Carlos E. Solivérez ii

CarlosE.Solivérezii

ThethecoverwasdrawnusingtheWolframDemonstrationsProjectEllipsoidbyJeffBryant.

ELECTROSTATICSANDMAGNETOSTATICSOFPOLARIZEDELLIPSOIDALBODIES:

THEDEPOLARIZATIONTENSORMETHOD

CarlosE.Solivérez

CarlosE.Solivérezii

ThisworkistheEnglishtranslationbytheauthorofthefirsteditionofhisbookinSpanish,Propiedadeselectrostáticasymagnetostáticasdecuerposelipsoidales:formalismodeltensordepolarización.Intheprocessoftranslationafewerrorswerecorrectedandanewchapter,severalproblemsandmanyappendiceswereaddedconcerningtheactualcalculationofthevaluesofthedepolarizationtensor.

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dulyacknowledgedinthefollowingversionofthetext.

1st.Spanishedition:Bariloche(RíoNegro,Argentina),June28,2012,32.467words,147pages.

1st.Englishedition:Bariloche(RíoNegro,Argentina),July15,2016,39.030words,197pages.ISBN978-987-28304-0-3.

Depolarizationtensormethod iii

Abstract

Astudyismadeofthebehaviourofellipsoidalbodies,withnofreecurrentsandcharges,underuniformappliedelectrostaticandmagnetostaticfields.Themethodisvalidforallsortsofsolidhomogenousisotropicoranisotropicmaterials:dielectric,ferroelectric,diamagnetic,paramagnetic,ferromagnetic,conducting,superconducting...Expressionsaregiven,inallcases,forthebody'sinternalandexternalfields,aswellasforthefreeelectromagneticenergyandthetorquesderivedfromit.Apartfromthebody'svolumeandtheelectromagneticpropertiesofthematerial,theseexpressionsdependonlyonthedepolarizationtensorndeterminedbythetwoaspectratiosoftheellipsoid.Explicitexpressionsaregivenforn—bothinfieldpointsinternalandexternaltothebody—intermsofelementaryfunctionsexceptforthetriaxialellipsoidwheretheyareLegendre’sellipticfunctions.Inthenon-linearrangetheelectricormagneticpolarizationisanimplicitfunctionoftheappliedfieldandtheanisotropyofthematerial,whiletheexternalfieldisanexplicitfunctionofthepolarization.Inthelinearrangeboththepolarizationandtheinternalfieldareexplicitfunctionsoftheconstantinternalvalueofninsidethebodyandtheisotropicoranisotropicsusceptibilitytensorχ ,andtheexternalfieldisanexplicitfunctionofthepolarizationandn.Intheisotropiccasethelimitvaluesχ=∞andχ=-1(SIunits)fullydescribethebehaviourofconductorandsuperconductorellipsoids.Adiscussionismadeofsomecommonerrorsinthetreatmentofelectromagneticsingularitiesandofthebehaviourofbodiesofinfiniteextensionandellipsoidalcavities.

CarlosE.Solivéreziv

Depolarizationtensormethod v

IndexAbstract iiiChapter1:Fundamentalconcepts 1Origin................................................................................................................................................................1History..............................................................................................................................................................1Applications...................................................................................................................................................3Requirements................................................................................................................................................5Limitationsofthemethod.......................................................................................................................5Dealingwithsingularities........................................................................................................................5Pointchargetypesingularity............................................................................................................6Stepdiscontinuities...............................................................................................................................9

Physicalunitsandmathematicalnotation....................................................................................10Organizationofthebook.......................................................................................................................11

Chapter2:EllipsoidsinElectricandMagneticFields 15Electricpolarization................................................................................................................................15Basicequations....................................................................................................................................15Permanentelectricpolarization...................................................................................................18Inducedelectricpolarization.........................................................................................................20Inducedelectricpolarizationoftwointeractingatoms.................................................................21Dielectrics..........................................................................................................................................................24

Magnetization............................................................................................................................................26Basicequations....................................................................................................................................26Permanentmagnetization...............................................................................................................28Inducedmagnetization.....................................................................................................................29

Conductors..................................................................................................................................................30Equivalentpolarization....................................................................................................................33

Superconductors.......................................................................................................................................34Magnetizationmodel.....................................................................................................................................35Surfaceconductioncurrentmodel..........................................................................................................36

Summaryofintegro-differentialequations..................................................................................38Summaryofinducedpolarizationequations...............................................................................39Generalformulationofthemethod..................................................................................................40Solvingtheintegro-differentialequationsbyiteration......................................................42

Chapter3:Thedepolarizationtensor:basictreatment 45Definition.....................................................................................................................................................45Generalpropertiesofn..........................................................................................................................47Symmetrictensor................................................................................................................................47Trace.........................................................................................................................................................48Orthogonaltransformations..........................................................................................................48Symmetries............................................................................................................................................49

nasasurfaceintegral.............................................................................................................................50Surfacestepdiscontinuity....................................................................................................................51Surfacechargedensity......................................................................................................................51Surfacestepdiscontinuity...............................................................................................................52

CalculationofnusingelectrostaticGauss’sLaw.........................................................................54Sheetofconstantthicknessandinfiniteextension..............................................................54Rightcircularcylinderofinfinitelength...................................................................................56Sphere......................................................................................................................................................58

OtherpropertiesoftheinternaldepolarizationtensorN......................................................59IntegralexpressionsoftheeigenvaluesofN..........................................................................59Diagonalizationandinversion.......................................................................................................60InfinitesemiaxesandtheinverseofN.......................................................................................60

CarlosE.Solivérezvi

Nisinvariantforsimilarellipsoids.............................................................................................62N(0)isdeterminedbyaspectratios...........................................................................................63

Chapter4:Thedepolarizationtensor:advancedtreatment 65Calculationsofnfromgravitationalpotentials...........................................................................65Triaxialellipsoid..................................................................................................................................65

InternaldepolarizationtensorN.......................................................................................................67ExpressionswithnormalellipticintegralsEandF..............................................................67Nellipticcylinder................................................................................................................................68Noblatespheroid................................................................................................................................69Nprolatespheroid..............................................................................................................................74UnifiedtreatmentofNforspheroids.........................................................................................77GraphsofNa,NbandNc....................................................................................................................79Na............................................................................................................................................................................80Nb...........................................................................................................................................................................80Nc............................................................................................................................................................................80Combinedgraph..............................................................................................................................................84

Externaldepolarizationtensor...........................................................................................................85ObtentionoftheexternalgravitationalpotentialbyIvory’smethod..........................85Nearandfarawaypointapproximationstonext....................................................................87Generalexpressionofexternaln..................................................................................................87Traces...................................................................................................................................................................90Verificacionforthesphere.........................................................................................................................91Ellipticcylinder................................................................................................................................................92Oblatespheroid...............................................................................................................................................95Prolatespheroid..............................................................................................................................................96Triaxialellipsoid.............................................................................................................................................97

Chapter5:Energy,forcesandcavities 99Thermodynamicsofelectrostaticandmagnetostaticenergy...............................................99Basicconcepts......................................................................................................................................99Thermodynamicsofelectromagnetism.................................................................................101

Anisotropyenergy.................................................................................................................................103Originoftorquesexertedonabody..............................................................................................105Torqueexertedonanellipsoidandthestateofequilibrium.............................................106Dielectrics............................................................................................................................................106Permanentpolarization............................................................................................................................106Inducedpolarization..................................................................................................................................108Anisotropicsusceptibility........................................................................................................................109

Magneticmaterials..........................................................................................................................110Permanentmagnetization.......................................................................................................................110Inducedmagnetization..............................................................................................................................111Magnetictorqueexperiments................................................................................................................112

Conductors..........................................................................................................................................112Superconductors..............................................................................................................................113

Forceonanellipsoidinanon-uniformfield.............................................................................114Infiniteandinfinitesimalbodies.....................................................................................................114Cavities.......................................................................................................................................................116Standardtreatment.........................................................................................................................116Somespecificcavities:orhomoeoids......................................................................................117Thinshellsandthedipolelayer.................................................................................................118

Chapter6:Selectedproblemas 121Electricpolarization.............................................................................................................................121Problem01:Autoconsistentelectricpolarizationoftwoatoms.................................121Problem02:Shapeorcrystallineanisotropy?....................................................................121Problem03:Dielectricsphere....................................................................................................122

Depolarizationtensormethod vii

Magneticpolarization..........................................................................................................................124Problem04:Derivingnfromthemagneticvectorpotential........................................124Problem05:Permanentlymagnetizedinfiniterightcircularcylinder....................124Problem06:Isotropicmagneticsphere.................................................................................126

Conductors...............................................................................................................................................126Problem07:Solveellipsoidalconductorsusingequivalentepolarization.............126Problem08:Aconductorisaperfectdielectric.................................................................127Problem09:Polarizationofasphericalconductor...........................................................127Problem10:Externalfieldofasphericalconductor........................................................128Problem11:DirectionsforwhichPisparalleltoE0intriaxialellipsoids...............129Problem12:Electricfieldsgeneratedbysharppoints....................................................129

Superconductors....................................................................................................................................130Problem13:Superconductinginfinitecylinderandsheet............................................130Problem14:Magnetizedsuperconductingsphere............................................................130Problem15:Superconductivityasperfectdiamagnetism.............................................131Problem16:Magneticmomentofasuperconductingellipsoid..................................132Problem17:DescriptionofMeissnerEffectusingmagnetizationcurrents...........133Problem18:Approximatesolutionfornonellipsoidalbodies....................................133

Depolarizationtensor..........................................................................................................................133Problem19:ComponentsofNinsphericalsymmetry....................................................133Problem20:Singleeigenvaluemaydeterminethetypeofspheroid........................134Problem21:Eigenvaluesandaspectratios..........................................................................134Problem22:Sequenceofellipsoidsoversomeaspectratiocurves..........................135Problem23:Eigenvaluesoftriaxialellipsoids....................................................................135Problem24:Depolarizationtensorofaverylongrightcircularcylinder..............136Problem25:Depolarizationtensorofasphere..................................................................136Problem26:AlternativewayofobtainingcertaineigenvaluesofN.........................136Problem27:SolvingtheprolatespheroidwithLegendre’sellipticintegrals.......136Problem28:Sphericalshell.........................................................................................................137Problem29.Unitvectornormaltotheellipsoid'ssurface.............................................139Problem30:Surfacestepdiscontinuityofthedepolarizationtensor......................139Problem31:Ellipsoidalconductorwithnetcharge.........................................................140Problem32:ApplicationofIvory’smethodtothesphere.............................................140Problem33:Expressionforfarawayfieldsobtainedfromnext...................................141Problem34:Fieldontheexternalsideofthesurfaceofapolarizedsphere.........142Problem35:Microscopicoriginofdepolarizationtensors...........................................143

Energy,forcesandtorques................................................................................................................143Problem36:Energyofadielectricsphere............................................................................143Problem37:Fermi’scontactinteraction...............................................................................143Problem38:Torquesonspheroids..........................................................................................144Problem39:Torqueexertedoverananisotropicdielectricsphere..........................145Problem40:Torqueonamagnetizeddisk...........................................................................145Problem41:Torqueonaverylongsuperconductingrightcircularcylinder.......146Problem42:Depolarizationtensorofinifinitesimallythinshells..............................146Problem43:Applicationsofthinshells..................................................................................146

Appendix1:Electromagneticunits 147Appendix2:VectorialOperators 151Appendix3:Integraltheoremsofvectorcalculus 153Laplacianofapointcharge’spotential........................................................................................153Divergence,gradientandrotortheorems..................................................................................155Extensionofthedivergencetheorem...........................................................................................155Pointchargetypesingularities..................................................................................................156

CarlosE.Solivérezviii

Stepdiscontinuities.........................................................................................................................157Generalizeddivergencetheorem..........................................................................................................157Generalizedrotortheorem......................................................................................................................158Polarizedbodies...........................................................................................................................................158

Appendix4:Fieldofanelectricdipole 161Generalexpression...............................................................................................................................161Dipolarfield'senergy...........................................................................................................................162

Appendix5:Simmetriesofthesusceptibilitytensorsofsinglecrystals 165Appendix6:Dyadics 167Appendix7:Ellipsoids 169Equationsandmainfeatures............................................................................................................169Equations,sections,volumeandsurface...............................................................................169Aspectratios.......................................................................................................................................170

Normalstothesurfaceandcentraldistances...........................................................................171Summaryofproperties..................................................................................................................173

Confocalellipsoids................................................................................................................................174Physical-geometricinterpretationofκ..................................................................................174Correspondingpoints.....................................................................................................................175

Appendix8:Usefulintegrals 179

ds

A+ s B + s( )2∫ ..................................................................................................................................179

ds

A+ s( )3/2 B + s( )∫ .................................................................................................................................180

Appendix9:Legendre’sellipticintegrals 181Definitions................................................................................................................................................181Notationusedinthemoreimportantreferences....................................................................181Specialvaluesandparametricgraphs..........................................................................................182Reductiontonormalformoftheellipticintegralsofinterest...........................................184

Mainreferences 187Alphabeticindex 189Abouttheauthor 193

Depolarizationtensormethod 1

Chapter1:Fundamentalconcepts

OriginThetopicsdiscussedinthisbookweredevelopedbytheauthor,withlong

intervals,sincethedecadeof1970.ThefirstoneswereinitiallypresentedinthetheoryofelectromagnetismcourseforstudentsofphysicsattheBalseiroInstitute(NationalAtomicEnergyComission–NationalUniversityofCuyo,Argentina)incombinationwithgeneraltechnicsforsolvingMaxwell'sdifferentialequations.Inthedecadeof1990afullrevisionwasmadeinordertoadapttheformulationtothelevelofafirstcourseonelectricityandmagnetismforengineerswithbasicknowledgeofvectoranalysis(BarilocheCampus,NationalUniversityofComahue,Argentina).Onlytwosmallportionsofthiswork1,2werepublishedininternationalscientificmagazineswithpeerreview,becauseeditorsconsideredthefulltextunsuitableforpublicationbecauseofitslengthandpedagogicalorientation.Athoroughrevisionwasmadeofallthematerialthusgathered,revisionthatwillcontinueaslongasreadersarewillingtomakecorrectionsandsuggestmodifications,andtheauthorisabletoanalizeandincorporatethem.

Thedepolarizationtensormethodprovidesaneficientandcompactwayofsolvingthecaseofellipsoidalbodiespolarizedbyconstantelectricandmagneticfields.Itisthuswellsuitedfortechnicalapplications,butalsoforresearchonanisotropicpropertiesofmaterials,forwhichnogeneralmethodis,totheknowledgeoftheauthor,presentlyavailable.Fromthepointofviewofthefoundationsofelectromagnetism,itisoneofthesimplestandclearestexampleoftheinextricablerelationshipbetweengeometricandphysicalpropertiesofmaterials.Thisrelationshipisexpressedbytheboundaryconditionsthatdeterminethesolutionofthediferentialequations,butthisisneversoclearlyandexplicitlyexpressedasbythedepolarizationtensor.Itisalsoagoodillustrationofthepowerofintegralmethodsforsolvingelectromagneticproblems.Themaincontributionoftheauthorinthisfield,asshownbythecurrentliteratureonthesubject,isthegeneralizationoftheuseofthistensorforallstaticelectromagneticproblems,thetreatmentofanisotropicmaterialsandthecalculationoffieldsoutsidethebody.

HistoryTheuseofwhatistodayidentifiedasdepolarizationtensorisveryoldinthe

analysisofpermanentlymagnetizedmaterials.ThefirsttoderiveitexplicityfromapotentialwasprobablyMaxwell3.TothatendheusedPoisson'sproof(seeeq.

1 Solivérez(1981),pp.1363-1364.2 Solivérez(2008),pp.203-207.3 Maxwell,pp.66-69.

CarlosE.Solivérez2

2.2)thatthecalculationofthefieldcreatedbyauniformlymagnetizedbodyismathematicallyequivalenttotakingthedirectionalderivativeofthegravitationalattractionofanhomogenousmassdistributionwiththesameshape.Themathematicalanalysisofthesubjectthusbenefitsfromthenumerousstudiesofgravitationalpotential(proportionaltotheintegralfdefinedbyeq.3.8)madesinceNewton'stimes.Itisinthisfieldofphysicsthatellipsoidalbodieswereandstillareintensivelystudied4.

Maxwellprovedthatellipsoidalbodiesareuniformlypolarizedwhenimmersedinauniformstaticappliedfield(aconstantvectorintheregionoccupiedbythebody).Hepointsoutforintegralf(therecalledV5)

...theonlycaseswithwhichweareacquaintedinwhichVisaquadraticfunctionofthecoordinateswithinthebodyarethoseinwhichthebodyisboundedbyacompletesurfaceoftheseconddegree,andtheonlycasesinwhichsuchabodyisoffinitedimensionsiswhenitisanellipsoid.

Theoriginofthisbehaviour—whichisanalizedinChapter2—isthatinternalfields,andthepolarizationstheyproduce,areproportionaltosecondderivativesoff.SoonafterwardsThomsonandTaitprovedellipsoidstobetheonlyfinitebodieswiththisproperty6.TheprincipalvaluesofMaxwell'sentityarefrequentlycalleddemagnetizingor

demagnetizationfactorsorcoefficients,andthestudyofitshistorybeganasearlyas18967.Theauthorofthisbookhasnotbeenabletodeterminewho,whenandwhererebaptizeditasdepolarizationtensor,anamethatrightlyincludesbothelectricandmagneticphenomena8.In1941,Stratton'sclassicaltextonelectromagnetismintroducedtheprincipalvaluesofthedepolarizationtensorinsidedielectricsbysolvingLaplace'sequationinellipsoidalcoordinates.Hecalledthemdepolarizationfactors9butheneithermentionstheirtensorialcharacter(whichMaxwellapparentlydidnotdetect),norappliesthemtoconductingormagneticmaterials.Twodistinguishedrussiantheoreticalphysicists,L.LandauandE.Lifschitz,generalizedin1969theuseofthesetensorialcoefficientstothecaseofdielectrics,conductorsandsuperconductorsinvolume8oftheirmonumentalCourseonTheoreticalPhysics10,introducingtheminasimilarfashion

4 N.R.Libovitz,Themathematicaldevelopmentoftheclassicalellipsoids,InternationalJournalof

EngineeringSciencevol.36,pp.1402-1420(1998).5 Maxwell,p.67.6 W.Thomson,W&P.G.Tait,P.G.,TreatiseonNaturalPhilosophy,vol.2,CambridgeUniversity

PressCambridge(England),2009(newprintingofthe1883edition).P.Dive,Attractiondesellipsoïdeshomogènesetréciproqued'unthéorèmedeNewton,Bull.Soc.Math.Francevol.59,pp.128-140(1931).W.Nikliborc,EineBemerkunguberdieVolumpotentiale,Math.Z.vol.35,625-631(1932).

7 H.DuBois,TheMagneticCircuit,Longmans(1896).8 ThenamewasusedbyVanVleck,chapterIV,in1932.9 Stratton,pp.206and213.10 LandauandLifchitz:conductors(pp.7,25,28),dielectrics(pp.42-45,54),ferromagnets

(p.163)andsuperconductors(p.170).


asStratton11.Theyseemtobethefirstonestopointoutthemathematicalequivalenceofthebehaviourofconductorsanddielectricswithinfinitesusceptibility12.In1945Stonergavegraphsandtablesofthedemagnetizingfactorsforspheroidsandgeneralellipsoids,withoutmentioningthetensor'sgeneralproperties.Almostsimultaneously,Osborngaveanumberofusefulformulas,tablesandgraphsforobtainingthevaluesofthedemagnetizingcoefficients.In1966MoskowitzandDellaTorreanalizedsomegeneralpropertiesofthedepolarizationtensor13.In1981thisauthorproved,forthemagneticcase,thatthistensoralsosolvestheanisotropiccaseandprovidesthenon-uniformvalueoftheinducedfieldoutsidethebody14.In2006alternativemethodswereproposedforthecalculationofthetensor15.In2008thisauthordiscussedtheaplicationtotheelectriccase,includingelectretsandconductors16.

ApplicationsElementarytextsonelectricityandmagnetismsolveonlythebehaviourofthe

followingfictivebodies(theequivalentellipsoidsareidentifiedbetweenparenthesis):

• pointcharge(ellipsoidwith3equalandverysmallsemi-axes);• sheetofuniformthicknessandinfiniteextension(ellipsoidwith1finitesemi-axisand2verylargesemi-axes);

• cylinderofcircularcross-sectionandinfinitelength(ellipsoidwith2equalsemi-axesand1verylargesemi-axis);

• volumeofinfiniteextension(illdefinedproblemdiscussedinChapter5).

Thesphere—thefinitebodyofhighestsimmetry—issolvedonlyinadvancedcoursesofelectromagnetismbyreducingLaplace'sequationtoasetofordinarydifferentialequationswiththecanonicalmethodofseparationofvariablesinsphericalcoordinates17.Solvingtriaxialellipsoids(threedifferentsemi-axes)withthesamemethodrequiresadvancedknowledgeofmetricpropertiesandtheuseofanuncommonandcomplexsistemofcurvilinearcoordinates18.Ontheotherhand,thedepolarizationtensormethodrequiresonlyabasic

knowledgeofvectoranalysisinordertosolvethewholefamilyofellipsoids,includingthesphere,sheetsofinfiniteextension,circularandellipticcylindersofinfinitelength.Theinternalfieldsmaythusbeexpressedintermsofconstantmatricespeculiarofeachbody,whosegeometricpartisafunctionoftwoparameterswhicharetheratiosoftwosemiaxestothethird(aspectratios).

11 LandauandLifchitz,p.20.12 LandauandLifchitz,p.40.13 MoskowitzandDellaTorre,pp.739-744.14 Solivérez(1981).15 M.Beleggia,M.DeGraefandY.Millev,Demagnetizationfactorsofthegeneralellipsoid:An

alternativetotheMaxwellapproach,PhilosophicalMagazinevol.86,pp.2451-2466,2006.16 Solivérez(2008).17 Jackson,pp.156-160;Reitz,pp.93-94;Stratton,pp.205-207.18 LandauandLifchitz,pp.20-27;Strattonpp.58-59,211-213,257-258.

CarlosE.Solivérez4

Onlylimitedtothehomogeneouscase,themethodisvalidforanyofthefollowingtypesofmaterials:

• isotropicones,likeamorphousandpolicrystallinesolids;• anisotropicones,likesinglecrystals;• permanentandinducedelectric(electrets,ferroelectrics...)andmagnetic(ferromagnets,antiferromagnets...)polarizations;

• conductors(metals)andinsulators(dielectrics);• superconductors.Themethodallowstoeasilysolvealargerangeofpracticalproblemsfor

realisticfinitebodiesbyusingmostlyelementarymathematicalfunctionandabasicknowledgeofvectoranalysis.Theexceptionaretriaxialellipsoidswhich—duetotheappearanceofellipticfunctions—requiretheuseoftablesorsoftwarelikeMathematica®orMaple®.Experimentaldeterminationofpolarizations19andofelectricandmagnetic

anisotropyconstantsareusuallymadebymeasuringforcesandtorquesonspheres,disksandcylindersplacedinknownuniformappliedfieldsorappropiatenon-uniformones20.Inversely,theseforcesandtorquesprovidethevalueofthefieldinwhichthebodiesareimmersed.Ellipsoidsincludeoraregoodapproximationstothosebodies21,alsoallowinganestimationoftheerrorscausedbydeparturesfromidealshapes.Theyarealsousedinsinglecrystalsforthecalculationoflatticesums22,themicroscopiccontributionsofatomicandmolecularelectricchargesanddipolemomentstothemacroscopicfield.Therearenumeroustechnologicalmethodsandproperties—likemagneticresonance23,electrostaticprecipitation,magneticcoercivityoftapesanddisks24—thatcanbemoreeasilyinterpretedwhenusingellipsoidalbodies.Also,someexperimentssuggestthatincertaincasesellipsoidalparticlesmayappearspontaneously25.Widelyusedanisotropicmaterials,seldomdiscussedintechnicaltextbooks26becauseofthedifficultiesofitsmathematicaltreatment27,maybeeasilydiscussedwiththedepolarizationtensormethod.Itis,therefore,surprisingthattheelectromagneticbehaviourofellipsoidal

bodiesisabsentfrommosttextbooksonelectricityandmagnetism,beingconfined

19 Inlinewiththeuseofthedenominationdepolarizationtensor,amagnetizationwillbecalled

magneticpolarizationinthisbook.Therefore,thetermpolarizationrefersbothtotheelectricandmagneticone.

20 H.Zijlstra,ExperimentalMethodsinmagnetism,vol.2,NorthHollandPublishingCo.,Amsterdam(1967).

21 M.Beleggiaetal.,J.Phys.D:Appl.Phys.39,pp.891-899(2006).22 M.Widom,Shape-AdaptedEwaldSummation.23 J.H.DuynandT.H.Barbara,Magn.Reson.Med.,72(1),pp.1-3(2014).24 Q.F.BrownJr.andA.H.Morrish,Phys.Rev.vol.105,p.1198(1957).25 G.R.Davies,Themechanismsofpiezoelectricityandpyroelectricityinpoly(vinylidenefluoride),

TheDielectricSociety1984Meeting,AbstractsofInvitedPapers.26 AnotableexceptionisthebookbyLandauandLifchitz.27 See,forinstance,theReportoftheCoulomb'sLawCommitteeoftheAmericanAssociationof

PhysicsTeachers,Am.J.Phys.vol.18,p.1(1950).


totheoreticaltreatisesandarticlesinspecializedjournals.Thisbooksintendstofillthisgap,collectingscatteredformulasandproperties,generalizingtheformulasandaddingnewoneswheneverpossible.

RequirementsThereaderisrequiredtobefamiliarwiththedifferentialandintegralphysical

lawsrelatingelectricandmagneticfieldsandpotentialswiththeirsources:electricchargedensityandpolarization,currentdensityandmagnetization.Therequiredknowledgeofmathematicsincludesoperationswithmatrices,derivatives,surfaceandvolumeintegrals,thecombinedapplicationofvectorialoperatorsandtheirintegraltheorems.Aspreviouslymentioned,itisnotnecessarytosolveLaplace'sequation,anunavoidablerequisiteunlessthedepolarizationtensormethodisused.

LimitationsofthemethodThedepolarizationtensormethodisvalidonlywhen:

• Theappliedfieldisuniformoverthewholebody,butnotnecessarilyelsewhere.InthemagneticcasetheconditioncanbeeasilyobtainedwithHelmholtzcoils28.

• Thebody'smaterialmusthaveastableellipsoidalshape—whichrulesoutliquids)—anduniformelectricormagneticpropertiesinamacroscopicscale.Otherwise,thematerialmaybeisotropic(amorphousormicro-policrystalline)oranisotropic(singlecrystalortexturedpolicrystalline29).Theappliedfielddoesnotneedtobefixed(unmodifiablebytheintroductionof

thebody),exceptforsimplicityinthecalculationoftheenergyandtheresultingforcesandtorques.Themethodisvalidfortorqueexperimentswheneverthevalueoftheappliedfieldcanbemeasuredorcalculatedaftertheintroductionoftheellipsoidalbody.

DealingwithsingularitiesElectromagneticphenomenaarefullydescribedbyMaxwell'sequationsandthe

auxiliarconstitutiveequationswhichdescribematerialproperties.Thestandardformulationmakesuseofvectoranalysisandtechniquesofresolutionofpartialdifferentialequations,butnotofintegro-differentialequationslikethosegivenbyeqs.2.31.Thedepolarizationtensorformulationdispenseswiththeuseofdifferentialequations,butmakesintensiveuseofvectoranalysisandofintegralrelationshipslikethedivergencetheorem,wherespecialprecautionsmustbetakenwhendealingwithsingularitieslikepointchargesandjumpdiscontinuitiesacrossthebody'ssurface:

28 W.Franzen,GenerationofUniformMagneticFieldsbyMeansofAir-CoreCoils,Rev.Sci.Instr.vol.

33,pp.933-938(1962).29 Apolicrystallinematerialistexturedwhenitsmicrocrystalshavepredominanceofcertaintype

offaces.Thisoftenhappensinlaminatedmetallicmateriales,becausethelaminationprocessfavourstheappearenceofcertaincrystalplanesanddisfavoursothers.

CarlosE.Solivérez6

pointchargetypesingularity: (1.1)

jumpdiscontinuity:F(r + )≠

F(r − ) through)surface)S. (1.2)

Thesesingularitiesappearrepeatedlyinsurfaceandvolumeintegrals,bothfortheelectricandmagneticcase.Astheyarisefromasimplifiedmathematicalrepresentationofphysicalfacts,tosolvethemisnecessarytotracethembacktothebasicphysicalphenomenatheydescribe.Thisphysicalanalysisismadeinthetwofollowingsections,whileinAppendix3theintegraltheoremsofvectoranalysisarecarefullyreformulatedinordertodealcorrectlywiththesesingularities,whereitmaynotbevalid,forinstance,tointerchangetheorderofderivationandintegration.Scalarandvectorialmagnitudeswithconstantvalueswithinaboundedregion

playanessentialroleinthemethod.Electricandmagneticpolarization,forinstance,haveafiniteandconstantvalueinsideandvanishoutsidetheellipsoidalvolumeV.AppliedelectricandmagneticfieldsareconstantonlyinregionthatbarelyexceedsV,andmayvarywidelyintheoutside.Thejumpdiscontinuitiesofpolarizationthroughthebody'ssurfacearethesourcesoftheinternalandexternalinducedfield.TheinvariabilityofappliedfieldsinVplayacrucialroleinthedeterminationoftheforcesandtorquesexertedontheellipsoidalbody(seeChapter5)

Itisthereforeconvenienttohaveacommondenominationforthesekindoffields,inordertoclearlydifferentiatethemfromtheconstantvectorsofmathematicsthatareinvariableoverallspace.Forthatreason,inthisbookavectorissaidtobeuniforminaregionVwhenitisconstantinV(invariablemagnitude,directionandsense),butnotnecessarilyoutside.

PointchargetypesingularityTheoriginofthepointchargetypesingularityistheexperimentalCoulomb's

law(thevalueofconstantk1indifferentsystemsofunitsisdiscussedinAppendix1),

(1.3)

whichgivestheforceexperiencedbyachargeqinpresenceofachargeq'.Theexperimentrequirestwodifferentcharges:thefield'ssourceq'andthechargeqwheretheforceisapplied(unlessoneiswillingtoinvokea"Münchauseneffect"30).Itisthereforenecessarytodifferentiatebetweenthefieldpoint

!r andthesourcepoint !r ' Thisrequirementisignoredinthemacroscopicformulationofelectromagnetismbecauseotherwisefieldswouldnotbedefinedinsidecontinuouschargedistributions.Thepriceonehastopayforthissimplificationisthattheresultingmacroscopicfielddiffersfromthemicroscopicfieldactuallyexperiencedbyatomsandmolecules.

30 ThemythicalMünchhausenbaronraisedhimselbypullingfromhisboot'sstraps.

!!1/r− r ' cuando r→ r ',

!!

F(r )= k1qq'

r− r 'r− r ' 3

= qE(r ),


Coulomb'slawgivesorigintothedefinitionofthedifferentialelectricfield

!!dE(r ) createdat !r byaninfinitesimalcharge !!dq= ρ(r ')d3r ' at !!r '. Thisfieldisexpresssedintermsofthevolumechargedensityρ by

(1.4)

wherefieldandsourcepointsareexplicit,aswillbealwaysdoneinallintegralsherein.

Thetotalelectricfieldcreatedbythecompletechargedistributionρisthusgivenby

(1.5)

Expressions1.3and1.5dolooksimilar,buttheydifferinafundamentalway.WhileforCoulomb'slawitisnotvalidtotake

!r = !r ', theelectricfieldexpressionrequiresitsvalidityforallvolumeandsurfaceintegrals.Thereareadditionalproblemsbecausethevolumechargedensityρforpointchargesisnotawellbehavedfunction,andthesamehappenswithsurfacechargedensities.Thesesingulardensitiesmaybegivenmathematicalrigourbytheuseofdistributiontheory,amethodthatwillnotbeusedherefortheevaluationofintegralsofthiskindbecausemostengineersarenotfamiliarwithit.Instead,appropiatecarewillbetakenfortheevaluationofsingularintegrandswithstandardmethods,anexampleofwhichisthecalculationofthetraceofthedepolarizationtensorineqs.3.15andA3.7.Thesimplestcaseofdealingwithpointchargetypesingularitiesisthatofeq.

1.5.Onemaytakeasystemofsphericalcoordinateswithoriginatpoint !r sothat

theintegralbecomes

!E(!r )= −k1 dϕ senθdθ R

R2ρ(ϕ ,θ ,R)R2dR

0

R(ϕ ;θ )

∫∫0

2π

∫ con !R = !r '− !r . (1.6)

ThesingularityturnsouttobeintegrablebecauseofthefactorR2inthedifferentialvolumeelement.Aswillbediscussedlateron,thisdoesnotmeanthatsuchsingularitiesdonotaffectthevalidityofthestandarddivergenceandcurltheoremsofvectorcalculus.

Let'sexaminethefirstofthesetheorems.Coulomb'sLaweq.1.3istheoriginofGauss’sLaw

!E(!r )id!S

S"∫∫ = 4πk1QS , (1.7)

whereQSisthechargeinsidetheclosedsurfaceS.Accordingtothedivergencetheoremeq.A3.8,

!E(!r )id!S

S"∫∫ = ∇ i

!E(!r )d3r

V∫∫∫ = 4πk1QS = 4πk1 ρdV

V∫∫∫ , (1.8)

!!dE(r )= k1

r− r 'r− r ' 3

dq'= k1r− r 'r− r ' 3

ρ(r ')d3r ',

!!

E(r )= k1

r− r 'r− r ' 3

ρ(r ')d3r 'V∫∫∫ .

CarlosE.Solivérez8

whereVistheregionboundedbyS.TheresultisoneofMaxwell'sequations(seeeq.A1.3)

∇ i!D(!r )= ε0∇ i

!E(!r )= 4πk1ε0ρ , sothat ∇ i

!E(!r )= 4πk1ρ. (1.9)

Thevolumechargedensityρofapointchargeeq.1.3wouldthenbe

ρ = 14πk1

∇ i!E(!r )= q'

4π ∇ i!r − !r '!r − !r '3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= q'4π

1!r − !r ' 3

∇ i(!r − !r ')+∇ 1!r − !r ' 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟i(!r − !r ')

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= q'4π

3!r − !r ' 3

− 3(!r − !r ')i(!r − !r ')!r − !r ' 5

⎛

⎝⎜⎜

⎞

⎠⎟⎟=0,

(1.10)

whereusehasbeenmadeofeq.A2.5.TheexperimentalvalidityofGauss'sLawhasbeenthoroughlyverified31,soit

followsthateq.1.9doesnothold.Someauthorstryto"solve"theproblembydenyingthevalidityofeq.1.10withdubiousmathematicalarguments32,buttheoriginoftheparadoxisthatthedivergencetheoremeq.A3.8isnotvalidforsingularfieldsliketheonegivenbyeq.1.3.Asimilarproblemariseswiththecalculationofthefollowingexpression,that

willbeusedlateronforthecalculationofthedepolarizationtensortrace:

∇ i∇ d3r '

!r− !r 'V∫∫∫ . (1.11)

Thisexpressioniswellknowninphysics,whereitisusuallysolvedbyusingDirac'sdeltafunction,thatis,invokingdistributiontheory.Someauthorsoftextsinelectromagnetismjustifyitsvaluebyjugglingwiththeorderofderivationandintegration,frequentlymakinginvalidmanipulations.ThetopicisdiscussedinAppendix3inordertomakeavalidderivationofeq.A3.7withnorecoursetoDirac'sdeltafunction.Onemaydealwithsingularfieldsintwodifferentways.Thefirstone,which

preservesMaxwell'sequationsevenforsingularchargedistributions,istoextendthedefinitionofcharge(andcurrent)densitiesinordertoincludepointcharges,surfaceandlinechargedistributions(andthecorrespondingcurrentdistributions).Thisrequiresdistributiontheory,acomplexmathematical

31 See,forinstance,YoungandFreedman,p.721.32 Reitz,p.45,ignoresthebasicruleofmakingallpossiblesimplificationsbeforetakinglimits,not

afterwards.


formalism33wellknowntophysicistsbutseldomusedbyengineers.Theotherway—discussedinAppendix3—istoextendthevalidityoftheintegraltheoremsofvectoranalysisforthecaseofsingularchargedistributionslikepointcharges.Thelatterapproachisusedbysometextsonelectromagnetism,butusuallynotexplicitystatedasageneralizationofthestandardtheorems.

StepdiscontinuitiesStepdiscontinuitiespresentsimilardifficulties.Inawellknowntexton

electromagnetismwhichtheauthorusedforhisfirststudiesofthesubject,thefollowingintegrationbypartsismade34(hererewriteninournotation):

∇Φ(!r )i !D(!r )− !D0(!r )( )d3rV '∫∫∫

= ∇ i Φ!D−!D0( )( )d3r

V '∫∫∫ − Φ(!r )∇ i

!D(!r )− !D0(!r )( )d3r

V '∫∫∫ ,

(1.12)

where Φisanelectricpotential, !D, !D0 arethedisplacementvectorsinthe

presenceandabsenceofthematerialbody,andusehasbeenmadeofeq.A2.5.TheintegrationvolumeV'includesboththedielectricbodyandthesourcesof Φ.Fromdivergencetheoremeq.A3.8,Jacksonarguesthat,astheintegrationsurfaceS'isoutsidethebody,itfollowsthatthere !!

D=D0 and

∇ i Φ!D−!D0( )( )d3r

V '∫∫∫ = Φ

!D−!D0( )d2!r =0,

S '"∫∫ (1.13)

where !!d2r isthedifferentialelementofareavectornormaltothesurface.Thestandarddivergencetheoremmaybeusedwhenthethefirstpartial

derivativesoftheintegrandarecontinuousinV'35,but !!D−D0 hasastep

discontinuityacrossthebody'ssurfaceS.OnemayavoidtheproblembydividingthevolumeofintegrationintwobythesurfaceSofstepdiscontinuities,themethodusedintheformulationofthegeneralizeddivergencetheoremeq.A3.18.Usingthistheoremeq.(1.12)gives

∇ i Φ!D−!D0( )( )d3r

V '∫∫∫ = Φ

!D−!D0( )d2!r

S '"∫∫ − Φ

!D+ −

!D−( )d2!r

S"∫∫ ≠0, (1.14)

becausethelastintegranddoesnotvanish.Thissortofproblems,whichappearfrequentlyintextsonelectromagnetism,

usuallyremainundetectedbecausesuchcomplexintegralsareseldomcomputedforrealdistributionsofmatter.Thiswasnotourcasebecauseitwasquiteeasyto

33 ForasimplifiedformulationbasedonL.Schwartz,Theoriedesdistributions(Hermann,Paris,1950)seeR.J.Gagnon,Am.J.Phys.vol.38,pp.879-891(1979).

34 Jackson,p.125.35 KornandKorn,p.163.

CarlosE.Solivérez10

usethedepolarizationtensormethodtocomparetheinitialexpressionforanellipsoidalbodywiththefinalone,findingnomatch.

PhysicalunitsandmathematicalnotationTextsusedbyphysicists—likeJackson's—tendtousetheGausssystemofunits

becauseit'smoresuitedforrelativisticanalysis,whileengineersuniformlyusestandardSIunits.Insteadofcommittingourselvestooneofthetwo,weusethegeneralsystemintroducedbyJacksonthroughconstantsk1,k2,k3,λandλ'definedandtabulatedinAppendix1forallcommonlyusedsystemsofunits36.

Themathematicalnotationusedinthisbookusesthefollowingconventions:

• Vectors(fields,polarizations...)andtensors(depolarization,susceptibilies...)arerepresentedintwoalternativeways:

1. Aslinearcombinationsofproductsofcomponentsandunitvectorssuchas

!!E(r )= Ez(

r )z. Variablesarealwayswrittenwithitalics,vectorsarealwayscrownedwitharrowsandunitvectorswith^.Theomissionofthearrowindicatesthevectormodule,

!!E = E. Secondranktensorsarerepresentedwith

italicsandboldletters,anditscomponentsaregivenindyadicnotation37(seeAppendix6)

n(!r )= nαβ(

!r )xα xββ∑

α∑ , (1.15)

thusmakingclearthatitsscalarproductwithavectorgivesanothervector.Thisintuitiveandcompactnotationsimplifiesthederivationofequations,butnotitsresolution.Forthelatteritismoreconvenientotousethefollowingalternativenotation.

2.Vectorsarealternativelyrepresentedascolumnorrowmatricesofdimensión1x3or3x1.Secondranktensorsarethenrepresentedassquare3x3matrices,asineq.3.66.Theonlydisadvantageofthisrepresentationisthatdependsonthechosencoordinatesystem,butsodoesthediagonalform(seebelow)ofthedepolarizationtensor.

• Vdenotesboththeregionoccupiedbytheellipsoidalbody(inintegrals)andthemagnitudeofitsvolume(inequations),conceptseasilydiscriminatedfromthecontext.

• SistheboundarysurfaceofregionV(thebody)and !!s(r S ) theoutgoingunit

vectornormaltoSatpoint !r S .

• x,y,zareorthogonalcartesiancoordinates.• Differentcoordinatesaregenericallycalledxα,whereα=x,y,zandxx=x,xy=y,xz=z.

36 Jackson,pp.613-618.37 MorseandFeshbach,pp.54-92.


• Unlessexplicitlystatedotherwise,thecartesiancoordinatesysteminuseistheonethatdiagonalizesthematrixrepresentationofthedepolarizationtensor,theellipsoidal'sbodyprincipalsystemofeq.A7.2.

• δαβ isKronecker'sdeltafunction(ormatrix)definedby!δαβ =

1if α=β0if α ≠ β⎧⎨⎩

.

Therefore,theunitvectorscorrespondingtoeachoftheorthogonalcoordinateaxisfulfillthecondition .

• Symbol isthegradientoperatornablasuchthatits

applicationtoascalarfunctiongivesavector.Thedivergenceoperatoriswrittenas∇·,thecurlas∇xandthelaplacianas∇·∇or Δ.

• isasourcepointforthefield(electriccharges,polarization,electriccurrent...),whichisalwaysexplicitydistinguishedfromthefieldpointwherethegeneratedfieldactsupon.

• d2r'isthescalardifferentialelementofareainasurfaceintegral.• isthevectorialdifferentialelementofareainasurfaceintegral.• d3r'isthescalardiferentialelementofvolumeinavolumeintegral.

OrganizationofthebookThreecategoriesofmaterialshaveverysimilarbehaviourbothintheelectric

andmagneticcase:spontaneouslypolarizedones(likeelectrets,ferroelectricsandferromagnets);thosepolarizedonlyinthepresenceoffields(likedielectrics,diamagnetsandparamagnets);thosewithvanishinginternalfields(conductorsandsuperconductors).Inordertofacilitatecomparisons,inChapter2thesecasesareseparatelydiscussedforelectricandmagneticsubstances.Inpermanentlypolarizedmaterialsauniformappliedfieldisnecessaryonlyfor

maintainingandorientingtheuniformpolarization.Thebody'sinternalandexternalfieldsarethenfullycharacterizedbythedepolarizationtensorn,bothwithorwithoutappliedfields.

Forsubstancesthatgetpolarizedonlyinthepresenceofappliedfields,thelinearapproximacionisvalidexceptinunusuallyhighfields.Inthisapproximationtheinducedpolarizationisproportionaltotheuniformappliedfieldthroughasusceptibilitythatisascalarintheisotropiccaseandasecondranktensorχ intheanisotropicone.Itisthenfoundthattheinducedpolarizationandtheinternalfield—bothuniformforellipsoidalbodies—maybefullyexpressedintermsofn,thesusceptibilityandtheappliedfield.Thisresultiswellknownforisotropicmaterials,buthasneverbeenfullydiscussedfortheanisotropiccase38.Inthelinearrangeatensorialrelationship—thebody'spolarizability,functionofnandχ39—maybestablishedbetweenthebody'selectricormagneticmomentandtheappliedfield,justasinthemolecularcase40.

38 SeethediscussionmadebyLandauandLifchitzinpp.58-61.39 LandauandLifchitz,p.7,introducethetensorbutdonotrelateitton.40 Kittel,p.459.

!!xα i xβ =δαβ

!!∇ = x ∂

∂x+ y ∂

∂ y+ z ∂

∂z

!!r '

!!d2r '


Accordingtothestandarddefinitionofmacroscopicfields,thereisnopolarizationinsideaconductororasuperconductor.However,afictivepolarization(inthisbookcalledequivalentpolarization)maybedefinedsuchthattheproblemturnsouttobeequivalenttothecaseofinducedpolarization.Theequivalentpolarizationsgivesboththerightvalueoftheinternalfieldsandthebody'selectricandmagneticmoments.Forconductorsthecorrespondstothelimitofaperfectdielectric,thatofinfiniteelectricsusceptibility.Forasuperconductor,itcorrespondstothelimitofperfectdiamagnetismwherethediamagneticsusceptibilityequals-1.

Theintegro-differentialequationsthatdetermineinducedfieldsandinternalpolarizationsarederivedforallcasesandcollectedinTable1.Anansatzcorrespondingtotheuniformsolutionsatisfiesalltheseequations.AttheendofChapter2aniterativemethodofsolvingtheintegro-differentialequationsisgiven,whichconfirmsthevalueoftheuniformsolutions.

AllalongChapter2thedepolarizationtensorandsomeofitspropertiesareusedwithoutproofinordertoshowitsconvenienceandwayofuse.ItsdefinitionandelementaryanalysisstartsatChapter3,wheresomeexpressionsaregivenforthetensor,aswellasasimplemethodfordeterminingitsprincipalvaluesinthecaseofthesphere,theinfinitelylongcircularcylinderandtheinfinitesheetofconstantthickness.Chapter4tacklesthemorecomplexcaseofthespheroids,uniaxialellipsoidandellipticcylinder,aswellasthestudyofthevalueofthedepolarizationtensoroutsidethebodyfortriaxialellipsoids.

Chapter5givesa(too)briefintroductiontothecalculationofelectromagneticenergies,fromwhicharederivedtheexpressionsforforcesandtorquesexertedonellipsoidalbodiesimmersedinuniformappliedfields.Infinitebodiesandcavitiesarediscussedattheendofthischapter,showingtheinconsistenciesintheirstandardanalysiswichusuallyinvokesa“rigid”behaviourofthepolarization.Bothtopicspresentdifficultiesrarelydiscussedinstandardtextbooks;theydeserveamorethoroughanalysisthatisbarelysketchedhere,probablyawholebook.TheinclusionofChapter6,whichgivesdetailedsolutionstoillustrative

exercisesonallpreviouslydiscussedtopics,requiresjustification.Thisbookisneitheratreatiseonelectromagnetism,noroneonthetheoryofpolarizedbodies.ItisanintroductiontoapracticalmethodofcalculationofstaticelectromagneticpropertiesofellipsoidalbodieswithnorecoursetotheresolutionofLaplace'sequation,thatis,tothetechniquesofresolutionofpartialdifferentialequations.Exerciseshave,therefore,acentralrole,butheretheyarenotdispersedthroughthetextasiscustomary.Twomainargumentsjustifygroupingtheminasinglechapter.

Thefirstoneisthatthespecificityofconcreteproblemsdistractsthereaderfromthemainlineofreasoning.Inthefewocasionswheretheauthorfeltthattheproblemisrelevanttothisreasoning,anactualcaseisdiscussedinthepertinentplace,butalinkisalgogiveninChapter6.Thenecessityofthislinkisaconsequenceofthesecondjustificationfortheexistenceofthischapter.

Thisbookisconceivedasacomplementtoordinarytextsonelectromagnetismbecauseoftheircustomarylackoftreatmentoffinitebodies,usuallyreservedtocourseswithadvancedknowledgeofpartialdifferentialequations.Therefore—althoughtheauthordoesnotrecommendit—itisexpectedthatitsusewillbe


occasionalandfragmentary,whichrequiresspecialprovisions.Thegivenorganizationmakestheselectionofexerciseseasier.Sodoestheredundancyinthediscussionofcentralconceptsandtheprovisionofadetailedalfabeticalindex.


Chapter2:EllipsoidsinElectricandMagneticFields

Thischapterdiscussestheuniformpolarizations—permanent,inducedandequivalent—andthefieldsgeneratedwhenhomogeneousellipsoidalbodiesareimmersedinuniformappliedfields.Resultsareexpressedintermsofthedepolarizationtensorn,leavingthesolutionofspecificcasesforChapter6.

Electricpolarization

BasicequationsThediscussionofthesubjectfirstmadebytheauthorin200841isheremodified

inordertomakeitsunderstandingeasier.TheelectricpolarizationofanellipsoidalbodyVisusuallyafunctionofthepositionandgeneratesanelectricfield

!E !P(!r ) derivablefromapotential42:

φP(!r )= k1

!P(!r ')i(!r − !r ´)!r− !r ´ 3

d3r 'V∫∫∫ , !EP(

!r )= −∇φP(!r ). (2.1)

Theformulaistheextensionofthepotentialofapointdipoletothecaseofacontinuousdistribution,themacroscopicrepresentationofacollectionofneutralatomswherethebaricentersofthenuclearandelectronicchargesdonotcoincide.Poissonwas,apparently,thefirsttonoticethat

φP(!r )= − !P i∇( )k1 1

!r− !r ´d3r '

V∫∫∫ , (2.2)

Thecharacterizationofauniformlypolarizedellipsoidalbodymaythusbederivedfromthatofauniformlychargedone,apropertythatwillbewidelyusedinthisbook.Asdiscussedinp.7,theintegrandineq.2.1hasanintegrablesingularity.Itis

frequentlysaidthatthemacroscopicelectromagneticfieldsaretheaverageofthemicroscopicfields,butithasbeenconvincinglyarguedthatoneshouldtake,

41 Solivérez(2008).42 Reitz,p.78eq.4-7.


instead,theaverageofthepotential43.Thesubject,notdiscussedinthisbook,ismentionedonlybecausethedepolarizationtensormayhaveanimportantrolefordeterminingtheconnectionbetweenthosetwoscales44.Atthesametime,agoodunderstandingoftheoriginoffundamentalexpressionsaseq.2.1(andsimilarmagneticones)isthebestguidefortheinterpretationandresolutionofmathematicalsingularitiesoriginatedinmathematicalidealizationsofactualnon-singulardistributionsofcharge.

Thepotentialeq.2.1mayberewrittenintwodifferentways.Thefirstonewillbeusedforthediscussionofanellipsoidalbodywithpermanentorinducedelectricpolarization.Thesecondone,forconductors.

Usingeqs.A2.2andA2.5theintegrandofeq.2.1mayberewritteninthefollowingfashion:

φP(!r )= k1

!P(!r ')i(!r − !r ´)!r − !r ´ 3

d3r 'V∫∫∫ = −k1

!P(!r ')i∇ 1

!r − !r ´⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ ,

−k1 ∇ i!P(!r ')!r − !r ´

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟d3r '

V∫∫∫ = −k1∇ i

!P(!r ')!r − !r ´

d3r 'V∫∫∫ .

(2.3)

Thelaststeprequiresapermutationoftheorderofderivationandintegration,anoperationthatmaynotbevalidforsingularintegrands.Theproblemhasbeenstudiedforthiskindofintegralsinthegravitationalcaseandproventobemathematicallyvalid45.

Thetotalelectricfield !!E(r ) isthevectorialsumoftheuniformappliedfield !!

E0

andtheinducedfield !EP :

!E(!r )= !E0 −∇φP(

!r )= !E0 +k1∇ ∇ i!P(!r ')!r − !r ´

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ , (2.4)

equationvalidinallspaceforallsortsofpolarizations.Thisequationwillbethestartingpointforthetreatmentbothofpermanentandinducedelectricpolarizations.When !

P isuniform—evenifit'safunction !!

P( E int ) oftheinternal

electricfield—thelastequationmayberewrittenas

43 M.C.Vanwormhoudt,Onthedefinitionofmacroscopicelectricandmagneticfields,Physicavol.

42,pp.439-446(1969).44 M.Widom,Shape-AdaptedEwaldSummation.Dekker,pp.141-144.Reitz,pp.81-83.Jackson

pp.115.45 MacMillan,pp.27-32.


!E(!r )= !E0 +k1∇ ∇ i

1!r − !r ´

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟!P( !E int )

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (2.5)

Fromeqs.3.3andA1.4,itfollowsthat

!E(!r )= !E0 − λ

ε0n(!r )i !P( !E int ) ∀!r , (2.6)

validbothinsideandoutsideV.Thedepolarizationtensornisdefinedbyeq.3.4,beinguniformonlyinsideellipsoidalbodies,aspointedoutinpage2.Inordertosolvethisequationitisnecessarytowriteseparateexpressionsfortheregionsinsideandoutsidethebody:

!E int =

!E0 − λ

ε0N i!P( !E int ) para !r ∈V ,

!E ext(!r )= !E0 − λ

ε0next(!r )i !P( !E int ) para !r ∉V ,

(2.7)

where !E int istheuniforminternalelectricfield, !!

E ext(r ) theexternalone,Nthe

internaldepolarizationtensor(uniformforellipsoids)and !!!next(r ) thenon-uniform

externalonewhichisalwaysafunctionofposition !r .TheeigenvaluesofmatrixN

arealwayspositive,makingtheinternalelectricfield(andthemagneticone,aswillbeseenlaterfortheferromagneticcase)smallerinmagnitudethantheappliedone,factthatgaveorigintothetensor’sname.Equations2.7arevalidbothforpermanentandinducedpolarizations,buteachcaseissolveddifferently.Whenanellipsoidalbodyisuniformlypolarized—statenotspontaneouslyacquiredforthepermanentcase—thefirstequationshowsthattheresultinginternalelectricfieldisalsouniform.

Potentialeq.2.3maybeexpressedintermsofpolarizationdensities.ThisrequiresadifferentuseofidentitiesA2.2andA2.5:

!P(!r ')i(!r − !r ´)!r − !r ´ 3

=!P(!r ')i∇' 1

!r − !r ´=∇'i

!P(!r ')!r − !r ´

⎛

⎝⎜

⎞

⎠⎟ −

∇'i !P(!r ')!r − !r ´

, (2.8)

where∇'operateson .Replacementoftheintegrandineq.2.3thengives

φP(!r )= k1 ∇'i

!P(!r ')!r− !r ´

⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ −k1

∇'i !P(!r ')!r − !r ´

d3r 'V∫∫∫ . (2.9)

Thefirstvolumeintegralappearswellsuitedfortheapplicationofdivergencetheoremeq.A3.8,andmosttextbooksapplyitwithoutfurtherado,replacingit

!!r '


withasurfaceintegral46.Infact,furtheranalysisisrequiredbecause !P hasastep

discontinuityonthebody'ssurfacebecauseisfiniteinsideandvanishesoutside.Thatis,thederivativesofvector !

P donotfulfillthestandardconditionsforthe

applicationofthetheorembecausetheymaybedifferentonbothsidesofthebody'ssurface47.Itturnsoutthatthisdoesnotaffectthevalueofthedivergenceinthevolumeintegralandofthefluxinthesurfaceintegral.TheproofissketchedinAppendix3(seethederivationofeq.A3.21).

Itisthusobtained

φP(!r )= k1

ρP(!r ')!r − !r ´

d3r 'V∫∫∫ +k1

σ P(!r ')d2r '!r − !r ´S

"∫∫ ,

where ρP(!r )= −∇ i

!P(!r ), σ P(

!r )= !P(!r )i s(!r ). (2.10)

where istheunitvectornormaltosurfaceSatpoint (seeeq.A7.8).Thesecharges,asrealastheconductionones,originateintherelativedisplacementofthebaricentersofnuclearandelectronicchargesofatomsandmolecules.Theexpressionwilllaterusedforsolvingthecaseofconductors.Asonlyuniformpolarizationswillbeconsideredinthisbook,ρPvanishesandtheonlycontributiontotheelectricfieldisthat

Figure1.Originofthedepolarizationeffect.

ofthesurfacedensityofchargeσP.Thisexplainsthefactthatinsidethebodythemagnitudeoftheelectricfieldissmallerthanintheoutside,asshowninFigure1.

Theelectricdisplacementvector !!D(r ) isintroducedinordertodiscriminatethe

polarizationcharges,boundtoatomsandmolecules,fromthefreeconductionones:

!D(!r )= ε0

!E(!r )+λ !P(!r ). (2.11)

Thisconstitutiveequationfortheelectriccasewillbeseldomusedhere,themagneticcasebeingquitedifferent.

Permanentelectricpolarization

Thenecessaryconditionforthepresenceofpermanentelectricpolarizationisforsomeatomsormoleculesofthematerial—ferroelectric,electretorthelikebelowtheircriticaltemperature—tohaveafiniteelectricdipolemomentintheabsenceofanappliedelectricfield( ).Thestateofpolarizationofthese

46 See,forinstance:Reitz,eq.4-12inp.78.47 J.ReyPastor,P.PiCalleja,andC.Trejo,AnálisisMatemáticovol.1,p.437(1959,4thedition).

!!s(r )

!r

!!E0 =0


materialsdependsonthebody'sinternalfield !!Eint(r ) ,butalsoonother

microscopicandmacroscopicphenomena.Thedominantonesamongtheformeraretheatomicandmolecularinteractions,whosemagnitudemayoutweighttheappliedfield.Amongthelatterthemostimportantonesaretheatomicandmolecularvibrations(temperature)andtheexistenceofdomains,localorderingofelectricmomentsthattendstominimizetheirinteractionenergy.Theinfluenceofdomainsisadifficulttopicbecausetheirsizeandstatisticaldistributionarecriticallydependentonthematerial'sthermalhistory,theirevolutionconsistingintransitionsbetweenmetaestablestates48.Thisleadstoirreversibility(hysteresis)andausuallynon-uniformpolarization.Areasonableapproximationtouniformpolarizationmaybeobtainedwitha

controlledprocessinvolvingacombinedvariationoftemperatureandtheapplicationofahighfield(saturation).Onceasufficientlyuniformpolarizationhasbeenobtained,onehastodetermineitsvalue.Themostcommonwayofdoingthisistosuspendthebodyinauniformfieldandmeasuretheappliedtorque.Therelationshipbetweenfieldandtorqueinvolvesthedepolarizationtensor(seethecorrespondingsectioninpage105).Inthisbooknofurtherdetailswillbegivenofthesetopics,whichare

thoroughlydiscussedinspecializedbooksandscientificjournals.Whatmattershereisthat—whateverthemethodused—thenecessaryconditionforauniformpermanentpolarization,althoughnotasufficienteone,isanellipsoidalshape.Edgesandcorners,forexample,areexamplesofshapesthatprecludeastateofuniformpolarizationeveninextremelyhighappliedfields.

Whenauniformpermanentpolarization !P isstablishedanditsvalueis

determined,thedepolarizationtensormethodcanbeusedtocalculateboththeinternalandexternalelectricfieldthrougheqs.2.7.Theseequations,rewritteninmatrixnotation,read

Eint = E0 − λε0N iP(Eint ) for !r ∈V ,

Eext(!r )= E0 − λε0next(!r )iP(Eint ) for !r ∉V .

(2.12)

Theexternalfield!!Eext cannotbedeterminedwithoutsolvingfirsttheequationforforEint andthenreplacingitsvalueinthesecondofeqs.2.12.Thisrequirestheknowledgeofthefunction !!

P( E int ) ,whichintheisotropiccaseisusuallytakentobe

Langevin'sfunction49.Problem06ofchapter5givesatentativegraphicalmethodforthisdetermination.Inanisotropicmaterialsthesolutionrequirestheseparate

48 Brown(1963).49 Dekker,pp.138and192.


determinationofanumberoffunctionsdeterminedbythesinglecrystalsymmetries(seeAppendix5).

InducedelectricpolarizationIntheabsenceofpermanentpolarizationtheinducedpolarizationisusually

onlyafunctionoftheappliedfield !E0 , theelectricsusceptibility!χe ofthe

magneticmaterial50andthebody'sshape.Thisissobecausethestateofpolarizationofanypointofthebodydependsontheinternalelectricfield,theadditionoftheappliedfieldandthatgeneratedbytherestofthebody.Theinducedfieldshould,therefore,bedeterminedinaself-consistentway,itsvalueateachpointdependingonthevalueatallotherpoints.Themathematicalexpressionofthisdependenceisclearlyexpressedbytheintegraleq.2.31.

Theappliedfield,thepolarizationandthetotalelectricfieldareingeneralnotparallelduetoacombinationoftheanisotropyofthesusceptibility(amaterialproperty)andtheshapeanisotropy,ageometricpropertycharacterizedbythedepolarizationtensor,linearonlyinthecaseofellipsoidalbodies.Thatis,thelinearityoftherelationshipbetweenthepolarizationandtheappliedfieldisnotamereconsequenceofthelinearityofthefieldequations,asinvokedbyLandauandLifschitztojustifytheintroductionofthebody'spolarizabilitytensor.Theyclaimtherethat51Sinceallthefieldequationsarelinear,itisevident…Infact,thereisacomplexrelationshipamongtheappliedfieldcomponents,theensuingpolarizationandtheresultingtotalfield,mediatedbythematerial'spropertiesandthebody'sgeometry.Inthetreatmentmadebystandardtextsonthetheoryofelectromagnetismtheserelationshipsarehiddenintheboundaryconditionsofdifferentialequations.Inthemethoddescribedbythisbookboththeanisotropyofthematerialandtheinfluenceofgeometryareexplicitlyexpressedbythesusceptibilityanddepolarizationtensors,whiletheinterrelationshipamongfieldsisexpressedintermsoflinearsystemsofequationsamongcomponents,clearlyvisibleandsolubleinmatrixrepresentation.Abetterunderstandingofthisabstractbehaviourmaybeobtainedfromaconcreteillustration.Twospecificexamplesaregiventothatend,thefirstoneinFigure2,thesecondinthefollowingsection.Figure2isastoryboardofthetimedsequenceofinteractionsoftwopolarizable

atomssubject,invacuum,toanappliedfield.Thesameproblemisnextmathematicallysolvedusingmatrices,inafashioncloselyresemblingthedepolarizationtensormethod,towhichitisanintroduction.

Thefigurereliesontheexperimentalfactthatfieldsdonotactinstantaneously,butpropagatewiththevelocitycofelectromagneticwaves.Theappliedfieldarrivestoatom1attimet1,inducingadipolemomentwhichacquiresitsfullvalueinalatertimet2,whenatom2isstillunpolarized.Attimet3theappliedfieldarrivestoatom2,butnotsotheonecreatedbythedipolemomentinducedonatom1.Molecule2acquiresitsinitialdipolemomentattimet4,butthisvaluewillchangewhentheinducedfieldofatom1reachesitslocation.Shortlyafterthat,at

50 Rememberthatthisisalowfieldapproximation,themostcommonbutnotuniversalcase.51 LandauandLifchitz,p.7.


timet5,themodifieddipolemomentofatom2isfullystablished.Buttheprocessdoesnotendhere,becauseatom2createsanewinducedfieldthatactsonatom1,modifyingagainitsdipolemomentattimet6.Thesuccesivechangesofpolarizationsandfieldstakeacertaintimetoreachavaluewithinasmallbutnon-zeroprecissionrange.Itisthusclearlyseenhowthevaluesobtainedarenecessarilyself-consistent,becausethedipolemomentofeachatomdependsonthatoftheotherthroughtheinducedfieldscreatedbyboth.

Figure2.Autoconsistentmutualpolarizationoftwoatoms.

InducedelectricpolarizationoftwointeractingatomsInordertofindtheequilibriumvaluesoftheinducedelectricdipolemoments

andtotalfielditwillbeassumedthatbothatomsareidentical,isotropicandhavenopermanentelectricmoments.Thishappens,forinstance,forclosedelectronicshells.Theatoms,adistancedapart,areimmersedinauniformappliedelectricfieldE0andtheinducedfieldswillbecalculatedusingthepointdipolemicroscopicmodel.

UndertheinfluenceofanapplieduniformfieldE0,electricdipolemomentsp(1)andp(2)areinducedontheatomsatr(1)andr(2.Inthelinearrange,foranisotropicatomjitsinduceddipolemomentisrelatedtothemicroscopicelectricfieldexperiencedby p(j)=γE(r(j)), (2.13)

whereconstantγisthesameforbothatoms.FieldE(r(1))istheadditionoftheappliedfieldE0andthefieldE(2)(r(1))generatedatr(1)bythedipolemomentoftheatomatr(2).Intherestofthissectionargumentsoffunctionsarealwaysfield


pointsandupperindicesidentifysourcepoints.Usingeq.A4.5andthefollowingpropertyoftheinteratomicdistance52

(2.14)

itturnsoutthatamatrixmmaybedefinedforbothatoms,intermsofvector !d ,

suchthat

E(1)(r(2))= k1m ip(1) , E(2)(r(1))= k1m ip(2). (2.15)

Fromeq.A4.5itisobtanined

m = 1d5

3dx2 −d2 3dxdy 3dxdz3dydx 3dy2 −d2 3dydz3dzdx 3dzdy 3dz2 −d2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

,

where d =!d = dx

2 +dy2 +d2z ,

(2.16)

!dα beingthecomponentsofvector !!d.

Fromeq.2.13,itfollowsthatthefieldsanddipolemomentsaredeterminedbythefollowingsystemofequations:

(2.17)

whereitisclearlyseenthatthefieldonatom1(upperequation)explicitlydependsofthefieldonatom2.Asthefieldonatom2explicitlydependsofthefieldonatom2(lowerequation),thisself-consistencyisexpressedbythefollowingpairofmatrixequations(thatis,asystemof6linearequations):

(2.18)

Asthetwoatomsareidentical,uponitsexchangethesystemremainsunchanged.Thatis,E(r(1))=E(r(2)).Thissymmetryargumentmaybeverifiedbysubstractingthesecondequationfromthefirstobtaining

(2.19)

where1istheunitmatrix.Thatis

52 Thenotation,whichappearstobeexcessivelydetailed,iswellsuitedforthecalculationof

latticesums,themicroscopicfieldscreatedinsinglecrystalsbyorderedatomsandmolecules.

!!d (1)(r (2))= r (2) − r (1) =

d = −

d (2)(r (1))= −(r (1) − r (2)),

!!!

E(r(1))= E0 +E(2)(r(1))= E0 +k1m ip(2) = E0 +k1m iγ E(r(2)),E(r(2))= E0 +E(1)(r(2))= E0 +k1m ip(1) = E0 +k1m iγ E(r(1)),

!!!E(r(1))= E0 +k1γm iE(r(2)), E(r(2))= E0 +k1γm iE(r(1)).

!!!E(r(1))−E(r(2))=1i E(r(1))−E(r(2))( ) = k1γm i E(r(2))−E(r(1))( ) ,


(2.20)

Astheleftmember'ssquarematrixisnon-singular(seebelow),thecolumnmatrixmustvanish,verifyingthat

(2.21)

Therefore,theoriginaltwoequationssystemreducestothesinglematrixequation(noticethesimilaritywiththesecondofeqs.2.7)

(2.22)

fromwhichEmaybeobtained,asfollows:

(2.23)

Matrix1-k1γ minverseiseasilyobtainedinthecoordinatesystemwhereitisdiagonal(itsprincipalone),usuallycorrespondingtotheatomicsystemhighestsymmetry.Inthiscasethehighestsymmetryisanarbitraryrotationaroundtheatomicaxisdefinedbyvector .Infact,ifonechoosesacoordinatesystemsuchthatbothatomsareonaxisz,andtheoriginisarbitrarilychosentobeonatom1,itisobtained

dx = dy =0, dz = d , m =−1/d3 0 00 −1/d3 00 0 2/d3

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟. (2.24)

Then

E =ExE y

Ez

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= 1−k1γm( )−1 iE0

=

1/1+k1γ /d3 0 00 1/1+k1γ /d3 00 0 1/1−2k1γ /d3

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

Ex0

Ey0

Ez0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

(2.25)

Choosingtheproportionalityconstantinsuchawaythatbothmembershavethesameunits(seeeq.A1.3),onemaydefinetheatomicelectricpolarizabilitytensor!αe by

p= γ E =

ε0λαe iE

0 , where αe =λε0γ 1−k1γm( )−1 . (2.26)

!!! 1−k1γm( )i E(r(1))−E(r(2))( ) =0.

!!E(r(1))= E(r(2))= E.

!!!E = E0 +k1γm iE,

!!! 1−k1γm( )iE = E0 , E = 1−k1γm( )−1 iE0.

!d


ThematrixCrelatingtheresultanttotheappliedfield,E=C·E0,hasthetypicalformofthetensorialpropertiesofuniaxialmaterials53,

C =

c⊥ 0 00 c⊥ 00 0 c

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

where c⊥ < c! ,

(2.27)

inthesinglecrystal'sprincipalcoordinatesystem.Asimpleestimationillustratesthatthedifferenceinvalueofthesetwocoefficientsiseasilymeasurable.k1γisoforderR3,Rbeingtheatomicradius54.Insolidstheinteratomicdistancedisororder2R.Therefore,k1γ/d3isabout0,1whichgives

(2.28)

theparallelcoefficientbeingabout37%largerthantheoneperpendiculartotheinteratomicaxis.

Thepreviouscalculationcorrespondstothelocalfieldofsolidstatephysics,themicroscopicelectricfieldexperiencedbyanatom.ThisfielddoesnotcoincidewiththemacroscopiconedescribedbyMaxwell'sequations,asuitableaverageoftheformeroritspotentialoverasinglecrystal'sunitcellorasufficientlylargenumberofatomsormoleculesinthepolicrystallineoramorphouscase.Animportantfeatureofmacroscopicfields,asdiscussedinpage6,isthatanyofthebody'sinternalpointsisbothafieldandasourcepoint.Thesingularitiesthuscreateddidnotariseintheprevioustreatmentbecausepointdipoleswhereused.Aconsequenceofthisapproximationisthatmatrixmcorrespondstotheexteriordepolarizationtensornext,asshownbyitszerotrace(seeeq.3.15).ThissuggeststhatitmaynotbefeasibletoobtaintheinternaldepolarizationtensorNasanaverageofmicroscopictensorslikem,unlesssignificativemathematicalandconceptualchangesaremadeinthetreatment.Ontheotherhand,atomsarenotgeometricalpoints,butnoconceptrelatedtothedepolarizationtensorseemstohavebeenintroducedforits(necessarily)quantumanalysis.

DielectricsAsinthecaseofpermanentpolarization,thestartingpointiseq.2.12.Inmost

dielectricsappliedfieldsaresignificantlysmallerthantheeffectiveelectricfieldsofquantuminteractionsamongelectrons.ItisthenvalidtokeeponlyfirstordertermsintheTaylor'sexpansionof :

!P(!r )= ε0

λχ e i!E(!r ). (2.29)

53 Nye,p.23Table3,seeAppendix5inthepresentbook.54 Dekker,p.135.C.Kittel,IntroductiontoSolidStatePhysics,WileyandSons,p.229,1979.

!!k1γd3

≈0,10, c⊥ ≈0,91, c ≈1,25,

!!P( E(r ))


Thereisnogeneralagreementontheconstantoneshoulduseforthedefinitionofelectricsusceptibility.TheadimensionaldefinitiongivenaboveandinthebookbyStratton55simplifiesourmainformulas,whileReitzconventionintroducesnofactorineq.2.2956.Thereadershouldcarefullyverifythedefinitionusedbyhisfavouriteauthor.Withtheaboveconventionthefollowingvalueofelectricpermeabilityε isobtained,

!D= ε0

!E +λ

!P = ε0

!E + ε0χ e i

!E = ε i

!E , where ε = ε0(1+ χ e ). (2.30)

Inhomogeneousmaterials,theonlyonesstudiedinthisbook,theadimensionalelectricsusceptibility isauniformandsymmetricrank2tensor57whichreducestoascalarinthecasesofnontexturedpolicrystallineandamorphousmaterials.Intheanisotropiccasesomeofitscomponentsmayberelated,dependingonthesymmetriesofthematerial'scrystallinestructure58.Fromeq.2.5,thetotalelectricfieldduetotheapplicationofauniformfieldtoa

dielectricwiththelinearpropertyeq.2.29satisfiestheequation

!E(!r )= !E0 + 1

4π ∇ ∇ iχ e i!E(!r ')!r− !r '

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ . (2.31)

Thissystemofintegro-diferentialequationsunivocallydeterminestheelectricfield59and,througheq.2.29,thepolarization.Inthepermanentpolarizationcasetheinternalfield'suniformitywasaconsequenceofadeliberatelystablisheduniformpolarization.Here,onthecontrary,theuniformcharacterofthefieldisaconsequenceofeq.2.31,fromwhichfollowsthatofthepolarization.

Auniforminternalelectricfield !!E int providesasolutiontoeqs.2.31ifthe

followingequationsaresatisfied(whichareeasilyobtainedinthesamewayaseqs.2.7):

!E int =

!E0 −N i χ e i

!E int , !P =

ε0λχ e i!E int for r∈V ,

!E ext(!r )= !E0 − λ

ε0next(!r )i !P for r∉V .

(2.32)

InmatrixrepresentationitiseasytoobtainEintfromthefirstequationfromwhichthevaluesofPandEextimmediatelyfollow:

55 Stratton,p.12eq.8.56 Reitz,p.86.57 LandauandLifchitz,p.58.58 Nye,p.23Table3:seeAppendix5ofthepresentbook.59 SeetheinitialdiscussioninSolvingtheintegro-differentialequationsbyiteration.

!χ e


Eint = 1+N i χe( )−1i E0 for r∈V ,

P=ε0λχ e i E

int =ε0λχ e i 1+N i χe( )−1i E0 ,

p=VP=ε0λαe i E

0 , αe =V χ e i 1+N i χe( )−1 ,Eext(!r )= E0 − λ

ε0next(!r )iP for r∉V ,

(2.33)

whereA-1istheinverseofmatrixA.TherelationshipbetweenthegeometryofellipsoidalbodiesandelectricpolarizationPwasstablishedbyLandauandLifschitzfortheisotropiccase60throughageneralization,whichisnotaproof,oftheresultsobtainedforthesphereandtheinfinitelylongcylinder:

Theexistenceofsucharelationshipfollowsfromtheformoftheboundaryconditions,aswesawaboveontheexamplesofthesphereandthecylinder.

Thegeometricconnection,previouslyextendedbythisauthortotheanisotropicelectriccase61,isbutasimpleconsequenceofthemethodthatallowstheintroductionof!αe , thebody’selectricpolarizabilitytensorwithunitsofvolume:

p=VP=

ε0λαe iE

0 , αe =Vχe i 1+N i χe( )−1 . (2.34)

ThistensorwasdefinedbyLandauandLifschitz62onlyforconductorsandmagneticmaterials.Noticethenon-casualsimilarityoftheformulawitheq.2.26.ThistensorwillbeusedinChapter5forthecalculationofthebody'senergyandappliedtorques.

Magnetization

Thebehaviourofellipsoidalbodiesunderapplieduniformmagneticfields !!H0—

thefirstworkpublishedbytheauthoronthedepolarizationtensormethod63—maybesolvedinasimilarwayastheelectriccase.

Basicequations

Unliketheelectriccase—wherethetreatmentofpolarizedmatterisdoneusingthefundamentalfield !

E—inthepermanentandinducedmagneticcasesuseis

60 LandauandLifchitz,p.44.61 Solivérez(2008),eq.17.62 LandauandLifchitz,pp.7and192.63 Solivérez(1981).


madeoftheauxiliarfield !!H. Themagneticinduction !

B willbeusedonlyinthe

caseofsuperconductors,whereusemustbemadeoftheconstitutiveeq.A1.10,

!B = µ0(

!H +λ ' !M). (2.35)

Thereasonforusingthemagneticfieldinsteadofthemagneticinductionisthatallequationscanthenbederivedasfortheelectriccasebecauseoftheexistenceofthefollowingmagneticpotential!φM

64:

!HM(!r )= −∇φM(

!r ), where φM(!r )= λ '

4π

!M(!r ')i(!r − !r ')!r− !r ' 3

d3r 'V∫∫∫ . (2.36)

Besides, !H playsadistinguishedroleinthestudyofpermanentemagnets.The

givenequationissimilartoeq.2.1oftheelectrostaticcase.Inthesamefashionasforthederivationofeq.2.5,oneobtains,uponadditionofanapplieduniformfield,thatthetotalmagneticfieldis

!H(!r )= !H0 + λ '

4π ∇ ∇ i1!r − !r ´

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟!M

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (2.37)

Thisequation,thebasisoftheanalysisofpermanentandinducedpolarizationsofellipsoidalbodies,isvalidinallspaceindependentlyofthesourceandspatialdependenceof !!

M(r ). When !

M isuniform,caseinwhichnoargumentiswritten,

thepreviousequationbecomes

!H(!r )= !H0 + λ '

4π ∇ ∇ i1!r − !r ´

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟!M

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (2.38)

Fromeq.3.3itfollowsthat

!H(!r )= !H0 −λ 'n(!r )i !M( !H) ∀!r . (2.39)

Aspreviouslydonefortheelectriccase,itismoreconvenienttorewritethisequationinordertoexplicitlyseparatethebody'sinternalandexternalregions:

!H int =

!H0 −λ 'N i

!M( !H int ) for !r ∈V ,

!Hext(!r )= !H0 −λ 'next(!r )i !M( !H int ) for !r ∉V .

(2.40)

Theseequationsarevalidbothforpermanentandinducedpolarizations,buttheyaresolveddifferentlyineachcase.Therefore,ifanellipsoidalbodyisuniformly

64 Adetailedreadingofpp.189-194ofReitz'sbookisrecommended.


magnetized(whichrequiresanontrivialpreparation),theresultinginternalmagneticfieldisalsouniform.

Forthetreatmentofsuperconductorsitisconvenienttoderivethemagneticfieldfromthesurfacedensityofcurrent,casethatbearsnoresemblancetothatofasurfacedensityofcharge(conductors).Tothatenditisconvenienttousethemagneticinduction,derivablefromthefollowingvectorpotential65:

!AM(!r )= µ0λ '

4π

!M(!r ')×(!r − !r ')!r− !r ' 3

d3r 'V∫∫∫ , !B(!r )=∇×

!AM(!r ), (2.41)

Theintegrandmaynowbetransformedusingeqs.A2.2andA2.8,obtaining

!AM(!r )= µ0λ '

4π

!M(!r ')×(!r − !r ')!r− !r ' 3

d3r 'V∫∫∫ =

µ0λ '4π

!M(!r ')×∇' 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫

=µ0λ '4π

∇'× !M(!r ')!r− !r '

d3r 'V∫∫∫ −

µ0λ '4π ∇'×

!M(!r ')!r− !r '

⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ .

(2.42)

Thelastvolumeintegralmaybetransformedintoasurfaceonebythecurltheoremeq.A3.10,wheredueatentionshouldbepaidtothestepdiscontinuitythroughthebody'ssurface(seeeq.A3.22).Itisthusobtained

!AM(!r )= µ0λ '

4π

!JM(!r ')

!r− !r 'd3r '

V∫∫∫ +

µ0λ '4π

!KM(!r )

!r− !r 'd2r '

S"∫∫ ,

where !JM(!r )=∇×

!M(!r ), !KM(

!r )= !M(!r )× s(!r ), (2.43)

where !!s(r ) istheunitvectornormaltosurfaceSatpoint !r (seeeq.A7.8), !

JM is

thevolumedensityofmagnetizationcurrentand !KM thesurfacedensity.

Thepreviousequationsshowhowmagnetizationmayoriginatefromcurrentslocalizedintheinteriorandonthesurfaceofthebody.Thevalidityofthisinterpretationforsuperconductorsisdiscussedintherelevantsectionandwillbealsousedtoshowthattheybehaveasperfectdiamagneticmaterials.

Permanentmagnetization

Thestudyofspontaneousmagnetizationandthephenomenaofsaturationandhysteresisprecededthatofferroelectricmaterials,whichtooktheirnamefromferromagnets.Asdiscussedintheferroelectriccase,amagnetizationisingeneralnotonlyafunctionoftheappliedmagneticfield,thematerialanditsshape,butalsoofitsownbuildupprocess.

65 Reitz,p.190eq.9-11.


Theresultingequations2.40aresimilartothoseobtainedinpage18forpermanentelectricpolarization.Writingtheminmatrixrepresentationitisobtained

Hint =H0 −λ 'N iM(Hint ) for !r ∈V ,Hext(!r )=H0 −λ 'next(!r )iM(Hint ) for !r ∉V . (2.44)

Thescalarandisotropicversionofthefirstequation,withM(H)proportionaltoBrillouin'sfunction66,iscustomarilyusedinthestudyofferromagnetism67.Theappliedfield,themagnetizationandthetotalfieldhaveingeneraldifferentdirections.Thesamethinghappensinisotropicmaterialsforallbutsphericalbodies,duetothegeometricalanisotropyintroducedbythedepolarizationtensorn(seeProblem06).

Inducedmagnetization

Unliketheelectriccase,inthemagneticonetherearetwodifferenttypesofinducedpolarization:diamagneticandparamagnetic.DiamagnetismisamanifestionofLenz'Law,achangeintheorbitalcirculationofelectronsthat,ashappenswithanycurrent,generatesafieldthatopposestheappliedone.Paramagnetismstemsfromtheorientationofthepermanentmagneticmomentsofatomswithoutclosedelectronicshells.Bothtypesarewelldescribedinthelinearrangeby

!M(!r )= 1

λ ' χm i!H(!r ), (2.45)

where!χm isthemagneticsusceptibility,asymmetricranktwotensor68.Asintheelectriccase,thereisnogeneralagreementonthedefinitionofmagneticsusceptibility,andthereadershouldchecktheoneusedbyhisfavouriteauthor,introducingthenecessaryadditionalconstants.Thedefinitiongivenbyeq.2.45simplifiestheformulaseqs.2.49.Themagneticsusceptibilityisanegativedefinitetensorinthediamagneticcaseandaositivedefiniteoneintheparamagneticcase,uniformforallthehomogeneousmaterialsstudiedinthisbookandascalarforpolicrystallinenon-texturedandamorphousmaterials.Inthesinglecrystalanisotropiccaseitscomponentshaverelationshipsdeterminedbythesymmetryofthematerial'slatticestructure69.

66 Dekker,pp.454-455.67 Dekker,p.466.68 LandauandLifchitz,p.58.69 Nye,p.23Table3.SeeAppendix5inthepresentbook.


Forinducedmagnetizationseq.2.37reducesto

!H(!r )= !H0 + 1

4π ∇ ∇ iχm i!H(!r ')!r− !r '

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ , (2.46)

systemofintegro-differentialequationsthatunivocallydeterminesthemagneticfield.When !!

H int isuniforminsidethebody—caseinwhichthespatialargumentis

omitted—thepreviousequationreducesto

!H int =

!H0 + 1

4π ∇ ∇ iχm i!H int

!r− !r 'd3r '

V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ . (2.47)

Fromeq.3.3itisthenobtained:

!H int =

!H0 −N i χm i

!H int , !H int = 1+N i χm( )−1 i

!H0 for !r ∈V ,

!Hext(!r )= !H0 −λ 'n(!r )i !M for !r ∉V ,!M = 1

λ ' χm i!H int = 1

λ ' χm i 1+N i χm( )−1 i!H0 ,

!m=V!M = 1

λ 'αm i!H0 , αm =V χm i 1+N i χm( )−1 .

(2.48)

!m isthemagneticdipolemomentand!αm istheellipsoidalbody'smagneticpolarizabilitytensor.

Inmatrixrepresentationthepreviousequationsarewritten

Hint = 1+N i χm( )−1 iH0 for !r ∈V ,Hext(!r )=H0 −λ 'n(!r )iM for !r ∉V ,

where M= 1λ 'χm i 1+N i χm( )−1 iH0 ,

m =VM= 1λ 'αm iH0 , αm =Vχm i 1+N i χm( )−1 .

(2.49)

ConductorsIfaconductingbodywithzeronetchargeisplacedinanapplieduniformfield,asurfacedensityofchargeσisinducedsuchthat,intheequilibriumstate,the

internalfieldvanishes.Thecancellationisnotinstantaneousbuttheculminationofadynamicprocessinwhichthebody'ssurfaceelectricchargesareredistributed.Electricforcesalonedonotsuffice,asshownbyEarnshaw'sTheorem70:electronsmoveinordertobeasfarawayaspossiblefromeachother,beingstoppedonlyby

70 Stratton,p.116.Thetheoremstablishesthatnochargemaybeinstableequilibriumunderthe

effectofonlyamacroscopicelectricfield.

!!E0


theatomicforcesthatpreventthemfromescapingfromthebodyatnormaltemperatures(athightemperaturesthermoionicemission—EdisonEffect—takesplace71).Thevanishingoftheinternalelectricfield—validforconductingbodiesofanyshape,notonlyellipsoids—is,therefore,anadditionalconditiontothelawsofelectrostatics,notaconsequenceofthem:

!E int =0. (2.50)

Thetotalelectricfieldmaybewrittenintermsoftheappliedoneandtheresulting—butyetunknown—surfacedensityofchargeσas72

!E(!r )= !E0 −k1∇

σ (!r ')!r − !r'

d2r'S"∫∫ , (2.51)

where !!σ (r ) and !!

E(r ) aremutuallydetermined.Thisequationmaybewritten

solelyintermsoftheelectricfieldbyusingthestepdiscontinuitythroughthebody'ssurfaceeq.3.35(readthesectionSurfacestepdiscontinuity):

k1σ (!r )= 1

4π!E + −

!E −( )i s(!r ), where !

E − =0, (2.52)

and !!s(r ) istheunitvectornormaltotheellipsoid'ssurfaceatpoint !r (seeeq.

A7.8).Therefore

!E(!r )= !E0 − 1

4π ∇!E +(!r ')id2!r '!r− !r 'S

"∫∫ , (2.53)

asystemofintegro-differentialequationsthatfullydeterminestheelectricfield.Theuseofthenotation !

E + stressesthestepdiscontinuitythroughthebody's

surface,crucialforthecorrectapplicationoftheintegraltheoremsofvectorcalculus.

Insidethebody !E(!r )=0,equationthatcanbesatisfiedonlyifthesecondterm

ofthefirstmemberisaconstant.Thisrequirementissatisfiedif !E + isuniform,

caseinwhicheq.3.29gives

14π ∇

!E + id2!r '!r− !r 'S

"∫∫ =N i!E + =

!E0 , (2.54)

whereNisthebody'sinternaldepolarizationtensor.Thisequationmaybeusedtoobtain !

E +wheneverNhasaninverse,whichistrueforthecaseofnonvanishing

71 Dekker,p.220.72 Reitz,p.31.


eigenvalues.Ifoneortwooftheeigenvaluesarezero,areducedinversemaybedefinedwhichsolvestheproblem,thelinksbeinggivenatpage60.

Theexternalfieldisthenobtainedasfollows:

!E int =

!E0 −N i

!E + =0, !E + =N −1 i

!E0 for !r ∈V ,

!E ext(!r )= !E0 −next(!r )i !E + for !r ∉V .

(2.55)

Theelectricdipolemoment !p oftheellipsoidalconductoris

!p = σ (!r ') !r 'd2r '

S"∫∫ , (2.56)

where,fromeq.2.52,thesurfacechargedensityis

σ (!r )=

ε0λs(!r )i !E + =

ε0λs(!r )iN −1 i

!E0. (2.57)

and !!s(r ) istheoutgoingunitvectornormaltothebody'ssurfaceatpoint !!r .

Therefore

!p =

ε0λ

!r 's(!r ')d2r 'S"∫∫

⎛

⎝⎜⎞

⎠⎟iN −1 i

!E0 =

ε0λ

!r 'd2!r 'S"∫∫

⎛

⎝⎜⎞

⎠⎟iN −1 i

!E0. (2.58)

Theintegralinsidetheparentesismaybeevaluatedbycomponentsusingthegradienttheoremeq.A3.9,asfollows:

xα xαα∑⎛⎝⎜

⎞⎠⎟d2!r

S"∫∫ = xα xαd

2!rS"∫∫

⎛

⎝⎜⎞

⎠⎟α∑ =

xα ∇xα( )d3rV∫∫∫

⎛

⎝⎜⎞

⎠⎟α∑ =V xα xα

α∑ =V 1.

(2.59)

where!!1istheunitdyadic,theonethatleavesunchangedanyvectorordyadic.Therefore

!p =

ε0λα e i!E0 , where α e =VN

−1 , (2.60)

!α c beingthepolarizabilitytensoroftheconductingbody.

Althoughitwillnotbeusedinthisbook(butseeproblemProblem31),itshouldbenoticedthatthesurfacechargedistribution !!σ (

r ) ofaconductingellipsoidwithnetchargeQmaybeexpressed—intheabsenceofappliedfield—intermsofthecentraldistanceofthatpoint(seep.172)asfollows73:

73 Stratton,p.209eq.12.


σ (!r )= d(!r )

4π abcQ, (2.61)

expressionthatshouldbecomparedwiththedensityinducedbyauniformfield,eq.3.32.Fromthereitfollowsthat

d(!r )d2r

S"∫∫ = 4π abc. (2.62)

Inmatrixrepresentationonegets

Eint= E0 −N iE+ =0, E+ =N−1 iE0 for !r ∈V ,Eext(!r )= E0 −next(!r )iE+ for !r ∉V

p=ε0λ

αc iE0 , where αc =VN

−1. (2.63)

Thevalueofthedipolemomentisthesameobtainedwiththepolarizationmodeleq.6.29.Exceptforthevanishinginternalfield,theequationsforaconductingellipsoid

aresimilartothoseofadielectric,wherevector correspondstotheuniformpolarization !

P (seeeq.2.10).Thissuggeststhatthetreatmentforbothtypesof

materialesmaybeunifiedbytheintroductionofasuitableequivalentepolarizationforconductors,atopicdiscussednext.

Equivalentpolarization

Thebehaviourofelectronsinconductorsmaybeconsideredasanextremecaseofpolarization.Indielectricsthedisplacementofelectronsundertheeffectofanappliedfieldisstrictlylimitedbythequantumenergiesbindingthemtothenuclei,whichrendersitmicroscopic,afractionoftheatomicormoleculardiameter74.Inconductors,asexplainedbybandtheory75,electronscaneasilymigratefromoneatomtoanother.Thismaybeinterpretedasamacroscopicpolarization,thatofamaterialwithinfiniteelectricsusceptibility,aswillbeshownbelow.

Asfollowsfromequations2.10,intheregionwhereapolarizationisuniformthevolumedensityofchargevanishesanditsonlyeffectcomesfromthestepdiscontinuitythroughthebody'ssurface.Onemay,then,definethefollowingelectricpotentialgeneratedbyanequivalentpolarization

!!Peq 76:

74 Dekker,p.135.75 Dekker,chapter10pp.238-274.76 Solivérez(2008),p.207.

!E +


φeq(!r )= k1

σ eq(!r ')d2r '!r − !r ´

⎛

⎝⎜

⎞

⎠⎟ = k1

!Peq(!r ')id2!r '!r − !r ´

⎛

⎝⎜

⎞

⎠⎟

S"∫∫ .

S"∫∫ (2.64)

Thispotentialgivesrisetotheelectricfield

!E(!r )= !E0 −k1∇

!Peqid

2!r '!r− !r 'S

"∫∫ , (2.65)

whichcoincideswitheq.2.53if

!E + = 4πk1

!Peq =

λε0

!Peq , (2.66)

whereeq.A1.4wasused.Thecaseofconductorsmaynowbesolvedinthesamefashionasthatofdielectricsif

!!Peq isgivenavaluesuchthattheinternalfield

vanishes.Thisvalue,asshowninProblem08atpage127,correspondstothepolarizationinducedinamaterialwithinfiniteelectricsusceptibility.

Superconductors

Ifamagneticinduction !!B0isappliedtoasuperconductingbody—inan

appropriaterangeoftemperatureandfieldintensity—surfacecurrentsaregeneratedsuchthattheresultinginternalvalueof !

B vanishes.ThisMeissnereffect

andthecancellationofthematerial'sresistivityarethemostnotoriousconsequencesofacomplexmicroscopicphenomenaexplainedbytheBCStheory(initialsofBardeen,CooperandSchrieffer)andphenomenologicallydescribedbytheLondonmodel.Thelattermakesuseofcurrentsconfinedwithinafinitebutverysmalldistanceofthebody'ssurface77.Fromthemacroscopicpointofviewthesecurrentsmaybecharacterizedaszerothicknesssurfacecurrents.Thedepolarizationtensormethodmaythenbeusedtomacroscopicallycharacterizesurfacecurrentsinsuperconductingellipsoidalbodiesinauniformmagneticinduction

!B 78.Asinthecaseofconductors,anadditionalcondition,theMeissner

effect,isrequired:

!B(!r ) = µ0

!H(!r )+λ ' !M(!r )( ) =0 for !r ∈V . (2.67)

Thiscondition—themagneticanalogousoftheinfiniteelectricsusceptibilityofconductors—maybeinterpretedasaperfectdiamagnetismthatcancelsanyinternalmagneticfield !!

H int . Alternately,onemayassumethat !

H and !

M vanish

77 Reitz,pp.325-328.78 Reitz,p.319.


insidethebodyand—theresistivitybeingzero—thattheMeissnereffectistheconsequenceofaperdurablesurfaceconductioncurrent.BothmodelsareusedtoanalysetheMeissnereffect.Althoughtheyaremutuallyincompatible,thecoherentuseofanyofthemfullydescribestheeffectfromtheelectromagneticpointofview79.Thedepolarizationtensormethodwill,therefore,beappliedtothesetwomodels,showingthatbothleadtoexactlythesameresults.

MagnetizationmodelEquations2.49describingthebehaviourofmagnetizableellipsoidalbodiesina

uniformfieldfield !!B0 mayberewrittenasfollows:

!B int = µ0

!H int +λ ' !M( ) , !H int =

!B0

µ0−λ 'N i

!M , for !r ∈V ,

!Bext(!r )= !B0 − µ0λ 'next(

!r )i !M for !r ∉V . (2.68)

Thevalueof !M maybeobtainedfromcondition2.67,

!B int = µ0

!H int +λ ' !M( ) =0, !M = − 1

λ '!H int , (2.69)

anequivalentemagneticpolarizationthatcorrespondstoaperfectdiamagneticsusceptibility,asdiscussedinProblem15.Replacingthisvalueinthesecondofequs.2.66,itisobtained

!H int =

!B0

µ0−λ 'N i

!M = −λ ' !M , !B0 = µ0 λ 'N i

!M −λ ' !M( ) = µ0λ ' N −1( )i !M. (2.70)

Rewritingthelastmembersothatthetensorispositivedefinite(hasaninverse)

!M = − 1

µ0λ '1−N( )−1 i

!B0. (2.71)

InthisinterpretationtheMeissnereffectistheconsequenceofamagnetizationthatcancelstheinternalmagneticfield !!

H int .Thismagnetizationgivesorigintoa

magneticmoment !m andanexternalmagneticinduction,bothfunctionsofthe

appliedfield:

!m=V ⋅!M = − V

µ0λ '1−N( )−1 i

!B0 = 1

µ0λ 'α s i!B0 , α s = −V 1−N( )−1

!Bext(!r )= !B0 − µ0λ 'n(

!r )i !M =!B0 +n(!r )i 1−N( )−1 i

!B0.

(2.72)

where!α s isthebody’spolarizabilitytensoroftheellipsoidalsuperconductor.

79 Reitz,chapter15pp.318-333.


Inmatrixnotation

Bint = µ0 Hint +λ 'M( ) =0, M= − 1

µ0λ '1−N( )−1 iB0 , for !r ∉V

Bext(!r )=B0 − µ0λ 'next(!r )iM for !r ∉V ,

where m =V ⋅M= 1µ0λ '

α s iB0 , α s = −V 1−N( )−1 .

(2.73)

Thesameresultmaybederivedfromeq.2.43,themagnetizationsurfacecurrentdiscussedinProblem17.InProblem14ourequationsarecomparedwiththosegivenbyReitz(eq.15-6)forthecaseofasuperconductingsphere.

SurfaceconductioncurrentmodelInthismodelofsuperconductivitytheMeissnereffectisderivedfromasurface

conductioncurrent.Thedifferencebetweenaconductioncurrent—thatofelectriccircuits—fromamagnetizationcurrentisthatthelatterisboundtosingleatomsormolecules,sothatnoelectronsaretransferredbetweenthem80.Magnetizationcurrentsareasrealasconductioncurrents,butonlytheformerarefeasibleininsulatingmaterials.

Thetotalmagneticinductionistheadditionoftheappliedone, !!B0 , andthat

generatedbyasurfaceconductioncurrent !K asgivenbyeq.2.43:

!B(!r )= !B0 +

!BK (!r ), !BK (

!r )=∇×!A(!r ),

!A(!r )=

µ0λ '4π

!KM(!r ')

!r− !r 'd2r ', !KM(

!r )= !M(!r )× s(!r ).S"∫∫

(2.74)

where !K(!r ) isavectorfieldsuchthatMeissner’seffecttakesplace.Itsvaluecanbe

foundbynotingthatanysurfacecurrent !K givesrisetoamagnetic'sinductionstep

discontinuitythroughthebody'ssurface81,

s(!r )× !B+(!r )− !B−(!r )( ) = s(!r )× !B+(!r )= µ0λ 'k3

!K(!r ) as !B−(!r )=0, (2.75)

where !!s(r ) istheoutgoingunitvectornormaltotheellipsoid'ssurfaceSatpoint

!r (seeeq.A7.8)andtheMeissnercondition

!B−(!r )=0 hasbeenused.Thus,taking

intoaccounteqs.A1.11andA7.8,theappropiatesurfacecurrentisgivenby

80 See,forinstance,Reitz,p.188Figure9-2.81 Stratton,p.246eq.7.


!K(!r )= k3

4πk2s(!r )× !B+(!r ), where s(!r )=

xa2x + y

b2y + z

c2z

x2

a4+ y

2

b4+ z

2

c4

. (2.76)

Replacementofthisvalueintheexpressionofthetotalinductiongives

!B(!r )= !B0 − 1

4π ∇×!B+(!r ')×d2!r '!r− !r 'S

"∫∫ , (2.77)

theintegro-differentialequationthatdeterminesthemagneticinductionforellipsoidalsuperconductor.UsingasanAnsatzauniformvalue

!B+, oneobtainsfor

thesecondterm

!BK (!r )= − 1

4π ∇×!B+×d2!r '!r− !r 'S

"∫∫ = − 14π ∇×

!B+× d2!r '

!r− !r 'S"∫∫

⎛

⎝⎜

⎞

⎠⎟

= − 14π ∇×

!B+× ∇' 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫

⎛

⎝⎜⎜

⎞

⎠⎟⎟= − 1

4π ∇× ∇ 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ ×

!B+

⎛

⎝⎜⎜

⎞

⎠⎟⎟, (2.78)

wherethecurltheoremeq.A3.9hasbeenused.Thelastterm—whichforthesakeofconcisioniswrittenintermsof !!f (

r ) (eqs.3.8and3.4)—maybetransformedaccordingtotheidentityeq.A2.9:

!BK (!r )=∇× ∇f (!r )× !B+( )

=!B+ i∇( )∇f (!r )− ∇f (!r )i∇( ) !B+ +∇f (!r ) ∇ i

!B+( )− !B+ ∇ i∇f (!r )( )

=!B+ i∇( )∇f (!r )− !B+Δf (!r ),

(2.79)

Theexpansionfollowsfromtheuniformcharacterof !B+ andthepropertyeq.A2.3

oftheLaplacian.Usingeqs.3.3andA3.7thelasttermmayberewrittenas

!BK (!r )= − 1

4π!B+ i∇( )∇ d3r '

!r− !r 'V∫∫∫ + 1

4π!B+Δ d3r '

!r− !r 'V∫∫∫

=N i!B+ −

!B+ for !r ∈V

next(!r )i !B+ for !r ∉V⎧⎨⎪

⎩⎪.

(2.80)

Therefore

!B(!r )= !B0 +

!BK (!r )=

!B0 − 1−N( )i !B++ for !r ∈V!B0 +next(!r )i !B+ for !r ∉V

⎧⎨⎪

⎩⎪. (2.81)

where !B+ isfullydeterminedbytheMeissnercondition


!B0 − 1−N( )i !B++ =0, !B+ = 1−N( )−1 i

!B0. (2.82)

AsshowninProblem16ofpage132,themagneticmomentgeneratedbythisdensity !

K ofsurfaceconductioncurrentisthesameasthatobtainedfromthe

magnetizacionmodeleq.2.73,

m = − 1

µ0λV!B+ = 1

µ0λ 'α s iB

0 , α s = −V 1−N( )−1 . (2.83)

Inmatrixrepresentation,

Bint =Hint =M=0, B+ = 1−N( )−1 iB0

Bext(!r )=B0 +n(!r )iB+ ,

m = 1µ0λ '

α s iB0 , α s = −V 1−N( )−1 ,

!K(!r )=

k34πk2

s(!r )× !B+ , where s(!r )=xa2x + y

b2y + z

c2z

x2

a4+ y

2

b4+ z

2

c4⎛

⎝⎜⎞

⎠⎟

.

(2.84)

Apartfromthedifferenceinelectron’slocalization,theonlydifferencebetweenthemagnetizationandthesurfacecurrentmodelsofsuperconductivityistheasignementoftheinternalvaluesof !

H and !!

M.

Summaryofintegro-differentialequations

Material Equation Eq.

dielectrics

!E(!r )= !E0 + 1

4π ∇ ∇ iχ e i!E(!r ')!r− !r '

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ . 2.31

diamagnetsparamagnets

!H(!r )= !H0 + 1

4π ∇ ∇ iχm i!H(!r ')!r− !r '

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ , 2.46

conductors

!E0 − 1

4π ∇!E +(!r ')id2!r '!r− !r '

=0 for !r ∈V .S"∫∫ 2.53

superconductors

!B(!r )= !B0 − 1

4π ∇×!B+(!r ')×d2!r '!r− !r 'S

"∫∫ , 2.77

Table1.Integro-differentialequationsderivedinthischapter.


Summaryofinducedpolarizationequations

Edep = −N i χe iE

int ,

Dielectrics:

Eint = 1+N i χe( )−1E0 for r∈V ,

P=ε0λχ e iE

int =ε0λχ e i 1+N i χe( )−1E0 ,

p=VP=ε0λαe iE

0 , αe =V χ e i 1+N i χe( )−1 ,Eext(!r )= E0 − λ

ε0next(!r )iP for r∉V .

(2.33)

H

dep = −N i χm iHint

Magneticmaterials:

Hint = 1+N i χm( )−1 iH0 for !r ∈V ,

M= 1λ 'χm i 1+N i χm( )−1 iH0 ,

m =VM= 1λ 'αm iH0 , αm =Vχm i 1+N i χm( )−1 ,

Hext(!r )=H0 −λ 'next(!r )iM for !r ∉V .

(2.49)

Edep = −N iE+

Conductors:

Eint = E0 −N iE+ =0, E+ =N−1 iE0 for !r ∈V ,

p=ε0λαc iE

0 , αc =VN−1 ,

Eext(!r )= E0 −next(!r )iE+ for !r ∉V .

(2.63)

Superconductors(magnetizationmodel):

Bint = µ0 Hint +λ 'M( ) =0, M= − 1

µ0λ '1−N( )−1 iB0 for∉V ,

m =V ⋅M= 1µ0λ '

α s iB0 , α s = −V 1-N( )−1 ,

Bext(!r )=B0 − µ0λ 'next(!r )iM for !r ∉V .

(2.73)


Superconductors(surfacecurrentmodel):

Bdep = −(1−N)iB+

(2.84)

GeneralformulationofthemethodAlongthesectionsofthischapterastudywasmadeoftheelectromagnetic

behaviourofsolidandhomogeneousellipsoidalbodies—eitherisotropicoranisotropic—underanappliedstaticanduniformelectricormagneticfield.Inwhatfollowsageneralformulationofthetreatmentisgiveninmatrixrepresentation,showingthattheensuingintegro-differentialequationsareessentiallyidenticalinallcasesandmayallbesolved,forthelinearcase,inexactlythesameway.Themaindiscussionisdevotedtoinducedpolarizationsbecausepermanentonesrequirespecificanalysis.

InallcasesavectorialfieldF—electricfield !!E , magneticfield !!

H ,ormagnetic

induction !B—isderivedfromastaticappliedoneF0throughavectorial

magnitudeQ(F)—electricpolarization,magnetizationorstepdiscontinuitythroughthebody'ssurface—whichisalinearfunctionofF.Q(F)fullydescribesthestateofuniformpolarizationofthebody,whichmaybealternatelycharacterizedbysomesurfacedensityofchargeorcurrent.ThelinearrelationshipbetweenQandFiseitherdescribedbyasusceptibility—caseofthelinearbutperphapsanisotropicrelationshipQ= χ ·F—orbyaconditionofcancellationoftheinternalfield.Volumeintegralsareinvolvedinmostcases,butconductorsandsuperconductorsinitiallyrequiresurfaceintegralsthatmaybetransformedintovolumeonesbydefininingappropriateequivalentpolarizations.Theensuingintegro-differentialequations,validoverallspace,areseparatelysolvedfortheregionoccupiedbythebodyandtheexternalone.Itisthenobtained

Fint = F0 −N iQ(Fint ) for !r ∈V ,Fext(!r )= F0 −next(!r )iQ(Fint ) for !r ∉V . (2.85)

Thedepolarizationtensorn,eq.3.4,isuniforminsideanellipsoidalbody(whereitspositionindependentvalueisN).!!Fint isthereforealsouniformthere,itsvaluebeingdeterminedbythefirstequation,aswellasthatof!!Q(F

int ). Thevalueoftheexternalfieldthenfollowsfromthesecondofeqs.2.85.ThegeometriccontributionisdeterminedbyNandnext,whilethematerial'spropertiesarecharacterizedbyQ.

Bint =Hint =0=M, B+ = 1−N( )−1 iB0 ,

m = 1µ0λ '

α s iB0 , α s = −V 1-N( )−1 ,

Bext(!r )=B0 +next(!r )i 1−N( )−1 iB0.


Theexplicitvaluesof!!Fint andQareobtainedindifferentwaysaccordingtothematerial'sproperties.Thefirstoneistheuseofthelinearrelationship

Q(Fint )= χ iFint , (2.86)

whereχ isasusceptibility.Inthiscase

Fint = 1+N i χ( )−1 iF0 , Q = χ iFint = χ i 1+N i χ( )−1 iF0. (2.87)

Thesecondcaseapplieswhentheinternalfieldvanishes(seeTable1):

F0 −N iQ =0 or (1−N)iQ = F0. (2.88)

Theserelationshipscorrespondtoconductorsandsuperconductors,wherethesusceptibilityobtainsitslargestvalue

Q = −N−1 iF0 or Q = (1−N)−1 iF0. (2.89)

ThesurfaceconductioncurrentmodelofsuperconductorsistheonlycaseforwhichnosingleQtensormaybedefinedforboththeinternalandexternalfield.Theoriginofthisasymmetryisthedifferencebetweentheconstitutiveequationsfortheelectricandthemagneticcase,eqs.2.11and2.35.

InthefollowingtablemagnitudesFandQareidentifiedforallcasesdiscussed,aswellasthelinearfunctionsandcancellationsrequiredforthevalidityofthetreatment.Althoughtheydonotsatisfyeqs.2.87to2.89,permanentpolarizationshavebeenincludedinordertogiveafullpanoramicviewofthemethod.

Material F Q χ Condition Eq.

Ferroelectric Eλε0P P=P(E

int ) 2.12

Dielectric Eλε0P χe

P=ε0λχe iE

int 2.33

Conductor(surfacecharge) E E+ E+ =N−1 iE0 2.63

Conductor(equivalentP) E !!!4πk1P χ=∞ !!P= χeEint 2.65

Ferromagnetic H λ'M !!M=M(Hint ) 2.44

Diamagneticandparamagnetic H λ'M !χm !!M= χm iHint 2.49

Superconductor(equivalentM) H λ'M !χm = −1 !!M= χmHint 2.73

Superconductor(surfaceconductioncurrent) B

!!Qint = (N−1 −1)iB+

Qext = −B+ Bint=0 2.84

Table2.Identificationofthemagnitudesinthegeneralfieldequationsfordifferentkindsofmaterials.


Solvingtheintegro-differentialequationsbyiteration

Thedielectriccase,eq.2.31,maybetakenasaprototypeoftheintegro-differentialequationswhichdeterminethefields:

!E(!r )= !E0 +O( !E), (2.90)

whereOisalinearoperatorwiththefollowingproperties(seeeq.3.3):

O!F(!r )( ) = k1∇ ∇ i

χ e i!F(!r ')!r− !r '

d3r 'V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ (2.92)

O k ⋅!F(!r )( ) = k ⋅O !F(!r )( ) ,

O!F(!r )+ !G(!r )( ) =O !F(!r )( )+O !G(!r )( ) ,O!F 0( ) = −4πN i χ e i

!F 0 if !r ∈V .

(2.92)

Thesecondmemberofthelastequationholdstrueonlywhen !!F 0 isauniform

vectorthroughoutthebody'svolumeV.

ThesolutionforeachoftheseequationswaspreviouslyfoundbyusingauniformAnsatz,asdiscussedinthedifferentsections.Theresultingfieldcoincides,inallknowncases,withthatobtainedbysolvingLaplace'sequationinellipsoidalcoordinatesbythemethodofseparationofvariables82.Theexamplesareonlyillustrativeandoneshouldprovethateq.2.90 hasauniquesolution,assketchedbelow.

Let !F (1) and

!F (2) betwodifferentsolutionsofeq.2.90,thatis

!F (1) =

!F 0 +O( !F (1)), !F (2) = !F 0 +O( !F (2)), thatis !F (1) − !F (2) = !G =O( !G). (2.93)

Byiterationitiseasilyprovedthat

!!G =On( G) for n =1,2,3..., (2.94)

equationthatissatisfiedonlyfor !!G =0, provingthatthesolutionisunique.

Theinhomogeneouseq.2.90maybesolvedbyiteration,succesivelyreplacingtheargumentoftheoperatorbythefullsecondmember.Thefirstreplacementgives

!E =!E0 +O

!E0 +O

!E( )( )

=O0 !E0( )+O1 !E0( )+O2 !E( ) = Oα!E0( )

α=0

1

∑ +O2 !E( ) , (2.95)

82 FullreferencestosolutionsaregiveninthesectionHistoryofChapter1.


whereOnisthenthpowerofO.

Theseconditerationgives

!E = Oα

!E0( )

α=0

1

∑ +O2 Oα!E0( )

α=0

1

∑ +O2 !E( )⎛⎝⎜

⎞⎠⎟

= Oα!E0( )

α=0

1

∑ + Oα+2 !E0( )α=0

1

∑ +O4 !E( ) = Oα!E0( )

α=0

3

∑ +O4 !E( ). (2.96)

Intheinfiniteiterationslimitonegets

!E = Oα

!E0( )

α=0

∞

∑ +O∞!E( ) ,

Oα!E0( ) = −4k1N i χ e( )α i

!E0 ,

(2.97)

wheretheoperatorseriescoincideswiththeTaylor'sexpansion

1

1+ x =1− x + x2 − x3 + x4 + ...= (−x)α

α=0

∞

∑ . (2.98)

Iftheseriesconvergesandthelasttermofeq.2.97tendsto0,itisobtained

!E(!r )= 1+N i χ( )−1 i

!E0. (2.99)

Theexpressionconcideswitheqs.2.33and2.87,showingthattheiterativeexpansionconvergesandcoincideswiththatobtainedusingauniformAnsatz.Althoughtheoreticalphysicistsfrequentlyuseseriesexpansionsofoperators,themethodisseldomusedbyengineers.Itiseasiertogivearigorousjustificationofthiskindofexpansionbyusingmatrices83.

Thesamemethodmaybeusedtosolvetheotherthreeintegro-differentialequationsinTable1.Thisiterativemethod,clumsierthantheuseofauniformAnsatz,ispresentedhereonlytostressthemathematicalconsistencyoftheuseofintegro-differentialequations.Itmight,perphaps,beusedasanapproximatemethodforstudyingthecaseofnonellipsoidalbodiesorequations2.85whenQisnotalinearfunctionofF.Theexplorationofthissubject,notmadebythetheauthor,islefttotheinterestedreader.

83 See,forinstance:G.GoertzelandN.Tralli;SomeMathematicalMethodsofPhysics;McGraw-Hill;

1960;chapters1,2and3,pp.7-49.


Chapter3:Thedepolarizationtensor:basictreatment

DefinitionThedepolarizationtensor n(

!r )arisesfromexpressionslike

∇ ∇I(!r )i !F( ) , !F i∇( ) ∇I(!r )( ) ,∇ ∇ i I(!r )!F( )and∇!F id2!r '!r − !r 'S

"∫∫ ,

whereI(!r )= d3r '!r − !r 'V

∫∫∫ and !F isuniforminV . (3.1)

TheintegrationismadeoverthevolumeVofanellipsoidalbodyoritsclosedsurfaceS,and isthepositionofafieldpointthatmaybeeitherinsideoroutsidethebody. isauniformvectorinsidetheregionofintegration—thebody’svolume,includingitsinteriorsurface—thatmaybeanelectricormagneticfieldorapolarization.Table1givesthelistoftheintegro-differentialequationsdiscussedinthisbookwheresuchexpressionsarise.Onlythefirsttwoexpressionswillbeanalyzednextasthethirdisverysimilar.

Thecasewherenisderivedfromasurfaceintegralisdiscussedatsectionnasasurfaceintegral.Theproofiseasiertofollowifonewrites and∇intermsofcomponentsandunitvectors,asfollows:

(3.2)

Then

∇ ∇I(!r )i !F( ) = xα∂

∂xαxβ

∂I(!r )∂xββ

∑ i xγ Fγγ∑⎛

⎝⎜⎞

⎠⎟α∑

= xα∂

∂xαxβ i xγ( )∂I(

!r )∂xβ

Fγβ ,γ∑⎛

⎝⎜⎞

⎠⎟α∑ = xα

∂∂xα

δβγ

∂I(!r )∂xβ

Fγβ ,γ∑⎛

⎝⎜⎞

⎠⎟α∑

= xα∂

∂xα

∂I(!r )∂xβ

Fββ∑⎛

⎝⎜⎞

⎠⎟= xα

∂2I(!r )∂xα ∂xβ

Fβα ,β∑ .

α∑

(3.3)

!r

!F

!F

∇ = xβ

∂∂xββ

∑ , !F = xγ Fγγ∑ , where xβ i xγ =δβγ .


Onemaynowdefinethecomponentsoftheranktwodepolarizationtensor

nαβ(!r )= − 1

4π∂2 I(!r )∂xα ∂xβ

= − 14π

∂2

∂xα ∂xβ

d3r '!r − !r 'V

∫∫∫ , (3.4)

thatindyadicrepresentation(seeAppendix6)maybewritten

n(!r )= xαnαβ(

!r )xβα ,β∑ . (3.5)

Forinstance

!F i∇( )∇I(!r )= Fα

∂∂xαα

∑⎛⎝⎜

⎞

⎠⎟∂I(!r )∂xβ

xββ∑ = Fα

∂2I(!r )∂xα ∂xβ

xβα ,β∑

= −4π !F i xαn(!r )αβ xβ = −

α ,β∑ 4πn(!r )i !F ,

(3.6)

asthedepolarizationtensorissymmetric(seeeq.3.12).

Therefore,

!F i∇( )∇I(!r )=∇ ∇ i I(!r )!F( ) =∇ ∇I(!r )i !F( ) = −∇

!F id2!r '!r − !r 'S

"∫∫= −4πn(!r )i !F , where n(!r )= xαnαβ xβ

α ,β∑ .

(3.7)

Thecalculationofthedepolarizationtensornisthereforereducedtothecalculationoftheintegral I(

!r ) eq.3.3fordiferentkindsofellipsoidalbodies.Forasolidellipsoidalbodythisintegralis,apartfromaconstantfactor,eitherthepotentialforauniformlychargedone,orthegravitationalpotentialforaconstantmassdensity.Inthischapterthefirstpotentialisused,asitismorefamiliartostudentsofengineering,andthesecondpotential—thelessfamiliaronewithmorecomplexderivation—isusedinthefollowingchaptertoderiveexpressionsforn.Itisconvenienttodefineanauxiliaryfunction f (

!r ) suchthateq.3.4reducesto

nαβ(!r )= ∂2 f (!r )

∂xα ∂xβ, where f (!r )= − 1

4πd!r '!r− !r '

= − φ(!r )4πkσV

∫∫∫ . (3.8)

Inthelasttermφisagravitationalorelectrostaticpotential,σtheconstantmassorchargedensityandkthegravitationalorelectrostaticconstantdeterminedbythechoiceofsystemofunits.Thatis,theyarethefactorssuchthat


1kσ

φ(!r )= d3r '!r− !r 'V

∫∫∫ (3.9)

Tensor n(!r ) isuniforminsideellipsoidalbodies,whereitwasoriginally

defined84.Theauthorofthisbookwasapparentlythefirstonetodefineitoutsidethesebodiesalthoughrestrictedtothemagneticcase85.Intheliteratureofmagnetismitsinternalvalueissometimescalleddemagnetizationtensor(apartfromapossibledenominator4π),itsprincipalvaluesbeingthewellknowndemagnetizingordemagnetizationcoefficientsorfactors86.Thetensorisindistinctlywrittenhereasadyadicnora3x3matrixn.Forapplicationsitisnecessarytoexplicitlyidentifytheregionsinsideand

outsidetheellipsoidalbody.TothatendN(dyadic)oN(matrix)denotetheuniforminternaldepolarizationtensorandnext( )onext( )thenon-uniformexternalone

N = xα xβNαβ

β∑

α∑ for !r ∈V , next(!r )= xα xβnαβ(

!r )β∑

α∑ for !r ∉V . (3.10)

N =

Nxx Nxy Nxz

Nxy N yy N yz

Nxz N yz Nzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

for !r ∈V ,

next(!r )=nxx(!r ) nxy(

!r ) nxz(!r )

nxy(!r ) nyy(

!r ) nyz(!r )

nxz(!r ) nyz(

!r ) nzz(!r )

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

for !r ∉V .

(3.11)

Thedifferencebetweenbothregionsisemphasizedbyexplicitlywritingthedependanceonpositionfortheexternalcaseandomittingitintheinternalcase,asforconstantsorparameters.

Generalpropertiesofn

Symmetrictensor

Astheorderofderivationmaybefreelyexchanged87,nturnsouttobeasymmetrictensor:

nαβ(!r )= nβα (

!r ), (3.12)

84 LandauandLifchitz,p.44.Maxwell,apparentlythefirsttointroduceit,ditnotidentifyits

tensorialcharacter.85 Solivérez(1981).86 MoskowitzandDellaTorre,p.739.87 MacMillan,pp.27-32.

!r !

r


propertythatwaspreviouslyusedineq.3.6.

Trace

Fromeq.3.4thetraceofn(sumofitsdiagonalelements)is

Tr n(!r )= nαα

α∑ (!r )= ∂2 f (!r )

∂x2αα∑ = Δ f (!r )= − 1

4π ∇ i∇ 1!r− !r '

d3r ',V∫∫∫ (3.13)

whereΔ istheLaplacianoperatoreq.A2.3.Thevalueofthisoperation,frequentlyusedinthetheoryofelectromagnetism,isanintegrablesingularityinsideVandvanishesoutsidethebody.Asthecalculationisoftenmadeinaquestionableway,inAppendix3(seeeq.A3.7)itisprovedthat88.

Δ d3r '

!r− !r '=

−4π for !r ∈V0 for !r ∉V⎧⎨⎩

.V∫∫∫ (3.14)

Itfollowsthat

Tr N =1, Tr next =0. (3.15)

Infact,theconstantsineq.3.8werechoseninordertomakethetraceofNtobe1.

Orthogonaltransformations

Thetransformationpropertiesofmatrixnarenowanalyzedunderanorthogonalchange(rotations,reflections,inversionsandallitscombinations)ofcoordinatesystem:

x 'α = Rαββ∑ xβ , r'=

x 'y 'z '

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

Rxx Rxy RxzRyx Ryy Ryz

Rzx Rzy Rzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

ixyz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=R ir. (3.16)

Thesetransformationsarethebasisoftheuseofsymmetrypropertiesinordertosimplifythecalculationofmanyphysicalproperties(seethesectionSymmetries,andProblem19).

SuchoperationsarealwaysrepresentedbyanorthogonalmatrixR,thatis,onesuchthatitsinversecoincideswithitstransposedmatrix:

R

−1 =R t , R iR t = R t iR =1, where Rαβt = Rβα . (3.17)

88 V.Hnizdo,Eur.J.Phys.vol.21,pp.L1-L3(2000)givesarigurousbutmorecomplexproof.


Theinversetransformationisthen

r =R−1 ir'=R t ir,xyz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

Rxx Ryx RzxRxy Ryy RzyRxz Ryz Rzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

ix 'y 'z '

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, xα = Rβα

β∑ x 'β . (3.18)

Inordertodeterminethecomponentsofninthenewcoordinatesystem,oneshouldusedefinitioneq.3.8,

nαβ(!r )= ∂2 f (!r )

∂xα ∂xβ, (3.19)

andexpressthevalueofthenewpartialderivativesintermsoftheoriginalones89.Intermsofthederivativeoperator:

∂∂x 'β

=∂

∂xα

∂xα∂x 'βα

∑ = Rβα

∂∂xα

.α∑ (3.20)

ItisthusobtainedforthenewmatrixN’

N 'αβ =

∂2 f∂x 'α ∂x 'β

= Rαγγ ,δ∑ ∂2 f

∂xγ ∂xδRβδ = R iN iR t( )

αβor N'=R iN iR t . (3.21)

whereeq.3.17wasused.

Byusingthistransformationpropertyonemaystablishrelationshipsamongcomponentsofthepolarizationtensor,topicdiscussedinthefollowingsection.Ifonedeterminestheeigenvaluesofninitsprincipalcoordinatesystem,the

valueofn'inacoordinatesystemrelatedtotheformerbyeq.3.21is

n'=R in iR t

=

Rxx Rxy RxzRyx Ryy Ryz

Rzx Rzy Rzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

nx(!r ) 0 00 ny(

!r ) 00 0 nz(

!r )

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

Rxx Ryx RzxRxy Ryy RzyRxz Ryz Rzz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

. (3.22)

SymmetriesTherelationshipsamongtheconstanteigenvaluesofNdeterminedbythe

body'ssymetries—quiteusefulforapplications—canbefoundbyusingthetransformationpropertieseq.3.22.Thesymmetriesoftheexternaldepolarizationtensornextaremoredifficulttotreatbecausesitscomponentsusuallyarecumbersomefunctionsofposition.Thiscaseisusuallyhandledbyusing

89 Santaló,pp.299-303.


appropiatecurvilinearcoordinatesystems(see,forinstance,thetreatmentofthesphereinpage58).

WhenanorthogonaltransformationRisasymmetryoftheellipsoidalbody,thevalueN'inthetransformedcoordinatesystemmustbeequaltotheoriginalone:

N'=R•N•Rt =N , or R•N =N•R , (3.23)

Fromapracticalpointofviewitissimplertousetheconmutationrelationshipthanthefirstone,asisillustratednext.Thecaseofspheroids(ellipsoidsofrevolution)isdiscussednowasanexample.

Iftheaxisofrevolutionistakenascoordinateaxisz,theexchangeofaxesxandy(areflectionplane)shouldleaveNunchanged.Therefore,for

R =0 1 01 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟,R iN =

0 Ny 0N x 0 00 0 Nz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=N iR =

0 N x 0Ny 0 00 0 Nz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

. (3.24)

ItthusturnsoutthatNx=Nyandnofurtherrelationshipsarefoundusingothersymmetryoperations.ThediagonalexpressionofNforspheroidsisthus

Nspheroid =

Ne 0 00 Ne 00 0 Np

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

(3.25)

Npbeingthepolareigenvaluecorrespondingtothespheroid'ssymmetryaxis,anNethatofthetwoequalequatorialones.

Inthecaseofthesphereanexchangeofaxisyandzisalsoasymmetry,whichleadstothethreeeigenvaluesbeingequal,asdiscussedinProblem19.Incombinationwiththetraceruleeqs.3.15,onethenobtains

Nx=Ny=Nz=N,TrN=3N=1andN=1/3. (3.26)

ThesphereistheonlytypeofellipsoidwheretheactualeigenvaluesofNmaybedeterminedbyusingonlythetraceruleandsymmetryarguments.Thereareonlytwoothercaseswherethiscanbedonewithoutfindingprimitivesofintegrals,butbothrequiretheknowledgeoftheintegralexpressionsofNα,topicdiscussedinsomesectionsofthischapterandthenext.

nasasurfaceintegralThedepolarizationtensornmaybeexpressedasasurfaceintegralincases

involvingsurfacedensitiesofchargeorcurrent.Tothatendoneshouldtakethescalarproductofeq.3.4withauniformvector !!

!C :


n(!r )i !C = xαnαβ(!r ) xβ

α ,β∑ i

!C = xαnαβ(

!r )Cβα ,β∑

= − 14π Cβ xα

∂∂xα

∂∂xβα ,β

∑ d3r '!r− !r 'V

∫∫∫ = 14π ∇ ∇' 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫

⎛

⎝⎜⎜

⎞

⎠⎟⎟i!C

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (3.27)

As (eq.3.8)isacontinuousfunction,thegradienttheoremeq.A3.9maybeusedtoobtain

∇' 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ = d2!r '

!r− !r 'S"∫∫ . (3.28)

As isanarbitraryvector,itfollowsthat

n(!r )= 1

4π ∇ d2 !r '!r − !r 'S"∫∫ , (3.29)

anexpressionpreviouslygivenbyMoskowitzandDellaTorre90.

SurfacestepdiscontinuityThedepolarizationtensorhasastepdiscontinuitythroughthebody’ssurface,

whosecharacteristicsarebetterunderstoodbyanalizingaconcretecase,heretakentobetheelectricone.Asdiscussednext,itturnsoutthatinthiscasethediscontinuityisequivalenttoachargedensity.Inthemagneticcase—wheretheanalysisisquitesimilar—itcorrespondstoasurfacecurrentdensity.InProblem30thereaderischallengedtofindtheexpressionforthesurfacediscontinuitywithoutusingthepropertiesofelectricormagneticfields.

Surfacechargedensity

Forauniformelectricpolarization —whetheritisspontaneous,inducedorequivalent—theresultingelectricfieldmayalwaysbewrittenas(seeeq.2.9)

(3.30)

As (eq.3.8)isacontinuousfunction,thegradienttheoremeq.A3.9maybeusedtoreducethevolumeintegraltothefollowingsurfaceintegraloverS:

90 MoskowitzandDellaTorre,p.740eq.17.

!!I(r )

!C

!P

!!

EP(r )= −∇φP(

r ) where φP(r )= k1 ∇' d

3r 'r− r 'V

∫∫∫⎛

⎝⎜⎜

⎞

⎠⎟⎟iP.

!!I(r )


(3.31)

ThelastintegralexpressesthepotentialcreatedbyasurfacechargedensityσP91.Usingeq.A7.8itisobtained

(3.32)

whichistheelectricchargedensitycreatedbythestepdiscontinuityonthebody’ssurface,includingtheequivalentpolarizationcaseinthecaseofconductors.Thisexpression,usefulforelllipsoidalconductors,isnotgiveninanyofthetextbooksusedforwritingofthisbook92.Thevalidityofeq.3.32maybeeasilyverifiedforthespecialcaseofthespherediscussedinProblem09.

Surfacestepdiscontinuity

Throughamaterialinterphasewithasurfacechargedensityσofnet(casenotconsideredinthisbook)orinducedchargethecomponentofanelectricfieldparalleltothesurfaceiscontinuous,butthenormalcomponentisnot.If s(

!rS ) istheunitvectornormaltothesurfaceatpoint

!rS ,thedecompositionofanyvectorincomponentsparallelandnormaltothatsurfaceis:

!F =!F"+!F⊥ ,

!F"= s i

!F( ) s = ss i !F , !F⊥ = 1− ss( )i !F , (3.33)

where1istheunitdyadicofeq.A6.4.

Thecontinuityoftheelectricfieldcomponenttangentialtothebody’ssurfaceisthenexpressedinthefollowingway93:

1− ss( )i !E + −!E −( ) = !E + −

!E −( )i 1− ss( ) =0. (3.34)

Thediscontinuityofthecomponentnormaltothebody'ssurfaceSis94:

91 Reitz,p.31eq.2-15.92 Stratton—theautorthatdiscussesinmoredetailtheelectromagneticpropertiesofellipsoidal

bodies—givesonlythesurfacedensityforchargedconductorsintheabsenceofanappliedfield,seeeq.12inp.209.

93 Reitz,p.90.94 Stratton,p.188.

!!

φP(r )= k1 ∇' d

3r 'r− r 'V

∫∫∫⎛

⎝⎜⎜

⎞

⎠⎟⎟iP = k1

d2r 'r− r 'S∫∫ i

P

= k1σ P(r ')d2r 'r− r 'S

∫∫ , where σ P(r ')= s(r )i P.

!!

σ P(r )= s(r )i P =

xPxa2

+yPyb2

+zPzc2

x2

a4+ y

2

b4+ z

2

c4

,


!E + −

!E −( )i s = 4πk1σ . (3.35)

Thesignsarechosensothatacomponentispositivewhenthevectorpointsoutgoingfromthepositivesideofthesurface(+),chosentobetheoneexternaltothebody.Forallthecasesdiscussedinthepreviouschapter—seeeqs.2.7andA1.4—the

followingrelationshipisfullfilled,

(3.36)

where isapointonsurfaceS.

Fromthisidentityandtheexpressionof eq.3.31,thefield'sdiscontinuitythroughthebody’ssurface,thedepolarizationtensor'sstepdiscontinuity,maybewrittenas (3.37)

Fortheequationtobevalidforarbitraryvaluesof !P —whetheritisa

spontaneous,inducedorequivalentpolarization—itshouldbe

s(!r S )s(!r S )i N −next(!r S )( ) = N −next(!r S )( )i s(!r S )s(!r S ). (3.38)

AnytensorTmaybedecomposedinthefollowingfashion

T = 1+ ss − ss( )iT = ss iT + 1− ss( )T =T

!+T⊥ , (3.39)

incomponentssuchthatwhenthescalarproductwithanarbitraryvectoristaken,thecomponentsparallelandnormaltothesurfacearerespectivelygivenbyeachterm.

Fromeqs.3.34and3.35onegets

N −next(!r S )= ss i N −next(!r S )( )+ 1− ss( )i N −next(!r S )( ) = ss , (3.40)

whichgivesthevalueonthebody’ssurfaceoftheexternaldepolarization.Thefollowingmatrixexpressionisthusobtained,validforalltypesofellipsoids:

next(!r S )=N − s(!r S )i s(!r S ). (3.41)

Fromeq.A7.8,

sα (!r S )sβ(

!r S )=xSα x

Sβ

xS( )2

a4+yS( )2b4

+ zS( )2

c4

. (3.42)

Atanelementaryleveltheexpressionmaybeverifiedforallellipsoidswherethecomponentsoftheexternaldepolarizationtensormaybeexpressedintermsof

!!!E + −

E − = 4πk1 N −next(r S )( )i P ,

!r S

!!s

!!!s i N −next(r S )( )i P = s i P.


elementaryfunctions,liketheinfinitecylinderandthesphere(seeProblem34).Lateron,itwillbeverifiedforalltypesofellipsoids,includingthetriaxialone(seeeq.4.66).Theformulaprovidesasimpleapproximationtothefieldsnearthesurfaceofanyellipsoid.Thisincludestheelectricfieldsatsharpmetallictips,atopicwhereexplicitformulasarerarelygiven(seeProblem12).

CalculationofnusingelectrostaticGauss’sLawThissectionisdevotedtothecalculationofnfromeq.3.8,whichrequiresthe

expressionoftheauxiliaryintegral f (!r ) .Asstatedthere,apartfroma

proportionalityconstantfcoincideswiththeelectricpotentialφgeneratedbyvolumeVwhenchargedwithuniformdensityρ95:

φ(!r )= k1ρ

d3r '!r− !r '

=V∫∫∫ −4πk1ρ f (

!r ), f (!r )= − φ(!r )4πk1ρ

. (3.43)

ThisexpressionisusefulfortheshapeswherethepotentialmaybederivedbyusingelectrostaticGauss’law,highsymmetrycaseswhicharefrequentlydiscussedinfirstuniversitycoursesonelectricityandmagnetismforphysicistandengineers.Inorderofincreasingdifficultytheseshapesare:

• a=b≫c:constantthicknesssheetofinfiniteextension;• a≫b=c:rightcircularcylinderofinfinitelength;• a=b=c:sphere.

Sheetofconstantthicknessandinfiniteextension

Thedepolarizationtensorofasheetofconstantthicknesse=2candinfiniteextension,asshowninthefigure,iscalculatedhere.Acartesiancoordinatesystemisusedsuchthatitszaxisisperpendiculartothesurfaceofthesheet,withoriginonitsmiddleplane.Thesimplestwaytosolvetheequivalent

electrostaticproblemofeq.3.43istoevaluatethe

Figure3.Infinitesheet.

electricfieldbyapplicationofelectrostaticGauss'sLaw,obtainingthenthepotentialbyintegration.Thismaybeeasilydoneduetothehighsymmetryoftheproblem.Asthesheetisinvariantunderarbitrarytranslationsalongtheaxesx,y,thefieldmayonlybeafunctionofz.Asitisalsoinvariantunderarbitraryrotationsaroundthezaxis,thefieldcannothavexandycomponents.Thelastsymmetryofinterestisthereflectionplaneparalleltothesheet'ssurfacesthroughitsmiddle,makingthefieldinoneofitssidesthespecularimageofthatintheother.Therefore

(3.44)

95 AsimilarexpressionappliesforfinthegravitationalcasediscussedbyMacmillanandKellog,

whereoneshouldtakek1=1andρtobethedensityofmass.

!E(−z)= − !E(z), !E(!r )= E(z)z , E(0)=0.


Figure4showsthesheet'scrosssectioningrayandthetwoclosedsurfaces,S1andS2,usedfortheapplicationofGauss’sLaw.Thesegaussianboxesarerightcircularcylinderssuchthattheirverticalfacesareparalleltoaxiszandthehorizontalones,ofareaA,arenormaltoit.Thefluxoftheelectricfieldvanishesbothonthebase(where

!E =0)andonthe

verticalface(whereitisnormaltothesurface).Thisreducesthecalculationtothefluxontheupperface,where

Figure4.Gaussianboxes

usedforsolvingtheinfinitesheet.

!E int id

"S

S1

#∫∫ = 4πk1Q1 , gives E int(z)⋅A= 4πk1ρ ⋅A⋅z , E int(z)= 4πk1ρ ⋅z ,!E ext id

"S

S2

#∫∫ = 4πk1Q2 , gives E ext(z)⋅A= 4πk1ρ ⋅A⋅e, E ext(z)= 4πk1ρ ⋅e. (3.45)

From andeq.3.43itisobtained

φ(z)= cte.−2πk1ρ ⋅z2 for !r ∈Vcte.−2πk1ρ ⋅e ⋅z for !r ∉V⎧⎨⎪

⎩⎪, f (z)=

12z

2 for !r ∈V ,12e ⋅z for !r ∉V

⎧

⎨⎪⎪

⎩⎪⎪

,. (3.46)

Fromthedefinitioneq.3.4itfollowsthat

forthesheet

N =0 0 00 0 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟, next =

0 0 00 0 00 0 0

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟. (3.47)

DuetotheinexistenceofN-1oneshouldnextanalyzetherelationshipofthefieldsthusobtainedwiththoseofthedepolarizationtensormethod.Equations2.33and2.49showthattherelevantmatrixforthecaseofinducedelectricandmagneticpolarizationsis

1+ χN( )−1 ==1 0 00 1 00 0 1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟+

0 0 00 0 00 0 χ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−1

=1 0 00 1 00 0 1+ χ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

−1

=1 0 00 1 00 0 1/ 1+ χ( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

(3.48)

whereitisclearnoproblemarises.Thisdoesnothappenforthecaseofconductorsandsuperconductors,eqs.2.63,2.73and2.84.Inthesecasesone

!!E(z)= − dφ(z)

dz


shouldgobacktotheoriginalequationthatwassolvedbytheinversionofamatrix.Forinstance,thefirstofeqs.2.63leadstotheequation

N iE+ = E0 , thatis E+

x =0,E+y =0,E+

z =E0z , (3.49)

whereE0istheapliedelectricfieldandE+isthetotalelectricfieldontheexternalsurfaceofthesheetduetothesurfacedensityofcharge96inducedbyE0.Thephysicalmeaningofthepreviousequationisthat,asregardsthe“polarization”effects—thosethathaveoriginintheredistributionofsurfacecharges—theonlycomponentoftheappliedfieldtobetakenintoaccountistheonenormaltothesheet,thezcomponent.Inthereducedspaceofthezaxis,matrixNhaswhatmaybecalleda“reduced”inverse(thisisbetterseeninthetwo-dimensionalcasediscussedattheendofnextsection).

RightcircularcylinderofinfinitelengthAsinthepreviousproblem,thevalueofnwillbe

derivedfromthecalculationoftheelectricfieldofauniformlychargedrightcircularcylinderofradiusb,infinitelengthanduniformelectricchargedensityρ.Bysymmetry,theelectricfieldliesontheplanenormaltothecylinderaxis,isangle-independentandradial,thatis,normaltothecylinder’ssurface.Itsvalue,easilyobtainedusingtheelectrostaticGauss’sLawis97:

Figure5.Infinitely

longcircularcylinder.

Er = −

∂φ(r)∂r

=2πk1ρ ⋅r for r ≤b

2πk1ρa2

rfor r ≥b

⎧⎨⎪

⎩⎪, r = y2 + z2 . (3.50)

Integratingthefieldwithrespecttoritisobtained

φ(r)= constant −πk1ρ ⋅r2 for r ≤b

constant −2πk1ρ ⋅b2 lnr for r ≥b

⎧⎨⎪

⎩⎪, (3.51)

f (!r )=14( y

2 + z2) for !r ∈V14b

2 ln( y2 + z2) for !r ∉V

⎧

⎨⎪⎪

⎩⎪⎪

. (3.52)

Fromeq.3.4itfollowsthat

96 ThereadershouldreadinfulldetailthesectionConductorsinthisbook.97 A.Halpern,3000SolvedProblemsinPhysics,Schaum'sOutlineSeries,McGraw-Hill,p.406

problem25.70;1988.


∂2 f∂x2

=0, ∂2 f∂ y2

= 12 ,∂2 f∂z2

= 12 ,∂2 f

∂xα ∂xβ=0 for xα ≠ xβ , r∈V , (3.53)

N =0 0 00 1/2 00 0 1/2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, (3.54)

whichsatisfiesthetraceruleeq.3.15.ThisvalueofNcoincides,apartfromtheexchangeofaxis,withthatobtainedbysymmetryarguments(seeeqs.3.25and3.73).

Outsidethebody

∂2 f∂x∂xα

= ∂2 f∂xα ∂x

=0 if xα = y ,z ,

∂2 f∂ y2

= − b2

2y2 − z2

r4, ∂2 f

∂z2= b

2

2y2 − z2

r4,

∂2 f∂ y∂z

= ∂2 f∂z∂ y

= − b2

22yzr4

.

(3.55)

Inasphericalcoordinatesystemwhere

x=r·cosϕ·sinθ,y=r·sinϕ·sinθ,z=r·cosθ, (3.56)

δ = x2 + y2 = r·sinθ ,

andRisthecylinders’sradius,thematrixexpressionsfornare

Cylinder

N =1/2 0 00 1/2 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

(3.57)

next(!r )= R2

2δ 2

−cos2ϕ −sin2ϕ 0−sin2ϕ cos2ϕ 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

(3.58)

Astherearenocomponentsalongthecylinder’szaxis,theproblemmaybesolvedinthereducedspaceofthex,ycomponentswherethereducedinverseN’is

N'= 1

21 00 1

⎛

⎝⎜⎞

⎠⎟, N'−1 =2 1 0

0 1⎛

⎝⎜⎞

⎠⎟, (3.59)


usingwhichthegeneralmatrixequationsmaybeappliedforallelectrostaticandmagnetostaticcases.

SphereThesphereisthelastbodyforwhichthe

electrostaticpotentialφmaybeobtainedfromtheelectrostaticGauss'sLaw.LetRbetheradiusofthesphere,Qthetotaluniformlydistributedchargeandtakeitscenterastheoriginofcoordinates.Fromthesymmetryofthebodytheelectricfieldhastoberadial,

thefollowingfunctionofthedistancertothecenterofthesphere.

Figura6.Sphere.

E(r)=k1QR3r for r ≤R,

k1Qr2

for r >R.

⎧

⎨⎪⎪

⎩⎪⎪

(3.60)

As

!E(!r )= E(r)r = −∇φ(r)= − d

drφ(r), then φ(r)= − E(r ')dr '

∞

r

∫ . (3.61)

Therefore

φ(!r )=cte.−k1

Q2R3 r

2 for !r ∈V

cte.+k1Qr

for !r ∉V

⎧

⎨⎪⎪

⎩⎪⎪

, (3.62)

As

(3.63)

itfollowsthattheauxiliaryfunctionf,eq.3.8,isgivenby

f (!r )=16r

2 for r ≤R

− R3

31rfor R ≤ r

⎧

⎨⎪⎪

⎩⎪⎪

. (3.64)

Fromeq.3.4and

(3.65)

whereδαβisKronecker'sdelta,onegetsthefollowingexpression.98

98 Solivérez(2008),p.205eq.12.

!!!E(!r )= E(r) !r ,

!!V = 4π3 R3 , ρ = Q

V= 3Q4πR3 , f (r)= − 1

4πk1ρφ(r)= − R3

3k1Qφ(r),

!!∂2r2

∂xα ∂xβ= ∂2(x2 + y2 + z2)

∂xα ∂xβ=2δαβ ,

∂2r −1

∂xα ∂xβ=3xα xβ −δαβr

2

r5,


Forthesphere N =

13 0 0

0 13 0

0 0 13

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

= 131,

next(!r )= − R3

3

3x2 − r2r5

3x ⋅ yr5

3x ⋅zr5

3y ⋅xr5

3y2 − r2r5

3y ⋅zr5

3z ⋅xr5

3z ⋅ yr5

3z2 − r2r5

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

.

(3.66)

Indyadicnotation

N = 131, n

ext(!r )= − R3

33!r!r −(!r i

!r )1r5

= − V4π3!r!r −(!r i

!r )1r5

, (3.67)

whereVisthevolumeofthesphere.Ageneralpropertyofthedepolarizationtensorforfinitebodiesishereputintoevidence:itsinternalvalueisindependentofthebody'svolumeV,whileitsexternalvalueisproportionaltoV.NoticethatTrN=1andTrnext=0,aspreviouslydiscussed.MatrixNcoincideswiththatobtainedinasimplerwayfromsymmetryarguments(seeProblem19).

OtherpropertiesoftheinternaldepolarizationtensorN

IntegralexpressionsoftheeigenvaluesofN

TheproofofmanyimportantpropertiesoftheinternaldepolarizationtensorNmaybeeasilyderivedfromitsgeneralintegralexpressions.Itisthereforeconvenienttousethevaluesgivenbyeqs.4.6,asreproducednext.NαandNα (0)aredifferentnotationsforthesameintegral.ThevaluesofNα (κ)withκ≠0arenecessaryforthecalculationofthedepolarizationtensorexternaltothebody.

Na =

abc2

ds

a2 + s( )3/2 b2 + s( ) c2 + s( )0

∞

∫ , wherea≥b≥ c , (3.68)

Nb =

abc2

ds

b2 + s( )3/2 a2 + s( ) c2 + s( )0

∞

∫ ,wherea≥b≥ c , (3.69)

Nc =

abc2

ds

c2 + s( )3/2 a2 + s( ) b2 + s( )0

∞

∫ ,wherea≥b≥ c. (3.70)


Theintegralsarealsohighlysymmetricasregardsthesemiaxesanditmaybeeasilyprovedthat

Nb(a,b,c)=Na(b,a,c),Nc(a,b,c)=Na(c,b,a). (3.71)

Thatis,onesinglefunctiongivesthethreeeigenvalues,aslongastheyaresortedinsuchawaythattheconditiona≥b≥cisfulfilled.Astheyareveryefficientmethodsforthenumericalevaluationofintegrals99,eqs.3.71maysimplifythenumeralcalculationofeigenvaluesthatarenecessaryforthegeneraltriaxialelipsoids.

Asthethreeintegralsdifferonlyinthesinglefactor1/(dα2+s)onemightassumethat

if a≥b≥ c , then 1

a2 + s≤ 1b2 + s

≤ 1c2 + s

and Na ≤Nb ≤Nc . (3.72)

Theconclusionistrue,buttheproofisflawed.Thismaybeseenbyusingasimilarargumenttoprovethatforconstantaandc,theeigenvaluesofNareadecreasingfunctionofb.AsseenfromFigure14,Figure15andFigure16,thisistrueforNb,butnotforNaandNc.Theoriginoftheparadoxisacombinationofthesymmetrypropertieseq.3.71andtheconstraintsimposedontherelativevaluesofa,bandcoriginatedinthemethodusedtoderivetheexpressionsofNαintermsofellipticfunctions.Inthiscasethesymmetrieseqs.3.71arelostandeacheigenvalueisgivenbyacompletelydifferentfunction(seeeqs.4.9).

DiagonalizationandinversionAsseenfromeq.4.6theinternaldepolarizationtensorNisdiagonalinthe

ellipsoid'sprincipalsystemofcoordinates,theonecoincidingwithitssemiaxes.Furthermore,fromtheintegralsthatdefineitseigenvalues(thediagonalmatrixelementsortensor'sprincipalvalues)itiseasilyprovedthattheyarenevernegativeanddonotvanishforfinitevaluesofthesemiaxes.Therefore,theNmatrixofallfiniteellipsoidsispositivedefiniteandalwayshasaninverse100N-1.Theonlyexceptionarethecaseswhenoneortwosemiaxesbecomeinfinite,wherethecorrespondingeigenvaluesvanish,asdiscussednext.ThecalculationoftheinternalandexternalfieldsforthesecasesisdiscussedatsectionsSheetofconstantthicknessandinfiniteextensionandRightcircularcylinderofinfinitelength.

InfinitesemiaxesandtheinverseofN

Theconstantthicknesssheetofinfiniteextensionandtheinfinitelylongrightcircularcylinderaretwocasesoftendiscussedinintroductorycoursesofelectromagnetismbecauseofthesimplicityoftheirtreatment.Theyarenotrealbodies,buttheyprovideusefulillustrationofimportantpropertiesandthefieldsthusobtainedaregoodaproximationestothoseofaverylongprolatespheirodneartheequator(forthecylinder’scase)andtoaveryshortoblatespheroidnear

99 Press,W.H.&Teukolsky,S.A.&Vetterling,W,T,&Flannery,B.P.;NumericalrecipesinC;

CambridgeUniversityPress(UnitedKingdom);1992.100 GoertzelandTralli,p.16.


theaxis(forthesheet’scase).Bothcasescanbetakentobethelimitsofspheroidalellipsoidswhereoneaxis(thesymmetryone)ortwoaxis(theequatorialones)areinfinitelylong.Athirdcasewillalsobeconsidered,theinfiniterightellipticcylinder,butonlyafterthetriaxialellipsoidsisdiscussed.

Whensemiaxisabecomesverylarge,thefollowinglimitisobtainedforitsintegraldefinitioneq.3.68:

Na(κ )= lima→∞

abc2

ds

a2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= lima→∞

bc2

ds

a4/3 + sa2/3

⎛⎝⎜

⎞⎠⎟

3/2

b2 + s( ) c2 + s( )κ

∞

∫ =0. (3.73)

Asthesamethinghappensforallcoordinateaxes,whenthevalueofsemiaxisα→,thecorrespondingprincipalvalueNαgoesto0.Oneshouldthencheckifthe

othertwoprincipalvalues,eqs.3.69and3.70,remainfinite,whichinourexampleareNbandNc.ForNb,

Nb(κ )= lima→∞

abc2

ds

b2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= lima→∞

bc2

ds

b2 + s( )3/2 1+ sa2

⎛⎝⎜

⎞⎠⎟c2 + s( )κ

∞

∫ = bc2ds

c2 + s b2 + s( )3/2κ

∞

∫ , (3.74)

wheretheintegrandisfiniteandnon-vanishing.

ThesamehappensforNc(λ),

Nc(κ )= lima→∞

abc2

ds

c2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= lima→∞

bc2

ds

c2 + s( )3/2 1+ sa2

⎛⎝⎜

⎞⎠⎟b2 + s( )κ

∞

∫ = bc2ds

b2 + s c2 + s( )3/2κ

∞

∫ . (3.75)

Thelasttwocasescorrespondtothesametypeofintegral,thatwillbesolvedwhendiscussingtheinfinitelylongellipticcylinderinnextchapter.

Thecaseb=ccorrespondstotheinfinitelylongrightcircularcylinder,where

Nb(κ )=Nc(κ )=

b2

2ds

b2 + s( )2κ∫ = − b2

2 b2 + s( )κ

∞

= 12

b2

b2 +κ( ) . (3.76)

∞


Thenextcaseforanalysisisthesheetofconstantthicknessandinfiniteextension.Tothatendconsideranellipsoidwithfinitesemiaxisc,lettingtheothertwogrowwithoutlimit.Inordertopreservetheirratiotheirvaluesarewrittenaskaandkb,andthelimitk→∞istaken.Then

limk→∞

Nc(κ )= limk→∞

ka( ) kb( )c2

ds

c2 + s( )3/2 k2a2 + s( ) k2b2 + s( )κ

∞

∫

= limk→∞

abc2

ds

c2 + s( )3/2 a2 + sk2

⎛⎝⎜

⎞⎠⎟b2 + s

k2⎛⎝⎜

⎞⎠⎟

κ

∞

∫ = c2ds

c2 + s( )3/2κ

∞

∫

= − c

c2 + s κ

∞

= c

c2 +κ.

(3.77)

TheinfinitelimitfortheintegralexpressionsforNx(κ)andNy(κ)iseasilyprovedtovanish.Thesameresultisobtainedatpage54byusingelectrostaticGauss’sLaw,wherethefieldoutsidethebodyisalsocalculatedshowingittobeindependentoftheratioa/b,sothatthereisnoellipticsheetofinfiniteextensión(althoughthereisarightellipticcylinderofinfinitelength,seeeqs.4.89and4.89).Theresultisconfirmedbytakingappropiatelimitsinthecasesoftheellipticcylindereq.4.16andtheoblatespheroideq.4.29withaspectratioγ=0.InallthesecasestheinverseofNdoesnotexistandoneshouldsolvethe

equationsthatrelatethecomponentsNα,theappliedfieldandthetotalfieldinsidethebody,asillustratedforthesheetofconstantthicknessandinfiniteextensionatpage55andfortherightcircularcylinderofinfinitelengthalpage57.

Nisinvariantforsimilarellipsoids

AnimportantpropertyoftheinternaldepolarizationtensorNisitsindependenceofthebody’svolume,oftheproblem'sscale.Thisisnotevidentfromamereinspectionofeq.4.6.Toprovetheassertionconsiderasimilarellipsoid,onewhosesemiaxesa’,b’,c’differfromtheoriginalonesbyafiniteconstantfactork:a'=k·a,b'=k·b,c'=k·c.Theprincipalvaluesofthissimilarellipsoidturnouttobe

Nα (κ |ka,kb,kc)=

k3abc2

ds

k2d2α + s( )2 k2a2 + s( ) k2a2 + s( ) k2a2 + s( ),

κ

∞

∫ (3.78)

apparentlydifferent.Achangeofintegrationvariables=k2s'doesnotmodifythelimitsofintegrationwhenκ=0andgivestheidentity


N(0|ka,kb,kc)= k3abc2

k2ds 'k2d2α +k

2s '( )2 k2a2 +k2s '( ) k2a2 +k2s '( ) k2a2 +k2s '( )0

∞

∫

= abc2ds '

d2α + s '( )2 a2 + s '( ) a2 + s '( ) a2 + s '( )0

∞

∫ =N(0|a,b,c) (3.79)

N 'α =k3abc2

k2ds

k2nα2 +k2s '( ) k2a2 +k2s '( ) k2b2 +k2s '( ) k2c2 +k2s '( )0

∞

∫

= abc2ds

nα2 + s '( ) a2 + s '( ) b2 + s '( ) c2 + s '( )0

∞

∫ =Nα .

Itfollowsthat

Nα(0|a,b,c)=Nα(0|k·a,k·b,k·c),whereα=x,y,z. (3.80)

Itshouldbestressedthatthepropertydoesnotapplyfortheintegralswithκ ≠ 0.

Therefore,theinternaldepolarizationtensorsofsimilarellipsoidsareequal.Thisisofimportanceforthediscussionofcavitiesinellipsoidalbodiesmadeinpage116.Itshouldbestressedherethattheexternaldepolarizationtensornextdoesnothavethisproperty;infact,itsvalueisproportionaltothebody’svolumeV.

N(0)isdeterminedbyaspectratiosTheprincipalvaluesofN(0)arenotafunctionofthethreesemiaxes,butofthe

ratiosoftwoofthemtoathird,theiraspectratios.Thisisacorollaryofthepreviouspropertywhenfactorkineqs.3.80istakentobetheinverseofthelargestsemiaxis,inthisbooktakentobea.

Nα (0)= 12aabaca

ds

dα /a( )2 + s⎛⎝

⎞⎠ a/a( )2 + s⎛

⎝⎞⎠ b/a( )2 + s⎛⎝

⎞⎠ c /a( )2 + s⎛⎝

⎞⎠

0

∞

∫

= 12βγ

ds

δα2 + s( ) 1+ s( ) β 2 + s( ) γ 2 + s( )0

∞

∫ ,

where α = x , y ,z , β = ba,γ = c

a, δ x =1, δ y = β , δ z = γ .

(3.82)

ThisistheconventionusedinthetabulationsofNα(0)forthetriaxialellipsoid,theonlytypewheretheeigenvaluescannotbeexpressedintermsofelementaryfunctions.


Forfurtherapplicationsitisconvenienttoidentifythevaluesofaspectratios βandγfordifferentcategoriesofellipsoids,asshowninTable3,where

a≥b≧c,1≥β≧γ . (3.83)

a b c β γ Typeofellipsoid

∞ ∞ c −101 0 sheetofconstantthicknessandinfiniteextension(seep.54)

∞ b b 0 0 rightcircularcylinderofinfinitelength(seep.56)

∞ b c<b 0 0 rightellipticcylinderofinfinitelength(seep.58)

a a c<a 1 c/a oblatespheroid(seep.69)

a b<a b b/a b/a prolatespheroid(seep.74)

a a a 1 1 Sphere(seep.58)

a b<a c<b b/a c/a triaxialellipsoid(seep.65)

Table3.Valuesofβ andγ correspondingtodifferenttypesofellipsoids.

101 Seeeq.3.77.


Chapter4:Thedepolarizationtensor:advancedtreatment

CalculationsofnfromgravitationalpotentialsThegravitationalpotentialofallellipsoidalshapeshavebeensolvedbothinside

andoutsidethebody.Theformulasobtainedareusefulnotonlyforthecalculationofthedepolarizationtensoroftheasymmetrictriaxialellipsoidbutalsoforthefollowingshapes:

• a>b>c:triaxialellipsoid;• a≫b>c:rightellipticcylinderofinfinitelength;• a=b>c:oblatespheroid;• a>b=c:prolatespheroid.

Theformulasgiveninthepreviouschapterfortheinfinitesheet,infinitecylinderandsphereprovideaverificationofthevalidityoftheformulasthatwillbeobtainednext.

Triaxialellipsoid

Thetriaxialellipsoidistheshapewhereallthesemiaxesaredifferent.Itsprincipalcoordinatesystem—asinallcases—istheonesuchthataxesx,y,zcoincidewiththeorderedsemiaxes a>b>c. (4.1)

Aswillbeseenshortly,thecomponentsofthedepolarizationtensornareheregivenbyellipticintegrals,non-elementarytrascendentalfunctionsunfamiliarformanyphysicistsandengineers.Astheirvaluesareusuallycalculatedfromseriesexpansionsortakenfromtables,

Figure7.Triaxial

ellipsoid.

thesefunctionsareusuallyexpressedintermsofafewcanonicalornormalones(Legendre’sellipticfunctions),twoofwhichappearhere.ItsdefinitionsandpropertiesintherangeofinterestaregivenatAppendix9.Nevertheless,aspointedoutinsectionIntegralexpressionsoftheeigenvaluesofN,itmightprovesimplertousethesingleintegraleq.3.68forthenumericalcalculationofinternaldepolarizationfactors.Forthesereasonthecomponentsofnwillbeexpressedintwodifferentways.Thecalculationof V(

!r ), theinternalgravitationalpotentialofanhomogeneoussolidellipsoidofmassdensityσcanbemadeusingeitherellipsoidal102or

102 Stratton,pp.207-211.


cartesiancoordinates103.Thelatter,usedherebecauseitisthemoreconciseapproach,wasmadein1839bythefrenchmathematicianGustaveLejeuneDirichlet104andwidelyreproducedafterwardsinbooksonpotentialtheory.TheversionofDirichlet’sproofgivenbyMacMillan105issummarizednext,wherethegravitationalconstantwaschosenbythisauthortobeunity106.Takingascoordinateaxestheprincipalonesoftheellipsoid,MacMillanchanges

toasystemofcoordinateswithoriginatfieldpoint Theradialpartofthevolumeintegralturnsouttohaveanupperlimitwhichisafunctionof Usingsymmetryarguments,heshowsthatseveraloftheresultingintegralsvanish,leadingtoaquadraticexpressioninthefieldpointcoordinates,

V(!r )=σ d3r '

!r − !r 'V∫∫∫ =C0 +Cxx

2 +C y y2 +Czz

2. (4.2)

Thefourcoefficientsarefunctionsoftheparametersa,bandcalone,andallmaybeexpressedintermsofasingleintegralanditsderivativesrespectofthedifferentsemiaxes107,

Cα =

1dα

∂C0∂dα

−C0dα2 , where α = x , y ,z , dx = a,dy = b,dz = c. (4.3)

Afterachangeofintegrationvariable108moresymmetricexpressionsforCx,Cy,andCz(seeeq.3.71)areobtained,asgivenbelow109.Asallthesecoefficientsareneededtoevaluatetheexternaldepolarizationtensornext(seesectionGeneralexpressionofexternaln),itisconvenienttousetheauxiliaryfunctionfeq.3.8:

f (!r )= −V(

!r )4πσ = abc4 −I0 + Ix x

2 + I y y2 + Izz

2( ) , (4.4)

where

I0(0)=ds

a2 + s( ) b2 + s( ) c2 + s( )0

∞

∫ , Iα =ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )0

∞

∫ ,

where α = x , y ,z and dx = a,dy = b,dz = c. (4.5)

103 MacMillan,pp.45-62.104 Stoner,p.807.105 MacMillan,section32,pp.45-49.106 MacMillan,p.24,eq.20.1.107 MacMillan,p.48eq.32.10.

108 Thesustitutionsinϕ = c / c2 + s replacesthepolarangleϕbythenewintegrationvariables,

andtheupperlimitofintegrationπ/2by∞.109 MacMillan,p.49eq.32.13.

!r . !r .


NoticethatthenotationIj(0)fortheintegralscomesaboutbecausesitturnsoutthattheyareaspecialcaseoftheintegralsIj(κ)discussedinsectionObtentionoftheexternalgravitationalpotentialbyIvory’smethod.

Fromeqs.4.4,4.5and3.4thedepolarizationfactorsN(0)aregivenby

Nα (0)= 12abc

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )0

∞

∫ ,

where α = x , y ,z and dx = a,dy = b,dz = c. (4.6)

Theintegralsmaybecheckedforthedepolarizationtensorofthesphere(a=b=c=R),where

(4.7)

coincidentwiththevaluespreviouslyderivedbothbysymmetryargumentsandbyusingGauss’sLaw(eq.3.66andProblem19).

InternaldepolarizationtensorN

ExpressionswithnormalellipticintegralsEandF

Theintegralseqs.4.4arereducedtoLegendre'sincompleteellipticintegralsofthefirstandsecondkindF(φ,k)andE(φ,k)(seeAppendix9)bythesustitution

s = a2

tan(t)2 −c2

sin(t)2 . (4.8)

Itisthusobtained110:

Nx(0)=Na(0)=β ⋅γ

1−β 2( ) 1−γ 2−E(φ ,k)+F(φ ,k)( ) ,

Ny(0)=Nb(0)= −γ 2

β 2 −γ 2 +β ⋅γ 1−γ 2

1−β 2( ) β 2 −γ 2( )E(φ ,k)−β ⋅γ

1−β 2( ) 1−γ 2F(φ ,k),

Nz(0)=Nc(0)=β 2

β 2 −γ 2( ) −β ⋅γ

β 2 −γ 2( ) 1−γ 2E(φ ,k),

(4.9)

110 MacMillan,pp.49-50;Osborn,eqs.2.1to2.6;Stoner,eqs.3.5–.

!!

NR =12R

3 ds

R2 + s( ) R2 + s( ) R2 + s( ) R2 + s( )0

∞

∫

= 12R

3 R2 + s( )−5/2ds0

∞

∫ = 12R

3 − 23 R2 + s( )−3/2⎡

⎣⎢

⎤

⎦⎥0

∞

= 13 ,


where β = b

a, γ = c

a, k = 1−β 2

1−γ 2 , 0≤φ = cos−1γ ≤ π2 . (4.10)

Theseprincipalvalueshavetheunittracepropertyeqs.3.15.Ifonelooksatthedenominators,formulasseemtobevalidonlyfor

β ≠1(b≠ a),β ≠γ (b≠ c)andγ ≠1(c ≠ a).

ThisturnsouttobefalseasmaybeseenfromFigure14,Figure15andFigure16.Thereasonisthat,whentakinglimits,expressionseqs.4.9convergetotherightvalues.

Nellipticcylinder

Theellipticcylinderisthetriaxialellipsoidsuchthatonesemiaxis—ainourordering—isextremelylarge.Itincludes,asaspecialcase,therightcircularcylinderofinfinitelength(b=cinourconvention).Fromsymmetryargumentsthepropertiesofthecylinderareindependentofcoordinatex,andthesystemreducestoanapparentlytwodimensionalone.Thus—asfollowsalsofromtakingthelimita→∞ineq.A7.2—theequationofthebody’ssurfaceis

Figure8.Ellipticcylinder.

y2

b2+ z

2

c2=1. (4.11)

TheexpressionsofNα(κ )111isgivenbyeqs.3.73,

Naec(κ )=0. (4.12)

Thetwoothertermsaregivenbyeqs.3.74and3.75,wherethevalueforNcmaybeobtainedfromthatofNbbypermutingawithb.Solvingtheintegralitisobtained112

N ecb (κ )=

bc2

ds

b2 + s( )3/2 c2 + sκ

∞

∫ = bc22

b2 − c2c2 + s

b2 + sκ

∞

= bcb2 − c2

1− c2 +κ

b2 +κ

⎛

⎝⎜

⎞

⎠⎟ .

(4.13)

111 AllintegralsIα(κ )areevaluatedforκ≠0becauseotherwisethecalculationswillhavetobeduplicatedinthesectiondevotedtothecalculationoftheExternaldepolarizationtensor.

112 Korn&Korn,p.941,eq.148.


Permutingbandcgives

N eca (κ )=0, N ec

b (κ )=bc

b2 − c2b2 +κ − c2 +κ

b2 +κ

⎛

⎝⎜

⎞

⎠⎟ ,

N ecc (κ )=

bcb2 − c2

b2 +κ − c2 +κ

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ .

(4.14)

Thedepolarizationfactorsareobtained113bysettingκ=0inthepreviousequations,giving

N eca (0)=0, N ce

b (0)=c

b+ c, N ce

c (0)=bb+ c

. (4.15)

Thevaluessatisfytheunittraceruleeq.3.15andtheorderNa≤Nb≤Na.NoticethattheseeigenvaluescannotbeobtainedfromthegraphsgivenbyFigure14,Figure15orFigure16.

Fortherightcircularcylinderthelasttwoeigenvaluesreducetothevalue½givenbyeq.3.57.Fromhereonemayobtaintheconstantthicknesssheetofinfiniteextension.Thelimitstotakeare

Nbcts(0)= lim

b→∞N ce

b (0)= limb→∞

cb+ c

=0,

Nccts(0)= lim

b→∞N ce

c (0)= limb→∞

bb+ c

= limb→∞

11+ c

b

=1, (4.16)

whichcoincidewiththecalculationsmadeusingGauss’sTheorem,eq.3.47.

Noblatespheroid

Anoblatespheroidis,inourconvention,theellipsoidofrevolutionsuchthatitspolarsemiaxisc,itsaxisofrotationalsymmetry,issmallerthantheequatorialones:a=b>c.TheequatorialeigenvalueNeoftheoblatespheroidisobtainedfromeq.4.6.Asthesameintegralswillbeusedlaterfortheexpressionoftheexternaltensor,thelowerlimitκistaken,thatfor

Figure9.Oblatespheroid.

theinternalcaseshouldbemadezero.Thus,

113 Osborn,eqs.2.17and2.18.MacMillan,p.71,usevalueofVwithκ=0.


No

e(κ )=Noa(κ )=No

b(κ )=a 2c2

ds

c2 + s a2 + s( )2κ

∞

∫ ,

(4.17)

TheprimitiveofthisintegralisgiveninAppendix8,theconstantsbeingA=c2,B=a2,Α<B.Therefore,eq.A8.5appliesgivingthefollowingequatorialterm.

Noe(κ )=

a2c2

c2 + sa2 − c2( ) a2 + s( ) +

1a2 − c2( )3/2

arctan c2 + sa2 − c2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

κ

∞

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2π /2−arctan c2 +κ

a2 − c2⎛

⎝⎜

⎞

⎠⎟ .

(4.18)

Using

tan π

2 −ϕ0⎛⎝⎜

⎞⎠⎟= 1tanϕ0

= a2 − c2

c2 +κ= tanϕ , (4.19)

Noe(κ )=No

a(κ )=Nob(κ )

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ .

(4.20)

Alternativeexpressionsexpressionsforangleϕ canbeobtainedasfollows114.

tanϕ = sinϕcosϕ , sinϕ = k a2 − c2 , cosϕ = k c2 +κ ,

sin2ϕ + cos2ϕ = k2 a2 +κ( ) =1, sothat k = a2 +κ( )−1/2 ,

therefore sinϕ = a2 − c2

a2 +κ<1, cosϕ = c2 +κ

a2 +κ<1.

(4.21)

114 MacMillan,p.62eq.39.2,usestheexpressionintermsofarcsin.


N oe(κ )=No

a(κ )=Nob(κ )

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2arccos c2 +κ

a2 +κ

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2arcsin a2 − c2

a2 +κ.

(4.22)

Forκ=0,theeigenvaluesofNmaybeexpressedintermsoftheaspectratioγasfollows.Noticethatβ=γ,sothateitheronemaybeusedintheequation.ThesecondischosensothatFigure10andFigure12havethesameindependentvariable.

N oe(0)=No

a(0)=Nob(0)

= − 12γ 2

1−γ 2( ) +12

γ

1−γ 2( )3/2arctan 1−γ 2

γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= − 12γ 2

1−γ 2( ) +12

γ

1−γ 2( )3/2arccos(γ )

= − 12γ 2

(1−γ 2)+12

γ(1−γ 2)3/2 arcsin 1−γ 2( ).

(4.23)

OsbornandMaxwellusetheexpressionintermsofarcsin115insteadofthesimpleroneintermsofarccos.Thepolartermmaybeobtainedsolvingthedefinitionintegraleq.4.6,whose

primitiveisgiveninAppendix8,wheretheconstantstakethevaluesA=c2,B=1sothatΑ<B.Therefore,eq.A8.11appliesgiving

115 Osborn,eq.2.19.Maxwell,p.69eqs.438.11and438.12,whereNp=-L/4π,Ne=-M/4π=-

N/4π,e2=1–γ2.


N op(κ )=No

c(κ )=a2c2

ds

c2 + s( )3/2 a2 + s( )κ

∞

∫ ,

= − a2c

c2 + s a2 + s( )− a2c c2 + sa2 − c2( ) a2 + s( ) −

a2c

a2 − c2( )3/2arctan c2 + s

a2 − c2⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

κ

∞

= a2c

a2 − c2( ) c2 +κ+ a2c

a2 − c2( )3/2arctan c2 +κ

a2 − c2⎛

⎝⎜

⎞

⎠⎟ −

π2

⎡

⎣⎢⎢

⎤

⎦⎥⎥.

(4.24)

Using tan ϕ −π /2( ) = −1/tan(ϕ0), (4.25)

andeqs.4.21,thefollowingexpressionsareobtainedforN∫c(κ).

N op(κ )=No

c(κ )

= a2c

a2 − c2( ) c2 +κ− a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

= a2c

a2 − c2( ) c2 +κ− a2c

a2 − c2( )3/2arccos c2 +κ

a2 +κ

= a2c

a2 − c2( ) c2 +κ− a2c

a2 − c2( )3/2arcsin a2 − c2

a2 +κ.

(4.26)

Forκ=0,thiseigenvalueofNmaybeexpressedintermsoftheaspectratioγ asfollows:

N op(0)=No

c(0)

= 11−γ 2( ) −

γ

1−γ 2( )3/2arctan 1−γ 2

γ

= 11−γ 2 −

γ(1−γ 2)3/2 arccos(γ )

= 11−γ 2 −

γ(1−γ 2)3/2 arcsin 1−γ 2( ).

(4.27)


Notice,byinspection,thateqs.4.23and4.27fulfilltheunittracerule

TrN =2Neo +Np

o =1. (4.28)

Ifγ≪1(veryflatoblatespheroid),theeigenvaluesmaybeapproximatedby116

Ne =Na =Nb ≅

π4 γ −γ 2 , Np =Nc ≅1−

π2γ +2γ 2 , (4.29)

whichshouldbecomparedwiththevaluesgivenatsectionSheetofconstantthicknessandinfiniteextension.

Figure10.NpandNeasafunctionofγ fortheoblatespheroid.

Figure10showsthevaluesoftheequatorialeigenvalueNeandthepolaroneNpfortheoblatespheroid.Thevaluesβ =γ=0 (a=b=∞)correspondtotheconstantthicknesssheetofinfiniteextensionwhereNp=1,Ne=0(Table3andeq.3.47);thevaluesofthesphereareβ=γ=1 (eq.3.66).BothfunctionswillturnouttobeimportantinthediscussionoftheeigenvaluesofNforthetriaxialellipsoid(seesectionGraphsofNa,NbandNc)wheretheyprovideupperorlowerboundsofcertaineigenvalues.

116 Osborn,eq.2.21.


Nprolatespheroid

Aprolatespheroidisanellipsoidofrevolution(spheroid)suchthat—inourconvention—thepolarsemiaxisNa,itsaxisofrotationalsymmetry,isgreaterthantheequatorialones:a>b=c.

TheequatorialeigenvaluesofNfortheprolatespheroidareobtainedfromeq.4.6.Asthesameintegralswillbeusedlaterfortheexpressionoftheexternaltensor,thelowerlimitκistaken,thatfortheinternalcaseshouldbemadezero.Usingtheupperindex“p”toidentifytheprolatespheroid,itisfoundthat

Figure11.Prolatespheroid.

Np

e(κ )=Npb(κ )=Np

c(κ )=ac2

2ds

a2 + s( ) c2 + s( )2κ

∞

∫ . (4.30)

Theintegraleq.4.30isdiscussedinAppendix8,wheretheconstantsareA=a2,B=c2,A>B,sothateq.A8.6applies.Itisthusfound

Npe(κ )=Np

b(κ )=Npc(κ )

= − 12γ 2 1+ s

1−γ 2( ) γ 2 + s( ) −14

γ 2

1−γ 2( )3/2ln 1+ s − 1−γ 2

1+ s + 1−γ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

∞

κ

= 12

ac2 a2 +κa2 − c2( ) c2 +κ( ) +

14

ac2

a2 − c2( )3/2ln a2 +κ − a2 − c2

a2 +κ + a2 − c2

⎛

⎝⎜

⎞

⎠⎟ .

(4.31)

Thevalueofthelogarithmcanbemadepositive(argument>1)andthisfunctioncanbereplacedbythehyperboliconesbyusingthesomeofthefollowingidentities:

ln a2 +κ − a2 − c2

a2 +κ + a2 − c2

⎛

⎝⎜

⎞

⎠⎟ = − ln a2 +κ + a2 − c2

a2 +κ − a2 − c2

⎛

⎝⎜

⎞

⎠⎟ = −2ln a2 +κ + a2 − c2( )

c2 +κ

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (4.32)


Thefollowingrelationship117isusedtoobtainalternativeexpressions:

if coshα = a2 +κc2 +κ

,

eα = coshα + sinhα = coshα + cosh2α −1 = a2 +κ + a2 − c2( )c2 +κ

. (4.33)

Npe(κ )=Np

b(κ )=Npc(κ )

= 12

ac2 a2 +κa2 − c2( ) c2 +κ( ) −

14

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2

⎛

⎝⎜

⎞

⎠⎟

= 12

ac2 a2 +κa2 − c2( ) c2 +κ( ) −

12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟

= 12

ac2 a2 +κa2 − c2( ) c2 +κ( ) −

12

ac2

a2 − c2( )3/2cosh−1 a2 +κ

c2 +κ.

(4.34)

ThecorrespondingeigenvaluesofNarefoundsettingκ=0ineqs.4.34andusingtheaspectratioγ.Noticethatβ=γ,sothateitheronemaybeusedintheequation,andthesecondischosensothatFigure10andFigure12havethesameindependentvariable118.Then

Npe(0)=Np

b(0)=Npc(0)

= 12

11−γ 2( ) −

14

γ 2

1−γ 2( )3/2ln 1+ 1−γ 2

1− 1−γ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 12

11−γ 2( ) −

12

γ 2

1−γ 2( )3/2ln 1+ 1−γ 2( )

γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 12

11−γ 2( ) −

12

γ 2

1−γ 2( )3/2cosh−1 1

γ⎛⎝⎜

⎞⎠⎟.

(4.35)

117 Korn&Korn,p.817eqs.21.2-37118 Osborn,eq.2.11.Macmillan,p.63eq.4withκ=0.


Theintegralexpressionforthepolartermmaybeobtainedfromeq.4.6giving

Np

p(κ )=Npa(κ )=

ac2

2ds

a2 + s( )3/2 c2 + s( )κ

∞

∫ (4.36)

TheprimitiveofthisintegralisgiveninAppendix8.AsA=a2,B=c2,A>B,eq.A8.12shouldbeusedobtaining

Npp(κ )=Na

p(κ )= ac2

a2 − c2( ) a2 + s+ 12

ac2

a2 − c2( )3/2ln a2 + s − a2 − c2

a2 + s + a2 − c2⎛

⎝⎜

⎞

⎠⎟

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

κ

∞

= − ac2

a2 − c2( ) a2 +κ+ 12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟ .

(4.37)

wheretheargumenthasbeeninvertedtomakepositivethevalueofthenaturallogarithm.Thesamealternativesasforeqs.4.34arevalidhere,giving

Npp(κ )=Na

p (κ )

= − ac2

a2 − c2( ) a2 +κ+ 12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟

= − ac2

a2 − c2( ) a2 +κ+ ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟

= − ac2

a2 − c2( ) a2 +κ+ ac2

a2 − c2( )3/2cosh−1 a2 +κ

c2 +κ.

(4.38)

TheeigenvaluesofN(0)intermsofaspectratioγ are

Npp(0)=Np

c(0)

= − γ 2

1−γ 2( ) +12

γ 2

1−γ 2( )3/2ln 1+ 1−γ 2

1− 1−γ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= − γ 2

1−γ 2( ) +γ 2

1−γ 2( )3/2ln 1+ 1−γ 2

γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= − γ 2

1−γ 2( ) +γ 2

1−γ 2( )3/2cosh−1 1

γ⎛⎝⎜

⎞⎠⎟.

(4.39)


Forveryslenderprolatespheroidsthefollowingapproximationsarevalid119,

Ne(0)=Nb(0)=Nc(0)≅ 12 1+γ 2( )− 12γ 2 ln(2/γ ),

Np(0)=Na(0)≅ −γ 2 +γ 2 ln(2/γ ), if γ ≪1. (4.40)

Theaspectratioγvanishesinthelimitofinfinitelylargea,whereupononegetstheeigenvaluesoftherightcircularclinderofinfinitelengthpreviouslyobtainedbyusingtheelectrostaticGauss’sLaw(seeeqs.3.57).

Figure12.NpandNeasafunctionofγ fortheprolatespheroid.

Figure12showsthevaluesofthepolareigenvalueNpandtheequatorialoneNefortheprolatespheroid.Thevaluesγ=0(a=∞),Np=0correspondtotheinfiniterightcircularcylinderwhereNe=½(Table3andeq.3.66);thevaluesβ�= γ=1,Na=Nb=Nc=1/3,tothesphere(eq.3.66).BothfunctionswillturnouttobeimportantinthediscussionoftheeigenvaluesofNforthetriaxialellipsoid(seesectionGraphsofNa,NbandNc)wheretheyprovideupperorlowerbounds.

UnifiedtreatmentofNforspheroids

AspointedoutinsectionIntegralexpressionsoftheeigenvaluesofN,asingleintegrallikeeq.3.68characterizesalleigenvaluesofN.Thesamethinghappenswithmorespecializedintegrals,liketheoneswithtwoequalsemiaxesthatgivethespheroid’seigenvalues.ThereasonforhavingdifferentintegralsforNa,Nb,Ncistheconventiona≥b≥c,wichunderlieseq.4.51.Thissubtlepointisdiscussednext.

Considertheintegralexpressionfortheequatorialeigenvalueoftheoblatespheroideq.4.17,wherethedomainofpositiverealnumbersisdividedintwo,

119 Osborn,eq.2.13.


Noe =

12γ

ds

1+ s( )2 γ 2 + s0

∞

∫ , where 0≤γ = ca≤1,

N ? = 12δ

ds

1+ s( )2 δ 2 + s0

∞

∫ , where 1≤δ = ca. (4.41)

Theaddedrangesareexpectedtogiveasinglecontinuouscurvewithcontinuousslope.Thecurvefor0≤γ≤1—thesamegiveninFigure10—isthedashedoneinthelefthalfofFigure13.N?(δ ),notrepresentedthere,isanincreasingfunctionofδwithminimumvalue1/3atδ=1andmaximumvalue1/2atδ=∞.Asthatdomainoftheindependentvariablemaynotberepresenteddirectly,itisconvenienttomakeachangeofvariableto1/δ,thatcorrespondstothedomain[1,0],itsgraphbeingthedashedcurveontherightsideofofFigure13.Thisresultingintegralis

N ? = 121δ

ds

1+ s( )2 1δ 2 + s

0

∞

∫ = 12

δ 4ds

δ 2 +δ 2s( )2 1+δ 2s0

∞

∫

= 12

δ 2ds 'δ 2 + s '( )2 1+ s '0

∞

∫ , where s'=δ 2s , 1≥ 1δ= ac≥0.

(4.42)

Thelastintegralcoincideswiththatfortheequatorialeigenvalueoftheprolatespheroid,eq.4.30.Itsgraphic—previouslygiveninFigure12—isthedashedlineattherightofFigure13.Thedefinitionδ=a/cseemstocontradictitsidentificationwiththeparameterγ=c/a.Asitwasinitiallyassumedthatc>a,bytheconventiona≥b≥ctheexchangeofnamesa↔cshouldbemade.ThatiswhyintherightsideofFigure13theindependentvariable,thatdoesnotrespecttheconvention,islabeledasa’/c’.

Therefore

N ? = 1

2γ2 ds

γ 2 + s( )2 1+ s0

∞

∫ =Npe (4.43)

istheequatorialeigenvalueoftheprolatespheroid,eq.4.30.Thereadershouldverify,inasimilarfashion,thattheextensiontoγ>1oftheexpressionofthepolareigenvalueoftheoblatespheroidgivesthepolareigenvalueoftheprolatespheroid,thecontinuouslineinFigure13.Thefunctionsthatgivethedependenceoftheeigenvaluesisverydifferentfor

theoblate(inversetrigonometricfunctions)andtheprolatespheroid(logarithmorinversehyperbolicfunctions),asseenineqs.A8.5andA8.6.TheoriginofthedifferenceisthesignofΔ=±(1-γ2),thatis,wethercissmallerorgreaterthana.


MacMillan120hasshownthattheuseofcomplexvariableunifiesbothsolutions(seeProblem26)

GraphsofNa,NbandNc

Figure13.Theequatorialandpolareigenvalues

ofspheroidsarecontinuouscurves.

ForaccuratecalculationsoftheeigenvaluesofNthebestchoiceistousemathematicalsoftwarethatincludestheevaluationoftheincompleteellipticfunctionsE(φ,k)andF(φ,k).Ontheotherhand,abetterunderstantingoftheirbehaviorisobtainedbytheuseofgraphs.AsdiscussedinsectionN(0)isdeterminedbyaspectratios,theeigenvaluesarefunctionsoftheratiosofsemiaxesβ=b/aandγ=c/a.Thebestgraphsknownbytheauthor,thoseofOsborn,depictthemassetsoffunctionsofγ ,whereeachofthe11curveshasfixedvaluesof0≤β≤1atintervalsof0.1.ThesegraphsforNa,NbandNcarereproducednextatfullpagewidth.Asshowninthecombinedgraphforthethreeeigenvalues,Figure17,forany

fixedvalueofγtheeigenvaluesareorderedasfollows:

Na≤Nb≤Nc. (4.44)

Twofiniteeigenvaluesmaybeequalonlyfortheequatorialcaseoftheprolateandoblatespheroids(seeFigure12andFigure10).ThelowestvaluesofNaaregivenbythecurveforthepolareigenvaluesoftheprolatespheroids(seeFigure12)andthelargestvaluesofNc(γ ,γ)forthepolareigenvaluesoftheoblatespheroids(see

120 MacMillan,p.63eq.39.4.


Figure10).Thein-betweencaseNbisboundedfrombelowbyNaandfromabovebyNc.

NaFunctionNa(β,γ)isshowninFigure14,whereitsvaluesareconfinedtothe

greyshadedregion,boundedbytheuppercurveNa(1,γ)(oblatespheroid,seeFigure10)andthelowestcurveNa(γ,γ)(prolatespheroid,seeFigure12)asfollowsfromTable3theorderingofsemiaxesconventioneq.3.83.Therefore

0≦Na(γ0,γ0 )≦Na(β,γ0)≦ Na(1,γ0)≦1/3. (4.45)

Forfixedγ0,asshownbytheconstantβcurves,Na isanincreasingfunctionofβ ;thatis ifβ2>β1,Na(β2,γ0)>Na(β1,γ0). (4.46)

Eachofthecurvesdrawncorrespondtoadifferentvalueoftheparameterβ=b/a,thehighestbeingtheonefortheoblatespheroid.TheuppervertexNa(1,1)=Nb(1,1)=Nc(1,1)=1/3correspondstothesphere,thetransitionvaluefromoblatetoprolatespheroids.Thelowestvertex,Na(β,0)=Na(1,0)=0,characterizestheconstantthicknesssheetofinfiniteextension(seeTable3).

NbFunctionNb(β,γ)isshowninFigure15,whereitsvaluesareconfinedtothe

greyshadedregion,boundedbythelowercurveNb(1,γ)(oblatespheroid,seeFigure10)andtheuppercurveNa(γ,γ )(prolatespheroid,seeFigure12)asfollowsfromTable3andtheorderingofsemiaxesconventioneq.3.83.Therefore

0≦Nb(1,γ0)≦Nb(β,γ0)≦ Nb(γ0,γ0 )≦1/2. (4.47)

Forfixedγ0,asshownbytheconstantβcurves,Nbisandecreasingfunctionofβ ;thatis ifβ2>β1 ,Na(β2,γ0)<Na(β1,γ0). (4.48)

Eachofthecurvesdrawncorrespondstoadifferentvalueoftheparameterβ=b/a,thehighestbeingtheonefortheprolatespheroid.TherighthandvertexNb(1,1)=Na(1,1)=Nc(1,1)=1/3correspondstothesphere,thecommonlimitoftheoblateandprolatespheroids.Thelowestleftvertex,Nb(β,0)=Nb(1,0)=0,characterizestheconstantthicknesssheetofinfiniteextension;theupperleftvertex,Nb(0,0),characterizestherightellipticcylinderofinfinitelength(seeTable3).Theverticallineβ=0correspondstothetheellipticcylinders,whosevaluescannotbeidentifiedfromthegraph,becauseallofthemhaveβ=γ=0inthisparametrization.

NcFunctionNc(β,γ)isshowninFigure16,whereitsvaluesareconfinedtothegrey

shadedregion,boundedbytheuppercurveNc(1,γ)(oblatespheroid,seeFigure10)andthelowestcurveNa(γ,γ )(prolatespheroid,seeFigure12)asfollowsfromTable3andtheorderingofsemiaxesconventioneq.3.83.


Figure14.Na(L/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure1).


Figure15.Nb(M/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure2).


Figure16.Nc(N/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure3).


Therefore 1/3≦Nc(1,γ0,γ0 )≦Nc(1,β,γ0)≦ Nc(1,1,γ0)≦1. (4.49)

Forfixedγ0,asshownbytheconstantβcurves,Ncisanincreasingfunctionofβ ;thatis ifβ2>β1,Na(β2,γ0)>Na(β1,γ0). (4.50)

Each of the curves drawn correspond to a different value of the parameterβ=b/a,thehighestbeingtheonefortheoblatespheroid.TherightvertexNc(1,1)=Nb(1,1)=Nc(1,1)=1/3correspondstothesphere,thecommonlimitoftheoblateandprolatespheroids.Thelowestleftvertex,Nc(0,0)=Nb(0,0)=1/2,characterizestherightellipticcylinderofinfinitelength;thehighestleftvertex,Nc(1,0)=1,theconstantthicknesssheetofinfiniteextension(seeTable3).

Combinedgraph

Figure17.DistributionofeigenvaluesofN.

InFigure17thegraphsforthethreeeigenvaluesarepiledup,identifyingwithdifferentcolorstheregionsandboundarycurvesforeachone121,where

121 AsimilaronewasgiveninStoner,Figure1.


eigenvaluesNpcorrespondtothepolaraxisandNetotheequatorialonesofthespheroids.Asanexample,thesegmentsofallowedvaluesforeachareshownforγ=0.25,illustratingtheorderinggivenbyeq.3.72.Theconstantβcurveshavebeenomittedforsimplicity.

ThereadershouldexerciseitsunderstandingofthegraphswithactivitiesliketheoneproposedinProblem22andincorporatedintoFigure17.Theellipsoidsthereanalyzedarerepresentedasseenfromadirectionatequalangleswiththethreesemiaxes,andthelengthofeach(cistheverticalone)isrepresentedasthecorrespondingfractionofa.Thevalueofγiskeptconstant(½)andthatofβis½,¾or1.ForNa,onthecurveforthepolareigenvalueoftheprolatespheroid,bisequaltoc(Nb=Nc).Whengoingupalongthelineγ=½thebsemiaxisincreasesforNauntilitisequaltoaonthecurvefortheequatorialeigenvalueoftheoblatespheroid(Na=Nb).ForNbthebsemiaxisdecreasesuntilisequaltoc(Nb=Nc)onthecurvefortheequatorialeigenvalueoftheprolatespheroid.ThebehaviourisagainreversedforNc,wherebincreasesuntilitisagainequaltoa(Na=Nb)onthecurveforthepolareigenvalueoftheoblatespheroid.

Therangeofthedifferenteigenvaluesis

0≤Na≤1/3,0≤Nb≤1/2,1/3≤Nc≤1, (4.51)sothatNcistheonlyeigenvaluethatmaybeunivocallyidentifiedfromitsvalueintherange(1/2,1](seeproblemProblem20).

Externaldepolarizationtensor

ObtentionoftheexternalgravitationalpotentialbyIvory’smethod

Intheprevioussectionsthevalueofthegravitationalpotential V(!r ) insidea

solidhomogeneousgeneralellipsoid,eq.4.2,wasusedtoderivethevaluesoftheinternaldepolarizationtensorN.Ifasimilarprocedurewereusedforobtainingthevalueoutsidethebody,next,theexpressionsobtainedwouldbeextremelycumbersome.Morethanacenturyago,in1809,theScottishmathematicianJamesIvory122gaveaconcisewayofderivingthepotential’sexternalvaluesfromtheinternalonesusingellipsoidsconfocalwiththebody’ssurfaceeq.A7.2.Theseconfocalellipsoidsareobtainedfromeq.A7.20,whereκ>0isusuallydefinedasthelargestalgebraicrootofthatequation123,anditsvalueisthesaneforall

!r onthatsurface124.Theseconfocalellipsoidalsurfacesarelargerthanthebody,encloseitcompletelyandhavethesamepotentialascertaincorrespondingpointsinsidethebody(seesectionConfocalellipsoidsofAppendix7).OnlythefinalexpressionsobtainedusingIvory’smethodaregivenhere.Readersinterestedin

122 JamesIvoryatEnglishWikipedia.123 Butseeeq.Problem20.124 ForabetterunderstandingofthemethodthereadershouldstartbyreadingsectionConfocal

ellipsoidsandtacklingafterwardsProblem32,itsapplicationfortheobtentionoftheexternalpotentialofasolidandhomogeneoussphericalmass.


learninghowtoobtainthemmaylookattherelevantsectionsofMacmillan’sbook125.

ThepotentialVandtheauxiliaryfunctionfeq.3.8arethengivenby126,

f (!r )= −V(

!r )4πσ = 1

2 −N0(κ )+Na(κ ) x2 +Nb(κ )y2 +Nc(κ )z2( ) , (4.52)

wherethecoefficientsNα (0 )(α=x,y,zora,b,c)aretheeigenvaluesoftheinternaldepolarizationtensor,butalsoappearintheexpressionoftheexternaloneforκ≠0.Thesecoefficientesaregivenbythefollowingintegrals

N0(κ )=

abc2

ds

a2 + s( ) b2 + s( ) c2 + s( )κ ( !r )

∞

∫ , (4.53)

Nα (κ )=abc2

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ ( !r )

∞

∫ ,

where dx = a,dy = b,dz = c , α = x , y ,z andx2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ=1.

(4.54)

TheexpressionsoftheintegralsintermsofLegendre’sellipticfunctionsaregivenattheendofAppendix9127,wheremoredetailsaregivenaboutthesefunctions.ThecorrespondingvaluesforNj(κ )are

N0(κ )=

abc

a2 − c2F φ(κ ),k( ) , (4.55)

Na(κ )=abc

a2 − c2 a2 −b2( )−E φ(κ ),k( )+F φ(κ ),k( )( ) , (4.56)

125 MacMillan,sections35-36,pp.52-58.126 MacMillan,p.56,eq.36.1.127 SeealsoBartczak&Breiter&Jusiel;Ellipsoids,materialpointsandmaterialsegments;Celestial

Mech.Dyn.Astr.;eq.6.


Nb(κ )= −abc c2 +κ

a2 +κ( ) b2 +κ( ) b2 − c2( )+ abc a2 − c2

a2 −b2( ) b2 − c2( )E φ(κ ),k( )

− abc

a2 − c2 a2 −b2( )F φ(κ ),k( ) ,

Nc(κ )=abc b2 +κ

a2 +κ( ) c2 +κ( ) b2 − c2( )− abc

a2 − c2( ) b2 − c2( )E φ(κ ),k( ) ,

k = a2 −b2

a2 − c2, x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ=1, φ(κ )= arcsin a2 − c2

a2 +κ

⎛

⎝⎜

⎞

⎠⎟ .

(4.56)

Noticethattheonlydifferencewiththeinteriorvaluesofthepotentialeq.4.4isthatthelowerlimit0oftheintegralsisherereplacedby κ(

!r ) ,originofthenotationNα (0).

Nearandfarawaypointapproximationstonext

Expressionsforthenearorfarawaypointsarethesimplestaproximationstotheexternalfieldofauniformlypolarizedellipsoidbody,bothwhenthepolarizationispermanentorinducedbyanappliedfield.Forfieldpointsveryneartothebody’ssurfacethevalueseqs.3.41and3.42

maybeused,coincidingwitheq.4.66forκ=0:

next(!r S )=N(0)−

x2 /a4w(!r )

x ⋅ y /a2b2w(!r )

x ⋅z /a2c2w(!r )

x ⋅ y /a2b2w(!r )

y2 /b4w(!r )

y ⋅z /b2c2w(!r )

x ⋅z /a2c2w(!r )

y ⋅z /b2c2w(!r )

z2 /c4w(!r )

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

,

where w(!r )= x2

a4+ y

2

b4+ z

2

c4.

(4.57)

Forpointsfarawayfromtheuniformlypolarizedbodyitisexpectedthatthefieldwillbethedipolarone(theextensionofthesecondofeqs.3.66)generatedbythedipolemomentofthebody,presumptionthatisconfirmedbyacalculationofthelimitvalueofκ andtheresultingexternaltensor(seeProblem33).

GeneralexpressionofexternalnFromeq.3.4thecomponentsofthedepolarizationtensor,eitherinsideor

outsidetheellipsoidalbody,aregivenby


nαβ

ext(!r )= ∂2 f (!r )∂xα ∂xβ

, where !r ∉V . (4.58)

Forthepurposeoftakingfirstderivativesitisconvenienttoexpressf,eqs.4.52and4.54,asthefollowingsingleintegral:

f (!r )= 12

x2

a2 + s+ y2

b2 + s+ z2

c2 + s−1

a2 + s( ) b2 + s( ) c2 + s( )

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟ds

κ ( !r )

∞

∫ = 12 Q(s ,!r )dsκ ( !r )

∞

∫ . (4.59)

Writteninthisfashiontheintegrand Q(s ,!r ) vanishesfortheupperandlowerlimit

oftheintegral(seeeq.A7.20),sothat

∂∂xβ

Q(s ,!r )dsκ ( !r )

∞

∫ = ∂∂κ

Q(s ,!r )dsκ ( !r )

∞

∫⎛

⎝⎜

⎞

⎠⎟∂κ∂xβ

+ ∂∂xβ

Q(s ,!r )dsκ ( !r )

∞

∫

= −Q s ,!r( )κ ( !r )

∞ ∂κ∂xβ

+ ∂∂xβ

Q(s ,!r )dsκ ( !r )

∞

∫ = ∂∂xβ

Q(s ,!r )dsκ ( !r )

∞

∫

= abc22xβ

(d2 + s) a2 + s( ) b2 + s( ) c2 + s( )ds

κ ( !r )

∞

∫ =Nβ κ )( ) xβ .

(4.60)

whereeq.4.54hasbeenused.

Thatis,

∂ f (!r )∂xβ

=Nβ κ(!r )( ) xβ , (4.61)

whereN0(κ )doesnotappear.Thecalculationofthesecondderivativesoffgives

∂∂xα

∂ f (!r )∂xβ

⎛

⎝⎜

⎞

⎠⎟ =

∂∂xα

Nβ κ(!r )( ) xβ( )=Nβ κ(!r )( )δαβ + xβ

∂∂κ

Nβ κ( )( )∂κ(!r )

∂xα, (4.62)

requiresthevalueof∂κ /∂xβ , whichmaybeobtainedfromitsimplicitdefinitioneq.A7.20.Theresultwillshedlightonthemeaningoftheexternaltensorandverifythebody’ssurfacevalueofnexteq.3.41.


Fromeq.A7.20,

∂∂xβ

x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ⎛

⎝⎜⎞

⎠⎟= ∂∂xβ

xα2

α=x , y,z∑ d2α +κ( )−1

=2xβd2β +κ

+ ∂∂κ

xα2

α∑ d2α +κ( )−1⎛

⎝⎜⎞⎠⎟∂κ∂xβ

=2xβd2β +κ

− xα2

α∑ d2α +κ( )−2⎛

⎝⎜⎞⎠⎟∂κ∂xβ

= ∂∂xβ

1=0.

(4.63)

Thatis,

∂κ∂xβ

=

2xβd2β +κ

x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, (4.64)

which,asmaybeseenfromeq.A7.8,definesthecomponentsofavectornormaltothesurfaceoftheconfocalellipsoidA7.20.

ThederivativesofNβgive

∂Nβ(κ )∂xα

=∂Nβ(κ )∂κ

∂κ∂xβ

= ∂∂κ

abc2

ds

dβ + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ ( !r )

∞

∫⎛

⎝

⎜⎜

⎞

⎠

⎟⎟∂κ∂xβ

= −abc

xαd2α +κ

⎛

⎝⎜

⎞

⎠⎟ /

x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

dβ +κ( ) a2 +κ( ) b2 +κ( ) c2 +κ( ).

(4.65)

Fromeqs.4.61,4.62and4.65itisthusobtained

nαβext(!r )= ∂2 f (!r )

∂xα ∂xβ=Nβ κ( )δαβ +

∂Nβ(κ )∂xα

xβ

=Nβ κ( )δαβ −abcsα (!r ,κ ) sβ(

!r ,κ )a2 +κ( ) b2 +κ( ) c2 +κ( )

.


Therefore128

next(!r )=N(κ )−abc s(!r |κ ) s(!r |κ )a2 +κ( ) b2 +κ( ) c2 +κ( )

,

where Nαβ(κ )=δαβ

abc2

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫ (4.66)

and s(!r |κ ) istheunitvectornormaltothesurfaceoftheκ confocalellipsoid

eq.A7.20attheexternalfieldpoint !r :

sα (!r |κ )=

xαd2α +κ

x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2,

where x2

a2 +κ+ y2

b222 +κ+ z2

c2 +κ

2

=1. (4.67)

NoticethatthefirsttermofnextisafieldpointdependentextensionoftheinternaldepolarizationtensorNandproportionaltoitsvolume.Inthesecondtermthenumeratorselectscomponentsofthepolarizationnormaltotheconfocalellipsoid,whilethedenominatorgivesadistanceoftheorderofr3(seeeq.A7.22).Forκ=0itisobtainedthevalueofnextatthesurfaceofthebodyeq.3.41.

TracesInwhatfollowsitisverifiedifeq.4.66fulfillsthezerotraceruleeqs.3.15.

Trnext(!r )= nαα

ext(!r )α∑ = Nαα (κ )−abc

s2α (!r |κ )

α∑( )/ s(!r |κ )2a2 +κ( ) b2 +κ( ) c2 +κ( )

.α∑ (4.68)

128 Barczak&t&Jusiel;Ellipsoids,materialpointsandmaterialsegments;CelestialMech.Dyn.Astr.;DOI10.1007/s10569-006-9017-x.ThevaluecoincideswiththeHessianofthegravitationalpotentialV,eqs.9and10.


FromthedefinitionsofNαeqs.4.54itfollowsthat

Nαα (κ )α∑ = a⋅b ⋅c2

1a2 + s

+ 1b2 + s

+ 1c2 + s

⎛⎝⎜

⎞⎠⎟

ds

a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= abc2b2 + s( ) c2 + s( )+ a2 + s( ) c2 + s( )+ a2 + s( ) b2 + s( )

a2 + s( ) b2 + s( ) c2 + s( )⎡⎣

⎤⎦3/2 ds

κ

∞

∫

= abc2

dds

a2 + s( ) b2 + s( ) c2 + s( )( )a2 + s( ) b2 + s( ) c2 + s( )⎡

⎣⎤⎦3/2 ds = −

abc

a2 + s( ) b2 + s( ) c2 + s( )κ

∞

κ

∞

∫

= abc

a2 +κ( ) b2 +κ( ) c2 +κ( ).

(4.69)

Thatis

TrN(κ )= abc

a2 +κ( ) b2 +κ( ) c2 +κ( ), (4.70)

which,asitseigenvaluesNa(κ ),Nb(κ )andNc(κ ),isoftheorderofr3(seeeq.A7.22)andmaybeusedtosimplifyitscalculation.

Fromeq.A7.18,

s2α (!r |κ )

α∑s(!r |κ )2

=1. (4.71)

Uponreplacementintheinitialexpressionforthetrace,itisverifiedthat

Tr next(!r )=0. (4.72)

Verificacionforthesphere

Equations4.66and4.67willnextbeverifiedusingthesolutionfoundforthesphereusingtheelectrostaticGauss’sLaweqs.3.66and3.67.Tothatendthedifferentcomponentesofthoseequationsareevaluatednext.

TheeigenvaluesofN(κ)are

Na(κ )=Nb(κ )=Nc(κ )=abc2

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= R3

2ds

R2 + s( )5/2κ

∞

∫ = − R3

3 R2 + s( )−3/2κ

∞

= R3

3 R2 +κ( )−3/2 , (4.73)


wherea=b=c=R.Thevalueofκ tobeusedisthesolutionoftheequation

x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ= r2

R2 +κ=1, κ = r2 −R2. (4.74)

Therefore

Na(κ )=Nb(κ )=Nc(κ )=

R3

3 R2 +κ( )3/2= 13R3

r3, (4.75)

itsvaluebeing1/3forthesphere’sinteriorwhereκ=0.Thelasttermofeq.4.66hasinthethenumerator

s(!r |κ )= sα (

!r |κ )xαα∑ =

xα xαrα

∑ =!rr, (4.76)

andthedenominatoris

a2 +κ( ) b2 +κ( ) c2 +κ( ) = r3 , (4.77)

giving

next(!r )= 1

3R3

r31−R3

!r!rr5

= 13R

3 1r31− 3!r!rr5

⎛

⎝⎜⎞

⎠⎟. (4.78)

Thisisthedyadicsuchthatbothforpermanentorinducedelectric(eqs.2.7and2.32a)andmagneticpolarization(eqs.2.40and2.48)givesthefollowingexternaldipolarfieldscontributions(seeeqs.3.66andA4.1):

!E !p(!r )= k1

3 !p i!r( )!r − r2!pr5

,!H !m(!r )= λ '

4π3 mi

!r( )!r − r2 !mr5

, (4.79)

whereeqs.A1.4andA1.11havebeenused.

Ellipticcylinder

Thefirsttermofeq.4.66maybeobtainedfromeqs.4.12,4.13and4.14129:

N eca (κ )=0,

N ecb (κ )=

bcb2 − c2

b2 +κ − c2 +κ

b2 +κ

⎛

⎝⎜

⎞

⎠⎟ , (4.80)

129 TheseexpressionscoincidewiththosederivedfromthepotentialgivenbyMacMillanatpage

71.


N ec

c (κ )=bc

b2 − c2b2 +κ − c2 +κ

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ . (4.80)

Takinglimitfora=∞inthesecondtermofeq.4.66gives

lima→∞

abc

a2 +κ( ) b2 +κ( ) c2 +κ( )= lim

a→∞

bc

1+ κa2

⎛⎝⎜

⎞⎠⎟b2 +κ( ) c2 +κ( )

= bc

b2 +κ( ) c2 +κ( ), (4.81)

s y(!r |κ )=

yb2 +κ

y2

b2 +κ( )2+ z2

c2 +κ( )2, sz(

!r |κ )=z

c2 +κy2

b2 +κ( )2+ z2

c2 +κ( )2. (4.82)

Ascoordinatexisabsent,thevalueofκisgivenhereby(seeeq.A7.23)

y2

b2 +κ+ z2

c2 +κ=1 (4.83)

Fromeq.4.70,

Tr N ec(κ )=N eca (κ )+N ec

b (κ )+N ecc (κ )

=0+ bcb2 − c2

b2 +κ − c2 +κ

b2 +κ+ b2 +κ − c2 +κ

c2 +κ

⎛

⎝⎜

⎞

⎠⎟

= bc

b2 +κ( ) c2 +κ( )= lim

a→∞

abc

a2 +κ( ) b2 +κ( ) c2 +κ( ).

(4.84)

Forb=ctherightcircularcylinderisobtained.Thevalueofκ isobtainedfrom

y2

b2 +κ+ z2

b2 +κ=1, sothat κ = r2 −b2. (4.85)

TheeigenvaluesofNshouldberecalculatedfromeqs.4.13and4.14,

N ecb (κ )=

bc2

ds

b2 + s( )3/2 c2 + sκ

∞

∫ = b2

2ds

b2 + s( )2κ

∞

∫

= − 12b2

b2 + sκ

∞

= 12b2

b2 +κ= 12b2

r2,

(4.86)

whereeq.4.85wasused.


ThecoefficientandthefactorsofmatrixSare

b2

b2 +κ= b

2

r2, sα (

!r |κ )=xα

b2 +κy2 + z2

b2 +κ( )2=xαr, (4.87)

sothatfortherightcircularcylinderofinifinitelength

next(!r )=N(κ )− bc

b2 +κ( ) c2 +κ( )S(!r |κ )

= b2

r2

0 0 00 1/2 00 0 1/2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟− b

2

r2

0 0 00 y2 /r2 yz /r20 yz /r2 z2 /r2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

= b2

r2

0 0 00 1/2− y2 /r2 − yz /r20 − yz /r2 1/2− z2 /r2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

(4.88)

theverysamezerotraceeq.3.58obtainedusingGauss’sLaw.

Summarizing,fortherightellipticcylinderofinfiniteextension:

next(!r )=N(κ )− bc

b2 +κ( ) c2 +κ( )S(!r ,κ ), N =

0 0 00 Nb

ec 00 0 Nc

ec

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

Nbec(κ )= bc

b2 − c2b2 +κ − c2 +κ

b2 +κ

⎛

⎝⎜

⎞

⎠⎟ , N ec

c (κ )=bc

b2 − c2b2 +κ − c2 +κ

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ ,

S(!r ,κ )=0 0 00 s2y s ysz0 s ysz s2z

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, sα (

!r ,λ)=

xαdα2 +κ

y2

b2 +κ( )2+ z2

c2 +κ( )2,

y2

b2 +κ+ z2

c2 +κ=1, α = y ,z , dy = b,dz = c.

(4.89)


Oblatespheroid

Foroblatespheroidsa=b,eq.4.66gives130

next(!r )=N(κ )−a2c s(!r ,κ ) s(!r ,κ )

a2 +κ( ) c2 +κ( ), (4.90)

where

x2 + y2

a2 +κ+ z2

c2 +κ=1, (4.91)

N(κ )=Ne(κ ) 0 00 Ne(κ ) 00 0 Np(κ )

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

. (4.92)

Fromeqs.4.20and4.26,

Noe(κ )=No

a(κ )=Nob(κ )

= − 12a2c c2 +κa2 − c2( ) a2 +κ( ) +

12

a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ . (4.93)

Nop(κ )=No

c(κ )

= a2c

a2 − c2( ) c2 +κ− a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟ . (4.94)

Fromeqs.and4.67andA7.27,

sα (!r ,κ )=

xx + yya2 +κ

+ zzc2 +κ

x2 + y2

a2 +κ( )2+ z2

c2 +κ( )2,

x2 + y2( )a2 +κ

+ z2

c2 +κ=1. (4.95)

Eqs.4.20and4.26reducetotheformulasfortheinfinitesheet(seeeq.3.47)fora=b→∞andtothoseofthesphere(seeeq.3.66)fora=b=c.

Thetraceeq.4.70issatisfiedbecause



TrN(κ )=Noa(κ )+N o

b(κ )+N ob(κ )=2No

e(κ )+N op(κ )

= − a2c c2 +κa2 − c2( ) a2 +κ( ) +

a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

⎛

⎝⎜

⎞

⎠⎟

+ a2c

a2 − c2( ) c2 +κ− a2c

a2 − c2( )3/2arctan a2 − c2

c2 +κ

= a2c

a2 − c2( ) c2 +κ− a2c c2 +κa2 − c2( ) a2 +κ( ) =

a2c

a2 +κ( ) c2 +κ= aac

a2 +κ( ) a2 +κ( ) c2 +κ( ).

(4.96)

ProlatespheroidForprolatespheroidsa>b=c,eq.4.66gives131

next(!r )=N p(κ )−ac2 s(!r ,κ ) s(!r ,κ )

a2 +κ( ) c2 +κ( ), (4.97)

wherex2

a2 +κ+ y

2 + z2

c2 +κ=1, (4.98)

Np(κ )=Np

p(κ ) 0 00 Ne

p(κ ) 00 0 Ne

p(κ )

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

. (4.99)

Fromeqs.4.31and4.37,

Npp(κ )=Np

a(κ )

= − ac2

a2 − c2( ) a2 +κ+ 12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟ , (4.100)

Npe(κ )=Np

b(κ )=Npc(κ )

= 12

ac2 a2 +κa2 − c2( ) c2 +κ( ) −

14

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟ . (4.101)



Finally,fromeqs.and4.67andA7.27,

sα (!r ,κ )=

xxa2 +κ

+ yy + zzc2 +κ

x2

a2 +κ( )2+ y2 + z2

c2 +κ( )2, where x2

a2 +κ+ y

2 + z2

c2 +κ=1. (4.102)

Thetraceeq.4.70issatisfiedbecause

TrN(κ )=Npa(κ )+Np

b(κ )+Npc(κ )=Np

p(κ +2Npe(κ )

= ac2 a2 +κa2 − c2( ) c2 +κ( ) −

12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟

− ac2

a2 − c2( ) a2 +κ+ 12

ac2

a2 − c2( )3/2ln a2 +κ + a2 − c2

a2 +κ − a2 − c2⎛

⎝⎜

⎞

⎠⎟

= ac2 a2 +κa2 − c2( ) c2 +κ( ) −

ac2

a2 − c2( ) a2 +κ= ac2

a2 +κ c2 +κ( )= acc

a2 +κ( ) c2 +κ( ) c2 +κ( ).

(4.103)

Eqs.4.37and4.31reducetotheformulasfortheinfinitecircularcylinderfora→∞,b=c,andtothoseofthespherefora=b=c.

Triaxialellipsoid

Fromeqs.A9.3,A9.5,A9.6andA9.7theeigenvaluesofN(κ)maybeexpressedintermsofLegendre'sellipticfuncionsasfollows.

Na(κ )=abc

a2 − c2 a2 −b2( ) −E(φ ,k)+F(φ ,k)( ) ,

Nb(κ )= −abc c2 +κ

a2 +κ( ) b2 +κ( ) b2 − c2( )

+ abc a2 − c2

a2 −b2( ) b2 − c2( )E(φ ,k)−abc

a2 − c2 a2 −b2( )F(φ ,k),

Nc(κ )=abc b2 +κ

a2 +κ( ) c2 +κ( ) b2 − c2( )− abc

a2 − c2 b2 − c2( )E(φ ,k),

where a≥b≥ c , k = a2 −b2

a2 − c2, sinφ = a2 − c2

a2 +κ,

x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ=1.

(4.104)


Chapter5:Energy,forcesandcavities

Thermodynamicsofelectrostaticandmagnetostaticenergy

Basicconcepts

Thermodynamicsstudiesthemacroscopicexchangesofenergy(work,heat,internalandradiationenergy...)betweenamaterialsystemanditssurroundings,andthevaluestakenbyitsstatevariablesintheprocess.Tothatendstatefunctions132aredefinedsuchthat—forreversibleprocesses—itsequilibriumvaluesareafunctiononlyofthesetoffinalvaluesofthestatevariablesandareindependentofthepathfollowedtoobtainthem133.Introductorycoursesofthermodynamicsdiscussonlythecaseofgases,wherethestatevariablesarevolumeV,pressurePandtemperatureT.Thefirsttwoareclearlyinsuficientforelasticsolids,andtheelectromagneticvariableshaveyettobeadded.Theperformanceandinterpretationofexperimentswithelectricandmagnetic

fieldsandpolarizationsrequireacarefulformulationofitsthermodynamicbehaviourthatisseldomdoneinintroductorytextsonelectromagnetism.Asmanydifficultiesarefoundintheway,ashortbutgeneraloverviewwillbegiveninthischapterinordertoidentifysomeimportantfeaturesoftheproblemaandwritedown—withoutproof—theprincipalexpressionsrequired,identifyingitssource.Theauthorisnotfamiliarwithanybookgivingafullanalysisofthethermodynamicsofelectromagnetism,butthereareseveralstudiesininternationaljournalsofphysics134.

Thestatevariablesrequiredforthefullthermodynamicalcharacterizationofelasticsolidsarelistedbelow.Theyaregroupedinpairs("cause"and"effect"),throughscalarandtensorialmaterial'sproperties,bythedifferentialrelationshipthatconnectstheminthelinearrange:

o temperatureTandentropyS,relatedbytheheatcapacityc:dS = c

TdT ;

o massmandgravitationalfield !!g, relatedbythespacemetric;

o straintensoreandstresstensorσ ,relatedbythecompliancetensorcofelasticmodulus:dσ = c •de;

132 LandauandLifschitzcallthemthermodinamicpotentials,anamemoreevocativeofitsproperties.Seepp.46-96.

133 Thischaracteristicisshared,forinstance,bytheelectrostaticpotential φ.134 See,forinstance,OnMagneticandElectrostaticEnergyandTheThermodynamicsof

MagnetizationbyE.A.Guggenheim,,Proc.Roy.Soc.(London)vol.A155,pp.49—101,1936.


o electricfield !E andelectricdisplacement !!

D, relatedbytheelectricpermeability

tensor ε : !!D= ε i

E;

o magneticfield !H andmagneticinduction !!

B , relatedbythemagnetic

permeabilitytensorµ : !!B = µ i

H.

Forthemagneticcaseequilibriumreferstothesteadystatewhereelectriccurrentsareconstant.Itmustalsobestressedthattheformulasgiveninthischapterarenotvalidfordynamicphenomenasuchasthoseinvolvedintheinteractionofelectromagneticradiationwithmatter,processesthataretypicallyadiabatic.

IrreversibleprocessesarecharacterizedbyanirreversibleincreaseofentropydS.IfdQisthequantityofheattransmittedinanelementaryprocess,then

dQ=T·dS. (5.1)

Conductioncurrentsinmetalsandmagnetichysteresisaretypicalelectromagneticirreversible(dissipative)processes.Theatomicandmolecularcurrentsgivingrisetothemagnetizationofsolidsarenotdissipative,noristhesuperconductorcase.Thetheoreticalcalculationofelectromagneticenergycomputesonlythereversibleworkrequiredtostablishaconfigurationoffieldsandpolarizationsinanidealnondissipativeprocess.Inrealprocesses—nomatterhowslowyoumakethem—irreversiblephenomenaarealwayspresent.Thus,workdWdoneonarealsystempartiallydissipatesininternal(caseoftheelectricalresistance)orexternalirreversibleprocesses.Therefore

dW=dWrev–T·dSirrev, (5.2)

andtheworkcomputedinelectromagneticcalculationsisusuallyonlythereversiblepartdWrev.ThemostimportantthermodynamicstatefunctionistheinternalenergyUthat

characterizesthebalancebetweenthequantityofheatdQdeliveredtothesystemandthemechanicalworkdWdonebythesystemonitssurroundings,

dU=dQ-dW. (5.3)

Forsolidstheworkexpressionusedforgases,dW=p·dV,mustbereplacedby

dW=c •de, (5.4)

towhichmustbeaddedthecontributionsofelectromagneticenergy.Newcomerstoelectromagnetismtrytocalculateinternalenergiesbyusing

forcestocomputetheexternalworkrequiredtobuildthesystem.Thismethod,originatedinmechanics,isreinforcedinthestudyofgases,wherepressuresareintroducedassurfacedistributionofforces.Thisisnotthecaseinelectromagnetism,wherethestartingpointforcalculatingthevariationsofenergyisnottheconceptofforceortorque.Onthecontrary—aswillbeseeninthischapter—thelatterareevaluatedastherateofchangeofelectromagneticenergyrespecttocertainlongitudesorangles.


Animportantpartoftheproblemisthatthevariationofelectromagneticenergydependsbothonitsspatialdistributionanditstimevariation.Ifatthispointthereaderarguesthatweareconsideringhereonlystaticfieldsandsteadystatecurrents,somevariationsshouldbeirrelevant,heshouldreadagaintheintroductiontoeq.5.2.Theelectromagneticenergycontainedinanydistributionofpolarizedmatteris,bydefinition,thatrequiredtobuildthatdistributionfromaninitialstatewherepolarizationsarezero.Thisnecessarilyrequiresatimevariationofpolarizationsandfields,processwhereinducedcurrentsmakefinitecontributionstotheenergyofthesystemandgenerateradiation,howeverslowlytheprocessismade.Moreover,fieldsarenotlimitedtomaterialbodiesbutextendthroughoutallspace.Accordingtothetypeofprocessamaterialsystemundergoes,itisconvenientto

describeitusingapeculiarstatefunctiondifferentfromU.Thisisdoneinawaysuchthatitsdifferentialexpressiondoesnotcontainthedifferentialofthestatevariableonewishestokeepconstant.Forinstance,Uisthemoreconvenientstatefunctionforthedescriptionofprocesseswherestrainscanbefixed,lettingtemperatureTandstressσ totakethevaluesthatfollowfromtheallowedheatexchange.Ofcourse,anyprocessmaybedescribedbyanystatefunction,butthetheoreticalanalysisissimplifiedwhenusing,ceterisparibus,anappropriatethermodynamicpotential.

ThermodynamicsofelectromagnetismThemeasurementofelectromagneticpropertiesofpolarizedbodiesis

frequently,thoughnotalways,madeinexperimentswheremechanicalworkisdone.Inthosecases—theonlydiscussedinthisbook—forcesandtorquesarerelatedtothevariationofparameterscharacterizingtheconfigurationoffieldsandpolarizations.Inaceterisparibusfashion,severalvariablesmustbekeptfixedthroughouttheprocess.

Thebody'sintrinsecgeometricconfiguration—therelativedistributionofpolarizablematter—isoneofthevariablesthatmustbekepconstant.Thisconfigurationis—inourcase—theellipsoidalshapeandsemiaxesthatdeterminethefields'svalues,usuallyinvariablewhende,eq.5.4,vanishes.Asmentionedbefore,itisthenconvenienttouseoftheinternalenergyU.Inpractice,strain'scontrolisnotalwayspossible,ashappensinthephenomenaofelectrostriction135,magnetostriction136,piezoelectricity137andpiezomagnetism,thatrequierespecialtreatment.

Electricandmagneticpolarizations—dependentonthemicroscopicbehaviourofatomsandmolecules—arefunctionsoftemperatureT,avariablethatwillbeusuallynecessarytokeepconstant.AsTisa“naturalvariable”ofU,itisconvenienttodefinethefollowingthermodynamicpotential

F=U−T·S, (5.5)

135 LandauandLifchitz,p.55.136 LandauandLifchitz,p.155.137 LandauandLifchitz,p.73.


calledHelmholtzfreeenergy,forwhich

dF=T·dS−dW− T·dS−S·dT=−dW− S·dT. (5.6)

Theexternalworkdoneinanisothermalprocessisthen

dW=dWrev−T·dSirrev.=−dF. (5.7)

Anysystemtendstoreorganizeitelementsinordertotominimizeitsenergyinthefinalstateofequilibrium.Whentherearenoexternalconstraints,FvariesuntilreachingaminimumdFvanishesandthesystemceasestodoexternalwork.Thereversibleworkdeliveredintheprocessofreacommodationis

ΔWrev = dWrev∫ = −(ΔF −T ⋅ΔSirrev. ), (5.8)

reasonwhyFcharacterizesthemaximumquantityofworkthatitispossibletoextractfromasystem138,139,conceptthatinchemistryiscalledafinity.TheinternalenergyUandthefreeenergyFincludeanelectromagnetic

contribution.ForFitsvalueFemistheamountofreversibleworknecessarytobuildupthefinalconfigurationofcharges,electriccurrentsandpolarizedmatter.Forthelinearcase(whosedescriptionrequiresseveralanisotropytensorsofranktwo)itsexpressionintermsoffieldsis140

Fem e=const .

= 12λ

!E(!r )i !D(!r )d3r

V∫∫∫ + 1

2λ '!B(!r )i !H(!r )d3r

V∫∫∫ , (5.9)

wheretheintegralistakenoveralltheregionscontainingthesourcesofthefields,theoneswhere !∇ i

D and !∇×

H arenon-vanishing.Therearenogeneral

expressionsforthenon-linearcase,whereeachspecificsituationhastobedealtwith.Forconductorsorsuperconductorsthesurfacedensityofchargeorcurrentshouldbemadeexplicitthroughappropiatemathematicaltransformations.

Someauthorsextendintegralstoallspace,inaccordancetotheconceptthatenergyiscontainedinthefields.AlthoughthisseemstomodifythevalueofFem,itisnotsobecausetherearepartialcancellationsofthecontributionsofthefieldsinsidethebodywiththoseoutsideit141.Specialcareshouldbetakenwiththecalculationoftheworknecessarytopolarizematterwithfieldsassumedtobefixed,assourcesprovideoreceiveenergyinordertokeepfieldsconstantwhenmatterisintroduced,evenforveryslowprocess142.

138 PanofskyandPhillips,p.90.139 LandauandLifchitz,pp.52-55,129-131.140 Reitz,pp.120and254.141 Stratton,pp.112-113,discussesthiscancelationbutgivesnoactualexamples.SeeProblem36.142 See,forinstance,V.GilinskyandD.Holliday,InteractionEnergyofaDielectricinanElectrostatic

Field,Am.J.Phys.vol.34,pp.1134-1138(1966).


Femdoesnoincludethecontributionfrombody’schangesofshape,whichisusuallydescribeusingthethermodynamicpotentialcalledGibbsfreeenergy143

dΦ=U–T·S+e·dσ , (5.10)

whereworkincludesthevariationsofeandstresstakesthefinalvaluecorrespondingtotheisothermalprocess.ThevariationofGibbsfreeenergycanbemeasuredinactualexperiments,butismoredifficulttocalculatethanHelmholtzfreeenergy.

FortheformulasgiveninthischapterprocessesareassumedtobeisothermalandHelmholtzfreeenergyFisevaluatedassumingaconstantstraintensore.Forinstance,actualexperimentsshouldprovideanefficientmeanoffasttransferenceofanyquantityofheatgenerated,sothattheisothermalconditionisfulfilled.Thesetwoconditionsareseldomfulfilledinirreversiblephenomenaashysteresisandinpiezoelectricandpiezomagneticprocessesthatmodifysignificantlythedimensionesofthecrystallattice,creatingalsoadditionalfields.Theuseofthedepolarizationtensormethodforthesekindofphenomena—aswellofotherspeculiarofsinglecrystales—willprobablyrequieremodificationsorrestrictionsthathavenotbeenstudiedbytheauthor144.

Dependingonthekindofanisotropy,itmaybeconvenienttousecoordinateaxesdifferentthantheellipsoid’ssemiaxeswhereNisdiagonal.Thegoalistosimplifytheinversematricesrequiredforthecase.ThisuseofacoordinatesystemfixedtothebodyshouldtakeintoaccountthecaseofrotatingsamplesdepictedinFigure21,bothforrotationsandtorques.

SolvedproblemsrelatedtotheenergyofpolarizableellipsoidalbodiesimmersedinappliedfieldsaresolvedinChapter6.Theyareonlyselectedexamplesofthewayinwhichthedepolarizationtensormethodmaysimplifytheanálisisofsomeexperiments.

AnisotropyenergyAbodyundertheactionofanappliedfieldexperiencesatorqueifitsenergy

dependsoftheorientationofthefieldrespecttothebody.Asphereofuntexturedpolicrystallineironhasminimumfreeenergywhen !

M and !!

H0 areparallel.Forany

otherorientation,thebody’smagneticdipoles(thatis, !M )experienceatorque

thantendstoalignthemwiththeappliedfield.Thisdoesnotgenerateatorqueonthebodywhenthematerialisisotropic,sothatthemagnetizationmayfreelyorientatewithnoexpenseofenergyapartfromdisipativeeffects.Thespherewillremaininequilibriumatanyorientationrespecttothefieldbecausethemagnetizationwillfollowthefield,notthebody.

143 LandauandLifchitz,p.62.144 Nye,chapterXpp.170-191,makesadetaileddiscussionoftheequilibriumthermodynamicsof

physicalpropertiesofsinglecrystals.Hisfigures10.1aand10.1bgiveaconciseperoquiteillustrativesummaryoftheproblem.


Thedirectionandmagnitudeofpolarization—andthereforethefreeenergy—maydependonthebody’sorientationrespecttotheappliedfield.Thisanisotropyofenergyhastwodifferentorigins,shapeandcristallineeffects.Shapeanisotropyofenergy145originates

fromtheeffectofthebody’sshapeontheinternalfieldandpolarization.ForellipsoidalbodiestheeffectisfullydescribedbytheinternaldepolarizationtensorN.Forthefamilyofellipsoidalbodies,theonlyonewithoutshapeanisotropyisthespherewherethethreeeigenvaluesareequal(seeeq.3.66).Fortherest,theenergyrequiredtoinducepolarizationalongtwodifferentdirectionswithintheellipsoidisingeneraldifferent.

Figure18.Deviationδoftheinternalfieldrespecttotheappliedone

Thegeneralequationsthatdescribethephenomenaofinducedpolarizationaregiveninsection2.85andreproducedbelow.Theupperindexthatidentifiestheinternalfieldhasbeensuppressedbecauseitwillbetheonlyfieldofinterestinwhatfollows.

F = F0 −N iQ(F), Q(F)= χ iF,F = 1+N i χ( )−1 iF0 , Q = χ iFint = χ i 1+N i χ( )−1 iF0.

(5.11)

Let’sanalyze,then,thebehaviourofanisotropictriaxialellipsoidalbodyontheplanexywheretheprincipalvaluesoftheinternaldepolarizationtensorareNaandNb.Thefieldandthepolarizationareexclusivelythoseinducedbythefixappliedfield !!

F 0. Inthebody’sprincipalsystemofcoordinatesthecomponentsontheplane

ofinterestare

Fy =ν yFy

0 , Fz =ν zFz0 , να =

11+ χNα

. (5.12)

Thedeviationangleδoftheinducedfieldrespecttotheappliedone(seeFigure18)canbefoundfromthescalarproduct

!F i!F 0 = cosδ ⋅F ⋅F 0 = FyFy0 +FzFz0

=ν y Fy0( )2 +ν z Fz

0( )2 = ν y sen2θ +ν z cos2θ( ) F 0( )2 . (5.13)

Themodulusoftheinternalfielddiffersfromtheappliedonebyafactordependingontheorientationofthelatterrespecttothebody:

F = F 2y +F2z = ν yFy

0( )2 + ν zFz0( )2

z= ν 2

y sen2θ +ν 2z cos2θ ⋅F 0. (5.14)

145 Brown(1963),p.106.


Thedeviationangleδisthengivenby

cosδ =

!F i!F 0

F ⋅F 0=

ν y sen2θ +ν z cos2θ

ν 2y sen2θ +ν 2

z cos2θ. (5.15)

!F and !!

F 0 areparallelonlyalongtheellipsoid’sejesprincipalaxes,whereits

magnitudediffersbythefactorνα.Duetoshapeanisotropytheminimumfreeenergyisobtainedwhenthebody´slargestorsmallersemiaxis—dependingonthekindofpolarization,asdiscussedinnextsection—isorientedalongtheappliedfield.Ifthisisnotso,thereisatorquethattendstoorientatethebodyalongthatdirection.Crystallineanisotropyenergyisoriginatedbythespatialorderofthematerial’s

atomsandmolecules.AsimpleexamplethatillustratesthephenomenaisthetwopolarizablemoleculesanalyzedinthesectionInducedelectricpolarizationoftwointeractingatoms.Theonlymaterialswithcrystallineanisotropyaresinglecrystalsortexturedpolicrystallineones.Inthelatterthesmallcrystalshavesomeofitsfacespreferentlyorientedalongcertaindirection,ashappensinlaminatedmetals.Powders,untexturedpolicrystallinematerialsandamorphoussolidshavenomacroscopicorderand,therefore,nocrystallineanisotropyenergy.Forthecaseofinducedpolarization—theonlyonediscussedhere—the

simmetriesthatgiveorigintocrystallineanisotropyarereflectedinthenon-scalaranisotropicsusceptibilitytensor.ThemainpropertiesofthistensorsaregiveninAppendix5.Itsdemonstration,notgiventhere,canbemadeinsimilarwayasforthecaseofthedepolarizationtensor(seesectionSymmetries)butrequiresaknowledgeofcrystallinestructuresthatcannotbeassumedforthereadersofthisbook.

OriginoftorquesexertedonabodyInthissectionitisdiscussedtheoriginofthemomentexertedonanellipsoid

outsidetheequilibriumorientation.Theexpressionoftheparresultanteforeachcaseisdiscussedinothersections.

AtFigure19anexampleisgivenofacommonexperimentwheretorqueappears.Aprolatespheroid,ellipsoidalshapecommonlyusedinexperiments,issuspendedsothatitcanrotatearoundanaxisnormaltoitssymmetryaxisx(correspondingtothelargestsemiaxisa).Therotationaxiszisoutgoingfromthebackofthefigure.TheminimumvalueoftheisothermalandisobaricfreeenergyFeisassumedtobeobtainedwhenthedirectionofthelargestsemiaxisacoincideswiththatofthefield

!F0. Ifthatsemiaxisisinitiallyanangleϕapartfromthefield,

acoupleisexertedonthebody,suchthattendstodiminishit.Thecouples’soriginisaforcedistributedoverthebody’svolumethat,dueto

theconstraintimposedbytherotation’saxis,manifestsasamomentofforcealongthataxis.Inthegivenconfiguration,wheretheplanexznormaltothepageisasymmetryplaneofthespheroid,theforceovereachdifferentialelementofvolumehasnocomponentnormaltoplaneyz.TheappearanceofthisforceisacommonexperimentalfactforallpotentialenergiesasFe.Thisissimilartowhathappens


withamass,wherethethattendstobringittothepositionofminimumgravitationalpotentialenergyistheforcedistributedoverallitsvolumethatwecallweight.Itisdiscussednexttherelationshipbetweentheforcecoupleandthevariationofthefreeenergy.

Figure19.Torqueexertedonellipsoidinappliedfield.

Attherightofthefigure !!f (r ) istheforceovertheelementofvolumedVat !r ,

whereristhedistancetotherotationaxisz.Angleϕistheusualoneofsphericalcoordinatesandtheforcehasbeendecomposedinitstangentialandradialcomponentsfϕandfr.TheworkdWdoneonthebodywhenitrotatesdϕtowardsthedirectionofstableequilibriumis

dW = −dϕ fϕrd

3rV∫∫∫ = −τ dϕ = dFe , (5.16)

whereτisthetorqueexertedonthebodyalongitssuspensionaxis(see,forinstance,eq.5.23).Therefore

τ = −

dFedϕ

. (5.17)

TorqueexertedonanellipsoidandthestateofequilibriumInwhatfollowsadiscussionismadeoftheproblemofequilibriumandtheforce

couplesexertedonellipsoidalbodiesmadewithdifferentkindsofmaterialsandsubjecttostaticelectricandmagneticfields.

Dielectrics

PermanentpolarizationTheanalysisismadehereoftheequilibriumconfigurationofapermanent

polarizationintheabsenceofappliedfield.Insuchcasetheelectrostaticcontributiontotheenergy,eq.5.9,becomes(seeeq.2.12)


Fe =

V2λ!EP i ε0

!EP +λ

!P( ) = − λV

2ε0Nα (1−Nα )P2α

α∑ . (5.18)

Asanillustration,theminimizationofenergywillbeanalizedforthecaseofspheroids,where

Fe = −λV2ε0

CeP2e +CpPp

2( ) , whereCe =Ne 1−Ne( ) ,Np =1−2Ne ,Cp =2Ne 1−2Ne( ) ,

(5.19)

wheresubindicespandeidentifythepolarandequatorialcomponentsoreigenvalues,respectivelyandthetraceruleeqs.3.15wasused.Figure20showstheirvaluesinarangethatincludestherightcircularcylinderofinfinitelength(pointA)tothesphere(pointB).Thesheetofconstantthicknessandinfiniteextension,notseenhere,hastheNp=1.TheverticallineatNe=1/3istheboundaryoftheregionsofprolateandoblatespheroids.

Figure20.

!!N ,C y C⊥asfunctionsof!!N⊥

.

ForoblatespheroidsthelargestcoefficientisCpsothatfreeenergyreachesitsmaximumnegativevaluewhenthepolarizationliesontheplaneoflargestsemiaxes,theequatorialones.ForprolatespheroidsthelargestcoefficientisCecorrespondingtothepolarcomponent(herethelargestsemiaxis),andthepolarizationtendstoalignalongthatdirection.Forthespherethepolarizationhasnopreferedalignmentbecausethecoefficientesarethesameforallcomponentsofpolarization.


Thetotalfreeenergyisnottheexpressioncustomarilyusedforthissortofproblemasitmaybesimplifiedeliminatingthe !

EP iEP (see,below,theanalogous

caseofpermanentemagnetization),perotheobtainedresultiscorrect.

InducedpolarizationAccordingtoStratton146,thetorque

τ exertedinthelinearcasebythefixed

field !!E0onanisotropicdielectricbodywhenitsorientationisnottheoneof

minimumfreeenergy,isobtainedfromthefollowinginteractionenergy,

F'e = −12

!P i!E0d3r

V∫∫∫ = −

Vε0χe

2λ EαintEα

0

α∑ = −

Vε0χe

2λ να Eα0( )2

α∑ ,

where να =1

1+ χeNα

, (5.20)

whereeq.2.33wasused.Themomentofdistributedforcesisthen147(seeeq.2.34):

!τ = !p×

!E0 , !p = ε0

λα e i!E0 , α e =V χ e i 1+N i χ e( )−1 . (5.21)

Foranisotropicdielectric

αe =V χe 1+N i χe( )−1 =V χe

11+ χeNa

0 0

0 11+ χeNb

0

0 0 11+ χeNc

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

. (5.22)

Whentheappliedfieldhasnozcomponent,asinFigure19,theonlytorquecomponentis

τ z =

ε0λχ 2eV

Nc −Nb

1+ χeNb( ) 1+ χeNc( )sin2ϕ!E0 i!E0. (5.23)

ThegeneralcaseofspheroidsisdiscussedatProblem38.Fortriaxialellipsoidswithsemiaxesordera>b>c,theorderoftheprincipalvaluesisNc>Nbandtheforcecouplehaspositivesign,correspondingtoarotationthattendstoalignthelargestsemiaxiswiththefield.Thisorientationistheoneofminimumenergy,asmaybeverifiedfromtheexpressionofF 'einitiallywritten.

146 Stratton,p.113eq.51.147 ComparewithStratton,p.216eq.53.


AnisotropicsusceptibilityTheelectricsusceptibilityofuniaxialcrystalshas—initsprincipalsystemof

coordinates,thatingeneraldoesnotcoincidewiththatofthebody—thefollowingexpression(seeAppendix5):

χe =

χ⊥ 0 00 χ⊥ 00 0 χ

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

. (5.24)

Themicroscopicoriginofthisanisotropyisnotdiscussedhere,buthasasimilaroriginasthatofthetwopolarizableatomsanalyzedatpage21,withtheadditionalcomplexityintroducedbytheregularorderofacrystallattice.Theforcecoupleexertedonasphericalbodyofthatmaterial—thatis,without

shapeanisotropy—doesnotdeterminethetwocomponentsbutitsdifference,asdiscussednextFromeqs.2.33and.3.66,thecomponentsoftheinternalfieldandthe

polarizationintheprincipalsystemofcoordinatesofthesusceptibilitytensorare:

Eαint =

E0α

1+ χα /3, Pα = χαEα

int =χαE

0α

1+ χα /3. (5.25)

Becauseoftherotationalsymmetryofthesusceptibility,theexperimentmustbedoneontheplanethatcontainsthesymmetryaxisandtheappliedfield.Whendescomposingallvectoresintheircomponentsparallelandperpendiculartothataxis,theinternalfieldisgiveby

E⊥int =

E0⊥

1+ χ⊥ /3, E

!int =

E0!

1+ χ!/3. (5.26)

Uponreplacementintheexpressionforthefreeenergyeq.5.20itisobtained

F 'e = −12

!P i!E0d3r

V∫∫∫ = −

ε02λV χαEα

intEα0

α∑

= −ε02λV χ⊥

E0⊥( )2

1+ χ⊥ /3+ χ

"

E"0( )2

1+ χ"/3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= −

ε02λV

E0⊥( )2

1/ χ⊥ +1/3+

E"0( )2

1/ χ"+1/3

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥. (5.27)

F'eisminimumalongthedirectionswithmaximumsusceptibility.For χ> χ⊥

theminimumisobtainedwhenthesymmetryaxisisparallelto !E0. For

!χ < χ⊥ , theminimumisobtainedwhenthesymmetryaxisisnormaltothefield.Althoughatfirstsightthisappearsindistinguisablefromashapeanisotropy,itisnotso(seeProblem02).ThecalculationofthetorqueexertedbytheappliedfieldonthebodyisdoneatProblem39.


Magneticmaterials

PermanentmagnetizationThemagnitudeofthespontaneousmagnetizationisusuallyconstantatconstant

temperature,independentlyofitsorientationrespecttothecrystalaxes148,149.Onemaythencomputetheenergyofspontaneouslymagnetizedmatterassuminginvariablethemagnitudeofthemicroscopicmagneticmoments.Thiscalculation—whichdoesnotincludetheenergynecessaryforbuildingupthemagneticatomicandmolecularmoments—gives150,151

Fm = −

µ02

!H int(!r )i !M(!r )d3r

V∫∫∫ . (5.28)

Forauniformlymagnetizedellipsoidalbodyeq.2.44gives

!H int = −λ 'N i

!M. (5.29)

Therefore

Fm = −

µ02 V!H int i

!M =

µ0λ '2 V

!M iN i

!M , (5.30)

expressionthatshouldbeminimizedwiththeconditionofconstantM.IntheprincipalsystemofcoordinatesofNthesphericalcomponentsof !

M are

Mx =Msenθ cosϕ , My =Msenθ senϕ , Mz =Mcosθ . (5.31)

Therefore

Fm =µ0λ '2 V Nα

α∑ M2

α

=µ0λ '2 V Nx sen2θ cos2ϕ +Ny sen2θ sen2ϕ +Nz cos2θ( )M2 ,

(5.32)

shouldbeminimized.Theminimumiseasilyfoundrealizingthattheparenthesisistheparametric

equation X = Nx senθ cosϕ , Y = Ny senθ senϕ , Z = Nz cosθ , (5.33)

oftheellipsoid

X 2

Nx( )2+ Y 2

Ny( )2+ Z2

Nz( )2=1, (5.34)

148 Kittel,p.533.149 LandauandLifchitz,p.146.150 Chikazumi,p.24eq.1.95.151 Stratton,p.130eq.63.


whosesquareradiusvectorR152hastobemaximized.TheminimumvalueofR2isobtainedalongtheaxiscorrespondingtothesmallestdenominator,thatis,thedirectionoftheellipsoid’slargestsemiaxis(seeeq.4.6).Thispropertyofpermanentlymagnetizatedprolatespheroidsexplainsthe

stabilityofthemagnetizationofacompass’sneedle.Inthiscasethereisnotorqueappliedonthebody,butanirreversiblerotationofmagnetization(hysteresis)thatalignsitinthedirectionofminimumisothermalandisobaricfreeenergy.Suchabodyhasapermanentmagneticdipolemomentgivenamongeqs.2.49anditsdirectionisthatofthepolarsemiaxis.Atconstanttemperaturethemagnetizationofthespheroidmaybeconsideredtobefixedbecauseinthepresenceoflowmagneticfieldsliketheterrestrialone,theinducedmagnetizationismuchlowerthanthespontaneousorpermanentone153.Forsuchacase154thefreeenergyofinteractionwithanappliedfieldandtheforcecoupleare

Fm = − !mi!B0 , !τ = !m×

!B0. (5.35)

Theseexpressionsdonotrequiredfurtheranálisis,asFmisaminimumwhen !m

hasthesamedirectionandsensethan !!B0 ,orientationgeneratedby thatvanishes

whenbothvectorsareparallel.

InducedmagnetizationForthecalculationoftheinteractionenergyofspontaneouselectricormagnetic

polarizationsitisnotnecessarytoincludetheenergyofitscreation.Itisnotsoforinducedpolarizations,wheretherequiredexpressionofthefreeenergyis155

Fm = −

µ02

!H0(!r )i !M(!r )d3r

V∫∫∫ = −

µ02 V!M i!H0 = −12

!mi!B0 , (5.36)

becausebothvectorsareuniform(seepage6).Thetorqueexertedbythemagneticinductionfieldovertheellipsoidalbodyis

thengivenby(seeeq.2.49)

!τ = !m×

!B0 , !m= 1

λ 'αm i!H0 , αm =V χm i 1+N i χm( )−1 . (5.37)

152 Radiusvectoristhedistanceofasurface’spointtoitscenter.153 Chikazumi,p.439.154 Jackson,p.150.Thefactor1/2isnotpresentwhenthemomentandthefieldare“rigid”or

fixed.155 Strattonpp.128eq.51and154eq.7.AnilluminatingdiscussionofthesubjectismadebyM.S.

PlessetandG.Venezian(Am.J.Phys.vol.32,pp.860-864,1964),wherethecalculationismadeofthetorqueappliedonanellipsoidalbodywithinducedmagnetization.SeealsoS.P.Puri,Am.J.Phys.vol.33,p.472,1965.

τ


MagnetictorqueexperimentsFigure21showsthemainpartsof

agoniometer,devicethatcanmeasurewithgoodprecissiontheforcecouplesexertedbyanappliedmagneticfieldoveramagnetizedbody.Theinstrumentisbasedonacalibratedtorsionwirethatcanberotatedinordertoreturnareflectedlightbeamtoitszero(positionintheabsenceofappliedfield).Theglasstubeblocksaircurrentsthatmaydisplacethebody.Thepaddlesimmersedinoilareadjustedinsizeandshapeinordertoprovidecriticaldampingforafastreturntozero.

Figure21.Elementarymeasurement

ofmagneticforcecouples.

Fortheisotropiccasetheforcecouplereducesto

!τ = !m×

!B0 =

µ0λ 'Vχm

!H0 i 1+ χmN( )−1 i

!H0. (5.38)

Intheellipsoid’sprincipalsystemofcoordinatesthecomponentsaregivenby

τ x =

µ0χ2mV

λ 'Nc −Nb

1+ χmNb( ) 1+ χmNc( )( )⎛

⎝⎜

⎞

⎠⎟Hy

0Hz0 , (5.39)

τ y =

µ0χ2mV

λ 'Na −Nc

1+ χmNa( ) 1+ χmNc( )( )⎛

⎝⎜

⎞

⎠⎟Hx

0Hz0 , (5.40)

τ z =

µ0χmVλ '

Nb −Na

1+ χmNa( ) 1+ χmNb( )( )⎛

⎝⎜

⎞

⎠⎟Hx

0Ht0. (5.41)

Bothfortheparamagnetic(𝜒m >0)andthediamagnetic(𝜒m <0,| 𝜒m|≪1)casedenominatorsarepositivenumbersandthesignisdeterminedbythedifferenceofprincipalvalues.Whentheappliedfieldhasonlyy,zcomponents,thetreatmentissimilartothatgivenfortheelectricone,eq.5.23.Thetorquetendsthentoalignthelargestsemiaxes(ainourconvention)withtheappliedfield,bothintheparamagneticanddiamagneticcase.

Conductors

TheenergyofasurfacechargedistributiononaconductorwithnetchargeQisgivenbythenextequation156.

156 Stratton,p.107eq.16.


Fe =

12λ φσ (!r )d2r

S"∫∫ = 1

2λ φS σ(!r )d2rS"∫∫ = 1

2λ φSQ =0. (5.42)

TheresultisobtainedbycomputingtheworkdonebybringingthechargesfrominfinitedistancestoaconductoratpotentialφSuntilthetotalchargeQisbuildup.Theformulaisnotvalidfortheseparationofchargesoverthesurfaceofanellipsoidwithzeronetcharge.Classicaltextsonelectromagnetism,likethoseofStrattonandJackson,restricttheiranalysistoconductingbodieswithnonvanishingnetchargewherethepreviouslygivenshape-independentformulaisvalid.TheexceptionisthatofLandauandLifshitz,whocomputethefollowingvaluefortheinteractionenergyofazeronetchargeconductorwithanappliedelectrostaticfield:

Fe = −

12!p i!E0 , (5.43)

where !p istheelectricdipolemomentofthebody.Forellipsoidalconductorsthis

momentisgivenbyeq.2.60,sothatintheprincipalsystemofcoordinates

Fe = −

12!E0 i!p = −

ε0V2λ!E0 iN −1 i

!E0 = −

ε0V2λ

1Nα

Eα0( )2

α∑ , (5.44)

coincidingwitheq.2.14ofLandau&Lifshitz.Theexpressionfortheforcecoupleexertedoveraconductingellipsoidisthe

oneforadielectric,eq.5.21,inthelimitofinfinitesusceptibility(seeProblem08).Itisthusobtained

τ x =

ε0Vλ

Nc −Nb

NbNc

E y0Ez

0 , τ y =ε0Vλ

Na −Nc

NaNc

Ex0Ez

0 , τ z =ε0Vλ

Nb −Na

NaNb

Ex0Ey

0 . (5.45)

Whenthethreedepolarizationfactorsareequal—thatis,forspheres—theforcecouplevanishes,ascorrespondstoitsshapeisotropy.Whenallthreearedifferent,asintriaxialellipsoids,theforcecoupletendstoalignthelargestsemiaxiswiththefield,ashappensinthedielectricandmagneticcase.

Superconductors

Asinthecaseofconductors,theforcecoupleexertedbytheappliedfieldonasuperconductingbodymaybederivedfromthecaseofinducedmagnetization,eq.5.37,usingtheconditionofperfectdiamagnetism!!χm = −1 (seeeq.2.73).

!τ = !m×

!B0 , !m= 1

µ0λ 'α s i!B0 , α s = −V 1−N( )−1 . (5.46)

Atypicalcomponentis

τ x =

Vµ0λ '

Nz −Ny

1−Ny( ) 1−Nz( )⎛

⎝⎜⎜

⎞

⎠⎟⎟By0Bz

0 , (5.47)


wheretheothercomponentsareobtainedfromacyclicpermutationofindices.

Forfinitevolumeellipsoidsthedenominatorisalwayspositiveandthebehaviourofasuperconductorbodyisthesameasthatofamagnetizedone,withtheminimumenergyobtainedwhenthelargestsemiaxisisalignedwiththeappliedfield.Degenerateellipsoids—asthesheetofinfiniteextensionandtheellypticcilinders—arenotvalidcasesfortheanalysisoftorquesthattheredivergewithvolume.

Forceonanellipsoidinanon-uniformfieldThedepolarizationtensormethodhasfewadvantagesforthetreatmentofthe

forcesexertedbynon-uniformfieldsonellipsoids.Nevertheless,duetoitsexperimentalrelevance,theexpressionsaregivenoftheforcesexertedonpolarizeddielectricandmagneticellipsoidsinquasi-uniformfields157,158:

!fe =V

!P i∇( ) !E0(!r )= !p i∇( ) !E0(!r )=

ε0λ!E0(!r )iα e i∇( ) !E0(!r ),

!fm =V

!M i∇( ) !B0(!r )= !mi∇( ) !B0(!r )= 1

µ0λ '!B0(!r )iαm i∇( ) !B0(!r ).

(5.48)

wherethebody’spolarizabilitytensors,eqs.2.33and2.49,havebeenused.Thevariationofthefieldinsidethebodyshouldbesmallenoughsothatthe

depolarizationtensormethodisvalidwithintheacceptablerangeoferror.Atthesametime,thevariationshouldbelargeenoughsothatthetotalforceismeasurable.Experimentsofthistype159makepossiblemeasurementsofdiamagneticsusceptibilitythatareimposiblewiththetorsionmethod(seeMagnetictorqueexperimentsatpage112).Thisissobecauseofthesmallvalueofthiskindofsusceptibilityandthefactthattorquesareproportionaltoitssquare;force,ontheotherhand,isdirectlyproportionaltothesusceptibility.

InfiniteandinfinitesimalbodiesEquation3.80showsthatsimilarellipsoidshaveequalinternaldepolarization

tensorsN.Therefore,ifonemakesallsemiaxesgrowwithoutlimitpreservingtheirrelativevalues,Ndoesnotchange.Onemustthenbecarefulwiththeuseofbodiesofinfiniteorindefiniteextension,subterfugeoftenusedfortheintroductionofdielectricormagneticmaterialsurroundingfinitebodies,unlessonespecifiestheirshape.Theseidealizationsarejustifiedonlyiftheyprovideagoodapproximationtoarealsituation,thatusuallyinvolvesfinitebodies.Thesphereofinfinitediametercanbeusedtoapproximateanisotropicdielectricenvironment,butonehasstilltodiscusstheeffectofthecavitythatcontainsthefinitebodyofinterest.Theellipticcylinderofinfinitelengthprovidesagoodapproximationtothemore

157 LandauandLifchitz,p.72eq.16.12.158 LandauandLifchitz,p.144eq.34.8.159 Faraday'smethod,seeH.Zjilstra,ExperimentalMethodsinmagnetism,vol.2,NorthHolland

Publishing,1967,p.94.


complexcalculationofthefieldneartheequatorofatriaxialellipsoid,buttherearebetteraproximationsthanthesheetofinfiniteextensiontothefieldnearthepolesofaveryflatoblatespheroid(seeeq.4.29).Polarizedmattercreatesfieldsintwodifferentways:bythespacevariationof

polarizationinsidethebody’svolumeandbythestepdiscontinuitythroughthebody’ssurface160.Foruniformpolarizationsastheonesstudiedhere,onlythesecondeffectappears,asillustratedfortheelectriccasebyeq.2.51,

!E(!r )= !E0 −k1∇

σ (!r ')!r − !r'

d2r'S"∫∫ .

ThecontributiontothefieldofsurfaceelementdSisσ dS/r2=σd𝛺(noticethatthegradientincreasesin1theexponentofthedenominator),whered𝛺isthesubtendedelementofsolidangle.Therefore,thecontributionofasurfacesectortothefieldataninternalpointisproportionaltothesubtendedsolidangle.Thisisillustratedbytheinfinitesheetandellipticcylinder,wherethedepolarizationfactors(principalvalues)vanishforthedirectionswherethesolidangletendsto0(thesemiaxistendsto∞).Thatis,verylargedistancesdonotensureverysmallcontributionsfromthesurfacesectortothefield,thatdependsonthesolidanglesubtendedbythatsector.Althoughatlargedistancesrthefieldofafinitepolarizedbody,whateveritsshape,hasadistancedependenceofr-3(seesectionNearandfarawaypointapproximationstonext),farawaysurfacechargesdonothavethisbehaviour.

Fromthemathematicalpointofviewtheproblemswithbodysofinfiniteextensionoriginatewhenthevalueoftheintegral

d3r '!r− !r 'V

∫∫∫ (5.49)

iscomputedbyextendingVoverallspace.Acarefulanalysisshowsthattheintegralissemi-convergent,thatitsvaluewhenvolumeVgrowsdependsonthepeculiarwaytheinfinitelimitistaken161.AninfinitenumberofdifferentlimitsmayexistdependingontheshapeofV,asillustratedbythepropertyeq.3.80ofthedepolarizationtensor.Thesameerrorofassumingallinfinitebodiestobeequalhasbeenmadewith

infinitesimalones.TheFermicontacttermisalittleknowncontributiontotheenergyofatomicelectronsoriginatedinitsinteractionwiththenucleus.Itscalculationwasmadeconsideringthenucleustobeapointcharge,implicitlyspherical.Thetermcontainsanintegralthatthisauthorprovedtobeoftheinternaldepolarizationtensortype162,thusrequiringabetterspecificationofthenuclearchargedistribution.Similarerrorsarerepeatedlymadeinthecalculations

160 Reitz,p.79eqs.4.13and4.14.161 MacMillan,pp.163-165.162C.E.Solivérez;Thecontacthyperfineinteraction:anilldefinedproblem;J.Phys.C:SolidSt.Phys.

vol.13;pp.L1017-L1019;1980.


ofmicroscopicfieldsincrystalswhenseriesofdipolarsarenumericallysummedwithouranalyzingitskindofconvergence.

Cavities

StandardtreatmentCavitieshavehadadistinguishedroleinthetheoryofelectricitysinceLord

Kelvinusedthemtodefine !E and !

D 163.Theproblemisthattheyaretreatedas

ellipsoidalbodieswiththeirsameproperties164,orasiftheextractionofanellipsoidalpieceofauniformlypolarizedbodydosnotmodifytheuniforminternalfields.Stratton,withoutpreviousdiscussion,treatscavitiesasiftheywerebodiesofthesameshapewiththedielectricpermittivityofvacuum165.Anotherreputedauthorismoreexplicitwhenhestates166:

FinallyweconsiderthefieldinsideacavityinamaterialmagnetizedtoanintensityI(Fig.1.16).Thefreepoledistributiononthesurfaceofthecavityisthesameasthatonthesurfaceofasolidbodywiththesameshapeasthecavity,andwiththesamemagnetizationasthematerialsurroundingthecavity,exceptthatthepolesareofoppositesign.Thismustbetrue,becauseifwesuperposethebodyandthehole,wehaveauniformlymagnetizedsolidwithoutfreepoles.

Aspreviouslystatedtheargumentinboldtypeistrueforuniformpolarization,wheretheonlycontributiontothefieldcomesfromsurfacepolarizationcharges167.Butthiscanonlybeequaltothepolarizationonthecavity’swallswhenthepolarizationisfixed(rigid,assomeauthorscallit),whichisfalse.Jacksonexplicitlysolvesthesphericalcavity168buthereplacesinthesolution

forthespheretheelectricpermittivitybyitsinversewiththepseudo-argument:Infact,inspectionofboundaryconditions(4.56)showsthattheresultsfromthecavitycanbeobtainedfromthoseforthespherebythereplacementeε→1/ ε.

Thealludedboundaryconditionsarethecontinuityofthetangentialcomponentoftheelectricfieldthroughthecavity’ssurfaceandthestepdiscontinuityofitsnormalcomponentduetothepolarizationchargeseq.3.35.Inaddition,hedoesnotspecifytheshapeofthedielectricbodywherethecavityismade.

VanVleck169,areputedspecialistonthepropertiesofpolarizedmatter,makesuseofcavitiesforthecalculationoflocalfields,buthiscavitiesarevirtual,notreal,amereartifactforpromotingfruitfuldiscussions.ThisisthecaseofthesocalledLorentzsphere,“cavity”usedtoestimatetheinternalorlocalelectricfieldatthe

163 See,forinstance,Stratton,p.214;Reitz,p.82.164 Stratton,pp.206and213.165 Stratton,p.206eq.31.166 Chikazumi,p.16. 167 Reitz,p.79eqs.4.13and4.14.168 Jackson,pp.114-115.169 VanVleck,chapterIV,analizesclasssicalcalculationsoffieldsinsidecavities.


moleculesofadielectric170,magnitudedifferentfromthemacroscopicfieldinMaxwell’sequations.The“cavity”isnotemptybutfilledwithmoleculeswhosefieldandpolarizationaretobecomputedindividuallyaccordingtotheirknowncrystallinepositions.

TheanalysismadeatsectionInducedelectricpolarizationoftwointeractingatomsillustratesthemutualinfluenceofdifferentportionsofmatteronthefinalpolarizationstate.Polarizationisnotautomaticallyuniform,noritremainsforeverfixedwhenthisstateisachieved.Shapeefectsarecrucialandonlyclosedsinglesurfacesoftheseconddegree(ellipsoids)havebeenprovedtobeamenableofuniformpolarization.Itisthereforeclearthatthefollowingtwoaspectshavetobecarefullyanalyzed:1. Theeffectoftheremovalofmaterialinsidetheellipsoidalbodyonthepolarizationstateoftherestofthebody.

2.Theeffectofthepolarizationchargesorcurrentsonthecavity’swallsoverthefieldsinsidethecavityandtheremainingpolarizedmatter.Itshouldalsobestressedthatauniformappliedandtotalfieldinsidethebody

isrequired,butthisisanecessaryconditionandnotaasufficientone.

Somespecificcavities:orhomoeoids

AcavityistheresultoftheremotionofavolumeV0ofpolarizedmatterfromalargeroneV.Thesimplestassumptionisthatbothpieceshaveellipsoidalshape,sothattheycan,inprinciple,beuniformlypolarized.Theintegralfromwhichthedepolarizationtensoristobecalculated(seethedefinitioneq.3.4)is

I '(!r )= d3r '

!r − !r'V−V0

∫∫∫ = d3r '!r − !r'V

∫∫∫ − d3r '!r − !r'V0

∫∫∫ = I(!r )− I0(!r ), (4.50)

whereawellknownmathematicalpropertyofintegralswasused.Usingthedefinitioneq.3.4onemayderivefromitthefollowingmathematicalrelationship

n'(!r )= n(!r )−n0(

!r ), (5.51)

wherenisthedepolarizationtensoroftheellipsoidalbodyofvolumeVandn0theoneofV0.Thisseemstoreducetheproblemoffieldstoaanalgebraiccombinationofknowncases,butitisnotso.Thesuperpositionprincipleappliestorigiddistributionsofcharges,dipolesorcurrents,butnottoinducedpolarizations.Althoughthisshouldbeclearenough,itisbesttosolvesomespecificcasetoillustratetheargument,whatisdonenext.

Shellsorhomoeoidsareoneofthemostsymmetriccaseofcavities,whentheremovedellipsoidalregionV0isconcentrictoasimilarellipsoidwithproportionalcorrespondingsemiaxes.Ifthefirstoftheequationsbelowdescribesthebody’ssurfaceandthesecondonethesurfaceofthecavity,theydetermineanhomoeoidorshell:

170 Dekker,pp.141-144.


x2

a2+ y

2

b2+ z

2

c2=1, x2

a2+ y

2

b2+ z

2

c2= k2 , where 0< k <1. (5.52)

Homoeoidshavebeenintensivelystudiedinthetheoryofgravitational171andelectromagneticpotentialsbecausetheyhavetheinterestingpropertythatthepotentialisconstantinsidetheshell,thatis,internalfieldsvanishasinsideconductorsbutthismaynotapplytopolarizationfields.

Problem28solvesthesimplecaseofansphericalshellusingtheelectrostaticGauss’sLaw.Thevaluesoftheauxiliaryfunction f (

!r )givenbyeq.6.72showthatinnoneofthethreeregionsdepictedbyFigure24itisaquadraticfunctionofcoordinates.Therefore,nowheremayadepolarizationtensorbedefinedand,presumably,nouniforminternalfieldandpolarizationmaybefound.Thisdoesnotruleoutthepossibilitythatthefieldinsidethecavityisuniform,itonlyshowsthatthedepolarizationtensormethodisnottheappropiateonetomakethecalculation.Ifitscavitiesaresmallenough,thepolarizationofanallipsoidalbodywillhave

smalldeparturesfromthatcalculatedfromitsdepolarizationtensor.Thereasonisthatthechangeinthefieldexternaltothecavityisproportionaltoitsdipolemoment,thatistothecavity’svolume.Thefieldaroundthecavity,ontheotherhand,wouldbeverydifferentfromtheoneinitsabsence.Themostcommonerrormadewiththetreatmentofcavitiesistheassumptionthatthepolarizationofthesurroundingmatterremainsinalterable,fixed,whichisfalsewithoutdoubt.

ThinshellsandthedipolelayerThelackofuniformityinthebody’sinteriordoesnotappearinan

infinitesimallythinshell,wherethebodyisreducedtoadoublesidedsurface,casethatisnowanalyzed.Thecaseofthesphericalhomoeoidisanalizedhere(seeproblemProblem28),whereforthethreedifferentregionsitisobtained

∂2 f1∂xα ∂xβ

=0 for 0≤ r ≤R1 , (5.53)


⎛

⎝⎜

⎞

⎠⎟

=

13+

R13

33x2 − r2r5

R13 x ⋅ yr5

R13 x ⋅zr5

R13 y ⋅xr5

13+

R13

33y2 − r2r5

R13 y ⋅zr5

R13 z ⋅xr5

R13 z ⋅ yr5

13+

R13

33z2 − r2r5

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

for R1 ≤ r ≤R2 , (5.54)

171 A.Gray,OntheAttractionsofSphericalandEllipsoidalShells,Proc.EdinburghMath.Soc.vol.23,

pp.91-100(1913-1914).



⎛

⎝⎜

⎞

⎠⎟

= −R23 −R1

3

3

3x2 − r2r5

3x ⋅ yr5

3x ⋅zr5

3y ⋅xr5

3y2 − r2r5

3y ⋅zr5

3z ⋅xr5

3z ⋅ yr5

3z2 − r2r5

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

for r ≥R2. (5.55)

TakingthelimitR2 →R1ineqs.5.54and5.55itisobtained

ncav =0, N =

x2

R12

xyR12

xzR12

yxR12

y2

R12

yzR12

zxR12

zyR12

z2

R12

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

= s(!r S1 )s(!r S1 ), next =0. (5.56)

Theexpression,validforallthinellipsoidalhomoeoids(seeProblem42),showsthatthedepolarizationtensorofathinshellisreducedtothediscontinuitythroughtheellipsoid’ssurfaceeq.3.38.Itsutilityhasyettobeasessed.


Chapter6:Selectedproblemas

Electricpolarization

Problem01:Autoconsistentelectricpolarizationoftwoatoms

Twoidenticalandisotropicatoms,withnopermanentelectricdipolemoment,areadistancedapartandimmersedinauniformelectricappliedfield.Asummingthattheinduceddipolemomentsareproportionaltothetotalelectricfieldateachatom,calculatetheirvaluesusingthepointdipolemodel.

Solution

SeesectionInducedelectricpolarizationoftwointeractingatoms.

Problem02:Shapeorcrystallineanisotropy?

Inalaboratory,astudentmadeanexperimentaldeterminationofthetwocoefficients thatrelatetheelectricpolarizationwiththeappliedfieldforanellipsoid:

λε0P⊥ = c⊥E⊥

0 , λε0P!= c

!E!0. (6.1)

Theabsentmindedstudentthatmadethemeasurementsforgottoregistertowhichofthetwofollowingbodiestheycorrespond:1)anisotropicdielectricspheroid;2)auniaxialdielectricsphere(seeAppendix5atpage165).Alargeamountoftime,workandresourceswereused,soitishighlydesirabletofindawaytoidentifytherightbody.Isitpossibletodifferentiatebetweenthetwocaseswithoutrepeatingthemeasurements?

SolutionIntheprincipalsystemofcoordinatesofaspheroid,theelectricpolarizationof

anisotropicmaterialwithsusceptibilityχplacedinauniformfieldhasthefollowingcomponents(seeeq.2.33)

λε0P⊥ =

χ1+N⊥χ

E0⊥ = c⊥E

0⊥ ,

λε0P!=

χ1+N

!χE0!= c

!E0!. (6.2)

Asthevalues !!N⊥ ,N ofthespheroidmaybeeasilycomputed,eachequation

provideavalueofχ,

χ⊥ =

c⊥1−N⊥c⊥

, χ!=

c!

1−N!c!

. (6.3)

!c⊥ ≠ c


If χ⊥ = χ

!= χ theproblemissolved.

Otherwisetheexpressionsforauniaxialdielectricspherehavetobesolved.Inthesusceptibilitytensor’sprincipalsystemtheequationsare

λε0P⊥ =

χ⊥

1+ χ⊥ /3E0

⊥ = c⊥E0⊥ ,

λε0P!=

χ!

1+ χ!/3E

0!= c

!E0!. (6.4)

Theeigenvaluesare

χ⊥ =

c⊥1− c⊥ /3

, χ!=

c!

1− c!/3 , (6.5)

wherethecoefficientsccannotbeequal.Thesusceptility’seigenvalueshavethesameorderasthecoefficients.Therefore,theshapeanisotropyofaspheroidhassomecharacteristicsthat

resemblethedielectricanisotropyofasphere,butothersaredifferent.

Problem03:Dielectricsphere

Give,inSIunits,theexpressionsoftheinternalandexternalelectricfieldofadielectricsphereunderanappliedfield.Computethebody’selectricdipolemomentandcomparetheresultswiththoseobtainedbyPanofskyandPhillips(PP)bysolvingLaplace’sdifferentialequation172.Identifythedepolarizationfactortheregivenandverifyifithastherightvalue.Solution

Forasphereofradiusa,inSIunits,eqs.2.33andA1.14give

Ezint = 3

(3ε0 + ε0χ)/ε0E0 = 3

κ +2E0 , (6.6)

whereκ=1+χisthedielectricconstant.FromthepotentialPP,eq.5-18,itisobtained

φ(!r )= − 3E

0

κ +2z , Ez = −∂φ(!r )∂z

= 3κ +2E

0 for r <a, (6.7)

coincidentconthepreviousequation.

Forthedepolarizationtensormethodtheelectricdipolemomentisrelatedtothepolarizationbyeqs.2.33:

!P = ε0χ

!Ezint =

3ε0χκ +2

!E0 =

3ε0(κ −1)κ +2

!E0 , !p =V !P = 4πε0a3

κ −1κ +2

!E0 , (6.8)

172 PanofskyandPhillips,pp.76-77.


thesamevaluegivenbyPP’seq.5-19.

Theexpressionfortheexternalfieldinthedepolarizationtensormethodisgivenbyeqs.2.33:

!E ext(!r )= !E0 − 1

ε0next(!r )i !P = !E0 −3κ −1

κ +2nsphereext (!r )i !E0 , (6.9)

wherensphereext isgivenbyeq.3.66.Thus

Eext(!r )= E0 +a3κ −1κ +2

3x2 − r2r5

3x ⋅ yr5

3x ⋅zr5

3y ⋅xr5

3y2 − r2r5

3y ⋅zr5

3z ⋅xr5

3z ⋅ yr5

3z2 − r2r5

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

i00E0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

=

a3κ −1κ +2

3x ⋅zr5

E0

a3κ −1κ +2

3y ⋅zr5

E0

E0 +a3κ −1κ +2

3z2 − r2r5

E0

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

(6.10)

ThepotentialgivenbyPP,eq.5-18,is

φ = a3κ −1

κ +2zr3E0 − zE0 for r >a. (6.11)

Thefield’scomponentsaretherefore

Ex = −∂φ∂x

= −a3κ −1κ +2zE

0 ∂∂x

r −3( ) = a3κ −1κ +2

3x ⋅zr5

E0 ,

Ey = −∂φ∂ y

= −a3κ −1κ +2zE

0 ∂∂ y

r −3( ) = a3κ −1κ +2

3y ⋅zr5

E0 ,

Ez = −∂φ∂z

= E0 −a3κ −1κ +2E

0 ∂∂z

zr −3( ) = E0 +a3κ −1κ +2

3z2 − r2r5

E0 ,

(6.12)

coincidentwiththevalueseqs.6.10.

ThedepolarizationfactordefinedbyporPP’seq.5.20(wherefactor!ε0 oftheusededitionshouldbeinthenumerator,notinthedenominator)


L= ε0

!E0 −

!E int

!P

=E0 − 3

κ +2E0

ε03ε0χκ +2E

0= κ −13(κ −1) =

13 , (6.13)

isthevalueofthethreeeigenvalueofthesphere’sunitdepolarizationtensor.

Magneticpolarization

Problem04:Derivingnfromthemagneticvectorpotential

Deriveeq.formagneticinductionintermsofnstartingfromthevectorpotentialforuniformlymagnetizedmatter !!

M : 173

!A(!r )=

µ0λ '4π

!M ×(!r − !r ')!r− !r ' 3

d3r '.V∫∫∫ (6.14)

Solution

UsandolasequationsA2.2and3.8seobtiene

!A(!r )=

µ0λ '4π ∇ 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫

⎛

⎝⎜⎜

⎞

⎠⎟⎟×!M =

µ0λ '4π ∇I(!r )× !M. (6.15)

Fromtheidentityeq.A2.9,

!B(!r )=∇×

!A(!r )=

µ0λ '4π ∇× ∇I(!r )× !M( )

∇× ∇I(!r )× !M( ) = !M i∇( )∇I(!r )− ∇I(!r )i∇( ) !M +∇I(!r ) ∇ i!M( )− !M ∇ i∇I(!r )( )

=!M i∇( )∇I(!r )− !MΔI(!r ).

(6.16)

Usingeqs.3.3andA3.7itisobtained

!B(!r )= µ0λ '

!M − µ0λ 'N i

!M for !r ∈V

−µ0λ 'next(!r )i !M for !r ∉V

⎧⎨⎪

⎩⎪, (6.17)

whichareeqs.2.35and2.44for !H0 =0.

Problem05:Permanentlymagnetizedinfiniterightcircularcylinder

AverylongrightcircularcylinderofradiusRhasapermanentuniformmagnetizationMforminganangleθ0withthesymmetryaxis.Evaluatetheinternal

173 Reitz,p.191eq.9-12.


andexternalmagneticfieldduetothemagnetizationandplottheexternalfieldlines.

SolutionTheinternalandexternalmagneticfieldsaregivenbythematrixequations

2.44.Thecoordinatesystemischosensothatthecylinder’saxiscorrespondstozandplanexzisparalleltovectorM.Fromthefirstofeqs.2.44,

Hint =

Hxint

Hyint

Hzint

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= −λ'N iM= −λ'

1/2 0 00 1/2 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i

Mx

0Mz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

−λ'Msinθ0 /200

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟. (6.18)

Thatis

Hxint = − λ'2 Msinθ0 , Hy

int =0, Hzint =0, H⊥ = −

λ'2 M⊥ , H! =0, (6.19)

whereinthelasttwoequationsthecomponentsaregiveninthedirectionsparallelandperpendiculartothecylinder’saxis.Itshouldbenoticedthattheinternalmagneticfield’scomponentparalleltothecylinder’saxisvanishesbecauseoftheshapeanisotropy.

Theexternalmagneticfieldisobtainedthesecondofeqs.2.44andeq.3.58:

Hext(!r )= −λ'next(!r )iMHxext

Hyext

Hzext

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= − λ'R

2

2ρ2

−cos2ϕ −sin2ϕ 0−sin2ϕ cos2ϕ 00 0 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i

M⊥

0M"

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

Hxext

Hyext

Hzext

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= λ'R22ρ2

M⊥ cos2ϕM⊥ sin2ϕ

0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

(6.20)

Theequationsofthefieldlinesare174

x2 + y2 = k ⋅ y , z = z0. (6.21)

wherekandz0are.ThegeometryoftheselinesisshownbyFigure22.Thelightergraycircumpherencesaretheequipotentallines,functiononlyofρ.Thekvaluesofthefieldlines,withtypicaldipolarshape,aresuccesivemultiplesof2,exceptforthehorizontaloneforwhichk=210.Linesarethesameforanyvalueofz.

Figure22.Fieldlinesofa

uniformlymagnetizedcylinder.

174 Kemmer,p.47Problem12.


Problem06:Isotropicmagneticsphere

AnisotropicferromagneticsphereisplacedinamagneticuniformfieldH0withfixedsources.AssumingthemagnetizationM(H)tobethefunctiongivenbytherightgraph—whereMsisthesaturationmagnetizationandM0thespontaneousone—determinetheresultingvaluesofthemagnetizationandthemagneticfield.Solution

Fromeq.2.44itshouldbe

Figure23.Graphicalsolutionfortheisotropicferromagneticsphere.

H =H0 − λ '

3 M(H), (6.22)

whereM(H)isthecurverepresentedinFigure23.Inthesamegraphisrepresentedtheline

g(H)= H

0 −Hλ '/3 , suchthat g(0)= g0 = H0

λ '/3 , g(H0)=0. (6.23)

ThedesiredvaluesofMandHaregivenbytheintersectionofthatlinewiththefunctionM(H),becauseatthatpoint

g(H)= H

0 −Hλ '/3 =M(H), thatis, H =H0 − λ '

3 M(H). (6.24)

Theexternalfieldisthegivenby

!H(!r )= !H0 − λ '

3 next(!r ) !M(!r ), (6.25)

wherethevalueofnextisthatofthesphere,eq.3.66.

Conductors

Problem07:SolveellipsoidalconductorsusingequivalentepolarizationDerivetheequationsthatsolvethecaseofanellipsoidalconductorinauniform

appliedfieldusingtheequivalentepolarizationmethod.Solution

Startingwiththepotentialeq.2.65,usetheexpressioneq.3.29forthedepolarizationtensorasummingthattheequivalentepolarizationisuniform.Itisthenobtained

!E(!r )= !E0 −k1∇

!Pf id

2!r '!r − !r 'S

"∫∫ =!E0 −4πk1n(

!r )i !Pf . (6.26)


Theequivalentepolarization !Pf isdeterminedbythevanishinginternalfield,

andtheexpressionoftheexternalfieldisobtainedinthesamewayasinProblem08.

Problem08:Aconductorisaperfectdielectric

Verifythatellipsoidalconductorsbehaveasdielectricswithinfinitescalarsusceptibility.Solution

Combiningeqs,2.12and2.33,foradielectricwithscalarχitisobtained

Eint = E0 − λε0N iPeq , where Peq =

ε0λχ 1+ χN( )−1E0 for r∈V ,

Eext(!r )= E0 − λε0next(!r )iPeq for r∉V .

(6.27)

Takingthelimit χ→∞intheprincipalsystemofcoordinatesofN,

χ 1+ χN( )−1 =χ 1+ χNx( )−1 0 0

0 χ 1+ χNy( )−1 0

0 0 χ 1+ χNz( )−1

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

,

limχ→∞

χ 1+ χNα( )−1 = limχ→∞

11χ +Nα

=N −1α , limχ→∞

χ 1+ χN( )−1 =N−1.

(6.28)

Fromthepreviousequationandeq.2.66:

!!!

Peq =ε0λN−1 iE0 , Eint = E0 − λ

ε0N iPeq =0,

Eext(r )= E0 − λε0next(r )iPeq , p=VPeq =

ε0λVN−1 iE0.

(6.29)

Theseequationsarethesameas2.63and2.66ifthefollowingassignationismade:

λε0Peq = E

+ . (6.30)

Problem09:PolarizationofasphericalconductorComparethechargedensityinducedonthesurfaceofasphericalconductor,

eq.2.57,thatobtainedbyusingtheequivalentpolarization !!Peq eq.2.66.


Solution175

Fromeq.2.57,thesurfacechargedensityσatpoint !r S is

σ (!r S )=

ε0λs(!r S )iN −1 i

!E0 , N = 131, s(

!r S )=!r S

R. (6.31)

Fromtheexpressionoftheelectricdipolemomentofthesphere,eq.2.60,

!p =

3ε0Vλ!E0 , σ (!r S )= 1

V

!p i!r S

R. (6.32)

Thesameexpressionisobtainedfromthelastofeqs.6.29fortheequivalentpolarizationmodel.Thisexpressionforσshouldbecomparedwiththatgivenbyeq.3.32.

Problem10:ExternalfieldofasphericalconductorProvethattheexternalfieldcreatedbyasphericalconductorimmersedina

uniformelectricfieldisthatofapointdipoleandfinditsmoment176.

SolutionThefieldofapolarizedellipsoidalconductorisgivenbythefirsttwoof

eqs.2.63,wheretheinternaldepolarizationtensorisN=(1/3)1.Thatis

!E ext(!r )= !E0 −next(!r )iN −1 i

!E0 = 1−3next(!r )( )i !E0 , (6.33)

wherenextforasphereisgivenbythesecondofeqs.3.67,

next(!r )= − R

3

33!r!r −(!r i

!r )1r5

. (6.34)

Therefore

!E ext(!r )= !E0 +R3 3

!r !r i!E0( )− r2 !E0

r5. (6.35)

Theelectricdipolemomentofthebodyisgivenbythelasttwoofeqs.2.63:

!p =

ε0VλN −1 i

!E0 =

4π ε0R3λ

!E0 = 1

k1R3!E0. (6.36)

Uponreplacementeintheexpressionfortheexternalelectricfielditisobtained

!E ext(!r )− !E0 = k1

3!r !r i!p( )− r2!pr5

, (6.37)

175 Solivérez(2008),p.207.176 Solivérez(2008),p.207.


whichistheexpressionforadipolarfield(seeeq.A4.1).

Problem11:DirectionsforwhichPisparalleltoE0intriaxialellipsoids

AmetallictriaxialellipsoidisplacedinanarbitraryuniformfieldE0.AssumingthethreeeigenvaluesofNareknown,determine:a)Theelectricdipolemomentpoftheellipsoid;b)thedirectionsalongwhichtheequivalentpolarizationPeqisparalleltotheappliedfieldE0;

c)theanglebetweenPandE0.SolutionFromeqs.2.63theelectricdipolemomentpintheellipsoid’sprincipalsystem

ofcoordinates—chosensothata>b>c—is

p=ε0VλN−1 iE0 =

ε0Vλ

N −1a 0 00 N −1

b 00 0 N −1

c

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

Ex0

Ey0

Ez0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, where 1

Na

< 1Nb

< 1Nc

. (6.38)

Theangleδ betweentheequivalentpolarizationPeq=p/VandthefieldE0isobtainedfromthescalarproductandnormofbothvectors:

cosδ =!P i!E0

P ⋅E0 ,!P i!E0 = PαEα

0

α∑ =

ε0λ

N −1α Eα

0( )2α∑ ,

P =ε0λ

Nα−1Eα

0( )2α∑ , E0 = Eα

0( )2α∑ .

(6.39)

Angleδvanisheswhenthenumeratorandthedenominatorofthefirstequationareequal,whichhappensonlyalongtheellipsoid’sprincipalaxes(seeeq.5.15andFigure18)where

!P i!E0 =

ε0λN −1

α Eα0( )2 , P = ε0

λNα

−1Eα0 , E0 = Eα , α = x , y ,z. (6.40)

Problem12:ElectricfieldsgeneratedbysharppointsAnalizethefieldsgeneratedinsharpportionsofconductorssubmittedto

uniformelectricfields.Tothatendcalculatethefieldintensityontheintersectionofthesurfaceofaverylongprolatespheroidwithsymmetryaxisxandcompareitwiththatonthespheroid’sequator.

Solution

Thefieldonthebody’ssurfacemaybeobtainedfromeqs.3.41andA7.8.Forthepoint(a,0,0)isobtained


s(a,0,0)= x , next(a,0,0)=Na −1 0 00 Nb 00 0 Nc

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟. (6.41)

Fromeq.2.63theelectricfieldatpoint(a,0,0)is

Eext(0,0,c)= E0 −next(0,0,c)iN−1iE0 , (6.42)

where

next(0,0,c)iN−1

=

Na −1 0 00 Nb 00 0 Nc

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i

Na−1 0 00 Nb

−1 00 0 Nc

−1

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=1−Na

−1 0 00 1 00 0 1

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟. (6.43)

Therefore,theelectricfieldhasthefollowingcomponentsnormalandparalledtothexaxis:

Exext(0,0,c)= 1

Nz

Ex0 , Ey

ext(0,0,c)= Ezext(0,0,c)=0. (6.44)

Whentheoblatespheroidbecomesverythina≫b=candNaisverysmall.Thexcomponentofthesurfacefieldbecomesverylarge,normalandoutgoingfromthebodyatpoint(a,0,0).Averysmallappliedelectricfield,liketheonesgeneratedduringanelectricstorm,maythusgenerateaverylargeexternalfieldwiththevaluegivenbyeq.6.44.

Theequatorialelectricfield,tangenttothesurface,vanishesastheinternalonebecauseofthecontinuityofthetangentialcomponentthroughanyinterphase.

Superconductors

Problem13:Superconductinginfinitecylinderandsheet

Showhowtosolvetheinfiniterightcircularcylinderofinfinitelengthandthesheetofconstantthicknessandinfiniteextensioninboththemagnetizationandsurfacecurrentmodel.Discusstheexistenceofa“reduced”inverseofmatrixN.

SolutionUsethesamemethodappliedforaconductinginfinitesheetinpage56.

Problem14:MagnetizedsuperconductingsphereEvaluatethemagnetizationandtheinternalandexternalfieldsgeneratedbya

superconductingsphereunderauniformappliedinduction!!B0z .ComparewiththeexpressionsgivenbyReitz177.

177 Reitz,p.eq.15-6.


Solution

TheexpressionsgivenbyReitz(inSIunits)are:

!B =0, !H = 3

2µ0B0z , !M = − 3

2µ0B0z for r ≤a. (6.45)

!B(!r )= µ0

!H(!r )= B0z − a

3

r3B0 cosθ r − 12

a3

r3B0 senθ θ ,

KM(a,θ ,ϕ)= −32µ0

B0 senθ ϕ. (6.46)

Fromeqs.2.73,whereN=(1/3)1,itisobtainedinSIunits(λ'=1)forr ≤a:

!B int = µ0

!H int +

!M( ) =0, !M = − 1

µ01− 13⎛⎝⎜

⎞⎠⎟

−1 !B0 = − 3

2µ0!B0 = −

!H int , (6.47)

asgivenbyReitz.

Equation3.66givesnextforthesphereshowingtheitisadipolarfield(seeeq.A4.1)forr≥a,

!Bext(!r )= !B0 − µ0n

ext(!r )i !M =!B0 − a

3

3 µ0r2!M −3( !M i

!r )!rr5

=!B0 + a

3

2

!B0 −3( !B0 i r)r

r3=!B0 − a

3

r3B0 cosθ r − 12

a3

r3B0senθ θ ,

as !B0 = B0z = B0 cosθ r −B0senθ θ ,

(6.48)

asgivenbyReitz.

Accordingtoeqs2.74andA7.8themagnetizationsurfacecurrentis

!KM(!r )= !M(!r )× s(!r )= − 3

2µ0!B0 × r = − 3

2µ0B0z × r = − 3

2µ0B0 senθ ϕ , (6.49)

whichcompletestheproofofperfectagreement.

Problem15:Superconductivityasperfectdiamagnetism

Verifythattheequationsdescribingsuperconductivitywiththemagnetizationmodelmayalsobeobtainedfromtheperfectdiamagneticcase,thatiswithsusceptibilityχm = −1.


Solution

Inducedmagnetizationinellipsoidsisdescribedbyeqs.2.49.ForHintandM,whereχm = −1, itis

Hint = 1−N( )−1 iH0 , M= 1

λ 'χmHint = − 1

λ 'Hint , Bint = µ0 H

int +λ 'M( ) =0. (6.50)

Theexternalfieldisthengiveby

Hext(!r )=H0 −λ 'n(!r )iM. (6.51)

Theexpressionsarethesamegivenbythemagnetizationmodelofsuperconductivity,eqs.2.73.

Problem16:MagneticmomentofasuperconductingellipsoidEvaluatethemagneticdipolemomentofasuperconductingtriaxialellipsoid

usingthesurfacecurrentmodel.Compareitsvaluewiththatgivenbythemagnetizationmodel.Solution

Fromeq.A1.12,themagneticdipolemomentofasurfacedistributionofcurrentis

!m=

k32

!r ×!K(!r )d2r.

S"∫∫ (6.52)

where iscurrentdensityvectorwhosevaluemaybeobtainedfromeqs.2.75andA1.11:

!K(!r )= 1

µ0λ 'k3s(!r )× !B+ , (6.53)

Itisthenobtained

!m= 1

2µ0λ '!r × d2!r ×

!B+( ).

S"∫∫ (6.54)

Theintegrand’sdoublevectorproductmaybeexpandedasfollows,

!r × d2!r ×

!B+( ) = !r i

!B+( )d2!r − !r id2!r( ) !B+ , (6.55)

whichreplacedintheintegralgives

!m= 1

2µ0λ '!r i!B+( )d2!r

S"∫∫ − 1

2µ0λ '!r id2!r

S"∫∫

⎛

⎝⎜⎞

⎠⎟!B+ . (6.56)

Thefirstsurfaceintegralmaybetransformedintoavolumeintegralbyusingthegradienttheoremeq.A3.9;thesecond,byusingthedivergencetheoremeq.A3.8.Thus

!K(!r )


!m= 12µ0λ '

∇ !r i!B+( )d3r

V∫∫∫ − 1

2µ0λ '∇ i!r( )d3r

V∫∫∫

⎛

⎝⎜⎞

⎠⎟!B+ ,

where ∇ !r i!B+( ) = !B+ , ∇ i

!r =3, d3rV∫∫∫ =V .

(6.57)

whereusehasbeenmadeof .Usingeq.2.82itisfinallyobtained

!m= − 1

µ0λV!B+ = 1

µ0λ 'α s i!B0 , α s = −V 1−N( )−1 . (6.58)

where isthebody’spolarizabilitytensorofthesuperconductor.Thisvalueof coincideswiththatgivenbyeq.2.72.

Problem17:DescriptionofMeissnerEffectusingmagnetizationcurrents

SolvetheMeissnerEffectinanellipsoidalsuperconductorusingonlythesurfacemagnetizationcurrent.

Solution

Thesolutionfollowsthesamestepsasinthederivationsurfacecurrentmodelofsuperconductivity,butusingeq.6.53.

Problem18:Approximatesolutionfornonellipsoidalbodies

FindanapproximatesolutionforthefieldgeneratedbyanonellipsoidalbodyinoneofthecasesofTable1.

TentativesolutionUsetheiterativemethoddiscussedattheendofsectionSolvingtheintegro-

differentialequationsbyiterationatpage42.Thisisanopenproblem.

Depolarizationtensor

Problem19:ComponentsofNinsphericalsymmetryAddtothesymmetryoperationsofspheroids(seeeq.3.25)asingleonethat

correspondstoasphere.UseittodeterminethethreeeigenvaluesofNincombinationwiththeunittraceruleeq.3.15.WhichisthematrixrepresentationofNinacoordinatesystemdifferentfromtheprincipalone?

Solution

Asymmetryshouldbeaddedthatexchangesthespheroid’srotationalaxisbyanyofthetwonormalaxes,forinstancearotationRin90°aroundcoordinateaxisx,asfollows.

!B+

!α s !m


R iN =

1 0 00 0 10 −1 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i

N⊥ 0 00 N⊥ 00 0 N

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=

N⊥ 0 00 0 N

!

0 −N⊥ 0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=N iR =

N⊥ 0 00 N⊥ 00 0 N

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

1 0 01 0 10 −1 0

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

N⊥ 0 00 0 N⊥

0 −N!

0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

.

(6.59)

fromwhich !!N⊥ =N =N. Iftheunittraceruleeq.3.15isapplied,itfollowsthat

3N=1,thatis,N=1/3.

AsNisproportionaltotheunitmatrix,itsrepresentationisinvariableinanycoordinatesystemduetothetransformationpropertyeq.3.23:

N'=R i(131)iR t = 1

31 (6.60)

wheretheorthogonalityrelationshipeq.3.17wasused.

Problem20:Singleeigenvaluemaydeterminethetypeofspheroid

AnellipsoidhasaneigenvalueN=0.16,theothertwobeingequal.Whatisthevalueofthelatter?Towhattypeofellipsoidtheypertain?Whichistheratioofthesmallersemiaxestothelargest?

SolutionThebodyisaspheroidwithpolareigenvalueNp=0.16andtwoequalequatorial

eigenvaluesNe.Fromthetraceruleeqs.3.15,Np+2Ne=1;therefore,Ne=(1−Np)/2=0.42,whereNp<Ne.Astheorderofsemiaxesistheinverseoftheorderofeigenvalues,thepolarsemiaxesisgreaterthentheequatorialones,correspondingtoaprolatespheroid(seeFigure20).Intheconventionalordera≥b≥c,Na=0.16,Nb=Nc=0.42.Theratioofsemiaxesβ=γmaybeobtainedfromeq.4.30orfromFigure14;thelatterbeingenoughforthegivenprecision.Byinspectionitiseasilyseenthatβ=γ ≅0.47,valuethatmaybecheckedusingFigure15forNb=Nc=0.42andadjustedforhigherprecissionusingeq.4.30.

Problem21:Eigenvaluesandaspectratios

Usingeachofthefollowingsetofdata,identifyalltheeigenvaluesofNandtheaspectratiosoftheellipsoidtowhichitpertain.a) N=0.65;γ=0.10.b) N=0.60;β=0.50.c) N1=N2=0.40.

Solutionsa)Therangeofvalues0.5-1.0pertainsonlytoNc.AsseenfromFigure16,for

γ=0.10thevalueNc=0.65isonthecurveβ=0.20.TheothereigenvaluesforthisaspectratioareNb=0.32(obtainedfromFigure15)andNa=0.03(Figure

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Problem24:Depolarizationtensorofaverylongrightcircularcylinder

Computethedepolarizationtensor,eq.3.4,ofarightcircularcylinderofinfinitelengthusingelectrostaticGauss’sLaw.Solution

SeesectionRightcircularcylinderofinfinitelengthofchapterChapter3.

Problem25:DepolarizationtensorofasphereComputethedepolarizationtensorofasphere,eq.3.4,usingelectrostatic

Gauss’sLaw.Solution

SeesectionSphereofchapter3.

Problem26:AlternativewayofobtainingcertaineigenvaluesofNExplorealternativeswaysofobtainingtheeigenvaluesofNnotgiveninthis

book.See,forinstance,MacMillan’sequations32.10and39.4.BearinmindthediscussionmadeatsectionUnifiedtreatmentofNforspheroids.Testyourapproachderivingthedepolarizationfactorsofthetwotypesofspheroids.

SolutionOpenproblem.

Problem27:SolvingtheprolatespheroidwithLegendre’sellipticintegrals

Showthattheprincipalvaluesfortheprolatespheroidsmaybeobtainedfromthegeneralexpressionintermsofnormalofellipticintegralseq.4.9.

SolutionTheprolatespheroidcorrespondstothecasek=1,forwhichEandFaregiven

byeq.A9.1,andφshouldbeexpressedintermsofβ=γ.

E(φ ,1)= sinφ , F(φ ,1)= dt

cost0

φ

∫ = ln tan φ /2+π /4( )( ) , cosφ = γ , (6.62)

whereusehasbeenmadeofthelastofeqs.4.9.Expressingthetangentintermsofcosine,itisobtained

tan φ /2+π /4( ) = sin φ /2+π /4( )cos φ /2+π /4( )

= sin(φ /2)cos(π /4)+ cos(φ /2)sin(π /4)cos(φ /2)cos(π /4)− sin(φ /2)sin(π /4) =sin(φ /2)+ cos(φ /2)cos(φ /2)− sin(φ /2) ,

(6.63)

assin(π/4)=cos(π/4).Takingintoaccountthefollowingtrigonometricidentity178,

178 Korn&Korn,p.810eq.21.2.9.


sin(φ /2)= 1− cosφ

2 , cos(φ /2)= 1+ cosφ2 for 0<φ < π2 , (6.64)

itisfinallyobtained

tan φ /2+π /4( ) = sin(φ /2)+ cos(φ /2)cos(φ /2)− sin(φ /2)

=

1− cosφ2 + 1+ cosφ

2⎛

⎝⎜

⎞

⎠⎟

1+ cosφ2 − 1− cosφ

2⎛

⎝⎜

⎞

⎠⎟

1+ cosφ2 + 1− cosφ

2⎛

⎝⎜

⎞

⎠⎟

1+ cosφ2 + 1− cosφ

2⎛

⎝⎜

⎞

⎠⎟

=1+ cosφ + 1− cosφ( )21+ cosφ − 1− cosφ( ) =

1+ 1− cos2φcosφ =

1+ 1−γ 2

γ.

(6.65)

Therefore

E(φ ,1)= 1−γ 2 , F(φ ,1)= ln 1+ 1−γ 2

γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟. (6.66)

Asβ=γ,replacementofthisvaluesinthefirstofeqs.4.9gives

Nx = −β ⋅γ ⋅E(φ ,1)1−β 2( ) 1−γ 2

+ β ⋅γ

1−β 2( ) 1−γ 2F(φ ,1)

= β 2

1−β 211−β 2

ln 1+ 1−β 2

β

⎛

⎝⎜

⎞

⎠⎟ −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟.

(6.67)

Thevaluecoincideswiththatofeq.4.30,obtainedbydirectintegrationofeq.3.82,whichservesatthesametimeofverificationoftheexpressioninnormalellipticintegralseq.A9.1.

N y = −γ 2

β 2 −γ 2 +β ⋅γ ⋅ 1−γ 2( )β 2 −γ 2( ) 1−β 2( ) −

β ⋅γ

1−β 2( ) 1−γ 2ln 1+ 1−γ 2

γ

⎛

⎝⎜⎜

⎞

⎠⎟⎟,

Nz =β 2

β 2 −γ 2( ) −β ⋅γ ⋅ 1−γ 2

β 2 −γ 2( ) 1−γ 2.

(6.68)

Problem28:Sphericalshell

Exploretheexistenceofadepolarizationtensorforasphericalshell(homoeoid),usingGauss’sLaw.Solution

Gauss’stheoremwillbeusedtofindthepotentialofthehomoeoidofFigure24wherethegrayregionishomogeneouslychargedmatter.The“depolarization


tensor”willthenbecalculatedbyderivationofthepotential.Theshellisassumedtohaveconstantvolumedensityofchargeρ.ForthecalculationofthefluxoftheelectricfieldthesurfacesS1,S2andS3willbeused,correspondingtothethreeregionsr<R1,R1<r<R2andR2<r.FromthesymmetryoftheproblemtheelectricfieldhasonlyradialcomponentE(r).OverS1thefluxis

!E ⋅d!S

S1

"∫∫ = 4πr2E = 4πk1Q1 =0, E =0. (5.69)

OverS2itis

!E id!S

S2

"∫∫ = 4πr2E = 4πk1Q2

= 4πk1ρ4π3 r3 −R31( ) ,

E =4πk1ρ3 r −

R13

r2⎛⎝⎜

⎞⎠⎟ for R1 ≤ r ≤R2.

(6.70)

Figure24.Sphericalhomoeoid.

OverS3itisobtained

!E id!S

S2

"∫∫ = 4πr2E = 4πk1Q3 = 4πk1ρ4π3 R2

3 −R31( ) ,

E =4πk1ρ3

R23 −R1

3

r2⎛⎝⎜

⎞⎠⎟ for R2 ≤ r.

(6.71)

Theelectricpotentialφisobtainedbyintegrationoverrofthepreviousexpressions,fromwhichitisobtainedf(seeeq.3.43):

f (!r )= − φ(r)4πk1ρ

= 14πk1ρ

E(r)dr∫ =

const. forr ≤R1 ,

const.+ 16r2 +R313 r

−1 for R1 ≤ r ≤R2 ,

const.+ R23 −R1

3

3 r −1 for R2 ≤ r.

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

(6.72)

valuesthatcoincidewiththosegivenbyMacMillan179.ThepotentialisthedifferenceofthepotentialsofthespheresofradiusR2andR1.Thedoublederivativesoff(seeeq.3.8)donotdefineavalid(constant)internaldepolarizationtensor.

179 MacMillan,p.40.


Problem29.Unitvectornormaltotheellipsoid'ssurface

Verifythatvectors , eq.A7.8,isnormaltothesurfaceattheintersectionswiththethreecartesianaxesandateverypointoftheellipsesdeterminedbytheintersectionwiththeplanesdefinedbyanarbitrarypairofcoordinateaxes.

Solution

Unitvector !!s(r ) isgivenby

s(!r )= x2

a4+ y

2

b4+ z

2

c4⎛

⎝⎜⎞

⎠⎟

−12 xa2x + y

b2y + z

c2z

⎛⎝⎜

⎞⎠⎟. (6.73)

Attheintersectionwithanycoordinateaxistheellipsoidalsurfaceistriviallynormaltoit.Lesstrivialistheconditionfortheellipticsectiondeterminedbyapairorcoordinateaxes,liketheplanexydiscussedbelow.Thecorrespondingellipsoid’scentralsectionforz=0is

x2

a2+ y

2

z2=1. (6.74)

Theequationofthestraightlinenormaltotheellipseatpoint(x1,y1)is180

y − y1x − x1

=a2 y1b2x1

. (6.75)

Vector!!s atthatpointis

s(!r1)=

x12

a4+y122

b4⎛

⎝⎜

⎞

⎠⎟

−12 x12

a2x +

y12

b2y

⎛

⎝⎜

⎞

⎠⎟ , (6.76)

whosecomponentshavetheratio

s y(!r1)

sx(!r1)

=y12

b2a2

x21, (6.77)

coincidentwiththeslopeofthestraightlinenormaltotheellipseatthatpoint.

Problem30:Surfacestepdiscontinuityofthedepolarizationtensor

Proveeq.3.38withoutmakingreferencetoanelectricormagneticfields.Solution

Hint:useeq.3.29.

180 KornandKorn,p.46.


Problem31:Ellipsoidalconductorwithnetcharge

Discussthefeasibilityofusingthedepolarizationtensormethodforobtainingthesurfacechargedistributiononellipsoidalconductorswithnetchargeinanappliedelectricfield.

Solution

Itisanopenproblemnotsolvedbytheauthor.Thedepolarizationtensormethodseemstorequirezeronetcharges.

Problem32:ApplicationofIvory’smethodtothesphereAsanillustrationofIvory’smethodgiveninsectionObtentionoftheexternal

gravitationalpotentialbyIvory’smethod,findtheinternalandexternalpotentialofahomogeneoussphericalmass.Solution

InternalvalueThepotentialinsidethebodyisgivenbyeqs.4.4and4.5,soitisnecessaryto

evaluatethefollowingintegrals.

ds

a2 + s( ) b2 + s( ) c2 + s( )0

∞

∫ = ds

R2 + s( )3/20

∞

∫ = −2 R2 + s( )−1/2∞

0= 2R, (6.78)

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )0

∞

∫ = ds

R2 + s( )5/2=

0

∞

∫ − 23 R2 + s( )−3/2

∞

0= 231R3. (6.79)

Thepotentialeq.4.2istherefore

φ(!r )=C0(λ)+Cx(λ)x2 +C y(λ)y2 +Cz(λ)z2 =M

12R − 1

2R3 r2⎛

⎝⎜⎞⎠⎟, (6.80)

whereMisthemassofthesphere.Theexpression,mutatismutandis,coincideswiththepotentialofauniformlychargedsphereinsidethebody(eq.3.62).

Externalvalue

Thevalueofκisdeterminedbysolvingeq.A7.20,thatforthesphereofradiusRbecomes

x2 + y2 + z2

R2 +κ= r2

R2 +κ=1, giving κ = r2 −R2. (6.81)

Thisvalueshouldbeusedineqs.4.52and4.54.Theintegralstoevaluateare

ds

a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫ = ds

R2 + s( )3/2r2−R2

∞

∫ = −2 s +R2( )−1/2r2−R2

∞

= 2r, (6.82)


ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫ = ds

R2 + s( )5/2r2−R2

∞

∫ = − 23 R2 + s( )−3/2

r2−R2

∞

= 231r3. (6.83)

Thepotentialeq.4.2atpointsexternaltothebodyistherefore

φ(!r )=C0(λ)+Cx(λ)x2 +C y(λ)y2 +Cz(λ)z2

=πσabc 21r− 23

x2 + y2 + z2

r3⎛

⎝⎜⎞

⎠⎟= 4πR

3σ3

1r= M4

1r, (6.84)

whereMisthemassofthesphere.Theexpression,mutatismutandis,coincideswiththepotentialofauniformlychargedsphereoutsidethebody(eq.3.62).

Problem33:ExpressionforfarawayfieldsobtainedfromnextUsingIvory’smethodgivetheexpressionofnextatlargedistancesfroma

uniformlypolarizedtriaxialelipsoid.Solution

Farawayfromthebodytheradialdistancer≫aandκ≫a2.Therefore,intheequationA7.20thatdeterminesκ itisvalidtodisregardthesquaredsemiaxesinthedenominatorsothat

x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ≅ r

2

κ≅1, κ ≅ r2. (6.85)

Thesevalueofκshouldnowbeusedintheexpressionfor nextαβ(!r ) ,eqs.4.66,with

Nα (κ )=abc2

ds

dα2 + s( ) a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫

= abc2dss5/2

= − abc3r2

∞

∫ s−3/2r2

∞= abc

31r3,

(6.86)

a2 +κ( ) b2 +κ( ) c2 +κ( ) = r3 , (6.87)

sα (!r |κ )=

xαd2α +κ

x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2=xαr. (6.88)

Therefore


next(!r )=N(κ )− abc

a2 +κ( ) b2 +κ( ) c2 +κ( )!s(!r |κ ) !s(!r |κ )

= 14π

V3r2

r51−3!r!rr5

⎛

⎝⎜⎞

⎠⎟,

(6.89)

whereV is the volume of the ellipsoid eq. A7.4. The tensor, equal to that of thesphereeq.3.66,correspondstoadipolarfieldwherethedipolemomentisVtimestheelectricormagneticpolarization(seeeq.6.36).

Problem34:Fieldontheexternalsideofthesurfaceofapolarizedsphere

Verifythevalidityofeq.3.41givingthevalueofnextonthesurfaceofasphere.

SolutionThequotedexpressionis

next(!r S )=N− s(!r S )s(!r S ), (6.90)

where,asforthespherea=b=c=R,

s(!r S )= x2

a4+ y

2

b4+ z

2

c4⎛

⎝⎜⎞

⎠⎟

−12 xa2x + y

b2y + z

c2z

⎛⎝⎜

⎞⎠⎟=!r S

R, where R= !r S . (6.91)

Fromeqs.3.66,NandthesurfacevalueoftheexternaldepolarizationtensorforthesphereofradiusRaregivenby

N =

13 0 0

0 13 0

0 0 13

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

, next(!r S )=

13−

x2

R2− x ⋅ yR2

− x ⋅zR2

− y ⋅xR2

13−

y2

R2− y ⋅zR2

− z ⋅xR2

− z ⋅ yR2

13−

z2

R2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

. (6.92)

Inmatrixnotation !!s(r )s(r ) is

s(!r S ) s(!r S )= 1R2

xyz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟i x y z( ) = 1

R2

x2 x ⋅ y x ⋅zy ⋅x y2 y ⋅zz ⋅x z ⋅ y z2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟. (6.93)

Therefore,thetwovaluescoincideand

!!!next(r S )=N− s(r S )s(r S ). (6.94)


Problem35:Microscopicoriginofdepolarizationtensors

Discussthederivationofdepolarizationtensorsfrommicroscopicmagnitudessuchasthetensormdiscussedatpage24.Solution

Apartfromtheargumentgivenatpage24,itisanopenproblemofinterestforthediscussionoftherelationshipbetweenmicroscopicandmacroscopicelectromagneticfields.

Energy,forcesandtorques

Problem36:Energyofadielectricsphere

AdielectricsphereofradiusRisuniformlypolarizedbyauniformappliedfield !!E0 .Verify,integrating !!ε0

EP iEP overallspace,thatthereisnocontributiontothe

totalfreeenergyfromthedepolarizationfield !!EP .

Solution

Theexternalfield !!EPext generatedbythesphere’spolarization !

P isthedipolar

oneeqs.6.10,A4.1and6.8:

!EPext(!r )= k1

3( !p i!r)!r − r2!pr5

, where !p =V !P =V3ε0(κ −1)κ +2

!E0. (6.95)

Thecontributiontotheenergyofthisexternalfieldis(seeeqs.A4.11andA1.4):

Feext =

ε02λ

!EPext(!r )i !EPext(

!r)d3rV '∫∫∫ =

ε0λ

4πk13

⎛

⎝⎜⎞

⎠⎟

2p2

V= 19

λε0V!P i!P (6.96)

Thecontributiontotheenergyoftheinternaldepolarizationfield !!EPint is(eqs.

2.12and3.66)

!EPint = − λ

ε0N i!P = − λ

3ε0!P , !DPint = ε0

!EPint +λ

!P = 23λ

!P ,

F inte = 12λ

!EPint i!DPintd3r

V∫∫∫ = V

2λ!EPint i!DPint( ) = −19

λε0V!P i!P , (6.97)

whichcancelstheexternalcontribution.

Problem37:Fermi’scontactinteraction

Foranatom,findtheenergyofinteractionbetweentheelectroniccurrentdensityandthemagnetizationofthenucleus.Approximatethelatterbyauniformlymagnetizedtriaxialellipsoid.ComparewiththestandardexpressionofFermi‘scontacttermUFanddiscusstheoriginofthedifference.


UF = −

132λµ0

!µn i!me , (6.98)

where !µn isthenuclearmagneticmomentand

!me theelectronicmagnetizationoverthenucleus181.

Problem38:TorquesonspheroidsComputethetorqueexertedoveranisotropicdielectricspheroidimmersedina

uniformappliedfield.Isitpossibletoobtainthevalueofthedielectricsusceptibilityonlyfromthisvalue?

Solution

Theforcecoupleisgivenbyeqs.5.21that,inthespheroid’sprincipalsystemwithzasthesymmetryaxis,gives

τ x =ε0λVχ 2

e

N!−N⊥

1+ χeN!( ) 1+ χeN⊥( )⎛

⎝⎜⎜

⎞

⎠⎟⎟E0yE

0z ,

τ y = −ε0λVχ 2

e

N!−N⊥

1+ χeN!( ) 1+ χeN⊥( )⎛

⎝⎜⎜

⎞

⎠⎟⎟E0zE

0x ,

τ z =ε0λVχ 2

e

N⊥ −N⊥

1+ χeN⊥( )2⎛

⎝⎜⎜

⎞

⎠⎟⎟E0xE

0y =0.

(6.99)

Theexpressionofthefieldcomponentsinsphericalcoordinatesis

E0x = E

0 senθ cosϕ , E0y = E

0 senθ senϕ , E0z = E

0 cosθ ,

τ x = k E0( )2senθ cosθ senϕ = k2 E0( )2sen2θ senϕ ,

τ y = −k E0( )2senθ cosθ cosϕ = − k2 E0( )2sen2θ cosϕ ,

!τ = k2 E0( )2sen2θ xsenϕ − ycosϕ( ) = k2 E0( )2sen2θ xcos(ϕ − π

2 )+ ysen(ϕ − π2 )

⎛⎝⎜

⎞⎠⎟,

k =ε0λVχ 2

e

N"−N⊥

1+ χeN"( ) 1+ χeN⊥( )⎛

⎝⎜⎜

⎞

⎠⎟⎟.

(6.100)

Thetorqueisnormaltotheplanedeterminedbythesymmetryaxisandthefield,andtendstoturnthelargestsemiaxistowardsthefield.Thevalueofkcanbe

181 C.E.Solivérez;Thecontacthyperfineinteraction:anilldefinedproblem;J.Phys.SolidSt.Phys.vol.13,L1017-L1019;1980.Althoughtheproblemisofquantumnature,thediscussionismadeintermsofclassicalelectromagnetism.


determinedfromtheamplitudeofthefunctionτ (θ).Asallotherparametersareknown,!χemaythenbefoundbysolvingaseconddegreeequation.

Problem39:Torqueexertedoverananisotropicdielectricsphere

Asphericalandanisotropicdielectricissuspendedinauniformandfixedelectricfield.Expressthetorqueexertedoverthesphereintermsoftheprincipalvaluesoftheelectricsusceptibilitytensor.

SolutionForasphereofanisotropicuniaxialdielectric(seeTable5)itissufficientto

studytheforcecoupleonanyplanecontainingthesymmetryaxis,usingascoordinatestheprincipalsystemofthesusceptibilitytensor.zaxisistakentobethesymmetryofrevolutiononeandplaneyzascontainingtheappliedfield

!E0.

Fromeqs.5.21and3.66itisthenobtained

!τ = !p×

!E0 , !p =

ε0λα e i!E0 , α e =V χ e i 1+ 3

1 χ e( )−1 , !E0 = E0y y +E

0z z ,

!p =ε0λV

χ⊥

1+ 31χ⊥

xx + yy( )+ χ"

1+ 31χ"

zz⎛

⎝⎜

⎞

⎠⎟ i E0

y y +E0z z( )

=ε0λV

χ⊥E0y

1+ 31χ⊥

y +χ"Ez0

1+ 31χ"

z⎛

⎝⎜

⎞

⎠⎟ .

(6.101)

Theforcecoupleisthen

!τ =

ε0λV

χ⊥E0y

1+ 31χ⊥

y +χ"Ez0

1+ 31χ"

z⎛

⎝⎜

⎞

⎠⎟ × E0

y y +E0z z( )

=ε0λV

χ⊥

1+ 31χ⊥

−χ"

1+ 31χ"

⎛

⎝⎜

⎞

⎠⎟ E0

yE0z x =

ε03λV

χ"− χ⊥

1+ 31χ⊥( ) 1+ 3

1χ"( )

⎛

⎝⎜⎜

⎞

⎠⎟⎟E0yE

0z x. (6.102)

Foranappliedfieldinquadrantxy,theexertedtorquemakesthebodyalignitssmallestsusceptibilityeigenvaluewiththeappliedfield.

Problem40:Torqueonamagnetizeddisk

AconstantthicknesscirculardiskofFe-Sialloyispermanentlymagnetizedinauniformway.Thediskissuspendedfromitsedgeinauniformandhorizontalmagneticfield.Determinethevalueofitsmagnetizationfromthetorqueexertedbythefieldonthedisk.SeetheexperimentalconfigurationatFigure21.

SolutionAcirculardiskofthicknessmuchsmallerthanitsdiametermaybeaproximated

byanoblatespheroidofpolarsemiaxisequaltohalfitsthicknessandanequatorialsemiaxisequaltoitsradius.Fromthediscussionatpage111aboutmagnetizationofacompass’sneedle,themagnetizationvectorshouldlieontheequatorialplane.


Ifthediskcanrotatefreely,Themagnetizationwillalignwiththefieldsothatthefreeenergyeq.5.35hasitminimumvalue.

Thecoordinatesystemischosensothatthesuspensionisparalleltothezaxis,themagnetizationliesontheplanexyandthefieldliesalongthedirectionx.Then

!τ (ϕ)= !m×

!B0 =VM cosϕ x + sinϕ y( )×B0 x= −VMB0 sinϕ z. (6.103)

ThevalueofMisobtainedfromtheamplitudeofthetorqueasafunctionofϕ.

Problem41:Torqueonaverylongsuperconductingrightcircularcylinder

Computethetorqueexertedoveraverylongcylindricalsuperconductorsuspendednormallytoitssymmetryaxisinhorizontalanduniformmagneticfield.

Solution

Thetorqueoverabodyoflargevolumeisverylarge,andlargebodiesareverydifficulttokeepatthetemperaturewheretheyaresuperconductors.Theproblemisunrealisticandwithoutpracticalinterest.

Problem42:Depolarizationtensorofinifinitesimallythinshells

Verifythatallinfinitesimallythinshellsofellipsoidalshapehaveadepolarizationtensorsimilartothatofthesphericaloneofeq.5.56:

N = s(!r S ) s(!r S ), next =0. (6.104)

where !!s(r S ) istheunitvectornormalatpoint !r S ofthebody’ssurface(seeeq.

A7.8).Solution

Usethesuperpositionprincipleforthepotentialofellipsoidswithconstantdensityofcharge.

Problem43:Applicationsofthinshells

Exploretheapplicationofthinellipsoidadshellsforsolvingapracticalproblem.Solution

Openanswer.


Appendix1:Electromagneticunits

Thedifferentsystemsofelectromagneticunitsarecharacterizedbytheconstantsappearinginthebasiclawsofelectromagnetism.WhilealmostwithoutexceptionengineersusetheInternationalSystemofUnits182(SI),physicistsoftenuseothersystemsmoreconvenientfortheirparticularfieldofstudies.Forthatreasonthesegeneralconstantsareusedinthisbook,followingJackson’sideas183.Afewotherconstantsarenextdiscussedthatdependonthedefinitionofcertainderivedbutrelevantmagnitudes,aselectricandmagneticsusceptibilitiesandmoments.

Coulomb’sLaw

(A1.1)

definesinsomesystemstheunitofelectriccharge,throughk1.TheconstantinGauss’sLawisaconsequenceofthisdefinition,

(A1.2)

Electricdisplacementisdefinedintermsofelectricfieldandelectricpolarizationdensity, (A1.3)

whereε0isvacuumpermittivity.λisanadimensionalconstantthatmakeselectricdisplacementamagnitudeofthesamespeciesthanelectricpolarization.

Forallthesystemsofunitsherediscussedithappensthat

4πk1ε0

λ=1. (A1.4)

Theelectricdipolemomentpoftwopointcharges+q,-qatdistancedis

182 ElectromagneticunitsattheBureauInternationaldesPoidsetMesures(BIPM),SImaintenance

agency.183 Jackson,pp.611-621.ThebookdoesnotanalyzetherelationshipsbetweenEandB,PandM

broughtbythetheoryofrelativity,whichsurelystablishesmorerelationshipsbetweenconstants,perphapsthosegivenbyeqs.A1.4andA1.10.

!!F = k1

qq'r2,

!!E idS

S∫∫ = 4πk1Q.

!!D= ε0

E +λ

P ,


p=

4πk1ε0λ

q·d = q ⋅d , (A1.5)

whereeq.A1.4wasused.

TheunitofelectriccurrentIisdefinedbytheforceperunitlengthexertedontwoparallelconductorsadistancedapart:

(A1.6)

FromMaxwell’sequationsA1.13,constantsk1andk2arerelatedbythevelocityoflightc,thatofpropagationofelectromagneticwavesinvacuum,

k1k2

= c2. (A1.7)

TheunitofmagneticinductionisdefinedbythemagneticpartoftheLorentzforceexertedoverachargeqmovingatvelocity :

!F = k3q

!v ×!B. (A1.8)

Ampère’sLawisthemagneticanalogueofGauss’sLaw,

!B id!l

C"∫ =

4πk2k3

I. (A1.9)

Therelationshipbetweenmagneticinduction ,magneticfieldandmagnetizationis (A1.10)

whereµ0isvacuummagneticpermeabiliity.λ'isanadimensionalconstantthatmakesHandMmagnitudesofthesamespecies.Thesourceofmagneticfield

!H_ is

magnetizedmatter,reasonwhyitismoreoftenusedinthisbookthan !B._

Forthesystemsofunitsdiscussedhereithappensthats

4πk2µ0k

23λ '

=1. (A1.11)

ThemagneticdipolemomentofaplanecoilofwirewithelectriccurrentIthatencirclesanareaAis m= k3 I·A. (A1.12)

TheconstantsappearinginMaxwell’sequationsarethefollowing:

!!ΔFΔl

=2k2I ⋅I 'd.

!v

!B

!!B = µ0(

H +λ ' M),


∇ i!D= λ ρ , ∇×

!E = −k3

∂!B∂t,

∇!B =0, ∇×

!H = k3 λ '!J + ∂

!D∂t

⎛⎝⎜

⎞⎠⎟ , (A1.13)

whereρisthevolumedensityofelectricchargeand !J thesurfacedensityof

electriccurrent.FortheSIsystemthegivenconstantshavethefollowingvalues:

k1 =1

4πε0=10−7c2 kg ⋅m

3

C2s2, k2 =

µ04π = 10

−7kg ⋅mC2

, k3 =1,

ε0 =1074πc2

C2s2

kg ⋅m3 , λ =1, µ0 = 4π10−7 kg ⋅mC2

, λ '=1. (A1.14)

Thefollowingtablegivesthevaluesforallthecommonsystemsofelectromagneticunits:SIorrationalizedMKS,ESUorelectrostaticCGS,EMUorelectromagneticCGS,GaussandHLorHeaviside-Lorentz.Gausssystemistheonepreferedbyphysicistforthestudyofrelativistictransformationsbecauseitmakesexplicitthespeedoflightc.

System λ λ'4πk1ε0

λ

4πk2µ0k

23λ '

SI 1/4𝜋𝜀0 𝜇0/4𝜋 1 184 1 4𝜋10-7 1 1 1

ESU 1 1/c2 1 1 4π 1/c2 4π 1 1

EMU c2 1 1 1/c2 4π 1 4π 1 1

Gauss 1 1/c2 1/c 1 4π 1 4π 1 1

HL 1/4π 1/4π c2 1/c 1 1 1 1 1 1

Table4.Thefivecommonsystemsofelectromagneticunits.

184 Seeeq.A1.14.

!!k1 !!k2 !!k3 !ε0 !µ0


Appendix2:VectorialOperators

Thefollowingpropertiesofoperator∇(nablaordel)arefrequentlyusedinthisbook185.

(A2.1)

∇ 1!r− !r '

= −!r − !r '!r− !r ' 3

. (A2.2)

(A2.3)

(A2.4)

(A2.5)

∇ i!F(!r )× !G(!r )( ) = !G(!r )i∇×

!F(!r )− !F(!r )∇×

!G(!r ). (A2.6)

(A2.7)

∇× f (!r )!F(!r )( ) = f (!r ) ∇×

!F(!r )( )+ ∇f (!r )( )× !F(!r ). (A2.8)

∇×

!F ×!G( ) = !G i∇( ) !F − !F i∇( ) !G + !F ∇ i

!G( )− !G ∇ i

!F( ). (A2.9)

185 KornandKorn,pp.157-162.

!!∇ = x ∂

∂x+ y ∂

∂ y+ z ∂

∂z.

!!Δ =∇ i∇ = ∂2

∂x2αα∑ .

!!

∇F(r )i G(r )⎡⎣ ⎤⎦

=F(r )i∇( ) G(r )+ G(r )i∇( ) F(r )+ F(r )× ∇×

G(r )( )+ G(r )× ∇×

F(r )( ).

!!∇ i f (r ) F(r )( ) = f (r )∇ iF(r )+∇f (r )i F(r ).

!!∇×∇f (r )=0.


Appendix3:Integraltheoremsofvectorcalculus

Laplacianofapointcharge’spotentialEq.A3.7thatdefinesthetraceoftheinternaldepolarizationtensorusesthe

valueofthefollowingexpression:

∇ i∇ f (!r ')

!r− !r 'd3r '

V∫∫∫

⎛

⎝⎜

⎞

⎠⎟ = ΔI '(!r ), (A3.1)

whichcoincideswith !!ΔI(r ) cuando !!f (

r ')=1. For !!r ≠ r ' theorderofintegratingandtakingderivativesmaybepermuted.

Δ f (!r ')!r− !r '

d3r 'V∫∫∫ = f (!r ')Δ 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ d3r '

V∫∫∫ ,

where Δ 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ =∇ i∇ 1

!r− !r '⎛

⎝⎜

⎞

⎠⎟ = −∇ i

(!r− !r ')!r− !r ' 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= −∇ i(!r− !r ')!r− !r ' 3

−∇ 1!r− !r ' 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟i(!r− !r ')= − 3

!r− !r ' 3+3(!r− !r ')i(!r− !r ')!r− !r ' 5

=0 ∀ !r ≠ !r ',

(A3.2)

whereeq.A2.5wasused.Itisthusprovedthatthelaplacianvanishesfor !!r ≠ r '.

Thecasewhereafieldpointcoincideswithasourcepoint—onlypossibleinsidethematerialbody—ismoredifficult.Itisnotuncommontofindbooksonelectromagnetismwitherroneousproofsthatbymerechancegivetherightvalue186.Thatiswhyaproofisgivenherethatcircumventsthecommonpitfalls.

Theorderofcalculationoftheintegralandthederivativesmaybeexchangedonlywhentheintegrandhasintegrablesingularities187.Itisthereforeconvenient

186 See,forinstanceJackson,p.13andReitz,p.44,wheretheytakethedoublederivativeinsidethe

integral.V.Hnizdo,Eur.J.Phys.21,pp.L1-L3(2000)makeserrorslikeusingGauss’sLawasdifferentfromthedivergencetheorem.

187 Hnizdo,;Generalizedsecond-orderpartialderivativesof1/r:EuropeanJournalofPhysics,vol.32,pp.287-297;2011.


todividethebody’svolumeintwoparts:asphericalvolumeofradiusR0aroundthesingularity—radiusthatattheendwilltendto0—andtherestofthebody,volumeV'=V-V0,asillustratedatFigure25.

Thatis,for !r fixedbutarbitrary,

I '(!r )= f (!r ')!r− !r '

d3r 'V∫∫∫

= f (!r ')!r− !r '

d3r 'V−V0

∫∫∫ + f (!r ')!r− !r '

d3r 'V0

∫∫∫ . (A3.3)

Asshownbefore,thelaplacianofthefirstintegralvanishessothatthecalculationisreducedto

Figure25.Isolatingthe

singularityinsidethebody.

Δ f (!r ')!r− !r 'V

∫∫∫ d3r '= Δ f (!r ')!r− !r 'V0

∫∫∫ d3r '

=∇ i∇ f (!r ')!r− !r '

d3r 'V0

∫∫∫ = −∇ i f (!r ')∇' 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ d3r '

V0

∫∫∫ . (A3.4)

When !!I '(r ) (eq.3.8)isacontinuousfunctionwithwellbehavedderivatives,

whichhappensforanycontinuosanddifferentiablefunction !!f (r ') ,thegradient

theoremeq.A3.9maybeusedtoreducethelastvolumeintegraltoasurfaceonewithoutsingularities:

∇ i f (!r ')∇' 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ d3r '

V0

∫∫∫ =∇ i f (!r ') d2!r '!r− !r 'S0

"∫∫ = f (!r ')∇ id2!r '!r− !r '

⎛

⎝⎜

⎞

⎠⎟

S0

"∫∫

= f (!r ')∇ 1!r− !r '

⎛

⎝⎜

⎞

⎠⎟ id2!r '

S0

"∫∫ = − f (!r ')(!r− !r ')id2!r '

R03

S0

"∫∫ = f (!r ')d2r 'R02

S0

"∫∫ . (A3.5)

where d2!r ' isthevectorialelementofarea.

Thefollowinglimitisnexttaken:

limR0→0

f (!r ')d2r 'R02

S0

"∫∫ = f (!r )limR0→0

d2r 'R02

S0

"∫∫ = 4π f (!r ), (A3.6)

whereusewasmadeofeq.A2.5,ofthefactthatthevectorsinthescalarproduct

(!r− !r ')id2!r ' areantiparallel,andthedefinitionofsolidangle.Atallstepstheintegrandshaveintegrablesingularities.Therefore


Δ f (!r ')d3r '

!r −!r 'V∫∫∫ =

−4π f (!r ) for !r ∈V0 for !r ∉V⎧⎨⎩

, (A3.7)

Divergence,gradientandrotortheoremsThedivergenceorGauss-Ostrogradsky’stheoremstablishesthefollowing

identity,188,189 (A3.8)

wherer istheunitvectornormaltosurfaceSandoutgoingfromvolumeV.Initsstandardformulationthetheoremrequiresthatthefield

!F anditsfirstderivatives

havenosingularitiesintheregiónVofintegrationanditsboundaryS.Itisalsovalidforfieldswithintegrablesingularities.Suchconditionsarenotfulfilledforthepointchargefieldeq.1.3.

Thegradienttheoremisacorollaryofthedivergencetheorem190,191

(A3.9)

Anothercorollaryisthecurltheorem192,193:

∇×!F(!r )d3r

V∫∫∫ = −

!F(!r )×d2!r

S"∫∫ . (A3.10)

ExtensionofthedivergencetheoremThestandardformulationofthedivergencetheoremisinsufficientfor

applicationstoelectromagnetism,wherethefieldsourcesareoftensingularitieslikepointcharges,linesofcharge,surfacedensitiesofchargeandcurrent,andstepdiscontinuitiesofpolarizations.Theextensionofthedivergencetheoremisdonenextfortwotypesodsingularities;itmayalsobedoneforthelineofcharge194,butitisofnointeresthere.

188 KornandKorn,p.163.189 Kuntzmann,pp.329-332.190 KornandKorn,p.163.191 Kuntzmann,p.339.192 KornandKorn,p.163.193 Kuntzmann,p.340.194 Kuntzmann,p.371.

!!∇ iF(r )d3r

V∫∫∫ =

F(r )id2r

S∫∫ ,

!!∇f (r )d3r

V∫∫∫ = f (r )d2r

S∫∫ .


Pointchargetypesingularities

Theextensionofthedivergencetheoremforpointcharges195,usuallynotclaimedasdifferentfromtheoriginalone196,isknowninphysicsastheelectrostaticGauss’sLaw.Itisappliedtofieldsofthefollowingkind:

(A3.11)

InordertoapplythestandarddivergencetheoremitisnecessarytoexcludefromtheregionVofintegrationaninfinitesimalvolumeVjaroundeverysingularity.TheexclusionsareindicatedherebyreplacingvolumeVbyV',replacingtheoriginalintegralbywhatmathematicianscallprincipalvalueintegral.TheprocessissimilartothatpreviouslydepictedatFigure25,exceptthatonemustincludethefluxatthesurfacesSjthataretheboundariesofeachvolumeVj.Therefore

(A3.12)

wherethesum’snegativesigncomesfromthefactthatthevectoresareentranttothevolume,notsalientasrequiredbythedivergencetheorem.Exceptforthefactorqj,alltheintegralsinthesumareequalbecausetheysubtendthesamesolidangle197,

(A3.13)

whereasphericalcoordinatesystemisused.Resultafinalmente

(A3.14)

IntheformulationoftheelectrostaticGauss’stheorem, !F istheelectricfield !

E

andtheequationispresentedinthefollowingway:

!E(!r )id2!r

S"∫∫ = 4πk1 ρ(!r )d3r

V '∫∫∫ +4πk1 qj ,

j∑ (A3.15)

wheretherelationshipbetween !E andthevolumechargedensityρ hasbeen

used198.Themostgeneralexpressionoftherelationshipbetweenthefluxofthe

195 Kuntzmann,p.369.196 Reitz,p.35.197 Kemmer,p.71.198 Reitz,p.85eq.4-29.

!!

F(r )=

qj(r − rj )r − rj

3j∑ .

∇ iF(r )d 3r

V '∫∫∫ =

F(r ) i d 2r

S∫∫ −

qj (r − rj ) i S jr − rj

3 d 2r 'Sj∫∫

j∑ ,

(r − rj ) i dSj

r − rj3 = sen(θ )dθ dϕ

Sj∫∫ = 4π ,

Sj∫∫

∇ iF(r )d 3r

V '∫∫∫ =

F(r ) i d 2r

S∫∫ − 4π qj .

j∑


electricfieldandthedistributionsoffreechargeisobtaineduponadditionofthesurfacedensitiesofcharge(stepdiscontinuitiesdiscussedinnextsection)andlineardensitiesofcharge(notdonehere).Thepolarizationchargesandcurrentsarediscussedinthemaintext,casebycase

StepdiscontinuitiesGeneralizeddivergencetheorem

Thegeneralizationisgivenhereofthedivergencetheoremforthecaseinwhichthefieldhasastepdiscontinuitythroughasurface199,thatinthisbookwillbethebody’ssurface.Afieldissaidtohaveastepdiscontinuity

(alsocalledjumpdiscontinuityordiscontinuityofthefirstkind)throughasurfaceΣwhenitiscontinuousandfiniteinanyneighbourhoodofΣ butitslimitvaluesoverbothsidesofΣ aredifferent.DefiningconventionallyapositiveandanegativesideofΣ200(seeFigure26)andidentifyingeachlimitwiththecorrespondingupper+or-index,onehas

(A3.16)

Figure26.Step

discontinuitysurfaceΣ .

Thedivergencetheoremeq.A3.8cannotbeappliedtoaregionthancontainsastepdiscontinuity.Theproblemisnotthattheintegralsinvolvedcannotbecomputedforsuchsurfaces201,buttheomissionofthesurface.SuchintegralshavetobedividedandcomputedseparatelyforeachoftheregionsthathaveΣasaboundary.InthecaseofFigure26thedivergencetheoremshouldnotbeappliedforthewholeofvolumeV,buttothetworegionsdeterminedbyΣ.AmathematicallyrigorouswayofjustifyingthisistoexcludeaninfinitesimalvolumeV0aroundΣ,volumedelimitedbythesurfaceS0identifiedinthefigurewithdashedlines.

Theapplicationofthetheoremintheindicatedwaygives

(A3.17)

WhenthevectorfieldvectorialisfiniteinanyneighbourhoodofΣ,soitisthefluxontheedgeofsurfaceS0tambiénloisandtendsto0whenthethicknessofV0goesto0.Thatis,theprincipalvalueintegralshouldbecomputed,whichwillbeindicatedinthisbookwiththespecificationlimV0→0.Takingthislimitdoesnotmodifythevolumeintegral’svalue.

199 Kuntzmann,pp.371-372.200 EnestelibroΣislasuperficiedelcuerpoandelvalorpositivedelcampoiselexterioraella.201 See,forinstance,MorseandFeshbach,p.34.

F+ (r )−

F− (r ) ≠ 0.

∇ iF(r )d 3r

V−V0∫∫∫ =

F(r ) i d

S

S∫∫ +

F(r ) i d

S

S0∫∫ .


Intheaforesaidlimitthepreviousequationgives

Therefore,

∇ i!F(!r )d3r

V∫∫∫ =

!F(!r )id!S

S"∫∫ −

!F +(!r )− !F −(!r )( )id !Σ

Σ"∫∫ . (A3.18)

where !dΣ isavectoroutgoingfromthepositivesideofthesurface.Theequation

isthegeneralizationofthedivergencetheoremforthecasewherethefieldhasastepdiscontinuityoveraclosedoropensurfaceΣ.

Amorecompactand“elegant”demonstrationmaybegivenbyusingdistributiontheory,butthisrequiresmanyotherconcedpts,asanextensionofthederivativeoperation202whichexceedsbyfartherequisiteofafirstcourseofvectoranalysisimposedinthisbook.

Generalizedrotortheorem

IfintheoremA3.18 !F isreplacedby !

F × c ,where !

c isanarbitraryconstantvector,onegets

∇ i!F(!r )× !c( )d3r

V∫∫∫ = !c i ∇×

!F(!r )d3r

V∫∫∫

=!F(!r )× !c( )id!S

S"∫∫ −

!F +(!r )− !F −(!r )( )× !c id

!Σ

Σ"∫∫

= !c i d!S ×!F(!r )( )

S"∫∫ − !c i d

!Σ ×

!F +(!r )− !F −(!r )( )

Σ"∫∫ ,

(A3.19)

whereeq.A2.6andthecyclicpropertyofthescalartripleproductwereused.As !c

isanarbitraryvector,itisobtained

∇×!F(!r )d3r

V∫∫∫ = −

!F(!r )×d!S

S"∫∫ +

!F +(!r )− !F −(!r )( )×d !Σ

Σ"∫∫ , (A3.20)

Polarizedbodies

Allthebodiesstudiedinthisbookarecharacterizedbyapolarizationvector !Q

withthefollowingproperties:

• Itisuniformandnon-vanishinginsidethebodyV;• itvanishesoutsidethebody;

202 See,forinstance,Farassat,F.,IntroductiontoGeneralizedFunctionswithapplicationsin

AerodynamicsandAeroaccoustics,NASA,LangleyResearchCenter,April1996.

∇ iF(r )d 3r

V∫∫∫ = lim

V0→0∇ iF(r )d 3r

V−V0∫∫∫

=F(r ) i d

S

S∫∫ −

F+ (r ) i d

Σ

Σ∫∫ +

F− (r ) i d

Σ

Σ∫∫

=F(r ) i d

S

S∫∫ −

F+ (r )−

F− (r )( ) i d Σ

Σ∫∫


• ithasavariablestepdiscontinuitythroughthebody’sclosedsurfaceS,theSsurfaceoftheprevioustheorems;

IftheintegrationregionisV'withfrontierS'thatcontainsbothVandΣ,itisobtainedforeq.A3.18

∇ i!F(!r )d3r

V '∫∫∫ = ∇ i

!F(!r )d3r

V∫∫∫

=!F(!r )id!S +

S '"∫∫

!F −(!r )id !Σ

Σ"∫∫ =

!F −(!r )id !Σ

Σ"∫∫ ,

(A3.21)

because !∇ iF and !

F vanishoutsideV.

Inthesamesay,eq.A3.20reducesto

∇×!F(!r )d3r

V∫∫∫ = −

!F −(!r )×d !Σ

Σ"∫∫ . (A3.22)


Appendix4:Fieldofanelectricdipole

Generalexpression

Theelectricfield !!E(r ) generatedatpointfield !r byabodywithelectricdipole

moment !p placedattheoriginis

!E(!r )= k1

3( !p i!r )!r − r2 !pr5

. (A4.1)

Thefield !!E j generatedbyadipole locatedatsourcepoint is203

!E j(!r )= k1

3!d j(!r )

!d j(!r )i !pj( )−d j(r)2 !pjd j(r)5 ,

= k13!d j(!r )

!d j(!r )−d j(r)21d j(r)5 i

!pj , where d j(!r )= !r − !r j , (A4.2)

and1istheunitdyadicsuchthatforanarbitraryvector !C itis

1i!C =!C . (A4.3)

Theupperindexindicatesthesourcepointandtheargumentthefieldpoint(seep.11).

Inmatrixrepresentationthepreviousexpressionbecomes

(A4.4)

203 Reitz,p.39eq.2-36.

!!pj !!r j

!!!E(r )= k1mj ⋅pj ,


where

(A4.5)

Dipolarfield'senergy

ApolarizablesphereofradiusRimmersedinanapplieduniformfield !!C 0

generatesoutsideadipolarfield !Cq(!r ). Dependingonthekindofmaterial

(dielectric,conductor,diamagnetic,paramagneticorsuperconductor)andappliedfield(electricormagnetic)theconstantsandthedipolarmomentswillvary.Nevertheles,inallcases,whenthecenterofthesphereistheoriginofthecoordinatesystem,theenergystoredintheexternalfieldisproportionaltothefollowingintegral:

!Cq(!r )i !Cq(!r )d3r

V '∫∫∫ , (A4.6)

where V' is the complementary region of V (the one occupied by the sphericalbody)and

!Cq(!r )= 3(

!q i!r )!r − r2!qr5

. (A4.7)

Theintegrandisthen

!Cq(!r )i !Cq(!r )=

3(!q i!r )!r − r2!q( )i 3(!q i

!r )!r − r2!q( )r10

= 9r2(!q i!r )2 −6r2(!q i

!r )2 +q2r4r10

= 3(!q i!r )+q2r6

. (A4.8)

Inansphericalcoordinatesystemwithzaxisparallelto


V '∫∫∫ = dϕ senθdθ r2Cq(!r )2dr

R

∞

∫0

π

∫0

2π

∫

=2π senθdθ r −4 3(!q i r)2 +q2( )drR

∞

∫0

π

∫

=2π q2 3cos2θ +1( )senθdθ0

π

∫⎛

⎝⎜

⎞

⎠⎟ r −4dr

R

∞

∫⎛

⎝⎜

⎞

⎠⎟ .

(A4.9)

!!!

mj(r )=

3d jx(r )2 −d j(r )2d j(r )5

3d jx(r )d jy(

r )d j(r )5

3d jx(r )d jz(

r )d j(r )5

3d jy(r )d jx(

r )d j(r )5

3d jy(r )2−d j(r )2

d j(r )53d jy(

r )d jz(r )

d j(r )5

3d jz(r )d jx(

r )d j(r )5

3d jz(r )d jy(

r )d j(r )5

3d jz(r )2−d j(r )2d j(r )5

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟

.

!q


As

3cos2θ +1( )senθdθ0

π

∫ = 3cos2θ +1( )d(cosθ )π

0

∫

= cos3θ + cosθ⎡⎣ ⎤⎦π0= 4, r −4dr

R

∞

∫ = r−3

3⎤

⎦⎥∞

R

= 13R3 ,

(A4.10)

therefore


V '∫∫∫ = 23

4πR3q2 =2 4π

3⎛⎝⎜

⎞⎠⎟

2q2

V, (A4.11)

whereVisthevolumeofthesphereandqthedipolemomentnorm.


Appendix5:Simmetriesofthesusceptibilitytensorsofsinglecrystals

Electricandmagneticsusceptibilities,asanytensorialproperty,havesimmetriesthatreflectthoseofthesinglecrystaltheycharacterize.Asillustratedbythedepolarizationtensoreq.3.23,therelationshipbetweencomponentsareaconsequenceofthetensor’stransformationproperties.Thefollowingtablegivestherelationshipsforallcrystallinesystemas204ofthecomponentsofasinglecrystal’ssusceptibilitytensor205,wherethenumberofindependentonesaregivenbetweenparenthesesinthefirstcolumn.

cristallinesystem clasification principal

coordinatesmatrix

representation

Cubic(1)

isotropicor

anaxialanyone

χ 0 00 χ 00 0 χ

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

trigonaltetragonalhexagonal

(2)

uniaxial

symmetryofrevolutionaroundaxisofmaximumsymmetry

χ⊥ 0 00 χ⊥ 00 0 χ

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Orthorhombic(3) biaxial

coordinateaxesparalleltothethreebinaryaxis

χ⊥ 0 00 χ⊥ 00 0 χ

!

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Monoclinic(4) biaxial

1coordinateaxisparalleltothesinglebinaryaxis

χ11 0 χ130 χ22 0χ13 0 χ33

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Triclinic(6) biaxial none

χ11 χ12 χ31χ12 χ22 χ23χ31 χ23 χ33

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Table5.Simmetriesofthesusceptibilitytensor.

204 Dekker,Capítulo1.205 Nye,p.23Table3.


Appendix6:Dyadics

Adyadicisamathematicalentity—alinearcombinationofexternalproductsofunitvectors—suchthatitsscalarproductwithavectorisavectorDyadicsmaybeinterpretedasvectoroperatorssuchthatitsaction(scalarproduct)onavectorgivesanothervector.Inthisbookdyadicsarewritteninboldfaceitalics.Theexpressionofn,thedyadicmostoftenquotedhere,is

n(!r )= x jnjk(

!r )xkj ,k∑ . (A6.1)

Aparticularcaseofdyadicistheexternalproductoftwovectors:

c =!A!B = Aj x Bk xk .

j ,k∑ (A6.2)

Suchisthecaseofthedyadic!!ss ineq.3.33.Thescalarproductofavectorwithadyadicmaybetakenintwodifferentways:

ni!A= x jnjk xk

j ,k∑ i xl Al

l∑ = x jnjk xk i xl Al

j ,k ,l∑

= x jnjkδkl Alj ,k ,l∑ = x jnjkAk

j ,k∑ = x j njkAk

k∑

j∑ ;

!A in= Aj x j i xk xlnkl

k ,l∑ = Aj x j i xk xlnkl

j ,k ,l∑

j∑

= Ajδ jknkl xlj ,k ,l∑ = Ajnjl xl

j ,l∑ = Aknkj x j

j ,k∑ = x j nkjAk

k∑

j∑ .

(A6.3)

Thesymmetricunitdyadic,herewritten1,is

1= x j x j

j∑ = xx + yy + zz , (A6.4)

suchthat

1i!A= x j x j

j∑ i xkAk

k∑ = x j x j i xkAk

j ,k∑ = x jδ jk Ak

j ,k∑ = x j Aj

j∑ =

!A;

!A i1=

!A.

(A6.5)

Ingeneral ni

!A≠!A in (A6.6)


Inthesamefashion,exchangingtheorderofvectorsmodifiesthevalueofexternalproducts.Forinstance,

a = x j xkajk

j ,k∑ ≠ xk x jajk

j ,k∑ = x j xkakj

j ,k∑ = at , (A6.7)

wherethelastdyadicisthetransposedofthefirst.Adyadicaissymmetricwhenitisequaltoitstransposeddyadic,thatis,when

itselementsajkaresymmetric:

a = x j xkajk

j ,k∑ = at = x j xkakj

j ,k∑ si ajk = akj . (A6.8)

Thetwowaysoftakingthescalarproductcoincideonlyforsymmetricdyadics:

ni!A= x j njkAk

k∑

j∑ = x j nkjAk

k∑

j∑ =

!A in. (A6.9)

Whenadyadicisrepresentedbyasquarematrix—herewritteninromanboldface—thetwowaysoftakingthescalarproductcorrespondtodifferentrepresentationsofthevectors:

If a =

axx axy axzayx ayy ayzazx azy azz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, A =

AxAyAz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,

A t ia = Ax Ay Az( )iaxx axy axzayx ayy ayzazx azy azz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=

Axaxx + Ayayx + AzazxAxaxy + Ayayy + AzazyAxaxz + Ayayz + Azazz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

,

a iA =

axx axy axzayx ayy ayzazx azy azz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

i

AxAyAz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

axxAx +axyAy +axzAzayx Ax +ayyAy +ayzAzazx Ax +azyAy +azzAz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟.

(A6.10)

Thetraceofadyadic,ofspecialinterestforthedepolarizationtensor,isdefinedasthesumofitsdiagonalelements.Thatis

Tra = ajj

j∑ = axx +ayy +azz . (A6.11)

Fordyadicsdefinedastheexternalproductoftwovectores,thetracecoincideswiththescalarproduct:

If a =

!A!B , Tra = ajj

j∑ = AjBj =

j∑ AxBx + AyBy + AzBz =

!A i!B. (A6.12)


Appendix7:Ellipsoids

Equationsandmainfeatures

Equations,sections,volumeandsurfaceInanarbitraryorthogonalsystemofcoordinatesthematrixequationofan

ellipsoidalsurface206is207

rt iA−1 ir =1, where r =xyz

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟, A =

Axx Axy AxzAxy Ayy AyzAxz Ayz Azz

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, (A7.1)

whereAisapositivedefinitesymmetricmatrixwitheigenvalues1/a2,1/b2and1/c2.ThecoordinatesystemthatdiagonalizesA,herecalledtheprincipalsystem,istheonewithoriginatthecenteroftheellipsoidthatcoincideswithitsprincipalaxes208.EquationA7.1thenbecomes

x2

a2+ y

2

b2+ z

2

c2=1 (A7.2)

wherea,bandcarethelengthsoftheellipsoidsemiaxes.

Theparametricequationsoftheellipsoidare209

x = a⋅cosu⋅sinv , y = b ⋅sinu⋅sinv , z = c ⋅cosv ,

where 0≤u≤2π , 0≤ v ≤π , (A7.3)

wherethedefinitionoftheanglesu,vandtheirdomainsareidenticaltotheanglesϕandθofasphericalcoordinatesystem,butnotitsrelationshipswiththecomponentesofthepositionvector !r .

Theintersectionoftheellipsoidwithaplaneisalwaysanellipse210thatbecomestwooverlappingcircumferencesforspheroids.

206 Thosenotfamiliarwithactualtri-dimensionalellipsoidscanprofitfromtheinteractiveviews

givenbytheWolframDemonstrationProjectEllipsoidsbyJeffBryant.207 KornandKorn,pp.74-82.208 KornandKorn,p.43.209 EllipsoidinWolframMathWorld,eqs.3-5.


Theellipsoid'svolumeVis

!!V = 4π3 abc. (A7.4)

ThesurfaceareaSis

S =2π c2 +2π absenφ E(φ;k)sen2φ +F(φ;k)cos2φ( ) ,

donde a≥b≥ c , cosφ = ca, k2 = a

2(b2 − c2)b2(a2 − c2) ,

(A7.5)

whereF(φ,k)andE(φ,k)aretheLegendre'sincompleteellipticintegralsofthefirstandsecondkinddiscussedinAppendix9.

Aspectratios

Indailylife,asphereiscalledasphereregardlessoftheradius’value,becauseitsapparentsizevarieswiththedistancetotheobserver.Inmathematicallanguageitissaidthatallspheresaresimilar,thesingleparameterthatcharacterizethemdiffersbyanarbitrarymultiplicationfactork.Whentwoparameterscharacterizetheshape,asinarectangleofsidesaandb,anobserverseestwosimilarones(sidesk·aandk·b)ashavingthesameaspect.Ifa≤b,theratioβ=b/aiscalledtheaspectratio,wheretheequalsigncharacterizesthesquareasaparticularcase.Bydefinitionanaspectratiohasthefollowingrangeofvalues, 0≤β≤1. (A7.6)

Itmayseemthatthevalue0shouldbeexcluded,butitisconvenienttoincludeitinordertocharacterizethedegeneraterectanglewitha=∞,twoparallellines.

Inthecaseofellipsoids,threeparametersa,b,ccharacterizeitsshapes.Ifaa≥b≥c,therearetwoaspectratios,

0≤β=b/a≤1,0≤γ=c/a≤1. (A7.7)

ThedifferentshapesofellipsoidsaccordingtoitsaspectratiosareidentifiedinTable3.Noticethattheaspectratiosoftheellipsoiddonotsufficetotellapartarightcircularcylinderofinifinitelengthfromanellipticoner.Theelectrostaticandmagnetostaticpropertiesofellipsoidalbodiesofthesameaspectisexactlythesamebecausesimilarellipsoidshavethesamedepolarizationtensor(seesectionN(0)isdeterminedbyaspectratios).

210 G.F.Childe,SingularPropertiesoftheEllipsoidandAssociatedSurfacesoftheNthDegree,

Macmillan,1861,pp.133-137.


NormalstothesurfaceandcentraldistancesTheunitvector normaltosurfaceSatpoint !

!r maybeobtainedfromthegradient211oftheeq.A7.2:

s(!r )=∇ x2

a2+ y

2

b2+ z

2

c2⎛⎝⎜

⎞⎠⎟

∇ x2

a2+ y

2

b2+ z

2

c2⎛⎝⎜

⎞⎠⎟

=

xa2x + y

b2y + z

c2z

x2

a4+ y

2

b4+ z

2

c4

. (A7.8)

Itmay,forinstance,beeasilycheckedthatthisvectorisnormaltothesurfaceoftheellipsoidattheintersectionswitheachofthethreecartesianaxesandontheellipsesdeterminedbytheplanesdefinedbyanytwocartesianaxes212(seeProblem29).

Unitvectors hassomeusefulproperties,bestderivedwhenitiswrittenintermsofthefollowingnormalvector

!s(!r ) :

s(!r )=

!s(!r )!s(!r )

, where !s(!r )= xa2x + y

b2y + z

c2z. (A7.9)

Then,if !r isthepositionvectorofapointontheellipsoidalsurface,equationA7.2

isequivalenttothevectorequation

!!

r1 is(r1)= x1x + y1 y + z1z( )i x1

a2x +

y1b2y +

z1c2z

⎛

⎝⎜⎞

⎠⎟

=x12

a2+y21b2

+z12

c2=1 ∀r1 ∈S.

(A7.10)

Theequationoftheplanetangenttotheellipsoidalsurfaceatpoint !!r1 is213

!r i!s(!r1)=

x1a2x +

y1b2y +

z1c2z =1, (A7.11)

relationshipthatmaybeeasilycheckedtobetrueattheintersectionsofthethreeprincipalaxeswiththeellipsoidalsurface.

Inasimilarfashion,vectors !!r1 and !!

s(r1) determineafamilyofparallelplaneswhoseequationis

!r i!r1 × s(

!r1)= k , (A7.12)

211 KornandKorn,p.158eq.5.5-5.212 KornandKorn,p.46.213 Kemmer,p.11Problem16andp.196.

!!s(r )


wherekisconstantforeachplaneandequaltoitsdistancetotheorigin.Theplanethroughpoint !!

r1 satisfiestheequation

!r1 i!r1 ×!s(!r1)=

!r1 ×!r1 i!s(!r1)=0, (A7.13)

expressionthatshowsthat !r1 ,!s(!r1) andthecenterOoftheellipsoidlieonthe

sameplane.Inwhatfollowsacentralplaneoftheellipsoidwillbeanyplanethatcontainsits

center,theoriginitsprincipalsystemofcoordinates.Inthesamefashion,acentralsectionoftheellipsoidwillbeanyintersectionofthesurfacewithacentralplane,asillustratedinFigure27.

Thegeometricalmeaningofvector !!s(r ) maybeobtainedfromFigure27.Itisthereseenthatthedistance!

d j fromtheellipsoid’scenterOtothetheplanetangenttoSatpoint

!rj istheprojection

ofthatvectoronthedirectionof!!s j . Thisvalueisherecalledthecentraldistancecorrespondingtothegivenpointoftheellipsoidalsurface.Thedotlinesinthefigurecorrespondtotheintersectionsofthetangentplanesandtheplanethatdeterminesthedepictedcentralsection.

Figure27.Centraldistanceandchordinanellipsoid’scentralsection.

Thegeneralexpressionforthecentraldistancedofapoint !r ontheellipsoidal

surfaceis

d(!r )= rcosθ = !r i s(!r )=

!r i!s(!r )s(!r ) for !r ∈S. (A7.14)

Usingeqs.A7.8andA7.10,thecentraldistanceturnsouttobe

d(!r )= x2

a2+ y

2

b2+ z

2

c2⎛

⎝⎜⎞

⎠⎟x2

a4+ y

2

b4+ z

2

c4⎛

⎝⎜⎞

⎠⎟

−1/2

= x2

a4+ y

2

b4+ z

2

c4⎛

⎝⎜⎞

⎠⎟

−1/2

= 1s(!r)

= 1x12

a4+y12

b4+z12

c4

= a2b2c2

b4c4x21 +a4c4 y21 +a

4b4z21,

(A7.15)

EquationA7.15showsthatthenormofvector !s(!r ) istheinverseofthecentral

distanceofpoint !r ontheellipsoidalsurface.Asimpleverificationofthethe

formulaistoapplyittothesphere,wherea=b=c=R=d.

Eq.A7.14determinestheanglebetweenvector !!r1 ofapointP1ontheellipsoidal

surfaceandthenormaltoitatthatpoint:


cosθ1 =

!r1 i s(!r1)r1

. (A7.16)

Thefollowingdyadicisoneofthecomponentsoftheexternaldepolarizationtensoreq.4.66:

s(!r |κ )s(!r |κ )= x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1 xα xα xβ xβd2α +κ( ) d2β +κ( )α ,β

∑ ,

where α = x , y ,z , dx = a,dy = b,dz = c. (A7.17)

Thistensorhasthepropertythatwhenappliedtoavectorleavesonlythecomponentnormaltotheellipsoidalsurfaceeq.A7.20(seealsosectionSurfacestepdiscontinuity,inparticulareq.3.39).

Thefollowingpropertyisresponsibleofthevanishingtraceoftheexternaldepolarizationtensoreq.:

Tr s(!r |κ )s(!r |κ )= x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1 x2αd2α +κ( )2α

∑

= x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1x2

a2 +κ( )2+ y2

b2 +κ( )2+ z2

c2 +κ( )2⎛

⎝⎜⎜

⎞

⎠⎟⎟=1.

(A7.18)

Summaryofproperties

Thegivenpropertiesofnormalvector !s(!r ) arecollectednext.

!s(!r )= xa2x + y

b2y + z

c2z : !s(!r )⊥ S at !r ;

!r i!s(!r1)=

x1a2x +

y1b2y +

z1c2z =1 istheplanetangenttoS at!r1;

!r1 i!s(!r1)=

x12

a2+y21b2

+z12

c2=1 ∀!r1 ∈S;

1s(!r ) = d(

!r )= 1x12

a4+y12

b4+z12

c4

= a2b2c2

b4c4x21 +a4c4 y21 +a

4b4z21

isthedistancetotheellipsoid'scenteroftheplanetangentat!r ;ss isthedyadicthatprojectsfromanyvectorthecomponent

normaltothesurfaceS oftheellipsoid.

(A7.19)


Confocalellipsoids

Figure28.Centralsectionoftwoconfocalellipsoids.

Forthecalculationofthepotentialoutsidethebodyuseismadeofafamilyofellipsoidsconfocalwiththebody’ssurface,

x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ=1, where κ ≥0, !r ∉V . (A7.20)

Figure28showsthecentralsectionz=0oftwosuchellipsoids,wherea=5,b=10andκ=20.Noticethattheconfocalellipseisalwaysrounderthantheoriginaloneandthatthedistancebetweensurfacesislargestforthesmallestsemiaxes.κisusuallydeterminedasthelargestorpositivealgebraicrootofeq.A7.20,where

!r isanyfieldpointoutsidethesurfaceoftheellipsoidalbodyeq.A7.23.Totheauthor’sknowledgenoexplicitexpressionshavebeengivenfortheseroots,whichingeneralarecalculatedpointbypoint214bymethodsliketheNewton-Raphsoniterations215.

Physical-geometricinterpretationofκ

Theunitoftheconfocalparameterκislength2,whichsuggestsitsinterpretationasasquareddistance.Thisassumptionisconfirmedwheneq.A7.20issolvedfortheonlytypeofspheroidwhereanexplicitsolutionmaybeeasilyfound,asphereofradiusR(seeeq.6.81).Inthiscase

214 See,forinstance,atBarczak&Breit&Jusiel,Ellipsoids,materialpointsandmaterialsegments,

commentintheparagraphprecedingeq.70inanun-numberedpage.215 Korn&Korn,section20.2-1.


κ = r2 −R2. (A7.21)

Forthegeneralellipsoideq.A7.20,upperandlowerboundsforκmaybefound.Thesquareddistancefromitscentertoanysurface’spointsatisfiesthefollowinginequality, c

2 +κ ≤ r2 ≥a2 +κ , sothat r2 −a2 ≤κ ≤ r2 − c2. (A7.22)

If !r0 isapointonthebody’ssurface,itscoordinatessatisfytheequation

x02

a2+y02

b2+z02

c2=1, (A7.23)

itsconfocalellipsoidbeinggivenbyeq.A7.20.Themaximumabsolutevalueofeachvariableisitssemiaxis,whenthetwoothervariablesarezero.Forx,forinstance,itisthenobtained

κ = x2 − x02 = a2 +κ( )−a2 , (A7.24)

andasimilarresultisobtainedisobtainedforyandzsuggestingthatthevalueofκmayber

2 − r20 .Theconjectureistruewhenthecomponentsof !r and!r0 are

takentohaveequalanglesintheirparametricexpressionseqs.A7.3:

x02 = a2 ⋅cos2u0 ⋅sin2v0 , x2 = a2 +κ( )⋅cos2u0 ⋅sin2v0y02 = b2 ⋅sin2u0 ⋅sin2v0 , y2 = b2 +κ( )⋅sin2u0 ⋅sin2v0 ,z02 = c2 ⋅cos2v0 , z2 = c2 +κ( )⋅cos2v0.

(A7.25)

ApairofsuchvectorsaredepictedinFigure28,showingthatingeneraltheyarenotcolinearwiththeelipsoid’scenter(theoriginofcoordinates).Forcorrespondingpoints

r2 − r02 = x2 − x0

2 + y2 − y02 + z2 − z0

2

= a2 +κ −a2( )cos2u0 ⋅sin2v0 +κ ⋅sin2u0 ⋅sin2v0 +κ ⋅cos2v0=κ sin2v0 + cos2v0( ) =κ .

(A7.26)

Thatis,

κ = r2 − r20 =

!r i!r − !r0 i

!r0 , (A7.27)

Correspondingpoints

Anytwopointssatisfyingeqs.A7.25arecalledcorrespondingpointsoftheconfocalpairofellipsoids.Asimilarrelationshipanditspropertiesarevalidforanytwopairofconfocalellipsoidsκ,κ’,butthiswillnotbeneededhere.


Thecorrespondanceofpoints !r0 ,!r mayalsobeexpressedas

x0a= x

a2 +κ, y0b

= y

b2 +κ, z0c= z

c2 +κ, (A7.28)

asfollowsfromthefactthattheangularpartsofeachpairofcomponentsarethesame.Whenthusexpressedthecorrespondenceprovidesasimplewayofprovingtheimportantpropertiesthatfollow.

If !r0 ,!ρ0 areanytwopointsonellipsoidA7.23and

!r , !ρ itscorrespondingpointsontheconfocalellipsoideq.A7.20,then

!r0 i!ρ = !r i

!ρ0. (A7.29)

Thecorrespondencebetweenthecomponentsξ,ψ,ζofvectors !ρ0 and

!ρ (eqs.

A7.28)andeq.A7.27give

ξ0a= ξ

a2 +κ, ψ 0b

= ψ

b2 +κ, ζ0c= ζ

c2 +κ,

where !ρ i!ρ −!ρ0 i!ρ0 =κ .

(A7.30)

Theproofeasilyfollowsfromtheexpansionofthescalarproductandtherelationshipseqs.A7.27,A7.28andA7.30:

!r0 i!ρ = x0ζ + y0ψ + z0ζ = a2 +κ

ax0ζ0 +

b2 +κb

y0ψ 0 +c2 +κc

z0ζ0 ,

!r i!ρ0 = xξ0 + yψ 0 + zζ0 =

a2 +κa

x0ξ0 +b2 +κb

y0ψ 0 +c2 +κc

z0ζ0

= !r0 i!ρ.

(A7.31)

Acorollaryofeqs.A7.27andA7.29is

!r −!ρ0 =

!ρ − !r0 , (A7.32)

asfollowsfrom

!r −!ρ0( )i !r − !ρ0( )− !ρ − !r0( )i !ρ − !r0( )

= !r i!r − !r i

!ρ0 −!ρ0 i!r +!ρ0 i!ρ0 −!ρ i!ρ +!ρ i!r0 +!r0 i!ρ − !r0 i

!r0= !r i!r − !r0 i

!r0 −!ρ i!ρ −!ρ0 i!ρ0( )+2 !r0 i

!ρ − !r i

!ρ0( )

=κ −κ +2⋅0=0.

(A7.33)


ThispropertyisthefoundationofIvory’smethodforthecalculationofthegravitationalpotentialatpointsexternaltothebody216.

216 MacMillan,pp.54-57.


Appendix8:Usefulintegrals

ds

A+ s B + s( )2∫

Theintegralsthatgivetheequatorialeigenvaluesofbothtypeofspheroidsareofthistype.InordertosolveitthefirststepistoreducetheexponentofA+sto1usingtheformula217

dsX ⋅Y 2

= − XA−B( )Y − 1

2 A−B( )dsX ⋅Y∫∫ , (A8.1)

where X=A+s,Y=B+s. (A8.2)

Theprimitivesoftheremainingintegralare218

dsX ⋅Y∫ = 2

B − Aarctan X

B − A

⎛

⎝⎜

⎞

⎠⎟ if A<B , (A8.3)

dsX ⋅Y∫ = 1

A−Bln X − A−B

X + A−B

⎛

⎝⎜

⎞

⎠⎟ if A>B. (A8.4)

Therefore,

dsX ⋅Y 2∫ = X

B − A( )Y + 1B − A( )3/2

arctan XB − A

⎛

⎝⎜

⎞

⎠⎟ if A<B. (A8.5)

217 Korn&Korn,lastlineofp.938andeq.154ofp.941.218 Korn&Korn,p.941eq.149.


dsX ⋅Y 2∫ = − X

A−B( )Y − 121

A−B( )3/2ln X − A−B

X + A−B

⎛

⎝⎜

⎞

⎠⎟ if A>B ,

where X = A+ s ,Y = B + s. (A8.6)

ds

A+ s( )3/2 B + s( )∫

Thisintegral,alsofoundforspheroids,maybereducedtothepreviousonewhenintegratingbyparts.Usingthesamenotationeq.A8.2,

ds

A+ s( )3/2 B + s( )∫ = dsX 3/2Y∫ . (A8.7)

Uponintegrationbyparts

udv∫ =u⋅v − vdu∫ , where

u= 1Y, du= − ds

Y 2 , v = −2X −1/2 , dv = X −3/2 , (A8.8)

itisobtained

udv∫ = ds

X 3/2Y∫ = − 2X ⋅Y

−2 dsX ⋅Y 2∫ . (A8.9)

Therefore

dsX 3/2Y∫ = − 2

X ⋅Y−2 ds

X ⋅Y 2∫ . (A8.10)

Uponreplacementfromeqs.A8.5andA8.6itisobtained

dsX 3/2Y∫ = − 2

X ⋅Y− 2 XB − A( )Y − 2

B − A( )3/2arctan X

B − A

⎛

⎝⎜

⎞

⎠⎟ if A<B , (A8.11)

dsX 3/2Y∫ = 2

A−B( ) X+ 1A−B( )3/2

ln X − A−BX + A−B

⎛

⎝⎜

⎞

⎠⎟ if A>B. (A8.12)


Appendix9:Legendre’sellipticintegrals

Definitions

!!F(φ ,k) and!!E(φ ,k) areLegendre'sincompleteellipticintegrals219ofthefirstandsecondkind,respectively,consideredtobeageneralizationofthewellknowntrigonometricfunctions.Thoughnotoftenmentionedinthetheoryofelectromagnetism,theyappearincommonproblemssuchasthecomputationofthearclengthofanellipse(fromwhichtheygottheirname),thedependenceoftheperiodofapendulusontheamplitudeofoscillation220andthesurfaceareaofanellipsoid(seeeq.A7.5).

Thecanonicalexpressionsfortheellipticintegralsare221:

E(φ ,k)≡ 1−k2sin2t dt0

φ

∫ ≡ 1− sin2α ⋅sin2t dt0

φ

∫ =E(φ \α ),

F(φ ,k)≡ dt

1−k2sin2t0

φ

∫ ≡ dt

1− sin2α ⋅sin2t0

φ

∫ =F(φ \α ) (A9.1)

φistheamplitude,ktheellipticmodulusand αthemodularangleoftheseintegrals.TheirvaluescanbefoundindoubleentrytableslikethoseofAbramowitzandStegun(pp.613-618),andmaybecalculatedforarbitraryvaluesoftheirtwoargumentsbymathematicalsoftwarelikeMathematica™andMaple™.

NotationusedinthemoreimportantreferencesDifferentauthorsusedifferentconventionsforthearguments,bothinorderand

meaning.Thefollowingtableshowstherelationshipbetweenthenotationusedinthisbookandthatoftheauthorsmoreoftenquotedhere.Beforemakinganycalculationsthereadershouldcarefullycheckthoseusedbyitsfavouriteoravailableauthororsoftware,forwhichitisrecommendedtocheckoneormorevalues,asthosegiveninProblem21.

219 Completeellipticintegralshaveφ=π/2.220 See,forinstance.C.E.Solivérez,Comportamientodeunpéndulorealsinrozamiento.221 Korn&Korn,p.833,eqs.21.6-29a;p.834,eqs.21.6-29cand21.6-29d.Weisstein,EricW;

EllipticIntegraloftheFirstKind;MathWorld--AWolframWebResource.Abramowitz&Stegun,pp.589eqs.17.2.8and17.2.6.


Author Nx,Ny,Nz β γ k E(φ ,k),F(φ ,k) pp.

Abramowitz&Stegun ___222 — — sinα

E(ϕ\α),F(ϕ\α);E(ϕ|m),F(ϕ|m)223

589-590

Korn&Korn ___224 — — sinα

E(k,ϕ),F(k,ϕ);E(ϕ\α),F(ϕ\α) 833-836

MacMillan ___225 β γ k E(ω,k),F(ω,k) 58-60

OsbornL/4π,M/4π,N/4π

cosϑ cosϕ sinα E(k,ϑ),F(k,ϑ) 351-352

Stoner Da,Db,Dc β γ k E(k,φ),F(k,φ) 807-808

Mathematica ___ — — k EllipticE[ϕ,k2],EllipticF[ϕ,k2],

Table6.Notationusedbyseveralauthors.

Althoughallargumentsareherereal,forafullanalysisofthegeneralpropertiesofellipticintegralstheyshouldbetakentobecomplexnumbers.Inrespecttotheirinterestinthisbook—theexpressionsforthecomponentsofthedepolarizationtensorofthetriaxialellipsoid,eqs.4.9and—theirargumentsarereal,boundedandgivenbyratiosoftheellipsoid’ssemiaxes,asfollows:

0≤ sinφ = 1−γ 2 ≤1, 0≤ k = 1−β 2

1−γ 2 = sinα ≤1,

where β = ba,γ = c

a.

(A9.2)

SpecialvaluesandparametricgraphsSomespecialvaluesoftheellipticfunctionsaregiveninthefollowingtablefor

itsuseinthecalculationofdepolarizationtensors.InthecaseoftheconstantinteriortensorN,thevaluesoftheargumentsareconnectedinaonetoonerelationshipwiththoseofβandγ,asgivenintheexpressiónoftheauxiliaryfunctionfforthetriaxialellipsoideq.4.52.Thesevaluesarealsoshowninthetable,identifyingthetypeofellipsoidtheycharacterize.

222 Givesgeneralpropertiesandtablesofvalues.223 m=sin2α.224 Givesgeneralpropertiesandgraphs.225 Givesonlytheexpressionofthepotential.


φ k α β γ Ellipsoid’stype E(φ ,k) F(φ ,k)

0 k α 1 1 sphere 0226 0227

φ 0 0 1 γ oblatespheroid φ228 φ229

φ 1 π/2 β β prolatespheroid sinφ230 ln tan φ /2+π /4( )( ) 231

π/2 1 π/2 0 0 infiniteellipticcylinder232 1 ∞

π/2 0 0 1233 0 infinitesheetofconstantthickness π/2 π/2234

Table7.SpecialvaluesoftheincompleteellipticfunctionsE(φ ,k)andF(φ ,k).

Figure29andFigure30showparametricgraphsofE(φ\α)andF(φ\α).Inbothcasestheleftsidegraphshowsthedependenceonαforφtakenasaparameter(therewrittenϕ).Therightsideshowsthedependenceonφ,whereeachcurvecorrespondstoadiferentvalueofparameterα .

Figure29.ParametricgraphoftheincompleteellipticfunctionofthesecondkindE.235

226 Eq.A9.1.227 Eq.A9.1.228 Abramowitz&Stegun,eq.17.4.23.229 Abramowitz&Stegun,eq.17.4.19.230 Abramowitz&Stegun,eq.17.4.25.231 Abramowitz&Stegun,eq.17.4.21.232 MacMillan,p.70.233 SeethecommentsontheinfinitesheetinsectionInfinitesemiaxes.234 Abramowitz&Stegun,Fig.17.2.


Figure30.ParametricgraphoftheincompleteellipticfunctionofthefirstkindF(φ\α).236

AsseeninFigure29,E(φ\α)isnevernegative.Itshighestvalueintheintervalsdeterminedbyeqs.A9.2isE(π/2\0)=π/2237.Itslowestvalueisattainedattheorigin,whereE(0\0)=0.Forconstantφ,E(φ\α)isadecreasingfunctionofαwhichisalmostconstantforthelowestvaluesofφ.Forconstantαitisanincreasingfunctionofφ,linearforα=0(thelineE(φ\0)=φinTable7).

AsseeninFigure30,F(φ\α)isnevernegativeanddivergesatF(π/2\ π/2).Forfixedαitisalmostconstantforlowvaluesφ(ϕinthefigure).Forα=0,islinearinφ(thelineE(φ\0)=φininTable7).

ReductiontonormalformoftheellipticintegralsofinterestTheintegralsofinterestforthecalculationofthedepolarizationtensorare238

Fora≥b≥ c ,k = a2 −b2

a2 − c2, sinφ = a2 − c2

a2 +κ, x2

a2 +κ+ y2

b2 +κ+ z2

c2 +κ=1, (A9.3)

235 Abramowitz&Stegun,p.594,figures17.6and17.7.236 Abramowitz&Stegun,p.594,figures17.3and17.4.237 Thiscorrespondstothecompleteellipticintegralofthesecondkind.Itsgraphisgivenin

Abramowitz&Stegun,p.592Figure17.2.

238 MacMillan,pp.58-60,wheretheexpressionforfunctiondninp.60shouldbednvκ =

b2 +κ

a2 +κ,

asfollowsfromitsdefinitionatp58,andE(ωκ)isE(ωκ,k).


ds

a2 + s( ) b2 + s( ) c2 + s( )κ

∞

∫ = 2a2 − c2

F(φ ,k), (A9.4)

ds

a2 + s( )3/2 b2 + s( ) c2 + s( )κ

∞

∫ = 2a2 − c2 a2 −b2( ) −E(φ ,k)+F(φ ,k)⎡⎣ ⎤⎦ , (A9.5)

ds

b2 + s( )3/2 a2 + s( ) c2 + s( )κ

∞

∫ = − 2 c2 +κ

a2 +κ( ) b2 +κ( ) b2 − c2( )

+ 2 a2 − c2

a2 −b2( ) b2 − c2( )E(φ ,k)−2

a2 − c2 a2 −b2( )F(φ ,k), (A9.6)

ds

c2 + s( )3/2 a2 + s( ) b2 + s( )κ

∞

∫

= 2 b2 +κ

a2 +κ( ) c2 +κ( ) b2 − c2( )− 2

a2 − c2 b2 − c2( )E(φ ,k). (A9.7)


Mainreferences

Thefollowinglistidentifiesthesourcesquotedmorethanonceinthisbook.Thissimplifiesbothquotationsanditsidentificationbythesurnamesofthefirsttwoauthorsofeachwork.Allotherreferencesaregivenatthebottomofthepage.

• Abramowitz,M.&Stegun,I.A.(editors);HandbookofMathematicalFunctions;DoverPublications;1972(10thprinting).

• Brown,Jr.,W.F.;MagnetostaticPrinciplesinFerromagnetism;NorthHollandPublishing;Amsterdam,1962.

• Brown,Jr.,W.F.;Micromagnetics;IntersciencePublishers;1963.• Chikazumi,S.;PhysicsofFerromagnetism;OxfordUniversityPress;1997. • Dekker,AdrianusJ.;SolidStatePhysics;PrenticeHall;1962(6thprinting).• Goertzel,G.&Tralli,N.;SomeMathematicalMethodsofPhysics;McGraw-Hill;1960.

• Gray,A.;OntheAttractionsofSphericalandEllipsoidalShells;Proc.EdinburghMath.Soc.vol.23;1913-1914;pp.91-100.

• Jackson,J.D.;ClassicalElectrodynamics;JohnWileyandSons;1975(2ndedition).Oneofthethreetextbooksmostquotedinthisbook,usesGauss'ssystemofunits.

• Kemmer,N.;VectorAnalysis:Aphysicist'sguidetothemathematicsoffieldsinthreedimensions;CambridgeUniversityPress;1977.

• Kellogg,OliverDimon;FoundationsofPotentialTheory;VerlagvonJuliusSpringer;Berlin(Germany);1929.

• Kittel,C;IntroductiontoSolidStatePhysics;JohnWileyandSons;1971(4thedition).

• Korn,G.A.&Korn,T.M.;MathematicalHandbookforScientistsandEngineers;McGraw-Hill;1968.

• Kuntzmann,J.;MathématiquesdethePhysiqueetdetheTechnique;Hermann;1961.

• Landau,L.D.&Lifchitz,E.M.;ElectrodynamicsofContinuousMedia;PergamonPress;1960.

• MacMillan,W.D.;TheTheoryofthePotential;DoverPublications;1958.• Maxwell,JamesClerk;ATreatiseonelectricityandmagnetism,vol.2;DoverPublications;1954;sectiones437and438.

• Morse,P.M.&Feshbach,H.;MethodsofTheoreticalPhysicsvol.1;McGraw-Hill;1953.

• Moskowitz,R.&DellaTorre,E.;TheoreticalAspectsofDemagnetizationTensors;IEEETransactionsonMagneticsvol.2,Nº4;pp.739-744;1966.

• Nye,J.F.;PhysicalPropertiesofCrystals;OxfordUniversityPress;1979.• Panofsky,W.K.H.&Phillips,M.;Classicalelectricityandmagnetism;Addison-Wesley;1995.

• Osborn,J.A.;PhysicalReviewvol.67;DemagnetizingFactorsoftheGeneralEllipsoid;pp.351-357;1945.

• Reitz,J.R.&Milford,F.J.&Christy,R.W.;FoundationsofElectromagneticTheory;Addison-Wesley;1960.Oneofthethreemaintextbooksquotedinthisbook,usesSIunits.


• Santaló,L.A.;Vectoresytensoresconsusaplicaciones;EUDEBA;1964(3rdedition).

• Solivérez,CarlosE.;MagnetostaticsofAnisotropicEllipsoidalBodies;IEEETransactionsonMagneticsMag.vol.17,Nº3;pp.1363-1364;1981.

• Solivérez,CarlosE.;Camposeléctricosgeneradosporelipsoidesuniformementepolarizados;RevistaMexicanadeFísica,vol.E54,Nº2;pp.203-207;2008.

• Stratton,J.A.;ElectromagneticTheory;McGraw-Hill;1941.ItdealswithellipsoidsbysolvingLaplace'sequationinellipsoidalcoordinates(pp.58-59).Thetextdiscussesindetailconducting(pp.207-211),dielectric(pp.211-217)andmagneticellipsoids(pp.257-258),includingdegenerateonesandspheroids.Oneofthethreemaintextbooksquotedinthisbook,usesSIunits.

• VanVleck,J.H.;TheTheoryofElectricandMagneticSusceptibilities;OxfordUniversityPress;1932(1stedition).

• Weisstein,EricW;MathWorld;WolframWebResource.• Young,H.D.;&Freedman,R.A.;SearsandZemansky´sUniversityPhysicswithModernPhysics;Addison-Wesley;2000(10thedition).


Alphabeticindex

A

aspectratio:63,170auxiliaryfunctionf:46

B

bodyofinfiniteextension:114

C

cavity:116compass

stabilityoftheneedle’smagnetization:111conductor

asperfectdielectric:127discussion:30surfacechargedensity:52

constitutiveequationelectriccase:18magneticcase:27

D

demagnetizationcoefficient:seedepolarizationtensor

demagnetizationfactor:seedepolarizationtensordemagnetizationtensor:seedepolarizationtensordemagnetizingcoefficient:seedepolarization

tensordemagnetizingfactor:seedepolarizationtensordepolarization:63,67depolarizationfactors:seedepolarizationtensordepolarizationtensor

body’ssurfacevalue:53cylinder,circular:56cylinder,elliptic:68,93definition:46eigenvalueNa:80,81eigenvalueNb:80,82eigenvalueNc:80eigenvalues,graphs:79eigenvalues,integralexpressions:67,179eigenvalues,order:84expressionsfromwhichisderived:46externalvalue:47,85history:1infinitesemiaxes:60internalvalue:47method,applications:3,4

method,limitations:2,5,103microscopicorigin:24,143Nforsimilarellipsoids:63N’sinverse:60N’s,values:3oblatespheroid:69,95obtentionfrompotentials:46principalcoordinatessystem:60prolatespheroid:74,96representationinarbitrarycartesian

coordinates:49sheet:54sheetofinfiniteextension:62sphere:50,58,133,136,142surfacestepdiscontinuity:51symmetrictensor:47symmetriesofN:50trace:48triaxialellipsoid:65,97,135

dielectric:24freeenergy:106

dielectricconstant:122Dirac'sdeltafunction:8Dirichlet,gravitationalpotentialofanellipsoidal

body:66dyadic

definitionandproperties:167notation:10

E

Earnshaw'stheorem:30electricdipolemoment

conductors:33induced:25

electricdisplacementvector:18electricfield

asasurfaceintegral:31asavolumeintegral:16atsharptips:129conductors:32generalpolarizationcase:17inducedpolarization:20,25integro-differentialequationforconductors:31integro-differentialequationfordielectrics:25onconductors'surface:31permanentpolarization:19pointdipole_:161

electricpermeability:25electricpolarization

equivalent_forconductors:33induced:20permanent:19

electricpotential


polarizationsurfacedensity:18polarizationvolumedensity:15,16

electricsusceptibilityapparentanisotropy:121definition:25

electromagneticunits:147electrostriction:101ellipsoid

normalvector:139ellipsoid

aspectratiosandtypeofellipsoid:64centraldistance:172centralplane:172centralsection:172confocalellipsoids:174correspondingpoints:175cylinder,circular:54cylinder,elliptic:65normalunitvector:171,173oblatespheroid:65principalcoordinatesystem:169prolatespheroid:65semiaxesorder:64sheet:54similarones:62sphere:54surfacearea:170tangentplane:171triaxialellipsoid:65volume:170

ellipticintegrals:170,181notation:181

energycrystallineanisotropy:105potential_ofanellipsoidalbody:66

equivalentpolarization:seeelectricandmagneticpolarization

F

fieldpoint:6,11,45example:21

forceelectriccase:114magneticcase:114

G

Gauss-Ostrogradsky’sintegraltheorem:155Gibbsfreeenergy:103

H

Helmholtzfreeenergydefinition:101linearcase:102

homoeoiddefinition:117spherical_:137

I

inducedelectricpolarization

atomiccase:21discussion:20

infiniterightcircularcylindermagnetized:125

integraltheoremscurltheorem:155,158divergencetheorem:155,156,158gradienttheorem:155singularities:5

irreversibleprocesses:100Ivory'smethod:85

J

jumpdiscontinuity:seesingularity

K

Kronecker'sdelta:11

L

laplacianofapointchargepotential’s:153latticesums:4,22localfield:24,116

M

macroscopicfield:6magnetic

polarizationinduced:30

magneticdipolemomentinduced:30permanent:29superconductor:35surfacecurrent:132torque:111

magneticdisk:145magneticfield

asavolumeintegral:27generalmagnetizationcase:27inducedmagnetizationcase:30integro-differentialequationforinduced

magnetization:30magneticinduction

constitutiveequation:27integro-differentialequationforthe

superconductingcase:37magnetizationmodel:36

magneticpolarizationdiscussion:26equivalent_forsuperconductors:35induced:29permanent:28,29

magneticscalarpotential:27magneticsusceptibility

definition:29magneticvectorpotential

magnetization:28magnetizationcurrent:28

magnetization:seemagneticpolarizationmagnetostriction:101


mathematicalnotation:10matrix:seematrixrepresentationmatrixrepresentation:10Meissnereffect:34microscopicfield:6

P

piezoelectricity:101piezomagnetism:101pointcharge:seesingularitypolarizabilitytensor

atomic:23body's_:20,114conductingbody:32,33dielectricbody:26magneticbody:30superconductingbody:35,133

principalcoordinatesystemdefinition:11

R

references:187

S

shell:seehomoeoidsingularity

integrable:15,48integraltheorems:156jumpdiscontinuity:6pointcharge:6,7stepdiscontinuity:9,157

sourcepoint:6,11example:21

sources:seereferencessphere

conductor:127,128dielectric:122ferromagnetic:126

superconductor:130statefunction:99statevariables:99stepdiscontinuity:seesingularitysuperconductivity:34

asperfectdiamagnetism:131superconductor

magnetizationmodel:35surfaceconductioncurrentmodel:36

surfacechargedensityconductor:31polarizeddielectrics:18

surfacecurrentsuperconductor:36

susceptibilitytensorcrystalsymmetries:165

T

texturedpolicrystals:5thermodynamicpotential:99thermodynamics:99thermodynamicsofelectromagnetism:101torque

anisotropicdielectric:109dielectrictriaxialellipsoid:108magnetizeddisk:145spheroid:144

V

vectornotation:10uniform:6unit:10

vectorialoperatorsnotation:11properties:151

volumemagnitude:10region:10


Abouttheauthor

CarlosEduardoSolivérez,bornin1939inSanSalvadordeJujuy(Argentina),hasaPh.D.inPhysicsfromtheUniversidadNacionaldethePlata(Argentina)andaSocialSciencesdiplomainEducationfromFacultadLatinoamericanadeCienciasSociales(FLACSO)—UniversidadAutónomadeMadrid(Spain).HemadegraduatestudiesonTheoreticalSolidStatePhysicsattheBerkeleycampusofthe

UniversityofCalifornia.HewasaresearchfellowattheTheoreticalPhysicsDepartamentofOxfordUniversity,visitingprofessoratFourierUniversity(Grenoble,France),staffmemberoftheCentroAtómicoBarilocheandteachedatseveralargentineuniversities,includingInstitutoBalseiro(UniversidadNacionaldeCuyo–CentroAtómicoBariloche,Argentina).HisspecialfieldofknowledgeinPhysicsisthetheoryofmagneticbehaviourofquasi-moleculargroups.Nowretired,heispresentlydevotedtothedivulgationofscientificandtechnologicalendeavoursinArgentinathroughthewikiEnciclopediadeCienciasyTecnologíasenArgentina—ofwhichheisthefounderandeditor—wherelinksaregiventohiswritings.

http://cyt-ar.com.ar/cyt-ar/index.php/P�gina_principal

http://cyt-ar.com.ar/cyt-ar/index.php/P�gina_principal

http://cyt-ar.com.ar/cyt-ar/index.php/Editor_ECyT-ar

Carlos E. Solivérez ii

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