Weber EM Fields Theory and Apps vol I (1950)

CO 00 S<OU_158460>5 OSMANIA UNIVERSITY LIBRARY CallNo.h>7'/toS7 A^ession No. Authory.yIv*' ,(-. .,,^o /I'.-- fc;Thisbookshould bereturned 6norbefore thedate lastmarked fielow. ELECTROMAGNETIC FIELDS Theory andApplications Volume IMapping ofFields ELECTROMAGNETIC FIELDS TheoryandApplications Volume IMapping ofFields ErnstWeber Professor ofElectrical Engineering PolytechnicInstitute ofBrooklyn John Wiley&Sons, Inc.,NewYork Chapman &Hall, Limited, London COPYRIGHT, 1950 BY JOHNWILEY &SONS, INC. AllRights Reserved Thia book oranypart thereof must not bereproduced inanyform without the written permission ofthepublisher. PRINTED INTHEUNITED STATES OFAMERICA To Oil.SONYA WEBER whose unselfish andinspiring attitude made thisbook possible PREFACE Thesubject ofelectromagnetic theory asformulated byJames Clerk Maxwell hasbecome classical, and itishardly possible to addbasically newmaterial. Yet,theastounding developments in physics and electrical engineering haveshown clearly that the utilization ofelectromagnetic phenomena hasnotreached the point ofsaturation. Forthisreason, abook giving arather com- prehensive survey ofthemethods ofanalysis and ofresults ob- tained withthem should prove ofvalue tothestudent andthe teacher inadvanced courses aswellastotheprofessional engineer andthephysicistintheresearch anddevelopment laboratory. Now, ithasbecome clear that thescopeofelectromagnetic theory and itsapplications toproblemsofinterest totheengineer, thephysicist, andtheapplied mathematician ismuch toogreat tobecovered inonevolume ofpractical size. Fortunately, the subject may berather naturally divided intotwofundamental branches: onedealing with static electric andmagnetic fields and leading tomethods ofsolving thefamilyofpotential equationsin various forms; theother dealing with thedynamic interaction of electric andmagneticfieldsandleading tomethods ofsolving the family ofwave equations invarious forms. This division hasbeen followed here,andthefirstvolume presents asurvey ofthemethods ofmapping thedistribution ofstatic electric andmagneticfields. Many authors whohave dealt with thissubject havehada tendency topresent aparticular version orviewpoint, orto emphasize oneparticular method ofanalysis. Admirable assuch treatises maybebythemselves, they arelesssuitable forusein graduate courses where emphasis must lieupon guidance toa basic understanding ofdifferent ways ofreasoning andofformulat- ingideas. Graduate study must concern itself primarily with basic concepts, ofwhich there arealways butafew;itshould demonstrate theconnection between them through generic prin- ciples andshould lead tothecritical understandingoftheir full implications. Onlywhen thisaimhasbeenreached illuminated viii Preface byconstructive applications canonespeak ofmastery ofthe subject. Inorder forthegraduate teacher inengineering orapplied science toachieve thisaim itisimperative that thebasic facts uponwhich theory restsandfromwhich itreceives support and confirmation bepresentedinbroad strokes; and itdemands a presentation notofmathematical detail ofexistence theorems, but ofillustrative examples which demonstrate thevariety offormula- tionsandapplicationsofthefewprinciples, sofrequently disguised under thenames ofspecific "laws." Ofcourse, asinanyquantita- tivetreatment, mathematics must beused asthemost precise andmost satisfying means ofexpression, and itisquite necessary torecognize, andconveniently refer to,theproofs ofexistence and ofuniquenessofsolutions which have been developed bypure mathematicians. Theburden ofthisgreat debt tothemathema- ticians hasbeen lightened onlybecause ofthetremendous stimula- tion ofmathematical research through theincessant need fornew solutions. The recognition ofthefundamental importance inelectrical engineering ofwell-founded field conceptsinalladvanced de- velopment anddesign, aswellasinresearch, hasledtotherequire- ment ofacourse inelectromagnetic theory innearlyallmajor graduate schools. Where thiscourse isgiven bytheDepartment ofPhysics, mathematical theory maypredominate, andwhere it isgiven bytheDepartment ofElectrical Engineering, design information may beemphasized. Inorder tocombine theem- phasis onthebasic aspects common toallpotential fields with a comprehensive treatment oftheavailable analytical and practical methods offield plotting, thisvolume hasbeen organized ina somewhat unconventional manner. Instead oftheusual vertical division into electrostatics, magnetostatics, andelectrokinetics, a horizontal division ofthesubject matter isused. Thus,allthe physical relationships arcestablishedfirst,andthemethods of actually obtaining static field distributions aredemonstrated sub- sequently. This avoids considerable repetition and leads toa clearer understanding ofthefactthatmethods ofanalysis arein- dependent ofthespecific branch ofapplication, andthatnomen- clature isfrequently accidental andbynomeans theessence of knowledge. Itis,ofcourse, assumed thatthereader possesses a general knowledgeoftheelectromagneticfield asnormally gained Preface ix inapertinent undergraduate course andthatheisfamiliar with theprinciplesofvector notation. Tobesure, thefield-mapping methods aregenerally formulated inspecific coordinate systems as conditioned bythegeometryofthefields studied; butthevector notation proves ofdefinite advantage forthepresentation ofthe basic relations inelectric, magnetic, andotherfields, astreated inthe firstthree chapters. Following thesummary ofthebasic physical relations, thecom- parative physical quantitiesinsixbranches ofphysics andengineer- ingarelisted intable 9.1,which serves asthekeyforthetranslation offield solutions inanyonebranch into solutions ofanalogous problemsintheother branches. Chapter 4deals with thesimple applications,ofthesuperposition principle, such assystems of point and linecharges, line currents, andsimple geometries of spatially distributed charges andcurrents. Formore complicated geometries,itisfrequently though notalways simplest tomap thefield distributions experimentally; theexperimental methods thathave been used successfully aredescribed inChapter 5, including theanalogies utilized intheelectrolytic trough. Asal- ternatives totheexperimental procedure, graphical andnumerical field-plotting methods aretaken upinChapter 6withemphasis onthepractical phases ofactual applications; rather extensive treatments aregiven oftheuses ofelectrical andmagnetic images and ofinversion methods which arenotalways sufficiently em- phasized. Next, theuseofanalytic functions forthesolutions of two-dimensional fieldproblemsisshown inChapter 7,and in particulartheextremely powerful methods of"conjugate" functions and ofconformal mapping, which areamply demonstrated. Finally, Chapter 8gives themathematical treatment ofthree- dimensional field problems, involving bynecessity athorough discussion oforthogonal coordinate systems that issupported by many illustrations which itishoped willmake iteasier to visualize clearly thegeometrical aspects. Inorder toaidateacher inorganizing thematerial intofeasible courses, several suggestions areoffered inlinewith courses which havebeentaught bytheauthor. Forthe firstpart ofacourse on Electromagnetic Theory dealing with static fields onemight com- bineChapters1and2andsection 8ofChapter 3with selected examples fromChapters 4and6andsection 25ofChapter7.For aone-semester course inApplications ofFunctions ofaComplex x Preface Variable onemight takethematerial ofChapter 7,sections 25to 28.Again, foracourse inClassical Boundary Value Problems dealing with thepotential equation, onemight combine section 9withthetwo-dimensional applications insection 29andChapter 8onthree-dimensional applications. Tosatisfy individual re- quirements,stillother combinations arepossible. Fortunately,itisnolonger necessary toapologize fortheuse oftherationalized MKS system ofunits inabook dealing with electromagnetic theory and itsapplications. There might, how- ever,becriticism ofthefactthattheengineering notationVl=j hasbeen carried into the classical realm ofanalytic functions. This factshould notbeconstrued asaserious offense, fornotation isnottheessence; rather,itshould betaken forwhat itis,a choice necessitated bythesevere conflict ofi=V^l with the symbol fortheinstantaneous value ofcurrent i=Imsinat,which isinternationally standardized andcustomarily defined asthe imaginary part ofIme3at ,with theeffective (rootmean square) value /=Im/^/2,allofwhich willoccur frequentlyinVolume II. The original suggestionofasmallvolume onmappingoffields wasmade in1935bythelateV.Karapetoff, Cornell University, as Chairman ofaSub-Committee onMonographs oftheCommittee onElectrical Insulation, National Research Council. Acrude draft ofthemanuscript hadthebenefit ofhiscriticisms aswellas those ofJ.F.H.Douglas, Marquette University, andH.Poritzky, Schenectady. World War IIinterfered with theplans forthis monograph. Furthermore, thevarious graduate courses given bytheauthor atthePolytechnic Institute ofBrooklyn slowly changed theoriginal conception ofthemonograph totherather different one ofthisvolume. Thecontact withmany graduate students hashadastrong educational influence upon me,and I wish toacknowledge tothemmydeep appreciation. Certainly through their persistent questioning andtheir gratifying response, theyhavemade teaching thedelightful professionitis. Iam greatly indebted alsotoanumber ofmycolleagues, especially to Paul Mariotti, who assisted inthepreparationofthedrawings; toProfessor William R.MacLean, whoread parts ofthemanu- script andmade constructive suggestions; andtoProfessor Charles A.Hachemeister, whoreadmost oftheproof andmademany helpful comments. Asthepreface occupies aprominent place in Preface xi thebook, Iamveryhappy andgrateful that Icould enlist forits composition theinvaluable assistance ofProfessor LeoE.Saidla, head oftheDepartmentofEnglish. Finally,Itakegreat pleasure inacknowledging theencouragement andsupport which Ireceived from President Harry S.Rogersinwriting thisbook. Ernst Weber Brooklyn, NewYork April, 1950 CONTENTS 1.THEELECTROSTATIC FIELD 1.Fundamental Relations intheElectrostatic Field 1 2.Analytical Theory oftheElectrostatic Field 6 3.Energy andForces intheElectrostatic Field 13 4.Critical Field Values 23 Problems 36 2.THEMAGNETOSTATIC FIELD 5.Fundamental Relations intheMagnetostatic Field 39 6.Analytical Theory oftheMagnetostatic Field 43 7.Energy andForces intheMagnetostatic Field 56 Problems 63 3.GENERAL FIELD ANALOGIES 8.The Electric Current Field 66 9.Other Physical Fields 71 Problems 80 4.FIELDS OFSIMPLE GEOMETRIES 10.Systems ofPoint Charges 82 11.Quasi Point Charges 97 12.LineCharges andQuasi LineCharges 106 13.LineCurrents andQuasi LineCurrents 1^9 14.Simple Systems ofDistributed Charges 144 15.Simple Systems ofDistributed Currents 152 Problems 165 5.EXPERIMENTAL MAPPING METHODS 16.Experimental Mapping ofElectrostatic Fields 169 17.Experimental Mapping ofMagnetic Fields 177 18.Utilization ofField Analogies 183 Problems 195 6.FIELD PLOTTING METHODS 19.Graphical Plotting ofElectrostatic Fields 197 20.Graphical Plotting ofMagnetostatic Fields 206 21.Method ofElectrical Images 215 22.Method ofMagnetic Images 233 23.Method ofInversion 244 24.Numerical Methods 259 Problems 270 xiv Contents 7.TWO-DIMENSIONAL ANALYTIC SOLUTIONS 25.Conjugate Functions 277 26.Conformal Mapping301 27.Conformal MappingofStraight-Line Polygons325 28.General LaplacianPotential Problems andConformal Mapping 362 29.Two-Dimensional Harmonic Function Systems383 Problems408 8.THREE-DIMENSIONAL ANALYTIC SOLUTIONS 30.Axially SymmetricalPotential Fields 414 31.General Orthogonal Coordinate Systems435 32.Cylindrical Coordinate andFunction Systems454 33.Confocal SpheroidalCoordinate andFunction Systems474 34.UseofGreen's Functions 517 Problems526 APPENDICES 1.Letter SymbolsforElectrical Quantities531 2.Conversion Tables forUnits 534 3.Review ofFundamentals ofVector Analysis538 4.General Bibliography547 5.OnBessel Functions 556 6.OnLegendre Functions 565 INDEX577 NOTES FORTHEREADER Thesymbols offield quantities aretabulated inAppendix1. Totransform therelations from therationalized MKS unit system toother unitsystems, consult Appendix2. Abrief review ofvector analysisisgiveninAppendix3. Equations arenumbered consecutively ineach section; refer- ences toequations indifferent sections carry thesection number, thus (5-4)means equation (4)insection 5. TheBibliography inAppendix 4listsonlybooks towhich several references aremade inthetext; such references, e.g.Attwood,A2 p.243, givethepageandtheauthor, thesuperscript indicating number 2ofsectionAoftheBibliography. 1.THEELECTROSTATIC FIELD 1-FUNDAMENTAL RELATIONS ,INTHEELECTROSTATIC FIELD From primitive observations, electrostatics divides allmaterials intoonlytwogroups, conductors andinsulators. The firstgroup isendowed with infinite mobility ofelectric charges such thatany redistribution occurs inanunobscrvably short time. Thesecond group haszeromobility ofelectric charges; any redistribution occurs inanuninterestingly long time. Admittedly, this isaradi- caldivision, but itleads toamuch simpler theory oftheelectro- static fieldthanwould bepossible otherwise. Inaddition, the results areofdirect practical value, anddeviations inspecific cases canreadily beindicated. Thebasic quantitative relationship ofelectrostatics isCoulomb's lawofforce action between twocharges QiandQ2, Thecharges areassumed tobeconfined toverysmall regions (point charges) sothat thedistance rcanbeidentified asthedistance between centers, and Eistheabsolute dielectric constant ofthe homogeneous infinitely extended medium inwhich theforceFeis measured; oneusually expresses e=ever,where EVistheabsolute dielectric constant offreespace (vacuum) (seeAppendix 2for unitrelations). The relative dielectric constant eristhenumeric value generally found inthetables ofmaterial constants. Through- outthisvolume, only isotropic dielectric media willbeconsidered, sothat eisalways assumed tobeindependent ofdirection. 1 2 TheElectrostatic Field [Ch. 1 Thestudy ofelectrostatics, then,isprimarily concerned with theequilibrium distribution ofcharges onthevarious conductors comprising aparticular system, under theinfluence ofthisCoulomb force. Ifthecharge Q2isvery small, sothat itcauses anegligible andonly local distortion ofthefield ofcharge Qi,itcanbeused as aprobe fortheexploration oftheforce field ofcharge Qi.From (1),thelimit value forvanishing Q2 --F* .E-JLQi^i- ,.ji w isthen interpreted asthe electric intensity orfield strength of charge Qi- Inthecase ofasingle positive point charge, thefield strength Ehas radial, outward direction,invector notation (see Appendix 3forabrief review ofvector analysis) 10, T31^^/QA where r/rserves toindicate theradial direction. Inthecase of anygeneral distribution ofatotal charge Q,onecansubdivide it intosmall elements Qajconsider each tobeapoint charge, andby useoftheprinciple ofsuperposition obtain theresultant field vectorEatanypointP 1nQa where theraaretheradius vectors from thecharges Qatothe point P. Ifoneplaces avery small chargeQintotheelectric field ofany number ofcharges Qa,and ifone ispermitted todisregard theeffect ofQupon thecharge distribution oftheQajthensuchasmall charge isagain called aprobe charge, since itcanwell serve toprobe orexplore the electric field ofthecharge assembly bymeans oftheforce action upon it,which isgivenby(2)as,(JE. Left freetomove, atverylowspeed, thisprobe chargewilltrace the direction ofthevectorEinspace andthepath described iscalled a,fieldlineoralso lineofforce;ithasthevectorEeverywhere as tangent. Defining thepathelement asds,itscomponents dxtdy t dzmust beproportional tothose ofE,sothat dx_dy_dz Sec. 1] Fundamental Relations which isthedifferential equation ofthefield lines. Since forany point charge thefield lines diverge radially forpositive signand converge radially fornegative sign, there canbenoclosed field lines. Carrying asmall charge Q2overanyfinite pathP\P% within Fio. 11Electrostatic Field ofaSingle PointCharge. thefield ofasingle point charge located attheorigin asinFig. 11requires thework Wpp-i=I /P!Fe-ds=Q2IE-ds(6) However, Ehasonly radial direction, sothatE-ds=Edrand hence withtheuseof(2) Thework isthusindependentofthepath;itdepends onlyonthe endpoints, and istherefore zero foraclosed path. Onecanim- mediately generalizethisfactbecause of(4)andcharacterize the electrostatic field asaconservative field. Thismeansalso, asseen from (6),thatthelineintegralofthevectorEvanishes forevery closed path;allfield linesemanate from,andterminate on,charges. 4 TheElectrostatic Field [Ch. 1 Ontheother hand, alineintegralisindependent ofthepathif theintegrand represents acomplete differential. This requires thatthecomponentsofEcanbeidentified asthederivatives of asingle, scalar function <,theelectrostatic potential. Vectorially, E=-grad $=-V* (7) andforthesingle point charge thepotential function becomes at oncefrom theabove -f-i(8) 47T T where risthedistance from thecharge center. Since (7)defines only thederivatives of$,anyarbitrary constant could beadded in(8).Foranynumber ofpoint chargesinasinglemediumE, superposition again holds andonehas *=j-E^(9) 4irea=1ra subject tosome arbitrary constant. Obviously, thescalar sum- mation involved in(9)ismore convenient than thevector sum requiredin(4).The surfaces obtained forconstant values of potential arecalled equipotential surfaces and areequally as characteristic forthefieldstructure asthefield lines; infact,they form with thelatter anorthogonal system ofsurfaces and lines. The objective offieldmappingisprecisely theevaluation ofthis orthogonal fieldgeometry inquantitative terms. Returning totheconceptsofconductors andinsulators inthe ideal sense,itmustbeclear atoncethatconductors canhavecharges onlyonthesurface andmusthave constant potential throughout their interior; anypotential variation would cause afieldvector and, therefore, aforce action until asurface charge distribution is established which maintains constant potential. Conversely, any chargeintheinterior oftheconductor would beasource offield lineswhich could bemaintained onlybyapotential difference. Anyconductor surfaceis,therefore, anequipotential surface, and thefield linesterminate perpendicularly toit. Aninsulator ordielectric, ontheother hand,willnormally not carryanycharges atall;itwillserve primarily toseparate charged conductors. Incertain instances, space charges produced by Sec. 1] Fundamental Relations 5 thermionic orother emission, byglow discharges, orbyarcscan exist within insulators. Assume again asingle point charge Qin ahomogeneous dielectric; then (2)willgivethefieldstrength as depending onthedielectric constant s.However, thequantity eE=Disindependentofthedielectric andappears asdensityof thecharge were itdistributed uniformly overthesurface ofasphere ofradius r.Itisdesignated asavector called dielectric fluxdensity (orelectric displacement), D=eE (10) forhomogeneous dielectrics forwhich eisaconstant. Again generalizing formany point charges, theintegral ofD-noverany closed surface Sgives thenthesum ofallcharges contained within thissurface (Gauss's dielectric fluxtheorem), >-ndS =ZQa (11) nomatter what their distribution. Foracontinuous space charge distribution offinitevolume density p,theright-hand side of(11) isbetter written astheintegral overthevolume Tbounded bythe closed surface S.Transforming alsotheleft-hand surface integral, onehasthen Applying thisrelation toverysmall dimensions oneconcludes that divD=V-D=P (12) oranyspace chargeisasource orsink ofthevectorDindependent ofthedielectric medium. Inisotrbpic dielectrics, with nospace charge, divD= andthevectors EandDhave thesame direction according to (10), sothatthefield lines ofthevectorEcanalsobeinterpreted asdielectric flux lines, being tangential tothevectorDatevery point. Since thetotal dielectric fluxcoming from acharge Qis numerically equaltothecharge, onecanconceive ofachosen number offlux lines torepresent thecharge value. Inthecase ofseveral charges, thefluxlines willthen quantitatively represent thedielectric fluxdistribution. Forconductors ofarbitrary shape 6 TheElectrostatic Field [Ch. 1 inauniform dielectric, Disnormal tothesurface and itsvalue is identical with thesurface density ofcharge, D=Dn=* (13) This follows from (11) sincenoelectric fieldcanexist within the conductor. The flux linesbounding afinite surface element dS which carries acharge<rdS=5Qform aflux tubewhich willlead toanelement 8S'onanother conductor where itdelimits acharge (SQ)=a'8S'. These fluxtubes areavaluable aidinthevis- ualization ofthefieldgeometryifnospace chargeispresent (see Fig.3-1). 2-ANALYTICAL THEORY OFTHEELECTROSTATIC FIELD Onthebasis ofsection1,thegeneral problemofelectrostatics canbeformulated astheevaluation ofthe field distribution in dielectrics and ofthesurface charge distribution onconductors subject tocertain known potentialorfield strength values des- ignated asboundary conditions. Actually, potential values as such arearbitrary, aspointed outinsection 1;only potential differences canbemeasured, sothat, toanysolution oftheelec- trostatic potential function, anarbitrary constant could beadded. Usually, onechooses some reference conductor such asground to beofzero potentialinorder tosimplify numerical computations. Asalready indicated, solution ofelectrostatic fieldproblems usually becomes more convenient with theuseofthescalar elec- trostatic potential. Asdefined in(1-7), theelectric fieldstrength Ecanbeexpressed asthenegative gradientofthepotential. Introducing thisinto relation (1-10) andthen substituting into (1-12), onehas V-D=-V-(eV$)=p (I) oralso [seeAppendix 3,(21)] V*-Ve+eV2$=-p (la) This represents themost general differential equation foranin- homogeneous isotropic dielectric, wherein thevariation ofemustbe known. Thoughthisgeneral casehas little practical value,it readily permits specializationforseveral important cases. Sec. 2] Electrostatic Problem 7 Special Cases oftheElectrostatic Problem, a.Ifthe dielectric ishomogeneous (e=cons), andwithout space charge, thedifferential equation (1)becomes Laplace's equation, orsimply thepotential equation V2*=(2) This special case isthemostimportant oneandadmits quite readily ofanalytical, graphical, aswellasexperimental, solutions; most ofthemapping methods pertain toit. Ifoneconsiders asingle dielectric bounded entirely byconductor surfaces with charge distributionscr,itispossible toconceive of theindividual surface charge elements adSaspoint charges inthe sense of(19)andtowrite atonceaformal solution of(2)inthe form ofthe*integral 1/vjO /O\dS(3) which hastobeextended over alltheconductor surfaces. Since onconductor surfaces, asseenfrom thedielectric,cr=Dn=eEn inaccordance with (1-13), onecanwrite (3)alsointheform (30) 4?rJJ r lirJJdn r which shows thattheentire potential distribution isdetermined by theknowledge ofthenormal potential gradient ontheconductor surfaces! Apotential probleminwhich thevalues ofthenormal componentofthe field gradient aregiven onthebounding con- ductor surfaces iscalled aboundary value problem ofthesecondkind, and (3a)represents theexplicit solution forthespecial casethat En^0.Though (3)and (3a) areformal expressions ofgreat value intheanalytical theory oftheelectrostatic field, asfor exampleingeneral existence proofsofsolutions inpotential theory, theydonothave comparable practical value because boundary values arerarely specifiedintheabove manner. However, (3a) points outthat Laplacian potential functions have exceptional qualitiesofregularity. Indeed, anyfunction <E>(z, y,z)which hascontinuous second order derivatives inx,y, and zthat satisfy theLaplace equation (2) iscalled aharmonic function within theregion where that istrue. Harmonic functions can, therefore, always beinterpreted aspotential solutions, and 8 TheElectrostatic Field [Ch. 1 anypotentialsolution must beaharmonic function orafinite or even infinite sum ofharmonic functions. Moreover, such func- tions canbeexpanded nearanypoint within theregion oftheir definition intoconvergent power series, which characterizes them asanalytic Junctions, sothat theLaplace equation (2)canhave only analytic solutions! Itisthisgreat regularity ofbehavior which hasledtothevarious powerful methods ofpotential theory; formathematical details seeKellogg.010 6.Iftheldielectric ishomogeneous (E=cons), butwith space charge, thedifferential equation (la)reduces to V2*=--(4) e which iscalled Poisson's equation. Thistype ofdifferential equa- tion finds itsmost useful applicationinvacuum tube orgaseous discharge problems. Aformal solution ofitisrepresented bythe superposition ofavolume integral over allspace charge elements pdrconceived aspoint charges upon anysolution ofLaplace's equation, forexample intheform (3)with theknown surface charge distribution a where risthedistance from thepoint atwhich*isbeingcomputed tothecharge elements. Though (5)canbeevaluated insomevery simple cases,inmost instances that isnotpossible.Ifthespace charge density pisgiven asanexplicit function ofthespace coordi- nates, then thesolution isfound best asthesuperposition ofa Laplacian potential function andaparticular integral ofthein- homogeneous differential equation. Inthepractically important problems, however, thespace charge density pisafunction of thepotential itself, sothat (4)becomes anon-linear differential equation; these problems willnotbeconsidered further since their scope goes farbeyond conventional potential theory. c.Ifthedielectric isinhomogeneous andwithout space charge, thegeneral differential equation (la)reduces to Ve-VS=(6) which canbesolved insimpler cases ifthevariation ofeisgiven. Sec. 2] Boundary Conditions 9 This differential equationisofimportance incable problems where thedielectric might besubjected totemperature gradients causing avariation ofthedielectric constant, aswellasincertain capaci- tance measurements where humidity andpressure variations might cause avariation ofthedielectric constant. TheBoundary Conditions oftheElectrostatic Field. As shown insection1,allconductor surfaces inanelectrostatic field must beequipotential surfaces; thespecification ofthepotential values ontheconductors, therefore, constitutes aconvenient set ofboundary conditions inthecase ofasingledielectric. Problems ofthistype aregenerally designated asboundary value problems of thefirstkind. Instead ofthepotential values, onecould alsoassign thetotal charge values fortheconductors, orgivepotential values forsomeandtotal charge values fortheremaining conductors. Aspointed outabove inconnection with (3a),knowledge ofthe normal potential gradient ontheboundaries oftheelectrostatic fieldregion alsodefines thepotential distribution uniquely except foranadditive constant $which canbeinterpreted asanabsolute reference potential andwhich, forconvenience, canbechosen as zero. Boundary value problems which specify thenormal gradient value over theboundary surface ofthe field region under con- sideration aresaid tobeofthesecond kind. Instead ofnormal gradient values, onecould also specify charge densities oncon- ductor surfaces. The specification ofpotential values over certain areas ofthe boundary surface ofthe field region under consideration and of normal gradient values over theother areas leads toaboundary value problem ofthemixed kind; though these areinfrequent in purely electrostatic fieldproblems, they arise often inconnection with stationary current flowandsimilar flowproblems (seesection 9). Ifseveral different dielectrics arepresent, then itisnecessary tosolve the differential equation (la) (orthepertinent special forms) foreach individual dielectric. Inaddition totheabove boundary conditions ontheconductor surfaces, continuity condi- tions attheboundary surfaces ofanytwo dielectrics have tobe satisfied inorder tolink allindividual solutions soastoform the complete solution oftheelectrostatic field distribution. Application ofGauss's dielectric fluxtheorem (111)tothesmall cylinder ofheight dh >enclosing thecharged surface element dS 10 TheElectrostatic Field [Ch. 1 inFig. 2-1,which might betheboundary surface between two different dielectrics, leads to -ndS=Dn2dS2-Dnl dS Inthelimit ofvanishing dh Dn2-D nl=<r(7) which isonegeneral boundary condition fordielectrics. Usually, FIG. 21Boundary Condition fortheDielectric FluxDensity. nosurface charge exists, sothat continuity ofthenormal com- ponent ofthedielectric fluxdensity isrequired. The existence ofthescalar poten- tialfunction <f>in(1-7) wasinferred from thefactthatthelineintegral of theelectric fieldstrength Evanishes forany closed path. Ifthis rela- tion isapplied toaboundary sur- face oftwo dielectrics andchoosing thepath ofintegration asshown in Fig. 2-2, oneimmediately obtains asdh-> En=Et2 (8) FIG.22Boundary Condition fortheElectric Field Strength, thesecond general boundary condi- Sec. 2] Electric Polarization 11 tion.Etstands forthefieldcomponents parallel totheboundary surface atthepoint P. Adielectric completely surrounded byother dielectrics must satisfy, therefore, boundary conditions involving only thederiva- tives ofthepotential function. Ifnosurface charge exists, the combination ofthetwoboundary conditions (7)and (8)leads to Et2 2Eti or tani=tan 2 (9) 2 iftheangles ofthefieldvectors withthenormals totheboundary surface aredesignated bya2andaitrespectively. Relation (9) isanalogoustoSnell's lawofrefraction inoptics and isfrequently called thelaw ofrefraction oftheelectrostatic field lines. Itis ofparticular value inthegraphical field plotting methods. Electric Polarization. The characterization ofadielectric medium bytheconstant eissatisfactory aslong asnoinquiryis made intothestructural aspectsofthemedium which might be responsiblefor e.Toobtain ahypothetical concept ofthenature ofadielectric, onecanseparate thedielectric fluxdensity (1-10) intotwocomponents, onewhich could bethought ofasexisting infreespace, theother astheparticular modification caused by dielectric matter, namely, D=EE+(e-)E=eE+P (10) where thequantity Pisdesignated aselectric polarization. The effect ofthisseparation upon thedifferential equationfor thepotentialinauniform dielectric isobtained byusing (10) in (1-12): divD=p=edivE+divP Introducing E=-V*from (1-7) p=-eV2*+divP onehasalso V2*=--(P-divP) o The dielectric cantherefore beinterpreted asafictitious space 12 TheElectrostatic Field [Ch. 1 chargedistribution ofvolume density p=divPexisting infree space.Ifoneexcludes any realspace charge bysetting p=0, then theunchargeddielectric medium canberepresented onlyby adistribution ofvery small dipoles (seesection 10),quadripoles, etc.,which forsmall finitevolume elements always have zero total charge, butwhich produce locally very strong dielectric fluxdensi- ties. Without anexternally appliedelectric field,itisassumed that these elemental units which aregenerally identified as molecules have random orientation sothat over finite small volume elements alsoP=0.Theapplicationofastatic electric field causes orientation ofthedipoles andappearance ofP.Ob- viously, without firstdefining theunderlying structure, Pcannot be evaluated. Ifoneconsiders, then, auniform dielectric ofvolume Tandcon- stant einfreespace exposed toanelectric fieldE ,onecanrepresent thedielectric bythesame volume Tfilled with fictitious space charge p=divP.Since theactual structure canonlyinclude satiated charge complexeslikedipoles, there remains onthesurface ofthevolume Talayerof"bound" charge withadensity awhich canbedefined from thecontinuity condition (7)ifcombined with (10), Dn2-D nl=<r=z(En2-Enl)+Pn2-Pnl Asseenfrom (3a), thenormal component Enonconductors, or correspondinglythedifference ofthenormal components inthe twoadjoining dielectrics, defines thepotential values; thus, with theabove, En2-E nl=-[*- (Pn2~Pnl)]=-(*+</) (12) EO o anda=(Pn2Pni)-Intheabsence oftruecharge, thetotal contribution totheresultant potential existing outside orinside thedielectric istherefore similar to(5) Obviously, Pmustdepend onE,theimpressed field;ifthis field Eisnotrigidly fixed, butsubject tomodification bythepresence ofthedielectric, thenmere superposition doesnotholdand(13) isonlyafirststepinthesolution. Sec. 3] Energy andForces 13 3-ENERGY ANDFORCES INTHEELECTROSTATIC FIELD Foranyfinite assemblage (orsystem) ofelectrical charges, the total algebraic sumcanbeeither zero ordifferent from zero. In the first case, thesystemiscalled acomplete system,allthefield linesterminate oncharges within thesystem, nofield linesgointo infinity, andthetotal dielectric fluxthrough anyclosed surface surrounding thesystemiszero. Inthesecondcase, thesystem isincompleteinfinite space; however, onecanassume anyvery large spherical surface ascarrying theopposite andequal ofthe resultant chargeofthefinite system since thetotal dielectric flux through anyclosed surface surrounding thesystem willbeequal tothecharge enclosed. Thesystem together with theclosed surface then willagain becomplete, andthefield linesgoing into "infinity" areusually identified with stray capacitances. TheIdeal Condenser. Thesimplest complete systemisthat oftwoconductors ininfinite space withequalandopposite charges. Ifthepotentialdifference between theconductors isgiven,V12= *i *2,with *i>$2,and ifthecharges ontheconductors are Qi>0,andQ2=Qiithen thecapacitance ofthesystemis defined astheratio V\2 ^l ^2 This capacitanceisapurely geometric characteristic oftheelec- trostatic fieldand itsdistribution, and isindicative ofthelinear relationship between thefield quantities. Thearrangement, called a"condenser," stores anelectrostatic fieldenergy equal tothework required tobuildupthecharges on theconductors. Since thetransfer ofacharge element dQrequires anamount ofwork givenby(16)and (18), dW=($1-$2)dQ (2) thetotalworkbecomes with (1)above =&1(h2 CdQ= 2C andtherefore thefieldenergy TTe=-=-Qi7i2= 2cvu (3) 14 TheElectrostatic Field [Ch. 1 Inthisexpression,allquantities areintegral quantities, directly amenable tomeasurement. TheInfluence ofGround. Generally, theconcept oftheideal condenser isabstract because oftheinevitable surroundings which willexert influence upon thefield distribution. Consider firstthe simplest case, theinfluence ofground upon thecharge distribution ontwoconductors. Obviously,ifthegroundisassumed, asis usual inelectrostatic problems, tobeanidealconductor of FIG.3-1The Influence ofGround upon theElectrostatic Field ofTwo Conductors. tialzero (seebeginning ofsection 2),chargeswillbeinduced init, and field lines willspanbetween ground andtheother twocon- ductors. Each conductor, therefore, willcarry atotal charge which willbebound partially bytheother twoconductors (seeFig.3la). Forconductor1,forexample, thecharge willbe Qi=Qio+Qi2- Ofcourse, theconceptofground could bere- placed bythat ofavery large closed shieldrepresenting the inevitable surroundings andmaking thesystem acomplete one. Onecannow define only capacitance coefficients (orpartial capacitances oralso direct capacitances asdefined byCampbell seefootnote 2,p.17)such as Qio Ql2 Q02 $0-$2(4) andbecause oftheindicated charge values inFig.3-la,onewill immediately conclude thatCi=CIQ,CQ2=C2Q,Ci2=C2i,or Sec. 3]Conductors inaHomogeneous Dielectric 15 thatthe"matrix" ofpossible capacitance coefficients CoiCY /~1 f\ /^i /3\ issymmetrical about themain diagonal. Physically, onlyI- )= \2/ 3!=3independent capacitance coefficients1 exist; this 2t!(o2t)! corresponds tothecircuit equivalent ofthree condensers asshown inFig.3-16. The total charge onconductor 1cannow readily beexpressed asthesum ofthepartial charges bound byallpossible potential differences from conductor 1totheothertwoconductors, namely, Qi=Qio+Qi2=Ci(*i-*)+C12(*i-*2) (5) andthealgebraic sign ofthepartial charge isthat ofthepotential difference. The total fieldenergy stored intheelectrostatic fieldbecomes thesum total ofthat ofthethree individual condensers, namely, we=y2cwvw2+y2c12y122+y2c02v022 (6) Systems ofConductors inaHomogeneous Dielectric. Assume asystem ofnconductors inauniform, homogeneous dielec- tricwithground asthe(n+l)stconductor with index 0.The matrix ofcapacitance coefficients (7ojgwillnowhave (n+1)rows andcolumns andbeadirect extension ofthatgiven fortwocon- ductors andground. The totalnumber ofdifferent capacitance (n+1)! n(n+1)coefficients isnow- =---with allCaa=0,and 2i\\n 1jI 2t thetotal charge onanyoneconductor inthesystem canbefound bythesuperposition oftheproducts ofallthemutual capacitance 1Thenumber isgiven asthecombination ofndifferent things taken r atatimewithout reference totheir order; this isnCr=- :-=(- 11 rl(n-r)\ \r/ orthenumber ofcombinations; seeEshbach: Handbook ofEngineering Fundamentals, p.2-21; John Wiley,NewYork, 1936. 16 TheElectrostatic Field [Ch. 1 coefficients andtherespective potential differences, n Qa=ECap(3>a-$0), a=0,1,2, -,n (7) 0=0 indirect analogyto(5). This expression alsoshows howthe mutual capacitancecoefficients ofasystem canbedetermined experimentally bymeans ofballistic galvanometers. Thecharges aremeasured afteraknown potentialdifference hasbeenimpressed between oneconductor andground towhich alltheremaining con- ductors areconnected. Starting withconductor1,andtaking the ground potential $=0,thecharge values willbefrom (7) Qi=ECIB&I; Qa=-Cai$i fora=0,2,3, -,n 0=0 Measurement ofthen+1charges (after disconnecting from groundsoastoavoid charge redistribution inthesystem) gives (n_|_i)capacitancecoefficients. Repetition oftheprocedure with rotational selection ofconductors 2,3, ,etc.,willgiveall other capacitancecoefficients. The total electrostatic energy ofthesystemisagain thesum ofthefield energiesofallindividual partial condensers. Because ofthefactthatthematrix terms totheright ofthemain diagonal compriseallthedifferent capacitance coefficients, onecanwrite thissumasinn -^n n This relation ismost useful asitcontains onlymeasurable quanti- tiesandcanbedirectly applied toengineering problems. Inquasi-electrostatic fields, with potentials applied tothecon- ductors, which aresinusoidally varying intime, thecharges will alsovary sinusoidally. Since thetime rate ofchange ofacharge isequivalenttoacurrent, oneobtains bydifferentiation of(7) with respect totime theconcept ofpartial charging currents, dQandn la=~T~=ECap-~ (< $0)=E^a)3 (9) at0=odl 0=o Thecharging currents Iacanbemeasured very readily, andthus, withknown applied potentials, aneasyexperimental determination Sec. 3] Maxwell's Coefficients 17 ofthecapacitancecoefficients ispossible.2Theprocedureisquite similar tothatoutlined above. Maxwell's Coefficients ofInduction andofPotential. Forsome applicationsitisconvenient toreformulate thelinear relationship between charges andpotential differences, asgiven in(7), asrelations between charges and individual conductor potentials, eventhough these potential values arenotabsolutely known. Thus, (7)canberewritten Qa= \0=0 / 0=0 =Zka0*0 9a=0,1,2,---,n (10) 0=0 where the fcaj9arethecoefficients ofinduction originally defined by Maxwell/17Vol. I,p.108.From (10)onecantaketherelations 0=0 fca/3=-C^ a=0,1,2,- -,n (lOa) (05*a) Thekaaarethecoefficients ofself-induction, ortheself-capaci- tances oftheconductors, whereas thekaparethecoefficients of mutual induction andalwayshave anegative signbecause they characterize induced charge values. Because thesystemiscomplete,thetotal charge onground (or ontheenclosing shield) must begivenby Qo=-Q a=l Introducing hereQafrom (10)and alsousing (10)witha=0, Qo= onecanexpressthecoefficients ofinduction between ground and thenconductors 2K.W.Wagner, E.T.Z., 33,635(1912); G.A.Campbell,BellSystem Techn. Jl., 1,18(1922);alsoinCollected Papers byG.A.Campbell, p.169; American Telephone andTelegraph Company, New York, 1937. 18 TheElectrostatic Field [Ch. 1 Thesystem (10) contains, therefore, onlynunknown charges but (n+1)unknown potentials. However, absolute values of potentialsareunknowable, sothatoneusually introduces here $0=andredefines thepotentials $aaspotential differences to ground. Thisreduces (10)to n Qa=Lkafit, a=1,2, ,n (11) 0=1 with-(n+1)different coefficients ofinduction because kap=kpa. 2i Thesystem (11)canreadily beinverted, i.e.,thepotentials can beexpressed interms ofthecharges *a=SaisQ*a=0,1,2,- ,n(12) 0=1 where thesajgarethecoefficients ofpotential originally defined by Maxwell/17Vol.I,p.108.The coefficient systems kapand sa$ aremutually related ascoefficients ofsystems oflinear equations; they arebestexpressed bymeans ofdeterminants where theA(s)and A(fc) arethecomplete coefficient determinants ofthe sajgandkap,andwhere theMa/garetherespective minors obtained from theAbycancelling theathrowand /3thcolumn. Onecanalsoexpress theelectrostatic fieldenergy ofthiscom- plete system interms ofthepotential andcharge values. In- creasing thecharge value oftheathconductor atpotential *a bybringing asmall increment dQafrom zeropotential requires a work according to(2)ofvaluedWa=$adQa.Applying small charge increments toallconductors ofthesystem bytaking them from zeropotential gives dW e=*dQa (14) a=l orwith (12) E(i: sa,3dQ^\ Qf>=Qpd^ (15) 0=1\a-l / fl-1 Sec. 3] Integral Forms forElectrostatic Energy 19 Thetwoforms (14)and(15)aresummations overthesame range andcantherefore becombined togive dW e=IE[*dQa+Qad*a]=IEd(*aQa)&a=l *a=l sothat We=\E*aQa (16)*a=l Bymeans of(10)and(11)onecanreadily show theidentity with relation (8).Ontheother hand, using (10) or(12),onealsohas We=5EEfcafl*a*0=EESa0QaQ0 (17) Integral Forms forElectrostatic Energy.Inmore general cases ofsystemsofconductors inaspace with various insulating media aswell asspace charges, integral relations forthetotal energy canbedeveloped. Onthebasis of(14)onecandefine for aspace charge dQ=pdr,and forasurface charge dQ=adSt andthus replace thesummation in(16)byintegrations: tobetaken over theentire fieldspace andover allconductor surfaces.<isthelocal value oftheelectrostatic potential.If oneknows thecharge andpotential distributions,itisthusfairly easy tocompute thetotal electrostatic energy. Forasystem of conductors inahomogeneousdielectric without space charge, this expression reduces obviously to(16)since theconductor potentials areconstant. Inthespecificcase ofafinite system ofconductors within a single uniform dielectric bounded byavery large spherical surface, /d$\ thesurface integralof(e$ Itaken over theentire dielectric \dn/ canbetransformed byGreen's theorem (seeAppendix 3), jfjf(e* )dS= jfjfjf[sV*V*+e*V2*]dr(19) intoavolume integral throughout thedielectric. Because atvery largedistance *>0as1/r,andd$/dn>as1/r2 ,thesurface 20 TheElectrostatic Field [Ch. 1 integral willvanish forthevery large sphere andleave only 'the integrals over theconductor surfaces withnormals pointing into these conductors. Introducing -V*=E, -eV$=D, e andreversing thenormal direction tobeoutward with respect to theconductors, oneobtains SinceDn=aontheconductor surfaces, comparison with (18) leads atonce tothealternative form fortheelectrostatic field energy -dr .(20) where theintegral hastobeextended overtheentire space occupied bythe electrostatic field. This form permits, according to Maxwell's point ofview, theinterpretation asifthefieldenergy bedistributed throughout space with alocal density HE'D, entirely determined bythefieldvectors EandD;thehypothetical nature ofthisinterpretation hastobekept inmind, however. Itcanbeshown3thatequation (20)holds foranyelectrostatic system, whatever thenature andnumber ofdifferent dielectrics may be,aslong asthevectors EandDsatisfy alltheboundary conditions andE=grad$ineachmedium. Forces inaSystem ofConductors. Assumingfirstanideal condenser with charge, potential, andenergy relations given by (1)and (3),onecanconsider two specific cases ofmechanical action, theoneinwhich thechargesarekept constant, andthe other inwhich thepotentials arekept constant. The firstcase arises when theconductors, after receiving their charges, are isolated from thesource; anydecrease oftheir effective distance expressed asanincrease ofthecapacitance willreduce thepoten- tialdifference and, therefore, decrease thestored energy. This means thattheconductors,ifleftfreetomove,willtend toconvert fieldenergy intomechanical workbyanattractive force supplied bythesystem; conversely, anexternal force acting toincrease the 3Seereferences LivensA16andStrattonA23inAppendix4. Sec. 3] Forces inaSystem ofConductors 21 distance will alsoincrease thepotential difference andthe field energy. Thesecond caseabove arises ifthetwoconductors remain connected toasource ofconstant potential difference; anydecrease oftheir effective distance willnowincrease thecharge accumulation and therefore increase the field energy. This means, however, thatthesource hastosupply notonly thisincrease infieldenergy butalsothemechanical work needed tomove theconductors with respect toeach other; bythelawofconservation ofenergy this mechanical work isequal totheincrease infield energy, since for afreely movable conductor itwould besupplied bythesystem itself outofitsfieldenergy.4 Ifnowanisolated system ofrigid conductors isgiven withknown charge values, therequired force ortorque tocause anychange of ageometric positional element5rj(linear orangular displacement) ofanyconductor canbecalculated bytheprinciple ofvirtual work, expressing therate ofmechanical work asthenegative rate of changeofthestored potential energyWe.Forfixedcharge values oneusesbestthesecond form of(17)andobtains directly forthe rate ofworkdonebythesystem which represents aforce action inthedirection of577if8rjisalinear displacement, oratorqueif5rjisanangular increment. If dW e/driispositive, external forces ortorques have todeliver work (negative);ifdW e/drj isnegative, theelectrostatic system con- verts partofthe fieldenergy intomechanical work (positive). Using anyother expression fortheelectrostatic fieldenergy and observing thecondition offixed charge values, onewillobtain the same result (21). Ontheother hand,ifthepotentials ofasystem ofconductors remain fixedbypermanent connections oftheconductors toenergy sources andachange 8rjofageometrical positional elementTJ (distance orangle) takes place, then inorder tomaintain these potentials, energyW8(positive) hastobesupplied bythesources attherate =oona a=00=0 4Fordetail illustrations seeAttwood,A2 p.191. 22 TheElectrostatic Field [Ch. 1 ofwhich onehalf isused tocover theincrease infieldenergy and theother halftocover therate ofwork needed toproduce the change 17.IfdW e/dri isnegative, thendW a/drjindicates the rate ofenergydelivered back tothesource. Theactual force or torque producing 5rjis,ofcourse, thesame asin(21), since the same initial potential-chargerelations arepresumed,sothat oralso (23) Analternative form to(22)isobtained byusing expression (8) fortheelectrostatic fieldenergy, 1. (,._^,=S (24) . 2dv\ 017 Ja=0/9=a _|_i orj which involves themorecommonly usedmutual capacitance coeffi- cients; thesigns ofWaassource energy andH7mechasmechanical work ofthesystem arethesame asin(21)and (22). Electro- static instruments relying upon force actions between suitably arranged fixedandmovable conductors present awide field of pertinent practical applications.Inmany instances, simplifying approximationsarepossible byinspectionoftheactual field dis- tribution. Stresses inthe Electrostatic Field. Asvisualized by Faraday andanalytically formulated byMaxwell, theforce action intheelectrostatic fieldcanbedirectly related tothefield lines. Thus, onecanintroduce astress perunit area ofmagnitude J^E-Dinthedirection ofthe field linesupon asurface element taken perpendiculartothefield lines,andapressure perunitarea ofthesame magnitude perpendiculartothe field linesupon a surface element paralleltothefield lines. Thisimagined system offorces accounts forCoulomb's force lawandpermits ready evaluation offorce actions ontheboundary surfaces ofdifferent media. Inthespecialcase ofaconductor, thevectorEisalways per- pendiculartothesurface oftheconductor, sothat there willbe Sec. 4] Critical Field Values 23 onlyanormal stress perunitarea ofmagnitude /=%ED, the same value asthelocal density oftheelectrostatic energy stored inthefield. Obviously, there willbenoresulting force inthecase ofspheres andcylinders withuniform charge distribution. Foraboundary surface between two insulators, thenormal force perunitarea willbegivenbythesum ofthedifferences of normal stress andnormal pressure onthetwosides oftheboundary surface. Onethushas(without surface charge) ,n-E2nD2n)+(EltDlt-fia(Dai) -ea)ErE2 (25) ifuse ismade oftheboundary conditions (2-7)and (2-8). This force urgesmedium 1towards2,andthetotal force isreadily found bytheproper surface integral. These forms donottake intoaccountelectrostriction, the propertyofcertain dielectric materials toexpand orcontract in anelectric field.Arather complete account ofthemoreadvanced theoryisfound inthereferences LivensA15and StrattonA23 , Appendix4. 4-CRITICAL FIELD VALUES One ofthemost obvious objectivesinelectrostatic designisto obtain forms ofinsulators andelectrodes which willwithstand all theelectrical stresses thatsound operation andoccasional fault may impose. Inorder todecide upon thereliability andfactor ofsafety from thispoint ofview,anydesign must bechecked with respect tocriteria pertaining tothe critical values ofelectric fieldstrength anddielectric losses. Foruniform field distributions asexistbetween plane electrodes, more orlessdefinite values of breakdown fieldstrength canbefound experimentally; table 4-1 gives asummary ofthese values foranumber ofgases, liquids, andsolids ofgeneralinterest. Fornon-uniform fielddistributions, thefieldstrength values though significant arenotbythemselves decisive criteria ofbreakdown. Theories have been developed, however, which attempt theformulation ofgenerally useful criteria andwhich takethehomogeneousfieldusually asthestarting point. Itwill, therefore, beofinterest tosummarize briefly therelation- ships established fortheoccurrence ofionization, corona, and breakdown ofthevarious insulating materials. 24 TheElectrostatic Field [Ch. 1 seg1^1-I I00O XX coooX CO I I0002222 xXXXXOo CD^HrHN Q *H<> +* h- I pSH <J^ & Oss :gI S2 'Si i00O 5||3 t>- CVIS i-jiO00^O CO TfHO(N Sec.4] Critical Field Values 25 o Il-loo I^" I I1XX' 'o ra IM TI<in-^ eqOOOOOOOOi-Hli-Hl |r-|^H^H 1-|^H^Hx'x1'xxxxxx 10 26 TheElectrostatic Field Vacuum. Themost ideal insulator istruevacuumwhich, entirely devoid ofelectricity aswellasofmatter,issubject neither toionization nortobreakdown. There are,however, thepossi- bilities ofelectric andthermionic emission ofelectrons orions from thesurfaces ofthesurrounding conductors1and insulators. According toSchottky's theory,2thepurely electrostatic removal (cold emission) ofanelectron from themetal shouldrequire a fieldstrength atthesurface ofthemetal ofabout 108volt/cm, a tremendously high value. Experimental evidence ofcoldemis- sion3indicates qualitative agreement; quantitative relationsare, however, very difficult toestablish onaccount oftheextreme sensi- tivity ofthemeasurements tocontamination ofthevacuum bygas absorbed inthemetal surface.4 Thermionic emission ofelectrons from ametal surface requires athermal energy larger than acritical value called the"work function," which depends onthelocation oftheelectron within themetal. If,therefore, ametal isheated invacuum, acurrent willbeobserved whose saturation value depends ontheabsolute temperature ofthemetal.5 Ingeneral, breakdown ofvacuum asaninsulator willbecaused bythesimultaneous action ofthermionic and electric fieldemis- sion; thelatter willbeusually small, except where inhomogeneities orimproper designmay raise thelocal fieldstrength toexcessive values.6 Gases. Even under normal conditions, air,themost natural insulator, shows afairly constant ioncontent withslight local 1Seetheexcellent summary, "Electron Emission," byJ.W.McNall, in Industrial Electronics Reference Book, Chapter 2;John Wiley, New York, 1948. 2W.Schottky,Zeits.f.Physik, 14,p.63(1923); R.H.Fowler andL.W. Nordheim,Proc. Royal Soc., (A)119, p.173(1928). 3Forexample, R.A.Millikan andC.C.Lauritsen, Phys. Rev., 33,p.598 (1929); F.Rother, Ann. d.Physik, 81,p.317(1926); A.J.Ahearn, Phys. Rev., 44,p.277(1933); C.M.Slack andL.F.Ehrke, Jl.Appl. Phys., 12,p. 165(1941). 4A.J.Ahearn, Phys. Rev., 60,p.238(1936); E.W.Miiller, Zeits.f.Physik, 106, p.541(1937). 6O.W.Richardson: TheEmission ofElectricity fromHotBodies; Longmans Green,NewYork, 1916; A.L.Reimann: Thermionic Emission; John Wiley, NewYork, 1934; T.J.Jones: Thermionic Emission; Methuen, London, 1936. 8H.W.Anderson, Trans. A./.#.#., 64,p.1315 (1935); discussion66, p.831(1936). Sec. 4] Critical Field Values inGases 27 variations. Because ofthenormal process ofrecombination, one must assume adefinite andconstant rate ofionproduction; measurements haveshown thistobeapproximately7 Over land 8.1to9ions/cm3/aec Over sea 4.3" Inbrick buildings 12to14" Thecauses forthisionization arevarious; theymayberadium emanation ory-radiation from theinterior oftheearth, orthey maybecosmic radiation from theuniverse. Anelectric fieldwill, therefore, acttoaccelerate these ionsorelectrons andcause addi- tional ionization. Aslong astherate ofrecombination equals the rate ofionproduction, astable condition willpersist. Ontheother hand,ifthisequilibriumisdisturbed byasudden increase ofthe electric field,bystrongly ionizing impurities, orbyanyother factors increasing therate ofionization, adischarge current willformwith dark glow, possibly leading tocorona andeventual breakdown. Foraplane, uniform condenser with distance dbetween the electrodes, J.S.Townsend8found that,ifabetherate ofioniza- tion(i.e.,thenumber ofionpairs created byanionmoving over aunitpath)fornegative ionsorelectrons, and/3that forpositive ions, theionization current density Jcanberelated tothesatura- tioncurrent density J$that exists before ionization takes placeby Infinite current orbreakdown willthenoccurwhen thedenomina- torbecomes zero, or _._ad o |8rf /O^OLE.]0 \&) Now, therate ofnegative ionization, a,canbeshown todepend primarily upon thefreepathoftheelectrons andthefieldstrength.9 Since thefreepathisinversely proportional tothepressureofthe gas,onecandeduce thesemiempirical formula , /B\ , -=Aexp^-^7^ )ionpairs/cm (3) 'McLennan, Phil. Mag., 24,p.520(1912); seealsoSchumann,837 p.8 andThomson andThomson,1340 p.156. 8J.S.Townsend, inHandbuch derRadiologie, Vol.1,1919; seealsoSchu- mann^37Cobine,B33andparticularly Maxfield andBenedict,1136 p.277. 9Formeasurements seeF.H.Sanders, Phys. Rev., 44,p.1020 (1934). 28 TheElectrostatic Field [Ch. 1 inwhich theconstants AandBcanbedetermined experimentally for allgases. Forairthenumerical values are10A=13.2,B=0.278, ifPisgiven inmillimeters ofmercury, andEinkilovolts/centi- meter. Ontheother hand, ionization bypositive ionstakes place only close tothecathode surface, sothat itisbetter described bya surface ionization number 7.11Thebreakdown criterion(2)for plane electrodes cantherefore bewritten inthemore practical form wherePisthepressure ofthegasinmillimeters ofmercury, and dthedistance oftheelectrodes incentimeters; (a/P)isgivenby (3),andInf1H 1isshown inFig.4-1asafunction ofP/E. Relation (4)permits thegeneral evaluation ofeither thebreak- down fieldstrength Eorthecritical distance dforplane electrodes with satisfactory results. Table 4-1gives measured values of thebreakdown strength forplane electrodes forseveral gases. Onthebasis ofTownsend's theory andavery greatnumber of experimental data,W.O.SchumannB37was able todeduce a valuable andcompletely general empirical criterion forbreakdown valid alsofornon-uniform field distributions. Hefound thatthe integral ofa,therate ofnegative ionization along anyfield linein agivenfield configuration, must belessthan aconstant value to insure stable operation. Applicationofthis criterion toseveral simple electrode configurations hasgiven results12inexcellent agreement withexperimental data. Forpractical numerical com- putations, especiallyinstudying theinfluence ofvarious geometric parametersoftheelectrode arrangement, thiscriterion canbeused fornormal pressure andtemperatureinthesimplified form13 r(E- t/a(5) 10Knoll, Ollendorff, andRompe,B34seeTable g29, p.70;seealsocurve in "Electrical Conduction inGases," byD.E.Marshall, Chapter 4inIndustrial Electronics Reference Book; John Wiley,NewYork, 1948. 11W.Schottky, Zeita.f. Physik, 14,63(1923); Maxfield andBenedict,836 p.292. 12D.W.VerPlanck, Trans. A.I.E.E., 60,p.99(1941); J.G.Hutton, Trans. A.I.E.E., 66,p.1674 (1947). .13B.Davis, Proc. A.I.E.E., 33,p.528(1914). Sec. 4] Critical Field Values inGases 29 whereE=24.5kv/cm,K=47.6(kv)2/cm,andEismeasured inkilovolts/centimeter. The integration hastobeperformed only over that partaarofthe"most dangerous"field line, for whichE^EQ.Theaccuracy ofthenumerical results isquite 30 25 20 I15 10 10 15 20 25 ~E FIG. 41Plot oftheFunction;30 - Jagainst P/E.P=pressure in millimeter ofmercury; E=field strength inkilovolts per centimeter. (Redrawn bypermissionfrom GasentladungsiabeLlen, byM.Knoll, F.Ollen- dorff, andU.Rompe;J.Springer, Berlin, 1935.) satisfactory formost practical purposes, especially since all measurements ofdielectric strength aresubject toamodifying factor ofprobability, explained byRogowski14asthesudden change from lower tohigher current densities intheconducting path preparing thebreakdown ofagas. Using thecriterion(5), onewould plotasabscissa thedistance along afield linechosen so astogive thelargest contribution totheintegral, then plot 14W.Rogowski, Arch.f.Elektrot., 26,p.643(1932); seealsoG.L.Nord, Trans. A.I.E.E., 64,p.955(1935). 30 TheElectrostatic Field (E #o)2asordinates, andintegrate overtheareatothepoint where thiscurve intersects theaxis ofabscissa. Ifthis isdone forvarious voltage values, interpolationwilllead tothevoltage which satisfies equalityin(5)andthus constitutes the critical voltageforbreakdown. Forafewsimple geometriesinair,more direct empirical relations havebeendeveloped fortheonset ofvisual corona andbreakdown. TABLE 4-2 CRITICAL FIELD STRENGTH VALUES OFAIRINSIMPLE GEOMETRIES Critical Field Strength Values (kv/cm) for Electrode Arrangement Two Parallel LikeWiresVisual Corona Sparkover (E") 2c Concentric Cylinders-<30,a -^30,a 30/'=Ei' V Slfl 4.38^EJ\=31 I1+ 7=-IVVa.V~a/ TwoLike Spheres s< 0.54V/ oEI"> <2a, Table 4-2summarizes these criteria forairasgiven byPeek815 fornormal atmospheric conditions. Ifthe field strength atthe point1oftheelectrode arrangement reaches thevalue correspond- ingtoEI'inthetable, visual corona must beexpected, and ifit reaches avalue corresponding toE^',sparkover must beexpected; forspheres with spacings>2a,thesparkover occurs atthesame critical fieldstrength EIasfors=2a. Sec. 4] Critical Field Values inLiquids 31 Ifasolid insulator(porcelain, glass)isused inconjunction with air,itssurface isboundary surface oftwomedia ofconsiderably different dielectric constants. Theevaluation ofthedistribution oftheelectrostatic field isthenmore complex, andtheadditional danger offlashover15 occurs, i.e.,breakdown oftheairbetween theelectrodes along thesurface ofthesolid insulator. Thisdanger ismostpronouncedifthe field lines areparallel totheboundary surface.16Itis,therefore, advisable, incombinations ofairand solidinsulators, todesign theboundary surfaces sothatthefield lines areperpendicular tothese boundary surfaces, atleast inthe proximity ofthemetal conductors. Though most relations havebeendeducedspecifically forair,they arevalid for allgases inthesamemanner except foranappro- priate changeoftheconstants (seeKnoll-Ollendorff-Rompe334 ). Liquids.Ifasteady potential difference isapplied toaliquid orsolidinsulator, acurrent willresult which isvery small ingood insulators andbecomesfairly large inpoor insulators. Obviously, then, there isnostrictly electrostaticfield,butrather acombina- tion oftheelectrostatic andtheelectric conductionfields, which will slightly modify theanalytic solution ofthepotential dis- tribution. Forlowvoltages, however, especially intherange in which theinsulators arebeing used inelectrical apparatus, this distortion ofthetrueelectrostatic field issmallandcanbeneglected. Athigher potential differences theresulting distortions become important anddetermine thebehavior oftheinsulator. Acloser investigation ofthecurrent shows that shortly after ad-cvoltage hasbeen applied toaninsulator, thecurrent will decrease atfirstfairly rapidly and practically exponentially; further decrease isslower andadefinite finalvalue isreached only after considerable time.17Toexplain theobserved time variation ofthecurrent, several theories have been advanced; themost plausible one,which hasbeen confirmed byexperiments onliquid 16SecPeekB1B ;J.J.Torok andW.G.C.Ramberg, Trans. A.I.E.E., 48, p.239(1929); SchwaigerB17 . 16C.V.Fields andC.L.Caldwell, Trans. A.I.E.E., 66,p.656(1946); W.W.Pendleton, Trans. A.I.E.E., 66,p.1324 (1947). 17J.B.Whitehead and II.Marvin, Trans. A.I.E.E., 48,p.299(1929); J.B.Whitehead, Trans. A.I.E.E., 60,p.692(1931); A.F.Joff6, Ann. d. Physik, 72,p.461(1923); H.Schiller, Zeits.f. techn. Physik, 6,p.589(1925); Arch.f.Elektrot., 17,p.600(1927); H.Schiller, Ann. d.Physik, 83,p.137 (1927); A.Gemant, E.T.Z., 64,p.468(1933); alsoGemantB14 . 32 TheElectrostatic Field [Ch. 1 and solid insulators, assumes aprogressive establishment ofspace charge neartheelectrodes sothatthepotential distribution becomes distorted. Astable condition isreached when thelocal ionization andthe local reaction ofthespace charge areinequilibrium. This effect isknown aspolarization andgivesrisetotheso-called absorption current. Onewould expectthispolarization todecay after theexternal potential difference hasbeenremoved, sothat asimilar component ofthedischarge current should bemeasured. Although many solids indeed show arelease ofthetotal collected charge, giving theimpressionofareversible absorption current, most ofthecommon liquids aswell asanumber ofsolids return lesscharge thanthey receive, andshow thecharacteristics ofanon- reversible absorption current. This isexplained byanelectrolytic cleaning-up process;itisassumed thatduring thecharging period thespace charges reach asaturation value, regular electrolysis sets in,andtheelectrolytic products aredissipated, thuspurifying the insulator andrenderingitofhigher dielectric strength. The final current isdue either tooccasional free electrons (present fromsome external cause ofionization, asX-rays or cosmic rays), ortopartial internal dissociation producing ions (partial electrolysis caused byinhomogeneities). It isvery characteristic oftheelectrolytic type ofconductivity inliquid and solid insulators thattheequivalent conductivity increases withthe temperature andcauses higher losses athigh temperatures, a factwhich isimportant inelectrical design. Itcanbeexplained bytheincreased thermal agitation, causing ahigher rate ofdis- sociation which finally canresult inactual decomposition.18 Asingases, soalso inliquids, there willalways bepresent an initial ionization caused byextraneous sources. Astrong elec- trostatic field willmaintain andincrease thisionization bycollision oftheswiftly moving positive ornegative ionswith themolecules andatoms ofthe liquid. Ionization bycollision isusually accompanied byluminosity (corona), since theoccasional recom- bination ofanelectron andapositive ionreleases theionization energy intheform ofinfrared oreven visual radiation. Experi- ments onoilbyGemant19demonstrate the scintillations by photographic record inanelectrophotograph. Therate ofioniza- uJ.B.Whitehead andE.E.Miner, Phys., 6,p.380(1935); J.B.White- headandB.P.Kang, Jl.Appl Phys., 11,p.596(1940). 19A.Gemant, Zeits.f.techn. Physik, 9,p.398(1928) and 13,p.184(1932). Sec. 4] Critical Field Values inLiquids 33 tion, atlower field strength atleast, seems tobeproportional to thefield strength,20 a=C(E-E) (6) whereEQisthelower limit offield strength required forioniza- tion,andCisaconstant depending upon shape andspacing ofthe electrodes. For sufficiently highfield strengths, ionization by collision increases very rapidly, "avalanche"-like, andcanreach astable condition byforming aspace charge close totheanode, aswasverified experimentally byGemant.21 Iftheionization becomes progressive, theconductivity might increase without limitandbreakdown might occur. Ageneral direct criterion forthis stability ofionization isnotknown, al- thoughit*seems well established that thebuilding upofthe space chargeisamain contributory factor tothe final break- down ofaliquid. Acriterion developed byDreyfus22forsharp- edged copper electrodes states that, inorder toavoid breakdown, theintegral along any field line ofthe field strength from the electrode with thehighestfieldstrength tothepoint awhere the fieldstrengthfallsbelow theasymptotic value forplane electrodes oflarge spacing must belessthan acritical voltage characteristic fortheliquid, namely, rEsds<ycrit (7) Although this criterion isbased upon theelectrostatic field dis- tribution and, therefore, does nottake intoaccount thespace charge reaction,itimplies thegeneral experience thatnotthelocal field strength, but, rather, afield zone,ischaracteristic forthe electrical stability. Without aspecific knowledge ofthevalue Vcrit,relation (7)willprimarily lead totheformulation ofthe dependenceofbreakdown upon theshape andspacing ofthe electrodes. Afew critical measurements willthen rapidly lead toaknowledgeofycritforthespecific material andconfiguration, making ageneral design criterion possible. Allpractical insulating liquids have acertain content ofmoisture 20A.Nikuradse, Ann. d.Physik, 13,p.851(1932); seealsoGcmantBU f p.122. 21A.Gemant, Phys. Zeits., 30,p.33(1929); J.Slepian, Electr. World, 91, p.761(1928); J.B.Whitchead,Electr. World, 94,p.1083 (1929). 22L.Dreyfus, Arch.f.Elektrot., 13,p.121(1924). 34 TheElectrostatic Field [Ch. 1 and air,each actinginadifferent manner toreduce thedielectric strength oftheliquid. Since water hasavery high dielectric constant, small droplets willpolarize andmove intothedensest regionoftheelectrostatic field;ifthere aremany small droplets ofwater, theymayform dielectric bridges from oneelectrode to theother,23which, ofcourse,isequivalent tobreakdown on account ofthemuch higher conductivity ofwater ascompared with other liquids. Airinclusions tend toionize very rapidly24because thelow dielectric constant ofaircauses avery high local field strength. Asfoci ofprogressive ionization, they notonlycanspread an avalanche ofionsthrough theliquid, butalsocause local heating anddistortion ofthefield distribution, aswellasinitiate chemical changes,25which reduce considerably the dielectric strength of theinsulating liquid. Solids. Insolid insulators theproblem ofionization ises- sentially identical with that ofelectrical breakdown, since any appreciable ionization willbeprogressive. However, thebreak- down ofasolid insulator canbedescribed aselectric orthermic according totheprevailing characteristics summarized below. 23Seereferences onhigh-voltage cables, particularly DunsheathB4 . 24F.W.Peek, Gen. Elec.Rev., 18,p.821(1915); A.Gemant, Wiss.Veroff. a.d.Siemens-Konzern, 7,part 2,p.305(1929); P.Dunsheath,Jl.I.E.E., 73,p.321(1933). 25J.Slepian, Electr. World, 91,p.761(1928); J.B.Whitehead, Electr. World, 94,p.1083 (1929). Sec. 4] Critical Field Values inSolids 35 Thetheory ofthermal breakdown isbased onthethermal instability ofcertain inhomogeneities intheinsulator inwhich heat isgenerated bythedielectric losses atalarger ratethancan betransferred bytheinsulator toitssurroundings. Afirst approximation assumed aconducting canal ofinhomogeneities26 through theinsulator, fromwhich heatwastransferred totheelec- trodes only; amore complete treatment assumed aninfinitely extended homogeneousthininsulator withheatconduction tothe electrodes only,whereby theelectrodes might have equal ordif- ferent temperatures andthus impose atemperature gradient upon the insulator.27The resulting formulas forthehighest permissible voltage applied toaninsulating plate giveproportion- alitywith thethickness forhighfieldstrengths, andvalues inde- pendentofthickness forfairly lowvalues offield strengths. A very largenumber ofexperimental data check thequantitative results ofthetheory.28For applied a-cvoltages, secondary effects have tobeconsidered also,such asvariation oftheapparent conductivity,29non-linearity incurrent voltage relations, and periodicdielectric losses which areindependent oftemperature. Thetheory ofelectric breakdown isbased onamechanical breakdown ofthecrystal structure caused byexcessive local field strengths. Since theideal crystal structure didnotgive satis- factory results,30thehypothesis ofinhomogeneities was intro- duced, especially theassumption offinecleavage openings inthe crystal. This assumption ledtorather satisfactory values for the electrical breakdown voltage31and itsvariation with the thickness ofthesubstance. Onecanconclude frommany experi- ments ontypically inhomogeneous crystals that there isadefinite breakdown fieldstrength which isareproduceable constant ofthe material and isoftheorder of2to5X105volts/cm forporcelain andsodium chloride, respectively. 26K.W.Wagner, Trans. A.I.E.E., 41,p.288(1922). 27W.Rogowski, Arch. f.Elektrot., 13,p.153(1924); Th.Karman, Arch, f.Elektrot., 13,p.174(1924); V.Fock, Arch. f.Elektrot., 19,p.71(1927); P.H.Moon, Trans. A.I.E.E., 50,p.1008 (1931). 28L.Inge,N.Semenoff, andA.Walther, Zeits.f.Physik, 32,p.273(1925); same, Arch.f.Elektrot., 17,p.433(1926); SchwaigerBl7 ;V.M.Montsinger, Trans. A.I.E.E., 64,p.1300 (1935). 29T.W.Dakin, Trans. A.I.E.E., 67,p.113(1948). 30W.Rogowski, Arch.f.Elektrot., 18,p.123(1927); A.Smekal, Arch.f. Elektrot., 18,p.525(1927); seealsoSchwaigerm7 . 31G.E.Horowitz, Arch. f.Elektrot., 18,p.535(1927). 36 TheElectrostatic Field [Ch. 1 Forentirely homogeneous crystals andamorphous substances anionization theory similar toTownsend's theory forgases was developed byJoffe"32andshows satisfactory agreement with experimentaltests. Thebreakdown field strength inthiscase is higher, but stillaconstant forthesubstance, and oftheorder of 1to3X106volts/cm forglass33andquartz, respectively. Since testdataonbreakdown ofsolid dielectrics always show aconsiderable spread, probability considerations havebeen intro- duced inorder topredict withreasonable safety breakdown oflarge areasamples from testsoncomparatively small samples.34 PROBLEMS 1.Find acharge arrangement which produces thepotential distribution *=T-^- ;isthesolution unique? (Stratton,A23 p.162.) 47TS r 2.Demonstrate thevalidity of(2-3) and (2-3a) bymeans ofGreen's theorem (Appendix 3).Hint: take inGreen's theorem *as1/randexclude r=byavery small sphere, counting itssurface asonepartoftheboundary surface oftheuniform dielectric. 3.Extend thedemonstration in2tothevalidity of(2-5). What restric- tionsmust beplaced upon thespace charge density p? 4.Compute theelectric field inside andoutside asphere ofradius awhich isuniformly polarized. The electric fieldcausing thepolarization Pishomo- geneous throughout space (assume freespace), hasthesame direction asP, andhasvalue EQ. Establish theequivalence withauniform dielectric sphere inthehomogeneousfieldEQ(seesection 21)andfindtheequivalent relative dielectric constant ofthesphereinterms ofEQandP. 5.Asphereofradius ainfreespace carries onitssurface adouble layer ofelectric charge, i.e.,ithasonthetwofaces ofitsboundary surface equal andopposite charge densities a.What isthepotential outside and inside thesphere? Hint: consider theradial distance between thecharge densities asfiaandvery small compared with allfinite distances; introduce the solid angle dtlsubtended byasurface element dSatthepoint ofobservation P. 6.Demonstrate thatthepotential ofadouble layer ofcharge density a, small surface dS,andcharge separation daisgiven infreespace by5*=ada Si}/47re v,where Sflisthesolid angle subtended bythesurface SSatthepoint ofobservation. Applythisasapproximationtothepotentialofaparallel 32A.Joflfo, J.Kurchaloff, andK.Sinjelmkoff, Publ. ofM.I.T., No. 117, Vol. 62,1927. 33N.D.Kenney, A.M.Luery, and J.D.Moriaty, Trans. A.I.E.E., 61, p.404(1932). 34M.C.Holmes, Jl.Franklin InsL, 211, p.777(1931); L.R.HillandP. L.Schmidt, Trans. A.I.E.E., 67,p.442(1948). Problems 37 plate condenser atvery large distances from it.Show theanalogy tothe electric dipoleinsection 10.Show thatthepotential difference between the faces ofthedouble layerisgiven by*i *2=o-5a/e v. 7.What isthemaximum charge that asmooth conducting sphereof radius acanhold inairunder normal conditions without exhibiting corona effects? 8.Accepting astress perunitarea ofmagnitude J^E-D inthedirection of theelectric field lines, show thattwo likecharges ofopposite sign attract each other inaccordance with Coulomb's law (1-1). Hint: utilize the symmetry ofthefield distribution; secalsosection 10. 9.Accepting apressure perunitarea ofmagnitude J^E-D normal tothe direction ofthefieldlines,show thattwo likechargesofsame sign repel each other inaccordance withCoulomb's law (1-1). Hint: utilize thesymmetry ofthefielddistribution; seealsosection 10. 10.Given afixedsystem ofnconductors andground (orgrounded envelop- ingshield) inasingle, uniform dielectric, assume thatpotentials4>aareapplied totheindividual conductors with respect toground andthatcharges Qaare measured. Ifthen potentials<ba'arcapplied, charges Qarwillresult whereby a(Green's reciprocation theorem). Prove this relation; utilize (3-11) or(3-12). 11.Inorder todetermine thecharge induced byanelectron inoneofthe electrodes ofavacuum tube, onecanapply Green's reciprocation theorem from thepreceding problem tothefollowing twoconditions: (a)allelectrodes aregrounded except theoneinquestion,towhich voltage Visapplied with respect toground andasmall conducting butuncharged sphereisplaced at thepositionoftheelectron; (6)allelectrodes aregrounded, andtheelectron chargeeisappliedtothesmall conducting sphere. Show thattheinduced charge ontheelectrode inquestionisQ'=e*/F, where*isthepotential existing under (a)onthesmall sphere. Applythistoaplane parallel diode. Applyittoacoaxial cylindrical diode. 12.Theuniqueness theorem states thatthepotential function asaharmonic function isuniquely determined within aclosed, regular region rofadielectric byitsvalues ontheboundary surface ofthisregion. Prove thisbyapplying Green's theorem (Appendix 3)tothedifference oftwopotential functions *i #2,eachfunction satisfying theLaplacian (orPoisson) differential equation and taking onthesame value *oontheboundary surface. 13.Extend theproofinproblem12toafinitenumber offinite conductor surfaces embedded inahomogeneous andisotropic dielectric ofinfinite extent without space charge. 14.Thepotential function *asaharmonic function isuniquely defined (except foranadditive constant) within aclosed, regular regionrofadielectric bythevalues ofitsnormal derivative ontheboundary surface. Prove this, following theoutline given inproblem12. 15.Extend theproof inproblem14toafinitenumber offinite conductor surfaces embedded inahomogeneous andisotropicdielectric ofinfinite extent without space charge. 16.Extend theproofinproblem14toafinitenumber offinite conduc- 38 TheElectrostatic Field [Ch. 1 torsembedded inseveral different dielectric media without space charge. (Smythc,A22 p.57.) 17.Inanelectrostatic field, theelectric charges onfixedconductors embed- dedinanisotropicdielectric offixed earesodistributed over their surfaces thattheelectrostatic fieldenergyisaminimum (Thomson's theorem). Prove thisbyapplying (3-20) totwodifferent sets*,E,Dand *',E',D',satisfying divD=divD'=p,andmaintaining thesame totalcharge oneachconductor; thefirst setaselectrostatic field solution must satisfy inaddition curlE=0, orE=V$and*=consonallconductors. Hint :takethedifference of therespectivefield energies anddemonstrate itasanessentially positive quantity. (Abraham Becker,A1 p.89.) 18.Theelectrostatic potential cannot haveamaximum oraminimum value atanypointofthefield freeofelectric charge. (Observe that analytic func- tions satisfy ananalogous condition; seesection 25). Demonstrate thisby applicationof(111).(Smythe,A22 p.13.) 19.Earnshaw's theorem asserts thatanelectric charge, subject only to electric forces, cannot beinstable equilibrium. Demonstrate thisbymeans oftheprooffor19.(Smythe,A22 p.13.) 20.Demonstrate thattheintroduction ofanuncharged conductor intoan electrostatic fieldproduced byasystem ofconductors with fixed charges decreases thefield energy. Hint :useasimilar approach tothat inproblem 17withboth field setscorresponding totrue electrostatic fields butextended over slightlydifferent volumes. (Stratton,A23 p.117.) 21.Computethepotential distribution everywhere infreespace produced byauniform space charge distribution confined toasphereofradius a.Can onedefine acapacitanceofthissphere? 22.Demonstrate that forahomogeneous sphere ofradiusa,theratio of surface potential tototal chargeisindependent ofthemanner inwhich the chargeisdistributed radially throughout thesphere (assuming thatthecharge densityisonlyafunction oftheradius andthatthedielectric constants eiand eforinside andoutside medium, respectively, areconstant). 23.Two conductors above ground andisolated from itform acondenser. Ifapotentialdifference V=$1 *2isapplied between them, what arethe individual potentials toground interms ofMaxwell's potential coefficients? Interms ofthecapacitancecoefficients? 24.Two conductors above ground areconnected andhave apotential difference Vapplied between them andground. What aretheindividual charges collected onthese conductors interms ofMaxwell's potentialcoeffi- cients? Interms ofthecapacitancecoefficients? 25.Given twoconductors 1and2above ground, three measurements are made: (a)voltage Visapplied between 2andground with 1isolated and charge Qziaregistered; (b)2isdisconnected and loftisolated, Visapplied between 1andground, andcharge Q\ isregistered; (c)1isnowdisconnected and leftisolated, Visagain applied between 2andground, andthenewcharge Qz'f isregistered. Taking ground atzeropotential,find allthepotential coefficients interms ofthecharges. Find allthecapacitance coefficients. Find theinduced potential values inthethree experiments. THEMAGNETOSTATIC FIELD 5-FUNDAMENTAL RELATIONS INTHEMAGNETOSTATIC FIELD Although themagnetic effects were studied first inconnection with natural oresandloadstones, thebasic relations canbemore readily formulated quantitatively bystudying themagnetic effects produced bysteady current flow. Inthis sense, then, Ampere's lawofforce action between currents becomes thebasis ofmagneto- statics comparable inimportance totheCoulomb lawofelectro- statics. Indeed, Ampere's law ismost suitable topoint outthe basically different physical aspects ofthemagnetostatic field; in itssimplest form fortwoparallel currents itdefines anattractive orrepulsive force ofvalue1 where conventionally thepositive signischosen forlikecurrents which attract each other; itiswelltoobserve that this iscontrary totheconvention established fortheforce between two electric charges. Inusing (1),thecurrents areassumed tobeconfined tovery thinwires parallel over thelengthIwhich itself islarge compared with thedistance rbetween thecenters ofthewires; Histheabsolute permeabilityofthehomogeneous, infinitely extended medium inwhich theforceFmismeasured (seeAppendix 2forunit relations). Oneusually expresses /i=HVHT,where nv istheabsolute permeabilityoffreespace (vacuum), and\LTthe 1Seetheinteresting account byA.M.Ampere, Ann. dechimie etdephys., 16,pp.59,170(1820). 39 40 TheMagnetos tatic Field [Ch.2 relative permeability;thelatter isthenumeric value generally found inthetables ofmaterial constants. Thebasic arrangement inthismagneticforce experimentisessentially two-dimensional, andtheinteracting currents areparallel vectors inspace;this explains thepresenceofthefactor 2w[ascompared with 4*-in (1-1) forthetruly three-dimensional case]andexplains thevaria- tionwith inverse distance (ascompared with inverse squareof distance intheCoulomb law). Throughoutthebook, only isotropic magnetic media willbeconsidered, sothatIJLcanalways beassumed tobeindependentofdirection. Ifthecurrent 72isvery small, sothat itcauses anegligible and only local distortion ofthefield ofcurrent I\,itcanbeused asa probefortheexplorationoftheforce field created bycurrent /i. From (1),thelimit value forvanishing 72andunitlength ,.bmF MI\v/\=---=Bl (2) should beinterpreted asthemagneticfield strength2ofthevery longlinecurrent /i;actually,itismore usually called magnetic fluxdensity. Asavector,itsdirection isnormal toboththecurrent vector Iiandtheradius vector rfrom thecurrent tothepointP (seeFig.5-1)andforms with these intheorder given aright- handed orthogonal triplet. One can, therefore,alsowrite vccto- rially f B=^- 2Ixr (3) 2irr where r/rserves toindicate theradial direction. Forthesingle linecurrent, this willdefine thevector Beverywhere tangential tocircles with ascenters. Thevector character hasbeen as- sociated hereandthroughout thissection directly withthecurrent because ofthevery small cross-section oftheconductor; more precisely, onecould introduce aseparate unitvector toemphasize thecurrent asscalar cross-sectional integral ofcurrent density as isdone inthenext section. 2That thevector Benters into allforce relations ofthemagneticfieldhas beenrepeatedly pointed outinsome oftheadvanced books onelectromagnetic theory; seeLivensA15andStrattonA23inthereference listofAppendix 4, alsoR.W.King: Electromagnetic Engineering, Vol. I;McGraw-Hill, New York, 1945. Sec. 5] Fundamental Relations 41 Now, quite differently from theelectrostatic case, theforce action ofthefieldBIupon current I2isactually perpendicular to both these vectors andforms anorthogonal triplet withthem inthe right-handed order I2,BlfFm,sothat vectorially onecanwrite for(1)with (2)ifonerefers theaction tounitlength =12'Bx (4) leading toattraction along thecenter line ifboth currents have the same direction, andtorepulsionifthey flow inoppositedirections. \ FIG. 51Magnetic Field ofaSingle Line Current. Instead ofanalogy between electrostatics andmagnetostatics one finds hereastrong difference whichis,ofcourse, related tothefact that charges arescalar quantities, whereas currents arespace vectors; charges arecenters ofconvergence ordivergenceofelec- trostatic field lines, whereas currents areaxes ofcirculation of magneticfield lines. Moving thesmall current I2very slowly parallel toitself over apathPiP 2inthefield ofthecurrentIi,while maintainingI2and itssource constant, requires thework Wf*Pi= JP!Fm-ds=Zl I2*Brds (5) ThelengthZcould beinterpreted asvector inthesame direction 42 TheMagnctostatic Field [Ch.2 asI2,sothatonecanrewrite (5)as l<Bl'AB=I&m (6) where$misdefined asthemagnetic flux *m= fjjdsxl-B!=ffBndS(7) through thearea described bythemotion oftheconductor. This magnetic fluxvanishes forany closed surface, which is demonstrated inthesimplest manner byintegrating (3)overthe surface ofasector ofacylindrical annulus. Thus, thefield isof theconservative type; motion overanyclosed pathmust give zero result forwork done. If,however, themagnetic fluxthrough any (reducible) closed surfaces vanishes, then thevector Bcannot haveanysources or sinks, or V-B=divB=(8) Forasystem ofnparallel linecurrents ofvery great length in aninfinite insulating medium theresultant vectorBcanreadily be evaluated bymeans ofsuperposition oftheindividual current contributions Iafrom (3) where theraaretheperpendicular vectors from thelinecurrents tothepoint ofobservation P.Such asystem is,ofcourse, two- dimensional innature, i.e.,the field distribution isthesame in anyplane orthogonal tothesystem.Ifonethen defines field lines asthecurves which have atevery point thevectorBastangent, onehas w=f<10> rfy&x asthedifferential equation forthetwo-dimensional case. Since the field lines circle around theconductors, andsince thevector Bhasnodivergence, there willbeonly closed field lines. Most materials have apermeabilityclose tothat offreespace; diamagnetic materials have permeabilities slightly smaller, weakly paramagnetic materials have permeabilities slightly larger, than Sec. 6] TheMagnetostatic Potential 43 that offreespace. Thereis,however, averyimportant group of metals andtheir alloys which haveveryhigh permeabilities. Since iron istheoutstanding representativeofthisgroup, theyhave been referred toasferromagnetic; recently, certain alloys ofweakly paramagneticmetals have been found alsotopossess high per- meabilities. Thiswhole group will, therefore, bedesignated better asstrongly paramagnetic. Formany practical purposesoffield mapping,itappears desirable toconsider asharp distinction between highly magnetic andnon-magneticmaterials and toassume aninfinite value of permeabilityforthe firstgroup, andthevalue forfreespace for thesecond group which combines both thediamagnetic andweakly paramagneticmaterials. Thisseems themuch more advisable, because allthehighly magnetic materials show strong non- linearity oftherelation betweenHandB,(theso-called saturation effects) and, additionally, exhibit strong influence ofthepast magnetic history oftheparticular sample, makingitwell-nigh impossible totreat these materials analytically. 6-ANALYTICAL THEORY OFTHEMAGNETOSTATIC FIELD The general magnctostatic problemistheevaluation ofthe magneticfield distribution produced bygiven configurations of theelectric current. Themagneticfield itself ischaracterized by thevector B,which hasphysical properties quite different from those ofthecorresponding vectorEoftheelectrostatic field. In thisbooknoattemptwillbemade totreat thenon-linear aspects ofmagnetic phenomena. TheMagnetostaticPotential. Asshown insection 5and indicated inFig.5-1,themagneticfield lines ofasingle very long linecurrent are circles. The lineintegralofthevectorBalong acircular field lineofradius ris,therefore, using (5-2), fflds=fP^rd0=M71 (1)Jc 2irJ0=o r and, indeed, anyother simply reducible (seeAppendix 3)path linking with thecurrent willgivethesame result; thevectorB isthus oftherotational (circuital) type. Since thelineintegral doesnotvanish, onecannot introduce ageneral scalar function ascorrelated potentialfunction.If,however, aclosed path C' 44 TheMagnetostatic Field [Ch.2 ischosen which doesnotencircle thecurrent, thelineintegral along C'does vanish. Onecanthus rescue thescalar potential concept ifonemakes surebyproper choice ofa"barrier" surface thatno possible path ofintegration canlinkwith thecurrent. Fora partial current loop this barrier surface isindicated inFig.6-1; itisprohibited ever tocross thisdouble surface. Allthe field lines arethen conceived to"start" attheside ofhigher potential value andtoterminate atthesideofthelower potentialvalue. Fio.6-1 Barrier Surface oftheMagnetostatic Potential Inintroducing thus arestricted definition ofascalar potential, onemight aswelltakecognizanceofthefactthatthelineintegral (1)depends onthemagnetic characteristic ofthemedium, namely, thepermeability /*.Itisconvenient, then, todefine anewvector H R O\ -B (4) usually called themagnetizing force (thoughitcertainlyisnota forceandnoteven directly responsible formechanical force actions) ormagnetic intensity. With this definition, (1)becomes LHds=/ (3) where 7stands fortheentire current flowthrough theclosed path C. Sec. 6] TheMagnetostatic Potential 45 (Frequently onewrites theright-hand sideasNIandmeans the totalnumber ofturns, each carrying thesame current 7.) Inregions outside ofcurrents andproperly provided with barrier surfaces over allcurrent loops, onecanthen define H=-grad7=-V3r ,JH-ds=3: l-32 (4) where 3risthemagnetostatic potential analogous to$,theelec- trostatic potential,defined by(1-7). Themagnetostatic poten- tialdifference (IFi 3^)isfrequently called magnetomotive force ormmf inanalogy totheelectrostatic use; forsingly closed line integralsofthetype (1),thismmfbecomes identical withthetotal current linked bytheclosed path.Itsvalue isindependent of thepathifitlinks thecurrent onlyonce orifitisreplaced bya lineintegral with terminal points onthetwosides ofthebarrier surface. Because of(5-8)and (2)and (4)above, onecannowdeduce VB=-V-(/iV!F)=(5) oralso V7 V/i+/iV23"=(5a) This represents themost generaldifferential equation forinhomo- geneous media, wherein thevariation ofMmust beknown. (Only thecase ofmagnetically isotropic media istreated here, othermedia being omitted asbeyond thescope ofthismonograph.) Com- parison with (21a) indicates thecloseanalogy between 3"andthe electrostatic potential $inmedia without space charge. Formagnetically homogeneous media thepermeabilityiscon- stant, and (5a)reduces totheLaplace equation V2?=(6) which isidentical with (2-2) fortheelectrostatic potential and, asthere,isthemostimportant caseadmitting readily ofanalytical, graphical,aswell^as experimental, solutions; most ofthemapping methods pertaintoit.Any solution of(6)must again beahar- monic function, andmust beanalytic just astheelectrostatic potentialfunction insection 2intheregions outside ofthe current-carrying conductors andtheproperly constructed barrier surfaces. Unlike theelectrostatic case,however, aformal solution TheMagnetostatic Field [Ch.2 cannot begiven readily interms ofsurface orvolume integrals of physically observable magnetic distribution functions. TheBoundary Conditions oftheMagnctostatic Field. Assuming aboundary surface between twomagnetically different materials asindicated inFig. 62,onecanapply therelation (5-7) totheclosed surface presented bythevery small cylinder ofheight dh Oneobtains0. =BnzdS2BnldSi= andinthelimit forvanishing FIG.62Boundary Condition forthe dh, Magnetic Flux Density. PBnl (7) This states thatthenormal componentofthemagnetic fluxdensity iscontinuous through anyboundary surface. The applicationof(3)toavery small rectangular path of integration across theboundary surface asindicated inFig.6-3 leads to H-ds=HtldsiHt2ds2=Jdhxds(8) ifHtdesignates thetangential component, and ifonedisregards thecontributions ofthenormal componentsofHbecause dh >0. Theright-hand side isagain thetotal current flow through the closed pathCand willvanish asdhismade tovanish unless the current density intheboundary surface itself isinfinitely large. Inthelatter case,onedefines lim(Jdh)=K (9)dh>0 asdensity ofthecurrentsheet, aconcept analogous tosurface charge density andparticularly convenient insimplifying thedescription ofdistributed windings inmachines, thininductancecoils,andthe like. With theconcept (9),thesecond general boundary condi- tionbecomes TT TT_ 1(10) Sec. 6] Boundary Conditions 47 whereKpisthecomponent normal toHtintheboundary surface. Inmagnetostatic problems, theboundary conditions usually pertain tothe field vectors andonly rarely involve given values ofthemagnetostatic potential; theyappear therefore frequently intheform ofgeneral boundary value problems. ForK=0,in Fro. 63Boundary Conditions fortheMagnetizing Force. theabsence ofacurrent sheet along theboundary surface, one cancombine (7)and (10)bytaking theratios onbothsides, n j) tiny_t>ni IIt%nti Ifoneintroduces therespective permeabilities andtheangles <*i, 2ofthefield vectors with thesurfacenormals, oneobtains tanai=tana2 (11) M2 thelawofrefractionofmagnetostatic field lines. This isofpar- ticular value forgraphicalfieldplotting. Ifoneapplies thelawofrefraction toaboundary surface between ahighly magnetic andanon-magnetic material ofpermeabilities IL\and/i2respectively,ftanaiJwillbeavery small quantity 48 TheMagnetostatic Field [Ch.2 forpracticallyallangles a\lessthan 7r/4. Thismeans, inturn, thattana2and, therefore, a2willbevery small, orthefield lines inthenon-magnetic material willbenearly perpendicular tothe surface ofthehighly magnetic material. Inpractical problems onefrequently assumes then thesurfaces ofhighly magnetic materials asequipotential surfaces forwhich ff=cons,much like theconductor surfaces inelectrostatics. This isfurther supported bythefactthatthecontribution ofahighly magnetic material to theline integral (3) isvery slight forreasonable values offlux densities. Onethuscanformulate certain magnetos taticproblems asboundary value problems ofthefirstkindanddirectly substitute electrostatic problems, forwhich solutions might already be known. TheMagnetic Vector Potential. Since thevector Bcannot have sources orsinks under any conditions,itispossible (see Appendix 3)toassociate with itavector potential Asuch that VxA=curlA=B(12) This vector potential can, inturn, berelated tocurrent density Jifonerewrites (3)interms ofsurface integrals. The lineintegral canbetransformed byStokes's theorem (seeAppendix 3),and thecurrent Ithrough theclosed curveCcanbedefined astheflux ofthecurrent density vectorJ, jfjf(curlH)dS =j[jfJ Because this relation toStokes's theorem holds foranysimple reducible surface,theintegrands themselves must beequal, giving with (12) curlH =J=Vx(-VxAl(13) oralso forgeneral inhomogeneous media (non-isotropic media willagain beconsidered beyond thescopeofthismonograph), -VxAxV I- J+-VxVxA=J (14) Comparingthisvector differential equation with thescalar poten- tialequations (2-1)and(6-5), oneappreciatesallefforts todefine, eventhough forlimited useonly, themagnetostatic potential CF. Sec. 6] TheMagnetic Vector Potential 49 Ofcourse, forhomogeneous media (14)reduces to VxVxA=nj (15) thevector equivalent ofPoisson's equation (2-4). Ifwenow select theCartesian coordinate system, because itistheonlyonein which thethree unitvectors arecompletely symmetrical, each of constant magnitude anddirection, wecaninterpret VxVxA=V(V-A)-(V-V)A (16) where V-V=V2istheusual Laplacian operator. Itiscustomary atthispoint tostipulate V-A=divA=(17) because A'isonlymathematically defined, notmeasurable asa physical quantity and,infact, onlyknown throughitscoordinate derivatives by(12). Thecondition (17) essentially amounts to adjustment ofAbyaddition ofthegradient ofanarbitrary scalar function SF;thisdoesnotaffect therelation (12)sinceVxvy=0. With (16)and(17), thebasic differential equation forthevector potential becomes intheCartesian coordinate system V2(iAx+jA y+k4z)=-/i(i/ x+}Jy+kJz) (18) simply asetofthree independent scalar potential equations of thePoisson type. Each oneofthese cannowbetreated exactly like (2-4), andtheformal solution results *'+ff**B(19) where Jisthecross-sectional density andKthecurrent sheet density; this integral expression can, ofcourse, again beused independently oftheparticular coordinate system. Forgiven current distributions inaninfinite medium ofconstant permea- bility, onecan therefore find thevector potential bydirect integration (see Figs. 6-4and6-5) inside thecurrent-carrying conductors aswellasoutside. Onecanalsofindsolutions bysuper- position ofthesolution ofthehomogeneous vector differential equation andanyparticular integral oftheinhomogeneous one. Itmust beborne inmind that (18)holds only intheCartesian system; inanyother coordinate system onemust return tothe more general form (15)asalsoemphasized inAppendix 3.Since 50 TheMagnetostatic Field [Ch.2 thevector potential equation (15)isnotgenerally separable with respect tothecomponents innon-Cartesian coordinates, the FIG.6-4 Vector Potential Produced byCurrent Filament ofVolume Distribution. FIG.65Vector Potential Produced byCurrent Filament ofaCurrent Sheet. integralsolutions (19)areofparticular significance. Forgeneral methods oftreating theform (15) seeSmythe,A22 p.260. Ifthere areseveral magnetically different media, onehasto find analytic solutions ofthePoisson equations (18), ormore Sec. 6] Current Filaments 51 generallyof(15) foreachmediumindependently, derive thefield vectors BandH,and satisfy theboundary conditions(7)and (10). Because oftheformal analogy between(19)andthescalar integral expression (2-4), onecanreadily formulate theboundary conditions forthevector potential itself asrequiring thecontinuity ofboth thenormal andthetangential components Anl=An2,Atl=At2 (20) This follows from thefactthat thescalar potential function is continuous everywhere except atalayer ofdipoles, across which itassumes afinite discontinuity; similarly, (20) willhold atall boundary surfaces except where theequivalent ofamagnetic dipole layerexists. Current Filaments. Inmany applications,itispossible to define thevolume elements drin(19) asthin filaments parallel tothedirection ofthecurrent density J,namely dr=dSds, where bothdSanddspoint inthedirection ofJandwhere dS defines asurface element normal toJ,whereas dsisthefilament length. Without current sheets, thesolution (19)canthusbe written byinterchanging thepositions ofJandds Ifthen thecross section oftheconductor isvery small compared withanydistance rfrom thepointofobservation P(seeFig.6-4), onecanfurther evaluate thecross-sectionintegral giving thetotal current /,andonehas A=f-/ (21)4wJr amuch simplerlineintegral indicating thattheelemental contribu- tiontothevector potential gives avectordAinthesame direction asthefilament. Onemust expect, ofcourse, thatA >ooasone comes very close tothefilamentary conductor. Theform (21) lends itself readily tothedirect evaluation of themagnetic fluxdensity B.With thecurloperation applied to bothsides,itcanbetaken under theintegral signbecause all quantities arecontinuous, B=VxA= I<fVx 4?rJ r 52 TheMagnetostatic Field [Ch.2 NowVmeans differentiation with respect to(x,y,z),whereas the integrationvariables are(z',y1 ',z7 )anddscontains onlythelatter set.Thus dsA\ ..-r V*=V I-Jxds=+dsx^ r \r/r3 Onethusobtains thegeneralized form ofthelawofBiotandSavart Aswiththevector potential, soalsowiththemagnetic fluxdensity: asoneapproachesthefilament, Bwillbecome infinite. The expressions (21)and (22)cantherefore bewellused tofindthe field quantitiesatsome distance from theconductors considered asfilaments, butonecannot actually admit zero cross section. Flux Linkages.Inaformal manner, onecangetthetotal magneticfluxthrough aclosed filament loopCproduced byits owncurrent with (12)as =(VxA) dS= dB (23) thelineintegralofthevector potential extended over theloop. Forafilament ofzero cross section thisexpressionwillhave little value,sinceA >ooalong thepathofintegration. If,onthe other hand, oneadmits thefinite cross section oftherealconductor thenthefluxconcept becomes indefinite, atleast intheform given in(23). However, forafinite cross section oracurrent sheet onecan compute thevector potential Aaccording to(19)anywhere in spacewith finite values. Ifonenowsubdivides thecurrent flow intofilaments JdSasinFig.64orKdsasinFig.65andcom- putesforeachonethefluxaccording to(23), theintegrals overthe respectivecross sections willthen constitute themagnetic flux linkages Ainthetwocases dSor' A'dsds' (24) where theinner integral remains afunction ofthelocation ofthe filaments within theconductors. The division bytherespective Sec. 6] Magnetization 53 currents 7and /'isnecessary torestore proper dimension, since (24) really signifies anaverage value ofthemagnetic flux. Inasimilar manner, thefluxofthevectorBthrough thefilament loopCcanbeused fortheinner integral in(24),ifBitself isthe total fluxdensity produced bytheconductor offinite cross section. This requires return tothemore general solution(19)forthevector potential andapplication ofthecurloperation toit.Asdone in thederivation of(22),onecantakeVxunder theintegral signand applyitto(1/r) only, since neither thevector densities JandK northeelements dranddSdepend onthecoordinates ofthepoint ofobservation whereBisevaluated. Thus, onehas *> where, ofcourse, r/rcanbereplaced bytheunit vector inthe direction ofr. Magnetization. Thecharacterization ofamagnetic medium bytheconstant /iissatisfactory aslong asnoinquiryismade into thestructural aspectsofthemedium thatmight beresponsible forIL.Toobtain ahypothetical concept ofthenature ofamag- netic material, onecanseparate inthemagnetic intensity Hthe contribution which canbethought oftoexist infreespace from that thought tobecaused bythepresence ofthemagnetic material. Unlike theelectrostatic case,however, onehashere -B-M(26) MO inaccordance with (21),where inmost instancesju>ju ,butwhere occasionally M<Mo(diamagnetic substances). SinceBdefines thedirectly observable force actions,itisHthat logically carries theinfluence ofthemedium;Misdesignated themagnetization ormagnetic polarization analogous toP,theelectric polarization (seesection 2). The effect ofthisseparation upon thedifferential equation for themagnetostatic potential function 7inauniform magnetic medium isobtained bytaking thedivergenceof(26)andobserving (5-8) divH=-divM sothatfrom (4)onehasatonce V2y=divM (27) 54 TheMagnetostatic Field [Ch.2 orthemagnetization vector provides asource field forthemagne- tostatic potential, much asdivPprovides fortheelectrostatic potential $in(2-11). Themagnetic material cantherefore be interpreted asafictitious distribution ofvolume density ofmag- netism pm'=divMexistinginfree space. Since there isno observable freemagnetic quantity, pmrcanonlymean adistribu- tion ofvery small dipoles (seesection 13)which forsmall finite volume elements always represents zero total magnetism, but which produces locally very strong magnetic intensities. These dipoles areequivalenttovery small current loops which are assumed tohaverandom orientation when noexternal magnetic field isapplied, sothat over finite small volume elements alsoM=0.Theapplication ofastatic magneticfieldcauses succes- siveorientation1ofsmall domains ofdipoles andappearance of M. Obviously, without first defining theunderlying structure,Mcannot beevaluated. Ifoneconsiders, then, anisotropic magnetic material ofvolume Tandconstant /iinfreespace andexposed toamagnetic fieldB0| onecanrepresent thismaterial bythesamevolume rfilled with fictitious magnetismofdensity pmr=divM.The orientation ofthedipoles caused byBwillalsoleave onthesurface ofthe volume Tanextra fieldwhich appears tocome from afictitious surface density ofmagnetism<rm' .Thiscanbedefined from the normal componentsofthemagnetic intensity Hinthesamemanner astheelectrostatic field gradient Edefines thefictitious surface chargeofpolarization, since thenormal component ofthegradient completely specifies thepotential distribution asshown in(2-3a). From (26)onehas,observing (7), Hn2-Hm=- (Mn2-Mni)=<rm' (28) The total contribution totheresultant magnetostatic potential existing outside orinside thematerial is,therefore, inanalogy to (2-13), Obviously, Mmustdepend onB ,theimpressed field, sothat (29) isonlyaformal solution inthegeneralcase. 1F.T.Bitter: Introduction toFerromagnetism; McGraw-Hill, NewYork, 1937; S.R.Williams: Magnetic Phenomena; McGraw-Hill, NewYork, 1931. Sec. 6] Magnetization 55 Theseparation (26)ofthecontributions tothemagnetic intensity canbeused alsowith thevector potential Afrom (12)and(19). Applying thecurloperation Vxto(26),onehaswith (14) curlH=J=VxB-VxM Mo sothatwith (12)oneobtains =VxB=/ioj+MOcurlM(30) Comparison with (15)shows atoncethattheeffect ofthemagnetic medium canberepresented asanequivalent current density J'=curlM,distributed throughout thevolume ofthemagnetic material. Theinterchange between pm',thevolume density of magnetism. inthemagnetostatic potential field,andthecurrent density J'inthevector potentialfield illustrates oncemore the ready conversion oftherespective concepts. Considering asbefore anisotropic magnetic material ofvolume randconstant/xinfreespace andexposed toamagnetic fieldB, onecanrepresent thismaterial bythesamevolume ^filled with fictitious current flow ofdensity J'=curlM.Foraformal solution onecanthen use(19),inwhich/ijistobereplaced by /ij'asindicated by(30)andwherepKmust bereplaced bythe surface discontinuityofthetangential components ofBinaccord- ancewith condition (10). From (26)onehas Ht2-Htl=-(Bt2-Btl)-(M tz- Mo =-(B*-Btl)-K' MO andsincenorealcurrent sheet density exists, theleft-hand side must bezero. This leads, therefore, tothefictitious current sheet densityK7andthus to (31) asthetotal contribution ofthematerial totheresultant vector potential existing outsideor_inside themagnetic material. Again, J7andK7must depend onB,sothat (31)canonly represenj formal solution inthegeneral casewhere themagnetic ma canreactupon theimpressed field,B .Ifonecanassume 56 TheMagnetostatic Field [Ch.2 stant magnetization Mthroughout thevolumeT,then (31)reduces tothesurface integral; amagnetized cylinder cantherefore be considered theequivalent ofathin cylindrical coilcarrying current ofsheet density K'.The direction ofK'issuch thatfrom its vectortipthedifferenceMt2Mtl,ifpositive, hascounterclock- wise direction. 7-ENERGY ANDFORCES INTHEMAGNETOSTATIC FIELD Considering anysystemofsteady current distributions, thenthe algebraic sum total ofcurrents through avery large cross-sectional surface (plane orcurved inspace) canbezero ordifferent from zero. Inthe firstcase, thesystem iscalled acomplete system, and allthecurrents flow inclosed loops andpermit definitions of fluxes andenergies infinite terms; thesecond casepresumes as partofthesystem wires ofinfinite length withnoreturn, a physically impossible arrangement which willnotbeconsidered further. Asemphasized previously, only linear relationships between currents andmagnetic fields willbeconsidered here; inallthe following relations, Mwilltherefore beassumed independent of thecurrent. Theextension tothenon-linear relationshipinan elementary manner isreadily possible and isgiven in AttwoodA2 ;however, theeval- uation ofthenon-linear field distributions isextremelydiffi- cult. TheSingle Current Loop. Thesimplest complete current systemisasingle current loop ofsimple geometry andarbi- FIG.7-1 Single Current Loop, Ideal-traiTconductor cross section, ized. Since current hastobesup- pliedbyasource,itisneces- sary toeffect anarrangement which minimizes themagnetic field oftheleads, asforexample twistingofbifilar wires oracoaxial cable asindicated inFig.7-1insimplelinodrawing. Themagnetic fluxlinkages aregiven byeither expression in (6-24) andareproportional totheloop current Ibecause ofthe Sec. 7] TheSingle Current Loop 57 linear relation (6-21) between current andvector potential. The ratio j=L (1) iscalled theinductance oftheloopand isapurely geometric characteristic ofthemagneticfieldand itsdistribution. Onecan actually giveanexplicit integral form ifheobserves that in (6-24) thetwosuccessive integrations areperformed inmutually perpendicular directions andcantherefore becombined intoa volume orasurface integral, respectively, overtheconductor r SinceAisitself given asanintegral overthesame cross section by (6-19), onehas - Knowing thecurrent distribution, onecan thereforedirectly evaluate theinductance oftheloop. This loopiscapableofstoring amagneticfieldenergy equal tothework required tobuildupthemagneticfield. Assuming asmall virtual displacement 5softheloop,andapplyingitfirst toone ofitsfilaments exposed toafield B,oneobtains with adaptation of(5-6) forthevirtual work onthisfilament 8W=(JdS)8$m or (Kds')83>m where 8$misthefluxthrough thesmall area covered bythe filament initstranslation 8s.Forthevirtual work onthetotal looponehastointegrate overtheconductor cross section andhas 8W=75A(4) utilizing (6-24).Ifthemagneticfield isproduced bythecurrent oftheloop itself, then6Acanbeproduced onlybyavariation of thecurrent with time sothatonemust introduce theinduction lawandaccount forthelosses. The totalmagneticfieldenergy, 58 TheMagnetostatic Field [Ch.2 however, canbeevaluated from (4),sothatwith (1),barring any deformation, Wm=1 IL81=Y2LI*=H/A=^7A2 (5) /= quite analogous totheelectrostatic relations in(3-3). Again as there,allthequantities arein- tegral quantities andareamen- able tomeasurement. Two Current Loops.If twocurrent loops with currents /iand72areinclose proximity, magnetic flux ofone will link withtheother. Computed any- where inspace, theexpression fortheresultant vector poten- tialAwillcontain oneterm de- pendent on/iandanother de- pending on72inaccordance withthesuperposition principle oflinear forms. The fluxlink- ages forthetwoloopswillthen beoftheform =L\I\+Z/12/2 FIG.7-2 Resultant Magnetic Field ofTwoCurrent Loops.(6) Onedesignates LIandL2asself- inductances ofthetwoloopsand LI2=LZI=Mastheirmutual inductance. These inductances aredenned by(3)ifone sets formally J=Ji+J2andidenti- fiestheresulting fourterms astheappropriateselfandmutual in- ductances. LI2and Z/2iareidentical because their definitions differ onlybytheinterchangeintheorder ofintegration. To emphasize thepartial linkage, onecanrewrite thepair ofrelations (6)intheform A!=(Lt-M)/x+M(h+72)| f (7)A2=(L2-M)7 a+M(h+/2) Sec. 7] TwoCurrent Loops 59 MLand define (LiM)=Si,(L2M)=S2,asprimary and secondary leakage inductance, respectively, considering the arrangement astheprototypeofatwo-winding transformer. M(Ii+72)=Amisthen called theusefully linkedflux, ormain fluxlinkage. Forthesimple case oftwocircular loops, Fig.72might repre- sent atypical resultant field distribution forassumed current values 11and72.Itisquite customary totake the field lines closing around onlyone conductor asrepresenting the leakage fluxlines of(7)andto take thelines passing through both loops asrepresenting the main flux lines of(7). This interpretation is,however, in- correct,1since thegeometryof thefield lines atevery point de- pends onboth currents simul- taneously, asdoes thevector potential; afewgraphs would readily bearoutthat fordiffer- entcurrent ratios theresultant field distribution changeslittle whereas thefluxcontributions (7)change rapidly. InInteraction ofTwoCurrent Loops.in{<)cnange rapiaiy. inFlG7.3 order torestore correlation be- tween fielddistribution andthe fluxlinkages according to(7),onemust consider onecurrent ata time, asforexample I2inFig. 7-3,andrelate ittothesecond equationof(7). The total magneticfieldenergy inthesystemisgiven bythe sum ofthetwoloop energies Wm= MIJ 2 (8) Thecenter term isthemutual energy. Itsvalue canbecomputed readily even iftheloops degenerate intocurrent filaments, since 1E.Weber, "What isLeakage?" Elektrot. undMasch., 48,p.943(1930); alsoE.T.Z., 61,pp.1221and1267 (1930). 60 TheMagnetostatic Field [Ch.2 thecontribution tothevector potential byonefilamentis,accord- ingto(6-21), AMrfdSlAl=S7lJ~7 whereas thefluxthrough thesecond filament isby(6-23) $12=/Ai-ds 2=M/! (9) i/C2 theintegral being taken over thesecond filament. The self- energies can,however, becomputed only forvolume orsurface distributions ofcurrents. Itshould benoted thatL\andL2,the self-inductances, are quite independentofthepresenceoftheother loop. This willbe soinallcases where thecurrent distribution isassumed tobe known. System ofLoops inHomogeneous Medium. Thegenerali- zation fromtwoloops tonloopsisnow readily made. The flux linkagesforloopawillbe Aa=ZW>,a=1,2,---,71 (10)=1 andthetotalmagneticfieldenergy becomes Wm=L7aAa=ELapIJ & (11) aquadraticfunction oftheloop currents. SinceLa=Lpa,there willben(n+l)/2 different inductance values. The force ortorque action insuch asystem caused byany changeofageometric element 5rj(linearorangular displacement) canbecalculated bytheprincipleofvirtual work. According to (4),themechanical work8Wcanbeexpressed directly asthechange ofthemagneticfieldenergy 75Aforasingle filament. Forfixed current values inthesystem (11),onehasthen forthemechanical action thepositive rate ofmagneticfieldenergy aTFmeeh,dWm 1ff dLag T=+-r-=+9^^^ ,W dri ay Aa=1/9=1 ch? whereas intheelectrostatic casethenegativerate ofenergy has tobetaken asseen in(3-26). This isrelated tothefactthat Sec. 7] Integral Forms ofEnergy 61 currents ofopposite sign repel each other, andthatmagnetic field energy haskinetic rather than potential character. Integral Forms ofMagnetostatic Energy. Themagnetic fieldenergy forafinite system ofloops ofarbitrary individual cross sections inahomogeneous medium canbeexpressed either by(11) or,introducing theexpressions (2)forthefluxlinkages, alsoas where theintegrals have tobeextended over allthecurrent loop volumes andcurrent loop sheets ofthesystem. Theform of(13) canreadily becompared with theanalogous electrostatic energy expression (3-18), where scalar potential andscalar densities take theplace ofthecorresponding vectors in(13). Thevector poten- tial itself isobtained by(6-19), quite analogous totheexpression forthescalar potential (2-5). Bymeans ofthevector analogueofGreen's theorem [Appendix 3,(32)], onecantransform (13)intoaverysimple volume integral. Letinthetheorem V=W=A,thevector potential, andobserve VxA=B, VxVxA=MJ andmultiply byl/2/i; then ityields Thesurface integral istobetaken overtheinfinite sphere bound- ingthemedium and allcurrent loop sheets which represent internal boundary surfaces forthemedium. Thevolume integrals aretobetaken over allspace within thisvery large sphere, whereby thesecond onewillcontribute only atplaces where J^0. Since, however, B >as1/r2 ,andA >as1/r,atvery large distance from the finite loop system, thesurfaceintegral canbe restricted tothecurrent loop sheets. Onthese,AandBnwillbe continuous, whereasHtwillhave adiscontinuity according to (6-10) ofvalue K,thecurrent sheet density; integrating over thesurface ofthesheet, therefore, onlyHtwillcontribute jfjf(AxB)dS->jfjf(AK)dS 62 TheMagnetostatic Field [Ch.2 andbycomparison with (13)onefinds thealternative expression forthemagneticfieldenergy tobetaken over allspace. Thisform again permits, according toMaxwell's point ofview, theinterpretation as ifthe field energy were distributed through space withalocal density J^H B, entirely determined bythefield vectors. This expression canbe shown tobevalid foranymagnetostatic system inwhich there arenopermanent magnets. Stresses intheMagnetostatic Field. Asintheelectrostatic field, sohere inthemagnetostaticfieldFaraday's visualization of force action asassociated with theconfiguration offield lineswas formulated byMaxwell, whointroduced astress perunitarea of magnitude YflB inthedirection ofthefield lines,andapressure perunit area ofthesame magnitude perpendicular tothe field linesupon asurface element parallel tothe field lines. This imagined system offorces accounts forAmpere's force lawand permits evaluation offorce actions ontheboundary surfaces of different magnetic materials. Thus, theforcenormal toabound- arysurface urging medium 1towards medium 2willbe(without current sheet) thedifference ofthenormal stresses plusthedif- ference ofthenormal pressures onthetwosides oftheboundary surface: fn= %(Hi nBinH2nB2n) Because ofthecontinuity ofBnandHtacross theboundary, one cantransform thisinto M2 = ^(MI-M2)HrH 2 (16) These forms donottake intoaccount thesecondary effects of magnetostriction anddonotapply strictly forferromagnetic materials. Arather complete account ofthemore advanced theoryisgiveninthereferences LivensA15and Stratton,A23 Appendix4. Problems 63 PROBLEMS 1.The barrier surface inFig.6-1hasamagnetostatic potential difference7172=Ibetween itsfaces. Show that itcanbeinterpreted, therefore, asafictitious magnetic shell (magnetic double layer) ofmoment /=amda perunit area,where <rmisthefictitious magnetic charge density andSathe very small charge separation. Hint :refer toproblem 6ofchapter1anduse (6-4). 2.Demonstrate theuniqueness theorem forthemagnetostatic potential function ofanynumber ofcurrent loops infreespace, eachloopfurnished with anappropriate barrier surface. Hint :note thepreceding problem andapply themethod ofproblem 12inchapter1. 3.Compute themagnetic field inside andoutside auniformly magnetized sphere ofradius a.Themagnetic field causing themagnetization Mis homogeneous throughout space (assume freespace), hasthesame direction asM,andhasvalue BQ. Establish theequivalence withauniform magnetic sphere intheliomogcncous fieldBOandfindtheequivalent relative permea- bility ofthesphereinterms ofMandBQ(seeproblem 4inchapter 1). 4.Themeasurement oftheforce action between thenearends oftwolong barmagnets leads to"Coulomb's force lawformagnetic poles." Show that thislawmusthave theformFm=AiQmiQm2/4irr2 ,whereQmarethemagnetic quantities measuring thepole strengths, Mtheabsolute permeability ofthe medium inwhich themeasurement ismade, and Tthecenter distance ofthe magnetic poles. Hint: deduce theconcept of"field strength" asinCoulomb's lawforelectric charges andobserve (5-4) aswell asproblem1above; see alsomagnetic dipole insection 13. 5.Demonstrate that theformal solution (6-19) satisfies thecondition (6-17) forfinite distributions ofJandK. 6.Give thederivation of(6-25) from theformal solution (6-19) forthe vector potential. 7.Inaregion freeofcurrent flowandbounded byaclosed surface S,the vector potentialisuniquely defined byitsvalues ontheboundary surface; demonstrate theuniqueness theorem forthevector potential. Hint: apply thevector analogue toGreen's theorem (Appendix 3)withP=Q=AA7 , whereAandA7aretwo different solutions each satisfying thedifferential equation VxVxA=andtheboundary condition. (Stratton,A23 p.256.) 8.Accepting astress perunitarea ofmagnitude >^H-B inthedirection ofthemagnetic fieldlines,show thattwoparallellikecurrents flowing inthe same direction attract each other inaccordance withAmpere's law (5-1). Hint :utilize thesymmetry ofthefield distribution. 9.Accepting apressure perunitarea ofmagnitude J^H-B normal tothe direction ofthe field lines, show thattwoparallellikecurrents flowing in opposite directionsrepel each other inaccordance withAmpere's law (5-1). Hint: utilize thesymmetry ofthefield distribution. 10.Given afixedsystem ofnconductor loopsinahomogeneous and iso- tropic medium, assume thatcurrents Iaareflowing intheindividual loopand that fluxlinkages Aaaremeasured. Ifthennewcurrents Ia'areflowing, the corresponding fluxlinkages Aa'arerelated by /aAa'=/a'Aa(analogue 64 TheMagnetostatic Field [Ch.2 toGreen's reciprocation theorem; seeproblem 10inchapter 1).Prove this relation; utilize (7-10). 11.Find theexpressionforthetorque exerted upon asingle plane filament loopofcurrent 7inauniform magnetic fieldBOiftheplaneoftheloopmakes theangle with thedirection ofBQ.Express therelation invectorial form byintroducing themagnetic moment oftheequivalent magnetic shell. 12.Find thegeneral expressionsfortheforceandthetorque exerted upon asmall circular filament loopofcurrent /inanon-uniform magnetic field B. Using theconceptoftheequivalent magnetic shell, convert theexpressions into vectorial form; compare with theanalogous electric dipole problemin section 10. 13.Find theforce action between twoidentical filament loopsofopposite current 7,ifthey areplaced parallel toeach other atvery small distance Sa.Observe thedirection oftheforceandcompare with theanalogous elec- trostatic problemoftwocharged conducting loops. 14.Any filament loopofcurrent 7canberepresentedasanetwork of elementary filament loops, thecontour ofeach ofwhich carries thesame current 7.Theforceoneachelementary loopofareadSisgivenbydFasfound in problem12.Demonstrate thatthetotal forceupon theactual loopcanalso beexpressed byF=7<t>dlxB,where dlisthevector lineelement oftheloop. (Smythe,A22 p.276.) 15.Onthebasis oftheexperimentally confirmed force action (5-4) one canassume theforce onanyelement 7dlofafilament loop tobegiven by dF=7dlxB. Using this,demonstrate thevalidity of(7-4) forafinite current loop offinite cross section S.Hint :divide thecurrent volume intofilaments JdS. 16.Formulate thegeneral boundary conditions pertaining tothenormal andtangentialderivatives ofthevector potential, excluding thepossibility ofamagneticshell intheboundary surface. 17.Formulate theboundary condition forthemagnetic vector potential iftheboundary surface isamagneticshell ofmoment miperunit area. *18.Themagneticforces acttoincrease themagneticfieldenergy asshown in(7-12), sothatthelatter isfrequently interpretedasanalogous tokinetic energy; consideringitaspotential energy, onemust define itbyU=Wm. Showbymeans ofFaraday's lawofinduction V=5*m/Mthat thework done inasmall actual (not virtual) displacementofasingle filament loop, keeping thecurrent 7constant,isexactly compensated bytheenergyfur- nished bythesupply voltage; thetotalwork doneonthecircuit isthus zero. (Stratum,A23 p.119.) 19.Show thatafreely movable filament loop carrying current 7willbe instable equilibriuminamagnetic field iftheloop links thegreatest possible magneticflux. 20.Given twofilament loops carrying currents 7iand72inarbitraryrela- tiveposition andwith individual supply voltages V\andVz.Ifthetwoloops attract each other and ifthecurrents arekept constant, show thatonehalf oftheenergy supplied bythesources isused forthemechanical work. Note problem 18. (Smythe,A22 p.306. ) Problems 65 21.Find thevector potential andmagneticfieldproduced byaplane current sheet ofinfinite extent withuniformparallel current flow ofdensity K. 22.Find themagneticfield inaninfinite plane slab ofmagnetic material withpermeability M2ifonitstwofaces thincurrent sheets areapplied carry- ingcurrents ofdensities Kinopposite directions. Find thefield outside theslab ifthepermeabilitythere ispi. 23.Find themagnetic field farfrom athin cylindrical barmagnetized uniformly paralleltoitsaxis iftheradius isaandthelength21.Define its magnetic moment andshow theequivalence toabarmagnet. 24.Demonstrate that forfinite current distributions (7-4) can alsobe expressedasthevolume integral ofH-5B. Hint: usethe firstform of(7-2) with fixed current value; observe (6-13) andVxfiA=5B. 25.Show that inferromagnetic materials thehysteresislosspercycleand perunitvolume canberepresented bytheintegralIM-dB taken over onecomplete? cycle ofmagnetization. GENERAL FIELD ANALOGIES 8-THEELECTRIC CURRENT FIELD Chapter1hasdealt with theelectrostatic field ininsulators (dielectrics). Inconductors, thepresence ofaconstant electric field causes acontinuous migration ofcharges, usually electrons inmetals, andions inelectrolytes. Theflow rate ofcharges or thecurrent '-2 asmeasured through astationary surface hasthecharacteristics oftheflow ofanincompressible fluid, namely, that neither source norsinkcanexist within thefluid itself. Interms ofthedensity oftheflowthrough unit area, designated bythevectorJ,incom- pressibility means IIT_jof\ /n\ItJ-ndb=(2)JJs orinaccordance withGauss's theorem (seeAppendix 3) divJ=V-J=(3) Forstationary currents, Ohm's lawexpresses theexperimental factthatthepotentialdifference $1$2(voltage) applied atthe ends ofalongconductor isproportionaltotheresulting current within theconductor, withRdefined asresistance. Forlong conductors (wires) of 66 Sec. 8] TheElectric Current Field 67 uniform cross section S,theresistance issimply related tothe geometry, R= ^s(5) where yistheuniform conductivity ofthematerial and Ithelength oftheconductor. This experimental factcanbetranslated into FIG.81Differential Form ofOhm's Law. avector relationship byconsidering avolume element inan extended conductor ofarbitrary shape. Referring toFig.8-1,the elemental potentialdifference inthedirection ofcurrent flowcan befound byaTaylorseries approximation as (d$/dl) dl,the current oftheelement asJdS,andtheresistance from (5)as -(dl/dS). Thus, (4)leads to 7 dl oringeneral vector relation -grad $=dl ydSJdS -;'(6) (7) Thiscanbereadily verified bywriting (6)forthethree orthogonal directions ofacoordinate system andtaking thevector sum. 68 General Field Analogies [Ch.3 Thenegative gradientoftheelectric potential *canbedenned astheelectric fieldstrength Ewithin conductors inthesamemanner aswithin dielectrics by(1-7), andoneobtains J=7E (8) the differential form ofOhm's law. Combining (7)with (3) yields, then, V-J=-V-(7V*)= oralso V$>-V7+7V2*=(9) thegeneral differential equationforaninhomogeneous conductor, wherein thevariation of7must beknown. (Only thecase of isotropic conductors istreated here.) Forhomogeneous media, 7willbeaconstant and (9)reduces to Laplace's equation V2*=(10) identical with (2-2)fortheelectrostatic field inhomogeneous dielectrics. Inaboundary value problemofthe firstkindwith onlytwoprescribed boundary potential values ofasinglemedium (seesection 2),thesolution fortheelectric field strength Ewill beidentical, therefore, whether thissinglemedium beadielectric oraconductor; inthe first case,which istheideal condenser of section 3,theelectric vectorEwillberesponsibleforadielectric fluxdensityD=eE,whereas inthesecond case,itwillberesponsi- bleforacurrent density J=7E.The total dielectric flux between thetwoboundary potentialswillbe jfjfDdS=ejjEdS whereas thetotal current flow fortheconducting medium is jdS=TffE dS=(*!-*a) jfjf sothat forthesamegeometry onehas CR=-(11) 7 Solving acondenser probleminelectrostatics, onecanimmediately obtain theresistance between thesame electrodes byapplication of(11). Sec. 8]Boundary Conditions ofElectric Field 69 TheBoundary Conditions oftheElectric Current Field. Ifseveral different conducting media arepresent, then itisneces- sary tosolve thedifferential equation (10), orthemore general form (9),foreach individual conductor andtolinkthese solutions bycontinuity conditions along theboundary surfaces. FIG.8-2 Continuity ofCurrent Flow across aBoundary Surface between TwoConductors. Thebasic condition (2),ifapplied toasmall cylinder ofheight dh >0,asindicated inFig.82,leads atonce to JndS=Jn%dS% JnldSi= Inthelimit forvanishing dh Jn,=Jni (12) which isonegeneral boundary condition and states thecon- tinuity ofcurrent flowacross aboundary surface under stationary conditions. From thefactthat theelectric field strengthisderived from thescalar potential $inthesamemanner asinelectrostatics, one candeduce asin(2-8) Et2=Etl (13) thesecond general boundary condition. Etistherespective field component tangentialtotheboundary surface. 70 General Field Analogies [Ch.3 Thecombination ofthetwoconditions (12)and(13)leads to Et%_72 Eti Enz~ 7iEnl or 7i tana\=tana2 (14) 72 ifa\and e*2designate, respectively, theangles ofthefieldvectors with thenormals totheboundary surface. Relation (14)isof particularvalue ingraphicalfield plotting, defining therefraction offield lines. Dissipation intoHeat. Since current isidentified with the migrationofcharges, onecancompute thework associated with current flowthrough aconductor. Moving asmall charge dQ through thepotentialdifference ($1<2)requires thework dW=(<!>!-4>2)dQ asindicated in(3-2). Thetime rate ofwork, orpower, canbe expressed with (1)as dWP=-T-=V12I=RI* (15)at ifonealso utilizes Ohm's law (4). Thispower must beexpended tomaintain thecurrent flowthrough theconductor, and itappears asheat created bythe"resistance" tothemigration oftheele- mentary charges. Theexperimental proof wasgiven byJoule, and (15)isusually called Joule's law. Onecanreadily express (15) interms ofthecharacteristic field vectors ifheappliesthisrelation tothevolume element shown in Fig. 8-1. With theresistance andcurrent values asdefined for (6),onehasforthepower lossinthevolume element dP=-^- (JdS)2=-J2dT=E-]dr (16)7dS 7 andtherefore forthetotalpower dissipated inaconductor Jdr (17) which isvalid forallmedia, even fornon-isotropic media. This form permits theinterpretation asifthedissipation would take Sec. 9] Other Physical Fields 71 place withavolume density EJ,entirely determined bythefield vectors Eand J;thehypothetical nature ofthisinterpretation hastobekept inmind, however. Concept oftheSemiconductor orSemidielectric. Though forpurposes ofanalysisitisconvenient toadmit onlytwoclasses ofmedia, namely, ideal dielectrics andpure conductors, many materials exhibit asignificant combination ofboth characteristics; such materials arethen called semiconductors orsemidielectrics, depending upon thecharacteristic onewishes tostress. The electric field distribution insemidielectrics isfound by solving theLaplace differential equation forthepotential *and satisfyingalltheboundary conditions. However, thefieldvector Enow causes simultaneously electrostatic polarization andcon- duction current, sothat D=eE, J=7E (18) both apply. Attheboundary surface oftwosemiconductors the conditions (12)and (13)have tobesatisfied; thecurrent flow must becontinuous, since otherwise unlimited accumulation of charge would occur, contrary tothecondition ofstationary flow. Because oftheexistence ofD,therespective boundary condition (2-7) A,2-An=o- (19) alsomustbesatisfied, i.e.,asurface charge density amustappear ofvalue Dn2-Dnl=--Jn (20) \72 7i/ obtained bycombination of(18)and (12). Onlyifbychance 72 7i' willthissurface charge disappear. 9-OTHER PHYSICAL FIELDS Theconcept ofastationary fieldoccurs inseveral other branches ofphysics andengineering, such asaerodynamics andhydrody- namics, conduction ofheat,andgravitational theory.1Asfaras 1E.Weber, "MappingofFields,"Electr. Eng., 63,p.1563 (1934). See also listofreferences inAppendix 4,C. 72 General Field Analogies [Ch.3 TABLE CORRESPONDENCE OPQUANTITIES QuantityElectrostatic Field Magnetostatic Field Potential function Potential difference Equipotential surface Potential gradient Characteristic constant ofmedium Associated fieldvector Flux oftheassociated vector Total flux* Divergenceofassociated fieldvector Basic differential equa- tion ofthepotential function Field transmittance,electrostatic potential ',magnetostatic poten- tial ds=F, voltage (electromotive force) *=cons (conductor surfaces) E=grad *,electric fieldstrength :,absolute dielectric constant D=eE,dielectric flux density (displacement vector) ,dS,dielectric *= flux DndS=Q,electric charge within S divD=p,space charge density * p/emagnetomotive force "=cons(usually iron surfaces) H=grad [F,mag- netizing force i,absolute permeability B=jxH,magnetic flux density (magnetic in- duction) *m=JBndS,magnetic flux dS= divB= =0 tcapacitance permeance *Total flux isdefined astheintegral overaclosed surface. Sec. 9] Other Physical Fields 73 9-1 INSCALAR POTENTIAL FIELDS 74 General Field Analogies [Ch.3 thephenomenaadmit thedefinition ofscalar potential functions, theirmathematical treatment ismuch alike;thismakes itpossible todeduce analogies andtotranslate solutions fromanyone field intoanyother field ofapplications. Inthismanner, though this book isprimarily concerned with electric andmagnetic fields, thesolutions given canreadily beinterpreted forapplications to other fieldproblems. Inorder toassist inthistranslation, table 91presents asurvey ofanumber ofbranches ofphysics which admit ofaunified mathematical treatment, utilizing thefieldconcept. Each oneof thebranches ofphysicsischaracterized byafundamental scalar satisfying thePoisson orLaplacedifferential equation, anda derived fieldvector which isdefined asthe(positive ornegative) gradientofthat scalar. Asthetable outlines indetail, there are additional analogous concepts foreach branch, andthequantities inanyonecolumn canbeconsidered entirely equivalent tothe respective quantities (inthesame line), forexample,oftheelec- trostatic field. Itisnecessary onlytostudy indetail thesolution ofaprobleminonebranch inorder tobeable topredict forevery other branch thesimilar solution with proper transposition of terms. Thecolumn oftable 91headed Electrostatic Field presents a summary oftherelations discussed insection 2;thecolumn headed MagnetostaticField summarizes therelations discussed in section 6;andthecolumn headed Electric Current Field sum- marizes the relations discussed insection 8.The relations presentedintheother columns willnowbebriefly discussed in order toprovide abetter understanding oftheterminology. Stationary Temperature Field.2Under stationary condi- tions, theflow ofheatpowerisvery similar tothestationary flow ofelectric current inconductors. Heat powerwillalways flow from points ofhigher temperature topointsoflower temperature; itcan, therefore, becharacterized byavector density Jwhich is measured inpower perunit area. Thevector Jmust, ofcourse, pointintothedirection ofgreatest temperature fall;itisindeed proportional tothetemperature gradient, J kgradT=-k(VT) (1) 2Seereferences inAppendix 4,C,b.Foranalogies toelectrical problems seeM.Avrami andV.Paschkis, Trans. A.I.C.E., 38,p.631(1942). Sec. 9] Stationary Temperature Field 75 where kisthethermal conductivity oftheconductor. Again, under stationary conditions, heatpower canneither accumulate norvanish locally (unless there areextraneous sources ofheat). The flux integralofJover aclosed surface must, therefore, be zero,andthevector Jitself cannot haveanydivergence, divJ=V-J=(2) Combination of(1)and (2)leads atonce totheLaplacian differen- tialequation V2r=(3) pointingtotemperature Tasthescalar function analogous tothe electrostatic potential$.Ausual problem inheat conduction assumes certain metallic surfaces asisothermal, i.e., ofconstant temperature, andendeavors tofindthetotal heatpower flowfrom hightemperature T\tolowtemperature T2through thethermal insulation between themetals. This isaboundary value problem ofthe firstkind, exactly likecurrent flowbetween twoequipoten- tialsurfaces. Ifonedefines thetotalpower flow Q=ffjndS (4) through thecross section S,onecanthenevaluate athermal resis- tance *,h=(5) andmeasure3itinthermal ohms (actually C/watt intheGiorgi system). The reciprocal quantity, thermal conductance,4is included intable 91. Inmany thermal problems relating tonatural cooling ofbodies byradiation, convection, andconduction ofheat totheambient medium, theboundary condition onthecooling surface defines the heat transfer totheambient interms ofNewton's condition -*(),--Ta) (6) 3C.Hering, Metal, andChem. Engg., 9,p.13(1011);Electrical Engineers' Handbook, Electric Power, edited byH.Fender, W.A.DelMar, andK. Mcllwain, p.14-193; John Wiley,NewYork, 1930. 4More generally usedandmeasured inwatt/C orBtu/F; seeW.II. McAdams: Heat Transmission; McGraw-Hill, NewYork, 1942. 76 General Field Analogies [Ch.3 Here, 7\isthetemperature and (dT/dri)iitsnormal derivative ontheboundary surface butwithin thecooling body; Taisthe ambient temperature (assumed constant); and <xtistheheat transfer coefficient. SinceTaisconstant, onecanintroduce the temperature riseabove ambient =T-Ta (7) asmain variable, rather than theabsolute temperature values. Thischanges (3)to V2=(8) with theboundary condition onthecooling surface =afl (9) Thistypeofproblem, usually referred toasaboundary value prob- lem ofthethird kind, hasnoanalogueintheelectric ormagnetic fields; therefore, nodirect solutions willbegiven here. Finally, there arethermal problems, ofconsiderable importance forelectrical design,inwhich distributed heat sources occur. Thecomputation ofthetemperature riseofelectrical conductors which carry current belongs inthisgroup. Here, theflowvector Jhasassource theJoule heat hcreated bytheelectric current inunitvolume; thus, divJ=V-J=h(10) andthecombination with (1)leadsnowto vr=-,(ii) aPoisson differential equation ofthesame type astheelectrostatic space charge equation (2-4). FluidDynamic Fields.5Theflow ofincompressible fluidsand gases without internal frictionis,infirstapproximation, again very similar totheelectric current flow.(Historically, thelaws ofelectric current flowwere patterned after therelations offluid flow.) Theflow density through unit area isdefined atanypoint in space astheproduct ofmass density pandvelocity vofthefluid F=pv (12) 6Seereferences inAppendix 4,C,c. Sec. 9] FluidDynamic Fields 77 Foranincompressible fluid withnosources orsinks, thetotal fluxthrough aclosed surface must vanish, i.e.,thevector Fcannot have adivergence divF=V-F=(13) This relation becomes inCartesian coordinates (PVX)+ (pVy)+ (PV2)= dx dy dz and isthewell-known equation ofcontinuity. Ifthe fluid is alsohomogeneous, sothat piseverywhere thesame, onehasfrom (13) divv=V-v=(14) Inirrotational flow, defined by curlv=Vxv=(15) onecanintroduce ascalar potential function. Itiscustomary to define avelocity potential <,such that velocity becomes thepositive gradient v=+grad$=+V* (16) andtoconsider theflowdensity vector Fasanassociated vector. Thecombination of(13)with (16)leads then atonce totheLaplace differential equationforthevelocity potential V2$=(17) For fluid flowsbounded bysolid guides, theboundary conditions arequite analogous tothose oftheelectric current flow, namely, thenormal component ofFmust vanish onthewalls. The lines towhich thevelocity vector viseverywhere tangential arecalled stream lines; theyform with respect tothecquipotential surfaces anorthogonal system ofgradient lines. Ifsources orsinks areincluded inthefieldregion, thenthetotal flux ofvectorFthrough aclosed surface inclosing asource becomes theeffluxE dS=(18) andlocally thedivergence ofFwillnotvanish. Asintheelectro- 78 General Field Analogies [Ch.3 static case, thepotential equation thenbecomes thePoisson type V2*=+-(19) p where eisthevolume density oftheefflux E.Usually, oneassumes point orlinesources, introducing them assingularities inthesame manner aspoint charges and linecharges inelectrostatics. Incompressible fluidmotion canalsoinclude rotation about an axis orabout asolid body. Inthis case, curlvisstillzero at every point, butthelineintegral ofthevector vinclosing theaxis ofrotation isobviously notzero. This isexactly analogous to themagneticfield oflinecurrents treated insection 6.Asindicated there, onecansalvage theconcept ofthescalar potential function byappropriately defining barrier surfaces asdouble layers of sources andsinks; nopath ofintegration canpenetrate thislayer. One willproceedinasimilar manner influiddynamics, defining thecirculating flow alsoasvortexflow. General solutions influid dynamics consist, therefore, ofasuperposition ofelectrostatic and magnetostatic field solutions. Gravitational Fields.6Thebasic theory ofelectrostatic and magnetostatic phenomena wasdeveloped byGauss inanalogy to thegravitational theory,7since allthree rested upon force actions varying astheinverse square ofthedistance. Oneusually writes theattractive forcebetween twomasses miandm2 f9n\(20) whereGistheuniversal gravitational constant, ofvalue 6.664X 1CT11m3/kgsec2intheMKS system ofunits. Similarly tothe point chargeinelectrostatics (section 1),onecanletm2beaprobe mass ofvanishing dimensions anddeduce afieldstrength g,orthe gravitational acceleration, as g=limJL=-(7^r(21) mz_o^2 r gisquite analogoustofield strength Ein(1-3), except forthe 6Seereferences inAppendix 4,C,d. 7K.F.Gauss: "General Theorems Concerning Attractive andRepulsive Forces Varying astheInverse Square oftheDistance" (originaltitleinLatin), 1826; seeCollected Works. Sec. 9] Gravitational Fields 79 negative sign indirection, which isoccasioned bythefactthat all masses have thesame signandattract, whereas likecharges repel each other. Thevector gdefines theforce lines aseverywhere tangential tog,giving thusadirect visualization ofthegravitational force field. Asforanyconservative force, thelineintegral ofFoveraclosed pathmust vanish, which infers theexistence ofascalar potential function Usuch that g=+gradU=+VC7 (22) Now, forasingle mass pointm\with itsradial forcelines, onecan form theflux ofgthrough aconcentric spherical surface, andone obtains with (21) j>ggn-dS=-04! (23) Itiscustomary toabsorb theuniversal factorGintheassociated vector f=g (24) andthusobtain generally fortheflux ofthisnewvector fndS=-47rM(25) whereMisthetotalmass within theclosed reducible surface of integration. ByGauss's theorem onecandeduce from (25) divf=V-f=-47Tp (26) ifpisthemass density ofanyarbitrary spatial distribution. This relation isagain quite analogous to(1-12), except forthenegative sign. Thecombination of(22)with (24)and(26)leads to V2t7=-^G P (27) thePoisson equation ofthegravitational potential function valid within regions occupied bydistributed mass. Outside ofmasses, (27) reduces,ofcourse, totheLaplacian differential equation. Electrostatic problems can, therefore, very readily beinterpreted asgravitational problems andviceversa. 80 General Field Analogies [Ch. 3 PROBLEMS 1.The electric resistance ofavolume element isfrequently written inthe formdR dl/ydS. Why should thisform bedeprecated? Howmust it beinterpreted andused inorder toyield correct results? 2.Prove thattheelectric current density inagiven conductor distributes itself sothat thejoule lossordissipation intoheat isaminimum. Hint: assume two sets ofcurrent densities J=yE, J'=ylL',both satisfying the condition ofzerodivergence toprevent accumulation ofcharge; the first set satisfies additionally E=-grad 0.(See alsoSmythe,A22 p.228.) 3.Since theelectric current fieldcanbederived from aLaplacian potential function, theuniqueness ofthepotential solution canbedemonstrated inthe samemanner asinelectrostatics. Formulate thetheorem corresponding to problem12inchapter1foraconductor andproveit. 4.Formulate theuniqueness theorem foraconductor corresponding tothe analogous theorem foradielectric defined inproblem 14inchapter1andprove it. 5.Ifonedefines conductance Gastheinverse ofresistance R,(8-11) can bewritten (C/e)=(G/y}. Introduce and define conductance coefficients forasystemofperfectly conducting electrodes within ahomogeneous iso- tropic medium ofconductivity yinanalogy tothecapacitance coefficients defined in(3-7); showhowtheycanbedetermined bysimple measurements. 6.Derive alumped resistance analogue toanelectrostatic problem ofn conductors embedded inauniform dielectric; notethepreceding problem. 7.Derive theanalogue toGreen's reciprocation theorem inproblem 10, chapter 1,forasystem ofperfectly conducting electrodes within ahomo- geneous andisotropic medium ofconductivity y. 8.The integral expression (8-17) fordissipated power inaconductor of arbitrary shapeisanalogous to(3-20) forthe electrostatic field energy. Convert itintotheformP=I2Rbymeans ofGreen's theorem, Appendix 3. 9.Demonstrate thatanyassumed current distribution forafixed total current inaconductor leads toahigher resistance than thecurrent distri- bution that follows from thesolution ofLaplace's equation (8-10). 10.Demonstrate thattheintroduction ofaperfectly conducting element of volume TOintoaconductor ofconductivity ydecreases theresistance. Hint : assume two sets]=a^EandJ;=<rEfderived from complete solutions of therespective Laplacian differential equations, the first setgiving*forthe original volume T,thesecond setgiving *'forthevolume (T TO);notethe similarity toproblem 20inchapter1. 11.Since thetemperaturefield inaregion without sources isasolution of theLaplaciandifferential equation,itsuniqueness canbedemonstrated in thesamemanner asinelectrostatics. Formulate thetheorem corresponding toproblem 12inchapter1forafinite regular region Tinauniform thermal medium andproveit. 12.Formulate theuniqueness theorem forthetemperaturefieldinathermal medium corresponding totheanalogous theorem foradielectric defined in problem 14ofchapter1andproveit. 13.Theheatexchange byconduction inasingle thermal medium between several sources, each ofdifferent uniform surface temperature Ta,canberep- Problems 81 resented bythermal conductance coefficients inanalogy tothecapacitance coefficients defined in(3-7); introduce anddefine these coefficients andshow howtheycanbedetermined bysimple measurements. 14.Derive alumped resistance analogue tothethermal system described inthepreceding problem. 15.Extend theproofinproblem 12toafinitenumber offinite heatsources, eachwith adifferent uniform fixedtemperature Taalongitssurface, embedded inahomogeneous andisotropic thermal medium ofinfinite extent inwhich no generationofheattakesplace. 16.Extend theproofinproblem 15toafinitenumber offinite heat sources embedded inseveral different thermal media inwhich nogeneration ofheat takesplace; consider thoboundary conditions atallinterfaces ofthethermal media astheanalogues to(8-12) and(8-13). 17.Since thevelocity potential inaregion without sources isasolution of theLaplaciandifferentialequation,itsuniqueness canbedemonstrated in thesamemanner asinelectrostatics. Formulate thetheoremcorresponding toproblem 13inchapter1forafiniteregular regionTinanincompressible fluidandproveit. 18.Theflow ofanincompressiblefluidbetween several sources andsinks ofdifferent butindividually constant values ofvelocity potentials $ffcanbe represented byhydraulic conductance coefficients inanalogy tothecapaci- tance coefficients defined in(3-7); introduce and define thesecoefficients; derive theanalogue interms oflumped electrical resistances. 19.Formulate theuniqueness theorem fortheflowpattern ofanincompres- sible fluid corresponding totheanalogous theorem foradielectric defined in problem14ofchapter 1,andproveit. 20.Find thegravitational potential distribution everywhere inspace pro- duced byasphereofuniform mass density pandofradius a. 21.Find thegravitational potential distribution everywhere inspace pro- duced byasphereoftotalmassMofradiusa,andwith adensity p(r)which isonly afunction oftheradial distance. Demonstrate that thepotential external tothesphereisthesame asifthemassMwere concentrated atthe center ofthesphere (point mass). Check theanalogous results forelectric charges. 22.Derive forthegravitational potential theintegral expression analogous to(2-5) fortheelectrostatic field. Demonstrate thevalidity bymeans of Green's theorem fromAppendix 3;observe theanalogous problems 2and3in chapter1. 23.Find thegravitational potential distribution everywhere inspace pro- duced byasphereofradius awhich iscomposed oftwohemispheres ofdif- ferent mass densities piand P2- 24.Deduce thegeneral boundary conditions forthevector fofthegravita- tional fielddefined by(9-24) and(9-26). 4-FIELDS OFSIMPLE GEOMETRIES Manyofthevery simplefield solutions arealsogiven inthe elementary textbooks andtreatises. Forthesake ofcompleteness, mention willbemade ofthese simple cases with references tothe appropriateliterature. 10SYSTEMS OFPOINT CHARGES The single point charge producesradial field lineswithafield vector "s?'< asoriginally deduced fromCoulomb's lawinsection 1.Ofcourse, both this fieldvector andtheassociated potential t-f W 47TET takeoninfinite values asr-sothatstrictly nophysical reality canbeattached tothisconceptof"point charge." However, the vanishing dimension eliminates theproblem ofcharge distribution andthusmakes readily possible theevaluation offielddistributions forsystemsofpoint charges bydirect superposition oftheindi- vidual contributions. TwoPoint Charges. Assume twopoint chargesofvalues Qi and$2;theresultant fieldvector isthevector sum i/Q. Q, where TIand r2aretheradius vectors from therespective charges 82 sec. 1UJ TwoFeint Charges 83 tothepoint ofobservation P.Since the field distribution must besymmetrical about thelineconnecting thecharges,itiscon- venient tochoose acylindrical coordinate system asinFig.10-1 FIG. 10-1TwoPoint Charges. with itsz-axis through thecharges andwith p=(x2+y2)^as theperpendiculardistance from it. The fieldlines arethen defined inaccordance with (1-5) by dz dp Theradius vectors are TI=uz(z+c)+upP ,T2=MZ(Z-c)+(4) (5) withuzandupasunitvectors intherespective directions. Intro- ducing (3)into (4)andomitting thefactor l/47re since itcancels outyield dz dp Crosswise multiplication and collection ofterms with thesame ra3inthedenominator leadto ^[pdz-(z+c)dp]+^f[Pdz-(z-c)dp]=(6) But = 2[pdz-(z c)dp] P/ P 84 Fields ofSimple Geometries[Ch.4 sothat(6)becomes also which canbeidentified asthecomplete differential of r2 theintegral ofthefield lines. Choice oftheconstant kleads to theindividual field lines. Fortheactual evaluation ofthe field lines, thiscanbechanged toamore convenient formbyobserving that cos 0!=(z+c)/ri and cos 2=(z-c)/r 2,sothat(8) becomes Qicos0j+Q2cos 2=k(9) Starting with aparticular pointPinthez-p-plane, onecan measure theangles 0iand 2andthusdetermine/c;onecanthen follow the field linethrough Pbychoosing different values ofel andbycomputing thenecessary angles 2from(9)forthespecific k.Intersection oftheradius vectors willgivethesuccessive points ofthefield line. Theequipotential lines inthez-p-plane aredirectly given by Assuming aspecific value ofthepotential<S>andchoosing adis- tanceri,onecancompute thenecessary distance r2as * 7-2= Thus, individual points along theequipotential linecanreadily beconstructed. Obviously, thefamily oftheequipotential lines andthat ofthe field linesmust bemutually orthogonal atall points, which usuallyisawelcome check. Good graphs ofequipotential and field lines forQ\=Q2and Qi=+Q2, respectively, canbefound inAttwood,A2 Figs. 1-22 to1-24; inJeans,A1 Figs. 17and15,16; inHarnwell,A9 Figs. 1-23and 1-22; inRamsay,A21 pp.36and37;and inSmythe,A22 Figs.108aand 108b. Inboth charge arrangements, there is alsosymmetry about theplane z= ofFig. 10-1. For like Sec. 10] TwoPoint Charges 85 charges, theequipotential surfaces close tothecharges arealmost spheres, become pear-shaped, merge intoanhour-glass shape, and finally approximate spheres again butwith centers at0,asthe values $decrease in(10). Foropposite charges, theequipotential surfaces close tothecharges arealsoalmost spheres, thenbecome pear-shaped butwith thepointed sections outward; theplane perpendicular toand bisecting thecenter line isalsoanequi- potentialsurface. One finds alsographs ofcharge arrangements Qi=4Q2in Jeans,A1 Figs. 19,20;inMaxwellA17asanexcellent plate atthe endofthebook; andinRamsay,A21 p.38;ofQi=2Q2andQi= 2Q2inAttwood,A2Figs. 1-25and1-27, respectively; and of Qi=+4Q 2again asanexcellent plate inMaxwell.A17Inall cases ofunequal charge values, onefinds asingular point onthe axisatwhich theelectric fieldEvanishes, or,since along theaxis onlyEzcan exist, whereEz=0.Forapoint P',Fig. 10-1, TI=c+z1 ', \r2 \=c z and, therefore, atthispoint, using (3)and(5), ForEz=onecansolve thequadratic formandobtain z'= (rjq=vV-1)c (13) with thedefinition rn, forlikecharges K1 Qi-Q2 t\=- -foropposite charges Qi+Q-2 Theupper sign in(13)holds forlikecharges where zf<c,whereas thelower signholds foropposite charges where z'>c;inthe case|Qi|<\Q2 \onecanusethesame definitions of77butmust reverse thesignsofthesquare root in(13). Atthesingular points, noforce action cantake place onaprobe charge located there; however,itiseasily shown, asinJeans,A1that these are points ofunstable equilibrium, nostable equilibrium being pos- sible inapurelyelectrostatic force field(Earnshaw's theorem, see 86 Fields ofSimple Geometries [Ch.4 SmytheA22 ).Actually, fortwo positive charges thefieldvector along theaxispoints from both directions towards thesingular point, whereas perpendicular totheaxis itpoints radially out; fortwonegative point charges justthereverse istrue. Fortwo opposite charges, thesingular point always occurs ontheside of thesmaller charge;ifthis isthepositive one,then thefieldvector behaves asinthecase oftwopositive charges;ifitisthenegative one,then thefieldvector behaves asinthecase oftwonegative charges. Fromthis,onededuces thatthepotential must have a saddle point. Thismeans thatthepotential goesthrough amini- mum along theaxis ifthefieldvectors there point towards thesingu- larpoint, andsimultaneouslyitgoesthrough amaximum inthe direction normal totheaxis. Thereverse istrue ifthefieldvectors along theaxispointaway from thesingular point. Conducting Planes andPoint Charges. The field distribu- tion oftwopoint chargesofequal magnitude andopposite sign, aspointed outabove, includes theplane ofsymmetry between the point charges asanequipotcntial surface. Conversely, then, one concludes thatthefieldbetween aconducting plane andapoint charge must bethesame asbetween twoequal point charges of opposite signseparated bydouble thedistance between plane and given point charge; thegeneralized utilization ofsuch analogies iscalled themethod ofimages1andwillbemore extensively treated insection 21.Fortheparticular case ofaconducting plane, Fig. 10-2 indicates the field distribution infront oftheconducting plane asonehalf ofthefieldbetween thegiven charge+Qand theimage Q.Thepotential distribution intheright-hand half spaceisgivenby where*isanarbitrary constant available toadjust theabsolute potentialoftheconducting plane. Along theplane 2=0,the fieldvector hasthevalue *-*--?(16) Thenegative sign defines thedirection intotheconducting plane, 1W.Thomson: Papers onElectrostatics andMagnetism, p.73;Macmillan, London, 1872; firstpublished inCambridge andDublin Math. Jl.,1848. Sec. 10] Conducting Planes andPoint Charges 87 and r2=h2+p2 ,where pisthenormal distance from thez-axis. Theinduced charge density ontheplaneisgivenby Q 3 ^ ' tn I andtheintegral overtheentire plane z=isreadily shown tobe Q.Rather complete treatment ofthisand similar uses dis- Fio. 10-2 Conducting Plane andPoint Charge. cussed below isfound inBennett and Crothers,A3 p.184; in Jeans,A1 p.185; inMason andWeaver,A1Gpp.109-112; in Maxwell/17Vol.I,p.252, etc.;andinRamsay,A21 p.116, etc. Theforce action onthegiven charge Qcaused byitsowninduc- ingaction upon theconducting plane canreadily becomputed as theforcebetween itand itsimage, iQ2 (2>02 (18) Itisthisforce action which hastobeovercome intheemission of electrons (where signs arejustreversed) from metal surfaces by either thermionic orfield forces;itwas firstintroduced asimage forcebyW.Schottky2inthecomputationofthework function 2W.Schottky, Zeits.f.Physik, 14,p.63(1923); seealsoA.L.Reimann: Thermionic Emission; John Wiley,NewYork, 1934. 88 Fields ofSimple Geometries [Ch.4 ofelectrons. Obviously, thisimage force canhave significance onlyfordistances from themetal surface, forwhich itappears approximatelylikeamathematical plane. Theextension toapoint charge inametallic corner asshown inFig.10-3isstraightforward. Inorder tomake theintersecting FIG. 10-3 Point Charge andTwo Intersecting Conducting Planes. planesAandBofthesame potential,itisnecessary toplace three image charges asindicated. Theresultant potential atapointP willthenbe (19) Obviously, theinduced charge distribution willhavetwomaxima, oneoneach plane almost opposite thelocation ofQandslightly shifted away from thecorner. The total charges onplanesA andBare,respectively, QA=Qtan"1-! IT a=-- 7T(20) andthesum total isagain Q.The forceaction, too,canbe found bysuperposition ofallthree image forces. Actually, this exampleisonlyaspecial case ofthemore general oneoftwometal- licplanes intersecting atanangle TT/TI,where nbeanyarbitrary Sec. 10] Sphere andPoint Charge 89 integer; thenumber ofimage chargesisthen (2n 1),sym- metrically located along thecircle through thegiven charge.If nisnotaninteger, aninfinite number ofimages results (seesection 26). Sphere andPoint Charge. Theequipotential surfaces oftwo point charges with opposite signs always include onesphere sur- rounding thesmaller charge. Thiscanbeseenfrom (10)and JL. Fig.10-1 ifonetakes$=0, yielding Qx 9* Qi(21) which defines aspherical sur- FIG.10-4 Point Charge andSphere, face. Referring toFig. 10-4, then, onecanreadily determine theradiusRbyselecting pointsPf andP"forwhich, respectively, //A' R-a(rz\"_W" b-R'W b+R Equating these expressions givesforthesphere r>2 uR*=ab,a=-^= +lb(22) (23) Onecansolve forthelocation andradius ofthespherealsodirectly interms ofthecharge ratioaandthedistance 2cbetween the charges, a=2cri' 1-(24) wherefrom a<6,ifa<1,i.e.,thesphere surrounds thesmaller charge. Reversing theprocess, onecandefine the field distribution between agrounded sphere ofradiusRandapoint charge Qias described bythe fieldbetween twopoint chargesofvalues Q2 andQi,whereQ2astheimageofQtwithrespect tothesurface of thesphere, hasthevalue (26) 90 Fields ofSimple Geometries [Ch.4 and islocated ontheaxis ata=R2/bfrom thecenter ofthe sphere towards Qi.Figure 105thus illustrates thefield distribu- tion foracharge+Qlocated atadistance equal totheradiusR from thesurface ofthesphere; inthis case, theimage chargeis J^Qaccording to(25)and islocated ata=R/2totheleftof C,thecenter ofthesphere. Thesingular pointofvanishingfield strengthislocated atz=(3+Vg)c inaccordance with (13) FIG,10-5 Field Lines between Point Charge andSphere for6=2R. and (14),whereby 2c=ba=%R isthedistance between the given point charge and itsimage. Good treatments ofthisand similar problems arefound inAttwood/2 pp.153-156; inJeans,A1 p.189; inMason andWeaver/16 p.112; inMaxwell/17Vol.I, p.245; inRamsay/21 p.117; inSmythe/22 p.114;and in Stratton/23 p.201. From theresultant potential distribution (26) obtained bycombining (25)and (10), onecanalsoobtain the Sec. 10] Sphere andPoint Charge 91 charge distribution onthesurface ofthesphere, expressed interms oftheangle6 R(b2+It*+2Rbcos(27) Integrated over thesphere, thisgives exactly Q2=(R/b)Q in accordance with (25).Maximum andminimum charge densities arefound for9=and= ?r,respectively, and their ratio becomes Even forb=1QR, this will stillbe(11/9)3=1.82, indicating the strong local fieldconcentration caused bypoint charges inthree- dimensional geometries. Ifthesphereisnotgrounded, butcarries anarbitrary charge Qfwith acorresponding potential value different fromzero, one canusedirect superposition oftheradial field ofanother point charge Q"-(Q'+Qijlocated atthecenter ofsphere. The total potential anywhere inspace willthenbethecombination of(26)andofthepotential produced byQ",namely, 47TE where Qiisthevalue oftheexternal point charge, r\and r2are asindicated inFig. 10-4,and risthedistance from thecenter of thesphere. Onthesurface ofthesphere, thisgivesnowthepoten- tial 5<* which, ofcourse, could beprescribed instead ofthecharge Q'. Thecharge density onthesurface ofthesphereisthesuperposition of(27)and oftheuniform density Q"/4vR2produced byQ'1 . Forthespecial casethatthesphereisinsulated,itscharge must remain zero, Q'=0,sothatQ"=(R/b)Q\. Thesurface charge ispartly negative opposite thepositive point charge Qiand partly positive.3Onecanreadily findtheangle 0owhich separates 8SeeAttwood/2p.154.andRamsay,A21p.117. 92 Fields ofSimple Geometries [Ch.4 thesetwozones, byputting theresultant charge density equal to zero, resulting in cos=-JL{1+a2-(1-*2 )% } (31) witha=VcL/b=R/bfrom (23). Forsmall values ofa,the binomial expansionofthesecond term yields theverysimple form cos=Y^OL, or=9413'. Asthepoint charge recedes, the angle defining theneutral zoneapproaches 90. The Electric Dipole.Iftwoequal chargesofopposite sign approach each other indefinitely without merging, then r\and r2 inFig.10-1 canbeclosely approximated bytheradius vector from thecenter $nd,with reference toFig.106, ^_ J; > (32)nr2r-1/2cos r+1/2cosOr2 Thus, thepotential function (10)becomes <i>=Q_ije =i (33) 4?r r* 4?rr ifonedefines p=Ql (34) asthedipole moment, avector pointing outofthepositive charge alongtheaxis ofthedipole. The field vector Efollows inthe spherical coordinate system from (33) directly as[Appendix 3, (38)] 6$ p2cos 1d* psin Unlike apoint charge, thedipole hasanalmost entirely local influence,itsfield lines concentrate between thetwocharges, and thefieldvectorEdecreases with thethirdpower ofthedistance. Figure 10-6 indicates alsothe field distribution, where the field lines aredefined by drrde ErEg which, upon integration, gives r=ksin26 (36) Sec. 10] TheElectric Dipole 93 where kisanarbitrary constantdefining anyindividual field line. Theequipotential surfaces are,from(33), (37) FIG. 10-6 Field Distribution ofanElectricDipole. Solid lines=field lines; dashed lines=equipotentiallines. Choosing asetofvalues for6,onethuscomputes thesetofr- values defining thelocus pointsforanyconstant value of<. Inauniform electric fieldE,thedipole doesnotexperience a resultant force; however, atorque T=pxE (38) willbeexerted, trying toalign thedipole moment pwith the 94 Fields ofSimple Geometries [Ch.4 electric field linethrough itscenter. Thispropertyisutilized in thevisualization ofelectric field linesbymeans ofsmall crystallic needles (seesection 16). Inanon-uniformfield, there willalso bearesultant force action ofvalue F=(p-V)E (39) since onlythevectorial difference ofthefieldvectorEattheends ofthedipole cancontribute. Good treatments oftheelectric dipole aregiven inHarnwell,A9 p.60;inJeans,A1 p.51; inMason andWeaver,A16 p.18;in Smythe,A22pp.6-10; and inStratton,A23 p.175. O \Qa FIG. 107Several Collinear Point Charges. Several Point Charges.Ifseveral point charges arearranged along aline, orcollinearly, thefield willbesymmetrical about the lineasaxis. The resultant potential *and field vector Eare readily found bysuperposition 47TE(40) Thegeneral evaluation offield lines isreadily possible inthesame manner asfortwopoint charges. Following (4)with thetwo componentsofEtaken from (40) assums (see also Fig. 10-7), onecandeduce arelation like (6)byagain collecting terms with thesame ra3 ,namely, (41) Sec. 10] Several Point Charges 95 Integration yields then ZQacosea=k(42)w theequation ofthefield lines. Thepractical useofthisequation is,however, ratherlimited, sothat field distributions areactually found bygraphical means; seeforexample AttwoodA2for(+Q, -MO, -MQ), Fig.1-26,andfor(+Q,-Q,+Q,-Q), Fig.1-28. Atlarge distances, thepotential function ofthese collinear point charges canbeexpressed interms ofthequantities r,6,thecoordi- nates ofthepointPwith respect tothearbitrary origin 0.One has, forlarge valuesr,approximately ra=rl-2cos8 + =rl-cos(? (43) sothat (40)becomes *=--Qa(1+-cosB\(44)47rsr(a) \ r / Ifonenowchooses theorigin sothatwith respect toitQa%a= () 0,i.e.,sothat itisidentical with thecenterofgravity ofthepoint charges, then thepotential canbesimplified to ^T^-EQa (45)47rer(a) Thus, atlarge distances, thepotential ofthecollinear charges can befound asthat ofasingle point charge equal tothesum total ofallcharges Qa,andlocated atthecenter ofgravity which can bedetermined intheconventional manner with respect toan arbitrary origin by (46) (a), where thezaaremeasured from theorigin withproper algebraic sign.IftheQa=0,thennocenter ofgravity exists; onecan, () however, findthetwocenters ofgravity, oneforallpositive charges andtheother for allnegative charges andshow that equal and oppositenointchargesLQ+=LQ~, located attherespective 96 Fields ofSimple Geometries [Ch.4 centers ofgravity, represent adipole which approximates the actual field atlarge distances. Foranyarbitrary complex ofpoint charges, (40)will stillhold, butnogeneral solution offield linescanbegiven. Anexcellent graphofthree charges (+15, -12,+20)isfound inplateIVof FIG. 10-8 Typical Geometric Relations forSpace ArrangementofPoint Charges. Maxwell.A17Atlarge distances from thecomplex, thepotential in(40)canagain beexpressed most simplyinterms ofthecenter ofgravity ofthepoint charges,sincefrom Fig.10-8 (47) ra=r1-2 cos9a+[ andthus=rl-cos However, a,astheangle between twovectors from theorigin, follows thecompositionoftherespectivedirection cosines,4so that axaxyayzaz cosea=--- 1---- 1--- rQarr0ar zQar 4Seeforexample: Eshbach, Handbook ofEngineering Fundamentals, p. 2-71; John Wiley,NewYork, 1936. Sec. 11] Quasi Point Charges 97 Choosing thereference point such thatwith reference toit = then oneobtains (45)again with thesame interpretation. The center ofgravity with respect toanarbitrarily chosen origin has thenthecoordinates5* f~ EG.*" 11-QUASI POINT CHARGES The concentration ofthe electric charge inamathematical pointisamatter ofcomputational convenience; nothing is changedtfsfarastheoutside field isconcerned ifoneassumes a small, finite radius aofthecharge andadistribution either uni- formly overthesurface withthedensityo-=Q/lira2 ,oruniformly overthevolume, with thedensity p=3Q/47ra3 .Ineither case, thepotential onthesurface ofthesphere hasnowassociated with itadefinite value obtained from (10-2) byputtingr=a *.= The ratio ofthecharge toitspotential canbedefined asthe capacitanceofthesphere, C=Q-=4irea (2) *8 and isdirectly proportional totheradius ofthesphere. Since the potential vanishes asr oo 7onecanalsoconsider thisascapaci- tance toaninfinitely large sphere, inthesense ofanideal con- denser. Formany practical purposes onecantreat widely separated charges aspoint chargesinorder toobtain theoverall field dis- tribution inmathematically simple formbydirect superposition. Yet, close tothecharges onecanavoid theexcessive values of potential and field strength, normally associated withtheconcept ofthepoint charge, bydefining quasi point charges, i.e.,small but finite charged spheres. 6SeeAttwood,A2pp.42-43. 98 Fields ofSimple Geometries [Ch.4 TwoQuasi Point Charges. Thepotential anywhere inspace surrounding twosmall spheres with Qi=Qz=Qandwith ai^2c,a2<C2c(seeFig.111)isapproximately that oftwopoint charges located atthecenters ofthespheres. Thus, inaccordance with (10-10), r\/^ i\ (3) FIQ. 11-1TwoSmall Spheres asQuasi Point Charges. Onthesurfaces ofthetwospheres, onehas,respectively, r==2 cr $=-(---\ri c,r2-a2, 2~^^ ^J sothatthepotential difference becomes F,2=Q/I 11\ $2= I 1 I 47TE\ai a2c/ Onecanthus define thecapacitance between these spheres as Q 47TE=47TE- a2(4) (5) (6) (7) Forequal radii, thecapacitance reduces toonehalfthat ofa single sphere, sothatonecaninterpret thisasaseries combination Sec. 11] TwoQuasi Point Charges 99 ofthecapacitance ofsphere1totheinfinite sphere andfrom there tosphere2. Though onewould construct thefieldpicture exactly inaccord- ance with section 10fortwopoint charges, theassumption of finite radiipermits theevaluation ofcapacitance coefficients which would bemanifestly impossible forpoint charges. Inorder to determine theerror of(4)and(5)onemight observe that this system canbedescribed interms ofMaxwell's potential coefficients givenin(3-12), namely, *i=SnQi+S12Q2 (8)*2=Said+S22Q2 from where, bycomparison with (4)and(5), S12=82i=r- S22=--(9) 4irea2 Forexample, thecontribution ofchargeQ2topotential $1actually varies fromamaximum atpoint PI'toaminimum atpointP\' (seeFig.111).Thus, actually, <4xeS 12< (10)2c+ax> 2c constitute thelimits ofvariation. Forequalandopposite charges, and forvalues (ai/c) ^0.2,one finds aresultant maximum potentialvariation of+1percent to 1.1percent referred to themedian potential value computed with512=l/47re2c. The approximations are,therefore, quite satisfactory aslong as2cis larger than tentimes theradius ofthelarger sphere. Fortheevaluation ofthecharge distribution onthesphere a\ t onecanalsousetherelations from section 10onsphere andpoint charge. Thus, acharge Q2,taken asapoint charge, induces on thegrounded sphere aiacharge Q2'=-|Q2=-iQ3 (ID where thenotation of(10-25) hasbeen translated intotheappro- priate oneindicated byFig. 11-1. The distribution over the sphere c^causes acharge density given by(10-27) with &replaced LOO Fields ofSimple Geometries [Ch.4 -jy (TT__0)because theinducing point chargeistotheright; igain translating, thisbecomes infirstapproximation -2*!cos 3icos nrhere ai=ai/2c, andwhere thebinomial expansion wasinvoked. Since thesphere a!isnotgrounded, butrather carries atotal charge Qi,onehastolocate atitscenter another charge Qi"=Qi~Q*=Qi+iQ2 which isuniformly distributed over thesphere's surface with density a/'=Qi"'/4irai2 .The total charge density becomes, bhen, Vcos6 (13) Theintegral overthetotal sphere gives Qi,asrequired; thesecond term isthenon-uniformity caused bytheproximityofQ2.The effect ofQ2depends upon(i2 );forQi=Q2and <*i=0.1 (the limit oftheapproximate treatment), thecharge density has maximum deviations of3percentfrom uniformity. Togetthecharge distribution onsphere o2,onewould reverse theprocess andobtain S*i. S\ 5a22cos6 (14) where 6isalways counted counterclockwise from thepositivez- axis. Obviously, the field distribution between twosmall spheres as inFig.11-1 issymmetrical about thei/-z-plane. One can,there- fore,usethelower halfofthearrangementtosimulate halfspheres inconductive ground (orelectrolyte) andevaluate theresistance between them asasimple grounding problem. Usingrelation (8-11), onehasatoncefrom (7)forthehalfspace yC/2 Thestream lines oftheelectric current intheearth areidentical Sec. 11]Conducting Planes andQuasi Point Charges 101 with theelectrostatic fieldlines, andthecurrent densities onthe spheres canbeevaluated from thecharge densities (13)and (14). Conducting Planes andQuasi Point Charges. Thesame approximate treatment isapplicable toasingle small spherical charge ofradius aatadistance h>5afrom aperfectly conducting plane. Bytheprincipleofimages (seesection 10andFig.10-2) onecanreplace theeffect oftheplanebyalikesphere atdistance 2/iwith opposite charge. With ai=a2=a,(7)willgive the capacitanceoftwoequal spheres; observing that thepotential difference between thetwospheres must bedoubled tomaintain thesame potential value ontheplane, oneobtains C=Trea (16) forthecapacitance between sphere andplane. The distribution oftheresulting fieldandoftheinduced charge density intheplane canbeobtained inthesamemanner asforthepoint charge and plane (seesection 10); themechanical force canbefound by (10-18). Thecharge distribution onthesphere itself isobtained from (14)withQi=Q2=Qandomission ofallindices, as (r=-^-2 (1-3a2cos6) (17) where a=a/2h inappropriate modification. This treatment can, ofcourse, beextended toallcases where thesolution forpoint charges haspreviously been giveninsec- tion 10. Fortwoquasi point charges opposite aconducting plane, as shown inFig. 11-2, onecanreadily substitute theappropriate images andfindtheresultant potentials as __LV +J_(!' rJ2/ 47re\a 2 2/i2/ ?rei2 rJ2/ 7re\a 2i2 where ri2=l(2c)2+(h2-AO2 ]*,r12'=[(2c)2+(h2+hrf]* (20) The factors toQiandQ2canbeidentified by(312)oralso (8) above astheMaxwell potential coefficients Sapforthetwoquasi point chargesinthepresence oftheconducting plane.Ifnow 102 Fields ofSimple Geometries [Ch.4 Qi=Qz=+Q, sothatthetwocharges represent acondenser arrangement inthepresence ofground, then their capacitance can easily becomputed from Comparison ofthisformwith (6)readily indicates thatthesecond FIG. 11-2TwoQuasi Point Charges above Ground. parenthesis stands fortheinfluence ofground upon thecapacitance ofthetwo spheres. Again, theinduced charge distribution on theconducting plane canbefound bytreating +Qand Qas actual point charges; similarly, mutual force actions canbe evaluated. Inorder toobtain thetotal charge distribution ononesphere, sayd2,onecanusethesuperposition onebyoneoftheeffects of each oftheother point charges. Thus, theeffect oftheactual charge+Q isagain givenbythesecond term in(14)withQi=Q, butwith 0'counted from thecenter lineasindicated inFig. 11-2. Fortheeffect oftheimage of-\-Qonewould count 0"from the diagonal center lineasindicated inFig. 11-2, and, finally, forthe Sec. 11] Sphere andQuasi Point Charge 103 image ofQoneintroduces 0'7/ .The resultant chargedistri- bution becomes, therefore, Q (72= (22) Sphere andQuasi Point Charge. Inavery similar manner, theresults ofthesection onpoint charge andsphere canbemodi- fiedtoallow forafinite, though small, radiusiofthequasi point charge andthuspermitdefinition ofcapacitance coefficients. Thus,forthegrounded sphere ofarbitrary radiusRinFig.10-4, onecanreplace theeffect onthesphere ofapoint charge ofvalue Q2=(R/b)Qi andlocated atadistance 2c=b(R2 /b)from Qi.Thepotential onaimustnowbe *i= ( ~r^~ 47TE\a 162c sothatthecapacitanceofthesmall sphere aiinthepresence of thegrounded sphere becomes c=?i= R/b b1-(fl/6)2 T n. R/h ~l (23) Since thelimitation a\/(b R)^0.1seems appropriate (alittle more severe than fortwopoint charges),itappears thattheincrease incapacitanceislimited tolessthan 5percent. Thecharge dis- tribution onthesmall sphere a\canbefound from (13)byreplacing Q2and2caccording tothedefinitions above. Inthegeneral case ofanyarbitrary chargeQaonthelarge sphere, thegeneral potential distribution isgiven by(10-29). Onecan readily deduce thepotential value onthesmall sphere ai,bylet- ting TI=ai,r2=2c,r=b,andalsoQ1=Qa.This leads after ordering to ,.-LFI(I'YU+J-ift (24, 47reLi o\2c O/J 4ire 104 Fields ofSimple Geometries [Ch.4 whereas thepotentialofthelarge sphereisdirectly from (10-30) 4-Treb 47TER Again, (24)and (25) define Maxwell's potential coefficients in accordance with (8) ;from these, onecan, ofcourse, compute the induction andcapacitance coefficients inaccordance with (3-13) and(3-11). Ifthetwocharges areequal and opposite, Qi=Qs=Q, then thespheres form acondenser ofdirect capacitance C=Q 47T aiR l-(R/b)(26) which forsmall values (R/b) atonceapproaches (6),thecapaci- tance oftwosmall spheres withthe appropriate changes innotation. Several Collinear Quasi Point Charges.Ifseveral small spheres ofradius aarearranged collin- early withequal spacings d>10a, the potential distribution can readily befound asthat ofpoint charges atthe centers ofthe spheres (sec Fig. 11-3). Forn spheres there arencharges andn potential values, sothat atotal ofnquantities must beprescribed topermit evaluation oftheother nunknowns. One practical case isobtained byassuming the firstsphere grounded, $1=0,thelastonecarrying thetotal voltage<f>n=V, andthe(n 2)spheres inbetween with floating potentials, i.e., insulated sothatQ2=Qa='''Qni=0.Thecharges Qiand Qnarethen related tothegiven potential values bythesimple formsFIG.11-3 Several Collinear Quasi Point Charges. *i=+SlnQn=(27) +SnnQn=V (28) Sec. 11] Several Collinear Quasi Point Charges 105 sincenoother chargesexist. Inamanner similar totwoquasi point charges, thepotential coefficients are 1 1 11=Snn Thecharges become,ifoneobserves (SuS nn) (n-(29) (30) \ \(b) -A \Qn= 1.0 0.9 0.8 0.7 0.6 *1^0.5 0.4 0.3 0.2 0.1 FIG.11-4 Distribution ofInduced Potentials over SixInsulated Collinear Quasi Point Charges: (a)oneendgrounded, (6)symmetrical distribution. andthefloating potentials forallvalues r=2,3, ,(n 1).Thedistribution foraseries ofn=6spheresisshown asline (a)inFig.11-4 fora/d=0.1; obviously, there islittle difference from thefreepotential distribu- tion ofasingle point chargelocated atthecenter ofQneventhough theactual field picture would bevery complex. F.Ollendorff1 hasused thissimplifiedfield picture asamodel toapproximate the field distribution about achain ofhigh-voltage suspension insulators, assuming themetal cap oftheuppermost insulator (next tothecrossarm) tohaveground potential, andthemetal 1F.Ollendorff, Arch.f. Elektrot., 16,p.261(1927) ;17,pp.79and242(1927). 106 Fields ofSimple Geometries [Ch.4 suspenderofthelowest one(connected totheconductor) tohave linepotential. Healsocomputed thepotential distribution over thesurface ofaninsulator, which checked satisfactorily with measured values. Ifoneassumes thepotentials $1= <=V,andagain insulated spheresinbetween withQ2=Qa='''Qn-i=0,then (27)and (28) willleadto V Ql=Qn=o.o^11TOin andfortheinduced potentials onefinds inaccordance with (31) This potential distribution over thespheresisshown asline(6) inFig.11-4 fora/d=0.1; asexpected,itisasymmetrical dis- tribution, again dropping sharply between theoutermost members ofthechain. Thisemphasizes thatthegreatest electrical stresses occur intheimmediate neighborhood ofthehigh-voltage terminal andthatonly controlled potential surfaces (rather than floating ones) canbringrelief. 12LINECHARGES ANDQUASI LINECHARGES Forknown distributions ofcharge along simple geometrical lines, itispossible toevaluate thepotential distribution bythedirect integration 1Xdsrf-47Tt/ T where Xisthelinecharge density, dsthelineclement, andrthe distance between thecharge element andthepoint ofobservation P.Expression (1) is,ofcourse, thelimit ofthesum (1-9)ofthe point charges (Xds). Forpractical applicationsitisdisconcerting thatthepotential and field strength values atthechargedlinebecome infinitely high. One can,however, frequently approximate agiven con- ductor geometry byquasilinecharges; thatis,onecancompute thegeneralfield distribution interms ofcharged lines, butthen select anappropriate equip otential surface close tothecharged Sec. 12] Finite Straight Line;RodElectrode 107 lineasagood representation oftheactual given geometry. This provides forfinite field values ontheequipotentialsurface and permits evaluation ofcapacitance coefficients andcharge densities. Finite Straight Line; Rod Electrode. The finite straight line inFig. 12-1may carry auniform charge distribution of \ FIG. 121Finite Straight LinewithUniform Charge Density. density Q/2c. beexpressedBecause ofaxialsymmetry, theintegral (1)can 1Qrt=+c 47TE 2cJl;=-c -f)2+p2 Q,(x+c)+ In (x-c)+r2(2) with thenotations from Fig. 12-1. The equipotential surfaces areconfocal rotational (prolate) ellipsoids, with thefixed fociFI andF2attheends ofthecharged line,which itself isadegenerate ellipsoidofvanishing minor axis. Forlarger distances, major and minor axes2aand26become nearly equal; theequipotential sur- faces approach spheres. The field lines aregiven bytheorthog- onalsystem ofconfocal hyperbolae. Details onthesimpler com- putations, aswell asrelated applications, arefound inAbraham 108 Fields ofSimple Geometries [Ch.4 andBecker,A1 p.62; inAttwood,A2pp.81-84;inBennett and Crothers,A3 p.192;inBreisig,A4 p.77. Sinceanyequipotential surface canbetaken asanewconductor surface, (2)gives alsothepotential distributionsurrounding a prolate ellipsoidal surface with total charge Qatconstant potential 3>fl.Ifonehasgiven themajor andminor axes 2a,2b,then c=Vo2b2 jchoosing pointPonthesurface oftheellipsoid, sayatP",thenx=0,TI=r2=a,and 47TS2ca c sothat thecapacitance follows, using thehyperbolic function instead ofthelogarithm, Itwillalways belessthan thecapacitance ofthesphere with diameter equal tothemajor axis. The fieldvector canbefound byuseof(2),andwithsimplifications thisyields Ex=-=+-(sina2-sinai)dx 4irs2c p d4> Q 1Ep=-=+-(COSai-COSa2) (6) dp 4ire2c p Thoughitisnotpossible todevelop asimple general expression forthecharge density ontheellipsoid here (seesection 31for that), themaximum andminimum values canreadily begiven. AtP',with ri=a+c,r2=ac,oneobtains from (5) tfmax=tEx .,2 (7) whereas atP",with p=b,cosai=c/a, cosaa=c/a,it follows from (6) (8) sothat theratio ofmaximum tominimum charge densityis Sec. 12] Finite Straight Line;RodElectrode 109 exactly theratio ofmajor tominor axis, a/6. This ratio also holds forthepertinentfield gradients, sothatthemaximum dielec- tricstressmust beexpected attheapex ofthemajor axis. Com- bining (7)with(4),onededuces c/b max 6tanh"1 (c/a)6 where thefactorFisafunction ofonly6/a=(9) (10) Figure 12-2 gives agraph of\/F\itindicates that, astheratio decreases, thegradient #maxincreases very rapidly indeed. 1.0 0.8 0.6 H*. 0.4 0.2 0.1 0.2 0.4 0.6 0.8 1.0 13=1 FIG.12-2 Factor l/F fortheMaximum Field Gradient oftheEllipsoid in Fig.12-1. Foralarge ratio a/6,theellipsoid canbemade toapproximate theshapeofacylindricalrod. Sincenoreasonable solution ofthe potential problem forafinitely long cylinder ofnon-vanishing diameter dandlengthIisknown,ithasbecome customary to substitute theellipsoidalrod. Forratios d/l<0.1,andchoosing 2a= I,26=d,theapproximationisvalid c= 110 Fields ofSimple Geometries [Ch.4 Introducingthisinto (3)gives sothatthecapacitanceoftherod-like antenna becomes c' This isdesignatedC'because other choices ofequivalent param- eters arepossible.Instead ofinscribing theellipsoid, giving the smallest equivalent, onecould justcircumscribe therodwithit, givingthelargest equivalent. Forfixedfoci,onemusthave (ffl")-(b)>=(a')2-(i/)2= - Inaddition, theellipse must passthrough x=1/2=a!,p=d/2 =b',so that innormal form Thecombination shows a"=aV2,b"=b'\/2, sothat the logarithmic term in(12)willnotbeinfluenced; however, For practical purposesitmight bemost advisable toselect an average value between (12)and (13),such as(1+V2)/2=1.2, and Thesame solution canalsobeapplied toallstationary flow problems, such ascurrent flowfrom avertical grounding rod, as inFig.12-3a,intotheground,orfrom ahorizontal tube lying on Sec. 12] Finite Straight LineaboveGround 111 thesurface ofground,1asinFig.12-36. Forthevertical rodone must take itslength within ground as1/2inorder tohave thesur- face ofground asplaneofsymmetry. Inboth cases, thevertical aswell asthehorizontal rod,onlyonehalf ofthetotal spaceis FIG.12-3 Thin EllipsoidsasGrounding Rods: (a)vertical, (b)horizontal, arrangement. occupied bythecurrent flow; theresistance istherefore twice thevalue obtained from thecapacitance expression (12),namely, (15) Though theoverall resistance isthesame, thecurrent distribution is,ofcourse, quite different, and, inparticular, thefieldgradient along thesurface ofthegroundismuch higher forthehorizontal rod. Finite Straight Line above Ground. The proximity of groundforavertical, uniformly charged line oflength 2ccanbe taken intoaccount bytheimage linebelow ground. With the same axialsymmetry asinFig.121,theintegration (2)overboth chargedlines (actual andimage) leads, with thedesignations in Fig.12-4a,to J_Q 'c)+ (x-h-c)+ra' -In(x+h+c)+ (x+h-c)+(16) Theequipotential surfaces have asort ofovalshape andinclude, ofcourse, theplane x=0.Forathinrodofmean diameter d 1Ollendorff,Al8 p.96; foragoodsummary ofgrounding problemsseeR. W.Ryder,Jl.I.E.E., 96,part III, p.175(1948); alsoR.Rudenberg, Electr. Engg., 64,p.1(1945). 112 Fields ofSimple Geometries [Ch.4 andlengthI^d,onecanfindtheapproximate capacitance, as influenced byground, byevaluating (16)atx= /i;thisyields in theneighborhoodofthecharged linewith psmall inallradius vectors Specifically,for2p=d,thediameter oftherod,thevalue of <i>s results, which isthepotential onthesurface oftherod. Putting p=oin(16),onecanthen solve thatrelation withthesame poten- tialvalue forx2and x\,theintersection oftheequipotential surface with the aj-axis; thedifference x2xi=Imust bethelengthof therod.Aslongas(d/l) <C1,oneobtains ingoodapproximation (17)2H+l/2 asthecapacitanceofafinite rodlocated perpendicular toacon- ducting plane. Thisbecomes forH *<*> torforasingle rodby itself, identical with (12); forH >0,onededuces (18) thecapacitanceofavertical rodofpotential3>8directly onthe surface oftheconducting plane ofpotential *=0,which gives thelargest possible value. Thecapacitance between thetworods isonehalfthevalue resulting from (17), because thepotential difference is(2<t a);onecould alsoconsider thetwo halfspaces connected inseries. Thecharge density induced intheplane*=canbeobtained from (5)with thepotential solution (16). Along x=0,onlythe component Exexists, and, asseenfrom Fig.12-4a, TI=r2"= [H2+p2 ]*,n"=r2'=[(H+O2+P2 ]M ,sothat ff=#z/z=0 i i n (19) Sec. 12] Finite Straight LineaboveGround 113 ifone alsouses 2c=I.Themaximum value obtains directly under therodatp=0, -Q'- w whereas forthesingle point charge above ground, (10-17) gave (pJThe variation with distance pisquite similar in both cases; thepoint charge, however, hasaslightly stronger local P(*,P) FIG.12-4 Finite Rod above Ground: (a)vertical, (6)horizontal, arrangement. effect, i.e.,theinduced charge densityisslightly higher closeto, andslightly lower farfrom, thepoint charge, than isthecase for thecharged finite rod. Thesame solution canbeapplied toallflowproblems, thermal, hydrodynamical, orelectrical, byuseofthegeneral table 9-1. Forexample, the electrical resistance between two cylindrical electrodes offinite lengthisgivenby a where thefactor 2accounts forthepotential difference (2<> s) between therods. 114 Fields ofSimple Geometries [Ch.4 Instead ofauniform charge distribution, onecanalsoassume some arbitrary function /()with -r2cJ-c sothat thetotal charge remains asQ.Theevaluation ofthe resulting integral canbemade simple with theproper choice of Forahorizontal, uniformly charged straight lineabove ground, asinFig. 12-46, noaxialsymmetry will exist. Thetreatment withtheimage linebelow ground canbecarried through insimilar manner asbefore2and willyield 1Q!",_[(*+c)2+(y-h)2+z*\A+(x+c) K\9 i/ i i\9 i 9-iI/I i/ \x c)+(y+hy+z*Y+(x c)_ which canalsobewritten interms oftheradius vectors r/,r/', r%' ,andr^ ',except thatnoAVthecoordinate zofthepointPmust beincluded. Theequipotential surfaces close tothecharged lines areslightly flattened ellipsoids which canagain readily betaken toapproximate afinite cylindrical rodoflengthIanddiameter d asindicated inFig.12-46. Evaluating (22)atx=z=andin theneighborhood ofthechargedlinewithy=hp,where pis small, oneobtains 1 (c2- Specifically, for2p=d,thediameter oftherod,thevalue$s results, which isthepotential onthesurface oftherod.Putting y=h,z= in(22), onecansolve itwith thesame potential value*Bforthevalue x;thiscorresponds totheintersection of theequipotential surface withtheliney=handdefines thelength 1/2.Aslong as(d/l)<^1,oneobtains ingood approximation (23)* * 2Foradifferent approximation seeF.L.ReQua, Trans. A.I.E.E., 64,p. 724(1945). Sec. 12] VeryLong Straight Line 115 asthecapacitance ofafinite rodparallel toaconducting plane. Thecapacitance between thetworodsthemselves willagain be onehalfthevalue given by(23)because thetotal potential dif- ference is(2$a). Foraverylongrodabove ground such that(4/i/Z):1,expres- sion (23) simplifies to -$-*$ '"7 Inthiscase(and thiscaseonly),itispossible todefine acapacitance perunit length ofthecharged lineorrodtoground '"7 which isindependent ofthelength oftherod. Thismeans that endeffects become anegligible part oftheelectrical fieldconfigura- tion, sothat forpractical purposes theimportant region ofthe fieldbetween thetwocharged lines, orbetween lineandground, canbeconsidered two-dimensional, depending onlyonthecross- sectional dimensions ofthesystem andnotonitslength (see further below). Asintheother cases above, thissolution canagain beapplied toallflowproblems. Forexample, onecanconsider theplane x=asthesurface ofanelectrolyte extending tox>intowhich twoelectrodes arcimmersed, formed bytheright halves oftherod and its"image." Theresistance between these electrodes canthen becomputed from (23)as where thefactor 4accounts forcurrent flowfrom onlyonehalf thetotal ellipsoidal surface (forx>0)andforthepotential dif- ference (2$ a)between therods. VeryLong Straight Line. Ifone lets c oin(2),he should obtain thepotential function ofavery long straight line. Obviously, unless onedefines Q/2c=Xasafinite charge perunit length, onecould notattach much sense toc*oo ;conversely, 116 Fields ofSimple Geometries [Ch.4 Q=2cXwill itselfbecome infinite with c,sothatatruly infinitely long linerepresentsdifficulties ofrealization.3 Assuming axialsymmetry andindependence with respect to coordinate x(end effects aredisregarded because ofthegreat length), then onecanmore readily deduce thedielectric flux densityDfrom theapplication ofGauss's theorem (1-11) toa concentric cylindrical surface ofradius pandunitlength Dp=(27) since theflux lines areradial. Thepotential canthenbefound bydirect integration f*a r\U \ /.\ (28) This value isindependentofthepath ofintegration, andpxisa conveniently chosen reference point atwhich oneassumes *=0. Such acompromiseiscustomary, since thelogarithmic potential function becomes infinite atboth limits p=andp=oo.The equipotentialsurfaces aretheconcentric cylinders p=cons; any oneofthese could bechosen asaconductor surface andbeassigned avalue$which could beadded onin(28). However, forthe single conductor itisnotpossible todefine acapacitance value because ofthelogarithmic nature ofthepotential variation. Two Parallel VeryLong Straight Lines. The resultant potentialfunction oftwoparallellines issimply thesuperposition \ /\ A /^ *li Ia\\ ,^2 , Ia2 <f>= Inl 1+ In 27TEVi/ 27TE where \iandX2arethelinear charge densities, and a\anda2 arbitrary constants corresponding toPIin(28). Forequal and opposite charge densities, Xi=X2=X,theexpression canbe simplified to ^X ,(^ where riand r2arethedistances from thecharged lines asindicated inFig.125.Thearbitrary constant $serves toadjust absolute potentialvalues when desired. The equipotential surfaces are 3SeeAttwood/2 p.76;Kellogg,010p.62;andSmythe,A22 p.62. Sec. 12] Cylinder andParallel Straight Line 117 given byr2/ri=k1 ':they arcthefamily ofexcentric cylinders with their axesMparallel tothelinecharges inthez-z-plane in Fig. 12-5; the field lines aregiven by 2<i=k":they are theorthogonal family ofcircles passing through thelinecharges andhave their centersNalong the7/-z-plane inFig.12-5. Details ofthecomputations andgraphical fieldpictures aregiveninmany books, such asAttwood,A2 pp.85-88; Bennett andCrothers,A3 p.140; Jeans,A1 p.195; Kupfmuller,A14pp.70-76; Mason and f=Constant FIG. 12-5Two Parallel VeryLong Straight Lines withEqual andOpposite Charges. Weaver,A1G p.136;Ramsay,A21 pp.41,140; Bewley,Dl p.43; andpracticallyallreferences listed inAppendix 4,B,aand4,B,b. Cylinder andParallel Straight Line. Since theequipoten- tialsurfaces oftwo parallel straight lines arecircularcylinders, (29)must alsodescribe thepotential distribution between afinite cylinder ofradiusR2inFig.12-5andalinecharge (+X). The cylinderwillcarry thetotal charge (X)perunitlength and will have apotential defined bythespecial values ofr2and TIalong itssurface. Thus, forthepoint P', n'=c+(m2-B2),r2=c-(m2-R2) (30) wherem2isthedistance oftheaxis ofthecylinder from theori- gin0.From thetriangleOMT inFig.12-5, onealsotakes R22=m22c2=(m2+c)(m 2 c)=ba(31) arelationship which permits interpretation ofthelinecharge (X) 118 Fields ofSimple Geometries [Ch.4 astheimage ofthelinecharge (+X) with respect tothecylinder R2.Therelation (31) defines thelocation of(X)whenR2and 6aregiven, or,conversely, locates thecenter ofR2withrespect to thetwoequal andoppositelinecharges. With (30)and (31), the general form (29)yieldsnow $0canbesochosen that,ifthecylinderisgrounded,itspotential value becomes zero. The electric fieldvector canmost readily becomputedintheCar- tesian coordinate system chosen inFig. 12-5. Using thegeneral form (29) withn=[(x+c)2+ i,2 ]*,r2=[(x-c)2+y8 ]*, onefinds X(x+cx----/nn ^03) 2 2dx 27TE\ri r2 Themaximum field strengthwillcertainly exist along thex-axis between thecylinder R2andthelinecharge (+X). Because of y=o,onlyExwillexist there and, inaccordance withtheassumed charges,willpoint inthepositive ^-direction EXA(_1___ L.\=Ac 27T\C X C+X/7TSC2 Onthelinecharge, where x >(c),thefieldstrengthwillapproach infinite value asexpected; onthecylinder, themaximum value willbe where a.=R2/b.Fora >0,thisbecomes consistent with (27). Bymeans ofatransformation ofcoordinates from(z,y)to (r,0),withtheaxis ofthecylinder R2ascenter, x=m2+rcos6, y=rsin Sec. 12] Two Parallel Cylinders 119 onecanevaluate Er=-(d$/dr) from (29)'and obtain theinduced surface charge density onthecylinder (negative, because ETis directed towards thesurface ofthecylinder), i++coB(36) Themaximum exists atP'for6= TT,theminimum atP"for6= 0,andtheir ratio is b-R Itissignificant tocompare this result with that forpoint charge andsphere insection 10andtoobserve thelarger inhomogeneity inthelatter case. Ifthecylinder R2isinsulated, then itcannot acquire any resultant charge. Theplacement ofalinecharge (+X) intothe axis ofthecylinder contributes aconstant potential onitssurface aswellasanadditional uniform charge density a'=A/27r# 2,but reduces thetotal charge tozero. The resultant charge density willbenegative closest totheinducing linecharge (+X), and positive ontheopposite side. The neutral zone exists where (o-)=a',which gives with (36)thevalue cos= a.For a=0.1,onefinds =9544/; theneutral zonemoves rapidly tolarger angles6asaincreases. Two Parallel Cylinders withEqual andOpposite Charges. Selecting anytwocylinders from thefamily oftheequipotential surfaces, theycanatoncebeconsidered ascarrying opposite and equal charges andasforming acondenser. Ifonecylinder is outside theother asinFig. 12-Ga,andRI,R2jDarethegiven parameters, onemust first locate theequivalent line charges. From thetriangles OP^Mi andOM 2P2onetakes therelations ,m^=c2+Ri2 ,m22=c2+R22 (38) sothat mi2mz2=(mi+m2)(mim2)=Ri2R22 Defining Ri/D= rji,R2/D= r)2,andobserving (mi+m2)=D, onereadily finds mi=[1+fa8- r,22 )],m3=[1-On2-,a2 )](39) 120 Fields ofSimple Geometries [Ch.4 sothattheorigin canbelocated. Combining now (38)and(39), onealsoobtains 2c=D(l-2(1,!"+,22 )+(m2- ,,22 )2]* (40) Thepotential values onthetwocylinders canbecomputed from O\R FIG.12-6Two Parallel Cylinders: (a)oneoutside theother, (b)oneinside theother. thegeneral form (29) inthesamemanner asindicated for(32) whereby forthepoint PI'oncylinder Ri (ri')i=c(mi #1), (r2)i=c+(mi RI) whereas forthepointP2oncylinder R2 (fi')2=c+(m2-R2), (r2')2=c-(m2-R2) Thepotential onR\ispositive, thatonR2negative, sothatthe capacitance perunitlength becomes4 (41) Rarcosh-^ +cosh-1 ^-2 ]L ^1 ^2J Thecharge distribution oneach cylinderisgivenbytheproperly modified form (36), using forthecylinder R2thevalue a2= R2/b2=R2/(c+ra2),and forthecylinder #1thevalue ai= 4A.E.Kennelly, Proc.Am.Phil. Soc., 48,p.142(1909); also Electr. World, 66,p.1000 (1910); C.L.Dawes, Phys,, 4,p.81(1933); alsomany oftheref- erences inAppendix 4,A;andSchwaiger,B17 p.68. Sec. 12] Two Parallel Cylinders 121 Ifthecylinders have equal radii, RI=R2=R,then (39) indi- catesmi=ra2=D/2, thecapacitance takes themuch simpler form C,, cosh'- andcharge and field distributions areperfectly symmetrical. Finally,ifone ofthecylinders encloses theother, asinFig. 12-66,relations (38)are stillvalid, butnow(w/ ?w2)=D,so that mi1=|[(V2-WO+1], >2=I[(V2- T722 )- 1] (43) This locates theorigin totheleftofthecylinders; thevalue for cremains thesame asin(40)andlocates thelinecharge (X). The potentials onthecylinders areevaluated asabove for(41). Selecting PI'andP2'asindicated inFig. 12-66 leads tothe capacitance perunitlength Ci= =- _(44) Rl' R*cosh"1^-cosh-1^L KZ K\J which increases beyondalllimits asR\approaches R2. YOTfinite butsmall radii ofthetwowires inFig.12-6awith the respective potentials $tand*2,onecanapply (29)tothesurfaces ofthewires with theapproximations X /> ri=RI,r2=2c=D, $1= In+$ 2?r /LI _o,~n -R *_\^ sothatthecapacitance perunitlength follows atonce ~ X ire(45) (46) , In Actually, ofcourse, thepotential contribution ofconductor 2over thesurface ofconductor 1isnotquite constant; theapproxima- 122 Fields ofSimple Geometries [Ch.4 tions in(45)for<t>iand ^>2areaccurate tobetter than 1percent ifD/R^10,which isgenerally trueforaerial transmission systems ofparallel wires. Thesame result isobtained from(41),ifone letsTJIandrj2in(39)and (40)become very small. Thecharge distribution can stillbeevaluated from (36);ifa2<^l, the simpler expression maybeused foreither RIorR2. Conducting Plane andParallel Straight LineorCylinder. The effect oftheconducting planeupon thefield distribution ofa single very long straightlineofcharge density (+A) canagain be replaced bythat oftheimage ofthestraight line,sothattheproblem reduces tothecase oftwo parallel very long straight lines (see references, p.116)andthepotentialisgiven by(29),where< isnow thepotentialoftheplane.6Thecharge induced onthe plane canbefound from (33)ifoneuses thenotation ofFig. 12-5, as X 2C /oxa=tEx=- 2 2 (48) 27TE<r+yz Thenegative signarises from thefactthat the field strengthis directed towards theplane. Forathin wire, therelations (45)and(46)arevalid withRI= R2.Thecapacitance perunitlengthofthewirewith respect to theconducting plane (forexample, ground) becomes from (46) C.=2(49) where thefactor 2accounts foronehalf ofthepotential difference ($!_$2)between wireandplane. Forthecharge distribution onthewireonecanuse(47)ifD/R^10.Foracylinder oflarger radius, theprecise form (42)must beused forthecapacitance per unit length, again inserting thefactor 2asin(49). Obviously, thecharge distribution ontheconducting planewillalways be BForaninteresting application toheat flowproblemsinconnection with the"heatpump"seeCh.H.Coogan, Paper No. 3,Engg. Exp. Station, Univ. ofConn., June 1948. Sec. 12] Dipole Line 123 givenby(48)since theequivalent linecharge and itsimage donot change. Ifthecharged line isparallel totheedge oftwointersecting conducting planes, thesame considerations apply asintheanalo- gous case forpoint charges treated insection 10;thenumber of necessary imagesisalways equal to(2n 1)iftheplanes intersect atananglew/nandnisaninteger. P(r,0) FIG. 12-7 Dipole Line (Small butFinite Spacing). Dipole Line. Ifthedistance 2c=Ibetween thetwocharged linesbecomes infinitesimally small, then thepotential function- (29)canbeapproximated (seeFig.12-7), withr!=r(1/2) cos0, r2=r+(i/2) cos0,by (50)r 27TS r or,defining thedipolemoment ofthetwo lines asavector, p=XI(51) 124 Fields ofSimple Geometries [Ch.4 along theaxis ofthedipole andoutofthepositive charged line, also -r (52) Theequipotential surfaces arecylinders with their axes parallel tothedipoleline inthea^z-plane and allpassing between the chargedlines (see Fig. 12-7). The field vector Eisgiven [Appendix 3,(38)]by d* x l Id* \ I. sothatthefield lines arefound byintegrating drEr-= -=coterddEe astheorthogonal family ofcircles sin=krwith centers along the?/-z-plane and allpassing through thez-axis. System ofParallel ThinWires above Ground. Inasystem ofnparallel thinwires above ground asinFig.12-8, onecanwrite thegeneral relationship inaccordance with (3-12) *a=SaiXl+Saz\2+----hSan\n (54) where $aisthepotential oftheathwire, Xi,X2, ,Anarethe individual linecharge densities, and saarethemutual potential coefficients which include the effect oftheimage. When the mutual distances arealllargecompared with theradii ofallwires, these coefficients canreadily beobtained bysimilar approxima- tions, asused for(45). Thus, theeffect ofwire and itsimage upon wireaisgivenby(29) s^= 2TslnSr(55) where rapisthedistance from center ofwireatocenter ofthe image of/3,whereas rapisthecenter distance ofthetwowires directly. Fortheself-coefficient onehas saa\a=^\n^(56) ZTTB rta where haistheheight above ground, 2hathedistance totheimage Sec. 12] Circular Ring ofCharge 125 ofwire a,andRaitsradius. Allthepotential coefficients can readily becomputedifthegeometryiscompletely given. By(3-13) and(3-11) onecanalsocompute themutual capaci- tance coefficients which, however, arealways rather complicated expressions, since theymust involve thecomplete determinant of thepotential coefficients aswellasthepertinent minors; ingeneral, 4 :feU* I'/t* -X2 -*71 FIG.12-8 System ofParallel Thin Wires above Ground. nosimplifications canbepermitted. Fortwowires above ground forming atransmission system with \i=X2=X,oneobtains, with thedesignationsofFig.12-8 forthecapacitance perunit lengthinthepresenceofground, _X1_2ire <$!<2Sns22~~ Many special applications arefound inthereferences listed in Appendix 4,B,a,aswellasinOllendorff,A18pp.123-143. Circular Ring ofCharge. Foratotal charge Quniformly distributed overacircular ring ofradiusa,thelinedensity will beQ/2ira, andthepotentialisobtained bydirect integration asin (1). Referring toFig.12-9, onecanchoose thepoint ofobserva- tionP(p, z)along thez-z-plane because oftheaxialsymmetry. Onethenhasthelineelement ds=ad<j>andthus ad<f) 47TE27ra/0=o[(p acos0)2+(asin 126 Fields ofSimple Geometries [Ch.4 Introducing thechange ofvariables (011endorff,A18 p.101-104), cos</>=2sin2 |9 1, theintegral reduces tothenormal form Q_2=-2 i 47TE IT[(pdj3 FIG. 12-9 Circular Ring ofCharge. ofthecomplete elliptic integral6ofthefirstkindFf-ifc J=K(k) with themodulus 4pa fc2 (p+a)2+z2 (59) Adifferent treatment withanexpansion intoaninfinite series of Legendre polynomialsisgiveninSmythe,A22 p.137. Along theaxis p=0,sothat fc2=0,andsinceK(0)=ir/2, onehas '<m aresult thatcanbeobtained more simply bydirect integration. 6SeeJahnkc andEmdc: Tables ofFunctions; reprinted byDover Publi- cations, NewYork, 1943; originally published byB.G.Teubner, Leipzig, 1938. This reference contains alsoextensive tables andgraphsofelliptic integrals andfunctions. Sec. 12] Circular Ring ofCharge aboveGround 127 The fieldstrength along theaxis is andhasamaximum value atz=a/A/2. Attwood,A2 p.65,gives asimple treatment andgraph ofthefield distribution. Inorder topermit adefinition ofcapacitance, onehastosub- stitute again aquasi linecharge, i.e.,admit asmall butfinite diam- eterd<^aofthecharge distribution asindicated atSinFig. 12-9. Onthesurface ofthisthin toroid k2approaches unity, so that itismore convenient tousethecomplementarymodulus A/2=1 /c2 ,forwhich oneobtains with (59) (P+a)2 (62) since thenumerator isexactly theequation ofthesmall cross- sectional circle; thesimplification inthedenominator isbased on / d\ f d\la -I<P< Ia+- 1andd<&a. Thecomplete elliptic inte- \ */ \ 2/ gralK(k) canbeexpressed7inascending powers ofA;'2 ;using only the firstterm, K(k)=In(4/A/)=In(16a/d), and (58)becomes *.ln(63)4ir TT2a d sothatthecapacitance ofthethin circular ring ofcharge follows as (64) anapproximation tobetter than2percentford/a^0.1. Further details must beleftforsection 33,dealing with toroidal coordinates. Circular Ring ofCharge above Ground. The effect of ground canreadily bereplaced bythat oftheimage ring ofcharge below ground, asindicated inFig. 12-10. The potential value anywhere inspaceisthedifference oftwoexpressions obtained from (58), onewith zreplaced by (z h)andrepresenting the contribution oftheactual charged line; theother with zreplaced 7Jahnke andEmde,loc.cit., p.73. 128 Fields ofSimple Geometries .4 by(z+h)andrepresenting thecontribution oftheimage. Thus Fia.12-10 Circular RingofCharge above Ground. where again K(ki) andK(k 2)arethecomplete elliptic integrals ofthe firstkindwith 4pa (p+a)2+(t-h)2 4pa(66) 2 (p+a)+(+A)a , From (65)onecanalsoevaluate the fieldvectorEbydirect dif- ferentiation. Inparticular, oneobtains forthecharge density induced inthesurface ofground for2=0: __Q_2 2/i <T-z/z=0^^^+^2_ where B(fc)isthecomplete elliptic integral [1-k2sin2 |S] Sec. 13] LineCurrents andQuasi LineCurrents 129 andwhere themodulus kiseither kior 7c2from (66)with 2=0. Theexpression (67) gives thecharge distribution asafunction ofpandshowsmaximum value close top=a,andaminimum atp=ofvalue Q 2h *>=-~ 47T(a2+h2)* Forlarge values ofh,thisdensity becomes identical with the maximum density induced byapoint charge atheight habove ground asseenfrom (10-17). Thepotential onthesurface oftheringcanbefound bysuper- position asthesum of(63)and ofthecontribution oftheimage according to(58)ifonereplaces p a,z 2h.Thus, (68) where now /c2follows from (59) forthesame values pand z k*= (2a)2+(2/t)2= 1+(fc/a)2 (69) From (68), thecapacitance with respect totheimageisdirectly C=Q/$3j whereas thecapacitance toground must betwice this value because onlyonehalfthepotential difference exists between ringandground. 13-LINECURRENTS AND QUASI LINECURRENTS Forcurrents concentrated inmathematical lines, themagnetic fieldBcanbeevaluated either bydirect integrationinaccordance with (6-22) orfrom thevector potential A,which itself isfound bytheline integral (6-21), namely, A=-^/fB=curlA(2)r Ineither case, only closed lineintegrals have physical significance, since steady currents canbemaintained onlyinclosed circuits. 130 Fields ofSimple Geometries [Ch.4 Because oftheconcentration ofthecurrent inamathematical line,both thevector potential Aandthemagnetic fluxdensity B approachinfinite values asoneapproaches thecurrent line; this hasalsobeen pointed outinsection 6.Inorder toavoid these infinite values, oneusually substitutes quasilinecurrents, i.e.,one admits finite andusually circular cross sections which aresmall compared with allother physical dimensions ofthesystem. Within theconductor oneassumes uniform current distribution andthesame permeability /iasthesurrounding medium, usually air. FIG. 131Representative Cross Section ofQuasi Line Current. Proximity ofQuasi Line Current. Ifoneconcentrates on theimmediate neighborhoodofastraight conductor ofsmall circular cross section with radius a,onemay disregard themag- netic field effects ofanyother current carrying parts ofthesystem ifthey arefarremoved; indeed, onecanconsider thepiece of quasilinecurrent astaken from avery long straight wireand consider itentirely byitself. Assume, asindicated above, uniform current density andtherefore circular magnetic field lines con- centric with theaxis oftheconductor, asinFig. 13-1. Applica- tion ofthelineintegral (6-3) toafield linewithin theconductor gives -'/, H/- whereas thesame lineintegral along afield lineoutside thecon- Sec. 13] Rectangular Current Loop 131 ductor gives 2pff/'=I,HS=/-(36) ZiTTp Onthesurface oftheconductor, continuity ofH^issatisfied;it reaches there itslargest value. Since themagnetic intensity H+" outside thewire decreases as1/p,theexternal magnetic fluxlinked with thetotal current overalengthIofthewire hasnophysical meaning unless theupper limit hasadefinite finite value. This serves toemphasize that only closed current loops canhave physical significance, eventhough onehasalready avoided theinfinities caused bylinecurrents. The necessity ofconsidering only closed current loops brings with itthefactthatthefield asfound in(3)willactually never exist inandnearquasilinecurrents; there willstrictly always be distortion caused byother parts ofthesystem. If,however, the radius aissmall enough, theactual field lines within thecon- ductor willbesoclose toconcentric circles thatonecanretain (3a) asafirst-order approximation. This gives, then, fortheinternal magneticfieldenergy perunitlength from (7-15) -< This yields bydefinition (7-5) theinternal inductance perunit length independentoftheradius oftheconductor andthesimplest expression obtainable foranyshape ofcross section. Rectangular Current Loop. Foracurrent loop ofcon- ductors ofsmall cross section intheshape oftherectangle in Fig.13-2andfedinamanner asindicated inFig. 7-1,onecan evaluate thevector potential atapoint P(x, y,z)byperforming theintegration prescribed by(2)along theaxis ofthewire. Because thedirection ofAisthesame asthat oftheelement ds, 132 Fields ofSimple Geometries [Ch.4 one willgenerally have atPthecomponent Axcontributed by theloop sides oflength 2a,andthecomponent Aybytheloop sides oflength 26. Therefore, _MT[r=+adsfs=~adsl"S'U 7"J.-4- r"777] where (r')2=(x-s)2+(y+6)2+z2 ;(6) FIG. 132Rectangular Current Loop. The integrals areofelementary typeandleadto A**71Al+O+Sr3~Q+AAx= 7In I---- -- )4ir \r2a+xr4+a+x/... (8) where n,r2,r3,and r4arethedistances from thepointPtothe individual vertices ofthe rectangle, respectively. Along the center plane y=0,onehas TI=r4,andr2=r3,sothatAx=0. Insimilar manner, onefinds From these expressionsforthecomponents ofthevector poten- Sec. 13] Rectangular Current Loop 133 tialonecanreadily deduce the fieldcomponents bythesecond relation(2), pdAyR-+dA*RdAv dAx BX='By+ Dz=---7dz dz dx dy The indicated differentiations arevery easily performed, since they are allofthetype where uiseither (x a),(y b),orzbut isnever contained in k.Intheplane oftherectangle, Bx=By=and =1,2,3,4 (11) with Xi=z4=(x+a),x2=x3=(x-a); y\=y2=y+b, 2/3=2/4=y b;and ra=[za2+ 2/a2 ]^- Obviously, B2becomes infinitely largeontheconductor loop. Inorder toobtain thetotal inductance oftheloop, themagnetic field isdivided intoexternal andinternal regions. Onecomputes theexternal magnetic fluxasthat fully linked with theloop cur- rent,which forsmall cross section isgivenbytheintegral d d^= xl t/u=b=-- >dx a+rt/u=b+-. where thelimits aretheinnermost points oftheloopconductor, asforexample P"andPf"inFig. 13-2. Division bycurrent I yields theexternal inductance -bIn l+(12) Tothisonehastoaddtheinternal inductance, which with(5) issimply L{=2(2a+26)Lfl=-(a+6) 134 Fields ofSimple Geometries [Ch.4 wheremistheabsolute permeabilityoftheconductor material. Onecan, ofcourse, alsoevaluate themagneticfielddirectly by applicationof(1)rather than firstdetermining thevector poten- tialasin(2). Frequently, however, theintegrations involved in (2)aresimpler toperform than those in(1),andthedifferentia- tions arereadily carried through. Forafinite length ofastraight wire, thecomputationsofthemagneticfield aregiveninmany elementary books such asAttwood,A2 p.276,who alsoapplies the results toarectangular loop; Bennett and Crothers,A3 p.424; andCullwick,A6 p.184.More general treatments arefound in Hague,344and particularlyinGrover.B43Itisimportant to observe thatinductance ofapiece ofwirenotforming aclosed loop hasnomeaning, since thedefinition ofexternal inductance rests upon that ofmagnetic fluxthrough adefined area,andthecurrent linkingit. TwoLong Parallel Straight Line Currents. Intransmis- sion lineproblems, theapproximate rectangular loops formed by theparallel wires permit theassumption 2a 26,sothat the contributions ofthetwosmall sidescanbedisregarded. Actually, in(8)onecanapproximatefor\x\ n=[(*+a)2+(y+b)2+z2]*= where pi=[(y+b)2+z2}^isthenormal distance ofPfrom the leftwire; analogously one finds r4with p2=[(y b)2+z2 ]H taking theplaceofPl.Asabecomes very large, but\x\remains small compared with a.theratio- ;-->1.Ontheother r4+a+x hand, andanalogouslyforr3;here, however,--->( )>so 7*2 a-\-x\pi/ that (8)takes theform M (y-b)2+ 2Iln~~x7,!V2 ,4?r (y+b)2+ where p2and piarethenormal distances ofPfrom thewiresand Sec. 13]TwoLong Parallel Straight LineCurrents 135 arethesame radii asaredesignated r2and ritrespectively, in Fig.12-5. Thesame approximations yield atonceAy=in(9), sothatthemagneticfieldbecomestwo-dimensional, independent ofdistance x.With (10)and (13)onederives dz dAx Itshould beemphasized that thisrequires \x\^a,or,physically, thatonekeeps very farfrom theends ofthetransmission line. The field configuration caneasily beevaluated, since the field lines aredefined by oralso dAx SAXdz+ dy=dAxdz dy=0,Ax=cons(15) Ifthemagnetic field istwo-dimensional, then asingle component ofthevector potential exists, andthelinesA=consbecome the field lines. Forthetwo parallel linecurrents, (13) indicates for thefield lines p2/pi=cons, orthesame condition asfound forthe equipotential surfaces oftwoparallel, very longanduniformly charged lines following (12-29). Thus, magnetic and electro- static field lines arethemutually orthogonal families ofcircles inFig.125.1This isnottrueforfinite cross sections ofthewires, though forvery small cross sections, orquasi linecurrents, itcan beassumed asareasonable approximation. Theexternal inductance fortwowires ofsmall diameters diand d2isobtained perunitlength from theflux,which canbeevaluated bestbyuseof(6-23) applied totherectangle abed inFig.13-3, (16) 1AlaoAttwood,A2p.269. forfurther details. 136 Fields ofSimple Geometries [Ch.4 sothat (17) since both diandd2must besmall compared with 26tojustify theuse ofthe fieldproduced byline currents. The internal inductance istwice thevalue (5),once foreach conductor. Thissame result isobtained bySmythe,A22 p.317,byusing expression (7-13)forthe field energy oftwo parallel con- ductors; theresults areactu- ally rigorousforany, even small, spacing 26ofthecon- ductors asshown in(15-13). Dipole LineCurrents. If thedistance 26ofthetwopar- allel linecurrents decreases to very small values, onehasthe analogous case tothat oftheelectrostatic dipoleline. Again, onecanthenapproximateasshown inFig.13-4 (analogousto Fig.12-7) Pi~r+6cos</>, p2=r 6cos < sothatthevector potential from (13)becomes, with 6FIG.13-3 External Flux ofTwo Long Parallel Quasi Line Currents. 2irr-6cos M6 r- ---/-cos r+bcos</>TTr Thenegative signarises from thefactthat thevector potential pointsinthesame direction asthenearest current;fory>and z>thenearer current isinthenegativedirection inaccordance with theFig. 13-2, which underlies theexpression (13). The magneticfieldcomponents arefound asin(14) or,better, using theright-handed cylindrical coordinatesr, </>,andxandAppendix 3,(37), Sec. 13] Dipole LineCurrents 137 The field lines arecircles through theorigin andaretheorthogonal family totheelectrostatic field lines ifoneconsiders thecurrents asrepresenting electrostatically equal andopposite chargedlines asinFig.127,butwith reversedsigns. P(r,0) FIG. 134Dipole Line Currents. Changing tothecomplementary angle6=ir/2 interpretationofAxinterms ofavector product where mi=I2bnpermits the (18) (19) isthemagnetic dipolemoment perunitlength ofthedipole current line,defined bythearea oftheloopperunitlength times thecur- rentbordering it,anddirected sothat asseenfrom itstipthe current flows counterclockwise. Theform (18)isvery similar to thedefinition ofthescalar electrostatic potential (12-52). In fact,onecould aswellintroduce thescalar magnetostatic potential function fromwhich themagnetic intensity components Hr,H0follow in exact analogy totheelectrostatic fieldcomponents (12-53). 138 Fields ofSimple Geometries [Ch.4 Since themagneticfieldcomponents decrease as1/r2 ,thedipole linecurrents represent amore local fieldthan conductors with finite separation; use ismade ofthisfact inbifilarwindings. Two Pairs ofLong Parallel Line Currents. Foreach pair ofparallel thinwires l'2'andl"2" (seeFig.13-5) with currents /'and/",respectively, theexternal vector potential isgiven by (13)andtherespectiveself-inductance by(17), using theprimed orthedouble-primed quantities from Fig. 13-5. FIG. 135Mutual Inductance ofTwo Pairs ofLong Parallel Wires. Toobtain themutual effect forthin wires, onecansubstitute ingoodapproximation thetotal fluxlinked bythecenter filaments ofthewires\"2"andproduced bythepairl'2'. Using thesame method asin(16)andintroducing thevector potential Ax'from (13),with p2= (ft,PI=Piatthecenter ofl"andwith p2= (fe, Pi=7>2atthecenter of2",onehas (20) utual=f/'[in^-In*1=f/'In* 2?r LPi P2J27Tq2pi Itisobviously possible toarrange thefourwires insuchaway that qip2=q2Pi, sothatnoresultant mutual linkage exists, or thatmagnetic interference isavoided. Though this isfeasible for rigid installations, thevariable spacing ofaerial transmission lines normally prevents utilization ofthisrelation, andconsequently recourse istaken toproper alternating transposition ofthewires.2 2H.S.Osborne, Trans. A.I.E.E., 37,p.897(1918); CorbettfB2 p.30, andAppendices. Sec. 13]Systems ofParallel Straight LineCurrents 139 Relation (20)isthen thebasic relation forevaluating the"cross talk" orinductive interference ofparallel pairs ofthin wires. Themagnetic field lines areobtained fromAx=cons, asin (17),whereAxisnowthetotal vector potential atanypointP inspace, (21) Pi with thep'sdesignating thenormal distances ofthepointPfrom therespective wires. Thegeometryisvery complex, depending onthecurrent ratio inthetwopairs ofwires. Iflinkage between theparallel wire pairsisdesired, then (20) gives theuseful mutual fluxandthemutual inductance M= ^mutuai/^'j whereas thedifference between self-inductance ofone pairand thisvalueMgives theleakage inductance inconformity with (7-7). Systems ofLong Parallel Straight Line Currents. For anynumber nofparallel wires isolated from ground, thesum of allcurrents must bezero inorder toconstitute aphysically pos- sible system.Itnowbecomes necessary todistinguish between "wire currents" and"loop currents" inthesense thattheformer aretheobserved currents laintheindividual wires, whereas the latter arethecurrents /associated with the definition and measurement offluxlinkages andtherefore ofinductances asin (7-10). Since theinductances aregeometric quantities, an accurate knowledgeoftheloop currents isnotnecessary fortheir evaluation; anyconvenient temporary choice issatisfactory. Thus, onecanchoose anyfourwires asacombination oftwopairs andatonceusethemethods outlined above. Sincenwires canbearranged innr=-(n 1)different pairs 2i orloops, andthese n'loops again innff=(n! 1)loop pairs, 2i there will, ingeneral, ben1=%(n+l)n(n \)(n 2)dif- ferent mutual loop inductances. Thisnumber n'includes loop pairs withoneconductor incommon, i.e.,arrangements inthree- conductor groups. InFig. 13-4, this latter case exists ifl"and l/ / l/V l'become identical and therefore p\ #1>(2b' 1- 140 Fields ofSimple Geometries [Ch.4 Thus, themutual inductanceis,from (20), where d'isthediameter ofthecommon conductor. Inthree-phase systems withn=3wires, n'=3,andn"=3, there exist three loopsandthree mutual loopinductances, which incase ofgeometrical symmetry areidentical andhave p2= q%=26'.Many practical applications arefound inWoodruff312 andClarke.31 With thesame assumptions asdiscussed fortwoparallel straight line currents, theresultant external vector potential ofthen wires canbewritten Ax=- -LIaInPa (23) *TTa=l where thepaarethenormal distances from thepoint ofobservation Ptothewire centers; thenegative signcould beavoided by writingIn(l/pa)asin(13). Thetwo-dimensional field distribu- tion isagain defined by(15),but itsanalytical interpretation is hardly feasible. The direct evaluation ofthemagnetic fieldhas beenshown in(5-9). Circular Loop. Assume anideal circular current produced byawireloopwith twisted leads asshown inFig. 7-1,and dis- regard the effect ofthese leads. The circular current isthen similar tothecircular ring ofcharge inFig. 12-9,and itsvector potential canbefound bydirect integration sin<frd<f>+u2acos<j>d$ (24)=o[(p-acos0)2+(asin0)2+ The firstcomponentisintheradial direction and iscancelled by thesymmetrical element( </>);thesecond component istangen- tialtothecircle and,upon changing variables asinsection 12for thecircular ring ofcharge, leads to *)2+*2]*Ed-Hfc2 )K(K)-(*)] (25) Sec. 13] Circular Loop 141 where K(k) andE(k) arethecomplete elliptic integrals offirst andsecond kinds,3 respectively. Themodulus kisthesame as (12-59). Themagneticfield vector Bisobtained bydifferentiation in ihecylindrical coordinate system [Appendix 3,(37)] ;(*)J (26) (a p)2+i with50=because oftheaxialsymmetry. Themagnetic field lines areplane curves inthemeridional planes andaredefined by BP/BZ=dp/dz, which leads with (26)tothetotal differential dp+ (PA)&=d(pA0)= dp dZ orto =cons (27) asthesimple equation offieldlines foraxially symmetrical fields. Thus, with (25), thefield linescanbefound numerically asoval- shaped closed curves surrounding thewire.Asimple graph is giveninAttwood,2 p.260;H.Lamb,C22 p.220, gives theanal- ogous graphofacircular vortex line. Extensive tables of (47ra z//i/)from (26)asfunctions ofp/aandz/ahavebeencom- puted byBlewett;4theFig.1ofthisreference alsogives graphs of theradial variation ofBzforthree different values ofz/a,which are utilized (same ref., p.979) tolocate twocircular coils ofdifferent currents, such astoproducecancellation oftheir fields overa limited region, foruseasflux coils insynchrotrons. 3SeeJahnke andEmde: Tables ofFunctions, p.73;reprinted byDover Publications, NewYork, 1943. 4J.P.Blewett, Jl.Appl. Phys., 18,p.968(1947). 142 Fields ofSimple Geometries [Ch.4 Ageneral analytical treatment isgiven in011endorff,A18 p.Ill; inSmythe,A22pp.266and270;andinZworykinetal.,m2 p.472; themagneticfieldBzintheplane oftheloop5where z=is giveninCullwick,AG p.140. Along theaxis forp=0,(26) reduces to B>= '*= i7^T^(28) which canbefound directly byuseof(1); seeAttwood,A2 p. 226; Harnwell,A9 p.288;andSpangenberg,329 p.400. Inorder toevaluate theinductance oftheloop,onehastoadmit afinite small diameter d,asinFig.129.Theexternal inductance isthenobtained byintegration ofA^along theinnermost filament, forwhich p=a (d/2),z=0,sothat Lex=2^p^=2Ma (l- )[(l-1/c2 )K(k)- fi(fc)](29) with/cfrom (12-59) as 2a- Since d<^a,onechanges advantageously tothecomplementary modulus k'asin(12-62); withK(k) ln(4//c') asthere, and E(k)=I,6oneobtains themuch simpler expression (Smythe,A22 p.316; Ollendorff,A18 p.113) (31) The internal inductance isclosely (2-jra) times thevalue given in (5). Magnetic Dipole. Foraverysmall circular loop, thedenomina- torof(24)canbewritten J The integral becomes, then, byexpanding thesquare root bi- 6SeealsoH.W.Reddick andF.H.Miller: Advanced Mathematics for Engineers, p.137;John Wiley,NewYork, 1938. 6Jahnke andEmde,loc.cit.,p.73. Sec. 13] Magnetic Dipole 143 nomially andtaking the firsttwoterms intothenumerator, MC2"a f~ ap"I aa2oA*=^-II-co801+-^cos0d0=f- 5/-^(32)4?r1/0=0 r L TO J 4r2r Using spherical coordinates, p/r=sin0,andthemagnetic field components are[Appendix 3,(41)] 2cos 4?r(33) These magnetic fieldcomponents have exactly thesame form as theelectric fieldcomponents (10-35) describing theelectricdipole, sothat itisproper toidentify thesmall circular current loop as theequivalent ofthemagnetic dipole,ofmagnetic moment(cfirl), which, asvector, points inthenormal direction fromwhich the current appears toflowcounterclockwise, m=na27r7(34) Themagnetic field lines aredefined inaccordance with (27)as /A\ j 2T (pA<t,)= IIJ= (TTT/-=cons4rVo/ 4?r r which isidentical with (10-36), indicating north andsouth mag- netic quantities tobetheequivalent ofpositive andnegative charges (Fig. 10-6). Onecan, ofcourse, nowconstruct ascalar magnetic potential inanalogy to(10-33), namely, > ifonedefines themagnetic moment as m=Qml(36) Qmrepresenting themagnetic.north quantity, and 1thecenter distance directed outofthenorth pole. Though thisanalogy is quite useful incertain respects,itisnecessary torealize that magnetic quantity hasnotbeen perceptibly isolated andthat its conceptisamathematical aiddevoid ofphysical reality. With theaidofthemagnetic dipole concept onecanalsocon- 144 Fields ofSimple Geometries [Ch.4 struct dipole layers astheequivalent offinitely large current loops. Thealignmentofthemagnetic dipole inauniform magnetic fieldBQiscaused byatorque T=m*B (37) Good treatments ofthemagnetic dipole aregiveninAttwood,A2 p.219;inSmythe,A22 p.266;andinStratton,A23 p.237. Two Circular Loops. Themagneticeffects oftwo circular loops areobtained bydirect superpositionoftheindividual vector potentials computed by(25) orofthemagneticfields computed by(26). Themutual linkageisreadily obtained byapplication of(29),choosing AQasproduced byoneloopandintegrating over thecenter line oftheother loop. Fortwocoaxial circular loops, thecomputations arestraight- forward. Iftheloop radii area'and a!1andtheir center distance h,then themutual inductanceis,from (25),with p=a",z=h, andareplaced bya', (38) where These forms/ aswellasmany others forarbitrary mutual location ofcircular loops,8aregiveninGrover,1343andintheform ofseries ofLegendre polynomialsalso inSmythe,A22pp.310-312. The mutual force action ofcoaxial loopsiscomputed bySmythe,A22 p.277 .9Simplefieldgraphsarefound10inAttwood/2 pp.226 and227. 14-SIMPLE SYSTEMS OFDISTRIBUTED CHARGES Thesimplest types ofcondenser arrangements aretwoparallel plates, two concentric cylinders, andtwo concentric spheres. 7S.Butterworth, Phil.Mag., 31,p.4439 (1916); alsoSdent. Papers Nail. Bur. ofStand., No.320, 1918. 8SeealsoProc. I.R.E., 32,p.620(1944). 9Fortabulated values, seeJahnke andEmde,loc.cit.,pp.86-89. 10SeealsoL.Fleischmann, Arch. f.Elektrot., 21,p.31(1929); Gianella, Revue gen.de1'elec., 22,pp.711and761(1927). Sec. 14] Parallel Plate Condenser 145 Their treatment forasingle dielectric isfound inany ofthe references inAppendix 4,A;4,B,a;and 4,B,basforexample in Attwood,A2pp.68-78, andwill, therefore, bevery briefly sum- marized with theemphasis ontheextensions tothe lessusual applications. Parallel Plate Condenser. For infinitely large plates of potentials 3>iand*n<*i(see Fig. 14-1 with EI=e2),the potential distribution must be linear inxandahomogeneous fieldgradient Eexists, x d Fl Ex= d FIG. 14-1 Parallel Plate Condenser withTwo Different Dielectrics. Area5-_L gdPractical condensers are, of course,offinitesize,sothat field fringing would have tobetaken intoaccount (see section 27). However,ifonesurrounds thefinite platesofarbitrary areaSby guard rings ofawidthWconsiderably larger than thedistance dandofthesame potentialsastheplates (see Fig. 14-2), thenthefieldbetween theactive condenser plates approximates theideal plate con- denser very closely aslong asthegap <7<Cd.Inwhat follows,itwillalways be assumed thatsystems which would require infinite extension tobeideally simple are approximated bysuitable guard arrange- ments. One can, ofcourse, alsoassume oneoftheelectrode potentials tobezero, which simplifies some oftheexpressions. The ideal parallel plate condenser of finite areaSandwith asingle dielectric ofconstant ecarries atotal positive charge onplate I,which follows from theuniform charge density 171= with (1)as*i d W>d FIG.14-2 Principle of Guard Rings forParallel Plate Condenser. Qi=+^(*i-(2) 146 Fields ofSimple Geometries [Ch.4 Thecapacitanceofthecondenseris,therefore, C=-,S (3)a With twodifferent uniform dielectrics separated byaplane paral- leltothecondenser planes, asshown inFig. 14-1, thepotential distribution ineach dielectric islinear (4) andtherespectivefieldvectors are Exl=- (4>i-*), EX2=(*- *n) (5)a o However, thepotential value <fmisunknown andmustbeevaluated from theboundary conditions attheplane ofseparationinaccord- ance with section 2.Continuity ofthe dielectric fluxdensity (normaltotheboundary surface) gives &iExl=2#z2,which leads with (5)to eibfri+2a3)n ,a^m=-r-(6) 16+E2a The significantanddistressing aspectisthefactthattheabove boundary condition stipulates ahigherfield gradient forthe dielectric material with lower constant (and generally lower dielectric strength) andvice versa, independent oftherelative thicknesses. Aslightairgapa<^binseries withasolid material willtherefore beoverstressed and willionize ifthefield gradient inthesolid ischosen ashigh asispermissiblefor italone; the only recourse isthecomplete elimination ofairandsubstitution byagood liquid insulator through somevacuum impregnation process;seePeek.3315 Thecharge onelectrode Iisagain found from theuniform charge density TJI=+Z>Xl/x=o;thecapacitanceofthetotal condenser isthen, with (5)and (6), (7) Sec. 14] Coaxial Cylinder Condenser 147 which canbeinterpreted astheseries combination ofthetwo partial condensers formed byassuming theboundary surface &m tobeaconducting surface. This ispossible here because the surface happens tobeequipotential. Should theboundary surface between thetwo dielectrics beofanyarbitrary shape, then the field distribution would nolonger behomogeneous and thepotential function would become rather complex. Forn>2dielectric slabs withboundary surfaces parallel to theelectrode surfaces, thesame procedure canbefollowed; the capacitance canbegiven atonce astheseries combination ofthe npartial capacitances, since z\Ei=eaEajsothat where daaretheindividual thicknesses, and Ethecorresponding dielectric constants. The field strength inanyone dielectric is found from (7)with (8) d\-1 (9) andthepotential value atany interface isthedifference of$i n and^daEauptothat interface. The totalsum daEais,of =1 course, thetotal potential difference. Inreal dielectrics, where theelectrical conductivities arenot negligibly small, theboundary conditions require continuity of thecurrent density, sothaty\E\=jaEaandthepotential dis- tribution isdetermined bytheconductivities. There willthen besurface charges onalltheinterfaces inaccordance with (820). Coaxial Cylinder Condenser. For infinitely long coaxial cylindersofpotentials $1and$n<<S>i,and ofradii#1andR2, respectively, andwith asingle dielectric, thepotential varies logarithmicallyasforasingle uniformly chargedlinein(12-28), sothat Er=*'"*" (10)UJlnfi 2/B Practical arrangements are, ofcourse,offinite length, sothat fieldfringing would have tobetaken intoaccount (seesection 30). However, asintheparallel plate condenser, onecanarrange guard 148 Fields ofSimple Geometries [Ch.4 electrodes cylinders ofsame radiiandsame potentials spaced a gapg<RIfrom thetestelectrodes which insure forthese the ideal coaxial cylinder field; thisprocedure willbeassumed throughout theremainder ofthis section, wherever precision re- quiresit. The ideal coaxial cylinder condenser oflength Lhasacharge ontheinner conductor Qi=2irRlLDT=Rl,which gives with(10) acapacitance c=Ql *!-*!! Since theradial electric field strengthislargest attheinner con- ductor,itrepresents thedesign criterion fortest electrodes,1for coaxial cables, and forbushings; ontheother hand, theouter radiusR2defines theoverall size. Forafixed value ofR2jthe inner radius RIcanbechosen soastolead tothelowest possible value ofET=RIbyminimizing ERI=R2 In n,2 intt2/HI with respect totheratioR2/R\= T/.Actually , (- )= for77=e dr)\ln r;/(13) where e=2.718--- isthebase ofnatural logarithms. Other considerations might alter slightly thisoptimum ratio, butfew designs deviate significantly (seereferences inAppendix 4,B,a, and4,B,b). The stationary flow ofheatbetween concentric cylinders of temperatures TIandTualsofollows therelations (10)withappro- priate useoftheanalysis pointed outinsection 9.Thus, thetotal heat flowperunitlengthisgivenby 27T/C_ where kisthethermal conductivity. Inacable inwhich the inner conductor carries acurrent /,andtheouter conductor represents aprotective sheath without current flow,Qthmust represent theheatgenerated bycurrent/,orQth=I2R,whereR istheelectrical resistance oftheinner conductor perunit length. 1Schwaiger,317andA.S.T.M., Tentative Standards forOilTesting, 1936. Sec. 14] Coaxial Cylinder Condenser 149 IfTU isthegiven ambient temperature, andTiselected asthe maximum permissible temperature oftheinner conductor inview oftheadjacent insulation, onecandeduce themaximum current rating /ofthecable (the current-carrying capacity). The simultaneous electric andthermal stresses ofthedielectric present themainprobleminefficient cabledesign. n<k FIG.14-3 Coaxial Cylinder Condenser withTwo Different Dielectrics. With twodifferentdielectrics inconcentric layers, asinFig.14-3, thepotential distribution ineach dielectric isgiven by(10)if appropriate substitutions aremade forradiiandpotential values pertinent tothedielectric layers. Theinterface potential *misde- termined bytheboundary condition z\Ei(r=R)=e2E2(r=R), or =2' which gives1RInR/R l*RInR2/R (EIInR2/R)$i+(e2In EIInR2/R+2InR/Ri(15) Asintheplane case, theboundary condition dictates adiscon- tinuity inradial electric field strength which tends tooverstress electrically anyairinclusions inbushings (seePeek,B15 p.316, 150 Fields ofSimple Geometries [Ch.4 andKarapetoff,A11 p.175). The overall capacitanceofthecon- denser istheseries combination ofthepartial capacitances ofthe twodielectric layers, R 1R2 which indicates theextension toanyarbitrary number nofcon- centric dielectric layers. Asseenfrom (10),each dielectric layer hasanon-uniform field gradient, varying from highest tolowest value intheratio ofthe bounding radii; furthermore, there isadiscontinuity attheinterface imposed bytheboundary conditions. Onecan,now, select theradii anddielectric constants ofsuccessive layers insuchamanner that thelargest gradient value becomes nearly equal foralllayers and consistent with therespectivecritical values. This process of "uniformization" ofthe electric potential distribution iscalled grading ofinsulation2and isgenerally desirable innon-uniform fields formost efficient useoftheinsulating material. Combining (10)and (11), onehas which holds inanylayer fortherespective dielectric constantE, sinceQiisafixed quantityforall.Keeping theproductEXrmin foreach layer tonearly thesame value improves theelectrical stress distribution markedly. Aspointed outbefore, however, thermal considerations may force compromises which vary with theactual characteristics ofinsulators. From (17), onecanalsodeduce theprincipleofthecondenser bushing3inwhich Eiskept constant, butwhere intheproduct (Lr) thelengthisstepped down inversely astheradius increases;this isachieved bymeans ofauxiliary electrodes. Obviously, inthe field distribution, fringing must betaken intoaccount. 2H.S.Osborne: "Potential Stresses inDielectrics," Dissertation atM.I.T., 1910; B.Hague, "Intersheath Stress," Electrician, 117,pp.161-163 (1936); Bennett andCrothers,A3 p.158; Schwaiger,1317 p.132; J.B.Whitehead, Trans. A.I.E.E., 64,p.555(1945). 3A.B.Rcynders,,71A.I.E.E., 28,p.209(1909); C.L.Fortescue and J.E.Mateer, Elec. JL,10,p.718(1913); E.E.Spracklen, D.E.Marshall, and P.O.Langguth, Trans. A.I.E.E., 47,p.684(1928); H.J.Lingal, H.L. Cole,andT.R.Watts, Trans. A.I.E.E., 62,p.269(1943). Sec. 14] Concentric Sphere Condenser 151 Concentric Sphere Condenser. Fortwoideally closed, con- centric spheresofradiiRiandR2andpotentials $1and$n<3% theradial field distribution isessentially thesame asforasingle quasi point charge ofsection (11), sothatwith satisfaction ofthe boundary values (18) Inmost practical cases, leadsmustbeused toapply thepotentials, andspacers areneeded between thespheres tomaintain con- centricity;itisassumed that infirstapproximation atleast these effects arenegligible. Itisalsopossible tousesections ofspherical surfaces with appropriate guard surfaces asindicated forthe cylindrical condenser. Thecharge ontheinner sphereisQi=47r#12Dr=/j1,which gives with (18)thecapacitance For finitely closed surfaces, asforthese concentric spheres, the capacitance value remainsfinite, even iftheouter surface recedes toinfinity. Since theradial electric field isstrongest attheinner sphere, onecanasforcylinders compute anoptimum ratioR2/Ri forwhich thelowest value of^=^1 exists withR2kept fixed. Minimizing $i-3n r?2 ERl=----- -(20)R2 TJ 1 with respect toi\=R2/R\leads to = for ,=2 (21) with theoptimum ratio offield strength values E (R2/Ri)2=4.Itishere, therefore,stillmore important than incylindrical arrangements tointroduce uniformization ofthe potential distribution. Theprocesses are, ofcourse, quite similar 152 Fields ofSimple Geometries [Ch.4 totheones forcoaxial cylinders except that itwillrarely bepos- sible tomaintain theideal condenser field. 15-SIMPLE SYSTEMS OFDISTRIBUTED CURRENTS Inmany practical applications involvingfinite currentdensities, thepermeability oftheconductor canbeassumed tobethesame asthat ofthesurrounding medium, usuallyair.Onecanthen evaluate bydirect integration either thevector potential by (6-19) orthemagnetic fieldbythegeneralized Biot-Savart law (6-22). Single LongConductor ofCircular Cross Section. Though thiscasewasused insection13,itisofvalue tosetdown the complete solution forlater applications. Assume forthemoment aconductor ofradius aandpermeability mandanexternal permea- bility ne\then themagnetic fieldvectoris,from(13-3), if7isthetotal current uniformly distributed overthecross section, and rthevariable distance from theaxis. Because ofaxialsym- metry, themagnetic field lines areconcentriccircles, andthefield depends onlyuponr.Itisdesirable alsotofindthevector poten- tialwhich canhave onlyacomponent parallel tothecurrent flow. Incylindrical coordinates wehave8$= from Appendixdr 3,(37), since allother components vanish. One can, therefore, directly integrate andobtain withDiandDeasintegration constants. Thesame result can beobtained bysolution ofthedifferential equation forthevector potential orbyapplication oftheintegral (619),asinSmythe,A22 p.317,andHague,B44 p.275.1There isnounique way indeter- mining theconstants, since thegeneral boundary conditions (6-7) and (610)apply only tothemagneticfield. Assuming continu- xButseealsodiscussion andcorrection: T.J.Higgins, Electr. Engg., 69, p.246(1940) andB.Hague, Electr. Engg. t69,p.479(1940). Sec. 15]TwoConductors ofCircular Cross Sections 153 ityofthevector potential according to(6-20), sothatAzi=AZe atr=a,onehasfrom (2) -^-1+/>.-=-^71na+De (3) Now, there isequally noreason why theadditive constant D; should contain/ie,norwhyDeshould contain/*,sothatthemost reasonable choice appears tobe A-=^/+^o,De=^I\na+AQ (4) whereA isanarbitrary constant which canaswellbetaken AQ= unless convenience suggests otherwise. One thus has (slightly atvariance withabove references) --['-'"-"<5> Themagneticfield lines areobtained bylettingAz=cons,which defines concentric circles inside andoutside theconductor. TwoParallel LongConductors ofCircular Cross Sections. Since normally noperceptible magnetic interaction ofsteady cur- rents, disturbing theuniform current distributions, occurs, the individual solutions ofthemagnetic field forthesingle conductors canbesuperimposed everywhere inspace. This isquite at variance with theelectrostatic caseanddestroys many analogies. Outside ofboth conductors theresultant magnetic field is(see Fig.15-1) thevectorial combination oftheindividual fields given by(1)andinaccordance with (5-3) B"=A(_L Ixr+ i.e.,itisthesame asfortwo linecurrents concentrated along the axes oftheconductors! Forequal andopposite currents, there- fore, themagneticfield lines outside theconductors willbethe family ofeccentric circles described insection 13;however, these field lines willnownotbeorthogonal totheelectrostatic fieldlines, orbeidentical with theelectrostatic equipotential lines, which arecircles generated bytwoequivalent charged lines (seesection 12)notidentical with theaxes oftheconductors; seeAttwood,A2 p.272. Onlyiftheradii oftheconductors areverysmallcompared 154 Fields ofSimple Geometries [Ch.4 with distance cantheapproximation bemade, identifying the magneticfield lineswith electrostatic equipotentiallines. Within conductor1,theresultant field is Ifthetwocurrents areequal andopposite, sothat I\=Iz=/, y FIG. 151Two Parallel Conductors ofCircular Cross Sections. there exists apointKIonthez-axis atwhich B\=0,namely, at asindicated inFig. 15-1. ThepointKI iscalled thekernel of conductor 1;itisalsocalled theconvergence center oftheresult- antfield lineswithin conductor 1.Thegeometry ofthefield lines isbestobtained bymeans ofthevector potential within1,which follows from(5)as Inthecylindrical coordinatesrj, </>iofconductor 1onecanexpress r2=[(2d)2+r-x2-cos Sec. 15]TwoConductors ofCircular Cross Sections 155 sothatAgl=cons gives -fe)2+ln [1+()2 -icos0i ]=cons takingallconstant terms inclusive ofIn(2d/R 2)ontheright-hand side. Obviously,forld QO ,theconcentric field lines ofthesingle conductor result. Ifdisreasonably large, oralso intheneigh- borhood ofthekernel KI,where rissmall, onecanapproximate In(1+u)=u,andthusobtain from (10) cons This istheequationofcircles with centers at /f^ 1\1R2 21 ^7>which isthelocation ofthekernel ifoneadmits thesame degree ofapproximationin(8).Themagneticfield lines inconductor 1(and similarly inconductor 2)start out, therefore, ascircles near thekernel, thenbecome deformed into oval-shaped curves, which, upon meeting theboundary surface, continue outside aseccentric circles. Figure 151shows two accurately computedfield lines asillustrations; AttwoodA2gives other illustrations, pp.272,393. The total inductance oftwovery long conductors forming a rectangular loopcanbefound perunitlength bydirect applica- tion of(7-1) and(7-2) Ll= 1=T2[ffJlAzi dSl where Ji=-\-I/irRi2anddSi=rdr\dfaarecurrent density and cross-section element ofconductor 1,andJ%,dS2correspondingly forconductor 2.Ifoneobserves (seeNo.523inB.O.Peirce: AShort Table ofIntegrals, Ginn&Co.,Boston, 1929) theintegrations in(11)with (9)and itsequivalentforconductor 2arereadily evaluated, leading to (13) 156 Fields ofSimple Geometries [Ch.4 which isidentical withtheform derived insection 13asapproxima- tion fortwowires atlarge distance 2d.Theexact form isgiven bySmythe,A22 p,318; Russel,B11 p.85,gives thecorresponding expression'fortwo parallel hollow cy- lindrical conductors. Theappearance ofthesimple logarithmic terms inthe final result ledMaxwellA17tothedefi- nition ofthe"geometric mean distance'1 Dofafinite cross section Sfrom a pointP S\nD =(14a) FIG.15-2 Coaxial Cylindri- calConductors.where risthedistance oftheelement dSfrom P,andtheintegrationisper- formed over theentire cross section. Theextension tothegeometric mean distance oftwoareas Siand 82with respect toeach other gives similarly InD= where risnowthemutual distance ofthetwosection elements dSianddS2',these integrals canberelated toinductance calcula- tions.2Because of(12), circular cross sections leadtoparticularly simple results. Coaxial Cylindrical Conductors. Inorder toprovide atwo- conductor system withnoexternal magnetic effects, onecanuse acoaxial cable orpair ofconductors, asshown inFig. 15-2, carrying currents /i=72=/.Themagneticfieldwithin the inner conductor andbetween thetwoconductors isthesame as given in(1)with appropriate changesinnotation: D I_i T B*toRf1' The field intheouter conductor is ,,2_p21 (16o) (156) 2SeeWoodruff312andClarkeB1 . Sec. 15] LongThin Rectangular Bars 157 reducing tozero atr=R2;Fig.15-2 indicates thevariation of thefield asafunction ofradius r. The total inductance ofthecable perunitlength canbestbe Computed from themagneticfieldenergyinterms ofthe field vectors asoutlined in(7-15). SinceHB=(l//i)Z?2 ,onecanuse relations (15) directly fortherespective zones andobtains the result 2Wm1Ui RQ + IfR2<1.257?o, thecontribution ofconductor 2canbeapproxi-D_r> mated by /i32- ;assuming alsothepermeabilities ofall O/LO three regions tobethesame, onehasthemuch simpler form asfound inBreisig,A4 p.161; Russel,B11 p.83,derives theinduct- ance oftwocoaxial hollow cylinders andthen reduces tothesolid inner conductor.3 LongThin Rectangular Bars. Formany practical applica- tions itispermissible toapproximate busbars asvery thinribbons ofrectangular cross section, asinFig.15-3; theadvantagewill beapparentinthenext subsection, where the finite rectangular cross section willbetreated. For infinitesimal thickness the current isdistributed inacurrent sheet ofuniform densityKz= I/2handthevector potential canbeevaluated bythesecond part of(6-19), where r=[x2+(y- 2/')2+ (z~2')*]^, y'varies over the width 2/i,and zover theinfinite length oftheconductor. The integralinz'leads to(2Inr'),with rrtheperpendtculardis- tance ofPfrom thefilament dy'asinthecase ofthevery long 3T.J.Higgins, Trans. A.I.E.E., 64,p.385(1945) gives anappraisal of existing literature. 158 Fields ofSimple Geometries [Ch.4 straight linecurrent (section 13); indeed, (Kzdy')could have been considered directly asastraightfilament. The further integration involves An (x2+u2 )du=uIn(x2+u2 )-2u+2xtsuT1- J x P(*,y) FIG. 153LongThin Rectangular BusBar. withu=(y y'). Introducing thelower limit HI=y+hand theupperlimitu2=yhtonehas (19) where r\,a\and r2,a2arethevalues designated inFig. 15-3; theconstant value [4h(lInh)]hasbeenadded inthebrackets inorder tomakeAz= atx=y=0,though this isstrictly arbitrary. Thevector potentialisfinite atallfinite points, and Az=cons defines themagneticfield lineswhich arevery nearly ellipses near theorigin; theybecome practicallycircles farfrom thebar,because then on=a2,n=r2=r,where risthedis- tance ofPfrom theorigin.4 Themagneticfieldvector isobtained bydifferentiation ofthe vector potential, and,withthesimplifying notation from Fig.15-3, 4H.B.Dwight,Electr. Reu., 70,p.1087 (1917); A.R.Stevenson andR.H. Park, Gen. Elec. Rev., 31,p.159(1928); alsoHague,1344 p.283. Sec. 15]Conductors ofRectangular Cross Section canbebrought intotheuseful forms B=- =- ln^ * dy 2w2h TI dAz I159 (20) These expressions canform thebasis ofgraphical field analysis fortwo-dimensional fields (seesection 20)andcanreadily beused fortheevaluation offorce actions between busbars.5 P(*,y) FIG. 154Long Conductor ofRectangular Cross Section. LongConductors ofRectangular Cross Section. Forfinite rectangular cross section, asinFig.15-4, thevector potential Az canbewritten A.=- -/rffInr'dx'dy' 2ir4a6JJ where r'=[(x x')2+(y y')2 ]y *,and theintegration ex- tends overtheentire cross section; here, /dxdy'/4ab canbecon- sidered astraight filament inaccordance with (13-23). The in- 6H.B.Dwight, Electr. World, 70,p.522(1917); Stevenson andPark, loc. cit.; alsoHague,1344 p.337;E.Weber, Wiss. Veroff.a.d.Siemens-Konzern, 8,p.166(1929). 160 Fields ofSimple Geometries [Ch.4 tegrations give,6with thedesignationsofFig. 15-4, A.--~ {(x-o)(y-fc)In^-(*+o)(y-6)ln^ +(x+o)(+b)ln^_ (X_a)(y+6)ln^ ' ir- I(z-a)-a4)+(x+a)2 (3- which iscertainly notsimple eventhough thegeometry isoneof thesimplest. Themagnetic field linesAz=cons arevery nearly ellipses (seeHague,344 p.281). Itissimpler tocompute themagneticfieldvector bythegeneral- izedBiot-Savart law(6.25) forthevolume distribution than to differentiate A2]results arefound inStrutt,loc.cit.,andHagueB44 . Extension offieldandinductance calculations totwo ormore rectangular conductors ispossible,7buttheevaluation ofin- ductances issimplified butnotmade simple bytheuseofthe geometric mean distances (GMD), asdefined in(14), which havebeencomputedforseveral arrangementsoflong solid rec- tangular conductors,8oftwo parallel very thin square tubular conductors,9oftwoparallel rectangular tubular conductors,10and also oftwocoaxial square tubular conductors ofequal cross sec- tions;11thismethod hasalsobeen applied tolongconductors of structural shape.12Byutilizing complex function theory, in particular some elements ofconformal mapping, themultiple integralsin(146) canbesimplified.13 6M.Strutt, Arch.f. Elektrot., 17,p.533,and 18,p.282(1928); Hague,044 p.280; A.H.M.Arnold, Jl.I.E.E., 70,p.579(1931). 7H.B.Dwight,Elec. JL,16,p.255(1919); Ed.lloth, Revue gen.deI'elect., 44,p.275(1938). 8E.B.Rosa, Bull. Nail. Bur. ofStand., 3,p.1(1907); T.J.Higgins, Jl Appl. Phys., 14,p.188(1943); H.B.Dwight, Trans. A.I.E.E., 66,p.536 (1946). 9H.B.Dwight andT.K.Wang, Trans. A.I.E.E., 67,p.762(1938). 10T.J.Higgins, Trans. A.I.E.E., 60,p.1046 (1941). 11H.P.Messinger andT.J.Higgins, Trans. A.I.E.E., 66,p.328(1945). 12T.J.Higgins, Trans. A.I.E.E., 62,p.53(1943) and65,p.893(1946). 13T.J.Higgins, Trans. A.I.E.E., 66,p.12(1947). Sec. 15] Cylindrical Coils 161 Theforce actions between conductors ofsolid rectangular cross sections canbecomputed bydirect integration oftheforces be- tween very thinrectangular bars,14ashasbeenborne outbyex- perimental measurements.15Similar computations weremade for conductors ofstructural shape.16 Cylindrical Coils. Consider ahelical current filament as shown inFig.15-5, starting atAinthez-z-plane andforming an integral numberNofturns ofradius aandpitch p.ApointPonthis helix isthen defined by x=acos</>, y=asin0, 2=a</>tana if</>iscounted fromA,and iftana= p/2ira, withatheslope; theline ele- mentdsatPhasthecomponents dx=asin<t>d<j), dy=+acosd<t>, dz=atanad0 Inorder tofindthemagnetic field at anypointMontheaxis, atadistance cfrom theorigin intheplane ofA, onebestuses thegeneralized Biot-Savart law (6-22). Forthe axial component, observing that theradius vector risdirected from thecurrent clement topointM,thisgives ydx+xdyFIG. 15-5 Helical Current Filament. [x2+y2+(z-c)2 ]* 4?r/0=o a[l+(0tanac/a)2]^(22) With thesubstitution u= (</>tanac/a), theintegral canbe easily evaluated. Itsimplifies further tousep=2iratana,and 14O.R.Schurig andM.F.Sayre,Jl.A.I.E.E., 44,p.365(1925); also Hague,B44 p.338. 16C.J.Barrow, Trans. A.I.E.E., 30,p.392(1911). 16T.J.Higgins, Trans. A.I.E.E., 62,p.659(1943) and63,p.710(1944). 162 Fields ofSimple Geometries [Ch.4 tomeasure thedistance ofMfrom thecenter ofthehelix as 5=(AT/2)p c;theresult is M/r Np/2+b Np/2-b IU"~ 2p[.(a2+(Np/2+b)2]*+ [a2+(tfp/2-ft)']*J Atthecenter ofthehelix 6=and M AT/ Bzo= 2[a2+(JVp/2)2]^=" [1 which reduces foraninfinitely longcoiltotheuniform value /i//p=M#Z- The fieldalong theaxishasinthegeneral case of theshort coilalsocomponentsinx-and^-directions and isnot completely axially symmetrical because ofthe helical pitch. Smythe,A22 p.272,indicates theevaluation ofthese components;17 more details arefound inGrover843 . Asa 0,thepitch palsoapproacheszero. Onecanobtain, however, thefield ofauniform cylindrical current sheetfrom (23), ifonedefines Np/2=Iwith 21designating thelengthofthe coil. Because ofthesymmetry, Bzisnowtheonlycomponent and (24) givesitsvalue atthecenter ofthehelix as * (25) For Ia,along coil, thisreduces toBZQ=nNI/2a, given in many references; forthesimpler treatment, seeAttwood,A2 p. 263;Bennett and Crothers,A3 p.457;Mason andWeaver,A16 p.208;Maxwell,A17 II,p.310;Harnwell,A9 p.288; Cullwick,AG p.142;andStratton,A23 p.232. For this latter case, particularlyforclosely wound cylindrical coils, inductance calculations have beenmade; they establish thelinkage between one ofthecircular loops with another and integrate overthelengthofthecoil,avoiding infinities byassuming finite, butsmall, radius ofthewire.The integrals becomeelliptic, asinsection 14,andcan also involve Bessel functions. See Russel,B11pp. 108, 113,butparticularly GrovcrB43andRosa andGrover.18Thesamemethod isdirectly applicable tothe computationofthemutual inductance andforce actions oftwo 17SeealsoA.Russel, Proc. Phys. Soc.London, 20,p.476(1907). 18E.B.RosaandF.W.Grover,Bull. Natl. Bur. ofStand., 8,p.1(1912). Sec. 15] Toroidal Coil 163 very thin coaxial coils,19aswell astocoils ofrectangular cross section, either coaxial orparallel.20Forasingle coilofsmall but finite cross section, specific simplifications arepossible inevaluat- ingthemagnetic field, which arevaluable forsearch coilsand similar applications.21 Insome applications, particularly where magnetic effects are tobeobserved onparticles orsample materials,itisimportant to have aclosely uniform magnetic fieldoveragiven volume. One cancompute, then, thenecessary arrangement ofwindings ofnon- uniform coils.22 *b (a) FIG.15-6 Toroidal Coil: (a)rectangular crosssection, (6)circular cross section. Toroidal Coil.Atoroidal core ofmagnetic material ofpermea- bility ^canbewound withwire ideally sothatthewinding repre- sents auniform current sheath circulating about thecore in meridional planes asinFig.15-6. Inthisideal case, themagnetic field isentirely confined within thecore, themagneticfield lines areconcentric circles about thez-axis, andeach linelinkswith the entire current volume, which might becalled 2-irRiKi=27rR2K2=- NI,ifthere areNidealized turns each carrying thecurrent 7. Themagnetic fluxdensityisthen, irrespective ofthecross section 19T.H.Havclock, Phil. Mag., (6),15,p.332(1908). 20II.B.Dwight: Electrical CoilsandConductors, Their Characteristics and Theory, McGraw-Hill, New York, 1945; seealsoGroverB43andHakB4B . 21H.B.Dwight andG.O.Peters, Trans. A.I.E.E., 63,p.684(1944). 22L.W.McKcehan, Rev. Scient. Instr., 1,p.150(1936) and 19,p.475 (1948); J.Hak, Arch.f.Elketrot., 30,p.736(1936). 164 Fields ofSimple Geometries [Ch.4 ofthecore, givenbythevalue ofitslineintegral along acircle 2irr=NI (26) Thismeans that8$varies inversely asthedistance from the z-axis sothatthemagnetic flux isnotuniformly distributed over thecross section ofthecore. Fortherectangularcross section inFig.15-6aonecanfindthe fluxlinkages bydirect integrationinsimple manner, A=Nf*B+adr=N2IaIn|?JR\ 27r HI(27) since allthefield lines arecompletely linked with alltheNturns. Dividing by/,onereadily getstheinductance Onlyforb :R\canoneapproximate thelogarithm byb/Riand thusexpress Lproportionallytothecross-sectional area. Forthecircular cross section inFig.15-6b,theintegrationisa littlemore involved because theheight oftheindividual slice dr isvariable, namely, rRm+a \=N B2[a2-(Rm-r)2]*drURma Introducing (26)andtaking thesquare rootintothedenominator bysimply multiplying numerator anddenominator byitgive three terms which inthelimits reduce tothesimple form A=nN2I[Rm-VRm2-a2 ] (28) ifoneobserves sin"1 (1)=T/2, sin"1(-1)=-ir/2. Again,if a<&Rmjonecanapproximate thebracket bya2/2Rm,leading to proportionalitywith thecross-sectional area. These simple cases arealsotreated inSmythe,A22 p.288,andRussel,B11 p.71. Theinductance hasalsobeencomputed foracore ofrectangular cross section andawinding ofvariable andcomparatively large thickness;23inthiscasetheincomplete linkage ofthe field lines within thewinding hastobetaken intoaccount, andtheresult is bynomeans briefthough relatively simple. 28H.B.Dwight, Trans. A.I.E.E., 64,p.805(1945). Problems 165 PROBLEMS 1.Toevaluate theearth resistance between twoground electrodes a distance 2capart onemight replace theelectrodes bysemispheres ofradius aiand02asinFig. 11-1, where they-z-plane might represent theboundary between ground and air.Assuming uniform conductivity 7,findtheamount ofcurrent between theelectrodes from thesurface tothevariable depth h below ground along thex-y-plane. Atwhat depth willthecurrent have reached halfofthetotal value? 2.Compute theamount ofcharge induced within acircular area ofradius h ofaninfinite conducting plane byapoint charge -\-Qlocated atadistance hfrom theplane. 3.Find thefield distribution andtheinduced charge densities forapoint charge+Qlocated midway between twoconducting planes intersecting at anangle ofir/3. Verify thateach conducting plane willhave induced in it-Q/2. 4.Compute theforce exerted upon apoint charge+Qbyaninsulated sphere ofradiusRifthepoint chargeislocated atdistance b>Rfrom the center ofthesphere. What willbetheforce ifthesphere carries acharge Qi? 5.Find theforceandtorque upon anelectric dipole located atadistance b>Rfrom thecenter ofagrounded sphere,if(a)thedipole has itsmoment pdirected along aradius vector from thecenter ofthesphere; (b)thedipole moment isatright angle totheradius vector. 6.Verify theforce action (10-39) uponanelectric dipole inanon-uniform electric field; derive thetorque exerted upon thedipole inanon-uniform electric field. 7.Find thecharge density induced inaninsulated sphere ofradiusRby anelectric dipole located at6>Rfrom thecenter ofthesphere, forthetwo principal directions ofthedipole moment given inproblem 5. 8.Anelectric dipole ofmoment phas itsaxisdirected atanangle 4*against aninfinite conducting plane. Find thecharge density induced intheplane. 9.Inproblem 8,findtheforceandtorque exerted upon thedipole. 10.Referring toFig. 11-2, assume thesmall sphere ofradius aitohave a voltage Vapplied between itandground. Find thepotential induced onthe insulated small sphere ofradius a^with respect toground. 11.Assume thetwosmall spheresinFig.11-2torepresent source andsink, respectively, fortheflowofanincompressible fluidbounded bythey-z-plane. Compute thehydraulic resistance. Interpret theproblem asastationary electric current problem. 12.Find theapproximate distribution oftheinduced charge onasmall sphere ofradius alocated ontheplaneofsymmetry between twoorthogonally intersecting conducting planes andadistance h>5afromthem. Show that themaximum densityisgreater bythefactor (\/2 ^)than forasingle plane atthesame distance. 13.Assume three likesmall spheres ofradius alocated symmetrically with respect toeach other atdistances h>5a.Compute themutual capaci- tance coefficients. Find theapproximate distribution oftheinduced charge density oneach sphere. 166 Fields ofSimple Geometries [Ch.4 14.The finite lineinFig.12-1 carries atotal charge Qdistributed with a linedensity proportionaltotheabsolute distance from thecenter oftheline. Find thepotentialdistribution inspace. What arethepotential values along p=0?Find theequipotentiallines atlarge distance from thecharged line. 15.The finite line inFig.12-1 carries acharge distribution with aline density directly proportional todistance from thecenter, positivefor> andnegative for<0,sothatthetotal chargeiszero. Find thepotential distribution inspace. What arethepotential values along p=0?Find theequipotentiallines atlarge distance from thecharged line. 16.The finite uniformly charged lineofFig. 1246isparallel totwocon- ducting planes which intersect orthogonally and islocated intheplane of symmetryatadistance hfrom theplanes. Determine thecapacitance with respect totheconducting planes. Utilize (12-43). 17.Find theforce exerted bytheconducting plane (orground) upon the uniformly charged lineofFig.1246. 18.Why cantheexpression (3-20) fortheelectrostatic fieldenergy notbe appliedtoasingle verylong straight linecarrying auniform charge density X? 19.What isthecapacitance toground oftwo identical parallel charged rods asinFig. 12-46, each carrying thecharge Q/2andboth located atthe same height habove ground? How does itdiffer from thecapacitanceofan identical single rodatthesame height above ground? 20.Compute theforce perunitlength between twoparallel infinitely long cylinders ofradiiRIandR2<Riwiththedistance 2c>(R]_+#2)between their axes. Show thesimplificationsif(a)2c (Ri+#2), or(6)RI=R2j or(c)Ri=R2and2c Ri. 21.Find thepotential distribution caused byauniformly charged very thin circular disk ofradius a. 22.Anelectrostatic voltmeter canbeconstructed based ontheforce action between two finite, charged cylinders enclosing each other. Find theforce perunitlength forthearrangement inFig.1266. 23.Oftwosemi-infinite coaxial cylinders ofradiiRI>R2the firstextends from2=0 to2= ooand isfixed; thesecond extends from z= cto 2=+ooandcanmoveparallel totheaxis. Ifthecylinders have potentials <$iand*2icompute theforce action between them. Hint: usetheprinciple ofvirtual work. 24.Threeparallel very long wires ofequalradiiRareatthesame height above ground. Find thecapacitance coefficients forthewires iftheir dis- tances 2c R. 25.Three parallel verylongwires above ground form athree-phase trans- mission line.What conditions must besatisfied inorder topermit thedefini- tion ofarealcapacitance perwire astheratio oftotal charge perunitlength ofthewire toitsphase voltage? 26.Find thepotential distribution atvery large distance fromnparallel wires constituting apower transmission linesystem. 27.Find theaverage capacitance togroundofonewire ofatransmission system, taking intoaccount itssagbetween twosupport towers. 28.Find thecapacitance between asmall sphereofradius 6located onthe Problems 167 center lineofathin circular ringofcharge andthis ring. Assume thecircular loop asinFig.129andthedistance ofthesphere asz=h. 29.Find theratio ofmaximum tominimum charge density forthecircular ringofcharge inFig.129. 30.Acircular loop ofradius acarrying current I\islocated midway between two parallel wires spaced 2c>2aapart andcanrotate about its diameter parallel toand inthesame plane with thewires. Find thetorque asafunction oftheangle between theplane oftheloopandtheplane ofthe wires ifthewires carry currents /2-Which istheposition ofstable equi- librium? 31.Find themagnetic field distribution atlarge distance fromnparallel wires, which form acomplete transmission system. Demonstrate that the fieldcanbeapproximated bythat ofanequivalent dipole lineandgivethe location ofthelatter. 32.Give themagnetic field distribution farfrom therectangular current loop infig. 13-2. Demonstrate theequivalence with thefield ofamagnetic dipole whose moment is4a&7. 33.Find theinductance ofathin elliptical current loop ofmajor and minor axisaandb,respectively, andofwirediameter d b. 34.Find themutual inductance between two parallel pairs ofdipole line currents asafunction oftheangle between their respective planes. 35.Prove thatthere isnomutual inductance between twopairsofparallel linecurrents d=/iand /2spaced 2aand 25,respectively, andcrossing or- thogonally. Assume that theplanes ofthewire pairs intersect along aline parallel tothe firstpairandatadistance 2afrom thenearer wire. 36.Demonstrate theequivalenceofthecircular current loopwithamag- netic shell ofdipolemoment n/perunit area. Find themagnetic field ofthe magnetic shellandshow theidentity with (13-26). 37.Find themutual inductance oftwoidentical circular loops ofradii a lyinginparallel planes ofsmall spacingc a. 38.The space between two parallel conducting planesisfilled with a dielectric whose dielectric constant varies linearly along thenormal tothe parallel planes from aminimum value eion*ito62on*2-Find thecapaci- tance perunitarea ofthiscondenser. 39.Inacoaxial cylindrical system, theinner solidmetal cylinder ofradius Riiskept attemperature TIbyjouleheatfrom acontinuous current; theouter metal cylinder (sheath)iskept attemperature T%.Find thetemperature dis- tribution andthethermal resistance ifthethermal conductivity oftheinsula- tionvaries linearly from alarger value kiatRItoasmaller value kzatR2. 40.Aparallel plate condenser ofspacing dbetween theconducting plates isfilledwith amedium ofdielectric constant e=e'+(ei e')exp (x/d). Find thecapacitance perunit area. Give thecharge density forapotential difference $1 *2applied totheplates. 41.Inacoaxial cylinder condenser two different imperfectdielectrics are used inconcentric layers asinFig.143,where theinner layer haselectrical conductivity 71andtheouter layer 72-Find thetotal current flow ifapo- tential difference *j*nisapplied. Find thepotential distribution and thesurface charges. 168 Fields ofSimple Geometries [Ch.4 42.Inacoaxial cable oneincreases theinductance inorder toimprove the transmission characteristics bywrapping amagnetic tape ofhighpermea- bilityupon theinner conductor. Assuming auniform layer ofpermeability Hzandofthickness tontheinner conductor ofradius RiinFig.15-2,what is theincrease ininductance perunitlength? 43.Wrapping athinmagnetic tape ofhigh permeability M2upon thetwo conductors ofFig.15-1withRI=R2,what willbetheapproximate increase ofinductance perunitlength? 44.Two parallel identical, long thinrectangular bars (Fig. 15-3) arear- rangedinparallel planes. Find theforce action between them iftheir distance isc<2/i, 45.Find theinductance ofthetwobars ofproblem44. 46.Findtheforce action between thetwoparallel conductors ofFig.15-1. 47.Athin flatpancakecoilcanbemade intwolayers soconnected that thecurrent flows inboth layers inthesame direction; theleads canthenbe ideally twisted sothatthecoilcanbereplaced byuniformly distributed circular currents. Find themagnetic field distribution ofthispancakecoil ifthe inner radius isR\andtheouter radius Rz- 48.Find themutual inductance oftwo parallel coaxial identical pancake coils asinproblem 47iftheir center distance ish. EXPERIMENTAL MAPPING METHODS The analytical expressions forthe field quantities insimple geometries arefairly simple themselves, sothat their usehas become reasonably common. Inmany instances, theycanbeused asfirstorqualitative approximations formore complex field dis- tributions. Where, however, quantitative values ofgreater accuracy arerequired,itbecomes necessary toobtain solutions fortheexact geometry with theattendant complications of analytical treatment. Toescape therigor ofadvanced mathemat- icalmethods, many experimental methods have been developed, inmost instances forspecific applications. These experimental methods are,ofcourse, also ofgreat value inaiding thevisualiza- tion offield distributions andaschecks onanalytical solutions. 16EXPERIMENTAL MAPPING OFELECTROSTATIC FIELDS Forthequantitative mapping ofelectrostatic fields,itsuffices tohave amap either ofthepotential distribution orofthe field lines. Two-dimensional geometries orthose with axialsymmetry aresimplest torepresent, because onesingle plane section gives alltheinformation needed. Forgeneral three-dimensional field distributions, oneneeds several tomany plane sections and, in addition, acareful interpretation oftheindividual maps inorder toconceive theactual field picture. Mapping ofPotential Distributions. By electrostatic induction, anisolated uncharged small metallic probe brought intoanelectrostaticfield, asshown inFig. 16-1, willexperience acharge separation butretain zero resultant charge;itwillalso 169 170 Experimental Mapping Methods [Ch. 5 assume thelocal potential value that existed, before itsinsertion, orapproximately theaverage value over itssurface ifitssize cannot bedisregarded. Connecting anelectrostatic voltmeter V totheprobe, asshown inthedottedline,willplace thecapacitance Cofthevoltmeter inparallel with thecapacitance C\pexisting between probe andconductor1,draw offaconsiderable partof thenegative induced charge oftheprobe, andleave itessentially positively charged, thus severely distorting theoriginalfield dis- tribution andaltering thelocal potential. Opportunity must be FIG. 161Potential Measurement byProbe. given, therefore, toexpel thesurplus positive charge, sothat<>p isidentical with theundisturbed local potential, before measure- ment canbemade. Then, withproper provision andwithsome experience, theprobe canbemoved soastokeep thispotential constant;itwillthus describe anequipotential surface. Of course, theleadtotheprobe can itself actdisturbingly; usually, local shieldingoftheleadwithanisolated braid avoids anyserious effects. The simplest probe arrangementistheuseofasmall metal sphere andconnection toground atAsoastorelease some ofthe surplus induced charge; seeMaxwell,A16 I,p.340. Although themethod issatisfactory todetermine therelative potential at theparticular point,itisnotapplicable when ground potential isused elsewhere inthesystem. Insuch cases, onecanusea smallBunsen burner asinPohl,A2 p.65,whereby theflame acts astheprobe andthehotgases provide anautomatic dissipator ofthefreeinduced charge; theburner itselfassumes thepotential atthepoint atwhich itislocated. Asimilar principleisinvolved Sec. 16] Mapping ofPotential Distributions 171 inthelessconvenient water-drop probe, inwhich water dripping continuously through ametal tube ataslow rate dissipates the surplus charge.1Caremust betaken, byappropriately shielding theprobe, toavoid field distortion byit. Another typeistheemission probe forfields invacuum.2In this caseasmall metal plate, properly coated with emissive material,isused asprobe, heated byaseparate electric heating coiltoatemperature highenough tocause thermionic emission andthus release thesurplus induced charges. Since thermionic emission isprimarily electronic, theelectrostatic voltmeter ofFig. 16-1must beconnected tothenegative conductor. Thisprobe hasbeen used extensively toexplore the field distribution near electrodes, particularly grids, within vacuum envelopes oringas discharges. Special vacuum- tight seals ofsimple construction must beprovided toallow foradjustment ofprobe location. The practical useofthisprobe requires experience, since theemitted electrons may collect asspace charge close tothemetal probe and cause distortion, especially inregionsofweak electric fields. Similarly, onemust guard against theemitted charges condensing upon one ofthemain electrode surfaces andupon dielectric supportsortheenvelope, producing considerable distortion ofthe potentialdistribution. Measurements aresomewhat slow, since itrequires appreciable time forthethermionic probe toacquire thelocal potential. Foracoaxial cylindrical diode, thepotential distribution has beenmeasured with averyfinetungsten-wire probe parallel to theequipotentialsurfaces.3Theanode iscoated onitsinner surface withafluorescent substance(e.g., willemite) which glows under thebombardment oftheelectrons emitted from thecathode. Iftheprobe wirehasthesame potential aslocal exists before insertion oftheprobe, theelectron stream from thecathode remains uniform, andtheanode illuminates uniformly; otherwise, theprobe wire causes ashadow ontheanode which isreadily observable withopen construction ofthetube. Thus, thepoten- tialoftheprobe canbeadjusted fordisappearanceoftheanode 1C.H.Lees, Proc. Royal Soc.,A91, p.440(1915); alsoA.Wigand, Ann. d. Physik, 76,p.279(1924), and85,p.333(1928). 2I.Langmuir,Jl.FranklinInst., 196, p.751(1923); alsoN.Semenoff and A.Walther, Zeits.f.Physik, 17,p.67(1923); A.Walther andL.Inge. Zeits.f. Physik, 19,p.192(1923). 3D,E.Kenyon,Rev. Scient. Instr., 11,p.308(1940). 172 Experimental Mapping Methods shadow. Inaparticular diode, theprobe wirewasstrung ina pivoted frame, allowing exploration ofthepotential distribution under operating conditions. Comparisonofthetheoretical dis- tribution forconditions oftemperature limitation andspace charge limitation with themeasurements wassatisfactory. Thismethod can, ofcourse, beused onlywhere thepotential distribution is constant along thelengthofthewire. Forlowaudiofrequencies, theratio ofthecapacitances C\pand C2p,between theprobe andthemain electrodes, canbetaken as FIG. 162Capacitance Probe forPotential Measurements. ameasure oftheprobe potential anddirectly indicated,4asshown inFig.16-2. Thecalibrated potentiometerissettosome definite ratio ofitstworesistance sections. Silence inthetelephone of theamplifier circuit willoccur iftheprobeisataposition sothat theratio ofthetwopartial capacitances with respect tothetwo electrodes becomes equal totheresistance ratio ofthepotentiom- eter; toavoid extraneous influences, thelead totheprobe must again becarefully shielded. Forbest sensitivity thecapacity of theprobe ought tobefairly large; this,however, must berecon- ciledwith thefactthattheprobe itselfmust besmall soasnotto distort the field distribution. Thefrequency isadvantageously chosen between about 500and1000 cycles persecond, although with aproper amplifier even commercial power frequencies are employable. Forvery high voltages thepotential distribution over the surface ofaxially symmetrical insulators canbedetermined con- 4N.Semenoff andA.Walther, Zeits./.Physik, 19,p.136(1923). Sec. 16] Utilization ofPotential Maps 173 veniently according tomethods developed byA.Schwaiger,317 p. 184.5Awireloopisplaced around theinsulator andacalibrated spark gapconnected between thiswireandtheoneelectrode of theinsulator; varying thespark gap setting orthepotential applied totheinsulator untilbreakdown occurs gives thepoten- tialdifference between thewire (orthelocal point onthesurface oftheinsulator) andtheelectrode.If,ontheother hand, the spark gapisconnected between thewireloopandthecenter tap ofacalibrated potentiometer, asinFig. 16-3, anullmethod can FIG. 163Measurement ofPotential Distribution withSpark Gap. bearranged. Thus, onecanvary thepotentiometer tapuntil thespark gapelectrodes canbebrought very close together without spark; theneeded potentiometer setting indicates thevalue of thelocal potential. Another method proposed bySchwaigerB17forextremely high voltages uses theprinciple oftheelectroscope. Small cotton or silkfibers, orpaper pieces, arefixed toanisolated wirelooponthe insulator;ifthevoltageisapplied totheinsulator, theelectro- static forces willcause these fibers tomake anangle with the insulator surface which canbeobserved withatelescope.Ifthe same angleisthenreproduced with aknownvoltage applied to thewire probe, thisvoltage willindicate thelocal potential on theinsulator surface. Forconvenience andrapidity ofmeasure- ments anumber ofexploring wire loops with indicators canbe used simultaneously. Utilization ofPotential Maps. The direct measurement of thepotential distribution leads toaplot oftheequipotential lines; inorder tocomplete thefieldpicture,itisthen necessary toplotthe field lines asthefamily oforthogonal curves. No 6Elektrot. undMasch., 37,p.569(1919); alsoA.Fontvieille, Revue gen. deI'elec., 10,p.599(1921). 174 Experimental Mapping Methods [Ch.5 difficulty should beencountered iftheequipotentiallines originally were chosen close enough. Inorder toobtain quantitative values forthefield strength,it isbest toplotonaseparate graph asabscissa distances along a Distance alongfield line FIG. 164Field Strength Distribution Obtained from Potential Graph. particularfield line(stretching this field lineintoastraight line), and asordinate theobserved potential values, asinFig. 16-4. Theapproximate potential distribution isobtained bydrawing a smooth curve through these distinct points. UsingE=(5*/5s), theaverage value ofthe field strength caneasily becomputed foreach oftheintervals 5s;inFig.16-4 these values areindicated atthecenter pointsoftheintervals 5s.Anapproximatefield strength distribution isobtained byagain drawing asmooth line through these distinct points. Extrapolation tothesurfaces of theelectrodes gives theapproximatefieldstrength values there. Sec. 16] Measurement ofSurface Charges 175 Knowing theelectric fieldstrength distribution, onecaneasily obtain thedisplacement vector ordielectric fluxdensity bysimply multiplying the field strength values with theabsolute dielectric constant ofthemedium. This gives alsothelocalcharge densities onthesurfaces oftheconductors, since they areequal tothe magnitudesofthedisplacement vector atthesurface ofthecon- ductor. Measurement ofSurface Charge Distributions. The local charge density onconductor surfaces canbestbedetermined by direct contact ofanisolated small metallic diskprobe with the conductor surface, sothat itassumes itspotential and carries the localcharge density according totheequilibrium distribution.If, then, theprobeiscarefully removed perpendicular tothesurface, thecharge remaining onitisequal tothecharge over thesame area oftheconductor, anddivision bythissmall area gives the charge density ingood approximation. Obviously, theaccuracy willdepend onthemanipulation andontherelative size ofthe probe, aswellasonitsshape. Themost suitable form ofprobeisasmall disk, preferably of thesame local surface curvature astheconductor, andwithan insulated handle. Thedisadvantagesoffitting andhandling such probe arc,however, considerable. Using, then, asmall flat circular diskprobe ofradius randthickness,Maxwell,A17 I,p. 344,derived therelation where amisthemeasured and <rthetruevalue ofthesurface charge density ascorrected forthefinite thickness oftheprobe. Fora small sphere ofradius aasprobe, Maxwell (loc. cit.)investigated thelocal field distortion producedifthissmall sphere beincontact with thesurface oftheconductor which hasaradius ofcurvature batthepointofcontact. The localcharge density afollows from themeasured charge qofthesphere as .--^T (2)tar Theknowledge ofthecharge distribution onthesurface of conductors isequivalent toknowledge ofthedielectric fluxdensity andthus ofthefieldstrength atthesurface oftheconductor. The 176 Experimental Mapping Methods [Ch. 5 latter isofparticular interest whenpredicting corona andbreak- down limits. Thevalues ofsurface field strength obtained by direct measurement canbecompared withtheextrapolated values from thepotential graph. Mapping ofField Lines. Visual records offield line dis- tributions areobtained inasimple manner bycutting theelectrodes oftinfoil,pasting them inproper relationship onsmooth paper, andthenpouring freshly powdered gypsum crystals onthepaper; tapping thepaper after thevoltage hasbeen applied tothetin foils willassist inhaving theneedle-like gypsum particles arrange themselves inthedirection ofthefield lines.6Instructive photo- graphs ofsimple geometries aregiven inPohl,A2ChapterII. Only freshpowder should beusedbecause gypsumishygroscopic. Insimilar manner, onecanusecotton fibers,7small pieces oflight paper, orsmall silkpieces asillustrated bySchwaiger,B17 p.184. Very interesting also istheuseofJ^percent crystalline quinine sulphate inturpentine, leading toasedimentation ofthecrystals along thefield lines.8Here, theelectrodes aremetal pieces ina shallow tank. Suspensions ofshortandcoarse artificial silkfibers incarbon- tetrachloride have been used togetphotographs oftheentire fieldgeometry onlarge-scale models.9Improved photographs were obtained withatank illuminated frombelow and filledwith twoliquids, carbon tetrachloride andeocene, separated bygravity, with thesilkfibers floatingintheplane ofseparation, thuspermit- tingasharp focussing ofthecamera. Itisimportant toselect a proper voltage, since toohigh avoltage willcause thefibers to drift rather quickly. Forhigh voltages andanytype ofelectric fieldwith axial symmetry, amethod developed byM.Toepler10isadvantageous. Theprobe consists here ofasmall pieceofstraw about 1in.long, provided with asteel needle axis ofabout J^in.suspended ona 6C.Fischer, Phys. Zeits., 9,p.221(1908). 7D.Robertson, Edinburgh Proc., 22,p.361(1889); A.Pen-in, Bull. Soc. Internationale desElectriciens, 6,p.83(1889). 8M.Seddig, Phys. Zeits., 6,p.403(1904); Ann. d.Physik, 11,p.815(1903), where anexcellentbibliographyisgiven. 9R.H.George, K.A.Oplinger, andC.F.Harding, Butt,No. 29,Engg. Exp. Station, Purdue Univ., Lafayette, Ind., 1927. 10V.Regerbis, E.T.Z., 46,pp.298,336(1925); thisreference gives several excellent fieldpicture reproductions andagood bibliography. Sec. 17] Mapping ofField Vector B 177 silkthread, sothatthestraw canrotate inavertical plane asshown inFig.16-5andassume thedirection ofthefield line. Theprojec- tions ofthevarious positions ofthestrawupon ameridional plane (most conveniently obtained bytracing with pencil theshadow pro- duced byparallel light) giveanarray offield lineelements which easily canbecomposed intocomplete field lines. Theadvantage of themethod istherapidity withwhich thefield lineelements canbe obtained, although thecomposition ofthe field picture requires experience. With allthemethods outlinedabove, oneobtains only thegeometry ofthefield linesandhastocompute thevalues ofthe field strength byconstructing theorthog- onal potential linesandthen using the same method, asshown inFig. 16-4. 17-EXPERIMENTAL MAPPING OFMAGNETIC FIELDS Many experimental methods have con-FlG 16.5straw Probe centrated onthedirect measurement ofthe forField Mapping, magnetic field vector B,since thevector potential Aisnotinitself amenable tomeasurement, indeed, is notanobservable physical quantity. SinceBcanconveniently bemeasured directly intheambient medium (incontrast tothe electric field vector E),problems ofcoildesign fordesired field distributions, ofcoredesign inferromagnetic circuits, andofproper linkage incoupled circuits have been solved frequently bythe construction andextensive study ofmodels asfarasapplicable. Theunfortunate fact ofvariable permeability ofmost magnetic materials hasmade imperative field exploration forprecise per- formance predictions. Mapping ofField Vector B.Themostcommon method of measuring the field vector Bisbymeans ofasmall search coil connected toaballistic galvanometer bymeans ofbifilar leads so astoavoid uncertain orvariable magnetic linkage over part of the circuit. Inexploring magnetic fields ofpermanent magnets, thesearch coil isquickly removed from thetestposition1toa final position 2,andthemaximum reading ofthegalvanometer is recorded astheintegral oftheelectric current intheclosed circuit. This current isgivenbyi=v/(R+Rg\where v=-N(d3> m/dt) 178 Experimental Mapping Methods [Ch.5 istheinduced voltage, andRandRgarethecoilandgalvanometer resistances, respectively; Nisthenumber ofturns ofthesearch coil,and $>mtheaverage magnetic fluxlinked withaturn.The maximum deflection ofthegalvanometer records effectively *' -_ AW/, ~ N/1\ FTt.T\BHh*ifthetimeconstant ofthecircuit isconsiderably smaller than that ofthegalvanometer. Ifthecoil is removed from thetestposition with flux$mltoaposition ofzeromag- neticfield, then thegalvanometer indicates directly the local com- ponentBnnormal tothecoilarea; ifthe coilcanbeflipped inplace, then thegalvanometer indicates 2Bn.Assume, asinFig.17- 1,that FIG.17-1 Average Linkage ofthecoil iscylindrical ofinner radius Search Coil inMagnetic Field. aouter radiusb,andheight h}then theturns perunitareaaregiven by N/h(b a);theaverage fluxlinkage forlocally uniform fieldBn isthen7 k .1 A=,^^-J^+rf+a,(2) sothatonecanalso define aneffective area ir/3(b2+ab+a2 ) ofthe coil. Turning thecoilinthree mutually orthogonal direc- tions, onecangetthethree coordinate system components ofB. Ontheother hand, onecanattempt tofindthedirection ofmaxi- mum indication which isorthogonal tothefield lineatthepoint ofmeasurement. Ofcourse, thecoilareamust bechosen smallenoughinorder to justify theassumption oflocally uniform fields. Forelectron optical systems, search coilsassmall as26=h=0.04cm,N=100 turns, with wire of0.002-cm diameter, have beenused1and di- mensions of2b=0.1cmarerather frequent; usually,inelectron lenses itisnecessary only tomeasure the fieldalong theaxis of symmetry, sothat themanipulationissimplified; seeAppendix 4,B,c,andalsosection 30.Formeasurements onlarger magnetic 1J.Dosse, Zeits.f. Physik, 117, p.437(1941). Sec. 17] Mapping ofField Vector B 179 systems onechooses conveniently aneffective area of1cm2 ;then thefluxvalue isidentical with thevalueBnin(2). Ingeneral, direct determination ofthedirection offield lines isnotvery satisfactory:itisusually more timeconsuming than measuring inthree mutually orthogonal directions; ontheother hand, the limited sensitivity ofthe ballistic galvanometer canintroduce a serious error forlowvalues ofthefieldcomponents. Itis,there- fore, advisable tocheck thefield distributions bymeans ofiron filings asindicated below. Inthecase ofelectromagnets,itisnotnecessary tomove the search coil;theexcitation current ofthemagnet canbeturned on oroff(over ashunt resistance toavoid arcing). Forsmaller units, theexcitation current canbereversed, leading then totwice the value ofQin(1),since&mz=~$mi-Ina~cmagnets, thesearch coilhasinduced initana-ccurrent which canbeamplified and readonavacuum tubevoltmeter orobserved onanoscilloscope; calibration isusually necessary tominimize errors. However, in thiscase, thecoilcanreadily beturned untilmaximum indication occurs, defining then thedirection ofthefield lines inrather con- venient manner. Toincrease thesensitivity ofthesearch coilarrangement in stationary magnetic fields, onecanprovide forrotation about an axispreferably normal tothedirection ofthe field lines. The fluxlinkage then varies sinusoidally andcauses ana-ccurrent in the coilcircuit, which canagain beamplified electronically and readonavacuum tubevoltmeter. Aninteresting andvery precise arrangement wasused inthemagneticfieldmeasurements pre- liminary tothedesign ofsynchrotron magnets.2Two coils of 26=0.3cmwere driven bythesame lucite spindle at1750rpm, oneexposed tothefieldtobemeasured, theother inthefield ofan auxiliary electromagnet with rotatable axis. The coilswere con- nected inseries opposition sothat differential readings resulted which were minimized byrotating theauxiliary electromagnet. Theoutput gave, then, thechange insearch coil fieldascompared with thefixedandopposing auxiliary coilfield,andtheangle of rotation ofthemagnet indicated thechange insearch coil field direction. Differential changes equivalent to0.1percent ofthe fieldvalue could bemeasured reliably. 2W.C.Parkinson, G.M.Grover, andH.R.Crane, Rev. Sclent.Instr., 18, p.734(1947). 180 Experimental Mapping Methods [Ch.5 Anentirely different method ofmeasuring Bisbymeans of thechangeofelectrical resistance which certain metals like bismuth, antimony, andtellurium experienceinamagneticfield.3 Thelargest effect isobserved inbismuth;since itcanbeproduced inthin wires andwound inspirals,ithasbeen usedmost fre- quently,4thoughitscharacteristics aresomewhat dependent on ambient factors such astemperature, stresses, and orientation. Itis,therefore, advisable tocalibrate these spirals before andafter use inorder toassure reliability ofthemeasurement. Their very simple useasonearm ofaWheatstone bridge makes them valuable tools forquick surveys ofrelatively strong magnetic fields.Amore elaborate andautomatically temperature-com- pensated bridge-type fluxmeter hasbeen developed byG.S. Smith.5 TheuseoftheHall effect inasmallgermanium probe forthe measurements ofmedium-range magnetic fields hasbeen de- scribed recently.6 Mapping ofMagnetic Field Lines. Forthestudy ofthe overall geometryofmagnetic fields, which canbesignificantly represented inplane sections such asintwo-dimensional geometries orgeometries with axialsymmetry, theuseofironfilings onpaper isindispensable. Excellent reproductions ofsimple fields are found inPohl,A2Chapters I,III,andV. Toobtain apermanent record ofthefieldlines, onecanplace a white carton coated with paraffin between heavy metal blocks constituting amodel ofthemagnetic andconducting materials. Pouring theironfilings ontheparaffin andletting them orient in themagnetic field, onecanthenheat theparaffin superficially so thatthefilings sink into itssurface. Thismethod hasbeen ex- tensively used forthestudy ofmagnetic field distributions in electrical machines7under varying conditions ofexcitation ofpole andarmature windings. Theiron filings give, ofcourse, onlytheoverall geometry ofthe 3L.L.Campbell:Galvanomagnetic andThermomagnetic Effects; Longmans, Green,NewYork, 1923. 4G.Bublitz, Arch. f.techn. Messen, No. 83,V391-2, May 1938. *Electr. Engg., 56,pp.441,475(1937); also Bull. No. 103,Engg. Exp. Station, Univ. ofWashington, Seattle,1940. 6G.L.Pearson, Rev. Scient.Instr., 19,p.263(1948). 7E.Roth, Bull. soc.franc. 6lec., 7,p.13(1937); some reproductions in Elektr. undMasch., 65,p.338(1937). Sec. 17]Measurement ofMagnetomotive Force 181 field; theydonotdirectly indicate themagnitude ofthe field vector B.Since, however, outside ofcurrent-carrying conductors, theconcept ofthemagnetostatic potential canbeused, asshown insection 6,itispossible toconstruct theorthogonal equipotential lines. Fortwo-dimensional and axially symmetrical fields, one canthenobtain quantitative values byusing thesame construction asisindicated insection 16fortheelectrostatic field. Toascribe definite values totheequipotential lines, onemustbeabletoestab- lishanabsolute scalesomewhere inthefield, asneeds tobedone also inthegraphicalfield plots explained insection 20orinthe experimental methods described below. Ballistic galvanometer FIG.17-2 Double-layer Coil forMeasurement ofMagnetostatic Potential Difference: (a)general view, (b)connection between layers at1. Measurement ofMagnetostatic Potential Differences. With aspecially constructed double-layer coil ofconsiderable lengthIbutvery small cross section, asindicated inFig.172,one canmeasure themagnetostatic potential difference ormagneto- motive force produced byanarbitrary conductor arrangement.8 Theinner layerisacontinuous helicalcoil,wound from2towards1, whereas theouter layer ontheleft-hand side iswound from 1 towards thecenter; Fig. 17-26 indicates thecontinuity ofthe wirefrom inner layeritoouter layero.Theouter layer onthe right-hand side isalsowound from 2towards thecenter, where thetwoends serve asbifilar leads toaballistic galvanometer.If thetwoends 1and2touch, thecoilforms geometrically acircular loop; however, there isnometallic contact between 1and 2,and 8W.Rogowski andW.Steinhaus, Arch.f.Elektrot., 1,p.141(1912); see also Pohl,A2Chapter IV, forexcellent demonstrations ofitsuses; also KupfmuUer fA14 p.143. 182 Experimental Mapping Methods [Ch. 5 anycurrent inthecircuit closed through thegalvanometer flows inthetwolayersiand oinopposite directions, thusproducing nonetmagneticfield. Ifthis coil isbrought intothefield ofacurrent andthecurrent isinterrupted, theballistic galvanometerwillindicate thechange inmagneticfluxlinked bythecoilasin(1).Ifthenumber of turns perlayer perunitlengthisnythen thecoillengthdlhasa fluxlinkage dA=SBn2ndl where5istheaverage area ofinnerandouter coilsection, andBn thecomponentofthemagnetic fieldnormal totheelement dl. / FIG. 173Measurement ofmmfProduced byCurrent Loop. The ballistic galvanometer measures thetotal fluxlinkage or,in accordance with(1), where (6-4) hasbeen introduced. Thus, thisdouble-layer coil measures directly themagnetostatic potential difference, in- dependentofitsownshape, between anytwopoints ofspaceitis capableofreaching. Bending thecoilintoacircle linkingitwitha circular current loop, asinFig.17-3,stillmeasures (JF23\)=/ theresult oftheline integralofHcarried right tothebarrier surface ofFig.6-1, since theends 1and2ofthecoildonotmake metallic contact. Ofcourse,itisnotpermissible tobend thecoil intoadouble loop circling thecurrent Itwice, sincethen itphysi- cally penetrates thebarrier surface;thiswould voidtheuniqueness condition ofpotential values. Sec. 18] Two-dimensional Current Flow 183 With oneendkept fixed inspace, theother coilendcanbeused tomapthepotential distribution relative tothefirstpointandthus introduce theabsolute scaleneeded forthequantitative interpre- tation offield linedistributions (seeabove). Measurement ofFlux Linkage. Inorder tocheck linkage orleakage calculations,itisfrequently desired tomeasure flux linkages. Themost accurate results incircuits without ironare obtained byplacing afinewire assearch coilright alongside the windingforwhich thelinkage should bemeasured andusing the ballistic galvanometer method, asforexample forhigh-frequency alternators.9Ifiron ispresent,itssaturation characteristics as well aseddy current effects have tobetaken intoaccount orat least qualitatively kept inmind. Ina-cmagnetic circuits, thesearch coilcanmeasure linkages under direct operating conditions, asforexample inslots of electrical machines,10and ithasbeenused asavoltmeter loop in high-voltage transformers after appropriate calibration. 18-UTILIZATION OFFIELD ANALOGIES Aspointed outinChapter 3,several other fieldphenomena besides electrostatics andmagnetostatics show thesame basic relationships between thecharacteristic field vectors, sothat close analogies canbeestablished, assummarized intable 9-1.Any solution foroneofthefieldtypes canreadily betranslated intoa solution fortheother field types. Intheexperimental investiga- tion, thispermits welcome substitutions ininstances where the originalfield isdifficult,ifnotimpossible, toexplore. Two-dimensional Current Flow. Current flow inthinplane conducting sheets (oruniform thin metallic films) isgenuinely two-dimensional andcanreadily beused inaccordance with section 8torepresent electrostatic ormagnetostatic field distributions in geometries which overthecenter portions atleastcanbeconsidered astwo-dimensional (see specifically sections 12to15forillustra- tions). Asanexample, take the dielectric fieldbetween two parallel cylindrical conductors within agrounded sheath, asin Fig. 18-1; assume alsotwo differentdielectrics, gutta-percha of 9N.M.Oboukhoff, Engg. Exp. Station Publ. No.40,Oklahoma Agricultural andMech. College, Stillwater, Oklahoma, June 1939. 10H.Rothcrt, Arch.f.Elektrot., 32,pp.306and372(1938). 184 Experimental Mapping Methods [Ch. 5 relative dielectric constant er=4close totheconductors and rubber with er=2.5asfiller. Inorder tomeasure the field strength distribution, onecanconveniently useanoversize model with thesame geometric proportions. Torepresent the two dielectrics indirect contact, one selects twometals ofthe Sheath $= FIG.18-1 Model ofTwo-conductor Cable withTwo Dielectrics. same ratio ofconductivities, saycopper andaluminum. Accord- ingtoAttwood,A2 p.118,onehasasratio ofresistivities PCu PAI1.915 3.14=0.61 ascompared with 2.5/4=0.625 forthe dielectric constants. Using thinmetal disks ofcopper andfitting these withgood contact (preferably brazing) intoanequally thinsheet ofaluminum, one hasthetwo-dimensional model ofthe dielectrics. Placing the composite sheet between copper blocks ofabout J^-in. length, representing theconductors inproportional sizes, completes the overall model. Ifonenow applies apotential difference with grounded center point between thecable conductors andconnects thesheath tothiscenter point, onecanexplore thepotential lines inthecurrent sheetbymeans ofaneedle contact andthusobtain practically thesame result asintheoriginal dielectric field. Sec. 18] Two-dimensional Current Flow 185 Onecanthen either tracebyhand theflowlines, which are, of course, orthogonal totheequipotential lines, orusethesame thin metal s^eet composition toexplore the field lines astheequi- potentiallines intheconjugate electrode arrangement. Inorder todothelatter, onehastoplace electrodes along properly selected field linesand restrict thecurrent flowalong theformer electrode surfaces tosatisfy theboundary conditions. Intheexample of Fig.18-1onewould cutoutthethinmetal sheet along thetwo circles $iand<nconstituting thecable conductors, andalsocut along theradius pofthesheet; this willmake these circular peripheries flow lines (previously equipotential surfaces), since the current cannot have anormal component there pointing outof themetal sheet. Onewould thenclamp thesheet between thin vertical" electrodes along thelines 3'-2', I'-l", 2//-3// ,applying tothecenter oneapositive potential *iandtotheoutertwothe negative potential *n; oronecould putanarrow slitinthemetal sheet along thelinel'-l" andapply attheupper edge4>Tandalong thelower edge$n.Theconjugate electrode arrangementwill generally giveabetter graph oftheflowlines, particularly forthe singular lines, than afree-hand plotcanprovide. Measuring thetotal current permits evaluation oftheresistance perunit thickness ofthesheet, which canbeconverted into ca- pacitance perunit length byusing (8-11), namely, C=e/7/2, where eandyrefer tothesame setofequivalent materials, either gutta-percha andcopper orrubber andaluminum. Theproof is thesame asthat for(8-11). Thismethod,ofcourse,isapplicable only tomodels oftwo- dimensional fields but israther convenient forsingle dielectrics. One difficulty incomposite fields isfinding metals ofconductivities bearing thesame ratio asthedielectric constants;itisalsoim- portant toavoid contact potentials andbesure ofsolid contact atallpointsofanyboundary. Space charge problems cannot be represented bythismethod. Magneticfields canbemodelled inasimilar manner,ifonecan define surfaces ofconstant magnetostatic potential. Torepresent, forexample, themagneticfieldproduced bytwoparallel wires of arbitrary andlarge cross section isnotpossible, since themag- netostatic potentialisnotknown ingeneral along thesurface, andwithin theconductor does noteven exist. Forthin con- ductors, however,itispossible torepresent themagneticfieldby 186 Experimental Mapping Methods [Ch. 5 utilizing thebarrier surface asindicated inFig. 6-1.Referring toFig.18-2, onetakes athinconductor sheet, punches thecircular holes corresponding tothetwoparallel wires+/L,/L,andmakes anarrow slitconnecting these holes. Ifthentwometal plates areplaced attheedges ofthis slitandthepotential difference is applied between them, current canflowonly inthesheet around theholes andtheflow lines willbenearly identical with the magnetic field lines. From aplot oftheequipotential linesone canreadily construct theorthogonal flow linesandcompute the FIG.18-2 Model forMagnetic Field ofTwo Parallel Long Wires. local densities asindicated insection16,oronecansecure the flow linesbytheconjugate electrode arrangement outlined above fortheelectrostatic field. Measuring thetotal current fflowing between theelectrodes permits theevaluation oftheresistance Rperunit thickness of thesheet. This canbeconverted intopermeance ffperunit length inthesamemanner asintocapacitance Cforthedielectric above, namely, -- yR(1) where/iistheabsolute permeability ofthemedium surrounding theconductors /L-This follows directly from table 9-1for corresponding quantities;itcanalsobeshown directly byestab- lishing theflux-current relations. Bringing amagnetic barorcore ofvery great length and of constant andhigh permeability near the parallel conductors presents thesame problem asistreated above fortwo different dielectrics; onehastofindtwometals ofabout thesame ratio of conductivities asthat ofthepermeabilities. This usually means Sec. 18] TheElectrolytic Trough 187 thatonewillusecopper torepresent thehigh permeability and apoor conductor torepresent air.Theonly seriousdifficulty withmodels ofmagnetic fields istheproper interpretation ofthe magnetostatic potential values onsurfaces where onemustknow thisvalue inorder tosetuptheproblem. The electric flow linescanalsobemade directly visible byusing blotting paper soaked with asolution ofcopper sulphate andthin copper stripsofproper shapes torepresent theelectrodes. As thewater evaporates, theelectrolytic action causes thecopper to precipitate along theelectric field linesandgives very striking reproductionsofthem.1 TheElectrolytic Trough. Amore general utilization ofthe analogy ofelectric fields inconductors toelectrostaticfields, or anypotential fields,isbymeans ofelectrolytic current distribu- tions either atd-cvoltages oratlower audiofrequencies where the magnetic induction effects areslight. Thearrangementisusually referred toasanelectrolytic trough and consistsessentially of alarge tank, preferably ofglass orimpregnated wood lined with copper oroflava slabs filled with distilled water andaslight amount offresh spring water, inorder toobtain aproper degree ofconductivity. Frequently, onecanuseordinary tapwater; occasionallyitmay bepreferable touseaveryweak solution of copper sulphate. The electrodes, usually made ofcopper, are immersed intheelectrolyte, andaprobe, usually ashort piece of nickel orplatinum wire ofabout 0.02-cm diameter,isused to indicate thelocal potential. Theprobe must beinsulated over itsentire length, except forabout 1cmorlessonitsextreme end; itcanbesealed inglassandshould have ametal sheath onits outside forshielding purposes. The electric circuit (seeFig.18-3)isessentially aWheatstone bridge, withtwoarms formed bytheprobe andtheelectrodes Iand II;theothertwoarms areADandBDonthecalibrated potentiometer. Theprobeismoved until itspotential isequal totheselected value onthetapDofthepotentiometer asindicated bythedetector. The position oftheprobe istransmitted toa stylus resting onadrafting table either byacarriage system fixed totherimofthetankandpermitting freemotion intwoperpen- dicular directions orbyapantograph asshown inFig. 18-3. Usually, forafixed position Donthepotentiometer onetraces LK.Molin, Fysisk Tidsskrift, 18,p.3(1919). 188 Experimental Mapping Methods thecomplete equipotentiallineinaparticular plane. Whenever balance isachieved, thestylus canbepressed intotherecording paper, resultinginaseries ofpoints more orlessclosely spaced. Either thedetection ofbalance isobtained byasensitive tele- phone2orvacuum tube voltmeter,3or itisautomatically recorded bymeans ofanamplifier andsolenoid which actsupon theindicating pencil whenever thescanning probe reaches apoint with theselected potential value.4Aspecialcircuit forincreased 400 to 1000 cps FIG. 18-3 Block DiagramofElectrolytic Trough. sensitivity, using atuned amplifier andcompoundrectifier and triode,isemployed byZworykinetaZ.,B32 p.393;itgives maxi- mum reading atbalance rather than zero indication. Completely automatic plottingofalldesired equipotentiallines inatwo-dimensional oraxially symmetricalfieldcanbeachieved5 bydriving theprobe atconstant speed along onedirection and adjustingitspositionintheorthogonal direction bymeans ofa servomechanism which corrects tozerodifference inprobe potential with respecttothepotentiallinetobemapped. Here, thepanto- graphwilltrace acontinuous line,with slight jitter where the probemotion needs considerable adjustment. Attheendofeach travel ontheborder ofthemapping region onecan lettheservo- mechanism select thepositionoftheprobe forthenext equi- 2W.Estorff, E.T.Z., 37,pp.60,76(1916). 3R.G.E.Hutter,Jl.Appl. Physics, 18,p.800(1947). 4J.A.Simpson, jr.,Rev. Scient. Instr., 12,p.37(1941). 6P.E.Green, jr.,Rev. Sclent. Instr., 19,p.646(1948). Sec. 18] TheElectrolytic Trough 189 potentiallinebefore thereturn travel isinitiated; inthismanner, complete regions canbemapped automatically atconsiderable saving intimeeventhough thesmoothed-outequipotential curves must bedrawn byhand. The electrolytic trough was firstproposed byFortescue,6who used d-cvoltage. Thedisadvantage ofpolarization effects inthe electrolyte ledtothemodification introduced byEstorff(loc. cit.),whoused a-cpotentials fromlowpower frequencies uptoabout 500cycles persecond tostudy thepotential distribution between twolarge spheres. Higher frequencies upto1500 cpshave been used; they usually have thedisadvantage ofincreasing capacitive effects notpermitting azerobalance, andthusreducing thesensi- tivity ofthedetector; according toZschaage,7thezeroreading can berestored bycoupling thedetector circuitinductively tothe oscillator circuit; another proposal istoparallel thepotentiometer branches bysmall capacitances.8Itis,ofcourse, important to keep theelectrode surfaces very clean because slight oxidation cancause arapid increase inthelocal surface resistance. Theadvantageoftheelectrolytic trough method isthepossi- bility ofreproducing practically anythree-dimensional field dis- tribution inauniform medium. Forhighaccuracyitmight be necessary togotovery large tanks inorder toreduce theerrors introduced bythewalls, whether they bemetal orinsulating material. Fortwo-dimensionalfields,itisusually besttoletthe planeofthefield coincide with thesurface ofthewater andthe electrode structures restonthefloor ofthetank, which should be coated with insulating cement orpaint. The current flow will then retain itstwo-dimensionalcharacter; studies offields in multiconductor cables9andontransmission lines10wereconducted inthisway.Aconductive sidewallcanbeutilized asrepre- sentation ofperfect ground, whereas aninsulated sidewallcan beused asplane ofsymmetry withsimplification oftheelectrode structure. Toobtain plane electron tubemodels, anode and cathode may berepresented asheavy metal plates across the trough, andgrids spaced such thatthesidewall coincides witha 6C.L.Fortescue and S.W.Farnsworth, Trans. A.I.E.E., 32,p.893(1913).7W.Zschaage, E.T.Z., 46,p.1215 (1925). 8J.F.II.Douglas, Trans.A.I.E.E., 43,p.982(1924). 9R.W.Atkinson, Trans. A.I.E.E., 38,p.971(1919) and43,p.966(1924); alsoSemenoff andWalther,B18 p.29. 10W.Zschaage.loc. tit. 190 Experimental Mapping Methods [Ch.5 plane ofsymmetry either between twogridwires orthrough one gridwire.11 The electrolytic trough canequally wellbeadapted toaxially symmetrical geometries. Themost obvious use isasemicylindri- caltrough with allelectrodes asrespective semicylinders; measure- FIG.18-4 Potential MapofConical Electrode System. ments canbemade along thesurface ofthewater. This can, of course, bereduced toaquarter cylinder, and, infact, justtoa wedge-shaped trough, either bytilting thefloor ofthetank orby tilting thewhole tank.12Itisusually satisfactory, then, touse plane electrodes anddisregard theactualslight curvature ofthe electrodes. Forexploration offields close totheaxis, asneeded 11H.Barkhausen and J.Bruck, E.T.Z., 54,p.175(1933); Spangenborg,B29 p.75. 12Barkhauaen andBruck,loc.cit., p.176;M.Bowman-Manifold andF.H. Nicoll, Nature, 140, p.39(1938); Zworykinetai.,B32 p.392; Myers,3327 p. 95;Cosslett,1122 p.27;Hutter,loc.cit.,p.801. Sec. 18] TheElectrolytic Trough 191 inelectron optical systems, oneshould usevery large-scale models inorder toavoid thecapillary riseoftheelectrolyte ontheprobe, whichmaycause considerable error invery shallow water. Figure 184gives thepotential distribution inaconical electrode system used inthestudy ofemission from asmall spherical area. Since theelectrolytic trough leads toapotential graph,itis necessary either totrace thefield linesbyhand astheorthogonal system ofcurves, ortousetheconjugate electrode arrangementin which theequipotential linesbecome identical with theoriginal field lines asoutlined intheprevious subsection. Torestrict the current flow intheelectrolyte,itisnecessary only toprovidein- sulating boundaries; replacing, therefore,allelectrodes inthe original set-up byinsulating material ofexactly thesame shape will satisfy theflowboundary conditions. Onecanthen place electrodes alongfield linesandapply appropriate potential values asoutlined before. Attempts havebeenmade toreproduce theeffect oftwodifferent dielectric materials inthefield, asforthestudy offielddistribution onporcelain insulators surrounded byair.Mixtures ofgraphite andbinder13were selected torepresent porcelain andtheconductiv- ityofthewater wasvaried bysaltadditions, makingitpossible toobtain reasonably goodfield distributions. Asimpler method consisted invarying thedepth ofwater intheratio ofdielectric constants, essentially substituting increased volume forincreased conductivity.14Neither method canbevery accurate. Italso hasnotbeen possible toadapt theelectrolytic trough totheex- plorationofspace charge fields, which would beofgreat value in manyvacuum tubeproblems. Ontheother hand, onecanmeasure theindividual resistances between anytwo desired electrodes orappropriately isolated electrode sections andthusobtain directly themutual(orpartial) capacitance coefficients inthesamemanner asdescribed inthe previous subsection andreferred toin(3-9). Asanillustrative application, take themodel ofatriode asshown inFig. 18-5. Applying thedesired potentials bymeans ofthepotentiometer asbefore, onecanconnect theends ofaslide wirepotentiometer totwoelectrodes, sayAandG,andconnect atelephone asdetector 13W.Estorff, E.T.Z., 39,pp.53,62,76(1918). 14R.H.George, K.A.Oplinger, andC.F.Harding, Bull.No.29,Engg. Experiment Station, Purdue Univ., Lafayette, Indiana, p.23. 192 Experimental Mapping Methods [Ch. 5 between thethird electrode Candthemoving contact, thus re- producing again aWheatstone bridge. Fornosound inthe telephone, thepartial resistances are ACV CGV whereby Imisthecurrent through theslide wirewhich must also bemeasured. The partial capacitancesforthevacuum tube itself follow atonceagain from (8-11) byC=e/yR with 7 theconductivity ofthe elec- trolyte. Inthisparticular ex- ampleitispossible toobtain another characteristic number, theamplification factor/*,as PotentiometerTank CAC RGC GC FIG.18-5 Measurement ofPartial Capacitances withElectrolytic Trough.directly astheratio oftheslide wirelengths, not necessitat- inganyother measurement.15 Actually, forthedetermina- tion oftheamplification factor alone, onecould usethemain potentiometer itself, connecting the telephone between Candthevariabletap,andadjusting thelatter forzero tone. Magnetostatic fields canbemodelled inasimilar manner to electrostatic fields ifonecandefine surfaces ofconstant magneto- static potential asoutlined intheprevious subsection. The field lines ofacircular current loop, forexample, canbemeasured by using themodel ofFig.18-2 inthewedge-type tank, letting the wettinglinecoincide with theaxis ofrevolution, placing anin- sulating slabbetween thetwopotential electrodes, andrepresent- ingthecircular conductor ILbyaninsulating rodinorder toestab- lishtheproper flowboundary. Inapplications topermanent magnets, asoccur ininstruments and inelectron optical systems, onecanfrequently assume the 16Y.Kusonoae, Proc. I.R.E., 17,p.1726 (1929); alsoBarkhausen and Bruck,loc.cit.,p.176. Sec. 18] TheRubber Membrane 193 magnetic material tobeofinfinite permeability16andtoascribe toitamagnetostatic potential difference which isgiven bythe lineintegralofHacross theairgapwithin theuniform section of the field distribution. Where the finite permeability must be taken intoaccount, theelectrolytic trough cangenerally notbe used inanyconvenient manner. TheRubber Membrane. Avery effective means forthe representationoftwo-dimensional potential fields isarubber membrane stretched with practically uniform tension overagiven FIG. 18-6. Section ofRubber-membrane Model ofFig.18-1 foraSingle Dielectric; Radial Scale Compressed. electrode arrangement inwhich height above areference planeis proportionaltothepotential value. Figure 18-6 indicates the representation byarubber membrane ofacross section along 3'-3" ofFig. 18-1, with theelectrode potentials 3>iand$n symmetrical about*=0,thesheath potential, andwithauniform dielectric. Actually, thedifferential equation oftheelastic membrane is17 which reduces totheLaplacian differential equationif(dz/dx)2<^1, and (dz/dy)2<1.These conditions cangenerally besatisfied ifonekeeps thetangent plane atanypoint towithin 15ofthe horizontal plane.18This requires rather large models ofsmall 16ZworykinetaZ.,B32 p.477. 17P.H.J.A.Kleynen, Philips Techn. Rev., 2,p.338(1937); alsoStrutt,B3 II,p.4. 18Zworykinetoi.,B32 p.419. 194 Experimental Mapping Methods [Ch.5 height differences. Oneusesconveniently asurgical rubber sheet, about 0.1cmthick, which isspread over theelectrode surfaces andeither pulled overawooden frame andfastened toitorlaced toalargersteel ring. Toassure uniform adherence tothelower electrode surfaces, counter weights arefrequently provided as indicated inFig. 18-6, inwhich theradial scale hasbeen con- siderably compressed tomake abetter picture. The electrode material isusually leadorsheet aluminum. Therubber membrane hasbeenused extensively forthedesign ofplane electron optical systems, since itlends itself inanunusual manner tothesolution ofcomplicated electron trajectories asin thebeampower tube19andintheelectrostatic electron multiplier;20 but ithasalsofound excellent application totwo-dimensional electric andmagnetic problemsincables andmachines.21 Hydraulic Analogies.Ithasbeen pointed outinsection 9 thattheconditions (divE=0)and(divB=0)canbeinterpreted ascharacteristic forincompressible flowphenomenaifEorBcan beidentified with thevelocity vector. Thishasledtoamethod which shows themagneticfield lines intheairgapofmachines by means offinely distributed, colored glycerin forced intowater flowing between glass plates; very clear photographs canbe obtained inthismanner.22Arecent adaptation ofthis fluid flow analogy usesaplaster slabandaparallel glass platebetween which clearwater flows; crystalsofpotassium permanganate aresprin- kledontheslabmodel tovisualize flow lines. Excellent photo- graphs have beenmade ofsource and sink flows confined by variously shaped barriers.23 Conversely, many studies offlow lines inhydrodynamics have direct applicability toelectric andmagneticfieldproblems; see particularly Prandtl andTietjens024andEckC2 . 190.II.Schade, Proc. I.R.E., 26,p.137(1938). 20V.K.Zworykin and J.A.Rajchman, Proc. I.R.E., 27,p.558(1939); E.G.Ramberg andG.A.Morton, Jl.Appl. Phys., 10,p.465(1939). 21M.Krondl,Elektr. undMasch., 67,p.543(1939). 22H.S.Hele-Shaw andA.Hay, Phil. Trans., A196, p.303(1900); H.S. Hele-Shaw, A.Hay,andP.H.Powell,Jl.I.E.E., 34,p.21(1904); W.M. Thornton, Electrician, 66,p.959(1906). 23A.D.Moore, Jl.Appl. Phys., 20,p.790(1949). Problems 195 PROBLEMS 1.InFig.16-1assume firstonly thetwoconductors *jand$nwith the isolated small probe; show thattherelative probe potentialis($!*p)= VCzp/(Ci p+Czp).Connect next the electrostatic voltmeter asshown; assume thattheoapacitiveeffect ofthenewleads benegligible andthatthe voltmeter besufficiently removed soasnottoinfluence thefield ofthemain conductors. Ifthevoltmeter capacitance beCv,show that thenewprobe potentialis(*i-*/)=VC2p/(Cip+C2p+Cw),i.e., lessthan before. Demonstrate thatconnection toground atAwillrestore theoriginal probe potential. What willbetheindication ofthevoltmeter? 2.Assume apoint charge Qlocated attheorigin 0.Introduce anisolated sphere asprobe electrode with center atPandradius R.Demonstrate that thepotential onthesurface ofthespherical probeisidentical invalue withthe potential value that existed atPbefore theprobe wasplaced there, inde- pendentlyoftheradius R.Show that this isstilltrue ifthefield atPispro- duced byanynumber ofpoint charges. 3.Thestraw probe inFig.16-5 issubject tothegravitational force. Find theerror inindicating thedirection oftheelectric field lines. Hint :assume athincylinder shell ofuniform polarization andfindtheequivalent dipole. 4.Find theaverage fluxlinkage forthesearch coilinFig.171ifthemag- netic fieldvaries linearly across thecoilarea. Compute theeffective coilarea. 5.Itisstated that (17-1) holds ifthetime constant ofthecircuit iscon- siderably smaller than that ofthegalvanometer. What modification would have tobemade ifthatwerenotthecase? What errorwould oneexpect in using (17-1) nevertheless? 6.What istheinfluence ofthemagneticfieldproduced bythecoilcurrent itselfupon theaccuracy ofrelation (17-1)? 7.Design atwo-dimensional current flowmodel forthemagnetic field ofa three-phase and (a)three-wire, (6)four-wire, transmission system. 8.Design atwo-dimensional current flowmodel oftheheat flowfrom the conductors ofathree-phase three-conductor cable tothesheath, assuming a single uniform dielectric medium, circular cross section oftheconductors and ofthesheath, andaconstant (steady-state) temperature ofthesheath. 9.Design thetwo-dimensional current flowmodel oftheelectrostatic field ofatriode whichmay alsobeassumed astwo-dimensional. Show thedetermi- nation ofthecapacitance coefficients between cathode, grid,andanode by means ofcurrent measurements. 10.Demonstrate thatonecansimulate theelectrostatic field ofthegeome- tryinFig.18-1byusing copper sheaths ofdifferent thicknesses forthetwo different dielectrics. The error ofcurrent redistribution atthetransitions canbemade smallbyusing aphysically larger model. 11.Show thearrangement ofelectrodes inanelectrolytic trough torepre- sent thetwo-dimensional electrostatic field inapentode. Demonstrate the current measurements necessary todetermine allthemutual capacitance coefficients. 12.Evaluate thenecessary sizeoftheelectrolytic trough inorder tomeasure thefielddistribution between twospheres ofunequalradiiRIandRZ-Assume 196 Experimental Mapping Methods [Ch.5 thatoneuseshemispheres andmeasures thepotential distribution along the water surface. 13.Design theelectrode arrangement inanelectrolytic trough inorder to represent themagnetic fieldproduced bytwocoaxial circular loops ofradiia andbandsmall wire radii piand pz.Find themutual inductance bysimple current measurement. 14.Design theelectrode arrangement inanelectrolytic trough inorder to measure themutual capacitances ofathree-wire transmission lineabove ground. 15.Design theelectrode arrangementinanelectrolytic trough tomeasure themutual inductances ofathree-wire aerial transmission line. 6*FIELD PLOTTING METHODS Asanalternative toexperimental methods, anumber ofgraphi- calandsemigraphical methods (requiring simple computations) have been developed. Itiscertain thatquick orientation with respect toamore complexfield structure canbeobtained best byasimple graphical construction; ontheother hand,ifhigher accuracyisdemanded, many trials ofsuccessively better ap- proximation areneeded sothatexperimental means thenbecome more economical. 19GRAPHICAL PLOTTING OFELECTROSTATIC FIELDS Althoughitisrelatively simple toobtain qualitative information about field lines inauniform dielectric bythepowder patterns offreshly ground gypsum crystals (see section 16), there is nosimpler method than the graphical onewhich furnishes quantitativeinformation. Of course, acombination ofapow- derpattern withgraphical quan- titative interpretation, where such ispossible,will give theFIG.19-1 Dielectric FluxTube, speediestresults. Thefoundation ofmost graphical methods istheconcept ofthe dielectric fluxtubeformed bythevectorDwhich has itsbaseona positive surface charge element dQ=<r'8S' onsome conductor, andwhich terminates ontheequal andopposite surface charge element 5Q=<r"bS" onsome other conductor. Everywhere between, thedielectric fluxelement DdSremains constant and directed from positive tonegative charge, even through dielectric 197 198 Field Plotting Methods [Ch.6 boundary surfaces aslong asthey areuncharged, which isthe usual case (see sections 1and 2). Selecting, then, two closely spaced equipotentiallines asinFig.19-1, with *"=*'+?!=*'-m (i) dl permitsthedefinition oflocal capacitanceas 5Q DSS which isaunique value since fortheentire volume element STthe values (DBS) and (E8l) remain constant. One can, therefore, choose some arbitrary representative point within thiselement and, since atanypointofauniform homogeneous medium D/E=e, obtain ac=e^(3) 01 inexact accordance with (14-3) giving thecapacitance ofafinite parallel plate condenser. Ifone,moreover, selects dlnumerically equalto5*S,thespace becomes subdivided into cube-like units bounded byslightly curved surfaces exactly analogous tothetrue cubes intheparallel plate condenser, and, asthere, onecannow simply count thenumber ofcubical units inseries between two electrodes toestablish thefinite numerical value ofthedenominator ofthetotal capacitance, andcount thenumber ofcubical units distributed overthesurface ofone oftheelectrodes toestablish therespective numerical value ofthenumerator. Thismethod canbeappliedtoevaluate thepartial capacitance coefficients in systemsofconductors aswellasthetotal capacitanceoftwocon- ductors forming acondenser, andcanbeextended toanynumber ofdielectrics inthefield ifproper account istaken oftherefraction oftheflux lines asdefined by(2-9). Field Plots forLine Charges. Forasinglelinecharge of great length, asshown insection 12,the field distribution ises- sentially two-dimensional and isaxially symmetrical, with radial field linesandcircular equipotentiallines. Inorder torepresent thefield quantitatively, onechooses unitlength inthedirection perpendiculartothepaper inFig.19-2andthus hasfor(3) dS=r50,dl=dr.Because ofaxialsymmetry onemight choose Sec. 19] Field Plots forLineCharges 199 60as27T/n, where nisthenumber offlux lines tobedrawn. As- sume n=16,then 50=22.5=0.393 radian; thusfrom(2), with8Sand 81numerically equal, 8r/r=8(Inr)=2ir/n; theequi- potentiallinesmust beselected sothattheratio ofsuccessive radii isTZ/TI=e2vlnorr2=1.481^. Theradius rmrwhich actually satisfies relation (3)isfound byapplying (3)tothesubdivision ofABCD, namely,In(r2/rm')=In(rm'A*i)=ir/n\ thisshows that rm'=(rir2)^isthegeometric mean oftheradii TIand r2. FIG.19-2 Flux Plot forSingle LineCharge. Thegraphical construction forrm' iscarried through inFig.192as wellasthefurther subdivision forrm" '.Forthesingle linecharge, thegraph canbeextended outward toinfinity andinward tozero radius, finding inboth directions noterminal; thisdifficulty was already pointed outinconnection with (1228).Ascribing asmall butfinite radius atothewireremoves thedifficulty thereandper- mitsintroduction ofanabsolute scale. Thecharge perunitlength contained within adielectric fluxtube is8q=D8S=(X/27r)50 = X/n,ifXisthetotal charge ontheconductor perunitlength. Thepotential difference between successive potential linesmustbe, from (2),5$=8q/8C=X/en, since8Sand 81in(3)have been chosen numerically equal. With afixed potential value <f>on theconductor surface,itisnow possible tolabel thepotential lines. Thechoice ofthenumber nofrepresentative flux linesim- mediately determines alltheprincipal quantities. Fortwo ormore parallel long linecharges, aresultant field 200 Field Plotting Methods [Ch.6 graph canbeobtained byutilizing theprincipleofsuperposition. Taketwo linechargesofvalues(2X)and(+3X) perunitlength asshown inFig. 19-3. Having chosen n=16forthecharge (2X),onemust choose n=24for(+3X) inorder tohaveeach fluxtube carry thesame dielectric flux; inturn, thismeans that theequipotential circles forthecharge (+3X) arespacedinthe FIG.19-3 Field Plot ofTwoLineCharges: upperhalf fieldlines, lower halfpotentiallines. ratio e'262=1.3.Theupper half ofFig. 19-3shows the in- dividual fluxlines foreachcharge andtheircombination inbroken straight lines asafirstapproximation totheresultant fluxtubes. Itisobvious thatvery close tothecharged lines theindividual fluxdistribution willremain practically unchanged; sinceanytwo successive flux lines delimit thesame fluxelement, thediagonals ofthequadrilateralswillapproximately point inthedirection of theresultant vectorDasindicated inthefigures byP-P1 '.This approximationwillbecloser thelarger thevaluenischosen;it wasproposed byMaxwell,A17 I,p.183, forpoint charges, but applies equally welltolinecharges andlinecurrents.1Theneces- sarysmoothing oftheflux linesshould beguided bytheexistence 1H.Ebert: Magnetische Kraftfelder; J.A.Earth, Leipzig, 1905. Sec. 19]Curvilinear Squares Electrostatic Fields 201 ofacenter ofgravityofthecharges found by(10-46), which de- termines thecharacter ofthe field atlarge distance; bytheex- istence ofsingular points whereE=0,asatx=5dinthe exampleif2disthedistance between thecharged lines;andbythe potential graph. Thepotential graphisshown onthelower half ofFig.19-3 asthecombination ofthetwoindividual families of equipotentialcircles. Since allthe circles arespaced atequal intervals 5$inpotential,infact,equal decrease ofpositive ornega- tivevalues asonerecedes from(+3X) and(2X), respectively, one finds constant potential values byproceeding from one intersection ofcircles tothat ofsucceeding circles oflarger radius asindicated byR-R'. Forcomparison, aselected field line is shown astheresult ofafirstsmoothing ontheupper half,andas orthogonallinetothepotential graph onthelowerhalf, ofFig.19-3. Ifthelinecharges hadbeen chosen ofthesamesign,(+2X) and(+3X), then thecombination ofthefield lineswould have to proceedinthedirection oftheother possible diagonal P-P" indicated byreversing thevectorD_2atthepointPinFig.19-3 inaccordance with positive fluxfrom (+2X). Thecombination oftheequipotential lines likewise would bechanged, sincenow increasing radiimean forboth charges decrease ofpositive po- tential. Thus, from oneintersection oftwo circles onehasto proceed tothat ofthenext larger circle belonging to(+3X), with thenext smaller belonging to(2X),asfromRtoR" . Formore thantwocharged lines itispossible first tocombine thefieldgraphs oftwoandthencombine thisresultant with the third individual fieldgraph, etc.; ofcourse, considerable effort is usually spent before onearrives atathoroughly satisfactory final graph which also satisfies (3). Curvilinear Squares forTwo-dimensional Field Plots. Forgeneral two-dimensional geometries, onecannotstart, aswith thelinecharge, from aknown dielectric fluxelement. Theap- plication ofconstant values 8Casdefined in(3)becomes amatter oftrialanderrorwith successive stages ofsystematic improvement aftersome experience. Since forunitlength normal tothegraph paper thesurface element becomes 8S=15s,therelation (3) reduces to (4) 202 Field Plotting Methods [Ch.6 where 5cislocal capacitance element perunit length,5stheline element normal totheflux lines,and 81theorthogonallineelement along theflux lines. Ifonenow chooses 5s=5Z,one isledto curvilinear squares,2asindicated inFig. 19-4, fromwhich this method obtained itsname. Many details andexamples ofplots aregiveninAttwood,A2pp.178-185, sothatonlyabriefsummary need begiven here. Obviously,thismethod applies toany potentialfield observing the analogiesoftable 9-1,where the capacitanceof(3)or(4)isre- placed bytheappropriate con- ductance orpermeance ofthe other typesoffields. Inapractical electrostatic problem withasingle dielectric, onewillhave given atleasttwo conductor surfaces withknown potentials, and either bysym- metry orextrapolation into homogeneous fields one will know atleastone field lineand theapproximate potential vari- ation alongit.With thisasa basis, onecanstart inFig.194, forexample, with field line ABandsubdivide alongitthe potentialdifference ($i $n)uniformly intoasmall number, say,rasubdivisions. From thisbaseline,onecannowproceed to sketch theapproximate equipotentiallinesandtoselect orthogonal field lines atsuch intervals that curvilinear squares result. Of course,this firstsketch willshowweak points oftheplot, asin some places oneortheother ofthetwomajor conditions 5s=dl, andorthogonalityoffieldandequipotentiallines might notbe satisfied. Theplotmust thenberepeated untilboth conditions aresatisfied everywhere, whichmay require considerable further subdivisions togain indetail accuracy. Each subdivision should 2A.D.Moore: Fundamentals ofElectrical Design, McGraw-Hill, NewYork, 1927; alsoA.D.Moore, Elec. JL,23,p.355(1926); Schwaiger,m7 p.181; H.Poritzky, Trans. A.I.E.E., 67,p.727(1938).FIG.19-4 Method ofCurvilinear Squares. Sec. 19]Curvilinear Squares Electrostatic Fields 203 proceed along thecheck lines 6sanddl,leading tosmaller curvi- linear squares which bythemselves must satisfy 5s=51.With thefinal plotachieved, say, fortwoconductors inthefield, one cangetthetotal capacitance between these twoconductors by counting thenumber ofsquares along afieldline, say,m,andthe number ofsquares along theconductor surfaceI,say, n,asC= sn/ra. With thegiven potential difference, thisgives thecharge onconductor IasQi=C($i $>n).Ifthere areseveral con- ductors intheelectrostaticfield, only partial capacitances canbe evaluated (seesection 3).Inthis case,mapwould represent the number ofsquares along afield linebetween thetwoconductors a.and0,andna$would bethenumber ofsquares along thesurface ofconductor alying between thetwo field lines limiting thetotal mutual dielectricflux; onewould haveCa/3=enap/map.The partial charge onconductor abound byconductor isthen Qa/3=C0(*a-* ft). Ifthesubdivision byequipotential lines isreasonably close, one canevaluate thefieldstrength Ealong any field lineastheslope ofthepotential graph bythesamemethod shown inFig. 16-4. Plotting, asthere, distance along afield lineasabscissa andthese fieldstrength values attherespective points asordinates, onecan extrapolate thevalues atthesurface oftheconductors. The graph with distance along aconductor surface asabscissa and dielectric fluxdensity orfieldstrength values attheterminal points oftheflux lines asordinates should again lead toasmooth curve; itgives thecharge distribution ontheconductor andpermits the evaluation ofthetotal electrostatic force exerted upon thecon- ductor (seesection 3). Once asatisfactoryfield plothasbeen developed foragiven electrode arrangement, onecanuse itinmany other ways. Thus, onecaninterchangefield linesandequipotential surfaces andobtain theconjugatefield distribution, which might bedirectly useful ormight needsome adjustments tomake itphysically realizable either asanother electrostatic field orasanyother type ofpo- tential fieldfrom table 9-1. Onecanalsointroduce ametallic surface along anyequipotentiallineandthusobtain solutions toa different anduseful electrode arrangement. Ifthere aretwo ormore dielectric materials inthefield, the procedure hastobesuitably modified. Selecting forthemedium 204 Field Plotting Methods [Ch.6 occupying thelargest space curvilinear squares sothatfrom (4)8c=EI,onemust then construct curvilinear rectangles in medium 2,since there 8c=ei( );onecan either make \EI81/ (2/ei)5s=81orvice versa, depending onconvenience. Inaddi- tion, ateach dielectric boundary theconditions ofrefraction given in(2-9) must besatisfied. These problems count among the most difficult ones; yetthegraphical method isactually theonly feasible approach, since an- alytically thedifficulties arein- superable, andexperimentally almost so.3 Field Plots ofAxially Symmetric Systems. The general relation (3)applied to axially symmetric systems gives FIG. 19-5 Flux Plotting inAxially Symmetric Systems. (5) ifpisthedistance from theaxis ofrotation ofthepointPwithin thevolume element (secFig.19-5) and 5sand dltheorthogonal curvilinear lineelements. Tokeep 5Cconstant asinthetwo- dimensional casemeans theselection ofequal numerical values forp8sand 81.Forasingle homogeneous dielectric medium, this condition requires foranassumed 5sincreasing length81ofthe curvilinear rectangles withincreasing distance from theaxis. The field plotisingeneral more difficult toobtain than inthetwo- dimensional casebecausep,thescalefactor, changes withtheshape andlocation ofthecurvilinearrectangle.4 Inapractical problemitisconvenient tostartfrom asection where the fieldcanbeapproximated either bythelogarithmic cylindrical potential distribution, asinFig. 19-2, orbythat ofa point charge ordipole, asinsection 10.Ingeneral itisadvisable firsttoobtain arough field sketch asiftheproblem were two- dimensional andthen tocorrect thesketch until thetwomajor 3P.D.Grout: "TheDetermination ofFields Satisfying Laplace's, Poisson's, andAssociated Equations byFlux Plotting," Radiation Laboratory Report No. 1047. Seealso hisextension toelectric fields inmagnetrons, Jl.AppL Phys., 18,p.348(1947). 4M.G.Leonard, Elec. Jl.,32,p.31(1935). Sec. 19] Field Plots forPoint Charges 205 conditions aresatisfied everywhere: orthogonalityofflux linesand equipotential lines,andconstancy of8Cin(5).Theevaluation of total capacitances, etc., follows thesame outline asforthetwo- dimensional case. Themost exacting exampleisfound intheoriginal treatise on thismethod,5where thefielddistribution about abushingofrather complex form isplotted; another example isthefieldgraphofa charged grid consisting ofparallel, equidistant coaxial circular rings.6 Field Plots forPoint Charges. Though onecould treat a single point charge oranynumber ofcollinear point charges by themethod just described, since theyform anaxial symmetric system,ithasbecome customary tofollow themethod ofMax- well^17 I,p.183(seealsoAttwood,A2 p.27),which refers tothe spherical coordinate system. Forthesingle point charge, the equipotential surfaces arespheres andthefield linespoint radially. Thesurface element ofasphere ofradius ris8S=2irr2sin686 and 81= fir,sothatfrom (3) 89 8C=27rer2sin6-(6)8r inwhich now rand 6areindependent coordinates. Tokeep8C constant andequal to27re,onecantherefore splitthecondition (6) intotwoselection rules, keeping sin686aswell as8r/r2constant, oralso 5(cos 6)- The first relation determines theselection ofthe fieldlines; one canconveniently subdivide theradius ofany circle along the assumed axisofrotationinto, say,nequal parts; theradius vectors through theintersections oftheordinates atthese points with the circle givethen the field linesbounding annular cones ofequal dielectric flux. Thesecond relation determines theselection of theequipotential lines. Itisbest toplotthefunction 1/rand, starting from anarbitrary base radius, assume equal intervals of ordinates. BK.Kuhlmann, Arch.f.Elektrot., 3,p.203 (1914); seealaoRoth,B16 pp.39and200,andSchwaiger,Bl7 p.183. flH.L.Poritzky, Trans. A.I.E.E., 67,727(1938). 206 Field Plotting Methods [Ch.6 Theresultant field oftwopoint charges canbereadily obtained bysuperpositionoftheindividual graphs offield linesandequi- potential lines inexactly thesamemanner aswasdone fortwoline chargesinFig. 19-3. Examples ofsuch combination plots are frequently found, inMaxwell,A17 p.183;inHarnwell,A9 p.35; and inothers. AttwoodA2givesmany graphs fortwo ormore collinear point charges with helpful guides forfield sketching. Using thesame principle, Maxwell,A17 I,p.180, also gives the combination plotofapoint charge inahomogeneous electrostatic field. 20-GRAPHICAL PLOTTING OFMAGNETOSTATIC FIELDS Fortherepresentation ofmagnetostatic fields outside ofcon- ductors, twobasically different methods areavailable, using either thescalar magnetostatic potential orthevector potential, Only theformer belongs totable 91ofanalogies because ofitsmathe- matical kinship totheelectrostatic potential function; itsuse, however, needs caution. Selecting thevector Basrepresenting themagnetic flux density, onecanform magnetic fluxtubes in analogy tothedielectric fluxtubes inFig.19-1, enclosing 5$m= B8S flux lines anywhere inspace. Their terminals must be created, however, asmagnetic potential double sheets inallcases where the field isdirectly produced bycurrents (seeFig.6-1as example) inorder tosecure uniqueness ofvalues. The total magnetic potential difference willthenalways bethevalue ofthe exciting current (orampere-turns), JFi IFn=/.Fortwo closely spaced magnetic equipotential surfaces J1and3"onehas thelocalpermeance ~y- y"mi asaunique value fortheentire volume element asinFig.19-1 for thecapacitance. Choosing some arbitrary representative point within theelement andobserving B=nHforauniform homo- geneous medium, oneobtains -,* < inexact analogy to(19-3) andtherefore subject tothesame interpretation forgraphical field plotting methods. Sec. 20] Field Plots forLineCurrents 207 Ifoneconsiders thefield outside ofhighly permeable magnetic materials forwhich onecanassume/x=oo ,and ifnocurrents are inthespace where themagnetic field isdesired, onecanascribe tothesurfaces ofthemagnetic materials constant magnetic po- tential values, establishing complete analogy totheelectrostatic field. This willbetrue forthestudy ofthemagnetic field inthe airgapofpermanent magnets orelectromagnets, ofelectrical machines, ofrelays, andinsimilar arrangements. Theuseofthevector potential forgraphical purposesisgenerally restricted toparallellinecurrents ofgreat length, since ithasa simple form only insuch applications. Itisnotadirectly observ- ablephysical quantity andtherefore notofprimary interest. Field Plots forLine Currents. Forasingle linecurrent of value+/and ofgreat length asshown insection 13,the field distribution isessentially two-dimensional and isaxially sym- metrical with circular field lines. Torepresent the field quanti- tatively, onechooses unitlengthintheaxial direction andhasthus forthesurface element normal totheflux lines8S=8r,andalso 51=r50;both elements areinterchanged ascompared with the electrostatic fluxplotinFig. 192,sothatthemagnetic field isthe exact conjugateoftheelectric field. One actually canproceed asthere,make dSequal numerically to81,and select8<f>=2ir/n, except that thismeans now subdividing themagnetic potential difference orcurrent /bynandconsequently selecting thecircular field lines intheratio ofradii e27r/n ,leading tothesamegeometry asFig.19-2 shows. Tohave thevectorBpoint inthedirection ofincreasing3<onemust choose themagnetostatic potential values such astoidentify=withIand=2irwith 0.The permeanceinanyvolume element sochosen is89*= /z,andthe magnetic fluxwithin thetubebecomes 8$m=5(?87=/i(7/n), sothat itispossible tocount thenumber offluxelements between anytwopoints along aradial line inorder toobtain thetotal magneticfluxbetween these points. Forthesingle linecurrent, thegraph canbeextended outward toinfinity andinward tozero radius, having noterminal ineitherdirection; thisdifficulty was pointed outinconnection with (13-1). Ascribing, asthere, a small butfinite radius atothewireremoves oneofthedifficulties; theinternal magnetic field isthengivenby(13-3). Fortwo ormore parallel long linecurrents, aresultant field graph canbeobtained byutilizing theprinciple ofsuperposition 208 Field Plotting Methods [Ch.6 asshown byMaxwell,A17 I,p.287.lAssume twolinecurrents of value(27)and(+37) asanalogous totheelectrostatic example, Fig. 19-3. Ifonechooses n=16forthecurrent(21),then onemust choose n=24for(+37) inorder tomark equal potential differences between successive straight lines intheupper half of Fig.19-3; this, inturn,means that for(+37) thefield linecircles arenowspaced closer, namely, intheratio e2T/24=1.3. The absolute values ofthemagnetostatic potentials canbe chosen inseveral ways, depending onwhere oneplaces thedis- continuity barrier. Itissimplest toretain for(+37) thechoice asforthesingle conductor, i.e.,leave thediscontinuity totheright 3=-2I+^I=--27 +31 y= FIG.201Choice ofMagnetostatic Potential Values forTwoLine Currents. ofthecurrent, andtochoose for(27)thediscontinuity totheleft ofthecurrent, leaving thespace between thetwocurrents con- tinuous inpotential values asshown inFig.20-1. Thecombination oftheequipotential lines intheupper half of Fig.19-3 forconstant resultant values isguided bythebroken straight lines, whereas thecombination ofthecircular field lines proceeds along theintersections ofsuccessive circles inthedirection oftheresultant fieldvector B.Onecanreadfrom thegraph at once thefluxlinkage that, forexample, arectangular linear loop would experienceifitslong sideswere oflengthIandplaced parallel tothelinecurrents atR'andRn ',respectively. Between Rr andH"areexactly twotubes offlux, each containing 5$m= n(2I/n), sincen=16wasreferred tothecurrent(27) ;thetotal fluxlinkageistherefore $m=2/i(27/16)Z=0^/8)27, sothatthe mutual inductance becomesM=pl/8. Hadthelinecurrents been chosen (+27) and(+37), then the same modification would have tobemade asindicated indetail fortheelectrostatic analogue. This graphical combination ofline current fields isvery satisfactory; many excellent examples of more complicatedfield distributions inthepresence ofironand involving theory ofimages (seesection 23fordetails) aregiven inHague344 . 1Extensive application wasmade byH.Ebert: Magnetische Kraftfelder; J.A.Barth, Leipzig, 1905; seealsoHague,1344 p.351. Sec. 20]Curvilinear Squares Magnetostatic Fields 209 Theinconvenience ofusing potential barriers forthefield lines inconnection withthemagnetostatic potential suggests useofthe vector potential A,which reduces forlinecurrents toasingle component Ae=-Jlnp (3) asshown insection 13,where itwasalsodemonstrated that in two-dimensional problems thelines ofconstant valueAzareidenti- calwith the field lines. Inaddition, (6-23) gives themagnetic fluxasthelineintegral ofA2,which reduces here forunitlength and forthe single conductor to5$m=Oi/27r)/ In(r 2/ri),if 8r=r2 TI,since only integration parallel totheaxis gives a contribution. Toselect, then, flux linessuch astogiveconstant values 5#msimply means tokeep (/Inr^/r^) constant, which is exactly thecondition forselecting thefield lineswiththemagneto- static potential above. Theuseofthevector potentialisthen fully equivalent asfarasselection offlux lines isconcerned, i.e., itwilllead inthecombination ofthetwo linecurrents above to thelower half ofFig.19-3;itwill,however, notgivetheupper half,which isnottoouseful except asacheck ontheflux lines. Curvilinear Squares forTwo-dimensional Field Plots. For general two-dimensional magneticfields outside ofconductors andwith aknown distribution ofmagnetostatic potential values along given surfaces, onecanconstruct elements ofconstant value 9*asdefined in(2)bytrialanderror with successive stages of improvement. Forunit length normal tothegraph paper, the surface element 8S=15s,and (2)becomes 5p=M^(4) where 5pisthelocalpermeance perunitlength,5sthelineelement normal totheflux lines, and dltheorthogonal lineelement along theflux lines. Selecting further 5s=5Z,onearrives atcurvilinear squares2asoutlined fortheelectrostatic field; seealso Fig. 19-4. 2Thismethod was firstintroduced byTh.Lehmann, E.T.Z., 30,pp.995 and1015 (1909); seealsoRichter,349Vol.I,1924; A.D.Moore: Fundamen- talsofElectrical Design; McGraw-Hill, New York, 1927; A.R.Stevenson andR.H.Park, Trans. A.I.E.E., 46,p.112(1927); Hague,B44 p.268;and Bewley,D1 p.167. 210 Field Plotting Methods [Ch.6 Verymuch effort hasbeen spent ontheevaluation ofthefield distribution intheairgaps ofelectrical machines with salient poles inorder todetermine accurately useful fluxlinkages aswell as leakage reactances.3Afirstassumption makes theopposite mag- netic materials, polefaceononesideandsmooth armature onthe other, surfaces ofconstant magnetostatic potentials withp=oo andwith adifference 5"i3^=Hg,whereH isthevalue of magnetizing force intheuniform part ofthefieldandgthere- spectiveairgaplength. Refinement isintroduced byletting the magnetostatic potential vary along thesmooth armature (keeping H=oo)inaccordance with thedistributed armature winding; see forexample Bewley,D1 p.176. Inallthese cases there exists a regionofuniform field inthepole center similar totheregion belowABinFig. 19-4, sothattheactual fluxplotcanbecarried forth asdescribed fortheelectrostaticfield, leading toatotal permeance &=nn/mifmisthenumber ofcurvilinear squares along afield lineacross thegap,andnisthenumber ofsquares along thearmature surface within onehalfpolepitch. Itisreadily seen that, even with thesimplifications made, anexperimental investigationintheelectrolytic trough would beutterly difficult because ofthevarying potential values onboundary surfaces. Thegraphical method hasseemed togivemost satisfactory results, leading tothecomposite fieldpicture intheform ofanorthogonal net. Two-dimensional Field Plots Including Current-carrying Regions. Inmany applications, thecurrent regions cannot be excluded from consideration, since they directly affect the field distribution astheexciter windings onmost electromagnets and onthepoles ofsynchronous andd-cmachines. Inside current- carrying regions, certainly themagnetostatic potential doesnot exist, sothatthemethod ofcurvilinear squares cannot beused. However, asdemonstrated fortwoconductors oflarge circular cross section in(15-8), there exists a"kernel" intowhich the field lines shrink. Since theboundary conditions onthesurface ofaconductor with finite current density require continuity ofthe magneticfieldwithout anyrefraction (seesection6),onecancon- 8Seeanyadvanced book onelectrical machinery butparticularly R.E. Doherty andC.A.Nickle, Trans. A.I.E.E., 46,p.912(1926); Stevenson and Park,loc. cit.;R.W.Wieseman, Trans. A.I.E.E., 46,p.141(1927); and Bewley fm p.167. Sec. 20]Two-dimensional Current-carrying Regions 211 tinue theorthogonallines aswell, finding thattheyconverge into thiskernelKasinFig.20-2. Since theycannownotbecalled potential lines ofany sort,4butsince they areorthogonal tothe fieldlines, they areusually called5"lines ofnowork." Actually, thefield lines suffer achangeinradius ofcurvature, i.e.,asecond- order effect inpassing across theboundary ofacurrent-carrying region. Inorder tocontinue graphical construction oftheorthogonal netinto current-carrying regions, thecurvilinear elements have FIG.202Kernel andField Lines within Current-carrying Regions. tobemodified. Applying the lineintegral ofthemagnetizing force inaccordance with (6-3) totheclosed path (AKBA),and assuming uniform current density J,oneobtains theresult HP8l=SJ(5) where Sisthearea(MKN) within theconductor. Obviously, onlydlmakes acontribution tothelineintegral, since theother path elements areeverywhere normal toH.Applying thesame lineintegral totheclosed path (A1KB1 A')gives Hp'81'=S'J(6) wherenowS'isthearea (A'KB'). With 5s'selected forconstant fluxelement andwith thesame permeability /iinside andoutside 4Thismethod wasdeveloped byTh.Lehmann, Revue gin.del'&ec.t14, pp.347and397(1923); seealsoStevenson andPark,loc.cit.;Hague,B44 p.270;andBewley,D1 p.169. 6Stevenson andPark,loc.cit.; alsoinGen. Elec. Rev., 31,pp.99and153 (1928). 212 Field Plotting Methods [Ch.6 theconductor, onehas5$m=pHpSs=nHp'ds'. Combining this lastrelation with (5)and (6)leads tothecondition Inside aconductor, therefore, thecurvilinear elements arerec- tangles ofdecreasing areaastheyapproach thekernel. Ingeneral configurations, theexact location ofthekernelis,however, not asimple matter andfrequently compromise methods arechosen.6 Usually oneplots first the fieldgraph without regard forthe current-carrying regions andguided onlybythesurfaces ofknown (orassumed) magnet ostatic potential values asoutlined above. Onethen introduces themodification caused bytheconductor as acorrection, estimating thelocation ofthekernel andchecking theadjustmentofthecurvilinear element with relation (7). This might leadtoslight changes oftheassumed magnetostatic potential onironsurfaces close totheconductor. Asanalternative method onecanusethesuperposition oftwo graphs7each ofwhich issimpler toconstruct than theresultant. Intwo-dimensionalfields, onlythecomponent ofvector potential normal tothefieldexists, say,Az',and itsatisfies (6-18) aPoisson differential equation within theconductor. Themag- netic field isthen Onecannowlook foravery simple solution of(8),preferably in onecoordinate only, which vanishes onatleast oneconductor surface asfarastheintegration constants permit andplotthe resulting field lines within theconductor. Onecanthen con- struct anormal Laplacian field plotbycurvilinear squares for Bx",By"which satisfies theusual conditions ofthemagnetostatic potential values andinaddition provides together withBxrand Bytherequired continuity ofthemagneticfieldvector across the 6Th.Lehmann, Revue gen.deI'elec., 31,p.171(1932) and34,p.351(1933). 7J.F.H.Douglas, Electr. Engg., 64,p.959(1935); H.Poritzky, Trans. A.I.E.E., 67,p.727(1938). Sec. 20] Conductors ofArbitrary Section 213 conductor boundaries. Obviously, thismethod requires very care- fuljudgment andweighing ofalternatives, but itcangiveexcellent results particularlyiftheconductor isindirect contact with iron surfaces, which simplifies thesatisfaction ofboundary conditions. Instead ofseparating thesolution forthecurrent-carrying region asabove, onecanusesuperposition ofthecompletefieldproduced bytheconductor alone, both inside andoutside itsboundaries, astaken fromsome analytical solution andplotitintotheregion inwhich theresultant field distribution isrequired, determine thecorrection needed attheboundaries ofthefieldregion tosatisfy theboundary conditions there, and construct acurvilinear y field plot forthis correction. Thecombination willthenbea complete solution oftheprob- lem. Fortheapplication of thismethod toarectangular conductor within arectangu- lararmature slotseePoritzky (loc. cit.). Two-dimensional Field Plot orConductors of Arbitrary Section. Ifone desires the field distribution surrounding alongconductor ofarbitrary large cross section butuniform current distribution, then itisfrequently difficult to apply themethod ofcurvilinear squares from thestart. Onecan utilize theknown field distribution ofavery thinrectangular strip asgiven in(15-20), whichis,referred todesignations inFig.20-3,FIG. 20-3 Evaluation ofMagnetic Field ofConductor ofLarge Section. B,--(*,-.PIa,rna+-In 2ir lana(10) andapplyittotheindividual strips ofequal widthwintowhich theconductor may conveniently bedivided.8The current in each stripisgiven asIa=Ila/^la,sothat aconstant factor (^-/ jcanbedeleted. Atapoint P,thecontributions ofthe 8W.Kramer, E.T.Z., 63,p.9(1932). 214 Field Plotting Methods [Ch.6 strips can allbeadded andtheresultant magnitude andthedi- rection ofBestablished. Theorthogonal direction oftheequi- potentiallinecanalsobenoted. Themethod israther rapid, since thesubdivision neednotbemade very fine,andwillyetgive good results except very close totheconductor surface. Having thus established several field linesandequipotential lines, one cancontinue with themethod ofcurvilinear squares andproceed intotheconductor asoutlined above. =-%NI 7=0 FIG.20.4 Magnetostatic Potential Barriers forCylindrical Coil. Field Plots ofAxially Symmetric Systems. The general relation (2)applied toaxially symmetric systems gives asin(19-5) 61 ifpisthedistance from theaxisofrotation ofthepointPwithin thevolume element asinFig. 19-5. The utilization of(11) follows exactly theelectrostatic case. Inpractical problems, which usually involve cylindrical current coils, thefield lines close totheaxisarenearly parallel, sothat dl along thelines isreasonably constant and 6s=dp.The field lines should thusbeselected inaccordance with (pSp)=H^(p2 )=cons, sothat their spacingisessentially asfo/ri)** for5r=r2 rim Atlarger distances, thefield ofcoilsapproximates that ofamag- netic dipole givenin(13-33). The values ofthemagneto- static potentials have tobeassumed inbestagreement with the Sec. 21]Images Plane Conducting Boundaries 215 geometry andwith therequirement ofpreventing completely closed field lines. Foracylindrical coil ofNturns, apossible choice istheplacementof%Nl atthetwoendfaces ofthecoil volume andadouble cylindrical mantle along which thepotential difference varies tozerovalue attheneutral zone z=asshown inFig.20-4.Aplotofthefield lines forthistype ofcoilwithand without ironcore isgiven inthereference toPoritzky (loc. cit.). Thecomputation ofinductance values from these graphsisin goodagreement with analytical results. 21-METHOD OFELECTRICAL IMAGES The solution oftheelectrostatic field distribution caused by point charges andlinecharges inthepresence ofsimple conductor ordielectric surfaces canfrequently beobtained without analytical means bythemethod ofelectrical images introduced byW.Thom- son.1Itisbased ontheconcept ofimaginary point orlinecharges notlocated within theregion offieldevaluation butsochosen that together with theoriginal point orlinechargesallboundary con- ditions inthisregion canbesatisfied. Though these imaginary orimage charges havenorealexistence, theycanbeused asifreal inorder toconstruct thefinal fieldbyanyoneofthesimple graphi- calmethods, tocompute theforce actions ontheoriginal charges, andtoconstruct models oranalogies forother types ofpotential fields. Some verysimple examples areincluded insections 10and 11,sincethey follow there rather naturally, and, indeed, havebeen thesource ofideas forthegeneralizationofthemethod. The material ishere organized according toplane, cylindrical, -and spherical boundaries,ofconductors and dielectrics. Noimage theory exists forspherical dielectric boundaries.2 Images with Respect toPlane Conducting Boundaries. The solutions forfields ofpoint and linecharges nearaninfinite plane conducting surface have been given analytically insection 10andextended toconductors ofvery small radii insection 11, sothatcharge distributions andcapacitances could beevaluated. Frequently, onecaninterpret ground orwalls ofbuildings for 1SeeW-Thomson :Papers onElectrostatics andMagnetism, p.73;Macmillan, London, 1872;firstpublished 1848; alsoMaxwell,A17 I,p.244; Jeans,A1 p. 186;Mason andWeaver,A16 p.110;andRamsay,A21p.114. 2Smythe fA22 p.115; Stratton,A23 p.204. 216 Field Plotting Methods [Ch.6 electrostatic purposes asconducting planes, sothatthese solutions apply directly tomany transmission lineandrelated problems. Apoint orlinecharge between parallel grounded metal planes asinFig.21-1 requires two infinite sequencesofimages,3which aresummarized intable 21-1, giving signs aswellaslocations of thecharges. Forapoint charge+Qthere issymmetry about the +Q $= FIG.21-1 Point orLineCharge between Two Parallel Conducting Planes. x-axis, andthetotal potential atPwillbethesum ofallimage contributions with ra2=p2+(x xa)2 , 47TE(1) whereby Qa=Qinaccordance with table 21-1,fromwhere also thexamust betaken. Theinduced surface charge oneither plane canbecomputed by(10-17) asthesum ofcontributions from the pairsofimages with respect toeach plane. Obviously, theseries must converge toafinite value, since thetotal charge onplanes IandIItogether must be(Q). Itistherefore possible totake apartial sumasanapproximation. Forthespecial case ofa=b=c,thecontributions ofallthe point charge images tothepotentialattheorigin become, deleting thefactor ^TTE, QL(-i)a 2c recognizing theseries expansion forIn2.Admitting nowasmall 8Maxwell,A17 I,p.273. Sec. 21]Images Plane Conducting Boundaries 217 but finite diameter dofthepoint charge, thetotal potential onits surface becomes ingood approximationifd<Cc, (2)_(?>,2) 47TE\d C / where the firstterm isthepotentialoftheoriginal small sphere by itself, andwhere thesecond partistheinfluence ofthetwo TABLE 21-1 LOCATION OFIMAGE CHARGES FORAPOINT OHLINECHARGE BETWEEN TwoPARALLEL CONDUCTING PLANES Location of Images reI +2bxa a Location of Order of Images ren Image +2n(o+6) +[2n(o+6)+26]2n -[2n(a +6)+2o] grounded planes. ThecapacitanceisgivenbyQ/$. Themaxi- mum induced charge density atthecenter ofeither plane is,from (10-17), with r=h=c,3c,5c,etc., fortheimage pairs, 21rc225 *-*-?=(3) orabout 92percent ofthemaximum density induced ifonlyone planeispresent, whereas onemight have expected reduction to 218 Field Plotting Methods [Ch.6 50percent. The larger value indicates acompression ofthein- duced charge distribution near theaxis asthemost pronounced effect ofthesecond plane. This solution canalsobeinterpreted interms ofcurrent flow from asmall spherical source, orfrom asemispherical source through onehalfthespace between theplanes. Inthelatter casetheresistance between thehemisphere andplane becomes R= J=^d(l-i1*2 )(4) where 7istheconductivity ofthemedium, andthefactor 2 entered because thetotal current /leaves through onlyonehalf thespherical surface.4 Forthelinecharge between theplanes atwo-dimensional field results withnodependence onz.Inthiscase, $P=-r ^aInra (5) 27TE(a) which canbeinterpreted aslogarithmoftheinfinite product of ravalues and identified with aclosed expression.5Using the alternation oftheimage signs, thiscanalsobeexpressed asthe logarithmoftheinfinite product ofratios oftwo ravalues, thus making thelogarithmand apurenumeric. The closed form can beobtained insimpler formbyconformal mapping insection 27. Alinecharge within arectangular channel, obtained byadding twoconducting planes parallel tothex-z-plane inFig.21-1, leads toinfinite arrays along both xandydirections.6Thecomplete solution involving elliptic integralsisalsofound more readily by conformal mapping, section 27. Images withRespect toPlane Dielectric Boundaries. As- sumeanynumber ofpointandlinechargesinadielectric medium EI atdistances hafrom asemi-infinite dielectric e2,asinFig.212. Whatever theactual charge distribution inEI,itsfieldmust beso arranged that across theboundary plane x=continuity of potential values andnormal dielectric fluxdensity arepreserved. Iftheactual charge distribution produces potentials &a(hax,y,z) 4 011endorff,A18 p.326. 6J.Kunz andP.L.Bagley,P%s.flev,,SeriesII,17,p.l47 (1921); Smythe,A22 p.84. 6C.M.Herbert, Phys. Rev., SeriesII,17,p.157(1921). Sec. 21] Images Plane Dielectric Boundaries 219 andtheeffect ofthedielectric E2istoberepresented byimage charges, thentheymust belocated atgeometrical image points inorder topermit satisfying theboundary conditions, i.e.,they must produce potentials$>a" (ha+x,y,z),which atx=have theidentical dependence on(?/,z)asthe $>avalues. FIG.212Point orLineCharge andPlane Dielectric Boundary. Foranyonepoint orlinecharge (seeFig.21-2), thetotal po- tential inEIisthusassumed tobe -X,I/,Z)+Qa"<J>(h a+X, (6) whereas thepotential inmedium s2isassumed tobeproduced by acharge Qafatthelocation oftheoriginal charge, thus -X, (7) Tocondense notation, Qahasbeen chosen inthissubsection for both pointand linecharge, though thelatter hasbeen designated Aelsewhere because itisacharge perunit length. Thefunctions <areidentical with l/4ireiri aandl/4irsir 2a ,respectively, fora point charge; andwith^andrespectively, for alinecharge, where rischosen arbitrarily asascale reference value, which might beidentical with hajsince itrepresents an additive constant tothepotential asin(12-28). Applying the boundary conditions anddefining </>x=o=<o( )=?ogive\dx/x -o 220 Field Plotting Methods [Ch.6 thetworelations 00+Qa"00=Qa'</>0 o-Q"?o) =e2Qc/?o fromwhich theimage charges canbeevaluated <2a'=^-Qa,Qa"=^pQ0) i+2 ei+2 Onecannowconstruct theresultant fieldpicture bysuperimposing thefields ofthepoint orlinecharges appropriateforeach region. Thus, inmedium EIforx>0,onedraws theresultant ofQa andQa/; ,uptotheplane x=0;inmedium e2forx<0,onedraws theradial field picture ofthesingle point orlinecharge Qaf '.In drawing theresultant field plot, auniform dielectric material ei must beassumed throughout space, since thedifference indielectric constants hasbeenaccounted forinthevalues oftheimage charges. Good fieldgraphs ordetail derivations aregiven inAttwood,A2 p.163, forthelinecharge; andforthepoint chargeinAbraham andBecker,A1 p.77;inJeans,A1 p.200; inMason andWeaver,A1G p.148; inHarnwell,A9 p.66;andinRamsay,A21 p.134;Smythe,A22 p.113, gives thegeneralized derivation. For E!<E2,theimage charge Qa" isnegative and for e2>ooapproaches (Q), the image value foraconducting plane;inthislatter caseQa' >as itshould. The force action upon apoint charge Qaiscomputed asthe interaction withQa",since their combined action defines the resultantfield, _1_OaQ"=J_i- 2/Q<A2 47TE! (2/ia)24! l+E2\2fcJ( ' which means attraction tothedielectric for EI<s2.Two equal charges placed symmetrically with respect tox= inthetwo dielectricswill, therefore, notreact with equal forces upon each other. Assuming charge Qatoreside onasmall sphere ofradius a,then itspotential canbefound from (6)as -2- iJLl(in a 47TE1 Thecapacitance Ca=Qa/$aislarger for s2>EIinthepresence ofthedielectric s2thanwithout it.Thecharge distribution can Sec. 21] Images Plane Dielectric Boundaries 221 readily befound byapplying (1113)with theappropriate substi- tution ofvalues. Theforce action perunitlength upon alinecharge canbecom- puted asQaEa"jwhereQaisthecharge perunitlength andEa" isthevalue ofthe field strength atx=haproduced byimage charge Qa".This yields which means attraction tothe dielectric for ei<s2.With r=hainthepotential expression, $ia=attheorigin. WithQa FIG.21-3 Point orLineCharge Midway between TwoPlane Dielectric Boundaries. assumed toreside onacylinder ofsmall radiusa,then, with re- spect totheorigin, theconductor hasthepotential value (13) Thecharge distribution canbefound byappropriate application of(12-36). Apoint orlinecharge between two parallel plane dielectric boundaries requires two infinite sets ofimages tosatisfy the boundary conditions atboth surfaces. Assume asinFig.21-3 apoint orlinecharge inmedium EImidway between thetwo like dielectrics 2,then table 21-2 gives thenecessary locations of theimage charges for EInowsymmetrically distributed. The first-order images, iQ", satisfy theboundary conditions onthe surfaces next tothem butnotonthefarther surfaces, sothat theymust betaken asneworiginal charges leading tosecond-order 222 Field Plotting Methods .6 images, 2Q",andsoforth. Ifthefactors in(9)areintroduced as 2ei , E!e2 rt 2 +2,-..v (14) then i<3"=i/'Q, 2<2"=(n"?Q, nQ"=(V')"0 (15) The fictitious charges iQfserve todefine thefields inIIandIII andbythemselves neednofurther compensation, since their effects TABLE 21-2 LOCATION orIMAGE CHARGES FORAPOINT OHLINECHARGE MIDWAY BETWEEN TwoPARALLEL PLANE DIELECTRIC BOUNDARIES Charges Defining Field inU(a2)Charges Defining Field inICharges Defining Field inm(ij) Original Qatx= donotappear within I.Again, these charges arerelated tothe nQ"values, andonehas iC'-fl'Q,&-vitf'=jv"Q, nQ'=-n'(n"r-1Q(16) Foreach section, theresultant fieldcanreadily bedetermined by theinfinite series which arecertain toconverge; onewillassume a uniform medium EIforthispurpose, since theeffects ofthedi- electrics areaccounted forbytheimage charges. Foraquasi point charge Qatx=ofsmall radius a,onecan determine thepotential onitssurface asthesum ofcontributions oftheoriginal charge and alltheimage charges defining thefield inI.With a<h,onecantake thedistances directly from table 21-2andobtain r~loo(fi"y L n=i 2n/i(17) Sec. 21]Images Cylindrical Conductor Boundaries 223 since theabsolute value ofif'iscertainly lessthan unity. For E2>EIthisindicates anincrease incapacitance Q/<caused bythe presence ofthetwodielectrics, ascompared with(11.2) forthe single quasi point charge. For e2->ooneobtains(2),thesame result asfortwoconducting planes, asitshould be. Maxwell/17 I,p.443, treats themore involved case ofapoint source within amedium ofconductance 71atadistance hfroman infinite plane parallel slab ofthickness a>hand ofconductance 72followed byaninfinite-extent medium ofconductance 73. Byanalogy, thiscanbetranslated intotheelectrostatic problem7 ofapoint charge inEIinfront ofaninfinite slab ofs2,followed by aninfinite-extent medium s3.Two infinite series ofimages are necessary; their locations arefound inthesamemanner asfor conducting planes andtheir charge values byappropriate applica- tion of(9). Images with Respect toCylindrical Conductor Bounda- ries. Thesolution foralinecharge parallel to,andlocated atdis- tance bfrom, theaxis ofaconducting cylinder surface ofradiusR wastreated insection12,locating theopposite andequal image linecharge onthecenter linewithin thecylinder atadistance d= R2/bfrom theaxis.Apoint charge close toaconducting cylin- dercannot betreated bysimple image theory, since itspotential function isincompatible with thelogarithmic potential function ofthetwo-dimensional cylinder (seesection32).Themethod of images canbeapplied alsotothepotential solution fortwoparallel cylinders offinite cross section either excluding orincluding each other8 (seesection12). Theapplication toaconducting cylinder withtwosymmetrically located opposite linecharges asinFig.21-4canreadily bemade. The location oftheimage linechargesisgiven byd=R2 /b; thefield outside istheresultant ofthefour linecharges, andthe field inside isofcourse zero. Ifnowthetwo linecharges recede toinfinity, b >oo fthetwoimages approach symmetrically the origin. Inthelimit onehasthecase ofaconducting cylinder in auniform electric fieldproduced bythetwo linecharges, which canbetaken from(12-33)atz= 2/=andwith creplaced by6as (18) 7SeealsoSrnythe,A22 p.181,whouses direct analytical methods. 8Attwood,A2 p.149; A.Russel, Jl.I.E.E., 64,p.238(1925). 224 Field Plotting Methods 1.6 where bothX*ooandb> to'produce afinite field. The field outside isnowgivenasthesuperposition ofEandthedipole line formed bytheimage charges, which have adipole moment per unitlength p=+x2yi=2^R2Ei (19) alsodirected along thepositive z-axis. The field ofthedipole FIG. 21.4 Conducting Cylinder andTwo Line Charges. Limiting case for b oo:conducting cylinder inuniform fieldE. lines isgiven directly by(12-53) incylindrical coordinates, in which theuniform fieldhasthecomponents Er=Ecos0,Ee=-E sin6 Thedirect addition ofthefieldsanduseof(19)givetheresultant field Er=E(l+^cos0,=-E(l-^\sisine(20) which isthesolution usually obtained byexpansion into circular harmonics; seeSmythe,A22 p.65.The fieldgraph canvery readily bedrawn asgraphical combination ofthe circles and parallel lines oftheindividual fields, leading tolocalconvergence upon the conducting cylinder with proper orthogonality there. Asseen from (20), theelectrical fieldstrength doubles atthesurface ofthe conductor forr=R,6=0.Thesame solution occurs, ofcourse, inhydraulics, with interchange offield linesandpotential lines, asforexample Bewley,D1 p.32,andreferences inAppendix 4,C,c. Sec. 21]Images Cylindrical Conductor Boundaries 225 Thetwo-conductor cableshown inFig.21-5canbetreated in thesamewaybythemethod ofimagesiftheconductor radii aresmall compared with theradius ofthesheath, a<CR.The images ofthegivenlinecharges with respect tothesheath are located atdistance 6=R2/dfrom theorigin,if2disthecenter FIG.21-5 Two-conductor Cable. distance ofthetwo conductors. Combined with the original charges theyproduce thepotential atP X iri' r2= In- r 27TE(21) which gives zerovalue onthesheath, asiseasily demonstrated forpointP1 .Toestablish thecapacitance oftheconductor pair inthepresence ofthesheath, onecanform thepotential difference 3>i $11byintroducing into (21) thedistances tothecenters oftheconductors, except that TI=afor$iand r2=afor$H. This leads to <j,_$=Aln('i 1" 27Tn \J or,ifoneuses 6=B2/d $ -InT 7T6Vo+aa/ -d2 2R+(22) from which thecapacitance perunit length canbeobtained9 9Attwood,A2 p.144. 226 Field Plotting Methods [Ch.6 asX/($i $11).Ifoneplots theresultant fieldandequipotential linesbygraphical superposition ofthelinecharge fields, onecan findabetter approximation, particularly forlarger radiia,by shifting theconductor centers slightly away from thelinecharge location towards thesheath.10Onecanalsoobtain closer ap- proximations bytaking more images with respect totheconductor cylinders andimaging these inturnonthesheath. With this method, two-andfour-conductor cables havebeen treated.11 Ifthetwoconductors aretransmission wires suspended within thecylindrical sheath, thenthey canbespaced sothatthey ex- perience noforce. Onlead Itheresultant forcewould be Af(-x)_(-*)_(+M 1 2Trz[_(b-d) 2d (b+d)] where thesigns account forforce directions andsigns ofcharges. Forvanishing force onecomputes atonce2d=ftVVS 2. Images withRespect toDielectric Cylindrical Boundaries. Theproblem oflinecharges parallel toadielectric cylinder is very similar tothat ofaplane dielectric boundary, consider- ingthelatter asacylinder ofinfinite radius. One expects, there fore, that alinecharge (+X)inFig. 21-6, located inmedium eioutside thedielectric cylindere2ofradiusRrequires animage charge X"=7/"Xfrom (14)atthegeometric image point d=R2/b with respect tothecylinder surface inorder todescribe thefield external tothecylinder. However, thatputs effectively aline charge within amedium thatmust remain uncharged, sothat another linecharge (X")isnecessary attheaxis; thisneutralizes the firstimage charge and, being spaced from itadistance d, produces theeffect ofadipole line. The total external potential is,therefore, thecombination ofthree linecharges where 3oaridInRareconstants. The fieldwithin thedielectric cylinderisdetermined, asintheplane case,byalinecharge X7= rj'Xfrom (14) located attheplace oftheoriginal charge. IftheradiusR >o ,thesolution fortheplane dielectric boundary 10 Breisig,A4 p.68,alsogives good field graph. 11H.Meinke, E.N.T., 17,p.42(1940); F.Sommer, E.N.T., 17,p.281 (1940). Sec. 21]Images Dielectric Cylindrical Boundaries 227 results. Ageneral verification oftheimage arrangementisgiven bymeans ofcircular harmonic functions inSmythe,A22 p.67. Theextension totwosymmetrically located linecharges (X) and(+X) asinFig.21-6 israther obvious; thecompensating charges attheaxis arenotnecessary, since theimages X" already neutralize the dielectric cylinder.Ifthetwo external linecharges recede toinfinity,b,theimages(X") approach A FIG.21-6 Dielectric Cylinder andTwo Line Charges. Limiting case for b >oo:dielectric cylinder inuniform fieldE. symmetrically theorigin. Thecase isentirely analogous tothat oftheconducting cylinder with theonlyadjustment invalue of linecharges!Inanalogy,inthelimit b >oo ,theproblemisthe oneofthedielectric cylinder inauniform fieldandtheresultant external field incylindrical coordinates isgiven by(20)with the extra factorr/',forthedipolelinecontribution, andtheadjust- ment insign Brin=-Bin9(24) Ifonelets 2 internal field> ,then rj">(1)and(20) results again. The isdefined bythetwosymmetricallinecharges 228 Field Plotting Methods [Gh.6 (X; )located atthesame place astheoriginal charged lines, which, however, have receded toinfinity. Thus, inside thedi- electric cylinder, auniform fieldremains, weakened bythefactor i/,or Ein=T/# (25) This solution (24)and (25)isthesame asobtained bymeans of circular harmonics bySmythe,A22 p.67. IfEI>s2,the field strength inside thecylinder becomes larger thanEand inthe limit canreach twice that value. Since frequently thedielectric oflower dielectric constant hasalsolower breakdownstrength, suchaphysical combination israther unfortunate. Ifalinechargeisplaced within thedielectric cylindere2,say, XatA"inFig.21-6, then itrequires animage linecharge X"at Atodescribe thefield inside thecylinder, butnofurther neutraliz- ingcharge; here, ofcourse, X"=(EZ ei/e24-i)Xbecause of theinterchange ofrelative positions. The field outside the di- electric cylinderisagain given byX'=(2Ei/Ei+2)Xlocated atA"andabalancing linecharge X"attheorigin, sothatthe effective charge within thecylinder remains X'+X"=X,asit should. Itiseasily demonstrated that theboundary conditions requiring continuity ofDTandEQinthecylindrical coordinates aresatisfied ifoneselects apoint ontheperiphery ofthecylinder andequates withproper algebraic signs thesum ofthelocalcom- ponents oneither side ofr=R.Again,ifR >o,theorigin moves also toinfinity andthesolution oftheplane dielectric boundaryresults. Forthecompositionoftheresultant outside fieldgraph from theindividual linecharges onemust assume uni- form space ofEI ;conversely, forthecomposition oftheinside field onemust assume uniform space ofE2.Thismethod hasbeen applied tothecomputation ofcable capacitances totake into account theinfluence ofthedielectric constants.12 Images withRespect toSpherical Conductor Boundaries. The effect ofasingle point charge upon asphere ofradiusRhas been extensively treated insection 10andforagiven point charge ofsmall finite radius insection 11. Iftheactual point chargeis located adistance bfrom thecenter ofthesphere, thentheimage point charge ofvalue (j~Qj=Q'lieswithin thesphere 12H.H.Meinke, E.N.T., 17,p.108(1940). Sec. 21]Images Spherical Conductor Boundaries 229 along thecenter lineatadistance d=R2/bfrom thecenter. Theapplication toaconducting sphere withtwosymmetrically located opposite point charges, asinFig. 21-7, canreadily be made; their images aresymmetrically located atd=R2/bfrom theorigin andtheir values are=FQR/b, respectively. The field outside thesphereisaxially symmetric and istheresultant ofthe fourpoint charges ;thefield insideis,ofcourse, zero. Because of symmetry, thespherewillhave zero potential andzero resultant FIG.21-7 Conducting Sphere andTwo Point Charges. Limiting case for 6 >oo;conducting sphereinuniform fieldE. charge. Onecanaddanyarbitrary point charge attheorigin without disturbing thesymmetry ortheboundary conditions. Ifnow, fortheuncharged sphere, thetwopoint chargesinFig. 217recede toinfinity,6 ><*> fandthetwoimage charges ap- proach symmetrically theorigin. Inthelimit onehasthecase ofanuncharged conducting sphere inauniform electric field pro- duced bythetwopoint charges, which canbetaken from (10-12) attheorigin (z=0)as Q (26) where bothQ>ooand b >toproduce afinite uniform field. Thetwoimage charges form adipole ofdipolcmoment R-.(27) 230 Field Plotting Methods [Ch.6 alsodirected along thepositive z-axis. The field ofthedipoleis directly given by(10-35) inspherical coordinates, inwhich the uniform fieldEhasthecomponents ErQ=Ecos9,Ee=-E sin The direct superposition ofthetwo field expressions anduseof (27)givefortheresultant field (2/?3\ / 723\ 1+=-3-Jcos0,Ee=-E (1- -^-Jsin (28) which indicates theconvergence ofthefield linesupon thesphere toterminate thereon orthogonally. Thecomplete details and T FIG.21-8 Grounded Sphere andUniformly Charged Wire. graphofthe field lines arefound inJeans,A1 p.192;and in Ramsay,A21 p.132.Asseenfrom (28), thefieldstrength hasthe largest value atr=R,6=0,atthepositive "pole" ofthesphere, where itreaches 3#. Again, theanalogous problem occurs in hydraulics withaspherical obstacle intheuniform flow ofanin- compressible fluid,known astheDirichlet problem; seereferences inAppendix 4,C,c. Extension oftheimage theory totheeffect ofauniformly charged lineoffinite length upon agrounded sphereispossible bydividing thelineintopoint charge elements (Q/2l)dx asinFig.21-8. Each element hasassociated animage element dQ' ,whereby 6=(c2+z2)* d=y,dQ'--(|d*)f(29) The totalimage charge within thesphereisalsothecharge Q' Sec. 21]linages Spherical Conductor Boundaries 231 induced onthesphere and isobtained bydirect integration; see Ramsay,A21 p.123: (30)=-ib I c which reduces totheexpressionforasingle point chargeifl/cis very small sothatsinh"1 l/c='l/c. The electrostatic field oftwofinite conducting spheres canbe described onlybyaninfinite sequence ofimages.Ifthetwo spheres have radiiRIandR2,potentials $1and 3>2,andacenter distance 2c,oneusestheprinciplesoflinear superposition toevalu- atetherespective charges QiandQ%which willaccumulate onthese spheres. One firstassumes $1onsphere1withafictitious point charge Qi=47re#i3>i atitscenter, which would produce this potential were thissphere alone. The presence ofthesecond sphere canbeaccounted foronlyifithaspotential zerobyplacing animage point charge (722/2c)Qi atadistance from itscenter R22/2ctowards sphere1.However, thisrequires anewimage charge within sphere1whose charge andlocation follow from the elementary image theory; thisprocess goesonadinfinitum but withquick convergence ofthecharge values sothatafewterms aregenerallysufficient. Complete details forthegeneral case aregiveninMaxwell,A17 I,p.270,who alsocomputes theforce action and inKirchhoff,A13 p.64,based onanearlier paper;13a verycomplete account isfound inRussel,B11 I,p.236,who also givesmany numerical values fortheimage series andcomputes maximum field strength andforces based onearlier papers,14to which Jeans,A1 p.196,also refers. Atreatment canalsobegiven interms ofadifference equation leading tosolutions interms of hyperbolic functions, asinSmythe,A22 p.117,andinOllendorff,A18 p.266. Fortwospheresofequal radii, orplaneandsphere, therelations aresomewhat simpler; seeinaddition toabove references Att- wood,A2 p.147;Schwaiger,B17 p.87,gives arather comprehensive treatment inconnection with thepractical application ascali- brated sphere gapforhigh-voltage measurements. Itisimportant tonote thattheactual charges onthespheres andtherefore the 13B.Kirchhoff, Crelle's JL,69,p.89(1861). 14A.Rusael, Phil. Mag., VI, 6,p.237(1906); Proc. Phys. Soc., 87,p.485 (1912), 24,p.22(1913), and 97,p.120(1920). 232 Field Plotting Methods [Ch.6 field strength ontheir surface and inthespace between them depend onthevalues ofthepotentials assigned tothespheres, so that thedistributions $1=+7/2,<J>2=-V/2, or<i>i=V, $2= willgivedifferent results. The successive images define theinduction coefficients /cajginaccordance with table 21-3; only TABLE 21-3 POTENTIALS ANDCHARGES ONTwo FINITE CONDUCTING SPHERES thesymmetrical potentialdistribution permits thegeneral con- cept ofcapacitanceforeachconductor, andonlyifthetwospheres arealike, sothatQi=Q2,doesacapacitance ofthesystem exist, inwhich case *nC==2(*n-fc12) IsinhftX![sinh (2n- 71=1(31) 1-1 (32) =-Rsinhft [sinh2n/ 71=1 Sec. 22]Magnetic Images Ideal Plane Boundaries 233 where/3isdefined bycosh|3=R/c,withRtheradius ofthespheres and2ctheir center distance. 22METHOD OFMAGNETIC IMAGES The solution ofthemagneticfield distribution caused byline currents inthepresenceofmagnetic materials hasnotbeen as generally common asthecorresponding electrostatic case.Though theprinciples canbeformulated inrather similar manner, the actual application frequently doesnotlend itself tosimple transfer ofaknown electrostatic solution because oftherestrictions which onemust impose upon thescalar magnetostatic potential (see section 6).Itmust beborne inmind, too,thatthemirror image oftheelectrostatic field ofapositive chargeisagain that ofa positive charge because ofitsessential source nature; themirror imageofthemagneticfield ofapositive currentis,however, that ofanegative current, since thecirculation ofthefield lines reverses inthemirror, sothat forgeometrical imaging onemust substitute negative current values. Finally, since there isnomagnetic conductor analogous toelectrical conductors, aboundary ofa magnetic material, even ifofinfinite permeability, need notbean equipotential surface;ifitshould beone,itmust bespecified explicitlyinorder tostate theboundary conditions inanun- ambiguous way. Images withRespect toIdealPlane Equipotential Bound- aries. Thesolution forthemagneticfield ofany linecurrents in air(confined tomathematical lines but ofarbitrary geometry) nearaplane equipotential boundary ofmagnetic material ofinfinite permeabilityisfound bysubstituting thegeometric image ofthe lineconductors, with thecurrents flowinginthesame direction as intheoriginal inplace ofthemagnetic material, andfinding the combined magneticfield inair. This isanalogous totheprocedure onaplane conductor surface inelectrostatics withtheappropriate changeinsign ofthesource image, and, indeed, themagneticfield lines willbeidentical with theequipotentiallines ofthepositive linechargesofthesame geometry, placing thepositive image behind theboundary plane1which, ofcourse, cannot bethen a conductor. Themagneticfield willnotextend intothemagnetic 1Hague,1344 p.93;also S.P.Thompson andMiles Walker, Phil. Mag., V, 39,p.213(1895), and II.Ebert: Magnetische Kraftfelder;J.A.Earth, Leipzig, 1905. 234 Field Plotting Methods [Ch.6 oo,asthelaw ofrefraction (6-11) material because of/* indicates. Ifonelong straightlinecurrent flows incloseproximity ofsuch anidealmagnetic boundary, asinFig.22-1, thefield inairisthe combination oftwo likeandequal linecurrents which canreadily beobtained bythesimple graphical methods ofsection 20;see alsoAttwood/2 p.395,andRussel,511 p.448. Theplacement ofthepotential barrier issubject tochoice;itwould, however, FIG.22-1 Long LineCurrent Parallel toIdeal Plane Magnetic Boundary. cause difficulty intheboundary planeifonechose theimage bar- riertotherightoftheimage current. Foraparallel pair oflong transmission line wires, Fig.22-2 shows thearrangementofthepairofimage currents; agood field graph canbefound inAttwood,A2 p.398.Theproximity ofthe magnetic material increases the self-inductance oftheline; this increase canbecomputed from thefluxproduced bytheimage conductors andlinked withtheoriginal loop. Thus, from (1320) itfollows atonceperunitlength (1) where theappropriate values from Fig.22-2 were substituted. Obviously, asddecreases, ALiincreases tothemaximum external inductance oftheoriginal loopwhen ittouches thesurface; one usually assumes doubling oftheentire self-inductance since the Sec. 22]Magnetic Images Ideal Plane Boundaries 235 internal inductance isaverysmall amount. Thesame procedure canbeused foranylinear current loop, sothattheforms ofsection 13become directly applicable. Foruniform current densities, thismethod canbeextended toconductors offinite cross sections, subdividing them into current elements JdSandapplying the imaging method toeach inturn. Intersecting plane boundaries ofmagnetic materials canbe FIG.22-2TwoLong Transmission-line Wires Parallel to Ideal Magnetic Boundary. treated similarly. Assume along straight linecurrent inair between planesU=which intersect atanyangle ir/n,where n isaninteger. Thegeometrical location ofthe(2n 1)imagesis thesame asforpoint charges discussed insection 10,but allthe image currents have thesame direction asthe original. The resultant field distribution canreadily becomposed asthesuper- positionofthetotal2nlinecurrents. Theimage location isgiven inHague,244 p.100,andaresultant fieldgraph forright-angle intersection inAttwood,A2 p.397,andinHague,344 p.102;other resultant graphs equally applicable to2n-conductor cables with symmetrical arrangementinRussel,1311 p.462. Along straight linecurrent inairbetween two parallel ideal magnetic equipotential surfaces similar toFig.211requires an 236 Field Plotting Methods [Ch.6 infinite series ofimages which arelocated exactly asshown in table 211forthecorrespondingelectric arrangement, except that againalltheimage currents arepositive here; seeHague,D44 pp.172and 169,andalsoBewley,D1 pp.158and 137. Itisad- vantageous tousethevector potential fortheoriginal andimage linecurrents, since themagnetostatic potential becomes somewhat unwieldy. According to(13-23), theresultant vector potentialis Az=-f/mr a (2) 27r() quite similar totheelectrostatic potential (215)fortheanalogous problem; withnfor1/eandIforthealternating charges, onecan get(2)from (21-5). Now ra=[(x-xa)2+y'2 ]1A ,withxafrom table 21-1,andthetwo infinite products canbeidentified inclosed form; thiscanbeobtained more simply byconformal mapping (seesection 27). Graphs aregiven inHague,B44 p.169,andin Bewley,D1 p.137;Attwood,A2 p.400,shows related ones ofseveral currents between two ideal ironboundary surfaces. Should one feeluneasy about thelogarithmsofdistances in(2),then one could introduce some fixed distance Rasreference andwrite Inra/R ;however,allthese InRterms would collect intoanadditive constant in(2)whose value would remain unknowable since the vector potentialitself isnotobservable. Because thefield vectors, asthederivatives ofthevector potential, would innocasecontain these arbitrary constant terms, thevector potential willgenerally bewritten intheform (2). Along straightlinecurrent within arectangular channel in iron ofinfinite permeability andbounded byfcquipotential surfaces leads toinfinite arrays ofpositive image currents;2thesolution canbeobtained more simply byconformal mapping. Images with Respect toPlane Magnetic Boundaries. Assume anynumber ofparallel straightlinecurrents Iainmedium Piatdistances hafrom theplane boundary ofthesemi-infinite medium ^2similar toFig.21-2. Whatever thelinecurrent dis- tribution invi,thetotal fieldmust satisfy theboundary conditions (6-7)and(6-10), excluding thepresence ofacurrent sheet inthe boundary plane x=0. Iftheindividual linecurrents produce two- dimensional vector potentials with onlyz-components, AZa(ha x,y),andtheeffect ofthemedium /i2istoberepresented by 2B.Hague, World Power, 6,pp.124and205(1926). Sec. 22] Magnetic Images Plane Boundaries 237 imageline currents, then theymust belocated atgeometrical image points inorder topermit satisfying theboundary conditions, i.e.,theymust produce vector potentials AZan (ha+x,y)which have atx=theidentical dependence onythat theAZavalues have. Foranyonelinecurrent thetotal vector potentialinmisthen assumed tobe ^Ua(1)=iula*(*- x,y)+mla"-*(h a+x,y) (3) whereas thevector potential injn2isassumed tobeproduced bya linecurrent Iaeatthelocation oftheoriginallinecurrent, AZa=Mla''*(ha-X,y) (4) Thefunctions M>areidentical, respectively, with (l/27r) ln(l/r la), and (l/27r) In(l/r 2a),where rlaand r2aaredesignated inFig. 21-2; themore general functional form ischosen toindicate the possible extension tothemore general arrangements. Inthe chosen coordinate system, theboundary conditions require con- tinuity ofBx=+(6A,/dy) andHy=-(\/^)(dA z/dx). De- fining (d^/dy) x==^i,and(d^/dx) x=Q=S^n,theboundary conditions givewith (3)and (4) (5) fromwhich theimage currents canbeevaluated 2/ii , ,/_M2-MI ,_la . . M2 M2H-Ml The field picture inmedium /iiforx> isobtained by(3)as theresultant ofthegiven linecurrent Iaandtheimage current/</', which, ascomparison with (21-9) shows, hasagain theopposite signoftheimage chargeQa"intheelectrostatic case. Inmedium H2with vector potential by(4)thefield lines aredrawn ascoming from alinecurrent /'located atthesame place asthegiven original.3Inamore general geometry oflinecurrents,allthree componentsofthevector potential must beused asinSmythe,A22 p.282;thefinal resultis,however, exactly thesame as(6)since both tangential components HyandHzgive identical equations. Itisnowseen that, for/i2>Mi,theimage Ia"willbepositive; 8G.F.C.Searle, Electrician, 40,p.453(1898). 238 Field Plotting Methods [Ch.6 forM2<Mili-e->ifthelinecurrent wereimbedded iniron, the image Ia"would benegative. As/i2* ,/a" /aand/' >0, asused inthepreceding subsection; conversely, when/*i>w, /</' (Ia)and7a' >2/a,acasewhich willbetaken uplater inmore detail. Excellent graphs forasingle linecurrent are given inAttwood,A2pp.403,405; inHague,B44pp.105,107;and inMoullin,B48 p.224. Theforce action onthesingle linecurrent Iacanbecomputed byAmpere's law (5-1). Since theresultant field inmedium pi isthecombined action ofIaandIan ',onecanassume theactual force tobethatbetween thesetwocurrents, sothatperunitlength .=MIIJa"=MiM2 MiJg2 ,-^*m" 27T2ha~ 27rM2+Mi2fcU; exactly analogous to(12), giving theforce action between two parallellinecharges. Thesame result canbeobtained byusing theforce expression interms ofthemagneticfieldBa" produced byId'.There willbeattraction totheironfor/*2>MI,since the force ispositive. Forapair oflong transmission linewires inairparallel toa plane ironsurface ofM,asshown inFig.22-2, theresultant field inairisdetermined bytheoriginal current andtheimage Ia" from (6).Theproximity oftheironincreases theself-inductance oftheline,andthisincrease canbefound asintheprevious sub- section, except thatthefactor from (6)enters, AT MOMMo, f. ./c\2 ~] /0xALl=9~~T~ln \l+b) @) 2?rM+MoLWJ Fordecreasing distanced,thetotal external inductance increases toamaximum value onthesurface when image and original be- come geometrically identical (9) whereLexlistobetaken from (13-17). Thisform(9)holds for anywireloopplaced onthesurface ofasemi-infinite ironblock. Asingle very long straight linecurrent inairbetween two infinite-extent blocks ofironadistance 2hapart, analogous to Fig. 21-3, requires two infinite scries ofimages tosatisfy the boundary conditions. Forsymmetrical arrangement thegeo- Sec. 22] Magnetic Images Plane Boundaries 239 metrical location ofalltheimages canbetaken from table 21-2, andtheimage current values canbefound quite analogously to theelectrostatic problem. Itisimportant tocombine thevector potentials oftheindividual images with theproper permeability ineachmedium asindicated by(2)and (3);this isdifferent from theelectrostatic case. Thus,inaironehas .=-f2 /[inr+ (r"T ln(r, lBrUIjlA*L =i J(10) where inanalogy to(2114)butwiththepertinent modifications 2^o / M-Mn . .--=T ,--=r (LL)M+MO MTMo andwhere rlln2=(x+2nA)2+ </2 ,rlUn2=(x-2nh)2+y* Fortwo parallel very long wires arranged asinFig.22-2but midway between twoblocks ofironadistance 2hapart, onehas from (10) with the radii topositive andnegative currents asindicated. The firstterm isthevector potentialoftheoriginal current pair exactly asin(1313) ;thesummation term istheeffect oftheimage pairs. The total inductance oftheloopcanbefound exactly as in(13-16), taking thedifference ofthevector potential values at thetwo wires. Again the firstterm gives thenormal external inductance inair(13-17)iftheloopisbyitself, sothat thein- crease ininductance becomes 03) sinceonconductor +/ 1-H.+=rlu+=2nh, m,-=rllln-=[(2nA)2+(2c)2]" andviceversa onconductor 7.Comparing (13)with(8),one recognizes theadditive effect ofalltheimage pairs ofcurrents. Obviously, this principle canbeextended atonce toanyplane linear current loopbetween twoblocks ofiron. Thesame result canalsobeused fortwoparallel wiresimbedded 240 Field Plotting Methods [Ch.6 inthecenter plane ofaninfinite slab ofiron ofthickness 2h. Interchanging Mand MO,oneobtains andrecognizes that there isadecrease ininductance onaccount ofthefinite thickness ofiron, since (14)isanalternating series andthe firstand largest term isnegative. Again, thiscanbe extended toanyplane linear current loop. Images withRespect toCylindrical Magnetic Boundaries. Theproblemoflong straight linecurrents parallel toacylinder ofmagnetic material issimilar tothat ofaplane magnetic boundary andcompletely analogous totheelectrostatic case insection 21. Forasingle long linecurrent +/asinFig.22-3, located atA inairwith MO,atadistance 6from thecenter ofthemagnetic cylinder with M>the field inairwillbedescribed byplacing an image linecurrent (+/") intothegeometric image lined=R2/b from theaxis; additionally, onemust place another linecurrent (/")along theaxis inorder toneutralize the firstimage line. The total external vector potential atapointPwilltherefore be with (11) (15) The field within themagnetic cylinder willbethat ofasingle linecurrent (+/)' located attheplaceoftheoriginal current; thefield lines willtherefore becircular arcsandthevector potential ByinterchangeofMand MO,oneobtains thesolution forastraight cylindrical tunnel inironwiththelinecurrent placed intheiron. Excellent fieldgraphs forboth alternatives aregiven inHague,B44 p.115. Theextension totwosymmetrically located linecurrents 7 asinFig.22-3 israther obvious; thecompensating linecurrents T/" attheorigin arenownotneeded. Ifthetwo linecurrents 7recede toinfinity asb >oo,theimages(7") approach the origin asd=(R2 /b)>0,sothattheyform adipole linecurrent. Inthelimit onehasthecase ofamagnetic cylinder inauniform Sec. 22]Magnetic Images Cylindrical Boundaries 241 magneticfieldproduced bythetwo linecurrents, which canbe taken from (1316)atx=y=as (17) where both />ooandb ootoproduce afinitefield, quite simi- larto(21-18). The field outside isgiven asthesuperposition of eDT-' Fia.22-3 Magnetic Cylinder andTwo Line Currents. Limiting case for b >QO :magnetic cylinder inuniform fieldB. thisuniform fieldandthecurrent dipole lineformed bythe image currents, which have adipole moment perunit length from (13-19) -R2Bxi (18) Mo directed along thepositivez-axis. The field ofthedipole line is 242 Field Plotting Methods [Ch.6 giveninsection 13inthecylindrical coordinates r,0,z,inwhich theuniform fieldBxhasthecomponents BT=BXQcose,Be=-Bxsin6 Thedirect addition ofthefieldsanduseof(18)givetheresultant field cose,Be=-BXQl-r" smB(19) which checks with thesolution obtained with circular harmonics; seealsoMoullin,348 p.198. Ifone lets/i ,thenr" 1and theforms become analogous to(21-20), which describe aconduct- ingcylinderinauniform electric field; thefield lines willtherefore benormal tothecylinder surface. Theinternal field isgivenby thetwosymmetricallinecurrents (7') which receded likethe originals toinfinity andtherefore produce auniform fieldgiven by(17),butwithpfor/x , (20) 2/i which ismuch stronger than theoriginal onebythefactor M+Mo AsM* >the field inside willapproach twice theoriginal uni- form field. Ifalinecurrent +7 isplaced within themagnetic cylinder M, say, atA"inFig. 22.3, then itrequires, asintheelectrostatic analogue, animage linecurrent (+/") atA,thegeometrical image pointofAntodescribe thefieldwithin thecylinder, where now I"=(Mo-^/(MO+M)7=-T"7from (11)because oftheinter- changeinrelative position ;thefield outside themagnetic cylinder isgivenbyacurrent I1=2ju/(M+Mo)^=r'latA",thelocation oftheoriginal, butnow itrequires another linecurrent ofvalue I'1attheaxis inorder tohave theexternal fielddetermined by the effective current 7=7'+7"within the cylinder. The demonstration that these images satisfyalltheboundary condi- tions, i.e.,continuity ofBTandHeincylindrical coordinates,is simply given byselecting apoint attheperiphery andequating withproper algebraic signs thecomponentalcontributions oneither side. Hague,644 p.Ill,gives considerable details andalsoshows excellent graphsfortheabove case aswell asthereverse, aline current inacylindricalairtunnel inablock ofiron. Inthelatter Sec. 22]Currents within IdealMagnetic Materials 243 application, nand /imust beinterchanged andthemechanical force perunitlength upon thecurrent issimply theinteraction of itandtheimage determining thefield intheairtunnel, namely, ,=Mo J// 2irb-dMO 27T-MO H-MO&~d2J ifdisdistance ofthecurrent from theaxis(A" inFig.22-3). Theforce isattraction totheiron, asthepositive sign indicates. Currents within IdealMagnetic Materials. Ifalinecurrent isim- bedded inamagnetic material ofin- finite permeability M= >then the imagerelations (6)and(11)place the negative image current intotheam- bientmedium atthegeometric image pointinorder todescribe the field within themagneticmaterial. In thiscase, then, theelectrostatic image solutions apply directly with inter- changeofelectric equipotentiallines tomagneticfield linesandviceversa. Inthismanner, along straightline current within ju=oandparallel to aplane boundary surface, asinFig. 22-4, hasamagneticfield given by theequipotentiallines ofalong straight chargedline parallel toaplane conducting surface; theywillbethefamily ofcircles described in(13-15). Thus, theequipotential boundary surface ofthe electrostatic analogue becomes afield line surface in themagnetic material, eventhoughitisboundary toamedium M=oo.The field inairissupposedly given byalinecurrent /'=21attheplaceoftheoriginal; thiswould represent circles inairquiteinconsistent with thefactthat theboundary surface itself coincides with field lines. Oneusually disregards theex- ternal fieldcompletely andsuppresses theimage/! Foralong straightlinecurrent within amagnetic cylinder of infinite permeability,thefield lines willagain bewholly contained within thecylinder and willbeidentical with theequipotential lines oftwoeccentric cylindrical conductors enclosing each other; see(12-43).FIG.224Long LineCurrent within IdealMagnetic Material Parallel toPlane Boundary. 244 Field Plotting Methods [Ch.6 23-METHOD OFINVERSION Themethod ofelectrical images hasbeenexpanded intoamore general toolbyusing thegeometrical processofimaging forthe transformation ofcertain given geometriesinto simpler ones. This isofparticular value inthree-dimensional problems involving spherical surfaces;itisoflessimportanceintwodimensions where onehasavailable thevery powerful method ofconformal repre- sentation. FIG.231Inversion inaSphere. The Kelvin Transformation. Geometrically, theinverse point toAwith respect tothesphere ofradiusRtcenter inFig. R2 231,ispoint A'with radial distance rAr=andthecoordinates TA R2 =2*A (1) since XA'/XA=rA'/rA,etc.;similarly forother pointsBandC. Ifthethree pointslieonanyclosed surface wholly outside the sphere R,then theinverse inthespherewillagain beaclosed surface inwhich thesuccessive points arearranged asthemir- rorimages ofthepoints ofthe original. Inparticular, one canshow thatspheres remain spheres ordegenerate intoplanes asspecial cases ofspheres. Thus, asphere S2ofradiusm andcenterMinFig.23-2 tangent tothesphere ofinversion Sec. 23] TheKelvin Transformation 245 atTbecomes again asphere S2' tangent atTbut ofradius m'=Rd=R(l R/b). Asm >oo,thesphere S2be- comes theplane S\and itsinverse becomes thesphere /of radius R/2passing through theorigin 0;thispoint becomes obviously theinverse ofthepointPasitmoves into infinity. Topreserve one-to-one relationship,itisconventional toconsider infinity asasingle point, astheinverse oftheorigin. Thisthen Sir 2' FIG.232Inversion ofSpheresinaSphere. alsomeans thatanysphere through theorigin and ofradius p<Rhasasinverse aplane atadistance b=R2/2pfrom the origin andnormal tothecenter line. Spheres which intersect thesphere ofinversion orthogonally aretransformed intothemselves. Take thesphere through point BinFig.231with radiusmandcenterM .Itsequationis (x-xM)2+y2+z2=r2-2xMx+xM2-m2 (2) Introducingr=R2/rfandx=xr2/R2 gives, upon reordering intothesameform as(2), ,/No 2xMR2 f ,R* XM2-(3) butXMZm2=R2form theright triangle MSO, sothat (3)can 246 Field Plotting Methods [Ch.6 atoncebewritten inidentical form as(2)with(r')2andxrinstead ofr2and x.Therefore, thesegment SB'S' ofthespherical surface within thesphereofinversion willbepoint bypoint theinverse ofthesphericalsurface SBS' outside. Onecanusethisproperty toshow insimple manner that thistransformation byinversion on asphereisconformed; theangle between anytwo lineelements is thesame asthatbetween their images. Oneneed only consider thelineelements dsiandds2aselements ofgreat circles ofspheres intersecting orthogonally thesphere ofinversion;then theirimages willbeelements ofthesame great circles intersecting atthesame point andatitsinverse. Ithasbeenshown1that there areno more general possibilitiesofconformal transformations inspace than theKelvin transformation. Suppose that itisdesired tofindapotential function $(z, y,z) forsome given conductor configuration withknown surface po- tentials. Introducing arbitrarily aconvenient sphere ofinversion ofradius R,onecanfindtheinverse geometry oftheconductors. The potential function solving theproblem intheinverse co- ordinates isthengiven by (4) Toshow this,onemight transform theLaplacian differential equa- tionfor$from coordinates (z,y,2)tothose (x1 ',y',z'). Following thegeneraltransformation equation (31-27), onehasforexample forthez-coordinate because ofthesymmetry oftheCartesian coordinate system where dx=(E2 /r'2 )dx'from (1)defines theuniform scale factor h=R2 /r'2 .Butonecanwrite ia2* a* The lastterm in(6),when taken with thecorresponding terms in yfandx,gives theLaplacian of(I//) which must bezero since 1Kellogg,010 p.235,who refers toBlaachke: Vorlesungen uber Differential- geometrie,Vol. I;J.Springer, Berlin, 1924. Sec. 23] Intersecting Spheres 247 (1/r') defines thepotential ofasingle point charge; see(10-2). The firsttwoterms in(6),however, arealso 'ri -u~ (i\\~'-L(.L\ r |_r'2dx'2+ r'dx'dx'V7J~T dx'V23*7 This leads with (5),adding thecorresponding expressions inthe other coordinates, to V2 *(*, y,z)=Q5 (V')2 [**(x, y, z)]=(7) where V'means differentiation with respect totheinverse coordi- natesandwhere ofcourse theoriginal coordinates mustbeexpressed interms oftheinverse ones asin(4). Omitting theextra factor (r'/R}5 ,oneseesthat (4)willbethepotential solution forthe inverse function, satisfying theboundary conditions intheinverse geometry2ifthere thecharge values aremultiplied by(R/rf ). Theentire potential problemistherefore transformed, notactually solved, bythisKelvin inversion inasphere. Itwillbehelpful inallcases where inversion reduces theproblem toonealready solved atleast inpart, orreadily solvable bymeans ofimages. Thetransformation ratio ofalinecharge density Xcanbe found bynoting that lineelements transform asin(1),andcharges transform intheratio R/r'=r/R, sothat Ingeneralitwillbecome avariable charge density unless risa constant. Surface charge densities transform intheratio 9 r(r\iR\3A7-H+ (R)- (r)= (R) Intersecting Spheres. Assume twospheres ofradiiRIand #2intersecting orthogonally asinFig.23-3, carrying atotalcharge Q.Byinversion onasphere selected with center at intheinter- section ofthegiven spheres and ofradius 2R2,thetwospheres become planes intersecting normally atUftheinverse point toU. Now,iftheoriginal conductor hadacharge Q,allthefield lines 2W.Thomson: Papers onElectrostatics andMagnetism; Macmillan, London, 1872;firstpublishedinJLdemath., 12,p.256(1847). SeealsoKirchhoff,A13 p.53;Maxwell,A17 I,p.253; Ollehdorff,A18 p.335; Kellogg,010p.232;and Murnaghan,013p.141. 248 Field Plotting Methods 1.6 from itgotoinfinity, which inthesense ofinversion isapointand must bethelocation ofapoint charge (Q).This point charge hasbeen transferred tothepoint with respect tothetwoplanes, but itsvaluemust bechanged, thoughitcannot bedetermined in theusual manner. Assume that itsvalue be(Q')- FIG.233TwoOrthogonally Intersecting Spheres. Intheinverse geometry theproblemisnowtofindthesolution ofapoint charge (Q')inthespace between twoconducting planes intersecting atright angles, aproblem treated insection 10,Fig. 10-3. Placing image charges atPI,P2,P3asshown in Fig.23-3, onecanatonce findthepotential anywhere inthespace between thetwoplanes, which by(4)canbetransformed intothe potential solution outside thetwointersecting spheres. Onecan alsosolve directly intheoriginal geometryifonenow inverts all theimage charges onthesphere ofinversion, asshown intable 231. Sec. 23] Intersecting Spheres 249 CO CO K_S gI s2OP (N Q? s 3IH OP OOHO fl'C l-HO 5 c3*" 6?H <5H <5 250 Field Plotting Methods [Ch.6 Thetotal charge ontheintersecting spheresisthesum ofthethree inverse images Q=Qi'+Q2+Q3'=tf(l+T)-S*"-do) where T=Rz/Ri, andthepotential ofthespheres canbede- termined asthesuperpositionofthethree point charge potentials foranyonepointofthesurface, say,U,where ' ,Q2'Q ifoneuses thevalues oftable 23-1. Thecapacitanceisthen, with (10), (12) When T >oo ,thecapacitance approaches that ofthelarger sphere RZ,and ifT >0,itapproaches that ofsphere R\.The surface charge density canbeevaluated byinversion ofthecharge dis- tribution ontheplanes, using relation(9).The fieldvector and thefieldgraph canbestbefound asthesuperposition ofthethree point charges. Afieldgraph andthecomplete solutionoriginally givenbyW.Thomson(loc. cit.)aregiven inMaxwell,A17 I,p.261, andinFig.IVthere; abrieftreatment isinRamsay,A21 p.128; seealsoSmythe,A22 p.123,andMurnaghan,Cl3 p.152. Two spheres intersecting atanyangle v/n,where nisinteger, canbetreated bythesamemethod, theinversion leading toplanes intersecting atangles 7r/n; seeMaxwell,Al7 I,p.261.From Fig. 23-3 itisalsoseenthataspherical lens asformed bytheover- lapping dotted spherical segments between andUisinverted intothespace between thedotted continuations oftheplanes containing thepointP2.The field distribution desired isnow theone outside thedotted right-angle plane corner with the point charge (Q')at0,which cannot beobtained bytheimage method butrequires theconstruction ofGreen's function3 (see section 34). Iftheintersecting spheres aretobeconsidered isolated and under theinfluence ofanexternal point charge Q ,then inaddition 8Bateman^1 p.472. Sec. 23] Segments ofaSphere 251 tothesolution above, onehastosolve theplane geometry forthe effect oftheinverse chargeQ',i-e.,addanother setofthree images zhQo', and transfer these back into theintersecting spheres. Thesum total ofallcharges within these spheres must bezero, which determines thevalue ofthecharge Qrfrom above. The potentialisthatobtained bythesuperposition ofallpoint charges within finite distance oftheorigin 0. FIG.23-4TwoSpheresinContact andPoint Charge Q. Asaspecial caseonemight consider twospheres ofradiiRI andR2contacting each other asinFig.23-4under theinfluence ofapoint charge Q.Assume thespheres tobegrounded andat zero potential; then inversion onasphere with center atthepoint ofcontact andradius 2R2produces two parallel planes with the inverse Q'ofthepoint charge between them. The solution of thisproblem leads toaninfinite number ofimage charges andhas been indicated insection 21(seealsoMaxwell,A17 I,p.274). For equal spheres R\=R%=Randthepoint charge inthecenter plane, thesolution issymmetrical and2c=4R. Maxwell,A17 I,p.263, considers alsothree spheres intersecting orthogonally andgives thecharge distribution. Segments ofaSphere.Ifthesegment ofaspherical surface oraspherical bowl isgiven likel-T-2 inFig.23-2 ofsphere 252 Field Plotting Methods Si',thenonecanchoose asphere ofinversion (asshown) which willtransform Siintotheplane >Siandthespherical bowl into thecircular disk 1/-27 .Assume thecharge onthebowl as+Q; then itsfield linesgooutintoinfinity defined asapoint forpurposes ofinversion, andterminate there onapoint charge (Q).The inversion brings thischarge intothepoint butofvalue(Q') asintheprevious subsection. Thus intheinverse geometry one hastofindthesolution ofacircular diskexposed totheinfluence FIG.235Spherical Bowl. ofasingle point charge (Q')located at0;thiscannot besolved byusual image methods. Itisseen, however, thatthesideofthe circular disk facing thepoint chargewillhave thelarger charge density induced; byinversion thisbecomes theconvex side ofthe spherical bowlwhich willtherefore carry thelarger charge. The field ofacharged spherical bowl canbefound alsoby superpositionofpartial solutions satisfying theboundary condi- tionsonthesurface ofthebowlandthat ofthesphere S'inFig. 23-5 obtained byinversion ofthecircular areaSonthesphere ofwhich thebowl ispart. With$asthepotential ofthespher- icalbowl,Ritsradius, andQitssegmental angle, onefinds thetotal charge onitsinnerandouter surface, respectively, as Qiie^(1-cosfl)I (13) For 12>andQe-ITTZR&Q, asthey should. The total Sec. 23] Stereographic Projection 253 charge defines thenthecapacitance C=Qi+Qe=4efl(sinfi+fl) (14) *o Extensive details arefound intheoriginal treatise byW.Thomson, loc.cit.y p.178,who gives specific applications forseveral values offl;also inMaxwell,A17 I,p.276; inJeans,Al p.250;and in Kirchhoff,A13 p.58. 011endorff,A18 p.366, gives averythorough treatment andapplies theresults tothecomputation oftheca- pacitanceofsuspension insulators; healsousestheinversion of thisproblem tosolve acircular diskandaplate withacircular holeunder theinfluence ofapoint charge. Stereographic Projection. Asdiscussed inconnection with Fig.23-2, theplane Siandthespherical surface Siarerelated asmutual images byinversion onthesphere R,center 0.A different interpretationispossible, stating thatanypoint1on sphere Si' isprojected from thecenter onto theplane Siina one-to-one relationship such that angles arepreserved. Infact, ifapotential distribution onthespherical surface S\r satisfies the two-dimensional Laplace equation [seeAppendix 3,(40)] sino where 6measures thecolatitude and thelongitude; then the transformed relation interms ofthepolar coordinates pand intheplaneisagain thepertinent form ofLaplace's differential equation.4Designating theradius R/2=a,then d d/dd\d p=2atan->p= 2 dp sothat (15)becomes Any solution ofLaplace's differential equation intheplane can therefore,ifexpressed inpolar coordinates, beprojected directly 4Maxwell,Al7 1,p.286; Kirchhoff,A13 p.139;Smythe,A22 p.239. 254 Field Plotting Methods [Ch.6 upon thespherical surface andconstitute asolution ofLaplace's equation onthesphere. This interpretationiscalled stenographic projection and isparticularly useful forthesolution ofcurrent flow problems inthinspherical shells andbowls. Assume twopoint electrodes onthesphere with apotential difference Vandinlocations Q\tfaand 2,</>2,constituting current entry and exitpoints which, ofcourse, canreadily bemade very small circular areas tokeep densities finite. Intheprojection on theplane, thelocations become ft ft Pi=2atan^>fa, p2=2otan^>2 (17)2 2 andtheproblem isidentical withthetwo-dimensional oneoffinding theelectric field distribution between two parallel long straight lines solved in(12-29). Introducing there byanalogy thecur- rent7forX,theconductivity yfore,onehas ri tan2-+tan2-2tan-tan^cos(0 fa) =^In 1 12 1(18) tan2-+tan2772tan-tan cos(< </>2) 2t 222 where riand r2,theradius vectors from thesource points tothe point ofobservation intheplane P(p, </>),havebeen expressed in terms ofthespherical coordinates. Special choices ofthevalues (17)permit simplifications. Forexample,5 0i=2= ,fa= 02=T/2,representing twoelectrodes ononegreat circle ofthe sphere, leads to n tan2-tan2- *(0, 4>)=;rcoth"1 2ir7 Ba. 2tan-tan-sin</> Ixu_!1-COS PLCOS</)cothx (19) 2iry sinasm sin < 5Smythe,A22 p.240. Sec. 23] Two-dimensional Inversion 255 Assume twosmall diameters d\andd2fortheactual electrodes; thetotal resistance canbefound byanalogy from (12-46),ifone uses (8-11) andadmits asmall finite thickness t R= = In (20.) where thedistance Dbetween theelectrode centers intheplaneis [o01 902 01 02 v~|W tan2htan22tan tan cos(fa <fo) 2222 J ->4atan- (21) The firstform isthegeneral expression, andthesecond form holds forthespecial arrangement above. Thediameters d\andd2in theplane areapproximately given inthespecial casea<ir/2as di=di(1-tan2- )d2=d2(1-tan2 5)(22) \ 2/ \ 2/ ifoneprojects theends ofthediameters upon theplane anduses infirstapproximationd=a8a,andtan(a+8a/2)~tan(a/2)+ (5a/2) (1-tan2a/2). Combining (21)and(22)with (20),one finally has (23) Thesamemethod canbeapplied toasegment ofaspherical surface, such asthespherical bowl l-T-2 Fig.23-2. The pro- jection ontheplane Siisnowacircular area ofdiameter l'-2' andtheflowproblem hastobesolved within the circle, usually withboundary condition preventing flowoutofthecircle. However, onecould assume aheavy ringasborder ofthebowlandasingle electrode contacting thesurface atsome point. Allthese problems ntheplane canbesolved bestbymeans ofsuitable conformal transformations ofthecircular areaasshown insections 26and28. Two-dimensional Inversion. Quite analogously tothethree- limensional inversion with respect toasphere, onecanformulate in- version oftwo-dimensional fields withrespect toacylinder which is different from themethod ofconformal representation.6Geomet- 6Forexample Smythe,A22p.87. 256 Field Plotting Methods [Ch.6 rically, theinverse point toAinFig.231(allfigures canreadily beinterpreted geometrically asapplying tospheres orcylinders) with respect tothecylinder ofradius R,axis0,ispointArwith axial distance TA=RZ/TAandthecoordinates /R2/R2 XA= 2XA, VA=~ 22M (24) Aswith spheres, sowith cylinders; upon inversion, cylinders remain cylindersordegenerate intoplanes andviceversa. Thus, acylinder $2ofradiusra,axisMinFig.232tangent tothecylinder ofinversion atT,becomes again acylinder $2'tangent atTbut ofradius ra'=R(l R/b). Asm >oo ,cylinder S%becomes the plane Si,and itsinverse becomes thecylinder ofradius R/2 passing through theaxis0,which isobviously theinverse ofthe infinitely distant cylinder. Topreserve one-to-one relationship, itisconventional toconsider two-dimensional infinity asaline, astheinverse oftheaxisat0.Thisthenmeans thatanycylinder through theaxis and ofradius p<Rhasasinverse aplane at adistance b=R2/2pfrom theaxis andnormal tothecenter line. Cylinders which intersect thecylinder ofinversion orthogonally aretransformed intothemselves; theproofisexactly thesame as in(2), (3),with theomission ofthecoordinate z.Thus, inFig. 23.1, thesegment SB'S' ofthecylindrical surface within the cylinderofinversion willbepoint bypoint theinverse ofthe cylindrical surface SBS' outside. Thetransformation isconformed. Angles between lineelements inthex-i/-plane arethesame as between theinverted lineelements, which could readily bedemon- strated byconsidering these lineelements asbelonging tocylinders intersecting thecylinderofinversion orthogonally. Intwodimensions, thetransfer oftheLaplacian differential equation totheinverse geometry canbedone inthesamemanner asintheKelvin transformation, only thatnow hi=h2=has before, and /i3=1,sincenochange takes place inthez-direction. Asconsequence, *V,')-*(;*',, (25) thatmeans that thesame function solves thepotential problem inthetwoinversely related geometries; noadjustment ofcharge Sec. 23] Two-dimensional Inversion 257 values becomes necessary. However, surface charge densities will transform intheratio ofthesurface elements; since thedepthis uniform, surface elements willtransform inthesamemanner as lineelements in(24), and, therefore, (26) Asanexample, take asingle conductor formed oftwoorthog- onally intersecting cylinders asinFig.233,carrying acharge X perunit length. Byinversion onacylinder selected with axis at intheintersection ofthetwocylinders and ofradius 2R2, thecylinders become theorthogonally intersecting planes.Ifthe original conductor carries thecharge X,itsfield lines gointo infinity andterminate there onalinecharge(X) which by inversion isdistributed ontheaxis. Theproblem intheinverse geometry is,therefore, that ofalinecharge (X)inthecorner formed bytheorthogonally intersecting conducting planes, asin section 12.The solution requires three image charges (+X), (X),(+X) atP%,Pa,P4,respectively; andtransferring these back intotheoriginal geometry, their locations aregiven intable 23-1, where d\,d2jdHare,respectively,their distances from the axis0.Thesum ofthese three image chargesis(+X), which is alsothecharge ontheintersecting cylinders, andthepotential intheoutside spaceissimply thesuperposition ofthethree line charges. The actual fieldgraph can alsobeobtained bycor- responding graphical superposition. The potential oftheinter- secting cylinders themselves presents thesame difficulty asthat ofasingle wire (1228),sothat itisnotpossible todefine aunique value ofcapacitance. However, thedistribution ofthesurface charge canbeobtained byinverting thesurface charge densities onthetwoplanes bymeans of(26)ontothecylinders. With theaddition ofalinecharge (X)parallel toandlocated atPIinFig.23-3 asthereturn wire, the field linesfrom the intersecting cylinderswill allterminate onthischarge (X).The inversion leadsnowtothetwoplanes asbefore with theinverse of(X)located atPI',which isalso 2-The solution ofthe problem with theplanes nowrequires locations ofthethree images asshown intable 23-2. The total charge ontheintersecting cylindersisagain (+X); their potentialisthesuperposition of thefour linecharge potentials given intheright-hand column. 258 Field Plotting Methods [Ch.6 i d i i-^ c,|J Il~ !~* sS I3^p- ^|a^p- ife IIS ife as o o - ?l;*18^*^ GO i !l-ilx Sec. 24] Numerical Methods 259 Forexample, forpoint forwhich thedistances arethose given inthetable, onehas 16r2X 9(1+16r2 ) ^(27) andthecapacitance perunitlengthisthensimply x 4*e (0).9U+16T2 )c=1 Two cylinders intersecting atanyangle TT/TI,where nisan integer, canbetreated inasimilar manner. The orthogonal intersection ofthree cylinders leads toarectangular metallic slot intheinversion. 24-NUMERICAL METHODS Forvery complicated two-dimensional oraxially symmetrical boundaries ofelectrostatic ormagnetic fields, numerical iterative processes have been developed tosolve insuccessive approxima- tions thesystem ofdifference equations which canbesubstituted1 forthepartial differential equation ofthepotential. The syste- matic processofsatisfying thedifference equations only atdistinct pointsinthedesired fieldregion byreducing stepbystepthelocal error toaninappreciable value isnow called therelaxation method? because oftheearly application toproblems ofstress calculations inframe works where theerrors canbeconstrued asresidual un- wanted forces which aregradually relaxed3or"liquidated." These relaxation methods leadtoanetofpotential values through which equipotentiallinescanbedrawn; because ofthenumerical computationofthepotential values, thesolutions canbeobtained more accurately thanbythepurely graphical method ofcurvi- linear squares. 1L.F.Richardson, Phil Trans., A210, p.307(1910); seealsoBateman,01 p.144. 2R.V.Southwell: Relaxation Methods inTheoretical Physics; Oxford Uni- versity Press, England, 1946. 3R.V.Southwell, Proc. Roy. Soc.,A161, p.56(1935) andA163, p.41(1935) ; seealsoR.V.Southwell: Relaxation Methods inEngineering Science; Oxford University Press, England, 1940. 260 Field Plotting Methods [Ch.6 Relaxation Method forTwo-dimensional Potential Fields. TheLaplace differential equation fortheelectrostatic andmag- netostatic potentials, (2-2) and(6-6), respectively, canbesolved foracircular boundary withknown values onitbymeans ofthe Poisson integral (28-1) expressing thepotential value anywhere within thecircle interms oftheboundary values. Inparticular, atthecenter ofthecircle taken astheoriginofthecylindrical coordinates onehas riNi=lim-E*Jtf-UVn=l J(1) where $5(4) aretheboundary values asacontinuous function of angle andwhere thesummation isextended over discrete values along thecircular periphery. Relation (1)defines thevalue of thepotentialinaLaplacian field astheaverage ofalltheequi- distant values. Itisthisproperty thatcanbeused foranumerical trialanderror procedure bychoosing uponfirstinspection aset ofpotential values atequidistant points throughout the field region andthenapplying thecriterion (1)andnoting thedifferences between theassumed values andthose expected according to(1). Arevision ofthe first setmust thenbemade, guided bythedis- crepanciesinthe firstchoice, with asecond check byrelation (1).Afinal solution isobtained ifeverywhere inthefieldregion equation (1)issatisfied. Inapractical problem, onewill firstdraw atrather large scale thegiven arrangement ofknown potential boundaries, asfor example Fig.24-1.Selecting, tobegin with, arather widesquare net4with intersection points asindicated by0,1,2,3,4,one fills theentire areaandindicates ateach point aguessed-at potential value, preferably guided byacrude fieldplot. Obviously, one willstart inregions, like and,where thepotential distribution ispractically linear between boundaries. With asquare net, (1) reduces to (2) where fortheexact solution #(0)=0,butwhere fortheassumed distribution afinite residual72(0)isobtained which isindicative 4Instead ofasquare netofpoints, onecanchoose either hexagonal or triangular nets ofpoints, ofwhich onlythelatter have attained somepractical significance; seeSouthwell, footnote 2. Sec. 24] Relaxation Two-dimensional Fields 261 s+ + + gjfifSfafs s_s~i?!To o s^ -+Ti T+ ~S-LS2S_ I+ 262 Field Plotting Methods [Ch.6 ofthedegreeofapproximation obtained. Certainly 72(0)must gotozeroeventually, butanycorrection at itself willaffect the residuals atallitsneighbors aswell. Itrequires, therefore, some little experience toestimate thecorrections needed, and itis generally desirable tonote next totheassumed potential values theresiduals inbrackets asshown intheupper part ofFig.24 1. One willusethedistribution oftheresiduals forthesecond esti- mate, which might bestbeentered onaduplicate ofthepotential boundary sketch. Itistobeex- pected thatthelargest residuals willoccur intheregion ofgreatest non-uniform potential variation, hzasatand inFig.24-1, but itisalsoimportant tonote that hlQ ,.theprocedure isadefinitely con- vergent one,5even ifonestarts from arather crude first guess. Good results arerecorded by FIG. 24-2 General Spacing ofStrutt,B3 p.38,forelectron tube Potential Points. problems.6Themethod7isillus- trated inCosslett,B22 p.22,and ZworykinetaZ.,B32 p.386, forelectron optical problems, and in Southwell8forthemagnetic fluxdistribution inagenerator; many applications havebeenmade toelastic andheatproblems.9 Instead ofequidistant points, onecanchoose points inany desirable combination andderive relationscorresponding to(1) and (2)withappropriate coefficients. Assume thegeneral spacing offourpoints asinFig.24-2; then infirstapproximation h2(3) 6Bateman,01 p.147;R.Courant, K.Friedrichs, andH.Lewy, Math.Ann 100, p.32(1928). flSeealsoM.J.O.Strutt, Ann. d.Physik, 87,p.153(1928). 7Seeparticularly G.Shortley andR.Weller, JLAppL Phys. t9,p.334, (1938) andBull.No.107,Ohio State Univ. Engg. Exper. Station, 1942. 8R.V.Southwell: Relaxation Methods inTheoreticalPhysics, p.92and Figs. 37,38;OxfordUniversity Press, England, 1946. 9D.G.Christopherson andR.V.Southwell, Proc. Royal Soc., A168, p. 317(1938); R.Weller, G.Shortley, andB.Fried, JLAppL Phys., 11,p.283 (1940); M.M.Frocht andM.M.Leven, JLAppL Phys., 12,p.596(1941); andH.W.Emmons, Trans. A.S.M.E., 66,p.607(1943). Sec. 24] Relaxation Two-dimensional Fields 263 where thederivatives canbechosen, forbetter approximation, as theaverage values between theendpoints indicated bythesub- scripts. Thus, 2\o*c/ (4) dx/Q 2\dx2/3 Introducing (4)into (3)andadding thetwoforms (3)after divid- ing,respectively, byhi2and ft32 ,onehas Anidentical relation obtains for (d2$/dy2 )ifoneexpresses 42 and$4inamanner analogous to$1and$3above. Thesum of thesecond derivatives must vanish, being theLaplacian ofthe potential $(0). Ontheother hand, onecantakethesum of1/hi ofthe first linein(4)andl//i3ofthesecondline,andwith(3) express thefirstderivative interms ofthedistinct potential values, namely, andexactly analogous for(d$/di/)o bychanging subscripts1and 3to2and 4,respectively. Using these values forthederivatives intheLaplacian, onehasthesingle exact relation j-r+j-r)*(0)= Jl\tl$ Il2 + which canbeapplied inrectangular spacing with hi=h^and /i2=hjorinany local change inspacing, orincase ofpoints close toirregular boundary surfaces. Ifhi=h2=h3=h=h, (7)immediately goes over into (2)with#(0)=0;conversely, onecanwrite (7)intheform of(2)with theresidual 72(0) not necessarily zerobutapproachingit. Withsome experience, oneusually finds reasonably satisfactory 264 Field Plotting Methods [Ch.6 14 12residuals after about sixtoeight complete traverses ofthe field region, orafter thatmany approximations foraparticular point spacing. However, thismight notper- mitagood fieldgraph tobedrawn, so that closer spacing atleast inthe re- gions ofrapid potential variation might become necessary. Since halving ofthe spacing means fourfold slower conver- gence,itisadvisable tostart inany casewiththewider spacing. Onecan expedite convergence byafactor nwith the use ofimprovement formulas10 which give better potential values for blocks ofn2points interms ofthe bordering potential values, thussmooth- ingouttheeffect ofanyonechange upon theneighboring resid- uals. Forafour-block as(0,1,2,3)inFig.24-3onecanobtain animproved value atby15 10 8 13 FIG.24-3 Four-block Im- provement Relation. 2(<f> 6 andwith thisonecannowsuccessively improve(8) (9) where $and$1aretheimproved values from(8)and(9), respectively. Instead ofapplying suchimprovement formulas to thepotential values themselves, onecanapplythem withadvantage tothe differences insuccessive approximations, as8r <i>(0)= <ir+1(0)3>r (0),ifthesuperscript indicates theordernumber of theapproximation. Asillustration consider the42region ofvalues inthedotted boxCofFig. 24-1, which hasrather irregular potential values. The first setofvalues asshown inthefigure waschosen after drawing thefewrepresentative field lines freehand andwithno attempt tobeaccurate. Apparently, thechoice ofpotential values leftmuch tobedesired, because theresiduals inthecenter 10Shortley andWeller, footnote 7. Sec. 24] Relaxation Two-dimensional Fields 265 oftheC-block arerather large; forconvenient reference thevalues arereproduced here: (500) (500) 500-500- 500500- 500- 200-(350) (180) (400) (250) Theheavy lines indicate thefixed potential values oftheelectrodes; thenumbers inparentheses arethepotential values justoutside ofCjwhich areneeded tocompute theresiduals R(a). The latter aretheencircled numbers with theproper sign inaccordance with (2). Applying thesystem ofnumbering given inFig.24-3 to thesixteen points above, andcomputing theimproved potential values by(8)and (9)forthefourcenter points, leadtothisnew setofpotential values andresiduals: (500) (500) 480 (+11.5] -500- 500200- (250)(350) (180) 266 Field Plotting Methods [Ch.6 Though thechanges inpotential values inthecenter block arenot large, they aresomuch intheright sense thattheresiduals have become very small. Onecannowproceed totheadjoining 42 block toimprove thefourvalues shown inthedotted lineright above. After having reduced theresiduals everywhere torather uniformly small values,itbecomes necessary togotoafinermesh asindicated inFig.24-1bytheprimed points l'2'3'4'; again, the four-block improvement formulas (8)and (9)willbevery helpful. Since itiswellestablished thatthemethod gives aconvergent iteration forthepotential function $(x, ?/),onecaneither attempt toestablish directly thelimiting value11orattempt tominimize theerror inthesense oftheleast-squares method.12Theformer becomes necessarily verycomplicated foranypractical boundary geometry. Thelatter isofvalue after several steps ofapproxima- tionhave been carried through; onecanthen select forexample thebestcorrection ofthepotential atbyadding -Y2o[Ri+R2+RS+R4- 4B(0)] (10) whereRaaretheresiduals attheindicated points inaccordance with (2). This introduces asmoothed-over correction similar to theimprovement formulas(8)and(9)butdirectly interms of theresiduals. Thus, forexample, onecancorrect thevalue tothe left of inFig.24-1by5*=-1,that totheright of by5$=2;these small changes tend toreduce thelocal residual appreciably buttoaffect theneighboring residuals only little. Theextension oftherelaxation method tothePoisson equation oftheform (2-4) d23>d2$ p dx2dy2~ e iseasily made. Restricting attention touniform spacing, onehas with (5)andthecorresponding form for(32&/dy2 ) which istobeequal to(p/e). Thus, h2-=fl(0) (12) 11D.Moskowitz, Quart. Appl. Math., 2,p.148(1944). 12O.L.Bowie, Jl.Appl. Phys., 18,p.830(1947). Sec. 24] Plotting ofFieldGraph 267 constitutes themodified form taking theplace of(2).Knowing thevalue ofdensity p(x,y)asafunction ofthelocalcoordinates, onecanreadily carry through thesame procedure asabove. Therelaxation method canalsobeextended totwo-dimensional non-homogeneous andeven non-isotropic electric current fieldsby replacing thefield region byanetwork ofresistors withnodes at arbitrarily selected points.13 Plotting ofField Graph. Asatisfactory numerical plot of thepotential function asobtained bytherelaxation method gives amore orlessdense setofdiscrete point values. Itispossible todraw theequipotentiallinesbyinspection andthen construct thefield lines astheorthogonal curves; insuchcase,however, the accuracy needs tobechecked bythegraphical method ofcurvi- linear squarestogivereliable results. Itisbetter alsotocompute thefieldvector andthedirection ofthefield lines ateach point ofthefinalnumerical plotandthus determine more accurately thedirection oftheequipotential lines themselves. The fieldvector isessentially known byrelation (6) and itscorresponding expression forthe^-derivative /d$\ 1/hi ha (13) where allthepotential values arethe final numerical solutions ataselected point and itsneighbors asinFig.24-1. Theangle ofthefield linewith thez-axis isgiven bytan<f>=Ey/Ex. Intheregions ofuniform point raster, onehas allha=h,and thusmuch simpler sothatthevalue ofthefieldvector becomes14(14) E(0)=[(*3-*i)2+(**-*2)2]* (15) 13L.Tasny-Tschiassny,Jl.Appl Phys., 20,p.419(1949). 14B.VanderPol,JLI.E.E., 81,p.381(1937). 268 Field Plotting Methods [Ch.6 andthedirection ofthefield lines isdefined by Ev$4-4> 2tan</>==-- where istheangle with z-axis. Thus, theequipotential lines have directions defined by . ./ Xtan=-= <I?2 Theapplicationofthemethod tomagnetic fields withboundaries ofknown magnet ostatic potential 7,orinfacttoanyLaplacian potential field, follows bydirect analogy. Though intheCartesian coordinate system thevector potential outside ofelectric currents reduces intwo-dimensional problems toasingle component satisfying theLaplacian differential equation, application ofthe relaxation method needs considerable modification because the vector potentialdoesnotdefine agradient field. Relaxation Method forAxisymmetrical Potential Fields. Foraxially symmetric fields, thepotential equation hastheform where pisthedistance from theaxis. Foranyselected point asinFig. 24-2, with 1-0-3 parallel tothe axis,onecanagain evaluate thesecond derivatives exactly asinthe (3)to(6), except thatnowtheextra term with the firstderivative appears in(14)with theabsolute scale factor 1/p,which adjusts thescale ofpotentialvalues inaccordance with thedistance from theaxis. Calling thispfortheselected point 0,whichis,ofcourse, thesame also forthepoints1and3,onecanusetheanalogous form (6) appliedtopoints 2and4andobtains upon collection andordering ofallterms, /2p 2po+ fe4-fe2\ }=2PO ^'hl(hl which corresponds to(7)foranygeneral spacing ofthepoints neighboring on0. Sec. 24] Automatic Computing Aids 269 Foruniform spacing ofthepoints with allha=h,anddesignat- ingtheaxial distance p=mh,relation (17)reduces to (2m+1)$2+2m<J>3 +(2m-1)*4=fi(0) (18) wherefi(0)=fortheexact solution. Foraparticular problem, oneassumes again, asinthetwo-dimensional problem, asetof potential values throughout the field region andcomputes the residuals inaccordance with(18),noting these nexttotheoriginally assumed potentials. Arevision ofthefirst setmust thenbemade, guided bytheseresiduals, with asecond check by(18).Afinal solution isobtained ifequation (18)issatisfied everywhere. One canexpedite theconvergence ofthemethod bycertain improve- ment formulas15forwhich coefficients have beencomputed and tabulated.Again,itissimpler towork withthedifference values ofpotentials 5r*(0)=*r+1(0)-$r (0),where thesuperscripts indicate theorder oftheapproximation. The final result isanet ofpotential values through which equipotential lines canbedrawn. Itisadvisable, however, to compute thefieldvectoralso,which forelectrostatic problems is givenby --().<> ifpand zarethecoordinates taking, respectively, theplaces of xandyofthetwo-dimensional field. The expressions forthe fieldcomponents are,therefore, identical with(13)and(14),and thedirections ofthe field lines aswell astheequipotential lines aredefined inthesamemanner asthere. Theextension ofthemethod tothesolution ofPoisson's dif- ferential equation ismade inexactly thesameway asforthetwo- dimensional problem. Automatic Computing Aids. With theincreasing perfection ofmathematical machines,16numerical methods forthesolution ofpartial differential equations havebecome very economical in 16G.Shortley, R.Weller, P.Darby, andE.H.Gamble, JlAppl. Phys., 18,p.116(1947). 16F.J.Murray: TheTheory ofMathematical Machines; Columbia Uni- versity Press,NewYork, 1947; D.R.Hartree:Calculating Instruments and Machines; University ofIllinoisPress, 1948. 270 Field Plotting Methods [Ch.6 time, assuming thatamachine isavailable atthetime ofneed. Fortwo-dimensional potential problems, forexample, oneproceeds byfirstlaying outthesetofpoints andascribing some reasonable potentials tothem. Then, oneuses (2)or(18) directly forthe improved secondset,bycomputing thenew$(0), setting R(Q)=0. This process, which isnormally slowly convergent, becomes ef- ficient ifoneemploys automatic high-speed computing machines, such aspunched cardmachines,17orelectronic digital computers.18 Adifferent approachisbymeans ofanalogue computers. Net- work analogues forpartial differential equations have been developed19andmodels built20which allow theautomatic solution ofpotential problems oftheLaplace andPoissontype, aswellas ofthevarious types ofwave equations. PROBLEMS 1.Construct theresultant electrostatic field linesandequipotential lines fortwoparallellinecharges asinFig.19-3butwith linecharge densities (+2X) and(+3X). What willbethefieldpicture atgreat distance from the charged lines? 2.Assume three symmetrically located linecharges oflinear densities (+*) f(-3X), (+2\) infreespace. Construct theresultant equipotential linesand field lines. What willbethe field picture atgreat distance from thecharged lines? 3.Along cylindrical conductor oflarge radiusRextendsparallel totwo planes which intersect orthogonally. Construct thefieldplotifthecylinder hascenter distances 2Rand3R,respectively, from thetwoplanes, (a)Find themutual capacitance perunitlength. (6)Find thecharge density induced intheplanes, using arelative scale,(c)Find thebreakdown voltage for R=10cm. 4.Using thesame cross-sectional geometry asinthepreceding problem, assume anaxis ofrotation parallel tothelinewith center distance 3Rfrom thecircle andadistance 2Rfrom this line. Construct the field plot ofthe resulting geometry, i.e.,atoroid outside andcoaxial withacylinder ofradius 17W. J.Eckert: Punched Card Methods inScientific Computation; the Thomas J.Watson Astronomical Computing Bureau, ColumbiaUniversity, 1940. SeealsoM.Kormes, Rev. Scient.Instr., 14,p.248(1943). 18"Proceedings ofSymposium onLarge ScaleDigital Calculating Ma- chinery," Annals ofComputation Lab., HarvardUniv., 16(1948); C.F.West and J.E.DeTurk, Proc.I.R.E., 36,p.1452 (1948). 19G.Kron,Electr. Engg., 67,p.672(1948); S.A.Schelkunoff, BellSystem Techn. Jl.t27,p.489(1948). 20K.Spangenberg andG.Walters, "An Electrical Network fortheStudy ofElectromagnetic Fields," Techn. Report, No. 1,ONR, Contract N6-ORI- 106,Stanford Univ., 1947. Problems 271 2Rlocated above aplane orthogonal tothecylinder, (a)Find themutual capacitance. (6)Find thebreakdown voltage forR=10cm. 5.Along cylindrical conductor oflarge radiusRextends inairparallel totheboundary planeofasolid dielectric ofabsolute dielectric constant e. Construct thefield plotsforadistance 2Roftheaxisofthecylinder from the dielectric. Find thecharge density distribution onthecylinder, using a relative scale. Find theforce action upon thecylinder. 6.Using thesame cross-sectional geometry asinproblem 5,assume an axis ofrotation within thedielectric atadistance 2Rfrom theboundary lineand parallel toit.Construct the fieldplot fortheresulting geometry, i.e.,atoroid outside and coaxial with adielectric cylinder, (a)Find the capacitanceofthetoroid. (6)Find the critical voltage forappearance of corona ifR=10cm. 7.Construct thefield plot infreespaceoftwovery small spheres ofradii RIand^2=272i,with adistance ofthetwocenters d=10ft2,andwith charges Q2=3Qi. Find thesingular point. What isthe field picture at large distance from thespheres? Define thefieldmapsinterms ofcapacitance coefficients andgive their relative values. 8.Alongrectangular busbarofdimensions 2oand2b<2aextendsparallel with itsbroader sidetoaninfinite conducting plane atadistance 46from itsplane ofsymmetry. Construct thefield plotifa/6=5,and(a)findthe capacitance with respect totheplane, (6)findthebreakdown voltageif6= 2cm, (c)findthecapacitance andbreakdown voltage between two likebus barswith 6=2cmforwhich theinfinite planeistheplane ofsymmetry. 9.Three parallel longrectangular busbarsformathree-phase transmission system. Construct the field plot forthree identical barswitha/b=5and withmutual distances c=2aifthevoltages tothevery distant ground are, respectively, Vi,V2=-Vi cot15,V3=ViSm45 (a)Find the sin15 mutual capacitancecoefficients perunitlength; (6)findthecharges perunit length onallthree conductors; (c)findtheforces acting ontheconductors. 10.Assume thesame cross-sectional geometry asinproblem 8andtake the lineparallel totherectangle asaxis ofrotation. Construct the field plotif0=6 andfindthecapacitanceoftheannularring. 11.Construct theresultant magnetic field lines inairfortwoparallelline currents ofvalues I\and72=-3/i, byusing (a)thevectorpotential, (6)the magnetostatic potential function. Find the field atlarge distance from the wires. 12.Construct theresultant magneticfield lines forathree-wire, three- phase transmission system ofsymmetrical geometry carrying thecurrents (a)/!,72=-2/b73=7i;(6)/!,72=-4/i, /3=+3/i. Find the field atlarge distance from thewires. 13.Assume fourparallel wires inairsoarranged that inacross-sectional plane they arelocated atthevertices ofasquare. Find theresulting magnetic field plotifthecurrents intheupper twowires arc/i,inthelower two wires ^2/1, sothatthemagnetic fluxes oppose. Obtain themutually linked fluxperunitlength from theresulting fieldplotandcompare with theanalyti- cally predicted value. (See alsosection7.) 272 Field Plotting Methods [Ch.6 14.Construct themagneticfieldplot inthespace between two ideal magnetic boundary surfaces ofconstant magnetostatic potentialsifone surface isaninfinite plane andtheopposite surface hasaperpendicular distance 2irx varying asgo/cosjwhere goistheminimum distance ofthetwoboundaries, xthelinear distance along theplane surface, and Tthespatial period ofthe field distribution. Find themagnetic reluctance perunit length. 15.Thesymmetrical poles ofamagnet have pole faces ofwidth 2afrom which thesteel tapers linearly overaheight h=5atothelarger cross section ofthepole core ofwidth 26=5a.Taking itasatwo-dimensional field problem (ofgreat length normal tothecross section) between ideal magnetic surfaces ofconstant magnetostatic potentials, construct thefield plot foran airgap2g=a/2. Find themagnetic reluctance perunit length. Find the variation ofthemagnetic fluxdensity intheplane ofsymmetry halving the airgap. 16.Assume asingle longconductor ofsquare cross section carrying uni- formly distributed current. Construct thefieldplotboth inside andoutside theconductor byusing (20-10). Verify forseveral points along afieldline, thatAz=cons inaccordance with (15-21). Verify thevalidity of(20-7) inside theconductor. 17.Thevector potential ofalong thinrectangular bar isgiven by(15-19), andthelinesAz=cons represent themagneticfield lines. Apply thisto asingle conductor ofsquare cross section carrying uniformly distributed current inorder toobtain itsmagneticfield lines. Choose asubdivision into sixstrips andcheck several points bytheexact solution (15-21). 18.Apply themethod ofthepreceding problem totwolong parallel and identical conductors ofsquare cross sections with sides 2aandcenter spacing 6a.Find theresulting magnetic field lines forequal andopposite currents. Find thelocation ofthekernels andcheck thevalidity of(20-7) inside the conductors. 19.Apply themethod ofproblem 17totwolong parallel conductors with equalandopposite currents, ofcircular cross sections andofradiiRi,R2=2Ri t andwith center spacing 2Rz. Find themagneticfield linesandthelocation ofthekernels andcheck with theexact solution insection 15. 20.Theinductance ofaloopformed bytwolong parallel conductors of finite cross sections withequal andopposite currents isdefined by(7-1)and (7-2) andreiterated in(15-11). Assume thetwoconductors ofidentical square cross sections with sides 2aandcenter spacing 6aasinproblem 18. Knowing themagnetic vector potential values, onecanevaluate theintegrals (15-11) graphically;findtheinductance perunitlength ofthecurrentloop. 21.Asingle longconductor ofsquare cross section with sides 2acarrying uniformly distributed current extends parallel toanidealmagnetic boundary surface ofconstant magnetostatic potential. Find themagneticfield lines andthelocation ofthekernel foradistance 4aoftheplane from thecenter oftheconductor; compare thelocation ofthekernel withthat inproblem 16. 22.Inproblem 21assume thecross section oftheconductor oriented with itsdiagonal normal tothemagnetic boundary plane, keeping thesame center distance. Find themagnetic field linesandthelocation ofthekernel. Problems 273 23.Acylindrical coilofmean diameter 2Rcarries atotal current-turns valueNIuniformly distributed overarectangular cross section ofsmall radial width Saandofheight 26=2R. Construct themagnetic fieldplot, utilizing thesuperposition ofsolutions foracircular loop ofcurrent given insection 13. Demonstrate thevalidity ofthesuperposition. Find theinductance ofthe cylindrical coil. 24.Acircular cylinder ofmagnetic steelwith relative permeability Mr=200, radius R,andheight h=2R,carries acircularloop ofwireonitssurface in theorthogonal planeofsymmetry. Construct the fieldplot. Find the inductance ifthewireloophasasmall radius a. 25.Aflatpancakecoilhasamean radius R,asmall height 56,awidth 2a=R,and carries atotal current-turns valueNIuniformly distributed over itscross section. Construct themagnetic fieldplot, utilizing the superpositionofsolutions foracircularloop ofcurrent given insection 13. Demonstrate thevalidity ofthesuperposition. Find theinductance ofthe coil. 26.Two identical flatpancake coils asdefined inproblem 25arearranged coaxially with acenter distance 2c=R\.Find themutualinductance, utilizing thefield plot ofproblem 25. 27.Two thin flatpancake coils ofmean radiiRiandRZ=%Ri, small heights 5&i=562,andwidths 2a\=%R\, 2az=Ri,arearranged coaxially withacenter distance 2c=R^Find themutual inductance ifthenumbers of turns areJViandNz,respectively. Utilize theresult ofproblem 25. 28.Acircular cylinder ofmagnetic stocl ofrelativepermeability /ur=100, radius RI,andheight h=2Rcarries onitssurface athin cylindrical coilof thesame height handwith atotal current-turns value NI. Construct the magneticfield plot. Find theinductance ofthecoil ifthelayer ofwires is thinbut finite. 29.Two small spheres ofequalradii parelocated atlarge distance 2cfrom eachother intheplane x=ofFig.21-1. Assume oneofthespheres tobe thesource ofcurrent /andtheother tobethesinkwithin theinfinite stratum ofconductivity 7andthickness (a+6). Find theresistance Rifthespace outside thestratum isnon-conductive. 30.Consider thesame geometry asinproblem 29butassume thespheres tocarry charges Q,respectively, and tobeinairbounded bytwoconducting planes. Find thecapacitance between thespheres asInfluenced bytheprox- imity oftheconducting planes. Find theinduced charge densities onthe planes. 31.Find infirstapproximation thecharge distribution onthesmall sphere ofFig.21-1withaT*b;assume thena=6. 32.Find theforce action between thetwospheres ofproblem 30;demon- strate thattheforce follows Coulomb's law ifonedefinesappropriately an equivalent center distance rf . 33.Along thin wire islocated midway between oneconducting plane andoneplane dielectric boundary, asforexampleinFig.21-3,ifthemedium IIisreplaced byaperfect conductor. Find thecapacitance perunitlength between thewire ofradius aandtheconducting plane. Find thecharge density induced intheconducting plane. 274 Field Plotting Methods [Ch.6 34.Two parallel very long wires ofsmall radiiaandwithmutual center distance 2carelocated with their axis intheboundary plane between two media ofconductivities 71and72and dielectric constants eiand e2-Find thecapacitance perunitlength between thewires; findthecurrent perunit length between thewires iftheir potential difference isV. 35.Twohomogeneous thincylinders ofradiiaandtemperatures T\andTZ extend paralleltotheplane boundary oftwomedia ofthermal conductivities kiand kz',thecylinder atTIislocated inmedium 1with itsaxisadistance 10afrom theboundary plane, thecylinder atTZislocated inmedium 2with itsaxisadistance 20afrom theboundary plane. Find theheat flow (ex- change) between thetwocylinders perunit length. 36.Asmall dipole oflargemoment pinarbitrary direction islocated be- tween thetwoplates ofaninfinite plane condenser similar toFig.21-1,if oneassumes apotential difference Vbetween theplates. Find force and torque action upon thedipole. 37.Find theinduced charge density inthecylinder ofradiusRinFig.21-4 ifitisgrounded andexposed tothetwo parallel symmetrically located line charges A.Demonstrate theidentity ofthesolution with (21-20) for r=Rifthelinecharges recede to . 38.Inathree-phase, three-conductor cable, thethree cylindrical conductors arelocated within thegrounded sheath symmetrically with respect toeach other atdistance dfrom thecentral axis. Find theapproximate potential distribution iftheradii oftheconductors areaandtheir potentials with re- spect tothesheath V\,%V\, %V\. Find allthecapacitance coefficients. Check thedegreeofapproximation bycomputing theresultant potential values overthesurfaces oftheconductors. 39.Show that inproblem 38onecangetabetter approximationifthe equivalentlinecharges oftheconductors arenotlocated intheir axisbut shifted slightly radially towards thecentral axis. 40.Athinlongwire ofradius aandlinear charge density +Xextends paral- leltoadielectric cylinder with distance bbetween axes asinFig.21-6. Find infirstapproximation thecharge distribution over thesurface ofthewire anddemonstrate theinfluence oftheproximity ofthedielectric cylinder. Plotthemaximum charge density asafunction ofthedielectric constant tz ofthecylinder. 41.Forthesame geometry asinproblem 40,compute themaximum charge density onthewire asafunction oftheradiusRofthedielectric cylinder, keeping constant thedistance from theaxisofthewiretothenearest surface pointofthedielectric cylinder. Show that forR >oneobtains thesame value asforawire parallel toaninfinite plane dielectric boundary. 42.Two metal pins aremolded intoaplastic cylinder giving thesame geometryasinFig.216withthepincenters atA"andB",theplastic cylinder ofdielectric constant e.zsurrounded byair.Find thepotential distribution between thepinsiftheyhave apotential difference Vapplied between them. Find thecapacitance perunitlength andcompareitwith thevalue inair alone. 43.Athin circular ring ofcharge iscoaxial with agrounded sphere of radius R,i.e.,theaxis oftheringpasses through thecenter ofthesphere. Problems 275 Find thecapacitanceoftheringwith respect tothesphereiftheplane ofthe ringisatdistance cfrom thecenter ofthesphere, and ifitswire radius ais small compared with allother dimensions. 44.Find theincrease inexternal inductance forathin circular loop of current lying parallelto(a)anideal equipotential magnetic boundary plane, (6)aboundary plane ofmagnetic material ofpermeability /*.Find theforce upon theloop inboth cases. 45.Assume theplaneofthetwo-wire transmission line inFig.22-2 to make anangle with themagnetic boundary plane. Find thetotal external inductance ifthemagnetic material haspermeability /i.Plotthevariation ofthisinductance asafunction oftheangle 0,keeping thedistance dconstant andequal toc,onehalfthespacing ofthewires. 46.Asingle thinwirecarrying current 7liesinaslotformed bytwoparallel solid blocks ofironadistance 2hapart andclosed byanorthogonal block. Find themagneticfield intheslot iftheaxisofthewirehasdistances aand b=2h afrom theparallel boundaries, anddistance cfrom thebase block. Find theforceupon thewire. 47.Along thinwire carrying current +/extends parallel toamagnetic cylinderofrelative permeability HT=200asshown inFig.223.Construct themagneticfield plotifthedistance b=2R.Find theforceupon thewire. 48.Acylindricalshell ofmagnetic material ofpermeability nhasouter radius RIandinner radius RZand isbrought intoauniform magneticfieldBQ. Find theresultant magneticfield iftheaxis oftheshell isorthogonal tothe magneticfield. 49.Athinlong wire carrying current /islocated halfway inthespace between asolidmagnetic cylinder ofradius R\andacoaxial shell ofinner radius #2andouter radius ^3.Construct themagneticfield plotifthe magneticmaterials have thesame relative permeability /ir=100and if R3=2R2=3fli. 50.Theendconnections ofwindings inelectrical machines might becon- sidered asrectangular loops extending normal totheiron core. Assume, then, athinwirecarrying current /andforming arectangular loop ofsidesanormal to,and2b=5aparallel to,aninfinite plane magnetic boundary. Find the inductance oftheloop section inair ifthepermeability oftheiron isMand theradius oftheround wire isf. 51.Two isolated andconducting spheres areintersecting orthogonally as inFig. 23-3. Find theelectric field distribution ifapoint charge+Q is located attheintersection ofthelineUOwith thesphere ofinversion onthe oppositesideofU.Find theforceupon thepoint charge. Find thecapaci- tance ofasmall sphereofradius acarrying thecharge+Qandhaving its center atthelocation ofthepoint charge. 52.Two spheres contacting each other asinFig.23-4 aregrounded and under theinfluence ofapoint charge+Qlocated attheintersection ofthe x-axis with thesphereofinversion. Find the electric field distribution. Find theinduced charge density onthespheres. 53.Usethesame cross-sectional geometry asinproblem 52,butsolve it asatwo-dimensional problem withalinecharge+Xreplacing thepoint charge andcylinders replacing thespheres. 276 Field Plotting Methods [Cb.6 54.Two parallel cylinders intersecting at7r/3form asingle conductor carrying acharge+Xperunitlength. Find thepotential distribution. Find thecharge distribution overthecylinder surfaces. 55.Avery thinhemispherical shell ofradiusRhastwopoint electrodes applied toitssurface atdiametrically opposite points oftheparallel circle ofradius R/2. Findtheresistance oftheshell iftheelectrodes canbeassumed assmall equipotentialcircles ofradiiaand iftheshellhasthickness tand conductivity 7.Find thecurrent distribution intheshell. 56.Assume that thehemispherical shell ofproblem 55isreduced toa zonebycuttingoffthesection below theparallel circle ofradius R/2sothat theelectrodes areapplied with their centers ontherimofthezone. Find the resistance. 57.Alongcylinder ofradiusRiscoaxial witharectangular sheath ofsmall sides 2a=4/2andlarge sides 26=10/2. Find thecapacitance between the two conductors. Hint: solve forthepotential distribution firstbythe relaxation method, assuming thepotential difference V=100volts; then construct curvilinear squarestoobtain thecapacitance. 58.Demonstrate thevalidity oftheimprovement formulas (24-8) and (24-9). 59.Two identical longbusbars ofrectangular cross sections arearranged parallel with their larger sides 26atacenter distance 2c=6a,where 2ais thesmaller side. Find thepotential distribution between thembythere- laxation method ifapotential difference Visapplied. Find thecapacitance. 60.Find thepotential distribution between thetwodeflecting plates ofan oscilloscope inclined symmetrically atanangle of20with respect tothecenter plane. Assume theplates asvery thin ofwidth 26,withaminimum distance 0.26,and infinitely longnormal tothecross-sectional plane. Find thevalue ofthetransverse electric fieldvector along theaxis ifthepotential difference isV.Find thecapacitance perunitlength. 61.Find themagneticfield distribution forthetwo-dimensional geometry shown inFig.27-13c ifthefinite distances aredefined asfollows: 6'-6"=g, 4^5=10g,3-4=200,2'-2"=400. Establish theboundary line forhomo- geneous field distribution. Find thereluctance perunitlength forthein- homogeneous part ofthe field. 7.TWO-DIMENSIONAL ANALYTIC SOLUTIONS Formany purposesitisdesirable tosecure analytic solutions of field problems, since they permit deduction ofbroad design principles aslong astheyremain manageable. Two-dimensional potential theory hashadthegreat benefit ofthebranch ofmathe- matics known as"TheoryofFunctions ofaComplex Variable" (seeAppendix 4,D),which hasledtomany rigorous solutions in singularly simple form particularly welladapted tothe inter- pretationofthefieldgeometry. 25-CONJUGATE FUNCTIONS Tofixanypoint inaplane, two realcoordinates have tobe given. With anorthogonal coordinate system, anynumber pair, asforexample (z,y)fortheplane Cartesian, or(r,$)fortheplane polar system, signifies apoint P.Such anumber pairorapoint can alsobedenoted incomplex form1byz=(x+jy)with j=V^l, andj2=-1. Asoneletsxandytakeonallpossible realvalues between( ) and(+oo),zcovers theentire plane, usually then referred toas thecomplex z-plane. From Fig.25-1onealsotakes forpointP z=x+jy=rcos</>+jrsin=re-70(1) where r= \z\=(x2+ 2/2)^istheabsolute value ofthecomplex numberz,sometimes also called modulus, andwhere isthe 1Mathematical texts normally designate complex numbers byz=x+iy with i=V 1;inelectrical engineering,ithasbecome customary touse jfortheimaginary unit inorder toavoid confusion with thesymbol for electric current which traditionallyischosen asi, 277 278 Two-dimensional Analytic Solutions [Ch. 7 argumentofz;ey0canbeinterpreted asadirection factor, having theabsolute value unity, similar toaunitvector invector analysis. However, there arebasic differences between plane vector analysis andcomplex function theory which make thelatter vastly more powerful asamathematical method ofanalysis. Obviously, theabsolute value ofacomplex number canalso beobtained bywriting |z|2=(x+jy)(x jy)=zz=x2+y2(2) ifagain j2=Iisobserved. The combination xjy=z, which inthez-plane leads totheimage point (x, y)ofthepoint (b) FIG.25-1 RepresentationofFunctions ofaComplex Variable: (a)complex z-plane, (6)complex w(z)-plane. (x,y)with respect tothez-axis,iscalled theconjugate complex number ofz. Theproductoftwocomplex numbers isagain acomplex number (3)asisalsothesquareofacomplex number z2=(*+jy?=(z2-y2 )+w and, indeed, anyconceivable functional operationwillalways again result inacomplex number (ofwhich real orimaginary numbers arethen only special cases). Complex numbers thus form aclosed number system.Itis,therefore, possibletointerpret anyfunction /(z)=wagain ascovering aplane withw=u+jv, asinFig. 25-1, inwhich anypointwisthen theimageofthe generating pointzofthez-plane. The possible relationships Sec. 25]Analytic Functions ofaComplex Variable 279 between wand zplanes aretheprimary object offunction theory; seeparticularly thereferences inAppendix 4,D,b,andKellogg,C1 chapter XII. Analytic Functions ofaComplex Variable. Forapplica- tions tolinear field problems, functions must beregular within theregions considered andtheymayhave prescribed discontinuities ontheboundaries corresponding tophysical sources likecharges, currents, etc. Itistherefore natural torestrict study ofcomplex functions toregular oranalytic functions inthesame sense aswith real functions, i.e., require single valuedness, continuity, and differentiability within theregions ofinterest. Single valuedness canusually bemetbyproper restriction ofthevariables andintro- duction ofbarriers asalready done inthecase ofthemagnetostatic potential,section 6;forcomplex functions onemust require that, toevery pointzachosen intheneighborhood ofzinFig.25la, there corresponds oneandonlyonepointwaf(za)=ua+jva intheneighborhoodofwinFig.25-16. Continuity requires that thepointwacanbemade tomove arbitrarily close towbyselecting zaproperlyclose tozandthat, inthelimit,wagoes intowasza goesintoz,nomatter inwhat direction thelatter isdone. With respect todifferentiability onehastoconsider that w=f(z)=u(x,y)+jv(x, y) (4) isthecomplex combination oftwofunctions each ofwhich depends onthetworealvariables xandy.Differentiability means, there- fore, theexistence ofthecontinuous first partial derivatives of uand vwithrespect toxandyaswellasthat limAw/Az dw/dz Az>0 exists, i.e.,hasthesame value atapoint z,nomatter howAz >0. Whereas with areal variable onlytwoopposite directions are possibleinapproaching apoint x,withacomplex variable zthere areinfinitely many directions inwhich toapproach thispointz. Choosing forconvenience once thex-andonce the^-direction, thepartial derivatives follow with theuseof(4)as dw_du.dv dw_du.dv Hx= ~dx+3dx' ~dj^= djy ~djy() Assuming theexistence ofthecontinuous partial derivatives, both results must have thesame value, sothatupon equating and 280 Two-dimensional Analytic Solutions [Ch. 7 separating theexpressionsinuand vinto realandimaginary parts onefinds du_dv du_dv dx~ dy' dy~~ dx thefundamental Cauchy-Riemann differential equations, which bring tolight theinherent regularityoftheanalytic functions and constitute thenecessary andsufficient conditions foranycomplex function tobeanalytic atapoint P(x, y).Thesufficiency follows from thefact that,ifonenowformulates thegeneral expression forthederivative lim = Az- Az Az+j&y anduses relations (6),onereproduces theexpressions (5). Itisalsodesirable toassure integrabilityoftheanalytic function /(z)inthecomplex plane. With (4)onehas J/(z)dz=f(u+jv)(dx+jdy} =f(udx-vdy)+jf(udy+vdx) (7) Foraclosed regular path (which hasnocross overs ordiscon- tinuities) intherealz-y-plane, each ofthelineintegrals canbe transformed2intoasurface integral ofthepartial derivatives of uand vwhich already have beenassumed toexistandtobe continuous, namely, (8) However, fortheanalytic function, theCauchy-Riemann equations (6)make theintegrands ontheright-handsidevanish atevery 2This isessentially thedivergence theorem (Gauss's theorem) intwo di- mensions; forthe specific form seeanybook onadvancedcalculus, like Doherty andKeller,03 p.247; Sokolnikoff andSokolmkoff,D9 p.173; aswell asallreferences onfunctions ofacomplex variable. Sec. 25]Conjugate Functions andPotential Fields 281 regular point, sothat foranyclosed regular path entirely within a regular region onehas " <fe=(9) Therefore theintegral overanyopenpath inaregular region cannot depend onthepathitself butonlyontheendpoints (Cauchy's integral theorem) and will itself beananalytic function ofeither oneofthelimits, since itsderivative is/(z)which wasassumed to beanalytic inthe first place.Itfurther follows that foranalytic functions derivatives ofanyorder existandthat, inturn, they are allanalytic functions. Conjugate Functions andPotential Fields. From the above itisassured thatthehigher partial derivatives ofu(x, y) andv(x,y)exist. Differentiating, therefore, thefirst ofequations (6)with respect toxandthesecond with respect toyandadding both, or,conversely, differentiating the firstwithrespect toyand thesecond with respect toxandsubtracting both, oneobtains d2ud2u d2vd2v Both therealandimaginary partofw=f(z), considered asfunc- tions oftheordinary realcoordinates xandy,satisfy theLaplacian differential equation andthus areharmonic functions andsolutions ofpotential problems (seesection 2). Asrealfunctions ofthetwovariables xandytu=cons, aswell asv=cons, defines families ofcurves inthereal x-7/-plane asin Fig.25-2; theslopesofthesetwofamilies arerelated, asdivision oftheCauchy-Riemann differential equations (6)reveals du/dx=dv/dy du/dy dv/dx(} i.e.,thetwofamilies ofplane curves aremutually orthogonal. Itis customary toidentify the realpart ofthecomplex harmonic function wasthepotential function; then intheformw=u+jv, viscalled theconjugate* function ortheharmonic conjugate tou\ intheform(jw)=vju,(u)iscalled theconjugate func- tion, orharmonic conjugate tov.Sinceuandvcanbeconjugate to 3This should notbeconfused with thedefinition ofconjugate complex numbers given inconnection with (2). 282 Two-dimensional Analytic Solutions each other, onefrequently designates them as"conjugate func- tions." Because ofthemutual orthogonality, theconjugate function defines thegradient lines orfield lines ofthepotential field, sothat thecomplex harmonic function gives atonce the entire orthogonal fieldgeometry without necessitating further com- putations. Ontheother hand,ifonly u(x,y)isgiven asareal harmonic function(satisfying theLaplacian differential equation), x FIG.25-2 UseofConjugate Functions inElectrostatic Field. thenonecanconstruct ananalytic function (u+jv),wherebyv isfound forexample from(6)byintegration ofthederivatives ofu(x,y) Itisself-evident thatthesum oftwocomplex harmonic functions (wi -\-w2)isagain aharmonic function andtherefore sums of conjugate functions areagain conjugate functions. Itisalso readily shown that,if t=r+jsisananalytic function of w=u+jv,and thisinturnananalytic function ofz=x+jy, then tisalsoananalytic function of(x+jy)and rand sarecon- jugate functions ofxandy.Since dr_drdu drdv dxdudx dvdx andtheCauchy-Riemann equation arevalid, onehas dr_dsdv dsdu_ds dx dvdydudy dy Sec. 25]Conjugate Functions andPotential Fields 283 showing thevalidity oftheCauchy-Riemann equations forrand sinterms ofxandyandtherefore their conjugate relationship. Theuseofconjugate functions willbediscussed astheyapply tothemapping ofelectrostatic potential fields, buttransfer toany other fieldproblem canreadily bemade bymeans oftable 9-1. Assume, then, inthecomplex harmonic function w=u+jv,the realpart u(x, y)=<f>asthepotential function; thefieldstrength vectorEfollows asgradientintherealx-y-plane du du x~ dx'y~ dy andthecomplex combination, taking intoaccount thesecond relation (6),gives4 Ex+jE y=E=-+j (14) Comparison with the first relation in(5)shows that thiscanalso beexpressed as *" (15)dz sothat theabsolute value ofthecomplex derivative isadirect measure ofthefieldstrength andtheconjugate complex derivative istheequivalent ofthetwo-dimensional gradient ofvector analysis. The dielectric fluxperunitlength inatwo-dimensional Cartesian system between anytwopoints inthefield isgivenby dS=(Dxdy-Dydx) Using (13)butsubstituting thederivatives ofvfrom (6),onehas orthedielectric fluxbetween 1and2ismeasured bythedifference ofthevalues oftheconjugate function atthetwopoints. There- 4Inwhat follows, anyvector fieldquantity appearing asacomplex number willbedesignated without indices orother distinctive marks; theabsolute value willbedesignated bytwovertical barsandanycomponent byasuitable subscript. 284 Two-dimensional Analytic Solutions [Ch. 7 fore, onefrequentlycalls theconjugate function alsotheflux function (orstream function inhydrodynamic problems). Onecanalsoestablish therelationship tothemethod ofcurvi- linear squares (section 19). Selecting twoequipotential lines Ui andu2inFig.252with apotential difference A*=MI-u2=\E\As1=Ex&x'+Eyky' (17) andtwoflux lines Viand v2togivethesame numerical difference vl-v2=\E\te"=Exby"-Ey&x" (18) thenonemusthave by"=As', Az"=-Ay' thetwoelements As'andAs"must beofequal length and, of course, orthogonal toeach other. Oralso, with thepotential difference between electrodes divided intoequal increments and fluxlines selected bychoosing values ofvwiththesame increments, oneobtains atonce theanalytic equivalent ofthecurvilinear squares. Thecapacitance oftheindividual curvilinear squareis thenagaineasinsection 19. Given twoelectrode surfaces ofpotentials $i=HIand$n= uiijwith values offluxfunction viandvumeasuring thetotal flux ofthevectorEbetween these electrodes (orthose parts of interest), thecapacitance perunitlength (total orpartial)isthen C,-.^^ (19) - Thiscanatoncebetranslated into allother potential fieldsby means oftable 9-1. Finally, onecantakefrom (14)with(5) du dvdwdw=-+J-=-=^ or,taking thelogarithmofboth sides, \In(Ex*+EJ)+j(*-tan"1 |j)=In(^=P+jQ (20) The lines ofconstant field strength arethus defined5byP= 6Re(u>) means therealpart ofthecomplex function, Im(w) theimaginary part. Sec. 25] LineCharges andLineCurrents 285 Re (In)iandthelines ofconstant direction offield linesbyVdz/ Q=Im(in JThis isofparticular interest inflowproblems, buthassignificanceinalldesign problems. Aninvestigationofallanalytic functionswill, therefore, leadto acorresponding array ofpotential solutions, whereby again simple typesoffields canbesuperimposed togive solutions formore complexcases. Line Charges andLine Currents (Source andVortex Lines). One ofthemost widely used functions isw=FInz, whereFisaconstant toadjust forphysical scale quantities. Assuming Ftobereal, then, w=FInz=FInr+jF<t>=u+jv (21) representingconcentric circular cquipotential linesandradial field lines asthechargedline(12-28). Tostaywithin physical inter- pretability, i.e.,make Inzsingle valued, theangle<must be restricted to^ tf><2ir,laying abarrier plane at=2ir,for example. Asananalytic function, Inzisregular intheentire z-plane except attheorigin 2=0,where thederivative l/zbecomes infinite, corresponding tothelocation ofthe line charge. To avoid thissingularity onecanadmit avery small butfinite radius ofthecharged line,makingitaquasilinecharge, asindicated in section 12.The total dielectric fluxperunitlength from theline according to(16)isthedifference between theextreme values of thefluxfunction along ancquipotential line, i.e., =X (22) sothatF=X/2ire asnoted intable 25-1. The fieldvector is according to(15)_ (23) and isdirected radially outward. Interchanging potential andflux lines (withVreal) w=-jVInz=F0-jVInr=u+jv (24) onehasthemagneticfield ofalinecurrent (vortex line) asin (13-36),forwhich themagnetostatic potential CF=u.The 286 Two-dimensional Analytic Solutions [Ch.7 i -ds Q^ OL,. 75oft"So^ '5rtw.I ^ *ondgw^^SSg Sp.8S rgf-ftd dd-a2^ -^.P^Q^ I I Sec. 25] LineCharges andLineCurrents 287 8 I t; II IIi N?1 J -J i+ iin 3s> 288 Two-dimensional Analytic Solutions [Gh. 7 potential values mustnowbemade unique byintroducing the same barrier plane asbefore, restricting ^<<2ir]asseen, the potential increases with angle <,sothatthefield lines aredirected clockwise. Theconstant Vmustnowbedetermined from the factthatthelineintegral ofthe fieldvector which equals thepo- tential difference across thebarrier isalsothevalue ofthecurrent causing thefield; thus, integrating inthemathematically positive sense, -27r)=7 (25) sothatV=7/27T. Toavoid thisnegative sign,which just expresses thefactthat clockwise field lines belong toacurrent inthenegative third axisdirection ofaright-handed coordinate system, onecould, ofcourse, choose apositive sign in(24), but then either thepotential values would benegative oronewould have tointroduce u=V(2ir 0)aspotential function; allthese possibilities havebeen used. Inhydrodynamics, where thecon- cept ofvortex lineoriginated, thisdifficulty doesnotarisebecause thevelocity vector isusually defined aspositive gradient ofthe potential function (table 25-1). The single vortex linecanalsobeused torepresent the field between twocoplanar potential surfaces withaninfinitesimal gap between them asshown inFig.25-4a. Inthis case, theconstant Vistobechosen as($! $2)Aandoneadds theconstant <f>2 which isalways possible, obtaining w=j Inz+ <J2 *l"*2Inr(26) 7T asthecomplete solution fortheupper halfz-plane. Superposition oftwoequal linecharges with opposite sign, i.e., ofasource andasink line, leads tothesame results asinsection 12,permitting thesame general useforfinite cylinders. Table 251gives thefunction aswellasseveral references usingitand showing graphs; thenotation isillustrated inFig.25-3 foracon- venient choice ofcoordinates. Ifthetwocharged lines recede symmetrically toinfinity, onehas 27TEM_\zM) VTTEM/ Sec. 25] LineCharges andLineCurrents 289 auniform field ofgradient Einthez-direction (anon-essential constant hasbeen dropped).Ifthetwo linecharges approach symmetrically very closely theaxis 0,onehas wX /.z+a\X2apilim(In)> = 27TE_\za/ 2ire z 2-jrs. z(28) adipole linecharge asin(12-52) with thedipole moment p= X-2a inthenegativez-direction. P(*,y) I*" (a) FIG.25-3 Several Line Charges (Source Lines): (a)source and sink, (6)source pairandsinkpair. Thecombinations ofthehomogeneousfield (25)with linecur- rent (24)andwith dipole line(28)arelisted intable 25-1; further combinations arefound particularlyinhydrodynamic flow studies. jy *,>*jy |02*,H (a) (b) FIG.25-4 Vortex Flow asSolution ofCoplanar Potential Surfaces: (a) single vortex, (b)vortexpair. Superpositionoftwo parallel, equal linecurrents ofopposite direction leads tothesame results asin(13-16); seetable 25-1. This superposition canalsobeused torepresent thefieldbetween three coplanar potential surfaces asindicated inFig.25-46. To obtain thecorrect constants forthepotential function w=V(<f>i 290 Two-dimensional Analytic Solutions [Ch. 7 otherwise vanish. Then, forfa= TT,fa=0,oneshould have u=<t>2which requires V=(3>2$i)/?r; forfa=fa=TTone finds then $1again. Thus, $1 $2 ,2+a.^w=+.7 In h$1 TT 2-a r$!= L-(0i-fa)+$1\+j In-(29)J IT r2 isthecomplete solution fortheupper halfz-plane. Formore thantwosource orvortexlines, thesame process of superposition canbefollowed. Solutions have been given forN coplanar charged lines6andforNequally charged lines equally spaced onacylindrical surface ofradiusR2andparallel toitsaxis. Inthelatter case,onecanwrite thesum -^ZIn(z-O=-^Mn(ZN-zaN ) (30) since z=Rgexp(j2ira/N) aresimply thenunitroots multiplied bytheconstant radiusRg.Close totheindividualwires, the potential lines arepractically radial; atadistance alittlemore than themutual spacing, thepotential linesmerge intopractically concentric circles; seeBewley,Dl p.53,forN=6. Superimposing ontothecylindrical array (30)aconcentric field byplacing alinecharge qcintothecenter ofthecylinder, onehas themodel ofacylindrical vacuum triode, with =-AcInz-\gIn(ZN-zaN )-2irev3>Q (31) where <t>isarealconstant toadjust potential values. Making theassumption thatthegridwirespacingissmall compared with theradial distances ofthecenters ofthegridwiresfrombothanode andcathode, onecantake thecontributions ofthegridwire potentials aspractically constant over these electrodes. The conditionsare,therefore (seealsoFig.25-5), u=$= oncathode where z= u=$=Vgongridwires where z=za+Pgej* ^(32) u=$=Vaonanode where z=Raej* 6W.H.Barkas, Phys. Rev., 49,p.627(1936). Sec. 25]Infinite Arrays ofLineCharges orCurrents 291 WithRaN^RgN^RCNonecansimplify therationalization of In(ZNzaN ),sothatcorresponding to(32)onehas =-\cInRc-\gInRgN- =-\ cInRa-\gInRaN-(33) fromwhich onecanreadily evaluate thelinecharge values. The amplification factor, defined astheratio ofthepartial capacitance FIG.255Schematic ofaTriode with Cylindrical Structure. between gridandcathode tothatbetween anode andcathode, follows then as7 d\/dV gN\nRJRg d\cfdVa\nRg/NPg(34) Using adouble gridoflinecharges,Npositive onaninner and Nnegative onanouter cylinder atthesame equal angular spacing, oneobtains close toeach doublet potential lineswhich closely approximate cable conductors.8ForN=3andN=4complete solutions forelectrical andthermal characteristics aregiven inthe references. Infinite Arrays ofLineCharges orCurrents. Theperiodic functions ofthemeromorphic type, i.e.,functions which behave likerational functions anywhere inthe finite z-plane (excluding 7W.Schottky, Arch. /.Elektrot., 8,p.1(1919) andM.v.Laue, Ann. d. Physik, 69,p.465(1919); seealsoRothe etaJ.,D8 p.89;Ollendorff,A18 p.156; ChaffeefB21 p.173;Dow,B23 p.39;andSpangenberg,B29 p.138. 8G.Mic, E.T.Z., 26,p.1(1905); OllendorfffAl8 p.134. 292 Two-dimensional Analytic Solutions [Ch.7 thepoint atinfinity), generally representinfinite plane arrays of linecharges orline currents. Thus, with proper restriction to single values ofthelogarithm itself, w= Inexp(2ir- J1(35) isnon-analytic atzn nja, atperiodic intervals along the imaginaryaxis. Itbehaves near these singularities like Inz, i.e.,represents positiveline charges, since onecanwrite there (2?r/a)z=2irjn+(27r/a)f, with f= i\+jasmall, sothat --Incosh f+sinh f-1 27TS L o a J The field linesfrom thepositive gridgotowards+> ,where w=-(X/e)(z/a); there isnofield atz>-<.Thus, (35) represents aninfinite plane array ofpositive linecharges witha superimposed uniform electric fieldparallel tothepositive x- direction and largeenough tocancel thegridfield asz o. Anexcellent graphisshown inMaxwell,A17 I,Fig.XIII. Fornumerical application, oneneeds therealpotential function, which isobtained byexpanding /z\ /27TX\f27T7/ ,. .2wy]expI2ir- J=expI- ]cos--h3sin-\a/ \a/L a oj and, expressing thelogarithm inpolar form, $=u(x,y)= Fordistances from thegridarray along thepositive z-direction of theorder ofa,thegridwire spacing, onecandisregardallbutthe firstterm, since e2* 500,andonehasleftauniform fielddirected inthepositive z-direction with X4 X- A-- =--s(38)47TEa ea Sec. 25] Infinite Arrays ofLineCharges orCurrents 293 with fieldgradient7=X/ea. Closer tothewires onecanwrite /a \X[27TX <!>(-<z<a)- \- \2 / 47T6Ia indicating aslight variation inthe^/-direction, producing awavy potentialline. Forvalues of|z|^0.2(a/27r), onecanusethe approximation (36),which means thatonecould usethesame field picture forfinite wire radii oftheorder of(0.2a/2ir) orless. Assume nowaconducting plane ofground potential placed at distance h^aparallel totheplaneofthegrating asinFig.25-6a andaddtothepotential function (37)anarbitrary constant * - Then, with (38)and (36),onehas, respectively, *x=h=Q=-h+*Q, *,r,-p=---In +So (39)ed ZTTZ a where$(|f|=p)isthegridwire potential. Eliminating $o>one has which defines thecapacitance ofthe"Maxwell" grating with respect toground perunitlength andforonesection as This field distribution canbeused forthecapacitance ofparallel antenna wires,9forthegrid-cathode capacitanceifthegridis very close tothecathode, andfortheapproximation ofthewire effect inhigh-voltage windings.10With thesuperposition ofa uniform field itapproximates alsoaninfinite gridbetween two distant planes andhasbeen used tocompute theeffect ofgrid wires intriodes onelectron paths.11 9P.O.Pedersen, Zeits.f.Hochfrequemtechnik, 7,p.434(1913). 10W.Grosser, Arch.f.Elektrot., 26,p.193(1931). 11K.Spangenberg, Proc. I.R.E., 28,p.226(1940). 294 Two-dimensional Analytic Solutions [Ch. 7 Similarly, thefunction to= Insin (42) 2-jrs. a with restriction totheprincipal partofthelogarithm toassure single valuedness,isnon-analytic atzn=na. Settinginthe neighborhoodofthese singular points z=na+f,onehas again equal positivelinecharges spaced atintervals aalong the x-axis. Thepotentialfunction isupon rationalization $=u(x,y)=-Incosh -cos + In2 (43) 4ire L a aJ 47TE which forvalues\y\^(a/2)ispractically aconstantpotential, sothatonecanintroduce symmetrically located plane equipoten- tialsurfaces andthushaveaninfinite gridmidway between two parallel conducting planes. Proceeding asintheprevious applica- tion,onefinds fromwhich again thecapacitance canbecomputed,12ortheresist- ancebyuseof(8-11). Interchanging potential and field linesby using Wi=jwin(42), onehasthesolution forthemagnetic field ofalinecurrent between twoparallel magnetic equipotential surfaces ofinfinite permeability andadistance aapart asshown indotted lines inthe leftpart ofFig.25-66; seeHague,844 p. 167,andWalker,D1 p.69.Foraninfinite array ofvortex lines seeLamb,022 p.207. Adding asecond gridofnegative wires as inFig.25-6c,onehas w= Insin- (z j6)Insin-(z+j6) (45) 2?re\_ a a J which gives attheplane y=thepotentialf>=andrepresents aninfinite gridvery close toagrounded conductor plane sothat theimage gridhastobeused/3orthecurrent flow inathinmetal 12Indifferent form, butsame result seeOllendorff,A18 p.158. 13J.H.Fremlin, Phil.Mag., 27,p.709(1939); alsoSpangenberg,B29 p.162. Sec. 25]Infinite Arrays ofLineCharges orCurrents 295 sheet ofwidth abetween two cylindrical electrodes14ofsmall radius pasintheshaded area inFig.25-6c.Thecapacitance between two oftheopposite gridwires within oneoftheperiodic +X-X l--i l-t-l1p^ I-1 . t_ FIG.25-6 Infinite Plane Arrays ofLine Charges: (a)Maxwell grating, (6)grating midway between parallel conducting planes, (c)parallel positive andnegative gratings, (d)alternating grating, (e)dipole grating. strips canbeobtained from (44)ifonesubstitutes 26forh.The resistance between thetwoelectrodes ifthemetal foilhasthickness tisthen given by iro^k r,n (46) Interchanging thepotential and field lines in(45)leads tothe magneticfield oftwooppositelinecurrents midway inanairgap oflength abetween twoinfinitely permeable magnetic pole faces; seeHague,544 p.177,andBewley,01pp.158and 137.Hague alsousestwocoplanar infinite plane gridarrays oflikesignto simulate theeffect ofmagnetic imagingifasinglelinecurrent is inarbitrary position between ideal pole faces. Allcomputations follow thesame pattern asgiven above. Itdoes notmatter 14F.Ollendorff, Arch.f.Elektrot., 19,p.123(1927); also Ollendorff,A1* p. 159,andRothe etaZ.,08p.96. 296 Two-dimensional Analytic Solutions [Ch. 7 whether oneusestrigonometric orhyperbolic functions, whether sinzorcosz;ascomplex functions they differ onlybyconstants orirrelevant shifts oftheorigin. Analternating array oflinechargesisrepresented by A ./TTZ\A/.TTZ .TTZ\w= In Itan I= - (Insm Incos I(47) 27re \a/ 2xe\ a a/ which canatonce beconsidered asuperpositionoftwoarrays (42) interlaced sothat thesingular points arenowspaced a/2 apart. Thepotential function isfound byrationalization as _ 2irx cosh--cos- cosh--hcos- a a(48) Good graphs ofthefield distribution arefound inBewley,01 p. 55,andRothe etaZ.,D8 p.96. Itcanrepresent athin infinitely longmetal strip ofwidth a/2withtwothinelectrodes atopposite sides, as(B)inFig.25-60",oritcanrepresent athin cylinder midway between parallel conducting planes, asat(A) inFig. 25-6dand alsotreated insection 21.Inthelatter case, the potential vanishes atx=(a/4); onthecylinder ofradius p itbecomes A irp \ a *I.I-P-In=+ In 27TS a 27T wp because tan(KZ/O) (?rz/a) near theorigin. Thecapacitance perunitlengthisthen (49) Onecanaswell select anyoneoftheequipotential surfaces which areofovalshape forexample tosimulate thecross-sectional shape ofpolesinalarge generator and findtheleakagefluxbetween adjacent poles.15 Aninfinite plane arrayofdipolelines isdescribed bythefunction (50)a/ 15B.Hague, Jl.I.E.E., 61,p.1072 (1923). Sec. 25] Infinite Arrays ofLineCharges orCurrents 297 forwhich thepotential function becomes usmh- cosh--cos- a a The analytic function has singularities atzn=.na\ intheir neighborhood, onehasz=na+f,sothat which represents adipoleline ofdipole momentXCL/TT perunit length directed inthepositive y-direction asindicated inFig. 25-6e.-Superimposing auniform field&inthepositive y- direction with apotential *=E?yproduces resultant field lines convergent intheneighborhood upon thedipole linesand remaining uniform atlarger distance from thegrid, asinthecase ofthesingle dipoleline(21-20); this isreadily seenbyletting y become large in(51). There will existanalmost circular equi- potentiallinesurrounding each dipole, which canbeused todefine acomparatively large cylinder andsimulate theproblem ofa grating with radii pnotnegligible compared withmutual distance a.This radius canbedetermined byfinding thepoint along the z-axis whereEv=0,where theimposed uniform field Z2equals theopposingfield oftheindividual dipole. The fieldvector of thedipole grating canbefound from (50), using (15), /eto\.Xirf.a/TAV1 E= I 1=-j---am2 I I \dz/J2na\_ \a/J Along thez-axis, y=0,sothat pcanbedetermined from '-5;;[*(?)]"-> oronecanselect thedipolemoment foragivenradius p.Assuming theresultant potential (*+*)tobedefined asshown inFig. 25-6e,namely,3>r=+(7/2) aty=-hand*r=-(7/2) at y=+h,onehaswithh>atheconditions -=zbtf/i =F-^-tanh-- 298 Two-dimensional Analytic Solutions [Ch. 7 With (52)these relations permit nowtheevaluation ofthecapac- itance. Thecharge perunit areaontheconductor +(7/2)is (eE), since auniform fieldgradient exists onit;thecapacitance oftheparallel plate condenser perunitdepth and foralength Na^2/iwiththegrating becomes with (52) istherefore increased considerably aspincreases. Lamb,C22 p. 68,hastreated theflow ofanincompressiblefluidthrough sucha gratingoffinite diameters. Ollendorff,A18 p.165,hasused the interchange ofpotential and field lines tosimulate theeffect of round holes intransformer laminations upon themagnetic flux, assuming theuniform -magneticfield lines parallel tothez-axis. Elliptic Geometries. Theinverse trigonometric orhyperbolic functions leadtoconfocalconies. Because these aremany valued, itisnecessary todefine theprincipal values carefully.16The function H+' z=sinw=sinucosh v+jcosu givesupon separationofrealandimaginary parts x=sinucoshv, y=cosusinh v from which, byelimination ofu,andrespectivelyvtonefinds =1 * shv)cosh v \sinh v(55) These relations define thelinesu=cons asconfocal hyperbolas andthose v=cons asconfocal ellipses. Onecaninterpret the field asthatproduced bytwocoplanar equipotential planes with agapofwidth 2between their edges, asshown inFig.25-7. In- terchangeoftheconfocal families byusing MI(Z)=jw=jsin"1z=sinh"1 (jz) (56) 16H.B.Dwight, Trans. A.I.E.E., 61,p.851(1942). Sec. 25J Construction ofConjugate Functions 299 leads toelliptic cylinders asequipotential surfaces; seeMaxwell,A17 I,p.290,andalsoFig.X;inthelimit thesebecome aplane strip, thereverse ofFig. 25-7, which hasbeen used toevaluate the capacitanceofbusbars.17Flow ofanincompressiblefluidthrough FIG.257Geometry ofsin~1z. aslitasinFig.25-7 oraround aplane stripistreated inLamb,C22 p.69. Asaspecial caseonemight consider w" which leadsupon rationalization to(57) (58) constituting confocal parabolas andgiving theflow ofanincom- pressiblefluid around asingle plate, astheleft-hand one in Fig.25-7; seePrandtl-Tietjens,024 p.157. Construction ofConjugate Functions. Itwould, ofcourse, bedesirable toconstruct foragiven equipotential surface f(Xjy)=directly that analytic function which isthecomplex potential solution with f(x,y)=asboundary. This, however, canbedone onlyforunicursal curves which areuniquely defined byasingle parameter. Tofindthecomplex potential w(z), describe theconductor surface f(x,y)=incomplex form as (59) 17D.Gabor, Arch.f.Elektrot., 14,p.247(1924). 300 Two-dimensional Analytic Solutions [Ch. 7 where pistherealparameter. Ontheother hand, theinversion ofw(z)would leadto z=x+jy=x(u, v)+jy(u, v) Ifoneletshereu=andselects x=x*,y=i/*,heagain describes theconductor surface incomplex form with potential u=and parameterv=p.Theanalytic potential function w(z)isthen actually found bytheinverse function z(w)=x*(-jw) +jy*(-jw) (60) oressentially byreplacingin(59)therealparameter p=vby theconjugate functions vju=jw,which assures analyticity ofz(w)andreduces to(59) foru=0.Expressing from (60)w asafunction ofzgives thefinal explicit solution;18 this,however, isfrequently notpossible. Assume anelliptical cylinder asconductor, withmajor axis2a andminor axis 26.Theexpression fortheellipse innormal and parametric forms is I4- f \ 1 h' fl/^ where theparameter pisactually thegenerating angleofthe ellipse from thetwoboundingcircles ofradiiaand 6.Incomplex form thisgives2*=aCOsp+#sinp and, replacing pby jw,onehas z=acos(jw)+jbsin(jw) (61) Thiscanbetransformed into w=cosh-1 (z[a2-62]-H )-tanlr1-(62) aninverse hyperbolic function, asonewould expect from the preceding subsection; Jeans,A1 p.270.The equipotential lines canbefound more directly from (61)as /xcoshaY,/ysinhaY_* _fu-if^\ \acosh (u+a) \bsinh (u+a)/'\a/ (63) 18Jeans,Alp.269;S.Higuchi, Technology Reports ofTohoku Univ., Sendai, Japan, 10,No. 4,p.38(1932); Smythe,A22p.78. Sec. 26] Conformal Mapping 301 Fromthis,onecanfindthepotential u\ofanother elliptic cylinder ofmajor axisa\=acosh (u\+a)/cosh a;with thefluxfunction from (62)onecanthen evaluate thecapacitance by(19). For other unicursal curves such asthecycloid, epicycloid, catenary, andsome spirals seeHiguchi,loc. cit. 26CONFORMAL MAPPING Thediscussion ofproperties offunctions ofacomplex variable w=/(z)hasalready introduced inFig.25- 1theconcept ofaone- to-one relationshipofpointsinthew=u4-jvplane tothose in thez=x+jyplane andvice versa. Oneconsiders thew-plane O' z-Plane'w-Plane FIG.261Conformal Mapping byAnalytic Functions. amap orrepresentation ortransformation ofthez-plane. The existence ofanon-vanishing derivative of/(z)atandintheneigh- borhood ofz,which assures thatthefunction isanalytic there, has unique geometric consequences. Consider Fig.26-1; letw\and w2beinthisorder theimages ofz\and z2,andassume their re- spective distances fromwand ztobetheinfinitesimal andcor- responding elements dw,dwz,and dz\,dz2.Obviously, dw\must bethemap ofdz\anddw2that ofdz2and, inpolar form. dz2=(1) Now, because oftheexistence ofanon-vanishing derivative, the 302 Two-dimensional Analytic Solutions [Ch. 7 ratios ofthetwopairs ofelements musthave thesame value, so that ds2 BothAf= andpcandepend onlyonthelocation ofthepoint P(x, y)oronthevalue ofz.Comparison ofthearguments shows also A\AZ=Oi\a2=a orthattheangle between thetwoelements dz\and cfe2isthesame asthatbetween dwianddw2inmagnitude and insense. Thus, going from thez-tothew-plane, thewhole infinitesimal neighbor- hood ofthepoint zisrotated through adefinite angle juandenlarged orreduced inadefinite ratio according tothescale factorM^1; inother words, thetransformation maintains infinitesimal pro- portionality: theregions about thecorresponding points z=x+ jyandw=u+jvareinfinitesimally similar. Thismeans further thatangles between intersecting curves inthez-plane arepreserved between thecorresponding curves inthew-plane; inparticular, orthogonal families ofcurves inthez-plane remain orthogonal whenmapped intothew-plane, however much theymayappear distorted infinite dimensions. Atransformation ofthiskind is called conformed transformation orconformal representation orcon- formal mapping. Any analytic function provides conformal mapping atallits regular points where itsderivative does notvanish. Itcanbe shown alsothat theconverse istrue, that functions u(x fy)and v(Xjy)which provide conformal mapping canalways becombined intoananalytic function u-\-jv=/(z). Transformation ofPotential Problems. Potential fields satisfying theLaplacian differential equation inthez-?/-plane are described byharmonic functions (see section2),oranalytic functions inthecomplex domain. Tofindasolution inthex- y-plane onecaneither useconjugate functions asdescribed in section 25oronecanconformally map thegeometry ofthex- i/-plane ontoati-0-plane bymeans ofananalytic function w=/(z) andthen solve thepotential problem inthetransformed geometry. Inthe firstcaseoneidentifies thepotential function $(x,y)with Sec. 26]Transformation ofPotential Problems 303 therealpart u(x, y)ofthecomplex function w(z)\ inthelatter case,now tobediscussed, onehastofindthepotential solution *asafunction ofthenewcoordinatesu,v.Onecanobviously findorconstruct acomplex potential solution P(w)=*(u, v)+jH(u, v) (3) whereSistheconjugate to$(u, v)and istheelectric fluxfunction, sothatthedielectric fluxbetween anytwopoints becomes *=e(Si-Ha) (4) asin(25-16); thecurves $(w, v)=consandE(u tv)=cons will again form curvilinear squares inthew-y-planeifoneselects equal increments. Thetransformation oftheLaplacian differential equation for$ from thez-y-coordinates tothe-w-u-coordinates (which areassumed tobeharmonic functions ofz,ybecause oftheanalyticity ofthe mapping function) canbeperformed directly, since dd&dud&dv dx dudx dvdX d2^=d*d2ud2^/du\ad*d*v <&$fdv\2 dx2~ dudx2+du2\dx) dvdx2+ dv2\dx) andsimilarly forthe^/-derivatives. Inthesumrepresenting the Laplacianinx,y,the first derivatives d^/du andd<b/dv appear multiplied bytheLaplaciansofuandv,respectively; both of these vanish because of(25-10). Usingfortheremaining terms theCauchy-Riemann equations (25-6), onefinds a'* a2*/a2*a2*\r/a u Thefactor totheLaplacianinu,vistheabsolute value = c/xj because of(25-5) andonthebasis oftheexistence ofthe dz derivative ofw(z) ;but itmustbefurther required thatdw/dz 7* inorder toobtain for<t(i6, v)again theLaplacian potential equation. The Laplacian potential equationistherefore in- variant toconformal mappingsofthegeometry atallregular points 304 Two-dimensional Analytic Solutions [Ch.7 oftheanalytic mapping function where thederivative does notvanish. The fieldvector inthew-v-planeisgiven asin(25-15) by Itcanreadily betransformed intotheoriginal geometry inthe z-y-plane by sincePasanalytic function inthew-planeisalsoanalytic inthe z-plane (section 25). Indeed, onecanevaluate itinthez-plane directly from themapping function without firstfindingitinthe tu-plane, aslong asonehasevaluated thecomplex potential solution P.Theabsolute value ofthefieldvector isfrom (7) \E\M=\E\M-^ (8)dz andtransforms inthedirect geometric transformation ratio at each point, sothatintegrations ofcharge densities overcorrespond- ingconductor surfaces inthetwoplanes give thesame result. Therefore, capacitances, inductances, resistances evaluated inthe w-plane geometry andexpressed interms ofz-plane dimensions have thesame values asifevaluated directly inthez-plane from thepotential distribution there. Thiscanalsobeseen ifoneconsiders the field energy. The energy density asgiven in(3-20) canbeexpressed byusing (8) andobserving thatonaccount of(2)fortwoorthogonal elements \dw\2 dudv=\\dxdy; thus \dz I J.ISlw'efadhf-s-Ww'b T=5 (9) demonstrating that itisinvariant toconformal transformations. The total fieldenergy incorresponding fieldspaces willtherefore bethesame. Onemight inquire intothetransformation possibilities ofspace charge problems. Since fieldenergy density canbeexpressed in Sec. 26]Points ofNon-conformality ofMapping 305 thealternate form (3-18) andthepotentialisinvariant, onehas dudv,. _ _ .... (p)wdxdy=(p)wU^f=(P)W dudv(1Q) \fe\ oronemust transform space charge densities intheratio P^l \dz[ from thex-y-plane intothei^-v-plane inorder topreserve invariance ofthePoisson equation, which nowbecomes with theLaplacian from (5) i2'i2du* dv* e Thesameis,ofcourse, true forthetwo-dimensional magnetic fieldproblems involving current distributions andbeing described byonecomponent ofthevector potential,1 Points ofNon-conformality ofMapping. The fact that conformalityismaintained only atregular pointsofthemapping function atwhich itsderivative doesnotvanish requires abrief and reassuring examination oftheproperties ofanalytic func- tions and their derivatives. Alldiscussions refer toone-valued functions ortotheproperly restricted domains ofmany-valued functions. Assume /(z)tobeanalytic atevery point inaregionRofthe z-y-plane around apointz .Then /(z)/zzwillalsobeanalytic there andeven very close toz=z,except directly atz=z , where thederivative doesnotexist; onemight exclude thispoint zbyasmall circle C'asinFig.26-2. Theintegralof/(z') over anyclosed curve Taround zandwith points z7 wholly withinR vanishes because ofCauchy's integral theorem (25-9). Now the integral of/(z')/(z' z)overthesame closed curve Twillnotso vanish; butbecause of(25-9)itcanbecontracted intothesmall circleCf along which onecanexpress (z7-z)=pej+ 9 dz=jpert <ty, z'=*+rf* (12) Thisthen gives //(z')dz' \J=j ^C"Z ZQ(13) 1Foraninteresting application toinductance calculations ofrectangular bars seeT.J.Higgins,Jl.Math, andPhys., 21,p.159(1942). 306 Two-dimensional Analytic Solutions [Ch. 7 where pcanbemade sovery small that /(z') >/(z ).Relation (13)leads toCauchy's integral, /(zo)=(14) which states thatthevalue ofananalytic function atanypointz inaregular regionRcanalways beexpressedinterms oftheknown values along aregular closed curve withinRsurrounding that jy FIG.262Series ExpansionofAnalytic Functions. point. Notonly thefunctionitself, butalso itsderivatives at ZQ,canbeexpressed byintegrals because oftheassumedregularity of/(*'), (15) Designating thedistance|z'2 |=/,thenonecanseeatonce thatthederivatives have asupper bound (16) Sec. 26]Points ofNon-conformality ofMapping 307 Thevalue ofthefunction /(z) atapointzintheregionR (Fig.262)cannowbeexpressed inapower series with thepoint zascenter andwitharange ofvalidity, say, tocircle C.Since 1 1 z'-z (z'-z)-(z-z) Z-Z Z-ZQ isabsolutely convergent because|z2 1<jz'z|,onecan multiply eachtermby/(z7 )dz'andintegrate over thecircle C, which gives with (14)and (15)atonce /CO-/(zo)+/'(zo)(z-z)+^/"(zo)(z-*o)2+ =i:aa(z-zr(17)a=0 This isaTaylor series expansion; breakingoffatafinite value a, onecanestimate theremainder by(16). The series (17)is definitely convergent within any circleCwithin which the function /(z)isanalytic atevery point; thisform illustrates that thederivatives arealsogiven byconvergent Taylorseries expan- sions. For rational functions, theTaylor series reduces toa polynomial withnoderivatives ofhigher order than thepoly- nomial. IfintheTaylor series (17) the first coefficient vanishes, or a=/(z )=0,then zisarootofthefirst order;inthis case, /(z)hasavanishing derivative ofthe first order andcanbe written /(z)=(z-zo)[ai+a2(z-z)+ ]=(z-zo)<7(z) (18) where nowg(z)iswithout root atz=z .Ifthe firstmcoeffi- cients arezero, then zisaroot ofrath order, afactor (z zQ)m canbeisolated, andthefirst raderivatives vanish. Itisimportant tonote, however, thatarootoccurs only atanisolated point,since (18), forexample, gives non-vanishing values foranyzexcept exactly z=z .Indeed, onecanshow (forexample Kellogg,010 p.352) that,if/(z) should vanish ininfinitely many points within R,then itmust vanish allthrough R.Thismeans for conformal mapping that within anyfinite regular regionofa 308 Two-dimensional Analytic Solutions [Ch.7 mapping function there canbeonlyafinitenumber ofzeros orroots ofthefunction where thederivative vanishes, that allthese roots occur atisolated points andcanbeexcluded byextremely small circles around each ofthem, andthat intheimmediate neighbor- hood oftherootvalues conditions ofconformality exist. Should f(z)beregular everywhere withinRinFig.26-2 except atthepoint where i'(ZQ)>or,better, where thederivative /'(ZG) doesnotexist, then theTaylor series expansion cannot be used, since itsvery basis (14)does notapply.Itispossible, however, togiveapower scries expansion fortheregion bounded byContheoutside andbythesmall circleCrontheinside,ifone extends theintegral (14)overCandCrandconnects thesetwo circles bytheclosely spaced parallel lines abandcdinFig.26-2 toprovide essentially one single continuous andcompletely analytic pathwithoutencirclingZQitself.2Because thecontribu- tions ofaband ofcdareequal andopposite, onecanreally dis- regard themandconsider onlyCand C'.OnConeproceeds as fortheTaylor series; forz'onC'onedevelops, because ofopposite direction ofintegration, 1 1 Z-Z(Z ZQ)-(z'-ZQ) Z 3 This againisabsolutely convergent because now\zr ZQ\< \zzQ\alongCf .Multiplying thisexpansion termbytermby /(z')dzandintegrating overC'giveitscontribution totheclosed integral. Combining thislatter with theTaylor series forCgives thetotal result /(z)=T(z-z)+-^+b* a+---(19) Z ZQ (Z ZQ)* where T(z ZQ)istheright-hand side of(17), portraying the regular behavior ofthefunction farfromZQ,whereas theextra terms portray thesingularity existing atZQ.Thevalues ofthe coefficients baarefrom theabove 2Thedirection ofintegration along theboundary ofanyregion shallalways besuch thattheregionliestotheleftofthepath ofintegration. Sec. 26]Points ofNon-conformality ofMapping 309 Ifonly &iisdifferent fromzero, thefunction f(z)issaidtohavea pole ofthefirstorder atz=z;itisobvious from (19)that the derivative doesnotexist atzz,infact, that itmust have a poleofsecond order there;indeed,allhigher derivatives musthave polesofoneorder higher than their order. Thefunction (z z) /(z)=g(z) is,ofcourse, regular intheentire regionRandcan again berepresented byapositive power series like (17).Ifthe highest order non-vanishing coefficient isbm,then zisapole of therathorder andonecansegregate afactor(zZQ)". Itis again important tonote, however, thatapoleoccurs only atan isolated point,since (19), forexample, gives non-infinite values for any zexcept z=ZQ.Thismeans forconformal mapping that within anyfinite regular region ofamapping function there can beonlya,finitenumber ofpoles ofthefunction, that allthese poles occur atisolated points andcanbeexcluded byextremely small circles around each ofthem, andthat intheimmediate neighbor- hood ofthepoles conditions ofconformality exist. Ifthenumber ofcoefficients baisunending atapoint ZQ,then itiscalled anessentially singular point, and itcanbeshown that initsneighborhood there exists aninfinite sequence ofpoints (a distinct point set)atwhich /(z)either takes onthesame value or hasapoleoffirst order. Theneighborhood ofessentially singular pointsistherefore unsuited forconformal mappings andmust be carefully avoided. Whether ornotatransformation canbeconsidered conformal atz=ooisamatter ofconvention. Mathematically,ithas become customary toattach toz=oothecharacteristics ofa point because byatransformation theregionz= ofthez-planeistransformed intoadefinite point,t=(origin),ofthei-plane. Infact, onedefines the characteristic ofafunction atthepoint z=ooasidentical with thecharacter ofthesame function ofargumentt=1/zat t=0. Thusw=z2hasapole ofsecond order atz=oo tbecause w*= l/t2hasapole ofsecond order at t=0.Infact,allpositive power polynomials areregular intheentire z-plane andhaveapole atz=ooofthesame order asthehighest power ofzindicates.If, 310 Two-dimensional Analytic Solutions [Ch. 7 therefore, ananalytic function isregular everywhere, including z=oo,itcanonlybeaconstant (theorem ofSturm-Liouville). Simple Linear Mapping Functions. The only class of functions that assures one-to-one relationship between theentire z-andtheentire w-plane without restriction except forazero or apoleistheclass oflinear functions. Itillustrates rather wella great variety ofpossibilities insimple form. Toidentify cor- responding points andregions,itisadvisable tousenumbered or lettered coordinate lines ofeither uniform square mesh orofpolar type. Thefunction w=z+z,u=x+xQ,v=y+7/0 (21) isasimpletranslation orshift oftheorigin. Thefunction w=mz= \m\ejttre^=|m|re/(*+")(22) isfor|ra|=1apure rotation ofthez-plane byanangle /x;for H=apure scalechange byafactor|m|,uniform inalldirections, acontraction for\m\<1andadilation for|m|>1;andforthe generalcaseacombined rotation andscalechange which could be done intwosteps z\=\m\Zj w=e3lizi Thecombination of(21)and(22)superimposes alsoashift ofthe origin, oratranslation. These transformations have notintro- duced anyfinite distortion, circles remaincircles, andstraight lines remain straightlines. Thetransformation by w=-=-e~j*(23) isofthetype ofinversion;however, because ofthechange insign oftheargument, angles aremirrored. Inaddition toinversion attheunit circle there isalsoinversion atthereal axis. Figure 26-3 indicates therelationship between z-andw-planes. Any potential problem intheborder-shaded infiniteregion outside the quartercircle 1-4inthew-plane has itscounterpart within the small region 0-1-4-0 inthez-plane. Thefunction 1/zisanalytic everywhere intheplane except attheorigin z=0,where ithas apoleofthefirstorder; thus, theoriginistobeexcluded from the Sec. 26] Simple Linear Mapping Functions 311 mapping region byavery small circle, correspondinginthew- plane toavery large circle, excluding w=oasa"point." Any potential solution can, therefore, notbeconsidered toremain regular inthepointz= itself. Thiscanbeseen atonce ifone considers current flow inathinmetal sheet shaped asthequarter circle 0140 inthez-plane andthinrodelectrodes applied, aposi- tiveoneat1andanegative oneat4. Ifthepointisadmitted, thecurrent density would havetwodirections, onefrom 1towards andtheother from towards 4;thiscondition doesnotexist ever soclose to0,itholds only forthemathematical pointz=0; theexclusion of0,which resolves this difficulty, canbeinterpreted iy FIG.26-3 Mapping byw=1/z,Complex Inversion. asadmission thatnophysical metal sheet could everactually possess themathematically sharp corner. Thesameargument will applyinallcases ofpolesandroots ofamapping function. Adifferent interpretationofthesame mapping function is obtained bywritingit B-=2?2-32!2 C24) AsFig.26-4 illustrates, thesquare netoflines parallel tou-and 0-axes inthew-plane corresponds tocircles passing through the origininthez-plane. Several corresponding areas areindicated; ifthey arecutoutintheshapes shown tobethemapsofthe shaded squaresofthew-plane, they will allpresent exactly the same resistance between electrodes placedatopposite corners, such as24-1, 2-13, 6-19, 23-1; inthelastcase,Iitself isactually excluded butonecangoever soclose. 312 Two-dimensional Analytic Solutions Thismapping function canbeused tosolveanyproblem involv- ingorthogonally intersecting cylinders. Thetwocylinders drawn indotted lines inthez-plane andmarked u=+l tv=2be- come thetwocorresponding planes inthetu-plane, theexterior of thecylinders corresponding tothearea intheright angle. Itisas ifthecylinder surfaces hadbeen separated atIandstraightened outintoplanes andstretched toinfinity atthesame time, preserv- ingtheoriginal right angle atA.Placing alinecharge+Xany- where outside thecylinders andparallel tothem onecanfindtheir capacitance bysolving theproblemofalinecharge+Xbetween orthogonally intersecting planes inthew-plane. Specifically, take jy -2-1u=0u=l FIG.26-4 Mapping byw=l/z. thediameter ofthelarger cylinder asd;then that ofthesmaller is d/2; locate thelinecharge (+X) atpoint 7with x=0,y=d. Theintroduction ofdiameter dforunity inthez-plane means that theunit circle isreplaced byacircle ofradiusd,andthusWi= d2 /z,themapping function instead of(24); this isreadily evident from (23). Thegeometric image ofthelinecharge1inthew- plane hasthen distance dfrom both planes;itselectrical images aretherefore atthecorners ofthesquareofside 2d,center A. Ascribing zeropotential tothecylinders andasmall radius pto thelinecharge (+X)inthez-plane gives zero potential tothe planes andradius p'tothegeometric image, namely, dzp=(25) Sec. 26] Simple Linear Mapping Functions 313 using thebasic relation (2)forsmall distances andrealizing that circles remain circles andthatonthecircle ofinversion noscale change takes place. Thepotential onthesurface ofpfinthew- plane willbetheresultant ofthefour linecharges and willbe identical with that oftheoriginal wire inthez-plane <ly=*p=-^-(-In p+2In2d-In2\/2d) (26) sothat themutual capacitance between wireandcylinders per unitlengthis (27) Thetreatment issomewhat similarto,butverymuch more straightforward than, inversion onthecylinderinsection 23.The samemethod canbeused ifthetwocylinders areofdielectric material ewith thelinecharge at7;inthiscaseonewould have to usetheimage theory, section 21,tosolve thelinecharge between twoorthogonally intersecting dielectric plane boundaries. Many other similar problems canbesimplified infieldgeometry bythis elementary mapping function. Asafurther example takethecylindrical surface ofdiameter d (sothatagain wi=d2/zisused) inthe2-plane passing through 7 and Iandmarked asv=1inFig.26-4 asaconductor, splitit along 7and Iwithavery small gapandapply potential $1to theright halfcontaining point 8,and <S>2<$1tothelefthalfcon- taining point 6;findthecapacitance and field distribution. The map inthew-plane willbetheline v2=dwith point 7onthe u-axis andpoint Iatinfinity. These arenowtwocoplanar equi- potential surfaces ofthesame arrangement asinFig.25-4 for which thecomplex potential solution isknown. Toprovide identity, onehastouseashift oforigin andarotation through TT, sothatby(21)and(22) = (7 314 Two-dimensional Analytic Solutions [Ch. 7 gives theexact z-plane arrangement ofFig.25-4. For this,the solution inform ofthecomplex potential function (3)isfrom (25-26) P=*+jE=-j lnw 2+$2 (28) 7T Upon simple rationalization onefinds thecomplete solution (.,). s.-i. (29) ,y)=-5i?ln* Along thetwoconducting semicircles onehasyd=(x2+y*\so thattan"1 !]inthepotential willbezero forx>and ITforx< thusdescribing theelectrode potentials. Thecapacitance perunit length canbefoundbydividing thedielectric flux e(Hi 2)by thepotential difference. Choosing ontherighthand semicylinder point1asx=-\-g,y=dandpoint 2asx=+g yy=(avoid- ingthepole), onehasimmediately Ci=-In-(30) 9 withdthediameter ofthecylinder and2g:dthegapbetween thecylinder halves. Asolution forthepotential function alone bymeans ofcircular harmonics isgiven inZworykineta.,B32 p. 371 .The electric fieldcanreadily becomputed by(7).Actually the field lines areidentical with themagnetic field lines oftwo equal andopposite linecurrents placed at7and I. TheBilinear Transformation. Thegeneral linear function Az+B -Dw+B withADBCT isthemost general transformation mapping thewhole ofthez-plane inone-to-one relationship upon thewhole ofthew-planc, preserving conformity atallpoints except z= (B/A), which isaroot,and z=(D/C\ which isapole of first order ofthemapping function. Circles and straight lines areagainmapped into circles andstraight lines, andany series Sec. 26] TheBilinear Transformation 315 oflinear transformations willmaintain thesame type; thelinear transformations formagroup. Thetransformation (31)hasas special cases alltheprevious simpler linear functions and, infact, canbemadeupinthree distinct steps, ,D 1 AAD-BC Wl=z+->*.-, w--+___^ This represents firstashift oftheorigin ortranslation, thenan inversion, and finally acombination oftranslation, rotation, and scale change. Obviously,ifADBC=0,thelaststepwould contract thewholew2-plane intoasingle pointA/Cand istherefore excluded assingular. Three arbitrary constants areavailable which cangenerally be chosen -totransform three specified points inthez-plane (for example acircle) intothree specified points intheiy-plane. This isalsoapparent from the firstand laststeps above. The con- stants arebestevaluated byusing theform of(31) B+Az-Dw-Czw=(32) whereBappears asasuperfluous additive constant, which canbe divided outin(31). Instead ofthree points, onecanselect one pointandanassigned direction throughthispointinbothplanes. Onecanalsomap theupper halfplane upon itself inaninfinite number ofways; inthiscasetheconstants must beallreal. If w=u+jv,zx+jy,rationalization of(31)andelimination ofugivethecondition v[(Cx 4-D}2+(Cy)2 ]=(AD-BC)y (33) which means that positive 7/-values willbecome positive v-values onlyifonehasADBC>0.Other interesting characteristics arediscussed inBateman,cipp.270and274. Aparticularly important applicationisthetransformation of theunit circleupon theupper halfw-plane. Take thefunction 1 z iww=j> Z=T- (34) 1+Z 3+W Ontheunit circle z=lej "*,andintroducing thisinto (34) simplifies itto sin</> _.w=y (35) 1+cos<J> 316 Two-dimensional Analytic Solutions [Ch. 7 showing thatwisrealandthepoints correspond asshown inFig. 26-5;theorigin2=becomes w=j.Concentric circles inthe 2-plane become excentric circles about 0'which areexactly like thepotential surfaces between twocharged lines, onelocated at 0'andonesymmetrically located inthelower halfplane. Corre- spondingly, radial linesbecome theorthogonal familyofcircles through 0'.Thefunction (34)hasonerootvalue atz=+1, where noconformity exists;itcorresponds totheorigin inthe w-plane which cannot bepart oftheanalytic field solution. There isalsoapoleatz=1,corresponding tothepoint infinity inthe wj-plane; themapping will, therefore, notbeconformal there; Jy FIG.26-5 MappingofUnit Circle uponUpper Half Plane. however, ashasbeen stressed, onecangoarbitrarily close. Ifone chooses in(34) r^1butkeeps thecorrespondenceofw-points andz-points, thenthemapping function hastobemodified to w.r z :7IJr+zz=jw 3+w(36) Thispermits thesolution ofanyfield distribution within thecylin- der\z\<rbysolving thecorresponding problem intheupper half plane. Inparticular, thecaseabove ofthesplit cylinder canbe transformed upon theupper halfplane with thesplitalong 1-0-3 toutilize thesingularities which have tobeexcluded inany case. Theunit circle canbemapped upon itself inthemost general way3bythefunction wza 1-az(37) 8Bateman,01 p.280. Sec. 26] TheBilinear Transformation 317 whereftisarealangle,aanycomplex number, andaitsconjugate complex value. This linear function maps theinterior ofunit circle inthez-plane upon theinterior ofunit circle intheu;-plane; thepointz=abecomes theorigininthew-plane, andtheangle means ageneral rotation. Apertinent example ofthetransformation (34)isacircular thin metal disk ofradius owithtwocurrent-carrying electrodes and twoseparate potential-measuringelectrodes tominimize contact effects. With theradius aasscale factor, thedistances areas indicated inFig. 26-5. The electrodes areshown placed along thediameter 2-4which inthew-plane becomes unit circle asseen from (34).Ifonesets z=jy,onefinds, indeed, .1-jy 2y+j(i-y2 } i+jyi+ 2/2 \w\=1,argw=tan"1 This gives thetransformed location oftheelectrodes inthe w-plane 2 1.2 2 2 7;=tan"1a~ > 9=tan"1^(38)2ab 2ac Thecurrent flowmustberestricted totheunit circle intheoriginal, ortotheupperhalf tu-plane. This isaccomplished byplacing image electrodes below thew-axis ofthesame signasthose above the axis. Taking Arasthepositive, andE'asthenegative electrodes, thetotalcomplex potentialisthesuperposition ofthat ofthetwosource linesAfand itsmirror image with respect tothe it-axis, andthat ofthetwosink linesErand itssimilar mirror image. Using appropriatelytable 25-1, onehas (39) Rationalization gives *(u, i>)or,with (34), $(x, y).Thepotential atpointTr isthenegativeofthat atS'because ofsymmetry; the potential difference between these points, which istobemeasured, isobtained incomplex form withw=e3from (39). With 318 Two-dimensional Analytic Solutions [Ch. 7 obvious simplifications, therealpartbecomes I 6(fl2+c2)+c(a2+b2 )- in2 2(40) Theresistance forathickness toftheplateis4 V/It. Rational Mapping Functions. Rational functions ofzare essentially fractions oftwopolynomials with asmany zeros asthe numerator polynomial andwith asmany poles asthere arezeros ofthedenominator polynomial. Thesimplest function w=zn=rVn*z=wlln(41) forinteger values ofnmultiplies angles inthez-plane bynand stretches theradial scale. Oneobviously hastorestrict regions in thez-planeinsuchmanner thatthemapping onthew-plane does notcover thew-plane more than once, i.e.,onehastolimit the regions inthez-plane sothatthey arecontained within2-ir/n. Thefunction (41)hasazeroofnthorder atz=0,andapole ofthe nthorder atz=o(byconvention). Asimple exampleisaline charge inthespace defined bytwoorthogonally intersecting dielec- tricboundaries. Thefunction w=z2willstretch these boundaries intoasingle plane boundary forwhich thesolution isknown bythe method ofimages; seesection 21. The electrostatic field inacylindrical triode asinFig.255can betreated byselecting asector 2ir/N inthez-plane, preferably with thegridwire centrally located, andapplying toitthetrans- formation w=ZN ,which willspreaditoutintotheentire w-plane where nowtheconcentric cylinders ofanode andcathode have radiiRaNandRcN yrespectively, andwhere thesingle gridwire alsohasenlarged buthaslost itscircular section. Only forrather small gridwires such thatNpg/R gir<%willtheapproximation byacircle befeasible. Thefurther treatment canconsider the cathode, which hasshrunk appreciably, andthegridwire astwo linecharges within thelarge anode cylinder. The final solution isfound either bymeans ofimages with respect totheanode cylinder, orbytransforming theanode by(36)ontotheupper half plane. Many treatments ofthisimportant problem have been 4J.H.Awberry, Phil. Mag., 13,p.674(1932); themapping used there ia different andconsiderably more complicated. Sec. 26] Rational Mapping Functions 319 given;infirstapproximation theresults arethesame as(34).5 Thesame mapping procedure hasbeen used toevaluate the magneticfield ofalonglinecurrentparallel totwointersecting planes.6 Thefunction w=z+, z=^J(f) -a2(42) withareal isalso listed intable 25-1 asthesolution byconjugate 17? \/-- \ Fia.26-6 MappingofEllipses into Circles. functions foraconducting cylinder inauniform electrostatic field. Asmapping function, with z (^a*\ + (a*\u=IrHIcos<. v=Ir 1si VT/ \r)sin (43) ittransforms theconcentric circles andradial lines inthez-plane intoconfocal ellipses andhyperbolas inthew-plane, asindicated inFig.26-6. From (43)onehasbyelimination ofeither ror^ (V=iV-a/r/' 2acos<-! (44) BM.Abraham, Arch. f.Elektrot., 8,p.42(1919); R.W.King, Phys. Rev., 16,p.256(1920); F.B.Vodges andF.R.Elder, Phys. Rev., 24,p.683(1924); B.Salzberg, Thesis M.E.E., Polytechnic Institute ofBrooklyn, 1932; also Proc. I.R.E., 30,p.134(1942); Dow,B23p.45;andSpangenberg,B29 p.135. 8J.Kucera, Revue gen.deI'elec., 43,p.355(1938). 320 Two-dimensional Analytic Solutions [Ch. 7 thenormal forms oftheconies. The circle r=atransforms into theplane strip ofwidth 4ainthew-plane with theareas outside corresponding toeach other. Obviously, thefunction (42)is two-valued, since itisnecessary torestrict themapping to |z|^ainthez-plane foronecomplete coverage ofthew-plane. Ifonewanted tomap theinterior ofcircle r=a,then|z|^a, andonewould obtain asecond coverage againby(43), inwhich positive values ofvcorrespond tonegative values of0;thenumber- ingontheplane stripwould bereversed, butotherwise onewould have thesame conic geometry. Onemust think, therefore, of circle r=ainthe2-plane asabarrier. The singularities ofthe mapping function (42) areapole offirstorder atz=which, however, doesnotoccur intheregion ofzvalues|z|^a;anda pole ofsecond order atinfinity, which canconveniently beex- cluded, since itoccurs inthew-plane alsoatinfinity. There are alsotworootvalues z=jaatwhich theconformality doesnot hold; these occur onthesurface ofthecylinder andcanbeavoided byvery small circles around them. Having established theproper regions forone-to-one relation- ship, onecanusethez-plane tosolve problems involving conic boundaries. Thecapacity oftwoconfocal elliptical cylinders, the larger withmajor axisMandminor axisN,thesmaller with M'andN1 ,canbefound from theconcentric cylinder geometry. The axes define (2a)2=M2+AT2 ,thenecessary width ofthe limit stripandthebase radius inthez-plane; further, from (43) forpointA u=b+ v=0, 6=(M+VM2-4a2 ) theradius oftheimage circle; onlythepositive signcanbeused forthesquare root since6>a.Similarly, onegets5'forM'. Forthecoaxial cylinders thecapacitanceistaken from (14-11), andwith band b'onehasperunitlength fortheelliptic cylinders7 <> Iftheinner elliptic cylinder degenerates intotheplane strip, its capacitanceis Ci=27TETin^-(M+VM2-4a2)l* (46) 7Ktipfmuller,A14p.101. Sec. 26] Transcendental Functions 321 which might alsobeused foraplane strip inacircular cylinder of large radiusMforwhich C1^2 (47) l-^ } Inasimilar manner, onecantransfer solutions foradielectric cylindrical shell ofradii 6andaintothesolution ofadielectric elliptical cylinder,asSmythe,A22 p.93.Onecanalsostart inthe z-plane with thesolution ofaconducting cylinder ofradius b inavertical uniform fieldandtransfer theupper half ofthefield, which isacircular cylindrical mound onaconducting plane8 orground, into thew-plane where itbecomes aflat elliptical mound9onground; oronecanrotate thefieldbyir/2andhave a steep elliptical mound. Thesame function hasbeenused extensively asmapping func- tion inhydrodynamics,10 particularly intheform w-2a This canbeobtained from (42)byusing thesecond form (2z it;)2=(w24a2 ),dividing by(w+2a)2 ,andsubstituting ontheleft-hand sidewfrom the firstform (42). Ofparticular interest isthemappingofexcentric circles from the2-plane, which yield circular arcs inthew-plane andbyfurther treatment thefamous wing profiles ofKutta-Joukowski and others; see references inAppendix 4,C,c,andBateman,01 p.311. Transcendental Functions. Themapping function w=Inz=Inr+j< (49) transforms theentire 2-plane inonesingle stripofwidth ^v^2ir, whereby theinterior oftheunit circle becomes thenegative half of thestripandtheexterior itspositive half, asinFig.26-7. To have aone-to-one correspondence, onemust limit $ </>$2ir andplace abarrier onthepositive z-axis;itsupper sidebecomes theline v=0,itslower sidetheline v=2ir,forming thefunda- mental regionofthemapping function. Anycurve drawn inthe 8Bateman,01p.261. 9Ollendorff,A18 p.185. 10Prandtl andTietjens,024p.173;Rothe etal.,mp.115;andBateman,01 322 Two-dimensional Analytic Solutions [Ch. 7 2-plane across thebarrier would obviously beentirely discontinuous inthew-plane. Should itbeinconvenient tohave thebarrier along thepositive z-axis,itcanberotated atwill, since itsonly function istoprevent ambiguityinmapping. Onecanimagine thatthemapping proceeds bycutting along thepositive realaxis (oranycorresponding barrier) soastoseparate thetwosides ofit which areindicated inFig. 26-7, rotating thelower sideand simultaneously rushing towards negative infinity, about getting therewhen thelower side ofthez-axis reaches theposition parallel totheupper sideand 2irabove it.Thiscanindeed beused to solve theproblemofadouble potential plane along thepositive I?.5' %2 \I__L ^\ J5= o. FIG.267Mapping ofz-plane intoStripofty-plane. real axiswith potentials +(F/2) onitsupper and (F/2) on itslower side,forming anextremely thincondenser forwhich the external orstrayfield isdesired. Inthew-plane thisbecomes a uniform electric fieldbetween two infinite parallel planes. Since thederivative isdw/dz=1/2,there isonlyonesingularity at2=0,andthis isabranch point, thepoint around which the 2-plane could berotated infinitely oftenwere itnotforthebarrier. Theneighborhood of2=must beexcluded from themapping region; thiscanbedone usually without sacrifice intheregion of realinterest. Considering onlytheupperhalf2-plane, ^<^TT, itbecomes the infinite parallel strip ^v^ITinthew-plane. Therectangle 1-5-6-3-1 inthew-plane corresponds totheannular ringwith likenumbers. Having aninfinite plane parallel strip inthew-plane parallel tothev-axis such asthecontinuation of 1-5-5 '-!',then onecanconsider themapping function (49) to Sec. 26] Transcendental Functions 323 wind thisstripontothez-plane around z=intothecorrespond- ingannular ringaninfinite number oftimes. Ifthestrip inthe w-plane hasaperiodic pattern adjusted toperiod 2ir,theneach layer inthe2-planeisidentical andonecangoback tothefunda- mental region tosolve thephysical problem. Aplanar triode with equally spaced gridwires canbetreated thisway,11reducing theproblem inthe2-plane tothesame asthemapping function zn inthecylindrical triode above. Similarly, onecantreat agrid of thin parallel strips, individually inclined atanyangle against the plane through their centers.12With asuperimposed uniform electric fieldonecancompute theamplification factor infirst approximation as ._ (50).,,.. In(d/irc) where histhedistance between anode andgridplanes, dthespacing between thegrid strip centers, and2ctheir individual lengths. Since theinverse function z ewisclosely related tothecom- plex trigonometric andhyperbolic functions, oneexpects inall cases infinite periodicity, which requires thedefinition offunda- mental regionsforuseful applications. Thus, w=cosh z=coshxsiny+jsinhxcosy (51) givesupon elimination ofeither xory These areagain confocal ellipses andhyperbolas asin(44)above, but itisasingleinfinite strip ^y^2irwhich maps upon the entire lu-plane; seeSokolnikoff andSokolnikoff,09 p.445,who appliesthisfunction toexamine seepage under adam. Thefunction z w+c_w=jccot-or-=e3Z(53)2 w-c transforms thesquare netofthe2-plane into thebiaxial circles identical with theequipotentialcircles and field linesaround two 11Dow,B23 p.24,givesmuch detail andmany graphs. 12S.D.Daymond andL.Rosenhead, Quart. Jl.Math., OxfordSeries, 9, p.89(1938). 324 Two-dimensional Analytic Solutions [Ch. 7 parallel equalandopposite linecharges, adistance 2capart. One has,then,13asinFig.125, y=Inw w cr2 In Graphical Superposition ofMaps. Inmany instances,it may notreadily bepossible tohandle more complex mapping functions analytically, sothat graphical methods ofsuperposition become desirable. Since thesum oftwo analytic functions is again analytic, and allanalytic functions provide conformal mappings, onecanmake useof w=wi+w2=(m+u2)+j(vi+v2) (54) bycombining plots ofsimple functions. Taking forexample MI=Inzfrom (49)andw2=In(z 2)/(z+2)asinversion of (53),numbering theuand vvalues ineach individual graph as pertaining tothesame value ofz,onecanthen construct points with coordinates which arethesums oftheindividual coordinates, again noting thespecific values ofz=re3*.Intheexample, (55)-- I. ^v^jY'1-- I2 wi=Inr,M2=-In-r- =-In 2 r2+4-4rcos< 2 rt __1rsin <-irsin 7 2l+rcos0 2+rcos0l 2 The analytical addition ofthetwofunctions isobviously rather difficult; butthegraphical combination14canbemade simple iftheindividual graphs have been prepared carefully forre- peateduse. Ifw(z)isafunction which canbeseparated intotwofunctional relations, asforexample thebilinear function(31), thenanother typeofgraphical combination ispossible. Having aplot of w2=l/(z+d),which isadipole lineshifted with itsorigin to z=d,andanother graph ofw=a+bw2,which isasimple linear operation, oneselects thew2-plane toplot initw=u+jv with curves u=cons; one also plots inthesame plane z=(l/w 2)d=x+jywith curves x=consandy=cons. 13Bateman,01 p.260. 14Y.Ikcda, JL.Faculty ofScience, Hokkaido Univ., Series II(Physics), 2, p.1(1938); seeparticularly Fig. 48. Sec. 27] Single Vertex 325 Intersections ofthefamilies ofx(w2)andy(w2)with thefamily u(w 2)give foreach particular w2thevalues u(x,y)which canbe transposed intoanewgraph presenting w(z) directly. Applica- tion ofthismethod tow=l/(ez+1)bysplittingitintow2=ez , andw=l/(w 2+1)isshown byY.Ikeda (loc. cit.),who also constructs w=\/z3 (l z),giving theflow ofwater overaplane with ashort inclined wallrepresenting aweir; other examples can befound there. 27-CONFORMAL MAPPING OFSTRAIGHT-LINE POLYGONS Asystematic method ofmapping polygonal regions bounded by straight linesupon theupper halfplane hasbeen developed z-Plane w-Plane FIG.271MappingofPolygon with Single Vertex uponUpper Half Plane. independently bySchwarz andChrist offel.1Itisprobably the most powerful method forthesolution ofLaplacian potential problemsintwodimensions. Single Vertex. Asshown insection 26,themapping function w-WQ=A(z-z)n=|A|rnej(a+n< (1) requires therestriction 0^0^ 2ir/n inorder toleadtoaone-to- onerelationship between z-andw-planes. Theinverse function, withnrealbutofanyvalue, z-2=k(w-wQ)lln(2) where k=A~llnmaps theupper halfw-plane intothesector ir/n ofthez-plane, asshown inFig.27-1. Onehas r=(p/\A\ )lln and= (\l/ a)/n, which uniquely determine onepointPofthe z-plane asthecorresponding onetoapoint P'inthew-plane, and 1H.A.Schwarz, Crelle's Jl.,70,p.105(1869); E.B.Christoffel, Ann. di mat., (2),1(1867); alsoGottinger Nachrichten, 1870. 326 Two-dimensional Analytic Solutions [Ch. 7 conversely, forvalues%^n^QO.The function z(w) is analytic intheentire plane except atw=WQforn>Iandat w=ooforn<1asthederivative (3)awn indicates; forn=I,thetransformation isofthelinear type andneednotbediscussed here. AsapointinFig.271inthew-plane travels along theboundary (a)where u<UQand v=0,thecorresponding point inthez-plane travels along thestraight line (a)towards ZQ.When thepoint in thew-plane changes tothesection (6),where u>UQand v=0, thecorresponding pointinthez-plane changesitsdirection of travel bytheangle (-I)-*1/"- 1)=ej>(1~1/n) ,asshown inFig.27-1. This isalsoseenfrom (3),sincedwand(w WQ)arerealalong the w-axis andthefactor (ww)onlychanges signasthetravelling point passesWQ.Onemayintroduce theangle yir-0-3- asdefining thechange inthedirection ofprogression along the boundaryofthecorresponding regions. Atthevertexitself, the mappingwillnever beconformal except forn=1,since thede- rivative either vanishes (forn<1)ordoesnotexist (forn>1). Avery small circle excluding thepointw=WQsuffices torelieve this difficulty asemphasizedinsection 26. Similarly, oneneeds toexclude thepointw= <*> tastheconventional investigation for theinverse variable demonstrates, except that thederivative vanishes forn>1anddoesnotexist forn<1. Assume theboundaries (a)and (b)inthez-plane tobeconductive planes andtohave thesame electrostatic potential 4>oandplace alinecharge (+X) atpoint P;then theproblem inthew-plane issimply that ofaninfinite conductive plane with alinecharge (+X) atPf .Thecomplex potential solution inthew-planeisin accordance with table 25-1,line 3, 27T6WW where w'isthelocation ofthepoint P'andw'that ofitselectro- static image below v=0.Thevalue ofw1canbefound from the Sec. 27] Single Vertex 327 location(r, </>)ofthelinechargeinthez-plane, andwcanbe expressed interms ofzby(1),sothat thecomplete solution is obtained atonce foranyvalue oftheangle (ir/n). Insection 12, themethod ofimages wasconvenient onlyifnwasaninteger; no such restriction exists here aslong asn^3/.With (5)onecan determine thepotential distribution aswellasthecapacitance ofthe linechargeifoneadmits asmall but finite diameter dfor it.On w thesurface ofthiswireonecantakeww-- 2 (26-25), andw-wr=2v',sothat.similar to dz andthecapacitance with respect totheconducting planes becomes C= 8$0=27TE-1 (6) Forthespecial casethat z=andtherefore WQ=0,themapping function (1)reduces tow=Azn ,where\A\isarbitrary andaso selected that (a)and (b)become thenegative andpositive portions ofthe it-axis, respectively. If,further, thelinechargeislocated intheplane ofsymmetry inthez-geometry atadistance a^>d from thecorner, theimageofthelinecharge inthew-planeis located onthev-axis atv'=\A\an .With thecapacitanceisnow C=dz=n\A\\z\ 27TE In(4aVdnan )In(4a/nd)(7) foranyvaluen^}/. Forseveral special values, Fig.27-2shows theregions ofthe z-plane corresponding totheupper halfw-plane. Forn=J^one hasaninfinitesimally thin plate, whose upper andlower sides become sections (a)and (6)oftheit-axis, respectively; oronecan interpretitasaninfinitesimally thin slitintheinfinite z-plane. Fortheexample with theparallel quasi linecharge ofdiameter d onecanalsofindthecharge density induced onthetwosides ofthe 328 Two-dimensional Analytic Solutions [Ch. 7 platebytheuseof(26-7). Thecasen=%withaparallel line charge wasonewhich could notbetreated byimage theory (see section 21); thesolution canreadily befound with this mapping method. The casen=2is,ofcourse, thesame as treated insection 12. Aparticularly important case isobtained byletting n >oo ; thisgivesfrom (3)withk/n=A/, --JL_(8)dwww Ifk'=1,theactual mapping function isofthesame type as (26-49) andrepresents, asshown inFig.26-7, themapping ofan y///////////////, '7= '//. FIG.27-2 Special Cases ofSingle Vertex Polygons. infinite strip ofwidth ITinthez-plane upon theupper halfw-plane. Forkr=b/ir,with breal, thefunction maps thestripofwidth b(real) upon theupper halfw-plane. Obviously, thiscannot be obtained from theintegral functions (1)or(2),demonstrating themore powerful treatment bymeans of(3). Placing aline charge between theplanes ofthez-geometry, onehasthesame problem aspresentedin(21-5);itisreadily solved hereby(5) inconjunction withthemapping function. Thismethod hasbeen usedbySmythe,A22 p.83,forasingleline charge,2andby Frankel3forone-, two-, and three-line-charge arrangements in order tofindthecharacteristic impedanceoftransmission lines. 2SeealsoE.Kehren, Dissertation, Tech. Hochschule Aachen; J.A.Earth, Leipzig, 1932. 3S.Frankel, Proc. I.R.E., 30,p.182(1942). Sec. 27] Mapping ofInside ofClosed Polygons Itisinteresting tonotethat (3)alsogives 1_ dw\dw) WWQWWQ329 (9) aform completely independent ofthescaleandrotation factor k andcontaining onlytheoutside angle7andthevertex location WQ. 2-Plane FIG.273MappingofInside ofPolygon uponUpper Half Plane. Mapping ofInside ofClosed Polygons. Inanalogy to(3) onecannowconstruct theexpression4 =C(w-wi)-yi (u>-w2)~r2 -''(w-wvry"' '' dw which, asindicated inFig.27-3,maps therealaxis ofthew-plane intothebroken lineofthepolygon inthez-plane. Ateachvertex onlythepertinent factor (wwa)changes sign, causing dzto change angle byexactly yairinthedirection indicated bythe algebraic sign ofya.Theconstant Cactsasscaleandrotation factor andmust bedetermined bythecorrelation ofone ofthe polygon sides (za za+1)with theimage (wawa+i)which,of 4IIistheconventional product notation denning aproductofsimilar terms with ordernumbers a. 330 Two-dimensional Analytic Solutions [Ch.7 course, requires theintegration z=CCll(w-waryadw+Ci (11)J() Thefurther integration constant Ciessentially locates theorigin. Application of(9)gives thesum (in }= dw\dw/(12) -wa which hasbeentaken asthestarting point forthegeneral proof of theuniqueness of(11) asmapping function oftheinside regions ofpolygons. Actually, (11)isanalytic everywhere except possibly atvertices (those forwhich ya>0)and therefore conformal everywhere except atallvertices asseenfrom (10). Forproofs onecanconsult theoriginal articles (loc. cit.)andmost ofthe advanced books inAppendix 4,D,aswell asKellogg,010 p.370, andBateman,01 p.296. Very close toafinite vertex imagewv,onecanapproximate wwawvwafor alla^y,and elect polar coordinates referred towvsuch that w-wv=Pvej +,dw=tf*dpv+jPwe?+d$ (13) The integral (11) gives then along asmall circle intheto-plane with center atwv where theintegration constant C\canbeselected aszvinaccordance with (2)forthesingle vertex. This gives z-2,=7--P,1"7"'n(w,-wary"eMl-">(15)* which foranyvalue(l)^y v<(+l) represents asmall circle with zvascenter, vanishing aspv>0.Foratotal variation of\l/in thelu-plane between andTT,theargument of(15)changes over therange zeroto(1yv}ir= j3^7r,ortheinternal polygon angle, asitshould be.Foryv=+1,theintegral (14)becomes z=[cn(w,-warya ]&+Ci (i6) Sec. 27] Mapping ofInside ofClosed Polygons 331 which isastraight lineatright angles tothedirections ofpro- gressionbefore and after thevertex zv.Asshown inFig.27-3, theangle yv=+1represents avertex atz= ,ortheinter- section oftwoparallellines asatz4,andtheir distance isdefined by ^=atz4"and^=ITatz4';thusfrom (16) Dv=zv"-z/=-jw(CII(w,-wa)-ya ] (17) CL*V That avertex ofthis singular type canbeadmitted isreadily appreciated from thefact that inthecomplex planez= is defined asapoint andcanbetransformed into finite distance by inversion (seesection 25). Foranyclosed polygonofNvertices, thesumtotal oftheinternal anglesis(N 2)?r; thisalsomeans T.=(1-A.)=N-(N-2)=2 (18) a=l a=l which isvaluable asacheck. Ifone ofthevertex images wvislocated atw=<,oralso Wv'=-\-aOjWj,"= ooastheopposite ends oftherealu=axis, (10) willnotcontain thefactor (wwv)because ofthemore basic form (12)inwhich thecorresponding additive term yv/(wwv) vanishes. Nearwvonecanthenapproximatein(10)wvwa= wv=pe?*with p.The integral (11)becomes, therefore, with (18) z=CC(pej+r(2~y^PJej+d*+d (19) since in(10)onlyyvismissing. Performing theintegration gives, withproperchoice oftheconstant Ci, (20) which foranyvalue (I)^TV<(+!) represents asmall circle with zvascenter, vanishing asp><.Foratotal variation of \l/inthew-plane between ITand 0,theargument of(20)changes overtherange (1TV)TT=j3v7rtozero, asitshould be.For yv=+1,theintegral (19)reduces to z=C#+C! (21) which isastraightlineatright angles tothedirections ofpro- 332 Two-dimensional Analytic Solutions [Ch. 7 gression before and after zvasabove in(16). Thedistance of thetwo parallellines isdefined nowby \l/=forz/and\l/=TT forZ,,",sothatfrom (21) Dv=zv"-z/=jirC (22) leading tothedirect evaluation oftheintegration constant C. For practical applicationsitisdesirable tomake useofall simplificationsinthemapping function that arepossible. With reference toFig.27-3, thefollowing points should beobserved for besteconomy: a.Theorder ofthevertex pointsinthez-plane and oftheir imagesinthew-plane must bethesameandsuch that,inthesense ofprogression, theregion tobetransformed isatthe left. b.Allangles arecounted positive inthecounterclockwise sense. c.Foranyconformal representation upon theupper halfplane, three ofthevertex imageswacanbechosen freely (seesection 26) ; thechoice should besoastomake theintegral (11) ofsimplest typeandofstandard form. d.Ifavertex imageislocated atwa= <*>,thecorresponding factor (wwa}doesnotappear inthemapping integral (11); oneshould choose thatvertex atwa=owhich leads togreatest simplification. e.Thesum ofallvertex exponents yaisequal totwo; thisshould beused asacheckwhen tabulating theindividual factors. /.Iftheneighborhood ofwaisthemap oftheregion between two parallellines inthez-plane, then their distance isgiven by (17);iftheneighborhood ofw=QOisthemap oftheregion between two parallel lines inthez-plane, then their distance is givenby(22). g.Themappingoftheupper halfw-plane upon thepolygon in thez-planeisconformal atallpoints except atthevertices them- selves. These vertices are,however, isolated points ofnon- conformality andcanbeapproached arbitrarily closely. h.Since three valueswacanbechosenarbitrarily, andsince the totalnumber ofconstants inthemapping integralis(N+2), namely, theNvertex images waandthetwo integration con- stantsCandCi,theremust beestablished (N 1)independent relations ofthetype (17) or(22) orsimilar integrals inorder to solve themapping problem completely. i.Itisadvisable totabulate therelations between correspond- Sec. 27] Parallel Plate Condenser 333 ingvertices inthe2-plane andimages inthew-plane inasystematic manner such as: Vertex location inz-plane z\ 22 za Change indirection ofprogression atvertex ynr y%ir yair Exponentinmappingfunction 71 72 Tffya=2) Location ofvertex image w\<wz<wa< Inthelast line,three values canbeassumed arbitrarily, theremain- ing(N 3)values enter asunknown constants intothemapping function (11). Itmight alsobeemphasized here that theintegral (11)is actually areal integral, since itistaken along therealaxis ofthe w-plane. However, depending upon therelative valuewina particular section oftherealaxis, several ofthefactors might assume complex values. Tobesure ofthecorrect values ofthe generally multivalued terms, oneshould bring theintegrand into suchform that allfactors with\wwa \<arewritten (w wa)~Ta(~1)~7a ;themethods ofintegration ofrealfunctions will then suffice fortheproper evaluation. Aswtakes oncomplex values intheinterior oftheupper halfplane, continuity inz(w) canbechecked bylettingv >andchecking thecorrectness of Parallel Plate Condenser. Asanillustrative example ofa complete solution take the classical problem ofevaluating the fringing fluxfortheparallel plate condenser. Assuming twovery thin plates asinFig.27-4aofinfinite extension andutilizing the symmetry ofthe fieldandpotential distribution, onehasFig. 27-46 asthez-plane geometry tobemapped upon thew-plane (27-4c). Themapping tableis,ifoneobserves (a)above, z-plane12 3 Vertex location/,'{+ ~f~j? +ja jj'l" 1^+jo o[ Ya7T 2lT IT +7T 7 +2 -1 +1(S7a=2) wa oo-l where allthree pointswacanbeselected freely. Because vertex 1hasthehighest coefficient 71,itisbestchosen atwi=oo ; vertex 3separates thetwopotential values, soitisbestchosen at wz= inorder tolead toastandard probleminthew-plane; 334 Two-dimensional Analytic Solutions [Ch.7 vertex 2must beonthenegative ii-axis, andonecannormalize thegeometryinthew-plane byselecting w2=I.Themapping function isthus defined by(10)andinaccordance with (d)above as or,integrated,=C(wdw =C(w+In(23) (24) 1" 23B VKH FIG.27-4 Parallel Plate Condenser: (a)actual geometry, (6)z-plane, (c)iw-plane. Selecting w=\w\tf*,with ^ \l/^TT,andrestricting Inwtothe fundamental region, sothat Inw=\o\w\+j\l/,make therelation (24)one-valued andsuitable fordetermination oftheconstants. Thus, (17)gives forvertex v=3andw3= -ja=-jirC, C=- Sec. 27] Parallel Plate Condenser 335 andthelocation ofvertex 2defines C\from (24)as -(-l+JT) +Ci fCi=- 7T 7T sothat inthefinalform z=-(1+w+Inw) (25) Itisdesirable tocheck theexact correspondence ofboundaries by letting ztravel along thedistinct sections inthez-geometry and verifying thatwtravels along theit-axis within thecorresponding limits orviceversa. Forexample, as(1)<w<with v=0, sothatu=\u\,onehas z=-(1- \u \+\n\u\+JTT)=ja+-(1+\n\u\- \u\) 7T 7T orx<0,y=a;thisdescribes inthez-plane theboundary from 2to3'asrequired. The potential solution intheupper halfw-plane isnowvery simply givenby(25-26) with theappropriate change innotation asin(26-28) P=$+jH=--($2-$')Inw+$; (26) where equipotential lines areconcentric circles andthefield lines aretheradial lines from theorigin. Itwould, ofcourse, be desirable tointroduce into(26)wasanexplicit function ofzand thus findthe fieldgeometry directly inthez-plane; but this is usually notpossible. The fieldvectorEcanbeobtained from (25-7) as Along theboundary, w=uisrealandthefieldvector isalways parallel totheimaginary axis, ornormal totheboundary;itis positive foru<(-1), negative foru>(-1), andbecomes infinitely large asu >(1)asinthecase ofanyconvex corner ofthepolygon. 336 Two-dimensional Analytic Solutions [Ch. 7 Since EQistheuniform value offield strength between the parallel plates, onecaneasily determine apointAtodefine the practical limit oftheuniform fieldbyfinding thevalue ufrom (27) forwhich\E\=T\EQandbychoosing forexample ij=1+ 5/100 for8percent tolerance. Oneobtains UA=5/100, and thusfrom (25) (28) jo-2 [in100+-(1+In )] For 5=1percent, this gives XA=1.144a, thelocation of pointAinFig.27-4a; generally, theend effects penetrate into homogeneousfield regions toadistance ofthesame order asthe length oftheuniform field line. Since the field lines inthew- plane arecircles, onecanfindpointBofFig.274aandcbyusing (25)withWB=u=1,namely, ZB=2a/ir. The total dielectric fluxfromAtothecorner 2isgivenby(26-4) *A,Z=t(*A-H2)=-(*a-*')In- TT U2 =-(* 2-*')ln^2(29) 7T 5 Fortheidealized condenser withuniform fielduptothecorner 2,thecorrespondingdielectric fluxwould be^1,2=^O|^A|with XAfrom (28). Comparisonofthis latter with (29)shows that fringing results inanactual increase ofdielectric fluxover the idealized condition ofamount 1-5/100~'In100+5/100- (l+ln5) Figure 27-5 gives thevalue(a^l/afrom (28)asafunction ofthe tolerance value 5and also/(5), thecorrection factor in(30)to theidealized dielectric flux^,2inorder toaccount forthefring- Sec. 27] Parallel Plate Condenser 337 ingfrom theunderside oftheupper condenser plate. From Fig. 27-5 onecan alsotake that, for|x^|/a=1,theactual field strengthis1.016J,andthat fringing increases thefluxcontribu- tioncomputed onthebasis ofEby31.1percent. The firsttreatment ofthisproblem byconformal mapping is duetoKirchhoff,A13 p.104,5who alsoappliedittocompute the edge correction ofcircular condenser plates;6subdividing the total space into three regions, Kirchhoff assumed homogeneous fieldbetween theplates uptoA,fringing field ascomputed above 1-5K V^ji.o 0.9 0.8 0.7 0.6- 123456789 10 FIG.275Fringing Correction forParallel Plate Condenser. extending toA',andthen aspacefield asproduced bytwouni- formly and oppositely charged circular disks ofinfinitesimal spacing. Anexcellent graphofthefringingfield distribution is giveninMaxwell,A17 I,Fig.XII,whoused, however, Helmholtz's approach byconjugate functions. Good treatments arealsofound inJeans,A1 p.272; inOllendorff,A18 p.212; inRothe etaZ.,D8 p.138; inReddick andMiller,07 p.377;andinBewley,D1 p.121. Hydrodynamic applications aregiven inPrandtl andTietjens,024 p.179,andinLamb,C22 p.70. WritinginFig.27-4cw=|w|ej ^,then oneobtains the field lines forconstant\w\,andtheequipotential lines forconstant ^. 5SeealsoMonats. d.Akad. d.Wissenschaften, Berlin, p.144,March 1877. 6Forextensive study ofedge corrections seeA.H.Scott andH.L.Curtis, JLResearch Nail. Bur. ofStand., 22,p.747(1939). 338 Two-dimensional Analytic Solutions [Ch. 7 Translation intothez-planeissimplest by(25) x=-[1+\w \cos^+In\w\] y=-[\l/+\w\sin\fr](31) Onecanthen investigate thefieldstrength distribution along any particular equipotential lineandfind^=7r/2asthelargest value of\l/forwhich thefield isnowherelarger thanE .Thecorrespond- ingconductor shapeisusually called theRogowski electrode;it assures thatbreakdown occurs inthehomogeneousfield E$,which permits thedefinition ofthebreakdown strength ofgases and liquids.7 Polygons with Parallel Boundaries. More general cases of boundaries madeupofparallel lines areshown inFigs. 276ato 27-6c.Thearrangement Fig.27-6acanbeused forfringing problems asinBateman,cl p.300,who alsocomputes thecharge distribution,orinGrosser8fortheevaluation ofelectric fields in high-voltage transformer shellwindings ofunequal height. The mapping function isdefined bythetable z-plane1234 i'r_oo 4-io Vf- Vertex location j//_^T +;a* j_I -r , ,o +3lT -7T +1T 7T 7 +3 -1 +1 -1(2T=2) Wa -1 +T where thechangeoftheangle ofprogression atpoint1must be chosen as3winorder torotate direction 4-l'intothat ofl"-2, since arotation by2ironlyproduces aparallel line ofsame sense of direction (seebelow under c).From thetable onehas =- (w+i)(w T) (32) sothat z=C\^-+(1-T)W-TInwl+Ci (33)L2 J 7W.Rogowski, Arch.f.Elektrot., 12,p.1(1923); alsoH.Rengier andW. Rogowski,Arch.f.Elektrot., 16,p.73(1926) andRengier, ibid., p.76. 8W.Grosser, Arch./. Elektrot., 26,p.193(1931). Sec. 27] Polygons with Parallel Boundaries 339 Theunknowns areC,Ci,andrforwhich three relations canbe established, oneforthedistance 3'-3" inaccordance with (17), FIG.276Several Arrangements ofTwo Parallel Conducting Planes. andtwoforthecorrespondence ofthepointszzandz4andtheir images w2andw4,respectively. Thus, by(17), +l)(u;-7)^=0=H-JTrrC (34) (35)za-23=-a=- givingC=-a/7TT. Further, 22=+ja=C%-(1-T)-TJT]+ This illustrates afrequent difficulty even forcomparatively simple mapping functions, namely, thedefinition ofrinterms ofatran- 340 Two-dimensional Analytic Solutions [Ch. 7 scendental equation obtained bysubtraction ofthetwoequa- tions (35) b Asimple graphical solution canreadily begiven. The total potential solution inthew-plane isagain (26)with $1replacing $'there. The electric field isthusby(25-7) E=+jr[(w+1)(w-r)]-1 (37)a indicating infinite values atbothsharp corners 2and 4. Two oppositely charged coplanar planes asinFig.27-66 lead / 1\ tothemapping function z=-lw -\ )Asconjugate function 2\w/ pair, thisgaveinsection 25thesolution foraconducting cylinder inauniform electrostatic field. Here,intheupper halfw-plane, however, thesolution isgivenby(26)with$1replacing $'there; seealsoSmythe/22 p.90,and Ollendorff,A18 p.203. Inserting themapping function into (26) requires theinversion w=-=b a - ) 1,where theupper signmust bechosen tohave point fl/ B, i.e., 2= located atw=+j.One then obtains with \n(t+Vt2-1)=cosh"1 t, p=$+js=-3-($2-fcj)cosh-1-+ <f>! (38) asthedirect solution forthecomplex potential inthez-plane. In theupper halfz-plane onehasthusatypical potential solution fortwocoplanar planes with agap ofwidth 2abetween their parallel edges. This solution willfrequently beneeded;itleads tothesame geometry asFig.25-7. Byinterchangingfieldand equipotential lines, oneobtains thefield ofasingle infinitely thin strip ofwidth 2a;thus, multiplying (38)byjand replacing ($2~"$1)byX/27re forasingle conductor, P=-^cosh-1(-}27T \a/(38a)v ' Sec. 27] Polygons with Parallel Boundaries 341 This gives thetotal dielectric fluxXfortheslab, since forzreal and z<a,cosh"1(- J=In-+j\/l [-} W La\ \a> sothat Atz=+a,tan"1(+0)=TT;atz=-a,tan"1(-0)=-TT; therefore (S+aS_ )=X/e. Fortwoparallel planes asinFig.276c,themapping function9 becomes z=C\w+-+(1-T)InwI+Ci (39) Lw J with theupper halfplaneidentical with case a;thereference also gives thefieldstrength near corner 2andalong theequipotential linew=\w\e. Forthree parallel planesinsymmetrical arrangement asin Fig.27-7a,themapping function contains twounknown param- eterspandq.Applicationof(22) topoint 5gives atonce jb=jwC, orC=b/ir; application ofthecorresponding relation (17) topoint1gives fromwhich pq=1.Integration anduseofthecorrespondence ofpoints z-i,z^and u>2, w>4,respectively, give finally (40) where pmust bedetermined graphically from a_I-p2 b*~ 2p- withp<1;values areshown inFig.27-8. The fieldstrength is, by(26-7), E=-j-b (w-p)(w-- \ p/ 9E.Kehren, footnote 2Fp.378. 342 Two-dimensional Analytic Solutions 1* *2>*1 |ft[Ch.7 5V FIG.277Several Arrangements forThree Parallel Conducting Planes. 0.2 0.4 0.6 0.8 1.0 0.8 0.6 04 0.2 * *-t FIG.27-8 Parameters fortheGeometry Shown inFig.27-7a. Sec.27TwoRight Angles andOneScaleParameter 343 and itsvalue along thecenter linex=0,which istheunit circle inthew-plane, canbefound withw=ej*.Specifically, forthe pointAonehasw=1,sothat **=' whereE= (<S>2$i)/6 andf(p)isshown inFig.27-8 asa function of2a/6; asisevident, thepresence ofthegap2alowers thefieldvalue atAbuthas little influence forratios 2a/b<0.3. Anapproximation tothissolution isgiven bySmythe,A22 p.90, bysuperimposing auniform field inFig.27-66. Inasimilar manner, thefringing from thecenter plateinFig. 27-7610canbeevaluated, aswell astheelectrostatic field dis- tribution forthree parallel plates arranged asinFig. 27-7c.u Theextension tomore thantwodifferent potential values requires amore general solution forthepotential intheupper halfw- plane asshown insection 28,particularly (28-8). Polygons withTwoRight Angles andOneScaleParameter. Thesimple right corner opposite aplane, asinFig.27-9a,is mapped ontheupper halfw-plane according tothetable z-plane123 Vertex iocation !*- .+JO -- -i wa -fl sothatthemapping function becomes (42a)aww "1z=2C[Vw-1-tan"1Vw-1]+Ci (426) The constant C=a/ITisdetermined byapplying (17)topoint v=1,andCi=abytheuse ofthecorrespondence between 10Handbuch derExperimental Physik, Vol. 19,p.29; J.Springer, Berlin, 1935. 11W.Grosser, Arch.f.Elektrot., 25,p.193(1931). 344 Two-dimensional Analytic Solutions [Ch. 7 points 22andw2.Thecorrespondence oftheboundaries canbe checked readily; forexample between points1and2,where <w<1,onecanwrite better 1 1+Vl-w"\^--ln/-=+a2 1-VI-uJ with theterms inbrackets realand negative. Asw >0,the logarithmic termapproaches In2~ (/2)=In2-In(w/2)=+ _>(> w/2 w_>o Itisimportant toconsider thelogarithmofafraction asthedif- ference oftwologarithms inorder topreserve thecorrectsign. InthearrangementofFig. 27-9atheconductor <J>2might represent thegrounded core ofahigh-voltage transformer, and $!thenegative end ofthehigh-voltage winding, sothat (26)is applicable totheupper halfw-plane. Onecanthen findtheend pointAofthemost dangerousfield linebysetting w= Iin (42).Computing theelectric fieldvector along various equipoten- tiallines, onecanestimate theeffect ofroundingoffthesharp corner asinRothe eta/.,D8 p.130.Onethus finds thatalong theequi- potentialline ofvalue [$i+0.05($2 *i)] thesmallest radius ofcurvature ispmax=0.052a andthemaximum field strength Emax=2.75#,if#o=(*a-*i)/; andon[<f>!+0.1(*2-*i)] onehaspm!tx=O.lOSa and 7max=2.0# -Actually, thisgeom- etrywas firstusedbyCarter12toevaluate thefringing flux from amagnetic polewith airgapainanelectrical machine, as alsotreated inBewlcy,D1 p.130,where good graphs areshown. Since theheight 2-3' isunlimited, thefringing fluxcanbedefined onlywithin arbitrary limits, asinthecase oftheplate condenser, Fig.27-4. The slot ofinfinite depth inFig.27-96 leads tothemapping function z=a+2j-{Vl-w2+Inw-In[1+Vl-w2 ]} (43) 7T with (26) assolution fortheupper half w-plane. PointAis defined byw=+jorz=j(b/v)[V2-ln(l+A/2)]=jO.346 12F.W.Carter, Jl.I.E.E., 29,part 146, p.925(1900). Sec. 27]TwoRight Angles andOneScaleParameter 345 andfrequently serves toseparate toothtipfluxfrom theactual slot flux. Brief treatments13aregiven inFrank andMises,C6 II, p.664,and inBateman,cl p.300. Onecould,ofcourse,restrict themapping region tooneofthesymmetrical halves; theupper FIG.27-9 Polygons withTwo Right Angles andOne Scale Parameter. halfw-plane would then present theproblemofFig.27-2 for The semi-infinite strip, Fig.27-9c,hasthemapping function (44)26xz= coshw 7T 13SeealsoR.Gans inVol.V,part 2,ofEncyclopedicderMathematischen Wissenschaften; B.G.Teubner, Leipzig, 1906; J.Kucera, Elektrot. und Masch., 58,p.329(1940). 346 Two-dimensional Analytic Solutions [Ch. 7 which isquite similar to(38);ifl"-2 carries potential $2and3-1/ potential $1,then thez-plane represents asemi-infinite ideal parallel plane condenser with 2-3asfield line; seeSmythe,A22 p. 88,andRothe etaZ.,D8 p.143. Assuming thestrip ofvery thin conducting material andplacing asource lineattheorigin of thez-plane, thenonehasinthew-plane theradial flow lines of asingle source lineatorigin ofthew-plane; seeWalker,D1 p. 66,forgraph. Interchanging inthis latter geometry flow lines andequipotential lines, oneobtains themagneticfield ofaline current midway intheairgapbetween two infinitely permeable iron blocks, asinWalker,D1 p.71,and inBewley,01 p.136. Finally, assuming inthew-plane auniform field parallel tothe 24-axis, oneobtains inthez-plane theflowbetween asource line atl"andasink lineatl'asinWalker,010 p.46,andinReddick and Miller,D7 p.376.Aninfinite grating oflikecharged strips ofwidth 26<2alocated along thei/-axis with center spacing 26 istreated bySmythe,A22 p.89,bymapping onesample asinFig. 279c.Inthetu-plane onehasasingle stripontheu-axis andthe potential solution is(38a) withw/bforz/a. Arectangular step intheboundary asinFig.27-9dhasthe mapping function z=-[Vw21cosh"1w]+ja (45) IthasbeenusedbyOllendorff,A18 p.199,14tocompute theeffect ofavertical riseinground (walls ortrees) upon thecapacitance ofparallel communication lines inamanner asshown in(6). Considering athinconducting sheet ofthisshape andapplying potential $2along l"-2, andpotential 3>ialong 3-1," giveaflow pattern forwhich (38)isthesolution inthetu-plane.15 Polygons withTwoRight Angles andTwo Scale Pa- rameters. Avery thin plate inaright-angle corner asinFig. 2710arequires amapping function (46) 14Also F.Ollendorff, E.N.T., 4,p.405(1927). 16Y.Ikeda, Jl.Faculty ofSciences, HokkaidoUniv., Series II(Physics), 2,p.1(1938) ;secparticularly Fig. 33. Sec. 27]TwoRight Angles; TwoScale Parameters wheremmust bedetermined (graphically) from347 which results from thecorrespondence ofthepointsz4andw4. Theother constants havebeendetermined inaccordance with the FIG.27-10 Polygons withTwoRight Angles andTwo Scale Parameters. previousillustrations. Detail computations ofthe field distribu- tionweremade byWalker,010 p.88whoapplied thisgeometry to leakage problems between poleandarmature ofelectrical machines; healsointroduced oneoftheequipotential surfaces as afeasible poleshoegeometry, shown dotted inFig.2710a. Avery widely usedgeometryisthat oftheslot, Fig.27106. 348 Two-dimensional Analytic Solutions [Ch. 7 Restricting themapping region totheright half slot,oneincludes aspart2-3' oftheboundary afield linesothatthew-plane requires assolution thecomplex potential function (38)withw/p replac- ingz/a.Themapping upon thew-planeisaccording tothetable z-plane12 3 Vertex location^{"I* ,I?~V ll+jO6[bjo yair -{-IT +- +TT -- +i-5 Wa - -p +P +1 defined bytheexpression dz_ dwwp Applicationof(22) topoint1gives atonceC=CL/TT]and (17) appliedtopoint 3givesp=a2 /(a2+262 ),sothatthemajor con- stants are alldetermined. Integration gives then whereR=[(w+p)/(w-1}}Y\q=(1-p)/2p=(6/a)2 ,and where Ci=0,asthecorrespondenceofpoints z%andwzdemon- strates. Theform (47)isobviously more difficult todealwith than previous forms, which istobeexpected asthegeometry becomes more involved. Inapplyingthisgeometry toarmature tooth-slot combinations ofelectrical machines, thepotential values *should bereplaced bythemagnetostatic potential JFandthesolution inthew-plane isfrom (38) P=y+js=_1(ya_y,)cosh-1-+7i (48) 7T p where Sisthemathematical fluxfunction. Since thepotential 3actually definesH }themagnetic field vectorBisthenfrom (25-7) Sec. 27]TwoRight Angles; TwoScale Parameters 349 whereBQ=(M/a)(7 2^i)istheuniform magneticfield inthe airgapfarfrom theslot. Along l"-2onehasw= \u\, \u\>p, sothat /U_i_ (50) isdirected normal tothepole surface l"-2andhasaminimum value atpoint 2given by\u\=p,namely, Bmin=JB( iftheslotwidth 26a,asusuallyisthecase. Toevaluate the effect pffringing onecandefine apointAatwhich\B\=0.98 andform theratio oftheactual magnetic fluxleaving between points 2andAtotheidealized magnetic fluxbetween CandA withuniform fieldvalueBQ.Thus, from (50), \UA \=0.96/0.04= 24,andtherefore RA=V(-u A+P)/(-U A-1)VuA/(uA+1)=0.98, sothatfrom (47) zAla. 1.98 2 l/no6\ -^=--In-+-tan"1 (0.98- )6 TT60.02 IT \ a/ which isplotted asfunction ofa/6inFig.27-11. Obviously,in order tobeapplicable tofinite tooth widths, theslotpitchmust certainly belarger than 2z^.Theactual magnetic fluxbetween points 2andAis,from (48)withw2=\vv\=p,WA= \UA\,andobserving cosh"1f J=jv+In[\u\Vu2p2 ], *OT=M(S2-Ex)=--(^2-ffi) Inftffl+2(-}7TILW whereVu2p21^(p/ii)2hasbeenused forsimplification. The idealized magnetic flux is3>o=B(zA 6);thenegative sign derives from thenegative direction ofthemagnetic field, having assumed [F2>IFi.Thefringing factor isthus 1aIn48+m[l+2(b/q)2 ] (52) 350 Two-dimensional Analytic Solutions [Ch. 7 which isalsoshown asafunction ofa/6inFig.27-11. This factor agrees invalue with others computed onthebasis ofcomparable assumptions;ithastheadvantage that itapplies withuniform accuracy inallcaseswhere theslotpitchr>2zA-The ratio of \3- FIG.27-11 Fringing Factor forArmature Slot inElectrical Machines. actual magnetic fluxforonefullslotpitch totheidealized magnetic fluxforonetooth canbecomputed with thedesignations ofFig. 27-11 as total (r-2b)B (r/2)-6(53) where f/isthevalue from (52). Evaluations ofasimilar type were firstmade byCarter16interms ofanequivalent airgap; rather complete treatments ofthe slotaregiven inWalker,D1 16F.W.Carter,Jl.I.E.E., 29,part 146, p.925(1900) forpole leakage, andF.W.Carter,Electr. World andEngr., 38,p.884(1901) forslotfringing; seealsotheextensive recent treatment inJ.Kucera, Elektrot. undMasch., 68, p.329(1940). Sec. 27]TwoRight Angles; TwoScale Parameters 351 p.81;inSmythe,A22 p.294; in011endorff,A18 p.216;andBewley,D1 p.139.Thesame geometry hasbeenused toevaluate thetem- perature fieldandheatflowbetween conductor andslotinelectrical machines.17 Thesamemapping function (47) isapplied totheproblem of theright-angle bend obtained byletting 0-3' inthez-plane of Fig.27-106 alsotake thepotential<J>2-Inthew-plane onehas then onlytwopotentials andthecomplex potential solution is given by(26)withanappropriate shift oftheorigin. JeansfA1 p.277,hasused thisfortheevaluation oftheelectric field inthe Leyden jar;alsoBewley,D1 p.126,who gives agood fieldgraph. Ithasalsobeen applied topoleleakage byWalker,010 p.73,tothe elastic torsion problem ofanidealized Lsteel bar,18andbyinter- change offieldand equipotential lines inthew-plane tothe magnetic fluxinatransformer core.19 Thearrangement inFig.27-lOcleads tothemapping function b[ 2u>-(p+l) a(p+l)w-2p\ ,_ ,z=-cosh1-- -coshL- -- --(6a) TTL pl b (pl)w _\ (54) where p=(b/a)2 .Ithasbeenused torepresent thefield distri- bution inlarge cable end sections;20byinterchange ofequi- potential and fieldlines, oneobtains either flow inachannel oftwo different widths, asWalker,D1 p.53,whogivesmuch detail anda good graph, orthecurrent flow inavery thinsheet, asSmythe,A22 p.230,andBewley,D1 p.125. The finite plate thickness ofaparallel plate condenser (see Fig.27-4) canbetaken intoaccount asshown inFig.27-10d. Themapping function with theassumed location ofcorresponding points becomes Vp Vpi-R(55) where R=[(w+!)/(>+p)]5 *,p=-1+2kVk2- 1,and k=1+b/a; thesign ofthesquare root inpissochosen that 17W.W.Peters, Wiss. Verdff. a.d.Siemens-Konzern, 4,p.197(1925). 18E.Trefftz, Math. Annalen, 82,p.97(1921); C.Dassen, Zeits. angew. Math, undMech., 3,p.258(1923). 19G.M.Stein, Trans. A.I.E.E., part I,67,p.95(1948). 20P.Andronescu, Arch.f.Elektrot., 14,p.379(1925). 352 Two-dimensional Analytic Solutions [Ch. 7 p>I.The field linesfrom thelower side3-4'donotspread as much asinthecase oftheinfinitely thin plate; forexample,if b/a=%>then pointAinFig.27-10d hasadistance 0.403a compared with distance (2/ir)a=0.636a forpointBinFig. 27-4. The fringingfieldbecomes particularly importantifthe condenser represents thedeflection platesofacathode-ray tube, since itcaninfluence theelectron path configuration markedly; ananalogouseffect ofthemagnetic fringingfieldupon thepathof ions exists inmass spectrometers.21 Polygons withTwoRight Angles andThree Scale Pa- rameters. InFig.2712apotential $2designates ahigh-voltage winding, $1thelow-voltage winding, aswellasthecore5"-l' ofa transformer.22Because ofthethree pairs ofparallel lines itis possible todetermine allparameters explicitly without performing theintegrationofthemapping derivative;onehas /A2//)\2c-~, 3=-+vw2+(0'p=(;)9 wherem=%[(a/b)2+1 (c/6)2 ].Thesolution inthelu-plane isgivenby(26), sothattheelectric field distribution canreadily becomputed. Thesame geometry inthez-plane canrepresent two other applicationsifoneconsiders thesymmetry ofFig.27-126. Asa simple electric lenssystem,23onecantake f>iasanaperture (usually very thinbutthenmore difficult tomap) and <$2asthe firstanode; thecenter line isthenafield line,andthesolution in theupper halfw-planeisagain (38)withwreplacing z/athere. Asabove, theparameters canbeevaluated withoutintegration, giving Vm2+4n], q=%[+m+Vm2+4n] wherem=(2/b2)(a2-c2 ),n=1+(2/b2)(a2+c2 ).The elec- tric field isby(25-7) /dp/dz\"WdW=" 21N.D.Coggeshall,Jl.Appl. Phys., 18,p.855(1947). 22L.Dreyfus, Arch.f.Elektrot., 13,p.125(1924). 23R.Herzog, Arch.f.Elektrot., 29,p.790(1935). Sec. 27]TwoRight Angles; Three Scale Parameters 353 whereEQ=(1/6) ($2 $1)- Inelectron optics one ismostly interested inthe field along the axis, forwhich intheiy-plane w=u,and(1)<u<+1. Therefore, <57) 'dw^w(w+p) FIG.27-12 Polygons withTwoRight Angles andThree Scale Parameters. This fieldhasamaximum atu=-(a-c)/(a+c).Theexact correspondenceofpointsinthetwoplanes can,however, onlybe established after integration, which isstraightforward butbecomes rather unwieldy. Theother applicationofFig.27-126 istoopposing stator and rotor slots ofelectrical machines, assuming $1=Fiasthe magnetostatic potentialoftherotorand$2=^2asthat ofthe stator;24thelineofsymmetryisthenamagneticfield linethrough 24J,Kucera,Elektrot. undMasch., 58,p.328(1940). 354 Two-dimensional Analytic Solutions [Gh. 7 thecenters oftheopposingslots atthemoment where these coincide. Thereference gives extensive treatment ofthevarious parts ofslotreactance. Another transformer problem25isillustrated inFig. 27-12c, where thethree pairs ofparallel linesagain permit direct evaluation oftheunknown parameters inverymuch thesamemanner as above. Polygons withMorethanTwoRight Angles. Since every right angle contributes asquare root factor intheexpression for thederivative ofthemapping function, theintegrations formore thantworight angleswillleadinvariably toelliptic andhyperel- liptic functions.26Thesimplest case istherectangle withuniform field asshown inFig.2713a. Because ofthesymmetryinthew- plane, themapping function canbewritten z=kC[(I-w2 )(l-k2w2)]~1Adw+Ci= kCF(k, w)+d t/O (58) where thelimits oftheintegral arechosen soastoidentifyitwith thestandard (Legendre) elliptic integral ofthefirstkind F(k,w), which istabulated forrealvalues ofw;kisthemodulus which must bedetermined from point-by-point correspondence in z-andw-planes. Thelength afrom z2toz3corresponds bysym- metry to a=2kCC[(1-w2 )(l-k2w2)]-*dw=2kCK(k)t/O where K(k)isthecomplete elliptic integral ofthe firstkind. The length jhfrom z3to24corresponds to jh=kC[F(k,Q-F(k,1)1=jkCK(k') ,=K(k)-K(k'\ where k'=Vl-k2 .From since 26L.Dreyfus,loc. cit. 26Forgood treatment seePierrepont,1516fornumerical values Jahnke and Emde: Tables ofFunctions; reprinted byDover Publications, New York, 1943. More extensive treatises areH.Hancock: Elliptic Integrals; John Wiley,NewYork, 1917; A.G.Greenhill: Applications ofElliptic Functions; Macmillan, London, 1892; andA.Hurwitz andR.Courant: Vorlesungen liber allgemeine Funktionentheorie undelliptische Funktionen; J.Springer, Berlin, 1929. Sec. 27]Polygons withMoreThanTwoRight Angles 355 thesetworelations onehas -, ~ 2kK(k)' 2h~ K(k') sothat foragiven l/kinthew-plane onecandetermine theratio a/h,orviceversa. Thevalue ofCiisbestobtained byidentifying w=with z=a/2bysymmetry, which gives from (58)im- mediately Ci=a/2. Thus, (58)becomes 2K(k)(60) Bateman,01 p.302, givesthis solution, andIkeda27gives agood fieldgfaphinthew-plane. Thecomplex potential solution inthe2-planeisbyinspection P=*2-Eoz, E=<t>2~*l (61)a Introducing (60) into thisform yields actually thecomplex potential solution forthew-plane directly, which willbeused as oneofthestandard solutions, namely, +5<*'+*'> (62) Thisis,ofcourse, also thecomplete solution oftwocoplanar parallel strips;28byinterchange offield linesandequipotential lines itbecomes thesolution ofthree coplanar strips, thecenter oneoffinite width 2,thesymmetrically located outer onesextend- ingtoinfinity.29 Tofindthesolution forother potential distributions inthe rectangle, one firstmaps therectangle geometrically by(60)upon 27Y.Ikeda,Jl.Faculty ofSciences, Hokkaido Univ., Series II(Physics), 2, p.1(1938). 28Forgraphsofthe field distribution forthecases k=sin10, sin45, sin80seeY.Ikeda andM.Kuwaori, Scient. Papers Inst. ofPhys. andChem. Research, 26,p.208(1935); seealsoF.Cap,Oesterr. Ing.-Archiv, 2,p.207 (1948) forthecasek=0.1. 29H.Petersohn, Zeits.f. Physik, 38,p.727(1926), whoalsostudies mappings byseveral typesofelliptic functions; also J.J.Thomson: Recent Researches inElectricity andMagnetism; Oxford University Press, 1893. 356 Two-dimensional Analytic Solutions [Ch.7 theupper halfw-plane andthentransforms thisupper halfplane uponitself soastoidentify thepotential problem withoneofthe three standard forms (26), (38), or(62). Iftherectangleisathin conducting sheet with potentials $1and <J>2applied over small sections oftheperiphery, thesolution30requires twomappings of Fia.27-13 Polygons withMore thanTwoRight Angles. thetype (60). Wires inrectangular ducts ofeither conducting31 ordielectric material canbetreated inthissame manner, the mapping function (60) leading toawireabove conducting, or dielectric,halfspace inthew-plane. Two parallel finite strips constituting aparallel plate condenser asinFig.27136have potential*'=H(*2+$1)attheplane of 80H.F.Moulton, Proc.London Math. iSoc., 3,p.104; alsoJeans,A1p.354. 31C.M.Herbert, Pfcya. Rev., II,17,p.157(1921); alsoStruttFB3 p.36. Sec. 27]Polygons withMoreThanTwoRight Angles 357 symmetry. Mappingofonequadrantinaccordance with the table z-plane1234 5 Vertex location/,'( ~J~JJ jb-a+jb jb 1[+jv Sir ITT TT 7air~2 2 2"^ 2 +1+i+i-1+l(S-=2) .J-i+1 +P leads totheintegral z=kCflK1-*W-fc22)!"*dw+d (63)Jow+l//c Thisformcanberesolved intoasum ofstandard elliptic integrals32 which arealsoinvolved intheevaluation oftheparameters kand p,aswellasoftheconstants CandCi.Thecomplex potential solution inthew-planeisgivenby(62). Thefactthatl/kinthe location ofthevertex pointsinthetu-planeisleftundetermined makes thechoice ofsymmetry equivalent tothedefinite choice of onemore vertex. One could, ofcourse, have chosen wi=dz soastoreduce theorder oftheintegral;inthatcasethemapping function would lead totheWeierstrass type ofelliptic integrals which would then alsoappearinthesolution intheupper half w-plane asinFrank andMises,C6 p.668;33Ktaian andBurgers,021 p.83,applythisand similar mappings toaerodynamic flow problems. Theabove solution isobviously identical with that forasingle stripabove aninfinite conducting plane ofpotential <>';Kehren (loc. cit.)hasextended thistothecase ofthesingle strip inaright-angle corner asinaLeydenjar,andY.Ikeda and M.Kuwaori (loc. cit.)have extended ittooneandtwo parallel strips midway between parallelinfinite planes andnormal to these, aswell asother arrangements.Ifthetwo parallel strips have thesame potential, then theline ofsymmetry between the stripsisafield lineandthesamemapping function leads toasingle 32SeePierrepont,016 p.384. 33Asalso inE.Kehren, Dissertation; J.A.Earth, Leipzig, 1932,andH.B. Palmer,Electr. Engg. t66,p.363(1937). 358 Two-dimensional Analytic Solutions [Ch. 7 charged strip3-5intheupper halftu-plane, asshown inFig.27136, forwhich thecomplex potential solution is(38a) ;seeFrank and Mises,00 p.668,Case II. Two semi-infinite rectangular electrodes asinFig.2713c,with theplaneofsymmetry ofpotential<S>'=^($2+$1)1canbe mapped byconsidering theright half oftheH-shaped region. Themapping function isthensymmetrical andbecomes z=r(i_/cV)H(l-w2)-*dw+d=^ E(k,kJo K (64) with thestandard form ofthe(Legendre) elliptic integral ofthe second kind. Inthew-plane, thecomplex potential solution's (38) with (z/a) replaced byw.Onecould alsohave chosen onequad- rantbounded byI/'-2-3-0 andthepositive z-axis; inthiscase integrals oftheWeierstrass typewould again beencountered, though thegeometryoftheupper w-plane remains thesame. Treatments arefound inFrank andMisesco II,p.664,and in Bateman,01 p.304; forextensive details ofnumerical computations and ofelectric resistance ormagnetic reluctance seeDavy.34 Thecurrent distribution inathinconducting sheet intheshape of anHisfound bythesamemapping function; forpotential $2 applied alongA-2and 3>ialong theopposite side ofthebridge, theupper halfw-plane hasthesame geometry asinFig.2713a, sothat (62)canbeused.Agraph ofthefield lines (orflow lines inthecorresponding hydrodynamic application)isagain given in Ikeda andKuwaori (loc. cit.). Thearmature slots inelectrical machines have actually the form ofFig.27-13d; with thesimplification ofinfinite depth, the mapping function involves standardelliptic integrals.35Without simplifying assumptions, themapping function becomes ahyperel- liptic integral involving sixright angles, which canonlybeap- proximated byelliptic integrals; thesame applies toonequadrant ofarectangular transformer core.36Salient polemachines have 34N.Davy, Phil. Mag., (7),36,p.819(1944) ;forgraphs seealsoY.Ikeda andM.Kuwaori,loc. cit. 85R.Cans, Arch.f.Elektrot., 9,p.231(1920) ;R.Frey, Vol.IVofArbeiten ausdemElektrotechnischen Institut Karlsruhe; J.Springer, Berlin, 1925. 86S.Bergmann, Math. Zeits.t19,p.8(1923) ;Zeite. ang&uo. Math, undMech., 6,p.319(1925). Sec. 27]Polygons withOther thanRight Angles 359 pole shoes which canberepresented asshown37inFig. 27-13e. Assume thecenter line l/-2/between neighboring poles ofthesame magnetic potential5"'asthearmature 6''-I/,andthemagnetic pole ofpotential ^2,themapping function involves elliptic integrals and leads intheupper w-plane totwocoplanar planes with infinitesimal gap,which hasthesolution (26). With thearmature 1"^0,A* r i7" 'l FIG.27-14 Polygons withOther thanRight Angles. 6"-l' omitted, thesame geometry hasbeenused tofind indetail themagneticfield distribution incyclotron magnets38with "shims" orpole shoes. Polygons withOther thanRight Angles. Thesymmetrical arrangementofFig.2714a ismapped upon theupper halfw-plane by Ci (65)=CCu which isanintegraloftheEuler type.39Theonly scaleparameter 37I.A.Terry andE.G.Keller, JLI.E.E., 83,p.845(1938). 38M.E.Rose, Phys. Rev., 63,p.715(1938). 39Jahnkc andEmde, Tables ofFunctions; reprinted byDover Publications, NewYork, 1943; originally published byB.G.Teubner, Leipzig, 1938. 360 Two-dimensional Analytic Solutions [Ch. 7 aisdetermined byintegration between 2and3,leading to rt (66) where r(q+1)=q\,sothatCcanbedetermined. Thesolution in theupper w-planeisgivenby(38)with (2w-1)replacing z/a; theshift inorigin wasintroduced inorder togivein(66)astandard form ofintegral. Thisgeometry hasbeenused tostudy thebreak- down ofoil40 experimentally andtheoretically. Special cases include a=%,oraninfinite platewithonesharp-edged electrode. Extension tonon-symmetrical alignment oftheelectrodes, particularly forthecase ofvanishing angles aand/3wasmadeby Kehren.41 Alargenumber ofmappings ofthetypeshown inFig.27146 havebeen published inJapan.42These includeparticularly a= 7T/3,anda=ir/4forFig.27-146, applications toregions formed bythepositive z-axis and2-3-1', andsolutions offlowproblems intriangular regions. Mapping ofRegions Outside ofPolygons. Ifinthegeneral Fig.27-3 itisdesired tomap theoutside region ofthestraight- linepolygon upon theupper halfw-plane,itisnecessary toreverse thedirection ofprogression along thepolygon inorder tosatisfy theconvention thattheregion tobemapped betotheleft;butin addition onehastoconsider thattheinfinite point ofthez-plane isnowapoint oftheregion tobemapped andthat there the function willcertainly notbeanalytic. Itcanbeshown that this results inamapping functionslightly modified ascompared with (11),namely, z=cfn(w-warya^ ^ (67)t/ (a) (WW)*(W 1U) where the,firstproduct istobeextended over allvertices ofthe given polygon inthez-plane, andwherewistheimage ofz=oo f WQitsconjugate complex value; seeBateman,01 p.305,and 40L.Dreyfus, Arch.f.Elektrot., 13,p.123(1924); also OllendorffA* p.209. 41E.Kehren, Dissertation; J.A.Earth, Leipzig, 1932. 42Y.Ikeda, Jl.Faculty ofScience, Hokkaido Univ., Series II(Physics), 2, p.1(1938); A.Migadzu, Technology Reports, Tohoku Imperial Univ., Sendai 10,No.4,p.51(1932). Sec. 27]Mapping ofRegions Outside ofPolygons 361 Kellogg,010 p.374.The choice ofWQisgenerally important because ofthecorresponding non-conformality ofthemapping in thez-plane. Thesum ofthevalues yaisnow 2incontra- distinction to(18), because thesum oftheoutside angles ofa closed polygonis(N+2)ir. Anexampleisthesimple straight linecutinthez-plane asin Fig.2715a.The contributions totheproduct function follow from thetable z-plane1 3 () Vertex location b +b TaTTIT IT (27a=2) T -1 -1 Wa 1 +1 WO j and'the location ofz QOmight bechosen atWQ=j,sothat 1 wdw+Cl=-c +Cl (68) / Thecorrespondence ofthepoints1and3leads toC=26, Ci=0.Assuming thecuttorepresent aflat strip conductor withatotal charge Xperunit length, thenthefield linesgoing out toinfinity canbepresumed toterminate there on(X).Inthe w-plane theproblem nowbecomes oneofalinecharge (X)at w=jabove aplane conducting surface; thesolution isfrom table 25-1,line3, w if$oisthepotential ontheconductor surface. The electric field strength is,by(25-7),__ itbecomes infinitely large atw=1.From (69)oneobtains thesurface charge density as&Ewithw=u,real.Byinversion, thestraight linecutcanbetransformed intoacircular arcasin Bateman,cl p.306,where applications tohydrodynamic problems alsoaretobefound. Arectangular hole inaninfinitely extended thinconducting sheet canbemapped upon theupper w-plane asinFig.27156, where againz= >ismapped atw=j.Iftheelectrodes areat 162 Two-dimensional Analytic Solutions [Ch. 7 rerylarge distance, onecanconsider them atz=ooand inthe ;-plane theyappear asadipole lineatw=j.Inorder tosatisfy heboundary conditions onthew-axis, i.e.,tomake itafield line, 2 1-1-4' |3 >5L=C- 2o A' FIG.27-15 Mapping ofRegions Outside ofPolygons. isecond dipole linemust belocated atw=j,sothat, inaccord- incewith table 25-1, line6,thesolution inthew-planeis c,1 1\ 21 aP=-2aS[ J=-j 5\w jw+j/ Tryw*+I GENERAL LAPLACIAN POTENTIAL PROBLEMS ANDCONFORMAL MAPPING(70) 28- Forcases ofamore general geometryitbecomes desirable to lave assurance ofreaching adefinite solution ofthepotential problem. Ithasbeenshown that theinterior ofanysimply connected region1bounded byregular curves canbemapped upon 1Aregion inwhich anysimple closed curve (without double points) canbe shrunk toapoint without leaving theregion; Kellogg,010 p.74. Sec. 28]Boundary Value Problems ofFirstKind 363 theinterior oftheunit circle inaone-to-one conformal manner;2 this isRiemann's fundamental theorem. Itis,ofcourse, difficult tofindtheparticular mapping function foranygeneral configura- tion oftheoriginal boundary curve, sothat inpractice several mappings might have tobeperformed orapproximations bymeans ofpolynomials might have tobeemployed; seeBateman,01 p.322. Several ofthemore general cases willbebriefly outlined here asfarastheyhavereached practical significance. FIG.28-1 Solution ofFirstBoundary Value Problem onUnit Circle, Solution ofBoundary Value Problems oftheFirst Kind. Ifbysomemeans themapping into theunit circle hasbeen accomplished, then itispossible tosolveanypotential problem fortheunit circle,ifthepotential values onitsperiphery aregiven (boundary value problemofthefirstkind orDirichlet problem), bymeans ofPoisson's integral, *-sJf i-U-')+r**(0/)*' (1) where(r, </>)defines apointPwithin unitcircle, and</>'apoint on theunit circle asinFig.281 ;seeBateman,01 p.238,oranyofthe references inAppendix 4,D,b.Forapplications inthecomplex z-plane onecanalsousethecomplex potential function ofSchwarz (2) inwhich therealpartistheform (1),since z=re3*. 2B.Riemann, Inaugural Dissertation, Gottingen, 1851; P.Koebe, Math. Annalen, 67,p.146(1909) and Jl.ofMath., 146, p.177(1915); Frank and Mises,cfl I,p.718;Bateman,01p.275. 364 Two-dimensional Analytic Solutions [Gh. 7 If,inparticular, there arenpotential values sectionally constant over theperiphery oftheunit circle, then foranyonepotential 3>aextending from Ba'toa" (Q\> istaken closest tozero and n"=2ir+0/),theintegral gives Foratotal ofnsuccessive different potential sections, thetotal solution isthen3 P=-*o+-f;(*+!-*)In[ej6""- z]+2*! (3) TT=1 where inthesummation $n+i=$\.The lasttermcompensates forsummation interms ofa",theendangleofthesection, and *0=+ ft,"-O* (4)^a=l isthemean value ofthepotential over theperiphery oftheunit circle and,according toGauss, identical withthepotential atthe center ofthecircle (Gauss's mean value theorem). Thus, for $=$1along ^<//^TTand$=$1alongTT^0'^27r, onehasfrom (3)thesimple result 2 2-h1p=_j_4> 1ln^lT+2<l>1 7T21 Near thepoints ofdiscontinuity ofpotential ontheunitcircle, thepotential function(3)behaves like (jInf)iff=ejea" w; thisis,inaccordance with table 25-1, line 2,thecomplex potential ofthemagnetic field ofalinecurrent, sothatthepotential value right atthediscontinuity isnotanalytic, but isregular inany arbitrarily close neighborhood. Mostmapping problems aresimplerifmapping upon theupper halfw-plane canbeachieved, rather thanupon unit circle. Since thefunction .1-2w=i-J maps theunit circle ofthe2-plane upon theupper halfw-plane, 3Bateman,01 p.242; alsoH.Villat, Bull, desoc.math, deFrance, 39,p.443 (1911). Sec. 28]Boundary Value Problems ofFirstKind 365 onecantransform Schwarz's complex potential solution (2)with u'-j ,., 2du' z=wj w+j into P=-u'w --(1+u'2)(w-u')72(5) (6) which isthegeneral solution ofthe firstboundary value problem intheupper half lu-plane. In(6),u'denotes theintegration variable along therealit-axis, andwisthearbitrary point where f, *2 $3 FIG.28-2 Solution ofFirstBoundary Value Problem inUpper HalfPlane. thepotential Pexists. The realpart of(6)leads totheequivalent ofPoisson's integral ontheunitcircle, (1),namely, *ir+ v=- / 7- 7irJ-*> (uu+v*(') du' permitting thedirect evaluation oftherealpotential distribution. If,again, there arenpotential values sectionally constant along theu-axis asindicated inFig.282,then foranyonepotential $a extending fromua'toua",theintegral in(6)canbeseparated into twosimple integrals with theresult du' *r=*a L" iw-u .1 i1+-In- 7-+-In- w-ua 2 1+ Thesum ofthencontributions canbecontracted, since ua'r= ',intotherather simple form o'n P=- WUa(8) 366 Two-dimensional Analytic Solutions [Ch. 7 Use hasbeenmade ofthe fact that foru\>(o)the term In,>0,whereas forun" >(+) thetermw u\ m"*"Un>ln(1)=JTT.Foronlytwopotential values,w-un namely, *=*ifor oo<u<0,and*= <2for<u<+, (8)reduces immediately tothestandard form (25-26) or(27-26) which hasbeenused extensively. Theapplicationtothree parallel thinlayersoftransformer wind- ingsasinFig.277cisnowstraightforward. Assume symmetryof potential distribution, namely, $1= 3>3=V,and <J>2=0; then (8)becomes if ^/2 Vl4-a2 ~l=*-71n-^- +7In-M- TT|_w+I w-qJ Theequipotentiallines inthew-plane areactually twofamilies of hyperbolas, onediverging from thepoint 3andtheother from the point5.Transposingthese into thez-plane shows that the outermost layer hasthestrongestfield concentration near it4 andmust therefore beparticularlywell insulated. The field vector canbefound againby(25-7) andbecomes with (9)andthe derivative ofthemapping function from Fig.27-7c, The fieldstrengthisinfinitely high atthethree sharp corners2,4, and 6;itiszero forw=%(q-1), i.e.,onthecenter layer, the exact location depending ontherelative geometric distances ofthe windings. Inasimilar manner canbetreated Figs. 2712aandc, aswell asproblemsinhigh-voltagetransformers involving three separate windings ofhigh, medium, andlowvoltage andthe groundedcore.5 Actually, thepotential along windingsisnotconstant butmay vary linearly; similarly, themagnetostatic potential varies fre- 4W.Grosser, Arch.f.Elektrot., 25,p.225(1931). 5J.Labus, Arch.f. Elektrot., 19,p.82(1927). Sec. 28]Second andMixed Boundary Value Problems 367 quently along theiron surface either because ofsaturation or because ofexciting windings. Insuch cases onecanmake useof (6)directlyifthevariation ofpotential hasbeen transposed from theoriginal z-plane totheupper halfw-plane, sothat<f>(u') isknown asafunction ofu.Unfortunately, theintegrations can becarried outonly inthesimplest cases, sothat either approxi- mations orgraphical ornumerical methods become .necessary. A rather simple illustration istheevaluation ofthemagneticfield distribution inthe airgap ofanelectrical machine iftherotor surface hasamagnetomotive force distribution which isconstant directly opposite thestator poleanddecreases linearly from the edgeofthepole totheplaneofsymmetry between poles.6 Solution ofSecond andMixed Boundary Value Problems. If,again, themapping ofasimply connected region into unit circle hasbeenaccomplished, buttheboundary conditions prescribe thevalues ofthenormal component ofthe field gradient, Er=d</dr, theboundary value problem issaidtobeofthe second kind (orNeumann problem);iftheboundary conditions prescribe over certain sections oftheperiphery ofunit circle the potential values andover theremaining sections thenormal componentofthegradient, then theboundary value problemis saidtobeofthemixed kind. Itisnotpossible inthese cases to deduce ageneral theorem ofpractical value comparable with the Poisson integral forthefirstboundary value problem ;infact,few problemsofthistypecanbesolved satisfactorily byconformal mapping alone. Themethod oftwo-dimensional harmonics (seesection 29)willgenerally prove toleadmost quickly tothe desired results. Inthespecial cases where theboundaryisformed partly by fixed potential values andpartly byfield lines along which the normal component En=d$/dn=0,conformal mapping gives quicksolutions ifthefinalmapupon theupper halfplane corre- sponds either toFig.27-6bwith (27-38) ascomplex potential solution, ortoitscorrelate with equipotential linesand field lines interchanged where (27-38a) gives thecomplex potential, or, finally, toFig.27-13o with (27-62) ascomplex potential solution. Itis,therefore, advisable toutilize symmetries which define at least one field linesothatoneofthese standard solutions canbe applied. 6T.Nakamura, Elektrotechn. Jl.t3,p.6(1939). 368 Two-dimensional Analytic Solutions [Ch. 7 Anexampleofageneral mixed boundary value problemisawire carrying charge Xperunitlength andlocated atPwithin aslotted cylindrical conductor7ofpotential $asshown inFig.28-3;itis desired tofindthefielddistribution around theslot3-5. Since the dielectric constant isthesame within andoutside thecylinder, one canonly stipulate continuity ofthe electric fieldvector across thecircular arc3-5.Mapping bythelinear function produces aone-to-one correspondence between theentire z-and w-planes developing thecylinder intoaflatstripofwidth 2p= (2sin0)/(I cos0), where 26istheslotangle. Thepoint P, location ofthewire,isimaged at i-2cosa+1 \/t/ R Further mapping by 1(-\ =cos"1-tw=pcosf (12) transforms theentire w-plane intoasemi-infinite strip ofwidth 27r,relating thefourquadrantsofthew-plane tothefour semi- infinite strips ofwidthir/2,each asindicated inFig.28-3, sothat theupper side oftheflatstripappears as(IT)<<0,thelower sideas<<TT.Inthef-plane theproblemisnowthat ofan infinitely periodic gridofwires spaced2irapart above aconduct- ingplane, sothatthecomplex potential solution becomes identical with (25-45) with appropriate changeofnotation, namely, P=- -[InsinH(f-fp)-InsinH(f-fp)] (13) ZTTE Here, fp=cos"1(wP/p)withWPfrom (11),and{>isthecon- jugate complex value offP;thefactor J^arises from thefactthat sinfhasperiod TT,whereas theproblem needs period2ir. Ifthe 7Ch.Snow,Scient. Papers Bur. Stand., 21,p.631(1926). Sec. 28]Second andMixed Boundary Value Problems 369 wire islocated atthecenter asanapproximation tocertain photo- electric arrangements,8then wP fP=cos1 (14) Thesame general solution (13)applies tothecasewhere thewire isoutside thecylinder;ifitmoves toz= > fthesolution forthe slotted cylinderisgiveninBateman,01 p.306. u,,,i !r JKu>(+p) u<0 ru<(~P) 'mages urn" f-Plane FIG.28-3 Charged Wire within aSlotted Cylindrical Conductor. Asimilar treatment solves theproblem ofadielectric cylinder carrying oneormore conducting layers onitssurface,9 Fig.284. The linear mapping function 1-zeiot transforms again theentire s-plane intotheentiretu-plane, but 8Th.C.Fry,Am.Math. Monthly, 39,p.199(1932); alsoBell Tel.Mono- graph No.671. 9J.Hodgkinson, Quart. Jl.Math., Oxfordseries, 9,p.5(1938). 370 Two-dimensional Analytic Solutions [Ch.7 now thecircular arccarrying theconducting layer with total charge Xperunitlengthisstretched intothepositive u-axis, the upper halfplaneisfilled with dielectric3,thelower halfplane with dielectric EI.The infinite point z=oismapped at(eja )= ey(a+ir)^ an(jsincephysically thecharged layer sends itsfield lines into z= ,ftwillnow represent achargedlinecarrying (X);itiscertainly apoint ofnonconformality. Afurther map- pingupon af-plane byw=f2reduces thearrangement tothe upper halff-plane withalinecharge (X)ineiabove aconduct- FIG.284Dielectric Cylinder withaConducting Layer. ingplane and infront ofaninfinite plane boundary ofdielectric e2.Theproblem inthef-planeisthusreduced toaconventional image problem. Thereference gives applications totwoconduct- inglayers ofeither likeoropposite sign. MappingofPolygons Bounded byCircular Arcs.Astudy ofmappings obtained byvarious analytic functions discloses transformations from regions bounded bycircular arcsintoregions bounded bystraight lines, sothatfurther transformationsutilizing theSchwarz-Christoffel mapping functions leadtothefinalsolution ofapotential problem. Thus, twoconducting cylinders incon- tactcarrying atotal charge Xperunitlength10andwith radii a and6asinFig.285canbemapped bycomplex inversion w= 10E.P.Adams, Am. Philos. Soc. Proc., Philadelphia, 76,1,p.11(1935). Sec. 28] Polygons Bounded byCircular Arcs 371 2j/zupon theupperhalfw-plane. Since theconductors have the same potential, thefield lines inthez-planewillgotoz=o 7and inthew-plane theywillconverge upon itsorigin0.Itisnecessary only toconsider theupper halfz-plane which ismapped intoa semi-infinite rectangular stripinthew-plane. Further mapping by j..n (16) produces aflatstrip inthef-plane forwhich thesolution was given in(27-38a). With theappropriateshift oftheorigin to jy w-Plane i-H234 * f-Plane FIG.285Two Freely Charged Conducting Cylinders inContact. (1p)/2andobserving that thetotal width ofthe stripis (1+p),onehas p(17) The electric field strength inthez-plane canbeevaluated asin (25-7), except thatnowtwosuccessive mappings areinvolved, sothat Here, dw/dz=(2j/z2 )=j(w2/2)hasbeen introduced inthe latter form butneeds theknowledge ofthecomplete mapping 372 Two-dimensional Analytic Solutions [Ch. 7 function(16).Many other examples arefound inthereference given, such astwoparallel cylinders notincontact butconnected byaconducting sheet along their center plane, andconductors madeupofthree orfour intersecting cylinders. The general theory forthemappingofpolygons bounded by circular arcsupon theupper halfplane was originally developed bySchwarz;11theresulting differential equation is,however, of second orderandnon-linear, sothat rigorous solutions cannot be obtained inapractical manner. Iftheproblem involves theround- ing offofanoriginally sharp corner, onecansolve thefield dis- tribution forthesharp corner bythemapping procedure for straight linepolygons insection 27andthen approximate the rounded corner byaproperly selected equipotential lineclose to jv h-PH 1 FIG.286Mapping ofRounded Corner. thesharp corner with thedesired smallest radius ofcurvature. Thishasbeendone fortherectangular corner opposite aninfinite plane (seeFig.27-9) inRothe etaZ.,D8 p.136,andforavery thin plane winding opposite aninfinite plane which isthesameproblem astheparallel plane condenser (Fig. 27-46), byGrosser.12It hastobeobserved, though, thattheequipotential linegenerally hasashape different from that oftheoriginal electrodes, which maynotmake itasatisfactory approximation. Amuch better procedure forreplacing asharp edgebyacircular cylinder asinFig.28-6 isthesubstitution13of [(w-p)~ya+*(w- q)-ya ] (19) forthevertex factor (w Ua)~yaintheconventional polygonal mapping function (27-10). Intheform (19), thecorresponding 11H.A.Schwarz, Crelle's JL,70,p.105(1869); alsoBateman,cl p.504. 12W.GrSsser, Arch.f.Elektrot., 26,p.211(1931). 13J.Herlitz,referred toinL.Dreyfus, Arch.f.Elektrot., 13,p.131(1923). Sec. 28] Polygons Bounded byCircular Arcs 373 change indirection ofprogression inthez-plane between w<p andw>qisyair,thesame asproduced bythenormal vertex factor (wua)~ya ;however, thechangeisnowgradual rather than abrupt, and ifXisfreely available forchoice, canbemade toapproximate acircle rather closely. Thelocations pandqof theimages ofAandBhave tobefound bythecorrespondence of points inthez-andw-plane, asestablished bythemapping func- tion. Instead of(19)onecould alsousethefactor, [w+\Vw2-!]-* (20) where theendpoints A,Bofthecircular arcarechosen at w=1. 0, 2'Tor * +7Tjo FIG.28-7 Rounded Corner andInfinite Plane. Thesolution fortherounded right-angle corner ofFig.287can befound byform (19)andthistable: _r. . .. Vertex locations la*1'f+^|Q+.+JO2 2'/O-joJ3 ajb5 a-\-b 37T 2 Themapping function follows as /-/-=[Vw-p+XVw-q] (21) where thefactor Vty 1of(27-42a) hasbeen replaced bythe 374 Two-dimensional Analytic Solutions [Ch.7 form (19). Integration leads tothemapping function \/-r~ i lwP z=2Ci\Vw-p-Vptanl+1- I Vp_ +XVw-q-Vqtan"1JW~~9+C2(22) Theconstant Cicanbedetermined byapplying (27-17) topoint 2,observing theequivalent definitions toestablish one-valued branches ofthefunctions, tan 4-0(from right),-In-->+jo for ti>->-0(from left), ^In-y-r-*+j*>+2\w\& Onethus obtains */T (24)+xVg] Atpoint 23onehasw=pandtherefore a-jb=2/XCirv^^-V^tanh"1 x/^-1^]+C2(25) from where, since Ciisreal,onetakesC2=a.Atpoint 24one hasw=qandtherefore a+6=2dvT17"?-V^tan"1^^+^2 (26) The relations (25)and (26)connect p,#,andX,sothatonecan either choose thelocation ofathird point onthew-axis orselect avalue Xwhich gives thebestapproximation toacircular arc. Since thelatter israther difficult, onemight choose q-p=1, q=1+p (27) since that simplifies (25)and(26)appreciably. Figure 28-8 gives theresultant values ofpandXasfunctions of6/a,thesignificant Sec. 28] Polygons Bounded byCircular Arcs 375 geometric ratio ofradius ofcurvature 6todistance oftheparallel planesa.Asbgets smaller, thevalue ofpbecomes large and thusapproaches that ofg,since forb >thecase ofthesharp corner should result,inwhich pandqmerge.Itisofinterest FIG.28-8 Rounded Corner andInfinite Plate. nowtocheck theactual contour described bythemapping function (22). Figure 28-9 shows thecontour forb=a/8andacor- respondingX=0.787; although notstrictly circular, thedeviation from the circular arc isnowhere larger than 0.16.Abetter approximation might beobtained byselecting bytrialanderror aspecific value ofXforagiven ratio b/aandleaving pandqtobe determined from (25)and (26). Thisis,ofcourse, atedious 376 Two-dimensional Analytic Solutions [Ch.7 process, since theentire computation must berepeated foreach ofthevalues ofX. Inthew-plane thecomplex potential solution isgivenby(2726), sothat theelectric field strength follows from (25-7) with (21) and (24)as (dP\=-( }=-j\dz/Vp+\Vq (28) Itsvalue increases along thevertical plane from theuniform field EQ=($2*i)/a somewhat below point 3andreaches amaxi- Fio.28-9 Comparison between Actual Contour andCircular Arc;Geometry Fig.28-7,Mapping Function (28-22). mum value atpoint 3where thecircular arcbegins; itthen decreases again along therounded corner butreaches EQonly somewhere along thehorizontal plane unless theradius b>0.38a. Themaximum value atw=pfollows from (28) (29) Itisplotted inFig.288alsoasafunction ofb/a; inorder tokeep themaximum value to2EQorless,theradius ofcurvature must Sec. 28] Polygons Bounded byCircular Arcs 377 beatleast b^0.15a. Thevalue ofthefieldstrength atpoint 5, theendpoint ofthecircular arc,follows withw=qfrom (28) and isactually and istherefore larger thanEQforb^0.38a. Similar computa- tionshavebeenmade toevaluate theoretically theelectric break- down between electrodes under oil.14 ijy *1 3'*2>, FIG.28-10 Rounded Corner andRectangular Corner. Forarounded corner inarectangular corner asinFig.28-10 themapping function isfound from dz toC1' (w-p)(30) where again (19)wasused toreplace thevertex factor ofthesharp corner. Acomplete discussion ofthemapping foravalue X= V(q+l)/(g 1),chosen because ofmost uniform distribution ofthefieldstrength overthearc,isgivenbyWalker,D1 p.108;a graph shows alsotheactual contour, which issimilar toFig.28-9. Plane gratings withvery large cylindrical wires canbetreated bythesame method.15Take onequarter oftheperiodic strips shown shaded inFig.28-11 asthemapping region; then the mapping derivative dwl [(w(31) 14L.Dreyfus, Arch.f.ElektroL, 13,p.131(1923). "Richmond, Proc. London Math. Soc., Series2,22,p.389(1923); also Smythe,A22 p.98. 378 Two-dimensional Analytic Solutions [Ch.7 transforms itintotheupper halfw-plane. Since therounding only affects thecorner 4butleaves theright angles atpoints 3 and5,thefactors (w+1)and(w 1)appear twice, once forthe existing right angles at3and5andthesecond time inadditive tiv I"kl 1 21 FIG.28-11 Plane Grating ofLarge Cylindrical Wires. combination toreplace (wu)y*inaccordance with (19). Separation of(31)intotwoterms andintegration give 2a -[tanh-'^Y*x)L \v>+p/ +Xtanh"1 (32) where wasdetermined byapplying (26-22) to 7T(1+X) point 1,andC2=wasfound bycorrespondenceofpointz2 w=p,observing /w-b1\H 2tanh"1 I)=In((w 1)+(w+p)\w-fp/ +2[(iy l)(w+p)]^}In(p 1) astheproper definition forone-valuedness. Having chosen the locations ofthree points inthew-planc, onemust findthevalues ofXandpfrom thecorrespondenceofthepoints 3and 4.In this case, thedeviation oftheactual contour from thequarter circle islessthan 0.02b. Thesolution inthew-plane willdepend onthestipulated boundary conditions;ifthecylinders are all isolated andcarry likepotentials, then thecontours l"-2-3 and 5-1' arefield linesandthesolution inthew-planeisgivenbythe complex potential function ofaflatstrip (2738a).Ifthecylinders carry alternatingly positive andnegative charges, then contours Sec. 28]Hydrodynamic Applications; theHodograph 379 2-3and5-1'arefield Ifnes, contour l"-2 isanequipotential line ofzero potential, andthecomplex potential inthew-plane will beanelliptic integral which canbeofthestandard form (27-62)if afurther transformation tothesymmetrical arrangement ofFig. 2713a ismade. Superposition ofauniform fieldandgoodgraphs canbefound inRichmond (loc. cit.). Hydrodynamic Applications ;theHodograph. With table 9-1 itisrelatively simple totranslate allLaplacian potential problems into solutions ofhydrodynamic problems. There are, however, problems which involve "free" surfaces, such asflow through various typesoforifices with jetformation which cannot betreated asconventional boundary value problems. Inideal fluids without effects ofgravity, theBernoulli equation16must hold along eachstream line : p+y&pv2=cons (33) where pisthestatic pressure, pthemass density, and vthetotal velocity atanyonepoint. Forafreesurface itisassumed that pressure pisconstant, usually atmospheric pressure, sothat (33) also requires aconstant velocity. Itispossible tosolve two- dimensional flowproblems involving free surfaces bymeans of conformal mapping ofthevelocity plane, orhodograph plane, rather than theactual geometry; seeLamb,022 p.69;andFrank andMises,06 II,p.417;andRothe etaZ.,D8 p.122. Consider twoplanes PIandP2inFig.2812inclined towards each other withanangle QTTinthez-plane. Theideal fluidissuing from theorificeACwillform ajetofunknown surface butwith constant velocity VQonitssurface. Ifthecomplex potential solution P=$+jEforthez-plane wereknown, theconjugate complex value ofthevelocityvcould befound as v==vx-jvv (34) where thepositive signhasbeenchosen forthepotential gradient inaccordance with prevailing custom inhydrodynamics.Ifone now defines anewcomplex quantity jTa(+jvy} (35) 18Forexample Eshbach: Handbook ofEngineering Fundamentals,p.6-19; John Wiley,NewYork, 1936. 380 Two-dimensional Analytic Solutions [Ch. 7 itwillhave thesame direction butinverse value ofthevelocity at each point inspace, andarepresentationinthe f-plane canat least fixtheboundaries ofthehodograph map. Along thetwo planes PIandP2thevelocity willhave thedirection oftheplanes andvary invalue from vattheorifice pointsAandCtovalue zero atinfinity onaccount ofthedivergenceoftheplanes. Plotting inthef-plane thelocus off,asdefined in(35), gives thedirections OAandOC,with thepoints Aandfcofradial distance I/VQand theinfinite points corresponding to\v\=0.The freesurface of jy {-Plane FIG.28-12 Flow ofIdeal Fluid through Orifice with JetFormation. the jetmust then berepresented bythe circle|f|=l/v ,the infinitely distant pointB'B" ofthez-plane corresponding toB onthenegative f-axis ofthef-plane. Thedirection ofthevelocity vector istowards inthe{"-plane inaccordance with flowfrom infinity towards AandC. Since allendpointsoffasrepresentativeofthevelocity vlie within theshaded areaand itsmirror image below therealaxis, onecanfindthesolution forthevelocity plane, orhodograph plane, byconformal mapping. Thus, bythetransformation f'=fnonecanchange theangle (1 T)TTtoTT,i.e.,compress or expand intoahalfplane fromwhich theinterior ofthecircle l/vQ isexcluded. Theangleyvisintheconventional sense thechange Sec. 28]Hydrodynamic Applications; theHodograph 381 inprogression turning from plane PItoplanePZinthez-plane and isnegativeifintheclockwise sense; thisgivesn=!/(! 7).In order tohaveOCinthef-plane coincide with thepositive -axis, onemust rotate thef-plane by (1+7)->andinorder tomake 2 thesemicircle ofradius1,onemust multiply fbyVQ,sothat as better transformation onehas f"=Lrf *J (36) This leadsnowtotheupper halff"-plane with unit circle excluded. Thefurther transformation f'"=Inf"gives, asshown inconnec- tionwith (26-49) and inFig. 26-7, asemi-infinite strip inthe z"Vplaneofwidth TT.Finally, onecantransform this strip into thecomplete upper half -plane by t=coshf'"=iif''+4/1 (37) IntheJ-plane theproblemisthat ofasink linelocated atB,the terminal ofthe free jet.Thecomplex potential solutionis, therefore, from table 25-1, line1, P=$+jS=-lnt (38) where Qisthetotal quantityofflow (per unitdepth) taken positive.Inparticular, thefree jetsurface isgiven bythepart oftherealr-axis between(1)and(+1). Totransfer thesolution (38)back intothez-planeitisnecessary tointegrate (34),which canbewritten with (35)to(37), (39)ar v VQ Since from (38) onecanintegrate either with respect toPorwith respect to t. Thegeneral integration cannot beperformed, butseveral special caseshavebeen evaluated. 382 Two-dimensional Analytic Solutions [Ch. 7 Thus, onehas for7=orn=la slitinaninfinite plate, the simplest typeoforifice, andtheintegral becomes -\\+C2,for\t\>1(41a) T~=7)]+C2, for\t\<1(416) Toassure one-valuedness, onehastoobserve carefully thesign oft. Thevalue ofCi=Q/2w follows from (40)and (39); thevalue of 2hastobedetermined from correspondence ofpoints in z-and J-planes. For t=+1,zc=jCl(+l)+C2 t=-1, zA=jC,(-l -*)+C2 sothat,ifonelocates theoriginofthez-plane asinFig.2812 midway between Aand C,thevalue ofC2=j(v/2)C\. The asymptotic width ofthejetisdetermined by t=+0,ZB"=oo+C2 t=-0,ZB'=oo-jVd+C2 sothatthecontraction coefficient becomes ZB" ZB'_* Good details ofthesolution with graphs aregiven inLamb,C22 p.90,andBewley,D1 p.146; seealsoFrank andMises,C6 II,p.425. For7= 1orn=J^oneobtains theBorda mouthpiece, a longtube thrust deep intothefluid tank; good details areagain found inLamb,022 p.88;Bewley, p.143;andFrank andMises,C6 p.424; seealsoRothe etaZ.,D8 p.122.Many other examples canbefound inthese references. Since thegeneral map ofthehodographinthef-plane upon the (-plane by(37)isindependent oftheoriginal geometry inthe z-plane, onecanalsostudy different types ofcomplex potential solutions intheJ-plane such ascombination ofsource lines or vortex linesandtransfer these back intothez-plane bymeans of Sec. 29]Two-dimensional Harmonic Functions 383 (39). Inthismanner very interesting solutions forflowpatterns inchannels have been obtained byMigadzu.17Ifashift ofthe origin inthef-planeismade, curved profiles ofchannels result. Forelectrical applications onemight observe thatthesolutions describe thecurrent distribution inthinconducting sheets; the freesurface canbeinterpreted asaboundary along which constant current densityismaintained. 29-TWO-DIMENSIONAL HARMONIC FUNCTION SYSTEMS Though two-dimensional Laplacian potential problems can formally always besolved byconformal mapping andreduction to standard boundary value problems fortheunit circle asindicated insection28,thepractical difficulties become rather greatwhen theboundary conditions involve potential values thatvary along theboundary (still afirstboundary value problem), orinvolve potential values aswellasconditions upon thefieldvector (mixed boundary value problems). Inthese latter problems itisfre- quently simpler toexpress thesolutions interms ofinfinite series of"orthogonal" functions generated bythe differential equa- tions fortheparticular type ofcoordinate system best suited fortheproblem. The firststepwillalways beaseparation ofthe twovariables, say, u,v,andconsequent reduction ofthepartial differential equation totwoordinary differential equations inu andv,respectively; practically anyofthereferences inAppen- dix4,C,a,describes themethod andgives illustrations which will bepresented here inconnection with theindividual coordinate systems. Each oftheordinary differential equations willbeofthesecond order and,ifuisoneofthegeneral variables, willhave theform A*)+rii(u)f(u) +ha(u)+Xn3(i01/(u)=(1) where171, 772,and773arefactors arising from thegeneral coordinate relations andwhere Xisanunknown constant appearing inthe processofseparation ofvariables (seethelater examples); the derivatives aredesignated bytheprimes. Actually, onecan rewrite (1)bymultiplying through withw(u)=exp(/r?i(w)du\ 17A.Migadzu, Technology Report ofTohoku Imperial Univ., Sendai, Japan, 10,No. 4,p.51(1932). 384 Two-dimensional Analytic Solutions [Gh.7 andcombining the firsttwoterms more conveniently as iq(u}+xp(w)] '(w)=(2) Any solution ofthisequation hastosatisfy boundary conditions attheextreme values aandbwhich utakes onwithin theregion ofthestated problem. Assuming homogeneous1boundary condi- tions suchthat &tu=a,aif(a)+a2/'(a)=0) (3) B,tu=b,&!/(&)+62/(&)=OJ then these cover allpossible typesofhomogeneous boundary value problems ofthe firstkind (with az=bz=0),ofthesecond kind (with 0,1=bi=0),and ofthethird kind (with none ofthe coefficients zero); seeKellogg,010 pp.236, 246,314. Ingeneral, the satisfaction ofthehomogeneous boundary conditions ispossible only forselected values oftheparameter X, the characteristic numbers (oreigen values) \aleading tothe characteristic functions (oreigen functions)2 <j>a(u). There exists, however, usually aninfinite sequenceofvalues Xa,adiscrete spectrum, andsince anequation ofthetype (2)hastobesolved forthesecond coordinatev,there willalsobeaninfinite number of corresponding functions $a(v).Each product <t>a(u)\l/ a(v) represents asolution oftheLaplacian differential equation and therefore aharmonic function (seesection 2),sothatthegeneral solution ofthepotential appears intheform *=LAa<t>a(u)*a(v) (4) a=l where thecoefficients Aahave tobedetermined from theaddi- tional boundary conditions pertaining totheboundaries v=c and v=d. Thehomogeneous differential equation (2),together with the homogeneous boundary conditions (3),iscalled aSturm-Liouville 1Homogeneous boundary conditions aredefined inthesamemanner as homogeneous linear differential equations, i.e.,eachterm islinear inthe unknown function oroneofitsderivatives. 2Atabulation ofthe lessusual function systems, associated differential equations, andcharacteristic numbers isgiven inE.Madelung: DieMathema- tischen HilfsmitteldesPhysikers; reprinted byDover Publications, NewYork, 1943. Sec. 29]Two-dimensional Harmonic Functions 385 problem inhonor oftheoriginal investigators, and itleads toa function system </>(u)which isorthogonal, ascanbedemonstrated quite readily. Introducing into (2)successively two ofthe characteristic functions</>aand</>0forf(u)andforming thediffer- ence oftheproducts onecanseparate thiswith thecomplete right-hand sides of(2) into (X-X0)p(u) <hxfr=^M'</>0-V</>a] (5) Integration intheboundary limits aandbofthevariable ugives ontheright-hand side of(5) -*/(a)*a(a)]= which vanishes ifonesubstitutes for a'and</>/thevalues result- ingfrom (3). Thus, since XajX0,theintegral ontheleft-hand side of(5)must vanish p(w)*a(w)*0(u) du=0,OL* (6) a which constitutes thecondition oforthogonality ofthefunction system <t>a(u)withp(u) asweight function. One could, ofcourse, define adifferent function system ha(u)=VrtT) *(u) (7) inwhich casetheweight function isabsorbed inha(u),and(6) reduces to "6 fca(u)^(ii)du=0,<**0 (8) a Thevalue oftheintegrals (6)or(8)fora=0,namely, Jia2M^=Na (9) isaconstant depending onaandcalled thenorm ofthefunction 386 Two-dimensional Analytic Solutions [Ch. 7 system.Ifoneusesthemodified functions or (10)VNaVN a theintegral (9)takes unitvalue; thefunctions (10)formthenan orthonormal system:they arenormalized. Thelatter modification is,ofcourse, notnecessary, but itcanresult insimplification. Good treatments oforthonormal function systems and their applications toboundary value problems arefound3inWebster,016 inByerly,C2inCourant and Hilbert,04inBateman,clin Churchill,03inMurnaghan.013Theadvantage oforthonormal function systemsisthefactthatanyreasonable function G(u) canberepresented within theinterval a^u^6uniquely in terms ofageneralized Fourier series G(u)= ca<t>a(u)=Caha(u) (11) a=l a=\ where Naca=fU G(u)4> a(u)p(u) du-, NaCa=G(u)h a(u)du (12)A=a with theassurance thateverywhere inthis interval the series converges towards G(u), andthatanyfirstncoefficients represent thebestapproximation inthemean toG(u) inthesense ofleast squares. Onecanalsoshow thatanysuchorthonormal function systemiscomplete, i.e.,that there isnofunction forwhich all coefficients vanish andwhich isyetdifferent from zero.Finally, evaluating thedeviation integrals for a(^)andha(u), p(u)\G(u)-caa(ii)l du; a L a=l J ~b du(13) with theaidof(6), (8),and (12),andletting n,oneobtains 8SeealsoL.Bieberbach :Theorie derDifferentialgleichungen; Dover Publi- cations, New York, 1944; originally J.Springer, Berlin, 1930; E.L.Ince: Ordinary Differential Equations, Dover Publications, NewYork, 1944. Sec. 29]Fourier Series inCartesian Coordinates 387 theParseval theorem fdu; #C2=[G(u)]2du (14)a=l i/u=o Returning now tothegeneral solution (4)oftheLaplacian potential problem, onecandemonstrate that this infinite series of harmonic functions represents aconvergent solution ifonecan apply tothefunction series\l/a(v)thesameargument asthat just presented forthefunction series</>a(u).Theactual demonstration forproblems ofdirect physical significance isrelatively simple, since onecanrestrict arguments toessentially analytic functions with only isolatedsingularities aspointed outinsection 27.For details ofexistence andconvergence proofs seeKellogg,010Chapter X;Courant andHilbert,04Vol. II;Frank andMises,C6Vol. I; andEvans.05 Fourier Series inCartesian Coordinates. IntheLaplacian differential equation T~2 T-2= dx2dy2 thevariables canbeseparated bydefining <>(z, y)=X(x)Y(y) asaproduct offunctions ofonlyonevariable each, since (15) becomes XfrYff X"Y+XY"=0 or =-(16)A Y Since inthelastform theleft-hand sidecandepend onlyonxand theright-hand sideonlyonyforanycombination ofxandy whatsoever, nonecancontain thevariable butmustbeaconstant, say,m2 ,sothat dx2' dy2 Forthefunction X(x), comparison with(2)showsw=1,q=0, p=1,X=m2 .Theobvious solutions aresinraz,cosmx;for Y(y) thefunctions sinhmy,coshmyaresolutions, sothat the harmonic function XY=(Cisinmx+C2cosmx)(Disinhmy+D2coshmy) (18) 388 Two-dimensional Analytic Solutions [Ch.7 aswellasanysum ofthese products willsatisfy (15). The selec- tion ofthespectrum ofm-valuesis,however, possible onlyby specifying theboundary conditions. Since (15) contains only thesecond derivatives,itisalways possible toaddterms ofthe type (ki+kzx+k^y)ifrequired bytheconditions oftheprob- lem. Consider therectangular regionO^z^a, ^?/^6shown inFig.29-lawith potential values asindicated there. Forthe variable xbothboundary conditions archomogeneous, ofthe type (3),requiringin(18) JT(0)=X(a)=(19o) Along x=only sinmxvanishes, sothatC2=0;along x=a itrequires sinma=0,m=ia=1, 2,- (196)a Thus, theconventional Fourier sine series constitutes thenatural orthogonal function system forCartesian coordinates infinite regions, withunity weight function, characteristic numbers ma2=Xa,and anorm from (9) rx=a .2/w\ aNa=Isin2 Ix}dx=- tA=o \a / 2(20) which isinthiscasenotdependent ontheordernumber a.It isgenerally notcustomary tonormalize thisFourier series;if desirable,itcanbedonebyusing amplitude factors V2/a. Since misknown by(19), theonehomogeneous boundary condition y(6)=0leads in(18)to Dalsinhmjb+Da2coshmab=0,-zp=cothmab sothat thepotential solution takes theform inaccordance with (4) sinh air(b-y)/a (21)_,,~ . .(x\ si *(x,y)=ZAasin (<w-1 ai \a/ sinh airofa where thecoefficients CaiandDazhavebeenmerged intoAaand thenegative values ofahave been suppressed, since they leave Sec. 29]Fourier Series inCartesian Coordinates 389 thefunction unchanged except forsign. The finalboundary condition requires ,0)=Aasinair-=G(x) i a(22) or,essentially, thattheAabetheregular coefficients ofaFourier sine series representing thegiven function G(x) intheinterval $x^a,or Aa=-I**'*0(x) sinair-dx (22a)at/c=o a inaccordance with (12)and (20). This, ofcourse, requires that G(x)canbesoexpanded, demonstrating that thisboundary value y=6 *= FIG.291Potential Solution inaRectangle. problem canbesolved inallcaseswhere G(x) permits representa- tion interms ofaFourier sine series. Thisproblem isusedby Churchill,03 p.137, toillustrate theproof ofuniqueness ofthe solution; asaheatflowproblem withtheidentical boundary condi- tions intemperatureitisalsosolved byChurchill,03 p.114,and byByerly,02 p.102. Thoughthisproblem appears tobearatherspecial onebecause ofthesimple boundary conditions, anyarbitrary potential distri- bution forexample along x=acanbetreated inthesamemanner, namely, assuming $>(a, y)=H(y)and*=onallother sides; thesolution forsimultaneously assuming thiscondition andG(x) along y=issimply thesum ofthetwoindependently found solutions according totheprinciple ofsuperposition valid forall linear problems. 390 Two-dimensional Analytic Solutions [Ch.7 Changing theboundary conditions tothemixed kind ofFig. 29-16 leaves thesolution X(x) with theidentical conditions (19a) andwith thesame series ofcharacteristic numbers (196)- The boundary condition ony=bisagain homogeneous butoftype Y'(b)=0,sothatwith (18) maDalcoshmab+maDa2sinhmjb=0,-^=tanhmab Dot andthepotential solution becomes ,,xAA.(Acosh air(b-y)/a $(z,y)=LAasin Iair- } (23)a=l \a/ cosairb/a Aty=thesameboundary condition asin(22) results. Physi- cally, Fig.29-16 canrepresent thestator ofanelectrical machine developed intoaplane structure ofheight 6,with pole pitch a andneutral zones atx=andx=aifG(x)isasymmetrical distribution ofthemagnet ostatic potential 7along theairgap. Again, theprinciple ofsuperposition canbeapplied inorder to satisfy more complicated boundary conditions. Thus Zworykin,032 p.369, applies this solution toaplane section oftheelectron multiplier, with oneconstant potential ontwojoining sides of therectangle, andwith adifferent potential ontheother pair of joiningsides. Itis,ofcourse, alsopossible tojoinseveral regions, within each ofwhich thepotential solution hasbeenfound ingeneral terms, byassuring continuityofthe electric potential values (orthe tangential components ofE)andthenormal components ofD across theboundaries. Intwo-dimensional magnetic problems, themagnetic vector potential reduces toasingle component paral- leltothecurrent flow (seesection 6)andintheCartesian system satisfies theLaplacian equation inregions free ofcurrent andthe Poisson differential equation inregions with current flow. Inthe latter case, foruniform current density, thesolution willbethe sum oftheLaplacian solution and ofaparticular integral which normally canbeobtained byinspection. Attheboundaries it isthen required that theconditions (6-20) or(6-7)and (6-10) besatisfied. Many applications torectangular current regions have beenmade inconnection with leakage computations on Sec. 29]Fourier Series inCartesian Coordinates 391 transformer windings,4onconductors inslots ofelectrical ma- chines,5andonpolewindings located intheinterpole space.6 Figure 29lacanalsorepresent cooling ofafinwith fixedtem- perature TOalong y=andtheboundary conditions dTk+fT=(24)an along x=0,x=a,andy=b,ifnisthenormal direction onany ofthese surfaces, kthethermal conductivity, and/theheat transfer coefficient forunit area. Actually, because ofsymmetry, onecanstatedT/dx=Oatz=a/2asamore convenient boundary condition replacing (24) atx=a.Take again thegeneral form (18); theconditions which X(x}must satisfy are atx=0, fcX'(O)+/X(0)==mkd+fC2 a__,a\ a.a atx=-yX I- )==Cicosm-C2sinm- 2 \2/ 2 2 fromwhich C2=-mCivtanm=--^- (25)/ 2 mk Thesecond relation defines thecharacteristic numbers maas solutions ofatranscendental equation, which isobtained bestby graphical construction, finding theintersections ofatangent graph with thehyperbola ontheright-hand side oftanq=(af/2kq). With (25)onehasthen Xa(x)=Cal(sinmax+cotma-cosmax J _cosm a(s-q/2)^i- :-m- smraaa/2 Inspite ofthefactthatthemavalues arenotharmonically related asintheconventional Fourier series, thefunction system (26) isorthogonal, ascanbeshown byapplying either (6)with unit 4W.Rogowski, Mitt. Forsch. V.D.I., No.71(1909); Bewley,D1 p.73;E. Roth, Revue gen.deI'elec., 23,p.773(1928); E.RothandG.Kouskoff, Revue gen.deI'tlec., 23,p.1061 (1928); Hague,B44p.302. 6E.Roth, Revue gen.deUtlec., 22,p.417(1927) and24,pp.137and179 (1928); Bewley,Dl p.81;Hague,844 p.314. 6A.R.Stevenson andR.H.Park, Gen. Elec. Rev.t31,p.101(1928); Hague,844 p.310. 392 Two-dimensional Analytic Solutions [Ch. 7 weight function, or(8),andbyobserving thesecond relation (25) intheresult; onehas / a\ / a\ cosm Ix- ]cosn[x- ]dx o \ 2/ \ 2/ (0forra 7*m ") / \ I a/ i+sinma\^ forn=mf(27)2\ ma/ J where nandmaretwovalues ofma.Thehomogeneous condition (24)aty=bgives theresult JfcY'(b)+fY(b)=Di(mk coshmb+/sinh mb) +D2(mksinhmb+fcoshmb)= fromwhich theratioDi/D 2isfound. Thetemperature aty= isthen subject tothefinalboundary condition " Acosm a(x-a/2)=^ whereAa=CalDa2asbefore. Theexpansion ofTintothenon- conventional Fourier series follows exactly (12), sothat 2T.2aAa= snrma-maNa 2 The finalform ofthetemperature distribution7is akcoshma(b y) -\-fsinhma(b y) o2-r . ., a=1 ma/ccoshmab-+-/sinhmao sinraaa/2cosma(x a/2) ; (28)maa+sinmaa Onaccount oftheboundary conditions (24), thisproblem could notbesolved byconformal mapping inanysimpler manner. Astheheight oftherectangle6 >inFig.29-la,Di/D 2> (1),sothatthesolution (21)goesover into *(*,y)=Aae-a*vfasinair-(29) =i a Thisform ofsolution hasbeen used8tocompute themagnetic 7Bateman,01p.213,where cosh(sroi/)isamisprint ofcos(smy)inthefinal solution. 8R.Rudenberg, E.T.Z., 27,p.109(1906); alsoOllendorff,Aiapp.227,235. Sec. 29]Fourier Integral inCartesian Coordinates 393 field distribution inarmatures ofinfinite height, joining themag- netostatic potential aty=tothat oftheairgapalong which single- ormultiphase current layers areassumed distributed. The needed excitation canbefound, aswellasatheoretical shape of thepoleform insynchronousjnachines. Similarly canbeevaluated theleakage field distribution surrounding transformer coils9or extending intothetransformer core. Fourier Integral inCartesian Coordinates.If,inFig. 29la,thelength oftherectangle a >oo fitaffects thecharacteris- ticnumbers; indeed,ifthesemi-infinite stripisconsidered, X(Q)= stillinsures 2=in(18),butnoother condition isavailable, since sinmxremains finite forx >oo.Thehomogeneous condi- tion7(6)=givesfrom (18) = cothmbD2 sothattheproduct (18)becomes, withCiD 2replaced byA, ____ .. sinhm(b y)XY=Asinmx- . , (30)sinmb Here, anyvalue ofmispossible; instead ofadiscrete spectrum of characteristic numbers onehasnowacontinuous spectrum. The boundary condition along y=requires thus therepresentation ofG(x) over theinfinite interval ^x^oointerms ofsinmx which ispossible withuniqueness bymeans oftheFourierintegral10 ifG(x)isbounded, atleast sectionally continuous, and if exists. Thus X=o_o[U(m) sinmx+W(m) cosmx]dm (31) 9W.Rogowaki, Mitt. Forsch. V.D.I., No.71(1909); Ollendorff,A18 p.257; A.R.Stevenson, Gen. Elec.Rev., 29,p.797(1926); Bewley,DLp.73. 10Fordetails seeparticularly H.B.Carslaw: Introduction totheTheory of Fourier Series andIntegrals; Macmillan, London, 1921; E.T.Whittaker and G.N.Watson: Modern Analysis; Cambridge University Press, 1935; N. Wiener: TheFourier Integral andCertain ofitsApplications; Cambridge University Press, 1933; E.C.Titchmarsh: Introduction totheTheory ofFourier Integrals; Oxford University Press, 1937. Forsimpler accounts refer toalmost anybook inAppendix 4,C,a. 394 Two-dimensional Analytic Solutions [Ch. 7 where thecoefficient functions U(m) andW(m) aregiven inturn bytherelations 1r* U(m)=-I G(x) sinmxdx, TTi/r=M W(m)=-C" G(x)cosmxdx (32) 7Ti/c=-eo quite analogous totheFourier series (22)andjustaspecial case oftheorthogonal function systems (11)and (12). Inparticular, U(m)istheFourier coefficient ofanoddfunction inx,andW(m) that ofaneven function inxtand, inturn,U(m) itself isanodd function inmandW(m) aneven function. Theparticular form (30)implies anoddfunction ofxwithW(m)=which might as wellbeassumed, since x< isoutside theregion oftheproblem. Comparisonof(30)fory=with (31)shows because ofthe uniquenessthatA=U(m), andthat thecomplete solution for thepotentialfunction asthemost general superposition ofall possiblesolutions musthave theform *(z, )=t"U(m) sinmx^"f~^dm (33)Jm=0 smhmo withU(m) from (32). The direct evaluation ofthis integral might bepossibleifU(m)isactually known. Onemight also introduce (32)withachangeofvariable toxdirectly into (33) andinterchange theorder ofintegration 2r^, t^ ,/r* ,sinhm(b y)-2r^, t^ ,/r* ,sinhm(b y) , $(x,v)=-IG(x)dxIsinrazsmraz-- --dm 7rt/z'=o Jm=Q smhrafr Here thelower limit inxfhasbeen replaced byzero, anda factor 2applied because oftheassumed oddcharacter ofG(x'). Theinner integral canthenbewritten intheform (seeByerly,C2 p.80,etc.). X"sinhq\m_-cos03mdm= =osinhqzm r sin(Tql/q2) 2q2cosh(TTq3/q2)+cos(TTqi/q 2) where qi=by,q2=b,q$=(x x'). This yields forthe potential_,1.mi G^dx' Sec. 29]Fourier Integral inCartesian Coordinates 395 where theoriginal twoterms were contracted intooneonthe basis thatG(xf )isassumed odd. Neither (33)nor(34)isgenerally ofgreat practical value; both constitute formal solutions which areamenable tonumerical ormachine computations. Anumber ofexamples arefound inByerly02 ;some ofthem canbehandled more simply byconformal mapping (section 26). Forpractical applications,itisadvantageous tousethecomplex form oftheFourier integral relationships G(z)=J-Cp(m) ejmxdm (36) 2-7Tt/m= GO where thecoefficient function F(m)=r~+ G(x )e-imxdx(36a) \JX= 00 isdefinitely complex. Actually, since G(x)isarealfunction, one canexpande3mxin(36a)andcompare thisrelation with (32) /+* /+ F(m)=IG(x) cosmxdx jIG(x) sinmxdx / 00 t/ 03 )] (37) finding F(m) simply acomplex combination oftherealFourier coefficients. Introducing thisexpression forF(m) into (36)gives asrealpart directly (31); theimaginary parts[W(m) sinmx U(m) cosmx}vanish when integrated inthelimits( )to (+00) because both ofthese areoddfunctions ofm,asapparent from (31)and (32). Theform (37) alsoindicates thatthereal partofF(m) must beaneven function ofmandtheimaginary partanoddfunction, sothat onecanfurthermore state: the absolute value|F(m)|isalways aneven function ofm,andthe argument tan"1[ImF(ra)/Re F(m)] isalways anoddfunction ofm. Thecomplex form oftheFourier integral hastheadvantage thatextensive tables11areavailablelisting thedualFourier integral 11Particularly G.A.Campbell andR.M.Foster: Fourier Integrals for Practical Applications; D.VanNostrand, New York, 1947; firstpublished asMonograph B-584, BellTelephone Laboratories, NewYork, 1931. These tables willbereferred toasC.-F. tables. 396 Two-dimensional Analytic Solutions [Ch. 7 coefficients incorresponding columns. From (36a)itisobvious thatF(m)willactually beafunction ofjm,since this istheonly parameterintheintegrand; theC.-F. tables (abbreviation for reference,loc.cit.)introduce therefore jm=pasanewvariable and listF(m) asafunction ofp.Infact, theevaluation ofmost oftheintegrals (36)issimplified bycompletely changing tothe variable p,thus G(x)=-. F(p)e**dp (38) 2irjJ-j* Inthisform, theintegral canbetreated either asthat ofareal variable along theimaginary axis or,byconsidering pasacom- plex variable, asanintegral inthecomplex p-plane. Thelatter in- terpretation leads directly intothetheory ofanalytic functions andpermits extensive useoftheCauchy integral theorem (2614). Assume thatF(p)isanalytic intheentire p-plane except ata finite orpossibly countably infinite12number ofpoints where it haspoles ofthe first order; then itcanberepresented asafinite orinfinite sum oflinear fractions (39) where thepaarethelocations ofthepoles. The basis ofthis expansionisGauss's fundamental theorem ofalgebraifF(p)is arational fraction,13orWeiers trass' product representation of trigonometric andhyperbolic functions; seeany ofthereferences, Appendix 4,D,b.ThevaluesRacanbeobtained either bydirect comparisonofcoefficients onboth sides of(39), orusually in simpler formbywriting F(p) asaproper fraction ofpositive power functions N(p)/D(p) andthen14 12Aseries ofpoints spacedatdefinite, known finite intervals, eventhough infinite innumber,iscalled countablyinfinite. 18Seeanycollege textbook onalgebra. 14SeeanybookonLaplace transforms; forexample M.F.Gardner and J.L.Barnes: Transients inLinear Systems; John Wiley, NewYork, 1942, Vol. I,p.155. Sec. 29]Fourier Integral inCartesian Coordinates 397 Under theassumed conditions thevalue oftheintegral (38)canbe shown toremain unchangedifthepathisclosed over theright- hand infinitely largesemicircle (with reversed direction) for x<0,andovertheleftinfinitely large semicircle forx>0;each ofthese closed integrals canfurther becontracted intovery small circles surrounding each pole asin(26-13), andtheresult isa sum ofresidues (with proper sign)ofthetype (26-14), namely, thevalue oftheintegrandin(38) exclusive oftheroot factor (P-Pa}taken atp=pa\orwith (39)and(40) 0(x) \dp(N(p)\ iD^e L IfF(p) possesses poles oforder higher than the first, themodifica- tions arethose leading totheforms (26-15) ateachsuch pole. LI I *2 FIG.292Potential Solution inInfinite Strip. Asanexample consider Fig.292,withtwopotentials along the x-axis separated byaninfinitesimal gapattheorigin, andwith d$/dy=ony=b.The basic solution oftheLaplaciandif- ferential equationis(18),andtherefore ony=b Y'(b)==mDicoshmb+mD2sinhmb,j^=tanhmb This givestheproduct solution coshmb where thex-variation isassumed inthecomplex form inantici- 398 Two-dimensional Analytic Solutions [Ch. 7 pation oftheuse ofintegral (36). Since coshmb=cosjmb= cospb,onecanreadily write thepotential function asaFourier integralintheform of(38), cospb Inorder todetermine A(p) onemust compare $(z, 0)with the given boundary values. Inturn, thisrequires arepresentation ofthepotential distribution along x=asaFourier integral. One can, ofcourse, always add <tiasageneral constant and define thepotential aszeroalong x<0,asabrupt step ofvalue ($1 $2)atx=0,andconstant atthisvalue forx>0.Thus G(x)=*x-(*!-*a)S_i(z) (43) where S_i(x)istheunitstep oftheC.-F. tables inpair415with thecoefficient F(p)=I/p. Therefore, aty=0,thepotential musthave theform G(x)=*(z,0)=*!-(*!-*a)-^-.r+'i ZTTJJ-] p'dp andcomparison with (42)aty=indicates theneed oftheaddi- tiveconstant $1aswellasA(p)=l/p.The finalsolutionis,then, (44)cospb using thesymbol 9Hor"mate" forthecumbersome integral nota- tion. The Fourier "mate" canfortunately befound inC.-F. tables aspair618,giving inclosed form withG(x)from (43). One easily verifies thisas complete solutionsatisfyingallboundary conditions. The C.-F. tables contain several similar forms intable II,section 2. Ifthepotential distribution along thex-axis isgiven as$1for x<and$ie~yxforx>0,thenonecanwrite 0(x) p(p Sec. 29] Circular Harmonics 399 using coefficient pair448 oftheC.-F. tables. The potential solution becomes now which cannot befound inthetables inclosed form. However, thefunction hasonlyfirst-order poles located atp=0,p=7, andp=(2v-l)r/2b with v=1,2, ,sothat (41) applies. Thesum oftheresidues atthepositive realpoles taken withnega- tivesigntomaintain positive sense ofintegration constitutes then thesolution forx<0,whereas thesum oftheresidues atthe negative realpolesandatp=constitutes thesolution forx>0. Forboundary conditions which prescribe potentials over finite sections oftheboundary andtangential flowovertheremainder, conformal mapping inaccordance with section 27canbeemployed totransform thegeometryoftheproblem sothat theboundary conditions canbemore readily satisfied. Circular Harmonics. TheLaplacian differential equation in polar coordinates permits direct separation ofvariables bydefining 3>(p, </>)= R(p)F(0)asaproductoffunctions ofonlyonevariable each. One obtains 3" and, dividing byRF,onecanargue asfor(16), sothat P2R"+PR'-m2R=0,F"+m2F=(48) with thegeneral solutions R=C1Pm+C2p~m ,F=DIsinm0+D2cosm</> (49) The selection ofthespectrumofm-values isagain only possible byspecifying theboundary conditions. Byinspection of(46) it isseenthatonecanaddtoanyproduct RF,orsum ofsuchprod- ucts,terms ofthetype ki+k2<l>+/c3Inp+ fc4Inp (50) 400 Two-dimensional Analytic Solutions [Ch. 7 asspecial solutions ifrequired; these lastterms correspond to m=0. Forinteger values ofm,thesolutions (49) arecalled circular harmonics; forpconstant, thefunctions Fm(<t>)represent thecon- ventional Fourier series foracircle andpermit expansion of arbitrarily given bounded functions ofphysical significance inthe samemanner as(22)inaplane strip. Forexample, thesolution oftheLaplacian potential within unit circle forgiven potential values *(^) along unit circle isfrom (49) <S>(p, </>)=ki+ pm (amsinm<+bmcosmtf) (51) 771=1 withC2=toavoid thesingularity atp=andwithC\Di andCiD 2contracted intoamand bm,respectively; these latter coefficients aredetermined inconventional manner astheFourier coefficients along unitcircle, 1/>27r\f*2vam=-/$(^) sinm\t/d$,bm=-I$(^) cosm\l/ d\l/ (52) unw Introducing these expressions into (51), onecanestablish the identities 1+2pmcosm(</>- iA)=Re[l+2(Pey(*"^)m ] =Rep= |_1 pe3^~*J1-f-p 2pcos(0^) andthusdemonstrate that (51)with (52)represents actually the Poisson integral solution (28-1)inexpanded form. Acylindrical conductor covered with adielectric layer ofcon- stant 2and offinite thickness surrounded byairasinFig.29-3 might beexposed toauniform electric fieldE .The potential corresponding toEQis Themodification ofthepotential distribution 3>iinairbythe presenceofthedielectric 2isgiveningeneral formby(49)and soisthepotential $2within e2,namely, cosm0, *2=i;(a2mpm+&2TOp-m )cosm0 (53) Sec. 29] Circular Harmonics 401 Since theeffect ofthedielectric must vanish atinfinity, onlynega- tivepowers inphavebeen retained in$lfandinboth cases the sineterms have beendropped because oftheevensymmetry in $o-Theboundary conditions thathave tobesatisfied are Fio.29-3 Cylindrical Conductor Covered with Dielectric Layer. From theconditions atp=bitisobvious that onlyterms for m=1canoccur asdenned by*o ;thesolution isthen bf 'kp (54) with k=[e2(&2+a2 )+i(&2-a2 )].This case istreated by Smythe/22 p.65;itreduces fora=atonce toasolid dielectric cylinderinauniform field asin(2124),asolution obtained bythe method ofimages. Inaquite similar manner could betreated the cylindrical dielectric shellwith dielectric EIinthecorep<a,except that theboundary conditions atp=awould belikethose at p=b.Forthemagnetic cylindrical shell thesolution isfound in Moullin,D48 p.198;thesolution forthemagnetic solid cylinder obtained fora=isthesame as(22-20). Slightly non-circular coaxial cables havebeen treated byassum- 402 Two-dimensional Analytic Solutions [Ch. 7 ingtheouter conductor boundary asaperiodic function ofangle 6(^)andcomputing theeffect upon capacitance atleast infirst approximation.15Themagnetic field distribution inunsaturated stators ofelectrical machines orintheairspace with rotorremoved hasbeen evaluated bythegeneral solutions (49), assuming a sinusoidal distribution oftheradial magneticfield along theair gapboundary.16Smythe,A22 p.275, also gives theaxialcom- = FIG.29-4 Single Line Current. ponent ofthevector potential produced byacurrent distribution inathin cylindrical shell, thecurrent flowing only parallel tothe cylinder axis. Themagnetic vector potential ofasingle linecurrent atp , 0ofrom theoriginofacoordinate system asshown inFig.29*4 is givenby(13-23) as Az=- IInR &1T =-7In[p2+p2-2PPOcos(0-(55) where thelastform takes asreference theorigin instead ofthe current location. Onecanwrite thelogarithmand alsoas Po2 |~1+(-Y-2-5-cos(*-o)l=p2 (1-<?)(!-g)L \PO/ Po J where q=(P/PO) exp [j(<f> <fo)]>andqistheconjugate complex value. Thus, in(55), In[p2+p2-2pPocos(0- =2Inpo+In(1-g)+In(1-q) 16P.Parzen,Jl.Appl. Phys., 18,p.774(1947). 16M.Schenkel, Elektrot. undMasch., 27,p.201(1909); alsoRichter,B I,p.162. Sec. 29] Circular Harmonics 403 andsince\q\<1,onecanexpand thelasttwologarithmic terms intoapower series, add likepowers ofthetwoconjugate complex numbers, andobtain Az=- -I[inpo--(-\m cosm(*-)1(56o) Z7TL m\PO/ J which isvalid forp^po,andbytheappropriate modification Az=- I[inp--C^Vcosm(*-*,)!(5) 2irL wi\P/ J which isvalid forp^p.With these forms themagnetic fields oflinecurrents canbetreated ifcylindrical iron shells orsheaths ,y X FIG.29-5 Shielding Effect ofMagnetic Cylindrical Shell. arepresent, since outside ofcurrent regions thesingle vector potential component Azinthetwo-dimensional polar coordinates also satisfies theLaplacian differential equation (46), ascanbe verified fromAppendix 3,(37). Theuseofthescalar magnetic potential, asinHague,B44 p.120, requires somewhat more care because ofthenecessary potential barrier (seesection 6). Onecanfindthemagnetic shielding effect ofacylindrical sheath within which two parallel wires arelocated asindicated inFig. 295bysuperimposing forregion1thetwolinecurrent potentials from (56)andaLaplacian potential solution ofthetype (51)with sineterms omitted because (56) willnotcontain them. Inusing (56a) or(566), onemust choose for <thevalues fa=TTand 404 Two-dimensional Analytic Solutions [Ch.7 fa=0,respectively, forthetwoconductors, andalso PI=P2=c instead ofPQ.Forregion 2,thevector potential without sources hasthecharacter ofthecomplete right-hand solution in(53), whereas inregion 3,outside thesheath, onewould have the left- handform of(53). Thecontinuity conditions atbothboundaries p=aandp=bapply tothenormal component B=1z " Pa* andthetangential component 1*A.HA=*Mdp The final result forthefieldjustoutside thesheath atp=bisthen Ml)2-(M2-Ml)2(l 771=1 /c\2m-l X(- 1 sin(2m-1)0 (57) rIE (M2+Ml)2-(M2-Ml)2 (7 X(7)cos(2m-1)0 W Obviously, theshieldingwillbemost effective when a :6and c^6;thepermeability influences the field only linearly. For brieftreatments seeSmythe,A22 p.284;ZworykinetaZ.,B32 p.482; andMoullin,348 p.209.Asimilar treatment forlinecurrents in acylindrical airspace between asolid inner magnetic cylinder and anouter magnetic cylindrical shellhasbeen used extensively by HagueB44tosimulate the field conditions inairgaps ofelec- trical machines andtocompute force actions onsingle coilsand windings. Elliptic Cylinder Coordinates. Asshown in(25-55), the inverse hyperbolic ortrigonometric sinefunction ofthecomplex variable zdefines anorthogonal ellipticfieldgeometry. Onecan therefore actually usethese functions todefineelliptic cylinder coordinates;ithasbeen customary, however, touserather the Sec. 29] Elliptic Cylinder Coordinates 405 analytic function z=fcosh fforthispurpose, where f={+jv and x=fcosh cos17, y=fsinh sinrj (58) oralso COSTJ=(59) which aretheequationsoftheconfocal ellipses andhyperbolas in terms ofthedistances from thetwo fociFIandF2inFig.296. Inthesamemanner asinsection 26onecandemonstrate the FIG,29-6 Elliptic Cylinder Coordinates. transformation oftheLaplacian potential equation from the x-y-coordinate system totheorthogonal {--^-coordinate system and find 1 dx2dy2~ /2(cosh2- Since thismust hold foranyvalue ofandTJwhatsoever andthe first factor cannot vanish, oneobtains again theLaplacian dif- ferential equationinterms of,t\andthuscansolve itinjustthe samemanner as(18) forxandyintheCartesian system. Thesimplest problemisthat oftwoconfocal elliptic cylinders of constant potentials.Ifthemajor andminor axes ofonecylinder areaiand bi,those ofthesecond a2andb2,then/=Vai2 b\2 defines thefocal length, which must bethesame forboth. The surfaces ofthecylinders aredefined from (59) ascoshj=ai//, cosh 2=a2/f,asonefinds fortheapexAofthemajor axis; orby fi=In(ai+bi)/f, 2=In(a2+b2)/f,asonefindsfrom (58) 406 Two-dimensional Analytic Solutions [Ch. 7 forthepointsAandB.Because ofthesimple boundary condi- tion,namely, $=$1onfiand$=$2on 2,thesolution ofthe problemis *=*!-(*!- *2)f1^-(61) 2 1 analogous to(14-1) fortheparallel plate condenser. The field vector hasonlyacomponent inthe{-direction ;itsvaluemust also befound bymeans ofthetransformation equations (58)andcan bebestexpressed as 7j_ ..,,,sinceEx=--=------ but =0; similarly for dx dfdx di\dx drj Ey.From (58)onehas =/sinh cosrj, =/cosh sinrj (62) df 6^ sothatwith (61) Et=*l~' [cosh2-cos2 ,,]-* (63) Thecharge density oncylinder fiwith potential $1is where77isvariable. Thetotal chargeistheintegral of<TIoverthe circumference oftheellipse andpermits thedefinition ofcapaci- tance perunitdepth forwhich theform isidentical with (26-45), namely, Iftheinner elliptic cylinder reduces toaflatstrip ofwidth2/, then 1=andthecharge density results from (64),with (58) foreachside, as =" /2sin77" V/2-x2In[(oa+ba)//I Sec. 29] Bipolar Coordinates 407 Itobviously becomes infinitely large at77=and77= TT,thetwo ends,andmusthave thesame signonupper andlower surface. Foranarbitrary potential distribution onone ofthe elliptic cylinders, aninfinite series oftheFourier typeinfunctions e"* sinmt\ispossible, asin(29). For details seeBateman,cl p.257, where alsoanapplicationisgiven toalinecharge paralleling an elliptic cylinder. Adielectric el- liptic cylinder exposed toauni- form electric field istreated in Ollendorff,A18 p.182. Itistoberecognized that the use"ofthese coordinates leads to more convenient expressionsfor thesolutions andpermits formula- tion oftheboundary conditions interms ofsimple parameters; theinterpretation ofthe field structure is,however, usually against aCartesian coordinate system asbackground unless one hasprepared anelliptical orthogonal netonwhich hecanread directly-and^-values. Parabolic Cylinder Coordinates. Parabolic cylinder co- ordinates (Stratton,A23 p.54,andBateman,cl p.486) arebased upon thegeometry defined bytheanalytic function z=2f2 asin(25-57), where f=+jyandFIG. 29-7. Parabolic Cylinder Coordinates. =V2p cos =Vx2+y2+x 17= oralso=Vx2+y2-x y=Cn(65) (66) Thus, constant values oforofrjlead tofamilies oforthogonal parabolic cylinders asinFig.29-7. Bipolar Coordinates. Bipolar coordinates (Stratton,A23 p.55, andBateman,01 p.260) arebased ontheanalytic function z=jccot(f/2) asin(26-53), where =0212 In (67) 408 Two-dimensional Analytic Solutions [Ch.7 Referring toFig. 12-5,=consand77=cons arethetwofam- ilies oforthogonalcircles which represent thepotential solution fortwoparallel wires offinite radii.17 PROBLEMS 1.Ncoplanar positively charged quasi lines, eachwith charge density X anddiameter d,areuniformly spaced adistance 2capart andarelocated a height habove agrounded conducting plane. Find thecapacitance ofthis finite grid. LetN > panddemonstrate thatoneobtains thesolution for the"Maxwell grating." 2.Inthetriode ofFig.255findthedistribution oftheradial electric field along thegrid circle|z|=Rgbetween two grid wires. AssumeN 20, Npg/RQ=0.1,Ra/Rg=4,Rg/R c=2and (a)Va=100volts,Vg=8volts; (6)Va=100volts,Va=-8volts. Observe that zaN=RN . 3.Find themutual capacitance coefficients foratetrode withtwo grids whose individual gridwires arelying along thesame radius vectors. 4.Find themutual capacitance coefficients foratetrode withtwogridsif theindividual wires oftheonegrid arelyingmidway between those ofthe other gridand (a)along thesame circle, (b)along two different circles. 5.IfthefieldvectorEonthecathode surface isdirected away from the cathode, noelectrons canleave. Find theconditions forthiscut-off ofemis- sionfrom partsofthecathode surface forthetriode inproblem 2interms of grid-cathode spacing. 6.Thegeometry inFig.255might represent athincopper sheet withsmall circular perforations andwith radial current flowfrom anelectrode forming theouter circular boundary toanother concentric electrode forming theinner one. Find thetotal resistance tocurrent flow iftheconductivity is7and thesmall thickness t.Assume uniform current densities attheelectrodes. 7.Sixwires areuniformly arranged onacircle toform acylindrical grid. Find theelectrostatic field distribution ifsuccessive wires alternatingly carry potentials V/2. Assume thewire radii small compared withspacing, but finite. 8.Inathree-phase four-wire transmission system, thethree-phase wires arearranged inaplane parallel toground, withmutual spacing 2b.The ground wire islocated aheight habove thecenter phase wire. Find the mutual linkages forunbalanced current flowwith currents /i, /i/2, /i/3 inthephase wires. 9.Athinrectangular copper sheet ofarea2oX25hascircular perforations along itscenter lineparalleltothelonger side 2a.Twoheavy electrodes are applied along thesides 2awith apotential difference V.Find thecurrent, iftheNperforations have equal spacing, and iftheoutermost oneshave their centers a/Nfrom theshorter sides ofthesheet. 10.Averylongandthincopper sheet ofwidth 2ahasapplied twoelectrodes ofsmall circular cross sections inalinetransverse tothesheet andatdis- tances a/2from theedges. Find theresistance forasmall thickness t. 17Foraninteresting application toatwo-wire problem seeG.Mie,Ann. d. Physik, 2,p.201(1900). Problems 409 11.Themagnetic sheets (laminations) foranelectromagnet areofrec- tangular shape andcarry 2bolt holes across thenarrow side ofwidth 2a. Find themagnetic reluctance iftheholes arespaced a/2from theedges ofthe sheet and ifthelength ofthesheet is10a, itssmall thickness t. 12.Ifthecontrol gridwires inavacuum tube arelocated very close tothe cathode, onecantreat theelectric field distribution asatwo-dimensional plane problem. Assume thegridwires asinFig.256awithaspacing h<a andcarrying anegative linecharge \g;assume theanode plane atadistance bfrom thecathode andcarrying apositive potential Vawith respect tothe cathode. Find the field strength Ealong thecathode surface. Find the mutual capacitancecoefficients. 13.Asingle long wire carrying current 7islocated between two parallel idealmagnetic boundary planes atdistance 2aandofpotentials 171andIF2- Find thevariation ofthemagnetic fluxdensity Balong thecloser surface. Find thevariation ofthemaximum value ofBasthewireapproaches oneof thesurfaces. 1'4.Discuss thepossiblefield solutions rendered bythefunction 15.Discuss thepossiblefield solutions rendered bythefunction In (tanh^\- 16.Nparallel long wires each carrying current 7arelocated inaplane paralleltotwoidealmagnetic boundary planes atdistance 2aandofpotentials yiandy2-Find thevariation ofthemagnetic fluxdensity Balong the closer boundary surface ifthespacing between thewires isa/4. 17.Athincoaxial annular ring ofcopperisslitalong oneradius andheavy electrodes areapplied there, impressing apotential difference Vbetween the twooppositefaces ofthat radius. Find thecurrent distribution. Find the resistance ofthesheet forasmall thickness t. 18.InFig.26-5 findthecurrent distribution along thediameter 1-3. z+a 19.Discuss theconformal mapping obtained bythefunction w=In- za 20.Discuss theconformal mapping obtained bythefunction 21.Discuss theconformal mapping obtained byw=tana 22.Athinring ofcopper sheet isbounded bytwoeccentric circles. Find theresistance iftwocircular electrodes ofsmall area areapplied with centers onthelarger circle attheends ofthediameter bisecting thering. 23.Consider along cylindrical duct ofsemicircular cross section with radius R]within theductextend twoparallel wires ofsmall radii pforming a transmission system. Find thecapacitance ofthesystemifthewires are located (a)symmetrical with respect tothecenter plane oftheduct, atR/2 from itandclose totheceiling; (6)above each other inaplane normal to theplane base oftheduct. 410 Two-dimensional Analytic Solutions [Ch. 7 24.Asolid cylindrical plastic basehassixmetal pinsembedded, symmetri- cally spaced, along acoaxial cylindrical surface. Find themutual capacitances perunitlength between thepins. 25.Acylindrical cable hasNconductors, each ofsmall circular crosssection, symmetrically distributed along acylindrical surface coaxial withthegrounded sheath. Find themutual capacitance coefficients. 26.Assume inFig.2766thegap2atobearectangular orifice fortheflow ofanideal fluidfrom large radial distance ontheupper halftolarge radial distance onthelower half ofthez-plane. Find thevelocity distribution. 27.Assume inFig.27-66 thetwocoplanar conducting planes tohave the same potential *=andaddalinecharge+Xatpoint B.Compute the surface charges induced inthetwo planes. Show thatthetotal charge on eachconducting planeis X/2. 28.Inproblem 27,ifthelinecharge resides onathinwire ofradiusp, computeitscapacitance with respect totheconducting planes. 29.Two cylindrical electrodes ofsmall radius pareplaced upon athinsheet ofcopper oftheshape asshown inFig.27-9c; electrode Aofpotential *2is centered at Zandelectrode Bofpotential *i<*2islocated with itscenter atdistance 2afromOzalong thex-axis. Compute theresistance ofthecopper sheet ifitssmall thickness ist. 30.Thelower halfofthez-plane inFig.2766might represent aninfinite- extent dielectric medium ofdielectric constante,covered for\x\>abytwo grounded thinmetal foils. Find thecapacitanceofawire ofradius plocated along they-axis atheight habove theboundary plane. 31.Forthesymmetrical arrangement inFig.27-76 findtheendpoint of thefield lineemanating from theedge 2. 32.Consider aparallel thinwire ofradius plocated aty=26inthe geometry ofFig.27-7a. Find itscapacitancecoefficients with respect tothe two coplanar planes assumed atground potential, andwith respect to theplane y=assumed tohave potential difference Vapplied between itand thewire. 33.Athincopper sheet might haveanabrupt change ofwidth asinFig. 27-96. Assume oneelectrode located across thenarrow part atadistance from thediscontinuity where thecurrent distribution isuniform towithin 1%; assume thesecond electrode ofscmicylindrical shape and ofsuch radius thatalong itsperiphery thecurrent density isuniform within 1%. Find theresistance between theelectrodes. 34.InFig.2796assume thetworight-angle electrodes tohave thesame potential <&iandtohave athird plane electrode ofpotential *2along the center plane from y=down toy= a.Find the field distribution. Find thepartial capacitance ofthecenter plane forthesections from y=+o toy=-a. 35.Atwo-wire transmission line islocated attheheight aabove theplane x<inFig.27-9candatthedistance x=afrom thediscontinuity. Find thecapacitance ofthelineperunit length, assuming theentire contour to haveground potential. 36.Plot inFig.27lOothepotential linesandselect agoodapproximation toapoleshoe configuration inelectrical machines. Find thefield linetermi- Problems 411 nating atpoint 2toseparatefield lines entering thearmature surface y= from those passing totheneighboring pole shoe. 37.Thegeometry ofFig.27-10a might beconsidered astheflow ofan ideal fluidfrom thechannel between y=andy=aintotheright corner and around theguide plate 3"-4r-l' intothelarger space above. Find theveloc- itydistribution along theequipotential lineextending from thecorner point 2. 38.Find theresistance ofathincopper sheet having theshape oftheright- anglebend inFig.27-106. Oneelectrode isapplied across thevertical branch atadistance from theorigin where thecurrent densityisuniform towithin itl%;theother electrode isapplied across thehorizontal branch atadistance determined inthesame manner. Find theresistance ofthecopper sheet of small thickness t. 39.Find thebreakdown fieldstrength forashellwinding ofatransformer ifitcanberepresented asinFig.27Wd,assuming basthethickness ofthe winding with 6=2o,andtaking theplane y=asthegrounded core. 40.Taking theplane y= inFig.2710dasaplane ofsymmetry, the figure represents theupper halfoftwoparallel long plates offinite thickness. Find thevariation ofthefieldvectorEalong theplane ofsymmetry y=for thecondition b=a/4.Compare these field-strength values with thecase b=0,shown inFig.274. 41.Carry through themappingofthegeometry, Fig.27-126,iftheopposing right-angle equipotential surfaces arcideal magnetic boundary surfaces of potentials [Fiand CF2-Find the field lines starting atthecorners 2and 4. Compute theindividual fluxvalues bounded bythese field lines. Determine thefield linebetween 2-3'and3"-4 along which thefieldvectorBiswithin 2%oftheuniform value (JFi-3r 2)/6- 42.Find theelectric field distribution within therectangle ofFig.27-13a bydirect conformal transformation,ifpotential *iisapplied tothetwojoin- ingsides 1-2and2-3,andpotential $2totheother twojoining sides3-4and 4^1. (Section ofplane electronmultiplier, Zworykin,032 p.369). 43.Find thecharge distribution over thecoplanar parallel strips inthe w-plane ofFig.2713a. 44.Find thecurrent distribution inalarge thincopper sheet iftwostrip electrodes areapplied asinthez-plane ofFig.27-136. Find theresistance forsmall thickness tofthecopper sheet, assuming theelectrodes tohave equipotential contours. 45.Find thecurrent distribution between thetwocoplanar strips ofthe w-plane ofFig.27136. Find theresistance between thestrips. 46.Athinwire ofcircular cross section carrying alinear charge density X islocated inarectangular tunnel within agrounded conducting material. Find thecapacitance perunitlength ofthewire ofsmall radius pwithin the tunnel. Find theforceupon thewire. 47.Replace theconducting material inproblem 46byadielectric material. Find theforce action upon thewire. 48.Inthez-plane ofFig.2713cconsider theboundary lineoftheshaded region asrepresenting ground with arectangular long ditch. Assume athin wire ofpotential difference Vtoground located intheshaded areaandfind itscapacitance toground. 412 Two-dimensional Analytic Solutions [Ch. 7 49.Intherectangular channel ofFig.29laassume thepotential *=*o along thebase plate y=0,and*=along theother three sides. Find the potential distribution within thechannel. Find thecharge density along all four sides. 50.Assume inFig.29lathatthechannel ismadeupoftwosections with *=$oalong thesides y=andx=a,andwith*=*oalong x= andy=b.Find thepotential distribution. Find the fieldline, starting at thecorner y=andx=a. 51.Find thecurrent distribution inathinrectangular copper sheet ifone electrode isapplied along y=andtheother electrode along x=a,andthe potential difference isV.Find theresistance ofthecopper sheet forasmall thickness t. 52.Thebase plateandthefacex=aofarectangular bararekept atcon- stant temperature TO]thetopfacelosesheat sothatthetemperature gradient isproportional tothelocal temperature (asin29-24); thefacex= ia insulated sothatonitdT/dn =0.Find thethermal resistance ofthebar perunit length. 53.Athin rectangular conducting sheet isonehalfcopper andonehalf aluminum. Find theresistance ifinFig.29laoneelectrode isapplied over thelefthalf ofy=0,which isofcopper, andtheother electrode isapplied over theright half ofy=b,which isofaluminum. Disregard contact po- tentials andassume both materials ofthesame small thickness t. 54.Thearmature ofanelectrical machine canbedeveloped intoaninfinite slab ofmagnetic material ofhigh permeability /iextending asinFig.29-2. Assume, asafirstmodel, thatthemagrietostatic potential along yis constant andofvalueyifor a<x<+a, isconstant andofvaluey2= ^1for 3a<x<aandfora<x<3a,andcontinue ininfinite alter- nation with theperiod 4a;because ofthehigh permeability, onecanassume aty=bthatdF/dn =0.Find themagnetic reluctance perunitlength for any periodic section. Find thedistribution ofthemagnetic fluxdensity along y=0. 55.Assume inproblem 54that themagnetic field lines arerefracted at y=bandextend intotheinfinite airspace above. Find themagnetic re- luctance perunitlength foranyperiodic section. Find thedistribution of themagnetic fluxdensity along y=andalong y=6. 56.Assume inproblem 54that themagnetostatic potential varies line- arlyalong y=with thesame period 4a,forexample, having value7= M(x+a)/a for<x<-2a,andvalue3=M(a x)/a for<x<2a. Find thedistribution ofthemagnetic fluxdensity along y=0. 57.Aninfinite strip ofthincopper sheet ofwidth basinFig.292hasone electrode ofpotential V/2applied atitslower edgealong 2a<x<aand asecond electrode ofpotential V/2along a<x<2a.Find theresistance ofthecopper sheet ifthesmall thickness ist.Describe thisasatwo-dimen- sional hydraulic flowproblem. 58.Assume thecylindrical shell inFig.29-5 torepresent thestator ofan electrical machine with inner radius R\andouter radius #2-Ontheinner surface, themagnetostatic potentialisconstant andofvalue 171for<< v/2andv<<37T/2,andofvalue$2=IFiovertheothertwoquadrants; Problems 413 attheouter surface dIF/dr =0.Find thereluctance perunitlength forone periodic section. Find thedistribution ofthemagnetic fluxdensity along theinner surface. 59.Inproblem 58,findthemagnetic field distribution intheairspace for r<R\.Find thereluctance perunitlength oftheairspace foraperiodic section. 60.Ifinproblem 58thecondition d!7/dr =ontheouter surface is relaxed andreplaced bytheusual magnetic boundary conditions ofrefraction, findthedistribution ofthemagnetic fluxdensity justoutside themagnetic shell. Find thevalue ofthemagnetic fluxdensity atlarge distance from the shell. 61.Avery long conductor oflarge rectangular cross section 2aX25 carries theuniformly distributed current /and isplaced snugly atthebottom ofaninfinite rectangularslotformed bytwoparallel blocks ofironspaced 2a. Find thedistribution ofthemagneticfield ifingood approximation thefield linescanbetaken asnormal toalliron surfaces. 62.Aninfinite block ofironcarries onitsplane surface aninfinitely periodic alternation oflikeconductors with large rectangular cross section, each carrying thesame total current /but inalternatingly opposite directions. Find themagnetic field distribution within theconductors andtheairspace outside, assuming thatthemagnetic field lines enter theironblock perpen- dicularly. 63.Athin circular cylindrical shell isslotted sothat itsarc isSTT/Sand carries potential V.Find thepotential distribution bytwo-dimensional inversion. Find thecharge distribution ontheslotted cylinder. 64.Athincopper sheet ofelliptical areawithmajor axis2aandminor axis26hastwoelectrodes ofsmall circular areas appliedatthefociFIandF2 (see Fig. 29-6). Find theresistance forasmall thickness tofthesheet. Hint: intheneighborhood ofF\ tfissmallandt\isclose tow\intheneighbor- hood ofFZ,fissmall andTJissmall. Satisfy *=+V/2 fort\=ITpi, l>=F/2 forrj=P2,where p\andp2arethesmall radii oftheelectrodes. Forthefieldvector observe (31-24). Check theresult byconformal mapping. 65.Along solidbarhasascross section theright half oftheellipse inFig. 29-6 withmajor axis2aandminor axis 26.Thebase77=ir/2iskept at temperature TI,andthecylinder surface iscooled sothat itstemperatureis TZ<TI.Find theheat flowtransmitted through thecylinder surface per unit length. 66.Inproblem 65assume that thetemperature ofthecylinder surface varies linearly from TIatthebase toTZ<TIatA.Find theheat flow transmitted through thecylinder surface perunit length. 67.Transform thetwo-dimensional Laplacian differential equation from cartesian to(a)parabolic cylinder coordinates; (6)bipolarcoordinates. 68.InFig.29-7assume theinfinite parabolic cylinder surfacerj=2to represent ground andtohave aparallellinecharge ofdensity Xlocated at =0, 77=4.Find, byconformal mapping,thelocation oftheimage line charge andthedistribution oftheinduced chargeinground. 8-THREE-DIMENSIONAL ANALYTIC SOLUTIONS Admittedly among themost difficult group ofboundary value problems, three-dimensional potential distributions require ac- quaintance with thelessusual function systems, many ofwhich have notbeen asextensively tabulated asmight bedesirable. Itisseldom possible toarrive atsolutions inclosed forms, and, actually, most ofthese simpler caseshavebeen treated insections 14and 15.Inpracticallyallcases treated here, therefore, infinite series expansions arenecessary sothat onecanonly speak of formally exact solutions ifthese arefeasible atall;forallpractical cases onemust accept theapproximations byfinite sums. This holds alsofortheaxially symmetricalfield distributions, which are sometimes called two-dimensional because theaxialsymmetry eliminates one ofthethree variables; they belong, however, definitely tothethree-dimensional class ofsolutions, involving thesame typesoffunction systems. 30-AXIALLY SYMMETRICAL POTENTIAL FIELDS Interms ofcylindrical coordinates thepotential equation with axialsymmetry hastheform [Appendix 3,(37)] Il(p^+^.0(1) andpermits readily separation ofthevariables byassuming $=R(p)Z(z\ whereRandZarefunctions ofonlyonevariable each. Introducing thisproduct into (1)anddividing byitgive 11d{dR\ ld2Z 414 Sec. 30] Axially Symmetrical Potential Fields 415 arguing asin(29-16) thateachtermcanatmost beafunction of theindicated variable, andsince theequation must hold forany combination oftheindependent variables, eachtermmust actually beaconstant. The possible values ofmareselected bythe boundary conditions andcanform either adiscrete oracontinuous spectrum, asshown insection 29. Thefactthatonlytwovariables appear inthepotential equation (1)just asinthetwo-dimensional case ledearly toattempts for utilization oftwo-dimensional field solutions andgraphs. Ithas been shown,1however, that theonly field geometries that are common forboth types ofproblems aretheorthogonal, confocal, conic sections, including circles; noother solutions canbe translated. Anapproximate utilization oftwo-dimensional solutions for axially symmetricalfields farfrom the axiswasshown by Maxwell,A17 I,p.305. Assume that the analytic function W=f(z) represents thecomplex solution ofapotential problem inthex-y-pl&ne bythemethod ofconjugate functions asoutlined insection 25. Ifw=u+jv,thenu(x,y)istherealpotential solution and satisfies theLaplacian differential equation d2uS2u Ifitisdesired tofindthesolution forthesame cross section of electrodes butrotated about anaxis parallel tothe ?/-axis andyQ tothe left ofit,then u(x,y) must satisfy (1)with (y+y) forpandxfor z.Expanded, thisbecomes d2ud2u_1du dy2dx2 7/0+ydy where usewasmade ofd/dy=d/d(y+ T/O),sothattheorigin need notbeshifted. This equation (3)hastheform ofaspace charge potential equation (3-4) withspace charge density which canbetaken as"correction." Obviously, inserting in(4) thetwo-dimensional solution u(x, T/)cannot giveanexactsolution; 1W.Gauster, Arch.f.ElektroL, 16,p.89(1926). 416 Three-dimensional Analytic Solutions [Ch.8 however,ifyQisconsiderably larger than theregion ofyforwhich thefield distribution isofreal interest, areasonably good approxi- mation canbehad. Onecanfurther simplify byapproximating du/dy toleadtosimple results. Thismethod canbestbeused to evaluate thecapacitance, since forthat itisnecessary only to compute thetotal space charge andadd ittothesurface charge ofthesamesign. The total charge then defines thetotal capaci- tance fortheaxially symmetrical system ofthesame potential difference. Maxwell applied thisprocedure toevaluate theeffect oftheguard ring forcircular electrodes from thetwo-dimensional solution (27-40) referring toFig.27-7a.Healsoconverted the end effect attheedge ofaplate parallel toandbetween two infinite plates, asinFig. 27-76, into asolution forconcentric cylinders byrotation about anaxisparallel tothe ?/-axis, andinto asolution forcircular disksbyrotation about anaxis parallel tothex-axis. Field Expansions near Axis. Inelectron optical field problems one ismainly concerned with thepotential and field values nearandontheaxis ofsymmetry. Since thepotential must befinite andcontinuous along theaxis ifitbelongs tothe fieldregion andmust beaneven function ofp,onecansolve (1) bymeans ofthepower series *(P, )-/2(Z)P2<" (5)a=0 where $(0, z)=/o(z), thepotential value along theaxis. Intro- ducing (5)into (1),oneobtains therecursion formula (2a+2)2/2a+2 (z)+/2a"(z)=(6) foranypower p2a .Thus,allthecoefficients /2a(z)in(5)canbe expressedinterms of/o(z), sothat .,,ft,*"(0,g)(p/2)2$IV(Q,z)( P/2)4 ,Z)=$(0, Z)---- 1-- 2-- '"(') where theprimes denote differentiations with respect toz;see Bateman,01 p.406; Briiche and Scherzer,B2 p.66;Spangen- berg,B29 p.339;andothers. Themain problem istherefore the evaluation ofthepotential orofthefieldgradient EZ(Q,z)=/O'(z) along theaxis either analytically,ifthat ispossible, ormost expeditiously withtheelectrolytic trough (section 18). Sec. 30] Field Expansions near Axis 417 Instead ofthepower series expansion inp,onecanuseLaplace's expression *(p, z)=-rW fQ(z+jpcosf)d* (8) 7T/^=0 where /isagain thepotential function along theaxis,butwith z replaced by (zH-jpsin^). This isverified byaTaylor series expansion of/about p=andintegration termbyterm, which leads to(7);Bateman,01 p.406,andalsoMyers,327 p.89. Though thepotential function must becontinuous along the axis,itcanpossess isolated singular points where thefieldvector vanishes, asdiscussed insection 10.Because ofthecontinuity, onecandevelop $(0, z)=/(z)atanypoint zontheaxisintoa Taylor series *(0, z)=/(z)=2 ,... andintroduce thisforthe firstterm in(7);thesecond derivative with respect toznear zbecomes -*o)+... andusing thisinthesecond term of(7),oneobtains near ZQ /o(2o)+/o'(zo)(z-Zo) (9) ifallterms involving higher than second derivatives inzare discarded. Along anequipotential linenear theaxisonemust thenhave cM>(p, Z)==/Q(ZQ)dz+/Q(ZQ)(Z ZQ)C?Z/^/(/'C^o) P^P (10) which gives fortheslope dp_/</(b)+/o"(go)(g-go) ~~^.. Asoneapproaches thepointzontheaxis,z >zandp>0,so thatdp/dz>ooatallregular pointsAofFig.30-1,asitmust be because oftheaxialsymmetry. Atasingular point B,however, 418 Three-dimensional Analytic Solutions [Ch.8 = onefinds /rfp\UAtan lim2/o%u) (*- /O"(O)P byde1'Hospital'srule. Thus, only saddle points canoccur as singular points, andatanysuch singularity thepair ofequip oten- tiallines intersects theaxisatangles tan"1(2)=5444'; see Myers,B27 p.95,andZworykinetaZ.,B32 p.377. This isquite different from the two-dimensional fielddistribution, forwhich thegeneral expansion cor- responding to(7)intheneighbor- hood ofanaxis ofsymmetry, chosen asx-axis,isgivenby 7 FIG.30-1 Potential Values near theAxis forAxially Sym- metrical System.(11) with/ (z)=*(0, x)denoting thepotential value along theaxis. Usingfor itthesame Taylor scries near apoint xasabove, introducingitinto(11),andestablishing theequipotential near xanalogous to(10)givenow d*(y, x)==[/(/(xo)+H/o"(x )(x-x)]dx-y2f"(x)ydy From this,theslopebecomes d3/^2/o/(xo)+/o"(xo)(x-x ) dxjo\XQ)y which again shows theorthogonality oftheequipotential lines to theaxis,butgives atasingular point lim(dy/dx)s=1;theinter- section oftheaxis ofsymmetry bytheequipotential lines ata singular point occurs atangles 45; seealsoZworykinetal.,B32 p.375. This demonstrates clearly that substitution oftwo- dimensional fields fortheaxially symmetrical fieldnear theaxis isbound togivepoorapproximations. Axially symmetrical magnetic fields arecompletely defined by onlyonecomponent ofthemagnetic vector potential; since cur- Sec. 30] Field Expansions near Axis 419 rents producing axially symmetrical fields must flow circularly around theaxis,onlyA^willexist, as(13-25) shows. Inregions free ofcurrent, asisusually truenear theaxis oftheelectron optical systems, thecomponent A^will satisfy thedifferential equation dp[_p dp which isobtained fromAppendix 3,(37). Inanalogy to(5)one canassume asolution neartheaxis oftheform =0 where onlyoddpowersofpcanappear because A$encircles the axis. Introducing (13) into(12), oneobtains therecursion formula i"(z)=(14) foranypower p2"" 1 .Thus,allthecoefficients /2a+i(z)in(13) canbeexpressedinterms ofderivatives offi(z)sothat Onecaninterpret thephysical meaning off\(z)ifonealsocon- siders thefieldvectorBwhose components aregiven asin(13-26) by This gives with (15) *,=2/,(,)- where itisnowapparent that 2/1(2)=B(0, z)represents the 420 Three-dimensional Analytic Solutions [Ch.8 axialcomponentofthemagnet fieldalong theaxis. With (16), thegeneral form (15)becomes (17) Thisform permits theutilization ofexperimental data;ifone finds agood analytical approximation tothemeasured field dis- tribution along theaxis,onecan construct acomplete solution anduse itfordetermination of electron paths oranyother de- sired information. This ispar- FIG.30-2 Potential Values nearthe. jrf j tant formagneticAxis forSpherical System.J* .& fields because theanalytical com- putations quicklylead into difficult functions,2aspointed outin section 13.Onecan, ofcourse, alsousethemagnetostatic poten- tialfunction 7which leads toforms quite similar to(7)and (8) asinZworykinetaZ.,B32 p.474. Occasionallyitisalso ofinterest toknow potential solutions ina spherical systemforsmall angles ofopening asindicated inFig. 30-2. From Appendix 3,(40)onehasforaxialsymmetry ina sphericalcoordinate system (18)' Forsmall angles onecanassume thesolution ofthetype *(r,0)=L/WW*2" (19)a=0 Approximatingin(18) sin0,andcollecting coefficients ofthe samepowersin0,onededuces therecursion formula T[r2/2*]+WWW=(20)dr 2SeeW.Glaser,Zeits.f.Physik, 118, p.264(1941). Sec. 30] Equidiameter Coaxial Cylinders 421 which yields because /(r)=*(r,0)thepotential along theaxis, *(r, 0)=*(r,0)- 2 *'(r, 0)] where theprimes denote differentiations with respect tor.This developmentisparticularly applicable toconical fields asexist in cathode-ray tubes andsimilar applications. Two Finite Equidiameter Coaxial Cylinders. Two finite coaxial cylinders ofequal diameters asinFig.30-3with potentials Z,, oo FIG.30-3Two Coaxial Cylinders ofEqual Radii (Two-cylinder Lens orMirror). $1and $2,respectively, constitute arather common electron lens ofsimple type. Their lengths might beLIandL2,withverysmall separation attheplanez=and their endfaces z=L\of potential <f>iand z=+L 2ofpotential4>2,disregarding anysmall apertures thatmight exist inthese planes. The solution ofthe potential distribution canbefound from(2)where thevariables have been separated. Fortheupper, positive sign ofm2onehas atonce asforX(x) in(29-17) and(29-18) Z(z)=Cisinmz+C2cosmz whereas thefunction R(p)must satisfy d?RIdR dp2pp(216) (22) 422 Three-dimensional Analytic Solutions [Ch.8 which isthenormal form ofthedifferential equationformodified Bessel functions ofzeroth order3[Appendix 5,(24)] B(p)=Di/ (ifip)+D2KQ(mp) (23) Thepotential function must becontinuous atallpoints except along therimp=aintheplane z=0,where there exists an isolated singularityofthesame type asinconformal mappingat thevertices ofstraightlinepolygons (section 27). This excludes thesecond term in(23)asapossible solution, since thefunction K(mp} hasalogarithmic singularityatp=0.Thepotential solution istherefore thegeneral product (Cisinmz+ 2cosraz)/O(WP) (24) towhich canbeadded byinspectionof(1)theparticular integrals *i+k2z (25) Theselection ofthespectrum ofmvaluesis,asalways, simplest with homogeneous boundary conditions ofthetype (29-3). Though these arenotdirectly specified, onecanallocate the in- homogeneous boundary conditions byplacing theburden ofsatisfy- ingtheconstant potential values atz=L\and z=+L2upon theparticular integrals (25) ki k2Li=$1, ki+k2L2=$2 or _^ _ ^x LI+L2 L2+LI andthus requiringof(24)thehomogeneous conditions Z(-Iu)=Z(+L 2)= 3Brief reviews ofBessel functions aregiven inSmythe,A22 p.168;Churchill,03 Chapter VIII; andalmost anybookonadvanced calculus. Extensive treatises areGray, Matthews andMacRobert;07Byerly;02N.W.McLachlan: Bessel Functions forEngineers; Oxford University Press, 1934;andG.N.Watson: Theory ofBessel Functions; Cambridge University Press, 1922. Fortables see Jahnke andEmde: Tables ofFunctions; reprinted byDover Publications, New York, 1943; originally byB.G.Teubner, Leipzig, 1938. See also Appendix5. Sec. 30] Equidiameter Coaxial Cylinders 423 This yields upon combination ofthetwoequations thecharacteris- ticequation sinra(L^snrai 2C2-7= 'or' sinmLi (LI+L2) a=1,2,... (27) andtherefore (28) where thenegative values ofahave been suppressed, since they leadtothesame functional expressions. Forp=a,thesum (28) isaconventional Fourier series inzandmust represent theactual potentialdistribution onp=aaswell astheparticular integral values. The coefficients Aaaretherefore determined by (aira\ L!+L2) sm Lt+z =La/Li+z\ "I ($2 ki/c2z)sin Iair Idz o \LI+L2/J analogousto(29-22) and(29-22a), with (Li+L2)asthehalf period.Thisexpansionisdefinitely permissible andconvergent, since thesinefunctions formanorthogonal system andsince the potentialvalues arebounded. The integralsin(29)canreadily beevaluated andactually reduce to . 2,. airLiAa=($1 $2)cos OC7T LI ~\~L2 424 Three-dimensional Analytic Solutions [Ch.8 sothatthefinal solution forthepotential becomes (Lt+2)*2+(L,-)*i *(P, -*(*!-*) S(W^L-) T a=1l<* V^l+L2/ (carp\ a(T.,1 \/>l+L<2/ (30) Inthecase ofsymmetry LI=L2=L,theFourier series will contain only theterms forwhich aiseven, since cos(a7r/2)= foraodd; inthis case, theplane ofsymmetry2=becomes an equipotentialsurface ofpotential ^(^i+$2)- Should, onthe other hand, potential $!vary linearly orinanyfashion along p=afrom avalue zero atz=LIidentified ascathode surface, toavalue 3>iatz=asintheelectrostatic image tube,4thenthe first integralin(29)would have tobeappropriately modified by using theknown function $1(2) instead oftheconstant value $1- TwoEquidiamcter Coaxial Cylinders, OneInfinitely Long. IfL2islargecompared with thediameter 2a,itmight aswellbe assumed infinitely longwith theeffect thattheFourier series goes over intoaFourier integral. Maintaining thesame boundary conditions asinFig.30-3, except thatL2= ,onecanspecify Z(Li)= for(21), using thepotential value $1asadditive constant tosatisfy thecondition atz=LI;thisgives r-t cotmLiL2 Therefore inaccordance with (24) Cz8inOT(Ll +Z)'O<P) dm(31)sinraLi since nodiscrete spectrumofm-values exists. Theunknown coefficient C2must beobtained byrepresenting thepotential value 4V.K.Zworykin andG.A.Morton,Jl.Optical Soc.Am., 26,p.181(1936); ZworykinetaJ.,Ba2 p.46; alaoE.G.Ramberg andG.A.Morton, Jl.Appl. Phys., 10,p.465(1939). Sec. 30]TwoEquidiameter Coaxial Cylinders 425 along p=ainFourier integral form analogous to(29-31) and (29-32). Introducing achange ofvariable tof=z+LI,sothat theorigin offisintheplane oftheendface,onehasfrom (31) *(a, f)=3i+rC2Smm rr7(ma)dm (32)Jm=osinraLi whereas thedirect Fourier representation by(29-36) or(29-38) would read inthesimpler complex form -i-f" F(m)e'mn*dm 2lTJm= ao dp,f>0 (33) Thepotential values arereferred to3>iandaretherefore zero for <f<LIandequal to($2 $1)forf>LI.However, this does notspecify thecharacter ofthepotential distribution for f<0;since (32) implies anoddfunction U(m) ascomparison with (29-31) indicates, onemust assume opposite potentials at symmetrical locations with respect tof=0.Theevaluation of F(m) orF(p) cannowbemade, keepinginmind achangeinsign forf<asnoted; thedirect integration asin(29-36a) with p=jmgives ' (*2-*i)e~pr*+f9 (*2-*i)e-pfdf /Li (epLl+e-pLl )=^($2-*i)coswLi (34)~fU a From thisonecangetthefunction U(m) byidentifyingitin accordance with (29-37) asrelated totheimaginary part ofF(m), sothat 1 23>i $o U(m)=--ImF(m)=---cosmLi IT TTm Thismustnowbeidentical with theintegrand in(32)except for sinrafandyields Cz_2$1 $2cosraLi sinraLiITm /o(ma) 426 Three-dimensional Analytic Solutions [Ch.8 which finally gives for(31)thesolution 2 p,z) 7T /cosmLi . /o(rap) I-sinra(Li+2)- -dm (35)Jm=om /o(ma) Comparison ofthisFourier integral withtheFourier series solution forfinite values ofL2in(30)demonstrates thevery close similarity between them. Inmany instances, (35)canreadily beobtained bynumerical orgraphical methods withLIandaasparameters, andzandpasultimate variables. Ifagain thepotential onp=a varies linearly over thedistance(LI)<z<0,asonemight assume intheelectrostatic image tube,5oneneed only tomodify theintegral (33)byintroducing thevariation along thedistances <f<LI. The analytical evaluation oftheintegral (35)isachieved best byreplacing therealvariable mbyp=jmandinterpreting the integral asoneinthecomplex p-planc aspointed outinconnection with (29-38). Thepoles oftheintegrand arelocated atp=and atIQ(ma)=jQ(jma)=Jo(pa)=0,thelatter being theBessel function offirstkindandgiving aninfinite number ofsymmetrically located root values, ofwhich the first sixare Pia=2.4048 p4a=11.7915 pza=5.5201 p5a=14.9309 p3a=8.6537 p6a=18.0711 Thus,inthecomplex form (33)with (34) |z)=*i-(Si-*a)' dp,.5(-id (36) xr J(pa) where thetotal potential values change sign asalready assumed intheintegral (34). Combining theexponentials into epzand eP(2Li+z)^ twointegralsofthetype (29-38) arise, eachwith poles offirstorder along positive andnegative realaxes. Inaccordance 6G.A.Morton andE.G.Ramberg, Phys., 7,p.451(1936); alsoZworykin etaJ.,B32p.381. Sec. 30]TwoEquidiameter Coaxial Cylinders 427 with (29-41) andobserving (d/dp) J(pa)=-aJ1(pa)onecan nowwrite thesums ofresidues inthefollowing groups Bi=1-,J'(pa?e-* validfor z> (paa)Ji(p aa) (37)Ru=+E, ,* valid forz< ^ (pao) e-Pa(2L1+z) valid for2>(-2Li) iv,.IV ^ (ptta)Ji(p aa) valid for2<(-2LO where allsums areextended onlyoverthepositive rootvalues of J(paa)listed inthetable above. The total solution forthe potentialisthus +#m] valid forz> validfor (-Li)<z<(38) theother ranges areofnointerest, lyingbeyond thedesired field region. Thepotentialreduces to$1atz=LI,since thesums in#nandRm cancel, and ittakes theproper values along p=a asseenfrom (36)where onlythepositiveunit step atz=can beconsidered for z>(-Li). The unit stepatz=-2Li is inverted because oftheoddsymmetryofpotential values; its effect istherefore infiiv-Thevalues ofthecompleteseries inRI and#11havebeencomputed andtabulated6inconnection witha general attempttosolve thepotential distribution fortwoequi- diameter cylinders with afinite separation 2dasshown inFig. 30-4. Onecanconsider theright half ofthisarrangement as equivalenttotheabove case, except thatthepotential function is actually unknown forp=aalong <z<d;theassumption of 6S.Bertram,Jl.Appl Phys., 13,p.496(1942); tabulation ofvalues for p/ainstepsof0.1,forz/ainstepsof0.05upto1.75,beyond which exponential approximationispossible. 428 Three-dimensional Analytic Solutions [Ch.8 linear variation ofthepotential along thisdistance leads toresults which check rather closely withdataobtained withtheelectrolytic trough. Analytically, oneneed onlymodify (34) inaccordance with theassumed potential variation andenter thisin(36)asa modification ofthe firstfactor under theintegral sign. FIG.30-4Two Coaxial Equidiameter Cylinders with Finite Separation. Two Infinite Coaxial Cylinders. Asimpler result obtains in thesymmetrical case ofFig.30-3, where thelengthsofboth equi- diameter cylinders LIandL2areinfinite. Oneneed onlyconsider theright halfwith potential H(*i+$2)intheplane2=and $=$2along p=a.Though onecould again usetheFourier integral method,itissimpler toformulate thesolution directly in terms oftheorthogonal Bessel function series. Forthispurpose take thelower sign ofra2in(2)where thevariables have been separated. ForZ(z) onehasthen hyperbolic or,betterstill, exponential functions assolution Because ofthe infinite extension forz>0,only thenegative exponential function canbeaccepted assolution. Thefunction R(p)mustnow satisfy d2R 1dR , 2_ TT+"T+mR dp* pdp(40) which isthenormal form ofthe differential equation forthe Sec. 30] Two Infinite Coaxial Cylinders 429 Bessel function jQ(mp) ofzeroth order7 B(p)=DiJofap) +D2N(mp) (41) Again,NQ(mp)hasalogarithmic singularity atp=0,sothat it cannot beadmitted assolution. Thus, onepotential solution is thegeneral product ce~mzJQ(mp) (42) where Z)iC 2hasbeencombined into c.Tosatisfy theboundary conditions, onecanaddaconstant 3>2asparticular integral and require of(42)that itvanish forp=a,which leads atoncetothe root values (maa)tabulated as(paa)for(36). Thefunctions /o(map)nowform anorthogonal system ofaSturm-Liouville problem, since thedifferential equation (40)canberewritten in theform (29-2), namely, n +m2PR(p)=(43) which defines thecharacteristic numbers X=m2 ,theweight function p(p)=p,with respect towhich orthogonality exists, and gives thenormNbytheintegration =0 pJo(m ap)JO(^P) dp o fO fora7*ft P-Ji2(roa)=Na fora=ft I2 Onecould,ofcourse, normalize these Bessel functions asin(2910) bydividing by\/]V^, andonecanexpand anybounded function intoaFourier-Bessel series inaccordance with (29-11); forthe more general forms seeAppendix5. The total potential solution isnow *(P|z)=*2+ECat-"2JQ(maP) (45)a=l where thecoefficients caarefound from theremaining boundary 7Se.fireferences, footnote 3. 430 Three-dimensional Analytic Solutions [Ch.8 condition which stipulates that $(p, 0)beconstant andequal to themedian potential value, *(p,0)=<J>2+EcaJQ(map)=i (<DL+*a) a=i * Applying the firstform of(29-12) inorder tofindthecoefficients ca,onehas 1 rp=a1 aNaca=-($!-<S>2)IpJ(map)dp=-(*i-$2)J\(m aa)& /p=0 J 7Wa WithWafrom (44) thisyields thefinalform forthepotential *(P, 2)=*2+(*i-*)L Thesummation isidentical with theoneoccurringinR\of(37) and istabulated asreferred above. Since onecandeduce 9=lAaJl(\a) * ifXaaretherootvalues ofJQ(\)=0,onecanreadily show that Inelectron optical problems, one ismainly concerned with the value ofthepotential and itsderivatives along theaxis p=as outlined inthe firstpart ofthis section. Thismakes ananalytic expression practically necessary, yetmakes itdesirable tohave a simple form toenter intothedifferential equation fortheelectron trajectories. Introducing p=into (46)gives *(0, 2)=*2-(*a-*i)Ze-"2 [(wiaa) Ji(m aa)]-1 (48) fortheaxial potential variation forthetwoinfinitely long cqui- diameter cylinders ofFig.303withLIooandL2>o.The sum in(48)canberepresented withverygood accuracy bythe much simpler form8 Ze~az[(maa)J^ntaO,)]-1Y2(\-tanhwz) (49) a=l 8F.Gray, BellSystem Techn. Jl.t18,p.25(1939); also S.Bertram, Proc. I.R.E., 28,p.418(1940). Sec. 30] Circular Aperture 431 where w=1.32/a, sothat ,z)-*i)tanh 02 (50) Forgraph ofthisandthe firsttwoderivatives seeZworykin etaZ.,B32 p.379. This approximation canbeused withsome modifications inother cases aswell. Ifthecylinders arenot ofequal diameters, theanalytical method becomes well-nigh impossible, andapproximations bya FIG.30-5 Symmetrical Circular Aperture. perturbation method using Green's function remain theonly recourse. Theelectrolytic trough hasbeen calledupon extensively insuch cases; seeSpangenberg,329 p.345. Circular Aperture. Assume acircular hole ofradius ain aninfinite conducting planeofpotential $1asinFig.305,which iscalled acircular aperture inelectron optics, then the field lines must leave theconducting planeatright angles andtendtobecome parallel tothe2-axis ofrevolution; atlarge distance thefieldmust benearly uniform sothatonecanplace parallel planes ofpotentials $2atsymmetrical distances dfrom theaperture plane. Inorder todescribe theboundary condition ontheaperture planeinthe simplest terms one selects theorthogonal oblate spheroidal coordinate system from section 33with aslight modification. Using instead of(3360)thesame transformation (32-52)asfor theelliptic cylinder andintroducing theauxiliary coordinates, sinh {, sin17 (51) 432 Three-dimensional Analytic Solutions [Ch.8 then thecylindrical coordinates pand zcanbeexpressed as zasinh sinTJ=auv, p=acosh cost\=aVu2+IVl v2 (52) where aistheradius oftheaperture andidentical with thefocal distance /ofthesystem. Thecoordinates andrjaresimilar to the elliptic cylinder coordinates in(29-59), and, indeed, con- stant values ofuand vimply constant values offand77andthus mean confocal ellipses andhyperbolas ofthesame shape as Fig.29-6butwith different selection rules. Inparticular,v= orr2 TI=2adescribes theequipotential surface $1, i.e.,the plane with thecircular aperture, and v=1orry=Tr/2orr2=n describes thez-axis. Ontheother hand, u=orr2+n=2a describes theaperture itselfandincreasing ugives theellipses of increasing axes. Because oftheaxialsymmetry ofthesolution, onecanintroduce (52)and(51)directly intotheLaplacian potential equation (332) andwith suppression ofthesecond derivative in</>obtain (53)du\' Solutions of(53)canbefound readily byseparation ofvariables; assuming<f>=M(u)-N(v) andintroducing into(53), onehas Several solutions arefeasible forspecific values ofm2which are ofthetype ofparticular integrals, since theboundary conditions arenotyetutilized. Thus, form2=2onefinds assuggested by and-tan"1u=(1+u2 au -tanh-1v=(1- dv that MN=[du+C2[utan"1u+1]}- [DlV+D2[vtanh-1v-1]} (55) Sec. 30] Circular Aperture 433 satisfies (53). Since theboundary condition requires<J>=$1on v=0,onecanchooseD2= in(55)andaddtheconstant $1; since v=1along the z-axis, thepotential remains finite there. Thepotential function is,therefore, *(u, v)=v[Ciu+C2[utan"1u+I]}+$1 withDIdiscarded assuperfluous. Actually, noother boundary conditions areavailable, but itisnecessary thatthesolution be symmetrical totheplanez=0,sothatonemust takeCi=0.It isalso inthenature oftheproblem thatanearly uniform field should result forlarge values ofz.Since tan"1u*(7r/2) as u*oo ,thepotential becomes forlarge values ofu lim*(u, v)->C2v(u+1)+*! C2-+$1 u > \2 / 2a ifonedisregards thevalue 1and utilizes (52). Introducing 2 a 2 f>$2at2=dgives atonceC2=($1 $2) ~;=EoQ, TT a ?r if# istheuniform field gradient atlarge distance from the aperture. Thus, 2 $(w, v)=$1 a|# |v[utan"1w+1] (56) andalong thez-axis where v=1andtherefore u=z/afrom (52), incylindrical coordinates pand z *(0, z)=*i--a\EQ \\-tan'1-+ll(57) TT \_a a J Thepotential atthesaddle point with z=is 2, ,=$x a# 7T andcanbemade tovanish withproper choice of$1.Good graphs ofthissymmetrical potential distribution arefound inSpangen- berg,B29 p.347,andinZworykinetaZ.,B32 p.384. Theproblem is solved withmore difficult notation inOllendorff,A18 p.295,andin Bruche and Scherzer,B2 p.69; seealsoLamb,C22 p.142, for hydrodynamic applications totheflow ofanideal fluidthrough a circular aperture. 434 Three-dimensional Analytic Solutions [Ch.8 I* 1s *l'Inorder toestimate thedegreeofapproximation, onecancon- sider thattan"112=85.4 leads to(utan"1 u)=17.86 1;this, however, requires adistance along thez-axis of(z/d)= u=12,ord=I2a inac- cordance with (52), where v=1.Admitting (utan"1 u) ^10astolerable approxima- tionrequires d^la. One can also achieve so- lutions forunsymmetrical po- tential distributions bysuper- imposing auniform electric fieldgradient E\,weakeningto (#o Ei),andstrengthening to(E+EI), therespective sectional gradients. With the designations ofFig.30-6, tak- ingallgradients with absolute values toavoid difficulties with signs,andobserving (52) forz,onehas 2 tjv)=$1 a\EQ\v[utan"1u+1]+a\Ei\uv(58) Thepotentials attheelectrode plates aregiven as $2=(\EQ\+\Ei\)d2+$1 andFIG. 30-6 Unsymmetrical Aperture Field. theydepend onthedistances dzanddzandcannot bechosen freely, since thesolution isapproximated bysuperposition and is notanexact one. For|-Bi|=\EQ\onehasafield freespace to theright oftheaperture plane andtheequipotential linesbulge through theaperture.9Good graphs arefound inSpangenberg,629 p.348,andinZworykin,B32 p.384. 9Th.C.Fry,Am.Math. Monthly, 39,p.199(1932); also Bell Tel.Lab. Monograph No.B-671; Ollendorff/18 p.296;andSmythe,A22 p.161. Sec. 31] Cartesian Coordinate System 435 31-GENERAL ORTHOGONAL COORDINATE SYSTEMS Forthesolution ofgeneral potential problems inthree-dimen- sional spaceitisdesirable tochoose coordinate systems which permit thesimplest formulation oftheboundary conditions, as pointed outpreviously. However, thecoordinate system in- fluences theform ofthebasic differential equations ofpotential, so that onlysuch coordinate systems areofpractical value which keep thisformamenable topresent-day mathematical treatments. This hasrestricted thechoice toorthogonal coordinate systems in which theunitvectors inthethree coordinate directions atanyone point aremutually orthogonal, or,differently stated, inwhich the three families ofsurfaces defined bykeeping thevalue ofeach coordinate constant inturn aremutually orthogonal. Itiscustomary toselect theCartesian system asfundamental, since initthethree coordinates play exactly equal rolesand all relations involve thethree coordinates inexactly symmetrical manner, sothatanycyclic1interchangewillnotaffect theform of anyboundary value problem. CARTESIAN COORDINATE SYSTEM TheLaplaciandifferential equation forthethree-dimensional case isgivenby Inorder toeffect asolution ingeneral terms onecanreadily separate thevariables byassuming aproduct function *(x,y,t)=X(x)Y(y)Z(z) (2) inwhich each factor isafunction ofonlyonevariable; this is obviously adirect extension ofthetwo-dimensional case in section 29.Introducing (2)into (1)anddividing through bythe product (2)willgive X~1X"+Y~1Y"+Z~1Z"=(3) where thedouble primes indicate thesecond derivatives with respecttothepertinentvariable. In(3)thevariables arealready * 1Acyclic interchangeisoneinwhich theorder ofsuccession oftheelements ispreserved,asforexample (x,y,z)to(y,z,z)to(z,x,y}. 436 Three-dimensional Analytic Solutions [Ch.8 separated, sothat inorder tobeanequation foranycombination ofthevariables z,y,andz,eachtermmustbyitselfbeaconstant, which isusually designated asseparation constant, because it enters onaccount ofthereduction toordinary differential equa- tions. Onehas, forexample, X"=-m2X, Y"=-n2Y,Z"=(m2+n2}Z (4) wherem2andn2arethecharacteristic numbers whose spectra are defined bytheboundary conditions;ifthelatter arehomogeneous, thisleads totheclassical Sturm-Liouville problem discussed more FIQ. 311Potential Distribution inRectangular Parallelepiped. extensively insection 29.Ofcourse, onecan associate the characteristic numbers m2andn2withany ofthetwovariables above; butthethird onemust thenaccept thenegative sum ofthe two. Toillustrate theprocedure, determine thepotential within a rectangular boxwith thedimensions shown inFig.31-1anda potential distribution $=G(x, y)ontheface z=and*= ontheother fivefaces. Thisdetermines theboundary conditions ashomogeneous inx-andy-directions, sothatmandncanbe found readily. Thetypical solution inthese variables isfrom(4) X=Cisinmx+C2cosmx, Y=DIsinny+D2cosny (5) with X(0)=X(a)=0, F(0)=(6) Introducing these conditions into (5)yields C2=0,sinma=0,D2=0,sinnb=(7) Sec. 31] Cartesian Coordinate System 437 andtherefore thecharacteristic numbers ma=np=-^ <*,ft=1,2,3,oo(8)a o Thenegative values ofmandnaresuppressed, since they leadto nonewfunctional forms. Thetypical solution forZisfrom(4) Z=P!sinhVma2+np2z+P2coshVma2+n^22(9) andsince atz=conemust satisfy Z(c)=0,thisgives TJ~=tanhVma2+np2c "i with raffandrapknown from (8).With allhomogeneous boundary conditions satisfied, thepotential hastheform a+V(c-z). <P(x, y,z)=2*L,ra,p. =smm axsmnpya coshVnia+npc (10) where thesumsmust extend over allthevalues ofmaandn$as defined in(8);thecoefficient CiaandDiphavebeenmerged with PIwhich therefore depends onaandftasindicated bythesub- scripts in(10). Inamore general caseonemight have toadd to(10) the solutions which correspond tothe singular casesm=and/or n=0. Ifonlym=0,then (4)givesX=C\x+C2,Yasbefore in(5)andZasin(9),butwith argument (npz);thisleads then tosingle summation inft. If onlyn=0,acorresponding single summation inawill result. Ifbothm=n=0,then theproduct (Cix+C2)(Diy+D2)(P 1z+P2) will occur. Inthepresent problemallthese.possibilities are excluded bythehomogeneous boundary conditions inxandy. The solution (10)represents adouble Fourier series inthetwo variables xandyasisnecessary inorder toexpress thegiven distribution G(x fy)defined over the finite area <x<a, <y<b.The extension from theone-dimensional Fourier series ofsection 29isstraightforward; general details onsuch 438 Three-dimensional Analytic Solutions [Ch.8 series arefound2inChurchill,03 p.116; inByerly,C2 p.139;and inCarslawC17inconjunction withproblemsofconduction ofheat. Assuming G(z,y)anoddperiodic function inxandy,bounded forallvalues ofxandyintheregionofdefinition andsatisfying theDirichlet conditions, thenonecanrepresentitas G(x, I/)=EZAatpsinmaxsinn&y (11) a |9 where inturnthecoefficients Aa,paredefined by 22rx=ia/^=b Aa,p=-'TI I G(XJ y)sinmaxsinnpydxdy (12) CLU/x=0t/i/=0 Sincea=1and=1in(8)define aand 6asrespective half fundamental periodsofthe distribution, G(x,y) canwellbe assumed oddwhen extended beyonditsregionofdefinition. To satisfy theboundary condition atz=0,comparison of(10) at z=with (11) yields atonce -Pa,i9tanhVm sothatthecomplete solution is N~~AsmhVma2+np2 (c-z) m $(x, y,z)= Aa,psinmaxsmn&y a ft smhVma+npc (13) withAa,pfrom (12). Any other boundary conditions with respect tothepotential canbehandled inanalogous manner. Ifthepotential isgiven overtwoormore ofthefaces, then theprinciple ofsuperposition canbeapplied, solvingforonly oneinhomogeneous boundary condition atatime asabove andthen taking thesum total ofall partialsolutions. Inaddition tothespectral solutions determined byseparation ofthevariables inclusive oftheirregular casesm=and/or n=0,there areaconsiderable number ofparticular integrals which attimes might lead tosimpler overall solutions. Thus, 2SeealsoH.S.Carslaw: Fourier Series and Integrals; Cambridge Uni- versity Press, Cambridge, 1930. Sec. 31] Cartesian Coordinate System 439 anyadditive combination with suitable individual constants of theterms *2-2/2 ,y2-*2 ,s2-*2 (14) andothers presents apossible solution, aswell asanygeneral solution ofthetwo-dimensional Laplacian differential equations inxandyoryand zorzand x\obviously, aconstant 3>can always beadded. Theselection ofthemost expeditious approach toanewproblemisstillanart,andtheonlyreassurance thata solution does indeed existandthat asolution isthecorrect and onlyonecomes from theexistence anduniqueness theorems ofpure andapplied mathematics, asfound inKellogg,clinCourant and Hilbert,04and inFrank andMises.C6Ifasolution satisfies the differential equation and alltheboundary conditions insofaras these arecompatible (or,perhaps better, correspond tosome physical reality), thesolution isthecorrect andonlyonenomatter how ithasbeen found. Ifintheabove problemc*oo fsothat therectangular box becomes arectangular semi-infinite prism, then thesolution (9) must bereplaced bytheexponential form Z=Plexp(-Vm a2+np2 z) (15) inorder toprovide regularity atz=GO.Keeping thesame boundary conditions asbefore results then in $(z, y,z)= Aa&exp (Vma2+np2 z)sinmaxsinnpy (16) where thecoefficients Aa,pareagain determined by(12). Boundary value problems involving themagnetic vector poten- tialcanbesolved with thesame facility, because intheCartesian coordinate system andinthisalone theidentity (6-16) holds, VxVxA =V(V-A)-(V-V)A withV-V=V2the conventional Laplacian operator. Since divA=aspostulated in(6-17), theproblem offinding solutions forthevector potential reduces tosolving thescalar Laplacian differential equations forthecomponents Ax,Ay,andAz,which is thesame procedure asjust illustrated forthe electrostatic potential. 440 Three-dimensional Analytic Solutions 1.8 GENERAL ORTHOGONAL COORDINATE SYSTEMS Transformation ofScalar Potential Problems toGeneral Orthogonal Coordinates. Assume ageneral orthogonal system ofcoordinatesui,u2,u$asgiveninFig.31-2; themutual relation- P" FIG.31-2 Orthogonal Curvilinear Coordinate System. ships between these coordinates andaCartesian system canbe expressed interms ofthefunctional relations ua=ua(x,y,z), a=1,2,3 (17) andtheinverse ones x=fi(ui, u2)M3),y=fz(ui,u2,1*3), z=/3(ui, 1/2,1*3) (18) Obviously, these reciprocal relations must beone-valued orat least restricted tosingle values within theapplicable ranges in order toprovide thenecessary uniqueness; they alsomusthaveno singularities within theranges used. Forthetransformation ofdifferential relations fromonesystem totheother, onetakes from (17) dua dua dua dx dy dz andconversely,*^tt i "f*a i*'f*a -i rtr / \= dx+ dy+ dz, a=1,2,3 (19) dx= ^dua, aduady=Er-5-dua, adua=^dua(20) Since thegeneral lineelement inCartesian coordinates isexpressed invector form (seeAppendix 3) ds=idx+jdy+kdz onehasforitsabsolute value ds2=ds-ds=dx2+dy2+dz2 Sec. 31]General Orthogonal Coordinate Systems 441 Introducing into (20)thevector representation, asforexample, dfi a/i d/idx=idx=idui+i-=-du2+i-^-du3 (21) andsimilarly fortheother two coordinate directions, one canagain form thescalar product ds-ds andobtains onaccount ofthepresumed orthogonality ofthegeneralized curvilinear unit vectors, thenormal form +h22du22+h32du32 where Ifnowdscoincides successively with thecoordinatedirections, so thatonlyoneofthedua^0,itgives thelinear arcelements in Fig.31-2 dsa=hadua (23) where hacanbeafunction ofallthree coordinates ua,butusually isarather simple expression adjusting forexample inasimple case anangular coordinate toalinear measure. Insome instances one canread thehavalues directly from theexpression fortheline element ds2 ;usually onehastoevaluate thehafrom (22)with the actual transformation equations (20). Good treatments ofthe transformation relations arefound inMason andWeaver,A16 p.116; inPlanck,A19 p.59;inStratton,A23 p.38;inKellogg,010 p.178; inHobson,09 p.1;inByerly,C2 p.238; inWebster,016 p.299;and inMurnaghan,C13 p.102; aswell asintheadvanced books on vector andtensor analysis. Though many authors, likeSmytheA22 andStratton,A23usethedefinition ofhaasgiven in(22), about again asmany usetheexact reciprocal ofit;caution istherefore necessary incomparing similar-looking forms. The expressions forthe first-order vector differentiations in generalized coordinates canbestbeobtained from theoriginal definitions. Thus, thegradient ofthescalar potential asthe linear rate ofchangeofthepotentialis d$V$=grad$=aa 442 Three-dimensional Analytic Solutions [Ch.8 and itscomponents arewith (23) =a-i+ + (24) Thedivergence ofavector canbededuced from Gauss's theorem (Appendix 3)applied toarectangular curvilinear parallelepiped formed bythecoordinate surfaces asinFig.31-2, The flux, for example ofvector D,through theopposite faces orthogonal toMI,isbytheuseoffirst-order linear approximation [Dids2ds3+ (Didszds3)dsi\DIds2ds3 dsi J where ds2ds3istheelemental areaandwhere thebracket gives thefluxoutofthefaceP-P"-p-Pr'r ]itis,ofcourse, important toobserve thevariation ofthelineelements ds2andds3asdefined by(23), along with that ofthevector component DI.Because ofthemutual independence ofdui,du2janddu3jtheresultant fluxcontribution becomes forthei/i-direction V-D=divD= _uUi OU2-(h2h3Di)du2du3dui du\ andanalogouslyfortheother two directions. Thesum total of thisflux isthen divDdr,where thevolume element dr=dsids2ds3=hih2h$duidu2du3 sothat (25) Thegeneraldifferential equation fortheelectrostatic potential isdeduced from divD=div(eE)=p asgiven in(2-1) andfollows with thedefinition E=grad$if oneintroduces therespective components from (24) into (25)1(MA)dM3 J " a/Ma d$\ a/MI a<E\ athji* a^>\"[=__ du\ hiEduj du2\h2duj du3\h3*duj]P (26) Sec. 31] Vector Potential Problems 443 Forconstant easinhomogeneous and isotropic media, onecan take itoutside and,iffurther nospace chargeispresent, onehas fortheLaplacian ofthescalar potentialI> vag=l ["d(h*h*j&\+J_(^i**\+JJhhiiYL hih2h3\_dui\ hidui/ du2\h2du2/du3\h%du^/j (27) Transformation ofVector Potential Problems toGeneral Orthogonal Coordinates. Applying thetheorem ofStokes (AppendixS)to theinfinitesimal curvilinear rectangle 0-p"-P'-p'" inFig.31-2, onehasforthecontribution tothelineintegral ofthe vectorVinthemathematically positive sense V2ds2-\V2ds2+/-(V2ds2)ds3]\L d3 JJ 8ds3+-(78ds3)- [ which reduces, because ofthemutual independence ofthedua,to -(^3^3) dusdu2-(h2V2)du2du3du2 dus Thismust becurliVintegrated overtheinfinitesimal areads2ds3, sothatupon division byds2ds3t curliV=7-(fc37 8)-~(fc,7 a) (28a) andwith cyclic rotation oftheindices oneobtains theother two components, namely, cur!2V=-i-f^-(/nFO-^-(fcs7 a)l (286)h3hi\_du 3 dui J cur!3V=-i-[/-(h2V2)- - (fc.70](28c) All/l2 \_OUi OU2 J Forthemagnetic vector potential Aoneactually needs the operation VxVxA,which isobtained inthesimplest manner by applying operation (28)once again tothecomponents (28).No further general simplificationispossible even ifoneassumes VA=ascustomary, since thesegregation VxVxA =V(V-A)-(V-V)A 444 Three-dimensional Analytic Solutions [Ch.8 asin(6-16) canbemeaningful only fortheCartesian system3if onereserves V-V=V2fortheconventional Laplacian operator, asappears thelogical choice. Inany case,itisnecessary to reduce thevector equations toscalar differential equations in vector components tomakethemamenable toprocesses ofsolution similar tothose employed fortheLaplacian differential equation ofthescalar potential. Separation ofVariables. Special solutions ofthepotential equation (27)cansometimes beobtained byinspection, butthe systematic approachisthereduction tosetsofordinary differential equations interms ofsingle variables. Thiscanbeachieved best bythemethod ofseparation ofvariables, assumingfirstthatthe potential function canbeexpressed astheproduct *(ui, 1*2,1*3)=F(Ul)G(u 2)H(u 3) (29) similar tothesimpler two-dimensional analogue insection 29. TheLaplacian differential equationisthenfrom (27)and,dividing through byFGH, r-i ^+or-iio+ff-iiH=0 dUi\hi f du2\h2I du3\h3) (30) since thedifferentiations pertain only tooneofthethree factors. Itdepends nowprimarily upon themetric factors hawhether or notcomplete separationispossible. Assume, forexample, thateach haisonlyaproduct function ofthecoordinates, fca=C(Wl)l?a(U2)ra(u 3) (31) then JL(W*F>\ =*M* d/kfadF\ dui\/ii / -n\ fidu\f!duj and similarly foreach oftheother terms in(30). This will permit theseparationofvariables ifalso =3(^1), 173(1*2)=171(1*2), fifaa)=ftfas) (32) 3Theidentity RXQXP=RPQR'QPisestablished only forvectors andneed notanddoesnothold fortheabove triple product involving the differential operator V. Sec. 31] Separation ofVariables 445 because then (30)reduces to l-(Hf=Q(33) Introducingasin(3)twoseparation constants byequating the lastterm in(33) tora2andthemiddle term ton2gives three ordinarydifferential equations, each oftheSturm-Liouville type (29-2)inwhich theboundary conditions willdefine thespectral selection ofthevalues raand n.Stratton,A23 p.198, gives a similar deduction withtheassumption that fta=Ma{(i*i)ii(ii a)r(i*3) (34) whereMadoesnotcontain uabutmight beanyfunction ofthe othertwovariables; Smythe,A22 p.124,findsaform similar to(33) for axially symmetrical potential problems. More specific criteria fortheseparabilitywillbeestablished inthefollowing two sections dealing with specific groupsofcoordinate systems. Thesystems permitting separationofthethree space variables with present-day methods canbegroupedinaccordance with their principal geometric aspects into Cartesian coordinate system (section 31),onlysystem symmetrical inallthree coordinates Cylindricalcoordinate systems (section 32)with conic sections normal totheaxis Circular cylinder Ellipticorhyperbolic cylinder Parabolic cylinder Confocal conicoid systems with axialsymmetry (section 33)andwith conic sections inthethree Cartesian coordinate planes Spherical system (and bipolar system) Prolate spheroidal system (and possible inverse) Oblate spheroidal system (and possible inverse) Paraboloidal system Toroidal system (and inverse ofcircular cylinder system) Systems involving elliptic functions (section 31) Ellipsoidal coordinates Annular coordinates (with possible inverse) 446 Three-dimensional Analytic Solutions [Ch.8 Thus, there areeleven distinct, separable, orthogonal coordi- natesystems (orsixteen, counting inverse andrelated systems) useful forthesolution ofpotential problems. Because oftherather involved mathematical apparatus needed forthetreatment ofthelastgroupofcoordinate systems, abrief summary ofthesimpler relations willbegiven here, whereas the twolarger andbyfarmore widely usedgroupsofcylindrical and axially symmetrical confocal systems willbetaken upinseparate sections. Orthogonal coordinate systems inwhich thevariables cannot be completely separated arestilluseful, butwithpresent-day methods solutions canbeobtained only inseries form notidentifiable with orthogonal function systems, sothatexamination ofconvergence becomes aprimary concern. Agood illustration isthebiaxial cylindrical coordinate system, which canbeused intwodimensions (see section 29)butdoes notpermit inclusion oftheaxial z- coordinate4without lossofseparability ofthevariables. ELLIPSOIDAL COORDINATE SYSTEM Theequationofageneral ellipsoid asinFig.31-3with thesemi- axesa>b>calong the z-,y- }2-directions, respectively,isin normal form Onecandescribe afamily oforthogonal andconfocal ellipsoids andhyperboloidsinanalogous manner tothetwo-dimensional conic sections byintroducing aparameter psuch that /j.2-.2 -2^+^+^=1 <35> This gives for+>p>(c2 ): ellipsoids for(-c2 )>p>(-b2 ): hyperboloidsofonesheet for(b2 )>p>(a2 ): hyperboloidsoftwosheets The ellipsoids areconfocal; setting z=in(35),onehasellipses ofhalf focal distance /i=(a2-62 )H ,therefore fixed; setting 4G.Mie,Ann. d.Physik, series IV, 2,p.201(1900). Sec. 31] Ellipsoidal Coordinate System 447 x=0,onehasellipses with/2=(b2c2)^;andsetting y=0, onehasellipses with/3=(a2c2)^>/IBOnecanconsider p>ascoordinate, defining uniquely anyparticular ellipsoid of thisconfocal family, andtostress itsrange ofvalues rewrite (35) y2 a2+ + +=1, >(-c (36a) As?>(c2 ),onemust alsohave z0,i.e.,oneobtains an infinitely thinelliptical disk inthex-y-planeofsemiaxes /3and FIG.31-3 Ellipsoidal Coordinates. /2</a-Thehyperboloidsofonesheet cansimilarly bedescribed bythecoordinate77 -.2 2 2 =1,(-c2 )> 17>(-62 )(366)a2+ 77^ 62+ 77-(c2+77) These hyperboloids arealso confocal andhave thesame focal lengths astheellipsoids. Setting x=ory=in(366), one hashyperbolas; butsettingz=0,onehasellipses ofmajor axis a'forwhich /3>a>/i,ofminor axis 6'</2,and offocal dis- tance 2/i. Thisshows thehyperboloids tobeofonesheetandto intersect the x-i/-plane wholly within thelimiting elliptic disk = c2 ;their limit is77 62andtherefore y*0,aplane strip bounded bythehyperbolasinthez-z-plane =1 448 Three-dimensional Analytic Solutions [Ch.8 Finally, thehyperboloids oftwo sheets aredescribed bythe coordinate f They areobviously confocal, and fory=and z=givehyper- bolas inthez-z-andz-?/-planes, respectively; forx=0,however, they giveimaginary intersection ofthet/-z-plane which isthusthe plane ofsymmetry. These hyperboloids intersect thez-axis for x<fiand inthelimiting case asf>62andy>0,become infinitely thin pencils andidentical with thesection ofthez-axis forwhich\x\>}\. Solving forthecoordinatesz,y,zfrom thethree relations (36) bydirect elimination, oneobtains x2=(/3/i)-2 [(a2+{)(a2+ )(a2+f)] V2=(/I/a)"2[>2+f)(&2+>7)(-b2-f)] (37) z2=(/a/a)"3 t(c2+)(-c2-i)(-ca-f)l with thefocal distances faasdefined above andwith allfactors positivewithin theproper ranges of,rj,ffrom(36). Differentiat- ingboth sides ofthe first linein(37),onehas 2zdx=(/a/I)'2 [(a2+ri(a?+f) +(a2+f)(a2+)dn+ (a2+f)(a2+if)*] (38) inwhich xcanbereintroduced from(37),andonethushasthe explicit form (20); similarly fordyand dz.Inaccordance with (22)onecanthenformulate themetric factors hajwhich areafter some considerable rearrangement5anduseof(47)from below, 47i22-tt- 11)(if-D02~2 0?); (39) =[(a"+m-&2-f)(-c2-f)]H 5For details seeparticularly Webster,Clflp.331; Hobson,09p.454; Murnaghan,013p.155;andByerly,c2 p.251. Sec. 31] Conducting Ellipsoid 449 This yields thenfrom (27) fortheLaplacian potential equation, ifonedivides through with (hih 2h^)asindicated andobserves the product character ofha, -rr1 -ftfo) +[(n-f)(6-f)]-1 (73(f)[ft(f)I?l=(40) dfi_ ofj Separation ofthevariables ispossible andleads tothesystem ofLam functions orellipsoidal harmonics which, ingeneral, invoLve elliptic integrals. Abrief treatment ofthese isgiven in Jeans,A1 p.244,and inWebster,016 p.333;more extensive treat- ments arefound inHobson,C9 p.459; inByerly,C2 p.254;and in advanced treatises onelliptic functions. Conducting Ellipsoid. Simple solutions result ifthepotential isdependent ononlyasingle variable, forexample f,which describes theconfocal ellipsoids.Ifaconducting ellipsoid ofsemiaxes a, 6,ciskept atapotential$>,then (40)reduces fortheoutside fieldto ar,xd$~| a* A which yields with (39)theelliptic integral oftheWeierstrass type * " (42) The limits havebeen chosen soastosecure thestandard form of theintegral;6thenegative signaccounts for appearing inthe lower limit. Ifoneselects *=forf=w,then =0.The constant Acanbedetermined bestfrom thetotal charge justasin thecase ofanysingle conductor (seesection 10or11).The field vector isfound from (24)with (39)and(41) 8Forasummary ofrelations andsome numerical values seeE.Jahnke and F.Emde: Tables ofFunctions, p.98;reprinted byDover Publications, New York, 1943; originally published byB.G.Teubner, Leipzig, 1938. 450 Three-dimensional Analytic Solutions [Gh.8 Forlarge values ofonecandisregard rjand,since their values are definitely limited by(366)and (36c), sothat 2A limE >-- -> Since onealsohasfrom (36a) for a2 , onefinds that atlarge distance thefieldvector varies as1/r2 ,as inthecase ofthesingle point charge (10-1), andonecanthere- foredetermine theconstant Aas This gives asfinal solution (c*+r* (43)7jf"t( with thefieldvector fromabove as f)r* (44) Onthesurface oftheconductor =andtherespective potential *odetermines thecapacitance oftheellipsoid Jo Thecharge density distribution isthen 2 (46)vy where thelasttransformation isobtained byforming [(z/a2 )2 -}- (2//Z>2 )2+(z/c2 )2 ]for= in(37), multiplying outthecor- responding right-hand sides, collecting terms, andobserving that ay_/bY/CY_ Jzfl)" \/l/2/+ \/2J^/"(47) Sec. 31] Axiaily Symmetrical Ellipsoids 451 Forthenumerical computations onecanreduce theelliptic integrals totheLegendre type.7 Treatments oftheconducting ellipsoid arefound inJeans,A1 p.247; inKirchoff,A13 p.34;inMason andWeaver/16 p.126; in Smythe,A22 p.Ill;andinStratton,A23 p.207, allofwhom deduce several ofthespecial cases below; also inKellogg,010 p.188; in Murnaghan,013 p.155;inByerly,C2 p.258;and inLamb,C22 p. 141,whoconsiders hydrodynamic applications. Application totheconducting ellipsoid inauniform electric field ismade inStratton,A23 p.209; thedielectric ellipsoid ina uniform electric field isalsotreated there(p.211), aswell asin Jeans,A1 p.253,and inMason andWeaver/16 p.156.The analogous solution forthemagnetic ellipsoid inauniform mag- netic-field isgiveninMaxwell,A17 II,p.66,and inFrank and Mises,06 II,p.720,andforfluid flowproblems inLamb,C22 p.143. For c= in(43), oneobtains thepotential produced bythe infinitely thinelliptic disk intheplane z=0.Thecapacitance canbeobtained from (45)aselliptic integral. Thecharge density follows from (46)bytakingcintothesquare root Q wherenowthe firsttwoterms vanish withc,whereas thelastone must bereplaced byitsexpression from (36a) with=0,sothat Thisbecomes infinitely large attherimofthedisk, asonewould expect. Axiaily Symmetrical Ellipsoids. Foraxialsymmetry about thez-axis inFig.31-3onehasa=6,anoblate spheroid, and this reduces alltheintegrals toelementary ones. Thepotential (43) becomes Qr* / l(+ ) 7SeeJahnke andEmde, loc.dt., p.59,andthereference there listed: J. Honel: Recueil deformulesetdetables num&riques; Gauthier-Villars, Paris, 1901. 452 Three-dimensional Analytic Solutions Thecapacitanceisreadily obtained as Q[Ch.8 (50) *t=o tan~V(o/c)2- andthecharge density,ifoneintroduces x2+y2=p2 ,becomes Q (7=(51) Ifc 0,onehastheinfinitely thin circular diskwith *= (a2-^-tan-1^ (52) andfrom thisforthecapacitance *(0)=8so (53) Thecharge density follows directly from (48)with a=band x2+y2=p2 , r\IA~ (54) Thisvalue holds, ofcourse, foreach side ofthedisk; inthecenter where p=onehasthesame density asonauniformly charged sphereofradius a.Thevalue ofcanreadily beexpressed in terms ofCartesian coordinates ifoneintroduces thesame simpli- fications into (36a). For axialsymmetry about thex-axis inFig.31-3 onehas b=c}aprolate spheroid, and thisagain reduces allintegrals to elementary ones. Thepotential (43)becomes Thecapacitanceisbydefinition from this Q (0) tanh-1VI-(6/a)2(56) Sec. 31] Annular Coordinates 453 which isidentical with (12-4), found therebydirect integrationin theCartesian coordinate system. Thecharge density becomes,if oneintroduces y2+z2=p2and 6=cinto (46), Q 47ra&2[a4b(57) Though theextreme values forx=0,p=bandx=a,p= hadbeen giveninsection 12,thisgeneral expression could notbe Fia.31-4 Annular Coordinates. found there inanysimple way. Theapproximationsforathin rodwith b<ahave been discussed insection 12andneed not berepeated. ANNULAR COORDINATES The circular annulus ofinner radius bandouter radius ain Fig.31-4canbeused asbasis ofanorthogonal coordinate system with axialsymmetry,inwhich therelations between,77,onthe onehand, andz,poftheunderlying cylindrical coordinate system, ontheother hand, aregiven bythe elliptic functions.8The sphereofradius \/ob isonemember ofthefamily ofsurfaces 7;=cons, intersecting thez-axis atright angles andterminating orthogonally ontheannulus proper. Theconfocal surfaces = conshavedoughnut-like shapes surrounding theannulus. Separa- 8Ch.Snow: TheHypergeometric andLegendre Functions with Applications toIntegral Equations andPotential Theory, p.295;National Bureau ofStandards, Washington, D.C., 1942. 454 Three-dimensional Analytic Solutions [Ch.8 tion ofthevariables,17,and<,thelongitude angle,ispossible, andtheensuing function systems arediscussed inthereference.9 Abilinear conformal transformation ofthemeridian plane w=z+jpintowr=c(w c)/(w+c)bends theaxis 2=into acircle and therefore theannulus into aspherical zone. This canagain betaken asbasis ofanorthogonal coordinate system whichis,infact, theinverse totheannular system andhasthe same function systems assolutions ofpotential problems. Iftheannulus shrinks intoacircularline, sothata=b,then thesystem describes thetoroidal coordinates (section 33);ifon theother hand,6=0,theannulus becomes thecircular disk, basis oftheoblate spheroidal system, andtreated in(52) as special case oftheellipsoidal coordinate system with axialsym- metry with respect tothe2-axis. Theannular coordinate system istherefore themost general axially symmetrical coordinate system permitting separationofthevariables. 32-CYLINDRICAL COORDINATE ANDFUNCTION SYSTEMS Asagroup, thecylindrical coordinate systems arecharacterized bythefactthatanycoordinate plane z=cons, with ztaken parallel tothecylindrical surfaces, intersects theother twocoordinate surfaces along conic sections. These arecircles and radial lines forthe circular, ellipses andhyperbolas fortheelliptical, and parabolas fortheparabolic cylinder systems.Ifthere isno variation ofpotential along thez-axis, thecorresponding two- dimensional cases result (seesection 29). Separability ofVariables. Itisofinterest toascertain the conditions ofseparability ofthevariables because itwill also serve asjustification thatonlythethree coordinate systems treated herehave attained practical significance. Foranycylindrical coordinate system, thethird coordinate is thelongitudinal oraxial coordinatez,sothatfrom (31-23) one infers atonce Ji3=1.Therequirement oforthogonality inthe z-i/-plane canbeinterpreted asmeaning thatanyother plane coordinate pair(,TJ)mustbetheresult ofaconformal transforma- tion w=x+jy=w(S ), f=+jrj (1) 9Ch.Snow, he. cit.; alsoN.Lebedev, Techn. Physics ofUSSR, 4,p.3 (1937). Sec. 32] Separability ofVariables 455 sothat (31-18) reduces to ;, y=j2Ui "n) where xandyareconjugate functions ofand77(seesection 25). This, inturn, implies that theCauchy-Riemann equations hold forxand?/,and, therefore, that (31-23) reduces tothesimple form ,9 ,9 ,9dw2 ni2=h22=h2=(2) TheLaplacianofthescalar potential (31-27) becomes thus Since hmust beindependent oftheaxial coordinate z,onecan introduce nowtheproduct function (Z) (4) andrewrite (3)upon dividing through by(4),withprimes denot- ingdifferentiation with respect tothepertinent variable, [tmtl|_|"-] 7"V+TT]+T- Thispermits atonce separation ofthelastterm 2"=m2Z,} Z=DIsinhmz+D2coshmz} where ra2canbeanyconstantvalue,realorcomplex. This leaves then E-iS+H-W =-mW (6) Ithasbeenshown1thatthenecessary and sufficient condition offurther separabilityisthefactthat Aa(,u)=ffitt)+ff2(i) (7) where g\andg2arefunctions ofonly and77,respectively. This dw' means, that mustitself beseparable intoasum offunctions 1Ch.Snow :TheHyper geometric andLegendre Functions withApplications toIntegral Equations andPotential Theory, p.202; National Bureau ofStand- ards, Washington, D.C., 1942. Reference ismade there toG.Haentzschel : Studien iLber dieReduktion derPotentialgleichung aufgewohnliche Differential Gleichungen; G.Reimer, Berlin, 1893. 456 Three-dimensional Analytic Solutions [Ch.8 each ofonlyonevariable. This obviously limits thechoice of practical cylinder coordinate systems totheconical sections, since only forthefamily oftrigonometric (including exponential and hyperbolic) functions onehasaclear separation asforexample 2 sinf=cos2+cosh2 77 1 Asarather special case (parabolic cylinder), onealsohas dw=4 + Introducing (7)into (6),oneobtains upon separation thetwo ordinarydifferential equations oftheSturm-Liouville type (8) TIT+ +P2]H= where p2isthesecond separation constant. CIRCULAR CYLINDER COORDINATES The axialsymmetryofthecircular cylinder makes itsimpler to proceed with thespecific coordinate relations rather than toapply thepreceding general deduction. Ofcourse, onecanemploy the conformal mapping function w=e~randobtain (3)and (8)as shown interms ofthecoordinates and77;onecanalso define p=e~*with oo<<+,$=77assuitable coordinates andsystematically obtain thegoverning equations (8)interms of themore usual coordinates pand\l/. Conventionally, however, onechooses ascoordinates directly thenormal distance pfrom thecylinder axis,theangle^counted from thez-axis oftheunderlying Cartesian system andthedis- tance zalong thez-axis fromanassumed origin 0.Thecoordinate surfaces arep=cons, giving coaxial right circularcylinders, \l/=cons, yielding planes through thez-axis, and z=cons, yield- ingplanes normal tothe axis. The lineelements inthethree coordinate directions are,forthepointPinFig.321, dp,=pd\fr, dz(9) Sec. 32J Circular Cylinder Coordinates 457 sothatbycomparison with (31-23) hi=1, h%=pj h$=1 (10) andthus forthecomponentsofthe field vector inaccordance with (31-24) d& _ ia* _ as,__Ep= fE+=1Ez=(11) Thepotential equation (31-27) becomes with (10)above =(12) Introducing theproduct function ^D/\D/lN 'Z/ \ ^1Q\<p=n>\p) r\y/) \z) \*-&) anddividing through byit,onecanreadily separate thevariables, A^ P" ds FIG. 321Circular Cylinder Coordinates. ifonestarts with thelastterm, leading to(5).Theremaining partin(12)nowreads oralso &-lTVTdp\ dp which permits further separation byassuming P"=-n2P P=BIsinnif/+B2cosnty(14) (15) 458 Three-dimensional Analytic Solutions [Ch.8 With(-n2 )forthelastterm, (14)gives finally which hasassolution theBessel functions2offirstandsecond kind ofordern R=CiJnOnp) +C2Nn(mp) (17) These functions canform orthogonal systems ofdifferent types which canbeused forexpansions ofinhomogeneous boundary values. Asspecial cases, onehastoconsider n=0,forwhich the dependence onzremains asin(5)butP=B\$+B2reduces to alinear form, andtheBessel functions become ofzeroth order, astreated in(30-41) foraxially symmetricalfields.If,onthe other hand,m=0,thenZ=DIZ -fD2from (5),thedependence on^remains thesame as(15),but(16)nowreduces tothe first oftheforms (29-48), resulting inthetwo-dimensional circular harmonics.If,finally,m=n=0,thesolution of(16)degenerates intothelogarithmic function, andthetotal contribution tothe potential solution becomes (CiInp+C2)(Dl2+ Hollow Cylindrical Ring. The hollow cylindrical ring of Fig.322with theindicated boundary potentials in(a)hasaxially symmetrical potential distribution, sothatindependence of\l/can bepresumed,orn=0.Thehomogeneous radial boundary con- ditions require from (17), sincen=0, CiJ(ma)+C2N(ma)=CiJ Q(mb)+C2N(mb)=(18) which canonlybesatisfied bynon-trivial values ofC\andC2 (non-vanishing)iftheir coefficient determinant vanishes, or J(ma) A/oM>)-J(mb)N(ma)=(19) 2Brief reviews ofBessel functions aregiven inSmythe,A22 p.168;Churchill,03 Chapter VIII;andalmost anybookonadvanced calculus. Extensive treatises areGray, Matthews, andMacRobert07 ;Byerly02 ;N.W.McLachlan: Bessel Functions forEngineers; Oxford University Press, 1934; andG.N. Watson: Theory ofBessel Functions; Cambridge University Press, 1922. Fortables seeJahnke andEmde: Tables ofFunctions; reprinted byDover Publications, New York, 1943; originally published byB.G.Teubner, Leipzig, 1938.Abriefsummary ofimportant relations isgiveninAppendix 5. Sec. 32] Hollow Cylindrical Ring 459 Setting ma=x,mb=ma(b/a)=kx,the first sixroots ofthis relation foralarge range ofvalues fcaregiven inJahnke and Emde,3 pp.204-209. Since from (18) also N(maa)_JV(mab) JQ(maa)J(mab)a=1,2,3, (20) fortheroot valuesmaascomputed from thetables, and since furtherD2= in(5)because of *=atz=0,thesolution for thepotentialisatthisstage =5ZCasinhmaz at J(mab")[' N(maP)(a) (21) Theonlyremaining boundary con- dition tobesatisfied isforz=b, where itisrequired toexpand the given function G(p) intoanor- thogonal system ofBessel func- tioncombinations ascontained in thebrackets of(21). This can,7.0. 1= I26 $=0 FIG.32-2 Hollow Circular Cy- lindrical Ring withTwo Typical Potential Applications: (a)radial, (6)longitudinal. indeed, bedonebecause thehomogeneous boundary conditions in pspecify theproblem asoftheSturm-Liouville type; seesection 29. Actually, with theabbreviation R(map)forthebracketed function in(21), the coefficients Caaredefined by[Appendix 5,(43)and5,(40)] sinhmacCpG(p)R (map)dp(22) t/p=a where the firstterm{ }=2JVa,withNathenorm oftheRQ functions asgiveninAppendix 5,(40).Themost salientdifficulty with theBessel functions isthelack ofknown integrals inclosed form, sothatmany expressions like (22)remain purely formal unless numerical ormachine computations are feasible. This 3E.Jahnke andF.Emde: Tables ofFunctions; reprinted byDover Publica- tions,NewYork, 1943; originally published byG.B.Teubner, Leipzig. 460 Three-dimensional Analytic Solutions [Ch.8 solution isgiveninSmythe,A22 p.183,and inByerly,C2 p.230, fortheequivalent temperature problem; Kellogg,010 p.203,treats thesimilar casewitha=0,thehollow finite cylinder, andagain Byerly,C2 p.226,gives thelatter solution fortheequivalent tem- perature distribution problem, modifyingitalsofordT/dp=on p=bandfordT/dp+hT=onp=b.Churchill,03Chapter VIII, solves several ofthesimpler problems involving time varia- tion. If,inthesame problem, Fig. 32-2, theboundary potential distribution (6)isselected, then theboundary conditions inz arehomogeneous, indicating trigonometric functions inz.Itis therefore preferable tochoose anegative sign in(5),sothat Z11=-m2z } \(23) Z=DIsinmz+D2cosmzJ leading totheconditions Z(0)=D2=0, Z(c)=DIsinme= with thespectrumofm-values airma= >a=1,2,---oo(24)c Since axialsymmetry prevails, n=and (16)becomes (25) with thesolution interms ofBessel functions ofimaginary argu- ment R=CiJoO'rap) + oralso intheform ofthemodified Bessel functions* R=Ai/(mp)+A2KQ(mp) (26) where these functions aredefined sothattheytakeonrealvalues; this ismerely amatter ofconvenience inorder tokeep thecon- 4Nouniformity exists with respect tothedefinition ofthemodified Bessel function ofthesecond kind; seeAppendix 5fortheinterrelations between current usages. Forthepresent exampleitdoesnotmatter which definition forKQischosen. Sec. 32] Finite Conducting Cylinder 461 stantsAiandA2torealvalues, since thephysical problem can tolerate only areal solution. The condition atp=arequires R=in(26), sothatthetotal solution takes theform aKQ(maP)siK(maa) Jsinmaz (27) Thismust thenrepresent theconventional Fourier series expansion ofG(z) atp=b,sothatthecoefficients arefound by KQ(ma b)^ 2r*==- / G(z) sinmazdz(28) Ci/z=0(waa) 2 c Thesolution isgiven inSmythe,A22 p.195,andalso inByerly,C2 p.232, fortheequivalent temperature problem; inboth instances, thespecial casea=isdeduced bysimply dropping themodified Bessel function ofthesecond kind, since ithasalogarithmic singularity atp=andcannot contribute tothesolution. Again,iftheboundary conditions require given potential varia- tions over several oftheboundary surface parts, theneach one condition canbecombined with zeropotential over allother parts tomake upatypical problem asillustrated. Thesum total of allindividual solutions willconstitute thecomplete solution by superposition. Finite Conducting Cylinder. Afinite conducting cylinder of length 2c,diameter 2a,andconductivity y,asshown inFig.32-3, hasapplied two electrodes atz= 6forcurrent supply and collection; thewidth ofthese electrodes is5,and itisassumed that thecurrent density normal totheelectrode areas canbedefined as //27ra5. Theflowmust beconfined within thecylinder, so thatonallsurfaces thenormal electric fieldmust vanish except over thetwobands where ithasthespecified value I/2iray8. The solution forthepotential andcurrent distribution willbe axially symmetrical, sothatn= in(8),andbecause ofthe finite lengthofthecylinderitwillbepreferable tochoose (23) fortheexpression ofZ.Theboundary conditions inzare homogeneous and ofthesecond kind, requiring attheends Z'(-c)=Z'(+c)=0,sothat m(Di cosmc+D 2sinrnc)=m(Di cosmeD2sinme)=0 (29) 462 Three-dimensional Analytic Solutions Thiscanbesatisfied onlyifD2 and cosmc-0, ma=(2c*+ 1)7r , =0,1,2, U\j.8 (30) The solution forR(p) willagain begiven by(26), butonly the firstkind ofthemodified Besscl function canbeadmitted, since Diagram of peripheral current density FIG.32-3 Current Flow within Finite Cylinder. hasalogarithmic singularity atp=0.The solution is therefore, uptothispoint, givenby )=AaIQ(map)sinmaz(31) Inorder todetermine thecoefficients Aa,onemust expand the assumed peripheral current distribution into theconventional Fourier series Jp(p=a)=Jasinmaz(32) at where, because oftheoddsymmetry, thecoefficients aregivenby 2.smmazdz /sinma5/2 .--T^vacma&/2y N (33)^ ' Sec. 32] Finite Conducting Cylinder 463 From thepotential distribution (31)onehastheradial current density P T dpT amaalm"PSmm*Z andcomparing thecoefficients ofthisFourier series atp=awith (33),onehas =*/sinm5/2\ sinmab tracy\mad/2 )mJi(maa) Itisseen that as5 thefactor inparentheses approaches unity sothatnoloss ofgenerality results ifoneassumes 5=0, though justification would beneeded fortheapplication ofthe Fourier series. Since thepotential difference isreadily given from (31)as V=$(p=0|Z=b)-*(p=0ig=_5)=2Aa/(mao)sinraab(35) onecanwrite forthetotal resistance with (34) *_2_/o(77l aq),^xxiigyi<*\ .2i/q/^Nl~*cy^ (maa)A(m aa)1-"> /smm-^b^ Thisproblem wastreated bySmythe,A22 p.236,andasimilar method wasusedbyOllcndorff,A18 p.341,tocompute theampli- fication factor ofatriodc with ahelicalgrid. Iftheelectrodes inFig.32-3donotcover theentire circum- ference butextend only from ^=(ir/q) to^=-\-(v/q) t where qmaybeanarbitrary realnumber, thentheaxialsymmetry willnolonger holdandthepotential function willbethedouble summation *(p, *,2)=ZE(Aniasmnt+B ntacosn\fr)I n(map)smm az (37) where thecoefficients BIandB2of(15)weremerged withDIof (23)andAIof(26)togiveAniCtandBUiaandwhere themaare thesame asin(24). Thisnowrepresents adouble Fourier series ofsame type as(31-10), and itscoefficients must bedetermined by comparisonoftheexpression fortheradial current density from 464 Three-dimensional Analytic Solutions [Ch.8 (37),withthedouble Fourier series representing thegiven current density. From (37)onehasatonce 6$ JP=-7 =-7L[An.a sinnty+Bn,acosn^]- OP na-- In(map)+/n-i(wap) masinmaz(38)map J The double Fourier expansion ofthegiven current density, defined as (ql/2ira8) overtheelectrode surface,isformally JP(p=<0=LJn>acosn^sinwaz(39) na where thecoefficients aredetermined forn^1bythedouble integral 4/ir/Q *=6+5/2g/Jna=--Id^I - -cosn\l/sinmazdzcvJ+=Q Jz=b-6i2 2iraS 21/sinnv/q\ /sinma8/2\ . --- '- smm\/s 1' (/\ 7rac\ n7r/g /\ma8/2 Usehasbeenmade ofthetwosymmetries, namely, thatJpisan even function in^andanoddfunction in z.Comparison of (38)atp=awith (39)indicates now An.a=0,Bn,a=/7i-i(m aa)--In(m aa}-^ L maa Jmay n21(41) where Jn,aistobetaken from (40). Thesums in(37), (38),and (39)mustbetaken fromn=ton= <x> ;however, theexpression (40)holds onlyforn^1because forn= 2TTr>+*/2 ql.Joa=--- I ^-sinmaga2 CTTq //sinm a5/2\. -- (-^ Jsmmab(42)TracVma6/2/v y which must beused for# ,a-Tneresistance between theelec- trodes canthenbedetermined asbefore. Point Charges and Dielectric Plate. Theproblem ofa single point chargeQlocated infront ofafinitely thick dielectric Sec. 32] Point Charges andDielectric Plate 465 plate canbetreated bythemethod ofimages asinsection 21; however, thisbecomes verycumbersome andtheresults arenotin practical form.Adifferent approachistheexpressionofthe point chargefield interms ofcylindrical coordinates andsatisfying theboundary conditions asintheconventional boundary value problem. Thepoint chargefieldalone isgivenby Q Q (43) inaccordance with Fig.32-4. Thiscanbeexpressed asaFourier integral *(p, z)=~f"J(mp)e-'*'dm 47TnAn=0(44) listed aspair557intheC.-F. tables5with rafortheintegration variable gthere. The total field inthethree regions must bebuiltupinterms of solutions (5), (15),and (17). Starting withn=because ofthe obvious axial symmetry, and rejecting in(17) thesecond term because itslogarithmic singularity ontheaxis p=0, onehas leftonlyCiJ(p), which must bethesame forall three regions except fordifferent constants. Thesolution of(5) musthaveD2=+Di forregion 1toprovide decreasing values forz<0,musthaveDI=D2 forregion 3toprovide decreas- ingvalues for z>0,and willQB (3) FIG.32-4 Point Charge andDielec- tricPlate. contain both constants forre- gion2.Sincenospectralselec- tion ofmvalues ispossible,allfinal solutions must beinterms of Fourier integrals. Thus, thetotal solution forregion1with su- perpositionof(44) fortheactual point charge there, andthose BSeereference, footnote 11onp.395; alsoBateman,cl p.409. 466 Three-dimensional Analytic Solutions [Ch.8 fortheother regions are ~|JQ(mp)dm =o (45) =0 where theconstants Chave beenmerged with theDconstants. Because theFourier integrals areunique representations, onecan satisfy theboundary conditions interms oftheintegrands in brackets. Continuity ofthepotentials andthenormal components ofdielectric fluxdensity requires (46) fromwhich onecansolve fortheconstants, forexample, e"2"16-kl22e~2mdQW= '" Inthese expressions, fci2= -fc23=!i^-e (48) EO+e canbedefined asreflection coefficients inanalogy tooptical problemsortotransmission linetheory. With theconstants from (47), theintegrals in(45) canactually beevaluated bythe theorem ofresidues orbyexpansion into partial fractions leading toinfinite sums related totheresults obtained byimage theory, though inmuch simpler form. Thisproblemistreated bySmythe,A22 p.181.With afinite radius ofthepoint charge onecanthencompute thecapacitance asinfluenced bythepresence ofthedielectricplate oronecan translate thisintoacurrent flowproblem exchanging dielectric Sec. 32] Point Charges andDielectric Plate 467 constants against conductivities asinMaxwell/17 I,p.443. For thepoint source located atAinFig.32-4,andassuming medium 1 tobenon-conductiveair,Smythe,A22 p.237, gives thesolution to thecurrent flowproblem; this isofpractical value ingeophysical problems exploring thestratification oftheearthbymeasurement ofthepotential distribution onitssurface between two point electrodes.6 Onecaninsimilar manner solve forthepotential distribution ofa circular ring ofcharge found bydirectintegration in(12-58) and inthepresence ofground in(12-65). Observe thatthepotential values ofthepoint charge along theaxisareobtained with p= from (44)andthat thepotential values along theaxis ofthe circular ring ofradius aasfound in(12-60) canberepresented by introducing p=ain(44),whereby 1 r 2=JJQ(ma)^1*1dm(49) Therefore, thepotential anywhere inspace should be Qr* M $(p, z)=-IJQ(ma)JQ(mp)e~m^dm(50) introducing thesame factor J(mp) asforthepoint charge; see Bateman,01pp.410, 417. Theform (50)cannowbeused to satisfy boundary conditions inanalogous procedure asforthe point charge. Forthecircular diskwithuniform charge distribution onecan simply integrate (50)with respect toafrom zero totheradius 6 ofthedisk. Thus, ifthetotal chargeisnow Q,then forthe elemental circular ringonehas dQ=~- bIT andtherefore (51) 6S.Stefanesco andC.andM.Schlumberger, Jl.dephysique, 1,p.132(1930). 468 Three-dimensional Analytic Solutions [Ch.8 This result7canagain beused forthesolution ofproblems involv- ingadielectric plate orplates parallel totheface ofthediskas above. VeryThin Cylindrical Coils. The electric fields ofthin cylindrical coilscanbecomputed bysolving thescalar potential inside andoutside interms oftheproduct functions (5), (15), and (17),where inside thecoilonlythefirstkind ofBessel function canbeadmitted, whereas intheoutside space thetwoforms in (17)combine intotheHankel function togivevanishing results asp>oo.Ollendorff,A18 p.337,applies thistoashort cylindrical coilwith thesimplifying assumption thattheelectric field inthe endfaces ispurely radial. Thevector potential inidealized thin cylindrical coilswithno axial current flowhasonlyaperipheral component A#,which, even foraxial symmetry, does notsatisfy Laplace's differential equation. Smythe,A22 p.290, finds themagneticfield distribution within thewindows ofanidealized shell-type transformer with very thin cylindrical windings andassuming theironasinfinitely permeable; healsogives several goodfield graphs, indicating the effect ofthepositioningofathincylindrical coilwithin thewindow. ELLIPTIC CYLINDER COORDINATES Here itisdefinitely advantageous tofollow thegeneralized relations atthebeginningofthesection. Utilizing theconformal transformation (see26-51) w=x+jy=fcosh(+jrj) (52) oneobtains x=fcosh cos17, y=fsinh sin17 (53) which represent confocal ellipses andhyperbolas with thefocal distance 2/asshown inFig.29-6. From (53), /V ,/yV=j(__Y _(yV_, \/cosh/ ^V/sinhf/' V/cosr,/ V/sinr,/ (54) The firstrelation describes theellipses withsemiaxes a=fcosh, b=fsinh f;thesecond relation gives thehyperbolas ofsemiaxes a=fcosrjjb=fsinrj.Specifically, fisanalogous totheradial distance pofthecircular cylinder, and f=istheellipse which 7A.Gray, Phil. Mag., Series6,38,p.201(1919); alsoBateman,01 p.410. Sec. 32] Elliptic Cylinder Coordinates 469 hasdegenerated intothefocal lineF1F2;17=andrj=2irare thehyperbolas which have degenerated intothepositive z-axis fromF2totheright,andrj=TTisthehyperbola which hasdegener- ated intothenegative z-axis from FItothe left; TJ=T/2 isthe plane ofsymmetry or2/-z-plane intheunderlying Cartesian system. Inaccordance with (2)onehasfrom (52) ~\=f2 (cosh2 J-cos2 i,) (55) sothatthecomponents ofthe fieldvector become from (31-24), with/is=1, E***--(cosh2-cos2 TJ)~^ En=--(cosh2-cos2i\Ty* ' /<? f dz E*i z(56) Defining in(55) 0l()=/2COsh2 f, 2(r])=-/2COS2 TJ, onehasdirectly from (8) ~p+(m2/2cosh2 f-p2)S=(57) J2|_|-V+(p2-m2/2cos2,,)H=(58) arj Both functionssatisfy, therefore, differential equations ofthe sametypewhich degenerateform intothestandard differential equations forhyperbolic andtrigonometric functions. Themore general form (58)withm5^ possesses solutions which arecalled Mathieu functions;8these solutions areperiodic in?].with period 2irasrequired fortheelliptic cylinder,ifp2is selected foranygiven valueminaccordance withadeterminantal 8Brief treatments ofMathieu functions aregiven inE.T.Whittaker andG. N.Watson: Modern Analysis, FourthEdition, Chapter XIX; Cambridge University Press, 1927; inStratton,A23 p.376;and inInce: Ordinary Dif- ferential Equations, Chapter XX;Longmans, 1927. Further details aregiven inM.J.O.Strutt: Lamesche, Mathieusche undverwandte Funktionen inPhysik undTechnik; J.Springer, Berlin, 1932.Asummaryoffunctional relations andgraphical representations aregiven inE.Jahnke andF.Einde: Tables ofFunctions; reprinted byDover'Publications, NewYork, 1943; originally published byB.G.Teubner, Leipzig, 1939; thenotation ofJahnke andEmde hasbeenusedhere 470 Three-dimensional Analytic Solutions 1.8 equation which leads toadenumerably infinite setofpnvalues. Aswiththedegenerate solution form=0,there areevenandodd functions, sothatthegeneral solution isofthetype Hn(n)=Bi(n)cen(i, fm)+B2se^,m) (59) where nisanordernumber starting from forthecefunctions (elliptic cosines) andfrom 1forthesefunctions (elliptic sines). TABLE 32-1 COMPARATIVE NOTATION FORELLIPTICAL COORDINATES *Same asJahnke andEmde,loc.cit.,p.283. tActually, theordernumbers nandmappearininterchanged positions in thisreference. Asperiodic functions, they can, ofcourse, alsobeexpanded into conventional Fourier series forwhich therecurrence formulas are found inJahnke andEmde,loc. cit.These ceand sefunctions form acomplete orthogonal system which canbenormalized in thesame general manner asthetrigonometric functions. Foranysolution Hw(r?)with theparameters mandpnthere exists asolution E7l(f)of(57), called associated radial Mathieu functions byStratton,A23 p.378, ormodified Mathieu functions,9 which areexpressible asinfinite sums ofBessel functions; choosing Bessel functions offirst, second, orthird (Hankel) kind, onehas therespective kinds ofassociated radial Mathieu functions. 9H.Jeffreys, Proc.London Math. Soc., Series 2,23,pp.437and455(1925); also P.Humbert: Fonctions deLam6 etfonctions deMathieu; Gauthiers- Villars, Paris, 1926. Sec. 32] Elliptic Cylinder Coordinates 471 Thecomparative notation ofsome references isgiveninTable 32-1. Asasimple illustration takethesplit elliptic cylinder ofinfinite lengthinFig.32-5. Because ofhomogeneityintheaxial direc- FIG.32-5TwoHalf Elliptic Cylinders. tion,nodependence onzwill exist, sothatm=andthesolutions of(57), (58)become g(f)=Cisinhp+C2coshp (60)H(77) Isinpri+B2cos There aretwosymmetry conditions which itisalways good to utilize; namely, themajor axis=must beafield line, sothat alongit ^H(S)=(pC lcoshp|-HpC2sinh LO Jt=o which yields Ci=0;andalong theminor axisTJ=ir/2and 17=37T/2 thepotentialisconstant andequal tothemedianvalue, namely zero, sothat (61)This leaves then forthepotential *(> i})=Bpcoshpcosprj P iftheremaining constants C2,B2aremerged andmade dependent onp.The finalboundary condition requires thepotentials on 472 Three-dimensional Analytic Solutions [Ch.8 =oasgiven inFig.325,which canobviously besatisfied by considering (61)aFourier series expansion inrj.From Fig.32-5 itisseen that$=$for-(ir/2)<17<(r/2) and$=-<f for(x/2)<T;<(3^/2). The conventional Fourier series for thissymmetrical rectangular function is cos(2n ,Tra^n -+- sothatcomparison with(61) yields p=2n+Iandasfinal solution cos (2ra+1)J? (62) n=o2n+1cosh (2n+1)v' Thevalue offisdetermined bythegiven axes oftheellipse, since from (46) forpointAonehasy=0,x=acorresponding to TJ=0,=o,andsimilarly forthepoint B,sothat a=/cosh0j 6=/sinhCo, Co=tanh"1-(63)a The fieldvector canbecomputed from (56)andwith itthecharge densities andcapacitance forasmall but finite gapbetween the halves. PARABOLIC CYLINDER COORDINATES Thecoordinates inthez-y-plane arechosen todescribe orthogo- nalparabolas asinthecase ofthetwo-dimensional parabolic co- ordinates insection 29with theadditions ofthethird coordinate z. Asseen inFig. 29-7, thetwo families ofparabolas canbede- finedby C=V2~p cos- TI=V2~Psin(64)Z 2 Specifically, C=istheparabola which hasdegenerated intothe negative z-axis and17=istheorthogonal parabola which has degenerated into the positive x-axis. Thecommon focus is located attheorigin 0,andthesigns ofandrjareuniquely defined by^in(64). Interms ofaconformal transformation one canexpress (64)by w=x+jy=ytf? (65) which gives parabolas asshown insection 29and inparticular Pcos^, y=fr=psin^ (66) Sec. 32] Parabolic Cylinder Coordinates 473 Using (65) in(2),oneobtains atonce **+, (67) Thecomponents ofthe field vector are, therefore, from (31-24) Et=-a2+fry^>*,=-(?+,2rM^.df dri d3>E,=-^(68)dz Separatingin(67), theindividual differential equations (8)become inthiscase ^4+(m2?-p2)Z=(69) 5-+(raV+p2)H=0 (70) dij2 Again, both functions satisfy differential equations ofthesame typewhich degenerateforra intothestandard differential equationforthehyperbolic andtrigonometric functions. Themore generalforms withm^ lead totheorthogonal function systems oftheparabolic cylinder; thus, bydefining in (69)anew variable s=\/2jm, and selecting fortheavailable constant p2=2jm(w+Vi],the differential equation results which hasassolution theparabolic cylinder functions10 exp-Hn(s) (72) 10Introduced byH.Weber, Math. Annalen, 1,p.1(1869); brieftreatment inE.T.Whittaker andG.N.Watson: Modern Analysis, Fourth Edition, p. 347; Cambridge University Press, 1927; and inBateman,clp.488.A summary offunctional relations andcurves aregiven inE.Jahnke andF. Emde: Tables ofFunctions; reprinted byDover Publications, New York, 1943; originally published byB.G.Teubner, Leipzig, 1939, whose notation hasbeenused here. 474 Three-dimensional Analytic Solutions [Ch.8 The coefficients aresochosen that^n(s)becomes normalized for real sintheranges= QOtos=+<;thefunctions Hn(s) aretheHermite polynomials11defined bytherelation (73) asthey areused in(72). Inanalogous manner oneobtains as solution of(70) Hn(i)s*n(js) (74) since (70)becomes identical inform with (69)ifonereplaces nbyjV Thecomparative notation ofsome references isgiveninTable 32-2. TABLE 32-2 COMPARATIVE NOTATION FORPARABOLIC COORDINATES Coordinate ThisBook BatemanciStrattonA28 ui 7; uz -n -n u3 z z z Hermite polynomial Hn(s) Un(s) (Generating exponential) expfJexp(s2 ) *Same asJahnke andEmde, loc. cit.tp.32. 33-CONFOCAL SPHEROIDAL COORDINATE ANDFUNCTION SYSTEMS The confocal spheroidal coordinate systems arecharacterized asagroup bythefactthateach oftheir coordinate surfaces is intersected bythethree Cartesian coordinate planes x=0, y=0,and 2=along conic sections. Since thegeneral ellip- soidal coordinate systemisdiscussed insection 31,onlycoordinate 11Because oftheir importanceinquantum mechanics, theHermite poly- nomials withexp(s2 )instead ofexpfJaretreated inpractically any introduction tothis topic, such asV.Rojansky: Introductory Quantum Me- chanics; Prentice-Hall, NewYork, 1942; andL.Pauling andE.B.Wilson: Introduction toQuantum Mechanics; McGraw-Hill, NewYork, 1935; seealso E.Madelung: Mathematical Tools forthePhysicist, p.59;reprinted byDover Publications, NewYork, 1943; originally published byJ.Springer, Berlin, 1936. Sec. 33] Separability ofVariables 475 systems with axialsymmetry willoccur here; this willpermit further generalization with respect totheseparation ofvariables. Itmight bestressed thatsymmetry ofthecoordinate system does notimply symmetry ofthepotential fields. Separability ofVariables. Onecanestablish basic conditions ofseparability quite similar tothose demonstrated insection 32 andthus justify again therelatively smallnumber ofcoordinate systems thathave attained practical significance. Forany coordinate system with axial symmetry one will choose asonecoordinate theangle ofrotation about theaxis ofsymmetry. Since thecircular cylinder coordinate system has inameridian plane thesame rectangular reference grid asthe Cartesian system normal toitsz-axis, onecanuse itasbackground system and, indeed, introduce thecomplex notation w=z+jp and consider any other orthogonal meridianal coordinate pair (,TJ)asreferred toitbyaconformal transformation(Fig. 331) Thedependence ofthegeometric scale inthemeridian planeupon thedistance pfrom theaxis ofrevolution isindicated inthe Laplacian potential equation ofthe circular cylinder bythe appearanceofthe firstderivative inp,namely, from (32-12) op pop pd(h dz Itisconvenient forthegeneral discussion todefine amodified potential function v'p$andtoseparate atonce thedependence on</>,sothatoneintroduces U(p, z)-F((f>) =V^*(P, 0,z) (3) into (2)which yields upon division byUF =(4) uyi Separationofthelastterm gives, therefore, sinra</>+A2cosm<j> 476 Three-dimensional Analytic Solutions [Ch.8 wheremisnormally aninteger, permitting conventional Fourier series expansions in0.Though thecoordinate systems inthis section areaxially symmetrical intheir coordinate surfaces,it does notfollow that allpo- tential solutions musthave the samesymmetry! =consThereduced potential equa- tionpertaining tothemeridi- onal distribution cannowbe written for the cylindrical system d2U d2U dp2+ dz2'cons FIG.33-1 General Coordinate Sys- temwith AxialSymmetry.=(6) which willbeused asthe rectangular background sys- tem.Any other pair ofmeridian plane coordinates (,rj)must berelated to(z,p)by(1),which defines 2=/i(,7?) Ph(i 17) asconjugate functions (seesection 25)inthesame sense asinany two-dimensional geometry. Since theCauchy-Riemann equations must hold forzandp,thetwo-dimensional metric factor from (31-23) becomes UJ2 (7) TheLaplacian inpand zinequation (6)transforms inaccordance with (26-5)if(z,y)isreplaced by (f,17)here, sothat(6)changes to flu2/ P2 Forfurther separation ofthevariables, oneintroduces nowthe product function which yields /i \1.2 2 0) Sec. 33] Spherical Coordinate System 477 Ithasbeenshown1thatthenecessary and sufficient condition offurther separabilityisthefactthat where 0iandg2arefunctions ofonly and77,respectively, This means thatdw2 itselfmust yieldafactor p2andtheremainder must beseparableintothesum oftwoindividual functions ofthe variables. This obviously imposes severe limitations upon the choice oforthogonal families ofsurfaces which canserve asorthog- onal coordinate systems with separability ofthe variables! Again, asinthecylindrical coordinate systems,itisprimarily thefamily ofconic sections which allows clear separability inthe mapping function;there isanadditional system employing elliptic functions fortherelationship (z,p)to({, 17)which leads tothe annular coordinate system briefly discussed insection 31. Ifthen (10)isvalid, theseparation of(9)leads tothetwo ordinary differential equationsoftheSturm-Liouville type (see section 29) dr,2 where p2isthesecond separationconstant andcanhaveanyreal orcomplex value. Inaddition tothesolutions interms oforthog- onalfunction systems, onecanalways findparticular solutions by inspection, such asindicated fortheCartesian system insection 31. SPHERICAL COORDINATE SYSTEM The sphericalorpolarcoordinate system possesses suchsym- metry that itissimplertoproceed with theconventional and specificcoordinate relations rather than toapply theabove systematic approach. Ofcourse, onecanemploy theconformal mappingfunction w=e~f=e~f(cos 77jsin17)andobtain (7) and (11)asshown; onecanalsointroduce themore usual coordi- 1Ch.Snow: TheHyper geometric andLegendre Functions withApplications toIntegral Equations andPotential Theory, p.202; National Bureau ofStand- ards, Washington, D.C., 1942; seealsoHobson,09Chapter X. 478 Three-dimensional Analytic Solutions [Ch.8 nates ofFig.33-1, namely,e*=rwith o><+ooand 6= r\andsystematically obtain theequations (11) interms ofthese newcoordinates. Conventionally, however, onechooses asmeridian coordinates directly theradial distance rfrom theorigin (pole)ofthesystem andthecolatitudc measured from thepositive direction ofthe axis ofrevolution sothat<6<IT.The coordinate surfaces r=cons arethen theconcentric spheres with center at0,and those =cons arethecoaxial cones with apices at0;<=cons FIG.33-2TwoConducting Hemispherical Shells. are, ofcourse, themeridian planes asoutlined previously. The lineelements inthethree coordinate directions areforthepoint Pin Fig.33-2 dsi=dr, ds2=rdd,ds3=pd</>=rsin6d^ (12) sothatbycomparison with (31-23) hi=1, h2=r,h3=rsin (13) andthus forthecomponentsofthefieldvector by(31-24) ,.^_(14) Thepotential equation (31-27) becomes with(13)anddeleting thefactor( Sec. 33] Spherical Coordinate System 479 Introducing theproduct function dividing through byitandmultiplying bysin26permit atonce theseparation ofthelastterm asin(5). There remains, then, sin2R'14(r2^} fromwhich onecanseparate the firstterm after clearing sin2 0, sothat * Thiscanbesatisfied byrnwhich yields n(n+1)=p2 (18) forn^0;oralsobyr~(n+1)withnj Iwhich alsoyields (18), so thatoneuses (18) asdefinition ofthesecond separation constant forinteger values ofnandhasasgeneral solution of(17) R=drn+C2r-<"+1) ,n=0,1,2,- (19) The differential equation forT(0)thusbecomes sm dd\ de/ L sin2 (20)v ' which hasassolution theassociated Legendre functions2ortesseral harmonics offirstandsecond kind, oforder n,and ofdegreem^n T(d)=D!P nm (cos 6)+D2Qnm (cos 0), n>m(21) Onefrequently denotes cos 6=^(oralso x)because ofthe simpler forms that result infunctional relations; inparticular, 2These functions arerather uniformly designated inthemanner indicated; good treatments canbefound inpractically anyoneofthereferences in Appendix 4,C,a,aswellasinSmythe,A22 p.128; inStratton,A23 p.172;and inJeans,A1 p.206. Excellent summaries ofdefinitions andinterrelations as well asgraphs andnumerical values aregiven inE.Jahnke andF.Emde: Tables ofFunctions; reprinted byDover Publications, New York, 1943; originally published byB.G.Teubner, Leipzig, 1939. Some ofthesimpler relations aregiven inAppendix6. 480 Three-dimensional Analytic Solutions [Ch.8 form=thisgives forthedifferential equation (20) +n(n+1)TOO=(22)-f|~(1- d\_ J which isclearly oftheSturm-Liouville type (29-2) with character- isticnumber X=n(n+1)andweight function p(n)=1.The solutions of(22)arevalid forproblems with axialsymmetry and areoftwokinds, theLegendre polynomialsorzonal harmonics of the firstkindPn(cos 0),which arecontinuous for allvalues 0$0$ir or 1$/i$l; andthezonal harmonics ofthesecond kindOn(cos 0),which have logarithmic singularitiesat=and =irorjLi==hl,sothatthey cannot constitute solutions for problems which include theaxis ofrevolution. TheLegendre polynomials PW(M)areorthogonal polynomialsin/iforallvalues ofthevariable;intherange (-1)$p$+1theycanbeused torepresent anybounded function interms ofaLegendre series, asshown inAppendix 6,(24)to6,(28). Form^andwith M=cos0,thedifferential equation (20)becomes dn]_ "MJTOO=(23) leading totheassociated Legendre functions which alsoareoftwo kinds. Inparticular, thefunctions Pn(iJ.)ofthe firstkind are again orthogonal withweight function unity intherange (1)^ H^(+1).Ifthese associated Legendrefunctions ofthe first kind arecombined with their respective trigonometric factors in tf>from (5),namely, Snm (0,0)=Pnw(cos 0)[Aisin ra</>+A2cosra</>] (24) they arefrequentlycalled surface harmonics ortesseral harmonics and constitute anorthogonal function system with respect to both ordernumbers nandm. Conducting SphericalShells. Iftwohemispherical shells ofinfinitesimal thickness andpotentials $1and $>2aregiven as inFig. 33-2, axialsymmetrywill prevail. Inaccordance with (16),thegeneral typeofsolution must betheproduct of(19) and(21)form=0, +C2r"^+1) ][DiPnfcos 0)+D2Qn(cos 0)] (25) Sec. 33] Conducting Spherical Shells 481 Fortheinterior spacer<aonecannot admit negative powers in r,and fortheexterior space r>anopositive powers inrcan appear; furthermore, thesecond kind ofLegendre function must bediscarded because ofitssingularity along the axis. This reduces thesolutions withappropriate contraction oftheamplitude factors to n=0 (26) Onthespherer=athese expressions must represent theexpansion ofthegiven potential values $=$1for$6<7r/2,and*= $2for7T/2<6$TTintoaLegendre series forwhich thecoefficients fortheinternal potential arenowfound with cos 6=/*from Appendix 6,(24)and 6,(25), n2n+1 Because ofthegeneral integral form[Appendix 6,(21)](27) (2n+ /n P^l(Ml)] (28) thevalues oftheintegrals in(27)canreadily bedetermined. For neven, say,n=2k,Appendix 6,(5)and6,(6)give +l (29) andfornodd, say,n2k+1, P2fc+i(0)=0,P2fc+i(+l)=-P2fc+i(-l)=+1 (30) One finds, therefore, thatn=gives theonlyeven contribution, and forn>onlyoddfunctions remain, just asonewould expect with theconventional Fourier series. Introducing the 482 Three-dimensional Analytic Solutions [Ch.8 results into(26),oneobtains \(*i+*s)-(*i-*>) PI 7 *(r, )= \ 7(31) This solution holds also forthetemperature distribution within asolid sphereifthetemperatureiskept constant overeach ofthe twohemispheric caps, (seeByerly,C2 p.173);italsodescribes the current distribution through asolid conducting sphere withhemi- spherical electrode caps. Since theplane9=ir/2isanequi- potential plane with <i>=^($1+$2),onecanusethesame solution fortheinternal potential distribution between onehemi- sphereshell r=aand ^6<ir/2 ofpotential <tiandthe circular base plateAB inFig.33-2 ofpotential H(*i+^2)- Choosing $2=0igives tothebase plate thepotential zero. Similar applications canbemade toconcentric spherical shells ofarbitrary potential distributions; inthis case thecomplete solution forR(r)in(19)must beused asinByerly,02 p.176. The case ofauniformly charged circular ringwithin aclosed sphericalshell istreated inSmythe,A22 p.138,byfinding the potential produced bytheringalong thespherical surface and compensatingitbyasolution ofthetype (26) forr^asoasto produce aconstant potential forr=a.Thismethod canbeused where theoriginal charge distribution isafixed one,asinthecase oflinecharges, and isnotdisturbed bythepresence ofother con- ductors. Solid Spherical Conductor. Assume twosmall electrodes tobring current to,andtocollect current from, asolid sphere as inFig. 33-3. Ifthese electrodes arelocated atAandB,at diametrically opposite points, thecurrent distribution willhave axialsymmetry andthepotential solution within thesphere will begivenbythefirst linein(26). Theboundary conditions require avanishing normal component ofthefieldvector over theentire Sec. 33] Solid Spherical Conductor 483 surface except for6^Tand for(TT T)<6<TT,where itmust have thevalue necessary tomaintain thecurrent density atthe electrodes. Inaccordance with(14), theradial fieldvector com- ponentisfrom (26) a*Er=-=-LnMnrn-lPn(Cos6) (32)or n=Q andatr=athismust betheLegendre series expansion [Appendix 6,(24)], sothatthecoefficients become similar to(27) 2n[nMnan-l ]= .--COBT J --C08Tj xv=+lj -PnGi)**-/-P( 17 t/M=OOBT -V(33) FIG.333Current Distribution inSolid Conducting Sphere, where Jisthecurrent density (positive,ifradially out) ,i (or)2* ifTisasmall angle. The integrals areevaluated again by(28), since thecurrent densities areconstant;3because of(29)and(30) only theoddfunctions contribute, sothatonecan restrict n= 2fc+1andthus [nMntt"-1 ]=+- .[P2k(cosT)- 7r)l (34) 3Smythe,A22 p.234, treats thesame problem butassumes in(33) P/iO*) Pn(l)which leads toinfinite potentials attheelectrodes anddoesnotpermit evaluation oftheresistance between electrodes. 484 Three-dimensional Analytic Solutions [Ch.8 Thecomplete potential solutionis,therefore, [P2fc(cosT)-P2fc+2(cosr)]P 2fc+i(cos 0)(35) Defining thevoltage between theelectrodes V=fc(a,=0)- *(o,=IT)=RI andobserving (30)aswellasthedefinition ofthecurrent density, (35) yields fortheresistance Rbetween theelectrodes 2fcTI?[P2/fc(cST)~~P2fc+2(cos T)l Ifthesecond electrode isshifted fromBat8=TTto5'at =a,theaxialsymmetry nolonger holds andthepotential solu- tionbecomes *(r f0,*)=IEMm ,nr"Pn(cos 0)cos 7710 n=0Lm=l (37) where the lastterm holds forra=andwhere advantage has beentaken oftheevensymmetry inwith respect totheplane through thecenters ofthetwoelectrodes bydropping thesine terms. The coefficients must bedetermined from theboundary conditions ontheelectrodes aspreviously. There isnochange at A,where theelectrode isdefined by<6<T,<<2?r; however, atBftheelectrode cannot easily bedescribed asround: itismore convenient todefine itasasmall square by(a+T)> 8>(a T)and r<</><rwithanarea (2ar)2 ,which one could, ofcourse, make equal tothat atA .Computing theradial derivative ofthepotential (37)andlettingitbezeroeverywhere except ontheelectrode surfaces where constant values areassumed, onehasforthecoefficients expressions similar to(33). Since no such simple integral relations exist fortheassociated Legendre functions as(28)fortheLegendre polynomials, furthersimplifying assumptions become necessary. Dielectric Spheres. The dielectric spherical shell offinite thickness inauniform electric fieldEQasinFig.33-4 isavery Sec. 33] Dielectric Spheres 485 simple application oftheLegendre functions. Expressing the given field asinsection 21by ETQcos0,Ee*=-E* sin (38) itbecomes obvious that thepotential functions forthevarious regions canonly contain terms n andn=Iinthegeneral FIG.33-4 Dielectric Spherical Shell inUniform Electric Field. axially symmetrical solution (26), sinceboundary conditions would render allother coefficients equal tozero. Forthethree regions onewould therefore have (39)forr2*>:*i=NQr-lP(cos 0)+N^P fora^r^6: <t>2(Coi+CW'1 )P(cos 0) +(C11r+C12r-2)P 1 forr$a:$3=MP(cos0)+MirP!(cos andasboundary conditions rdd r60 atr=a: e3= e2 dr dr r60 r60(40) 486 Three-dimensional Analytic Solutions [Ch.8 Introducing thegradients from (39)andtheexternal fieldfrom (38), theconstants can allbeevaluated, giving C02=NO= CQI=M=0,and [2e2-eie2-e3r /a "l9~^ ^~LU l where terms havebeen collected toleadtothissimpler expression. Since theelectric field inside thesphericalshell follows from (39)as E=-Mi cos6,Ee=+Mx sin itisdefinitely auniform field liketheimpressedfield (38)and hasthesame direction asE. Itsintensityisdecreased bythe factor within thebrackets of(41), sothat thisfactor /cadefines directly theshielding efficiencyofthe dielectric shell. Ifone assumes ei=e3=EQand s2= ,then Foravalue e=5eandb/a=2,onehask8=(1.62)"1=0.617, sothat dielectric shielding canbemade effective onlywith special materials with very large dielectric constants. Solutions are given briefly inSmythe,A22 p.139,andmore extensively in Ollendorff,A18 p.55;Maxwell,A17 I,p.438, solves theanalogous current distribution forconductors oflikegeometry andconduc- tivities 7i,72,73,respectively. Thecompletely analogous case ofamagneticshell inauniform magnetic field isobtained byappropriate substitutions;itistreated inMaxwell,A17 II,p.59;inMoullin,048 p.205; inSmythe,A22 p. 288;and inFrank andMises,C6 II,p.718. Ifone letsa >0,thesphericalshellbecomes asolid sphere. Thesolution for$1remains thesame asin(39) ;$2reduces to *2=CVPi (cos 0)=Curcos (43) Ithas, therefore, thesameform as$3before, andonefinds 2 Sec. 33] Dielectric Spheres 487 again auniform fieldthroughout thesphere. Theapplication to theanalogous magnetic case isobvious. The dielectric sphere is treated well inJeans,A1 p.228; inHarnwell,A9p.67;inMason andWeaver,A16 p.151;inRamsay/21 p.135;andinStratton,A23 p.205; themagnetic sphere inMoullin,B48p.205;inPlanck,A19 p.99;andinFrank andMises,Ce II,p.716. Ifone lets 2- , thesolution becomes identical with that oftheconducting sphere inauniform field (seesection 21). Itisworth noting that at thepole 9=theelectric field strength hasthelargest radial value; forthedielectric itfollows from (43) dr s2+2si andforairbecause ofthecontinuity ofthedielectric fluxdensity D, EQ (44) 4- Ifei 2,thenE(2)->%&,and ife2 &i,#(1)->3#; thedielec- tricoflower dielectric constant always carries alarger local field strength than theimpressed uniform fieldE\ Spherical air bubbles intransformer oilcorrespond tothe first alternative, and water drops intransformer oiltothesecond alternative; bothcan readily ionize under fieldstrength values considered moderate for the oil. Ithadbeen stressed insection 21thatnoimage treatment exists forapoint charge andadielectric sphere. Assume thepoint charge located atQasinFig.33-5; then itspotentialisgiven by (45) Inorder tobeable tosatisfy theboundary conditions onthe surface ofthesphere, thepotential mustbeexpressed interms of thespherical coordinates rand0,which canbedonebytheclassical expansion (r')-1=[r2+b2-2rbcosBT* " Pn(coB0), r<b (46)ifrybn-oW 488 Three-dimensional Analytic Solutions [Ch.8 which hasledtothedesignation Legendre coefficients forthepoly- nomials Pn(cos 0).Taking forthetotal external potential the combination ($1+*Q),where $1isthereaction potential ofthe dielectric sphere and isidentical with thesecond line in(26), P(r,0) FIG.33-5 Dielectric Sphere andPoint Charge. andfortheinternal potential $2the first line in(26), onecan satisfy theboundary conditions d d atr=a: ei ($1+*Q)=e2$2,or OT(47) andactually finds asinStratton,A23 p.204, forthecoefficients of (26): Q-n(e 2- 2n+1 Mn=?-2n+ 1(48) [n(e2 As e2* ,theinner potential<S>2becomes aconstant andthe potential solution $1canbeshown tobeidentical withaLegendre series ofthetype (46) forapoint charge Q(a/6) located ata distance d=a?/bfrom thecenter ofthesphere;ittherefore reduces tothesolution ofapoint charge andanisolated conducting sphere given insection 21.Nosuch simple interpretation is possibleforthedielectric sphere. Admitting asmall but finite radius a\ofthepoint charge Q permits thedetermination ofitscapacitance asinfluenced bythe presence ofthedielectric sphere. Thetotal potential onthesurface ofthegiven quasi point chargeisnowthevalue of$Qfrom (45) Sec. 33] Uniformly Charged Circle andDisk atr'=GIandthat of<f>iatrband=0,since withNnfrom (48)489 <&6;thus sincePn(l)=1.For e2><*>this expression forcapacitance becomes identical with (11-23), thecapacitance ofaquasi point FIG.33-6 Circular Ring ofCharge. charge Qnearanisolated conducting sphere. Numerically, the effect upon thecapacitance ofthequasi point charge caused by thedielectric sphereismuch smaller than that caused bythe conducting sphere, though bothtend toincrease it. Uniformly Charged Circle andDisk. Though thecircular ringofcharge hasbeen treated insection 12,amore convenient formulation canbeobtained bytheuseofLegendre polynomials. Asobtained in(12-60), thepotential along theaxiswhere = canbewritten Q/4?rer/ ,where rr isthedistance ofanypoint on thecircle ofcharge tothepoint ofobservation Aontheaxis, as indicated inFig.33-6. Butonecanexpand (r')-1=[z2+c2-2zccosa]'* intotheLegendre series (46), sothatthepotential along theaxis isalso z<c (50) r-- [(:Y**( a)Pn(cos9)1 ; 4lTECLn=0\C/ Jr<c 1f"/C\n+1"1 T-~M-) Pn(COSa)P(cOS0) 47TEcLn=OVY Jr>c490 Three-dimensional Analytic Solutions [Ch.8 where the firstbracket isused forz<c,andthesecond bracket forz>c.From (50)onecanatonce construct thesolution anywhereinspace byreplacingzbyrandadding thefactors Pn(cos 6),which formally reproduces theexpansions (26)butnow withknown coefficients basically obtained bycoefficient com- parison along theaxis ofrotation. That thismethod isgenerally applicableinsystems with axialsymmetry where theaxisbelongs completelytothe field regionisdemonstrated inByerly,02 p. 157; inKellogg,010 p.255; inWebster,016 p.346;and inBate- man,02 p.406.Thesolution forthepotential becomes thus (51) Shifting theorigin inFig.33-6 tothecenterMofthecircle, where a=ir/2and r'=Vz2+c2 ,oneseesthat theexpansion follows thebinomial theorem. Themore general forms (50)and (51) are,however, useful, since they permit extension todisks, cylinders, spherical capsand zones, bysimply integrating the axial potentialofthecircle over thegiven geometry interms of cand a.Forgravitational potentials such applications aregiven inByerly02 ;Jeans,A1 p.226,solves theuniformly charged spheri- calcap. Thus theextension toauniformly charged circular disk ofradius aisreadily made byfirstdetermining thepotential along the axis. Integrating thepotential produced along theaxisbythe circle above, with theorigin chosen atthecenter ofthedisk, one hassimply 4irea2 forexample from Attwood,A2 p.67.Expanding intopositive or negative powersofzbythebinomial theorem, replacing zbyr andapplying theappropriate Legendre factor, onecanthen con- Sec. 33] Circular Currents 491 struct thecomplete solution andsimilarly forr<a,asgiven also inWebster,016 p.346,and inChurchill,03 p.198,forthegravitational analogue. Circular Currents. Foraxially symmetrical magnetic fields, thevector potential Areduces tothesingle component A^parallel tothecircular currents; evenso,itdoesnotsatisfy theLaplacian differential equation inthespherical coordinate system, butrather from (6-15) withJ=theequation VXVXA= which isobtained byapplying twice Appendix 3,(41). Upon separationofvariables, onefinds thesolution inridentical with (19), but in6oneobtains theassociated Legendre function of order nand first degree (ra=1).Thus, thegeneral solution becomes, disregarding Qnlassingular ontheaxis, A*=L(Clnrn+Canr-<"+n )Pnl (cos 0) (54) n Theradial magneticfieldcomponentisfromAppendix 3,(41) Br--(sin0A )rsm0 dd Because onehasfromAppendix 6,(31), Pn(cos 0)=sin-- -Pn(cos 0)=-Pn(cos 0)acos aB onecanusethis in(54)andintroduce itfordifferentiation into BT.This leads to 492 Three-dimensional Analytic Solutions [Ch.8 onaccount of(20)form=0,sincePnisjustaspecial case ofthe general spherical harmonic T(6). Thus, theradial fieldcomponent becomes BT=n(n+IJCCmi^1+C2nr-n-2)Pn(cosfl) (55a) n=0 Themeridian component Be is,then,fromAppendix 3,(41)and using (54) directly, --[(*+ OCinf""1-rAnr-^lPn1 (cosfl) (556) n=0 Assuming nowaninfinitely thin circular current located asin Fig.33-6, onecanevaluate the field distribution byusing (54) inamanner similar to(26). Forthisonewillexpand thelocally concentrated current distribution along thespherer=cintoa series ofassociated Legendre functions andthen satisfy the boundary conditions which require (seesection 6) atr=c:BTl=J5r2,B62-Bei=i^K^ (56) whereK^=I/c&d,ifthetotal current ofthecircular loopisI. Obviously,inallthree forms (54)and (55), onehastouseonly thepositive powersofrforr<c,andonly thenegative powers forr>c.Introducing therespective partsof(55a) intothe firstboundary condition (56)gives atonce foreachvalue n, r.cn~l--TO c~n~2 v.'ln1' ^2nc Introducing therespective partsof(55b) intothesecond boundary condition (56)gives n"2Pn1 (COS 6) ..7 Cn~~lPnl(COS0) This requires now theexpansion ofK^intoasimilar series of associated functions Pn1 (cos 0), Sec. 33] Conical Boundaries 493 where thecoefficients canbefound asforanyorthogonal function system by(29-12), with thenormNn(l)=-- ,+ ,;.2nH-1(n1)! fromAppendix 6,(48)andweight function p=1.Thus, observ- ingthatthecurrent isrestricted to60at6=a,oneobtains in goodapproximation Nn(l)D n=fV K+Pn1 (cos 0)sin9dB Pnl(cosa)sina50 t/0-0 COu With thisexpression forthecoefficients inK$onecanreduce the second boundary condition toindividual relations foreach value n\thisalsopermits theevaluation oftheconstants C\nandC2n- The fieldcomponents become, then, finally, Pn1 (coso)Pn(cos 0),r^a (57) Rr/c)"-11/n Pn1 (cosa)Pn1 (cosff),ra where thesignsandterms inthebrackets arerelated totheranges ofrasindicated ontheright. Obviously, these expressions could beconverted tocylindrical coordinates andcompared with the elliptic integrals in(1326).Themajor advantage oftheformula- tionwith Legendre functions liesinthefactthatonecannow again integrate with respect tocandaover various current dis- tributions oncylindrical4orspherical surfaces. Solutions forthe circular looparegiven inMaxwell,A17 II,p.304,andinSmythe,A22 pp.263and270,who alsoconsiders general spherical surface dis- tributions. Conical Boundaries. Ifitisdesired tosolve thepotential distribution inaconical space asinFig.33-7 with thegiven boundary values, thenonehasinahomogeneous boundary value problem ofaxialsymmetry, sothatthesolution of(22)issubject to D^Pn(cosa)+D2Qn(cosa)=DtPn(cos3)+D2Qn(cos3)= 4SeeH.B.Dwight, Trans. A.I.E.E., 61,p.327(1942) forcomparative practical forms offieldexpressions forcylindrical coils. 494 Three-dimensional Analytic Solutions [Gh.8 This condition canbesatisfied onlyifthedeterminant ofthe Legendre functions vanishes Pn(cos OL)Qn(cosa)-Pn(cos |9)Qn(cos0)=(58) which inturnmeans that this relation defines theordernasa realbutnon-integral number, since theLegendre functions canbe considered asanalytic andcontinuous functions oftheir order numbers; seeparticularly Hobson,C9Chapter IX,onthediscus- FIG.33-7 Conical Boundaries. sion ofthezeros ofLegendre functions. Having theordernum- bers,onecanthen write thepotential *(r, )=(cos8)- n(,COSa)Qn(cos 0) (59) andmust determine thecoefficients Dnsuch that forr=aone has*=$o- Ontheother hand,iftheboundary value problemishomoge- neous with respect totwospherical surfaces r=aand r=b,then from (19)follows 1)=dbn+C26~(n+1)= which defines theordernumber as flb-(+i)=a-=lnl~ Ina/6 andadmitting In1=In(exp j'2irp), onehasthecomplex order Sec. 33] Dipole Coordinates 495 numbers n=J/+jqwith q=7rp/(ln a/6), leading tothecone functions.5 Dipole Coordinates. Thefunction w=e~*hasbeenshown todefine thespherical coordinate system; theconformal trans- formation totheu/-plane w= alsogives lnI /. j. ;7T7''=/ coth 2=*'+jp' u>+f,TI w'-f or^asshown inFig.33-8andalready discussed in(26-53), the biaxial family ofcircles such asthefieldpicture oftwoparallel line FIG.33-8 Dipolar Coordinates. chargesX.Because oftheaxial symmetry, thiscoordinate system actually provides twofamilies oforthogonal spheres and iscalled thedipolar coordinate system. Solving forthecoordinates ofthew'-plane, onehas sinh sin cosh fcos77 cosh cos 6Introduced byF.G.Mchler, Math. Ann. 18,p.161(1881); seealsorather extensive treatment inHeine,08 II,p.217,andinHobson,09p.444. 496 Three-dimensional Analytic Solutions andfrom (7)[Ch.8 h'*dw' Thus,(cosh cos h'2 -75=[sinr P defines thesystem asclearly separable initscoordinates inaccord- ancewith thecondition (10). Itisdefinitely related tothepolar orspherical coordinate system andleads alsototesseral harmonics. Theproblem oftwo finite spheres, which wastreated withan infinite number ofimagesinsection 21,hasbeen solved bymeans 77=cons FIG.33-9 Oblate Spheroidal Coordinates. ofthiscoordinate system byHobson,C9 p.448. Further details arefound inSnow,loc.cit.jp.235. OBLATE SPHEROIDAL COORDINATES Theoblate spheroidal coordinate system isanaxially symmetrical ellipsoidal system inwhich theminor axis istheaxis ofrevolution asshown inFig.33-9. Inaccordance withthegeneral discussion atthebeginning ofthesection, oneobtains themeridian co- ordinatesf, 77bytheconformal transformation from theunder- lying cylinder system =z+JP=fsinh({+JTJ) (60) which issimilar totheoneused fortheelliptic cylinder in(32-52). From (60)oneobtains z=fsinh cosTJ, P="=/cosh sin77 (61) Sec. 33] Oblate Spheroidal Coordinates 497 which represent confocalellipses andhyperbolas with 2/asthe focal distance. This isshown explicitly by J+ \Jcosh{/lf\/cosJ+\/sinJ \/sinh, The first relation describes theellipsoids=conswithsemiaxes a=fcosh along thep-direction andminor axes 6=/sinh along theaxis ofrevolution; these degenerate intothecircular area p^/inthe z=plane for=0.Thesecond relation gives theorthogonal andconfocalhyperboloids TJ=cons, degener- ating intotheplane z=withacircular hole forTJ=ir/2.The ranges ofvalues are<<QO ,<T\<IT,quite analogous to thespherical coordinate pair r,9. Inaccordance with(7)onehas h2=H2 =f2|CQsh2f|=f2(cogh2_gin2^ (62) rffI sothat h2cosh2sin2 77 1 1_ _ p2cosh2sin2 77 sin2 rjcosh2 andtherefore (63) Onecanthus writedown atonce theseparated Sturm-Liouville equations (11). Itisadvantageous atthispoint totransform these equations (11)byachangeofvariables andredefinition of thefunctions, namely, u=sinh,S()=Vcosh Wi(u) } \(64) V=COS?), H(TJ)=Vsinr\W2(v) ) intoequationsofthetype (20) or(22),namely, [<"+ (65) 498 Three-dimensional Analytic Solutions [Ch.8 The solutions forWzaretheconventional associated Legendre functions ifonewrites (p2J)=(pH)(P+H)anddefines (p Vti=n}sothat W2(v)=DiPnm (cos )+D2Qnm(cos ,) (66) Since asubstitution juforwmakes thedifferential equation forW\ identical with that forW2,onecanwrite with thesame values of (P2%)=n(n+1)thesolutions forWiinanalogy to(66) (seealsoAppendix 6) Wl(u)=C1Pnm (jsinh$+C2Qnm (jsinh$ (67) Itis,however, customary todefine these modified associated Legendre functions ofimaginary (orcomplex) argument, say i=r+j8fby[Appendix 6,(33)and 6,(39)] h(68) dtmn \t\in\Vn(2t)n+l which assures that forimaginary argument thecombinations exp(-jn^)Pnm (js) , jexp(+jn^)Qnm (js) \ A/ \ */ takeonrealvalues. Theasymptotic expressions in(68)demon- strate theanalogy ofthese functions totheradial functions (19) forthespherical coordinate system. Returning totheoriginal definition oftheproduct solution for thepotentialin(3)andintroducing (8), (61),and (64)yieldnow (69) where theconstant f~* can, ofcourse, beabsorbed intheother constants. Treatments ofthiscoordinate system, solutions init, and inparticular discussion ofthevarious system functions are found inseveral references, andforconvenience table 331gives thecomparative notations used. Conducting Spheroids.Ascribe toasolid conducting spheroidofscmiaxes aand 6theconstant potential $;then the Sec. 33] Conducting Spheroids 499 potential distribution inspace willbeaxially symmetrical sothat m=andW\ }W^reduce totheplain Legendre functions. The surface ofthespheroidisdefined bythesemiaxes as a=/cosho, b=fsinh f , o=tanh"1 (- )\a/ TABLE 33-1 COMPARATIVE NOTATION FOBOBLATE SPHEROIDAL COORDINATES This Coordinate Book Bateman01Byerlyc2HobsonC9Lamb022SmytheA22 MI u=sinh =sinhrjtanhrjf sinh ^f=sinh77 f uz v=cosr; fj.=sin tan/ cos9p=cos8 113 </</></>< cj Focal dis- tance 2/ 2k 2f 2c 2k 2ci Distance from axis p u Distance along axis z z z z z x Onaccount oftheasymptotic behavior ofthefunctions Pnmand Qnmgivenin(68), theformer must beexcluded because the potential must atleast remain finite atinfinite distance. One thushas left *(, i)=[DmPn (cos r?)+D2nQn(cos -n)]Qn(jsinh) 71=0 This, then,must represent theexpansion ofthepotential function for J= into theconventional Legendre series; because Qn hasalogarithmic singularity for77=0,TTorforcos77=1,it alsomust beexcluded ifthez-axis belongs tothe field region. Fortheassumed constant potential theseries thus reduces toa constant, i.e.,n=0,sothat (,= =coth"1 (jsinh g) Qo(j sinh J) coth-1 (jsinh fo) cot"1(sinh {) cot'1(sinh f)(70) 500 Three-dimensional Analytic Solutions [Ch.8 where thedefinition of(JofromAppendix 6hasbeen used; see Byerly,02 p.247. The charge distribution isreadily found from thenormal component ofthefield vector, which isfrom thegeneral definition (31-24) with (62) EI=-\=3>[/cosh \/cosh2-sin2 TJcot"1 (sinh fo)]"1 hd =*[/W+iVu2+v2cot"1UoT1(71) where onemight useforquicker computation from (61)and (64) z=fuv, p=fVu2+iVl v2 Onthesurface ofthespheroid onehasu=usinhQIsothat v2cot"1u]-1 (72) The total charge canbefound bestasfortheellipsoidinsection 31 byletting fbecome very large, sothattheequipotential surfaces approach spheres. From (61)onehas z=fucosTJ, pfusinrj,z2+p2=r2J2u2 andforthefieldvector in(71) thisgives limE*~$Q\fu2cot i*OJ 2,_. r*cotUQ which isthesame asthat ofapoint charge Qattheorigin. The value ofthechargeitself isfound byintegrating (e^)overalarge sphere. Thecapacitanceofthespheroid follows then as C=Q_=_^_ (73) <P COt UQ With thecharge value from (73)onecanreplace $intheexpres- sions offield vector andcharge density. Themaximum charge exists foru=UQand v=andtheminimum charge forv=1, sothat (72) yields tQ lirab'mm 4ira2 ortheratio is(a/6), directly theratio ofthesemiaxes ofthe spheroid. Sec. 33] Dielectric Spheroids 501 Inthelimit as o 0>thespheroid becomes acircular disk of radius/.Since cot"1=ir/2,onegetsatonce thesame values forcharge density and capacitance asfound in(31-54) and (31-53), respectively. 011endorff,A18 p.280, solves thiscaseand appliesittothecapacitanceofanumbrella antenna above ground. Smythe,A22 p.160,treats theuncharged circular disk inauniform electric field ofarbitrary angle with theplane ofthediskandalso computes thetorque exerted on it.Byerly,C2 p.153, expresses thepotential along thez-axis bysetting 77= in(61),andby replacing sinh=z//in (70),where also =0, Replacing now znbyrnandmultiplying eachtermbyPn(cos 6), onehasthealternative form forthepotential *(r, o)=*-1*oEirr^f^p(cos*)' r<a (74) 2 (l)n/A2n+1 -$0E (-)P2n(cos0), r>a TT n2n+1V/ where thevalue forr>afollows from thecorresponding expan- sion ofthepotential. Ifthepotential along= isaprescribed function, forexample fortwohemispheroidalshells where $= <t>ifor<i\<ir/2 and$=$2for7r/2< ?;<TT,asinthecorresponding case of twohemispheresinFig. 33-2, onecantake thedevelopment fortheoutside potential directly from (27), replacing [Mnan ]by [DinQn(jsinh o)l-Fortheinside fieldonewould have tosubsti- tutePnm (jsinh )forthesecond kind ofthemodified Legendre function; seeByerly,C2 p.248,who solves theanalogous case ofa temperaturefieldbetween twohemispheroidal caps. Thehydro- dynamic problemofaspheroid moving through aninfinite ideal fluid istreated inLamb,C22 p.135. Dielectric Spheroids.Inanalogy tothedielectric sphere ina uniform electric field, onecantreat thedielectric spheroid=f inauniform electric field; here, however, onehastoobserve the direction oftheimpressedfieldandcanobviously consider two principalorientations :parallel totheaxisofrevolution andnormal toit.The first case isbyfarthesimpler one, since itretains 502 Three-dimensional Analytic Solutions [Ch.8 axialsymmetryinthe field distribution. Expressing theim- pressed potential as $=-Ez--Efuv=-E/sinhcosi? (75) andforming thelocal potential solutions $1outside thespheroid indielectric constant siand$2inside thespheroid ofdielectric constant e2,thenonehasfrom (69)withm=andusinguand v asabbreviations from (64), ,if)=EA*P.dO-Qn(Ju), M>UQ (76) U<U wherePnissuppressedintheoutside potential because itincreases beyondalllimits as><*> ,and <3nissuppressed in$2since it hasalogarithmic singularityat 1.Asinthespherical case, the form of(75)requires similar forms of(76)because oftheboundary conditions. Noting cost\in(75) restricts thesums ton=1, since onlyPI(cos 77)=cos17.Theboundary conditions can nowbesetdown as +-. (77, where inthesecond form thefactor l/hhasbeen omitted. With thefunctional forms ofAppendix 6,(4)and 6,(16)onefindsfrom theboundary conditions (77), observing d/d=(du/d)(d/du)= -**-Elfuv= 1+--1)(V+1)(1-ucor1 tio) F sinh (tanh1-(78) The fieldwithin thespheroidisagain uniform inthesame direction asEand for 2>EIweaker than theimpressed field. For b/a=J/2,onefindsA=1+0.53(e 2/ei 1),andas6decreases, A2/21? sothat foravery flatspheroidal disktheinner field strength becomes EiEQ si/e2.This solution isgiven with Sec. 33] Inverse Coordinate System 503 considerable detail inOllendorff,A18 p.289; healso treats the second orientation ofEnormal tothe axis,which requires the use oftheassociated Legendre function ofdegreem=Ibut otherwise isyetsimple. Inthecase e2>oneobtains again thesolution oftheconducting spheroid inauniform electric field. Obviously, thisanalysis canbetransposed tosolve theanalo- gousproblem ofanironspheroid inauniform magneticfield. Inturn, onecansolve fortheproper azimuthal current distribution inaspheroidalcoiltogiveauniform magnetic field within.6 Inverse Coordinate System. Bytheinversion w (w- where ZQ=/sinh,thew-planeistransformed sothat the p-axis inFig.339isbent intoacircle, andthepartOFibecomes a =cons Fio.3310Inverse toOblate Spheroidal Coordinate System. finite circular arcwhich upon rotation about thez-axis forms a spherical cap.The confocal ellipses then transform into shells about thisspherical capandabout apole attheorigin where the base circle intersects. For z=0,thep-axis inverts into itself withOFitransforming intothecomplementary part ofthep-axis, thus leading tothecircular apertureinaninfinite conductive plane butwith thepole atthecenter oftheaperture asshown in Fig.3310.Thefunction systems involved inthesolution are identical with those above, since thesame metric factor applies, sothat itcanbeclassed withtheoblate spheroidal system. 6J.P.Blewett, Jl.Appl. Phys., 18,p.968(1947). 504 Three-dimensional Analytic Solutions PROLATE SPHEROIDAL COORDINATES1.8 Theprolate spheroidal coordinate systemisanaxially symmetri- calellipsoidal system inwhich themajor axis istheaxis ofrevolu- tionasshown inFig.33-11. Themeridian coordinates{, t\are TJ=cons Jt. FIG.33-11 Prolate Spheroidal Coordinates. obtained bythesame conformal transformation asintheelliptic cylinder coordinate system w=2+jp=fcosh(f+j-rj) sothat 2=/cosh cosrjj P=fsinh sint\ (79) which represents again confocal ellipses andhyperbolas. Spe- cifically, andasintheelliptical cylinder, !/ Z\2/ \fcos77/\/sir_ cosh sinh sinT\ Theprolate spheroids generated bytherevolution oftheellipses about their major axisdegenerate intothefocal lineFiF zfor =andapproach spheres as>>.Thetwo-sheeted hyper- boloids generated bytherevolution ofthehyperbolas about the 2-axis degenerateforrj=andTJ=ITintotherespective sections ofthe2-axis outside thefocal points FIandF2)andbecome identical with theplane ofsymmetry2= forT/=ir/2.The rangesofvalues0<<QQ,0<i7<7rare quite analogous tothe spherical coordinate pair r,6. Inaccordance with (7)themetric coefficient h?isidentical with (32-55), sothat h2_cosh2cos2 t\ 1. 1 7" sinh2 f-sin2 t\sinh2sin2 Tj Sec. 33] Prolate Spheroidal Coordinates 505 andtherefore This permits oneimmediately toutilize theseparated Sturm- Liouville equations (11 );butasinthecase oftheoblate spheroidal coordinates itisadvantageous totransform these equations bya changeofvariables andredefinition offunctions u=cosh, H(f)=Vsinh Wi(u) (81) v=cosij, H(T?)=VsinTJW2(v) This results inthedifferential equations (82) Setting again p234=w(n+1),both equations areofthe type (20)or(22), sothatforW2onehasthesolution (66),whereas forWibecause ofu2>1,onemust choose themodified associated Legendre functions (seeAppendix 6),namely, W,(U}=CJV1 (Cosh {)+C2Qnm(CQSh ) (83) Returning totheoriginal definition oftheproduct solution forthe TABLE 33-2 COMPARATIVE NOTATION FORPROLATE SPHEROIDAL COORDINATES Coordinate Dateman01Byerly02Hobson09Lamb022SmytheA22 u\ u=cosh 9=cosh77coth /cosh77f=cosh77 77 112 v=cos77 /i=cos tanhr?/ cos9/x=cos u3 u Focal dis- tance %f 2k y 2c 2fc 2c2 Distance from axis p uVx2+j/2p S p Distance along axis z z z z x z 506 Three-dimensional Analytic Solutions [Ch.8 potentialin(3)andintroducing (8), (79),and (81), thisyields now *(f, n,*)=EEr*Wi (cosh f)W2(cos )Ffo) (84) where theconstant f~** can, ofcourse, beabsorbed intheother constants. Treatments ofthiscoordinate system, solutions init, anddiscussions ofthefunction systems appearinginthese solutions arefound inseveral references, and forconvenience table 33-2 gives thecomparative notations used. Conducting Spheroids. With thesuitable modifications, the applicationsoftheoblate spheroidal system canreadily betrans- posed intosolutions fortheprolate spheroids. Since theasymp- toticforms (68)apply alsoforrealarguments \p\>1,thepoten- tialoutside aconducting spheroidofsemiaxes a=fcosh, b=fsinhfo;isbytransposition of(70) 5o(cosh {)^coth-1(cosh )^s9o~~^~~~~^^~~~~~^^^^~ Qo(cosh f) coth"1(cosh ) InIn(u+l)/(u-1)m where thedefinition of^fromAppendix 6hasbeen used. The normal componentofthefieldvector isfrom thegeneral definition (31-24) withh2from (32-55) Et=7=$o[/sinhJVcosh2cos2 ijcoth"1(cosh fo)]""1 hd% =$(,[/VV-v2Vu2-1coth"1wo]"1 (86) where onemight useforquicker computations z=fuv, p=fVu2-1Vl-v2 (87) Onthesurface ofthespheroid u=UQ=cosh,sothat the charge density becomes The total charge canbefound inanalogous manner asfor(73), sothat thecapacitancefollows: ~*t (89) UQ Sec. 33] Dielectric Spheroids 507 With thecharge value from thisexpression onecanreplace $ in(86) forthefieldvector and (88) forthecharge density. The maximum andminimum values ofcharge density exist foru=UQ and v=1,v=0,respectively, sothat (88) yields Q Q ortheratio isagain a/b,asfortheoblate spheroid; however, the respective maximum values aswell astheminimum values inthe twocases bearthesame ratios a/6,withthelarger values occurring intheprolate spheroid. Inthelimit asf 0,thespheroid becomes anellipsoidal rod oflength 2/which hasbeen treated rather completely insection 12. Ageneral solution with therodasspecial case isgiven inKirch- hoff,A13 p.37;inOllendorff,Al8 p.308,who alsoapplies thesolution tothecurrent flowfrom agrounding electrode reaching below the level ofground water; andinByerly,C2 p.155, interms ofzonal Legendre harmonics, andp.250forgravitational potential applica- tions. Smythe,A22 p.167, solves the fieldnear asemispheroidal mound onaninfinite ground plane; seealsoBateman,01 p.436. Ifthepotential along={isaprescribed function, then it canbereadily expressed from (84) asanormal Legendre series if axialsymmetry prevails, orasseries ofsurface harmonics (24) in themore generalcase. Dielectric Spheroids. The prolate dielectric spheroid ina uniform electric fieldEcanbetreated inexact analogy tothe oblate spheroid. Two principal orientations arepossible; the electric fieldcanbeeither parallel totheaxis zornormal toit. The first case retains axialsymmetry, and itssolution proceeds exactly aswith theoblate spheroid, leading again totheuniform internal field ofvalue Ei= A=1+(--1)(uQ2-l)(uo coth-1UQ-1)(90)A Vs! / where UQ=cosh f=a/f-Hereonefinds forb/a=%thevalue A=1+0.177 (e2/ei 1),andasbdecreases, A >1,sothatthe very thinrodhasauniform inner fieldwhich isequal totheim- pressed outer field,ElEQ .However, thecontinuity ofthe normal componentofDrequires then that atthepole ofthe 508 Three-dimensional Analytic Solutions [Ch.8 spheroid alocal field strength exist ofvalue causing avery strong concentration ofthe electric fieldondi- electric objects, asmasts, poles, orsharp mountain peaks. This solution isgiven fortheanalogous magnetic casewith considerable detail in011endorff,A18 p.315;Lamb,C22 p.132, alsosolves the motion ofaspheroid through anideal fluid. Iftheuniform electric field isoriented normal totheaxis zand parallel totheplane=0,then itspotential canbeexpressed in theform <3>o=-Ex=-#pcos0 =-EfVu2-iVl-v2cos0, u<UQ(91) where thez-direction istaken from theCartesian system, con- verted tocylindrical coordinatesp,0,andwith (87) finally tothe spheroidal system. Now, thegeneral solution (84)hastosatisfy theboundary conditions which areidentical with (77)andwhich relate onlytooru;onesurmises that in(5)onlym=1canoccur andonlythecosine term; further, that in(66)onlyn=1canlead totherequisite termVl v2=sin17andthatthesecond kind ofassociated Legendre functions must beavoided, since theaxis y=Belongs tothe field region. Onecantherefore write for thelocal potential solutions inside andoutside *i(f, u,0)=DtPSWQSM cos0, u>UQ (92) *2(f, i?,0)=CiPS (u)/Y (u)cos0, u<UQ With thefunctional forms fromAppendix 6,onefinds then the inner potential where El= *A=1+I 1 )^No (u>Q2 1)coth"1 UQ] (93) There isagain auniform field inside thespheroid inthesame direc- tionastheimpressed field,andfor e2>EIweaker than it.For b/a=Ytone finds hereA=1+0.412(e2/ei-1),and asb Sec. 33] Inverse Coordinate System 509 decreases, A >H(2/i+1)-This solution isgiven indetail by Ollendorff,A18 p.319, fortheanalogous magnetic problem; he appliesitalsotoevaluate theerror inlong-range radio navigation caused bythebody oftheship. Conducting Hyperboloids. Selecting oneofthehyperboloids inFig.3311asaconductor surface ofpotential3>incombination with either thesymmetrical oneofpotential $,orthecenter plane2=with zero potential, gives solutions forneedle elec- trodes7which might beapplied tohigh-voltage rectifiers.8Assume axialsymmetry; then variation ofthepotential occurs onlywith v=cosv],andtheanalogous solution to(85)isinthiscase , ,Qo(cos 17) tanh"1 (cos 77) Qo(cos 770) tanh1 (cos TJO) where770=tan"1b/a isdefined bythesemiaxes ofthehyperboloid electrode. The field vector canbeevaluated from thegeneral definition (30-24) with h2from(32-55), sothat =--=*o[/sin r,\/cosh2 f-cos2 77tanrT1 (cos rjo)]"1 nO7) (95) Themaximum field strength exists for=andT;= TJO,atthe apex ofthehyperboloid; itsvalue becomes from (95) where thefocal distance /=(a2b2)^; thiscanbeplotted entirely asafunctioa ofa/forb/f,indicating therapid increase of -Emax withthedecrease oftheangle 770- Inverse Coordinate System. Bytheinversion W-ZQ where z=/cosh f,thew-planeistransformed sothatthefocal length 2/moves along theaxistopoints F\andF2'andtheorigin becomes apole, theimageofz=oo.Theconfocalellipses thus transform intoodd-shaped surfaces ofrevolution andinthelimit 7J.Miillcr, Arch.f.Elektrot., 29,568(1935). 8R.Strigel, Fachberichte, V.D.E., 1929; seealsoOllendorff,Aiap.311. 510 Three-dimensional Analytic Solutions 1.8 forz represent essentially hyperboloids about thez-axis with ellipsoid-shaped surfaces close tothepoleandsurroundingitas shown inFig. 33-12. The function systems involved inthe FIG. 3312Inverse toProlate Spheroidal Coordinate System. solution areidentical with those above; thiscoordinate system canbeclassed therefore with theprolate spheroidal coordinate system. PARABOLOIDAL COORDINATES Theparaboloidal coordinate system canbeconsidered asingular case ofthespheroidal system where oneofthefocihasmoved into infinity. One obtains themeridian coordinates,TJasinthe parabolic cylinder coordinates bytheconformal transformation sothat (96) These represent confocal parabolas with focus attheorigin as shown inFig.3313.Theranges ofthevariables are<<oo and<T)<oo. Inaccordance with (7)onehas 2dw222^=~~~=\ T^?/dc andtherefore from (10)with (96) 0itt)=72 (97) Sec. 33] Paraboloidal Conductors 511 Onecantherefore write down atonce theseparated Sturm- Liouville equations (11)andidentify them asbelonging tothe class ofBessel equations with solutions9 (98) H(u)= where pisanarbitrary separation parameter, andJmtNmarethe conventional andIm,Kmthemodified Bessel functions offirst andsecond kind; seealsoAppendix 5. FIG.33-13 Paraboloidal Coordinates. Returning totheoriginal definition oftheproduct solution for thepotentialin(3)andintroducing (8), (96),and (98)yieldnow TOp Treatments ofthiscoordinate system arelessfrequent; see,how- ever,Bateman,01 p.449,andreferences there. Paraboloidal Conductors. Ascribe toasolid conducting paraboloid=?othepotential4> ;then thepotential distribution must beaxially symmetrical and,moreover, candepend onlyon .Forthissingular case,in=p=0,sothatbothequations (11) reduce tothesameform 9Jahnke andEmde, loc.cit., p.146. 512 Three-dimensional Analytic Solutions [Ch.8 which hasassolution for B(t)=V(d+C2In{), B!=Ci+C2In andsimilarly for17 HO?)=V^(Di+D2In77), 2=!*!+D2In77 sothatthepotential function (99)becomes, suppressing thevaria- tionwith77, Ci+Caln* (100) Defining theboundary conditions $(o)=3>oand$(1)=0,and choosing filargeenough sothattheparaboloid almost becomes a plane asaconvenient reference, then The fieldstrengthisfrom thegeneral definition (31-24) with h2 fromabove which has itsmaximum value at77=0,=-Ollendorff,A18 p.204,hasused thissystem torepresent amodel ofapininsulator. Another special case isindicated inFig.3313,where the paraboloid generated byAPcarries potential $1andthatgenerated byPBpotential3>2>$1.The internal potential distribution willshow axialsymmetry, sothatm=0.Since thesecond kind ofBessel functions possess logarithmic singularities attheorigin which isincluded inthefield region, theymust besuppressed and thesolution becomes, from (99)with (98), Tosatisfy theboundary conditions onemight bestaddthecon- stant $1tothepotential solution andthusmake thecondition at 77=770homogeneous, namely, $=0,This requires then i/o(pio)= leading toaninfinite number ofroot values pnasdiscussed in sections 30and32.Thepotential solution isnow o)=(102) Sec. 33] Toroidal Coordinates 513 andthecoefficients Cnmust beevaluated byinterpreting the summation astheFourier-Bessel series expansion ofthepotential ($2 *i)at=ointerms oftheorthogonal system J^pnii) between thelimits <t\<r).From Appendix 5,(43)and 5,(40) thecoefficients are,therefore ~$l)jf* where thebrackets ontheleft-hand sideindicate thenorm2Nn. Since ($1$2)isconstant, theintegration canbeperformed and gives simply (rjo/p n)/i(pn*7o), sothatthefinalcomplete solution fortheinternal potentialis *\V-on on -*i)E r/>x7-TT7- ^n^o(Pnto) (,PnT?o) ^UPnT/o) Because oftheaxialsymmetry, thiscanrepresent aparaboloidal electron lenssystem with focussing action forwhich the field vector isfound bytheapplicationof(31-24) asbefore. For potential distributions inexternal regionsitisusually necessary toformulate Fourier integrals rather than Fourier series intheparameter p,since theranges ofboth variables andT/ extend toinfinity. TOROIDAL COORDINATES Thetoroidal coordinate system represents inanymeridian plane thesame cross section asthetwo-dimensional biaxial system with twoorthogonal families ofcircles asproduced bythepotential distribution between twoparallel charged lines insection 12.One obtains themeridian coordinates andrjbytheconformal trans- formation discussed in(26-53) andagain inconnection with the dipolar coordinates inthissection withtheappropriate modification forthedifferent axis ofrotation w=z+jp=jfcoth [J^(+jrj)] (104) Explicitly, from thisonehas sinT/ sinh z=f p=/cosh cost\ cosh cost\ which represent thecircles shown inFig.33-14, namely, z2+(p-/coth)2=(-^-T)i(z-/cot r,)2+p2= sm 514 Three-dimensional Analytic Solutions [Ch.8 The firstrelation describes thecircles =conswith centers along thep-axis and radii a=//sinh {,degenerating for=intothe z-axis andfor= >intothepoint F;byrotation, each circle generates atoroid, shrinking tothecircle ofradius /for=<*>. Thesecond relation gives theorthogonal circular arcst\=cons, which generate spherical capswiththecircle ofradius/ ascommon =cons FIG.3314Toroidal Coordinates. base; thelineOFdescribes abarrier surface where thevalues 97=TTandrj=-\-Tjoinback toback. Inaccordance with (7)onehas dw (cosh Jcos77)2 sothatwith (105) h2 sinh2(106) (107) One can, therefore, writedown atonce theseparated Sturm- Liouville equations (11). Since 02W=0,thesolutions for H(T?)aresimply trigonometric functions, H(??)=DIsinprj+D2cospi (108) Sec. 33] Conducting Toroid 515 Thevalue ofg\()isthesame asin(80) fortheprolate spheroids sothatthesame substitution forH() asin(81)willbeindicated; thesolutionis,therefore, given by(83), oralso,withn=p J^, asdefined thereandin(66), g(f)=Vsinhftt^cosh?) =Vshml [CiPp_Hm (cosh )+C2QP_H" (cosh {)] (109) Returning totheoriginaldefinition oftheproduct solution forthe potentialin(3)andintroducing (105), (108)and(109) yieldnow Vcosh {-cosT, (110) where, ofcourse, /~"Hcanbeabsorbed intheother constants. Treatments ofthiscoordinate system, solutions init,anddiscus- sions ofthefunction systems arefound inseveral references, and forconvenience, table 33-3 gives thecomparative notations used. TABLE 33-3 COMPARATIVE NOTATION FORTOROIDAL COORDINATES Coordinate ThisBook Bateman01Byerly02Hobson09 u\ ^cosh"1u a=cosh"1s at\ u2 ij t p U* * T Radius ofbase circle / a a c Distance from axis p p r p Distance along axis z z z z Conducting Toroid. Ascribe toasolid conducting toroid as inFig.3314ofcenter diameter 2bandofcross-sectional radius a apotential <t>ojthepotentialdistribution inspace willbeaxially symmetrical, sothatm=0.Todetermine thevalue Jforthe surface ofthetoroid, onecanusetherelations from thedefinition ofthecircles above sothata={> b=OM=/coth fsmh n cosh o=~' a 516 Three-dimensional Analytic Solutions [Ch.8 Fortheexternal potential > onecanuseonly thesecond kind ofmodified Legendre function in(109) inaccordance with theasymptotic forms (68); then (110) reduces tothesingle sum- mation Vcosh -cosij i.psinprj+D2iPcosPTJ) <2P_H(cosh )(111) Since at=othepotential must givetheconstant value $o>the series (111)must actually represent there theFourier expansion of3>o(cosh J cos17)~^ ;which permits coefficient comparison andcomplete solution oftheproblem. SinceVcosh cost\is even symmetrical about17=0,only cosine terms willoccur, so that inconventional Fourier coefficient determination (112),oVcosh f cos r Foranyother potential variation withrjthemodification ofthe integralisrather obvious. Ifthepotential isalsoafunction of 0,then adouble Fourier series results. Thetoroidal coordinates were introduced byC.Neumann;10brief treatments areinBate- man,01 p.461,andinByerly,C2 p.266; alsoinHobson,C9 p.433, who calls thespecial type ofLegendre functions occurring here "ring functions." Obviously, asa ->oneapproaches thesolu- tion forthecircular ring ofchargein(12-58). The fieldvector andcharge density canagain befound bythe applicationof(31-24) with (106). Related Coordinate Systems. The toroidal coordinate system canbeconsidered aspecial case oftheannular coordinate systeminsection 31ifinFig.31-4onetakes b=a=/;theoval rings thengoover intothecircular ones ofFig.33-14. If,ontheother hand, onelets/-> inFig.3314,thecircles allpassthrough theorigin, asinthepotential solution forthe dipole line insection 12,Fig. 12-7. This coordinate systemis then actually theinverse tothecircular cylindrical coordinate 10C.Neumann: Theorie derElektrizitdts- und derWarme-Verteilung in einem Ringe; Halle, 1864; seealsoG.Szego, Bull.Am.Math. Soc., 61,325 (1945). Sec. 34] UseofGreen's Functions 517 system ofsection 32andleads tothesame function systems, thus demonstrating theclose innerrelationshipofallthese coordinate systems. 34-USEOFGREEN'S FUNCTIONS Starting with thesecond identity ofGreen (seeAppendix 3), fffJJJrdr= *-*\dS (1)dn dn/ where both*and^areharmonic orpotential functions andwhere thevolume Tisbounded byaregular surface S(which canbecon- tracted toapoint without intersecting itself), choose for^=1/r with rthedistance from anarbitrary point P(x, y,z)within the volume T.Obviously, ^ >ooasr >0,sothatthepointPmust be.excluded byavery small volume T'ofsurface AS'.Since V2(l/r)= asthesolution oftheLaolacian. potential equation, theidentity (1)becomes Since S'isvery small andcanaswellbeassumed tobeasmall sphere around pointPatr=0,onehas also, observing thatthe outward normal onSr isdirected towards point P, a/i\ i a*a* .2J -- 1- )=-- 2>--=>dS'=r2dtt dn\r/r2dn dr with dtitheelement ofthesolid angle from P,sothat thelast integral becomes rra* //r-JJn dr Here, <J>(P)istheaverage potential value overthesurface S',and asr0,itbecomes identical with thepotential value atPitself; theremaining integral vanishes asr >0,assuming thatd$/dr remains finite asinanyregular region. 518 Three-dimensional Analytic Solutions [Gh.8 Theidentity (2)yieldsnow finally anexpressionforthepotential itself, lir With thegeneral interpretation (seesection 2)(again observing theproper direction ofthenormal) the firstterm constitutes theintegral expression fortheelectro- static potential asgivenin(2-5) interms ofspace charge within thevolume T.Having assumed Stobearegular surface, the othertwointegralsin(3)constitute fictitious charge effects; the firstrepresents thepotential ofanequivalent surface charge dis- tribution aontheinside ofSasin(2-3), thesecond thepotential ofanequivalent dipole moment distribution withmoment e<i>dS perelement dS(directed intothevolume T)ascomparison with (12-33) shows. Thus thepotential insideSisdefined completely bytheactual space charge within Tandbycharge distributions ontheinside ofSwhich replace theeffect ofallcharges actually located outside ofSandwhich reduce thepotential outside ofS everywhere tozero.Anexcellent detailed interpretation ofthis integral representation (3)andthephysical meaning ofitsparts isgiveninStratton,A23pp.185-192. Therepresentation (3)can stillbemaintained ifconductors are located within thesurface S;inthiscase,however, theconductor surfaces must beconsidered aspart ofthebounding surface ofT andthefirstsurface integral in(3)willthen include therealcharge densities onthese conductor surfaces. GREEN'S FUNCTION FORTHE FIRST BOUNDARY VALUE POTENTIAL PROBLEM Itisseenthatthepotential atanypoint ofaregular region can befound byintegrations over theboundary oftheregion. Ifno space chargeispresent, the firstterm in(3)disappears. If, furthermore, only surfaces withknown potential values form the Sec. 34] Green's Function 519 boundaries oftheelectrostaticfield, sothattheboundary condi- tions state*=$>a,&$--onsurfaces a,/3, ,then theboundary value problemiscalled ofthefirstkind, asstated insections 2and 6and,more particularly, section 28.The potential solution is then inintegral form Ingeneral,if$andVaresolutions oftheLaplacian potential problem, theleft-hand side of(1)vanishes completely, andone alsohas Adding (4)and (5)shows that,ifonecanselect \Finsuchamanner thatanywhere onthesurface Sthevalue of =(6)onS thesecond terms drop out,andonehas where G(P,Q)iscalled Green' sfunction ofthefirstkindand isthe combination (6).Itwill, therefore, generally beafunction of thepoint P(x, y,z)ofobservation >where thepotential value is tobefound, andofthepoint Q(x, y,z)onthesurface Swhere the potentialvalue isknown. SeeKellogg,010 p.236;Bateman,01 p. 240;andmany other textbooks onadvanced mathematics. TheuseofGreen's function forthesolution ofpotential problems ofthe first kind, then, requires thatoneassume atapointPin theregionrbounded bysurfaces ofknown potential values, apoint chargeofunitcharge value andwith 1/rasreduced potential function (strippedofallconstant factors), andthatonefindthe suitable setofimage charges with respect tothebounding surfaces which renders allofthem ofzeropotential value. Thesum total ofpotentialsoftheoriginal and allimage charges then constitutes G(P, Q),which canbeused tofindthepotential functions by(7) foranypointPintheregionr.Itisclear thattheuseofGreen's function demands thecomplete solution ofarelated boundary 520 Three-dimensional Analytic Solutions 1.8 value problem, even ifnotquite ascomplex asthewhole original problem might be!Moreover,itthen requires afurther surface integration. For thisreason, there areonly afewinstances in which Green's function actually hasbeen used forthepotential evaluation; however, many specific Green's functions ofthe first kindhavebeencomputed:indeed, every oneoftheimage problems involving apoint charge canbeinterpreted asaGreen function. +2 FIG.34-1 Green's Function forthePlane z=0. Green's Function forPlanes. Fortheinfinite conducting plane located atz=0,thepotential solution (10-15) gives Green's function when referred toFig.10-2 as G(P,Q)=-- Tp Tp:(8) where TPand rp>arethedistances between anypoint P(x, y,z) oritsimage pointP1 '(x, y, z)andanypoint intheconducting plane Q(x, y,0)asinFig.34-1, sothat rP2=[(*-x)2+(y- y)2+(-z)2 ], rP>2=((x-x)2+(y- y)2+(+z)2 ](9) since theimage must belocated symmetrically with respect tothe planez=0.Thenormal derivative atz=isthen -G(P,Q)=-2z[(z-x)2+(y-y)2+ oz(10) Sec. 34]Green's Function forSpherical Surfaces 521 sothat foranygeneral potential distribution intheplane 2= onehasinaccordance with (7)thepotential solution /*+ /+. *(*,*,*) =5-Idx [(z-z)2 27Tt/- i/-- +06-y?+z2]-*(x, y)dy (11) Complete results for$=3>owithin therectangle a<x<+a, 6<y<+&,and$=outside ofit,aregiven inByerly,C2 p. 138. Thismethod can, ofcourse, readily beextended toametallic corner formed bytwoperpendicularly intersecting conducting planes, asby(10-19), orforintersection atanyother angle, ir/n, where nisinteger. Obviously, theamount oflabor rapidly grows prohibitiveifonekeeps inmind theintegration (7),which might beperformed numerically orbymachine methods inspecial cases. Fortwoparallel planes, Green's function canberepresented by aFourier integral1inaxial distancep,since axialsymmetry pre- vails; therelation tothesolution bymeans ofimages asinsection 21isgiven inBateman,ci p.414,who also gives, p.472,Green's function foraconducting wedge, i.e.,intheoutside space oftwo conducting planes intersecting atir/nwithn>1andinteger as well forthesemi-infinite single plane. Smythe,A22 p.210, gives theresults forarectangular prism andarectangular boxinterms ofFourier double series expansions. Green's Function forSpherical Surfaces. Forthesingle sphereofradius athepotential solution inthepresence ofapoint chargeisgivenby(10-26), where bisthedistance ofthepoint charge from thecenter ofthesphere. Inorder togetGreen's function onehastomake thespherical coordinates ofPmore general, say,P(r, 0,<)andthose ofQ(r=a,0,#)asinFig.34-2. Thus onehas where, then rp*=[r2+f2-2rfcos7], (13) 1C.Fox, Phil. Mag., 6,7,p.994(1928); alsoBateman,cl p.413. 522 Three-dimensional Analytic Solutions [Ch.8 aretherespectivedistances ofthepoint charge atPand itsimage atP'jwhich islocated onthesame radius vector butatadis- tance a2 /rfromthecenter. Theangle between theradius vectors toP(or P')andQisgivenby cos7=cos cos9+sin sin5cos(c/> ?) (14) Thenormal derivatives with respecttofcannowbeevaluated and give d^/l\ rcos7-r tj/1\_(Q2 /y)cos7-f ~drW" rPA' dr\rP,) rP? P(r,e,<t>) FIG.342Green's Function fortheSphere. Sincenow forf=a,G(P, Q)=asboundary condition, onecan replace rp>from (12)sothat andtherefore -r~~a(^**" ddC^* -^$($,$)a2sinddd? (15) 4ira /jaro ^-o TP where rP2=r2+a2-2racos7 (16) Sec. 34] Green's Function fortheCylinder 523 andcos7from (14). Theintegral (15)isthecelebrated Poisson integral forthesphere; seeKellogg,010 p.241;Bateman,ci p.367; andmany other textbooks onadvanced mathematics. Green's functions foracircular diskandaspherical bowl are alsogiveninBateman,ci p.465. The solution forthepotential ofapoint charge inanearthed cone ofsemiopening6=acanbeobtained inspherical coordinates bymeans ofthegeneral product functional expansion (33-16). Thepoint charge, which isconfined tothevolume element dr, mustbeexpressedasadouble Fourier series inand0.Smythe,A22 p.154,gives thecomplete solution fortheconical space aswellas fortheconical box. Green's Function fortheCylinder. For the circular cylinder, thepotentialofapoint charge cannot befound byany image theory. Onehastousethecomplete functional solution from section 32andexpand thepoint charge, asdistributed overa small volume 5r,intoaFourier-Bessel series. Asolution which issymmetrical about theplane through thepoint charge andthe axis ofthecylinder andalsosymmetrical about theplane through thepoint charge andnormal totheaxis ofthecylinderisgivenby thedouble series (seesection 32) G(P,Q)=LLCn,me-^*-^ Jm(pnp)cosm(9-*,) (17) nm ifaistheradius ofthecylinder; thepoint chargeislocated at pointPwithcoordinates p<a,z=ZQ,and</>= </> ,andthepoint Qat(p=a,z,?).Thevalues pnareobtained from thecondition that Jm(pna)= Intheplane=<fothefield lineshavenoaxialcomponent except right atthepoint charge, which isassumed asaninfinitesimal area8S=poM^, overwhich theintegral gives onehalfthetotal electric fluxinthepositive z-direction andonehalf inthenegative z-direction. The Fourier coefficients Cn,minthe derivative (dG/dz) are,therefore, or 524 Three-dimensional Analytic Solutions [Ch.8 where thebracket ontheleft-hand side isthenorm oftheBessel function oforderra,andwhere ontheright-hand side I/TTholds form>0,1/27T form=0.Inaccordance with thedefinition of G(P,Q)onehas,then, -(19)2 sothatwith thisand (18),oneobtains +2EJm^np)'Jm(V^cosm(0_0)1 (20)m=l [Jm+l(Pna)\ } This function, asrequired,issymmetrical inthecoordinates ofthe pointsPandQ;actually thezerosubscripts have been dropped, since they arenolonger necessary. Touse (20), onemust differentiate with respect top;intheresult setp=aandintroduce itagain intothegeneral form (7),which becomes here )ad (21)1/*- /-*+*r^G(P Q)~\ *(P,0,z)=--/dzI\J^ ITJO J* Ldp Jp TheGreen function isdeduced2inSmythe,A22 p.174, forthe cylinder aswell asforthecylindrical box. Forthesomewhat simpler case ofapoint charge ontheaxisthesolution canbe given interms ofaFourier integral towhich reasonable approxi- mations3canbemade soastoallow further integrations. GREEN'S FUNCTIONS FOROTHER BOUNDARY VALUE POTENTIAL PROBLEMS Potential Problems oftheSecond Kind. Ifthenormal componentofthe field gradient orthecharge distribution is specified ontheboundary surfaces, rather than thepotential values, one calls theboundary value problem ofthesecondkind, asstated insections 2and6and,moreparticularly, insection 28. Starting again from thetwo relations (4)and (5),oneobserves 2C.J.Bouwkamp andN.G.dcBruijn, JlAppl. Phys., 18,p.573(1947). 3E.Weber, JlAppl. Phys., 10,p.663(1939); alsoBouwkamp andde Bruijn,loc. cit. Sec. 34] Other Potential Problems 525 that,ifonecanfindafunction ^such thatthecombination G(2>(P,Q)= hasanormal derivative ontheboundary surfaces which isatmosta constant butpreferably zero, then thesum of(4)and(5)gives Thefunction (7(2)iscalled Green's function ofthesecond kind;it determines thepotential function except fortheconstant second term (kisanarbitrary constant), which isasexpected, since only thenormal derivative ofthepotential isknown; seeKellogg,010 p.246. Rather fewexplicit solutions have been given forthissecond kind ofGreen's function, sothat theusual terminology "Green's function" without qualification isassumed toapply tothe first kind only. Fortheinterior ofthesphere ofradiusa,thesecond kind of Green's function interms ofthecoordinates ofP(r, 0,0),P'(r'= a2 /r,0,c/>), andQ(f=a,0,J)is )=+-+ln_-(23)rP rrP>aa2+rrp.-rfcos7v' where rPand rP,aredefined asin(13)butwith r<a,andcos7 isgiven by(14). Onthesurface ofthesphere r=a,and rp>=(a/r}r p,sothattheform(22)becomesexplicitly X+-In ;- --ad$+$(0) (24) \_rPaa+rp-rcos7J' v' where $(0) isthepotential atthecenter ofthesphere inaccord- ancewith themean theorem ofGauss (Bateman,01 p.369). The above solution isgiven inKellogg,010 p.247. Other Potential Problems. Itispossible toconstruct formally Green's function formany other types ofproblems, notably those inwhich boundary conditions ofthe firstand second kind aremixed.Similarly, onecandeduce aformulation 526 Three-dimensional Analytic Solutions [Ch.8 forthethird kind ofboundary value problem, asinBateman,cl p.141. Inmany cases, however, the direct solution ofthe boundary value problemislessinvolved. Onemight surmise that Green's function fortwo-dimensional problems would bemuch simpler toformulate and that, indeed, itshould have close relation tothecomplex potential theory. Asa'matter offact, there exists aunique relationship between Green's function foraregular region andtheconformal transfor- mation ofthatregion upon theunit circle which wasdiscussed in section 28.Whenever onecanperform the latter, onehasthe explicit solution fortheformer andvice versa, sothatGreen's function willgivenoadvantage. Forthedetails seeKellogg,010 p.365. PROBLEMS 1.Two infinitely long coaxial cylinders ofequal diameters andpotentials *=and*=Vhave finite separation 2dasinFig.304.Assuming the potential tovary asl/^V sin(wz/2d) along p=aacross thegap, findthe potentialdistribution near theaxisp=0. 2.Asemi-infinite cylindrical barofcircular cross section with radius a extends forz>0.Thebase at2= iskept atahightemperature Tand heat istransferred from thecylindrical boundary surface inaccordance with (29-24). Find thetemperature gradient along theaxis p=0.Find the amount ofheat transferred totheambient medium. 3.Express thesolution (30-46) fortwosemi-infinite coaxial cylinders near theaxisp=intheseries form (30-7) andidentify the first three terms. 4.Formulate thesolution forthepotential distribution between twosemi- infinite coaxial cylinders ofdifferent radiiR\andRI>R\,both starting at z andforming there anelectron lenssimilar toFig.30-3. Point outthe basic difficulty ofanexact solution. 5.Demonstrate thevalidity oftheexpressions (30-16) and(30-17) for vector potential andmagnetic fluxdensity near theaxis ofasingle circular loop ofcurrent. 6.Find thebestspacing ofthree coaxial circular current loops lying in parallel planes inorder toproduce nearly uniform magneticfield close tothe axis, if(a)theloops areidentical andcarry thesame current7,(b)theloops areidentical butcarry conveniently chosen different currents. 7.Forthetwosemi-infinite coaxial cylindersinFig.30-3determine the values ofzforwhich thepotential along theaxis iswithin 2%oftherespective cylinder potential. Check thiswith theapproximation form (30-50). 8.Avery largenumber ofcoaxial cylinders ofequal diameters arearranged with infinitesimal gaps similar tothetwocylinders shown inFig.30-3. As- suming thatthevoltage increment between anytwoneighboring cylindersis AV,findthepotential distribution along the axis. Choose thelength to diameter ratioL/2a sothat thepotential increases nearly linearly along theaxis. Problems 527 9.Acircular cylinder oflength L,radius a,andcompletely closed except foracoaxial circular apertureofradius b<ainoneoftheendfaces canbe considered acollector ofelectrons orions. Find thepotential distribution inside,ifthecylinderisatground potential andtheaperture hasanarbitrary radial potential distribution *(r).Plotsome equipotential surfaces if6=a/2 andtheaperture potentialisalinear function oftheradius. 10.Inasolidcube ofside a,oneface iskept attemperature To,theopposite face isideally insulated, and alltheother faces transfer heat totheambient medium ofzerotemperatureinaccordance with (29-24), namely, k(dT/dn) + fT=0.Find thethermal resistance ofthecube. 11.Arectangular metal boxasshown inFig.31-1hasthefacex=kept atpotentialdifference Vwith respect toallother faces. Find thepotential distribution. Find thesurface charge onthefacex=a. 12.Assume inproblem 11thatthetwoopposite faces x=andx=aare kept atthesame potential difference with respect toalltheother faces. Find thepotential distribution. Find thecharge distribution onfacex and itstot'al charge. 13.The solidconducting rectangular block inFig.31-1hastwoelectrodes with potential difference Vapplied, onecovering thelefthalftopfacex=*a, theother covering theright halfofthelower facex=0.Find theresistance oftheblock. Hint: divide theblock intotwohalves bytheplane z=c/2 andestablish theboundary conditions inthisplane. 14.Find thecapacitance between two confocalellipsoids ofsemiaxes A>B>C,anda>b>c,respectively. 15.Find thegravitational potential produced byanellipsoid ofmass density pandwithsemiaxes a>b>c.Show that atlarge distance themass canbeconsidered asconcentrated atthecenter oftheellipsoid. 16.Adielectric ellipsoid withsemiaxes a>b>candofdielectric constant ebecomes uniformly polarizedinauniform electric field#(0)which isparallel tothelargestaxis. IfthepolarizationisP,parallel to(0)and inopposite direction, findtheresultant potential distribution. Plotsome resultant equi- potential and field lines. Find theresultant potential atlarge distance from theellipsoid. Find theequivalentdielectric constant interms ofPand#(0) . 17.Aconducting ellipsoid ofsemiaxes a>b>cisintroduced intoauni- form electric fieldE(Q)with itsaxis 6parallel toit.Find theresultant po- tential distribution. Find thecharge distribution over thesurface ofthe ellipsoid. 18.Find theapproximationsforathinlongrodforwhich a2>bandb=c byutilizing thesolution fortheconducting ellipsoid. Verify theresults with thevalues obtained insection 12. 19.Deduce theLaplacian differential equation and itsseparation into ordinary differential equations forthecircular cylinder coordinates bymeans ofthegeneralized theory insection 32. 20.Asolid cylindrical ring asinFig.322hasthebase z=ckept athigh temperature T\,theopposite endface z= iscooled toalowtemperature TO,andthecylinder surfaces p=aandp=btransfer heat tothesurrounding medium inaccordance with (29-24) intheformA;(dT/d7i) +f(Ti-T2)=0, whereT2isthefixed temperature oftheambient and2\>T2>TO-Find 528 Three-dimensional Analytic Solutions [Ch.8 thetemperature distribution inthering. Find thetotalheat transferred from theheated base z=c. 21.Allthewalls ofahollow cylindrical ringasinFig.32-2 arekept atzero potential except anannular ringa\<p<b\ontheface2=0, which isat potential Vandseparated byinfinitesimal gapsfrom therestofthesurface. Find thecharge distribution ontheannular ring. Find itscapacitance with respect tothewalls ofthecylinder. 22.Asolid cylindrical ringasinFig.322hastwothinringelectrodes ap- pliedinitscenter planez=c/2,oneattheouter surface p=b,theother at theinner surface p=a.Find thecurrent distribution ifthetotal current enteringis/,thesmall width oftheelectrodes w,andthecurrent distribution canbeassumed asuniform. Find theresistance ofthering. 23.Asolid cylinder offinite lengthisheated internally bydistributed sources ofspace density hsuch asjoule heat. Find thetemperature distri- bution iftheheat lossonallsurfaces isgiven by(29-24) intheformk(dT/dn) +f(T To)=0,where TOistheambient temperature. Find themaximum temperature. Find thetemperature distribution along theaxis. 24.Ahollow cylinder offinite lengthisgrounded. Find thepotential dis- tribution ifaquasi sphere withcharge Qisplaced ontheaxisatthecenter of thecylinder. Find itscapacitance. Hint: divide thecylinder space bythe planeofsymmetry intotwohalves andconsider that inthisplane thenormal dielectric flux isinjected likecurrent fromanelectrode. (SeealsoSmythe,A22 p.175.) 25.Averylongconducting cylinder ofradius aiscovered withadielectric ofthickness a/2. Find thedistribution ofthepotential andtheelectric field within thedielectric iftheconductor surface hasapotential value given by Vsin(2ire/L) andtheouter surface ofthedielectric iskept atzero potential. 26.Asolidconducting cylinder ofradius aand finite length hasonecylin- drical electrode ofsmall radius papplied atoneendfacewith center atr=a/2 andthesecond, likeelectrode attheopposite faceagain with center atr=a/2 butindiametrically opposite position. Findtheresistance tothecurrent flow. 27.Ahollow cylinder ofradius aand finite length Lhasanarrow slotcut initscylindrical surface parallel totheaxisandoflength L/2. Assuming that thecylinderisgrounded andthattheslot iscovered withastrip ofpotential V,findthecapacitanceofthestripwith respect tothecylinder walls. Assume theslotsymmetrically located andthestrip fitting intotheslotwith infini- tesimal clearances. 28.Apoint chargeQislocated atadistance dfromaninfinite plane dielec- tricboundary. Find thepotential distribution bymeans of(32-44) forthe potentialofthepoint charge andthecomplete solution oftheboundary value problem. Demonstrate that theresult canbeinterpreted interms ofthe image method insection 21. 29.Find thecapacitance between twosmall spheres ofradii p\and PI carrying charges Qandbeing located symmetrically with respect tothe dielectric plateinFig.32-4. 30.Acircular loop ofcurrent ofradiusRislocated inaplane parallel to aninfinite magnetic plateofthickness aandpermeability /*fanalogous to Fig.32-4. Find theinductance oftheloop. Problems 529 31.Acircular ringofchargeofloopradiusRandsmall wire radius pislo- cated inaplane parallel toaninfinite planedielectric boundary andatdistance afrom it.Find itscapacitance. Find thecharge distribution onitssurface. 32.Twosmall semispherical electrodes with centers inthesurface ofground andburied initareadistance 2capart. Find theresistance between them if theground hasuniform conductivity 71toadepth a\,anduniform conduc- tivity 72fortheadditional depth a^beyond which theconductivityissolarge that itcanbeassumed infinite. 33.Verify thesolution (32-62) forthetwohalfelliptic cylinders inFig. 325bymeans ofconformal mapping. 34.Find thecharge distribution onthetwohalfcylinders ofFig.325and evaluate thecapacitance. 35.Deduce theLaplacian differential equation andtheequivalent ordinary differential equations inthecoordinates rand forthespherical coordinate system from thegeneral theory atthebeginning ofsection 33. 36.Find thetemperature distribution inasolid sphere ofradius aifthe twodiametrically opposite caps<9<Tr/4and 3?r/4<6<ITarekept at hightemperature TIandthezone ofthesurface ir/4<<37T/4 iskept at lowtemperature TQ.Find thethermal resistance. 37.Asolid hemisphere oflarge radius aissetwith itsflatsurface upon conducting ground ofpotential *=0.Find thecurrent distribution inthe sphere,ifanelectrode ofpotential Visapplied over <<ir/Qwith pole=located onthenormal toground. 38.Asolid conducting sphere hastwoelectrodes ofsmall areaSapplied atthepointsAandB'asinFig.33-3. Find theresistance between the electrodes. Hint: usefortheassociated Legendre functions aTaylor series approximation nearBr , 39.Aconducting thin hemispherical shell ofradius aisplaced with its large circle asmall distance above aconducting plane. Find thecapacitance between theshellandtheplane. Find thecharge distribution induced inthe planeiftheshell carries atotal charge Q. 40.The dielectric spherical shell offinite thickness asinFig.33-4with e2=eand ea=ecarries twohemispherical electrodes ofpotential differenceVonitsouter surface. Find theinternal capacitance between thehemispheres. Compareitwith thecase ofasingle dielectric ofconstant e . 41.Auniformly charged circular ringofradiusRiscoaxial withadielectric sphere withconstant e2andradius a.Find theresultant field distribution if theplane ofthering isatdistance bfrom thecenter ofthesphere. Find the capacitance ofthering forasmall wireradius p. 42.Find thepotential solution forapoint chargeQlocated within thedi- electric sphere at6<ainFig.335.Find itscapacitance forasmall radius p.Find theapproximate charge distribution onthesurface ofthesphere. 43.Find themutual inductance oftwocoaxial parallel circular current loops ofequal radiiaandcenter distance dinterms ofspherical harmonics. 44.Acircular current loop ofradiusRiscoaxial withamagnetic sphere of permeability Mandradius a.Find theresultant magnetic field distribution iftheplane oftheloop isatdistance bfrom thecenter ofthesphere. Find theinductance oftheloop forasmall wireradius p. 530 Three-dimensional Analytic Solutions [Ch.8 45.Find thepotentialdistribution within thecone =w/6and r^oif thepotentialdifference between thecone surface andthespherical zone isV. Find thecapacitance. 46.Apoint chargeQislocated attheaxis ofagrounded conewithangle =7T/6. Find theinduced charge distribution onthecone. Assuming the charge toreside onasmall sphere ofradius p,find itscapacitance. 47.Athin circular metallic disk ofradiusRislocated atthecenter ofan oblate conducting spheroid ofsemiaxes aandb=3a/4. Find thepotential distribution ifthediskbelongs tothefamily ofspheroids. Find thecapaci- tance ofthedisk. 48.Findthepotential distribution within adielectric, oblate spheroidal shell offinite thickness inauniform electric fieldEparallel totheaxisofrotation. 49.Avery small sphere withcharge Qislocated inthecenter ofacircular apertureofaninfinite conducting plane. Find thefield distribution. Find thecapacitance ofthesphere. 50.Athin metallic rodoflength 2cislocated atthecenter ofaprolate conducting spheroid ofsemiaxes aand6=3o/4. Find thepotential distribu- tion iftherodbelongs tothefamily ofthespheroids. Find thecapacitance of therodforasmall radius p. 51.Find thepotential distribution within adielectric prolate spheroidal shell offinite thickness inauniform fieldEparallel totheaxis ofrotation. 52.The inside ofatank ofinsulating material filled withconducting fluid canbeapproximated byaprolate spheroid ofsemiaxes aandb=a/2.Two electrodes areinserted attheopposite endsalong theaxisofrevolution;their lengths area/10.Find thetotal resistance ofthefluid iftheuniform con- ductivityisy. 53.Asmall sphereislocated with itscenter inthesurface plane ofground. Atadistance cdirectly below thesphereisavery long thinrodextending perpendiculartothesurface oftheground. Find theresistance between the sphere andtherod ifapotentialdifference Visapplied and iftheconductivity ofground canbeassumed uniform. Usetheinverse totheprolate spheroidal coordinate system. 54.Show thattheintegrals in(33-112) arereducible toelliptic integrals. Demonstrate that thesolution in(33-111) actually becomes that forthe circular ring ofchargeifaisvery small and b>/.Observe that as obe- comes very large, onecanapproximate (cosh ocos17)^+Vcosh o (^cosij/Vcosh fo). 55.The infinite planez=haszero potential everywhere except fora circular area ofradiusRwhere thepotentialisV.Findthepotentialdistribu- tion fori>0.Find thecharge distribution intheplanez=0. 56.Twoconducting planes intersect atanangle ir/6. Find thepotential distribution between their halves ifthey arebisected byaplane normal to both, andonehalf oftheintersecting planes carries potential zero, theother halfpotentialV.Find thecharge distributions ontheplanes. 57.Find Green's function fortheinterior ofacubical box. 58.Athin sphericalshell isbisected intotwohemisphericalshells witha potential difference Vbetween them. Find thepotential distribution inside theshellsbyGreen's function, andverify thesolution (33-31). Appendix 1 LETTER SYMBOLS FOR ELECTRICAL QUANTITIES The letter symbolsforelectrical quantities havebeenchosen inclose correspondence withthelatest"Proposed American Standard," prepared in1947bytheCommittee Z10.8 onLetter Symbols forElectrical Quantities oftheAmerican Standards Association under thechairman- shipofProfessor Edward Bennett. Tomake reference more con- venient, table 1-2gives analphabeticallistofthequantities, their symbols, and their units inthenowmost frequently used rationalized MKSC system (theextended Giorgi system ofunits), which hasas fundamental units themeter, kilogram-mass, second, andcoulomb. Where thestandards proposal allows alternative symbols ordesignations, achoice hasbeenmade herewhich leads tominimum conflicts. The onlymajor discrepancyisthesymbol K,usedhere forthecurrent sheet density instead ofA,since thelatter would conflict with A,thesymbol TABLE 1-1 ALPHABETICAL LISTOFSYMBOLS WITH ITEMNUMBER orTABLE 1-2 ItemNo. Symbol Table 1-2 7 7 e 11 er 13 ev 12 A 20 X 3 E 24 p 27 ^ 30 5 36,26 Q 2 ^ 32 F 21 <ft 39 M 31 G 6 R 40 p 5 H 25 S 16 <r 4 / 8 V 17 * 35 J 9 W 42,18 *m 19 K 10 * 15 L 23 531 532 Appendix 1 formagnetic vector potential, andboth quantities occur simultaneously inseveral oftherelations insection 6. Asanadditional assistance fortheidentification ofthesymbols used, table 11gives thealphabetic listing ofthesymbols withtherespective itemnumbers oftable 1-2. TABLE 1-2 ALPHABETICAL LISTOFTHENAMES OFQUANTITIES WITH THEIR SYMBOLS ANDUNITS Letter Symbols forElectrical Quantities 533 TABLE 1-2 Continued Item Quantity Symbol MKSC Unit Ampere-turn Coulomb-meter Weber-meter Henry Henry permeterRemarks Note 4onsub- scripts Orweberper ampere-turn Volt Ampere-turn Weber permeter Watt Ampere-turn per weber Ohm Meterpersecond JouleNote 4onsub- scripts Notes1,2 Notes1,2 Note 1.Quantities perunitlength, area, orvolume aregenerally designated bythecapitalletters from thetable with thesubscript1unless aspecific symbolislisted inthetable. Note 2.Formutual coefficients(partial capacitances, inductances, resist- ances, etc.) double subscripts areused inthesense ofdeterminant notation, i.e.,the firstindex indicates row, thesecond column ofthesquare arrayof coefficients. Note 3.Current sheet density cannot bedesignated byAasproposedin thestandards, since itoccurs inthesame equation with A,themagnetic vector potential; thenotation KifalsousedbyStratton.A4a Note 4-Potential differences usually carry adoublesubscript, theorder indicatingthedirection inwhich thedifference istobetaken. Appendix 2 CONVERSION TABLES FORUNITS Fortheconversion ofunits fromonesystem toanother itiswellto keepinmind afewbasic concepts pertaining tophysical quantities1 which tend tominimize misinterpretations. Anymathematical equation defines arelation between numerical values, whereas physical laws relate physical quantities whose values areexpressed with reference tospecifically chosen units. Aphysical quantity Qisbestconceived astheproduct ofanumerical valueN andthechosen unit U, Q=NU (I) which merely reiterates thefact thatmeasurement isbasic toany quantitative knowledge about thephysical quantity Q.Conversion from aunitU\toanother unit C72, Q=NiUi=NZU2 (2) involves theknowledge oftheconversion factor Ui=Nl2Uz (3) which relates relative magnitudes ofunits butwhich obviously itself must beapurenumber foranytwoconsistent unitsystems; thus, Q=NiUi=(NiNiz)U 2,Nz=NiNiz (4) Thefollowing conversion table gives these values N\awith theMKSC system ofunits chosen assystem 1,since ithasbeen usedthroughout thismonograph; a=2ischosen astheCGSelectrostatic, a=3the CGS electromagnetic, anda=4thesymmetrical Gaussian, system of units, respectively. Any relation between physical quantities given intheMKSC system ofunits, asforexample equation (6-3) H-ds=7MKSC units (6) 1Handbook ofEngineering Fundamentals, section 3,"Physical Units and Standards," edited byO.Eshbach; published byJohn Wiley,NewYork, 1936. 534 Conversion Tables forUnits 535 willretain exactly thesame form inanyother unitsystem inwhich UHU8=U1 orinwhich allunits areconnected by"unitary" relations. There are, however, very fewsuch desirably consistent and logical unitsystems besides theMKSC system which utilize well-established units. All thesystemsintable 2-1fora=2,3,4contain several unitswhich are rather arbitrarily denned andtherefore lead toextra numerical factors inequationslike (5)which must becommitted tomemory. Toestablish thegeneral procedure ofconverting relations like (5) fromoneunitsystem1toanother system a,assume asimple equation giveninsystem1 A-B=C (6) where A,B,Carephysical quantities denned by(1),sothat insystems fand a,respectively, A=NfUf, B=NJUS, C (7)A=NaaUaa ,B=WC7 a6 ,C=NacUac with conversion factors N\asuch aslisted intable 2-1. Insystem1 forwhich (6)isvalid,itisobvious thatwith (7) NfNS =tfic ,UfUJ =C/!c(8) Inthesystema.suchanassumptionisnotgenerally warranted, and theunits might berelated by Ua*Ua*=kUa (9) where kmust beanumeric forany self-consistent unitsystem which claims tobeuseful fordimensional analysis ormodel theory. But in order tomaintain (6)asequation,itmustnowread A()()=^C(a) (10)K sothatwith theunits ofsystem athenumerical values arecorrected forthenon-unitary relation (9).Todetermine koneneeds only to convert (9)tosystem1bytherelations, l/i=Nla*Ua*,US=Nla*Ua, t/!'=Nla<Ua sothat Nla>AT,.' 536 o s s s I H0200 CO 8 SgItem TableAppendix 2 K/ s$iOOHP 32 a?CHQ gfi*&OOHP oooo 2CO_._j ,_,i-lN i3Si I000000 II II ^oo^ e f=xxo:ll II II ooooooo bbttb xxxxxxxxxxxx CqW ^K l4\r^\H\F^F\t x II ?f I Conversion Tables forUnits 537 andwith theunitequation (8)thisgivesatonce Nla*1=Nla*Nlab NiSNtj' kNla** } tobeused in(10). Applyingthisconversion toequation (5)andexpressingitinthe CGS electromagnetic system, onehas,witha=3from table 2-1, -ds=47r/(CGS emu) Similarly, forthesymmetrical Gaussian system witha=4,follows .4, kNu13X10+93X1010 H-ds= /(CGS- Gaussian). whereV=3X1010isthevalue ofthevelocity oflightbutnotthe velocityitself. Thistreatment canreadily beextended toanymixture ofunits such asthepoor compromises thatweremade rather early inmagnetic computations byexpressing theleft-hand side in(5)inCGS electro- magnetic units andtheright-hand side in"practical" ornowMKSC units. Oneobtains H-ds=/, H,sinemu; /inMKS and itshould befeltastheauthor's obligation toindicate clearly the hybrid units used. Appendix 3 REVIEW OFFUNDAMENTALS OF VECTOR ANALYSIS Adistinction ismade between aphysical quantity which isuniquely given bynumerical value andunitandcalled ascalarquantity, andone that requires inaddition thespecification ofdirection inspace, avector quantity. Thenotation ofavector, V,therefore implies thefactthat onemustknow allthree components inacoordinate system, sayVxt Vv,VfinCartesian coordinates, inorder tobeable toconstruct the vector. Ifthere isassociated with every point inspace avector quantity describing aphysical phenomenon, such spaceiscalled avectorfield; the electrostatic field, forexample, isdescribed bythe electric field strength. Vector Algebra. AvectorVmay bedisplaced parallel toitself inspace aslong asitretains both length and direction. Avector ofthesame length butopposite direction isdesignated as FIG.A-l Addition and Sub-*henegative vector-V .Thelength of traction ofVectors. tnevector isusually designated as absolute value|v|=V:itisascalar value; thedirection isusually designated byur=V/V, theunit vector oflength1.Avector ofthesame direction asVbutoflength V~lis called theinverse vectorV"1toV. TwovectorsVandWreferred toacommon starting point determine aplane. Thesum ofthetwovectors isthedirected diagonal ofthe parallelogram formed bythem asinFig.A-lwith thesamestarting point0.Thedifference (V-W)oftwovectors isobtained byaddingWtoV;itisthesecond diagonal inthesameparallelogram. For more thantwovectors thecorresponding extensions hold, since one canalways combine twovectors atonetime, their resultant with the 538 Fundamentals ofVector Analysis 539 third, etc.Theorder isirrelevant, since thecommutative, associative, anddistributive laws ofalgebra arevalid. Though vector operations assuch areentirely independent ofspecific coordinate systems,itisconvenient tointroduce orthogonal reference coordinates andthesimplestistherectangular Cartesian system asin Fig.A-2.Thevector canthen beexpressed V=iVx+JV V+kV, (1) whereVx,VV)andV2aretheprojections ofVupon thethree coordinate axes. Thesumanddifference oftwovectorsWandVarethensimply VW=i(V,Ws)+j(VvWy)+k(7, W.) (2) indicating theobvious extension toanynumber ofvectors. FIG.A-2Cartesian Reference Coordinates. The scalar ordotproduct oftwovectors (also inner product)isdefined V-W=VWcos7=V(W cos7)=W(V cos7) =(VXWX+VVWV+V,W S)(3) Itisascalar andcanbeinterpretedintwodifferent ways:astheproduct ofVandtheprojection ofWuponVorviceversa, valuable when work istobecomputed; and asthesum oftheproducts ofcorresponding vector components, valuable asconcept because ofitsfeasible extension tondimensions oreven infinite orthogonal systems (see sections 29 and 31). Application totheunit vectors, already used in(3),gives =J-J=k-k =1, i-J (4) 540 Appendix 3 The vector orcross productoftwovectors (also outer product)ia defined as VxWiJk VXVyVZWxWyWz=nVW sin7i(VyW Z~VZWy) +J(V ZWX-VXWZ) y-VyW x)(5) Itisavector directed normal totheplane denned byVandWandso that itforms thethird direction inaright-handed triplet withthem; obviously interchange ofVandWchanges thedirection ofn.Ifthe crossproductoftwovectors vanishes, theymusthave thesame direction. Onecaninterpret themagnitudeof(5)asthearea oftheparallelogram formed byVandW. Applicationtotheunitvectors gives iX i=jXj=kXk=iXj=k,jXk= i,kXi=j(6) Several significant productsofthree ormore vectors aresummarized below; theproofs canreadily begiven bydirect expansion with (4) and (6): U.(VxW)=V-(WxU) =W-(UxV) cyclic change (7) UX(VXW)=(U-W)V-(U-V)W (8) (TXU)-(VXW)=(T-V)(U-W)-(T-W)(U-V) (9) Formulation ofvector algebrainany other specific orthogonal coordinate system requires primarily thepertinent definition ofthe unit vectors. Thegeneralized forms forthese aregiven insection 31. Vector Differentiation. Inphysical problems, vectors arefunc- tions ofthespace coordinates which enter asscalar variables intothe definition ofthevector components. Differentiation with respect to oneofthese scalar variables follows exactly therules ofdifferentiation ofscalar quantities. Ontheother hand, inthevector field itisimportant toobtain the differential variation with respect toallthree space variables. Itis convenient andeconomical, then, tointroduce avectorial combination ofthederivative symbolsintheform ofthevector differential operator V=("del" or"nabla") =i+j+k-(10)dx dy dz Applicationofthisoperatortoascalar space function $(x, y,z)gives d<2> d<2> d$V*=i^+j^+k^=grad* (U)dx dy dz which isknown asthegradient ofthescalar function;itisactually a vector pointing everywhere inthedirection oflargest space variation Fundamentals ofVector Analysis 541 of<S>(z, y,z)andtherefore isalways normal tothesurfaces $(z, y,z)= cons, which arecalled niveau orlevel surfaces of<. Asavector, Vcanbeapplied toafield vector either inscalar orin vector product form inaccordance with (3)and (5), respectively. The results inthese twocases are ijk <L*LL dxdydz V.VvV,if^-^1\By dz/ dV,(12) curlV (13) The physical significance of(12) isobvious from the fact that divV= isnothing butthelawofcontinuity offluid flow, characteriz- ingthefluid asincompressible ;ifthefluid iscompressible, divVisthen related tothelocalchange inmass density. Thephysical significance of(13)canalsobeseen bestbyconsidering curlV=0,inwhich case eachcomponent must bezero,which canonlybeifinturn V-T>dx oralso ifVy=,yz=,V=grad' vxdx+ydi/+yzd2=d$(14) isacomplete differential whose integral overany closed pathmust vanish. IfVistaken asaforce vector, then (14)expresses thelawof conservation ofenergy and curlV=characterizes theforce fieldVas aconservative onewith aforce function or"potential" <f>(x, y,z). Since integration over aclosed pathisinvolved, one calls thevalue curlValsoameasure ofthe"circulation" ofthevector V. With thedefinitions given, onecannowdeduce v'=++~=V2*=Laplace V-(VxV)=0 Vx(V$)= Inaformal sense onecanwrite with (8) Vx(VxV)=(V-V)V-(V-V)V(16) (17) (18) 542 Appendix 3 which isnotvery sensible; butonecaninterpret bydirect expansion in theCartesian system the firstpart asmeaning V(V-V) =grad divV, andthesecond part asV^. However, thelatter contraction isper- missible only intheCartesian system, where itcanbeidentified with theLaplacian from (15);innoother coordinate systemisany explicit definition possible, andonecertainly must beware ofconfusingitwith thewell-established operation (15). Again, with (7)and (8)onecaninterpret Vapplied toproducts, such as (19) +Wx(VxV) (20) (21) (22) (23)V(V-W)=(V-V)W+(W-V)V+V V-(*V)=(V$)-V+$(V-V) V.(VxW)=W-(VxV)-V-(VxW) FIG.A-3Line Integral ofaVector; Stokes's Theorem. Vector Integral Theorems. The line integral ofavectorVis defined as (24) TVds =rV.ds=f*Vcos7da Ifthepathisaclosed oneandregular,sothat itcanbecontracted to apoint without intersecting itself, asforexample inFig.A-3,thenone cantransform v'dsff(VxV)'ds vds (25) Fundamentals ofVector Analysis 543 This isStokes' stheorem, andSisthesurface bounded bytheclosed path withdS=ndSsochosen that,from thetopofthenormal direction, theclosed Line integral appears counterclockwise; thesurface Scan have anyconvenient shape whatsoever. Iftheclosed line integral vanishes foranyregular path inacontinuous volumer,thenfrom (25)and (17) andin V-ds=0,VxV=curlV= V=V$=grad$(26) Thismeans that avector field with nocirculation isderivable in accordance with (17)from ascalar function usually called potential: it isapotential field;itisalso called alamellar fieldbecause theequi- potentialsurfaces subdivide space intonon-intersecting lamellas. .Thesurface integral ofavectorVisdefined asfluxofthevector, :V-dS=ffv ndS=Cfv cos7dS (27)JJs JJs where dS=ndSisthevector representation ofthesurface element as shown inFig.A-4. Ifthesurface Sbecomes theclosed oneS'which FIQ.A-4Surface Integral ofaVector; Gauss's Theorem. isregular,sothat itcanbecontracted toapoint without intersecting itself, then onecantransform where Tisthevolume bounded bytheclosed surface S'.This isusually called Gauss's theorem or,better, thedivergence theorem. Iftheclosed surface integral vanishes foranyclosed surface within avolumeT', 544 Appendix 3 thenfrom (28)and (16) inr':<fv-dS=0,V-V=divV= and' (29) V=VxA=curlA Thismeans that avector fieldwithout divergenceisderivable from another vector function A,usually called vector potential;itisasolenoidal fieldbecause thevector hasnosources andnosinks; the field lines defined aseverywhere tangential tothevector areclosed lines. Substituting in(28) forthevectorVtheproduct <f>(V>I>) andusing (21) intheright-hand integral, onehas dS=CCCV*-V* dr+fff <f>V2*dr (30) This isGreen's firsttheorem, which canbeused formany formal deduc- tions inthetheory ofpotential fields. Interchanging3>and^and subtracting thissecond relation from theabove, oneobtains (since the center term cancels) ndS=ff/Vv2*-*V2$]dr (31) which isGreen's second theorem. Substitutingin(28) forthevectorVthevector product Vx(VxW) and using theidentity (22) V-Vx(VxW) =(VxW)-(VxV) -V- [Vx(VxW)] onehas (VxVxW)-dS=f(T(VxW)-(VxV) dr-fffV-(VxVxW) dr(32) which isthevector analogue toGreen's firsttheorem (30) ;seeStratton,A23 p.250.Byinterchange ofVandWandsubtracting thissecond relation from (32),oneobtains [Vx(VxW)-Wx(VxV)]-dS = [V-(VxVxW)-W-(VxVxV)] dr(33) which isthevector analogue toGreen's second theorem (31). General Curvilinear Coordinates. Allthevector operations can readily beexpressed inanyorthogonal coordinate system byusing the general forms ofmetric factors deduced insection 31and specifically Fundamentals ofVector Analysis 545 illustrated inthevarious coordinate systems ofsignificanceinapplica- tions. Because offrequent references inthetext tothecylindrical and spherical coordinate systems, themost important vector operations arerepeated below forthese systems only. Cylindrical coordinatesp, \l/,z: __ /a* ia*a$\,,..V$=I i-i)(34) \dp pd\// dz/ 1dId&\ I32$32$V-V*=V2*=-(p ]+ +- pdp\P dp) p2dt'^dz*(35) (36) .(37) pp pd$/J Spherical coordinatesr,6,<p: /a* ia* ia*\V<p= I j---i--I (oo; \drr36 rsin30/ V-V=if(r^ T)+-i-J(sin07e)+-^^(39) r23r rsin^a0 rsin6dc/> TT)+ (40) rsin6\_dOd<j> I|-_l_^ -f(,7,)!,iff(ryf,-^11(41) rLsm a0 ar JrL^^ ^JJ References. Practicallyallthereferences inAppendix 4,A,where useofvector notation ismentioned, alsogive considerable detail on vector analysis;inaddition, many books onadvanced calculus contain chapters onvector analysis.Particular references notmentioned in Appendix4are : L.Brand: Vector andTensor Analysis; JohnWiley&Sons,NewYork, 1947. J.F.Coffin: Vector Analysis, Second Edition; JohnWiley&Sons,NewYork, 1924." 546 Appendix 3 J.W.Gibbs: Vector Analysis, edited byE.B.Wilson; YaleUniversity Press, NewHaven, 1901. L.PageandN.I.Adams: Electrodynamics, Chapter I;D.VanNostrand Co., NewYork, 1940. H.B.Phillips: Vector Analysis; JohnWiley&Sons,NewYork, 1933. H.H.Skilling: Fundamentals ofElectric Waves, Second Edition; JohnWiley &Sons,NewYork, 1948. J.Spielrein: Vektorrechnung; C.Wittwer, Stuttgart, 1927. A.P.Wills: Vector Analysis withanIntroduction toTensor Analysis; Prentice- Hall,NewYork, 1931. Appendix 4 GENERAL BIBLIOGRAPHY A.BOOKS ONELECTROMAGNETIC THEORY WITH PARTICULAR REFERENCE TOELECTRIC AND MAGNETIC FIELD PROBLEMS 1.M.Abraham andR.Becker: TheClassical Theory ofElectricity; Blackie andSons, London, 1932. Most readable presentation oftheclassical theoryinvector notation; usesCGS units. 2.S.S.Attwood: Electric andMagnetic Fields, Third Edition;John Wiley &Sons,NewYork, 1949. Very good introduction with excellent illustrations and fieldmapsof simpler types; usesMKS units. 3.E.Bennett andH.M.Crothers: Introductory Electrodynamics forEngi- neers; McGraw-Hill, NewYork, 1926. Very clear expositionofbasic facts, hypotheses, anddeductions, with many applications; usesapractical unitsystem. 4.F.Breisig: Theoretische Telegraphie; F.Vicweg &Son,Braunschweig, 1924. Givesmany practical applications ofsimple field problems; uses vector notation andCGS units. 5.E.Cohn: Daselektromagnetische Feld; J.Springer, Berlin, 1927. Very goodand clear treatment withmany applications; usesvector notation andapractical unitsystem. 6.E.G.Cullwick: TheFundamentals ofElectromagnetism; Macmillan, New York, 1939. 7.J.Fischer: Einfiihrung indieklassische Elektrodynamik; J.Springer, Berlin, 1936. Good presentation withmany practical examples; usesvector nota- tionandapractical unitsystem. 8.Handbuch derPhysik, Vol. 12:Theorien derElektrizitdt, Elektrostatik (1927); Vol. 15:Magnetismus, Elektromagnetisches Feld (1927); J. Springer, Berlin. Very comprehensive treatment with extensivebibliographies. 9.G.P.Harnwell: Principles ofElectricity andMagnetism; McGraw-Hill, NewYork, 1938. Very good introduction, using vector notation andMKS units. 10.J.Jeans: Electricity andMagnetism; Cambridge University Press, 1927. Very comprehensive treatment, particularly ofelectrostatics; uses longnotation andCGS units. 547 548 Appendix4 11.V.Karapetoff:TheElectric Circuit; McGraw-Hill, NewYork, 1910. Simple treatment, from engineering viewpoint; usesapractical unit system. 12.V.Karapetoff: TheMagnetic Circuit; McGraw-Hill, NewYork, 1910. Simple treatment from engineering viewpoint; usesapractical unit system. 13.G.Kirchhoff: Vorlesungen uber Elektrizitdt undMagnetismua; B.G. Teubner, Leipzig, 1891. Many detailed solutions ofstatic fields inlong notation; usesCGS units. 14.K.Kiipfmuller: Einfiihrung indietheoretische Elektrotechnik; J.Springer, Berlin, 1932. Very clear treatment withemphasis ontechnical applications; uses vector notation andapractical unitsystem. 15.G.H.Livens: TheTheory ofElectricity; Cambridge University Press, London, 1926. Verythorough andrigorous presentation oftheory invector notation; usesCGS units; nopractical applications. 16.M.Mason andW.Weaver: TheElectromagnetic Field; Chicago University Press, 1929. Excellent advanced presentationofstatic fields inmedia; usesvector notation andCGS units. 17.J.C.Maxwell: ATreatise onElectricity andMagnetism, Third Edition; Clarendon Press, Oxford, 1892. The original presentation; many detailed solutions ofstatic field problems with excellent field graphs; useslong notation andCGS units. 18.F.Ollendorff :Potentialfelder derElektrotechnik; J.Springer, Berlin, 1932. Largecollection ofsolutions offieldproblemsindifferent coordinate systems; usesvector notation andapractical unitsystem. 19.M.Planck: Theory ofElectricity andMagnetism; Macmillan, London, 1932. Very clear basic presentation, fewapplications; usesvector notation andCGS units. 20.R.W.Pohl: Physical Principles ofElectricity andMagnetism; Blackie andSon,London, 1933. Best presentationofexperimental evidence andvery clear exposition ofthephysical concepts; usesapractical unitsystem. 21.A.S.Ramsay: Electricity andMagnetism; Cambridge University Press, London, 1937. Clear presentation with simpler applications; usesvector notation andCGS units. 22.W.R.Smythe: Static andDynamic Electricity; McGraw-Hill, NewYork, 1939. Very extensive mathematical treatment andmany applications; largest collection ofproblems; usesvector notation andCGS units. 23.J.A.Stratton: Electromagnetic Theory; McGraw-Hill, NewYork, 1941. Veryadvanced treatment, very clearandcomprehensive; usesvector notation andMKS units. General Bibliography 549 24. J.B.Whitehead: Electricity andMagnetism; McGraw-Hill, NewYork, 1939. B.BOOKS ONAPPLICATIONS OFELECTRIC ANDMAGNETIC FIELDS (a)Fields inCables andLines 1.E.Clarke: Circuit Analysis ofA-CPower Systems, Vol. I;JohnWiley<fc Sons,NewYork, 1943. 2.L.J.Corbett: Inductive Coordination ofElectric Power andCommunica- tionCircuits; J.H.Neblett PressRoom, SanFrancisco, 1936. Discusses allphases oflineinterference. 3.W.A.DelMar; Electric Cables; McGraw-Hill, NewYork, 1934. Excellent bibliography. 4.P.Dunsheath: High Voltage Cables; I.Pitman &Sons, London, 1929. 5.H.B.Dwight: Transmission Line Formulas; D.VanNostrand, New York, 1925. Gives derivations ofcapacitance andinductance formulas forcon- ventional linesandcables. 6.L.Emanueli: High Voltage Cables; JohnWiley&Sons,NewYork, 1930. 7.A.E.Kennelly:Applications ofHyperbolic Functions toElectrical Engineer- ing;NewYork, 1912. 8.W.Nesbit: Electrical Characteristics ofTransmission Circuits; E.Pitts- burgh, 1926. 9.F.E.Pernot: Electrical Phenomena inParallel Conductors: NewYork, 1918. 10.D.M.Robinson: Dielectric Phenomena inHigh Voltage Cables; Instru- ments Publishing Company, Pittsburgh, 1936. Verygood bibliography; descriptive. 11.A.Russel: TheTheory ofAlternating Currents; Cambridge University Press, England, 1914. 12.L.F.Woodruff: Principles ofElectric Power Transmission, Second Edition; John Wiley&Sons,NewYork, 1938. Gives derivations ofcapacitances andinductances forconventional andn-vvire linesandcables. (b)General Dielectric Fields 13.A.Gemant: Elektrophysik derIsolierstoffe;J.Springer, Berlin, 1931. Verygood presentation ofphysicsofdielectrics. 14.A.Gemant: Liquid Dielectrics; John Wiley&Sons,New York. 1933. Monograph onphysicalcharacteristics ofliquid insulators. 15.F.W.Peek, Jr.: Dielectric Phenomena inHigh-Voltage Engineering; McGraw-Hill, NewYork, 1929. Extensive empirical datafrom engineering viewpoint. 16.A.Roth: Hochspannungstechnik;J.Springer, Berlin, 1927. Most comprehensive treatment oftheory anddesign ofinsulating materials; extensive bibliography. 17.A.Schwaiger: Theory ofDielectrics, Second Edition, translated byR.W. Sorensen; JohnWiley&Sons,NewYork, 1932. 550 Appendix 4 Contains many solutions ofpracticalfieldproblems andcorrelation withbreak-down data. 18.N.Semenoff andA.Walter: Diephysikalischen Grundlagen derelektrischen Festigkeitslehre; J.Springer, Berlin, 1928. Very goodsurvey ofexperimental methods. 19.J.B.Whitehead: Lectures onDielectric Theory andInsulation; McGraw- Hill,NewYork, 1923. (c)Electrons inElectric andMagnetic Fields 20.E.Briiche andO.Scherzer: Geometrische Elektronenoptik; J.Springer, Berlin, 1934. Original treatise onthesubject; rather comprehensive. 21.E.L.Chaffee: Theory ofThermionic Vacuum Tubes; McGraw-Hill, NewYork, 1933. Classical treatise onvacuum tube characteristics with several elec- trostatic fieldproblems. 22.V.E.Cosslett: Introduction toElectron Optics; Oxford University Press, England, 1946. Good introductory presentationofprinciples andapplications. 23.W.G.Dow :Fundamentals ofEngineering Electronics; JohnWiley&Sons, NewYork, 1937. Very detailed treatment ofbasic concepts; good bibliography. 24.O.Klemperer: Electron Optics; Cambridge University Press, England, 1939. Very concise andadvanced monograph. 25. I.G.Maloff andD.W.Epstein: Electron Optics inTelevision; McGraw- Hill,NewYork, 1938. Good basic treatment with particular applications tocathode-ray tubes. 26. J.Millman and S.Seely: Electronics; McGraw-Hill, NewYork, 1941. Good general presentation ofapplications. 27.L.M.Myers: Electron Optics; D.VanNostrand, NewYork, 1939. Very comprehensive treatment offield solutions andelectron trajec- tories; very extensive bibliography. 28. J.Picht: Einfuhrung indieTheorie derElektronenoptik; J.A.Earth, Leipzig, 1939. Concise andveryadvanced treatment. 29.K.R.Spangenberg: Vacuum Tubes; McGraw-Hill, New York, 1948. Very extensive treatment ofelectric fields invacuum tubes ofall types, including space charge effects andelectronoptics. 30.M.J.O.Strutt: ModerneMehrgitter Elektronenrdhren,Vo\.2; J.Springer, Berlin, 1938. Excellent butbrief treatise offieldproblems. 31.V.K.Zworykin andG.A.Morton: Television; The Electronics ofImage Transmission; JohnWiley&Sons,NewYork, 1940. Excellent expositionofprinciples ofaxially symmetrical fields and electron trajectories; good bibliography. General Bibliography 551 32.V.K.Zworykin, G.A.Morton, E.G.Ramberg, J.Hillier, andA.W. Vance: Electron Optics andtheElectron Microscope; JohnWiley&Sons,NewYork, 1945. Very comprehensive andauthoritative treatise withmany design principles andillustrative applications. (d)Electric Discharges inGases 33.J.D.Cobine: Gaseous Conductors; McGraw-Hill, NewYork, 1941. 34.M.Knoll, F.Ollendorff, andR.Rompe: Gasentladungstabellen; J.Springer, Berlin, 1935. Comprehensive tables ofallphysical quantities relating togaseous conduction. 35.L.B.Loeb: Fundamental Processes ofElectrical Discharge inGases; John Wiley&Sons,NewYork, 1939. Good basic treatment. 36.F.A.Maxfield andR.R.Benedict: Theory ofGaseous Conduction and Electronics; McGraw-Hill, NewYork, 1941. Clear presentation ofbasicprinciples; usesMKS units. 37.W.O.Schumann: ElektrischeDurchbruchsfeldstdrke vonGasen; J.Springer, Berlin, 1923. 38.R.Seeliger: Einfuhrung indiePhysik derGasentladungen; J.Springer, Berlin, 1933. 39.J.Slepian: Conduction ofElectricity inGases; Educ. Dept., Westinghouse Electric Corporation, 1933. Good advanced treatment. 40.J.J.Thomson andG.P.Thomson: Conduction ofElectricity through Gases; Cambridge University Press, 1928. Classical treatise onsubject, very detailed. 41. J.S.Townsend: Motions ofElectrons inGases; Oxford University Press 1923. (e)Magnetic Fields 42.A.M.Gray: Electrical Machine Design; McGraw-Hill, NewYork, 1926. Givespractical details ofmagnetic circuit computations. 43.F.W.Grover: Inductance Calculations; D.VanNostrand, NewYork 1948. Large collection offormulas fortheinductances ofsimple circuits and coilswithmany tables. 44.B.Hague: Electromagnetic Problems inElectricalEngineering; Oxford University Press, London, 1929. Presents basic theory andmany advanced solutions ofmagnetic field problems involving iron; alsogives good field graphs. 45.J.Hak: EisenloseDrosselspulen; K.F.Koehler, Leipzig, 1938. Givesmany computations ofinductances ofcoilswithout ironanda verycomprehensive bibliography. 46.E.Jasse: DieElektromagnete; J.Springer, Berlin, 1930. Magnetic circuit andforce actions aretreated from designer's view- point. 552 Appendix 4 47.M.Liwschitz: Dieeleklrischen Maschinen, Vol. 3:Design Principles; J. Springer, Berlin, 1934. Gives practical design principles withvery clear engineering view- point. 48.E.B.Moullin: ThePrinciples ofElectromagnetism; Oxford University Press, London, 1932. Good introduction withmany practical solutions ofsimpler problems. 49.R.Richter: Elektrische Maschinen, Vol. 1:Fundamentals andD-cMachines (1924); Vol. 2:Synchronous Machines andConverters (1930); Vol. 3: Transformers (1932); Vol. 4:Asynchronous Machines (1936); J. Springer,Berlin. Ineachvolume extensive computationsofmagnetic circuits and field distributions areincluded; treatment from viewpoint ofdevelopment engineer. C.BOOKS ONPOTENTIAL THEORY (a)General Potential Theory 1.H.Bateman: Partial Differential Equations ofMathematical Physics; Dover Publications, NewYork, 1944. Generalized treatment ofboundary value problems inmany coordinate systems; applications toallfields ofphysics. 2.W.E.Byerly:Fourier's Series andSpherical, Cylindrical, andEllipsoidal Harmonics; Ginn, Boston, 1902. Givesmany physical applications, particularly toproblems ofelectric andtemperaturefields. 3.R.V.Churchill: Fourier Series andBoundary Value Problems; McGraw- Hill,NewYork, 1941. Excellent introduction intosolution ofboundary value problems from allfields ofphysics. 4.R.Courant andD.Hilbert: Methoden dermathematischen Physik tVol. I(1931); Vol. II(1937); J.Springer, Berlin. Very comprehensive andrigorous mathematical treatise withmany applicationstophysical problems. 5.G.C.Evans: TheLogarithmic Potential, Discontinuous Dirichlet, and Neumann Problems; American Mathematical Society, Colloquium Publi- cations, Vol.VI,NewYork, 1927. 6.Ph.Frank andR.V.Mises: DieDifferential- undIntegralgleichungen der Mechanik undPhysik, Vol. I(1930); Vol.II(1935); F.Vieweg &Sohn, Braunschweig. Very comprehensive volumes ofapplications toallfields ofphysics. 7.A.Gray, G.B.Matthews, andT.M.MacRobert: ATreatise onBessel Functions andTheir Applications inPhysics; Macmillan, London, 1931. Many applicationstotemperaturefields. 8.E.Heine: Anwendungen derKugelfunktionen; Berlin, 1881. Many applications involving spherical, ellipsoidal, andBessel har- General Bibliography 553 9.E.W.Hobson: Spherical andEllipsoidal Harmonics; Cambridge Uni- versity Press, Cambridge, 1931. Most extensive treatise onthese harmonic functions withmany applicationstopotential problems. 10.O.D.Kellogg: Foundations ofPotential Theory; J.Springer, Berlin, 1929. Classical volume onpotential theory inallfields ofphysics; rigorous establishment ofmethods ofsolutions. 11.A.Korn: Lehrbuch derPotentialtheorie; Berlin, 1899. 12.T.M.MacRobert: Spherical Harmonics; E.P.Button, NewYork, 1927. Gives alsoapplicationstoelectrical problems. 13.F.D.Murnaghan: Introduction toApplied Mathematics; John Wiley& Sons,NewYork, 1948. Modern advanced treatment ofpotential equation with application toelectrostatics, andofgeneral boundary value problems byGreen's function andbyintegral equations. 14.B.O.Peirce: Newtonian Potential Function; Ginn, Boston, 1902. Gives themathematical theory ofthegravitational potential with some applicationstoelectrostatics. 15.W.Sternberg: Potentialtheorie; W.deGruyter, Leipzig, 1925. Brief mathematical treatise onexistence ofsolutions. 16.A.G.Webster: Partial Differential Equations; G.B.Teubner, Leipzig, 1927. General exposition ofmethods ofsolution withmany applications to allfields ofphysics. (b)Temperature Fields 17.H.S.Carslaw and J.C.Jaeger: ConductionofHeat inSolids; Oxford University Press,NewYork, 1947. Successor toIntroduction toMathematical Theory oftheConduction of Heat inSolids byH.S.Carslaw, aclassical reference onheatboundary value problems withmany solutions ofvalue inpractical applications. 18.J.B.J.Fourier: Theorie analytique delachaleur; Paris, 1822; English translation byFreeman, Cambridge University Press, England, 1878. Originaltreatise formulating thetheory ofheatwithmany illustra- tiveapplications. 19.L.R.Ingersoll andO.J.Zobel: Mathematical Theory ofHeat Conduction withEngineering andGeological Applications; Ginn, Boston, 1913. Excellent treatment byrigorous andapproximation methods with much practical information onconduction ofheat inmaterials. (c)FluidDynamic Fields 20.B.Eck: Einfuhrung indietechnische Stromungslehre, Vol. I:Theory (1935); Vol. II:Laboratory Methods (1936); J.Springer, Berlin. Excellent studies offlow lines. 21.Th.V.Karman and J.M.Burgers: General Aerodynamic Theory ,Perfect Fluids, Vol. IIofAerodynamic Theory, edited byW.F.Durand; J. Springer, Berlin, 1935. Excellent advanced theory offluid flowwithmany applications. 554 Appendix 4 22.H.Lamb: Hydrodynamics, Sixth Edition; Cambridge University Press, England, 1932. Advanced classical treatise withmany practical solutions. 23.W.Mtiller: Mathematische Stromungslehre; J.Springer, Berlin, 1928. Basic treatise withmany illustrative graphs. 24.L.Prandtl and O.G.Tietjens: Applied Hydro- andAeromechanics; McGraw-Hill, NewYork, 1934. Excellent andcomprehensive treatment. 25.T.G.Whitlock: Elementary Applied Aerodynamics; Oxford University Press, London, 1931. (d)Gravitational Fields 26.A.R.Clarke: Geodesy; Oxford, 1880. 27.G.Kirchhoff :Vorlesungen uberMechanik; B.G.Teubner, Leipzig, 1897. Many detailed solutions. 28.R.B.Lindsay: Physical Mechanics; D.VanNostrand, NewYork, 1933. Seealsoreferences 2,10,14,and16ofsection a. (e)Elastic Potential Problems 29.A.Clebsch: Theorie derElastizitdt fester Korper; Leipzig, 1862; French translation bySt.Venant andFlamant, Paris, 1883. Comprehensive andbasic treatise withadvanced solutions. 30.A.E.H.Love: Theory ofElasticity, Fifth Edition; Cambridge University Press, England, 1934. Advanced classical treatise ontheory ofelasticity. 31.A.Nadai: Dieelastischen Flatten; J.Springer, Berlin, 1925. 32.S.Timoshenko: Theory ofElasticity; McGraw-Hill, NewYork, 1934. D.BOOKS ONCOMPLEX FUNCTION THEORY ANDCONFORMAL MAPPING (a)BriefandIntroductory Books 1.L.V.Bewley: Two-dimensional Fields inElectrical Engineering; Mac- millan, NewYork, 1948. 2.L.Bieberbach: Einfuhrung indiekonforme Abbildung; Sammlung Goschen, Leipzig, 1915. 3.R.E.Doherty andE.G.Keller: Mathematics ofModern Engineering, Vol.1,Chapter IV,p.242;JohnWiley&Sons,NewYork, 1936. 4.S.L.Green: TheTheory andUseoftheComplex Variable; I.Pitman & Sons, London, 1939. 5.K.Knopp: Funktionentheorie; Sammlung Goschen, Leipzig, 1918. 6.L.Lewent: Konforme Abbildung; B.G.Teubner, Leipzig, 1912. 7.H.W.Reddick andF.H.Miller: Advanced Mathematics forEngineers, Chapter X,Second Edition; JohnWiley&Sons,NewYork, 1947. 8.R.Rothe, F.Ollendorff, andK.Pohlhausen: Theory ofFunctions as Applied toEngineering Problems; Technology Press, Cambridge, Mass., 1933. 9.I.S.Sokolnikoff andE.S.Sokolnikoff: Higher Mathematics forEngineers andPhysicists, Chapter XV; McGraw-Hill, NewYork, 1934. General Bibliography 555 10.M.Walker: Conjugate Functions forEngineers; Oxford University Press, 1933. (b)Extensive andAdvanced Books 11.L.Bieberbach: Lehrbuch derFunktionentheorie, 2Vols.;reprint byChelsea Publishing Company, New York, 1945; originally published byB.G. Teubner, Leipzig. 12.E.Borel :Lemonssur lesfonctions entieres; Paris, 1900. 13.E.Goursat: Cours d'analyse mathematique; A.Hermann, Paris, 1910, 1911. 14.A.Hurwitz: Vorlesungen uber allgemeine Funktionentheorie; J.Springer, Berlin, 1929. 15.W.F.Osgood: Lehrbuch derFunktionentheorie; B.G.Teubner, Leipzig, 1912. 16.J.Pierpont: Functions ofaComplex Variable; Ginn, Boston, 1914. 17.E.Study andW.Blaschke: Konforme Abbildung einfach zusammenhangen- derBereiche; B.G.Teubncr, Leipzig, 1913. 18.E.C.Titchmarsh: Theory ofFunctions; Oxford University Press, 1932. 19.E.T.Whittaker andG.N.Watson: ACourse ofModern Analysis, Chapters 5and6onanalytic functions; Cambridge University Press, 1927. Appendix5 ONBESSEL FUNCTIONS TheBessel differential equation pf(rp?V (mV-pi)*- (1) dp\ dpj canbesolved byapowerseries inrap=xmultiplied by(mp)p ,sothat themost direct result forrealargument xandanyrealvalue p> is T^_ i_, _,JpW~ p! 1IKP+D2!(p+l)(p +2) P> (2) Actually,this solution, the firstkind ofBessel functions, canbecon- tinued intothecomplex domain asJp(z)byreplacingxin(2)bythe complexvariable z=x+jy.Thefunction Jp(x) isregular atx= andatx=oandpossesses aninfinite number ofzeros forrealvalues oftheargument which arenotharmonically spaced butapproacha spacingofTTforvalues oftheargument which arelargecomparedwith theorder number p.Forsmall and large values oftheargument, theapproximationshold r _n (4) Forinteger values noftheorder, thefunctional values arereal for positiveornegative values oftheargument,aswell asoforder, and actually J(-x)=(-DV n(x)=J-n(x), n>0,integer (5) Fornon-integer values p>oneinterprets p!=r(p+l) 556 OnBessel Functions 557 where T(p+1)isthegamma function1ofEuler. Fornegative values oftheargument, thefunction takes oninthiscasecomplex values, which canbewritten best JP(J2mx)=j2mpJP(x) (6) Fornegative, non-integer orders, theBessel function (2)becomes T-" P>0 (7) where pw p!sinPTT(8) This Bessel function approachesinfinite values asx >0,sothat for small values oftheargument withp=n+ t\,nbeing thenearest integer, ~/O\B (9) Thegeneral solution oftheequation (1)could, therefore, berepresented by AJp(x)+BJ.p(x) aslong aspisnon-integer; forinteger values, (5)shows thatJ-n(x) isnotadifferent solution fromJn(x). Inorder tohave amore general second solution,itiscustomary todefine aBessel function ofthesecond kind(Neumann function) =-Wcospir--/-,(*) sinpir which clearlyisrelated to(7)forpnon-integer;thissolution canbe continued intothecomplex domain asNp(z)inthesamemanner as Jp(z).Forinteger values oftheordernumber nonetakes =limsm(n+ which isformally written =J. sm /ITT 1E.Jahnke andF.Emde, Tables ofFunctions, p.9;reprinted byDover Publications, NewYork, 1943; originally published byB.G.Teubner, Leipzig, 1938. 558 Appendix 5 andcanbeexpressed asarather unwieldy series expansion2bythe usual process ofevaluating indeterminate forms. This function always approaches infinite values asx >0,infact forsmall values ofxtthefollowing approximations hold : AT(x)--In (13) TTyx 1J7Tn<p<n +l(15) where In7=C=0.5772, theEuler constant, andwhere with P=n+ -n, n)+*(n-i|)L *()= -- Forvery large values oftheargument, Asinfx- NP(x)/sinx-(p+H)-' x>p (16) Fornegative order numbers, onehaswith (10) AT /x J-v(x)cosPTT+/p(z) T,. . ,T,vN-p(x)= =Jp(x)smpir+Np(x)cosPTT sinpir (17) sothatnonewsolution results whatever thevalue ofpmay be.For integer values noftheorder, thefunctional values areallrealand (17) gives verysimply N-n(x)=(-l)"N n(x)=Nn(-x) (18) Thegeneral solution oftheBessel equation (1) is,therefore, Rp(mp)=AJp(mp)+BNp(mp) (19) which reduces tothe firstterm iftheaxisp= ofthecylindrical systemisincluded intheregion ofthesolution, sinceNpisnotregular forp=0. Bessel functions ofthethirdkind(Hankel functions) arereally aspecial combination oftheBessel functions ofthe firstandsecond kind; how- 2Jahnke andEmde, loc.tit.,p.132. OnBessel Functions 559 ever, forapplications inboundary value problems theyhave particular usefulness. They aregivenbythedefinitions B,<(x)-J,(x)+jN f(x) (20) ff,'(x) -/,(*) -jN p(x) (21) foranyvalue ofp>andcanbeextended tocomplex argument inthe samemanner asthetwoindividual functions Jp(z)andNp(z).For negative values oftheargument onehas H-p(1)(z)=e*"ffp<(z), H-pW(x)=<r'**Hp(x) (22) Both functions aresingular atx=because ofNp(x)\ their values for xparereadily given bythecomplex combination of(4)and (16), exp ;'x-(p+ P(23) _11 -(P TheHankel functions aretherefore related tothefirstandsecond kind ofBessel functions asthecomplex exponential tothecosine and sine functions. Indeed,ifoneexpands the differential equation (1), divides byp2 ,and letsp><*> ,itreduces tothedifferential equation of thetrigonometric orcomplex exponential functions, indicating thatthe Bessel solutions degenerateinto thesimpler harmonic series forthe plane boundary value problem. Forcomplex argument, neither Jp(z)norNp(z)remains finite as z oobecause ofthecomplex trigonometric functions; however,Hp(l)(z)willvanish asz > ifIm(z)>0,andHp(2)()similarlyif Im(z)<asseenfrom (23). This factaccounts fortheuseinbound- aryvalue problems where vanishing values atinfinity arerequired. Modified Bessel Functions. Inmany problems, the differ- ential equation (1)might have(in2 )replace (-p-m2 ),sothat (24) Thesolution isthengivenbythesame group ofBessel functions, butof imaginary argument jmp=jx.Theapproximations (4),(16),and(24) forrealarguments x^>psuggest characteristics likethehyperbolic and realexponential functions forimaginary arguments jx,if|x| p. 560 Appendix 5 Thishasprompted theintroduction ofthe"modified11Bessel functions which have realfunction values. From (2)one seesthat allterms inthebrackets remain realfor imaginary argument, sothatthemodified Bessel function ofthefirstkind (s/2)2 p! [l!(p+l) 2!(p-I-l)(p+2) p> (25) defines arealsolution of(24), which, however, canreadily beextended intothecomplex domain asIp(z)byreplacing xin(25)byz=x+jy. Forsmall values oftherealargument, theapproximations hold p\I(X) 1; x 1 (26) whereas forlarge values ofxthefunction grows beyond alllimits. Fornegative, non-integer orders, onecandefine inanalogy to(7)the modified Bessel function (27) asarealsolution of(24)with(p)!from (8). But, again, forinteger orders nthisgives nonewfunction, butrather 7_n(z)=7n(x)=(-i)/ n(-x), n>0,integer (28) Itistherefore customary toconstruct amodified Bessel function ofthe second kind inclose analogyto(10), - 2Lsinp?r J which isclearly related to(27)butcarries theextra factor Tr/2. For integer values noftheorder number, onetakes asin(11) Kn(t )=1Lfen-(n+,)(*)-W*n(3Q) which isformally written as(29)withnreplacing pandwhich canbe expressedasanunwieldy series expansion bytheprocess ofevaluating indeterminate forms. Introducing (10) into (20)andcombining thecoefficients ofJP(x), OnBessel Functions 561 onehasforimaginary argument - smpirTT BOthat forlargearguments x,byuseof(23), (32) The general solution ofthemodified Bessel equation (24) is,therefore, flp(rap)=AIp(mp)+BKp(mp) (33) which reduces tothefirstterm iftheaxisp=ofthecylindrical system isincluded intheregionofthesolution, becauseKpisnotregular there; andwhich reduces tothesecond term ifthepoint p=QOisincluded in'theregion ofthesolution, since Ipisnotregular there. Notation ofBessel Functions. Though thenotation forthe Bessel functions ofthe firstkindandforthemodified Bessel functions hasremained rather wellstandardized since their introduction intothe mathematical literature, notevenseeming uniformity hasbeenachieved with respect totheBessel functions ofthesecond kind. Table 5-1 gives thecomparative notation asnowfound intheliterature, and Jahnke andEmde's first edition, p.173 (seebelow), should becon- sulted forthenotations anddefinitions offunctions used intheearlier literature. Itismost unfortunate that very fewauthors arecon- siderate enoughtorelate theirownnotation atleast tothat ofstandard works. TheBessel function ofthesecond kindNp(x)defined in(10) isfre- quently designatedasYp(x);however,this isalsothenotation intro- duced byC.Neumann in1867 forafunction defined by J*,<)+/, In*(34) where In7=C=0.5772 istheEuler constant. Toavoid confusion, some authors useKp(x)forthisfunction which, however,isthestandard designation forthemodified Bessel function ofthesecond kind. Itis, therefore, imperative toascertain thedefining equationsforeach ofthe function symbols used before starting comparison ofsolutions. TheOrthogonal Function System. Themost general solution oftheBessel equation canbetaken as(19), since theHankel functions by(20)and (21) arecovered bythespecial constants B=jA,and since themodified functions differ onlybyconstants from standard Bessel functions asshown by(25)and (31). One can, therefore, discuss allgeneral relations directly interms ofR(mp)=R(x). 562 Appendix 5 H^pQ 35 9I*I IoIs-S..B jS1eJ . ttj NKM IIS PQ OnBessel Functions 563 Useful relations areforrealarguments x=mp +RP+i(x)=^Rp(x) (35)x -Rp-l(x)-Rp^(x)=B,(x) (36)Z Z ax f ,/pp) (37) dp -p^ftpii (mp) (38) Inthelasttworelations upper andlower signshave tobetaken cor- respondingly. Because theBessel equation (1)isoftheSturm-Liouville type dis- cussed insection 29,namely, with characteristic numbers X=raa2 ,andweight function p(p)=p, theBessel functions formanorthogonal function system within arange Pi^P^P2forhomogeneous boundary conditions. One finds, then, forthenorm with (37)and(38)andintegrating byparts -Rp-i(m ap)Rp+l(map)\\ (39) Jlpi which reduces forthehomogeneousfirstboundary value problem with (35) to if RP(map2)=RP(mapi)=(40) and forthehomogeneous second boundary value problem with (35) and (36) to |[(w ap2)2-p2][RP(map2)]2 (41) 564 Appendix 5 Any integrable function G(p) canthenbeanalyzed interms ofthe Fourier-Bessel series G(p)=AaRp(m ttp) (42)a=l where thecoefficients Aahave tobedetermined bytheintegral pG(P)Rp(map)dp (43) Unfortunately, these integrations canbeperformed inclosed form for veryfewfunctions G(p) ;seeWatson (ref.below) forthemost complete collection ofintegral relations. References. Most texts onadvanced calculus have achapter devoted toBessel functions; they areusually restricted tothe first kind, however, asforexample Churchill,03Reddick and Miller,D7and Woods (see6below). More complete relations aregiveninSmythe,A22 andother references cited intable 5-1,aswellasinthefollowing books: 1.E.Jahnke andF.Emde: TablesofFunctions; reprinted byDover Publi- cations, New York, 1943; originally published byB.G.Teubner, Leipzig, 1909 (First Edition) ;1938 (Third Edition). 2.Th. v.Karman andM.A.Biot: Mathematical Methods inEngineering, Chapter II;McGraw-Hill, NewYork, 1940. 3.N.W.McLachlan: Bessel Functions forEngineers; Oxford University Press, 1934. 4.N.Nielsen: ZyUnder funktionen; B.G.Teubner, Leipzig, 1904. 5.G.N.Watson: Bessel Functions; Cambridge University Press, 1922. 6.F.S.Woods: Advanced Calculus; Giiin, Boston, 1926. Asaconvenient collection ofreferences totabulated values ofthe Bessel functions seeA.Fletcher, J.C.P.Miller, and L.Rosenhead: AnIndexofMathematical Tables; McGraw-Hill, NewYork, 1946, p. 244. Appendix 6 ONLEGENDRE FUNCTIONS TheLegendre differential equation ^4(sinB^ )+n(n+1)T=0, n=integer (1) sin o0\a0/ canbesolved most readily interms ofapower series incosB With theintroduction of/zinto (1),theequation transforms into |T(1-M2 )?]+n(n+1)T=0, d/xL d/iJ(2) and forinteger values ofn,asassumed, thesolutions actually become polynomials n(n-1)^T T) n(n-l)(n-2)(n-3) _1 2-4-(2n- l)(2n- 3) J ofwhich the firstfewhave theexplicit forms PoGO=1 PiGO-M (4) Pa(/0 These arevariously called Legendre1scoefficients, Legendre's polynomials, orLegendre Junctions ofthefirst kind. Because oftheir polynomial nature, these functions actually exist inthecomplex domain asPn(z) byreplacing /xin(4)bythecomplex variable z=x+jy;they are regularintheentire z-plane with theexception ofz= ,where they have apole oftheorder n.Onereadily has p..(o)M-.)-"3 25 t;6;2-". PWO.-O I (5) Pnd)=1] aswellas P(-)=(-D"P n() (6) 565 566 Appendix 6 Legendre Functions oftheSecond Kind. Asecond andlinearly independent solution ofthedifferential equation (2)forinteger values ofnisgiven bythesecond kind ofLegendre functions where=P.GOh -W.-I(M), M=cosfl (7)2 1/i =E-Pm-lOOPn-.O*) (8)m=i rn isapolynomial ofthe(n l)stdegree; the firstterm in(7),however, haslogarithmic singularities at/i= 1.The general solution ofthe Legendre equationisthen which reduces tothe firstterm iftheaxisp d=lofthespherical problemisincluded intheregion ofthesolution because ofthelogarith- micsingularities ofQn(/x). Thefunctions ofthe firstfeworders areexplicitly defined as QoGO= ^Ini--Qi(M)=PiMQoGO-1 (10) LI)-- 2+- 2M 3 showing theeven order functions tobeodd in/x,andconversely; one alsohasforthisreason 0, l-3-5---(2n-1) Because ofthelogarithmic term, Qd)=(12) Extension ofthesolution (7)intothecomplex domain aswellasto realvalues x>1requires amodification inthelogarithmic term, namely, 5.W-5P.GO In^i-Wn-,(z) (13)^ 2 1 OnLegendre Functions 567 sothat forrealvalues x>1thefunction remains real. Expanding (13) intoapower aeries, oneobtains On]fl(*+D(n +2) 1 Qn(Z) 2-4-(2n2(2n+3) zn+8 1 1 zn+*" J Since inthecomplex domain z= 1represents branch pointsofthe function (13),onemust introduce abranch cutorbarrier along thereal axisconnectingz=-J-land z= 1inorder tomake thelogarithmic term inQn(z)one-valued. One defines, then, 2+1=Plem ,z-1=P2e>*2 with< </>i<2-7T,IT<02<+ir; this gives different values just above andjustbelow thebranch cutandactually defines Q(M) in(7) ashalfthesum ofthevalues Qn(n+jO)andQn(M-JO). Forvalues\z\^1,onecanapproximate (14)bythefirstterm, " 1.3.5..n+l) Since T^n-i(z)isobtained from (8)byreplacing jubythecomplex variable z=x+jy,onecanusethe explicit forms (10) with the appropriate changeforQo(z), sothat 221 Forpurely imaginary arguments onecanalsousetheidentity(16) 2jy-1 TheOrthogonal Function System forInteger Values n.The differential equation (2)isdefinitely oftheSturm-Liouville typewith thecharacteristic numbers X=n(n+1)which areinteger because n isinteger, andwithweight function p(^)=1.Fortherealvariable jit=cos6and|/i|^1,thegeneralsolution isgivenby(9);forthecom- plexvariable z**Mthegeneral solution isgivenby Tn(z)=APn(z)+BQn(z) (17) 568 Appendix 6 since thedefinition (3)ofPncanbedirectly extended intothecomplex domain. Several useful relations are (n+l)!Tn(M)+nSV-iOO=(2n+1)iiT.OO (18) TViGi)=(2n+DTnGO (19) ,d (1- /i2 )3-Tn(n)=(n+l)[/iT n(/i)-TVnGO] (20) d/i (2w+1)frn(jLi)d/i=Tn+i(/i)-Tn-i(M) (21) which alsohold forTn(z)if/uisconsistently replaced by2. Ifonenowwrites theequation (2)fortwodifferent values ofn,say, n=aandn=j3,multiplies the firstbyT0andthesecond byTaand subtracts them, onehas Integration between inandMagivesontheright-hand sidetheform /v> ./M" with afactor thatcanvanish only fora=/3;theleft-hand side is directly Butnotwofunctions oftheseriesPnorQnortheir derivatives canvanish atthesame value oftheargument n7*(observe that intheBessel functions anadjustable parameter mwasavailable); even atju= onlyeven-ordered Pnandodd-ordered Qnvanish. Itistherefore not possible toconstruct anorthogonal system offunctionssatisfying the firstorsecond boundary value problem, except intheinterval MI= 1 to/i2=+1. However, atthose values Qn(/0possesses logarithmic singularities, soithastobeexcluded also. OnlyPn(pOcanforman orthogonal function system andonly intheinterval 1^/u^+1; i.e., +i ffora*j3= ] (22) (Nnfor OL==nr+ IJ~l OnLegendre Functions 569 where thenormNnisfound byintegration byparts anduseof(20), (21)08 *=(23) Thesame consideration shows thatTn(z)cannot formanyorthogonal system, noteven fortherealvariable 1^x^< ,sincePn(z)* asx >oo fandQn(x)>coasx 1. Inorder toexpand anybounded function G(/i) with atmost afinite number ofdiscontinuities intotheorthogonal Legendre series, 000=AnPn(M) (24)n=0 onedetermines thecoefficients inaccordance with (29-12) and (23) above f+ C/-1dp (25)* Ifthefunction tobeexpandedisgiveninterms ofthecolatitude 6,then G(0)=AnPn(cos0) (26)n=0 andtheLegendre functions (4)canbeconverted into functions of multiples oftheangle 6,namely, Po(cos0)=1 Pi(cos0) =cos0 (27) P2(cos6)=Ji(3 cos20+1)JMcos 0)=^(5 cos30+3cos0) The coefficients in(26) arethen bestdetermined by An=?!LJ:r(7(0)p n(cos0) sin0d0 (28) Associated Legendre Functions oftheFirst Kind. Themore general Legendre differential equation forinteger values nandm (n,m)=integer (29) canalsobesolved most readily interms ofapower series in/x=cos0. Introducing JJLinto (29), thistransforms to 1 =cos0 (30) A[(i_ )J~|+[(+1)_^?!_"|r_o,dMLd/*JL 1-M2J 570 Appendix 6 andforinteger values ofnandmasassumed, thesolutions arerelated totheLegendre polynomials, namely, P.-GO =(1-M2 )W2-^P.GO, M=cos* (31) dju which arecalled associated Legendre functions ofthefirstkind, ofordern anddegree m.Because of(3)itisseenthatthese associated functions existonly form^naslong asbothnandmareinteger. Forthefirst fewvalues n,m(31)gives theexplicit forms =3(1- (32) which arevalid for |p\^1and realvalues of/*. Theextension torealvalues x>1and togeneral complex values z=x+jyiscustomarily donebydefining themodified functions ?,-() =(*2-l)"^pn(i) (33)dzm which areregular polynomials intheentire z-plane formeven, buthave branch points atz= 1formodd. Itistherefore necessary tointro- duce abranch cutorbarrier from z= 1toz=+1along thereal axis inorder tomakePnm (z)single-valuedinthez-plane. Actually, then, thevaluePnm(cos6)in(31) willbeonehalfthesum ofthevalues justabove andjustbelow thereal axis, adjusted by (l)m/2ascom- parisonof(33)and (31) indicates, Pn-(cos 19)=%[j-mPnm(cos+JO)+j+mPnm(cosd-JO)] (34) Onereadily verifies that Pnm(0)= for(n+m)=odd* pmffU-(1\M(n-m)1'3'5'"(n Pn (0)"(1}2-4.6..- (n-m) for(n+m)=even(35) whereas P.-(-M) =(-l)"+'"Pn'"(M) (36) which alsoholds forthemodified function fnm (z). OnLegendre Functions 571 Associated Legendre Functions oftheSecond Kind. Asecond and linearly independent solution ofthedifferential equation (30)is givenby Q"(M) =(1-M2 )W2 ;^;Qn(M),M=cos (9 (37) dju whereQn(n)istheLegendre function ofthesecond kind defined in(7). Since thelogarithmic term remains inQnm ,itscharacteristics willbe essentially dictated bythose ofQn.Forthe firsttwovalues ofn,m onehasexplicitly QI'(M)=QO(M)+p Q^M)=\3vQM +y~ fl(1-M2 )H(38) L1-M2J Q.00 where QO(M)istaken from (10). Theextension torealvalues x>1and togeneral complex values z=x+jyiscustomarily donebydefining modified functions related to(13),namely, Qnm (z)=(z2-1)W2^;Q(*) 09) dzm which decrease tozeroasz<*> fsothattheycanbeused forpotential solutions outside ofaclosed surface. Asinthecase ofQn(z) tonemust introduce abranch cutorbarrier between z= 1and z+1inorder torender thefunction one-valued. Actually, then, therelation (34) canbeused forQnminidentical manner. Onereadily verifies that Qnm(0)= for(n+m)=even 1.i1-3-5- (n+m) for(n+m)=odd whereas Qm(-M)=(-l)n+wl+IQnm(M) (41) which alsoholds forthemodified functions Qnm(z\ TheOrthogonal Function System forInteger Values nandm. The differential equation (30)isagain oftheSturm-Liouville typewith 572 Appendix 6 twopossible sets ofcharacteristic numbers andweight functions (see section 29): X=n(n+1),- jLt (42) or X=-m2 , p(/i)=-- -q(n)=n(n+1) 1-M2 Foreach realvariable M=cos6,thegeneral solution isgiven by Tn-(M)=APn-(M) +BQn-Oi) (43) which canbeextended tothecomplex plane byusing themodified functions_ __ __?"()=AP nm (z)+BQnm (z) (44) Some generally useful relations are (n-m+IJT^GO +(n+w)T 7l_1m (M) =(2n+l) MT,r(M) (45) -(n-m+DZWi-fo) +(n+DM^-0*) =(l_M2)Arnm (/i) (46) d/i which alsohold forTnm (z)ifMisconsistently replaced byz. Ifonewrites equation (30) fortwo pairs ofvalues nandm,say, n=a,m=r\n= /3,m=s;multiplies the firstonebyTfandthe second onebyTar ,andsubtracts them, onehasforthecase r=s=m, Bythesame reasoning asfortheLegendre functions onefinds atonce thatonlytheassociated functions ofthe firstkindcanformanorthogo- nalsystem andonly intheinterval 1$/i^+1 ;i.e., . (47)Nn(m)fora= j9=n where thenormNn(m)pertainstoafixed degreemandvariable ordern and isfound as Nn(m)=>i!L^(48)2n+1(n-w)! OnLegendre Functions 573 Forthesecond caseaj9=n}onehas Repeating thereasoning butnow pertaining tothedegreesrand s, onefinds thatagain onlytheassociated functions ofthe firstkindcan formanorthogonal system andonly intheinterval 1^/i^+1 ;i.e., (49) forr=s=m where thenormNm(n)pertains toafixed order nandvariable degree mand isfound as m(nm)\ Which ofthetwoalternatives arises inapplications depends onthe nature oftheproblem; however, theorthogonalization (47)and (48) with respect toorder forfixed degreesisthenatural oneforspherical coordinates. Ifforexample adistribution function onaspherical surface isgiven asG(0, </>),where 9isthecolatitude and <thelongitude, then thisfunction canberepresented as 0(0, 4>)=E{AnPn(cos0) n-Om=n +L[Anmcosra0+flrimsinm0]P nm(cos0)l (51) m=l where thecoefficients Anpertain toanaxially symmetrical part ofthe distribution function andaredetermined by(25),ifonedefines ;rr*G(d,<l>)d<t> *TTJo(52) astheaverage value ofG(6,0).The coefficients AnmandBnmare found, respectively, bythecombination ofthepertinent Fourier series coefficient integration andtheintegration correspondingto(28)but with thenorm (48),namely, Anm \=12n+ 1(n-m)\ Bnm lr' 2 (n+m)! X2' d0f0(0, <j>)Pnm(cos8)r08m<t> \sinddB(53) .-o Jfl=o Ismm0J 574 Appendix 6 Non-integral Legendre Functions. Both theLegendre polyno- mials andtheassociated Legendre functions ofthe firstkind could be made orthogonal function systems overaninterval pi^ JLL^1ifone could assure eitherPn(jui)=0,orP,,m (jLii)=0.Thismeans, however, selecting anorder number nsuch that azero ismade tooccur at H=/Hi;thisrequires definitions ofLegendre functions fornon-integral orders. Though thishasbeen donebyrelating these generalized Legendre functions tothehypergeometric functions (seeparticularly Hobson9 ),thelack ofadequate tables makes their usemore formal than practical. Notation ofLegendre Functions. Though theLegendre functions have generally been lesssubject toconfusing notation, there isenough variety tomake thecomparative table 6-1 desirable. Themost difficult feature isthefactthatanumber ofauthors usethesamesymbol forthefunctions ofrealargument |/i|^1andcomplex argument 2, eventhough thefunctional forms andtherefore some oftherecursion formulas differ. Certainly, onecanremember these ifone deals frequently with these functions; forstudy purposesitisnotconvenient. References. Many texts onadvanced calculus andadvanced electromagnetic theory have atleast brief chapters ontheLegendre polynomials; Churchill03andMurnaghan013also deal with the Legendre functions ofthesecond kind and realargument |M|^1; extension totheassociated Legendre functions ofthe firstkind with realargument |/i|^1isshown inKellogg,010 Stratton,A23and Webster010 .The generalized treatment isfound inthereferences cited intable 6-1, inparticular alsointhefollowing references: 1.N.M.Ferrers: Spherical Harmonics, London, 1877. 2.E.Jahnke andF.Emde: Tables ofFunctions, Third Edition; reprinted by Dover Publications, New York, 1943; originally published byB.G. Teubner, Leipzig, 1938. 3.C.Snow: TheHypergeometric andLegendre Functions withApplicationsto Integral Equations ofPotential Theory; National Bureau ofStandards, Washington, B.C., 1942. 4.W.J.Sternberg andT.L.Smith: TheTheory ofPotential andSpherical Harmonics; University ofToronto Press, Canada, 1946. 5.A.Wangerin: Theorie desPotentiates und derKugelfunktionen; B.G. Teubner, Leipzig, 1909. Asaconvenient collection ofreferences totabulated values ofthe Legendre functions seeA.Fletcher, J.C.P.Miller, andL.Rosenhead: AnIndex ofMathematical Tables; McGraw-Hill, New York, 1946, p.232. OnLegendre Functions TABLE 6-1 COMPARATIVE NOTATION OPLEGENDRE FUNCTIONSS75 References (1):Jahnke andEmde(ref. p.574); Heine08used thesame symbolsforassociated Legendre functions with interchange ofnandmand with extra factors; sincesome relations arenotquite correctly stated, consult Hobson.09 References (2):Bateman01 ;SmytheA22usesMthroughout forthevariable; MacRobertC12usesxthroughout forthevariable, occasionally replacingit byM=cos0,and alsousesTnm (x)for(31);Whittaker andWatson019use zthroughoutforthevariable, occasionally replacingitbycos forzrealand M$i. References (3);Snow (ref. p.574) ;(*)thissymbol was firstusedbyFerrers (ref. p.574). References (4):Hobson09 ;(*)heuses, however, (l)mrnm (/i)and calls that alsoPn*"(/0; (t)heuses(-l)mQnn (p)from firstcolumn and also calls itQ,T (M). INDEX (Problem numbers areitalicized andfollow thepagenumber andacolon) Absolute dielectric constant, 1,72, 536 Absolute permeability, 39,72,536 Absorption current, inliquids, 32 non-reversible, 32 reversible, 32 Ampere's law,39 Amplification factor, measurement in electrolytic trough, 192 oftriode, cylindrical, 291 with grid strips, 323 Analytic function, 8 ofcomplex variable, 279-281 inconformal mapping, 302 continuity of,279 differentiability of,279,280 atinfinity, 309 integrability of,280 series expansion of,307 Laurent, 308 Taylor, 307 single-valuedness of,279 singularity of,308 essential, 309 Annular coordinates, 453,454 Aperture, circular, 431-434 unsymmetrical, 434 Associated Legendre functions, 479, 498,569 comparative notation, 575 differential equation of,479, 480, 569 offirstkind, 479,480,492,569 modified, 498,569 modified, 498, 502, 505, 508,515 asorthogonal system, 572,573 relations between, 572 ofsecond kind, 479,480,570 modified, 498,499,506,570 Barrier surface, 44,78; see also Potential, magnetostaticBarrier surface,forcylindrical coil, 214 forparallel wires, 208 Bessel functions, 422,429,458,556 comparative notation of,561,562 differential equation of,556 general solution of,558 expansion in,564 offirstkind, 426, 458, 512,556 series for,556 zeros of,426 normof,429 asorthogonal system, 429 relations between, 563 ofsecond kind, 458,459, 557,558 seriesfor,558 ofthird kind, 558,559 Bessel modified functions, 421, 422, 460, 511,559 comparative notation of,561,562 differential equation of,559 general solution of,561 offirstkind, 560 series for,560 ofsecond kind, 560 Bifilar wires, 56 Bilinear transformation, 314-318 Biot-Savart law, 52,129 Bipolar coordinates, 407 Borda mouthpiece, 382 Boundary conditions, fordielectric fluxdensity, 10 forelectric current, 69 forelectric field strength, 10,69 formagnetic fluxdensity, 46 formagnetic vector potential, 51 formagnetizing force, 47 Boundary value problem, offirst kind, 9 inplane, 363 ofmixed kind, 9 inplane, 367 577 578 Index Boundary value problem,ofsecond kind, 7,9 inplane, 367 ofthird kind, 76 Breakdown, electric, ofgases, 27 ofliquids, 31 ofsolids, 34,35 ofvacuum, 26 thermal, ofsolids, 34,35 Capacitance, 13,72 ofcircular disk,452 ofcircular ring ofcharge, 127 above ground, 129 ofcondenser, cylindrical, 148 two dielectrics, 150 plane, 146 two dielectrics, 146 spherical, 151 from curvilinear squares, 203 ofellipsoid, 450 oblate, 452,500 prolate, 108,452,506 byinversion, 248-253 measurement of,bycurrent model, 185 byelectrolytic trough, 192 ofquasi point charge, 97 between planes, 217 near sphere, conducting, 98,103, 104,489 dielectric, 489 ofrod,antenna, 110 horizontal, above ground, 114 vertical, above ground, 112 ofspheres, conducting, 232 intersecting, 250 ofspherical bowl, 253 ofspheroid, oblate, 452,500 prolate, 108,452,506 perunit length, ofcable, coaxial, 148 two-conductor, 225 between cylinders, concentric, 148 elliptic, 320,406 parallel, 120,121 enclosing each other, 121Capacitance, perunitlength, ofMax- well grating, 293 ofplane strip, 320,321 between semicylinders, 314 between wire,andground, 115, 122 andintersecting cylinders, 259, 313 andintersecting planes, 327 between wires andground, 125 Capacitance coefficients, 15; seealso Capacitances, partial matrixof,15 measurement of,16 Capacitances, direct, 14 partial, 14 from curvilinear squares, 203 measurement of,16,203 stray, 13 Cauchy's integral, 306 Cauchy's integral theorem, 281 useinFourier integral, 396 Cauchy-Ricmann equations, 280 Center ofcharges, 95-97 Characteristic functions, 384 Characteristic numbers, 384 continuous spectrum of,393,424 discrete spectrum of,384,423, 437, 460 bytangent graph, 391 Charge, electric, 1,72; seealsoIn- duced charge; Line charge; Point charge image, 215 Charge density, electric; seeElectric charge density magnetic, fictitious, 54 Charging currents, 16 Circular aperture, 431-434 unsymmetrical, 434 Circular cylinder coordinates, 456, 457 Circular disk, 452,501 charge density, on,452 with uniform, 467, 490,491 gravitational potential of,491 Circular harmonics, 399 Circular ring ofcharge, 125,467,489 Index 579 Circular ringofcharge, above ground, 127 potential of,489 Circulation ofvector, 541 Complete system,inelectrostatics, 13 inmagnetostatics, 56 Complex potential function, 303 forcoplanar planes, 335 with finite gap, 340,348 forcoplanar strips, 355 forhalfplane, 364,365 forlinecharge above ground, 326 forplane strip, 340,341 forsemicylinders, 314 forunit circle, 363,364 Complex variable, 277-278 "absolute value of,277 analytic function of,279-281 argument of,278 conjugate complex of,280 modulus of,277 Condenser, 13 coaxial cylinder, 147 concentric sphere,151 elliptic cylinder, 320,406 ideal, 13 plane, 145 conformal mapping of,333-338 fringing flux in,336 Condenser bushing, 150 Conductance, electric, 73 hydraulic, 73 thermal, 73,75 Conductance coefficients, electric, 80: 5 hydraulic,80:18 thermal, 80:13 Conductivity, electric, 67 thermal, 73,75 Conductor, electrostatic, 1,4 Conductors, system of,electrostatic energy of,16,18,19 inelectrostatic field, 15 forces in,21,22 Cone functions, 495 Conformal mapping, byanalytic functions, 301,302 linear, 310-318Conformal mapping, byanalytic func- tions, rational, 318-321 transcendental, 321-323 ofcondenser, parallel plate, 333- 338 with thick plates, 351 ofcylinder grating, 377,378 ofcylinders, elliptic, 319-321 intersecting circular, 312 bygraphical superposition, 324 ofhydrodynamic problems, 379- 383 non-conformality of,305-310 ofpolygons, closed, inside of,329- 333 outside of,360-362 rules for,332 with circular arcs,370-379 with parallel lines, 338-343 ofrectangle, 354-355 ofrounded corners, 372-376 ofslots, rectangular, 344,347,348- 350,356 opposing, 352,353 ofstrip, rectangular, 345-351 ofstrips, coplanar, 356,357 ofunit circle, onhalf plane, 315, 316 onitself, 316 ofvertex, single, 325-328 Conformal representation, 302; see alsoConformal mapping Conformal transformation, 246, 253, 302; see also Conformal mapping byinversion, 246-259 inthree dimensions, 246 intwodimensions, 256 bystereographic projection, 253 Conjugate functions, 281 forcylinder, elliptic, 298,300 inuniform field, 287 forlinecharges, 285,286 array of,cylindrical, 290 plane, alternating, 296 dipole, 296-298 uniform, 291-295 dipole, 286,288 580 Index Conjugate functions, forlinecharges, pair of,286,288 forlinecurrent, 285,286,288 pair,286 inuniform field, 287 forMaxwell grating, 292 forplanes, coplanar, 288,290 forsource line, 285,286 forunicursal curves, 299,300 foruniform field, 286,288 forvortex line, 285,286,288 Conservative electrostatic field, 3 Continuity, ofelectric current, 69 equation, hydraulic, 77 Coordinates, annular, 453,454 bipolar, 407 cartesian, plane, 387 three-dimensional, 435 confocal spheroidal, 474,475 cylinder, circular, 456,457 elliptic, two-dimensional, 404,405 three-dimensional, 431, 432, 468,469 general, 454,455 parabolic, two-dimensional, 407 three-dimensional, 472,473 dipolar, 495,496 orthogonal, general, 440-442 paraboloidal, 510,511 polar, 399 spherical, 420,477-479 spheroidal, oblate, 496,497 inverse to,503 prolate, 504,505 inverse to,509 toroidal, 513,514 Coulomb's law,1 for"magnetic poles," 63:4 Critical fieldstrengthofair,30 forbreakdown, 30 forcorona, 30 Cross productofvectors, 540 Curl, 541 incylindrical coordinates, 545 inorthogonal coordinates, 443 inspherical coordinates, 545 Current, electric, 66,73;seealsoLine current; quasilinecurrentCurrent filament, 51 helical, 161 Current loop, 56,144 circular, 140,492 rectangular, 131 Current loops, system of,60 magnetic energy of,60,61 Current sheet, 46 model, 183-187 Current density, 46,66,73 incylinder, finite, 462,464 equivalent,formagnetization, 55 sheet, 46 equivalent, 55 insphere, 483,484 Curvilinear coordinates, 440, 545; seealsoCoordinates Curvilinear squares, inelectric fields, 201 inmagnetic fields, 209 Cylinder, conducting, current dis- tribution in,461-464 linecharges parallel to,118,119, 223 inuniform field, 224,286,288 with dielectric, 400,401 dielectric, conducting strip on,369, 370 line charges parallel to,226, 228 inuniform field, 227,401 inversion in,255-259 magnetic, inuniform field, 241, 242 Cylinders, coaxial, equidiameter, 421- 431 with finite gap, 427,428 infinitely long, 428,429 contacting, 370,371 Cylindrical coil, electric field of, 468 magnetic fieldof,493 Cylindrical ring, hollow, 458,459 Cylindrical shell, dielectric, inuni- formfield, 401 magnetic, 404 shielding effect of,403,404 inuniform field, 401 Index 581 Diamagnetic materials, 42 Dielectric breakdown strength ofin- sulators, 24,25 Dielectric flux, 5,72 Dielectric fluxdensity, 5,72 oflong line,116 Dielectric flux lines, 5,197 Dielectric fluxtube, 6,197 Dipolar coordinates, 495 Dipole, electric, 92 magnetic, 143 Dipole linecharge, 123,286,288 grating, 295-298 inuniform field, 297,298 Dipole linecurrent, 136 magnetic moment of,137 Dipole moment, electric, 92 ofdipole line, 123,289 Dirichlet boundary value problems, 363 Disk,seeCircular disk; Ellipticdisk Dissipationintoheat, 70 asaminimum, 80:2 Divergence, 541 incylindrical coordinates, 545 offield vector, 72,73 inorthogonal coordinates, 442 inspherical coordinates, 545 Divergence theorem, 543 intwodimensions, 280 Dotproductofvectors, 539,540 Earnshaw's theorem, 38:19,85 Eigen functions, 384 Eigen values, 384; seealsoCharac- teristic numbers Electric charge density, "bound," 12 from curvilinear squares, 203 fictitious, 12 foraxisymmetrical systems, 415 line,116 ininverse system, 247 measurement of,175 forsemiconductors, 71 surface, 6,71; seealsoInduced charge density oncircular disk,452 onelliptic cylinder, 406Electric charge density, surface, on elliptic disk,451 onellipsoid, 450 oblate, 452,500 prolate, 108, 109,453,506 ininverse system, 247,257 onparallel cylinders, 122 volume, 5,72 inconformal mapping, 305 Electric fieldlines, 2 differential equation of,2,3 inaxisymmetrical system, 83,92 fordipole, 92 line,124 forlines, parallel, 117,200 mapping of,176, 177,201,204 inaxisymmetrical system, 204 byrelaxation method, 268,269 byconformal mapping, 303,304 byconjugate functions, 282 bycurrent sheet model, 183-185 bycurvilinear squares, 201 byelectrolytic trough, 191 byhydraulic flow lines, 194 byimages, 216-229 byrelaxation method, 267,268 bystraw probe, 177 forpoint charges, 84,205 nearplane, conducting, 87 near sphere, conducting, 90 Electric field strength, 2,72,73 forcondenser, plane, 335 with several dielectrics, 146,147 from conformal mapping, 304 from conjugate functions, 283 forcylinder, and line,118 inuniform field,224 forcylinders, coaxial, 147 optimum valueof,148 fordipole, 92 line,124 forellipsoid, prolate, 108 forpoint charges, 82 byrelaxation method, 267 nearrounded corner, 375,376 forspheres, concentric, 151 Electric intensity, 2;seealsoField strength, electric 582 Index Electrolytic trough, 187-193 useforaxialsymmetry, 190 Electromotive force, 73 Electronlens,421 paraboloidal, 513 two-cylinder, 421 Electronoptical field, inaperture, circular, 431-433 electric, 416,417 magnetic, 419,420 Electrostatic equipotential surfaces, 4,72 ofdipole, 93 oflinecharge, finite, 107 oflinecharges, parallel, 117 mapping of,170-173 inaxisymmetrical system, 204 byconformal mapping, 303,304 byconjugate functions, 281 bycurrent sheet model, 183-185 bycurvilinear squares, 201 byelectrolytic trough, 189-192 byimages, 216-229 byrelaxation method, 267 byrubber membrane, 193 ofpoint charges, 84 Electrostatic potential, 4,72 inaperture, circular, 433,434 inaxisymmetrical system, 416,420 byrelaxation method, 268,269 singular points of,417,418 bycircular harmonics, 399-401 ofcircularring, 125, 126,467, 489, 490 inconical space, 494 byconjugate functions, 281,285 ofcylinders, coaxial, 147,149 incylindrical lens,423-431 incylindrical ring,458-461 differential equation, of,6 formal solutionof,8,12 ofdipole, 92 line, 123,124 ofdisk, circular, 452,501; 467,490 ofdoublelayer, 36:6 ofellipsoids, 449-452 byFourierintegral, 393-399 byFourierseries, 388,389,437,438Electrostaticpotential, byGreen's functions, 519-525 ofhyperboloids, 509 ininverse system, 247 oflinecharges, 106, 107, 116,285- 288 ofgratings, 290,292-298 above ground, 111, 114,116 betweenplanes, 216 maximum value of,38:18 ofMaxwell grating, 292 measurement of,170-174 incurrent sheet model, 183-185 inelectrolytic trough, 189-192 ofparaboloid, 512,513 inparallelepiped, 436-438 ofpoint charges, 4,82,84,95 inLegendre functions, 487 between planes, 88,216 nearsphere, 90,91 ofquasi point charges, 105 inrectangle, 388,389 byrelaxation method, 260-270 ofspheres, concentric, 151 intersecting, 250 ofspherical shells, 254,480,481 ofspheroid, conducting, oblate, 499 prolate, 506 dielectric, oblate, 502 prolate, 508 oftoroid, conducting, 516 uniqueness theoremof,37:12 Ellipsoidal coordinates, 446-448 Laplace equation in,449 Elliptic cylinder, 298,300 capacitance of,perunitlength, 320, 406 conformal mapping of,319,320 coordinates, 404-407, 432,46&-472 comparative notation, 470 split, potential in,471,472 Elliptic disk, 451 charge density on,451 Equipotential surfaces, seeElectro- staticequipotential surfaces Fieldanalogies, 72,73 Fieldenergy, electrostatic, 19,20 Index 583 Field energy, electrostatic,ofcon- denser, 13,15 asminimum, 38:17 magnetostatic, 61,62 ofcurrent loops, 57,59 system of,60 Fieldlines, seeElectric field lines; Magneticfield lines Field strength, electric, 2,72,73;see alwElectric fieldstrength magnetic, 40;seealsoMagnetic fluxdensity Fluid dynamic field, 73,76 Flux function, 284,303 forsource line, 285,286 mforvortex line, 285,286 Flux tube, dielectric, 6,197 magnetic, 206 Force, electric, onconductors, 21,22 ondipole, 94 onlinecharge, bydielectric, 221 onpoint charges,1 near dielectric plane, 220 magnetic, oncurrent loop, 64:14 onlinecurrents, 39 inmagnetic duct, 243 nearmagnetic plane, 238 Force function, 73 Fourier integral, 393-399, 424 Campbell-Foster tables of,395 incartesian coordinates, 393,394 coefficients, 395,396 complex form of,395 incylinder coordinates, 424,465 evaluation byresidues, 396,397 ofunit step,398 Fourier series, 387-392, 422, 437, 461 incartesian coordinates, 387-389, 436,437 incylinder coordinates, 422, 423, 461,462 elliptic, 472 double, 437, 438,464 generalized, 386 norm of,488 asorthogonal system, 388 inpolar coordinates, 399,400Fringing flux,incondenser, plane, 336,338 correction factor for,336,337 inmachines, 344,349,350 correction factor for,349,350 Gauss's fluxtheorem, 5 Gauss's theorem, 543 Geometric mean distance, 156 Gradient, 540 incylinder coordinates, 545 elliptic, 469 parabolic, 473 inorthogonal coordinates, 441,442 ofpotential, 4,72,73 inspherical coordinates, 545 Gradingofinsulation, 150 Grating ofline charges, cylindrical, 291 plane, infinite, 292,295-298 Gravitational field, 73,78 Green's function, offirstkind, 519 forcylinder, 523,524 forplane, 520 forsphere, 521 ofsecond kind, 525 forsphere, 525 fortwo-dimensional problems,526 Green's reciprocation theorem, 37:10 electric current analogue, 80:7 magnetic analogue, 63:10 Green's theorem, first,544 second, 544 vector analogue, 544 Ground inelectrostatics, 14,102,122, 125 Grounding rods, 111,113 Grounding spheres, 100 Guard rings, 145,148 Hankel functions, 558,559 Harmonic function, 7 Harmonics, circular, 399 surface, 480,573 tesseral, 480,573 zonal, 480 Heatpower flow, 75 incoaxial cable, 148 584 Index Hermite polynomials, 474 Hodograph, 379,380 Hydraulic flow lines, 194 Image, oflinecharge, incylinder, 118, 119,223 dielectric, 226,228 inplanes, 216,221 dielectric, 219 ofline currents, incylinder, 240, 242,243 inplanes, 234, 236,238,243 intersecting, 235 parallel, 236,239 inplate, 240 ofpoint charges,inplanes, 86 dielectric, 219 intersecting, 88 parallel, 216,221 inspheres, grounded, 89 insulated, 91 intersecting, 248,249 ofring, inplane, 127,128 ofrod,horizontal, inground, 111 vertical, inground, 114 ofwire insphere, 230 Image force, 87 Induced charge, byelectron,indiode, 37:11 bypoint charge, inplane, 88 insphere, 91 bywire, charged,insphere, 231 Induced charge density, oncylinder, bylinecharge, 119 onground, byrod,112 onplane, bycylinder, 122 bypoint charge, 87 byring ofcharge, 128 on.sphere, bypoint charge, 91, 100 Inductance, ofcoil, cylindrical, 162 toroidal, 164 ofloop, 67 circular, 142 rectangular, 133 perunitlength,ofconductors, 165 coaxial, 157 external, ofwires, 136Inductance, perunit length, ofwires, near magnetic plane, 234, 238 between magnetic planes, 239 inmagnetic plate, 240 internal, ofround wire, 131 from vector potential, 57 Inductances, leakage, 59 loop, ofsystem ofwires, 139 mutual, ofloops, 58-60 circular, 144 self, ofloops, 58 Insulators, 1,4 breakdown strength of,24,25 dielectric properties of,24,25 resistivity of,surface, 24,25 volume, 24,25 Inversion,incomplex plane, 310,311 incylinder, 255-259 ofcylinders, intersecting, 257 insphere, 244-253 ofpotential values, 247 ofsphere, 244,245 ofspherical bowl, 252 ofspheres, contacting, 251 intersecting, 247-250 lonization, ofgases, 27,28 ofliquids, 32,33 ofsolids, 36 Irrotational fluid flow, 77 Isotherms, 73,75 Isotropic medium, dielectric, 1 magnetic, 40,45 Joule's law,70 Kelvin transformation, 244 "Kernel" ofmagnetic field,210 Lamellar field,543 Laplace transform, 396 ofunit step,398 Laplace's equation, 7,45,68,75,77, 541,545 foraxisymmetrical fields, 268, 414, 420,422 solution, approximate, 415 byrelaxation, 268,269 Index 585 Laplace's equation,incartesian co- ordinates, 387, 435,541 forconjugate functions, 281 incylinder coordinates, 414,455 circular, 457,475,545 elliptic, 405,432 general, 455 forelectric currents, 68 forelectrostatics, 7 inellipsoidal coordinates, 449 forfluiddynamics, 77 invariance of,303 ininverse system, 247,256 formagnetostatics, 45 inorthogonal coordinates, 442,443 inpolar coordinates, 399 solution of,bycircular harmonics, 399-402 byconjugate functions, 281-284 byFourier integral, 397,398 byFourierseries, 388-390 byorthogonal functions, 384 byrelaxation method, 260-266 onsphere, surface of,253 inspherical coordinates, 420, 478, 545 inspheroidal coordinates, 475,476 forstereographic projection, 253 fortemperature, 75 transformationof,byanalytic functions, 303 inorthogonal coordinates, 440- 443 Leakage, magnetic, 59 Legendre functions, associated, 479, 480; see also Associated Legendre functions comparative notationof,575 differential equation of,480,565 general solution, 566,567 offirstkind, 480, 485, 489,565 explicit formof,565 modified, 502 non-integral, 574 normof,569 asorthogonal system, 568,569 relations between, 568 ofsecond kind, 480,494,566Legendre functions, ofsecond kind, explicit formof,566 modified, 499,502, 566,667 Legendre polynomials, 480, 488,565; seealsoLegendre functions Legendre series, 480,569 expansion into, 481,569 Line charge, 106, 198,205 incylinder, slotted, 368,369 field plot of,199 pair, 116,286,288 field plot of,200 parallel tocylinder, conducting, 223 dielectric, 227 paralleltocylinder, conducting 117-119 dielectric, 226,228 paralleltocylinders, intersecting, 257, 258,312 paralleltoplane, conducting, 122 dielectric, 219 paralleltoplanes, conducting, 216, 326 paralleltoplate, dielectric, 221 Line charges, grating, cylindrical, 291 plane, 291-298 system of,124,125 Line current, 40,285-288, 402,403 inchannel, magnetic, 236 incylinder, magnetic, 242,243 dipole, 136 field, plot of,207 inuniform, 287 pair, 134,286 incylindrical shell, 403,404 field plot of,200 parallel toplanes, magnetic, 234, 238,239 parallel toplate, magnetic, 239 paralleltocylinder, magnetic, 240 parallel toplane, magnetic, 234, 236,243 paralleltoplanes, magnetic, 236, 238,346 Line currents, systemofparallel, 139 586 Index Magneticfield lines, 42,135,141 foraxisymmetrical fields, 141 forconductors, circular section, 153-155 differential equation of,42,135 ofdipole, 143 "kernel" of,210,211 mapping of,inaxisymmetrical fields, 214 incurrent-carrying regions, 210 bycurrent sheet model, 187 bycurvilinear squares, 209 inelectrolytic trough, 192 byhydraulic flow lines, 194 byimages, 234-243 byironfilings, 180 forlarge cross sections, 213 forlinecurrents, 207,208 bysuperposition, 212,213 fortwo-dimensional fields, 135 Magnetic field strength, 40; seealso Magnetic fluxdensity Magnetic flux, 42,72 ofloop, filament, 52 rectangular, 133 mutual, ofloops, 59,60,138 usefully linked, 59,60 Magnetic fluxdensity, 40,72 inaxisymmetrical fields, 419, 491, 492 near axis,419 ofbar, thin, 159 from Biot-Savart law, 52,53,129 ofcoil, cylindrical, 162 ofcylinder, magnetic, inuniform field, 242 ofcylindrical coaxial conductors, 156 ofcylindrical shell inuniformfield, 401 ofdipole, 143 ofdipole linecurrent, 136 offilament, helical, 162 oflarge cross section, circular, 152, 153 oflinecurrent, 40,41 pair, 42,135 ofloop, circular, 141,493Magnetic fluxdensity, measurement of,177-180 ofquasi linecurrent, 130,131 Magnetic fluxlinkages, 52 ofcoils, -search, 178 toroidal, 164 from fieldplot,208 ofloops, 57,58 system of,60 measurement of,183 Magnetic induction,seeMagnetic fluxdensity Magnetic intensity, 44;seealsoMag- netizing force Magnetic moment ofdipole, 143 linecurrent, 137 Magnetic North quantity, 143 Magnetic scalar potential, seeMag- netostatic potential Magnetic shell, 63: 1 Magnetic vector potential, 48 inaxisymmetrical fields, 419, 491, 492 near axis, 419,420 ofbar, thin, 158 differential equation of,48 inorthogonal coordinates, 443 ofdipole, 143 linecurrent, 136,137 offilament, 50,51,60,129 forfluxplotting, 209 oflarge cross section, circular, 152, 153 rectangular, 160 ofline current, incircular har- monics, 402,40? pair, 134 parallel magnetic, cylinder, 240 planes, 239 parallel magnetic, cylinder, 240 plane, 237 planes, 239 ofloop, circular, 132,133 rectangular, 140 inorthogonal coordinates, 443 solutionfor,formal, 49,55 uniqueness theorem for,63:7 Index 587 Magnetization, 53 Magnetizing force, 44,72 Magnetomotive force, 45,72 measurement of,181 Magnetostatic potential, 45,72 "barrier" surface for,44 differential equation of,45 formal solutionof,54 ofdipole, 143 linecurrent, 137 forfield plotting, 208 oflinecurrent, 285 ofshell, magnetic, 63: 1 uniqueness theorem for,63:2 Magnetostatic potential difference, 45 measurement of,181,182 bycurrent sheet model, 185-187 inelectrolytic trough, 189-193 byrubber membrane, 193 Magnetostriction, 62 Main fluxlinkage, 59 Mathieu functions, 469,470 differential equation of,469 radial, 470 Maxwell grating, 292 capacitance of,293 Maxwell's coefficients, ofinduction, 17 fortwospheres, 232 ofpotential, 18 forquasi point charges, 99 nearplane, 101 nearsphere, 104 forwires, system of,125 Measurement, ofcharge, surface dis- tribution, 175 ofelectric potential, 169-174 with probe, 170-172 withspark gap,173 ofmagnetic fluxdensity, 177-180 byHall effect, 180 byresistance change, 180 bysearchcoil,177-179 ofmagnetic fluxlinkage, 183 Nabla, 540 crossproduct of,541,542 dotproduct of,541,542 Neumann function, 557,561Neumann's problem, 367 Norm, ofafunction, 385,386 ofassociated Legendre functions, 572,573 ofBessel functions, 429,563 ofFourierseries, 488 ofLegendre functions, 569 Normalized functions, 387 Ohm's law,66 differential formof,68 Orthogonal coordinate systems, 435, 440-446 listof,445 Orthogonal function system, 385 Fourier series as,388 Orthogonality, ofassociated Legendre functions, 571 ofBessel functions, 429,459,563 condition of,385 ofLegendre functions, 568 Orthonormal functions, 386 Fourier sines as,388 Orthonormal system, 386 Parabolic cylinder coordinates, 407, 472 comparative notation of,474 Parabolic cylinder functions, 473 Paraboloidal coordinates, 510,511 Paraboloidal electron lens, 512,513 Paramagnetic materials, 42,43 Parseval theorem, 387 Permeance, 72 from curvilinear squares, 210 measurement of,186,192 bycurrent sheet model, 186 inelectrolytic trough, 192,193 Point charge, 2,82,83 nearconducting plane, 86 near conducting planes, intersect- ing,88 parallel, 216 nearconducting sphere, grounded, 89 insulated, 91,489 nearconducting spheres, intersect- ing250,251 588 Index Point charge, near dielectric plane, 219 near dielectric plate, 221, 465, 466 near dielectric sphere, 487,488 electrostatic field of,3,82 fieldplot of,205 inFourier integral form, 465 quasi, 97;seealsoQuasi point charge Point charges, collinear, 94 fieldplot for,206 quasi, 98; seealsoQuasi point charges near sphere, conducting, 229 Poisson's equation, 8 inconformal geometry, 305 forfluiddynamic field,78 forgravitational field, 79 solution for,formal, 8 byrelaxation method, 206 fortemperature, 76 vector equivalent of,49 formal solution of,49 Poisson's integral, 363 forhalfplane, 365 forsphere, 522,523 Polarization, electric, 11 magnetic, 53 Pole ofcomplex function, 309 Potential, from conjugate functions, 281-284 electric current, 68,73 incylinder, 462,463 insphere, 483,484 uniqueness theorem for,80:3 electrostatic,seeElectrostatic po- tential gravitational, 73,79 magnetic vector,seeMagnetic vec- torpotential magnetostatic,seeMagnetostatic potential byrelaxation method, 260-269 velocity, 73,77 uniqueness theorem for,80:17 Potential difference, 13,66,72,73 Potential gradient, 72Probe, capacitance, 172 charge, 2 current, 40 emission, 71 formeasurement, ofcharge density, 175 ofpotential, 170,195: 1 straw, 177 tungsten wire, 171 Quasi linecharge, 106 circular ring as,127 Quasi linecurrent, 130 Quasi point charge, 97 capacitance of,97 near plane, conducting, 101 dielectric, 220 Quasi point charges, collinear, 105 nearground, 101 two,98 Refraction, ofcurrent lines, electric, 70 offieldlines, electric, 11 magnetic, 47 Regular function, 279; see also Analytic function Regular path, 280,542 Regular point, 280 Regular region, 306 Regular surface, 543 Relative dielectric constant,1 ofinsulators, 24 Relative permeability, 39,40 Relaxation method, 259 foraxisymmetrical fields, 268,269 computingaidsfor,269,270 improvement formulafor,264 forLaplace's equation, 260-266 forPoisson's equation, 266 two-dimensional, 260 Resistance, electric, 66 capacitance, relation to,68 ofcylinder, finite, 463 between hemisphere and plane, 218 ofplane strip, 295 ofrod, electrodes, 113,115 grounding, 111 Index 589 Resistance, electric, ofsphere, 484 betweenspheres, 100 ofspherical shell, 255 thermal, 75 Rubber membrane model, 193 Saddle point ofpotential, 86,418,433 inaperture, circular, 433 inaxisymmetrical fields, 418 fortwopoint charges, 86,90 Scalar, 538 gradient of,540 Scalarproduct, 539 Schumann'scriterion, 28 Schwarz-Christoffelfunction, 325, 329 Schwarz's complex potential, 363 Search coil,177-178 Selfcapacitances, 17 Selfinductances, 58 Semiconductor, 71 Semidielectric, 71 Separability ofvariables, 444,445 incylindrical coordinates, 454r- 456 inspheroidal coordinates, 475,476 Separation ofvariables, 444,445 incartesiancoordinates, plane, 387 three-dimensional, 435,436 incylinder coordinates, 454-456 axisymmetrical, 414, 421,428 elliptic, 432,469 parabolic, 473 inorthogonal coordinates, 444,445 inpolar coordinates, 399 inspherical coordinates, 478,479 inspheroidal coordinates, 475-477 Singularity ofcomplex function, 308 branch point as,322 essential, 309 isolated, 309 pole as,309 Singular point ofpotential, seeSaddle point ofpotential Solenoidalfield,544 Source line, 285,286 pair, 286,288 Sourcelines, 286Source lines, grating of,cylindrical, 290 plane, 291-298 Southwell's relaxation method, see Relaxation method Space charge, electric, 5,8 density, see Electric charge density fictitious, frompolarization, 11, lfc magnetic, fictitious, 54 Spectrum ofcharacteristic numbers, continuous, 393 discrete, 384 Sphere, conducting, current in,482- 484 andpoint charges, 89,91,229 andquasi point charge, 103 inuniformfield, 486,487 andwire, finite, 230 dielectric, andpoint charge, 487, 488 andquasi point charge, 488,489 inuniformfield, 486,487 inversionin,244-253; seealso Inversion magnetic, inuniformfield, 486,487 Spheres, concentric, 151 contacting, 251 andpoint charge, 251 induction coefficientsof,231 intersecting, 247-250 andpoint charge, 250,251 Spherical bowl, 252 stereographic projection of,255 Spherical coordinates, 477-479 with axialsymmetry, 420 Spherical shell, conducting, current in,254 potential of,481,482 stereographic projection of,253 dielectric, shielding efficiency of, 486 inuniformfield, 484,485 magnetic, inuniformfield, 486 Spheroid, conducting, oblate, 451, 498-500 prolate, 452,506 590 Index Spheroid,dielectric inuniformfield, oblate, 501,502 prolate, 507,508 Spheroidal coordinates, 474,475,496, 504 oblate, 496,497 comparative notation of,499 inverse to,503 prolate, 504,505 comparative notation of,505 inverseto,509 Stereographic projection, 253 ofspherical, bowl, 255 shell, 254,255 Stokes's theorem, 542,543 Stream function, 284; seealsoFlux function Stream lines, 77 Stresses, infield, electrostatic, 22 magnetostatic, 62 onsurface,ofconductor, 23 ofdielectric, 23 ofmagnetic materials, 62 Sturm-Liouville, problem of,384, 385 Sturm-Liouville theorem, 310 Surface harmonics, 480 expansion into,573 Temperature distribution,incable, 148,149 inplane rectangle, 391,392 insphere, 482 Temperature field, 73,74 uniqueness theorem for,80:11 Temperature gradient, 73,74 Tesseral harmonics, 480 Thermal ohm, 75 Thomson's theorem, 38:17 Toroid, conducting, 515,516 Toroidal coil, 163,178 Toroidal coordinates, 513,514 comparative notation of,515 Torque, oncurrent loop, 60 onelectric dipole, 93Torque, onelectrostatic conductor, 21,22 onmagnetic dipole, 144 Townsend's theory, 27,28 Triode, amplification factor of,291 byconjugate functions, 291 measurement of,192 cylindrical, 290,291 conformal mapping of,318 potential in,290 Two-conductor cable, 225 capacitance of,225 Uniqueness theorem forpotential, 37:12 Unitary relations, 535 Units,MKSC, 532,533 conversion factors from, 534,536 toCOS electromagnetic units, 536 toCGS electrostatic units, 536 toGaussian units, 536 Unit vectors, 538 incartesian coordinates, 539 Vector, 538 Vector algebra, 538 Vector differential operator, 540 Vector differentiation, 540,541 Vector integral theorems, 542-545 Vector potential, 48,544; seealso Magnetic vector potential Vector product, 540 Vortex flow,78 Vortex line, 285, 286, 288,289 circular, 141 pair, 286,289 Weight function, 385 Work, oncurrent element, 41 onpoint charge, 3 Zonal harmonics, 480; see also Legendre functions