bispherical two spheres

Revi,taMexicanadeFísica42,No.•(1996)663-67. Ontheevaluationofthecapacitanceofbispherical capacitors A.GÓNGORA-T. ANDE.LEY-Koo InstitutodeFísica,Universidad NacionalAutónoma deMéxico Apartadopostal20-364,01000México,D.F.,México Recibido el6dediciembre de1994;aceptado el20defebrerode1996 ABSTRACT. Theelectrostatic fieldandthecapacitance ofspherical capacitors withconcentric electrodes isstudiedintheintroductory physicscourses. Thestudyofthesameproblem when theelectrodes arenotconcentric requiresahigherlevelofelectrical andmathematical knowledge. Maxwell inhisTreatisesolvedtheproblem bythemethodofimagesandbyexpansions inspherical harmonics, butbothsolutions presentslowconvergence difliculties. Inthispaperthesolution of theproblem isconstructed byusingbispherical coordinates andshowing thatthecorresponding expansionsinbisphericalharmonicshavenoconvergence difficulties. RESUMEN. Elcampoelectrostático ylacapacitancia decondensadores esféricos conelectrodos concéntricos seestudiaenloscursosintroductorios defísica.Elestudiodelmismoproblema cuando loselectrodosnosonconcéntricos requierenivelesmásavanzadosdeconocimientos eléctricosy matemáticos. Maxwell ensutratadoresolvióelproblema porelmétododeimágenes ypormedio dedesarrollos enarmónicos esféricos, peroambassoluciones presentan dificultades deconvergencia lenta.Enesteartículo lasolución delproblema seconstruye usandocoordenadas biesféricas y mostrando quelosdesarrollos correspondientes enarmónicos biesféricos notienendificultades de convergencia. PAes:41.10.Dq 1.INTRODUCTION Theelectrostatic fieldandthecapacitance ofacharged conducting sphereisastandard topicinthebeginning ofthestudyofelectricity [1-31.Thecorresponding properties fora spherical capacitor withconcentric electrodes followimmediately asanapplication ofthe previous resultsandthesuperposition principie [1-31.Theinquisitive readersmaywonder whathappens tothefieldandthecapacitance whenthespherical electrodes areoffcenter. Theirsearchforananswerintheabovereferences andeveninmoreadvanced textbooks willverylikelyprovefrustrating. Inthispaperweprovide aguidetothefewreferences ontheproblem andpresent ourownquantitative solution. Maxwell studied theproblem oftwononintersecting spheres bythemethod ofimages andbyexpansions inspherical harmonics centered ineachsphere, obtaining infinitese- riesrepresentations forthecapacity coefficients [4).However, bothsolutions present slow convergence difficulties astheiruseincurrent research ontheelectrostatic response of bidimensional arrangements ofmicrospheres hasshown151.MoonandSpencer 16Jstudied theproblem oftwoequalspheres usingbispherical coordinates, whileArfken[7]assignsthe problem ofasphereandaplaneasanexercise intheapplication ofthesamecoordinates. 664 A.GÓNGORA-T. ANDE.LEy-Koo Inthisworkthegeneralproblemofbisphericalcapacitorsisformulated andsolved.In Sect.2weintroducethebispherical coordinates asthenaturalcoordinates todescribe sucheapacitorsandtheireleetrostatie fields;theemphasisisinthegeometryanditmay beassimilated bystudentsintheintermediate leve!.Section3givestheformalanalytic treatmentoftheproblemconstructing theelectrostatic potentialfunctionusingtheGreen functiontechnique[8]'andobtainingfromheretheelectricintensityfield,thechargedis- tributionandtotalchargesontheelectrodes,andfinallythecapacitance ofthecapacitors. Specialattentionispaidtotheanalysisoftheseriesrepresenting thecapacitance forthe differentgeometrical configurations, inordertoexhibitthattherearenoconvergence dif- ficulties.TheGreenfunctionandsornerelevantintegralsareexplieitlyconstructed inthe Appendix.Advancedlevelknowledgeofelectrostatics andmathematics arerequiredtoap- preciatethedifficultiesandfinepointsinthistreatment.Section4containsadiscussion ofsornedetailsofdidacticinterest. 2.BISPllERICAL COORDINATES TODESCRIBE B1SPllERICAL CAPACITORS Thebispherical coordinates arerelatedtothecartesiancoordinates throughtheequa- tions[81 asin~cos'1'x=------cosh1)-cos~asin~sin'1'y=cosh1)-cos~asinh1) z=------.cosh1)-cos~(1) Thesurfaceswithfixedvaluesof~correspond tosurfacesofrevolutionaroundthez-axis withmeridiancrosssectionsthatarecirculararcswitharadiusacsc~andcentersonthe xy-planeadistanceacot~fromthez-axis;allthesesurfacesmeetat(x=O,Y=O,:1:a), andincludetheexternalpartofthez-axis(x=O,Y=O,Izl>a)corresponding to~=O, thesphereofradillsacenteredattheorigincorrespouding to~=;,andthecentralpart ofthez-axis(x=O,Y=O,Izl<a)corresponding to~=1[.Thesurfaceswithfixedvalues of1)correspond tospherescenteredat(x=O,Y=O,z=acoth1))andradiusalesch1)1;the limitingpositions1)=:1:00correspond tothepoints(x=O,Y=O,z=:1:a),respectively, and1)=Ocorresponds tothexy-plane.Thesurfaceswithfixedvaluesof'1'correspond to thefamiliarmeridiallhalf-planes meetingatthez-axis.Thesecoordinates areillustrated inFig.1. Nowwecanpointouthowthebisphericalcoordinates canservetodescribetheeapac- itorsundereonsideration. Anypairofsphericalelectrodeseanbedefinedthroughtheir respectivesphericalcoordinates '11and1)2.Figure2illllstratesthreedifferentpossiblesit- uations:a)onesphereinsideanother1)1>1)2>O,b)sphereandplane1)1>Oand1)2=O, whichisexplicitlystudiedin[71;amIc)spheresoutsideeachother'11>Oand1)2<O, wherethesituationoftwoequalspherescorresponds to'12=-1)1[61.Theintermediate levelstudentshollldnothaveanydifficllltiesindetermining thevaluesofa,1)1andTJ2for twospheresdefinedbytheirradiiandthedistancebetweentheircentersasinRef.[4]. Theorthogonality ofthebispherical coordinates canbeestablished byevaluatingthe infinitesimal displacement fromEq.(1)andidentifying therespectivesealefactorsand ONTHEEVALUATION OFTHECAPACITANCE OFBISPIIERICAL CAPACITORS 665 .,,=-00--'-. ""- \ \ \ \ II I / I / /./\ \~.2! \2 I J // /tz -/ ./ / / / / / I I \ \ \ \ "- "-"- " FIGURE1.Meridiancrosssectionoíbisphericalcoordinates(~,'1,'1'):~=constantarecircular aresmeeting at(x=O,Y=O,z=:f:aL1]=constant arecirc1eswithcenters 00thez-axis,and '1'=constantarehalf-planesmeetingatthez-axis. unitvectors: h=h= u__ ,"cosh'1-coseIusin~,- 'P-cosll1/-COS~(2) (3) ~=(cosh'1cos~-1)(icos'i'+jsin'1')-ksinh1)sin~ cosh1)-cos~ -sinh1)sin~(icos'l'+jsin'i') -k(coshIICOS~ -1) cosh11-cos~(4) - .<j;=-isen'i'+jcos'1'. 666 A.GÓNGORA-T. ANDE.LEY-Koo o FIGURE2.MeridiancrosssectionsoCbisphericalcapacitors:a)onesphereinsideanother,b)sphere andplaneandc)spheresoutsideeachother. 3.ANALYTICAL DESCRIPTION OFELECTROSTATIC FIELD,SOURCES ANDCAPACITANCE Theelectrostatic potentialfunctionmustsatisfytheLaplaceequation {(cosh7]-cos03 a2sin{[8sin{88sin{8] 8{cosh7]-cos{8{+87]cosh7]-cos{87] (cosh7]-COS0282}"'«)=O+2'2<82'P,,7],<P ,asm,'1'(5) andalsotheboundaryconditionsattherespectiveelectrodes </J({,7]=7]1,'1')=VI, </J(C7]=1]2,'1')=V2=O. Equation(5)isR-separable [9),havinggeneralsolutionsoftheform 00l </J(C7],<p) =(coSh7]-COS01/2¿::¿::[AlmPF'(CoSO+BlmQl'(cosO] I=Om=O(6) (7) Thepresenceofthesquare-root ofthebinomialfactoristhesignoCtheRseparability, whichmakesthefulfillmentoftheboundaryconditionofEq.(6)anontrivialmatter. ONTlIEEVALUATION OFTlIECAPACITANCE OFBISPlIERICAL CAPACITORS 667 ForthisreasonweresorttotheGreenfunctiontechniquetoconstructtheelectrostatic potentialfunction[8]: withDirichletboundaryconditions Gn(f,f')ls=O, 1>(f)=-~ida'1>(f,)8Gn(f,f'), 4"ls 8n'(9) (10) (11) wheren'isthedisplacement perpendicular totheboundarysurfaceS.TheGreenfunction forthebisphericalcapacitorsisconstructedintheAppendix.Itsnormalderivativeatthe sphere'11,Eq.(35),issubstitutedinEq.(11)toobtain _VI 1/2~.¿....sinh[(l+~)('1-'12»)1>(C'1,'1')--(cosh'l-cosO~~Ylm(C '1').h[(l+1)(_)1 a 1=0m=-I sm2'11'12 l'1h1hed( h<p'drp'hYI:"((' '1")(cosh'11-cos()1/2.oo '1' ( )1/2~sinh[(l+~)('1-'12)1=VIcosh'l-cos(~----,'I~- __ 1=0sinh[(l+2)('11-'12») ]\'n()lo'd('sin('NIP£( cosel xIFIcos(------ __, O(cosh'll-cos(')1/2(12) wheretheexplicitformsofthescalefactors,Eqs.(3),havebecnused.Heretheintegration overtheazimuthalangleselectsthetermswithm=Oonly,reflectingthesymmetry ofthesystemunderrotationsaroundthez-axis,andNIisthenormalization factorof theLegendrcpolynomials. ItisrecognizedthatEq.(12)isaspecialcaseofEq.(8), satisfyingobviouslytheboundaryconditionofEq.(7).Theintegralover('isrepresented asGI(cosh'1¡)andevaluatedintheAppcndix.Thefulfillmentoftheboundarycondition ofEq.(6)isnotsoimrnediately obvious,butitcanbeverifiedasfollows: lo,d('sinh('=1>((,'1='lI,rp)=VI(cosh'l¡-cosOI/2 (h )1/2LNIPI(cosONIPt(cos() . OCOS'11-COS('£=0 lo,d('sinh(' =VI(cosh'11-cosOI/2 (h )1/2ó(cOS('-cosO=VI'(13)ocos'11-COS(' 668 A.GÓNGORA-T. ANDE.LEY-Koo Thesummation overecanbeidentified astherepresentation oftheDirac-delta function thentheintegration canbedoneimmediately, andthecancellation ofthesquarerootof thebinomialleads tothepotential ofEq.(6).Thesameresultcanbeestablished byusing Eqs.(36)and(37)directly. Theelectric intensity fieldisobtained bytakingthenegative gradient ofEq.(12): E(~,TJ,cp)=-[~h:a{+iih:a,+'\:a,,]tI>(CTJ,cp) =_V¡(cosh'1_COS03/2fC/(coshTJ¡) a /=0sinh[(e+t)(TJI-'12)] {~[N/dP/t;¡SO+2(COShs~n~ cosO'N/P/(cosO] sinh[(e+t)(TJ-'12)] -[(e!)h[(e!)(_)]sinhTJSinh[(e+t)(TJ-TJ2))]+'1+2COS +2'1'12+2(h <)cos'1-cos, xN/PI(COSO}. (14) Itisobvious thatthedirection oftheelectric fieldlinesatthesphere1/2isthatofvector iiperpendicular totheelectrode.Itisnotsoobvious, butthesameholdsatthesphere'11 asthefollowing examination ofthe~component oftheelectric intensity shows: '1 00 -~ '1 3/2'"~.E(~,TJ=TJ1,CP) =--(coshTJ¡-cosO L..,C/(coshTJ¡) a 1=0 [NdP/(coshO sin~ "P(<)] /----+ ------11(/ lcos~.~ 2(cosh'11-COSO According toEq.(37)thesuminthesecondterminthisexpression is .~C/(cosTJI)N/P/(cosO sin~SIn~L----'----'-------'--~ =--------, /=02(cosh'11-COSO2(cosh'11-COS03/2 whilethesuminthefirsttermis(15) (16) 00 dP/(cos{) d 1 sin( (17)"'c(coshTJ)N/-~~ --------- ---------L..,/1 d~-d~(cosh'11-COS{)1/2-2(cosh'11-cos03/2'/=0 andtheresultisthemutualcancellation ofbothsums.Therefore theelectric intensity at thesphere'11hasonlyitsiicomponent perpendicular totheelectrode. ONTHEEVALUATION OFTHECAPACITANCE OFBISPIIERICAL CAPACITORS 669 Theelectricchargedistribution ontheelectrodesisevaluatedbyusingGauss'law appliedtoEq.(14)attherespectivespheres: ThetotalchargesontheelectrodesareobtaincdbyintegratingEqs.(18)-(19)overthe respectivespheres: Q2=r(ha(e'1,",)h{d~h'i'd",1JoJo '1;112 V¡27ra2 f:(l+!)C/(coshr¡¡) r~sin~N/P/(cosO =-41Ta /=osinh[(l+ !)(r¡¡-'72)]Jo(COShr¡2-COSOI/2' =_V¡af:(l+!)C/(coshr¡¡)C/(cosh'12) 2/=0sinh[(l+!)(r¡l-'12)] Q¡=r(ha(~,r¡,,,,)h{ h'i'~d",1JoJo f7=TJ1 _V1a~ C/(coshr¡¡) - L ¡ 2/=0sinh[(l+2)(1)1-'12)] {(l+!)COSh[(l+!)(r¡l -'12)1rd~sin~N/P/(cosO 22Jo(coshr¡¡-cos01/2 .hl(l1)()]lo'd~sinh~N/P/( cosOsinhr¡¡}+SIn+-r¡¡-'12 --------- 2 o2(cosh'11-COS01/2 V¡a~ C/(coshr¡¡) {I 1 =-2L.h[l1)()J(l+2)cosh[(l+2)('11-r¡2)]C/(coshr¡¡)l=OSIn(+'i'11-'12 .h[(l1)()1dC/(cosh r¡¡)}-SIn+2'11-'12 d . '11(20) (21) 670 A.GÓNGORA-T. ANDE.LEy-Koo Inthelastlinetheintegralsover~areidentifiedwithC/(cosh1)¡)anditsderivativeby usingEg.(36).PhysicallythemagnitudesofthechargesinEgs.(20)and(21)areexpected tobeegua!.Theadvancedlevelstudentsmayprovethattheexpressioninsidethecurly bracketsisinfactC/(cosh1)2). ThecapacitanceofthebisphericalcapacitorfollowsfromeitherEgs.(20),(21), a00 C=-¿)e+!)C/(cosh1)1)C/(cosh1)2)csch[(l+!)(1)1-1/2)1. 2/=0(22) Itisinstructivetoanalyzetheconvergence oftheseriesinEg.(22)forsorneparticular cases. a)1£oneofthesphereshasaverysmallradius,say1)1-00,Eg.(38)indicatesthat thesummationinEg.(22)containsonlythee=oterm,andthecapacitance is reducedto (23) b)1£theotherspherealsohasasmallradiusandenelosesthefirstone1)2-00,with 1)2:s:1)1 2ae-ry,/2e-'I2/2 e=~_ll..=.12. e2-e22a e711-e'1J21 [ 1 acsch'71-acschrn1=---.!._.1.'TI r2(24) reproducing thewcll-knownresultofconcentricspheres[1-3]. c)1£oneofthesphereshasaverylargeradiussay1)2-0+,Eg.(39)indicatesthat thesummationinEg.(22)willhavetoineludemanyterms, buttheserieswillconvergefairlyrapidlybecausethefactorC/(cosh1)[)hasavalue smallerthanthatofEg.(39),andthehyperboliccosecantfactorwilldecrease exponentially aseincreases.Obviously,thecaseofFig.2bwith1)2=Oisa particularcaseofEg.(25). d)Theparallelplatecapacitorcorresponds tothelimit1)1=-1)2-O, 00 C=aLcsch[(2l+1)1)1], /=0(26) withaninfinitecapacityduetotheinfinitcarea. e)Thecaseofspheresexternaltoeachothcrdoesnotpresentanyconvcrgence problembecausethehyperboliccosecantfactorinEg.(22)willexponentially decreasewithemorerapidlythaninEq.(25). ONTHEEVALUATION OFTHECAPACITANCE OFBlSPHERICAL CAPACITORS 671 4.DISCUSSION Theelectrostatic fieldforbispherical capacitorshasbeendescribedinbispherical coor- dinates,throughitselectrostatic potentialfllnetion,Eqs.(12)and(36),anditselectrie intensityfield,Eq.(14).Theehargedistributions andtotalehargesontheeleetrodes, Eqs.(18)-(19),and(20)-(21),respectively, andtheeapacitance, Eq.(22),werealsooh- tained.AHthesequantitiesaregivenassllperpositions ofaninfinitenumberofbispherieal harmonieeontriblltions. ThereasonbehindthisinfinitenumberistheR-separability of theLaplaeeequation,Eqs.(5)and(8).Theeonvergenee oftheseriesfortheeapaci- tanee,Eq.(22),wasalsoexplicitlyanalyzedanditsnumeriealimplementation forspeeific spheresshouldnotmeetwithanydifficulties. Thiseanbeeontrastedwiththedifficlllties ofMaxweH'ssolutionsmentionedintheIntroduetion [4,5].Sornereadersmayrecognize thesimilarityanddifferencesofthisproblemandthoseofthetoroidalandspherieal-eap eleetrodeeapaeitors[10,11J;itisveryinstruetive toreadthesepapersinsuecessionin ordertoappreciatetheirrelationships andpeculiarities. Fromthedidaeticpointofview thesllggestedorderforseniorundergraduate orgraduatestudentsistostartwiththe presentpaperandfoHowwiththeothertwo,onaeeountoftheclosenessinformofthe bispherical harmoniestothefamiliarsphericalharmonics, andtheincreasingremoteness ofthetoroidalandspheriealeapharmonics. ApPENDIX Theeonstruetion oftheGreenfunetionrequiresthesolutionofEq.(9)inbispherieal eoordinates, forwhiehtheLaplacianhastheexplieitformofEq.(5),andtheDirae-delta funetioniswrittenas (27) Hereweusetheeompleteness ofthebi-spherieal harmonicsintheangles({,'{!)torepresent theeorresponding Dirae-delta functions. TheDirichletboundaryeonditionontheGreenfunetion,Eq.(10),takestheexplieit form Co({,r¡=r¡¡,<P;{',11',<P') =O,Co(Cr¡=112,<p;{','1',<p')=O. (28) ThecombinedanalysisofEqs.(9)and(5)withtheexplicitformsofEqs.(8)and(27) suggeststheharmonicexpansionoftheGreenfunetion, Co(C'1,<p;(','1',<p')=(cosh'1'-eos(')1/2(cosl171-eos01/2 00 t XLLYt:,,({', <p')Ytm({, <p)9t(11, '1'), (29) (=-0m:-t 672 A.GÓNGORA-T. ANDE.LEY-Koo wherewehavealsoincorporated thesymmetryundertheexchangeofthefield-pointand thesource-point, and gt(r¡,r¡')=Atsinh[(f+t)(r¡¡-r¡»]sinh[(f+t)(r¡<-r¡z)J, (30) guaranteeing thattheboundaryconditionsofEq.(28)aresatisfied,andthattheGreen functioniscontinuous atr¡=r¡'.Thereremainstodeterminethecoefficients At.Thisis accomplished bysubstitution ofEqs.(27)and(29)-(30)intheexplicitformofEq.(9), makinguseoftheR-separability oftheLaplaceequation,andofthelinearindependence ofthebispherical harmonics, toobtain [d2 12],411"(,--(f+-)gt(r¡,r¡)=--{)r¡-r¡).dr¡2 2 a(31) Theintegration ofthisequationleadsintumtothediscontinuity inther¡-derivatives of theGreenfunction 411" a(32) UseoftheexplicitformofEq.(30)givesthevalueofthecoefficients, 411" At=a(f+t)sinh[(f+ t)(r¡¡-r¡2)]' TheexplicitformoftheGreenfunction,Eqs.(29)-(30),becomes GD((,'1,'1';(,r¡','1")=411"(coshr¡'-COS()1/2(coshr¡ -coS01/2 a 00 t XLLYt:"((,<p')Ytm((, '1') l=Om=-l(33) xsinh[(f+t)(r¡¡-r¡»]sinh[(f+t)(r¡<-r¡z)] (34) (f+t)sinh[(f+t)(r¡¡-r¡2)] Theconstruction oftheelectrostatic potentialfunctionrequiresthenormalderivativeat thesphericalelectrode'11: 1DGol--- h,D'fJTJ11'=111=_1_411"(coshr¡_COS01/2 h1]'a 00t•,, sinh[(f+t)(r¡-r¡2)] LLYtm(C<p )Ytm((,'1').h[(f+!)(-)]/=0m;-t SID2r¡¡r¡2 ONTHEEVALllATION OFTHECAPACITANCE OFB1SPIIERICAL CAPACITORS 673 (35) (36){-(COShTi'-COS()I/2cosh[(£+~)(r¡1-r¡')j +11(COShr¡'-COS()-1/2Sinhr¡'Sinh[(£+~)(r¡I-r¡')1} 2(£+2) ~'=~l =__1_411"(coshr¡_coSOI/2 h~,a 001" sinh[(£+~)(r¡-TJ2)] ~nE/I:"(( ,<p)Ylm((,<p)sinh[(£+~)(r¡1-r¡2)] x(coshr¡¡-cos()1/2 TheintegralappearinginEq.(12), lo,df,'sineNIPI(cosh(')-------- =GI(coshr¡¡)o(coshr¡1-cosf)I/2 1(h)1+1F(1113.£3.h2)=21NIsecr¡12212+"2+" +2,secr¡1, canbeinterpreted asthecoefficients intheLegendrepolynomial expansionoftheinverse ofthesquarerootofthebinomial, 1 (coshr¡1-cos(')1/200 =l:GI(coshr¡¡)NIPI( cose)l=O _~"n( )1(h)l+lF(1 113.£3.h2)-~1YlrtCOS( INsecr¡l2212+"2+" +2,secr¡1'1=0 2l (37) Theexplicitformofthisexpansioncanbeobtainedbyusingsuccessively thebinomial expansion,expressingthepowersofcoseasalinearcombination ofLegendrepolynomials, exchanging theorderofthesummations andidentifyingoneofthemasthehypergeometric function2Fl'Obviously,theexplicitformoftheintegralinEq.(36)iswrittenbyreading itofffromEq.(37). ThenumericalvaluesoftheGI(coshr¡)functionsvarymonotonically between and ()1-1(11l3.3.)_2Glcosh71 -2lN2F¡2"+4'2+,,£+2,1-v'U+T"~ol 2l+1 whentheradiusgoesfromzerotoinfinity.(39) 674 A.GÓNGORA-T. ANDE.LEY-Koo ACKNOWLEDGEMENTS Thewritingofthispaperhadbeencontemplated forthefuturebytheauthors.However, thepresentation oftheworkonthespherical-cap-electrode capacitors[11)attheCanadian- American-Mexican PhysicsMeeting,motivatedthewritingatthepresenttime.\Vethank Dr.JorgeJoséofNortheastern UniversityforlettingliSknowabontthecurrentinterest intheproblemandforRefs.[4,6]. 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