computational-electromagnetics-lecture-notes-prather.pdf

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1、EE840COMPUTATIONALELECTROMAGNETICS–FALL2002Lecture#1:FundamentalConceptsofElectromagneticTheory1OriginofMaxwell’sEquations:Theclassicfieldofelectromagneticshasalongandillustrioushistory.Itsrangeofapplicationsextendsfromdistributedcircuittheoryupthroughmodernoptics.Tothisenditcanbeusedtoanalyze

2、passivecomponents,suchascircuitsandantennas,aswellasactivedevicessuchaslasersandmicrowavesources.Inessencetheentirefieldofelectromagneticscanbeconciselysummedupintoasetoffourequationsthatthwerecompiledintoaself-consistentsetofequationsbyJamesClerkMaxwellinthe19century.Asaresult,theywerecollect

3、ivelyreferredtoasMaxwell’sequations.Fortimedependent,orelectrodynamic,phenomenaeachequationinthesetbecomescoupledtoeveryotherequation,inthesensethattosolvethecompleteboundaryvalueproblemyoumustaccountforeachequationanditseffectontheothers.Fortimedependent,orelectrostatic,phenomena,eachequation

4、inthesetbecomesdecoupledandcanthereforebesolvedforindependently.InthecontextofthisclasswewillfocusontheelectrodynamicphenomenaandthereforewillderivesolutionmethodsforthecompletesetofMaxwell’sequations.WebeginwithareviewoftheoriginofMaxwell’sequations.Eachequationinthesetcantraceitsorigintooneo

5、ftheempiricallawsdiscoveredbyGauss,Ampere,andFaraday,whicharediscussedbelow.1.1Gauss’sLawGauss’slaw(electricandmagneticrelatesthefluxflowingoutofaclosedsurfacetothechargeenclosed:∫D⋅dS=Q,whereDisthefluxdensity(electricinthiscase),dSisanincrementalsurfaceelement,andQisthechargeenclosed.WEcanals

6、orepresentthechargeenclosedasavolumeintegral,∫D⋅dS=∫ρ(r)dV,VwhichallowsustoapplyGauss’slaw,orthedivergencetheorem,∫A⋅dS=∫∇⋅Adv.(1)Proof:FromFig.1wecanevaluateEq.(1)overtheelementalvolumeof∆x,∆y,∆z,asA⋅ds=A∆y∆z+A∆x∆z+A∆x∆y∫xyz∂∂∂∇AdV=A+A+A∆x∆y∆z∫xyz∂x∂y∂zAAAxyz=++∆x∆y∆z∆x∆y∆z=A∆y∆z+

7、A∆x∆z+A∆x∆y.xyz1zˆz∆x,∆y∆x,∆zyˆy∆y,∆zxˆxFigure1:Elementalvolumeandsurfacecontaininganelectriccharge.Therefore,accordingtoGauss’slaw∫∫D⋅dS=∇⋅Ddv=∫ρ(r)dV=QVVormoresuccinctly,∇⋅D=ρ(r)InasimilarfashionwecanapplyGauss’slawtothemagneticfieldt