关闭

关闭

关闭

封号提示

内容

首页 02 格林定理.pdf

02 格林定理.pdf

02 格林定理.pdf

上传者: zenger2017 2018-03-03 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《02 格林定理pdf》,可适用于自然科学领域,主题内容包含Green#sTheoremsGreen#sTheoremsForthedeterminationoftheelectrostaticpotenti符等。

Green#sTheoremsGreen#sTheoremsForthedeterminationoftheelectrostaticpotentialweencounteredtherelation()()||dVrrrr()Thisrelationcanbeemployedifthechargedistribution()risknownBrequentlyinelectrostaticstheproblemisadifferentoneBesideschargedistributions()ralsothepotentialdistributions()rordistributionsoftheelectricfield()Eroncertainsurfacesandbodiesaregiven(seeFigure)ThechargedistributionsproducingthemaremostlynotknownTheseboundaryconditionsfixthepotentialatallpointsinspaceForthesolutionofthisboundaryvalueproblemthesocalledGreen#stheoremsarerequired,whichwillbederivedinthefollowingLet()Ardenoteavectorfield,andlet()rand()rbetwoscalarfieldsLetthevectorfieldAbeconstructedoutoftheandfieldsinthefollowingway:A()Thus,theproductruleyields()A()WestartfromGauss#theoremVSdVdaAAn()andreplaceAbyequation()Furthermore,wewrite()nAnnthederivativeofthescalarfieldinthedirectionofthenormalvectortothesurfaceTheprimesmeanthatthederivativesaretakenwithrespecttothecoordinateswhicharealsointegratedWiththisreplacementthefirstGreen#stheoremfollowsfromGauss#theorem:()()()()()()VSdVdanrrrrrr()Ingeneral,thechargedensity()randthepotentialdistribution()r,ortheelectricfield()EronasurfaceS(positionvectorr)aregivenThesecondGreen#stheoremisderivedbyinterchangingthearbitrarilychosenscalarfieldsin()andsubsequentsubtractionoftheequationobtainedfromequation()ThesecondGreen#stheoremthenreads(with):()()()()()()()()VSdVdannrrrrrrrr()FromGreen#stheoremsandemployingPoisson#sdifferentialequationwewanttofindnowanintegralrepresentationofthepotentialmoregeneralthantherelation()Forthearbitrarilychosenscalarfieldsweset()||rrrand()()rr(potential)()Then()||rrrr()and()()rr()Withtheserelationsweget,accordingtoequation()()()()||VdVrrrrrr()()||||Sdannrrrrrr()Ifthepointofobservationrliesinsidetheintegrationvolume,thenthefirsttermonthelefthandsideyieldsthepotentialatr:()()()()||||||VSdVdannrrrrrrrrrr()NowwetreattwospecialcasesofthisequationAtfirst,weshifttheintegrationsurfaceStoinfinityThen,theintegralgoesfastertozero,eg,forapointcharge:()~||nrrrrthanthesurfaceelementtendstoinfinityConsequently,thesurfaceintegralvanishes,andtheknownform()remains:()()||dVrrrrOntheotherhand,iftheintegrationvolumeisfreeofcharges,thenthefirsttermofequation()becomeszero,andthepotentialisdeterminedonlybythevaluesofthepotentialandthevaluesofitsderivativesattheboundaryoftheintegrationregion(thesurfaceS)OneshouldnotethattheintegrationvolumealwayshastocontainthepointrinordertogetasolutionforthepotentialAccordingtoequation()thepotential()risdefinedbythepotentialanditsnormalderivativesontheboundarysurfaceBut,bygivingbothvaluesSandnStheproblemisoverdetermined:aswewillseesoon,itisenoughtogiveoneconditiontofixthepotentialTheboundaryconditionsarecalledDirichletboundaryconditionsifisgivenattheboundary,andNeumannboundaryconditionsifthenormalderivativen(thatisthenormalcomponentoftheelectricfieldintensity)isgiventhereUniquenessofthesolutionsWewillnowshowthatthepotentialisdetermineduniquelybygivingDirichletorNeumannboundaryconditionsThetwosolutions()rand()r,assumedtobedifferentfromeachother,obeythePoissonequation(ortheLaplaceequation):()()()rrr,thatis,()()AttheboundarybothfunctionssatisfythesameconditionsSSornSnS()NowwesetuandusethefirstGreen#stheoremwithu,sothat()()()()()()VSuuuuudVudanrrrrrr()Duetothecondition()thefirsttermofthevolumeintegralvanishesThesurfaceintegralbecomeszerosinceeitheru(Dirichlet)orun(Neumann)vanishonthesurfaceduetotheboundaryconditionsThismeans:()()VuudVrror()ur()Thus,uisconstantthroughoutthevolumeVAsforDirichletboundaryconditions,uiszeroattheboundary,andwehaveueverywhere,thatis,,sothattheuniquenessisdemonstratedForNeumannboundaryconditions,bothsolutionsfordifferatmostinaninsignificantadditiveconstantWiththeproveduniquenessofthesolutionofthepotentialproblemitbecomesclearthatthepotentialisfixeduniquelybyeitherDirichletorNeumannboundaryconditionsIfSandnSaregivenasrequiredinequation()theproblemisoverdeterminedInotherwords,SandnSdependoneachotherGreenfunctionWiththehelpofthesecondGreen#stheoremwecancalculatenowthesolutionofthePoissonequationortheLaplaceequationwithinacertainboundedvolumewithknownDirichletorNeumannboundaryconditionsTogettheequations()and(),wechoose||rrcorrespondingtothepotentialofaunitchargeforwhichholds()||rrrr()ThisfunctionisoneoftheclassoffunctionsGdependingon,rr,andforwhichwehave(,)()Grrrrwith(,)(,)||GFrrrrrr()Here,FhastofulfilltheLaplaceequationFGiscalledtheGreenfunction,anditissymmetricinrandr:(,)(,)GGrrrrItshouldbestatedagainthattheparticularsolution(,)||rrrrobeysPoisson#sequation(),butitdoesnotsatisfyDirichletorNeumannboundaryconditions,exceptifthesurfaceliesatinfinityFortheGreenfunction(,)Grrtheboundaryconditionscanbetakenintoaccountviathefunctions(,)Frrinequation()WewillcontinuetofollowthisideaLetusconsiderequation(),whichisaconditionalequationfor()rButitstillcontainsbothconditions,()Sraswellas()nSrDuetotheGreenfunctionandthefreedomcontainedinit,itispossibletochooseafunction(,)Frrinsuchawaythatoneofthesurfaceintegralsvanishesand,thus,anequationwitheitherDirichletorNeumannboundaryconditionisobtainedSettingand(,)Grrin(),andtakingintoaccounttherelationsfoundabovefortheGreenfunction,wefindinanalogytoequation()()(,)()()(,)(,)()VSGGdVGdannrrrrrrrrrr()Byanappropriatechoiceof(,)Grroneortheotherofthesurfaceintegralscanbeeliminatedinthefollowingway:forDirichletboundaryconditions,weset(,)DGrrifrliesonthesurfaceSThen,theproblemisformulatedonlyforDirichletboundaryconditionssincetheequationgivenabovebecomes(,)()()(,)()DDVSGGdVdanrrrrrrr()ForNeumannboundaryconditionstheobviousansatz(,)NGnrrleadstoawrongresultsinceitdoesnotfulfilltherequirementofGauss#law(foraunitcharge):(,)(,)(,)NNNSSVGdaGdaGdVnrrrrnrr()Therefore,thesimplestansatzis(,)NGnSrr()ifSistheentiresurfaceandrliesonthesurfaceThenweobtain()()()(,)(,)SNNVSGdVGdanrrrrrrr()Here,()SSSdaristheaveragevalueofthepotentialatthesurfaceThisaveragevaluecanbeabsorbedalwaysintotheadditiveconstantinwhichthepotentialisarbitraryIftheentiresurfaceStendstoinfinity,thenitvanishesif()rdecreasesfasterthanr,consideringrHere,thephysicalmeaningof(,)Frrshouldbementioned(,)FrrsolvestheLaplaceequationinsideVHence,thefunction(,)FrrdenotesthepotentialofachargedistributionoutsidethevolumeVsothat,togetherwiththepotential||rrofthepointchargeatthepointr,theGreenfunctioncanjustreachthevalues(,)DGrror(,)NGnSrrattheboundarysurface(Sr)ItisclearthattheexternalchargedistributiondependsonthepointchargeinthevolumewhosepotentialorthenormalderivativeofwhosepotentialithastocompensateforSrThismeansthat(,)Frrdependsontheparameterr,whichgivesthepositionofthepointchargedistributioninthevolumeThemethodofimagesthatisusedinthefollowingexamplesisbasedonthisknowledge(Figure)Example:ThegroundedconductingsphereinthefieldofachargeConsideraconductingsphereofradiusaheldatgroundpotential()Groundedmeansthatthepotentialisthesameasthatofthesurfaceoftheearth,andtherefore,thesameasveryfarfromthesphere,atinfinityByywedenotethevectorfromthecenterofthespheretothepositionofthechargeq,andbyrthepointofobservation(seeFigure)Theboundaryconditionsare(||)(||)arrThisisaDirichletproblemFirst,wedeterminetheGreenfunctionDGobeyingtheboundarycondition,usingthemethodofelectricimagesThatis,wetrytoachievetheboundaryconditionbyplacingasecondchargeofmagnitudeqatanappropriateposition(eg,atyinthefigure)suchthatbysuperpositionwiththefirstchargeqtheboundaryconditionissatisfiedAswillbeshownbythecalculation,itissufficienttouseoneimagechargeqpossessing,forsymmetryreasons,apositionvectoryparalleltoyWechoosethecenterofthesphereastheoriginTodeterminethemagnitudeandposition,wesetupthecommonpotentialofbothchargesandtrytofulfilltheboundaryconditionsThepotentialofthetwochargesis(,)||||qqryryryIfrnrandyynyyaretheunitvectorsinrandydirection,respectively,then(,)||||qqryryrynnnnFromthisexpression,byfactoringoutroryatthepointraweobtainthefollowingrelation:(||,)qqayaayayrynnnncoscosqqyyaaayaayyaccordingtotheassumption(DirichletboundaryconditionfortheGreenfunction)ThisequationiscorrectforallcosnnonlyifqqayandyaayThus,thechargeandthedimensionoftheimageareaqqyayyThereisanother,moretrivial,solutionqq,yya,butherethechargewouldvanishWecanseethatqandyapproachzeroifybecomeslarge,thatis,thechargevanishesatinfinityFurthermore,qandqbecomeoppositeandequalwhen||yytendstozeroWiththesevaluesforqandy,thepotentialofthechargeqofagroundedsphereis(,)coscosqqaqqaryryryyayryaryryyrThisholdsfortheregionexternaltothesphere,ie,forraNamely,thisregionisthechargedvolumeVofequation()ThesurfaceofthesphereistheboundingsurfaceSSinceaccordingtoourgeneralconsiderationsthesolutionoftheboundaryvalueproblemisunique,thesolutionfoundfortheproblemistheonlyoneTheGreenfunctionforthispotentialdistributionis(,)(,)coscosDGqryryryayraryryItmaybecheckedeasilythat(,)DGayaswellas(,)DGravanishasrequiredFurthermore,wehave(,)(,)DDGGryyr,asalwaysTodeterminethechargeinducedbyqatthesphere,westartfromthestatementthatthejumpofthefieldintensityacrosschargedsurfacesisequaltothechargedensityatthesurface()nEEnEnnbecauseEsincethefieldsinsideahomogeneousconductingspherebreakdownTheelectronsoftheconductorshiftuntilnofurtherforcesareacting,thatis,()ErforraThisfollowsfromthefactthatthewholesphere,includingitsinteriorregion,hasthepotentialHence,thesystemissphericallysymmetric()()()cosaqyrararanrayaayywhereistheanglebetweenrandyDrawingthedensityofthesurfacechargeagainsttheanglefortworatiosya,oneobtainsFigureThetotalinducedchargeQis(,)(,)sinSaQyaddyadqqyjustequaltotheimagecharge,asitmustbeaccordingtoGauss#lawThenextquestiontobeansweredconcernsthestrengthoftheattractiveforcebetweenthechargeqandthechargeinducedbyitatthesurfaceofthesphereWecouldperformthiscalculationbyintegrationusingthechargedensityofthesurfacefoundaboveButtheinducedcharge,aswehaveseen,behavesjustlikeachargeqsittingatthepositiony,soitiseasiertocomputetheforcebetweentheoriginalchargeandtheimagecharge:||aqqqyFayyyyHence,thetwochargesalwaysattracteachotherForsmallerdistanceswehaveyaandtherefore()()()()qayqayqqFyayayayadThequantitydyadenotesthedistancebetweenthechargeqandthesphericalsurfaceInthevicinityofthesurfaceofthespheretheforcedecreasesasd,baroutsideitvariesasy,similartotheforceinadipolefieldThisisthetypicalbehaviorofthechargeandtheinducedchargeObviously,ifachargeleavesthesurfaceofthesphereandfurthermovestoinfinityworkhastobedoneagainsttheattractiveforceoftheinducedchargeThisistheworkfunctionItisthereasonthat,ingeneral,thechargesremainonaconductor(thesphereinourcase)anddonotleaveittomovetoinfinityDuetotherepulsionoflikechargesonewouldnotexpectthis,atfirstBut,whenachargemovesoffthesurfacethecorrespondingimagechargeappearsimmediately,attractingtheoriginalcharge(togetitback)Example:AconductingungroundedsphereinthefieldofachargeLetaconductingbutungroundedspherebeinthevicinityofapointchargeqIfthissphere,inthechargefreespace,hasatotalchargeQdistributeduniformlyoveritssurface,thenthischargeisconservedifqislocatedintheexteriorregionbecausethechargecannotdrainoffOfcourse,achargeqwillbedistributedoverthesurfaceinsuchawaythatthesurfacechargedistributioncalculatedinExamplewillbeobservedTheresidualchargeQqwillbedistributeduniformlyoverthespherebecausetheforcesofqarebalancedalreadybyqaqyattheposition()ayyyyTherefore,thechargeQqwillactlikeitisconcentratedatthecenterHence,intheexteriorregionthepotentialfoundintheExamplewillgetanadditionaltermoftheform()Qqr:(,)aqaqQqyyayryryrrySimilarly,theattractiveforcebetweenqandthespherecanbedeterminedagain()()grad(,)aqaaqQyyyqqayryryryFyEyryrrry()()()qqayaQyyyayy()Thedivergentselfinteractionofthechargeqrepresentedforrybythefirstterminthepotential(,)ryhasbeenomittedhereWhiletheforcebetweenthechargeandthecorrespondingimagechargeisalwaysattractive,forsmalldistancesalwaysastrongattractiveforceappearsinthiscase,borQqthesphereandthechargerepeleachotheratlargerdistances(seeFigure)ThestrongattractiveforceoccurringinclosevicinityofthesurfaceandtheworkofescapetobedonetooverwhelmitareresponsibleforthefactthatlikechargesremainonthesurfaceofthesphereinspiteoftheelectrostaticrepulsionTheworkfunctionisindependentofthesignsofQandqIfthesesignsareequal,workcanbegainedbeyondacertaindistanceBut,ifthesphereisgrounded,workhastobedoneuptoinfinityExample:AconductingspherekeptatapotentialinthefieldofachargeLetapointchargeqbeinthevicinityofaconductingspherewhosesurfaceiskeptataconstantpotentialVThisproblemcanbereducedtotheformerexampleifwebearinmindthatachargeofmagnitudeVasittinginthecenterofthesphereproducestherequiredpotentialVatthesurfaceHence,therequiredequationsareobtainedbyreplacingQqbyVaintheexpressionsofthepreviousexample:(,)()qaqVayayryryrryand()()qqayVayyayyFyTheseequationsdescribethepotentialandtheforcesoutsidethesphere,thatis,forraThepotentialVarsatisfies()Varoutsidethesphere,soitisaparticularsolution(,)FrrofthehomogeneousPoissonequation,thatis,oftheLaplaceequationItischoseninsuchawaythattheboundaryconditionisfulfilledatthesphericalsurfaceSince(,)ryobeysPoisson#sequation(,)()qryryaswellastheboundarycondition,thepotentialconstructedinthismannerisunique(compareequations()())Hence,thesolution(,)ryfoundistheonlyoneAscanbeseen,attractiveforcesoccurinthevicinityofthesurfaceofthesphereinthiscase,tooExample:AconductingsphereinanuniformelectricfieldLetaconductingspherebeplacedinanuniformelectricfieldasinFigureAswewillseelater,itisnotnecessarytodistinguishbetweenagroundedsphereandanungroundedsphereinthiscaseThiscaseisreducedtoourfirstexamplebyassumingthattheelectricfieldisgeneratedbytwoequalandoppositechargesqandqlocatedatequaldistancesonoppositesidesofthesphereThesetwochargesgenerateafieldofmagnitudeqRattheoriginTogetauniformfield,weassumethatqandRtendtoinfinityinsuchawaythatthemagnitudeofthefieldintensityEqRisalwaysconservedThetotalpotentialisthesuperpositionofthepotentialsofthetwochargesgeneratingthefieldandtheirimagechargesThemagnitudeandthepositionoftheimagechargescanbetakenfromExample,andthusweobtain(,)aaqqqqyyryryryryrySettingyR,thenduetoaayyRyyyythepotentialis(,,)coscosqqrRRrrRRrrRcoscosqaqaaaaaRrrRrrRRRRcoscosqqrrrrRrRrRRRRcoscosqaqaaaaaaaaaRrRrrRrRrRrRIfrRtherootcanbeexpanded,andtakingintoaccountthetermlinearinrRweobtain(,)coscoscosqqaarrErRRrrTheexpressioncosErEristhepotentialintheuniformfield,whilecoscosaEaErrobviouslycomesfromtheimagechargesThetwoimagechargesqandqformadipolewiththedipolemomentqaaqpqyaaERRRThesecondtermintheexpressionforthepotentialcalculatedaboveisnothingbutthepotentialofthedipoleJustasinthefirstexamplehere,theinducedsurfacechargedensitycanbecalculated,too:()()cosraErThen,thetotalinducedchargeis()()sinSQdaaddSincenoadditionalchargeisinduced,thereisnoneedtodistinguishbetweenagroundedandanungroundedsphereinthisexampleExample:TheinversionofapotentialwithrespecttoasphereIftwopointcharges(,)iqisittingatthepositions(,,)iiircreatethepotential(,,)ratthepoint(,,)r,thenitisclaimedthatthepotentialoriginatingfromthecharges()iiiqarqinvertedwithrespecttoasphereofradiusaandthussittingatthepositions(,,)iiiarisgivenby(,,),,aarrrIftheinvertedcharges,definedinthisway,arenegative,ie,()iiiqarq,thentheyareidenticalwiththeimagechargesgeneratedataconductinggroundedsphereofradiusa(seeFigure)Ifiistheanglebetweenrandir,thenthepotentialofthechargeslyingatthepositionsiris(,,)cosiiiiiqrrrrrandbecauseofiiweobtainforthepotentialoftheinvertedchargesiq()iiarqatthepositions(,,)iiiiarrthefollowingexpression:(,,)coscosiiiiiiiiiiiaqqrrrrrraarrrr,,cosiiiiiqaaarrraarrrrBymeansofthisrelationwecancalculateeasilythepotentialoftheimagechargeswhenthepotentialoftheinitialchargeisknownThismethodiscalledinversionwithrespecttoaspherethecenterofthesphereiscalledthecenterofinversionThismethodhasbeenpresentedforpointcharges,butitcanbeappliedeasilytoextendedchargedistributionsExercise:PointchargesinfrontofaconductingplaneLetapointchargeqbeatadistanceainfrontofaninfinitelyextendingconductingwallWhatchargedensitywillbeinducedatthewallWhatisthemagnitudeofthetotalchargeoftheplaneTreattheproblembyintroducinganimagecharge(SeeFigure)SolutionSinceqisapointcharge,intheabsenceoftheconductingplaneitselectricfieldwouldhavetobederivedfromthepotentialqrBut,bynomeansdoesthispotentialfulfilltheconditiontobeconstantattheconductingplaneInastrictmathematicalmanner,acertainsolution()HroftheLaplaceequationHhastobesuperposedtothepotentialofthepointchargetofulfillthisconditionWeobtainafieldforwhichtheplaneisanequipotentialsurface,andhencefulfillstheboundaryconditioniftheoppositechargeq,arrangedmirrorsymmetricatthedistanceafromtheplane,isassociatedwiththechargeqIfristhedistanceofthepointofobservationfromtheimagecharge,then()()()HqqrrrrrrepresentsthepotentialofthetotalfieldItiszeroattheboundaryplanesincehererrThis()rsatisfiesthePoissonequation()()xqarreandtheboundaryconditionDuetotheuniquenessofthesolution(seeequations()and())thisistheonlysolutionThenthecomponentofthefieldintensitynormaltotheboundaryplaneisnxxqqExxrr()FromthePythagoreantheorem,()raxy,()raxyandwegetaxxrr,axxrrSubstitutingthesevaluesinto()andtakingintoaccountthatrratthepointx,weobtainnxxqqaxaxqaEqxrrrrrSinceaccordingtoGauss#law(forfluxthroughthesurface)nE,theinducedsurfacedensityisobtainedbysolvingfor:nEqarHere,rliesontheplaneThechargeisdistributedovertheplanesurfaceoftheconductingwallinsuchawaythatthesurfacedensityisinverselyproportionaltothethirdpowerofthedistancefromtheelectricpointchargeToobtainthetotalchargeoftheplaneweintroducepolarcoordinates(,)Thesurfaceelementisdadd,andraThetotalchargeisobtainedbythefollowingintegration:()qaddadqaSo,weobtaintheresultthatintheplaneanequalbutoppositechargeisinducedThisisnotsurprising,keepinginmindourknowledgeaboutthechargesinducedataconductingsphere(Example)Example:TheGreenfunctionforasphere:GeneralsolutionofthespecialpotentialproblemAccordingtoourgeneralconsiderationsinthischapter,thesolutionofthepotentialofapointchargenearagroundedsphere(atthesurface)obtainedinExampleisjusttheGreenfunctionofthePoissonequationforDirichletboundaryconditions,(,)DaGayyryryyrHere,theexteriorregionofthesphereandthesphericalsurfacewhosenormalvectornpointstothecenterofthesphere(seeFigure)arethevolumeVandthesur

用户评论(0)

0/200

精彩专题

上传我的资料

每篇奖励 +2积分

资料评价:

/24
0下载券 下载 加入VIP, 送下载券

意见
反馈

立即扫码关注

爱问共享资料微信公众号

返回
顶部