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February 3, 2016 CH 25 Capacitance
I.
Capacitance
A. Figurebelowshowsthebasicelementsofanycapacitor–two
isolatedconductorsofanyshapewithaninsulatingmaterial
betweenthem.Theinsulatingmaterialcanbeair.Thetwoconductors
nomatterwhattheirgeometryarecalledPlates.
B. Whenacapacitorischarged,itsplateshavechargesofequal
magnitudesbutoppositesigns:q+andq‐.However,werefertothe
chargeofacapacitorasjustbeingq,theabsolutevalueofthese
chargesontheplates.
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February 3, 2016 C. ThechargeqandthepotentialdifferenceVforacapacitorare
proportionaltoeachother:
D. TheproportionalityconstantCiscalledthe___________________________
ofthecapacitor.Itsvaluedependsonlyonthegeometryoftheplates
andnotontheirchargeorpotentialdifference.
E. TheSIunitiscalledthefarad(F):
II.
ChargingaCapacitor
A. ThecircuitshownbelowisincompletebecauseswitchSisopen;
thatis,theswitchdoesnotelectricallyconnectthewiresattachedto
it.Whentheswitchisclosed,electricallyconnectingthosewires,the
circuitiscompleteandchargecanthenflowthroughtheswitchand
thewires.
B. Astheplatesbecomeoppositelycharged,thatpotentialdifference
increasesuntilitequalsthepotentialdifferenceVbetweenthe
terminalsofthebattery.Withtheelectricfieldzero,thereisno
furtherdriveofelectrons.Thecapacitoristhensaidtobefully
charged,withapotentialdifferenceVandchargeq.
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February 3, 2016 III.
CalculatingtheCapacitance
A. TorelatetheelectricfieldEbetweentheplatesofacapacitorto
thechargeqoneitherplate,weuseGauss’law:
B. HereqisthechargeenclosedbyaGaussiansurfaceand
isthenetelectricfluxthroughthatsurface.Inourspecial
caseinthefigure,
inwhichAistheareaofthatpartoftheGaussiansurfacethrough
whichthereisaflux.
C. Thepotentialdifferencebetweentheplatesofacapacitoris
relatedtothefieldEby
1.
IfVisthedifferenceVf‐Vi,
where, Page3
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February 3, 2016 D. ACylindricalCapacitor(proofisinthebook)
1.
Wherebistheouterplateradiusanda
istheinnerplateradius.
2.
Listheheight(length)ofthecylinder.
E. ASphericalCapacitor(proofisinthebook)
1.
Wherebistheouterradius,
2.
Andaistheinnerradiusasshown.
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February 3, 2016 F. CalculatingtheCapacitance:AnIsolatedSphere:
1.
Wecanassignacapacitancetoasingleisolatedsphericalconductorof
radiusRbyassumingthatthe“missingplate”isaconductingsphereofinfinite
radius.
2.
Thefieldlinesthatleavethesurfaceofapositivelychargedisolated
conductormustendsomewhere;thewallsoftheroominwhichtheconductor
ishousedcanserveeffectivelyasoursphereofinfiniteradius.
3.
Tofindthecapacitanceoftheconductor,wefirstrewritethe
capacitanceas:
4.
Nowlettingb→∞,andsubstitutingRfora,
G. SampleProblems:
1.
(a)Findthecapacitanceofanair‐filledparallel‐platecapacitorwith
squareplatesofedgelengths0.80cmandplateseparationd=0.30mm.(Later
inthechapter,capacitorsmightbefilledwithadielectricmaterial,buthere
we'lluseonlyair.)
(b)Whatischargeonthiscapacitorifweconnectedittoyourcar
battery(12V)?
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February 3, 2016 2.
Theplatesofasphericalcapacitorhaveradii r1  38.0 mm and
r2  40.0 mm .
a)
Calculatethecapacitance.
b)
Whatmustbetheplateareaofaparallel‐platecapacitorwiththe
sameplateseparationandcapacitance?
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February 3, 2016 IV.
CapacitorsinParallel
A. WhenapotentialdifferenceVisappliedacrossseveralcapacitors
connectedinparallel,thatpotentialdifferenceVisappliedacross
eachcapacitor.Thetotalchargeqstoredonthecapacitorsisthesum
ofthechargesstoredonallthecapacitors.
B. Capacitorsconnectedinparallelcanbereplacedwithan
equivalentcapacitorthathasthesametotalchargeqandthesame
potentialdifferenceVastheactualcapacitors.
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February 3, 2016 V.
CapacitorsinSeries
A. WhenapotentialdifferenceVisappliedacrossseveralcapacitors
connectedinseries,thecapacitorshaveidenticalchargeq.Thesumof
thepotentialdifferencesacrossallthecapacitorsisequaltothe
appliedpotentialdifferenceV.
B. Capacitorsthatareconnectedinseriescanbereplacedwithan
equivalentcapacitorthathasthesamechargeqandthesametotal
potentialdifferenceVastheactualseriescapacitors.
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February 3, 2016 VI.
CapacitorsinParallelandinSeries
A. Sampleproblems:
1.
FindtheequivalentCapacitanceforthebelowcircuitassumingall
capacitorsare12µF.
2.
FindVoltageonCapacitorC2iftheBatteryisa12voltscarbattery.
3.
FindchargeonC1
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February 3, 2016 B. Inthecircuitshown,whatisthevoltagefeltacrossthe9.0μF
capacitorandthechargeonthe3.0 μFcapacitor?
9µF 5µF 150.0 V Show all work: Page
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February 3, 2016 VII.
EnergyStoredinanElectricField:
A. Thepotentialenergyofachargedcapacitormaybeviewedas
beingstoredintheElectricFieldbetweenitsplates.
B. Supposethat,atagiveninstant,achargeq’hasbeentransferred
fromoneplateofacapacitortotheother.ThepotentialdifferenceV’
betweentheplatesatthatinstantwillbeq’/C.Ifanextraincrementof
chargedq’isthentransferred,theincrementofworkrequiredwillbe,
C. Theworkrequiredtobringthetotalcapacitorchargeuptoafinal
valueqis
D. ThisworkisstoredaspotentialenergyUinthecapacitor,sothat,
E. Thiscanalsobeexpressedas:
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February 3, 2016 VIII. EnergyDensity:
A. Inaparallel‐platecapacitor,neglectingfringing,theelectricfield
hasthesamevalueatallpointsbetweentheplates.Thus,theenergy
densityu—thatis,thepotentialenergyperunitvolumebetweenthe
plates—shouldalsobeuniform.
B. Wecanfindubydividingthetotalpotentialenergybythevolume
Adofthespacebetweentheplates.
1.
Butsince(C= A/d),thisresultbecomes
0
C. However,(E=‐V/s),V/dequalstheelectricfieldmagnitudeE.
Therefore.
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February 3, 2016 IX.
EnergySampleproblem:
A. Theparallelplatesinacapacitor,withaplateareaof8.50cm2and
anair‐filledseparationof3.00mm,arechargedbya6.00Vbattery.
Theyarethendisconnectedfromthebatteryandpulledapart
(withoutdischarge)toaseparationof8.00mm.Neglectingfringing,
find(a)thepotentialdifferencebetweentheplates,(b)theinitial
storedenergy,(c)thefinalstoredenergy,and(d)theworkrequired
toseparatetheplates.
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February 3, 2016 X.
CapacitorwithaDielectric:
A. Adielectric,isaninsulatingmaterialsuchasmineraloilorplastic,
andischaracterizedbyanumericalfactor,calledthedielectric
constantofthematerial.
1.
Somedielectrics,suchasstrontiumtitanate,canincreasethe
capacitancebymorethantwoordersofmagnitude.
B. Inaregioncompletelyfilledbyadielectricmaterialofdielectric
constant,allelectrostaticequationscontainingpermittivity
constante0aretobemodifiedbyreplacinge0withe0
C. Theintroductionofadielectricalsolimitsthepotentialdifference
thatcanbeappliedbetweentheplatestoacertainvalueVmax,called
thebreakdownpotential.Everydielectricmaterialhasacharacteristic
dielectricstrength,whichisthemaximumvalueoftheelectricfield
thatitcantoleratewithoutbreakdown.
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February 3, 2016 D. SampleproblemswithDielectric
1.
Aparallel‐platecapacitorhastwosquaremetalplates.Thesidesof
eachplateare3.0cmlongandtheplatesareseparatedby5.0mm.Thespace
betweentheplatesisfilledwithTeflon,whichhasadielectricconstant= 2.1.
Thecapacitanceofthiscapacitorisclosestto:
A. 1.6 pF
B. 33 pF
C. 14 pF
D. 2.1 pF
E. 3.3 pF Show all work: Page
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February 3, 2016 2.
InFig.below,howmuchchargeisstoredontheparallel‐plate
capacitorsbythe12.0Vbattery?Oneisfilledwithair,andtheotherisfilled
withadielectricforwhichκ=3.00;bothcapacitorshaveaplateareaof
5.00×10‐3m2andaplateseparationof2.00mm.
Show all work: Page
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