TheMaxwellequationsareasetoffourlaws:Faraday’slaw,Ampere’slaw,andthe electricandmagneticformsofGauss’slaw.Eachofthesetoffourequationsmay, inturn,bewritteninthefollowingthreedifferentforms:integral,differential,and boundaryforms.Ithasbeenargued[1]thattheintegralformsofMaxwell’sequations arethemostfundamentalinthesensethatallotherformsderivefromthem.Wedeal hereonlywithtime-harmonicproblemsandassumethatallsourceandfieldquantities varyas e j ω t ,effectivelyreplacingtheoperator ∂/∂ t by j ω inthetime-domainforms ofMaxwell’sequations[2].
2.1.1IntegralformofMaxwell’sequations
TheintegralformofMaxwell’sequationsforexp(j ω t )timedependenceis
where S isan open surfacewith closed boundary C .Ontheotherhand,if S isa closed surface,enclosingavolume V ,wecanalsowrite
Theelectricandmagneticfieldstrengthquantities,(E , H ),arerelatedtothecorrespondingfluxdensityquantities(D, B)viathe constitutiveequations involvingthe localpermittivityandpermeability,(ε , μ),respectively:
Intheproblemsconsideredhere,weassumethatthematerialparameters(ε , μ)are linear,piecewisehomogeneous,andisotropic.InthefirsttwoMaxwell’sequations, asshowninFigure2.1, S hasaunitnormal ˆ n andtheclosedcurve C hasaunit tangent ˆ chosensuchthattheunitvector ˆ u = ˆ ׈n isbothnormalto C andpoints awayfrom S .Inthelasttwoequations, S isa closed surfacewithunitnormal ˆ n,the boundaryofavolume V .Volumetricelectricandmagneticsourcecurrents J and M , withunits[A /m 2 ]and[V /m 2 ]appearin(2.1)and(2.2),respectively.Whilethere isnoexperimentalevidencefortheexistenceofmagneticmonopolesorcurrents, magneticcurrentsandchargesnotonlygiveanelegantsymmetrytoMaxwell’sequationsbutalsoprovideflexibleandmathematicallyconvenientmeansforrepresenting electromagneticfields.Magneticandelectriccurrentsmerelycomprisemagneticand electricchargesinmotion;thevolumechargedensitiesofthosechargeswedefineas m [Wb/m 3 ]and q [C/m 3 ],respectively.Theconservationofchargeprinciplestates thatineveryboundedregionofspace,electricchargeisconserved;forbothconvenienceandmathematicalsymmetries,weassumethatthissameconservationlawalso holdsformagneticcharges.Thus,thetotalchargeforeitherchargetypechangesina regiononlyaschargescrosstheregion’sboundaries(i.e.,enterorleavetheregion). Therate(in[C/s]),forexample,atwhichelectricchargesdecreaseinaregionmust
Figure2.1ThesurfaceSwithunitnormal ˆ n isboundedbythecurve ∂ S = C.The unitvectors ˆ and ˆ u lieinthetangentplaneofS; ˆ isalsotangenttoC while ˆ u isnormaltoCandpointsawayfromS.Thethreeunitvectors satisfy ˆ u × ˆ =ˆn andformamutuallyorthogonalright-handedtriad alongC.
equalthenetelectriccurrentflux(in[C/s])exitingtheregion’sboundaries.Magnetic chargesandcurrentsaresimilarlyrelated;bothresultsaresuccinctlysummarizedby the continuityequations,
where,fortime-harmonicquantities, j ω playstheroleof ∂/∂ t ,withbothsidesof (2.7)and(2.8)representingratesofdecreaseofthetotalchargein V .Thoughthecontinuityequationsfollowfromphysics,independentofMaxwell’sequations,thelatter areconsistentwiththem.Forexample,ifwereplacethefirstsurfaceintegralin(2.2) withthe closed boundarysurfaceof(2.3),thenthecontourintegralof(2.2)vanishes (i.e.,surface S nolongerhasaboundarycontour C ),and(2.7)follows.Similarly, (2.8)followsfromapplyingboth(2.1)and(2.4)toacommonclosedsurface S
Rectangular path for deriving the boundary form of Ampere's law. (a)(b) Pillbox surface for the boundary form of Gauss's law.
Figure2.2GeometriesusedtoderiveboundaryformsofMaxwell’sequations pathwhoselongestsidesareparalleltoandonoppositesidesofamaterialboundary relatethefieldintensitycomponents(E , H )onoppositesidestoany surfacecurrent flux(Js , Ms )throughtherectanglethatmightbepresentontheboundary;fluxintegral contributionsfromfinitevolumetriccurrents(J , M )orthefluxfields(B, D)donot contributetothelimitastheareaenclosedbythepathvanishes,leavingonlysurface currentfluxcontributions.Thisresultsintheequations(2.15)and(2.16)belowwhere ˆ n isnormaltotheinterfaceandpointsfromthe tothe + regionassociatedwitheach sideoftheinterface.Forthesecondpair,twosuchrectangularpathsperpendicularto oneanotherarethecross-sectionsofacylindricalpillboxofinfinitesimalheightsides andradius a (h << a)whosecircularendfacesareparalleltoandonoppositesides oftheinterfaceofthetwomedia.IntegratingthetwoGausslawsoverthevolume enclosedcapturesanysurfacechargedensitycontributionsfromtheinterface;volume chargeandcylindricalfacecontributionsvanishinthelimitastheheightvanishes, whereasthearea(π a2 )commontothetwofluxfaceandchargedensityintegrals cancels,resultingin(2.17)and(2.18)below.
s = (E + E ) ׈n,(2.15) Js =ˆn × (H + H ),(2.16) qs =ˆn (D+ D ),(2.17) ms =ˆn · (B+ B ),(2.18)
where Ms and Js aremagneticandelectricsurfacecurrentdensitiesand qs and ms are electricandmagneticsurfacecurrentdensities,respectively.Similarly,theboundary formofthecontinuityequationsforvolumetriccurrents J and M ataninterfaceis obtainedas
ˆ n (J + J ) + ∇ s Js =−j ω qs ,(2.19)
ˆ n · (M + M ) + ∇ s · Ms =−j ω ms ,(2.20)
where ∇s · isthesurfacedivergenceoperator.Weremindthereaderthatthetwoline integralsontheLHSoftheintegralformsofFaraday’sandAmpere’slawshavethe units(V)and(A),respectively,asdothemagneticandelectricline(filament)currents K and I passingthroughanenclosedpath;hence,theunitsof surface currentdensities Ms and Js mustbe[V /m]and[A /m],while volumetric currentdensityunitsfor M and J are[V /m 2 ]and[A /m 2 ].Notealsothatadifferentialvolumetricintegralover afilamentcurrentproducesa dipolemoment result, Id ,where d isthedifferential lengthofthefilamentthatfallsinsidethedifferentialvolume.Thus,inthesense thattheydonotradiate,pointsourcesofcurrentareessentiallynon-existentbecause aradiatingcurrentelementmusthaveatleastsomenon-vanishinglength, d > 0. Sinceadipole’sorientationisalsoimportant,itsmomentisoftenexpressedinvector formas Id or Id .
Wehavealreadynotedthatif ω = 0,thetwoGausslawsareimpliedbythedifferential formsofFaraday’sandAmpere’slawsplusthecontinuityequations.Togetherwith theirbehavioratinfinity,thetwocurlequationsshouldbesufficienttodetermine (E , H )foragivenpairofcurrentsources(J , M ).Onestepinthisdirectionisto eliminateeither E or H betweenthetwo.Forexample,theeliminationofthemagnetic fieldiseasilyaccomplishedbytakingthecurlofbothsidesofFaraday’slaw,(2.9),and substitutingfromAmpere’slaw,(2.10),toeliminatethemagneticfield.Thisyields theso-calledvectorHelmholtzequationfortheelectricfield:
where k = ω √με = 2π/λ isthe wavenumber andwhere λ isthewavelengthofa wavepropagatinginthematerialifboth μ and ε arebothreal(i.e.,themedium islossless).Applyingthesameprocedure,butreversingtherolesofFaraday’sand Ampere’slaws,andeliminatingtheelectricfieldleadstothemagneticformofthe vectorHelmholtzequation:
Unfortunately,itstillremainsdifficulttofindgeneralsolutionsof(2.21)and(2.22). Itturnsoutthat potential representationsofthefieldsarebettersuitedtofinding solutionssincetheycanbechosentoautomaticallysatisfyconditionssuchasthe vanishingofacurlordivergence.Tothisend,wefirstnotethatsinceMaxwell’s equationsarelinear,thesourcepair(J , M )canbepartitionedintoasumofsimpler sourcepairsetsas(J , 0) + (0, M ),eachofwhichinvolvesbuta single currentspecies (electricormagnetic,respectively).Onceweobtainthefieldsduetoeachsingle currentspeciesactingalone,weobtainsolutionsforbothpresentbysuperposingthe resultsforeachspecies.Forexample,forelectriccurrentsonly,weset M = 0 and m = 0in(2.9)and(2.12),respectively,anddeterminethefields(E J , H J ),withthe superscriptindicatingthesourcetype.Forthiscase,andunderthemildhypothesisof derivativecontinuity[3],themagneticformofGauss’slaw, ∇ ∇· BJ = 0,impliesthat BJ =∇ ∇ ∇× A (andviceversa)foravector A wecallthe magneticvectorpotential since
weuseittodirectlydeterminemagneticfieldquantities, H J = 1 μ BJ = 1 μ ∇ ∇ ∇× A. A is notuniquelyspecified,however,sincebothitscurlanddivergencecanbespecified independently,accordingtotheHelmholtzdecompositiontheorem[4].Laterwewill alsospecify ∇ ∇ ∇· A soastosimplifythedefiningequationfor A;forthemoment, however,wesubstitutethisrepresentationof H into(2.9),andrearrangeto ∇ ∇ ∇× (E J + j ω A) = 0.Theidentity ∇ ∇ ∇× (∇ ∇ ∇ ) = 0 impliesthatif E J hastheform E J = j ω A −∇ ∇ ∇ ,itwouldautomaticallysatisfythiszerocurlcondition;butitcanalsobe shownthatunderrathermildrestrictions,every E J canalwaysbesorepresented[3]. Thenewlyintroducedfunction iscalledthe electricscalarpotential andthenegative signischosentoagreewiththefamiliarelectrostatic(i.e., ω → 0)representationfor E J involvingscalarpotentialonly.Insummary,wehavenowdevelopedthepotential fieldrepresentation(E J , H J ) = ( j ω A −∇ ∇ ∇ , 1 μ ∇ ∇ ∇× A)forthefieldsduetoelectric currentsources J actingalone.Similarly,formagneticcurrents M actingalone,the potentialfieldrepresentationis(E M , H M ) = ( 1 ε ∇ ∇ ∇× F , j ω F −∇ ∇ ∇ ),involving the electricvectorpotential F and magneticscalarpotential .Superposingthese results,weobtainfinallytheso-called mixedpotential representationof(E , H ) = (E J + E M , H J + H M ):
Substitutingthepotentialrepresentation(2.25)for(E J , H J )intoAmpere’slaw,using theidentity ∇ ∇ ∇× (∇ ∇× A) =∇ ∇ (∇ ∇ ∇·
2 A,andrearranging,weobtain
Similarly,substituting(E M , H M ),(2.26),intoFaraday’slaw,wefind
Despitestrongsimilaritiesbetweentheequationpair(2.21)and(2.22)andthepair (2.27)and(2.28),thelatterprovideanimportantopportunityforsimplificationsince thedivergencesof A and F remainyettobespecified.Ofmanypossibleso-called gauge choiceshere,theso-calledLorenzgauges,areobviousones:
sothat(2.27)and(2.28)immediatelysimplifyto ∇ 2 A + k 2 A =−μJ ,(2.31)
2 F + k 2 F =−ε M ,(2.32)
respectively.Equationsforthescalarpotentials and areeasilyobtainedbysubstituting(2.25)and(2.26)intothetwoGausslaws,(2.11)and(2.12),respectively, yielding
2 ψ + k 2 ψ =−f (r ),(2.35) forsomepotential ψ withscalarvolumesourcedensity f (r ).Wenotethesourceterm f (r )canbewrittenasasuperpositionorconvolutionintegral
f (r ) = V f (r )δ (r r ) dV ,(2.36)
where V includesatleastthe support of f (r ),supp (f (r )),thatis,thesmallestclosed regionoutsideofwhichthefunction f (r )vanishesidentically.Thus,thesetsupp (f (r )) iscompletelycontainedwithin V .Thethree-dimensionaldeltafunction δ (r r ) = δ (x x )δ (y y )δ (z z )representsa pointsourceofunitdensityat r ;noteits unitsare[1/m 3 ].Wewishtoexpress ψ asasource-weightedsumoverpointsource solutions,andtodosowefirstintroduceascalarfunction G (r , r ),asolutionofthe point-sourceequation
∇ 2 G + k 2 G =−δ (r r ),(2.37)
where G (r , r )representsagenericpotentialfieldat r duetoaunitdensitysource at r .Sincea(scalar)pointsourceisbothnon-directionaland(2.37)iscoordinate systemindependent,weshouldalsoexpectthesameof G ,i.e.,thatitdependsonly ontheradialseparation R =|r r | betweentheobservationpoint r andsourcepoint r .Wealsonotethatinalocalsphericalcoordinatesystemwithanoriginat r , ∇ 2 G = 1 R2 d dR R2 dG dR ,fromwhichoneeasilyverifiesthat
G (r , r ) = e jkR 4π R (2.38) satisfies(2.37)everywhereexceptpossiblyat R = 0(sincetheexpressionfor ∇ 2 involvesdivisionbyzerothere).Toalsoverifytheunitvolumeintegralbehaviorof the3Ddeltafunctionat R = 0,wemustexaminetheequalityofvolumeintegralsover
Integralequationsforreal-lifemultiscaleelectromagneticproblems termsonbothsidesof(2.37).Thenon-directionalnatureoftheproblemalsosuggests integratingoveravolumetrictestsphere, Va ,ofradius a centeredat R = 0,noting that ∇ 2 = ∇ · ∇ ,andevaluatingtheintegraloftheLaplaciantermviathedivergencetheorem.Indeed,onefinds Va (
Indeed,that(2.39)isasolutionof(2.35)iseasilyverifiedbydirectsubstitutionand useof(2.37)and(2.36).Weremarkthat,byimplication,theintegralin(2.39)must alsoincludeanycontributionsfromsurface,filament,orpointsourcespresent,which onewritesintheforms S G (r , r )fs (r ) dS , C G (r , r )f (r ) d ,or
● Notetheextenttowhichwerelyonlinearityandsuperposition:thefields(E , H ) arevectorcomponentsums,eachtermofwhichisasumofrealandimaginary parts,andconstructedasalinearsuperpositionofavectorpotential,thevectorvaluedgradientofascalarpotential,andthecurlofavectorpotential,i.e.,a
sumoverbothcurrentandchargeterms,respectively.Eachofthese,inturn,is aconvolutionintegral,i.e.,asumofapoint-sourceresponses G (r , r )weighted bysourcedensitiesoftheappropriatechargeorcurrentspeciesdefinedon V . Lateron,indiscretizingaproblem,wefurthersubdividethesourcedomainsinto collectionsofsimplersubdomains,eachwithsimplifiedlocalsourcerepresentations.Inthelattercase,however,thenewlevelofsumsistypicallyrepresented asmatrix–vectorproducts.
2.1.5Farfieldsandfarpotentials
Inthefarfield,itisconvenienttoexpressafieldobservationpoint r inspherical coordinatesas r = r (cos φ sin θ ˆ x + sin φ sin θ ˆ y + cos θ ˆ z )whoseoriginisat r = 0, assumednotfarfromorevenwithinthesourceregion.Atlargeradialdistances r =|r | fromallsourcepoints r ,theapproximation R =|r r |≈ r −ˆ r r where ˆ r = r /r , canbeusedinthelocallyvaryingphasefactorterminthenumeratorof G (r , r )while R ≈ r canbeusedinthedenominatortoapproximate G (r , r )as e jkr 4π r e jk r r inthefar vectorpotentials,yielding
Similarly,thefarscalarpotentialsare
Butsince
,wemaysimplyreplace
r inthefar field.CombinedwiththeLorenzgaugeconditions,itthenfollowsthatthegradients ofthescalarpotentialsareentirelyradiallydirectedandcompletelycancelradial componentsofthevectorpotentialcontributionstothefarfield;thus,inthefarfield, fields(E , H )becomecompletelytransversetotheobservationvector r andcanbe writtenas
