BASICELECTROMAGNETICTHEORY

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BASICELECTROMAGNETICTHEORYFieldTheoryFoundationsandStructure

JAMESBABINGTON

MERCURYLEARNINGANDINFORMATION

Dulles,VirginiaBoston,Massachusetts

NewDelhi

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Originaltitleandcopyright:BasicElectromagneticTheory:FieldTheoryFoundationsandStructure.Copyright©2013byJamesBabington.Allrightsreserved.PublishedbyThePantanetoPress.

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CONTENTS

Preface

1 Introduction1.1 GeneralRemarks

2 MathematicalTopics2.1 SummaryofNecessaryMathematicalFormulae

2.1.1 SomeBasicGroupTheory2.1.2 OperatorsandEigenvalueBusiness2.1.3 DifferentialVectorCalculus2.1.4 IntegralVectorCalculus2.1.5 Functions,Distributions,andAllThingsFourier

2.2 CoordinateSystemsandTransformations2.3 PhysicalConstantsandUsefulNumbers

3 Maxwell’sEquationsPartI3.1 TheLawofCoulomb-ElectricCharge,Force,andBasicMeasurement3.2 TheLawofGauss-TheElectricFieldandSomeBasicMath3.3 The“Magnetic”LawortheLawwithNoName3.4 VoltageandCapacitance3.5 Summary

4 Maxwell’sEquationsPartII4.1 TheConservationofCharge4.2 ElectricCurrentsandtheLawofAmpère

4.2.1 TheMagneticFluxDensityandCurrentCoupling4.3 TheLawofFaraday4.4 TheDisplacementCurrent4.5 Summary

5 PhysicalDegreesofFreedom5.1 TheWaveEquationandSolutions5.2 BasicWaveSolutions-ScalarFieldTheory

5.2.1 SolutionsinSphericalCoordinates5.2.2 SolutionsinCylindricalCoordinates5.2.3 Comments

5.3 DegreesofFreedom-Polarization5.3.1 PolarizationinCartesianCoordinates5.3.2 PolarizationinSphericalCoordinates-TEandTMModes

5.4 GaugeInvarianceandPotentials5.5 Summary

6 PhysicalObservables6.1 TheRLCCircuit6.2 FieldEnergy,Momentum,andAngularMomentum

6.2.1 TheStressTensorandConservationofMomentum6.2.2 ThePoyntingVectorandEnergyConservation6.2.3 ConservationofAngularMomentum

6.3 Polarization(Again)CourtesyofStokes6.4 Summary

7 DistributionsofCharge,MacroscopicMatter,andBoundaryConditions7.1 Multipoles

7.1.1 InterludeonElectricandMagneticDipoles7.2 BoundaryConditions

7.3 ElectromagneticInteractionswithMatter7.4 ElectricalConductivity7.5 FieldEquationswithGeneralMatter7.6 Summary

8 ConclusionsandOmissions

ExercisesandSolutions

Bibliography

Index

PREFACE

Thisbookisintendedasashortintroductiontothefoundationsofelectromagneticfieldtheory.Ithas been based on my experience both as a former field theorist (working on quantum fieldtheories, quantum electrodynamics and Casimir physics) and currently as an applied opticalphysicist.Indeed,mythoughtswereinthelastcoupleofyearstosimplywritedownwhatIthoughtI knew and try to critically examine this. What has been very interesting to see is that, in myopinion,anappreciationofitshistorytogetherwiththecharactersinvolvedisanecessitytogettogripswiththefoundationsofthesubject,andthereforeunderstandingatadeeperlevel.

I was fortunate enough as an undergraduate to be exposed to some quite advancedelectromagneticfieldtheoryearlyon.Mythentutor,RobinDevenish,atHertfordCollege,Oxford,setmesomeveryinterestingproblemsonelectromagneticfieldswritteninthefour-vectornotationofspecialrelativityattheendofmyfirstyear.Thiswasrealclassical fieldtheorythatonewouldtypically study beforemoving on to quantum fields. It came about because I actually wanted tostudy thegeneral theoryof relativity, and Ihadaskedhimhow toget into it. Inhindsight, thesewereexactlytherightproblemstoworkonoverasummervacation,andIthankhimgreatlyforhisdirectionhere.

My prejudice is clearly going to be skewed towards the field theory aspects. However, andimportantly,Ihavenotshiedawayfromexperimentalandmeasurementissuesthatultimatelyarethereasonforintroducingafieldtheoryinthefirstplace.Inparticular,thehistoricaldemonstrationofaprinciple,orthequantificationofalawtakesfirstplace.Themathematicsshouldbetheretocapturetheessenceofanyexperimentalstudiesandprovidealanguagewithwhichtoaskfurtherquestions.With this inmind, I am assuming on the part of the reader a genuine interest in thesubjectandpresentthematerialfromthisstandpoint.

Ihaveincludedsomeproblemsattheendwithsolutions;thesearemeanttoilluminatepartsofthetextwitheitheradefinitecalculationmethod,ortojustseesomenumbers.Inadditiontothese,exercisesarescatteredthroughoutthetexttogivealittlepoketothereadertointeract.

Finally, Iwould like todedicate this book tomywifeSerenaandmy sonAlbertwith the verygreatestaffection.

JamesBabington

June2016

CHAPTER 1INTRODUCTION

Electromagnetismisperhapsthemostbasictheoryweseearoundusinthenaturalworld.Itisresponsibleforthecolorwesee,theforcesthatbindustogetherandforanytechnologybeyondthesteam engine. It is the theorypar excellence in that it is responsible for the macroscopic worldaroundusthatwearepartofandinteractwith.Butitgoesfarbeyondthisasitisalsothegatewayto the sub-atomic world that tries to ultimately explain the existence of matter and forces. Fromtheseconceptsmustemergethestructuressuchastheatomicnucleithatarenecessarytogetanyformoflifegoing.Modernparticlephysicstheoriesthattrytoexplainthenatureofquarksandtheverysmall,andtheevolutionanddynamicsoftheuniverseontheother,takeastheirstartingpointsomething that looksquitesimilar toelectromagnetism(that is thecentral roleofgauge theoriesandsymmetryasagoverningprinciple).

Oneofthemainthrustsofthisbookwastoseehowthetheoryofelectromagnetismis(orshouldbe) consistent, together with making noapriori assumptions about the nature of matter and itsinteractions. Inparticular, Ihave taken theoppositeapproach from thatof the infamousBleaneyandBleaney[5]andabandonedthemodernatomicviewpointfromtheoutset.Obviouslyonehastohavegoodreasonfordoingthisanditisthefollowingquestion;howcanyoubesurethatwhatyouaremeasuringisthegenuinephysicalvariableandnotsomethingthatrequiressomeextratheoryon top or refers back to itself? This is an important point because I suspect many seeminginconsistenciesthatoneencountersinadvancedworkwouldsimplynotappearifthecorrespondingmeasurement theoryhadbeen correctly setupandapplied.All this being said, there is obviouslygreatmeritintakingtheatomicviewpointongoodfaithfromscratchasitspeedsupthetimetobeabletoperformcalculationsandgettogripswithmorematerial.Both[8]and[5]areclassicworksonelectromagnetismandessentialreading;thecurrentbooksimplytriestoaimatamacroscopicandfoundationallevel,andshouldthereforebeconsideredascomplementarymaterial.

Theoutlineofthisbookisasfollows.Inthecurrentchapter,wedescribeinfairlygeneraltermsfieldtheorywithacertainamountofflowerylanguagetotryandgivethebiggerpicture.FollowingonfromthisinChapter2,welistinanoverviewlikewaythenecessarymaththatwillbeusedlateron throughout the book. We do not prove theorem’s here, but rather state them. The form issomewhat terse but gives the notation and conventions used throughout. It gives the necessarylanguagewithwhichtobeabletounderstandthelatterchapters.

Chapter3derivesthefirsttwoofMaxwell’sequations.Boththeelectricandmagneticfieldsareintroducedandthenanequationofmotionforeachisdeduced.Thisisaconstructivechapterandaims firstly tostart theconstructionofa field theory,andsecondly, toshowtheparallelbetweenelectricandmagneticphenomena.

Chapter4 formulates the remaining Maxwell equations, together with the continuity equationandtheLorentzforcelawfordynamicchargesmovinginbackgroundfields.Currentsareincluded,boththemorestandardelectriccurrentandthedisplacementcurrentdeducedbyMaxwellfromtheconsistencyofthefieldtheory.Withtheseinplaceonhasthegoverningdynamicsofthetheory.

Chapter5 describes the attributes of the fields from a mathematical standpoint. This involvessolutions to the wave equations in different coordinate systems. From here one finds thatpolarization is anecessary consequence tokeep trackof thegenuinedegreesof freedom. It alsosurfaces thenwith the realization thatwhatwearedealingwith isagauge theoryand sogaugetransformationsandfixingarediscussed.

Chapter6givesanaccountofwhatwecanmeasureandthecorrespondingquantitiesinthefieldtheory. This is a complicated topic, but essential understanding for the foundations ofelectromagnetismaswellasapplications(suchasphotometry,polarimetry,etc).

InChapter7,wediscusshowfieldsinteractwithrealmatter.Thetwodistinctaspectsherehowchargesassemblethemselvesintomultipolesandhowthedynamicsofthesegiverisetoresponsefunctions.Asummaryandomissionschapterthenfollows.

I have included short exercises throughout the text as a way for the reader to try and assistassimilation.Thesearenotdifficultproblems,rathertheyarejustmeanttofillinthegapstypicallywith the derivation of a result. At the end of the book there are some problems with workedsolutions.Theseareabitmoredifficultandcoversomematerialthatwasnotincludedinthemainchapters.

Awordofcautionaboutnotationusedhereisalsoappropriate.Ihavefoundovertheyearsthatone tends to switch between notation, and this is certainly what the student will encounter inskippingbetweendifferentbooksandpapers.Myadviceonthisfrontisthatitshouldbeclearfromthecontextinwhichtheequationsarepresented.Thusoneshouldnotbealarmedatseeingx,x,xaallmeaning thesamepositionvectorbecause itdependson thesettingwhichonegives thebestaesthetic.Likewise,seeingE(x)writtendownshouldn’t leadthereadertothinkthefieldisonlyafunction of the x Cartesian coordinate. Common sense should prevail here and if there is anyconsternation,thebestsolutionistowritedownoneselftheoffendingexpressioninamorepleasingform.

1.1 GENERALREMARKS

Fieldtheoryisacomplicatedbusiness.TheoriginalconceptionduetoMichaelFaradayandthelatermathematicalsynthesisduetoJamesClerkMaxwellhasremainedfairlywellintactforoveracentury.Initscurrentform,theelectromagneticfieldequationsserveasthebasisforinvestigatinganymicroornanostructure,attheappliedlevel,andistheDNAblueprintformoderntheoryofthesub-atomicworld.

The impression one gets, certainly for school level (in particular A-level in the UK), is thatelectromagnetism isquiteaneatand tidyaffair, and that the subjectdoesn’t requireany furtherinvestigation (save for in applications). I would like to challenge this view, that instead not justbecauseof itshistory,butbecauseasyoustarttoappreciateitsfoundations, it isricherbutmorecomplex.

Thereare couple of points that Iwould like the reader tohaveat thebackof themindwhileusingthisbook.Oneisofmeasurablesandtheotherisofdisturbances.Bothofthesearestronglyinterconnectedandcannotreallybeconsideredbythemselves.Sincethisisaquantitativescience,itisobviouslynecessarytobeabletocalculatequantitiesthatcanthenbemeasured.Infactthisisthepurposeofthetheoryinsomerealsense(thisdependsverymuchphilosophicalschoolonehassubscribedto).Themeasurableswillthereforerequireasetofdevicesthataresensitivetoandcanrespondaccordinglytothequantitybeingmeasured.Thismeansthatthereiscouplingbetweenthedeviceandtheelectromagneticfieldquantitybeingmeasuredthatnecessarilyhasacertainamountofextrastructureandassumptionsbuiltintoit.Clearlythiswilldisturbtheoriginalconfiguration,which may well be small, but there is necessity to be able to quantify this. Admittedly, this is adifficultprogrammetopursuerigorouslyandaheuristicapproachwilloftenberequiredtomakeprogress. Nevertheless, for those with the predisposition, I would like to advocate a rigorousapproach as a topic for further research. My hope with the current book is that it can stimulatethoughtsinthisdirection,aswellasintroducingthebasicsubjectmatteratanintroductorylevel.

A closing comment concerns the history of the subject. Obviously with a history as long aselectricity and magnetism, this is going to be varied, full and sometimes confusing. It seemed ashamenottogivesomeoftheoriginalprotagonistsalittlebitmorecharacter.Tothisend,IhavemadesomehistoricalinclusionswhereIfeltitilluminatingorworthyacknowledgementthatfitsthegeneraldevelopmentofthetopicsaddressedherein.

CHAPTER2MATHEMATICALTOPICS

In this first chapter,wewill provide essentially a look up table ofmathematical formulae andconceptsthatfeatureprominentlythroughoutthebook.Itwillconsistofbasicallyadefinitionorastatementaboutmathematicalquantitiesthatareneededinthedescriptionandmanipulationofthefield equations. The key topics are vector calculus, a bit on Fourier transforms, and coordinatesystems.Afewsidetopics,suchasgrouptheory,arealsoincludedtoprovidesomeextralanguageforsomeofthemathwewillencounterlateron.Thisisintendedonlytobroadenthevocabularyofthe reader. In fact each of these topics sit in a more general theory, which has in part beenborrowed from. For example, vector calculus and coordinate systems naturally lie in differentialgeometry,themathematicalstructurethatforoneunderpinsgeneralrelativity.Theobjectiveofthischapter is therefore tooutline thenotationused throughout the followingchaptersand tocollecttogetherrelevantmathematicalresults.Attheendweprovideatableofsomeusefulnumbersandnotationaldescriptionsthatarefairlystandard.See[15]forgoodintroductorymaterialand[2]foramoresophisticatedtreatment.

2.1 SUMMARYOFNECESSARYMATHEMATICALFORMULAE

Wewillfocusonherefieldsthatlivenaturallyonournormalthreedimensionalspace.Alloftheresultsgeneralizestraightforwardly toann-dimensionalspace.Ascalar field is defined simply assomefunction f(x),where and Avector field is defined as a vector valued functionV(x),orintermsofitsthreecomponentsVa(x).Thethreecomponentscanbespecifiedwithrespecttoanorthonormalbasisofvectors(orforthatmatteranyvectorsthatspanthespace),ea(x),suchthat

(2.1)

whereinthelastlinetheEinsteinsummationconventionbeenadopted.Thismeansthatwhenthereis one upstairs and one downstairs index that are the same letter, they are summed over (alsoknownasbeingcontracted).Wewillassumeinthisbookthattheorthonormalbasisvectorsareselfevident and from the stand point of performing operations on vector fields, simply make theidentification

(2.2)

Thisisjustanotationalconvenience.Thereisanimportantdistinctionbetweenanupstairsindexandandownstairsindex.Anupperindexisreferredtoasbeingatangentvectorindex(orsimplyavectorindex),whileadownstairsindexisreferredtoascovectorindex.Notethatthisistrueforthecomponentsratherthanthebasisvectorsindex.

Ifanobjecthasmorethanoneindexthenit isreferredtoasatensor fieldandingeneralwilllooklike

Two important second rank tensors that are the underpinnings of all operations are theKroneckerdeltaandthemetrictensor.Theyaredefinedinthefollowingway

TheKroneckerDelta

(2.3)

Thiscanbethoughtofasanidentitymatrix(andinourcaseis3×3).

Themetrictensorgab:

(2.4)

where ds2 is the square distance of a small line element and dxa are small distances in therespectivecoordinate.Ametricisthenatensorfieldthatallowsonetocomputedistancesbetweenpoints.Thinkingof thisalsoasamatrixallowsus todefine its inverse,gabwithnow two indicesupstairs,

(2.5)

Thekeypropertyofthemetricanditsinverseisthatitisamachinethatallowsonetoraiseorlowerindicesonanytensorfield.

Thescalarproductbetweentwovectorfieldsisdefinedby

(2.6)

(2.7)

Notethatthescalarproductonlymakessensewhenthetwovectorfieldsareevaluatedatthesamepoint.

Thedivergenceofavectorisdefinedas

(2.8)

where the gradient operator is given by the partial derivatives Moreexactly,thistypeofderivativeisreferredtoasacovariantderivative,whichmeansitshouldnotactonanyfactorofthemetric(themetricissaidtobecovariantlyconstant).Sothepartialderivativehastopromotedtoaccommodatethis.Thiswouldberelevantforexampleifyouwantedtodoyourcalculusonthesurfaceofasphere.

Thewedge(orcross)productoftwocovectorfieldsisgivenby

(2.9)andthecurlofacovectorfieldisdefinedas

(2.10)where istheLevi-Civitatensor.Itisdefinedby

(2.11)(2.12)

NotethatthisdefinitionrequiresthattheLevi-Civitatensorhasitsindicesraisedandthereforeinvolvesthreefactorsoftheinversemetric.It isanunfortunateaccidentofthreedimensionsthattheLevi-Civitatensorprovidesonewithamapfromwhatappeartobevectorstovectors, i.e.wetake the curl of a vector and get another vector. Aswith the covariant derivative, there ismoretakingplace.Thederivativeofacovector fieldgivesa two indextensor (with indicesdownstairs)and thencontracting thiswith theLevi-Civita tensor (with indices)upstairs (knownas taking theHodgedualindifferentialgeometry)leadstoatensorwithoneindexupstairs,i.e.avector.Thisiswhyweseeminglygetanothervectorquantity.

ConsideringatwoindextensorFabforsimplicity,atensorissaidtobesymmetricif

(2.13)

andantisymmetricif

(2.14)

The metric tensor is a symmetric tensor (gab = gba), while the Levi-Civita is totally anti-symmetric on its indices. If a tensor satisfies such conditions, then the number of independentcomponents is reduced. For example a second rank tensor has 3 = 9 independent components,whereasasymmetrictensorhas3×2=6independentcomponents.

2.1.1 SomeBasicGroupTheoryTheideasofgrouptheoryaresousefulthatitisworthcollectinghereafewbasicconceptsand

definitions.Thiswilladdclarityinparticulartosomeofthemathematicalissuesencounteredwith

Maxwell’sequations.

A group is defined as a set of objectsG on which we can impose a rule combining pairs ofelements to form other objects This rule of combination must satisfy certainaxioms to make it useful. It might be helpful to keep in mind rotations of a body in threedimensionalspace.Thesetofallsuchisthenthegroupofrotations.Theaxiomsare:

1. Combingelementsisanassociativeoperation,whichmeansthat itissuchthata·(b·c)=(a·b)·c.

2. Forall thereexistsaunitelement definedbyg·e=e·g=g.

3. Thereexistsforevery aninverseelement definedbythepropertythatg·g−1=g−1·g=e.

Asanexampleof agroup, thegroupof rotations in threedimensions iswrittenasG=SO(3)whichmeans inwords“specialorthogonal transformations in threedimensions”.Elementsof thisgrouparesimply3×3matricesRthatperformarotationonapositionvector.Theorthogonalbitmeansthattheysatisfytheconstraintthatanyrotationpreservesthelengthofthepositionvectorand thespecialbitdiscountsreflections (orparity transformations).Anothergroup thatoccurs inelectromagnetism is the groupG =U(1) which is the one dimensional unitary group. This is anexample of an abelian group, where the order of operations is irrelevant, while SO(3) is a non-abeliangroup;performingtworotationsaboutdifferentaxesisdifferentdependingonwhichoneisdonefirst.Thishasthetechnicalnameofcommutation.Therotationsjustdescribedareexamplesofoperatorsifwestartthinkingmoregenerally.Fortwooperators and onecanformanotheroperatorgivenbythecommutatorthatisdefinedby

(2.15)

Twooperatorsaresaidtocommuteif Soabeliangroupshavecommutatorsthatarezero, and non-abelian groups have nonzero commutators. One last remark; the groups we areconsideringhereareall continuousgroupsrather thandiscretegroups (anexampleofadiscretegrouparethepermutationsofasetofnumbers).

2.1.2 OperatorsandEigenvalueBusinessIngeneral,anoperatorisanythingthatactsonanotherobject,forexampleapartialderivative

onafunctionoramatrixonavector.Theeigenvaluesλnandeigenvectors(oreigenfunctions)Vnofanoperator aredefinedbytherelation

(2.16)

Eigenvaluesareuseful inconnectionwithgrouptheorybecausetheycanbeusedto labelhowthe group theory is realized at the practical level of finding solutions (this goes by the name ofrepresentation theory). In turn, the eigenvectors are the indivisible units that the group theoryforbidsfromturningintooneanother.

Theeigenvaluesandeigenvectors(oreigenfunctions)satisfyveryneatrelations inperhapsthemostusefulcasewhentheoperatorisHermitian.AnHermitianoperatorisgivenbythecondition

wherethedaggerindicatestheHermitianconjugateoperation.Inthistheeigenvaluesarereal If the eigenvectors and functions are normalized to unity, then they both satisfy theorthonormalityconditions

(2.17)

foreigenvectorsand

(2.18)

foreigenfunctionsdefinedontheinterval[a,b].Notforbothfunctionsandvectors,theconceptofvector space can be applied to both which the allows eigenvalue problems to be viewed from aunifiedperspective.

2.1.3 DifferentialVectorCalculusWenowlistasetof identitiesthatareuseful inthesimplificationofvectorcalculusequations.

Forscalarfieldsf(x)andg(x)andvectorfieldsVa(x)andWa(x),thereareasetofonederivative-twofieldrelations

(2.19)

(2.20)(2.21)

(2.22)(2.23)

(2.24)

Therearealsoausefulsetoftwoderivative-onefieldrelations

(2.25)(2.26)(2.27)(2.28)

Aninterestingpointtonoteisthatintheabovetheresultingexpressionswilldependexplicitlyonthemetricused.

One can express the total timederivative of a function in termsof partial derivatives and theinstantaneousvelocity

(2.29)

2.1.4 IntegralVectorCalculusItisnecessarytobeabletointegrateoverdifferenttypesofsurfaces.Theseareonedimensional

curves, two dimensional surfaces, and three dimensional volumes. Therefore a knowledge of thedifferentintegrationmeasuresisrequiredsothatonecanswitchbetweendifferentcoordinatesandmetricswithease.

Thelengthofthecurve isobtainedfromthelineelementinEquation(2.4)by

(2.30)

Itisinthisformthatthemetriccanbeseenasthemeansbywhichdistancesbetweenpointsarecalculated.

ThefollowingintegrationmeasuresaredefinedwithrespecttotheLevi-Civitatensor,

(2.31)

(2.32)

where the tensor hab is obtained from the metric gab by restricting it to the two-dimensionalsubspacewheretheareaintegrationistobecarriedout.Thismaybeassimpleaschoosingoneofthecoordinatevaluestobeaconstantvalue.

There are two key integral relations for vector fields. They allow one to transform betweenvolume and surface integrals, and separately between surface integrals and line integrals whentherearederivativesinvolved.Becauseofthedifferentdimensionalityoftheintegrationmeasure,derivativesofthevectorfieldsfeatureintheserelations

Foravolume thatisboundedbythesurface thedivergencetheoremisgivenby

(2.33)

Foranarea boundedbytheloop Stokes’stheoremisgivenby

(2.34)

2.1.5 Functions,Distributions,andallThingsFourierNextweconsidersomebasicpropertiesoffunctionsanddistributions.Thisisnecessarybecause

havingwrittenthefieldequationsfromthepreviousvectorcalculustechniques,weneedtobeabletoanalyzeandextractinformationoutofthem.

AFourierseriesisaninfinitesumoftrigonometricfunctionsgivenbythefollowingexpression

(2.35)

where Inparticular,thef(x)herewillbeaperiodicfunction,sothatforsomeL,f(x+L)=f(x).

AFourier transform of a function in one dimensional space, say f(x) where to anotherfunctioninreciprocalspace(givenbyk)isdefinedby

(2.36)

(2.37)

TheFouriertransformisthecontinuumversionofaFourierseries,wherewelettheintegerthatdefinesthesummationturnintoacontinuousvariable,i.e. ItisasimplemattertoperformaFourier transform in three dimensions (or indeed in any number of dimensions). The above thenbecome

(2.38)

(2.39)

The Dirac delta function, δ3(x), in three dimensions is defined with respect to an arbitraryfunctionas

(2.40)

It isnotreallya functionbutratheradistributionwhichmeansthat itshouldbesat insideanintegral.Itisusedsooftenthatonecannotreallydowithoutit.Anintegralrepresentationofthethedeltafunctioninthreedimensionsisgivenby

(2.41)

Theconvolutionoftwofunctionsf(x)andg(x)isdefinedby

(2.42)

UsingFourier transforms in thisdefinition results in theconvolution theorem; ifF(k)andG(k)aretheFouriertransformsoff(x)andg(x),thentheirconvolutionsatisfiesthesimplemultiplicativeproperty

(2.43)

TwousefulexpressionsforwhatareGreenfunctionsoftheLaplaceequationare

(2.44)

(2.45)

(2.46)

AnintegralrepresentationoftheaboveGreenfunctionisgivenby

(2.47)

Afunctionf(x)issaidtobesquare-normalizableif

(2.48)

Thisshowsuptimetotimewhenitisnecessarytoensurethatphysicalfieldsarenotendingupwithinfiniteenergy.

2.2 COORDINATESYSTEMSANDTRANSFORMATIONS

The vectors and tensor quantities encountered so far require us to specify both a set ofcoordinatesxawhichlabelspointsinthespaceweareworkinginandavectorspacewithwhichatensor can be expanded in. The components of a tensor transform under a coordinatetransformationaccording tohowdifferentialsorderivatives transformbyapplicationof thechainruleofdifferentiation.Forexample,thelineelementdxacanserveasabasisoforthogonalvectors(ifthemetricisdiagonal)inwhichaco-vectorfieldcanbeexpandedin.Inthenewcoordinatesx′awehavethetransformationlaw

(2.49)

Sincewehaveanupstairsindexhere,thisservesasthetransformationlawforavectorfield

(2.50)

Inasimilarfashion,ageneralcoordinatetransformationonaderivativeoperatorisgivenagainbythechainruleas

(2.51)

Thelowerindexheregivesnowthetransformationlawforaco-vectorfieldas

(2.52)

Giventhenaparticularcoordinatetransformationthetransformationmatrix canthenbecalculatedexplicitly.Whilethisgivesasetofvectorsthatareatleastlocallyorthogonal,itremainsto normalize them to unity in order to obtain an orthonormal basis. Themajority of expressionsencounteredatleastatabasiclevelareexpressedinanorthonormalbasis.

StandardCartesiancoordinatesaregivenby(x,y,z)togetherwiththeirlineelement.Withournumberingconventionthesearegivenby

(2.53)(2.54)(2.55)

ThelineelementisjustthefamiliaronebasedonPythagoras’stheorem

(2.56)

TheLaplacianoperator isgivenby

(2.57)

Thepositionvectorxcanbewrittenintermsofanorthonormalbasisas(2.58)

Thecomponentsofthewedgeproductwithrespecttotheorthonormalbasis(ex,ey,ez)are

(2.59)

(2.60)

(2.61)

Spherical coordinates are defined by the coordinate transformation from Cartesiancoordinatesby

(2.62)(2.63)

(2.64)withr≥0,0<θ<π, Noticethatthereisadegeneracyinthecoordinatesattheorigin;atr=0,θand canbeanythingandstilllabelthesamepoint.Thecorrespondinglineelementis

(2.65)

FollowingfromthistheLaplacianoperatorisgivenby

(2.66)

Thecomponentsofthewedgeproductwithrespecttotheorthonormalbasis are

(2.67)

(2.68)

(2.69)

Cylindricalcoordinates(r,θ,z)aredefinedbyanothercoordinatetransformationfromCartesiancoordinatesby

(2.70)(2.71)

(2.72)withr≥0,0<θ<2π,0<z<∞.Thelineelementisgivenby

(2.73)

TheLaplacianoperatoristhen

(2.74)

Thecomponentsofthewedgeproductwithrespecttotheorthonormalbasis(er,eθ,ez)are

(2.75)

(2.76)

(2.77)

Thesearethemostoftenusedcoordinatesystemsinelectromagnetism,butthereareothers.Itlargelydependsontheproblemathandthatistryingtobesolved.

2.3 PHYSICALCONSTANTSANDUSEFULNUMBERS

In this last section, we list a number of physical constants, useful numbers, and somemathematical notation used throughout. The standard SI units are used whenever a numericalcalculationisperformed.

PhysicalConstant,Property/Notation Symbol Value

Speedoflight c 2.998×108ms−1

Permittivityoffreespace 8.854×10−12C2m−2N−1

Permeabilityoffreespace μ0 4π×10−7NA−2

Elementaryelectriccharge e 1.602×10−19C

GravityofEarth g 9.81ms−2

Earth’smagneticfieldatequator 3×10−5T

IntensityofsolarradiationatEarth’ssurface ≈1.0kWm−2

Typicalrefractiveindexofglass ntypical ≈1.5

Typicallaboratoryelectricfield Etypical ≈104Vm−1

Typicallaboratorymagneticfield Btypical ≈10T

Opticalfrequency fopt. ≈1015Hz

Theintegers

Therealnumbers

Thecomplexnumbers

CHAPTER3MAXWELL’SEQUATIONSPARTI

Inthischapter,wewillthinkabouttheformulationoftwoofthefieldequationsfromthelargerset collectively known as “Maxwell’s equations”. They areGauss’s law and themagnetic versionwhichissometimesreferredtoasGauss’smagneticlaw.ThisiswheretheelectricfieldEaandthemagneticfluxdensityBawillmaketheirfirstappearances.Theelectricfieldisintertwinedfromtheoutsetwiththenotionofelectriccharge,whilethemagneticfluxdensityhasnosuchcounterpart.ThesequantitiesareintroducedasawayofaccountingforobservedstaticforcesFaonmacroscopicbodies.Fortheelectric field it isduetoacquiringanetchargebysomemeans; for themagneticfluxdensityitismorecomplicated.

3.1 THELAWOFCOULOMB-ELECTRICCHARGE,FORCE,ANDBASICMEASUREMENT

Letusconsiderfromfirstprinciplesanumberofobservationsandexperimentsthatleadtotheidea of electric charge and the electric field. Much of this may be considered to be somewhatelementary material. It is interesting, however, to have an historical perspective and to clearlydelineate the foundations of the subject. Ultimately we must be able to trace back any usefuldefinitions or theoretical abstractions to a set of basic physicalmeasurables. For an illuminatingandusefuldiscussionofthishistoryandthebasicsetofexperimentssee[11].

Experiment1(Greekantiquity-circa1700s)Taketwobodies(forexampleglassandrubber)thatareinitiallyinmechanicalequilibrium.Nextapplyfrictionalforcestotheirsurfaceswithathirdmaterial(forexamplerubthemwithasilkclothorapieceoffur).Whathappensatsomelatertimetothetwobodies?

Observation 1At some later time they are no longer in equilibrium and will feel attractive orrepulsiveforcesFatooneanotherbysomeactionatadistanceagent.

Normallytoproduceamechanicalforceonabodytherewouldhavetobesomephysicalcontactfor this tohappen.This is not the case in theabove (the same is also true forgravity -muchofhistoricalelectrostaticstookitsinspirationformNewtoniangravity).

Thales of Miletus (ca 624 BC–547 BC), according to Bertrand Russell,“Western philosophy begins with Thales”, and so does the path toelectromagnetism.SomeoftheearliestobservationsofstaticelectricitywereduetoThales(around600BC).Thebasicobservationinthisepochwasthatthefossilmaterialamberwouldattractobjectswhenithadbeenrubbedbywool.

This first experiment andobservationgivesus the idea that somequantity canbe transferredbetween bodies and depending on how much and its type will lead to different amounts ofobservableforces.Wearethusledtothequantitywhichwecalltheelectricchargeofabodywhichmight be in the form of discrete point like units that are sat at particular points, or distributed

continuouslyinsomefashionthroughoutabodygivenbyachargedensity.AkeyobservationofthisbehaviorwasmadebyBenjaminFranklin.Heobservedafundamentalbookkeepingprincipletakingplacewhichwenowcallthelawofconservationofcharge:-

Principle 1 - In any system, electric charge is neither created nor destroyed but is simplyredistributedamongstthesubsystems.

Reconsidering our first experimentwhere the frictional forces cause charge to be transferredfromonebodytotheother,weseethateachmusthaveanequalbutoppositeamounttosatisfytheconservationofcharge(giventhatthenetchargeatthestartwaszero).Notealsothatwehaven’tspecifiedanythingaboutthedynamicsofhowthechargemoved,orwhattypeofquantityitis(oneshouldthinkscalar,vector,tensor…here).

BenjaminFranklin (1706–1790), the quintessential polymath, dabbling in ahuge number of areas such as printing, authoring, science, and politicaltheory, aswell beingPresident of theState of Philadelphia and one of thefounding fathers. It is amazing he had any time to do any experimentalphysicsatall.

It isnownecessarytoputtheobservedforcesonamorequantitativefootingbysystematicallyvarying thepreviouslydescribedsystems.Thebasicparametersare theamountofchargeon thetwobodies,theirseparationandshapesandalsothetypesofmaterialtheyaremadeof.Whatwewillbeabletomeasurethenisthemacroscopicforceonthebodiesthatwaspreviouslyencounteredqualitatively. Thiswas investigatedbyCoulomb in the late1700swhich resulted in thenowwellknownCoulomb’slawofelectrostatics:-

Experiment2(Coulomb)Takingtwosphericalbodies(forsimplicity),varytheseparationbetweenthemandtheamountofthequantitytransferredinthepreviousexperiment.

Observation2Theforcebetweenthemisfoundvaryastheinversesquareoftheseparationandisproportional to the amount of the quantity residing on/in the body due to the previous transferprocess.

Anecessarypointtoappreciatehereishowthisforcecanbemeasured.Onewaytomeasuretheelectricforceonabodyisbybalancingitwithaknownmechanicalforce.Wecanalsodothesamebalancing with torques if the geometry of the apparatus is arranged appropriately. The formoriginallyusedbyCoulombisjustsuchasystem,theapparatusbeingknownasatorsionbalance(seeFigure3.1).

Charles-AugustindeCoulomb(1736–1806),asaCaptainreturningtoFrancefromtheWestIndies,Coulombdiscoveredtheinversesquarelawforchargedbodies(1785).Inadditiontothishealsoestablishedforconductorsthatthechargeresidedonthesurface.

ThismethodwasusedearlierbyCavendish tomeasuregravitational forcesbetweenbodies. Itcomprises of a bar suspendedat itsmidpoint by a thin fiber.At one end is a bodywhich canbechargedtoaknownreferencelevel.Onesuchwayistochargeanidenticalbodyoutsideandthenbringthetwointocontact-bygeometrythechargewilldistributedevenlybetweenthetwoandwehave thus defined the reference point. When a second charged body is brought into proximityrepulsionorattractionresults.Thebartwistsandbecausethefiberhasaweakspringconstantonecanobtaina largedeflectionangleof thebar, thusensuringthatweakforcesmapto largeeasilyobserved angles.When the torque due to the electric force balances the restoring torque of thefiber, attaining an equilibrium, we can read off the observed force by the angular deflection.Thereforethetorsionbalancephysicallymeasurestheforceandfromthisanamountofchargecanbededuced,withrespecttothereferencelevel.

FIGURE3.1:Coulomb’storsionbalanceexperiment(1785)TheleftpictureistakenfromCoulomb’s1785memoirsoftheFrenchacademyofsciences.Therightfigureshowstheprinciples

ofoperation,asdescribedinthemaintext.

It is straightforward to codify Coulomb’s law mathematically. Suppose we have two bodieslabelledby Iand Jandthat theyhavepositionsxa(I)andxa(J).Bypositionsweeithermean theircenterofmass,forfinitesizeobjects,orregardthemaspointlikeobjectswhentheirseparationismuch larger than their respective length scales. It is observed that charge is a scalar quantity -varyingtheorientationhasnoeffectonthemutualforce.ThenCoulomb’slawstatesthattheforceonchargeqIduetochargeqJisgivenby

(3.1)

A constant of proportionality has been introduced and we need to think about the units ofmeasure and howmore generallywe shouldmeasure the electric charge on a body. In fact thisturnsouttobeasubtle issuebecausewewill inadditionneedto introduceadynamicalelement.Clearly,boththeobservedforcesandseparationsarehonesttogoodnessmeasurablequantities.Ifwecanidentifythecorrectsymmetryinaproblemthathasbeensetupwiththisinmind,wewillbeable find the ratio of the charges. But ultimately the observed forceswill have to bemadewithrespecttotheundeterminedconstant whichiscalledthepermittivityoffreespace.Ifchargewasinfinitely divisible we would need to define a basic unit, whereas if (as is the actual case forelectronsandprotons) it is found tohaveaminimumdiscreteunit,wecanuse this asourbasicnaturalunitofcharge.Theupshotisthat,atthispoint,wewillnotascribeto aparticularnumberandreturntoitsnumericalvaluelateron.

Anaturalquestiontoaskiswhatistheeffectofanumberofchargedbodiesononeanother.Letusdoanotherexperiment.

Experiment3Supposeweintroduceotherelectricallychargedbodiesinadditiontotheprevioustwoalreadyconsidered.Whatistheeffectononeofthebodies?

Observation3(Theprincipleoflinearsuperposition)Afterthisintroductionandalongenoughtimeforallthebodiestosettledown,theresultantforceonthebodyisthevectorsumduetoalltheotherbodies.

Inconnectionwiththisprinciple,itisusefultointroducetheideaofatestchargeasthisiswhatwewerealluding to in thediscussionofCoulomb’s law.Supposewehaveanumberofbodiesallchargedinsomestaticconfiguration.Ifweintroduceanotherbodythatdoesn’tchangetheexistingconfiguration, we then have a measure of the local force felt there; it doesn’t disrupt theconfiguration and only feels the measurable force acting on it. We will assume that for anymacroscopicconfigurationofbodiesconsidered,atestchargecanalwaysbefound.

Building on Coulomb’s law, it is necessary to think about how tomeasure electric charge onbodies in more detail and quantitatively. To that end, we consider some very basic materialpropertiesandmakeastatementabout thetypesofmaterials involved,aspartof theirdefinitionand grouping relies on the movement of charge (which will anticipate the following chapter, inparticular theconsequencesofcurrents).Wewillnotworryat thispointabout thedetailsofhowchargemovesapartfromsayingthatitdoes.Amaterialissaidtobeaconductorifchargecanmove

freelyinitandaninsulatorifitisimpededandsoremainsstatic.Forinsulators,chargecanbebuiltupinlocalregionsonabodywhereasforconductorsthechargeisredistributedonthesurfaceofthebody.Thenotionofthemovementofchargeisaveryimportantoneasitisthemechanismbywhichsystemscanfindnewelectrostaticequilibriawhenwechangethesystem.Ofcourse,thisisasliding scale; we know charge doesmove and can expect differentmaterials to display differingamounts of resistance to charge movement. The definitions introduced are simply a convenientdivision.Thesedefinitionsresultfromhowdifferentmaterialsareabletotransferchargefromoneanothere.g. for twoconductorsbringingonecharged into contactwith theotherunchargedwillresult in the totalchargebeingsharedeasilybetweenthe two.We lastlydefineground tobe theinfinitereservoirofchargethatwhenabodyisconnectedbyaconductortotheearth(terrafirma),thechargeflowsfromthebodytothereservoir thusbecomingchargeneutral - thisreliesontheearththereforebeingconductive.

Onecanmeasuretheelectricchargeonabodybyusinganelectrometeroranelectroscope.Atitssimplestlevelanelectroscopeconsistsofastripofmetalfoilthatishungoverametalbarwhichisinturnconnectedtotheoutsideworld.Chargeistransferredontothefoilwhichcanthenmoveupwardssymmetricallyuntiltheelectrostaticforceisbalancedbythedownwardgravitationalforce.Theangleofseparationisthenameasureoftheelectricforceandsincetheseparationisknown,wecancalculate thecharge.Anelectrometeron the theotherhand isoftenamechanicaldevicesimilar in spirit to the torsion balance that requires an auxiliary store of charge to operate. InFigure 3.2, a quadrant electrometer is shown. Its principle feature is that the charge can bedeterminedtoahighdegreeofaccuracy.

Weperformonelastexperimentbeforegettingdowntosomemathandcollectivelybuildingupaschemewithwhichtoperformcalculations.Itwasmentionedearlierthatforconductorswithanetcharge, all the charge resides on the surface. Given that it is now possible to measure electriccharge,wearenowinapositiontoobservethis.

Experiment4Supposewelowerachargedmetalspheresuspendbyaninsulatingthreadintoanuncharged hollow conductor that is insulated from the ground. What happens to the hollowconductor?

Observation4Itisfoundthatchargeisinducedontothesurfaceofthehollowconductorbutnotintoitsinterior.

Thisbasic observationwasoriginally foundbyanumberof scientists (Franklin,Coloumb, andFaraday) in slightly different ways. An electrometer is connected to the outside of the metalcontainerasinFigure3.3.Asthesphereisloweredinside,theelectrometerregistersacharge.Thispersistswhenthespheretouchestheinsideofthecontainer.Takingthesphereout,onefindsnowthatitnolongerhasanychargeonit.Theinsideofthecontaineralsohaslostitscharge,whereasthe outside still retains its original charge which is equal to the original charge on the sphere.Experimentallythenwehavethebasicfactthatforconductors,anychargemustresideonitsoutersurface when the system is in equilibrium. The interested reader can find in [17] a goodcommentaryaboutthesepoints.Wewill return to thispoint lateron in thischapterwhenwearethinkingintermsoffields.

FIGURE3.2:ThequadrantelectrometerdevelopedbyLordKelvininthe1860s.Analuminiumvanewithaknownamountofcharge(thatisstoredviaaLeydenjaratthebottomoftheapparatus)issuspendedbyathinfiberwhichhasamirrorattached.Thevaneinturnissuspendedinsideacircularmetalboxlikestructurethatisdividedintoquadrants.Eachquadrantisconnectedtothe

diagonallyoppositeonewithaconductorsothattheywillhavethesamecharge.Onepairofquadrantsarepositivelycharged,theremainingtwoaregrounded.Asthequadrantsarechargedduetoanexternalsourcethatisbeinginvestigated,thevanerotatesandthereforesodoesthemirror.Shiningabeamoflightoffthemirrorprovidesameasureofthedeflectionactionthus

establishingtheamountofchargeonthesample.

FIGURE3.3:Faraday’sicepailexperiment,fordemonstratingthatelectrostaticchargeresidesonthesurfaceofaconductor,notinitsinterior.Itconsistsofametalcontainerintowhichachargedsphereislowered.Themetalcontainerisconnectedtoanelectrometer,shownonthe

right.

3.2 THELAWOFGAUSS-THEELECTRICFIELDANDSOMEBASICMATH

Forces arewhatwe canmeasure, aspreviously seen, butwhat exactly is there locatedat thepointwherethechargeis?Aconceptualleapisrequiredheretoassertthattheotherchargedbodyproducessomething at the pointwhere the original charge is located, regardless ofwhether thechargeisthereornot.Acouplingisthereforetakingplacebetweentheelectriccharge(whichwehavealreadysaid isascalarquantity)andavectorquantitydue to theotherchargedbody,suchthatonejustmultipliestheother.WecallthisvectorquantitytheelectricfieldEaanditisgivenbythedefiningrelation

(3.2)

foradiscretechargeqlocatedatx.WecanextendthistoacollectionofNchargesbyuseofthelinearsuperpositionprinciple,sothatonehasbysimplesummationthetotalforcegivenby

(3.3)

Thisreadilyextendstoacontinuousdistributionofchargeas

(3.4)

whereρ(x) is the charge density of a body of volumeV. These are the defining relations for theconceptof theelectric field.There isan interesting interplaybetween thecontinuousordiscretenature of the electric charge1 whichwill depend on the length scalewe are choosing to do ourmeasurementsorperformcalculations.Forexample, ifweareworkingwithdiscretechargesthathavesomecharacteristicseparationd,andworkingonalengthscale thenwecantreatthediscretesetofchargesasacontinuumi.e.

The use of the test charge introduced earlier should now be apparent. It provides a localmeasureoftheelectricfieldstrengthatapoint,sinceitwillnotdisruptthelocalfieldthere.Takingtheideaofatestchargefurther,letusassumewehavesomechargeonasmallinsulatingball.Nowimaginewemaketheradiussmallerandsmallerandwedecreasethechargeonthere.Eitherwewillreachapracticallimitorafundamentallimittohowmanytimesthisreductioncanbedone2.WhenthisisdoneCoulomb’slawseemstoholdateachreduction.Thisisfortunateaswearenowabletogiveexpressionsfortheelectricfieldintermsofitssourcecharge.ByusingEquation(3.1)togetherwiththelinearsuperpositionprinciple,thefieldatxisthen

(3.5)

Itisstraightforwardtoconvertthistoacontinuumversionaswell

(3.6)

The reader will notice just how similar these expressions are to what is encountered inNewtoniangravity.Infactthestaticelectricfieldisalsoacentralforcefieldanditiseasilyverifiedthatitsatisfies

(3.7)

(3.8)

(Exercise:verifythis)Aswewillseeinthenextchapter,thisonlyholdsforstaticfields.

Together, Equations (3.6) and (3.7) constitute the equations of electrostatics. To solve theseequationswehavetospecifyforagivensystemitsgeometryandtheboundaryconditionsthefieldmustsatisfy.WecanwriteEquation(3.7)alsoinintegralform

(3.9)

(3.10)

foratwodimensionalarea (withthetopologyofadisk).ThiscanbeconvertedtoalineintegralbytheuseofStokestheoremEquation(2.34)suchthat

(3.11)

Fromthedefinitionoftheelectricfield,weseethatthisisastatementabouttheworkdoneinmovingourtestchargearoundaclosedloop.Inthiscasenoworkisdone,andinparticular,leadstotheresultthattheworkdonebytheelectrostaticforcebetweentwopointsispathindependentforacentralforce.Itisamathematicalresultthatanysuchvectorfieldcanbewrittenasthegradientofascalarfunctionsothattheelectricfieldcanberecastas

(3.12)(Exercise:verifythisispathindependent)

Thescalarfunction isknownas theelectrostaticpotential.Thisquantitywillbe importantwhenwe start thinkingabout voltagesas anobservable.The reader ismore likely tobe familiarwiththeideaofvoltagesfromstandardschoolphysicsandelectronicscourses.

Havingintroducedtheconceptofanelectricfield,wecannowturnthequestionaroundandaskforastatementabouthowtheelectric field isproducedorsourcedbyotherelectriccharges.Weknowalreadythatitisduetothebodyhavinganetchargeofthesametypeasthefirst.Thattheyshouldbeproportional isalsoevident.FromCoulomb’sobservationofan inversesquare law,onecanseethatifweweretosurroundthesourcebodywithaspherethatwascenteredonwherethesourcebodywas,thentheproductoftheareaofthesphereandthesizeoftheelectricfieldshouldbeanumericalconstant.Infactitshouldalsobeproportionaltohowthechargeisdistributedonthesourcebody.ThisisallencapsulatedinGauss’slaw.

Carl Friedrich Gauss (1777–1855), one of the strongest mathematicians ofthe19thcentury(hewasnamedthe“PrinceofMathematicians”).Heworkedon a huge number of different areas in mathematics and physics. Gausspublished this particular result in 1838; however it had already beendiscovered earlier byGeorgeGreen in 1828 andwas later rediscoveredbyLordKelvinin1842.

Wedefinetheelectricflux throughaclosedtwodimensionalsurface whichistheboundaryofthevolume tobe:-

(3.13)

Inintegralform,Gauss’slawstatesthattheelectricfluxforaclosedsurfaceareathatenclosesacertainamountofchargeisproportionaltotheenclosedcharge:-

(3.14)

Theconstant is thepermittivityof freespacementionedearlierso thatwhenweconsiderasinglepointchargeanditselectricfieldatsomedistancewegetbackCoulomb’slawEquation(3.1).

(Exercise:verifythis)

Sinceitisanintegralequationwearemakingaglobalstatementaboutthesystem.However,wecanalsomakealocalstatement(thatis,recasttheequationsothatitholdsateachpointinspace)

byusingoneofourusefulintegralrelations.Thetwo-dimensionalsurfaceintegralcanbeconvertedtothevolumeintegralbythedivergencetheoremEquation(2.33)whence,

(3.15)

We have now replaced here the discrete charge summation with a continuous distribution ofchargedensity integrated,as inearlierexamples.Bydroppingthe integralsGauss’s lawbecomeslocal,soithasadifferentialform.Onefinds

(3.16)

(3.17)

Gauss’slawisthefirstofthefourofMaxwell’sequationandweshallrefertoitthroughoutthetext asMaxwellI.We can also deriveGauss’s Law simply by taking the divergence ofEquation(3.6)andusingEquations(2.44)and(2.46)sothat

(3.18)

Notealsofromthenatureoftheexperimentsandobservationspreviouslydiscussedthatwearedealing initiallywithstaticphenomena; thebodies thathavebeensubjected to forcesviaelectricchargesononeanotherhaveallbeenallowedto“settledown”afterhavingbeendisturbedandsowedonotseeanytimedependenceinourequations.

Twoimportantequationsfollowbytheuseofthescalarpotentialthatisvalidwhendealingwithstaticphenomena.IfweusetheresultEquation(3.12),onefindsthatthescalarfield satisfiesbyvirtueofEquation(3.17)

(3.19)

whichisknownasthePoissonequation.Thisisforaregionofspacewherethereisadistributionofcharge. In fact, this equation can be inverted to express the potential as a function of chargedistribution.ItresultsinascalarformofEquation(3.6)bycombiningitwithEquations(2.44)and(3.12)togive

(3.20)

Ifthereisnochargepresentsuchthatρ(x)=0then

(3.21)which is called theLaplace equation. The solution of these two second order partial differentialequationsisoneofthekeyaspectsinstudyingelectrostaticphenomena.

It isworthwhileputtingapicturetothe ideasofelectricvector fieldsandchargesthatactassources.Figure3.4showshowironfilingsalignthemselvesbetweenapositiveandnegativechargeand also two positive charges. The filings serve to illuminate (not literally) the direction of theelectric vector field at a particular point in space (it doesn’t say anything about its magnitudebecausethiswouldrequiresometypeofmovementtoshowtheforce).Itisconventionaltoattachanarrowtooneofthesefieldlinesthatpointsawayoutwardforapositivechargeandinwardsforanegative charge. A field line therefore either starts on a charge or terminates on it. Thecorrespondingcaseformagnetismisquitedifferent!

AsafinalremarkasconcernsFigure3.4,thereisaneatrelationshipwiththescalarpotential.Supposewe consider an equipotential surface given by Then, from the definition of theelectricfieldasagradientofthescalarpotential,theelectricfieldlinesmustbeorthogonaltotheequipotential surfaces. To see this, first take the differential of the scalar potential, which is

Now,onanequipotentialsurface mustalsoequalzeroanddxamustlieinthelocalareaelementof theequipotentialsurface.Therefore mustbenormaltotheequipotentialsurface.InthiswayinFigure3.4wecoulddrawinlinesthatcutfieldlinesnormally.

FIGURE3.4:Apictorialrepresentationoftheelectricfieldintheregionoftwocharges.

3.3 THE“MAGNETIC”LAWORTHELAWWITHNONAME

Atthispointwehavearrivedattheideaofelectricchargesthatproduceandcoupletoelectricfields. These then do the same in return. These systems are therefore always coupled and theinteractioncannotbeturnedoff3.Wenowturnourattentiontothephenomenaofmagnetism.Somematerials (e.g. ironormagnetite)whensuspended in freeairwillnaturallyalign themselves inapreferential direction directed from theNorth to the South pole along a great circle. This basicproperty is theprinciplebywhichthemagneticcompassoperates,apieceof technologythathasbeenaroundasanavigationalaidsincethe11thcentury.Amaterialthatshowsthisalignmentalongagreatcircleissaidtobemagnetized.Itisalsopossibletomagnetizeabodywhichpreviouslydidnotdisplayanymagnetization (that is, for thosematerialswhichcandisplaymagnetization).Thisleadstothefollowingexperiment:-

Experiment5Taketwomagneticbodiesintheformofironbarmagnets.ForasinglebarmagnetlabeloneendpositiveandtheothernegativesothatthepositivepointstotheNorthPoleandthenegativetotheSouthPole.Nowbringthemintocloseproximity.Whathappenstothebarmagnetssometimelater?

Observation5Thebarmagnetsendseitherrepelifapositive(negative)endfromonebarisclosetoapositive(negative)oftheotherend,ortheyattractoneanotheriftheendsareapositiveandanegative.Thatislikesignendsrepelwhileunlikeendsattract.

This looks very much like the phenomena encountered earlier in electrostatics (at leastqualitatively) where the positive or negative ends corresponds to the the two types of electriccharge. Indeed, looking at Figure 3.5 that shows a bar magnet, one sees that the fields linesdisplayedbytheironfilingslookquitesimilartotheelectricfieldlinesbetweenthetwoequalandoppositeelectricchargesshowninFigure3.4.Anaturalquestionarisesnowastowhetherwecanisolate thecorresponding“charges”ateitherendof thebarmagnet, such thatwemight tryandbuildupapicturesimilartothatofelectriccharges.Considerthefollowingexperiment.

Experiment6Takeonesuchbarmagnetanddivideitintotwobycuttingitacrossitsaxiswherethepositiveandnegativechargesreside.Whathappens?

Observation 6 We find two new (smaller) bar magnets have been created. Where we haveperformedthecutwefindapositiveandnegativechargeoneithersidethatgivesrisetothetwonewbarmagnets.

Nomatter howmany timeswe subdivide the barmagnetwewill continually find smaller andsmallerbarmagnets.Weareofcoursetryingtoinvestigatemagnetisminananalogouswaytotheelectricalforcespreviouslyfound.Havingintroducedanelectricchargeearlierwehavejustfoundthattheirappearstobenocomparable“magnetic”chargeformagnetizedbodies4.However,theydoappeartoshareacommonformofinteractionbetweenthemselves.

FIGURE3.5:IronFilingslininguparoundabarmagnet.Thefilingsshowthelocaldirectionofthemagneticfieldinthevicinityofthemagnet’spoles.

It’sfrustratingcuttingbarmagnets.

Giventhattheideaandintroductionofanelectricfieldprovedsousefulbeforehandtoaccountforobservablestaticforces,itisclearweshouldthendosomethingsimilarformagnetism.TothisendweintroduceamagneticfieldBa(x)thatabarmagnetwillcoupletosuchthatitfeelsaforceand aligns itself with. Not only does it couple to themagnetic field but it is also a source of amagnetic field.Thus theEarthproducesamagnetic field,whichwecan regardas abackgroundfield inasimilarway to thebarmagnets,albeitonamuch largerscale.Notealso thatwhile theelectric charge is a scalar quantity, for the bar magnet a vector type charge can be associatedconnectingthenorthandsouthpoles.

It is necessary now to setup some basic field theory that is an equivalent of Gauss’s lawapplicabletomagneticbodies.Wedefinethereforethemagneticfluxas

(3.22)

Intheelectriccasetheelectricfluxthroughaclosedsurfacewasfoundtobeproportionaltotheenclosedcharge.Sincethereappearstobenofreemagneticchargethemagneticfluxmustbezero

(3.23)

ToseethisgeometricallytakethebarmagnetshowninFigure3.5andsurrounditbyasphere.Foreverysmallareaonthespherewherethemagneticfieldpiercesitpointingoutwards,thereisanothersmallareasomewhereelsewherethemagneticfieldpiercesthespherepointinginwards.Asbefore,thedivergencetheoremcanbeappliedtothis,resultinginanintegraloverthevolume.Clearlythedifferentialversionisjust

(3.24)(Exercise:verifythis)

Weshall refer to thisasMaxwellIIand is thesecondof the fourMaxwell’sequations.Onanhistorical note as the title of this section suggests,Equation (3.24) does not have a name as theothersdo.ItseemsMaxwellwroteitdownfirstbasedonearlierworkofLordKelvinandFaraday’sideaofclosedmagneticlinesofforce[6,17].ItmightbetterbecalledtheMaxwell-Kelvin-Faradaylaw,thoughthisisabitofamouthful.

At this juncture,wehave introducedthe ideaof twodistinct fieldsandonesource.MaxwellItellsusthatachargedensityρ(x)isasourceoftheelectricfieldandthatitwillfeelaforceduetothepresenceofanelectric field there.Themagnetic fieldhoweverdoesnothaveasinglesourceterm but rather a positive and negative end separated by some distance. To account for theobservedforcesonelectricallyormagneticallychargedbodies,anelectricfieldhastovanishonacharge;thereisadefinitesingleterminationpoint.Amagneticfieldalwayscomesasloop;thereisnostartnorendpoint.

3.4 VOLTAGEANDCAPACITANCE

Itisnowpossibletodefinetwobasicquantitiesforconductorsthatarealreadyprobablyfamiliartothereader,butareessentialinanyformofmeasurementtheoryandtheconstructionofsimpleelectroniccircuits.Theyarevoltageandcapacitance.

Definition1The voltageV of a conductor is thepotential therewith respect to some referencepointorspace.

For example, ifwe think of an electrical circuitwherewires are essentially straight lines, thevoltage is the potential difference between two points, typically with a component in between.Comparethiswiththecaseofachargedmetalspherewhereweoftenchoosethedifferencetobebetween the surface of the metal sphere and a sphere at infinity. The potential can also bemeasuredbyuseoftheelectrometerencounteredearlierthatwasusedtomeasurecharge;itactsasalocalprobeofthepotentialatsomepoint.Thisfollowsbecausehavingdeterminedthechargeonabody,wejustneedtomeasurethedistancetothelocalprobetofindthepotential.Weassumein addition that we can identify the traditional term voltage associated with a battery with thispotentialdifferenceandwillthereforeregardthemassynonymous.Thereadercanconsult[17]and[11]formoreonitsequivalenceandhistoricalfinding.Fromnowonwewilltakethevoltagetobeamacroscopicphysicalmeasurablethatcanusedtomeasureotherquantities.Nowletusturntothecapacitance.

Definition2Thecapacitanceofabodyistheabilityofaconductororcollectionofconductorstostore electrical charge. It is defined as the ratio of the electrical charge on a conductor to thevoltageonit.Onewritesitschematicallyas

Forthecaseofasingleconductorthisisaconstantthatdependsonitssizeandgeometry.Itisevident,however, that foramoregeneralsituationofmanybodies it ismorecomplicated.This isbecause the introduction of a new body (remember it is not a test body!) disrupts the previouselectrostaticequilibrium;all thechargesare shiftedaroundas thenewbody is introducedandanewelectrostaticequilibriumhastobefound.Forexample,supposewehavesomeconductorwithacharge on it andwe knowwhat it’s equilibriumpotential is.Nowbring a second conductor intoproximitywiththefirstbody.Chargeswillbeinducedinthesecond.Obviouslyithasrequiredworktoseparatethecharges,whichcanonlycomefromthepotentialoftheoriginalbody.Thereforethepotential must have lowered, thereby increasing the capacitance since the charge on it hasremainedconstant.Sayingitinslightlydifferentway,foracollectionofconductorsimagineweputknownamountsofchargeonthemandthenelectricallyisolateeachone.Clearlythereispotentialdue to thisdistributionofchargeandthiswilldependontheoverallgeometryof thesystem.Wehave already encountered an early form of capacitor in connection with the electrometer. TheLeydenjaremployedtherewasusedforthesimplepurposeofstoringaknownamountofelectriccharge-thiswastheoriginalformofthecapacitor[11].Togetanideaoftheformacapacitancetakes,considerthefollowingsimpleexample.

Example3.1AsolidmetalsphereofradiusRcarriesachargeQonit.Calculatethecapacitanceofthesphere.

Since this problem involves spherical geometry we clearly need to work in spherical polarcoordinates.WecanapplyMaxwellItothisintheformofEquation(3.14).Bysphericalsymmetrytheelectricfieldonlyhasaradialcomponentandisgivenby

(3.25)

(3.26)

From this we can calculate the potential difference between the surface of the sphere and asphereatinfinity,

(3.27)(3.28)

(3.29)

Thecapacitanceistherefore

(3.30)

Another important example that we would like to start thinking about is the parallel platecapacitor in the formof twodisks eachof radiusa separatedby a small distanced.What is thecapacitanceof this system? It is in factquiteadifficultproblem tocalculateexactlybecause theedgeofthediskscauseacomplicationinthemath.Oneshouldnoteherethatthissystemwillplayan important part for other aspects of the field theorywe are investigating.Wewill return to itagainandagainasthereisreallyalotofphysicstakingplaceunderthehood.

Thestandardgeometryoftheparallelplatediskcapacitor.Itconsistsoftwometal disks, each of radiusa and a separationd. Each has beenconnected to a wire and ends on some type of module labelled as“Operation”Thiscouldbeasimpleswitch,abatterycellorsomethingelse.

Notwithstandingtheabove,letuscalculateanexpressionforthecapacitance(perunitarea)inthelimitthat Thisamountstosayingthatwehavetwoinfiniteplatesandthatoneachwehaveaconstantchargedensityσ.Oneplatehas+σ theotherhas−σ.What then is thepotentialdifference between the two plates? Let the separation d be in the z coordinate of the standardCartesian coordinates so that the plates lie in the x, y plane. By symmetry the electric field isconstant-usingMaxwellIonefinds

(3.31)

(3.32)

Using the linearsuperpositionprinciple theelectric field in the tworegionsoutside theplatescancel,whereasfortheregioninbetweenthefieldsaddtogether.Thepotentialdifferencebetweenthetwosurfacesis

(3.33)

(3.34)

(3.35)

Fromthisitisstraightforwardtoderivethecapacitanceperunitareaas

(3.36)

Thisisausefulexpressiontobearinmindasitgivesusatleastafirstapproximationtothefinitegeometrydiskcase.Thecomplicationssetinforfinitegeometriesbecauseoffringingeffectsattheboundariesandtypicallythiswillrequireanumericaltypeofsolution.

3.5 SUMMARY

Wehave seen in this chapter that toaccount forobservationsof static forcesbetweenbodies,charges and fields have had to be introduced. Bymeasuring amechanical force we can in turnmeasure an electric charge on another bodywith respect to someknown reference amount. Theintroduction of a scalar potential together with Gauss’s law turns electrostatics into a precisemathematicaltheorywhereoneneedstosolveeitherthePoissonequationinthepresenceofstaticcharges,ortheLaplaceequationinfreespace.Formagneticinteractionswehavefoundaphysicalfieldbutnotasimpleequivalentofelectriccharge.Thiswouldhavemadelifesimpler(oratleastmoresymmetric)asonewouldperhapsliketohaveasimilarsituationtotheelectriccase.Standardreferencesonelectrostaticsare; [8] very completeand rigorous; [5]useful butwith a condensedmatter/atomic bent; and [7], always a joy to read and full ofmany interesting side lines. For an

historicalpointofviewonMaxwellII,thereadercanconsult[17]asastartingpoint.

NOTES1Ofcourseweknow that there isa lower limitwhereelectricchargebecomesdiscreteat theatomic leveland that it cannotbefurtherreduced.TheoildropexperimentduetoMillikangivesacleardemonstrationofthisbutreliesonextrastructuresofarnotintroduced.Wearejusttryingtokeepeachmatterlogicallyseparate2Thisfollowsonfromthepreviouspoint.Theanswertothisinourenlightenedageiswellestablishedandstandardschooltextbookfactsthatarecommittedtomemory.Wesimplywishtokeepthingsasmacroscopicandbottomupaspossibletobeginwith.3Onecanseethisisadifficultpathtotread;ifwecan’tturntheinteractionsoff,canwebesurewearemeasuringtherightthing?And does the very act of measuring the system disturb the system too much such that we do not obtain genuine results? Notsurprisinglythereisalargeoverlapherewithquantummechanics.4ThisinparticularmakesMaxwell’sequationsasymmetricalwithregardtotheirmattercontent;weshalldiscussthismorelater.

CHAPTER4MAXWELL’SEQUATIONSPARTII

We have encountered the two fundamental physical fields Ea and Ba that arise in staticexperimentalsituations.Theyareconstructsthatareusedtoquantifytheforces found inelectricandmagneticsituationsaftertheyhavesettleddownintoanequilibriumstate.Theelectricchargedistributionsandmagneticbarshavebeenstaticand thereforebydefinition,notbeenmoving.Anatural question to ask is exactly this :- what if they domove in some simple fashion (constantvelocityorconstantacceleration)?Bystudying thenatureof this timedependentphenomena,wewillarriveattheastonishingfactthattheelectricandmagneticfieldsareintertwined.Notonlythattheyarereallytwosidesofthesamecoin.Asonevariesintimeandspacethereisalwaystheotherpresentandcanneverbeswitchedoff.Itisinthissensethetopicistrulycalledelectromagnetism.Originallyelectricityandmagnetismwereconsideredseparatephenomenaanditwasonlywiththediscoveryand introductionof thegalvanic cell that currentswere linked tomagnetismandunitywasbroughtforward.Forsomehistoricalbackgroundonthispoint,see[11]and[6].

4.1 THECONSERVATIONOFCHARGE

Inthepreviouschapter,weencounteredthelawofconservationofcharge.Itisabookkeepingdevicethatdemandsthatafinalsysteminsomestatemusthavethesamechargeasthesysteminits initialstate.To formulate thismoregenerallyweneedto thinkabout themovementofchargewhichnaturally leads to the ideaofacurrent.Wewillwrite it in the formofwhat isknownasacontinuityequation.Consideravolume boundedbythesurfacearea andwithinwhichsitsacertainamountof chargeQ(t) at time t.A current I(t) is simply the flowof chargeperunit timethroughsomesurface Thiscouldconsistofasetofdiscretechargesmovingontheirowntrajectoriesoracontinuumwherethereisaflowateachpoint.Supposethischargenowstartstomoveabout.Howmuchchargewill therebe in at some later time?Whatweneed to thinkaboutistherateofchangeofcharge withinthevolumeasafunctionoftime.Letusalsocut up the boundary into lots of small areas If we know the velocity va(t, x) of a smallamountofchargeΔQ(t) inasmallvolumeΔV thathaspartof itsboundariesasoneof theareas,thenwecancalculatetheoutflowofchargethroughthisarea.

Avolume withaboundary enclosesachargeQ(t)which is afunctionoftime.ThedashedcylinderindicatesthesmallvolumeΔVinsidethemainvolumeandjustbelowtheboundarysurface.Oneofthe cylinders end faces (with area ) forms a small part of theboundary.

WedefineacurrentdensityJa(t,x)tobetheflowofchargeperunitareaperunittime.Thenthedynamicstatementoftheconservationofchargeis

(4.1)

Given a velocity vector fieldva(t,x) and a charge densitywe canwrite an expression for thecurrentdensityintermsofthesevariables,thatis

(4.2)

(Exercise:checkyouagreewiththedimensionsofthisexpression.)

AswiththeearlierMaxwell’sIandIIequations,thiscanbewrittenindifferentialformwhenthechargedistributioniscontinuous

(4.3)

Thisistheequationofcontinuityandalthoughitisapplyingheretoelectriccharge,italsoshowsupinmanyotherareasofphysics(notablyfluiddynamics).Afurtherpointthatwewilldiscussinthe next chapter is to do with symmetry. The conservation of charge has a mathematicalconsequencethatthefieldequationshaveasymmetrybuiltintothem.

Howdoes one generate a current in the first place? There are two straightforwardways thatfollowfromitsdefinition.Oneissimplytomoveawayfromachargedobjectataconstantvelocity.Relativetoyouthechargewillthusappeartobemoving.Thesecondistheuseofasimplebatterycell.Thisinvolvessomechemistrythatconvertschemicalenergyintoelectricalenergyintheformofasustainedpotentialdifferenceattwoterminals-negativechargebuildsupatoneend,positivechargeattheother[11].Aslongasthereisapotentialdifferenceinaregionofspacethenweknowthatachargewillcouple to the localelectric fieldand feela force. Itwill thereforebe ina localstateofacceleration.Althoughdischargingacapacitordoesindeedproduceacurrent,itisnotveryusefulbecauseitmanifestsitselfasasuddenburstofuncontrolledchargemovement.

At this juncturewe are interested in steady currentsmeaning constant velocities sowemustthink about a further element. In defining the electrical properties of materials we defined aneffectivedivisionbetweenconductorsandinsulators.Ofcoursethisboundaryisnotsharpandthereis a whole spectrum of behavior. Depending on the material, the charge will be subjected todifferent amounts of resistive forces when a potential difference is applied across two regions.When the two balance out, wewill attain a steady flow and hence a steady current. It is usefultherefore to consider how the electrical current and voltage are related. The relationshipconsideredhereisaneffectivedescriptionthatweareestablishingphenomenologicallyasausefulwayofcharacterisingmaterialswhich in turncanbeused inapplyingelectromagnetism toothersystems.Thisisbecauseaprioriwedonotknowwhathappensonasmalllengthscale.Thebasicidea is thatwewantanequationofmotion for theelectriccharge,butnot tokeep trackofeachindividual charge. Since this type of scenario invariably involves wires, the equation of motionshouldbeforthecurrent,ratherthananindividualelectriccharge,sothatwearealwaysworkingatacoarsegrainedlevelwherewehaveachanceofmakingdecentmeasurements.

Across-sectionalviewofapieceofwirethathasbeenmagnifiedtoshowthebasicmaterial configuration. It takes the formof ametallic cylinderwhichhasapotentialV(t)appliedacrossitsendfacesofareaA.Thisthenmust set up a movement of charge within the volume, the constantvelocitycontributionbeingthemacroscopiccurrentI(t).

ForapotentialdifferenceV(t)betweentwoendfacesofmaterialwemakeaguessatageneralpolynomialoforderNtypebehaviorfortheequationofmotion

(4.4)

Herewehaven’tincludedanyhigherorderderivativetermssimplybecauseequationsofmotiontendtohaveonlyasecondtimederivativeastheirhighest-higherderivativesleadstoallsortsof

mathematicalandphysicaldifficulties,notablyimproperlybehavedsolutionsandghosts.Obviouslyinorder tomapoutwhich termsare important inEquation (4.4)weneed tobeable tomeasurecurrents as well as voltages. In principle, we could use the electrometer to measure both theelectricchargeand itsrateof flow,butonemightanticipatepracticaldifficulties inthis. Itwouldthereforebegoodtohaveanindependentmeanswithwhichtomeasurejustthecurrent.Todothiswe need to bring back onto the scene the magnetic compass and its transformation into agalvanometer.

HansChristianOersted(1777–1851),aprofessoratCopenhagen,submittedasetofinsightfulobservationsoncurrentsandmagnetismtoaleadinggroupofphysicistswhichwerelargelyignored.Workinginbothphysicsandchemistry,he was the first physicist to name and explicitly describe the “thoughtexperiment”concept.

A key observation was made by Hans Christian Oersted in 1820 (during a lecture), that acompass needle was deflected when the current from a battery was switched on or off. Moregenerallyhe found that thecompassneedlewasdeflectedwhen itwasplaced in thevicinityofacurrent carrying wire. Building on this Ampère aligned a current carrying wire (produced via abattery) inaNorth-Southdirectionandthensimplyplacedamagneticcompasscloseto ithavingfound that the needles’ plane of rotation should be perpendicular to the Earth’s magnetic fluxdensityforitnottobeinfluencedbyit.Thedegreeofthedeflectionisthusameasureofthelocalcurrentflowingintheconductor.Ifthedeflectionsettlesdowntoaconstantvaluethelocalcurrentmusthavereachedasteadystate.Wehavealreadyintroducedthemagneticfluxdensityinthelastchapter. What must be taking place here is a balance between the torque produced due to thebackgroundmagnetic fluxdensityof theEarthanda localmagnetic fluxdensityproducedby thecurrentcarryingwireingeneral -currentsareasourceofmagneticfluxdensity!Thuswehaveaconsistentpictureandalocalprobewithwhichtoperformmeasurements.

Andrè-Marie Ampère (1775–1836), Maxwell described him as the “Newton” ofelectricity. With a flurry of experimental and theoretical activity in 1820 Ampèrelinked firmly themagnetic field tosteadycurrentsources.He isalsocreditedwiththebasicinventionofthe“Galvanometer”.

Giventhatwecannowmeasurecurrentaswellasvoltagebytwoindependentmeans,wecanreturntoEquation(4.4)andtrytoestablishaphenomenologicalrelationship.ThisisexactlywhatOhmdidandestablishedthewellknownphenomenologicallaw

(4.5)

GeorgSimonOhm(1789–1854),foundthefamiliarOhm’slawbyvaryingthelength,area and material of different wire type configurations. This was then furtherdevelopedbytheuseofFourierseriesanalysisandanalogiestothermalheatflow.

Forthediagramdepictingthecross-sectionalviewofapieceofwireonecanseethatthereisanewmacroscopiccomponentbeingconceptualised - theresistor.Thissimply limits theamountofcurrentthatcanflowinacircuitthatwouldotherwiseflowwithouthindrance.Sincethematerialsresistance has a dissipative effect on the current flow, the component will naturally turn thisabsorbedenergyintoheat,thustransformingtheenergyinsomefashion.

4.2 ELECTRICCURRENTSANDTHELAWOFAMPÈRE

Nowthatweknowwhatcurrentisandhowtomeasureit,wearegoingtotryandplaythesamegameaswedidforelectriccharge.Tothatend,wewill thinkabouthowsteadycurrents interactandthatthemagneticfieldistheagentbywhichforcesareaffected.See[6]formoredetailsonthehistoryandexperimentaldiscussion.

Experiment7(Ampère,1820–25) -Take twowirescarryingsteadycurrents in the formof two

separatecurrentloopsandbringthemintoproximitywithoneanother.Whathappenstotheloopsofwire?

Observation7Eachofthemfeelsaforceandcanbeeitherattractiveorrepulsivedependingupontheorientationofthetwoloopsrelativetooneanother.Itisproportionaltoeachofthecurrentsandvariesinverselywiththesquareoftheseparation.

Oneshouldinfactbecarefulhereaboutthesquareoftheseparationobservationbecausewithtwo different current loops there are obviously other length scales that can enter. This will beclearerwhenwewritetheforcedown.Wenowtrytocodifyasimilarequationforthecurrentsaswe did in Equation (3.1) for the electric charges. There is a striking difference that can beimmediatelyseenbetweenchargesandcurrents.Whilechargesreferred tosmall local regions inspace,steadycurrentsrequireloopsofaconductortosustainthemselves(i.e.weneedchargetobecontinuallymovingthroughsomeregion).

One of Ampère’s experimental arrangements. This one gives therepulsionofcollinearelectriccurrents.

TheequivalentofCoulomb’slawforelectriccurrentsisAmpèreslawforcurrentloopsMandN:-

(4.6)

AswithCoulomb’slaw,anarbitraryconstanthasbeenintroducedandwilldependonthesystemofunitsused.WewilluseEquation(4.6)todefinethesystemofunitsandthusthereferencepoint.Notealso theplethoraof indices - it isquiteeasy for theseexpressions tobecomeunwieldy.Theampèrewillbethestandardunitofcurrent,sothatoneampèreiswrittenas1A.Itisofcourseanarbitrarydefinition,justasthekilogramormeteraresimplydefinedtobesomething.Thisallowsustodefinetheunitofchargeasthecoulombsothat1A=1Cs−1.Thentheconstantofproportionalityisdefinedtobeμ0=4π×10−7NA−2,and iscalledthepermeabilityof freespace.UsingAmpère’slaw,thistellsusthattwoparallelwireseachcarrying1Aofcurrent1mapartwillfeelaforceperunit length of 2×10−7N. So a base unit of current has been defined in terms of a measurablemechanicalforce.

(Exercise:verifythis.)

4.2.1 TheMagneticFluxDensityandCurrentCouplingSinceweareplayingthesamegameaswedidforelectricfields,wewanttounderstandhowthe

currentloopisasourceofmagneticfield.TheobservationsofOerstedgaveusthebasiccouplingbetweenthefieldandthecurrentandAmpère’slawgivesusthecouplingbetweenthecurrents.Letustakeastepbackandaskhowweshouldtrytorunourmagneticinvestigationinasimilarwaytothetheelectriccase.Supposewedoanotherexperiment;

Experiment8Replaceoneof thecurrent loopswithabarmagnet.Whathappens to thecurrentloopandbarmagnet?

Observation8Eachofthemfeelsaforcethatcanbeeitherattractiveorrepulsivedependinguponthe orientation of the two and basically behaves in a similar manner as if we had the previouscurrentloops(atlargeenoughseparations).

Thecurrentloopsandbarmagnetsinteractinapparentlythesameway,soweconcludethattheyarethesamesources-currents(themovementofcharge).WhatthenisthemagneticequivalentofEquation(3.2)?ClearlyweneedacurrentloopfromAmpère’slawsothebasiccouplingmustlooklike

(4.7)

Note thedifferencewith theelectriccase, thatwereallyneedthink in termsofcurrent loops.Forthesustainedandconstantcurrent,aloopofwireandabatterycellarerequired.Thereforetheobservableeffectisnecessarilyontheentireloopandwilleffectivelybeatitscenterofmass-theeffectisnon-local.Ifthegeometryandconfigurationpermitsit,wemaybeabletodroptheintegralsuchthat

(4.8)

wherethereisacurrentinasmallelementoflength,butthisformwillnotalwaysmakesense.

Asitstands,thecurrentsarenaturallyconfinedtothewiresinwhichtheyhavebeensetup.IfwecanidentifyindividualchargesasinEquation(3.2), that isofapoint likenature,thenwecanrewriteEquation(4.7)intermsofitsvelocityv(x)ofthechargeqsothat

(4.9)

Notehereanotherdifferencebetweenamagneticandelectricfield.Anelectricfieldisavectorquantitywhilethemagneticfieldisanaxial-vector.Inbriefthismeansthatunderareflectiontheelectric fieldpoints in theoppositedirection,while theaxial vector remainsunchanged.Onecanseethisfromthewedgeproduct-thevelocityandthewedgeproductwouldyieldaforcethatwasanaxialvector,wereitnotforthefactthatthemagneticfluxdensity isalsoanaxialvector.Howthese vectors are classified is based on the group theory for how the fields transform undersymmetry transformations. In particular, these definitions follow from considering how the fieldstransformundertherotationgroupandtheparitygroup(writtenasO(3)).

(⇐Left)Jean-BaptisteBiot(1774–1862)-acriticofAmpèreandextremelyconservative-moreLaplacianthanLaplacehimself!

(Right⇒)FelixSavart(1791–1841)-wenttoParisin1819toseeBiotandendedupputtingforwardtheBiot-Savartlawin1820.

TheexperimentsofBiotandSavartestablishedthedependencyofthethemagneticfluxdensityonthecurrent[6].ReturningtoEquations(4.6)and(4.7)weseethatthemagneticfluxdensityduetoasteadycurrentloopistherefore

(4.10)

Whenthechargeiscontinuoussothatthecurrentbecomesacurrentdensity(asisusedinthecontinuityequation),wemustformacontinuumversionoftheabovewiththeresult

(4.11)

RememberthatalthoughthecurrentIisthephysicalmeasurable,anyconductingwirewillhavefinite cross sectional area. This means that while it will be the current density that enters ourequations,thiswillhavetobeintegratedoverthewireareatogivethephysicalmeasurableI.TheunitsofthemagneticfluxdensityaretheTesla,T,where1T=1NA−1m−1.

Itisalsoworthwritingthisincomponents

(4.12)

anddrawingattentiontoaslightlymathematicalpoint.Inthisformitisclearthatthemagneticfluxdensity is metric dependent and similarly for the equivalent electric field expression, sincedistancesenterexplicitly.However,thesecorrespondtoaparticularsolutionofthefieldequations.Whatisindependentofthemetricistheabsenceofanymagneticmonopoles.ThisisreflectedintheformofMaxwellIIwherethereisnosourcetermpresentthatisakintotheelectriccharge.

4.3 THELAWOFFARADAY

Michael Faraday (1791–1867). The influence of Faraday’s work on physics andelectromagnetism cannot be understated; electrolysis, polarization and

electromagneticinductiontonamejustafew.Hisachievementsareallthemoreremarkableashestartedhiscareerasabookbinderwithlittleformaleducation.

Upuntilthispoint,wehaveinvestigatedelectricandmagneticphenomenathathavemostlybeendecoupled.This isbecausehistorically the twooriginatedseparatelyandwhileelectrostaticswasseeminglywellunderstoodmathematicallyandexperimentally,sourcesofcurrentwerelessso;theywere shrouded in the ethos of galvanism [11, 17, 6]. The observation of Oersted of changingcurrentsproducingmagnetic fieldsofcoursestillhadavoltage(fromthebattery) toproducethecurrentinthefirstplace,sothestateofdecouplingisnotabsolute.Thetypeofcouplingmanifestsitselfintheforcelaws

(4.13)

(4.14)

forcontinuouschargedistributions.ThecorrespondingforcelawforathepointliketestchargeqisgivenbytheLorentzforcelaw

(4.15)

Byacouplingwetypicallymeanasimplemultiplicationofthetestbodychargeorcurrentandthefieldatthatpointduetosomethingelse.Thefirstoftheseisasimplescalarcoupling,whilethesecond(forthemagneticcase)isatensorcoupling.Further,themagneticfieldhasnopointsource,arising as they do from the steady movement of charge. It is this very fact that suggests thefollowing:giventhattheyoriginatefromthesamebasicsource(electricalcharge),itmightwellbethatthemagneticandelectricfieldsaretwosidesofthesamecoin.IfIchangeoneintime,willIseeamanifestationoftheother?Thisisexactlywhathappens.Timeforanotherexperiment.

Experiment9(Faraday,1831)Supposewemoveabarmagnet througha loop of conductingwire.Whathappensintheloop?

Observation9Weobserve a timedependent current is induced in the loop. It is found that theinducedvoltagethatproducesthiscurrentisproportionaltotherateofchangeofmagneticflux.

Faraday’soriginalexperimentusedanothercoilofwireinsteadofabarmagnetthroughwhichatimevaryingcurrentpassed.Thishadtheeffectofcreatingatimevaryingmagneticflux,someofwhichcutthecoilsinteriorinasimilarfashiontotheabovemovingbarmagnet.

Wecancapturetheaboveexperimentintermsoftheelectricandmagneticfieldsandlinkthemdirectly forthefirst time.ThevoltageV(t) inour loop is just the line integralof theelectric fieldwhichitselfmustnowbetimedependent

(4.16)

FIGURE4.1:Faraday’sinductionexperiment,1820.Whenthebarmagnetismovedtowardsorawayfromthecoil(initsvicinity)atimevaryingmagneticfluxisgeneratedfromstandpointofthe

coil.Thiscutsthetheloopformedbythecoilandthegalvanometerisobservedtodeflect.Thisisthesameobservableeffectaswhenatimevaryingvoltageissetupinthecoil.

Thissatisfiesbasedontheaboveobservation

(4.17)

Thequestionatthispointiswhattheconstantofproportionalityshouldbe.Rememberthatupuntil this experiment the electric and magnetic fields have been considered separate. Electricchargehasbeenmeasuredwithrespecttoanarbitrarysource,andcurrenthasbeenmeasuredwitha particular compass. The next input on this law was to fix the relative sign by essentiallyperforminganumberofexperiments.ThiswasdonebyEmilLenz [6]andcanbesummarisedasfollows:-

Principle 2 (Lenz’s law) - the induced current takes the route such that the magnetic field itproducesopposesthechangeinfluxthroughtheloopthatinitiatesit.

Thebestthatwecandoatthispointthereforeistowritedown

(4.18)

wheretheminussigniscapturingtheresultofLenz’law.

Heinrich FriedrichEmil Lenz (1804–1865). Lenz arrived at his law based on theexamples of induction that Faraday had considered, and some of his ownexperiments.

The fact that the constant of proportionality appears to be undetermined requires us toimposesomefurtherconstraintsorsymmetries.Wehavedefinedtheunitofcurrent(theAmpère)asbeingaflowof1Cs−1.SowenowtaketheCoulombtobetheunitofchargeusedinCoulomb’sforcelawEquation(3.1).Thisfixestheunitssuchthat becomesadimensionlessnumberandforthepermittivityoffreespace tohaveunitsthatfeaturetheColoumb.

(Exercise:verifytheconstantofproportionalityisadimensionlessnumberandcheckthedimensionsofthe )

Wecangofurtherandevaluate bythinkingabouttherelativemotionoftheloopwithrespecttothebarmagnetandtheconsequenceofassumingGalileaninvariance.

Principle3Physical lawsmustbe invariantunderaGalileantransformation. Inparticular, forcesfeltbybodiesmustbe thesame inany inertial frame that ismovingrelative tooneanotherwithconstantvelocity.

AGalileantransformationisatimedependentmappingbetweentwocoordinatesystemsthataremovingrelativetooneanotherwithsomeconstantvelocityua.If(t,xa)and(t′,x′a)arecoordinatesin the original andmoving frames of reference, then the transformation is given by the definingrelations

(4.19)(4.20)

Fromthis,itissimpletoseehowthevelocityinthetwoframesarerelated,viz

(4.21)

NowconsideragainEquation(4.15)inviewofGalileaninvariance.Supposethatatestchargeisat rest and subjected to fields E(x) andB(x) in the unprimed reference frame1. In the primedsystem, since the fields are vector quantities, we expect them to also undergo a GalileantransformationtothefieldsE′(x′)andB′(x′).Galilean invariancesays that the forceshouldbe thesameinbothreferenceframessothat

(4.22)

TocompletethetransformationlawsweshouldalsocalculatewhattheB(x)transformstoo.This,however,isnotnecessaryforthetaskathandofdetermining 2.WritingEquation(4.18)inintegralformwehave

(4.23)

wheretheloopofwireisalongthecontour andthemagneticfluxdensityofthebarmagnetcutsthearea Thereadercanprobablyseetheroadahead. If thebarmagnet ismovingsothat theloop is stationary, the time derivative just acts on the field. In the Galilean transformed framehowever,theloophassomeconstantrelativevelocity(andsothereforedoesthearea),suchthatweneed to invoke Equation (2.29). On this point we should remember the Leibnitz rule fordifferentiatingundertheintegralsign

(4.24)

Itbecomes

(4.25)

ThiscanbesimplifiedbyusingEquation(2.24)andMaxwellIIto

(4.26)

Finally,byusingStoke’stheoremthelasttermcanbeconvertedtoalineintegralandregroupedwiththeelectricfield

(4.27)

For Galilean invariance to hold the line integral quantity in the above must be equal to thetransformedelectricfieldfoundinEquation(4.22)sothat Equation(4.27)becomes

(4.28)

Sotheconstantofproportionalityhasbeendetermined.

Forthesituationwheretheloopitselfisnotvaryingintime,wecanwriteFaraday’slawinlocaldifferential form, as we did for Gauss’s law. By using Stoke’s theorem Equation (2.34) we canconvertthelineintegralintoasurfaceintegral

(4.29)

Thisreducesto

(4.30)

Itisworthdrawingthereadersattentionheretoacoupleofpointsaboutthisequation.Thefirstis that it isastatementabout thebasicelectromagnetic fieldswithnomattercontentandall theaction is happening in the vacuum of free space. The electric and magnetic fields are alwayscoupledinthiswayregardlessofwhatchargesandcurrentsareonthescene.Thesecondpointisamathematicalone.Thisequationiswhatistermedindifferentialgeometryasexact,whichmeansthatlocallywecanreplacetheelectricandmagneticfieldswithpotentials(seeforexample[2]forafullerdiscussionofthesetypesofmathematicalissues).Wehavealreadypartlydonethiswiththescalarpotentialfortheelectricfield.Inthenextchapterwewilllookatthismorefullyandhowitworksforbothfields.

Faraday’slawisofcrucial importancebecauseit linkselectricandmagneticfieldsforthefirsttime.Ifyouhaveamagneticfieldthatvariesintime,thereisanelectricfieldpresentatthesametime.WewillrefertothisequationasMaxwellIII.Itshouldnotcomeastoomuchofasurprise,however, since the original experimental means of detecting a current was from the magnetic

deflectionofacompass,thatwasinitiatedwithanelectricpotential!AveryreadablediscussionofFaraday’slawcanbefoundin[7].

One immediateuse ofMaxwellIII is in the operation of an inductor and its inductance.Notsurprisingly this features a coiled conducting structure, taking the form of something like asquashed helix. The key aspect to this component is that we have a conducting pathway thatcirculatesaboutanaxissothatamagneticfieldcanbeproducedbyasteadycurrent.Whatthenisthephenomenologicalparameterthatisliketheresistanceforaresistorandthecapacitanceforacapacitor?

Thebasicgeometryofaninductorconsistsofalooptypespiralstructure.Thisallowscharge to circulate around a common orthogonal axis and thereby couple to amagneticfluxdensityinthisdirection.

Consideringthenasimpleinductorgeometrywhereweapplyatimevaryingpotentialacrossit,MaxwellIIItakestheform

(4.31)

togetherwiththeBiot-SavartLawEquation(4.10)thatthemagneticfluxdensityisproportionaltothecurrentI(t),wecanwritetheaboveas

(4.32)(4.33)

(4.34)

In the above, L is the inductance for the inducting component. Typically the loop integral isevaluatedbyoverlayinganumberofcirclesontopofoneanotherleadingNnumberofcoilturns,ratherthanevaluatingthecomplicatedhelicalarrangement.However,forcertainapplicationsthisdistinctionmaybenecessary.

4.4 THEDISPLACEMENTCURRENT

WenowreturntoAmpère’sstudyofthemagneticfieldproducedbyasteadycurrent ina longstraightwire.Letus reconsider theexpressionEquation (4.10)whichgives themagnetic field atsomepointduetoasteadycurrent.Clearlythemagneticfieldisinadirectionnormaltothecurrentflow. It is alsonormal to thedifferencevectorwhich, togetherwith thedirectionof current flow,define a plane. Themagnetic field has only one component and it is normal to this plane. Ifwerotatethedifferencevectorabouttheaxisofthecurrent,themagneticfieldisunchangedandwehaveanhonesttogoodnesssymmetry.

Ouraimhereistotryandinvertthisrelationship-wewantthecurrentintermsofthemagneticfield.ThesimplestwaytoproceediswithEquation(4.12);wewanttogetridoftheintegralovervolumeontherighthandside.TothatendweknowthatifwecanextractaDiracdeltafunction,theintegralwillcollapseandwewillhaveanotherlocalequation.ThisisfurthersupportedbyrecallingGauss’s law, where we have a partial derivative acting on the electric field that is equal to thechargedensity.SoconsidertakingthecurlofEquation(4.12);

(4.35)

To make progress on simplifying the right hand side we use Equation (2.44) so that we canrewriteEquation(4.35)as

(4.36)

We can simplify the above double curl using the vector identity, where we now write ourexpressionsincomponents

(4.37)

Thisgivesincomponents

(4.38)

Thesecondtermintheaboveisstraightforwardtoevaluate,sinceitjustgivesadeltafunctionusingEquation(2.46),so

(4.39)

Thelasttermcanbefurthersimplifiedto

(4.40)

wherewe have used Equation (2.46) and have performed an integration by parts (we have alsoassumedthattheboundarytermvanishesbythecurrentvanishingon ).Inthecaseofmagneto-staticswherewehavesteadycurrents,

(4.41)

WecanthereforeoperateontheintegrandandagainmakeuseofEquation(2.46)

(4.42)

whichcanbereadilyconvertedtoasurfaceintegralanddropped,sinceinthelimittheboundarytendstoinfinity,theintegrandvanishes.Thuswefindthat

(4.43)

This can be readily converted to an expression involving the current by integrating the righthandsideoveranarea

(4.44)

whilethelefthandside,byStoke’stheorem,canbeconvertedintoalineintegral

(4.45)

ThisequationisknownasAmpères’circuit lawanditspracticalmerit isthat itcanbeusedtocalculatethemagneticfieldwhenthesystemhassomemanifestsymmetriesinit[8].

As it stands, Ampère’s circuit law is valid when the current is in a steady state. This is not,however, the whole story. If we think of the parallel plate capacitor we encountered in the lastchapter, it was assumed we built up the charge by some static transfer process. By pushing acurrent into it, the plates can similarly build up an amount of charge inwell controlledmanner,ratherthanthelumpinessinstaticcase.Also,castingaglanceatEquation(4.40)tellsusthatifthechargedensityisageneralfunctionoftime

(4.46)

sothatthelasttermingeneraldoesnotvanish.Toaidusinbetterunderstandingthisequation,letusdoanotherexperiment.

Experiment 10Suppose we apply a time varying voltage across a parallel plate capacitor andconsiderthetimeperiodwhenitischargingup.Whathappenstothemagneticfield?

Observation 10 The magnetic field surrounding the current in the wires is continuous in theseparationregionofthecapacitorplates,asdetectedbyagalvanometer.

Theparallelplatecapacitoragain.Avoltageisappliedacrosstheplatesso

that a current flowsand chargeaccumulates on thedisksuntil it saturates.Thisisatimedependentprocess.

Notethatnocurrenttravelsbetweenthetwocapacitorplates.Sincechargeisbuildingupontheplates,theelectricfieldinbetweenwillbevaryingasafunctionoftime.IfwetakethedivergenceofEquation(4.43),wefindthatitisinconsistentwithconservationofcharge

(4.47)

ThelefthandsideisidenticallyzeroduetothevectoridentityEquation(2.28).Maxwellgavethesolution to this problem by introducing the displacement current (see for instance [6, 8]) toaccountfortheconservationofcharge.Itisdefinedas

(4.48)

IfweaddthistothecurrentinEquation(4.47)thenwerestorethecontinuityequation.WiththisthenAmpère’slawbecomes

(4.49)

Notethatwehavemodifiedthisfieldequationatthelocallevel.Itisinterestingtoseehowthisarisesattheglobal level(intermsofthebasicAmpèreanintegrals)asfollows.Goingbacktoourobservation that themagnetic field is continuousbetween themetal plates,weknow there is nophysicalmovement of charge between the plates. But is nonzero on the plates. Therefore

isnonzerobetween theplatesandup toa constant factoranda must be the requireddisplacementcurrent.WecanseethisisexactlywhatwashappeninginEquation(4.46)ifwenowuseMaxwellI

(4.50)

Togetridoftheintegralasbeforewepulloutthetimederivativeandperformanintegrationbyparts.Thisgives

(4.51)

This integral is more complicated to evaluate because of the tensorial nature of the twoderivatives, but it must reduce to a Dirac delta function in some fashion. We shall defer itsevaluationuntil thenextchapterwhenweunderstandtheconceptofgaugepotentialsandgaugeinvariance.Acceptingthisthenfornow,thefinalformofAmpère’soriginallawaftertheadditionofthedisplacementcurrentwhichmustcoincidewiththelocalversionpreviouslyfoundis

(4.52)

ThisisthelastofMaxwell’sequationsreferredtoasMaxwellIV.

James ClerkMaxwell (1831–1879). The fact that Einstein had pictures of bothIsaac Newton andMaxwell hanging on his wall gives a fair idea of Maxwell’scontribution.

Wecannowreturntotheissueofthenumericalvalueofthepermittivityoffreespacethatwas

encountered in the previous chapter. In Equation (4.52) the displacement current has thepermittivityoffreespace multiplyingtheelectricfieldterm.Thisequationalsolinkstheelectricandmagnetic field in the sameway that Faraday’s law does except that nowwe have amattercontententering.Becausethemattercontentisexplicit,thepermittivityhastoenterintoit.Soanumericalvalueforthepermittivitycannowbefoundbecauseweknowhowtomeasurecurrentsand electric/magnetic fields. This required in the first instance the arbitrary definition of μ0,otherwisethiswouldnotbepossible.Thenumericalvalueofthepermittivityoffreespacecannowbedeterminedby againperforming simple types of force experiments (for example on a parallelplate capacitor configuration) or just the capacitance.OriginallyWeber andKohlrausch [6]madethismeasurementbydischargingaLeydenjarthoughagalvanometer;theLeydenjar’schargewasmeasuredbefore and afterwith an electrometer. The key aspect is to ensure that charge is nowmeasuredinCoulombs,sincecurrentsaregivenintermsofCoulombspersecond.Itisfoundtobe

Itisnowpossibletodefineboththeunitofvoltageandtheunitofelectricfield,sincewehaveadefinitionoftheCoulombunit.Theunitofvoltageisthevoltwhere1V=1JC−1,whiletheunitofelectricfieldistheVm−1.

(⇐Left)WilhelmWeber (1804–1891)andRudolfKohlrausch (1809–1858) (right ⇒) succeeded in 1856 in measuring the ratio ofelectrostatictomagnetostaticforces.Thisnumberhasthedimensionoftheinverseofvelocitysquared

4.5 SUMMARY

WecollecthereMaxwell’sequationforcompleteness

(4.53)

(4.54)(4.55)

(4.56)

These field equations tell us how charges and currents are sources for electric andmagneticfieldsandhowelectricandmagneticfieldsarecoupledtogether.Goingbacktohowtheelectricandmagnetic fieldswere deduced, in addition to the fourMaxwell equationswe have the continuityequationandtheLorentzforcelaw

(4.57)

(4.58)

whichtellsushowindividualchargesmoveinthepresenceofelectricandmagneticfields.

Afewmoreremarksareinorderhereaboutthetheorywehavesofar.Thereisthemathematicalproblemofhowtosolvetheseequationsandthephysicalproblemofchoosingboundaryconditionsand the sources. In searching for the most common solutions (not the vulgar type), perhaps ofpracticalinterestandthesimplestmathematically,itisoftenthecasethatthefieldsandthesourcesaredecoupledinaveryparticularway.Aninspectionoftheaboveequationswouldsuggestthatifwe specify the sources then we can try and calculate the corresponding fields. In the inversescenario they also suggest that if we specify the fields thenwe can calculate the trajectories ofchargedparticlesorbodies.Butofcoursethisdivisionisonethatwehavemadeforconvenience.Assoonastheonequantityisspecified,thedynamicalvariablewearetryingtocalculatewillhaveareturneffect(orback-reaction)onthesystem.Thisisadifficultproblemtogettogripswithandtheinterestedreadercanconsult[8]forfurtherdiscussion.Standardreferencesforthischapterare[8]and[7].

NOTES1Question:howdoyouknowthetestchargeisatrest?2Infact,asonecanseebydimensionalanalysis,themagneticfluxdensitytransformationwillhaveafactorinvolvingtherelativevelocityandthesquareofthespeedoflight,leadingtoamuchsmallercontributionforsmallspeeds.

CHAPTER5PHYSICALDEGREESOFFREEDOM

PerhapsthemoststartlingofalltheconsequencestoemergefromMaxwell’sEquationsisthattheyadmitwavesolutionswhichinthevacuumpropagatewiththespeedoflight-theyarelight!Bydecouplingthefieldequations,thatis,separatingtheelectricandmagneticfieldsfromoneanotherbyperformingafewmathematicaloperations,wefindfieldsthatsatisfywaveequations.However,thepictureiscomplicatedbythefactthatthereissomeredundancyinthesystem(therearetwophysicalpolarizationstatesoflightthatthereaderprobablyalreadyknowsabout).Itwillthereforebe necessary to be careful and to ensure that we do not have any spurious degrees of freedomrunningaroundinthesystem.

This chapter is largelymathematical.Having found theequations thatgovernelectromagneticfields and simple charged matter, it is obviously necessary to investigate them to begin tounderstandthefieldconceptindetail.Itisnowthatonewantstobringthefullmachineryofvectorcalculus (and more generally differential geometry) into the arena in order to manipulate themathematicalstructuressofarfound.

5.1 THEWAVEEQUATIONANDSOLUTIONS

OurstartingpointistheMaxwellEquations(4.53)–(4.56).Supposethatinsomeregionofspacetherearenocurrentsorchargesources,whatwecall thevacuum1.Whatare the typeofelectricandmagneticfieldsthatcanexisthere?InfreespaceEquations(4.53)–(4.56)aresimply

(5.1)(5.2)(5.3)(5.4)

IfwenowtakethecurlofEquations(5.3)and(5.4)weobtain

(5.5)(5.6)

Thepartialderivativeswithrespecttotimeandspacecommute,i.e. sothat

(5.7)(5.8)

ByusingEquations(5.3)and(5.4)afurthertimewecanachievethedecouplingoftheelectricandmagneticfields,sothat

(5.9)(5.10)

Inthiswaythefieldshavedecoupledintotwoseparateequationsofmotion.TosimplifythetwowedgeproductswehavetousethevectoridentityEquation(2.28),where

(5.11)(5.12)

Withthis,thesourcefreeMaxwellEquationsfinallyreduceto

(5.13)

(5.14)(5.15)(5.16)

(5.17)

where we have introduced the second order differential operator which is known as theD’Alembertianoperator.Itssignificanceisthatitisawaveoperatorandwhatwehavejustwrittendown are the three dimensionalwave equations. The reader should already be familiarwith thewaveequationinonespatialdimension,wherethesolutionsingenerallooklikef(x±vt),withvthespeedof propagationalong thex direction. So electric andmagnetic fields that arewave like innatureareadmittedassolutionsanddescribepropagation infreespace(thevacuum)ataspeed

Inthepreviouschapterwesawthatµ0wasadefinedquantityandthat wasmeasuredrelativetothis.Historicallyitwasfoundthatbytakingthevaluesstatedtherethennumericallythevalueofcwasfoundtobeclosetothespeedoflightc≈3.0×108ms−1.

TheFizeau-Foucault experimentwas away ofmeasuring the speedoflightbymechanicalmeans.Itwasdevisedsuchthatthetimeparameteris linked to a rotational speed. Light is shone on amirror at a knowndistance from the source. In between is a toothed wheel that rotateswithaknownangularvelocity.Thebeamoflightisthusconvertedintoaseriesofpulses.Makingobservationsclose to thesourceandknowingthenumberofteethonthewheelallowsusmeasurethetransittimeofthebeamoflight.Fizeauarrivedatavalueofc≈3.1×108ms−1.

The great insight of Maxwell was to realize that light is exactly this type of electromagneticdisturbance and that it is one part of infinite spectrum of different types of an electromagneticradiation [17,6].Measuring the speed of light of course had been performed in an independentfashionpreviously.Thekeyparametertomeasureisatransittimeforlighttopropagatefromonepointtoanother,giventhatweknowtheseparationaccurately.

(Left ⇐) Armand Hippolyte Louis Fizeau (1819–1896), inaddition to the measurement of the speed of light alsopredicted red shifting of electromagnetic waves and thediscoveryoftheDopplerEffect.HisnameisinscribedontheEiffelTower.

(Right ⇒) Jean Bernard Léon Foucault (1819–1868) alsoinventedaratherspecialpendulumthatshowsthattheEarthisrotating,ashisnamesuggests.

5.2 BASICWAVESOLUTIONS-SCALARFIELDTHEORY

What do the fields look like that satisfy the wave equation? Looking at Equations (5.13) and(5.15) one sees thatweare solving the samebasicpartial differential equation,but that the twovector fields are not independent. We will need to ensure that the solutions we obtain areconsistent.Tothisend,wewillstartoff just lookingat theelectric fieldsolutionsandthenchecklaterontoseewhattheconsistencyconditionsdemand.

Toobtainasolutionwemustconsiderthecoordinatesystem.Sofarwehavenotspecifiedanyatall,buttoobtainananalyticalexpressionitisobviousthatachoicehastobemade.Ingeneral,thechoiceof a coordinate systemshouldbebasedonanygeometrical symmetries in theproblem, ifindeedanyexistatall.Forexample,itwouldbeunwisetochooseoblatespheroidalcoordinatesto

solveaprobleminvolvingasinglelongcylinder.Wewilldevelopsolutionsinthethreesimplestandperhapsmost useful coordinate systems: (i)Cartesian coordinates, (ii) spherical coordinates, and(iii) cylindrical coordinates. Our aim here is to arrive at the solutions without completemathematicalrigor,butrathertohaveanappreciationofthetypeofsolutionsandwhattheymeanwithrespecttooneanother.

Since we are choosing to consider wave like solutions the time dependency will clearly beoscillatory. Regardless of the spatial coordinates used, we can write down the functionaldependenceontime.Itissimply

(5.18)

forasinglefrequencymodeor

(5.19)

whenthereiscontinuousspectrumoffrequencies(ormodes).Noteheretheobservationaboutthispart of the solution, thatwehaveassumedaFourier transformof the solution;weare thereforeturning the operator in time (that is the second partial derivative in time) into an algebraicexpression in frequency space. This is quite the common theme in searching for solutions to thewaveequation.Foracontinuousspace (suchas the timecoordinate) thestrategy is to look foraFouriertransformsolution,whileforaperiodic(compact)spacethestrategyistolookforaFourierseriestypesolution.

SubstitutionofEquation(5.18)intotheEquation(5.13)gives

(5.20)

ThisequationisknownasthescalarHelmholtzequation,andforthetimebeingwepostponethecomplicationthattheelectricfieldisaconstrainedvectorfieldandtreatitasascalarquantitywithanindex.

HermannLudwigFerdinandvonHelmholtz (1821–1894).Bothaphysicistandaphysician, he worked on a number of areas ranging from color vision, visualperception, thermodynamics, electrodynamics, plus a good smattering ofphilosophy.

To solve this equation we will use the method of separation of variables to obtain analyticexpressionsof thecoordinatesused. In factwehavedonethis implicitly inEquation(5.18);writethesolutionasEa(t,x)=Ea(x)T(t)andthenrearrangeintoseparatespatialandtimepieces,sothat

(5.21)

Theterminsquarebracketsmustbeaconstant, sothatthespatialpartisindependentoftime. The solutions are exponentials in time. If this constant is negative, we have oscillatingexponentialswhileif itpositivethesolutionsexponentiallygrowordecay.Ingeneralitcouldbeacomplexnumber, Thechoicemade isdictatedbywhat typeofboundaryconditionsone isputting on the problem and also what type of source they are originating from. Of immediateinterestisofsometypeofoscillatingsourcesothatwechoosetheoscillatingsolutionsfrom

(5.22)

ThisleadsdirectlytotheHelmholtzequationEquation(5.20).

The simplest set of coordinates to work in are Cartesian coordinates wherein the Helmholtzequationis

(5.23)

This has the well known plane wave set of solutions which can be obtained by separation ofvariablesifdesired.Eachcoordinatespanstheentirerealline soitwillbeaFouriertransform

solutionthatweseek.NotealsothatthereisnodifferencebetweenanupstairsandadownstairsindexinthesecoordinatesbecausethemetricissimplytheKroneckerDeltatensor.Onefinds

(5.24)(5.25)(5.26)

TheconstantsEa are just thecomplexamplitudesof theelectric field,whichare thearbitraryintegration constants. That they are complex should not worry the reader, as it is only the fullelectricfieldthatneedstobereal.Withthisinmindthefullelectricfieldsolutionreads

(5.27)

(Exercise:verifythisisasolution)Theplanewavesolutionsarealsoeigenfunctionsofthedifferentdifferentialoperatorsthatmake

uptheHelmholtzoperator,withtherespectivewavevectorsbeingtheeigenvalues.Thisisofcourseintimatelylinkedwiththeseparationofvariablesmethodwhichwecanillustratethus.Recallthatforanoperator(oramatrix) witheigenvaluesλnandeigenfunctions(orvectors) onehas

(5.28)

Apply this to Equation (5.23); ifwe take then the eigenfunction is with theeigenvalueλn=−k1k1.Equation(5.23)thenbecomes

(5.29)

where the electric field has the eigenfunction multiplying the remaining part of the solution(separationofvariables). In thismannerwe thenbuildup the remainingpartsof thesolution. Inothercoordinatesystemsthesamestrategycanbeadopted.

One last thing that we can see here is that in general there may be many wave vectorscontributingtotheelectricfieldinwhichcaseweshouldsumoverthemwithanappropriateweightforeachmode

(5.30)

ThisisexactlyaFouriertransforminthreedimensionalspace,astobeexpectedbecausewearesolvingawaveequation.Inadditiontothis,allofthespatialdimensionsareinfinite(non-compact).TheHelmholtzequationthenservestoputaconstraintonthewavevectorsintheformofEquation(5.25).

5.2.1 SolutionsinSphericalCoordinatesTurning now to spherical coordinates one can try to build up a general solution using

sphericalwaves,theequivalentofplanewavesinthepreviousdiscussion.Thereisasubtletynowthatsince insphericalcoordinates themetric isnon-trivial,wewouldneedtoensurethatweareusing thecorrect formof theLaplacianoperator toactof theelectricvector field.TheLaplacianoperatorgiveninEquation(2.66)isforascalarfield.Wewillassumefornowthatascalarfieldwillbe sufficient but it is worth checking under what conditions this is valid or is a well definedapproximation.

TheHelmholtzoperatorinsphericalcoordinatesactingonascalarfieldε(x)isgivenby

(5.31)

Assumingagaintheseparationofvariablesmethodsothatthesolutioninsphericalcoordinatestaketheform onecanstartthereductionoffagain(wehaveincludedafactorofkintheradialsolutionsothatitsargumentisdimensionless).ThesimplestterminthisLaplacianisthe partialderivativeandsothisistheplacetostartwith

(5.32)

Insphericalcoordinates,anysolutionmustbeinvariantunderafullrotationofthe coordinate

sothat leavesanysolutioninvariant.Thisforcesthetwoconditionsthatm2>0,sothatthe solutions are not exponentially growing or decaying, and that so that thesolutionsareperiodic.Itisastraightforwarddifferentialequationsolve,withthetwosolutions

(5.33)

UsingthistosimplifyEquation(5.31)resultsin

(5.34)

Sinceeachterminvolvefactorsofr2,oneneeds toensure that there isaresultingdifferentialequationinrhasnootherfactorsinvolvingtheothertwocoordinatesinit.Sothetwotermsthathavetheθfactorsinshouldbesetequaltosomeconstant

(5.35)

wheretheconstanthasbeenchosentobethevalue−l(l+1)forlaterconvenience.Indeed,giventheearlierdiscussionabouteigenvalues,onemaywellbetemptedtosuspectthatthelgivenhereissomethingtodowiththis-anditcertainlyis.ItisthereforehelpfultorewriteEquations(5.34)and(5.35)inthefollowingform

(5.36)

(5.37)

(Exercise:verifythisistrue.)The problem has now been reduced to finding solutions for two separate second order

differentialequations.Thesetwoequationsoccursooftenintheoreticalphysicsthattheyhavebeennamed.Equation(5.36)isknownasthesphericalBesselequation,whileEquation(5.37)isknownas the associated Legendre equation. A standard piece of mathematical analysis to solve theseequations is tomake a power series substitution into the differential equations and then deducerecurrence relations amongst the coefficients. Any standard textbook on mathematical methodsdescribe this procedure, so we shall not pursue this avenue further (see for example [15]). TheradialpartofthesolutionthenconsistsofthesphericalBesselfunctionsjl(kr)(whichareregularattheorigin)andthesphericalNeumann functionsnl(kr) (whichdivergeat theorigin).Theangularpart of the solution consists of the associated Legendre polynomials Pl,m(cos θ). The generalsolutionthenlookslike

(5.38)

wherethesummationrunsoveralll, andthemsummationisboundedby−l≤m≤l.Inadditiontotheconstrainedsummation,itisnecessarytoimposearealityconditiononthesolutions,sothatthephysicalfieldasafunctionoftime(ratherthanfrequency)isrealvalued.Thisisbecauseboththesolutionsarecomplexvalued.Notethatsummationsoverthediscreteangularmomentumnumbershavereplacedthecontinuousintegrationsoverplanewavevectors.Thefactthattheseareeigenfunctionswitheigenvaluesisperhapsalittlelessclearhere(afterallwehaven’texplainedit)butitisthereandmuchmoreinterestinginthiscase.

To finish off this subsection, it is useful to write down recursion relations for generating therespectivesolutionsfromsimplerwellknownelementaryfunctions(ratherthanjustapowerseriesexpansion).

(5.39)

(5.40)

(5.41)

(5.42)

The introducedabovearecalledthesphericalharmonicsandplayafundamentalroleinspherical coordinates. They are the eigenfunctions on the twodimensional sphere.Note also thedifferent nature of the dispersion relation here compared with the plane wave solutions. Inspherical coordinates, via the spherical Bessel functions, the constant k defines the length scaleoverwhich the radial solutions vary. The angular form, however, is unconstrained by it and onlycoupledthebytheleigenvalue.

5.2.2 SolutionsinCylindricalCoordinatesThe other standard set of coordinates that every undergraduate has to play with are the

cylindrical coordinates (r, θ, z). The method of solution proceeds analogously with sphericalcoordinates.NotethatthisisthehalfwayhousebetweenCartesianandspherical;onedimensioniscompact(θ),oneissemi-infinite(r)andtheotherisinfinite(z).ThefirstthingtodoiswritedowntheHelmholtzequationincylindricalcoordinates

(5.43)

Boththeθandzvariablesdonotpresentmuchdifficultynowinsolvingaswehavealreadycomeacrosstheirtypesofsolutionearlier.Theyarebothcomplexexponentialswitheitheracontinuousordiscreteseparationconstant(thinkeigenvalue!)thatlabelsthesolution.Theylooklike

(5.44)(5.45)

With these solutions the Helmholtz equation reduces to a simple second order differentialequationgivenby

(5.46)

ThisisjustthestandardBesselequationwithsolutionsgivenbytheBesselfunctionsJm(x)andNm(x).TheyarerelatedsimplytoprevioussphericalBesselandNeumannfunctionsdefinedby

(5.47)

(5.48)

The full solution then will require a combination of a sum over the discrete index and anintegrationoverthecontinuousone.Thus,

(5.49)

Aswith the other solutions found for theHelmholtz equation, it is necessary impose a realityconstraintsothatthephysicalelectricfieldsarerealvalued.

5.2.3 CommentsA fewremarksare inorderhereabout thescalar fieldsolutions found in the threecoordinate

systems. The first comment is that it is only in spherical coordinates that the solutions can be

square normalizable (see Equation (2.48)). This is intimately related to the fact that two of thecoordinates are compact (the θ and ) and the type of source that is ultimately producing theelectricfield.Theimplicationhere isthatthesourcehasafinitespatialextent inalldirectionsofspace.Onecanthereforeputaclosedsurfacearound it.Thiscanonlybedone insphericalpolarcoordinateswherethisclosedsurfacewouldbeasphereofsuitablesizeradius.Asecondremarkfollowingonfromthisisthattheenergyassociatedwiththesewaveswillthenbefinite.Whilewehavenotdefinedsofartheenergycontainedinafluctuatingfield,itwillturnouttheenergydensitylooks like the product of two fields together. In the other two coordinate systems this is not thecase.Anecessaryconditionistomaintainthegeometryoftheproblem.InCartesiancoordinateswewould be considering an infinite plane (or surface) source giving rise to the plane wave typesolutions. Similarly in cylindrical coordinates it would be an infinite line (or curve) source thatwouldproducecylindricalwavetypesolution.Afinalremarkisthatfordistancesmuchlargerthanthe corresponding wavelength (given by ) one can often work, at least locally, in Cartesiancoordinatesasanapproximationtooneoftheothers.Thisisbecauselocallyacurvedsurfacecanbe approximated by a flat plane.Onemust remember then not to calculate outside its region ofapplicability.Usefuldiscussionsonrelatedmatterscanbefoundinboth[8]and[7].

5.3 DEGREESOFFREEDOM-POLARIZATION

Wehave just seen that theelectricandmagnetic fields satisfy separatelywaveequations,andonemightconcludehastily that therearesixdegreesof freedom(the threeelectricEaandthreemagneticBaundeterminedamplitudes).Thatthis isnotthecasewillbeseenbybeingabitmoreexplicitwiththetypeofwavesolutionsweencounterandacloserlookathowtheyaresatisfyingMaxwell’sequations.Itshouldbesaidthatthisisadifficulttopic-wearetryingtofindconsistentsolutions to the full vector field equations. This is necessary however to understand whatpolarizationreallyis.

5.3.1 PolarizationinCartesianCoordinatesLet us consider again the plane wave solutions in Cartesian coordinates for a a particular

wavevector.Wenowalsoincludeasimilarsolutionforthemagneticfieldandaskthequestionhowtheyareconnected.Thegeneralvectorsolutionswilllooklike

(5.50)(5.51)

wherethewavevectorkagivesthedirectionofpropagation(andwavelength)andbothamplitudesare in general constant complex numbers.Wemust remember now that to derive the twowaveequations we had to perform certain vector calculus operations on the first order Maxwell’sequations to disentangle the electric and magnetic fields. So we should really substitute thesolutionsintotheoriginalequations.Doingsoyields

(5.52)(5.53)(5.54)

(5.55)

Thefirsttwoequationssaythatonecomponentoftheelectricandmagneticfieldareredundant.Foragivenka,oneofthephysicalfieldvectorcomponentscanbefoundintermsoftheothertwo.Sowe lose a degree of freedom for the electric andmagnetic fields. Similarly, for the next twoequations,itcanbeseenthatweloseanotherdegreeoffreedomfromeach.Wehavethereforelostfourdegreesoffreedom,bringingdownthephysicaldegreesoffreedomfromsixtotwo.

Thisnowtakesusneatlyontothepolarizationofanelectromagneticplanewave.Equation(5.54)saysthatthemagneticfieldisnormaltothewavevectorandtheelectricfield,whileEquation(5.55)saysthattheelectricfieldisnormaltothewavevectorandthemagneticfield.Sothethreeformamutuallyorthogonalsetofvectors.Withoutanylossofgenerality,imaginethewaveispropagatingin the +z direction. The electric field will then have in general the components (Ex, Ey). Thequestion we now ask is what is the allowed solution space? Let us return to our wave solutionEquation(5.50)andwriteitoutincomponents

(5.56)(5.57)

Wehaveallowedhereforaphasedifference betweenthetwocomponents,with andpositive.The strategynow is toeliminate the timedependenceso that find the required solutionspace.Usingthedoubleangleformulaintheabovewehave

(5.58)(5.59)

ThetimedependencecannowbeeliminatedfromEquation(5.59)

(5.60)

Thisleadstotheequationofanellipse

(5.61)

wheretheellipseisrotatedbysomeanglewithrespecttothex,ycoordinates.Iteasytoworkoutwhatthisangleis-justperformarotationonthex,yelectricfieldcomponents.

(Exercise:calculatetheangleαtheellipseisrotatedbyintheabove.)

Following on from this general state of polarization, it is necessary to catalogue the differentstatesofpolarizationthatalldescendfromtheellipticalcase,sincetheseareoftenusedinpractice.

Linearpolarization-thetwocomponentsareeithercompletelyinphaseoroutofphasewithoneanotherby with Theelectricfieldoscillatesinastraighttiltedlinethatisdeterminedbythetwoamplitudes.Circularpolarization-heretheamplitudesareequal,E1=E2,butnowthephaserelationshipisgivenby with Thepositivesignisdefinedtobeleft-circularlypolarized,whilethenegativesignisdefinedtoberight-circularlypolarized.Ellipticalpolarization-Thisiswhentheamplitudesandphasetakeongeneralvaluesasalreadydescribed.TheangleoftiltαoftheellipsewithrespecttothecoordinatesystemofEx,Eyisgivenby

What about polarization in the other coordinate systems?We know that either thewaves areoutgoingor incomingwithrespecttotheircoordinateorigin.Theyaremadeupofeigenfunctionsthatrespectthesymmetriesofthecoordinatesystemabouttheircoordinateorigin.Themathofitismore complicated (though quite interesting) so only a sketch of the solutions are given. Theinterestedreadercanpursuethisfurtherin[8]and[13].

5.3.2 PolarizationinSphericalCoordinates-TEandTMModesInsphericalcoordinatesthefullvectorHelmholtzequationneedstobesolved.Theseneedtobe

disentangled into two polarization states known as the transverse electric (TE) and transversemagnetic(TM)modes.Heavyuseismadeoftheangularmomentumoperator Wegiveherejustabriefstatementoftheformtheytake.Thegeneralsolutiontakestheform

(5.62)

(5.63)

where

(5.64)

(5.65)

(5.66)

WhatthisshowsisthatthesolutionsconsistofsolutionstothescalarHelmholtzequationwhichare then acted on by suitable operators to generate the full vector field solutions. From theboundaryconditionsandthesourceconditionsalloftheintegrationconstantscanbedeterminedinasasetofmultipolefields.Itishelpfultojustwritedowntheequivalentpolarizationconditionsfor

thesphericalcase.ThepolarizationstategivenbyjusttheTEmodessatisfy

(5.67)

(5.68)

whilethepolarizationstateofjusttheTMmodessatisfy

(5.69)(5.70)

Multipole fields, while being somewhat more fiddly in a mathematical sense, are extremelyuseful.For example, if youwant to consider light scattering fromadielectric ormetallic sphere,thesearethesolutionstypicallyrequiredtoevaluatephysicalobservables.ComparingthiswiththemorestandardCartesiancoordinatespolarizationwherewehadanellipseofpossibilities,weseesomething simplerhere.The reader is referred to [8] for a fuller discussion. In a similar fashionpolarizationincylindricalcoordinatescanbeconstructed,andisquiterelevantforstructuressuchaswaveguides[8].

5.4 GAUGEINVARIANCEANDPOTENTIALS

Earlierweencounteredascalarpotentialthatwasusedtodescribethestaticelectricfield.Bytakingthegradientofthiswecouldrecovertheelectricfield.Anaturalquestiontoaskiswhatisthecorrespondingpotentialforthemagneticfield?RecallthatMaxwellIIisastatementthattherearenomagneticmonopoles

(5.71)

Justaswefoundearlierthatascalarpotentialsolvesthecurlfreeconditionontheelectricfield our question now is what is the corresponding potential for the above. It is a

mathematical result (see for instance [2]) that a vector potential Aa solves the divergence freeconditiononthevectorfieldBasuchthat

(5.72)

(Exercise:verifythissolvesMaxwellII.)IfwenowputthisintoMaxwellIIIwefindanewstatementfortheelectricfield,thatis

(5.73)

Equation(3.7)hasbeenmodifiedtoincludethetimedependenteffectsofMaxwellIII.However,itisonceagainintheformofthecurlofsomenewvectorequallingzero,soweknowthatitcanbewrittenasthegradientofsomescalarfunctionviz,

(5.74)

Justas themagnetic fieldhasbeenwritten in termsofapotential,sohastheelectric field (intermsof thesamevectorpotential),butalsoanadditional scalarpotential.Onecannothelpbutnoticethe lackofsymmetryherebetweentheelectricandmagnetic fields.Wealsoseemtohavecreatedaproblemforourselvesintermsofthenumberofdegreesoffreedomwenowhaveinthesystem.Itwasfoundpreviouslythattherewereonlytwophysicalpolarizationstates,yetnowweseemtohaveatotaloffour(scalarplusvector)degreesoffreedom.Whythemismatch?

Whatwehavejusttoucheduponistheconceptofgaugeinvariance,firstexpoundedbyHermannWeylinconnectionwithcoordinatetransformationsingeneralrelativity.Theastutereaderwillhavenoticedthattherehasalwaysbeensomethingtodowithsymmetrylurkinginthebackground.Thebasicpointthatwewillseeisthatthepotentialswehavejustdiscoveredarenotunique-wecanaddarbitraryfunctions(infacttwo)tothemandthefieldequationswillbeunchangedbecausethephysical fields themselvesare invariantunder this transformation.This typeofdeepsymmetry isthebasisformuchofmodernparticlephysicsandtheStandardModel.Foranhistoricalperspectiveonclassicalgaugetheorysee[9].

Hermann Klaus Hugo Weyl (1885–1955), mathematician, theoretical physicistandphilosopher,heworkedonalargenumberofareasincludingnumbertheory,group theory, and the geometry associated with combining electromagnetism

withgeneralrelativity.

Tothisend,weconsidersubstitutingEquations(5.72)and(5.74)intothefieldEquations(4.53)–(4.56).Onefindsthat

(5.75)(5.76)(5.77)

(5.78)

Immediatelyitisclearthattheseequationswillsimplify.Equation(5.76)isidenticallyzero,whilein Equation (5.77) a simple cancellation of the vector potential occurs together with the vectoridentitythatthecurlofadivergenceiszero.Thisleavesthetworemainingequationsas

(5.79)

(5.80)

HerewehaveusedEquation (2.24) togetherwitha smallamountof rearrangement.Now it istime to use the gauge invariance. A short calculation reveals that if we use instead the gaugepotentials thatarerelatedtotheoriginalonesinEquation(5.79)andEquation(5.80)by

(5.81)(5.82)

where is an arbitrary function, then the electric and magnetic fields remain unchanged.Therefore,thefieldequationsforthepotentialsEquation(5.79)andEquation(5.80)willalsoremaininvariantwiththischoiceofpotentials.Intechnical languagetheyaresaidtobegaugeinvariant.This means that we can choose the arbitrary function so that andAa satisfy some condition,therebychoosingagauge.LookingatEquation(5.80)onecanseethatitisnearlyawaveequation,wejustneedtogetridofthelastterm.Wecandothisbychoosingthegauge

(5.83)

TheabovegaugeconditionisinfactknownastheLorentzgauge.Bymakingthischoiceoneofthefourpotentialsbecomesafunctionoftheotherthree.Forexample,performanintegrationovertimesuchthat

(5.84)

Herethescalarpotentialbecomesafunctionalofthevectorpotential.

Withthischoiceofconstraintonthegaugepotentialthetwofieldequationsreduceto

(5.85)

(5.86)

Itshouldbeapparentthatwehavenotquitefinishedaswestillappeartohavethreedegreesoffreedom in thesystem.Weneed to imposea furtherconstraint tobesure thatwehaveonly twophysical degrees of freedom, which we know to be true from the two states of polarizationpreviouslyfound.Wheredoesthiscomefrom?

Let us re-examine the Lorentz gauge condition Equation (5.83) that we imposed. Clearly we

choseitbecausethefieldequationsforthescalarandvectorpotentialswoulddisentanglefromoneanother ratherneatly,butapart from that it lookedquitearbitrary.This is thepointaboutgaugetransformationsthattheyarearbitrary.Oncewechooseagaugeweneedtoensuretwoimportantproperties; one is thatwehave removedall spuriousdegreesof freedom from the system;andasecondisthatallthephysicalobservablesandpropertiesremaingaugeinvariant.Letuspindownthe firstpoint.Whathappens to theLorentzgauge ifwe ifwedoanothergauge transformation?Obviously the electric and magnetic fields are the same, whereas the Lorentz gauge conditionbecomes

(5.87)

where is another arbitrary function that specifies the gauge transformation. The Lorentzgaugeispreservedintheaboveprovidedthenewgaugetransformationparametersatisfies

(5.88)

thatis,itmustsatisfyascalarwaveequation.Sothisarbitraryscalarcanthenbechosentosetoneofthecomponentsofthevectorpotentialtozero.ForexampleinCartesiancoordinatesifwehaveawavepropagatinginthez-direction,wecanchooseAz(t,x,y,z)=0byasuitablechoiceofsatisfying the wave equation. At this point we arrive at two components of the vector potentialrepresentingthetwophysicaldegreesoffreedomoftheelectromagneticfield.Noteanotherfeatureofthisprocess.Byperformingagaugetransformationonthegaugeconditionitself,wewereabletoeliminateanotherspuriousdegreeoffreedom.Onemightaskifitispossibletodothesamethingagain and finishwith only onedegree of freedom.The answer is an emphatic “no”, because anysubsequent gauge transformation will have a gauge function that also satisfies the scalar waveequationand(bythesuperpositionofsolutionfor linearequation)thissecondgaugefunctioncanbeabsorbedintothefirsttransformation.

AnothercommonlyencounteredgaugechoiceistheCoulombgaugewhichisgivenby

(5.89)

Withthisgaugeconditionthefieldequationsbecome

(5.90)

(5.91)

FortheconnectionsbetweentheLorentzandCoulombgauges,see[10].

Given now that we have written the field equations in terms of potentials, what are theimmediate quantities of interest? Certainly, it would be useful to have the gauge potential as afunctionalofthecurrent,aswecanthensimplyderivethemagneticfield.Inadditiontothis, theLorentzforcelawEquation(4.15)whenexpressedintermsofgaugepotentialsisalsoanobservableof interest. As a final point,we should return to displacement current encountered in the globalsettingandtrytoevaluateEquation(4.51).

Firstly, in thecaseofmagnetostatics,where thedisplacementcurrentdoesnotcontribute,wecanrewriteAmpère’slawintermsofthevectorpotentialas

(5.92)

wherewehave takenthevectorpotential tobe in theCoulombgaugeandhave just inverted theLaplacianoperator inexactly thesamewayas in thePoissonequation.Movingon to theLorentzforceequationoneseesthat

(5.93)

whichrearrangesto

(5.94)

TheinterestingobservationhereisthatthechargedparticlereceivesacontributionofqAatoitsmomentum(inclassicalmechanics,notablywhenoneworksintermsofhamiltonians,onehastobecareful to distinguish between velocity and conjugatemomenta). It looks like that themomenta

isagaugedependentquantity.

(Exercise:CheckthattheEquation(5.94)writtenintermsofthegaugepotentialremainsgaugeinvariant.)

Finallywereconsiderthedisplacementcurrentusinggaugepotentials.IfweconsiderevaluatingEquation(4.51)bywritingtheelectricfieldintermsofthegaugepotentialsonefinds

(5.95)

IfwechoosetheCoulombgaugethenthe term integratessimply,while theA termvanishesafteranintegrationbyparts.Therefore

(5.96)

Thefinalstepistorealizethattheconservationofchargeresultsfromtakingthedivergenceoftheaboveequation;therefore,wecansimplyaddtheColoumbgaugeconditiontothedivergenceoftheabove.Asafieldequationitmustbegaugeinvariantsowemakethereplacement

(5.97)

whichisnowagaugeinvariantequation.

The discussion of physical observables remaining gauge invariant is simple provided we areworking with the fields and not the gauge potentials. So far we have built up the field theoryimplicitlyassumingsomesymmetriesinthesystem.ConsiderthecoordinateswherethefieldissatatsayEa(t,x).Herewehavetakentimettobejustaparameter,butthespatialcoordinatesxhavebeen implicitly assume to transform under the rotation group in three dimensions. If we havechosenagaugeforexamplewhereAz(t,x)=0,thenonecanseethatthissymmetrygrouphasnowbeenbroken.Oneneedstoensurethenthatanymeasurablewillnotsufferfromthisissue(indeedthis goes into making sure that any quantum theory of electrodynamics remains internallyconsistent).

5.5 SUMMARY

Wehavefound inthischaptersomeof thebasicphysicalpropertiesthatemergeby lookingatthemathematicaldetailsofthethefieldequations.Theequationsadmitwavelikesolutionsinfreespace,whichwehavefocusedonasthephysicalsolutionsofthefieldequations.Gooddiscussionscanbefoundin[8]and[15].Itiscomplicatedbythefactthatthephysicalpropagatingmodes(twodegrees of freedom) are fewer in number than the number of components that constitute theelectric and magnetic fields. These are hidden by the original fields and the correspondingpotentialsfromwhichtheyarederived.This isthecentralfeatureofgaugeinvariance,andisthekey mathematical structure around which electromagnetism and more general field theoriesrevolve. A fuller discussion of the historical roots of gauge invariance and how they feature inclassicalelectrodynamicsaregivenin[9]and[10].

NOTE1Thevacuumisasubtleconcept.Inquantumtheoriesofelectromagneticfields,forexample,forcescanstillariseduetofluctuationsinthevacuum.Thisisarichbutsomewhatadvancedtopic.

CHAPTER 6PHYSICALOBSERVABLES

Thephysicalobservableswehaveencounteredthusfarhavebeenforcesthatwecouldmeasuremechanically. Electric forces were turned into voltage measurements by using an electrometer.Havingunderstoodtheconnectionbetweencurrentsandmagneticfields,galvanometershavebeenconstructedthatprovideameasureforacurrent.Eachtimeamechanicalforceortorquehasbeentraded for a voltage or current. These are then what we would regard as macroscopic physicalobservables. They can be used to quantify measurements of different electromagnetic structureswhenatestbodyisintroducedasalocalprobeofthesystem.

Anotherfeaturethatisapparentisthatthefieldtheoryhasbeeninvestigatedatthelinearlevel.Forexample,whenweconsideredtheforceonachargeorcurrentithasalwaysbeenproportionaltothefieldinalinearway.Thisisnotsurprisingbecauseofhowthefieldsaredefinedinthefirstplace. But we can now go a bit further and ask, since this is a dynamical system, what are theobservable properties like energy or momentum of the fields themselves. To answer this type ofquestion, it is necessary to go to the next order and considerbilinear field terms. This is wheremuch of the dynamics and observables reside. A detector will only register a small piece ofinformationaboutthefieldinthiscontext.

Before getting on to this, however, we first make a short consolidation of the physicalobservablesthatareassociatednotwiththefieldsthemselves,butwiththemovementofcharge.Afieldisappliedandthechargesmove.Thismovementunderdifferentconditionsproducesdistincteffects. We shall therefore consider the students favourite electrical circuit as a means tounderstandsomeoftheintricaciesinvolved.

6.1 THERLCCIRCUIT

Every student of physics knows and loves (or hates) this circuit. It is the Resistor-Inductor-Capacitor circuit and it is quite ubiquitous in its dynamical form. Thinking of it mechanically, itdescribes an oscillating body that is subjected to a frictional force. The basic observable in thiscircuit is thecurrentandanappliedvoltage is the source thatdrives thecircuit.Letus startoffanalyzing this circuit asa tubeofmetal (thewire)which lies in thez-direction.Theendshaveacrosssectionalarea acrosswhichweapplyatimedependentvoltageV(t).Thecurrentacrosstheplanez=0isgivenby

(6.1)

Wewillassume thatatanyothercrosssectionalplane to the tube thecurrent is thesame. Inprinciple Ja is a function of position along the tube. The essential constraint is obviouslyconservation of charge. The three components we can immediately insert are the parallel platecapacitor (given in Section 3.4), the resistor (given in Section 4.1), and the inductor (given isSection4.3).Intermsofthetubegeometry,acutismadeinthetubeandasmallsectionremoved.Thenthefacesaremadeoversizesothatthediameterofthefaceismuchlargerthanthediameterof the tube - this is the insertionofaparallelplatediskcapacitor.Wemakeasimilarcut for theresistorandinsertadifferentpieceofmaterial,thatneitherconductstoomuch,nortoolittle.Forthefinalcomponent(theinductor)wetakeasectionoftube,squashitandthencoilitupsothatwehaveahelical structurewith anaxis anda radius about it.Wenow take the tube tobe small incomparisonwiththecomponents,thatis Tofinishoffthecircuitweassumethatitsshapeisrectangular with four corners as shown in Figure (6.1). The question now to ask is what is thedynamicalequationthatgovernsthissystem?

FIGURE6.1:TheRLCcircuit-thebasicformofthisequationappearsinlargenumberofdifferentareas.Apendulumwithfriction,amaterialwithabsorptionandadecayingparticlestate

allsharethiscommontypeofstructure.

The two constraints that determine the equation of motion are the conservation of current,togetherwiththatthepotentialdifferencesacrosseachcomponentmustsumtotheappliedvoltageV(t).Itisgivenbythewellknownequation

(6.2)

SinceweknowV(t),wecandifferentiateittoobtainanexpressioninvolvingthecurrent,viz

(6.3)

Sowhathavewelearned?ByapplyingwhatisoftencalledatestfunctionV(t),wecandrivethiscircuitsothatitresponds.Theresponseisthemovementofchargeandthephysicalobservableisthe current as a functionof time (asmeasuredby agalvanometer).Froma simple knowledgeofdifferential equations, we recognize two solutions to the above - the transient and the particularintegral.IfthevoltageV(t)isswitchedoffthenthesolutionrevertstothetransientsolutionwherethe constants of integration are determined from its behavior at the switching off time. Thetransient solution looks like exponential functions which have a real and imaginary component.Theyare

(6.4)(6.5)

(6.6)

where areconstantsofintegration.Thereisalwaysanexponentiallydecayingcomponenttothesesolutionsbutthesemaywellbeoscillatingaswell.Whenthetestfunctionison,thesolutionistheparticularintegralplusthetransientsolutionthatisdeterminedfromearlierbehavior.NotealsothatthisisperhapsthesimplestexampleoftheuseofGreenfunctionsthatonemayencounter,andservesasabasisformorecomplicatedproblems.TosolveusingaGreenfunctionwesimplywritethecurrentas

(6.7)

We are making here an explicit causal link where a source of oscillation at an earlier timepropagatesandhasitsresultingeffectatalatertime.WiththisEquation(6.3)istransformedto

(6.8)

Itmaynotbeobvious,butitisexpedienttoperformaFouriertransformonthisequationsothatwecantradetimederivativesformultiplesoffrequency Onefinds

(6.9)

The time dependent Green function then is obtained by transforming back into the timecoordinate

(6.10)

(6.11)

WiththisGreenfunctionwehavethesystemresponse(thecurrent)asafunctionaloftheinputtestfunction(thevoltage)viz

(6.12)

InadditiontothemovementofchargethrougheachcomponentgivenbyEquation(6.3),thereisalso their energy/power behavior as a function of time. For each component there is a potentialdifference(andacorrespondingforce)acrossitthatdoesworkonthechargetomoveitbetweenthetwopointsp1andp2.Thismovementofchargeisgivenexactlybythecurrent.Therefore,theinstantaneousrateofdoingworkWisforeachofthethreecomponents

(6.13)

With this result,wecanconsidereachcomponent in turn.The resistor isparticularly simple -justuseOhm’slawforthevoltageintermsofthecurrenttoreach

(6.14)

This is theenergy lost in theresistorwhich isconverted intoheating in thecomponentwhichwould then be given off as thermal radiation (another topic in itself). Similarly, for the inductorusingEquation(4.33)wehave

(6.15)

Thisexpressionrepresentshowtheinductorreleasesandstoresitsenergy.Forthecapacitor,wehavetoworkatthelevelofchargeratherthancurrentsuchthat

(6.16)

Again,thisgiveshowthecapacitorstoresandreleasesenergyintothecircuit.Forthemostpartthisisstandardelectricaltheoryandthereadercanconsultanystandardtext(forexample[7])formorediscussion.

Ofcourse,inordertoderivethesemacroscopicobservables,informationhasbeenthrownawaytoreachtheseresults.Wehavetradedthe localcurrentdensity foracurrentandcoarsegrainedoverthegeometryofthecircuit.Itsvirtueliesinitsapplicabilityandutility.Allofelectronicsisbuiltupfromsimilarconsiderations.Thisisperhapsonethesimplestexamplesofphysicalobservables-it sits on the matter side and is linear. We now turn to similar considerations for the fieldsthemselvesandtheirbilinearnature.

6.2 FIELDENERGY,MOMENTUM,ANDANGULARMOMENTUM

Thefieldsthatweoriginallyencounteredwereduetothepresenceofchargedbodiesorsteadycurrents.Observed static forceson similar chargesor currentswereour reason to introduce thefieldsandnecessarilyinalinearway.Wealsohavefoundthatelectromagneticfieldscanpropagateinthevacuum.Thenaturalquestiontothenaskisifsuchpropagatingfieldscancarryenergywiththemandcanproducephysicalforcesonotherbodies.Toanswerthis,wewillhavetoworkatthenextlevelofcomplicationandconsiderquantitiesthatarebilinearinthefields.

AsimpleillustrationofthiscanbemadebyconsideringtheLorentzforcelaw.Supposeweaskwhattheforceonachargeqisduetoapassingelectromagneticwavethatislinearlypolarized.Foranoscillatingelectricfield,theLorentzforcelawlookslike

(6.17)

Takingasareferencepointanopticalfrequencyintheabovewithanorderofmagnitudevalueofoneseesthatthechargewillbeforcedtooscillateextremelyrapidly.Ifwecanagree

thatthisfluctuation(thatisthedisplacementcausedbytheforce)istoofasttobemeasured,thenitbecomesnecessarytoemployanaveragingprocedure.Insuchcaseswetakethetimeaverageover

oneperiodTofoscillationwiththeresultthat

(6.18)

(6.19)

Sophysicalquantities thatare linear in the fieldswhentheyareoscillatingaveragetozero inthisscenario. It is thennecessary toworkwithhigherorderpolynomial fieldquantities toobtainnon-vanishingphysicalmeasurables.

6.2.1 TheStressTensorandConservationofMomentumFirstly, let us formulate the conservationofmomentumby startingwith theLorentz force law

appliedtoacontinuumchargedensityenclosedinavolume

(6.20)

Conservation laws in physics are ultimately the consequence of symmetries. From classicalmechanics,oneofthekeyresultsisthattheequationsofmotiondonotchangeunderasymmetrytransformationandthisleadstoaconstantofmotion(thiscanbeexpressedmostelegantlyintheLagrangianformalismwheretheconceptofaNoethercurrentisintroduced).Inthisway,invarianceundertimetranslationsleadstoconservationofenergy;invarianceunderspatialtranslationsleadsto conservation of momentum; and invariance under rotations leads to conservation of angularmomentum.Wewillbeginwithaconsiderationofthefieldequations.Symmetriesalsogiverisetoaconserved“charge”,wheretheideaofchargeisusedinaverygeneralsense.Historicallyitarosefromelectricchargewheretheunderlyinggaugesymmetryofthefieldequationsleadstothis.Itisalsoworthremarkingonthetensornatureoftheencounteredcharges.IfonerecallsEquation(4.3)for the conservation of electric charge we see a scalar and a vector quantity present. This isbecausewearelookingatthetransportofascalarquantity,theelectriccharge.Ifwenowwishtothinkabouttheconservationofmomentum,itshouldnotbeasurprisetoseetheintroductionofadifferenttensorialobject.

Our strategy then is to eliminate the matter fields by the use Maxwell’s Equations (4.53)and(4.56).Thenwewillhaveanexpressionthatisbilinearinthefields.Sincewehaveaboundingarea

of the volume, we are thus looking to find a stress on this area. It is therefore necessary toconvertthevolumeintegralintoasurfaceintegral.

(6.21)

Notingthat

(6.22)

andusingthisintheaboveallowstheforcetobewrittenas

(6.23)

Beforemakingfurthersimplificationsletusunderstandwhatwehavesofar.Thelefthandsideisclearly a mechanical quantity. It is the sum of all the forces on the charge distribution and sothereforecanbewrittenusingNewton’ssecondlawas

(6.24)

Ontherighthandsidewecanalsoseeatotalderivativewithrespecttotime.Thefactthatitiscross product between the electric and magnetic field means that for wave like solutions, it ispointinginthedirectionofpropagation.Italsohastheunitsofmomentumwhichsuggeststhatweidentifyitwithfieldmomentum,viz

(6.25)

Theremainingtermsarebilinear inthe fieldsandalsohaveaderivative. Ifwecollect thetwotypesofmomentumtogetherthenasmallrearrangementusingEquation(4.55)gives

(6.26)

Inthisformweseethatthetimederivativeofthetotalmomentummustequaltheforceactinguponit.Sotherighthandsidemustbetheforceofthephysicalfields.WecanalsomaketherighthandsidesymmetricalinthefieldsbyaddingintoitMaxwellII(Equation(4.54))sothat

(6.27)

Afinalstepinthesimplificationistousethetensoridentity

(6.28)

Usingthisresultonefinds

(6.29)

(Exercise:verifythis.)

From the right hand side we see that we are taking the divergence of a symmetric two-indextensor.Itisinfactthestresstensorandfromtheabovewedefineittobe

(6.30)

TocheckwhatitmeanswejustrewritetheaboveconservationofmomentumlawandmakeuseofGauss’stheorem

(6.31)

Ifweintegratethestresstensoroveranareawefindthetotalforceonthatarea.Bymeasuringthe forceonthisboundarysurface,wearethen indeedmeasuringthestress tensorTabandso itrepresentsaphysicalobservable.

Notealsoherethat it isreally thetotalmomentumthat is themeaningfulquantityandthat ingeneral it will not be possible to make a sharp split between the two. Consider again Equation(6.29); the right hand describes the flow of field momentum across the bounding surface of thevolume.The left had sidehasbotha changing fieldandmechanicalpart.How themomentum isdistributedbetween these two inside thevolume isnotspecified.This isnotsomuchofan issueheresinceitisthestresstensorevaluatedontheboundingsurfacethatcanbemeasured(thetotalchangeinmomentummustbetheappliedforce).

Onesuchmeasurementoftheelectromagneticstressistheradiationpressureonasurface.Thisistheexperimentalobservablethatpavedthewaytothrowingthecorpusculartheoryoflightintosomeconfusion(seeforinstance[12]).

6.2.2 ThePoyntingVectorandEnergyConservationCarryingonfromthepreviousmatterandfieldssystem,whereforcesareactingonthematter,

wecanalsoaskiftheydoworkonthem.Ifthisistruethenbecausethisisatimevaryingsystem,wecancalculatetherateofdoingworkonthem.Itisonlytheelectricfieldthatcandoworkontheassembledcharges;theymusthaveavelocityandfromtheLorentzforcelawEquation(4.58), themagneticfielddoesn’tcontributesinceitisorthogonal.Ifthefieldsaredoingworkonthechargesthenenergymustbelostfromthefieldandanaccompanyingflowofenergymustresult.ThisisthebasicstatementofPoynting’stheorem.

JohnHenryPoynting (1852–1914) worked with Maxwell in the late 1870s andformulatedhisconservationlawin1874.

Therateofdoingworkonmatterbythephysicalfieldsisgivenby

(6.32)

and not surprisingly the algebraic manipulations are similar to before. We use Maxwell IV(Equation(4.56))toexpressthecurrentdensityintermsoftheelectricandmagneticfieldssothat

(6.33)

Forthelaststep,wejusthavetouseMaxwellIII(Equation(4.55))forthefinalsimplification

(6.34)

(6.35)

where

(6.36)

(6.37)

WeseethenthatUrepresentstheenergydensityofthefieldwhileSa istheenergyfluxandiscalled thePoynting vector. This is the statement of conservation of energy; if fields do work oncharges thenthe fieldsenergydensitymustchangetogetherwitha fluxoutof/into theenclosingvolume.Inthecasewherenoworkisdoneontheinternalchargesweseeconservationofenergyjustforthefield.ThePoyntingvectorwhenintegratedoversomeclosedsurfacerepresentsthetotalpowerofthefieldthathasflowedintooroutofthatboundedvolume.

Samuel Pierpont Langley (1834–1906) first invented the bolometer in 1878 inconnectionwithhisworkinastronomy.

Thetypeofmeasurementrequiredforthepowerisabitdifferentfromthepreviousconservationofmomentumobservables.Inthatcaseweweremeasuringaforce,whichcouldbeachievedbyamechanical type balancing procedure. For a power measurement we need to employ a differentapproach.Essentiallyweneedtheincomingradiationtobedepositedcompletelyinthematteritselfso that the fields do work on the charges. One such a device is a bolometer (first invented bySamuelLangley). Itmeasures incomingelectromagneticpowerbyturningthedepositedpower inthe detector into heat. This in turn can be converted phenomenologically into a change of oftemperature,whichinturncanbeconvertedtoanincidentpower.

Notethataswithmeasuringthestresstensor,theobservablehashadtobetraded(orbalanced)with a similar quantity (in this case a thermal energy). There is a second more subtle pointconnectedwiththeprocessofdepositingenergyandEquation(6.35).Howcanonebesurethatalloftheincidentpowerfluxisconvertedintomechanicalpowerandtherebyheat?ThelefthandsideofEquation (6.35)hasbothamechanicalanda fieldpart,bothofwhicharescalarquantities. Inprincipletheincidentfluxthatentersthevolumecanbedistributedbetweenthetwocontributionscompletelyarbitrarily.Theotherpointisthatwearehavingtoinvokesomeoutsidephenomenologyin order to make the measurement. Therefore it is necessary to use thermodynamics and coarsegrainedmaterialproperties to try tocapturethe field information.Thereadermight like to thinkovertheseissuesatalaterdate.

AbolometerbeingusedtomeasurethemagnitudeofthePoyntingvector.Ifall of the incident flux is deposited into the absorbing material and thisturneddirectlyintoheat,thenameasurementofthedepositedpowercanbemade. A measured temperature difference can be then be turned into apower, as the weak thermal link to the cold bath is designed to ensure asteadystateconfiguration.

6.2.3 ConservationofAngularMomentumWe have seen two conservation laws at work in basic electromagnetic field theory, the

conservationofenergyandtheconservationofmomentum.Asanymanonthestreetwilltellyou,itisonlynaturaltoconsidertheconservationofangularmomentumaswell.Justasinthecaseoftheconservationofmomentum,whereaforcecouplestoadisplacementofcharge,weneedtoconsiderhow a torque couples to an angular change. If we consider Equation (6.21) again but drop theintegralsignwehaveatthepointxthat

(6.38)

LetussupposeforthetimebeingweareworkinginstandardCartesiancoordinates.Thenx isalsothepositionvectorfromthecoordinatesystemorigin-thismeansthatwecanuseittoformthetorqueδΓa(x)atthepointxbythestandardwedgeproduct

(6.39)

It is possible to save oneself some work here by recalling the calculation of the stress tensorEquations(6.29)and(6.30).Basicallywecanformthesimpleproductofxawiththestresstensorandothercomponentsbecauseoftworeasons.Firstlythepartialtimederivativedoesn’tactonthexasoitcanbemovedaroundfreely.Secondly,thespatialderivativeoperatorswillalsoactonxainasimpleway.Summingupallthecontributionstothetorquebyreinstatingtheintegral,wefindtheequivalentofEquation(6.27)is

(6.40)

Whatweseehereisexactlytheangularequivalentoftheconservationofmomentum.ThefirstpointtonoteinEquation(6.40)isthatfromsimplemechanicstheappliedtorqueonabodyisequaltotherateofchangeofangularmomentum,viz

(6.41)

Thefirsttermontherightwasoriginallythefieldmomentum,butnowithastheextrafactorofxainvolved.Thismustbethecorrespondingfieldangularmomentum,sothat

(6.42)

ThiscansimplifiedusingthetensoridentityEquation(6.28)to

(6.43)

TheremainingtermsinEquation(6.40)canbewrittenagainintermsofthestresstensorsothat

(6.44)

SincethestresstensorissymmetricwehavebeenabletoputthexainsidethebracketsasititdifferentiatestogiveaKroneckerdelta.Thisinturnmeansthestresstensorindicescontractwiththe givinganullresult.Sinceitisatotalderivativewecanagainformasurfaceintegralsothat

(6.45)

We have arrived at another physical measurable and conservation law. The above is thestatementoftheconservationofangularmomentum;thelefthandsideistherateofchangeofthetotal(fieldplusmechanical)angularmomentumandthatthismustequalthetotaltorqueduetothefields which is the measurable. The tensor quantity on the right is the angular equivalent of thestresstensor.

Canwemeasuretheangularmomentumofthefield?Itturnsouttobepossiblebutinvolvesanextra degree of sophistication. Essentially (again) it requires transferring angular momentum tomatter.Theinterestedreadercanconsult[4]formoredetailsofthisadvancedtopic.

6.3 POLARIZATION(AGAIN)COURTESYOFSTOKES

InChapter5,weencounteredthepolarizationoftheelectromagneticfield - thefactthattherearetwophysicaldegreesoffreedomthathavetobeextractedfromtheelectricandmagneticfields.In that case, where we considered plane wave type solutions, it was enough to know whichcomponents of the electric and magnetic field were nonzero with respect to the direction ofpropagation given by the wave vector. We return to this topic now from the point of view of thephysicalmeasurablesthatthischapterhasbeenfocusedon.

Experiment11(Étienne-LouisMalus)Placeaspecialmaterialinfrontofalightsourcecalledapolarizer and a second similar polarizer (sometimes called an analyzer) after the first polarizer.Nowrotateonewithrespecttotheother.Whathappens?

Observation11The intensityofobserved light (asseenby theeye forexample)dependson therelativeanglebetweenthepolarizerandtheanalyzer.ItsfunctionalformisI(θ)=I(0)cos2(θ).ThisisknownasMalus’law.

Étienne-Louis Malus (1775–1812), formerly a military engineer in the army ofNapoleon, first published his law in 1809. He also has his name on the EiffelTower.

As we have already mentioned, the electric or magnetic field by itself is not measurable forrapidlyoscillatingfieldssimplybecauseitistoofastforadetectortorespondtoit.Whatwewanttodoforthestateofpolarizationistradeupforabilinearobject,similartothestresstensororthePoyntingvectorthathasalreadybeenfound.Thenwecanapplythesameaveragingproceduretoobtainaphysicalmeasurable.Soconsiderthefollowingobject

(6.46)

For the plane-wave like solutions we know already that the time average will be nonzero forcomponents that contain the directions of polarization and zero otherwise. What we didn’t seeearlieraretwoimportantaspects,firstlythatthereisaparameterspacethatdescribesthestateof

polarizationevenforrapidlyoscillatingfields.Inadditiontothis,duetoalackofinformationaboutthe source that sometimes occur (e.g. natural light), the polarization state may have a randomelementtoitleadingtoanincompletespecification.

We now define the Stokes parameters for a plane wave propagating in the z-direction (inCartesiancoordinates).Fourparametersare introduced todescribe thestateofpolarizationsuchthat

(6.47)(6.48)(6.49)

(6.50)

Recallthatthevariable isthephasedifferencebetweenthexandyelectricfieldcomponentsasgiveninEquation(5.59),butwehavenowallowedittohaveageneraltimedependence.OnecanimmediatelyverifythatwiththesedefinitionstheStokesparameterssatisfy

(6.51)

forfullypolarizedlighti.e.wherethereisdefiniteknownamountofeachcomponentsuchthat isaconstant.

Fortotallyunpolarizedlight,bydefinition

(6.52)

ThisimpliesthattheStokesparametersthentaketheform

(6.53)

GeorgeGabriel Stokes (1819–1903), mathematician, physicist, theologian, andpolitician. The parameters which bear his name were published in 1852 andthen were forgotten for some time. He also held the Lucasian Professor ofMathematicschairatCambridge.

There is a nice way of thinking about the state of polarization geometrically. The degree ofpolarization isgivenbyapoint that liesonor ina threedimensionalball. If thepoint liesonthesurface, called the Poincaré sphere, the state is completely polarized. Inside the ball we have ingeneralapartiallypolarizedstate-amixtureofunpolarizedandpolarized.Atthecenterthestateiscompletely unpolarized. Collectively then any degree of polarization can be represented by theinequality

(6.54)

Thesecanberelatedtothepolarizationstatesassociatedwithplanewaves(linear,circular,andelliptical)encounteredinChapter5.ToaidinthepresentationwegroupthefourStokesparametersintotheobject withthefollowingstatements:

Linearpolarization-thisiswhen Thepositivesignindicateslinearpolarizationinthexdirection,whiletheminussigncorrespondstoydirectionpolarization.Circularpolarization-herewerequirethat Thepositiveisleftcircularlypolarized,thenegativeisrightcircularlypolarized.Ellipticalpolarization-thisisthemostgeneralstateandanyparticularpointonthespherewillcorrespondtoanellipticallypolarizedstate.

FIGURE6.2:ThePoincarésphere-Thedifferentstatesofpolarizationaregivenbyapointontheboundingspherewhilepointintheinteriorareonlypartiallypolarized.

InFigure6.2, one can see graphically how the different polarization states sit on the sphere.There is the further geometrically interesting property that rotations on the Poincaré spherecorrespond to transformations of the state of polarization. Thus we can, with the appropriatedefinitions,starttobuildthisintoavectorspaceandconsiderotheroperationssuchasadditions.Basicgroup theorycanbeused todescribemanipulationsofpolarizedbeams.See [14] forsomeusefuldiscussion.

Jules Henri Poincaré (1854–1912), another polymath in the history ofscience. He excelled in all areas, contributing to topology, chaos, andmathematical physics. The Poincaré group is the symmetry which is builtintoallrelativisticfieldtheories.

6.4 SUMMARY

Theaimofthischapterwastoestablishthebasicfieldtheorymeasurablesandtheprinciplesoftheirmeasurement.Inthefirstinstancevoltagesandcurrentsthatwehaveappliedexternallywereused to understand the coarse grained dynamics of simple circuits. Then we have seen that toobserve the field properties (energy, momentum, angular momentum and its polarization) it isnecessary to consider bilinear field quantities as the observables. These characterize theelectromagnetic field as a dynamical quantity very much like what one encounter in classicalmechanics.Startingpointfortopicsdiscussedinthischaptercanbefoundin[8]and[7].

CHAPTER7DISTRIBUTIONSOFCHARGE,MACROSCOPICMATTER,ANDBOUNDARYCONDITIONS

Inthischapter,wewillconsidertheimplicationsofhowrealmatterinteractswiththephysicalelectromagneticfield.Therearemanysubtleproblemsassociatedintryingtounderstandthebasicmechanisms.Wewanttobeabletothinkaboutthiswithoutresortingtoanysortof“fundamental”microscopic theory. The principle difficulty one encounters is in applying electromagnetism toaccountforstructureofmatter.ThequestionisthatifCoulomblawholdsatthemicroscopiclevelthenwhatistheunderlyingstabilityofmatter-whydoesn’teverythingjustcollapseinonitselforblowapart.Asthereaderisnodoubtaware,itisinthearenaofquantummechanicsthatthishastobesortedout.However,thisbookisontheclassicaltheoryofelectromagneticfields.Onemustbecarefulnottotreadinotherpastures,lestwenottreatthetopicconsistently.

Thinkingaboutthisthensomewhatfurther,onecanaskhowthefieldschangewhentheyenterintoorareconfinedinnormalsolidmatter.Insuchcircumstancesoneneedstoknowwhathappensat theseparating intermediate surfacebetween twosuchmedia. Ifwewere to regardCoulomb’slaw as something very basic that would continue to be true atmicroscopic distances (so that itaccounts for the structure ofmatter), then aswe look at smaller and smaller length scales, howdoes the distribution of charge appear?What happens to this charge distribution as we changelengthscale?Necessarily,thiswillhavetobeaninpartphenomenologicalapproachbecausewedonotknowthemicroscopictheory.Sothebasicphysicalobservablesandmeasurementsshouldnotlosesightofthephenomenologicalassumptions.

Asinpreviouschapters,thesetupandformalismissuchthattheelectricandmagneticcasesaretreatedinananalogousmirrorlikefashion.Theastutereaderhasprobablybecomeawarethatthemagneticcasestendtobealittlebitmorecomplicatedthantheequivalentelectricone.Thebasicreasonforthis,inageneralsense,isthatforthemagneticfieldwearealwaysfindingarotationalangularmomentumtypevariablethatisintertwinedwiththemagneticfield.Intheelectriccase,itistypicallyalineardistanceorvelocitythathastheequivalentconnection.

7.1 MULTIPOLES

Thus far, we have developed the theory and measurement of electromagnetic fields bythemselves. Of course the measurement process requires that these fields interact with ourmeasuringdevices,butthattheythemselvesarenotreallydisturbed.However,theyareofcoursemade from realmatter so the question now turns to how fields interactwith realmatter. Let usrevisittheparallelplatecapacitoràlaFaraday[6]andconsiderthefollowingexperiment.

Experiment 12 (Faraday, 1836) Take the parallel plate capacitor, apply a voltage across theplatesandmeasureitscapacitancewithanelectrometer.Whathappenswhenweinsertaninsulatorinbetweentheplates?

Observation12The capacitance is found to increase as the themeasured charge on the platesincreases.

TheexperimentalapparatusFaradayusedin1836toquantitativelystudytheeffectofdielectricsinbetweencapacitorplates.Inthisversion,thesphericalcapacitorplatesgeometry removed any doubts about the particular geometric configurations (e.g.edgeeffects).

Fromthisexperimentweestablishthefactthatthechargedensityhaschangedontheplates.Sohow should we interpret this? If the charge density on the plates has changed then the chargedensity in the insulatormustalsohavechanged toproduce thiseffect.But itcan’tbe thesimpleaddition or subtraction of charge because this would lead to a violation of the conservation ofcharge. It would give an observable net Coulomb like force outside the capacitor. There areobviously charges inside the insulator, but theymust be in the form ofbound states - groups ofchargethat formastableunit.Whenthe insulator isplacedbetweentheplates, theelectric fieldduetothepotentialdifferencemustcausethechargesintheinsulatortoberedistributed.Intermsoftheboundstatetheyaredisplacedwithrespecttooneanother,butrelaxbacktotheiroriginalconfigurationwhentheyaretakenoutfromthecapacitorplates.Thisthenisthepicturewehaveofthemechanismbywhichtheinteractiontakesplace.

Letustryandcapturethisintermsofthefieldequations[8].ConsiderEquation(4.53)(MaxwellI)whereachargedensity isasourceforanelectricfield.Wecanactuallythinkaboutseparatingthechargedensityintodifferentcontributions,dependingonalengthscaleandwhetheritisfreechargeorispartofaboundstate.Ifwedothisthen

(7.1)

By free chargewemean charge that can run around in thematerial subject only to resistiveforces(collisions,butwhatreallyarecollisions?)thatwilleventuallybringittoastop.Foraboundstatecharge,aphenomenologicalmodelcouldbeapairofequalandoppositechargesthathaveanharmonic potential binding them together (a spring). This is perhaps the simplest example of aboundstatechargedistribution.Notetherethatthespringconstantwouldhavetobeintroducedbyhand-itisaphenomenologicalparameterthatisn’tcalculatedbutwouldhavetobemeasured.Ingeneral,however, thedistributionofcharge inaboundstatewillbemorecomplicatedandsowenowaddressthispoint.

Considerthefollowingmodelofacollectionofcharges.TostartoffwithletusassumetheyarepointlikesothatthechargedensitycanbeexpressedintermsofanumberofDiracdeltafunctionsandasetofcenterofmasscoordinates.Asbeforewemakeadistinctionbetweenmobilechargeandboundcharge.Letusassumethatthematerialweareconsideringhasbothmobileandboundchargespresent,whereforthetimebeingwefocusontheboundcharges.Next, letusdividethechargesupintogroups(orsystemsofchargethatweshallcolorfullyrefertoas“molecules”)thatweindexbyN.Aroundeachofthesewecansurroundthemwithasmallvolumeandregarditasanobjectinitsownright(thisisexactlyhowthemolecularstructureofmatterisdeveloped).Sothisgroupinglookslike

(7.2)

wherexN is thecenterofmasscoordinate for themoleculeN.Next,weuse theseemingly trivialequation

(7.3)

(7.4)

Sincewearethinkingabouta largenumberofsystems, that isanensembleofdistinctchargedistributions,thissystemstartstotakeonastatisticalcharacter.Tothatenditissensibletoreplacethe above equations with spatially averaged versions. This means replacing the delta functiondistributionintheabovewithastatisticaltestfunctionthatlocalizesthefunctioninasimilarwaybut as applied to a large number of charge groups. We define a spatial averaging, to berealizedbyatestfunctionπ(x)(asmoothandfiniteversionofaDiracdeltafunction)suchthat

(7.5)

It has the effect of localizing any function about some preferred point. As it is a statisticalfunction, the length scale which the averaging is done over should be much larger than themolecular dimension. Next we consider the charge density about the individual center of masscoordinatesusingtheprevioustestfunction.ThenEquation(7.4)becomes

(7.6)

(7.7)

(7.8)

Thespatiallyaveragedchargedensityhasbeendecomposed intoasumofcontributionsabouteachgroupscenterofmassindicatedbytheindexi.Sincethespatialaveragingoccursonalengthscale much bigger than x(i,N) (that is π(x − xN) is slowly varying on the length scale of theinterchange separations), it is a simplematter toperformaTaylor seriesexpansion inpowersofx(i,N)

(7.9)

fromwhichwedefinethemultipolemomentsoftheN-thmoleculeas

(7.10)(7.11)

(7.12)

Thefactorof three in thequadrupolemoment isaconvenientdefinition fornormalizationonlyandnothingfundamental.Giventhedefinitionofthetestfunction,togetherwiththedeltafunctionnatureofthechargedistributions,thiscanberecastas

(7.13)

ItisnowstraightforwardtosumoveralltheNmoleculestoobtainthemacroscopicpolarization.Equation(7.1)hastohavethechargedensitysourcesreplacedwiththespatiallyaveragedversionsandcanbewrittenas

(7.14)

(7.15)

Nowwemakethefollowingdefinitions

(7.16)

(7.17)

(7.18)

(7.19)

Withthesedefinitions,theresultingfieldequationis

(7.20)

(7.21)

where it is customary to denote the combination of the bare electric field and the multipoleadditionsasthedisplacementfield,definedas

(7.22)

In relation to Equations (7.21) and (7.22) it is clear what this means physically if thedisplacement field is linear in theelectric field: themeasuredcharge isgreater thanwithout thedielectric,sotheelectricfieldmustbemodifiedinsidethedielectric.Ifthereareonlypointsourceswhichcanattractorrepelthentheymustbeintheformofboundstates,thatisstablecollectionsofspatiallydistributedcharge,andtheseareexactlythemultipolemomentswehavealreadyworkedout. A further point tomention is thatwe have been deliberately vague about the length scalesinvolved. If we assume the atomic theory of matter andmolecular theory we can indeed put innumbers here. Instead, what we have shown here is how one can go about establishing such atheory.Byusingamacroscopictheoryplussomephenomenology,onecanstartbuildingupsuchamodel.

Inanexactlysimilarfashion,theanalogoussituationforthemagneticfieldcanbecarriedout.Equation (4.56) (Maxwell IV) has to be modified for both electric and magnetic fields. TheequivalentofEquation(7.1)forboundstatecurrentsis

(7.23)

SimilarlytoEquation(7.2),theboundstatecurrentdensityis

(7.24)

Thestartingpointtoturnthisintotheequivalentof(7.21)istoreconsiderthecurrentdensityinthelightofEquation(7.7)suchthat

(7.25)

TheexpressionforJNtakesthesameformasforchargedensity(i.e.,theargumentiscenteredaroundthecenterofmasscoordinateandthechargetocenterseparation),butinadditiontothisweshouldnotlosesightofthevelocitiesofthecollectionofchargesthatcollectivelyconstitutethebasiccurrent.Onehas

(7.26)

(7.27)

Asbefore,itisnecessarytoexpandthespatialaveragingfunctioninapowerseriesinx(iN).Tosimplifymattersweshallperformtheexpansiononlytolinearorderinx(iN).Doingthisonefinds

(7.28)

Thereasonforwritingtheaboveinthisformatisthateachlinehasaseparateinterpretationandcan therefore rewritten in simpler terms.The first line inEquation (7.28) is the average currentdensityofmoleculeN

(7.29)

The second line in Equation (7.28) gives zero since on average there is no current densityassociatedwiththerelativevelocities,sothat

(7.30)

LookingnowatEquation(7.9),thepartialtimederivativeofthedipoletermisjustthethirdlineabove

(7.31)

sothatthethirdlineinEquation(7.9)is

(7.32)

Thelastlinethatinvolvesbothrelativeseparationsandvelocitiesrequiresfurtherworkoncewehavesimplifiedthecurrentdensity.ReturningtoEquation(7.23)onecannowsubstituteinthethemoleculardensityasfollows

(7.33)

Theaboveimmediatelyshowsthatthereisaclearcontributiontothecurrentdensityinthefirstline.Inthesecondline,thepartialtimederivativeofthedipolemomentbecomesthepartialtimederivativeofthepolarizationdensitywhichwehavealreadyencountered.Thiscanbeputtogetherwith the electric field contribution to form the displacement field. The last term presents thegenuinelynewterm.Wewant to factor this inwith theBa field,but todo this requires theLevi-Civitatensor.ItisthereforenecessarytousetheidentityEquation(6.28)againsothat

(7.34)

However, the molecular separations and velocities are orthogonal so the last term abovevanishes.We now have two wedge products, one of which can be used in conjunction with themagneticfield.UsingthisinEquation(7.33)onefinds

(7.35)

Themagnetizationandthemacroscopiccurrentarethusdefinedas

(7.36)

(7.37)

(7.38)

Therefore,Equation(7.35)canberecastas

(7.39)

(7.40)

wherewehaveonlyincludedthemacroscopicmagnetizationinthemagneticfieldstrengthHa.

Fromthedefinitionsofthetwomacroscopicfieldswecannowdefinethetwoimportantmaterialproperties.Thesearethepermittivitytensor andthepermeabilitytensor andaregivenby

(7.41)

(7.42)

The above relations constitute the manner in which the physical fields interact withphenomenologicalmodelsofmatter.Assuch,theyserveasbackgroundfieldswhichdeterminehowtheelectromagneticfieldsthemselvesbehaveinmatter.

7.1.1 InterludeonElectricandMagneticDipolesThemultipoleexpressionsjustencounteredarequitegeneral.Itisoftenthecase,however,that

thedipoletermsrepresentthemostinterestingpartofthetotalexpressions.Thisissobecausetheyrepresent the largest contribution to the series. The next set of observables one would want toconsiderare thecouplingsof theelectricandmagneticdipoles toexternalelectricandmagneticfields.Thenaturalsetofobservablestoconsiderareobviouslytheobservedforces,butinaddition,sinceadistancescaleisinvolved(thespatialseparationofthechargeortheradiusofthecurrentloop),atorqueisalsonowpossible.

Toderiveexpressionsfortheforcesandtorquesforasimple(molecular)electricandmagneticdipole one must appreciate that this is two particle problem. Considering Equations (7.11) and(7.38)foratwoparticlesystemrevealsthefollowing.Theelectricdipoleconsistsoftwoequalandoppositecharges±qwithsomeseparationbetweenthem(weconsiderherethespecialcaseoftwoequalmassestomaketheboundstatesnicesymmetricobjectsandsimplifythecalculations),whilethemagneticdipoleistwoequalandoppositechargescirculatingaroundacommoncenterofmass.Wecan thereforewritedown twoequationsofmotion (TheLorentz force law) thatgovernseachchargeinthepresenceofspatiallyvaryingelectricandmagneticfields.Firstly,considertheelectricdipole

(7.43)

(7.44)

WenowexpandeachpositionvectorofthechargesaboutthecenterofmasscoordinateX inapowerseriesintheirseparationvectorx,whicharedefinedby

(7.45)(7.46)

Replacingtheargumentsoftheexternalfieldswiththecoordinates(X,x)givesapowerseriesinx

(7.47)

(7.48)

(7.49)

(7.50)

Therefore,addingtogetherthetwoforcesonthedipoleresultsin

(7.51)

(7.52)

(7.53)

One can see that in a uniform electric field, the dipolar force vanishes to a first orderapproximation.

Thetorqueissimplyevaluatednowaboutthecentermasscoordinatetobe

(7.54)

(7.55)

(7.56)

Soforaconstantelectricfield,adipolewillbeactedupontoproduceatorqueuntilthevectordaisalignedwiththefield.

Turning our attention now to themagnetic dipole we start again with the Lorentz Force lawequationswhereaspatiallyvaryingmagneticfieldispresent

(7.57)

(7.58)

Asbeforeweexpandthemagneticfieldaboutthecenterofmasscoordinatessothat

(7.59)

(7.60)

(7.61)

(7.62)

For the symmetrical pair of circulating chargeswe are considering,v(1)(X)=v(2)(X)=v. Thetotalforceonthemagneticdipoleisjustthesumoftheabovetwo

(7.63)(7.64)

(7.65)

WeknowthattoformthemagneticdipolemomentwerequireatleastanextraLevi-Civitatensorso we use Equation (6.28) to introduce the necessary wedge product. This leads to, after somecalculation

(7.66)

(7.67)

Thelasttermiszeroforthespecialcaseofthecircularsymmetricdipole.ThiscanbeseenbytakingX = 0 andworking in cylindrical polar coordinates. In this case the velocity only has aθcomponent,whileB(0)andxare independentofθ.Theforceonthemagneticdipolethenfurthersimplifiesto

(7.68)(7.69)

(7.70)

As one can see, themagnetic case is considerablymore awkward to deal with. However, the

torqueonamagneticdipole isniceandsimple.Evaluatingthis in thecenterofmasscoordinatesandusingEquation(7.54)onefinds

(7.71)(7.72)

Ifweweretoworkatthenextorderofapproximationi.e.,quadrupoles,thetypeandnumberofmanipulationswillbefargreater.Forotherderivationssee[8]and[5].

7.2 BOUNDARYCONDITIONS

As the field equations stand, we could in principle try to find solutions for the electric andmagneticfieldswhentheyareinamaterialregionasdescribedbythepermittivityandpermeabilitytensors.Sofar,wehavereallybeenthinkingaboutacontinuousmedium-aregionofspacewherethematerialpropertiesaredescribedbysmoothlyvaryingfunctions.Butwhathappensifwecomeacrossasharpboundary, forexampleaglass-air interface?Weknowthat thematerialpropertieswillchangediscontinuouslyacrosssuchaspace.Whatistherightwayofhandlingthis?

ConsideringthefourMaxwell’sequations,weareaskingforsolutionsofthetwobarefieldsEa

andBaandthetwodressedfieldsDaandHa, inthetwodifferentregions.Weshouldcertainlybeabletofindseparatesolutionsforthefieldsinthetwoseparateregions.Sincethefieldequationsinvolvederivativesoffields,wecanseethatwewillpotentiallyrunintotroubleusingthembecauseofthediscontinuities.However,ifwerecastthemintheirintegralformwearemorelikelytohavesuccessasanumberofthespatialderivativeswillvanish.

To that end, let us start with the scalar equations,Maxwell II and the modifiedMaxwell I,Equation(7.21),andwritethemintheirintegralform

(7.73)

(7.74)

Supposewe consider a small volume that is located on the boundary of two distinctmaterialregionsthatintersectsbothregions.Iftheboxis“thin”thenthefacesthatareparalleltothetheseparatingregionspacewillhavethebiggerareas.Wecanapproximatetheintegralabovebyjustasumovertheareasofthetopandbottomfaces.

The continuity conditions are obtainedby considering a cylinderwithareaA(withnormalvectorn)upperandlowersurfacesandthicknesss.Itisathincylinderwhichmeans ThevectorVhasanegligiblescalarproductwiththesideareanormal.

Ofcourse,westillhaveto formthescalarproductwiththesurfacenormalwhich leadsto theresult

(7.75)

wherena is the surface normal to the interface region. By the exact same construction we canderiveananalogousresultforthedisplacementfield

(7.76)whereσisthesurfacechargeresidingontheinterfaceregion.Becausewehavetakenthelimitthatthevolumeisthin,thechargedensityisthendistributedinavolumethatsitsontheinterface.Sothevolumechargedensitybecomeslocalizedtoasurfacechargedensity.

Ifwenowconsider thevector fieldequationswecanadopt thesameprocedureasbeforeandwritethemintheirintegralform

(7.77)

(7.78)

It isnecessarynowtocuttheinterfaceregioninadifferentway.Sincewehavealineintegralandanarea,wecanseethatweneedtoformasmallloopthatcutsbothregionsandthatformsasmallarea.Thesmallsidesarenormaltotheinterfacesurfaceandthelongsidesparalleltoit.

The continuity conditions are obtained by considering a rectangularloopwithlengthLandthicknesssthatintersectsbothregions.Itisasmall loopwhichmeans The vectorV has a negligible scalarproductwiththesideelementvectors.

As the small sides tend to zero the area also tends to zero so that the area integralsfeaturingthetimederivativeoftheBandD fieldsvanish.Theloop integralcanbesplit intofourpieces.The small sideelementswhichare in thedirectionof thenormal to the interface surfacevanish i.e., the scalar product projection with the normal. This means it is only the orthogonalprojectionofthenormalgivenbythewedgeproductthatisnonzero.Therefore

(7.79)

To obtain the final continuity equationwe need the analogous situation of the surface chargepreviouslyencountered.Onemustincludeasurfacecurrentdensitywsothat

(7.80)

With these equations one can now glue solutions in two distinct material regions at theircommonboundary.Gooddiscussionscanbefoundin[8]and[5].

7.3 ELECTROMAGNETICINTERACTIONSWITHMATTER

Afundamentaltopictoaddressisthenatureofhowandtheeffectelectromagneticfieldshaveonrealphysicalmatter.Indirectlythistopicisimplicitinallofthediscussionssofar,asthecontrolofchargeandcurrenthasbeenperformedbyususingphysicalmatter.Wenowwishtogofurtherandask in what way thematter reacts in the presence of physical fields, be they oscillating, static,large,or small.Wewill assume thatamixtureofmacroscopicandmicroscopicmodellingofbulkrealmatterwillbe sufficient for thispurpose. In reality,wewould reallyneed to thinkabout thequantumnature ofmatter in order toget a consistent calculation scheme inwhich to talk aboutatoms and their coupling to electromagnetic fields. Turning this around, however, we can try topushthesemodelsasfarastheycangoasawaytobetterunderstandthenecessityofanyquantumscheme.Youhavetostartsomewhere!

Ourfirstmacroscopicmodelforbulkmatterthenwillbetoassumethatitconsistsofsometypeoflatticeofboundstatecharges,thatoverallhasnonetelectricalcharge.Supposeweconsideronesuchpair(i.e.welookatthelocalchargedistributionatsomepoint)andaskwhatistheeffectofapplyinganelectricfieldtoit.Thechargesmustbedisplacedbysomesmalldistance,whereweareassumingthattheboundingpotentialcanadmitanewequilibriumposition.Indeed,thinkingaboutthe simple molecular models encountered earlier in the chapter, we can imagine that they areflexibleinthepresenceofappliedfields.Alternatively,thinkofthesampleasbeingoverallchargeneutralbutthatweknowithaspositiveandnegativechargedegreesoffreedominsideit.Supposetheyare sittingon topofoneanother.Byapplyinganexternal field to thesamplewe force suchpairstoseparate.

TheelectricpolarizationδPa(t,x)foracollectionofaboundstateofcharges(the±qpairoftheelectricdipole)isthendefinedtobe

(7.81)

wheren(t,x)isnumberdensitydistributionforthecollectionofboundstates(forthetimebeingweset andreinstateitattheend)andδxa(t,x)istheincreaseinseparationfromequilibriumofthechargepair.Thebracketsindicatethatsincetheboundstateisamicroscopicstructureandweareapplyingthefieldonamacroscopicsample,weshouldthinkofitinsomestatisticalsense(ifwewereusingquantummechanicshere, thiswouldsimplybetheexpectationvaluebetweengroundstates). In complete generality this happens due to the application of some electric field at anearlier time and some other point in space δEb(t′,x′). There is then a polynomial relationshipbetweenthetwo

(7.82)

Thefirsttermdefinesthelineartheory,whilethesecondandhigherordersdefinethenon-lineartheory.Thequantities andξabcarecalled ingeneralresponse functions.Theyareclearlynon-local in timeand in spaceand tell ushowstrongly a system reacts togiven input (or test) field.Therefore,theyalsorepresentphysicalobservables.See[1]foranicediscussion.

Wewillnowmaketwosimplifyingassumptionswhichwillallowustoderiveusefulrelationsandthatalsohavephysicalrelevance.Thefirstisthatweshallworkatthelinearlevel.Notsurprisingly,thismakesmanycalculationstractable.Secondly,wewillconsiderresponse functionsthatdonotdependonspacesothatEquation(7.82)canbewrittenas

(7.83)

Clearly in this form the right hand side is a convolution integral. It therefore affords simplemultiplicativeformwhenFouriertransformed,bytheuseoftheconvolutiontheorem.Infrequencyspaceitbecomes

(7.84)andthus

(7.85)

Ifwecancalculatethethepolarizationasafunctionofelectricfield,wecanextracttheresponsefunction.Thisparticularresponsefunctionisknownastheelectricsusceptibility.WhatmaynotbeobviousisthatitisbasicallytheGreen’sfunctionofthemodel.IfweknowtheequationofmotionofthechargethenwecaninvertthisandobtaintheGreenfunction.ThisisexactlythesameaswhatweencounteredfortheRLCcircuitandsocarriesoverstraightforwardly.Wewilldothisnowforasimplemodelofadampedharmonicoscillator.

Recall the Lorentz force law, Equation (4.15). For an harmonic oscillator that is damped, theequationofmotionofthechargeqwithdisplacementvectorδxa(t)takestheform

(7.86)

Hereagainonthelefthandsideweareconsideringtheseparationofchargetobeastatisticalquantity.Sowejusthaveinverttheoperator toobtainthesusceptibility.Ashortcutispossibleherebyjustsubstitutinginharmonicoscillatortypesolutions,i.e.

(7.87)

(7.88)

SubstitutingtheseintoEquation(7.86)andassumingaconstantnumberdensityn(t,x)=Ngives

(7.89)

(7.90)

Wethereforefindtheresponsefunctiontobe

(7.91)

Ourmodel is thus describing the following behavior. Amacroscopic sample has an oscillatingelectric field applied to it. The field can see the local charge difference of the bound states thatconstitutethemodelanddrivesthemtooscillate.Thiswillthenultimatelydependonthenatureoftheboundstateforcesastohowstrongorweaktheirsubsequentoscillationswillbe.Theresponsefunction represents another physical observable, this time about the macroscopic sample beingprobed. The local charge oscillations cause the input field to be deviated (or scattered) in somefashionandwewillobservethisasanalteredoutputfield.Thisisbasisforexplainingthelawsofrefractionintermsofarefractiveindexbylinkingittotheelectricsusceptibility.

Abriefcommentisalsoinorderabouttheequivalentmagneticsystem,whereafluctuatingtestmagneticfieldgivesrisetoafluctuatingcurrentloop.Intheelectriccase,theappliedtestelectricfield caused two equal and opposite charges to separate some distance.Upon application of themagneticfield,thechargesthatareseparatedstartrotatingabouttheircenterofmass.Justastheelectricfieldwillbereplacedbythemagneticfield,theperturbedseparationwillbereplacedbytheperturbed angular momentum. In this fashion, we can calculate a response function for themagneticcase.

7.4 ELECTRICALCONDUCTIVITY

It is also possible to obtain a response function for how well a material conducts electricity.Ohm’slawthatonetypicallylearnsatschoolisthefirsttermintheseriesofEquation(4.4),thatis

(7.92)

This is a coarse grained phenomenological model. We can also, however, turn this into acontinuum version. The current needs to be replaced by the current density and thepotentialdifferenceturnsintothetheelectricfield ThisshouldnotbetooalarmingbecausetheoriginalformofOhm’slawhadlostallthedetailsaboutthegeometryandmicroscopicbehavior.Bothofthethesearevectorquantitiessoingeneraltheywillberelatedbyatwoindextensor

(7.93)

where is defined as the conductivity tensor. This is yet another constitutive relation that isnecessarytospecifywhensolvingthefullfieldequationsincertaincircumstances.Itisaresponsefunctionandsoexactlyparallels thestructureencounteredearlier for thepolarization.Since inaconductor thecharge isnotbound,a simplealternative toEquation (7.86) is todrop thebindingpotentialterm,whence

(7.94)

Ifweperformthesamesubstitutionfortheelectricfieldandthedisplacementvector,theaboveinfrequencyspacebecomes

(7.95)

(7.96)

whichisknownastheDrudemodel,andforfrequenciesthataresolargethatdampingisnegligible

(7.97)

(7.98)

whichisknownasthePlasmamodel.

Paul Karl Ludwig Drude (1863–1906), made advances in connecting opticalphenomenawithelectromagnetismandthethermalpropertiesofsolids.

Wecanrelatetheconductivitytothesusceptibilityinthefollowingway(tosimplifythenotationwe will take them to be scalars and promote them to tensors at the end). Firstly, write downEquation(7.39),

(7.99)

Nowsubstituteinthethefrequencyspacesusceptibilityandconductivity

(7.100)

If we consider setting the system then has some conductivity. Now set σ = 0 andsubstitutetheDrudemodelinforthesusceptibility.Theimaginarypartagainhasappearedtogiveusaconductivity.Therefore,wecancalculatetheconductivityfromsusceptibilityby

(7.101)

Whatonecansee then is that for there isnorealdifferencebetweenadielectricandaconductor, because ultimately they are both describing the movement of charge subjected tobindingpotentials.

7.5 FIELDEQUATIONSWITHGENERALMATTER

Given that we have simple models for macroscopic matter in terms their underlying microstructure, it is instructivetomakethecorrespondingsubstitutionsintothefieldequations.Afirstobviouspoint isthatthefieldequationsingeneralarenolongerlocal intime,sincetheresponsefunction isnon-local intime(seeEquation(7.82)).Theassumptionwith this typeof interaction isthat the field arrives where the dipole (or some microstructure) is located, it then disturbs thesystem,andfinallythesystemtriestoreturntoitsequilibriumstate,therebygivinguptheenergyjust acquired. This interactionmust take place over a a certain amount of time that allows thisdisturbanceandthenrelaxationtotakeplace.Whatthismeansforthefieldequationsisthatit issimplertoworkwiththefrequencycomponentsofthefieldsandthematter;ifitisnecessary,onecanFouriertransformtothetimedifferencecoordinatesattheend.

Letusconsiderthenamacroscopicmediumwithapermittivity apermeability andaconductivity Thentheconstitutiverelationsaregivenby

(7.102)(7.103)

(7.104)

Thefieldequationsinfrequencyspaceare

(7.105)(7.106)(7.107)

(7.108)

Tosimplify,wemultiplyEquation(7.107)bytheinverseofthepermeabilitytensorandtakethecurl,sothat

(7.109)

WenowuseEquation(7.108)tosimplifythesecondterm

(7.110)

Finally,usingtheconstitutiverelationsEquations(7.102)and(7.104)oneobtains

(7.111)

(7.112)

ThisisthemostgeneralvectorHelmholtztypeequationthatoneislikelytoencounter.ThereisasimilarequationforthemagneticfieldHa.

(Exercise:WhatisthecorrespondingvectorHelmholtzequationthatHamustsatisfy?)Itisworthcommentingonsomeaspectsoftheformoftheseequations.Thefirstpointisdueto

the tensorial nature of material properties, in general wave propagation will be different in alldirections.Nextitwillalsobeinhomogeneousasthetensorsvaryinposition.Theywilldependonthe frequency of the fields and due to the real and imaginary parts wewill find both wave like

solutionscombinedwithgrowingordecayingmodes.Thesewillbefixedbytheparticularboundaryconditionsimposed.

7.6 SUMMARY

Thischapterhasfocusedonhowrealmatterinteractswiththeelectromagneticfield.Maxwell’sfield equations have beengeneralized tomaterialmedia and the basic conditions have been laiddownthatarenecessary for the fields toobey.Response functionshavebeenderivedthatsimplematerialmodelsgive,aswellastheideaofsmallstructures(multipoles)thatcausethebarefieldstobecomedressed.Fromhereonecanstart toconsider themanyapplicationselectromagnetismhastoeverythingaroundus.Discussionsofrealmattercanbefoundin[8]and[7].

CHAPTER8CONCLUSIONSANDOMISSIONS

Alas,wehavereachedjourney’send.Thisbookhasbeenintendedforthereadertogainsomeinsightintothefoundationsofelectromagnetismandfieldtheory.Inthissenseitisacomplimentarytextbookandshouldhopefullyservetothrowupquestionsforthereadertofurtherpursue.

Inanysuchwork(andonafiniteandshorttimescale)itisnotpossibletobecomplete.Ihavenotcoveredmanytopicswhicharealsonecessarytohaveafullunderstanding(whateverthatmightmean).Listednowareahostoftopicsthatmayturnouttobeasecondvolumetothisbook,orcanbepursuedbythereaderattheirleisure.MostofthetopicsIhavenotcoveredareeitherfirmlyinthe applications arena, or else sufficiently advanced that I thought it would be an unnecessarydetourfromtheobjectiveofthebook.

Theradiationofmovingcharges,suchasfoundinantennaeorinacceleratingbeamsofchargeshasbeenomitted.Thisisanimportanttopicandthereadercandonobetterthanconsulting[7]tomakeastarthere.InparticulartheLiénard-Wiechertpotentialswouldbeanappropriateplacetostarttogetherwithsimpledipoleradiation.

Specialrelativityandtheformulationofelectromagnetismasarelativisticfieldtheoryhasbeencompletely omitted. While this is a topic necessary to understand advanced field theory (forexample, toget togripswithmodernquantumfield theories), I felt itwouldbetoodistractingtoenter in the current volume. In particular, one must be quite comfortable with Lagrangian andHamiltonianmechanicstoappreciateitsfullsignificance,andthesetopicsinthemselvescanquitehappilyfillatextbook.Itisalsothecaseherethatoneistreadingamoremathematicalpath.Forexamplethenatureofhowfieldsinteractwithatomicmattercanbequitemathematicallytechnical[16] and one might also then think about group theory issues and the spinor formulation ofelectrodynamics[3].Theseareindeedinterestingadvancedtopics,butsincetheyhavealesswellestablishedmeasurementtheorybasis,Ihavenotpushedatthesedoors.

Afinalcoupleoftopicsworthmentioningandworthyoffurtherdiscussionarethescatteringofelectromagneticwaves and diffraction,which can be viewed from a unified rather than separateapproach. These are certainly basic topics and correlate strongly with the philosophy here of ameasurementcentricapproach,buttodothemfulljusticewouldrequireenoughmaterialtofattenoutasecondbook.Theinterestedreadercanconsult[8]or[13]tomakeastartonthistopic.

Thisbookhastriedtodiscusselectromagnetismfromthepointofviewofnottakinganythingonfaith.Coupledtogetherwiththishasbeentheconstructivebottomupapproachtodevelopingthefieldtheoryconceptsthatgivethegoverningdynamicsofthesystem.Asaclosingsetofthoughts,onemightjustsaythatitisnotaclosedbook.Whileelectrodynamicsisawelldevelopedandtestedtheory, there do occur conceptual problems as well that do not have universally agreed uponsolutions. Radiation reaction, self force, and pre-acceleration are some of the difficulties oneencounterswhenchargesinteractwiththefieldstheythemselvesproduce.Inasimilarspirit,itisnot always clear which stress tensor to use when considering interactions within materialsthemselves(forexamplethestresstensorwehavedefinedortheMinkowskiversion).Theseissuesindicate thatevenclassically thereareconsistency issues thatmayneed further thoughtperhapswith the help of some quantum input. This is a good task for the interested reader to furtherresearchandagoodplacetoend.

EXERCISESANDSOLUTIONS

EXERCISES

Exercise1Takeanaveragepartyballoonthatisfullyinflatedandchargeitupbyrubbingitonyourhair.

Whatisatypicalvoltagethattheballoonacquiresduetothiselectrostaticchargingprocess?

Exercise2Considerthefollowingelectrometerconfiguration.Adischasanareaof0.02m2thatisseparated

froma fixedplatebydistance0.2mm.Thepotentialdifferencebetween the twoplates isheldat200V. What is the force between the two plates and how does this depend on the potentialdifference?howsensitiveisthistothevoltageacrosstheplates?Supposetheupperdiscisattachedtoaspring.Ifinequilibriumtheseparationofthetwoplatesisd(whenthepotentialdifferenceisV)or a (when the potential difference is zero), deduce a criterion for stable equilibrium for theseparationdconfiguration.

Exercise3Supposewehavefourverylongmetallicplatesthatformawaveguidetypestructure,thatis,two

longstripsofacertainwidthandparallel tooneanother,andanother two longstripsofanotherwidth and also parallel two one another. These are then glued together to form a rectangularstructurethatisextendedinonedirection.Supposenowwechoosetoholdthedifferentplatesatdifferentelectrostaticpotentials.Twooppositefacesareheldatzeropotential,whiletheothertwoareheldataconstantvalueV.Whatisthepotentialinsidethewave-guide?Thewave-guidehasalengthLthatismuchlargerthantheseparationsofthefaces.

Exercise4Consideroneofthediskplatesthatformaparallelplatediskcapacitor.Supposeithasastatic

chargedensityonitthatvariesasaquadraticpoweroftheradialcoordinate.Whatisthepotentialandtheelectricfieldofthischargedistributiondirectlyabovethecenterofthedisk?Whatdoesitlooklikefarawayandclosetothedisk?

Exercise5Find an expression for themagnetic field of an elliptical current loop at a general pointwith

respecttoitscenterduetoasteadycurrent,inthelimitthatthepointisfarawayfromtheellipse.Do this by considering the vector potential first from which the magnetic field can be derived.Evaluatethecorrespondingexpressionwhentheellipseiscircularandnearlycircular.

Exercise6ConsiderthegaugetransformationthatrelatestheColoumbgaugetotheLorentzgauge.What

equation must the corresponding gauge function satisfy? What does this look like in reciprocalspace?

Exercise7Consider firstly theGreen function thatsolves thescalarHelmholtzequation. In termsof this,

whatistheGreentensorthatsolvesthevectorHelmholtzequation?Whattransverseconditiondoesitsatisfy?FindexpressionsfortheGreentensorinbothwavevectorspaceandpositionspace.Asthewavelengthoftheradiationisvaried,whichtermsintheGreentensorarethemostimportant?

Exercise8Aperfectlyabsorbingsheetofmetalhas200Wof light incidentonitnormallyfor10s.Whatis

thetotallinearmomentumtransferredtothesheet?IfthemagnitudeofthePoyntingvectoratthetop of Earth’s atmosphere is ≈ 1.4kWm−2, what is the average radiation pressure on the abovemetalsheet?Ifwewantedtousethisasasolarsail,whatsizeofsheetisnecessarytocapture1Nofforce?

Exercise9Consider the expression for the energy, momentum, and angular momentum of the

electromagneticfield.Supposeintothesewesubstituteplanewavesolutions.Whataretheparticlelikepropertiesfeaturingintheseexpressions?

Exercise10Supposemagneticmonopolesdidexist.WhatformwouldMaxwell’sequationstake?Discussthe

symmetrybetweentheelectricandmagneticcharges.WhatformwouldtheLorentzforcelawnowtake?Howwouldyoureconcileusing

SOLUTIONS

Solution1This is one of those orders of magnitude, guesstimate questions based partly on common

experience.Ifthisisnotcommonknowledge,youcaneasilydothissimpleexperimentyourself!Tomeasure thechargeon theballoonandsodetermine itsvoltage,weneedanotherballoon that issimilarlycharged.Wecanthenbalancethecorrespondingrepulsiveforcewithanothermechanicalforce.Itisasimplemattertobalancethisagainstgravity.Suspendeachballoonfromapointbytwopiecesofthreadthatarethesamelength.Becausetheballoonshaveaboutthesameamountofnetchargeofthesamekind,theywillrepel.Theangleαthethreadwillmakewiththeverticalwillbeintherange Onenow justresolves forcesat thecenterofeachballoontobalanceoffgravityandtherepulsiveforces.

Puttinginsomenumbers,atypicalballoonhasamassm≈5g,whileasuspensionangleofα=30° and separation of their centers of 50cmare also typical. The downward force is thus≈5×10−2Nandthereforethehorizontalforceis≈5sin(30)≈2.5×10−2N.WecanthusequatethistotherepulsiveCoulombforce

Assumingnowthatthechargehasbeendistributeduniformlyoverthesphere,allweneedisthecapacitanceofaspheretogetthevoltage-andweknowthis.TheradiusoftheballoonisR≈15cmso

Solution2For this problem we can assume the parallel plates are well described by the infinite plane

idealization since the separations ismuch smaller than the square root of thediscs areaA.Thepotentialdifferenceandthereforetheforceacrosstheplates,rememberingthelowerplateisfixed,isgivenby(fromChapter3)

Thereforeputtinginthenumbers,oneobtainsfortheforcebetweentheplatesthatF≈1.77N.togetthesensitivityatthisvoltage,wejustneedtodifferentiatewithrespecttothevoltage,resultingin

Sincewehavenowattachedaspringtotheupperdisc,thelinearrestoringforcenowneedstobeincludedintheforcesactingonthedisc.Supposetheplateshavesomeseparationx,then

WhenV=0,x=asothereforeatthesecondequilibriumx0=a.Attheotherequilibriumonehas

Forthispositiontobeastableequilibriumrequiresthat andso

Solution3ObviouslyweneedtoworkinCartesiancoordinates,soletthelongdirectionbez.Wedefinethe

facesatthepotentialVtobex=±aandthezeropotentialfacestobey=0,b.ThisisaboundaryproblemassociatedwithsolvingLaplace’sequationinCartesiancoordinates

In thecase thatweregardL asbeingvery long,wecandrop thez-dependence such that theabovereduces to the two-dimensionalLaplaceequation in thex,ycoordinates.Theseparationofvariables for this equation leads to both exponential and sinusoidal solutions. The boundaryconditionsrequirethatthepotentialvanishesintwoplaneswhichcanonlybefulfilledbythesineorcosineeigenfunction.Writingthesolutionas

onehas

Theboundaryconditionstellusthefollowing;X(x)mustbesymmetricinxsincethepotentialisthesameontheplanesx=±a.Thismeansa1=0.Next,sincethepotentialvanishesaty=0,wemusthaveb2=0;andfinallyaty=b,theonlywayforthesinefunctiontovanishthereisifkb=nπandthereforethat

where Sothesolutionlookssofarlike

thesumbeingperformedoverallpositivensincethisisalinearequationandwemustingeneral

superposeallpossiblesolutions.Thenegativeintegersdonotgiveanynewsolutionswhichiswhythenhavebeenrestrictedtoonlythepositivevalues.Todeterminethecoefficientsweusethelastpieceofinformationavailable,thatweknowthepotentialatx=a

This is just a standardFourier series iny; oneuses theorthogonality of theeigenfunctions toinverttheaboveequation

Sothecoefficientsareonlynonzeroforoddvaluesofn.

Solution4ThestartingpointforthisproblemisgoingtobePoisson’sequationinitsinvertedform.Wewill

calculate the potential first from which we can obtain the electric field by differentiation. Thepotentialisgivenintermsofthechargedensityby

Working in cylindrical coordinates, let the radius of the disk be a and the charge density beThepotentialthenbecomes

Ifwemakethesubstitutionr′2=Xtheintegralthentakesastandardintegralformthatonecanfindinabookinintegrals(orjustdoityourself)

Thisintegratesto

Theelectricfieldthenhasonlyaz-componentandis

Finallyweshouldconsiderthelimitswhen andalso Thepotential in these limitsreducesto

Fromtheseapproximationsweseeonaxisthattheelectricfieldvanishesjustabovethedisc,andfallsoffas farawayfromthedisc;thisistobeexpectedsinceatlargedistancesawayfromthedisconecannotresolveanystructureanditisonlythetotalchargethatismeaningful.

Solution5To evaluate the magnetic field we will need to use the Biot-Savart law since we have an

expressionforthecurrentI.Inthiscaseittakesthelineintegralform

wheredlisthelineelementoftheellipseand(x−x′)isthevectorbetweenthepointx′wherethelineelementislocatedandxwhereweareevaluatingthefield.Itwillclearlybenecessarytousethe equation of the ellipse to be able to perform the integration. Since we are dealing with amagnetostaticsituationitissimplertocalculatethevectorpotentialfirstandfromthiscalculatethefield(thiswayremovesaninitialwedgeproduct).Thevectorpotentialisgivenby

Sowecanseeherethatsincethevectorpotentialonlyhasarotationalcomponentthefieldwillbe out of the plane.Wework in spherical polar coordinateswith the origin at the center of theellipse,withtheellipseitselflyinginthe plane.Tostartwithwewillevaluatethefieldatthegeneraloffaxispoint so thatwecandifferentiate it.After thiswewill restrict it toonaxis.Thepointsontheellipsearegivenby sothat

Themoreinvolvedpartisthelengthvectorofthecurrentelementitself.Itisthevariable thatparametrizestheellipse,butnowr′isafunctionof (forthecircler′isofcourseaconstant).Thepositionvectorx′liesinthex,yplaneoftheequivalentCartesiancoordinates.Intermsofthisbasisthenthepositionvector isgivenby Thereforethelineelement inthisbasis isgivenby

FromthisonethenhasanexpressionforthevectorpotentialintheCartesiancoordinatebasis.Onemustbecarefulheretonotloosesightofthedifferentcoordinatesystemsandvariables.Thevectorpotentialcomponentsthenare

Fromtheseexpressionsthemagnetic fieldcanbecalculated in fullgenerality,eventhoughwecanevaluatethemexplicitlyincertainlimits.Wewillevaluatetheminthetwolimits, and

Firstlyfor

It is now time to include the equation of the ellipse to evaluate the integral. In standard twodimensionalCartesiancoordinatestheellipseisparameterizedbyforconstantsaandb

If the ellipse is nearly circular then where is small number that will serve as anexpansionparameter.Therefore

Withtheseexpressionswecannowevaluatethegaugepotentialcomponents

By inspectionwe can simplify theseby remembering the rules for integrating sine and cosinefunctions.Withthisinmindtheybecome

where termshavebeendropped.Theintegralscannowbeevaluatedstraightforwardlysincetheytaketheformofproductsofsineandcosinefunctionsoncethedoubleanglehasbeenexpanded.Withthisinmind,theyreduceto

To finally obtain themagnetic field in the z direction we need to partially differentiate theseexpressions with respect x and y. So we write the above expressions back into Cartesiancoordinates

Themagneticfieldinthezdirectionisjust so

Thedifferencethenfromthecircularcurrentloopwillbeseenatpointsoffthezaxis.

Solution6LetusassumethatweknowthepotentialsintheCoulombgauge,givenby andsothatwe

will perform a gauge transformation on these to the Lorentz gauge with potentials TheLorentzconditionis

SincethesepotentialsarerelatedtotheCoulombgaugepotentialsbyagaugefunction wehave

wheretheCoulombgaugecondition hasbeenused.IfwenowFouriertransformboththegaugefunctionandthescalarpotentialthenanexpressioninreciprocalspacecanalsobeobtained,sothat

Thegaugefunctionthendependsexplicitlyonthechargecontentandwillbeproportionaltothevelocityofthesecharges.

Solution7ThescalarGreenfunctionΔ(x,y)thatsatisfiesthescalarHelmholtzequationisgivenby

ThetensorGreenfunctionΔab(x,y)thatsatisfiesthevectorHelmholtzequationissimilarlygivenby

ThetrickhereistotryandwritetheGreentensorinsuchawaythatwhenitissubstitutedintothevectorHelmholtzequation, itreducestothescalarequation.Tothatendwetrythefollowingdecomposition

whereαandβareconstantstobedetermined.Thedoublewedgeofthederivativeoperatorcanbewrittenas Thisleadsdirectlytothethetwoconditions

ThereforetheGreentensoris

It is a simplematter to see that this satisfies a transverse constraint. Ifwe act on theGreentensorwiththederivativeoperatoronefinds

ThisresulthasitsrootinGauss’slawandisthebasicstatementthatforpropagatingfieldsthereisalocalplaneoffluctuationsthatarenormaltothedirectionofpropagation.

ItisasimplemattertoobtainanexpressionfortheGreenfunctioninwavevectorspace;simplyperformaFouriertransform

StraightforwarddifferentiationnowgivesGreentensorforthevectorHelmholtzequation

We can also calculate what the Green function looks like in position space by evaluating theabove integrals.Todo this requiresperformingan integral in thecomplexplane.Since this topiclies outside themain drive of this book,wewill assume the reader knows how to perform suchintegrals (i.e. method of residues etc). For the scalar Green function a is added to thedenominator that gives a prescription of how to enclose the poles in the complex plane.Havingdonethistheintegralcanthenbeevaluatedbyfindingtheresiduesoftheintegrand.Thisresultsinthetwoexpressions

The+signprescriptiongivestheretardedsolution(whereeffectfollowscause)andthe−signgivestheadvancedsolution(wherecausefollowseffect),amountingtoeitherincomingoroutgoingwaves.FromthiswecansimplydeducethepositionspaceversionoftheGreentensor,whichisfortheretardedsolution

TheexpressionfortheGreentensorhasthereforeorganizeditselfintoincreasingpowersinthedenominatorofthedimensionlessvariable The iscalledthenearfieldterm,sincethisdominateswhenforradiationofwavelengthλonehas The termiscalledthe intermediate field when and therefore dominates the expression. Finally theremaining term is called the far field piece which is the most important whenInterestingly, the intermediate field is exactly out of phase with the far field and near fieldGreentensorcontributions.

Solution8Assumingthatthe incidentradiationispropagatinginthezdirectionandthatthemetalsheet

spansthex,yplane,theconservationofmomentumequationgivesthattheradiationpressureρinthezdirectionisjustthestresstensorcomponentTzz.Usingtheformofthestresstensorintermsofthefields,this is just Inanamountof timeΔt,theenergyΔE incidentonthesheetistherefore whichisnumericallyjustthepowertimesthetime,ΔE=2000J.On theother side,multiplying the radiationpressurebyΔt and integratingover thesheetgivesusthetotalchangeinlinearmomentumΔPwhichistherefore

For a planewavewe take themagnitudes of the electric andmagnetic field to be relatedby (this can be verified by just making a plane wave substitution into Maxwell’s field

equations)whichallowsustotaketheradiationpressuretobe Sonumericallywefindthatonthemetalsheetρ≈4.7×10−6Nm−2.Toproduce1Nofforceonasolarsailthenwouldrequireanareaof≈2.1×106m2.Ifthiswasasquaresailthenthelengthofasidewouldbeapproximately1kmlong.

Solution9Byconsiderplanewavesolutionsof the fieldequationsandsummingovertheircorresponding

eigenvalues,weareattemptingtogenerateasolutionthatistypicallyafluctuation.Tobedefinite,wewill look at field fluctuations in amatter free region of space and start from the scalar andvectorpotentials.LetusfixthegaugetobetheColoumbgaugesothat and Themodeexpansioninplanewavesofthegaugefieldthentakestheform

Thisiseasilyverifiedasasolutiontothewaveequationwiththeconstraintthat Thegaugeconditionrequiresthatk·a=0.Tothatendwecanintroducetwopolarizationvectorsthatcharacterisethemodesofvibration.Writethemase(λ)(k),whereλ=1,2arethetwopolarizationstate labels.Usingthese thevectorFouriermodesab(k)are thenreplacedwithascalarquantitythat has the polarization mode attached, that is a(λ)(k). With these two quantities the gaugepotentialnowlookslike

Obviously,thepolarizationvectorswillberequiredtosatisfycertainconstraintsthatdescribethebasicchoiceoftheColoumbgauge.Theyare

the second condition just ensuring that the polarization vectors form a local set of orthonormalbasisvectorsthat,fromthefirstcondition,aretransversetothewavevector.Thisisthenthebasicfieldanditsdecomposition.Letusnowturntoevaluatingthemomentumofthefieldgivenby

ThefieldsEandBcanbewrittenintermsofthemodedecompositionas

Ifwenowsubstitute these into theexpression for themomentumandperform the integrationoverthespatialcoordinate,wewillobtaindeltafunctionsinthewavevectors.Asanexample,letusevaluatethefirsttermintheelectricfieldwiththesecondterminthemagneticfield,

Theargumentofmomentumintheabove(1,2)denotesthecrosstermspecification.Performingthe integral over space gives the delta function Then the integral overk′ can be donesimply,alsogiving Thetwowedgeproductthensimplifiesto

Theabovethensimplifiesto

Weobtainasimilarresultfortheothercrossterminvolvingana(λ)(k)anditscomplexconjugate,while theother two terms integrate to zero.This canbe seenas follows for the first term in theelectricandmagneticfieldcrossproduct

sincetheintegrandisanoddfunctionandthereforeintegratestozero.Thefinalresultwereareleftwiththenis

TheenergyofthefieldEisgivenby

Withsimilarmanipulationstobeforethisreducesto

The angular momentum of the field is a little bit more tricky because we have the explicitappearanceofthepositionvectorintheintegral,thatis

Again we substitute in the field decompositions as before, but in addition, to generate the

positionvectorweneedtopartiallydifferentiatetheexponentialbythewavevector,i.e.

Usingthisinthesimplificationresultsinthefieldangularmomentum

Lookingatthethreeresultsweseethatthecommonfactor actsasanoverallnormalizationfor the modes and the density of these modes are given by the bilinear a(λ)(k)a(λ)(k)*. Themacroscopicphysicalobservables(P,E,L)maptothemodeparameters whichuptoaconstantfactorarethemodevaluesofthemomentum,energyandangularmomentum.Infactin thequantum theory this constant factor isPlanck’s constantand thesemodes thendescribeaparticlelikeexcitation.

Solution10Ifmagneticmonopolesdidexistwewouldhavetoascribetothemanewmagneticchargegor

magneticchargedensityρM(x).ThiswouldrequireMaxwellIItotakethenewform

Inadditiontothis,assumingthatitcanmoveinmuchthesamewayasanelectriccharge,onenow has amagnetic current. It is therefore necessary tomodify Faraday’s law (Maxwell III) toincludethissothat

For the other twoMaxwell equations it is sensible to label the electric charge and current asρE(x)andJE. Ifwenowformacomplexchargegivenbyρ+ iρMandacomplexcurrentJE+ iJMthenthereexistsanewsymmetryofthefieldequations.StartingwiththechargethenewversionofMaxwellIreads

Itissimpleenoughtoseenowthatthetransformations(infactaU(1)Abeliangrouprotation)

leavetheaboveGausslawinvariantandisthereforeasymmetryofthisequation.Wealsoneedtochecktheremainingcomplexfieldequationwhichis

With as before, the generalized field equations are invariantunder this symmetry transformation.This isperhaps thesimplestexampleofwhat isknownasaduality relation, and physically it is describing how we can interchange electric and magneticchargeswithoutanyeffectonthephysics-whatyoumeanbyelectricormagneticchargedependsonhowyoulookatit.

For electric charges the Lorentz force law stays the same, but we now have in addition amagneticLorentzforcelawforthemagneticchargegwhichis

Noteatthispointthereisnodualitysymmetryhere;iftherearetwotypesofcharges,youshouldbeable todistinguish them(bydefinition)byapplyingelectricandmagnetic fields.However, ifaparticlehasbothtypesofcharge(knownasadyon)thentheabovebecomes

whichisinvariantunderthesymmetrytransformation.Toseethistheaboveneedstobewrittenintermsof thecomplexchargeQ :=q+ igandthecomplex fieldG :=E+ iB, togetherwith theircomplexconjugates.Onefinds

whereeachpiecenowisseparatelyinvariant.

Turning our attention now to the vector potential, we were able to introduce this originallybecause wasexactlyzeroandtherefore wasamathematicalresult.Ifwenowhaveamonopolesourcetheninsphericalcoordinates

Canweobtainthisfromagaugepotential?thecurlofthegaugefieldhascomponents

ThesimplestsolutionwecanlookforiswhenAr=0andAθ=0.Thefirstequationthentellsusthat should be inversely proportional to sin θ, while the second requires that it is inverselyproportional to r. The first equation also tells us that we need a cos θ (plus a constant) in thenumeratorsothatwhendifferentiateditwillcancelofftheremainingsinθinthedenominator.Onecanseethereforethatthegaugepotential

gives the monopole magnetic field. However, there is a problem because this has a singularbehavior at θ = π. It is this singular aspect that causes the standard vector calculus identity

tobreakdown,asitneedstoifwegoingtoextractamonopolecharge.Infactitdoesn’tmatter at all that the gauge field has a singularity provided the magnetic field shows no suchbehavior.Wecansidestep thesingularity in thepotentialby introducinga secondpotential.Theideahereisthateachpotentialisonlytobeusedinasubspacewhereitdoesn’thaveasingularity.Togethertheythencancoverthewholespace.Anexamplechoicewouldbe

theplusreferringtotheupperhemisphereandtheminussign,tothelowerhemisphere(the isasmallangle).Providedoneusestheseontheircorrectcoordinatepatchwehaveasolutionforthemagneticfieldthatiscorrecteverywhere.

Asafinalpoint,toseehowallthistiestogether,whathappenstothetwogaugepotentialsintheoverlap region, for example at They aremagically related by a gauge transformation asfollows

sothegaugefunctionis

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INDEX

Ampère,50Angularmomentum,108AssociatedLegendreequation,78

Biot,53Biot-Savartlaw,53Bolometer,107

Capacitance,40Coloumbgauge,90Commutator,9Continuityequation,46Convolution,14Convolutiontheorem,14Coulomb,24Coulomb’slaw,25Crossproduct,7

D’Alembertian,71Diracdeltafunction,13Displacementcurrent,65DivergenceTheorem,12Drude,136Drudemodel,136

Eigenvalues,10Eigenvectors,10Electricdipole,125ElectricField,29Electricsusceptibility,133Electrostaticpotential,31Energydensity,106Equipotential,34

Faraday,55Fizeau,72Fizeau-Foucaultexperiment,71Foucault,72FourierSeries,13,73Fouriertransform,13Franklin,23

Gaugetransformation,89Gauss,32Grouptheory,9

Helmholtz,74Helmholtzequation,73

Inductance,60

Kohlrausch,67

Langley,106Laplace’sequation,34Lenz,57Lenz’slaw,57Levi-Civitatensor,8Lineelement,12Lorentzforcelaw,55Lorentzgauge,88,89

Magneticdipole,127Malus,110Maxwell,66Maxwell’sequations,66Multipoles,116–128

Oersted,49Ohm,50

Poincaré,113Poincarésphere,113Poisson’sequation,33Poynting,105Poyntingvector,106

Radiationpressure,104Resistance,50Responsefunctions,133

Savart,53Scalarfield,6Scalarpotential,31Scalarproduct,7Separationofvariables,74SphericalBesselequation,78Stokes,112Stokesparameters,111Stokes’stheorem,12Stresstensor,103Superpositionprinciple,29

Tensor,6Thales,22

Vectorfield,6Vectorpotential,86

Waveequation,71Weber,67Weyl,87