《计算电磁学》第九讲

《计算电磁学》PartII：矩量法Dr.PingDU(杜平)SchoolofElectronicScienceandAppliedPhysics,HefeiUniversityofTechnologyE-mail:[e-mail]2ElectrostaticFields(静电场)Dec.5,20112Outline§2.1OperatorFormulation(算子描述)§2.2ChargedConductingPlate(含电荷的导电平板)3§2.1OperatorFormulationThestaticelectricintensityEisconvenientlyfoundfromanelectrostaticpotential,whichisE(2-1)wheredenotesthegradientoperator.Inaregionofconstantpermittivityandvolumechargedensity,theelectrostaticpotentialsatisfies2(2-2)istheLaplacianoperator(拉普拉斯算子).24Foruniquesolution,theboundaryconditionsonareneeded.Inotherwords,thedomainoftheoperatormustbespecified.Fornow,considerfieldsfromchargesinunboundedspace,inwhichcaseconstantasrrr(2-3)whereristhedistancefromthecoordinateorigin(坐标原点),foreveryoffiniteextent.ThedifferentialoperatorformulationisL(2-4)where2L(2-5)5ThedomainofListhosefunctionswhoseLaplacianexistsandhaveboundedatinfinityaccordingto(2-3).Thesolutiontothisproblemis(',',')(,,)'''4xyzxyzdxdydzR(2-6)whereisthedistancebetweenthesourcepoint222(')(')(')Rxxyyzz()andthefieldpoint().',','xyz,,xyzHence,theinverseoperatortoLis11'''4LdxdydzR(2-7)Notethat(2-7)isinverseto(2-5)onlyforboundaryconditions(2-3).Iftheboundaryconditionsarechanged,changes.1L6Asuitableinnerproductforelectrostaticproblems(constant)isThat(2-8)satisfiestherequiredpostulates(1-2),(1-3)and(1-4)iseasilyverified.,(,,)(,,)xyzxyzdxdydz(2-8)wheretheintegrationisoverallspace.LetusanalyzethepropertiesoftheoperatorL.Forthis,formtheleftsideof(1-5),2,()Ld(2-9)whereddxdydz7Green’sidentityis22VSddsnn(2-10)whereSisthesurfaceboundingthevolumeVandnisoutwarddirectionnormaltoS.LetSbeasphereofradiusr,sothatinthelimitthevolumeVincludesallspace.rForandsatisfyingboundaryconditions(2-3),and1/Cr2Cnras.rHenceas.Similarlyfor.3CnrrnSinceincreasesonlyas,therightsideof(2-10)vanishesas2sindsrdd2rr.Equation(2-10)thenreducesto22dd(2-11)8ItisevidentthattheadjointoperatorisaL2aLL(2-12)SincethedomainofisthatofL,theoperatorLisself-adjoint(自伴的).aLThemathematicalconceptofself-adjointnessinthiscaseisrelatedtothephysicalconceptofreciprocity.Itisevidentfrom(2-5)and(2-7)thatLandarerealoperators.1LTheyarealsopositivedefinitebecausetheysatisfy(1-6).ForL,form2*,*()Ld(2-13)andusethevectoridentityplusthedivergence2()Theorem(散度定理).9Theresultis***,VSLdds(2-14)whereSboundsV.AgaintakeSasphereofradiusr.Forsatisfying(2-3),thelasttermof(2-14)vanishesas.rThen2*,||Ld(2-15)and,forrealand,Lispositivedefinite.Inthiscase,positive0definitenessofLisrelatedtotheconceptofelectrostaticenergy(静电能）.10§2.2ChargedConductingPlate(含电荷的导电平板)Considerasquareconducting2ameteronasideandlyingontheplane0zwithcenterattheoriginasshowninFig.2-1.XZY2a2a2b2bConductingplateFig.2-1.Squareconductingplateandsubsections11Letrepresentthesurfacechargedensityontheplate.Here,weassumethatthethicknessiszero.(,)xyTheelectrostaticpotentialatanypointinspaceis(',')(,,)''4aaaaxyxyzdxdyR(2-16)where222(')(')(')RxxyyzzTheboundaryconditionis(constant)ontheplate.VTheintegralequationfortheproblemis222(',')''4(')(')(')aaaaxyVdxdyxxyyzz(2-17)12where,.xayaTheunknowntobedeterminedisthechargedensity.(,)xyAparameterofinterestisthecapacitanceoftheplate1(,)aaaaqCdxdyxyVV(2-18)whichisacontinuouslinearfunctionalof.Letusfirstgothroughasimplesubsectionandpoint-matchingsolution,andlaterinterpretitintermsofmoregeneralconcepts.ConsidertheplatedividedintoNsquaresubsections,asshowninFig.2-1.Definebasisfunctionsnm1ons0onallothersnf(2-19)13Thusthechargedensitycanberepresentedby1(,)Nnnnxyf(2-20)Substituting(2-20)in(2-17),andsatisfyingtheresultantequationatthemid-pointofeach,weobtainthesetofequations(,)mmxyms1NmnnnVl1,2,,mN(2-21)where221''4(')(')nnmnxymmldxdyxxyy(2-22)14Notethatisthepotentialatthecenterofduetoauniformmnlmschargedensityofunitamplitudeover.nsAsolutiontotheset(2-21)givestheintermsofwhichthechargedensitymisapproximatedby(2-20).Thecorrespondingcapacitanceoftheplate,approximating(2-18),is111NnnmnnnmnCslsV(2-23)Totranslatetheaboveresultsintothelanguageoflinearspacesandthemethodofmoments(MoM),let15(,)(,)fxyxy(,)gxyVxaya222(',')()''4(')(')(')aaaafxyLfdxdyxxyyzz(2-24),(2-25)(2-26)Thenisequivalentto(2-17).()LfgAsuitableinnerproduct,satisfying(1-2)to(1-4),forwhichLisself-adjoint,is,(,)(,)aaaafgdxdyfxygxy(2-27)Wechoosethefunctions(2-19)asasubsectionalbasis.16Thetestingfunctionsaredefinedas()()mmmwxxyy(2-28)Thisisthetwo-dimensionalDiracdeltafunction.Theelementsofthe[l]matrix(1-25)arethoseof(2-22),andthe[g]matrixof(1-26)is[]mVVgV(2-29)Thematrixequationequation(1-24)isidenticaltothesetofequations(2-21).17Intermsoftheinnerproduct(2-27),thecapacitance(2-18)canbewritten2,CVFornumericalresults,theof(2-22)mustbeevaluated.mnlLetdenotethesidelengthofeach.22/baNnsThepotentialatthecenterof