Capacitance-and-Dielectrics-Home-The-University-of-电容与介电层里大学课件

CapacitorsDeviceforstoringelectricalenergywhichcanthenbereleasedinacontrolledmannerConsistsoftwoconductors,carryingchargesofqand–q,thatareseparated,usuallybyanonconductingmaterial-aninsulatorSymbolincircuitsisIttakeswork,whichisthenstoredaspotentialenergyintheelectricfieldthatissetupbetweenthetwoplates,toplacechargesontheconductingplatesofthecapacitorSincethereisanelectricfieldbetweentheplatesthereisalsoapotentialdifferencebetweentheplatesCapacitorsWeusuallytalkaboutcapacitorsintermsofparallelconductingplatesTheyinfactcanbeanytwoconductingobjectsCapacitanceUnitsareThecapacitanceisdefinedtobetheratiooftheamountofchargethatisonthecapacitortothepotentialdifferencebetweentheplatesatthispointCalculatingtheCapacitanceWestartwiththesimplestform–twoparallelconductingplatesseparatedbyvacuumLettheconductingplateshaveareaAandbeseparatedbyadistancedThemagnitudeoftheelectricfieldbetweenthetwoplatesisgivenbyWetreatthefieldasbeinguniformallowingustowriteCalculatingtheCapacitancePuttingthisalltogether,wehaveforthecapacitanceThecapacitanceisonlydependentuponthegeometryofthecapacitor1faradCapacitorGivena1faradparallelplatecapacitorhavingaplateseparationof1mm.Whatistheareaoftheplates?WestartwithAndrearrangetosolveforA,givingThiscorrespondstoasquareabout10kmonaside!SeriesorParallelCapacitorsSometimesinordertoobtainneededvaluesofcapacitance,capacitorsarecombinedineitherorSeriesParallelCapacitorsinSeriesCapacitorsareoftencombinedinseriesandthequestionthenbecomeswhatistheequivalentcapacitance?GivenwhatisWestartbyputtingavoltage,Vab,acrossthecapacitorsCapacitorsinSeriesCapacitorsbecomechargedbecauseofVab

IfupperplateofC1getsachargeof+Q,ThenthelowerplateofC1getsachargeof-QWhathappenswithC2?Sincethereisnosourceofchargeatpointc,andwehaveeffectivelyputachargeof–QonthelowerplateofC1,theupperplateofC2getsachargeof+QThisthenmeansthatlowerplateofC2hasachargeof-QChargeConservationWealsohavetohavethatthepotentialacrossC1plusthepotentialacrossC2shouldequalthepotentialdropacrossthetwocapacitorsWehaveThenDividingthroughbyQ,wehaveCapacitorsinSeriesThelefthandsideistheinverseofthedefinitionofcapacitanceSowethenhavefortheequivalentcapacitanceIftherearemorethantwocapacitorsinseries,theresultantcapacitanceisgivenbyTheequivalentcapacitorwillalsohavethesamevoltageacrossitCapacitorsinSeriesCapacitorsinParallelCapacitorscanalsobeconnectedinparallelGivenwhatisAgainwestartbyputtingavoltageacrossaandbTheupperplatesofbothcapacitorsareatthesamepotentialLikewiseforthebottomplatesWehavethatNowCapacitorsinParallelTheequivalentcapacitorwillhavethesamevoltageacrossit,asdothecapacitorsinparallelButwhataboutthechargeontheequivalentcapacitor?TheequivalentcapacitorwillhavethesametotalchargeUsingthiswethenhaveCapacitorsinParallelTheequivalentcapacitanceisjustthesumofthetwocapacitorsIfwehavemorethantwo,theresultantcapacitanceisjustthesumoftheindividualcapacitancesCapacitorsinParallelExample1CC1C2C3ababºÞWheredowestart?RecognizethatC1andC2areparallelwitheachotherandcombinethesetogetC12ThisC12

istheninserieswithwithC3

TheresultantcapacitanceisthengivenbyWhichoftheseconfigurationshasthelowestoverallcapacitance?Threeconfigurationsareconstructedusingidenticalcapacitorsa)

ConfigurationA

b)

ConfigurationB

c)

ConfigurationCCCCCCConfigurationAConfigurationBConfigurationCExample2ThenetcapacitanceforAisjustCInB,thecapsareinseriesandtheresultantisgivenbyInC,thecapsareinparallelandtheresultantisgivenbyAcircuitconsistsofthreeunequalcapacitorsC1,C2,andC3whichareconnectedtoabatteryofemfE.ThecapacitorsobtainchargesQ1

Q2,Q3,andhavevoltagesacrosstheirplatesV1,V2,andV3.Ceqistheequivalentcapacitanceofthecircuit.Checkallofthefollowingthatapply:a)

Q1=Q2

b)

Q2=Q3 c)

V2=V3

d)

E=V1

e)

V1<V2

f)

Ceq>C1Example3Adetailedworksheetisavailabledetailingtheanswers(a)

Ceq

=(3/2)C(b)

Ceq

=(2/3)C(c)

Ceq

=3CooCCCCeqCCCCC1Example4Whatistheequivalentcapacitance,Ceq,ofthecombinationshown?EnergyStoredinaCapacitorElectricalPotentialenergyisstoredinacapacitorTheenergycomesfromtheworkthatisdoneinchargingthecapacitorLetqandvbetheintermediatechargeandpotentialonthecapacitorTheincrementalworkdoneinbringinganincrementalcharge,dq,tothecapacitoristhengivenbyEnergyStoredinaCapacitorThetotalworkdoneisjusttheintegralofthisequationfrom0toQUsingtherelationshipbetweencapacitance,voltageandchargewealsoobtainwhereUisthestoredpotentialenergyExample5dA-----++++Nowsupposeyoupulltheplatesfurtherapartsothatthefinalseparationisd1WhichofthequantitiesQ,C,V,U,Echange?

Howdothesequantitieschange?Answers:SupposethecapacitorshownhereischargedtoQandthenthebatteryisdisconnectedChargeonthecapacitordoesnotchangeCapacitanceDecreasesVoltageIncreasesPotentialEnergyIncreasesQ:C:V:U:E:ElectricFielddoesnotchangeSupposethebattery(V)iskeptattachedtothecapacitor

Againpulltheplatesapartfromdtod1Nowwhichquantities,ifany,change?Howmuchdothesequantitieschange?Answers:Example6Q:C:V:U:E:VoltageoncapacitordoesnotchangeCapacitanceDecreasesChargeDecreases

PotentialEnergyDecreasesElectricFieldDecreasesElectricFieldEnergyDensityThepotentialenergythatisstoredinthecapacitorcanbethoughtofasbeingstoredintheelectricfieldthatisintheregionbetweenthetwoplatesofthecapacitorThequantitythatisofinterestisinfacttheenergydensitywhereAanddaretheareaofthecapacitorplatesandtheirseparation,respectivelyElectricFieldEnergyDensityUsingandwethenhaveEventhoughweusedtherelationshipforaparallelcapacitor,thisresultholdsforallcapacitorsregardlessofconfigurationThisrepresentstheenergydensityoftheelectricfieldingeneralDielectricsMostcapacitorshaveanonconductingmaterialbetweentheirplatesThisnonconductingmaterial,adielectric,accomplishesthreethings1)Solvesmechanicalproblemofkeepingtheplatesseparated2)Increasesthemaximumpotentialdifferenceallowedbetweentheplates3)IncreasesthecapacitanceofagivencapacitoroverwhatitwouldbewithoutthedielectricDielectricsSupposewehaveacapacitorofvalueC0thatischargedtoapotentialdifferenceofV0andthenremovedfromthechargingsourceWewouldthenfindthatithasachargeofWenowinsertthedielectricmaterialintothecapacitorWefindthatthepotentialdifferencedecreasesbyafactorKOrequivalentlythecapacitancehasincreasedbyafactorofKThisconstantKisknownasthedielectricconstantandisdependentuponthematerialusedandisanumbergreaterthan1PolarizationWithoutthedielectricinthecapacitor,wehaveTheelectricfieldpointsundiminishedfromthepositivetothenegativeplateQ+++++++++++++++---------------E0V0WiththedielectricinplacewehaveTheelectricfieldbetweentheplatesofthecapacitorisreducedbecausesomeofthematerialwithinthedielectricrearrangessothattheirnegativechargesareorientedtowardsthepositiveplate---------------+