induction generat ing an electric Field ; and (3) electrodynamics where the electric and magnetic fields are of equal importance resulting in radi ating waves. Wherever possible, electrodynamic solutions are examined in various limits to illustrate the appropriateness of the previously developed quasi-static circuit Theory approxi mations.

10 Many of my students and graduate teaching assistants have helped in checking the text and exercise solutions and have assisted in preparing some of the Field plots. Markus Zahn Notes to the Student Vii and Instructor A NOTE TO THE STUDENT In this text I have tried to make it as simple as possible for an interested student to learn the difficult subject of electromag netic Field Theory by presenting many worked examples emphasizing physical processes, devices, and models. The problems at the back of each chapter are grouped by chapter sections and extend the text material. To avoid tedium, most integrals needed for problem solution are supplied as hints.