电磁场与电磁波第11讲焦耳定律边界条件电阻计算及第5章复习

FieldandWaveElectromagnetic电磁场与电磁波第11讲1作业情况1班：人合计：人情况:21.CurrentDensityandOhm’sLawReview33.EquationofContinuityandKirchhoff’sCurrentLaw2.ElectromotiveForceandKirchhoff’sVoltageLawOutsidethesourceInsidethesourceEConductingmediumPNEImpressedsourceEi4Twofieldsarefoundtobe

verysimilar

insource-freeregion.

SteadyElectricCurrentField

ElectrostaticField

The

electric

currentdensity

J

correspondstothe

electric

fieldintensity

E,andtheelectric

current

linestotheelectricfield

lines.Insomecases,sincethesteadyelectriccurrentfieldiseasytobeconstructedandmeasured,theelectrostaticfieldcanbeinvestigatedbasedonthesteadyelectriccurrentfieldwiththesameboundaryconditions,andthismethodiscalled

electrostaticsimulation.5CapacitanceJEResistanceBasedontheequationsfortwofields,wecanfindtheresistanceandconductancebetweentwoelectrodesas6Incertainsituations,electrostaticandsteady-currentproblemsarenotexactlyanalogous,evenwhenthegeometricalconfigurationsarethesame.Thisisbecausecurrentflowcanbeconfinedstrictlywithinaconductor(whichhasaverylargeincomparisontothatofthesurroundingmedium),whereaselectricfluxusuallycannotbecontainedwithinadielectricslaboffinitedimensions.Therangeofthedielectricconstantofavailablematerialsisverylimited,andthefluxfringingaroundconductoredgesmakesthecomputationofcapacitancelessaccurate.7MaintopicSteadyElectricCurrents3.ResistanceCalculations1.PowerDissipationandJoule’sLaw2.BoundaryConditionsforCurrentDensity81.PowerDissipationandJoule’sLawWehaveindicatedthatundertheinfluenceofanelectricfield,conductionelectronsinaconductorundergoadriftmotionmacroscopically.Microscopicallytheseelectronscollidewithatomsonlatticesites（格点）.Energyisthustransmittedfromtheelectricfieldtotheatomsinthermalvibration.TheworkwdonebyanelectricfieldEinmovingachargeqadistanceℓisqE·ℓ,whichcorrespondstoapowerWhereuisthedriftvelocity.Thetotalpowerdeliveredtoallthechargecarriersinvolumedvis:9ThetotalelectricpowerconvertedintoheatinvolumeV:ThisisknownasJoule’slaw.ThepointfunctionE·Jisapowerdensityundersteady-currentconditions.Inaconductorofaconstantcrosssection,wecanwrittenas102.BoundaryConditionsforCurrentDensityWhencurrentobliquelycrossesaninterfacebetweentwomediawithdifferentconductivities(1≠2),thecurrentdensityvectorchangesbothindirectionandinmagnitude.AsetofboundaryconditionscanbederivedforJinawaysimilartothatusedinSection3-9forobtainingtheboundaryconditionsforDandE.ThegoverningequationsforsteadycurrentdensityJintheabsenceofnon-conservativeenergysourcesareDifferentialformIntegralformGoverningEquationsforSteadyCurrentDensity11E2E1

2

1atwhacdban2hS

2

1an2D1D2s12J2J1

2

1atwhacdban2hS2

1an2J1J2sthenormalcomponentofcurrentdensityvector

J

beingcontinuous.theratioofthetangentialcomponentsofcurrentdensityvectorJattwosidesofaninterfaceisequaltotheratiooftheconductivities.13

1,1an2E2,D2,J2s

2,2E1,D1,J1Whenasteady-currentflowsacrosstheboundarybetweentwodifferentlossydielectrics:14ForahomogeneousconductingmediumWeknowthatacurl-freevectorfieldcanbeexpressedasthegradientofascalarpotentialfield.LetuswriteSubstitutionofthisequationintoyieldsaLaplace’sequationin;thatis15Example5-4P214:

Anemf

isappliedacrossaparallel-platecapacitorofareaS.Thespacebetweentheconductiveplatesisfilledwithtwodifferentlossydielectricsofthicknessd1

andd2,permittivity1

and2

,andconductivities1

and2

respectively.Determine(a)thecurrentdensitybetweentheplates,(b)theelectricfieldintensitiesinbothdielectrics,and(c)thesurfacechargedensitiesontheplatesandattheinterface.x12++++++++++++---------------yo161.x12++++++++++++---------------yo2.MethodoneOrmethodtwo173.181.Chooseanappropriatecoordinatesystemforthegivengeometry.2.AssumeapotentialdifferenceV0betweenconductorterminals.3.FindEfromE=-V(2V=0),orotherrelations.4.FindthetotalcurrentwhereSisthecross-sectionalareaoverwhichIflows.5.FindresistanceRbytakingtheratioV0/I.Theprocedureforcomputingtheresistanceofapieceofconductingmaterialbetweenspecifiedequipotentialsurfaces(orterminals)isasfollows:3.ResistanceCalculations19Example5-6.Aconductingmaterialofuniformthicknesshandconductivityhasshapeofaquarterofaflatcircularwasher,withinnerradiusaandouterradiusb,asshowninthefigure.Calculatetheresistancebetweentwoendfaces.思路V0yxhabr0(r,)020Solution:Thecylindricalcoordinatesystemshouldbeselected.Assumetheelectricpotentialdifferencebetweentwoendfacesis

V0,andlet

SincetheelectricpotentialV

isrelatedtotheangle,itshouldsatisfythefollowingequationThegeneralsolutionisTheelectricpotentialatTheelectricpotentialatV0yxhabr0(r,)021

Basedonthegivenboundaryconditions,wefindThecurrentdensity

J

intheconductingmediumis

ThenthecurrentI

flowingintotheconductingmediumacrosstheendfaceatisConsequently,theresistance

R

betweentwoendfacesis22summary1.PowerDissipationandJoule’sLaw3.ResistanceCalculationselectrostaticsimulation232.BoundaryConditionsforCurrentDensityDifferentialformIntegralformGoverningEquationsforSteadyCurrentDensityWhenasteady-currentflowsacrosstheboundarybetweentwodifferentlossydielectrics:24homeworkThankyou!Bye-bye!答疑安排时间：周一

下午14：00~16：00地点：1401，1