ELECTROSTATICSANDMAGNETOSTATICSOFPOLARIZEDELLIPSOIDALBODIES:
THEDEPOLARIZATIONTENSORMETHOD
CarlosE.Solivérez
CarlosE.Solivérezii
ThisworkistheEnglishtranslationbytheauthorofthefirsteditionofhisbookinSpanish,Propiedadeselectrostáticasymagnetostáticasdecuerposelipsoidales:formalismodeltensordepolarización.Intheprocessoftranslationafewerrorswerecorrectedandanewchapter,severalproblemsandmanyappendiceswereaddedconcerningtheactualcalculationofthevaluesofthedepolarizationtensor.
Free Scientific Information
ElectroniceditionsbyCarlosE.Solivérez
Theauthoradvocatesfreeaccesstoallscientificandtechnologicalinformationobtainedwithfullorpartialgovernment'ssupport.Thisinformationshouldbeconsideredacommonsbelongingtothepeoplewhocreates,supportsandgivesmeaningtoit.Fromyear2012onwards,theseauthorpublishesallhisworksinInternet,includingthepresentbook,foritsfreeuseanddistributionundera
Attribution-NonCommercial-ShareAlike3.0Unported(CCBY-NC-SA3.0)license.
Thisworkwillbemodifiedanytimetheauthorfindsanerrororconsiders
convenienttomakefurtherelaborationsofitscontenttomakeitmoreusefuloreasiertoundersand.Allvaluablecontributionsofreaders,speciallyerrors,willbe
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.
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).
Depolarizationtensormethod 3
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).
Depolarizationtensormethod 5
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 ),
Depolarizationtensormethod 7
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.
Depolarizationtensormethod 9
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.
Depolarizationtensormethod 11
• 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 '
CarlosE.Solivérez12
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
Depolarizationtensormethod 13
occasionalandfragmentary,whichrequiresspecialprovisions.Thegivenorganizationmakestheselectionofexerciseseasier.Sodoestheredundancyinthediscussionofcentralconceptsandtheprovisionofadetailedalfabeticalindex.
Depolarizationtensormethod 15
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.
CarlosE.Solivérez16
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.
Depolarizationtensormethod 17
!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 '
CarlosE.Solivérez18
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
Depolarizationtensormethod 19
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.
CarlosE.Solivérez20
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.
Depolarizationtensormethod 21
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
CarlosE.Solivérez22
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))( ) ,
Depolarizationtensormethod 23
(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
CarlosE.Solivérez24
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 ))
Depolarizationtensormethod 25
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
CarlosE.Solivérez26
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).
Depolarizationtensormethod 27
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.
CarlosE.Solivérez28
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.
Depolarizationtensormethod 29
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.
CarlosE.Solivérez30
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
Depolarizationtensormethod 31
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.
CarlosE.Solivérez32
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.
Depolarizationtensormethod 33
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 +
CarlosE.Solivérez34
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.
Depolarizationtensormethod 35
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.
CarlosE.Solivérez36
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.
Depolarizationtensormethod 37
!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
CarlosE.Solivérez38
!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.
Depolarizationtensormethod 39
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)
CarlosE.Solivérez40
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.
Depolarizationtensormethod 41
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.
CarlosE.Solivérez42
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.
Depolarizationtensormethod 43
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.
Depolarizationtensormethod 45
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 .
CarlosE.Solivérez46
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
Depolarizationtensormethod 47
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
CarlosE.Solivérez48
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.
Depolarizationtensormethod 49
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.
CarlosE.Solivérez50
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 :
Depolarizationtensormethod 51
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 )
CarlosE.Solivérez52
(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
,
Depolarizationtensormethod 53
!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.
CarlosE.Solivérez54
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.
Depolarizationtensormethod 55
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
CarlosE.Solivérez56
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.
Depolarizationtensormethod 57
∂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)
CarlosE.Solivérez58
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,
Depolarizationtensormethod 59
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)
CarlosE.Solivérez60
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.
Depolarizationtensormethod 61
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)
∞
CarlosE.Solivérez62
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
Depolarizationtensormethod 63
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.
CarlosE.Solivérez64
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.
Depolarizationtensormethod 65
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.
CarlosE.Solivérez66
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 .
Depolarizationtensormethod 67
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 ,
CarlosE.Solivérez68
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.
Depolarizationtensormethod 69
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.
CarlosE.Solivérez70
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.
Depolarizationtensormethod 71
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.
CarlosE.Solivérez72
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)
Depolarizationtensormethod 73
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.
CarlosE.Solivérez74
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)
Depolarizationtensormethod 75
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.
CarlosE.Solivérez76
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)
Depolarizationtensormethod 77
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.
CarlosE.Solivérez78
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.
Depolarizationtensormethod 79
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.
CarlosE.Solivérez80
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.
Depolarizationtensormethod 81
Figure14.Na(L/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure1).
CarlosE.Solivérez82
Figure15.Nb(M/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure2).
Depolarizationtensormethod 83
Figure16.Nc(N/4π)asafunctionofc/a(γ ),whereeachcurvecorrespondstoadifferentvalueoftheparameterb/a (β )(takenfromOsborn,Figure3).
CarlosE.Solivérez84
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.
Depolarizationtensormethod 85
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.
CarlosE.Solivérez86
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.
Depolarizationtensormethod 87
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
CarlosE.Solivérez88
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.
Depolarizationtensormethod 89
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( )
.
CarlosE.Solivérez90
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.
Depolarizationtensormethod 91
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)
CarlosE.Solivérez92
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.
Depolarizationtensormethod 93
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.
CarlosE.Solivérez94
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)
Depolarizationtensormethod 95
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
130 MacMillan,p.62eq.39.2.
CarlosE.Solivérez96
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)
131 MacMillan,p.62eq.39.2.
Depolarizationtensormethod 97
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)
Depolarizationtensormethod 99
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.
CarlosE.Solivérez100
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.
Depolarizationtensormethod 101
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.
CarlosE.Solivérez102
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).
Depolarizationtensormethod 103
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.
CarlosE.Solivérez104
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.
Depolarizationtensormethod 105
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
CarlosE.Solivérez106
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)
Depolarizationtensormethod 107
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.
CarlosE.Solivérez108
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.
Depolarizationtensormethod 109
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.
CarlosE.Solivérez110
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.
Depolarizationtensormethod 111
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
CarlosE.Solivérez112
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.
Depolarizationtensormethod 113
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)
CarlosE.Solivérez114
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.
Depolarizationtensormethod 115
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.
CarlosE.Solivérez116
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.
Depolarizationtensormethod 117
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.
CarlosE.Solivérez118
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)
∂2 f2∂xα ∂xβ
⎛
⎝⎜
⎞
⎠⎟
=
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).
Depolarizationtensormethod 119
∂2 f3∂xα ∂xβ
⎛
⎝⎜
⎞
⎠⎟
= −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.
Depolarizationtensormethod 121
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
CarlosE.Solivérez122
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.
Depolarizationtensormethod 123
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)
CarlosE.Solivérez124
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.
Depolarizationtensormethod 125
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.
CarlosE.Solivérez126
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)
Depolarizationtensormethod 127
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.
CarlosE.Solivérez128
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.
Depolarizationtensormethod 129
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
CarlosE.Solivérez130
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.
Depolarizationtensormethod 131
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.
CarlosE.Solivérez132
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 )
Depolarizationtensormethod 133
!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
CarlosE.Solivérez134
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
!"#$%&'()&*($+,*"+-$',."*/$0, 94C,
9;^6,
2^,F,-(.(%&',*/(+?,/&##"+-,(+,*/(-,A&-"W,P/"'",9-,<,76E7,&+0,',<,76C7W,P/"'",I'$.,J(?B'",9E,(*,(-,I$B+0,),,<,76=C6,K/",*P$,'".&(+(+?,"(?"+N&%B"-,&'",9;,<,76,=8,\$2*&(+"0,I'$.,J(?B'",9C^,&+0,9+,<,7699,\J(?B'",9;^6,
A^,K/",$+%S,#$--(2%",-#/"'$(0,(-,*/",#'$%&*",$+",P(*/,9;,<,9-,<,76;7W,',<,),,<,76CG,&+0,9+,<,76=76,
S+8(7$6&993&H$U5$"-$&8F&$77.1)8.#)&8B$+&)86$&,)1$-*&+,*.8&-5+B$)&
eN"',*/",%(+",),<,76CW,&+&%()",*/",A/&+?"-,$I,-/&#",$I,*/","%%(#-$(0,A$''"-#$+0(+?,*$,9+W,9; &+0,9- &*,*/'"",0(II"'"+*,N&%B"-,$I,'6 !"N(-",&,-(.#%",?"$."*'(A,'"#'"-"+*&*($+,*/&*,-/$P-,A%"&'%S,*/",A/&+?"-,(+,*/",'&*($,$I,-".(&_"-6,H875*.8"&
' ) 2*% 2>% 2)% :77.1)8.#& &
76C7, 76C7, 769G, 76;9, 76;9, #'$%&*",-#/"'$(0,,
76GC, 76C7, 76=9, 7649, 76;5, *'(&_(&%,,
9677, 76C7, 76=;, 76=;, 76C9, $2%&*",-#/"'$(0,,
S+8(7$6 9>3 :.<$"B,75$)&8F *+.,%.,7&$77.1)8.#)
[-(+?,*/",0&*&,?(N"+,2S,e-2$'+,&+0,M*$+"',A&%AB%",*/",#'(+A(#&%,N&%B"-,$I,V,I$',&+,"%%(#-$(0,-BA/,*/&*,(*-,-".(&_"-,-&*(-IS,*/",'"%&*($+-/(#-,9,3,=,3,46,
H875*.8"&
K/",N&%B"-,A&+,2",$2*&(+"0,I'$.,*/",*&2%"-,&+0,?'&#/-,$I,e-2$'+",&+0,M*$+"'6,F-,2$*/,B-",*/",A$+N"+*($+,+,+,;,+,-W,P",.&S,A/$$-",+,<,4W,;,<,=W,-,<9W,P(*/,&-#"A*,'&*($- ),<,-y+,<,7W44444n,',<,;y+,<,7WEEEEE6,
J'$.,e-2$'+"Y-,K&2%",X,*/",I$%%$P(+?,&#'$_(.&*",N&%B"-,&'",$2*&(+"03,
-y+ <,7W4;=7= &+0,;y+ <,7WE5;9;3, ,9+,<,7W9E9GEn,9;,<,7W=E=77n,9-,<7WCGE=;6,, \E6E9^,, -y+,<,7W4;=7=,&+0,;y+,<,7WC899C3, ,
9+ <,7W9;5;7n,9; <,7W47;;7n,9- <7WC;E=76
J'$.,M*$+"'d-,?'&#/W,*/&*,?(N"-,%&'?"',"''$'-W,$+",$2*&(+-,9+,<,7W9Cn,9;,<,7W=8n,9;,<,7WCC6,
CarlosE.Solivérez136
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.
Depolarizationtensormethod 137
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
CarlosE.Solivérez138
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.
Depolarizationtensormethod 139
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.
CarlosE.Solivérez140
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)
Depolarizationtensormethod 141
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
CarlosE.Solivérez142
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)
Depolarizationtensormethod 143
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.
CarlosE.Solivérez144
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.
Depolarizationtensormethod 145
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.
CarlosE.Solivérez146
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.
Depolarizationtensormethod 147
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 ,
CarlosE.Solivérez148
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),
Depolarizationtensormethod 149
∇ 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
Depolarizationtensormethod 151
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.
Depolarizationtensormethod 153
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.
CarlosE.Solivérez154
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
Depolarizationtensormethod 155
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∫∫ .
CarlosE.Solivérez156
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∑
Depolarizationtensormethod 157
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∫∫ .
CarlosE.Solivérez158
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∫∫
Depolarizationtensormethod 159
• 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)
Depolarizationtensormethod 161
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 ,
CarlosE.Solivérez162
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
!Cq(!r )i !Cq(!r )d3r
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
Depolarizationtensormethod 163
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
!Cq(!r )i !Cq(!r )d3r
V '∫∫∫ = 23
4πR3q2 =2 4π
3⎛⎝⎜
⎞⎠⎟
2q2
V, (A4.11)
whereVisthevolumeofthesphereandqthedipolemomentnorm.
Depolarizationtensormethod 165
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.
Depolarizationtensormethod 167
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)
CarlosE.Solivérez168
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)
Depolarizationtensormethod 169
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.
CarlosE.Solivérez170
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.
Depolarizationtensormethod 171
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 )
CarlosE.Solivérez172
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:
Depolarizationtensormethod 173
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)
CarlosE.Solivérez174
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.
Depolarizationtensormethod 175
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.
CarlosE.Solivérez176
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)
Depolarizationtensormethod 177
ThispropertyisthefoundationofIvory’smethodforthecalculationofthegravitationalpotentialatpointsexternaltothebody216.
216 MacMillan,pp.54-57.
Depolarizationtensormethod 179
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.
CarlosE.Solivérez180
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)
Depolarizationtensormethod 181
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.
CarlosE.Solivérez182
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.
Depolarizationtensormethod 183
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.
CarlosE.Solivérez184
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).
Depolarizationtensormethod 185
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)
Depolarizationtensormethod 187
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.
CarlosE.Solivérez188
• 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).
Depolarizationtensormethod 189
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
CarlosE.Solivérez190
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
Depolarizationtensormethod 191
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
Depolarizationtensormethod 193
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.
Top Related