the history of maxwells equations

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Te History Of Maxwell's EquationsLindsay Guilmee (Class of 2012)Sacred Heart University, [email protected]

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[Typetext] [Typetext] [Typetext]

TheHistoryofMaxwellsEquations

Fall

2011

LindsayGuilmette

SacredHeartUniversity


3/17

TheHistoryofMaxwellsEquations 1

JamesClerkMaxwellwasbornin1831inScotlandtoafamilyofFellowsoftheRoyalSociety,an

eliteorganizationofthetopscientistsofalldisciplinesinGreatBritain(TheRoyalSociety,2011).Maxwell

beganhisacademiccareerquiteearly.HepresentedhisfirstpaperOvalCurvestotheRoyalSocietyof

Edinburghwhenhewasfourteen(Forfar,1995).MaxwellbeganhisundergraduatestudiesatEdinburgh

UniversityatagesixteenandenteredgraduateschoolatCambridgeUniversityatagenineteen.After

graduation,hewasafellowandprofessoratavarietyofcollegesintheUnitedKingdom.Maxwellwas

inductedasaFellowofTheRoyalSocietyofEdinburghwhenhewas25,andpromotedtoaFellowofThe

RoyalSocietyatage30.Afterafruitfulcareer,JamesMaxwellpassedawayattheageof48ofstomach

cancer,whichwasoddlythesamecauseandtimingofhismothersdeathwhenMaxwellwaseightyears

old(Forfar,1995).

In1819,twelveyearsbeforeMaxwellwasborn,thefieldofelectromagnetismwasborn.Hans

ChristianOerstedofDenmarkhadsuspectedthatelectricityandmagnetismwererelatedforsometime

andwasfinallyabletocreatesomeevidenceforit.Inhisexperiment,Oerstedheldawireabovea

compass.Whenacurrenttravelledthroughthewire,henoticedthattheneedleofthecompassmoved.

Thislaunchedaninvestigationofhowelectricityandmagnetismwererelated.AndrAmprewasso

excitedafterlearningofOerstedsdiscoverythathecreatedarough

mathematicaltheorythatdescribeditinoneweekin1820,and

publishedhisfinishedequationin1821(Peters,2000).Ampres

equationbecomesthefourthofMaxwellsfamousequations.Ayear

later,thefirstgalvanometerwasmade,whichwasgreatlybeneficial to

themeasurementofelectriccurrent(TheEncyclopediaAmericana

Corporation,1918).AgalvanometerreliesonthefindingsofOersteds

experiment:whenacurrenttravelsthroughacoil,anearby

magnetizedneedleisdeflected.AswasfoundafterOersted,theneedle Agalvanometer1


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isdeflectedinproportiontothestrengthofthecurrent(FloridaStateUniversity,LosAlamosNational

Laboratory,UniversityofFlorida,2011).

ThemostproductivescientistinelectromagnetismbeforeMaxwellwasMichaelFaraday,whowas

actuallyachemist.Inthecourseofhisresearch,whichlastedfrom1831to1855(TheEncyclopedia

AmericanaCorporation,1918),heproducedover16,000perfectlyorganizedentriesinhislaboratory

notes(Peters,2000).TheresultsofhisexperimentsformedthebasisforMaxwellselectromagnetic

theory.AlthoughFaradayhadanamazingphysicalintuitionthatguidedhimthroughhisexperimentsto

successfullycreateanearlycompletemodelofelectromagneticphenomena,sincehewasnotatalla

mathematician,hewasonlyabletodescribethismodelinwordsanddidnotunifyhisresultsintoatheory

(TheEncyclopediaAmericanaCorporation,1918).Sincethephysicistsofthetimeweremathematically

trained,theyhardlypaidFaradaysresultsanyattention.Theysawthelackofmathematicsasalackof

sophistication(Hunt,1983).

AlthoughFaradaycompletedanamazingnumberofexperiments,therearetwoseriesthatare

mostimportanttoconcretelyunderstandhiscontributionstoMaxwellstheory.Thefirstsetof

experimentsallowedFaradaytodiscovertheprincipleofelectromagneticinduction.Hewoundtwo

separatewiresaroundawoodenbobbinandinsulatedthemfromeachother.Tooneheconnecteda

battery,andtotheotherheconnectedagalvanometer.Hediscoveredthatbybreakingorreconnecting

thecircuitconnectedtothebattery,acurrentoftheoppositedirectionwasinducedintheotherwire.

Thisamazingdiscoveryformsthebasisofanydevicethatuseselectricity,includingtelephones,which

wereinventedlaterinthe19th

century.Faradayalsofoundthatsimplymovingamagnetcloserorfarther

awayfromaclosedcircuitwouldinduceacurrent(TheEncyclopediaAmericanaCorporation,1918).This

discoveryallowedFaradaytodesignandbuildageneratoraswellasthefirsttransformerandthefirst

electricmotor.Whenaskedbyapoliticianwhattheuseofthisworkwas,Faradayrepliedatpresent,I


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dontknow,butonedayyouwillbeabletotaxthem(Peters,2000,p.6).Electromagnetic

inductionisatrulyincrediblephenomenonthatredefinedhowthesourcesofelectricitywere

thoughtof.Thesecondexperimentisactuallyonethatmanyelementaryschoolchildren

carryoutintheirclassrooms.Itinvolvesspreadingironfilingsonasurfaceandsimplyplacing

amagnetonthem.Thiscausestheirontobemovedbythemagneticfieldinapatternthatloopsfrom

oneendofthemagnettotheother.Faradaysawthispatternandthoughtthattheselinesofforce,as

hecalledthem,mustbethekeytoelectromagnetism.Whenheinducedanelectriccurrentinawirewith

amagnet,hesawthislineofforcecurvingaroundthewire,aswenowknowisthecase.

ThefirstpaperthatJamesMaxwellpublishedafterearninghisgraduatedegreewasa75page

analysisofFaradaysconceptoflinesofforcepublishedin1855andwassuitablytitledOnFaradays

LinesofForce(Forfar,1995).MaxwelltranslatedFaradaysideasintomathematics.Maxwellcreated

vectorstodescribethemainplayersofelectromagnetism:E,theelectricfieldintensity,H,themagnetic

fieldintensity,B,themagneticfluxdensity,andItheelectriccurrentdensity.EandHareforcesandBandIarefluxes(linesofforce)producedbytheforces(Peters,2000,p.9).Awaytopicturefluxistoimaginehavingasquareloopofwireinaflowingriver.Thefluxofthevelocityofthewaterwouldbelike

consideringhowmuchwaterwillflowthroughtheloop.Thefluxofanelectricfieldisproportionaltothe

numberofelectricfieldlinesthatgothroughsuchaloop(Sciolla,2004).

NoticethatMaxwellhadtransformedFaradayslinesofforceintomathematicalconceptsthatwestill

usetoday.Fieldsareusedinmanydisciplinesacrossmathematics.ThefirstmentionofafieldinMaxwells

ATreatiseonElectricityandMagnetismwasofanelectricfield,whichhedescribedas

Ironfilings2

Asquareloopofwireinflowingwater3


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theportionofspaceintheneighborhoodofelectrifiedbodies,consideredwithreferenceto

electricphenomena.Itmaybeoccupiedbyairorotherbodies,oritmaybeasocalledvacuum,

fromwhichwehavewithdrawneverysubstancewhichwecanactuponwiththemeansatour

disposal.Ifanelectrifiedbodybeplacedatanypartoftheelectricfielditwill,ingeneral,produce

asensibledisturbanceintheelectrificationoftheotherbodies(Maxwell,1892,pp.4748).

Withthesevectors,Maxwellcreatedcountlessequationstodescribehiselectromagnetictheory(Maxwell,

1892).DespitethepublishingofATreatiseonElectricityandMagnetism,whichwasahandbookstarting

fromsquareoneforeverythingonecouldknowaboutelectromagnetismatMaxwellstime,notmuch

attentionwaspaidtoMaxwellandhisrevolutionaryideas.ItwasnotuntilHeinrichHertzfound

experimentalevidenceforMaxwellsconceptofafieldin1887bydiscoveringelectromagneticwavesin

spacedidthescientificcommunitybegintolookatMaxwellstheory.

BeforeweexplorethebasicequationsofMaxwellstheoryofelectromagnetism,wemustlearn

twooperations.Firstrecallthataderivativeofafunctionfistheproportionofthechangeinf(x)and

thechangeinx: .Theproportioniscalledapartialderivativebecauseitisusedwhenafunctionisofmorethanonevariableandissimplythederivativeofthatfunctionwithrespecttox,

treatingallothervariablesasconstants.Whenafunctionisofmorethanonevariable,suchasf(x,y,z),we

have

.Thedeloperator,,isapseudovectorofthepartialderivativesoff:

andisusedinthetwofollowingoperations(Sciolla,2004).

Thefirstoperationiscalleddivergence,andisdenotedby wherevisa

functionofx,y,andz.Thisoperationusesthedotproduct.Recallthatthedotproductoftwovectors

, , and , , is andyieldsascalar.Thedivergenceofafunctionrepresentshowmuchafunctionspreadsaroundapoint(Sciolla,2004).If,,representsan


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electricfield,thenthedivergenceofthefirstfunctionbelowshowsthatanyelectronsinthefieldare

beingrepelledawayfromtheorigin(Sciolla,2004).

Thesecondoperationiscalledcurl,whichdescribeshowmuchafunctioncurlsaroundapoint,

(Sciolla,2004)andisdenotedby .Anexampleofthecalculationofthecurlofafunctionrepresentingamagneticfield,,isshownbelow.Thedeterminantofthematrixbelowisapseudo

crossproduct: 0

0

0 0 0 0 1 1 2 whichissometimeswrittenas2.

Withtheseoperationsinmind,wearereadytolookattheevolutionofMaxwellsequations.

Divergenceoffunctionsrepresentingelectricfields4

Thecurlofafunctionrepresentingamagneticfield5


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In1861,Maxwellpublishedthebeginningsofhisfamousequations.Itwaswhiledevelopingthese

equationsthathediscoveredthatlightissimilartoelectricityandmagnetism.Sincetheseequationswere

ofadifferentformthanweseethemtoday,tobetterappreciateMaxwellsdistillationofFaradays

theory,wewilllookattheforminwhichmathematicianandphysicistOliverHeavisidewrotethema

coupleofdecadesafterthepublicationofMaxwellstreatise.Twoaremostnotable(Darrigol,2005):

/

/

whereMisthemagneticcurrent,whichdescribeshowamagneticfieldseemstoflowfromonepoleto

theother,jistheelectricalcurrent,and/isthedisplacementcurrent,whichisbasicallyapotentialforelectriccurrenttoflow.Theremainingvariablesaredescribedabove.

Electricpotentialwasanewconceptthatgrewtochangeelectromagnetictheorycompletely.It

wasFaradaythathadfirstthoughtofthisparadigmbutcouldnotexpressitmathematically(Peters,2000).

Maxwellactuallycreatedanewformofderivativeinordertoachievethisexpression:DY/Dtisthe

convectivederivativedefinedbyDY/Dt=Y/t(vY)+v(Y),wherevisthevelocityofthestates(Darrigol,2005,p.2)whichareE,H,D,andB.SinceDiselectricflux,DD/Dtrepresentsvaryingelectric

currentpotential.ThisandDB/DtwereaddedbyMaxwelltotheoriginalequationswrittenbyHeinrich

Lenz(thefirst)andAndrAmpre(thesecond)(Darrigol,2005,p.2).

ATreatiseonElectricityandMagnetism,publishedin1873,recordsthetransformationof

Maxwellscompletetheoryofelectromagnetismintotwelveequations(Maxwell,1892).Itisanamazing

featforonescientisttowritetheequationsthatsummarizedanentirefieldofresearch.Theselftaught

mathematicianandphysicistOliverHeaviside,alsofromGreatBritain,furthercondensedMaxwells

theoryintothefourequationsthataremostknowntoday:(Heaviside,1893).


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0

/

Thefirstequationrelateselectricfluxtothevolumetricchargedensity,.Thesecondequationsaysthatamagneticfielddoesnotdivergebecausetherearenomonopoles(Sciolla,2004).Thethird

equationsummarizestheresultfromFaradayselectromagneticinduction:thevoltageinducedina

circuitwasproportionaltothetimerateofchangeofmagneticfluxthroughthecircuit(Peters,2000,p.

9).ThefourthisfromAmpre:themagneticfieldinaclosedlooparoundacurrentwas,infact,equalto

thecurrentthroughtheloop(Peters,2000,p.9).

Themodernpartialderivativeversionoftheseequationsisverysimilar,butconstantshavebeen

addedtosometermsinordertoconformtostandardsystemsofunit.Herewewilllookattheversionin

cgs(centimeters,grams,seconds cgswasactuallyfirstintroducedayearafterthepublicationof

Maxwellstreatise(WolframResearchProducts,2007)):

4

0

1

4 1

wherecisthespeedoflight(Sciolla,2004).


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Today,wealsosometimesfinditconvenienttouseMaxwellsequationsintheirintegralform.The

followingareinSIunits(Fowler,2009):

/

0

whereqistheelectriccharge,istheelectricconstant,andisthemagneticconstant.Thefirstequationsaysthattheintegraloftheoutgoingelectricfieldoveranareaenclosingavolumeequalsthe

totalchargeinside(Fowler,2009).Thesecondisstillanotherwayofsayingthattherearenomagnetic

monopoles.Thethirdandfourthequationshaveapathintegralthatsimplydenotesintegrationarounda

closedpath.Inthiscase,thatpathisusuallyawire.Theleftsideoftheequationgivesthetotalvoltage

changearoundthecircuit,whichisgeneratedbyavaryingmagneticfieldthreadingthroughthecircuit

(Fowler,2009).Thefourthequationgivesthetotalmagneticforcearoundacircuitintermsofthecurrent

throughthecircuit,plusanyvaryingelectricfieldthroughthecircuit(thatsthedisplacementcurrent)

(Fowler,2009).

Maxwellsequationsareamazingbecausetheysummarizethefundamentalphenomenarelated

toelectromagnetisminfourconciseequations.However,hisequationshadmoreofaneffecton

electromagnetismthanadditionalconvenience.Maxwellextended,connectedandtranslatedFaradays

conceptsofelectromagnetismintoafullfledgedmathematicaltheory,whichputitonthetablefor

comparisontotheleadingelectromagnetictheoryatthetime.Wewillcomparethesetheoriesusingthe


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LeydenJar,adevicestillusedinMaxwellstime,aspictured.The

LeydenJarhastwocoatingsofaconductivemetalthatdonot

toucheachother,oneontheinsideofaninsulatingglass,andthe

otherontheoutside.Theinnercoatingischargedwithametalrod

andthenthemetalrodisremoved.Whenascientisttouchesthe

outsideofthejar,thescientistreceivesanelectricshock(Wolfram

ResearchProducts,2007).Thissurprisedscientistsbecausethey

thoughtonlytheinnercoatinghadbeencharged.Thedifferingexplanationsofthisphenomenon

encompasstheheartofthedifferencebetweenMaxwellstheoryandtheleadingelectromagnetictheory

(TheEncyclopediaAmericanaCorporation,1918).

Thisprevalenttheorywascalledactionatadistance.Thistheoryassumesthatinsulators

completelyblockelectriccurrent.Therefore,thereasonthatachargecouldbereceivedfromtheouter

coatingisthatelectricityworksatadistance,asgravitydoes.Whileelectricitywasbuildinguponthe

innercoating,anelectricityofanoppositekindwasattractedtotheoutercoating.Therefore,when

touched,theoutercoatingcoulddeliverashock(TheEncyclopediaAmericanaCorporation,1918).

MaxwellsawthephenomenonoftheLeydenJarandwasinspiredbyFaradayslinesofforce.

MaxwellexpandeduponFaradaysconceptionofelectricitytoassertthatnotonlydoeselectricitytravel

alonglinesofforce,butalsorequiresamedium.Hesawinsulatorssuchasglassasbeingresistiveto

electriccurrent,notimmunetoit.Hesawtheprocessofchargingtheinnercoatingtobelikedisplacing

electricityinthewaythatyouwoulddisplaceaspringbystretchingitawayfromitsequilibrium.The

resistiveforceoftheglassblockstheelectricityfromreturningtoitsequilibriumuntilthechargingforce

overcomesit.Inthespringexample,thechargingforcewouldbetheforceofthespringandtheresistive

forcewouldbetheforceofthehandsthatarepullingthespringapart.Whentheforceofthespring

TheLe denJar6


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overcomestheforceexertedbythehands,thespringsnapsbackintoequilibrium.ThisishowMaxwell

sawthephenomenonoftheLeydenjar:whenthechargingforceovercomestheresistiveforce,the

electricitysnapsbackintoequilibriumthroughanelectriccurrent(TheEncyclopediaAmericana

Corporation,1918).Thischangeinconceptionwouldgreatlysimplifythemathematicsinvolvedinthe

studyofelectromagnetismaswellasmoreaccuratelyexplainelectromagneticphenomena(Darrigol,

2005).

Itisamazinghowquicklyelectromagnetismdevelopedinsuchashorttimefromitsbeginnings

withOerstedsexperimentin1819.However,thisquickpacewasnotjustcoincidence,butwasrelatedto

theculturalchangesthatGreatBritainwasgoingthroughduringthe19th

century.Duringthemid19th

century,theideathattheprincipleoffreetradewouldbringpeaceandprosperitybegantopercolate

throughEuropeansociety.Forsometime,thedominatingeconomicphilosophyhadbeenmercantilism,

whichwasadoctrineofstrictprotectionofastatesresources,hightariffsontrade,andcolonial

domination.Theseideasofthebenefitsoffreetradeintheintellectualculturehadaplaceinthescientific

cultureaswell.Ratherthanworkingindependentlyononesreputationandtheory,scientistsbegantosee

thebenefitsofdevelopingupontheirpeersideasanddiscussingtherelativemeritofscientifictheoriesas

acommunity(Turner,1980).Maxwellspentthemajorityofhisworkinelectromagnetismsimply

convertingFaradaystheoryintomathematics,butthecombinationoftheirrespectivetalentscreatedan

incrediblecontributiontoscience.ItwasthisideaoffreetradethatmadeMaxwellwanttosharehis

theorywithEurope,notjusttodominatethefield,buttopeacefullyofferacomparisontotheleading

theory(Forfar,1995).Duringthemid19thcentury,theBritishscientificculturealsofullyequatedsocietys

progresswithscientificprogress.Theyhadseensciencecreatethetelegraph,seenthetelegraph

revolutionizecommunicationandEuropeansociety,andseenthescientificadvancementofthetelegraph

continuetorevolutionizesociety.Theysawthepowerofsciencetochangetheworld(Turner,1980). For


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physicistsspecifically,itwasthepowerofthetelegraphthatwasamajormotivationtounderstand

electromagnetismcompletelytoadvancecommunication(Maxwell,1892).

Itwasbecauseofthisbeliefthatscientificadvancementwasimportantforsocietysprogressthat

scientistsbeganwantingtoinfluencepublicdecisionsandhavepublicfunding.However,thepeoplewho

wereinthepublicforefrontatthetime,politiciansandbusinessmen,dismissedtheideathatscientific

workhadanythingtodowiththeirwork.Inanattempttowinsocietyover,manyscientistsbeganto

emphasizehowimportantsciencewastothesecurity,economiccompetency,andstabilityofGreat

Britain.ThisstrategywasdrawingonthefactthatBritishscienceandBritishpoliticsandbusinesshada

commonenemy:Germany.Bythelate19th

century,GermanybecameanindustrialthreattoGreat

Britainseconomicdominance,andwasslowlybecomingamilitarythreataswell.Germanyhadalsobeen

athorninthesidesofBritishscientistsforyears.Germanyspublicfundingandpublicrecognitionoftheir

scientistsresearchhadmadeGermanscienceflourishinthe19thcentury,andwasmuchmoreinfluential

andprosperousthanBritishscience(Turner,1980).Infact,theleadingtheoryofelectromagnetism,

actionatadistance,wasdevelopedinGermanyanditwastheseGermanscientistswholookeddown

uponandrefusedtoacknowledgeMaxwellstheory.Frustratedwiththelackofresponseofthescientific

communitytohispaperOnFaradaysLinesofForce,Maxwelltransformedhiscomprehensivetheory

intotwelveequations,agreatfeatevenfortodayspaceofscientificprogress(Forfar,1995).However,the

Germanscientistsstillrefusedtorelinquishtheirmonopolyonelectromagnetism,andsimplydismissed

Maxwellstheoryassomethingthatjustworksoutonpaper(Turner,1980).OliverHeavisideseditingof

Maxwellsequationsbegantoturnthetide,butitwasnotuntilHeinrichHertz,aGermanphysicist,

demonstratedin1887theexistenceofafieldofelectromagnetismbyfindingelectromagneticwavesin

spacethatGermanscientistsbegantoacceptMaxwellstheory(Hunt,1983).


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Maxwellstheoryofelectromagnetismwasrevolutionary.Itcompletelychangedthestudyof

electromagnetismaswellasEuropeansociety.AfterHertzdiscoveredelectromagneticwavesinspace,the

wirelesstelegraphwasinvented,whichmadefastcommunicationpossibleacrossoceansforthefirsttime.

Maxwellstheoryalsomadelongdistanceelectricalwireseffective,somethingtheactionatadistance

theorywasnotabletodo(Hunt,1983).Stilltoday,weoweanenormousamountoftechnologyto

Maxwellstheoryofelectromagnetismandhisperseverancethroughculturalobstaclestoadvocatehis

talentofmathematicalinterpretation.


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ReferencesDarrigol,O.(2005).TheGenisisoftheTheoryofRelativity.RetrievedOctober12,2011,fromSminaire

Poincar:http://www.bourbaphy.fr/darrigol2.pdf

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http://www.magnet.fsu.edu/education/tutorials/java/galvanometer/index.html

Forfar,D.O.(1995,July).JamesClerkMaxwell:MakerofWaves.RetrievedNovember30,2011,fromJamesClerkMaxwellFoundation:http://www.clerkmaxwellfoundation.org/Maker_of_Waves.pdf

Fowler,M.(2009,May).Maxwell'sEquationsandElectromagneticWaves.RetrievedNovember30,2011,fromGalileoandEinstein:

http://galileo.phys.virginia.edu/classes/109N/more_stuff/Maxwell_Eq.html

Heaviside,O.(1893).ElectromagneticTheory.Boston:"TheElectrician"printingandpublishingcompany,limited.

Hunt,B.J.(1983,September)."Practicevs.Theory":TheBritishElectricalDebate,18881891.Isis,74(3),341355.

Maxwell,J.C.(1892).ATreatiseonElectricityandMagnetism.Oxford:ClarendonPress.Peters,R.A.(2000,April5).ABriefOutlineoftheHistoryofElectromagnetism.RetrievedOctober12,

2011,fromRichardAlanPetersII,Ph.D.:

http://www.vuse.vanderbilt.edu/~rap2/papers/em_history.pdf

Sciolla,G.(2004).PhysicsII:ElectricityandMagnetism.RetrievedOctober7,2011,fromMITOpenCourseWare:http://ocw.mit.edu/courses/physics/8022physicsiielectricityand

magnetismfall2004/

TheEncyclopediaAmericanaCorporation.(1918).Electricity,ItsHistoryandProgress.InTheEncyclopediaAmericana(Vol.10,pp.175180).NewYork:J.B.LyonCompany.RetrievedNovember2011,from

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TheRoyalSociety.(2011).Fellowship.RetrievedDecember2,2011,fromTheRoyalSociety:Excellencein

Science:http://royalsociety.org/aboutus/fellowship/

Turner,F.M.(1980,December).PublicScienceinBritain,18801919.Isis,71(4),589608.WalterLewin,J.B.(2010).PhysicsII:ElectricityandMagnetism.RetrievedOctober7,2011,fromMIT

OpenCourseWare:http://ocw.mit.edu/courses/physics/802scphysicsiielectricityand

magnetismfall2010/


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WolframResearchProducts.(2007).cgs.RetrievedDecember7,2011,fromEricWeisstein'sWorldof

Physics:http://scienceworld.wolfram.com/physics/cgs.html

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Endnotes1. http://hyperphysics.phyastr.gsu.edu/hbase/magnetic/galvan.html2.

http://www.cyberphysics.co.uk/My_Notes/Magnets1.htm

3. http://ocw.mit.edu/courses/physics/8022physicsiielectricityandmagnetismfall2004/lecturenotes/lecture2.pdf



6. http://electronics.howstuffworks.com/capacitor.htm/printable

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