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SCHAUM'S Tee portect mad kee Carta price! CO a Coa aa el et ry ae ew baw hy ow pe eng Fl pater pater dew be sdb le bp 1 fe ew es SCHAUM'S OUTLINE OF THEORY AND PROBLEMS ELECTROMAGNETICS Second Edition SCHAUM’S OUTLINE SERIES McGRAW-HILL New York Son Prenciaco Washington, D.C. Auckland Bogoté Coracas Lisbon London Madrid Merico Cty Milan Montreat New Delt San Juan Singapore Sydney Tokyo Toronto JOSEPH A, EDMINISTER is currently Director of Corporate Relations for the College of Engineering at Cornell University. In 1984 he held an TEEE Congressional Fellowship in the office of Congressman Dennis E. Eckast (D-H), He received BEE, MSE and JD degrecs from the University of Akron. He served as professor of electrical engineering, scting department head of electrical engineering, assistant dean and acting dean of engincering, all ut the University of Akron. He is an attorney in the state of Ohio and a registered patent attorney. He taught electric circuit analysis and electromagnetic theory throughout his academic career. He is a Professor Emeritus of Electrical Engineering from The Univesity of Akron, Argon Binet opyrightd ©1905 by McGraw il Ine ane Math, le ‘Schees Outine of Theory and Prblens ot ELECTRONAGNETICS ‘copy © 1955, 199 by The MeGemm Alogi nl grr Pathe Une Sikes of Arron cet sprite user he Copy Act 19a pt of pibbencn iy bere of dtd ay fray em esc aa ‘resin yen, wit Ge ir writen pemison of pb 567189101112 1314 15 16171419 20 PRS PRS ISBN 0-07-021234~1 Formerly published under ISBN 0-07-018993-5) ‘Spomoting Eton: Davi Bekwith Proce Superior AI Riker ing Supervue: Paty Andrews Front Mater Eater Mareen Waker nea of Congres Catling in Pebltion Dots inter. Soph Schau cline of ey and puobie of lestomat by Joseph A. Eine —2e Peri (Shats line sens) ected (Sen Ouran TElecomagnesn. 1, Tle, 1. Ue: Tay na pee cedrommgroa. i. Sete. Ocaess i988 percent saa McGraw-Hill -ALieson of THeM Gran HAC Preface ‘The second edition of Schaum's Outline of Electromagnetics offers three new Cchapters—in transmission lines, waveguides, and antennas. These have been included to make the book a more pawerfu 1001 for students and practitioners of electromagnetic field theory. I take pleasure here in thanking my colleagues M. L. Kult and K_F. Lee for their contribution of this valuable material ‘The basie approach of the frst edition has been retained: “As in other Schaum's Outines the emphasis is on how to solve problems. Each chapter ‘consists of an ample set of problems with detailed solutions, and a further set of problems with answers, preceded by 1 simplified outline of the principles and facts needed to understand the problems and their solutions. Throughout the ‘book the mathematics has heen Kept ac simple as possble, and an abstract approach has heen avoided. Conezete examples are liberally used and numerous ‘graphs and sketches are given, [have found in many years of teaching that the solution of most problems begins witha carefully drawn sketch.” Once again i is to my students—my former students—that I wish to dedicate this book. SJosern A. Enwunisren i Contents VECTOR ANALYSIS snsssennnininninnnnesiennnnsnanns LI Introduction 1.2 Veor Nolan 13 Vector Algebra 14 Cooedinate Systeme 1.5 Differential Volume, Surfoce, and Line Elements i COULOMB FORCES AND ELECTRIC FIELD INTENSITY 21 Coulomb's Law 22 Electric Field Imensity 23 Charge Disoibuions 244 Standard Charge Configurations ELECTRIC FLUX AND GAUSS’ LAW sn ‘3 Net Change in «Region 32 Fleer Flue and Fux Density 33 Gaus Liw 34 Relation Between Flux Density and Electric Field Intensity 155 Specat Gaui Surteces, DIVERGENCE AND THE DIVERGENCE THEOREM 41 Divergence 42 Divergence in Caron Coordinates 4.3 Divergence f D {$4 The Del Oper 4.5 The Divergence Tharer, THE ELECTROSTATIC FIELD: WORK, ENERGY, AND POTENTIAL .. 5.1 Work Done in Moving s Polat Charge 52 Conservative Property of the Electrostatic Feld 5.3 Elevrie Potential Between Two Points 5.4 Potential of» Point Charge $5 Potential of » Charge Distibution §.6 Gradient” 8.7 Rete tionship Between B and V8.8 Bneray in Static Electric Fields Cuopter (CURRENT, CURRENT DENSITY, AND CONDUCTORS .. G1 fottdvcion 62 Charges in Motion 6.3 Convention Curremt Density 3 64 Conducion Current Density 6.5 Conducinity 66 Corent F 67 Resistance RG Carseat sheet Dewsity K 6.9 Contimity of Corot (6:10 Conductor-Dielecee Bowndary Conditions CAPACITANCE AND DIELECTRIC MATERIALS. 17 Polarization P snd Relative Permitunty <, 7.2 Capacitance 7-3 Maltipl Dieleciric Capacitors. 7.4 Energy Stored in » Capacitor 7-5 Fixed Voluge D and E. 7.6 Fixsd Charge D and E 7-7 Boundary Conditions atthe Smtectace of Two Diclecuis, LAPLACE’S EQUATION .. 2 Intodacton £2 Poisons Equation and Laplace's Equation 6 Exp Forms of Laplace’ Equation 84 Uniqueness Theorem 8.5 Mean Value snd Masioum Value ‘Theorems 6 Cartesian Solution in One Vatable 182 Cartesian Product Solution 8.8 Cyfincriesl Product Solution 8.9 Spherical Product Slaton 4 Quapter 9 CONTENTS. AMPERE’S LAW AND THE MAGNETIC FIELD... ‘94 Introduction 9:2 Biot-Suvart Law 9.3 Ampére's Law 4 Curl 9.5 Rela. tionship of # and H_ 9.6 Magnetic Fax Density B 9.7 Vector Magnetic Polen al A 918 Stokes! Theorem mnumnnnene 138 FORCES AND TORQUES IN MAGNETIC FIELDS .. - 10.1 Magnetic Force on Parties 10.2 Electric and Magnetic Fields Com bined 10.3 Magnetic Force on a Current Element 106 Work aad. Power 105 Torque 10.6 Magnetic Moment of Planar Coit INDUCTANCE AND MAGNETIC CIRCUITS. Wt tnductanee 112 Standaed Conductor Conigusions 113 Faraday’s Law andSelf-lnductance 11.4 Internal Inductance 15 Mutualfadctance 11.6 Mag. fei: Greate 11.7 The H=ft Curve 1L# Ampere’ Law for Magnetic Circuits 11.9 Coreewith Ait aps 1110 Mulple Coil 1.11 Parallel Magnetic Gicaits Chapter 12 DISPLACEMENT CURRENT AND INDUCED EME 121 Displacement Curent 122 Ratio of J. 10 Jp 12:3 Formays Law and Lenz Law 124 Conductors in Motion Through Tine-Independent Fick 1255 Conductors in Motion Theough Time Dependent Fels MAXWELL'S EQUATIONS AND BOUNDARY CONDITIONS eves 205 B31 tawodacion 132 Boundary Relations for MogneticFielés 13:3 Cureeat ‘Shect at the Boundary 134 Summary of Boundary Conditions 13.5 Maxwell's Equations Chapter 14 ELECTROMAGNETIC WAVES . 141 Iotroducion M2 Www Equations 163 Solutions in Carteran Coord. inates "144 Solutions for Partially Conducting Media 5 Sobitions fr Perfect Dielectrics 1.6 Solutions for Good Conductors; Skin Depth 147 laterace Conditions at Normal Incidence 14.8 Oblique Tacidence and Sael's Laws 14.9 Perpendicslar Polarization 18.10 Parallel Polerration 14.11 Standing Waves HLI2 Power and the Poynting Vector TRANSMISSION LINES ss senennannentemnnese BST 1511 Invoduction 15.2 Distribvied Parameters 15.3 loeremental Model, VOI tages and Canrents 154 Sinusoidal Steady-State Excitation 15.5 The Sath Chart 15.6 Impedance Matching 15.7 Single-Stub Matching 15.8 Dooble- Stub Matching 15.9 impedence Measurement 15.10 Transients in Loses. ines Quapter 16 WAVEGUIDES sneer 16.1 Introduction 162 Transverse and Axil Fields 16. TE and TM Modes, ‘Wine Impedances 16.4 Determination of the Axial Fields 16.5 Mode Cutoff Frequencies 166 Dominant Mode 16.7 Power Transmitted ina Losess Wineguide 164 Power Dasipaion ia a Lossy Waveguide (Clupter 17 ANTENNAS wssnmnnenemnsnnies 2B 171 Jntrodocton 17-2 Cormene Source and the Band H Fields 17-3 Eletic (Hertzian) Dipole Antenna 17.4 Antenna Parameters 17.5 Small Ciesla ‘Loop Antenna’ 17.6 Finite Length Dipole 17.7 Monopole Artemia 17.8 Self ‘and: Mtual Impedances 17.9'The Receiving Antenta 17-10 Linear Arrays TT Reflector Appendix A... Appendix B .seeceveeeereseeeree Chapter 1 Vector Analysis 1L1_ INTRODUCTION ‘Vectors ate introduced in physics and mathematics courses primarily in the cartesian coordinate system, Although cylindrical coordinates may be found in calculus texts, the spherical coordinate ‘stem is eldom presented. All three coordinate systems must be used in electromagnetics. AS the notation, both for the vectors and the coordinate systems, difers from one text to another, & ‘thorough understanding ofthe notation employed herein is essential for setting up the problems and ‘obtaining solutions. 1.2. VECTOR NOTATION In order to distinguish vectors (quantities having magnitude und direction) from scalars (quantities having magnitude only) the vectors are denoted by boliface symbols. A unit vector, one ‘of absolute value (or magnitude or length) 1, will in ehis book always be indicated by a boldface, lowercase a. The unit vector in the direction of @ vector A is determined by dividing A by its absolute value: A wa oF By use ofthe unit vectors te,» ay along the , y, and z axes of « cartesian coordinate system, an anbitrary vector ean be written i component form: Re Ag tA tA tm terms of components, the absolute value of a vector is defined by Win A= VATS AEA 1.3 VECTOR ALGEBRA 1. Vectors may be added and subtracted, AEB=(A.0,+ 4,9, 44,0.) #(B.a,+ Bye, + Bm = Art Beda +A, $B, + (As Bee, 2. The associative, distributive, and commutative laws apply. A+ BHO) (A+B)+E HASB)AKA+EB (hy HRJA= RATE A At+B=B+A 3. The dot product of two vectors is, by defsition, AsB=ABcos6 (read “A dot B”) where 6 is the smaller angle between A and B. In Example 1 its shown that ASBHAB+A;B ABs which gives, in partiolar, l= VATA. 1 2 VECTOR ANALYSIS [onar.1 EXAMPLE 1. The dot product obeys the ditbutive and scalar multiplication laws AMBtOQ=ASBHACC ‘This being the cate, ABR (AR HARTA, 1+ 8,4, + 8) #ABslae2 0+ AByla,- 4) + ADA) FADie ra) 4- 0 ABO) However, 9.18 =8°—aca,—1 Because the cos@ in the dot product ié unity when the angle is zo, And when 8= 97, cos Os 7070; hence al ther Jot products of the uit vectors are zero. Ths ABAAB, IAB, +48, 4, The cross product of two vectors i, by definition, AXB=(ABsin@)a, (read “A cross B") ‘where 0 is the smaller angle between A and B, snd a, isa unit vector normal to the plane determined by A and B when they ate drawn from a common point. There are to normals to the plane, so further specification is needed. The normal selected is the one in the irection of advance of a righthand sexew when A is turned toward Bt (Fig. 1-1). Because of this direction requirement, the commutative law does not apply 10 the cross product; instead, ax! BXA Fett Expanding the cross product in component form, AXB= (AG +A,8, + Ait) x (Bite + Bye, + 8,8.) (A,B, ~ A,B, + (A.B, ~ A,B. +(A.B,— A,B), which is conveniently expressed as a determinant: eae AL A, A, BBB AXB: CHAP. 1} VECTOR ANALYSIS 3 EXAMPLE2 Given A=24,445,-34, and 8: ASB = AVL) + X= +(-KO)= = 34, = 38,6, 14 COORDINATE SYSTEMS ‘A problem wich has cylindrical or spherical symmetry could be expressed and solved in the familiar cartesian coordinate system. However, the solution would fail to show the symmetry and in most cases would be needlessly complex. Therefore, throughout this book, in addition to the ‘cartesian coordinate system, the circular cylindrical and he spherical coordinate systems will be Used, All three will be examined together in order to ilustrate the similarities and the differences. "A point Pi described by three coordinates, in cartesian (x,y, 2), in circular eylindrical (, 2), and in spherical (¢, 0, @), a6 shown in Fig. 1-2. The order of specifying the coonfinates is important and should be carefully folowed. The angle isthe same angle in both the cylindrical and spherical sysiems, But, in the order of the coordinates, @ appears in the second postion in cylindrical, (Fz), and the thint postion in spherical, (F, 0, 6). The same symbol, , & used in {In eylindrical coordinates r measures the spherical system r measures the distance from the origin to the point. It should be elear from the context ofthe problema which ris intended. er cuedin Orinda er Sebetea Pett A point is also defined by the intersection of three orthogonal surfaces, a8 shown in Fig. 1-3. Ta ‘cartesian coordinates the surfuces are the infinite planes x= const, y~const., and z= const. In cylindrical coordinates, z=eonst. is the same infinite plane as in cartesian; = ‘const. is half plane with its edge along the z axis; r=const. is a eight circular cylinder. These three surfeces are orthogonal and their imterscetion locates point P. In spherical coordinates, $= ‘const. is the same Tall plane as in cylindrical; r= const. is a sphere with its center’ at the origin; @=const. is a right cicular cone whose axis is the z axis and whose vertex is at the loriia, Note that @ is imited to the range 0-= 0. Figure 1-4 stows the thiee unit vectors at pot P. a dae cartesian syste the wait vectors have {fixed directions, independent of the location of P. ‘This isnot tue forthe other two systems (except in the case of a,). Each unit vector is normal to its caordinate surface and is in the direction in which the coordinate increases. Notice that all these systems are right-handed: a Xaaa a kaye, aXe