UniversityPhysicsJune32025Part4Electromagnestism9.7Capacitor9.7.1CapacitanceofIsolatedConductors9.7.2Capacitanceofacapacitor9.7.3CalculatingtheCapacitanceofCapacitor9.7.4CapacitorsinParallelandSeriesContents:9.7.1CapacitanceofIsolatedConductorsApairofconductors,whetherseparatedbyemptyspaceorbyanonconductingmaterial,formsacapacitor.Unit：1F=1C/V,F,pFDefinitionTuningFunctionStoresenergyFilterClassificationExampleWhatisthecapacitanceofanisolatedconductingsphereofradiusR?Calculatethecapacitanceoftheearth(theradiusofearthisabout6400km)?Solution:Iftheearthhavebeenchargedwithq,thenthenpotentialisThecapacitanceis9.7.2CapacitanceofacapacitorThecapacitanceofacapacitorisdefinedastheratiooftheamountofelectricitychargedq(q>0)ononeplatetothepotentialdifferenceΔVbetweenthetwoplates,thatis,Thecapacitanceofacapacitordependsonthesize,shape,andrelativepositionoftheplatesandthepermittivityofthedielectricbetweentheplates,anditdoesn’tdependonwhetherthecapacitorischargedornot.極板極板+

Q-

Q

V9.7.3CalculatingtheCapacitanceofCapacitorSteps:

1.Assumeachargeqoneachplateofacapacitor.2.CalculatetheelectricfieldEbetweentheplatesintermsofthischarge,usingGauss’law,3.CalculatethepotentialdifferenceVbetweenthetwoplates,

4.CalculateCfrom

1.ParallelPlateCapacitorFindthecapacitanceofaparallel-platecapacitorwithplatesofareaAseparatedbyasmalldistanced.2．CylindricalCapacitorAcylindricalcapacitorisaconductorsetconsistingoftwocoaxialcylindricalconductorcylinders.andtheradiusoftheinnerandoutercylindricalconductorsareR1andR2,respectively.3.SphericalCapacitorAsphericalcapacitorconsistsoftwothinconcentricsphericalconductingshells,ofradiiaandb.Determinethecapacitanceofthiscapacitor.9.7.4CapacitorsinParallelandSeriesCapacitorsinParallelPotentialdifferencebetweenaandbCharges

Equivalentcapacitance

ThereforeEndurancevoltageisunchanged.CapacitorsinSeriesPotentialdifferencebetweenaandbChargesEquivalentcapacitanceThereforeEndurancevoltageincreases

CapacitorsinSeries9.8ElectrostaticEnergy9.8.1Electrostaticenergyofachargedsystem9.8.2ElectrostaticEnergyofAnElectricalFieldContents:9.8.1ElectrostaticenergyofachargedsystemEnergyinaChargedCapacitorTheenergystoredinacapacitorwillbeequaltotheworkdonetochargeit,Atagiveninstant

Ifanadditionalelementofchargedq

istransferred,theamountofworkrequiredisThetotalworkWneededtoincreasethecapacitorchargefromzerotoafinalvalueq9.8.2ElectrostaticEnergyofAnElectricalFieldEnergyinElectricFieldsElectricfieldenergydensityForanyfieldExample9.8.1

Inthesphericalcapacitor,thevolumebetweentheconcentricsphericalconductingshellsisfilledwithaninsulatingoilwithdielectricconstant

r.FindtheenergystoredinthiscapacitorwithachargeQ.Solution:AccordingtoGauss'slawindielectrics,wehaveThen,themagnitudeoftheelectricaldisplacementD

betweenthesphericalshellsisTheenergydensityoftheelectricfieldisThetotalenergystoredintheelectricfieldisSummaryCapacitanceCalculatingtheCapacitanceofCapacitor

1.Assumeachargeqoneachplateofacapacitor.3.CalculatethepotentialdifferenceVbetweenthetwoplates,

4.CalculateCfrom

2.CalculatetheelectricfieldEbetweenthe