1Electromagnetism
Radioisatechnologythatisbaseduponelectromagneticphenomenaandan understandingofelectromagnetictheoryiscrucialtotheunderstandingofradio.Ideas ofelectricityandmagnetismhavebeeninexistenceformanymillennia,butthetheory ofelectromagnetismwastheresultofasurgeinactivityoverthelastfourcenturies.The developmentofelectromagnetictheoryculminatedintheMaxwellequations,equations thatarecrucialtoourunderstandingofradiowaves.Radioisanexampleofthetriumphof theoreticalscienceinthatitwaspredictedthroughtheoryratherthanbeingdiscoveredby accident.Itistheaimofthecurrentchaptertodescribeelectromagnetictheorythroughits historicaldevelopment.Thechapterhasbeenwrittenforthosewithverylittleknowledge intheareaandsocanbeskippedbythosewhoalreadypossessagoodknowledgeofthe subject.However,itisexpectedthatsomereaderswillbealittlerustyonthetheoryand sothischapterwillserveasrevisionforthem.
1.1Electricity
ThefirstrecordedobservationsofelectricaleffectsgobacktotheGreeks.Inthesixth centuryBC,ThalesofMiletusobservedthatamber,whenrubbed,wouldattractlight objects.Thisphenomenonisexemplifiedbytheoldschoolboytrickofrubbingacomb onyourtrousersandthenseeingitliftsmallscrapsofpaper.Todayweknowthatmatter ismadeupofatomswhichcontainparticleswithpositiveelectriccharge(protons), negativeelectriccharge(electrons)andnocharge(neutrons).Further,thatlikecharges repeleachotherandthatunlikechargesattract.Asimplemodelofasingleatomconsists ofanumberofelectronsthatorbitaroundanucleusconsistingofthesamenumberof protonsandpossiblysomeneutrons(seeFigure1.1).Theelectronsarearrangedinshells aroundthenucleus,eachshellcontainingelectronsofapproximatelythesameenergy (theenergyincreaseswithradius)andaredesignated,inorderofenergy,asK,L,M,N, O,PandQ(itshouldbenotedthattheenergygapbetweentheseshellsismuchlarger thantherangeofenergieswithinashell).Duetoquantummechanicaleffects,theshells containonlyalimitednumberofelectrons(theKshellcancontainamaximumof2 electrons,theLshell8electrons,theMshell18electrons,theNshell32,etc.).Matter willconsistofalargecollectionofsuchatomswhich,undernormalcircumstances, willbeinoverallelectricalneutrality(thenumbersofelectronsandprotonsareequal). Undersomecircumstances,however,itispossibletoincrease,ordecrease,thenumber
Fig.1.1 Atomicstructureconsistsofelectronsorbitinganequalnumberofprotonsandpossiblysome neutrons.
ofelectronsandthematerialwillbecomeelectricallycharged.Thisiswhatisachieved intheaboverubbingprocess,sometimesknownasthetriboelectriceffect.Theessential conditionfortheeffecttoexististhatthematerialsbeingrubbedtogetherhavedifferent strengthsoftheforcethatbindtheirelectronstothenucleus(glasshasafarstronger bondthanrubberforexample).Whenthematerialsarebroughttogether,electronsinthe materialwiththeweakerforcewillbeattractedtothematerialwiththestrongerforce. Whenthematerialsarethenseparated,someofthetransferredelectronswillremainon thematerialwiththestrongerforceandbothmaterialswillbecharged,onepositively (theonewiththeweakerforce)andonenegatively(theonewiththestrongerforce).
Realmattercanbequitecomplexinstructure,withmanymaterialscomposedof moleculesthatarecomplexcombinationsofdifferentkindsofatoms.Theheavieratoms (thosewithalargenumberofprotons)canhavemanylayersofelectronssurroundingthe nucleusandthismeansthatthebondoftheouterelectronscanberelativelylow.Thiscan leadtohighelectronmobilityinmaterialscomposedofsuchatoms.Materialsforwhich theelectronsarehighlymobile,relativetotheprotons,areknownasconductorsandare exemplifiedbymetalssuchascopper,silverandgold.Materialswheretheelectronsare relativelyimmobileareknownasinsulators(glassandrubberbeingimportantexamples). Insulatorsandconductorsturnedouttobeofgreatimportanceinthedevelopmentof electricity.
Theseventeenthandeighteenthcenturieswereaperiodofgreatadvancesinour knowledgeofelectricaleffects,muchofitmadepossiblebyincreasinglysophisticated machinesfordevelopingchargedmaterialsthroughthetriboelectriceffect.Figure1.2 showsthebasicmechanismofsuchmachines.Therubberbeltrollsovertheglasscylinder andthiscauseselectricalchargetobuilduponthesecomponentsthroughthetriboelectric effect.Whenthecomponentsseparate,thebeltwillbenegativelychargedandthecylinder positivelycharged.Thenegativechargesonthebeltwilleventuallyreachaconducting brushthatsweepsthemupontoaconductingmetalwirealongwhichtheytraveluntil reachingaconductingsphereonwhichtheyaccumulate.Inasimilarfashion,thepositive chargetravelswiththecylinderuntilitreachesaconductingbrush.Atthisbrush,the positivechargeisneutralisedbynegativechargethathasbeendrawnfromthelower spherealongtheconductingwire.Inthisfashion,positivechargeaccumulatesonthe lowersphere.AsshowninFigure1.2,thechargeaccumulatesonopposingfacesofthe spheres.Thisoccursduetothemobilityofelectronsonconductorsandthefactthat
Fig.1.2 Abasicmachineforcreatingpositiveandnegativechargebythetriboelectriceffect.
opposingchargesattract.Themediumbetweenthespheresiscomposedofairandthis willtendtoactasaninsulatorandsothechargewilljustaccumulateonthespheres. Furthermore,thechargesontheopposingsphereswillbalanceeachotherout.
Ifachargedparticleisplacedbetweenthespheres,itwillbedrawntowardsthesphere withtheopposingchargeandrepulsedbythespherewiththesamecharge.Consequently, ifwewanttoincreasetheamountofnegativechargeontheupperspherebydirectly movingpositivechargetothelowersphere,thiswillrequireanexternalagencytodo somework.Thisbringsustotheimportantconceptof potentialdifference.Thepotential differencebetweentwopointsisdefinedtobetheworkdonebyanexternalforcein movingpositivechargebetweenthesepointsandismeasuredintermsofvolts(1voltis 1joulepercoulomb).Inordertoquantifythis,weneedtobeabletocalculatetheforce thatonechargeimposesuponanother.Theforce F imposedoncharge q bycharge Q is givenby Coulomb’slaw
where r isthedistancethatseparatesthechargesand 0 isknownasthe permittivity offreespace(i.e.spacethatisdevoidofmatter).Thisforceisrepulsiveifthecharges havethesamesignandattractiveifthesignisdifferent.Thelawwasformulatedby CharlesAugustindeCoulombin1784astheresultofmuchexperimentalwork.The unitsofchargeareknownascoulombs,withaprotonhavingacharge1.60219 × 10 19 coulombsandanelectronminusthatamount.Ifdistancesaremeasuredinmetresand theforceinnewtons, 0 = 8.85 × 10 12
Forceis vector innature,i.e.ithasbothmagnitudeanddirection.Consequently,we needsomeunderstandingofvectorquantities.Pictorially,wecanrepresentavectoras anarrowthatpointsinthedirectionofthevectorwithitslengthequaltothemagnitude (Figure1.3).Vectorsarenotonlyusefulfordescribingquantitiessuchasforce,butcan alsobeusedfordescribingthegeometricalconceptofposition.Thepositionofapoint canbedescribedbythevectorthatjoinssomearbitraryorigintothispoint,themagnitude beingthedistancefromtheorigintothepoint.Animportantconceptinvectorsisthatof
Fig.1.3 a)Vectorrepresentedgraphicallyasanarrowandb)anglebetweenvectorsforthevectordot product.
Fig.1.4 Theadditionofvectors.
the dotproduct oftwovectors a and b,writtenas a · b.Ifthetwovectorshavemagnitudes a and b,respectively,thedotproductisdefinedtobe ab cos θ where θ istheanglebetween thesevectors(seeFigure1.3).Itcannowbeseenthat a = √a · a and b = √b · b.(Note thatweoftenuse |x | asmathematicalshorthandformagnitude x = √x x ofthevector.) Thedotproductcanbeusedtofindthecomponentofaforce F inaparticulardirection. Let ˆ t beaunitvector(| ˆ t |= 1)inthedirectionofinterest,then ˆ t F isthecomponentof forceinthatdirection.
Animportantoperationwecanperformonavector p istomultiplyitbyascalar s to getanewvector sp thatpointsinthesamedirectionas p butnowhasthemagnitude sp. Anotherimportantoperationwhenwehavemultiplevectorsistheiraddition.Forthe vectors a and b,ifwejointhetipofthearrowrepresenting a tothebaseofthearrow representing b,thesum a + b isrepresentedbythearrowfromthebaseofthearrow representing a tothetipofthearrowrepresenting b (seeFigure1.4).
Intermsofvectors,Coulomb’slawcanberewrittenas
where ˆ r isaunitvector( ˆ r ˆ r = 1)inthedirectionfrom Q to q .Analternativewayof lookingatthisistoregardcharge Q ascreatingan electricfield (sometimesknownas the electricintensity)
thatpervadesspace.Whenacharge q isplacedinthisfield,itisacteduponbyaforce q E where E isthevalueofthefieldatthepositionofcharge q (E willhaveunitsofvolts
permetre).Theconceptofafieldthatexistsatallpointsofspacewasarevolutionin thinkingandwasanextremelyimportantstepinthedevelopmentofelectromagnetism. Onecannowaskwhatthefieldwillbewhentherearechargesatavarietyoflocations. Fortunately,itturnsoutthatthisfieldwillsimplyconsistofthesumofthefieldsdue totheindividualcharges.Consequently,ataposition r ,asystemof N chargeshasthe electricfield
where ri isthepositionofthe i th charge Qi and |r ri | isthedistancefrom ri to r Wenowreturntothequestionofthepotentialdifferencebetweenpoints rA and rB . Thisistheworkdoneinmovingaunitchargefromapoint rA toapoint rB .Ifthereisa constantelectricfield,theworkdoneinmovingfrompoint rA to rB is (rB rA ) · E (i.e. minusthefieldinthedirectionof rB from rA multipliedbythedistanceinthatdirection). Whenmovingthroughthefieldproducedbyafinitenumberofcharges,however,the forcewillvaryfrompointtopoint.Consequently,wewillneedtosplitthepathover whichtheunitchargemovesintoanumberofshortsegmentsoneachofwhichthe electricfieldcanberegardedasconstant(seeFigure1.5).Thepotentialdifferencewill nowbeapproximatedby
where M isthenumberofsegments.Takingthelimitwherethesegmentlengthstend tozero,theabovesumbecomesthemathematicaloperationofintegrationalongaline, thatis
Inthecaseofourfinitesystemofcharge,wewilldefinethe potentialV ofthesystem tobethepotentialdifferencewhenpoint rA isapointatinfinityand rB isthetestpoint r ,then
Fig.1.5 Pathforcalculatingworkdonewhendividedintosegments(
and
).
Fig.1.6 Fieldlinesandlinesofconstantpotentialforpositiveandnegativecharges.
Fig.1.7 Fieldlinesandlinesofconstantpotentialforadipole.
Wecanvisualiseafieldintermsofwhatareknownas fieldlines.Suchlineshave thepropertythat,atanypoint,theirtangentisinthedirectionofthefieldatthatpoint.
Figure1.6showsthefieldlinesforpositiveandnegativecharges,thefieldsruninthe radialdirection(outwardsandinwardsrespectively).Itwillbenotedthatthefieldlines spreadoutaswemoveawayfromthesourcesandsothedensityoffieldlinesatanypoint isanindicationofthestrengthofthefieldatthatpoint.Alsoshownarethesurfacesof constantpotential(sphericalsurfacesaroundthechargethataredepictedasbrokenlines).
Figure1.7showsthefieldlinesforpositiveandnegativechargesofequalmagnitude thatareseparatedbyafinitedistance d .Thiscombinationisoftenknownasadipoleand isimportantinthedevelopmentofradiotheory.Atgreatdistancesfromthedipolethe effectsofthechargeswillalmostbalanceoutandsothefieldwillbemuchweakerthan
Fig.1.8 a)Geometryofaparallelplatecapacitorandb)fieldlinesinachargedcapacitor. thatofasinglecharge.Atgreatdistances,thefieldwillhavetheform
where p = Q(r+ r ) isknownasthedipolemomentwith r+ and r thepositionsof thepositiveandnegativechargesrespectively.
WenowreturntotheconfigurationofFigure1.2andnotethatthemachinecausesthe accumulationequalnumbersofopposite-signedcharges,positiveonthelowersphere andnegativeontheuppersphere.Thespheresessentiallystorechargeandareanexample ofanelectricaldeviceknownasa capacitor.Itwillbenotedthatthepotentialoneach spheremustbeconstant.Thispropertyfollowsfromthefactthatchargescanmovefreely onaconductingsphereandsonofurtherworkisneededtomovethemaroundonthe sphere.Itturnsoutthatthecharge Q onthelowersphereisproportionaltothepotential difference V = V+Q V Q betweenthespheres.Theconstantofproportionality C is knownasthe capacitance (Q = CV )andismeasuredinfarads(coulombspervolt). Spheresarenottheonlycapacitorsandanimportantformofcapacitorisknownasthe parallelplatecapacitor(seeFigure1.8a).Inthisdevicethechargeisaccumulatedon opposingfacesoftwoparallelplates.Thefieldbetweentheplatesismainlyconstant (magnitude E = Q/ A),exceptattheedges,whereitadjuststothezerofieldoutside thecapacitor.Iftheplatesaredistance d apartandhavesurfacearea A,thecapacitance willbe C = 0 A/d .Thisvaluecanbeenhancedbyinsertinganinsulatingmaterial betweentheplates.Thecapacitancewillnowgivenby C = A/d where isknown asthe permittivity oftheinsulator.Whenaninsulatorisadded(seeFigure1.9a),the moleculesbecomepolarised(electronsaredrawntowardsthepositiveplateandprotons towardsthenegativeplate).Thematerialwillthenconsistofacollectionofdipolesthat areorientatedalongtheoriginalfieldlineandthiscausesanadditionalfieldthatpartially counterstheoriginalfield.Thereducedfieldinsidethedielectricwillthenresultinan increasedcapacitance.Thecapacitorisanimportantcomponentinelectroniccircuits andisrepresentedbythesymbolshowninFigure1.9b.
Ifweconnectthetwosidesofacapacitorbyaconductor,electronswillflowfromthe negativesidetothepositivesideuntilallthechargehasbeenneutralised.Foraperfect conductor,thiswillhappeninstantaneously.Inreality,however,conductorsareimperfect andtherewillbesomeresistancetotheflowduetocollisionsonthemolecularscale. TheflowthroughanimperfectconductorisdescribedbyOhm’slaw,accordingtowhich
Fig.1.9 a)Parallelcapacitorwithdielectricandb)symbolforcapacitor.
Fig.1.10 Resistorandacapacitordrainedbyaresistor.
thepotentialdrop V acrosstheconductorisproportionaltothecurrent I throughthe conductor.Currentistherateatwhichchargeflowsinaconductorandismeasuredin amperes(1ampereis1coulombpersecond).Somewhatconfusingly,currenthasalways beentakentobeflowofpositivechargefromhighertolowerpotential(theopposite directiontotherealityofelectronflow)andsoistherateofdecreaseofcharge Q onthe capacitorplate(I =−dQ/dt inthelanguageofcalculus).Theconstantofproportionality inOhm’slawisknownastheresistance R (V = RI )andhasunitsofohms(1ohmis 1amppervolt).GeorgeOhmproposedhisfamouslawin1827anditisanimportant relationincircuittheory.Inthecaseofawireoflength L andcross-sectionalarea A,the resistanceisgivenby R = L /Aσ where σ isamaterialpropertyknownasitsconductivity. Animperfectconductorisknownasaresistorandisanimportantcomponentin electroniccircuits.Aresistorisalossydeviceanddissipatesenergyasheatatarate RI 2 (thisisknownas Ohmicloss).Figure1.10bshowsasimplecircuitconsistingofa capacitorandaresistorthatdissipatestheenergystoredinthecapacitor(Figure1.10a showsthesymbolusedtorepresenttheresistor).Whentheswitchisthrown,acurrent I willflowthroughtheresistorandthevoltagedropacrossthecapacitorwillbegivenby V = RI .Astheresistordrainsthecapacitor,thevoltageacrossthecapacitorwilldrop sincethechargewillbesteadilydepleted(seeFigure1.10c).Since Q = C V wewillhave I =−C dV /dt andhence V =−RC dV /dt .Thisisanordinarydifferentialequationthat hasthesolution V = V0 exp( t /RC ) where V0 istheinitialvoltagedifferencebetween thecapacitorplatesand t isthetimeafterswitchon.
Muchoftheearlydevelopmentofthescienceofelectricitywashinderedbythe needtousemachines,suchasthatshowninFigure1.2,togenerateelectriccharge. In1794,however,thisprocesswasrevolutionisedthroughtheinventionofthe battery byAlessandroVolta,adevicethatcreateschargethroughachemicalprocessrather thanamechanicalprocess.Figure1.11showsasingle-cellversionofVolta’sbattery
Fig.1.11 Volta’sbattery.
(Voltainfactmadeastackoftheseinordertoproducelargepotentialdifferences).It consistsofalayerofcopper(theanode),alayeroffeltthatissoakedinamixtureof waterandsulphuricacid(theelectrolyte)andalayerofzinc(thecathode).Withinthe electrolyte,thesulphuricacidwilldisassociateintoSO2 4 andH2+ ions.Atthecopper plateelectronsaredrawnintotheelectrolytetocombinewithhydrogenionsandform hydrogengas,hencecausinganaccumulationofpositivecharge.Meanwhile,atthezinc plate,thisiscounterbalancedbyzincionsdissolvingintotheelectrolyte,hencecausing anaccumulationofnegativecharge.Thechemistrycanbesummarisedas
Animportantconceptinelectromagnetictheory(andmanyotherfieldtheories)isthe conceptof flux .Consideraflatsurfacewitharea A andunitnormal n .If G isaconstant vectorfield,itwillhaveaflux n GA acrossthesurface(i.e.thenormalcomponent ofthefieldmultipliedbytheareaofthesurface).Agoodillustrationofthenotionof fluxcomesfromthestudyoffluidflow.Suchamediumisusuallydescribedinterms ofitsvelocityfield,avectorfieldthatgivesthemagnitudeanddirectionofthefluid velocityatagivenpoint.Thefluxisthenthetotalvolumeoffluidcrossingthesurface inaunittime.Forageneralsurfacesurface S withunitnormal n ,the flux through S isdefinedbytheintegraloverthesurfaceofthenormalcomponentofthevectorfield, i.e. S G(r ) · n dS .Thesurfaceintegralisacalculusconceptthatcanbeunderstoodby approximatingthesurfacebyasetofsmallflatsurfaceelementsoneachofwhich n and G canbeapproximatedbyconstantvalues.Ifthe i thelementhasarea Si ,we approximate n byaconstantvector ni and G byaconstantvector Gi .Thetotalflux throughSisthenapproximatedbythesumofthefluxes G(ri ) · ni Si throughthese smallerelements,i.e. totalfluxthrough S ≈ N i =1 Gi ni Si .(1.10)
Inthelimitofthissumastheareasofthesurfaceelementstendtozero,theabovesum thenbecomesthesurfaceintegral S G(r ) · n dS .
Fig.1.12 Fluxsurfaceintegral.
Fig.1.13 Gauss’law.
Animportantpropertyoftheelectricfieldisthatthetotalfluxthroughaclosedsurface S isproportionaltothechargecontainedwithinthatsurface.Thisisknownas Gauss’ Law which,inmathematicalterms,isgivenby S E (r ) · n dS = totalchargewithin S ,(1.11)
where S isanarbitraryclosedsurfaceinspaceand n isunitnormalonthissurface.Gauss’ lawisoneofthefundamentallawsofelectromagnetism.Asimpleexampleisgivenby asinglechargelocatedattheoriginandasurface S thatconsistsofasphereofradius a withcentreatthecharge.ThefieldisgivenbyEq.1.3andfromwhich E · n = Q/4π 0 a2 since n isaunitvectorintheradialdirection(i.e.thefielddirection).Since E n is constant,wesimplymultiplybytheareaofthesphere(4π a2 )togettheintegralover thesphere.Asaconsequence S E (r ) n dS = q / 0 ,whichisGauss’law. 1.2Magnetism
Atthetimeoftheirdiscoveryofelectrostaticattraction,theGreekswerealsoaware thatthemineralmagnetite(theoxideofironFe3 O4 )couldattractpiecesofnon-oxide iron.Further,thattheironitselfcouldbemagnetisedbystokingwiththemagnetite. TheChinesewerealsoawarethatmagnetite(alsoknownaslodestone)wasanaturally occurring magnet thatcouldattractiron.Indeed,theChinesealsodiscoveredtheeffect
ofamagnetorientatingitselfwithrespecttoEarth.Bythetwelfthcentury,boththe ChineseandEuropeanswereusingcompassesintheformoflodestonesfornavigation. However,ittookuntil1600fortheEarthitselftoberecognisedashavingthepropertyof amagnet.ThiswasrecognisedbyWilliamGilbertinhisbook’DeMagnete’,oneofthe firstworksonmagnetism.TherecognitionoftheEarth’smagneticpropertiesledtothe designationofthetwoendsofamagnetasNorthandSouth.However,unlikeelectric sourceswherepositiveandnegativechargecanhaveseparateexistence,thesourcesof magneticfieldsarealwaysfoundinNorth/Southpairs.Becauseofthis,thefluxofa magneticfieldthroughaclosedsurface S iszero,i.e.
where B isthemagneticfield(sometimesknownasthe magneticfluxdensity).
SincethemagneticpolesalwaysappearinNorthandSouthpairs,thebasicsourceof magnetismisthemagneticdipole.Thishasafield
where M isknownasthedipolemomentand μ0 isaconstantthatisknownasthe permeabilityoffreespace.ThedipolewillhavethefieldlinesshowninFigure1.14 (alsoshownisthefieldlinesofEarth’smagneticfieldforwhichnorthisat79◦ latitude). Ifthebasicmagneticsourceisthedipole,howdoweinterpretthedipolemoment?It turnsoutthat,ifwesuspendamagneticdipoleofmoment m infield B ,thedipolewill experienceatorque
Thisisamorecomplexbehaviourthantheinteractionofanelectricchargewithan electricfield.Inparticular,itinvolvesa vectorproduct ,definedby a × b = ab sin(θ) ˆ n where θ istheanglebetweenthevectorsand ˆ n isaunitvectorthatisperpendicularto magneticnorth
a)b)
Fig.1.14 Magneticfieldlines.
Vectorproduct.
Fig.1.16 Themagneticfieldofacurrent-carryingwire.
both a and b (directiondefinedbytheright-handscrewruleasshowninFigure1.15).An importantconsequenceof1.14isthatamagnet,freelysuspendedinthemagneticfield ofEarth,willrotateuntilitalignswithEarth’sfieldlines(i.e.untilthetorquebecomes zero),aneffectthatisusedinnavigationintheguiseofacompass.
In1820,HansChristianOersteddiscoveredthemagneticeffectofcurrent.By observingthedeflectionsofacompass,heshowedthatalongstraightwirecarrying asteadycurrent I causedamagneticfieldthathadcircularfieldlinescentredonthewire (seeFigure1.16a)andamagnitude B thatdependeduponthedistance r fromthewire
where I isthecurrentinthewire.Theunitforthemagneticfieldisusuallythetesla,a quantitythatisonenewtonperamperepermetre.Further,insuchunits,thepermeability hasthevalue4π × 10 7 .
FromtheworkofOersted,itbecameclearthatthemovingchargehadtheabilityto causeamagneticfield.Further,thatacurrentcarryingwirecouldexperiencetheforceof amagneticfield.AccordingtoOersted,awireelementoflength L carryingacurrent I willsufferaforce
where t isaunitvectorinthedirectionofthecurrent.
Fig.1.15
a)currentloopb)unmagnetised c)magnetised SN
Fig.1.17 Thecurrentloopandthecurrentloopmodelofmagnetism.
Thefactthatmovingchargecouldproduceamagneticfieldledtofurtherillumination oftheconceptofamagneticdipole.Around1820,theformulaofOerstedwasfurther generalisedtoallowforanarbitrarycircuit C bytheworkofJean-BaptisteBiotand FelixSavart.TheBiot–Savartformulais
Forpointsatalargedistancefromaplanarcurrentloop,thisexpressionreducesto(1.13) with M = I An where n isaunitvectorperpendiculartotheplaneoftheloopand A isthe areaoftheloop.Themagneticdipolecanthusbepicturedasaloopofcurrent.Infact, atamolecularlevel,wecaninterpretthisasorbitingelectronsorspinningcharge.All matterwillconsistofmanysuchdipoles,butinmostmatterthesewillbeinarandom configurationandhencehavenoneteffect.However,formaterialssuchasmagnetite, thesedipolesarealignedwitheachotherandhencethematerialwillexhibitmagnetic properties.Materialssuchasironcanbemagnetisedwhenexternalfieldsaligntheir dipolesandmaterialssuchassteelcanretainthismagnetism.
Currentcanberegardedasastreamofchargetravellingdownawireandtheabove considerationssuggestthattheforce F thatactsuponacharge q willbe
where v isthevelocityvectorofthecharge.Thisforceisoftenknownasthe Lorentzforce andisimportantforunderstandingtheinteractionofmatterwiththeelectromagnetic field.
AgenerallawconnectingmagneticfieldsandcurrentwasdiscoveredbyAndre-Marie Ampèrein1823.Considerasurface S throughwhichcurrentpassesandwhichisbounded byacurve C . Ampère’slaw,initsmathematicalform,thenstatesthat
where I isthetotalcurrentpassingthroughthesurfaceS.Ifweconsiderthecaseofa longstraightwirecarryingcurrent I ,wecouldtakethecurve C tobeacircleofradius a thatiscentredonthewire.Inthiscase,themagneticfield B willbeconstanton C
Fig.1.18 Themagneticfieldofalongsolenoid.
andwewillhave C B (r ) dr = 2πμ0 aB .Substitutingfrom(1.15),theright-handside becomes μ0 I ,i.e.wehaveAmpère’slaw.
Ampère’slawisausefulresultfordeterminingcomplexmagneticfields.Consider theexampleofaninfinitelylongsolenoid(agoodapproximationtoalongsolenoid, asinFigure1.18a).Bysymmetry,theonlydependenceofthemagneticfieldisthe radialdistancerfromtheaxis.Further,byanalogywiththelimitingcaseofaninfinite wire,thefieldlinesoutsidethewirewillbecircularandcentredonthesolenoidaxis, i.e.thefieldwillpointintherotationaldirection.WefirstapplyAmpère’slawona circularcurve C withradius r andcentredontheaxis(thecurve1inFigure1.18b).On curve C themagneticfieldwillbeconstantandso C B (r ) · dr = 2π r B ,where B isthe componentmagneticfieldthatistangentto C .Consequently,ifcurrent I flowsthrough thesolenoid,Ampère’slawwillimply B = μ0 I /2π r .Insidethesolenoid,Ampère’slaw willimplythattherotationalcomponentofthefieldiszeroandso,asaconsequence, wetakethefieldtobeparalleltothesolenoidaxis.IfweapplyAmpère’slawonthe rectangularcurveinFigure1.18(curve2),wefindthat C B (r ) dr = DB where B isthe magnitudeofthemagneticfieldparalleltotheaxis.Thecurrentthroughtheloopwillbe nDI where n isthenumberofturnsperunitlengthonthesolenoid.Asaconsequence, Ampère’slawwillimplythat B = μ0 nI insidethesolenoid.Uptonow,wehaveimplicitly assumedthattheradialcomponentofthemagneticfieldiszero,butwecanverifythis using(1.12).Wetakethesurface S tobeacylinderofradius r ,andlengthD,with thesameaxisasthesolenoid.Themagneticfluxthroughthecylinderendswillcancel, butthecontributionfromthecurvedsurfacewillbe2π rDB ,where B isnowtheradial componentofthemagneticfield.Equation1.12willthenimplythatthisradialcomponent iszero.
Whatemergesfromourconsiderationsisthatthevariousintegralresults,suchas Ampère’sandGauss’laws,constituteapowerfulandself-containeddescriptionof electromagnetism.Inordertocompletethisdescription,however,weneedtointroduce onefurtherintegrallawandthisisthesubjectofthenextsection.
1.3Electromagnetism
Wenowconsidertheconsequencesofthevariationoffieldswithtime.Thisbringsusto Faraday’slaw,oneofthekeydiscoveriesinthedevelopmentofelectromagnetictheory.
In1830,MichaelFaradaydiscoveredmagneticinductionwhenhenotedthat,bymoving aloopofwireinandoutofamagneticfield,hecouldcauseacurrenttoflowinthe loop.Thiswasanimportantdiscoveryas,hitherto,thebatteryandthechargedcapacitor hadbeentheonlymeansofdrivingacurrentthroughanelectricalcircuit.Somewhat confusingly,theeffectivepotentialofthisnewsortofgeneratorcametobeknownasthe electromotiveforce (orEMFforshort).FaradayconcludedthattheEMFinducedina circuitwasproportionaltotherateofchangeofmagneticfluxthroughthatcircuit,aresult thatisknownas Faraday’slaw..Forasurface S withboundingcurve C (seeFigure1.19), thefluxisgivenby = S
andthelawhasthemathematicalform
Figure1.20showstwodifferentwaysinwhichmagneticfluxcanvaryinacircuit.If weconsidertheloopinFigure1.20atoberotatingatangularspeed ω ,theflux through theloopwillbe = AB sin(ω t ) where A istheareaoftheloopand B isthemagneticfield (weassumetheplaneoftheloopisparalleltothefieldwhen t = 0).Asaconsequence, anEMF AB cos(ω t ) willbegeneratedandthiscausesan alternatingcurrent toflowin theload.InFigure1.20baconductingbarwithloadmovesoverarectangularcircuitat speed v andsocausesthetotalareaofthecircuittochangeatrate vd .Asaconsequence, themagneticfluxwillincreaseatarate vd B andso,byFaraday’slaw,anEMFof vdB willbegeneratedinthecircuit(B isamagneticfieldorthogonaltotheloop).Wecan viewthislastexamplefromtheviewpointoftheLorentzforce.Aunitcharge,located ontheconductingbar,willsufferaLorentzforce vB intheclockwisedirectiondueto
GeometryforFaraday’slaw.
Fig.1.20 Magneticinduction.
Fig.1.19
theimposedmotiontransversetothebar.Consequently,integratingalongthebar,we obtainanEMFof vd B .
Atime-varyingcurrentbringsustotheconceptofmutualimpedance.Considera solenoidwithawireloopwrappedaroundit.Ifwenowdrivethesolenoidbyaalternating current I1 (t ),therewillbeamagneticfield B (t ) = μ0 nI1 (t ) throughtheloopandhence aflux = Aμ0 nI1 (t ) where A istheareaoftheloop.AccordingtoFaraday’slaw,this willgenerateanEMFof
(1.21)
inthewire,where L21 = Aμ0 n isknownasthe mutualinductance ofthewireloopand solenoid.Ifthesolenoidhasafinitelength l with N1 turnsthen n = N1 /l andthemutual inductancewillbegivenby L12 = μ0 N1 A/l .Further,andiftheloophas N2 turns,the mutualinductancewillnowbegivenby L12 = μ0 N1 N2 A/l .Adevicewithmutually interactingwindingsisknownasa transformer andisrepresentedbythesymbolshown inFigure1.21b.Thelongersolenoidisoftenknownastheprimaryandtheloopwinding asthesecondary.Ifcurrentflowsthroughthesecondary,itisclearthesecondaryitself willcauseadditionalfluxandso
where L22 isknownastheselfinductanceofthesecondarywinding.Itisclearthatthe primarywillalsoexperienceselfinductanceandthattheEMFgeneratedintheprimary willtaketheform
whereitshouldbenotedthat L12 = L21 .Forthesolenoid,itisobviousthatitwill induceaflux = Aμ0 I1 (t )Aμ0 N1 /l initselfandsoitwillhaveaselfinductance L11 = μ0 N 2 1 A/l .Likewise,thesecondarywillhaveaselfinductance
l TheunitofinductanceisthehenryandisnamedafterJosephHenrywhodiscovered magneticinductionindependentlyofFaradayandataboutthesametime.Aninductance of1henrywillresultinanEMFof1voltinaclosedloopforachangeof1ampinthe currentoveraperiodof1second.
Fromtheaboveconsiderations,itwillbenotedthatasolenoidthatcarriesacurrent I willalwayshaveaself-inducedEMF,evenifthesecondarywindingdoesnotexist. Forthisreason,suchadeviceisknownasan inductor andisanimportantcomponent inradiotechnology(itisrepresentedbythesymbolshowninFigure1.21a).Forsucha componentwewillhavetheself-inducedEMF
where I isthecurrentintheinductorand L istheselfinductance(L = μ0 N 2 A/l fora solenoidwithlength l andcross-sectionalarea A).Wecanenhancetheinductanceof thesolenoidbywindingitaroundacoremadeupofferromagneticmaterial(ironand cobaltforexample).AscanbeseeninFigure1.22,thesolenoidfieldcausesthecurrent loopswithinthecoretoalignandthiscausesanincreaseinthemagneticfluxdensity. Consequently,theinductanceinthesolenoidwillnowbecome L = μN 2 A/l ,where μ is knownasthe permeability ofthecore.Permeabilityisamaterialpropertyofthesolenoid coreandhasunitsofhenriespermetre.
WenowconsiderthecircuitshowninFigure1.23b,consistingofaseriescapacitor, resistor,inductorandswitch.Beforetheswitchisclosedweassumethecapacitortobe chargedtoavoltage V0 .Aftertheswitchisclosed,however,thecapacitordischarges andthepotentialdifferenceacrossthecapacitorwilldecay.
Magneticfluxenhancedbyaferromagneticmaterial.
Fig.1.22
WecananalysethecircuitbynotingthefamousKirchhoffcircuitlaws:
1.Thetotalcurrentintoanyjunctionisequaltothetotalcurrentout.
2.Thetotalvoltagedroparoundanycircuitloopiszero.
(Sincetheybothhaveunitsofvolts,voltageisaterminologyoftenusedforboth potentialdifferenceandEMF.)Fromthefirstlawweobtainthatthesamecurrent I flowsinandoutofallcomponentsandfromthesecondlawweobtainthat
Intermsofthecharge Q ontheuppercapacitorplate,wehavetheordinarydifferential equation(ODE)
whichcanbesolvedtoyield
where ω0 = 1/√LC and ζ = R√C /L /2.Itwillbenotedthatwhenthecapacitoris dischargedthroughaninductor,itwill ring,i.e.therewillbeoscillationsinthecircuit atanangularfrequencyof 1 ζ 2 ω0 .Further,theoscillationswilldecayataratethat isdependentupontheamountofresistance R inthecircuit.
Thefrequency ω atwhichacircuitringsisofgreatimportanttousinradio.Consider theparallelcombinationofacapacitor C ,aninductor L andaloadresistance R with theinductordrivenbyharmonicvoltagesource VS cos(ω t ) (seeFigure1.24).Bythe Kirchhoffcurrentlaw,wehavethat
andfromtheKirchoffvoltagelaw
Asaconsequence
and,dividingby LC ,weobtain
Afterthesourceisswitchedon,thesolutionwillsettledowntothesteadystate
,(1.32)
where ω0 = 1/√LC ,i.e.thevoltageinthecircuitoscillatesattheforcingfrequency ω .It willbenoted,however,thatasfrequency ω approaches ω0 ,theamplitudeofoscillations willincrease,reachingapeakvalueof QVS where Q = R/ω L .Thecircuitissaidto resonate atfrequency ω0 and Q isameasureofthestrengthofthisresonance.
1.4Maxwell’sEquations
Untilabout1860,Eqs.(1.11),(1.12),(1.19)and(1.20)werepresumedtocorrectlyreflect thecontentofelectromagnetictheory.Whilsttheseequationsimplythattime-varying magneticfluxwillcauseanelectricfield,theydonotimplythattime-varyingelectricflux willcauseamagneticfield.Around1860thephysicistJamesClerkMaxwellbecame convincedthattime-varyingelectricfluxshouldcauseamagneticfield.Indeed,thereare goodreasonsforbelievingthatAmpère’slawneedssomeformofmodification.Consider Ampère’slawinform
Weconsiderthecaseoftwochargedspheresthataremadetodischargebyconnecting themthroughaconductingwirethat,asaresult,carriesacurrent I .Referringto Figure1.25,ifweapplyEq.(1.33)usingsurface S1 weobtainthat C1 B (r ) · dr = μ0 I andusingsurface S2 weobtainthat C2 B (r ) dr = 0.Ifweaddthesetworesults,the pathintegralswillcancel(thecurvesareidenticalbuttheintegralsareevaluatedin oppositedirections)andthiswillimplythat I = 0.Thisclearlyposesaproblemfor electromagnetictheory.Maxwell’ssolutionwastoaddanothertermtoEq.(1.33),which hecalledthe displacementcurrent .Theresultingequationis
Another random document with no related content on Scribd:
chromosphere
chronic Chronicle chronicled chronicler chroniclers chronological chronologically chronology
Chu Chua
Chuang
Chugoku
chun
CHUNG
Chungking church churches
Churchill churchwardens churchyard
chwang
Chéng
Chêkiang
Chêng
Chêngtu
Chün cider
Cienfuegos cigarette cigarettes
Cincinnati Cindad
cipher
Cipriano circa Circle
circles circuit circuitous circuits circular circulars circulate circulated circulating circulation Circum circumference circumnavigated circumspect circumspection circumstance circumstances circumstantial Cisleithania Cisleithanian Cisneros cistern citadel cited cites Cities Citizen citizens citizenship city civic civil civilian civilians Civilis civilisation civilised
civilising Civilization civilizations civilize
CIVILIZED
civilly clad claim claimant claimants
Claimed claiming claims clair
Clam clamor clamored clamoring clamorous clamoured clan Clancy clandestinely clans clansmen clap Clara Clarence clarion Clark Clarke
Clary clash clashed clashing clasp Class
classed classes classical classically
classics classification classified classify
Classifying
Claude Clause
clauses clawed CLAY
claypits
Clayton clean cleaned cleaners cleaning cleanliness cleanse cleansing clear clearance cleared clearer clearest clearing Clearly clearness
cleavage cleave clemency
Clemens clement
Clements
clergy
clergyman
clergymen
Clerical
Clericalism
clericals
clerk clerks
Clery cleveite
CLEVELAND
clever client climate climatic
Climatically climax climb climbed clime
Clinch cling clinging clique cliques clock clog clos close
Closed closely closer closes closest closet closing clot
cloth
clothe
clothed clothes
Clothing clouded clouds
cloven Club Clubs clue clumsy clung
clustered clusters
Cnossos
Co coach coadjutor
Coahnila
COAL
coalfield coalfields coaling coalition coals
COAMO coarse Coast
Coastguard coasting
Coasts coastwise coat
coated coats
Cobham
Coblenz
Cobra
Coburg coccidium
Cockburn
cocked cockpit
Cockran
cocoa cocoanuts
cod
Code codes
codfish
codification
Codlin
Codman coeducational
coerce coerced coercion
coercive coeval
coffee
coffer coffers
coffin
coffins cognisance cognisant cognizable cognizance coherence coherent coherer
cohesion coil
coils
Coin coinage coincided coincident
Coincidentally coinciding coined coins
Coit
Coke cold coldly Coleman
COLENSO
Coler
Colesberg Colima collapse collapsed collar collars collate collateral colleague colleagues collect collected collecting collection Collections collective collectively collectivist collector collectors collects
COLLEGE COLLEGES
collegiate collier
colliers
Collins Collis collision
collisions
Colloquially
Cologan
Colomb Colombari
COLOMBIA
Colombian Colombo
Colon Colonel
colonelcy colonels
colonial colonials
Colonies colonisation
colonists colonization colonize
colonizing colonnade
Colony color
COLORADO
Colorados colored colors
colossal colossi
Colossus colour coloured colourless colours
cols
Colt Columbia Columbian COLUMBUS column columns
Colón Com Comanches combat combatant combatants combated combating combative combats combed combination combinations Combine combined combines combing combining combustible combustion Come comer comers comes comfort
comfortable comfortably comforting comforts coming comity
Command commandant
Commandants commanded commandeer commandeered
Commandeering Commander commanders
Commanding COMMANDO commandos commands comme commemorate commemorated commemorates commence commenced commencement commences
Commencing commend commendable commendation commended commending commensurate comment commented
Commenting
comments
COMMERCE
Commercial
Commerciale commercialism
Commercially Commerford commettant comminuted commissariat commissaries
Commissary Commission
COMMISSIONED commissioner commissioners
Commissions commit committal committed committee committees committing commodities commodity
Commodore Common Commoners commonly Commons commonwealth commonwealths commotion communal communes communicate communicated
communicates communicating communication Communications communicative communion communions communique communities community commutation commuted comp compact compacted compactly Compagnie companies companion companions company comparable comparative comparatively Compare Compared compares comparing Comparison comparisons compartments compass compassed Compassionate compatible compatriots compel
Compelled compelling compels compendium compensate compensates compensating COMPENSATION compensations
Compensatory compete competed competence competency competent competing Competition competitive competitively competitor competitors compilation compile compiled compilers complacency complacent complain complainant complained complaining complaint Complaints complement COMPLETE completed completely
completeness completes completing completion complex complexities complexity compliance complicate complicated complication complications complicity complied complies complimenting Compliments comply complying component comport comports compos compose composed composer composing composite composition compound compounded compounds comprehend comprehended comprehending comprehension comprehensive
compressed compression compressor comprise comprised comprises comprising Compromis compromise compromised compromises compromising Comptroller compulsion
COMPULSORY compunction computation computed comrade comrades Comte Comus Concas conceal concealed concealing concealment conceals concede conceded concedes conceding conceit conceivable conceivably conceive conceived
Conceiving concentrate concentrated concentrating concentration concentrations concentric
Concepcion conception
Conceptions concern concerned
Concerning concerns
Concert concerted concerts
Concession concessionaire concessionary concessions conciliate conciliated conciliating Conciliation conciliators conciliatory concisely conclude concluded concludes concluding Conclusion conclusions conclusive conclusively concocted
concomitant Concord Concordat concourse concrete concubine concupiscence concur concurred concurrence concurrent concurrently concurring concurs condamnant condemn condemnable condemnation condemnatory condemned condemning condemns condensation condenser condensers condescension condign condition conditional conditionally conditioned conditions
CONDOMINIUM conduce conduces conducive conduct
conducted conducting conductive conductivity conductor conductors conducts conduits
Condé confederacies
CONFEDERATE
Confederated Confederation confer Conference conferences conferred conferring confers confess confessed confessedly confesses confessing
Confession confessions confessor confessors confide confided
Confidence
Confident confidential confidentially confidently
Confiding configuration
confine confined confinement confines confining confirm Confirmation confirmatory confirmed confirming confirms confiscated conflagration conflict
Conflicting conflicts confluence conform conformable conformably conformation conformed conformer conforming conformity conformément confounded confounding confront confronted confronts Confucianism Confucius confuse confused confusing confusion
