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Numerical Techniques in Electromagnetics Second Edition Matthew N. O. Sadiku, Ph.D. Numerical Techniques in Electromagnetics Second Edition Boca Raton London New York Washington, D.C. CRC Press

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NumericalTechniques inElectromagneticsSecond EditionMatthew N. O. Sadiku, Ph.D.NumericalTechniques inElectromagneticsSecond EditionBoca Raton London New York Washington, D.C.CRC Press This book contains information obtained from authentic and highly regarded sources. Reprinted materialis quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonableefforts have been made to publish reliable data and information, but the author and the publisher cannotassume responsibility for the validity of all materials or for the consequences of their use.Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, microlming, and recording, or by any information storage orretrieval system, without prior permission in writing from the publisher.The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, forcreating new works, or for resale. Specic permission must be obtained in writing from CRC Press LLCfor such copying.Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and areused only for identication and explanation, without intent to infringe. 2001 by CRC Press LLCNo claim to original U.S. Government worksInternational Standard Book Number 0-8493-1395-3Library of Congress Card Number 00-026823Printed in the United States of America 1 2 3 4 5 6 7 8 9 0Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Sadiku, Matthew N. O.Numerical techniques in electromagnetics/Matthew N.O. Sadiku.[2nd ed.].p. cm.Includes bibliographical references and index.ISBN 0-8493-1395-3 (alk. paper)1. Electromagnetism. 2. Numerical analysis. I. Title.QC760 .S242000537 .01 515dc21 00-026823 CIP PrefaceTheartofcomputationofelectromagnetic(EM)problemshasgrownexponentiallyforthreedecadesduetotheavailabilityofpowerfulcomputerresources.Inspiteofthis,theEMcommunityhassufferedwithoutasuitabletextoncomputationaltechniquescommonlyusedinsolvingEM-relatedproblems.Althoughtherehavebeenmonographsononeparticulartechniqueortheother,themonographsarewrittenfortheexpertsratherthanstudents.Onlyafewtextscoverthemajortechniquesanddothatinamannersuitableforclassroomuse.Itseemsexpertsinthisareaarefamiliarwithoneorfewtechniquesandnotmanyexpertsseemtobefamiliarwithallthecommontechniques.Thistextattemptstollthegap.Thetextisintendedforseniorsorgraduatestudentsandmaybeusedforaone-semesterortwo-semestercourse.ThemainrequirementsforstudentstakingacoursebasedonthistextareintroductoryEMcoursesandaknowledgeofahigh-levelcomputerlanguage,preferablyFORTRANorC.SoftwarepackagessuchasMatlabandMathcadmaybehelpfultools.Althoughfamiliaritywithlinearalgebraandnumericalanalysisisuseful,itisnotrequired.Inwritingthisbook,threemajorobjectiveswereborneinmind.First,thebookisintendedtoteachstudentshowtopose,numericallyanalyze,andsolveEMproblems.Second,itisdesignedtogivethemtheabilitytoexpandtheirproblemsolvingskillsusingavarietyofavailablenumericalmethods.Third,itismeanttopreparegraduatestudentsforresearchinEM.Theaimthroughouthasbeensimplicityofpresentationsothatthetextcanbeusefulforbothteachingandself-study.Instrivingaftersimplicity,however,thereaderisreferredtothereferencesformoreinformation.Towardtheendofeachchapter,thetechniquescoveredinthechapterareappliedtoreallifeproblems.SincetheapplicationofthetechniqueisasvastasEMandauthorsexperienceislimited,thechoiceofapplicationisselective.Chapter1coverssomefundamentalconceptsinEM.Chapter2isintendedtoputnumericalmethodsinaproperperspective.Analyticalmethodssuchasseparationofvariablesandseriesexpansionarecovered.Chapter3discussesthenitediffer-encemethodsandbeginswiththederivationofdifferenceequationfromapartialdifferentialequation(PDE)usingforward,backward,andcentraldifferences.Thenite-differencetime-domain(FDTD)techniqueinvolvingYeesalgorithmispre- vsentedandappliedtoscatteringproblems.Numericalintegrationiscoveredusingtrapezoidal,Simpsons,Newton-Cotesrules,andGaussianquadratures.Chapter4onvariationalmethodsservesasapreparatorygroundforthenexttwomajortopics:momentmethodsandniteelementmethods.Basicconceptssuchasinnerproduct,self-adjointoperator,functionals,andEulerequationarecovered.Chapter5onmomentmethodsfocusesonthesolutionofintegralequations.Chap-ter6onniteelementmethodcoversthebasicstepsinvolvedinusingtheniteelementmethod.SolutionsofLaplaces,Poissons,andwaveequationsusingtheniteelementmethodarecovered.Chapter7isdevotedtotransmission-linematrixormodeling(TLM).Themethodisappliedtodiffusionandscatteringproblems.Chapter8isonMonteCarlomethods,whileChapter9isonthemethodoflines.Sincethepublicationoftherstedition,therehasbeenanincreasedawarenessandutilizationofnumericaltechniques.ManygraduatecurriculanowincludecoursesinnumericalanalysisofEMproblems.However,notmuchhaschangedincompu-tationalelectromagnetics.AmajornoticeablechangeisintheFDTDmethod.Themethodseemstohaveattractedmuchattentionandmanyimprovementsarebeingmadetothestandardalgorithm.Thiseditionaddsthenoticeablechangeinincorpo-ratingabsorbingboundaryconditionsinFDTD,FEM,andTLM.Chapter9isanewchapteronthemethodoflines.AcknowledgementsIamgreatlyindebtedtoTempleUniversityforgrantingmeasabbaticalinFall1998duringwhichIwasabletodomostoftherevision.Ispecicallywouldliketothankmydean,Dr.KeyaSadeghipour,andmychairman,Dr.JohnHelferty,fortheirsupport.SpecialthanksareduetoRaymondGarciaofGeorgiaTechforwritingAppendicesCandDinC++.IamdeeplygratefultoDr.ArthurD.SnideroftheUniversityofSouthFloridaandMohammadR.ZunoubiofMississippiStateUniversityfortakingthetimetosendmethelistoferrorsintherstedition.IthankDr.ReinholdPreglaforhelpinginclarifyingconceptsinChapter9onthemethodoflines.Iexpressmydeepestgratitudetomywife,Chris,andourdaughters,AnnandJoyce,fortheirpatience,sacrices,andprayers.ANotetoStudentsBeforeyouembarkonwritingyourowncomputerprogramorusingtheonesinthistext,youshouldtrytounderstandallrelevanttheoreticalbackgrounds.Acomputerisnomorethanatoolusedintheanalysisofaprogram.Forthisreason,youshouldbeasclearaspossiblewhatthemachineisreallybeingaskedtodobeforesettingitoffonseveralhoursofexpensivecomputations.IthasbeenwellsaidbyA.C.DoylethatItisacapitalmistaketotheorizebeforeyouhavealltheevidence.Itbiasesthejudgment.Therefore,youshouldnevertrusttheresultsofanumericalcomputationunlesstheyarevalidated,atleastinpart.Youvalidatetheresultsbycomparingthemwiththoseobtainedbypreviousinvestigatorsorwithsimilarresultsobtainedusingadifferentapproachwhichmaybeanalyticalornumerical.Forthisreason,itisadvisablethatyoubecomefamiliarwithasmanynumericaltechniquesaspossible.Thereferencesprovidedattheendofeachchapterarebynomeansexhaustivebutaremeanttoserveasthestartingpointforfurtherreading.Contents1 Fundamental Concepts1.1 Introduction 1.2 Review of Electromagnetic Theory 1.2.1 Electrostatic Fields 1.2.2 Magnetostatic Fields 1.2.3 Time-varying Fields 1.2.4 Boundary Conditions 1.2.5 Wave Equations 1.2.6 Time-varying Potentials 1.2.7 Time-harmonic Fields1.3 Classication of EM Problems1.3.1 Classication of Solution Regions 1.3.2 Classication of Differential Equations 1.4 Some Important Theorems 1.4.1 Superposition Principle 1.4.2 Uniqueness Theorem References Problems 2 Analytical Methods2.1 Introduction 2.2 Separation of Variables 2.3 Separation of Variables in Rectangular Coordinates 2.3.1 Laplaces Equations 2.3.2 Wave Equation 2.4 Separation of Variables in Cylindrical Coordinates 2.4.1 Laplaces Equation 2.4.2 Wave Equation 2.5 Separation of Variables in Spherical Coordinates2.5.1 Laplaces Equation 2.5.2 Wave Equation 2.6 Some Useful Orthogonal Functions 2.7 Series Expansion 2.7.1 Poissons Equation in a Cube 2.7.2 Poissons Equation in a Cylinder 2.7.3 Strip Transmission Line2.8 Practical Applications 2.8.1 Scattering by Dielectric Sphere2.8.2 Scattering Cross Sections 2.9 Attenuation Due to Raindrops2.10 Concluding Remarks References Problems3 Finite Difference Methods3.1 Introduction3.2 Finite Difference Schemes 3.3 Finite Differencing of Parabolic PDEs 3.4 Finite Differencing of Hyperbolic PDEs 3.5 Finite Differencing of Elliptic PDEs3.5.1 Band Matrix Method 3.5.2 Iterative Methods 3.6 Accuracy and Stability of FD Solutions 3.7 Practical Applications I Guided Structures3.7.1 Transmission Lines 3.7.2 Waveguides3.8 Practical Applications II Wave Scattering (FDTD) 3.8.1 Yees Finite Difference Algorithm3.8.2 Accuracy and Stability3.8.3 Lattice Truncation Conditions3.8.4 Initial Fields 3.8.5 Programming Aspects3.9 Absorbing Boundary Conditions for FDTD 3.10 Finite Differencing for Nonrectangular Systems3.10.1 Cylindrical Coordinates 3.10.2 Spherical Coordinates 3.11 Numerical Integration 3.11.1 Eulers Rule3.11.2 Trapezoidal Rule 3.11.3 Simpsons Rule 3.11.4 Newton-Cotes Rules3.11.5 Gaussian Rules 3.11.6 Multiple Integration 3.12 Concluding Remarks References Problems 4 Variational Methods4.1 Introduction 4.2 Operators in Linear Spaces 4.3 Calculus of Variations4.4 Construction of Functionals from PDEs 4.5 Rayleigh-Ritz Method 4.6 Weighted Residual Method 4.6.1 Collocation Method4.6.2 Subdomain Method4.6.3 Galerkin Method4.6.4 Least Squares Method4.7 Eigenvalue Problems 4.8 Practical Applications4.9 Concluding RemarksReferencesProblems 5 Moment Methods5.1 Introduction5.2 Integral Equations5.2.1 Classication of Integral Equations5.2.2 Connection Between Differential and Integral Equations 5.3 Greens Functions 5.3.1 For Free Space 5.3.2 For Domain with Conducting Boundaries 5.4 Applications I Quasi-Static Problems5.5 Applications II Scattering Problems 5.5.1 Scattering by Conducting Cylinder 5.5.2 Scattering by an Arbitrary Array of Parallel Wires5.6 Applications III Radiation Problems5.6.1 Hallens Integral Equation 5.6.2 Pocklingtons Integral Equation 5.6.3 Expansion and Weighting Functions5.7 Applications IV EM Absorption in the Human Body 5.7.1 Derivation of Integral Equations 5.7.2 Transformation to Matrix Equation (Discretization) 5.7.3 Evaluation of Matrix Elements 5.7.4 Solution of the Matrix Equation 5.8 Concluding Remarks References Problems 6 Finite Element Method6.1 Introduction 6.2 Solution of Laplaces Equation 6.2.1 Finite Element Discretization6.2.2 Element Governing Equations 6.2.3 Assembling of All Elements 6.2.4 Solving the Resulting Equations 6.3 Solution of Poissons Equation 6.3.1 Deriving Element-governing Equations 6.3.2 Solving the Resulting Equations 6.4 Solution of the Wave Equation 6.5 Automatic Mesh Generation I Rectangular Domains 6.6 Automatic Mesh Generation II Arbitrary Domains 6.6.1 Denition of Blocks 6.6.2 Subdivision of Each Block 6.6.3 Connection of Individual Blocks 6.7 Bandwidth Reduction 6.8 Higher Order Elements6.8.1 Pascal Triangle6.8.2 Local Coordinates 6.8.3 Shape Functions6.8.4 Fundamental Matrices 6.9 Three-Dimensional Elements 6.10 Finite Element Methods for Exterior Problems 6.10.1 Innite Element Method 6.10.2 Boundary Element Method6.10.3 Absorbing Boundary Conditions 6.11 Concluding RemarksReferencesProblems7 Transmission-line-matrix Method7.1 Introduction 7.2 Transmission-line Equations 7.3 Solution of Diffusion Equation 7.4 Solution of Wave Equations 7.4.1 Equivalence Between Network and Field Parameters 7.4.2 Dispersion Relation of Propagation Velocity 7.4.3 Scattering Matrix 7.4.4 Boundary Representation 7.4.5 Computation of Fields and Frequency Response 7.4.6 Output Response and Accuracy of Results 7.5 Inhomogeneous and Lossy Media in TLM 7.5.1 General Two-Dimensional Shunt Node 7.5.2 Scattering Matrix 7.5.3 Representation of Lossy Boundaries 7.6 Three-Dimensional TLM Mesh 7.6.1 Series Nodes 7.6.2 Three-Dimensional Node 7.6.3 Boundary Conditions 7.7 Error Sources and Correction 7.7.1 Truncation Error 7.7.2 Coarseness Error 7.7.3 Velocity Error 7.7.4 Misalignment Error 7.8 Absorbing Boundary Conditions 7.9 Concluding Remarks ReferencesProblems 8 Monte Carlo Methods8.1 Introduction 8.2 Generation of Random Numbers and Variables 8.3 Evaluation of Error8.4 Numerical Integration 8.4.1 Crude Monte Carlo Integration 8.4.2 Monte Carlo Integration with Antithetic Variates8.4.3 Improper Integrals8.5 Solution of Potential Problems 8.5.1 Fixed Random Walk8.5.2 Floating Random Walk 8.5.3 Exodus Method 8.6 Regional Monte Carlo Methods 8.7 Concluding RemarksReferences Problems9 Method of Lines9.1 Introduction 9.2 Solution of Laplaces Equation 9.2.1 Rectangular Coordinates9.2.2 Cylindrical Coordinates 9.3 Solution of Wave Equation 9.3.1 Planar Microstrip Structures 9.3.2 Cylindrical Microstrip Structures 9.4 Time-Domain Solution 9.5 Concluding Remarks References Problems A Vector RelationsA.1 Vector IdentitiesA.2 Vector Theorems A.3 Orthogonal Coordinates B Solving Electromagnetic Problems Using C++B.1 Introduction B.2 A Brief Description of C++ B.3 Object-Orientation B.4 C++ Object-Oriented Language Features B.5 A Final Note References C Numerical Techniques in C++D Solution of Simultaneous EquationsD.1 Elimination Methods D.1.1 Gausss Method D.1.2 Choleskys Method D.2 Iterative Methods D.2.1 Jacobis Method D.2.2 Gauss-Seidel Method D.2.3 Relaxation Method D.2.4 Gradient Methods ....D.3 Matrix Inversion D.4 Eigenvalue Problems D.4.1 Iteration (or Power) MethodD.4.2 Jacobis Method E Answers to Odd-Numbered ProblemsTomyteacherCarlA.VentriceandmyparentsAyisatandSolomonSadikuChapter1FundamentalConceptsScienceknowsnocountrybecauseknowledgebelongstohumanityandisthetorchwhichilluminatestheworld.Scienceisthehighestpersonicationofthenationbecausethatnationwillremaintherstwhichcarriesthefurthesttheworksofthoughtsandintelligence.LouisPasteur1.1IntroductionScientistsandengineersuseseveraltechniquesinsolvingcontinuumoreldprob-lems.Looselyspeaking,thesetechniquescanbeclassiedasexperimental,analyti-cal,ornumerical.Experimentsareexpensive,timeconsuming,sometimeshazardous,andusuallydonotallowmuchexibilityinparametervariation.However,everynu-mericalmethod,asweshallsee,involvesananalyticsimplicationtothepointwhereitiseasytoapplythenumericalmethod.Notwithstandingthisfact,thefollowingmethodsareamongthemostcommonlyusedinelectromagnetics(EM).A.Analyticalmethods(exactsolutions)(1)separationofvariables(2)seriesexpansion(3)conformalmapping(4)integralsolutions,e.g.,LaplaceandFouriertransforms(5)perturbationmethodsB.Numericalmethods(approximatesolutions)(1)nitedifferencemethod(2)methodofweightedresiduals(3)momentmethod(4)niteelementmethod 2001 by CRC PRESS LLC(5)transmission-linemodeling(6)MonteCarlomethod(7)methodoflinesApplicationofthesemethodsisnotlimitedtoEM-relatedproblems;theyndapplicationsinothercontinuumproblemssuchasinuid,heattransfer,andacous-tics[1].Asweshallsee,someofthenumericalmethodsarerelatedandtheyallgenerallygiveapproximatesolutionsofsufcientaccuracyforengineeringpurposes.Sinceourobjectiveistostudythesemethodsindetailinthesubsequentchapters,itmaybeprematuretosaymorethanthisatthispoint.TheneedfornumericalsolutionofelectromagneticproblemsisbestexpressedinthewordsofParisandHurd:Mostproblemsthatcanbesolvedformally(analyti-cally)havebeensolved.1Untilthe1940s,mostEMproblemsweresolvedusingtheclassicalmethodsofseparationofvariablesandintegralequationsolutions.Besidesthefactthatahighdegreeofingenuity,experience,andeffortwererequiredtoapplythosemethods,onlyanarrowrangeofpracticalproblemscouldbeinvestigatedduetothecomplexgeometriesdeningtheproblems.NumericalsolutionofEMproblemsstartedinthemid-1960swiththeavailabilityofmodernhigh-speeddigitalcomputers.Sincethen,considerableefforthasbeenexpendedonsolvingpractical,complexEM-relatedproblemsforwhichclosedformanalyticalsolutionsareeitherintractableordonotexist.Thenumericalapproachhastheadvantageofallowingtheactualworktobecarriedoutbyoperatorswithoutaknowledgeofhighermathematicsorphysics,witharesultingeconomyoflaboronthepartofthehighlytrainedpersonnel.BeforewesetouttostudythevarioustechniquesusedinanalyzingEMproblems,itisexpedienttoremindourselvesofthephysicallawsgoverningEMphenomenaingeneral.ThiswillbedoneinSection1.2.InSection1.3,weshallbeacquaintedwithdifferentwaysEMproblemsarecategorized.TheprincipleofsuperpositionanduniquenesstheoremwillbecoveredinSection1.4.1.2ReviewofElectromagneticTheoryThewholesubjectofEMunfoldsasalogicaldeductionfromeightpostulatedequations,namely,Maxwellsfoureldequationsandfourmedium-dependentequa-tions[2][4].Beforewebrieyreviewtheseequations,itmaybehelpfultostatetwoimportanttheoremscommonlyusedinEM.Thesearethedivergence(orGausss)1BasicElectromagneticTheory,D.T.ParisandF.K.Hurd,McGraw-Hill,NewYork,1969,p.166. 2001 by CRC PRESS LLCtheorem,

SFdS=

vFdv(1.1)andStokesstheorem

LFdl=

SFdS(1.2)PerhapsthebestwaytoreviewEMtheoryisbyusingthefundamentalconceptofelectriccharge.EMtheorycanberegardedasthestudyofeldsproducedbyelectricchargesatrestandinmotion.Electrostaticeldsareusuallyproducedbystaticelectriccharges,whereasmagnetostaticeldsareduetomotionofelectricchargeswithuniformvelocity(directcurrent).Dynamicortime-varyingeldsareusuallyduetoacceleratedchargesortime-varyingcurrents.1.2.1ElectrostaticFieldsThetwofundamentallawsgoverningtheseelectrostaticeldsareGaussslaw,

DdS=

vdv(1.3)whichisadirectconsequenceofCoulombsforcelaw,andthelawdescribingelec-trostaticeldsasconservative,

Edl=0(1.4)InEqs.(1.3)and(1.4),Distheelectricuxdensity(incoulombs/meter2),visthevolumechargedensity(incoulombs/meter3),andEistheelectriceldintensity(involts/meter).TheintegralformofthelawsinEqs.(1.3)and(1.4)canbeexpressedinthedifferentialformbyapplyingEq.(1.1)toEq.(1.3)andEq.(1.2)toEq.(1.4).WeobtainD=v(1.5)andE=0(1.6)ThevectoreldsDandEarerelatedasD=

E(1.7)where

isthedielectricpermittivity(infarads/meter)ofthemedium.IntermsoftheelectricpotentialV(involts),EisexpressedasE=V(1.8) 2001 by CRC PRESS LLCorV=

Edl(1.9)CombiningEqs.(1.5),(1.7),and(1.8)givesPoissonsequation:

V=v(1.10a)or,if

isconstant,2V=v

(1.10b)Whenv=0,Eq.(1.10)becomesLaplacesequation:

V=0(1.11a)orforconstant

2V=0(1.11b)1.2.2MagnetostaticFieldsThebasiclawsofmagnetostaticeldsareAmpereslaw

LHdl=

SJdS(1.12)whichisrelatedtoBiot-Savartlaw,andthelawofconservationofmagneticux(alsocalledGaussslawformagnetostatics)

BdS=0(1.13)whereHisthemagneticeldintensity(inamperes/meter),Jeistheelectriccur-rentdensity(inamperes/meter2),andBisthemagneticuxdensity(inteslaorwebers/meter2).ApplyingEq.(1.2)toEq.(1.12)andEq.(1.1)toEq.(1.13)yieldstheirdifferentialformasH=Je(1.14)andB=0(1.15)ThevectoreldsBandHarerelatedthroughthepermeability(inhenries/meter)ofthemediumasB=H(1.16) 2001 by CRC PRESS LLCAlso,JisrelatedtoEthroughtheconductivity(inmhos/meter)ofthemediumasJ=E(1.17)ThisisusuallyreferredtoaspointformofOhmslaw.IntermsofthemagneticvectorpotentialA(inWb/meter)B=A(1.18)Applyingthevectoridentity(F)=(F)2F(1.19)toEqs.(1.14)and(1.18)andassumingCoulombgaugecondition(A=0)leadstoPoissonsequationformagnetostaticelds:2A=J(1.20)WhenJ=0,Eq.(1.20)becomesLaplacesequation2A=0(1.21)1.2.3Time-varyingFieldsInthiscase,electricandmagneticeldsexistsimultaneously.Equations(1.5)and(1.15)remainthesamewhereasEqs.(1.6)and(1.14)requiresomemodicationfordynamicelds.ModicationofEq.(1.6)isnecessarytoincorporateFaradayslawofinduction,andthatofEq.(1.14)iswarrantedtoallowfordisplacementcurrent.Thetime-varyingEMeldsaregovernedbyphysicallawsexpressedmathematicallyasD=v(1.22a)B=0(1.22b)E=BtJm(1.22c)H=Je+Dt(1.22d)whereJm=Histhemagneticconductivecurrentdensity(involts/squaremeter)andisthemagneticresistivity(inohms/meter).TheseequationsarereferredtoasMaxwellsequationsinthegeneralizedform.Theyarerst-orderlinearcoupleddifferentialequationsrelatingthevectoreldquan- 2001 by CRC PRESS LLCtitiestoeachother.TheequivalentintegralformofEq.(1.22)is

SDdS=

vvdv(1.23a)

SBdS=0(1.23b)

LEdl=

S

Bt+Jm

dS(1.23c)

LHdl=

S

Je+Dt

dS(1.23d)InadditiontothesefourMaxwellsequations,therearefourmedium-dependentequations:D=

E(1.24a)B=H(1.24b)Je=E(1.24c)Jm=M(1.24d)Thesearecalledconstitutiverelationsforthemediuminwhichtheeldsexist.Equa-tions(1.22)and(1.24)formtheeightpostulatedequationsonwhichEMtheoryun-foldsitself.WemustnotethatintheregionwhereMaxwellianeldsexist,theeldsareassumedtobe:(1)singlevalued,(2)bounded,and(3)continuousfunctionsofspaceandtimewithcontinuousderivatives.Itisworthwhiletomentiontwootherfundamentalequationsthatgohand-in-handwithMaxwellsequations.OneistheLorentzforceequationF=Q(E+uB)(1.25)whereFistheforceexperiencedbyaparticlewithchargeQmovingatvelocityuinanEMeld;theLorentzforceequationconstitutesalinkbetweenEMandmechanics.Theotheristhecontinuityequation J=vt(1.26)whichexpressestheconservation(orindestructibility)ofelectriccharge.Theconti-nuityequationisimplicitinMaxwellsequations(seeExample1.2).Equation(1.26)isnotpeculiartoEM.Inuidmechanics,whereJcorrespondswithvelocityandvwithmass,Eq.(1.26)expressesthelawofconservationofmass. 2001 by CRC PRESS LLC1.2.4BoundaryConditionsThematerialmediuminwhichanEMeldexistsisusuallycharacterizedbyitsconstitutiveparameters,

,and.Themediumissaidtobelinearif,

,andareindependentofEandHornonlinearotherwise.Itishomogeneousif,

,andarenotfunctionsofspacevariablesorinhomogeneousotherwise.Itisisotropicif,

,andareindependentofdirection(scalars)oranisotropicotherwise.Figure1.1Interfacebetweentwomedia.Theboundaryconditionsattheinterfaceseparatingtwodifferentmedia1and2,withparameters(1,

1,1)and(2,

2,2)asshowninFig.1.1,areeasilyderivedfromtheintegralformofMaxwellsequations.TheyareE1t=E2tor(E1E2)an12=0(1.27a)H1tH21=Kor(H1H2)an12=K(1.27b)D1nD2n=Sor(D1D2)an12=S(1.27c)B1nB2n=0or(B2B1)an12=0(1.27d)wherean12isaunitnormalvectordirectedfrommedium1tomedium2,subscripts1and2denoteeldsinregions1and2,andsubscriptstandn,respectively,denotetangentandnormalcomponentsoftheelds.Equations(1.27a)and(1.27d)statethatthetangentialcomponentsofEandthenormalcomponentsofBarecontinuousacrosstheboundary.Equation(1.27b)statesthatthetangentialcomponentofHisdiscontinuousbythesurfacecurrentdensityKontheboundary.Equation(1.27c)statesthatthediscontinuityinthenormalcomponentofDisthesameasthesurfacechargedensitysontheboundary.Inpractice,onlytwoofMaxwellsequationsareused(Eqs.(1.22c)and(1.22d))whenamediumissource-free(J=0,v=0),sincetheothertwoareimplied(seeProblem1.3).Also,inpractice,itissufcienttomakethetangentialcomponentsoftheeldssatisfythenecessaryboundaryconditionssincethenormalcomponentsimplicitlysatisfytheircorrespondingboundaryconditions.1.2.5WaveEquationsAsmentionedearlier,Maxwellsequationsarecoupledrst-orderdifferentialequa-tionswhicharedifculttoapplywhensolvingboundary-valueproblems.Thedif-cultyisovercomebydecouplingtherst-orderequations,therebyobtainingthewaveequation,asecond-orderdifferentialequationwhichisusefulforsolvingproblems. 2001 by CRC PRESS LLCToobtainthewaveequationforalinear,isotropic,homogeneous,source-freemedium(v=0,J=0)fromEq.(1.22),wetakethecurlofbothsidesofEq.(1.22c).ThisgivesE=t(H)(1.28)From(1.22d),H=

EtsinceJ=0,sothatEq.(1.28)becomesE=

2Et2(1.29)ApplyingthevectoridentityF=(F)2F(1.30)inEq.(1.29),(E)2E=

2Et2Sincev=0,E=0fromEq.(1.22a),andhenceweobtain2E

2Et2=0(1.31)whichisthetime-dependentvectorHelmholtzequationorsimplywaveequation.IfwehadstartedthederivationwithEq.(1.22d),wewouldobtainthewaveequationforHas2H

2Ht2=0(1.32)Equations(1.31)and(1.32)aretheequationsofmotionofEMwavesinthemediumunderconsideration.Thevelocity(inm/s)ofwavepropagationisu=1

(1.33)whereu=c3108m/sinfreespace.Itshouldbenotedthateachofthevectorequationsin(1.31)and(1.32)hasthreescalarcomponents,sothataltogetherwehavesixscalarequationsforEx,Ey,Ez,Hx,Hy,andHz.Thuseachcomponentofthewaveequationshastheform2

1u22

t2=0(1.34)whichisthescalarwaveequation. 2001 by CRC PRESS LLC1.2.6Time-varyingPotentialsAlthoughweareofteninterestedinelectricandmagneticeldintensities(EandH),whicharephysicallymeasurablequantities,itisoftenconvenienttouseauxiliaryfunctionsinanalyzinganEMeld.TheseauxiliaryfunctionsarethescalarelectricpotentialVandvectormagneticpotentialA.Althoughthesepotentialfunctionsarearbitrary,theyarerequiredtosatisfyMaxwellsequations.Theirderivationisbasedontwofundamentalvectoridentities(seeProb.1.1),

=0(1.35)andF=0(1.36)whichanarbitraryscalareld

andvectoreldFmustsatisfy.Maxwellsequa-tion(1.22b)alongwithEq.(1.36)issatisedifwedeneAsuchthatB=A(1.37)SubstitutingthisintoEq.(1.22c)gives

E+At

=0SincethisequationhastobecompatiblewithEq.(1.35),wecanchoosethescalareldVsuchthatE+At=VorE=VAt(1.38)Thus,ifweknewthepotentialfunctionsVandA,theeldsEandBcouldbeobtainedfromEqs.(1.37)and(1.38).However,westillneedtondthesolutionforthepotentialfunctions.SubstitutingEqs.(1.37)and(1.38)intoEq.(1.22d)andassumingalinear,homogeneousmedium,A=J+

t

VAt

ApplyingthevectoridentityinEq.(1.30)leadsto2A(A)=J+

2At2+

Vt(1.39)SubstitutingEq.(1.38)intoEq.(1.22a)givesE= =2V(A)t 2001 by CRC PRESS LLCor2V+tA=v

(1.40)AccordingtotheHelmholtztheoremofvectoranalysis,avectorisuniquelydenedifandonlyifbothitscurlanddivergencearespecied.WehaveonlyspeciedthecurlofAinEq.(1.37);wemaychoosethedivergenceofAsothatthedifferentialequations(1.39)and(1.40)havethesimplestformspossible.Weachievethisintheso-calledLorentzcondition:A=

Vt(1.41)IncorporatingthisconditionintoEqs.(1.39)and(1.40)resultsin2A

2At2=J(1.42)and2V

2Vt2=v

(1.43)whichareinhomogeneouswaveequations.ThusMaxwellsequationsintermsofthepotentialsVandAreducetothethreeequations(1.41)to(1.43).Inotherwords,thethreeequationsareequivalenttotheordinaryformofMaxwellsequationsinthatpotentialssatisfyingtheseequationsalwaysleadtoasolutionofMaxwellsequationsforEandBwhenusedwithEqs.(1.37)and(1.38).IntegralsolutionstoEqs.(1.42)and(1.43)aretheso-calledretardedpotentialsA=

[J]dv4R(1.44)andV=

[v]dv4

R(1.45)whereRisthedistancefromthesourcepointtotheeldpoint,andthesquarebracketsdenotevandJarespeciedatatimeR(

)1/2earlierthanforwhichAorVisbeingdetermined.1.2.7Time-harmonicFieldsUptothispoint,wehaveconsideredthegeneralcaseofarbitrarytimevariationofEMelds.Inmanypracticalsituations,especiallyatlowfrequencies,itissufcienttodealwithonlythesteady-state(orequilibrium)solutionofEMeldswhenproduced 2001 by CRC PRESS LLCbysinusoidalcurrents.Sucheldsaresaidtobesinusoidaltime-varyingortime-harmonic,thatis,theyvaryatasinusoidalfrequency.Anarbitrarytime-dependenteldF(x,y,z,t)orF(r,t)canbeexpressedasF(r,t)=ReFs(r)ejt(1.46)whereFs(r)=Fs(x,y,z)isthephasorformofF(r,t)andisingeneralcomplex,Re[]indicatestakingtherealpartofquantityinbrackets,andistheangularfrequency(inrad/s)ofthesinusoidalexcitation.TheEMeldquantitiescanberepresentedinphasornotationas E(r,t)D(r,t)H(r,t)B(r,t) = Es(r)Ds(r)Hs(r)Bs(r) ejt(1.47)Usingthephasorrepresentationallowsustoreplacethetimederivations/tbyjsinceejtt=jejtThusMaxwellsequations,insinusoidalsteadystate,becomeDs=vs(1.48a)Bs=0(1.48b)Es=jBsJms(1.48c)Hs=Jes+jDs(1.48d)Weshouldobservethattheeffectofthetime-harmonicassumptionistoeliminatethetimedependencefromMaxwellsequations,therebyreducingthetime-spacedepen-dencetospacedependenceonly.Thissimplicationdoesnotexcludemoregeneraltime-varyingeldsifweconsidertobeoneelementofanentirefrequencyspec-trum,withalltheFouriercomponentssuperposed.Inotherwords,anonsinusoidaleldcanberepresentedasF(r,t)=Re

Fs(r,)ejtd(1.49)ThusthesolutionstoMaxwellsequationsforanonsinusoidaleldcanbeobtainedbysummingalltheFouriercomponentsFs(r,)over.Henceforth,wedropthesubscriptsdenotingphasorquantitywhennoconfusionresults.ReplacingthetimederivativeinEq.(1.34)by(j)2yieldsthescalarwaveequationinphasorrepresentationas2

+k2

=0(1.50) 2001 by CRC PRESS LLCwherekisthepropagationconstant(inrad/m),givenbyk= u=2fu=2(1.51)WerecallthatEqs.(1.31)to(1.34)wereobtainedassumingthatv=0=J.Ifv=0=J,Eq.(1.50)willhavethegeneralform(seeProb.1.4)2

+k2

=g(1.52)WenoticethatthisHelmholtzequationreducesto:(1)Poissonsequation2

=g(1.53)whenk=0(i.e.,=0forstaticcase).(2)Laplacesequation2

=0(1.54)whenk=0=g.ThusPoissonsandLaplacesequationsarespecialcasesoftheHelmholtzequation.Notethatfunction

issaidtobeharmonicifitsatisesLaplacesequation.Example1.1Fromthedivergencetheorem,deriveGreenstheorem

v

U2VV2U

dv=

S

UVnVUn

dSwhere

n=

anisthedirectionalderivationof

alongtheoutwardnormaltoS.SolutionInEq.(1.1),letF=UV,then

v(UV)dv=

SUVdS(1.55)But(UV)=UV+VU=U2V+UV 2001 by CRC PRESS LLCSubstitutingthisintoEq.(1.55)givesGreensrstidentity:

v

U2V+UV

dv=

SUVdS(1.56)ByinterchangingUandVinEq.(1.56),weobtain

v

V2U+VU

dv=

SVUdS(1.57)SubtractingEq.(1.57)fromEq.(1.56)leadstoGreenssecondidentityorGreenstheorem:

v

U2VV2U

dv=

S(UVVU)dSExample1.2Showthatthecontinuityequationisimplicit(orincorporated)inMaxwellsequations.SolutionAccordingtoEq.(1.36),thedivergenceofthecurlofanyvectoreldiszero.Hence,takingthedivergenceofEq.(1.22d)gives0=H=J+tDButD=vfromEq.(1.22a).Thus,0=J+vtwhichisthecontinuityequation.Example1.3Express:(a)E=10sin(tkz)ax+20cos(tkz)ayinphasorform.(b)Hs=(4j3)sinxax+ej10xazininstantaneousform.Solution(a)Wecanexpresssinascos(/2).Hence,E=10cos(tkz/2)ax+20cos(tkz)ay=Re

10ejkzej/2ax+20ejkzay

ejt=ReEsejt 2001 by CRC PRESS LLCThus,Es=10ejkzej/2ax+20ejkzay=

j10ax+20ay

ejkz(b)SinceH=ReHsejt=Re5sinxej(t36.87)ax+1 xej(t+10)az=5sinxcos(t36.87)ax+1 xcos(t+10)az1.3ClassicationofEMProblemsClassifyingEMproblemswillhelpuslatertoanswerthequestionofwhatmethodisbestforsolvingagivenproblem.Continuumproblemsarecategorizeddifferentlydependingontheparticularitemofinterest,whichcouldbeoneofthese:(1)thesolutionregionoftheproblem,(2)thenatureoftheequationdescribingtheproblem,or(3)theassociatedboundaryconditions.(Infact,theabovethreeitemsdeneaproblemuniquely.)Itwillsoonbecomeevidentthattheseclassicationsaresometimesnotindependentofeachother.1.3.1ClassicationofSolutionRegionsIntermsofthesolutionregionorproblemdomain,theproblemcouldbeaninteriorproblem,alsovariablycalledaninner,closed,orboundedproblem,oranexteriorproblem,alsovariablycalledanouter,open,orunboundedproblem.ConsiderthesolutionregionRwithboundaryS,asshowninFig.1.2.IfpartorallofSisatinnity,Risexterior/open,otherwiseRisinterior/closed.Forexample,wavepropagationinawaveguideisaninteriorproblem,whereaswhilewavepropagationinfreespacescatteringofEMwavesbyraindrops,andradiationfromadipoleantennaareexteriorproblems.Aproblemcanalsobeclassiedintermsoftheelectrical,constitutiveproperties(,

,)ofthesolutionregion.AsmentionedinSection1.2.4,thesolutionregioncouldbelinear(ornonlinear),homogeneous(orinhomogeneous),andisotropic(oranisotropic).Weshallbeconcerned,forthemostpart,withlinear,homogeneous,isotropicmediainthistext. 2001 by CRC PRESS LLCFigure1.2SolutionregionRwithboundaryS.1.3.2ClassicationofDifferentialEquationsEMproblemsareclassiedintermsoftheequationsdescribingthem.Theequa-tionscouldbedifferentialorintegralorboth.MostEMproblemscanbestatedintermsofanoperatorequationL

=g(1.58)whereLisanoperator(differential,integral,orintegro-differential),gistheknownexcitationorsource,and

istheunknownfunctiontobedetermined.AtypicalexampleistheelectrostaticprobleminvolvingPoissonsequation.Indifferentialform,Eq.(1.58)becomes2V=v

(1.59)sothatL=2istheLaplacianoperator,g=v/

isthesourceterm,and

=Vistheelectricpotential.Inintegralform,PoissonsequationisoftheformV=

vdv4

r2(1.60)sothatL=

dv4r2,g=V,and

=v/

Inthissection,weshalllimitourdiscussiontodifferentialequations;integralequa-tionswillbeconsideredindetailinChapter5.AsobservedinEqs.(1.52)to(1.54),EMproblemsinvolvelinear,second-orderdifferentialequations.Ingeneral,asecond-orderpartialdifferentialequation(PDE)isgivenbya2

x2+b2

xy+c2

y2+d

x+e

y+f

=g 2001 by CRC PRESS LLCorsimplya

xx+b

xy+c

yy+d

x+e

y+f

=g(1.61)Thecoefcients,a,bandcingeneralarefunctionsofxandy;theymayalsodependon

itself,inwhichcasethePDEissaidtobenonlinear.APDEinwhichg(x,y)inEq.(1.61)equalszeroistermedhomogeneous;itisinhomogeneousifg(x,y)=0.NoticethatEq.(1.61)hasthesameformasEq.(1.58),whereLisnowadifferentialoperatorgivenbyL=a2x2+b2xy+c2y2+dx+ey+f(1.62)APDEingeneralcanhavebothboundaryvaluesandinitialvalues.PDEswhoseboundaryconditionsarespeciedarecalledsteady-stateequations.Ifonlyinitialvaluesarespecied,theyarecalledtransientequations.Anylinearsecond-orderPDEcanbeclassiedaselliptic,hyperbolic,orparabolicdependingonthecoefcientsa,b,andc.Equation(1.61)issaidtobe:ellipticifb24ac0parabolicifb24ac=0(1.63)Thetermshyperbolic,parabolic,andellipticarederivedfromthefactthatthequadraticequationax2+bxy+cy2+dx+ey+f=0representsahyperbola,parabola,orellipseifb24acispositive,zero,ornegative,respectively.Ineachofthesecategories,therearePDEsthatmodelcertainphysicalphenomena.SuchphenomenaarenotlimitedtoEMbutextendtoalmostallareasofscienceandengineering.ThusthemathematicalmodelspeciedinEq.(1.61)arisesinproblemsinvolvingheattransfer,boundary-layerow,vibrations,elasticity,electrostatic,wavepropagation,andsoon.EllipticPDEsareassociatedwithsteady-statephenomena,i.e.,boundary-valueproblems.TypicalexamplesofthistypeofPDEincludeLaplacesequation2

x2+2

y2=0(1.64)andPoissonsequation2

x2+2

y2=g(x,y)(1.65)whereinbothcasesa=c=1,b=0.AnellipticPDEusuallymodelsaninteriorproblem,andhencethesolutionregionisusuallyclosedorboundedasinFig.1.3(a). 2001 by CRC PRESS LLCFigure1.3(a)Elliptic,(b)parabolic,orhyperbolicproblem.HyperbolicPDEsariseinpropagationproblems.Thesolutionregionisusuallyopensothatasolutionadvancesoutwardindenitelyfrominitialconditionswhilealwayssatisfyingspeciedboundaryconditions.AtypicalexampleofhyperbolicPDEisthewaveequationinonedimension2

x2=1u22

t2(1.66)wherea=u2,b=0,c=1.Noticethatthewaveequationin(1.50)isnothyper-bolicbutelliptic,sincethetime-dependencehasbeensuppressedandtheequationismerelythesteady-statesolutionofEq.(1.34).ParabolicPDEsaregenerallyassociatedwithproblemsinwhichthequantityofinterestvariesslowlyincomparisonwiththerandommotionswhichproducethevariations.ThemostcommonparabolicPDEisthediffusion(orheat)equationinonedimension2

x2=k

t(1.67)wherea=1,b=0=c.LikehyperbolicPDE,thesolutionregionforparabolicPDEisusuallyopen,asinFig.1.3(b).Theinitialandboundaryconditionstypicallyassociatedwithparabolicequationsresemblethoseforhyperbolicproblemsexceptthatonlyoneinitialconditionatt=0isnecessarysinceEq.(1.67)isonlyrstorderintime.Also,parabolicandhyperbolicequationsaresolvedusingsimilartechniques,whereasellipticequationsareusuallymoredifcultandrequiredifferenttechniques. 2001 by CRC PRESS LLCNotethat:(1)sincethecoefcientsa,b,andcareingeneralfunctionsofxandy,theclassicationofEq.(1.61)maychangefrompointtopointinthesolutionregion,and(2)PDEswithmorethantwoindependentvariables(x,y,z,t,...)maynottasneatlyintotheclassicationabove.Asummaryofourdiscussionsofarinthissection is shown in Table 1.1.Table1.1ClassicationofPartialDifferentialEquationsTypeSignofExampleSolutionregionb24acEllipticLaplacesequation:Closed

xx+

yy=0Hyperbolic+Waveequation:Openu2

xx=

ttParabolic0Diffusionequation:Open

xx=k

tThetypeofproblemrepresentedbyEq.(1.58)issaidtobedeterministic,sincethequantityofinterestcanbedetermineddirectly.Anothertypeofproblemwherethequantityisfoundindirectlyiscallednondeterministicoreigenvalue.ThestandardeigenproblemisoftheformL

=

(1.68)wherethesourceterminEq.(1.58)hasbeenreplacedby

.AmoregeneralversionisthegeneralizedeigenproblemhavingtheformL

=M

(1.69)whereM,likeL,isalinearoperatorforEMproblems.InEqs.(1.68)and(1.69),onlysomeparticularvaluesofcalledeigenvaluesarepermissible;associatedwiththesevaluesarethecorrespondingsolutions

calledeigenfunctions.Eigenproblemsareusuallyencounteredinvibrationandwaveguideproblemswheretheeigenval-uescorrespondtophysicalquantitiessuchasresonanceandcutofffrequencies,respectively.1.3.3ClassicationofBoundaryConditionsOurproblemconsistsofndingtheunknownfunction

ofapartialdifferentialequation.Inadditiontothefactthat

satisesEq.(1.58)withinaprescribedsolutionregionR,

mustsatisfycertainconditionsonS,theboundaryofR.UsuallytheseboundaryconditionsareoftheDirichletandNeumanntypes.Whereaboundaryhasboth,amixedboundaryconditionissaidtoexist.(1)Dirichletboundarycondition:

(r)=0,ronS(1.70) 2001 by CRC PRESS LLC(2)Neumannboundarycondition:

(r)n=0,ronS,(1.71)i.e.,thenormalderivativeof

vanishesonS.(3)Mixedboundarycondition:

(r)n+h(r)

(r)=0,ronS,(1.72)whereh(r)isaknownfunctionand

nisthedirectionalderivativeof

alongtheoutwardnormaltotheboundaryS,i.e.,

n=

an(1.73)whereanisaunitnormaldirectedoutofR,asshowninFig.1.2.NotethattheNeumannboundaryconditionisaspecialcaseofthemixedconditionwithh(r)=0.TheconditionsinEq.(1.70)to(1.72)arecalledhomogeneousboundaryconditions.Themoregeneralonesaretheinhomogeneous:Dirichlet:

(r)=p(r),ronS(1.74)Neumann:

(r)n=q(r),ronS(1.75)Mixed:

(r)n+h(r)

(r)=w(r),ronS(1.76)wherep(r),q(r),andw(r)areexplicitlyknownfunctionsontheboundaryS.Forexample,

(0)=1isaninhomogeneousDirichletboundarycondition,andtheassociatedhomogeneouscounterpartis

(0)=0.Also

(1)=2and

(1)=0are,respectively,inhomogeneousandhomogeneousNeumannboundaryconditions.Inelectrostatics,forexample,ifthevalueofelectricpotentialisspeciedonS,wehaveDirichletboundarycondition,whereasifthesurfacecharge(s=Dn=

Vn)isspecied,theboundaryconditionisNeumann.Theproblemofndingafunction

thatisharmonicinaregioniscalledDirichletproblem(orNeumannproblem)if

(or

n)isprescribedontheboundaryoftheregion.Itisworthobservingthatthetermhomogeneoushasbeenusedtomeandifferentthings.Thesolutionregioncouldbehomogeneousmeaningthat,

,andareconstantwithinR;thePDEcouldbehomogeneousifg=0sothatL

=0;andtheboundaryconditionsarehomogeneouswhenp(r)=q(r)=w(r)=0.Example1.4Classifytheseequationsaselliptic,hyperbolic,orparabolic: 2001 by CRC PRESS LLC(a)4

xx+2

x+

y+x+y=0(b)ex2Vx2+cosy2Vxy2Vy2=0.Statewhethertheequationsarehomogeneousorinhomogeneous.Solution(a)InthisPDE,a=4,b=0=c.Henceb24ac=0,i.e.,thePDEisparabolic.Sinceg=xy,thePDEisinhomogeneous.(b)ForthisPDE,a=ex,b=cosy,c=1.Henceb24ac=cos2y+4ex>0andthePDEishyperbolic.Sinceg=0,thePDEishomogeneous.1.4SomeImportantTheoremsTwotheoremsareoffundamentalimportanceinsolvingEMproblems.Thesearetheprincipleofsuperpositionandtheuniquenesstheorem.1.4.1SuperpositionPrincipleTheprincipleofsuperpositionisappliedinseveralways.Weshallconsidertwoofthese.Ifeachmemberofasetoffunctions

n,n=1,2,...,N,isasolutiontothePDEL

=0withsomeprescribedboundaryconditions,thenalinearcombination

N=

0+N n=1an

n(1.77)alsosatisesL

=g.GivenaproblemdescribedbythePDEL

=g(1.78) 2001 by CRC PRESS LLCsubjecttotheboundaryconditionsM1(s)=h1M2(s)=h2. . .MN(s)=hN,(1.79)aslongasLislinear,wemaydividetheproblemintoaseriesofproblemsasfollows:L

0=gL

1=0L

N=0M1(s)=0M1(s)=h1M1(s)=0M2(s)=0M2(s)=0M2(s)=0. . .. . .. . .MN(s)=0MN(s)=0MN(s)=hN(1.80)where

0,

1,...,

Narethesolutionstothereducedproblems,whichareeasiertosolvethantheoriginalproblem.Thesolutiontotheoriginalproblemisgivenby

=N n=0

n(1.81)1.4.2UniquenessTheoremThistheoremguaranteesthatthesolutionobtainedforaPDEwithsomeprescribedboundaryconditionsistheonlyonepossible.ForEMproblems,thetheoremmaybestatedasfollows:Ifinanywayasetofelds(E,H)isfoundwhichsatisessimultaneouslyMaxwellsequationsandtheprescribedboundaryconditions,thissetisunique.Therefore,aeldisuniquelyspeciedbythesources(v,J)withinthemediumplusthetangentialcomponentsofEorHovertheboundary.Toprovetheuniquenesstheorem,supposethereexisttwosolutions(withsubscripts1and2)thatsatisfyMaxwellsequations

E1,2=v(1.82a)H1,2=0(1.82b)E1,2=H1,2t(1.82c)H1,2=J+E1,2+

E1,2t(1.82d)Ifwedenotethedifferenceofthetwoeldsas

E=E2E1and

H=H2H1, 2001 by CRC PRESS LLC

Eand

Hmustsatisfythesource-freeMaxwellsequations,i.e.,

E=0(1.83a)

H=0(1.83b)

E=

Ht(1.83c)

H=

E+

Et(1.83d)DottingbothsidesofEq.(1.83d)with

Egives

E

H=|

E|2+

E

Et(1.84)UsingthevectoridentityA(B)=B(A)(AB)andEq.(1.83c),Eq.(1.84)becomes(

E

H)=1 2t

|

H|2+

|

E|2

|

E|2IntegratingovervolumevboundedbysurfaceSandapplyingdivergencetheoremtotheleft-handside,weobtain

S(

E

H)dS=t

v1 2

|

E|2+1 2|

H|2dv

v|

E|dv(1.85)showingthat

Eand

HsatisfythePoyntingtheoremjustasE1,2andH1,2.Onlythetangentialcomponentsof

Eand

HcontributetothesurfaceintegralontheleftsideofEq.(1.85).Therefore,ifthetangentialcomponentsofE1andE2orH1andH2areequaloverS(therebysatisfyingEq.(1.27)),thetangentialcomponentsof

Eand

HvanishonS.Consequently,thesurfaceintegralinEq.(1.85)isidenticallyzero,andhencetherightsideoftheequationmustvanishalso.Itfollowsthat

E=0duetothesecondintegralontherightsideandhencealso

H=0throughoutthevolume.ThusE1=E2andH1=H2,conrmingthatthesolutionisunique.Thetheoremjustprovedfortime-varyingeldsalsoholdsforstaticeldsasaspecialcase.IntermsofelectrostaticpotentialV,theuniquenesstheoremmaybestatedasfollows:Asolutionto2V=0isuniquelydeterminedbyspecifyingeitherthevalueofVorthenormalcomponentofVateachpointontheboundarysurface.Foramagnetostaticeld,thetheorembecomes:Asolutionof2A=0(andA=0)isuniquelydeterminedbyspecifyingthevalueofAorthetangentialcomponentofB=(A)ateachpointontheboundarysurface. 2001 by CRC PRESS LLCReferences[1]K.H.HuebnerandE.A.Thornton,TheFiniteElementMethodforEngineers.NewYork:JohnWileyandSons,1982,Chap.3,pp.62107.[2]J.A.Kong,ElectromagneticWaveTheory.NewYork:JohnWileyandSons,1986,Chap.1,pp.141.[3]R.E.Collins,FoundationsofMicrowaveEngineering.NewYork:McGraw-Hill,1966,Chap.2,pp.1163.[4]M.N.O.Sadiku,ElementsofElectromagnetics.NewYork:OxfordUniv.Press,1994,Chap.9,pp.409452.Problems1.1Inacoordinatesystemofyourchoice,provethat:(a)

=0,(b)F=0,(c)F=(F)2F,where

andFarescalarandvectorelds,respectively.1.2IfUandVarescalarelds,showthat

LUVdl=

LVUdl1.3Showthatinasource-freeregion(J=0,v=0),Maxwellsequationscanbereducedtothetwocurlequations.1.4Inderivingthewaveequations(1.31)and(1.32),weassumedasource-freemedium(J=0,v=0).Showthatifv=0,J=0,theequationsbecome2E1c22Et2=(v/

)+Jt,2H1c22Ht2=JWhatassumptionshaveyoumadetoarriveattheseexpressions? 2001 by CRC PRESS LLC1.5DeterminewhethertheeldsE=20sin(tkz)ax10cos(t+kz)ayH=ko10cos(t+kz)ax+20sin(tkz)ay,wherek=o

o,satisfyMaxwellsequations.1.6Infreespace,theelectricuxdensityisgivenbyD=D0cos(t+z)axUseMaxwellsequationtondH.1.7Infreespace,asourceradiatesthemagneticeldHs=H0ejawhere=0

0.DetermineEs.1.8AnelectricdipoleoflengthLinfreespacehasaradicaleldgiveninsphericalsystem(r,,)asHs=IL4rsin

1 r+j

ejraFindEsusingMaxwellsequations.1.9ShowthattheelectriceldEs=20sin(kxx)cos(kyy)az,wherek2 x+k2 y=20

0,canberepresentedasthesuperpositionoffourpropagatingplanewaves.FindthecorrespondingHseld.1.10(a)ExpressIs=ejzsinxcosyininstantaneousform.(b)DeterminethephasorformofV=20sin(t2x)10cos(t4x)1.11Foreachofthefollowingphasors,determinethecorrespondinginstantaneousform:(a)As=(ax+jay)e2jz(b)Bs=j10sinxax+5ej12z/4az(c)Cs=2 jej3xcos2x+e3xj4x1.12Showthatatime-harmonicEMeldinaconductingmedium(>>

)satisesthediffusionequation2EsjEs=0 2001 by CRC PRESS LLC1.13Showthatinaninhomogeneousmedium,thewaveequationsbecome

1jEs

+j

Es=0,

1j

Hs

+jHs=01.14Showthatthetime-harmonicpotentialfunctionVsandAssatisfythefollowinginhomogeneouswaveequation2Vs+k2Vs=vs

2As+k2As=Jswherek2=2

.1.15ClassifythefollowingPDEsaselliptic,parabolic,orhyperbolic.(a)

xx+2

xy+5

yy=0(b)(y2+1)

xx+(x2+1)

yy=0(c)

xx2cosx

xy(3+sin2x)

yyy

y=0(d)x2

xx2xy

xy+y2

yy+x

x+y

y=01.16RepeatProb.1.15forthefollowingPDEs:(a)2

x2=

x+

t(,=constant)whichiscalledconvectiveheatequation.(b)2+

=0whichistheHelmholtzequation.(c)2

+[(x)]

=0whichisthetime-independentSchrodingerequation. 2001 by CRC PRESS LLC 2001 by CRC PRESS LLCChapter2AnalyticalMethodsIvelearnedthatabout90percentofthethingsthathappentomearegoodandonlyabout10percentarebad.Tobehappy,Ijusthavetofocusonthe90percent.Anonymous2.1IntroductionThemostsatisfactorysolutionofaeldproblemisanexactmathematicalone.Althoughinmanypracticalcasessuchananalyticalsolutioncannotbeobtainedandwemustresorttonumericalapproximatesolution,analyticalsolutionisusefulincheckingsolutionsobtainedfromnumericalmethods.Also,onewouldhardlyappreciatetheneedfornumericalmethodswithoutrstseeingthelimitationsoftheclassicalanalyticalmethods.Henceourobjectiveinthischapteristobrieyexaminethecommonanalyticalmethodsandtherebyputnumericalmethodsinproperperspective.ThemostcommonlyusedanalyticalmethodsinsolvingEM-relatedproblemsin-clude:(1)separationofvariables(2)seriesexpansion(3)conformalmapping(4)integralmethodsPerhapsthemostpowerfulanalyticalmethodistheseparationofvariables;itisthemethodthatwillbeemphasizedinthischapter.SincetheapplicationofconformalmappingisrestrictedtocertainEMproblems,itwillnotbediscussedhere.TheinterestedreaderisreferredtoGibbs[1].TheintegralmethodswillbecoveredinChapter5,andfullydiscussedin[2]. 2001 by CRC PRESS LLC2.2SeparationofVariablesThemethodofseparationofvariables(sometimescalledthemethodofFourier)isaconvenientmethodforsolvingapartialdifferentialequation(PDE).Basically,itentailsseekingasolutionwhichbreaksupintoaproductoffunctions,eachofwhichinvolvesonlyoneofthevariables.Forexample,ifweareseekingasolution

(x,y,z,t)tosomePDE,werequirethatithastheproductform

(x,y,z,t)=X(x)Y(y)Z(z)T(t)(2.1)AsolutionoftheforminEq.(2.1)issaidtobeseparableinx,y,z,andt.Forexample,considerthefunctions(1)x2yzsin10t,(2)xy2+2 t,(3)(2x+y2)zcos10t.(1)iscompletelyseparable,(2)isnotseparable,while(3)isseparableonlyinzandt.Todeterminewhetherthemethodofindependentseparationofvariablescanbeap-pliedtoagivenphysicalproblem,wemustconsiderthePDEdescribingtheproblem,theshapeofthesolutionregion,andtheboundaryconditionsthethreeelementsthatuniquelydeneaproblem.Forexample,toapplythemethodtoaprobleminvolvingtwovariablesxandy(orand,etc.),threethingsmustbeconsidered[3]:(i)ThedifferentialoperatorLmustbeseparable,i.e.,itmustbeafunctionof

(x,y)suchthatL{X(x)Y(y)}

(x,y)X(x)Y(y)isasumofafunctionofxonlyandafunctionofyonly.(ii)Allinitialandboundaryconditionsmustbeonconstant-coordinatesurfaces,i.e.,x=constant,y=constant.(iii)Thelinearoperatorsdeningtheboundaryconditionsatx=constant(ory=constant)mustinvolvenopartialderivativesof

withrespecttoy(orx),andtheircoefcientmustbeindependentofy(orx).Forexample,theoperatorequationL

=2

x2+2

xy+2

y2violates(i).IfthesolutionregionRisnotarectanglewithsidesparalleltothexandyaxes,(ii)isviolated.Withaboundarycondition

=0onapartofx=0and

/x=0onanotherpart,(iii)isviolated.Withthispreliminarydiscussion,wewillnowapplythemethodofseparationofvariablestoPDEsinrectangular,circularcylindrical,andsphericalcoordinatesystems.Ineachoftheseapplications,weshallalwaystakethesethreemajorsteps: 2001 by CRC PRESS LLC(1)separatethe(independent)variables(2)ndparticularsolutionsoftheseparatedequations,whichsatisfysomeoftheboundaryconditions(3)combinethesesolutionstosatisfytheremainingboundaryconditionsWebegintheapplicationofseparationofvariablesbyndingtheproductsolutionofthehomogeneousscalarwaveequation2

1c22

t2=0(2.2)SolutiontoLaplacesequationcanbederivedasaspecialcaseofthewaveequation.Diffusionandheatequationcanbehandledinthesamemanneraswewilltreatwaveequation.TosolveEq.(2.2),itisexpedientthatwerstseparatethetimedependence.Welet

(r,t)=U(r)T(t)(2.3)SubstitutingthisinEq.(2.2),T2U1c2UT

=0DividingbyUTgives2UU=T

c2T(2.4)Theleftsideisindependentoft,whiletherightsideisindependentofr;theequalitycanbetrueonlyifeachsideisindependentofbothvariables.Ifweletanarbitraryconstantk2bethecommonvalueofthetwosides,Eq.(2.4)reducestoT

+c2k2T=0,(2.5a)2U+k2U=0(2.5b)Thuswehavebeenabletoseparatethespacevariablerfromthetimevariablet.Thearbitraryconstantk2introducedinthecourseoftheseparationofvariablesiscalledtheseparationconstant.Weshallseethatingeneralthetotalnumberofindependentseparationconstantsinagivenproblemisonelessthanthenumberofindependentvariablesinvolved.Equation(2.5a)isanordinarydifferentialequationwithsolutionT(t)=a1ejckt+a2ejckt(2.6a)orT(t)=b1cos(ckt)+b2sin(ckt)(2.6b)Sincethetimedependencedoesnotchangewithacoordinatesystem,thetimede-pendenceexpressedinEq.(2.6)isthesameforallcoordinatesystems.Therefore,weshallhenceforthrestrictourefforttoseekingsolutiontoEq.(2.5b).Noticethatifk=0,thetimedependencedisappearsandEq.(2.5b)becomesLaplacesequation. 2001 by CRC PRESS LLC2.3SeparationofVariablesinRectangularCoordinatesInordernottocomplicatethings,weshallrstconsiderLaplacesequationintwodimensionsandlaterextendtheideatowaveequationsinthreedimensions.2.3.1LaplacesEquationsConsidertheDirichletproblemofaninnitelylongrectangularconductingtroughwhose cross section is shown in Fig. 2.1. For simplicity, let three of its sides beFigure2.1Crosssectionoftherectangularconductingtrough.maintainedatzeropotentialwhilethefourthsideisataxedpotentialVo.Thisisaboundaryvalueproblem.ThePDEtobesolvedis2Vx2+2Vy2=0(2.7)subjectto(Dirichlet)boundaryconditionsV(0,y)=0(2.8a)V(a,y)=0(2.8b)V(x,0)=0(2.8c)V(x,b)=Vo(2.8d)WeletV(x,y)=X(x)Y(y)(2.9)SubstitutethisintoEq.(2.7)anddividebyXY.ThisleadstoX

X+Y

Y=0 2001 by CRC PRESS LLCorX

X=Y

Y=(2.10)whereistheseparationconstant.ThustheseparatedequationsareX

X=0(2.11)Y

+Y=0(2.12)Tosolvetheordinarydifferentialequations(2.11)and(2.12),wemustimposetheboundaryconditionsinEq.(2.8).However,theseboundaryconditionsmustbetrans-formedsothattheycanbeapplieddirectlytotheseparatedequations.SinceV=XY,V(0,y)=0X(0)=0(2.13a)V(a,y)=0X(a)=0(2.13b)V(x,0)=0Y(0)=0(2.13c)V(x,b)=VoX(x)Y(b)=Vo(2.13d)Noticethatonlythehomogeneousconditionsareseparable.TosolveEq.(2.11),wedistinguishthethreepossiblecases:=0,>0,and0,say=2,Eq.(2.11)becomesX

2X=0(2.16)withthecorrespondingauxiliaryequationsm22=0orm=.HencethegeneralsolutionisX=b1ex+b2ex(2.17) 2001 by CRC PRESS LLCorX=b3sinhx+b4coshx(2.18)Theboundaryconditionsareappliedtodetermineb3andb4.X(0)=0b4=0X(a)=0b3=0sincesinhxisneverzerofor>0.HenceX(x)=0,atrivialsolution,andweconcludethatcase>0isnotvalid.Case3:If