64Introduction to Tensor Calculus and Continuum Me

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=N
m2iscalledthepermittivityinavacuum,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)wheredisanelementofvolume.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,dordwithr=j~rjandber=~rr:Iftheelectriceldisconservative,thenr~E=0,andsoitisderivablefromapotentialfunctionVbytakingthenegativeofthegradientofVand~E=−rV:(2:6:7)FortheseconditionsnotethatrV
d~r=−~E
d~risanexactdierentialsothatthepotentialfunctioncanberepresentedbythelineintegralV=V(P)=−ZP~E
d~r(2:6:8)whereissomereferencepoint(usuallyinnity,whereV(1)=0).ForaconservativeelectriceldthelineintegralwillbeindependentofthepathconnectinganytwopointsaandbsothatV(b)−V(a)=−Zb~E
d~r−−Za~E
d~r=−Zba~E
d~r=ZbarV
d~r:(2:6:9)Let=1inequation(2.6.8),thenthepotentialfunctionassociatedwithapointchargemovingintheradialdirectionberisV(r)=−Zr1~E
d~r=−q40Zr11r2dr=q401rjr1=q40r:327Bysuperposition,thepotentialatapointPforacontinuousvolumedistributionofchargesisgivenbyV(P)=140ZZZVrdandforasurfacedistributionofchargesV(P)=140ZZSrdandforalinedistributionofchargesV(P)=140ZCrds;andforadiscretedistributionofpointchargesV(P)=140NXi=1qiri.Whenthepotentialfunctionsaredenedfromacommonreferencepoint,thentheprincipalofsuperpositionapplies.ThepotentialfunctionVisrelatedtotheworkdoneWinmovingachargewithintheelectriceld.TheworkdoneinmovingatestchargeQfrompointatopointbisanintegraloftheforcetimesdistancemoved.TheelectricforceonatestchargeQis~F=Q~Eandsotheforce~F=−Q~Eisinoppositiontothisforceasyoumovethetestcharge.TheworkdoneisW=Zba~F
d~r=Zba−Q~E
d~r=QZbarV
d~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