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FieldandWaveElectromagnetic電磁場與電磁波第11講1作業(yè)情況1班:人合計(jì):人情況: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(格點(diǎn)).Energyisthustransmittedfromtheelectricfieldtotheatomsinthermalvibration.TheworkwdonebyanelectricfieldEinmovingachargeqadistance?isqE·?,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!答疑安排時(shí)間:周一
下午14:00~16:00地點(diǎn):1401,1
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