01 静电学的基本概念

101IntroductionandFundamentalConceptsCoulomb'slawIntheinvestigationofthepropertiesofchargedbodiesatrestthefollowingresultshavebeenobtainedexperimentally:chargedbodies(charges)exertaforceoneachother.Therearetwokindsofcharges,positiveandnegativeones.Unlikechargesattracteachother,likechargesrepeleachother.Theforcebetweentwocharges1qand2qisproportionaltotheirproduct:1212~Fqq.Theforcedecreaseswiththesquareofthemutualdistance,thatis,122121~||Frr.Theelectrostaticforcesarecentralforces.Thus,fortheforceexertedbythecharge2onthecharge1wecanwrite121212312||kqqrrFrr.(1.1)kisaconstantofproportionalitystilltobefixed.SeeFigure1.1.ThisequationfortheforceactingbetweentwochargesiscalledCoulomb'slaw.Furthermore,theprincipleofsuperpositionisvalid:Theelectricforcesexertedbyseveralcharges2q,34,,qqonatestcharge1qsuperposeeachotherundisturbedwithouttheforcebetween1qandacertaincharge(e.g.,2q)beingchangedduetothepresenceofothercharges.Inparticular,thisimpliesthattheforcesbetweenchargescanbemerelytwo-bodyforces;many-bodyforcesdonotoccur.Formany-bodyforcestheforcebet-weentwobodies1and2dependsalsoonthepositionsoftheotherbodies34,,rr.Forexample.athree-bodyforcewouldbe12121232312123123||()1||skqqqqqrrFrrrrrr.(1.2)Here,sristhecenterofgravitybetween1qand2q.SeeFigure1.2.Thisthree-bodyforcewouldtendtoatwo-bodyforce(1.1)as3r,asshouldbe.2Microscopically,onecanimaginethattheforce(i.e.,aforcefield)originatesfromthevirtualexchangeofparticles.Thesearethrownbackandforthbetweenthecenters,liketennisballs,andinthiswaytheybindthecenterstoeachother.Fortwo-bodyforces,thisexchangeproceedsbetweentwocentersonly;forthree-(many-)bodyforces,adetourviathethirdcenter(orseveralcenters)occurs.(SeeFigure1.3.)IntheCoulombinteractionphotonsareexchanged;intheweakinteraction,Z-andW-bosons;inthegravitationalinteraction,gravitons;andinthestrong(nuclear)interaction,π-mesons(or,onadeeperlevel,gluons).Thephotonsandgravitonshavearestmassequaltozero.Therefore,theseforcesareofinfiniterange.Ontheotherhand,theshortrangeofthestronginteraction(13~2fm210cm)isbasedonthefiniterestmassoftheπ-meson.Nowadays,weknowthatpionsandnucleonsarebuiltupoutofquarks.Thequarksinteractbytheexchangeofheavyphotons(interactingintensivelyamongeachotherandcouplingtotheso-calledglue-balls).Theseheavyphotonsarecalledgluons.FortheordinaryCoulombforceinthepresenceoffurtherchargesiqtheforceexertedoncharge1qreads11321||NiiiikqqrrFrr.(1.3)Inthisform,Coulomb'slawisvalidexactlyonlyforpointchargesandforuniformlychargedsphericalbodies.Forchargesofarbitraryshape,deviationsappearwhichwillbediscussedlateron.Nevertheless,oneshouldwonderaboutthe21r-dependenceoftheCoulombforce.Thisparticularforcelawisrelatedtothefactthattherestmassofthephotonsexchangedbythechargesiszero.AccordingtoHeisenberg'suncertaintyrelationtheycanthenbeproducedvirtuallywithalongrangeR.Theuncertaintyrelationstates~Et,Euncertaintyofenergy2c~tE,tuncertaintyoftime2c.(1.4)Thelatterexpressiongivesthelifetimeofavirtualparticlehavingtherestmass2Ec,andthustherangeis~cRctE.ThisexplainsthelongrangeoftheCoulombforce.Ifthephotonhadtherestmass,thentheCoulombpotential(comparethefollowingpages)wouldhavetobeoftheYukawatype,namely,()~reVrr.(1.5)3Here,2hccwouldbetheso-calledComptonwavelengthofaphotonhavingtherestmass.For0oneobtainstheCoulombpotentialofapointcharge.Nowadays,thebestprecisionmeasurementsofthephotonmassyieldavalueof216510eVc.1SystemsofunitsThechargerepresentsanewphysicalpropertyofbodies.Now,wemayintroduceanewdimensionforthecharge,orexpressitintermsofthedimensionsmass,length,andtimeusedinmechanics.Theproduct12kqqisfixedinequation(1.1).Subjecttothisconditionthedimensionofthesinglefactors,chargeandconstantofpropor-tionality,canstillbechosenfreely.Dependingonthechoiceofkwegetdifferentsystemsofunits.Intextbooks,mainlytwodifferentsystemsofunitsarestillusednowadays,theGaussianandtherationalizedsystemofunits.IntheGaussiansystemofunitstheconstantofproportionalityktakesthenumericalvalue1andremainsnondimensional.Then,thechargeisnolongeranindependentunit.IntheCGS-system,oneobtainsfromequation(1.1)theunit321211cmgsforthecharge,whichisalsodenotedtheelectrostaticunit(esu)orstatCoulomb.Thisexplicittracingbackofelectromagneticquantitiestomechanicalunitscanbefoundvirtuallyonlyinoldertextbooks;inmorerecenttextbooksonatomicandnuclearphysicsorquantummechanicsusingtheGaussiansystemofunitsthechargeishandledlikeanindepen-dentunit;therebythephysicalinterrelationsoftenbecomeclearer.Setting12||rrr,equation(1.1)takesthesimpleform:122qqFr.(1.6)Theoppositelineistakenintheso-calledrationalizedsystemofunits.Here,theunitisfixedbythecharge.Itsvalueisdeterminedbymeasuringtheforceexertedbytwocurrent-carryingconductorsoneachother.Accordingtothedefinition,whenacurrentofoneAmpereisflowingthroughtwoparallel,rectilinear,infinitelylongconductorsplacedatadistanceofonemeterfromeachother,aforceof7Ne21on0wtpermeteroftheirlengthactsbetweenthem.Theproductofcurrentandtimegivesthequantityofcharge:1Coulomb(C)1Amperesecond(As).Bythis(arbitrary)definition,theconstantofproportionalityktakesadimensionaswellasafixednumericalvalue;onesets014k.(1.7)Theconstant0iscalledthepermittivityofvacuum;ithasthevalue1209As1As8.85410Vm4910Vm.(1.8)Forthemoment,weconsidertheunitvolt(V)asanabbreviationof12311V1Amkgs1NmC.Intheframeworkofthissystemofunits,Coulomb'slawreads122014qqFr.(1.9)Acomparisonof(1.9)and(1.6)thenyieldstherelationbetweenthechargesintheGaussiansystemofunits(q)andintherationalizedsystemofunits(q)(alsocalledthemksA—(meterkilogramsecondAmpere)—system).Itreads014qq.(1.10)Theunitofqis1Coulomb(C)1Amperesecond(As).IntheGaussiansystem,thiscorrespondsto290911Vm1Coulomb1As1As910As41As44910Vm33991822mkgcmg910VmAs910910ss99310ergcm310cgschargeunit9310statCoulomb.(1.11)Inmacroscopicphysicsandexperimentalphysics,therationalizedsystemofunitsisusedpredominantly.Inatomicphysics,nuclearphysics,andmanytextbooksintheoreticalphysics,theGaussiansystemofunitsisusedmostly.Here,wewilluseexclusivelytheGaussiansystemofunits.1TheelectricfieldintensityToexplainthenotionoftheelectricfieldintensity,westartfromtheforceFexertedbyacharge1qonatestchargeqthatisassmallaspossible.Thefieldintensitycausedby1qatthepositionrofthechargeqisdefinedbythequotient:()qFEr.(1.12)Since,ingeneral,theelectricfieldisalteredbythetestchargeq,wetakethelimitofaninfinitelysmallcharge:0limqdqdqFFE(1.13)WithCoulomb'slaw(1.1)theelectricfieldofapointcharge1qis1131()()||qrrErrr.(1.14)Theelectricfieldvector()Erofapositivepointchargeisdirectedradiallyoutward;thatofanegativepointchargeisdirectedradiallyinward,asdepictedinFigure1.4.Accordingtotheprincipleofsuperposition,forasystemofpointchargeswehave3()()()||iiiiiiqrrErErrr.(1.15)2ThisisshowninFigure1.5.Inthepresenceofacontinuouschargedistribution,wehavetogofromasummationoverthepointchargestoanintegrationoverthespatialdistribution(seeFigure1.6).Insteadofthepointchargeiqwehavetoinsertthechargeelement()dVr.Here,anddVarethechargedensityandthevolumeelement,respectively:3()()||dVrrErrrr.(1.16)Tacitly,wehaveassumedtheprincipleofsuperpositiontobevalid.Butitsvalidityisnotself-evident.Aswewillseelater,itisidenticaltotheassumptionthatthefundamentalelectromagneticequationsarelinearequations.1Gauss'lawWecandeterminethefluxofthefieldintensityproducedbyapointchargethroughasurfaceSenclosingthischarge.Supposethatthepointchargeqisplacedattheoriginofthecoordinatesystem;thenthefluxthroughthesurfaceelementdaisgivenby2qdadarrrEnn(1.17)wherenisthenormalvectortothesurface(seeFigure1.7).But,thefieldintensitycrossingtheareadanisequaltothatcrossingtheareacosda.Expressedintermsofthesolidangle,thisimplies(Figures1.8and1.9):2cosdard.(1.18)Therefore,222cosqqdadardqdrrEn.(1.19)Thefluxcrossingthesurfaceisobtainedbyintegration4SSSdaqdqdqEn(1.20)becausetheintegrationoverthesolidangleyields4.So,weobtaintheresult4forinsidetheclosedsurface,0foroutsidetheclosedsurface.SqqdaqEn(1.21)TherelationderivedhereforapointchargeiscalledGauss'law.Ifthesurfaceen-closesseveralcharges,thenaccordingtotheprincipleofsuperposition,4iiiiiiSSSdadadaqEnEnEn(1.22)2orforacontinuouschargedistribution4()SVdadVEnr.(1.23)Thesurfaceintegralontheleft-handsideistransformedintoavolumeintegralwiththeaidoftheGausstheorem:divSVdadVEnE.(1.24)Hence,div4()VVdVdVEr(1.25)or[div4()]0VdVEr.(1.26)Sincethisisvalidforanarbitraryvolume,theintegrandhastobezero,andweobtaintherelationdiv()4()Err(1.27)betweenthefieldintensityandthechargedistributionproducingit.So,inspacethechargesarethesources(positivecharges)andsinks(negativecharges)oftheelectricfield.1TheelectricpotentialNowwedemonstratethattheelectricfieldcanbewrittenasthegradientofapotential(seeFigure1.10).Wehave3()()||dVrrErrrr.(1.28)Differentiatingtheexpression1||rrwithrespecttotheunprimedcoordinater,thenweseethat223111||||||||||||rrrrrrrrrrrrrrrr.(1.29)Fromthisrelation,oneobtainsforthefieldintensity1()()()||||dVdVrErrrrrr.(1.30)Thus,thefieldintensitycanbederivedasthegradientofapotential.Thepotential()risobtainedastheintegralovertheentirechargedistribution:()()||dVrrrr.(1.31)Withthisdefinition,wecanwriteforthefieldintensity()grad()()Errr.(1.32)Sincethecurlofagradientalwaysvanishes(0),weobtaincurl0EE.(1.33)So,wehaveshownthattheelectrostaticfieldcanbedescribedbythetwodifferentialequations4E(1.34)0E.(1.35)Thesecondequationimpliesthatelectrostaticforcesareconservativeforces.Inotherwords,theelectrostaticfieldisirrotational.Taking(1.32)intoaccount,weobtainfromthedivergenceofequation(1.34)2()4()rEr(1.36)or()4()rr.(1.37)ThisequationiscalledPoisson'sequation;foracharge-freeregion0,Poisson'sequationreducestotheLaplaceequation()0r.(1.38)Nowwewillshowthatapotentialoftheform1()||rrr(1.39)(apointcharge1qhasbeenchosen)satisfiesthePoissonequationforapointcharge.Forthat,weplacethechargeattheoriginofthecoordinatesystem(1()rr)2andapplytheLaplaceoperatortoit.Forapointchargetheproblemissphericallysymmetric,andwehavetoconsiderther-coordinateonly.For0rweobtainbysimplearithmetic2223363111111330rrrrrrrrrrrrrrrrror2222111110rrrrrrrrrr.(1.40)Butfor0rtheexpressionisnotdefined.Therefore,wehavetoperformalimitingprocedure:weintegrateinaneighborhoodof0randtransformthisvolumeintegralbymeansofGauss'theoremintoasurfaceintegralindependentofr:222111114dVdVdadardrrrrrrrnn.(1.41)So,wehaveshownthat10rfor0r(1.42)andthatforthevolumeintegralthefollowingrelationisvalid:14dVr.(1.43)1Mathematicalsupplement:Theδ-functionAtthispoint,itisusefultointroduceDirac's-function.Diracintroducedthe-functioninanalogytoKronecker'sik-symbol,asageneralizationforcontinuousindices()()()fafxxadx.(1.44)Hence,bythe-function()xa,thefunctionvalue()faatthepointxaisassignedtothefunction()fx;bytheexpressiongivenaboveitisdefinedonlyasafunctional.Quantitieslike()xcannotberegardedasfunctionsintheusualsense.TheyarenotintegrableintheframeworkofRiemann'snotionoftheintegral.The-functionistreatedinamathematicallyexactmannerwithinthetheoryofdistributions.Distributionmeansageneralizationofthenotionoffunctioninfunctionalanalysis;linearfunctionaloncertainabstractspaces.TheDirac-functionforexample,althoughveryimportantintheoreticalphysics,isadistribution,notafunction.ThetheoryofdistributionwasdevelopedbyL.Schwartzbetween1945and1950.Sincethen,ithasbeenmadeuseofinmanyfieldsofanalysis,e.g.,inthetheoryofdiffe-rentialequations,andinmodemphysics.Here,werestrictourselvestogivingsomepropertiesofthe-functionwithaheuristicproof.Thequantity()xcanbeviewedintermsofthelimitoffunctionshavingthepropertythatitvanisheseverywhereandbecomessingularatthepoint0xinsuchawaythat00()1xdx,0.(1.45)The-functionisthendefinedbythefollowingproperties:()0xawhenxa(1.46)and21121for,()0otherwise.bbbabxadx(1.47)Fortheproductofanarbitraryfunctionandthederivativeofthe-functiononeobtains()()()()()()()fxxadxfxxafxxadxfa.(1.48)Justthesameeasywayonefinds()(1||)()axax,since1()for0,||1()()11()()for0.||aaaaaaaazdzaaaxdxzdzazdzzdzaaa(1.49)Therefore,ingeneral,()()||axxa.Ifthe-functioncontainsafunction()fxoftheindependentvariablexintheargumentthenonecantransforminthefollowingway:1(())()|()|iiifxxxfx(1.50)if()fxhassimplezerosatixx(seeFigure1.11).Inthevicinityofazeroix,thesameargumentsapply.Thefactoraisreplacedby()ifxbecauseinthenear-estneighborhoodofixthefunction()fxcanbeapproximatedby()()iifxxx.Thiscanbeprovedmathematicallyinmoredetailinthefollowingway.2Letthefunction()fxhaveNsimplezeros(1,,)ixiN,thatis,()0ifx,()0ifx.The-functioncontributestotheintegral()(())Igxfxdxonlyifitsargumentvanishes.Henceitissufficienttoconsideronlythosecontributionstotheintegralgivenaboveresultingfrom(arbitrarilysmall)neighborhoodsofthein-dividualzerosixof()fx:1()(())iixaNixaIgxfxdx.(1.51)If()0ifx,itisalsononzeroinanintervalaboutix(let()fxbecontinuous).But,letabechosensosmallthat()0fxforall(,)iixxaxa,1,,in.Then,thefunctionisinvertibleinanyneighborhood(,)iixaxa;lettheinversefunctionbe1()ify.Withthehelpofthesubstitutionformulawethenobtain()1111111()((0))()(())()(())|((0))||()|iifxaNNNiiiiiiiiifxagfgxdyIgfyyffyfffx(1.52)since1(0)iifx.Theabsolutevalueappearssince()0()()iiifxfxafxathatis,thelimitsofintegrationhavetobeinterchangedintheintegral.Butthisexpre-ssioncanbewrittenas11()()|()|NiiiIxxgxdxfx.(1.53)Distributionscanbeunderstoodalsoasalimitoffunctions.Forexample,weconsiderthefamilyoffunctions221(,)xfxe(1.54)whereisaparameter.Theintegralofthisfunctionis22222111(,)xyxyfxdxedxedyedxdy20121ed(1.55)thatis,itisindependentof.Nowforanypoint0xthelimitofthesefunctionsapproacheszeroas0.Hence,22001lim(,)lim()xfxex.(1.56)Suchrepresentationsofdistributionsintermsoflimitsoffunctionsareoftenveryusefultoidentifythepropertiesofdistributions.AnotherapproachisofferedbythenotionofWeyl'seigendifferentials,withwhichwewillbeconcernedinmoredetailinthelecturesonquantummechanics.3Forspatialandmultidimensionalproblemswewritethe-functionwithavectorialargumentbyformingthe-functionoftheindividualCartesiancomponents:()()()()xyzxayazara(1.57)isafunctionalvanishingeverywhereexceptra.Forexample,bymeansofthe-functionwecandescribeasequenceofpointcharges,likeacontinuouschargedistri-bution.Achargedensityoftheform1()()Niiiqrra(1.58)describesNpointchargesofthemagnitudeiqsittingatthepointsia.Inthesameway,withthehelpofthe-functionwecanwritethetworelations(1.42)and(1.43)inaclosedform:14()rr.(1.59)Ifthechargeisnotplacedattheorigin,onehascorrespondingly14()||rrrr.(1.60)Ifthefieldisproducedbyacontinuouschargedistribution()r,then()1()()||||dVdVrrrrrrr4()()4()dVrrrr(1.61)correspondingexactlytothePoissonequation(1.37).1ThepotentialenergyofachargeinanelectricfieldWhenachargeqismovedinanelectricfieldofintensity()ErfromAtoB,theforceqFEactsonit.Theworkdoneduringthismotionisgivenbythepathintegralovertheforce(Figure1.12)()()BBAAWdqdFrrErr.(1.62)Theminussignappearsbecauseworkdoneagainstthefieldisconsideredtobepositive.Ifinthisequationthefieldintensityisexpressedbythepotential,then()(()())BBAAWqdqdqBArr(1.63)thatis,theworkdonealongthepathfromAtoBcorrespondsexactlytothedifferenceinthepotentialenergyatthesetwopoints.Thus,theworkisindependentofthepath.注.此处的W是指外力所做,且被储存为势能的那部分功.1ThefieldintensityacrosschargedsurfacesWenowconsiderthebehaviorofthecomponentsoftheelectricfieldintensityacrossachargedsurface.Thesurfacechargeisdescribedbythechargedensity,havingthedimensionofchargeperarea:0()()limaqarr.(1.64)Anareaelementacarriesthecharge()aqdar.TheareaelementashouldlieinsidethevolumeelementV.ApplyingGauss'law,weobtain44()SadaqdaEnr(1.65)wheretheintegralextendsoverthesurfaceSofthevolumeVenclosingtheareaelementa.Now,wechoosethevolumeelementtobeacuboidofnegligiblethickness;thenthefluxoftheE-fieldthroughthelateralsurfacescanbeneglected,andweobtainthefluxthroughtheareabyintegration.Forbookkeepingpurposes,wewilllabelthefieldintensitybeforeandaftercrossingthesurfacebytheindices1and2,respectively.Hence,oneobtains(Figure1.13)21()4()SdaaaEnEnEnr.Bycancellationandfactoringout,wefurtherobtain214()EnEnr(1.66)thatis,thenormalcomponentoftheE-fieldchangesbytheamount4incross-ingachargedsurface.Sincethefieldintensitycanbederivedfromthepotential,weknowthatcurlcurlgrad0E,andthecirculationintegralvanishesduetotheStokestheorem:curl0ddaErEn.WeintegratenowalongtheclosedpathparalleltothecomponentsoftheE-fieldtangentialtothesurfaceSofthevolumeelementVwhereweassumethatthethicknessofthevolumeelementcanbeneglected(seeFigure1.14).Thenthecontributionsfromthefrontsurfacescanbealsoneglected,andweobtain21()0ttdErEErforallr.(1.67)Hence,21ttEE;thatis,thetangentialcomponentoftheE-fielddoesnotchangeacrossachargedsurface.Therefractionoftheelectricfieldcrossinga(positively)chargedlayerisillustratedinFigure1.13.1Example1.1:Theparallel-platecapacitorAsanexamplefortheapplicationofGauss'law,wewillcalculatethefieldintensityinaparallel-platecapacitor.Thisdeviceconsistsoftwoconductingplatesarrangedparalleltoeachotherataseparationd(seeFigures1.15and1.16).Thechargedensityontheplatesisconstant:qaorqa,whereqrepresentstheentirechargeononeplate.Tosimplifytheproblem,wewillneglectthestrayfieldsattheedgeofthecapacitorandassumethatthereisnofieldoutsidethecapacitor;then,wecancalculatethefieldintensityEfromequation(1.66)byapplyingGauss'lawforoneplate:44qEaEn0n.nisanormalvectortotheplatesandpointsfromthepositivelychargedplatetothenegativelychargedplate;asdoestheelectricfieldE.Thefieldintensityinsidethecapacitorisconstant.Forthepotentialdifferencebetweentheplatesweobtain210000dddddddEdxEdrEr.InsertingtherelationfoundforEyieldsthevoltageV:2112()44qVEddda.Thevoltageisthedifferenceofthepotentialsatthepositivelychargedplate(1)andthenegativelychargedplate(2).Now,wedefinethecapacitanceCofacapacitorbytheequationqCV.ThistellsusthechargeqthecapacitorcancarryforagivenvoltageV.Thus,thecapacitanceofaparallel-platecapacitoris4aCd.(1.68)1Exercise1.2:ThesphericalcapacitorCalculatethecapacitanceofacapacitorconsistingoftwoconcentricsphericalshells(seeFigure1.17).Eachspherecarriesauniformlydistributedcharge.Theradiiofthespheresare1rand2r.Theouterspherehasthenegativechargeq,andtheinnerspherehasthepositivechargeq.SolutionBecauseofthesphericallysymmetricarrangement,thefielddistributionisalsosphericallysymmetric.Usingsphericalcoordinates,weneedtoconsideronlyther-component;thefieldhasnoothercomponents.Theinteriorofachargedsphericalshellisfield-free.ThisfollowsimmediatelyfromGauss'law,sincenochargeispresentinsidethesphere,andalways22440nnSSdEdarErEEawherergivestheradiusofasphericalsurfacelyinginsidethechargedistribution.Forthesphericalcapacitor,thismeansthattheelectricfieldEoriginatesonlyfromtheinnersphere.Placingasphericalsurfaceofradiusrbetweenthesphericalshells,thenaccordingtoGauss'law:22444nnSSdEdarErEqEa.Hence,thefieldintensitydependsquadraticallyontheradius:2()()nqErErr.(1.69)Thepotentialdifferencebetweentheinnerandtheouterspheresis22222111111221211rrrrrnrrrrrqVdddEdrdrqrrrrEr.So,weobtainforthecapacitanceofasphericalcapacitor1221rrqCVrr.(1.70)Concerningtheresultforthefieldintensity,wenotethatthefieldintensityisthesameasthatofapointchargeofequalstrengthplacedatthecenterofthesphere.Thisresultisvalidforallsphericallysymmetricchargedistributions.1Exercise1.3:ThecylindricalcapacitorCalculatethecapacitanceofacapacitorconsistingoftwocoaxialcylindersofheighthandradii1rand2r(seeFigure1.18).Ne