工程电磁场第三章

EngineeringElectromagneticsW.H.HaytJr.andJ.A.BuckChapter3:ElectricFluxDensity,Gauss’Law,andDivergenceFaradayExperimentHestartedwithapairofmetalspheresofdifferentsizes;thelargeroneconsistedoftwohemispheresthatcouldbeassembledaroundthesmallersphere+QFaradayApparatus,BeforeGroundingTheinnercharge,Q,inducesanequalandoppositecharge,-Q,ontheinsidesurfaceoftheoutersphere,byattractingfreeelectronsintheoutermaterialtowardthepositivecharge.Thismeansthatbeforetheoutersphereisgrounded,charge+Qresidesontheoutsidesurfaceoftheouterconductor.FaradayApparatus,AfterGroundingq=0groundattachedAttachingthegroundconnectstheoutersurfacetoanunlimitedsupplyoffreeelectrons,whichthenneutralizethepositivechargelayer.Thenetchargeontheoutersphereisthenthechargeontheinnerlayer,or-Q.InterpretationoftheFaradayExperimentq=0Faradayconcludedthatthereoccurredacharge“displacement”fromtheinnerspheretotheoutersphere.Displacementinvolvesafloworflux,existingwithinthedielectric,andwhosemagnitudeisequivalenttotheamountof“displaced”charge.Specifically:ElectricFluxDensityq=0Thedensityoffluxattheinnerspheresurfaceisequivalenttothedensityofchargethere(inCoul/m2)VectorFieldDescriptionofFluxDensityq=0Avectorfieldisestablishedwhichpointsinthedirectionofthe“flow”ordisplacement.Inthiscase,thedirectionistheoutwardradialdirectioninsphericalcoordinates.Ateachsurface,wewouldhave:Radially-DependentElectricFluxDensityq=0rAtageneralradiusrbetweenspheres,wewouldhave:ExpressedinunitsofCoulombs/m2,anddefinedovertherange(a≤r≤b)D(r)PointChargeFieldsIfwenowlettheinnersphereradiusreducetoapoint,whilemaintainingthesamecharge,andlettheoutersphereradiusapproachinfinity,wehaveapointcharge.Theelectricfluxdensityisunchanged,butisdefinedoverallspace:C/m2(0r∞)Wecomparethistotheelectricfieldintensityinfreespace:V/m(0r∞)..andweseethat:FindingEandDfromChargeDistributionsWelearnedinChapter2that:Itnowfollowsthat:Gauss’LawTheelectricfluxpassingthroughanyclosedsurfaceisequaltothetotalchargeenclosedbythatsurfaceDevelopmentofGauss’LawWedefinethedifferentialsurfacearea(avector)aswherenistheunitoutwardnormalvectortothesurface,andwheredSistheareaofthedifferentialspotonthesurfaceMathematicalStatementofGauss’LawLinecharge:Surfacecharge:Volumecharge:inwhichthechargecanexistintheformofpointcharges:Foravolumecharge,wewouldhave:oracontinuouschargedistribution:UsingGauss’LawtoSolveforDEvaluatedataSurfaceKnowingQ,weneedtosolveforD,usingGauss’Law:Thesolutioniseasyifwecanchooseasurface,S,overwhichtointegrate(Gaussiansurface)thatsatisfiesthefollowingtwoconditions:Theintegralnowsimplfies:Sothat:whereExample:PointChargeFieldBeginwiththeradialfluxdensity:andconsiderasphericalsurfaceofradiusathatsurroundsthecharge,onwhich:Onthesurface,thedifferentialareais:andthis,combinedwiththeoutwardunitnormalvectoris:PointChargeApplication(continued)Now,theintegrandbecomes:andtheintegralissetupas:==AnotherExample:LineChargeFieldConsideralinechargeofuniformchargedensityLonthezaxisthatextendsovertherangezWeneedtochooseanappropriateGaussiansurface,beingmindfuloftheseconsiderations:Weknowfromsymmetrythatthefieldwillberadially-directed(normaltothezaxis)incylindricalcoordinates:andthatthefieldwillvarywithradiusonly:Sowechooseacylindricalsurfaceofradius,andoflengthL.LineChargeField(continued)Giving:Sothatfinally:AnotherExample:CoaxialTransmissionLineWehavetwoconcentriccylinders,withthezaxisdowntheircenters.SurfacechargeofdensitySexistsontheoutersurfaceoftheinnercylinder.A-directedfieldisexpected,andthisshouldvaryonlywith(likealinecharge).WethereforechooseacylindricalGaussiansurfaceoflengthLandofradius,whereab.ThelefthandsideofGauss’Lawiswritten:…andtherighthandsidebecomes:CoaxialTransmissionLine(continued)Wemaynowsolveforthefluxdensity:andtheelectricfieldintensitybecomes:CoaxialTransmissionLine:ExteriorFieldIfaGaussiancylindricalsurfaceisdrawnoutside(b),atotalchargeofzeroisenclosed,leadingtotheconclusionthator:ElectricFluxWithinaDifferentialVolumeElementTakingthefrontsurface,forexample,wehave:ElectricFluxWithinaDifferentialVolumeElementWenowhave:andinasimilarmanner:Therefore:minussignbecauseDx0isinwardfluxthroughthebacksurface.ChargeWithinaDifferentialVolumeElementNow,byasimilarprocess,wefindthat:andAllresultsareassembledtoyield:v=Q(byGauss’Law)whereQisthechargeenclosedwithinvolumevDivergenceandMaxwell’sFirstEquationMathematically,thisis:Applyingourpreviousresult,wehave:divA=andwhenthevectorfieldistheelectricfluxdensity:=divDMaxwell’sfirstequationDivergenceExpressionsintheThreeCoordinateSystemsTheDelOperator=divD=Thedeloperatorisavectordifferentialoperator,andisdefinedas:Notethat:DivergenceTheoremWenowhaveMaxwell’sfirstequation(orthepointformofGauss’Law)whichstates:andGauss’sLawinlarge-scaleformreads:leadingtotheDivergenceTheorem:StatementoftheDivergenceTheorem