Jeans, The Mathematical Theory of Electricity and Magnetism, 5th Edition

=o =—.if é =< =o 4 3 — ==s =a THEMATHEMATICAL THEORY OF ELECTRICITY ANDMAGNETISM CAMBRIDGE UNIVERSITY PRESS LONDON :Fetter Lane w. THEMATHEMATICAL THEORY OF ELECTRICITY ANDMAGNETISM BY J.H.JEANS, D.Sc, LL.D., F.R.S. FORMERLY STOKES LECTURER INAPPLIED MATHEMATICS INTHEUNIVERSITY OFCAMBRIDGE; SOMETIME PROFESSOR OFAPPLIED MATHEMATICS INPRINCETON UNIVERSITY FIFTH EDITION A> CAMBRIDGE ATTHEUNIVERSITY PRESS 1927 43 13at First Edition 1908 Second Edition 1911 Third Edition 1915 iW*A Edition 1920(reprinted 1923) i^A^cfe'ftbra 1925( „ 1927) PRINTED INGREAT BRITAIN PREFACE [TOTHEFIRST EDITION] THEREisacertain well-definedrangeinElectromagnetic Theory,which everystudent ofphysics maybeexpectedtohave covered, withmore orlessofthoroughness,beforeproceedingtothestudyofspecialbranches ofdevelopmentsofthesubject. Thepresentbook isintended togivethe mathematicaltheoryofthisrangeofelectromagnetism, togetherwith the mathematicalanalysis requiredinitstreatment. Therangeisveryapproximatelythat ofMaxwell'soriginal Treatise, but thepresentbook isinmany respectsmore elementarythan that ofMaxwell. Maxwell's Treatise waswritten forthefully-equipped mathematician: the presentbook iswritten moreespeciallyforthestudent, and forthephysicist oflimited mathematical attainments. Thequestionsofmathematicalanalysiswhich aretreated inthetexthave been inserted intheplaceswheretheyare firstneeded forthedevelopment ofthephysical theory,inthebelief that, inmany cases, themathematical and physicaltheories illuminate oneanother bybeingstudiedsimultaneously. Forexample,brief sketches ofthetheories ofspherical,zonalandellipsoidal harmonics aregiveninthechapteronSpecialProblems inElectrostatics, interwoven with thestudyofharmonicpotentialsand electricalapplications: Stokes' Theorem issimilarly giveninconnection with themagneticvector- potential,andsoon.One result ofthisarrangementistodestroy,atleast in appearance,thebalance oftheamounts ofspaceallotted tothedifferentparts ofthesubject. Forinstance, more than halfthebookappearstobedevoted toElectrostatics, but thisspace will,perhaps,notseem excessive when itis noticed howmanyofthepagesintheElectrostaticpartofthebook aredevoted tonon-electricalsubjectsinappliedmathematics(potential-theory, theoryof stress, etc.),orinpuremathematics (Green's Theorem, harmonicanalysis, complex variable, Fourier's series, conjugate functions, curvilinear coordinates, etc.). Anumber ofexamples, takenmainlyfrom theusualCambridge examina- tionpapers,areinserted. These mayprovide problemsforthemathematical student, but itishopedthattheymayalsoformasortofcompendiumofresults forthephysicist, shewing whattypesofproblemadmit ofexact mathematical solution. Itisagainapleasuretorecordmythanks tothe officials oftheUniversity Press fortheirunfailing vigilance andhelpduringtheprintingofthebook. J.H.JEANS. Princeton, December, 1907. vi Preface [TOTHESECONDEDITION] Thesecond Edition willbefound todifferonlyvery slightlyfrom the first inalle.\ceptthelastfewchapters. ThechapteronElectromagnetic Theory ofLight has,however, beenlargelyrewritten andconsiderably amplified,and twonewchapters appearinthepresent edition, ontheMotion ofElectrons andontheGeneralEquationsoftheElectromagneticField. These lastchapters attempttogiveanintroduction tothemore recentdevelopmentsofthesubject. Theydonotaimatanythinglikecompletenessoftreatment, even inthesmall partsofthesubjects withwhichthey deal,but itishoped theywillform a useful introduction tomorecompleteandspecialisedworks andmonographs. J.H.JEANS. Cambridge, August,1911. [TOTHETHIRDEDITION] Inpreparingathird Edition Ihavemadeonlyafewchangesinthelatter chapters,which werenecessarytobringthebookuptodate. J.H.JEANS. London, November, 1914. [TOTHEFOURTH EDITION] Itwillbefound that themainchangesinthefourth Edition consist ina rearrangementofthelaterchapters andtheaddition ofawholly newchapter ontheTheoryofRelativity.Itneedhardlybesaidthatnoattemptismade togiveafullaccount oftheTheory;Ihave tried topresentitsbroad outlines inthesimplest possible way,andinstrivingaftersimplicityIhaveintentionally omitted allelaboration and detail. Itishopedthatthenewchapterwillpro- videasuitable introduction totheTheoryofRelativityforthestudent who approachesthesubjectforthe first time, equippedwith suchknowledgeof generalelectricaltheoryascanbegained from therestofthebook. J.H.JEANS. Dorking, December, 1919. Preface vii [TOTHEFIFTH EDITION] InpreparingaFifth Edition Ihave introduced thechangesthatseemed tobecalled forbythenowestablishedpositionofthenewtheories ofrelativity andquanta.Ihave notattempted anydetailed account ofthetheoryof quantabuthaveadded achapteron"The Electrical Structure ofMatter" which willintroduce thereader tothistheory. Itisapleasuretorecordmythanks tofriends andcorrespondents who havehelped mebymaking suggestionsandpointingouterrors andmisprints inearlier editions. Mythanks areespeciallydue toDrA.Russell, F.R.S., DrHaroldJeffreys, F.R.S., Professor E.P.Adams, DrR.E.Baynes, Mr L.A.PassandDrH.L.Curtis. J.H.JEANS. Dorking, March, 1925. CONTENTS INTRODUCTION Thethree divisions ofElectromagnetismPAGE 1 ELECTROSTATICS ANDCURRENT ELECTRICITY CIIAP. I. II. III. IV.Physical Principles..... TheElectrostatic Field ofForce Conductors andCondensers.... SystemsofConductors V.Dielectrics andInductive Capacity, VI.TheState oftheMedium intheElectrostatic Field VII. GeneralAnalytical Theorems.... VIII. Methods fortheSolution ofSpecial Problems . IX.Steady Currents inLinear Conductors X.SteadyCurrents inContinuous Media5 24 66 88 115 140 156 185 300 341 MAGNETISM XLPermanent Magnetism XII. Induced Magnetism364 408 ELECTROMAGNETISM XIII. TheMagnetic Field produced byElectric Currents XIV. Induction ofCurrents inLinear Circuits . XV. Induction ofCurrents inContinuous Media XVI. Dynamical TheoryofCurrents.... XVII. Displacement Currents andElectromagnetic Waves XVIII. TheElectromagnetic TheoryofLight XIX. TheMotion ofElectrons XX.TheTheoryofRelativity XXI. TheElectrical Structure ofMatter . Index425 452 473 485 510 532 559 593 629 647 INTRODUCTION TEETHREE DIVISIONS OFELECTROMAGNETISM 1.The factthatapieceofamber, onbeing rubbed, attracted toitself other small bodies, wasknown totheGreeks, thediscoveryofthis factbeing- attributed toThales ofMiletus (640-548 B.C.).Asecond fact,namely,that acertain mineral ore(lodestone) possessedthepropertyofattracting iron, ismentioned byLucretius. These two facts have formed thebasis from which themodern science ofElectromagnetismhasgrown.Ithasbeen found thatthetwophenomenaarenotisolated, butareinsignificantunits in avastandintricate series ofphenomena.Tostudy,andasfaraspossible interpret,thesephenomenaistheprovinceofElectromagnetism.And the mathematical developmentofthesubject must aim atbringingaslarge anumber ofthephenomenaaspossiblewithin thepowerofexact mathe- matical treatment. 2.The firstgreatbranch ofthescience ofElectromagnetismisknown asElectrostatics. Thesecond branch iscommonly spokenofasMagnetism, but ismoreaccuratelydescribed asMagnetostatics. Wemay saythat Electrostatics hasbeendevelopedfrom thesingle propertyofamberalready mentioned, andthatMagnetostaticshasbeendeveloped from thesingle propertyofthelodestone. These twobranches ofElectromagnetismdeal solelywith states ofrest, notwith motion orchangesofstate, and are therefore concernedonlywithphenomenawhich canbedescribed asstatical. Thedevelopmentsofthetwostatical branches ofElectromagnetism, namely Electrostatics andMagnetostatics,areentirely independentofoneanother. The science ofElectrostatics could havebeendevelopedifthepropertiesof thelodestone hadnever been discovered, andsimilarlythe science of Magnetostaticscould have beendevelopedwithout anyknowledgeofthe propertiesofamber. The third branch ofElectromagnetism, namely, Electrodynamics,deals with themotion ofelectricity andmagnetism, and itisinthedevelopment ofthisbranch thatwe first find that thetwogroupsofphenomenaof electricityandmagnetismarerelated toone another. The relation is J. 1 2 Introduction areciprocalrelation: itisfound thatmagnetsinmotionproducethesame effects aselectricityatrest, whileelectricityinmotionproducesthesame effects ismagnetsatrest. The third division ofElectromagnetism, then, connects thetwoformer divisions ofElectrostatics andMagnetostatics, and isinasensesymmetrically placedwithregardtothem.Perhaps wemay comparethewhole structure ofElectromagnetismtoanarchmade ofthree stones. Thetwosidestones canbeplacedinposition independently,neither inanywayrestingontheother, butthethird cannot beplacedinposition until tietwoside stones aresecurelyfixed. The third stone restsequally onthetwoother stones andforms aconnection between them. 3.Inthepresentbook these three divisions willbedevelopedinthe order inwhichtheyhavebeen mentioned, namely Electrostatics, Magneto- statics, Electromagnetism.The earlierchapterswillgiveanexplanationof thephysicalideasadopted byMaxwell inhisTreatise onElectricity and Magnetismsidebysidewith apurelymathematicaltheory. Maxwell's treat- ment ofElectrical Science wasdifferentiated from that ofother writers by hisinsistence onFaraday's conceptionofelectric andmagnetic energyas residinginthemedium.Accordingtothisview, theforcesactingonelectrified ormagnetisedbodies didnotform thewholesystemofforces inaction, but servedonlytoreveal thepresenceofavastly more intricatesystemofforces, which acted throughouttheetherbywhich thematerial bodies weresupposed tobesurrounded. Itwasonlythroughthepresenceofmatter thatthesup- posed systemofforces becameperceptibletohuman observation, sothat it wasnecessarytotrytoreconstruct thewholesystemofforces from nodata exceptthosegiven bytheresultant effect oftheforces onmatter, where matter waspresent.Asmightbeexpected,these dataprovedinsufficient to givefullanddefinite knowledgeofthesystemofethereal forces; itwasfound thatagreatnumber ofsystemsofethereal forces could beconstructed, each ofwhich would producethesame effects onmatter asareobserved. Ofthese systems, however, asingleoneseemed soverymuch moreprobablethanany oftheothers, that itwasunhesitatingly adoptedbothbyMaxwell andby Faraday. Assoon asthestephadbeen taken ofattributingthemechanical forces actingonmatter toasystemofforcesacting throughoutthewhole ether, afurtherphysical developmentwasmade notonly possiblebutalsonecessary. Astress intheether mightbesupposedtorepresenteither anelectric ora magnetic force, butcould notbeboth. Faraday supposedastress intheether tobeidentical with electrostatic force. There wasnolonger anypossibility, inthisscheme oftheuniverse, ofregarding magnetostaticforces asevidence ofsimplestresses intheether. The three divisions ofElectromagnetism 3 Ithas,however, been said thatmagnetostaticforces arefound tobe produced bythemotion ofelectriccharges. Now ifelectricchargesatrest produce simplestresses intheether, themotion ofelectriccharges must obviouslybeaccompanied bychangesinthestresses intheether. Itaccord- inglybecamepossibletoidentify magnetostaticforce withchangeinthe systemofstresses intheether. Thisinterpretationofmagneticforceformed anessentialpartofMaxwell'stheory. Comparingtheether toanelastic material medium, wemaysaythattheelectric forces wereinterpretedasthe staticalpressuresand strains whichaccompaniedthecompression,dilatation ordisplacementofthemedium, whilemagneticforces wereinterpretedasthe pressuresandstrains inthemedium causedbyitsmotion andmomentum. Thus electrostaticenergy wasregardedasthepotential energyofthemedium, whilemagnetic energywasregardedasitskineticenergy.Maxwell shewed thatthewhole series ofknown electrostatic andmagnetostatic phenomena mightbeconsistently interpretedasphenomena produced bythestresses andmotion ofamedium, thismotionbeinginconformitywith thelaws of dynamics.Thishypothesisisexamined intheearlierchaptersofthebook, although,aswillbeseen later, recentdevelopmentscallforatleastadrastic modification, andmoreprobablyforthecomplete abandonment ofthewhole hypothesis. 4.Theobservational factthatmagnetostaticforces wereproduced bythe motion ofelectriccharges inevitablyraised thequestionoftheinterpretation ofgeneral magnetic phenomenainelectrical terms.Asolution oftheproblem suggested byAmpere andWeber needs but little modification torepresent theanswer towhich moderninvestigationshave led.Recentexperimental researches shew that allmatter must besupposedtoconsistsolelyofelectrically charged particles, and itseemshighly probablethat allmagnetic phenomena canbeexplained bythemotion ofthesecharges.Ifthemotion ofthecharges isgoverned byaregularityofacertain kind, thebodyasawhole willshew magnetic properties.Ifthisregularitydoes notobtain, themagneticforces produced bythemotions oftheindividual chargeswillonthewhole neutralise oneanother, andthebodywillappeartobenon-magnetic. Onthisview the electricityandmagnetismwhich atfirstsight appearedtoexistindependently intheuniverse, areresolved intoelectricityalone—electricity andmagnetism becomeelectricityatrestandelectricityinmotion. Thisdiscoveryoftheultimateidentityofelectricity andmagnetismisby nomeans the lastword ofthescience ofElectromagnetism. Asfarback as thetime ofMaxwell andFaraday,itwasrecognisedthattheforces atwork inchemicalphenomena must beregarded largely,ifnotentirely,aselectrical forces. Later, Maxwell shewedlighttobeanelectromagnetic phenomenon, sothatthewhole science ofOpticsbecame abranch ofElectromagnetism. 1—2 4 Introduction Graduallytheconviction grewthat allphysical forces, with thepossible exceptionofGravitation, wouldprovetobeultimatelyofElectromagnetic origin,;othatbytheendofthenineteenthcenturymost scientists believed that thescience ofElectromagnetismwould advancealongtheroadopened outbyMaxwell until thewholephysicaluniverse hadbeenexplainedinthe terms ofelectromagnetic theory. Recentlythis belief hasexperiencedtwo verysevere checks. If,asMaxwell believed, theultimate seatofelectromagneticandoptical phenomenaistheether, itoughttobepossibletofindoutsomethingabout theether byelectromagneticandopticalmeans. Itought,forinstance, at least tobepossibletodetermine thevelocitywithwhich wemovethroughthe ether.Aseries ofexperimentsdevised tothisendhaveoneand allfailed to disclose thisvelocity. Toevery experimental enquiry, Nature seems togive theanswer either that there isnoether orthat natural phenomena goon exactlyasifthere werenoether. Ifthisview isfinally established, andat presentthere seemsonlyaverymeagrechance ofanyalternative, Maxwell's theoryoftheelectromagneticethermustnecessarilyfalloutofscience;itwill have served itspurposeasascaffoldingwhich willhaveenabled thestructure ofelectromagnetic theorytohavebeen built inperfect form, but itwillnot bepartofthat structure. Nevertheless thetime forfinally deciding how much ofMaxwell'stheoryisscaffoldingandhowmuch ispartoftheessential structure hashardly yetcome, sothat inthepresent bookweshall first developthetheory alongthegenerallines initiatedbyMaxwell, andthen shall devote achaptertothedevelopmentofamoremoderntheory andtoa discussion ofhow fartheexistence ofanether isessential toelectromagnetic theory. Thesecond check toMaxwell'stheoryhasoriginatedfrom thestudyof radiation andtheultimate electrical structure ofmatter; phenomenaof primary importance havebeen found nottobereconcileable with Maxwell's original theory. Inasense thenew factshardlycutattheroots ofthetheory; theymust rather bethoughtofasrestrictingthespreadofthebranches. There isnoquestionthattheelectrical phenomenaofeveryday life,thunder- storms, telephones anddynamos,are allgoverned byMaxwell's laws;itis onlywhenwepasstothephenomena arisingfrom themost intimate electrical structure ofmatter thatMaxwell's lawsappeartobeinadequate. Our final chapterwillcontain anexplanationofthefailure ofMaxwell'sElectrodynamics todealwith theseproblems, andaverybrief introduction tothenewtheory which hastaken itsplace. CHAPTER I PHYSICAL PRINCIPLES TheFundamental Conceptions ofElectrostatics I.State ofElectrification ofaBody. 5.Weproceedtoadiscussion ofthefundamentalconceptionswhich form thebasis ofElectrostatics. The first ofthese isthat ofastate of electrification ofabody. When apieceofamber hasbeenrubbed sothat it attracts small bodies toitself,wesaythat itisinastate ofelectrification — or,moreshortly,that itiselectrified. Other bodies besides amberpossessthepowerofattractingsmall bodies afterbeing rubbed, andarethereforesusceptibleofelectrification. Indeed itisfound that allbodiespossessthisproperty, althoughitislesseasily recognisedinthecase ofmost bodies, than inthecase ofamber. For instance abrass rodwith aglass handle, ifrubbed onapieceofsilkorcloth, willshew thepowertoamarkeddegree. The electrification here resides in thebrass;aswillbeexplained immediately,theinterpositionofglassor some similar substance between thebrassandthehand isnecessaryinorder that thebrassmayretain itspowerforasufficient time toenable usto observe it.Ifwehold theinstrument bythebrass rodandrubtheglass handle wefindthatthesamepowerisacquired bytheglass. II.Conductors and Insulators. 6.Letusnowsupposethatweholdtheelectrified brass rodinonehand byitsglass handle, andthatwetouch itwith theother hand.Wefindthat aftertouchingititspowerofattractingsmall bodies willhavecompletely disappeared.Ifweimmerse itinastream ofwater orpassitthrougha flame wefindthesame result. Ifontheother handwetouch itwith apieceofsilkorarodofglass,orstand itinacurrent ofair,wefind that itspowerofattractingsmall bodies remains unimpaired,atanyrate foratime. Itappearstherefore that thehumanbody,aflame orwater 6 Electrostatics —Physical Principles [ch.i have thepowerofdestroyingtheelectrification ofthebrass rodwhenplaced incontact withit,while silkandglassand airdonotpossessthisproperty. Itisforthisreason that inhandlingtheelectrified brass rod,thesubstance indirect contact with thebrass hasbeensupposedtobeglassandnotthe hand. Inthiswaywearrive attheidea ofdividingallsubstances intotwo classesaccordingastheydoordonotremove theelectrification when touch- ingthe electrifiedbody. The class which remove the electrification are called (onductors, forasweshall see later, theyconduct theelectrification awayfrom the electrified bodyrather thandestroyitaltogether; theclass which allow the electrified bodytoretain itselectrification arecalled non- conductors orinsulators. The classification ofbodies intoconductors and insulatorsappearstohave been first discovered byStephen Gray (1696- 1736). Atthesame time itmust beexplainedthat thedifference between insulators andconductors isoneofdegree only.Ifourelectrified brass rod were leftstandingforaweek incontactonlywith theairsurroundingitand theglassofitshandle, weshould find ithard todetect traces ofelectrifica- tion after thistime—theelectrification would havebeenconducted awayby theairandtheglass. Soalso ifwehadbeen able toimmerse therodina flame forabillionth ofasecondonly,wemighthave found that itretained considerable traces ofelectrification. Itistherefore morelogicaltospeakof goodconductors andbadconductors than tospeakofconductors andinsula- tors. Nevertheless thedifference between agoodandabadconductor isso enormous, that forourpresent purpose weneedhardlytake intoaccount the feebleconducting powerofabadconductor, andmaywithout serious incon- sistency, speakofabadconductor asaninsulator. Thereis,ofcourse, nothing topreventusimagininganideal substance which hasnoconducting power atall. Itwill oftensimplifytheargumenttoimaginesuch asubstance, although wecannot realise itinnature. Itmaybementioned herethat ofallsubstances themetals arebyvery much thebestconductors. Nextcome solutions ofsaltsandacids, andlastly asverybadconductors (and therefore asgood insulators) come oils,waxes, silk, glassandsuch substances assealing wax, shellac, indiarubber. Gases underordinaryconditions aregoodinsulators. Indeed itisworthnoticing that ifthishadnotbeen so,weshouldprobablynever havebecomeacquainted with electric phenomenaatall,forallelectricitywould becarried away by conductionthroughtheairassoon asitwasgenerated. Flames, however, conduct well, and, forreasons which willbeexplained later, allgasesbecome goodconductors when inthepresenceofradium orofso-called radio-active substances. Distilled water isanalmostperfect insulator, butanyother sampleofwater willcontainimpurities whichgenerallycause ittoconduct 6,7]TheFundamental Conceptions ofElectrostatics 7 tolerably well,andhence awetbodyisgenerallyabadinsulator. Soalsoan electrified bodysuspendedinairloses itselectrification much morerapidlyin dampweather than indry,owingtoconduction bywater-particlesinthe air. When thebodyisincontact with insulatorsonly,itissaid tobe "insulated." The insulation issaid tobegoodwhen the electrifiedbody retains itselectrification foralonginterval oftime, and issaid tobepoor when theelectrificationdisappears rapidly. Good insulation willenable a bodytoretain most ofitselectrification forsomedays,while withpoorinsula- tiontheelectrification will lastonlyforafewminutes orseconds. III. Quantity ofElectricity. 7.Wepassnext totheconceptionofadefinitequantityofelectricity, thisquantity measuringthedegreeofelectrification ofthebodywithwhich itisassociated. Itisfound thatthequantityofelectricityassociated with anybodyremains constantexceptinsofarasitisconducted awaybycon- ductors. Toillustrate, and tosome extent toprovethis law,wemayuse aninstrument known asthegold-leaf electroscope.This consists ofaglass vessel, throughthetopofwhich ametal rod ispassed, supportingatitslower endtwogold-leaveswhich under normal conditions hangflatsidebyside, touchingoneanotherthroughouttheirlength. When anelectrifiedbody touches orisbroughtnear tothebrass rod,thetwogold-leavesareseen to separate,forreasons which willbecome clear later(§21),sothat theinstru- ment canbeused toexamine whether ornotabodyiselectrified. Letusfixametal vessel onthetopofthebrass rod,thevesselbeing closed buthavingalidthroughwhich bodies canbein- serted. The lidmust besuppliedwith aninsulating handle for itsmanipulation. Supposethatwehave electrified somepieceofmatter—tomake thepicture definite, supposethatwehave electrified asmall brass rodbyrubbingitonsilk—and letussuspendthisbody inside the vessel byaninsulatingthread insuch a manner that itdoes nottouch thesides ofthe vessel. Letusclose the lidofthe vessel, sothat the vessel entirelysurrounds the electrifiedbody,andnote the amount ofseparationofthegold-leavesoftheelectro- scope. Letustrytheexperiment anynumber oftimes, placingthe electrified bodyindifferentpositionsinside theclosed vessel, takingcareonlythat itdoes notcome into contact with thesides ofthe vessel orwith any other conductors. We shall findthat ineverycase theseparationofthe gold-leavesisexactlythesame.Via. l. 8 Electrostatics —Physical Principles [ch.i Inthiswaythen,wegettheidea ofadefinitequantityofelectrification associated with thebrass rod, thisquantity being independentoftheposition ofther>dinside theclosed vessel oftheelectroscope. We find, further, that thedivergenceofthegold-leavesisnotonlyindependentofthepositionof therodinside thevessel, but isindependentofanychangesofstate which therodmayhaveexperiencedbetween successive insertions inthe vessel, provided onlythat ithasnotbeen touchedbyconductingbodies. We mightforinstance heat therod, or,ifitwassufficiently thin,wemight bend itintoadifferentshape, andonreplacingitinside the vessel we should findthat itproduced exactlythesame deviation ofthegold-leaves asbefoie. Wemay, then, regardtheelectricalpropertiesoftherodasbeing duetoaquantityofelectricityassociated withtherod,thisquantity remaining permanentlythesame, exceptinsofarastheoriginal chargeislessened by contact with conductors, orincreased byafreshsupply. 8.Wecanregardtheelectroscopeasgivinganindication ofthemagni- tude ofaquantityofelectricity,twocharges being equalwhentheyproduce thesamedivergenceoftheleaves oftheelectroscope. Inthesamewaywecanregardaspring-balanceasgivinganindication ofthemagnitudeofaweight,twoweights being equalwhentheyproduce thesame extension ofthespring. Thequestionoftheactualquantitative measurement ofaquantityof electricityasamultipleofaspecifiedunithasnotyetbeen touched. We can,however, easilydevise means fortheexactquantitative measurement ofelectricityinterms ofaunit.Wecanchargeabrass rodtoanydegree weplease,andagreethat thechargeonthisrod istobetaken tobethe standard unitcharge. Byrubbinganumber ofrods until eachproduces exactlythesamedivergenceoftheelectroscopeasthestandardcharge, we canprepareanumber ofunitcharges,andwecannowsaythatachargeis equaltonunits, ifitproducesthesame deviation oftheelectroscopeas would beproduced bynunits allinserted inthevessel oftheelectroscope atonce. Thismethod ofmeasuringanelectricchargeisofcourse notone thatanyrationalbeingwouldapplyinpractice,buttheobjectofthe present explanationistoelucidate thefundamentalprinciples, andnotto giveanaccount ofpracticalmethods. 9.Positive andNegative Electricity. Letussupposethatweinsert in thevessel oftheelectroscopethepieceofsilkonwhich oneofthebrass rods hasbeensupposedtohavebeen rubbed inorder toproduceitsunit charge. Weshall findthat thesilkproducesadivergenceoftheleaves of theelectroscope,andfurther that thisdivergenceisexactly equaltothat which isproduced byinsertingthebrass rodalone intothevessel ofthe electroscope. If,however, weinsert thebrass rodandthesilktogetherinto theelectroscope,nodeviation oftheleaves canbedetected. 7-11] TheFundamental Conceptions ofElectrostatics 9 Again,letussupposethatwechargeabrass rodAwithachargewhich thedivergenceoftheleaves shews tobenunits. Letusrubasecond brass rodBwithapieceofsilkGuntil ithasacharge,asindicatedbytheelectro- scope,ofmunits,mbeingsmaller than n.Ifweinsert thetwobrass rods together,theelectroscope will, asalready explained, giveadivergencecorre- spondington+munits. If,however, weinsert therodAandthesilkG together,thedeviation willbefound tocorrespondton—munits. Inthiswayitisfound thatachargeofelectricity must besupposedto havesignaswell asmagnitude. Asamatter ofconvention, weagreeto speakofthemunits ofchargeonthesilkasmpositive units, ormorebriefly asacharge+m,while wespeakofthechargeonthebrass asmnegative units, oracharge—w. 10. GenerationofElectricity.Itisfound tobeagenerallawthat,on rubbingtwobodies which areinitially uncharged, equal quantitiesofpositive andnegative electricityareproducedonthetwobodies, sothat the total charge generated,measuredalgebraically,isnil. Wehave seen thattheelectroscopedoesnotdetermine thesignofthe charge placedinside theclosed vessel, butonlyitsmagnitude. Wecan, however, determine both thesignandmagnitude bytwoobservations. Let usfirst insert thecharged bodyalone intothevessel. Then ifthedivergence oftheleavescorrespondstomunits,weknow that the.chargeiseither+m or—m,and ifwenowinsert thebodyincompanywithanothercharged body, ofwhich thechargeisknown tobe+n,then thecharge weareattempting tomeasure willbe+mor—maccordingasthedivergenceoftheleaves indicates n+morn~munits. With more elaborate instruments tobe described later(electrometers)itispossibletodetermine both themagnitude andsignofacharge byoneobservation. 11. Ifwehadrubbed arodofglass,instead ofoneofbrass, onthe silk, weshould have found thatthesilkhadanegative charge,andtheglassof course anequal 'positive charge.Itthereforeappearsthat thesignofthe charge producedonabodybyfrictiondependsnotonlyonthenature ofthe body itself, but alsoonthenature ofthebodywithwhich ithasbeen rubbed. Thefollowingisfound tobeagenerallaw :Ifrubbing asubstance Aon asecond substance BchargesApositivelyandBnegatively,and ifrubbing thesubstance Bonathird substance GchargesBpositivelyandGnegatively, thenrubbingthesubstance Aonthesubstance GwillchargeApositively andGnegatively. Itisthereforepossibletoarrange anynumber ofsubstances inalistsuch thatasubstance ischargedwithpositiveornegative electricity when rubbed 10 Electrostatics —Physical Principles [on.I with asecond substance, accordingasthe firstsubstance stands above or below thesecond substance onthe list.Thefollowingisalistofthiskind, which 'ncludes some ofthemostimportantsubstances : Cat's skin, Glass, Ivory, Silk,Rockcrystal, TheHand, Wood, Sulphur, Flannel, Cotton, Shellac, Caoutchouc, Resins, Guttapercha, Metals, Guncotton. Asubstance issaid tobeelectropositiveorelectronegativetoasecond substance accordingasitstands above orbelow itonalistofthis kind. Thus ofanypairofsubstances one isalways electropositivetotheother, the other being electronegativetothe first.Twosubstances, although chemically thesame, must beregardedasdistinct forthepurposesofalistsuch asthe above, iftheirphysicalconditions aredifferent;forinstance, itisfound that ahotbodymust beplacedlower onthe listthan acoldbodyofthesame chemicalcomposition. IV. Attraction andRepulsion ofElectricCharges. 12.Asmall ball ofpith,orsomesimilarly light substance, coated with gold-leafandsuspended byaninsulating thread, forms aconvenient instru- ment forinvestigatingtheforces, ifany,which arebroughtintoplaybythe presenceofelectriccharges.Letuselectrifyapithballofthiskindpositively andsuspenditfrom afixedpoint.Weshall findthatwhen webringa second small body chargedwithpositive electricitynear tothis firstbody thetwobodies tend torepeloneanother, whereas ifwebringanegatively charged bodynear toit,thetwobodies tend toattract oneanother. From thisandsimilar experimentsitisfound thattwosmall bodieschargedwith electricityofthesamesign repeloneanother, andthattwosmall bodies chargedwithelectricityofdifferentsignsattract oneanother. Thislawcanbewell illustrated bytying togetherafewlightsilkthreads bytheir ends, sothattheyform atassel, andallowingthethreads tohang vertically.Ifwenowstroke thethreads with thehand, orbrush them with abrush ofanykind, thethreads allbecomepositively electrified, andthere- forerepeloneanother. They consequentlynolonger hang verticallybut spread themselves outintoacone.Asimilar phenomenoncanoften be noticed onbrushingthehair indryweather. The hairsbecomepositively electrified and sotend tostand outfrom thehead. 13.Onshaking upamixture ofpowderedredleadandyellow sulphur, theparticlesofredlead willbecomepositively electrified, andthose ofthe sulphurwillbecomenegatively electrified, astheresult ofthefriction which hasoccurred between thetwo sets ofparticlesintheshaking.Ifsome of thispowderisnowdusted ontoapositivelyelectrified body,theparticlesof sulphurwillbeattracted andthose ofredleadrepelled.Theredlead will therefore fall off,orbeeasily removed byabreath ofair,while thesulphur 11-15] TheFundamental Conceptions ofElectrostatics 11 particleswillberetained. Thepositivelyelectrified bodywill therefore assume ayellowcolour onbeing dusted with thepowder, andsimilarlya negativelyelectrified bodywould become red. Itmaysometimes becon- venient tousethismethod ofdetermining whether the electrification ofa bodyispositiveornegative. 14.The attraction andrepulsionoftwochargedbodies isinmany respectsdifferent from theforce between onecharged andoneuncharged body. The latter force, aswehaveexplained, wasknown totheGreeks :it must beattributed, asweshall see,towhat isknown as"electric induction," and isinvariablyattractive. The forces between twobodies both ofwhich arecharged,forces whichmaybeeither attractive orrepulsive, seem hardly tohavebeen noticed until theeighteenth century. The observations ofRobert Symmer (1759) onthe attractions and repulsionsofchargedbodies areatleastamusing. Hewasinthehabit ofwearingtwopairsofstockings simultaneously,aworstedpairforcomfort andasilkpairforappearance.Inpullingoffhisstockings henoticed that theygaveacrackling noise, andsometimes thattheyeven emittedsparks when taken offinthedark. Ontakingthetwostockingsofftogetherfrom thefootandthendrawingtheonefrom inside theother, hefound thatboth became inflated soastoreproducetheshapeofthe foot,andexhibited attractions andrepulsionsatadistance ofasmuch asafootandahalf. "When thisexperimentisperformedwithtwoblackstockingsinone hand, andtwowhite intheother, itexhibits averycuriousspectacle;the repulsionofthose ofthesame colour, andtheattraction ofthose ofdifferent colours, throws them intoanagitationthat isnotunentertaining,and makes them catch each atthat ofitsopposite colour, and atagreater distance than onewouldexpect. When allowed tocometogether theyall unite inonemass. Whenseparated, theyresume their formerappearance, andadmit oftherepetitionoftheexperimentasoften asyou please,till theirelectricity, gradually wasting,stands inneed ofbeingrecruited." TheLawofForce between chargedParticles. 15.TheTorsion Balance. Coulomb (1785)devised aninstrument known astheTorsion Balance, which enabled himnotonlytoverifythelaws of attraction andrepulsion qualitatively,butalso toformanestimate ofthe actualmagnitudeofthese forces. Theapparatusconsists essentiallyoftwolightballsA,C,fixed atthetwo ends ofarodwhich issuspendedatitsmiddlepointBbyaveryfinethread ofsilver, quartzorother material. Theupperendofthethread isfastened toamovable head JD,sothat thethread andtherodcanbemade to rotate byscrewingthehead. Iftherod isacted ononlybyitsweight,the 12 Electrostatics —Physical Prin ciples [ch.l condition forequilibriumisobviouslythat there shall benotorsion in thethread. If,however, wefixathird small ballEinthesameplaneas theother two,and ifthethree balls areelec- trified, theforces between thefixed balland themovable ones willexert acoupleonthe moving rod,andthecondition forequilibrium isthat thiscoupleshallexactlybalance, that duetothetorsion. Coulomb found thatthe coupleexerted bythetorsion ofthethread wasexactly proportionaltotheangle through which oneendofthethread hadbeenturned relativelytotheother, andinthiswaywas enabled tomeasure hiselectric forces. In Coulomb's experimentsoneonlyofthetwo movable ballswas electrified, thesecond serv- ingmerelyasacounterpoise,andthefixed ballwasatthesame distance from thetorsion thread asthetwomovable balls. Fig. 2. Supposethat thehead ofthethread is turned tosuchapositionthattheballswhen unchargedrestinequilibrium, justtouchingoneanother Avithoutpressure.Lettheballs receivecharges e,e',and lettherepulsionbetween them result inthebarturning through anangle6.Thecoupleexerted onthebarbythetorsion ofthethread isproportionalto0,andmaytherefore betaken tobek6. Ifaisthe radius ofthecircle described bythemovable ball,wemayregardthecouple actingontherodfrom theelectric forces asmadeupofaforce F,equal tothe force ofrepulsionbetween thetwo balls, multiplied byacos\Qt thearm ofthemoment. The condition forequilibriumisaccordingly aFcos\B=k9. Letusnowsupposethatthetorsion head isturnedthrough anangle insuch adirection astomake thetwochargedballsapproach each other; after theturninghasceased, letussupposethat theballs areallowed to come torest. Inthenewpositionofequilibrium,letussuppose that the twochargedballs subtend anangle6'atthecentre, instead oftheformer angle6.Thecoupleexerted bythetoi'sion thread isnowk{9'+<f>),sothat ifF'isthenew force ofrepulsion wemust have aF'cos\&=k(6'+0). Byobservingthevalue of <f>requiredtogivedefinite values to&wecan calculate values ofF'correspondingtoanyseries ofvalues of6'.From a series ofexperimentsofthiskind itisfound that solongasthechargeson thetwo balls remain thesame, F'isproportionaltocosec2 ^#',fromwhich itiseasilyseen tofollow that theforce ofrepulsion variesinverselyasthe 15,16]TheFundamental Conceptions ofElectrostatics 13 squareofthedistance. Andwhen thechargesonthetwo balls arevaried itisfound thattheforce varies astheproductofthetwocharges,solongas their distanceapartremains thesame. Astheresult ofaseries ofexperi- ments conducted inthiswayCoulomb wasable toenunciate thelaw : Theforcebetween twosmall chargedbodies isproportionaltotheproduct oftheir charges, and isinversely proportionaltothesquare oftheir distance apart,theforce beingoneofrepulsionorattraction accordingasthetwo chargesareofthesame orofoppositekinds. 16.Inmathematicallanguage wemaysaythat there isaforce ofrepul- sion ofamount cee' /y*Z•(1) where e,e'arethecharges,rtheir distanceapart,and cisapositive constant. Ife,eareofopposite signstheproductee'isnegative,andanegative repulsionmust beinterpretedasanattraction. Althoughthislawwas firstpublished byCoulomb,itsubsequently appearedthat ithadbeen discovered atanearlier datebyCavendish, whose experimentsweremuch more refined than those ofCoulomb. Caven- dishwasable tosatisfyhimself that thelawwascertainlyintermediate between theinverse 2+^and2—^thpowerofthedistance(seebelow, §§46—48). Unfortunatelyhisresearches remained unknown until his manuscripts werepublishedin1879byClerk Maxwell. TheexperimentsofCoulomb andCavendish, itneedhardly besaid, wereveryrough comparedwith those which arerenderedpossible bymodern refinements oftheory andpractice,sothat these experimentsarenolonger thejustificationforusingthelawexpressed byformula(1)asthebasis of theMathematicalTheoryofElectricity. More delicateexperimentswith the apparatususedbyCavendish, which willbeexplained later, have, however, been found togiveacompleteconfirmation ofCoulomb's Law, solongas thechargedbodies maybothberegardedasinfinitelysmall comparedwith their distanceapart. Anydeviation from thelawofCoulomb must accord- inglybeattributed tothe finite sizes ofthebodies whichcarrythecharges. As itisonlyinthecase ofinfinitelysmall bodies that thesymbolrof formula(1)hashadanymeaning assignedtoit,wemayregardthelaw(1) asabsolutely true, atanyrate solongasrislargeenoughtobeameasurable quantity. 14 Electrostatics —Physical Principles [ch.i TheUnitofElectricity. 17.ThelawofCoulomb suppliesuswith aconvenient unit inwhich tomeasure electric charges. Theunit ofmass, thepoundorgramme,isapurely arbitrary unit,and allquantitiesofmass aremeasured simply bycomparisonwith this unit. Thesame istrue oftheunit ofspace.Ifitwerepossibletokeepacharge ofelectricity unimpaired throughalltimewemighttakeanarbitrary charge ofelectricityasstandard, andmeasure allcharges bycomparisonwith this onestandard charge,inthewaysuggestedin§8.Asitisnotpossibletodo this,wefind itconvenient tomeasureelectricitywith reference totheunits ofmass, lengthandtime ofwhich wearealreadyinpossession,andCoulomb's Lawenables ustodothis.Wedefine astheunitchargeachargesuch that when twounitchargesareplacedoneoneach oftwosmallparticlesat adistance ofacentimetreapart,theforce ofrepulsion between theparticles isonedyne. With this definition itisclear that thequantitycinthe formula(1)becomes equaltounity,solongasthe c.G.S. systemofunits isused. Inasimilarway,ifthemass ofabodydidnotremain constant, wemight have todefine theunit ofmass with reference tothose oftimeandlength bysayingthatamass isaunitmassprovidedthat tAvosuch masses, placed ataunitdistanceapart, produceineach other bytheir mutualgravitational attraction anacceleration ofacentimetrepersecondpersecond. Inthis caseweshould have thegravitationalacceleration fgiven byanequation oftheform /-£(2). and thisequationwould determine theunit ofmass. 18. Physicaldimensions. Iftheunit ofmass were determinedby equation (2),mwouldappeartohave thedimensions ofanacceleration multiplied bythesquareofadistance, andtherefore dimensions DT~\ Asamatter offact,however, weknow thatmass issomething entirely apart fromlength andtime, exceptinsofarasitisconnected withthemthrough thelawofgravitation. Thecomplete gravitational acceleration isgiven by where yistheso-called" gravitation constant." Byourproposed definition ofunitmassweshould havemade thevalue of7numerically equaltounity; but itsphysical dimensions arenotthose of 17,18]TheFundamental Conceptions ofElectrostatics 15 amere number, sothatwecannotneglectthefactor ywhenequating physicaldimensions onthetwo sides oftheequation. Soalso intheformula F=~(8) wecananddochoose ourunit ofchargeinsuch awaythat thenumerical value ofcisunity,sothat thenumericalequation becomes *=%(4), butwemustremember thatthefactor cstillretains itsphysical dimensions. Electricityissomething entirely apartfrom mass, lengthandtime, and it follows thatweoughttotreat thedimensions ofequation (3),byintroducing anewunit ofelectricity Eandsayingthat cisofthedimensions ofaforce divided byE2 /r*andtherefore ofdimensions MLsE-2T-\ If,however, wecomparedimensions inequation (4),neglectingtotake account ofthephysical dimensions ofthesuppressedfactor c,itappearsas thoughachargeofelectricitycanbeexpi'essedinterms oftheunits of mass, lengthandtime, justasitmight appearfromequation (2)asthough amass could beexpressedinterms oftheunits oflength and time. The apparentdimensions ofachargeofelectricityarenow MWT-1 (5). Itwillbereadilyunderstood that these dimensions aremerely apparent andnotinanyway real,when itisstated that othersystemsofunits are also inuse,and that theapparent physicaldimensions ofachargeof electricityarefound tobedifferent inthedifferentsystemsofunits. The systemwhich wehavejust described, inwhich theunit isdenned as thechargewhich makes cnumerically equaltounityinequation (3),is known astheElectrostatic systemofunits. There willbedifferent electrostaticsystemsofunitscorrespondingto different units oflength,mass andtime. Inthe c.G.s.systemthese units aretaken tobethecentimetre, grammeandsecond. Inpassingfrom one systemofunits toanother theunit ofelectricitywillchangeasifitwere aphysical quantity having dimensions M^L*T~X ,solongaswehold tothe agreementthatequation (4)istobenumerically true, i.e.solongasthe units remain electrostatic. Thisgivesacertain importancetotheapparent dimensions oftheunit ofelectricity,asexpressedinformula(5). 16 Electrostatics —Physical Principles [ch.i V.Electrification byInduction. 19.Letussuspendametal rodbyinsulating supports. Suppose that therod isoriginally uncharged,andthatwebringasmallbody charged withelectricitynear tooneendoftherod,withoutallowingthetwobodies totouch Weshall findonsprinklingtherodwith electrified powderofthe kindpreviouslydescribed(§13),that therod isnowelectrified, thesignsof thechargesatthetwoendsbeingdifferent. This electrification isknown as electriiication byinduction. Wespeakoftheelectricity ontherodasan induce Icharge,andthatontheoriginallyelectrifiedbodyastheinducingor exciting charge. We.find that theinducedchargeattheendoftherod nearest totheinducing chargeisofsign oppositetothat oftheinducing charge,that atthefurther endoftherodbeingofthesamesignasthe inducing charge.Iftheinducing chargeisremoved toagreatdistance fromtherod,wefindthattheinducedcharges disappear completely,therod resumingitsoriginalunelectrified state. Iftherod isarrangedsothat itcanbedivided intotwoparts,wecan separatethetwopartsbeforeremovingtheinducing charge, andinthisway canretain thetwopartsoftheinducedchargeforfurther examination. Ifweinsert thetwoinducedchargesintothevessel oftheelectroscope, wefind that the total electrification isnil: ingenerating electricity by induction, asingeneratingitbyfriction, wecanonly generate equal quantitiesofpositiveandnegative electricity; wecannot alter thealgebraic totalcharge. Thus thegenerationofelectricity byinduction isinnoway aviolation ofthelawthat thetotalchargeonabodyremains unaltered exceptinsofarasitisremoved byconduction. 20. Iftheinducing chargeisplacedonasufficiently light conductor, we notice aviolent attraction between itandtherodwhich carries theinduced charge. This, however, asweshallnowshew, isonlyinaccordance with Coulomb's Law. Let us,forthesake ofargument, supposethat the inducing chargeisapositive chargee.Letusdivide upthatpartofthe ABC CB'A' ( ) Pig. 3. rodwhich isnegatively chargedintosmallpartsAB,BG,...,beginningfrom theendAwhich isnearest totheinducing charge I,insuchawaythateach partcontains thesame smallcharge—e,ofnegative electricity.Letus similarlydivide upthepartoftherodwhich ispositively chargedinto 19-22] TheFundamental Conceptions ofElectrostatics 17 sections A'B',B'C,...,beginningfrom thefurther end,andsuchthateach of thesepartscontains acharge-feofpositive electricity. Since thetotal inducedchargeiszero, thenumber ofpositively chargedsections A'B', B'C, ...must beexactly equaltothenumber ofnegatively chargedsections AB,BO, Thewhole series ofsections cantherefore bedivided intoa series ofpairsABandA'B';BCandB'C;etc. such that thetwosections ofanypaircontainequalandopposite charges. ThechargeonA'B' beingofthesamesignastheinducing charge e,repels thebody/which carries thischarge,while thechargeonAB,beingofthe samesignasthechargeoni",attracts /.SinceABisnearer toIthan A'B', itfollows from Coulomb's Law that theattractive forceee/r2between AB and/isnumerically greaterthan therepulsiveforceee/r2between A'B'and I,sothattheresultant action ofthepairofsections AB,A'B'upon/isan attraction. Obviouslyasimilar result istrue foreveryotherpairofsections, sothatwearrive attheresult thatthewhole forcebetween thetwobodies isattractive. This resultfullyaccounts forthefundamentalpropertyofacharged body toattract small bodies towhich nochargehasbeengiven. Theproximityof thecharged bodyinduceschargesofdifferentsignsonthosepartsofthebody which arenearer to,andfurther away from, theinducing charge,andalthough thetotalinducedchargeiszero,yettheattractions willalways outweighthe repulsions,sothattheresultant force isalwaysoneofattraction. 21.Thesameconceptions explainthedivergenceofthegold-leavesof theelectroscopewhich occurs when acharged bodyisbroughtnear tothe plateoftheelectroscopeorintroduced intoaclosed vesselstandingonthis plate.Alltheconducting partsoftheelectroscope —gold-leaves, rod,plate and vessel ifany—mayberegardedasasingle conductor, andofthisthe gold-leavesform thepartfurthest removed from thecharged body. The leavesaccordingly become charged byinduction withelectricityofthesame signasthat ofthecharged body, andasthechargesonthetwogold-leaves areofsimilarsign,they repeloneanother. 22.Onseparatingthetwopartsofaconductor while aninducedcharge isonit,andthenremoving both from theinfluence oftheinducedcharge, wegaintwochargesofelectricity without anydiminution oftheinducing charge. Wecanstore orutilise thesechargesinanywayandonreplacing thetwopartsoftheconductor inposition, weshallagainobtain aninduced charge.Thisagainmaybeutilised orstored, andsoonindefinitely.There istherefore nolimit tothemagnitudeofthechargeswhich canbeobtained fromasmall initialcharge byrepeatingtheprocessofinduction. Thisprincipleunderlies theaction oftheElectrophorus. Acake ofresin iselectrifiedbyfriction, and forconvenience isplacedwith itselectrified j. 2 I 18 Electrostatics —Physical Principles [oh.i surface uppermostonahorizontal table.Ametal disc isheldbyaninsulating handleparalleltothecake ofresin andataslightdistance above it.The operat>rthen touches theuppersurface ofthediscwith hisfinger. When theprocesshasreached thisstage,themetal disc,thebodyoftheoperator andtheearth itself formoneconductor. Thenegative electricityontheresin induces apositive chargeonthenearerpartsofthisconductor—primarily onthemetal disc—andanegative chargeonthemore remotepartsofthe conductor—thefurtherregionoftheearth. When theoperatorremoves his fi]ger,thedisc isleftinsulated andinpossessionofapositive charge. Asalready explained,thischarge maybeusedandtheprocess repeated indefinitely. Inallitsessentials, theprincipleutilised inthegenerationofelectricity bythe"influence machines" ofVoss, Holtz, Wimshurst andothers isidentical with that oftheelectrophorus. Themachines arearrangedsothatbythe turningofahandle, thevariousstagesoftheprocessarerepeated cyclically time after time. 23. Electric Equilibrium. Returningtotheapparatusillustrated in fig. 3,p.16, itisfound that ifweremove theinducing chargewithout allowingtheconductingrodtocome into contact with other conductors, thechargeontheroddisappears graduallyastheinducing charge recedes, positiveandnegative electricity combininginequal quantitiesandneutral- isingoneanother. Thisshews that theinducing charge must besupposed toactupontheelectricityoftheinducedcharge,rather thanuponthe matter oftheconductor. Uponthesameprinciple,thevariouspartsofthe inducedcharge mustbesupposedtoactdirectly upononeanother. Moreover, inaconductorchargedwithelectricityatrest, there isnoreaction between matter aDdelectricity tendingtopreventthepassageofelectricity through theconductor. For ifthere were, itwould bepossibleforpartsoftheinduced chargetoberetained, after theinducing chargehadbeen removed, theparts oftheinducedcharge beingretained inposition bytheir reaction with the matter oftheconductor. Nothingofthiskind isobserved tooccur. We conclude then thattheelements ofelectricalchargeonaconductor areeach inequilibriumunder theinfluencesolelyoftheforces exertedbytheremaining elements ofcharge. 24.Anexceptionoccurs when theelectricityisactuallyatthesurface oftheconductor. Here there isanobvious reaction between matter and electricity—thereaction whichprevents theelectricityfromleavingthe surface oftheconductor.Clearlythisreaction willbenormal tothesurface, sothat theforcesacting upontheelectricityindirections which lieinthe tangent planetothesurface must beentirelyforces from otherchargesof electricity, andthese must beinequilibrium. Tobalance theaction ofthe matter ontheelectricitythere must beanequalandoppositereaction of 22-27] Theories ofElectrical Phenomena 19 electricityonmatter. This, then, willactnormally outwards atthesurface of theconductor. Experimentallyitisbestputinevidence bytheelectrification ofsoap-bubbles. Asoap-bubble when electrified isobserved toexpand, the normal reaction betweenelectricityandmatter atitssurfacedrivingthe surface outwards untilequilibriumisreestablished(seebelow, §94). 25.Alsowhen twoconductors ofdifferent material areplacedincon- tact, electric phenomenaarefound tooccur which havebeenexplained by Helmholtz astheresult oftheoperationofreactions betweenelectricity and matter atthesurfaces oftheconductors. Thus, although electricitycanpass quite freelyover thedifferentpartsofthesame conductor,itisnotstrictly true tosaythatelectricitycanpass freelyfrom oneconductor toanother of different material withwhich itisincontact. Compared, however, with the forces withwhich weshall ingeneralbedealinginelectrostatics, itwillbe legitimatetodisregard entirely anyforces ofthekindjustdescribed. We shall therefore neglectthedifference between thematerials ofdifferent con- ductors, sothatanynumber ofconductorsplacedincontact mayberegarded asasingleconductor. Theories toexplain Electrical Phenomena. 26. One-fluid Theory. Franklin, asfarback as1751, tried toinclude allthe electrical phenomenawithwhich hewasacquaintedinonesimple explanation.Hesuggestedthat allthesephenomenacould beexplained by supposingtheexistence ofanindestructible"electric fluid," which could be associated with matter indifferentdegrees. Correspondingtothenormal state ofmatter, inwhich noelectricalpropertiesareexhibited, there is adefinite normal amount of"electric fluid." When abody wascharged withpositive electricity,Franklinexplainedthat there wasanexcess of "electric fluid"above thenormal amount, andsimilarlyachargeofnegative electricity representedadeficiencyofelectric fluid. Thegenerationofequal quantitiesofpositiveandnegative electricity wasnowexplained:forinstance, inrubbingtwobodiestogether wesimplytransfer"electric fluid"fromone totheother. Toexplaintheattractions andrepulsionsofelectrified bodies, Franklin supposedthattheparticlesofordinarymatterrepelledoneanother, while attractingthe"electric fluid." Inthenormal state ofmatter the quantitiesof"electric fluid"andordinarymatter werejustbalanced, sothat therewasneither attraction norrepulsion between bodies inthenormal state. Accordingtoalater modification ofthetheorytheattractionsjustout-balanced therepulsionsinthenormal state, theresidual forceaccountingforgravitation. 27.Two-fluid Theory. Afurtherattempttoexplainelectric phenomena wasmade bythetwo-fluidtheory.Inthisthere were threethings concerned, ordinarymatter andtwoelectric fluids—positive andnegative. Thedegree ofelectrification wassupposedtobethemeasure oftheexcess ofpositive 2—2 20 Electrostatics —Physical Principles [oh.i electricityovernegative,orofnegativeoverpositive, accordingtothesign ofthe electrification. Thetwokinds ofelectricityattracted andrepelled, electa 3itiesofthesame kindrepelling,andofoppositekindsattracting,and inthiswaytheobserved attractions andrepulsionsofelectrified bodies were explainedwithout havingrecourse tosystemsofforces betweenelectricity andordinarymatter. Itis,however, obvious that thetwo-fluidtheorywas tooelaborate forthe facts. Onthistheory ordinary matter devoid ofboth kinds ofelectricitywould bephysicallydifferent from matterpossessing equal quantitiesofthetwokinds ofelectricity, althoughboth bodies would equally shew anabsence ofelectrification. There isnoevidence that itis possibletoestablish anyphysicaldifference ofthiskind betweentotally unelectrified bodies, sothat thetwo-fluidtheory must bedismissed as explainingmore than there istobeexplained. 28.Modern viewofElectricity. Thetwotheories which havejustbeen mentioned rested onnoexperimentalevidenceexceptsuch asisrequired toestablish thephenomenawith whichtheyaredirectlyconcerned. The modern view ofelectricity,ontheother hand, isbased onanenormous mass ofexperimental evidence, towhich contributions aremade, notonlybythe phenomenaofelectrostatics, but alsobythephenomenaofalmostevery branch ofphysicsandchemistry. Themodernexplanationofelectricityis found tobear averyclose resemblance totheolderexplanationoftheone- fluidtheory—somuch sothat itwillbeconvenient toexplain themodern view ofelectricity simply bymakingtheappropriatemodifications ofthe one-fluidtheory. Wesupposethe"electric-fluid" oftheone-fluidtheory replaced bya crowd ofsmallparticles—"electrons," itwillbeconvenient tocallthem—all exactly similar, andeachhaving exactlythesamechargeofnegative electricity permanentlyattached toit.Accordingtothebestrecent determinations, the amount ofthischargeis4,l774x10-10electrostatic units, while themass of each electron is9'00x10-28grammes.These determinations, which aredue toMillikan andBucherer, areprobablyaccurate toabout onepartinathou- sand. Toalowerdegreeofaccuracytheradius oftheelectron isprobably about 2x10-13cms.Wecanformsomeconceptionoftheintense concentra- tionofmassandelectrification intheelectron bynoticingthatagrammeof electrons, crammedtogetherincubicalpiling,wouldoccupy only7x10-11 cubic centimetres, while twogrammesofelectronsplacedatadistance ofa metreapartwouldrepeloneanother with aforceequaltotheweightof 3x1022tons. The electric force ofrepulsion outweighsthegravitationalforce ofattraction intheratio of4*2x1042toone. Apieceofordinary matter initsunelectrified state contains acertain number ofelectrons ofthiskind, and thisnumber isjustsuch thattwo piecesofmatter each inthisstate exert noelectrical forces ononeanother— 27.28] Modem View ofElectricity 21 thiscondition infactdefines theunelectrified state.Apieceofmatterappears tobechargedwithnegativeorpositive electricity accordingasthenumber of negatively-chargedelectrons itpossessesisinexcess ordefect ofthenumber itwouldpossessinitsunelectrified state. From this itfollows thatwecannotgoondividingachargeofelectricity indefinitely—anatural limit isimposed bythechargeofoneelectron, justas inchemistry wesupposeanatural limit tobeimposedonthedivisibilityof matterbythemass ofanatom. Themodern view ofelectricity maythenbe justlydescribed asan"atomic" view.And ofalltheexperimental evidence whichsupportsthisviewnone ismorestrikingthan thecircumstance that these"atoms"continually reappearinexperimentsofthemost varied kinds, and thattheatomicchargeofelectricity appears alwaystobepreciselythesame. Italsofollows that inchargingabodywithelectricity weeither addto orsubtract from itsmassaccordingaswechargeitwithnegative electricity (i.e.,addtoitanumber ofelectrons),orchargeitwithpositive electricity {i.e.,remove from itanumber ofelectrons). Since themass ofanelectron is sominute incomparisonwith thechargeitcarries, itwillreadilybeseen thatthechangeinitsmass isverymuch toosmall tobeperceptible byany methods ofmeasurement which areatourdisposal.Maxwell mentions, as anexampleofabody possessinganelectriccharge largecomparedwith its mass, thecaseofagrammeofgold,whichmaybebeaten intoagold-leafone squaremetre inarea,andcan, inthis state, holdachargeof60,000 electro- static units ofnegative electricity. Themass ofthenumber ofnegatively electrified electronsnecessarytocarrythischargewillbefound, astheresult ofabrief calculation from thedataalready given,tobeabout 10~13grammes. Thechangeofweight byelectrification istherefore onewhich itisfarbeyond thepowerofthemost sensitive balance todetect. Onthisview ofelectricity,theelectrons mustrepel oneanother, and must beattracted bymatter which isdevoid ofelectrons, orinwhich there is adeficiencyofelectrons. Theelectrons move aboutfreely through conductors, butnotthroughinsulators. The reactions which, aswehave seen,must be supposedtooccur atthesurface ofchargedconductors between "matter" and "electricity,"cannowbeinterpreted simplyassystemsofforces between the electrons andtheremainder ofthematter. Uptoacertain extent these forces willrestrain theelectrons fromleavingtheconductor, but iftheelectric forcesacting ontheelectrons exceed acertain limit, theywillovercome the forcesacting between theelectrons andtheremainder oftheconductor, and anelectricdischargetakesplacefrom thesurface oftheconductor. Thus anessential feature ofthemodern view ofelectricityisthat it regardstheflowofelectricityasamaterial flowofchargedelectrons. Good conductors andgoodinsulators arenowseen tomeansimplysubstances in which theelectrons move withextreme easeandextremedifficulty respectively. 22 Electrostatics —Physical Principles [ch.i Thelawthatequal quantitiesofpositive andnegative electricityaregenerated simultaneouslymeans that electrons mayflowabout, butcannot becreated orannihilated. Themodern view enables usalsotogiveasimple physical interpretation tothephenomenonofinduction. Apositive charge placednearaconductor willattract theelectrons intheconductor, andthese willflowthroughthe conductor towards thechargeuntil electricalequilibriumisestablished. There willbethenanexcess ofnegativeelectrons intheregionsnear the posit-vecharge,and thisexcess willappearasaninducednegative charge. Thedeficiencyofelectrons inthemore remotepartsoftheconductor will appearasaninducedpositive charge.Iftheinducing chargeisnegative, theflowofelectrons willbeintheopposite direction, sothatthesignsofthe inducedchargeswillbereversed. Inaninsulator, noflowofelectrons cantake place,sothatthephenomenonofelectrification byinduction doesnotoccur. Onthisview ofelectricity, negative electricityisessentiallydifferent in itsnature frompositive electricity:thedifference issomethingmore funda- mental thanamere difference ofsign. Experimental proofofthisdifference isnotwanting, e.g.,asharply pointedconductor canholdagreater chargeof positivethan ofnegative electricitybeforereachingthelimit atwhich a discharge beginstotakeplacefrom itssurface. Butuntilwecome tothose partsofelectrictheoryinwhich theflow ofelectricityhastobedefinitely regardedasaflowofelectrons, thisessential difference betweenpositive and negative electricitywillnotappear,andthedifference between thetwo will beadequately represented byadifference ofsign. Inthelastchapterofthebook, itwillbeexplained howrecentexperi- mental work hastraced thisessential difference betweenpositive andnegative electricity down toitssource. We shall seethat thepositive electricity occursonlyinthecentral cores or"nuclei"oftheatom ofwhich matter is constituted, while theouterregionsofthese atoms consist ofnegatively- charged particles,the"electrons" alreadydescribed. For this reason the negative electricitycanrunabout from oneatom toanother, andevenfrom oneconductor toanother, butthepositive electricity necessarilyremainsper- manentlyassociated with thesame atoms ofmatter. Summary. 29. Itwillbeuseful toconclude thechapter byasummaryoftheresults which arearrived atbyexperiment, independentlyofallhypothesesastothe nature ofelectricity. These havebeen stated byMaxwell intheform oflaws, asfollows: Law I.The total electrification ofabody,orsystemofbodies, remainsalwaysthesame, exceptinsofarasitreceives electrification from orgiveselectrification toother bodies. 28,29] Maxwells Laws 23 Law II.When onebodyelectrifies another byconduction, the total electrification ofthetwobodies remains thesame;thatis,the oneloses asmuchpositiveorgainsasmuchnegativeelectrification as theothergainsofpositiveorloses ofnegativeelectrification. Law III.When electrification isproduced byfriction, orbyany otherknown method, equal quantitiesofpositiveandnegativeelectrifi- cation areproduced. Definition. The electrostatic unit ofelectricityisthatquantityof positive electricity which, whenplacedatunit distance fromanequal quantity, repelsitwith unit offorce. LawIV.Therepulsion between twosmall bodiescharged respect- ivelywith eand e'units ofelectricityisnumerically equaltothe productofthechargesdividedbythesquareofthedistance. These aretheforms inwhich thelaws aregiven byMaxwell. LawI,it willbeseen, includes IIand III.AsregardstheDefinition andLawIV, itisnecessarytospecifythemedium inwhich thesmall bodies areplaced, since, asweshall seelater, theforce isdifferent when thebodies areinair, orinavacuum, orsurrounded byothernon-conductingmedia. Itisusual toassume, forpurposesoftheDefinition andLawIV,thatthebodies arein air.For strict scientific exactness, weoughtfurther tospecifythedensity, thetemperature,andtheexact chemicalcompositionofthe air. Alsowe have seen thatwhen theelectricityisnotinsulated onsmall bodies, but is free tomove onconductors, theforces ofLawIVmust beregardedasacting onthechargesofelectricitythemselves. When theelectricityisnotfreeto move, there isanaction andreaction between theelectricityandmatter, so thattheforces whichreallyactontheelectricity appeartoactonthebodies themselves whichcarrythecharges. CHAPTER II THEELECTROSTATIC FIELD OFFORCE Conceptions used intheSurvey ofaField ofForce I.TheIntensityatapoint. 30.Thespaceintheneighbourhoodofchargesofelectricity,considered with reference totheelectric phenomena occurringinthisspace,isspokenof astheelectric field. Anewchargeofelectricity, placedatanypointinanelectric field, willexperienceattractions orrepulsionsfrom allthechargesinthe field. Theintroduction ofanewchargewillingeneraldisturb thearrangement ofthechargesonalltheconductors inthe fieldbyaprocessofinduction. If,however, thenewchargeissupposedtobeinfinitesimal, the effects of induction willbenegligible,sothattheforcesactingonthenewcharge may besupposedtoarisefrom thechargesoftheoriginalfield. Letussupposethatweintroduce aninfinitesimalchargeeonaninfinitely small conductor. Anycharge e^inthefield atadistance r,from thepoint willrepelthechargewith aforceee^r?. Thechargeewillexperiencea similarrepulsionfromevery chargeinthe field, sothateachrepulsionwillbe proportionaltoe. The resultant ofthese forces, obtained bytheusual rules forthecom- positionofforces, willbeaforceproportionaltoe—sayaforceReinsome direction OP.Wedefine the electricintensityat tobeaforce ofwhich themagnitudeisR,andthedirection isOP. Thus The electricintensityatanypointisgiven,inmagnitude and direction, by theforceperunitcharge which would actonacharged particle placedatthis point,thecharge ontheparticle being supposedsosmall that thedistribution ofelectricityontheconductors inthefieldisnotaffected byitspresence. The electricintensityat0,defined inthisway,depends onlyonthe permanentfield offorce, andhasnothingtodowith thecharge,orthe size, oreven theexistence ofthesmall conductor which hasbeen used toexplain 30,31]Lines ofForce 25 themeaningoftheelectricintensity.There willbeadefiniteintensityat every pointoftheelectric field, quite independentlyofthepresence ofsmall chargedbodies. Asmall charged body might, however, conveniently beused forexploring theelectric fieldanddetermining experimentallythedirection oftheelectric intensityatanypointinthe field. For ifwesupposethebody carryinga chargeetobeheldbyaninsulating thread, both thebodyandthreadbeing solightthat theirweights maybeneglected, thenclearlyalltheforces actingonthecharged bodymaybereduced totwo:— (i)AforceReinthedirection oftheelectricintensityatthepoint occupied by e, (ii) thetension ofthethreadacting alongthethread. Forequilibriumthese two forces must beequal andopposite. Hence the direction oftheintensityatthepoint occupied bythesmall charged bodyis obtained atoncebyproducingthedirection ofthethreadthroughthecharged body. And ifwetietheother endofthethread toadelicatespring balance, wecanmeasure thetension ofthespring,andsince this isnumerically equal toRe,weshould beable todetermine Rifewereknown. Wemightin thiswaydetermine themagnitude anddirection oftheelectricintensityat anypointinthe field. Inasimilar way,afloat attheendofafishing-line might beused todetermine the strength anddirection ofthecurrent atanypointonasmall lake. And, justaswith the electricintensity, weshouldonly getthetrue direction ofthecurrent bysupposingthe float tobeofinfinitesimal size.Wecould notimaginethedirection ofthecurrent obtained byanchoring abattleshipinthelake,because thepresenceoftheshipwould disturb thewholesystemofcurrents. II.LinesofForce. 31.Letusstart atanypointintheelectric field,andmove ashort distance OPinthedirection oftheelectricintensityat0.StartingfromP letusmove ashort distance PQinthedirection oftheintensityatP, Q Fio. 4. and soon.Inthiswayweobtain abroken path OPQR..., formed of anumber ofsmall rectilinear elements. Letusnowpasstothelimiting case inwhich each oftheelements OP,PQ,QR,...isinfinitelysmall. Thebroken pathbecomes acontinuous curve, and ithasthepropertythat atevery pointonittheelectricintensityisinthedirection ofthetangent 26 Electrostatics —Field ofForce[oh.ii tothecurve atthatpoint.Such acurve iscalled aLine ofForce. We maytherefore define aline offorce asfollows :— AAneofforceisacurve intheelectricfield,such that thetangentatevery pointi?inthedirection oftheelectricintensityatthatpoint. Ifwesupposethemotion ofacharged particletobesomuch retarded byfrictional resistance that itcannot acquire anyappreciable momentum, then acharged particleset freeintheelectric fieldwould trace outalineofforce. Inthesameway,weshould have lines ofcurrent onthesurface ofalake,such thatthetangenttoalineofcurrent atany pointioincided with thedirection ofthecurrent, andasmall float setfreeonthelake would describe acurrent-line. 32.The resultant ofanumber ofknown forces hasadefinite direction, sothat there isasingledirection fortheelectricintensityatevery pointof the field. Itfollows thattwo lines offorce cannever intersect;forifthey didthere would betwodirections fortheelectricintensityatthepointof intersection (namely,thetwotangentstothelines offorce atthispoint)so that theresultant ofanumber ofknown forces would beactingintwo directions atonce.Anexception occurs, asweshall see,when theresultant intensityvanishes atanypoint. Theintensity Rmayberegardedascompoundedofthreecomponents X,Y,Z,paralleltothreerectangularaxes Ox,Oy,Oz. Themagnitudeoftheelectricintensityisthengiven by R*=X2+F2+Z\ andthedirection cosines ofitsdirection are XY£ R'R'R' These, therefore, arealsothedirection cosines ofthetangentatx,y,z tothe line offorcethroughthepoint. The differentialequationofthe systemoflines offorce isaccordingly dx_dy_dzX~T~~Z' III. ThePotential 33.Inmovingthesmalltest-chargeeabout inthe field,wemayeither have todoworkagainstelectric forces, orwemayfind that these forces willdowork forus.Asmallcharged particlewhich hasbeenplacedata pointinthe electric fieldmayberegardedasastore ofenergy,this energy being equaltothework(positiveornegative)which hasbeendone intakingthechargeto inoppositiontotherepulsionsandattractions of the field. Theenergycanbereclaimedbyallowingtheparticletoretrace itspath. Assume thecharge onthemoving particletobesosmall that 31-33] ThePotential 27 thedistribution ofelectricityontheconductors inthe field isnotaffected byit.Then thework done inbringingthechargeetoapointispro- portionaltoe,andmaybetaken tobeVe.Theamount ofworkdone will ofcourse dependonthepositionfromwhich thecharged particlestarted. Itisconvenient, inmeasuring Ve,tosupposethat theparticlestarted ata pointoutside the fieldaltogether,i.e.from apointsofarremoved from all thechargesofthe field that their effect atthispointisinappreciable —for brevity, wemay saythepointatinfinity. Wenow defineVtobethe potentialatthepoint0.Thus Thepotentialatanypointinthefieldistheworkperunitcharge which has tobedone onacharged particletobringittothatpoint,thechargeonthe particle being supposedsosmall that thedistribution ofelectricityonthe conductors inthefieldisnotaffected byitspresence. Inmovingthesmallchargeefrom x,y,ztox+dx,y+dy,z+dz,we shall have toperformanamount ofwork -(Xdx+Ydy+Zdz) e, sothat inbringingthechargeeintopositionatx,y,zfrom outside thefield altogether, wedoanamount ofwork -ej(XdtIx+Ydy+Zdz), where theintegralistakenalongthepathfollowed bye. Denotingthework done onthechargeeinbringingittoanypoint x,y,zinthe electric fieldbyVe,weclearly have y=-fXS (Xdx+Ydy+Zdz) (6), "00 givingamathematicalexpressionforthepotentialatthepoint x,y,z. Thesame result canbeputinadifferent form. Ifdsisanyelement of thepath,and iftheintensity Rattheextremityofthiselement makes an angle6with ds,then thecomponentoftheforceactingonewhenmoving along ds,resolved inthedirection ofmotion ofe,isRecos6.Thework done inmovingealongtheelement dsisaccordingly —Recosdds, sothatthewhole work inbringingefrominfinitytox,y,zis [x,y,z—el Rcos dds, Joo andsince this isequal, bydefinition, toVe,wemusthave V=-fX'y'ZRcos6ds (7). 28 Electrostatics— Field ofForce [ch.ii Weseeatonce thatthetwoexpressions (6)and(7)justobtained forV areidentical, onnoticingthat istheanglebetween twolines ofwhich the directioi .cosines arerespectively XYZ ,dxdy dz R'R'R ds' ds' ds' aXdx Ydy ZdzWethereiore have ™*6=rTs+Rds+Rds> sothat Rcos0ds=Xdx+Ydy+Zdz, andtheidentityofthetwoexpressionsbecomes obvious. IftheTheorem oftheConservation ofEnergyistrue intheElectro- static Field, thework done inbringingasmallchargeefrominfinitytoany pointPmust bethesame whatever pathtoPwechoose. For ifthe amounts ofwork were different ontwo differentpaths,letthese amounts beVPeandVP'e,and lettheformer bethegreater.Then bytakingthe chargefromPtoinfinity bytheformer pathandbringingitbackbythe latter, weshould gainanamount ofwork{VP—VP')e,which would be contrarytotheConservation ofEnergy.ThusVPandVPmust beequal, andthepotentialatPisthesame, nomatter bywhatpathwereach P. ThepotentialatPwillaccordingly depend onlyonthecoordinates x,y,z ofP. Assoon asweintroduce thespeciallawoftheinversesquare, weshall find that thepotentialmust beasingle-valuedfunction ofx,y,z,asa consequenceofthislaw(§39),andhence shall beable toprovethat the Theorem ofConservation ofEnergyistrue inanElectrostatic field. For themoment, however, weassume this. 34.Letusdenote byWthework done inmovingachargeefromP toQ.InbringingthechargefrominfinitytoP,wedoanamount ofwork Fio. 5. which bydefinition isequaltoVPewhereVPdenotes thevalue ofVatthe pointP.Hence intakingitfrominfinitytoQ,wedoatotalamount of workVPe+W. This, however,isalsoequal bydefinition toVqe.Hence wehave Vpe+W=VQe, or W=(VQ-VJ>)e (8). 33-86]ThePotential 29 35.Definition. Asurfaceintheelectricfieldsuch that atevery point onitthepotentialhasthesame value, iscalled anEquipotential Surface. Indiscussing thephenomenaoftheelectrostatic field,itisconvenient tothink ofthe whole field asmappedoutbysystemsofequipotentialsurfaces and lines offorce, justas ingeographywethink oftheearth's surface asdivided upbyparallelsoflatitude andof longitude. Amore exactparallelisobtained ifwethink oftheearth's surface asmapped outby"contour-lines" ofequal height above sea-level, andbylines ofgreatest slope. These reproduceallthepropertiesofequipotentialsandlines offorce, forinpointoffact theyareactual equipotentialsandlines offorce forthegravitationalfield offorce. Theorem. Equipotential surfacescutlinesofforceatright angles. LetPbeanypointintheelectric field,and letQbeanadjacent point onthesameequipotentialasP.Then, bydefinition, Vp=Vq,sothatby equation (8)W=0,Wbeingtheamount ofwork done inmovingachargee fromPtoQ.IfRistheintensityatQ,and6theanglewhich itsdirection makes withQP,theamount ofthisworkmust be—RecosxPQ,sothat Recos=0. Hence cos0=0, sothat the lineofforce cuts theequipotentialatright angles. Asinaformer theorem, anexceptionhastobemade infavour of thecase inwhichP=0. 36.Instead ofP,Qbeingonthesameequipotential,letthemnowbe onalineparalleltotheaxis ofx,their coordinatesbeing x,y,zandx+dx, y,zrespectively.InmovingthechargeefromPtoQthework done is —Xedx, andbyequation (8)itisalso(Vq—Vp)e.Hence -Xdx=VQ-VP. SinceQandPareadjacent, wehave, from thedefinition ofadifferential coefficient, dV_V Q-VP dx dx hence wehave therelations *--&--?--fo* results which areofcourse obvious ondifferentiating equation (6)with respecttox,yand zrespectively. Similarly,ifweimagine P,Qtobetwopointsonthesame lineofforce weobtain BVR== -~ds-' where ^-denotes differentiationalongalineofforce. SinceRisnecessarily ... gp- , positive,itfollows that-j-~isnegative,i.e.Vdecreases assincreases, orthe OS 30 Electrostatics —Field ofForce[oh.ii intensityisinthedirection ofVdecreasing. Thus the lines offorce run fromhighertolower values ofV,and, aswehavealready seen, cut all equipot•ntials atright angles. 37.Atapointwhich isoccupied byconducting material, theelectric charges,ashasalreadybeen said,must beinequilibrium under theaction of theforces from alltheotherchargesinthe field. Theresultant forcefrom allthesechargesonanyelement ofchargeeishowever Re,sothatwemust haveB=0.HenceX=Y=Z=0,sothat dec dydz Inother words,Vmust beconstantthroughoutaconductor forelectro- staticequilibriumtobepossible. And inparticular the surface ofa conductor must beanequipotential surface, orpartofone.Theequi- potentialofwhich thesurface ofaconductor isparthasthepeculiarity ofbeingthree-dimensional instead oftwo-dimensional, for itoccupiesthe whole interior aswell asthesurface oftheconductor. Inthesame way,inconsidering theanalogous arrangement ofcontour-lines andlines ofgreatest slope onamapoftheearth's surface, wefindthattheedgeofalake orsea must beacontour-line, butthat instrictness thisparticular contour must beregardedas two-dimensional rather than one-dimensional, since itcoincides with thewhole surface of thelakeorsea. IfVisnotconstant inanyconductor, theintensityisinthedirection of Vdecreasing. Hencepositive electricitytends toflow inthedirection ofV decreasing, andnegative electricityinthedirection ofVincreasing.Iftwo conductors inwhich thepotential hasdifferent values arejoined byathird conductor, theintensityinthethird conductor willbeindirection from theconductor athigher potentialtothatatlowerpotential. Electricitywill flowthroughthisconductor, andwillcontinue toflowuntil theredistribution ofpotentialcausedbythetransfer ofthiselectricityissuch thatthepotential isthesame atallpointsoftheconductors, which maynowberegardedas formingonesingleconductor. Thusalthough thepotential hasbeen definedonlywith reference to single points,itispossibletospeakofthepotentialofawhole conductor. Infact,themathematicalexpressionofthecondition thatequilibriumshall bepossibleforagiven systemofchargesissimplythat thepotentialshall beconstantthroughout each conductor. Andwhen electric contact is established between twoconductors, eitherbyjoining them byawire orby other means, thenewcondition forequilibrium which ismadenecessary by thenewphysicalcondition introduced, issimplythat thepotentialsofthe twoconductors shall beequal. 36-38]ThePotential 31 Theearth isaconductor, and istherefore atthesamepotential through- out. Inallpractical applicationsofelectrostatics, itwillbelegitimate to regardthepotentialoftheearth aszero, adistantpoint ontheearth's surface replacingtheimaginary pointatinfinity,with reference towhich potentialshave sofarbeen measured. Thus anyconductor canbereduced topotentialzerobyjoiningitbyametallic wire totheearth. Mathematical expressions oftheLawoftheInverse Square. I.Values ofPotential andIntensity. 38.Wenow discuss thevalues ofthepotential andcomponentsof electric intensity when thespacebetween theconductors isair,sothat theelectric forces aredetermined byCoulomb's Law. Ifwehave asingle point chargeexatapoint P,thevalue ofR,the resultant intensityatanypoint 0,is PO*' and itsdirection isthat ofPO.Hence if6istheangle between OPand Fig. 6. 00',thelinejoiningtoanadjacent point 0',thework done inmovinga chargeefrom to0' =eRcos .00' =eR(OP-0'P)=—eRdr, whereOP=r,O'P=r+dr.Hence thework doneagainsttherepulsion ofthechargeexinbringingefrominfinityto0'byanypathis -e Rdr=-e \dr=—\ where rx=O'P. Ifthere areotherchargese2>^s> •••thework doneagainstallthe repulsionsinbringingachargeeto0'willbethesum ofterms such asthe above, say \ri r2rs J 32 Electrostatics —Field ofForce[ch.n where r2,r3,...arethedistances from 0'toe2,e3,...,sothatbydefinition F=-x+^+-3+ (10).nr2r3 39. Itisnow clear thatthepotentialatanypoint depends onlyonthe coordinates ofthepoint,sothatthework done inbringingasmallcharge from infinitytoapointPisalwaysthesame, nomatter whatpathwe choose, theresult assumed in§33. Itfollows thatwecannot alter theamount ofenergyinthe fieldby moving chargesabout insuch awaythat thefinal state ofthe field isthe same astheoriginalstate. Inother words, theConservation ofEnergyis true oftheElectrostatic Field. 40. Analytically,letussupposethat thechargee1isatx[,y1}zx\e2at x2,y2,z2;and soon.Therepulsiononasmallchargeeatx,y,zresulting from thepresenceofexatxltyXizxis exe (x-x.y+iy-y.y +iz-z.r andthedirection-cosines ofthedirection inwhich this force actsonthe charge e,are ^Ifl VjlVletc [(x-x,f+(y- 2/x)2+(z-zrf]i' [O-xxf+(y-yxy+(z-ttf$' Hence thecomponent paralleltotheaxis ofxis e^e(x—Xi) [(x-xy +iy-yrf +iz-zj-f Byaddingallsuchcomponents, weobtain asthecomponentofthe electricintensityatx,y,z, Z=26l(x~Xl) l (11),[(x-xj +iy-yj +iz-zj]? andthere aresimilarequationsforYand Z. Wehave asthevalue ofVatx,y,z,byequation (6), V=-\*(Xdx+Ydy+Zdz)JCO _^r.r,y,zv gi{(x-Xl)dx+(y—yr)dy+(z—zz)dz] J00 =2[(x-x,y+(y-y,f+(z-gffl [{x-x.f +iy-y^+^-z^ givingthesame result asequation (10). 38-42]Gauss' Theorem 33 41. Iftheelectric distribution isnotconfined topoints, wecanimagine itdivided intosmall elements whichmaybetreated aspoint charges.For instance iftheelectricityisspread throughoutavolume, letthechargeon anyelement ofvolumedx'dy'dzbepdx'dy'dzsothat pmaybespokenofas the" density"ofelectricityatx,y,z .Then informula(11)wecanreplace #ibypdx'dy'dz', andxltyltzx,byx,y',z'.Instead ofsummingthecharges 6j,...weofcourseintegrate pdx'dy'dz' throughallthosepartsofthespace which contain electricalcharges.Inthiswayweobtain p(x—x')dx'dy'dz [(«-O'+(y-2/')= +(*-0sI-III—f- 3,etc., and V=((fpdx'dy'dz' JJJ[(x-x'f+(y- y'f+(z-zjf Theseequationsareoneform ofmathematicalexpressionofthelawof theinversesquareofthedistance. Anattempttoperformtheintegration, ineven afewsimple cases, willspeedilyconvince thestudent that theform isnotonewhich lends itself torapid progress. Asecond form ofmathe- maticalexpressionofthelawoftheinversesquareissupplied byaTheorem ofGauss which weshallnowprove, and itisthisexpressionofthelawwhich willform thebasis ofourdevelopmentofelectrostaticaltheory. II. Gauss' Theorem. 42.Theorem. Ifanyclosedsurfaceistaken intheelectricfield,and ifNdenotes thecomponent oftheelectricintensityatanypoint ofthissurface inthedirectionoftheoutward normal, then SINdS=4>ttE, where theintegration extends over thewholeofthesurface, andEisthetotal chargeenclosedbythesurface. Letussupposethechargesinthe field,both inside andoutside theclosed surface, tobee1at%,e2atli,and soon.Theintensityatanypointis theresultant oftheintensities duetothecharges separately,sothat atany pointofthesurface, wemaywrite N=N,+N2+ (12), whereNlfi\T2,...arethenormalcomponentsofintensity due toe1}e2,... separately. Instead ofattemptingtocalculate 11NdSdirectly, weshall calculate separatelythevalues ofllNidS,jJN'2dS, ....Thevalue ofJJNdSwill, byequation (12), bethesum oftheseintegrals. j. 3 34 Electrostatics— Field ofForce [ch.n Letustakeanysmall element dSoftheclosed surface intheneighbour- hood ofapointQonthesurface andjoineachpointofitsboundarytothe point &,Letthesmall cone soformed cutoffanelement ofareadafrom Fig. 7. asphere drawn through Qwith i?ascentre, andanelement ofarea dcofrom asphereofunit radius drawn aboutPxascentre. Letthenormal tothe closed surface atQinthedirection awayfrom 7^make anangle6withPXQ. TheintensityatQduetothechargeexatPxiseJRQ2inthedirection P}Q,sothatthecomponentoftheintensity alongthenormal tothesurface inthedirectionawayfrom i?is cos6 nor Thecontribution to IjJSfjdSfrom theelement ofsurface isaccordingly ±nk»cos6dS, the+or—signbeingtakenaccordingasthenormal atQinthedirection awayfromPtistheoutward orinward normal tothesurface. Now cos6dSisequaltoda;theprojectionofdSonthesphere through Q having /?ascentre, forthetwonormals todSand d<rareinclined atan angle6.Alsoda=P1Q2doy.Fordo;doaretheareas cutoffbythesame coneonspheresofradiiP XQandunity respectively. Hence P%™a6d8=e -p§-=e>dl°' IfPxisinside theclosed surface, alinefrom P{toanypointontheunit sphere surrounding i?mayeither cuttheclosed surfaceonlyonce asat Q(fig. 8)—inwhich casethenormal tothesurface atQinthedirection awayfrom i?istheoutward normal tothesurface—oritmaycutthree times, asatQ',Q",Q'"—inwhich casetwoofthenormals awayfrom i?(those atQ',Q'"infig.8)areoutward normals tothesurface, while thethird normal awayfromPt(thatatQ"inthefigure)isaninward normal—oritmay 42]Gauss' Theorem 35 cut five, seven, oranyoddnumber oftimes. Thus aconethroughasmall element ofareadwonaunitsphereaboutPxmaycuttheclosed surface any oddnumber oftimes. However manytimes itcuts, the firstsmall areacut offwillcontribute exd(oto\\NxdS,thesecond andthird small areas ifthey Fig. 8. occur willcontribute —exdcoand+exdcorespectively,thefourth and fifth if theyoccur willcontribute —exdajand+exdwrespectively, and soon.The total contribution from theconesurroundingdco is,inevery case,+exdu>. Fio. 9. •Summingover allcones which canbedrawn inthiswaythrough Txweobtain thewhole value of IJNxdS,which isthusseen tobesimplyexmultiplied by •thetotal surface area oftheunitsphereround i?,andtherefore 4nrex. 3—2 36 Electrostatics —Field ofForce[ch.n Ontheother hand if7?isoutside theclosed surface, asinfig. 9,the conethrough anyelement ofarea dcoontheunitsphere mayeither notcut the clcjedsurface atall,ormaycuttwice, orfour, sixoranyevennumber oftimes. Iftheconethroughdcointersects thesurface atall,the firstpair ofelements ofsurface which arecutoffbythecone contribute —e^dcoand \-e xdcorespectivelyto I\NydS. Thesecondpair,ifthey occur,make asimilar contribution and soon.Ineverycasethetotal contribution fromanysmall conethrough i?isnil.Bysummingover allsuch conesweshall include thecontributions from allpartsoftheclosed surface, sothat ifPxisoutside thesurface I\NxdSisequaltozero. Wehavenowseen that llN-^dS isequaltokrrexwhen thechargeexis inside theclosed surface, and isequaltozerowhen thechargeexisoutside theclosed surface. Hence (JxfdS= (JN,dS+(JN2dS+... =4ttx(thesumofallthechargesinside thesurface) whichprovesthetheorem. Obviouslv thetheorem istrue alsowhen there isacontinuous distribution ofelectricityinaddition toanumber ofpoint charges.Forclearly wecan divide upthecontinuous distribution intoanumber ofsmall elements and treat each asapoint charge. dV Since N,thenormal componentofintensity,isequal by§36to—-~—, where =-denotes differentiationalongtheoutward normal,itappearsthat wecanalsoexpressGauss' Theorem intheform "dV SIdS=- 4>ttE. on Gauss' theorem forms themost convenient method atourdisposal,of expressingthelawoftheinversesquare. Wecanobtain apreliminary conceptionofthephysical meaningunder- lyingthetheorembynoticingthat ifthesurface contains nochargeatall, thetheoremexpressesthattheaveragenormalintensityisnil. Ifthere is anegative chargeinside thesurface, thetheorem shews that theaverage normalintensityisnegative,sothatapositively charged particle placedat apointontheimaginarysurface willbelikelytoexperienceanattraction to theinterior ofthesurface rather than arepulsion awayfromit,and vice versa ifthesurface contains apositive charge. 42-46] Gauss' Theorem 37 Corollaries toGauss' Theorem. 43.Theorem. Ifaclosed surfacehedrawn, such thatevery pointonit isoccupied byconducting material, thetotalchargeinside itisnil. Wehave seen that atanypoint occupied byconducting material, the electricintensity must vanish. Hence atevery pointoftheclosed surface, N=0,sothat 11NdS=0,andtherefore, byGauss' Theorem, thetotalcharge inside theclosed surface must vanish. Thetwofollowing specialcases ofthistheorem areofthegreatest importance. 44.Theorem. There isnochargeatanypoint which isoccupied bycon- ducting material, unless thispointisonthesurface ofaconductor. For ifthepointisnotonthesurface, itwillbepossibletosurround the point byasmallsphere,such thatevery pointofthissphereisinside the conductor. Bytheprecedingtheorem thechargeinside thissphereisnil, hence there isnochargeatthepointinquestion. Thistheorem isoften stated bysaying:— Thecharge ofaconductor resides onitssurface. 45.Theorem. Ifwehave ahollow closed conductor, andplace any number ofchargedbodies insideit,thechargeonitsinner surfaceluillbeequal inmagnitudebutoppositeinsign,tothetotalchargeonthebodies inside. Forwecandraw aclosed surfaceentirelyinside thematerial ofthe conductor, andbythetheorem of§43,thewholechargeinside this surface must benil.Thiswholecharge is,however, thesum of(i)thechargeonthe inner surface oftheconductor, and(ii)thechargesonthebodies inside the conductor. Hence these twomust beequalandopposite. This resultexplainsthepropertyoftheelectroscopewhich ledustothe conceptionofadefinitequantityofelectricity. The vesselplaced onthe plateoftheelectroscopeformed ahollow closed conductor. Thechargeon theinner surface ofthisconductor, wenow see,must beequal andopposite tothetotalcharge inside, andsince thetotalchargeonthisconductor isnil, thechargeonitsouter surface must beequal andoppositetothatonthe inner surface, andthereforeexactly equaltothesum ofthecharges placed inside, independentlyofthepositionofthesecharges. TheCavendish Proof oftheLawoftheInverseSquare. 46.Wehave deduced from thelawoftheinversesquare,that the chargeinside aclosed conductor iszero.We shallnowshew that the converse theorem isalso true. Hence, intheknown fact,revealed bythe 38 Electrostatics— Field ofForce[ch.n observations ofCavendish andMaxwell, that thechargeinside aclosed conductor iszero,wehaveexperimental proofofthelawoftheinverse square-/hich admits ofmuchgreater accuracythan theexperimental proof ofCoulomb. Thetheorem that ifthere isnochargeinside asphericalconductor the lawofforcemust bethat oftheinversesquareisduetoLaplace. Weneed consider thisconverse theoremonlyinitsapplicationtoaspherical conductor, thisbengtheactual form ofconductor usedbyCavendish. Theapparatus illustrated infig.10isnotthat usedbyCavendish, but isanimproved formdesigned byMaxwell, whorepeatedCavendish's experimentinamore delicate form. Twosphericalshells arefixedbyaringofebonite soastobeconcentric with oneanother, andinsulated from oneanother. Electrical contact canbeestablished between thetwo byletting down thesmalltrap-door Bthroughwhich awirepasses,thewirebeingofsuch alengthasjust toestablish contact when thetrap-doorisclosed. The experimentisconductedbyelectrifyingtheouter shell, openingthetrap-door byaninsulatingthread withoutdischargingtheconductor, afterwards dis- chargingtheouter conductor andtestingwhetherany chargeistobefound ontheinner shellbyplacingit inelectrical contact with adelicateelectroscope by means ofaconductingwire insertedthroughthetrap- door. Itisfound thatthere arenotraces ofacharge ontheinnersphere. FlG -10 -47.Suppose westart tofindthelawofelectric force such thatthere shall benochargeontheinner sphere. Letusassume alawofforce such that therepulsionbetween two charges e,eatdistance rapartisee'<j>(r). Thepotential,calculated as explainedin§33, is Zef(f>(r)dr (13), Jr where thesummation extends over allthechargesinthe field. Letuscalculate thepotentialatapointinside thesphereduetoachargeEspread entirelyoverthesurface ofthesphere.Ifthesphereisofradius a, thearea ofitssurface is47ra2 ,sothattheamount ofcharge perunitarea is EJ^na2 ,andtheexpressionforthepotential becomes V'= ll^{\~Mr)dr}a*smed0d4>(14), thesummation ofexpression (13)beingnowreplaced byanintegrationwhich 46,47] Cavendish's Proof ofLaw ofForce 39 extends overthewholesphere.Inthisexpressionristhedistance from the pointatwhich thepotentialisevaluated, totheelement a2sin0ddd<j>of sphericalsurface. Ifweagreetoevaluate thepotentialatapointsituated ontheaxis0=0 atadistance cfrom thecentre, wemaywrite r2=a2+c2-2accos9. Since cisaconstant, weobtain astherelation between drand dO,by differentiation ofthis lastequation, rdr=acsinddd (15). Ifweintegrate expression (14)withrespectto<£,thelimitsbeingof course<f)=and <f>=2tt,weobtain F=\EJ6" (f<f>(r)dr}sinOdd, or,onchangingthevariable from 6tor,bythehelpofrelation(15) ~r=a+c /f« \rdr rr=a+c //"=\ V=%E<j>(r)dr)Jr=a-c \Jr 'ao Ifweintroduce anewfunctionf(r),definedby /(r)=[y 4>{r)dr}rdr, weobtain asthevalue ofV, V=^c[f{a+c)-f{a-G)]' Iftheinner andouterspheresareinelectrical contact, theirpotentials arethesame;andif,asexperiment shews tobethecase, there isnocharge ontheinnersphere,then thewholepotentialmust bethatjustfound. This expression must, accordingly,have thesame value whether crepresentsthe radius oftheoutersphereorthat oftheinner. Since this istruewhatever theradius oftheinnersphere may be,theexpressionmust bethesame for allvalues ofc.Wemustaccordinglyhave 2acV., v /./ v —=r-=7(«+c)-J(a- c), whereVisthesame forallvalues ofc.Differentiatingthisequationtwice withrespecttoc,weobtain 0=/"(a +c)-/"(a-c). Since bydefinition, /(r)depends onlyonthelawofforce, andnotonaorc itfollows from therelation /"(a+c)=/"(a- o), thatf"(r)must beaconstant, sayG. 40 Electrostatics —Field ofForce[ch.n Hence f(r)=A+Br+\Gr\ andb\definition f(r)=1(1 <f>(r)dr) rdr, sotha •,onequatingthetwovalues of/" (r), B+Cr=r I <f>(r)dr. r°° B There fore (f>(r)dr=C +- , Jr r sothatthelawofforce isthat oftheinversesquare. 48.Maxwell hasexamined whatchargewould beproducedontheinner sphere if,instead ofthelawofforcebeing accurately B/?°2 ,itwere ofthe formB/r2+ v,whereqissome smallquantity.Inthiswayhefound that ifq wereeven sogreatasYi6M>^necnargeontheinnerspherewould havebeen toogreattoescapeobservation. Aswehave seen, thelimitwhich Cavendish wasable toassigntoqwas£$. Itmaybeurgedthat theform Bjr2+<*isnotasufficiently general lawofforce toassume. TothisMaxwell hasrepliedthat itisthemost generallawunder which conductors which areofdifferent sizesbutgeometri- callysimilar canbeelectrifiedsimilarly,whileexperimentshews that inpoint offactgeometricallysimilar conductors areelectrifiedsimilarly. Wemay saythen with confidence that theerror inthelawoftheinversesquare,if any,isextremelysmall. Itshould, however, beclearlyunderstood that experimenthasonlyprovedthelawB/r2forvalues ofrwhich aregreat enoughtoadmit ofobservation. Thelawofforce between two electric chargeswhich areatverysmall distances from oneanother stillremains entirely unknown tous. ^—__^ III. TheEquations ofPoisson andLaplace. 49.There isstillathirdwayofexpressingthelawoftheinverse square, andthiscanbededuced mostreadilyfrom Gauss' Theorem. Letusexamine thesmallrectangular parallel- epiped,ofvolumedxdydz,which isboundedby thesixplanefaces x=£±\dx, y=v±\dy, z=K± \&z- Weshallsupposethat thiselement doesnotcon- tainanypoint chargesofelectricity,orpartof Fig. 11. anycharged surface, but forthesake ofgenerality weshallsupposethatthewholespaceischargedA 47-49] Equations ofLaplace andPoisson 41 with acontinuous distribution ofelectricity,thevolume-densityofelectrifi- cation intheneighbourhoodofthesmall element under considerationbeing p.Thewholechargecontainedbytheelement ofvolume isaccordingly pdxdydz,sothat Gauss' Theorem assumes theform NdS=^irpdxdydz (16). Thesurfaceintegralisthesum ofsixcontributions, onefrom each faceof theparallelepiped. Thecontribution from that facewhich liesintheplane x=i~—\dxisequaltodydz,thearea ofthe face,multiplied bythemean value ofNover this face. Toasufficientapproximation,thismaybe supposedtobethevalue ofJV"atthecentre ofthe face, i.e.atthepoint £—\dx, 7),£,and thisagainmaybewritten ftl)\dx J' ,,' sothatthecontribution to IINdS from thisface is <dV\ dydz\dx) Similarlythecontribution from theoppositeface is "dydz(a?) thesignbeingdifferent because theoutward normal isnowthepositiveaxis ofx,whereasformerlyitwasthenegativeaxis.Thesumofthecontributions from thetwofacesperpendiculartotheaxis ofxistherefore -*M«U„, -£),.„„„!<' dVTheexpressioninside curled brackets istheincrement inthefunction— when xundergoesasmall increment dx.Thisweknow isdx^- (-^— J,so thatexpression (17)canbeputintheform - fa?dxdydz. Thewhole value of jJNdSisaccordingly /d'V d*v d2V\ ., ,U?+w+1*)dxdydz ' dy andequation (16)nowassumes theform d*V d*V d*V . ,1QN^+w+^=~^p (}' 42 Electrostatics —Field ofForce[ch.n This isknown asPoisson's Equation; clearlyifweknow thevalue ofthe potentialatevery point,itenables ustofindthecharges bywhich this potentiilisproduced. 50.Infreespace,where there arenoelectriccharges,theequation assumes theform 927 opy&y andths isknown asLaplace's Equation. Weshall denote theoperator d*_ &_ d*_ da?By2dz1 byV2 ,mthatLaplace's equation maybewritten intheabbreviated form V2F=0 (20). Equations (18)and(20) expressthesame fact asGauss' Theorem, but expressitintheform ofadifferentialequation. Equation (20)shews that inaregioninwhich nocharges exist, thepotentialsatisfies adifferential equation which isindependentofthechargesoutside thisregion bywhich thepotentialisproduced.Itwilleasilybeverified bydirect differentiation thatthevalue ofVgiveninequation (10)isasolution ofequation (20). Wecanobtain anidea ofthephysical meaningofthis differential equationasfollows. LetustakeanypointOandconstruct asphereofradius rabout this point. Themean value ofVaveragedoverthesurface ofthesphereis ^ 1 4>7T)VdS =-^ ffvsin0ddd<f>, wherer,0, (f>arepolar coordinates, having Oasorigin.Ifwechangethe radius ofthisspherefrom rtor+dr,therateofchangeofVis BV 1ffdV. dVdS 4nrr2JJdr =0,byGauss' Theorem, shewingthatVisindependentoftheradius rofthesphere. Takingr=0, thevalue ofVisseen tobeequaltothepotentialattheorigin0. Thisgivesthefollowing interpretationofthedifferentialequation: Vvaries from pointtopointinsuchawaythat theaveragevalueofV taken overanysphere surrounding anypointisequaltothevalue ofVat0. 49-54] Maxima andMinima ofPotential 43 Deductions fromLawofInverse Square. 51.Theorem. Thepotentialcannot haveamaximum oraminimum value atanypointinspacexuhich isnotoccupied byanelectriccharge. For ifthepotentialistobeamaximum atanypoint 0,thepotentialat every pointonasphereofsmall radius rsurrounding must belessthan that at0.Hence theaveragevalue ofthepotentialonasmallsphere surroundingmust belessthan thevalue at0,aresult inoppositionto that ofthe last section. Asimilarproofshews thatthevalue ofVcannot beaminimum. 52.Asecondproofofthistheorem isobtained atoncefromLaplace's equation. Regarding Vsimplyasafunction ofx,y,z,anecessarycondition . . d-V d2V d2V forVtohave amaximum value atanypointisthat-~-j,-^-jand-=-yshall eachbenegativeatthepointinquestion,acondition which isinconsistent withLaplace's equation dx* ay2dz2 Soalso forVtobeaminimum, thethree differential coefficients would have tobeallpositive,andthisagainwould beinconsistent withLaplace's equation. 53. IfVisamaximum atanypoint 0,which aswehavejustseen dVmust beoccupied byanelectriccharge,then thevalue of-^-must be negativeaswecross asphereofsmall radius r.Thus 11-~-dS isnegative where theintegrationistaken over asmallsphere surrounding 0,andby Gauss' Theorem thevalue ofthesurfaceintegralis—kire,where eisthe totalchargeinside thesphere. Thus emust bepositive, andsimilarlyifV isaminimum, emust benegative.Thus : IfVisamaximum atanypoint,thepointmust beoccupied byapositive charge, andifVisaminimum atanypoint,thepointmust beoccupied bya negative charge. 54.Wehave seen(§36)that inmoving alongaline offorceweare moving,atevery point,fromhighertolowerpotential,sothatthepotential continuallydecreases aswemovealongalineofforce. Hence aline of force canendonlyatapointatwhich thepotentialisaminimum, and similarly bytracingalineofforce backwards, weseethat itcanbegin only atapointofwhich thepotentialisamaximum.Combiningthis result with that oftheprevious theorem, itfollows that : Lines offorce canbegin onlyon-positive charges, andcanendonlyon negative charges. 44 Electrostatics —Field ofForce[oh.ii Itisofcoursepossibleforaline offorce tobegin onapositive charge, andgotoinfinity,thepotential decreasingalltheway,inwhich casethe line offorce has, strictly speaking,noendatall.Soalso,alineofforcemay come frominfinity,andendonanegative charge. Obviouslyalineofforce cannotbeginandendonthesame conductor, forifitdid so,thepotentialatitstwoendswould bethesame. Hence there canbenolines offorce intheinterior ofahollow conductor which contains nocharges;consequentlythere canbenochargesonitsinner surface. TubesofForce. 55,Letusselect anysmall areadSinthe field,and letusdraw the lines offorce through every pointoftheboundaryofthissmall area. If dSistakensufficiently small, wecansupposetheelectricintensitytobethe same inmagnitudeanddirection atevery pointofdS,sothatthedirections ofthelines offorce atallthepointsontheboundarywillbeapproximately allparallel. Bydrawingthelines offorce, then,weshall obtain a"tubular" surface—i.e.,asurface such that intheneighbourhoodofanypointthe surface mayberegardedascylindrical. The surface obtained inthisway iscalled a"tube offorce." Anormal cross-section ofa"tube offorce" isa section which cuts allthelines offorcethroughitsboundaryatright angles. Ittherefore formspartofanequipotentialsurface. 56.Theorem. Iftaltw2betheareasoftwonormal cross-sections ofthe same tubeofforce, andRltR2theintensities atthese sections, then R1(01=R2(02. Consider theclosed surface formed bythetwo cross-sections ofareas o)1,w2,andofthepartofthetube offorce joiningthem. There isnochargeinside this surface, sothatbyGauss' theorem, ljNdS=0. Ifthedirection ofthelines offorce isfrom «!too)2,then theoutward normalintensity Fig. 12. over &>2isRitsothatthecontribution from this area tothesurfaceintegralisR2(o2.Soalso over Wjtheoutward normalintensityis—Rusothat a^givesacontribution —RiCO!. Over therestofthesurface, theoutward normal isperpendicularto theelectricintensity,sothat JVr=0,andthispartofthesurface contributes nothingtoIjNdS.Thewhole value ofthisintegral, then, is R2(i)i—RiCOx, andsince this, aswehave seen,must vanish, thetheorem isproved. 54-58] Tubes ofForce 45 57.Coulomb's Law. IfRistheoutwardintensityatapoint just outside aconductor, thenR=4>7ra, where aisthesurface density ofelectri- fication ontheconductor. Wehave already seen that thewhole electrification ofaconductor must reside onthesurface. Therefore wenolongerdealwith avolumedensity ofelectrificationp,such thatthechargeintheelement ofvolumedxdydzis pdxdydz,butwith asurface-densityofelectrification asuch thatthecharge onanelement dSofthesurface oftheconductor isadS. Thesurface oftheconductor, aswehave seen, isanequipotential,sothat bythetheorem ofp.29,theintensityisinadirection normal tothe surface. Letusdrawperpendicularstothesurface atevery pointontheboundaryofasmall element ofarea dS,theseper- pendicularseachextendingasmall distance intotheconductor inonedirection andasmall distance awayfrom theconductor intheother direction. Wecanclose thecylindricalsurface so formed, bytwosmallplane areas, eachequalandparalleltothe originalelement ofareadS.LetusnowapplyGauss' Theorem tothisclosed surface. Thenormalintensityiszerooverevery partofthis surface exceptover thecapofareadSwhich is outside theconductor. Over thiscaptheoutward normal in- tensityisR,sothatthevalue ofthesurfaceintegralofnormal intensitytaken over theclosed surface, consists ofthesingletermRdS. The totalchargeinside thesurface isadS, sothatbyGauss' Theorem, RdS=4,7r*dS(21), andCoulomb's Law follows ondividing bydS. 58.Letusdraw thecompletetube offorce which isformed bythe lines offorcestartingfrompointsontheboundaryoftheelement dSofthe surface oftheconductor. Letussupposethat thesurfacedensityonthis element ispositive,sothat theareadSforms thenormal cross-section atFig. 13. Fig. 14. thepositive end, orbeginning,ofthetube offorce. Letussupposethat at thenegativeendofthetube offorce, thenormal cross-section isdS',that 46 Electrostatics —Field ofForce [oh.ii thesurface densityofelectrification isa-',a'beingofcoursenegative,and thattheintensityinthedirection ofthelines offorce isR'.Then, asin equati )n(21), R'dS'=-4>ir*'dS', since \,heoutward intensityisnow-R'. Since R,R'aretheintensities attwopointsinthesame tube offorce atwhich thenormal cross-sections aredS,dS', itfollows from thetheorem of553,chat RdS=R'dS' andhence, oncomparingthevaluesjustfound forRdSandR'dS', that crdS=—cr'dS'. Since crdSand a-'dS' arerespectivelythechargesofelectricityfromwhich thetubebeginsandonwhich itterminates, weseethat : Thenegative charge ofelectricityonwhich atubeofforceterminates is numerically equaltothepositive charge from which itstarts. Ifweclose theends ofthetube offorcebytwosmall capsinside the conductors, asinfig.14,wehave aclosed surface such that thenormal intensityvanishes atevery point. Thus, byGauss' Theorem, the total chargeinside must vanish, givingtheresult atonce. 59.Thenumerical value ofeither ofthechargesattheends ofa tube offorcemayconvenientlybespokenofasthestrengthofthetube.A tube ofunitstrengthisspokenofbymanywriters asaunit tubeofforce. Thestrengthofatube offorce is<rdS inthenotationalready used,and this,byCoulomb's Law, isequalto-j—RdSwhereRistheintensityatthe 47T enddSofthetube. Bythetheorem of§56,RdS isequaltoRlw1where _Rj,«!aretheintensityandcross-section atanypointofthetube. Hence jRjO)!=47Ttimes thestrengthofthetube. Itfollows that : Theintensityatanypointisequalto4>tttimes theaggregate strength per unit areaofthetubes which cross aplane drawn atright anglestothe directionoftheintensity. Interms ofunit tubes offorce, wemay saythat theintensityis4nr times thenumber ofunit tubesperunitareawhich cross aplane drawn at right anglestotheintensity. Theconceptionoftubes offorce isdue toFaraday:indeed itformed almost hisonlyinstrument forpicturingtohimself thephenomenaofthe Electric Field. Itwillbefound thatanumber oftheorems connected with theelectric fieldbecome almost obvious wheninterpretedwith thehelpof theconceptionoftubes offorce. Forinstance weprovedonp.37that ..(22),58-62] Tubes ofForce 47 when anumber ofchargedbodies areplacedinside ahollow conductor, they induce onitsinner surface acharge equal andoppositetothesum ofall theircharges.Thismaynowberegardedasaspecialcase oftheobvious theorem that thetotalchargeassociated with thebeginnings andtermi- nations ofanynumber oftubes offorce, none ofwhichpasstoinfinity, must benil. Examples ofFields ofForce. * 60. Itwillbeofadvantagetostudyafewparticularfields ofelectric forcebymeans ofdrawingtheir lines offorceandequipotentialsurfaces. I.TwoEqualPointCharges. 61.LetA,Bbetwoequal point charges, sayatthepoints x=—a,+a. Theequationsofthe lines offorce which areintheplaneofx,yare easilyfound tobe a#=F= y dx~X~ (PB3-PA*x+a{TWTPA3 wherePisthepoint x,y. Thisequationadmits ofintegrationintheform x+ax—a /nn. -pj-+-p]f=cons(23). From thisequationthelines offorcecanbedrawn, andwillbefound tolie asinfig.15. 62.There are,however, onlyafew cases inwhich the differential equationsofthelines offorce canbeintegrated,and itisfrequently simplest toobtain thepropertiesofthelines offorcedirectlyfrom thedifferential equation. Thefollowingtreatment illustrates themethod oftreatinglines offorce withoutintegratingthedifferentialequation. Fromequation (22)weseethatobvious lines offorce are dy (i)y=0,^-=0,givingtheaxisAB; (ii)x=0,PA=PB, ^=oo,givingthe linewhich bisectsABat right angles. These lines intersect atG,themiddlepointofAB. Atthispoint, then, ^-hastwo values, andsince Jf-=^ >itfollows thatwemust haveX=0,dx oxA F=0. Inother words, thepointCisapointofequilibrium,asisotherwise obvious. 48 Electrostatics —Field ofForce[ch.n Thesame result canbeseen inanother way.Ifwestart fromAand draw asmall tube surroundingthelineAB,itisclear thatthecross-section ofthetube,nomatter howsmall itwasinitially,willhavebecome infinite bythetime itreaches theplanewhich bisectsABatright angles—infact thecress-section isidentical with theinfiniteplane. Since theproductof thecross-section andthenormalintensityisconstantthroughoutatube, it follows that atthepoint G,theintensity must vanish. Fig. 15. Atagreatdistance Rfrom thepointsAandB,thefraction PB*-PA* TB*+PA* vanishes totheorder of1/R,sothat dxx' exceptforterms oftheorder of1/ifr Thus atinfinitythelines offorce becomeasymptotictostraightlinespassing throughtheorigin. Letussupposethatalineofforce starts fromAmakinganangle6with BAproduced,and isasymptoticatinfinitytoalinethrough Cwhich makes anangle <f>withBAproduced. Byrotatingthis line offorce about the axisABweobtain asurface which mayberegardedastheboundaryof abundle oftubes offorce. This surface cuts offanarea 2tt(1-cos6)r* 62] Charges +e,+e 49 from asmallsphereofradius rdrawn about A,and atevery pointof thisspheretheintensityise/r2normal tothesphere. The surfaceagain cuts offanarea 2tt(1-cos<f>)R2 from asphereofvery greatradiusRdrawn about G,and atevery point ofthisspheretheintensityis2e/R2 .Hence, applyingGauss' Theorem tothepartofthe field enclosed bythetwospheresofradii randR, andthesurface formed bytherevolution ofthelineofforce about AB, weobtain 2tt(1-cos6)r2x- 2-2?r(1-cos<j>)R*xj|=0, fromwhich follows therelation sin\Q=\/2sin\</>. Inparticular,thelineofforcewhich leaves J.inadirectionperpendicular toAB isbentthrough anangleof30°before itreaches itsasymptoteat infinity. The sections oftheequipotentials made bytheplaneofxyforthiscase areshewn infig.16which isdrawn onthesame scale asfig.15.Theequa- tions ofthese curves areofcourse +=cons.,PA'PB curves ofthesixthdegree. Theequipotentialwhichpasses through Gis ofinterest, asitintersects itself atthepointG.This isanecessaryconse- Fig. 16. Indeed theconditionsquenceofthefactthatGisapointofequilibrium, forapointofequilibrium, namely dZ=o?I=o,8-?=o,dx dydz maybeinterpretedasthecondition that theequipotential (V= constant) throughthepointshould have adoubletangent planeoratangenteone at thepoint. j. 4 50 Electrostatics —Field ofForce[ch.u II.Pointcharges +e,—e. 6J.Letcharges ±ebeatthepoints x=±a(A,B)respectively. The diffeientialequationsofthelines offorce arefound tobe dy_Y_ y cte~X==/PB3+PA3 \' and iheintegralofthis is x+ax—a PAPB The lines offorce areshewn infig.17=cons. Fig. 17. III. Electric Douhlet. 64.Animportantcase occurs whenwehavetwolarge charges-fe,—e, equalandoppositeinsign,atasmall distanceapart. Takiag Cartesian coordinates, letussuppose wehave thecharge +eata,0,andthecharge—eat—a,0,0,sothatthedistance ofthechargesis2a. Thepotentialis e e V(a;-ay+y2+z2V(#+a)2+y%+z%' andwhen aisvery small, sothatsquaresandhigher powersofamaybe neglected,thisbecomes 2eax (x2+y2+z2 )* Ifaismade tovanish, while ebecomes infinite, insuch awaythat 2ea retains the finite value/x,thesystemisdescribed asanelectric i 63,64] Charges +e,—e 51 doublet ofstrength //,havingforitsdirection thepositiveaxis ofx.Its potentialis fix (a?+y-+z2)$' Fig. 18. or,ifweturn topolarcoordinates andwritex—rcos9,ig ficos9 .(24). The lines offorce areshewn infig.18.Obviouslythe lines atthe centre ofthisfigure become identical with those shewn infig.17,ifthe latter areshrunkindefinitelyinsize. 4-2 52 Electrostatics —Field ofForce[ch.II 65.IV. Point charges +4e,—e. Fig.19representsthedistribution ofthelines offorcewhen the electric field isproduced bytwopoint charges, +4eatAand—eatB. Atinfinitytheresultant force willbe3e/r2 ,where risthedistance from apointnear toAandB.Thedirection ofthisforce isoutwards. Thus no lines offorce canarrive atBfrominfinity,sothat allthelines offorce which enterBmustcome fromA.Theremaininglines offorcefromAgo toinfinity.Thetubes offorce fromAtoBform abundle ofaggregate Fig. 19. strength e,while those fromAtoinfinityhaveaggregate strengthSe.The twobundles oftubes offorce areseparated bythelines offorcethroughG. AtGthedirection oftheresultant force isclearly indeterminate, sothatG isapointofequilibrium. Asthecondition thatGisapointofequilibrium wehave AC BG* SothatAB=BG.AtGthetwo. lines offorce fromAcoalesce andthen separate outintotwodistinct lines offorce, onefromGtoB,andtheother fromCtoinfinityinthedirectionoppositetoGB. Theequipotentials inthis field, thesystemofcurves 4J_PAPB"cons-' arerepresentedinfig.20,which isdrawn onthesame scale asfig.19. 65] Charges +4e,—e 53 SinceGisapointofequilibriumtheequipotential throughthepointG must ofcourse cutitself atG.AtGthepotential 4e GAe GBAB' sinceGA=2GB. From theloopofthisequipotentialwhich surrounds B, thepotential must fallcontinuouslyto—ooasweapproach B,since, bythe theorem of§51,there canbenomaxima orminima ofpotentialbetween thisloopandthepointB.Alsonoequipotentialcanintersect itself since there areobviously nopointsofequilibrium exceptG.One oftheinter- Fm. 20. mediateequipotentialsisofspecial interest, namelythat overwhich the potentialiszero. This isthelocus ofthepointPgiven by A 1=0,PAPB and istherefore asphere.This isrepresented bytheouter ofthetwo closed curves which surround Binthefigure. Inthesamewayweseethattheotherloopoftheequipotential through Gmust beoccupied byequipotentialsforwhich thepotentialrises steadily tothevalue+ooatA.Soalsooutside theequipotential through G,the potentialfallssteadilytothevalue zero atinfinity.Thus thezeroequi- potentialconsists oftwospheres—thesphereatinfinityandthesphere surrounding Bwhich hasalreadybeen mentioned. 54 Electrostatics —Field ofForce[ch.II V.Three equal chargesatthecorners ofanequilateral triangle. 66.Asafurther example wemayexamine thedispositionofequi- poteibialswhen the field isproduced bythreepoint chargesatthecorners ofanequilateral triangle. The intersection ofthese bytheplaneinwhich thechargeslieisrepresentedinfig.21,inwhich A,B,Garethepointsat which thechargesareplaced,and I)isthecentre ofthetriangle ABG. ]twillbefound that there arethreepointsofequilibrium, oneoneach ofthelinesAD,BD,CD. TakingAD=a,thedistance ofeachpointof equilibriumfromDisjustlessthan£a.Thesameequipotential passes throughallthreepointsofequilibrium.Ifthechargeateach ofthepoints Fig. 21. A,B,Gistaken tobeunity,thisequipotentialhasapotential304 aThe equipotentialhasthreeloops surroundingthepoints A,B,G.Ineach of theseloopstheequipotentialsareclosed curves, whichfinallyreduce to small circlessurroundingthepoints A,B,G.Those drawncorrespondto 325 35375 ,4 ,— , ,and- .a a a athepotentials 304Outside theequipotential ,theequipotentialsareclosed curvesa 66] Charges +e,+e,+e 55 surroundingtheformerequipotential, andfinally reducingtocircles atin- 22*25 2*5 ^"75 finity. Thecurves drawn correspond topotentials- , ,— ,and-— . r raaa a There remains theregion between thepointBandtheequipotentiall AtBthepotentialis,sothatthepotentialfalls aswerecede from thea a 304 equipotential andreaches itsminimum value atB.Thepotential ata Bisofcourse notaminimum foralldirections inspace:forthepotential increases aswemoveawayfromBindirections which areintheplane ABG, butobviously decreases aswemoveawayfromBinadirectionper- Fm. 22. pendiculartothisplane. TakingBasorigin, andtheplaneABG asplane ofxy,itwillbefound thatnearDthepotentialis a4a3 ° Thus theequipotential through Disshapedlikearightcircular cone in theimmediateneighbourhoodofthepointB.From theequation just found, itisobvious thatnearBthesections oftheequipotentials bythe planeABG willbecirclessurroundingB. 56 Electrostatics —Field ofForce[ch.n From astudyofthesection oftheequipotentialsasshewn infig.21,itis easytoconstruct thecompletesurfaces. Weseethateachequipotentialfor whichVhasaveryhighvalue consists ofthree smallspheres surroundingthe point.,A,B,0.Forsmaller values ofV,which must, however, begreater than,eachequipotentialstillconsists ofthree closed surfaces surround- ingA,B,C,butthese surfaces arenolonger spherical,eachonebulgingout towards thepointD.AsVdecreases, thesurfaces continue toswell out, 304 until,whenV= ,thesurfaces touch oneanothersimultaneously,ina (Ju waywhich willreadilybeunderstood onexaminingthesection ofthisequi- potentialasshewn infig.21. Itwillbeseen that thisequipotentialis shapedlikeaflower ofthreepetalsfromwhich thecentre hasbeen cutaway. 3AsVdecreases further thesurfaces continue toswell, andwhenV=-,thea spaceatthecentre becomes filledup.For stillsmaller values ofVthe equipotentialsareclosedsingly-connected surfaces, whichfinally become spheresatinfinity correspondingtothepotential V=0. The sections oftheequipotentials byaplane through DAperpendicular totheplaneABO areshewn infig.22. Special Properties ofEquipotentials andLines ofForce. TheEquipotentials andLinesofForce atinfinity. 67.In§40,weobtained thegeneral equation 7=2e _l [(a?-^)a+(y-yiT+(z-z*yf' Ifrdenotes thedistance ofx,y,zfrom theorigin,andrxthedistance of #i>2/i>2i>fr°mtneorigin, wemaywrite thisintheform [r2-2{ccx 1+yVl+zzx)+rfft' Atagreatdistance from theoriginthismaybeexpandedindescending powersofthedistance, intheform t^_^Mi ,xasi+yyi+zzi ,3(^ 1+y<y1+-g-g1)2In3 ) y--r\1+ J3 +2 r< 2>+"7" Theterm oforder -is— - .r r Theterm oforder-is-%ex{xx l+yyx+zzx). Thus bytakingtheoriginatthiscentroid, theterm oforder-willr266-68] Equipotentials andLines ofForce 57 Iftheoriginistaken atthecentroid ofexat<&,,yltzue2atxity%>z2,etc., wehave Xe xxx=0,texyx=0,Xe1z1=0. bytaking disappear. Theterm oforder—is r3 3 1 ^32ex(xx x+yy x+zzx)*-~texrx\ LetA,B,G,bethemoments ofinertia about theaxes, ofexatx1,yx,zx, etc.,and letIbethemoment ofinertia about thelinejoiningtheoriginto x,y,z\then2W =^(A+B+C), 2e x(xx x+yy x+tutf=r2(Xe^-1), andtheterms oforder-become A+B+C-3I 2r3 Thustakingthecentroid ofthechargesasorigin,thepotentialatagreat distance from theorigincanbeexpandedintheform F_SeA+B+G-SI Thusexcept when thetotalcharge 2evanishes, the field atinfinityis thesame asifthe totalcharge %ewere collected atthecentroid ofthe charges. Thus theequipotentials approximatetospheres havingthispoint ascentre, andtheasymptotestothelines offorce areradiidrawnthrough the centroid. These results areillustrated inthespecialfields offorce considered in§§61—66. TheLines ofForcefromcollinear charges. 68.When thefield isproduced solely bychargesallinthesamestraight line,theequipotentialsareobviouslysurfaces ofrevolution about this line, while thelines offorce lieentirelyinplanes throughthis line. Inthis important case, theequationofthelines offorceadmits ofdirectintegration. Let%,1%,B,...bethepositionsofthechargesex,e2,e3,—Let Q,Q' beanytwoadjacent points onalineofforce. Let iV^bethefootofthe perpendicular fromQtotheaxisi?^,...,and letacircle bedrawnperpen- dicular tothis axiswith centreNandradius QN. This circle subtends ati?asolidangle 2tt(1-cosex), 58 Electrostatics —Field ofForce[ch.n where^istheangle Qi?iV. Thus thesurfaceintegralofnormal force arisingfrome,,taken over the circle QN,is 273-0! (1—cos#j) andthetotal surfaceintegralofnormal force taken over thissurface is 2-77-26! (1—cos#i). Ifwedraw thesimilar circle through Q',weobtain aclosed surface bounded bythese two circles andbythesurface formedbytherevolution Fig. 23. ofQQ\ This contains noelectriccharge,sothat thesurfaceintegralof normal force taken over itmust benil.Hence theintegralofforce over the circleQNmust bethesame asthat over thesimilar circle drawn through Q'.Thisgivestheequationsofthelines offorce intheform (integralofnormal forcethroughcircle such asQN)=constant, which aswehave seen,becomes 2e xcos#,=constant. Analytically,letthepoint i?have coordinates a^,0,0,let^have coordinates a2,0,0,etc.and letQbethepoint x,y,z.Then cos6X=ijjWl V(#-arf+y2+z*' andtheequationofthesurfaces formed bytherevolution ofthelines of force is 2x^ -=constant. V(#- tfj)2+y2+z* Itwilleasily beverifiedbydifferentiation that this isanintegralofthe differentialequation dxX' 68,69] Equipotentials andLines ofForce 59 Equipotentialswhich intersect themselves. 69.Wehave seen that, ingeneral,theequipotential through anypoint ofequilibriummust intersect itself atthepointofequilibrium. Let x,y,zbeapointofequilibrium, and letthepotentialatthispointbe denoted byV .Letthepotentialatanadjacent pointx+f,y-f77,z+£,be denoted byVtjr,,(.ByTaylor's Theorem, if/(#, y,z)isanyfunction of x,y,z,wehave where the differential coefficients of/areevaluated atx,y,z.Taking f(x, y,z)tobethepotentialatx,y,z,thisofcoursebeingafunction ofthe variables x,y,z,theforegoing equation becomes ay+*&+H*d^+^vdxTy Ifx,y,zisapointofequilibrium,TrydVdV„dV ,.(y?rV _„32F \/orx aF=ar=8F=03#3^/3.z /wy &y\ /3217- 32y%Referred toa;,7/,zasorigin,thecoordinates ofthepointx+%,y+ 77, +£become£,77,£,andtheequationoftheequipotential V=G becomes d2V.«fc32F Intheneighbourhoodofthepointofequilibrium,thevalues of£,77,£are small, sothat ingeneralthetermscontaining powersof£,77,£higherthan squares maybeneglected,and theequationoftheequipotential V=C becomes fry gay Inparticulartheequipotential V=V becomes identical, intheneighbourhood ofthepointofequilibrium,with thecone 3a;2 ^ 3#3?/ Letthiscone, referred toitsprincipal axes,become a¥*+b7)'*+c?a=*0 (26), then, since thesum ofthecoefficients ofthesquaresofthevariables isan invariant, 32F327 d2VA 60 Electrostatics —Field ofForce[ch.ii Now a+b+c= isthecondition that thecone shall have threeper- pendicular generators. Hence weseethat atthepointatwhich an equipoential cuts itself,wecanalwaysfindthreeperpendicular tangentsto theequipotential.Moreover wecanfindtheseperpendicular tangentsinan infinite number ofways. Intheparticularcase inwhich thecone isoneofrevolution(e.g.,ifthe whole field issymmetricalabout anaxis, asinfigures16and20),the equationoftheconemust become p+v'2_2£'»=0, where theaxisof£'istheaxisofsymmetry. Thesection oftheequipotential made byanyplane throughtheaxis,saythatof£'§",mustnowbecome £/s_2£'2=o intheneighbourhoodofthepointofequilibrium, and thisshews that the tangentstotheequipotentialseachmake aconstantangletan-1\/2(=54°44') with theaxisofsymmetry. Inthemoregeneralcases inwhich there isnotsymmetry about anaxis, thetwobranches ofthesurface willingeneralintersect inaline,andthe conereduces totwoplanes,theequation being dp+br}'2=0, where theaxisof£'isthelineofintersection. Wenowhave a+b=0,so thatthetangent planestotheequipotentialintersect atright angles. Ananalogoustheorem canbeproved when nsheets ofanequipotential intersect atapoint. Thetheorem states that thensheets makeequal angles 7r/nwithoneanother. (Rankin's Theorem, seeMaxwell'sElectricity andMagnetism, §115, orThomson andTait's NaturalPhilosophy, §780.) 70.Aconductor isalwaysanequipotential,andcanbeconstructed soas tocut itself atanyangleweplease.Itwillbeseen that theforegoing theorems can faileitherthroughthea,band cofequation (24)allvanishing, orthroughtheir allbecominginfinite. Intheformer casethepotential near apointatwhich theconductor cuts itself, isoftheform(cf.equation (25)), /d3V cPV \ **<-K+t(pg+«P,&5+...). sothatthecomponentsofintensityareoftheforms -*»(* aF+2&5S*+" Theintensitynearthepointofequilibriumistherefore asmallquantityof thesecond order, andsincebyCoulomb's LawR=4nrcr, itfollows thatthe 69-71] Equipotentials andLines ofForce 61 surface densityiszeroalongthelineofintersection, and isproportionalto thesquareofthedistance from thelineofintersection atadjacent points. If,however, a,band care allinfinite, wehave theelectricintensityalso infinite, andtherefore thesurfacedensityisinfinitealongthelineofinter- section. Itisclear that thesurfacedensitywillvanish when theconducting surface cuts itself insuch awaythattheanglelessthantworight angles isexternal totheconductor; andthat thesurfacedensitywillbecome infinite when theangle greaterthantworight anglesisexternal tothe conductor. This becomes obvious onexaminingthearrangementofthe lines offorce intheneighbourhoodoftheangle. Fio. 24.Angle greater thantworight angles external toconductor. Fig. 25.Angle lessthan tworight angles external toconductor. 71.Thearrangementshewn infig.25issuch aswillbefound atthe pointofalightningconductor. Theobjectofthelightningconductor is toensure thattheintensityshall begreateratitspointthanonanypart ofthebuildingsitisdesignedtoprotect. Thedischargewilltherefore take 62 Electrostatics— Field ofForce[ch.n placefromthepointofthelightningconductor sooner thanfromanypartof thebuilding,andbyputtingtheconductor ingoodelectrical communication with tieearth, itispossibletoensure thatnoharm shallbedone tothe mainbraidings bytheelectrical discharge. Anapplicationofthesame principlewillexplainthedangertoahuman beingoranimal ofstandingintheopenairinthepresenceofathunder cloud, orofstandingunder anisolated tree. Theupward point,whether thehead ofmar oranimal, orthesummit ofthetree,tends tocollect thelines offorce whichpassfrom thecloud totheground,sothatadischargeofelectricity willtakeplacefrom thehead ortreerather thanfrom theground. 72.Thepropertyoflines offorce ofclustering togetherinthiswayis utilised alsointhemanufacture ofelectrical instruments. Acageofwire is Fio. 27. placedround theinstrument andalmost allthe lines offorce fromany chargeswhich theremaybeoutside theinstrument willclustertogether on theconvex surfaces ofthewire. Veryfewlines offorceescape throughthis cage,sothat theinstrument inside thecageishardlyaffected atallbyany electric phenomenawhich maytakeplaceoutside it.Fig.27shews the wayinwhich lines offorce areabsorbed byawiregrating.Itisdrawn to representthelines offorce ofauniform fieldmeetingaplane grating placed atright anglestothefield offorce. 71,72] Examples 63 Theprotectionofawirecageisnotadequateforthemost sensitive in- struments, and itisusual toenclose thementirelyinametal case, except onlyforonesmallwindowthroughwhichreadingscanbetaken. When this arrangementisadopted,nolines offorce atallcanpassfrom externalcharges totheinstrument inside themetal caseexceptforaninfinitesimal number passing throughthewindow. Lines offorcewhich encounter thecasetermi- nateonitwithout inanywayaffectingtheelectric field inside, andthein- strument isalmostperfectlyscreened fromanyexternal electric field.(Cf. §114 below.) EXAMPLES. 1.Twoparticleseach ofmassmandcharged with eunits ofelectricityofthesame sign aresuspended bystrings each oflength afrom thesame point; prove that the inclination 6ofeach stringtothevertical isgiven bytheequation imga?sin36=e2cos6. 2.Charges +4e,—eareplacedatthepoints A,B,andCisthepointofequilibrium. Prove thatthelineofforcewhich passes through CmeetsABatanangleof60°atAand atright anglesatC. 3.Find theangleatA(question 2)between ABandthelineofforcewhich leavesB atright anglestoAB. 4.Two positive chargesexand e2areplacedatthepointsAandBrespectively. Shew thatthetangentatinfinitytothelineofforcewhich starts from exmaking anangle awithBAproduced, makes anangle 2«n-i(\/-S-«ii^\\ e!+e22J withBA,andpasses throughthepointGinABsuch that AC :CB=e2:ev 5.Point charges +e,—eareplacedatthepoints A,B.Thelineofforcewhich leaves Amaking anangleawithABmeets theplanewhich bisects ABatright angles,inP. Shew that .a/5.PABsin-=N/2sin^—. 6.Ifanyclosed surface bedrawn notenclosing acharged bodyoranypartofone, shew that atevery pointofacertain closed lineonthesurface itintersects theequi- potentialsurface through thepointatright angles. 7.Thepotentialisgivenatfourpointsnear each other andnot allinoneplane. Obtain anapproximate construction forthedirection ofthe field intheir neighbourhood. 64 Electrostatics —Field ofForce[ch.u 8.The potentialsatthefourcorners ofasmall tetrahedron A,B,C,DareVlyF2, V3,Virespectively. Gisthecentre ofgravityofmassesMxatA,i/2atB,M3atC, J/4at0.Shew thatthepotentialatOis MlV1+M2V%+M3V3+MiVtl Mt+Ms+Mi+Mi 9.Charges Ze,—e,—eareplacedatA,B,Crespectively, whereBisthemiddle pointofAC.Draw arough diagramoftheHues offorce; shew thatalineofforcewhich starts fromAmaking anangleawithAB>cos~1 (—£)willnotreachBor0,andshew thattheasymptoteofthelineofforce forwhich a=cos-1 (—§)isatright anglestoAC. 10. Ifthere arethree electrified points A,B,Cinastraight line,such thatAC=f, f —6CLBC—-7,andthecharges aree,—^—andVarespectively, shew that there isalwaysa spherical equipotential surface, anddiscuss thepositionofthepointsofequilibrium on thelineABCwhenV=e—r<.andwhenV=e- 11.AandCarespherical conductors withcharges e+e'and—erespectively. Shew that there iseither apoint oralineofequilibrium, depending ontherelative sizeand positionsofthespheres, andone'/e.Draw adiagramforeach casegivingthelines of forceandthesections oftheequipotentials byaplane through thecentres. 12.Anelectrified bodyisplacedinthevicinityofaconductor intheform ofa surface ofanticlastic curvature. Shew that atthatpointofanylineofforce passing from thebodytotheconductor, atwhich theforce isaminimum, theprincipalcurvatures of theequipotentialsurface areequal andopposite. 13.Shew that itisnotpossibleforevery familyofnon-intersectingsurfaces infree spacetobeafamilyofequipotentials, andthatthecondition thatthefamilyofsurfaces /(X, x,y,z)=0 shallbecapableofbeing equipotentialsisthat a^x a^x a^x dx2dy2dz2 \ox/ \pyj \czj shallbeafunction ofXonly. 14.Inthelastquestion, ifthecondition issatisfied findthepotential. 15.Shew thattheconfocalellipsoids *2 +J^+*=1aHA^ +Ac2+A' canformasystem ofequipotentials, andexpress thepotential asafunction ofA. 16. Iftwocharged concentric shells beconnected byawire, theinner one iswholly discharged.Ifthelawofforcewere-3^, prove thatthere would beachargeBonthe inner shellsuch that ifAwere thecharge ontheoutershell,and/,gthesumand differ- ence oftheradii, 2gB=-Ap{{f-g) log(f+g) -flogf+glogg} approximately. Examples 65 17.Three infiniteparallelwires cutaplane perpendiculartothem intheangular points A,B,Cofanequilateral triangle, andhavecharges e,e,—e'perunitlength respectively.Prove that theextreme lines offorce whichpassfromAtoCmake at 2g 5g' 2e+a' starting angles—^—-irand———nwithAC,provided thate'^>2e.6e 6e 18.Anegative point charge—e2liesbetween twopositive point chargesexand e3on thelinejoining them andatdistances a,/3fromthemrespectively. Shewthat,ifthe magnitudesofthecharges aregiven by «i_«3_e2\3 aa+/3,and if1<X2<m> there isacircle atevery pointofwhich theforce vanishes. Determine thegeneral form oftheequipotentialsurface onwhich this circle lies. 19.Chargesofelectricityelt—e2,e3,(e3>ei) areplacedinastraight line, the negative charge being midway between theother two. Shew that,if4e2Hebetween (e33-e^)3and(e33+e^)3 ,thenumber ofunittubes offorce thatpassfrom extoe2is *(«i+e2-e3)+-^=(e3i-ef)(«j*-2**,*+e£fi.4V2 CHAPTER III CONDUCTORS ANDCONDENSERS 73.Byaconductor, aspreviously explained,ismeantanybodyor systemofbodies, such thatelectricitycanflowfreelyoverthewhole. When electricityisatrestonsuchaconductor, wehave seen(§44)thatthecharge will resideentirelyontheouter surface, and(§37)that thepotentialwill beconstant over this surface. Aconductor maybeused forthestorageofelectricity,but itisfound thatamuch more efficientarrangementisobtainedbytaking twoormore conductors—generallythinplatesofmetal—andarranging them inacertain way.Thisarrangementforstoring electricityisspokenofasa"con- denser." Inthepresent Chapter weshall discuss thetheoryofsingle conductors andofcondensers, workingoutinfullthetheoryofsome ofthe simplercases. Conductors. ASphericalConductor. 74.Thesimplest exampleofaconductor issupplied byasphere,it being supposedthatthesphereissofarremoved from allother bodies that their influence maybeneglected.Inthiscase itisobvious fromsymmetry thatthechargewillspreaditselfuniformlyover thesurface. Thus ifeis thecharge,andatheradius, thesurfacedensity<risgiven by totalchargee total area ofsurface 4ura2* The electricintensityatthesurfacebeing,aswehave seen, equalto 47ro-, ise/a\ From symmetrythedirection oftheintensityatanypointoutside the spheremust beinadirectionpassing throughthecentre. Tofindthe amount ofthisintensityatadistance rfrom thecentre, letusdraw asphere ofradius r,concentric with theconductor. Atevery pointofthissphere theamount oftheoutward electricintensityisbysymmetrythesame, sayR, 73-75] Spheres andCylinders 67 and itsdirection aswehave seen isnormal tothesurface.ApplyingGauss' Theorem tothissphere, wefindthatthesurfaceintegralofnormalintensity \\NdS becomes simplyRmultiplied bythearea ofthesurface 47rra ,sothat 4sirr-R=4<7re, or R— ~.2 Thisbecomese/a2atthe surface, agreeingwith thevaluepreviously obtained. Thus theelectric force atanypointisthesame asifthecharged sphere werereplaced byapoint charge e,atthecentre ofthesphere. And, just asinthecaseofasingle point charge e,thepotentialatapointoutside the sphere,distant rfrom itscentre, is J*r2r a sothat atthesurface ofthespherethepotentialis- Inside thesphere,ashasbeenprovedin§37,thepotentialisconstant, andthereforeequaltoe/a,itsvalue atthesurface, while theelectricintensity vanishes. Aswegradually charge uptheconductor,itappears that thepotential atthesurface isalways proportionaltothechargeoftheconductor. Itiscustomarytospeakofthepotentialatthesurface ofaconductor as "thepotentialoftheconductor," andtheratio ofthechargetothispotential isdefined tobethe" capacity"oftheconductor. From ageneral theorem, which weshall soon arrive at,itwillbeseen that theratio ofchargeto potential remains thesamethroughouttheprocessofcharging anyconductor orcondenser, sothat ineverycasethecapacity depends onlyontheshape and sizeoftheconductor orcondenser inquestion. Forasphere,aswe have seen, charge e capacity =— -—^— r=-=a, 1 J potentiale a sothatthecapacityofasphereisequaltoitsradius. ACylindrical Conductor, 75.Letusnext consider thedistribution ofelectricity onacircular cylinder, thecylindereitherextendingtoinfinity,orelsehavingitsends so faraway from thepartsunder consideration that their influence maybe neglected. Asinthecase ofthesphere,thechargedistributes itselfsymmetrically, 5—2 68 Conductors andCondensers[ch.Ill sothat ifaistheradius ofthecylinder,and ifithasachargeeperunit length, wehave Vrra Tofindtheintensityatanypointoutside theconductor, construct aGauss' surfacebyfirstdrawingacylinderofradius r,coaxal with theoriginal cylinder, andthencuttingoffaunitlength bytwoparallel planesat unit distanceapart, perpendiculartothe axis. Fromsym- metrytheforce atevery pointisperpendiculartotheaxis ofthecylinder,sothat thenormalintensityvanishes at every pointoftheplaneends ofthisGauss' surface. The surfaceintegralofnormalintensitywill therefore consist entirelyofthecontributions from thecurvedpartofthe surface, and thiscurvedpartconsists ofacircular band, of unitwidth andradius r—hence ofarea 2irr. IfRisthe outwardintensityatevery pointofthiscurved surface, Gauss' Theoremsuppliestherelation 2irrR=4nre, sothatrFig. 28. This,wenotice, isindependentofa,sothattheintensityisthesame as itwould beifawerevery small, i.e.,asifwehadafinewire electrified with achargeeperunitlength. Intheforegoing, wemustsupposertobesosmall, that atadistance r from thecylindertheinfluence oftheends isstillnegligibleincomparison with that ofthenearerpartsofthecylinder,sothattheinvestigationdoes nothold forlarge values ofr.Itfollows thatwecannot findthepotential byintegratingtheintensityfrominfinity,ashasbeen done inthecases of thepoint charge and ofthesphere. Wehave, however, thegeneral differential equation dV dr=-R, sothat inthepresent case, solongasrremainssufficientlysmall dV 2e or r giving upon integration V=C-2e\ogr. Theconstant ofintegration Ccannot bedetermined without aknowledge oftheconditions attheends ofthecylinder.Thus foralong cylinder,the intensityatpointsnear thecylinderisindependentoftheconditions atthe ends, butthepotential andcapacity dependonthese conditions, and are therefore notinvestigatedhere. 75-77] InfinitePlane, 69 AnInfinite Plane. 76.Suppose wehave aplane extendingtoinfinityinalldirections, and electrified with achargeaperunit area. Fromsymmetryitisobvious that thelines offorce willbeperpendiculartotheplaneatevery point,sothat thetubes offorce willbeofuniform cross-section. Letustake asGauss' surface thetube offorce which hasascross-sectionanyelement wofarea ofthecharged plane,thistubebeingclosedbytwo cross-sections each of area <uatdistance rfrom theplane.IfRistheintensityover either of these cross-sections thecontribution ofeach cross-section toGauss'integral isRco,sothatGauss' Theoremgivesatonce 2Ro>=4sTT<JGi, whence R=2tto-. Theintensityistherefore thesame atalldistances from theplane. The result that atthesurface oftheplanetheintensityislira-,mayat firstseem tobeinoppositiontoCoulomb's Theorem(§57)which states that theintensityatthesurface ofaconductor is47r<x. Itwill,however, beseen from theproofofthis theorem, that itdealsonlywith conductors in which theconductingmatter isoffinite thickness;ifwewish toregard the electrified planeasaconductor ofthiskindwemustregardthe total electrification asbeingdivided between thetwo faces, the surface density being\aoneach,andCoulomb's Theorem thengivesthecorrect result. Iftheplaneisnotactually infinite, theresult obtained foraninfinite planewillholdwithin aregionwhich issufficientlynear totheplaneforthe edgestohavenoinfluence. Asintheformer case ofthecylinder, wecan obtain thepotentialwithin thisregion byintegration.Ifrmeasures the perpendiculardistance from theplane -!^=i2=27nr,or sothat V=G—27rcrr, and, asbefore, theconstant ofintegrationcannot bedetermined without aknowledgeoftheconditions attheedges. 77. Itisinstructive tocomparethethr^ee expressionswhich havebeen obtained fortheelectricintensityatpointsoutside acharged sphere, cylinder andplane respectively. Takingrtobethedistance from thecentre ofthe 70 Conductors andCondensers[ch.Ill sphere,from theaxis ofthecylinder,andfrom theplane, respectively, we have found that outside thesphere, Risproportionalto— , outside thecylinder, Risproportionalto- , outside theplane, Risconstant. From thepointofview oftubes offorce, these results areobvious enough deductions from thetheorem thattheintensityvariesinverselyasthecross- section ofatube offorce. The lines offorcefromasphere meet inapoint, thecentre ofthesphere,sothat thetubes offorce arecones, with cross- sectionproportionaltothesquareofthedistance from thevertex. The lines offorcefrom acylinderallmeet aline,theaxisofthecylinder,atright angles,sothat thetubes offorce arewedges, with cross-sectionproportional tothedistance from theedge. Andthelines offorcefromaplaneallmeet theplaneatright angles,sothatthetubes offorce areprisms,ofwhich the cross-section isconstant. 78.Wemayalsoexamine theresults from thepointofview which regardstheelectricintensityastheresultant oftheattractions orrepulsions from different elements ofthechargedsurface. Letusfirst consider thecharged plane. LetP,P'betwopointsat distances r,rfrom theplane, and letQbethe footoftheperpendicular from either ontothe plane.IfPisnear toQ,itwillbeseen that almost thewhole oftheintensityatPisdue tothechargesintheimmediateneighbourhood ofQ.Themore distantpartscontribute forces which makeangleswithQPnearly equaltoa right angle,and afterbeingresolvedalongQP these forceshardlycontributeanythingtothe resultantintensityatP. Owingtothegreaterdistance ofthepoint P', theforces fromgivenelements oftheplaneare smaller atP'than atP,buthave toberesolved throughasmallerangle. The forces from the regionsnearQaregreatlydiminished from the former cause andarehardlyaffectedbythelatter. The forces fromremoteregionsarehardlyaffected bytheformer circumstance, bufc their effect is greatlyincreasedbythe latter. Thus onmoving Fig. 29. 77-79] Spherical Condenser 71 fromPtoP'theforces exerted byregionsnearQdecrease inefficiency, while those exerted bymore remoteregions gain. The result that the total resultantintensityisthesame atP'asatP,shews that the decrease oftheonejustbalances thegainoftheother. Ifwereplacetheinfiniteplane byasphere, wefindthat theforce at anearpointPisasbefore contributed almostentirely bythechargesinthe neighbourhoodofQ.OnmovingfromP toP',these forces arediminishedjustas before, butthenumber ofdistant elements[ Qj: ofareawhich nowaddcontributions to theintensityatP' ismuch lessthan before. Thus thegaininthecontributions Fig. 30. from these elements does not suffice to balance thediminution inthecontributions from theregionsnearQ,sothat theresultantintensityfalls offonwithdrawingfromPtoP' Thecase ofacylinderisofcourse intermediate between thatofaplane andthat ofasphere. Condensers. SphericalCondenser. 79.Supposethatweenclose thesphericalconductor ofradius adis- cussed in§74,inside asecondsphericalconductor ofinternal radiusb,the twoconductors being placedsoastobeconcentric andinsulated fromone another. Itagain appearsfromsymmetrythattheintensityatevery pointmust beinadirectionpassing throughthecommon centre ofthetwospheres,and must bethesame inamount atevery pointofanysphereconcentric with thetwoconducting spheres.Letusimagineaconcentricsphereofradius r drawn between thetwoconductors, andwhen thechargeontheinnersphere ise,lettheintensityatevery pointoftheimaginary sphereofradius rbe R.Then, asbefore, Gauss' Theorem,appliedtothesphereofradius r,gives therelation 4nrr2R=4nre, Q sothat R—- ,r2 Thisonlyholds forvalues ofrintermediate between aandb,sothat to obtain thepotential wecannotintegratefrominfinity, butmust usethe differentialequation.This is or r-.«» 72 Conductors andCondensers[ch.m whichupon integration gives V=C+-(27).r Wecandetermine theconstant ofintegrationassoon asweknow the potentialofeither ofthespheres. Supposeforinstance that theouter sphereisputtoearth sothatV=0 overthespherer=b,thenweobtain at oncefromequation (27) sothatG=—e/b,andequation (27)becomes rb Ontakingr=a,wefindthatthepotentialoftheinnersphereise(— rj, and itschargeise,sothatthecapacityofthecondenser is 1 ah or11 b-a' ab 80.Inthemoregeneralcase inwhich theoutersphereisnotputto earth, letussupposethatVa,Vbarethepotentialsofthetwospheresof radii aand b,sothat, fromequation (27) a F6=C+|. Thenwehaveonsubtraction «-*>-(H)' sothatthecapacityis ~— y.. The lines offorcewhich startfrom theinnersphere must allendonthe inner surface oftheoutersphere, andeach lineofforce hasequal and opposite chargesatitstwoends. Thus ifthechargeontheinnersphereis e,thatontheinner surface oftheouterspheremust be—e.Wecanthere- foreregardthecapacityofthecondenser asbeingthechargeoneither of thetwospheresdivided bythedifference ofpotential,thefractionbeing taken always positive. Onthisview, however, weleave outofaccountany chargewhich theremaybeontheouter surface oftheoutersphere:this isnotregardedaspartofthechargeofthecondenser. 79-82] Cylindrical Condenser 73 Anexamination oftheexpressionforthecapacity, ab willshew that itcanbemade aslargeasweplease bymakingb—a sufficientlysmall. Thisexplains whyacondenser issomuch more efficient forthestorageofelectricitythan asingle conductor. 81.Bytakingmore thantwospheres wecanformmorecomplicated condensers.Suppose,forinstance, wetake concentricspheresofradii a,b,cinascendingorder ofmagnitude, andconnect both thespheresof radiiaand ctoearth, that ofradius bremaininginsulated. LetVbethe potentialofthemiddlesphere,and letexand e2bethetotalchargesonits inner andouter surfaces.Regardingtheinner surface ofthemiddlesphere andthesurface oftheinnermostsphereasformingasingle spherical condenser, wehave Vab 6l~b-a' andagain regardingtheouter surface ofthemiddlesphereandtheoutermost sphereasformingasecondspherical condenser, wehave Vbc c—b Hence thetotalchargeEofthemiddle sheet isgiven by E=ex+e2 ab be=V- 7+ b—a c—b sothatregardedasasingle condenser, thesystemofthreesphereshasa capacity ab be+ b—a c—b' which isequaltothesum ofthecapacitiesofthetwoconstituent condensers intowhich wehave resolved thesystem.This isaspecialcase ofageneral theorem tobegivenlater(§85). CoaxalCylinders. 82.Aconductingcircularcylinderofradius asurrounded byasecond coaxalcylinderofinternal radius bwillform acondenser. Ifeisthecharge ontheinnercylinder perunitlength,and ifVisthepotentialatanypoint between thetwocylindersatadistance rfrom theircommon axis,wehave, asin§75, V=C-2e\ogr, 74 Conductors andCondensers [ch.in and itisnowpossibletodetermine theconstant Cassoon asthepotentialof eithercylinderisknown. LetVa,Vt,bethepotentialsoftheinner andoutercylinders,sothat Va=C-2eloga, Vb=C-2e\ogb. Bysubtraction Va-Vb=2elogf- J, sothatthecapacityis ., perunitlength. Parallel Plate Condenser. 83.This condenser consists oftwoparallel plates facingoneanother, sayatdistance dapart.Lines offorce willpassfrom theinner faceofone totheinner faceoftheother, and inregions sufficientlyfarremoved from theedgesoftheplatethese lines offorce willbeperpendiculartotheplate throughouttheirlength.Ifcristhesurfacedensityofelectrification ofone plate,that oftheother willbe—a.Since thecross-section ofatube remains thesamethroughoutitslength,and since the electricintensity varies asthecross-section, itfollows that theintensitymust bethesame throughoutthewholelengthofatube, and this,byCoulomb's Theorem, willbe47ro-, itsvalue atthesurface ofeitherplate.Hence thedifference of potentialbetween thetwoplates,obtained byintegratingtheintensity4tto- alongalineofforce, willbe Arrrcrd. Thecapacity perunit area isequaltothecharge perunit areaa divided bythis difference ofpotential,and istherefore 1 4>nd' Thecapacityofacondenser formed oftwoparallel plates, each ofareaA, istherefore A 4>ird' exceptforacorrectionrequired bytheirregularitiesinthelines offorce near theedgesoftheplates. InductiveCapacity. 84. Itwas found byCavendish, and afterwardsindependently by Faraday,that thecapacityofacondenserdependsnotonlyontheshape and sizeoftheconducting platesbut alsoonthenature oftheinsulating material, ordielectric touseFaraday's word, bywhichtheyareseparated. 82-85]Series ofCondensers 75 Itisfurther found thatonreplacingairbysome other dielectric, the capacityofacondenser isaltered inaratio which isindependentofthe shape and sizeofthecondenser, andwhichdepends onlyonthedielectric itself. This constant ratio iscalled thespecific inductivecapacityofthe dielectric, theinductivecapacityofairbeingtaken tobeunity. Weshall discuss thetheoryofdielectrics inalaterChapter.Atpresent itwillbeenoughtoknow that ifCisthecapacityofacondenser when its platesareseparated byair,then itscapacity, when theplatesareseparated byany dielectric, willbeKG,whereKistheinductivecapacityofthe particulardielectric used. Thecapacitiescalculated inthisChapterhave all been calculated onthesuppositionthat there isairbetween theplates,so thatwhen thedielectric isdifferent from aireachcapacity must bemulti- pliedbyK. The following table willgivesome idea ofthevalues ofEactually observed for different dielectrics. Foragreatmanysubstances thevalue ofKisfound tovarywidely fordifferent specimensofthematerial andfordifferentphysicalconditions. Sulphur 76 Conducto7,sandCondensers[ch.m andthetotalchargeEisgiven by E=e1+e2+... =(C 1+C2+...)(F 1-F). Thus thesystemofcondensers behaves likeasinglecondenser ofcapacity Cx+C7a+C3+ .... Itwillbenoticed that thecompound condenser discussed in§81con- sistedvirtuallyoftwosimple spherical condensers connected inparallel. Condensers inCascade. 86.Wemight, however, connect thelowpotential plateofthe first to thehigh potential plateofthesecond, thelowpotential plateofthesecond tothehigh potential plateofthe third, and soon. This isknown as arrangingthecondensers incascade. Fig. 32. Supposethatthehigh potential plateofthe firsthasachargee.This induces acharge—eonthelowpotential plate, andsince thisplate together with thehigh potential plateofthesecond condenser nowform asingle insulated conductor, theremust beacharge+eonthehigh potential plate ofthesecond condenser. This induces acharge-eonthelowpotential plateofthiscondenser, andsoonindefinitely;eachhigh potential platewill haveacharge+e,eachlowpotential plateacharge—e. Thus thedifference ofpotentialofthetwoplatesofthe firstcondenser willbee/C lythat ofthesecond condenser willbee/C 2,andsoon,sothatthe total fallofpotentialfrom thehigh potential plateofthe first tothelow potential plateofthelastwillbe 1 1 ...). Weseethatthearrangementacts likeasinglecondenser ofcapacity 1 JL_ _1 85-89] TheLeyden Jar 77 Pkactical Condensers. Practical Units. 87.As willbeexplained morefully later, thepractical units of electricians areentirelydifferent from thetheoretical units inwhich we have sofarsupposed measurements tobemade. Thepractical unit of capacityiscalled thefarad, and isequal, veryapproximately,to9x10"times thetheoretical C.G.S. electrostatic unit, i.e., isequaltotheactualcapacity ofasphereofradius 9x10ucms. This unit istoolargeformostpurposes, sothat itisconvenient tointroduce asubsidiaryunit—themicrofarad— equaltoamillionth ofthefarad, andtherefore to9x105C.G.S. electrostatic units. Standard condensers canbeobtained ofwhich thecapacityisequal toagiven fraction, frequentlyone-third orone-fifth, ofthemicrofarad. TheLeydenJar. 88.Forexperimental purposesthecommonest form ofcondenser isthe LeydenJar. This consistsessentiallyofaglass vessel, bottle-shaped,of which thegreater partofthesurface iscoated inside andoutside with tinfoil. Thetwocoatings form thetwoplatesofthecondenser, contact with theinnercoating beingestablished byabrass rodwhich comesthroughtheneck ofthebottle, thelower endhavingattached toitachain which rests ontheinner coating oftinfoil.O csssai^p Fig. 33.Toform aroughnumerical estimate ofthe capacityofaLeyden Jar, letussupposethatthe thickness oftheglassis\cm.,that itsspecific inductivecapacityis7,andthattheareacovered with tinfoil is400sq.cms. Neglectingcorrectionsrequired bytheirregu- larities inthelines offorce attheedgesandatthesharp anglesatthe bottom ofthejar,andregardingthewholesystemasasingle parallel plate condenser, weobtain asanapproximatevalue forthecapacity KA 4nrdelectrostatic units, inwhich wemustputK=7,A=400andd=|.Onsubstitutingthese values thecapacityisfound tobeapproximately450 electrostatic units, orabout^q$microfarad. 89.Parallel Plates. Amore convenient condenser forsomepurposesisamodification of theparallel platecondenser. Letussupposethatwearrangenplates,each 78 Conductors andCondensers[ch.m ofareaA,paralleltooneanother, thedistance between anytwoadjacent plates beingd.Ifalternateplatesarejoined togethersoastobeinelectrical contact thespacebetween eachadjacent pairofplates mayberegardedas Fig. 34. KA formingasingle parallel platecondenser ofcapacity j—-% ,sothatthecapacity ofthecompoundcondenser is(n—1)KA/4nrd. Bymakingnlargeandd small, wecanmake thiscapacity large withoutcausingtheapparatusto occupyanunduly largeamount ofspace. For thisreason standard con- densers areusuallymade ofthispattern. 90.Guard Ring. Inboth thecondensers described thecapacitycan onlybecalculatedapproximately. Lord Kelvin hasdevised amodification oftheparallel platecondenser inwhich theerror caused bytheirregularities ofthelines offorce near theedgesisdispensed with, sothat itispossible accuratelytocalculate thecapacityfrommeasurements oftheplates. Theprincipleconsists inmakingoneplateBofthecondenserlarger than thesecondplateA,theremainder ofthespace opposite Bbeing occupied by a"guard ring"Gwhich fitsAsocloselyasalmost totouch, and isinthe sameplanewith it.Theguard ringGandtheplate A,ifatthesame potential, maywithout serious error beregardedasformingasingle plateof aparallel platecondenser ofwhich theotherplateisB.Theirregularities inthetubes offorcenowoccur attheouteredgeoftheguard ring G,while thelines offorcefromAtoBareperfectly straight anduniform. Thus ifA isthearea oftheplateAitscapacity maybesupposed,withgreat accuracy, tobe 4ivd' where disthedistance between theplatesAandB. 89-92]Mechanical Force 79 Submarine Gables. 91.Unfortunatelyforpractical electricians, asubmarine cable forms acondenser, ofwhich thecapacityisfrequently veryconsiderable. The effect ofthisuponthetransmission ofsignalswillbediscussed later.Acable consistsgenerallyofacoreofstrands ofcopperwiresurroundedbyalayerof insulating material, thewholebeingenclosed inasheathingofiron wire. Thisarrangementacts asacondenser ofthetypeofthecoaxalcylinders investigatedin§82,thecoreformingtheinnercylinderwhilst theiron sheathingandtheseaoutside form theoutercylinder. Inthecapacityformula obtained in§82,namely K »•© letussupposethat b=2a,andthatK=3'2,thisbeingabout thevalue for theinsulatingmaterialgenerallyused. Usingthevalueloge2="69315, we findacapacityof231 electrostatic unitsperunitlength. Thus acable 2000 miles inlengthhasacapacity equaltothat ofasphereofradius 2000x231 miles, i.e.,ofasphere greaterthan theearth. Inpractical units, thecapacityofsuchacable would beabout 827microfarads. Mechanical Force onaConducting Surface. 92.LetQbeanypointonthesurface ofaconductor, and letthe surface-densityatthepointQbe <x.Letusdrawanysmall areadS Fig. 36. enclosing Q.Bytaking dSsufficiently small, wemay regardthearea as perfectly plane,andthechargeonthearea willbeadS. Theelectricityon theremainder oftheconductor willexert forces ofattraction orrepulsionon thecharge <rdS,andthese forces willshew themselves asamechanical force actingontheelement ofareadSoftheconductor. Werequiretofindthe amount ofthismechanical force. 80 Conductors andCondensers [ch.ih The electricintensityatapointnearQandjustoutside theconductor is 4-7TO-,byCoulomb's Law,and itsdirection isnormally awayfrom thesurface. Ofthisintensity, partarises from thechargeondSitself, andpartfrom the chargesontheremainder oftheconductor. Asregardsthe firstpart,which arises from thechargeondSitself,wemaynotice thatwhen wearecon- sideringapoint sufficientlyclose tothesurface, theelement dSmaybe treated asaninfinite electrifiedplane,the electrificationbeingofuniform density<r.Theintensity arisingfrom theelectrification ofdSatsuch a pointisaccordinglyanintensity2ircrnormally awayfrom thesurface. Since thetotalintensityis4>tt<tnormally awayfrom thesurface, itfollows thatthe intensity arisingfrom theelectrification ofthepartsoftheconductor other thandSmust alsobe2iranormally awayfrom thesurface. Itistheforces composingthisintensitywhichproducethemechanical action ondS. ThechargeondSbeing adS, thetotal force willbe2ira-dSnormally away from thesurface. Thusperunit areathere isaforce 2tto-2tendingtorepel thecharge normally awayfrom thesurface. Thechargeispreventedfrom leavingthesurface oftheconductor bytheaction betweenelectricityand matter which hasalreadybeenexplained.Action andreactionbeing equal andopposite,itfollows that there isamechanical force 27rcr2perunitarea acting normallyoutwards onthematerial surface oftheconductor. RememberingthatR=4"7rcr,wefind thatthemechanical force canalso R2 beexpressedas^—perunit area. 07T 93.Letustrytoformsome estimate ofthemagnitudeofthismechanical force ascomparedwith other mechanical forces withwhich wearemore familiar. Wehavealready mentioned Maxwell's estimate thatagrammeof gold,beaten intoagold-leafonesquaremetre inarea, canholdachargeof 60,000 electrostatic units. Thisgives3unitspersquarecentimetre asthe chargeoneach face,givingfortheintensityatthesurface, R=4nra=38C.G.S. units, and forthemechanical force i?2 2-77-cr2=^—=56dynes persq.cm. Lord Kelvin, however, found that airwascapableofsustaininga tension of9600grainswt.per sq.foot, orabout 700dynes per sq.cm. ThisgivesR=130,a=10. 7?3 TakingR=100 asalarge value ofR,wefind^—=400dynes per sq.cm.Thepressureofanormalatmosphereis 1,013,570 dynes persq.cm., 92-94] Electrified Soap-Bubble 81 sothat theforce ontheconductingsurface would beonlyabout^^ofan atmosphere:say*3mm. ofmercury. Ifagold-leafisbeaten sothin that 1gm.occupies1sq.metre ofarea, theweightofthis is'0981dyne per sq.cm. Inorder that 2-rra2maybe equalto'0981, wemust have <r=-1249. Thus asmallpieceofgold-leaf would beliftedupfrom achargedsurface onwhich itrested assoon asthe surfaceacquiredachargeofabout|ofaunitper sq.cm. Electrified Soap-Bubble. 94.Ashasalreadybeen said, thismechanical forceshews itself wellon electrifyingasoap-bubble. Letusfirstsupposeaclosedsoap-bubble blown, ofradius a.Ifthe atmospheric pressureisIT,thepressureinside willbesomewhatgreater than II,theresultingoutward forcebeing justbalancedbythetension ofthe surface ofthebubble. If,however, thebubble iselectrified there willbean additional forceacting normallyoutwards onthesurface ofthebubble, namely theforce ofamount lira3,perunitareajust investigated,andthebubble will expanduntilequilibriumisreached between thisandtheother forcesacting onthesurface. Astheelectrification andconsequentlytheradiuschange,thepressure inside willvary inverselyasthevolume, andthereforeinverselyasa3 .Let Fig. 37. us,then, supposethepressuretobe«/a3 .Consider theequilibriumofthe small element ofsurface cutoffbyacircular conethroughthecentre, ofsmall semi-verticalangle6.Thiselement isacircle ofradius a6,andtherefore ofarea ira}Q% .The forcesactingare : (i)Theatmospheric pressureIT7ra2#2normallyinwards. (ii)Theinternalpressure—ird262normallyoutwards. j. 6 82 Conductors andCondensers[ch.m (iii)Themechanical forcedue toelectrification, 2ira2x7ra22normally outwards. (iv)Thesystemoftensionsactinginthesurface ofthebubble across theboundaryoftheelement. IfTisthetensionperunitlength,thetension acrossanyelement of lengthdsofthesmall circle willbeTdsactingatanangle6withthetangent planeatP,thecentre ofthe circle. Thismayberesolved intoTdscos6in thetangent plane,andTdssin6alongPO. Combiningtheforces allround thesmall circle ofcircumference 2ira6,wefindthat thecomponentsinthe tangent plane destroyoneanother, while thosealongPOcombine into a resultant 2irad xTsin 6.Toasufficientapproximationthismaybewritten as2ira6*T. Theequationofequilibriumoftheelement ofarea isaccordingly n-Tra2^2--,Tra2^-27r<727ra22+2-rradiT=0,a3 k 2T or,simplifying,II---2ira2+—-0(28). Letabetheradius when thebubble isuncharged, and lettheradius be a,when thebubble hasachargee,sothat Then n-—,+—=0,a3 tt* e22T .n-—- g—t+—=o. Wecanwithout serious errorassume Ttobethesame inthetwocases. Ifweeliminate Tfrom these twoequations, weobtain II(ax-a)-k(—.:-— 2}= ^Ox2a2 /87TO!3' givingthechargeinterms oftheradii inthechargedandunchargedstates. 95.Wehave seen(§93)that themaximum pressureonthesurface which electrification canproduceisonlyabout^^atmosphere:thus itis notpossibleforelectrification tochangethepressureinside bymore than about ^^y atmosphere,sothat theincrease inthe sizeofthebubble is necessarily very slight. If,however, thebubble isblown onatube which isopentothe air, equation (28)becomes 7TCT2=T a' 94-97] Energy 83 Asarough approximation,wemaystillregardthebubble asauniformly charged sphere,sothat ifVisitspotential, o-=V/4nra, andtherelation is V2=1§ttcl T, givingVinterms oftheradius ofthebubble, ifthetension Tisknown. In thiscase theelectrification canbemade toproducealarge changeinthe radius, byusingfilms forwhichTisverysmall. Energy ofDischarge. 96.Ondischargingaconductor orcondenser, acertain amount of energyissetfree. Thismayshew itself invariousways, e.g.asasparkor sound (asinlightningandthunder), theheatingofawire, orthepiercing ofaholethroughasolid dielectric. Theenergythus liberated hasbeen previouslystored upinchargingtheconductor orcondenser. Tocalculate theamount ofthisenergy,letussupposethatoneplateof acondenser istoearth, andthattheotherplatehasachargeeand isat potential V,sothat ifCisthecapacityofthecondenser, e=CV(29). Ifwebring upanadditionalchargedefrominfinity,thework tobe doneis,inaccordance with thedefinition ofpotential,Vde. This isequal todW,whereWdenotes thetotalwork done inchargingthecondenserup tothisstage,sothat dW= Vde =-jrbyequation (29). Onintegrationweobtain W=k%(30), noconstant ofintegration being added sinceWmust vanish when e=0. Thisexpression givesthework done inchargingacondenser, andtherefore givesalsotheenergyofdischarge,which maybeused increatingaspark, inheatingawire, etc. Clearlyanexactlysimilarinvestigationwillapplytoasingle conductor, sothatexpression (30) givestheenergyeither ofacondenser orofasingle conductor.Usingtherelation e=CV,theenergy maybeexpressedinany oneoftheforms e- W=\^=\eV=\GV*(31). 97.Asanexampleoftheuseofthisformula, letussupposethatwe have aparallel plate condenser, thearea ofeachplate being A,andthe 6—2 84 Conductors andCondensers[oh.in distance oftheplates being d,sothatG=A/4nrd, by§83.Letabethe surface densityofthehigh potential plate,sothat e=aA.Letthelow potential platebeatzeropotential,then thepotentialofthehigh potential plateis V=^=4nrda, \j andtheelectrical energyis W=±eV=27rda2A. Now letuspulltheplates apart,sothatdisincreased tod'.The electrical energyisnow 2ird'a2A,sothat there hasbeen anincrease of electrical energyofamount 2tt<t*A(d'-d). Itiseasytoseethat thisexactly representsthework done inseparating thetwoplates.Themechanical force oneitherplateis2ira2perunit area, sothat thetotal mechanical force onaplateis2tt<t-A.Obviously, then, theabove isthework done inseparatingtheplates throughadistance d'-d. Itappearsfrom thisthataparallel platecondenser affords aready means ofobtainingelectricalenergyattheexpenseofmechanical. Amore valuable propertyofsuch acondenser isthat itenables ustoincrease aninitial difference ofpotential. The initial difference ofpotential Girder isincreased, bytheseparation,to 4f7rd'cr. Bytaking dsmall and d'large,aninitial small difference ofpotential maybemultipliedalmostindefinitely, andapotentialdifference which is toosmall toobserve maybeincreased until itissufficiently greattoaffect aninstrument. Bymakinguseofthisprinciple,Volta firstsucceeded in detectingthedifference ofelectrostaticpotentialbetween thetwoterminals ofanelectricbattery. There arepracticaldifficulties which restrict theapplicationoftheprinciple. For iftheinitial distance dismade toosmall thecondenser maydischargeitself byaspark passing directlybetween theplates,while ifd!ismadelargecom- paredwiththesizeoftheplatestheformulae wehaveused arenolongertrue. EXAMPLES. 1.Thetwoplatesofaparallel plate condenser areeach ofareaA,andthedistance between them isd,thisdistance being small compared with thesizeoftheplates. Find theattraction between them when charged topotential differenceV,neglecting the irregularitiescaused bytheedgesoftheplates. Find alsotheenergysetfreewhen the platesareconnected byawire. 97] Examples 85 2.Asheet ofmetal ofthickness tisintroduced between thetwoplatesofaparallel plate condenser which areatadistance dapart, and isplacedsoastobeparalleltothe plates. Shew thatthecapacityofthecondenser isincreased byanamount t 4nd(d-t) perunit area. Examine thecase inwhich tisverynearly equaltod. 3.Ahigh-pressure main consists first ofacentral conductor, which isacopper tube ofinnerandouter diameters of^and\%inches. Theouter conductor isasecond copper tube coaxal with thefirst,fromwhich itisseparated byinsulating material, and of diameters1§£and\\%inches. Outside this ismore insulating material, andenclosing thewhole isanirontube ofinternal diameter 2^inches. Thecapacityoftheconductor isfound tobe"367microfaradpermile :calculate theinductivecapacityoftheinsulating material. 4.Aninfinite planeischargedtosurfacedensity o-,andPisapointdistant halfan inchfrom theplane. Shew that ofthetotalintensity271-0-atP,half isduetothecharges atpoints which arewithin oneinch ofP,andhalftothecharges beyond. 5.Adiscofvulcanite(non-conducting)ofradius 5inches,ischargedtoauniform surfacedensity<rbyfriction. Find theelectric intensities atpointsontheaxisofthe discdistantrespectively 1,3,5,7inches from thesurface. 6.Acondenser consists ofasphereofradius asurrounded byaconcentricspherical shell ofradius b.Theinner sphereisputtoearth, andtheouter shell isinsulated. b2 Shew that thecapacityofthecondenser soformed isy—-. 7.Fourequal large conducting plates A,B,C,Darefixedparalleltooneanother. AandDareconnected toearth,BhasachargeEperunit area,andCacharge E'per unit area. Thedistance between AandBisa,between BandCisb,andbetween Cand Disc.Find thepotentialsofBand C. 8.Acirculargold-leafofradius bislaidonthesurface ofacharged conducting sphereofradiusa,abeing largecomparedtob.Prove that thelossofelectrical energy inremoving theleaffrom theconductor—assuming that itcarries awayitswhole charge— isapproximately \b2E2 \o?,whereEisthechargeoftheconductor, andthecapacityofthe leaf iscomparabletob. 9.Twocondensers ofcapacities ClandC2,andpossessing initially charges #1andQit areconnected inparallel. Shew thatthere isalossofenergyofamount 2C1Ca(C1+C2)' 10.TwoLeyden Jars A,Bhavecapacities ClfC-2respectively. Aischarged anda spark taken :itisthen chargedasbefore andaspark passedbetween theknobs of AandB.AandBarethenseparated andareeach discharged byaspark. Shew that theenergiesofthefoursparksareintheratio (C1+C2)2 :(Ci+C2)tf2=Ci~Wi- 11.Assuming anadequate number ofcondensers ofequal capacity C,shewhowa compound condenser canbeformed ofequivalent capacity 8C,where 6isanyrational number. 8G Conductors andCondensers [ch.in 12.Three insulated concentricspherical conductors, whose radii inascendingorder ofmagnitudearea,b,c,have chargeset,e2,e3respectively,find theirpotentials andshew that iftheinnermostspherebeconnected toearth thepotentialoftheoutermost is diminished by «/fi+£2+£3 13.Aconducting sphereofradius aissurrounded bytwothinconcentricspherical conductingshells ofradii bandc,theintervening spaces beingfilled with dielectrics of inductivecapacities EandLrespectively.Iftheshell breceives acharge E,theother twobeing uncharged, determine thelossofenergy andthepotential atanypointwhen thespheres AandCareconnected byawire. 34.Three thin conductingsheets areintheform ofconcentric spheresofradii a+d,a,a—crespectively. The dielectric between theouter andmiddle sheet isof inductive capacity E,thatbetween themiddle andinner sheet isair.Atfirsttheouter sheet isuninsulated, theinner sheet isuncharged and insulated, themiddle sheet is chargedtopotential Vand insulated. The inner sheet isnow uninsulated without connection with themiddle sheet. Prove thatthepotentialofthemiddle sheet falls to EVc(a+d) Ec(a+d)+d(a-c)' 15.Two insulated conductors AandBaregeometrically similar, theratio oftheir linear dimensions beingasLtoL'.Theconductors areplacedsoastobeoutofeach other's field ofinduction. ThepotentialofAisVand itschargeisE,thepotential ofBisVand itschargeisE'.Theconductors arethen connected byathin wire. Provethat,after electrostaticequilibriumhasbeen restored, the loss ofelectrostatic energyis ,(EL'-E'L)(Y- V) *Z+L' 16. Iftwosurfaces betaken inanyfamilyofequipotentialsinfreespace, andtwo metal conductors formed soastooccupytheirpositions, then thecapacityofthe CC-condenser thus formed isn1^-,where d,C2arethecapacitiesoftheexternal and internal conductors whenexisting alone inaninfinite field. 17.Aconductor (B)withoneinternalcavityofradius biskeptatpotentialU.A conducting sphere (A),ofradiusa,atgreat height aboveBcontains inacavity water which leaksdown avery thinwirepassing without contact intothecavityofBthrough ahole inthetopofB.Attheendofthewirespherical dropsareformed, concentric with thecavity ;and,when ofradiusd,theyfallpassing without contact through asmall hole inthebottom ofB,andarereceived inacavityofathird conductor (C)ofcapacityc atagreat distance below B.Initially,before leaking commences, theconductors AandC areuncharged. Prove that after therthdrophasfallen thepotentialofCis far(b-dy ,\an. \(ab+bd-ad)r\c' where thedisturbingeffect ofthewireandholeonthecapacitiesisneglected. 18.Aninsulatedspherical conductor, formed oftwohemisphericalshells incontact, whose inner andouter radii arebandb',haswithin itaconcentric sphericalconductor of radius a,andwithout itanotherspherical conductor ofwhich theinternal radius isc. These twoconductors areearth-connected andthemiddle onereceives acharge. Shew that thetwo shells willnotseparateif 2ac>bc +b'a. Examples 8*7 19.Outside aspherical charged conductor there isaconcentric insulated butun- charged conducting spherical shell, which consists oftwosegments. Prove thatthetwo segments willnotseparateifthedistance oftheseparating plane from thecentre isless than ab (a2+J2)J' wherea,baretheinternal andexternal radii oftheshell. 20.Asoap-bubbleofradius aisformed byafilm oftension T,theexternal atmospheric pressure beingII.Thebubble istouched byawirefrom alarge conductor atpotential V,andthefilm isanelectrical conductor. Prove that itsradius increases to r,given by n(r»-a")+2T(r2-a2 )=-^. 21. Iftheradius andtension ofaspherical soap-bubble beaandTrespectively, shew thatthechargeofelectricity requiredtoexpandthebubble totwice itslinear dimensions would be nbeingtheatmospheric pressure. 22.Athinspherical conducting envelope,oftension Tfor allmagnitudesofits radius, andwithnoairinside oroutside,isinsulated andcharged with aquantity Qof electricity. Prove thatthetotal gaininmechanical energyinvolved inbringingacharge qfromaninfinite distance andplacingitontheenvelope,which bothinitially andfinally isinmechanicalequilibrium,is 23.Aspherical soap-bubbleisblown inside another concentric withit,andthe former hasachargeEofelectricity,thelatter being originally uncharged. The latter nowhasasmallcharge given toit.Shew that ifaand2awere theoriginal radii,the new radii willbeapproximately a+x,2a+y,where / 101 7E2\ \2y{Ua+T)=x{2AUa +^rT+±—^% wherenistheatmospheric pressure, andTisthesurface-tension ofeach bubble. 24.Shew that the electriccapacityofaconductor islessthan that ofanyother conductor which cancompletelysurround it. 25. Iftheinnersphereofaconcentricspherical condenser ismovedslightly outof position,sothat thetwospheresarenolonger concentric, shew that thecapacityis increased. CHAPTEE IV SYSTEMS OFCONDUCTORS 98.Inthepresent Chapter wediscuss thegeneral theoryofanelectro- static field inwhich there areanynumber ofconductors. Thechargeon eachconductor willofcourse influence thedistribution ofchargesontheother conductors byinduction, andtheproblemistoinvestigatethedistributions ofelectricitywhich are tobeexpectedafterallowingforthismutual induction. Wehave seen that inanelectrostatic field thepotentialcannot bea maximum oraminimumexceptatpointswhere electricchargesoccur. It follows thatthehighest potentialinthefieldmust occur onaconductor, or elseatinfinity,thelatter caseoccurring onlywhen thepotentialofevery conductor isnegative. Excludingthiscase forthemoment, there must be oneconductor ofwhich thepotentialishigherthan thatanywhereelsein the field. Since lines offorcerunonlyfromhighertolowerpotential (§36), itfollows thatnolines offorce canenter thisconductor, therebeingno higher potentialfromwhichtheycancome, sothat lines offorcemust leave itatevery pointofitssurface. Inother words, itselectrification must be positiveatevery point. Soalso,except when thepotentialofeveryconductor ispositive,there must beoneconductor ofwhich thepotentialislower than thatanywhere elseinthe field,andtheelectrification atevery pointofthisconductor must benegative. Ifthetotalchargeonaconductor isnil,thetotalstrengthofthetubes offorce which enter itmust beexactly equaltothetotalstrengthofthe tubes which leave it.There must therefore beboth tubes which enter and tubes which leave itssurface, sothat itspotential must beintermediate between thehighest andlowestpotentialsinthe field. For ifitspotential were thehighestinthe field,notubes could enterit,and viceversa. On anysuchconductor theregionsofpositiveelectrification areseparatedfrom regionsofnegativeelectrification by"lines ofnoelectrification," these lines beinglocialongwhich a=0.Ingeneraltheresultantintensityatany 98,99] Systems ofConductors 89 pointofaconductor is4nrcr. Atanypointofaline ofnoelectrification, thisintensity vanishes, sothatevery pointofa"line ofnoelectrification" isalsoapointofequilibrium. Atapointofequilibriumwehavealreadyseen that theequipotential throughthepointcuts itself. Aline ofnoelectrification, however, lies entirelyonasingle equipotential,sothat thisequipotential must cutitself alongthelineofnoelectrification. Moreover, by§69,itmust cut itself at right angles, except when itconsists ofmore thantwosheets. 99.Wecanprovethetwofollowing propositions: I.Ifthepotential ofeveryconductor inthefieldisgiven,there isonly onedistribution ofelectric charges which willproducethis distributionof potential. II.Ifthetotal charge ofeveryconductor inthefieldisgiven,there is onlyonewayinwhich these charges candistribute themselves soastobein equilibrium. IfpropositionI.isnottrue, letussupposethatthere aretwodifferent distributions ofelectricitywhich willproducetherequired potentials.Let <rdenote thesurfacedensityatanypointinthe first distribution, andain thesecond. Consider animaginarydistribution ofelectricitysuch that the surfacedensityatanypointis<r—a.Thepotentialofthis distribution atanypointPis where theintegrationextends over thesurfaces ofalltheconductors, and risthedistance fromPtotheelement dS. IfPisapointonthesurface ofanyconductor, ff°dSand\\~dS arebyhypothesis equal,eachbeing equaltothegiven potentialofthe conductor onwhichPlies. Thus v-lfe** -!!'*>-*. sothatthesupposeddistribution ofdensity a—a'issuch thatthepotential vanishes over allthesurfaces oftheconductors. There cantherefore beno lines offorce, sothat there canbenocharges,i.e., cr—a'=everywhere,so thatthetwodistributions arethesame. Andagain,ifpropositionII.isnottrue, letussupposethat there are two different distributions a-andasuch that the total chargeoneach conductor hastheassignedvalue.Adistribution cr—cr'nowgiveszero asthe totalchargeoneach conductor. Itfollows, asin§98,that the 90 Systems ofConductors[ch.iv potentialofeveryconductor must beintermediate between thehighestand lowestpotentialsinthe field, aconclusion which isobviously absurd, as itprevents everyconductor fromhavingeither thehighestorthelowest potential.Itfollows thatthepotentialsofalltheconductors must beequal, sothatagainthere canbenolines offorceandnochargesatany point, i.e.,g=cr'everywhere. Itisclearfrom thisthatthedistribution ofelectricityinthefield isfully specifiedwhenweknow either (i)thetotalchargeoneach conductor, or(ii)thepotentialofeach conductor. SUPEEPOSITION OFEFFECTS. 100. Suppose wehavetwoequilibriumdistributions: (i)Adistribution ofwhich thesurfacedensityis o-atany point, givingtotalchargesElfE2,...onthedifferent conductors, andpotentials '1 >'2>•••• (ii)Adistribution ofsurfacedensity a,givingtotalchargesEx',E2,... andpotentials V/,V2,.... Consider adistribution ofsurfacedensity a+<r'.Clearlythe total chargesontheconductors willbeEj+E^,E2+E2,...,and ifVPisthe potentialatanypoint P, W/'-T^ where thenotation isthesame asbefore. IfPisonthe first conductor, however, weknow that // //-dS=Vltr -dS^V/,r sothatVp=Tf+VI' ;andsimilarly whenPisonanyother conductor. Thus theimaginarydistribution ofsurfacedensityisanequilibrium distribution, since itmakes thesurface ofeach conductor anequipotential,andthe potentialsare K+K, v+v/, .... The totalcharges,aswehave seen, areEx+E/,E2+E2,...,andfrom theproposition previously proved,itfollows thatthedistribution ofsurface- density<r+a'istheonlydistributioncorrespondingtothesecharges. Wehaveaccordinglyarrived atthefollowing proposition: IfchargesEltE2>...giverise topotentials K>V2>...,andifcharges 99-101] Superposition ofEffects 91 Ex,E2,...giverise topotentials Vx,V2',...,thenchargesEx+Ex,E2+E2,... willgiverise topotentials VX+Vx,V2+V2,.... Inwords: ifwesuperposetwosystemsofcharges,thepotentials produced canbeobtained byadding togetherthepotentials correspondingtothetwo component systems. Clearlythepropositioncanbeextended soastoapplytothesuperposition ofanynumber ofsystems. Wecanobviouslydeduce thefollowing: IfchargesEx,E2,...giverise topotentials Vx,V, ...,thenchargesKEX,KE2,...giverise topotentials KVX,KV2,.... 101.Suppose now thatwehave nconductors fixed inposition and uncharged. Letusrefer tothese conductors asconductor(1),conductor(2), etc.Supposethat theresult ofplacingunitcharge onconductor(1)and leavingtheothersunchargedistoproduce potentials ^11jPm••-Pin, onthenconductorsrespectively,then theresult ofplacingEXon(1)and havingtheothersunchargedistoproduce potentials P\\EX,p12E1}...pmEx. Similarly,ifplacingunitchargeon(2)andleavingtheothersuncharged gives potentials P2U P?2>•'•Pint thenplacingE2on(2)andleavingtheothersuncharged gives potentials p2XE2,p2iE 2,...pmE2. Inthesamewaywecancalculate theresult ofplacingE3on(3),Eton (4),andsoon. Ifwenowsuperposethesolutions wehave obtained, wefind that the effect ofsimultaneouschargesEx,E2,...Enistogive potentials VX,V2,...Vn, where Vx=pnEx+p 2XE2+p 31E3+...' V2=px2Ex+p 22E2+p 3iE3+...I(32). etc. Theseequations givethepotentialsinterms ofthecharges. The coefficients pxx,p2X,...donotdependoneither thepotentialsorcharges, being purely geometrical quantities,whichdependonthe size,shapeand positionofthedifferent conductors. 92 Systems ofConductors[ch.iv Green's Reciprocation Theorem. 102. LetussupposethatchargeseP,eQ,...onelements ofconducting surfaces atP,Q,...produce potentials VP,VQ,...atP,Q,...,and that similarly chargeseP',eQ',...produce potentials VP,VQ',....Then Green's Theorem states that Jll&pVp~—^*&p Vpy thesummationextendingineach caseover allthechargesinthe field. Toprovethetheorem, weneedonlynotice that Vp~ZPQ' thesummation extendingover allcharges excepteP,sothat in^ePVPthe coefficient of-p-^iseP'eQfrom theterm ePVP,and ePe <i'from theterm eQ'VQ.Thus ePeQ'+eQeP PQ^e'V-S2Fq =2ePTp',fromsymmetry. 103. Thefollowingtheorem follows atonce : Iftotal chargesEltE2ontheseparateconductors ofasystem produce potentials Tf,V2>...,andifcharges E-[,E2',...produce potentials Vx', V/, ...,then 2EV'=ZE'V (33), thesummationextendingineach caseover alltheconductors. Toseethetruth ofthis,weneedonlydivide upthechargesElyE2,... into smallchargeseP,eQ,...onthedifferent small elements ofthesurfaces oftheconductors, andthepropositionbecomes identical with thatjust proved. 104. Letusnowconsider thespecialcaseinwhich E,=l,Ea=Es=E4=...=Q, sothat ^=Pn,K=Pvnetc-5 and #/=0,Et'=l,E3'=EJ=...=0. sothat K'=Pn, V/=P-22>etc. Then'SEV=p 21andHE'V=p 12,sothatthetheoremjustprovedbecomes Pi2=Pa- inwords: thepotentialtowhich(1)israised byputtingunitchargeon (2),alltheother conductors being uncharged,isequaltothepotentialto which (2)israisedbyputtingunitchargeon(1),alltheother conductors being uncharged. 102-105] Coefficients ofPotential 93 Asaspecial case, letusreduce conductor(2)toapoint P,andsuppose thatthesystemcontains inadditiononlyoneother conductor(1).Then Thepotentialtowhich theconductor israisedbyplacing aunitcharge atP,theconductoritself being uncharged,isequaltothepotentialatPwhen unitchargeisplacedontheconductor. Forinstance, lettheconductor beasphere, and letthepointPbeata distance rfrom itscentre. Unitchargeonthesphere produces potential -atP,sothat unitchargeatPraises thespheretopotential-. Coefficients ofPotential, Capacity and Induction. 105. The relations p12=p2l,etc.reduce thenumber ofthecoefficients Pn,Pn>•••Pnn>which occur inequations (32),to%n(n+l).These coeffi- cients arecalled thecoefficients ofpotentialofthenconductors. Knowing thevalues ofthese coefficients, equations (31) givethepotentialsinterms ofthecharges. Ifweknow thepotentials V1}V2,...,wecanobtain thevalues ofthe charges bysolving equations (32).Weobtain asystemofequationsof theform ^=guK+g nK+—1 E*=quK+q22V2+ etc..(34). Thevalues oftheq'sobtained byactual solution oftheequations (32),are "A 7^22 7-'32 P'23P-33Pm Pn3 wherePinPzn• •Pnn#21Pzi P-2ZP33Pm Pn3 pinPan•••Pnn PnP21•••Pni P12P-22•••Pni•(35), PinPin••'Pnn Thusqrsistheco-factor ofprsinA,divided byA. Therelationqrs=q^ follows asanalgebraical consequenceoftherelation prs=psr ,orisatonce obvious from therelation 2EV'=2E'V, andequations (34),ontakingthesame setsofvalues asin§104. 94 Systems ofConductors[ch.iv There arencoefficients ofthetype qn,q22)...qnn-These areknown as coefficients ofcapacity.There are^n(n—1)coefficients ofthetype qrs,and these areknown ascoefficients ofinduction. Fromequations (34),itisclear thatquisthevalue ofExwhen Tf=l,Vi=Vi=...=0.This leads toanextended definition ofthe capacityofaconductor, inwhich account istaken oftheinfluence ofthe other conductors inthe field.Wedefine thecapacityoftheconductor 1, when inthepresenceofconductors 2,3,4,...,tobeqn,namely,thecharge requiredtoraise conductor 1tounitpotential,alltheother conductorsbeing puttoearth. Energy ofaSystem ofcharged Conductors. 106. Suppose werequiretofindtheenergyofasystemofconductors, theircharges beingEuE2)•••En,sothat theirpotentialsareVlf%,...Vn given byequations (32). LetWdenote theenergy when thechargesarekEltkE2,...JcEn. Correspondingtothesecharges,thepotentialswillbekV1}kV2,...kVn.It webring upanadditional smallcharge dk .Exfrominfinitytoconductor 1, thework tobedone willbedkEx.kV[;ifwebring updkE2toconductor 2 thework willbedkE2kV2andsoon.Letusnowbring charges dkExto1, dkE2to2,dkE3to3,...dkEnton.The totalwork done is kdk(E1Vl+E.zV2+...+E nVn) (36), andthefinalchargesare (k+dk)Ely(k+dk)E2,...(k+dk)En. Theenergyinthisstate isthesame function ofk+dkasWisofk,andmay therefore beexpressedas W+d -J^dk. dW Expression (36),theincrease inenergy,isthereforeequalto-^-rdk,whence d^=k(E lVl+E2V2+...+EnVu), sothatonintegration W=W(E&+E.2V2+...+EnVn). Noconstant ofintegrationisadded, sinceWmust vanish when k=0. Takingk=1,weobtain theenergy correspondingtothe finalcharges Ei,Ei}...En,intheform W=&EV,(37). 105-109] Energy 95 Ifwesubstitute fortheV'stheir values interms ofthechargesasgiven by equations (31),weobtain W=$(puEii+2puE1E2+p22E2*+...) (38), andsimilarlyfromequations (34), W=$(qnV?+2q 12VX+q*J?+...) ....(39). 107. IfWisexpressedasafunction oftheE's,weobtainbydiffer- entiation of(38), ^r=pnE1+p12E2+...+pmEn =Vi,byequation (32). This result isclearfrom other considerations. Ifweincrease thechargedWonconductor 1bydE1}theincrease ofenergyis^rrdEltand isalsoV^dE x since this istheworkdoneonbringing upanewchargedExtopotential~VX. Thus ondividing bydEx,weget wrv~ (40)- Soalso W=El (41) asisatonceobvious ondifferentiation of(39). 108. InchangingthechargesfromEX,E 2,...toE-f,E2',...letussuppose that thepotentials changefromVltV2,...toVi,V 2,.... Thework done,W—W,isgiven byW-W=\%(E'Y'-EV). Since, however, by§103,1EV=HE'V, thisexpressionforthework done caneither bewritten intheform \2[E'V-EV-(EV-E'V)}, which leads atonce to W-W=^(E'-E)(V'+ V) (42); orintheform |2{E'V-EV+(EV-E'V)}, which leads toW-W=%%(V- V)(E'+E) (43). 109. Ifthechangesinthechargesareonly small, wemayreplace E'byE+dE,andfindthatequation (42)reduces to dW=ZVdE, fromwhichequation (40)isobvious, whileequation (43)reduces to dW=$EdV, leadingatonce to(41). 96 Systems ofConductors [oh.iv 110. Itisworthnoticingthatthecoefficients ofpotential, capacity and induction canbeexpressedasdifferential coefficients oftheenergy ;thus _32TT Pr*~dE rdEa> andsoon. The lasttwoequations giveindependent proofsoftherelations Prs==Psr> Qrs~ Qsr* Properties oftheCoefficients. 111.Acertain number ofpropertiescanbededuced atoncefrom the factthattheenergymustalwaysbepositive.Forinstance since thevalue ofWgiven byequation (38)ispositiveforallvalues ofEuE2,...En,it follows atonce that Pn, P-2-2,p33,•••arepositive, thatpnp^-Puispositive,that PuPkPiz PviP^p-aispositive .P13.P23.P33 andsoon. Similarlyfromequation (39),itfollows that tfn. #22, £33,•••arepositive, andthere areother relations similar tothose above. 112. More valuableproperties can,however, beobtained from acon- sideration ofthedistribution ofthelines offorce inthe field. Letusfirstconsider thefieldwhen E1=l,E2=Ez=...=0. Thepotentialsare V1=p 11,V2=pn,etc. Since conductors 2,3,...areuncharged,theirpotentials must beinter- mediate between thehighest andlowestpotentialsinthe field. Thus the potentialof1must beeither thehighestorthelowest inthe field, theother extreme potential beingatinfinity.Itisimpossibleforthepotentialof1 tobethelowest inthe field;forifitwere, lines offorcewould enter inat every point,and itscharge would benegative. Thus thehighest potential inthe fieldmust bethat ofconductor1,andtheotherpotentials must all 110-114] Properties oftheCoefficients 97 beintermediate between thispotentialandthepotentialatinfinity, and must therefore allbepositive.Thuspn>p12,pa,...pmareallpositive and thefirstisthegreatest. Next letusput Vx=l,V2=V,= ...=0, sothatthechargesare qu,ql2,q^,•••<?m- Thehighest potentialinthe field isthatofconductor 1.Thus lines of force leave butdonotenter conductor 1.The linesmayeithergotothe other conductors ortoinfinity. Nolines canleave theother conductors. Thus thechargeon1mustbepositive,andthechargeson2,3,...allnegative, i.e.,qnispositiveandql2,q13,...areallnegative.Moreover thetotalstrength ofthetubesarrivingatinfinityisqn+qu+qi3+•••+5i»isothat thismust bepositive. 113. Tosumup,wehave seenthat (i)Allthecoefficients ofpotential (pn,pi 2,•••)arepositive, (ii)Allthecoefficients ofcapacity (qn,q^, ...)arepositive, (iii)Allthecoefficients ofinduction(q12,q13,...)arenegative, andwehave obtained therelations (pu—p12)ispositive, (qn+<?i2+ • •+,qm)ispositive. Inlimitingcases itisofcoursepossibleforanyofthequantities which havebeen described asalways positiveoralways negative,tovanish. Values oftheCoefficients inSpecial Cases. ElectricScreening. 114. The first case inwhich weshall consider thevalues ofthe coefficients isthat inwhich oneconductor, say 1,iscompletely surrounded byasecond conductor 2. Fig. 38. IfEz=0,theconductor 2becomes aclosed conductor withnocharge inside, sothat thepotentialinitsinterior isconstant, andtherefore K—K- PuttingE1= >therelation ~%=V2givestheequation (P12-P22)E2+(p13-p, 3)Es+ ...=0. J. 7 98 Systems ofConductors[en.iv Thisbeingtrue forallvalues ofE2,E3,...wemust have Pm=P22, ?Jis=J02s>etc. Next letusputunitcharge on1,leaving theother conductorsuncharged. Theenergyis%pn.Ifwejoin1and2byawire, theconductors 1and2 form asingle conductor, sothattheelectricitywill allflow totheouter surface. Thiswiremaynowberemoved, andtheenergyinthesystemis\p^. Energy must, however, havebeen lostintheflow ofelectricity,sothatp22 must belessthanpn. Sincewehavealreadyseen thatpVi=p 22andpn—p12cannot benegative, itisclear thatpwcannot begreaterthanpnTheforegoing argument, however, goesfurther andenables ustoprove thatpn—p.^isactually positive. Letusnextsupposethatconductor 2isputtoearth, sothatV2=0. Then ifE1=0,itfollows thatT[=0. Hence from theequations E1=q11V1+qliV%+...+qmVn (44) weobtain inthisspecialcasethat qi3Vi+quV±+•••+qmV a=o. This istrue,whatever thevalues ofVs,Vi}....sothat Supposethatconductor 1israised tounitpotential while alltheother conductors areputtoearth. Theaggregate strengthofthetubes offorce which gotoinfinity, namely qn+q12+...+qm(§112),isinthiscase zero, so thatql2=-qn- Thesystemofequations (44)nowreduces, whenV2=0,to E1=qllV1 (45), E2=ql2Vi+q^V i+q.2iVi+(46), Es=qS3Vi+q~iVi+... Ei=qZiVi+qiiVi+...(47) Equations (47)shew that therelations betweenchargesandpotential outside 2arequite independentofthe electrical conditions which obtain inside 2.Soalsotheconditions inside 2arenotaffected bythose outside 2, asisobvious fromequation (45). These results become obvious whenwe consider thatnolines offorce cancross conductor 2,andthatthere isnoway except bycrossingconductor 2foralineofforce topassfrom theconductors outside 2tothose inside 2. Anelectric system which iscompletely surrounded byaconductor at potentialzero issaid tobe" electricallyscreened"from allelectricsystems 114,115] Coefficients forSpherical Condenser 99 outside thisconductor;forchargesoutside this"screen"cannot affect the screened system.Theprincipleofelectricscreeningisutilised inelectro- static instruments, inorder that theinstrument maynotbeaffected by external electric actions other than those which itisrequiredtoobserve. As acompleteconductor wouldpreventobservation oftheworkingofthe instrument, acageofwire isfrequentlyused asascreen, thisbeing very nearlyasefficient asacompletelyclosed conductor(see §72). Inmore delicate instruments thescreening maybecomplete exceptforasmall window toadmit ofobservation ofthe interior. SphericalCondenser. 115. Letusapplythemethods ofthisChaptertothesphericalcon- denser described in§79.Lettheinnersphereofradius abetaken tobe conductor 1,andtheoutersphereofradius bbetaken tobeconductor 2. Theequations connecting potentials andchargesare V1=pllE1+p 21E2, V,=p 12E1+p 22E2. Aunitcharge placedon2raises both 1and 2topotential 1/6,sothaton puttingE1=0,E2=1,wemust haveV1=V2=1/6.Hence itfollows that 1 Pi\P& 7• Ifweleave 2unchargedandplaceunitchargeon1,thefield offorce isthat investigatedin§79,sothat V[=1/a,V2=l/b.Hence 1 1 Pn= af Pl*= b' These results exemplify (i)thegeneralrelation p12=p.21, (ii)therelationpeculiartoelectricscreening, pi2=p, 2. Theequationsnowbecome Vl~ a b' v*b+ b SolvingforExandE2interms ofVxandV2,weobtain a6_ ab_ b—a b—a b—a b—a L. ab ab h-sothat?ii=^.<Z"=^=-F3- a>*-TrS' 7—2 100 Systems ofConductors[ch.iv Wenotice thatq12=q21,that thevalue ofeach isnegative,andthat qu=—qn,inaccordance with§113. The value ofqnisthecapacityof sphere1when 2istoearth, and isinagreementwith theresult of§79. b2 Thecapacityof2when 1istoearth, q22,isseen tobe ^.This can alsobeseenbyregardingthesystemascomposedoftwocondensers, the innersphereandtheinner surface oftheoutersphereform asingle spherical condenser ofcapacityj,while theouter surface oftheouterspherehas capacityb.The totalcapacity accordingly ab . b2 +b b—a b—a' Twospheresatagreatdistanceapart. 116. Suppose wehavetwospheres,radii a,b,placedwith their centres atagreatdistance capart. Letusfirstplaceunitchargeontheformer, the Fig. 39. charge being placedsothatthesurfacedensityisconstant. This willnot produceuniformpotentialover 2 ;atapointdistant rfrom thecentre of1 itwillproduce potential 1/r.Wecan,however, adjustthispotentialtothe uniform value1/cbyplacingonthesurface of2adistribution ofelectricity such that itproducesapotentialover thissurface. Take B,thecentre ofthesecondsphere,asorigin, andABasaxisofx. Thenwemaywrite 11r—ex c 1= =— .asiaras— . cr cr c2c2 Let crbethesurfacedensity requiredtoproducethispotential, then clearlyaisanoddfunction ofx,andtherefore thetotalcharge,thevalue of aintegratedover thesphere,vanishes. Thus thepotentialof2canbe adjustedtotheuniform value 1/cwithoutalteringthetotalcharge on2 from zero, neglecting 1/c3 .Thenew surfacedensity beingoftheorder of 1/c2 ,theadditionalpotential producedon1byitwillbeatmost oforder1/c3 , sothat ifweneglect 1/c3wehave found anequilibrium arrangement which makes ^=1,L\=0,V^1 -,K=l. & c 115-117] Coefficients fortwodistantSpheres 101 Substitutingthese values intheequations V2=p 12E1+p 22E2, wefindatonce 102 Systems ofConductors [ch.iv givingtheratio inwhich thechargeEwill distribute itself between the twoconductors 1and 2.Iftheconductors 3,4,...areeither absent or uncharged, E2 apu-pn' which isindependentofEandalways positive.Itistobenoticed thatEl vanishesonlyifp&^piz, i.e.,if2entirelysurrounds 1. Mechanical Forces onConductors. 118.Wehavealreadyseen that themechanical forceonaconductor is theresultant ofasystemoftensions over itssurface ofamount 27rcr2perunit area. The results ofthepresent Chapterenable ustofindtheresultant forceonanyconductor interms oftheelectrical coefficients ofthesystem. Supposethat thepositionsoftheconductors arespecified byanyco- ordinates£i,f2>•••>sothatpn,pw,•••,<7u> qvt, •••>andconsequentlyalsoW, arefunctions ofthef's.If^isincreased to£x+d%uwithout thechargeson dW theconductorsbeing altered, theincrease inelectrical energyis-^rd% x,and thisincrease mustrepresentmechanical workdone inmovingtheconductors. The forcetendingtoincrease £isaccordingly %' Since thechargesontheconductors aretobekept constant, itwill of course bemost convenient tousetheform ofWgiven byequation (38),and theforce isobtained intheform -t(^Ef+2&*BlE,+ ...)(48). Itishoweverpossible, byjoiningtheconductors totheterminals of electric batteries, tokeeptheirpotentialsconstant. Inthis case, however, wemust notusetheexpression (39)forW,andsoobtain fortheforce -i{w/'+*WJ>V'+~)(49) ' forthebatteries arenowcapableofsupplying energy,andanincrease of electrical energydoesnotnecessarily mean anequal expenditureofmechanical energy,forwemust notneglectthework donebythebatteries. Since the resultant mechanical forceonanyconductor mayberegardedastheresultant oftensions 27ro-2perunit areaactingover itssurface, itisclear that this resultant force inanyposition depends solelyonthechargesinthisposition. Itistherefore thesamewhether thechargesorpotentialsarekept constant, andexpression (48)willgivethis forcewhether theconductors areconnected tobatteries ornot. 117-120]Mechanical forces 103 119. Asanillustration, wemayconsider the force between thetwo charged spheresdiscussed in§116. dWThe forcetendingtoincrease c,namely—=— ,is _i(dpnF2,93P™FP,dpa™A 2l~e7Al+2~a7^2+*^2J' andsubstitutingthevalues 1 .1 pn=~+terms in- ,a c3 1 Pl2=-+ c 104 Systems ofConductors [en.iv andsince thisequationistrue foralldisplacementsandtherefore forall values of§£,Sf2>...»itfollows thateach coefficient must vanishseparately. Thus||=0,or dWedWr_( dWAswehave seen,—^isthemechanical forcetendingtoincrease f1( dW and thishasnowbeenshewn tobeequalto-^,which isexpression (49) with thesignreversed. Thus themechanical force, whether thechargesor thepotentialsarekept constant, is »(§g*+»f|MSK+ ...)(52), aformwhich isconvenient whenweknow thepotentials,butnotthe charges,ofthesystem. Inmakingasmalldisplacementofthesystemsuch thatfxischanged dW intofj+d^i,themechanical work done is-^d% x.Ifthepotentialsare keptconstant theincrease inelectricalenergyis-~d%x.The difference of theseexpressions, namely fiWy d_K\ represents energy supplied bythebatteries. Fromequation (51),itappears dWrthat thisexpressionisequalto2-^ifd%nsothatthebatteriessupply energy equaltotwice theincrease intheelectrical energyofthesystem,andofthis energyhalfgoestoanincrease ofthefinal electricalenergy,while half is expendedasmechanical work inthemotion oftheconductors. Introduction ofanewconductor into thefield. 121.When anewconductor isintroduced intothe field, thecoefficients Pn>Pn> •••» tfn, <?i2>•••arenaturallyaltered. Letussupposethenewconductor introduced ininfinitesimalpieces, which arebroughtintothefieldunchargedandplacedinpositionsothat theyareineverywayintheir finalplaces exceptthat electric communication isnotestablished between thedifferentpieces.Sofarnowork hasbeen doneandtheelectrical energyofthe fieldremains unaltered. Now letelectric communication beestablished between thedifferent pieces,sothatthewhole structure becomes asingleconductor. Theseparate 120-122] TheAttracted Disc Electrometer 105 pieces, originallyatdifferentpotentials,arenowbroughttothesame potential bytheflow ofelectricityover the surface oftheconductor. Electricitycanonlyflowfromplacesofhighertoplacesoflowerpotential, sothat electrical energyislostinthis flow. Thus theintroduction ofthe newconductor hasdiminished theelectricenergyofthe field. Ifwenowputthenewconductor toearth there isingeneralafurther flow ofelectricity,sothattheenergyisstillfurther diminished. Thus theelectric energyofanyfield isdiminished bytheintroduction of anewconductor, whether insulated ornot. Consider thecase inwhich thenewconductor remains insulated. Let theenergyofthefield before theintroduction ofthenewconductor be $(pnE1i+2p 12E1E2+...+p nnEn>).... (53). After introduction, theenergy maybetaken tobe h(PiiW +2pi*EiE 3+...+ Pnn'Er?) (54), wherepn',etc.,arethenew coefficients ofpotential.Further coefficients of thetype Pi,n+i,p-2,n+i> "->Pn+i,n+iareofcoursebroughtinto existence, butdo notenter intotheexpressionfortheenergy,sincebyhypothesis En+1=0. Since expression (54)islessthanexpression (53),itfollows that (Pn-Pn)Ef+2(p12-pa')E1E2+... ispositiveforallvalues ofEltE2, Hence pa—pnispositive, andother relations maybeobtained, asin§111. Electrometers. I.TheAttracted Disc Electrometer. Fig. 40. 122. Thisinstrument is,asregardsitsessentialprinciple,abalance in which thebeam hasaweightfixed atoneendandadiscsuspendedfrom theother. Under normal conditions thefixedweightissufficiently heavy 106 Systems ofConductors[ch.iv tooutweighthe disc. Inusingtheinstrument thedisc ismade tobecome oneplateofaparallel plate condenser, ofwhich thesecondplateisadjusted until theelectric attraction between thetwoplatesofthecondenser isjust sufficient torestore thebalance. Theinequalitiesinthedistribution ofthelines offorce which would otherwise occur attheedgesofthediscareavoided bytheuseofaguard- ring (§90),soarrangedthatwhen thebeam ofthebalance ishorizontal theguard-ringand disc areexactlyinoneplane, and fitascloselyasis practicable. Letussupposethatthedisc isofareaAandthatthediscandguard- ringareraised topotentialV.Letthesecondplateofthecondenser be placed paralleltothediscatadistance hfromit,andputtoearth. Then theintensity between thediscandlowerplateisuniform andequaltoVfk, sothatthesurfacedensityonthelower faceofthedisc isa=Vj^irh. The mechanical forceactingonthedisc istherefore aforce lira^A orV2A/87rh* acting vertically downwards throughthecentre ofthe disc. Ifthisjust suffices tokeepthebeam horizontal, itmust beexactly equaltotheweight, sayW,which would have tobeplacedonthisdisctomaintainequilibrium ifitwereuncharged.Thisweightisaconstant oftheinstrument, sothat theequation V-A 8tt/i2 enables ustodetermine Vinterms ofknownquantities byobservingh. Theinstrument isarrangedsothat thelowerplatecanbemovedparallel toitself byamicrometer screw, thereadingofwhichgiveshwithgreat accuracy. Wecanaccordinglydetermine Vinabsolute units, from the equation A' Ifwewish todetermine adifference ofpotential wecanraise theupper platetoonepotential Vl;andthelowerplatetothesecondpotential V2, andwethen have f8TrW A Amore accurate method ofdeterminingadifference ofpotentialistokeep thediscataconstantpotential v,and raise thelowerplate successivelyto potentials VxandV2.Ifh^andh2arethevalues ofhwhichbringthediscto itsstandardpositionwhen thepotentialsofthelowerplateareVxandJ£,we haveV1-Vi=hJi v-Vx=hlsJ8ttW A' /HitW 122,123] TheQuadrant Electrometer 107 sothatV^V-<Jh-K*J8ttW Itisnowonlynecessarytomeasure h^—h2,thedistancethrough which thelowerplateismoved forward, and thiscanbedetermined withgreat accuracy,asitdepends solelyonthemotion ofthemicrometer screw. II.TheQuadrant Electrometer. 123. MeasurementofPotentialDifference.This instrument ismore delicate than thediscelectrometer just described, butenables usonlyto compare twopotentials,orpotentialdiffer- ences; wecannot measure asingle potential interms ofknown units. Theprincipal partoftheinstrument consists ofametalcylinderofheightsmall comparedwith itsradius, divided into four quadrants A,B,C,Dbytwodiameters at right angles.Thesequadrantsareinsulated separately,andthenopposite quadrants areconnected inpairs,twobywiresjoined toapointEandtwobywiresjoinedto some otherpointF. Theinside ofthecylinderishollow and inside thisametal discor"needle" isfree tomove, being suspended byadelicate fibre, sothat itcanrotate withouttouching thequadrants.Beforeusingtheinstrument theneedle ischargedtoahigh potential, say v,either bymeans ofthe fibre,ifthis isaconductor, orbyasmallconducting wirehangingfrom theneedle whichpasses throughthebottom ofthe cylinder. The fibre isadjustedsothatwhen thequadrantsareatthesame potentialtheneedle rests, asshewn inthefigure,inasymmetrical position withrespecttothequadrants.Inthis state either surface oftheneedle andtheoppositefaces ofthequadrants mayberegardedasformingaparallel platecondenser. If,however, thepotentialofthetwoquadrants joinedtoEisdifferent from that ofthetwoquadrants joinedtoF,there isanelectrical force tendingtodragtheneedle under thatpairofquadrantsofwhich thepotential ismorenearly equaltov.Theneedleaccordingly moves inthis direction until theelectric forces areinequilibriumwith thetorsion ofthe fibre,and anobservation oftheangle throughwhich theneedle turns willgiveanFig. 41. 108 Systems ofConductors [ch.iv indication ofthedifference ofpotentialbetween thetwopairsofquadrants. Thisangleismosteasilyobserved byattachingasmall mirror tothefibre justabove thepointatwhich itemergesfrom thequadrants. Letussupposethatwhen theneedle hasturnedthroughanangle 6, thetotal areaAoftheneedle isplacedsothatanareaSisinside thepair ofquadrantsatpotential K,andanareaA—Sinside thepairatpotential V>.Lethbetheperpendiculardistance from either face oftheneedle to the faces ofthequadrants.Then thesystem mayberegardedastwo parallel platecondensers ofarea S,distance h,and difference ofpotential v—V[,andtwoparallel platecondensers forwhich thesequantitieshave the valuesA—S,h,v—Vz.There aretwocondensers ofeach kind because there aretwo faces, upperandlower, totheneedle. The electricalenergy ofthissystemisaccordingly (v-vys (v-vy(A-S) 4>Trh 4nrh Theenergyhereappearsasaquadraticfunction ofthethreepotentials concerned: itisexpressedinthesame form astheWrof§120. The mechanical forcetendingtoincrease 6,i.e.,themoment ofthecouple tending toturn theneedle inthedirection of6increasing,istherefore-^-.Now inWvtheonlyterm inthecoefficients ofthepotentialswhich varies with 6 isS,sothatondifferentiation weobtain Wy_(v-V,f-(v-TQ2 d_S dd~ 4ttA d0° Ifristheradius oftheneedle—measured from itscentre, which isunder thelineofdivision ofthequadrants—weclearly haver^=r2 ,sothatwecan write theequation justobtained intheform 9TTr(2t>-K-K)G?-K) d9" 4ttAr\ Inequilibriumthiscoupleisbalancedbythetorsioncoupleofthe fibre, which tends todecrease 6.Thiscouple maybetaken tobek6,where kisa constant, sothattheequationofequilibriumis W4^h{°b)- Forsmalldisplacementsoftheneedle, r2maybereplaced bya2 ,the radius oftheneedle atitscentre line. Also visgenerally large compared withKandV2.The lastequation accordingly assumes thesimplerform 123,124] TheQuadrant Electrometer 109 shewingthat 6is,forsmalldisplacementsoftheneedle, approximately proportionaltothedifference ofpotentialofthetwopairsofquadrants. Theinstrument canbemadeextraordinarilysensitive owingtothepossibility ofobtaining quartz-fibresforwhich thevalue ofkisverysmall. Ifthedifference ofpotentialtobemeasured islarge,wemaychargethe needlesimply byjoiningittooneofthepairsofquadrants, saythepairat potential J£.Wethenhavev=V,,andequation (55)becomes kff~ 4tt/*' sothat isnowproportionaltothesquareofthepotentialdifference tobe measured. a2 Writing „— j-%=C,sothatGisaconstant oftheinstrument, wehave, when vislarge e=Cv{V l-Vi) (56), when v=V2, e^iciv.-vy (57). 124. Measurementofcharge. Letusspeakofthepairsofquadrants atpotentials Yx,V2asconductors 1,2respectively,and lettheneedle be conductor 3.When thequadrantsare toearth and theneedle isat potential T^,thechargeEinduced onthe firstpairofquadrants bythe chargeontheneedle willbegiven by whereq13isthecoefficient ofinduction. This coefficient isafunction ofthe angle6which defined thepositionoftheneedle. Iftheinstrument is adjustedsothat=when bothpairsofquadrantsaretoearth, wemust usethevalue ofq13correspondingto6=0,say{ql3\,sothat E=(qiz\V 3 (58). Nowsupposethatthe firstpairofquadrantsisinsulated andreceives anadditionalcharge Q,thesecondpairbeingstill toearth. Lettheneedle bedeflected throughanangle6inconsequence.Since thechargeonthe firstpairofquadrantsisnowE+Q,wehave E+Q=(qn)eV 1+(q13)eV3. Onsubtracting equation (58)from thisweobtain If6issmall thismaybewritten 110 Systems ofConductors[ch.iv whereqn,-^aresupposedcalculated for6=0.SinceV2=0,wehave from equation (56),=OV%Ylt sothat <3=(^+^3 ^)^' shewingthat forsmall values of0,Qisdirectly proportionalto0. Letussupposethatwejointhe firstpairofquadrants (conductor 1) toacondenser ofknown capacity Twhich isentirelyoutside theelectro- meter. Since theneedle(3)isentirely screened bythequadrantsthevalue ofq13remains unaltered, whileqnwillbecomequ+V.If0'isnow the deflection oftheneedle, wehave 'qn+Tdqv*-{*&+%*)' sothat,bycombination with thelastequation, wehave If0"isthedeflection obtained byjoiningthepairsofquadrantstothe terminals ofabatteryofknownpotentialdifference D,wehave from equation (56), CVy.D' andonsubstitutingthisvalue forGV3,ourequation becomes Q= 0"0"' l'~~0 giving Qinterms oftheknownquantities V,Dandthethreereadings 0,0'and 0". Anordinary quadrantelectrometer willmeasure differences ofpotential down toaboutj^Welectrostatic units. Thus inspiteofitssomewhathigh capacityofabout 50electrostatic units,itforms anextremelyefficient instru- ment forthemeasurement ordetection ofsmall electriccharges. Animprovedform oftheinstrument hasrecentlybeen introduced by Dolazalek, inwhich theelectrostaticcapacityisverysmall. This iscapable ofmeasuring potentialdifferences down totoo'oooelectrostatic units, and is correspondinglymore sensitive forthemeasurement ofcharges. 124] Examples 111 EXAMPLES. 1.Ifthealgebraic sum ofthecharges onasystem ofconductors bepositive, thenon oneatleast thesurface densityiseverywhere positive. 2.There areanumber ofinsulated conductors ingivenfixedpositions. The capacitiesofanytwoofthem intheir given positions areC\andC2,and their mutual coefficient ofinduction isB.Prove that ifthese conductors bejoined byathin wire, the capacityofthecombined conductor is Ci+Ca+25. 3.Asystemofinsulated conductors having beenchargedinanymanner, charges are transferred from oneconductor toanother tilltheyareallbroughttothesamepotentialV. Shew that V=Eft*+2s 2), where«i,s2arethealgebraic sums ofthecoefficients ofcapacity andinductionrespectively, andEisthesum ofthecharges. 4.Prove thattheeffect oftheoperationdescribed inthelastquestionisadecrease oftheelectrostatic energy equaltowhatwould betheenergyofthesystemifeach ofthe original potentials werediminished byV. 5.Two equal similar condensers, each consistingoftwospherical shells, radiia,b, areinsulated andplacedatagreat distance rapart. Charges e,e'aregiventotheinner shells. Iftheouter surfaces arenowjoined byawire,shew that thelossofenergyis approximately 6.Acondenser isformed oftwothinconcentricspherical shells, radii a,b.Asmall hole exists intheouter sheet through which aninsulated wire passes connectingthe inner sheet with athird conductor ofcapacity c,atagreat distance rfrom thecondenser. Theouter sheet ofthecondenser isputtoearth, andthecharge onthetwoconnected conductors isE.Prove thatapproximatelytheforceonthethird conductor is "Ai-"Yr». 7.Two closed equipotentials VlfVQaresuch that Vicontains V,andVPisthe potentialatanypointPbetween them. IfnowachargeEbeputatP,andboth equipotentialsbereplaced byconductingshells andearth-connected, then thecharges Ei,Einduced onthetwosurfaces aregiven by Ey Eq E 8.Aconductor ischarged fromanelectrophorus byrepeated contacts with aplate, which after each contact isrecharged with aquantity Eofelectricity from theelectro- phorus.Prove that ifeisthechargeoftheconductor after the firstoperation,the ultimate chargeis Ee E-e 112Systems ofConductors [ch.iv 9.Four equal unchargedinsulated conductors areplaced symmetricallyatthecorners ofaregular tetrahedron, andaretouched inturnbyamoving spherical conductor atthe points nearest tothecentre ofthetetrahedron, receiving chargeseue2,e3,et.Shew that thecharges areingeometrical progression. 10.Inquestion 9replace"tetrahedron"by"square," andprove that Oi-*>) (e^3-e22 )=ex(e2e3-e^). 11.Shew that ifthedistance xbetween twoconductors issogreatascompared with thelinear dimensions ofeither, thatthesquareoftheratio ofthese linear dimensions to xmaybeneglected,then thecoefficient ofinduction between them is-CO'lx, where C,C arethecapacitiesoftheconductors when isolated. 12.Two insulated fixed condensers areatgiven potentials when alone intheelectric fieldandcharged with quantities EltE2ofelectricity. Their coefficients ofpotentialare pn,pn,P22-But iftheyaresurrounded byaspherical conductor ofvery large radiusR atpotentialzerowith itscentre nearthem, thetwoconductorsrequire charges E{,EJto producethegiven potentials. Prove, neglecting -^ ,that E2'-E2Pn-pu' 13.Shew thatthelocus ofthepositions, inwhich aunitcharge willinduce agiven charge onagivenuninsulated conductor, isanequipotential surface ofthatconductor supposed freelyelectrified. 14.Prove(i)that ifaconductor, insulated infreespace andraised tounitpotential, produceatanyexternal pointPapotential denoted by(P),then aunitcharge placedat Pinthepresenceofthisconductor uninsulated willinduce onitacharge—(P); (ii)that ifthepotential atapointQduetotheinduced charge bedenoted by(PQ), then(PQ)isasymmetricalfunction ofthepositionsofPand Q. 15.Two small uninsulated spheres areplaced near together between twolarge parallel planes, oneofwhich ischarged, andtheother connected toearth. Shew by figuresthenature ofthedisturbance soproducedintheuniformfield,when thelineof centres is(i)perpendicular, (ii)paralleltotheplanes. 16.Ahollow conductor Aisatzeropotential, andcontains initscavity twoother insulated conductors, Band C,which aremutually external :Bhasapositive charge, and Cisuncharged. Analyse thedifferenttypesoflines offorce within thecavity which are possible, classifying with respecttotheconductor from which thelinestarts, andthe conductor atwhich itends,andprovingtheimpossibilityofthegeometrically possible types which arerejected. Hence provethatBandCareatpositive potentials, thepotentialofGbeing lessthan thatofB. 17.AportionPofaconductor, thecapacityofwhich isC,canbeseparated from the conductor. Thecapacityofthisportion, when atalong distance from otherbodies,isc. Theconductor isinsulated, andthepartPwhen ataconsiderable distance from the remainder ischarged with aquantityeandallowed tomove under themutual attraction uptoit;describe andexplain thechanges which takeplaceintheelectricalenergyofthe system. Examples 113 18.Aconductor having acharge $1issurrounded byasecond conductor withcharge Q2.Theinner isconnected byawire toavery distant uncharged conductor. Itisthen disconnected, andtheouter conductor connected. Shew thatthecharges Qi,Q2,arenow m+n+mn'' m+n' where G,C(l+m) arethecoefficients ofcapacityofthenearconductors, andCn isthe capacityofthedistant one. 19. Ifoneconductor contains alltheothers, andthere aren+l inall,shew that there aren+1relations between either thecoefficients ofpotential orthecoefficients of induction, and ifthepotentialofthelargest beV,andthat oftheothers VuV2,...V„, thenthemost general expressionfortheenergyis^GV2increased byaquadratic function ofVt-V,V2—V,...Vn—V;where Cisadefinite constant forallpositionsofthe inner conductors. 20.Theinner sphereofaspherical condenser(radii a,b)hasaconstant charge E, andtheouter conductor isatpotentialzero. Under theinternal forces theouter conductor contracts from radius btoradius by.Prove that thework done bythe electric forces is 21.If,inthelastquestion,theinner conductor hasaconstantpotential V,itscharge being variable, shew thattheworkdone is *(bi-a)(b-a)' andinvestigate thequantityofenergy supplied bythebattery. 22.With theusualnotation, prove that Pn+P23>Pl2+Pl3 PllP23>Pl2Pl3- 23.Shew that ifp„.,pr3,pssbethree coefficients before theintroduction ofanew conductor, andp„\pra',pa8'thesame coefficients afterwards, then (PrrPsa-Prr'pss) <t(Pn~Pn?- 24.Asystemconsists ofp+q+2conductors, AltA2,...Ap,BuB2,...Bq,C,D.Prove thatwhen thecharges ontheA'sandonC,andthepotentialsoftheB'sandofCare known, there cannot bemore than onepossibledistribution inequilibrium,unlessCis electricallyscreened from D. 25.A,B,C,Darefourconductors, ofwhichBsurrounds AandDsurrounds C. Given thecoefficients ofcapacity andinduction (i)ofAandBwhenCandDareremoved, (ii)ofCandDwhenAandBareremoved, (iii)ofBandDwhenAandGareremoved, determine those forthecomplete systemoffourconductors. 26.Twoequal andsimilar conductors AandBarecharged andplaced symmetrically withregardtoeach other;athird moveable conductor Cisearned soastooccupy j.• 8 114 Systems ofConductors [oh.iv successively twopositions,onepractically whollywithin A,theother within B,the positions being similar andsuch that thecoefficients ofpotential ofCineitherposition arep,q,rinascendingorder ofmagnitude. Ineachposition Cisinturnconnected with theconductor surrounding it,puttoearth, andthen insulated. Determine thecharges ontheconductors after anynumber ofcyclesofsuchoperations,andshew thatthey ultimatelylead totheratios l:-0:/9»-l, where /3isthepositiverootof rx2—qx+p-r=0. 27.Two conductors areofcapacities GxandC2,when each isalone inthe field. Theyareboth inthe field atpotentials VxandF2respectively,atagreat distance r apart. Prove that therepulsion between theconductors is Ci^(rr 1-c2r2)(rr 2-c1r1) Asfaraswhatpowerof-isthisresult accurate ? 28.Twoequal andsimilar insulated conductors areplaced symmetricallywith regard toeach other, oneofthem being uncharged. Another insulated conductor ismade to touch themalternatelyinasymmetrical manner, beginning with theonewhich hasa charge.Ifelte2betheircharges when ithastouched eachonce,shew that theircharges, when ithastouched each rtimes, arerespectively ex2 r to_«.\2r-l-J .2 2ej-e2m^tt- «&{'-<??;} 29.Three conductors Ai}A2andAsaresuch thatA3ispracticallyinside A%.Axis alternately connected withA2andA3bymeans ofafinewire, the firstcontact being with A3.AihasachargeEinitially, A2andA3being uncharged. Prove thatthecharge on Axafter ithasbeenconnected ntimes withA2is M. fi ,°(y-/3) /"-H3Yt-1 ) a+/S\ ^I3(a+y)\a+yj J» wherea,ftystand forpn-pi 2,P22—P12 andJO33-p\ 2respectively. 30.Twospheres,radiia,b,have their centres atadistance capart. Shew that neglecting (a/c)6and(6/c)6 ,lb31la3 *u=£~?;fn=VP22=b~1 CHAPTER Y DIELECTRICS ANDINDUCTIVE CAPACITY 125.Mention hasalreadybeenmade(§84)ofthe fact, discovered originally byCavendish, andafterwards rediscovered byFaraday,that the capacityofaconductordependsonthenature ofthedielectric substance between itsplates. Letusimaginethatwehavetwoparallel plate condensers, similar inall respects exceptthatonehasnothing butairbetween itsplateswhile inthe other thisspaceisfilled with adielectric ofinductivecapacity K.Letus supposethatthetwohigh-potential platesareconnected byawire,and also thetwolow-potential plates.Letthecondensers becharged,thepotential ofthehigh-potential plates being Tf,andthat ofthelow-potential plates beingV . Then itisfound thatthecharges possessed bythetwocondensers arenot equal.Thecapacity perunitarea oftheair-condenser islj&ird;that ofthe other condenser isfound tobeKj^ird. Hence thecharges perunit area ofthetwocondensers arerespectively ^andKVl~ 4nrd 4,-rrd Theworkdone intakingunitchargefromthe low-potential platetothehigh-potential plateis thesame ineither condenser, namely T^— To,so that theintensity between theplatesineither condenser isthesame, namely d' Intheair-condenser thisintensity mayberegardedastheresultant ofthe attraction ofthenegatively charged plateandtherepulsionofthepositively charged plate,thelawofattraction orrepulsion beingCoulomb's law— .Fig. 42. 8—2 116 Dielectrics andInductive Capacity [ch.v Itis,however, obvious that ifwewere tocalculate theintensityinthe second condenser from this law,then thevalue obtained would beKtimes V-V that inthe first condenser, andwould therefore beK1 ,° .Inpointof V—V fact,theactual value oftheintensityisknown tobe* ,° . ThusFaraday's discoveryshews thatCoulomb's lawofforce isnotof universalvalidity:thelawhasonlybeenproved experimentallyforair,and itisnowfound nottobetrue fordielectrics ofwhich theinductivecapacity isdifferent fromunity. Thisdiscoveryhasfar-reachingeffects onthedevelopmentofthemathe- maticaltheoryofelectricity.Inthepresent book, Coulomb's lawwas introduced in§88,andformed thebasis ofallsubsequent investigations. Thuseverytheorem which hasbeenprovedinthepresentbook from§38 onwardsrequiresreconsideration. 126.We shall followFaradayintreatingthewholesubjectfrom the pointofview oflines offorce. Theconceptionsofpotential,ofintensity,and oflines offorce areentirely independentofCoulomb's law,andinthepresent book havebeen discussed(§§30—37)before thelawwasintroduced. The conceptionofatube offorce follows atonce from that ofalineofforce, onimagininglines offorcedrawnthroughthedifferentpointsonasmall closed curve. Letusextend todielectrics oneform ofthedefinition ofthe strengthofatube offorcewhich hasalreadybeen used foratube inair,and agreethatthestrengthofatube istobemeasuredbythechargeenclosed byitspositive end,whether inairordielectric. Inthedielectric condenser, thesurfacedensityonthepositive plateis V-VK——-^,and this,bydefinition, isalso theaggregate strengthofthe tubesperunit area ofcross-section. Theintensityinthe dielectric is V—V— j—- ,sothat inthedielectric theintensityisnolonger,asinair,equal to4-7Ttimes theaggregate strengthoftubesperunit area,but isequalto 4nrJKtimes thisamount. Thus ifPistheaggregate strengthofthetubesperunit area ofcross- section, theintensity Risrelated toPbytheequation £=XP(59) inthedielectric, instead ofbytheequation P=4ttP(60) which wasfound tohold inair. 125-128J Experimental Basis 117 127. Equation (59)hasbeenprovedtobetheappropriate generalisation ofequation (60)onlyinavery specialcase.Faraday, however, believed the relationexpressed byequation (59) tobeuniversally true,andtheresults obtained onthissuppositionarefound tobeincomplete agreement with experiment. Henceequation (59),orsomeequationofthesamesignificance, isuniversallytaken asthebasis ofthemathematicaltheoryofdielectrics. Weaccordingly proceed byassumingtheuniversal truth ofequation (59), anassumptionforwhich ajustificationwillbefoundwhenwecome tostudy themolecular constitution ofdielectrics. Itisconvenient tohave asingleword toexpresstheaggregate strength oftubesperunitarea ofcross-section, thequantity which hasbeendenoted byP.Weshall speakofthisquantityasthe" polarisation,"atermdueto; Faraday.Maxwell'sexplanationofthemeaningoftheterm" polarisation" isthat"anelementary portionofabodymaybesaid tobepolarised when itacquires equalandopposite propertiesontwooppositesides."Faraday explainedthepropertiesofdielectrics bymeans ofhisconceptionthatthe molecules ofthedielectric were inapolarised state, andthequantity P isfound tomeasure theamount ofthepolarisationatanypointinthe dielectric. Weshallcome tothisphysical interpretationofthequantity P atalaterstage:forthepresent wesimplyusetheterm" polarisation"as aname forthemathematicalquantityP. Thissamequantityiscalled the"displacement" byMaxwell, andunder- lyingtheuseofthisterm also, there isaphysical interpretationwhich we shallcome uponlater. 128.Wenow have asthe basis ofourmathematical theorythe following: Definition. Thestrength ofatubeofforceisdefinedtobethecharge enclosed bythepositiveendofthetube. Definition. Thepolarisationatanypointisdefinedtobetheaggregate strength oftubes offorce perunitarea ofcross- section. Experimental Law. Theintensityatanypointis4nr/Ktimes the polarisation,whereKistheinductive capacity ofthedielectric atthepoint. Inthis lastrelation, wemeasure theintensity alongalineofforce, while thepolarisationismeasured byconsideringthefluxoftubes offorce across asmall areaperpendiculartothelines offorce. Suppose, however, thatwe takesome direction 00'makingananglewith that ofthelines offorce. The aggregate strengthofthetubes offorce which cross anareadS perpendicularto00' willbePcosOdS, forthese tubes areexactlythose which cross anareadScos6perpendiculartothe lines offorce. Thus, consistentlywith thedefinition ofpolarisation,wemaysaythatthepolari- sation inthedirection 00' isequaltoPcosd. Since thepolarisationin 118 Dielectrics andInductive Capacity [ch.v anydirection isequaltoPmultiplied bythecosine oftheanglebetween thisdirection andthat ofthelines offorce, itisclear thatthepolarisation mayberegardedasavector, ofwhich thedirection isthat ofthelines of force, andofwhich themagnitudeisP. Thepolarisation havingbeen seen tobeavector, wemayspeakofits components /,g,h.Clearly/isthenumber oftubesperunit areawhich cross aplane perpendiculartotheaxis ofx,and soon. Theresultjustobtained maybeexpressed analytically bytheequations J4nr*4?r 4tt 129. Thepolarisation Pbeing measured bytheaggregate strengthof tubesperunit area ofcross-section, itfollows that ifa>isthecross-section atanypointofatube ofstrength e,wehave e=&>P.Nowwehave defined thestrengthofatube offorce asbeing equaltothechargeatitspositive end, sothatbydefinition thestrengtheofatube doesnotvaryfrompoint topointofthetube. Thus theproduct&>P isconstantalongatube, or <oKR isconstantalongatube, replacingtheresult that coR isconstant inair(§56). Thevalue oftheproductcoPatanypointofatube, being equalto —— ,depends onlyonthephysicalconditionsprevailingatthepoint0. Itis,however, known tobeequaltothechargeatthepositiveendofthe tube. Hence itmust also,fromsymmetry,beequaltominus thechargeat thenegativeendofthetube. Thus thechargesatthetwoends ofatube, whether inthesame orindifferent dielectrics, willbeequal andopposite, andthenumerical value ofeither isthestrengthofthetube. Gauss' Theorem. 130. LetSbeanyclosed surface, and letebetheangle between the direction oftheoutward normal toanyelement ofsurface dSandthedirection ofthelines offorce attheelement. Theaggregate strengthofthetubes of forcewhich cross theelement ofareadSisPcosedS,andtheintegral fjPcosedS, whichmaybecalled thesurfaceintegralofnormalpolarisation,willmeasure theaggregate strengthofallthetubes which cross thesurface S,thestrength ofatubebeingestimated aspositive when itcrosses thesurface from inside tooutside, andasnegativewhen itcrosses inthereverse direction. Atubewhich enters thesurface from outside, andwhich, aftercrossing 128-1 31J Gauss' Theorem 119 thespaceenclosed bythesurface, leaves itagain,willaddnocontribution to 11PcosedS, itsstrength beingcountednegatively where itenters the surface, andpositively where itemerges. Atubewhich starts from orends onachargeeinside thesurface Swill,however, supplyacontribution to IIPcosedSoncrossingthesurface. Ifeispositive,thestrengthofthe tube ise;and, asitcrosses from inside tooutside, itiscountedpositively, andthecontribution totheintegralise.Again,ifeisnegative,thestrength ofthetube is—e,andthis iscountednegatively,sothatthecontribution is againe. Thus onsummingforalltubes, Pcos€dS=E,// whereEisthetotalchargeinside thesurface. Theleft-hand member is simplythealgebraical sumofthestrengthsofthetubes whichbeginorend inside thesurface;theright-hand member isthealgebraical sum ofthe chargesonwhich these tubesbeginorend.Putting theequation becomes 11KB,cosedS= 4nrE. Thequantity Rcoseis,however, thecomponentofintensity alongthe outward normal, thequantitywhich hasbeenpreviouslydenoted byN,so thatwearrive attheequation f(KNdS=4>7rE (61). When thedielectric was air,Gauss' theorem wasobtained intheform //NdS=4tt#. Equation (61)istherefore thegeneralisedform ofGauss' Theorem which must beusedwhen theinductivecapacityisdifferent fromunity. Since dVN=—-x—,theequation maybewritten intheform dVK^dS=-4>7rE. on 131. Theform ofthisequationshews atonce thatagreatmanyresults which have beenshewn tobetrue forairaretrue also fordielectrics other than air. Itisobvious, forinstance, thatVcannot beamaximum oraminimum atapointinadielectric which isnotoccupied byanelectric charge:as 120 Dielectrics andInductive Capacity [ch.v aconsequencealllines offorce mustbegin andendoncharged bodies, aresult which wastacitly assumed indefiningthestrengthofatube of force. Anumber oftheorems were obtained inthediscussion oftheelectrostatic field inair,bytakingaGauss' Surface, partlyinairandpartlyinacon- ductor. Gauss' Theorem wasused intheform ffNdS=4*7rE, butwenow seethat iftheinductivecapacityoftheconductor were not equaltounity,thisequation oughttobereplaced byequation (61).Itis, however, clear that thedifference cannot affect the final result;Niszero inside aconductor, sothat itdoesnotmatter whetherNismultiplied byK ornot. Thus results obtained forsystemsofconductors inairupontheassumption thatCoulomb's lawofforce holdsthroughoutthefield areseen tobetrue whether theinductivecapacityinside theconductors isequaltounityornot. TheEquations ofPoisson andLaplace. 132. In§49,weappliedGauss' theorem toasurface which wasformed byasmallrectangular parallelepiped,ofedges dx,dy,dz,paralleltothe axes ofcoordinates. Ifweapplythetheoremexpressed byequation (61)to thesame element ofvolume, weobtain dy\dy wherepisthevolumedensityofelectrification. This, then, isthegeneralised form ofPoisson'sequation:thegeneralisedform ofLaplace's equationis obtained atonceonputting p=0. Interms ofthecomponentsofpolarisation, equation (62)maybewritten dfdgdh_ dx+ dy+dz~p Cbd) ' while ifthedielectric isuncharged,4(*£K(*8 *K(*£)--*->m. 1+1+1=°<«*) ElectricChargesinaninfinite homogeneousDielectric. 133. Consider achargeeplaced byitself inaninfinite dielectric. If thedielectric ishomogeneous,itfollows from considerations ofsymmetry thatthelines offorcemust beradial, astheywould beinair.Byapplication 131-135]Gauss' Theorem 121 ofequation (61) toasphereofradius r,havingthepoint chargeascentre, it isfound thattheintensityatadistance rfrom thechargeis Kr*' The forcebetween twopoint charges e,e,atdistance rapartinahomo- geneous unbounded dielectric istherefore ee' •(65),Kr* andthepotentialofanynumber ofcharges,obtained byintegrationofthis expression,is v=T^l(66)' Coulomb's Equation. 134. Thestrengthofatubebeingmeasured bythechargeatitsend, it follows that atapoint justoutside aconductor, P,theaggregate strength ofthetubesperunit ofcross-section, becomesnumerically equaltocr,the surfacedensity. Wehave alsothegeneralrelation R-—P andonreplacing Pbya,wearrive atthegeneralisedform ofCoulomb's equation, R=~(67), inwhichKistheinductivecapacityatthepointunder consideration. Conditions tobesatisfied attheBoundary ofaDielectric. 135. Letusexamine theconditions which willobtain ataboundaryat which theinductivecapacity changes abruptlyfromK1toK2. Thepotential must becontinuous incrossingtheboundary,forifP,Q, aretwoinfinitelynearpointsonoppositesides oftheboundary,theAvorkdone inbringingasmallchargetoPmust bethesame asthatdone inbringing ittoQ.Asaconsequenceofthepotential being continuous, itfollows that thetangential componentsoftheintensity must alsobecontinuous. For if P,Qaretwoverynearpointsondifferent sides oftheboundary,and P',Q' asimilarpairofpointsatasmall distanceaway, wehaveVe=VQ,and VP'=Vq,sothat PP' QQ'' Theexpressions onthetwo sides ofthisequation are,however, thetwo intensities inthedirection PP', onthetwo sides oftheboundary,which establishes the result. 122 Dielectrics andInductive Capacity [ch.v Also, ifthere isnochargeontheboundary,theaggregate strengthof thetubes which meet theboundaryinanysmall areaonthisboundaryis thesamewhether estimated intheone dielectric ortheother, forthetubes donotalter theirstrengthincrossingtheboundary,andnone canbeginor endintheboundary.Thus thenormalcomponentofthepolarisationis continuous. 136. IfRiistheintensityinthe firstmedium ofinductivecapacityKxt measured atapointclose totheboundary, and ifexistheanglewhich the lines offorcemake with thenormal totheboundaryatthispoint,then the normalpolarisationinthe firstmedium is -j—KxCOS6X. 47T Similarly,that inthesecond medium is -7-Mocos e2, sothat KiRx cos€i=KoR 2cose2 (G8). Since, inthenotationalready used, is^cos e1=N1=—-^,on theequation justobtained maybeputineither oftheforms K&^KJST, (69), *s-*s<70>- Intheseequations,itisamatter ofindifference whether thenormal is drawn from the firstmedium tothesecond orinthereverse direction;itis only necessarythat thesame normal should betaken onboth sides ofthe equation.Relation(70)isobtained atonceonapplyingthegeneralised form ofGauss' theorem toasmallcylinder having parallelends atinfinitesimal distanceapart,oneineachmedium. 137.Tosumup,wehave found that inpassingfrom onedielectric to another, thesurface ofseparation being uncharged: (i)thetangential components ofintensityhave thesame values onthe twosidesoftheboundary, (ii)thenormalcomponents ofpolarisationhave thesame values. Or,interms ofthepotential, (i)Viscontinuous, (ii)Kd-Iiscontinuou,dn 135-138J Boundary Conditions 123 Refraction ofthelinesofforce. 133.From thecontinuityofthetangential componentsofintensity,it follows : (i) thatthedirections ofR^andR2,theintensities onthetwosides of theboundary, must lieinaplane containingthenormal, and (ii) thatRxsinex=R2sine2. Combiningthelastrelation withequation (68),weobtain Kxcotex=K2cote2 (71). From thisrelation, itappearsthat ifKxisgreaterthanK2,then€jisgreater than e2,and vice versa. Thus inpassingfrom asmaller value ofKtoa greatervalue ofK,thelines arebentawayfrom thenormal. Inillustration ofthis, fig.43shews thearrangementoflines offorcewhen apoint charge isplacedinfront ofaninfinite slabofdielectric (K=7). Fig. 43. 124 Dielectrics andInductive Capacity [ch.v Asmall charged particle placedatanypointofthis field willexperience aforce ofwhich thedirection isalongthetangenttothelineofforcethrough thepoint.The force isproduced bythepoint charge,but itsdirection will notingeneral passthroughthepoint charge. Thusweconclude that in afield inwhich theinductive capacityisnotuniform theforcebetween two point chargesdoesnotingeneralactalongthelinejoining them. 139.Asanexampleoftheaction ofadielectric letusimagineaparallel platecondenser inwhich aslabofdielectric ofthickness tisplacedbetween theplates,itstwofacesbeing paralleltotheplatesand atdistances a,bfromthem, sothata+b+1=d,where disthedistance between theplates. Itisobvious fromsymmetrythat thelines offorce arestraight throughouttheirpath, equation (71)being satisfiedbye1=e2=0. Let <tbethecharge perunit area, sothatthepolari- sation isequaltoaeverywhere. Theintensity, by equation (67),is R=47rcr inair, and4>7TR=-~aindielectric.Fig. 44. Hence thedifference ofpotentialbetween theplates,orthework done in takingunitchargefromoneplatetotheother inoppositiontotheelectric intensity, 4"7r=47TO- .a+-j=a .t+47TCT .6 = 47ro-je*-(l-i)*j, andthecapacity perunitarea is Thus theintroduction oftheslab ofdielectric hasthesame effect as movingtheplates adistance (1— -^ )tnearertogether. Supposenow that theslab ispartlyoutside thecondenser andpartly between theplates. Ofthetotal areaAofthecondenser, letanareaBbe occupied bytheslab ofdielectric, anareaA—Bhaving onlyairbetween theplates. 138-141] Boundary Conditions 125 The lines offorce willbestraight, exceptforthose whichpassnear tothe edgeofthedielectric slab.Neglectingasmall correctionrequired bythe curvature ofthese lines, thecapacity Gofthecondenser isgiven by c=B A-B M 4rf{d-(l-£)« aquantity which increases asBincreases. IfVisthepotentialdifference andEthecharge,theelectricalenergy E2=401/2=1— Ifwekeepthecharge constant, the electricalenergyincreases asthe slab iswithdrawn. There must therefore beamechanical forcetendingto resist withdrawal :theslab ofdielectric willbesucked inbetween theplates ofthecondenser. This, aswillbeseen later,isaparticularcaseofageneral theorem thatanypieceofdielectric isacted onbyforces which tend to dragitfrom theweaker tothestronger partsofanelectric field offorce. Charge ontheSurface ofaDielectric. 140. LetdSbeanysmall area ofasurface whichseparates twomedia ofinductivecapacities K1}K2,and letthisboundingsurface have achargeof electricity,thesurfacedensityoverdSbeingo\Ifweapply Gauss' Theorem toasmallcylinder circumscribing dSweobtain *£+*£—***<72>- where—ineither medium denotes differentiation withrespect tothenormal drawn awayfromdSintothedielectric. 141. Aswehave seen, thesurface ofadielectric maybe charged byfriction. Amoreinteresting wayisbyutilising theconducting powersofaflame.PlQ '45- Letusplaceachargeeinfront ofaslab ofdielectric asinfig.43. Aflameissuingfrom ametallampheld inthehandmayberegardedas aconductor atpotentialzero. Onallowingtheflame toplayover the surface ofthe dielectric, thissurface isreduced topotential zero,andthe distribution ofthelines offorce isnowexactlythesame asifthefaceof thedielectric werereplaced byaconducting planeatpotentialzero. The 126 Dielectrics andInductive Capacity [ch.v lines offorce from thepoint chargeterminate onthisplane,sothat there must beatotalcharge—espreadover it.Iftheplanewereactuallya conductor thiswould besimplyaninducedcharge. If,however, theplane istheboundaryofadielectric, thechargediffers fromaninducedchargeon aconductor inthat itcannotdisappeariftheoriginal chargeeisremoved. Forthisreason, Faradaydescribed itasa"bound" charge. Thechargehas ofcourse come tothedielectricthroughtheconductingflame. Molecular Action inaDielectric. 142.From theobserved influence ofthestructure ofadielectricupon theelectric phenomena occurringinafield inwhich itwasJ3laced, Faraday was ledtosupposethattheparticlesofthedielectric themselves tookpart inthis electric action. Afterdescribinghisresearches onthe electric action—"induction"tousehisownterm—inaspace occupied bydielectric hesays*: "Thus inductionappearstobeessentiallyanaction ofcontiguous parti- cles,throughtheintermediation ofwhich theelectric force, originatingor appearingatacertainplace,ispropagatedtoorsustained atadistance...." "Induction appearstoconsist inacertainpolarisedstate oftheparticles, intowhichtheyarethrown bytheelectrified body sustainingtheaction, the particles assuming positiveandnegative pointsorparts...." "Withrespecttothetermpolarity...,Imean atpresent...adisposition offorcebywhich thesame moleculeacquires opposite powersondifferent parts." Andagain,laterf, "Idonotconsider thepowers whendeveloped bythepolarisationas limited totwodistinctpointsorspots onthesurface ofeachparticletobe considered asthepolesofanaxis,butasresident onlarge portionsofthat surface, astheyareuponthesurface ofaconductor ofsensible sizewhen it isthrown intoapolarstate." "Insuch solid bodies asglass, lac,sulphur, etc.,theparticles appearto beable tobecomepolarisedinalldirections, foramasswhenexperimented uponsoastoascertain itsinductivecapacityinthree ormoredirections, givesnoindication ofadifference. Now, astheparticlesarefixed inthe mass, andasthedirection oftheinductionthrough them mustchangewith itschargerelative tothemass, theconstant effect indicates thattheycan bepolarised electricallyinanydirection." *Experimental Researches, 1295, 1298, 1304. (Nov. 1837.) tExperimental Researches, 1686, 1688, 1679. (June, 1838.) 141-143]Molecular Theory 127 "Theparticlesofaninsulatingdielectric whilst under induction maybe compared...toaseries ofsmall insulated conductors. Ifthespace round acharged globewere filled with amixture ofaninsulatingdielectric and smallglobular conductors, the latterbeingatalittle distance from each other, soastobeinsulated, then these would intheir condition andaction exactlyresemble what Iconsider tobethecondition andaction ofthe particlesoftheinsulatingdielectric itself. Iftheglobe werecharged, these little conductors would allbepolar;iftheglobeweredischarged, theywould allreturn totheirnormal state, tobepolarised again upontherecharging oftheglobe...." Asregardsthequestionofwhatactuallytheparticlesarewhichundergo thispolarisation, Faraday says*: "Animportant inquiry regardingtheelectricpolarityoftheparticlesof aninsulating dielectric, is,whether itbethemolecules oftheparticular substance acted on,orthecomponentorultimateparticles, which thus act thepartofinsulated conducting polarising portions." "The conclusion Ihave arrived atis,that itisthemolecules ofthe substance whichpolariseaswholes; and thathowevercomplicatedthe compositionofabodymay be,allthoseparticlesoratoms which areheld together bychemicalaffinitytoform onemolecule oftheresulting body actasoneconductingmass orparticle when inductive phenomenaand polarisationareproducedinthesubstance ofwhich itisapart." 143.Amathematical discussion oftheaction ofadielectric constructed asimagined byFaraday,hasbeengiven byMossotti, who utilised amathe- matical method which hadbeendeveloped byPoisson fortheexamination of asimilar questioninmagnetism.For this discussion themolecules are represented provisionallyasconductors ofelectricity. Toobtain afirst idea ofthe effect ofanelectric fieldonadielectric of thekindpictured byFaraday,letusconsider aparallel plate condenser, + ++ + -+ Fig. 46. *Experimental Researches, 1699, 1700. 128 Dielectrics andInductive Capacity [ch.v havinganumber ofinsulated uncharged conducting molecules inthespace between theplates. Imagineatube ofstrengthemeetingamolecule. At thepointwhere this occurs, thetubeterminates bymeetingaconductor, so that there must beacharge—eonthesurface ofthemolecule. Since the totalchargeonthemolecule isniltheremust beacorresponding chargeon theopposite surface, andthischarge mayberegardedasapointofrestarting ofthetube. Thetubethenmaybesupposedtobecontinually stoppedand restarted bymolecules asitcrosses from oneplateofthecondenser tothe other. Ateachencounter with amolecule there areinducedcharges—e,+e onthesurface ofthemolecule. Anysuchpairofcharges, beingatonlya small distanceapart,mayberegardedasformingasmall doublet, ofthekind ofwhich thefield offorcewasinvestigatedin§64. 144.Wehavenowreplacedthedielectric byaseries ofconductors, the medium between which maybesupposedtobeairorether. Inthespace between these conductors thelawofforce willbethat oftheinversesquare. Incalculatingtheintensityatanypointfrom thislawwehave toreckon theforces from thedoublets aswell astheforces from theoriginal charges onthecondenser-plates. Aglanceatfig.46willshew thattheforces from thedoublets actinoppositiontotheoriginalforces. Thus forgiven charges onthecondenser-platestheintensityatanypoint between theplatesis lessened bythepresenceofconductingmolecules. Thisgeneralresult canbeseen atoncefrom thetheorem of§121. The introduction ofnew conductors(the molecules)lessens theenergycor- respondingtogiven chargesontheplates,i.e.increases thecapacityofthe condenser, andsolessens theintensity between theplates. 145. Incalculatingthatpartoftheintensitywhich arises from the doublets, itwillbeconvenient todivide thedielectric intoconcentricspherical shells havingascentre thepointatwhich theintensityisrequired. The volume oftheshell ofradii randr+dris47rr2dr,sothatthenumber of doublets included initwillcontain r2drasafactor. Thepotential produced HiCOS byanydoublet atapointdistant rfrom itis-—-— ,sothattheintensity willcontain afactor— .Thus theintensity arisingfrom allthedoublets in the shell ofradii r,r+drwilldependonrthroughthefactor -.r-dr dr or— r Theimportanceofthedifferent shells isaccordinglythesame, asregards comparativeorders ofmagnitude,asthat ofthecorrespondingcontributions fdv totheintegral/— .Thevalue ofthisintegralislogr+aconstant, and this 143-146] Molecular Theory129 isinfinite when r=andwhen r=oo.Thus theimportantcontributions come fromverysmall andvery largevalues ofr.Itcanhowever beseen thatthecontributions fromlargevalues ofrneutralise oneanother, forthe term cos6inthepotentialsofthedifferent doublets willbejustasoften positiveasnegative. Hence itisnecessary onlytoconsider thecontributions from shells for which risvery small, sothatthewhole field atanypointmayberegarded asarising entirelyfrom thedoublets intheimmediateneighbourhoodofthe point. The force willobviously varyaswemove inandoutamongstthe molecules, depending largelyonthenearness andpositionofthenearest molecules. If,however, weaveragethisforcethroughoutasmall volume, we shall obtain anaverage intensityofthe fieldproduced bythedoublets, and this willdepend onlyonthestrengthandnumber ofthedoublets inand near tothiselement ofvolume. Obviouslythisaverage intensitynearany pointwillbeexactly proportionaltotheaverage strengthofthedoublets nearthepoint, andthisagainwillbeexactly proportionaltothestrengthof theinducingfieldbywhich thedoublets areproduced,sothat atanypoint wemaysaythattheaveragefield ofthedoublets stands tothetotal field in aratiowhichdepends onlyonthestructure ofthemedium atthepoint. 146.Nowsupposethatourmeasurements arenotsufficientlyrefined to enable ustotakeaccount oftherapid changesofintensityoftheelectric fieldwhich must occur within small distances ofmolecular order ofmagnitude. Letussuppose,aswelegitimately may,that theforces which wemeasure areforcesaveraged throughadistance which contains agreat number of molecules. Then theforcewhich wemeasure willconsist ofthesum ofthe averageforceproduced bythedoublets, and oftheforceproduced bythe external field. The fieldwhich weobserve may accordinglyberegardedas thesuperpositionoftwo fields, orwhat amounts tothesamething,the observedintensity Rmayberegardedastheresultant oftwo intensities RltM2,where JR.!istheaverage intensity arisingfrom theneighbouring doublets, R2istheintensitydue tothechargesoutside thedielectric, and to thedistant doublets inthe dielectric. These forces, aswehave seen,must beproportionaltooneanother, so thateachmust beproportionaltothepolarisationP. Itfollows thatPis proportionaltoR,theratiodepending onlyonthestructure ofthemedium atthepoint.Ifwetaketherelation tobe R=^P • (73), thenKistheinductivecapacityatthepoint, andtherelation between R andPisexactlytherelationuponwhich ourwholetheoryhasbeen based, j. 9 130 Dielectrics midInductive Capacity [ch.v 147. Thetheorycouldaccordinglybebased onMossotti'stheory,instead ofonFaraday's assumption,andfrom thehypothesisofmolecularpolarisa- tionweshould beable todeduce alltheresults ofthetheory, byfirst deducing equation (73)from Mossotti'shypothesis,andthen therequired results fromequation (73)inthewayinwhichtheyhavebeendeduced in thepresent chapter. Thus theinfluence oftheconductingmoleculesproduces physicallythe same result asifthepropertiesofthemedium were altered intheway suggested byFaraday, andmathematicallythepropertiesofthemedium are ineither caserepresented bythepresenceofthefactorKinequation (73). Relation between InductiveCapacity andStructureofMedium. 148. The electrostatic unit offorcewasdefined insuchawaythatthe inductivecapacityofairwastaken asunity.Itisnowobvious that itwould havebeenmore scientific tohave taken ether asstandard medium, sothat theinductivecapacityofeverymedium would havebeengreaterthanunity. Unfortunately,thepracticeofreferringallinductivecapacitiestoairas standard hasbecome toofirmlyestablished forthis tobepossible. The difference between thetwostandards isvery slight,theinductivecapacity ofnormal airinterms ofether being 1*000590. Thus theinductivecapacity ofavacuum maybetaken tobe"99941 referred toair. Solongasthemolecules areatdistancesapartwhich aregreat compared with their linear dimensions, wemayneglecttheinteraction ofthecharges induced onthedifferent molecules, and treat their effects asadditive. It follows that inagasK—K,whereKistheinductivecapacityoffreeether, oughttobeproportionaltothedensityofthegas.Thislaw isfound tobe inexact agreementwithexperiment*. 149. Itis,however, possibletogofurther andcalculate theactual value ofthe ratio ofK—Ktothedensity. Wehave seen that this willbe aconstant foragiven substance, sothatweshalldetermine itsvalue inthe simplestcase :weshall consider athin slab ofthedielectricplacedina parallel plate condenser, asdescribed in§139. Letthisslabbeofthicknesse, and letitcoincide with theplaneofyz.Letthedielectric contain nmole- culesperunitvolume. Theelement dydzwillcontain nedydzmolecules. Ifeach ofthese is adoublet ofstrength fju,theelement dydzwillhave afieldwhich willbe equivalentatalldistantpointstothat ofasingledoublet ofstrength nyuedydz.This isexactlythe fieldwhich would beproducedifthetwo faces oftheslabwerechargedwithelectricityofsurfacedensity ±?i/t. *Boltzmann, Wiener Sitzungsber. 69,p.812. 147-149]Molecular Theory 131 Wecanaccordinglyatonce findthefieldproduced bythese doublets—it isthesame asthat ofaparallel plate condenser, inwhich theplatesareat distance eapartand arechargedtosurfacedensity +n/t.There isno intensity exceptbetween theplates,andhere theintensityofthe field is Thus ifRisthetotalintensityoutside the slab, that inside willbe R— 4<7rn/jb.IfKistheinductivecapacityofthematerial oftheslab,and Kthat ofthefreeether outside the slab,wehave KR=K(R-4i7rnfi), sothat —T^=~R^ )" Itremains todetermine theratio/j,/R. Thepotentialofadoublet is —while that ofthefieldRmaybetak,en tobe—Rx+G.Thus thetotal potentialofasingledoublet andtheexternal field is f^-Rx +C, and thismakes thesurface r=aanequipotentialif— 3=R.Thus the surfaces ofthemolecules willbeequipotentialsifweimaginethemolecules tobespheresofradius a,andthecentres ofthedoublets tocoincide with thecentres ofthespheres,thestrengthofeachdoubletbeingRa?. Putting (j,=Ra3 ,equation (74)becomes* K-K K=47rna8 . Now inunitvolume ofdielectric, thespace occupied bythenmolecules 4-7T . K—K is-5-na?.Callingthisquantity 6,wehave—^r--=SO,or,since ourcalcu- lationsonlyholdonthehypothesisthat6issmall, §r=1+30(75). Ifthelines offorcewentstraightacross fromoneplateofthecondenser *Clausius(Mech. Wdrmetheorie, 2,p.94)hasobtained therelation K-Kp _4w 3K+2K Q~ 3Ua' byconsidering thefield inside asphere ofdielectric. Thevalue ofKmust ofcourse beinde- pendentoftheshape ofthepieceofthedielectric considered. Theapparent discrepancyinthe twovalues ofKobtained, isremoved assoon aswereflect thatboth proceed ontheassumption thatK-Kissmall, fortheresults agree asfarasfirstpowersofK-Kq. Pagliani (Accad.dei Lined, 2,p.48)finds that inpointoffacttheequation—7r-^=47rna3A agrees better withexperiment than theformula ofClausius. 9—2 132 Dielectrics andInductive Capacity [CH.V totheother, theproportionofthelengthofeachwhich would beinside a conductor would, ontheaverage,be6.Since there isnofallofapotential inside aconductor, thetotal fallofpotentialfrom oneplatetotheother would beonly1—6times what itwould beifthemolecules were absent, andthe ratioK/Kwould be1/(1—0)or, if issmall, 1+6.Since, however, thelines offorce tend torunthroughconductors whereverpossible, there ismoreshorteningoflines offorce than isshewn bythissimple calculation. Equation (75)shews thatwhen themolecules aresphericalthe effect isthree times thatgiven bythissimplecalculation. Forothershapes ofmolecules themultiplyingfactormightofcourse bedifferent. Equation (75) givesatonce amethod ofdeterminingforsubstances forwhich 6issmall, namely gases, but,owingtotheunwarranted assumption that themolecules arespherical,theresults willbetrue asregardsorder of magnitude only.Ifthe dielectric isagasatatmospheric pressure,the value ofnisknown, beingabout 2-705 x1019 ,andthisenables ustocalculate thevalue ofa. 150. Thefollowingtablegivesseries ofvalues of-~forgasesatatmo- -fio spheric pressure: Gas 149-151] Molecular Theory 133 The lasttwocolumnsgive respectivelythevalues ofacalculated from equation (75),andthevalue ofagiven bytheTheoryofGases. Thetwo sets ofvalues donotagree exactly—thiscould notbeexpected whenwe remember themagnitudeoftheerrors introduced intreatingthemolecules asspherical. Butwhatagreementthere issupplies very significant evidence astothetruth ofthetheoryofmolecularpolarisation. 151. Itstillremains toexplain whatphysical propertyofthemolecule justifies usintreatingitssurface asaperfectconductor. Ithasalready beenexplainedthat allmatter contains anumber ofnegatively charged par- ticles orelectrons. These form theouterlayersoftheatoms andmolecules and itisbytheirmotion thattheconduction ofelectricityiseffected. Inadielectric there isnoconduction, sothat each electron must remainpermanently associated with thesame molecule. Thereis,however, plentyofevidence that theelectrons arenotrigidlyfixed tothemolecules butarefreetomove within certain limits. Themolecule mayberegardedasconsisting partially orwhollyofacluster ofelectrons, normallyatrestinpositionsofequilibrium under thevarious attractions andrepulsions present,butcapableofvibrating about thesepositions. Under theinfluence ofanexternal field offorce, theelectrons willmoveslightlyfrom theirequilibrium positions—wemay imaginethatakind oftidalmotion ofelectrons takesplaceinthemolecule. Obviously, bythetime thatequilibriumisattained, theouter surface ofthe molecule must beanequipotential. This, however, isexactly what isrequired forMossotti'shypothesis. Wemay accordingly abandon theconceptionof conducting spheres,which wasonly requiredtomake thesurface ofthe molecule anequipotential,andmay,withoutimpairingthepowerofMossotti's explanation, replacetheseconducting spheres byshells ofelectrons. Ifin somewaywecanfurtherreplacethese shells byringsofelectrons inrapid orbital motion, themodifiedhypothesiswillbeinverycloseagreementwith modern beliefs astothestructure ofmatter. Onthisview, thequantityatabulated inthesixthcolumn ofthetable onp.132, willmeasure theradius oftheoutermost shell ofelectrons. Even outside thisoutermost shell, however, there willbeanappreciablefield of force, sothatwhen twomolecules ofagascollide there willingeneralbea considerable distance between their outermostlayersofelectrons. Thus if thecollisions ofmolecules inagasaretoberegardedasthecollisions of elasticspheres,theradius ofthesespheres must besupposedtobecon- siderably greaterthan a.Now itistheradius oftheseimaginaryelastic spheres which wecalculate intheKineticTheoryofGases :there istherefore nodifficultyinunderstandingthedifferences between thetwosetsofvalues foragiveninthetable ofp.132. Itisknown thatmolecules arenotingeneral sphericalinshape, but, as weshall seebelow, there isnodifficultyinextendingMossotti'stheoryto cover thecase ofnon-sphericalmolecules. 134: Dielectrics andInductive Capacity [CH.V Anisotropic Media. 152. There aresome dielectrics, generallyofcrystalline structure, in whichFaraday'srelation betweenpolarisationandintensityisfound not tobetrue. Thepolarisationinsuch dielectrics isnot, ingeneral,inthe same direction astheintensity,andtheanglebetween thepolarisationand intensityand alsotheratio ofthesequantitiesarefound todependonthe direction ofthefieldrelativelytotheaxes ofthecrystal. Weshall findthat theconceptionofmolecular action accounts forthesepeculiaritiesofcrystalline dielectrics. Letusconsider anextreme case inwhich thesphericalmolecules of fig.46arereplaced byanumber ofveryelongatedorneedle-shapedbodies. The lines offorce willhave their effectivelengthsshortened byanamount which dependsonwhether much orlittle ofthem fallswithin thematerial of theneedle-shaped molecules, and, asin§149, there willbeanequationof theform where 6istheaggregatevolume ofthenumber ofmolecules which occur in aunitvolume ofthegas,and sisanumericalmultiplier. But itisatonce clear that thevalue ofswilldependnotonlyontheshapebutalsoonthe orientation ofthemolecules. Clearlythevalue ofswillbegreatest when theneedles areplacedsothat theirgreatest lengthliesinthedirection of 4 + ++ + -++ + Fig. 46a.i-f+ t-1+ i-1+ Fio.46b. Flo.46c, thelines offorce, asinfig.46a,and willbeleastwhen theneedles lieat right anglestothisposition,asinfig.46b.Ortoputthematter inanother way,apieceofdielectric inwhich themolecules areneedle-shapedand parallelwillexhibit different values ofKaccordingasthe field offorce is paralleloratright anglestothelengthsoftheneedles. 152] Anisotropic Media 135 This extreme case illustrates thefundamentalpropertyofcrystalline dielectrics, but itoughttobeunderstood that inactual substances thevalues ofKdonotdiffer somuch fordifferent directions asthisextreme casemight besupposedtosuggest.Forinstance forquartz,oneofthesubstances in which thedifference ismost marked, Curie finds theextreme values ofKto be4-55and4-49. Beforeattemptingtoconstruct amathematicaltheoryofthebehaviour ofacrystallinedielectric wemayexamine thecase ofadielectrichaving needle-shapedmoleculesplaced paralleltooneanother, butsoastomake anyanglewith thedirection ofthelines offorce, asinfig.46c. Itisatonce clear thatnotonlyaretheeffectivelengthsofthelines of force shortened bythepresenceofthemolecules, but alsothedirections of thelines offorce aretwisted. Itfollows thatthepolarisation, regardedasa vector asin§128,must ingeneralhaveadirection different from that ofthe average intensity Rofthe field. Toanalysesuch acaseweshall, asin§146,regardthe fieldnearany pointasthesuperpositionoftwo fields : (i)the fieldwhich arises from thedoublets ontheneighbouring molecules, sayafield ofcomponentsofintensity X1}YlyZ1; (ii)thefieldcaused bythedoubletsarisingfrom thedistant molecules andfrom thechargesoutside the dielectric, sayafield ofcomponentsof intensity XitY2,Z2. Clearlyinthecasewearenowconsidering,theintensities R1}R2of these fields willnotbeinthesame direction. Thecomponentsofintensityofthewhole field aregiven by X=Xa+X2,etc. Todiscuss the firstpartofthe field, letusregardthewhole field as thesuperpositionofthree fields, having respectively components (X, 0,0), (0,Y,0)and(0,0,Z).Ifthemolecules arespherical,orif,notbeing spherical,their orientations inspacearedistributed atrandom, thenclearly the field ofcomponents (X, 0,0)willinduce doublets which willproduce simplyafield ofcomponents (K'X, 0,0)where K'isaconstant. But ifthe molecules areneithersphericalinshapenorarrangedatrandom asregards •their orientations inspace,itwillbenecessarytoassume that theinduced doubletsgiverise toafield ofcomponents K'nX, K\,X, K\ 3X. 136 Dielectrics andInductive Capacity [ch.v Onsuperposingthedoublets inducedbythethree fields (X, 0,0), (0,Y,0)and(0,0,Z),weobtain X,=K'nX+K'aY+K'S1Z\ Yl=K'uX+K'mY+K' nZ- (76). Zx=K'13X+K'wY+K'33ZJ Thuswehave relations oftheform 4,Trf=K uX+K21Y+K slZ\ I expressingtherelations betweenpolarisationandintensity.4tt#=Kl2X+KUY+K32Z | (77) 4?r/i=Kl3X+KwY+K33Z These arethegeneral equationsforcrystallinemedia. "Weshallshortly prove (§176)that K12=K21,K23=K32,KS1=K13 (78), sothatthere arenotnine,butonly six,independentconstants. Non-spherical Molecules. 152a.Amedium inwhich themolecules arenotspherical butareoriented atrandom canbediscussed inasimilarway. Thewhole field(X,Y,Z)may beregardedasthesuperpositionofthree fields(X, 0,0),(0,Y,0)and(0,0,Z). Theinduced doubletsproduced bythe first field willproduceafield ofcom- ponents (K'X, 0,0), thecomponents along OyandOznecessarily vanishingonaccount ofthe random orientation ofthemolecules. The other fieldssimilarly produce induced fields (0,K'Y, 0)and(0,0,K'Z), whence wereadilyobtainequationsoftheform 4>Trf=KX >4>7ig=KY,^h=KZ. Thus Mossotti'stheorycanreadilybeextended tonon-spherical molecules, butthedifficulty remains thataccordingtomodern views, amolecule does notconsist oflayersofelectrons atrest,butofsystemsofelectrons inorbital motion. Itwillnotbepossibletomake theappropriate modification inthe theoryuntil theexact nature ofthis orbital motion isknown. 152] Examples 137 EXAMPLES. 1.Aspherical condenser, radii a,b,hasairinthespace between thespheres. The inner spherereceives acoat ofpaintofuniform thickness tandofamaterial ofwhich theinductive capacityisK.Find thechange producedinthecapacityofthecondenser. 2.Aconductor hasacharge e,andV1}V2arethepotentialsoftwoequipotential surfaces completely surroundingit(Vx>V2).Thespace between these twosurfaces is now filled with adielectric ofinductivecapacity K.Shew that thechangeinthe energyofthesystemis \e{J x-n){K-\)lK. 3.Thesurfaces ofanair-condenser areconcentricspheres.Ifhalfthespace between thespheresbefilled with solid dielectric ofspecificinductive capacity K,thedividing surface between thesolidandtheairbeing aplane through thecentre ofthespheres, shew thatthecapacitywillbethesame asthough thewhole dielectric were ofuniform specificinductive capacity £(1+K). 4.The radii oftheinner andouter shells oftwoequal spherical condensers, remote from each other andimmersed inaninfinite dielectric ofinductivecapacity K,are respectivelyaand b,andtheinductivecapacitiesofthedielectric inside thecondensers areKitK2.Both surfaces ofthe firstcondenser areinsulated andcharged,thesecond being uncharged. Theinner surface ofthesecond condenser isnowconnected toearth, andtheouter surface isconnected totheouter surface ofthe firstcondenser byawire ofnegligible capacity. Shew thatthelossofenergyis Q*{2(b-a)K+aK 2} 2Kb{{b-a)K+aK 2y where Qisthequantityofelectricitywhich flowsalong thewire. 5.Theouter coatingofalong cylindrical condenser isathin shell ofradiusa,and thedielectric between thecylinders hasinductivecapacity Kononeside ofaplane throughthe axis,andK'ontheother side. Shew thatwhen theinnercylinderis connected toearth, andtheouter hasacharge qperunitlength, theresultant force on theoutercylinderis 4f(K-K') 7ra(A'+A") perunitlength. 6.Aheterogeneousdielectric isformed ofnconcentricspherical layersofspecific inductivecapacities A'j,A"2,...A'n,starting from theinnermostdielectric, which forms a solidsphere ;alsotheoutermost dielectric extends toinfinity. The radii ofthespherical boundarysurfaces areaua2,...a„_! respectively. Prove that thepotential cluetoa quantity Qofelectricityatthecentre ofthespheresatapoint distant rfrom thecentre inthedielectric K,is A,\r ajA8+1\a, atJrJ Anah 138 Dielectrics andInductive Capacity [ch.v 7.Acondenser isformed bytworectangular parallel conducting platesofbreadth bandareaAatdistance dfrom each other. Also aparallelslab ofadielectric ofthickness tandofthesame area isbetween theplates. This slab ispulled alongitslength from between theplates,sothatonlyalength xisbetween theplates.Prove thattheelectric force sucking theslabback toitsoriginal positionis 27r£2dbt'(d-t') {A{d-t')+xbt'}*' where t'=t(K- 1)1K,Kisthespecificinductivecapacityoftheslab,Eisthecharge, and thedisturbances produced bytheedgesareneglected. 8.Three closed surfaces1,2,3areequipotentialsinanelectric field. Ifthespace between 1and2isfilled with adielectric K,andthatbetween 2and3isfilled witha dielectric K\shew thatthecapacityofacondenser having1and3forfaces isC,given by C~AK+BE" where A,Barethecapacitiesofair-condensers having asfaces thesurfaces 1,2and2,3 respectively. 9.The surface separatingtwo dielectrics (Ki,E2)hasanactual charge<rperunit area. The electric forces onthetwosides oftheboundaryareF1}F2atangles c^,c2with thecommon normal. Shewhow todetermine F2,andprovethat E2cotc2=E1cotCi(1 FiFi cosCiJ' 10.Thespacebetween twoconcentricspheresradiia,bwhich arekeptatpotentials A,B,isfilled with aheterogeneousdielectric ofwhich theinductivecapacity varies as the?ithpowerofthedistance from theircommon centre. Shew that thepotentialatany pointbetween thesurfaces is Aan+1-Bbn+1an+1bn+1A-B an+1_ftn+1rn+1an+1_£n+1' 11.Acondenser isformed oftwoparallel plates, distant hapart, oneofwhich is atzeropotential.Thespace between theplatesisfilled withadielectric whose inductive capacityincreases uniformly fromoneplatetotheother. Shew thatthecapacity perunit area is K2—K1 AirhlogZ2/Z1' whereKxandE2arethevalues oftheinductive capacityatthesurfaces oftheplate. The inequalitiesofdistribution attheedgesoftheplatesareneglected. 12.Aspherical conductor ofradius aissurrounded byaconcentricspherical conductingshellwhose internal radius isb,andtheintervening spaceisoccupied bya dielectric whosespecific inductive capacityatadistance rfrom thecentre is .Ifthe innersphereisinsulated andhasacharge E,theshell being connected with theearth,E b(c+r) provethatthepotentialinthedielectric atadistance rfrom thecentre is—log—}^. Examples 139 13.Aspherical conductor ofradius aissurrounded byaconcentricsphericalshell of radiusb,andthespace between them isfilled with adielectric ofwhich theinductive capacityatdistance rfrom thecentre isfie~p2p~3where p=rja. Prove thatthecapacity ofthecondenser soformed is hl 2/^a(ea2-e)_1 . _r 14. Ifthespecificinductive capacityvaries ase<*,where risthedistance from a fixedpointinthemedium, verifythatasolution ofthedifferentialequationsatisfied by thepotentialis /a\2rlr ri-] andhence determine thepotentialatanypointofasphere, whose inductivecapacityis theabove function ofthedistance from thecentre, when placedinauniform field of force. 15.Shew thatthecapacityofacondenser consistingoftheconducting spheres r=a, r=6,andaheterogeneousdielectric ofinductive capacity K=f(d, (p),is i^^ fff(&,&sin0d8dtp. 16.Inanimaginary crystalline medium themolecules arediscsplacedsoastobe allparalleltotheplaneofocy.Shew thatthecomponentsofintensity andpolarisation areconnected byequationsoftheform AttJ=KnX+K2iY ;4ng=KnX+K 22'ir )4rrh=K33Z. CHAPTER VI THESTATE OFTHEMEDIUM INTHEELECTROSTATIC FIELD 153.Thewhole electrostatictheoryhassofarbeen basedsimply upon Coulomb's Law oftheinversesquareofthedistance. Wehavesupposed that onechargeofelectricityexerts certain forces uponasecond distant charge,butnothinghasbeen saidastothemechanism bywhich thisaction takesplace.Inhandlingthisquestionthere aretwopossibilities open.We mayeither assume"action atadistance"asanultimateexplanation—i.e. simplyassert thattwobodies actononeanother across theintervening space,withoutattemptingtogoanyfurther towards anexplanationofhow such action isbrought about—orwemay tentatively assume thatsome medium connects theonebodywith theother, andexamine whether itis possibletoascribepropertiestothismedium, such that theobserved action willbetransmitted bythemedium.Faraday andMaxwell followed thelatter course. Theyrefused toadmit"action atadistance"asanultimateexplana- tionofelectricphenomena, findingsuch action unthinkable unless transmitted byanintervening medium. 154. Itisworthenquiring whether there isanyvalid apriori argument which compels ustoresort toaction through amedium. Some writers have attemptedtouse thephenomenonofInductive Capacitytoprovethat theenergyofacondenser must reside inthespace between thecharged plates,rather thanontheplatesthemselves —for, they say,changethemedium between theplates, keepingtheplatesinthesame condition, andtheenergyischanged. AstudyofFaraday's molecular explanationoftheaction in adielectric willshew that thisargument proves nothingastotherealquestionatissue. Itgoessofarastoprovethatwhen there aremolecules placed between electriccharges, these molecules themselves acquire charges, andsomaybesaidtobenewstores ofenergy, but itleaves untouched thequestionofwhether theenergyresides inthecharges onthe molecules orintheether between them. Again,thephenomenonofinduction issometimes quoted against action atadistance— asmall conductorplacedatapointPinanelectrostatic fieldshews phenomenawhich dependontheelectric intensityatP.This istaken toshew thatthestate oftheether atthepointPbefore theintroduction oftheconductor was insomewaydifferent from what itwould havebeen iftherehadnotbeen electricchargesintheneighbourhood. But allthat isprovedisthatthestate ofthepointPaftertheintroduction oftheconductor 153, 154]TheStateoftheMedium intheElectrostatic Field 141 willbedifferent fromwhat itwould have been ifthere hadnotbeen electriccharges in theneighbourhood, andthiscanbeexplained equallywelleither byaction atadistance or actionthrough amedium. Thenewconductor isacollection ofpositive andnegative charges:thephenomena under question areproduced bythesecharges being acted upon bytheother chargesinthefield,butwhether thisaction isaction atadistance oraction through amedium cannot betold. Indeed,itwillbeseenthat,viewed inthelight oftheelectron-theory andofFaraday's theoryofdielectricpolarisation, electrical action stands onjust thesame level as gravitational action. Ineach casethesystem offorces tobeexplained mayberegarded asasystem offorces between indestructible centres, whether ofelectricity orofmatter, andthelawofforce isthelawoftheinversesquare, independentlyofthestate ofthe space between thecentres. Nownoscientist would claim thatthere isanyApriori proof thatgravitationistransmitted through amedium—indeed thetrend ofopinionatpresent isquiteintheopposite direction—and this fact initself suffices toshew thatthere isno apriori means ofestablishing that electrical action istransmitted through amedium. Failing anapriori argument, anattempt maybemade todisprove action atadistance, orrather tomake itimprobable, byanappealtoexperience. Itmaybeargued that as alltheforces ofwhich wehaveexperienceinevery-daylifeareforces between substances incontact, therefore itfollows byanalogy that forces ofgravitation, electricity and magnetism, mustultimately reduce toforces between substances incontact—i.e.must be transmitted throughamedium. Upon analysis, however,itwillbeseenthat thisargument divides allforces intotwoclasses : (a)Forces ofgravitation, electricity andmagnetism, which appeartoactata distance. (/3)Forces ofpressure andimpact between solid bodies, hydrostatic pressure,etc. which appeartoactthroughamedium. Theargumentisnowseen tobethatbecause class(/3)appeartoactthrough amedium, therefore class(a)must inrealityactthrough amedium. Theargument could, withequal logical force, beused intheexactly oppositedirection :indeed ithasbeen sousedbythe followers ofBoscovitch. TheNewtoniandiscoveryofgravitation, andofapparent action atadistance, sooccupiedtheattention ofscientists atthetime ofBoscovitch that it seemed natural toregard action atadistance astheultimate basis offorce, and to trytointerpret action through amedium interms ofaction atadistance. Thereversion from thisviewcame, ashasbeensaid, withFaraday. Hertz's subsequent discoveryofthe finitevelocityofpropagationofelectric action, which hadpreviously beenpredicted byMaxwell'stheory, came tothesupportofFaraday's view. Toseeexactly what ismeant bythis finitevelocityofpropagation,letusimagine thatweplace twouncharged conductors A,Batadistance rfrom oneanother. By charging A,andsoperforming work atA,wecaninduce charges onconductor B,and when thishasbeen done, there willbeanattraction between conductors AandB.We cansuppose thatconductor Aisheldfast,andthat conductor Bisallowed tomove towards A,work being performed bytheattraction from conductor A.Wearenow recovering fromBworkwhich wasoriginally performedatA.TheexperimentsofHertz shew thatafinite time isrequired before anyoftheworkspentatAbecomes available atB.Anatural explanationistosupposethatwork spent onAassumes theform of energy whichspreadsitself outthrough thewhole ofspace, andthat the finite time observed before energy becomes available atBisthetime requiredforthe firstpartof theadvancing energytotravel fromAtoB.This explanation involves regarding energy 142TheState oftheMedium intheElectrostatic Field[on.vi asadefinitephysical entity, capableofbeinglocalised inspace.Itoughttobenoticed thatoursenses giveusnoknowledgeofenergyasaphysical entity:weexperience force, notenergy. Andthefactthatenergy appearstobepropagated through space with finite velocitydoesnotjustify usinconcludingthat ithasarealphysical existence, for,aswe shallsee,thepotential appearstobepropagatedinthesame way,andthepotential can onlyberegardedasaconvenient mathematical fiction. 155. Althoughnosufficient reason hasbeen foundcompellingusto ascribe electric action tothepresenceofanintervening medium, weare still freetoassume, asahypothesis,thatsuch amedium exists andthat electric action istransmitted throughthismedium. Asvarious electric andelectro- magnetic phenomenaarediscussed weshallexamine whatpropertieswould have tobeattributed tothemedium toaccount fortheseproperties.Ifitis found thatcontradictory properties would have tobeascribed tothemedium, then thehypothesisofactionthroughanintervening medium willhave tobe abandoned. Ifthepropertiesarefound tobeconsistent, thenthehypotheses ofaction atadistance andactionthroughamedium are stillboth inthe field,butthelatter becomes more orlessprobable justinproportionasthe propertiesofthehypothetical medium seemprobableorimprobable. Weshall return tothegeneral questionoftheexistence ofamedium inChapterXX. 156. Since electric action takesplace even across themostcomplete vacuum obtainable, weconclude that ifthis action istransmitted bya medium, thismedium must bethe ether.Assumingthat theaction is transmittedbytheether, wemustsupposethat atanypointintheelectro- static field there willbeanaction andreaction between thetwopartsofthe ether atoppositesides ofthepoint. The ether, inother words, isinastate ofstress atevery pointinthe electrostatic field. Beforediscussingthe particular systemofstressesappropriatetoanelectrostatic field,weshall investigatethegeneral theoryofstresses inamedium atrest. General Theory ofStresses inamedium atrest. 157. Letustakeasmall areadSinthemediumperpendiculartothe axisofx.Letusspeakofthatpartofthemedium near todSforwhich x isgreaterthan itsvalue overdSasx+,andthat forwhich xislessthan this value asa?_,sothat theareadSseparatesthetworegions x+andx_. Thosepartsofthemediumbywhich these tworegionsareoccupied exert forces upon oneanother across dS,andthissystemofforces isspokenofas thestress across dS.Obviouslythis stress will consist ofanaction and reaction, thetwobeing equal andopposite.Also itisclear thattheamount ofthisstress willbeproportionaltodS. Letusassume thattheforce exertedbyx+on#_hascomponents Ixxdk, JxydS, lzZdS. 154-157]General Theory ofStress 143 then theforce exerted byx_onx+willhavecomponents —PxxdS, —T^ydS,—PxzdS. Thequantities Pxx ,Pxy,Pxzarespokenofasthecomponentsofstress perpendiculartoOx. Similarlythere willbecomponentsofstressPyx>Pyy, PyZperpendiculartoOy,andcomponentsofstressPzx ,Pzy,Pzzperpendicular toOz. Letusnexttakeasmallparallelepipedinthe medium, bounded byplanes x=£,x=£+dx; y=v,y=v+dy; z=%, z=£+dz. The stressacting upontheparallelepiped across theface ofareadydzintheplane x=g willhavecomponents -(Pxx)z-t dydz,-(Pxy)x^dydz,-(Pxz)x^dydz, while thestress acting upontheparallelepipedacross theoppositeface will havecomponents (Pxz)x-z+dzdydz, (Pxy)x~s+dzdydz, (Pxz)x^+dXdydz. Compoundingthese two stresses, wefindthattheresultant ofthestresses acting upontheparallelepipedacross thepairoffacesparalleltotheplane ofyz,hascomponents dxFig. 47. dP—^dxdydz,000 dxdxdydz. Similarlyfrom theotherpairsoffaces,wegetresultant forces ofcom- ponents IX dxdydz,dPyy anddy dPzx dzdxdydz,dy dPzydxdydz,dPvyz dy dPzzdxdydz, dxdydz,—dxdydz. Forgenerality,letussupposethat inaddition totheaction ofthese stresses themedium isacted upon byforcesactingfrom adistance, of amount H,H,Zperunitvolume. Thecomponentsoftheforcesactingon theparallelepipedofvolume dxdydzwillbe Sdxdydz, Hdxdydz, Zdxdydz. Compoundingalltheforces which havebeen obtained, weobtain asequations ofequilibrium andtwosimilarequations.dxV-tyX ,V-*ZX n+ Bydz.(79) 144TheState oftheMedium intheElectrostatic Field[ch.vt 158. These threeequationsensure that themedium shall have no motion oftranslation, but forequilibriumitisalsonecessarythat there should benorotation. Toafirstapproximation, thestress across anyface maybesupposedtoactatthecentre ofthe face,andtheforce B,H,Zat thecentre oftheparallelepiped. Taking moments about alinethroughthe centreparalleltotheaxisofOx,weobtain astheequationofequilibrium Pyz-Hy=(80). Thisandthetwosimilarequationsobtainedbytakingmoments about linesparalleltoOy,Ozensure thatthere shallbenorotation ofthemedium. Thus thenecessaryandsufficient condition fortheequilibriumofthemedium isexpressed bythreeequationsoftheform of(79),andthreeequationsofthe form of(80). 159.Supposenext thatwetake asmall areadSanywhereinthe medium. Letthedirection cosines ofthenormal todSbe±I,±m,±n.Letthepartsofthe medium close todSandonthetwosides ofitbe spokenofasS+and$_,thesebeing named so thatalinedrawn fromdSwith direction cosines +1,+m,+nwillbedrawn into S+,andone with direction cosines — I,—m,—nwillbedrawn into&_. Letthe force exerted byS+on&_ across theareadShavecomponents FdS, GdS, HdS, Fig. 48. then the force exertedby#_onS+willhave components -FdS, -GdS, -HdS. Thequantities F,G,Harespokenofasthecomponentsofstress across aplaneofdirection cosinesI,m,n. Tofindthevalues ofF,G,H,letusdraw asmall tetrahedronhaving three facesparalleltothecoordinateplanes andafourth havingdirection cosines I,m,n.IfdSisthearea ofthe last face, theareas oftheother faces are IdS,mdS, ndSand thevolume oftheparallelepipedis %\f2lmn (dS)2 .Resolving paralleltoOx,wehave, since themedium inside thistetrahedron isinequilibrium, £V2K (dS)%H-ldSP xx-mdSPyx-ndSP 2X+FdS=0, giving,sincedSissupposed vanishingly small, F=lP xx+mPyx+nPzx (81) andthere aretwosimilarequationstodetermine GandH. 158-160] General Theory ofStress 145 160. Assumingthatequation (80)andthetwosimilarequations are satisfied, thenormal componentofstress across theplaneofwhich the direction cosines areI,m,nis IF+mQ+nH=PP xx+m2Pyy+n2Pzz+2mnPyz+2nlPzx+2lmPxy. Thequadric x2Pxx+y2Pyy+z2Pzz+2yzP yz+2zxPzx+2xyP xy=l(82) iscalled thestress-quadric.Ifristhelengthofitsradius vector drawn in thedirectionI,m,n,wehave r2(l*Pxx+m\y+n*Pzz+2mnPyz+2nlPzx+2lmPxy)=1. Itisnow clear thatthenormal stress acrossanyplane I,m,nismeasured bythereciprocalofthesquareoftheradius vector ofwhich thedirection cosines areI,m,n.Moreover thedirection ofthestress acrossanyplane I,m,nisthat ofthenormal tothestress-quadricattheextremityofthis radius vector. Forrbeingthelengthofthisradius vector, thecoordinates ofitsextremitywillberl,rm,rn.The direction cosines ofthenormal at thispointareintheratio rlPxx+rmPxy+mPzx:rlPxy+rmPyy+rnPyz:rlPzx+rmPyZ+rnPzz orF :G :H,whichprovestheresult. Thestress-quadrichasthreeprincipal axes,andthedirections ofthese arespokenofastheaxes ofthe stress. Thus thestress atanypointhas three axes,andthese arealwaysatright anglestooneanother. Ifasmall areabetakenperpendiculartoastress axis atanypoint,thestress across thisarea willbenormal tothearea. Iftheamounts ofthese stresses are P1}Pi,P3)then theequationofthestress-quadricreferred toitsprincipal ixes willbe Clearlyapositive principalstress isasimple tension, andanegative principalstress isasimple pressure. Assimpleillustrations ofthistheory,itmaybenoticed that (i)Forasimple hydrostatic pressure P,thestress-quadric becomes animaginary sphere p(e+ri2+{2)=-i: Thepressureisthesame inalldirections, andthepressureacross anyplaneisatright anglestotheplane (forthetangent planetoasphereisatright anglestotheradius vector). (ii)Forasimple pull,asinarope, thestress-quadric degeneratesintotwoparallel planes P£2=l. J. 10 146TheState oftheMedium intheElectrostatic Field[ch.vi TheStresses inanElectrostatic Field. 161. Ifaninfinitesimal charged particleisintroduced intotheelectric field atanypoint,thephenomenaexhibited byitmust, onthepresentview ofelectric action, depend solelyonthestate ofstress atthepoint.The phenomenamust therefore bededucible from aknowledgeofthe stress- quadricatthepoint. Theonlyphenomenonobserved isamechanical force tendingtodragtheparticleinacertain direction—namely,inthedirection ofthelineofforcethroughthepoint.Thus frominspectionofthestress- quadric,itmust bepossibletosingleoutthisonedirection. Weconclude thatthestress-quadricmust beasurface ofrevolution, havingthisdirection foritsaxis. Theequationofthestress-quadricatany point,referred to itsprincipal axes,mustaccordinglybe Z?r+W+£2 )=1 (83), where theaxisoffcoincides with thelineofforcethroughthepoint.Thus thesystemofstresses must consist ofatension Ptalongthelines offorce, andatension ^perpendiculartothelines offorce—and ifeither ofthe quantities i?orP2isfound tobenegative,thetension must beinterpreted asapressure. Since theelectrical phenomenaatanypoint depend onlyonthestress- quadric,itfollows thatRmust bededucible from aknowledgeofi?and I\. Moreover, theonlyphenomena known arethose whichdependonthe magnitudeofR,sothat itisreasonable tosupposethattheonlyquantity which canbededuced from aknowledgeofPxand P^isthequantity R— inother words, that 7?and P%arefunctions ofRonly.We shall forthe presentassume thisasaprovisional hypothesis,toberejectedifitisfound tobeincapableofexplainingthefacts. 162. Theexpressionofi?asafunction ofRcanbeobtained atonce byconsideringtheforcesactingonachargedconductor. Anyelement dS R2 ofsurfaceexperiencesaforce—dSurgingitnormally awayfrom thecon- ductor. Onthepresentview oftheoriginoftheforces intheelectric field, wemustinterpretthisforce astheresultant oftheether-stresses onitstwo sides. Thus, resolving normallytotheconductor, wemust have ^rdS=(P_) BdS-(P 1)dS, where(P)e, (7?)denote thevalues ofi?when theintensityisRand respectively.Inside theconductor there isnointensity,sothat the stress-quadrics becomespheres,forthere isnothingtodifferentiate one direction from another. Anyvalue which(/?)„mayhaveaccordinglyarises 1(51-164]Stresses inElectrostatic Field 147 simplyfrom ahydrostatic pressureortensionthroughout themedium, and thiscannot influence theforces onconductors.Leaving anysuchhydrostatic pressureoutofaccount, wemaytake(i?)=0,and soobtain (P1)Rinthe form R2 «=8^ •.<84>- 163.Wecanmosteasilyarrive atthefunction ofRwhich must be taken toexpressthevalue ofR2,byconsideringaspecialcase. Consider asphericalcondenser formed ofspheresofradii a,b.Ifthis condenser iscutintotwoequalhalves byaplane throughitscentre, the twohalves willrepeloneanother. This action mustnowbeascribed tothe stresses inthemedium across theplaneofsection. Since thelines offorce areradial these stresses areperpendiculartothelines offorce, andwesee atonce that thestressperpendiculartothelines offorce isapressure. To calculate thefunction ofRwhichexpressesthispressure, wemay suppose b—aequaltosomeverysmallquantity c,sothatRmayberegardedas constantalongthelengthofaline offorce. The area overwhich this pressureacts isit(b2—a2 ),and since thepressure perunit area inthe mediumperpendiculartoaline offorce is—R,the totalrepulsion between thetwohalves ofthecondenser willbe—R2Tr(b2—a2 ). Thewhole forceoneither halfofthecondenser ishowever aforce 2tt<t2 perunitareaovereachhemisphere,normal toitssurface. Theresultant of allthe forcesactingontheinnerhemisphereisira2x2ira2 ,orputting 2ira2a=E,sothatEisthechargeoneitherhemisphere,this force isE2/2a2 . Similarly,theforce onthehemisphereofradius bisE2/2b2 .Thus there- '1 1\ sultantrepulsiononthecompletehalfofthecondenser is\E2 (-—yA.Since thishasbeen seen tobealsoequalto—R.7T(b2—a2 ),wehaveW,R2 ^=--^7-„ =-27T(7 ontakinga=binthelimit. Thus inorder that theobserved actions maybeaccountedfor,itis necessarythatwehave 7?a R3P—zLiP—-— ^"Stt'*~8tt' Moreover,ifthese stresses exist, theywillaccount foralltheobserved mechanical action onconductors, forthestresses result inamechanical force 27ro-2perunitareaonthesurface ofeveryconductor. 164. Itremains toexamine whether these stresses aresuch ascanbe transmitted byanether atrest. 10—2 148TheState oftheMedium intheElectrostatic Field[oh.vi Asapreliminarywemust findthevalues ofthestress-components Pxx , Pxy,...referred tofixed axes Ox,Oy,Oz. Thestress-quadricatanypointintheether, referred toitsprincipal axes, isseenoncomparisonwithequation (83)tobe £(p-^2-n=i (85).07T Here theaxis offisinthedirection oftheline offorce atthepoint. Letthedirection-cosines ofthisdirection be lltmui\.Then ontransforming toaxes Ox,Oy,Ozwemay replace fbylxx+mxy+nxz. Equation (85)maybereplaced by 07T andontransformingaxes £2+rf+£2transforms into a?+y2+z2 .Thus the transformed equationofthestress-quadricis R2^{2(l,x+my+n&f-(x2+y2+z2 )}=1. Comparingwithequation (82),weobtain £.«=?- (2V-1) (86), Pxy=^(2limi) (87),07T andsimilar values fortheremaining componentsofstress. Oragain,since X=l^R,Y=rn^R,Z=^R, theseequations maybeexpressedintheform Pxx=^{X2-Y2-Z2 ),07T _XY xy4, Inthissystemofstress-components, therelations Pxy=Pyxaresatisfied, asofcoursetheymust besince thesystemofstresses hasbeen derived by assumingtheexistence ofastress-quadric. Thus thestresses donotsetup rotations intheether(cf.equation (80)). Inorder that theremaybealsonotendencytotranslation, thestress- components mustsatisfy equationsofthetype Optx |OPcy,0*xz a /OQ\ -dx~+ ~dy-+~dF=°(88)' expressingthatnoforces beyond these stresses arerequiredtokeepthe ether atrest(cf.equation (79)). 164-166]Stresses inElectrostatic Field 149 Onsubstitutingthevalues ofthestress-components,wehave dx dydz ' -ssIs{X'-T'-^+1<2Xr>+&<***>} =J.\2X(—+-+~\+2F(^-d-I)+2Z(^-d -l)\ 8tt\ \dx dy dz) \dydxJ' \dz dx))' Onputting Y-d-Z V-JZ 7-JJ. *~dx'*' dy1dz* wefindatonce that dX_dY =&V_92F_ dy dx dxdy dxdy' dX_dZ_ &V_d*V dz dx dxdz dxdz' aXdYd_Z=_/cPV d^v&Z\= dx dydz \dx2dy2dz'2)' shewingthatequation (88)issatisfied. 165. Thus, torecapitulate, wehave found that asystemofstresses consistingof (i)atension-6—perunitarea inthedirection ofthelines offorce, (ii)apressure—perunitareaperpendiculartothelines offorce, isonewhich canbetransmitted bythemedium, inthat itdoesnottend to setupmotions intheether, and isonewhich willexplaintheobserved forces intheelectrostatic field. Moreover itistheonlysystemofstresses capableofdoing this,which issuch that thestress atapoint depends only ontheelectricintensityatthatpoint. Examples ofStress. 166.Assumingthissystemofstresses toexist, itisofvalue totryto picture theactual stresses inthefield inafewsimplecases. Consider firstthefieldsurroundingapoint charge.Thetubes offorce arecones. Letusconsider theequilibriumoftheether enclosed bya frustum ofone ofthese cones which isbounded bytwoendsp,q.If G)p,o)qaretheareas ofthese ends,wefind that there aretensions of 150TheState oftheMedium intheElectrostatic Field[ch.vi amountsBpWp Bg'COg 8tt'8tt sothat the forces onthetwoends have as resultant aforcetendingtomove theether inwards towards thecharge.Thistendency- isofcourse balancedbythepressures acting onthecurved surface, each ofwhich hasa component tendingtopresstheether inside thefrustum awayfrom thecharge.SinceRpa>p=Rqcoq,theformer isthegreater. Fig. 49.167.Amorecomplex exampleisafforded bytwoequal point charges,ofwhich the lines offorce areshewn in fig.50. Fia. 50. The lines offorce oneitherchargefallthickest ontheside furthest removed from theothercharge,sothat their resultant action onthecharges amounts toatraction onthesurface ofeachtendingtodragitawayfrom theother, andthistractionappearstousasarepulsion between thebodies. Wecanexamine thematter inadifferent waybyconsideringtheaction andreaction across thetwosides oftheplanewhich bisects thelinejoining thetwocharges. Nolines offorce cross thisplane,which isaccordingly made upentirelyofthesidewalls oftubes offorce. Thus there isapressure R?—perunitareaactingacross thisplaneatevery point. The resultant of o7T allthesepressures,after transmission bytheether from theplanetothe charges immersed intheether, appearsasaforce ofrepulsionexerted by thechargesononeanother. 166-169] EnergyintheElectrostatic Field 151 Energy intheMedium. 168. Insetting upthesystemofstresses inamediumoriginallyun- stressed, work must bedone, analogoustothework done incompressing agas.Thisworkmustrepresenttheenergyofthestressed medium, and this inturnmustrepresenttheenergyoftheelectrostatic field.Clearly, from theform ofthe stresses, theenergy perunitvolume ofthemedium atanypoint must beafunction ofRonly. Todetermine theform ofthis function, wemayexamine thesimplecase ofaparallel plate condenser, R2 andwefind atonce that thefunction must be5—.07T Wehavenow toexamine whether theenergyofanyelectrostatic field R2 canberegardedasmade upofacontribution ofamount 5—perunitvolume o7T fromevery partofthe field. Infig.51,letPQbeatube offorce ofstrength e,passingfromPat potential VPtoQatpotential VQ.The ether inside thistube offorce R2 being supposedtopossess energy—perunitvolume, thetotalenergyenclosed bythetube willbe cods,Lp87T where wisthecross section atany point,andthe integrationisalongthetube. Since Rco=4nre, thisexpression =\e IRds =-*e ]PTsds =ie(V f-VQ). This, however, isexactlythecontribution madebythecharges+eat P,Qtotheexpression ^2eF. Thus onsummingover alltubes offorce,we findthat thetotal energyofthe field\SeV maybeobtainedexactly, by R2 assigning energytotheether attherateof3—perunitvolume. EnergyinaDielectric. 169.Byimaginingtheparallel platecondenser of§168 filled with dielectric ofinductivecapacity K,andcalculatingtheenergywhen charged,KR2 wefind that theenergy,ifspread throughthe dielectric, must be—— perunit volume. 152TheState oftheMedium intheElectrostatic Field[ch.vi Letusnowexamine whether thetotalenergyofanyfieldcanberegarded asarising from acontribution ofthisamountperunitvolume. Theenergy contained inasingletube offorce, with thenotationalready used, willbe QKR2 , o)as,IP8?rKR or,since -r—=P,wherePisthepolarisation,thisenergy47T =|\RPcods rQ=Ie\Rds Jp =2e(Kp— ' Ka)j sothatthetotalenergyis\XeV, asbefore. Thus adistribution ofenergyof 7T7?2 amount -=-—perunitvolume willaccount fortheenergyofanyfield. Crystallinedielectrics. 170.Wehave seen(§152) that inacrystalline dielectric, thecom- ponentsofpolarisation andofelectricintensitywillbeconnected byequations oftheform 4tt/=KUX+K2lY+K31Z\ 4tt#=K12X+K22Y+K32Z\(89). 4ttA=KIZX+KsiY+K3SZJ Theenergyofanydistribution ofelectricity,nomatter what thedielectric may be,willbe^%EV. IfV1}V2arethepotentialsatthetwoends of aunit tube, thepartofthissumwhich iscontributed bythechargesatthe ends ofthistube willbe£(Fx—TQ.If9/dsdenote differentiationalongthe fdV. fdV tube, thismaybewritten —\\—ds,oragain—\I—Pcods,wherePisthe polarisation, and a>thecross section ofthetube. Thus theenergy maybe dV supposedtobedistributed attherateof—\-^-Pperunitvolume. Ifeisthe anglebetween thedirection ofthepolarisationandthat ofthe electric dV intensity, wehave—-~-=Rcose,sothat theenergy perunitvolume =\RPcose=$(/X+gY+hZ) (90). Inaslightincrease totheelectriccharges,thechangeintheenergyof thesystem is,by§109,equalto1,V8E, sothatthechangeintheenergy per unitvolume ofthemedium is BW=XSf+ Y8g+Z8h. TllUS ~df=X>Tg=Y>lh=Z<91>- 169-171] Maxwell's Displacement Theory 153 From formulae(89)and(90),wemust have W=%(fX+gY+hZ) =i{KUX*+(K12+K21)XY+...}, fromwhich \x=~L[KnX+*(/fi2+z2i)Y+*(^13+*>z)- Wemust alsohave 8JF 8TT_§£dWdg_dWdh dXdfdX+ dgdX+dhdX =^{KuX+K21Y+K3lZ}. Comparingtheseexpressions, weseethatwemust have -**-12=-ft-nt -^-13=-"L31> -^-23==-^32« Theenergy perunitvolume isnow W=± r(KnX*+2K12XY+...) (92). Maxwell's Displacement Theory. 171. Maxwellattemptedtoconstruct apictureofthephenomena occurringintheelectric fieldbymeans ofhisconceptionof"electric dis- placement."Electricintensity, accordingtoMaxwell, actinginanymedium— whether thismedium beaconductor, aninsulator, orfreeether—produces amotion ofelectricity through themedium. Itisclear that Maxwell's conceptionofelectricity,ashere used, must bewider than thatwhich we haveuptothepresent beenusing,forelectricity,aswehave sofarunder- stood it,isincapable ofmoving throughinsulators orfreeether. Maxwell's motion ofelectricityinconductors isthatwithwhichwearealreadyfamiliar. Aswehave seen, themotion willcontinue solongastheelectricintensity continues toexist.AccordingtoMaxwell, there isalsoamotion inan insulator orinfreeether, butwith thedifference that theelectricitycannot travelindefinitely throughthese media, but issimply displacedasmall distance within themedium inthedirection oftheelectricintensity,the extent ofthedisplacementinisotropic mediabeing exactly proportional totheintensity, andinthesame direction. Theconceptionwillpei-haps beunderstood moreclearly oncomparing aconductor to aliquid andaninsulator toanelastic solid.Asmallparticle immersed inaliquidwill continue tomove through theliquid solongasthere isaforce acting onit,butaparticle immersed inanelastic solid willbemerely "displaced" byaforce acting on it.The amount ofthisdisplacementwillbeproportionaltotheforceacting, andwhen theforce isremoved, theparticlewillreturn toitsoriginal position. 154TheState oftheMedium intheElectrostatic Field[ch.vi Thus atanypointinanymedium thedisplacementhasmagnitudeand direction. Thedisplacement, then, isavector, and itscomponentinany direction maybemeasured bythetotalquantityofelectricity perunit area which hascrossed asmall areaperpendiculartothis direction, thequantity beingmeasured fromatime atwhich noelectricintensitywasacting. 172.Suppose, now, thatanelectric field isgradually broughtinto existence, the field atanyinstantbeing exactlysimilar tothe final field exceptthat theintensityateachpointislessthan the finalintensityin some definite ratio k.Letthedisplacementbectimes theintensity,so thatwhen theintensityatanypointiskR,thedisplacementisc/cR. The direction ofthisdisplacementisalongthe lines offorce, sothat the electricity mayberegardedasmoving throughthetubes offorce :thelines offorcebecome identical nowwith thecurrent-lines ofastream, towhich theyhavealreadybeencompared. Letusconsider asmall element ofvolume cut offbytwoadjacent equipotentials andatube offorce. Letthecross section ofthetube of force beco,andthenormal distance between theequipotentialswherethey meet thetube offorce beds,sothat theelement under consideration isofvolume cods.Onincreasingtheintensity fromkRto(k+d/c)R,there isanincrease ofdisplacement fromckR toc(k+c?«)R,and therefore anadditional dis- placementofelectricityofamount cRdicperunit area. Thus oftheelectricity originallyinside thesmall element ofvolume, aquantitycRcodic flows outacross oneofthe bounding equipotentials,whilst anequal quantityflows in across the other. LetVi;V.Abethepotentialsofthese surfaces, then thewhole work done indisplacingtheelectricity originally inside theelement ofvolume cods, isexactlythework oftransferringa quantitycRd/c ofelectricityfrompotential Vxtopotential V%. It is therefore cRco(V l—V^dic and, sinceV2—V1=/cRds, thismaybewritten as cR2cods/cdK. Thus astheintensityisincreased from toR,thetotalwork spentindisplacingtheelectricityintheelement ofvolume cods =IcR2 (cods)Kdic=%cR2 .cods. Jo This work, onMaxwell'stheory,issimplytheenergystored upinthe nnrl t.he» r^lST>ln.PAmpn+•. n.+.nnv 47T-ds- R2 element ofvolume cods ofthemedium, and isthereforeequalto3—cods. 07T Thus cmust betakenequaltoj—,andthedisplacementatanypointis measuredby R_ 47T* 171-174]MaxwelVs Displacement Theory 155 Iftheelement ofvolume istaken inadielectric ofinductivecapacity K, 77D2 77" theenergyis-=— ,sothat c=j—,andthedisplacementis KR 4nr' 173. Itisnowevident thatMaxwell's"displacement"isidentical in magnitudeand direction withFaraday's "polarisation"introduced in Chap.v. Denotingeitherquantity byP,wehadtherelation //PcosedS=E(93), expressingthat thenormalcomponentofPintegratedoveranyclosed surface isequaltothe totalchargeinside. OnMaxwell'sinterpretationof thequantity P,thesurfaceintegral11PcosedSsimplymeasures thetotal quantityofelectricitywhich hascrossed thesurface from inside tooutside. Thusequation (93) expressesthat thetotaloutward displacementacross any closedsurfaceisequaltothetotalchargeinside. Ifwenow follow Maxwell insupposingthatelectricityisoftwokinds, (i)thekindwhichappearsasachargeonanelectrifiedbody, (ii)thekindwhich Maxwellimaginestooccupythewhole ofspace, andto undergo displacement when electric action takesplace, then itappearsthatanyincrease ofelectricityofkind(i)insideanyclosed surface isaccompanied byanexactly equaldecrease ofelectricityofkind(ii). Inother words thesum total ofthetwokinds ofelectricityinside anyclosed surface remains constant. 174. Itwillbeunderstood thatMaxwell'stheoryofelectricaldisplace- mentattemptstogiveaphysical pictureoftheprocessesoftheelectric field,., butthatthetruth ofthepictureisbynomeans essential tothemathematical theoryofelectricity. Thedisplacement theoryishistorically importantbecause itledMaxwell tothehypothesisofdisplacementcurrents which form the~ foundation ofhiselectromagnetic theoryoflight (Chap. xvn). Butweshall seelater thatthegeneral electromagnetic theorycanbedevelopedwithoutv thepreliminary displacement theory. Thedisplacement theoryhasserved as partofthescaffolding bywhich theelectromagnetic theorywasconstructed; whether thescaffolding ought now tobediscarded remains anopen question. CHAPTER VII GENERAL ANALYTICAL THEOREMS Green's Theorem. 175.Atheorem, firstgiven byGreen, andcommonlycalled after him, enables ustoexpressanintegraltaken over thesurfaces ofanumber of bodies asanintegraltaken throughthespacebetween them. Thistheorem naturallyhasmany applicationstoElectrostaticTheory.Itsuppliesameans ofhandling analyticallytheproblemswhichFaradaytreatedgeometrically with thehelpofhisconceptionoftubes offorce. 176. Theorem. Ifu,v,warecontinuousfunctions oftheCartesian coordinates x,y,z,then 1 ll(lu+mv+nw)dS=— l\\[^-+^~+^-jdxdydz (94). Here2denotes that thesurfaceintegralsaresummed overanynumber of closed surfaces, whichmayinclude asspecialcases either (i)oneoffinite sizewhich encloses alltheothers, or (ii)animaginary sphereofinfinite radius, andI,m,narethedirection-cosines ofthenormal drawn ineverycasefrom theelement dSintothespacebetween thesurfaces. Thevolumeintegralis takenthroughoutthespacebetween thesurfaces. rr^ Consider firstthevalue of Ihr-dxdydz. Takeanysmallprismwith its axisparalleltothat ofx,andofcross sectiondydz. Let itmeet thesurfaces atP,Q,R,S,T,U,...(fig. 53),cuttingoffareasdSP,dSQ,dSR,.... Thecontribution ofthisprismto IM^-dxdydzisdydzI»-dx,where the integralistaken overthosepartsoftheprismwhich arebetween thesurfaces. du , fQduThus \^dx=\ ^dx+ ~dx+... JOX Jpox }Rox pI/O/ JR =—Up+UQ—UR+US' 175-177] Green's Theorem157 where uP,uQ,uR>...arethevalues ofuatP,Q,R,.... Also, since thepro- jectionofeach oftheareasdSP,dSQ)...ontheplaneofyzisdydz,wehave dydz=lPdSP=-lQdSQ=lRdSR=..., where lP,lQ,lR,...arethevalues of IatP,Q,R,.... Thesignsinfront of lp,Iq,Ir>-"arealternately positive andnegative, because, asweproceed alongPQR..., thenormal drawn intothespace between thesurfaces makes angleswhich arealternatelyacute andobtuse with thepositive axisofx. Fig. 53. Thus dydz\—dx=dydz(—Up +uQ—uR+...) =—IpUpdSp—lQUQdS Q—lRuRdSR—(95), andonaddingthesimilarequationsobtained foralltheprismsweobtain fffd £dxdydz=-zfjludS(96), theterms ontheright-handsides ofequationsofthetype (95)combiningso asexactlytogivethetermontheright-handsideof(96). Wecantreat thefunctions vandwsimilarly,andsoobtainaltogether ///(is+S+i)dxdydz=~2 /i(lu+mv+ nw)dS- provingthetheorem. 177. Ifu,v,warethethreecomponentsofanyvector F,then the expression du dvdw dxdydz isdenoted, forreasons which willbecome clear later,bydivF.IfNisthe componentofthevector inthedirection ofthenormal(I,m,n)todS,theD N=he+mv+ww. 158 General Analytical Theorems[ch.vh Thus Green's Theorem assumes theform jffdiyFdxdydz=-^ff]STdS (97). AvectorFwhich issuch that divF=atevery pointwithin acertain regionissaid tobe"solenoidal"within thatregion.IfFissolenoidal within any region,Green's Theorem shews that jJNdS=0, where theintegralistaken overanyclosed surface inside theregionwithin whichFissolenoidal. Two instances ofasolenoidal vector have sofar occurred inthisbook—theelectricintensityinfreespace,andthepolarisa- tioninanunchargeddielectric. 178. Integration through spaceexternal toclosedsurfaces.Let the outer surface beasphereatinfinity, sayasphereofradius r,where ris tobemade infinite inthe limit. Thevalue of jj(lu+mv+nw)dS taken over thisspherewillvanish ifu,v,andwvanish morerapidlyat infinitythan— .Thus, ifthiscondition issatisfied, wehave that ///—+—+^— jdxdydz=—%jj(lu+rnv+nw)dS, where thevolumeintegrationistakenthroughallspaceexternal tocertain closed surfaces, andthesurfaceintegrationistaken over these surfaces, I,m,nbeingthedirection-cosines oftheoutward normal. 179. Integration throughtheinteriorofaclosedsurface.Lettheinner surfaces infig.53alldisappear,thenwehave ///(^+1+Tz)dxdydz= "/I{lu+mv+nw)dS' where thevolumeintegrationisthroughoutthespaceinside aclosed surface, andthesurfaceintegrationisover this area, I,m,nbeingthedirection- cosines oftheinward normal tothesurface. 180. Integration through aregioninwhich u,v,warediscontinuous. Theonlycase ofdiscontinuityofu,v,wwhichpossesses anyphysical import- ance isthat inwhich u,v,wchange discontinuouslyinvalue incrossing certain surfaces, thesebeingfinite innumber. Totreat this case,weenclose each surface ofdiscontinuityinside asurface drawn soastofititcloselyon 177-180]Green's Theorem 159 both sides. Inthespace left, after theinteriors ofsuch closed surfaces have been excluded, thefunctions u,v,warecontinuous. Wemay accordingly applyGreen's Theorem, andobtain fdu.dv.dw\ 3cc///'+=-+~- Jdxdydz=-% ll(lu+mv+nw)dS -2'jj(lu+mv+nw)dS (98), where 2denotes summation over theclosed surfaces bywhich theoriginal spacewas limited, and2'denotes summation over thenew closed surfaces which surround surfaces ofdiscontinuityofu,v,w.Now correspondingtoanyelement ofareadSonasurface ofdis- continuitythere willbetwoelements ofarea oftheenclosing surface. Letthedirection-cosines ofthetwonormals todSbe lx,mx,nxand l2,m2in2,sothat lx=—12,mx=—m2,and nx=—?vLet these direction-cosines bethose ofnormals drawn fromdStothetwo sides ofthesurface, which weshall denote by1and 2,and letthevalues ofu,v,wonthetwo sides ofthesurface ofdiscontinuityattheelement dSbe ui>Vi>wiandu2,v2)w2.Thenclearlythetwoelements of theenclosing surface, which fitagainsttheelement dSof theoriginalsurface ofdiscontinuity,willcontribute toFig. 54. VIj(lu+mv+nw)dS anamount ordS[(l xiix+m$ x+nxwx)+(l2u2+m2v2+n2w2)] [li(ux-u2)+mx(vx-v^+rh(wx—w2))dS. theformThus thewhole value of1' 11(lu+mv+nw)dSmaybeexpressedin form 2" 11{h(«i-u2)+m,(vx-v2)+nx(w x-w2)}dS, where theintegrationisnowovertheactual surfaces ofdiscontinuity.Thus Green's Theorem becomes =—2 I\{lu+mv+nw)dS - ^"\\{k(Ui~«0+™i(vi~%)+*h(Pi~ w«)]dS (99). 160 General Analytical Theorems[ch.vh Special FormofGreen's Theorem. 181.Animportantcase ofthetheorem occurs when u,v,whave the special values U= ^dx~> dy' where <I>andM*areanyfunctions ofx,yand z.Thevalue of(lu+mv+w) isnow dV d^ &P\ By *dn-> where s-denotes differentiation along thenormal, ofwhich thedirection- on cosines areI,m,n. We alsohave ^> +^=ika +M$alu dxdydzdx\ dx) dy {dy dz\ dz) 8<£9^d®dV d<&d^ ,fd2X¥d^VdHr \ dxdx dydydzdz \dx2dy2dz*) Thus thetheorem becomes Thistheorem istrue forallvalues of<Pand'SP,sothatwemayinter- change<£>andy,andtheequationremains true.Subtractingtheequation soobtained fromequation (100), weget [jT(<£V*¥-^V2 $>)dxdydz=-2!!( <5>|?-¥^)dS(101). Applications ofGreen's Theorem. 182. Inequation (101), put<£>=1and^=7,whereFdenotes the electrostaticpotential. Weobtain [fjv*Vdxdydz=-zff^dS(102). 181-183]Greerts Theorem 101 Letusdivide thesumontherightintoIlttheintegraloverasingle closed surfaceenclosing anynumber ofconductors, and72,theintegrals over thesurfaces oftheconductors. Thus '-//!> where r-denotes differentiation along thenormal drawn intothesurface. on dV .Thus—-=— isequaltothecomponentofintensity alongthisnormal, and therefore to—iV,whereMisthecomponent alongtheoutward normal. Hence I^-ffNdS. dVAtthesurface ofaconductor -^—=—4nro-, sothatan //'J2=47rS IIotZaS overconductors =4-7Txtotalchargeonconductors. Ifthere isanyvolume electrification, V2V=— 4>irp,sothat 11]V2Vdccdydz=—4-7T 11\pdxdydz, andtheintegralontheright representsthetotalvolume electrification. Thusequation (102) becomes \NdS=4nrx(total chargeonconductors +totalvolumeelectrification), sothatthetheorem reduces toGauss' Theorem. 183.NextputOand"^eachequaltoV.Thenequation (100) becomes Take thesurfaces now tobethesurfaces ofconductors, andasphere of 1o"rr radius ratinfinity. Atinfinity Visoforder - ,sothat -*- isoforderr on 1 oV— ,andhence V-~-,integratedoverthesphereatinfinity, vanishes(§178).* (Jit Theequationbecomes -4ttfjfpVdxdydz4-jjJR'dxdydz-4ttffVadS=0. j. 11 162 General Analytical Theorems[ch.vn The firstand lasttermstogether give—47rxSeT7 ",where eisany element ofcharge,either ofvolume-electrification orsurface-electrification. Thus thewholeequationbecomes \JLeV= IIItt- dxdydz, shewingthat theenergy mayberegardedasdistributedthroughthespace 7?2 outside theconductors, totheamount 5—perunitvolume—the result 07T alreadyobtained in§168. 184. InGreen's Theorem, take Vox 2/=3>if— —(*£) HereKisultimatelytobetaken tobetheinductivecapacity, which mayvarydiscontinuouslyoncrossingtheboundary between two dielectrics. Weaccordingly suppose u,v,wtobediscontinuous, anduseGreen's Theorem intheformgivenin§180.Wehavethen {oxdx oyoyozoz)uHi -*//»(«S+-5+-S)->- -%SKK^+K^W)dS(io3) - where r—,z—have themeanings assigned tothem in$140. Ifweput<E>=1,^=V,inthisequation,itreduces, asin§130, to ffr)V \\K^-dS=—4>7rxtotalchargeinside surface, sothattheresult isthat oftheextension ofGauss' Theorem.Again,ifwe put<&="^=V,theequation becomes KR* dxdydz=^XeV }8tt andtheresult isthat of§1G9. 183-187] Uniqueness ofSolution 163 Greens 'Reciprocation Theorem. 185. Inequation (101), put<£=V,V=V', whereVisthepotential ofonedistribution ofelectricity, andVisthat ofasecond andindependent distribution. Theequationbecomes fff(pV'-P'V)dxdydz +zJf(*V'-<T'V)dS=Q, which issimplythetheorem of§102,namely XeV'=Xe'V (104). Ifweassignthesame values to<f>,^inequation (103), weagainobtain equation (104), which isnowseen tobeapplicable when dielectrics are present. Uniqueness ofSolution. 186.WecanuseGreen's Theorem toobtainanalytical proofsofthe theorems already givenin§99. Theorem. Ifthevalueofthepotential Visknown atevery pointon anumber ofclosed surfaces bywhich aspaceisboundedinternally and externally,there isonly onevalue forVatevery point ofthisintervening space,whichsatisfiesthecondition thatV2Veither vanishes orhasanassigned value, atevery point ofthisspace. For, ifpossible,letV,Vdenote twovalues ofthepotential,both ofwhich satisfytherequisiteconditions. ThenV—V=0 atevery pointofthe surfaces, andV2(F'—V)=atevery pointofthespace. Putting<&andrF eachequaltoV—Finequation (100), weobtain and thisintegral, beingasum ofsquares,canonlyvanishthroughthe vanishingofeach term.Wemust therefore have 5<V-r>-4<7-F)-5<F'.-F)-0(105), orV—Vequaltoaconstant. And sinceV—Vvanishes atthesurfaces, thisconstant must bezero, sothatV=Veverywhere,i.e.thetwosolutions VandVareidentical :there isonlyonesolution. dV . 187. Theorem. Given thevalueof^atevery point ofanumber of closedsurfaces,there isonlyonepossiblevalueforV(except foradditive constants),ateachpoint oftheintervening space, subjecttothecondition that V2V=throughoutthisspace,orhasanassignedvalue ateachpoint. 11—2 164 General Analytical Theorems[CH.VII Theproofisalmost identical with that ofthe lasttheorem, theonly differencebeingthat atever}' pointofthesurfaces wehave J;<r-F)-«. instead oftheformer conditionV—V=0.We stillhave xfj(V'-V)?- n(V'-V)dS=0, sothatequation (105)istrue,andtheresult follows asbefore, exceptthat VandVmaynow differ byaconstant. 188. Theoremsexactlysimilar tothese lasttwotheorems areeasily seen tobetruewhen thedielectric isdifferent from air. For, letV,Vbetwosolutions, such that atallpointsofthespace,and atthe surface eitherV—V=0,or ) (V-V')=0. dn ByGreen's Theorem d(v-v')y fi{V-V) y\d(V-V) dx Bydzdxdydz =-///<r-F'>[l{Kl(y-r >l+£{*4(f-r >} +ai*5<F-^]dxdydz dS =byhypothesis. Equation (105)now follows asbefore, sothattheresult isproved. Comparisons ofdifferent fields. 189. Theorem, i/anynumberofsurfacesarefixedinposition, anda given chargeisplacedoneachsurface,then theenergyisaminimum when thecharges areplacedsothatevery surfaceisanequipotential. LetVbethe actualpotentialatany pointofthe field, andV thepotential when theelectricityisarrangedsothat each surface is 187-190] Comparisons ofdifferent Fields 165 anequipotential. Callingthecorresponding energiesWandW,we have Ifweput3>= 7,¥=V- V,inequation (100), wefindthat the last integral becomes 47rJJ \dn dn or,sinceVisbyhypothesisconstant overeach conductor, andthisvanishes since each totalcharge\\cr'dS isthesame asthecorre- spondingtotalcharge11adS. Thus ,r-w'-s///K-«),+••(***• Thisintegralisessentially positive,sothatWisgreaterthanW,which provesthetheorem. Ifanydistribution issuddenlysetfreeandallowed toflow sothat the surface ofeach conductor becomes anequipotential,the lossofenergyW—Wisseen tobeequaltotheenergyofafield ofpotentialV—Vat anypoint. 190. Theorem. Theintroductionofanewconductor lessens theenergy ofthefield. Letaccentedsymbolsrefer tothefield after anewconductor 8hasbeen introduced, insulated anduncharged. Then W—W=— IIjB?dxdydz throughthefield beforeSisintroduced ~o~HIR'2dxdydz throughthefield afterSisintroduced =q-IIIB?dxdydz throughthespace ultimately occupied by# +g-\\\{B;2-R'2 )throughthefield afterSisintroduced. 166 General Analytical Theorems[ch.vii The lastintegral and this, asinthelasttheorem,isequalto kilj\^-j^)'+ --}d°:dyd° .£WZ(£-©« , where2denotes summation over allconductors, includingS. This lastsumofsurfaceintegrals vanishes, sothat W-W'=~ IffR2dxdydzthrough S +-Q~llj\(^-5— )+•••}dxdydz throughthe field after $hasbeen introduced. ThusW—Wisessentially positive,whichprovesthetheorem. Onputtingthenewconductor totheearth, itfollows from thepreceding theorem thattheenergyisstillfurther lessened. 191. Theorem. Anyincrease intheinductivecapacity ofthedielectric hetween conductors lessens theenergy ofthefield. Lettheconductors ofthe field besupposedfixed inposition and in- sulated, sothat their totalchargeremains unaltered. Lettheinductive capacityatanypoint changefromKtoK+8K,andasaconsequencelet thepotential changefromVtoV+SV, andthetotalenergyofthe field fromWtoW+BW. IfE1}E2,...denote thetotalchargesoftheconductors, V1}V2i...their potentials,andpthevolumedensityabanypoint, W=%XEV+\fffpVdxdydz, sothat, since theE'sandpremain unalteredbychangesinK,wehave hW=\ZEhV+%[\LhV dxdydz (106). Wealsohave sothat*--isi!«m+m^)hd^ 190-192] Earnshavfs Theorem 167 ByGreen's Theorem, thelastline thesummation ofsurfaceintegrals beingover the surfaces ofallthe conductors, =jfjpBVdxdydz+2ffaSVdS - ffjpBVdxdydz+XEBV =28IF byequation (106). Thusequation (107) becomes BW=-L fjJR'BKdxdydz-28W, sothat BW=-~ IjJR2BKdcc dydz. ThusBW isnecessarily negativeif8K ispositive, provingthetheorem. Itisworthnoticing that,onthemolecular theoryofdielectrics, theincrease inthe inductivecapacityofthedielectric atanypointwillbemostreadily accomplished by introducing newmolecules.If,asinChap, v,these molecules areregarded asuncharged conductors, thetheoremjustproved becomes identical with that of§190. Earnshaw's Theorem. 192. Theorem. Acharged body placedinanelectricfield offorce cannot restinstable equilibrium under theinfluence oftheelectric forces alone. Letussupposethecharged bodyAtobeinany position,inthe field offorceproduced byother bodies B,B', Firstsupposeallthe elec- tricity onA,B,B', ...tobefixed inposition onthese conductors. Let Vdenote thepotential,atanypointofthe field, oftheelectricityon B,B',—Let x,y,zbethecoordinates ofanydefinitepointinA,sayits centre ofgravity, and letx+a,y+b,z+cbethecoordinates ofanyother point. Thepotential energyofanyelement ofchargeeatx+a,y+b,z+c iseV,whereVisevaluated atx+a,y+b,z+c.DenotingeVbyw,we clearlyhave sinceVisasolution ofLaplace's equation. 168 General Analytical Theorems [ch.vii LetWbethetotal energyofthebodyAinthe field offorce from B,B', ....ThenW=Xw, and therefore d2w d2wd*w A 1 1= dx*^ dy2dz*' i.e.thesumW=Sw satisfiesLaplace's equation,because thisequationis satisfied bytheterms ofthesumseparately.Itfollows from thisequation, asin§52,thatWcannot beatruemaximum oratrueminimum forany values ofx,y,z.Thus, whatever thepositionofthebodyA,itwillalways bepossibletofindadisplacement —i.e.achangeinthevalues ofx,y,z—for whichWdecreases. If,after thisdisplacement,theelectricityonthecon- ductors A,B,B',...issetfree, sothateach surface becomes anequipotential, itfollows from§189that theenergyofthe field isstillfurther lessened. Thus adisplacementofthebodyAhasbeen found which lessens theenergy ofthe field,andtherefore thebodyAcannot restinstableequilibrium. Onephysical applicationofEarnshaw's Theorem isofextreme importance. The theorem shews thatanelectron cannot rest instable equilibriumunder theforces of attraction andrepulsion from other charges,solongasthese forces aresupposedtoobey thelawoftheinverse square ofthedistance. Thus,ifamolecule istoberegarded asa cluster ofelectrons andpositive charges,asin§151,thenthelawofforcemust besome- thingdifferent from that oftheinverse square. There seems tobenodifficulty about thesuppositionthat atverysmall distances the lawofforce isdifferent from theinversesquare. Onthecontrary,there would beavery realdifficultyinsupposingthatthelawl/?-2helddown tozerovalues ofr.Fortheforce between twochargesatzerodistance would beinfinite;weshould have chargesofoppo- sitesigncontinually rushing together and,when oncetogether, noforcewould beadequate toseparate them. Thus theuniverse would intime consist onlyofdoublets, each consistingofpermanentlyinterlockedpositive andnegative charges.Ifthelaw1/r2 helddown tozerovalues ofr,thedistanceapartofthecharges would bezero, sothat thestrengthofeach doublet would benil,andthere would benowayofdetectingits presence. Thus thematter intheuniverse would tend toshrink intonothing orto diminishindefinitelyinsize. Theobserved permanenceofmatterprecludes anysuch hypothesis. Earnshaw's Theoremaccordinglylimits ustotwo alternatives. Either themolecule doesnotconsist ofacluster ofelectrons inrelativerest, orelsethelawoftheinverse squarefails atmolecular distances. Eecent experimental investigations decidevery definitely against thesecond alternative andinfavour ofthe first. Recent experiments onthedeflection ofthepositively charged a-particles bymatter indicate thatthelawoftheinverse square holdsdown todistances oftheorder of10~ucms., adistance which islessthanathousandthpartoftheradius of thehydrogen atom, andalarge mass ofother evidencesuggests, with aprobability approximatingtocertainty,thattheelectrons inanatom ormolecule must beinrapid orbital motion. Thus theproblemofthestructure ofthemolecule isremoved from the provinceofEarnshaw's Theorem. 192,193] Stresses intheMedium 169 Stresses intheMedium. 193. Letustakeanysurface Sinthemedium, enclosing anynumber ofchargesatpointsandonsurfaces 8ltS2, LetI,m,nbethe direction-cosines ofthenormal atanypointof SltS2,...orS,thenormalbeing supposed drawn, asinGreen's Theorem, into thespace between thesurfaces. The total mechanical forceactingonallthematter inside thissurface iscompoundedofaforceeRinthedirection oftheintensity actingonevery point chargeorelement ofvolume-chargee,andaforce 2ira2or^aR per unitareaoneachelement ofconductingsurface. IfX,Y,Zarethecom- ponents paralleltotheaxes ofthetotal mechanical force, X=ZeX+%UaXdS =fffpXdxdydz +2(Uo-XdS, where thesurfaceintegralistaken over allconductors Si,S2,...inside the surface S,andthevolumeintegral throughoutthespace between Sandthese surfaces.Substitutingforpand<r, 1[f[fd*Vd2Vd2V\dV ,,,x=^jj){w+w+w)tedxdydz ByGreen's Theorem, ///|I|?dxdydz=hfffl(^jdxdyd* "—isili^), «w-i//i^), «. IHw IF*"*"**=-JSSwh(^)dxdydt dy2dxyJJJdydy fly""^ JJ""fafiy xjl^vwu-fU*™*. Now ///^h(S)dxdydz= /(/*IQ2 dxdydz 170 General Analytical Theorems sothatthelastequation becomes[CH.VII —m +Mi.s*dVdV dxdy) dVdV andthere isasimilar value for d*vdv dxdydz.—m dxdy\dS dS, dz*dx Substitutingthese values, equation (108) becomes _iq*lj/Z?ZY-(— Y-8F9F aF3F) oxdy ox02 dxJ\dyJ Sincewehave atevery pointofthesurface ofaconductor d_VdVd_V dx_dydz Im n itfollows that theintegralover each conductor vanishes, leaving onlythe integralwithrespecttodS,whichgives•(109), X=(lPxx+mPzy+nPxz)d$, where %,=£-(x*-Y*-z*) t07T 1 Ifwewrite also P=—TZ theresultant forceparalleltotheaxisofYwillbe Y-- jj(lPxy+mPyy+nPyz)dS, andthere isasimilar value forZ.The action istherefore thesame(cf. §159) asifthere wasasystemofstresses ofcomponents PPPPPPLxxiLyyy*zz>Jyz>xzx>J-xy> given bytheaboveequations:i.e.thesemayberegardedasthestresses of themedium. 193,194]Stresses intheMedium 171 194. Itremains toinvestigatethecouplesonthesysteminside S.If L,M,Narethemoments oftheresultantcouple about theaxes ofx,y,z, wehave L= Jffp(yZ-zY) dxdydz +\ttt*(yZ-zY) dS 1[f[fd*V ,d*V ,d*V\ (dV dV\. _ _ NowSSSd^{yd^-zw)dxdydz = -\\\diL{yYz-zd^)dxdydz ^ffjdVf dVdV\,„ ffjdVf dV dV\7e -% )\lTx\y^-z^)dS-\\lte\y^-2^)d*> 4>7rl]]\dxdxVdzZ -by) dydy\VdzZ dy) dVd(dVdV\],,7sothat L dzdzVdzdyJj -^t !!{l dx-+m^+n^){^~Z dy-)dS ifff 7dv dV dV\(dV dv\;o ,„„ x The firstterm inthisexpression j_r/rftdvd*v.dvd*vbv&t 47T[Hi(dv^X_ w&v_dVd*v\ i]j\y[deedxdz+ dydydz+ dzdz2) Udxdydz'dVd-VdVd2VdVd"V J)xdxdy dydy*dzdydz J) =-L\\\{yd^-zd -w)dxdydz =1-2ff(ynR*-zmR*) dS+-L ff(ynIP-zm&) d$(111). Thesecond term inexpression (110)forLmay,invirtue oftherelations (109), beexpressedintheform -i-2IkynK-zmR2 )dS, which isexactlycancelled bythe firstterm inexpression (111). 172 General Analytical Theorems[ch.vii Weareaccordinglyleftwith -if/H— >-«('S^-5F+-©('E-'*)}* =-\\{y (lP*z+mPyz+nP„)-z{lPxy+mPyy+nPy2)}dS, verifyingthatthecouplesarealsoaccounted forbythesupposed systemof ether-stresses. 195. Thus the stresses intheether areidentical with thosealready found inChapter vi,and these, aswehave seen,maybesupposedto consist ofatension 3—perunit area across the lines offorce, anda 07T pressure ^—perunit area indirectionsperpendiculartothe lines offorce. 07T Mechanical Forces onDielectrics intheField. 196. Letusbegin byconsideringafield inwhich there arenosurface charges,andnodiscontinuities inthestructure ofthedielectrics. Weshall afterwards beable totreatsurface-charges and discontinuities aslimiting cases. Letussupposethatthemechanical forces onmaterial bodies are3,H,Z perunitvolume atanytypical point x,y,zofthis field. Letusdisplacethematerial bodies inthe field insuch awaythatthe point x,y,zcomes tothepointx+Bx,y+By,z+8z.Thework done in thewhole field willbe =-[j[(BSx+my+Z8z)dxdydz (112), and thismust shew itself inanequalincrease intheelectricenergy. The electric energyWcanbeputineither oftheforms W=W1= ±fffPVdxdydz, When thedisplacementtakesplace,there willbeaslightvariation in thedistribution ofelectricityand aslightalteration ofthepotential. There isalso aslight changeinthevalue ofKatanypoint owingto themotion ofthedielectrics inthe field. Thus wecanput BW^SWi^iBW^ +(BW 1)y, BW=BW2=(BW i)K+(BW 2)v, where (BW^p denotes thechange producedinthefunction W[bythevaria- 194-196] Mechanical Forces onDielectrics 173 tion ofelectricaldensity alone, (BW^ythatproduced bythevariation of potential alone, and soon. Wehave (SWJr=kfffp8Vdxdydz, ByGreen's Theorem, thelastexpressiontransforms into ^-c///^e(*is+4('a l)+£('©}*** =I\\p8V dxdydz, sothat 2(W x)v={hW 2)r. Weaccordinglyhave 8W= 28W,-8W2=2(8W,),-(BW 2)K, thevariationproduced byalterations inVnolonger appearing. Now(8W,),=\[[ftpVdxdydz, sothat 8W= ftf{v8p-^8K\dxdydz (113). Thechangeinpisduetotwocauses. Inthe firstplace,theelectrifica- tion at'x, y,zwasoriginallyatx—8x,y—8y,z—8z,sothat8phasaspart ofitsvalue -&«-!*-ifc<114>- Again,theelement ofvolume dxdydz becomeschanged bydisplacement intoanelement TX+dx^dx \\dy+ dy^dy \\dz+ dz^dz \* 7 7,/,d8x d8y d8z\ ,,,-x «*•*<«•(1+-5T+^+ 17,)<115>- sothat, even ifthere were nomotion oftranslation, anoriginal charge pdxdydz would afterdisplacement occupythevolumegiven byexpression (115), and thiswouldgiveanincrease inpofamount -'(£+?+©<116>- 174 General Analytical Theorems[ch.vii CombiningthetwopartsofBpgiven byexpressions (114) and(115), wefind *.~g<P*0+4G>*>+ !((.8O}. ThechangeinKisalsoduetotwocauses. Inthe firstplacethepoint which inthedisplaced positionisatx,y,zwasoriginallyatx—Bxty—By, z—Bz. Hence aspartofthevalue inBKwehave dK ,dK .dK fi Also, with thedisplacement,thedensityofthemedium ischanged,so that itsmolecular structure ischanged,andthere isacorresponding change inK. Ifwedenote thedensityofthemediumbyt,andtheincrease int produced bythedisplacement byBt,theincrease inKdue tothiscause willbe OT andweknow, asinequation (116), that 'dSx dBy dBz\ dxdydzJ'~. UViV VOL OOZ\or=-r (^— Wenowhave, asthetotal value ofBK, SEr dK&dK .dK « dxdyJdz dKfdBx dBy d_Bz\ drVdxdy dzJ' andhence, onsubstitutinginequation (113)forBpandBK, *W— jJfV^P^+d^+d J^^dxdydz [[[R*(dK BdK* dKe\,,.+ jjl8^{-dx-8x+ dy-By+ -dz-**)dxdydz [f[R* dKfdBx dBy dBz\, .,+j]j^T^{-dx-+ -dJ+Wjdxdyd2 - Integrating byparts,thisbecomes sw=\\\^p*x +\y:phy+%phz)dxdydz 196-198]Stresses inDielectrics 175 or,rearrangingtheterms, BW=dVB?fdK \dx)" dx 8-7TVdxJdx\8-rr' drJd(^rdK\Sx+ J[(I Comparingwithexpression (112),weobtain dVB^dK d_(R8y+ i- 1 P +dKTdx 8irdx'dx\8tt' dr etc.,givingthebodyforcesactingonthematter ofthedielectric.Bzvdxdydz. .(117), 197. Thismaybewritten intheform R^d_K d_(R?_dK\ 8ttdx dxV87T dr) Thus inaddition totheforce ofcomponents (pX, pY,pZ)actingonthe chargesofthedielectric, there isanadditional force ofcomponents _R?_dK R^dK _&d_K 87rdx'8rrdy'8irdz arisingfrom variations inK,andalsoaforce ofcomponents dx\8ttT dr)'dyV8ttT dr)'dzWT dr)' which occurs when either theintensityofthe field orthestructure ofthe dielectric varies frompointtopoint. Stresses inDielectric Media. 198. Replacing pbyitsvalue, asgiven byLaplace's equation, weobtain equation (117)intheform - 1Kk^+IU^+Kk^)2^ 8tt [dx\dx\ dxj' dy\ dyJ'dz j d_K dxdihm+mdz) dx\ dr 1_ 8ttd_ dxKm^hm dVd'dV\Kd_(dVV dxdx\ dxJ dx\dx) K-(—dx\dy dxdy\dy ndVd(TsdV\ Tjrd(dV\*+2 dx-dz{K te)+K Tx[-dF) dx\ dr)\ 176 General AnalyticalTheorems [ch.vii 8tt\dx«€J<h?r Ifweput <-xxKf/3F\*/dV\' idV\'\R<dK1R. p--efa3y'et0 (119) ' dPcx .dixy.dPx2 thisbecomes B=W+ ~dJ+dz' Letussupposethat amedium issubjectedtoasystemofinternal stresses Pxx ,Pxy,etc.; and let itbefound that asystemofbodyforces ofcomponents B',H',Z'isjustsufficient tokeepthemedium atrest when under theaction ofthese stresses. Then fromequation (79)we must have dPxx ,dPxy.dPXig/=_l"_£Z+^B+dx dydz Thus ifPxx ,Pxy>etc.have thevaluesgiven byequations (118) and(119), wehave H'=-B, etc. Thisshews that themechanical force H,H,Zreversed wouldjustbe inequilibrium with thesystemofstresses Pxx ,Pxy,etc.given byequations (118) and(119). Inother words, themechanical forces which have been found toactonadielectric canexactlybeaccounted forbyasystemof stresses inthemedium, these stressesbeing given byequations (118) and (119). 199. Thesystemofstressesgiven byequations (118) and(119)canbe regardedasthesuperpositionoftwosystems: I.Asysteminwhich II.Asysteminwhich '"""Sirdr' tXy=*yz=*ZX=U. 198-200]Stresses inDielectric Media 177 The firstsystemisexactlyKtimes thesystem which hasbeen found to occur infreeether, while thesecondsystem representsahydrostatic pressure ofamount)&dK 8ttT dr' (Ingeneral=-willbepositive,sothat thispressurewillbenegative, and must beinterpretedasatension.) Hence, asin§165,thesystemofstresses maybesupposedtoconsist of: ten2 (i)atension-5—perunitarea inthedirection ofthelines offorce;07T (ii)apressure-~—perunitareaperpendiculartothelines offorce; (iii)ahydrostatic pressureofamount —5—t^r—inalldirections. 07T OT Thesystemofstresses wehave obtained was firstgiven byHelmholtz. Thesystem differs from thatgiven byMaxwell byincluding thepressure-—r-=- .Theneglect of thispressure byMaxwell, andbyother writers whohave followed him, doesnotappearto bedefensible. Helmholtz hasshewn that stillfurther terms arerequiredifthedielectric issuch thatthevalue ofKchanges when themedium issubjectedtodistortion without changeofvolume. 200. Thissystemofstresses hasnotbeenprovedtobetheonlysystem ofstresses bywhich themechanical forces canbereplaced, and, aswehave seen, itisnotcertain thatthemechanical forces must beregardedasarising from asystemofstresses atall,rather thanfrom action atadistance. Itmaybenoticed, however, thatwhether ornotthese stressesactually exist, theresultant force onanypieceofdielectric must beexactlythe same asitwould beifthestressesactuallyexisted. Fortheresultant force onany pieceofdielectric hasacomponent Xparalleltotheaxis ofx,given by X= IlEidxdydz =-IklPxx+mPxy+nPx2)dS byGreen's Theorem, and thisshews that theactual force isidentical with what itwould beifthese stresses existed(cf.§193). J. 12 178 General Analytical Theorems[ch.vii Force onachargedconductor. 201. The mechanical force onthe surface ofachargedconductor immersed inadielectric canbeobtained atoncebyregardingitas produced bythe stresses intheether. There willbenostresses inthe interior oftheconductor, sothattheforce onitssurface mayberegarded asdue tothetensions ofthetubes offorce inthedielectric. Thetension isaccordinglyofamount KB?E2dK 87T 8-7T dr perunit area,anexpressionwhich canbewritten inthesimplerform R2dIV \ Force atboundary ofadielectric. 202. Letusconsider theequilibriumofadielectric atasurface of discontinuity,atwhich the lines offorceundergorefraction onpassing from onemedium ofinductivecapacityKltoasecond ofinductive capacityK2. Letaxesbetaken sothat theboundaryistheplaneofxy,while the lines offorce atthepointunder consideration lie intheplaneofxz. Letthecomponentsof intensityinthe firstmedium be(X x,0,Zj),while thecorresponding quantitiesinthesecond medium are(X2,0,Z2).Theboundaryconditions ob- tained in§137requirethat Xx=X2,K\Z^=K2Z2—4nrh, where histhenormal componentofpolarisation. Inview ofalaterphysical interpretationof the forces,itwillbeconvenient toregard these forces asdivided upinto thetwosystemsmentioned in§199,andtoconsider thecontributions from thesesystems separately. Asregardsthecontribution from thefirstsystem,theforceperunitarea actingonthedielectric from the firstmedium hascomponents *«. o,gw-zA while thatfrom thesecond medium hascomponents K K 4JX2Z2, 0,-J(Zi-Xi). 201,202] Stresses inDielectric Media 179 SinceKXXXZX=K2X2Z2,itfollows that the resultant force onthe boundaryisparalleltoOz—i.e. isnormal tothe surface. Itsamount, measured asatensiondraggingthesurface inthedirection frommedium 1 tomedium 2 which aftersimplificationcanbeshewn tobeequalto >X? 2irh*X?2-7r/i2\.„ _. This isalways positiveifKY>K2.Thus this forceinvariablytends to dragthesurface from themedium inwhichKisgreater,tothat inwhichKisless—i.e.toincrease theregioninwhichKislargeattheexpenseof theregioninwhichKissmall. Thisnormal force isexactlysimilar tothe normal force onthesurface ofaconductor, which tends toincrease the volume oftheregionenclosed bytheconductingsurface. OnMaxwell'sTheory, theforces which havenowbeen considered aretheonlyones in existence, sothataccordingtothistheorythetotal mechanical force isthatjustfound, andtheboundaryforces ought alwaystotend toincrease theregioninwhichKislarge. Thistheory,aswehavesaid,isincomplete,sothat itisnotsurprisingthattheresultjust stated isnotconfirmed byexperiment. Wenowproceedtoconsider theaction ofthesecondsystemofforces— thesystemofnegative hydrostatic pressures.There arepressures perunit area ofamounts RldK, R?dK% 87r13tx'8ir23t2 acting respectivelyonthetwosides oftheboundary.There isaccordingly aresultant tension ofamount 1/ dK, dK2 perunit area,tendingtodragtheboundarysurface fromregion1toregion2. Thus thetotal tensionperunit area,draggingthesurface intoregion 1,is fe+^J^'-^-sP^-^id(120)- In§139, inconsideringaparallel platecondenser with amovable dielectric slab,wediscovered theexistence ofamechanical forcetending todragthedielectric inbetween theplates.This force isidentical with the mechanical forcejustdiscussed. Butwehavenowarrived atamechanical interpretationofthis force, forwecanregardthepullonthedielectric as theresultant ofthepullsofthetubes offorce atthedifferentpartsofthe surface ofthedielectric. 12—2 180 General Analytical Theorems[ch.vii Letusattempttoassign physical interpretationstotheterms ofex- pression (120) byconsideringtheirsignificanceinthisparticularinstance. Consider firstaregioninthecondenser sofarremoved from theedgesof thecondenser and oftheslab ofdielectric, that the fieldmaybetreated 4<7r/t asabsolutelyuniform(cf. fig.44,p.124).WeputK2=l,X1—0,Ry=-^r- inexpression (120) andobtain 2rf(¥-£i)<121> astheforce perunit areaoneither face ofthedielectric, acting normally outwards. The forces will ofcourse actinsuch adirection thattheytend to decrease theelectrostaticenergyofthe field.Now thisenergyismadeup ofcontributions 27rA2perunitvolume from air,and^=- perunitvolume from the dielectric. From theconditions oftheproblemhmust remain unaltered. Thus thetotalenergycanbedecreased ineither oftwoways— byincreasingthevolumeoccupied bydielectric anddecreasingthatoccupied byair,orbyincreasingthevalue ofKinthedielectric. There willtherefore beatendencyfortheboundaryofthedielectric tomove insuchadirection astoincrease thevolumeoccupied bydielectric, andalsoatendencyforthis boundarytomove sothatKwillbeincreased bytheconsequent change ofdensity.These twotendencies arerepresented bythetwoterms of expression (121). If—isnegative,anexpansionofthedielectric willboth increase the OT volumeoccupied bythe dielectric, and will alsoincrease thevalue ofK inside the dielectric. Inthis case, then, both tendencies acttowards an expansionofthe dielectric, andweaccordinglyfind thatboth terms in expression (121) arepositive. dK If-r— ispositive,thetendencytoexpansion, represented bythe first (positive)term ofexpression (121)ischecked byatendencytocontraction (toincrease t,andtherefore K)represented bythesecond (now negative) term ofexpression (121).If—isnotonly positive,but isnumerically large, expression (121)maybenegativeandthedielectric will contract. In thiscasethedecrease inenergy resultingontheincrease ofKproduced by contraction willmore thanoutweighthegain resultingfrom thediminution ofthevolumeoccupied bydielectric. 202,203] Stresses inDielectric Media 181 These considerations enable ustoseethephysical significance ofallthe X2 terms inexpression (120), exceptthe firstterm-^-{K x—1).Tointerpret thistermwemust examine theconditions near theedgeofthedielectric slab, for itisonlyhere thatX1hasavalue different from zero.Weseeat once that thistermrepresentsapullatandnear theedgeofthedielectric, tendingtosuck thedielectric further between theplates—infactthisforce alonegivesrise tothetendencytomotion oftheslab asawhole, which was discovered in§139. Keturningtothegeneral systemsofforces of§199,wemay saythat the firstsystem (whichaswehave seenalwaystends todragthesurface ofthedielectric intotheregioninwhichKhasthegreater value) represents thetendencyforthesystemtodecrease itsenergy byincreasingthevolume occupied bydielectrics oflargeinductivecapacity,whilst thesecondsystem (which tends tocompressorexpandthedielectric insuchawayastoincrease itsinductivecapacity) representsthetendencyofthesystemtodecrease its energy byincreasingtheinductivecapacityofitsdielectrics. Thatany increase intheinductivecapacityisinvariably accompanied byadecrease ofenergyhasalreadybeenprovedin§191. Electrostriction. 203. Itwillnowbeclear thattheaction ofthevarious tractions onthe surface ofadielectric mustalwaysbeaccompaniednotonlybyatendency forthedielectric tomove asawhole, butalsobyaslight changeinshape anddimensions ofthedielectric asthisyieldstotheforcesactingon it. This latterphenomenonisknown aselectrostriction. Ithasbeen observed experimentally byQuincke and others. Aconvenient wayofshewingits existence isto fillthebulb ofathermometer-tube withliquid, andplace thewhole inanelectric field. Thepullsonthesurface oftheglassresult inanincrease inthevolume ofthebulb, andtheliquidisobserved to fallinthetube. From what hasalreadybeen said itwillbeclear that adielectric mayeitherexpandorcontract under theinfluence ofelectric forces. The stresses intheinterior ofadielectric, asgivenin§199,mayalso beaccompanied bymechanical deformation. Thus ithasbeen observedby Kerr and others, that apieceofnon-crystalline glass acquires crystalline properties whenplacedinanelectric field. Such apieceofglassreflects lightlikeauniaxalcrystalofwhich theopticaxis isinthedirection ofthe lines offorce. 182 General Analytical Theorems[ch.vii Green's Equivalent Stratum. 204. LetSbeanyclosed surfaceenclosinganumber ofelectriccharges, and letPbeanypointoutside it.ThepotentialatPduetothecharges insideSis Vp=111-dxdydz, .P Fio. 56. where risthedistance fromPtotheelement dxdydz, andtheintegration extendsthroughoutS.ByGreen's Theorem(equation (101)) IJJ(UWV-VWU)dxdydz=IJ(ud^-Vd £)dS, where thenormal isnowdrawn outwards from thesurface S. Inthisequation, putU=-,then, sinceV2F=— 4<7rp,wehave asthe value ofthe first term, fjfuWdccdydz=-4>ttVp. And sinceV2Z7=0,thesecond term vanishes. Theequation accordingly becomes -**-Bf£)-r*®}«:(m)- 205.Suppose, first, thatthesurface Sisanequipotential. Then =vfffv*fydxdydz =0, sothatequation (122) becomes VP=.U—±L±l dS(123). 204-207]Green's Equivalent Stratum 183 Thus thepotentialofanysystemofchargesisthesame atevery point outside anyselectedequipotentialwhich surrounds allthecharges,asthat ofachargeofelectricity spreadover thisequipotential, andhavingsurface 197 density—j—~— .Obviously,infact, iftheequipotentialisreplaced bya conductor, this willbethedensityonitsouter surface. 206. Ifthesurface isnotanequipotential,theterm //V=-(-)dS willnotvanish. Since, however,jj,^-(-]isthepotentialofadoublet of strength /u,and direction that oftheoutward normal, itfollows that 117^-(-}dSisthepotentialofasystemofdoubletsarrangedover the surface S,thedirection atevery point beingthat oftheoutward normal, and thetotalstrengthofdoubletsperunit area atanypoint beingV. Thus thepotential Vpmayberegardedasdue tothepresence onthe surface Sof 1dV (i)asurfacedensityofelectricity—j—-~—; V (ii)adistribution ofelectric doublets, ofstrength-—perunit area, anddirection that oftheoutward normal. 207. Equation (122) expressesthepotentialatanypointinthespace dV outside Sinterms ofthevalues ofVand -~-overtheboundaryofthisspace. Wehave seen, however, thatthevalue ofthepotentialisuniquely determined dV bythevalues either ofVoroi—overtheboundaryofthespace. Inactual electrostatic problems,theboundaries aregenerally conductors, andtherefore equipotentials.Inthis caseequation (123) expressesthevalues ofthe dV potentialinterms of-~-only, amountinginfactsimplyto rP=jfUs. What isgenerally requiredisaknowledgeofthevalue ofVPinterras ofthe values ofVover theboundaries, and thisthepresent method isunable to give. Forspecial shapesofboundary,solutions have been obtained by variousspecial methods, andthese itisproposedtodiscuss inthenext chapter. 184 General Analytical Theorems[ch. VII EXAMPLES. 1.Iftheelectricityinthefield isconfined toagiven system ofconductors atgiven potentials, andtheinductive capacityofthedielectric isslightlyalteredaccordingtoany lawsuch that atnopointisitdiminished, andsuch thatthedifferential coefficients ofthe increment arealsosmall atallpoints, prove thattheenergyofthefield isincreased. 2.Aslab ofdielectric ofinductive capacityKandofthickness xisplaced inside a parallel platecondenser soastobeparalleltotheplates. Shew thatthesurface ofthe slabexperiencesatension 3.ForagasK=-l+8p,wherepisthedensity and6issmall.Aconductor is immersed inthegas:shew that if62isneglectedthemechanical force ontheconductor is2jr(72perunit area. Giveaphysical interpretationofthis result. CHAPTER VIII METHODS FORTHESOLUTION OFSPECIAL PROBLEMS TheMethod ofImages. Charge induced onaninfinite uninsulatedplane. 208.ThepotentialatPofchargeseatapointAand-eatanother point A'is V=——APA'P.(124), and thisvanishes ifPisontheplane which bisects AA'atright angles. Call thisplane theplaneS.Then theabove value ofVgivesV=0 over theplane S,V= atinfinity, and satisfiesLaplace's equationintheregion totherightof8,exceptatthepoint A,atwhich itgivesapoint chargee. i * V x '/'V \ \•'/.»' "X » XX \'I'' S <!i\V '/ \'Sx ' I VVx Fig. 57. Theseconditions, however, areexactlythose which would have tobesatisfied bythepotential ontherightofSifSwere aconducting planeatzero potential under theinfluence ofachargeeatA.These conditions amount toaknowledgeofthevalue ofthepotentialatevery pointontheboundary ofacertainregion—namely,that totherightoftheplaneS—andofthe charges inside thisregion. There is,asweknow, onlyonevalue ofthe 186 Methods fortheSolution ofSpecial Problems[oh.viii potentialinside thisregionwhich satisfies these conditions(cf.§186),sothat thisvalue must bethatgiven byequation (124). Totherightof8thepotentialisthesame, whether wehave the charge—eatA'orthechargeontheconducting plane8.Tothe leftofS inthelatter casethere isnoelectric field. Hence thelines offorce,when theplaneSisaconductor, areentirelytotherightofS,andarethesame asintheoriginalfield inwhich thetwopoint-chargeswerepresent.The linesendontheplane S,terminatingofcourse onthechargeinduced onS. Wecanfindtheamount ofthisinducedchargeatanypartoftheplane byCoulomb's Law. Takingtheplanetobetheplaneofyz,andthepointA tobethepoint (a,0,0)ontheaxisofx,wehave 47TO-=R=——- ex —li!• dx(V(ic-a)2+y"+z"\/(x+af+y-+z2 where thelastlinehastobecalculated atthepointontheplaneSatwhich werequirethedensity. Wemust thereforeputx=after differentiation, andsoobtain forthedensityatthepoint 0,y,zontheplane S, 2ae 4>7rcr=— (a2+f+z*f' or,ifa2+y1+z-=r2 ,sothatristhedistance ofthepointontheplaneS from thepointA, ae 2irr •125 •099 •044 •021 •012 •007Thus thesurfacedensityfalls offinverselyasthecube ofthedistance from thepoint A.The distribution ofelectricityonthe planeisrepresented graphicallyinfig.58,inwhich the thickness oftheshadedpartisproportionaltothesurface densityofelectricity. Thenegative electricityis,soto speak, heaped upnear thepointAunder theinfluence oftheattraction ofthechargeatA.The fieldproduced bythis distribution ofelectricityontheplaneSatany pointtotherightof8is,asweknow, exactlythesame as would beproduced bythepoint charge—eatA'. 209. Thisproblemaffords thesimplestillustration ofa general method forthesolution ofelectrostaticproblems, which isknown asthe"method ofimages." Theprinciple underlyingthismethod isthat offindingasystemofelectric chargessuch thatacertain surface, ultimatelytobemade intoaconductor, iscaused tocoincide with theequipotential V=0.We thenreplacethechargesinside thisequipotential bytheGreen'sequivalentFio. 5S. 208-210] Images 187 stratum onitssurface(cf.§204). Asthis surface isanequipotential, we canimagineittobereplaced byaconductor andthechargesonitwillbe inequilibrium. Thesecharges nowbecomecharges induced onaconductor atpotentialzerobychargesoutside thisconductor. From theanalogywithoptical imagesinamirror, thesystemofpoint chargeswhich have tobecombined with theoriginal chargestoproducezero potentialoveraconductor arespokenofasthe"electricalimages"ofthe original charges. Forinstance, intheexample already discussed, thefield is produced partly bythechargeatA,partly bythecharge induced onthe infiniteplane:themethod ofimagesenables ustoreplacethewholecharge induced ontheplane byasingle point chargeatA'.Soalso, ifAwere a candleplacedinfront ofaninfiniteplane mirror, theillumination infront of themirror would beproduced partly bythecandle atA,partly bythelight reflected from theinfinite mirror;themethod ofoptical imagesenables usto replacethewhole ofthisreflectedlightbythelightfromasinglesource atA'. 210. Inanelectrostatic fieldproduced byanynumber ofpoint charges, wecan,aswehave seen, select anyequipotentialandreplaceitbyacon- ductor. Thechargesoneither side ofthisequipotentialarethen the "images"ofthose ontheother side. Thus ifwecanwrite theequationofanysurface intheform -+^+C+.-.=0 (125), where risthedistance fromapointoutside thesurface, andr',r",...arethe distances frompointsinside thesurface, thenwemay saythatcharges e',e", ...atthese latterpointsaretheimagesofachargeeattheformer point. Themethod ofimages maybeappliedinasimilar waytotwo-dimensional problems. Supposethattheequationofacylindricalsurface canbeexpressed intheform o-2elogr-2e'logr'-2e"logr"—...-0, where ristheperpendiculardistance from afixed lineononeside ofthe surface, andr',r",...areperpendiculardistances from fixed linesontheother side.Thenline-chargesofline-densitiese',e",...atthese latter linesmaybe taken tobetheimageofaline-chargeofline-densityeattheformer line. Illustrations oftheuseofimagesinthree dimensions aregivenin §§211—219.Anillustration oftheuseofatwo-dimensional imagewill befound in§220. 188 Methods fortheSolution ofSpecial Problems[ch.viii Charges induced onIntersecting planes, 211. Itwillbefound thatcharges eat x, y, 0, —eat—x, y, 0, —eat x,—y, 0, -er __^eat—x,—y, givezeropotentialovertheplanesx=0,y=0. Thepotentialofthesechargesistherefore the same, inthequadrantinwhich x,yareboth positive,asiftheboundaryofthisquadrant were aconductorputtoearth under thein- fluence ofachargeeatthepoint x,y,0. Itwillbefound thataconductorconsisting ofthreeplanes intersectingatright anglescan betreated inthesameway. 212. Themethod ofimagesalsosuppliesasolution when theconductorFig. 59. 7T consists oftwoplanes intersectingatanyangleoftheform— ,where nis anypositive integer.Ifwetakepolar coordinates, sothat thetwoplanes 7T are6=0,6=- ,andsupposethechargetobeachargeeatthepoint r,6, weshall findthatcharges eat(r, 6),(r,B+^), (r.+^),..., _eat(,.,_*),(r,-(«+^)). (l-(«+£))..... givezeropotentialovertheplanes =0,$=-. 211-213] Images 189 Charge induced onasphere. 213. Themost obvious case, other than theinfiniteplane, ofasurface whoseequationcanbeexpressedintheform(125),isasphere. Fig. 61. IfR,Qareanytwoinversepointsinthesphere, andPanypoint onthe surface, wehave RP:PQ=00:OQ, sothatOQ PQPR°Ji=0. 00Thus theimageofachargeeatQisacharge—e^atR,orthe imageofanypointatadistance /from thecentre ofasphereofradius a ect isacharge-jattheinversepoint,i.e.atapointonthesame radius a2 distant-jfrom thecentre. Letustakepolar coordinates, havingthecentre ofthespherefororigin andthelineOQas=0.Our result isthat atanypoint8outside the sphere, thepotentialduetoachargeeatQandthechargeinduced onthe surface ofthesphere, supposed puttoearth, is ea ~ QSRS e ea Vr2+/2-2/rcos0 /„a< a2a /<V+7^7rc wherer,6arethecoordinates ofS. 190 Methods fortheSolution ofSpecial Problems[ch.viii 214.Wecannow findthesurface-densityoftheinducedcharge.For atanypointonthesphere =B_ l_dV 4nr 47rdr* inwhich wehave toputr=a after differentiation.Clearly dv drea\r—jcos 1 (r2+/2-2/rcosfffi,(,a4_a-n\$'e(r—/cos #) Puttingr—aweobtain —/cos# a8/8-a8/cos (a2+/2-2/acos6>)t (a2/2+a4-2a3/cos0)$!4?rVa2+f*-i e { a—f2/a \ 4tt((a2+y2_2/acos6>)ti e(f2-a2 ) 4tta.£Q3' Thus thesurface-densityvariesinverselyasSQ*, sothat itisgreatestat Cand falls offcontinuallyaswerecede from theradius OC.The total on chargeonthesphereis—j,ascanbeseen atoncebyconsideringthatthe totalstrengthofthetubes offorce which endonitisjustthesame aswould Fig. 62. 214-216] Images 191 bethetotalstrengthofthetubes ending ontheimageatRiftheconductor were notpresent. Figure62shews thelines offorcewhen thestrengthoftheimageisa quarterofthat oftheoriginal charge,sothatf=4>a. Itisobtained from fig.19byreplacingthespherical equipotential byaconductor, andannihi- latingthefield inside. Superposition ofFields. 215.Wehave seen thatbyaddingthepotentialsoftwoseparatefields atevery point,weobtain thepotential produced bycharges equaltothetotal chargesinthetwo fields. Inthiswaywecanarrive atthe fieldproduced byanynumber ofpoint chargesanduninsulated conductors ofthekindwe have described. Thepotentialofeachconductor iszero inthefinal solution because itiszero foreachseparatefield. There isalsoanothertypeoffield which may beadded tothat obtainedbythemethod ofimages, namelythefieldproduced byraisingthe conductor orconductors togiven potentials,without othercharges being present. Bysuperposingafield ofthiskindwecanfindtheeffect ofpoint chargeswThentheconductors areatanypotential. 216. Forinstance, suppose that, asinfig.62,wehave apoint chargee andtheconductor atpotential0.Letussuperposeontothefield offorce already found, thefieldwhich isobtainedbyraisingtheconductor topotentialVwhen thepoint chargeisabsent. Thechargeonthesphereinthesecond field isaV,sothatthetotalchargeis itea aV-j. Bygivingdifferent values toV,wecanobtain thetotal field,when the sphere hasanygiven chargeorpotential. Ifthesphereistobeuncharged, wemust haveV=-^,sothatapoint charge placedatadistance /from thecentre ofanuncharged sphereraises ittopotential-,,aresult which isalsoobvious from thetheorem of§104. 192 Methods fortheSolution ofSpecial Problems[ch.viii Sphereinauniform field offorce. 2YJ.Auniform field offorce ofwhich thelines areparalleltotheaxis ofxmayberegardedasduetoaninfinitechargeEatx=R,andacharge—EatX——R,when inthelimitEandRbothbecome infinite. The intensityatanypointis 2E R* paralleltotheaxis ofx,sothat toproduceauniform field inwhich the intensityisFparalleltothe axis ofx,wemustsuppose EandRto become infinite insuch awaythat -R>=-R dV Since, inthis case,F=—~—,thepotentialofsuch afield willclearly be-Fx+G. "Supposethatasphereisplacedinauniform field offorce ofthiskind, itscentre beingattheorigin. WecansupposethechargeEatx=Rto haveanimageofstrength Ea_a?"XatX~R> while theotherchargehasanimage Ea aa These twoimages mayberegardedasadoublet(cf.§64)ofstrength -p-x-p,andofdirectionparalleltothenegativeaxisofx.Thestrengthit -it -TP—W- Thuswemaysaythattheimageofauniform field offorce ofstrength F isadoublet ofstrength Fa3andofdirectionparalleltothat oftheintensity oftheuniform field. Thepotentialofthisdoublet is Fa3cos6 r»' andthat ofthefield oforiginalfield offorce is -Fx+C, or,inpolar coordinates,—FrcosQ+G, 217] Images sothatthepotentialofthewhole field =—Fcos (r193 a)=)+o .(126). a Fig. 63. As itought,thisgivesaconstantpotential Gover thesurface ofthe sphere. Fig.U. The lines offorce oftheuniform fieldFdisturbedbythepresenceofa doublet ofstrength Fa3areshewn infig.63.Onobliteratingallthelines offorce inside asphereofradiusa,weobtainfig.64,whichaccordingly shews thelines offorcewhen asphereofradius aisplacedinafield of intensityF.Thesefiguresaretaken fromThomson'sReprint ofPaperson Electrostatics andMagnetism (pp.488,489)*. *Iamindebted toLord Kelvin forpermissiontousethese figures. 13 194 Methods fortheSolution ofSpecial Problems[ch.viii 218. Line ofnoelectrification. Thetheoryoflines ofnoelectrification hasalreadybeenbriefly givenin§98.Wehave seenthatonanyconductor onwhich thetotalchargeiszero,andwhich isnotentirelyscreened from anelectric field, there must besomepointsatwhich thesurface-densitya- ispositive,andsomepointsatwhich itisnegative. Theregionsinwhich a ispositiveandthose inwhich c-isnegative mustbeseparated byalineor systemoflinesontheconductor, atevery pointofwhich a=0.These lines areknown aslinesofnoelectrification. IfRistheresultantintensity, wehave atanypoint onalineofno electrification,R=4tto-=0, sothatevery pointofalineofnoelectrification isapointofequilibrium. Atsuchapointtheequipotentialintersects itself, andthere aretwoormore lines offorce. Iftheconductorpossessesasingle tangent planeatapointonalineof noelectrification, thenonesheet oftheequipotential throughthispointwill betheconductor itself: bythetheorem of§69,thesecond sheet must intersect theconductor atright angles. These results areillustrated inthefield offig.64.Clearlythelineofno electrification onthesphereisthegreatcircle inaplane perpendicularto thedirection ofthe field. Theequipotential which intersects itselfalong theline ofnoelectrification(V=G)consists ofthesphereitself andthe plane containingthelineofnoelectrification. Indeed, from formula(126), itisobvious that thepotentialisequaltoC,either when 6=— ,or when r=a. The intersection ofthelines offorcealongthelineofnoelectrification isshewnclearlyinfig.64. Planefacewithhemisphericalboss. 219. Ifweregardthewholeequipotential V=Casaconductor, we obtain thedistribution ofelectricityonaplaneconductor onwhich there isahemisphericalboss ofradius a.Ifwetake theplanetobe a;=0,we have, byformula (126), V-C=-Fcos0(r-^)=-Fx(l-^). Atapointontheplane, 1(dV\ Ff,as ' 4tt\dxJ,=0 4ttJ1 r3 j' andonthehemisphere 47r\drJr=a4-7T' 218-220] Images 195 Thewholechargeonthehemisphereisfound onintegrationtobe Iff-3cos6 )lira? sind6=fFa2 ,=V47T while, ifthehemisphere were notpresent,thechargeonthepartofthe plane nowcovered bythebase ofthehemispherewould be (s)m'=ift' Thus thepresenceoftheboss results intherebeingthree times asmuch electricity onthispartoftheplaneasthere would otherwise be :this is compensated bythediminution ofsurface-densityonthosepartsoftheplane whichimmediatelysurround theboss. Capacity ofatelegraph-wire. 220.Animportant practical applicationofthemethod ofimagesisthe determination ofthecapacityofalong straightwireplaced paralleltoan infiniteplaneatpotential zero, atadistance hfrom theplane.Thismaybe supposedtorepresentatelegraph-wireatheighthabove thesurface ofthe earth. Letussupposethat thewire hasachargeeperunitlength. Tofind thefield offorceweimagineanimage chargedwith acharge—eperunit lengthatadistance hbelow theearth's surface. Thepotentialatapointat distancesr,r'from thewireandimage respectively is,by§§75and100, C—2elogr+2elog r', andforthistovanish attheearth's surface wemust takeC= 0.Thus the potentialis 2elog- .°r Atasmall distance afromtheline-chargewhichrepresentsthetelegraph- wire,wemayputr'=2/i,sothatthepotentialis 2elog— ,°a fromwhich itappearsthat acylinderofsmall radius asurroundingthe wire isanequipotential. Wemaynowsupposethewire tohave afinite radius a,andtocoincide with thisequipotential. Thus thecapacityofthe wireperunitlengthis «£? 13—2 196 Methods fortheSolution ofSpecialProblems [ch.vm Infiniteseries ofImages. 221. Suppose wehavetwospheres,centres A,Bandradii a,b,ofwhich thecentres areatdistance capart,andthatwerequiretofindthefieldwhen Fig. 65. both arecharged. Wecanobtain this fieldbysuperposinganinfinite series ofseparatefields(cf.§116). SupposefirstthatAisatpotential VwhileBisatpotentialzero. Asa first fieldwecantake that ofacharge VaatA.Thisgivesauniform potential VoverA,butdoesnotgivezeropotentialoverB.Wecanreduce thepotentialoverBtozerobysuperposingasecond fieldarisingfrom theimageoftheoriginal chargeinsphere B,namelyachargeatB',c b2 where BB'=— .Thisnew field has,however, disturbed thepotentialover A.Toreduce this toitsoriginalvaluewesuperposeanew fieldarising from theimageofthechargeatB'inA,namelyacharge.?jatA',C c c whereAA'=^.This field inturn disturbs thepotentialoverB,andso c c wesuperposeanother field, and soonindefinitely. Thestrengthsofthe various fields, however, continually diminish, sothatalthough wegetan infinite series toexpressthepotential,thisseries isconvergent. Asweshall see,this series canbesummed asadefiniteintegral,oritmaybethatagood approximationwillbeobtainedbytaking onlyafinitenumber ofterms. The totalchargeonAisclearlythesum oftheoriginal charge Vaplus thestrengthsoftheimages A',A",...etc., forthissum measures the aggregate strengthofthetubes offorce which endonA.Similarlythe chargeonBisthesum ofthestrengthsoftheimagesatB',B", Toobtain thefieldcorrespondingtogiven potentialsofbothAandBwe superposeontothefieldalready found, thesimilar field obtained byraisingBtotherequired potential while that ofAremains zero. 221,222] Images 197 If9ii, 922,Quarethecoefficients ofcapacity andinduction, thetotalcharge onAwhenBistoearth andV—1isqn;similarlythatonBisqu.Inthis waywecanfindthe coefficientsqu>q12from theseries ofimages already obtained. The result isfound tobe o?b asb* qn-a+ c,_fta+ (c,_&2)2_^+..., ab a"-b* ?12~ c c(c2-62-a2 )+-' andfromsymmetry v*-b+^z^-2+ (c2_a,f_b,ci+ •-., Asfaras— ,these resultsclearly agreewith those of§116. 222.The series forqn,q12,q22havebeenputinamore manageable formbyPoisson andKirchhoff. LetA,denote thepositionofthesthoftheseries ofpoints A',A", ...,andBgthesth oftheseries£',B",...;thenAgistheimageofBtinthesphereofradius a,andsimilarly B,istheimage ofAt_\inthesphereofradius b.Letag=AA, ybg=BBg,and letthe chargesatA„Babeea,e'grespectively. Then ag(c—b,)=a2sinceA,istheimageofBg, 68(c-a,_!)=62„B3„ „At_v Further, bycomparingthestrengthsofacharge and itsimage, (127), 198 Methods fortheSolution ofSpecialProblems[oh.vin Theproductofthese roots isunity,sothat ifaistherootwhich islessthanunity, we cansuppose sothat andsimilarly Wenowhavee,=AiP+B' *-S A> 8A'cP+B" 08a8 g„=a+e1+e2+...=ct+2Aa-,s+g, Todetermine A,B,wehave ^+5 a=a, a26 ^^+.8 c2-62' AB 1 sothat where Thus and 7ll=a<l-a{ I^+TZ^a+1-^4+ ...}. TodetermineJ',5',wehave-ei«(i-£2 )' aa'(l-gg)e»~l_^2a28» el=^a& ^'a2+5' ^=-77a«6» 4'a4+.B'c(c2-a2~62)' from which, inthesameway, ?12=__(l- a2)j—+_+i__ 6+ ...J. Thevalue ofq™canofcourse bewritten downbysymmetry from that ofqn. as The coefficients eachdepend onasum ofthetype2-—pr~^ a•This series hasbeen J.gU expressedinterms ofdefiniteintegrals byPoisson. From theknown formularsinpt_,feP+l] _1_ Joe2,T'-l_* tep-lj 2/> weobtain, onputting jd=log£2a2 ", "8_i.°'ft/"a'sin(log£'«»)* ,,, l_^2 a28-fa log|2 a««~Jo^^ From thisfollows sa*_1y f°2a'sin(2logg+2sloga)* l-£2a282(l-a) 21og£+2slog«W Je2rt-l =_ir?*dt«rsin(2<logg)-asin(2<log£/«),, 2(l-a) jol-a2'+1Jo(e27rJ-l)[l-2acos(2doga)+a2J^i 222,223] Images 199 The series hasalsobeen expressedinfinite terms byE.W.Barnes(Quart. Journ. Math. 138(1903), p.155)interms ofDouble Gamma Functions, butneither ofthese forms is convenient fornumerical computation. A.Russell(Proc. Phys.Soc.23(1911), p.352)hasshewn howtheoriginalseries canbe rearrangedinarapidly convergentform. Ifnisaninteger,tobechosensubsequently, arS=co 2 „1_£2„23*=01—5a3=71-1gS s=n-l=2a+2a«(2£2Pa2 P*)1=71 ^P=0' P=co (gan)2p p=01a 3=0W2a28 Thelarger nischosen tobethemorerapidlythesecond series converges, althoughof course largevalues fornrequire thecomputationofalargenumber(n)ofterms inthe originalseries. Asanexample, given byRussell, supposethata=7r, b=r,c=10r;itis sufficient totaken—\andtheseries arefound tobe gr11=7r+fr{l+0-0O89509 +O'0OOO929 +0-0000009 +...}=7-5765970r, -£i2=0-7r +T%r{l +0-0003580 +0-0000001 +...}=08143266r, ?22=ri601124r. Asasecond exampleRussell takes a=98r,6=10- 8r,andc=a+b+0'2r, sothatthe spheresarealmost incontact;thevalues ofthecoefficients areobtained tosevenfigures ontaking n=4andcomputingseven terms ofthesecond series. 223. Havingcalculated the coefficients, wecanobtain the relations between thechargesandpotentials,andcanfind alsothemechanical force between thespheres.Ifthisforce isaforce ofrepulsion F,wehave dWE_Ldpn dpi2FF,dp2,„ oragaindc dc dc oc Thefollowing table, applicabletotwospheresofequal radius, taken tobeunity,is compiled from materials given byLord Kelvin*. c 200 Methods fortheSolution ofSpecial Problems[oh.vni Imagesindielectrics. 224. Themethod ofimagescan alsobeappliedtofind the field produced bypoint charges when half ofthe field isoccupied bydielectric, theboundaryofthedielectricbeinganinfiniteplane. Webegin byconsideringthefieldproduced byasingle chargeeatP,it being possibletoobtain themostgeneralfieldbythesuperpositionofsimple fields ofthiskind. Weshallshew thatthe field inairisthesame asthatduetoacharge eatPandacertainchargee'atP',theimageofP,while the field inthe dielectric isthesame asthatdue toacertainchargee"atP,ifthewhole fieldwereoccupied byair. Fig. 66. LetPP'betaken foraxis ofx,theorigin beingintheboundary ofthedielectric, and letOP=a.Thenwehave toshew thatthepotential YAinairis V.=e+e' */(x+af+y2+z2VO-a)2+y2+z%' while that inthedielectric is V(a?+a)2+f+z*' Thesepotentials, wenotice, satisfy Laplace's equationineachmedium, everywhere exceptatthepoint P,andtheyarise from adistribution of chargeswhich consists ofasingle point chargeeatP.Thepotentialinair atthepoint 0,y,zontheboundaryis VA=e+e Va2+y2+z*' 224,225] Images 201 while that inthedielectric atthesamepointis 7„= Va2+y2+z2 Thus thecondition that thepotentialshall becontinuous ateachpoint oftheboundarycanbesatisfiedbytaking >/e"=e+e(129). Theremainingcondition tobesatisfied isthat atevery pointofthe dV . . dV boundary, ^—inairshall beequaltoK-^-inthedielectric;i.e.that K-—=-^ twhen a;=0.ox ox Now,whenx=0, RdVD^Ke"a dx(a2+y2+z2f dVA ea e'a+ dos(a2+y1+z*f (a3+y2+z2f' sothat this lastcondition issatisfiedbytaking Ke"=e-e'(130). Thus theconditions oftheproblemarecompletelysatisfied bygiving e,e"values such aswillsatisfyrelations (129) and(130);i.e.bytaking 2 ^*"=TTK* K-l.(131). e=-TTKe ) 225. Thepullonthedielectric isthatduetothetensions ofthelines offorcewhich cross itsboundary.Inairthese lines offorce arethesame asifwehadcharges e,e'atP,P'entirelyinair,sothat thewhole tension inthedirection PJP ofthelines offorce inairis ee' pp'2' #(K-l) 4a2 (/iT+iy Thissystemoftensions shews itself asanattraction between the dielectric andthepoint charge.Ifthe dielectric isfree tomove and thepoint charge fixed, the dielectric willbedrawn towards thepoint charge bythis force, andconverselyifthe dielectric isfixed thepoint chargewillbeattracted towards thedielectric bythis force. 202 Methods fortheSolution ofSpecialProblems [oh.vm Inversion. 226. Thegeometricalmethod ofinversion maysometimes beused to deduce thesolution ofoneproblemfrom that ofanother problemofwhich thesolution isalreadyknown. Geometrical Theory. 227. Let beanypointwhich weshall callthecentre ofinversion, and Fig. 67. letABbeaspheredrawn about with aradiusKwhich weshall callthe radius ofinversion. CorrespondingtoanypointPwecanfindasecondpoint P',theinverse toPinthesphere.These twopointsareonthesame radius atdistances from such thatOP .OP'=K\ AsPdescribesanysurfacePQ ...,P'willdescribe some other surface PQ'..., eachpoint Q'onthesecond surfacebeingtheinverse ofsomepoint Qontheoriginalsurface. This second surface issaid tobetheinverse oftheoriginal surface, andtheprocessofdeducingthesecond surface from the first isdescribed asinvertingthe first surface. Itisclear that ifP'Q'...istheinverse ofPQ..., then theinverse of P'Q'..- willbePQ.... Ifthepolar equationofasurface referred tothecentre ofinversion asoriginbe/(r,8,<p)=0,then theequationofitsinverse will be f[— ,0,<£J=0.Forthepolar equationoftheinverse surface isby definition /(r, 0, </>)=0,where rr'=K-forallvalues of6and</>. 226,227] Inversion 203 Inverseofasphere. Letchords PP', QQ',...ofasphere meet in (fig. 68).Then 0P.0P' =0Q.0Q'=... =t\ where tisthelengthofthetangentfrom tothesphere. Thus, iftisthe radius ofinversion, thesurface PQ...istheinverse ofP'Q'..., i.e.thesphere Fig. 68. isitsown inverse. With some other radius ofinversion K,letP"Q".. theinverse ofPQ ....then 0P.0P"=0Q.0Q'=... =K\ OP" OQ" IP-•••- ^be sothat Thus theinverse ofaOP' OQ'" andthelocus ofP",Q",...isseen tobeasphere sphereisalwaysanothersphere. Aspecial investigationisneeded when thesphere passes through0.Let OSbethediameterthrough 0,and let 8'bethepointinverse toS.Then,if P'istheinverse ofanypointPonthe circle, 0P.0P' =0S.0S\ OP_OS' or0S~0P" sothatPOS, S'OP' aresimilartriangles. SinceOPS isaright angle,itfollows thatOS'P' isaright angle,sothat the locus ofP'isaplane throughS'perpen- dicular toOS'. Thus theinverse ofa spherewhichpasses throughthecentre ofinversion isaplane, and, conversely,theinverse ofanyplaneisasphere whichpasses throughthecentre ofinversion.Fig. 69. 204 Methods fortheSolution ofSpecial Problems[ch.viii 228. IfP,Qareadjacent points onasurface, and P',Q'arethecorre- sponding pointsonitsinverse, thenOPQ, OQ'P' aresimilartriangles,sothatPQ, P'Q'makeequal angleswithOPP'. By making PQ coincide, wefind that the tangent planeatPtothesurfacePQ andthetangent planeatP'tothesur- faceP'Q'makeequal angleswithOPP'. Hence,ifweinvert twosurfaces which intersect inP,wefind that theangle between thetwoinverse'surfaces atP'isequaltotheangle between the originalsurfaces atP,i.e.anangle ofintersection isnotalteredbyinversion. Also, ifasmall conethroughcuts offareas dS,dS'from thesurface PQ... and itsinverseP'Q'...,itfollows that d#OP' dS'- OP'*'Fig. 70. ElectricalApplications. 229. LetPP',QQ'betwopairsofinversepoints (fig. 70). Letacharge eatQproduce potential VpatP,and letachargeeatQ'produce potential VpatP',sothat VP'= then Take thenP~PQ*~ P'Q' Il-i ?Q-iop VPe-P'Q'~ e'OQ" e~OQ~ K' VpOPK VPK~OF' Now letQbeapointofaconducting surface, andreplaceebycrdS, thechargeontheelement ofsurface dSatQ.LetVpdenote thepotential ofthewhole surface atP,and letVpdenote thepotentialatP'duetoa chargee'oneachelement dS'oftheinverse surface, such that e'OQ' adSKK Then, since Vp=Vp-^p,foreachelement ofcharge, wehavebyaddition VP'=VvK Thus chargese'ondS', etc.produceapotential VPK OP'atP'. 228-230] Inversion 205 NowsupposethatPisapointontheconducting surfaceQ,sothat VPbecomessimplythepotentialofthissurface, sayV.Thechargeseon dS', etc.nowproduceapotential Qpiatjt, sothat ifwith thesecharges wecombine acharge—VKat0,thepotential producedatP'iszero. Thus thegiven systemofcharges spread overthe surface P'Q' ...,togetherwith acharge—VK attheorigin, make the surface P'Q'...anequipotentialofpotentialzero. Inother words, from a knowledgeofthedistribution which raises PQ... topotential V,wecan findthedistribution ontheinverse surface P'Q'...when itisputtoearth under theinfluence ofacharge—VKatthecentre ofinversion. Ife,e'arethechargesoncorresponding elements dS,dS'atQ,Q',we have seenthat e'a'dS'KOQ' /OQ' "~0Q~K"Vi eadS0Q~K~VOQ' dS' OQ'"whlledS=W „ a(0Q'\-% K3„ocl,Hence7-(w)=W>(132) ' givingtheratio ofthesurface densities onthetwoconductors. Conversely,ifweknow thedistribution induced onaconductor PQ...at potentialzerobyaunitchargeatapoint 0,thenbyinversion about we obtain thedistribution ontheinverse conductor P'Q'... when raised to potential -^..Asbefore, theratio ofthedensities isgiven byequation (132). Examples ofInversion. 230. Sphere. Thesimplestelectricalproblemofwhich weknow the solution isthat ofasphereraised toagiven potential.Letusexamine what this solution becomes oninversion. Ifweinvert withrespecttoapointPoutside thesphere, weobtain the distribution onanothersphere whenputtoearth under theinfluence ofa point chargeP.This distribution hasalreadybeen obtained in§214by themethod ofimages. The result there obtained, that thesurface-density variesinverselyasthecube ofthedistance fromP,cannowbeseen atonce fromequation (132). Soalso, ifPisinside thesphere, weobtain thedistribution onan uninsulatedsphere produced byapoint chargeinside it,aresult which can againbeobtained bythemethod ofimages. WhenPisonthesphere, weobtain thedistribution onanuninsulated plane, alreadyobtained in§208. 206 Methods fortheSolution ofSpecial Problems[ch.vm 231.IntersectingPlanes. Asamorecomplicated exampleofinversion, letusinvert theresults obtained in§212.Wethere shewed how tofind Fig. 71. 7T thedistribution ontwoplanes cuttingatanangle— ,whenputtoearth lb under theinfluence ofapoint charge anywhereintheacuteanglebetween them. Ifweinvert thesolution weobtain thedistribution ontwospheres, cuttingatanangle nr\n,raised toagiven potential. Byasuitable choice oftheradius andoriginofinversion, wecangiveanyradiiwelike tothe twospheres. Ifwetake theradius ofonetobeinfinite, wegetthedistribution ona planewith anexcrescence intheform ofapieceofasphere:inthepar- ticular case ofn=2,thisexcrescence ishemispherical,andweobtain the distribution ofelectricity onaplanefacewith ahemisphericalboss. This can,however, beobtained moredirectly bythemethod of§219. Spherical Harmonics. 232. Theproblemoffindingthesolution ofanyelectrostaticproblemis equivalenttothat offindingasolution ofLaplace's equation throughoutthespacenotoccupied byconductors, such asshallsatisfycertain conditions attheboundaries ofthisspace—i.e.atinfinity andonthesurfaces ofconductors. Thetheoryofsphericalharmonicsattemptstoprovidea generalsolution oftheequation V2F=0. This isnoconvenientgeneralsolution infinite terms :wetherefore examine solutionsexpressedasaninfinite series. Ifeach term ofsuch aseries isasolution oftheequation,thesum oftheseries isnecessarily asolution. 231-233] Spherical Harmonics 207 233. Letustakespherical polar coordinates r,6, <f>,andsearch for solutions oftheform V=RS, whereRisafunction ofronly,andSisafunction of6and$only. Laplace's equation, expressedinspherical polars, canbeobtainedanalyti- callyfrom theequation d2Vd-v a2r dx*+ df+d?~° bychangingvariables from x,y,ztor,6,<f>,but ismosteasilyobtainedby applyingGauss' Theorem tothesmall element ofvolume boundedbythe spheresrandr+dr,thecones 6and6+d6,andthediametralplanes (j>and <f>+dj>.Theequationisfound tobe r»dr\dr)+ r*sin6ddV™ d0J+r2sin2 d(f>*"' SubstitutingthevalueF=RS,weobtain^/ 2^\_R_ d_(.„dS\ _E_c^S r2drV3rJr2sin6dd\&m dd)+ r*sin26d<fr" ' or,simplifying, i9/,as\ la/ ..as\ ,jia^_ JR3rVdr)+8sin6d0[8m dd)+Ssin26d<f>* The firstterm isafunction ofronly,while thelasttwoterms areinde- pendentofr.Thus theequationcanonlybesatisfied bytaking 1d(3>R> RdrX*Tr)=K<133>' 13/ .adS\ ,1 82#„/1Q ., whereKisaconstant.Equation (133), regardedasadifferentialequation forR,canbesolved, thesolutionbeing &-*** +£»(135), where A,Barearbitrary constants, andn(n+1)=K.Aftersimplification equation (134) becomes ^Hl)+S5n>5+"<"+1>s=°<186> Anysolution ofthisequationwillbedenotedbySn,thesolutionbeinga function ofnaswellasof6and <f>.The solution ofLaplace's equation we have obtained isnow V=RS=(Ar»+J^Sn, andbytheaddition ofsuch solutions, themostgeneralsolution ofLaplace's equation maybereached. 208 Methods fortheSolution ofSpecial Problems[oh.Tin 234. Definitions. Anysolution ofLaplace's equationissaid tobea spherical harmonic. Asolution which ishomogeneousinx,y,zofdimensions nissaid tobea spherical harmonicofdegreen. Asphericalharmonic ofdegreenmust beoftheform rnmultiplied by afunction of6and$>,itmust therefore beoftheformArnSn,whereSn isasolution ofequation (136). Anysolution 8nofequation (136)issaid tobeasurface-harmonic of degreen. 235. Theorem. IfVisany spherical harmonic ofdegree n,then yjrm+ifaasphericalharmonicofdegree—(n+1). ForVmust beoftheformArnSn,sothat VASn rzn+irn+i which isknown tobeasolution ofLaplace's equation, and isofdimensions —(n+1)inr.ConverselyifVisasphericalharmonic ofdegree—(n+1), then r2n+1Visasphericalharmonic ofdegreen. 236. Theorem. IfVisanyspherical harmonicofdegree n,then fis+t+uy dafdyW where s,t,anduareanyintegers,isaspherical harmonic ofdegree n—s—t—u. d*v dn-vd*v AFora?+5E+^"a ' sothatondifferentiation stimes withrespecttox,ttimes withrespecttoy, andutimes withrespecttoz, gs+t+u+2"|7 fls+t+u+2y fis+t+u+^y daf+2dytdzu+ dx*dyt+*dzu+dxsdytdzu+2= ' °r V' [dtfdyw)=°' which provesthetheorem. 237. Theorem. IfSm,Snaretwosurfaceharmonicsofdifferent degrees m,n,then \\SnSmda)=0, where theintegrationisover thesurface ofaunitsphere. InGreen's Theorem(§181), (<£V2¥--¥V«<D) dxdydz=- \\(<P^-¥^)dS,dn dnJ put<£=rnSn>^=rmSm,andtakethesurface tobetheunitsphere. 234-239] Spherical Harmonics 209 ThenV23>=0,Va¥=0,^-=-5-=-nrn-1#n,and^-=-mrm-1/Sm.on dr 9n Thus thevolumeintegral vanishes, andtheequation becomes [j(nrm+n-ig nSm_mrm+n-i^ n/Sfm)da)=0, or,sincenis,,byhypothesis,notequaltom, onomd(o=0.# Harmonics ofIntegral Degree. 238. Thefollowingtable ofexamplesofharmonics ofintegral degrees 7i=0, —1,-2, +1,istaken fromThomson andTait's NaturalPhilosophy. _,.?/ ,r+3 .», r+2rz(x2-y2 )2rxyzn-0. 1,tan-1^,log ,tan"^log ,,\,*./,..,,%,,.#' r-2 a;°r—z(x2+y2 )2(V+y2 )2 Also ifVisanyoneofthese harmonics, -~-^, -^-^,-~-^areharmonics ofdegree—1,so that r-tt-^,r-yr-^, r-^~ areharmonics ofdegree zero. Asexamples ofharmonics derivedoxdyoz inthiswaymaybegiven rxryzxzy x x x2~+y2> X2+1J2'X2+1J2'X2+y2'T+l'T^z' Bydifferentiating anyharmonic Vanynumber *oftimes, multiplying byr2,_1and differentiating agains-1times, weobtain more harmonics ofdegreezero. n=—1.Anyharmonic ofdegreezerodivided byrordifferentiated withrespectto x,yorz,e.g. 1 1,.y1.r+z x x ,-tan1- ,-logr'r x' r°r—z'x2+y2' r(r+z)' n=-2.Bydifferentiating harmonics ofdegree—1with respecttox,yorzweobtain harmonics ofdegree—2,e.g. x y z z ,.y z,r+z -q> H-> ~%,-,tan-1 '2 -,-log. 11=1.Multiplying harmonics ofdegree—2byr3 ,weobtain harmonics ofdegree 1,e.g .y ,r-\-z n x,y,z,atan-1-,slog--—2?\'^ ' a? r—z Rational Integral Harmonics. 239.Animportantclass ofharmonic consists ofrationalintegral algebraic functions ofx,y,z.Inthemostgeneral homogeneousfunction ofx,y,zof degree nthere are\(n+1)(n+2)coefficients. IfweoperatewithV2we areleftwith ahomogeneousfunction ofx,y,zofdegree n—2,andtherefore possessing \n(n—1)coefficients. Fortheoriginalfunction tobeaspherical harmonic, these%n(n— 1)coefficients must allvanish, sothatwemust have^n(n—1)relations between theoriginal ^(w+l)(?i+2)coefficients. j 14 210 Methods jortheSolution ofSpecial Problems[ch.viii Thus thenumber ofcoefficients which mayberegardedasindependentin theoriginal function, subjecttothecondition ofitsbeingaharmonic, is ±(n+I)(n+2)-4 in(n-l), or2n+1.This, then, isthenumber ofindependentrational harmonics of degreen. For instance, when n=1themostgeneralharmonic is Ax+By+Cz, possessingthreeindependent arbitrary constants, andsorepresentingthree independentharmonics which mayconvenientlybetaken tobex,yand z. When n=2,themostgeneralharmonic is ax2+by2+cz2+dyz+ezx-Yfxy, where a,b,caresubjecttoa+b+c=0.The fiveindependentharmonics may convenientlybetaken tobe yz, zx, xy,x2—y2 ,x2—z2 . When n=0,2n+1=1.Thus there isonlyoneharmonic ofdegree zero, andthismaybetaken tobeV— 1. Correspondingtoarationalintegralharmonic Vnofpositive degree n, y there istheharmonic-—^ofdegree—(n+1).These harmonics ofdegree —(n+1)areaccordingly2/i+1innumber. Thus theonlyharmonic of thiskindandofdegree—1is Consider nowthevariousexpressionsofthetype gs+t+u /J> .(137),da?dyfdzu\r where s+t+u=n. These, asweknow, areharmonics ofdegree—(n+1),andfrom§235 y itisobvious thattheymust beoftheform~^i,whereVnisarational integralharmonic ofdegreen.Since -isharmonic, V2(- J=0,sothat d2[l\ fd2d2\fl\ /loox Themostgeneralharmonic obtained bycombiningtheharmonics of type (137)is 2^u3^a^(r)(139)' butbyequation (138)thiscanbereduced atonce totheform dzpqda?By*\rjpqdx*dy* \r)' 239,240] Spherical Harmonics 211 wherep+q=n—1andp+q=n.Thisagainmaybereplaced by dz„=0pdxPdyf^1-P\rJ PZpdxPoyn~v\rJ' sothat there are2w+1arbitraryconstants inall,and itisobvious onexamination that the harmonics, multiplied byallthe coefficients Bp,...Bp',...areindependent. Thus, bydifferentiating-ntimes, wehave arrived at2n+1independentrationalintegral harmonics, and itisknown that this isasmanyasthere are. ExpansioninRational Integral Harmonics. 240. Theorem*. The valueofany finite single-valued function of position onaspherical surfacecan heexpressed,atevery point ofthe surfaceatwhich thefunctioniscontinuous, asaseriesofrationalintegral harmonics, providedthefunctionhasonlyafinite numberoflinesandpoints ofdiscontinuity andofmaxima andminima onthesurface. LetFbethearbitraryfunction ofpositiononthesphere,and letthe spherebesupposedofradius a.LetPbeanypointoutside thesphereata distance /from itscentre 0,and letQbeanypoint onthesurface of thesphere. p iiG. 72. LetPQbeequaltoR,sothat R*=f*+a2-2a/cosPOQ. Wehave theidentity f2-a*[fdS_a.(140),4>wa JJR3f"" where theintegrationistaken over thesurface ofthesphere,aresult which itiseasytoprove byintegration. Apoint chargeeplacedatPinduces surface density--—„3onthesurface of thesphere (§214), andthe total induced chargeis-~i- Theidentityistherefore obvious from electrostatic principles. *Theproof ofthistheorem isstated intheformwhich seems bestsuited totherequirements ofthestudent ofelectricity andmakes nopretenceatabsolute mathematical rigour. 14—2 212 Methods fortheSolution ofSpecial Problems[ch.vm Now introduce aquantity udenned by f*-a? CfFdSu='a- 4<ira[[FdS.(141), sothatuisafunction ofthepositionofP.IfPisveryclose tothe sphere,/2—a2issmall, andtheimportantcontributions totheintegralarise from those terms forwhichRisverysmall :i.e.fromelements near toP. Ifthevalue ofFdoes notchange abruptlynear tothepoint P,or oscillate with infinitefrequency, wecansupposethat asPapproachesthe sphere,allelements onthespherefrom which thecontribution tothe integral (141)areofimportance,have thesame F.This value ofFwillof course bethevalue atthepointatwhichPultimatelytouches thesphere, sayFp.Thus inthelimitwehave (/2-a2 )FPreds 4>ira JR*u=.(142), a=Fp-f,byequation (140), =FP, when inthelimit/becomesequaltoa. Ifthevalue ofFoscillates with infinite frequencynear tothepoint P,weobviously may nottakeFoutside thesignofintegrationinpassingfromequation (141)to equation (142). Ifthevalue ofFisdiscontinuous atthepointPofthespherewithwhichP ultimately coincides, weagain cannot takeFoutside thesignofintegration. Suppose, however, thatwetakecoordinatesp,3toexpressthepositionofapointP'onthesurface ofthesphere verynear toP,thecoordinatepbeingthedistance PP", and3beingthe angle whichPP'makes withanylinethroughPinthetangent planeatP.ThenF mayberegardedasafunction ofp,3,andthefactthatFisdiscontinuous atPisexpressed bysayingthat asweapproachthelimitp=0,thelimiting value ofF(assuming sucha limit toexist)isafunction of3—i.e.dependsonthepathbywhichPisapproached. LetF(3)denote this limit. Then u--_/2-aa fF(3) Pdpd3 AnaiM 4-rra Aira Ztt]F(3)r/2-/rlidS~13 d3 1^ 2ttF(3) (-.)d3,byequation (140). Onpassingtothelimitandputting a—f,wefindthat u=±fF(S)43.... •(143), 240] Spherical Harmonics 213 i.e.uistheaverage value ofFtaken onasmall circle ofinfinitesimal radius surroundingO P.Inparticular,ifFchanges abruptly oncrossingacertain linethroughP yhavinga value Fiononeside,andavalueF2ontheother, then thelimiting value ofuis u=$(F l+F2). Ifwetake todenote theanglePOQ, -^=(/2-2a/cos0+a2)-^ 1/ a2-2afcos 6\-h 1 7L,a2-2a/cos6 /a2-2a/cos0\3 •1- o »z r-gI~ I— f" P or,arrangingindescending powers of/ .(144), inwhich i?,P±,R,...arefunctions of6,being obviouslyrationalintegral functions ofcos6.When 6=0, andwhen =ir, sothatwhen 6=0, andwhen #=ir,•••Ji1/_aa p=p— —l —P—P=—P=-—1 Itisclear, therefore, that theseries (144)isconvergentfor=and 6=7r,andaconsideration ofthegeometrical interpretationofthis series willshew that itmust beconvergentforallintermediate values*. Differentiating equation (144) withrespectto/weget 1 dR acos6—f~R3dfa a2 Wys-Z% Ti-(145). Ifwemultiplythisequation by2/andaddcorrespondingsides to equation (144), weobtain F Multiplyingthisequation by—-r— ,andintegratingoverthesurface ofthe sphere, weobtain p-a? [[FdS_™2n +l 47ra R3o4ttFRa1 fn+ldS, *Being apowerseries incos6itcanonlyhave asingle radius ofconvergence, and this cannot bebetween cos=1andcos0=-l. 214 Methods fortheSolution ofSpecial Problems[ch.vin or,byequation (141), *=i^!<2"+i>/M7Hrf& Ifthefunction Fiscontinuous andnon-oscillatoryatthepoint P,then onpassingtothelimitandputting f=a,weobtain ^i{2n +l)ffFPndS(146). *'J 4vra2 IfFisdiscontinuous andnon-oscillatory, then thevalue oftheseries ontherightis notF,but isthefunction defined inequation (143). Now itisknown that1/risaspherical harmonic, sothatwehave where thedifferentiation iswithrespecttothecoordinates ofQ.Hence. 1/R must beoftheform(cf.§233) 1«,/ , .B h=x {At"+?&)8* <147^ whereSnisasurface harmonic oforder n.Comparingwithequation (144), andrememberingthatainthisequationisthesame astherofequation (147), weseethatPlX,regardedasafunction ofthepositionofQ,isasurface harmonic oforder n,andwehavealreadyseen that itisaseries ofpowers CO ofcos9,orof- ,thehighest power beingthenth,sothatrnPnisarational integral harmonic oforder n.Itfollows that FrnPndS, beingthesum ofanumber ofterms each oftheform rnPn,isalsoarational integralharmonic oforder n,sayVn.Onthesurface ofthesphere Vn=an fJFPndS, sothatequation (146) becomes '-as!5^7- (148 >' which establishes theresult inquestion. 241.Theorem. Theexpansion ofanarbitrary function ofposition onthe surface ofasphereasaseriesofrationalintegralharmonics isunique. For ifpossibleletthesame function Fbeexpandedintwoways, say F=tW n (149),F=XW n' (150), whereWn,Wn'arerationalintegralharmonics oforder n.Then thefunction u=2(W;i-Wn) 240-243] Spherical Harmonics 215 isaspherical harmonic, which vanishes atevery pointofthesphere. Since V-u=atevery pointinside thesphereitisimpossibleforutohave either amaximum oraminimum value inside thesphere (cf.§52),sothatu= atevery pointinside thesphere.SinceWtl—Wn' isaharmonic oforder n, itmust beoftheform rn8n,where Snisasurface harmonic, sothat u=lrnSn=0. Thus uisapowerseries inrwhich vanishes forallvalues ofrfrom r= tor=a.ThusSn= for allvalues ofn.HenceWn=Wn',andthetwo expansions (149) and(150) areseen tobeidentical. 242. Itisclear that inelectrostatics weshall ingeneral onlybe concerned with functions which arefinite andsingle-valuedatevery point, andofwhich thediscontinuities arefinite innumber. Thus theonlyclasses ofharmonics which areofimportancearerationalintegral harmonics, and in future weconfine ourattention tothese.Wehave found that (i)The rationalintegralharmonics ofdegree nare(2w+1)innumber, andmayallbederived from theharmonic -bydifferentiation. (ii)Anyfunction ofpositiononaspherical surface, which satisfies the conditions which obtain inaphysical problem,canbe expandedasaseries ofrationalintegral harmonics, p"p P' and thiscanbedoneonlyinoneway. 243. Beforeconsideringthese harmonics indetail, wemay trytoformsome idea ofthephysical concep- tionswhich lead tothem mostdirectly. The function -isthepotentialofaunitcharge attheorigin. If,asin§64,weconsider twocharges +eatpoints 0',0"atequalsmall distances a,—a from theorigin alongtheaxisofx,weobtain asthe potentialatP, e e e eO'OO' Fia.73. V=OP0"P~~OP" OP' =-e.PP 1(1dx\r axisIfwetake-e .PP"=1,wehave adoublet ofstrength-1paralleltothe r)/I\ ofx,andthepotentialatPis^-f- J.Infactthispotentialisexactly x thesame as— 3alreadyfound in§64. 216 Methods fortheSolution ofSpecial Problems[ch.vm Thus thethree harmonics oforder—1obtainedbydividingtherational integralharmonics oforder 1byr3 ,namely ^-(-J.k~(-),k~(-),are simplythepotentialsofthree doublets each ofunitstrength, parallelto thenegativeaxes ofx,y,zrespectively. Ifinfig.73wereplacethechargeeat0'byadoublet ofstrengthe paralleltothenegativeaxis ofx,andthecharge—eat0"byadoublet ofstrength—eparalleltothenegativeaxis ofx,weobtain apotential -<-).dx2\r/ Ifinstead ofthedoubletsbeing paralleltotheaxis ofx,wetakethem paralleltotheaxisofy,weobtain apotential a2/i> dxdy \r, Sowecangoonindefinitely,forondifferentiatingthepotentialof asystemwithrespecttoxwegetthepotentialofasystemobtained byreplacingeach unitchargeoftheoriginal system byadoublet ofunit strength paralleltotheaxis ofx.Thus allharmonics oftype fiS+t+Uf I) dxsdytdzwV? (cf.§236)canberegardedaspotentialsofsystemsofdoublets attheorigin, and, aswehave seen(§239),itisthesepotentials whichgiverisetothe rationalintegralharmonics. 244. Forinstance infinding asystemtogivepotential ~-^(- j,wemay replace the chargeinfig.73byacharge—atdistance 2afrom and-—at0.Thechargeat0' maybesimilarly treated, sothatthewhole systemisseen toconsist ofcharges E,-2E, E, atthepointsx=—b,0,bwhere 6=2a,andE2= j-2. Asystemofthiskindplaced along each axisgives acharge-6E attheorigin and achargeEateach corner ofaregular octahedron having theorigin ascentre. The potential ~ dx2\rj 8y2\rjdz2\r) =0, sothatsuch asystem sends outnolines offorce. 245. Themostimportantclass ofrationalintegralharmonics isformed byharmonics which aresymmetricalabout anaxis, saythat ofx.There is oneharmonic ofeachdegree n,namelythatderived from thefunction —(-)dxn\r)' These harmonics weproceedtoinvestigate. 243-247] Spherical Harmonics 217 Legendre's Coefficients. 246. Thefunction ,- n=—(151)Va2-2arcos^+r2 can,aswehavealreadyseen(cf.equation (144)),beexpandedinaconvergent series intheform 1 1 r r2rn va2-2arcos <9+r2axa2a3an+1 ifaisgreaterthan r.Here thecoefficients %,!%,... arefunctions ofcos0, and areknown asLegendre'scoefficients. When wewish tospecifythe particularvalue ofcos0,wewriteF^,asPn(cos 0). Interchangingrandainequation (152)wefind that,ifr>a, =-+pA+P^+ (1-53).Va2-2arcos+r2r r2r3 Wehave alreadyseen that thefunctions I{,P^,...aresurface harmonics, eachterm oftheequations (152) and(153) separately satisfying Laplace's equation. Theequationsatisfied bythegeneralsurface harmonic Snof degreen.namely equation (136),is dSn\ d2SnHw)+;J*b+"<"+i>*-asm6d0\ d0Jsm2 0d<f>2 Inthepresentcasei^isindependentof<p,sothatthedifferentialequation satisfied byPnis or,ifwewritefiforcos0, rA(1-^? £}+Hn+1)^° (15i)- Thisequationisknown asLegendre's equation. 247.Byactualexpansionofexpression (151) sothatonpickingoutthecoefficient ofrn ,weobtain D_1.3...2w-1 1.3... 2n~3n_21.3...2n-5n_4_ ^~JH^~~27(n-2)\*+ 2.4.(ti-4)!/""" (155). ThusPnisaneven oroddfunction offxaccordingasniseven orodd. It will readilybeverified thatexpression (155)isasolution inseries of equation (154). 218 Methods fortheSolution ofSpecial Problems[ch.viii Letustakeaxes Ox,Oy,Oz,theaxisOxtocoincide with theline=0, then\xr=rcos6=x.Then itappearsthatPnrnisarationalintegralfunction ofx,y,andzofdegree n,and,beingasolution ofLaplace's equation,itmust bearationalintegralharmonic ofdegreen.Wehave seen that there can onlybeoneharmonic ofthistypewhich isalsosymmetricalabout anaxis; this, then,must bePnrn . 248. Ifwewrite (a2-2ar/M+r2 )-^=/(a) wehave,byMaclaurin's Theorem, /(a)=/(0) +adm da+a2p/(a)l a=2 da2+... a=0.(156). IfPisthepointwhosepolarcoordinates are a,and Qisthepoint r,6,thenf{a)= -p-~.TheCartesian co- ordinates ofPmaybetaken tobea,0, ;letthose ofQbe #,y,z.Then/(a)=,,sothatasregards V(a?-a)2+2/2+.z2 differentiation of/(a), 3_ 8a9_ dx'Fig. 74. Thus 3»(aart Ja=oV' 1dx-Uo 9*n(0) 3^nV«2+y2+£2 jp/r 9#nvr, sothatequation (156) becomes 1d_(l\a2^/laBx\r)+2 !&Z2Vr andoncomparisonwithexpansion (153), weseethat nn!dxn\r)' givingtheform forPnwhich wehavealreadyfound toexist in§245. 249.Amore convenient form forPncanbeobtained asfollows. Let 1-hy=(1-2h/j,+A2 )* (157), sothaty=fM+hy2-\ .(158). 247-251] Spherical Harmonics 219 From this relation wecanexpand ybyLagrange's Theorem(cf.Edwards, Differential Calculus, §517) intheform y=/1+Aa_+...+_y(V)+-- Differentiating withrespecttofx, Fromequation (157), however, wefind jp=(1- 2hfj,+h*)-*=l +hP1+...+hnPn+.... Equatingthecoefficients ofhninthetwoexpansions, wefind .fi-Rn©V-i>-ass). 250. This lastformulasuppliestheeasiest wayofcalculatingactual values of/£.Thevalues ofi?,P2,...P 7arefound tobe ^(A4)=A*t i^(^)=1(3^-1), /£(/*)=M°>8-3/*)» J30»)=£(35/^-3(y +3), />(/.)=£(63^-70^+15//), P6O)=J B(23V-B15fi*+105/a"-5), P7(fi)=TV(42V-693/i6+315/a8-35/*). 251. Theequation (/i2—l)n=has2nreal roots, ofwhich nmaybe regardedascoincidingat/*=1,andwat/*=-1.Byawell-known theorem, thefirstderivedequation, !>'-i)"=o, willhave 2n—1real rootsseparatingthose oftheoriginal equation. Passingtothenthderivedequation, wefindthat theequation |>-iy=o hasnreal roots, andthat these must allliebetween/u.=—1andp.==+1. The roots are allseparate,fortwo roots couldonlybecoincident ifthe original equation (/*2—l)n=hadn+1coincident roots. Thus thenroots oftheequation Pn(fx,)=areallrealandseparateand liebetween/x=—1and/m=+1. 220 Methods fortheSolution ofSpecial Problems[en.vm 252. Putting jx=1,weobtain l+P 1h+PJi2+...=\fl-2h +h2 =l+h+h2+..., sothati?=i?=...=1.Similarly, when/a=-1,wefind(cf. §240)that _£=+£=-£=... =-1. Wecannowshew thatthroughouttherangefrom/*=—1to/t=+l, thenumerical value ofi^isnevergreaterthanunity. Wehave (1-2hcos6+h?)'i=(1-heie)-$(1-Ae"ie )"^ x (1+\zhe~id+i^|/i2e-2ifi+ ...), sothatonpickingout coefficients ofhn , D1.3...2»-1a11.3...2n-3, ON. ,P-= 2.4...2n2coBwg +2-2.4...2n-22cOB(n-2)g+"" Everycoefficient ispositive,sothatPnisnumerically greatestwhen each cosine isequaltounity,i.e.when 6=0.ThusPnisnevergreaterthan unity. Fig.75shews thegraphsofP1}P^,Pz,P*,fromp.=—1to/*=+1,the value of6beingtaken asabscissa. »=o6=1 6=3n 0=7, y-+ i \\\^v1 \\'V^ /\ 252,253J Spherical Harmonics 221 Relations betweencoefficients ofdifferent orders. 253.Wehave (l-2h(j. +h*)-i=l+2hnPn (160). Differentiatingwithregardtoh, (fx-h)(l-2h fM+hi)-*=C knhn-'Pn (161), i 00 sothat(ji-h)(1+^hnPn)=(1-2hfi+h?)tnhn^Pn. i i Equatingcoefficients ofhn ,weobtain (n+l)P n+1+nPn_1=(2n+l) fiPn (162). This isthedifferenceequationsatisfied bythree successive coefficients. Again,ifwedifferentiateequation (160) withrespecttop. op sothat,bycombiningwith(161), 1 Ofl Equatingcoefficients ofhn , r> 0-tnOrn—\ /Inn\nPn= f*d^--d]T(16o >- Differentiating (162), weobtain op Eliminating /x-^from thisand (163), (fc.+l)J»-?gi-^(164). Byintegrationofthisweobtain /sco**-w£;t*0t) <165>- whilst bytheaddition ofsuccessiveequationsofthetypeof(16-1), we obtain 3-Pn=(2n-l)P n-l+(2n-5)P n^+ (166). 0(1 222 Methods fortheSolution ofSpecial Problems[ch.viii 254.Wehavehadthegeneraltheorem(§237) jjsnsmda>=o, from which thetheorem JJpn(ji)Pm(M)dOi= follows asaspecialcase. Orsince dco=sin6d0d<j>=—d/idfy, j+1Pn^)Pm{ji)d li=Q(167). Tofind IFtf (/Jb)dfi,letussquaretheequation o multiply byd/x,andintegratefrom/*=—1toyu.=+1. The result is r+i<*> J-l"+1<» -l allproductsoftheformPnPmvanishingonintegration, byequation (167). Thus IPndfiisthecoefficient ofA2Kin [+1dp J-i1- i.e.in2/i/x+A»' 1,1-/i 9 and this coefficient iseasily seen tobe— Weaccordinglyhave £{««}•*- £^1d<58). 255.Wecanobtain thistheorem inanother way,and inamoregenera] form, by using theexpansionof§240,namely FpBSdd>!(2s+1}jjFF*(cos^dS> where 6istheangle between thepointPandtheelement dSonthesphere. This expansionistrue foranyfunction Fsubjecttocertain restrictions. TakingFtobea surface harmonic Snofordern,weobtain (^„)p=^-2T(25+1)([snPt(cos 6)dS «=o In 4^fJSnPn(cos 6)dS, 254-25G] Spherical Harmonics 223 allotherintegrals vanishing bythetheorem of§237.Thus //AM^i^OU*-! orJfsnPn(fjL)da>=~-(S n)^ l (169). This isthegeneral theorem, ofwhichequation (168) expressesaparticular case. To passtothisparticular case,wereplace SnbyFn(/x)andobtain, instead ofequation (169), ff{Pn(M)Fsindd6d4>=~Pn(1), or,afterintegratingwithrespectto<f>, agreeing withequation (168). ExpansionsinLegendre's Coefficients. 256. Theorem. Thevalueofanyfunction of6,which isfiniteand single-valued from6= to6=ir,andwhich hasonlyafinite numberof discontinuities and ofmaxima andminima within thisrange,can be expressed, forevery value ofwithin thisrange forwhich thefunctionis continuous, asaseries ofLegendre's Coefficients. This issimplyaparticularcaseofthetheorem of§240. Itistherefore unnecessarytogiveaseparate proofofthetheorem. Theexpansioniseasilyfound. Assume ittobe /(//,)=a+a1Pl+a2P2+...+asPs+ (170), thenonmultiplying byP„,{ii)dfi, andintegratingfrom/*=—1to/x,=+l, weobtain r+i s=oo r+i Ipn(a0/(/*) dfil=Za8Ps(p)Pn(/*) dfj, s=0 J—l —1 S= J—1 la, 2n+1' every integral vanishing, exceptthat forwhich s=n.Thus 2w+1r+i a*jy.i^fi^dfM(i7i),2 givingthecoefficients intheexpansion. Iff(fjb)hasadiscontinuity when/jl= /j, ,thevalue assumedbythe series(168) onputting /j,= /*<,is,asin§240, equalto iWW+/«} (172), wherefi(/J> ),/3(/x)arethevalues off(/j.)onthetwosides ofthediscon- tinuity. 224 Methods fortheSolution ofSpecial Problems[oh.viii Harmonic Potentials. 257.Wearenowinapositiontoapplytheresults obtained toproblems ofelectrostatics. Consider firstasphere havingasurface densityofelectricity Sn.The potentialatanyinternal pointPis 'Snds rr sndsVP=f[SndS_[[ JJPQJJV5 -//2—2arcos6+r2 ^(1+-£(cos 6)+-*£(cos 0)+...)dS b7T In+1an+1 4tt rnSn0,2~^+i(Sn)coso=i> bythetheorems of§§237and255, .(173),2n+1a™ thisexpression beingevaluated atP. SimilarlythepotentialatanyexternalpointPis 4,7ran+2Snp (2n+l)r»+1' Thesepotentialsareobviouslysolutions ofLaplace's equation, and itis easytoverifythattheycorrespondtothegivensurfacedensity,for \3^/outside \<W inside Thisgivesusthefundamentalpropertyofharmonics, onwhich their applicationtopotential-problems depends•Adistributionofsurface density Snonasphere givesrise toapotential which atevery pointisproportional toSn. 258. Thedensityofthemostgeneralsurface distribution can,bythe theorem of§240,beexpressedasasum ofsurface harmonics, say a—S+Si+S.j+..., inwhichSisofcoursesimplyaconstant. Thepotential, bytheresults of thelast section, is (S/r\S/r\2 )V=4<'7ra\S +~{-j +-^i-j+...Y ataninternalpoint ...(174), =4?rtt\S(-)+oM-)+-f(-) +•••[atanexternalpoint ...(175). 257-259] Spherical Harmonics 225 Examples oftheuseofHarmonic Potentials. I.Potentialofspherical capand circularring. 259. Asafirstexample,letusfindthepotential ofaspherical cap ofanglea—i.e.thesurface cutfromasphere by arightcircular cone ofsemiverticalanglea— electrified toauniform surfacedensity<x . Wecanregardthis asacomplete sphere electrified tosurfacedensity a,where a-=afrom 6= to6=a, a=0 from 6=ato6=ir. Thevalue ofabeing symmetrical about the axis 6=0,letusassume forthevalue ofa expandedinharmonics a=a+axI^(cos6)+aiJ^(cos6)+... then,byequation (171), 2n+lf9=°Fig. 76. an= 2 2w+laPn(cos6)d(cos6) [0=Pn(cos6)d(cos6) J9=a =J<r[Pn-i (cosa)-Pn+1(cos a)} byequation (165), except when n=0. Forthiscasewehave = «o=\°"ofd(cos6)=Jo-(1—cosa).=a Thus h<To (1-cosa)+2JPn-i(cos a)-i^+1(cosa)[•2J(cos6) j»=H J J Itisofinterest tonotice thatwhen 6=a,thevalue ofagiven by this series isa=^cr 0)asitoughttobe(cf.expression (172)). Thepotentialatanexternalpointmaynowbewritten down inthe a-cos.)(?)+Tu»^(»^) £p.(cos9)- (176),"form V=2iraa andthat ataninternalpointis V=2iraa(i-cos«)+TJL-E"«>-*«(««> fry tt=l Ztt+1W .(177). 15 226 Methods fortheSolution ofSpecial Problems[oh.viii Ondifferentiating withrespecttoa,weobtain thepotentialofaringof linedensity<radcc. Atapointatwhich r>a,wedifferentiateexpression (176), andobtain -1+2i^(cosa)sina(-] i£(cos#) I, or,puttingacrda=tandsimplifying, F=2ttt 2Pn(cosa)sina(-)Pn(cos6) .(178).n= v*/ Obviouslythepotentialatapointatwhich r<acanbeobtained on replacing(^byQ. 260. These last results canbeobtained moredirectly byconsidering that atanypointontheaxis=thepotentialis 2iraT sina or,ifr>a,V=-,„vr2H-a2—2arcosa 27rar sinan=°°YP» (cos«)(£)*,n=0 v/ andexpression (178)istheonlyexpansioninLagrange'scoefficients which satisfiesLaplace's equation andagreeswith thisexpression when 6=0. II. Uninsulatedsphereinfield offorce. 261. Themethod ofharmonics enables ustofind the field offorce producedwhen aconducting sphereisintroduced intoanypermanentfield offorce. Letussupposefirstthat thesphereisuninsulated. Fig. 77. 259-261] Spherical Harmonics 227 Letthespherebeofradius a.Round thecentre ofthe field describe aslightly larger sphereofradius a,sosmall asnottoenclose anyofthe fixedcharges bywhich thepermanentfield offorce isproduced.Between these twospheresthepotentialofthe field willbecapableofexpressionin aseries ofrationalintegral harmonics, say V=V+V1+Vi+ (179). Theproblemistosuperposeonthis apotential, produced bythe induced electrification onthesphere,which shallgiveatotalpotential equaltozero over thespherer=a.Clearlytheonlyformpossiblefor thisnewpotentialis r—*®-*®'-*@'-<180>- Thus thetotalpotentialbetween thespheresr=aandr=a'is Putting Vn=rnSn,thesurfacedensityofelectrification onthesphere is, byCoulomb's Law, S(2ti+1)K.4<7ra This result isindeed obvious from§258,onconsideringthat the surface electrification mustgiverise tothepotential (180). Ifnisdifferent from zero, fjsoSndS=Q, where theintegrationisoveraDysphere,sothat JJsndS=(n^O), andffVndS=(n^O) (181). Thus thetotal chargeonthesphere =--jV .kira?=-Va, 4>7ra andT^wasthepotentialoftheoriginalfield atthecentre ofthesphere. 15—2 228 Methods J"ortheSolution ofSpecial Problems[ch.viii 262.Incidentally wemay notice, asaconsequenceof(181), that the mean value ofapotential averagedover thesurface ofanysphere which does not include anyelectricchargeisequaltothepotentialatthe centre(cf.§50). Ifthesphereisintroduced insulated, wesuperpose ontothe field already given,the field ofachargeEspread uniformlyover thesurface of E thesphere,andthepotentialofthis field is— .Weobtain theparticular case ofanuncharged sphere bytakingE=VQa,andthepotentialofthis field, namely Io(- )>justannihilates the firstterm inexpression (180),to which ithastobeadded. Itwilleasilybeverified that, ontakingthepotentialoftheoriginal field tobeVi=Fx, wearrive attheresultsalreadyobtained in§217. III. Dielectricsphereinafield offorce. 263.Ananalogoustreatment willgivethesolution when ahomo- geneousdielectricsphereisplacedinapermanentfield offorce. The treatment will, perhaps,besufficiently exemplified byconsideringthecase ofthesimplefield ofpotential V1=Fx=rS1. Letusassume forthepotential VQoutside thesphere Fig. 78. and forthepotential Viinside thesphere Vi=l3rS 1} doterm oftheform-Jbeingincluded inVi,asitwouldgiveinfinite 262-264] Spherical Harmonics 229 potentialattheorigin. The constants a,/3aretobedetermined from theconditions Vi=X I ira^==a^Utr=a. dr drJ Thesegive a+-=/3a, (Jj a3 whence a=--^—~a3 ,/3= sothat V=Fx\l-jK+2™'H~K+2' K-l[c£3^ r, V<=KT2F*- Thus thelines offorce inside thedielectric are allparalleltothose of theoriginal field,buttheintensityisdiminished intheratio^—=.The field isshewn infig,78. IV.Nearly spherical surfaces. 264. Ifr=a,thesurface r=a+x>where%isafunction of6and <f>,will representasurface which isnearly sphericalif^issmall. Inthiscase% mayberegardedasafunction ofposition onthesurface ofthespherer=a, andexpandedinaseries ofrationalintegral harmonics intheform X=S+S1+S.i+... inwhich SltS2,...areallsmall. Thevolume enclosed bythissurface is £jjr3do) =i(a3+Sa*x)dec 47ra3 00 4}ira3 .„~=—s-+4-Tra2S . If$o=0,thevolume isthat oftheoriginal spherer=a. 230 Methods fortheSolution ofSpecial Problems[en.vin Thefollowing specialcases areofimportance: r=a+eP^Toobtain theform ofthissurface, wepassadistance ecos6 alongtheradius ateachpointofthespherer=a.Itiseasilyseen that when eissmall thelocus ofthepointssoobtained isasphereofradius a, ofwhich thecentre isatadistance efrom theorigin. rsa+ajSj. Themostgeneralform fora^ isIx+rny+nz,and this maybeexpressedasaecos6,where 6isnowmeasured from the line of which thedirection cosines areinthe ratio I:m :n.Thus thesurface is thesame asbefore. r=a+S2.Since risnearly equaltoa,thismaybewritten r2=a2+2aS 2 2=a2+-r2S2,a or a?+y2+z2=a2+anexpressionoftheseconddegree. Thus thesurface isanellipsoidofwhich thecentre isattheorigin.Itwill easilybefound thatr=a+eP2representsaspheroidofsemi-axes a-j-e,a—^, 3eandtherefore ofellipticity~-. 265.Wecantreat thesenearly sphericalsurfaces inthesamewayin whichsphericalsurfaces have been treated, neglectingthesquaresofthe small harmonics astheyoccur. 266. Asanexample, supposethesurface r=a+Sntobeaconductor, raised tounitpotential. Weassume anexternalpotential A™fa\n+1 r \rj whereAandBhave tobefound from thecondition thatV=l when r=a+8 n-Neglecting squaresofSn,thisgives A/Sn a\ a) sothat A=a,B=~, ct aan and V=-+—-T-.Sn. /v%,^71+1 ByapplyingGauss' Theorem toasphereofradiusgreaterthan awe readilyfind that the totalchargeisa,the coefficient of-.Thus the 264-267] Spherical Harmonics 231 capacityoftheconductor isdifferent from that ofthesphere only by terms inSn2 ,butthesurface distribution isdifferent, for a dVW•<• 1 uo.4)7ro-=—^—=—-z—,itweneglect £L2 >on or° -£+(«+*) ;Ib«• a2\ a/Va2 1TO-1 a a"5 thesurfacedensity becoming uniform, asitought,when n=l,i.e.when the conductor isstillspherical. 267. Asasecondexample,letusexamine the field inside aspherical condenser when thetwospheresarenotquiteconcentric.Takingthecentre oftheinner asorigin,lettheequationsofthetwospheresbe r=a, Wehave tofindapotentialwhich shall have, say,unitvalue overr=a, andshall vanish overr=b+ei?.Assume V^+i^+C +DZr,r r2 whenBandDaresmall, thenwemust have Theseequations must betrue allover thespheres,sothat thecoefficients ofi?andtheterms which donotinvolve T\must vanishseparately. Thus -+C-1=0;-+Da =0;a a2 j+g=°>-e4+¥+Db=o - From the firsttwoequations abA=- , b—a and thisbeingthecoefficient of-inthepotential,isthecapacityofthe condenser. Thus toafirstapproximation,thecapacityofthecondenser remains unaltered, butsinceBandDdonotvanish, thesurface distribution isaltered. 232 Methods fortheSolution ofSpecial Problems[ch.vni . Y.CollectionofElectricCharges. 267 a.Ifacollection ofelectricchargesarearrangedinany"way whateversubject onlytothecondition thatnone ofthem lieoutside the sphere r=a,then thepotentialatanypointoutside thesphere must be -p.eSiS2 '— "1 «"I ;T•••J where eisthetotalchargeinside thesphere (cf.§266)andSlt&>,...are surface harmonics whichdependonthearrangementofthechargesinside thesphere. Ifthetotalchargeisnot zero, thepotentialcanalsobetreated asin §67,andoncomparingthetwoexpressionsobtained forthepotential, we canidentifytheharmonics S1}S2,....Wefindthat and itwillbeeasilyverified bydifferentiation thattheexpressions onthe rightareharmonics. Thisexampleisofsome interest inconnection with theelectron-theoryofmatter, for acollection ofpositive andnegative chargesallcollected within adistance aofacentre may givesome representationofthestructure ofamolecule. The total charge ona molecule iszero,sothatwemust take e=0,andthepotentialbecomes Themostgeneral form forStis(cf.§239)-(Ax+By +Cz),orncos6,where 6isthe angle between thelinesfrom theorigintothepoint x,y,zandthat tothepointAtB, &ndixisJ(A2+B2+C2 ). . ,.ucos6 ,Thus theterm which isimportantinthepotentialwhen rislargeis-— ^— ?shewing that atasufficient distance themolecule hasthesame fieldofforce asacertain doublet of strength /*.Clearly whenfihasanyvalue different from zero, themolecule is"polarised" (cf.§142)inFaraday'ssense. If/x=0,the potentialbecomes shewing thattheforcenow falls offastheinverse fourth powerofthedistance. Itisworthnoticingthattheaverageforce atanydistance risalways zero, sothat to obtain forces whichare,ontheaverage, repulsive, wehave toassume thepresence of terms inthepotentialwhich donotsatisfy Laplace's equation,andwhich accordingly arenotderivable from forces obeyingthesimple law e/r2 (cf.§192). 267a-269] Spherical Harmonics 233 Further Analytical Theory ofHarmonics. General Theory ofZonal Harmonics. 268. Thegeneral equationsatisfied byasurface harmonic oforder n, which issymmetricalabout anaxis, hasalreadybeen seen tobe k{(1-^}+n(n+1)Sn=0 (182)' One solution isknown tobePn,sothatwecan find theotherby aknown method. Assume Sn=Pnuasasolution, where uisafunction ofix.Theequationbecomes (l-", )|i{i"+-p»|}-2'i{|"+p»|}+»(»+l)'p»M=-(183)' and, sincePnisitself asolution, (W')|;(g)-V§+»(»+l)ii=0. Multiplyingthisbyuandsubtractingfrom(183),weare leftwith or,multiplying byPnandrearranging, oragain£{<!-*>*)g+{d-rf>SI1g)-0. Onintegrationthisbecomes (1-u?)Pn*^=constant. OfM Wemaytherefore take inwhich thelimitsmaybeanyweplease.Ifwewrite «»=<c™(184)- thecompletesolution ofequation (182)is Sn=Pnu=APn+BQn. 269. Thetwosolutions PnandQncanbeobtained directly bysolving theoriginal equation (182)inaseries ofpowersof/a. Assume asolution Sn=bofS+blfx^+b^+*+..., 234 Methods fortheSolution ofSpecial Problems[ch.viii substitute inequation (182), andequatetozero the coefficients ofthe differentpowersoffi.The first coefficient isfound tobebr(r— 1),so that ifthis istovanish wemust have r=orr=1.Thevalue r=leads tothesolution n(w+l) , ,(n-2)w(n+l)(w+3) ..4Uo-i--12fi+1.2.3.4^'" while thevalue r=1leads tothesolution (n-1)(n+2) (n-3)(n-1)(n+2)(n+4) Wl_/tEO^+1.2.3.4.5**-• Thecompletesolution oftheequationistherefore au+/3WJ. Ifnisintegraloneofthetwo series terminates, while theother does not. Ifniseven theseries uterminates, while ifnisoddtheterminating series isw,.Butwehavealreadyfound oneterminatingseries which is asolution oftheoriginal equation, namelyPn.Hence ineither case the terminatingseries must beproportionalto1^,and therefore the infinite series must beproportionaltoQn. 270.Wecanobtain amore useful form forQnfromexpression (184). The roots ofi^(fi)=are,aswehave seen,ninnumber, allrealand separate,andlyingbetween —1and+1.Letustake these roots tobe a1}a2,...an.Then 1u+1 via-a.(u,- a.)Vv fi-1fi+1 \fi-a, (jj,-asyj onresolvingintopartialfractions.Putting fj,—+1and—1,wefindatonce thata=\,&=—£. Inthegeneralfraction 1 1 D{x—al){x—ai)...i letussupposeallthefactors inthedenominator tobedistinct, sothatwe maywrite C-% Co+——+....Dx—Oxx—a2 Onputting#=&!,weobtain atonce 0i= {ax-a,)(a,-a,)(a,-a4)...' 1 C2= ;r-— - ,etC (a2—a2)(a,-a3)(a2—a4)... Spherical Harmonics 235 Now letaaand <zabecomevery nearly equal, sayaa=ax+dux>then 1c,=- da-i(a1—a3)(a 1—a4)...' while The fractions nowcombine intoCo= da,!(a2-a3)(<x2—a4)...' Cl ,C2 (ci+c2)#- (cia2-c^ ) andonputtingthisequalto Cn Co+x—ax(x—a-,)-' itisclear thatthevalue ofc/must betaken tobecx+c2.Now 2 da-L((a2-a3)( 1(3-iif1 I 1 a2—a4)...(oj—a3)(«i—at)...) da-,[dx\(#-a,)(a?-a4).../»=«, 3*1£JW andthisremains truehowever manyoftheroots a3,a4...,coincide among themselves, solongastheydonotcoincide with theroot Oj.Thus, in expression (185), thevalue ofcgis 3 fQ*-«.)»] Putting wefindthat•%(/*) /*-«•=£(/*)» c8=.if. a/.l(i-^){iswpj „,„, s«,Ki- «.•)(B(«.))•;• Since(/a—ag)i£(/*)isasolution ofequation (182), wefindthat ^[(1-^R00+0*-«.)^}]+n(n+1)0*-«.)-B=0. Onputting fi—a8,thisreduces to da.{(1 aa>)B(a 8)}+(l-« s>)d4^=0,das giving,onmultiplication byR(as), ~[d-^){R(as)Y]=0. Hence cs=0. 236 Methods fortheSolution ofSpecial Problems[ch.vui Equation (185)nowbecomes 1 0*-i){&00}--] sothat,onintegration,fl—1fM+1+xd„ F=£log^ +£ds .(186),(/*»-1){Pn(ft)}* »"-°/*-l- /*-<% Onmultiplying byPn{fj) }weobtain fromequation (184), whereWn-!isarationalintegralfunction of/j,ofdegree n—1. Itisnowclear thatQn(fi)isfinite andcontinuous from/*=—1to/a=+1, butbecomes infinite attheactual valuesfi=+1. Tofindthevalue ofT^_iwesubstitute expression (186)inLegendre's equation,ofwhich itisknown tobeasolution, andobtain 9 {(1-^)^1+^+1)1^ 3/4 --$&-»& ^^og^)yn(n+l)^Pn(^log^±\ =2 dp, =2{(2n-l)P n-1+(2n-5)P n-3+...} (187). SinceWn-iisarationalintegral algebraicfunction of/xofdegreen—1,it canbeexpandedintheform "n— l=&\Pn—\ H"a2-*n— 2+•••5 sothat 91(1-^)^1 +n(n+1)1^ =%ag1 {(1-^)%s j+n(n+1)i?,_ s _9yU, ( 3/1 =2a«(n(n+1)—(w-s)(n-5+1)}i^_ g. Comparingwith (187), wefindthatas=when sisodd,and isequalto 2(2?i-2s+l) s(2n-s +l) when siseven. Thus w2n-lp2tt-5p2n-9 and Qn=£-& (/*)log'"_I'3(n-l) /t+12ft-1 /*+ 1.1*-*w— l5(»-2) 2w-5 3(w-l)-*Tl— 3t ••• 270-273] Spherical Harmonics 237 271.When wearedealingwithcomplete spheresitisimpossiblefor thesolution Qntooccur. Ifthespaceislimited insuch awaythat the infinities oftheQnharmonic areexcluded, itmaybenecessarytotake intoaccount both the l?nandQnharmonics. Aninstance ofsuch acase occurs inconsideringthepotentialatpointsoutside aconductor ofwhich theshapeisthat ofacompletecone. Tesseral Harmonics. 272. Theequationsatisfied bythegeneralsurface harmonic Snis sin090 V oUJsm26d<f>- Asasolution, letusexamine sn=©s>, where©isafunction of6only,and <l>isafunction of$only.On substitutingthisvalue intheequation,anddividing by@<£/sin2 6,weobtain sin6d(.ad®\132<E> ,, ,-x.,aA Wemust therefore have Id2®_ <£>8</>2"*' sin6d(..9©\,.-,x•2/, The solution oftheformerequationissinglevaluedonlywhen kisofthe form—m2 ,wheremisaninteger.Inthiscase <E>=Cmcosmc/>+Dmsinm<£, and©isgiven by 1d(.aa©\f,,xm2 )_ _ sin-^(Sm^8^)+r(?l+1)-sin^|=O> or,interms offi, 4^-^S+ i"(-+1)-Ale-° (188)' anequationwhich reduces toLegendre's equation whenm=0. 273. Toobtain thegeneralsolution ofequation (188),consider the differentialequation (1-^)^+2/^=(189), ofwhich thesolution isreadilyseen tobe z=C(\-ti?)n(190). Ifwedifferentiateequation (189)stimesweobtain 238 Methods fortheSolution ofSpecial Problems[ch.vin Ifinthisweputs=n,andagaindifferentiate withrespecttop,we obtain hfr-K®}***1*®-(192)' dnzwhich isLegendre's equationwith^—-asvariable. Thus asolution ofthis equationisseen tobe givingatonce theform forPnalreadyobtained in§249. Thegeneral solution ofequation (192) weknow tobe d^n=APn+BQn. Ifwenow differentiate (192)mtimes, theresult isthesame asthat of differentiating (189)m+n+1 times, and istherefore obtainedbyputting s=m+n+1in(191). Thisgives (l-/^|-^-2(m+l)^^^m or,multiplying by(1— /a2 )2 , 0--V?)* o,.m+n+2-2(m+1)/*(1- /x2 )2- !»2m+n » +(m+ri+l)(7i-m)(l-^2 )2^T- n=(193). Let (l-/x2 )2|—--=». Then*(l-^.£-* ^(i-^. *r"'^H1-^p^-(2™+2)M-^5; 1^a^£{a-io£}-(i-^JS-^ +^a-^K 2; »_/) 3m+n-. ffl(l-/i')s-^(l -JU2 )2 =-i;j(m+n+1)(n-m)+m-,^7— ,f>byequation (193), =—vUi(n+ 1)—m+n 2,,2 ?n2 1-/rj Thus vsatisfies andthis isthesame asequation (188), which issatisfied by®. 273,274] Spherical Harmonics 239 274. Thesolution ofequation (188) hasnowbeen seen tobe ©=(1-a2)2- - where .^—n=APn+BQn. mdmP -?mOHence .=^(1-^)^+5(1-^^ Thefunctions(W)2^,(W»)2^? areknown astheassociatedLegendrianfunctions ofthe firstandsecond kinds, andaregenerallydenoted byP% (/x),Q™(ft).Asregardstheformer wemayreplacePn,fromequation (159), by 1dn———(u?-1V» andobtain thefunction intheform 1 VI7m+np"(^)=2^(i-^6^2-i)n (194)- Itisclear from thisform that thefunction vanishes ifm+n>2n, i.e.if m>n.Itisalso clear that itisarationalintegralfunction ofsin6and cos 6.From theform ofQn(/j.),which isnotarationalintegralfunction ofll, itisclear thatQ™(/x)cannot bearationalintegralfunction ofsin6and cos6. Thus ofthesolution wehave obtained forSn,onlythepart P£(fi)(Cmcosm<f>+Dmsinm$)% givesrise torationalintegralharmonics. ThetermsP™(/x)cosm<f)and P™(/a)sinm(f>areknown astesseral harmonics. Clearlythere are(2n+1)tesseral harmonics ofdegree n,namely Pn(fi),cos<f>Pl(fi),sin4>P\(ji),...cosw^PJO*),sinn#P£(/4 These mayberegardedasthe(2n+1)independentrationalintegralhar- monics ofdegreenofwhich theexistence hasalreadybeenprovedin§239. Usingtheformula andsubstitutingthevalue obtained in§247 forPn(fi) (cf.equation (155)), weobtainP™(/x)intheform (2n)!sin- fl fn_m_(n-m)(n-m-l) n_m_2 ^W= 2»W!(n-m)ilC°S * 2(2»-l) (»-to)(n-to-1)(n-m-2)(n-m-3) B_jn_4._1+2.4(2»-l)(2w-3) "T 240 Methods fortheSolution ofSpecial Problems[ch.viii Thevalues ofthetesseral harmonics ofthefirstfourorders aregivenin thefollowingtable. Order 1. cos0,sin cos0,sin sin0. Order 2.£(3cos3—1),3sin cos cos0,3sin cos sin0, 3sin2cos20,3sin2sin20. Order 3. £(5cos3-3cos0),fsin(5cos2-1)cos0, •Isin(5cos2—1)sin0,15sin2cos cos20, 15sin2cos sin20,15sin3cos30, 15sin3sin30. Order 4.£(35cos"0-30 cos2+3),§sin(7cos3-3cos0)cos0, |sin(7cos3-3cos0)sin0, jy*sin2 (7cos20-1)cos20, -V5-sin2(7cos2-1)sin20,105sin3cos cos30, 105sin3cos sin30, 105sin4cos40, 105sin4sin40. 275.Wehavenowfound thatthemostgeneralrationalintegralsurface harmonic isoftheform n Sn=%P% (ft)(Amcosm0+Bmsinm0), o inwhich P™(/a)istobeinterpretedtomeani^(/i), whenm=0. Letusdenoteanytesseral harmonics ofthetype P™(/i)(.4cosm0 +Psinm0) byS™. Thenby§237,jlS%fl™,day= ifn={=«'.Ifw=w',then JJ8Sfl*= //*T(A*)*?'00(4»cosm0+flmsinm0) (.4m'cosm+i?^'sinm'0)d&>, andthisvanishesexcept whenm=m'. When n=n'andm=ra'thevalue of 11S%S%fdwclearly dependson that of I[P™ (fi)}2 dfi,andthiswenowproceedtoobtain. Wehave r+l r+l/pirnp\2 J.i1^0*)N/'=j_ i(i-/*, )B(y** (1-^)' 274-276] Spherical Harmonics 241 dnz Since^—-=i?tisasolution ofequation (191),weobtain, ontakings=to+n inthisequation, andmultiplying throughout by(1— fM2 )m~\ dm~1P+(n+to)(w-to+1)(1- ya2 )"1"1 g-^?, which, again,maybewritten Inequation (195)the firsttermontheright-hand vanishes, sothat f+lr+l /^m-ip\2 J_t{P- (/*)}*dp=(n+to)(n-to+1)J^(1-^r-1 (-g-^Jrf/* '=(n+to)(n-to+1)J**{P™-1 0")}2^, areduction formula fromwhich wereadilyobtain " (p»«)«,=( (^™| ;/*'(p„«}>*. 2(n+m)! 2/«+1(n—to)!" These results enable ustofindanyintegralofthetypeJ1$nS'nc?a>. Biaxal Harmonics. 276. Itisoften convenient tobeable toexpresszonal harmonics referred tooneaxis interms ofharmonics referred toother axes—i.e.to beable tochangetheaxes ofreference ofzonal harmonics. Let^beaharmonic having OPasaxis. AtQthevalue ofthis is Pn(cos 7),where 7istheangle PQ,andourproblemistoexpressthis harmonic oforder nasasum ofzonal and tesseral harmonics referred to other axes. With reference tothese axes, letthecoordinates ofQbe6,ty, letthose ofPbe©,<£,and letusassume aseries ofthetype s=nPn(cos7)=2P"n(cos6)(Ascossty+Bssinsty). Letusmultiply byPs n(cos6)cosstyandintegrateover thesurface ofaunit sphere. Weobtain J[pn(cos7)[Ps n(cos6)cossty)do=A,JJ{Pn(cos 0)}2cos2 stydco. j. 16 242 Methods fortheSolution ofSpecial Problems[ch.viii Byequation (169), JJPn(cos7){Ps n(cos6)coss<J>]dco=^^{P*(cos 0)coss<£jy=0 =2^1P»(cos®)coss®, and J|{P;(cos 0)}2cos2 s<f>do= j+l {P°n(/x)}2dp[^cos250d<j> Thus2?r(n+g)! 2n+1(n-s)!' (r>—<AI -4.=2i_^;pj(cos0)cos5$,(w+s)! andsimilarly ^=2 (^|-;P«(cos@)SmS«>. Thisanalysisneeds modification when s=0,but itisreadilyfound that 4o=£(cos0), Pn=0, sothat P»(cos7)=Pn(cos0)P„(cos0)+T2^—4-;P*(cos0)P^(cos0)coss(6-<£)«=i (?i+s)! (196). General Theory ofCurvilinear Coordinates. 277. Letuswrite 0> 2/,*)=\ yfr(x,y,z)=p, X0>V>z)=v> where $,yjr,^denoteanyfunctions ofx,y,z.Thenwemaysupposeapoint inspace specified bythevalues ofX, /j,,vatthepoint,i.e.byaknowledgeof thosemembers ofthethree families ofsurfaces </>(x,y,z)—cons.;^(x,y,z)=cons.;%(x,y,z)=cons. whichpassthroughit. Thevalues ofX,ft,varecalled"curvilinear coordinates"ofthepoint.Agreat simplificationisintroduced into theanalysisconnected with curvilinear coordinates, ifthethree families ofsurfaces arechosen insuch awaythattheycutorthogonallyatevery point.Inwhat follows weshall supposethis tobethecase—thecoordinates willbe" orthogonalcurvilinear coordinates." Thepoints X, /x,vandX+dX,p,vwillbeadjacent points,andthe distance between them willbeequaltodXmultiplied byafunction of 276-278] General Curvilinear Coordinates 243 X, /a,and v—letusassume itequaltoy-•Similarly,letthedistance ft] fromA,, fx,vtoX,/x+d/x,vbe-7- ,and letthedistance from X,/x,vto rfra 7 i_'^" A,,u,*>+afbe-j—. Then thedistance dsfrom X, /x,vtoX+d\, /x+d/x,v+dvwillbe given by thisbeingthediagonalofarectangular parallelepipedofedges dXd/x 1dv fti' ft2 ^3 Laplace's equationincurvilinear coordinates isobtained mostreadily by applyingGauss' Theorem tothesmallrectangular parallelepipedofwhich theedgesaretheeight points X±\dX,1^+2 dfi> v+\dv. Inthiswayweobtain therelation •dV intheform// n<®=(197) dX\h. 2h3oXJdix\h 3hidfx) dvX^h* dvJ andaswehavealreadyseen thatequation (197)isexactly equivalentto Laplace's equation V2V=0,itappearsthatequation (198) mustrepresent Laplace's equationtransformed intocurvilinear coordinates. Inanyparticular systemofcurvilinear coordinates themethod ofpro- cedure istoexpress h^,h2,h3interms ofX,/xandv,andthentrytoobtain solutions ofequation (198), givingVasafunction ofX,fxand v. Spherical Polar Coordinates. 278. Thesystemofsurfaces r=cons., 6=cons.,cf>=cons, inspherical polarcoordinatesgivesasystemoforthogonalcurvilinear coordinates. In these coordinatesequation (198) assumes theform drVdr)+ sin6BO\Sm dOJ+ sinsdp~ ' alreadyobtained in§233,which hasbeen found tolead tothetheoryof spherical harmonics. 1G—2 244 Methods fortheSolution ofSpecial Problems[oh.vm Confocal Coordinates. 279. Afterspherical polar coordinates, thesystemofcurvilinear coordi- nates which comes next inorder ofsimplicity andimportanceisthat in which thesurfaces areconfocalellipsoids andhyperboloidsofoneandtwo sheets. Thissystemwillnowbeexamined. Takingtheellipsoid asastandard, theconicoid x2l+l2-^1 "d") 1-1^+^=1(200)a-+ b2+6c2+ willbeconfocal with thestandardellipsoid whatever value 6may have,and allconfocal conicoids arerepresentedinturnbythisequationas8passes from—ooto+co . Ifthevalues ofx,y,zaregiven, equation (200)isacubicequationin6. Itcanbeshewn thatthethree roots in6are allreal, sothatthree confocals passthrough anypointinspace, and itcanfurther beshewn that atevery pointthese three confocals areorthogonal.Itcanalsobeshewn that of these confocals one isanellipsoid,oneahyperboloidofonesheet, andone ahyperboloidoftwo sheets. Let A,fi,vbethethree values of6whichsatisfy equation (200)atany point,and letA,fi,vreferrespectivelytotheellipsoid, hyperboloidofone sheet, andhyperboloidoftwo sheets. Then A, /u,,vmaybetaken tobe orthogonalcurvilinear coordinates, thefamilies ofsurfaces A=cons.,lc=cons., v=cons,being respectivelythesystemofellipsoids, hyperboloidsofone sheet, andhyperboloidsoftwo sheets, which areconfocal with thestandard ellipsoid (199). 280. The firstproblem,asalready explained,istofindthequantities which havebeen denoted in§277byh1}7i2,hs.Asasteptowardsthis,we begin byexpressing x,y,zasfunctions ofthecurvilinear coordinates A,lc,v. Theexpression ^2/i»2 /y2 isclearlyarationalintegralfunction of6ofdegree 3,thecoefficient of6s being—1.Itvanishes when 6isequaltoA.,/xorv,thesebeingthecurvi- linear coordinates ofthepoint x,y,z.Hence theexpression must beequal, identically,to-(0-\)(0- fi)(6- v). Putting6=—a2intheidentityobtained inthisway,wegettherelation x2 (b2-a2 )(c2-a2 )=(a2+A)(a2+fi)(a2+v), 279-282] Confocal Coordinates 245 sothat x,y,zaregivenasfunctions ofA,/x,vbytherelations .(a2+A)(a2+/z)(a2+^) "- (b2-a2)(o2-a2 )GtC<201>- 281. Toexaminechangesaswemovealongthenormal tothesurface A=cons.,wemustkeep /*and vconstant. Thuswehave, onlogarithmic differentiation ofequation (201), „dx_dX *j—== xa2+A,' andthere areofcourse similarequations giving dyand dz.Thus forthe lengthdsofanelement ofthenormal toX=constant, wehave (ds)2=(dx)2+(dy)*+(dz)* tV\t c(a2+VK&2-a2)(c2-a2 ) =i/V7\ Y>(A-^)(A-y)tK > (a2+A)(62+A)(c2+A)' Thequantitydsis,however, identical with thequantitycalled -=—in §277, sothatwehave 4(q2+A)(62+A)(C2+A)*~ (X-f,)(X-v){^2) > andclearlyh2andA3canbeobtained bycyclic interchangeofthe letters A,fiand v. 282. Ifforbrevity wewrite AA=V(a2+A)(b*+A)(c2+A), wefindthat Ms2AMA, sothatbysubstitution inequation (198), Laplace's equationinthepresent coordinates isseen tobe ^r-^A^^}+a7r-x)A^^r3-4(^~^A^^r° (203). Onmultiplying throughout byAAAMA„,thisequationbecomes (204). 246 Methods fortheSolution ofSpecial Problems[ch.viii Letusnowintroduce new variables a,/3,7,given by "KdX A*' *-/'£' f"dv f") A.' thenwehave —=AA— ; andequation (204) becomes d2V ?2V ?PV (/1_ 1/)|^+(l/_x)|Z+(x_ /i)|I=(205). Distribution ofElectricityonafreely-charged Ellipsoid. 283. Beforediscussingthegeneralsolution ofLaplace's equation,itwill beadvantageoustoexamine afewspecial problems. Inthe firstplace,itisclear thataparticularsolution ofequation (205)is V=A+Ba (206), where A,Barearbitraryconstants. Theequipotentialsarethesurfaces a=constant, andaretherefore confocalellipsoids.Thuswecan,from this solution, obtain thefieldwhen anellipsoidalconductor isfreelyelectrified. Forinstance, iftheellipsoid x2 1/2z2 , h— -I—=1a2b2c2 israised tounitpotential,thepotentialatanyexternalpointwillbegiven byequation (206) provided wechooseAandBsoastohaveV=1when A,=0,andV=0 whenX=00 .Inthiswayweobtain ["dX V=^-^ (207). Joa; Thesurfacedensityatanypointontheellipsoidisgiven by -J—--?Z?±--h,— dn d\dn dX /,dX aI dX abc -r-AA.(208). 282-285] Confocal Coordinates 247 Thus thesurfacedensityatdifferentpointsoftheellipsoidisproportional tohy. 284. Thequantity h^admits ofasimple geometrical interpretation. LetI,m,nbethedirection-cosines ofthetangent planetotheellipsoidat Fig. 79. anypoint X,fi,v,and letpbetheperpendicularfrom theoriginontothis tangent plane. Then from thegeometryoftheellipsoid wehave p2=(a2+X)l2+(b2+X)m2+(c2+X)n2(209). Moving alongthenormal, weshallcome tothepointX+dX, fi,v.The tangent planeatthispointhasthesame direction-cosinesI,m,nasbefore, buttheperpendicularfrom theoriginwillbep+dp,where dp=-r-.To obtain dpwedifferentiateequation (209), allowingXalone tovary,andso have 2pdp=dX(I2+m2+n2 )=dX. Comparingthiswithdp= -j-,weseethat hx=2p. Thus thesurfacedensityatanypointisproportionaltotheperpendicular from thecentre ontothetangent planeatthepoint. Infig.79,thethickness oftheshadingatanypointisproportionalto theperpendicularfrom thecentre ontothetangent plane,sothat the shading representsthedistribution ofelectricityonafreelyelectrified ellipsoid. Itwillbeeasilyverified that theouterboundaryofthisshading must beanellipsoid,similar toandconcentric with theoriginal ellipsoid. 285.Replacing h^by2pinequation (208),wefind forthetotalchargeE ontheellipsoid, ZTTUOC -r- JoAA Since IIpdSisthree times thevolume oftheellipsoid,andtherefore equalto4nra.bc, thisreduces to e=2 fJo 248 Methods fortheSolution ofSpecial Problems[ch.vm Since theellipsoidissupposedtoberaised tounitpotential,thisquantityEgivesthecapacityofanellipsoidalconductor electrified infreespace. Thecapacitycanhowever beobtained morereadily byexaminingthe form ofthepotentialatinfinity. Atpointswhich areatadistance r from thecentre oftheellipsoidsogreatthat a,b,cmaybeneglectedin comparisonwith r,X,becomesequaltor2 ,sothatAK=r2 ,and dX_2 Thus atinfinitythelimitingformassumed byequation (207)is 2/; Vf^dx' JoAl E andsince thevalue ofVatinfinity must be—thevalue ofEfollows atT once. Afreely -charged spheroid. 286. TheintegralI-r- isintegrableifanytwo ofthesemi-axes JoA\ becomeequaltooneanother. Ifb=c,theellipsoidisaprolate spheroid,and itscapacityisfound tobe 2 2aeEr^—,kg(i±^y where eistheeccentricity. Ifa=b,theellipsoidisanoblatespheroid,and itscapacityisfound tobe 2 aeE= r-Jo(a?dX sin_1e (a2+\)(c2+X,)i EllipticDisc. 287. Inthepreceding analysis,letabecomevanishingly small, then theconductor becomes anellipticdisc ofsemi-axes band c. Theperpendicularfrom theoriginontothetangent-planeisgiven,asin theellipsoid, by p"2= x2y2z*h— -I— a4b* c4 285-289] Confocal Coordinates 249 andwhen aismadeverysmall inthelimit, thisbecomes 1_a- a*V2— \-yi-*L2' sothatthesurfacedensityatanypoint x,yinthedisc isproportionalto b**, (210). (l-y--Z -) Circular Disc. 288.Onfurthersimplifying byputtingb=c,wearrive atthecase ofa circular disc. Thedensityofelectrification isseen atoncefromexpression (210)tobeproportionalto 1- c--* and therefore variesinverselyastheshortest chord which canbedrawn throughthepoint. Moreover, when a=and b=c,wehaveAx=(c2+X)Va,,sothat r^tan-^andf ^JaAAcWxJ JoAxIT C Thus thecapacityofacircular disc is— ,andwhen thedisc israised to potential unity,thepotentialatanyexternalpointis 2-tan-1 ,tVvV where\isthepositiverootof tf£+£» X c2+X 289. Lord Kelvin* quotes some interesting experiments byCoulomb onthedensity atdifferent points onacircularplateofradius 5inches. The results aregiveninthe followingtable : Distances from the plate's edge 250 Methods fortheSolution ofSpecial Problems[ch.vni Much more remarkable isCavendish'sexperimentaldetermination ofthecapacityofa circular disc. Cavendish found this tobe-=-^=times that ofasphereofequal radius, whiletheory shews thetrue value ofthedenominator tobetjor1-5708! 290.Byinvertingthedistribution ofelectricityonacircular disc,taking theoriginofinversion tobeapointintheplaneofthedisc,Kelvin* has obtained thedistribution ofelectricityonadiscinfluenced byapoint charge initsplane,aproblem previouslysolved byanother method byGreen. The generalGreen's function foracircular dischasbeen obtained byHobsonf. Spherical Bowl. 291. Lord Kelvin has also,byinversion, obtained thesolution fora sphericalbowl ofanyangle freelyelectrified. Letthebowl beapieceof asphereofdiameter/.Letthedistance from the middle pointofthebowl toanypointofthebowl ber,and letthegreatestvalue ofr,i.e.thedis- tance from apointontheedgetothemiddlepoint ofthebowl, bea.Then Kelvin finds fortheelec- tricdensities inside andoutside thebowl : PiV 2tt2/Pa' a?r2—tan-ra- a-)• Po= Pi+V 2*/- Some numerical results calculated from these formulae areofinterest. The sixvalues inthefollowingtables refer tothemiddlepoint andthefivepoints dividingthearcfrom themiddle pointtotheedgeinto sixequal parts. Plane disc Curved discarc10° Curved discarc20° 1-00 1-01 1-06 1-15 1-34 1-81Po 289-292] Ellipsoidal Harmonics 251 Bowl arc270° Bowl arc340° Pi •013 •014 •018 •025 •045 •120Po 252 Methods fortheSolution ofSpecial Problems[ch.viii must bethesame function. Byasimilarprocedure,itfollows thatfmust alsobethesame function, sothattheequationcanbewritten (fi-v)f(X) +(v-X)/(/i) +(X-fi)f(v)=0. Tofindtheform ofthefunction /weputX=andobtain /00-/(o) =/fr)-/«>\ flv Thus afunction of/x,isequaltothesame function ofv,sothateachmust beaconstant.CallingthisB,andwritingAfor/(0), wefindthat f(X)=A+BX. 293.Restoringitsvalue to/(A)weseethatwemust have ~=(A+B\)L (212), andsimilarequations,with thesame constants AandB,must besatisfied byMandN. Equation (212), onsubstitutingforainterms ofX,becomes adifferentialequationofthesecond order inX,whileMandNsatisfy equationswhich areidenticalexceptthatfjuand varethevariables., 7(A,£j=(A+BX)L (213), The solution ofequation (213)isknown asaLamp's function, orellip- soidal harmonic. The function iscommonlywritten asE^(X), wherep,n arenewarbitrary constants, connected with theconstants AandBbythe relations n(n+l)=B,and(b2+c2)p=-A. ThusEl(A.)isasolution of d ^={n(n+l)X-p(b* +c>)}L, andasolution ofequation (211)is V=*X2E>i{\)E><ji)E>i(v) (214). pn 294. Equation (213) beingofthesecond order, must have twoinde- pendentsolutions.DenotingonebyL,lettheother besupposedtobeLu. Thenwemust have dot 82(Lu) da?^=(A+BX)L, =(A+Bx)Lu; 293-295] Ellipsoidal Harmonics 253 sothatonmultiplyingtheformerequation byu,andsubtractingfrom the latter, rd-u _dLdu _ da? dotda mi [do. fd\Thus u=L2JZ2AX' andthecompletesolution isseen tobe OL+DLffa, whereCandDarearbitraryconstants. Accordingly,thecompletesolution ofequation (211) canbewritten as V^(GnpE^ )+DnpE^)J{^^^ (cnp"E*(v) +DnP"Ei{v)\ {E^)Y^. Thiscorresponds exactlytothegeneralsolution inrationalintegral spherical harmonics, namely V=XZ(G npr"+Dnpr-^) pn (Onp'e?p*+Dnp'erW) (Cnp"P»(cos0)+D np"P*(cos6)). Ellipsoidinuniform field offorce. 295. Asanillustration oftheuseofconfocal coordinates, letusexamine thefieldproduced byplacinganuninsulatedellipsoidinauniform field of force. Thepotentialoftheundisturbed field offorcemaybetaken tobeV=Fx, orinconfocal coordinates(cf.equation (201)) V(b*~-a2 )(c2-a2 ) This isoftheform V=GLMN, whereGistheconstant F(62—a2 )~~ -(c2—a2 )~* ,andL,M,Narefunctions of A.only, fxonlyandvonly, respectively, namely L=va2+\,etc. SinceV=LMN isasolution ofLaplace's equation,there must, asin§294, beasecond solution V—Lu .MN,where dX fdXuD\KJ(a2+X)Ax 254 Methods fortheSolution ofSpecial Problems[ch.viii Theupperlimit ofintegrationisarbitrary:ifwetake ittobeinfinite, bothuandLuwillvanish atinfinity,whileMandJVareinanycase finite atinfinity. ThusLu .MN isapotential which vanishes atinfinityand is proportional (sinceuisafunction ofXonly)atevery pointofanyoneofthe surfaces X=cons., tothepotentialoftheoriginalfield. Thus thesolution V=CLMN+DLu.MN.(215) canbemade togivezeropotentialoveranyoneofthesurfaces X=cons., by asuitable choice oftheconstant D. Forinstance iftheconductor isX=0,wehave, ontheconductor, dXuH(a2!+X)A> Thus ontheconductor wehave V=LMN(g+dT ,,d\A). V Jo(a2+X)A A/ Thecondition forthistovanishgivesthevalue ofD,andonsubstituting thisvalue ofD,equation (215) becomes V=CLMN fl- KId\ o(a2+X)A X/ dX JK(a3+X)A,=Jtx\ 1— dX o(a2+X)A A/ dX =^(a2+X)A. (gl6) Jo(a2+X)AA Thisgivesthefieldwhen theoriginalfield isparalleltothemajoraxis oftheellipsoid.Iftheoriginalfield isinanyother direction wecanresolve itinto three fieldsparalleltothethree axes oftheellipsoid,andthe final field isthenfound bythesuperpositionofthree fields ofthetypeofthat given byequation (216). Spheroidal Harmonics. 296.When anytwosemi-axes ofthestandardellipsoidbecomeequal themethod ofconfocal coordinates breaks down. Fortheequation +^+^=i (2ma2+6&2+e cn~+e 295-297] Ellipsoidal Harmonics 255 reduces toaquadratic,andhasthereforeonlytwo roots, say\,ft.The surfaces \=cons,and/j,=cons, arenowconfocalellipsoids andhyperboloids ofrevolution, butobviouslyathirdfamilyofsurfaces isrequiredbefore the positionofapointcanbefixed. Such afamilyofsurfaces, orthogonalto thetwopresent families, issupplied bythesystemofdiametralplanes throughtheaxis ofrevolution ofthestandardellipsoid. Thetwocases inwhich thestandardellipsoidisaprolate spheroid and anoblatespheroid require separate examination. ProlateSpheroids. 297. Letthestandard surface betheprolate spheroid a2_t" 62~ ' inwhich a>b. Ifwewrite y=•orcos(j), z=-STsin<f>, then thecurvilinear coordinates maybetaken tobe\,/u, <£>,where X, fj,are theroots of x*+7^^=1 (218).a2+e fr+ Inthisequation, puttf—fr^c* anda2+0=c262 ,then theequation becomes x2 . CT2 If£2 ,rfaretheroots ofthisequationin0'2 ,wereadilyfindthat ,x2=£2 t?2c2 , sothatwemaytake x=c%v (219), ct=cv/ (1-P)(7 ?2-1) (220) inwhichr\istaken tobethegreaterofthetwo roots. The surfaces £=cons., 77=cons, areidentical with thesurfaces #=cons., andareaccordinglyconfocalellipsoidsandhyperboloids.The coordinates £,tj, <f>maynowbetaken tobeorthogonalcurvilinear coordinates. Itiseasilyfound that hA/TEE j,-I/ZZI h1 fromwhichLaplace's equationisobtained intheform 8inw3Fld\n *,dr ) 1"'~?' 3°Fn 256 Methods fortheSolution ofSpecial Problems[ch.vin 298. Letussearch forsolutions oftheform F=EH3>, where 3,H,<£aresolutionssolelyoff,rjand <f>respectively. Onsubstituting thistentative solution andsimplifying,weobtain a-r)(T-i)iJU(W')i}-4i>-^ +i?5=o. 772-p L3S£l 9£) Ha77{w ' 877JJ4>302 Asinthetheoryofspherical harmonics, theonly possiblesolution results fromtaking where—m2isaconstant, andmmust beanintegerifthesolution istobe singlevalued. Thesolution is <I>=Gcosmfy+Dsinm<fi (221). Wemustnowhave 11in_«dM+11k.-1)!5l=m'("'~^) 33fr«;8fJ+H3,lW 1;S,f (l-f>)(,= _l) m- m*+"1-P ^-1' andthiscanonlybesatisfied bytaking togetherwith Jj^-^S-S^-(223)- Equations (222) and(223)areidentical with theequation alreadydis- cussed in§§273, 274. The solutions areknown tobe B=AP^) +BQ^), n=A'P%( v)+B'Q%( v), where s=n(n+1)andP™,Q%aretheassociatedLegendrianfunctions already investigated. Combiningthevaluesjustobtained for3,Hwith thevalue for <£>given byequation (221), weobtain thegeneralsolution F=2S3H<S>mn =XS{AP™{%) +BQ£(®} {A'P2( V)+B'Q:( V)}{Ccosmtf>+Dsinm(/>}.mn Atinfinityitiseasilyfound that 77=00,f= .- =COS0,vV+ot2 while attheorigin <q=1,f=0. Thus inthespaceoutside anyspheroid,thesolution P™(£) Q™(>/)isfinite everywhere, while, inthespace inside, thefinite solution isPjj'(£)P,"l (rj). 298-301] Problems intwoDimensions 257 OblateSpheroids. 299. Foranoblatespheroid,a2-b2isnegative, sothat inequation (218)wereplaceb2-a2by«2 ,sothatk=ic,andobtain, inplaceofequations (219) and(220), x=K^irj, &=KV(l-f2 )(1- rf). Replacingivby£wemaytake£,fand<f>asrealorthogonal curvilinear coordinates, connected with Cartesian coordinatesbytherelations x=«f£ vr=*V(l-£2)(l+£2 ). Weproceedtosearch forsolutions ofthetype F=EZ<D, andfindthatH,3>mustsatisfythesameequations asbefore, whileZmust satisfy -||(1+^|}-rfi2Z+7l(ri+1)Z=a Thesolution ofthis is Z=A'P™(iO +B'Q%(ia andthemostgeneralsolution maynowbewritten down asbefore. Problems intwoDimensions. 300. Often when asolution ofathree-dimensionalproblemcannot be obtained, itisfoundpossibletosolve asimilar butsimpler two-dimensional problem,and toinfer themainphysicalfeatures ofthethree-dimensional problemfrom those ofthetwo-dimensionalproblem. Weareaccordingly ledtoexamine methods forthesolution ofelectrostaticproblemsintwo dimensions. Attheoutset wenotice that theunit isnolongerthepoint-charge, but theuniformline-charge,aline-chargeofline-densitycrhavingapotential (cf.§75) (7—2crlogr. MethodofImages. 301. Themethod ofimagesisavailable intwodimensions, butpresents nospecialfeatures. Anexampleofitsusehasalreadybeengivenin§220. j. 17 258 Methods fortheSolution ofSpecial Problems[ch.vin Method ofInversion. 302. Intwodimensions theinversion isofcourse about aline. Letthis berepresented bythepointinfig.81. LetPP',QQ'betwopairsofinversepoints. Letaline-chargeeatQ produce potential VpatP,and leta line-chargee'atQproduce potential Vp atP',sothat VP=C-2e\ogPQ; Vjy=C'-2e'\ogP'Q'. Ifwetake e=e',weobtainFlG> 81> Vj,-VP,=C"-2e\og^ =C"-2e\og^(224). LetPbeapointonanequipotential when there arechargesexatQl} e2atQ2,etc.,and letVdenote thepotentialofthisequipotential.LetV denote thepotentialatP'under theinfluence ofchargese1}e2,•••a^the inversepointsofQ1}Q2,....Then, bysummation ofequationssuch as(224), V-V=-S(2elogOP')+2(2elogOQ)+constants, or V=constants- 2(Xe)logOP' (225). ThepotentialatP'ofchargese1}e2,...attheinversepointsofQltQ2,... plusacharge—2eat is V+C+2($e)\og0P', and thisbyequation (225)isaconstant. This resultgivesthemethod of inversion intwodimensions : Ifasurface Sisanequipotential under theinfluence ofline-charges elfe2,...atQ1}Q2>...,then thesurface which istheinverseofSabout aline will beanequipotentialunder theinfluence ofline-chargese1}e2,... onthelines inverse toQ1}Q2,...togetherwithacharge—Xeattheline 0. Two-dimensiona IHarmonics. 303.Asolution ofLaplace's equationcanbeobtained which isthe analogueintwodimensions ofthethree-dimensional solution inspherical harmonics. Intwodimensions wehave two coordinates, r,6,thesebecoming identical withordinarytwo-dimensionalpolarcoordinates.Laplace's equa- tionbecomes ld_foV\d*V 302-304] Problems intwoDimensions 259 andonassumingtheform inwhichRisafunction ofronly,and©afunction of6only,weobtain the solution intheform V="5°°(Arn+—J(Ccosn<J>+Dsinn<£). M=0V?/ Thus the"harmonic-functions"intwodimensions arethefamiliar sine andcosine functions. The functions whichcorrespondtorationalintegral harmonics arethefunctions rnsinnd,rncosn6. Inx,ycoordinates these areobviouslyrationalintegralfunctions ofx andyofdegreen. Correspondingtothetheorem of§240, thatanyfunction ofposition onthesurface ofaspherecan(subjecttocertainrestrictions) beexpanded inaseries ofrationalintegral harmonics, wehave thefamous theorem of Fourier, thatanyfunction ofpositiononthecircumference ofacircle can (subjecttocertain restrictions) beexpandedinaseries ofsinesandcosines. Intheproofwhich follows(asalsointheproofof§240),noattemptismade atabsolute mathematicalrigour:asbefore, theform ofproof givenisthat which seems bestsuited totheneeds ofthestudent ofelectricaltheory. Fourier sTheorem. 304. Thevalue ofanyfunction Fofposition onthecircumference ofa circle can beexpressed,atevery point ofthecircumferenceatwhich the functioniscontinuous, asaseriesofsinesand cosines, providedthefunctionis single-valued,andhasonlyafinitenumberofdiscontinuities andofmaxima andminima onthecircumference ofthe circle. LetP(/, a.)beanypointoutside thecircle, then ifRisthedistance fromPtotheelement dsofthe circle r^^p^/>a) (a,6)wehave /2iraRia ds=1. This result caneasily beobtained byinte- gration,orcanbeseen atonce from physical considerations, fortheintegrandisthecharge induced onaconducting cylinder byunit line- chargeatP, Fig. 82. 17—2 260 Methods fortheSolution ofSpecial Problems[ch.viii Letusnowintroduce afunction udefinedby u=p-o? [Fhds.(226).2ira JR2 Then, subjecttotheconditions stated forFwefind, asin§240,thaton thecircumference ofthecircle, thefunction ubecomes identical with F. Alsowehave 1_1 B?~p+a?-2a/cos(0-a) 1 (/-ael<e~a >)(/-ae-^e-a) ) f2-ai\f-aei{e~a)a-f&{e~a) J =7^2 ii+2!(7rcosw('-a) }- Hence u==—/F\1+22 (4 )cosw(0- a)[c?s 2ttJ 0=o77"1\/nr8=2ir e=oFcosn(0-a)dd, andonpassingtothelimit andputting a=f,thisbecomes ^=^-|^d<?+-$ Fcosn(0-ct)d0 (227), expressing Fasaseries ofsines andcosines ofmultiplesofcl Wecanputthis result intheform 00F=F+X(ancosnot+bnsinnot), where1f2lT an=- IFcosnddO, ttjo hn=-\**Famnddd, •2tt - 1 /"2,r and F=^-\ Fd9, sothatFisthemean value ofF. IfFhasadiscontinuityatanypoint=ftofthecircle, and ifF1}i£are thevalues ofFatthediscontinuity,then obviouslyatthepoint=fion the circle, equation (226) becomes u=^(F 1+F2), sothat thevalue oftheseries(227)atadiscontinuityisthearithmetic mean ofthetwovalues ofFatthediscontinuity (cf. §256). 304-307] Conjugate Functions 261 305.Wecouldgoontodevelopthetheoryofellipsoidal harmonics etc. intwodimensions, but allsuch theories aresimply particularcases ofavery general theorywhich willnowbeexplained. Conjugate Functions. GeneralTheory. 306. Intwo-dimensionalproblems,theequationtobesatisfied bythe potentialis fty^yw+W=0 (228); and thishasageneralsolution infinite terms, namely V=f(x +iy)+F(x-iy) (229), where /andFarearbitrary functions, inwhich the coefficients mayof course involve theimaginaryi. ForVtobewholly real,Fmust bethefunction obtained fromfon changingiinto—*.Letf(x+iy)beequaltou+ivwhere uandvare real,thenF(x+iy)must beequaltou—iv,sothatwemust haveV=2u. Ifweintroduce asecond function Uequalto—2v,wehave U+iV=-2v +2iu =2i(u+iv) =2if(x+iy) =j>(x+iy) (230), where<f>(x+iy)isacompletely generalfunction ofthesinglevariable x+iy. Thus themostgeneralform ofthepotentialwhich iswholly real,canbe derived from themostgeneral arbitraryfunction ofthesinglevariable x+iy, ontakingthepotentialtobetheimaginary partofthisfunction. 307. If (f)(x-fiy)isafunction ofx+iy,theni<j)(x+iy)will alsobe afunction, andtheimaginary partofthisfunction will alsogiveapossible potential. Wehave, however, fromequation (230), i<j>(x+iy)=i(U+iV) =-V+iU, shewingthatUisapossible potential. Thuswhenwehave arelation ofthetype expressed byequation (230), eitherUorVwillbeapossible potential. 262 Methods fortheSolution ofSpecial Problems[ch.vm 308. Taking Vtobethepotential, wehavebydifferentiation of equation (230), dU ,.dV .,,.., andhence .fd_U.d_V \dx dx_dU,dV " dy dy' Equatingrealandimaginary partsintheaboveequation, weobtain dU=d_V dxdy' dU=_d_V dy dx' sothat djjdv dUd_y dxdx dydy0..(231). This however isthecondition that thefamilies ofcurvesU=cons., V=cons., should cutorthogonallyatevery point.Thus the curves JJ=cons, aretheorthogonal trajectoriesoftheequipotentials —i.e.are thelines offorce. 309.Representation ofcomplex quantities. Ifwewrite z=x+iy sothat zisacomplex quantity, wecansuppose thepositionofthepointPindicatedbythevalue ofthesingle complexvariable z.Ifzisexpressed inDemoivre's form z=reie=r(cos6+isin6), thenwefindthatr=*/x2+y2and=tan-1y~.Thex Fig. 83. quantityrisknown asthemodulus ofzand isdenotedby\z\,while 6is known astheargumentofzand isdenotedbyargz.Therepresentationof acomplex quantityinaplaneinthiswayisknown asanArgand diagram. 310. Additionofcomplex quantities. LetPbez=x+iy,and letP'be z=x'+iy'.Thevalue ofz+zis(x+x')+i(y+y'),sothat ifQrepresents thevalue z+zitisclear thatOPQP' willbeaparallelogram. Thus to addtogetherthecomplex quantitieszandzwecompletetheparallelogram OPP', andthefourthpointofthisparallelogramwillrepresentz+z' . 308-311] Conjugate Functions 263 Thematter maybeputmoresimply bysupposingthecomplex quantity z=x+iyrepresented bythedirection andlengthofaline,such that its projectionsontworectangularaxes are x,y.Forinstance infig.83,the value ofzwillberepresented equally byeitherOPorP'Q.Wenowhave thefollowingrule fortheaddition ofcomplex quantities. Tofindz+z,describe apathfrom theorigin representingzinmagnitude anddirection, andfrom theextremityofthisdescribe apath representingz. The linejoiningtheorigintotheextremityofthissecondpathwillrepre- sent z+z' 311. Multiplication ofcomplex quantities.If z=x+iy=r(cos6+isin6), and z'=x'+iy'=r(cos&+isin0'), then, bymultiplication zz'=rr{cos(0+0')+ism(0+0')}, sothat |zz' |=rr'= \z |\z'\, arg{zz')=6+6'=argz+arg z', andclearly wecanextend this result toanynumber offactors. Thuswe have theimportantrules : Themodulus ofaproductistheproduct ofthemoduli ofthefactors. Theargument ofaproductisthesumofthearguments ofthefactors. There isageometrical interpretationofmultiplication. Infig.84,letOA=1,OP-*,OP'=*'andOQ=zz\ Then theangles QOA,P'OAbeing equalto6+0'and 9'respectively, theangle QOP' must beequalto6,andtherefore toPOA. Moreover OQOP OP'~OA' each ratio being equaltor,sothatthetriangles QOP' andPOA aresimilar. Thus tomultiply thevector OP'bythevector OP,wesimply construct onOP'atrianglesimilar toAOP. Thesame result canbemoreshortlyex- pressed bysayingthat tomultiply /(=OP')by z(=OP),wemultiplythelength OP'by |z \and turn itthroughanangle argz. Soalso todividebyz,wedivide thelength ofthelinerepresentingthedividendby |z \and turnthroughanangle—argz.Ineither caseanangleispositive when theturningisinthedirection whichbringsusfrom theaxisxtothat ofyafter anangle tt/2. 264 Methods fortheSolution ofSpecialProblems [ch.vni Gonformal Representation. 312.Wecannow consider morefullythemeaningoftherelation JJ+iV= <f>(as+iy). Let uswrite z=x+iy,andW=U+iV,zandWbeing complex imaginaries,which wemustnowsupposeinaccordance withequation (230) tobeconnected bytherelation W=<f>(z) (232). Wecanrepresentvalues ofzinoneArgand diagram,andvalues ofWin another. Theplaneinwhich values ofzarerepresentedwillbecalled the 2-plane,theother willbecalled theW-plane. Any pointPinthe.z-plane correspondstoadefinite value ofzand this,byequation (232),maygiveone ormore values ofW,accordingas<pisorisnotasingle-valuedfunction. IfQisapointintheW-planewhichrepresentsoneofthese values ofW, thepointsPandQaresaid tocorrespond. AsPdescribes anycurveSinthe2-plane,thepointQintheTT-plane which correspondstoPwilldescribe some curveTintheW-plane,andthe curveTissaid tocorrespondtothecurve S.Inparticular, corresponding toanyinfinitesimal linearpathPP' inthes-plane,there willcorrespond asmall linear element QQ'intheTf-plane.IfOP,OP'representthevalues z,z+dzrespectively,then theelement PP' willrepresentdz. Similarlythe dW element QQ'willrepresentdWor—,—dz. Hence wecangettheelement QQ'from theelement PP'onmultiplying itby-T- ,i.e.by^-<f>(z),orby <f>'(x+iy).Thismultiplier depends solely&Z oz onthepositionofthepointPinthe2-plane,andnotonthelengthor dW direction oftheelement dz. Ifweexpress -5—or<£'(x+iy)intheform dW -j-=$'\x+iy)=p(cos%+isinx), wefind that theelement dWcanbeobtained from thecorrespondingdW element dzbymultiplyingitslength bypordz dW,andturningitthrough anangle %,orargf^- ).Itfollows thatanyelement ofarea inthe2-plane isrepresentedintheW-plane byanelement ofarea ofwhich theshape isexactlysimilar tothat oftheoriginal element, thelinear dimensions are ptimes asgreat,andtheorientation isobtained byturningtheoriginal element throughanangle %. 312-315] Conjugate Functions 265 From thecircumstance that theshapesoftwocorrespondingelements inthetwoplanesarethesame, theprocessofpassingfrom oneplaneto theother isknown asconformed representation. 313. Letusexamine thevalue ofthequantity pwhich, aswehave seen,measures thelinearmagnification producedinasmall areaonpassing from the^-planetotheIP-plane.dWWehave p(cos%+isin%)=~y—= <f>'(x+iy) =du.d_v dx dx dv.dV dy dx dV.dV dydx oxJ\dyJ dW iscalled the"modulus oftransformation.sothatp= Thequantity p,or Wenow seethat ifVisthepotential,thismodulus measures theelectric /TdVy /dVy intensity R,ora/ f-^— J-Mj— J.SinceR=4nra, thiscircumstancepro- vides asimple means offinding <r,thesurface-densityofelectricityat anypointofaconductingsurface. 314. If jr-denote differentiationalongthesurface ofaconductor, on which thepotential Visconstant, wehave dW dz ds' sothat <r=-j—.ft=-j—-~- . 47T 47TOS The totalchargeonastripofunitwidth betweenanytwopoints P,Qof theconductor isaccordinglyhs=llQMds=l^-u^ <233>- 315. If,onequatingrealandimaginary partsofanytransformation of theform U+iV=cf>(x +iy) (234), itisfound that thecurvef{x, y)=correspondstotheconstant value V=C,thenclearlythegeneralvalue ofVobtained fromequation (234) willbeasolution ofLaplace's equation subjecttothecondition ofhaving theconstant valueV=Cover theboundary /(x,y)=0.Itwill therefore bethepotentialinanelectrostatic field inwhich thecurvefix, y)=may betaken tobeaconductor raised topotentialC. 266 Methods fortheSolution ofSpecialProblems[ch.viii 316.From agiventransformation itisobviously always possibleto deduce thecorrespondingelectrostatic field, butonbeing giventhecon- ductors andpotentialsinthe field, itisbynomeansalways possibleto deduce therequired transformation. We shallbegin bytheexamination of afewfields which aregiven bysimple known transformations. Special Transformations. I.W=z11 . 317. Consideringthetransformation W=z11 ,wehave U+iV=(x+iy)n=rn(cosnd+isinnd), sothatV=rnsinnd.Thusanyoneofthesurfaces rnsinnd—constant maybesupposedtobeanequipotential, includingasaspecialcase rnsinnd=0, IT inwhich theequipotentialconsists oftwoplanes cuttingatanangle- . This transformation canbefurther discussed byassigning particular values ton. n=1.Thisgives simplyV—x,auniform field offorce. n=2.ThisgivesV=2xy,sothat theequipotentialsarerectangular hyperbolic cylinders, includingasaspecialcasetwoplanes intersecting atright angles (fig. 85). Fig. 85. Fig. 86. 316,317] ConjugateFunctions 267 This transformationgivesthe field intheimmediate neighbourhoodof twoconducting planes meetingatright anglesinanyfield offorce. Italso givesthefieldbetween twocoaxalrectangular hyperbolas. Fig. 87. n=\.Thisgivesx+iy=(U+iV)2 ,sothat x=U*-V\ y=2UV, andoneliminating Uweobtain 2/2=4F2(x+V2 ). Thus theequipotentialsareconfocal andcoaxalparabolic cylinders,in- cludingasaspecialcase(V=0)asemi-infiniteplane bounded bytheline offoci. This transformation clearly givesthe field intheimmediateneighbour- hood ofaconducting sharp straight edgeinanyfield offorce(fig. 86). n=—1.Thisgives U+iV=-(cos-1sinV r andtheequipotentialsare rF=sin<9 or^+^--^=0. Thus theequipotentialsareaseries ofcircularcylinders,alltouching theplane y=alongtheaxisx=0,y=(fig. 87). 268 Methods fortheSolution ofSpecial Problems[ch.vm II. IF=log* 318. Thetransformation W=logz gives U+iV=\ogr +i0, sothattheequipotentialsaretheplanes6=constant, asystemofplanesall intersectinginthesame line. Asaspecial case,wemaytake=and 6=irtobetheconductors, andobtain thefieldwhen thetwohalves ofa planeareraised todifferentpotentials. The lines offorce,U=constant, are circles(fig. 88). Fig. 88. IfwetakeUtobethepotential,theequipotentialsareconcentric circularcylinders, andthe field isseen tobesimplythatdue toauniform line-charge,oruniformlyelectrifiedcylinder. Itmaybenoticed thatthetransformation W=log(z—a) givesthetransformationappropriatetoaline-chargeatz=a. Alsowenotice that z—aF=logz+a givesafieldequivalenttothesuperpositionofthefieldsgiven by W=log(z-a)andW=—log(z+a). Thistransformation isaccordinglythatappropriatetotwoequal andopposite •line-charges alongtheparallellines z=aandz=—a. This lasttransformationgivesU=when y=0,sothat itgives the transformation foraline-chargeinfront ofaparallelinfiniteplane. 318-320] Conjugate Functions 269 General Methods. I.Unicursal Curves. 319. Supposethat thecoordinates ofapoint onaconductor canhe expressedasrealfunctions ofarealparameter, which varies asthepoint moves over theconductor, insuch awaythatthewholerangeofvariation oftheparameter just correspondstomotion overthewhole conductor. In other words, supposethatthecoordinates x,ycanbeexpressedintheform x=f(p), y=F(p)> andthat allrealvalues ofpgive pointsontheconductor, while, conversely, allpointsontheconductorcorrespondtorealvalues ofp. Then thetransformation z=f(W) +iF(W) (235) willgiveV= overtheconductor. ForonputtingV= inequation (235) weobtain x+iy=f(U) +iF(U), sothat x=f{U\ y=F(U), andbyhypothesistheelimination ofUwilllead totheequationofthe conductor. 320. Forexample,consider theparabola (referred toitsfocus asorigin), t/2=4a(x+a). Wecanwrite thecoordinates ofanypointonthisparabolaintheform x+a=am2 ,y=2am, andthetransformation isseen tobe z=aW2-a+2aiW=a(W-if, or W-i= (£)* agreeingwith thatwhich hasalreadybeen seen in§317 togiveaparabola asapossible equipotential. 270 Methods fortheSolution ofSpecial Problems[ch.vni 321. Asasecondexampleofthismethod, letusconsider theellipse a?+ 62"i* Thecoordinates ofapointontheellipse maybeexpressedintheform x=acos<£,y=bsin(f>, andthetransformation isseen tobe z=acosW+ibsinW. Fio. 89. Wecantakea=ccosh a,b=csinh a,where c^=a7—52 ,andthetrans* formation becomes z=ccos(W+ia)=ccos{U+i(F+a)}. Thesame transformation maybeexpressedinthebetter known form z=ccoshW. Theequipotentialsaretheconfocalellipses a?y2 + -1.a2+X62+X while the lines offorce areconfocalhyperbolic cylinders. Ontaking 7 asthepotential,wegetafield inwhich theequipotentialsareconfocal hyperbolic cylinders. 321,322] Conjugate Functions 271 II.Schwarz'sTransformation. 322. Schwarz hasshewn how toobtain atransformation inwhich one equipotentialcanbeanylinearpolygon. Atanyangleofapolygonitisclear thatthepropertythatsmall elements remainunchangedinshapecannolongerhold. Thereason iseasilyseen to bethatthemodulus oftransformation iseither infinite orzero(cf. figs.24 and25,p.61). Thus, attheanglesofanypolygon, dW dz=oroo . Thesame result isevident from electrostatic considerations. Atanangle ofa conductor, thesurface-density<riseither infinite orzero(§70),while wehave the relation(§313),dW 47T 4ff dz Letussupposethatthepolygoninthe2-planeistocorrespondtothe lineV=intheW-plane,and lettheangular points correspondto U=uJ}JJ=u2,etc. Then,when W=ultW=w2>etc., dz -r-jjrmust either vanish orbecome infinite. Wemustaccordingly have dz ^^FiW-u^iW-u^(236), where Xi, X-,,...arenumbers which maybepositiveornegative,whileF denotes afunction, atpresent unknown, ofW. Suppose that, aswemovealongthepolygon,thevalues ofUatthe angular pointsoccur intheorder ultu2,....Then, onpassing alongthe side ofthepolygonwhichjoinsthetwoanglesU=u1}U=u2,wepassalong arangeforwhichV=0,andv^kUku^. Thus, alongthis side ofthe polygon,W—ultW—u2,W—u3,etc.arerealquantities; positiveornegative, which retain thesamesignalongthewhole ofthisedge.Itfollows that, as wepass alongthisedge,thechangeinthevalue ofarg (-ttjt),asgiven KdWJ byequation (236),isequaltothechangeinargF,theargumentsofthe factors (W-u^{W-u^... undergoingnochange. Nowarg[-T^n]measures theinclination oftheaxisV=totheedgeof thepolygonatany point,sothat ifthepotygonistoberectilinear, this must remain constant aswepassalong anyedge.Itfollows thattheremust benochangeinargFaswepassalong anysideofthepolygon. 272 Methods fortheSolution ofSpecial Problems[oh.vm This condition canbesatisfied bysupposing Ftobeapurenumerical constant. Takingittobereal,wehave, fromequation (236), arg\dw)=Xl&TS(W-Wi)+X2arg(TT-m2)+ (237). Onpassing throughtheangular pointatwhichW=u2,thequantitiesW—Ui,W—u3,etc.remain ofthesamesign,while thesingle quantityW—u2changes sign. Thusarg (W—u2)increases by tt,whence, byequa- tion(237), arg(-t™-)increases by"Kir. The axisV=0 does notturn inthe TT-planeonpassing throughthe valueW=u2,whilearg (-Trs-)measures theinclination oftheelement of thepolygoninthe^-planetothecorrespondingelement oftheaxisV=in theJF-plane. Hence, onpassing throughthevalueW=u2,theperimeterofthe polygoninthe^-planemust turnthroughanangle equaltotheincrease in arg (-TTiv),namelyX27r,thedirection ofturning beingfromOxtoOy.Thus Xx7r,XjTt,...must betheexterioranglesofthepolygon,thesebeing positive when thepolygonisconvex totheaxisOx. Or, ifa1}ct2,...aretheinterior angles,reckonedpositivewhen thepolygonisconcave totheaxis ofx,we must have X,=——1,etc. 7T Thus thetransformation requiredforapolygon havinginternalangles OfiyGt'zj•••IS ^=G{W-u iy~\W-u 2y-X (238), where Wj,u2,...arerealquantities,whichgivethevalues ofUattheangular points. 323.Asanillustration oftheuseofSchwarz's transformation, suppose theconducting systemtoconsist ofasemi-innniteplane placed paralleltoan infiniteplane. Infig.90,lettheconductor besupposedtobeapolygon ABODE, which isdescribed byfollowingthedotted lineinthedirection ofthearrows. The points A,B,0,Eare allsupposedtobeatinfinity,thepointsBand coinciding.LetustakeAtobeW=—oo,BorCtobeW=0,jDtobe W=1and^tobeW=+oo .Theanglesofthepolygonarezero at(BO) and 2iratD.Thus thetransformation is dz_GW-1 dW W 322-325] ConjugateFunctions 273 giving upon integration z=C{W-\ogW+D}(239), where C,Dareconstants ofintegrationwhich maybeobtained from the E>VV=+co <=. <-.„w=+i c\ ->- >~~*s W=-oo Fro. 90. condition that thetwoplanesaretobe,say,y=andy=h.From these conditions weobtain G=— ,D=iir,sothatthetransformation is z=-{W-\ogW+iir} (240). IT Onreplacingz,Wby—z,—W,thetransformation assumes thesimplerform z=-(W+\ogW) (241). 7T III. SuccessiveTransformations. 324. If£= <f>(z),W=f(0areanytwotransformations, thenbyelimi- nation of£,arelation W=F(z) (242) isobtained, whichmayberegardedasanewtransformation. Wemayregardtherelation £= </>(z)asexpressingatransformation from the2-planeintoa£-plane,while thesecond relation W=f(%) expressesa further transformation from the£-planeintoaT7-plane.Thus the final transformation (242)mayberegardedastheresult oftwosuccessive trans- formations. Twouses ofsuccessive transformations areofparticular importance. 325. Conductorinfluenced byline-charge. Thetransformation gives,aswehave seen(§318) thesolution when aline-chargeisplacedat £=ainfront oftheplane represented bytherealaxisoff,Letthefurther transformation £=f(z)transform therealaxisof£intoasurface S,andthe point f=aintothepointz=z,sothata=/(^ )«Then thetransformation j. 18 274 Methods fortheSolution ofSpecial Problems[ch.vm givesthesolution when aline-chargeisplacedatz=zva.thepresenceof thesurface S.Inthistransformation itmust beremembered that U,and notV,isthepotential (cf.§318). 326. Conductors atdifferent potentials. Letussupposethatthetrans- formation%=<f)(z)transforms aconductor into the real axis of£.The further transformation W=G+Dlog£(§318)willgivethesolution when thetwopartsofthisplaneondifferent sides oftheoriginareraised to differentpotentials GandC+ttD. Thus thetransformation obtainedbyelimination of£,namely W=G+D\ogcf>(z), willtransform twopartsofthesame conductor intotwoparallel planes, andsowillgivethesolution ofaprobleminwhich twopartsofthesame conductor areraised todifferentpotentials. Examples oftheuseofConjugate Functions. 327.Twoexamplesofpractical importancewillnowbegiventoillus- trate theuseofthemethods ofconjugate functions. ExampleI.Parallel Plate Condenser. 328. Thetransformation *=^(t-log£-MV) hasbeen found totransform thetwoplatesinfig.90intothepositive and negative partsofthereal axis of£.Thefurther transformation W=log£ givesthesolution when these twopartsoftherealaxisof£areatpotentials and itrespectively (§326). Thus thetransformation obtained bytheelimination of£,namely z=-(ew-W+iir)TT•(243), willtransform thetwoplanesoffig.90—oneinfinite andonesemi-infinite — intotwo infiniteparallel planes. Thusequation (243) gives the trans- formation suitable tothecase ofasemi-infiniteplaneatdistance hfrom aparallelinfiniteplane,thepotentialdifferencebeingit. Bytheprincipleofimagesitisobvious that thedistribution onthe iipper plateisthesame asitwould beifthelowerplatewere asemi- infinite planeatdistance 2/iinstead ofaninfiniteplaneatdistance h.The equipotentialsandlines offorce foreitherproblemareshewn infig.91. 325-328] Conjugate Functions 275 Separatingrealandimaginary partsinequation (243), x IT h(eucosV-U), y=-(eusmV-V+tt).IT Thus theequipotential V— isthe liney=h,theequipotential V=itis theliney—0. Fig. 91. Ontheformerequipotential,therelation between xandUis h TT.(244). When TJ——co,a;=+co;ast/increases, xdecreases until itreaches a minimum value x—h/irwhenU= ;and asJ7further increasesthrough positive values xagain increases, reaching x=qc when Z7=+oo .Thus as Uvaries whileV=0, thepathdescribed isthepathPQR infig.91. Theintensityatanypointis \dWR= dz h\eITWII" Atapointonthe'equipotential V=0,thesurface-densityis R_1a~47r~4A(e^-iy IS— 2 276 Methods fortheSolution ofSpecial Problems[ch.vni AtP,U=—oo,sothat <7= -tj;asweapproach Q,aincreases andfinally becomes infinite atQ,while afterpassing Qandmoving along QR,theupper sideoftheplate,adecreases, andultimatelyvanishes totheorder ofe~u . The totalchargewithinanyrangeU1}U2is,byequation (233), Itthereforeappearsthatthetotalchargeontheupper partoftheplateQR isinfinite. Let us,however, consider thechargesonthetwo sides ofastripofthe plateofwidth Ifrom Q,i.e.thestripbetween x— hjirandx=l+K\tt. The twovalues ofUcorrespondingtothepointsintheupperandlower faces at which thisstrip terminates, arefromequation (244) thetworealroots of l+hJ±{ev_U) (245). Ofthese rootsweknow that one,sayUltisnegative andtheother(U^) ispositive.If Iislarge, wefindthat thenegativerootU^is,toafirsfr approximation, equalto ir,'h and this isitsactual valuewhen Iisvery large. Thus thechargeonthe lowerplatewithin alargedistance Ioftheedgeis h/.h andtherefore thedisturbance inthedistribution ofelectricityasweapproach Qresults inanincrease onthechargeofthelowerplate equaltowhatwould bethechargeonastripofwidthk/irintheundisturbed state. If Iislargethepositiveroot ofequation (245)is «t.-i*(i+t). sothatthetotalchargeonastripofwidth Ioftheupper plate approximates, when Iislarge,to loo- [1+_1 4tt°V hj' Thusalthoughthechargeontheupper plateisinfinite,itvanishes in comparisonwith thatonthelowerplate. 328,329] Conjugate Functions 277 ExampleIT.BendofaLeydenJar. 329. Themethod ofconjugatefunctions enables ustoapproximateto thecorrectionrequiredintheformula forthecapacityofaLeyden Jar,on account ofthepresenceofthesharp bend intheplates. A ^=-aF f=&_D B Fig. 92. Asapreliminary,letusfindthecapacityofatwo-dimensional condenser formed oftwoconductors, each ofwhich consists ofaninfiniteplate, bent intoanL-shape,thetwo L'sbeingfitted intooneanother asinfig.92. Letusassume the fivepoints A,B,(CD)/E, Ftobe£=—oo,—a,0, +b,+oorespectively,and letusforconveniencesuppose thepotential difference which occurs onpassing throughthevalue £*=0 tobe ir.Then thetransformation is whereW=log£(cf.§326). Tointegrate, weputu=(%+a)~- (£—b)^,andobtain-/M^W^. .(246), whereCisaconstant ofintegration. TomakeCvanish, wemust have z=Qwhen u=0,i.e.atthepointE. WeshallaccordinglytakeEasorigin,sothatG=0. 278 Methods fortheSolution ofSpecial Problems[oh.viii AtB,wenowhave £=—a,u=oo,andtherefore z=±ITAa/-+17tJ..Va Thus thedistances between thepairsofarms are ita/-AandirA respectively. LetPbeanypointinEFwhich isatadistance fromEgreat compared withEB. Letthevalue of£atPbe£p,sothatt,Pispositive andgreater than b. Wehave Tf=U+iV=log£sothatalongtheconductor FED,V= andU=log£ The totalcharge perunitwidth onthestripEPis,byformula(233), jydS=±(U P-UE)=±(}og!; P-\ogb) (247). IfPisfarremoved fromE,thevalue of£Pisvery great,andsince ?=~(248), thevalue ofit?willbenearly equaltounityatP. Fromequation (246), z=-2AJ-tan-1 lS/ju+2Alog(1+u)-Alog(1-u% sothatlog(1-t<2 )=2log(1+«)-2a/-tan"1^/|m- -|(249), inwhich thetermslog(1—u2 ),—z/A,arelargeatPincomparisonwith the others.Again, fromequation (248),wehave log£=log(em2+&)-log(l-w2 ) (250), inwhichlog £,log(1—v?)arelargeatP,incomparisonwith theterm log(ait2+b).Combining equations (249) and(250), log£=log(av?+b)-2log(1+«)+2^tan"1y^it+-J (251), inwhich thetermslog£and-jarelargeatPincomparisonwith theother terms. AtPwemayputu=1inalltermsexcept log£andz/A,andobtain asanapproximation log&=log(o+6)-2log2+2Jj/|tan"1 /y/|+§. 329,330] Multiple-valued Potentials 279 Thevalue ofzP-isofcourse xP+iyP,orEP. Thus, from theequation just obtained, equation (247)maybethrown intotheform /p1 <rds=—(log£,-log&) =ISi1+?)-21og2+2^tan-J\+f}...(252). Ifthelines offorcewere notdisturbed bythebend,weshould have ads!1fEP\ 4>tt\A )' Equation (252) shews that Iads isgreater than this,byanamount JE IK i1+1)--ios2+2\/ltan"'v7 ?}(253>- Letusdenote thedistances between theplates, namelyttAa/-andirA, byhand A;respectively,sothata/-=- .Expression (253)nowbecomes sothatthechargeontheplateEP isthesame asitwould beinaparallel platecondenser inwhich thebreadth ofthestripwasgreaterthanEPby When h=k,thisbecomes £(|-log,2)or-279A. Multiple-valued Potentials. 330. There aremany problemstowhich mathematicalanalysis yields more than onesolution, althoughitmaybefound thatonlyoneofthese solutions willultimately satisfytheactual data oftheproblem. Insuch acase itwill often beofinterest toexamine whatinterpretationhasto begiventotherejectedsolutions. Theproblemofdeterminingthepotential when theboundaryconditions aregivenisnotofthis class, forithasalreadybeenshewn(§§186—188) that, subjecttospecified boundary conditions, thetermination ofthepoten- tial isabsolutely unique. But itmayhappen that, insearchingforthe required solution, wecomeuponamultiple-valuedsolution ofLaplace's equation. Onlyonevalue cansatisfytheboundary conditions, butthe interpretationoftheother values isofinterest, andinthiswaywearrive atthestudyofmultiple-valued potentials. 280 Methods fortheSolution ofSpecial Problems[ch.viii ConjugateFunctions onaRiemann'sSurface. 331.Anobvious case ofamultiple-valued potentialarises from the conjugatefunction transformation W=<f>(z) (254), when(f)isnotasingle-valuedfunction ofz.Such cases havealready occurred in§§317, 320, 323, etc. Themeaningofthemultiple-valued potentialbecomes clear assoon asweconstruct aRiemann's surface onwhich $(z)canberepresentedas asingle-valuedfunction ofposition. OnepointonthisRiemann's surface mustnowcorrespondtoeach value ofW,andtherefore toeachpointin theIT-plane.Thusweseethat thetransformation (254)transforms the complete TT-planeinto acompleteRiemann's surface. Correspondingto agivenvalue ofztheremaybemanyvalues ofthepotential,butthese values will refer tothedifferent sheets oftheRiemann's surface. Ifany regiononthis surface isselected, which doesnotcontainanybranchpoints orlines,wecanregardthisregionasarealtwo-dimensionalregion, andthe correspondingvalue ofthepotential,asgiven byequation (254),willgive thesolution ofanelectrostaticproblem. 332. Toillustrate thisbyaconcrete case, consider thetransformation F= ..(255), _-a TF-plane. Fig. 93.z-surface-7?' which hasalready been considered in§317. TheRiemann's surfaceappro- priatefortherepresentationofthetwo-valued function z*maybesupposed tobeasurface oftwo infinite sheets connectedalongabranch linewhich extends over thepositivehalf ofthereal axis ofz. Toregardthissurface asadeformation oftheTF-plane, wemustsuppose that aslit iscutalongthelineOB(fig.93)intheTF-plane, andthat the 331-333] Multiple-valuedPotentials 281 twoedgesofthe slitaretaken andturned sothattheangle lir,whichthey originallyenclosed intheW-plane,isincreased to4>ir,afterwhich theedges areagain joined together. Theuppersheet oftheRiemann's surface soformed willnowrepresent theupperhalf oftheW-plane,while thelower sheet willrepresentthelower half.Twopoints i?,B„whichrepresent equalandoppositevalues ofW, say±Wn,will(byequation (255)) berepresented bypointsatwhich zhas thesame value; theyareaccordinglythetwopointsontheupperand lower sheetrespectivelyforwhich zhasthevalueW2 . Acircularpath pqrs surroundingintheW-planebecomes adouble circle onthe^-surface, one circlebeingontheuppersheet andoneon thelower, andthepath beingcontinuous since itcrosses from onesheet totheother each time itmeets thebranch-line. Aline a/3intheupperhalf oftheW-plane becomes, aswehave seen, aparabola a/3ontheuppersheet ofthe^-surface.Similarlyaline a/3'in thelower half oftheW-planewillbecome aparabola a'/3'onthelower sheet ofthe^-surface. Thespaceoutside theparabola a/3ontheuppersheet of the^-surface transforms intoaspaceintheW-planebounded bythelinea/3 andthelineatinfinity. Consequentlythetransformation under consideration givesthesolution oftheelectrostaticproblem,inwhich thefield isbounded onlybyaconducting parabolaandtheregionatinfinity. Thesame isnot true ofthespaceinside theparabola a/3,forthistransforms intoaspacein theW-planebounded byboth thelinea/3andtheaxisAOB. Itisnow clear that thetransformation hasnoapplicationtoproblemsinwhich the electrostatic field isthespaceinside aparabola. Ingeneralitwillbeseenthattwopoints,which areclose tooneanother ononesheet ofthe^-surface, butareonoppositesides, ofabranch-line, willtransform intotwopointswhich arenotadjacenttooneanother inthe W-plane,andwhich thereforecorrespondtodifferentpotentials.Conse- quently wecannot solve aproblem byatransformation whichrequiresa branch-line tobeintroduced intothatpartoftheRiemann's surface which representstheelectrostatic field. ImagesonaRiemann's Surface. 333. Inthetheoryofelectricalimages,asystemofimaginary chargesis placedinaregionwhich doesnotformpartoftheactual electrostatic field. When atwo-dimensional problemissolved byaconjugatefunction trans- formation, theelectrostatic field must, aswehave seen, berepresented by aregiononasinglesheet ofthecorrespondingRiemann's surface, and this regionmust notbebroken bybranch-lines. Thesame, however, isnottrue ofthepartofthe field inwhich theimaginary imagesareplaced,forthis 282 Methods fortheSolution ofSpecialProblems[ch.viii mayberepresented byaregionononeoftheother sheets oftheRiemann's surface. Totakethesimplest possible illustration, supposethat inthe£-plane we have aline-chargeealongthelinerepresented bythepoint P,infront of f-planez-surface P»+eP»(upper sheet) A O B O A P'*- e P•(lower sliect) Fig. 94. theuninsulatedconducting plane represented bythereal axisAB. The solution, asweknow, isobtained byplacingacharge—eatthepoint P', which istheimageofPinAOB. Thevalue ofthepotential (U)isgiven, asin§818,by U+iV=A log£-&» r-w Letusnowtransform thissolution bymeans ofthetransformation £=z$ (256). Theconducting planeAOBtransforms intoasemi-infiniteplane OB,which maybetaken tocoincide with thebranch-line oftheRiemann's surface. ThechargeeatPbecomes achargeatapointPontheuppersheet ofthe surface, while theimageatP'becomes achargeatapointP'onthelower sheet. Thuswecanreplacethesemi-infinite conductor OBinthe2-plane byanimageatapointP'onthelower sheet ofaRiemann's surface, andwe obtain the fieldduetoaline-chargeandasemi-infinite con clactor inan ordinarytwo-dimensionalspace. From thetransformation used, thepotentialisfound tobegiven by \/z—Va.U+iV=A\ozo\Jz—"J- a inwhichUisthepotential,z—aisthepoint (a,a)ontheupper sheet, and z=—aistheimageonthelower sheet. IncalculatingapotentialonaRiemann's surface, wemust notassume thepotentialofaline-chargeeatthepoint (a,a)tobe 0-2e\ogR (257), whereRisthedistance from thepoint (a,a).Infact, thispotentialwould obviously haveaninfinity both atthepoint (a,a)ontheupper sheet, and alsoatthepoint {a,a)onthelower sheet, and would bethepotentialof twoline-charges,oneatthepoint (a,a)oneach sheet. 333-335] MuUiple-valued Potentials 283 Theappropriate potential-functionforasingle chargecaneasilybe found. Asintheproblem just discussed, itisclear thatthepotential duetothe single line-chargeat(a,a)ontheuppersheet isthevalue ofUgiven by U+ iV=G+Alog(VJ-Va) =6'+;!log(rM-a*e¥ ) ==G+Alog\(Vrcos -x—V«cos- J+i(Vrsin-—Vasin- JI, sothat U=G+%Alog-IfVrcos^—Vacos~]+(Vrsin-—Vasin- =C+\Alog{r-2Varcos^(0-a)+a}, and ifthis istobethepotentialdue toaline-charge e,itisclear, on examiningthevalue ofUnearthepoint (a,a),thatthevalue ofAmust be —2e.Thus thepotentialfunction must be C-elog {r-2Varcos£(0- a)+a} (258), instead ofthatgiven byexpression (257), namely, C-elog{r2-2ar cos(#-«) +a2 } (259). Itwillbenoticed thatbothexpressionsaresingle-valued forgivenvalues of(r,6),butthat foragivenvalue ofz,expression (258)hastwovalues, correspondingtotwovalues of6differing by2tt,whileexpression (259) has onlyonevalue. Or,tostate thesamethinginother words, theexpression (259)isperiodicin with aperiod 2tt,whileexpression (258)isperiodic with aperiod4)ir. Potential inaRiemann's Space. 334. Sommerfeld* hasextended these ideas soastoprovidethesolution ofproblemsinthree-dimensionalspace. Hismethod rests onthedetermination ofamultiple-valued potential function, thefunction being capableofrepresentationasasingle-valued function ofpositionina"Riemann'sspace,"thisspace beinganimaginary spacewhich bears thesame relation torealthree-dimensionalspaceasa Riemann's surface bears toaplane. 335. Thebest introduction tothismethod willbefound inastudyof thesimplest possible example,and this willbeobtained byconsideringthe three-dimensional problem analogoustothetwo-dimensionalproblem already discussed in§333. *"Ueber verzweigte Potentiate imRaum," Froc. Lund. Math. Soc. 28,p.395,and 30,p.161. 284 Methods fortheSolution ofSpecial Problems[oh.viii Wesupposethatwehave asingle point-chargeinthepresenceofan uninsulatedconductingsemi-infiniteplane bounded byastraight edge.Let ustakecylindricalcoordinates r,6,z,takingtheedgeoftheplanetobe r=0,theplaneitself tobe8=0,andtheplane throughthechargeatright anglestotheedgeoftheconductor tobez=0.Letthecoordinates ofthe point-chargebea,a,0. TheRiemann'sspaceistobetheexactanalogueoftheRiemann's surface described in§332. That istosay,itistobesuch thatonerevolu- tionround theliner—takes usfrom one"sheet"totheother ofthe space,while tworevolutionsbringusback tothestarting-point. Thus, for afunction tobeasingle-valuedfunction ofpositioninthisspace,itmust be aperiodicfunction ofofperiod4nr. Letusdenote byf(r, 6,z,a,a,0)afunction ofr,6,andzwhich isto satisfythefollowingconditions : (i)itmustbeasolution ofLaplace's equation; (ii)itmustbeacontinuous andsingle-valuedfunction ofpositionin theRiemann'sspace; (iii)itmust have oneandonlyoneinfinity,thisbeingatthepoint a, cl,onthe first"sheet"ofthespace,andthefunction approximatingnear thepointtothefunction-p,where B,is thedistance from thispoint; (iv)itmust vanish when r=co . Itcanbeshewn, byamethodexactlysimilar tothatused in§186,that there canbeonlyonefunctionsatisfyingthese conditions. Hence thefunc- tionf(r, 0,z,a,a,0)canbeuniquely determined, andwhen found itwillbe thepotentialintheRiemann'sspaceofapoint-chargeofunitstrengthatthe point a,a,0. Consider nowthefunction f(r, 0,z,a,a,0)-/(r, 6,z,a,-a,0) (260), which isofcourse thepotentialofequalandopposite point-chargesatthe point a,a,0,and atitsimageintheplane6=0,namely,thepoint a,—a,0. This function, byconditions(i)and(iv),satisfiesLaplace's equationand vanishes atinfinity. Onthe first sheet ofthesurface, onwhich avaries from to2ir(orfrom 4>ttto67r, etc.),ithasonlyoneinfinity, namely,at a,a,0,atwhich itassumes thevalue-5. Jlv From theconditions which itsatisfies, thefunction/(r, 0,z,a,a,0)must clearlyinvolve 6andaonlythrough6—a,andmust moreover beaneven function of6-cr.Itfollows that,when 6=0,expression (260)vanishes. 335,336] Multiple-valued Potentials 285 Again,since thefunction fisperiodicin6with aperiod 2tt, itfollows that,when 6=—2ir,expression (2G0)maybewritten intheform f(r, 2tt, z,a,a,0)-/(r,-2tt, z,a,-a,0), and thisclearlyvanishes. Thusexpression (260) vanishes when =and when 6=2tt.That istosay,itvanishes onboth sides ofthesemi-infinite conducting plane. Itisnow clear thatexpression (260)satisfies alltheconditions which have tobesatisfied bythepotential. Theproblemisaccordinglyreduced tothat ofthedetermination ofthefunction/(?*, 6,z,a,a,0). 336. Letuswrite r=ep ,a=ep ', then thedistanceRfromr,0,ztoa,a,isgiven by R*_r2_2arcos(6-a)+a2+z" =2ar{cosi(p—p)—cos(0— a)]+z\ Takenewfunctions R'and/(w) given byKl=2ar(cosi(p-p)-cos(0- it)}+z\ J\' gilt gta* Thefunction f(u)hasinfinities whenu=a,a±2ir,a±4ur, ...,itsresidue being unityateachinfinity. Also,whenu=a,thevalue ofR'becomes R. Hence theintegral £,f(u)du(261), where theintegralistaken roundanyclosed contour intheit-plane which surrounds thevalue u=a,butnoother oftheinfinitiesoff(ii)}willhave as itsvalue 2iirx^.Weaccordinglyhave 1 1fl eiu S=^JB'i=37>(262). Theintegral justfoundgivesaform forthepotentialfunction inordinary space which, asweshallnow see,caneasilybemodified soastogivethe potentialfunction intheRiern ami'sspacewhich wearenowconsidering. Wenotice firstthatp>,regardedasafunction ofr,6,andz,isasolution ofLaplace's equation, whatever value umayhave. Hence theintegral (261) willbeasolution ofLaplace's equationforallvalues off(u),foreachterm oftheintegrandwillsatisfytheequation separately. Ifwetake/: e2—e2 286 Methods fortheSolution ofSpecial Problems[ch.vin weseethattheinfinities of/(w)occur when u—a,a± 4<7r,a±S-n-, etc.,and theresidue ateach isunity. Hence, ifwetake theintegralround one infinity only, sayu=a,thevalue of is/iff/M*1 (263) willbecome identical with -~atthepointatwhich R'=0.Moreover, expression (263) is,aswehave seen, asolution ofLaplace's equation:it isseen oninspectiontobeasingle-valuedfunction ofpositiononthe Biemann's surface, andtobeperiodicin9withperiod4nr.Hence itisthe potential-functionofwhich weareinsearch. Thus iu f(r, 6,z,a,a,0)= 4vr I£_•»Vr2-2arcos{6-u) +a?+z* The details oftheintegrationcanbefound inSommerfeld'spaper.The value oftheintegralisfound tobe 12,_,/^+T -^-tanla/, where t=cos\(<fi—a),a=cosh(p—p1 ). Othersystemsofcoordinates canbetreated inthesameway ;details will befound inthepaperstowhich reference hasalreadybeenmade. 337. Thepresent chapterhasattemptedtogiveanaccount ofthe principalmethods available forthesolution ofelectrostaticproblems. Afew examples havebeengivenofeachmethod, butnoattempthasbeenmade to enumerate alltheproblems which canbesolved. Thereader whowishes to study particular problems morefullymaybereferred tothefollowingworks : Sir"W.Thomson (Lord Kelvin). PapersonElectricity andMagnetism. Inparticular anumber ofexamplesofimages andinversion willbefound here, with mmierical calculations. Maxwell.Electricity andMagnetism.Vol. I.(3rd Edn.). InChap.ix.thetheoryofspherical harmonics isdeveloped, andtheproblem ofthe distribution ofelectricity onanearly spherical conductor free inspace,asalsothatona nearly sj^herical conductor enclosed inanearly spherical andnearly concentric conduct- ingvessel, aresolved indetail. The coefficients ofcapacity andinduction oftwospherical conductors areinvestigated byspherical harmonics.Chapterxi.containsexamplesofthe method ofimages andinversion. Chapter xn.contains anumber ofexamples ofconjugate functions, some beingofspecial importanceinthetheoryofelectrostatic instruments. J.J.Thomson. Recent Researches inElectricity andMagnetism. Chapterin.contains important examplesofconjugate function transformations. In particular problemsaresolved which enable ustoestimate theeffect onthecapacityofa condenser produced bythe slitbetween aguard ringandthemoveableplate ofthecon- denser. Transformations aregiven which solve theproblemsof(i)acondenser formedby 336,337] Examples287 twoparallel andequal platesoffinite breadth;(ii)acondenser formed bytwoparalleland equal strips placed inthesame plane;(iii)apileofplates;(iv)asystemof2nplates arranged radiallyatangles ir/nwithoneanother, alternateplates beingatthesame potential. Kirchhoff. Gesammelte abhandlungen. Aformula isgivenforthecapacityoftwocircularplatesofanuniform thicknessplaced coaxiallyatanydistanceapart. EXAMPLES. 1.Aninfinite conducting planeatzeropotentialisunder theinfluence ofachargeof electricityatapoint0.Shew thatthecharge onanyarea oftheplaneisproportionalto theangleitsubtends at0. 2.Acharged particleisplacedinthespace between twouninsulated planes which intersect atright angles. Sketch thesections oftheequipotentials madebyanimaginary plane throughthecharged particle,atright anglestotheplanes. 3.Inquestion 2,lettheparticlehaveacharge e,andbeequidistant from theplanes. Shew thatthetotalcharge onastrip,ofwhich oneedgeisthelineofintersection ofthe planes, andofwhich thewidth isequaltothedistance oftheparticle from this lineof intersection,is—\e. 4.Inquestion 3,thestripisinsulated from theremainder oftheplanes,these being still toearth, andtheparticleisremoved. Find thepotentialatthepoint formerly occupied bytheparticle, produced byraisingthestriptopotentialV. 5.Iftwo infiniteplaneuninsulated conductors meet atanangleof60°,andthere isa chargeeatapoint equidistant from each,anddistant rfrom thelineofintersection, find theelectrification atanypointoftheplanes. Shew that atapointinaprincipal plane throughthecharged pointatadistance r^/3from thelineofintersection, thesurface densityis 3 1+,47rr2\4 7J7 6.Two small pith balls, each ofmass m,areconnected byalight insulating rod. Therod issupported byparallel threads, andhangsinahorizontalpositioninfront ofan infinite vertical planeatpotentialzero. Ifthe ballswhen charged with eunits of electricityareatadistance afrom theplate, equaltohalfthelengthoftherod,shew thattheinclination 6ofthestringstothevertical isgivenby e2 tan<9=-s-(1A' Amga2 \2*/ 7.What istheleastpositive chargethatmust begiventoaspherical conductor, insulated andinfluenced byanexternalpoint-chargeeatdistance rfrom itscentre,in order thatthesurface density maybeeverywhere positive? 8.Anuninsulated conducting sphereisunder theinfluence ofanexternal electric charge;findtheratio inwhich theinduced chargeisdivided between thepartofits surface indirect view oftheexternalcharge andtheremaining part. 9.Apoint-chargeeisbrought near toasphericalconductor ofradius ahaving a charge E.Shew thattheparticlewillberepelled bythesphere,unless itsdistance from thenearestpointofitssurface islessthan\a*/~p%approximately. 288 Methods fortheSolution ofSpecial Problems[ch.viii 10.Ahollow conductor hastheform ofaquarterofasphere bounded bytwo perpendicular diametral planes.Find theimageofacharge placedatanypoint inside. 11.Aconductingsurface consists oftwo infinite planes which meet atright angles, andaquarterofasphereofradius afitted intotheright angle.Iftheconductor isatzero potential, andapoint-chargeeissymmetrically placedwithregardtotheplanesandthe spherical surface atagreatdistance /from thecentre, shew that thecharge induced on thespherical portionisapproximately—beaPjirf3 . 12.Apoint-chargeisplacedinfront ofaninfinite slab ofdielectric, bounded bya planeface. Theangle between alineofforce inthedielectric andthenormal totheface oftheslab isa;theangle between thesametwolines intheimmediate neighbourhoodof thechargeis/3.Prove thata,/3areconnected bytherelation .13 /2k .aSm 2=VmSm 2' 13.Anelectrifiedparticleisplacedinfront ofaninfinitelythickplateofdielectric. Shew thattheparticleisurged towards theplatebyaforce k+14J2' where disthedistance ofthepoint from theplate. 14.Two dielectrics ofinductivecapacities kjandk2areseparated byaninfinite plane face. Chargeseue2areplacedatpoints onalineatright anglestotheplane, each ata distance afrom theplane.Find theforces onthetwocharges, andexplain whytheyare unequal. 15.Two conductors ofcapacitiescx,c2inairareonthesame normal totheplane boundary between twodielectricskj,k2,atgreat distances a,bfrom theboundary. They areconnected byathinwireandcharged. Prove thatthechargeisdistributed between them approximatelyintheratio Kj—k2 2kj J Kl tc22b(Kt+K2)(K1+K2)(a+b)rK2 \cC!2a(n 1+n2) (Kl+K2)(a+b)} 16.Athinplane conducting lamina ofanyshape and size isunder theinfluence ofa fixed electrical distribution ononesideofit.If<ribethedensityoftheinduced charge atapointPontheside ofthelamina facing thefixed distribution, and <r2that atthe corresponding point ontheother side,provethat <n—<r2=cr,where o-isthedensityatP ofthedistribution induced onaninfinite plane conductor coinciding with thelamina. 17.Aninfiniteplate with ahemisphericalboss ofradius aisatzeropotential under theinfluence ofapoint-chargeeontheaxisofthebossdistant /fromtheplate. Find the surface densityatanypointoftheplate,andshew thatthechargeisattracted towards theplate with aforce e24e2a3/3 4/2(/4-a4 )2* 18.Aconductor isformed bytheouter surfaces oftwoequal spheres, theangle between their radii atapointofintersection being 277/3. Shew that thecapacityofthe conductor soformed is 5^/3-4 where aistheradius ofeithersphere.2N/3"> Examples 289 19.Within asphericalhollow inaconductor connected toearth, equal point-charges eareplaced atequal distances /from thecentre, onthesame diameter. Shew thateach isacted onbyaforce equal to r_4«^3_i-i L(«4-/4 )2+ 4/2j' 20.Ahollowsphere ofsulphur (ofinductivecapacity 3)whose inner radius ishalf its outer isintroduced intoauniform field ofelectric force. Prove that theintensityofthe field inthehollow willbelessthan that oftheoriginalfield intheratio 27 :34. 21.Aconducting sphericalshell ofradius aisplaced, insulated andwithoutcharge, inauniform field ofelectric force ofintensity F.Shew that ifthesphere becutintotwo hemispheres byaplane perpendiculartothefield, these hemispheres tend toseparate and require forces equalto-^cPF2tokeepthemtogether. 22.Anuncharged insulated conductor formed oftwoequal spheresofradius a cutting oneanother atright angles,isplacedinauniform field offorce ofintensity F, with thelinejoining thecentresparalleltothelines offorce. Prove that thecharges induced onthetwospheresare^Fa2and—^Fa2 . 23.Aconducting plane hasahemispherical boss ofradiusa,andatadistance /from thecentre ofthebossandalongitsaxisthere isapoint-chargee.Iftheplane andthe bossbekeptatzeropotential, provethatthecharge induced ontheboss is -Ji._/l=gLl 24.Aconductor isbounded bythelarger portionsoftwoequal spheresofradius a cuttingatanangle ^tt,and ofathirdsphereofradius ccutting thetwoformer orthogonally. Shew that thecapacityoftheconductor is c+a(f-I V3)-«c{2(a2+c2)-^-2(a2+3c2)-*+(a2+4c2)~^}. 25.Aspherical conductor ofinternal radius6,which isuncharged andinsulated, surrounds aspherical conductor ofradiusa,thedistance between their centresbeing c, which issmall. Thecharge ontheinner conductor isE.Find thepotential function forpoints between theconductors, andshew that thesurfacedensityatapointPonthe inner conductor is E_/I_3ccos6\ 4n\a2b*-a3)' where 8istheangle thattheradius through Pmakes with thelineofcentres, andterms inc2areneglected. 26. Ifaparticle charged withaquantityeofelectricity beplacedatthemiddlepoint ofthelinejoining thecentres oftwoequal spherical conductors keptatzeropotential, shew that thecharge induced oneachsphereis -2em(l-m+m2-3m3-f4m4 ), neglecting higher powersofm,which istheratio oftheradius tothedistance between the centres ofthespheres. 27.Two insulated conducting spheresofradii a,b,thedistance cofwhose centres islargecompared withaandb,havechargesEuE2respectively. Shew thatthepotential energyisapproximately J. 19 290 Methods fortheSolution ofSpecialProblems[en.vm 28.Shew thattheforcebetween twoinsulated sphericalconductors ofradius aplaced inanelectric field ofuniform intensity Fperpendiculartotheir lineofcentres is c4\c3c° cbeing thedistance between their centres. 29.Twounchargedinsulatedspheres,radii a,b,areplacedinauniform field offorce sothat their lineofcentres isparalleltothelines offorce, thedistance cbetween their centres being great comparedwithaand b.Prove that thesurface densityatthepoint atwhich thelineofcentres cutsthefirstsphere (a)isapproximately F ( GZ>315a&328a26357a3631&y+-jT+-?-+-ji-+-jr-+"'r 30.Aconducting sphereofradius aisembedded inadielectric (K)whose outer boundaryisaconcentric sphereofradius 2a.Shew that ifthesystembeplacedin auniform field offorce F,equal quantitiesofpositive andnegative electricityare separated ofamount 9Fa*K 5/i+7' 31.Asphereofglassofradius aisheld inairwith itscentre atadistance cfrom a pointatwhich there isapositive chargee.Prove thattheresultant attraction is where/9=(A-1)/(A+1). 32.Aconducting sphericalshell ofradius aisplaced,insulated andwithoutcharge, inauniform field offorce ofintensityF.Shew that ifthespherebecutintotwo hemispheres byaplane perpendiculartothefield, aforce^a2F2*3requiredtoprevent thehemispheresfromseparating. 33.Aspherical shell, ofradiia,bandinductivecapacity K,isplacedinauniform field offorce F.Shew thattheforce inside theshell isuniform andequalto 9KF 9A-2(A-l)-2(63/a3-l)* 34.Thesurface ofaconductor being oneofrevolution whose equationis 417_ r+ r~ 12' wherer,r'arethedistances ofanypoint fromtwo fixed pointsatdistance 8apart,find theelectric densityateither vertex when theconductor hasagiven charge. 35.Thecurve 9afa+x a—x11 when rotated round theaxis ofxgeneratesasingleclosed surface, which ismade the boundingsurface ofaconductor. Shew that itscapacitywillbea,andthatthesurface densityattheendoftheaxis willbee/dna2 ,where eisthetotalcharge. 36.Twoequal sphereseach ofradius aareincontact. Shew thatthecapacityofthe conductor soformed is2alog e2. Examples 291 37.Two spheresofradiia,bareincontact, abeing largecompared with b.Shew that iftheconductor soformed israised topotential V,thecharges onthetwospheresare Va 1-——-jrandla -r-—-^. \6(a+6)V \G(a+b)y 33.Aconducting sphereofradius aisincontact withaninfiniteconducting plane. Shew that ifaunitpoint-chargebeplaced beyond thesphere andonthediameterthrough thepointofcontact atdistance cfrom thatpoint,thecharges induced ontheplane and sphere are KCL,TT<X,TTtt .TTCL cot—and—cot 1. c c c c 39.Prove that ifthecentres oftwoequal uninsulatedsphericalconductors ofradius abeatadistance 2capart, thecharge induced oneachbyaunitchargeatapoint midway between them is where c=acosh a.2(-l)nsech«a, l 40.Shew thatthecapacityofaspherical conductor ofradiusa,with itscentre ata distance cfromaninfinite conducting plane,is QO asinha2cosechwa, i where c=acosh a. 41Aninsulated conducting sphereofradius aisplaced midway between two parallelinfinite uninsulatedplanesatagreatdistance 2capart. Neglecting(- J,shew that thecapacityofthesphereisapproximately a|l+|log2J. 42Twospheresofradii r1}r2touch each other, andtheircapacitiesinthisposition arecx,c2.Shew that fool ool col) where /=—— .Jrx+r2 43.Aconducting sphereofradius aisplacedinair,with itscentre atadistance e from theplanefaceofaninfinite dielectric. Shew that itscapacityis oo/X-IV1-1 asinhaT(t^— ; Icosech na,7\A+1/ where a=cja. 44.Apoint-chargeeisplaced between twoparalleluninsulated infinite conducting planes,atdistances aand bfromthemrespectively. Shew thatthepotentialatapoint between theplanes which isatadistance zfrom thecharge and isonthelinethroughthe charge perpendiculartotheplanesis \2a+2b) \2a+2b) \2a+2bJ' V2a+2bJ\+ ;„.. .:+ {l \2a+2bJ \2a+2bJ \2a+2bJ \ 19—2 292 Methods fortheSolution ofSpecial Problems[ch.viii 45.Aspherical conductor ofradius aissurrounded byauniform dielectricA",which isbounded byasphereofradius bhavingitscentre atasmall distance yfrom thecentre oftheconductor. Prove that ifthepotentialoftheconductor isV,andthere areno other conductors inthe field, thesurface densityatapoint where theradius makes an angle 6with theline ofcentres is KVbf 6(iT-l)ya8cos0 }• 47ra{(ff-l)o +6}\^2(K-l)a3+(K+2)b3 . 46.Ashell ofglassofinductive capacity A,which isbounded byconcentricspherical surfaces ofradiia,b(a<b\ surrounds anelectrifiedparticlewith chargeEwhich isata pointQatasmall distance cfrom0,thecentre ofthespheres. Shew thatthepotential atapointPoutside theshell atadistance rfromQisapproximately E 2Eo(b3-a3)(K-lfcos (9 r+2a3(K-lf-b3(K+2) (2K+1) r2' where 6istheangle whichQPmakes withOQproduced. 47. Ifthecentres ofthetwo shells ofaspherical condenser beseparated byasmall distance d,provethatthecapacityisapproximately ab ( abd2 "1 b^a\+ (b-a){V-a?))' b 48.Acondenser isformed oftwospherical conducting sheets, one ofradius b surrounding theother ofradius a.Thedistance between thecentres isc,thisbeingso small that(c/a)2maybeneglected. The surface densities ontheinner conductor atthe extremities oftheaxis ofsymmetryoftheinstrument are<ri, 0-2,andthemean surface densityovertheinner conductor isa.Prove that o"2~o"i 6ca2 cr b3- 49.Theequationofthesurface ofaconductor isr=a(1+ePn),where eisvery small, andtheconductor isplacedinauniform field offorceFparalleltotheaxis ofharmonics. Shew thatthesurface densityoftheinduced chargeatanypointisgreater than itwould beifthesurface wereperfectly spherical, bytheamount 4<r8 (2n+l){(w+1)i>^ 1+(w-2)i>"-|}- 50.Aconductor atpotential Vwhose surface isoftheformr=a(l+ePn)issur- rounded byadielectric(A")whose boundaryisthesurface r=b(1+rjP n),andoutside this thedielectric isair.Shew thatthepotentialintheairatadistance rfrom theoriginis KabV 1(2n+l)eanb2n+1+(K-l)r 1bn{nb2n+1+(n+l)a'in+l }Pn (K-l)a +b\_i- (l+n+»A')6s*+1+(A'-l)(n +l)oa»+l r where squares andhigher powersofeand77areneglected.«]. 51.Thesurface ofaconductor isnearly spherical,itsequation being r=a(1+oSy, where eissmall. Shew that iftheconductor isuninsulated, thecharge induced on itbyaunitchargeatadistance /from theorigin audofangularcoordinates6, <f>is approximately Examples 293 52.Auniform circular wire ofradius acharged withelectricityoflinedensitye surrounds anuninsulated concentric sphericalconductor ofradius c;prove that the electrical densityatanypointofthesurface oftheconductor is 53.Adielectricsphereissurrounded byathin circular wire oflarger radius b carryingacharge E.Prove that thepotentialwithin thesphereis ^LS, 1Vl,1+4^ 1.3.5...2n-l /r\*»\ *1i~ 1+2»(1 +Z)2.4.6...2«W2n J* 54. Ifwithin aconductor formed byacone ofsemi-vertical anglecos-1 fxandtwo sphericalsurfaces r=a, r=bwith centres atthevertex ofthecone, acharge qontheaxis atdistance r'from thevertexgives potential V,and ifwewrite *a r=ae~\ V=Ue2 ,Xo=log-^> mnun thesummation withrespecttomextendingtoallpositive integers, andthatwith respect tontoallnumbersintegralorfractional forwhichPn(^)=0,determine Amn.Effecting thesummation with respecttom,shew thatwhen r</, andthatwhen r>r', 55.Asphericalshell ofradius awithalittle hole initisfreelyelectrified topotential V.Prove thatthecharge onitsinner surface islessthanVS/8ira, whereSisthearea of thehole. 56.Athinspherical conductingshellfromwhich anyportions havebeenremoved is freelyelectrified. Prove thatthedifference ofdensities inside andoutside atanypointis constant. 57. Electricityisinduced onanuninsulatedspherical conductor ofradiusa,bya uniform surface distribution, density <r,overanexternal concentric non-conducting spherical segmentofradius c.Prove that thesurfacedensityatthepointAofthe conductor atthenearer endoftheaxis ofthesegmentis whereBisthepointofthesegment onitsaxis,andDisanypoint onitsedge. 58.Twoconductingdiscs ofradiia,a'arefixed atright anglestothelinewhich joinstheir centres, thelengthofthis linebeing r,largecompared with a.Ifthe first havepotential Vandthesecond isuninsulated, prove that thecharge onthe first is 2anr9-V 7r2r2—4aa'' 59.Aspherical conductor ofdiameter aiskeptatzeropotentialinthepresenceofa fineuniformwire,intheform ofacircle ofradius cinatangent planetothespherewith 294 Methods fortheSolution ofSpecial Problems[ch.vm itscentre atthepoint ofcontact, which hasachargeEofelectricity; provethat the electricaldensity induced onthesphereatapoint whose direction from thecentre ofthe ringmakes anangle -tywiththenormal totheplaneis c2Esec3itsf2*ki-/,Y (a2+c*sec2ylt-2aetan4,cos6)"*dO. 60.Prove thatthecapacityofahemisphericalshell ofradius ais 61.Prove that thecapacityofanelliptic plateofsmalleccentricityeandareaAis approximately x/®§(^*«)- 62.Acircular disc ofradius aisunder theinfluence ofacharge qatapointinits planeatdistance bfrom thecentre ofthedisc. Shew that thedensityoftheinduced distribution atapoint onthedisc is q/&-<& 2n2R2Va2-/-*' wherer,Rarethedistances ofthepoint from thecentre ofthediscandthecharge. 63.Anellipsoidal conductor differs but little from asphere.Itsvolume isequalto that ofasphereofradiusr,itsaxes are2r(l+a),2r(l+/3), 2r(l+y). Shew thatneg- lecting cubes ofa,/3,y,itscapacityis 64.Aprolate conducting spheroid,semi-axesa,b,hasachargeEofelectricity. Shew thatrepulsion between thetwohalves intowhich itisdivided byitsdiametralplaneis E2 ,a log- 4(a2-62 )&6" Determine thevalue oftheforce inthecase ofasphere. 65.One faceofacondenser isacircularplateofradius a :theother isasegment of asphereofradius R,Rbeingsolargethat theplateisalmost flat. Shew that the capacityis^KR\ogtiltwhereti,tarethethickness ofdielectric atthemiddle andedge ofthecondenser. Determine alsothedistribution ofthecharge. 66.Athin circular discofradius aiselectrified withchargeEandsurroundedbya spheroidalconductor with chargeE1,placedsothattheedgeofthedisc isthelocus ofthe focusSofthegenerating ellipse. Shew thattheenergyofthesystemis 2a 2a Bbeing anextremityofthepolaraxisofthespheroid, andGthecentre. 67. Ifthetwosurfaces ofacondenser areconcentric andcoaxial oblatespheroidsof small ellipticitieseand <'andpolar axes 2cand2c',prove thatthecapacityis CC'(C-c)-*{c'-C+$ (ec'- e'c)}, neglecting squaresoftheellipticities ;and findthedistribution ofelectricityoneach Burface tothesame order .ofapproximation. Examples 295 68.Anaccumulator isformed oftwoconfocalprolate spheroids, andthespecific inductive capacityofthedielectric isA7/ur, where cristhedistance ofanypoint from the axis. Prove thatthecapacityoftheaccumulator is where a,bandaubtarethesemi-axes ofthegenerating ellipses. 69.Athinsphericalbowl isformed bytheportionofthesphere #2+y2+z2=as !*>&qji, £j2 bounded byandlyingwithin thecone— - 2+t»=- 2,and isputinconnection with theearth (X" 0" C" byafinewire. istheorigin, and C,diametrically oppositeto0,isthevertex ofthe bowl;Qisanypointontherim,andPisanypoint onthegreatcircle arcCQ.Shew thatthesurface densityinduced atPbyachargeEplacedat is Ec CQ where47ra&/0P"(0Pa-0$2)4' d6 Jo(a2(a2sin2(9+62cos2 (9)^' 70.Three longthin wires, equally electrified, areplaced paralleltoeach other sothat theyarecutbyaplane perpendiculartothem intheangular pointsofanequilateral triangleofside sJZc ;shew thatthepolar equationofanequipotentialcurve drawn onthe planeis r6+c6-2r3c3cos3$=constant, thepolebeingatthecentre ofthetriangle andtheinitial linepassing through oneofthe wires. 71.Aflatpieceofcorrugated metal(y=asinmx)ischarged withelectricity.Find thesurface densityatany point,andshew that itexceeds theaverage density approxi- matelyintheratiomy:1. 72.Alonghollowcylindricalconductor isdivided intotwopartsbyaplane through theaxis,andthepartsareseparated byasmall interval. Ifthetwopartsarekeptat potentials VxandV2,thepotentialatanypointwithin thecylinderis 2 7T a2-?-2 where risthedistance from theaxis,and <9istheangle between theplane joiningthe pointtotheaxisandtheplane through theaxisnormal totheplaneofseparation. 73.Shew that thecapacity perunitlengthofatelegraphwire ofradius aatheighth above thesurface oftheearth is 71Anelectrified linewithchargeeperunitlengthisparalleltoacircular cylinder ofradius aandinductive capacity K,thedistance ofthewirefrom thecentre ofthe cylinder beingc.Shew that theforce onthewire perunit lengthis K-l 4aV K+l c(c2-a2 )' 75.Acylindricalconductor ofinfinite length, whose cross-section istheouter boundaryofthree equal orthogonalcircles ofradiusa,hasachargeeperunit length. Prove that theelectric densityatdistance rfrom theaxis is e(3r2+a2)(3r2-a2-\/6ar)(3r2-a2+v/ 6a/-) 6^a r2(9r4-3a2r2+a4 ) 296 Methods fortheSolution ofSpecial Problems[ch.viii 76. Ifthecylinder xi+yi=aibefreely charged, shew that infreespace theresultant force varies as / a8\~* (r4+2a4cos4<9+-^), wherex=r cos6,y=rsin8;andthat itsdirection makes with theaxisofxanangle r4—a* \2 -. -.tan 2(9 r4+a4 77. If(f)+i\ls=f(x +iy),andthecurves forwhich<£=constant beclosed, shew that thecapacity Cofacondenser withboundarysurfaces =<£i, <£=$oi3 KM 4tt(0i-0o) perunitlength, where[\^]istheincrement ofv|/-onpassing onceround a0-curve. 78.Using thetransformation x+iy=ccot\(U+iV), shew that thecapacity Cper unitlengthofacondenser formed bytworightcircular cylinders (radii a,b),oneinside theother, withparallel axes atadistance dapart,isgiven by ^= 2-"-'(t) 79.Aplane infinite electric gratingismade ofequal andequidistant parallelthin metalplates, thedistance between their successive central linesbeing tt,andthebreadth ofeachplate 2sin~l(-= j.Shew thatwhen thegratingiselectrified toconstant potential, thepotential andchargefunctionsV,Uinthesurrounding spacearegiven bytheequation sin(U+iV)—Ksin(x+iy). Deduce that,when thegratingistoearth and isplacedinauniform field offorce ofunit intensityatright angles toitsplane, thecharge andpotential functions oftheportion of thefieldwhichpenetrates throughthegrating areexpressed by U+iV-(x+ty), andexpandthepotentialinthelatter probleminaFourier Series. 80.Acylinder whose cross-section isonebranch ofarectangular hyperbolais maintained atzero potential under theinfluence ofaline-charge paralleltoitsaxis andontheconcave side. Prove thattheimageconsists ofthree such linecharges, and hence findthedensityoftheinduced distribution. 81.Acylindrical spaceisbounded bytwocoaxial andconfocal parabolic cylinders, whose latera recta are4aand4b,andauniformlyelectrified linewhich isparalleltothe generatorsofthecylinderintersects theaxeswhichpassthroughthefociinpointsdistant cfromthem(a>c> b).Shew thatthepotential throughout thespaceis Alogwr"cos-77- 1r-sin- cosh"—cos •J 7T7-cos-77- 1vsin-+c^-a- -b* 11 cosh- +cos x,|c^-64a?-b* J wherer,arepolarcoordinates ofasection, thefocus being thepole. Determine Ain terms oftheelectrification perunitlength oftheline. Examples 297 82.Aninfinitely long elliptic cylinderofinductivecapacity K,given byg=awhere x+iy=ccosh(£+ it]),isinauniform fieldPparalleltothemajor axis ofanysection. Shew that thepotentialatanypointinside thecylinderis p1-fcotha A-fcoth a* 83.Two insulated unchargedcircular cylinders outside eachother, given by rj=aand r/=—8where x+iy=ctsm ^(i+ir]),areplacedinauniform field offorce ofpotential Fx. Shew thatthepotential duetothedistribution onthecylindersis ofv,xnew(,,"a)sinhn/3 +e-w(,?+^sinhna . _ xN 'sinh?i(a +/3)s 84Two circularcylindersoutside eachother, given byr/=aandt)=—8where a;+t>=ctanf (f+iij), areputtoearth under theinfluence ofaline-charge Eonthelinex=0,y=0.Shew that thepotentialoftheinduced chargeoutside thecylindersis .r,v1e~wasinhn(n+8)+e~nsinhrc (a-n) -4A2, £, ,s\"cosn£+constant,n sinhn(a+8)s ' thesummationbeing taken foralloddpositive integral values ofn. 85.Thecross-sections oftwoinfinitely longmetalliccylinders arethecurves (#2+#2+c2 )2-4e2.£2=a4and(.r2+y2+c2 )2-4c2a72=&*, whereh>a>c. Iftheyarekeptatpotentials V1andV2respectively, theintervening space beingfilled withair,provethat thesurface densitiesperunitlengthofthe electricity ontheopposed surfaces are Vl~V \J^+tf andFg~Y \v^+? 47ra2log-47r62los;- respectively. 86.What problemsaresolved bythetransformation 1 where a>1? 87.What probleminElectrostatics issolved bythetransformation x+iy=en(4>+i^), where\^istaken asthepotential function, <£being thefunctionconjugatetoit? 88.Onehalfofahyperbolic cylinderisgiven by17=±tj 1,where |»?i |<^-,and£, rjare giveninterms oftheCartesian coordinatesx,yofaprincipalsection bythetrans- formation x+iy=ccosh(£+irj). Thehalf-cylinderisuninsulated andunder theinfluence ofachargeofdensityEperunit length placed along thelineofinternal foci Prove that thesuri'acedensityatanypoint ofthecylinderis -Elij^cm cosh^-Vcosh 2£-COS2ny 298 Methods fortheSolution ofSpecial Problems[ch.vm 89.Verify that,ifr,sberealpositive constants, z=x+iy,a=pe1 ^,-=-+-,the C i s neld offorce outside theconductors x2+y2+2sx=0, x'z+y2-2rx=0 duetoadoublet at thepointz=a,outside both thecircles, ofstrength p,and inclination atothe axis,is given byputting ^+iF=^|e*-(--2«COtCff(I-iVe-««-^COtC ff(I-I)}, wherez=a istheinversepointto2=awithregardtoeither ofthecircles. 90.Avery thinindefinitely great conducting planeisbounded byastraight edge of indefinitelength, and isconnected with theearth.AunitchargeisplacedatapointP. Prove thatthepotential atanypointQduetothechargeatPandtheelectricity induced ontheconducting planeis 11 _,/10-4A 11 _,/1<f>+4>'pn-cos-M cos' --7377; -cos~M COS Ptyn\o-2JPQTT\o-2 where P'istheimageofPintheplane,thecylindricalcoordinates ofQandPare (r,<p,z),(r', <p',z1 ),thestraight edgeistheaxisofz,theangles <f>, <f>'liebetween and2ir, (f>=ontheconductor, f(r+r')2+(2-/)2l* *-\—4& j' andthose values oftheinverse functions aretaken which liebetween\irand jr. 91.Asemi-infinite conducting planeisatzeropotential under theinfluence ofan electriccharge qatapointQoutside it.Shew that thepotentialatanypointPis given by 1 '2/7"{cosh,-cos(*-«,)}*tan-i^cQsh|;;_co3|^ /^,«-!, ,/cosh in+COSA(6+d{)~\- {cosh,-cos(^^)} Han-^-^—A^^t wherer,8,zarethecylindricalcoordinates ofthepoint P,(rj,dlf0)ofthepoint Q,8-0 istheequationoftheconducting plane, and 2/Tjcosht)—r3+rx2+*2 . Hence obtain thepotentialatanypoint duetoaspherical bowl atconstantpotential, andshew thatthecapacityofthebowl is -jl+^4,7T[Silla) where aistheradius oftheaperture, andaistheangle subtended bythisradius atthe centre ofthesphereofwhich thebowl isapart. 92.Athin circular conductingdisc isconnected toearth and isunder theinfluence ofacharge qofelectricityatanexternalpointP.ThepositionofanypointQisdenoted bytheperi-polarcoordinatesp,8,0,wherepisthelogarithmoftheratio ofthedistances fromQtothetwopoints R,Sinwhich aplaneQBSthrough theaxisofthedisccuts its rim, 8istheangle RQS, and<pistheangletheplaneQRSmakes with afixedplane through theaxis ofthedisc, thecoordinate 8havingvalues between -nand+ir,and changingfrom+irto—ttinpassing throughthedisc. Prove orverifythatthepotential ofthechargeinduced onthediscatanypointQ(p,8,<p)is ~QP2~nsiu~1^os^6~^seoh^~QP'l-+-sin-1{-cos£(# +#o)sech!«} > Examples 299 wherep,6,cpQarethecoordinates ofP,6being positive, thepoint P'istheoptical imageofPinthe disc, aisgiven bytheequation cosa=coshpcoshp—sinhpsinhpcos((p— <p), andthesmallest values oftheinverse functions aretobetaken. Prove thatthetotal charge onthedisc is- qdojir'. Explain howtoadapttheformula forthepotentialtothecase inwhich thecircular disc isreplaced byaspherical bowlwith thesame rim. 93.Shew thatthepotentialatanypointPofacircular bowl, electrified topotential G,is Cf. .AB OA . .(OP AB\\ *\amAP+BP+0P8m~\0A•APTBP)\> where isthecentre ofthebowl,andA,Barethepointsinwhich aplane through P andtheaxisofthebowl cutsthecircular rim. Find thedensityofelectricityatapoint oneither sideofthebowlandshew thatthe capacityis a . . .—(a+sina),IT where aistheradius ofthesphere, and2aistheangle subtended atthecentre. 94.Twospheresarechargedtopotentials VQandVx.The ratio ofthedistances of anypointfrom thetwolimiting pointsofthespheres being denoted byevandtheangle between themby£,provethatthepotentialatthepoint £, 77is sinh(n+|)(/3+»?) +VXJ{2(cosh ,-cosQ)2sinh\n+llp+l}P*(«»• *". where77=a, ij=- /3aretheequationsofthespheres. Hence findthecharge oneither sphere.f.^c*«—eijffl&Slfcg'-.cfl'-^ CHAPTER IX STEADY CURRENTS INLINEAR CONDUCTORS Physical Principles. 338. Iftwoconductors chargedwithelectricitytodifferentpotentials areconnected byaconducting wire,weknow thataflowofelectricitywill takeplace alongthewire. This flow willtend toequalisethepotentials ofthetwoconductors, andwhen thesepotentials becomeequaltheflow of electricitywill cease. Ifwehadsomemeans bywhich thechargesonthe conductors could bereplenishedasquicklyastheywere carried away by conductionthroughthewire, then thecurrent would never cease. Thecon- ductors would remain permanentlyatdifferentpotentials, andthere would beasteadyflowofelectricityfrom onetotheother. Means areknown by which twoconductors canbekeptpermanentlyatdifferentpotentials,sothat asteadyflowofelectricitytakesplace through anyconductor orconductors joiningthem.Weaccordinglyhave todiscuss themathematicaltheoryof such currents ofelectricity. We shallbegin bytheconsideration oftheflowofelectricityinlinear conductors, byalinear conductorbeing meant onewhich hasadefinite cross-section atevery point. Thecommonest instance ofalinear conductor isawire. 339. Definition. Thestrength ofacurrent atanypointinawire or other linear conductor, ismeasuredbythenumberofunitsofelectricitywhich flowacross anycross-sectionoftheconductor perunit time. Iftheunits ofelectricityaremeasured inElectrostatic Units, then the current also willbemeasured inElectrostatic Units. These, however, aswill beexplained later, arenottheunits inwhich currents areusually measured inpractice. LetP,Qbetwocross-sections ofalinear conductor inwhich asteady current isflowing,and letussupposethatnoother conductors touch this conductor between Pand Q.Then, since thecurrent is,byhypothesis, steady,there must benoaccumulation ofelectricityintheregionofthe 338-341] Physical Principles 301 conductor between PandQ.Hence therateofflowintothesection ofthe conductor acrossPmust beexactly equaltotherate offlowoutofthis section across Q.Or,thecurrents atPandQmust beequal. Hence we speakofthecurrent inaconductor, rather than ofthecurrent atapointin aconductor. For, aswepassalongaconductor, thecurrent cannotchange exceptatpointsatwhich theconductor istouched byother conductors. Ohm's Law. 340. Inalinear conductor inwhich acurrent isflowing, wehave electricityinmotion atevery point,andhence must have acontinuous variation inpotentialaswepass alongtheconductor. This isnot in oppositiontotheresultpreviouslyobtained inElectrostatics, forinthe previous analysisithad tobeassumed that theelectricitywas atrest. Inthepresent instance, theelectricityisnotatrest,beinginfactkept inmotion bythedifference ofpotentialunder discussion. Theanalogy between potentialandheightofwater willperhaps help.Alake in which thewater isatrest isanalogoustoaconductor inwhichelectricityisinequi- librium. Thetheorem thatthepotentialisconstant overaconductor inwhichelectricity isinequilibrium,isanalogoustothehydrostatictheorem thatthesurface ofstillwater must allbeatthesame level.Aconductor through which acurrent ofelectricityis flowingfinds itsanalogueinastream ofrunningwater. Here thelevel isnotthesame at allpointsoftheriver—itisthedifference oflevelwhich causes thewater toflow. The water will flowmore rapidlyinariver inwhich thegradientislarge than inonein which itissmall. The electrical analogytothis isexpressed byOhm's Law. Ohm's Law. Thedifference ofpotentialbetween anytwopoints ofawire orother linear conductor inwhich acurrent isflowing,stands tothecurrent flowing throughtheconductor inaconstant ratio, which iscalled theresistance between thetwopoints. Itishereassumed that there isnojunctionwith other conductors between these twopoints,sothat thecurrent throughtheconductor is adefinitequantity. 341. Thus ifCisthecurrent flowingbetween twopoints P,Qatwhich thepotentialsareVP,VQ,wehave VP-VQ=CR (264), whereRistheresistance between thepointsPand Q.Verydelicate experimentshave failed todetect anyvariation intheratio (fallofpotential)/(current), asthecurrent isvaried, andthisjustifiesusinspeakingoftheresistance as adefinitequantityassociated with theconductor. The resistance depends naturallyonthepositionsofthetwopoints bywhich thecurrent enters and leaves theconductor, butwhen once these twopointsarefixed theresistance 302 Steady Currents inLinear Conductors[ch.ix isindependentoftheamount ofcurrent. Ingeneral, however, theresistance ofaconductor varies with thetemperature,and forsome substances, ofwhich selenium isanotableexample,itvaries with theamount oflight fallingon theconductor. TheVoltaic Cell. 342. Thesimplest arrangement bywhich asteadyflowofelectricitycan beproducedisthatknown asaVoltaic Cell. This isrepresented diagram- maticallyinFig.95.Avoltaic cellconsistsessentiallyoftwoconductors Fig. 95. A,Bofdifferent materials, placedinaliquidwhich actschemicallyonat least oneofthem. Onestablishingelectrical contact between thetwoends oftheconductors which areoutoftheliquid,itisfound thatacontinuous current flows round thecircuit which isformed bythetwoconductors and theliquid,theenergywhich isrequiredtomaintain thecurrentbeing derived from chemical action inthe cell. Toexplaintheaction ofthe cell, itwillbenecessarytotouch onasubject ofwhich afullaccount would beoutofplaceinthepresentbook. Asan experimentalfact itisfound thattwoconductors ofdissimilar material, when placedincontact, have differentpotentials when there isnoflowofelectricity from onetotheother*, althoughofcourse thepotentialoverthewhole of either conductor must beconstant. Inthelightofthisexperimental fact, letusconsider theconditionsprevailinginthevoltaic cellbefore thetwo ends a,boftheconductors arejoined. Solongasthetwoconductors A,BandtheliquidCdonotformaclosed circuit, there canbenoflowofelectricity.Thus there iselectricequilibrium, •Foralongtime there hasbeen adivergence ofopinion astowhether this difference of potentialisnotdue tothechemical change atthesurfaces oftheconductors, and therefore dependent onthepresenceofalayerofairorotherthud substance between theconductors. It seems now tobealmost certain that this isthecase, butthequestionisnotone ofvital importanceasregardsthemathematical theoryofelectric currents. 341-344] Physical Principles 303 andthethree conductors have definitepotentials VA,VB,VC.Thedifference ofpotential between thetwo"terminals" a,bisVA—VB,butthepeculiarity ofthevoltaic cell isthat this difference ofpotentialisnotequaltothe difference ofpotential between thetwoconductors whentheyareplaced incontact and areinelectricalequilibrium without thepresenceofthe liquidG.Thus onelectrically joiningthepoints a,binthevoltaic cell electricalequilibriumisanimpossibility,andacurrent isestablished inthe circuit which willcontinue until thephysicalconditions becomechangedor thesupplyofchemicalenergyisexhausted. Electromotive Force. 343. LetA,B,Gbeanythree conductorsarrangedsoastoform aclosed circuit. LetVABbethecontact difference ofpotential between AandBwhen there iselectricequilibrium, and letVBC ,VCAhave similarmeanings. Ifthethree substances canbeplacedinaclosed circuit withoutany currentflowing,thenwecanhaveequilibriuminwhich thethree conductors willhavepotentials VA,VB,V,such that VA-VB=VAB ;VB-VC=VBC ;VC-VA=VGA. Thuswemust have vAB+vBO+vCA=o, aresult known asVolta's Law. If,however, thethree conductors form avoltaiccell,theexpression on theleft-hand oftheaboveequationdoesnotvanish, and itsvalue iscalled theelectromotiveforceofthe cell.Denotingtheelectromotive forcebyE, wehave VAB+VB0+VCA=E(265). Weaccordingly have thefollowingdefinition : Definition. TheElectromotive Force ofacell isthealgebraic sumofthe discontinuitiesofpotential encountered inpassinginorderthroughtheseries ofconductorsofwhich the cell iscomposed. Clearlyanelectromotive force hasdirection aswell asmagnitude.It isusual tospeakofthetwoconductors whichpassintotheliquidasthe high-potentialterminal andthelow-potential terminal, orsometimes asthe positiveandnegativeterminals.Knowingwhich isthepositiveorhigh- potential terminal, weshall ofcourse know thedirection oftheelectromotive force. 344. Iftheconductors G,Aofavoltaic cellABG areseparated,and thenjoined byafourth conductor D,such that there isnochemical action between Dandtheconductors GorA,itwilleasilybeseen thatthesum of thediscontinuities inthenew circuit isthesame asintheold. 304 Steady Currents inLinear Conductors [ch.ix Forbyhypothesis CDAcanform aclosed circuit inwhich nochemical action canoccur, andtherefore inwhich there canbeelectricequilibrium. Hence wemust have rcD+VDA+VAC=0(266). Moreover thesumofallthediscontinuities inthecircuit is VAB+VBC+VCD+VDA =VAB+VBC-VAC,byequation (266) =Vab+Vbc+Vca =E,byequation (265), provingtheresult.Asimilar proofshews thatwemayintroduceanyseries ofconductors between thetwoterminals ofacell,andsolongasthere isno chemical action inwhich thesenewconductors areinvolved, thesumofallthe discontinuities inthecircuit willbeconstant, andequaltotheelectromotive force ofthe cell. LetABC...MNbeanyseries ofconductors, includingavoltaiccell, and letthematerial ofNbethesame asthatofA.UNandAarejoined weobtain aclosed circuit ofelectromotive force E,such that VAB+V£C+...+VJIN+VNA=E. Moreover VNA=0,since thematerial ofNandAisthesame. Thus the relation mayberewritten as VAB+V£C+...+VM„=E (267). Intheopenseries ofconductors ABC ...MN, there canbenocurrent, so thateach conductor must beatadefinite uniformpotential.Ifwedenote thepotentials byVA,VB,...VM>VN,wehave *A~'B~ *AB> *M~'N=Kmy Henceequation (267) becomes VA-VN=E. Wenow seethat theelectromotiveforce ofacell isthedifference of potentialbetween theendsofthe cellwhen thecellforms anopen circuit, and thematerialsofthetwoends arethesame. Aseries ofcells, joinedinseries sothat thehigh-potentialterminal of one isinelectrical contact with thelow-potentialterminal ofthenext,and soon,iscalled abatteryofcells, oran"electricbattery" arrangedinseries. Itwillbeclear fromwhat hasjustbeenproved,that theelectromotive force ofsuchabatteryofcells isequaltothesum oftheelectromotive forces oftheseparatecells oftheseries. 344,345] Units 305 Units. 345.Ontheelectrostatic system,aunit current hasbeen defined tobe acurrent such thatanelectrostatic unit ofelectricitycrossesanyselected cross-section ofaconductor inunit time. Forpractical purposes,adifferent unit,known astheampere,isinuse.Theampereisequal veryapproximately to3x109electrostatic units ofcurrent (seebelow, §587). Toformsome idea oftheactual magnitudeofthis unit,itmaybestated that the amount ofcurrent requiredtoringanelectric bell isabout halfanampere. About the sameamount isrequiredtolighta50c.P.100-voltmetallic filament incandescent lamp. Asanelectromotive force isofthesamephysicalnature asadifference ofpotential,theelectrostatic unit ofelectromotive force istaken tobethe same asthat ofpotential.Thepracticalunit isabout3^oftheelectrostatic unit,and isknown asthevolt(seebelow, §587). Itmaybementioned that theelectromotive force ofasinglevoltaic cell isgenerally intermediate between oneandtwo volts;theelectromotive force whichproducesa perceptible shock inthehuman bodyisabout 30volts, while anelectromotive force of500volts ormore isdangeroustolife. Both ofthese latterquantities, however, vary enormously with thecondition ofthebody, andparticularlywith thestate ofdryness ormoisture ofthe skin. The electoomotive force used towork anelectric bell is commonly6or8volts, while anelectric lightinstallation willgenerally have avoltage ofabout 100or200 volts. Theunit ofresistance, inallsystemsofunits, istaken tobearesistance such that unit difference ofpotentialbetween itsextremitiesproducesunit currentthroughtheconductor. Wethen have,byOhm's Law, difference ofpotential atextremities ,„„. current =r .x (268).resistancev ' Inthepractical systemofunits, theunit ofresistance iscalled theohm. From what hasalreadybeen said, itfollows thatwhen twopoints havinga potential-differenceofonevoltareconnected byaresistance ofoneohm, the currentflowing throughthisresistance willbeoneampere.Inthiscasethe difference ofpotentialis^^electrostatic units, andthecurrent is3x109 electrostatic units, sothatbyrelation(268),itfollows thatoneohmmust be equalto=— =-^melectrostatic units ofresistance(seebelow, §587). Some idea oftheamount ofthis unitmay begathered from thestatement that theresistance ofamile ofordinary telegraphwire isabout 10ohms. The resistance ofagood telegraphinsulator maybebillions ofohms. 20 306Steady Currents inLinear Conductors[oh.ix Physical Theoeies ofConduction. Electron-theory ofconduction. 345 a.Ashasbeenalready explained (§28),themodern view of electricity regardsacurrent ofelectricityasamaterial flow ofelectric charges.In allconductorsexceptasmall classknown aselectrolytic conductors (see below, §345b),thesecharged bodies arebelieved tobe identical with theelectrons. Inasolidsome oftheelectrons aresupposedtobepermanently bound to particularatoms ormolecules, whilst others, spokenofas"free" electrons, move about intheinterstices ofthe solid, continually havingtheir courses changed bycollisions with themolecules. Both kinds ofelectrons willbe influenced bythepresenceofanelectric field. Itisprobablethat the restricted motions ofthe"bound"electrons account forthephenomenonof inductivecapacity (§151)whilst theunrestricted motion ofthefreeelectrons explainsthephenomenonofelectricconductivity. Evenwhen noelectric forces areapplied,thefreeelectrons move about throughasolid, buttheymove atrandom inalldirections, sothat asmany electrons move fromrighttoleftasfrom lefttorightandtheresultant current isnil. Ifanelectric force isappliedtotheconductor, each electron hassuperposedontoitsrandom motion amotionimpressedonitbythe electric force, andtheelectrons asawhole aredriventhroughtheconductor bythecontinued action oftheelectric force. Ifitwere notfortheir collisions with themolecules oftheconductor, theelectrons wouldgain indefinitelyin momentum under theaction oftheimpressedelectric force, buttheeffect of collisions iscontinuallytocheck thisgrowthofmomentum. Letussupposethat there areNelectronsperunitlengthofthe conductor, andthat atanymoment these haveanaverageforwardvelocity uthroughthematerial oftheconductor. Ifmisthemass ofeach electron, thetotalmomentum ofthemovingelectrons willbeNmu. The rate at which this totalmomentum ischecked bycollisions willbeproportionalto Nandtou,andmaybetaken tobeNyu. Therateatwhich themomentum isincreased bytheelectric forcesactingisNXe, whereXistheelectric intensity and eisthecharge,measuredpositively,ofeach electron. Thus wehave theequation j-(Nmu)=NXe-Nya(a). Inunittime thenumber ofelectrons whichpassanyfixedpointinthe conductor isNu,sothatthetotal flow ofelectricity perunittimepastany pointisNeu. This isbydefinitionequaltothecurrent intheconductor, so that ifwecallthisi,wehave Neu=i(b). 345a,345b] Electrolytic Conduction 307 This enables ustoreduceequation (a)totheform *N"x-^i) (o).dtmVNe2 Theequationshews that ifasteadyelectric force isapplied,such that theintensityatanypointisX,thecurrent willnotincreaseindefinitely butwillremainstationaryafter ithasreached avalue igiven by i= X. 7 IfVisthepotentialatanypointofaconducting wire, and ifsisa dV coordinate measuredalongthewire,wehaveX=—— ,sothat ds~Ne*1' Integratingbetween anytwopointsPandQoftheconductor, wehave This istheelectron-theory interpretationofequation (264), andexplains howthetruth ofOhm's Law isinvolved inthemodernconceptionofthe nature ofanelectric current. Itwillbenoticed thatonthisview ofthe matter, Ohm's Law isonlytrue forsteadycurrents. Wenotice that theresistance oftheconductor, onthistheory,is<y/Ne2 perunitlength. Thus, generally speaking,bodies inwhich there aremany freeelectronsoughttobegood conductors, andconversely. Thecharge ontheelectron being 4*774xlO-10electrostatic units, wemaynotice that acurrent ofoneampere (3x109electrostatic units ofcurrent)isoneinwhich 6%3x1018electrons passanygiven pointoftheconductor every second. Consider a conductor inwhich thenumber ofelectronspercubic centimetre is1021 (cf.§615,below). Then inawire of1square mm. cross-section there are1019electrons perunitlength, so thattheaverage velocityofthesewhen thewire isconveyingacurrent of1ampereisof theorder ofonecm.persec. Thisaverage velocityissuperposed ontoarandomvelocity which isknown tobeoftheorder ofmagnitudeof107cms.persec,sothattheadditional velocity produced byeven astrongcurrent isonly very slightincomparison with the normalvelocityofagitationoftheelectrons. Electrolyticconduction. 345 b.Besides thetypeofelectric conductionjust explained,there isa second, andentirelydifferenttype,known asElectrolytic conduction, the distinguishingcharacteristic ofwhich isthat thepassageofacurrent is accompanied bychemicalchangeintheconductor. For instance, ifacurrent ispassed throughasolution ofpotassium chloride inwater, itwillbefound thatsome ofthesalt isdivided upbythe passageofthecurrent into itschemical constituents, andthatthepotassium 20—2 308 Steady Currents inLinear Conductors[ch.ix appears solelyatthepointatwhich thecurrent leaves theliquid,while the chlorinesimilarly appearsatthepointatwhich thecurrent enters. Itthus appearsthatduringthepassageofanelectric current, there isanactual transportofmatterthroughtheliquid,chlorinemovinginonedirection and potassiumintheother. Itismoreover foundbyexperimentthatthetotal amount, whether ofpotassiumorchlorine, which isliberatedbyanycurrent isexactly proportionaltotheamount ofelectricity which hasflowedthrough theelectrolyte. These and other factssuggestedtoFaradaytheexplanation, now universally accepted,that thecarriers ofthecurrent areidentical with the matter which istransported throughtheelectrolyte. For instance, inthe foregoing illustration, eachatom ofpotassiumcarries apositive chargetothe pointwhere thecurrent leaves theliquid,while eachatom ofchlorine, movinginthedirectionoppositetothat ofthecurrent, carries anegative charge. Theprocessisperhaps explained moreclearly byregardingthetotal current asmadeupoftwoparts,firstapositivecurrent andsecond anegative currentflowinginthereverse direction. Then theatoms ofchlorine arethe carriers ofthenegative current, andtheatoms ofpotassiumarethecarriers ofthepositivecurrent. Electrolytes maybesolid, liquid,orgaseous,but inmost cases of importance theyareliquids, beingsolutions ofsalts oracids. Thetwoparts intowhich themolecule oftheelectrolyteisdivided arecalled theions (Icov),thatwhich carries thepositivecurrentbeingcalled thepositive ion, andtheotherbeingcalled thenegativeion.Thepointatwhich thecurrent enters theelectrolyteiscalled theanode, thepointatwhich itleaves is called thecathode. Thetwo ions are also called theanion orcation accordingasthey giveuptheirchargesattheanode orcathoderespectively. Thuswehave The anion carries —charge against current, and delivers itatthe anode, The cation carries +chargewith current, and delivers itatthe cathode. Whenpotassiumchloride istheelectrolyte,thepotassiumatom isthe cation, andthechlorine atom istheanion. Ifexperimentsareperformed with different chlorides(sayofpotassium, sodium, and lithium),itwillbe found thattheamount ofchlorine liberated byagivencurrent isinevery casethesame, while theamounts ofpotassium, sodium, orlithium, being exactlythoserequiredtocombine with this fixedamount ofchlorine, are necessarily proportionaltotheir atomicweights.Thissuggeststhat each atom ofchlorine, nomatter what theelectrolyte maybeinwhich itoccurs, alwayscarries thesamenegative charge, say—e,while eachatom ofpotassium, 345 b,345c] Electrolytic Conduction 309 sodium, orlithium carries thesamepositive charge, say+E.Moreover E and emust beequal,orelseeach undissociated molecule oftheelectrolyte would have tobesupposedtocarryachargeE—e,whereas itschargeis known tobenil. Itisfound tobeageneralrule thateveryanion which ischemically monovalent carries thesamecharge —e,whileevery monovalent cation carries acharge-fe.Moreover divalent ionscarry charges+2e,trivalent ionscarry charges+Se,and soon. Asregardstheactualcharges carried, itisfound that oneampereof currentflowingforonesecondthroughasaltofsilver liberates 0'001118 grammesofsilver. Silver ismonovalent and itsatomicweightis107-92 (referredto=16),sothattheamount ofanyother monovalent element of atomicweightmdeposited bythesame current willbe-00001036 xm grammes.Itfollows thatthepassageofoneelectrostatic unit ofelectricity •nu•nru*-0-00001036 xm _ ,cwillresult intheliberation of—— ,or345x10~15xmgrammes ofthesubstance. Wecancalculate from these datahowmanyionsaredeposited byoneunit ofcurrent, andhence theamount ofchargecarriedbyeach ion. Itisfound that, towithin thelimits ofexperimental error, thenegative chargecarried byeachmonovalent anion isexactly equaltothechargecarried bytheelectron. Itfollows thateachmonovalent anion hasassociated with itoneelectron inexcess ofthenumberrequiredtogiveitzerocharge,while eachmonovalent cation hasadeficiencyofone electron;divalent ionshave anexcess or deficiencyoftwo electrons, and soon. 345 c.Ohm's Lawappears,ingeneral,tobestrictlytrue fortheresist- ance ofelectrolytes.InthelightoftheexplanationofOhm's Lawgivenin §345 a,this willbeseen tosuggestthattheionsarefreetomove assoon as anelectricintensity,nomatter how small, beginstoactonthem.They must therefore bealreadyinastate ofdissociation;nopartoftheelectric intensityisrequiredtoeffect theseparationofthemolecule into ions. Other facts confirm thisconclusion, such asforinstance thefactthatvariousphysical properties—electricconductivity, colour, optical rotatory power,etc.—areadditive inthe sense that theamountpossessed bythewholeelectrolyteisthesum oftheamounts known tobepossessed bytheseparateions. Wemaythereforesupposethat assoon asanelectric forcebeginstoact, allthepositiveionsbegintomove inthedirection oftheelectric force, while allthenegativeionsbegintomove intheoppositedirection. Letussuppose theaveragevelocities ofthepositiveandnegativeions tobeu,vrespectively, and letussupposethatthere areNofeachperunitlengthoftheelectrolyte measuredalongthepathofthecurrent. Then acrossanycross-section ofthe electrolytetherepassinunittimeNupositiveionseachcarryingachargese 310Steady Currents inLinear Conductors[ch.ix inthedirection inwhich thecurrent ismeasured, andNvnegativeionseach carryingacharge-seinthereverse direction, sbeingthevalencyofeach ion. Itfollows thatthetotal current isgiven by i=Nse(u+v) (d). Each unit oftimeNupositiveions cross across-section close tothe anode, havingstarted frompositions between this cross-section andthe anode. Thus each unit oftimeNumolecules areseparatedintheneigh- bourhood oftheanode, andsimilarly Nvmolecules areseparatedinthe neighbourhoodofthecathode. Theconcentration ofthe salt isaccordingly weakened both attheanode andatthecathode, andtheratio oftheamounts ofthese weakeningsisthat ofu :v.Thisprovidesamethod ofdetermining theratio ofu :v. Alsoequation (d)providesamethod ofdetermining u+v,for icanbe readily measured, andNse isthetotalcharge which must bepassed through theelectrolytetoliberate theions inunitlength, and thiscanbeeasily determined. Knowingu+vandtheratiou :v,itispossibletodetermine uand v. Thefollowingtablegivesresults oftheexperimentsofKohlrausch onthree chlorides ofalkali metals, fordifferent concentrations, thecurrent ineach casebeingsuch astogiveapotentialfallof1voltpercentimetre. Concentration 3-15C-346] Kirchhoff's Laws 311 Conduction through gases. 345 d.Inagasinitsnormal state, anelectric current cannot becarried ineither ofthewayswhich arepossibleinasolid oraliquid, and itis consequentlyfound that agasunder ordinaryconditions conductselectricity onlyinaveryfeebledegree.Ifhowever Rontgen raysarepassed through thegas,orultra-violetlightofveryshort wave-length,orastream ofthe raysfromradium oroneoftheradio-active metals, then itisfound thatthe gasacquiresconsiderable conducting powers,foratime atleast. For this kind ofconduction itisfound thatOhm's Law isnotobeyed,therelation between thecurrent andthepotential-gradient beinganextremely complex one. Thecomplicated phenomenaofconduction through gasescan allbe explainedonthehypothesisthatthegasisconducting onlywhen"ionised," andthefunction oftheRontgen rays,ultra-violetlight,etc. issupposedto bethat ofdividing upsome ofthemolecules into their componentions. Thesubjectofconduction through gasesistooextensive tobetreated here. Inwhat follows itisassumed that theconductors under discussion arenot gases,sothatOhm's Law willbeassumed tobeobeyed throughout. Kirchhoff's Laws. 346. Problems occur inwhich theflow ofelectricityisnotthrough asinglecontinuous series ofconductors :theremaybejunctionsofthree or more conductors atwhich thecurrent ofelectricityisfreetodistribute itself between differentpaths,and itmaybeimportanttodetermine howthe electricitywillpassthroughanetwork ofconductorscontaining junctions. The firstprincipletobeused isthat, since thecurrents aresupposed steady,there canbenoaccumulation ofelectricityatanypoint,sothatthe sum ofallthecurrents which enteranyjunctionmust beequaltothesum ofallthecurrents which leave it.Or,ifweintroduce theconvention that currents flowingintoajunctionaretobecounted aspositive,while those leavingitaretobereckonednegative,thenwemaystate theprinciplein theform : Thealgebraic sumofthecurrents atanyjunctionmust bezero. From thislaw itfollows thatanynetwork ofcurrents, nomatter how complicated,canberegardedasmadeupofanumber ofclosed currents, each ofuniform strength throughoutitslength.Insome conductors, twoormore ofthese currents mayofcourse besuperposed. Letthevarious junctionsbedenoted byA,B,C,...,and lettheir potentialsbeVA,VB,V,....LetRABbetheresistance ofanysinglecon- ductorconnectingtwojunctions AandB,and letGABbethecurrent flowing 312 Steady Currents inLinear Conductors[ch.ix throughitfromAtoB.Letusselect anypath throughthenetwork of conductors, such astostartfrom ajunction andbringusback tothestarting point, sayABC...NA. Then onapplying Ohm's Law totheseparatecon- ductors ofwhich thispathisformed, weobtain(§341) V—V—H 7? rArB—^AB^ABi *B~*C=^BC-^BOt Byaddition weobtain 2(7^=0(269), where thesummation istaken over alltheconductors which form theclosed circuit. Inthisinvestigationithasbeenassumed thatthere arenodiscontinuities ofpotential, and therefore nobatteries, intheselected circuit. Ifdis- continuities occur, aslightmodification willhave tobemade.We shall treatpointsatwhich discontinuities occur asjunctions,and ifAisajunction ofthiskind, thepotentialsatAonthetwosides ofthesurface ofseparation between thetwoconductors willbedenoted byVAandVA.Then, byOhm's Law,weobtain forthe falls ofpotentialinthedifferent conductors ofthe circuit, VA'-VB=GABRAB , VB—Va=zL>BCKBG ,etc., andbyaddition oftheseequations $(VA'-VA)=tCR. The left-hand member issimplythesum ofallthediscontinuities of potential met inpassinground the circuit, eachbeing measured with its proper sign.Itisthereforeequaltothesumoftheelectromotive forces of allthebatteries inthe circuit, these alsobeing measured with theirproper signs. Thuswemaywrite %CR=t£(270), where thesummation ineachterm istaken round anyclosed circuit of conductors, and thisequation, togetherwith 2C=0 (271), inwhich thesummation now refers toallthecurrentsenteringorleavinga single junction,suffices todetermine thecurrent ineach conductor ofthe network. Equation (271) expresses what isknown asKirchhoff's FirstLaw, while equation (270) expressestheSecond Law. 346-348] Kirchhoff's Laws 313 Conductors inSeries. 347.When alltheconductors formasingleclosedcircuit, thecurrent througheachconductor isthesame, sayC,sothatequation (270) becomes CXR=IE. Thesum2i2 isspokenofasthe"resistance ofthecircuit," sothatthe current inthecircuit isequaltothetotal electromotive force dividedbythe total resistance. Conductorsarrangedinsuchawaythatthewhole current passes througheach ofthem insuccession aresaid tobearranged"in series." Conductors inParallel. 348. Itispossibletoconnect anytwopoints A,Bbyanumber of conductors insuch awaythat thecurrent divides itself between allthese Fig. 96. conductors onitsjourneyfromAtoB,nopartofitpassing through more than oneconductor. Conductors placedinthiswayaresaid tobearranged "inparallel." Letussupposethatthetwopoints A,Bareconnected byanumber of conductorsarrangedinparallel.LetR1}R2,...betheresistances ofthe conductors, andCu(72,...thecurrentsflowing throughthem. Then ifVA,VB arethepotentialsatAandB,wehave,byOhm's Law, VA—VB=C1R1=C2R2=.... The total current which enters atAisC1+Ca+...,sayC.Thuswe have K-V-2l-£l ==G VAVB—— -...— RiR2 R\R% Thearrangementofconductors inparallelistherefore seen tooffer the same resistance tothecurrent asasingleconductor ofresistance 1 D+r>T••• Mi JX^ Thereciprocal,oftheresistance ofaconductor iscalled the"conductivity" oftheconductor. Theconductivityofthesystemofconductors arranged inparallelis-ft+rr+-", and istherefore equaltothesum ofthe 314 Steady Currents inLinear Conductors[ch.ix conductivities oftheseparateconductors. Alsowehave seen that the current divides itself between thedifferent conductors intheratio oftheir conductivities Measurements. TheMeasurementofCurrent. 349. Theinstrument used formeasuringthecurrentpassinginacircuit atanygiveninstant iscalled agalvanometer.Thetheoryofthisinstrument willbegiveninalaterchapter (Chap. xiu). Formeasuringthetotal quantityofelectricity passingwithin agiven timeaninstrument called avoltameter issometimes used. The current, inpassing throughthevoltameter, encounters anumber ofdiscontinuities ofpotentialincrossingwhich electricalenergybecomes transformed into chemicalenergy.Thus avoltameter ispracticallyavoltaic cellrunback- wards. Onmeasuringtheamount ofchemicalenergywhich hasbeen stored inthevoltameter, weobtain ameasure ofthetotalquantityofelectricity which haspassed throughtheinstrument. TheMeasurement ofResistance. 350. TheResistance Box.Aresistance box isapieceofapparatus which consistsessentiallyofacollection ofcoils ofwire ofknown resistances, arrangedsothatanycombination ofthese coilscanbearrangedinseries. Themost usual arrangementisoneinwhich thetwoextremities ofeach coilarebroughttotheuppersurface ofthebox,andarethere connected toathick band ofcopperwhich runs overthesurface ofthebox. This Fig. 97, band ofcopperiscontinuous, exceptbetween thetwoterminals ofeachcoil, andintheseplacesthecopperiscutawayinsuchawaythatacopper plug canbemade tofitexactlyintothegap,andsoputthetwosides ofthegap inelectrical contact throughtheplug. Thearrangementisshewndiagram- maticallyinfig.97.When theplugisinserted inanygapDE,theplug andthecoilbeneath thegapDEformtwoconductors inparallel connecting 348-351] Measurements 315 thepointsDandE.Denotingtheresistances ofthecoilandplugbyRC)Rp, theresistance between DandEwillbe RcRp and sinceRpisvery small, thismaybeneglected. When theplugis removed, theresistance fromDtoEmaybetaken tobetheresistance of the coil. Thus theresistance ofthewhole box willbethesum ofthe resistances ofallthe coils ofwhich theplugshavebeen removed. 351. TheWheatstoneBridge.This isanarrangement bywhich itis possibletocomparethe resistances ofconductors, and sodetermine an unknown resistance interms ofknown resistances. The" bridge" isrepresented diagrammaticallyinfig.98.Thecurrent enters itatAandleaves itatD,thesepoints beingconnected bythelines ABD,ACDarrangedinparallel. The lineAD iscomposedoftwocon- ductors AB,BDofresistances RlyR2,andthelineACD issimilarly composed oftwoconductors AG,CDofresistances R3,R4. Ifcurrent isallowed toflowthroughthisarrangementofconductors, it willnotingeneral happenthat thepointsBandCwillbeatthesame potential,sothat ifBandGareconnected byanew conductor, there will usually beacurrentflowing through BG. Themethod ofusingthe Wheatstonebridgeconsists invaryingtheresistances ofoneormore ofthe conductors R1}R2,R3,R4untilnocurrent flowsthroughtheconductor BG. When thebridgeisadjustedinthisway,thepoints B,Gmust beatthe samepotential, sayv.LetVA,VDdenote thepotentialsatAandD,and letthecurrentthroughABD beG.Then, byOhm's Law, VA-v=CR1,v-Vj>=GR 2, ,,, RiXi-vsothat -^=Tr.R2v-Vj) From asimilar consideration oftheflow inAGD, weobtain R3_VA-v R*v-VD' 7? 7? sothatwemust have W=-d"3 (272), 316 Steady Currents inLinear Conductors[oh.ix asthecondition tobesatisfied between theresistances when there isno current inBC. Clearly byadjustingthebridgeinthiswaywecandetermine anunknown resistance R^interms ofknown resistances R2,R3,R4.Inthesimplest form ofWheatstone'sbridge,thelineAGD isasingleuniform wire,andthe positionofthepointcanbevaried bymovinga"slidingcontact"along thewire. The ratio oftheresistances Rs:R4isinthiscasesimplytheratio ofthetwolengths AG,CDofthewire, sothattheratioR^:R2canbefound byslidingthecontact GalongthewireAGD until there isobserved tobe nocurrent inBG,andthenreadingthelengthsAGandGD. Examples ofCurrents inaNetwork. I.Wheatstone's Bridgenotinadjustment. 352. Thecondition that there shallbenocurrent inthe" bridge"BG infig.98hasbeen seen tobethatgiven byequation (272). B Supposethat thiscondition isnot satisfied, and letusexamine theflow ofcurrents which then takesplaceinthenetwork ofconductors. Letthe conductors AB,BD,AG,GDasbefore beofresistances R1}R2,R3,Ri}and letthecurrentsflowing throughthem bedenoted byx1}x2,x3,x4.Letthe bridgeBGbeofresistance Rb,and letthecurrentflowing throughitfrom BtoGbexb. From Kirchhoff 'sLaws, weobtain thefollowing equations: (Law I,point B) x1—x2—xb=(273), (Law I,point G)x3—x4+xb=(274), (Law II,circuit ABG) x^+x bRb—x3R3=(275), (Law II,circuit BCD) xbRb+x4R4-x2R,=(276). These fourequationsenable ustodetermine theratios ofthefivecurrents xx,x2,x3,x4,xb.Wemaybegin byeliminating#2andxtfromequations (273), (274) and(276), andobtain xb(Rb+R2+R4)+x3R4—xxR2=0, andfrom thisandequation (275), Xb X3 Xi R%R 3—RxRtR1(Rb+R2+R4)+RbRaR3(Rb-fR3+R4)+RbR4 (277). 351-353] Flow ofCurrents inaNetwork 317 The ratios oftheother currents canbewritten down fromsymmetry. Ifthetotal currententeringatAisdenoted byX,wehaveX=xx+x3. Thus ifeach ofthefractions ofequations (277)isdenoted by6, X=6{(R,+R3)(R,+R4)+Rb(R 1+R2+Rs+R4)} (278), andthisgives 6,andhence theactual values ofthecurrents, interras ofthe total currententeringatA. The fallofpotentialfromAtoDisgiven by VA-Vj)=Rxxx+R2x2} andfromequations (277)this isfound toreduce to vA-Vj>=\e, where X=RXR3(Ri+R4)+R2Ri(R3+Ri)+Rb(R1R3+R^Ri+RiR t+R2R3), sothat\isthesum oftheproductsofthefiveresistances taken three at atime, omittingthetwoproductsofthethree resistances which meet atthe pointsBandC. There isnowacurrentXflowing throughthenetwork, andhavinga fallofpotential VA—Vj).Hence theequivalentresistance ofthenetwork Va-Vj, xd x (R,+R3)(R 2+R4)+Rb(R,+R2+RS+R4)' byequation (278). II.Telegraphwire withfaults. 353. Asamorecomplex exampleoftheflow ofelectricityinasystem oflinear conductors, wemayexamine thecase ofatelegraph wire, inwhich there areanumber ofconnexions throughwhich thecurrent canleak to earth. Such leaks aretechnically known as"faults." FiFo To R;F«-j R, R. Fio. 100^ »+i B LetABbethewire,and letFltF2)...Fn-ltFnbethepointsonitat which faults occur, theresistances throughthese faults beingRuR2>... 318 Steady Currents inLinear Conductors[oh.ix Rn-i,Rn>andtheresistances ofthesections AFltF1F2,...Fn_rFnandFnB beingrur2,...rn,rn+1.LettheendBbesupposed puttoearth, and letthe current besupposedtobegenerated byabatteryofwhich oneterminal is connected toAwhile theother end istoearth. Theequivalentresistance ofthewhole network ofconductors fromAto earth canbefound inaverysimple way.CurrentarrivingatFnfrom the sectionFn-xFnpassestoearth throughtwoconductorsarrangedinparallel, ofwhich theresistances areRnandrn+1.Hence theresistance fromFnto earth is 1 -K-n rn+i andtheresistance fromFn^toearth, throughFn)is 1 .(279). f t*"n rn+1 Current reaching 2^_, can,however, passtoearthbytwopaths,either throughthefault atFn_ltorpastFn.Thesepaths mayberegardedas arrangedinparallel,their resistances beingRn-randexpression (279) respectively.Thus theequivalentresistance fromFn_xis 1 +K-' itn_!rn+ or,written asacontinued fraction, 1 1 Wecancontinue inthisway,untilfinally wefindasthewhole resistance fromAtoearth, J_1J_111 Tl+ jRf1+n+R2-1+'"+rn+Ru'1+rn+1' Ifthecurrents orpotentialsarerequired,itwillbefound best toattack theprobleminadifferent manner. LetVA ,Vx,V2,...bethepotentialsatthepoints A,F1}F2,...,then, by Ohm's Law, thecurrent fromFs^toFgrs „FstoFsK-KS'S+l S+l r,S+l Fgthroughthefault=-5. Rg' 353,354] Flow ofCurrents inaNetwork 319 Hence, byKirckhoff's first law, Z-V... V=0,V—VV—V or T^+1rs+f1-Vs(Rs->+rr1+n+r1 )+Vs.,rg~l=0, andfrom thisandthesystemofsimilarequations,thepotentials maybe found. IfalltheR'sarethesame, andalso allthe r'sarethesame, theequation reduces toadifferenceequationwith constant coefficients. These conditions mightariseapproximatelyifthelineweresupported byaseries ofsimilar imperfectinsulators atequaldistancesapart. The differenceequationisin thiscaseseen tobe TSh-E^+IQ+IU-O. and ifweput1+—=cosha, thesolution isknown tobe l^=^coshsa +Bsinhsa(280), inwhichAandBareconstants which must bedetermined from the conditions attheends ofthe line. Forinstance toexpressthat theendB istoearth, wehaveVn+1=0,andtherefore A=-Btanh(n+l)a. III.Submarine cableimperfectlyinsulated. 354. Ifwepasstothelimitingcase ofaninfinite number offaults, we have theanalysis appropriatetoalinefromwhich there isleakageatevery point. Theconditions nowcontemplated maybesupposedtoberealised in asubmarine cable inwhich, owingtotheimperfectionoftheinsulating sheath, thecurrent leaksthroughtotheseaatevery point. Theprobleminthisform canalsobeattacked bythemethods ofthe infinitesimal calculus. LetVbethepotentialatadistance xalongthe cable,Vnowbeing regardedasacontinuous function ofx.Letthe resistance ofthecable besupposedtobeRperunitlength,then there- sistance fromxtox+dxwillbeRdx. Theresistance oftheinsulation from o xtox+dx,being inversely proportionaltodx,maybesupposedtobe-r- . LetGbethecurrent inthecable atthepoint x,sothattheleakfrom dC dxdCthecablebetween thepointsxandx+dxis— -j-dx.This leak isacurrent 320 Steady Currents inLinear Conductors[ch.ix dxwhich flowsthrougharesistance-5-with afallofpotential V.Henceby Ohm's Law, V=-^dx(®-)dx \dxj' 35— iff(281)- dV Also, the fallofpotential alongthecable fromxtox+dxis—-r—dx,the (too current isC,andtheresistance isRdx. HencebyOhm's Law, dV-(^~=RC(282).dx Eliminating Gfromequations (281) and(282), wefindasthedifferential equationsatisfied byV, d_(IdV\V dx\Rdx)~ S' IfRandShave thesame values atallpointsofthecable, thesolution ofthisequationis V=Acosh^/ -<=x+Bsinh*/ -5x, which iseasilyseen tobethelimitingformassumed byequation (280). Generation ofHeat inConductors. TheJouleEffect. 355. LetP,Qbeanytwopointsinalinear conductor, letVp,Vqbe thepotentialsatthesepoints,Rtheresistance between them, andxthe current flowingfromPtoQ.Then, byOhm's Law, Vp-Vq=Rx '.(283). Inmovingasingleunit ofelectricityfromQtoPanamount ofwork is doneagainstthe electric fieldequaltoVp—Vq.Hence when aunit of electricity passesfromPtoQ,there isworkdoneonitbytheelectric field ofamount Vp—Vq.Theenergy represented bythework shews itself in aheatingoftheconductor. Theelectrontheory gives asimple explanationofthemechanism ofthistransforma- tionofenergy. The electric forces dowork ontheelectrons indriving them through the field. The total kinetic energyoftheelectronscan,aswehave seen(§345a),beregarded asmadeupoftwoparts,theenergy ofrandom motion andtheenergyofforward motion. Thework donebytheelectric field goes directly towards increasingthissecondpartof thekineticenergyoftheelectrons. But afteranumber ofcollisions thedirection ofthe velocityofforward motion iscompletely changed, andtheenergyofthismotion has become indistinguishable from theenergyoftherandom motion oftheelectrons. Thus thecollisions arecontinually transforming forward motion intorandom motion, orwhat isthesamething,into heat. 354-356] Generation ofHeat 321 Wearesupposingthatxunits ofelectricity pass perunittime from PtoQ.Hence thework donebytheelectric fieldperunittime within the regionPQisx(Vp- VQ),andthisagain, byequation (283),isequaltoRx2 . Thus inunit time, theheatgeneratedinthesectionPQofthecon- ductorrepresents Rx2units ofmechanicalenergy. Each unit ofenergyis equalto -junits ofheat, whereJisthe"mechanicalequivalent ofheat." Thus thenumber ofheat-unitsdevelopedinunittime intheconductor PQ willbe Rx2 -J-(284). Itisimportanttonotice that inthisformula xandRaremeasured in electrostatic units. Ifthevalues oftheresistance andcurrent aregivenin practical units,wemust transform toelectrostatic units beforeusingformula (284). Lettheresistance ofaconductor beR'ohms, and letthecurrentflowing throughit bexfamperes. Then,inelectrostatic units, thevalues oftheresistance Randthecurrent xaregiven by /?'R=x-^r-r.andx=3x I09a/.Ox1011 Thus thenumber ofheat-units produced perunittime is R*(3x103)2 J~9x10". J' andonsubstitutingforJitsvalue 4-2x107inc.G.s.-centigrade units, thisbecomes 0-24fi'A Generation ofHeat aminimum. 356. Ingeneralthesolution ofanyphysical problemisarrived atbythe solution ofasystemofequations,thenumber oftheseequations being equal tothenumber ofunknownquantitiesintheproblem. Thecondition that anyfunction inwhich theseunknownquantitiesenter asvariables shall bea maximum oraminimum, isalsoarrived atbythesolution ofanequal number ofequations.Ifitispossibletodiscover afunction oftheunknown quantities such thatthetwosystemsofequationsbecome identical,—i.e.if theequations whichexpressthatthefunction isamaximum oraminimum arethesame asthose which contain thesolution ofthephysical problem— thenwemaysaythatthesolution oftheproblemiscontained inthesingle statement thatthefunction inquestionisamaximum oraminimum. Examplesoffunctions which serve thispurposearenothard tofind. In §189,weprovedthatwhen anelectrostatic systemisinequilibrium,its potential energyisaminimum. Thus thesolution ofanyelectrostatic problemiscontained inthesinglestatement that thefunction which j. 21 322 Steady Currents inLinear Conductors [ch.ix expressesthepotential energyisaminimum.Again,thesolution ofany dynamical problemiscontained inthestatement that the"action" isa minimum, while inthermodynamicstheequilibriumstate ofanysystem canbeexpressed bythecondition thatthe"entropy"shall beamaximum. Itwillnowbeshewn that thefunction whichexpressesthetotal rate of generationofheatplaysasimilar r61e inthetheoryofsteadyelectric currents. 357. Theorem. When asteadycurrent flows through anetwork of conductors inwhich nodiscontinuities ofpotentialoccur {and which, therefore, contains nobatteries),thecurrents aredistributed insuchawaythat therateof generation ofheatinthenetwork isaminimum, subject onlytotheconditions imposed byKirchhoff's firstlaw;andconversely. Toprove this, letusselectanyclosed circuitPQR...Pinthenetwork, and letthecurrents andresistances inthesections PQ,QR,...bexl,x%,... andR1}R2, Letthecurrents andresistances inthose sections ofthenet- work Avhich arenotincluded inthis closed circuit bedenoted byxa,%i,... andRa,Rf,, ....Then thetotal rateofproductionofheat is XRa^ +tR^ (285). Adifferentarrangementofcurrents, andonemoreover which does not violate Kirchhoff's first law,canbeobtained inimagination bysupposingall thecurrents inthecircuitPQR...Pincreasedbythesameamount e.The total rateofproductionofheat isnow XRaXa* +^R1Oz+*)V andthisexceeds theactual rateofproductionofheat, asgiven byexpression (285), by2R1(2x 1e+e*) (286). Now iftheoriginaldistribution ofcurrents isthatwhichactuallyoccurs innature, then XR.x,=0, byKirchhoff's second law. Thus therate ofproductionofheat,under the newimaginarydistribution ofcurrents, exceeds that intheactual distribu- tionbye'XRi, anessentially positive quantity. Themostgeneralalteration which canbesupposed made totheoriginal systemofcurrents, consistentlywith Kirchhoff's firstlawremaining satisfied, willconsist insuperposing uponthissystemanumber ofcurrentsflowing inclosed circuits inthenetwork. One such current istypified bythe currente,alreadydiscussed. Ifwehaveanynumber ofsuch currents, the resultingincrease intherateofheat-production =XR,(^+e+e'+e"+ ...)2-^RlXl\ 356-358] Generation ofHeat 323 wheree,e',e",...aretheadditional currentsflowing throughtheresistance i2j.Asbefore thisexpression =2%R1x1(e+e'+e"+...)+XR, (e+/+e"+...)2 =2^1(e+e/+e,/+...)3 , byKirchhoff's second law. This isanessentially positive quantity,sothat anyalteration inthedistribution ofthecurrents increases therate ofheat- production.Inother words, theoriginaldistribution wasthat inwhich the ratewasaminimum. Toprovetheconverse itissufficient tonotice that iftherate ofheat- productionisgiventobeaminimum, thenexpression (286) must vanish as farasthe firstpowerofe,sothatwehave tR.x,=0, andofcourse similarequationsforallotherpossibleclosed circuits. These, however, areknown tobetheequations which determine theactual dis- tribution. 358. Theorem. When asystem ofsteadycurrents flows throughanet- workofconductors ofresistances RltR2,...,containingbatteries ofelectromotive forcesE1}E2)...,thecurrents x1}x2,...aredistributed insuchawaythat the function ZRx*-2'$Ex (287) isaminimum, subjecttotheconditions imposed byKirchhoff's firstlaw;and, conversely. Asbefore, wecanimaginethemostgeneralvariationpossibletoconsist ofthesuperpositionofsmall currents e,e,e",...flowinginclosed circuits. Theincrease inthefunction (287) produced bythisvariation is 2i2[(x+e+e'+...)2-a2 ]-2XE [(x+e+ e'+...)- x] =2e .(%Rx-IE)+2e'(...)+... +XR(e+e'+...y (288). Ifthesystemofcurrents x,x,...isthenaturalsystem,then the first line ofthisexpressionvanishes byKirchhoff's second law(cf.equations (270)), andtheincrease inheat-productionistheessentially positive quantity 2E(e+e'+...)V shewingthattheoriginalvalue offunction (287) must havebeen aminimum. Conversely,iftheoriginalvalue offunction (287) wasgiventobea minimum, thenexpression (288) must vanish asfarasfirstpowersofe,e,..., sothatwemust have 2ifo=E,etc., shewingthatthecurrents x,x,...must bethenaturalsystemofcurrents. 21—2 324 Steady Currents inLinear Conductors[ch.ix 359. Theorem. IftwopointsA,Bareconnected byanetwork ofcon- ductors, adecrease intheresistance ofanyoneofthese conductors willdecrease (or,inspecial cases, leave unaltered)theequivalentresistance fromAtoB. LetxbethecurrentflowingfromAtoB,Rtheequivalentresistance of thenetwork, andVA—VBthe fallofpotential. Thegenerationofheatper unit timerepresentstheenergyset freebyxunitsmoving througha potential-difference VA-V£.Thus therate ofgenerationofheat is *(VA-VS), or,sinceVi—VB=Rx,therateofgenerationofheat willbeRx2 . Lettheresistance ofanysingleconductor inthenetwork besupposed decreased fromRxtoRJ,and letx1bethecurrentoriginally flowing through thenetwork. Ifweimaginethecurrents toremain unaltered inspiteofthe changeintheresistance ofthisconductor, then there willbeadecrease in therateofheat-production equalto(R 1—R/)xf.Thecurrents nowflowing arenotthenatural currents, but ifweallow thecurrententeringthenetwork todistribute itself inthenaturalway,there is,by§357,afurther decrease intherate ofheat-production.Thus adecrease intheresistance ofthe singleconductor hasresulted inadecrease inthenatural rate ofheat- production. IfR,R'aretheequivalentresistances before andafter thechange,the two rates ofheat-productionareRx2andR'x2 .Wehaveprovedthat R'x2<Rx2 ,sothatR'<R,provingthetheorem. General Theory ofaNetwork. 360. Inaddition todependingontheresistances oftheconductors, the flowofcurrentsthroughanetworkdependsontheorder inwhich thecon- ductors areconnectedtogether,butnotonthegeometrical shapes, positions ordistances oftheconductors. Thuswecanobtain themostgeneralcaseof flowthrough anynetwork byconsideringanumber ofpoints 1,2,...n,con- nected inpairsbyconductors ofgeneralresistances which maybedenoted by R12)R23, If,inanyspecial problem, anytwopoints P,Qarenotjoined byaconductor, wemustsimply supposeRPQtobeinfinite. Discontinuities ofpotential must notbeexcluded, soweshallsupposethat inpassing through theconductor FQ,wepassover discontinuities ofalgebraic sumEpq.This isthesame assupposingthat there arebatteries inthearmPQoftotal electromotive forceEPQ.Weshallsupposethat thecurrentflowinginPQ fromPtoQisxPQ)andshall denote thepotentialsatthepoints 1,2,...by The total fallofpotentialfromPtoQisVP—VQ,butofthisanamount 359,360] General Theory ofaNetwork 325 —EPQiscontributed bydiscontinuities, sothattheaggregatefallfromPto Qwhich arises from thesteady potential gradientinconductors willbe VP-VQ+EPQ. Hence, byOhm's Law, Vp— Vq+HjpQ=KpqXpQ. Ifweintroduce asymbolKPQtodenote theconductivity -p— ,wehave thecurrentgiven by xPQ=Kpq(Vp-V q+EpQ) (289). Supposethat currentsXx,X2,...enter thesystemfrom outside atthe points 1,2,...,thenwemust have 4i=#12T#13T#14T•••t since there istobenoaccumulation ofelectricityatthepoint 1,andsoon forthepoints 2,3,....Substitutingfromequations (289)intotheright hand ofthisequation, Zi-Kn(K-K+El2)+K1S(K-V3+E13)+... =V1(K12+K13+...) -(K 12V2+K13V3+...)+K12E12+K13El3+ (290). ThesymbolKPPhassofarhadnomeaning assignedtoit.Letususe it todenote—(KP1+KP2+KP3+...);thenequation (290)maybewritten in themore concise form X1=-(K 11V1+K12V3+...)+K12EW+K13E19+ (291). There arenequationsofthistype,but itiseasilyseen thattheyarenot allindependent.For ifweaddcorresponding members weobtain Zi+X2+...+Xn=-ZV1(KU+K12+...+Km)+22(KPQEPQ+KQPEQP). i The firsttermontherightvanishes onaccount ofthemeaningwhich hasbeen assignedtoKu,etc.; while thesecond term vanishes becauseEPQ=—EQP, whileKPQ=Kqp.Thus theequationreduces to X1+X2+...+X n=0, whichsimply expressesthatthetotal flowintothenetwork isequaltothe total flowoutofit,acondition which must besatisfiedbyXuX2,...Xnat theoutset. Thuswearrive attheconclusion thattheequationsofsystem (291) arenotindependent. This isasitshouldbe,for iftheequations wereindependent, weshould have nequations from which itwould bepossibletodetermine thevalues of1^,V2,...in terms ofX1}X2,...;whereasclearly from aknowledgeofthecurrents entering the network, wemust beable todeterminedifferencesofpotential only,andnotabsolute values. 326 Steady Currents inLinear Conductors[ch.ix Totheright-handsideofequation (291),letusaddtheexpression (Kn+Kls+...+Km)Vn, ofwhich thevalue iszerobythedefinition ofKn.Theequationbecomes Kn(V 1-Vn)+Kia(Vi-Vn)+...+K1>n-1(Vn-1-Vn) =—XY+KnE12+Kl3E13+...+KmEm. There arenequationsofthistypeinall.Ofthese the first(n—1)may beregardedasasystemofequations determining V-VV—VV—V That theseequationsareindependentwillbeseenaposteriori from thefact thattheyenable ustodetermine thevalues ofthen—1independent quantitiesV—VV-V V,-V'i 'n> '2 'n> ••'> 'n—l 'n- Solvingtheseequations, wehave —X\+K\iE\i +•••+KmEm , —X2+K2lE21 +...+KmE2n, it22> 1123)-"-i,n-i *»2,71—1 —-^n— l*r"-^ti—1,1-^n— 1,1+•••+J^-n~z,n ^Jn—\,ni -**-n—i,a>-l*-n— i,3j •••> -"-n— i,ti—l Km Ki3>K13) ••>'M.n-i it2l> -^22) -^23) •'•>^ 2,71—1 -**-n— 1,1jJ-^-n— 1,2>i*7i—1,3> •••>1*-n—l,n—l Thecurrentflowinginconductor Infollows atoncefromequation (289), andthecurrents intheother conductors canbewritten down from symmetry. Ifwedenote thedeterminant inthedenominator oftheforegoing equation byA,andtheminor ofthetermKpqbyAP<2,wefindthat the value ofVy—V,xcanbeexpressedintheform K-K= (-X 1+K12E12+...+KmEm)^ +(-X 2+K21E21+...+K2nE2n)^+ (292). 361.Supposefirstthat thewhole systemofcurrents inthenetwork is produced byacurrentXenteringatPandleavingatQ,therebeing no batteries inthenetwork. Then alltheE'svanish, and alltheX'svanish exceptXPandXQ,thesebeing given by AP=—2Ln=.A . 3G0-362]General Theory ofaNetwork 327 Equation (292)nowbecomes VV—YApiY^Q1 "l—Yn=~Ap-^-—Ac-£- =^(\*~APl)' sothat K-K=(X-K)-0£-K) =^(AQ1-Ag2-AP1+Ap,) (293). Replacing 1,2byP,QandP,Qby1,2,wefindthat ifacurrentX enters thenetwork at1andleaves itat2,the fallofpotentialfromP toQis VP-VQ=^(A2P-A2Q-A1P+A1Q) (294), andsinceArg=Ag,., itisclear that theright-handmembers ofequations (293) and(294)areidentical. From thiswehave thetheorem : Thepotential-fall fromAtoBwhen unit current traverses thenetwork fromGtoDisthesame asthepotential-fall fromGtoDwhen unit current traverses thenetwork fromAtoB. 362. Let itnowbesupposedthat thewhole flow ofcurrent inthe network isproduced byabatteryofelectromotive forceEplacedinthe conductor PQ.Wenowtake alltheZ'sequaltozero inequation (292) and alltheE'sequaltozeroexceptEPQwhich weputequaltoE,and EQPwhich weputequalto—E.Wethen have Ap,.rr„AC AVl-Vn=KPQEPQ~p+EQPEQP- =^(AP1-Ayi). Hence K-K=^%^(AP1-Ap,-AQ1+A^) (295), and,byequation (289), thecurrent flowinginthearm12is a12=K»K*E (Api-Ap,-AQ1+AQ2) (296). Thisexpressionremains unaltered ifwereplace 1,2byP,QandP,Qby 1,2.From thiswededuce thetheorem : Thecurrent whichflowsfromAtoBwhen anelectromotive forceEis introduced intothearmGDofthenetwork, isequaltothecurrent whichflows fromGtoDwhen thesame electromotive forceisintroduced into the armAB. 328 Steady Currents inLinear Conductors [ch.ix ConjugateConductors. 363. Thesameexpressionoccurs asafactor intheright-handmembers ofeach oftheequations (293), (294), (295), and(296), namely, Api+A^-A^-Apa (297). Ifthisexpression vanishes, thetwoconductors 12andPQaresaid tobe " conjugate." Byexaminingtheform assumed byequations (293)to(296), when expression (297) vanishes, weobtain thefollowingtheorems. Theorem I.Iftheconductors ABandCDareconjugate,acurrent enteringatAandleavingatBwillproducenocurrent inCD. Similarly, acurrententeringatCandleavingatDwillproducenocurrent inAB. Theorem II.Iftheconductors ABandCDareconjugate,abattery introduced intothearmABproducesnocurrent inCD.Similarly,abattery introduced into thearmCDproducesnocurrent inAB. Asanillustration oftwoconductors which areconjugate,itmaybe noticed thatwhen theWheatstone'sBridge (§352)isinadjustment,the conductors ADandBCareconjugate. Equations expressedinSymmetricalForm. 364. Thedeterminant Aisnotinform asymmetricfunction ofthe npoints 1,2,...,n,sothatequationsandconditions which mustnecessarily involve these npoints symmetricallyhave notyetbeen expressedin symmetricalform. Wehave, forinstance, A„=M3li.21,"&)-*£24)-"-25>•••>" 2,n—1 •"-31 > "-32 J -"-34 JA35, •••>-^3,n—1 -"-n-i,i>AM_1,2>An-i,4) -K-n—i.S) •••>&-n—\,n—\ inwhich thepointswhich enterunsymmetricallyarenotonly1and3,but also n.Similarly, wehave Au=-jfi2l) -^22>A23, A25, •••; -"2,71— 1 A31,A32,A33,A35, •••> **-3,n—l -tt-n-i,i>An_i)2,J^n-i,3> An_i>6,•••>-K-n- l,n-l sothat,onsubtraction, A13-A14=A21, A22, ^23+^24, A31,A32,A33+A34,#25, ^35,...,IV2>n—1 •••>**-3,n—i ii»_i,ij An-l,2, An-1,3+An_li4,iln_ii5,...,An_lin_i 363,364] General Theory ofaNetwork 329 From therelation KP1+KP2+...+KPin_l+KPin=(298), itfollows that thesum ofalltheterms inthe firstrowoftheabove deter- minant isequalto—K2>n,thesumofalltheterms inthesecond row isequal to—K 3)n,andsoon.Thus theequation maybereplaced by AU-AM«(-!)» -^21> -»*22> A%5> A-31) -^32) -*^3S>...,A2,n—1,J^-2,Jl ...tA 3>jj,—! ,A 3>n Am— i,D -ft-n— i,2> -"^n— 1,5) •••>-&n— i,n— 1>-K-n— i,n andsimilarly, ^28A24=(— •) -*Mli -&12) -^15) »••>A1>n -^31) -"-32) ^35> •••>-^3,n Aw— 1,1»An— 1,2>An— 1,5) •••>-^n-i,n These twodeterminants differonlyintheir first row, sothatonsub- traction, (Au-A^-CA^-AaO =(-!)»Kn+K^,Kl2+K22,K16+K2i,...,Khn+K2>n K3i, -»^32> ^35) •••) -"-3,n -n-n-1,1) A3I) ^32> -A-n—1,2> An— 1,5) •••> -^-n— l,n ...,A;3,71 -&n-i,i) A«— 1,2j-"-n— 1,5) •••> -Kn—1,1 -^n,i> ^n,2, -&n,5> •••>&~n,n•(299), thelasttransformation beingeffected bytheuseofrelation (298). Therelation which hasnowbeen obtained isinasymmetrical shape.If Disasymmetricaldeterminant given by D=-^lD -*M2> -"-13) •••)Ai.n A2i,-&22> -^-23) •••) -"-2,n Z*-n,\> -^n,2> An,3) »••)A«,n then thedeterminant ontheright-handofequation (299)isobtained from Dbystrikingoutthelinesandcolumns which contain thetermsKnandK24. Thusequation (299)maybewritten intheform A13+A24-A^-A14=dK lsdK13UJJ.24 330 Steady Currents inLinear Conductors Againthedeterminant Agiven by[CH.IX A=•*^n> -^12) -^13> •••> -"-i,n—1 ^21) -^22> -^-23) •••> -^-2,n— 2 -^n-i,i> -"-n— 1,2> -**n-i,3> •••>-^-n— i,n—l.(300) maybewritten intheform A=dD dKntn' This isnotofsymmetrical form, forthepointnentersunsymmetrically. Wecan,however, easily shew thatthevalue ofAissymmetrical, althoughits form isunsymmetrical. Byapplicationofrelation(298), wecantransformequation (300)into A=-**n,l> -"n,2>-**-n,3>•••>-"-n,n—l **21>"22> **23J "•>" 2,n—1 "n— i,i> **-«— 1,2> -ti-n— 1,3» •••> -"-n— i,n— l =(— -I) -^21) -^22) -&23) •••> l*-2,n— 1 J-*-n— 1,1>'in— 1,2)-'i-n—1,3) •••)-K-n—i,n— 1 -&n,i) -^-n, 2) -^-n.S) •••» -^n,ra—1 -ft-22) -"-23) •••) -"^2,n— 1) -**-2,n -**-n—1,2)-**-n— 1,3) •••»-'i-n—i,n— l)^n-i,fl 7fn,2)#n,3)•..,ii 7i,n—l)#«,n 8Z) 3^ii* ThusAisthedifferential coefficient ofZ)withrespecttoeither ifuor Kn>n,orofcourse withrespecttoanyother oneoftheterms intheleading diagonalofP.Thus,ifKdenote anyterm intheleading diagonalofP, wehave andthisvirtually expresses Ainasymmetrical form. Wecannowexpressinsymmetricalform therelations which havebeen obtained in§§360to362, asfollows : I.(§362.) Theconductors 1,2andP,Qwill beconjugate if d-P dKhPdK.2tQ=0. 364-366] Slowly-varying Currents 331 II.(Equation 293.) Iftheconductors1,2andP,Qarenotconjugate, acurrentXenteringatPandleaving atQproduces in1,2afall of potential given by *-*—-*— a^—• dK III.(Equation 295.) Iftheconductors1,2andP,Qarenotconjugate, abattery ofelectromotive forceEplacedinthearmPQproduces in1,2afall ofpotential given by d2D V-V—K i?°KltpoK^ q K, y2—JlpqU— , dK andacurrent from1to2given by li12li-pQ Allthese results andformulae obtain illustration intheresultsalready- obtained fortheWheatstone's Bridgein§§351and352. Slowly-varying Currents. 365. Alltheanalysisofthepresent chapterhasproceeded uponthe assumptionthat thecurrents areabsolutely steady, shewingnovariation with thetime.Changesinthestrengthofelectric currents areingeneral accompanied byaseries ofphenomena,which maybespokenofas "induction phenomena,"ofwhich thediscussion isbeyondthescopeofthe present chapter. If,however, therate ofchangeofthestrengthofthe currents isvery small, theimportanceoftheinductionphenomenaalso becomesvery small, sothat ifthevariation ofthecurrents isslow, the analysisofthepresent chapterwillgiveacloseapproximationtothetruth. Thismethod ofdealingwithslowly-varyingcurrents willbeillustrated by twoexamples. I.Discharge ofaCondenser throughahighResistance. 366. Letthetwoplates A,Bofacondenser ofcapacity Cbeconnected byaconductor ofhighresistance R,and letthecondenser bedischarged by leakage throughthisconductor. Atanyinstant letthepotentialsofthetwo platesbeVA,VB,sothat thechargesontheseplateswillbe±G(VA—VB). Let ibethecurrent intheconductor, measured inthedirection fromAtoB. 332 Steady Currents inLinear Conductors[ch.ix Then, byOhm's Law, VA-VB=Ri, whence wefindthatthechargesonplatesAandBarerespectively +CRi and—CRi. Since iunits leaveplateAperunit time,wemust have adifferential equationofwhich thesolution is t i=ieCR , where iisthecurrent attime t=0.Thecondition that thestrengthof thecurrent shallonlyvary slowlyisnowseenaposterioritobethatOR shall belarge. Attime tthechargeontheplateAisCRi or t CRie~CR. Thismaybewritten as t where Qisthechargeattime t=0.Thus both thechargeandthecurrent areseen tofalloffexponentiallywiththetime,bothhavingthesamemodulus ofdecay CR. Later(§516)weshallexamine thesameproblembutwithout thelimita- tionthatthecurrentonlyvariesslowly. II.Transmission ofSignals along aCable. 367. Ithasalreadybeenmentioned thatacable actsasanelectrostatic condenser ofconsiderable capacity.This factretards thetransmission of signals,andinacable ofhigh-capacity,therate oftransmission maybeso slow that theanalysisofthepresent chaptercanbeused without serious error. Letxbeacoordinate which measures distancesalongthecable, letV,i bethepotentialatxandthecurrent inthedirection of^-increasing,and let KandRbethecapacityandresistance ofthecableperunitlength,these latterquantities being supposed independentofx. The section ofthecable betweenpointsAandBatdistances xand x+dxisacondenser ofcapacity Kdx, and isatthesame time aconductor 366-368] Transmission ofSignals 333 ofresistance Rdx.Thepotentialofthecondenser isV,sothat itschargeis VKdx. The fallofpotentialintheconductor is sothatbyOhm's Law, dV—=-dx=iRdx(301). Thecurrent enters thesectioni£atarate iunitsperunit time, and dxdileaves atarateofi+~-dxunitsperunit time. Hence thechargeinthis di section decreases atarate=-dxperunit time, sothatwemust have I(VKdx) =-^dx(302). Eliminatingifromequations (301) and(302),weobtain d2V dV 368. Thisequation, beingapartialdifferentialequationofthesecond order, must havetwoarbitraryfunctions initscompletesolution. We shall shew, however, that there isaparticularsolution inwhichVisafunction of thesinglevariablex\\Jt,and this solution willbefound togiveusallthe information werequire. Letusintroduce thenewvariable u,given byu=xf\ft,and letusassume provisionallythat there isasolution Vofequation (303) which isafunction ofuonly.Forthissolution wemust have d*V=1d*V dx2tdu2* dV^dVdu_ Lx_dV dt~dudt~2V*3du' sothatequation (303) becomes du? \2 ^/tsdu) =-lKRu~ (304).du The factthat thisequationinvolves Vanduonly,shews thatthere isan integraloftheoriginal equationforwhichVisafunction ofuonly.This integraliseasily obtained, forequation (304)canbeputintheform eO*S~wdu whence ^=Ce^KRn\du inwhichCisaconstant ofintegration. 334 Steady Currents inLinear Conductors[ch.ix Integrating this,wefindthat thesolution forVis V=cfU e-±KRu *du, inwhich thelower limit totheintegralisasecond constant ofintegration. Introducinganew variableysuch thaty2=\KRu2 ,andchangingthe constants ofintegration,wemaywrite thesolution intheform V=V+C e-y2 dy (305).JOO 369.Wemustremember that this isnotthegeneralsolution ofequa- tion(303), but issimplyoneparticularsolution. Thus thesolution cannot beadjustedtosatisfy anyinitial andboundaryconditions weplease, butwill represent onlythesolutioncorrespondingtoonedefinite setofinitial and boundaryconditions. Wenowproceedtoexamine what these conditions are. Attime t=0,thevalue ofx\*]tisinfiniteexceptatthepoint x=0. Thusexceptatthispoint, wehaveV=X when t=0.Atthispointthe value ofxjfjtisindeterminate attheactual instant t=0,butimmediately after thisinstant assumes thevalue zero,which itretainsthroughalltime. Thus atx=0,thepotentialhastheconstant value V^Vo+G'Te-y'dy, J00 or,say,V=Vuwhere 0'=2^~^. Atx=oo, thevalue ofVisV=V throughalltime. Thusequation (305) expressesthesolution foralineofinfinitelength which isinitiallyatpotential V—T£,andofwhich theendx—ooremains at thispotentialallthetime, while theendx= israised topotential Tfby being suddenlyconnected toabattery-terminalattheinstant t=0. Thecurrent atanyinstant isgiven by 1dV i=—-=-~-,fromequation (301), C"l/KR_^! , . ,onKN =—d9V~T~e 4t»irornequation (305), /~fF~KRx*=(^"T0V&rfe""*"(306)- Weseethat thecurrent vanishesonlywhen t—andwhen t=oo . Thus even within aninfinitesimal time ofmaking contact, there will, accordingtoequation (306), beacurrent atallpoints alongthewire. It must, however, beremembered thatequation (306)isonlyanapproxima- tion, holding solelyforslowly-varying currents, sothatwemust notapply 368,369] Transmission ofSignals 335 thesolution attheinstant t=atwhich thecurrents, asgiven byequation (306), varywith infiniterapidity.Forlargervalues oft,however, wemay supposethecurrentgiven byequation (306). Themaximum current atanypointisfound, ondifferentiating equation (306),tooccur attheinstantgiven by t=\KRx> (307), sothatthefurtheralongthewirewego,thelongerittakes forthecurrent toattain itsmaximum value. Themaximum value ofthis current, when it occurs, is <F->»\/s-/"! <308>' andsoisproportionalto- .Thus thefurther wegofrom theendx=0,the CC smaller themaximum current willbe. Wenotice thatKoccurs inexpression (307) butnotin(308). Thus the electrostaticcapacityofacable willnotinterfere with thestrengthofsignals sentalongacable, butwillinterfere withtherapidityoftheir transmission. Equation (307) expresseswhat iscommonlycalled the"KR law"—the retardingeffect isproportionaltotheproductofKandR.Thetheory just developediscommonly spokenofastheElectrostaticTheoryofpropagation ofsignals.Itwas firstgiven byLord Kelvin in1855 inapaper* which is notable ashavingestablished thetheoreticalfeasibilityofanAtlantic cable. Weshall discuss inalaterchapterthemoregeneral problemofthetrans- mission ofsignals alongawire ofanykind. Itwillthen bepossibleto estimate thedegreeoferror involved inthesimple assumptions ofthe ElectrostaticTheory. EXAMPLES. 1.Alength 4aofuniform wire isbent intotheform ofasquare, andtheopposite angular pointsarejoinedwithstraight piecesofthesame wire, which areincontact attheir intersection. Agivencurrent enters attheintersection ofthediagonals and leaves atanangular point:findthecurrent strengthinthevarious partsofthenetwork, andshew that itswhole resistance isequaltothat ofalength a\l\ 2^2+1 ofthewire. 2.Anetwork isformed ofuniform wire intheshapeofarectangleofsides 2a,3a, withparallel wires arrangedsoastodivide theinternal spaceinto sixsquaresofsides a, thecontact atpointsofintersection being perfect.Shew that ifacurrent enter the frameworkbyonecorner andleave itbytheopposite,theresistance isequivalenttothat ofalength 121a/69ofthewire. *"On theTheoryoftheElectrio Telegraph,"Proc. Roy.Soc.1855. 336 Steady Currents inLinear Conductors[ch.ix 3.Afault ofgivenearth-resistance developsinatelegraphline. Prove thatthe current atthereceiving end,generated byanassigned batteryatthesignalling end,is leastwhen thefault isatthemiddle oftheline. 4.The resistances ofthree wires BC,CA,AB, ofthesame uniform section and material, area,b,crespectively.Another wirefromAofconstant resistance dcanmake aslidingcontact withBC. Ifacurrent enter atAandleave atthepointofcontact withBC,shew that themaximum resistance ofthenetwork is (a+b+c)d a+b+c+4d' anddetermine theleast resistance. 5.Acertain kind ofcellhasaresistance of10ohms andanelectromotive force of 85ofavolt. Shew that thegreatestcurrent which canbeproducedinawirewhose resistance is22*5ohms, byabatteryoffivesuch cells arrangedinasingle series,of which anyelement iseither one cellorasetofcells inparallel,isexactly"06ofan ampere. 6.Sixpoints A,A',B,B',C,Careconnected tooneanother bycopperwirewhose lengthsinyardsareasfollows: AA'=16,BC=B'C=l, BC'=B'C'=2,AB=A'B'=G, AC'=A'C' =8.AlsoBandB'arejoined bywires, each ayardinlength,totheterminals ofabattery whose internal resistance isequaltothat ofryardsofthewire,and allthe wires areofthesame thickness. Shew thatthecurrent inthewireAA' isequaltothat which thebatterywould maintain inasimplecircuit consistingof31r+104yardsof thewire. 7.Twoplaces A,Bareconnected byatelegraphline ofwhich theend atAis connected tooneterminal ofabattery, andtheendatBtooneterminal ofareceiver, theother terminals ofthebattery andreceiver being connected toearth. AtapointC ofthelineafault isdeveloped,ofwhich theresistance isr.Iftheresistances ofAC,CB bep,qrespectively, shew thatthecurrent inthereceiver isdiminished intheratio r(p+q):qr+rp+pq, theresistances ofthebattery,receiver andearth circuit being neglected. 8.Two cells ofelectromotive forces ex,e2andresistances rx,r2areconnected in paralleltotheends ofawire ofresistance B.Shew thatthecurrent inthewire is ei^+yi rxR+r2R-\-r1r2' andfindtherates atwhich thecells areworking. 9.Anetwork ofconductors isintheform ofatetrahedron PQRS ;there isabattery ofelectromotive forceEinPQ,andtheresistance ofPQ,includingthebattery,isR. Iftheresistances inQR,RPareeachequaltor,andtheresistances inPS,RSareeach equaltoJr,andthat inQS=§r,findthecurrent ineachbranch. 10.A,B,C,Darethefourjunction pointsofaWheatstone's Bridge, and the resistancesc,/3,b,yinAB,BD,AC,CDrespectivelyaresuch thatthebatterysends no current through thegalvanometerinBC. Ifnowanewbatteryofelectromotive forceE beintroduced into thegalvanometer circuit, and soraise thetotal resistance inthat circuit toa,findthecurrent that willflowthroughthegalvanometer. 11.AcableAB,50miles inlength,isknown tohaveone fault, and itisnecessaryto localise it.IftheendAisattached toabattery,andhas itspotential maintained at200volts, while theother endBisinsulated,itisfound thatthepotentialofBwhen Examples 337 steadyis40volts. Similarly whenAisinsulated thepotential towhichBmust beraised togiveAasteady potentialof40volts is300volts. Shew thatthedistance ofthefault fromAis19-05miles. 12.Awire isinterpolatedinacircuit ofgiven resistance andelectromotive force. Find theresistance oftheinterpolated wire inorder that therateofgeneration ofheat mayheamaximum. 13.The resistances oftheoppositesides ofaTVheatstone's Bridge area,a'andb,b' respectively. Shew thatwhen thetwodiagonals which contain thebattery andgalvano- meter areinterchanged,EE_(a-a')(b-b')(G-R) GC aa'-bb'' whereGandCarethecurrents through thegalvanometer inthetwocases,GandRare theresistances ofthegalvanometer andbattery conductors, andEtheelectromotive force ofthebattery. 14.Acurrent Gisintroduced intoanetwork oflinear conductors atA,andtaken outatB,theheatgenerated being IT1.Ifthenetwork beclosed byjoining A,Bbya resistance rinwhich anelectromotive forceEisinserted, theheat generatedisH2. Prove that C2rE* 15.AnumberNofincandescent lamps, each ofresistancer,arefedbyamachine of resistance R(including theleads).Ifthelight emitted byanylampisproportionalto thesquareoftheheatproduced, prove that themost economical wayofarranging the lampsistoplacethem inparallel arc,each arccontaining nlamps, where nistheinteger nearest toV'JVR/r. 16.Abatteryofelectromotive forceEandofresistance Bisconnected with thetwo terminals oftwowiresarrangedinparallel. The firstwire includes avoltameter which contains discontinuities ofpotential such thataunit currentpassing throughitfora unittime doespunits ofwork Theresistance ofthefirstwire, including thevoltameter, isR:that ofthesecond isr.Shew that ifEisgreater thanp(B+r)jr,thecurrent throughthebatteryis E(R+r)-pr Rr+B(R+r)' 17.Asystemof30conductors ofequalresistance areconnected inthesamewayas theedgesofadodecahedron. Shew that theresistance ofthenetwork between apairof oppositecorners is£oftheresistance ofasingle conductor. 18.Inanetwork PA,PB,PG,PD,AB,BC,CD,DA,theresistances area,fty,8, y+8,d+a,a+/3,/3+y respectively. Shew that,ifADcontains abatteryofelectromotive force E,thecurrent inBCis P(ap+y8).E 2/>2<2+(/3S-ay)2' where P=a+p+y+d,§=/3y+ya+a/3+aS+/3S+yS. 19.Awireforms aregular hexagonandtheangular pointsarejoinedtothecentre bywires each ofwhich hasaresistance -oftheresistance ofaside ofthehexagon. Shew thattheresistance toacurrent enteringatoneangular pointofthehexagon and leavingitbytheopposite pointis 2(ra+3) (n+1)0+4) times theresistance ofasideofthehexagon. J. 22 338 Steady Currents inLinear Conductors[ch.ix 20.Twolongequal parallelwiresAB,A'B',oflength I,have their ends B,B'joined byawire ofnegligible resistance, while A,A'arejoinedtothepolesofacellwhose resistance isequaltothat ofalengthrofthewire.Asimilar cell isplacedasabridge across thewires atadistance xfrom A,A'.Shew thattheeffect ofthesecond cell isto increase thecurrent inBB' intheratio 2(21+r)(x+r)l{r(4l+r) +2x(2l-r)-4:X2 }. 21.There arenpoints 1,2,...n,joinedinpairsbylinear conductors. Onintroducing acurrent atelectrode 1andtakingitoutat2,thepotentialsofthese areV\,V2,...Pn. Ifx12istheactual current inthedirection 12,andxx2'anyother thatmerelysatisfies the conditions ofintroduction at1andabstraction at2,shew that 2(r12a?12a?i2')=(Pi-P2)C=2(r12x12-), andinterprettheresultphysically. Ifxtypifytheactual current when thecurrent enters at1andleaves at2,andy typifytheactual current when thecurrent enters at3andleaves at4,shew that 2(r12x12y12)=(Zs-X4)C=(l\-Y2)C, where theX'sarepotentials correspondingtocurrentsx,and theY'sarepotentials correspondingtocurrentsy. 22.A,B,Carethree stations onthesametelegraphwire.AnoperatoratAknows thatthere isafaultbetween AandB,andobserves thatthecurrent atAwhen heusesa given batteryisi,i'ori",accordingasBisinsulated andCtoearth,Btoearth,orB andCboth insulated. Shew thatthedistance ofthefaultfromAis {ka-k'b+{b-aft(ka-k'bft}j(k- k'), i" i"where AB=a,BC=b-a, k—-.—-nk'=- l i-i 23. Sixconductorsjoinfourpoints A,B,C,Dinpairs, andhave resistances a,a,b,/3,c,y,wherea,arefer toBC,ADrespectively, and soon. Ifthisnetwork beused asaresistancecoil,with A,Baselectrodes, shew that theresistance caunot lieoutside thelimits [^^r-[MGnr+(H)T]-i 24.Twoequal straight piecesofwireAAn,BBnareeach divided intonequal parts atthepointsAt...An_iandBi...B n_irespectively, theresistance ofeachpartand that ofAnBnbeing R.Thecorresponding pointsofeach wirefrom 1toninclusive arejoined bycross wires, andabatteryisplacedinAB .Shew that,ifthecurrent through each cross wire isthesame, theresistance ofthecross wireAaBais {(n-sY +(n-s)+\}R 25. Ifnpointsarejoined twoandtwobywires ofequalresistancer,andtwoof them areconnected totheelectrodes ofabatteryofelectromotive forceEandresistance R,shew thatthecurrent inthewirejoiningthetwopointsis 2E 2r+nR' 26.Sixpoints A,B,C,D,P,Qarejoined bynineconductors AB,AP,BC,BQ,PQ, QC,PD,DC,AD.Anelectromotive force isinserted intheconductor AD,and a galvanometerinPQ. Denoting theresistance ofanyconductor XYbyrXY ,shew that ifnocurrent passes through thegalvanometer, (i'bo+rBQ+rCQ)(rABrDP-rAPrD0)+rBC(rBQrDP-rAPrCQ)=0. Examples 339 27.Anetwork ismade byjoiningthe fivepoints 1,2,3,4,5byconductors inevery possible way. Shew thatthecondition thatconductors 23and14areconjugateis (A"15+^+E3b+E&)(E^E^-E13E2i) =Eb2(EbiEls-EuE,b)+Eb3(E2iEbl-E^E^), whereEr)isconductivityofconductor rs. 28.Two endless wires areeach divided intomnequal parts bythesuccessive terminals ofmnconnecting wires, theresistance ofeachpart being R.There isan identicallysimilarbatteryinevery mthconnecting wire, thetotal resistance ofeach being thesame, andtheresistance ofeach oftheothermn—nconnectingwires ish. Prove that thecurrent throughaconnecting wirewhich istherthfrom thenearest batteryis £C(l-tana)(tanra+tanm-ra)/(tana-tanm a), whereCisthecurrent through eachbattery, andsin2a=hj(k+R). 29.AlonglineoftelegraphwireAAXA2...AnAn+1issupported bynequidistant insulators atAltA2,...An.TheendAisconnected toonepoleofabatteryofelectro- motive forceEandresistance B,andtheotherpoleofthisbatteryisputtoearth, as alsotheother endAn+Xofthewire. The resistance ofeachportion AAX,AtA2,... AnAn+iisthesame, R.Inwetweather there isaleakagetoearth ateachinsulator, whose resistance maybetakenequaltor.Shew thatthecurrentstrengthinAPAP+xis Ecos\x(2n-2p +l)a Bcosh(2/i+1)a+\l~Rrsinh(2?i+2)a' where 2sinha=\]R\r. 30.Aregular polygon A1A2...A nisformed ofnpiecesofuniform wire, each of resistance<r,andthecentre isjoinedtoeachangular point byastraight pieceofthe same wire. Shew that,ifthepointismaintained atzeropotential, andthepointAx atpotential V,thecurrent that flows intheconductor ArAr+1is 2Fsinh asinh(n-2r+1)a acoshna' where aisgiven bytheequation 7rcosh2a=l+ sin— n 31.Aresistance network isconstructed of2nrectangular meshes formingatruncated cylinder of2nfaces, withtwoends each intheform ofaregular polygonof2nsides. Each ofthese sides isofresistancer,andtheotheredges ofresistance R.Ifthe electrodes betwoopposite corners, then theresistance is , ,,.tanh 6 where sinh26=^.la 32.Anetwork isformed byasystemofconductors joining every pairofasetof npoints, theresistances oftheconductors beingallequal, andthere isanelectromotive force intheconductorjoiningthepointsA1}A2.Shew that there isnocurrent inany conductorexcept those whichpassthrough AxorA2,and findthecurrent inthese conductors. 22—2 340 Steady Currents inLinear Conductors[ch.ix 33.Eachmember oftheseries ofnpointsAx,A<i,...A nisunited toitssuccessor byawire ofresistancep,andsimilarlyfortheseries ofnpointsBx,B2,...B n.Each pairofpoints correspondinginthetwoseries, such asArandBr,isunited byawire ofresistance R.Asteady current ienters thenetwork atAiandleaves itatBn.Shew thatthecurrent atA\divides itself between AxAiandA-J5 Xintheratio sinha+sinh(n—1)a+sinh{n-2)a :sinha+sinh{n—1)a-sinh(n—2)a, where cosh a=„ . 34.Anundergroundcable oflength aisbadly insulated sothat ithas faults throughoutitslength indefinitely near tooneanother anduniformlydistributed. The conductivityofthefaults is1/p'perunitlengthofcable, andtheresistance ofthe cable ispperunit length. Onepoleofabatteryisconnected tooneendofacable andtheotherpoleisearthed. Prove that thecurrent atthefarther end isthesame asifthecable were freefrom faults andoftotal resistance \/pp'smhTay'sj 35.Twoparallel conducting wires atunitdistance areconnected by7i41crosspieces ofthesame wire, soastoformnsquares. Acurrent enters byanouter corner ofthe first square, and leaves bythediagonally opposite corner ofthe last. Shew that,if theresistance isthat ofalength £?i+o„ofthewire, <*n+l: a„+2 36.A,Baretheends ofalong telegraphwirewithanumber offaults, andCis anintermediate point onthewire. The resistance toacurrent sentfromAisRwhen Cisearth connected, but ifCisnotearth connected theresistance isSorTaccording astheendBistoearth orinsulated. IfR\S',T'denote theresistances under similar circumstances when acurrent issentfromBtowards A,shew that T'(R-S) =R'(R-T). 37.Theinnerplatesoftwocondensers ofcapacities C,Carejoined bywires of resistances R,R'toapoint P,and their outerplates bywires ofnegligibleresistance toapoint Q.Iftheinnerplates bealsoconnected through agalvanometer, shew that theneedle will suffer nosudden deflection onjoining P,Qtothepoles ofabattery, ifCR=C'R'. 38.Aninfinite cable ofcapacity andresistance KandRperunit lengthisatzero potential. Attheinstant t—0oneend issuddenly connected toabatteryforan infinitesimal interval andthen insulated. Shew that, exceptforverysmall values oft, thepotentialatanyinstant atadistance xfrom thisendofthecable willbepro- portional to 1_^R^ 7te~« • CHAPTEE X STEADY CURRENTS INCONTINUOUS MEDIA Components ofCurrent. 370. Inthepresent chapter weshall considersteadycurrents ofelec- tricity flowing throughcontinuous two-andthree-dimensional conductors instead ofthrough systemsoflinear conductors. Wecan findthedirection offlow atanypointPinaconductorby imaginingthatwetake asmallplaneofareadSandturn itabout atthe pointPuntilwefindthepositioninwhich theamount ofelectricity crossing itperunittime isamaximum. Thenormal totheplane when inthis positionwillgivethedirection ofthecurrent atP,and ifthetotalamount ofelectricity crossingthisplane perunittimewhen inthispositionisCdS, thenCmaybedefined tobethestrengthofthecurrent atP. IfI,m,narethedirection-cosines ofthedirection ofthecurrent atP, then thecurrent Cmaybetreated asthesuperpositionofthree currents IC,mC,nCparalleltotheaxes. Toprovethisweneedonlynotice thatthe flowacross anareadSofwhich thenormal makes anangle6with thedirec- tionofthecurrent, andhasdirection-cosinesI',m', n',must beCdS cos0,or CdS{IV+mm!+nn'). The firstterm ofthisexpression mayberegardedasthecontribution from acurrent ICparalleltotheaxisOx,andsoon.Thequantities IC,mC,nC arecalled thecomponentsofthecurrent atthepoint P. Lines andTubesofFlow. 371. Definition. Alineofflowisalinedrawn inaconductor such thatatevery pointitstangentisinthedirectionofthecurrent atthepoint. Definition. Atubeofflowisatubularregion ofinfinitesimalcross- section, boundedbylinesofflow. 342 Steady Currents incontinuous Media[oh.x Itisclear that atevery pointonthesurface ofatube offlow,thecurrent istangentialtothesurface. Thus nocurrent crosses theboundaryofatube offlow,fromwhich itfollows that theaggregatecurrentflowingacross all cross-sections ofatube offlow willbethesame. Theamount ofthiscurrent willbecalled thestrengthofthetube. Thus ifGisthecurrent atanypointofatube offlow,and if&>isthe cross-section ofthetube atthatpoint,then Ceo isconstantthroughoutthe lengthofthetube,and isequaltothestrengthofthetube. There isanobvious analogy between tubes offlow incurrentelectricity andtubes offorce instaticalelectricity,thecurrent CcorrespondingtothepolarisationP. Incurrentelectricity,Ca> isconstant andequaltothestrengthofthetube offlow, while instaticalelectricity Pa isconstant andequaltothestrengthofthetube offorce (§129). SpecificResistance. 372.Thespecificresistance ofasubstance isdefined tobetheresistance ofacube ofunitedgeofthesubstance, thecurrententering byaperfectly conductingelectrode which extends overthewhole ofone face,andleaving byasimilar electrode ontheoppositeface. Thespecificresistances ofsome substances ofwhich conductors and insulators are frequently made aregiveninthefollowingtable. The units arethecentimetre and theohm. Dilute sulphuric acid(^acidat22°C.) 33. „ „ „(|acidat22°C.) 1-6. Glass(at200°C.) 2-27xl07 . „(at400°C.) 7-35x10*. Guttapercha, about 3xl014 . Iftisthespecificresistance ofanysubstance, theresistance ofawire It oflengthIandcross-section swillclearlybe— . sSilver ... 371-374] Ohm'sLaw 343 ofaparticleattheendofanysmall interval oftime iscompoundedofthevelocityat thebeginningoftheinterval together with thevelocity generated during theinterval. The lattervelocityisinthedirection oftheforcesacting ontheparticle, but isgenerally insignificantincomparisonwith theoriginal velocityoftheparticle. Intheparticular case inwhich theoriginal velocityoftheparticle wasvery small, thedirection ofmotion attheendofasmall interval willbethat oftheforceacting ontheparticle.Ifthe particle moves inaresisting medium,itmaybethat thevelocityoftheparticleiskept permanently verysmall bytheresistance ofthemedium :inthiscase thedirection of motion oftheparticleatevery instant, relativelytothemedium, maybethat ofthe forcesacting on it. Onthemodern view ofelectricity, acurrent ofelectricityiscomposedofelectrons which aredriven throughaconductor bythe electric forcesacting onthem, and in their motion experience frequentcollisions with themolecules oftheconductor. The effect ofthese collisions iscontinuallytocheck theforwardvelocityoftheelectrons, so that thisforwardvelocityiskeptsmalljustasiftheyweremoving through aresisting medium oftheordinary kind,andsoitcomes about that thedirection offlowofcurrent isinthedirection oftheelectricintensity (cf.§345a). 374. Letusselect anytube offorce ofsmall cross-section inside a conductor, and letP,Qbeanytwopointsonthistube offorce, atwhich thepotentialsareVPandVQ,theformerbeingthegreater. Letthese pointsbesoneartogetherthatthroughouttherangePQthecross-section ofthetube offorcemaybesupposedtohave aconstant valueco,while the specificresistance ofthematerial oftheconductor maybesupposedto have aconstant value t. From what hasbeen said in§373, itfollows thatthetube offorceunder consideration isalsoatube offlow. IfGdenotes thecurrent, then the current flowing throughthistube offlow inthedirection fromPtoQ willbeCeo. This current may,within therange PQ,beregardedasflowing throughaconductor ofcross-section coandofspecificresistance t.The PQ.Tresistance ofthisconductor fromPtoQisaccordingly — ,while the fall ofpotentialisVF-Vq.ThusbyOhm's Law CO sothatP pQQ—Gr. If ;r-denotes differentiation along thetube offorce, thefraction onthe ds leftoftheforegoing equation reduces, whenPandQaremade tocoincide, dV to—— ,sothattheequationassumes theform -d-^=Gr (309). OS 344 Steady Currents incontinuous Media[ch.x LetI,m,7ibethedirection-cosines ofthelineofflowatP,and letu,v,w bethecomponentsofthecurrent atP,sothatu=IG,etc.Then —-=I^—=—LUt=—ut,etc.,ox OS andweseethatequation (309)isequivalenttothethreeequations u= 374-377] Equation ofContinuity 345 Thesameequationcanbeobtained atonceonconsideringthecurrent- flowacross thedifferent faces ofasmallrectangular parallelepipedofedges dx,dy,dz(cf.§49). Equation (310)ofcourseexpressesthat thevectorCofwhich the componentsare u,v,w,must besolenoidal. Theequationofcontinuity- canaccordinglybeexpressedintheform divC=0. Equation satisfied bythePotential. 376.Onsubstitutinginequation (311) thevalues foru,v,wgiven by equations (310),weobtain dx\r dx)dy\rdy) dz\r dz) Thepotentialmustaccordinglybeasolution ofthis differentialequation. Theequationisthesame aswould besatisfiedbythepotentialinan unchargeddielectric inanelectrostatic field, providedtheinductivecapacity atevery pointisproportionalto-.Ifthespecificresistance ofthecon- ductor isthesamethroughout,thedifferential equationtobesatisfiedby thepotentialreduces to 377.Wemayforconveniencesupposethatthecurrent enters andleaves byperfectly conducting electrodes, andthattheconductorthroughwhich the current flows isbounded, exceptattheelectrodes, byperfectinsulators. Then, overthesurface ofcontact between theconductor andtheelectrodes, the potentialwillbeconstant. Over theremainingboundaries oftheconductor, thecondition tobesatisfied isthatthere shallbenoflowofcurrent, andthis dV isexpressed mathematically bythecondition that -~-shall vanish. Thus theproblemofdeterminingthecurrent-flow inaconductor amounts mathematicallytodeterminingafunctionVsuch thatequation (312)issatis- dV fiedthroughoutthevolume oftheconductor, while either—=0,orelseVhas aspecified value, ateachpointontheboundary. Bythemethod used in§188, itiseasily shewn thatthesolution ofthisproblemisunique. Itisonlyinaveryfewsimplecases thatanexact solution oftheproblem canbeobtained. There are,however, various artifices bywhich approxima- tionscanbereached, andvarious waysofregardingtheproblemfromwhich it maybepossibletoformsome ideas ofthephysical processeswhich determine thenature oftheflow inaconductor. Some ofthese willbediscussed later (§§386—394). Atpresent weconsidergeneralcharacteristics oftheflowof currentsthroughconductors. 346 Steady Currents incontinuous Media[ch.x Conditions tobesatisfied attheBoundary oftwo Conducting Media. 378. The conditions tobesatisfied ataboundaryatwhich thecurrent flows from oneconductor toanother areasfollows: (i)Since there must benoaccumulation ofelectricityattheboundary, thenormal flowacross theboundary must bethesame whether calculated in the firstmedium orthesecond. Inother words -^—must becontinuous,rdn where 5-denotes differentiationalongthenormal totheboundary.on (ii)Thetangentialforcemust becontinuous, orelsethepotentialwould notbecontinuous. Thus -7—must becontinuous,OS where =-denotes differentiationalong anylineintheboundary. Theseboundaryconditions arejustthesame aswould besatisfied inan electrostaticalproblemattheboundary between two dielectrics ofinductive capacities equaltothetwovalues of-.Thus theequipotentialsinthis electrostatic problemcoincide with theequipotentialsintheactual current problem,andthelines offorce intheelectrostaticproblem correspondwith thelines offlowinthecurrentproblem. Clearlythese results could bededuced atoncefrom thedifferential equation (312) on passingtothelimitandmakingrbecome discontinuous oncrossing aboundary. Refraction ofLines ofFlow. 379. Letanyline offlow cross theboundarybetween two different conductingmedia ofspecificresistances rlft2,making anglese1}e2with the normal atthepointatwhich itmeets theboundaryinthetwomedia respectively. The lines offlowsatisfythesame conditions aswould be satisfiedbyelectrostatic lines offorcecrossingtheboundarybetween two dielectrics ofinductivecapacities— ,— ,sothatwemust have(cf.equa- tion(71)) —cot6j=—cot e2. Ti T2 Hence rttan e1—r2tan e2, expressingthelawofrefraction oflines offlow. 378-381] Boundary Conditions 347 380. Asanexampleofrefraction oflines ofcurrent flow,wemay consider thecase ofasteadyuniform current inaconductorbeingdis- turbed bythepresenceofasphereofdifferent metal inside theconductor. The linesshewn infig.78willrepresentthe lines offlow ifthespecific resistance ofthesphereislessthan that ofthemain conductor. The lines offlowtend tocrowd intothesphere,thisbeingthebetter conductor—in thelanguageofpopular science, thecurrent tends totake thepathofleast resistance. ChargeonaSurface ofDiscontinuity. 381. Ifuisthenormal componentofcurrentflowingacross the boundarybetween two different conductors, wehavebyOhm's Law, txdn t2dn' where =-denotes differentiation along thenormal which isdrawn inthe dn direction inwhich uismeasured(sayfrom(1)to(2)),andVltV2arethe potentialsinthetwoconductors. Ifthere isnochargeontheboundarybetween thetwoconductors we must, fromequation (70), have therelation on on whereKltK2aretheinductivecapacitiesofthetwoconductors. This condition will,however, ingeneralbeinconsistent with thecondition which, aswehavejust seen, ismadenecessary bythecontinuityofu.Thus there willingeneralbeasurfacechargeontheboundary between twoconductors ofdifferent materials. Theamount ofthischargeisgivenatoncebyequation (72), p.125. Ifa denotes thesurfacedensityatanypoint,wehave on on =-(K lr1-R2r2)u (313). This surfacechargeisverysmall compared with thecharges which occur instatical electricity. Forinstance,ifwehave current of100amperes persq.cm.passing fromone metallic conductor toanother, wetake informula(313), u=!00arnperes=3x10uelectrostatic units, 10~6 r=10-6ohms=___ >} K=l, thelasttwobeing true asregardsorder ofmagnitude only. Thevalue ofAircr isofthe order ofmagnitudeofKtu,orJx10-6inelectrostatic units. Ashasbeen said, thevalue of4n-cratthesurface ofaconductor chargedashighlyaspossibleinairisoftheorder of100. 348 Steady Currents incontinuous Media[oh.x 382. Asanexampleofthedistribution ofasurfacecharge, wemay notice that thesurface-densityofthechargeonthesurface ofthesphere dV considered in§380 willbeproportionaltoeither value of— ,andtherefore tocos6,where 6istheanglebetween theradiusthroughthepoint andthe direction offlowoftheundisturbed current. Generation ofHeat. 383. Consider anysmall element ofatube offlow, length ds,cross- 1dV section &>.Thecurrentperunit areais,byequations (310), — ,so 1dV that thecurrentflowing throughthetube is——co.The resistance of tcIs theelement ofthetubeunder consideration is— .Hence, asin5355,thew> a> amount ofheatgenerated perunittime inthiselement is <ldV Vrds 1fdVV (ldV Vrds 1/dVy -a-o>—or-— \TOS jft) T\OS )ft)ds. . . . .l/dVyThus theheatgenerated perunittimeperunitvolume is— f -^— J,and thetotalgenerationofheatperunittime willbe mm+(%hm^ ™ Thus theheatgenerated perunittime is87rtimes theenergyofthe whole field intheanalogouselectrostatic problem (§169). Rateofgeneration ofheataminimum. 384. Itcanbeshewn that foragivencurrentflowing throughacon- ductor, therateofheatgenerationisaminimum when thecurrent distributes itself asdirected byOhm's Law. Todothiswehave tocomparetherate of heatgeneration justobtained with therate ofheatgeneration when the current distributes itself insome otherway. Letussupposethat thecomponentsofcurrent atanypoint have no longerthevalues 1dV 1dV_ldV tdx'tdy'tdz assignedtothembyOhm's Law,butthattheyhave different values ldV ldV ldV tox tdytdz 382-385]Generation ofHeat 349 Inorder thattheremaybenoaccumulation atanypoint under thisnew distribution, thecomponentsofcurrent mustsatisfytheequationofcon- tinuity,sothatwemust have die ,dvdw _ ,„,.~ 5-+^r+;T=(315).oxoyozv ' Bythesamereasoningasin§383,wefind fortherate atwhich heat is generatedunder thenewsystemofcurrents, ///T ((-VTx+ •)'+Hw+Vf+("rW+ <*)]*** which, onexpanding,isequalto -~2!IKud^+v%+wd^)dxdydz +[ffr(wi+v2+w2)dxdydz (316). Ontransforming byGreen's Theorem, thesecond term =2fjfvP£+^+d ^)dxdydz-2ffv(lu+mv+nw)dS. Thevolumeintegralvanishes byequation (315), theintegrandofthe surfaceintegralvanishes overeach electrode from thecondition thatthetotal flowofcurrent across theelectrode istoremain unaltered, andatevery point oftheinsulating boundaryfrom thecondition that there istobenoflow across thisboundary.Thus thenewrateofgenerationofheat isrepresented bythe firstandthird terms ofexpression (316). The firsttermrepresents theoldrate ofgenerationofheat, thethird term isanessentially positive quantity.Thus therate ofheatgenerationisincreased byanydeviation from thenatural distribution ofcurrents, provingtheresult. 385.Animmediate result ofthis isthatanyincrease ordecrease inthe specificresistance ofanypartofaconductor isaccompanied byanincrease ordecrease oftheresistance oftheconductor asawhole. Forondecreasing thevalue oftatanypointandkeepingthe distribution ofcurrents unaltered, therate ofheatproductionwillobviouslydecrease. Onallow- ingthecurrents toassume their natural distribution, the rate ofheat productionwillfurther decrease. Thus therate ofheatproductionwith a natural distribution ofcurrents islessened byanydecrease ofspecific resistance. But if/isthetotal current transmitted bytheconductor, and Rtheresistance oftheconductor, this rate ofheatproductionisRI-. ThusRdecreases when tisdecreased atany point,andobviouslythe converse must betrue(cf. §359). 350 Steady Currents incontinuous Media[ch.x TheSolution ofSpecial Problems. Current-flowinanInfiniteConductor. 386.Agood approximationtotheconditions ofelectric flow can occasionallybeobtained byneglectingthe restrictive influence ofthe boundaries ofaconductor, andregardingtheproblemasoneofflowbetween twoelectrodes inaninfinite conductor. Forsimplicity, weshall consider only thecase inwhich theconductor ishomogeneous. The conditions tobesatisfied bythepotential Vareasfollows. We must haveV=VXoveroneelectrode, andV=V2overthesecond electrode, dV 1 while—must vanish atinfinity toahigher order than—andthroughout theconductor wemust haveV2F=(§376).Wecaneasilysee(cf.§§186, 187)thatthese conditions determine Vuniquely. Consider nowananalogouselectrostaticproblem.Lettheconducting medium bereplaced byair,while theelectrodes remain conductors. Let theelectrodes receiveequalandopposite chargesofelectricityuntil their difference ofpotentialisVx—V2.Atthisstagelet-\}rdenote theelectro- staticpotentialatanypointinthe field. Letyfr1}^2bethevalues of\jrover thetwoelectrodes, sothat^—ty2=Vi—V2.Then there willbeaconstant C(namely K—^i)>sucn thatyjr+Cassumes thevaluesV1}Krespectively over thetwo electrodes. Moreover V2 -^=throughoutthe field, sothat V2 (-v^+(7)=throughoutthe field,and\|r=0atinfinity exceptforterms -I o in— (cf.§67),sothat ~-(^+C)vanishes atinfinitytoahigherorder than— . Hence-fy+Csatisfies theconditions which, aswehave seen,must be satisfied bythepotential Vinthecurrentproblem, andthese areknown to suffice todetermine Vuniquely.Itfollows that thevalue ofVmust be Tjr+C. Thus thelines offlow inthecurrentproblemareidentical with thelines offorcewhen thetwoelectrodes arechargedtodifferentpotentialsinair. Thenormal current-flow atanypointonthesurface ofanelectrode is ldV tdn' sothatthetotal flowofcurrent outwards from thiselectrode iff|ZdS=_If(|fcdS .on tJJon 386,387] Special Problems 351 IfEisthechargeonthiselectrode intheanalogouselectrostaticproblem wehave, byGauss' Theorem, -11%**-***> 4<7rE sothatthetotal flowofcurrent isseen tobe .T Ifpn>Pn>P-aarethecoefficients ofpotentialintheelectrostaticproblem f^PnE-puE, ^=p uE-p 22E, sothat Yi-V*=fi~^2=(Pn-%2+P22) E. If/isthe total current, andRtheequivalentresistance between the electrodes, wehavejustseen that T' sothat B-^^-j^CPu-apta+J?.)(317). Ifweregardthetwoelectrodes inairasformingacondenser, anddenote itscapacity by0,wehave sothat B=E^=ss<318> 387. Asinstances oftheapplicationsofformulae (317) and(318)to special problems, wehave thefollowing: I.The resistanceperunitlength between twoconcentriccylindersof radii a,b(as,forinstance, theresistance between thecore ofasubmarine cableandthesea), is,byformula(318), II.The resistanceperunitlengthbetween twostraight parallel cylindricalwires ofradii a,b,placedwith their centres atagreatdistance r apart,inaninfiniteconducting medium, is,byformula(317), rr—y-(loga—2logr+logb) r ,r2=2il0S56- 352 Steady Currents incontinuous Media[ch.x III.The resistance between twospherical electrodes, radii a,b,ata greatdistance rapart,inaninfinite conducting medium, is,byformula (317), 4>tt\ab 388. Iftwoelectrodes ofanyshapeareplacedinaninfinite medium at adistance rapart,which isgreat comparedwith their linear distances, we maytakep12informula (317) equal,toafirstapproximation,to- .This is small comparedwithpnandp&,sothat, toafirstapproximation,wemay replaceformula (317) by Itaccordingly appearsthattheresistance oftheinfinite medium maybe regardedasthesum oftworesistances—aresistance -~atthecrossingof TT) thecurrent from the first electrode tothemedium, andaresistance-f-=at thereturn ofthecurrent from themedium tothesecond electrode. Thus wemaylegitimately speakoftheresistance ofasingle junctionbetween an electrode andtheconducting medium surroundingit. Forinstance, supposeacircularplate ofradius aisburied deepintheearth, andacts aselectrode todistribute acurrent through theearth. Thevalue ofpnforadisc of radius ais„- ,sothattheresistance ofthejunctionis— .Soalso ifadisc ofradius a 7- isplacedontheearth's surface, theresistance atthejunctionis— ,andclearlythis also istheresistance iftheelectrode isasemicircle ofradius aburiedverticallyinthe earth with itsdiameter inthesurface. Flow inaPlane SheetofMetal. 389.When theflowtakesplaceinasheet ofmetal ofuniform thickness andstructure, sothatthecurrent atevery pointmayberegardedasflowing inaplane paralleltothesurface ofthesheet, thewholeproblembecomes two-dimensional. Ifx,yarerectangular coordinates, theproblemreduces to that offindingasolution of da?+ df dVwhich shallbesuch that eitherVhasagiven value, orelse-^—=0,atevery pointoftheboundary. Themethodsalready giveninChap,vin forobtain- ingtwo-dimensional solutions ofLaplace's equationaretherefore available forthepresent problem. Themethod ofgreatestvalue isthat ofConjugate Functions. 387-390] Special Problems 353 Iftheconducting medium extends toinfinity,orisboundedentirely by thetwo electrodes, thetransformations willbeidentical with thosealready discussed fortwoconductors atdifferentpotentials (§386). Ifthemedium dV hasalsoboundaries atwhich—=0,theprocedure must beslightlydifferent. Wemusttrytotransform thetwoelectrodes into linesV=constant, andthe other boundaries into linesU=constant, sothatthewhole ofthemedium becomes transformed intotheinterior ofarectangleintheU,Vplane. Let U+iV=f(x +iy) beatransformation whichgivestherequiredvalue forVoverboth electrodes, OXT andgives=—=over theboundaryofaconductor. ThenVwillbethe potentialatanypoint,thelinesV=constant willbetheequipotentials, and thelinesU=constant, beingtheorthogonal trajectoriesoftheequipotentials, willbethelines offlow. Atanypointthedirection ofthecurrent isnormal totheequipotential throughthepoint,andtheamount ofthecurrent isgiven by ton r\IT OTT O But—isequal to-=-,where^-denotes differentiation intheequipotential.on^os os Thus thecurrentflowingacrossanypiecePQofanequipotential [Q=1Gds QldU, 1 i:v**-\<«-<* IfP,Qareanytwopointsintheconductor, apathfromPtoQcanbe regardedasmadeupofapieceofanequipotential PN,andapieceofaline offlowJSTQ. The*flow across JSfQiszero, that across PiY is -(TJN-UP).T This isaccordinglythetotal flow across PQ,andsinceUN=UQ,itmay bewritten as ±(UQ-UP). 390. Asanillustration, letussupposethat theconducting plateisa polygon,two ormoreedges beingtheelectrodes. Wecantransform this intotherealaxis inthef-plane byatransformation ofthetype %=(s-<hY~\s-<hy~1 (3i9), 23 354 Steady Currents incontinuous Media[CH.X andthis realaxishastobetransformed intoarectangleformed(say)bythe linesV=V,V'=Vi,£7=0, U=GintheTT-plane. The transformation forthis willbe dW ar=[a-<K£-a P)(?-<>a:-<>r* •(320), where a,apandaq,ararethepointsonthereal axis of£which determine theends oftheelectrodes. Byelimination of£from theintegralsofequa- tions (319)and(320)weobtain thetransformationrequired. 391. Thefollowing exampleofthismethod istaken from apaper by H.F.Moulton (Proc.Lond. Math. Soc. in.p.104). »Q 2-plane. Fig. 101.B a TT-plano. Fig. 102. Infig.101, letABCD bearectangular plate,thepiecePQofoneormore sidesbeingoneelectrode, andthepieceRSofoneormore other sidesbeing theother electrode. Lettherectangle PQRS infig.102beitstransforma- tionintheTf-plane.Intheintermediate£-plane,letthepoints A,B,G,B transform to£=a,b,c,drespectively,and letthepoints P,Q,R,Stransform to£=£>, q,r,srespectively. Then thetransformations are dz ;«[(?-a)(£-&)(r-c)(r-d)]-*, Ifwewritedt; (b—c)(a—d) K,(q-r)(p-s) (a-c)(b- d) 2m=V(a-c)(6-d), theintegralsare=\(p-r)(q-s) 2m'=^(p-r)(q-s), y_a(b—d)-b(a—d)snimz(mod k) 'h-d—(n- d\sn2«j.*frnnrl „\ (321),b—d—(a—d)sn2mz(mod k) t>_P(q—s)—q(p—s)sn2m'W(mod X) q—s— (p—s)sn2m'W(mod X).(322). The sidesAB,AD ofthe first rectangle aretheperiods— . ofmm 390-392] Special Problems 355 snmz(mod k) ;thesidesPQ,PSofthesecondrectanglearetheperiodsin T'T' W,say— ,,—7,ofsnm'W(mod X). jjIntheTT-plane,thepotentialdifference ofthetwoelectrodes isPS,or— ,, 1 L' while thecurrent is-PQ,or—j-.Theequivalent resistance oftheplatet niTr isaccordingly tL'/L,sothatthequantitywearetryingtodetermine isL'JL. Letthecoordinates ofP,Q,R,Sinthe2-planebezx,z2,z3>z4.Inthe £-planethecoordinates ofthesepointsarep,q,r,s.Hence fromequations (321), wehave _a(b—d)—b(a—d)sn2mz1(mod k)^ (b—d)—{a—d)sn2mzx(mod k)' andsimilarequationsforq,r,s.The ratioL'jLofwhich weareinsearch isnowgiven by L'(q—r)(p—s)(sn2mz2—sn2mz3)(sn2mzl—sn2W24) L(p—r)(q—s)(sn2mz1—sn2mz3)(sn2mz2—sn2m^4)' thewhole beingtomodulus k.Thevalues ofsnmzcanbeobtained from Legendre'sTables. Moulton hascalculated theresistance ofasquaresheet with electrodes, each oflength equaltoone-fifth ofaside, inthefollowingfourcases : (1) Electrodes atmiddle oftwoopposite sides, Resistance =1/745.R, (2)Electrodes atends oftwooppositesides andfacingoneanother, Resistance =2-408.K, (3) Electrodes atends oftwooppositesides andnotfacingone another, Resistance =2'589.R, (4)Electrodes bentequallyround twooppositecorners ofsquare, Resistance =3027R, whereRistheresistance ofthesquare when thewhole oftwooppositesides form theelectrodes. Acomparisonoftheresults incases(2)and(3)shews howlargeapartoftheresistance isduetothecrowdinginofthelines of force near theelectrode, andhowsmall apartarises from theuncrowded partofthepath. Limits totheResistanceofaConductor. 392. The result obtained in§386enables ustoassignanupperand alower limit totheresistance ofaconductor, when thisresistance cannot be calculatedaccurately.For ifany partsoftheconductor aremade into perfect conductors, theresistance ofthewhole willbelessened, and itmay bepossibletochange partsoftheconductor intoperfectconductors insuch 23—2 356 Steady Currents incontinuous Media[ch.x awaythat theresistance ofthenewconductor canbecalculated. This resistance willthenbealower limit totheresistance oftheoriginalcon- ductor. Asanillustration, wemayexamine thecaseofastraightwire ofvariable cross-section S.Letusimaginethat atsmall distancesalongitslength we take cross-sections ofinfinitelysmall thickness, andmake these intoperfect conductors. Theresistance between twosuch sections atdistance dsapart, tds willbe-77- ,where8isthecross-section ofeither. Thus alower limit to theresistance issupplied bytheformula 'ds[ds IS' 393. Again,ifwereplace partsoftheconductorbyinsulators, socausing thecurrent toflowingiven channels, theresistance ofthewhole isincreased, andinthiswaywemaybeable toassignanupperlimit totheresistance ofaconductor. 394. Asaninstance ofaconductor totheresistance ofwhich both upperandlower limits canbeassigned,letusconsider thecase ofa cylindricalconductor ABterminatinginaninfinite conductor Gofthesame material. Thisexampleis ofpractical importanceinconnection withmercury resistance standards. Theappropriate analysis was firstgiven byLordRayleigh, discussingaparallel probleminthetheoryofsound. Let Ibethelengthandatheradius ofthetube. Toobtain alower limit totheresistance, weimagine aperfectly conducting planeinserted atB.The resistance then consists of theresistance tothisnewelectrode atB,plustheresistance from thiswith It theinfinite conductor G.Theformer resistance is,the latter, bv5388TT/12 'JO'TTCL* nr ist- .sothatalower limit tothewhole resistance is4a It t ira' 4a' irawhich istheresistance ofalengthI+— -ofthetube. Toobtain anupperlimit totheresistance, weimagine non-conducting tubesplacedinside themain tubeAB, sothat thecurrent isconstrained to flow inauniform streamparalleltotheaxis ofthemain tube until the endBisreached. After thisthecurrent flowsthroughthesemi-infinite conductor GasdirectedbyOhm's Law. 392-394] Special Problems 357 The resistance ofthetubeABis,asbefore,—-.Toobtain theresist- 7TCL- ance oftheconductor G,wemust examine thecorrespondingelectrostatic problem.If/isthetotal current, theflow ofcurrentperunit area over thecircular mouth atBisIJTra2 .Inorder that thepotentialsinthe electrostaticproblem maybethesame, wemust have auniform surface densityofelectricity r\/I\ rl onthesurface ofthe disc. TheheatgeneratedisI2R,whereRistheresistance oftheconductor C. Itisalsomm+Q+m^<-> taken throughtheconductor G.Now ifWistheelectrostaticenergyof rl adisc ofradius a,having auniform surface density a=,„„oneach side, wehave where theintegralistakenthroughallspace,oragain, where theintegralistakenthroughthesemi-infinitespaceononeside of the disc, i.e.throughthespace G,ifthedisc ismade tocoincide with the mouth B.Onsubstitutingforthevolumeintegralinexpression (323),we findthat 4ttT7PR=HLlL(324). Following Maxwell, weshall find itconvenient tocalculateWdirectly from thepotential.Ifadisc ofradius rhasauniform surfacedensity a oneach side, thepotentialatapointPonitsedgewillbe where theintegralistaken overoneside ofthe disc,andristhedistance fromPtotheelement dxdy. Taking polar coordinates, withPasorigin, theequationofthe circle willber=2bcos;wemay replace dxdy by rdrdd, andobtain rr=25cos 9re=- Vp=2a\\\drdd =8b<r. 358 Steady Currents incontinuous Media[ch.x Onincreasingtheradius ofthedisc tob+db,webring upacharge kirbadb frominfinitytopotential 8bcr, sothat thework done is dW=SMWdb, andintegratingfrom 6=tob=a,wefind forthepotential energyofthe completedisc ofradius a, Thus, fromequation (324), 4ttTT 1287r2a3<7JR=Ft 3Pr rl or,smce a-= R=47r2a2' 8t 3-7r2a Thusanupperlimit tothewhole resistance is It 8+T Tra237r2a' gwhich istheresistance ofalengthI+5—aofthetube. Thuswemay saythat theresistance ofthewhole isthat ofalength 1+actofthetube,where aisintermediate between Tand^— ,i.e.between 4 Sir •785and-849. LordRayleigh*, bymore elaborateanalysis,hasshewn that theupperlimit foramust belessthan '8242, andbelieves that thetrue value ofamust beprettyclose to"82. Thepassage ofElectricity through Dielectrics. 395. Since even thebest insulators arenotwhollydevoid ofconducting power,itisofimportancetoconsider theflowofelectricityindielectrics. Usingtheprevious notation, weshall denote thepotentialatanypoint inthedielectric byV,thespecificresistance byt,andtheinductivecapacity byK.Weshall considersteadyflow first. Iftheflow istobesteady,theequationofcontinuity, namely 1(Id-Z^+1(Id-L\+1(I?L\=(3^5)dx\tdx)dy\r oy)dz\r dz)^" must besatisfied. Also ifthere isavolumedensityofelectrificationp,the potentialmustsatisfy equation (62),namely !(*£K(*©-£(*$"*»<32e>- *Theory ofSound, Vol. 11.Appendix A. 394-396] Passage ofElectricity through Dielectrics 359 From acomparisonofequations (325) and(326),itisclear thatsteady- flow willnotgenerallybeconsistent withhaving p=0.Hence ifcurrents are startedflowing throughanuncharged dielectric, the dielectric will acquirevolumechargesbefore thecurrents becomesteady. When the currents have becomesteady,thevalue ofVwillbedetermined by equation (325) andtheboundary conditions, andthevalue ofpisthen given byequation (326). Fromequations (325) and(326), weobtain p=-^TT\d-Xd-x(Kt)+ *ydy{Kt)+-dldz{Kr) \-(327)- Thecondition thatpshall vanish, whatever thevalue ofV,isthatKrshall beconstantthroughoutthedielectric :ifthiscondition issatisfied thevalue ofpnecessarilyvanishes atevery pointforallsystemsofsteadycurrents. Themostimportantcase ofthisconditionbeingsatisfied occurs when the dielectric ishomogeneous throughout.IfKr isnotconstantthroughout thedielectric, equation (327) shews thatwecanhavep=atevery point providedthesurfaces F=cons. andKr=cons, cutoneanother atright anglesatevery point,i.e.provided Kr isconstantalong everylineofflow. Wehavealreadyhadanillustration(§381) oftheaccumulation of chargewhich occurs when thevalue ofKrvaries inpassing alongaline offlow Time ofRelaxation inaHomogeneousDielectric. 396. Letahomogeneousdielectric bechargedsothat thevolume densityatanypointisp. Ifanyclosed surface istaken inside the dielectric, the totalcharge inside this surface must be \\\pdxdydz, while therate atwhichelectricityflows intothesurfacewill, asin§375,be 11(lu+mv+nw)dS, where u,v,warethecomponentsofcurrent andI,m,narethedirection cosines ofthenormal drawn intothesurface. Since this rate offlow into thesurface must beequaltotherateatwhich thechargeinside thesurface increases, wemust have ll(lu+ mv+nw)dS=-r\\\pdxdydz = SSid dtdxdydz- 360 Steady Currents incontinuous Media[ch.x Theintegralontheleftmay,byGreen's Theorem, betransformed into -///(fs+S+lf)***' andthisagainisequal, byequations (310),to d2V d2Vd2V\ . .,m Thuswehave ff[[i fd2Vd*v d*v\dp),,,A andsince this istruewhatever surface istaken, eachintegrand must vanish separately,andwemust have, atevery pointofthedielectric, d2V d2Vd2V=dp dx2+ By2+dz2~T dt' Wehave also, asinequation (326), d2Vd2Vd2V=4tt/j dx2+ dy2+dz2~K' sothatTt=-KTp-dp_47T Theintegralofthisequationis "IT* wherepisthevalue ofpattime £=0. Thus thechargeatevery pointinthedielectric falls offexponentially 4>7T Ktwith thetime, themodulus ofdecay being -^^.Thetime -r— ,inwhichAt 47r allthechargesinthe dielectric arereduced to1/etimes theiroriginal value,iscalled the"time ofrelaxation," being analogoustothecorresponding quantityintheDynamical TheoryofGases*. The relaxation-time admits ofexperimental determination, and astis easily determined, thisgivesusameans ofdetermining Kexperimentally forconductors. Inthecase ofgood conductors, therelaxation-time istoo small tobeobserved withanyaccuracy,butthemethod hasbeenemployed byCohn andArons-f-todetermine theinductivecapacityofwater. The value obtained, A~=73- 6,isingoodagreementwith thevalues obtained in other ways (cf.§84). *Cf.Maxwell, Collected Works, n.p.681,orJeans, Dynamical Theory ofGases, p.294. tWied. Ann. xxviii. p.454. 396,397] Passage ofElectricity through Dielectrics 361 Discharge ofaCondenser. m 397. Letussupposethat acondenser ischarged uptoacertain potential, andthatacertain amount ofleakagetakesplace throughthe dielectric between thetwoplates. Then, aswehavejust seen, thedielectric will,exceptinvery special cases, becomechargedwithelectricity. Nowsupposethat thetwoplatesareconnected byawire, sothat, in ordinary language,thecondenser isdischarged. Conduction throughthe wire isaverymuchquicker processthan conductionthroughthedielectric, sothatwemaysupposethattheplatesofthecondenser arereduced tothe samepotentialbefore thecharges imprisonedinthedielectric havebegunto move. Forsimplicity,letussupposethat theplatesofthecondenser are both reduced topotentialzero. Then thesurface ofthedielectricmay, with fairaccuracy,beregardedasanequipotential surface, thepotential beingzero allover it.Itfollows thatthere canbenolines offorce outside thisequipotential:alllines offorce whichoriginateonthechargesim- prisonedinthe dielectric, andwhich donotterminate onsimilarcharges, must terminate onthesurface ofthe dielectric. Thus weshall have a systemofchargesonthesurface ofthedielectric, thesecharges being equal inmagnitude butoppositeinsigntothose oftheGreen's"equivalent stratum" correspondingtothesystemofcharges imprisonedinthedielectric. Thissystemofcharges onthesurface ofthedielectric isofthekindwhich Faraday would calla"bound" charge (cf.§141). Supposetheplatesofthecondenser tobeagaininsulated. Thesystem ofchargesinside thedielectric andatitssurface isnotanequilibriumdis- tribution, sothat currents willbesetupinthe dielectric, andageneral rearrangementofelectricitywilltakepkce. Thepotentials throughoutthe dielectric willchange, andinparticularthepotentialsofthecondenser-plates atthesurface ofthedielectric willchange.Inother words, thechargeon theseplatesisnolongera"bound" charge,butbecomes, atleastpartially,a "free"charge. Onjoiningthetwoplates byawire, anewdischargewill takeplace. This isMaxwell'sexplanationofthephenomenonof"residualdischarge." Itisfound that,some time after acondenser hasbeendischargedand insulated, asecond andsmallerdischargecanbeobtained onjoiningthe plates,after this athird, and soon,almostindefinitely.Itshould be noticed that,ontheexplanationwhich hasbeengiven,noresidualdischarge oughttotakeplaceifthedielectric isperfectly homogeneous.Itthusbecomes possibletotestthetheory byexperimentsonhomogeneousdielectrics. Rowland andNichols* testedcalcspar,which isaperfectly homogeneous crystal,andfound notrace ofresidualdischarge.Hertz-f-found traces ofa *Phil.Mag. [5]vol.n.p.414(1881). tWied. Ann. xx.(18S3), p.279. 362 Steady Currents incontinuous Media[ch.x residualdischargeinahomogeneous fluid, benzene, butfound thatthese dis- appearedasimpuritieswereremoved from the fluid; Arons* obtained the same result withparaffin. Finally Muraokaf experimented with various oils, paraffin, resin, turpentine andxylol.Residualdischargeswere notfound inthe oilssingly, butappearedassoon astwoormore weremixedtogether. These facts areinagreementwithMaxwell'stheoryofresidualdischarge and affordstrongconfirmation ofthetheory. Ontheother hand there area largenumber ofexperimentalfactswhich are difficult toexplaininterms ofMaxwell'stheory alone, andwhich seem tosuggestthat thetheoryis incomplete. EXAMPLES. 1.Theends ofarectangular conducting lamina ofbreadthc,length a,anduniform thicknessr,aremaintained atdifferentpotentials.Iff(x, y)bethespecific resistancep atapoint whose distances fromanendandasidearex,y,provethat theresistance of thelamina cannot belessthan,orgreater than T1°<fy' Jopdy /;a pdx 0J0r 2.Twolargevessels filled withmercury areconnected byacapillary tube ofuniform bore. Findsuperior andinferior limits totheconductivity. 3.Acylindricalcable consists ofaconductingcore ofcopper surrounded byathin insulating sheath ofmaterial ofgiven specific resistance. Shew that ifthesectional areas ofthecoreandsheath aregiven, theresistance tolateralleakageisgreatest when thesurfaces ofthetwomaterials arecoaxalrightcircularcylinders. 4.Prove that theproductoftheresistance toleakage perunit length between two practically infinitely long parallel wires insulated byauniform dielectric andatdifferent potentials, andthecapacity perunitlength,isEpjAir, whereKistheinductivecapacity andpthespecificresistance ofthedielectric. Prove alsothatthetime thatelapsesbefore thepotentialdifference sinks toagivenfraction ofitsoriginal value isindependentofthe sectional dimensions andrelativepositionsofthewires. 5.Iftherightsections ofthewires inthelastquestionaresemicircles described on oppositesides ofasquareasdiameters, andoutside thesquare,while thecylindrical space whose section isthesemicirclessimilarlydescribed ontheother twosides ofthesquareis filledupwith adielectric ofinfinitespecific resistance, and alltheneighbouring spaceis filledupwithadielectric ofresistancep,prove that theleakage perunitlengthinunit time is2Vjp,whereVisthepotentialdifference. 6.If<p+fy=f{x-\-iy),andthecurves forwhich$=cons. boclosedcurves, shew that theinsulation resistance between lengthsIofthesurfaces<p=<po, <p=<pi,is p(<ftx~ (ftp) where [^Jistheincrement of\j/onpassing onceround a0-curve, andpisthespecific resistance ofthe dielectric. *Wied. Ann. xxxv.(1888), p.291. fWied. Ann. xl.(1890), p.328. Examples 363 7.Current enters andleaves auniform circular discthrough twocircular wires of small radius ewhose central linespassthrough theedgeofthedisc attheextremities of achord oflengthd.Shew thatthetotal resistance ofthesheet is (2tr/»r) log(d/e). 8.Usingthetransformation log(x+iy)=i+ir], provethattheresistance ofaninfinitestripofuniform breadth itbetween twoelectrodes distant 2aapart,situated onthemiddle lineofthestripandhaving equalradii8,is |log(|tanh«)- 9.Shew thatthetransformation x'+itj=cosh it(x+iy)\'a enables ustoobtain thepotential due toanydistribution ofelectrodes upon athin conductor intheform ofthesemi-infinitestripbounded byy=0,y=a,andx=0. Ifthemargin beuninsulated, findthepotential andflowduetoasource atthepoint x=c,y=5.Shew that iftheflows across thethreeedges areequal, then7rc=acosh-12. 10.Equal andopposite electrodes areplaced attheextremities ofthebase ofan isosceles triangular lamina, thelengthofoneoftheequalsidesbeing a,andthevertical angle—Shew thatthelines offlowandequal potentialaregiven bya smh^-+1=^/3- ,2^1-enu where 35rQ)ua=TC£\TQVze«-aV andthemodulus ofenuissin75°,theorigin beingatthevertex. 11.Acircular sheet ofcopper,ofspecificresistanceo-jperunit area,isinserted ina very large sheet oftinfoil(<r ),andcurrents flow inthecomposite sheet, entering and leavingatelectrodes. Prove thatthecurrent-function inthetinfoilcorrespondingtoan electrode atwhich acurrent eenters the tinfoil isthecoefficient of*intheimaginary partof 2tt"log(*-c)+^log«1 where aistheradius ofthecopper sheet,2isacomplexvariable with itsoriginatthe centre ofthesheet, and cisthedistance oftheelectrode from theorigin,thereal axis passing throughtheelectrode. Generalise theexpressionforanypositionoftheelectrode inthecopperorinthe tinfoil, andinvestigate thecorresponding expressions determining thelines offlow inthe copper. 12.Auniform conducting sheet hastheform ofthecatenaryofrevolution w2+z2=c2cosh2- . c Prove that thepotentialatanypoint duetoanelectrode atXq,yo, Zq,introducing a currentC,is constant -^log(cosh*-*°-^0+ZZ °^°\onV+^)W+^) f> CHAPTER XI PERMANENT MAGNETISM Physical Phenomena. 398. Itisfound that certain bodies, known asmagnets,will attract or repeloneanother, while amagnetwill alsoexert forces onpiecesofiron orsteelwhich arenotthemselvesmagnets,these forcesbeing invariably attractive. Themost familiar fact ofmagnetism, namelythetendencyof amagneticneedle topointnorth andsouth,issimplyaparticularinstance ofthe first ofthesets ofphenomena justmentioned,itbeingfound that theearth itselfmayberegardedasavastaggregationofmagnets. Thesimplest pieceofapparatusused fortheexperimental studyof magnetismisthatknown asabar-magnet.This consists ofabarofsteel which shews thepropertyofattractingtoitself smallpiecesofsteel oriron. Usuallyitisfound that themagnetic propertiesofabar-magnetreside largelyorentirelyatitstwoends. Forinstance, ifthewhole bar isdipped intoacollection ofironfilings,itisfound thatthefilingsareattracted in greatnumbers toitstwoends, while there ishardly anyattraction tothe middleparts,sothatonliftingthebaroutfrom thecollection offilings, we shall findthatfilingscontinue tocluster round theends ofthebar,while themiddleregionswillbecomparativelyfree. Poles ofaMagnet. 399. Thetwoends ofamagnet—or,morestrictly,thetworegions inwhich themagnetic propertiesareconcentrated —arespokenofasthe "poles"ofthemagnet.Ifthemagnetisfreely suspended,itwillturn sothat the linejoiningthetwopoles points approximatelynorth and south. Thepolewhichplacesitself soastopointtowards thenorth is called the" north-seeking pole,"while theotherpole, pointingtothesouth, iscalled the" south-seeking pole." Byexperimentingwithtwoormoremagnets,itisfound tobeageneral lawthat similarpoles repeloneanother, while dissimilarpolesattract one another. 398-401] Permanent Magnetism 365 Theearthmayroughlyberegardedasasingle magnetofwhich thetwo magnetic polesareatpointsnear tothegeographicalnorth andsouthpoles. Since thenorthernmagnetic poleoftheearth attracts thenorth-seeking poleofasuspended bar-magnet,itisclear that thisnorthernmagnetic pole must beasouth-seeking pole;andsimilarlythesouthernpoleoftheearth must beanorth-seeking pole.LordKelvinspeaksofasouth-seeking poleas a"truenorth" pole—i.e.apoleofwhich themagnetismisofthekindfound inthenortherly regionsoftheearth. But forpurposesofmathematical theoryitwillbemost convenient todistinguishthetwokinds ofpoleby theentirelyneutral terms, positiveandnegative. And, asamatter of convention, weagreetocallthenorth-seeking pole positive. Thus we have thefollowing pairsofterms : North-seeking=True South=Positive, South-seeking=TrueNorth=Negative. LawofForce betiveen MagneticPoles. 400.Byexperimentswith historsion-balance, Coulomb established that theforcebetween twomagnetic polesvariesinverselyasthesquareofthe distance between them. Itwasfound alsotobeproportionaltotheproduct oftwoquantities spokenofasthe" strengths"ofthepoles. Thus ifFisthe repulsionbetween twopolesofstrengths m,w!atadistance rapart,wehave „cmm'F=—(328). Itisfound that cdependsonthemedium inwhich thepolesareplaced, but isotherwise constant.Clearlyifweagreethatthestrengthofpositive polesistobereckoned aspositive,while that ofnegative polesisreckoned negative,then cwillbeapositive quantity TheUnitMagnetic Pole. 401. Just asCoulomb's electrostatic lawofforcesuppliedaconvenient wayofmeasuringthestrengthofanelectriccharge,sothelawexpressed byequation (328) providesaconvenient wayofmeasuringthestrengthofa magnetic pole,andsogivesasystemofmagneticunits.Asystemofunits, analogoustotheelectrostaticsystem (§§17,18)isobtainedbydefining the unitpoletobesuch astomake c=1inequation (328). Thissystemis called theMagnetic (or,moregenerally, Electromagnetic) systemofunits. Wedefine aunitpole,inthissystem,tobeapoleofstrengthsuch that when placedatunit distance from apoleofequal strengththerepulsion between thetwopolesisoneofunit force. 366 Permanent Magnetism [ch.xi Thus theforceFbetween twopolesofstrengths m,m',measured inthe Electromagnetic systemofunits, isgiven by F=— (329).r2 Thephysicaldimensions ofthemagneticunitcanbediscussed injust thesamewayinwhich thephysicaldimensions oftheelectrostatic unit have alreadybeen discussed in§18. MomentofaLine-Magnet. 402. Itisfound that every positive polehasassociated with ita negative poleofexactly equal strength,and that these twopolesare alwaysinthesamepieceofmatter. Thus notonlyarepositiveandnegative magnetism necessarily brought intoexistence togetherandinequal quantities,asisthecasewithpositive andnegative electricity, but, further, itisimpossibletoseparatethepositive andnegative magnetismaftertheyhavebeenbroughtinto existence, andin thisrespect magnetismisunlikeelectricity. Itfollows that itisimpossibletohaveabody" chargedwithmagnetism" inthewayinwhich wecanhave abody chargedwithelectricity. Amag- netised bodymay possess anynumber ofpoles, andateachpolethereis,in asense, achargeofmagnetism;butthetotalchargeofmagnetisminthe bodywillalwaysbezero. Hence itfollows thatthesimplestandmostfundamentalpieceofmatter wecanimaginewhich isofinterest forthetheoryofmagnetism,isnota small body carryingachargeofmagnetism,butasmallbody carrying (so tospeak)twoequalandopposite chargesatacertain distanceapart. This leads ustointroduce theconceptionofaline-magnet. Aline- magnetisanidealbar-magnetofwhich thewidth isinfinitesimal, the length finite, andthepolesatthetwoextreme ends. Thusgeometrically theideal line-magnetisaline, while itspolesarepoints. .Thestrengthsofthetwopolesofaline-magnetarenecessarily equal andopposite.Theproductofthenumericalstrengthofeitherpoleandthe distance between thepolesiscalled the"moment"oftheline-magnet. Magnetic Particle. 403. Ifweimaginethedistance between thetwopolesofaline-magnet toshrink until itisinfinitesimal, themagnet becomes what isspokenofasa magnetic particle.If±marethestrengthsofitspolesanddsisthedistance between thetwopoles,themoment ofthemagnetic particleismds. 401-403] Physical Phenomena 367 Itiseasily shewn that, asregardsallphenomena occurringatafinite distanceaway,twomagnetic particleshave thesame effect iftheirmoments areequal ;theirlengthandthestrengthsoftheirpoles separatelyareofno importance.Toseethisweneedonlyconsider thecase oftwomagnetic particles,eachhaving poles+m,andlength ds,andtherefore moment mds. Clearlythese willproducethesame effect atfinite distances whetherthey areplacedendtoendorsidebyside. Inthelatter case,wehave amagnet oflength ds,poles+2m,while intheformer casethetwocontiguous poles, beingofopposite sign,neutralise oneanother, andthearrangementisin effect amagnetoflength2dsandpoles+m.Thus ineach casethemoment isthesame, namely 2mds, while thestrengthsofthepolesandtheir distances apartaredifferent. Ifweplacealargenumber nofsimilarmagnetic particles end toend, allthepoleswillneutralise oneanotherexceptthose attheextreme ends, sothatthearrangement producesthesame effect asaline-magnetoflength nds.Bytakingn=-r ,where Iisafinitelength, weseethat theeffect of aline-magnetoflengthIcanbeproduced exactly bynmagnetic particles oflengthds. Thetwoarrangementswillbeindistinguishable bytheirmagneticeffects atallexternalpoints. There is,however, awaybywhich itwould beeasy todistinguishthem. Ifthearrangementweresimply twopoles+m,atthe ends ofawire oflength I,thenoncuttingthewire intotwopieces, weshould have onepoleremainingineachpiece. If,however, thearrangementwere ± — +-+-+~+-+~±—+ ,..,.+—f—b 4--+-+--f-- ——————— —^—^———(in———————— — Fig. 104. that ofaseries ofmagnetic particles, weshould beable todivide theseries between twoparticles,andshould inthiswayobtain twocomplete magnets. Thepairofpolesonthetwo sides ofthepointofdivision which have sofar beenneutralisingoneanother nowfigureasindependent poles. Asamatter ofexperiment,itisnotonlyfound tobepossibletoproduce twocomplete magnets bycuttingasingle magnet between itspoles, but it isfound thattwonewmagnetsareproduced, nomatter atwhatpoint the cuttingtakesplace. The inference isnotonlythatanaturalmagnet must besupposedtoconsist ofmagnetic particles, butalsothat theseparticles aresosmall thatwhen themagnetiscutintwo,there isnopossibilityof 368 Permanent Magnetism [oh.xi cuttingamagnetic particleintwo, sothatonepoleisleftoneach side of thedivision. Inother words, wemustsupposethemagnetic particleseither tobeidentical with themolecules ofwhich thematter iscomposedorelse tobeeven smaller than these molecules. Atthesame time, itwillnot benecessarytolimit themagnetic particleofmathematicalanalysis by assigningthis definitemeaningtoit:anycollection ofmolecules, sosmall that thewholespace occupied byitmayberegardedasinfinitesimal, will bespokenofasamagnetic particle. 404. Axisofamagnetic particle. The axis ofamagnetic particleis defined tobethedirection ofalinedrawn from thenegativetothepositive poleoftheparticle. Itwillbeclear, fromwhat hasalreadybeen said, that the effect of amagnetic particleatallexternalpointsisknown whenweknow its position,axisandmoment. Intensity ofMagnetisation. 405. Inconsideringabar-magnet, which must besupposedtohave breadth aswell aslength, wehave toconsider themagnetic particlesas beingstacked sidebyside aswell asplaced end toend. Forclearness, let ussupposethat themagnetisarectangular parallelepiped,itslength being paralleltotheaxisofx,while itsheight andbreadth areparalleltothetwo other axes. Thepolesofthisbar-magnet maybesupposedtoconsist of auniform distribution ofinfinitesimalmagnetic polesovereach ofthetwo facesparalleltotheplaneofyz,letussayadistribution ofpolesofaggregate strength Iperunit area atthepositive pole,and—/perunit area atthe negative pole,sothat ifAisthearea ofeach ofthese faces, thepolesof themagnetareofstrengths+IA. Asafirststep,wemay regardthemagnetasmadeupofaninfinite number ofline-magnets placedsidebyside, eachline-magnet beinga rectangular prism paralleltothelengthofthemagnet, andofverysmall cross-section. Thus aprismofcross-section dydzmayberegardedasaline- magnet having poles +1dydz. Thisagainmayberegardedasmadeupof anumber ofmagnetic particles. Asatype,letusconsider aparticleof length dx,sothat thevolume ofthemagnet occupied bythisparticleis dxdydz. Thepolesofthisparticleareofstrength ±Idydz,sothat the moment oftheparticleis Idxdydz. Ifwetakeanysmall cluster oftheseparticles, occupyingasmall volume dv,thesum oftheirmoments isclearly Idv,andtheseproduce thesame magneticeffects atexternalpointsasasingle particleofmoment Idv. 403-407] TheMagnetic Field ofForce 3G9 Thequantity /iscalled the" intensityofmagnetisation"ofthemagnet. Themagnetisationhasdirection aswell asmagnitude.Inthepresent instance thedirection isthat oftheaxis ofx. 406. Ingeneral, wedefine theintensity anddirection ofmagnetisation asfollows : Theintensity ofmagnetisationatanypoint ofamagnetised bodyisdefined tobetheratioofthemagnetic momentofanysmallparticleatthispointto thevolumeoftheparticle. Thedirection ofmagnetisationatanypoint ofamagnetised bodyisdefined tobethedirection ofthemagneticaxisofasmallparticle ofmagneticmatter atthepoint. Instead ofspecifyingthemagnetisationofabodyinterms ofitspoles, itisbothmore convenient from themathematicalpointofview, andmore inaccordance with truth from thephysical pointofview, tospecifythe intensityatevery pointinmagnitudeand direction. Thus thebar-magnet which hasbeen under consideration would bespecified bythestatement that itsintensityofmagnetisationatevery pointis/paralleltotheaxis ofx.Abodysuch that theintensityisthesame atevery point, both in magnitude and direction, issaid tobeuniformly magnetised. TheMagnetic Field ofForce. 407. The field offorceproduced byacollection ofmagnetsisinmany respectssimilar toanelectrostatic fieldofforce, sothatthevariousconceptions which werefound ofuseinelectrostatictheorywillagainbeemployed. The first oftheseconceptions wasthat ofelectricintensityatapoint. Inelectrostatic theory,theintensityatanypointwasdefined tobethe force perunitchargewhich would actonasmallcharged particle placed atthepoint.Itwasnecessarytosupposethechargetobeofinfinitesimal amount, inorder that thechargesontheconductors inthe fieldmightnot bedisturbed byinduction. There is,asweshall see later, aphenomenonofmagnetic induction, which isinmany respectssimilar tothat ofelectrostatic induction, sothat indefining magnetic intensity wehaveagaintointroduce acondition to exclude effects ofinduction. Also, toavoid confusion between themagnetic intensityandtheintensity ofmagnetisationdefined in§406, itwillbeconvenient tospeakofmagnetic force atapoint,rather than ofmagnetic intensity. Weaccordingly have the following definition, analogoustothatgivenin§30. j. '24 370 Permanent Magnetism [ch.xi Themagnetic forceatanypointisgiven,inmagnitude and direction, bytheforce perunitstrength ofpole,which would actonamagnetic pole situated atthispoint,thestrength ofthepole being supposedsosmall that themagnetism ofthefieldisnotaffected byitspresence. 408. The otherquantitiesandconceptionsfollow inorder, asin ChapterII.Thuswehave thefollowingdefinitions: Alineofforceisacurve inthemagnetic field,such that thetangentat every pointisinthedirectionofthemagnetic forceatthatpoint (cf.§31). Thepotentialatanypointinthefieldistheworkperunitstrength ofpole which hastobedone onamagnetic poletobringittothatpointfrom infinity, thestrength ofthepole being supposedsosmall that themagnetism ofthefield isnotaffected byitspresence (cf.§33). LetOdenote themagnetic potentialand a,/3,ythecomponentsof magneticforce atanypoint x,y,z,thenwehave from this definition (cf.equation (6)), a=-r'V '\adx+^dy+r/dz) (330), andtherelations(cf.equations (9)), a=-^'^= -dy"7="^( }- Asurfaceinthemagnetic fieldsuch thatatevery pointonitthepotential hasthesame value, iscalled anEquipotential Surface (cf.§35). From this definition, asin§35,follows thetheorem : Equipotential Surfacescutlinesofforceatright angles. Thelawofforcebeingthesame asinelectrostatics, wehave asthevalue ofthepotential (cf.equation (10)), V=2™(332), wheremisthestrengthofany typical pole,andristhedistance from it tothepointatwhich thepotentialisbeingevaluated. Asin§42,wehave Gauss' Theorem : fJ~dS=-4>7r^m(333), where theintegrationisoveranyclosed surface, and%m isthesum of thestrengthsofallthepolesinside this surface. Ifthesurface isdrawn soasnottocutthrough anymagnetised matter, Smwillbetheaggregate strengthofthepolesofcomplete magnetic particles,andthereforeequal tozero. Thus forasurface drawn inthisway d ^dS=(334).//; 407-410] TheMagnetic Field ofForce 371 Ifthepositionofthesurface Sisdetermined bygeometricalconditions— if,forinstance, itistheboundaryofasmallrectangular element dxdydz— thenwecannotsupposeittocontainonlycomplete magnetic particles,and equation (334)willnotingeneralbetrue. Ifthere isnomagneticmatterpresentinacertainregion, equation (334) istrue foranysurface inthisregion,andonapplyingittothesurface ofthe smallrectangularelement dxdydz, weobtain, asin§50, 82oa2nd2n _/oorN^+ a^+^=° ^' thedifferentialequationsatisfied bythemagnetic potentialatevery point ofaregioninwhich there isnomagnetic matterpresent. TubesofForce. 409.Atubular surface boundedbylines offorce is,asinelectrostatics, called atube offorce. Letwl,tw2betheareas ofanytwonormal cross- sections ofathin tube offorce, and letHuH2bethevalues ofthe intensities atthesepoints. ByapplyingGauss' Theorem tothe closed surface formed bythetwo cross-sections and theportionofthetube which liesbetween them, weobtain, asin§56, H1&>i—i72o)2=0, providedthere isnomagneticmatter inside this closed surface. Thus infreespacetheproductHcoremains constant. Thevalue ofthis productiscalled thestrengthofthetube. Inelectrostatics, itwasfound convenient todefine aunittube tobeonewhich ended onaunitcharge,sothattheproductofintensity andcross-section wasnotequaltounity butto4n- PotentialofaMagneticParticle. 410. Letamagnetic particleconsist ofapoleofstrength—m,at0,and apoleofstrength+m xatP,thedistance OPbeing infinitesimal. ThepotentialatanypointQwillbe nQ=PQ~ot(336)' IfweputOQ=r,anddenote theangleP0Qby0, -»«;+ml thisbecomes Flo-1Q5. _m1(0Q-PQ) _<nhOPcos_ficos Uq~~ PQ.OQ~~ PQ.OQ~ ~1^~{6°n' where/x=ml.OP,themoment oftheparticle. 24—2 372 Permanent Magnetism [CH.XI Theanalysisheregivenandtheresult reached areexactlysimilar to those already givenforanelectric doublet in§64.Thesame result canalso beputinadifferent form. LetusputOP=ds,and let^-denote differentiation inthedirection of OP,theaxisoftheparticle.Thenequation (336) admits ofexpressionin theform °.-".*5©-*i(?)(338)- LetI,m,nbethedirection-cosines oftheaxis oftheparticle,then formula (338) canalsobewritten 8m -3/1N -dlM .(339), nQ= fj,dx\r3/1\+m„- -+ oy\rj3 where, indifferentiation, x,y,zaresupposedtobethecoordinates ofthe particle,andnotofthepoint Q. 411. Resolution ofamagnetic particle. Equation (339) shew* thatthe potentialofthesingle particle wehavebeenconsideringisthesame asthe potentialofthreeseparate particles,ofstrengths liI,fimandfin,andaxes in thedirections Ox,Oy,Ozrespectively Thus amagnetic particle maybe resolved intocomponents,andthisresolution follows theusual vector law. Thesame result canbeseengeometrically. Letusstartfrom andmove adistance Idsparalleltotheaxisofx,then adistance mdsparalleltotheaxisofy,andthen adistance ndsparalleltotheaxis ofz.This series ofmovementsbringsusfrom toP,a distance dsinthedirectionI,m,n.Letthe pathbeOqrPinfig.106.Themagnetic particle under consideration haspoles—mxatand+m, atP.Withoutalteringthe field,wecansuper- posetwoequalandopposite poles±mxatq,and alsotwoequalandopposite poles+ni^atr. The sixpolesnowinthe fieldcanbetaken inthreepairssoastoconstitute three doublets ofstrengths m^.Oq,m^.qrandm^rPrespec- tively along Oq,qrandrP. These, however, are doublets ofstrengths fil,fimand/xnparalleltothecoordinate axes. PotentialofaMagnetised Body. 412. LetIbetheintensityofmagnetisation atanypointofamag- netisedbody, and letI,in,nbethedirection-cosines ofthedirection of magnetisationatthispoint. 410-413] TheMagnetic Field ofForce 373 Thematteroccupying anyelement ofvolumedxdydzatthispointwill beamagnetic particleofwhich themoment isIdxdydz andtheaxis isin direction I,m,n.Byformula (339), thepotentialofthisparticleatanv externalpointis 1dx\r)' dy^{l)+n Fz{l)}dx^det sothat,byintegration, weobtain asthepotentialofthewholebodyatany externalpoint Q, nQ= dx\r) dy4©"I©}***<340> inwhich risthedistance fromQtotheelement dxdydz, andtheintegration extends overthewhole ofthemagnetised body. Ifweintroducequantities A,B,Gdefined by A=Il\ B=Im G=In thenequation (340) canbeputintheform 9/1\ .„3t\\ „d.(341), n«= A dx\r dy\r) dz-)>dxdydz. .(342). Thequantities A,B,Garecalled thecomponentsofmagnetisationatthe point x,y,z.Equation (342) shews thatthepotentialoftheoriginal magnet, ofmagnetisation I,isthesame asthepotentialofthreesuperposed magnets, ofintensities A,B,Gparalleltothethree axes. This isalsoobvious from thefactthattheparticleofstrength Idxdydz, whichoccupiestheelement of volume dxdydz, mayberesolved into threeparticles paralleltotheaxes, of which thestrengthswillbeAdxdydz, BdxdydzandGdxdydz,ifJ.,B,Gare given byequations (341). Potential ofauniformly Magnetised Body. 413. Ifthemagnetisationofanybodyisuniform, thevalues ofA,B,G arethesame atallpointsofthebody. Letthecoordinates ofthepointQinequation (342) bex',y',z',sothat i-[(*-x'f+(y-y')'+(s- *')']"K Then, clearly, |(I)=-1,(J),etc. 374 Permanent Magnetism [ch.xi Replacingdifferentiation withrespecttox,y,zbydifferentiation with respecttox,y',zinthisway,wefindthatequation (342) assumes theform n^-{Al+Bh+cM\\dxd^ (343)- 7) ri 7) thequantities A,B,Gandtheoperators r— ;,~—, ,=p,being taken outside the signofintegration,sincetheyarenotaffectedbychangesinx,y,z. IfVdenote thepotentialofauniform distribution ofelectricityofvolume density unity throughouttheregion occupied bythemagnet, wehave VQ=jji^dxdydz(344), sothatequation (343) becomes Q«=-^-#-<#(3«>. or nQ=AX+BY+CZ, where X,Y,ZarethecomponentsofelectricintensityatQproduced by this distribution. Oragainif^-,denotes differentiation withrespecttothecoordinates ofQ inadirectionparalleltothat ofthemagnetisationofthebody, namelythat ofdirection-cosinesI,m,n,equation (345) becomes "«=-^(346). 414. Yetanotherexpressionforthepotentialofauniformly magnetised bodyisobtained ontransforming equation (342) byGreen's Theorem. If V,m,n'arethedirection-cosines oftheoutward-drawn normal tothemagnet atanyelement dSofitssurface, theequationobtained after transformation is nQ= Jf(Al'+Bm'+Cri)ldS. Byequations (341), Al'+Bm'+On'=I(W+mm'+nn') =i"cos8, where 9istheanglebetween thedirection ofmagnetisationandtheoutward normal totheelement dSofsurface. Theequation nowbecomes '/cos 6 O,.//:dS(347), shewingthat thepotentialatanyexternalpointisthesame asthat ofa surface distribution ofmagnetic polesofdensity /cos6perunit area, spread overthesurface ofthemagnet. 413-416]TheMagnetic Field ofForce 375 This distribution isofcourse simplythe"Green'sEquivalent Stratum" (§204)which isnecessarytoproducetheobserved external field. Thebar-magnet alreadyconsidered in§405,providesanobvious illustra- tionofthese results. 415.Uniformly magnetised sphere. Asecond andinteresting example ofauniformly magnetised bodyisasphere, magnetisedwith uniform intensityI.Thisacquiresitsinterest from thefactthattheearthmay,to averyrough approximation,beregardedasauniformly magnetised sphere. Ifwefollow themethod of§313,weobtain forthevalue ofVq,defined byequation (344), where aistheradius ofthesphere.Ifwesupposethemagnetisationtobe inthedirection oftheaxisof#,wehave Thus thepotentialatanyexternalpointisthesame asthatofamagnetic particleofmoment%7ra3Iatthecentre ofthesphere. Totreat theproblem bythemethod of§414,wehave tocalculate the potentialofasurfacedensity 7cos6spreadover thesurface ofthesphere. Regardingcos6asthe first zonal harmonic Px(cos 6),theresult follows at oncefrom§257 Poisson's imaginary Magnetic Matter. 416. ifthemagnetisationofthebodyisnotuniform, thevalue ofQQ giveninequation (342)cannot betransformed intoasurfaceintegral,so that thepotentialofthemagnetcannot berepresentedasbeingdue toa surface chargeofmagneticmatter. IfweapplyGreen's Theorem tothe integralwhich occurs inequation (342), weobtain -- lll\(I+1+s)*** +//;<"+mB+^dS- whereI,m,narethedirection-cosines oftheoutward-drawn normal tothe element dSofsurface. 376 Permanent Magnetism [ch.xi Thus nQ=jjj^dxdydz+jj^dS(348), wherep,aaregiven by fdA dBdC\,_, ft. r=-[te+^+^)<349) ' a=LA+mB +nG (350). Thus thepotentialofthemagnetatanyexternalpointQisthesame as ifthere were adistribution ofmagnetic charges throughouttheinterior, of volume-density pgiven byequation (349), togetherwith adistribution over thesurface, ofsurface-densityagiven byequation (350). Potential ofaMagneticShell. 417.Amagnetised bodywhich issothinthat itsthickness atevery point maybetreated asinfinitesimal, iscalled a" magneticshell."Throughout thesmall thickness ofashellweshallsupposethemagnetisationtoremain constant inmagnitudeanddirection, sothat tospecifythemagnetisationof ashellwerequiretoknow thethickness oftheshellandtheintensity and direction ofthemagnetisationatevery point. Shells inwhich themagnetisationisinthedirection ofthenormal tothe surface oftheshell arespokenofas"normally-magnetisedshells." These form theonlyclass ofmagneticshells ofanyimportance,sothatweshall deal onlywithnormally-magnetised shells, and itwillbeunnecessarytorepeatin everycasethestatement thatnormalmagnetisationisintended. IfIistheintensityofmagnetisationatanypointinside ashell ofthis kind,and ifrisitsthickness atthispoint,theproductItisspokenofas the" strength"oftheshell atthispoint. Anyelement dSoftheshell will behave asamagnetic particleofmoment IrdS, sothatthestrengthofa shell isthemagnetic momentperunit area, justastheintensityofmagneti- sation ofabodyisthemagnetic momentperunitvolume. Anyelement dSofashellofstrength (/>behaves likeamagnetic particle of strength <fidSofwhich theaxis isnormal todS. Themagnetisationofamagneticshellmayoften beconveniently pictured asbeingduetothepresenceoflayersofpositive andnegative poles onits two faces.Clearlyif<f>isthestrength andtthethickness ofashell at any point,thesurface-densityofthesepolesmust betaken tobe+— ,T 418. Toobtain thepotentialofashell atanexternalpoint, weregard anyelement dSoftheshell asamagnetic particleofmoment<f>dSandaxis inthedirection ofthenormal totheshell atthispoint,itbeing agreed that thisnormal must bedrawn inthedirection ofmagnetisationofthe shell. 416-420]Potential Energy 377 Thepotentialoftheelement dSOxtheshell atapointQdistant rfromdS isthen *«4(;)- sothatthepotentialofthewhole shell atQisgiven by -//^« where istheanglebetween thenormal atdSandthelinejoining dStoP. Clearly dScos6istheprojectionoftheelement dSonaplane perpendicular tothelinejoining dStoPsothat—isthesolidanglesubtended by dSatQ.Denotingthisbydco,wehave thepotentialintheform nQ= ffij)dto(351). 419. Uniformshell. Iftheshell isofuniformstrength, $maybetaken outside chesignofintegrationinequation (351),sothatweobtain £lQ= (j>(!dw= (})£l (352), where 11isthetotal solidanglesubtendedbytheshell atQ. Potential Energy ofaMagnet inaField ofForce. 420. Thepotential energyofamagnetinanexternal field offorce is equaltotheworkdone inbringing upthemagnetfrominfinity,the field of force being supposedtoremain unalteredduringtheprocess. Consider firstthepotential energyofasingle particle, consistingofapole ofstrength—7nxatandapoleofstrength +mxatP.Let thepotentialofthefield offorce atbeHandatPbefLP. Then theamounts ofworkdoneonthetwopolesinbringing upthisparticlefrominfinityarerespectively—m^o and mjfip, sothat thepotential energyoftheparticle when in theposition OP =raj(HP—fi) =m1.OP -zr- ,inthenotationalready used, an / 7an anam ,_ s 378 Permanent Magnetism [CH.XI Thepotential energyofanymagnetised bodycanbefound byintegration ofexpression (353),thebody being regardedasanaggregationofmagnetic particles. 421. Equation (353) assumes aspecialform ifthemagneticfield isdue solelytothepresenceofasecond magnetic particle.Letthisbeofmoment fi,itsaxishavingdirection cosines V,m!,n',and itscentrehavingcoordinates x',y,z'.Thenwehave asthevalue ofX2,from§410, °-'»£M'»+<£+4)(?) : Substitutingthese values forOintheformulaejustobtained, wehave as themutualpotential energyofthetwomagnets, 32(V dsds' \r, 11+iA/V— +'—+'—V- dxdybzj\dx'dy' dz'J\rj This issymmetricalwith respecttothetwomagnets,asofcourse itoughttobe—itis immaterial whether webringthe firstmagnetintothefield ofthesecond, orthesecond intothe fieldofthe first. Ifwenowput r{(x-xy +(y-yj +(z-z'y}^ weobtain ondifferentiation, 3/1\_x—x' _x—x dx'\r)~ {(a._xy+(y_yy+(Z_zyfi~ r3 82/1\ 1S(x-x'Ysothat dx'dx' \rj r° r ^/l^_ S(x-x')(y-y') r5dydx' \r Hence weobtain asthevalue ofW, W=^{U'+mm'+nn'),etc. fy*d 2>fAIJL [l(x-x')+m(y-y')+n{z- z')}[V(x-x')+m'(y-y')+n'(z-z')}. Letusnowdenote theangle between theaxes ofthetwomagnets by e, andtheanglesbetween thelinejoiningthetwomagnets andtheaxes ofthe firstandsecond magnets respectively by8and 6'.Then cose—ll'+ mm!+nn', cos6=- {I(x~x')+m(y—y')+n(z- z')}, cosd'= ^{l'(x-x')+m'(y-y') +n'(z-z')} 420-422]Potential Energy379 sothatWcanbeexpressedintheform F=^(cose-3cos0cos6>')(354). Ifwetake thelinedrawn from the firstmagnettothesecond aspolein spherical polar coordinates, anddenote theazimuths oftheaxes ofthetwo magnets by yfr, yjr',then thepolarcoordinates ofthedirections oftheaxes of thetwomagnetswillbe0,y{rand0', yfr'respectively,andweshallhave cose=cos cos&+sin sin0'cos(yjr— yjr'). Onsubstitutingthisvalue forcoseinequation (354),weobtain W=&£{sinsinffcos(f-f)-2cos cosff) (355). 422. Knowingthemutualpotential energy W,wecanderive aknow- ledgeofallthemechanical forces bydifferentiation. For instance the repulsion between thetwomagnets,i.e.theforcetendingtoincreaser,is -wordW ,or r4{sinsin&cos(yjr— yjrf )—2cos cosff). Thus, whatever thepositionofthemagnets,theforce between them varies astheinverse fourth powerofthedistance. Ifthemagnetsareparalleltooneanother, =0'andyjr= yjr',sothatthe repulsion r*(sin20-2 cos2 0). Thuswhen #=0,i.e.when themagnetsliealongthelinejoining them, theforce isanattractive force-~- .When =k,sothatthemagnetsare q' atright anglestothelinejoining them, theforce isarepulsiveforce—— . Inpassingfrom theonepositiontotheother theforcechanges from oneof attraction tooneofrepulsion when sin2—2cos2=0,i.e.when =tan-1 *J2. Thecouplescanbefound inthesameway.If^isanyangle,thecouple tendingtoincrease theangle%is—-~— ,or -^-k- {sinsin&cos(^- •«//)-2cos cos0% sothat allthecouples vary inverselyasthecubeofthedistance. 380 Permanent Magnetism [ch.xi Forinstance, taking x*°Dethesame asi/r,wefind that thecouple tendingtorotate the firstmagnetabout thelinejoiningittothesecond, inthedirection oftyincreasing sothat thiscouplevanishes ifeither ofthemagnetsisalongthelinejoining them, oriftheyareinthesameplane,results which areobvious enough geometrically. Potential Energy ofaShell inaFieldofForce. 423. Consider ashell ofwhich thestrengthatanypointiscj),placed inafield ofpotentialO.Theelement dSoftheshell isamagnetic particle ofstrength <j>dS,sothat itspotential energyinthe field offorce will,by formula(353), be *<• where »-denotes differentiationalongthenormal tothe shell. Thus the potential energyofthewhole shell willbe W^JJ^dS(356). Iftheshell isofuniformstrength,thismaybereplaced by ^-//i<857>- Since thenormal componentofforce atapoint justoutside theshell andonitspositiveface is—^—,itisclear thatIj-^-dSisequaltominus thesurfaceintegralofnormal force taken overthepositiveface oftheshell, and thisagainisequaltominus thenumber ofunittubes offorcewhich emergefrom theshell onitspositiveface.Denotingthisnumber ofunit tubes byn,equation (357)maybeexpressedintheform W=-<f>n (358). Here itmust benoticed thatweareconcernedonlywith theoriginal field before theshell issupposed placedinposition. Or,inother terms, the number nisthenumber oftubes which would cross thespace occupied by the shell, iftheshell were annihilated. Since thetubes arecounted onthe positiveface ofthe shell,weseethatnmayberegardedasthenumber of unittubes oftheexternal fieldwhich cross theshell inthedirection ofits magnetisation. 1dxdydz u -^—\-m-~- +n^-)422-426] Force inside aMagnetised Body 381 424. Consider afieldconsisting onlyoftwoshells, each ofunitstrength. Let n^bethenumber oftubes from shell 1which cross theareaoccupied by2,and letn%bethenumber oftubes from shell 2which cross thearea occupied by1.Thepotential energyofthe fieldmayberegardedasbeing either theenergyofshell 1inthe field setupby2,orastheenergyof shell 2inthe field setupby1.Regardedinthe firstmanner, theenergy ofthe field isfound tobe—n2;regardedinthesecond manner, theenergy isfound tobe—Jij.Hence weseethatni=n2.This result, which is ofgreat importance,willbeobtainedagainlater(§446) byapurely geometrical method. Potential Energy ofanyMagnetised BodyinaMagnetic FieldofForce. 425. Let/betheintensityofmagnetisation andI,m,nthedirection- cosines ofthedirection ofmagnetisationatanypoint x,y,zofamagnetised body,andlet12bethepotential,atthispoint,ofanexternal field ofmagnetic force. Theelement dxdydzofthemagnetised bodyisamagnetic particle ofstrength Idxdydz,ofwhich theaxis isinthedirectionI,m,n.Thus its potential energyinthefield offorceis,byformula(353), dy andbyintegrationthepotentialofthewholemagnetis Force inside aMagnetised Body. 426 Sofarthemagneticforce hasbeen defined anddiscussedonlyin regionsnotoccupied bymagnetisedmatter :itisnownecessarytoconsider themore difficultquestionofthemeasurement offorce atpointsinside a magnetised body. Attheoutset weareconfronted with adifficultyofthesame kind as thatencountered indiscussingthemeasurement ofelectric force inside a dielectric, onthemolecularhypothesis explainedin§143.Wefound that themolecules ofadielectric could beregardedaseachpossessing twoequal andopposite chargesofelectricityontwooppositefaces. Ifwereplace " electricity" by"magnetism"thestate isverysimilar towhatwebelieve tobethestate oftheultimate magnetic particles.Intheelectricproblem adifficultyarose from thefactthat theelectric force inside matter varied rapidlyaswepassedfrom onemolecule toanother, because theintensityof the field setupbythechargesonthemolecules nearest toanypoint was 382 Permanent Magnetism [ch.xi comparablewith thewhole field.Asimilardifficultyarises inthemagnetic problem,butwillbehandled inaway slightlydifferent from thatpreviously adopted.There aretworeasons forthisdifference oftreatment—inthe first place, wearenotwillingtoidentifytheultimatemagnetic particleswith themolecules ofthematter, and inthesecondplace,wearenotwillingto assume that themagnetismofanultimateparticle maybelocalised inthe form ofchargesonthetwooppositefaces.Weshall follow amethod which rests onnoassumptionsastotheconnection between molecular structure andmagnetic properties, beyondthewell-established factthatoncutting amagnetnewmagnetic poles appearonthesurfaces createdbycutting. 427.Onewayofmeasuringtheforce atapointQinside amagnetwill betoimagineacavity scoopedoutofthemagneticmatter soastoenclose thepoint Q,andthen toimaginetheforce measured onapoleofunit strength placedatQ.Thismethod ofmeasurement willonlydetermine adefinite force atQifitcanbeshewn that theforce isindependentof theposition, shapeand sizeofthecavity, and this, aswillbeobvious from what follows, isnotgenerallythe case. 428. Letussuppose that, inorder toform acavityinwhich toplace theimaginaryunitpole,weremove asmallcylinderofmagnetic matter, the axis ofthiscylinder beinginthedirection ofmagnetisationatthepoint. LetthiscylinderbeoflengthIandcross-section S,and lettheintensityof magnetisationatthepointbe/.Letthesizeofthecylinder besupposedto beverygreatincomparisonwith thescale ofmolecular structure, although verysmall incomparisonwith thescale ofvariation inthemagnetisation ofthebody. Insteel oriron there areroughly1023molecules tothecubiccentimetre, sothata lengthof1millimetre mayberegardedaslargewhen measured bythemolecularscale, althoughinmostmagnetsthemagnetisation maybetreated asconstant within alength ofamillimetre. Atapointnear thecentre ofthiscavity weareatadistance from the nearestmagnetic particles,which is,byhypothesis, great compared with molecular dimensions. Hence, by§416,wemay regardthepotential at pointsnear thecentre ofthecavityasbeingthatdue tothefollowing distributions ofimaginary magneticmatter.— I.Adistribution ofsurface-density IA+mB+nC,spread over the surface ofevery magnet. II.Adistribution ofvolume-density fdA dBdodBdC\ dy+ dzj' \dx dy spread throughoutthewholespacewhich isoccupied bymagnetic matter after thecavityhasbeenscoopedout. 426-430] Force inside aMagnetised Body 383 III.Adistribution ofsurface-density IA+mB+nO,spreadover the walls ofthecavity. From thewayinwhich thecavityhasbeen chosen, itfollows that IA+mB+nCvanishes over the side-walls, and isequalto+1onthe two ends. The forceactingonanimaginaryunitpole placedatornear the centre ofthecavity mayberegardedasthe forcearisingfrom these three distributions. 429. The force from distribution IIIcanbemade tovanish bytaking thelengthofthecavitytobevery greatincomparisonwith thelinear dimensions ofitsends. Fortheends ofthecavity maythen betreated as points,andtheforce exerted byeither enduponaunitpoleplacedatthe centre ofthecavitywillbe SI andthis willvanish ifSissmallcomparedwith I2 .The resultant force will therefore arisesolelyfrom distributions Iand II. The forcearisingfrom distribution IImayberegardedasthe force arisingfrom adistribution ofvolume-density fd_AdBdC Vdxdy dz , spread throughoutthewhole ofthemagnetised matter, regardlessofthe existence ofthecavity, togetherwith adistribution ofvolume-density \dx dydz spread throughthespace occupied bythecavity. The force from this latter distribution vanishes inthelimitwhen the size ofthecavityis infinitesimal, sothat the force from distribution IImayberegardedas thatfrom avolume-density fd_Ad_B d_G \dxdydz spread throughalltheoriginal magnetisedmatter. Wehavenowarrived ataforcewhich isindependentoftheshape,size andpositionofthecavity, provided onlythat thesesatisfytheconditions which havealreadybeen laiddown. This forcewedefine tobethemagnetic force,atthepointunder discussion, inside themagnetised body. 430. Inthenotation of§416,theforcewhich hasjustbeen defined is duetoadistribution ofsurface-density a,andadistribution ofvolume-density 384 Permanent Magnetism [ch.xi pthroughoutthewhole magnetisedmatter. Thepotentialofthese distribu- tions is jj'dS+ jjfP-dxdych, orClQifweregardthis asdefined byequation (348). Thus, with this meaning assignedto£Iq,thecomponentsofforce atapointQinside a magnetic bodywillbe _d^Q _BJIqdnQ dx' dy*dz Atthesame time itmust beremembered thatQqhasnotbeenshewn to bethetruevalue ofthepotential except when thepointQisoutside the magneticmatter. The truepotentialinside magnetisedmatter willvary rapidlyaswepassfrom onemagnetic particletoanother. 431. LetusnextsupposethatthelengthIofthecylindrical cavityis verysmall comparedwith thelinear dimensions ofan end.The force, asbefore, isthatduetothedistributions I,IIandIIIof§428. The force from distribution III, however, willnolonger vanish, forthisdistribution con- sists ofdistributions +/over theends ofthecavity,Fid 1ORandtheforce from these isnotnownegligible. From analogywith thedistribution ofelectricityonaparallel plate condenser, it isclear that theforcearisingfrom distribution III isaforce 4>7rl inthe direction ofmagnetisation. The forces from distributions Iand IIare easilyseen tobethesame asintheformer case. Thus theforceonaunit poleplacedatapointQinside acavityofthekindwearenowconsidering istheresultant of (i)themagneticforce atQ,asdefined in§429, (ii)aforce 4nrlinthedirection oftheintensityofmagnetisationatQ. Theresultant ofthese forces iscalled themagnetic induction atQ. 432. Themagneticforce willbedenoted byH,and itscomponents bya,/3,7. Theinduction willbedenoted byB,and itscomponents bya,b,c. Wehave seen that theforceBistheresultant ofaforceHandaforce 47r/. Thecomponentsofthis latter force are\irA, 4nrB, 4nrG. Hence we have theequations a=a+4>7tA' 6=/3+4tt5J- (359). c=<y+4<7rG 430-434] Force inside aMagnetised Body385 Fig. 109.433. Letusnext consider theforce onaunitpoleinside acylindrical cavity when thecavityisdisc-shaped,asin§431,but its axis isnotinthedirection ofmagnetisation.The force can, asin§428,beregardedasarisingfrom three distributions. Distributions Iand IIarethesame asbefore, but distribution III willnow consist ofchargesboth onthe endandontheside- walls ofthecylinder. Bymakingthe lengthofthecylindersmall incomparisonwith thelinear dimensions ofitscross-section, theforce from the distri- bution intheside-walls canbemade tovanish. And if istheangle between theaxis ofthecavity and thedirection ofmagnetisation,the distribution ontheends isoneofdensity+Icos0.Thus theforcearising from distribution III isaforce 4nrlcos inthedirection oftheaxis of thecavity. Thus theforce onapoleplacedinside thiscavity mayberegardedas compoundedoftheforceH(arisingfrom distributions IandII),andaforce 4<7rlcos inthedirection ofmagnetisation, arising from distribution III. Let ebetheanglebetween thedirection oftheforceHandtheaxisof thecavity,thenthecomponentforce inthedirection oftheaxisofthecavity =Hcose+4s7rlcos0. IfI,m,narethedirection-cosines ofthis lastdirection, Hcose=la.+m/3+ny, 4-77-7cos=4nr(lA+ mB+nO), sothat,byequations (395), Hcose+477-1cos=la+mb+nc. Thus thecomponentoftheforce inthedirection oftheaxisofthecavity isthesame asthecomponent,inthesame direction, ofthemagneticinduc- tion,namelyla+mb+nc. 434.Wearenow inapositiontounderstand theimportanceofthe vector which hasbeen called theinduction. This arisesentirelyfrom the propertyoftheinduction which isexpressedinthefollowingtheorem : Theorem. Thesurface-integral ofthenormal component ofinduction, taken overanysurface whatever, vanishes, orinother words(cf.§177), field.Theinduction isasoleuoidal vector throughoutthewholeofthemagnetic 25 386 Permanent Magnetism [CH.XI Toprovethis letustakeanyclosed surface Sinthe field, this surface cutting anynumber ofmagnetisedbodies.Alongthosepartsofthesurface which areinside magnetic bodies, letusremove alayerofmatter, sothatthe surface nolonger actually passes through anymagnetic matter. Fig. 110. Then byGauss' Theorem(§409), ms=o.(360), where JSfisthecomponentofforce inthedirection oftheoutward normal to S,actingonaunitpoleplacedatanypointofthesurface S.This force, however, isexactlyidentical with that considered in§433,and itsnormal componenthasbeen seen tobeidentical with thenormal componentofthe induction. Thus iV,inequation (360),willbethenormal componentof induction, sothat thisequation provesthetheorem. Analytically,thetheorem maybestated intheform lf(la+mb+nc)dS=(361), and this,byGreen's Theorem(§179),isidentical with (362).II da db dc dxBydz 435. Definition. Byalineofinduction ismeant acurve inthe magnetic field such that thetangentatevery pointisinthedirectionof themagneticinduction atthatpoint. Definition. Atubeofinduction isatubularsurface ofsmall cross- section, which isboundedentirely bylinesofinduction. Byaproof exactlysimilar tothat of§409, itcanbeshewn that the productoftheinduction andcross-section ofatube retains aconstant value alongthetube. Thisconstant value iscalled thestrengthofthetube. 434-437] Force inside aMagnetised Body 387 Infreespacethelinesandtubes ofinduction become identical with the linesandtubes offorce, andtheforegoingdefinition ofthestrengthofatube ofinduction issuch astomake thestrengthsofthetubes alsobecome identical. 436. Atanypointofasurface letBbetheinduction, and letebethe anglebetween thedirection oftheinduction andthenormal tothesurface. Theaggregatecross-section ofallthetubes whichpassthroughanelement dSofthissurface isdScose,sothattheaggregate strengthofallthese tubes isBcosedS. SinceBcose=N,whereNisthenormal induction, thismay bewritten intheformNdS.Thus theaggregate strengthofthetubes of induction which crossanyarea isequalto NdS. This,wemay say,isthenumber ofunit-tubes ofinduction which cross thisarea. Thetheorem thatfjNdS=0, where theintegrationextends over aclosed surface, maynowbestated in theform thatthenumber oftubes which enteranyclosed surface isequal tothenumber which leave it.This istruenomatter where thesurface issituated, sothatweseethat tubes ofinduction canhave nobeginning orending. 437. Letustakeanyclosed circuit sinspace,and letnbethenumber oftubes ofinduction whichpassthroughthis circuit inaspecifieddirection. Then nwill alsobethenumber <3ftubes which cutanyareawhatever which isbounded bythecircuit s.IfSisanysuch area, thisnumber is known tobelllfdS, where theintegrationistaken overthearea S,sothat "*-//NdS. Thenumber n,however, depends onlyonthepositionofthecurve sby which theareaSisbounded, sothat itmust bepossibletoexpress nina formwhich depends onlyonthepositionofthecurves,andnotonthearea S. Inother words,itmust bepossibletoreplace11NdSbyanexpression which depends onlyontheboundaryofthearea s.Thisweareenabled todoby atheorem duetoStokes. 25—2 388 Permanent Magnetism [CH.XI Stokes' Theorem. 438. Theorem. IfX,Y,Zarecontinuousfunctions ofpositioninspace, then [(X^+Y^ +ZpldsJ\as as as/ 'fdZ3F\ dX dzD+-£-£)**•••<** where thelineintegralistaken round anyclosed curve inspace, andthesurface integralistaken overanyarea(orshell) boundedbythecontour. HereI,m,narethedirection-cosines ofthenormal tothesurface. A rule isneeded tofixthedirection inwhich thenormal istobedrawn. The followingisperhapsthesimplest. Imaginetheshell turned about inspace sothatthetangent planeatanypointPisparalleltotheplaneofxy,and sothatthedirection inwhich thelineintegralistaken round thecontour isthesame asthat ofturningfrom theaxis ofcctotheaxisofy.Then thenormal atPmust besupposed drawn inthedirection ofthepositive axisofz. 439. Toprovethetheorem, letusselectanytwopoints A,Bonthe contour, and letusintroduce aquantity /defined by IX/O U/o too//=B thepathfromAtoBbeingthesame asthat followed intheintegralof equation (363). Letusalso introduce aquantity Jequaltothesame Fig. ill. integraltaken fromAtoB,butalongtheopposite edgeoftheshell. Th thewhole integralontheleftofequation (363)isequaltoI—J.en 438,439] Stokes' Theorem 389 Itwillbepossibletoconnect AandBbyaseries ofnon-intersecting linesdrawn intheshell insuch awayastodivide thewhole shell into narrowstrips. Letusdenote these linesbytheletters a,b,...n,thelines beingtaken inorder across the shell, startingwith thelinenearest tothat alongwhich weintegrateincalculating/.Letusdenote thevalue of Bf„dx „dy „dz\ T1X-j'+Y-f+Z-j-idsas ds dsj /;( takenalongthelineabyIa. Then theleft-hand member ofequation (363) =I-J =(I-I a)+(T m-Ib)+(Ib-Ie)+ ...+(/„-J> Letusconsider thevalue ofanyterm ofthisseries, sayIa—Ib. Letustake eachpointonthelineaandcause ittoundergoaslight displacement,sothat thecoordinates ofanypoint x,y,zarechangedto x+8x,y+8y,z+8z. If8x,8y,8zarecontinuous functions ofx,y,zthe result willbetodisplacethelineaintosomeadjacent position,andbya suitable choice ofthevalues of8x,8y,8zthisdisplaced positionoflineacan bemade tocoincide with line b.Ifthis isdone, itisclear that thevalue of Ia,afterreplacing x,y,zbyx+8x,y +8y,z+8z,willbeIb.Hence ifwe denote thisnewvalue ofIabyIa+81,weshallhave Ia+8I=Ib, sothat Ia—Ib=—81 Ja\ds ds ds) andthevalue ofthisquantitycanbeobtained bytheordinaryrules ofthe calculus ofvariations. "Wehave rB fjT rB(jr rBrj 8X^ds= 8XCpds+ X~{8x)ds JAds JA ds JAdsx (B/dx ax dxs\dx , , =]AteSx+dtBy+te8 Vds-ds+X8x X8xBrBjx— I—r—8xds,A JAds B maybeomitted, andsince 8xvanishes both atAandB,theterm andthewholeexpression putequalto [B \(dAs—8dX-8zy \—-(———^+——)Sx\ds Ja\\dx dyydz Jds\dxdsdyds dzds)j 390 Permanent Magnetism oragain,onsimplifying,to[CH.XI /.AM (S,j*>-sM -d* fa*-S,p)\els. \dy \Jds dsj dz\ds dsj) ThismayDewritten intheform \jr~(tydx"~Bxdy)—-x—(&ccfo—&3cfcc)L /;.(364). Fig. 112. Now infig.112, letP,Q,P'bethepoints x,y,z;x+dx,y+dy,z+dz; andx+Sx,y+By,z+Bz.LetdSdenote thearea oftheparallelogram PQQ'P', and letI,m,nbethedirection-cosines ofthenormal toitsplane. Then theprojectionoftheparallelogramontheplaneofxywillbeofarea ndS, while thecoordinates ofthree ofitsangular pointswillbex,y\x+dx, y+dy;andx+8x,y+By.Usingtheusual formula forthearea,weobtain ndS=(Bydx—Bxdy), andusingthisrelation inexpression (364),weobtain 8JB X^ds=j(^ndS-^mdS)(365),Jy theintegral denotingsummation over allthose elements ofarea oftheshell which liebetween linesaand b. typeof(365),weobtain "BdxBysummation ofthreeequationsofthe [BdxIa—Ib=—8 IX-Y~ds—B Jads ds i +hrA dX dzBd? ZT-dsds ~)^dS+dxd 4)ndS where theintegrationhasthesamemeaningasbefore. Ifweaddasystem ofequationsofthistype,one foreachstrip,theleft-hand, asalready seen, becomes I—J,which isequaltotheleft-hand member ofequation (363), while theright-hand member ofthenewequationisalsotheright-hand member ofequation (363). Thisprovesthetheorem. 439-441] Stokes7Theorem 391 440. Stokes' Theorem canbereadily expressedinavector notation. If X,Y,Zarethecomponentsofanyvector F,itisusual todenotebycurlP thevector ofwhich thecomponentsare dZ_dY dX_dZ d_Y_dX dydz'dzd%' dady' Hence Stokes' Theorem assumes theform J'(componentofFalong ds)ds =/(componentsofcurlPalongnormal todS)dS. Thetheorem enables ustotransformanylineintegral taken round a closed circuit intoasurfaceintegraltaken overanyareabywhich thecircuit canbefilledup.Theconverseoperationofchangingasurfaceintegralinto alineintegral mayormaynotbepossible. 441. Theorem. Itwillbepossibletotransformthesurface integral \(lu+mv+niv)dS (366)//< intoalineintegral taken round thecontouroftheareaSif,andonly if, du dv div ai+^+^=(367) atevery point ofthearea S. Itiseasytoseethat thiscondition isanecessaryone.LetS'denoteany areahavingthesameboundaryasS,andbeing adjacenttoit,butnot coincidingwith it.Then ifIisthelineintegralintowhich thesurface integralcanbetransformed, wemust have I=jj(lu+mv+nw)dS (368), andalso I=(((I'u+m'v+n'w)dS'(369). Onequatingthese twovalues for i"weobtain anequationwhich maybe expressedintheform ff(lu+mv+niv)dS=(370), where theintegrationisover aclosed surface bounded bySand S',and I,m,narethedirection-cosines oftheoutward normal tothesurface atany point. Fromequation (370), thenecessityofcondition (367)follows atonce. Condition (367)ismosteasily provedtobesufficient byexhibitingan actual solution oftheproblem when thiscondition issatisfied. Wehave to 392 Permanent Magnetism [CH.XI shew that, subjecttocondition (367) being satisfied, there arefunctions X,Y,Zsuch that d_Z_d7 =\ dydz dX_d_Z dzdx dY_d_X dx dyj=vV =w•(371), forifthis isso,therequiredlineintegralis(IX+mY+ nZ)dS. Byinspectionasolution ofequations (371)isseen tobe X=fvdz, Y=-fudz,Z=(372), dudv\ 7fdw 7 7T- )dz=-^dz=w,dxdy! Jdzforitisobvious thatthe firsttwoequationsaresatisfied, andonsubstituting inthethird,weobtain d_Y_d_X_ dxdy shewingthattheproposedsolution satisfies alltheconditions. 442. Theabsence ofsymmetryfrom solution(372) suggeststhat this solution isnotthemostgeneralsolution. Themostgeneralsolution can, however, beeasilyfound. Ifweassume ittobe X=jvdz+X', Y=-judz+Y',Z=Z' thenwefind,onsubstitution inequations (371), thatwemust have ar_aF az;=az' dY'_dxr * dz'.(373), .(374),dydz'dz dx'dxdy and ifweintroduce anewvariable %defined by%=jX'dx,wefind atonce that dx' dy' dz' sothatthemostgeneralsolution ofequations (371)is *~h+&Y=~h^yzJi<375>- Substitutingthese values, thelineintegralisfound tobe dy ds+f^ds, ![{hz)-£-(hz )ds_ andthecondition that thisshallbeequaltothesurfaceintegralisthat orthatxshallbesingle-valued. 441-444] Vector-Potential 393 Thus ifxisanysingle-valued function, equations (375) representasolu- tion,andthemostgeneral solution, ofequations (371). Vector-Potential. 443. The discussion astothetransformation from surface tolineinte- gralsarose inconnection with theintegral jjJ^dSor I J(la+mb+nc)dS,in which a,b,carethecomponentsofmagneticinduction. Since thecondition da db dc_„ dxdydz issatisfiedthroughoutallspace,itmustalways bepossible (cf.§441)to transform thesurfaceintegralintoalineintegral byarelation oftheform Jfoa+mb+nc)dS=f(F^+G^+H^jds. Thevector ofwhich thecomponentsareF,G,Hisknown asthemagnetic vector--potential. From what hasbeen said in§442,itisclear thatthevector-potentialis notfullydetermined when themagneticfield isgiven. Ontheother hand, ifthevector-potentialisgiventhemagneticfield isfully determined", being given bytheequations =d_H_dG\ dydz dFdH dz dx _dG_dF dx dy> (376). We shall calculate somepossiblevalues ofthecomponentsofvector- potentialinafewsimplecases. Itmust beremembered that thevalues obtained, althoughsolutions ofequations (376),willnotbethemostgeneral solutions. MagneticParticle. 444. Letusfirstsupposethatthefield isproduced byasingle magnetic particleatthepoint x',y',zinfreespace, paralleltotheaxis ofz.Then, 7)/I\ byequation (338),Q,=fx^-> (- ],sothat atanypoint x,y,z, dn a2/i\ aj/ia=a=— andsimilarlydx dxdz'WJ dxdz\7' b= ^dyTz[-r)'C=^aT2lr 394 Permanent Magnetism [OH.XI Theequationstobesolved(equations (376)) are dH_dG =J!_/1N dydz dxdz \r. dF_d_H =_8*_/1> dz dx dydz\r, dG_dF_ &_(V dx dydz2\rj andthesimplest solution, similar tothatgiven byequations (372),is F=fi G=-f*dxH=0."' dy[r. Thecomponentsofvector-potentialforamagnet paralleltotheaxes of xorycanbewritten down fromsymmetry.Interms ofthecoordinates x',y,z'ofthemagnetic particle,thissolution maybeexpressedas F=- dy4)-G=/idx'H=0. 445. Letussuperposethe fields ofamagnetic particleofstrength l\i paralleltotheaxisofx,oneofstrength mpparalleltotheaxis ofy,and oneofstrength n/j,paralleltotheaxis ofz.Thenweobtain thevector- potentialatx,y,zduetoamagnetic particleofstrength fiandaxis(I,m,n) atx,y\z'intheforms 1_JL^1«.f—-n— dz dy)r \dz'dy' d_d dxF=—fx(m dz)r1/9 d\ H=H*7d d\i /,a a dydx]r \oydx1\ r i>r 1 rI....(377). Thenumber oflines ofinduction which cross thecircuit fromamagnetic particleis(§437) /(F^+G^+H^ds,ds whichmaybewritten intheform dx ds' I,ds dy ds3 m, d_ dx\r)' dy\r)'ds) dz ds n dz\rds, theintegral being taken round thecircuit inthedirection determined bythe rulegivenin§438(p.388). 444-446] Vector- Potential 395 Uniform MagneticShell. 446. Next letussupposethatthelines offorceproceedfrom auniform magnetic shell, supposedforsimplicitytobeofunitstrength. Let V,m,n' bethedirection-cosines ofthenormal toanyelement dS'ofthis shell. Then theelement dS' willbeamagnetic particleofmoment dS'andof direction-cosinesI',m',n.Theelementaccordinglycontributes toFaterm which, byequations (377),isseen tobe K-4)(£K dy'. where x,y',zarethecoordinates oftheelement dS'. Thus thewhole value ofFis *=//K4-4)(>'- This surfaceintegralsatisfies thecondition of§441, sothat itmust be possibletotransform itintoalineintegraloftheform Theequations giving /,g,hare Clearlyasolution isdh dy'' 396 Permanent Magnetism [ch.xi Ifeistheanglebetween thetwoelements ds,ds',thedirection ofthese elements beingtaken tobethat inwhich theintegrationtakesplace, then dxdxdydy' dzdz_ dsds dsds' dsds'' ffCOS6 sothat n=11 dsds'. From theruleastodirectionsgiven onp.388, itwillbeclear that ifthe integrationistaken inthesame direction round both circuits, then the direction inwhich thenlines cross thecircuit willbethatofthedirection ofmagnetisationoftheshell. Clearly nissymmetricalaoregardsthetwo circuits sands',sothatwe have theimportantresult : Thenumberoftubesofinductioncrossingthecircuit sfromashellofunit strength boundedbythecircuit s'isequaltothenumberoftubesofinduction crossingthecircuit s'fromashellofunitstrength boundedbythecircuit s. Herewehave arrived atapurely geometrical proofofthetheorem alreadyobtained fromdynamical principlesin§424. Energy ofaMagnetic Field. 447. Let a,b,c,...nbeasystemofmagnetised bodies, themagnetisation ofeachbeing permanent,and letussupposethat thetotalmagneticfield arisessolelyfrom these bodies. Letussupposethatthepotentialflatany pointisregardedasthesumofthepotentialsduetotheseparate magnets. Denotingthese byI2a,H&, ...fln,weshallhave Letusdenote thepotential energyofmagnet a,whenplacedinthefield offorce ofpotential H,by£2(a) ;ifplacedinthe field offorcearisingfrom magnetbalone, byH&(a),etc. Letusimaginethatweconstruct themagneticfieldbybringing upthe magnets a,b,c,...ninthisorder, frominfinitytotheir finalpositions. Wedonowork inbringing magnetaintoposition,forthere areno forcesagainstwhich work canbedone. After theoperationofplacingain position,thepotentialofthefield isf!a.Theoperationofbringing magnet afrominfinityhasofcourse beensimplythat ofmovingafield offorce of potentialflafrominfinity,where thissame field offorce hadpreviously existed. Onbringing upmagnet b,thework done isthatofplacing magnetbin afield offorce ofpotentialI2a.Theworkdone isaccordingly Qa(b). 446-448] Energy ofaMagnetic Field 397 Theworkdone inbringing upmagnetcisthat ofplacing magnetcina field offorce ofpotential£la+n6.Itistherefore fla(c)+n6(c). Continuingthisprocess wefindthatthetotalwork done, W,isgiven by w=na(b) +na(C)+n6(C) +aa(d)+nb(d)+nc(d)+etc. If,however, themagnetshadbeenbrought upinthereverse order, we should havehad W=nb(a)+Hc(a)+Qd(a)+...+nn(a) +nc{b)+nd(b)+...+n n(b) +£id(c)+...+n n(c) +etc. sothatbyaddition ofthese twovalues forW,wehave 2W= nb(a)+nc(a)+£l d(a)+...+nn(a) +aa(b) +nc(b)+nd(b)+...+nn(b) +na(c)+n6(c) +nd(c)+...+n„(c) +na(d)+nb(d)+nc(d) +...+nn(d) +etc. The first line isequaltoQ(a)exceptfortheabsence oftheterm Q,a(a), andsoonfortheother lines. Thuswehave 2W= ft(a)-n B(a) +H(6)-nb(b)+etc. =2f2(a)-2n a(a) (378). Thequantity Qa(a),thepotential energyofthemagnet ainitsown field offorce,ispurelyaconstant ofthemagnet a,being entirely independent ofthepropertiesorpositionsoftheothermagnets b,c,d,....Thus in equation (378), wemay regardthetermXHa(a)asaconstant, andmay replacetheequation by W-^tCl (a)+constant(379). 448. Ifwetake themagnets a,b,c,...ntobetheultimatemagnetic particles,thevalues ofOa(a),H6(6),...etc. allvanish, andtheirsum also vanishes. Thusequation (379) assumes theform W=\%£l{a) (380), where thestandardconfigurationfromwhichWismeasured isoneinwhich theultimateparticlesarescattered atinfinity. Thevalue ofH(a)fora single particleis(cf.§420) an an an> /7ai2 an en\ 398 Permanent Magnetism [ch.xi Onreplacing/u,byIdxdydz, wefind fortheenergyofasystemof magnetisedbodies -*£+J,f+a£)***<381>- theintegration beingtakenthroughoutallmagnetisedmatter. 449.Analternativeproofcanbegivenofequations (380) and(381), followingthemethod of§106, inwhich weobtained theenergyofasystem ofelectriccharges. Outofthemagneticmaterials scattered atinfinity,itwillbepossibleto construct nsystems,eachexactlysimilar asregards arrangementinspaceto thefinalsystem,butofonlyone-nth thestrengthofthe finalsystem.Ifn ismadevery great,itiseasilyseen that thework done inconstructinga single systemvanishes totheorder of— ,sothat, inthelimitwhennisvery great,thework done inconstructingtheseries ofnsystemsisinfinitesimal. Thus theenergyofthefinalsystem mayberegardedasthework done in superposingthis series ofnsystems. Letussupposesomanyofthecomponent systemstohavebeensuper- posed,that thesysteminpositionisktimes itsfinalstrength,where k isapositive quantitylessthanunity. Thepotentialofthe field atany pointwillbe /eft.Onbringing upanewsystemletussupposethatkis increased tok+die,sothat thestrengthofthenewsystemisd/ctimes that ofthe finalsystem.Inbringing upthenewsystem, weplaceamagnetof d/ctimes thestrengthofainafield offorce ofpotential /eft,andsoonwith theothermagnets.Thus theworkdone is die .«ft(a)+die .kCI(b)+..., andonintegrationoftheworkperformed, weobtain W=f/td*{fl(a)+ft(&)+...}Jo =iSft (a), agreeingwithequation (380), andleadingasbefore toequation (381). 450. Ifthemagneticmatter consistssolelyofnormally magnetised shells, wemay replace equation (381) by where dsdenotes thickness anddSanelement ofarea ofashell.Replacing Idsby <j>,sothat</>isthestrengthofashell,wehave w=^lhd£ds - 448-451] EnergyintheMedium 399 Foruniform shells,</>maybetaken outside thesignofintegration,and theequationbecomes (cf.§423),where nisthenumber oflines ofinduction which cross theshell. This calculation measures theenergyfrom astandardconfigurationin which themagneticmaterials are allscattered atinfinity. Tocalculate theenergymeasured from astandardconfigurationinwhich theshells have alreadybeen constructed andarescattered atinfinityascomplete shells, we useequation (378), namely W=±X{n(a)-n a(a)}, fromwhich weobtain TT=iS//*£<* where——denotes thevalues-—atthesurface ofanyshell iftheshell itself on dn issupposedannihilated. Ifalltheshells areuniform, thismayagainbewritten W=-%S<f>n' (382), where n'isthenumber oftubes offorce from theremaining shells, which cross theshell ofstrength <£.Anexampleofthishasalreadyoccurred in §424. Energy intheMedium. 451.Wehave seen that theenergyofamagneticfield isgiven by (cf.equation (381)) *"*1I+*S+*)***(383 >- theintegration beingtaken over allmagneticmatter. Asapreliminaryto transformingthis intoanintegraltaken throughallspace, weshallprove that flf(aGL+b/3+cy)dxdydz=(384), theintegration being throughallspace. Theintegralonthe leftcanbewritten as and this,byGreen's Theorem, maybetransformed into 111Hhp+=-+^-\dxdydz- jIO(la+mb+nc)dS, 400 Permanent Magnetism [ch.xi thelatterintegral beingtaken overasphereatinfinity. Now atinfinityO isoftheorder of— (cf. §67),while la+mb+ncvanishes, anddSisof theorder ofr2 ,sothatthesurfaceintegral vanishes onpassingtothelimit r=oo .Also thevolumeintegralvanishes since da db do_ dxdydz' andhence thetheorem isproved. Replacing a,b,cbytheir values, asgiven byequations (359),wefindthat equation (384) becomes ff[(a2+/32+72 )dxdydz +4tt j(Act+B/3+Cy)dxdydz=...(385). Bothintegralsaretakenthroughallspace,but sinceA—B=G=0 exceptinmagnetic matter, wecanregardthelatterintegralasbeingtaken onlyoverthespace occupied bymagneticmatter. Thisintegralistherefore equal, byequation (383),to—2W,sothatequation (385) becomes W= -^jf!(a?+/32+^)dxdydz (386), theintegral beingtakenthroughallspace. Thisexpressionisexactly analogoustothatwhich hasbeen obtained for theenergyofanelectrostaticsystem, namely, W=^ rjfj(X2+Y2+Z2 )dxdydz. And, asinthecase ofanelectrostaticsystem, equation (386)maybe interpretedasmeaningthattheenergy mayberegardedasspread through themedium atarate5—(a2+/32+y2 )perunitvolume. 07T Terrestrial Magnetism. 452. Themagnetismoftheearth isvery irregularlydistributed and is constantly changing.Thesimplestandroughest approximationofalltothe state oftheearth's magnetismisobtained byregardingitasabarmagnet, possessingtwopolesnear toitssurface, thepositionofthese in1906being asfollows : North Pole 70°30'N., 97°40'W. South Pole* 73°39'S., 146° 15'E. Another approximation,which isbetter inmany ways althoughstill very rough,isobtained byregardingtheearth asauniformly magnetised sphere. *SirE.Shackleton givesthepositionoftheSouth Pole in1909 as72°25'S.,155° 10'E. 451-454] Terrestrial Magnetism 401 With thehelpofacompass-needle,itwillbepossibletofind the direction ofthe lines offorce oftheearth's field atany point.Itwill alsobepossibletomeasure theintensityofthis field,bycomparingitwith knownmagnetic fields, orbymeasuringthe force withwhich itactson amagnetofknownstrength. 453. Atanypointontheearth, letussupposethattheangle between the line ofmagneticforce andthehorizontal is0,thisbeing reckoned positiveifthelineofforcepoints down intotheearth, and letthehorizontal projectionofthe line offorcemake anangle8with thegeographical meridianthroughthepoint,thisbeing reckonedpositiveifthis linepoints west ofnorth. Theanglefriscalled thedipatthepoint,theangle8is called thedeclination. LetHbethehorizontalcomponentofforce, then thetotal forcemaybe regardedasmadeupofthreecomponents: X=Hcos8,towards thenorth, Y=Hsin8,towards thewest, Z=Htan 6,vertically downwards. Ifflisthepotential due totheearth's field atapointotlatitudeI, longitude X,andatdistance rfrom thecentre, wehave(cf.equations (331)) y-_I» r=-la",Z=f(387).roi rcos IdX or7 Analysis ofPotentialofEarth'sfield. 454. SinceOisthepotentialofamagnetic system,thevalue ofHin regionsinwhich there isnomagnetisationmust(by §408)beasolution of Laplace's equation,andmust therefore(by §233)becapableofexpansionin theform 12= (^1+^+...)+(£/+&V+£,V+...) (388), inwhichSltS2,...S',$/>&/> •••aresurface harmonics, ofdegreesindicated bythesubscripts. Attheearth's surface, the firstterm isthepartofthepotentialwhich arises frommagnetisminside theearth, while thesecond term arises from magnetismoutside. Thesurface harmonic Sncan,asin§275,beexpandedintheform m=n Sn=SP%(sin I)(AniTncosm\+Bn<msinm\),m= sothat XIcanbeputintheform «=oom=n(JPm(sin Z)n=S2 ]n )l+1(Animcosm\+Bn<msinmX) +rnP%(sin I)(A'n>mcosm\+B'nt7nsinm\)[. 26 402 Permanent Magnetism [ch.xi Hence fromequations (387)weobtain thevalues ofX,Y,Zatanypoint interms ofthelongitudeandlatitude ofthepointandtheconstants such as xi.T^Tn} -Dji.wij-£«•n,m>&n,m,' Byobservingthevalues ofX,Y,Zatagreat number ofpoints,we obtain asystemofequationsbetween theconstants An<m,etc.,andon solvingtheseweobtain theactual values oftheconstants, andtherefore aknowledgeofthepotentialasexpressed byequation (388). Ifthemagneticfield aroseentirelyfrommagnetisminside theearth, weshould ofcourseexpecttofind#/=82'=••.=0,while ifthemagnetic field arose from magnetism entirelyoutside the earth, weshould find Sl=S2=...=0. 455. The resultsactuallyobtained areofextreme interest. Themag- netic field oftheearth, aswehave said, isconstantly changing.Inaddition toaslow, irregular,and so-called "secular"change,itisfound that there areperiodic changesofwhich theperiods are, ingeneral, recognisableas theperiodsofastronomical phenomena. Forinstance there isadaily period,ayearly period,aperiod equaltothelunar month, aperiodof about 26gdays (the periodofrotation oftheinner core ofthesun*), aperiodofabout 11years (the periodofsun-spot variations), aperiodof 19years (theperiodofthemotion ofthelunar nodes), and soon.Thus thepotentialcanbedividedupinto anumber ofperiodic partsanda residual constant, orslowlyandirregularly changing, part.Alltheperiodic partsareextremelysmall incomparisonwith the latter. Itisfound, on analysingthepotentialsofthese differentpartsofthe field, thattheconstant field arises frommagnetisationinside theearth, while thedailyvariation arises mainly frommagnetisationoutside the earth. The former result mighthavebeenanticipated,butthelatter could nothavebeenpredicted withanyconfidence. Forthevariationmighthaverepresented nothing more than achangeinthepermanent magnetismoftheearth due tothe coolingandheatingoftheearth's mass, ortothetides inthesolid matter of theearth produced bythesun's attraction. Thisdailyvariation isnotsuch ascould beexplained bythemagnetism ofthesun itself; Chreefhasfound that itcannot beexplained bythe coolingandheatingeither ofthe earth's mass, oroftheatmosphereas suggested byFaraday.Balfour StewartJputforward thehypothesisthatthe dailyvariation wasduemainlytoelectric currentscirculatingintheupper atmosphereasaresult oftheelectromotive forces induced bytheconnective *Theouter surface ofthesun isnotrigid,androtates atdifferent rates indifferent latitudes. Thus itisimpossibletodiscover theactual rate ofrotation oftheinner core except bysuch indirect methods asthat ofobserving periods ofmagneticvariation. fRoy.Soc. Phil. Trans., 202, p.335. JArt."Terrestrial Magnetism,"inthe9thEdn. oftheEncyc.Brit.(1882). 454-456]Terrestrial Magnetism 403 motion oftheatmosphereacross theearth'smagneticfield. Thishypothesis wasexamined anddeveloped bySchuster*, whoexamined thedailyvaria- tionsbythemethod ofharmonicanalysis, already explained. Schuster found theoriginofthemagneticfield tobemainlyexternal ;hesuggestedalsothat theconvection currents indicated bythediurnal barometricchanges were ulti- mately responsibleforthephenomenon, andfurther found thatasmallpart ofthe fieldmust beattributed tooriginsinside theearth :these itwas suggested mightbeasystemofcurrents induced intheearthbytheatmo- sphericcurrents above. Chapmanfhasrecentlyreexamined thequestion,andobtains results in substantialagreementwith Schuster'stheory. Hefinds thatthecontribution from inside theearth isabout 28percent, ofthetotal diurnal variation. It issupposedthattheconducting layerintheupper atmosphereinwhich the induced currents flow isthat ofwhich wealready have evidence inthepheno- menon ofthebendingofelectromagneticwaves round theearth; thislayeris alsotheseatoftheaurora borealis. Chapmanfinds thattheinternalmagnetic field ofinduced currents would beexplained byassuming that, beneath an uppernon-conductivelayerof150or200milesdepth,theearth hasaspecific resistance ofabout 4x10~nC.G.s. units. Besides thevariationjust considered, there isfound tobealunar diurnal variation, ofperiod equaltotheapparent periodofmotion ofthemoon. This appearstobetheresult ofasemi-diurnal tidal oscillation oftheatmosphere themechanism beingotherwise similar tothatalready explained. 456.Thenon-periodic partoftheearth's field isfound toariseentirely frommagnetisminside theearth, havingapotentialoftheform «9 Thismethod ofanalysingtheearth's field isduetoGauss, whocalculated thecoefficients, withsuchaccuracyaswasthenpossible,fortheyear1830. Themost complete analysisofthefieldwhichnow exists hasbeen calculated byNeumayerfortheyear 1885, usingobservations ofthe field at1800 pointsontheearth's surface. The firstfewcoefficients obtained byNeumayerareasfollows : (Ahl=-0248,41|D--3157 {Bhi=_.060S} A2<0=-0079 -I"2'1"".™? t2>2 [42fl=--0498,^l2i2=--0057, [B2A=-0130, J52>2=--0126, =--0<H4 {^,i=-039G, ^3, 2=--0279, ^3>3=--0033, U?3,i=-0074,B3>2=--0004, £3i3=--0055, A --0344 j^4,i=--0306,^4>2=-:01983Aii3=-0068, ^l4>4=--0008, 4,0 l#4,i=--0119, J94>2=-0071,£4)3=-0051,Bi>4=-0010. *Phil. Trans. A,180(1889), p.467,audA,203(1907), p.163. tPhil. Trans. A,213(1913), p.279,andA,218(1919), p.1. 26—2 404 Permanent Magnetism [ch.xi 457.Thesimplest approximationisofcourse obtained byignoringall harmonics beyondthe first. Thisgivesasthemagnetic potential XI=—-UMPi(sin I)+iY(sin I)(AhlcosX+2?MsinX)[ =- 1*3157sin I+cos I(-0248 cos\--0603 sinX)l. Theexpressioninbrackets isnecessarilyabiaxial harmonic oforderunity (cf. §276) ;itiseasilyfound tobeequalto*3224 cos7,where 7isthe angulardistance ofthepoint (I,X)from thepoint lat.78°20'N.,long.67°17'W(389). Thepotentialisnow £1=-3224 52i2 , which isthepotentialofauniformly magnetised sphere, havingasdirection ofmagnetisationtheradiusthroughthepoint (§415). Oragain,itisthe potentialofasingle magnetic particleatthecentre oftheearth, pointing inthissame direction. Itisnaturally impossibletodistinguish between these twopossibilities byasurveyofthe field outside theearth. Green's theorem hasalready shewn thatwecannot locate thesources ofafield inside aclosed surface byastudyofthefield outside thesurface. EXAMPLES. 1.Two small magnetsfloat horizontally onthesurface ofwater, onealongthe direction ofthestraightlinejoiningtheir centres, andtheother atright anglestoit. Prove thattheaction ofeachmagnet ontheother reduces toasingleforce atright angles tothestraightlinejoining thecentres, andmeetingthat lineatone-third ofitslength from thelongitudinal magnet. 2.Asmall magnet ACB,freetoturnabout itscentreC,isacted onbyasmall fixed magnet PQ. Prove that inequilibrium theaxisACB liesintheplanePQC, andthat tan#=—itan6',where6,&aretheangles which thetwomagnets make with theline joining them. 3.Three small magnets havingtheir centres attheangular pointsofanequilateral triangle ABC, andbeing freetomove about their centres, canrest inequilibrium with themagnet atAparallel toBC,andthose atBandCrespectivelyatright anglestoAB andAC. Prove thatthemagnetic moments areintheratios V3:4 :4. 4.The axis ofasmall magnet makes anangle <£withthenormal toaplane. Prove thatthelinefrom themagnettothepointintheplane where thenumber oflines of force crossingitperunit area isamaximum makes anangle 6with theaxis ofthe magnet,such that 2tan <9=3tan2($-0). Examples 405 5.Two small magnetslieinthesameplane, andmakeangles 6,ffwith theline joiningtheir centres. Shew thatthelineofaction oftheresultant force between them divides thelineofcentres intheratio tan ff+2tan6 :tan8+2tan ff 6.Two small magnets have their centres atdistance rapart, makeangles 8,ffwith thelinejoining them, andanangleewith each other. Shew thattheforceonthefirst magnetinitsowndirection is 3mm' ,„ „.——(5cos26cos8-cosff-2cosecos8). Shew that thecouple about thelinejoining them which themagnets exert onone another is mm' o?sini wheredistheshortest distance between their axesproduced. 7.Two magnetic needles ofmoments M.M'aresolderedtogether sothat their directions include ananglea.Shew thatwhen they aresuspended soastoswing freely inauniform horizontal magnetic field, their directions willmakeangles 8,&with the lines offorce, givenby sin6_sin6'_sina M'M (i/2+M>2+2MM' cosa)*' 8.Prove that ifthere aretwomagnetic molecules, ofmomentsMandM',with their centres fixed atAandB,whereAB=r, andoneofthemolecules swings freely, while the other isacted onbyagiven couple,sothatwhen thesystemisinequilibrium this molecule makes anangle 8withAB,then themoment ofthecoupleis fMM' sin28jr3(3cos25+1)*, where there isnoexternal field. 9.Twosmall equal magnets have their centresfixed, andcanturnabout them ina magneticfield ofuniformintensity H,whose direction isperpendiculartotheliner joining thecentres. Shew thattheposition inwhich themagnets bothpointinthe direction ofthelines offorce oftheuniform field isstableonlyif H>3M/r*. 10.Twomagnetic particlesofequalmoment arefixed with their axesparalleltothe axis of2,and inthesame direction, andwith their centres atthepoints +a,0,0.Shew that ifanother magnetic molecule isfreetoturnabout itscentre, which isfixed atthe point (0,y,z),itsaxis willrest intheplanex=0,and willmake with theaxisofzthe angle , Svztan-1—=——5.2zi-az—yi Examine which ofthetwopositionsofequilibriumisstable. 11.Prove that there arefourpositions inwhich agiven barmagnet maybeplaced soastodestroytheearth's control ofacompass-needle,sothat theneedle canpoint indifferentlyinalldirections. Ifthebar isshort compared with itsdistance from the needle, shew thatonepairofthese positions areabout1^times more distant than the otherpair. I406 Permanent Magnetism [ch.xi 12.Three small magnets,each ofmagnetic momentju,arefixed attheangular points ofanequilateral triangle ABC,sothat their northpoleslieinthedirections AC,AB,BC respectively. Another small magnet, moment//,isplacedatthecentre ofthetriangle, and isfreetomove about itscentre. Prove thattheperiodofasmall oscillation isthe same asthat ofapendulumoflength Ibzgj\l^blp.yL, where bisthelengthofasideofthe triangle, and/themoment ofinertia ofthemovable magnet about itscentre. 13.Three magnetic particlesofequal moments areplacedatthecorners ofan equilateral triangle, andcanturnabout thosepointssoastopointinanydirection inthe planeofthetriangle. Prove that there arefourandonlyfourpositionsofequilibrium such that theangles, measured inthesame sense ofrotation, between theaxes ofthe magnetsandthebisectors ofthecorresponding angles ofthetriangle areequal. Also provethat thetwosymmetrical positions areunstable. 14.Four smallequal magnets areplaced atthecorners ofasquare, and oscillate under theactions theyexert oneach other. Prove that thetimes ofvibration ofthe principaloscillations are Mk2cP)2 |m23(2+l/2v/2)J fMk*-d?}h (m2(3-1/2^/2)/' .(Mk2d32v/21* 277\—3^— r• wheremisthemagnetic moment, andM&2themoment ofinertia, ofamagnet, anddisa sideofthesquare. 15.Asystem ofmagnetsliesentirelyinoneplane and itisfound thatwhen the axis ofasmall needle travels round acontour intheplane that contains nomagnetic poles, theneedle turnscompletely round. Prove that thecontour contains atleast one equilibrium point. 16.Prove thatthepotential ofabody uniformly magnetised withintensity Iis,at anyexternalpoint, thesame asthatduetoacomplex magnetic shellcoinciding with the surface ofthebodyandofstrength Ix,where xisacoordinate measuredparallel tothe direction ofmasrnetisation. 17.Asphereofhard steel ismagnetised uniformlyinaconstant direction anda magnetic particleisheld atanexternalpoint with theaxis oftheparticle parallel tothe direction ofmagnetisation ofthesphere. Find thecouples acting onthesphere andon theparticle. 18.Aspherical magnetic shell ofradius aisnormally magnetised sothat itsstrength atanypointisSi,where Siisaspherical surface harmonic ofpositive order i.Shew thatthepotentialatadistance rfrom thecentre is -47rfirK«ywhen >-<«, 2i+li+1 Si(-) whenr>a. 19. Ifasmallspherical cavity bemade within amagnetised body, prove that the componentsofmagneticforce within thecavityare a+$A, (3+±B, y+£C/. Examples 407 20. Iftheearth were auniformly magnetised sphere, shew that thetangent ofthe dipatanypointwould beequaltotwice thetangent atthemagnetic latitude. 21.Prove that ifthehorizontal component,inthedirection ofthemeridian, ofthe earth's magneticforce wereknown allover itssurface,alltheother elements ofits magneticforce might betheoreticallydeduced. 22.From theprinciplethatthelineintegralofthemagnetic force round anycircuit ordinarily vanishes, shew thatthetwohorizontal componentsofthemagnetic force atany station maybededuced approximatelyfrom theknown values forthree other stations which liearound it.Shew thatthese sisknown elements arenotindependent, butmust satisfy oneequationofcondition. 23. Iftheearth were asphere, and itsmagnetism duetotwosmallstraight bar magnetsofthesame strength situated atthepoles, with their axes inthesame direction alongtheearth'saxis,provethatthedip8inlatitude Xwould begiven by /\x\ ,X „,X„, cx < 8cot I8+- j=cot--6tan--3tan2- . 24.Assuming thattheearth isasphereofradiusa,andthatthemagnetic potential Qisrepresented by -*©+*©'+* ©'+*©' shew that 12iscompletely determined byobservations ofhorizontalintensity,declination anddipatfour stations, andofdipatfourmore. 25.Assumingthat intheexpansionoftheearth's magnetic potential the fifthand higher harmonics may beneglected, shew that observations oftheresultantmagnetic force ateight points aresufficient todetermine thepotential everywhere. 26.Assuming thattheearth's magnetismisentirely duetointernalcauses, andthat inlatitude Xthenortherly componentofthehorizontal force isAcosX+Bcos3 X,prove that inthislatitude thevertical component reckoned downwards is 2(,4+fS)sinX-§flsin3X, CHAPTER XII INDUCED MAGNETISM Physical Phenomena. 458. Reference hasalreadybeenmade tothewell-known factthat amagnetwillattract smallpiecesofiron orsteelwhich arenotthemselves magnets. Herewehave aphenomenonwhich atfirstsightdoesnotseem tobeexplained bythelawoftheattractions andrepulsionsofmagnetic poles.Itisfound, however, that thephenomenonisdue toamagnetic "induction"ofakind almostexactlysimilar totheelectrostatic induction alreadydiscussed. Itcanbeshewn thatapieceofiron orsteel, placedin thepresenceofamagnet,will itselfbecomemagnetised. Temporarily,this pieceofironorsteel willbepossessedofmagnetic polesofitsown,andthe systemofattractions andrepulsions between these andthepolesofthe original permanent magnetwillaccount fortheforces which areobserved toactonthemetal. Ithas,however, been seen thatpairsofcorresponding positive and negative polescannot beseparated bymore than molecular distances, so thatweareledtosupposethateachparticleofthebodyinwhich magnetism isinduced must becomemagnetised,theadjacent poles neutralisingone another asinapermanent magnet. Takingthisview,itwillbeseen thattheattraction ofamagnetforan unmagnetised bodyisanalogoustotheattraction ofanelectrified bodyfor apieceofdielectric(§197), rather than toitsattraction foranuncharged conductor. The attraction ofacharged bodyforafragmentofadielectric hasbeen seen todepend uponamolecular phenomenon taking placeinthe dielectric. Each molecule becomes itself electrified onitsopposite faces, with chargesofopposite sign,thesecharges being equalandoppositesothatthe totalchargeonanymolecule isnil. Inthesameway,whenmagnetismis induced inanysubstance, eachmolecule ofthesubstance must besupposedto become amagnetic particle,thetotalchargeofmagnetismoneachparticle beingnil. Itfollows that theattraction ofamagnetforanon-magnetic bodyismerely theaggregateoftheattractive forcesactingonthedifferent individualparticlesofthebody. 459. Confirmation ofthisview isfound inthe factthat theintensity oftheattraction exertedbyamagnetonanon-magnetised bodydepends on 458-460]Induced Magnetism 409 thematerial ofthelatter. Thesignificanceofthis fact will,perhaps,bestbe realised bycomparingitwith thecorrespondingfactofelectrostatics. When anunchargedconductor isattracted byacharged body,thephenomenain theformer bodywhich lead tothisattraction aremass-phenomena:currents ofelectricityflowthroughthemass ofthebodyuntil itssurface becomes anequipotential.Thus theattractiondepends solely upontheshapeof thebodyandnotuponitsstructure. Ontheother hand, thephenomena which lead totheattraction ofafragmentofdielectric are,aswehave seen, molecular phenomena. Theyareconditioned bytheshapeandarrangement ofthemolecules, with theresult that thetotal forcedependsonthenature ofthedielectric material. Allmagnetic phenomena occurringinmaterial bodies must bemolecular, asaconsequenceofthefactthatcorresponding positive andnegative poles cannot beseparated bymore than molecular distances. Hence weshould naturally expecttofind, aswedofind, that allmagnetic phenomenain material bodies, and inparticulartheattraction ofunmagnetisedmatter byamagnet,would dependonthenature ofthematter. There would be arealdifficultyiftheattraction were found todepend onlyontheshape ofthebodies. 460. Theamount ofthe action due tomagneticinduction varies enormously more with thenature ofthematter than isthecasewith the correspondingelectric action. Among common substances thephenomenon ofmagneticinduction isnotatallwell-markedexceptinironand steel. These substances shew thephenomenontoadegreewhichappears very surprising when comparedwith thecorrespondingelectrostatic phenomenon. After these substances, thenext best forshewingthephenomenaofinduction arenickel and cobalt, althoughthese areveryinferior toironand steel. It isworthnoticingthattheatomicweightsofiron, nickel andcobalt arevery closetogether*,andthatthethree elements holdcorresponding positionsin thetable ofelementsarranged accordingtotheperiodiclaw. Ithasrecentlybeen found that certain rare metals shewmagnetic induction toanextentcomparablewith iron,andthatalloyscanbeformed toshewgreat powersofinductionalthoughtheelements ofwhich these alloysareformed arealmostentirely non-magneticf. Itappears probablethat allsubstancespossess somepowerofmagnetic induction, althoughthis isgenerally extremelyfeeble incomparison with that ofthesubstances alreadymentioned. Insome substances, the effect isoftheopposite signfrom that iniron, sothat afragmentofsuchmatter isrepelledfrom amagnetic pole.Substances inwhich theeffect isofthe *Iron=55-5, nickel=58-3, cobalt=58-56. tForanaccount ofthecomposition andpropertiesofHeusler's alloys,seeapaper by J.C.McLennan, Phijs. Review, Vol.24,p.449. 1 410 Induced Magnetism [ch.xii same kind asiniron arecalled'paramagnetic,while substances inwhich the effect isoftheoppositekind arecalled diamagnetic. Thephenomenonofmagneticinduction ismuch moremarked inpara- magnetic,than indiamagnetic,substances. Themostdiamagneticsubstance known isbismuth, and itscoefficient ofsusceptibility (§461,below)isonly about77T-ofthat ofthemostparamagnetic samplesofiron. 109 Coefficients ofSusceptibility andPermeability. 461.When abodywhichpossessesnopermanent magnetismofitsown isplacedinamagnetic field, eachelement ofitsvolume will, forthetime it remains under theinfluence ofthemagnetic field, beamagnetic particle. Ifthebodyisnon-crystallinethedirection oftheinducedmagnetisationat anypointwillbethat ofthemagneticforce atthepoint. Thus ifHdenote themagneticforce atanypoinf,wecansupposethattheinducedmagnetism, ofanintensity /,has itsdirection thesame asthat ofH. Thus ifa,/?, <yarethecomponentsofmagnetic force, andA,B,Cthe componentsofinducedmagnetisation, weshallhaveequationsoftheform A=/ca \ B=kJ3\ (390), C=*7J thequantitykbeingthesame ineachequationbecause thedirections ofI andHarethesame. Thequantitykiscalled themagnetic susceptibility. Ifthebodyhasnopermanent magnetisation,thewholecomponentsof magnetisationarethequantities A,B,Ggiven byequations (390), andthe componentsofinduction aregiven (cf.equations (359)) by a=a+4nrA=a(1+4>7tk), &=/3+47t£=/3(1+47™), c=7+4nrC=7(1+ 47J7C). Ifweput ^=1+4™(391), wehave a=fiot' b=fx/3>(392), c=fx<y, andfiiscalled themagnetic permeability. 462. Thequantitieskandfiarebynomeans constant foragiven substance. Their valuedepends largely uponthephysical conditions, particularlythetemperature,ofthesubstance, uponthestrengthofthe magneticfield inwhich thesubstance isplaced, andupontheprevious magnetic experiencesofthesubstance mquestion. 400-463] Physical Phenomena 411 Wepasstotheconsideration ofthewayinwhich themagneticcoefficients varywithsome ofthese circumstances. Askand/u,areconnected byasimple relation(equation (391)),itwillbesufficient todiscuss thevariations ofone ofthesequantities only,andthequantity /mwillbethemost convenient for thispurpose. Moreover, asthephenomenonofinducedmagnetisationis almostinsignificantinallsubstances exceptironand steel,itwillbesufficient toconsider themagnetic phenomenaofthese substancesonly. 463. Dependence ofponH.Thewayinwhich thevalue of//,depends onHis,initsmain features, thesame forallkinds ofiron. Forsmall forces, fiisaconstant, forlargerforcesfxincreases, finallyitreaches amaximum, and after thisdecreases insuch awaythatultimately fiHapproximatesto aconstant value, known asthe"saturation" value. This isrepresented graphicallyinatypicalcaseinfig.113,whichrepresentstheresults obtained byEwingfromexperimentsonapieceofiron wire. mH=1500.0 /iH=10000 MH=5000M=3000 ix=2000 ^=1000 H= Theabscissaerepresentvalues ofH,theordinate ofthethick curve the value offxH,andtheordinate ofthethincurve thevalue of//,.The corre- spondingnumerical values areasfollows : H 412 Induced Magnetism, [en.xii 464. Retentiveness andHysteresis.Itisfound that after themagnetising force isremoved from asampleofiron,theiron stillretains some ofitsmag- netism. Herewehaveaphenomenonsimilar totheelectrostatic phenomenon ofresidual charge alreadydescribed in§397. Fig.114 istaken from apaper byProf.Ewing (Phil. Trans. Roy.Soc. 1885). The abscissae representvalues ofH,and ordinates values ofB, theinduction. Themagneticfieldwasincreased fromH=toH=22, andasHincreased thevalue ofBincreased inthemanner shewn bythe curveOPofthegraph. Onagain diminishing HfromH=22toH=0, the graphforBwasfound tobethatgiven bythecurve PE. Thusduringthis operationthere wasalwaysmoremagnetisationthan atthecorresponding stageoftheoriginal operation,andfinally when theinducingfieldwas entirely removed, there wasmagnetisation left,ofintensity represented by OE. The fieldwasthen further decreased fromH=toH=—20,and then increasedagainfromH=—20toH=22.ThechangesinBare shewn inthegraph. >H 465. Dependence offiontemperature. Ashasalreadybeen said, the value of/j,dependstoalargeextent onthetemperatureofthemetal. In general,thevalue offjucontinuallyincreases asthetemperatureisraised, this increasebeingslow atfirstbutafterwards morerapid,until atemperature known asthe" temperatureofrecalescence" isreached. Thistemperature hasvaluesrangingfrom 600° to700° forsteelandfrom 700° to800° foriron. Thistemperature takes itsname from thecircumstance thatapieceofmetal cooling throughthistemperaturewillsink toadullglowbeforereaching it, andwillthenbecomebrighter againonpassing throughit. Afterpassingthetemperatureofrecalescence, thevalue offifallswith extremerapidity,and atatemperature onlyafewdegreesabove this temperature,ironappearstobealmostcompletely non-magnetic. '464-467] Mathematical Theory 413 Forparamagnetic substances, itappearstobeagenerallawthat the susceptibilitykvariesinverselyastheabsolutetemperature (Curie's Law). Mathematical Theory. 466. If£1isthemagnetic potential, supposedtobedefined atpoints insidemagneticmatter byequation (348), wehave, asinequations (341) (cf.§430), a=—-=-etc., sothat an , ao an Thequantities a,b,c,aswehave seen(§434), satisfy da db dc _ /«««*S+*+»-°<393> atevery point, and r \(la+mb+nc)dS=(394),//< where theintegrationistaken overanyclosed surface. Interms ofthe potential, equation (393) becomes ^^J+tyl^J+9l^fcJ=(39o)' whileequation (394) becomes //nd ^dS=(396). Iffx,isconstantthroughout anyvolume, equation (395) becomes V2D=0. Thus inside amass ofhomogeneousnon-magnetised matter, themagnetic potentialsatisfiesLaplace's Equation. 467.Atasurface atwhich thevalue of/j,changes abruptly wemay takeaclosed surface formed oftwoareasfitting closelyabout anelement dS oftheboundary,these twoareasbeingonoppositesides oftheboundary. Onapplying equation (396), weobtain fi1~-+fi2~-=(397), where a1}u2arethepermeabilities onthetwo sides, and=— .=-denote OVxov2 differentiations withrespecttonormals tothesurface drawn into thetwo mediarespectively. Equations (397) and(395) (or(396)), combined with thecondition that flmust becontinuous, suffice todetermine 12uniquely.Theequations 414 Induced Magnetism [ch.xii satisfied byO,themagnetic potential,areexactlythesame asthose which would besatisfied byV,theelectrostaticpotential,ifjawere theInductive Capacityofadielectric. Thus thelawofrefraction oflines ofmagnetic induction isexactlyidentical with thelawofrefraction oflines ofelectric forceinvestigatedin§138,andfigures (43)and(78)may equallywellbe taken torepresentlines ofmagneticinductionpassingfrom onemedium to asecond medium ofdifferentpermeability. 468.Atanyexternal point Q,themagnetic potentialofthemagnetisation induced inabodyinwhichfj,andkhave constant valuesis,byequation (342), Transforming byGreen's Theorem, yJJJr JJ\ Oxcy dzjr -.«//©>(399)' shewingthat thepotentialisthesame ifthere were alayerofmagnetic matter ofsurfacedensity—k-=-spreadoverthesurface ofthebody.This isPoisson'sexpressionforthepotentialduetoinducedmagnetism. Wecanalsotransformequation (398)into —/K©*9 (400)- shewingthatthepotentialatanyexternalpointQoftheinduced magnetism isthesame asifthere were amagneticshell ofstrength—fcflcoinciding with thesurface ofthebody. Bodyinwhichpermanentandinduced magnetismcoexist. 469. Ifapermanent magnethasapermeabilitydifferent fromunity, we shall have amagnetisation arising partlyfrompermanentandpartlyfrom inducedmagnetism.Ifkisthesusceptibilityand/theintensityofthe permanent magnetisationatanypoint,thecomponentsofthetotalmagnet- isation atanypointwillbe A=II+ko,etc (401), 467-471] Energy ofaMagnetic Field 415 andthecomponentsofinduction are a=a+4<7rA=4>ttII+fia,etc.(402). Forsuchasubstance,itisclear thatequations (395) and(396)willnot ingeneralbesatisfied. Energy ofaMagnetic Field. 470. Toobtain theenergyofamagneticfield inwhich bothpermanent andinducedmagnetism maybepresent, wereturn tothegeneral equation obtained in§451, I(aa+b/3+cy)dxdydz=(403). Onsubstitutingfora,b,cfromequations (402),thisbecomes 4tt\\\I(la+m&+ny)dxdydz +11 J/x(a2+/32+y*)dxdydz=...(404). Whether ornotinduced magnetismispresent,itisproved,in§448,thatthe energyofthefield is w=kISSt <?i§+m^+n!r)dxdydz > where theintegralistakenthroughallspace.This isequalto—3—times the firstterm inequation (404). Thus W=^ rjffp(tf+/32+r)dxdydz (405). This could havebeen foreseen from analogy with theformula W=i- fIfK(X2+Y*+Z2 )dxdydz, whichgives theenergyofanelectrostatic field. From formula (405)weseethat theenergyofamagneticfieldmaybe LlH2 supposed spread throughoutthemedium, atarate^—perunitvolume. Mechanical Forces intheField. 471. Themechanical forcesactingonapieceofmatter inamagnetic fieldcanberegardedasthesuperpositionoftwosystems—first,theforcesacting onthematter invirtue ofitspermanent magnetism (ifany), and, secondly, theforcesactingonthematter invirtue ofitsinducedmagnetism (ifany). Theproblemoffinding expressionsforthemechanical forces inamagnetic field ismathematicallyidentical with that offindingtheforces inanelectro- static field. This istheproblemofwhich thesolution hasalready been 416 Induced Magnetism [ch.xii givenin§196. The result oftheanalysistheregivenmayatoncebe appliedtothemagnetic problem. Inequation (117), p.175,wefound thevalue of2,the^-componentof themechanical forceperunitvolume, intheform dVR*dK d{R*dK\ dx 87r dec dec\87r 8t/* Totranslate this result tothemagnetic problem, wemustregard pas specifyingthedensityofmagnetic poles,Rmust bereplaced byH,the magnetic intensity,andKby /*,themagnetic permeability.Also the electrostaticpotential Vmust bereplaced bythemagnetic potentialX2.We then have, asthevalue of5inamagnetic field, a=-PTx-^r^+dx\MT d-r)(406)' Clearlythe firstterm inthevalue ofHisthatarisingfrom theper- manentmagnetismofthebody,while thesecond andthird terms arisefrom theinduced magnetism.The firstterm canbetransformed inthemanner already explainedinthelastchapter.Itiswith theremainingterms that weareatpresentconcerned. These willrepresenttheforceswhen noper- manent magnetismispresent. Denotingthecomponentsofthis forceby 2',H',Z\wehave .~=-^£+dx{^T£)<407> 472. Thisgeneralformula assumes aspecialform inacasewhich isof great importance, namely when themagnetic medium isafluid. Allliquid magneticmedia inwhich thesusceptibilityisatallmarked consist ofsolutions ofsalts ofiron,andthemagnetic propertiesoftheliquid arisefromthepresenceofthe salts insolution.AccordingtoQuincke, the solution havingthegreatest susceptibilityisasolution ofchloride ofiron in methyl alcohol, and forthisthevalue offi—1isabout toW*. Insuch a liquid,thefieldarisingfrom theinduced magnetismwillbesmallcompared with thatarisingfrom theoriginal field, sothatthemagnetisationofany single particleofthe saltinthesolution mayberegardedasproduced entirely bytheoriginalfield. Hence wehave conditions similar tothose which obtain electrostaticallyinagas.Theinduced fieldmayberegarded simplyastheaggregateofthe fieldsarisingfrom thedifferentparticlesof themagnetic medium, and isthereforejointly proportionaltothedensityof theseparticlesand tothestrengthoftheinducingfield. The latter fact shews that, foragiven densityofthemedium, fioughttobeindependentof H,aresult towhich weshall return later. Theformer factshews that, as *Cf.G.T.Walker,"Aberration"(Cambridge Univ. Press, 1900), p.7G. 471-474] Magnetostriction 417 thedensitytchanges, /j,—1oughttobeproportionaltot—aresultanalogous totheresult thatK—1isproportionaltothedensityinagas.Ithasbeen foundexperimentally byQuincke* thatp—1isapproximately proportional toT. Ingaseswehave conditionspreciselysimilar tothose which obtain when agasisplacedinanelectrostatic field. Hence/j,—1must, foragas,be proportionaltor,forexactlythesame reason forwhichK—1isproportional totThis result alsohasbeen verifiedbyQuincke f. Thuswemay saythat forfluid media, whetherliquidorgaseous, /a—1 is,ingeneral, proportionaltor,where tisthedensityofthemagnetic liquid, inthecase ofaliquidinsolution, orofthegas itself, inthecase ofagas. 473. Ifweassume therelation fi~l=cr(408), where cisaconstant, wefind thatexpression (407)maybeputinthe simplerform h-,a— lda=~—r- 87rdxm, shewingthatthewhole mechanical force isthesame aswould besetupbya hydrostatic pressureatevery pointofthemedium ofamount 8H* 7T IfHvaries frompointtopointofthe field, theeffect ofthispressurewill clearlybetourgethemedium tocongregateinthemore intensepartsofthe field. This hasbeen observed byMatteucciJforamediumconsistingof dropsofchloride ofiron dissolved inalcoholplacedinamedium ofolive oil. Thedropsofsolution were observed tomove towards thestrongest partsof the field. Magnetostriction. 474. Ifaliquidisplacedinamagnetic field,ityieldsunder the influence ofthemechanical forcesacting upon it,sothatwehave a phenomenonofmagnetostriction, analogoustothephenomenonofelectro- striction already explained (§203). Clearlytheliquidwillexpanduntil the pressureisdecreased byanamount^—H2ateachpoint,thenewpressure andthemechanical forcesresultingfrom themagneticfieldnowproducing equilibriuminthe fluid. Bymeasuringtheexpansionofaliquid placedin amagneticfieldQuincke hasbeen able toverifytheagreementbetween theoryandexperiment. *Wied. Ann. 24,p.347. XComptes Rendus, 36,p.917.+Wied. Ann. 34,p.401. J. 27 418 Induced Magnetism [ch.xii Molecular Theories. Poisson sMolecular Theory ofInducedMagnetism. 475. InChapter Vitwasfoundpossibletoaccount foralltheelectro- staticpropertiesofadielectric bysupposingittoconsist ofanumber of perfectly conductingmolecules. Poisson attemptedtoapplyasimilar explanationtothephenomenonofmagneticinduction. Poisson'stheory can,however, bedisprovedatonce,byaconsideration of thenumerical values obtained forthepermeability /a.Thisquantityis analogoustothequantityKofChapter V,sothat itsvaluemaybeestimated interms ofthemolecular structure ofthemagneticmatter. The factwith respecttowhich Poisson'stheorybreaks down istheexistence ofsubstances (namely,different kinds ofsoftiron)forwhich thevalue offiisvery large. Tounderstand thesignificanceoftheexistence ofsuch substances, letus consider the fieldproduced when auniform infinite slabofsuchasubstance isplacedinauniform field ofmagnetic force, sothatthefaceoftheslab is atright anglestothelines offorce. Ifthevalue of/*isvery large,the fall ofpotentialincrossingtheslab isverysmall.Throughoutthesupposed perfectly-conducting magneticmolecules thepotential would, onPoisson's theory,beconstant, sothat the fallofpotentialcould occuronlyinthe interstices between themolecules. Inthese interstices(cf. fig.46),the fallof potential perunitlengthwould becomparablewith that outside the slab. Hence avery largevalue of/acould beaccounted foronlybysupposingthe molecules tobepacked togethersocloselyastoleavehardly anyinterstices. Samplesofironcanbeobtained forwhich^isaslargeas4000;itisknown, from other evidence, thatthemolecules ofiron arenotsoclosetogetherthat suchavalue officould beaccounted forinthemannerproposed byPoisson. Itisworthnoticing, too,that Poisson'stheorydoesnotseem able,without modification, togiveanyreasonable account ofthephenomenaofsaturation, hysteresis,etc. Weber's Molecular Theory ofInducedMagnetism. 476.Atheory putforwardbyWeber shews much moreability than thetheoryofPoisson toexplainthe facts ofinducedmagnetism. Webersupposes that,even inasubstance which shews nomagnetisation, everymolecule isapermanent magnet,butthattheeffects ofthese different magnetscounteract oneanother, owingtotheir axesbeingscattered at random inalldirections. When thematter isplacedinamagneticfield each molecule tends, under theinfluence ofthe field, tosetitself sothat itsaxis isalongthe lines offorce, justasacompass-needle tends toset itselfalongthelines offorce oftheearth's magnetic field. Theaxes ofthe 475-477] Molecular Theories 419 molecules nolonger pointinalldirectionsindifferently,sothatthemagnetic fields ofthedifferent molecules nolonger destroy oneanother, andthebody asawhole shewsmagnetisation. This,onWeber'stheory,isthemagnetisa- tioninduced bytheexternal field offorce. Webersupposesthateach molecule, initsnormal state, isinaposition ofequilibriumunder theinfluence oftheforces from alltheneighbouring molecules, andthatwhen itismoved outofthisposition bytheaction of anexternal magnetic field, the forces from theother molecules tend to restore ittoitsoldposition.Itis,therefore, clear that solongasthe external field issmall, theangle throughwhich each axis isturned bythe action ofthefield willbeexactly proportionaltotheintensityofthe field, sothatthemagnetisationinduced inthebodywillbejust proportionalto thestrengthoftheinducingfield. Inother words, forsmall values ofH, [Xmust beindependentofH. There is,however, anatural limitimposed upontheintensityofthe inducedmagnetisation. Under theinfluence ofaveryintense field allthe molecules willsetthemselves sothat their axes arealongthelines offorce. Themagnetisationinduced inthebodyisnow ofaquitedefiniteintensity, andnoincrease oftheinducingfield can increase theintensityofthe inducedmagnetisation beyondthis limit. Thus Weber'stheoryaccounts quite satisfactorilyforthephenomenonofsaturation, aphenomenonwhich Poisson'stheorywasunable toexplain. 477. Inconnection with thisaspectofWeber'stheory, someexperi- ments ofBeetz areofgreat importance. Anarrow linewasscratched in acoatofvarnishcoveringasilver wire Thewirewasplacedinasolution ofasalt ofiron, arrangedsothat ironcould bedeposited electrolytically onthewire atthepointsatwhich thevarnish hadbeen scratchedaway. The effect wasofcourse todepositalongthin filament ofironalongthe scratch. If,however, theexperimentwasperformedinamagneticfield whose lines offorce were inthedirection ofthescratch, itwasfound not onlythatthefilament ofirondepositedonthewirewasmagnetised,but that itsmagnetisationwasveryintense. Moreover, oncausingapowerful magnetisingforce toactinthesame direction astheoriginal field, itwas found that theincrease intheintensityoftheinducedmagnetisation was very small, shewingthat themagnetisationhadpreviouslybeennearlyat thepointofsaturation. Nowif,asWebersupposed,themolecules ofironwerealready magnets before being depositedonthesilver wire, thenanymagneticforce sufficient toarrange them inorder onthewireoughttohaveproducedafilament in astate ofmagnetic saturation, whileif,asPoissonsupposed,themagnetism inthemolecules wasmerelyinduced bytheexternal magnetic field, then themagnetisationofthefilament oughttohave beenproportionaltothe 420 Induced Magnetism [CH.XII original field,andoughttohavedisappeared when the fieldwasdestroyed. Thus, asbetween these twohypotheses,theexperimentsdecideconclusively fortheformer. 478. Weber'stheoryisillustrated bythefollowing analysis. Consider amolecule which, inthenormal state ofthematter, has itsaxis inthedirection OP,and let the field offorce from theneigh- bouringmolecules beafield ofin- tensity D,thedirection ofthe lines offorcebeingofcourseparallelto OPNow letanexternal field of intensityHbeapplied,itsdirection beingadirection OAmakingan angleawith OP. The total field acting onthemolecule isnowcom- es poundedofDalongOPandH along OA.Fig. 115. Infig.115, letSO,OPrepresentHandDinmagnitudeand direction, thenSPwillrepresenttheresultant field, sothatthenewdirection ofthe axis ofthemolecule willbeSP. Supposethat there arenmoleculesper unitvolume, each ofmoment m.Originally, when theaxes ofthemolecules were scatteredindifferentlyinalldirections, thenumber forwhich the angleahadavalue between aanda4-dawashnsinada. These molecules nowhave their axespointinginthedirection SP,andtherefore makingan anglePSA(=6,say)with thedirection oftheexternalmagneticfield. The aggregate moment ofallthese molecules resolved inthedirection ofOA is accordingly hmn sinacos6da, andonintegrationtheaggregate moment ofallthemoleculesperunit volume, which isthesame astheintensityoftheinducedmagnetisationI, isgiven by ro=7r 1=1 ^mnsinacosdda(409).J<x=0 IfRisthevalue ofSP,measured onthesame scaleonwhichSOandOP representHandDrespectively,then B?=H* +Dn--2HD cosa. sothat,onchangingthevariable from atoR,wemust have therelation, obtainedbydifferentiation oftheaboveequation, RdR=HDsinada. 477-479] We alsohave cos6= sothatequation (409) becomes I=^mnIMolecular Theories 421 2RH R*+H*- Z)2 2H*DdR. Infig.115thelimits ofintegrationforRareR=D+HandR=D—H. If,however, H>I),then thepoint8falls outside thecircleAPR andthe limits forRareR=D+HandR=H-D. Onintegrating, 422 Induced Magnetism [CH.XII equilibrium.Itmight be,forinstance, thatoriginallythemolecule hadtwo possible positionsofequilibrium, OPandOQinfig.117.Supposethe molecule tobeinposition OPand tobe actedupon byagradually increasingforce insome direction OA.Atfirstthemolecule willturnfrom theposition OPtowards OA. But itmaybethat, assoon asthemolecule passes someposition OR, itsuddenly swings round andtakes upapositioninwhich itFlG -117# must beregardedasbeingdeflected from thepositionofequilibrium OQand notfromOP. Let itsnewpositionbeOS,then thedeflexionproducedis theangleSOP instead oftheangleROP which would begiven byWeber's theory. InMaxwell'soriginal discussion, nodistinction wasmade between the position OR,atwhich themagnetbrokeawayfrom itsoldpositionofequi- librium, andOS,thenewpositionofequilibrium. Maxwellaccordingly hadto assume that insomeunknownway,theforce ofrestitution broke down as soon asthemagnetreached theposition OR. Theimprovementofdistinguishing between theposition OR,thelimit of stabilityunder theoldpositionofequilibrium, andOS,thenewpositionof equilibrium, wasintroduced byEwing.InEwing'sform ofthetheory,no forces areneeded beyondthoseprovided bythemutual action ofthemagnets upononeanother. Oneither form ofthetheory,itisclear that theratio of/toHwill remainapproximatelyconstant until themoleculesbegintobreakawayfrom theiroriginal positionsofequilibrium. Assoon asthishappens,theinduced magnetismwill increase morerapidlythan theinducingforce—i.e./u,will increase withH,inagreement with observation. Ifthemagnetisingforce isnowremoved, themolecule intheposition OSwillnotreturn toitsoriginal position OP,but totheposition OQ. It willtherefore stillhave adeflexion QOP, called byMaxwell its"permanent set,"andthis willaccount forthe"retentiveness"ofthesubstance. Nomoleculartheoryofthiskind can,however, beregardedasatall complete. Weshall return tothediscussion ofmolecular theories ofmagne- tism inChapterxvi. EXAMPLES. 1.Asmall magnetisplacedatthecentre ofasphericalshell ofradiiaand b. Determine themagneticforce atanypoint outside the shell. 2.Asystemofpermanent magnetsissuch thatthedistribution inallplanes parallel toacertain planeisthesame. Prove that ifaright circular solidcylinder beplaced in thefieldwith itsaxisperpendiculartotheseplanes, thestrengthofthe field atanypoint inside thecylinderistherebyaltered inaconstant ratio. Examples423 3.Amagnetic particleofmoment mliesatadistance ainfront ofaninfinite block ofsoftironbounded byaplane face, towhich theaxis oftheparticleisperpendicular. Find theforce acting onthemagnet, andshew thatthepotential energyofthesystemis -m2(^-l)/8a3 (/x+l). 4.Thewhole ofthespace onthenegativeside oftheyzplaneisfilled with softiron, andamagnetic particleofmoment matthepoint (a,0,0)pointsinthedirection (cos a,0,sina).Prove thatthemagnetic potentialatthepoint x,y,zinside theiron is 2m zsina-(a—x)cosa 1+^{(a-xf+y^ +z^i• 5.AsmallmagnetofmomentMisheld inthepresenceofavery largefixedmass of softiron ofpermeability pwith avery large planeface :themagnetisatadistance a from theplanefaceandmakes anangle 6with theshortest distance from ittotheplane. Shew thatacertain force, andacouple -1)J/2sin6cos(9/8 (/*+1)a\ arerequiredtokeepthemagnetinposition. 6.Asmall sphereofradius bisplaced near acircuit which, when carryingunit current, would produce afield ofstrength ITatthepointwhere thecentre ofthesphereis placed. Shew that if<isthecoefficient ofmagneticinduction forthesphere,thepresence ofthesphereincreases theself-induction ofthewireby,approximately, 87r63K(3+27r*)7Z2 (3+4tt/c)2 7.Ifthemagneticfieldwithin abodyofpermeability /xbeuniform, shew thatany spherical portioncanberemoved andthecavityfilledupwith aconcentricspherical nucleus ofpermeability /xxandaconcentric shell ofpermeability /x2without affecting the external field,provided filiesbetween/xxandn2,andtheratio ofthevolume ofthenucleus tothat oftheshell isproperlychosen. Prove alsothat the field inside thenucleus is uniform, andthat itsintensityisgreaterorlessthan thatoutside accordingasp.isgreater orlessthanpv 8Asphereofradius ahasatanypoint (x,y,z)componentsofpermanent magneti- sation (Px, Qy, 0),theoriginofcoordinates beingatitscentre. Itissurrounded bya sphericalshell ofuniform permeability p,theboundingradii being aand b.Determine thevector potentialatanoutsidepoint. 9.Asphereofsoftiron ofradius aisplacedinafield ofuniform magneticforce paralleltotheaxisofz.Shew thatthelines offorce external tothespherelieonsurfaces ofrevolution, theequationofwhich isoftheform {*+2 -£#(")>^>=—. rbeingthedistance from thecentre ofthesphere. 10.Asphereofsoftironofpermeability p.isintroduced intoafield offorce inwhich thepotentialisahomogeneous polynomialofdegree ninx,y,z.Shew thatthepotential inside thesphereisreduced toitsoriginalvaluemultiplied by 2»+l rip+n+1" 11. Ifashell ofradii a,bisintroduced inplaceofthesphereinthelastquestion, shew thattheforce inside thecavityisaltered intheratio (2»+l)2 /x:(%/*+ra+l)(w/x+ra+/i)-»(» +J)(/i-l)2 (r) 424 Induced Magnetism [ch. xii 12.Aninfinitely longhollow ironcylinderofpermeability fx,thecross-sectionbeing concentric circles ofradiia,b,isplacedinauniform field ofmagnetic force thedirection ofwhich isperpendiculartothegenerators ofthecylinder. Shew thatthenumber of lines ofinduction through thespace occupied bythecylinderischanged byinserting the cylinderinthefield, intheratio 6»0»+l)«-o^0*-l)«:2/i{6«G»+l)-a^0*-l)}. 13.Acylinderofiron ofpermeability fihasforcross-section thecurve r=a(l+ecos2<9), where e2maybeneglected. Find thedistribution ofpotential when thecylinderisplaced inafield offorce ofwhich thepotential before theintroduction ofthecylinder was Q.=Axy 14.Aninfiniteelliptic cylinderofsoftiron isplacedinauniform field ofpotential —(Xx+Yy),theequationofthecylinder being— 2+T2=l.Shew that thepotentialof theinduced magnetismatanyinternal pointis -(u-1)f-r^—Xx+—?—rYy 15.Asolidelliptic cylinder whoseequationis|=agiven by x+ty=ccosh(£+irj) isplacedinafield ofmagneticforce whosepotentialisA(x2—y2 ).Shew that inthe space external tothecylinderthepotentialoftheinduced magnetism is -^c2cosech2(a+i3)sin4ae2(a~^~ f)cos27?, where coth2/3isthepermeability. 16.Asolidellipsoidofsoftiron, semi-axesa,b,candpermeability p,isplacedina uniform field offorceXparalleltotheaxisofx,which isthemajoraxis.Verifythatthe internal andexternalpotentialsoftheinducedmagnetisation are Q1=PA1x,Q=PAox, where !r <**a=T- 1Jo(at+ylrfiW+yl,)* (<?+&)¥° J*(a2d^r (a2+^(b2+f)i (c2+^' J^(a2+yl,)§(b2+yl,)h(c2+,},)$' P=(^-l)X/{( F-l)A 1+2(abc)-^ anaXistheparameteroftheconfocal through thepoint considered. 17.Aunitmagnetic poleisplacedontheaxisofzatadistance /from thecentre of asphereofsoftiron ofradius a.Shew that thepotential oftheinduced magnetism at anvexternal pointis n -i- f+xdtd6 1fx-la? TT^+1/2 OJZ+luZcos6—dH^2' wherez,warethecylindricalcoordinates ofthepoint. Find alsothepotentialatan internal point. 18.Amagnetic poleofstrength misplacedinfront ofanironplateofpermeability /*andthickness c.Ifthispolebetheoriginofrectangular coordinatesx,y,and ifxbe perpendicularandyparalleltotheplate, shew that thepotential behind theplateis given bya-1O=m(l-p2 )jol_°p^Jtct,wherep= fi+1' CHAPTEE XIII THEMAGNETIC FIELD PRODUCED BYELECTRIC CURRENTS (1)Experimental Basis. 480. Sofarthesubjectsofelectricity andmagnetismhavebeendeveloped asentirely separate groupsofphysical phenomena. Althoughthemathe- matical treatment inthetwocases hasbeenonparallel lines,wehave not hadoccasion todealwithanyphysicallinksconnectingthetwo series of phenomena. The first definite link ofthekindwasdiscovered byOersted in1820. Oersted's discoverywasthe factthat acurrent ofelectricity produceda magneticfield initsneighbourhood. Thenature ofthis field canbeinvestigatedinasimplemanner. We firstdouble backonitself awire inwhich^ acurrent isflowing (fig. 118, 1).Itis found thatnomagneticfield isproduced. Next weopentheend into asmall plane loopPQRS (fig.118, 2).Itisfound that atdistances from theloopwhich are great comparedwith itslinear dimensions, such aloopexercises thesame magnetic forces asamagnetic particleofwhich the axis isperpendiculartotheplanePQRS, andthemoment isjointly proportionaltothestrengthofthecurrent and totheareaPQRS ThesinglecurrentflowinginthecircuitOPQRSTis obviously equivalenttotwocurrents ofequal strength,theoneflowingin thecircuitOPSTobtained byjoiningthepointsPand S,andtheother flowingintheclosed circuit PQRSP. The former current isshewn, by thepreliminary experiment,tohave nomagnetic effects, sothat thewhole magneticfieldmaybeascribed tothesmall closed circuit PQRS.Q i R (2) Fig. 118. Fig. 119.426TheMagnetic Fieldproduced byElectric Currents[en.xm 481. Instead ofregardingthis field asduetoaparticleofmomentjointly proportionaltotheareaPQRS and tothecurrent-strength, wemayregard itasdue toasmall magnetic shell, coincidingwith theareaPQRS, andof strength simply proportionaltothecurrentflowinginPQRS. 482. Next, letusconsider thecurrentflowinginaclosed circuit ofany shape weplease, andnotnecessarilyin oneplane.Letuscover intheclosed circuit byanarea ofanykindhavingthe circuit foritsboundary, and letuscut upthisarea intoinfinitelysmall meshes bytwosystemsoflines.Acurrent of strengthiflowinground theboundary circuit, isexactly equivalenttoacurrent ofstrengthiflowinground eachmesh in thesame direction asthecurrent inthe boundary. For,ifweimaginethislatter systemofcurrents inexistence, anyline such asABintheinterior willhave twocurrentsflowing through it,one from each ofthetwomeshes which itseparates,andthese currents will beequalbut inoppositedirections. Thus allthecurrents inthe lines which have been introduced intheinterior ofthe circuit annihilate one another asregardstotal effect, while thecurrents inthosepartsofthe meshes which coincide with theoriginalcircuitjustcombine toreproduce theoriginalcurrentflowinginthis circuit. Thus theoriginalcircuit isequivalent,asregards magnetic effect, toa systemofcurrents, oneineach mesh. Bytakingthemeshessufficiently small, wemayregardeachmesh asplane,sothat themagneticeffect ofa currentcirculatinginitisknown :themagneticeffect ofthecurrent ina singlemesh isthat ofamagneticshell ofstrength proportionaltothecurrent andcoincidinginpositionwith themesh. Thus, byaddition, wefind that thewholesystemofcurrents produces thesamemagneticeffects asasingle magneticshellcoincidingwith thesurface ofwhich theoriginalcurrent- circuit istheboundary,andofstrength proportionaltothecurrent. This shell, then, producesthesamemagneticeffect astheoriginal singlecurrent. Themagneticshell isspokenofasthe" equivalent magneticshell." Thuswehave obtained thefollowingresult : "Acurrentfloivinginanyclosed circuit producesthesamemagnetic field asacertain magnetic shell,known asthe' equivalent magneticshell.' This shellmagbetaken tobeanyshellhavingthecircuitforitsboundary,its strength being uniform andproportionaltothatofthecurrent" 481-484] Experimental Basis 427 LawofSigns.Ifanobserver isimaginedtostand onthat side ofthe "equivalent magneticshell" which contains thenegative poles,thecurrent flowsround him inthesame direction asthat inwhich thesunmoves round anobserverstandingontheearth's surface inthenorthernhemisphere. Wecan also state thelawbysayingthat todrive anordinary right- handed screw{e.g.acork-screw)inthedirection ofmagnetisationofthe shell, thescrew would have tobeturned inthe direction ofthe current.c <<_^. Current Thelawofsigns expresses afact ofnature, nota-J-+ -f-+ mathematical convention. Atthesametime,itmust be noticed thatthelawdoesnotexpress thatnature shews anypreference inthisrespectforright-handed over left-.' i-ij mi-iiii i Direction ofMagnetisationnancled screws. 1woconventions havealready beenmadef. , ,77 indeciding which aretobecalled thepositivedirections ofcurrent andofmagnetisation, and ifeither ofthese conventions hadbeendifferent, theword"right-handed"inthelawofsigns would have had tobereplaced by"left-handed.n 483. Since, by§346,anysystemofcurrents canberegardedasthe superpositionofanumber ofsimpleclosed currents, itfollows that the magneticfieldproduced byanysystemofcurrents canalwaysberegardedas thatproduced byanumber ofmagnetic shells, each ofuniformstrength. Electromagnetic UnitofCurrent. 484. Ifiisthestrengthofthecurrentflowinginacircuit, and<£the strengthoftheequivalent magnetic shell, then <f>=ki, where kisaconstant, which ispositiveifthelawofsigns juststated has beenobeyedindeterminingthesignsof c/>and i. Inthesystemofunitsknown asElectromagnetic, wetakek=l, and define aunitcurrent asonesuch that theequivalent magneticshell isof unitstrength. Thestrengthofacurrent, inthese units, istherefore measured byitsmagneticeffects.Obviouslythestrengthmeasured inthis waywillbeentirelydifferent from thestrength measuredbythenumber of electrostatic units ofelectricitywhichpassagiven point.This latter method ofmeasurement istheelectrostatic method. Afulldiscussion ofsystemsof units willbegivenlater(§585);atpresentitmaybestated that acurrent which isofunitstrength when measuredelectromagneticallyinc.G.s. units is ofstrength3x1010(very approximately) when measuredelectrostatically. The practicalunit ofcurrent, theampere, is,asalready stated, equalto3x109 electrostatic units ofcurrent, sothat theelectromagneticunit ofcurrent is equalto10amperes. 428TheMagnetic Fieldproduced byElecfric Currents[en.xm Aunitchargeofelectricityinelectromagneticunits willbetheamount ofelectricitythatpassesafixedpoint perunittime inacircuit inwhich an electromagneticunit ofcurrent isflowing.Itisthereforeequalto3x10 electrostatic units.10 Fig. 121. -S^.Work done inthreading aCircuit. 485. Infig.121 letthethick linerepresentacircuit inwhich acurrent isflowing,andletthethin linethrough ^ ._ thepointPrepresenttheoutline of /' 'x any equivalent magnetic shell,P being anypointintheshell. Letus imaginethatwethread thecircuit by( anyclosedpathbeginningandending \ atP,thispath being represented by thedotted lineinthefigure. Atevery \^ pointofthispath except P,wehavea "***--. fullknowledgeofthemagneticforces. Itwillbeconvenient toregardtheshell ashavingadefinite, although infinitesimal, thickness atP.LetP+,P.denote thepointsin which thepathintersects thepositiveandnegativefaces ofthe shell. Thenwemaysaythattheforces areknown atallpointsof thepath, exceptoverthesmallrangei+P. Theoriginalcurrent can,however, berepresented byany number ofequivalent magnetic shells, foranyshell iscapableof representingthecurrent, provided onlyithasasboundarythe circuit inwhich thecurrent isflowing. Letanyotherequivalentshell cutthepathinthepoints Q+Q-. From ourknowledgeoftheforces exerted bythis shell,wecandetermine the forces exerted bythecurrent atallpointsofthepath exceptthose within therangeofQ+Q_. Inparticular wecandetermine theforces overtherange P+FL,and itisatonceobvious thatonpassingtothelimit andmakingthe rangeP+P-infinitesimal, theforces atthepointsP+,P.,andatallpointsonthe infinitesimalrange P^R.must beequal. Obviouslytheforces arealso finite. Thework done onaunitpoleintakingitround thecompletecircuit fromP.back toP., isaccordinglythesame asthatdone intakingitfromP. round thepathtoP+.Thiscanbecalculated bysupposingtheforces tobe exerted bythe firstequivalent shell, forthepathisentirelyoutside this shell. Ifthepotentialdue totheshell isClPatP+and isHP_atP,the workdone isflP—flP_. Now n,thepotentialoftheshell atanypoint, is,asweknow(§419), equaltoia>,Avhere <oisthesolidanglesubtendedbytheshelland iistheFig. 122. 484-486] Magnetic Potential ofField 429 current, measured inelectromagneticunits. Thechangeinthesolidangle aswepassfromP_toP+is,asamatter ofgeometry, equalto4>ir.Thus ftp-fiP_=47Ti(410). Thework done intakingaunitpoleround thepathdescribed isaccord- ingly4nri. Magnetic Potential ofaField duetoCurrents. 486. Letusfixuponadefiniteequivalentshell torepresentacurrent of strengthi.Letusbringaunitpolefrom in- finitytoanypointA%byapathwhich cuts theequivalentshell inpoints P,Q,...Z.For simplicity,letusatfirstsupposethat ateach ofthesepointsthepath passesfrom the ^V——^//^*^^ :>^>^L positivetothenegativesideofthe shell, and letthepointsonthetwosides oftheshell be denoted, asbefore, byP+>P_;Q+,Q_;andFlG -123 - soon. Then, ifildenotes themagnetic potential due totheequivalent shell, theworkdone inbringingtheunitpolefrominfinitytoP+willbeftp.In+ thelimit i+andR.arecoincident, sothatthework intakingtheunitpole onfromi+toP_isinfinitesimal. IntakingitfromP_toQ+work isdone of amount £lQ—ftP_,fromQ+toQ_,thework isinfinitesimal, andsoon,until ultimately wearrive atA.Thus the totalwork done inbringingtheunit poletoAis aP+(n Q-Op )+(nR-nQ)+...+(ciA-nz), "T" +~"+" — or,rearranging,is &A+(^+~^P_)+(fy>+-&Q_)+•••• Now each oftheterms 0,P+—£lP_>£lQ—£lQ_,etc. isequal byequation (410)to4>iri, sothat ifnisthenumber ofthese terms, thewholeexpression isequalto £lA+4<7rni. Replacing QAbyiw,where wisthesolidanglesubtended bytheshell at A,wefind forthepotentialatAduetotheelectric current (co+4>7rn)i (411). Ifthepathcutstheequivalentshellntimes inthedirection from+to— , andmtimes intheopposite direction, thequantitynmust bereplaced by n—m. Expression (411) shews thatthepotentialatapointisnotasingle-valued function ofthecoordinates ofthepoint. The forces, which areobtainedby differentiation ofthispotential, are,however, single-valued. 430TheMagnetic Fieldproduced byElectric Currents[on.xin Fig. 124.Current ininfinite straightwire. 487. Asanillustration oftheresults obtained, letusconsider the magneticfieldproduced byacurrentflowinginastraightwirewhich isof suchgreat lengththat itmayberegardedasinfinite, thereturn current being entirelyatinfinity. Letustake theline itself foraxis ofz.Anysemi-infiniteplanetermi- nated bythislinemayberegardedasanequivalent magneticshell. Letus fixonanyplaneandtake itastheplaneofxz. Consider anypointPsuch thatOP,theshortest distance fromPto theaxis ofz,makes anangle6with Ox.Thecone through Pwhich issubtended bythe semi-infinite plane Ox, isbounded bytwoplanes—oneaplane through Pandtheaxisofz;theother aplane throughPparalleltotheplanezOx. These contain anangle 7r—6,sothatthesolidanglesubtended bytheplane zOx atPis2(-7T— 9).Givingthis value to coin formula(411),weobtain asthemagnetic potentialatP a={2(tt-6)+4wtt}i. rsry Since -^—= itisclear that there isnoradial magnetic force, andthe dr& force atanypointinthedirection of6increasing d£l_2i rdd r' This result isotherwise obvious. Ifthework done intakingaunitpole round acircle ofcircumference 27rr istobe4nri, thetangentialforce at every pointmust be—. 488- This result admits ofasimple experimentalconfirmation. LetPQR beadiscsuspendedinsuch awaythat theonlymotion of which itiscapableisoneofpurerotation about a long straightwire inwhich acurrent isflowing. Onthisdisc letussupposethatanimaginaryunit poleisplacedatadistance rfrom thewire. There willbeacouple tendingtoturn the disc, the 2imoment ofthiscouple being—xrov2i.Similarly ifweplaceaunitnegative poleonthediscthere is acouple—2i. Onplacingamagnetised bodyonthedisc, there willbeasystemofcouples consistingofone of moment 2iforevery positive poleandoneofmoment —2iforevery negative pole.Since thetotalchargeFig. 125. 487-489] MagneticPotential ofField 431 inanymagnetisnil, itappearsthat theresultantcouplemust vanish, so thatthedisc willshewnotendencytorotate. Thiscaneasilybeverified. Circular Current. 489. Letusfindthepotentialduetoacurrent ofstrengthiflowingina circle ofradius a.Theequivalent magneticshellmaybesupposedtobea hemisphereofradius abounded bythis circle. Thepotentialatanypointontheaxisofthe circle canreadilybefound. Foratapointontheaxis distant rfrom thecentre ofthe circle, thesolidangle wsubtendedbythe circle isgiven by <y=27r(l-cosa)=27r(l-r Va2+ sothatthepotentialatthispointis Q=2ni(1- sfaF+r'- Thisexpressioncanbeexpandedinpowersofr bythebinomial theorem. Weobtain thefollowing expansions: ifr<a. (a2a3 v2.4... 2n\aj ifr>a, _ft.fla2..w+11.3...2n- 1fa\2nFig. 126. 2r 2.4... 2n.(412), .(413). From this itispossibletodeduce thepotentialatanypointinspace. Letustakespherical polar coordinates, takingthecentre ofthe circle as origin,andtheaxisofthe circle asthe initial line6=0.Inside thesphere r=a,thepotentialisasolution ofV2f2=which issymmetrical about the axis 6=0,andremains finite attheorigin.Itisthereforecapableof expansionintheform O=XAnrnPn(cos 6). o Alongtheaxiswehave 6=0,sothat thisassumed value ofHbecomes n=ZAnrn , o andthecoefficients maybedeterminedbycomparisonwithequation (412). 432TheMagneticFieldproduced byElectric Currents[ch.xiii Thusweobtain forthepotentials, H=2m\1--P,(cos6)+\-P3(cos0)- ... +(-l)n+1 -)Pm+1(cos0)+...\...(414), when r<a,and [1a22.4...2w Va 3a4 ^^i?(cos6>)-g-^(cos^)-... f(-1)^1 ;3 •;•2n~1 (ffPm(cos0)+ ...}...(415),2.4 when r>a. ml Atpointssonear totheoriginthat—maybeneglected,thepotentialis a* Q=2iri(1 cosa»)-«(!-£). where 2;=rcos^,andthemagneticforce isauniform force—-~-= dz a paralleltothe axis. Fig. 127.Solenoids. 490Acylinder, wounduniformlywith wire throughwhich acurrent canbesent,iscalled a"solenoid." Consider firstacircularcylinderotradius aand height h,havingawire coiled round itattheuniform rate ofnturnsperunitlength,thewirecarryinga current iLet zbeacoordinatemeasuringthe distance ofanycross-section from thebase ofthe solenoid. Then thesmalllayerbetween zandz+dz, beingofthickness dz,willcontain ndzturns ofwire. Thecurrents flowinginallthese turnsmaybere- gardedasasinglecurrent nidzflowinginacircle, this circlebeingofradius aandatdistance zfrom thebase ofthesolenoid. Themagnetic potential ofthiscurrent maybewritten down from theformula ofthelastsection, and thepotentialofthewhole solenoid follows byintegration. 491. Endless Solenoid. Inthelimitingcase inwhich thesolenoid isof infinitelength (orinwhich theends aresofarawaythat thesolenoid may betreated asthoughitwere ofinfinitelength),the fieldcanbedetermined inasimplermanner. Consider firstthefield outside thesolenoid. Intakingaunitpoleround anypathoutside thesolenoid whichcompletelysurrounds thesolenoid, the work done is,by§485, 4nri.Thecurrentflowing perunitlengthofthe Faf QR' F3 R489-492] Galvanometers 433 solenoid isni.Ingeneral weareconcerned with cases inwhich this isfinite nbeing very largeand ibeing verysmall. Thequantity immayaccordingly beneglected,andwecansupposethat thework done intakingunitpole round thesolenoid iszero. Itfollows that theforce outside thesolenoid canhavenocomponent at right anglestoplanes throughtheaxis,andclearly, byasimilarargument, thesame must betrue inside thesolenoid. Hence thelines ofinduction must lieentirelyintheplanes throughtheaxisofthe solenoid. Fromsymmetry,there isnoreasonwhy thelines ofinduction atanypoint shouldconverge towards, rather thandiverge from, the axis, orvice versa. Hence thelines ofinduction willbeparallel totheaxis,andtheforce atevery pointwillbeentirely p paralleltotheaxis. Letthe linesPQR, P'Q'R' infig.128beradii meetingthe axis, the linesPF, QQ',RR'being paralleltotheaxisandeach oflengthe.LettheFlG^3 magneticforcesalongthese lines beFltF2andF3 respectively. Intakingunitpoleround theclosedpathPP'Q'QP thework done is i?e-F26, andsince thismust vanish, wemust haveF1=F2.Hence theforce atall pointsoutside thesolenoid must bethesame;itmust bethesame asthe force atinfinity andmustconsequentlyvanish. Thus there isnoforce atall outside thesolenoid. Intakingunitpoleround theclosedpathPP'R'RP, thework done is F3e,andthismust beequalto4nrnie, sothatwemust haveF3e=4tirni. Thus theforce atanypointinside thesolenoid isaforce 4nrniparalleltotheaxis. Thus the field offorcearising from aninfinite solenoid consists ofa uniform field ofstrength4nrni inside thesolenoid, therebeingnofield atall outside. Theconstruction ofasolenoidaccordingly suppliesasimple wayof obtainingauniformmagneticfield ofanyrequired strength. Galvanometers, 492.Agalvanometerisaninstrument formeasuringthestrengthofan electric current, themethod ofmeasurementusually beingtoobserve the strengthofthemagneticfieldproduced bythecurrent bynotingitsaction onasmall movablemagnet. There arenaturallyvarious classes andtypesofgalvanometers designed tofulfil variousspecial purposes. j. 23 434TheMagnetic Fieldproduced byElectric Currents[oh.xiii TheTangent Galvanometer, 493. Inthetangent galvanometerthecurrent flows inavertical circular coil, atthecentre ofwhich asmallmagneticneedle ispivoted soastobefree toturn inahorizontalplane. Before use,theinstrument isplacedsothattheplaneofthecoilcontains thelines ofmagneticforce oftheearth's field. Theneedleaccordinglyrests intheplaneofthe coil.When thecurrent isallowed toflow inthe coil anew field isoriginated,the lines offorcebeingatright anglestothe planeofthe coilandtheneedle willnowplaceitself soastobeinequi- librium under thefieldproduced bythesuperpositionofthetwo fields—the earth's fieldandthe fieldproduced bythecurrent. Astheneedle canonlymove inahorizontalplane, weneed consider onlythehorizontalcomponentsofthetwo fields. LetH,asusual, denote thehorizontal componentoftheearth's field. Let ibethecurrentflowing inthe coil,measured inelectromagnetic units, letabetheradius and letn bethenumber ofturns ofwire. Near thecentre ofthe coilthe field produced bythecurrentis,by§489,auniform field atright anglesto theplaneofthe coil, ofintensity.The total horizontal field istherefore compoundedofafield of strengthHintheplaneofthe coil,andafield of strengthatright anglestoit. Theresultant willmake anangle6with theplane ofthe coil,where /2-rrin\ tanfl= %' (416),xz andtheneedle will setitselfalongthelines offorce ofthe field. Thus the needle will, Avhen inequilibrium, make anangle6with theplaneofthe coil,where 6isgiven byequation (416).Ifweobserve 6wecandetermine ifromequation (416).Wehave i=?tan0 (417), where Gisaconstant, known asthegalvanometer constant, itsvalue being Theinstrument iscalled thetangent galvanometerfrom thecircum- stance that thecurrent isproportionaltothetangentoftheangle6. 493,494] Galvanometers 435 Thetangent galvanometerhastheadvantage that allcurrents, nomatter how small orhowgreat,canbemeasured withoutalteringtheadjustment oftheinstrument. Adisadvantageisthatthereadingsarenotverysensi- tivewhen thecurrents tobemeasured arelarge—onlyaverysmallchange inthereadingisproduced byaconsiderablechangeinthecurrent. Let thecurrent beincreased byanamount di,and letthecorresponding change in6bedd,thenfromequation (417), d0 sothat ifiislarge, -pissmall. Thus, althoughtheinstrument maybe used forthemeasurement oflarge currents, themeasurements cannot be effected withmuchaccuracy. Asecond defect oftheinstrument iscausedbythecircumstance that the fieldproduced bythecurrent isnotabsolutelyuniform near thecentre ofthe coil. Ifaistheradius ofthecoil,and bthedistance ofeitherpole ofthemagnetfrom itscentre, thepoleswillbeinapartofthe field in which theintensitydiffers from that atthecentre ofthe coilbyterms of b3 theorder of— .For instance, ifthemagnetisoneinchlong,while the Cb coilhasadiameter of10inches, theintensityofthefield willbedifferent from thatassumed, byterms oftheorder of(yV)3 >sothatthereadingwillbe subjecttoanerror ofabout onepartinathousand. Byreplacingthesinglecoilofthetangent galvanometer bytwoormore parallel coils, itispossibletomake the field intheregioninwhich the magnet moves, asuniform asweplease.Itisthereforepossible, although attheexpenseofgreat complication,tomake atangent galvanometer which shall read toanyrequired degreeofaccuracy TheSine Galvanometer. 494. The sinegalvanometerdiffers from thetangent galvanometerin havingitscoiladjustedsothat itcanbeturned about avertical axis. Before thecurrent issentthroughthe coil,theinstrument isturned until theneedle isatrestintheplaneofthe coil.The coil isthen inthedirec- tionoftheearth's field atthepoint. Assoon asacurrent issentthroughthe coil,theneedle isdeflected, as inthetangent galvanometer.The coil isnowslowlyturned inthedirection inwhich theneedle hasmoved, until itovertakes theneedle, andassoon astheneedle isagainatrest intheplaneofthe coil,areadingistaken, givingtheangle throughwhich the coilhasbeen turned. Let6bethis angle,then theearth's fieldmayberesolved intocomponents, Hcos6in 28—2 436TheMagnetic Fieldproduced byElectric Currents[ch.xiii theplaneofthe coilandHsin6atright anglestothisplane. Since the needle rests intheplaneofthe coil, thelatter component must bejust neutralized bythe field setupbythecurrent, thisbeing,aswehave seen, entirelyatright anglestotheplaneofthe coil.Weaccordinglyhave . 2irinHsinQ= ,a sothatwemust have FT" i=g-sm0(418), where G,thegalvanometer constant, hasthesamemeaningasbefore. Thisinstrument hasthedisadvantagethat itcannot beused tomeasure TT currentsgreaterthan -~ •^ is>however, sensitive over thewholerange throughwhich itcanbeused :ifd6istheincrease in6caused byachange diini,wehave d6=-^sec6di,11 sothatthegreaterthecurrent themore sensitive theinstrument. Thegreat advantageofthisform ofgalvanometer, however, isthatwhen thereadingistaken themagnetisalwaysinthesamepositionrelative tothe field setupbythecurrent inthe coil. Thus thedeviations from uniformityofintensityatthecentre ofthe fielddonotproduce anyerror inthereadingsobtained :theyresultonlyinthegalvanometerconstant havingavalue different from thatwhich ithassofarbeensupposedto have. Butwhen oncetherightvalue hasbeenassignedtotheconstant G, equation (418)willbetrueabsolutely,nomatter howlargethemovable needle maybeincomparisonwith the coil. Othergalvanometers. 495. There arevarious othertypesofgalvanometersinusetoserve variouspurposesother than theexact measurement ofacurrent. For full descriptionsofthese thereader maybereferred tobookstreatingthe theoryofelectricity andmagnetismfrom themoreexperimentalside. The following maybebriefly mentioned here• I.TheD'Arsonval Galvanometer. This instrument istypicalofaclass ofgalvanometerinwhich there isnomoving needle, themoving partbeing the coil itself, which isfree toturn inastrong magneticfield. The coil issuspended byatorsion fibrebetween thepolesofapowerfulhorseshoe magnet. When acurrent issentthroughthe coil,the coil itselfproduces thesame field asamagnetic shell, and sotends toset itself across the 494,495] Galvanometers 437 lines offorce ofthepermanent magnet,thismotionbeingresisted byno forcesexceptthetorsion ofthe fibre. II.TheMirror Galvanometer. This isagalvanometer originally designed byLord Kelvin forthemeasurement ofthesmall currents used inthetrans- mission ofsignals bysubmarine cables. Thedesign is,initsmain outlines, identical with thatofthetangent galvanometer, but, tomake theinstrument assensitive aspossible,the coil ismade ofagreatnumber ofturns offine wire,wound ascloselyaspossibleround thespaceinwhich theneedle moves, andtheneedle issuspendedasdelicatelyaspossible byafine torsion-thread. Tomake theinstrument stillmore sensitive, permanent magnetscanbearrangedsoastoneutralizepartoftheintensityofthe earth's field. Theinstrument isreadbyobservingthemotion ofarayof lightreflected from asmall mirror which moves with theneedle :itisfrom thisthattheinstrument takes itsname. Inthemost sensitive form ofthis instrument avisible motion ofthespotoflightcanbeproduced byacurrent of10~10amperes. III. The Ballistic Galvanometer. This instrument does notmeasure thecurrentpassingatagiven instant, butthe total flow ofelectricity whichpasses during aninfinitesimal interval. Iftheneedle isatrest in theplaneofthe coil,acurrent sentthroughthe coil will establish a magneticfieldtendingtoturn theneedle outofthisplane. Solongas theneedle isapproximatelyintheplaneofthe coil,thecouple acting on theneedle willbeproportionaltothecurrent inthe coil :letitbedenoted byci,where iisthecurrent. Then ifa>istheangular velocityoftheneedle atany instant, weshall haveanequationoftheform ink--j-=ci,at where mk* isthemoment ofinertia oftheneedle.Integrating throughthe small interval oftimeduringwhich thecurrent maybesupposedtoflow, weobtain mk2Q=c Iidt. HereOistheangular velocitywithwhich theneedle starts intomotion, and Jidtisthetotal current whichpasses throughthe coil. Thus thetotal flow Iidtcanbeobtained bymeasuring O,andthisagaincanbeobtained by observingtheangle throughwhich theneedleswingsbeforecomingtorest attheendofitsoscillation. 438TheMagnetic Fieldproduced byElectric Currents[ch.xiii Vector-potential ofaField duetoCurrents. 496.From theformulae obtained in§446 forthevector-potentialofa uniformmagnetic shell,wecanatonce writedownexpressionsforthevector- potentialofafieldduetocurrents. For,by§483,thefieldduetoanysystemofcurrents mayberegardedas thefieldduetoanumber ofshells ofuniformstrength,sothatthevector- potentialatanypointwillbethesum ofthevector-potentials duetothese different shells. Hence if<£, cf>',...arethestrengthsofthevarious shells, thevector-potentialatanypointPhascomponents (cf.§446) where thesummation isover alltheshells, anddx,ds'refer toanelement of theedgeofashell ofstrength <p,thiselementbeingatadistance rfrom the pointP. Theequations justfoundmay clearly bereplaced by F=(i-pdsJras .(419),Jrds J7'ds) where dsisnowanelement ofanywire orlinear conductor inwhich a current ofstrengthiisflowing, andtheintegrationisnowalongallthe conductors inthe field. Bytheuseofequations (376),wemayatonce obtain thecomponentsof magneticforce orinduction atanypoint x,y,z'intheforms _d_H_dGa" dijdz' /'£'©=-£©$**(420)- = i Mechanical Action intheField. Ampere'srulefortheforcefromacircuit. 497. Let(x,y,z)bethepositionofanyelement dsofacircuit, and letPbeanypoint {x, y',z')infreespace. From equations (420)itfollows that themagneticforce atPmaybe regardedasmadeupofcontributions from eachelement ofthecircuit such thatthecontribution from theelement dsathascomponents l'{a7®£-aT'0)l}^etc-'etc - 496-498] Mechanical Action 439 Onputtingr2=(x-a/)2+(y—y'f+(z- z'f,anddifferentiating,these componentsbecome ids\y—y'dzz—z'dy]ids [z—z'dxx—xdz) rdsdy) ids (z—z'dxx—xdz) Letusdenotex—xy—yz—z byh>whifhtthesebeingthedirection- cosines ofthelineOP,and let~ ,-~ ,-r- dsdsds bedenoted byl2,m2,n2,thesebeingthe direction-cosines ofds.Then thecom- ponentsofforce(421) become ids ids . . (nJt-nJJ, -^(^m2-4mi) ...(422). Clearlytheresultant isaforce atright angles both toOPandtods,and ofamount^(423), where 6istheanglebetween OPand ds. Thus thetotal force atPmayberegardedasmade upofcontributions such as(423) fromeachelement ofthe circuit. This isknown asAmpere's law. Mechanical action onacircuit. 498.Weareatpresent assumingthecurrents tobesteady,sothat action andreaction maybesupposedtobeequal andopposite.Itfollows that theforce exerted ataunitpoleatPuponthecircuit ofwhich the element dsispart,mayberegardedasmade upofforces ofamount isin perunitlength, actingatright anglestoOPandtods. Ifwehavepolesof strength matP,m'atP', etc.,theresultant force onthecircuit maybe regardedasmadeupofcontributions imsin6im'sinO' perunitlength.Theresultant ofthese forcesmaybeputintheform iHsinx whereHistheresultantmagnetic intensityat ofallthepoles m,m,etc., andxistheanglebetween thedirection ofthisintensityand ds.This resultant force acts atright anglestothedirections ofHand ofds. 440TheMagnetic Fieldproduced byElectric Currents[ch.xm 499.Wehave found thattheforcefrom awhole circuit isthesame asif eachelement idscontributed aforce idssind/r2 ,andthe force onawhole circuit isthesame asifeachelement were acted onbyaforceiHsin%.But solongaswearedealing onlywithcompleteclosed currents,itisimpossible todiscover what theactual forcefrom oronasingle element ofthecurrent will be.Inalaterchapter weshallregardacurrent asastream ofelectrons inmotion. Theelement idswillthenbetreated asasmallnumber ofmoving electrons, andweshallbeable toshew thattheactual forces associated with thesingleelement idsareexactlyidentical with thosejust found. Energy ofaSystem ofCircuits carrying Currents. 500. Theenergyofamagnetic field, aswehave seen(§470),is g^jjffi(cc+/32+ry*)dxdydz (424). Iftheenergyresides inthemedium, thisexpression mayberegardedas theenergyofthe field,nomatter how this field isproduced.Ifthefield is produced wholly bycurrents, expression (424)mayberegardedastheenergy ofthesystemofcurrents. Asweshallnow see, itcanbetransformed ina simple way,soastoexpresstheenergyofthefield interms ofthecurrents bywhich thefield isproduced. Theintegral throughallspace,asgiven byexpression (424),maybe regardedasthesumoftheintegralstaken over allthetubes ofinductionby whichspaceisfilled. The lines ofinduction, aswehave seen, willbeclosed curves, sothatthetubes areclosed tubularspaces. Ifdsisanelement oflength,anddSthecross-section atanypoint,ofa tube ofunitstrength, wemayreplace dxdydz bydSds, andinstead ofinte- gratingwithrespecttodSwemaysumover alltubes. Thusexpression (424) becomes ^XJV(a!+/32 -f-72)d£}tfs, where thesummation isover allunittubes ofinduction. IfH2=a.-+/32+72 , wehave,bythedefinition ofaunit tube,fiHdS=1,sothat li(a2+/3°-+r)dS=fiH'dS=H, andtheintegralbecomes Now IHds istheworkperformedonaunitpoleintakingitonceround thetube ofinduction, and thisweknow isequalto4nr1'i, where Ifi isthe sum ofallthecurrents threadedbythetube, taken each with itsproper sign. Thus theenergybecomes£2(2't). 499-501] Energy 441 This indicates that foreverytime thataunittube threads acurrenti, acontribution ^iisadded totheenergy Thus thewholeenergyis h^iN (425), where thesummation isover allthecurrents inthe field, andNisthe number ofunittubes which thread thecurrent i. 501.Wehave seen thatashell ofstrength <j)isequivalent,asregards thefieldproducedatallexternalpoints,toacurrenti,if<f>=i.Theenergy ofasystemofcurrents hashowever been found tobe^ZiN, whereas the energyofasystemofshells wasfound(§450) tobe -it<j)N (426). The difference ofsigncanreadilybeaccounted for.Letusconsider a singleshell ofstrength </>,and letdSbeanelement ofarea,anddnanelement oflengthinside theshellmeasurednormallytotheshell. Atanypoint just outside theshell, letthethreecomponentsofmagneticforce be a,j3,y,the firstbeingacomponent normal totheshell,andtheothersbeing components indirections which lieintheshell. Onpassingtotheinside oftheshell, the normal induction isdiscontinuousowingtothepermanent magnetismwhich must besupposedtoreside onthesurface oftheshell. Thus inside theshell, wemaysupposethecomponentsofforce tobeS+-, ft,y,wherefx,isthe permeabilityofthematter ofwhich the shell iscomposed,andSisthe forceoriginatingfrom thepermanent magnetismofthe shell. Thecontribution totheenergyofthe fieldwhich ismade bythespace inside theshell is where theintegralistakenthroughouttheinterior oftheshell;or Thiscanberegardedasthesumofthreeintegrals, (i)^rl\\'lStdndS \ 1f[[faU-^ +^dnaK- (427). (fi)8^111 (T+^+A^JdndS <iii>~ \ 442TheMagnetic Fieldproduced byElectric Currents[ch.xiii Onreducingthethickness oftheshellindefinitely, Sbecomesinfinite, for atanypointoftheshell, Sdn=—(differenceofpotentialbetween thetwoforces ofshell) =— 4>TT(j), sothatSbecomes infinite when thethickness vanishes. Thus onpassingtothelimit, the firstintegral becomes infinite. Thisquantity is,however, aconstant, foritrepresentsthe energy requiredtoseparatetheshell into infinitesimalpolesscattered at infinity. Thesecondintegralvanishes onpassingtothelimit, andsoneed notbe further considered. Thethirdintegralcanbesimplified. Wehave M!SttdndS=UNSdn )dS - Now ISdn=— 47r<£,while 11adS istheintegralofnormal induction over theshell, andmaytherefore bereplaced byN,thenumber ofunittubes of induction from theexternal field, whichpassthroughthe shell. Thus the thirdintegralisseen tobeequalto Incalculating expression (424)when theenergyisthat ofasystemof currents, thecontribution from thespace occupied bytheequivalent mag- netic shells isinfinitesimal. Thus alltheterms which wehave discussed representdifferences between theenergiesofshells andofcircuits. Terms such asthe firstintegralsofscheme (427) represent merelythat theenergiesaremeasured from different standardpositions.Inthecase of theshells, wesupposetheshells tohave apermanent existence, andmerely tobebroughtintoposition.The currents, ontheother hand, have tobe created, aswell asplacedinposition. Beyondthis difference, there isan outstandingdifference ofamount<£iVforeach circuit, and thisexactly accounts forthedifference betweenexpressions (425) and(426). 502. Letussupposethatwehave asystemofcircuits, which weshall denote bythenumbers 1,2,....Letussupposethatwhen aunitcurrent flows through 1,alltheother circuitsbeingdevoid ofcurrents, amagnetic field isproducedsuch that thenumbers oftubes ofinduction which cross circuits 1,2,3,...are "11 J-"12 >-"13 >•••• 501-503] Coefficients ofInduction 443 Similarly, when aunitcurrent flowsthrough 2,letthenumbers oftubes ofinduction be -"21 >-"22 >-"23 )•••• Thetheorem of§446shews atonce that '12coseL21= \\—dsds, etc(428).r Ifcurrents i1,i2,...flowthroughthecircuitssimultaneously, and ifthe numbers oftubes ofinduction which cutthecircuits areNuN2,N3,...,we haveNx=Luix+L12i2+L13i3+...}r••\~hJiU)•N2=L2lix+Z^+^23*3+••>etc.J Theenergyofthesystemofcurrents is =iXh(Zut'i+L12i2+...), =i£nVs+£12^2+%L 22i22+ (430). Coefficients ofInduction. 503.The coefficient Lniscommonlycalled thecoefficient ofself-induc- tion(or,morebriefly,theself-inductance)ofcircuit 1,whileL12iscalled thecoefficient ofmutual induction ofthetwocircuits 1and 2.The value ofL12foranypairofcircuits canbecalculated from formula(428). Asanexample,consider theimportantcase oftwo circular wires, radii a,a'inparallel planes,thelinejoiningtheir centresbeing perpendicularto theplanes andoflength d,b.Formula (428) gives ^f2*aa'cos(d-e')ddde' JV2 oJo[a2+a'2+b2-2aa' cos(6-6')f =2tt2naa'cosijrdty o[a2+a'2+b2-2aacos\jr]%' AiflffPut °°= («+ay+b»*-*(—+) andwereadilyfind t tt>\k fhlT2sin2(£-l ,,Ll2=^ir(aa)*c =deb Jo(l-c2sin2<^Y = 4>ir(aa'^l(^-c)K(o)-^E(c), whereK(c),E(c)arethecomplete ellipticfunctions tomodulus c. When thecirclesnearly coincide, bissmall andaand a'arenearly equal. Thus cisnearly equaltounity,andE(c) approximatestounity.Put c'-(1-c2)K 444TheMagnetic Fieldproduced byElectric Currents[oh.xiii sothat c'issmall, then Kic)^'£—r-/1*& j,-'o(l-c2sin2 </>)*Jo(cos2 </>+c'2sin2 (/>)i ofwhich theapproximatevalue isfound tobelog(4/c'). Ifristhenearest distanceapartofthetwo circles, wehave,when ris small, c=r/2a,sothat K(c)=\og(8a/r), L12=4<7ra(\og—-2\ (430a). and 504. Itmightbeexpectedthatwecould obtain thevalue ofLninany problem bymakingthetwocircuits 1and2coincide, butthisprovesnotto bethecase; thevalue oftheintegralinequation (428), where theintegralis taken twice round thesame circuit, isalwaysinfinite. Asaninstance, we maynotice thatonputtingr=intheformulajustobtained, wefindL12=oo . Wecanreadilyseewhythismust be.When there isonlyonecurrent flowing, wehave iZfch"=g£(//(«*+P2+Y2 )dxdydz, each sideofthisequation representingtheenergyofthecurrent. Near to thewire, atasmall distance rfromit,themagneticforce is2i\rsothat a2+/S2+72=4di /r2 .Thus theenergycontained within athinringformed of coaxalcylindersofradii rur2,bent soastofollow thewireconveyingthe current, willbe /*fff419 „rdrddds, ottJ JJr2 where theintegrationwithrespecttorisfrom rxtor2,thatwithrespectto 6isfrom to27r,andthatwithrespecttosisalongthewire.Integrating, wefindenergy fii2log{r^n) perunitlength, andontakingrx=0,theenergyisseen tobeinfinite. Supposethat thewire hasacircular cross-section ofradius a,andthat thecurrent isuniformlydistributed over thiscross-section. Acircle ofradius rinside thewire willenclose acurrent ir2/a2 ,sothatthemagneticforce at distance rfrom thecentre willbe2ir/a2 ,and (aj Onintegratingthisfrom r= tor=awefindthat there ismagnetic energyinside thewire ofamount£/A'2perunitlength,where p!isthe magnetic permeabilityofthematerial ofthewire. Hence thetotalenergy perunitlengthinside acylinderofradius r2enclosingthewire is ^'t'2+/u'2log(r 2/a) (4306). 503-505] Coefficients ofInduction 445 Evenwhen aisfinite this stillbecomes infinite when r2ismade infinite— i.e.when themagneticfieldextends toinfinity. Thus theself-inductionper unitlengthofastraightwire infreespaceisinfiniteexcept when the magneticfield islimitedbythepresenceofother conductors. Supposethat thereturn current iscarried by.aconcentriccylinderof radius bsurroundingthewire. The total flowofcurrentthroughacircle of radiusgreaterthan biszero, sothatthere willbenomagneticforce outside thecylindrical conductor, andthemagneticfield willbelimitedbythecylinder r=b.Theenergy perunitlengthisnowgiven byformula(4306) with r2put equaltob,sothatthecoefficient ofself-inductionperunitlengthis L=%fi+2filog(b/a) (430c), andthis isfinite forallfinite values ofband a. 505.Theenergyofthemagneticfieldproduced byacurrent iinawire willalwaysbethesum oftheenergiesofthemagneticfield inthewireand ofthemagneticfield outside thewire. Ifthecurrent isuniformlydistributed inthewire, theformerenergywillalwaysbe^fi'i2asin§504.Thus L,the self-induction ofawire oflengthI,willalwaysbeoftheform L=Wl+L' (430d), where theterm\\xlarises from the field inside thewire,andL'arises from thefield outside thewire. When thecircuit liesentirelyinoneplaneandtheradius ofcross-section ofthewire issmall asimplevalue canbeobtained forL'.Let8denote the curve formed bythecentres ofthecross-sections ofthewire,and letS'denote thecurve formed bytheinneredgeofthewire intheplaneinwhich thecircuit lies.Then itwillbeeasilyverified thatthemagneticforce atany pointinside S'isthesame asifthewhole current %flowedalongthecurve 8. Hence thenumber oftubes ofinduction which flowthroughS'when the current flows inthewire isthesame asifacurrent iflowed in8,andsois equaltoitimesL\ 2where L'Kisthecoefficient ofmutual induction between 8and 8'.Thus informula (430 d),L'willbethecoefficient ofmutual induc- tionbetween thecircuits Sand S'. Asanexample,letusfindthe coefficient ofself-induction inawire of lengthlirawhose cross-section isacircle ofradiusr,bent intoacircle of radius a.ThecurveSisacircle ofradius a,thecurve 8'isaconcentric circle ofradius a—r.Byformula (430 a), Z'=4™ (log—-2 8a sothat L=ira/jL+47ra flog2 j. 446TheMagneticFieldproduced byElectric Currents[ch.xiii Asasecondexample,letusfindthecoefficient ofself-induction ofarect- angularcircuit ofsides a,bmade ofwire ofcircular cross-section ofradius r. Inthiscasethecircuit Swillbearectangleofsides a,b,while thecircuit S' isacoplanarconcentric rectangleofsides a—r,b—r.Weevaluate L'the coefficient ofmutual induction ofSandS'from formula(428). There isno contribution frompairsofelements onsidesperpendiculartooneanother, since forthese cose=0;thewhole value ofL'iscontributedbyparallel pairs ofelements. Fortwoparallellines oflengthsI,Vatdistance hapart, wefind "dsds' [& ft1'dxdx hiJ-V'[(ac'-ocY +h*$ hi -hi•i00""~Jbsmn-1- dx af=-Wh =(I+V)sinh-1l-±l-{l- V)sinh-1^-[4A2+(l+Ijf +[4A*+(J_Z')»]1. OnmakingI—I'small, andreplacingsinh-1byitslogarithmic value, this becomes 21log*+(ft+A')*_2(Ji+&)*+2A. Byrepeateduseofthisformula wefind U=-8(a+b)+8(a2+62 )^-4alog[a+(a2+62 )*] -46log[6+(a2+t2 )-]+4(a+6)log— , andthecoefficient ofself-induction isnowgiven by L=(a+b)p,+L'. 505a.Formula (430 c),expressingtheself-inductionperunitlengthofa circular wirewith aconcentric return, canbeputintheformL=h/uf+L', where L'=2filog(b/a). IfKistheelectrostaticcapacity perunitlengthofthecondenser formed bythewireand itssurrounding cylinder, wehave, from§82, KK= 2log(b/a)' where kistheinductivecapacityoftheinsulatingmaterialsurroundingthe wire. Thus L'=^(430*). Itisnotamere accident that thissimplerelation holds.Suppose we solve theelectrostaticproblem bythemethod ofconjugatefunctions(§312). 505,505a] Examples 447 Theappropriatetransformation isreadily found tobe(cf.§318) U+iV= Cons.+2logr+1i0, where x=rcos6,y=rsin6.Inthistransformation Umaybetaken tobethe electrostaticpotentialduetounitcharge perunitlength, andVwillclearlybe themagnetic potentialduetounitcurrent. Itfollows atoncethatthevalue ofX2+Y2atanypointwhen there isunitcharge perunitlengthisthesame asthevalue ofa2+/32atthesamepointwhen there isunitcurrentflowing, andrelation(430 e)isatonceseen tobetrue. Theargumentcanbeapplied equallywelltoanyconjugate-function trans- formation whatever. Thus relation(430 a)isseen tobeuniversally true for anystraightconductoraccompanied byaparallelreturn. EXAMPLES. 1.Acurrent iflows inavery long straightwire. Find theforces andcouplesit exerts upon asmall magnet. Shew that ifthecentre ofthesmall magnetisfixed atadistance cfrom thewire,it hastwofreesmall oscillations about itspositionofequilibrium,ofequal period '-A/ whereMk2isthemoment ofinertia, and/uthemagnetic moment,ofthemagnet. 2.Twoparallel straightinfinite wires convey equal currents ofstrengthiinopposite directions,their distance apart being2a.Amagnetic particleofstrength pandmoment ofinertia ink"1isfreetoturnabout apivotatitscentre, distant cfrom each ofthewires. Shew thatthetime ofasmall oscillation isthat ofapendulumoflengthIgiven by 4:ialjx=mgk'icl . 3.Twoequal magnetic poles areobserved torepel each other withaforce of40dynes when atadecimetreapart.Acurrent isthen sentthrough 100metres ofthin wire wound intoacircular ringeightdecimetres indiameter andtheforce ononeofthepoles placedatthecentre is25dynes. Find thestrengthofthecurrent inamperes. 4.Regardingtheearth asauniformly and rigidly magnetised sphereofradiusa, anddenotingtheintensityofthemagneticfieldontheequator byH,shew thatawire surroundingtheearth along theparallelofsouth latitude A,andcarryingacurrent i from west toeast,wouldexperiencearesultant force towards thesouth poleofthe heavens ofamount QnaiH sinXcos2X. 5.Shew that atanypoint along alineofforce, thevector potentialduetoacurrent inacircle isinversely proportionaltothedistance between thecentre ofthecircle and thefootoftheperpendicular from thepoint ontotheplaneofthe circle. Hence trace thelines ofconstant vectorpotential. 6.Acurrent iflows inacircuit intheshapeofanellipseofareaAandlengthI. Shew that theforce atthecentre isnil/A. 448TheMagneticField produced byElectric Currents[ch.xiii 7.Acurrent iflows round acircle ofradius a,andacurrent i'flows inaverylong straightwire inthesameplane. Shew thatthemutual attraction is47m'(seca-1),where aistheanglesubtended bythecircle atthenearestpointofthestraight wire. 8.If,inthelastquestion,thecircle isplaced perpendiculartothestraight wirewith itscentre atdistance cfromit,shew thatthere isacouple tending tosetthetwowires in thesameplane,ofmoment 2irii'a2 lcor2nii'c, accordingasc>or<a. 9.Along straightcurrent intersects atright angles adiameter ofacircular current, andtheplaneofthecircle makes anacute angle awiththeplane throughthisdiameter andthestraightcurrent. Shew thatthecoefficient ofmutual induction is 4it{cseca-(c2sec2a-a2 )2}or47rctan (--- ), accordingasthestraightcurrentpasseswithin orwithout thecircle, abeing theradius of thecircle, and cthedistance ofthestraight current from itscentre. 10.Prove thatthecoefficient ofmutual induction between apairofinfinitely long straightwires andacircular oneofradius ainthesameplane andwith itscentre ata distance b(>a)fromeach ofthestraight wires,is 87r(6-V62-a2 ). 11.Acircuit contains astraight wire oflength 2aconveyingacurrent. Asecond straight wire,infinite inboth directions, makes anangleawith thefirst,and their common perpendicularisoflengthcandmeets the firstwire initsmiddlepoint. Prove thattheadditional electromagneticforces onthe firststraight wire,duetothepresence ofacurrent inthesecond wire, constitute awrench ofpitch „/.,asina\/._,asina 2asina-ctan~1 /sin2atan~1 ./ . _asincA/ (asina-ctan1 J/s 12.Two circular wires ofradiia,bhave acommon centre, andarefreetoturnonan insulatingaxiswhich isadiameter ofboth. Shew thatwhen thewirescarry currents i,i',acouple ofmagnitude *?('-*5)* isrequiredtoholdthem with their planesatright angles,itbeing assumed thatb\aisso small that itsfifthpowermaybeneglected. 13.Two circular circuits areinplanesatright anglestothelinejoiningtheir centres. Shew thatthecoefficient ofinduction =27r(a2-c2 )Pcos26d6 \/a2sin26+c2cos26 where a,carethelongest andshortest lineswhich canbedrawn from onecircuit tothe other. Find theforcebetween thecircuits. 14.Twocurrentsi,Vflowround twosquares each ofside a,placed with theiredges paralleltooneanother andatright anglestothedistance cbetween their centres. Shew thattheyattract withaforce ,j\/2a2+c2 ,a2+2c2 ow..,|W2a2+c2 ,a2+2c2 ) 1a2+c-cVa2+c2J V«2+< 15.Acurrent iflows inarectangularcircuit whose sides areoflengths 2a,2b,and thecircuit isfreetorotate about anaxisthroughitscentreparalleltothesides oflength 2a.Another current i'flows inalong straightwireparalleltotheaxisandatadistance Examples 449 dfrom it.Prove thatthecouple requiredtokeeptheplaneoftherectangleinclined at anangle <£totheplane throughitscentre andthestraight current is %ii'abd(b2+cP)sm(f> bi+di-2b2d2 cos2cf>' 16.Two circular wires liewith theirplanes parallel onthesamesphere, andcarry opposite currentsinversely proportionaltotheareas ofthecircuits. Asmall magnet has itscentre fixed atthecentre ofthesphere, andmovesfreely ahout it.Shew that itwill beinequilibrium when itsaxiseither isatright anglestotheplanesofthecircuits, or makes anangle tan-1^withthem. 17.Aninfinitely long straight wireconveys acurrent and liesinfront ofandparallel toaninfinite block ofsoftironbounded byaplaneface. Find themagnetic potentialat allpoints, andtheforcewhich tends todisplacethewire. 18.Asmallsphereofradius bisplacedintheneighbourhood ofacircuit, which whencarrying acurrent ofunitstrength would produce magneticforceHatthepoint where thecentre ofthesphereisplaced. Shewthat,ifkisthecoefficient ofinduced magnetizationforthesphere, thepresenceofthesphere increases thecoefficient ofself- induction ofthewirebyanamountapproximately equalto 87rb\(3 +27rK)E2 19.Acircular wire ofradius aisconcentric withasphericalshell ofsoftironofradii 6andcIfasteady unitcurrent flowround thewire,shew thatthepresence oftheiron increases thenumber oflines ofinduction throughthewireby 2ff2a4 (c3_&3)(^_!)(M+2) 63{(2M+1)Qx+2)Cs-20*-1)262} approximately,where aissmall compared with 6and c. 20.Arightcircularcylindrical cavityismade inaninfinite mass ofiron ofperme- ability nInthiscavity awirerunsparalleltotheaxisofthecylinder carrying asteady current ofstrength/.Prove thatthewire isattracted towards thenearestpartofthe surface ofthecavitywithaforceperunitlength equalto 2(,*-l)/2 0*+i)rf' wheredisthedistance ofthewirefrom itselectrostatic imageinthecylinder. 21.Asteadycurrent Cflows along onewireandback along another one, inside a long cylindrical tube ofsoftiron ofpermeability p,whose internal andexternal radii are axanda2,thewires being paralleltotheaxis ofthecylinder andatequaldistance aon oppositesides ofit.Shew thatthemagnetic potentialoutside thetube willbe F=^ sin6+-|sin38+^sin50 +..., Hence shew thatatube ofsoftiron, of150cm.radius and5cm.thickness, forwhich the effective value ofnis1200c.G.s., willreduce themagneticfield atadistance, duetothe current,tolessthan one-twentieth ofitsnatural strength. 9Q 450TheMagneticFieldproduced byElectric Currents[oh.xtii 22.Awire iswound inaspiralofangle aonthesurface ofaninsulating cylinderof radiusa,sothat itmakes ncomplete turns onthecylinder. Acurrent iflows through thewire. Prove thattheresultant magneticforce atthecentre ofthecylinderis 27rm a(l+7r2»2tan2a)i alongtheaxis. 23.Acurrent ofstrengthiflows along aninfinitely long straight wire,andreturns in aparallelwire. These wires areinsulated andtouchalong generators thesurface ofan infinite uniform circularcylinderofmaterial whose coefficient ofinduction ish.Prove that thecylinderbecomes magnetizedasalamellar magnet whosestrengthis2irkiJ{l+2Trk). 24.Afinewire covered with insulating material iswound intheform ofacircular disc,theendsbeingatthecentre andthecircumference. Acurrent issentthrough the wiresuch that/isthequantityofelectricitythat flowsperunittime across unitlength ofanyradius ofthedisc. Shew thatthemagneticforce atanypointontheaxisofthe disc is 27r/{cosh-1(seca)—sina}, where aistheangle subtended atthepoint byanyradius ofthedisc. 25. Coils ofwire intheform ofcircles oflatitude arewound upon asphere and produce amagnetic potential ArnPnatinternalpoints when acurrent issentthrough them. Find themode ofwinding andthepotentialatexternalpoints. 26.Atangent galvanometeristohave fiveturns ofcopper wire,and istobemade so thatthetangentoftheangleofdeflection istobeequaltothenumber ofamperes flowing inthe coil. Iftheearth's horizontal force is-18dynes, shew thattheradius ofthe coil must beabout 17"45cms. 27.Agivencurrent sentthrough atangent galvanometerdeflects themagnet through anangle6.Theplaneofthecoil isslowly rotated round thevertical axisthroughthe centre ofthemagnet. Prove that if6>\iv,themagnetwilldescribe completerevolu- tions, but if6<j7r,themagnotwilloscillate through ananglesin-1(tan#)oneach sideof themeridian. 28.Prove that,ifaslighterror ismade inreading theangleofdeflection ofatangent galvanometer,thepercentageerror inthededuced value ofthecurrent isaminimum ifthe angleofdeflection isj7r. 29.The circumference ofasinegalvanometeris1metre :theearth's horizontal magneticforce is"18c.G.s. units. Shew thatthegreatestcurrent which canbemeasured bythegalvanometeris4-56amperes approximately. 30.Thepolesofabattery (ofelectromotive force 2-9voltsandinternal resistance 4ohms) arejoinedtothose ofatangent galvanometer whose coilhas20turns ofwireand isofmean radius 10cms. :shew thatthedeflection ofthegalvanometerisapproximately 45°.The horizontalintensityoftheearth's magneticforce is1*8andtheresistance of thegalvanometeris16ohms. 31.Atangent galvanometerisincorrectly fixed, sothatequal andoppositecurrents giveangular readingsaand/3measured inthesame sense. Shew thattheplaneofthe coil,supposed vertical, makes anangleewith itsproper positionsuch that 2tan e=tana+tan/3. 32. Ifthere beanerror ainthedetermination ofthemagnetic meridian, findthe truestrength ofacurrent which isiasascertained bymeans ofasinegalvanometer. Examples 451 33.Inatangent galvanometer, thesensibilityismeasured bytheratio oftheincre- ment ofdeflection totheincrement ofcurrent, estimatedperunit current. Shew that thegalvanometerwillbemost sensitive when thedeflection is— ,andthat inmeasuring thecurrent given byagenerator whose electromotive force isE,andinternal resistance B,thegalvanometerwillbemost sensitive ifthere beplacedacross theterminals ashunt ofresistance _BRr_ E-H(R+rY where ristheresistance ofthegalvanometer, andHistheconstant oftheinstrument. What isthemeaningoftheresult ifthedenominator vanishes orisnegative? 34.Atangent galvanometer consists oftwoequalcircles ofradius 3cms. placed ona common axis8cms.apart.Asteadycurrent sent inoppositedirections throughthetwo circles deflects asmall needleplaced ontheaxismidway between thetwo circles through ananglea.Shew that iftheearth's horizontal magneticforce beRinc.G.s. units, then thestrengthofthecurrent inc.G.s. units willbe125.£ftana/367r. 35.Agalvanometercoilofnturns isintheform ofananchor-ring described bythe revolution ofacircle ofradius babout anaxis initsplanedistant afrom itscentre. Shew that theconstant ofthegalvanometer ~ aJK cn2wdn2udu (£=&/«) (8nj3k2a)[{l+k2 )E-(l-tf) K]. 29—2 CHAPTEE XIV INDUCTION OFCURRENTS INLINEAR CIRCUITS Physical Principles. 506. Ithasbeen seen that,onmovingamagnetic poleabout inthe presenceofelectric currents, there isacertain amount ofwork doneonthe polebytheforces ofthe field. Iftheconservation ofenergyistobetrue of afield ofthiskind, theworkdoneonthemagnetic polemust berepresented bythedisappearanceofanequalamount ofenergyinsome otherpartofthe field. Ifallthecurrents inthe fieldremainsteady,there isonlyonestore ofenergyfromwhich thisamount ofwork canbedrawn, namelytheenergy ofthebatteries which maintain thecurrents, sothat these batteries must, duringthemotion ofthemagnetic poles, giveupmore than sufficientenergy tomaintain thecurrents, theexcess amount ofenergy representing work performedonthepoles. Oragain,ifthebatteriessupply energyata uniform rate, partofthisenergy must beused inperforming work onthe moving poles,sothat thecurrents maintained inthecircuits willbeless thantheywould beifthemoving poleswere atrest. Letussupposethatwehaveanimaginary arrangement bywhich addi- tional electromotive forces canbeinserted into, orremoved from, each circuit asrequired,and letussupposethat thisarrangementismanipulatedsoasto keepeach current constant. Consider firstthecaseofasinglemovablepoleofstrength mandasingle circuit inwhich thecurrent ismaintained atauniformstrengthi.Ifa>is thesolidangle subtended bythecircuit atthepositionofthepoleatany instant, thepotential energyofthepoleinthefield ofthecurrent ismiw, so that inaninfinitesimal interval dtofthemotion ofthepole,theworkper- formed onthepolebytheforces ofthe field ismi-=-dt.Thecurrent which hasflowed inthistime isidt,sothattheextrawork donebytheadditional batteries isthesame asthat ofanadditional electromotive forcem-r- . at 506,507] Physical Principles 453 Thus themotion ofthepolemust have setupanadditional electromotive force inthecircuit ofamount —m-57 ,tocounteract which theadditional electromotive forces areneeded. The electromotive force—m-7-which at appearstobesetupbythemotion ofthemagnetsiscalled theelectromotive forceduetoinduction. Thenumber oftubes ofinduction which startfrom thepoleofstrength m is47rm,andofthese anumber mmpassthroughthecircuit. Thus ifnisthe number oftubes ofinduction whichpassthroughthecircuit atanyinstant, theelectromotive forcemaybeexpressedintheform—-=- . Soalso ifwehaveanynumber ofmagnetic poles,oranymagnetic system ofanykind,wefind,byaddition ofeffects such asthatjustconsidered, that dN there willbeanelectromotive force 7—arisingfrom themotion ofthe wholesystem,whereNisthetotalnumber oftubes ofinduction which cut thecircuit. Itwillbenoticed thattheargument wehavegiven supplies noreason fortakingNto bethenumber oftubes ofinduction rather than tubes offorce. But ifthenumber of tubes crossingthecircuit istodepend onlyontheboundaryofthecircuit wemust take tubes ofinduction andnottubes offorce,fortheinduction isasolenoidal vector while theforce,ingeneral,isnot. dN 507. The electromotive force ofinduction—-5—hasbeensupposedto bemeasured inthesame direction asthecurrent, andoncomparingthis with thelawofsigns previously givenin§483,weobtain therelation between thedirections oftheelectromotive force round the circuit, andof thelines ofinduction across the circuit. Themagnitudeanddirection of theelectromotive force aregiveninthetwofollowinglaws: Neumann's Law. Whenever thenumberoftubesofmagneticinduction which areenclosedbyacircuit ischanging,there isanelectromotiveforce acting round thecircuit, inaddition totheelectromotiveforce ofanybatteries whichmaybeinthecircuit, theamountofthisadditional electromotiveforce being equaltotherateofdiminutionofthenumberoftubes ofinduction enclosed bythecircuit. Lenz's Law. The'positivedirectionoftheelectromotiveforcefj—jand thedirection inwhich atubeofforcemust pass throughthecircuit inorder to becounted aspositive,arerelated inthesamewayastheforwardmotion and rotation ofaright-handedscrew. 454 Induction ofCurrents inLinear Circuits[ch.xiv Ifthere isnobatteryinthe circuit, thetotal electromotive force willbe dN— ,andthecurrentoriginated bythiselectromotive force isspokenofas CLZ an"induced"current. 508. Inorder thatthephenomenaofinduced currents maybeconsistent withtheconservation ofenergy,itmustobviouslybeamatter ofindifference whether wecause themagneticlines ofinduction tomove across thecircuit, orcause thecircuit tomove across thelines ofinduction. ThusNeumann's Lawmustapply equallytoacircuit atrestandacircuit inmotion. Soalso ifthecircuit isflexible, and istwisted about soastochangethenumber of lines ofinduction whichpassthrough it,there willbeaninduced current of which theamount willbegiven byNeumann's Law. 509. Forinstance ifametalringisspunabout adiameter, thenumber oflines ofinduction from theearth's fieldwhichpassthroughitwillchange continuously,sothat currents will flow init.Furthermore, energywillbe consumed bythese currents sothatworkmust beexpendedtokeepthering inrotation. Againthewheels andaxles oftwocars inmotion onthesame lineofrails, togetherwiththerails themselves, mayberegardedasforming aclosed circuit ofcontinually changingdimensions intheearth'smagnetic field. Thus there willbecurrentsflowinginthe circuit, andthere willbe electromagneticforcestendingtoretard oraccelerate themotions ofthecars. 510. If,aswehavebeen ledtobelieve, electromagnetic phenomenaare theeffect oftheaction ofthemediumitself, andnotofaction atadistance, itisclear thattheinduced current mustdependonthemotion ofthelines of force, andcannotdependonthemanner inwhich these lines offorce arepro- duced. Thus induction must occurjustthesamewhether themagneticfield originatesinactual magnetsorinelectric currents inotherpartsofthe field. Thisconsequenceofthehypothesisthattheaction ispropagated throughthe medium isconfirmed byexperiment—indeed inFaraday's original investiga- tionsoninduction, thefieldwasproduced byasecond current. 511. Letussupposethatwehavetwo circuits 1,2,ofwhich 1contains abattery andakeybywhich thecircuit canbeclosed andbroken, while circuit 2 remains permanently closed, andcontains a galvanometerbutnobattery. Onclosing thecircuit 1,acurrent flowsthroughcircuit 1,setting upamagneticfield. Some ofthe tubes ofinduction ofthis fieldpassthrough circuit 2,sothatthenumber ofthese tubes Key changesasthecurrent establishes itself inaery circuit 1,andthegalvanometerin2will• " accordinglyshew acurrent. When thecurrent in1hasreached itssteady 507-513] General Equations 455 value, asgiven byOhm's Law, thenumber oftubesthroughcircuit 2willno longer varywith thetime, sothat there willbenoelectromotive force in circuit2,andthegalvanometerwillshew nocurrent. Ifwebreak the circuit1,there isagainachangeinthenumber oftubes ofinductionpassing throughthesecond circuit, sothat thegalvanometerwillagain shew a momentarycurrent. General Equations ofInduction inLinear Circuits. 512. Letussupposethatwehaveanynumber ofcircuits1,2, Lettheir resistances beRuR2,...,letthem contain batteries ofelectro- motive forcesElfE2,...,and letthecurrentsflowinginthem atanyinstant DOtj,12)•••. Thenumbers oftubes ofinduction N1}J¥2,...which cross these circuits aregiven by(cf.equations (429)) J^j=Zuij.+L12i2+L13i3+ ...,etc. Incircuit 1there isanelectromotive forceExduetothebatteries, andan dN electromotive force r-1due toinduction. Thus thetotal electromotiveat force atanyinstant isE-^—-7—1 ,and this,byOhm's Law,must beequalto Rii x.Thuswehave theequation Ei--r (LnH+Ll2i2+L13i3+...)=Rih (431). Similarlyforthesecond circuit, E2-jt(Lnh.+L,2i2+L23iz+...)=R2i2(432), andsoonfortheother circuits. Equations (431), (432),...mayberegardedasdifferentialequationsfrom which wecanderive thecurrents i1}i2,...interms ofthetimeandthe initial conditions. Weshall consider variousspecialcases ofthisproblem. Induction inaSingle Circuit. 513. Ifthere isonlyasingle circuit, ofresistance Randself-induction L, equation (431) becomes E-j t(Li 1)=Ri1 (433). Letususethisequationfirst tofindtheeffect ofclosingacircuitpre- viouslybroken.Supposethat before thetime t=thecircuit hasbeen open, butthat atthisinstant itissuddenlyclosed with akey,sothatthe current isfreetoflowunder theaction oftheelectromotive force E. 456 Induction ofCurrents inLinear Circuits[ch.xiv The firststepwillbetodetermine theconditions immediatelyafter the d circuit isclosed. Since -n(Lii) is,byequation (433), afinitequantity,it follows thatLixmust increase ordecrease continuously,sothatimmediately afterclosingthecircuit thevalue ofLixmust bezero. Tofindthewayinwhich ixincreases, wehavenowtosolveequation (433), inwhich E,LandRare allconstants, subjecttotheinitial condition that ix=when t=0.Writingtheequationintheform weseethatthegeneralsolution is -5*E-Ri1=CeL whereGisaconstant, andinorder that ixmayvanish when t—0,wemust haveG=E,sothatthesolution is ;,=|(i-e-z<).(434). Itwillbeseen Fm. 131.Thegraphofixasafunction oftisshewn infig.131. that thecurrent risesgraduallytoitsfinal valueEjR given byOhm's Law, this rise being rapidifLissmall, butslow ifLis great.Thuswemaysaythattheincrease in thecurrent isretarded byitsself-induction. Wecanseewhythisshould be.Theenergy ofthecurrent i±is^Lif,andthis islargewhen Lislarge.Thisenergy representsworkper- formed bytheelectric forces: when thecurrent isi1}therate atwhich these forcesperform work isE\,aquantitywhich doesnotdependonL.ThuswhenLislarge,agreat time isrequiredfor theelectric forces toestablish thegreatamount ofenergyLi*. Asimple analogy maymake theeffect ofthis self-induction clearer. Lettheflowof thecurrent berepresented bytheturningofamill-wheel, theaction oftheelectric forces being represented bythefallingofthewaterbywhich themill-wheel isturned. Alarge value ofLmeanslarge energyforafinitecurrent, andmust therefore berepresented by supposingthemill-wheel tohavealargemoment ofinertia. Clearly awheel withasmall moment ofinertia willincrease itsspeed uptoitsmaximum speed with great rapidity, while forawheel withalargemoment ofinertia thespeedwillonlyincreaseslowly. AlternatingCurrent. 514. Letusnextsupposethattheelectromotive force inthecircuit is notproduced bybatteries, butbymovingthe circuit, orpartofthecircuit, inamagneticfield. IfNisthenumber oftubes ofinduction ofthe 513,514] Induction inaSingle Circuit 457 externalmagneticfieldwhich areenclosedbythecircuit atany instant, theequationis -j t(Li 1+N)=Ri1 (435). Thesimplestcase arises when iVisasimple-harmonic function ofthe time, proportionalletussaytocospt.Wecansimplifytheproblem bysup- posingthatNisoftheformC(cospt+isinpt).The realpartofNwill giverisetoarealvalue ofi1}andtheimaginary partofNtoanimaginary value ofvThus ifwetakeN=Geiptweshall obtain avalue fori,ofwhich therealpartwillbethetruevaluerequiredforilt Assuming N=G(cospt+isinpt)=Geipt ,theequation becomes ^jt{IA 1^G^)=Riu andclearlythesolution willbeproportionaltoeipt.Thus thedifferential operator-=rwillactonlyonafactor eipt ,and willaccordinglybeequivalentto multiplication byip.Wemayaccordinglywrite theequationas -ip(Li 1+Ceipt )=Ril, asimple algebraic equationofwhich thesolution is ._-piGeipt h~R+Lip' Letthemodulus andargumentofthisexpression bedenoted bypand%, sothatthevalue ofthewholeexpressionisp(cos%+*sin%).Thevalue of p,themodulus, isequal (§311) totheproductofthemoduli ofthefactors, so that pGP~ V.R8+Ly' while theargument x>being equal (§311) tothesumoftheargumentsof thefactors, isgiven by *=p*-\~tan_1(^)- Thesolutionrequiredfori,istherealtermpcos%,sothat h.=pcosX =—j£-^sinf^-tan-f^[ (436). Theelectromotive forceproduced bythechangeinthenumber oftubes oftheexternal field is ~~^=-jt(0cospt)=pGsinpt 458 Induction ofCurrents inLinear Circuits[cii.xiv Thus, ifself-induction wereneglected,thecurrent, asgiven byOhm's Law,would be pG. andthisofcourse wouldagreewith thatwhich would begiven byequation (436)ifLwere zero. Themodificationsproduced bytheexistence ofself-induction arerepre- sented bythepresenceofLinexpression (436), andaretwoinnumber. In the firstplacethephaseofthecurrentlagsbehind that oftheimpressed electromotive forcebytan-1-~ ,andinthesecondplacetheapparentresist- ance isincreased fromRto*JR2 -f-Dp1 . 515. The conditions assumed inthisproblemaresufficientlyclose to those which occur intheworkingofadynamotoillustrate thisworking. A coilwhich formspartofacompletecircuit iscaused torotaterapidlyina magneticfield insuchawayastocutavarying number oflines ofinduction. Thequantity £-maybesupposedtorepresentthenumber ofalterna- tionspersecond. Inthesimplecase ofatwo-polealternator this willbe equaltothenumber ofrevolutions oftheengine bywhich thedynamois driven, sothatthecurrent sentthroughthecircuit willbean" alternating" current offrequency equaltothat oftheengine.Intheexample given,the rate atwhich heat isgeneratedis(pcos%)2R,andtheaverage rate,averaged overalargenumber ofalternations, is\p~Ror 1p*C*R 2R*+iy This, then,would betherate atwhich theengine drivingthedynamo would have toperformthework. Discharge ofaCondenser. 516.Afurther exampleoftheeffect ofinduction inasinglecircuit which isofextreme interest issupplied bythephenomenonofthedischargeofa condenser. Letussupposethat thechargesonthetwoplatesatanyinstant areQ and—Q,theplates beingconnected byawire ofresistance Randofself- induction L.IfGisthecapacityofthecondenser, thedifference ofpotential ofthetwoplateswillbe^,and this willnowplaythesamepartasthe electromotive force ofabattery. Theequationisaccordingly §-|(2*)-JK(437). 514-516] Discharge ofaCondenser 459 Thequantities Qand iarenotindependent,forimeasures therate of flowofelectricitytoorfrom eitherplate, andtherefore therateofdiminution ofQ.Weaccordinglyhave%—-—-, andonsubstitutingthisexpressionfor i,equation (437) becomes d*Q dQQL-dF+R -dt+c=°- Thesolution isknown tobe Q^Ae-^t +Be-^*(438), where A,Barearbitrary constants, and\ltX2aretheroots of Lx*-Rx +^=(439). Ifthecircuit iscompletedattime t=0,thechargeoneachplate being initially Q ,wemust have, attime t=0, andthese conditions determine theconstants Aand B.Theequations givingthesequantitiesare A+B=Q ,A\+B\2=0. Iftheroots ofequation (439)are real, itisclear, since both theirsum andtheirproductarepositive, thattheymust themselves bepositive quanti- ties. Thus thevalue ofQgiven byequation (438)willgraduallysinkfrom Qtozero. Thecurrent atanyinstant is dQ dti=-^=A\1e-^t+B\2e-^ =A\e-^t(l-e-^-^t ), andthis starts bybeing zero, rises toamaximum andthen fallsagainto zero. Thecurrent isalwaysinthesame direction, sothatQisalwaysofthe samesign. Itis,however, possibleforequation (439)tohaveimaginaryroots. This willbethecase if itG 4<L isnegative. Denoting R2—~-,whennegative, by—*8 ,theroots willbe R±XK Xl '2=2L' 460 Induction ofCurrents inLinear Circuits[ch.xiv sothatthesolution (438) becomes _Rt Lvet_M Q=e*L(Ae°L+Be*L ) =e2LDcos (^y-«> where D,earenew constants. Inthiscasethedischargeisoscillatory.The charge Qchanges signatintervals,sothatthecharges surgebackwards andforwards from oneplatetotheother. Thepresenceoftheexponentialm e2Lshews thateachchargeislessthan thepreceding one, sothat the charges ultimatelydieaway.ThegraphsforQand iinthetwocases of 4X (i)R2>-77-(discharge continuous), 4X (ii)R2<-77-(discharge oscillatory), aregiveninfigs.132and133. Fig. 132. (i)discharge continuous. Fig. 133. (ii)discharge oscillatory. Theexistence oftheoscillatory dischargeisofinterest, asthepossibility ofadischargeofthistypewaspredictedonpurelytheoreticalgrounds by LordKelvin in1853. Fouryearslater theactual oscillations were observed byFeddersen. 516-519] Pair ofCircuits 461 517. Itisofvalue tocomparethephysical processesinthetwokinds of discharge. Letusconsider firstthecontinuousdischargeofwhich thegraphsare shewn infig.132. The firstpartofthedischargeissimilar totheflow alreadyconsidered in§513.Atfirstwecanimaginethatthecondenser is exactly equivalenttoabatteryofelectromotive forceE=-~>andtheactof dischargingisequivalenttocompletingacircuitcontainingthisbattery. After atime thedifference between thetwocases comes into effect. The batterywould maintain aconstant electromotive force, sothatthecurrent Ewould reach aconstant finalvalue-^,whereas thecondenser doesnotsupply aconstant electromotive force. Asthedischarge occurs, thepotentialdiffer- encebetween theplatesofthecondenser diminishes, andsotheelectromotive force, andconsequentlythecurrent, alsodiminish. Thus thegraphforiin fig.132,canberegardedasshewingagradualincrease towards thevalue p[whereE=-~\intheearlierstages,combined with agradual fallingoffof thecurrent, consequentonthediminution ofE,inthelaterstages. Fortheoscillatory dischargetooccur, thevalue ofLmust begreaterthan forthecontinuousdischarge. Theenergyofacurrent ofgiven amount is accordingly greater,while therateatwhich this isdissipated bythegenera- tionofheat,namely Hi2 ,remains unaltered bythegreatervalue ofL.Thus forsufficiently greatvalues ofLthecurrent may persist even after thecon- denser isfully discharged,acontinuation ofthecurrent meaningthat the condenser againbecomescharged,butwithelectricityofdifferentsignsfrom theoriginal charges.Inthiswaywegettheoscillatory discharge. Induction inaPair ofCircuits. 518. IfL,M,Narethecoefficients otinduction (Ln,L12,X22)ofapairof circuits ofresistances R,S,inwhich batteries ofelectromotive forcesEltE2 areplaced,thegeneral equationsbecome Ei-^iLh+MiJ^Bix (440), E2-j t(Mi 1+Ni2)=Si2 (441). SuddenCompleting ofCircuit. 519. Letusconsider theconditions which must holdwhen oneofthe circuits issuddenly completed,theprocess occupyingtheinfinitesimal inter- valfrom t=tot=t.Letthechangeswhich occur in^and i2duringthis 462 Induction ofCurrents inLinear Circuits[ch.xiv interval bedenoted byAt\andAi2.Equations (440) and(441) shew that duringtheinterval from t=Qtot=rthevalues of-r(Li x+Mi2)and of -j-(Mi x+Ni2)are finite, sothatwhen risinfinitesimal, thechangesin Guv Lix+Mi2andMix+Ni2must vanish. Thuswemust have LM,+MAi2=0, MAti+NM2=0. Exceptinthespecialcase inwhichLN—Mz=(acase ofimportance, which willbeconsideredlater), theseequationscanbesatisfiedonlyby Ai'x=At2=0.Thus thecurrents remain unalteredbysuddenly makinga circuit, andthechangeinthecurrents isgradual andnotinstantaneous. 520. Suppose,forinstance, that before theinstant t=circuit 2is closed butcontains nobattery,while circuit 1,containingabattery,isbroken. Let circuit 1beclosed attheinstant t=0,then the initial conditions are that attime t=0,ix=i2=0.Theequationstobesolved are [R+Li^+Mit^E- .(442), Ti/rdMdt^^(s+N-^i^O(443). Thesolution isknown tobe i2=Be~Kt+B'e-yt , where A,A\B,B'areconstants, andX,\'aretheroots of (R-L\)(S-N\)-if2\2=0, orof RS-(RN +SL)\+(LN-M2)Xi=(444). Theenergyofthecurrents, namely %(Li 1*+2Mi1i2+m2*), being positiveforallvalues ofixand i2,itfollows thatLN'—M2isnecessarily positive.SinceRSandRN+SLarealsonecessarily positive, weseethat all thecoefficients inequation (444)arepositive,sothattheroots \,\'areboth positive. When t=0,wemust have (i2)(,=B+B'=•(445), .(446), 519-521] Pair ofCircuits 463 andinorder thatequation (443)maybesatisfied atevery instant, wemust have -MAXe~Kt-MA'Xer™ +(S-NX)Be~M+(S-NX)B'e^'1=0, forallvalues oft,and forthistobesatisfied thecoefficients ofe~Ktande~yt must vanishseparately. Thuswemust have (S-NX)B=MAX(447), (S-NX)B'=MA'X(448), and ifthese relations aresatisfied, and X,Xaretheroots ofequation (444), thenequation (442)willbesatisfiedidentically. Fromequations (445), (446), (447) and(448),weobtain B=-F AX -A'X_-E,MMS-N\~~S-NX~RS(\-i-X-*y andthesolution isfound tobe (S-Nx)E, (S-NX)E,hBSX(X-i-X'-^Le)+ RXX'(X'-i-X-ij{1-e>> ME, u ME, RSiX-i-X-1 ) RS(x'~i-x-i)e * EWenotice thatthecurrent in1rises toitssteadyvalue^,therisebeing similar innature tothatwhenonlyasinglecircuit isconcerned(§513). The rise isquickifXandVarelarge—i.e.ifthe coefficients ofinduction are small, andconversely. Thecurrent in2isinitially zero, rises toamaximum andthen sinksagaintozero. Thechangesinthiscurrent arequickorslow accordingasthose ofcurrent 1arequickorslow. Sudden Breaking ofCircuit 521. Thebreakingofacircuit mayberepresented mathematically by supposingtheresistance tobecome infinite. Thus ifcircuit 1isbroken, the process occurringintheinterval from t=tot=r,thevalue ofRwill become infiniteduringthis interval, while thevalue of\becomes zero. The changesini,andi2are stilldeterminedbyequations (440) and(441), butwe cannolongertreatiJasa constant, andwecannot assert that intheinterval from totthevalue ofRixisalwaysfinite. Itfollows, however, fromequation (441)that-j-(Mi,+Ni2)remains finite throughouttheshort interval, sothatwehave, with thesame notation as before, MM,+NAi2=0. 464 Induction ofCurrents inLinear Circuits[ch.xiv Supposeforinstance thatbefore thecircuit 1wasbroken wehadasteadyE current -^incircuit 1,andnocurrent incircuit 2.Weshallthenhave sothatLix=-ElR* ME, NR* andtherefore immediatelyafter thebreak, theinitial current incircuit 2is .ME, %2-~NR' This currentsimply decays under theinfluence oftheresistance ofthe circuit. PuttingE%=andi,=inequation (441)weobtain du_S . dt~~Nh> andthesolution whichgivesi2=-^~ initiallyis h~NRe ' Thechangesinthecurrenti,duringtheinfinitesimal interval tareof interest. These aregoverned byequation (440),thevalue ofRnotbeing constant. Thevalue ofExisfinite, andmayaccordinglybeneglectedincomparison with theother terms ofequation (440), which arevery great duringthe interval oftransition. Thus theequation becomes, approximately, dd r-(Li,+Mi,)=-Ri, .(449). Thevalue of-j-(Mi, -f,ZW2)is,aswehavealready seen, finite, sothatwe M maysubtract-^times thisquantity from theleft-hand member ofequation (449) andtheequation remains true Bydoingthisweeliminate i2>and obtain /M\di, (L-Njdt=-Rh- Thesolution whichgivestoi,theinitial value(i,\is N!>dt givingthewayinAvhich thecurrent falls tozero.Wenotice that if LN—M2isvery small, thecurrent falls offatonce, while ifLN—Mn-islarge, thecurrent willpersistforalongertime. Intheformer casethebreaking ofthecircuit isaccompanied onlybyavery slight spark,inthelatter case byastronger spark. 521,522] Pair ofCircuits 465 One Circuitcontaining aPeriodic Electromotive Force. 522. Letussupposenext thatthecircuits contain nobatteries, butthat circuit 1isactedupon byaperiodicelectromotive force, sayEcospt,such as mightarise ifthis circuit contained adynamo. Asin§514, itissimplesttoassume anelectromotive forceEeipt:the solutionactually requiredwillbeobtainedbyultimately rejectingthe imaginaryterms inthesolution obtained. Theequationstobesolved arenow EeW-jiLh+M^Rix (450), -jt(Mi 1+Ni2)=iSi2 (451). Asbefore both itand i2,asgiven bytheseequations,willinvolve the dtdtimeonlythroughafactor eipt ,sothatwemay replace -j-byip,andthe equations become fromwhich weobtainRh.+Lipi-t+Mipi 2=Eeipt , Si2+Mipi!+Nipi 2=0, L Eeipt S+Nip-Mip (R+Lip)(S+Nip)+My-' Thecurrent ixintheprimaryisgiven, from theseequations, by Eeipt ii= n T.M-p1K+Lip+-rz ~rr- FS+Nip Eeipt ~Z~My{S-Nip) Eeipt R'+L'ip' SMy NMy-here jy_B+B_j^., V-L-^j^. The case ofnosecondarycircuitbeing presentisobtained atonceby putting S=oo,andthesolution for\isseen tobethesame asifno secondarycircuit werepresent, exceptthat R',L'arereplaced byRandL. Thus thecurrent intheprimarycircuit isaffected bythepresenceofthe secondaryinjustthesamewayasifitsresistance were increased from RtoR',and itscoefficient ofself-induction decreased from L'toL. j. 30 466 Induction ofCurrents inLinear Circuits[oh.xiv Theamplitudesofthetwocurrents are|t2|and|t2|,sothattheratio of theamplitudeofthecurrent inthesecondarytothat intheprimaryis |i21-Mip v (452). Thedifference ofphaseofthetwocurrents =argi2—argix =arg(h/h) -Mip\=arg8+Nip) =7r-tan_1 (i|)(453)- 523. Theanalysisisofpractical importanceinconnection with the theoryoftransformers. Insuchapplications,thecurrentusuallyisofvery high frequency,sothatpislarge,andwefindthatapproximatelytheratio .M . oftheamplitudes (cf.expression (452))is-^,while thedifference ofphase (cf.expression (453))isir.Theselimiting results, forthecaseofpinfinite, canbeobtained ataglancefromequation (451). Theright-hand member, Si2,isfinite, sothat*(Mi 1+Ni2)isfinite inspiteoftheinfinitely rapid variations initand i2separately.Inother words, wemust haveapproxi- mately Mix+Ni2constant, andclearlythevalue ofthisconstant must be zero, givingatonce thetwo resultsjustobtained. 524. Whatever thevalue ofp,theresultexpressedinequation (452) can bededuced atoncefrom theprincipleofenergy. Thecurrent intheprimary isthesame asitwould beifthesecondarycircuit wereremoved andR,L changedtoR',U.Thus therate atwhich thegenerator performs work is R'ix2 ,oraveragedoveragreatnumber ofperiods (sincetaisasimple-harmonic function ofthetime)is^R' \ix|2 .Ofthisanamount \R\i x|2isconsumed in theprimary,sothattherateatwhich work isperformedinthesecondaryis \{R!-R)\h\\ox iSMy.. This rate ofperformingwork isalsoknown tobe-^It^l2 ,andon equatingthese twoexpressions weobtain atonce the resultexpressed byequation (452). 522-526] Pair ofCircuits 467 Case inwhichLN—M2issmall. 525. Theenergyofcurrents ilti2inthetwocircuits is 1(^+21^+^) (454), andsince thismustalwaysbepositive,itfollows thatLN—M2must neces- sarilybepositive. Theresults obtained inthespecialcaseinwhichLN—M2 issosmall astobenegligibleincomparisonwith theotherquantities involved areofspecial interest, sothatweshallnowexamine whatspecial features areintroduced into theproblems whenLN—M2isverysmall. Expression (454) canbetransformed into ,/r.„,..,LN-M2 .,$(Li l+Mia)*+— 2J— •»"» sothatwhenLN—M2isneglectedtheenergy becomes ^{Lh+Mi^y, andthisvanishes forthespecialcase inwhich thecurrents areintheratio h/*a=—MjL.Thisenables ustofindthegeometrical meaningoftherelation LN—M2=0.Forsince theenergyofthecurrents, asin§501, is weseethat thisenergycanonlyvanish ifthemagneticforce vanishes at every point.Thisrequiresthattheequivalent magneticshells must coincide andbeofstrengthswhich areequal andopposite.Thus thetwo circuits must coincidegeometrically. Thenumber ofturns ofwire inthecircuits mayofcourse bedifferent :ifwehave rturns intheprimaryand sinthe secondary, wemust have LM_rM~N~s' andwhen thecurrents aresuch astogiveafield ofzeroenergy,each fraction isequalto—i2Jii> 526. Letusnextexamine themodifications introduced intotheanalysis bytheneglectofLN—M2inproblemsinwhich thevalue ofthisquantityis small. Wehave thegeneral equations (§518), Ex-j(Lh+Mij)=mi (455), E2-jt(Mi 1+Ni2)=Sl(45G). Ifwemultiply equation (455) byMandequation (456) byLandsub- tract,weobtain ME.-LE^RMn-SLi, (457), anequation which contains nodifferentials. 30—2 468 Induction ofCurrents inLinear Circuits[ch.xiv 527. Toillustrate, letusconsider thesudden makingofone circuit, discussed inthegeneralcasein§519. Thegeneral equationsthere obtained, namely LM,+M.Ai8=0, MAi,+NAi2=0, nowbecome identical. Wenolongercandeduce therelations At\=At2=0, buthaveonlythesingleinitial conditions £$—£ (458).At2Lv ' Butbysupposing equations (455) and(456) replaced byequations (455) and(457)wehaveonlyone differential coefficient andtherefore onlyone constant ofintegrationinthesolution, and thiscanbedetermined from theone initial conditionexpressed byequation (458). Let us,forinstance, consider the definiteproblemdiscussed(forthe general case)in§520. Circuit 2contains nobatterysothatE2=0,andat time t=circuit 1issuddenly closed, sothat theelectromotive forceEx comes intoplayinthe first circuit. The initial currents aregiven by (from equation (458)), L^+Mi^O (459), (from equation (457)), MEX=RMix-SLi2 (460),. i,% ME, ME,sothatM-LRM*+8L*L(RN +SL)^ Thus finite currents come into existence atonce, butthesystemof currents isoneofzeroenergy,sinceequation (459)issatisfied. Tofindthe subsequent changes, wemultiply equation (455) by-^andequation (456) by M -~(putting E2=0),andfindonaddition LE, ~{R+~s)di(Lil+Mi^=Lil+Mii ' R ofwhich thesolution, subjecttotheinitial condition Lix+Mi2=0,is T7? /-B-Sf ,NL'&i /i~-»y-ur..g ^ Li,+M^=^r(l-e'^^A From thisandequation (460)weobtain h.-4 E, to, _&xA-™ tRM+LS* andthese equations givethecurrents atanytime. 527] Examines 469 These results canofcourse bededuced alsobyexaminingthelimiting formassumedbythesolution of§520,whenLN'—if2vanishes. Theproblemofthebreakingofacircuit, discussed in§521, canbe examined inasimilar wayinthespecialcase inwhichLN—if2=0.The current£,inthebroken circuit isfound todisappear instantaneously,its energy immediately reappearingasthat ofacurrent Lix\Mincircuit(2) ; this latter current thendecaysunder theresistance ofthecircuit. EXAMPLES. 1.Acoil isrotated with constant angular velocitya>about anaxis initsplane ina uniform field offorce perpendiculartotheaxis ofrotation. Find thecurrent inthecoil atanytime, andshew that itisgreatest when theplane ofthe coilmakes anangle tan_1 (-o-jwith thelines ofmagneticforce. 2.The resistance andself-induction ofacoilareRandL,and itsendsAandBare connected with theelectrodes ofacondenser ofcapacity Cbywires ofnegligible resistance. There isacurrent Icosptinacircuitconnecting AandB,andthechargeofthecon- denser isinthesame phaseasthis current. Shew that thechargeatanytime is -^cospt,andthatG(R2+p2L2 )=L.Obtain alsothecurrent inthe coil. 3.Theends B,Dofawire(R,L)areconnected with theplates ofacondenser of capacityC.Thewire rotates aboutBDwhich isvertical with angular velocity o>,the areabetween thewireandBDbeing A.IfHisthehorizontal componentoftheearth's magnetism, shew that theaveragerate atwhich work must bedone tomaintain the rotation is £WAtCPRafill&CW +(1-GLc*2 )2 ]. 4.Aclosed solenoid consists ofalargenumber JVofcircular coils ofwire, each of radius a,wound uniformly upon acircular cylinderofheight 2h.Atthecentre ofthe cylinderisasmall magnet whose axis coincides with that ofthecylinder, andwhose moment isaperiodic quantity fisinpt.Shew thatacurrent flows inthesolenoid whose intensityisapproximately !—^-rsin{pt+a), {(a2+h2)(R2+Z2p2 )}% where R,Laretheresistance andself-induction ofthesolenoid, andtana=R/Lp. 5.Acircular coilofnturns,ofradius aandresistance R,spins withangular velocity <around avertical diameter intheearth's horizontal magneticfield IT:shew that the , , ,•,; x-•BhiWa'aR n. average electromagnetic damping couplewhich resists itsmotion isa/m<sj3\'{jlven H=Q-17, %=50,R=Iohm, a=10cm., andthat thecoilmakes 20turns persecond, expressthecoupleindyne-centimetres,andthemean squareofthecurrent inamperes. 6.Acondenser, capacity G,isdischarged through acircuit, resistance/?,induction L, containing aperiodicelectromotive forceEsinnt.Shew thatthe"forced"current inthe circuit is #sin(nt- ff)[r2+(nL-^A where tan6=(n2CL-l)/wCi2. 470 Induction ofCurrents inLinear Circuits[en.xiv 7.Two circuits, resistances RxandR2,coefficients ofinduction Z,M,N,lieneareach other, andanelectromotive forceEisswitched intooneofthem. Shew that thetotal quantityofelectricitythattraverses theother isEM/R 1R2. 8.Acurrent isinduced inacoilBbyacurrent I&mptinacoilA.Shew thatthe mean force tendingtoincrease anycoordinate ofposition 6is xPpiLM dM 2R2+L*p*dS' where L,M,Narethecoefficients ofinduction ofthe coils,andRistheresistance ofB 9.Aplane circuit, area S,rotates withuniformvelocitya>about theaxis otz,which liesinitsplaneatadistance hfrom thecentre ofgravityofthearea.Amagnetic molecule ofstrength pisfixed intheaxis ofxatagreatdistance afrom theorigin, pointinginthedirection Ox.Prove that thecurrent attime t'sapproximately „nC,Micos(»'- «)+,.„., ;To.acos(2o)i- v)> a8(/i2+ZW)i a4(£2+4Z2a>2 )£ whererj,earedeterminate constants. 10.Two points A,Barejoined byawire ofresistance Rwithout self-induction; Bisjoinedtoathird pointCbytwowires each ofresistance R,ofwhich one iswithout self-induction, andtheother hasacoefficient ofinduction L IftheendsA,Carekept ata-potentialdifference Ecospt,provethat thedifference ofpotentials atBandCwill beE'cos{pt- y),where 11.Acondenser, capacity C,charge Q,isdischarged through acircuit ofresistance R,there being another circuit ofresistance Sinthe field. IfLJy=M2 ,shew that there willbeinitial currents -NQjC{RN+SL)andMQjC(RN'+SL), and findthecurrents at anytime. 12.Two insulated wires A,Bofthesame resistance have thesame coefficient of self-induction L,while that ofmutual induction isslightlylessthan L.Theends ofB areconnected byawire ofsmall resistance, andthose ofJ.byabatteryofsmallresistance, andattheendofatime tacurrent iispassing through A.Prove thatexcept when tis very small, *=i(«o+**) approximately,where iisthepermanentcurrent inA,and i'isthecurrent ineach after atimet,when theends ofboth areconnected inmultiple arcbythebattery. 13.Theends ofacoilformingalong straight uniform solenoid ofinturns perunit lengthareconnected with ashort solenoidal coilofnturns andcross-section A,situated inside thesolenoid, sothatthewhole forms asingle completecircuit. The latter coilcan rotate freely about anaxisatright anglestothelengthofthesolenoid. Shew that infree motion without anyexternalfield,thecurrent iandtheangle6between thecross-sections ofthecoils aredetermined bytheequations Ri=—-j-{L1i+L2i+8TrmnAi cos6), I-y^+kirmnAV sin#=0, whereLuX2arethecoefficients ofself-induction ofthetwocoils,Iisthemoment of inertia oftherotating coil,Ristheresistance ofthewholecircuit, andtheeffect ofthe ends ofthelong solenoid isneglected. Examples 471 14.Two electrified conductors whose coefficients ofelectrostaticcapacityareyuy2,r areconnected through acoilofresistance Randlarge inductance L.Verifythat the frequencyoftheelectric oscillations thus established is J_/2r+yi+y21&\i 2tt\yiy2-r2L4L*J' 15.Anelectric circuit contains animpressed electromotive force which alternates inanarbitrary manner and alsoaninductance. Isitpossible, byconnecting the extremities oftheinductance tothepolesofacondenser,toarrange sothatthecurrent inthecircuit shallalways beinstepwith theelectromotive forceandproportionaltoit1 16.Two coils(resistances R,S;coefficients ofinduction L,AT,JV)arearrangedin parallelinsuchpositions thatwhen asteady current isdivided between thetwo,the resultant magneticforce vanishes atacertainsuspended galvanometer needle. Prove that ifthecurrents aresuddenlystarted bycompleting acircuitincluding thecoils, then the initial magneticforce ontheneedle willnotingeneral vanish, butthatthere willbe a"throw" oftheneedle, equaltothatwhich would beproduced bythesteady (final) current inthe firstwireflowing throughthat wire foratime interval M-LM-N R"8' 17.Acondenser ofcapacity Cisdischarged through twocircuits, oneofresistance R andself-induction L,andtheother ofresistance R'andcontaining acondenser ofcapacity C.Prove that ifQisthecharge onthecondenser atanytime, cVQ(LL ,RR\c?Q(RRMR'\dQ4-Qo 18.Acondenser ofcapacity Cisconnected byleads ofresistancer,soastobein parallel withacoilofself-induction L,theresistance ofthecoiland itsleadsbeing R.If thisarrangement forms partofacircuit inwhich there isanelectromotive force ofperiod— ,shew that itcanbereplaced byawirewithout self-induction if P (R2-LIC)=p2ZC(r2-LICr), andthattheresistance ofthisequivalentwiremust be(Rr+L/0)l(R^ rr). 19.Two coils, ofwhich thecoefficients ofself-andmutual-induction areZltZ2,M, andtheresistances RuR2,carry steadycurrentsG\,C2produced byconstant electro- motive forces inserted inthem. Shewhow tocalculate thetotal extra currents produced inthecoilsbyinserting agivenresistance inoneofthem, andthus alsoincreasingits coefficients ofinduction bygiven amounts. Intheprimary coil,supposed open,there isanelectromotive force which would produceasteadycurrent C,and inthesecondarycoilthere isnoelectromotive force. Prove thatthecurrent induced inthesecondary byclosing theprimaryisthesame, as regards itseffects onagalvanometerandanelectrodynamometer, andalsowithregard to theheatproduced byit,asasteadycurrent ofmagnitude CMRl 2R1L2+R2L1i i?iZ 2+/?2Zx lastingforatime —&R~R' while thecurrent induced inthesecondary bysuddenly breaking theprimarycircuit may berepresentedinthesame respects byasteadycurrent ofmagnitude CM[2L 2lastingfor atime2L%jR 2. 472 Induction ofCurrents inLinear Circuits[ch.xiv 20.Twoconductors ABD,ACD arearrangedinmultiplearc. Their resistances are R,Sandtheir coefficients ofself-andmutual-induction areL,N,andM.Prove that when placedinseries with leads conveying acurrent offrequency p,thetwo circuits producethesame effect asasinglecircuit whose coefficient ofself-induction is NR2+LS2+2MRS+p2(LN- J/2 )(L+iV- 2M) (L+lV-23f)2p2+(R+S)2 ' andwhose resistance is RS(S+R)+p2{R(F-M)2+S(L-3I)2 } (L+AT-2M)2p2+(R+S)2 21.Acondenser ofcapacity Ccontaining acharge Qisdischarged round acircuit in theneighbourhoodofasecond circuit. Theresistances ofthecircuits areR,S,andtheir coefficients ofinduction areL,M,N. Obtain equationstodetermine thecurrents atanymoment. Ifxisthecurrent intheprimary, andthedisturbance beover inatime lessthant, shew that {~+SINR2+^\+S2Lr\fT &dt=\^[CS2L+CSNR+m.andthat Examine how|'x2dtvaries with S. o CHAPTER XV INDUCTION OFCURRENTS INCONTINUOUS MEDIA General Equations. 528.Wehave seen thatwhen thenumber N,oftubes ofinduction, dN which crossany circuit, ischanging,there isanelectromotive force—-rr actinground the circuit. Thus achangeinthemagneticfieldbringsinto playcertain electric forces which would otherwise beabsent. Wehavenowabandoned theconceptionofaction atadistance, sothat wemustsupposethattheelectric force atanypoint depends solelyonthe changesinthemagneticfield atthatpoint. Thus atapointatwhich the magneticfield ischanging, weseethat there must beelectric forces setup bythechangesinthemagnetic field,andtheamount ofthese forces must be thesame whether thepoint happenstocoincide withanelement ofaclosed conductingcircuit ornot. Letdsbeanelement ofanyclosed circuit drawn inthe field, either ina conducting medium ornot,and letX,Y,Zdenote thecomponentsofelectric intensityatthispoint. Then thework donebytheelectric forces onaunit electric chargeintakingitround this circuit is zs+Fs+4><461>> dN and this,bytheprinciple just explained,must beequalto——rr whereNis thenumber oftubes ofinduction which cross this circuit. 529.Wehave(cf.§437) N=ff(la+vib+nc)dS (462), dN sothatonequating expression (461)to— —j-,wehave J('5+lri+*£)*-//('S+-»+-S)*"^ 474 Induction ofCurrents inContinuous Media[ch.xv Theleft-hand member isequal, byStokes' Theorem(§438),to /ii«®-s+-(S-©+-e-s)}^ theintegration beingover thesame area asthatontheright hand ofequa- tion(463). Hence wehave firax.&vi [[[-.(dZ dY da\ (dX dZ db\ (dY {\dydz dtj \dz doc dtj'\dx dydt. Thisequationistrue forevery surface, sothatnotonlymust each inte- grand vanish, but itmust vanish forallpossiblevalues ofI,m,n.Hence each coefficient ofI,m,nmust vanishseparately. Wemustaccordinglyhave dadZdY fAnA^ -Tt=W~te•-(6)' dbdXdZ /*ne\-*=w_ to(465) - dodYdX -Tt= d^--dy~(466)- 530. Thecomponents F,G,Hofthemagnetic vector-potentialare given,asinequations (376), by dHdG ,.„„,a=s7-al'etc<467>- Oncomparingtheseequations withequations (464)—(466),itisclear thatthesimplestsolution forthevector-potentialisgiven bytherelations £--*al=-F-£--*w IfF,G,Histhemostgeneral vector-potential, wemust have relations of theform(cf.equations (375)) dF dV where^isanarbitraryfunctionreplacingthe—^ofequations (375). 531.Writingthese relations intheform '--£-£ («* r=-f-f<«* "-£-£ <«* wehaveequations givingtheelectric forcesexplicitly. 529-533] General Equations 475 The function ^has, sofar,hadnophysical meaning assignedto it. Equations (470), (471), (472) shew thattheelectric force (X,Y,Z)canbe regardedascompoundedoftwoforces : (i)aforce (— j-,—-r- ,j-)arisingfrom thechangesinthemag- netic field; (ii) aforce ofcomponentsf—^—,—-=—,—-~— Jwhich ispresent when there arenomagnetic changes occurring. Wenow seethatthesecond force istheforcearisingfrom theordinary electrostatic field, sothatwemay identify"Wwith theelectrostaticpotential when nochangesareoccurring. ThemeaningtobeassignedtoSfwhen changesareinprogressisdiscussed below(Chapter xx). 532. Ifthemedium isaconducting medium, thepresenceoftheelectric forces setsupcurrents, andthecomponents u,v,wofthecurrent atany point are,asin§374,connected with thecurrents bytheequations X=TU, Y=TV, Z=T1V, theseequations beingtheexpressionofOhm's Law,where tisthespecific resistance oftheconductor atthepoint. Onsubstitutingthese values forX,Y,Zinequations (464)—(466)or (470)—(472), weobtain asystemofequations connectingthecurrents in theconductor with thechangesinthemagneticfield. 533. Thereis,however, afurthersystemofequations expressingrela- tionsbetween thecurrents andthemagneticfield.Wehave seen(§480) that acurrent setsupamagneticfield ofknownintensity, and since the wholemagneticfieldmust arise either from currents orfrompermanent magnets,this factgivesrise toasecondsystemofequations. Inafieldarising solelyfrompermanent magnetism, wecantakeaunit poleroundanyclosedpathinthe field,andthetotalwork done willbenil. Hence ontakingaunitpoleround aclosed circuit inthemostgeneral magnetic field, thework done willbethesame asifthere werenoperma- nentmagnetism,andthewhole fieldwereduetothecurrentspresent. The amount ofthiswork, aswehave seen, is4nrXi, where %iisthesum ofallthe currents which flowthroughthecircuit round which thepoleistaken. If u,v,warethecomponentsofcurrent atanypoint,wehave St=11(lu+mv+nw)dS, theintegration beingoveranyareawhich hastheclosedpathasboundary. Hence ourexperimentalfactleads totheequation 476 Induction ofCurrents inContinuous Media[ch.xv Transformingthelineintegralintoasurfaceintegral byStokes' Theorem (§438),weobtain theequationintheform 'd& da//K6-I-H-(M-irv)+n[^—= 4>7rtuI[dS—0.oxoy Aswith theintegralof§529,eachintegrandmust vanish forallvalues ofI,m,n,sothatwemust have 4™=i-i-<«3>- ^Ji-%(«5>- 534. Ifwedifferentiate these threeequationswithrespecttox,y,z respectivelyandadd,weobtain •(476),du dvdw _ 2-+~-+j-= ox oydz ofwhich themeaning (cf.§375,equation (311))isthatnoelectricityis destroyedorcreated orallowed toaccumulate intheconductor. Theinterpretationofthis result isnotthat itisaphysical impossibilityforelectricity toaccumulate inaconductor, butthattheassumptions uponwhich weareworking are notsufficiently generaltocover cases inwhich there issuchanaccumulation ofelectricity. Itiseasytoseedirectly how thishascome about. Thesupposition underlying our equationsisthattheworkdone intaking aunitpoleround acircuit isequaltoAntimes thetotal current flowthrough thecircuit. Itisonlywhenequation (476)issatisfied by thecurrent components thattheexpression"total flowthrough acircuit"hasadefinite significance:thecurrent flowacrosseveryareabounded bythecircuit must bethesame. Weshall seelater (Chapter xvn)howtheequations must bemodified tocover thecase ofanelectric flow inwhich thecondition isnotsatisfied. Forthepresent weproceed upon thesuppositionthatthecondition issatisfied. Currents inhomogeneousmedia. 535. Letusnowsupposethatweareconsideringthecurrents ina homogeneous non-magnetised medium. Wewrite a=fjLa, etc.,X=tu,etc., inwhichfxandtareconstant. Thesystemsofequationsof§§529and533 nowbecome da. (dw dv^ dt-A4j7=t I—~— I.etc - dydz 4,iru=——--, etc. oyoz.(477), .(478). 533-537]General Equations477 Differentiating equation (478) withrespecttothetime,weobtain du d(dy\d(d/3\^dt= Zy\?Tt)-dz\?Tt) f3fdv du\ d/dudw \dy\dx dy/dz\dz dx ~T \{dx~*+df+dzV dx\dx dy+ dz =tV2 u, invirtue ofequation (476). Similarequationsaresatisfied bytheothercurrent-components,sothat wehave thesystemofdifferentialequations \4777*du=V2 rdt~ 4>7ru dv_„ tdt> (479). 4t7radw„, tdt I Ifweeliminate thecurrent-componentsfrom thesystemofequations (477) and(478), weobtain *E£^=V*a (480),tdt andsimilarequationsaresatisfied byband c. 536. Theequationwhich hasbeen found tobesatisfied byu,v,w, a,/3and7isthewell-knownequationofconduction ofheat. Thus weseethatthecurrents induced inamass ofmetal, aswell asthecom- ponentsofthemagneticfield associated with these currents, will diffuse throughthemetal inthesamewayasheat diffusesthroughauniform conductor. Rapidly alternatingcurrents. 537.Theequationsassume aform ofspecialinterest when thecurrents arealternatingcurrents ofhigh frequency. Wemayassume eachcomponent ofcurrent tobeproportionaltoeipt(cf.§514), andmaythenreplacethe operator -j-bythemultiplier ip.Theequations nowassume theform in^E«-v>» (48i), ^wa=V2a>etCti 478 Induction ofCurrents inContinuous Media[oh.xv and ifpissolargethat itmaybetreated asinfinite, theseequations assume thesimpleform u=v=w=0, a=b=c=0. Thus forcurrents ofinfinitefrequency,there isneither current nor magneticfield inthe interior. The currents areconfined tothesurface, andtheonly partoftheconductor which comes intoplayatallisathin skinonthesurface. Equations (481)enable ustoformanestimate ofthethickness ofthis skinwhen thefrequencyofthecurrents isverygreatwithoutbeing actually infinite. Atapointonthesurface oftheconductor, letustake rectangular axes sothat thedirection ofthecurrent isthat ofOxwhile thenormal to thesurface isOz. Ifthethickness oftheskin isvezy small, weneed not consider anyregion exceptthat intheimmediate neighbourhoodofthe origin,sothat theproblemispracticallyidentical with that ofcurrent flowing paralleltoOxinaninfinite slab ofmetal havingtheplane Oxy foraboundary. Equation (481) reduces inthiscase to 4s7rfiip d2u ~T~u=d?> and ifweput—=«2 ,thesolution is u=Ae-KZ+BeK *. Thevalue ofkisfound tobe sothat u=Aev TevT+BeyTevT andthecondition that thecurrent istobeconfined toathin skinmaynow beexpressed bythecondition thatu=when z=oo,and isaccordinglyB=0.Themultiplier Aisindependentofz,but will ofcourse involve thetimethroughthefactor eipt ;letusputA=u eipt ,andwethen have thesolution -n/¥v("-v/2:f*) u=une e 537]General Equations 479 Rejectingtheimaginary part,weareleftwith therealsolution u=ueV tcos(pi—a/ z)> from which weseethat aswepassinwards from thesurface ofthecon- ductor, thephaseofthecurrentchangesatauniform rate,while itsamplitude decreasesexponentially. Wecanbestformanidea oftherate ofdecrease oftheamplitude byconsidering a concrete case. Forcopper wemaytake (inc.o.s. electromagnetic units) fi=l,t=1600. Thus foracurrent which alternates 1000timespersecond, wehave p=2nx 1000, a/——=5approximately. Itfollows that atadepthof1cm.thecurrent willbeonlye~5or-0067 times itsvalue atthesurface. Thus thecurrent ispractically confined toaskin ofthickness 1cm. rz= oo The total currentperunitwidth ofthesurface atatime tis Iudz,of Je=0 which thevalue isfound tobe ncos(pt—j 4>7T/U.p Thus, ifwedenote theamplitude oftheaggregatecurrent byU,the value ofuQwillbeUa/—. Theheatgenerated perunit time inastripofunitwidth andunit lengthis •t=l rz=cc Iu2dtdz t=QJ2= =hTU<? Jz=0e-V5^.-.-2x/2-^lz dz 2~/j,p T Thus theresistance oftheconductor isthesame aswould bethe resistance forsteadycurrents ofaskin ofdepth 2/ Theresults wehave obtained willsuffice toexplain whyitisthattheconductors used toconvey rapidly alternating currents aremade hollow, asalsowhyitisthatlightning conductors aremade ofstrips,rather thancylinders,ofruetaL 480 Induction ofCurrents inContinuous Media[ch.xv Plane Current-sheets. 538.Wenextexamine thephenomenonoftheinduction ofcurrents inaplanesheet ofmetal. Lettheplaneofthecurrent-sheet betaken tobez=0.Letusintroduce acurrent-function <I>,which istobedefined forevery pointinthesheetby thestatement thatthetotalstrengthofallthecurrents which flowbetween thepointandtheboundaryis<I>.Then thecurrents inthesheet areknown when thevalue of <E>isknown atevery pointofthesheet. Ifweassume thatnoelectricityisintroduced into, orremoved from, thecurrent-sheet, or allowed toaccumulate atanypointofit,thenclearly<E>willbeasingle- valued function ofpositiononthesheet. Theequationofthecurrent-lines willbe <l>=constant, andthe line <1>=willbetheboundaryofthecurrent-sheet. Between thelines <I>and <&+•d<&wehave acurrent ofstrengthd<&flowinginaclosed circuit. The magneticfieldproduced bythiscurrent isthesame asthatproduced by amagneticshell ofstrengthd<&coincidingwith thatpartofthecurrent- sheet which isenclosed bythis circuit, sothat themagneticeffect ofthe whole systemofcurrents inthesheet isthat ofashellcoincidingwith thesheet and ofvariablestrength<£.Thisagainmaybereplaced bya distribution ofmagnetic polesofsurfacedensity <l>/eonthepositivesideof the sheet, togetherwith adistribution ofsurfacedensity—<£/eonthe negative side ofthesheet, where eisthethickness ofthesheet. LetPdenote thepotentialatanypointofadistribution ofpolesof strength<E>,sothat P=Jjjdafdyf (482), where doc'dy'isanyelement ofthesheet. Themagnetic potentialatany pointoutside thecurrent-sheet ofthefieldproduced bythecurrents isthen dP tt=-gj(483). Ifaistheresistance ofaunitsquareofthesheet atany point,and u,vthecomponentsofcurrent, wehave, byOhm's Law, X=au, Y=cv. Thecomponents u,varereadilyfound tobegiven by 538,539] Plane Current-sheets 481 sothatwehave theequations z=^'¥=-"te<484) true atevery pointofthesheet. Hence, byequation (466), dc_dY dX /32<I> 323>\ The totalmagneticfield consists ofthepartofpotentialIIdue tothe currents andapartofpotential (say) O',duetothemagnetic system bywhich thecurrents areinduced. Thus thetotalmagnetic potentialisO+Of,and atapoint justoutside thecurrent-sheet(taking /x=1) dcd3.__.,. theequation (485) becomes M«°+«>— £+S)«"* Thefunction P(equation (482))isthepotentialofadistribution ofpoles ofsurfacedensity<t>onthesheet. HencePsatisfiesLaplace's equationat allpointsoutside thesheet, andatapoint justoutside thesheet andonits positiveface——=2tt<£>. 482 Induction ofCurrents inContinuous Media[ch.xv themostgeneralmotion oftheinducingfieldmaybereplaced bythecrea- tion ofaseries ofpoles.Thesimplest problemarises when theinducing field isproduced bythesudden creation ofasingle pole,andthesolution ofthemostgeneral problemcanbeobtained from thesolution ofthissimple problem byaddition. Fromequations (487) and(488)itisclear that-j-^-(fl+fl')remains finite onboth surfaces ofthesheetduringthesudden creation ofanew pole,sothat^-(H+O')remains unaltered invalue overthewhole surface ofthesheet. Lettheincrement in—(H+CI')atanypointinspacebe denoted byA,thenAisapotentialofwhich thepolesareknown inthe spaceoutside thesheet, andofwhich thevalue isknown tobezeroover thesurface ofthesheet. Themethods ofChapterviii areaccordingly available forthedetermination ofA :therequiredvalue ofAisthe electrostaticpotential when thecurrent-sheet isputtoearth inthe on' presenceofthepoint chargeswhich wouldgiveapotential-=— ifthesheet 02 were absent. Physically,the factthat^-(O+X2')remains unaltered over thewhole surface ofthesheet means that the field offorcejustoutside thesheet remains unaltered, andhence that currents areinstantaneouslyinduced in thesheet such that the lines offorce atthesurfaces ofthesheet remain unaltered. Theinduced currents canbefound foranyshapeofcurrent-sheet for which thecorrespondingelectrostaticproblemcanbesolved *,butingeneral theresults aretoocomplicatedtobeofphysicalinterest. Infinite Plane Current-sheet. 540. Letthecurrent-sheet beofinfinite extent, andoccupy thewhole oftheplaneofxz,and letthemoving magnetic system beintheregion inwhich zisnegative. Thenthroughout theregionforwhich zispositive thepotentialCl+Cl'hasnopoles, andhence thepotential dtdz{iL+n)2^3^ *Seeapaper bytheauthor, "FiniteCurrent-sheets," Proc. Land. Math. Soc. Vol. xxxi. p.151. 539,540] Plane Current-sheets 483 hasnopoles.Moreover thispotentialisasolution ofLaplace's equation, andvanishes over theboundaryoftheregion, namelyatinfinity andover theplanez=(cf.equation (487)). Hence itvanishesthroughoutthe wholeregion (cf.§186),sothatequation (487)must betrue atevery point intheregionforwhich zispositive. Wemay accordingly integratewith respecttozandobtain theequationintheform >+^f ^ noarbitraryfunction ofx,ybeing added because theequation must be satisfied atinfinity. Themotion ofthesystemofmagnetsonthenegativesideofthesheet maybereplaced,asin§539,bytheinstantaneous creation ofanumber of poles. Atthecreation ofasingle polecurrents aresetupinthesheet such that 12+12'remains unaltered(cf.equation (489)) onthepositiveside of thesheet. Thus these currents form amagneticscreen andshield thespace onthepositivesideofthesheet from theeffects ofthemagnetic changeson thenegativeside. Toexamine thewayinwhich these currentsdecayunder theinfluence ofresistance and self-induction, weputD,'=inequation (489), and find that IImust beasolution oftheequation dt 2-jtdz' Thegeneralsolution ofthisequationis andthiscorrespondstotheinitial value H=/(aj, y,z). Thus thedecayofthecurrents canbetraced bytakingthe field of potential12attime t=Qandmovingitparalleltotheaxis ofzwith a velocity jr-. 31— '2 484 Induction ofCurrents inContinuous Media[ch.xv EXAMPLES. 1.Prove thatthecurrents induced inasolid withaninfiniteplane face,owingto magnetic changesnear the face, circulateparalleltoit,andmayberegardedasdueto thediffusion intothesolid ofcurrent-sheets induced ateach instant onthesurface soas toscreen offthemagnetic changes from theinterior. Shew that forperiodic changes, thecurrentpenetratestoadepth proportionaltothe square root oftheperiod.Give asolution forthecase inwhich thestrengthofafixed inducing magnetvaries ascospt. 2.Amagnetic systemismoving towards aninfiniteplane conductingsheet with velocityw.Shew thatthemagnetic potentialontheother side ofthesheet isthesame asitwould beifthesheet wereaway, andthestrengthsofalltheelements ofthemagnetic system werechangedintheratioR/(R+w),where2nR isthespecificresistance ofthe sheetperunit area. Shew thattheresult isunaltered ifthesystemismoving away from thesheet, andexamine thecase ofw=—R. Ifthesystemisamagnetic particleofmassMandmoment m,with itsaxisperpen- dicular tothesheet, provethat iftheparticle hasbeenprojectedatright anglestothe sheet, thenwhen itisatadistance zfrom thesheet,itsvelocityzisgiven by pf(£-i2)2=C-m2 /823. 3.Asmall magnet horizontally magnetisedismoving with avelocity uparalleltoa thin horizontalplateofmetal. Shew thattheretardingforce onthemagnet duetothe currents induced intheplateis to2uR WfQiQ +R)' wheremisthemoment ofthemagnet,citsdistance above theplate,2nR theresistance ofasq.cm.oftheplate, andQ2=u2+R2 . 4.Aslowly alternatingcurrent Icosptistraversing asmall circular coilwhose magnetic moment foraunitcurrent isM.Athinspherical shell, ofradius aandspecific resistance<r,has itscentre ontheaxis ofthe coilatadistance /from thecentre ofthe coil.Shew thatthecurrents intheshellform circles round theaxis ofthecoil,andthat thestrengthofthecurrent inanycircle whose radius subtends ananglecos-1 /xatthe centre is ^-L-iLL2(2»+l)^ -^cose„coS(^-a , , (2n+Do-where tane n=- :— . 4:irpa 5.Aninfinite ironplateisbounded bytheparallel planes x=k,x——h; wire is wound uniformly round theplate, thelayersofwirebeing paralleltotheaxisofy.Ifan alternating current issentthrough thewireproducing outside theplate amagnetic force Hqcosptparalleltoz,prove thatH,themagneticforce intheplateatadistance xfrom thecentre,willbegiven by „„(cos\i2mx-\-Qos2mx\S^=H cosh2^ +cos2mA;«»(**+#' _sinhm(h+x)sinm(h-x) -sinhm(h-x) sinm(h+x) coshm(h+x)cosm{h-x)+coshm(h-x)cosin(h+x)' where m2=2irnpj<T. Discuss thespecial cases of(i)mhsmall, (ii)mhlarge. CHAPTER XVI DYNAMICAL THEORY OFCURRENTS General Theory ofDynamical Systems. 541.Wehave sofardevelopedthetheoryofelectromagnetism by- startingfromanumber ofsimpledatawhich arefurnished orconfirmedby- experiment,andexaminingthemathematical andphysical consequences which canbededuced from these data. There arealwaystwodirections inwhich itispossibleforatheoretical science toproceed.Itispossibletostartfrom thesimple experimentaldata andfrom these todeduce thetheoryofmorecomplex phenomena. And it mayalsobepossibletostartfrom theexperimentaldataandtoanalysethese intosomethingstillmoresimpleandfundamental. Wemay,infact, either advance fromsimple phenomenatocomplex,orwemay passbackwards from simple phenomenatophenomenawhich are stillsimpler,inthesense of beingmore fundamental. Asanexampleofatheoretical science ofwhich thedevelopmentisalmost entirelyofthesecond kindmaybementioned theDynamical Theoryof Gases. Thetheorystarts with certainsimple experimental data, such as theexistence ofpressureinagas,andtherelation ofthispressuretothe temperatureanddensityofagas.And thetheoryisdeveloped byshewing that thesephenomena mayberegardedasconsequencesofstillmore funda- mental phenomena, namelythemotion ofthemolecules ofthegas. Inourdevelopmentofelectromagnetic theorythere hassofarbeen but littleprogressinthissecond direction. Itistruethatwehave seen thatthe phenomenafromwhich westarted—such astheattractions andrepulsions ofelectriccharges,ortheinduction ofelectric currents—maybeinterpreted astheconsequencesofother andmore fundamental phenomena taking place intheether bywhich thematerialsystemsaresurrounded. Wehaveeven obtained formulae forthestresses andtheenergyintheether. But ithas notbeenpossibletoproceed anyfurther andtoexplaintheexistence ofthese stresses andenergyinterms oftheultimate mechanism oftheether. 486 Dynamical Theory ofCurrents[ch.xvi Thereason whywehave beenbroughttoahalt inthedevelopmentof electromagnetic theorywillbecome clear assoon aswecontrast thistheory with thetheoryofgases.Theultimate mechanism withwhich thetheoryof gasesisconcerned isthatofmolecules inmotion, andweknow(oratleast canprovisionallyassume thatweknow) theultimate lawsbywhich this motion isgoverned. Ontheother hand theultimate mechanism withwhich electromagnetic theoryisconcerned isthat ofaction intheether, andweare inutterignoranceoftheultimate lawswhichgovernaction intheether. Wedonotknowhowtheether behaves, andsocanmake noprogresstowards explaining electromagnetic phenomenainterms ofthebehaviour oftheether. 542. There isabranch ofdynamics whichattemptstoexplainthe relation between themotions ofcertain knownpartsofamechanism, even when thenature oftheremaining partsiscompletelyunknown. Weturn to thisbranch ofdynamicsforassistance inthepresent problem. Thewhole mechanism before usconsists ofasystemofcharged conductors, magnets, currents, etc.,andoftheether bywhich allthese areconnected. Ofthis mechanism onepart (themotion ofthematerial bodies) isknown tous,while theremainder (theflowofelectric currents, thetransmission ofactionbythe ether, etc.)isunknown tous,except indirectly byitseffect onthe firstpart ofthemechanism. 543.Ananalogy,firstsuggested byProfessor Clerk Maxwell, will ex- plainthewayinwhich wearenowattackingtheproblem. Imaginethatwehave acomplicated machine inaclosed room, theonly connection between thismachine andtheexterior oftheroombeing by means ofanumber ofropeswhich hang throughholes inthefloor intothe room beneath. Amanwhocannotgetintotheroom which contains the machine willhavenoopportunityofactually inspectingthemechanism, but hecanmanipulateittoacertain extent bypullingthedifferentropes. If, onpullingonerope,hefinds that others aresetintomotion, hewillunder- stand thattheropes must beconnectedbysome kind ofmechanism above, althoughhemaybeunable todiscover theexact nature ofthismechanism. Inthisanalogy,theconcealed mechanism issupposedtorepresent thosepartsofthe universe which donotdirectlyaffect oursenses—e.g.theether—while theropes represent thosepartsofwhich wecanobserve themotion—e.g.material bodies. Innature, there arecertain actswhich wecanperform (analogous tothepullingofcertainropes), andthese areinvariably followed bycertainconsequences (analogoustothemotion ofotherropes), buttheultimate mechanism bywhich thecause producesthe effect isunknown. For instance wecancloseanelectric circuit bypressing akey,andtheneedle ofadistant galvanometer maybesetintomotion. "Weinfer that there must besomemechanism connecting thetwo,butthenature ofthismechanism isalmostcompletely unknown. Suppose nowthatanobserver mayhandle theropes,butmaynotpene- trate intotheroom above toexamine themechanism towhichtheyare 541-545]Hamilton's Principle 487 attached. Hewillknow thatwhatever thismechanism may be,certain laws must governthemanipulationoftheropes, providedthatthemechanism is itselfsubjecttotheordinarylaws ofmechanics. Totakethesimplest illustration, supposethatthere aretworopes only,AandB,and thatwhenropeAispulled down adistance ofoneinch,itisfound thatropeBrises through twoinches. Themechanism connecting AandBmaybealever oranarrange- ment ofpulleysorofclockwork, orsomethingdifferent fromanyofthese. Butwhatever itis,providedthat itissubjecttothelaws ofdynamics,theexperimenterwillknow, from themechanicalprincipleof"virtual work," that thedownward motion ofropeA canberestrained onapplyingtoBaforce equaltohalf ofthat appliedtoA. 544. Thebranch ofdynamicsofwhich wearenowgoingtomake use enables ustopredictwhat relation thereoughttobebetween themotions of theaccessiblepartsofthemechanism. Ifthesepredictionsareborne outby experiment,then there willbeapresumptionthattheconcealed mechanism issubjecttothelaws ofdynamics.Ifthepredictionsarenotconfirmed by experiment,weshallknow thattheconcealed mechanism isnotgoverned by thelaws ofdynamics. Hamilton's Principle. 545. Suppose,first, thatwehave adynamical system composedofdis- creteparticles,each ofwhich moves inaccordance with Newton's Laws of Motion. Letany typical particleofmassmxhave atanyinstant tcoordi- nates xx,yXyzxandcomponentsofvelocity ux,vx,wx,and letitbeacted onby forces ofwhich theresultant hascomponents Xx>Yx,Zx.Then, since the motion oftheparticleisassumed tobegoverned byNewton's Laws, wehave «i^--*i(490), «i^*i(491), ™>^r=^ (492). Letuscomparethismotion with aslightlydifferent motion, inwhich Newton's Laws arenotobeyed.Attheinstant tletthecoordinates ofthis sameparticlebexx+8xX)yx+8yx,zx+8zxand letitscomponentsofvelocity beWi+Swj, vx+8vx,wx+8ivx.Letusmultiply equations (490), (491) and (492) by8xx,Bylt8zxrespectively,andadd.Weobtain mx(^8xx+^fyt+^W)=Xx8xx+Yx8yx+Zx8zx...(493). Now^8xx=^(ux8xx)-uXJt(8x x) = -y(u x8xx)— itj8ux. 488 Dynamical Theory ofCurrents[ch.xvi Ifwesumequation (493)foralltheparticlesofthesystem, replacingthe terms ontheleftbytheir values asjustobtained, wearrive attheequation -r-Smx (tijBx1+vx8yx+wxBzx)-Smj(uxBux+vxBv,+w1BwJ =S(X lScc1+Y 1By1+ZM) (494). LetTdenote thekinetic energyoftheactual motion, andT+BT that of theslightlyvaried motion, then sothat BT=%m x(i^Bu^+vxBvx+wxBw,), andthis isthevalue ofthesecond term inequation (494). IfWandW+BWarethepotential energiesofthetwoconfigurations (assumingtheforces toform aconservativesystem),wehave W=-S J(X,dxx+Y,dVl+Zxdzx), and 8F=-2(Z 1S«1+F 18y1+^18^1)J andsothevalue oftheright-handmember ofequation (494)is—BW. Wemaynowrewriteequation (494)intheform 8(T—W)= -jiSrax (wjBxx+vxByx+wxBzx). Thisequationistrue ateveryinstant ofthemotion. Letusintegrateit throughoutthewhole ofthemotion, sayfrom t—tot=r.Weobtain ~~\t=T B\\T-W)dt= Joini!(Wi&Ci+v1By1+WxBz^ t=o.(495). Thedisplacedmotion hasbeen supposedtobeanymotion which differsonly slightlyfrom theactual motion. Letusnow limit itbythe restriction that theconfigurationsatthebeginningandendofthemotion aretocoincide with those oftheactual motion, sothat thedisplacedmotion isnowtobeoneinwhich thesystemstarts from thesameconfigurationasin theactual motion attime t=0,and, afterpassing throughaseries ofcon- figurations slightlydifferent from those oftheactual motion, finallyends in thesameconfigurationattime t=tasthat oftheactual motion. Mathe- maticallythisnew restriction isexpressed bysayingthat attimes t=and t=rwemust have8x=8y=Bz= foreachparticle. Equation (495)now becomes :t\T-W)dt=(496). Jo 546.Speakingofthetwopartsofthemechanism under discussion asthe"accessible"and"concealed" parts,letussupposethat thekinetic andpotential energies TandWdepend onlyontheconfigurationofthe 545-548] Lagrange's Equations 489 accessiblepartsofthemechanism. Thenthroughout anyimaginarymotion oftheaccessiblepartsofthesystem weshallhave aknowledgeofTandW atevery instant, andhence shallbeable tocalculate thevalue of (\T-W)dt (497).Jo Wecanimagineaninfinite number ofmotions whichbringthesystem fromoneconfiguration Aattime t=toasecondconfiguration Battime t=t, andwecancalculate thevalue oftheintegralforeach.Equation (496) shews thatthose motions forwhich thevalue oftheintegralisstationary would be themotionsactually possibleforthesystem. Havingfound which these motions were,weshould have aknowledgeofthechangesintheaccessible partsofthesystem, althoughtheconcealedparts remained unknown tous, both asregardstheir nature andtheir motion. 547.Equation (496)hasbeenprovedtobetrueonlyforasystemcon- sistingofdiscrete materialparticles. Atthesame time theequationitself contains, initsform, noreference totheexistence ofdiscreteparticles.It isatleastpossiblethat theequation maybetheexpressionofageneral dynamical principle which istrue forallsystems whethertheyconsist of discreteparticlesornot.Wecannot ofcourse know whether ornotthis isso.What wehave todointhepresent chapteristoexamine whether thephenomenaofelectric currents areinaccordance with thisequation.Weshall findthatthey are,butweshall ofcourse have norighttodeduce from this factthattheultimate mechanism ofelectric currents istobefound inthemotion ofdiscreteparticles.Beforesettingtoworkonthisproblem, however, weshallexpress equation (496)inadifferent form. Lagrange's Equations forConservative Systems ofForces. 548. Letlt#2,...nbeasetofquantitiesassociated with amechanical systemsuch thatwhen their value isknown, theconfigurationofthesystem isfullydetermined. Then U 2,...nareknown asthegeneralisedcoordi- nates ofthesystem. Thevelocityofanymoving particleofthesystemwilldependonthevalues dBdd of-Tf, -tj,etc. Letusdenote thesequantities by lt62,etc. Letxbea Cartesian coordinate ofanymoving particle. Then byhypothesis xisa function ofl} 2,...,say x=f(01,02,...), sothatbydifferentiation, dx_dfa3/x Tt-Wi^de*2*"" 490 Dynamical Theory ofCurrents[CH.XVI Thus eachcomponentofvelocityofeachmoving particlewillbealinear function of1; 2,...,fromwhich itfollows thatthekinetic energyofmotion ofthesystem must beaquadraticfunction of0,, 2,...,thecoefficients inthis functionbeingofcourse functions of0,, 2,.... LetusdenoteT—WbyL,sothatLisafunction of0,, 2,... n, andof0„ 2,... n,say L=cf)(0u 2)...n,0„ 2,... n). IfL+BL isthevalue olLinthedisplaced configurationl+B01, 02+B0 2,•••8n+B0 n,wehave Ot>i OUn OUi sothatequation (496), whichmaybeputintheform BL=0,fJo nowassumes theform —80,+2^80, 30,i80! Wehave B0,=(0,+80,)-0,.(498). L30,sothat/;g^=/;gfw<a Hdt\deJ The lastterm vanishes since, hyhypothesis, Sftvanishes atthebeginning andendofthemotion, andequation (498)nowassumes theform Joi130,cftW) Letusdenote theintegrand, namely i(30,eftVd0,/J by/,sothattheequation becomes Idt=0. 548-550] Lagrange's Equations491 The varied motion isentirelyatourdisposal, exceptthat itmust be continuous andmust besuch that theconfigurationsinthevaried motion coincide with those intheactual motion attheinstants t=and t=t. Thus thevalues of801,802,...ateveryinstant maybeanywepleasewhich arepermitted bythemechanism ofthesystem, exceptthattheymust be continuous functions oftandmust vanish when t=andwhen t=r.Whatever series ofvalues weassignto801}802,...,wehave seen thattheequation Idt= o istrue. Hence thevalue of7must vanish atevery instant, andwemust have ^idL_didL\) ila^ dt\ddj) 549. Atthisstagethere aretwoalternatives tobeconsidered. Itmay bethatwhatever values areassignedto801}802,...80n,thenewconfigura- tion #i4-801} 2+$02,•••&n+80nwillbeapossible configuration —that isto say,willbeoneinwhich thesystemcanbeplacedwithoutviolatingthe constraints imposed bythemechanism ofthesystem.Inthiscaseequation (499)must betrue forallvalues of80lt892)...80n,sothateachtermmust vanishseparately,andwehave thesystemofequations dLd/dL Q)=0, (s=l,%...n) (500)d0sdt\dd^ There arenequationsbetween thenvariableslt 2,...nandthetime. Hence these equationsenable ustotrace thechangesin1} 2,...nandto expresstheir values asfunctions ofthetimeandofthe initial values of 0\,02,••@n> V\,02,••0n- 550. Next, supposethat certain constraints areimposedonthevalues of lf 2,... nbythemechanism ofthesystem.Letthese beminnumber, and letthem besuch thatthesmall increments 80lf802,...80nareconnected byequationsoftheform al801+a2802+...+a n80n=O (501), b1801+b2802+...+bn80n=O(502), etc. Thenequation (499) must betrue forallvalues of80x,802,...which are such asalsotosatisfy equations (501), (502),etc. Letusmultiply equations (501), (502),...by\fi,...andaddtoequation (499). Weobtain anequationoftheform X-[———(^)+Xa l+pb 1+...j89,=(503). 492 Dynamical Theory ofCurrents[ch.xvi Letusassign arbitraryvalues to80m+l ,80m+2 ,...80n,andthenassignto themquantities80!,802,...80mthevaluesgiven bythemequations (501), (502),etc. Inthiswayweobtain asystemofvalues for80l}802,...80n which ispermitted bytheconstraints ofthesystem. Themmultipliers X,fi,...areatourdisposal:letthese besupposedto bechosen sothatthemequations d-^-^-(~)+\as+fMbs+...=0, (s=l, 2,...m) (504)ddsat\9ty aresatisfied. Thenequation (503)reduces to m+i{ousatVdtv ) andsincearbitraryvalues havebeenassignedto80m+1,...80n,itfollows that each coefficient inthisequation must vanishseparately. Combiningthe systemofequationssoobtained withequations (504),weobtain thecomplete systemofequations ^-^(5T) +Xa *+/i&*+-"= ' (s==1>2>••») (506).oUsatVdty Lagrange's Equations forGeneral(including Non-conservative) Forces. 551. Ifthesystemofforces isnotaconservativesystem, wecannot replacetheexpression 2(Z xSai+F 1Sy1+Z1&1) in§545by—8WwhereWisthepotential energy. Wemay, however, still denote thisexpressionforbrevity by—{STF},nointerpretation being assigned tothissymbol, andequation (496)willassume theform [T (BT- {BW})dt=(507).Jo Bythetransformation used in§548,wemayreplaceI8Tdtby Jo iXde,dAddJ) Now—[8W] is,bydefinition, thework done inmovingthesystemfrom theconfiguration lf 2,...ntotheconfigurationX+80lf 2+802,...n+80n. Itistherefore alinear function of801}802,...80n ,andwemaywrite -{8W}=®1801+®2802+...+®n80n, where®ltS2,•••®narefunctions of1} 2,...6n. 550-552] Lagrange's Equations 493 Wenowhaveequation (507)intheform Vi 130! dt\ddj J Asbefore eachintegrandmust vanish. Wehave therefore ateveryinstant i(dd, dt\ddj \ Ifthecoordinates dly62,...9nare allcapableofindependent variation, thisleads atonce tothesystemofequations I©-f.="<"^--> (608)- while ifthevariations inlt#2,...areconnected bytheconstraintsimplied inequations (501), (502),...weobtain, asbefore, thesystemofequations d/dT\ dT dtr-±\-<LL=®g+Xas+pbs+..., (s=l,2,...n)...(509). Thequantities ©i,®2,...arecalled the"generalisedforces"correspond- ingtothecoordinatesi}62,.... Lagrange's Equations forImpulsiveForces. 552. Letusnow supposethat thesystemisacted onbyaseries of impulsive forces, these lasting throughtheinfinitesimal interval from t= tot=t.Ifwemultiply equations (508) bydtandintegrate throughoutthis interval weobtain 'dT' L90Jt=TfTdT fT_^-dt=%sdt. t=o Jodo* Jo 7)T The interval tistobeconsidered asinfinitesimal, and^-isfinite. Thus thesecond termmaybeneglectedandtheequation becomes dT fT changein—=®sdt (510). ddsJo We call\®sdtthegeneralised impulse correspondingtothegeneralised Jo force ©«,andthen, fromtheanalogybetweenequation (510) andtheequation changeinmomentum =impulse, 8JT coordinate 6wecall—thegeneralised momentum correspondingtothegeneralised dO. 494 Dynamical Theory ofCurrents[oh.xvi Application toElectromagnetic Phenomena. 553.Wehavealreadyobtainedexpressionsfortheenergyofanelectro- staticsystem,asystemofmagnets,ofcurrents, etc.,and ineverycase this energycanbeexpressedinterms ofcoordinates associated with"accessible" partsofthemechanism. Wecanalsofindtheworkdone inanysmallchange inthesystem,sothatwecanobtain thevalues ofthequantities denoted in thelastsection by©x,©2, Allthatremains tobedone before wecan it~ >_ i.: :„.:n_i~e c?za>7\ j._ ±1• i.. j. j_-c apply Lagrange's equations provisionally (cf.§547)totheinterpretationof electromagnetic phenomenaistodetermine whether thedifferent kinds of energyaretoberegardedaskineticenergyorpotential energy. Kinetic andPotentialEnergy. 554. At firstsightitmightbethoughtobvious that theenergyof electricchargesatrestand ofmagnetsatrestoughttobetreated as potential energy,while that ofelectricchargesormagnetsinmotionought tobetreated askinetic. Onthisview theenergyofasteadyelectric current, beingtheenergyofaseries ofchargesinmotion, oughttobe regardedaskineticenergy. Wehave alsoseen that thisenergyistobe regardedasbeing spread throughoutthemediumsurroundingthecircuit in which thecurrent flows, andnotasconcentrated inthecircuit itself. Thus wemustregardthemedium aspossessingkineticenergyatevery point,the amount ofthisenergy being,aswehave seen,^—perunitvolume. Butwehave alsobeen ledtosupposethat themedium isinjustthe same condition whether themagneticforce isproduced bysteadycurrents or bymagneticshells atrest. Thus, onthesimpleview which wearenow considering, wearedriven totreat theenergyofmagnetsatrestaskinetic— aresult which isinconsistent with thesimple conceptionsfromwhich we started. Havingarrived atthiscontradictory result, there isnojustification left fortreatingelectrostaticenergy, anymore thanmagnetostatic energy, aspotentialrather than kinetic. 555. Abandoningthissimplebutunsatisfactory hypothesis,letusturn ourattention inthe firstplacetothedefinite discussion ofthenature ofthe energyofasteadyelectric current. Letussupposethatwehavetwocurrentsi,i'flowinginsmall circuits at adistance rapart. Asamatter ofexperiment weknow that these circuits exert mechanical forcesupononeanother asiftheyweremagneticshells of strengthsi,%'.LetussupposethataforceRisrequiredtokeepthemapart, sothatinitiallythecircuits attracted oneanother with aforce R,butare 553-555]Kinetic andPotential Energy 495 now inequilibriumunder theaction oftheirmutual attraction andthis force Ractinginthedirection ofrincreasing. cose IfMisthequantity11 dsds,weknow thatthevalue ofRis R=-H'd -§(511), thisvaluebeingfounddirectlyfrom theexperimentalfactthatthecircuits attract liketheirequivalent magneticshell(cf.§499). Theenergyofthetwocurrents isknown tobe E=±(Li2+2Mii'+Ni'2 ) (512). Letussuppose,forthesake ofgenerality,that this consists ofkinetic energy Tandpotential energy W.Then, assumingforthemoment thatthe mechanism ofthese currents isdynamical,inthesense thatLagrange's equations maybeapplied, weshall have adynamical systemofenergyT+W,andoneofthecoordinates maybetaken tober,thedistanceapart ofthecircuits. •(513),TheLagrangian equation correspondingtothecoordinate risfound to be(cf.equation (508)), dfdT\d(T-W) _R dt\dr] dr andsinceweknow that, intheequilibrium configuration, d(dT\ n p.,dM weobtain onsubstitution inequation (513), d(T-W) __..,m dr dr Fromequation (512)weseethat theright-hand member isthevalue of dEed(T+W) dWA, ,. , ^—,oror jr.Hence ourequation shews that -r—=0,fromwhich we deduce thatW=0.Inother words, assumingthat asystemofsteady currents forms adynamical system,theenergyofthissystem must be whollykinetic. This resultcompels usalso toacceptthat theenergyofasystemof magnetsatrestmust alsobewhollykinetic. Weshall discuss this result later. Forthepresent weconfine ourattention tothecase ofelectric phenomena only.Wehave found that ifthemechanism ofthesepheno- mena isdynamical (the hypothesis uponwhich wearegoingtowork), then theenergyofelectric currents must bekinetic. 496 Dynamical Theory ofCurrents[ch.xvi Induction ofCurrents. 556. Letusconsider anumber ofcurrentsflowinginclosed circuits. Letthestrengthsofthecurrents bei1}i2,...and letthenumber oftubes of induction which cross these circuits atanyinstant beN1}N2,...,sothat if themagneticfield arisesentirely from thecurrents, wehave(cf.§502) Ar_._.\ (514).J\2=L2li1+L22i2+...,etc.J Theenergyofthecurrents iswhollykinetic sothatwemaytake =%(L 11i*+2L12i1i2+...) asbefore(§503). Inthegeneral dynamical problem,itwillberemembered thatTwasa quadraticfunction ofthe velocities. Thus i1}i2,...mustnowbetreated as velocities andwemust take ascoordinatesquantities x1}x2,...,defined by •CciZ-j•CLu/2h=Wl2= ~dt'etc ' Clearly xxmeasures thequantityofelectricitywhich hasflowedpastany pointincircuit 1since agiven instant, andsoon.Thus interms ofthe coordinates x1}x2,...wehave T=^(L ux12+2Lni1d>a+...) (515). There isnopotential energyinthepresent system, butthesystemis acted onbyexternal forces, namelytheelectromotive forces inthebatteries andthereaction between thecurrents andthematerial ofthecircuits which shews itself intheresistance ofthe circuits. Wehave therefore toevaluate thegeneralisedforces®1}®2>•••• Consider asmallchangeinthesysteminwhich xxisincreasedby8xlfso that thecurrent ^flows foratime dtgiven byi1dt=Sx1.Theworkper- formed bythebatteryisE1Bx1,theworkperformed bythereaction with the matter ofthecircuit, being equalandoppositetotheheatgeneratedinthe circuit, is—R^dt. Thus ifXjisthegeneralisedforcecorrespondingtothe coordinate xlfwehave Xxhxx=E^8xt—R^ifdt, sothat X1=Ex—R^. TheLagrangian equation correspondingtothecoordinate x1is d_(d_T\_dT^ dt[dij dx,ly or ~(Lui1+L12i2+...)=E1-R1i1 (516), oragainjfc,—^-=K1%1. 556,556a]Induction ofCurrents 497 Theequations correspondingtothecoordinates x2,x3>•••are h2—=K2%2,etc. Thus theLagrangian equationsarefound tobeexactlyidentical with the equationsofcurrent-inductionalready obtained, shewingnotonlythat the phenomenonofinduction isconsistent with thehypothesisthat thewhole mechanism isadynamical system,but alsothat thisphenomenonfollows as adirectconsequenceofthishypothesis.Inthissystemtheaccessibleparts ofthemechanism arethecurrentsflowinginthewires; theinaccessible partsconsist oftheether which transmits theaction from one circuit to another. 556 a.Ontheelectrontheory,thekineticenergymust besupposedmade uppartlyofmagnetic energy,asbefore, andpartlyofthekinetic energyof themotion oftheelectrons bywhich thecurrent isproduced. Lettheaverageforwardvelocityoftheelectrons atanypointbeu(cf. §345 a),and letU+uQbetheactualvelocityofanysingle electron, sothat theaveragevalue ofuisnil.Thekineticenergyofmotion oftheelectrons, sayTe,isthen The firsttermrepresents partoftheheat-energyofthematter, andthis doesnotdependonthevalues ofthecurrents xx,x2,—Toevaluate the second termweuseequation (b)of§345 a, Neu=i=x, andobtain thekinetic energyo£theelectrons inthecomplete systemof currents intheform s+x2'j^ds+m Thus thetotal kineticenergy maystillbeexpressedintheform (515)if wetake '<n=L'n+j-jy-;ds,etc (517), and inthisthe firstterm isthecontribution from themagnetic energy (cf.§503),andthesecond term isthecontribution from thekineticenergyof theelectrons. Equation (516) assumes theform 498 Dynamical Theory ofCurrents[ch.xvi Iftheinduction terms onthe leftareomitted, wehave astheequationof acircuit inwhich induction isnegligible ft-Rih-ljjfeds dt=°' This, with thehelpoftheformulae of§345a,maybeexpressedinthe form jxds-i.j^ds-^f^ds-O, which inturn isseen tobeexactlyidentical withequation (c)of§345 a, integratedround thecircuit. Thusweseethattheanalysisof§556applies perfectlytotheelectron theoryofmatter, provided Ln,L^, ...aresupposedtohave thevaluesgiven byequation (517), andequation (517 a)isthen thegeneral equationof induction ofcurrents, when theinertia oftheelectrons istaken intoaccount. Electrokinetic Momentum. 557. Thegeneralised momentumcorrespondingtothecoordinate xxis =-rorNx.Thus thegeneralised momentacorrespondingtothecurrents OX\ inthedifferent circuits areNx,N2,...,thenumbers oftubes ofinduction which cross thecircuits. ThequantityNxisaccordingly sometimes called the electrokinetic momentum ofcircuit 1,andsoon. IfwegivetoLnthevalue obtained inequation (517)of§556 a,the value oftheelectrokinetic momentum is(cf.equations (514)) (L'nh+L12i2+...)+hj-y-ds, inwhichclearlythelastterm comes from themomentum oftheelectrons, andtheremainingterms from themomentum ofthemagneticfield. Examples. I.Discharge ofaCondenser. 558. Asafurther illustration ofthedynamical theory,letusconsider thedischargeofacondenser. LetQbethechargeonthepositive plate atany instant, and letthisbetaken asaLagrangian coordinate. The current iisgiven byi=--^-=—Q.Inthenotationalready employed (§516)wehave 556a-559]Electric Oscillations 499 andLagrange's equationis d fd_T\_dT d_W=_ dt\dQj dQ+dQ~*' which istheequation alreadyobtained in§516,andleads tothesolution alreadyfound. II. Oscillations inanetworkofconductors 559. Theequations governingthecurrentsflowinginanynetwork of conductors when induction istaken intoaccount canbeobtained from the general dynamical theory. Letussupposethat the currents inthe different conductors are h>h,•••inyand letthecorrespondingcoordinates bexltx2,...xn,these being given by\=-77- ,etc. Ifanyconductor, say 1,terminates ona condenserplate,letxxdenote theactualchargeontheplate, and letthe dxcurrent bemeasured towards theplate,sothat therelations ix=-~, etc. will still hold. Letconductor 1contain anelectromotive forceExandbe ofresistance Rr. Thequantitiesxltx2,...maybetaken asLagrangian coordinates, but theyarenot, ingeneral, independentcoordinates. Ifanynumber ofthe conductors, say 2,3,...smeet inapoint,thecondition fornoaccumulation ofelectricityatthepoint is,byKirchhoff 'sfirst law, i2±h±...±is=0, from which wefind that variations inx2,x3,...areconnectedbythe relations Bx2±Sx3+...+Sxs=0. Letussupposethat there aremjunctions. Thecorrespondingcon- straints onthevalues ofSxl}8x2>...canbeexpressed bymequationsof theform a18x1+a2Sx2+...+anBxn=0)..... \(518)> &!Sxx+b28x2+...+bnSxn=0> etc., inwhich each ofthecoefficients ax,a2)...an,b1}...hasforitsvalue either 0,+1or—1. ThekineticenergyTwill beaquadraticfunction ofxx,x2,etc.,while the potential energyW(arisingfrom thecharges,ifany,onthecondensers) will 32—2 500 Dynamical Theory ofCurrents fCH.XVI beaquadraticfunction ofx1;x2,....Thedynamical equationsarenownin number, thesebeingoftheform(cf.equations (509)) dfdT\ BTdW„„.,.,__ ./tin , a(a)-s+^-*-^+x^+^+ ";('-1^-"»)-"( 619)- Theseequations, togetherwith themequationsobtained byapplying Kirchhoff's firstlawtothedifferentjunctions, form asystemofm+n equa- tions, from which wecaneliminate themmultipliers \ /x,...,andthen determine thenvariables %1,x2,...xn. 560. Asanexampleoftheuseoftheseequations,letusimaginethata current /arrives atAanddivides intotwopartsilti2,which flowalongarms Ct ACB,ADB andreunite atB.Neglecting induction between these arms andtheleads toAandB,wemaysupposethatthepartofthekineticenergy which involvesijand i2is ±Li 1*+Mi1i%+lN'if. There arenobatteries andnocondenser inthearms inwhich the currents ixand %2flow. The currents are,however, connectedbythe relation ij+i2=I sothatthecorrespondingcoordinates xYandx2areconnectedby 8xi+Sx2=0. Thedynamical equationsarenowfound tobe(cf.equations (519)) d dt dt(Liy+Mi2)=-Rix+X, (Mii+Ni2)=-Si2+\. Ifwesubtract andreplacei2byI— i'i,weeliminate \andobtain di dT (L+N-2M)a£+(M-N)<^=SI-(R +S)i 1. IfIisgivenasafunction ofthetime, thisequation enables ustodeter- mine i1}andthence i2. 559-561] Electric Oscillations 501 Forinstance, supposethat thecurrent Iisanalternatingcurrent of frequency p\2ir.IfweputI=ieipt ,thesolution oftheequationis . S-(M-N)iph~(L+N-2M)ip +(R+S)' whilesimilarlyi2= (L+N-2M)ip +(R+S) Whenp—0,thesolution ofcourse reduces tothat forsteadycurrents. Aspincreases, wenotice that thethree currents il3i2and/become, in general,indifferentphases, and that theiramplitudes assume values whichdepend uponthecoefficients ofinduction aswell asontheresistances. Finally,forverygreatvalues ofp,thevalues ofixand i%aregiven by N^M=T^M=L+N-2M' shewingthatthecurrents arenow inthesamephaseandaredivided ina ratio whichdepends onlyontheir coefficients ofinduction. For instance, ifthearmsAGB,ABB arearrangedsoastohaveverylittle mutual induction (Mvery small), thecurrent willdistribute itself between thetwo arms intheinverse ratio ofthecoefficients ofself-induction. Itispossibletoarrangeforvalues forL,MandNsuch that thetwo currents itand i2shallbeofopposite sign.Insuch acasethecurrent inone atleast ofthebranches isgreaterthan that inthemain circuit. Let us,for instance, supposethatthebranches consist oftwo coilshavingrand sturns respectively, arrangedsoastohaveverylittlemagnetic leakage.Then LN—M2isnegligible (cf.§525)andwehaveapproximately r-rs s2' Theequations become h._*2 I s—r s—r' sothatthecurrents willflowinopposite directions, andeithermaybegreater than thecurrent inthemain circuit. Bymakingsnearly equaltorand keepingthemagnetic leakageassmall aspossible,wecanmake both currentslargecomparedwith theoriginalcurrent. III.Rapidly alternatingcurrents. 561. This lastproblemillustrates animportant pointinthegeneral theoryofrapidly alternatingcurrents. Inthegeneral equations (519), d(dT\ dT3W „D..,dmy^ 8+wrE'~Bela+Xaa+tih '+---' letussupposethatthewholesystemisoscillatingwithfrequency pj2-rr,which issogreatthat itmaybetreated asinfinite. Wemayassume thatevery 502 Dynamical Theory ofCurrents[ch.xvi d_ dtd variable isproportionaltoeipt ,andmayaccordingly replace-7-bythemulti plier ip.Theequations nowbecome .(dT\ dTdW„_.^. %P[^~)~?~+d— s=s+^s+'"' and alltheterms onthelefthandmaybeneglectedincomparisonwith the first,which contains thefactorip.Theterms ontherightcannotlegitimately beneglectedbecause \,fi,...areentirely undetermined, andmaybeofthe samelargeorder ofmagnitudeastheterms retained. Ifwereplace A,,fi,.. byipX', ipfi, ...,theequations become —+\'as+ix'b s4-...=0, etc.OXg inwhich V,//,...arenowundeterminedmultipliers. These, however, are exactlytheequationswhichexpressthatTisamaximum oraminimum forvalues of cb1}x2,...which areconsistent with therelations(cf.§559) necessarytosatisfyKirchhoff" sfirst law. SinceTcanbemade aslargeas weplease,thesolution mustclearly makeTaminimum. Thusweseethat Asthefrequency ofasystem ofalternatingcurrents becomesvery great, thecurrents tend todistribute themselves insuchawayastomake thekinetic energy ofthecurrents aminimumsubject onlytotherelations imposed by Kirchhoff'sfirstlaw. This resultmaybecomparedwith thatpreviouslyobtained(§357)for steadycurrents. Weseethatwhile thedistribution ofsteadycurrents is. determinedentirely bytheresistance oftheconductors, that ofrapidly alternatingcurrentsis,inthe limit inwhich thefrequencyisinfinite, determinedentirely bythecoefficients ofinduction. Itfollows that, inacontinuous medium ofanykind, thedistribution of rapidly alternatingcurrents willdepend onlyonthegeometricalrelations of themedium, andnotonitsconducting properties.Inpointoffact,wehave alreadyseen that thecurrent tends toflowentirelyinthesurface ofthe conductor(§537).Wenow obtain thefurther result that itwill, inthe limit, distribute itself inthesamewayover thesurface ofthisconductor, nomatter inwhatwaythespecificresistance varies frompointtopointof thesurface. IV.TransmissionofSignals along awire. 562. Imagineasignal beingsentalongawire, initiallyfreefrom all electrical disturbance. Atanyinstant letidenote thecurrent atapoint distant xfrom theendofthewire,and letqdenote thetotalquantityof electricitywhich hasflowedpastthispoint. Then iandqarefunctions of xand t. 561,562]Electric Oscillations 503 Letqbemeasured inelectrostatic units, butletibemeasured inelectro- magneticunits. Then therate offlowpastanypointwillbeiCelectrostatic unitspersecond, where Cdenotes thenumber ofelectrostatic units inone electromagneticunit(cf.§484). Thus L%df IfListheself-induction ofthewireperunitlength,thetotal kinetic energyofthecurrents is where theintegralistakenalongthewire. Inanyelement dxofthewire thechargeis—^dx,sothat ifKistheelectrostaticcapacityofthewire perunitlength,thepotential energyWisgiven by LetRbetheresistance ofthewireperunitlengthinelectromagnetic units, then therateofgenerationofheat is R\i2dx. The values ofqatdifferentpointsofthewiremay betaken as Lagrangian coordinates, fortheysuffice tospecify thepositionofeach element ofcurrent. TheLagrangian equation correspondingtothecoor- dinateqatanydistance xwillbe(cf.§556) dt[dq) dq+ 'dq~Mh mwhich wehave ~-r= -^r2qand—=. dWToevaluate -=—,letusimagine qchangedtoq+8qatevery pointofthe wire, subjecttoBqvanishingatthetwoends. Theincrement inW,say8W, isgiven by '(d(q+Bq)\*fdqV \dx j\dx)sw=mdx=Ti!a £iL(Sq)dx - and,onintegrating byparts,thisbecomes Thus atanypoint x,dW ld*q dq~Kdx2 504 Dynamical Theory ofCurrents[ch.xvi TheLagrangian equation accordinglybecomes O2dt*Kdx*~ dt' K.?2 SinceGi=-~,itisatonce seen thatthecurrent iatanypointsatisfies dt thesame differentialequation,and this isalsotrue ofthepotential V,since dx namely^-=—KV. Thusq,iandVallsatisfythesame differentialequation, Ek*±. +KR*±-»±(520) Thisequationisthegeneral equationforthetransmission ofelectric signals alongawire. Itiscalled the" Telegraphic equation" byPoincare andothers. Wehave seen in§505 a,how tocalculate theself-inductionperunit lengthofanywire. Ifthewire issufficientlythin incomparisonwith its distance from other conductors, theself-induction Lperunitlength becomes identical with thequantitydenoted byL'in§505,andweaccordinglyhave therelation(cf.equation (430 e)), KL=k/jl, where kisthedielectric constant, and/xthemagnetic permeabilityofthe insulatorsurroundingthewire. Letusput a?=— KfA sothatadepends onlyonthepropertiesoftheinsulating material, andthe telegraphic equationbecomes a?dt*+ dt" dx*' Forslowsignals,the firstterm inthisequation,which arises from the inertia oftheelectric current, maybeneglected. Theequationthen reduces toequation (303)ofChapter IXwhich wasobtained astheequationof transmission ofsignals alongasubmarine cable. Underpracticalconditions signals alongasubmarine cable aresoretardedbythehighelectrostatic capacityofthecable that this inertia termmay legitimatelybeneglected, buttheterm hastoberetained when theequationisappliedtotelegraph andtelephone problems. When thewire isfarremoved from other conductors, theelectrostatic capacityKwillbesmall. IfKisneglected entirely,theequation becomes d-6992 (f> dt" dx*' 562,563] Mechanical Action 505 Thesolution ofthisequationis $=f(oc-at)+<E>{x+at), where/,Oarearbitrary functions, andthesolution isseen torepresentthe transmission ofasignalwithout changeoftypeorlossofintensity,the velocityoftransmissionbeinga. Inpractical telephony andtelegraphyitisnotusualpossibletoneglect entirelythevalue ofKinthesecond term oftheequation.Solutions of thegeneralthree-termequationhave been obtained byHeaviside*, Poin- caref, PicardJ, Boussinesq§,andRiemann||. Itisfound thatthesignalisstilltransmitted with thesamevelocity a, butthat there isachangeoftypeand lossofintensity;there isalsoan electric fieldandcurrent lefttrailingbehind eachsignal;these would of course tend toconfuse thesucceeding signalifthesignalsaresentwithout sufficient interval. Thus forrapidtransmission orclearspeakingitisnecessarytoreduce thevalue ofKR(cf.§369); thesmaller thisterm ismade, thesmaller the amount ofblurringorindistinctness will be.Weseeatoncewhytelephone wires arekeptasfaraspossiblefrom other conductors, andcanunderstand thedifficultyofclearspeakingorrapid signalling throughasubmarine cable. Mechanical Force acting onaCircuit. 563.Let6beanygeometrical coordinate, and let©bethegeneralised force tendingtoincrease thecoordinate 6,sothat tokeepthesystemof circuits atrestwemustsupposeitacted onbyanexternal force—©.Then Lagrange's equationforthecoordinate 6is ±^T\JTdtVb0/ de andtherefore, when thesystemisinequilibrium, wemust have *-%<521>- Iftheenergyofthesystemwere wholly potentialandofamount W,the force©would begiven bydW©= de Thus themechanical forcesactingarejustthesame astheywould beif thesystemhadpotential energyofamount —T. *Phil.Mag. 1888and Coll. Papers. tc-R-117(1893), p.1027. tC.R.118(1894), p.16. §C.P.118(1894), p.162. ||Riemann-Weber, Diepartielle DifferentialgleichungenderMath. Physik,4thedn. (1901), ii.p.322. 506 Dynamical Theory ofCurrents[ch.xvi 564.Letussupposethatanygeometrical displacement takesplace,this resultinginincreasesWY,802,...inthegeometricalcoordinates1}d2,...,and letthecurrents inthecircuits remain unaltered, additionalenergy being supplied bythebatteries when needed. Theincrease inthekineticenergyofthesystemofcurrents is while thework donebytheelectrical forcesduring displacementis2©c£0 which, byequation (521),isalsoequalto These twoquantitieswould beequal andoppositeifthesystemwere aconservativedynamical systemacted onbynoexternal forces. Inpointof facttheyareseen tobeequal andofthesamesign.Theinference isthatthe batteriessupply duringthemotion anamount ofenergy equaltotwice the increase intheenergyofthesystem. Ofthissupplyofenergyhalfappears asanincrease intheenergyofthesystem,while theother half isused inthe performanceofmechanical work. This result should becomparedwith thatobtained in§120. 565. Asanexampleoftheuseofformula(521),letusexamine the forceactingonanelement ofacircuit. Letthe componentsofthemechanical forceactingonany element dsofacircuitcarryingacurrent ibede- noted byX,7,Z. Tofindthevalue ofX,wehave toconsider a displacementinwhich theelement dsisdisplaced adistance dxparalleltoitself, theremainder ofthe circuit beingleftunmoved. Letthecomponentofmagneticinduction perpendiculartotheplane containingdsanddxbedenoted byN,then if Tdenotes thekineticenergyofthewholesystem,theincrease inTcaused bydisplacementwillbeequaltoitimes theincrease inthenumber oftubes ofinduction enclosed bythecircuit, andtherefore dT=iNdsdcc. Thus, using equation (521), X=7T-=iNds, andthere aresimilarequations givingthevalues ofthecomponents Fand Z. IfBisthetotal induction and ifBcoseisthecomponentatright angles tods,then theresultant forceactingondsisseen tobeaforce ofamount iBcoseds,actingatright anglestotheplane containing Band ds,andin such adirection astoincrease thekineticenergyofthesystem.This isa generalisationoftheresultalreadyobtained in§498. 564-567] Magnetic Energy 507 Magnetic Energy. 566.Wehave seen that theenergyofthe field offorce setupbya systemofelectric currents must besupposedtobekineticenergy. We know alsothat this field isidentical with that setupbyacertainsystemof magnetsatrest. These two facts canbereconciledonlybysupposingthat theenergyofasystemofmagnetsatrest iskineticenergy—asuggestion originallyduetoAmpere. Weber'stheoryofmagnetism (§476)hasalreadyledustoregard any magnetic bodyasacollection ofpermanently magnetised particles. Ampere imaginedthemagnetismofeachparticletoarise fromanelectric current which flowedpermanentlyround anon-resistingcircuit intheinterior ofthe particle. Thephenomenaofmagnetism,onthishypothesis, become inall respectsidentical with those ofelectric currents, andinparticulartheenergy ofamagnetic bodymust beinterpretedasthekineticenergyofsystemsof electric currentscirculatingintheindividual molecules. Forinstance two magnetic polesofopposite signattract because twosystemsofcurrents flowinginoppositedirections attract. Wehave seen that themechanical forces inasystemofenergyEare BE . . BE— -jtq,etc., iftheenergyispotential,butare+^z,etc., iftheenergyis kinetic. Itmighttherefore bethoughtthattheacceptanceofthehypothesis that allmagnetic energyiskinetic wouldcompelustosupposeallmechanical forces inthemagnetic systemtobetheexactoppositesofwhatwehave previously supposed them tobe. This, however, isnot so,becauseaccepting thishypothesis compelsusalsotosupposetheenergytobeexactly opposite inamount towhatwepreviously supposedittobe.Instead ofsupposing BE thatwehavepotential energyEand forces—«—,etc.,wenowsupposethat wehave kineticenergy—Eandforces H =—- ,etc.,sothattheamounts of OOl/ theforces areunaltered. Tounderstand how itisthattheamount ofthemagnetic energy must be supposedtochange signassoon aswesupposeittooriginatefromaseries of molecular currents, weneedonlyreferback to§502. 567. Themolecular currents bywhich wearenowsupposing magnetism tobeoriginated must besupposedtobeacted onbynoresistance andbyno batteries, but iftheassemblageofcurrents istoconstitute atruedynamical system wemustsuppose themcapableofbeingacted upon byinduction whenever thenumber oftubes offorce orinduction which crosses them ischanged.Inthegeneral dynamical equation d(BT\dT 508 Dynamical Theory ofCurrents[ch.xvi wemayputEandReachequaltozero,and^-isalready known tovanish. 7\T Thus theequation expressesthat -^-.remains unaltered. Wenow seethatthestrengthsofthemolecular currents willbechanged byinduction insuchawaythattheelectrokinetic momentum ofeachremains unaltered. Ifthemolecule isplacedinamagneticfieldwhose lines offorce runinthesame direction asthose from themolecule, thentheeffect ofinduc- tion istodecrease thestrengthofthemolecule until theaggregate number oftubes offorcewhich cross itisequaltothenumberoriginally crossingit. This effect ofinduction isoftheoppositekindfrom thatrequiredtoexplain thephenomenonofinducedmagnetisminironandotherparamagneticsub- stances. Ithas,however, beensuggested byWeber that itmayaccount for thephenomenonofdiamagnetism. 568. Modern views astothestructure ofmattercompelustoabandon Ampere's conceptionofmolecular currents, but thisconceptioncanbere- placed byanother which isequally capableofaccountingformagnetic phenomena. Onthemodern view allelectric currents areexplainedasthe motion ofstreams ofelectrons. Theflow ofAmpere'smolecular current may accordinglybereplaced bythemotion ofringsofelectrons. The rotation ofoneormoreringsofelectrons wouldgiverisetoamagneticfieldexactly similar tothatwhich would beproduced bytheflowofacurrent ofelectricity inacircuit ofnoresistance. Itisonthese lines that itappears probablethatanexplanationof magnetic phenomenawillbefound inthefuture. Nocomplete explanation hassofarbeen obtained, forthesimple andsufficient reason thatthearrange- mentandbehaviour oftheelectrons inthemolecule oratom isstillunknown. EXAMPLES. 1.Two wires arearrangedinparallel,their resistances beingRandS,and their coefficients ofinduction being L,M,N.Shew that foranalternating current offrequency pthepairofwires actlikeasingle conductor ofresistance Randself-induction L,given by R RS(R+S)+p*{R(N-My +S(L-Mf} L 1 NR*+LS2+2MRS+p'i(LiV-M2)(L +tf-2M) {R+Sf+p-^L +N-lMf' 2.Aconductor ofconsiderable capacity Sisdischarged throughawire ofself-induc- tion L.Ataseries ofpoints along thewiredividingitintonequal parts, (n-\) equal conductors each ofcapacity*S"areattached. Find anequationtodetermine theperiods ofoscillations inthewire,andshew that iftheresistance ofthewiremaybeneglected theequation maybewritten 2tan^0(S-%S')=S'cot?«£, where thecurrent varies ase~,andsin2<£=>S"X2Z/4?j. Examples 509 3.A"Wheatstone bridge arrangementisused tocompare thecoefficient ofmutual inductionMoftwo coilswiththecoefficient ofself-induction Lofathird coil.Oneofthe coils ofthepairisplacedinthebatterycircuit AC,theother isconnected toB,Dasa shunt tothegalvanometer, andthethird coil isplacedinAD.Thebridgeisfirstbalanced forsteady currents, theresistances ofAB,BC,CD,DAbeing thenRuR2,R3,i?4:the resistance oftheshunt isaltered tillthere isnodeflection ofthegalvanometerneedle at make andbreak ofthebattery circuit, andthetotal resistance oftheshunt isthen R. Prove that LRR1=M{R 1+Rif. 4.Two circuits eachcontaining acondenser, having thesame natural frequency when atadistance, arebroughtclosetogether. Shewthat, unless themutual induction between thecircuits issmall, there willbeineach circuit twofundamentalperiodsofoscillation given by „_1 1 whereC1,C2arethecapacities, Lx,L2thecoefficients ofself-induction, andMthecoefficient ofmutual induction,ofthecircuits. 5.Letanetwork beformed ofconductors A,B,...arrangedinanyorder. Prove that when aperiodicelectromotive forceFcosptisplacedinAthecurrent inBisthesame in amplitude andphaseasthecurrent isinAwhen anelectromotive forceFcosptisplaced inB. CHAPTEK XVII DISPLACEMENT QUERENTS ANDELECTROMAGNETIC WAVES Maxwell's Equations. 569.Ourdevelopmentofthetheoryofelectromagnetismhasbeenbased upontheexperimentalfactthat thework done intakingaunitmagnetic poleround anyclosedpathinthefield isequalto4nrtimes theaggregate current enclosed bythispath.But ithasalreadybeen seen(§534) that this developmentofthetheoryisnotsufficiently generaltotake account of phenomenainwhich theflow ofcurrent isnotsteady:"theaggregatecurrent enclosed byapath"isanexpressionwhich hasadefinite meaning onlywhen theflow ofcurrent issteady.Beforeproceedingtoamoregeneral theory, which istocover allpossiblecases ofcurrent flow, itisnecessarytodeter- mine inwhatwaytheexperimentalbasis istobegeneralised,inorder to providematerial fortheconstruction ofamorecomplete theory. Theanswer tothisquestion hasbeenprovided byMaxwell.According toMaxwell'sdisplacement theory (§171),themotion ofelectricchargesis accompanied bya"displacement"ofthesurrounding medium. Themotion produced bythisdisplacementwillbespokenofasa"displacement-current," andwehave seen that thetotal flowwhich isobtained bycompoundingthe displacement-currentwith thecurrentproduced bythemotion ofelectric charges (whichwillbecalled theconduction-current),willbesuch thatthe total flow intoanyclosed surfaceis,under allcircumstances, zero. Thus if S1}$2areanytwosurfaces bounded bythesame closed path s,thetotal flowofcurrent across Siisthesame as thetotal flow, inthesame direction, acrossS2,sothat eithermaybetaken tobetheflowthroughthecircuit s. Maxwell'stheory proceedsonthesuppositionthat in anyflow ofcurrent, theworkdone intakingaunitmagnetic poleround sis equalto4>tttimes thetotal flowofcurrent, includingthedisplacement-current, throughs.Thejustificationforthissuppositionisobtained assoon asitis seenhow itbringsabout acomplete agreementbetweenelectromagnetic theory andinnumerable facts ofobservation. 570. Letusfirstputthehypothesisoftheexistence ofdisplacement- currents intomathematicallanguage.Let u,v,wbethecomponentsofthe 569,570] Disjilacement Currents 511 ordinarycurrent atanypointwhich isproduced bythemotion ofelectric charges,and letthisbemeasured, asbefore, inelectromagneticunits (cf.§484). Letthecomponentsofthedisplacement, which hasbeenshewn tobeidentical withFaraday's polarisation (§172), bedenoted asbefore by f,g,h.OnMaxwell'stheoryofdisplacement, f,g,harethequantitiesof electricityofthesecond kindwhich have crossed unitareasperpendicularto thecoordinate axes atanypoint. Thecorrespondingrates ofcurrent-flow, or quantitieswhich cross unitareaperunittime, areofcourse dfdgdh di'di' 'di' These areaccordinglythecomponentsofMaxwell's"displacement- current." They are,however, measured inelectrostatic units. Ifwesuppose there tobeCelectrostatic units ofchargeinoneelectromagnetic unit, the displacement current, measured inelectromagnetic units, willhave com- ponents ldfldg\dh Gdt' Cdt' Gdt{0Z >' andMaxwell's total current, measured inelectromagnetic units, willhave components 1df Ida 1dhU+Cdi>V+Cdt>W+Cdi' Maxwell'shypothesisisthat thework done intakingaunitmagnetic poleround aclosed circuit isequalto4)7rtimes thetotal currentflowing throughthat circuit. Thishypothesis is,aswehave seen, self-consistent, because the total current behaves likeanincompressible fluid, andconse- quentlythe total flowthroughacircuit hasadefinitemeaningwhich is independentoftheparticularsurface weselect, closing upthe circuit, over which tomeasure thecurrent. Thehypothesis may betransformed into mathematicallanguage by followingtheprocedureof§533. Itisfound toberepresented bythe equations .(1df\ dy d/3\ dy 4nr {V+Cdt)"'dz dx 1dh\ d/3 da.(523). VCdtJdxdy / These aretheequationswhich mustreplace equations (473)—(475)inthe mostgeneral motion ofelectricity.Ifwedifferentiate thethreeequations withrespecttox,y,zandadd,weobtain du dvdw_1d/df dgdh\ dw dvd«/_1d/df dgdh\ dx+ dy~fe~~Cdt[dx~+ dy"+ dz')' 512 Displacement Currents[ch.xvii Since, byequation (63), df_dgdh_ dxdydz" thismaybewritten intheform *(!:+!+!")=-!<** NowG(—+o~+o~)dxdydz simply expressestherateatwhich currents ofordinary electricity,measured inelectrostatic units, flowoutofasmall delement ofvolume dxdydz,andsoisnecessarily equalto—-y-(pdxdydz). Weaccordinglyseethatequation (524)istrue, quite independentlyof thetruth ofMaxwell'sdisplacement-theory.Itfollows thatequations (523) form aconsistent scheme, independentlyofthetruth ofthehypothesisfrom whichtheyhavebeen derived. Thedisplacement-theory mayberegarded merelyasscaffolding,andMaxwell'stheory mayberegardedasbeing simply thetheory expressed byequations (523), independentlyofanyphysicalin- terpretationthatmaybeassignedtothevarious terms intheseequations. Although wemay,ifweplease,discard Maxwell'sinterpretation,itwillbe convenient tocontinue tousethename" displacement-current"todesignate thevector whosecomponentsaregiven byformula(522). Weproceedtoexamine theconsequences impliedinMaxwell'sequations (523).Since thetruth oftheequationsmustultimatelyrestonsomething more substantial than thedisplacement-theory bythehelpofwhichthey were derived, itisimportanttoseizeevery opportunityofcomparingthe results ofthetheorywith observation. Maxwell's Equations foranon-conducting Medium. 571. Inanon-conducting medium there canbenoordinarycurrents of electricity,sothatweputu—v=w=0,andMaxwell'sequations assume the form 4Trdf_dy_d/3' Gdtdydz .(525).4>ttdg_da dy Gdt dzdx 4-7Tdh_3/3 da Gdtdxdyt Wenotice thatthewhole oftheleft-hand members ariseentirelyfrom the " displacement-current."Ifthedisplacement-currentwere omitted, weshould have dydz 570-572] Maxwells Equations 513 sothatthemagneticforces(a,/3,7)would bederivable from apotential,and theonlymagneticfield inadielectric inwhich nocurrents flowed would be onearisingfrompermanent magnetism. Maxwell'shypothesis,asexpressedinequations (525), impliesthat there willbeamagneticfield inadielectric whenever theelectric fieldchanges, andenables ustocalculate theforces inthis field. Magnetic Field ofaMoving Charge. 572. Asasimplebutimportant exampleoftheuseofMaxwell's equations (525)letuscalculate themagneticfieldproduced byasingle point chargeemovingwith avelocityU. Letthedirection ofmotion ofthechargeatanyinstant tbetaken for axisofx,thepositionofthecharge beingtaken fororigin. Let(fig. 137) bethe positionofthechargeattime t, and 0'itspositionattime t-dt; then0'0=udt. LetPbethepointatwhich wewish toevaluate themag- netic force. DrawPQparallel andequalto00'. Then the electric field atPattime t willbethesame astheelectric field atQattime t—dt,sothatFig. 137. theincrease intheelectr] 514 Displacement Currents[oh.xvii This solution isalsoseen tosatisfythe firstequationinvirtue ofthe relation(cf.Equation (64)), itistherefore therequiredsolution oftheproblem. Fortheelectric field ofasingle point charge, wehave* A J!eXAeV A 7eZ andonsubstitutingthese valuesfor/, g,h,thesolution becomes aa Uez UeV ,cnmO=0,P=~Q^,7=0^(526)- Theseequations givethecomponentsofmagneticforce atany point. The lines ofmagneticforce arecircles about thepathoftheelectron, and theintensityatdistance rfrom theelectron is %*¥<527>> where 6istheanglebetween thedistance randthedirection ofmotion. 572a.Ifasmall element dsofacircuit inwhich acurrent i(measured inelectromagnetic units)isflowingcontains Nds electronsmovingwithan averageforwardvelocity u0)wehave(cf.equation (b)of§345) NeuQ=Ci. Themagneticforce atdistance rproduced bythemotion oftheelectrons intheelement dsofthecircuit is(cf.expression 527)) eusin6 .,sin8Nds -~— or ids-G 7 ?- This isexactlyidentical with theforcegiven byAmpere's Law(§497). ButAmpere'sformula wasonlyprovedtobetruewhenintegratedround aclosed circuit, whereas itisnowseen thatMaxwell'stheory impliesthat theformula istrue foreveryelement ofacircuit. Experimental Confirmation. 573. Thepossibilitythat amovingelectriccharge might producea magneticfieldoccurred toFaraday andwasnoted byhiminhisExperimental Researches (1837); theeffect wasobserved byRowland in1876andagain by Rontgenin1885. Maxwell'sequations,aswehavejust seen, predictthe actual amount ofthis effect. Theonlyquantityother than themeasurable electricchargewhichappearsinMaxwell's formulae isG,theratio ofthe electric units, and thiscanbedetermined inother ways (cf.§582below),its value being found tobealmostexactly3x1010 . *This isnotquite accurate, forthemotion ofthemagneticfield(a, /3,7)induces anelectric fieldwhich ought tobetaken intoaccount inevaluating (/,g,h).Equations (526) are,however, ver}' nearly accurate exceptforvery rapidly moving charges. Theexact solution willbegiven later(cf.§§624, 647, 656). 572-574] Experimental Confirmation 515 The firstattempttomeasure theeffectquantitativelywasmadebyRowland andHutchinson in1889. Theyused discschargedtoapotentialof5000 volts, which weremade torotate at125revolutions asecond. Themotion ofthe chargeddiscsmayberegardedasthemotion ofasuccession ofelectriccharges, andthemagneticforcepredicted byMaxwell'stheorycanbecalculated from formula(527). Oncomparingtheobserved effect with thatpredicted by theory,values forGwerefound which varied from 2-26x1010to3*74x1010 ,the meanbeing3"19x1010 .More exactexperimentsofasimilartypeperformed byH.Pender in1901gaveforGanaveragevalue of3*05x1010 ;asecond set,withslightlymodifiedapparatus, gaveG=2-96x1010 .These values will beseen toagree very closelywith theknown value forC,3*00x1010 ,sothat theexperimentsnotonlyprovetheexistence ofthemagneticfieldproduced bymoving charges,butalsoconfirm Maxwell'stheory quantitatively. Itmaybeobjected thattheforegoing experiments onlytestthemagnetic fieldproduced byacontinuous chain ofelectriccharges movinginaclosed circuit, butthisobjection cannot beurged against experiments performed by E.P.Adams in1901. Intheseexperiments chargedbrassspheres weremade topassasuspended magneticneedle attherateofabout 800persecond and theapparatus wasarrangedsothat the effect ofonespherehadalmost disappearedbefore theneedle came under theinfluence ofthenext. From a series ofsuchexperiments Adams determined values forCrangingfrom 2-6x1010to3-1x1010 ,themeanbeing28x1010 . Further confirmation oftheexistence ofthedisplacement-currentispro- vided inagreatnumber ofindirectways, particularly throughtheelectro- magnetic theoryoflightandtheelectromagneticmass oftheelectron. For thepresent weshallassume thetruth ofMaxwell'shypothesis andproceedto examine itsconsequences. TheGeneral Equations oftheElectromagnetic Field. 574. In§529,weobtained thesystemofequations da=dZ_d7 dtdydz inwhich allthequantities wereexpressedinelectromagneticunits. Ifthe electric forces areexpressedinelectrostatic units, X,Y,Zmust bereplacedin theseequations byCX,CY,CZ,andthesystemofequations becomes 1da_dZ_dYGdtdy dz _^db_^_X_dZGdt dzdx }Ldc_dY_dXGdtdxdy, 33—2.(528). 516 DisplacementCurrents[ch.xvii These threeequations togetherwithequations (523), namely ,/ 1df\ dry d/3^ ldg\ da_dyu~nj+ )s~s™r \ />V'Gdtj dzdx (L^-?£_?? \Cdtj dxdy constitute asystemofsixequations givingtherateofchangesintheelectric andmagneticfields interms ofthefield atanyinstant. Withthemmaybe associated thetwoequations (63)and(362), namely jjf+ge+j*(530),dxdydzr p+^+d^=(531).dxdydz Theeight equations (528)—(531) form themostgeneral systemof equationsoftheelectromagneticfield. Inthese equations u,v,w,a,b,c, a,/3,7areexpressedinelectromagnetic units, whilef,g,h,X,Y,Zare expressedinelectrostatic units. Localisation andFlowofEnergy. 575.Wehavealreadyconsidered thehypothesisthatelectromagnetic energy maynotbeconfined totheregions occupied byelectriccharges, magnetsand currents, butmaybespread throughthewhole ofspace. On thishypothesisthekinetic(magnetic) energy Tandthepotential (electric) energyWofanisotropic medium aregiven by T=-^jfjfi(a2+j32+72 )dxdydz, W=^-jjJK(X2+F2+Z2 )dxdydz, andtheenergyissupposedtobelocalised inspaceinthewayindicated by theseintegrals. Knowingthekinetic andpotential energiesofthesystem, itoughttobepossibletodetermine itsequationsofmotion bythegeneral dynamicalmethodsexplainedinChapter XVI. Thequantities a,/3,<ywhich enter inthekinetic energymust befunda- mentallyofthenature ofvelocities. Letusdenote themby£,rj,£,sothat £,7],%maybetreated aspositionalcoordinates. Similarly u,v,wwhichexpresstherates offlow ofelectricityatany pointareofthenature ofvelocities. Ifqx,qy,qzdenote thetotalquantity ofelectricity,measured inelectrostatic units, which have crossed unit areas perpendiculartoOx,Oy,Ozatanypointsince aspecified instant, then Cu=qx,Cv=qy,Cw=qz. 574,575]Localisation andFlow ofEnergy Maxwell'sequations (529)nowassume theform 4tt/. df\ a£ drj. -G\^+dt)= d-y-dzetQ ' givingonintegrating,andreplacing 4<irfbyKX,517 .(532).dydz This relation connects thevariouspositionalcoordinatesqx,X(regarded asa" displacement"), £etc. Theprincipleofleast action canbeexpressed,asinequation (507),in theform (\ST-{BW})dt=0, Jo where thevalue of{8W]inthepresent problemis {8W}=8W+! i\(X8qx+Y8q y+Z8q z)dxdydz, whichagain, onsubstitutingforW,canbeputintheform {SW }=^jjj[X(K8X+4tt%)+Y(K8Y+ 4>7r8q y) +Z(K8Z+4nr8q z)]dxdydz i.- , ,.-,—^~ .dy dzJ \dz dx} \dxdyJ onusingrelations (532). Onfurthertransforming byGreen's Theorem, this becomes [8W]=^ff[X(m$S-n8v)+...]dSSL[[[[ YfiJX-dk) V(d3-dlt\, 7(d8rl_d8^ 4ttJJJ l\dy dzJ+\dz dx)+[dx dyjdxdydz -£///(S*-£*Kdxdydz. Similarlyonvarying T,wefind 8T=1-jfjlpaSa+/*{38/3+fiyBj] dxdydz =j-111[aS|+6S?7+c§£]dxdydz, giving TOT*=^-IffhaSZ+b8n+c8£)dxdydz1* —j— jdt JJj(d8^+b8r]+68%)dxdydz. Asin§545,wesupposethevalues of8%, 8r],S£all tovanish attheinstants t=and t=r,sothatthetoplineontherighthand vanishes. 518 DisplacementCurrents |_CH -xvn Collecting terms, wenowobtain r.(»-i.in)-i-£r-//i}«+s-©«? -^-[T dtt\\{{nY-mZ)^+...}d8. Ifoursuppositionsastothelocalisation ofthekinetic andpotential energiesarecorrect, then£,77,£mayberegardedasindependentcoordinates atevery pointofthe field. Thus thevariations&%, Stj,8£mayhave all possiblevalues atallpointsofthe field. Itfollows that their coefficients must vanishseparately;hence atevery pointofthe field,wemust have adZBY _ These aretheequationswhich theprincipleofleast actiongivesasthe equationsofmotion whenweassume Maxwell'sequations (529).Weseeat once thattheyareidentical withequations (528),sothat thetwo setsof equations (528) and(529)arerelatedthroughtheprincipleofleast action. Poynting'sTheorem. 576. Ifwestillassume theenergytobelocalised inthemedium inthe wayimagined byMaxwell, the totalenergyinanyclosedregionwillbe given by T+W= jjjiS(X2+F2+Z*)+£(a2+/32+72 )|dxdydz, whence, ondifferentiating,andreplacing fia.bya,KXby4nrf, etc., d(T+W) [[[[(^df „dg „dh\ 1fdaadb dc\) , . . Onsubstitutingfromequations (528) and(529),thisbecomes d(T+W) C[ffi^fdy d/B\ fdZ dY\ vfdy d/3\ fdZ dY\ ,,, dt4t7rJJJ\ \dy dzj \dydzJ*j -Gjjj(uX+vY+wZ) dxdydz. Inthisequation,the last linerepresents exactlytherate atwhich work isperformedorenergy dissipated bytheflow ofcurrents, sothatthe first linemustrepresenttherateatwhichenergyflows intotheregionfrom outside. 575-577] Poynting'sTheorem 519 ByGreen's Theorem(§179),the first line = ~§ir[NZ/3~7^+m(YV~Za)+n(Ya-X/3)}d# £,ra,nbeingthedirection-cosines ofthenormal inwards intotheregion. Thus ifweput TI^^iYy-ZP),etc (533), j itappearsthat thevalue of-j(T+W)isthesame asifthere were aflow ofenergyinthedirection I,m,nofamount ITIX+mUy+nU2.Thevector II ofwhich17a;,Uy,II2arecomponentsisofamount n=v(iv+iy+n/)=-£-rhsme, where R,Haretheelectric andmagneticintensities and6istheangle between them. The direction ofthevector IIisatright anglestobothR andH,andtheflow ofenergyinto oroutofthesurfaces isthesame asif there were aflowequaltoITinmagnitudeanddirection atevery pointof space.This vector IIiscalled the"Poyntingfluxofenergy." TheintegralofthePoyntingFlux over aclosed surfacegivesthetotal flowofenergyintooroutofasurface, but ithasnotbeenproved,andweare notentitled toassume, thatthere isanactual flowofenergyatevery point equaltothePoyntingFlux. Forinstance ifanelectrifiedsphereisplaced near toabarmagnet,this latter assumptionwouldrequireaperpetualflow ofenergyatevery pointinthefield exceptthespecial pointsatwhich the electric andmagneticlines offorce aretangentialtooneanother. Itisdifficult tobelieve that thispredictedcirculation ofenergycanhaveanyphysical reality. Ontheother hand itistobenoticed thatsuchacirculation ofenergy isalmost meaningless. The circulation ofafluid isadefinite oonception because itispossibletoidentifythedifferentparticlesofafluid;wecansay forinstance whether ornottheparticles enteringasmall element ofvolume areidentical ornotwithanequalnumber ofparticles coming out,butthe same isnottrueofenergy. Equations foraUniform Isotropic Dielectric. 577.Wereturn now tothegeneral equationsof§574,andproceedto examine theformtheyassume inauniformisotropicdielectric. Since there canbenoelectric current weputu=v=w=0.Wealsoput 47r/=KX etc.,a=fxaetc., 520 Displacement Cm-rents[CH.XVII andtheequationsassume theform Gdt dydz KdY=da_dy Gdt dzdxfjida. GdtdZ dydY\ dz •(A),lid/3_dX_d_Z> dzdx' Cdt.(B). KdZ=dJ3___da _^*y=^lr_?^Gdtdx dy) Gdt dxdyI From the firstequationofsystem (A),wehave Kjx<PX9_/>*y\d_(nd/3\ G2dt2~dy\Cdi) dz\Cdt)' andonsubstitutingthevalues of~ -j-and^-y-from thelasttwoequations ofsystem (B),thisequationbecomes K^d2X_ d_(d_Y_dX\ d_fd_X_dZ\G2dt2dy\dx dy)+dz\dz dx) d2X.d*X_d_(dYdZ\+dx\dy dz, dy2'dz2 Since themedium issupposedtobeuncharged,wehave axdY.dZ dxdycz d2X sothatthelasttermmaybereplaced by+-^-j,andtheequation becomes Kfid2X=V2XC2dt2 Byexactlysimilaranalysis wecanobtain thedifferentialequationsatis- fiedbyY,Z,a,/3and7,and ineach case this differentialequationisfound tobeidentical with that satisfied byX.Thus thethreecomponentsof electric forceandthethreecomponentsofmagneticforce allsatisfy exactly thesame differentialequation, namely ~J±=a2V2V dt2 (534), where astands forC/^K/u,. Thisequation,forreasons which willbeseen from itssolution, isknown asthe"equationofwave-propagation." d2vSolutions of-^=a2^2x Solution forsphericalwaves. 578. Thegeneralsolution oftheequationofwave-propagationisbest approached byconsideringthespecialformassumed when thesolution % isspherically symmetrical.Iftyisafunction ofronly,where risthe distance fromany point, wehave d-y „_, a2d(„dy^ —^=a2V2v=\r2— dt*Xr2drVdr 577-579] Equation ofWave-propagation 521 whichmaybetransformed into d?(rx)_d?(rX) «o-vS^'T andthesolution is r%=/(r-a0 +^(r+a(536)> where/and<I>arearbitraryfunctions. Theform ofsolution shews that thevalue ofXa^anyinstant overa sphereofanyradius rdepends uponitsvalues atatime tpreviousover twospheresofradii r—atandr+at.Inother words, theinfluence ofany value ofXispropagatedbackwards and forwards withvelocitya.For instance, ifattime t=thevalue ofxiszeroexceptoverthesurface ofa sphereofradius r,then attime tthevalue ofXiszeroeverywhere except overthesurfaces ofthetwospheresofradii r±at;wehave therefore two spherical waves, converginganddivergingwith thesamevelocitya. General solution (Liouville). 579. Thegeneralsolution oftheequationcanbeobtained inthe following manner, originallyduetoLiouville. Expressedinspherical polars, r,6and<£,theequationtobesolved is 1<*2*19AAA ,1 d(ziuedA|ia2*-Q a2dP~ r2dr\or) r2sin6dd\ ddj r2 d<t>2 Letusmultiply bysin6d6dcf) andintegratethisequationoverthesur- faceofasphereofradius rsurroundingtheorigin.Ifweput X=f[xsineddd(p(537), theequationbecomes a2dt2~ r2dr\ dr)' theremainingtermsvanishingonintegration.Thesolution ofthisequation (cf.equation (536))is X=-{f(at-r) +^(at+r)} (538). Forsmall values ofrthisassumes theform X=1{/(at)+4>(at)}-r[f(at)-4>'(at)}+^{/"(at)+3>"(at)}+.. .] (539). Inorder thatXmaybefinite attheorigin throughalltime,wemust have f(at)+3>(at)= atevery instant, sothat thefunction 4>must beidentical with-/.On puttingr=0,equation (539) becomes (\)r=0=-2/'(at)3 522 Displacement Currents[ch.xvii andfromequation (537), puttingr=0,wehave (V)r=o=47r(x),. =o, sothat47r(x) r=o=-2/,(aO (540). Equation (538)maynowbewritten as rX=/(at-r)—/(at+r). Ondifferentiatingthisequationwithrespecttorand trespectively, ^(r\)=-f(at-r)-/(at +r), ~(r\)- f(at-r)-f(at +r),adt andonaddition wehave -y(- +r)-|(A)+l|(rt.). Thisequationistrue forallvalues ofrand i:puttingt=0,wehave -2/'(r)=l(rX, =0)+^=0 asanequation which istrue forallvalues ofr.Givingtorthespecial value r=at,theequation becomes -2/(at)=jt(t\t=0)+tit= . The lefthand is.byequation (520), equalto47r(%)r=o- Ifweuse%,%to denote themean values of%an(^Xaveraged overasphereofradius atat anyinstant, theequation becomes (X)r=o= fo(txt=o)+i%t=o (541). Thus thevalue ofxa^anypoint (which weselect tobetheorigin)at anyinstant tdepends onlyonthevalues of%and%attime t=overa sphereofradius atsurroundingthispoint. The solution isofthesame nature asthatobtained in§578,but isnolongerlimited tosphericalwaves. General solution(Kirchhoff). 580.Astillmoregeneral form ofsolution hasbeengiven byKirchhoff. Let <£>and "SPbeanytwoindependentsolutions oftheoriginal equation,so that d2(& d2^ -a¥=a2V2®>V=a2V^(542)- ByGreen's Theorem(equation (101)) -XJJ(&^-V^dS=ff[(®V*V-VV*®)dxdydz r]]\ dn dnj dV .T.cZ3» a2Jflre-*f)-**579,580] Equation ofWave-propagation 523 byequations (542). Thevolumeintegrationsextendthroughtheinterior ofanyspace boundedbytheclosed surfaces S1}S2,...,andthenormals to SltS2,...aredrawn, asusual, intothespace.Ifweintegratetheequation justobtainedthroughouttheinterval oftime from t=—t'tot=+t",we obtain (543).-f SofarM*hasdenoted anysolution ofthedifferentialequation.Letus nowtake ittobe-F(r+at),thisbeingasolution(cf.equation (536)) what- ever function isdenotedbyF,and letF(x)beafunction ofxsuch that it and allitsdifferential coefficients vanish forallvalues ofxexceptx=0,while ('°F(x)dx=l. J—oo QSuchafunction, forinstance, isF(x)=Lt—7-= zr . c=ojt(^2+c2 ) Wecanchoose t'sothat, forallvalues ofrconsidered, thevalue of 1—at'isnegative. Thevalue ofr+at'' ispositiveift"ispositive.Thus F(r+at)and allitsdifferential coefficients vanish attheinstants t=t"and t=— t',sothat theright-handmember ofequation (543) vanishes, andthe equationbecomes rt" re,9^ 9<^-sJ-,*JJrar-*air;'w-(544)- Letusnowsupposethesurfaces overwhich thisintegralistaken tobe twoinnumber. First, asphereofinfinitesimal radius r,surroundingthe origin,which willbedenoted byS1}andsecond, asurface, asyetunspecified, which willbedenoted byS.Letusfirstcalculate thevalue ofthecontribu- tion toequation (544) from the first surface. Wehave, onthis first surface, V=LF(r +at), ^=-^=--,F(r +atH-F(r +at), sothatwhen rismade tovanish inthelimit,wehave ft•H-*T$«~*r*~.*<«ft andtherefore /w/(*£-*hdds>=-wr,*'-F(at)di 47T , a ^_ since theintegrandvanishesexcept when t=0. 524 Displacement Currents Thusequation (544) becomes[CH.XVII *'-?—;£:( *= 47rJ_# aeft dn c?i /•<" fcj>dr +*k&)F{r+at)-lF(-r+at)d^;}dt (545)- Integrating byparts,wehave, asthevalue ofthe firstterm under the timeintegral, /-%-F'(r+at)dt -t>ron' ardnF(r+ at)t=t" •<"1drd®t ft -v J-fciron at The firstterm vanishes atboth limits, andequation (545)nowbecomes Wecannowintegratewithrespecttothetime, forF(r+at)existsonly attheinstant t=—r/a.Thus theequationbecomes ia<£> dn «=o 47rJJ ar3w cfa dn\r) r tm-ZadS, givingthevalue of <E>atthetime t=interms ofthevalues of<&and 4> taken atpreviousinstants overanysurfacesurroundingthepoint. The solution reduces tothat ofLiouville ontakingthesurface Stobeasphere, sothatr-=——.on or Aswith theformer solutions, theresult obtainedclearlyindicatespropa- gationinalldirections withuniformvelocitya. Propagation ofElectromagnetic Waves. 581. Itisnow clear thatthesystemofequations C2dt" etc.,obtained in§577indicate that, inahomogeneous isotropic dielectric, all electromagneticeffectsoughttobepropagatedwith theuniformvelocity CThismaybecomparedwith theresult obtained in§562. Itwas s/Kf* there shewn that electricsignals propagated alongawirewould advance with C avelocity -7=where K, fxwere theinductivecapacity andmagneticVif/* 580-582] Electromagnetic Waves 525 permeabilityofthemediumsurroundingthewire. Itnowappearsthatthe velocityofsignals alongawire isidentical with thevelocityofwaves inthe medium outside thewire. Maxwell'sdisplacement theory givesasimple explanationofthis. Acurrentflowinginawire isaccompanied byadisplacement current in theether. This setsupamagneticfieldwhich ispropagatedwithvelocity G/wK/jlinthedielectric andthisinturninduces afurther current inthewire. Onthisview theactualprocessofpropagationtakesplaceinthemedium, thewire directs thepathoftheelectromagneticdisturbance andabsorbs some oftheenergy. Itistobenoticed that thevelocityofpropagation alongwires was obtained in§562before wehadintroduced theconceptionof" displacement- currents" atall.That the result isnot inconsistent with thevelocity obtained onthehypothesisofdisplacement-currentswillbeunderstood from theresult of§575. Numerical Values. 582.Wenotice that infree air,inwhichK=fi— 1,thevelocityofpro- pagationofelectric waves, whetheralongwires orinthe air,oughttobethe same asG,theratio oftheelectric units. This enables ustoapplyasevere testtothetruth ofthetheorywhich hassofarbeendeveloped,forboth the value ofGandthevelocityofpropagationofelectric waves admit ofdirect experimentaldetermination. Thebestdeterminations ofG,theratio ofthetwounits, arethefollowing: Rosa andDorsey (1907) 2*9971 x1010 Perot andFabry (1898) 2-9973 x1010 Hurmuzeseu(1896) 3-0010xl010 Abraham (1890) 2-9913xl010 The true value isprobably veryclose tothevalue obtainedbyRosa and Dorsey, namelyG=29971 x1010 . Recent determinations ofthevelocityofpropagationofelectromagnetic waves inairareasfollows : Maclean (1899) 2-991x1010 Saunders (1897) 2-997 x101C Trowbridge andDuane(1895)... 3-003 x1010 Themean ofthese values is2997 x1010 . Inthedeterminations ofSaunders and ofTrowbridgeandDuane the waves wereguided bycopper wires, while theexperimentsofMaclean dealt withwavespropagated throughairwithout wires. Theequalityofvelocities isofcourse aconsequence, and alsoaconfirmation, oftheresults obtained in§562. 526 Displacement Currents[ch.xvn Theratio oftheunits, G,isalsoequal,oratleastverynearly equal,tothe velocityoflightinair,and thisconfirmed Maxwell inhissuggestionthat light propagationisaspecialcase ofthepropagationofelectromagnetic waves. Out ofthissuggestion, amplyborne outbytheresults offurther experiments,hasgrowntheelectromagnetic theoryoflightofwhich ashort account isgiveninthenextchapter. Thebestdetermination ofthevelocity oflightinairatpresentavailable isthat ofMichelson whofinds(October 1924) forthevalue ofthisvelocity 299735 x1010cms.asecond, with aprobableerror ofonein22,000. Exceptforsmall differences, which arewellwithin theerrors ofthevarious experiments,thequantities previouslymentioned areseen toagreewith this invalue. Thuswemay say,that theratio ofunitsCisidentical with thevelocity ofpropagationofelectromagnetic waves, andthisagainisidentical with the velocityoflight. Equations foraUniform Isotropic Conductor. 583. Inanisotropicconductor thecurrent (u,v,w)isproportionalat every pointtotheelectric force (X,Y,Z).Wearesupposing u,v,wtobe measured inelectromagneticunits. Thevalues ofthecomponentsofelectric force, measured inelectromagnetic units, areGX,GY,GZ,thesebeingof course theforcesactingonanelectromagneticunit ofelectricalcharge. Thus byOhm's Law,GX.u= ,etc.T where risthespecificresistance measured inelectromagneticunits. Ifwe furtherput4nrf=KX, etc.,equations (529)become /4tt(7Kd\vBy 3/3 /K._- \rGdtj dydz andtwosimilarequations. Onreplacing equations (529) bythese, theequationsof§574become the general equationsofanisotropic conducting medium. Ifwedifferentiate thethreeequationsofthesystem (546) withrespect tox,y,zandadd,weobtain 4t7CKd\(dX dYdZ tGdtj\dx dydz Fromequation (530)wehave BXdYdZ_4,7rp dxdydzK')-a 582-583a] Isotropic Conductor 527 sothatourequation becomes dp__4tt02 dt~~K^9' Ifpisthevalue ofpattime=0,thesolution ofthisequationis _4ttC2 p=PoeK*\ shewingthatpfallsaway exponentially,nomatter what electric ormagnetic fieldsmaybeacting.Thisequationisidentical with thatalreadyobtained in§396,thefactorG2simply correspondingtoachangeofunits. Thus inside aconducting mediumanyinitialchargewillrapidly disappear, andwemay supposethat BXBYdZ=Q.=Q decdydz'" 583 a.Multiplyboth sides ofequation (546) by//,anddifferentiate with respecttothetime.Wefind Gdf-+tdt~dy\di) dz\di Theright-hand member ofthisequation maybyequations (528) be replaced by dy\d% dyJ dz\dzdxj or G dec\dcc dy dz]\ andthis isequaltoGV2X,invirtue oftherelation oxdydz Thus theequation becomes, ondividing through byG, Kp,d*X4,7rp,dX G2dt2rdt=V2X. ThisequationinvolvesXonly, and soisthedifferentialequationsatisfied byXwhenelectromagneticwaves arepropagatedinaconductor.Naturally Y,Zsatisfysimilarequations, andequations (528) shew thata,b,cora,/3,7 again satisfysimilarequations. Thus X,Y,Z,a,/3,7allsatisfythesame differentialequation, namely djXInr&dX dt2+Kr dt where astands forG/^/Kp,. Thecompletesolution ofthisequationhasbeen given byRiemann*. *Dievartielle Differ entialgleichung enderMath.Physik, 4thedu. (1901), n.p.399. 528Displacement Currents[en.xvn Wemaynotice that inadielectric, t=oo,sothatthesecond term dis- appears. Theequation then reduces, asitought,toequation (534) already obtained in§577. Inmany problems,thesecond term ismoreimportant than the first.When the firstterm isomitted, theequation reduces tothe well-knownequationofconduction ofheat, alreadyobtained in§535 (equation (480)). Toformanestimate oftherelativeimportanceofthetwoterms onthe left, letusexamine thecase ofanalternatingcurrent inwhich thetime- d factor iselpt.Wemayasusualreplace-r-byip,andtheequation becomesdt -p*+4?r(72 KttpW=a2V2 %. Theneglectofthe first term, which isofcourse thesamethingas neglectingthedisplacement-current,isclearly permissibleif4>ttG2/Ktpis numerically large. When this ratio isnotlarge,theerrorproduced bythe neglectofthe firstterm willbegreatestinproblemsinwhich tislarge (conductorsofhigh resistance) andinwhichpislarge (rapidly changing fields). Onsubstitutingnumerical values itwillbefound that inproblems ofconductionthrough metals, theneglectofthefactorRrpf^-rrC2produces aquite inappreciableerror unless piscomparablewith 1015—i.e.unless we aredealingwithoscillatingfields ofwhich thefrequencyiscomparablewith that oflight-waves.Thus theeffect ofthedisplacement-currentinmetals hasbeeninappreciableintheproblemssofardiscussed, sothattheneglect ofthis effectmayberegardedasjustifiable. Thematter standsdifferently asregardstheproblemstobediscussed inthenextchapter,inwhich the oscillations ofthefield areidentical with those oflight-waves. Units. 584.Wemayatthisstagesumupallthat hasbeen saidabout the differentsystemsofelectrical units. There arethree differentsystemsofunits tobeconsidered, ofwhich two aretheoreticalsystems,theelectrostatic andtheelectromagnetic, while the third isthepractical system. We shallbegin bydiscussingthetwo theoretical systemsand their relation tooneanother. 585. IntheElectrostatic Systemthefundamental unit istheunit of electriccharge,thisbeingdefined asachargesuch thattwosuchchargesat unitdistanceapartinairexert unit forceupononeanother. There will, of course, bedifferentsystemsofelectrostatic unitscorrespondingtodifferent units oflength,massandtime,buttheonlysystemwhich needbeconsidered 5830-585]Units 529 isthat inwhich these units aretaken tobethecentimetre, grammeand secondrespectively. IntheElectromagnetic Systemthefundamental unit istheunitmag- neticpole,thisbeingdefined tobesuch thattwosuchpolesatunit distance apartinairexert unit force upononeanother.Againtheonlysystem which need beconsidered isthat inwhich theunits oflength,mass and time arethecentimetre, grammeandsecond. From theunit ofelectricchargecanbederived other units— e.g.of electric force, ofelectricpotential,ofelectric current, etc.—inwhich to measurequantitieswhich occur inelectric phenomena.These units will ofcourse alsobeelectrostatic units, beingderived from thefundamental electrostatic unit. Soalsofrom theunitmagnetic polecanbederived other units— e.g.of magnetic force, ofmagnetic potential,ofstrengthofamagnetic shell, etc.— inwhich tomeasurequantitieswhich occur inmagnetic phenomena.These units willbelongtotheelectromagnetic system. Ifelectric phenomenawereentirelydissociated frommagnetic phenomena, thetwoentirelydifferent setsofunitswould benecessary,andthere could be noconnection between them. Butthediscoveryoftheconnection between electric currents andmagneticforces enables usatonce toform aconnection between thetwosetsofunits. Itenables ustomeasure electricquantities— e.g.thestrengthofacurrent—inelectromagnetic units, andconversely we canmeasuremagnetic quantitiesinelectrostatic units. We find, forinstance, thatamagneticshell ofunitstrength (inelectro- magnetic measure) producesthesame field asacurrent ofcertainstrength. Weaccordinglytake thestrengthofthiscurrent tobeunityinelectro- magnetic measure, andsoobtain anelectromagneticunit ofelectric current. We find, asamatter ofexperiment,that thisunit isnotthesame asthe electrostatic unit ofcurrent, and therefore denote itsmeasure inelectro- static units ofcurrent byG.This isthesame astakingtheelectromagnetic unit ofchargetobeGtimes theelectrostatic unit, forcurrent ismeasured in eithersystemofunits asachargeofelectricity perunit time. Inthesamewaywecanproceedtoconnect theother units inthetwo systems.Forinstance, theelectromagneticunit ofelectricintensitywillbe theintensityinafield inwhich anelectromagneticunitofcharge experiences aforce ofonedyne. Anelectrostatic unit ofchargeinthesame fieldwould ofcourseexperienceaforce ofI/Odynes,sothattheelectrostatic measure of theintensityinthis fieldwould be1/(7. Thus theelectromagneticunit of intensityis1/(7times theelectrostatic. Thefollowingtable oftheratios of theunits canbeconstructed inthisway: j. 34 530 Displacement Currents RatiosofUnits.[CH.XVII ChargeofElectricity. Electromotive Force. ElectricIntensity. Potential. Electric Polarisation. Capacity. Current. Resistance ofaconductor. Strengthofmagnetic pole. Magnetic Intensity. „ Induction. Inductive Capacity. Magnetic Permeability.Oneelectromag.unit=Celectrostat. units. >» j> I » " >> »=i/o ,, „ ;> >) / >' »—fl =C2 —c J) )) *-/^' » »» J> »==-'•/o „ }, _n :> »^j; )> » >>^I/O,, ,, =C2 =I/O2 586. Thevalue ofC,aswehave said, isequaltoabout 3x1010inc.G.s. units. Ifunits other than thecentimetre, grammeandsecond aretaken, the value ofCwillbedifferent. Sincewehave seenthatGrepresentsavelocity, itiseasytoobtain itsvalue inanysystemofunits. Forinstance avelocity 3x1010inc.G.s. units=671x108milesperhour, sothat if miles andhours aretaken asunits thevalue ofCwillbe6'71x108 . Practical Units. 587. Thepractical systemofunits isderived from theelectromagnetic system,eachpracticalunitdiffering onlyfrom thecorrespondingelectro- magneticunitbyacertain poweroften,thepower beingselected soas tomake theunit ofconvenient size. The actual measures ofthepractical units areasfollows : Forlegalandcommercialpurposes,theunits aredefined interms ofmaterial standards. Thus accordingtotheresolutions oftheInternational Conference of1908 thelegal (Inter- national) ohm isdefined tobetheresistance offered toasteadycurrent byauniformQuantity ChargeofElectricity Electromotive Force"! 585-588] Units 531 column ofmercuryoflength 106-300cms., thetemperature being 0°C, andthemass being14*4521 grammes,this resistancebeing equal, asnearlyascanbedetermined by experiment,to109electromagneticunits. Similarlythelegal (International) ampereis defined tobethecurrent which, whenpassed through asolution ofsilver nitrate inwater, depositssilver attherateof•00111800 grammes persecond. Physical Dimensions ofUnits. 588. Asexplainedin§18,alltheelectric andmagneticunits willhave apparent dimensions inmass, lengthand time. These areshewn inthe followingtable: CHAPTER XVIII THEELECTROMAGNETIC THEORY OFLIGHT Velocity ofLight inDifferent Media. 589. Ithasbeen seen that, ontheelectromagnetic theoryoflight,the propagationofwaves oflightinvacuooughttotakeplacewith avelocity equal,within limits ofexperimental error, totheactual observedvelocity oflight.Afurther testcanbeappliedtothetheory byexamining whether theobserved andcalculated velocities areinagreementinother media. Accordingtotheelectromagnetic theory,ifVisthevelocityinany medium, andVthevelocityinvacuo, weoughttohave therelation VJ_/1 n~VAV VI>' whereK,/iarefer tofreespace. Forfreespaceand allmedia which willbeconsidered, wemaytake/*=1. Also ifvistherefractive index foraplane wave. oflight passingfrom free spacetoanymedium, wehavefromoptical theorytherelation V sothat, accordingtotheelectromagnetic theory,therefractive index ofany mediumoughttobeconnected with itsinductivecapacity bytherelation "VZ: Onedifficulty appearsatonce.Accordingtothisequation thereoughtto beasingledefinite refractive index foreachmedium, whereas thephenomenon ofdispersionshews that therefractive index ofanymedium varies with the wave-lengthofthelight.Itiseasytotrace thisdifficultytoitssource. The phenomenonofdispersionissupposedtoarise from theperiodic motion of chargedelectrons associated with themolecules ofthemedium(cf. §610, below), whereas thetheoretical value which hasbeenobtained forthevelocity oflighthasbeen deduced onthesuppositionthat there arenomoving chargesatanypointofthedielectric(cf.§577).Acorrection tothevalue justobtained forvwillbeneeded torepresentthe effect ofthemotion of chargedelectrons inthemedium. When thismotion isinfinitely slow, the correctiondisappears,sothatourequation oughttogivethetruevalue ofv inthelimitingcaseoflight,orotherelectromagnetic waves, ofinfinite wave- length.Itisimpossibletodealexperimentallywithwaves ofinfinite wave- 589,590] Velocity ofLightinDifferent Media 533 length, butthefollowingtables* shew that asthewave-length increases, the refractive index vapproximatesto*JKjK Q. Water. Ethyl Alcohol. v£"V80=8-94.\fK~51. Wave-length (cms.) 534 TheElectromagnetic Theory ofLight [oh.xviii Waves ofLight innon-conducting Media. SolutionofDifferential Equation forPlane Waves. 591. Theequationofwave-propagation ^n,=a2VV dt2 * has,asaparticular solution, y=Agi* (lx+my+nz-at)(547) providedI2-fm2+v?—1.Thisvalue of%isacomplex quantityofwhich the realandimaginary parts separately mustbesolutions oftheoriginal equation. Thuswehave thetwosolutions %=Acoskilx+my+nz—at) (548), %=Asink(Ix -fmy+nz—at). Either ofthese solutionsrepresentsthepropagationofaplane wave. The direction-cosines ofthedirection ofpropagationareI,m,n,andthe velocityofpropagationisa.Usuallyitwillbefoundsimplesttotakethe value ofxgiven byequation (547)asthesolution oftheequation andreject imaginaryterms after theanal}Tsis iscompleted.Thisprocedurewillbe followed throughoutthepresent chapter;itwill ofcoursegivethesame result aswould beobtained bytaking equation (548)asthesolution ofthe differentialequation. Propagation ofaPlane Wave. 592. Letusnow consider indetail thepropagationofaplane wave of light,thedirection ofpropagation being taken, forsimplicity,tobetheaxis ofx.Thevalues ofX,Y,Z,a,@,7must allbesolutions ofthedifferential equation,eachbeingoftheform X=AeiK(x~at)(549). The sixvalues ofX,Y,Z,a,ft,<yarenotindependent, being connected by thesixequationsof§577,namely KdX Gdt KdY Cdt Kd2[ Cdtd<y dy da ds' dft dxdz dx da dy•(A),fida Cdi'' fidft Cdt" fidy G~didZ dy dX97 ' dz d_Zydz dx d_Y dxd_X•(B). 591-592a] Crystalline Media 535 From theform ofsolution(equation (549)),itisclear that allthe differ- entialoperators maybereplaced bymultipliers. Wemayput d . 3_. 3 3Adt~~tKa'd~x~%K> fy~"3*" Theequations nowbecome X= \ a=0 Kav__ \ V±q-C7 l (A'), Cp- "\ (B'). ^Z=fi/May=-7G Since Kfia?=C2 ,itisclear thatthesecond andthirdequationsin(A')are identical with thethirdandsecondequations respectivelyin(B'). SinceX=0,a=0,itappearsthatboth theelectric andmagneticforces are,atevery instant, atright anglestotheaxis ofx,i.e.tothedirection of propagation. From thelasttwoequationsofsystem (A')weobtain /3Y+yZ=0, shewingthattheelectric forceandthemagneticforce arealsoatright angles tooneanother. Oncomparingtheresults obtained from theelectromagnetic theoryof light,with those obtained fromphysical optics,itisfound thatthewave of lightwhich wehavebeenexaminingisaplane-polarised raywhoseplaneof polarisationistheplane containingthemagneticforceandthedirection of propagation. Thus themagneticforce isintheplaneofpolarisation,while theelectric force isatright anglestothisplane. Crystalline Dielectric Media. 592 a.Letusconsider thepropagationoflight,ontheelectromagnetic theory,inacrystalline medium inwhich theratio ofthepolarisationtothe electric force isdifferent indifferent directions. Byequation (92),theelectricenergyWperunitvolume insuchamedium isgiven by W=i(/fnZ2+2K12XY+ ...). Ifwetransform axes, takingasnewaxes ofreference theprincipalaxes of thequadricKnx2+2K12xy+...=1, then theenergy perunitvolume assumes theform W=i(K,X*+K2Y>+KZZ% 536 TheElectromagnetic Theory ofLight [ch.xviii Thecomponentsofpolarisationarenowgiven by(cf.equations (89)) 4tt/=K,X, 47r#=K2Y,4,-irh=KZZ, sothatthegeneral equations (529) and(528)of§574assume theforms Gdtdydz KzdY_da_dy Gdt~ dzdx? K,dZ Gdtd_l docda dy}•(A"),/j,dadZ 592a,592b] Mechanical Action 537 Ifwemultiply these threeequations by I,m,nrespectivelyandadd,we obtain IKXX+mK2Y+nK3Z=0, shewingthattheelectricpolarisationisinthewave-front. Thesystem (B")ofequationsreduce to V [xpa=mZ—nY, andtwosimilarequations, andonagain multiplying by I,m,nandadding, weobtain la+m/3+ny=0, which shews thatthemagneticforce also isinthewave-front. Weshall notdiscusscrystalline media indetail inthepresentbook since theirspecial peculiaritiesarethesame ontheelectromagneticasonany othertheoryoflight. Thediscussion ofthesepeculiaritiesisabranch ofthe science ofopticsrather than ofelectromagnetism. Mechanical Action. EnergyinLight-waves. 592 b.Forawave oflight propagated alongtheaxis ofOx,andhaving theelectric forceparalleltoOy,wehave(cf.§592) thesolution X=Z=0;Y=Y cosic(x-at), a=/8= ;y=ycosk(x—at), andthis satisfies alltheelectromagnetic equations, providedtheratio of<yto Yisgiven by jo_Ka_G_IK Y~G~ fia~VfM' Theenergy perunitvolume atthepointxis ~(KP+^72 )=i(KF2+/z7o2 )cos2k(x- at).07T 07T Since/xy2=KY2 ,itappearsthat the electricenergyisequaltothe magneticatevery pointofthewave. Theaveragevalue ofcos2k(x—at), averagedwithrespecteither toxortot,is£,sothattheaverage energy per unitvolume =KYf=wl 8tt 8tt' AsMaxwell haspointed out*, these formulae enable ustodetermine the magnitudeoftheelectric andmagneticforces involved inthepropagationof *Maxwell, Electricity andMagnetism (Third Edition), §793. Thus thetotalpressure perunitarea Sir 87Tcos2k(x— at). This isexactlytheexpression justfound fortheenergy perunitvolume Thusweseethat overeverywave-front thereought,ontheelectromagnetic theory,tobeapressureofamountperunit areaequaltotheenergyofthe waveperunitvolume atthatpoint.Theexistence ofthispressurehasbeen demonstratedexperimentally byLebedew*andbyNichols andHullf,and their resultsagree quantitativelywith thosepredicted byMaxwell'sTheory. Refraction andReflection. Conditions ataBoundarybetween twodifferent media. 593. Letusnext consider whathappens when awave meets aboundary between twodifferent dielectric media 1,2.Letthesuffix 1refer toquanti- tiesevaluated inthe firstmedium, andthesuffix 2toquantitiesevaluated in thesecond medium. Forsimplicityletussupposetheboundarytocoincide with theplaneofyz. *Annalen derPhysik, 6,p.433. tPhysical Review, 13,p.307.538 TheElectromagnetic Theory ofLight [ch.xvin light. Accordingtothedetermination ofLangley,themean energyofsun- light,afterallowingforpartial absorption bythe earth'satmosphere,is 4-3x10~5ergsperunitvolume. Thisgives,asthemaximum value ofthe electricintensity, T=*33C.G.S. electrostatic units=9'9voltspercentimetre, and, asthemaximum value ofthemagnetic force, 7o="033 c.G.S.electromagnetic units, which isabout one-sixth ofthehorizontal componentoftheearth's field in England. ThePressureofRadiation. 592 c.Invirtue oftheexistence oftheelectricintensity Y,there isinanyKY2 medium(§165) apressure——perunitarea atright anglestothelines of electric force. There istherefore apressureofthisamountperunit area overeach wave-front.Similarlythemagneticfield results(§471)inapres- o sure oiamount £-*perunit area. brr 5925-594] Refraction andReflection 539 Attheboundary,theconditions tobesatisfied are(§§137,467): (1) thetangential componentsofelectric forcemust becontinuous, (2)thenormalcomponentsofelectricpolarisationmust becontinuous, (3)thetangential componentsofmagneticforcemust becontinuous, (4)thenormalcomponentsofmagneticinduction must becontinuous. Analytically,these conditions areexpressed bytheequations K.X^K.X,, Yx=Y.2y Z,=Z, (550), /i1a1= /u2a2, /3i=/32, 7i=72 (551). Itwillbeatonce seen thatthese sixequationsarenotindependent:if thelasttwoofequations (550)aresatisfied, then the firstofequations (551) isnecessarilysatisfied also, asaconsequenceoftherelation pda=dZ_dYCdtdydz beingsatisfied ineachmedium, whilesimilarly,ifthe lasttwoofequations (551) aresatisfied, then the first ofequations (550)isnecessarilysatisfied. Thus there areonlyfourindependentconditions tobesatisfied atthe boundary, andeach ofthese must besatisfied forallvalues ofy,zand t. Itismost convenient tosupposethefourboundaryconditions tobethe continuityofY,Z,j3,7. Refraction ofaWavepolarisedinplane ofincidence. 594. Letusnowimagineawave oflighttobepropagated through medium(1),and tomeet theboun- dary,thiswavebeing supposed polar- ised intheplaneofincidence. Let theboundary,asbefore, betheplane ofyz,and lettheplaneofincidence besupposedtobetheplaneofxy. Since thewave issupposedtobepolar- ised intheplaneofincidence, the magneticforcemust beintheplane ofxy,andtheelectric forcemust be paralleltotheaxis ofz.Hence for thiswave,wemaytake X=F=0, Z=Z'ei<x(xcos8l+y sin9l~Fl*' a_a' gt'ici(xcos61+ysin0i-V\t) Q_Q'gi«i(a:cos0i+y sin9i-V\t) 7=0, 540 TheElectromagnetic Theory ofLight [ch.xviii and itisfound thatthesixequations (A),(B)ofp.534aresatisfied if of £' Z' (552).sin#i—cosv1 TheangleBxisseen tobethe" angleofincidence"ofthewave, namely, theanglebetween itsdirection ofpropagationandthenormal(Ox) tothe boundary. Letussupposethat inthesecond medium there isarefracted wave, given byX=Y=0, 2,=Z"g'^tecosdj+ysin02—V2t) tt_a" git2(xcos2+ysin0,—V2t) Q_Q"QiKi(xcos63+ysin2-Vd) 7=0, where, inorder thattheequationsofpropagation maybesatisfied, wemust have °" _.J8"-*"_ fM8)sin#2-cos ^2 //j..Wit Itwillbefound onsubstitution intheboundary equations (550) and (551) thatthepresenceofanincident andrefracted wave isnotsufficient to enable theseequationstobesatisfied. Theequations can,however, allbe satisfied ifwesupposethat inthe firstmedium, inaddition totheincident wave, there isareflected wavegiven by X=F=0, Z=Z'"e:K^xeose 3+ysine3-vit) a__ ft'" q%k%(xcos3+ysin3-VJ) Q_Q"ig«s(xcos8s+ysine,- Vit) 7=0, where, inorder that theequationsofpropagation maybesatisfied, wemust have /// QUI17/1/ 4^—S-g.- 4— (5M).sin63—cosvzixx Theboundaryconditions mustbesatisfied forallvalues ofyand t.Since yand tenteronlythrough exponentialsinthedifferent waves, thisrequires thatwehave «!sin#x=/£„sin62=k3sin6Z (555), k1V1=k2V2=k3V1 (556). 594,595] Refraction andReflection 541 From(556)wemust have k1=k3,andhence from(555),sin6X=sin3. Since 6Xand3must notbeidentical, wemust have 91=it— 3.Thus Theangle ofincidence isequaltotheangle ofreflection. Wefurther have, fromequations (555) and(556), sin^Vx^errrv(5o7X where vistheindex ofrefraction onpassingfrommedium 1tomedium2, sothat thesineoftheangle ofincidence isequaltovtimes thesineofthe angle ofrefraction. Thus thegeometricallaws ofreflection andrefraction canbededuced at oncefrom theelectromagnetic theory.These laws can,however, bededuced frompractically anyundidatory theoryoflight.Amore severe testofa theoryisitsabilitytopredict rightlytherelative intensities oftheincident, reflected andrefracted waves, andthiswenowproceedtoexamine. 595. Theonlyboundaryconditions tobesatisfied arethecontinuity,at theboundary,ofZand/3(cf.§593). Thuswemust have Z'+Z'"=Z"(558), /3'+£"'=/3" (559). Onsubstitutingfromequations (552), (553) and(554), the lastrelation becomes ^/^cos01(Z'-Z"')=^/^cos02Z"(560), sothat alltheboundaryconditions aresatisfied if Z' Z" Z"'T^^Y^T^u(561 >' K2Micos2dowhere u°=^k^i(S62)- For allmedia inwhichlightcanbepropagated, wemaytakefi=l,so that .2cos62_sinBxcos62tandx " cos6Xsin#2cos ti1tand2^°'' Thus theratio oftheamplitudeofthereflected totheincidentrayis Z'"_1—u_tan2—tanQx_sin(# 2—#i) 'W" 1+u~ tan2+tandt= sin(<92+0~)(^ Thispredictionofthetheoryisingood agreementwithexperiment. Z"Thisbeing so,thepredictedratio of-=>isnecessarilyinagreementwith ex- periment,since both intheory andexperimenttheenergyoftheincident wavemust beequaltothesum oftheenergiesofthereflected andrefracted waves. 542 TheElectromagnetic Theory ofLight [ch.xviii Totalreflection. 596.Wehave seen(equation (557))thattheangle62isgiven by sin2=-sin duv where vistheindex ofrefraction forlight passingfrommedium 1to medium 2.Ifvislessthanunity,thevalue of-sindYmaybeeither greaterorlessthanunity accordingas81>or<sin-1v.Intheformer case sin#2isgreaterthanunity,sothatthevalue of62isimaginary. Thiscircumstance doesnotaffect thevalue oftheforegoing analysisina case inwhich 61>sin-1 v,butthegeometrical interpretationnolongerholds. Letusdenote -sin6Xbyp,and\/p2—1byq.Then intheanalysis we mayreplacesin62byp,andcos62byiq,bothpandqbeingrealquantities. Theexponentialwhich occurs intherefracted wave isnow qik2(xcos8+ysin63-F^t ) =giK2(.iqx+py -Vit) =q-k&x Qix^PU-V2t) m Thus therefracted wave ispropagated paralleltotheaxis ofy,i.e. normal totheboundary,and itsmagnitude decreasesproportionallytothe factor e~K*qx .Atasmall distance from theboundarytherefracted wave becomesimperceptible. Algebraically,thevalues ofZ',Z"andZ'"arestillgiven byequations (561), butwenowhave /Kofi^ cos#2_.IKvfjL! qVyiiaif xCOS#!VyttaA'j cos6X' sothatuisanimaginary quantity, sayu=iv,and,fromequations (561), Z"'\-u 1-ivu Since visreal,wehaveZ'' 1+u1+iv' 1—iv 1+iv Z'"=Z'=1,sothatwemaytake /l—iv\where %=arg( J=—2tan-1 ?;. Inthereflected wave,wenowhave 2,—Z'" g**i(-«cos0,+y sin<9,-F,*) =Ze**1(~xc°s9i+y sine,-F 1t-2tau-1») 596-598] Refraction andReflection 543 Comparingwith theincident wave, inwhich Z=Z'eiKi(xcos6i+y sin°i~v^ weseethat reflection isnowaccompanied byachangeofphase—2ktan-1 v, buttheamplitudeofthewave remains unaltered, asobviouslyitmust from theprincipleofenergy. Refraction ofaWavepolarised 'perpendiculartoplane ofincidence. 597. Theanalysiswhich hasbeenalready givencaneasilybemodified soastoapplytothecase inwhich thepolarisationoftheincident wave is perpendiculartotheplaneofincidence. Allthat isnecessaryistointer- change correspondingelectric andmagnetic quantities:wethen have an incident wave inwhich themagneticforce isperpendiculartotheplaneof incidence, andthis iswhat isrequired. Clearlyallthegeometricallawswhich havealready been obtained will remain truewithout modification, andtheanalysisof§596(total reflection) willalsoholdwithout modification. Formula(563), givingtheamplitudeofthereflectedray, will,however, requirealteration. Wehave, asinequation (564),forthe ratio ofthe amplitudesoftheincident andreflectedrays, —-=r~i—(o65),714-u\ /» butthevalue ofu,instead ofbeing given byequation (563), mustnowbe supposedtobegiven by 2_ix2K^cos2 2K2fixcos2 #j' thisequation beingobtainedbytheinterchangeofelectric andmagnetic terms inequation (562). Taking /x2= /u,1=1,weobtain cos2sin2cos2sin202tt=vl2cos0!sin1cosQxsin2QX whence, fromequation (565), ry'" tan(d2-ex) .(566),7tan(0 a+0a) givingtheratio oftheamplitudesoftheincident andreflected waves. This result alsoagreeswellwithexperiment. 598.Wenotice that if1+62=90°,then7"'=0.Thus there isacertain angleofincidence such thatnolightisreflected. Beyondthisangley"is negative,sothat thereflectedlightwillshew anabrupt changeofphase of180°. Thisangleofincidence isknown asthepolarising angle, because if abeam ofnon-polarised lightisincident atthisangle,thereflected beam will 544 TheElectromagnetic Theory ofLight [ch.xviii consistentirelyoflight polarisedintheplaneofincidence, andwillaccordingly beplane-polarised light. Ithasbeen found byJamin thatformula (566)isnotquiteaccurate atandnear tothepolarising angle.Itappearsfromexperimentthat a certain small amount oflightisreflected atallangles,andthat instead of asudden changeofphaseof180°occurringatthisanglethere isagradual change, beginningatacertain distance ononeside ofthepolarising angle andnotreaching180° until acertain distance ontheother side. Lord Rayleighshewed that thisdiscrepancybetweentheory andexperiment canoften beattributedlargelytothepresenceofthin films ofgrease and other impuritiesonthereflectingsurface. Drude found that theout- standing discrepancycould beaccounted forbysupposingthephenomena ofreflection andrefraction tooccur, notactuallyatthesurface between the twomedia, butthroughoutasmall transitionlayerofwhich thethickness must besupposed finite, althoughsmall comparedwith thewave-lengthof thelight. Waves inMetallic andConducting Media. 599. Inametallic medium ofspecificresistance t,equations (A)of§592, namelyKdX_dy dJ3 Cdt~dydz{bb '>' etc.,must bereplaced (cf.equation (546)) by tCdt] dydz etc. Foraplanewave oflight,thetimemaybesupposedtoenterthroughthe complex imaginaryeiptandwemay replace -jbyip-Thus theleft-hand of equation (567) becomes —~-X,while theleft-hand ofequation (568) becomes [h—pi)X. Itaccordingly appearsthattheconducting powerofthe 4nrC2 medium canbeallowed forbyreplacingKbyK+—— . TT To 600. Inanon-conducting medium, theequation-~-^=V2 ^,satisfied by each ofthequantities X,Y,Z,a,yS,7(cf.§577),reduces to -p°-K[A _i—V=V*O%=v"% 598-601] Metallic andConducting Media 545 when thewave isoffrequency p/2ir. Thecorresponding equationforacon- ducting medium must, bywhat hasjustbeen said,be -'{%+%)*-"x ^ anequation which hasalreadybeen obtained in§583 a. Foraplane wavepropagatedinadirection which, forsimplicity, weshall supposetobetheaxisofx,thesolution ofthisequationwillbe ^=AeW e±(-Q+ir)x(570), where(q+irf=-^+*Z*t*(571). Clearlythesolution(570) representsthepropagationofwaves with a velocity Vequaltop/r,theamplitudeofthese wavesfallingoffwith a modulus ofdecay qperunitlength. Onequating imaginary partsofequation (571)weobtain qr=^(572), sothatqisgiven by 2^p=2ZLP> tr t 601. Foragood conductor tissmall, sothatqislarge, shewingthat good conductors arenecessarily badtransmitters oflight.Forawave of lightinsilver orcopper wemaytake asapproximatevalues inC.G.S. units (rememberingthattasgivenonp.342 ismeasured inpractical units) r=1-6x10~6ohms=16x103 (electromag.), /*=1,V=3x1010 , fromwhich weobtainq=1*2x108 .Itappears that, accordingtothistheory, arayoflightinagoodconductoroughttobealmostextinguishedbefore traversing more than asmallportionofawave-length.Thispredictionof thetheoryisnotborne outbyexperiment. Weshall seebelow(§600) thatthedifficultyistosome extent removed ontakingaccount ofthepresenceofelectrons inthemetal. Beforepassing tothemoregeneral theoryinwhich theelectrons aretaken intoaccount we shallexamine thephenomenonofmetallic reflectionaccordingtoourpresent simple theory, and shallagainfindthatthesimple theoryfails toagree with the facts. 35 546 TheElectromagnetic Theory ofLight [ch.xvm MetallicReflection. 602. Letussuppose,asinfig.138,thatwehave awave oflightinci- dent atanangle6lupontheboundarybetween twomedia, and letussuppose medium 2tobeaconducting medium ofinductivecapacityK2.Then(cf. §599)alltheanalysiswhich hasbeengivenin§§593—597will stillhold if wetakeK2tobeacomplex quantity given by K2=K2'+^-(574). SinceK2iscomplex,itfollows atonce thatV2iscomplex, being given by C2 K2= K.2^2 andhence thattheangle62iscomplex, being given (cf.equation (557)) by sin20osin2^ sin2^C2_.2ffKl/h K2'aK2Etfh—vlK2fJhi Thevalue ofuisnowgiven,fromequation (562), by.(575). u2—K2^cos2 2 ix2Kxcos2 X -^1sec2^1-^-1-tan2 6»1 (576) (cf.equation (575))forlight polarisedintheplaneofincidence. Forlight polarised perpendiculartotheplaneofincidence, thevalue ofuisfound, as before, byinterchangingelectric andmagnetic symbols. Onputtingu=a+i/3,wehave, asbefore(equation (564)), 2T_1-M_l-a-i/3 Z'~ 1+u~ 1+a+i/3' Ifweputthis fraction intheformpe%then the reflected wave is given by %=2!"eiKi(-xcos^+2/sin 6,-Vxt)_%*pgi*,(-zcos9 x+ysine^Vit+\), Comparingthiswith theincident wave, forwhich Z=Z'eiK1{xcose '+ysin9l~Vlt] weseethat there isachangeofphase k^xatreflection, andtheamplitude ischangedintheratio 1 :p.The electric force intherefracted wave is accompanied byasystemofcurrents, andthesedissipate energy,sothat theamplitudeofthereflected wave must belessthan that oftheincident wave. 1-a-i0Wehavepe'x= 1+a+i/3' 602-604] MetallicReflection 547 sothat n2=———=1 (577^P (1+«#+£» (l+a)2+/32 l°"; shewingthatp<1,asitoughttobe.Also y=-tan"1-£--tan"1=-£-=-tan"1-—^—- (578). A1-a 1+a l-a--/33 603.Experimental determinations ofthevalues ofpand^havebeen obtained, butonlyforlightincidentnormally, the firstmediumbeingair. Forthisreason weshallonlycanyontheanalysisforthecase of6=0.It isnowamatter ofindifference whether thelightispolarisedinoratright anglestotheplaneofincidence;indeed itiseasilyverified that thevalues givenforpand%byequations (577) and(578) arethesame ineither case. Takingforsimplicitytheanalysis appropriatetolight polarisedinthe planeofincidence, andputting=0,/*i=l,-5^=1,wehave fromequation (576) u2=—=b- , fi2 fju2ipr/x 2 and, sinceu=a+i/3,thisgives a3-/Sa=—' (579) fa a{3=-2-^(580). prfx, 604. Letusconsider theresults asappliedtolightofgreat wave-length, forwhich pisverysmall. Forsuch values ofp,a/3 isclearly very large comparedwith a2—/32 ,sothat aand(3arenearly equal numerically,andwe maysupposeasanapproximationthat(cf.equation (580)) a=- /3=v/2^(581). When aandf3areequal andlarge, equation (577) becomes 9 'pr/j. "!=1-i=1-2V^<582>- Letussupposethatanincident beam hasintensitydenotedby100,and that ofthisabeam ofintensity Risreflected from thesurface ofthemetal, while abeam ofintensity100—Renters themetal. ThenRmaybecalled thereflecting powerofthemetal. Theintensityoftheabsorbed beam is 100-.8=100(1 -p-) =2°°y^(583>- 35—2 548 TheElectromagnetic Theory ofLight [ch.xviii Wenotice that forwaves ofvery great wave-length (pvery small)R approximatesto100, sothat forwaves ofvery great wave-lengthallmetals becomeperfectreflectors. This isasitshould be,forthese waves ofvery longperiod mayultimatelybetreated asslowly-changingelectrostatic fields, andtheelectrons atthesurface ofthemetal screen itsinterior from the effects oftheelectric disturbancesfalling uponit(cf.§114). Equation (583) predictsthewayinwhich 100—Roughttoincrease aspincreases, andanextremely importantseries ofexperimentshavebeen conducted byHagen andRubens* totestthetruth oftheformula for lightofgreat wave-length. Thefollowingtable will illustrate theresults obtained f: 604-607] Electron Theory 549 Electron Theory. 606.Wehavenowreached astageinthedevelopmentofelectromagnetic theoryinwhich itisclear thatthesimple conceptionswhich have sofarbeen employedarenolonger adequatetogiveacomplete explanationofthe phenomena. Theconceptionsonwhich thepreceding analysis hasbeenbased havebeen theoriginal conceptionsofMaxwell'stheory:itisnatural now to examine inwhatwaythetheorycanbemodified orimproved bytheintro- duction ofthemoremodernconceptionsoftheelectrontheory.Instead of regardingacurrent asacontinuous flow ofelectricity, weshall take definite account ofthepresenceofelectrons. Weshall have toconsider two sets ofelectrons, the"free" and"bound" electrons of§345 a,thesebeingthe mechanismsrespectivelyofconduction andofinductivecapacity. Theapplicationofanelectric forceXwill result inamotion offree electrons similar tothatinvestigatedin§345 a,and inamotion ofthe bound electrons similar tothat discussed in§151. But ifXisvariable with thetime, theinertia oftheelectrons willcome intoplayandthe resultingmotions willbedifferent from thosegiven byOhm's lawand Faraday'slaw.We shallsupposethat atanyinstant thecurrentproduced bythemotion ofthefree electrons is«/,andthat thatproduced bythe motion ofthebound electrons isu^. 607.Wemayconsider firsttheevaluation ofUf.TakingNtobethe number offreeelectronsperunitvolume, andallowingforchangeofnotation, equation (c)of§345amaybere-written intheform cx=™'+l?t<584>- inwhich, asthroughoutthischapter,Xisexpressedinelectrostatic units, whileUfisinelectromagnetic units, and r'stands for7/iVe2 ,sothattbecomes identical with thespecificresistance twhen thecurrents aresteady. Thisequationisapplicabletoourpresent investigationifwesupposeXtobeperiodicinthetime offrequency p/2tt. TakingX=XQeipt ,the solution ofequation (584)is CTogft* ,,„_,u'= ,ra .(58D)- Thequantityrheremaydependonp,andwithout afullknowledgeofthe structure ofmatter itisimpossibletodecide howimportantthedependence oftonpmaybe.Wearethereforecompelledtoretain itasanunknown quantityinourequations, rememberingthat itbecomes identical with twhen p=0,and isprobably numerically comparablewith rforallvalues ofp. 550 TheElectromagnetic Theory ofLight [ch.xviii Wemaynote thattherealpartofthecurrent, correspondingtotheforce X=Xcospt,is CX .N ——cos{pt—e)cose,T inwhich tane=„„,,shewingthat theinertia oftheelectrons, asrepre- sented inthe lastterm ofequation (584),results inalageinthephase ofthecurrent, accompanied byachangeinamplitude.Therateofgeneration ofheatbythecurrentUf,being equaltotheaveragevalue ofUfXcospt, isfound tobeA—r^cos2eorA-,where T Tp Ti,=T'sec2e=T'+-p^:/ (586). Itisworthnoticingthat forlightofshortwave-lengththelastterm inrp maybemoreimportantthan the firstterm r.Thus tpmaybelargestfor good conductors, andsmallest forbadconductors. 608.Weturn totheevaluation ofi(b,thecurrent produced bythesmall excursions ofthebound electrons, astheyoscillate under theperiodicelectric forces. Weshallregardamolecule(oratom),asin§151,asacluster ofelectrons, andthese electrons willbesupposed capableofperformingsmall excursions about theirpositionsofequilibrium. Ashasalreadybeen said(§192)itis probablethat thisconceptionofthestructure ofthemoleculerepresents only ahalf-wayhouse towards thetruth, but itprovidesapictureormodel ofthe structure with thehelpofwhichmany properties maybeexplained. Let1,$2,...begeneralisedcoordinates(cf. §548) determiningthe positionsoftheelectrons inthemolecule, thesebeingchosen soastobe measured from thepositionofequilibrium. Solongasweconsideronly small vibrations, thekineticenergy Tandthepotential energyWofthe molecule canbeexpressedintheforms 2W= and,2+ZandA+a2A2+(587), 2T=bj?+2bJA+bj?+(588), inwhich thecoefficients an,a12,a™, ...,bn,...maybetreated asconstants. Byaknownalgebraic process, new variables<£1; <f>2,...canbefound, such thatequations (587), (588) whenexpressedinterms ofthese variables assume theforms 2IF=a4?+a,c£ 22+ (589), 2T=yS112+/S2^2+ (590), theseequations involving only squaresofthenew coordinates <f>1} (f>.2,.... Thecoordinates found inthiswayforanydynamical systemarespokenofas the"principalcoordinates" ofthesystem. 607-609] Electron Theory 551 Theequationofmotion ofthemolecule, when acted onbynoexternal forces, isreadily found tobe(cf.equations (500)) fis<j>s=-«s<f>s, (5=1,2,...) (591). Theseequationsareknown torepresent simply periodic changesin <f>1} </>2,...offrequencies nJ27r, n2/2ir,... given by "."-J;(592). Itispossiblethatwehave evidence ofthefrequenciesofmolecular vibra- tion incertain ofthelines ofthespectrumemitted bythesubstance under consideration; ifsoequations (592) connect thefrequenciesofthesespectral lineswith thecoefficients oftheprincipalcoordinates ofthemolecule. 609. Ifthemolecule isnowsupposedtovibrate under theinfluence of externally appliedforces (such,forinstance, aswould occur duringthe passageofawave oflight throughthemedium), equation (591) must be replaced (cf.equation (508)) by 0,4>,=-<*,<!),+ ®, (593), where <&gisthatpartofthe"generalisedforce"correspondingtothe coordinate<ps,whichoriginatesintheexternally appliedforces. IfXistheelectromotive force inthewave oflightatanyinstant, each electron willexperienceaforce Xe,andthere willbeacontribution ofthe form %sXeto^>g. Againtheelectrostatic fieldcreated bythedisplacementsoftheelectrons inthevariousneighbouring molecules willcontribute afurther term to<E>g. Thedisplacementofanyelectronthroughadistance fwillproducethesame field asthecreation ofadoublet ofstrength e|.Thus ifthere areM moleculesperunit volume, the totalstrengthofthedoubletsperunit volume, sayV,maybesupposedtobeoftheform r=ife(7l 1+72 2+...) (594), andthese willproduceanelectricintensityofwhich theaveragevaluemay betaken tobe(cf.§145)kY,which must beadded totheoriginal intensityXofthewave. The total value of<&sistherefore £se(X+kF),sothatonreplacingasby itsvalue fromequation (592), equation (593) becomes ^s(4>s+ns2 <f>s)=^e(X+KV) (595). IfwesupposeXtodependonthetimethroughthefactor eipt ,then <f>willclearly dependonthetimethroughthesame factor, andwemay replace §8by—p2 (ps.Equation (595)nowbecomes *•-A(«-*)(596) ' 552 TheElectromagnetic Theory ofLight [ch.xviii whence, byequation (594), T=M*%^lAX+kT) (597), and ifwewrite e-M'lfjr+0(598), thisgives,asthevalue ofT, r=T^ex (599> Thecurrentproduced bythemotion ofthebound electrons isubin electromagnetic,and therefore Cubinelectrostatic units. Itsvalue in electrostatic units isalso(cf.§345a)Neu orSe jr.,where thesummation istakenthroughaunitvolume, and this inturn isequaltoY.Thus _T_ipX Ub~C~l-K0 C The total current, expressedinelectromagnetic units, is ldf Incalculating fwemustremember that thepolarisation produced by themotion ofthebound electrons isalreadyallowed forinthepresence ofthetermub.Weaccordinglytake/equal simplytoX/47T, andon furtherreplacingubandUfbythevalues found forthem, thetotal current becomes ipX/_ 4tt0 \CX,,.AA. Inplaceofequation (569), theequationofpropagationis \ 'PV+N?''')) Asin§600,thesolution is x=AeiPte±®+ir)x(601), (7+,v,=-^(1+ i^)+_ilS-(602). Non-conducting media. 610. Foranon-conducting medium r'=oo,sothat the lastterm in equation (602) vanishes, andtheright-hand member becomeswhollyreal. Forcertain values of6,thisright-hand member isnegative,sothatq=0, shewingthatlightistransmitted without diminution; themedium is perfectly transparent. 609,610] Electron Theory 553 Fortransparent media wemaytakefi=1,andthevelocityofpropagationVisgiven by 1_r2 _l_f4tt<9 F2 _p»C3Vl-«0/ Ifvistherefractive index ofthemedium, ascomparedwith that ofa vacuum, V=CJv,sothat Arr-0 *°=1+i^(603)- whence$£-"-***=?(604)' inwmcn a=1,cs=— --.— ,sothataand c.areconstants. Clearly (cf. §609) thevalue ofacanbecalculated ifwemake assumptionsastothearrangementofthemolecules inthemedium. On assumingthat themolecules areregularly arrangedincubicalpiling,kis found tohave thevalue§tt,sothatabecomesequalto2. Formula (604) inwhich aisneglected altogetherbecomesexactlyidentical with thewell-knownSellmeyerorKetteler-Helmholtz formula forthe dispersionoflight,ofwhich theaccuracyisknown tobeveryconsiderable. Ifaisputequalto2,theformula becomes identical withdispersionformulae which havebeensuggested byLarmor andLorentz. IthasbeenshewnbyMaclaurin* thatformula(604)willgiveresults in almostperfect agreement withexperiment,atleast forcertain solids, ifais treated asanadjustable constant. Theagreementoftheformula issovery goodthat little doubt canbefeltthat itisfounded onatrue basis. Mac- laurin finds foravalueswidelydifferent from 2(forrocksalt a=5"51, for fiuorite a=l"04), thedifferences between these numbers and2pointing perhapstothecrystalline arrangementofthemolecules. Forliquids and gasesweshouldexpecttofindaequalto2. SinceMisproportionaltop,thedensityofthesubstance, formula(604) v'2—1 indicates thatoughttovary directlyaspwhenpvaries. This law, withatakenequalto2,wasannounced byH.A.Lorentz fofLeyden and L.Lorenz^:ofCopenhagenin1880. Itstruth hasbeen verified byvarious observers, and, inparticular, byMagri§foralarge rangeofdensities ofair. Fromequation (604)italsofollows thatthevalues of foramixture i,2-1ofliquidsorgases oughttobeequaltothesum ofthevalues of—forits *Proc. Roy. Soc.A,81,p.367(1908). jWied. Ann. 9,p.641(1880). %Wied. Ann. 11,p.70(1880). §Phys. Zeitschrift, 6,p.629(1905). 554 TheElectromagnetic Theory ofLight [oh.xviii ingredients,alawwhich isalsofound toagree closelywith observation on takinga=2. 611. Forcertain other values of6,therighthand ofequation (602) (in which t'istakeninfinite)isfound toberealandpositive. Wenowhave r= andthesolution (601) becomes x=AeiPte±9x(605), shewingthat there isnowave-motionproper,butsimplyextinction ofthe light. Thus there arecertainrangesofvalues ofp(namelythose whichmake (q+ir)2 positiveinequation (601))forwhichlightcannot betransmitted atall;these mustrepresent absorptionbands inthespectrumofthesub- stance. Clearly (q+ir)2becomespositive when 6islarge andnegative.Itwill benoticed that 6,asgiven byequation (598), becomes infinite when phas anyofthevalues n2,n2,...,changingfrom—ooto+ooasppasses through these values. Thus theabsorptionbands willoccur close tothefrequencies ofthenatural vibrations ofthemolecule. Butjustintheseregions wehave toconsider certain newphysical agencieswhich cannotlegitimatelybe neglected when phasvalues near tonltn2,...,although probably negligible inotherregionsofthespectrum. 612. Equation (593)isnotstrictlytruewiththevaluewehaveassigned to<l>s.For,inthe firstplacethevibrationsrepresented bythechangesin<ps aresubjecttodissipationonaccount oftheradiation oflight,andofthisno account hasbeen taken. Inthesecondplacethere must besudden forces actinginliquidsandgasesoccasioned bymolecularimpactsandrequiringthe addition ofterms to<E>Sthroughouttheshortperiodsoftheseimpacts.There must beanalogous changestobeconsidered inthecase ofasolid, although ourignoranceoftheprocessesofmolecular motion inasolidmakes itim- possibletospecify them withanyprecision. The effect oftheseagencies must betothrow the$/softhedifferent molecules outofphasewithoneanother andalsooutofphasewithXandY. Theanalysisof§609hasmade theratios ofX :T : <j>swhollyreal(cf.equa- tions (596) and(597)), indicatingthatX,Vand </>sareexactlyinthesame phase.Theconsiderationsjustbroughtforward shew thatthese ratios ought alsotocontain smallimaginary parts. Theprocessofseparatingrealandimaginary partsinequation (602)now becomes much morecomplicated,but itwillbeobvious that forallvalues of p,both qandrwillhavesome value different from zero. Thus there is always some extinction oflightandsome transmission, forallvalues ofp,and there isnolongerthesudden changefrom total extinction toperfecttrans- mission. Theedgesoftheabsorptionbandbecomegradualandnotsharp. 610-615] Electron Theory 555 Butthemolecular model now inuseprobablydoes notrepresentthedetails ofmolecular action with sufficient truthfulness tomake itworthtryingto representtheconditions nowunder discussion inexactanalysis. Conducting media. 613. Foraconducting medium weretain tinequation (602), andon equating imaginary partsweobtain, inplaceofequation (572)of§600, qr=- ./a ,mT.(606), T2+N2e4>P where tpisgiven byequation (586). Thusequation (573)of§600becomes replaced by q=2-^(607).Tp For visiblelightthisgivesaverymuch smaller value ofqthan that discussed in§600,andthevalue ofqwillobviouslybestillfurther modified bytheconsiderations mentioned in§612. 614.Oncomparingthetotal current, asgiven byformula(600),with the value--- -nassignedtoitintheanalysisof§§594—598,weseethat all this earlieranalysiswillapplytothepresent problemifwesupposeKtobe acomplex quantity given by 47rC2K=v2+ ,m ..(608), where visgiven byformula(603). If,asin§603,weput wefindW2=^?=(a-M73)2 , H>2 a--/32= flom4ttC2 v— Ne2TT r .(609), a27rG2ap= PTpftz sothat thereflecting powerB,ofametalmaybecalculated fromequation (577)interms ofrp. 615.Oncomparingformulae (609) withexperiment,thegeneralresult appearstoemerge, that, inorder toaccount fortheoptical propertiesof conductors inthisway,thenumber offree electrons inconductors must be comparablewith thenumber ofatoms.Accordingtoapaper bySchuster, 556 TheElectromagnetic Theory ofLight [ch.xviii publishedin1904*, theratio ofthenumber offreeelectrons toatoms ought torangefrom 1to3invarious substances; Nicholsonf,astheresult ofamore elaborateinvestigation,obtains values forthisratiorangingfrom 2to7. This result discloses adifficultyfromwhich theelectrontheory,inthe form inwhich wehave sofarconsidered it,hasshewn littlepowerofextri- catingitself. Specific Heats and ElectricalConductivity. 616. Accordingtothewell-known lawofDulong andPetit theatomic heats ofalargenumber ofelements have values which areapproximatelyall equal. Nernst andLindemann haverecentlydetermined thespecificheats of alargenumber ofelements, andhave found that, foralltheelementsthey have examined, theatomic heats measured forconstant volume{i.e.after correction forexpansion arisingoutofchangeoftemperature)have allthe same value 5*95.Now theatomic heatrepresentstheincreaseperunit rise oftemperatureintheenergyofthesolidmeasuredperatom ofitsstructure. Thisenergycanberegardedasthesum oftwocontributions, namelythe energyoftheatoms andtheenergyofthefreeelectrons. Theenergyofthe atoms canbecalculated bythewell-known methods oftheKinetic Theoryof matter, and itisfound that thisenergywillprovideacontribution tothe atomic heatequal exactlytothetotalamount oftheatomic heat,namely 5*95; inother words thecontribution from theenergyofthefreeelectrons is assmall astheexperimentalerror. Butthecontribution from agivennumber offreeelectrons alsoadmits oftheoretical calculation ifwemake theassump- tionthat their motion conforms totheordinary dynamicallaws. Ifthere were asmanyfreeelectrons asone-tenth ofthenumber ofatoms, thecontri- bution totheatomic heatwould be'30,sothatthetotalatomic heatwould be6'25, anumber much toolargetobereconciled with theexperimentsof Nernst andLindemann. 617. Theforegoing figuresreferonlytomatter atcomparatively high temperatures. Thespecificheats oftheelements havehowever been deter- minedbyNernst andLindemannthroughaverywiderangeoftemperatures, namely from normaltemperaturesdown tothelowesttemperatures now available inthelaboratory. And ithasrecentlybeenshewn byDebyethat theatomic heats foundbytheseexperiments are,atalltemperatures,almost exactly equaltothose tobeexpectedontheoretical groundsonthesupposition thatthefreeelectrons contribute nothingtothespecificheat. Theobserved atomic heatsagreesowellwith those calculated fromtheory,forallsubstances examined andatalltemperatures available, thattheconclusion seems tobe inevitable thatthenumber offreeelectrons isverysmallcomparedwith the number ofatoms. *Phil.Mag. February 1904. tPhil.Mag. Aug. 1911. 615-619]Electron Theory 557 618. Thusweareledtotheconclusion thatalthoughthepresentelectron theory mayshew acertain powerofexplainingtheoptical propertiesof metals, qualitativelyatleast, yetthisexplanationdemands thepresenceof afargreaternumber offreeelectrons than canbereconciled with thevalues ofthespecificheats. Ifthepresentelectrontheorywere inotherrespects satisfactory,the difficulty justrevealed mightbethoughttoconstitute aserious defect inthe electromagnetic theoryoflight.Butthepresentelectrontheoryisfarfrom satisfactoryinotherrespects;indeed adifficulty verysimilar tothatjust disclosed hasbeen found toarise inconnection with amuchsimpler pheno- menon, namelytheconductivityofmetals. Wehave seen(§345a)thatthe electrontheory requiresthat in'agoodconductor thenumber offreeelectrons should belarge; approximatelyhowlargeitmust beisamatter which can alsobedetermined byfurtheranalysis.Therequisite analysishasbeen given byDrude. 619.Wemaysuppose,asarough approximationtothetruth, thatina conductor each free electron movesfreelyforacertainlengthoftime t between twoconsecutive collisions with molecules. Inthenotationalready used in§345 a,themomentum gainedinthistime willbeXet. Ifwe supposethismomentum tobeentirelychecked ateach collision(cf.§§355, 373),theaverageforward momentum ofalltheelectrons atanyinstant will behXet, andsince this isequaltomuinthenotation of§345 a,wehave u=\—t (610),2m andhence(byequation (6),§345a) i=Neu=±—tX (611).2m Thus thequantity 7of§345ais,asregardsorder ofmagnitudeatleast, equalto— ,andthespecificresistance rofasubstance willbegiven by t 1 1/V>2 -=~—t (612),t2m whereNisthenumber offreeelectronspercubic centimetre. Now forsilver orcoppert=1'6x10-6ohms=l'8x10-18inelectrostatic units. Thevalue of 1e2 -it inelectrostatic units is1*26x108 ,andhence togivetotthevalue 2m appropriateforsilver orcopper wemust haveNt=5x109approximately.In silver orcopperthenumber ofatomspercubic centimetre isoftheorder of 1023 ,sothat iftheobserved values ofthespecificheats donotallow ofNbeing more thanone-hundredth partofthiswemust atmostsupposethatNisof theorder of1021 ,and thisrequiresttobecomparablewith 5x10~12atleast. 558 TheElectromagnetic Theory ofLight [ch.xviii Since theaverage velocityofthefreeelectrons isbelieved tobeabout 107cms. persecond(§345a),thiswouldrequireeach electron totravel anaverage distance of5xl0-scms.between consecutive violent collisions. Thisappears tobetoolargetobereconciled withpresentbeliefs astothestructure of matter. Thedifficultybecomes much worsewhenweconsider thephenomenonat lowtemperatures. Kamerlingh Onnes hasfound forsilver atatemperature of13'88° abs.aresistanceonlyequalto0"7percent, ofthat at0°C.Thus in silver atthislowtemperature wemust haveNtoftheorder of1012 ,sothat if wetakeN=1021asabove, t=10-9 .Thisvelocityoffreeelectrons atthislow temperatureisoftheorder of2x106 ,sothattheaveragedistance travelled would beabout-^cm. 620.Wehavenowfound that contradictions exist inconnection with theElectromagnetic TheoryofLight,thetheoryofSpecific Heats ofmetals, andthetheoryofElectricConductivity,solongaswetreat thesequestions interms ofordinary dynamicallawsandMaxwell'selectromagnetic equations.Alargeaccumulation ofevidence, ofwhich ourdiscussion hastouchedonlyon asmallfringe, suggeststhatanewsystemofdynamics andanew electron theoryisneeded. Sofarascanbeseenthespecialfeature ofthisnewtheory must bethattheinteraction between electrons andradiation isofanentirely different nature from thatimagined bytheclassical laws. Thenewtheoryis inexistence and isgenerally known astheQuantum-theory. Abrief intro- duction toitwillbefound inthelastchapterofthepresentbook. CHAPTER XIX THEMOTION OFELECTRONS General Equations. 621.Themotion ofanelectron orother electriccharge givesriseto asystemofdisplacement currents, which inturnproduceamagneticfield. Thechangesinthismagneticfieldgiverisetonew electric forces, andsoon. Thus themotion ofelectrons orotherchargesisaccompanied bymagnetic and electric fields, mutually interacting. Toexamine thenature and effects ofthese fields istheobjectofthepresent chapter. Thenecessary equationshavealreadybeen obtained in§574,butthe current u,v,wwillnowberegardedasproduced bythemotion ofcharged bodies. Ifatanypoint x,y,zthere isavolumedensity pofelectricity movingwithavelocityofcomponents u,v,w,then thecurrent atx,y,zhas components pu,pv,pwinelectrostatic units. Since u,v,winequations (529) aremeasured inelectromagnetic units, theymust bereplaced bypu/C, pv/C, pw/C,andtheequationsbecome 4tt/ df\ dy 3/3 /iMmx Equations (528), namelyIda dZdY -CTt=dy--di>etc<614)' remain unaltered, andthetwosetsofequations (613) and(614) provide the material forourpresentdiscussion. When wehadthesesameequationsunder review in§574,Gwasregarded merelyastheratio oftheunits.Wemaynowregard Casbeingthevelocity oflight,thisbeingalsothevelocityofanyotherelectromagnetic disturbance infreespace. 622. Ifwedifferentiateequations (613) withrespecttox,y,zandadd, weobtain9/v3/%,3/v dfdf da dh^ dxvydzrdt\dx dy dz, Wehave also, asanequationofcontinuity, expressing thattheincrease inpinanysmall element isaccounted forbytheflowofelectricityacross the facesbywhich theelement isbounded, 560 TheMotion ofElectrons[ch.xix Bycomparison with theequation justobtained, wehave d(df+dg+dh\dp dt\dx dydz) dt' ofwhich theintegralisourformerequation (63),namely di+ dy+dz-p (615)- Similarly,ondifferentiating equations (614) withrespecttox,y,zand adding,weobtain d(dadb dc\ dt\dx dy dz)'. ofwhich theintegralisourformerequation (362), namely da db dc _ /«,«xs+£+s-° •.(616)- 623.Atapointatwhich there isnoelectriccharge (p=0),equations (613) and(614)become identical with thesystemsofequations (A)and(B) of§577,andthequantities X,Y,Z,a,/3,7must allsatisfythedifferential equation (534), namely ^=«2V2% (617). Force ofaMoving Electron. 624. Consider afresh theproblemofwhich apreliminarydiscussion has alreadybeengivenin§572, ofasingleelectron movingwith avelocity u paralleltoOx. Since thefieldnecessarily moves with theelectron, therate ofchangeofanyquantity %aswefollow itinitsmotion must benil.Thus wemust have (d dt+ute)*-° sothat -rmaybereplaced by—u=-throughoutourequations. Ctt COG Equation (617) becomes dx" Xdx* dy"dz* or,sincea2=G2 JK/j,, {1~~u^)d^+ df+dF-°(618)- Alsoequations (613), (614) assume theforms ^O-B-g-S•<->' 5E-S-S<-> 622-627] Force onamoving Electron 561 625. Inmostproblems,thevelocityofmotion uissmall compared with thevelocityoflight,sothatu/Gmaybetreated asasmallquantity. Equation (619) shews thatthemagneticfield setupbyamoving charge may beregardedassmall ifu/Cissmall. Thesame isofcourse true ofthe field setupbyanynumber ofmoving charges providedalltheir velocities are small comparedwith thatoflight. When u/Gissmall, equation (620) shews that d_Z_dY dydz willbeasmallquantityofthesecond order. Letussuppose,until thecon- traryisstated, thatu/Gissosmall foreachmoving chargethatu2/C2may legitimatelybeneglected. Then dydz sothat theforces X,Y,Zarederivable from apotentialO.When u2/C2is neglected equation (618)reduces toV2^=0.Thisequationissatisfied by X,Y,Zseparately, and therefore alsobyO.Since X,Y,Zalsosatisfy equation (615),or dXdYZZ=^ dx dydz"' itisclear that thevalues ofX,Y,Zareexactlythesame asifthemoving chargewereinstantaneouslyatrest. 626. This isexactlytheassumption wemade in§572incalculatingthe magneticforcefrom amoving charge. The forces there calculated, namely a=0,P=-Cri>y=G^(621)' arenowseen tobeaccurateprovided u2/C2maybeneglected, butnot otherwise. TheForce acting onaMoving Electron. 627.Theassumption wehavemade thatu/Gissmall isthesame as assumingtoafirstapproximationthatGissogreatthatthemedium maybe supposedtoadjustitselfinstantaneouslytochanges occurringinit,just as anincompressiblefluidwould do.Thetime taken foraction topassfrom onepointtoanother maybeneglected. Wemayaccordingly assume that at anyinstant themechanical actions ofanytwopartsofthe fieldupon one another aresuch that action andreaction areequalandopposite. J. 36 562 TheMotion ofElectrons[CH.XIX Fromequations (621),itappearsthatanelectronmovingwithvelocity u,0,attheoriginwillexert aforce ofcomponents 0,-uemz uemy ~G~^ uponamagnetic poleofstrength matx,y,z.Itfollows thatamagnetic poleofstrength matx,y,zwillexert aforce ofcomponents 0,ueviz uemy .(622) uponthemovingelectron attheorigin. 628. Ifwehave anumber ofmagnetic poles,theresultant forceupon themovingelectron hascomponents 0,ue^mz ue^my .(623)G~"r3'Gr andthecomponentsofmagneticforce attheoriginaregiven by(cf.§408) ~mx <x=—Z—r,etc. r3 Thus theforceonthemovingelectron maybeputintheform ue 0,-^7,>.(624). Plainlytheforceontheelectron willbegiven byformulae(624), whether themagneticfield arises frompolesofpermanent magnetismornot. Itis clearlyaforce atright anglesboth tothedirection ofmotion oftheelectron, andtothemagneticforce a,/3,7atthepoint.IfHistheresultantmagnetic force, and6theanglebetween thedirections ofHandtheaxis ofx,-then theresultant ofthemechanical force isueH sin6/G. 629. Iftheelectron hascomponentsofvelocity u,V,w,thecomponent ofthemechanical forceonitwillbe jjiyv-pw), ^(aw-ryu), ^(@u-av)(625). Since themechanical force isalways perpendiculartothedirection of motion, itdoesnowork onthemoving particle;and, inparticular,ifa charged particle movesfreelyinamagnetic field, itsvelocityremains con- stant. Theexistence ofthisforceexplainsthemechanism bywhich aninduced current isset upinawiremoved across magneticlines offorce. The force(625) has itsdirection along thewireandsosetseach electron intomotion, producing acurrentproportional jointlyto thevelocity andstrengthofthefield—i.e.todN\dt. 627-632]Force, onamoving Electron 563 The"HallEffect" 630.Verydirect evidence oftheexistence ofthis force isprovided by the"Hall Effect." Hall* found thatwhen ametallic conductorconveying acurrent isplacedinamagnetic field, thelines offlowrearrangethemselves astheywould under asuperposedelectromotive force atright anglesboth tothedirection ofthecurrent andofthemagneticfield. Thesame effect hasalsobeen detected inelectrolytesandingases. TheHall Effect isofinterest asexhibitingadefinitepointofdivergence between Maxwell'soriginal theory andthemodern electrontheory.Accord- ingtoMaxwell'stheory,amagneticfield could actonlyonthematerial conductorconveyingacurrent, andnotonthecurrent itself, sothat ifthe conductor washeld atrestthelines offlowoughttoremain unalteredf. The electrontheory,confirmed bytheexperimentalevidence oftheHall Effect, shews that this isnot so,andthatthelines offlowmust bealtered inthepresenceofatransversemagneticfield. Motionofacharged 'particleinauniform magnetic field. 631.Letaparticleofchargeemovefreelyinauniformmagneticfield ofintensity H.Let itsvelocity beresolved intoacomponent Aparallelto thelines offorce, andacomponent Bintheplane perpendiculartothem. Bywhat hasjustbeen said(§629) bothAandBmust remain constant throughoutthemotion, andthere willbeaforceeHBjG actingontheparticle inadirectionperpendiculartothat ofB,andintheplane perpendicularto thelines offorce. Thus ifmisthemass oftheparticle,itsacceleration must beeHB/mCinthissame direction. Considering onlythemotion inaplane perpendiculartothelines offorce, wehave avelocity Bandanacceleration eHB/mC perpendiculartoit.This latter must beequaltoB2 Jp,wherepisthecurvature ofthepath. Thus p=— jj-,aconstant, shewingthatthemotion inquestioniscircular. Combiningthis circular motion with themotionparalleltothelines of forcewefindthatthecompleteorbit isacircular helix, ofradius BmGjeH, described about oneofthelines ofmagneticforce asaxis. 632.Bymeasuringthecurvature ofanorbit described inthismanner, itisfoundpossibletodetermine ejmexperimentallyforelectrons andother charged particles (cf.§665below). Incidentallythefactthat curvature is observed atallprovides experimentalconfirmation oftheexistence ofthe force actingonamovingelectron. *Phil.Mag. 9(1880), p.225. tMaxwell, Electricity andMagnetism, §501. 36—2 564 TheMotion ofElectrons[ch.xix TheZeemanEffect. 633.When asource oflight emittingaline-spectrumisplacedina strong magnetic field, thelines ofthespectrumareobserved toundergo certainstrikingmodifications. Thesimplestformassumedbythepheno- menon isasfollows. Ifthelightisexamined inadirectionparalleltothelines ofmagnetic force, each ofthespectrallinesappears splitintotwo lines, onoppositesides of,andequidistant from, thepositionoftheoriginal line,andthelightof these twolines isfound tobecircularly polarised,thedirection ofpolarisation beingdifferent forthetwo. Ifthelightisexamined across thelines offorce, these same two lines appear, accompaniednowbyalineattheoriginal positionofthe line, so that theoriginallinenowappears splitinto three. The side lines are observed tobeplane polarisedinaplane throughthelineofsightandthe lines offorce, while themiddle line isplane polarisedinaplane perpendicular tothelines offorce. 634. These various phenomenawere observedbyZeeman in1896, and anexplanationinterms oftheelectrontheory wasatoncesuggested by Lorentz. Letusfirstexamine asimpleartificial case inwhich thespectrum contains onelineonly,assumed tobeproduced bytheoscillations ofasingleelectron about apositionofequilibrium. Ifthefrequencyofthis oscillation isp/Ztt,theequationsofmotion ofthe electron must beoftheform d?xm-tt2=-mp2 x,etc(626), inwhich x,y,zarethecoordinates oftheelectron referred toitspositionof equilibrium. Next supposetheelectron tomove inafield offorce ofintensityH paralleltotheaxisofx.Inaddition totheforce ofrestitution ofcomponents-mp2 x,-mp2 y,—mp-z,theelectron willbeacted onbyaforce(cf.formulae (625))ofcomponents neHdz eHdy 'TT~dV ~~C~df Inplaceoftheformerequations, theequationsofmotion arenow &\rm dt2=~mP2x d?y ,eHdzmw-=-mpy+-cTt d2z eHdy.(627). 633-635] TheZeemanEffect Thesolutions oftheseequationsare x=Acos(pt— e), y=A1cos{qxt-ex)+A2cos(q2t-ea), z=Azsin(qxt- e,)+^2sin(g2£—ea), inwhich A,AltA2>e,e^eaareconstants, andqltq2aretheroots ofb§o —mq*=—mp*+eH c-1.(628). Foreven thestrongestfields which areavailable inthelaboratory,the value ofthe lastterm inthisequationissmallcomparedwith that ofthe other terms, sothatthesolution ofequation (628)maybetaken tobe eH Theoriginalvibrations oftheelectron,alloffrequency p,maynowbe replaced bythethreefollowingvibrations : I.x=Acos(pt— e), 2/=0,z=0. II.x=0,y=Aicos III.x=0,y—A2cosp+eH 2mC. eH\t (p-hrc)1-'*z= J.1sin z=—J..2sinp+ PeH 2mC eH 2mC.t t-e. Vibration Ioffrequency pisalinear motion oftheelectronparallel toOx,thedirection ofthelines ofmagneticforce. Themagneticforce in theemitted radiation isaccordingly always paralleltotheplaneofyzand vanishes immediatelybehind andinfront oftheelectron(cf.§618). Thus there isnoradiation emitted inthedirection oftheaxis ofx,andthe radiation emitted intheplaneofyzwillbepolarised (§592) inthisplane. Vibrations IIandIIIrepresentcircular motions intheplaneofyzof eH frequencies p±=—^.Clearlytheradiation emittedalongtheaxisofxwill becircularly polarised,while thatemitted intheplaneofyzwillbeplane polarisedinaplane throughthelineOxandthelineofsight (themotion alongtheline ofsight sendingnoradiation inthisdirection). Thus the observedappearancesareaccounted for. 635. Theanalysis justgiven explainstheobserved facts ofthenormal Zeeman Effect, butonlyinterms ofamodel which isknown nottobein accordance with theactual facts ofatomic structure. Aswaspointedoutby Larmor, theexplanation justgivencanbeeasily generalisedsothattheatomic model shall atleast accord better with thefacts ofnature than thatwehave justhadunder discussion. 566 TheMotion ofElectrons[ch.XIX Ifanelectron ismovinginafield ofmagneticforce ofintensityHparallel totheaxisofx,itsequationsofmotion willbe d-X=Fm •(629),mdf-~^ dry -r,eHdzmcd=F»+uit d2z_peHdymdP~z~Gdt) whereFx,Fy,Fzarethecomponentsoftheforcewhich actsontheelectron apartfrom thesuperimposed magneticfieldH.Theseequationsofcourse containequations (627) asaspecialcase. Ifx,y,zwere coordinates measured with reference toasystemofaxes rotatingwithuniformangular velocitycoabout theaxisofxinthedirection fromOytoOz,thecomponentofthevelocityofthepoint x,y,zinspace would begiven by .(630), 635,636] TheZeemanEffect 567 theequationsofmotion(629)oftheelectron inthesuperimposed magnetic fieldbecome m lTt=F*' UUTPmT^=Fy, mnt=F" which would bepreciselytheequationsofmotion oftheelectron referred to axes atrestwith themagneticfield non-existent. Thus thesuperposition ofthemagneticfieldparalleltotheaxisofxisseen tohavehadpreciselythe same effect onelectronic motion asthesettingoftheaxes inrotation with anangular velocity<odefinedbyequation (633). Before themagneticfield issuperposed,lettheelectron describe apath such thatwhen itscoordinates areresolved intosimple-harmonicterms by Fourier's theorem, oneoftheconstituentsimple-harmonicvibrations isof theform x=0,y=A1cos(pt—ej), z=Axsin(pt— e^). Assume thatoneofthelines inthespectrumoftheatomwhen initsnatural statecorrespondstoafrequency pj2ir. Thesuperpositionofamagnetic fieldhasthesame effect onthecoordinates x,y,zasthesettingoftheaxes inrotation withanangular velocity co,sothatwhen this field issuperposed thecoordinates oftheelectron maybetaken tobe x=0,y=A1cos[(p+(o)t—d], z=A^in^p +^t—e2J. Itisatonceseen thatthevibration isidentical ingeneral typewiththe vibration IIthatwefound in§634, sothatthediscussion ofthepolarisation andchangeoffrequencytheregivenwillapplytothepresentcase also. 636. The discussion ofthe last section isapplicabletoanyelectron describinganorbit such that itsmotion canberesolved into oscillations of definitefrequencies.Itshews thateachspectrallineoughtingeneraltobe resolved intoatripletofthreeequidistant lines, alineinitiallyatpgiving placetolines atp±8pwhere bp=^tch<634>- Thisrepresents what isnormally observed, andaformation oftripletsof thistypeiscommonly spokenofasthenormal Zeeman Effect. Certain lines separateoutinamorecomplex wayinthepresenceofamagnetic field, these lineshaving generally appearedasmultiplelines (doublets, triplets, etc.)even before themagneticfieldwasturned on.This isknown asthecomplexor abnormal Zeeman Effect, and isnotcoveredbyoursimple theory. 568 TheMotion ofElectrons[CH.XIX Inthenormal Zeeman Effect, thefrequency difference, predicted by- equation (634),isconstant forallthelines ofthespectrum. Observationally this isfound tobethecase,andequation (634) makes itpossibletodetermine avalue ofejmfrom theobservedseparationofspectrallines inamagnetic field ofknownstrength. Thevalue soobtainedprovestobeingood agree- ment with values forejmmeasuredbyother andmore direct methods. TheGeneral Equations ofMoving Electrons. 637.Wenowreturn tothegeneral equationsof§621, namely 4W df\ dry d/3 ,nnw .(636),dy Gdt~dydz' anddiscuss thefield setupbythemotion ofelectriccharges when there is norestriction astothesmallness oftheir velocities. Onmultiplyingboth sides ofequation (635) by/xanddifferentiatingwith respecttothetime,weobtain Gdt\pU+dt)~dt\dy~dz)' Usingrelations (636)wereadilyfindthattheright-hand member \_dy\dx dyJdz\dzdxj^ =Cv*x _a/SI3FdZ\ dx\dxdydz) Putting 47r/"=KXand-^+^—+-^=4nrp,thisbecomes V2Xdx Kud'Xdz 47T/id C*dt2C*, . 4-7Tdp 638. This isthedifferentialequationsatisfied byX.Similarequations areofcourse satisfied byFand Z.Ifwedivide both sides ofequation (636) by /j,anddifferentiate withrespecttothetime,wereadilyfindthatasatisfies thedifferentialequation Kfid'a4ttV2a-C2dt* G!<P*)-!<pr)]. Weshallshortlyobtain these differentialequationsinanotherway. 636-640] General Equations 569 Introduction ofthePotentials. 639.Withequations (636)wemaycombine therelation da db dc_ /ri>7\ dxdydz'' (equation (616)), and itfollows, asin§443,thatwecanfindavector-potential ofcomponents F,G,Hconnected with a,6,cbytherelations -?-£.«*<«»> andwithX,Y„Zbytherelations(cf.§530) x4f=-^-etc<639>- inwhich^isafunction, atpresent undetermined inthegeneral case,which becomes identical with theelectrostaticpotential when there isnomotion. 640.Wehave seen(§442) thatequations (638)arenotadequateto determine F,0,Hcompletely,andhence \Palso(cf.equation (639))isnot fullydetermined. LetF,G,H0yyVQbeanyspecialsetofvaluessatisfying equations (638) and(639). Then themostgeneralvalues ofF,G,Haregiven by(cf.§442) F=F+d ^,etc(640), where%isanyarbitrary single-valued function. Tofindthemostgeneralvalue ofW,wehavefromequation (639) dx+G\dt dxdtj dxGdxdt' sothat,onintegration, 13v^=^--,^+aconstant(641). From (640) and(641)weobtain dx+ dydz+Gdt dx dydz+GdtXG3dt* (642). Thefunction %isentirelyatourdisposal,sothat xC2dt* mayhaveanyvaluewepleasetoassigntoit.Letusagreetogivetoxsuch avalue, foreveryinstant oftimeand allvalues ofx,y,z,asshallmake the right-hand member ofequation (642) vanish. 570 TheMotion ofElectrons[oh.xix Thevalue of^isnow fixed, exceptforasetofvalues of^such that xc* dt* ateveryinstant andpoint,these values of^representingofcourse contribu- tions thatmightarise from asetofdisturbancespropagated throughthe medium from outside. Exceptforsuch additional values of%,thevalues ofF,G,H,tyarenow uniquelydetermined byequations (640) and(641). Thevector-potential willinfuture mean thespecialvector ofwhich these values ofF,G,Hare thecomponents,while thecorresponding specialvalue of^willbecalled the "Electric Potential." Fromequation (642)itfollows thatthevector-potential andtheelectric potentialareconnected bytherelation dFdGdH^^dV dx+ dy+dz~ adt(b4d> Differential Equations satisfied bythePotentials. 641. Ifwedifferentiateequations (639) withrespecttox,y,zandadd, weobtain /axbyd_z\id/a?do,atf\ VnF \dx+dy^dz)+Gdt\dx^ 8*/dzj~' which, onsubstitutingfromequations (643) and(639), becomes V*&dt'K(b*^' thedifferential equationsatisfied byM*.Wenotice that forasteadyfield it becomes identical with Poisson'sequation,while inregionsinwhich there are nochargesitbecomes identical with theequationofwave-propagation. 642.Toobtain the differentialequationsatisfiedbyF,wetransform equation (635)bytheuseofequations (638).Wehave 4nra f df\ dcdb dy\dx ayJdz\dz dxj mIP*+*Jt+*?)-?*dx\dx dydzJ whence, fromequations (643) and(639),w-*"-^-- <** the differentialequationsatisfiedbyF.Similarequationsareofcourse satisfiedbyGandH. 640-645] General Equations571 Differential Equations satisfied bytheForces. Kixd? 643. Operatingonequation (639) with theoperatorV'2—-~-y-2,we have KnffiX_ ldfr72Z,KpffiF\d(^ JrKf±cM> C2dt2 ^x_K^d^__l^(Kd,d^\_d_f C2dt2Gdt\ C2dt2)dx\ =^>>+l?l ^- This isthedifferentialequationsatisfied byX,andsimilarequationsare satisfied byYandZ.These sameequationswere obtainedbyamore direct method in§637. 644. Forthedifferentialequationsatisfied by a,/3,7wehave, from equations (638) and(645), G2dt2/A G2dt2)\dy dz)— ~CVTy W\(647)' andsimilarequationsfor/3and7.Theseequations agreewith thosealready obtained in§638. SolutionoftheDifferential Equations. 645. Itwillbeseen that allthedifferentialequationsareofthesame general form,namely Vs*-^=-4™<648>- where crarises from electriccharges,atrestorinmotion. Clearlythevalue of^mayberegardedasthesum ofcontributions from thevalues ofainthedifferent small elements ofvolume. Thesimplest solution for%isthatarising from adistribution of cratand close tothe origin,crbeingzeroeverywhereelse. Forthisspecialsolution %isafunction ofronly,which mustsatisfy xa2dt2 everywhere exceptattheorigin. Proceedingasin§578,andrejectingthe term whichrepresents convergent waves, ashavingnophysical importance, weobtain thesolution(cf.equation (536)) 'x=lf(r-at)(649), wherefissofaraperfectly arbitraryfunction. 572 TheMotion ofElectrons[ch.xix Close totheorigin,thisreduces to %-;/<-«*)(650), and itnowappearsthat inequation (648) themiddle termbecomesinsig- nificant near theoriginincomparisonwith the firsttermV2 ^.Thus close totheorigintheequationbecomes identical with Poisson'sequation, andthe integralis \<rdxdydz//.X=-^~ = ;(651), where theintegralistakenonlythroughtheelement ofvolume attheorigin inwhich crexists, andtrepresentstheintegralofatakenthroughthis element ofvolume. Oncomparingsolutions (650) and(651),both ofwhich aretruenearthe origin, wefindthat /(-aO-r (652), and thisdetermines thefunction /completely. Thegeneralsolution(649) isnowfullyknown, andbysummation ofsuch solutions thegeneralsolution ofequation (648)isobtained. LetP,Qbeanypointsdistant rapart;let tbeanyinstant oftime,and let tQdenote theinstant oftimer/aprevioustoit,sothat t=t—r/a. Clearlytistheinstant ofdeparturefromPofadisturbancereaching Qat t. Wemayspeakoftasthe"retarded time" atPcorrespondingtothetime tatQ. With thismeaning assignedtot,wehave f{r-at)=f\-a{t--^=f(-at«)=r, where risevaluated attime t(cf.equation (652)).Ifweagreetodenote by[<£]thevalue of</>estimated attheretarded time atthepointatwhich<£ occurs, then thisvalue oftwillbeexpressed by[t],and solution(649) becomes %=V(653). Themostgeneralsolution ofequation (648), obtainedbythesummation ofsolutions such as(653),is xrrrM^ =2[i](654) _ thelastformapplying when thedistribution ofaoccursonlyatpointsorin smallregionssosmall that thevariations oftheretardation oftimethrough eachregionarenegligible. TheanalogyofPoisson'sequationand itssolution inelectrostatics(cf. §§49,40,41)isobvious. 645-647]General Equations 573 646.Fromequations (644) and(645)itfollows that thepotentialsare given by *4///[«^(655), *-g///kg]<^y* .e* (656). Thesepotentialsarecommonly spokenofas"Retarded Potentials."They differ from theordinary potentials,inwhich thefinitevelocityofpropagation isnottaken intoaccount, onlyinthatthequantitiesinthenumerators must beevaluated attheretarded timesappropriatetothepoint. Thesolution ofequations (646)and(647)maybesimilarlywritten down, but itisusuallyeasier toevaluate the forces bydifferentiation ofthe potentials. Ifthemovingelectrons informula (656) areconveyingcurrents inlinear circuits, theformula becomes (ontaking p.=1) where thesummation isover the different circuits and ixdenotes the doc ^-componentofthecurrent, which mayalsobeexpressedasi-r-. This formula maybecomparedwith(419), fromwhich itdiffersonlyinthat it takes account ofthe finite timerequiredforthepropagationofelectro- magneticaction. TheField setupbyMovingElectrons. 647.Anelectron isachargeoftotalamount espread throughavery small volume. When weattempttoapplytheequations justobtained tothe motion ofelectrons, acomplicationarises.Wemust notintegrate porpv throughthespace occupied bytheelectron because theretarded time varies from onepartoftheelectron toanother. And thiscomplication does not disappearevenwhenwepasstothelimitandsupposetheelectron tobeof infinitesimal size. Lettheelectron bemovingwith avelocity (not necessarily uniform)of which thecomponentsatanyinstant areu,v,w.Suppose wewish to evaluate thepotentialsatx',y',zattime t. Let x,y,zbethepositionofanyelement oftheelectron attheretarded time t,definedby t=t-- where r2=(as'-x)2+(y'-y)2+(z'-z)\ Qj Wemayspeakofx,y,zastheeffectivepositionoftheelement ofthe electron under consideration, since theelement contributes tothepotentials weareinsearch of,onlywhen itisatx,y,z. 574 TheMotion ofElectrons[CH.XIX Theretarded time twillbedifferent fordifferentpartsoftheelectron. Let itsvalue atthecentre oftheelectron be6 .Letthepositionofthe element under consideration attime8bex0)y,z .Then theelement which isatx,y,zQattime 6hasmoved tox,y,zbytime t,sothat x=x+u(t-#o)+§u(t o-0o)2+...etc., where U,v,wrefer tothevelocityoftheelectron attime 6 . Rememberingthat tisafunction ofx,y,z,weobtain ondifferentiation withrespecttox, dr dt andsimilarly, lr-§{*+*&-*•> +lHt-ey+...}. Those elements oftheelectron which have their effectivepositionsinside asmall element ofvolume dxdydz occupyatthefixed time 6anelement of volume dxdydz .The ratio ofthese elements ofvolume isgiven bythe usual Jacobian determinant dxdydz dxdydz 647] General Equations 575 where allquantitiesareevaluated atthetime t=O,orsince fffpda} dyodz=e, e^= K1-*a wheresquarebracketssignifythat thequantityinside istobeevaluated attheretarded time asestimated attheelectron.Similarly equation (656) becomes* F= ~Cu v ir ar Supposeitisrequiredtocalculate the field atapointattime t.Let Ebethepositionofoneoftheelectrons inthe p £ field atatime tsuch that r t=t—— ,a where r=EO.Then thequantitiesinsquare brackets must becalculated forthiselectron in thepositionEatthetime t. Letthevelocityoftheelectron atthetime tQ6 beVinadirection EFmakinganangle6withFi8-139 - EO,and letEFbethedistance V(t—1)which theelectron would describe bythetime tifitsvelocityremained constant. IfFG istheperpendicularfromFontoEO,theintercept EG isgiven by EG=EFcos6=Vcosd(t-t). NowYcos6issimplythecomponent Yrofvelocity along EO,while t-t=rfa.ThusEG=rVrjaand VjOG=r-EG =r1aj Theformulae forthepotentialsnowbecome 1eV=KOG COG' IfsquaresofY/Careneglected,theangleFOE infigure 139 isasmall angleandOG isapproximately equaltoOF.Thus asfarasterms ofthefirst *These formulae for&andFwere firstgiven byLienard(L'Eclairage Electrique, 1898) and E.Wiechert (Arch. Neerland. 5,1900, p.549). Ourproof hasfollowed closely themethod given byLorentz (Theory ofElectrons, p.254) ;analternative proofisgiven bySchott (Electromagnetic Radiation, 1912, p.22). [CH.XIX 576 TheMotion ofElectrons order inVJCthepotentialsare KOF' COF' whereFisthepositionoftheelectron attheinstant atwhich thepotentials areevaluated, exceptforacorrectionarisingfrom accelerations orsudden changesinthemotion oftheelectron. 648. Inthecaseinwhich u,v,waretreated assmallwecanalsowrite down thepotentials directlyfromequations (655)and(656). Forinthiscase dxdydzbecomesequaltodxdydzandtheequations assume theforms 0] F_/4^T ]^~~n~ >•\Jr_ Kr' C where risthedistance fromthepoint x,y',z'atwhich theforces aremeasured totheeffectivepositionoftheelectron. Thus themagneticforces aregiven by a~a[dy' dz')-c{dy' r dzrJ'etC{b°7)' fj>\dy'dz'JG\dy Since [ew]isafunction oft—r/a,wehave 9r, 1 orL Ja sothatdt(ew)=--[ew],a 3[ew]_y'—yd[ew]_y'—y fl[ew] _[ew] } dy'r rdrr r[ar r'1 andonsubstitution inequations (657)weobtain formulae fora,j3,y. These formulae areseen tocontain terms both inr_1andr~\Atagreat distance from theelectron theformer alone areofimportance, andthecom- ponentsofforcebecome a=— y-y[ew]-Z —^-[ev]\,etc.aC ( Similarly wefind fortheelectric forces atagreatdistance X=-£C^],etc.(658). O.(659). Forasingleelectron moving alongtheaxis ofxwithanacceleration u, infreespaceforwhich/i=K=1,thecomponentsofforceassume thesimple forms .(660), thesebeingaccurateonlyatgreatdistances from theelectron. 647-650] Radiation ofEnergy 577 Radiation ofEnergy. 649.Wesawin§576that theflow ofenergyacross anyclosed surface isgiven by jj(lUx+mUy+nUz)dS (661), where n,=^.(F7-Z/3),etc. Inprovingthistheenergywasassumed tobelocalised inthemedium in thewayimagined byMaxwell, but ifweidentifyourclosed surface with a sphereatinfinitythisassumptionisnolonger necessary.Forindependently ofthisassumption,thetotalenergyinthewhole ofspaceisgiven by T+W=jjj\~(X2+F2+Z*)+^(a2+/32+74dxdydz andfrom thiswecandeduce formula (661) directly. Onassigningtoa,/3,y, X,Y,Zthevalue obtained inequations (660)fortheforces from asingle electron, wefind n*=o,uy=-^Xy, n,=^x/3, IUX+mUy+nUz={y'~y^rZf(^)a > whence theflow ofenergyacross asphereofinfinite radius isreadily found tobe ?** (662) This isLarmor's formula fortherate atwhich asingle movingelectron radiatesenergy. Wenotice thatasteady velocity ucontributesnothingto theradiation;energyisradiated awayfromanelectron which isundergoing acceleration butnotfromoneinsteadymotion. Itmust beadded that thenewdynamicsreferred toin§620seems to throw doubt onthisformula foremission ofradiation. Many physicists now question whetheranyemission ofradiation isproduced bytheacceleration of anelectron, except under certainspecialconditions.Bearingthiscaution in mind,wemayproceedtoexamine some oftheconsequencesoftheformulae justobtained. 650. Ifeach ofacluster ofelectrons issonear tothepoint x,y,zthat differences ofretardation oftimemaybeneglected throughoutthecluster, theradiation from thecluster iseasily seen tobethesame asthatfroma singleelectron ofchargeEmovingwithcomponentsofacceleration U,V,W, such that EU=%eu, etc. J. 37 578 TheMotion ofElectrons[en.xix Thecondition thatthere shallbenoradiation from suchacluster is 2eu=Sev=2ew=0. Ifthis condition isnot satisfied, therate ofemission ofradiation is (cf.formula(662)) ^s{(teuy+(Zevy+($ewy} (663). 651. Consider next thefieldproduced byaparticleofchargeEoscillating alongtheaxisofxwithsimpleharmonic motion, itscoordinate atanyinstant beingxcospt.Wehave Eu=- Ep^xocospt;[Eu]=-Ep"xcosp[t— -), andthefieldcanbewritten downbysubstitution informulae(660). From formula (662) theaveragerateofemission ofradiation isfound tobe 1p'E'a:* _16tt*E2x*C 3C3~~~3V where Xisthewave-lengthoftheemittedlight. Aparticle movinginthiswayisspokenofasasimpleHertzian vibrator. Itsmotion wastaken byHertz torepresenttheoscillatingflow ofcurrent in anoscillatory dischargeofacondenser. Such anoscillation formed thesource ofthewaves inHertz'soriginal experiments (1888), andforms thesource of thewaves used inmodern wirelesstelegraphy. 652.Acaseofgreatinterest isthat inwhich thevelocityofamoving electron undergoesaverysuddenchange,such aswould occurduringa collision with matter ofanykind. Letusrepresentsuch asuddenchange bysupposingthat eu,ev,ewvanishexcept throughaverysmall interval surroundingthetime t=0,duringwhichtheyarevery great. Atapointat distance r,[eu], [er]and[ew~\willvanishexcept throughasmall interval oftimesurroundingtheinstant t=r/a.Duringthis short interval, the electric andmagneticforces willbevery great;before andafter thisinterval theywillhave thesmaller valuesarisingfrom thesteadymotion ofthe electron. Thus thesudden check onthemotion oftheelectron results in theoutwardspreadofathinsheet ofelectric andmagnetic force, theforces being veryintense butonlyofbrief duration. Theradiation which isemitted whenrapidly movingelectrons impingeon matter isgenerallycalled X-radiation orRontgen-radiation.Itwassuggested byStokes that thisconsists ofthinsheets or"pulses"ofelectric andmagnetic force ofthetypewehavejust investigated. Althoughthere isnodoubt that this istrue inageneral way, yetthegrowthofthenewd}rnamicsalready referred tohasmade itclear that there isfarmore intheproblemof X-radiation thancanbeexplained bythetheories ofMaxwell andStokes. 650-653] Forces onMoving Charges 579 Mechanical Forces onMoving Charges. 653.Whether weassume Maxwell's localisation ofenergyinthemedium ornot,thetotalenergyofanelectromagnetic field, aswenoticed in§649, willbeT+W,where W=[\[^{X*+Y*+Z*)dxdydz (664), T=\[[^(^+^+^)dxdydz (665), andtheintegralsextendthroughthewhole ofspace. Letussuppose that,onaccount oftheelectromagneticforces atwork, eachelement ofcharge experiencesamechanical force ofcomponents E,H,Z perunitcharge. Wecanfindtheforces H,H,Zbythemethods of§196and thegeneral principleofleast action. Letusimagineasmalldisplacedmotion inwhich thecoordinates ofany point x,y,zaredisplacedtox+8x,y+8y,z+8z,while thecomponentsof electricpolarisationarechangedfromf,g,htof+8f,g+Sg,h+8Ii,these newcomponentsofpolarisationaswell astheoldsatisfyingrelation(615). Thus ifpisthedensityofelectricityatanypointintheoriginal motion and p+8pthecorresponding densityinthedisplaced motion, wemust have dx dydzr' o8f d8gd8h_g dxdydz"' Letusdenote thetotalworkperformed -bythemechanical forces inthis smalldisplacement by—{811} (cf.§551),sothat {8U\= ffJp(B8x+R8y +ZSz)dxdydz (666). Then theequationsofmotion arecontained in(cf.equation (507)) l\8T-8W-{8U})dt^O (667).Jo Wehave 8T=~ ffj{a8a+b8/3+c8y) onapplyingGreen's Theorem;andonfurtherusing equation (635), this becomes £r =c///ii's ('>t'+l)+es ('3,'+l)+/f8 ('J"'+S)}"^ 37—2 580 TheMotion ofElectrons[ch.XTX Let 8,-j-refer toapointfixed inspace, and letA,y-refer toapoint movingwiththemovingmaterial. Thenwehave thetwoformulae forAu, Au=-^8x=-j-8x+u«—8x+v^-8x+w— 8x,JJt dt oxoyoz ~du -.du*du«sAu=8u+^8x+~8y+-7r8*,3# 3y* 3.2 sothatoncomparison 8u=-r:8x+u^8x+v7-8x+w^-8x at oxoyozdusdu^ ,3*7. dx-Bx+dy^+ dz-Sz )• Wenowhave d 8(pu+f)=u8 P+p8u+Jt8f d=u8p+^(p8x+8f)-8xd£ (3 ,3 3\ r. /3c/ *3^7. 3f/' \ Onsubstitutingfordp/dt and8ptheir values(cf.§622) andsimplifying, weobtain *(pu+f)=^0>&s+8/)+1(prS*-Pf%)-1(ptffc -pir&O,dt whence 8T= ~ fffFJt(P%x+¥)dxdydz +terms inG,H +CIIIF^(Pv%x—puty)—^(Pu^z~Pw%x )\dxdydz +.... Transforming byGreen's Theorem, thesecond linein8Tbecomes clll{{dy~~ Tz)(Pw8V~Pv82)+ •••}dxdydz —-^\\\{p8x (cv—bw) +p8y(aw—cu)+p8z(bu—av)} dxdydz. Onintegratingwithrespecttothetime,andtransformingthe firstterm onintegration byparts,wehave 7 8Tdt=f dt-^!jjd~(p8x +8f)+p8x(cv-bw)+... dxdydz. 653] Forces onMoving Charges 581 Wehave from variation ofequation (664), 8W= jfj(X8f+Y8g+Z8h)dxdydz. Hence, freed from theintegrationwithrespecttothetime, equation (667) becomes clil—- rff(p&x+8f)+P&x(cv-bw)+•••dxdydz -fjf(X8f+Y8g+Z8h)dxdydz -fjfp(B8x+H8y+Z8z)dxdydz=0 (668). Wemaynotequatecoefficients ofthe differentials, for8f,8g,8hare notindependent, beingconnected by d8f ,B8g d8h _ 3 ,.xd, *,.d , ->.&+^+"^=sp=~35(/,&)"^(p8y)"ai(^z)- Wemultiplythisbyanundeterminedmultiplier ty,afunction ofx,y,z, andintegrate throughallspace.Weobtain or,afterintegration byparts, Addingthisintegraltothe lefthand ofequation (668), wemayequate coefficients, andobtain X=-C-dt-dx->etC (669)' ldFdV 1,.N*=-Cdi-dx-+c(CV-bw) =X+hcv-bw),etc (670). The firstequationissimply equation (639),ofwhich wehavenowobtained aproofdirect from theprincipleofleast action(cf.§575) ;thesecondgives usthemechanical forcesactingonmoving charges.Itwillbeseenthatthe forcesgiven byformula (670)areidentical with those obtained in§629,but theyhavenowbeen obtained without anylimitation astothesmallness or steadiness ofthevelocities. 582 TheMotion ofElectrons[ch.XIX Stresses intheMedium. 654.Wecannext evaluate thestresses inthemedium, followingthe method of§193andassumingthemedium tobefreeether. LetXbethetotal^-componentofforce actingonanyfiniteregionofthe medium, sothat X=japdxdydz=I\pXdocdydz +p\\livP7'~ftpw)dxdydz. OnsubstitutingforpV,p\vfromequations (635), thelasttermbecomes Onsubstitutingforpfromequation (615), and ford/3/dt, dy/dt from equations (636), andcollecting terms, thisbecomes 1X= b7T79XBYd_Z\x_ \dx dydz)^-<- 654-656] Motion withUniform Velocity 583 perunitvolume, thenequation (672) would becomeexactlytheequationof motion ofthismedium, ifitissupposedtobeacted onbyasystemofstresses defined by PXX=~(X2-P-Z2+a2-/3*-72 )etc.l 8?r I(674). P^=- ii-(ZF+a y8)etc. Thus themechanical action issuch ascanbetransmittedbyamedium inmotion, themomentumperunitvolumebeing given byformula(673). Thevector whosecomponentsaregiven byformula (673)iscommonlycalled the"electromagnetic momentum." Weseethat itisofamountequaltoI/O2 times thePoynting Flux, andinthesame direction. Foranelectrostatic ormagnetostaticfieldexisting alone, the electro- magnetic momentum vanishes, andthestresses reduce tothosepreviously found in§§193and471. Motion withUniform Velocity. 656. Letusagainreturn tothegeneral equations, andexamine thespecial formtheyassume forasystem movingwithuniformvelocity.Thismayfor convenience besupposedtobeavelocity uparalleltotheaxisofx. Asin§624wemayreplace-j-by—u-j-andthegeneral equation (648) becomes a?)ox2oyoz2 Letusnowwrite kfor f1 j,andtheequation becomes 7>dx2+dy2+ dz>4?r<7' or,ifwewrite xforkx, S+S+S--4-<6Y5>- Wemayconveniently speakofx',y,zasthe"contracted"coordinates correspondingtotheoriginalcoordinates x,y,z,since iftwosurfaces have thesameequation,oneinx',y,zandtheother inx,y,zcoordinates, the former willbeidentical with thelatter contracted intheratio1/kparallelto theaxisofx. Equation (675)isPoisson'sequationincontracted coordinates. Its solution is fffadx'dydz _fffadxdydz vt X~]JJ F~K Jj!S~=K~r" where rdenotes distance measured inthecontractedspace. 584 TheMotion ofElectrons[ch.xix Hence(cf.equations (644), (645))thevalues ofM*andF,G,Hare given by F=1^Xp,G=H=0•(676), sothatthepotentialsarethesame incontracted coordinates astheywould be inordinarycoordinates ifthesystemwere atrest,multiplied bythefactor k. Motionofauniformly electrified sphere. 657.Toillustrate themethodjustexplained, weshallexamine the field produced byauniformlyelectrifiedsphereofradius a,movingwithvelocityU. Thesurface inthecontractedspaceisasphereofradius a,sothatthat in theuncontractedspaceisaprolate spheroidofsemi-axesica,a,a,andthere- foreofeccentricity u/C.Tofindthedistribution ofelectricity, weimagine thechargeonthespheretobeuniformly spread between thespheresr=a andr=a+e,where eisinfinitesimal. Thechargeonthespheroidisnow seen tobeuniformly spreadbetween thespheroiditself andanother similar spheroidofsemi-axes k(a+e),a+e,a+e.Thus thedistribution ofelectricity inthespheroidintheuncontractedspaceisjustwhat itwould beifthe spheroidwere afreely charged conductor, and isgiven bytheanalysisof §§283,284. 658.The field hasbeen discussed indetail bySearle* andAbrahamf. The electric andmagnetic energiesWandTarefound tobegiven by we2[SC*-U\ G+u T-f- f°2+°* 1G+U 8a 1CugC-u 2a[u°C—uj while thetotalelectromagnetic momentum Ginthewhole ofspaceisgiven by e2(Q*+u2C+u 2 Va\Cu^gC^u u thisdirection ofGbeingofcourse that ofU. Motionofanysysteminequilibrium. 659.When amaterialsystem moves withanyvelocity u,theelectric field produced byitschargesisdifferent fromthefieldwhen atrest.Thedifference between these fieldsmustshew itself inasystemofforces which must acton themoving systemandinsomewaymodifyitsconfiguration. *Phil. Trans. A,187(1896), p.165. tPhys. Zeitschrift,5(1901), p.576, orTheorie derElektrizitat (2nd ed.), p.165. 656-660] Electromagnetic Mass 585 Letusconsider firstasimple systemwhich weshall callSinwhich all theforces areelectrostatic, and allthechargesaresupposedconcentrated in points (e.g. electrons). Letussupposethatwhen thesystemisatrestthere isequilibrium when achargeexisatx=xx,y=yx,z=zx\e2atx=x2,y=y2, z=z2,andsoon. Letuscomparethiswith asecondsystemS'consistingofthesame electrons butmovingwith auniformvelocity u,andhavingthechargesex atx=x1}y=yltz=zx\e2atx=x.2,y=yi,z—zi,etc., sothateach electron hasthepositioninthecontractedspacewhichcorrespondstoitsoriginal positionintheoriginal space. Then ifVdenotes theelectrostaticpotential intheoriginal system,thepotentialsinthemoving systemare(cf.equations (676)) W=KV,F=^%G=0,H=0, andtheforces inthemoving systemare dxCdt _?WvdF dxCdx _a^/K/iv*\ _id_v dx\ C2)~KOX' T=—^-=—k^- ,etc. dy dy Wenotice thattheelectrostatic forces inS'arel//ctimes those inSas regardstheir^-components,butktimes those inSasregardstheiry-com- ponents.Asaspecialcasewenotice that ifthesystem Swasinelectrical equilibrium,then S'will alsobeinelectricalequilibrium,sothat asystem which isinequilibrium when atrestcanregain equilibriumafterbeingsetin motion withvelocity ubycontractinginaratio1/k'. Electromagnetic Mass. 660. Consider acharged body,which willultimatelybeidentified withan electron, movingwith auniformvelocity Uparalleltotheaxisofx.Letus first consider thesimplecase inwhich uissosmall thatu2/C2maybe neglected. Themoving chargecreates amagneticfield. Ifthecharged bodyis supposedtobeasphereofradius a,whose surface isuniformlyelectrified to atotalcharge e,then there isnofield inside thesphere, andthecomponents ofmagneticforce outside thespherearegiven by ao uez UeV 586 TheMotion ofElectrons[ch.xix Ifweassume localisation ofenergyinthemedium, then atadistance r greaterthan afrom thecentre ofthespherethere willbemagnetic energy perunitvolume ofamount 1/,no ,ne*U2sin2e where 6denotes theanglebetween theradius randtheaxis of oc.Oninte- gration,thetotalenergyofthismagneticfield isfound tobe S//^^^*=§^(6"). This result isofcourseonlytrueprovidedwesupposetheenergytoreside inthemedium asimagined byMaxwell. Inthis case theenergy, being magnetic,must besupposedtobekinetic energy. Thus ifthecharged bodyissupposedtobeofmassm,thetotal kinetic energyofitsforward movement willbe 4("'«+!^>'<678>' inwhich the firstterm arises from theordinary mass ofthebodyandthe second from thekineticenergyofthemedium. Ananalogy fromhydrodynamicswill illustrate theresult atwhich wehave arrived. Suppose wehave aballoon ofmassmmovinginairwith avelocityvanddisplacing a mass m'ofair. Ifthevelocityvissmall comparedwith thevelocityofpropagation of waves inair,themotion oftheballoon willsetupcurrents intheairsurrounding it,such that thevelocityofthese currents willbeproportionaltovatevery point. Thewhole kinetic energyofthemotion willaccordingly be £(m+J/)v2 , thetermkmv2being contributed bythemotion ofthematter oftheballoonitself, andthe term£Mv"bytheaircurrents outside theballoon. Thevalue ofMiscomparable withm', themass ofairdisplaced—forinstance iftheballoon isspherical, and ifthemotion ofthe airisirrotational, thevalue ofMisknown tobe\m! (cf.Lamb, Hydrodynamics, §91). 661.Strictly speaking,formula(678)istrueonlywhen uremainssteady throughthemotion. Anychangeinthevalue ofuwillbeaccompanied by magneticdisturbances intheether whichspreadoutwithvelocity Gfrom thesphere. Anexamination ofintegral (677) will,however, shew that the energyisconcentrated round thesphere—theenergyoutside asphereof radiusRisonlyafraction a/Rofthewhole, and ifRistaken tobealarge multipleofathismaybedisregarded.Thetimerequiredfortheenergyto readjustitself after achangeofvelocityisnowcomparablewithRJC. Thus ifweexclude suddenchangesinu,and limit ourattention to gradual changes extendingoverperiods great comparedwithRjG,wemay takeexpression (678)torepresentthekineticenergy,both forsteady and variable motion. Theproblem gainsallitsimportance from itsapplicationtotheelectron. Forthis aisoftheorder of2x10~13cms. (seebelow, §6G6),sothat allexcept onepercent, ofthe 660-662] Electromagnetic Mass 587 magnetic energyiscontained within asphereofradiusR=2x10~ncms. SinceC=3x1010 , thetime ofreadjustmentofthisenergyis*66x10~21seconds, aninterval small enoughtobe disregardedinalmost allphysical problems. 662.We shallnow consider thesameprobleminadifferent manner, and shallremove therestriction thatu/Gistobeasmallquantity. The electron will stillbesupposedtomove with auniformvelocity u,v,wwhich maybeofanyamount. The fieldarisingfrom itsmotion maybecalculated asexplainedin§647. Solongastheelectron hasnoacceleration, theforces X,Y,Z,a,fi,7falloffatinfinityas1/r2 ,sothatthestresses defined by equations (67-i)falloffas1/r4 . Ifwenowapply equation (672)tothefield ofthesingle electron, allowing theclosed surface Storecede toinfinity,theequationbecomes X=-±j tffJnxda !dydz (679), where theintegralistaken throughthewhole ofspace.HereXwillnow representthe^-componentoftheponderomotiveforce ontheelectron from thefield setupbyitsownmotionthroughtheether. When theelectron moves withuniformvelocity,theintegralontheright retains aconstant value. Inthis caseX=Y=Z= ;there isnoresultant forceactingontheelectron from theether. Nowsupposethattheelectron hasnotonlyavelocity u,v,wbutalsoan acceleration U,v,w.The forces X,Y,Z,a,(3,ynowcontain terms in1/r, butthese depend onlyontheaccelerations. When thesurface Srecedes to infinityinequation (672),thesurfaceintegralswillnolonger vanish, butwill contain terms dependentonthesquaresandproductsoftheaccelerations. Ifwesupposetheaccelerations tobesosmall that theirsquares andproducts maybeneglected,thenequation (679) remains trueeven foranaccelerated electron. Wehave seenthat Ila.willdependonthevalues ofu,v,w,u,etc.,both attheinstant tunder consideration andalsoatprecedinginstants. Thuswe mayingeneral supposethat jj-Al\Uxdxdydz=f x{u,v,w,u,... u,...etc.). Each sideofthisequation representsthe^-componentofelectromagnetic momentum, andequation (679) assumes theform •dfx.dfx.dfx ••dfx du dv dw du(680). Itisclear thattheforceXwilldependonalltheaccelerations andtheir differential coefficients withrespecttothetime. 588 TheMotion ofElectrons [ch.xix Consider first thecase inwhich alltheaccelerations aresteadyand so small that theirsquares maybeneglected.Thenif,Vetc. allvanish and equation (680) reduces to X=6u dv dw.(681). Ingeneral dfx/duetc.maydependonU,V,w,but ifweagreethatsquares ofu,v,wmaybeneglectedincalculating X,thenwemaycalculatedfx/du etc.onthesuppositionthat U,v,wallvanish. Inother wordsfxetc.maybe calculated asifthemotion weresteady. When themotion issteadythewhole electromagnetic momentum Gis clearlyinthedirection ofthemotion and itsamount willdepend onlyonC, where C*=u'1+V2+w2 .Thuswemayput f=?G whereGisthewholeelectromagnetic momentum inthewhole ofspace,a function ofconly.Ondifferentiation, weobtain y.J.?!M. d£x=uvi(G\ etcduogBcKg)' dv cdc\o/' Nowsupposethewhole motion tobeinthedirection ofOx,sothatc=u, v=w=Q.Thethreeequationssuch as(680)nowassume theforms „ .dG Tr.G„ GX=-vK-,Y=-v~, Z--w-. dG G G When uexists alone, v=w=0,sothatY=Z=0.Thus theelectro- magneticfield exerts aforce ontheelectron inthedirectionoppositetoV. This force isthesame aswould beexerted iftheelectronpossessedanad- ditional massequaltodG/dc.This iscalled thelongitudinal electromagnetic mass oftheelectron. Anelectron ofmassm„willrespondtoaforce inthe direction ofitsmotion inthesamewayasanelectron, unencumbered byan electromagnetic field, ofmass dG dGm+u^ (682). SimilarlyifVexists, alongtheopposingforce oftheelectromagneticfield is—v(G/g). Byasimilarinterpretation, G/ciscalled thetransverse electro- magneticmass. Theelectron willrespondtoaforce transverse toitsmotion inthesamewayasanelectron, unencumbered byamagnetic field, ofmass m^%(683). 662-665] Electromagnetic Mass 589 663.Abrahamsuggestedin1904 that theelectron mightbetreated as arigid sphereofradius a,uniformlyelectrified over itssurface. Ifso,the longitudinaland transverse masses mjandmtwould begiven,from the formulae of§658,by e2C(2uG .C+u\ e2C/C'+uK C+u nu\ e2CfC'2+u\U+u .u\ mt=^¥A~C^l0gG^-u-2 G) 664. Lorentz broughtforward analternative conceptionoftheelectron accordingtowhich itissphericalinshape onlywhen atrest.Theelectricity isnotsupposedtoberigidlyfixed inaspherical configuration,sothatwhen theelectron issetinmotion with avelocity U,itcontracts, inaccordance with thetheorem of§659, intheratio 1:kinitsdirection ofmotion andso assumes theform ofanoblatespheroid. Againstthisconceptionofthe electron Abraham hasbroughttheobjectionthattheoriginalelectron cannot besimplyadistribution ofelectricchargesacted onbytheirownmutual repulsions;there must beother forces atwork tokeepthechargesfrom flying apart. When these other forces aretaken into account, there isno reason forsupposingthatthecontracted electron would beinequilibrium,or ifitwere inequilibrium,that theequilibriumwould bestable. We shall return tothispointlater. Theelectromagneticfield ofLorentz's electron isreadilycalculated bythe method of§656, fortheconfiguration,whenexpressedinterms ofcontracted coordinates, isspherically symmetrical. IfWistheelectrostatic energyofthesystemofchargeswhich constitute theelectron when atrest, itisreadilyfound thattheelectromagnetic momen- tumGofthecontracted electron movingwithvelocity uis *3(72' sothatthelongitudinalandtransverse masses are 4WA (684).m'=3^*3 4Wy 665.Theformulae forthetransverse mass canbetestedexperimentally. Itwasshewn in§631thatanelectron inauniform magneticfieldHwould describe apathofconstant curvature muC/eH,where Uisthevelocity perpendiculartothemagneticlines offorce. Whenelectromagnetic mass istaken intoaccount, minthisformula must bereplaced bym+mt,where misthemass oftheelectron apartfrom itselectromagneticmass.Experi- 590 TheMotion ofElectrons[ch.xix ments todetermine thevariation ofm+mtwith thevelocity were first undertakenbyKaufmann in1906. More recentexperiments byBucherer, Bestelmeyerandothers shew thatm+mtvariespreciselyas(1—u2/C-)~?ork. This isinexactagreementwith thetransverse mass oftheLorentz contractile electron ifmistaken tobezero—i.e.ifthemass oftheelectron issupposed tobewholly electromagnetic. 666. Allexperiments agreeingivingavalue fore/matzerovelocity very nearly equalto1*767x10vinElectromagnetic Units(Bucherer's value). Com- biningthiswith Millikan's value fore,namely 4774xlO-10inElectrostatic Units, wefind forthemass oftheelectron atrest m=9-00x10-28grammes. Themass oftheelectron atrestis,from formulae(684), m-3C,2. Ifthechargeeoftheelectron issupposed spread uniformlyoverthesurface ofasphereofradius a,thevalue ofW,theelectrostaticenergy,ise2 /2a, sothat m=s^<685> inagreementwith formula(678). Inthisequation weknow thevalues of m,eand C,socandeduce a=l-S74xlO-13cms. Thismust betheradius oftheelectron ifitschargeisspread uniformlyover thesurface ofasphere.Ifthechargeisspread uniformly throughthevolume ofasphere,W=Se2/oa,giving m=_JL_ ;a=2-249xl0"13cms. 5aU2 Other distributions ofchargewouldgiveother values forabutalways ofthesame order.Weconclude that thevalue ofaisoftheorder of 2xl0~13cms. TheInternal MechanicsoftheElectron. 667. Letusregardtheelectron asacontractilesphereofradius awhose surface isuniformly chargedwithelectricity. Thenmisgiven byformula (685), andtheelectromagnetic energyoftheelectron, whenmovingwith a velocity U,isfound tobe T=iiiC2(k-~ j+aconstant(686). 665-668] Electromagnetic Mass 591 Supposeanacceleration utooperateforaninstant dt.Since thelongitudinal mass isiuk3 ,theworkdonebytheforceproducingtheacceleration is m/c3vdt dtwhichmaybewritten as-j-(mC'2 /c)dt.Theincrement intheelectromagnetic energy (686) is,however, s^«-h(t)* andthis isnotequaltotheworkdoneontheelectron. Tosatisfytheconservation ofenergyitappearsthat inaddition toits electromagnetic energy Ttheelectron must haveenergy Uofsometype unknown butofamount 1mC2U=-i1-aconstant(687). TD fC ThenT+U=m/c +a,constant, andthework donebyexternal forces is equaltotheincrement ofT+U. 668. Ifachargeeisspreadoveraconducting sphereofradiusa,the forceperunitareaonitsconductingsurface is R=2ira*e2 87ra4' The electron isnotacharged conductor, buttheabove formula makes it clear thattheelectron atrestcould beheld inequilibrium bytheaction ofa normal tensionRofamount e2 /87ra4,perunit area. Poincare* hasshewn that theelectron initscontracted statewould stillbeinequilibriumiftensions of thisamount continued toactwhile theelectron wasinmotion. Now ifvis thevolume oftheelectron atanyinstant, thework doneonthese tensions as theelectron changes shapewillbeRdv.When theelectron ismovingwith velocity U,itsvolume is§7ra3 //c,sothat De21mC*Rv=£—=- . ba/c 4k Thus ifUistaken tobeRvinformula(687), theconservation ofenergywill beexactlysatisfied. There isnoevidence astowhether these tensions doordonotexist; the possibilityoftheir existencesuggestsamechanismbywhich theelectron can beheld inequilibriumatallvelocities, while itsmotion conforms tothecon- servation ofenergy. *Rendiconti delCircolo Matem. diPalermo, 21(1906), p.129. 592 TheMotion ofElectrons[ch.xix TheReaction onanAccelerated Electron. 669.Thewhole forceactingonamovingelectron isgiven byequation (680),inwhich wehave sofarneglectedallterms beyondthose in(J,v,w. Lorentz*hascalculated theeffect oftheterms inu,v,wetc.,andfinds that they giverise toaforceactingontheelectron ofcomponents Fx,Fy,Fz given by 2^=-!^s£etc (688). Lorentz alsogivesformulae fromwhich theremainingterms inequation (680) canbecalculated, butthese terms areoflittlephysicalinterest. The forcegiven byformula (688)mayberegardedasafrictional resistance opposingthemotion oftheelectron throughtheether. The rateatwhich theelectron doeswork toovercome thisforce is uFx+vFy+wFz, sothattheworkdonebytheelectron inaninterval from t=tot=rwillbe 2e2fT -oTTsj(UU+VV+WW)dt. Onintegrating byparts,thisbecomes 2e^ 3sUU+VV+ WW o^^Jo The lasttermrepresentstheradiation emitted bytheelectron ascalculated byLarmor's formula (662); thefirsttermmustrepresent changesintheenergy stored intheether. *TheTheory ofElectrons, p.251. CHAPTEE XX THETHEORY OFRELATIVITY Motion through theEther. TheMichelson-Morley Experiment. 670.When wehavespokenofasystematrestwehave sofarmeant, for allpractical purposes,asystematrestinourlaboratories. But ifwehavebeen rightinconjecturingthat allelectromagnetic phenomena have their seat in theether, then asystematrestwould mostnaturallybetaken tomean asystematrest intheether.Wehave sofarmade noclear distinction between theconceptionsofrestintheether andrestrelative tothewalls of alaboratory. Theviewwasatonetime held thatamoving body dragstheetheralong with it.Ifthiswere atrueview thedistinctionjustreferred towould not arise;abodyatrest relative tothewalls ofalaboratorywould alsobe atrestintheether. Butintime itwasfound that thiswasnotatrueview; itcould notbereconciledsimultaneouslywith results oflaboratory experiments such asFizeau's water-tube experiment (cf.§687below), andwith theastro- nomicaltheoryoftheaberration oflight* (cf.§689below). Finallyitbecame established thattheether, ifoneexisted atall,could notshare inthemotion ofmovingbodies;itmust bestagnant, andmovingbodies mustsimply move throughitwithoutsetting upmass-motions init. The earth'svelocityinitsorbit isabout 30kms. asecond, sothatthe velocityoftheearth relative tothesupposedethermust atsome season ofthe yearbeatleast30kms. asecond. Ifanether exists, theremust beastream ofetherflowing through every laboratorywhich must attain velocities atleast asgreatas30kms. asecond. Startingin1887, Michelson andMorley conductedexperimentswithaview tomeasuringtheactualvelocityofthissupposed stream ofether relative to theirlaboratory, or,what isthesamething,thevelocityoftheearththrough theether. Theprincipleoftheexperimentiseasily explained. Letthe laboratorybemovingwithvelocityuthroughtheether, then arayoflight travelling againstthestream ofether willmove withanactualvelocity Cin theether, andsowillhaveanapparent velocity C—uifmeasuredrelatively *Forafuller account thereader isreferred tospecial treatises—Larmor's Ether andMatter (Camb.Univ. Press, 1900)orCunningham's Relativity (Camb. Univ. Press, 1914). j. 38 594 TheTheory ofRelativity [ch.xx tothemoving laboratory. Similarlyarayoflightmade totravel inthe reverse direction willhave anapparent velocity C+u.Ifaraytravel over apathIand isthen reflected back toitsstarting-point,thetime £2taken willbegiven by t>"ffh+uk-"{1-$T(689)- Supposenext thatarayismade totravel adistance Lacross thedirection ofmotion andback toitsstarting-point,thesystem movingwithvelocityu asbefore. Letthewhole timebe t2,thenthedistance travelledbythesystem isuU.Theactualpathoftheraythroughtheether consists oftwoequal parts,onebefore reflection andoneafter;eachpartisthehypotenuseofa right-angled triangleofsidesLand\ut 2,andthetime ofdescribingeachpart is^t2.Hence %t2C=(L* +%%%*)?, 2Z/, u"-\~^whence i2=^1-J(690). From formulae (689) and(690)itappearsthatthetimes taken byarayof lighttotravel adistance Iandbereflected back, while thelaboratoryisin motionthroughtheether, willbedifferentaccordingasthepathoftherays isalongoracross thedirection ofmotion ofthesystem.Thistime difference admits ofmeasurement byoptical means, andfrom suchmeasurements it oughttobepossibletodetermine u. When theexperimentwasperformednotime difference could beobserved. Theobviousexplanationwould bethat, atthemoment ofperformingthe experiment,thelaboratorywasatrestintheether, butthisexplanation was notfound tobetenable, since notime difference could bediscovered atany season oftheyear. TheFitzgerald-LorentzContractionHypothesis. 671. Fitzgeraldin1893andLorentz in1895suggested independently that thereason whynotime difference wasobserved mightbebecause the arm Ioftheapparatuswhich moved withvelocityulongitudinally through theether wascontracted inaratio(1—u?/C2 )-asaresult ofitsmotion. In such acasethearm Iwould haveshrunk fromaninitiallengthlgiven by measured inthesystem when atrest. Equation (689), expressedinterms of /,nowbecomes h~c[ o\ andsoagreeswithformula (690). 670-673] Motion throughtheEther 595 Thus theFitzgerald-Lorentz contractionhypothesis would account com- pletelyforthenullresult oftheMichelson-Morley experiment. Thehypothesis initself isnotunreasonable, forwehavealreadyseen(§659)thatanelectrostatic systemsetinmotion withavelocityuwouldonlyregainitsequilibriumafter contracting longitudinallyinexactlytheratio(1—u2/C2 )-assumed bythe hypothesis.Itistruethatthearms ofsandstone andpineusedbyMichelson andMorleywerenotpurelyelectrostaticsystems.Butneither istheelectron (cf.§664),andyetLorentz'shypothesisthat thiscontractslongitudinallyin exactlythesame ratio isfound tolead toavalue fortheelectromagnetic masswhich isentirelyconfirmed byexperiment (§665). 672.Accordingtothecontractionhypothesis,theMichelson-Morley experimentfailed todetect thevelocityofmotionthroughtheether because thismotion wasexactlyconcealed bytheshrinkageoftheapparatus.Ifthis were so,thevelocity oughtofcourse tobecome measurable ifwecould inany waymeasure theamount ofthisshrinkage. Itisatonceobvious that theshrinkagecould notbemeasured, oreven detected, byanyprocessofdirect measurement, foranymaterialmeasuring- rodwould shrink inexactlythesame ratio astheapparatustobemeasured. Indirect meansmight, however, beexpectedtoreveal theamount ofshrinkage. 673. LordRayleigh* pointedoutthatanisotropic mediumoughtto becomeanisotropic when shrunk, sothatordinary transparentmatterought tobedoubly refractingforarayoflight crossingitinadirectionobliqueto itsmotionthroughtheether. Butnotrace ofdouble refraction wasfound eitherbyLord RayleighorbyBracej" whorepeatedtheexperimentwith apparatussosensitive thatafiftiethpartoftheexpectedeffect would have been detected. Followingasimilar train ofthought,Trouton andRankinej tried to detectchangesintheresistance ofabarofmetal asitwasturned invarious directions, butfound nomeasurablechange. Theseexperimentsdonotproveeither thatthere isnomotionthroughthe ether, orthattheFitzgerald-Lorentzcontraction doesnotoccur. They prove that ifthere ismotion throughanether, and ifthecontraction does occur, then theeffect ofthiscontraction issomehow veiled orcompensated bysome other effect. Thus Lorentzshewed§thatthenullresult oftheexperimentsof RayleighandBrace would beexactlyaccounted foronhisowntheoryofthe constitution oftheelectron, onwhich theelectrons would becontracted in justthesame ratio asthetransparentmatter. AndTrouton andRankine shewed, intheiroriginal paper,thatthenull result oftheirexperimentisan inevitableconsequenceoftheelectrontheoryofconductionthroughmatter *Phil.Mag. 4(1902), p.678. tIbid. 7(1904), p.317. JProc. B.S.80(1908), p.420. §Theory ofElectrons, p.217. 38—2 596 TheTheory ofRelativity [ch.xx (cf.§345a),providedtheelectron hasthetransverse andlongitudinal masses assignedtoitbyLorentz(§664). Thus theseexperiments, undertaken originallyinorder tofindvelocity throughtheether, resultedfinallyinpro- vidingconfirmation ofLorentz'stheoryoftheconstitution oftheelectron. Inother experiments,thecompensatoryeffect isstillmoreeasilydis- covered. Acharged bodymoving throughtheetheroughttosetupa magnetic field, sothatevery charged bodyinalaboratory oughttobesur- rounded byamagneticfieldproportionaltou/C. Everyothercharged body inthelaboratoryismovingacross thelines offorce ofthismagneticfield withvelocityicand sooughttobeacted onbyamechanical forcepro- portionaltou?/C2 .Trouton andNoble*suspendedaparallel plate condenser byatorsion thread andlooked foracouple, proportionaltou2/C2 ,tendingto turntheplates paralleltothedirection ofmotionthroughtheether.Nosuch couple wasobserved. Thenull result ofthisexperimentisreadily explainedasaconsequence oftheFitzgerald-Lorentzcontraction. Ashrinkageofthedistance between theplatesdecreases theenergyofthecondenser. There istherefore a mechanicalcouple tendingtoturnthesysteminto itspositionofminimum potential energy—i.e.intoapositioninwhich theplatesareatright angles tothedirection ofmotionthroughtheether. Itisreadilyverified that this couple exactlyneutralises thecoupleofmagnetic origin, which theoriginal experimenttried todetect. Indeed, grantedtheFitzgerald-Lorentz shrinkage, thetheorem provedin§659shews atonce thatthesystem would beinequi- librium inallorientations. TheRelativity-Condition. 674. These andsimilarexperimentshave oneand allfailed todetect motion throughanether. Theyhave notprovedthat there isnomotion throughanether, butshew that ifthismotion exists, itseffects areinevery caseveiled bysome other effect, and, inevery case, ithasproved possible to discover thisveilingeffect asaneffectpredicted bygeneral electromagnetic theory. Thequestionarises whether there mustalwaysand ofnecessity bea veilingeffect inevery experiment.Inother words, aretheelectromagnetic equationsofsuch anature that itisinherently impossibletodetect motion throughanetherbyelectromagneticmeans ? Itiswellknown thattheordinary Newtonianequationsofdynamicsare ofthisnature. Fortheequations ,t./• *Phil. Trans. A,202(1903), p.165andProc. R.S.72(1903), p.132. 673-675] TheRelativity -Condition 597 donotchangetheir formwhen referred toaxesmovingwithauniformvelocity u—i.e.when xisreplaced byx—ut.Thus allphenomena governed bythese equationsarethesame onanearthmovingwith auniformvelocityuasthey would beonanearth atrest, sothat itisnecessarilyfutile toattemptto determine theearth'svelocityinspace bymeans ofsuchphenomena. Systemsofequationsornatural lawswhich aresuch astomake itim- possibletodetermine absolute motion maybesaid tosatisfythe"Relativity- condition." The characteristic ofsuchequationswillbethattheydonot changetheir formwhen referred toaxesmovingwith auniformvelocit}^ relative totheaxes towhichtheywereoriginallyreferred. Wehaveseenthat theNewtonianequations satisfytherelativity-condition,andthecontinual failure ofexperimenttodetermine theearth'svelocity throughtheether leads ustoconsider whether theelectromagneticlawsmaynotalsosatisfythe relativity-condition. Ifwesimply changetheelectromagneticlawsbyreplacingxbyx—ut,it isatonce seen that achangeofform results. Butthehypothesisofthe Fitzgerald-Lorentzcontraction hasalready given groundsforsuspectingthat therequired change maynotbesosimpleasthis. Forinstance, itmaybe thatonchangingtomoving axes, alllengths paralleltothe#-axisoughtto becontracted intheratio (1—u2/C2 )~.Inthiscasethetransformation would befromxtoanewcoordinate xdefined byx=k(x—ut),where redenotes (1—ii2/C2)~•Theanalysisof§659hasalready shewn that allelectrostatic phenomenaconform totherelativity-condition when thistransformation is made. Thischange reallyamounts toachangeinthemeasurement oftheunit oflength,asregards lengths paralleltotheaxisofx,whenwechangethe velocityofmotionparalleltotheaxisofx.Followingamethodoriginated by Einstein* weproceedtoexamine whether similarchangesinalltheunits can result intheelectromagneticlawsconformingtotherelativity-condition. 675. Consider first thecondition that thesimple phenomenonofthe transmission ofalight-signalshallsatisfytherelativity-condition. Imagine anexperimenter &movingwithanunknown butuniformvelocity, andusing coordinates x,y,z,ttorecord theresult ofhisobservations. Ifthepheno- menon oflight-transmissionsatisfies therelativity-condition,asignal started fromtheoriginatanyinstant t=willaftertime thavereachedpoints lying onasphere x"+y2+z2-CH-=(691), whereCisthevelocityoflightdetermined bytheobserver S. Letasecond observer S'move with adifferentvelocity, and lethimuse coordinates x,y',z',t'torecord theresult ofhisobservations. Thesphere *Ann. d.Phyx. 17(1905), p.891. 598 TheTheory ofRelativity [ch.xx whoseequationis(691)for8willhaveanequation expressedinterms of x,y',z,t'forS',and iftherelativity-conditionissatisfied, thisequation must be x2+y'2+z'2-C'n'2=(092), where C"isthevelocityoflightdetermined by8'. IfS'changeshisunits oflengthortimehewillchangehisvalue ofC, which isthedistancelight appearstohim totravel inunit time.Wemay withoutanylossofgenerality suppose8'touseunits which makeCequal toa. Wemayalsosupposethatlightwillappear,both to8andto8',totravel instraightlineswithuniformvelocity*.Thus for8theequation connecting theposition x,y,zofalight-signalwith thetime tmust belinear inx,y,z and t.Thesimilarequationfor8'willbelinear inx,y',zand t'.Thus x,y',z and t'willnecessarilybelinear functions ofx,y,zand t.Andwehave already supposedthattheequationsoftransformation from x,y,z,t'tox,y,z,t must besuch thatequation (691)transforms intoequation (692), C"being equaltoC. Letusintroduce newvariables r,r'inplaceoft,t',thesebeing given by t=iCt, t'=iCt'where i=a/(- 1).Thenequations (691) and(692)become x2+y2+z2+t2=0, x'2+y'2+z'2+t'2=0. Therelativity-conditionissatisfied ifalinear transformation transforms theoneequationintotheother. Since theequationsoftransformation are linear thisrequiresthat x2+y2+z2+t2=k(x2+y'2+z'2+t'2 ) (693), where kisaconstant. Imagineafour-dimensionalspaceconstructed inwhich x,y,z,rare orthogonalrectilinear coordinates. Onaccount ofthelinearityoftheequations oftransformation, x,y',z',rmayalsoberegardedasrectilinear coordinates inthissamespace,butthese have notyetbeenrequiredtobeorthogonal. Now x2+y2+z2+r2isthesquareofthedistance ofthepoint x,y,z,tfrom theorigin whenexpressedinx,y,z,tcoordinates, sothat,byequation (693), k(x2+y'2+z'2+t'2 )must bethesquareofthedistance ofx',y',z,rfromthe origin.Itfollows atonce thatx,y',z,rmust beorthogonal coordinates; if *According toEinstein's theoryofgeneralised relativity,towhich weshall return below (§702), light does nottravel instraight lines inthepresenceofagravitationalfield. The assumption wehavejustmade marks thepartingoftheways between theoldphysical theory ofrelativity andthenewgeneralised theory. Onthenewtheory theassumption just made isstrictly true onlyataninfinite distance from allmatter;itmay nevertheless be regardedasavery accurate firstapproximation tothetruth exceptingravitationalfields enormously more intense thananyofwhich wehave experience. 675-677]TheRelativity -Condition 599 theywere notorthogonal,crossproducts x'y\x'r etc.would enter intothe expressionforthesquareofthedistance fromx,y',z,rtotheorigin. Thus theaxes ofx,y\z,tcanbeobtained from those ofx,y,z,rbyapure rotation inthefour-dimensionalspace. Wehavealreadyfixed theratio ofS"sunits oflengthandtimebymakingC=0.Ifwefurtherchangetheabsolute values ofthese units,wecanalter thevalue ofk,andwemayagreetofixthese absolute values sothatk—1. Thechangefrom coordinates x,y,z,ttox,y',z', r',orconversely,isnow effected byapure rigidbodyrotation oftheaxes. Wemaynotice inpassingthat iftherelativity-conditionissatisfied as regardsthetransmission oflight-signals,nosetofaxes x,y,z,rinthefour- dimensionalspaceisgeometricallymore fundamental thananyother.A changeofvelocityoftranslation ismerelyeffectedbyturningtheaxesabout, andnoobserver canclaim onpurely geometrical groundsthat hissystemof axesprovidesastandard setfromwhich allotherpositionsoftheaxesought tobemeasured. 676.Thesimplestcase ofrotation oftheaxes occurs whenevery point movesparalleltooneofthecoordinateplanes, sayx,t.Theformulae oftrans- formation thenassume thesimpleforms x=xcos+tsin0\ T'=TCos6-xsind\-(694). y'=y;z'=zJ Todetermine whatphysical meaningistobeassignedto0,wenotice that x=when x=-rtan6=-iCttand(695). Thus apointwhich theexperimenter Sregardsasmoving alongtheaxis ofxwithavelocity—iCtan willappearto8'tobeatrest. Inother words theaxes ofS'move relative tothose ofSwith avelocity—iGtan9alongthe axisofx.Letusput u=-iCtan6(696), then thetransformation(694)isthatappropriatetothecaseinwhich theaxes ofS'have avelocity (u,0,0)relative tothose ofS. thenk=cos0,andtheformulae oftransformation(694)become x=K(x-ut), y'=y,z'=z, {/=«U__J(697). 677.FollowingEinstein wehavefound that thetransformation relations (697) express thenecessary and sufficient condition thatthepropagationof light-signalsshall conform totherelativity-condition. Wehavealready 600 TheTheory ofRelativity [ch.xx noticed(§674) thatthe first relation x=k(x—ut)issimply anexpressionof theFitzgerald-Lorentzcontraction which isnecessaryiftheMichelson-Morley experimentistoconform totherelativity-condition. Wenowhave thefurther information that ifallexperimentsoflighttransmission aretosatisfythe relativity-condition, wemust have thefurther relation ,Ixu t=K [t~ JT2 Thetransformation (697), although wehave obtained itbyamethod due mainlytoEinstein, iscommonly known asLorentz's transformation. For Lorentz hadshewn*, before theappearanceofEinstein'spaper,thatprecisely thesame transformationexpressesthecondition that theordinaryelectro- dynamical equationsshallconform totherelativity-condition. 678. Beforeproving this, letusexamine some ofthepurelykinematical propertiesoftheLorentz transformationexpressed byequations (697). Transformingtoaxesmovingwith arelativevelocity uisequivalent,as wehave seen, toturningtheaxesthroughanangle6inthex,rplane, where 6isgiven byequation (696). Transformingtoaxesmovingwith avelocityu' relative tothesenewaxes isequivalenttoturning throughafurtherangle6' given by u'=-iCtan 6'. Butthese lastaxescanbeobtained from theoriginalaxesonturning throughanangle6+6',andwehave -iC(tan6+tan6') u+u'-iCtan(6+6')1—tan6tan 6'.,uu'1+Ci Thus thevelocityofthelast setofmovingaxes relative tothe first is notu+u;itisu,given by u=^±±, (698),UU 1+CJ andwenotice thatuisnecessarilylessthanu+u'when both uand u'are positive. Weshouldonlyhave arighttoexpectthatuwould beequalto u+uifboth8andS'measured theirlengths,times andvelocities insimilar ways,and this,under theLorentz transformation, theydonotdo. Asadirectconsequenceofequation (698), (C-u)(C-u')G-u= l+% sothat ifuand v!areeach lessthan C,thenuisnecessarilylessthan G. *Amsterdam Proc.(1901), p.809. 677-679] TheRelativity- Condition 601 Thus nopossible compoundingofvelocities lessthanGcanevergivea resultantvelocity ugreaterthan G.Asaspecialcase ifu'=Gthenu=G, regardlessofthevalue ofu;theresultant ofthevelocityoflightandany othervelocityisthevelocityoflight. Similaranalysiswillgivetheresult ofsuperposingtwo velocities notin thesame direction, buttherequiredformulae canbeobtained rather more directlyfrom theformulae oftransformation (697),asweshallnow see. 679.Letapointmove withvelocity u,v,wrelative totheaxesusedby S,sothat x=x+ut,y=y+vt, z=za+wt(699), and letthevelocityofthesamepointrelative totheaxes usedbyS'be u', v',w\sothat x'=xj+u't',y=y'+v't', z=z'+w't' (700). Inthese lastequations,letussubstitute Lorentz's values forx',y',z,t',as given byequations (697).Weobtain K(x—ut)—x+UKit— j^J, y=yd+v' K\t-XXI & andasimilarequationforz.Differentiate these threeequationswithrespect ctx tot,putting -j-—u, etc.,inaccordance withequations (699), andwefind dt u-u=u(l--^fj fromwhich follows directlyV=v'k(1- W=W/k(1- u=uu\ uu ~G'<) u'+u 1+uu ~& v' H w=.'uu W K1+.(701), .(702). uu 7F Intheseequations U,v,wmayberegardedastheresultantvelocity obtained bycompoundingvelocities u,0,and u',v', iv*. 602 TheTheory ofRelativity [CH.XX Fromequations (701)weobtaindirectly u—u u'=- 1-Ull IT* r'= *K") TT'=.IF UU ~C2.(703). These arealsoanecessary consequenceofequations (702),for if,v',w' isthevelocityobtained bycompoundingvelocities U,v,and—u,0,0. Electromagnetic Equations. 680.FollowingLorentz* andEinsteinf,letusnowproceedtotransform thegeneral electrodynamical equationsofChap.XIX(§§621, 622),namely 4-7T/ pU+df\ dy d/3 dt) dy5- ,etc. oz l_da_dZ_d_Y Gdt~dydz' df dgdh__ dxdydz^ dadb ?c_n docdydz tothenewvariables x,y',z',t'connected with x,y,z,tbyrelations(697). Ifvisanyfunction whatever ofx,y,z,twehave dx=hete_+heM_=Kfix_}Lbc\.(704), •(705), •(706), .(707), dx dx'dx dt'dx=Kdxc*dt dtdxdt^dt'dt \dt' dx' d_X=dJC.hc^hc dy dy''dz dz'>.(708). Thethreeequations (704)andequation (706) accordingly assume theform (709),47T ~C 47T ~C *I.e.ante.(df \dtpU+K[dxdy_d/3 dy'dz tAnn. d.Physik, 17(1905), p.916. 679-681] Electromagnetic Equations603 47T ~0,dh dhVK^di'-UM3/3u3/3 =*k-,- ,9a:' C'23*'9a .(711), Kdl+?l+M_KUdf_ 7 and Kat7+^+a?~^^"'=^ Ifweintroduce a,/3',7',/', #',h',defined by /'=/ 4^2). £'= *(/3+^)'ff'=K{^^) r/=K\y-^g),h'=K(h+u 4>irC•(713); (715).thenequations (710) and(711)maybewritten intheform 4-7T/dg'\_dady' ~C 4tt/ cZA'N_3#_da' C\pW+dt')~dx dy 681.We stillrequiretotransformpU,pV,pwtothenewcoordinates. Thedensity pinthenewcoordinates mustbesuch that pdxdy'dz=pdxdydz. Since thecoordinates x,y', z',rarederived from x,y,z,rbyapure rotation infour-dimensionalspace, wehave nil* 7/z t* )——,-^-L—'—.-£=1.ordx'dy'dz'dT =dxdi/dzdr. d{x,y,z,r)u J^ .(716), Thusdxdydz _dr' dt'_l dx'dy'dz ~fc= Tt=K {"C2 sothatp'=pic\l-.(717). Combiningthiswithequations (703), wefind atonce thatp(u—u)=p'(f, pv=p'v'andpw=p'w'.Thusequations (714) and(715) become 4-7T+dg'\_da'dy' dt' dz' dx•(718), .(719).C\pW+dt')~dx'dy' -Onmultiplying throughout bykandusingrelations(713), equation (709) becomes 4?r ~Gdf (df dadh\~] dyd$ which, bytheuseofequation (712), reduces further to xdf] dy' 3/3'4_7T C/cpdtdy'dz 604 HieTheory ofRelativity [ch.xx Usingtherelation p(u— u)=p'u'justobtained, andalsotherelation /=/', thisbecomes ^(p>v>+gi)=*L_W(72o).G[pU+ dt'J dy'dzl ; dy' Finally, again usingrelations (713), equation (712)transforms into df_1idg_dhf\_w_(dy_dj3\_kudfK dx7+K\dy'+ dz')+4ttG\dy' dz')C2df~p' Usingtherelation (720), which hasjustbeen obtained, thisbecomes K1,'dq'dh'\ /nUu+«w+w= p{1 -ir*j> doD'V G'\ or,multiplying throughout bykandusing equation (717), df djfdhf_ dx'+ dy'+dz'-p {'-i}- 682.Wehavenowseenthat ifthenewquantities a,/3',y'}f,g,h'are denned byequations (713), then theelectricequations (704) and(706),when transformed tocoordinates x,y',z, t',resume theiroriginalformexactly. Bypreciselysimilaranalysis wefindthat if a'=a, X'=X b'=K(b+%z), Y'=k(Y-^c .(722),G)' \ then themagnetic equations (705) and(707), when transformed tothenew coordinates x',y,z', t',willalsoresume theiroriginalformexactly. Thus itappearsthat therelativity-conditionwillbesatisfied byall electromagnetic phenomena,iftherelation between theforces asestimated bySandthose estimated byS'movingwith avelocity (u,0,0)relative toS canbesupposedtobethosegiven byrelations (713) and (722).Ifthese relations arefound, inactual fact, tobesatisfied,itwillbeimpossibleto determine absolute motionbyanyelectromagneticmeans whatever. 683. Consider firsttheformassumed bytherelations infreespace,for which wemaytakeK= /jl=1.Here a,b,cbecome identical with a,/3,7,and a', b',cwith a,ft,y'.Also/,g,hbecome thesame asX/4nr, Y/4nr, Z/4>ir andsimilarly for/', g',h'.Thetwosetsofequations (713) and(722)arenow seen tobecome identical, eachreducingto «'=«, X'=X \ *r-.(/»+5*). r-„(r-5,) y_«(7-£y),z--*(2+ jj/s)I 681-684] Electromagnetic Equations 605 IfiPjC-isneglected,kmaybeputequaltounity,andtheforcesX',T',Z' areexactlythose which wefound in§628 fortheforces onaunitcharge movingwithvelocity (u,0,0).Similarlytheforces a,j3\yareeasily shewn bythemethod of§572tobepreciselythose which would beacting onaunit magnetic polemovingwith avelocity (u,0,0).Thus there isdirectexperi- mental verification oftheseequations when u2/C2isneglected. When u2/C2isnotneglected,itisnaturally impossibletoobtain direct experimentalevidence oftheaccuracyoftheequations. Acomplicationarises from thefactthatSandS'areusingdifferent units oflengthinthedirection ofOx,andonallowingforthisandtreatingtheprobleminthemanner of §656,itisatoncefound thatthepresenceofthefactors kinequations (723) exactly representsthecomplicationintroducedbythefiniteness ofu2/G2 . Thus itappears, bywhat isnot farshort ofabsoluteproof,that the relativity-conditionissatisfied byallelectromagnetic phenomena. 684.Theproblem presented byphenomenaindielectric andmagnetic media isnaturallymorecomplex.Varioushypotheseshavebeenputforward astotherelation between a,b',c'anda',ft ',<y'inmoving magnetic media, as alsoregardingtherelation between f, g',h'andX', Y',Z'inmoving dielectrics. Some ofthesehypothesesareinagreementwith relations (713) and(722),while some arenot.Experimentshavebeenconductedbyvarious physicists,andinparticular byH.A.Wilson andA.Eichenwald, with aview todiscriminatingbetween these rivalhypotheses.Ineach casethevictorious hypothesisisfound tobeinconformitywithequations (713) and(722) above. Wilson* moved adielectric body throughamagneticfieldandfound that there wasanelectricpolarisation (f,g,ti)setupofwhich theamount agreed very closelywith thatdemanded byequations (713). AndEichenwaldf setapolariseddielectric inmotion andfound that itproducedamagnetic field similar tothatdemanded byequations (713). Thus there seems tobe experimentalconfirmation foreveryterm inequations (713). Experiments onmoving magneticmedia have notbeenperformed,butthere seems tobe littleroom fordoubt thattheywouldsimilarlyconfirmequations (722). If,onthestrengthofthisevidence, weassumeequations (713) and(722) tobefully confirmed, thenwehaveshewn thattheelectromagnetic equations conform totherelativity-condition.Inother words, allexperimentsto determinevelocity throughtheether arenecessarilyfutile. Ifforthemoment weassume thatwearemoving throughtheether with avelocity uina direction which wecallOx,thenwemayconsider thatweareplayingthe roleofourobserver S',while animaginaryobserver atrestintheethermay besupposedtobeplayingtheroleofourobserver S.Butwehave seen that *Phil. Trans. A,201(l'JOlj, p.121. fAnn. d.Phys.11(1904), p.121. 606 TheTheory ofRelativity [ch.xx allelectromagnetic phenomenawould beexactlythesame forusasforS.If wecould deduce avelocityuthroughtheether forourmotion, Swould necessarilydeduce avelocity uforhisownmotion, which would becontrary tothe facts. Bythisargument,hereputintheform ofareductio ad absurdum, weseetheimpossibilityofdeterminingourvelocity throughthe ether. Ifatanytimeequations (713) and(722)areprovedtobeuntrue—and, aswehave seen, theremaining opportunitiesforprovingtheseequations untrue areveryfew—then itwillbecomepossible,intheoryatleast, to determine ourvelocity throughthe ether. But forthepresent weshall assume, asaworking hypothesis,that itisinnoway possibletodetermine velocity throughtheether. This iscommonlycalled theHypothesisofRela- tivity.Weproceedtoexamine some oftheconsequencesofthishypothesis. TheRelativity Hypothesis. 685.Thishypothesiscommits us,generally speaking,toalltheequationsof thepresent chapter.Itdoesnotcommit ustoanyspecial physical interpreta- tions ofthem. Forinstance, the firstequationoftheLorentz transformation, namelyx=k(x—ut),mayifwepleasebeinterpretedinterms oftheFitz- gerald-Lorentz contraction-hypothesis;wemay postulateafixed ether and theequationisthen taken toshew thatanylength movingwith avelocity (w,0,0)throughtheether willbecontracted inthedirection ofthe#-axis intheratio1/k. Alternatively wemay interpretthesameequationinsuch awayasnottoassume afixed ether atall.Anyobserver Smeasures outa spherewhich remains atrestrelative tohim;toasecond observer S'moving relative toSwithavelocity {u,0,0),thisspherewillappeartobecontracted intheratio1/kalongOx. Inasimilar wayalltheotherequationscanbeinterpretedsoastohave noreference toafixed ether :theymaybetaken merelyasexpressingrelations betweenquantitiesasmeasured byoneobserver Sandanother observer S' movingwithavelocityurelative toS. Thekinematical relations ofEinstein, namely equations (702), may,on thisinterpretation,beregarded merelyaslaws forthecompositionofvelocities. Itappearsthat thesimplelaws ofcompositionofvelocities and ofvector- addition—theso-called" parallelogramofvelocities"—arenolongertrue if thehypothesisofrelativityistrue.Thesimplelawsaretrue ifu2/C2andu'2/C2 aresmall, butnototherwise. Startling thoughthisresultmayappear,there isalmost directexperimental confirmation ofit,asweshall soon see. 684-687] TheRelativity Hypothesis 607 Optical consequences oftheRelativity Hypothesis. 686.Therelativity hypothesismakes noclaim toexplainthenature of phenomena,itmerely proposes, tentatively,agenerallawofarestrictive nature, which sofarhasappearedtodominate allknownphenomena.All explanationsofphenomenawhich conform tothelimits ofthisrestriction are equally permitted bythishypothesis,butthehypothesisserves torule out, tentatively,allexplanationswhich donotconform tothecondition. Conse- quentlyitisonlyinrarecases thattheprincipleofrelativity byitself enables ustoobtain afullsolution ofaproblem. Asaninstance ofsuch acase,we have seen that itenables ustodetermine theelectric andmagneticforces in ponderablemedia. Other instances occur inoptical phenomena,andtothese wenow turn. Fizeaus Water Tube Experiment. 687. InFizeau's water tubeexperiment,astream ofwater wasmade to flowthroughatube, itsvelocityofflowbeingurelative totheearth, anda rayoflightwaspassed throughthewater inthedirection ofitsmotion. To anobserver movingwith thestream, thewater wouldappeartobeatrest, so thatthelightwould bepropagatedrelative tothisobserver withavelocityu' connected with therefractive-index vofthewaterbytherelation u'=C/v. Accordingtotheclassical laws ofkinematics, thelight oughttotravel, relative toanobserver atrestontheearth, with avelocityu+uor -+u (724).v Fizeau found itpossibletomeasure theactualvelocity byaninterference method andformula (724) wasnotconfirmed. Theformula £+«(1-*)(725) wasfound torepresentthevelocity accuratelyboth forwater andother trans- parentmedia. Asweshallnow see,formula (725)isnotonlyconsistent with thetheory ofrelativity,butcould alsohavebeenfullypredicted bythistheory. Forthe velocityinquestionissimplythatwhich results fromcompoundingthe velocities uandC/v,andtheresultantvelocity obtainedbytherelativity formula (702)is v Gu= ^-=-4u(G\ v,(uG Ifu2/C2isneglectedthisreduces theformula(725). Inthisexperiment wehaveverydirectexperimentalconfirmation ofEinstein's formula forthe compositionofvelocities. 608 TheTheory ofRelativity [ch.xx Reflection ofLightfrom aMoving Mirror andEmission from aMovingSource. 688. Accordingtotherelativity hypothesis,thevelocityoflightinfree spaceisalways equaltoG.Ifthelightbeobserved byanobserver moving with avelocity urelative tothesource, thevelocityisstillequaltoG,forwe have seen in§678that thevelocityobtained bycompoundingavelocity G withanyothervelocityuisitselfequaltoG. Thisconsequenceoftherelativity hypothesishasbeen tested byMajorana. Hefirstexamined thelightreflectedbyamovingmirror andfound itsvelocity tobeexactly equaltoGindependentlyofthevelocityofthemirror*. Ina laterinvestigation fhetested thevelocityoflightemitted byarapidly- movingsource andfound thistobeequaltoGindependentlyofthevelocity ofthesource. Theseexperimentalresults areofverygreat importance,foritwillbeseen that theMichelson-Morley experimentandtheexperimentsofMajorana taken incombination establish theLorentz transformationequations (697) asafactofobservation. TheMichelson-Morley experiment shewed thatthe averageto-and-frovelocityoflightreflected from amirror back tothesource wasthesame foralldirections inspace. TheMajorana experiments now shew thattheresult istrue fortheseparate pathsbefore andafter reflection, sothatthevelocityoflight,asmeasured byanyobserver, isthesame forall directions inspace.Wenowhave asanexperimentalfactthat,independently ofthevelocities ofthesource andobserver, thewave-surface isasphere having theobserver ascentre. This ispreciselythesuppositionfromwhich westarted in§675;itwasfound toleaddirectlytotheLorentz transformation(697). Aberration and theDoppler effect. 689.Asin§591,theequationofwave-propagationinfreespace, namely dt2 x> hasasolution X=Acos— =j(Ix+my+nz-Ct) (726), where P-fm2+n2=1,andthiscorrespondstothepropagationofaplanewave oflightoffrequencyvinadirectionI,m,n.Supposethat thesamerayof light appearstotheobserver S'tobeoffrequencyvand tobepropagated inadirectionV,m,n',sothat thesolution ofthewave-equationforS'willbe 2tt X=A'cos- 77^(l'x'+m'y' +n'z'-Ct') (727). *Phil.Mag. 35(1918), p.163. fPhil.Mag. 37(1919), p.145. 688,689]TheRelativity Hypothesis 609 Onsubstitutingforx',y,z,t'interms ofx,y,z,tfromequations (697), thisbecomes xu> x=Acos^I'k(x—ut)+m'y+n'z—Gicyt— C- Thisexpressionmust beidentical with(726),sothatbycomparison we obtain im'_n_k(C+Vu)_v mn G v.(728). Aberration. Equatingthe firstandfourth fractions inequations (728) wefind u l'+ IG l+l,u G.(729). Thismust, accordingtothehypothesisofrelativity,betheexact formula forastronomical aberration. Lettheobserver Sbeatrest relative toany systemofaxes inuniform motion, while $'moves relative tothese axeswith avelocity ualongOx.ThenlightwhichappearstoStoarrive inadirection I,m,nwillappearto8'toarrive inadirection V,m,n,whereI,I'arerelated byequation (729). Put I=cos$,I'=coscf>';then u cos(f)'—cos<£=I'—I=—sin2 <£' Cil+^cosfi IfujGissmall, thisreduces totheordinaryformula ofpractical astronomy, (£'-<£=sin0'Qy. Doppler Effect. Equatingthelasttwofractions inequations (728),wefind UCOS(£'N-=(1V G.(730). This isthe fullexpressionfortheDopplereffect. Ifu-JC2isneglected,the right-hand member reduces to u 1+^cos(f)', which istheDopplerfactorusuallyassumed. Iftheobserver ismoving directlytowards thesource oflightwithvelocity u,wehave cos0'=1,and equation (730) becomes v Vu 1+ 1-II J..(731). 39 610 TheTheory ofRelativity [ch.xx Acceleration, MassandForce. 690. Informulae (702)weobtainedequationsforthevelocity u,V,W obtainedbycompoundingavelocity u,0,with avelocity u', v',w'.When thevelocity u',v',w'issosmall that itssquare maybeneglectedincom- parisonwithC2 ,these formulae reduce to v=u+U-U=u+\ ,%1**> v=-; w=— K K I Supposethat u,0, isthevelocityrelative toSofamoving particleat aninstant t=0.Let itappeartoanobserver S',movingwith auniform velocity u,0,0,tohave accelerations du' dv' dw* W'ww these being measured inthecoordinates usedbyS'.Informulae (732)let usput ,dv',,. ,dv' .. .dw/ -, zw^^xu=Wdt'r= J7dt-^=Wdt (733 >' then u,v,wwillbethevelocities, asmeasured bySattheendofasmall interval dt'asmeasured byS'. Thetimest,t'usedbySand S'areconnectedbyequation (697), namely, sothatondifferentiation withrespecttotfollowingtheparticleinitsmotion, dt'/.,udx\ /_u2 \1 ,w~,x dt VOdt) VCV * Thus relations (733)maybereplaced by ,ldu',U=KWdt>etC» andequations (732)become 1du' k*dt'a"'" «2dt'""' "~ /C2dt1du'1±1dv'Jx 1dw'7 ,_ ,U=u+Zi-Mdt>v=Z>17dt>w=Z,ZWdt(735). IfT7, -77 ,-t-aretheaccelerations asmeasuredbyS,wemust have du _=u>+-j-dt,etc., whence bycomparisonwithequations (735), du_1du\ dv_1 LdY,dw_ldw dt'' k»dt''dt~ K*dt'' dt~K*W(736)t These formulaegivetheaccelerations asmeasured bySinterms ofthe accelerations asmeasuredbyanobserver S'movingwith theparticle. 690-692] Acceleration, MassandForce 611 691. Lettheaccelerations besupposedtooriginatefrom theaction ofa force. Since theparticleissupposedtobeatrest relative to3'atthein- stant t'—0,itsequationsofmotion, interms ofthecoordinates usedby>S", willbe du'j),dv'n, dw', dt'~'"" dif wheremisthemass, asestimated by3',and P',Q',R'arethecomponents oftheforce, alsoasestimated byS'. From theseequationsandequations (736), weobtain dv dt~~~'""" ~di andthese willbetheequationsofmotion asobserved byS. Iftheparticleisanelectron ofcharge e,thevalues ofP',Q',R'willbe eX', eY', eZ',whereX\ Y',Z'aregiven byequations (723). Substituting these,wefind fortheequationsofmotion oftheelectron asobserved by3,m/c—=Jrm 3=Q'; m/c2-T7=li dt.(737), rriK6 1)1K' mic*du dt dv dt dw ~di=eX =e/c =etcZ+?/3.(738). Theobserver Swillsupposetheelectron movingwithvelocity utohave longitudinalandtransverse masses viiandmt;and hisequationsofmotion fortheelectron willbe duv dvdv(u\ dw („u„\ Bycomparisonwithequations (738).(739). mi=mic*;mt=mK.(740). 692. These arepreciselyLorentz'sexpressionsforthelongitudinaland transverse mass ofamovingelectron(§664),which wehaveseen tobefully confirmed byexperiment (§665). Indeducingtheseexpressionsin§664 wesupposedtheinertia tobeproduced byamagneticfield intheether, so thatudenoted thevelocityrelative totheether, butthetheoryofrelativity shews thatUmay legitimatelybesupposedtomean merelythevelocity relative totheobserver bywhom theaccelerations aremeasured. Afurther difference between thetwocalculations mayalsobenoticed. When themass 39—2 612 TheTheory ofRelativity [CH.XX wasregardedasarisingfromanethereal magnetic field, itwaspossibleto estimate theradius oftheelectron from aknowledgeofthevalue ofm;the relativitycalculation doesnotmake anysuch estimatepossible. Wemaynotice that m/rdud dvd=T,(mfcu); mic-T.=-;,(m>cv), etc.,dt dt dt dt whence itfollows that theequationofmotion ofanelectronmovingwith anyvelocity U,V,Wrelative toanyobserver must be It(mKu)=e(x+^y-^^ d_ dt A dt where kisnowgiven by(mKV)=e^Y+-a-^ryJ (ni/cw)=e(z+^j3-~a\ T2j_tt-2J.t^2\—f.(741), 1 Oi ).(742). Toshew that these arethetrueequations,itissufficient tonotice that theyareinvariant asregardstransformations ofthex,y,zaxes,andreduce toequations (738)foronespecialdirection ofthese axes. Momentum andEneegy. 693.Theexpressionsontherightofequations (741)arethecomponents offorce onthemovingelectron astheywould bemeasuredbyS(cf.§§629, 653).Ifwedenote thesebyP,Q,R,and ifweregard vikV, vikV,micw asthecomponentsofmomentum ofthemoving electron, then theequations ofmotion (741) assume theform {Force)=(rede ofchange ofmomentum) (743). Therateatwhich work isdoneontheelectron willbe Pu+Qv+Rw =mUj-(KU)+mv-j~(/cv)+mw -j-(kw), andaftersimple algebraictransformation thisbecomes dPu+Qv+Rw=j(micC2 ). .(744). Thus ifwesupposetheenergyoftheelectron tobew/cC2+aconstant we have theequation (Rate ofdoing work)=(rate ofincreaseofenergy). 692-695] Mass, 31omentum andEnergy613 Intheequation, Energy=m/cC2+cons (745), theadditive constant isentirelyatourdisposal.Ifwetake itequalto—mC2 , theenergyisgiven by Energy=wC2(«-l) (746), andthisreduces totheNewtonian kineticenergy Jra(u2+V2+w2 )when the velocityissmall comparedwith that oflight. But itisgenerally more con- venient toputtheadditive constantequaltozero, sothattheenergyissimply ttikG2 .ThismayberegardedasrepresentingkineticenergyhiC2{k—1)and intrinsic electronic energymC2 . 694.Letusput r=mC2(l--)=mC2 7)T' then-=—=ttikU, etc.andequations (741)becomeC2 £(£H£©-** These areanalogoustotheclassicalLagrangian equationsofmotion ofa particle. Wemust note, however, that T'isnotthekineticenergy,but is equaltothekinetic energydividedbyk. Conservation ofMass,Momentum andEnergy. 695. Indenningtheenergyofanelectron tobemicC'2 ,wehavealready arranged, bydefinition, that conservation ofenergyshall hold. The total energyofasystemofelectrons isSm/cC2 ,andfromequation (744)itisatonce apparentthat ifnowork isdone from outside thetotalenergyremains constant. Asasystemofelectrons changetheir velocities under theirmutual inter- actions, thevalues ofkforthedifferent electrons willbecontinually changing. Ifthetotalenergyremains constant,itisclearfromequation (745)that2m# must remain constant. Thus ifinfuture weagreetodefine themass ofa movingelectron asra/e—the"transverse" mass of§664—then itappearsthat conservation ofenergywillimplyconservation ofmass. The fullprincipleofconservation ofenergystates that asenergyisinter- changed between different modes ofenergythesum total ofenergy always remains constant. Hence inorder thatthesum total ofmasses shallremain constant—i.e.tosecure complete conservation ofmass—itisnecessarythat all forms ofenergyshouldpossess mass,andenergyEofanykindwhatever must possessmass ofamount E/C2 . Forinstancesupposethatasystemofelectrons ofenergy E,andtherefore oftotalmass2m« equaltoEjC2 ,radiates away energyofamount R.The 614 TheTheory ofRelativity [ch.xx finalenergyoftheelectrons isE-R,sothat their finalmass isE/C2-R/C2 . Buttheradiationpossessing energyRmust alsopossessmassR/C2 ,sothat thetotalmass ofelectrons andradiation remains equaltoEIC2 ,andthetotal mass isentirelyconserved. 696.Ourtypicalelectron hasbeen supposedtomove withavelocityof components U,V,wrelative toanobserver S.Letussupposeitsvelocity relative toasecond observer S'tobeu',v'w',when 8'moves relative to£ with avelocity u,0,0.Then thetwo sets ofvelocities arerelatedbythe kinematical equations (702) and(703)of§679. Letuswrite =1-u2+v2+w^ C2 ,A,U'2+v'2+w'2\ •-T c2J Kq<*rZ>-k•(747), sothat tcisidentical with thekof§679.Onusingthevalues ofu',v',w' given byequations (703), wefind fll2\ f_u*+v2+w^ sothat,onraisingeach side tothepowTer—\, K'=KKe(l-^j(748), or,again usingthe firstofequations (703), k'u'—kk(u— ll). Multiplyingboth sidesbym,andsummingover alltheparticlesinthe field, l.'niKU' =kHm/cU— ukXniK. LetM, fixdenote thetotalmassandtotal^-momentum asobserved byS, and letM',/xx'denote thesamequantitiesasobserved byS'.Then our equation maybewritten fix=k ixx—ukM(749). Similarly,sinceSismovingrelative toS'with avelocity—u, fjix=Kfix'+ukM(750). 695-698] Mass,Momentum andEnergy 615 Ifthetotalenergyofthesystemremains constantthroughout anymotion,MandM'must remain constant, sothat/xxand/xxnecessarilyremain constant. Thus conservation ofenergy impliesconservation ofmomentum. 697.Asaparticular case, letussupposethevelocityoftheaxes ofS' chosen sothatfix'=0.Thus S'moves withthecentre ofgravityofthesystem, and thiscentre ofgravity moves relative toSwith avelocity ualongOx. Putting fxx=inequations (749) and(750), wefind fj,x=uM, M=kM\ Thus the^-momentum observed bySisutimes thetotalmass observed byS.The totalmass observed bySisktimes thetotal mass observed by S'.And, ifwetaketheenergy equaltoMC2 ,thetotalenergyobservedby5 isktimes thetotalenergyobserved byS'. 698. Returningtotheanalysisof§696, letussupposethatthesystem emits abeam ofradiationalongOx.Lettheenergyofthisbeam asestimated bySbeE,sothat itsmass isE/O2 ,and letitsmomentum asestimatedbyS beRxalongOx.Letaccented letters denote thesamequantitiesasestimated byS'.Then inorder thatequation (749)maybetrueboth before and after theemission oftheradiation, bothmassandmomentumbeing assumed tobe conserved, wemust have Rx'=kRx-uKoE/C2 . Thisequationistrue forallvalues ofu.Take uequaltoC,sothatS' moves with thebeam oflight. ThenRx=0,andtheequation becomes i4=f(751). Thus themomentum ofabeam oflight,asmeasuredbyanyobserver whatever, isequalto(I/O)times itsenergy. Or,again,themomentum is equaltoCtimes themass. IfWistheenergyofthebeamperunitvolume, themomentumperunitvolume willbeW/C,andtheflowofmomentumper unitarea ofcross-section willbeCtimes this,andsoequaltoW.Thus the pressureofradiation isequaltotheenergy perunitvolume. This result wasobtained in§592casaconsequenceofthehypothesisthat electric action wastransmitted byanether. Itnowappearsthattheresult is inaccordance with thehypothesisofrelativity,andcanbededuced asadirect consequenceofthishypothesis. 616 TheTheory ofRelativity TheEnergy andMomentum ofRadiation.[ch.XX 699. Infreespacethefundamentalequations (613) and(614)ofp.559 assume theform 4,-rrpU aldX_dy_dJ3 ^ ^^ .(753).C+Gdt''Bydz,etc \da_dZ__dY Cdi~dy~dz,QtQ Multiplythese sixequations byX,Y,Z,—a,— /3,— <yrespectivelyandadd correspondingsides.Weobtain ~p(Xu+Tv +Zyr)+~^ t(X'+Y' +Z'+ci'+l3'+7') Multiplyboth sidesbyC/4-7T andintegrate throughout anyclosedspace. Assumingthat thedistribution ofelectricdensity parisesentirelyfrom electrons, weobtain d ^///(X2+Yl+Z2+a2+@2+7")dxdydz Xe(Xu+Yv +Zw)+dt& G_ 4>7T In§693,weputl(Yy- Zj3)+m(Za-Xy)+n(X/3- Ya) vw e(X+^7— -^/3)=P,etc.dS.(754). Multiplyingthese relations byu,V,wandadding wefind, after afurther useofequation (744), e(Xu+ Yv+Zw)=Pu+Qv+Rw=j(vikC2 ). Thusequation (754) becomes d dtXm/cC2+^.jffiX2+Y*+Z*+a2+/32+r)dxdydz~\ =^((I(Yy-Z/3)+..^,dS... (755). Now lettheclosedspace beallowed toextend toinfinity,sothat the integrationisthroughthewhole ofspace. Thesurfaceintegralontheright ofequation (755)now vanishes, sothat theleft-hand member must also vanish. Inother words, throughoutthemotion ofthesystemofelectrons, 2m«(72+ >7T(Z2 -1-Y-+Z*+a2+yS2+r)dxdydz=cons. ...(756). 699,700] TheEnergy andMomentum ofRadiation (517 IfRistheenergyradiated awayfrom asystemofelectrons, wehave alreadyhadtherelation XmrcC2+R=cons., whence itappearsthatthevolume-integralin(756)mustrepresentradiated energy. Ifweassume theradiatedenergytobelocalised inspace accordingtothe distribution oftheintegral,then theflowofenergyintoanyclosed surface must berepresented bythesurface-integralontheright hand ofequation (755), andthisflow ispreciselythatgiven bythePoyntingFlux of§576. 700.Again,letusmultiplythesixequations (752) and(753) by0,7, -/3,0,Z,-Y andadd.Weobtain —adec 3/3dy+dJ3+dy\_x(dX+dY+d_Z\ dy dz)\dx dy dzj' \dx dydzJ\dx dy Dividing throughout by47r,andusing equations (615) and(616),this becomes P(x+l 618 TheTheory ofRelativity [ch.xx hasnowbeen obtained, independentlyoftheassumptionofanether, asa general expressionforthemomentum ofradiation. Ifweassume themomentum tobelocalised inspace accordingtothe distribution oftheintegral,thenequation (757) canbeinterpretedasshewing thatthere isaflow of#-momentumperunitarea atanypoint,whose com- ponentswillbePxx ,Pxy>Pxz ,where PXX=~(X>- Fa-^+aa-/S3-73 ), Pxy=~(XY+a/3),etc. These arepreciselythequantities weobtained in§655 torepresentthe componentsofstress inanassumed ether. Ontherelativity-theory, they appearinamuch moregeneral wayasrepresentingtheflow ofmomentum inspace. TheExistence ofanEther. 701.Throughouttheearlierchaptersofthisbook,wetreated theexistence ofanether asaworking hypothesis.Maxwell andFaraday appeartohave hadnodoubt thattheether hadarealobjective existence, butnoproofthat itexists outside ourownminds haseverbeen obtained, and itseems best to regarditmerelyasaworking hypothesis,tobediscarded ifitisfound tolead tocontradictoryorimpossible results, and toberetained ifitprovestobe useful aswell asself-consistent. The considerations which haveseemed tofavour thehypothesisofan objectiveether aremainlythefollowing: (i)Thatlightandother forms ofelectromagneticaction arepropagated withauniformvelocity C,which ismostnaturally interpretedasavelocityof propagationinamedium ofsome sort. (ii)That thehypothesisofanobjectiveetherexplainselectrical forces withcomparative simplicityasarisingfromsystemsofstresses transmittedby theether. (iii)That thehypothesisofanobjectiveethergivesasimpleaccount ofelectromagnetic energyasbeingtheenergyofamedium inastate ofstrain and stress. The force ofthe first consideration isverymuch weakened bythedis- covery, resultingfrom theMichelson-Morley experiment,thatthevelocityof propagationisthesame foranobserver moving throughthesupposedether as foroneatrest.Wehave seen inthepresent chapterthat this fact,whether we assume anether toexist ornot,requiresustoadopt systemsofkinematics anddynamicswhich aredifferent from theoldclassicalsystems. The con- sequencesofthesenewsystemsofkinematical anddynamicallaws arefound 700,701] TheExistence ofanEther 619 tobeconfirmed byexperiment.Iftheyhadnotbeen confirmed, weshould have reached animpasse ;thecircumstance thattheyareconfirmedprovides nodefinite information onthequestionoftheexistence ofanether. Butthe hypothesisofanether isweakened tothisextent, thatthetheoryofrelativity hasshewn thattheresults inquestion, although possibly admittingofanex- planationinterms ofanether, arenecessary consequencesofthesimpler suppositionthatphenomenaarethesame forallobservers, nomatter with whatvelocity theyaremoving. Andthesimplest wayofallofarrangingthat phenomenashallbethesame forallmoving observers, istosupposethatthere isnoether atall;allmovingobservers thennecessarilystand onthesame footing,forthere isnofixedframework bywhich theirmotion canbeestimated. Thehypothesisthatthere isanethermaygiveapossible explanationofthe phenomena,butthehypothesisthat there isnoetherprovidesanequally possibleandverymuchsimpler explanation. Ifwe stillwish toretain thehypothesisofanether throughwhichlight andelectromagnetic phenomenaarepropagated, wemustadjusttheproperties ofthisether toagreewithexperiment. Nowwehave seen(§688) that,no matter howanobserver andasource oflight move, thewave-surface formed bythelightemitted atanyinstant willbeasphere havingtheobserver asits centre. Iftheobserved constantvelocityoflightissimplytheconstantvelocity ofpropagation throughanethereal medium, itwould seem tofollow thateach observer mustcarryacompleteether about with him. This atleast robsthe ether ofthegreater partofitsreality. Wecannotquite gosofarastoassert thattheether isreduced toasubjective imagination,asasimple analogywill shew.Anumber oftravellers mayallseewhattheywould describe inordinary languageasbeingthesame rainbow. Theangleoftherainbow would bethe same foreach traveller, andnoamount oftravellingtowards therainbow would cause ittosubtend agreater angle.Ifthetravellers compared observations theywould have toconclude thateach traveller carried hisownrainbow about with him. Thiswould not,however, provetherainbow tobemerelyasub- jectiveillusion;when therainbowdisappearedforone traveller itwould disappearfor all.Considerations such aswehavementioned donotprovein strictness thatlightcannot bepropagated throughanether; whattheyprove isthat ifanether exists, itmust besomething verydifferent from the absolutely objectiveether imagined byMaxwell andFaraday. Thehypothesisofanether shewedgreat aptitudeforexplainingeither electric ormagneticforces insystemsatrest;asimple systemofpressures andtensions wasfound toaccountperfectlyfortheobserved forces. Onthe other hand thesame explanationcannot account forboth electric and magneticforces simultaneously. If,asisusually assumed, theelectric forces areaccounted forbysimple pressuresand tensions, thensome other ex- planationmust befound formagnetic forces, andthehypothesisofether- 620 TheTheory ofRelativity [ch.xx stresses loses itsprincipal advantage.Further ithasbeen found that the hypothesisofanether atrest failsentirelytoaccount forthe forces in systemsinmotion(§655). Toaccount forthese forces, itappears that the ethermust besupposed endowed withmomentum. Ontherelativity-theory alsowehave seen(§700) thattheforces canbeexplainedinterms ofaflow ofmomentum. Therelativity-theoryhasthusshewn thatwhat isessential to theetherealexplanationisnottheether butthemomentum withwhich itwas supposedtobeendowed. Itisquite easytoimagineaflowofmomentum without therebeinganether tocarry it,andtheconceptionofforces and pressures arisingfromaflowofmomentum isonewithwhich wehavebecome familiar inother branches ofphysics,asforexampletheKineticTheoryof Gases. Almost similar remarksapplytotheinterpretationofelectromagnetic energy.Stress intheether would accountquite simplyforeither electric or magnetic energy,butnotforboth.Usuallytheenergyofethereal stress is regardedaselectrostaticenergy,sothat kinetic energyoftheethermust be invoked toaccount formagnetic energy. Againtheether hastobesupposed endowed with motion, butthemotionrequisitetoaccount forthemagnetic energyissomething quitedifferent from thatcorrespondingtothemomentum requiredtoaccount forelectromagneticforces. Sofarfrom theetherproviding asimple explanationofallphenomena,itisfound thathighly complex pro- perties must beascribed toitinorder toaccount forelectrical andmagnetic properties simultaneously. Ifanether existed, itwouldprovideafixed setofaxes relative towhich all positionsandvelocities could bemeasured. Toaccount fortheresult ofthe Michelson-Morley experiment,itwould benecessarytopostulateareal shrinkageofallbodies moving throughtheether. Thisshrinkage could not bedetected bymechanical means, forameasuringrodwould shrink inprecisely thesame ratio asthebodytobemeasured, but itcould bedetected bygravi- tational means unlessevery gravitationalfield offorce shrunk injustsucha wayastoconceal theshrinkageofmatter. Forinstance, ifthegravitational field didnotshrink, thegeoid,orsurface ofmean sea-level ontheearth, might beagravitational equipotentialforsome onevelocity throughtheether, but could notremain anequipotentialastheearth'svelocity throughtheether changedfrompointtopointofitsorbit. Thuswemight anticipateseasonal anddailytidalsurgingsasaresult oftheearth's motionthroughtheether. Nosuch events areobserved. Itistrue thateven ifthese occurred the earth's motionthroughtheether mightnotbesufficiently rapidforthem to becapableofobservation, butthegeneralised theoryofrelativity, explained inthenext section, makes itclear thatsuch events could notbeobserved whatever the earth's motion mightbe.There isnolonger anyroom for reasonable doubt thatgravitational phenomenaconform totherelativity condition. 701,702]Generalised Relativity 021 If,then,wecontinue tobelieve intheexistence ofanether weare compelledtobelieve notonlythat allelectromagnetic phenomenaareina conspiracytoconceal from usthespeedofourmotionthroughtheether, but alsothatgravitational phenomena,which sofarasisknown havenothingto dowiththeether, arepartiestothesameconspiracy. Thesimpler viewseems tobethatthere isnoether. Ifweacceptthisview, there isnoconspiracyof concealment forthesimplereason thatthere isnolonger anythingtoconceal. Generalised Relativity. 702. Indiscussingthetransmission oflight-signalsin§675,wemade the assumptionthatlighttravelled instraightlines with uniformvelocity.If spaceandtimewereknown tobeuniformthroughouttheir whole extent, nosuchassumptionwould beneeded; theuniformityofvelocityandthe straightnessofpathwould bedirectconsequencesoftheuniformityoftime andspace. Thetheoryofrelativityhashoweverdevelopedinsuch adirection that spaceandtimecannolongerbesupposedtobeeverywhereuniform. Inthe early daysofthetheoryitwasnoticed thatNewton'sinverse-squarelawof gravitationdidnotconform totherelativity-condition,andin1915 Einstein putforward atheoryofgeneralised relativity accordingtowhich allgravita- tional phenomenaaretheconsequences merelyofdeparturesfromuniformity oftimeandspace. AccordingtoEinstein'stheory,thepropertiesofboth timeandspacein theneighbourhoodofagravitatingmass differ from those inregionsfar removed from allmatter. Thepropertiesofspaceinthelatterregionscanbe adequatelydescribed bythegeometryofEuclid, butthose intheneighbour- hood ofgravitatingmatter need anewgeometryfortheirdescription. Inordinary space,asdescribed byEuclid'sgeometry, parallellines never meet. Othergeometriesandotherspaces can,however, beimagined.For instance, twolinesdrawn upontheearth's surface, throughtwopointsonthe earth'sequator,bothrunningduenorth, and sorunning exactly parallelto oneanother, willultimatelymeet inapoint—theNorth Pole.Weseethat thegeometryofasphericalsurface isdifferent from that ofaplane,somuch sothatalmost alltheordinarytheorems ofEuclideangeometryfailwhen appliedtoasphericalsurface. Thegeometry required byEinstein'sgravitational theoryismuch less simplethan thespherical geometrywehavejustused asanillustration. Itis notconcerned withtwo-dimensional surfaces, orevenwith athree-dimensional space.Itisconcerned with afour-dimensional continuum, ofthetypecon- sidered in§675, inwhich threespace-coordinatesandonetime-coordinate areplotted paralleltofourrectangularaxes. Intheneighbourhoodofgravi- 622 TheTheory ofRelativity [ch.xx tatingmatter thisfour-dimensional continuum issupposedtobecurved somewhat inthewayinwhich theearth's two-dimensional surface iscurved. Acircle ofdiameter 1000 miles drawn round apointontheearth's surface willnothave acircumference of10007T miles, asitwould beifthecircle were drawn onaplane,butofonlyabout 997-77- miles. Inthesame way,according toEinstein'sgeometry,thecircumference ofacircle ofradius rdrawn about agravitatingmass isnotprecisely 2irr,but islessbyafraction which ispro- portionaltothemassand falls offaswerecede from it. Asaconsequenceofthespecial geometryofaspherical surface, itisnot possibletodraw amaponaplanesurface soastoshew allthepartsofthe earth's surface simultaneouslyintheirproper shapes andrelative sizes. This results from itsnotbeing possibletoselect coordinates x,yontheearth's surface such thattheelement oflengthdsisgiven by ds2=dx2+dtf (759) atallpointsofthesurface. Thesimplestcoordinates itispossibletoselect aretheordinary 6,</>ofspherical polar coordinates, interms ofwhich the element oflengthonthesurface ofasphereofunitradius isgiven by ds2=dd2+sin26dcf>2(7C0). AccordingtoEinstein'sgeneralised relativity,theelement oflengthinthe four-dimensional continuum canbeexpressedintheform* ds2=dx2+dy-+dz2+cZt2(761) only inregionswhich arefarremoved from allgravitatingmatter. Thisform fords2isofcourse analogoustoexpression (759)forthevalue ofds2onaplane surface. Asin§675,rstands foriCtwhere i=«/(— 1). Ifwereplacetbyitsvalue iCtandtransform from thespacecoordinates x.y,ztotheusualspherical polarcoordinates r,6,(f),equation (761) assumes theform ds2=dr2+r2dd2+r2sin2 6d(p2-C"-dt2(762). Einstein'stheory requiresthat intheneighbourhoodofagravitating particleofmass m,equation (762)shallbereplaced by ds2=dr l+r2dd2+r2sin2ddd>2-C2 (1-^)dt2 ...(763).27nof\ rO-/ l~7C2~ Aparticle describinga"geodesic"ormost directpath—defined by Sjds=—inthisspace,canbeshewn tochangeitscoordinates x,y,z,tin veryapproximatelythesamewayasaparticle describinganordinary ellipse orhyperbolainordinary spaceabout amassmunder alawofattractive forceym/r2 .Theagreement, however, isnotquite exact, andEinstein'stheory isfound torequirethreephenomenawhich were notpredicted by,andarein *Intechnical investigations onRelativity-tin*isusually written forourds3 . 702,703] Generalised Relativity 623 factinconsistent with, theclassical Newtoniantheory; first, theperiheliaof alltheplanets oughttoadvance ataratewhich should beeasilydetected in thecase ofMercury; second, light passingnear tothesunoughttoshew anappreciabledeflection;and third, thespectrallines emitted inastrong gravitationalfieldsuch asthat ofthesunoughttobeseen shiftedslightlyto theredwhencomparedwith thecorrespondinglines emitted intheweak gravitationalfield oftheearth. Ofthese threephenomena,the firsthadbeen observedbyLeverrierlongbeforeanyexplanation wasforthcoming,thesecond wasobserved assoon asitwaslookedfor,first inthesolareclipseof1919and subsequentlyinthat of1923, while thethird isstillindoubt, onaccount of theextremedifficultyoftheobservation andthesmallness ofthequantityto bemeasured. Butthequantitative agreementinthecase ofthe firsttwo phenomenaissogoodthatnodoubt isfeltastothesubstantial truth and accuracyofEinstein'stheory. Inbrief Einsteinsupposesaparticleinagravitationalfield todescribe a straight path throughacurvedspace* whereas Newton hadimaginedit todescribe acurvedpath throughastraight space. Newtonimaginedthe curvature ofpathtoresult from theaction of"forces" which emanated from thegravitating mass, and tried, althoughwithout success, tointerpretthese forces asstresses transmitted byagravitationalether. Einstein'stheory abolishes theconceptionofgravitational"force" and soescapes altogether thedilemma ofhavingtosupposethese forces either tobetransmitted throughamedium orbydirect action atadistance. WeyVs Electromagnetic Theory. 703. Thisdilemma ofactionthroughamedium oraction atadistance ispreciselythatwhich hasledtomost confusion inelectromagnetic theory (cf. §154).Ifitcanbeavoided ingravitational theory,itwould seem reasonable tohopethatanelectromagnetic theorycould beconstructed which should alsoavoid it. Thishasinactual factbeenattempted. Weylin1918, followedbyEcldington in1921, shewed that Einstein'sgeometryisfarfrombeingthemostgeneral geometryinwhich therelativity-conditionissatisfied. Intheexpression (763) whichspecifiestheelement oflengthdsonEinstein'stheory,the coefficients ofthedifferentials dr,rd0,rsin0d(f>anddtarefunctionssolelyof thepositionofdsinspace. With certain conventions astothemeaningand *Itmust always beremembered that thespace inquestionisfour-dimensional; otherwise thestatement appears nonsensical. Itwould beabsurd tosaythat theapproximate semicircle described bytheearth between perihelion andaphelionisthemost direct pathbetween these two points;itisonlywhen thesixmonths interval intime istaken intoaccount that thestatement beginstoappear reasonable. Wecangetridofthetime-interval bysupposingtheparticle to move with infinite velocityinwhich casethepathbecomes astraight lineeven inordinary three- dimensional space. 624 TheTheory ofRelativity [ch.xx method ofthemeasurement oflength*, wemaydeduce from thisexpression thatthelengthofameasuringrodwouldchangeasitwasmoved about from placetoplaceinagravitational field,butthat itslengthatanyinstant would depend solely onitspositioninspace.Indeed wemaybeevenmoreprecise, forthecoefficients ofthedifferentials inexpression (763)alldependonthe single quantity1j^which inturndepends onlyonthegravitational potential ym/i~,sothat thelengthinquestion depends onlyonthegravi- tationalpotentialattheplace. Inthemostgeneral geometry possibleinaspaceofcoordinates x,y,z,r, ameasuringrodoflengthImovedparalleltoitself throughadisplacement dx,dy,dz,drmaybeexpectedtoexperienceachangeoflengthdldefined by dl=l(Fdx+Gdy+IIdz+Kdr) (764), where F,G,H,Kmaybethemostgeneralfunctions ofthepositionofthe point.Iftherod ismoved fromonepointPtoanyotherpointQitswhole changeoflengthwillbegiven by \ogl -fi=[Q (Fdx+Gdy+Hdz +Kdr) (765). lp Jp InEinstein'sgeometry, Iqand lPdepend onlyonthepositionsofPandQ, sothattheintegrandontherightisnecessarilyaperfectdifferential. The condition that thisintegrandshall beaperfectdifferential isexpressed by thesixequations dH_dG =Q dydz dxdy'dz dr InWeyl's geometry,ontheother hand, theintegrandontheright hand isnotingeneralaperfect differential, andthesixquantities which constitute *Unfortunatelyitisround these conventions thatthedifficulties ofthesubject mainly centre. Itismeaningless tospeakofameasuring rodchangingitslength unless there issomething more absolute against which itcanbemeasured, andthecomplexities ofthetheory, especially of Weyl's theory, turnontheproperties oftheimaginary gauges ormeshes against which material objects maybemeasured. Itisimpossible togiveafulldiscussion inthepresent book. The student whowishes topursue thesubject further maybereferred to Eddington, Space, TimeandGravitation. Eddington, TheMathematical Theory ofRelativity. Weyl, Raum, Zeit, Materie (French Translation, Temps, Espace, Matiere). Itought tobeadded thatthebrief sketch inthepresent book follows theexposition ofEddington rather than that ofWeyl.dK 703,704] GeneralisedRelativity G25 theleft-hand members ofequations (766),instead ofvanishing,have values a,b,c,d,e,fsothat dGdH dy dF_dH =b dz dx dG dxdF_ dy~C>dK 626 TheTheory ofRelativity [ch.xx variousattemptswhich have beenmade toavoid it,whileretainingthe obviousadvantagesofthetheory,canhardlybediscussed here. 705.Fromequations (768) and(769)itiseasytodevelopthewhole of theclassicalelectromagnetic theory. Wenotice firstthatequations (768) areidentical with thesixequations (767) which form asystem symmetricalwithrespecttothefour coordinates x,y,z,t.This ofcourse ensures that theequations satisfytherelativity condition. The firstthree ofthissystemofsixequationscontainonlythe three coordinates x,y,z;tisentirelyabsent from thissetofthree. From the setofthree equationsfromwhich rwasabsent weobtainedequation (769). From thecorrespondingthree setsofequationsfromwhich x,yandzinturn areabsent, itisofcoursepossibletoobtain three otherequationsofsimilar type.These arefound tobe 1da 704,705] Generalised Relativity 627 ofwhich thesolution, obtained asin§645,is ^=mW\dxdydz (776)^ Since ourequations have allsatisfied therelativity condition, these formulae must remain true foranyrotation oftheaxes inthex,y,z,tspace. Writing 4dx*df"*" dz-C2dt*' wenotice thatV42interms ofthecoordinates x,y,zytassumes theform 4~ dx*+ dy"+ dz*+ dr*' which isobviouslyinvariant foranyrotation oftheaxes. Letusnowtakeequation (775) which, inx,y,z,rcoordinates, hasthe form V42^=-4ttp (777), andtransform toanew setoforthogonal axes, obtained byrotatingtheold axes inthe x,y,z,rspace. Ashasjustbeen seen theoperator V42willbe thesame inthenewaxes asintheold. M*isthecomponent alongtheaxis oftofthevector -iF,-iO,-%H, -iK, whilepisthecorresponding componentofthevector dx dy dz PTt>PTt'Pd~r'P orofthevector .u .v.w-1Pq>~~1Pq>~lPJj>P' where V,v,warethecomponentsofvelocityofp.Thus after transfor- mation, equation (777) becomes V,2(-ikF-il2G-il3H+l^Y)=-4tt(-ip^k-ip^l2-ip^k+p), where l1}l2,l3,I4,arethedirection cosines oftheoldaxisofr. Since thisequationmust betrue forallvalues of l1}l2)l3and l4,we deduce atonce V42 JP=-4 7rpg(778), andtwo similarequations. Returningtothecoordinatesx,y,z, t,these assume theform d*F d*F d2F 1d2F .u,,___. dx*+ df+^~VW=~P~G(l and similarequationsinGandH.Thesetogetherwithequation (775) constitute theequationsofpropagationofthefourpotentials F,G,H,AF. 40—2 628 TheTheory ofRelativity [ch.xx Bydirect solution ofequation (779)weobtain thevalue ofFintheform F= jjj[P u]dx^dydz(780X with similarequationsforGand 17. Using equations (779)and(768)wefind 1dn~V(&Fc*F&F\ ~~GdxJt~\d^+ dy2+ dz*J' and,byuseofequation (773),this ~ dx[dx+ dy dz) \da** dy*+ dz*J _d_(d_G_dF\ d_(dF_dE\ dy\bx dy) dz\dz dxJ =£-f(781), dydz givingMaxwell'sequationsofmagneticforce. Differentiatingthisandthetwosimilarequationswithrespecttox,y,z andadding, weobtain, bytheuseofequation (774), d,.d,.d,.dp_ which istheequationofcontinuity (cf.§622), shewingthat electriccharges have apermanentexistence. 706. Einstein'sgravitational theory wasacceptedassoon asthephenomena itpredictedinoppositiontothe classical Newtoniantheorywereactually observed. ItisimpossibleforWeyl's electromagnetic theorytoestablish itself inasimilar manner since thephenomenaitpredictsarepreciselyidentical with those oftheclassicaltheoryofMaxwell. Asadirect observational test isimpossible, Weyl's theorycanonlybejudged byitsinherentplausibility. Itmaybesaid tobetheonly theoryatpresentinthe field, Maxwell's mechanism ofstresses andstrains inanetherhaving,forallpractical purposes, received itsdeathblow bytheestablishment ofthe restrictedrelativity theory.Initsfavour maybesaid that itgivesaconsistent account of electromagnetic phenomenaonlines which, inview oftheconvincing experi- mental confirmation obtained fortheparallel theoryofgravitation, must be admitted tobeinaccordance with thegeneral workingsofnature. The principal objection which canbebrought againstithasalreadybeenmen- tioned(§704). CHAPTER XXI THEELECTRICAL STRUCTURE OFMATTER 707.Bytheendofthenineteenthcentury,itwasgenerally believed that allphysical phenomena, with thepossible exceptionofgravitation, were of electricorigin.Associated with thiswasthebelief thatmatter wasapurely electrical structure. Positive andnegative charges, arrangedincombination indifferentways,weresupposedtogiverisetoallthevarious kinds ofmatter intheuniverse, changesinthepositionsandarrangementsofthesecharges being regardedastheoriginofallthephenomenaofphysicsandchemistry. Variousconjectures weremade astotheactualarrangementofthepositive andnegative chargesinmatter, butpositive knowledgewasonly obtained when thenewexperimental methods made available bythediscoveryof radioactive substances werebroughtinto action. 708.Thespecial propertiesofradioactive substancesoriginatefrom their spontaneously andcontinuously emitting raysofvarious kinds. Ifabeam of theemittedraysisallowed totraverse astrong magnetic field, itisfound to besplitupintothree distinct beams, twoofwhich aredeflected inopposite directions, while thethirdpasses straighton.The threetypesofraysin these three beams areknown asarays,' /?raysand7rays respectively. We have seen(§631) thatachargedelectricparticle traversingamagneticfield willdescribe acircle ofradius vmC/eH. Thecurvature ofthepathsofthe araysisfound tobethesame asiftherayswerepositively charged particles, that ofthepathsofthe/3raysiscurved asthough theywerenegatively charged particles,while theabsence ofcurvature ofthepathsofthe7rays suggeststhattheyarenotcharged particlesatall.Thechargescanbemeasured byshootingtheraysintoanelectrometer. Itis,found that thearaysare rapidly moving particleseachwithapositive charge equaltotwice thecharge onanelectron, andwith amass almostexactly equaltothat ofthehelium atom. The/3raysprove simplytobenegativeelectronsmovingwith velocities comparablewith that oflight. The7raysarefound toberadiation ofthe samegeneral nature aslightorX-rays,butofexceedinglyshortwave-length. Ifathinpieceofmetal foil isplacedinthepathofabeam ofaparticles, themajorityoftheparticles passthroughwithout theirpaths shewing any appreciable deflection, butasmall fraction ofthetotalnumber aresubstantially deflected. Fromexperimentsunder varied conditions, thedeflections arefound tobesuch aswould beexpectedifonlyisolated small areas ofthe foilhad thepowerofappreciably deflectingtheparticles, andthenumber ofsuch areas isfound tobeequaltothenumber ofatoms inthe foil.Moreover the deflections observed arepreciselythose which would beexpectedifeach of 630 TheElectrical Structure ofMatter[ch.xxi these areas hadatitscentre afixedparticlewhichrepelledtheaparticle accordingtothelawoftheinversesquareofthedistance. InvestigationsofthistypeledSirE.Rutherford toputforward in1911a theoryofthestructure oftheatom, generally known asthenucleartheory, which hasstood thetest oftimeandhasnowwon universalacceptance. Accordingtothistheory,anatom consists ofapositively chargedcentral nucleus surrounded byanumber ofnegative electrons, thechargeonthe central nucleus beingsuch thatthetotalchargeoftheatom iszero. Thesimplestatom isthehydrogen atom, consistingofonlyoneelectron andthepositivenucleus. If—eisthechargeofanelectron, that ofthe positivenucleus ofthehydrogenatom isofcourse+e.Next inorder comes thehelium atomconsistingoftwoelectrons andapositivenucleus ofcharge +2e.Thispositivenucleus isfound tobeexactlyidentical with theaparticles ofradium radiations. Withinsignificant exceptionschemical elements areknown havingre- spectively 1,2,3,...electrons, andconsequentlynuclei ofcharges +e,+2e, 4-Se, ...,upto60electrons andanuclearcharge+60e(Neodymium).After thisgapsappearinthesequence,whichappearstoendaltogetheratUranium with92electrons andacharge +92e.Thenumber which fixes theposition ofanelement inthissequenceiscalled its"atomic number"; fortheelements oflowatomic number, theatomic number isapproximately equaltohalfthe atomicweight.The firstfewelements with their atomic numbers areas follows : Atomic Number 1 2 708-710] TheElectrical Structure ofMatter 631 thenucleus hasabout 1844 times themass oftheelectron. Inthehelium atom themass ofthenucleus isabout 7320 times themass ofasingleelectron and sooutweighsthetwoattendant electrons intheratio ofabout 3660 toone. Astheratio ofatomicweighttonumber ofelectrons isabout thesame inall elementsexcept hydrogen,itfollows that inallelements other thanhydrogen onlyabout onepartin3660 ofthetotal mass resides outside thecentral nucleus. Formostpurposes wemaythink ofthecentre ofgravityofanatom ascoincidingwith itsnucleus. Assumingthemass ofthenegativeelectron tobewholly electromagnetic, wehaveseen that itsradius must beoftheorder of2x10~13cms. Ifthemass ofthenucleus also iswholly electromagnetic,itsradius must bemuch smaller than thatofthenegative electron; that ofthenucleus ofthehydrogen atom, forinstance, would beabout 10-16cms. There isdirect evidence that the nucleus isexceedingly small, experimentsonthescatteringofarayshaving shewn that aparticlescanpasswithin 2x10-13cms. ofthecentre ofan atomic nucleus, andyetbedeflected inaccordance with theordinarylawof theinversesquare. Thesefigures shew thatboth nuclei andelectrons areverysmall incom- parisonwith atoms. Thehydrogenatom whose radius isapproximately 0*53x10-8cms. ismadeupofonlytwoconstituentparts,each ofradius 2x10-13cms. orless. Since apositiveandanegative chargecannot stand in staticalequilibriumatadistanceapart equaltoseveral thousands oftimes theradius ofeither, wemustsupposethat thetwochargesmaintain their distance asaconsequenceoforbital motion. Thenegativeelectron doesnot fallontothepositiveelectron forthesame reason forwhich theearth does not fallonto thesun. Inmany respectsanatommaybecomparedtoasolar system,theheavy positivenucleus atthecentre oftheatomrepresentingthe sunandtheelectronsrepresenting planets,thelawofforce between the nucleus andtheelectronsbeingthesame asthatbetween sunandplanets, namelyaforce ofattractionvaryingastheinversesquareofthedistance. 710. Accordingtotheanalysisof§650,anegativeelectrondescribingan orbit about anucleus must radiateenergy.IfE,earethechargesonthe nucleus andelectronrespectively,andmthemass oftheelectron, theac- celeration oftheelectron towards thenucleus isEe/mr2 ,while theacceleration ofthenucleus maybeneglectedincomparison,onaccount ofitsmuchgreater mass. From formula(663), therate ofemission ofradiationperunittime is ^[{Xeur+(tevy+^ewy}=w^?(782). Forthehydrogen atom, consisting onlyofoneelectron andanucleus ofequal charge, wemayput E=-e=4-774 x10"10el.stat. units, r=0"53x10~8cms., 632 TheElectrical Structure ofMatter[oh.xxi fromwhich therateofradiation isfound tobe0*46ergspersecond. Asa result ofthis lossofenergy,theradius oftheorbitoughttodecrease. When theradius isr,theenergyoftheorbit isreadilyfouud tobe—e2 J2r,sothat therateofdecrease ofenergyis e2dr ~2? di' Puttingthisequalto0"46ergsasecond, wefindthat—dr/dt must be equaltoabout 112cms.asecond. Since theradius oftheorbitinitiallyis only0'53x10-8cms., thedistance between thetwo constituents ofthe hydrogen atomoughttovanishaltogetherinafraction ofamillionth ofa second. Even inthelightofcommon sense such aconclusion ispreposterous;it ismore sointhelightofexactknowledge. Sofarasweknow allhydrogen atoms, nomatter how orwhere selected, areidentical structures, allgiving thesamespectrumand allhaving preciselythesame radiusexceptfora reservation which willshortly beexplained.There isnottheslightest indication ofanysecularchangeintheirproperties,andachangeofthe rapidityofthatjustcalculated isutterlyoutofthequestion. The conclusion towhich wearedriven isnotmerelythat anormal hydrogen atom ofthetypewehave beenconsideringdoes notradiate as rapidlyasispredicted byequation (782), butthat itdoes notradiate atall. Insomewaythewholetheory which hasledtotheconclusion thatan accelerated electron must radiateenergyisinneed ofamendment. 711. Thisdiscovery,ifitstood alone, would beextremely disconcerting. Inactual fact itdoes notstand alone;itisonlyone ofalongseries of discoveries, each ofwhich hasindicated, withverylittleroom fordoubt, that theclassical mechanics ofNewton andtheclassicalelectrodynamicsofMaxwell both failwhenappliedtoatomic phenomena.Since thebeginningofthe present centurytheneed hasbeenrecognisedforawholly newsystemof dynamics,such asshallbeapplicabletophysical phenomenaonatomic and sub-atomic scales, and shallmergeinto theNewtonian andMaxwellian dynamicsinthecase oflargerscalephenomena.Inspiteofmuch labour, thissystemofdynamicshasnotyetbeen found initsentirety. Fragments areknown with faircertainty, althoughitisofcourseimpossibletofeel absolute confidence inanypartofthesystemuntil thewhole hasbeenpieced togetherandseen toform aconsistent structure.Fortunatelytheparts which areknown with thenearestapproximationtocertaintyareprecisely those which arenecessaryinthediscussion ofthesubjectofthepresent chapter,theelectrical structure ofmatter. 712. Indiscussingthemotion ofadynamical systemaspredicted bythe classical mechanics, theusualprocedureistostart from thegeneral equations ofmotion, which aredifferentialequationsoftheseconddegree,andattempt 710-712] TheElectrical Structure ofMatter 633 inthe firstplacetodiscover oneormore firstintegralsoftheseequations. Inmany problems,forinstance, theequationsofenergyandoflinear and angular momentumfigureasfirstintegralsoftheequationsofmotion. Each time afirstintegralisderived from theequationsofmotion aconstant of integrationisintroduced, different values ofthisconstantrepresentingdifferent states ofthedynamical system. Under theclassical mechanics these constants ofintegrationcouldusuallyhaveanyvalue wechose toassigntothem, or, ifthiswasnotpossible,there wasatleast afinite continuousrangeofvalues opentoeach constant ofintegration. Thedistinguishingfeature ofthenewdynamicsisthatthere arenolonger continuousrangesofvaluesopentotheconstants ofintegration,butonly certain definite discrete values.Generally speaking,thevalues available for each constant ofintegrationareaninfinite setassociated with thenatural integers 1,2,3,...as,forinstance, the setofvalues obtained bytaking integral multiplesofagivenconstant. Thus theconstants ofintegration shew asortofatomicity. Aparallelcanbefound intheatomicityofelectricity.Inthe earlier chaptersofthisbook,wetreated electricchargesasbeing capableofcon- tinuous variation; forinstance, incalculatingtheenergyofacondenser in §97wetreated theelectric chargeeaschanging continuouslyandintegrated withrespecttode.Inactual factweknow that inchargingacondenser, the chargemustmove bywhole electrons atatime. Theprocedureoftreating thechargeascapableofcontinuous variation was,nevertheless, legitimateso longasweweredealingwithchargesofbillions ofelectrons; ourstepde could bequite insignificantincomparisonwith thetotal value ofe,although representing perhapsamillion electrons. Thesameprocedure would, how- ever, lead todisastrous errors ifitwere followed inproblemsofatomicphysics. Anatom normallyisanelectricallyneutral structure, thetotalchargeofthe positivenucleus andthenegativeelectronsbeingzero. Itispossibleto chargeitpositively bywithdrawing one,two,three ormore electrons. Thus theatom canhave apositive charge E,butEisrestricted tohavingthe discrete values e,2e,Se,...etc.;itmaynotberegardedascapableofcon- tinuous variation. Inpreciselythesameway, althoughforreasons notclearly understood, theconstants ofintegrationinthenewdynamicsarelimited todefinite discrete values which mayperhaps,inasimilar manner, beintegral multiples ofafundamental constant. Itmayforinstance bepossibleforaconstant ofintegrationtohave anyoneofthevalues 0,c,2c,3c, ...butnomore possibleforittohave thevalues heor|cthan itispossibleforanatom to carrythecharges \eor|e. There isafurther difference between thenewdynamicsandthe old. Under theolddynamicsaconstant ofintegrationwasatrue constant, and retained itsvalueabsolutely unchangeduntil theconditions oftheproblem 634 TheElectrical Structure ofMatter[ch.xxi altered. Thenewdynamicshasnoknowledgeofsuch absoluteconstancy. Ifaconstant ofintegrationcanhaveanyoneofthevaluesc,2c,3c,...and hasoneofthese values, say 2c,atagiven instant, there isapossibilityofits valuetaking ajumpfrom thevalue 2ctosome other value. Itisusual to think ofthesejumpsasoccurring absolutely spontaneously, althoughthis conceptionisprobably onlyacover forourignoranceofsomeunderlying mechanism. Thenewmechanics wasoriginally developed byPlanck andothers froma studyofthephenomenaofblack-bodyradiation. In1913 Prof.N.Bohr appliedthenewsystemtotheproblemofatomic motions andwas ledto atheoryofthenature ofthese motions whichgained immediateacceptance andwhich hasstood thetestoftime. Thistheory weshallnowexplain. Bohr's Theory. 713. Letusconsider thesimplestcase ofasingleelectron ofchargee describinganorbit about anucleus ofcharge E,which, onaccount ofits muchgreater mass, istreated asafixed centre offorce. Inordinary polar coordinates theequationsofmotion oftheelectron are eEm{r-r82 )=-~(783), ™j t(r2 0)=(784). Equation (784)atonceyieldstheintegral mr2-=cons(785), butinaccordance with theprinciples justexplained, wemaynotsupposeall values tobepermissiblefortheconstant ontheright.We shallsuppose that itisrestricted tobeinganintegral multipleofafundamental constant which, toagreewithanestablished notation, weshall denotebyhjlir. Thus equation (785)must bewritten intheform mr20=Th/2>ir (786), where risaninteger. Usingthisvalue for6toeliminate theangle from equation (783), weobtain 1frhy eE,HoH. r3m\27r/ r which, onintegration withrespecttor,yieldstheintegral •2 ,1fTh r2m\zireE-—=cons(788). Utilising equation (786),thisassumes theform eE £??i(r2+r22 )=cons(789), 712-715] Bohr's Theory 635 which isatonce seen tobetheintegralofenergy,butagaintheconstant on therightmust berestricted tocertain definite values, justaswasthecase with theright-hand member ofequation (785). 714. Beforeproceedingtothecomparatively complicated general case, letusconsider thesimple problemofcircular orbits.Assumingthat circular orbits arepossible, these willbeobtained byputtingr=inequation (787) giving T2 /l2 r=i^s<70O>- Ashasbeen seen, thehydrogen atom consists ofasingleelectrondescribing anorbit about anucleus ofcharge e,while thehelium atom consists oftwo electronsdescribingorbits about anucleus ofcharge2e.Thus onputtingE=e theforegoing analysisisapplicabletoanormalhydrogen atom, while onputtingE=2eitbecomesapplicabletoahelium atom fromwhich one electron hasbeen removed—i.e., toapositively chargedhelium atom of chargee. Thus thecircular orbits inboth these atoms areobtainedbygiving variousintegral values totinequation (790). The radii ofthevarious circular orbits which arepossibleforeither atom areseen tobeproportional tothesquaresofthenatural numbers, andsoto1,4,9,16,25, ...,while the radiipossibleforthepositively charged helium atom areexactlyhalfthose which arepossibleforthehydrogen atom. 715.Mention hasalreadybeenmade ofthepossibilityofwhatappearto bespontaneous jumps taking placeinthevalues oftheconstants ofintegra- tion,andtherefore also inthevalue oftinequation (790).Weproceedto discuss thesechanges. Correspondingtothevalues ofrand 6determinedbyequations (786) and(790), thenegative energyWofacircular orbit isfound tobegiven by W=-lmr*6* +—=^h%(791). Thevalues ofWformadiscrete series, proportionaltotheinversesquares ofthenatural numbers. Ifaspontaneous changeoccurs intitcanonlybe from ahighervalue ofttoalower value, sinceanychangeinthereverse direction would lessen thevalue ofWand soincrease theenergyofthe system.IfTaisgreaterthan t2,both Tjandt2being integral numbers, a spontaneous jumpfrom t=txtot=t2results inthesystem losing energyof amount 27r2e2#2m,/1 1\ fYrnex , .(792). AccordingtoBohr'stheorythis lostenergyleaves thesystemintheform ofradiation; indeed thetheory supposesthat theatoms wehave been 636 TheElectrical Structure ofMatter[ch.xxi consideringdonotemit radiation atallexceptontheoccasion ofjumpsof thekindwehavebeenconsidering. Achangeinthevalue oft,then, results inachangeintheamount of radiant energyinthespace surroundingtheatom.Now Planck, from his studyofblack-bodyradiation towhich wehavealready referred, hadconcluded thattheradiant energyofanenclosure, ifitchangedatall,mustchange by jumps. The radiation inanyspaceorenclosure canbeanalysed byFourier's theorem into trains ofwaves ofdifferentfrequencies, andtheenergyofthe radiation canberegardedasthesum oftheenergiesofthese trains ofwaves. The totalenergyofradiation mayaccordinglybethoughtofasthesum of thecontributions from disturbances ofdifferentfrequencies.Planck's con- clusion wasthattheenergyofradiation ofeachfrequency must beanintegral multipleofacertain unit, thisunitbeing equaltoafundamental constant h multiplied bythefrequencyvoftheradiation inquestion. Planck called this unita"quantum."Thus thequantumofenergyoffrequencyvwasequalto hv,andthetotalenergyoffrequency v,beinganintegral number ofquanta, wasrestricted tobeingofamount rhvwhere twasanintegral number. It followed thatanychangeinthefield ofradiation must consist ofajumpin theenergyoftheradiation ofadefinitefrequencyandmust beequalin amount toanintegral number ofquantaofthisradiation. Inview ofthese results ofPlanck, itwasnatural forBohr tosupposethat whenenergyoftheamountspecified byformula(792) wassetfreeintospace, itformed onequantumofenergy.Thissupposition byitself suffices tode- termine thefrequencyoftheradiatedenergy.Forhv,thequantumofenergy, must beequaltoexpression (792)inorder tosatisfytheprincipleofthe conservation ofenergy, andthisgivestherelation '-*&-£)(793>' where^=™>(794). Thus Bohr'stheoryrestricts theradiation emitted from ahydrogen atom, orfrom apositively chargedhelium atom, tooneofthefrequencies specified byformula(793) where t2andrxarepositive integers.Thisgivesaseries ofdetachedfrequencies,orwhat thespectroscopistcalls a"line-spectrum," whereas itiseasilyseen that theclassicalsystemofelectrodynamics would havepredictedacontinuousrangeoffrequenciesora"continuousspectrum." 716.Thespectrumofhydrogen,asobserved inthelightfromanordinary vacuum tube, consists ofaseries oflines, thestrongestofwhich(Ha)liesat theredendofthespectrumwhile theremainder(Hp,Hy,Hs,...)spread out, ateverdiminishing distances, towards the violet. Asfarback as1885Balmer hadfound thatthefrequenciesofthedifferent members ofthis series could 715-717] Bohr's Theory 637 berepresented,withvery great precision (about onepartin200,000), by givingtonthevalues 3,4,5,...intheformula —xik-xl(795)> ,22 withiV=32902 x1015 .Itisclear that ifwearefreetoassignthisvalue to theNwhich isdenned byequation (794), then Bohr's theoretical formula (793)willcontain Balmer's observational formula asthespecialseries oflines obtained ontakingt2=2.Wearenotfree toassign anyarbitraryvalue to theNofequation (794)since thevalue ofevery quantitywhich enters into Nisknown. The values ofeandmhavealreadybeengiven (§28);for hydrogen Eisequaltoe,andthevalue ofhcanbeobtained from astudyof thespectrumofblackbodyradiation andinavarietyofother ways. The bestdeterminations ofhgive h=6-545xlO-27 , andonsubstitutingthese values forh,eandm,thevalue ofiVgiven by equation (794)isfound tobe N=3-294xlO18 , whichagrees,towithin theerrors inthedetermination ofeand h,with the observed valueN**3'290 x1015 . Itis,then, clear thatNhaspreciselytherequiredvalue inequation (793), andthat Bohr's theoretical formula (793) includes Balmer's observational formula(795)asaspecialcase. Itwas this success ofBohr'stheorythat broughtabout itsimmediateacceptance bythemajorityofphysicists. The theory, however, requiresthatBalmer's series should beonlyoneofaninfinite number. Other series areobtained byputtingt2equalto1,3,4,5,...ooin equation (793), and,ifBohr'stheoryiscorrect, these seriesought equallyto appearinthehydrogen spectrum. Themajorityofthelines ofthese series wereunknown when Bohr'stheorywas firstpublished,but allthepredicted lineshavebeen found which lieintheregionofthespectrum which isacces- sible toobservation. 717.Accordingtothistheorythespectrumofpositively chargedhelium isthesame asthat ofhydrogen exceptthatEinequation (794) must be replaced by2einstead ofby e,asforhydrogen. Or, ifwedenote thevalue ofNforhydrogen byNH ,thevalue ofNforhelium willbe^NHandthe spectrumwillbegiven by "***&-& Inthisformula even values forboth t2andX]givevalues ofvwhich are identical with thewholesystemofvalues ofvgiven byequation (793)forthe hydrogen spectrum. Thespectrumofionised heliumought accordinglyto shew allthelines ofthenormalhydrogen spectrum and, inaddition, the 638 TheElectrical Structure ofMatter[en.xxi various lineswhich areobtained bygivingoddvalues tot2ort2orboth in theabove formula. This isinactual factfound tobethecase, exceptfora reservation which mustnowbeexplained. Indeducingformula (793)weassumed thenucleus tobesomassive that itcould betreated asafixed centre offorce. Inactual factthenucleus of thehydrogenatom hasabout 1844 times themass ofthenegative electron, sothat thisassumptionleads toanerror oftheorder ofonein1844 inthe value ofNforhydrogen.Forhelium theratio ofthetwomasses isabout 7300 toone,andthepredictedvalue ofNforhelium isinerrorbyabout onepartin7300.When allowance ismade forthese errors, thevalue ofiV forhelium isnotexactlyfourtimes thevalue forhydrogen,andthedifference isshewnspectroscopically bythehydrogen spectrumnotcoinciding exactly with thecorrespondinglines oftheheliumspectrum. Bymeasuringthe distance between correspondinglinesFowler deduced thevalue 1836 forthe mass-ratio hydrogennucleus andthenegativeelectron. Allowingforthe relativitycorrection andother refinements Paschen subsequentlyamended this to1843-7, avalue which isinexcellentagreementwith thevalues ofthis ratiodetermined byother methods. 718.Sofarwehave considered onlycircular orbits. The orbit ofan electron about anucleus is,however, innowayrestricted tobeing circular; asthelawofforce isthat oftheinversesquaretheorbitmaybeelliptic, parabolicorhyperbolic. But,justastheradius ofacircular orbit isrestricted tohavingcertain definite values, sotheeccentricityofanelliptic orbit, as well asitsmajor axis,isrestricted tohavingcertain values. Anadequatediscussion ofthemanner ofcalculatingthese restrictions wouldcarryustoofaroutside thescopeofthepresentbook. Thefollowing will,however, suffice fortheimmediatepurposeinhand. Letq1}q2)q3,...bethegeneralisedcoordinates ofanydynamical system, defined asin§548,and letplyp2,p3,...bethecorresponding momenta defined by !>!=!?etc (796), oqx whereEistheenergy expressedasafunction ofq1}q2,q3>•••andq1}q2,q3).... Forcertaindynamical systemsitispossibletodeduce, byordinary mechanics, anumber ofequationsofmotion such that onlyonecoordinate andthe corresponding momentum, e.g.,q1andp1}enter ineach. Asweshall atonce see,equation (784)isanequationofthistype involving onlythemomentum correspondingtothecoordinate 0,andequation (787)isanother, foritinvolves onlythecoordinate randthecorresponding momentum mr.Thesolution of suchanequationwillbethesame asifthewholesystemhadonlytheone degreeoffreedomcorrespondingtothisonecoordinate, sothat either the coordinate willincrease ordecrease beyond limit, orwill oscillaterepeatedly 717,718] Bohr's Theory 639 between twoconstant extreme values. Inthelatter case itisfound thatthe properrestriction toapplytothemotion ofanycoordinateqxisthatgiven by JPldqi=Th(797), where theintegrationextendsthroughoutawhole oscillation inthevalue ofq,histheconstantalready denned, andrisanyintegral number. Forinstance ifqxisthecoordinate 6intheorbit ofanelectron about a nucleus, themomentum, asdenned byequation (796),ispx=mr26.Equation (785)mayaccordinglybewritten intheform px=constant, which isoftherequired form, since itcontains nocoordinates ormomenta other thanpxandqx.Acompleteoscillation extends fromqi=toq1=27r, sothatequation (797) assumes theform 2irp 1=rh, or mr'26=rhj^ir, which isidentical withourequation (786). Similarly,ifq2isthecoordinate rofthesame orbit,p2=mrsothat equation (787)isoftherequiredform. The firstintegralofthisequationis equation (788), andthismaybewritten intheform eE=-w,£i+—(-mr-m\2i whereWisconstant, thenegative energyofthe orbit. Thisequation gives p£asaquadraticfunction of1/r;itmaybewritten intheform *'=\h)U" r)[r~ rj' where r1}r2aredetermined by (t)~r^ S)'S+S-* w Inthecourse ofacompleteoscillation rvaries from rxtor2andthenback torx.Thus thepathofintegrationinequation (797)maybetaken tobe twice therangefrom rxtor2,andtheequation assumes theform rh\[r* 2WJ,,dr=r'h.(799),1\/1 1\1 r)\r r2j__ where risanewinteger. Evaluatingtheintegral bythetransformation wereadilyobtain 2-=-cos2+-sin2 6>,rrx r2 r, 1 dr=IT V(n^)[Vr 2-VrJ2 . 640 TheElectrical Structure ofMatter[ch.xxi Equation (799)nowbecomes T 2V(nr 2)(V^-Vrj)2' andeach side isatonceseen tobeequalto T+T rx+r2' The elimination of1\andr2from thisequationandthetwoequations (798) gives 2ir2me2E2 w-r^iT'tt(800)- (T+TfIV Weobtain allpossiblevalues forWbygiving integralvalues torandr. Itisatonce seen thatnonewvalues areintroduced beyondthosealready discovered inequation (791). Byawell-known formula theeccentricityeoftheorbit isgiven by ^'tSS-p^ji(801)- and,since rand t'arenecessarily integrals,itisclear thatonlydefinite values arepermissiblefortheeccentricity. Thesemi-axes a,boftheorbit are related by --=1-a2e" sothatequation (801) shews that bjawillbecommensurable forallorbits which canbedescribed. Byawell-known theorem, theenergyofanellipticorbit isequaltothat ofacircular orbitwhose radius isequaltothesemi-major-axisoftheellipse. Itfollows fromequation (790) thatthesemi-major-axisaofanellipticorbit isgiven by nh°-a=4^><802>' where niswritten fort+t,while fromequation (801)theeccentricityis given by e2=l--, (803).n- The orbits n=1(t=1),n=2(t=1or2),n=3(t=1,2or3)andn=4 (t= 1,2,3or4)areshewn infig.140*. The distance ofclosestapproach tothenucleus isa(1—e)which isgiven by ^2a (-1~6)=A 2J?n(n~W-T2 ).4nr2eEm *Reproduced bypermission from apaper byBohr {Nature, July 7,1923). 718-720] Buhrs Theory 041 Theexpression ontherighthas itsminimum value when t=1and n=oo,namely Thus theelectron neverapproachesthenucleus towithin adistance less than halfoftheradius ofthesmallest circular orbit. Fig. 140. 719. Since theextension toellipticalorbits hasintroduced nonewvalues ofW,itfollows thatthespectrumwillconsist ofthose lineswhich werepre- dicted bythesimple theoryofcircular orbits andnoothers. Nevertheless thepossibilityofellipticalorbits hasintroduced anessential difference into thespectrum.Since anorbit ofanypermissible energy WjorW2cannowbe described inmore than oneway,itfollows that afallfromenergyW2to energyWxcanoccur inmore thanoneway,sothatthespectrallinewhich is produced byafallfromenergyW2toenergyWxmayappropriatelybethought ofasthesuperpositionofanumber oflines, allofwhich, although having preciselythesamefrequency,areproduced bydifferent events. 720. Itispossibletoseparateoutthese coincident lines inavarietyof ways. Perhapsthesimplestisbyplacingtheradiatingatoms inamagnetic field. Each electronic orbit isaffected bythe field,anddifferent orbits, even thoughofthesame energybefore the fieldwasput on,willbeaffected in different ways.Itfollows that thespectrallines which wereoriginally coincident willbedisplacedtodifferent extents, asisobserved intheZeeman effect. Itcanbeshewn that theexplanationofthenormal Zeeman effect which hasalreadybeengivenin§635holds valid evenwhen thequantum- restrictions areappliedtotheelectronic orbits. Theanomalous Zeeman .t 41 642 TheElectrical Structure ofMatter[ch.xxi effectpresentsamorecomplicated problemwhich canhardly yetbesaid to havebeensatisfactorilysolved. Theplacingoftheradiatingmatter inapowerfulelectrostatic field also results inaseparationoftheoriginallycoincident lines, thisbeingknown as theStark effect. Thedynamical theory justexplained providesacalculation oftheseparationstobeexpectedinthis case,andthepredicted separations arefound toagree very closelywith thoseactuallyobserved. 721.Butperhapsthemostinterestingfeature ofallisthat there isa slight separationevenwhen externalmagneticandelectric fields areentirely absent. Intheanalysisof§713,wetreatedmthemass oftheelectron asan absolute constant, although wealready knew(§660) thatthemass varies with thevelocityofmotion oftheelectron. Forcircular orbits thisdoesnotmatter much; themass ofcourse remainsstrictlyconstantthroughoutthedescription ofanysinglecircular orbit, although varying slightlyfromoneorbit toanother. Butinanellipticorbit themass varies fromoneparttoanother ofthesame orbit.When allowance ismade forthis,theorbit isnolonger strictly ellip- tical,andformula (800) only providesafirstapproximationtoitsenergy. When thenecessaryadditional terms areincluded, itisfound thatthevalue ofWnolonger depends solelyont+t',butontand r'separately.Itfollows thateach ofthelineswhich oursimple theorytreated asasuperpositionof coincident linesmust inactual factshew a"fine-structure" ofadjacent slightly separatedlines. Such "fine-structures" areeasilyobserved inapowerful spectroscope. Thetheoreticalseparationstobeexpectedhavebeen calculated bySommerfeld and others, andalthoughtheobservedseparationsareso small astomake exact measurementexceedinglydifficult, there seems no room fordoubt thattheyagreewith thosepredicted bytheory. Thedynamical theoryofthese phenomenaisnotgiveninthepresent book. Thereader whowishes tostudyitisreferred totheoriginal papersof Bohr, Sommerfeld and others, ortotheauthor's"Dynamical Theoryof Gases." 722.Thenewdynamics,ashasnowbeen seen, allots adefinite size to theatom and soprovidesamechanismbywhich atoms have apermanent existence, instead ofradiating awayalltheirenergyandcollapsing.Forthe hydrogenatom theminimumenergyisfound bytakingr+r=l inequation (800),andsince tcannot bezero, thisrequiresthatr=1andt=0.Thus theorbit ofminimumenergyisthecircular orbit ofradius(cf.equation (802)) A2 a=-=-£5- (804). The electrons inhydrogen atoms candescribe circular orbits ofradii 4,9,16, ...times thisandavarietyofnon-circular orbits aswell,but this equationdefines thehydrogen atom initsnormal state ofminimumenergy. 720-723]Bohr's Theory 643 Oninsertingthenumerical valuesalready given,wefinda=0'53x10-8cms., which isingoodagreementwith theradius ofthehydrogen atom asfound in otherways. Dewar found thedensityofsolidhydrogenat132° absolute tobe00763. Thus acubic centimetre ofhydrogenatthistemperaturehasmass-0763 grammesand consists ofatoms ofhydrogeneach ofwhich isknown tohave amass of1*662 x10~24grammes.Itfollows that thenumber ofhydrogen atoms inacubic centimetre ofsolidhydrogenis4-59x1022 ,sothatthespace occupied byeach is2-18x10~23cubic centimetres. This isthespacethat would beoccupiediftheatoms werespheres arrangedincubicalpacking, eachbeingofradius 1*40x10~8cms. In§150wefound that thedielectric constant ofhydrogenisthesame asifthemolecules werespheresofradius 0'916 x10-8cms.Asimilar calculation would havesuggestedthat the hydrogenatom mightberegardedashavingaradiusequaltol/\/2times thisor0*723 x10~8cms. Neither ofthese calculations canlayclaim togreat accuracy;afarmore accurate determination ofatomic dimensions isobtained from theKineticTheoryofGases. Ifthemolecules ofhydrogenareregarded asspheres,thethreephenomenaofviscosity,conduction ofheatanddiffusion agreeinassigningtothesespheresaradius of68x10~8cms., while obser- vations onthedeviations fromBoyle'slawsuggesttheslightlylower value of 0*64x10-8cms.Againthehydrogenatommaybesupposedtohave aradius equaltol/\/2 times that ofthemolecule, sothatthetwovalues oftheatomic radius arerespectively 054x10-8and051x10~8cms., incloseagreement with thevalue 0"53x10-8required bytheelectrical structure oftheatom. Itmust, however, benoticed thattheKineticTheory requirestheatom tooccupyathree-dimensional volume, whereas onBohr'stheorythehydrogen atom isatmost adisc. Ifweimaginethis disc, theorbit ofthenegative electron, tobecontinually changingitsorientation inspacewepassnaturally totheconceptionofthehydrogen atomreservingforitself, orperhaps clearing foritself, aspherical space equalinradius totheorbit oftheelectron. This conceptionisinaccordance with theknown facts ofcrystalstructure. 723.Thetheoryofstructures ofmore thantwoconstituentpartsisfar lessadvanced, nosatisfactorymechanismhaving yetbeen devised foreither thehelium atom orthehydrogenmolecule. Alargeamount ofconsistent evidencesuggeststhattheelectrons ofcomplexatoms arearrangedinshells orrings correspondingtodifferent quantum numbers, butthemethod of arrangementhasnotyetbeenbroughtwithin thescopeofmathematical treatment. Questions ofelectricalconductivityandoftheopticalanddispersive pro- pertiesofsubstances areclearly subjectsfortreatment bythenewdynamics, butonlymeagre progresshassofarbeen made. Thesameappliestothe problemofthenature ofradiation. There isatpresentadivergenceof 41—2 T/2T2 vtTVVry T3> andequations (805) and(806)nowshew that thepossible frequenciesof radiation aregiven by v=sn(807). Ifthemotion oftheelectron, describingitsorbit withfrequency n,had been analysed byFourier's theorem, andtheresultingradiation calculatedby theclassical electrodynamics, weshould havefound radiations offrequencies n,2n,Sn, ..., sothataccordingtotheclassicalelectrodynamics also,thefrequenciesofthe emitted radiation would begiven byequation (807). Itaccordingly appearsthatinthelimitingcase inwhich tand r'areboth large,theoldclassicalelectrodynamicsandthenewquantum dynamics agree inpredictingthesamefrequenciesforthespectrumofemitted radiation. In thislimiting case, theradius oftheelectron orbit isinfinite, successive radii644 TheElectrical Structure ofMatter[ch.xxi opinionastowhether radiation ispropagatedinaccordance with Maxwell's equationsorintheform of"atomic"packetsofenergywhich travelthrough spacewithoutspreadingoutinthemanner demandedbytheclassical electro- magnetic theory. TheCorrespondence Principle ofBohr. 724. Inconclusion wemayrefer toaprocedurewhich holds outsome hopeofbridgingthegulfbetween theclassicalelectrodynamics andthenew electrodynamicsofquanta. When anelectron describes acircular orbit about anucleus, thenumber ofrevolutionspersecond inthis orbit, n,isequaltoB\1it, whence, from equations (786)and(790), 47r2e2#*mn= T3/,3 (805> Theperiodofanellipticorbit isknown tobethesame asthat ofa circular orbit ofthesameenergy,sothat thesameequationwillgivethe frequencyofrevolution inanellipticorbit oftotalquantum number t. Thefrequenciesoftheradiation which canbeemitted ontheelectron droppingfrom this orbit toone oflower quantum number r'are,from equation (792), V=^~{^-^(806)- Iftheintegerstandrarebothlarge,anddifferonlybyasmallnumbers, which must ofcourse alsobeintegral,theapproximatevalue of(-^ -) willbe 723,724] TheCorrespondence Principle 645 onlydiffer byaninfinitesimal fraction ofeach,andorbits ofalleccentricities arepossible.Thus theelectron isjustonthevergeofbecomingafree electron. Thislimitingcaseprovidesabridge between theoldmechanics andthenew; ononeside ofthebridgetheclassicalelectrodynamicsholds undisputed sway,butaswecross thebridgeandadvance intotheterritory ontheother side, the additional restrictions imposed bythequantum dynamicsbecome evermoreimportantuntilfinally theymaybeconsidered togovernthewhole situation. Theexplorationoftheterritoryonthefar sideofthebridgewillprovidework foranewgenerationofmathematical physicists;thepresent workattempts onlytobringthereader asfarasthe bridge,and tomake clear tohim that ifhecrosses ithemustexpectto find different conditionsprevailingontheother side. INDEX Thenumbersrefertothepages, [pp.1—299, Electrostatic Problems,pp.300—end,Current andMagnetic.'] Aberration, 593,609 Abraham, 525,584,589 Absorption oflight, 545, 554,555 ,, bands, 554 Accelerated electron, 577,592 Action atadistance, 140 ,,mechanical, seeMechanical action, Mechanical force ,,principleofleast, 488,517,579 Adams, E.P.,515 Alternating currents, 456,465, 477, 501,528 Amber, electrification of,1 Ampere, 3,507 (unitofcurrent), 305, 530,531 Ampere's law(fieldofacurrent), 439,514 Angle ofconductor, lines offorce near, 61 Anion, 308 Anisotropic media, 134, 152,535 Anode, 308 Argand diagram, 262 Argument ofacomplex quantity, 262 Arons, 362 Atom, structureof,22,630,634;seealsoMolecule Atomic heat, 556 ,,nature ofelectricity, 21,309,631,633 ,,numbers, 630 Attracted-disc electrometer, 105 Ballistic galvanometer, 437 Balmer's Series, 636 Barnes, E.W.,199 Batteries, workdoneby,104,506 Biaxal harmonics, 241 Bohr, 634 ff.,644 Boscovitch, 141 Bound-charge, 126,361,552 Boundary-conditions,indielectrics (electro- static), 121,178 )> )) 648 Index Conformal representation, 264,280 Conjugate functions, 261-279, 286 ,, conductors, 328 Contact difference ofpotential,303 ,,conductors in,101, 303,317 Continuity, equation of,344,476,559 Contracted coordinates, 583 Contractile electron, 589,595 Contraction hypothesis (Lorentz-Fitzgerald), 594, 597,606 Correspondence principle, 614 Coulomb's torsion balance, 11,365 law(R=4t«t), 45,121 ,, (unitofcharge),530 Crystalline media, 134, 152,535 Current- sheets, 480 Currents ofelectricity, 22,300,306 ,,inlinear conductors, 300, 452, 496, 499,502 „ ,,continuous media, 311, 473, 502, 526,544, 555,557 ,, ,,dielectrics, 358,510,550 induction of,452,473,496,562 magneticfield of,425,438, 513,514, 573 ,,measurement of,305,314 ,, slowly-varying, 331 Curvilinear coordinates, 242 Cylindrical conductors andcondensers, 67,73, 187, 195,257-279 D'Arsonval galvanometer, 436 Debye's theory ofspecific heats, 556 Declination, magnetic, 401 Deformable electron, 589, 596,611 Diamaguetism, 410,505 Dielectrics, 74,115 ,, boundary of,121,178 ,, currents in,358, 510,550 ,, images in,200 ,,inductive capacity of,74,115,532 ,, molecular action in,126,551 ,,stresses andmechanical action in, 172-181, 201, 579,619 ,, time ofrelaxation of,359 Dip,magnetic,401 Disc, circular orelliptic, 248,249 Discharge ofcondenser, 88,331, 361,458,498 Dispersionoflight, 532, 553,643 Displacement (electrostatic), 117, 153,552 -currents, 155, 510,514,528 ,, -theory ofMaxwell, 153,510,514 Dolazalek electrometer, 110 Doppler-effeet, 609 Doublet, electric, 50,168, 193,215, 232,551 Drude's theory ofconduction, 557 Dynamical theory ofcurrents, 485>»Dynamo, action of,458,465 Earnshaw's theorem, 167 Eddington, 623,624 Eichenwald, 605 Einstein, 597, 598,599, 600, 602,621 Electric charges, force between, 11,12,13,37 ,, ,, equilibrium of,23,167 „currents, seeCurrents „ intensity, 24,31,117, 121,571,575 „ potential, 26,31,121, 569,570,575 ,,screening, 62,97,548 Electricity, measurement ofquantity of,8,77, 109,437 ,, positive andnegative, 8 ,, theories of,19,20 Electrification, 5 ,, atsurfaces andboundaries, 18, 21,45,61,194,347 byfriction, 1,9 byinduction, 16,125,186 ,,lineofzero, 88,194 Electrokinetic momentum, 498 Electrolytic conduction, 307 Electromagnetic field, general equations of,568, 602 ,,Weyl's theory ofthe,623 mass, 585,596, 611,613 momentum, 583,615,617,620 theory oflight, 3,526,532 ff. units, 427,528 ,, waves, 524,525 Electrometers, 105,107 Electromotive force, 303,453 Electron, charge andmassof,20,590 ,, internal mechanicsof,590 ,, motion of,inconduction, 306, 307, 320, 343, 496, 549, 557, 562,563 ,, ,, ,,infreespace, 559 ff. ,, sizeof,586, 590,612 ,, structureof,589,590,596 ,, theoryofconduction, 306.549,555,557 ,, ,, ,,dispersion, 553,554 Electrophorus, 17 Electropositive, electronegative, 10 Electroscope, gold-leaf, 7,17 Electrostriction, 181 Ellipsoidal analysis, 230, 244,251 ,, conductors, 246,253 ,, harmonics, 251 Elliptic cylinders, 270 „ disc,248 Energy, conservation of,28,32 „flow of,519,617 „localisation of,151,399,415,443,494, 516, 576,617,620>» »> »> >» »> Index 649 Energy, mass of,613 momentum of,615 ofconductors andcondensers, 83,106 ,,light-waves, 537,617 „magnetic field, 396,399, 415,507 ,,magnetised bodies, 377, 380,381 ,,systems ofcurrents, 443 Equilibrium, points of,59,167 Equipotential surfaces, 29,47-62, 370 Equivalent stratum(Green's), 182,361,375 Ewing, 422 Expansions inharmonics, 211 ,, ,,Legendre's coefficients, 223 ,, ,,sinesandcosines, 259 Farad(unitofcapacity), 77,530 Faraday, 3,74,115, 116, 126, 140, 155, 308, 402, 514,618 Finite current sheets, 481 Fitzgerald, 594 Fizeau's water-tube experiment, 593,607 Flame, conducting power of,6,125 Flux ofenergy, 519,617 Force, linesof,25,29,43,47-58, 62,370 ,,magnetic, 381 ,,mechanical, seeMechanical force „tubes of,44,47-58, 117,371 Fourier's theorem, 259 Franklin, 19 Fresnel, 536 Galvanometer, 433 Gases, conduction in,311 ,,inductive capacity of,132,532 ,,velocity oflight in,533 Gauss' theorem, 33,118,161, 162, 370,386 Generalised coordinates, 489 ,, forces, 493 „ momenta, 493 „ relativity, 598,621 Generation ofelectricity, 9 ,,heat, 320,348 Gravitation, 620,621,629 Green, analytical theorem of,156 equivalentstratum of,182,361,375 ,,reciprocationtheorem of,92,163 Guard-ring, 78,106 Hagen andKubens, 548 Hall effect, 563 Hamilton's principle, 487 Harmonic potential, 224 Harmonics, biaxal, 241 ,, ellipsoidal,251 spherical, 206-223, 233-242, 243 tesseral, 237 zonal, 233Harmonics, tablesof— harmonics ofintegral degrees, 258 Legendre's coefficients, 219 tesseral harmonics, 240 Heat, generation of,320,348 Heaviside, 505 Helmholtz, stresses indielectrics, 177 Hertzian vibrator, 578 Holtz influence machine, 18 Hurmuzescu, 525 Hydrogen atom, 631, 632,636,641 Hyperbolic cylinders, 267,270 Hysteresis, magnetic, 412 Images inelectrostatics, 185-201, 258,281,! Impulsive forces, 493 Induction, coefficients of(electrostatics), 93,96, 97 ,, ,, ,,(circuits),443 „electrification by,16,125,186 ,, magnetic, 384 ,, ofcurrents, 452,562 Inductive capacity ofdielectric, 74,115, 134, 532 ,, ,, ,,crystals,135 ,, „ ,,gases, 132,533 ,, „ ,,liquids, 75,360 ,, ,,interms ofmolecular struc- ture, 130, 134,553 Infinite conductors, resistance in,350 Infinity,field at,56 Insulators andconductors, 5,545 Intensity (electric), 24,32,33,117, 121, 571, 575 ,,ofmagnetisation, 368 Intersecting planes, 188.206 ,, spheres, 206 Inverse square, law of,13,31,37,168,365 Inversion, 202, 258,286 Ion,308 ,,velocity of,310 Ionisation, 311 Jamin, 544 Joule effect inconductors, 320 Kamerlingh Onnes, 558 Kaufmann, 590 Kelvin (Lord), 193, 199, 249, 250, 335, 365, 469 Ketteler-Helmholtz formula, 553 Kirchhoff, 198,287 ,,'sLaws, 311 ,, solution ofwave-equation, 522 Lagrange's equations, 489, 492,493 Lame's functions, 252 650 Index Laplace's equation, 40,42,120,243,245 ,, ,, solution insphericalhar- monics, 206 ,, ,, solution inellipsoidal har- monics, 251 ,, ,, solution inspheroidalhar- monics, 206 Larmor, 553, 565,577 Law offorce, 13,31,37,168,365 ,, ,,between current elements, 441 Least action, 488, 517,579 Lebedew, 538 Legendre's coefficients, 217, 225,231 Lenz's lawofinduction ofcurrents, 453 Leyden jar, 77,277 Lienard, 575 Light, electromagnetic theory of,3,526,532 „velocity of,526,532 ,,dispersion of,532,553,643 Lightning conductor, 61,479 Lindemann, 556 Lines offorce (electrostatic), 25,29,43,47,62 i, ), ,,(magnetic), 370 „„flow, 341 ,, ,,induction, 386 Liouville, solution ofwave-equation, 521 Lorentz(H.A.),553,575,589,592,594,600,602 Lorenz(L.),553 Maclaurin, 553 Maclean, 525 Magnetic field, 369 ,, ,,produced bycurrents, 425 „energy of,396,415, 494,507 ,, ,,ofmovingelectrons,514,561,573 ,,matter, Poisson's imaginary, 375 ,, particle, 366 ,, ,, potential of,372 ,, ,, potential energy of,377 ,, ,, resolutionof,372 ,, ,, vector-potential of,393 shell, 376,426 ,, ,,potential of,376 ,, ,,potential energy of,380 ,, ,,vector-potential of,395 Magnetised body, 367 ,, ,,potential of,372 ,, ,,potential energy of,381 ,, „measurement offorce inside a,381 Magnetism, physical facts of,364,408,425 ,, terrestrial, 400 theories of,3,418,508 Magnetostriction, 417 Magri, 553 Majorana, 608 Mass, electromagnetic, 585, 611,613>> »> »i o »)J» J)Matter, structure of,20,130, 134; seealso Electron andMolecule ,,imaginary magnetic, 375 Maxwell, 2,3etpassim ,,displacement theory, 153,510 ,, theoryofinduced magnetism, 421 theory oflight, 3,526,532 Measurements : charge ofelectricity, 8,77,109,437 current ofelectricity, 314,433 inductive capacity, 74,360 potential difference, 106,107 resistance, 314 Mechanical action intheether, 3,140,579,618 ,, ,,dielectrics, 172 ,, ,,magnetic media, 415 force onacircuit, 439,505 conductor, 102 ,,dielectric, 124,172 ,,moving electron, 561 ff., 579, 581,611 ,, ,, ,,surface, 79,178 Medium between conductors, 140,151 Metallic media, reflection and refraction of light in,546,555 ,, ,, absorption in,545 Michelson, 526 andMorley, 593, 618,620 Millikan, 20 Mirror galvanometer, 437 Molecular theoryofdielectric action, 126, 133, 361 „ „ ,,magnetism, 3,366, 409, 418, 421,508 ,, ,, ,,light propagation, 551 Molecule andAtom, structureof,133,168,232, 550,567, 630,643 ,, radiation oflight from, 577,632 Moment ofamagnet, 366 Momentum, electrokinetic, 498 „ electromagnetic, 583, 615,617,620 ,, generalised, 493 Mossotti's theory ofdielectric action, 127,168 Moving charge, field ofa,513,573 ,, ,, forceona,560 Multiple-valued potentials, 279,429 Muraoka, 362 Nernst andLindemann, 556 Network ofconductors, steady currents in,311, 316,322 ,, ,, ,, oscillations in,499 Neumann's lawofcurrent induction, 453 Nichols andHull, 538 Nicholson, 556 Oersted, 425 Index 651 Ohm (unit ofresistance), 305,530 Ohm's law, 301, 307, 309,343 Oscillations inanetwork ofconductors, 499 Oscillatory discharge ofacondenser, 460 Parabolic cylinders, 267,269 Parallel plate condenser, 77,115,272,274 Paramagnetism, 410,413 Particle, magnetic, .366, 372, 377,393 Pender, 515 Permeability, magnetic, 410 Perot andFabry, 525 Perrotin, 526 Physical dimensions ofelectric quantities, 14, 531 Picard, 505 Planck, 634,636 Plane conductors andcondensers, 69,185,194, 272 ,,current sheets, 480,482 ,,semi-infinite(electrified), 266,273,282 ,,waves oflight, 534 Poincare\ 505,591 Poisson's equation, 40,121 „imaginary magnetic matter, 375,418 ,,theoryofinduced magnetism, 127, 418 Polarisation(electrostatic), 117, 118, 126, 155, 232,536 oflight, 535, 543,565 Polarising angle oflight, 543 Polarityofmolecules, 126 Potential (electrostatic), 26,31,121,345 ,, ,, maxima andminima, 43,167 „ (electric), 570,625 (magnetic), 370, 413,429 (vector), 393,625 ,, coefficients of,93,96,97 Poyntings theorem, 518 Practical units, 530 Pressure ofradiation, 538,615 Principal coordinates, 550 Pulse ofelectric action, 578 Quadrant electrometer, 107 Quadric, stress-, 147 Quantity ofelectricity, 7,8,77,109,437 Quantum theory, 558 Quincke, 181,416,417 Radiant Energy, mass of,613, 614,616 „ ,,momentum of,615,616 ,, ,, nature of,644 Radiation ofEnergy, 577, 592,617 ,, pressure of,538,615 „ from electrons, 576,592Radius ofatom, 642,643 „molecule, 132,613 Rapidly alternating currents, 477,501 Rayleigh (Lord), 358,595 Recalescence, 412 Reciprocation theorem ofGreen, 92,163 Reflection oflight, 540,541, 542, 546,548 ,, coefficients ofmetals, 548 Rafraction oflight, 539, 541,543 ,, ,,lines offorce, 123 „„„flow, 346 Refractive index, 532,553 Relativity, theory of,593 ff. Relaxation, time of(foradielectric), 359 Residual discharge, 361 Resistance ofaconductor, 301,314,355,557 ,, measurement of,314 ,, specific, 342 -box, 314 Resolution ofamagnetic particle, 372 Retentiveness (magnetic), 412,422 Riemann, 527 ,,'ssurface, 280 Rontgen, 514 ,, rays, 311,578 RosaandDorsey, 525 Rowland, 514,515 ,,andNichols, 361 Russell, A.,199 Rutherford, 630 * Saturation (magnetic),411 Saunders, 525 Schott, 575 Schuster, 403,555 Schwarz's transformation, 271 Screening, electric, 62,97,548 Searle, 584 Self-induction, 456 Sellmeyer's dispersion formula, 553 Shell, magnetic,seeMagneticshell Signals, transmission of,332,502 Sine-galvanometer, 435 Soap-bubble,electrification of,81 Solenoid, magnetic, 432 Solenoidal vector, 158 Sommerfeld, 283 Specific heats, 556 Specific inductive capacity,seeInductive capacity Spherical conductors andcondensers, 66,71, 99,100, 189, 192, 196, 226, 228, 231,264 „ bowl, 250 „ harmonics(theory), 206, 233,243 ,, ,, (applications), 224,401 Spheroidal conductor, 248 652 Index Spheroidal harmonics, 254,257 Stark effect, 642 Stewart, Balfour, 402 Stokes, 578 Stokes' theorem, 388 Stresses, general theory of,142 ,, electrostatic, 146,169 ,, indielectrics, 175 ,,„electromagnetic field, 582, 618, 619 ,, ,,magnetic media, 415 Submarine cable, 79,319,332, 351,505 Superposition offields, 90,191 Surface-electrification inconductors, 18,21,37, 45,61,121,194 „ „ ,,dielectrics, 125 ,,harmonics, 208 Susceptibility, magnetic, 410 Tangent galvanometer, 434 Telegraph wire, capacity of,195 ,, ,,transmission ofsignals along, 317, 332,502 Telegraphic equation, 504 Terrestrial magnetism, 400 Tesseral harmonics, 237 Thomson, J.J.,286 Time ofrelaxation, 359 Torsion balance, 11,365 Transformer, theory of,465 Trouton andRankine, 595 Trowbridge andDuane, 525 Tubes offorce(electrostatic), 44,46,47,117Tubes offorce(magnetic),371 ,, ,,flow,341 „ ,,induction, 386 Unicuraal curves, 269 Uniformly magnetised body, 373 Uniqueness ofsolution, 89.163 Units, 14,77,305, 365,427, 522,528 „ ratio ofelectrical, 525, 528,530 Vector-potential, 393, 438, 474,625 Velocity ofelectromagnetic waves, 505,524,525 „light, 526,532 Volt(unitofpotential), 305,530 Volta's law,303 Voltaic cell,302 Voltmeter, 314 Water-tube experiment, 593,607 Wave-propagation, equation of,520,525, 534, 571 indielectrics, 524 ,,metals, 527,545 ,,crystalline media, 536 „ „ velocity of,524,525 Weber's theory ofmagnetism, 3,418,508 Weyl, 623 Wheatstone's bridge, 315,316 Wiechert, 575 Wilson, H.A.,605 Zeeman effect, 564,641 Zonal harmonics, 233l> PLEASE DONOTREMOVE CARDS ORSLIPSFROM THISPOCKET UNIVERSITY OFTORONTO LIBRARY 578 ENGW*. 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