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1ELECTRICITY131WEOBTAINTHATTHEFIELDEVALUATEDATTHECENTEROFCURVATUREDUETOACHARGEDDISTRIBUTIONONACIRCULARARCISGIVENBY0SIN4ERALONGTHESYMMETRYAXIS,WITHQ/RWITHINRADIANS弧度）INTHISPROBLEM,EACHCHARGEDQUARTERCIRCLEPRODUCESAFIELDOFMAGNITUDE/42/400|1122|SIN/244QQERRRTHATPRODUCEDBYTHEPOSITIVEQUARTERCIRCLEPOINTSAT45,ANDTHATOFTHENEGATIVEQUARTERCIRCLEPOINTSAT45ATHEMAGNITUDEOFTHENETFIELDISNET,22009221222122|14|2COS454489910NMC445010C206N/C50010MXQQERRBBYSYMMETRY,THENETFIELDPOINTSVERTICALLYDOWNWARDINTHEJDIRECTION,OR90COUNTERCLOCKWISEFROMTHEXAXIS132WEASSUMEQ0USINGTHENOTATIONQ/LWENOTETHATTHEINFINITESIMALCHARGEONANELEMENTDXOFTHERODCONTAINSCHARGEDQDXBYSYMMETRY,WECONCLUDETHATALLHORIZONTALFIELDCOMPONENTSDUETOTHEDQSCANCELANDWENEEDONLY“SUM”INTEGRATETHEVERTICALCOMPONENTSSYMMETRYALSOALLOWSUSTOINTEGRATETHESECONTRIBUTIONSOVERONLYHALFTHEROD0XL/2ANDTHENSIMPLYDOUBLETHERESULTINTHATREGARDWENOTETHATSINR/RWHERE22RXRATHEMAGNITUDEIS222222220000/223222202200022220022SIN4422212242LLLLDQDXYERXRXRQLRRDXXRXRXRQLQLRRLRLRWITH1278110CQ,0145MLANDR00600M,WEHAVE|124N/CEBASNOTEDABOVE,THEELECTRICFIELDEPOINTSINTHEYDIRECTION,OR90COUNTERCLOCKWISEFROMTHEXAXIS133THETOTALELECTRICFLUXTHROUGHTHECUBEISEDATHENETFLUXTHROUGHTHETWOFACESPARALLELTOTHEYZPLANEIS22112211132101130110241021661212YZYZXXYZYZYZEXXEXXDYDZDYDZDYDZSIMILARLY,THENETFLUXTHROUGHTHETWOFACESPARALLELTOTHEXZPLANEIS2211432111330XZXZYYEYYEYYDXDZDYDZ,ANDTHENETFLUXTHROUGHTHETWOFACESPARALLELTOTHEXYPLANEIS221141211032316XYXYZZEZZEZZDXDYDXDYBBBBAPPLYINGGAUSSLAW,WEOBTAINENC00006000120240XYXZYZQBWHICHIMPLIESTHATB200N/CM3134WEASSUMETHECHARGEDENSITYOFBOTHTHECONDUCTINGCYLINDERANDTHESHELLAREUNIFORM,ANDWENEGLECTFRINGINGEFFECTSYMMETRYCANBEUSEDTOSHOWTHATTHEELECTRICFIELDISRADIAL,BOTHBETWEENTHECYLINDERANDTHESHELLANDOUTSIDETHESHELLITISZERO,OFCOURSE,INSIDETHECYLINDERANDINSIDETHESHELLAWETAKETHEGAUSSIANSURFACETOBEACYLINDEROFLENGTHL,COAXIALWITHTHEGIVENCYLINDERSANDOFLARGERRADIUSRTHANEITHEROFTHEMTHEFLUXTHROUGHTHISSURFACEIS2,RLEWHEREEISTHEMAGNITUDEOFTHEFIELDATTHEGAUSSIANSURFACEWEMAYIGNOREANYFLUXTHROUGHTHEENDSNOW,THECHARGEENCLOSEDBYTHEGAUSSIANSURFACEISQENCQ1Q2Q13401012CCONSEQUENTLY,GAUSSLAWYIELDS0ENC2,RLEQOR12ENC12222034010C000214N/C,2288510C/NM110M20013010MQELROR|000214N/CEBTHENEGATIVESIGNINEINDICATESTHATTHEFIELDPOINTSINWARDCNEXT,FORR500R1,THECHARGEENCLOSEDBYTHEGAUSSIANSURFACEISQENCQ13401012CCONSEQUENTLY,GAUSSLAWYIELDS0ENC2,RLEQOR12ENC12222034010C000855N/C2288510C/NM110M50013010MQELRDTHEPOSITIVESIGNINDICATESTHATTHEFIELDPOINTSOUTWARDEWECONSIDERACYLINDRICALGAUSSIANSURFACEWHOSERADIUSPLACESITWITHINTHESHELLITSELFTHEELECTRICFIELDISZEROATALLPOINTSONTHESURFACESINCEANYFIELDWITHINACONDUCTINGMATERIALWOULDLEADTOCURRENTFLOWANDTHUSTOASITUATIONOTHERTHANTHEELECTROSTATICONESBEINGCONSIDEREDHERE,SOTHETOTALELECTRICFLUXTHROUGHTHEGAUSSIANSURFACEISZEROANDTHENETCHARGEWITHINITISZEROBYGAUSSLAWSINCETHECENTRALRODHASCHARGEQ1,THEINNERSURFACEOFTHESHELLMUSTHAVECHARGEQINQ13401012C4FSINCETHESHELLISKNOWNTOHAVETOTALCHARGEQ2200Q1,ITMUSTHAVECHARGEQOUTQ2QINQ13401012CONITSOUTERSURFACE135WEUSEAGAUSSIANSURFACEINTHEFORMOFABOXWITHRECTANGULARSIDESTHECROSSSECTIONISSHOWNWITHDASHEDLINESINTHEDIAGRAMBELOWITISCENTEREDATTHECENTRALPLANEOFTHESLAB,SOTHELEFTANDRIGHTFACESAREEACHADISTANCEXFROMTHECENTRALPLANEWETAKETHETHICKNESSOFTHERECTANGULARSOLIDTOBEA,THESAMEASITSLENGTH,SOTHELEFTANDRIGHTFACESARESQUARESTHEELECTRICFIELDISNORMALTOTHELEFTANDRIGHTFACESANDISUNIFORMOVERTHEMSINCE580FC/M3ISPOSITIVE,ITPOINTSOUTWARDATBOTHFACESTOWARDTHELEFTATTHELEFTFACEANDTOWARDTHERIGHTATTHERIGHTFACEFURTHERMORE,THEMAGNITUDEISTHESAMEATBOTHFACESTHEELECTRICFLUXTHROUGHEACHOFTHESEFACESISEA2THEFIELDISPARALLELTOTHEOTHERFACESOFTHEGAUSSIANSURFACEANDTHEFLUXTHROUGHTHEMISZEROTHETOTALFLUXTHROUGHTHEGAUSSIANSURFACEIS22EATHEVOLUMEENCLOSEDBYTHEGAUSSIANSURFACEIS2A2XANDTHECHARGECONTAINEDWITHINITIS22QAXGAUSSLAWYIELDS20EA22A2XWESOLVEFORTHEMAGNITUDEOFTHEELECTRICFIELD0/EXAFORX0,E0BFORX200MM200103M,153361222058010C/M20010M13110N/C88510C/NMXECFORXD/2470MM470103M,153361222058010C/M47010M30810N/C88510C/NMXE5DFORX260MM260102M,WETAKEAGAUSSIANSURFACEOFTHESAMESHAPEANDORIENTATION,BUTWITHXD/2,SOTHELEFTANDRIGHTFACESAREOUTSIDETHESLABTHETOTALFLUXTHROUGHTHESURFACEISAGAIN22EABUTTHECHARGEENCLOSEDISNOWQA2DGAUSSLAWYIELDS20EA2A2D,SO153361222058010C/M94010M30810N/C2288510C/NMDE136THEFIELDISZEROFOR0RATHUS,AE0ATR0,BE0ATRA/200,ANDFORARBTHEENCLOSEDCHARGEQENCFORARBISRELATEDTOTHEVOLUMEBYQRAENCFHGIKJ434333THEREFORE,THEELECTRICFIELDISEQRRRARARFHGIKJ144434330202330332ENCFORARBCFORR150A,WEHAVE33932122200150237518410C/M0100M2375732N/C31503225388510C/NM225AAAEADFORRB200A,THEELECTRICFIELDIS33932122200200718410C/M0100M7121N/C320034388510C/NM4AAAEAEFORRBWEHAVE2TOTAL/4EQROR63203BAERTHUS,FORR300B600A,THEELECTRICFIELDIS33932122200200718410C/M0100M7135N/C3600336388510C/NM36AAAEA137AWEINTEGRATETHEVOLUMECHARGEDENSITYOVERTHEVOLUMEANDREQUIRETHERESULTBEEQUALTOTHETOTALCHARGE204RDXDYDZDRRQSUBSTITUTINGTHEEXPRESSIONSR/R,WITHS141PC/M3,ANDPERFORMINGTHEINTEGRATIONLEADSTO444SRQROR312331514110C/M00560M77810CSQRBATR0,THEELECTRICFIELDISZEROE0SINCETHEENCLOSEDCHARGEISZEROATACERTAINPOINTWITHINTHESPHERE,ATSOMEDISTANCERFROMTHECENTER,THEFIELDISGIVENBYGAUSSLAWENC2014QERWHEREQENCISGIVENBYANINTEGRALSIMILARTOTHATWORKEDINPARTA42ENC0444RSRQDRRRTHEREFORE,7422001144SSRRERRRCFORRR/200,WHERER560CM,THEELECTRICFIELDIS2922123003/2001189910NMC14110C/M00560M4440040055810N/CSSRRERDFORRR,THEELECTRICFIELDIS2922123002189910NMC14110C/M00560M4422310N/CSSRRER138AFROMGAUSSLAW,WEGETERQRRRRRRBGCH1414433030330ENCLBTHECHARGEDISTRIBUTIONINTHISCASEISEQUIVALENTTOTHATOFAWHOLESPHEREOFCHARGEDENSITYPLUSASMALLERSPHEREOFCHARGEDENSITYWHICHFILLSTHEVOIDBYSUPERPOSITIONERRRAABGBG333000141CONSIDERANINFINITESIMALSEGMENTOFTHEROD,LOCATEDBETWEENXANDXDXITHASLENGTHDXANDCONTAINSCHARGEDQDXCXDXITSDISTANCEFROMP1ISDXANDTHEPOTENTIALITCREATESATP1ISDVDQDXCXDXDX141400TOFINDTHETOTALPOTENTIALATP1,WEINTEGRATEOVERTHELENGTHOFTHERODANDOBTAIN8000009221222LNLN14440120M89910NMC28910C/M0120M0030MLN10030M18610VLLCXDXCCLVXDXDLDDXD142ACONSIDERANINFINITESIMALSEGMENTOFTHERODFROMXTOXDXITSCONTRIBUTIONTOTHEPOTENTIALATPOINTP2ISDVXDXXYCXXYDX1414022022THUS,2222ROD0009221222224489910NMC49910C/M0100M00356M00356M31610VLPCXCVDVDXLYYXYBTHEYCOMPONENTOFTHEFIELDTHEREIS2222009221222214400356M89910NMC49910C/M10100M00356M0298N/CPYVCDCYELYYYDYLYCWEOBTAINEDABOVETHEVALUEOFTHEPOTENTIALATANYPOINTPSTRICTLYONTHEYAXISINORDERTOOBTAINEXX,YWENEEDTOFIRSTCALCULATEVX,YTHATIS,WEMUSTFINDTHEPOTENTIALFORANARBITRARYPOINTLOCATEDATX,YTHENEXX,YCANBEOBTAINEDFROM,/XEXYVXYX143ATHEELECTRICFIELDATTHESPHERICALSURFACEIS204EQER9WHILETHEPOTENTIALOFASPHERE,WITHV0ATIS04EQVRTHUSWEHAVE68150/638109610EVERVMMVBTHEDIFFERENCEINPOTENTIALBETWEENTHESURFACEANDMUSTBETHESAME,SOADDTHESAMECONSTANTTOBOTHLOCATIONS89610VV144FROMTHESPATIALDEPENDENCEOFTHEELECTRICPOTENTIAL,VX,Y,ZY22XY4XYZ,WEFINDTHECOMPONENTSOFTHEELECTRICFIELDFROMTHEPARTIALDERIVATIVESOFVEX2Y4YZEY2Y2X4XZEZ4XY221224EYZIYXXZJXYK145AAFTERTHECONNECTION,IFTHETWOSPHERESWEREATDIFFERENTPOTENTIALS,THENWOULDBEAFLOWOFCHARGEINTHEWIRETHUSTHEPOTENTIALSMUSTBETHESAMEBWEASSUMETHESPHERESARESOFARAPARTTHATTHEPOTENTIALOFONESPHEREATTHEOTHERSPHEREISESSENTIALLYZEROTHEINITIALPOTENTIALSARE0101/4VQR,020VAFTERTHECONNECTION,2QISTRANSFERREDTOTHESECONDSPHERE,SOWEHAVE12012202/4/4VQQRVQR,WHICHGIVES2212/QRQRR151THECHARGESONCAPACITORS2AND3ARETHESAME,SOTHESECAPACITORSMAYBEREPLACEDBYANEQUIVALENTCAPACITANCEDETERMINEDFROM111232323CCCCCCCEQTHUS,CEQC2C3/C2C3THECHARGEONTHEEQUIVALENTCAPACITORISTHESAMEASTHECHARGEONEITHEROFTHETWOCAPACITORSINTHECOMBINATIONANDTHEPOTENTIALDIFFERENCEACROSSTHEEQUIVALENTCAPACITORISGIVENBYQ2/CEQTHEPOTENTIALDIFFERENCEACROSSCAPACITOR1ISQ1/C1,WHEREQ1ISTHECHARGEONTHISCAPACITORTHEPOTENTIALDIFFERENCEACROSSTHECOMBINATIONOFCAPACITORS2AND3MUSTBETHESAMEASTHEPOTENTIALDIFFERENCEACROSSCAPACITOR1,SOQ1/C1Q2/CEQ10NOWSOMEOFTHECHARGEORIGINALLYONCAPACITOR1FLOWSTOTHECOMBINATIONOF2AND3IFQ0ISTHEORIGINALCHARGE,CONSERVATIONOFCHARGEYIELDSQ1Q2Q0C1V0,WHEREV0ISTHEORIGINALPOTENTIALDIFFERENCEACROSSCAPACITOR1ASOLVINGTHETWOEQUATIONS1212101EQ,QQQQCVCCFORQ1ANDQ2,WEOBTAIN22212301010123EQ1121323123CCCVCVCVQCCCCCCCCCCCCCWITHV0120V,C1400F,C2600FANDC3300F,WEFINDCEQ200FANDQ1320CBTHECHARGEONCAPACITORS2IS2101400F120V320C160CQCVQCTHECHARGEONCAPACITOR3ISTHESAMEASTHATONCAPACITOR23101400F120V320C160CQCVQ152FORMAXIMUMCAPACITANCETHETWOGROUPSOFPLATESMUST
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