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325x2.6ELECTRICANDMAGNETICFIELDSIntroductionInelectromagnetictheorythemkssystemofunitsandtheGaussiansystemofunitsaretheonesmostoftenencountered.Inthissectiontheequationswillbegiveninthemkssystemofunits.IfyouwanttheequationsintheGaussiansystemofunitsmakethereplacementsgiveninthecolumn3ofTable1.Table1.MKSANDGAUSSIANUNITSMKSsymbolMKSunitsReplacementsymbolGAUSSIANunits~E(Electric eld)volt=m~Estatvolt=cm~B(Magnetic eld)weber=m2~Bcgauss~D(Displacement eld)coulomb=m2~D4statcoulomb=cm2~H(AuxiliaryMagnetic eld)ampere=mc~H4oersted~J(Currentdensity)ampere=m2~Jstatampere=cm2~A(Vectorpotential)weber/m~Acgauss-cmV(Electricpotential)voltVstatvolt(Dielectricconstant)4(Magneticpermeability)4c2ElectrostaticsAbasicprobleminelectrostatictheoryistodeterminetheforce~FonachargeQplacedadistancerfromanotherchargeq.ThesolutiontothisproblemisCoulomb'slaw~F=140qQr2ber(2:6:1)whereq,Qaremeasuredincoulombs,0=8:8510−12coulomb2=Nm2iscalledthepermittivityinavacuum,risinmeters,[~F]hasunitsofNewtonsandberisaunitvectorpointingfromqtoQifq;QhavethesamesignorpointingfromQtoqifq;Qareofoppositesign.Thequantity~E=~F=Qiscalledtheelectric eldproducedbythecharges.InthespecialcaseQ=1,wehave~E=~FandsoQ=1iscalledatestcharge.Thistellsusthattheelectric eldatapointPcanbeviewedastheforceperunitchargeexertedonatestchargeQplacedatthepointP.ThetestchargeQisalwayspositiveandsoisrepulsedifqispositiveandattractedifqisnegative.Theelectric eldassociatedwithmanychargesisobtainedbytheprincipalofsuperposition.Forexample,letq1;q2;:::;qndenoten-chargeshavingrespectivelythedistancesr1;r2;:::;rnfromatestchargeQplacedatapointP:TheforceexertedonQis~F=~F1+~F2++~Fn~F=140q1Qr21ber1+q2Qr22ber2++qnQr2nbernor~E=~E(P)=~FQ=140nXi=1qir2iberi(2:6:2)326where~E=~E(P)istheelectric eldassociatedwiththesystemofcharges.Theequation(2.6.2)canbegen-eralizedtoothersituationsbyde ningothertypesofchargedistributions.Weintroducealinechargedensity,(coulomb=m),asurfacechargedensity,(coulomb=m2),avolumechargedensity,(coulomb=m3),thenwecancalculatetheelectric eldassociatedwiththeseothertypesofchargedistributions.Forexample,ifthereisachargedistribution=(s)alongacurveC,wheresisanarclengthparameter,thenwewouldhave~E(P)=140ZCberr2ds(2:6:3)astheelectric eldatapointPduetothischargedistribution.Theintegralinequation(2.6.3)beingalineintegralalongthecurveCandwheredsisanelementofarclength.Hereequation(2.6.3)representsacontinuoussummationofthechargesalongthecurveC.ForacontinuouschargedistributionoverasurfaceS,theelectric eldatapointPis~E(P)=140ZZSberr2d(2:6:4)wheredrepresentsanelementofsurfaceareaonS.Similarly,ifrepresentsacontinuouschargedistri-butionthroughoutavolumeV,thentheelectric eldisrepresented~E(P)=140ZZZVberr2d (2:6:5)whered isanelementofvolume.Intheequations(2.6.3),(2.6.4),(2.6.5)welet(x;y;z)denotethepositionofthetestchargeandlet(x0;y0;z0)denoteapointontheline,onthesurfaceorwithinthevolume,then~r=(x−x0)be1+(y−y0)be2+(z−z0)be3(2:6:6)representsthedistancefromthepointPtoanelementofchargeds,dord withr=j~rjandber=~rr:Iftheelectric eldisconservative,thenr~E=0,andsoitisderivablefromapotentialfunctionVbytakingthenegativeofthegradientofVand~E=−rV:(2:6:7)FortheseconditionsnotethatrVd~r=−~Ed~risanexactdi erentialsothatthepotentialfunctioncanberepresentedbythelineintegralV=V(P)=−ZP ~Ed~r(2:6:8)where issomereferencepoint(usuallyin nity,whereV(1)=0).Foraconservativeelectric eldthelineintegralwillbeindependentofthepathconnectinganytwopointsaandbsothatV(b)−V(a)=−Zb ~Ed~r−−Za ~Ed~r=−Zba~Ed~r=ZbarVd~r:(2:6:9)Let =1inequation(2.6.8),thenthepotentialfunctionassociatedwithapointchargemovingintheradialdirectionberisV(r)=−Zr1~Ed~r=−q40Zr11r2dr=q401rjr1=q40r:327Bysuperposition,thepotentialatapointPforacontinuousvolumedistributionofchargesisgivenbyV(P)=140ZZZVrd andforasurfacedistributionofchargesV(P)=140ZZSrdandforalinedistributionofchargesV(P)=140ZCrds;andforadiscretedistributionofpointchargesV(P)=140NXi=1qiri.Whenthepotentialfunctionsarede nedfromacommonreferencepoint,thentheprincipalofsuperpositionapplies.ThepotentialfunctionVisrelatedtotheworkdoneWinmovingachargewithintheelectric eld.TheworkdoneinmovingatestchargeQfrompointatopointbisanintegraloftheforcetimesdistancemoved.TheelectricforceonatestchargeQis~F=Q~Eandsotheforce~F=−Q~Eisinoppositiontothisforceasyoumovethetestcharge.TheworkdoneisW=Zba~Fd~r=Zba−Q~Ed~r=QZbarVd~r=Q[V(b)−V(a)]:(2:6:10)Theworkdoneisindependentofthepathjoiningthetwopointsanddependsonlyontheendpointsandthechangeinthepotential.IfonemovesQfromin nitytopointb,thentheabovebecomesW=QV(b):Anelectric eld~E=~E(P)isavector eldwhichcanberepresentedgraphicallybyconstructingvectorsatvariousselectedpointsinthespace.Suchaplotiscalledavector eldplot.A eldlineassociatedwithavector eldisacurvesuchthatthetangentvectortoapointonthecurvehasthesamedirectionasthevector eldatthatpoint.Fieldlinesareusedasanaidforvisualizationofanelectric eldandvector eldsingeneral.Thetangenttoa eldlineatapointhasthesamedirectionasthevector eld~Eatthat