Transcription of CAPACITANCEANDCAPACITORS - PHYSNET
1 Project PHYSNETP hysics Bldg. Michigan State University East Lansing, MI MISN-0-135 CAPACITANCEANDCAPACITORS + + ++ + ++++- - -- - ----ABCD1 CAPACITANCEANDCAPACITORSbyWilliamF. Faissler1. Introduction.. 1a. Why We StudyCapacitors.. 1b. CapacitanceDe ned.. 1c. A Summaryof WhatFollows .. 12. Capacitors..2a. TheParallelPlateCapacitor.. 2b. TheCylindricalCapacitor.. 3c. A Generalization.. 4d. TheUnitsof Capacitance.. 5e. CapacitorsAre Neutral.. 5f. TheVoltage-Current Relationin a Capacitor..5g. TheSymbol for a Capacitor.. 63. UsefulRelationships.. 6a. TheEnergyStoredin a ChargedCapacitor.. 6b. Capacitorsin Parallel.. 7c. Capacitorsin Series.. 84. [4. a .. 9b. ElectricFieldsin Dielectrics.. 9c. An AtomicModel .. 10d. DielectricBreakdown .. 11 Acknowledgments.. 12 Glossary.. 122ID Sheet: MISN-0-135 Title:Capacitanceand CapacitorsAuthor:WilliamFaissler, Physics,NortheasternUniversity,Boston, :2/28/2000 Evaluation:Stage0 Length:1 hr; 24 pagesInputSkills:1.]
2 ApplyGauss'slaw to determinethe electric eldproducedby agiven planar,spherical,or cylindricalstaticchargedistribution(MISN -0-132,MISN-0-133).2. Derive the expressionfor the potential di erencebetween theplatesof a parallelplatecapacitor(MISN-0-134).3. Describe how one reducesa complexcircuithavingone type of el-ement to a singleelement usingthe conceptofequivalentelementsand illustratingwithan example(MISN-0-119).OutputSkills(Knowled ge) :capacitance,capacitor,dielectric,dielec tricconstant, necapacitancein termsof chargeand Gauss'slaw to derive the capacitanceof cylindricaland , fromenergyand chargeconservationprinciples,the capac-itanceof capacitorsconnectedin , fromenergyand chargeconservationprinciples,the capac-itanceof capacitorsconnectedin (RuleApplication) capacitanceof a given cylindricalor (ProblemSolving):S1. Calculatethe capacitanceof a given seriesand/orparallelcombi-nationof A DEVELOPMENTAL-STAGE PUBLICATIONOF PROJECTPHYSNETThe goalof our project is to assista network of educatorsand scientistsintransferringphysicsfromone personto support manuscriptprocessingand distribution,alongwithcommunicationand alsowork withemployers to identify basicscienti cskillsas well as physicstopicsthatare neededin scienceandtechnology.
3 Anumber of our publicationsare aimedat assistingusersin designed:(i) to be updatedquickly in responseto eldtestsand newscienti cdevelopments; (ii) to be usedin both class-room andprofessionalsettings;(iii)to show the prerequisitedependen-ciesexistingamongth e variouschunksof physicsknowledgeandskill,as a guideboth to mental organizationand to use of the materials;and(iv) to be adaptedquickly to speci cuserneedsrangingfromsingle-skillinstruc tionto ,reviewers and eldtestersare DirectorADVISORY COMMITTEED. AlanBromleyYale UniversityE. LeonardJossemTheOhioStateUniversityA. A. StrassenburgS. U. N. Y., Stony BrookViewsexpressedin a moduleare thoseof the moduleauthor(s)and arenot necessarilythoseof otherproject 2001,PeterSignellfor Project PHYSNET ,Physics-Astronomy Bldg.,Mich. StateUniv.,E. Lansing,MI 48824;(517) our liberaluse policiessee: Faissler1. We ,capacitorsanda number of may ap-pear to you as if capacitanceand capacitorsare simplya digressionin theprocessof understandingthe electrostatic eld,capacitorsare one of themoreimportant buildingblocks usedin constructingmodernelectronicequipment.
4 Thus the materialyou learnfromthismoduleis actuallyusefulin conductorsseparatedfromeachotherby empty spaceor by an insulatorforma theword \capacitor"refersto a deviceconsistingof the two conductorsandwhatever is in the spacebetween good exampleis the ordinary\CableTelevision" this moduleit will be shown thatif one of acapacitor'sconductorshas a chargeof + Q on it and the otherhas a chargeof Q, thenthe potential di erence,V, between the two conductorsislinearlyproportionalto the charge:V=Q=C:(1)Theproportionality constant's inverse,C, is calledthe \capacitance"ofthe the sizesof the twoconductors,on theirrelative positions,and on whatis between independent of A Summaryof this modulewe will derive thefollowingresults,exceptthatwe will assumevacuumbetween the platesof capacitorssoK= 1:1. For a parallelplatecapacitor,the type commonlyfoundin electroniccircuits:C=KA4 ked(2)2.
5 For a cylindricalcapacitor,exempli edby a coaxialcable:C=KL2ke`n(R2=R1)(3)5 MISN-0-13523. For any capacitor,the voltage-current relationship:i=CdVdt(4)4. For any capacitor,the energyrequiredto chargeit:U=12Q2C=12QV=12CV2(5)5. For a number of capacitorsin parallel,the equivalent capacitance:CEQ=C1+C2+: : :+Cn:(6)and for the capacitorsin series:1 CEQ=1C1+1C2+: : :+1Cn(7)Finally, in the lastsectionof the module,the e ectof puttingadielectricmaterialbetween a capacitor'sconductors(makingK6= 1) willbe Capacitors2a. of all capacitorsis the parallelplatecapacitorillustratedin Figure1: we will hereshowthatits capacitanceisC=A=(4 ked). Thiscapacitorconsistsof twoparallelconductingplates,each withareaAand separatedby a distanced. For the present thisgap is assumedto be empty (vacuum);laterinthe modulewe will discusswhathappens in the otherusualcasewhere++++++------Plate APlate BdE` insulatingdielectricmaterialis put between the two plateshas a chargeof +Qwhilethe otherhas a chargeof Qandin this particularcasewe have assumedthatthe upper plateis positivewhilethe lower plateis negative.
6 By applyingGauss'slaw, you can easilysee thatthe magnitudeof theelectric eldin the gap between the two platesis:1E= 4 keQA(8)and the directionof the electric eldis fromthe upper plateto the lowerplateas shown in the cancalculatethe potential di erencebetween the twoplates,herelabeledAandB:V= ZBA~E d~`=ZBA4 keQAd` ;= 4 keQdAThus the voltageacrossthe capacitoris proportionalto the chargeonthe capacitor,withthe constant of proportionality beinga combinationof be writtenasV=Q=C[Eq. (1)] wherethe capacitanceCis given by:C=A4 ked(parallelplate)(9)You recognizethis as the resultadvertisedearlier[Eq. (2)] for the caseofvacuum(K= 1); laterwe will discussthe caseof a capacitormadewithan common\cylindrical"capacitoris illustratedin Fig. 2. It consistsof two concentric cylindersof lengthL, separatedby a now it is empty thecylindershas a chargeof +Qon it and the otherhas a chargeof Q; inthis particularcase,the positive chargehas been put on the innercylinder.
7 By applyingGauss'slaw you can easilysee thatthe magnitudeof theelectric eldin the gap between the two cylindersis:E= 2keQLr(R1 r R2)(10)1 See \Gauss'sLaw Appliedto Cylindricaland PlanarChargeDistributions"(MISN-0-133).7 MISN-0-1354+Q-QR1R2r+++++++-----------++ ++++++ the radiusof the point at which you are calculatingthe can calculatethe potential di erencebetween the innerandoutercylinders:V= ZBA~E d~r=ZR2R12keQLdrr= 2keQL(`n R2 `n R1)= 2keQ`n(R2=R1)L(11)Onceagain,the voltageis proportionalto the chargeon eithercylinderand the constant of proportionality is a collectionof nd:C=L2ke`n(R2=R1)(12)2c. A applyingGauss'slaw to the two con gu-rationsconsideredso far, it was necessaryto assumethatthe point atwhich the electric eldwas beingcalculatedwas so far fromthe edgesofthe platesor cylindersthattherewas no e ectdue to the variationof the eldsnearthe capacitor' whenthe capacitorplatesare so smallor the cylindersso shortthatthis cannotbe true?
8 Itcan be shown, althoughit is beyondthe scope of thismoduleto do so,thatthe formshown in Eq. (1) is truein all any arrangementof two isolatedconductorswithone havinga chargeof +Qand the other8 MISN-0-1355 Q, the potential di erencebetween the two conductorsis proportionaltoQand the constant of proportionality 1=Cdependsonlyon geometri-cal the simplecasesof two parallelplatesor two concentriccylinders,Eq. (9) or (12), respectively, can be usedto calculatethe other,less idealized,geometriesmorecomplexformulaem ustbe any case,the capacitanceCcan be measuredby electronicmeansand the validity of Eq. (1) is well establishedby measuredin unitsof farad(abbreviated\F");fromEq. (1) you can see thatif one coulombof chargeis placedon a one faradcapacitor,therewillbe a potentialof one volt acrossthe mostelectronicuses,the faradisa very largeunit;commonlyusedcapacitorsare measuredin unitsofmicrofarads,nanofaradsand even picofarads.
9 If you rearrangeEq. (1) you will ndthatcapacitancehas the dimen-sionsof \chargeper unitvoltage"(voltage= electrostaticpotential dif-ference). illustratedin Fig. 1, the normalcapacitoras a wholeis electricallyneutral;one plateof the capacitorhasa chargeof +Qon it whilethe otherhas a chargeof semantic problem,sincecapacitorsare com-monlysaidto \storecharge."If you are concentratingon onlyone plateof the capacitor,thenyou may well be ableto treatthe capacitoras if itis \storingcharge"but in generalthe capacitoras a wholeis say, \a capacitorhas a chargeQon it," we meanthatit has a chargeof +Qon one plateand Qon the TheVoltage-Current Relationin a ndoutwhathappens whenthe appliedvoltagechangeswithtime,we simplyrearrangeEq. (1) and di erentiateit:Q=C VdQdt=VdCdt+CdVdtdQdt=CdVdt:Thelast equationfollows becauseC, which dependsonlyon geometricalconstants, is not a functionof timesodC=dt= 0.
10 Finally,dQ=dt(therateof changeof the chargeon one of the plates)is the current for a the leadto thatplate:i=CdVdtThisrelationshipis oftenusedin determiningthe e ectsof TheSymbol for a for a capacitor,tobe usedin drawingschematicdiagrams,is illustratedin Fig. UsefulRelationships3a. The EnergyStoredin a is oftenusefulto know the amount of potential energystoredin a capacitorChavingchargeofq. Thepotential di erencebetween the two platesis given by Eq. (1):V=q=C. If a smallamountof charge,dq, is moved fromone plateto the other,thenthe amount ofwork doneis:dW=V dq=q dq=C:(13)Movingchargefromone plateto the otheris equivalent to any othermethod of chargingthe if you simplyintegrateEq. (13),startingwithzerochargeand endingwitha nalchargeQ, you will havecalculatedthe work neededto chargethe capacitorandhenceyou willhave calculatedthe potential energystoredin the chargedcapacitor:U=ZQOq dq=C=q22C Q0=Q22C=12V Q=12CV2wherethe last resultsfollow by usingEq.
