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March 10, 2016 CH 32 Maxwell’s Equations and Magnetism of Matter I.
Maxwell’sEquations
A.
Maxwell’sFirstEquation
1.
InChapter23youlearned:
2.
Theintegraloftheoutgoingelectricfieldoveranarea
enclosingavolumeequalsthetotalchargeinside,inappropriate
units.Thiscanberewrittenas:
a)
(Gauss’LawforElectricFields)
3.
ThisisMaxwell’sfirstequation.Itrepresentscompletelycoveringthe
surfacewithalargenumberoftinypatcheshavingareas.(Thelittleareasare
smallenoughtoberegardedasflat,thevectormagnitudedAisjustthevalue
ofthearea,thedirectionofthevectorisperpendiculartotheareaelement,
pointingoutwardsawayfromtheenclosedvolume.)Hencethedotproduct
withtheelectricfieldselectsthecomponentofthatfieldpointing
perpendicularlyoutwards(itwouldcountnegativelyifthefieldwerepointing
inwards)—thisistheonlycomponentofthefieldthatcontributestoactual
electricfluxacrossthesurface.(Rememberfluxjustmeansflow—thepicture
oftheelectricfieldinthiscontextislikeafluidflowingoutfromthecharges,
thefieldvectorrepresentingthedirectionandvelocityoftheflowingfluid.)
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B.
March 10, 2016 Maxwell’sSecondEquation
1.
ThesecondMaxwellequationistheanalogousoneforthemagnetic
field,whichhasnosourcesorsinks(nomagneticmonopoles,thefieldlines
justflowaroundinclosedcurves).Againthinkingoftheforcelinesas
representingakindoffluidflow,theso‐called"magneticflux",weseethatfor
aclosedsurface,asmuchmagneticfluxflowsintothesurfaceasflowsout—
sincetherearenosources.Thiscanperhapsbevisualizedmostclearlyby
takingagroupofneighboringlinesofforceformingaslendertube—the
"fluid"insidethistubeflowsroundandround,soasthetubegoesintothe
closedsurfacethencomesoutagain(maybemorethanonce)itiseasytosee
thatwhatflowsintotheclosedsurfaceatoneplaceflowsoutatanother.
Thereforethenetfluxoutoftheenclosedvolumeiszero,Maxwell’ssecond
equation:
a)
(Gauss’LawforMagneticFields)
b)
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c)
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d)
Somegrandunifiedtheoriespredicttheexistenceof
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e)
The___________________________theoryofmagneticchargestartedwitha
paperbythephysicistPaulA.M.Diracin1931.Inthispaper,Diracshowedthat
ifanymagneticmonopolesexistintheuniverse,thenallelectricchargeinthe
universemustbe___________________________________.Theelectricchargeis,infact,
quantized,whichsuggests(butdoesnotnecessarilyprove)thatmonopoles
exist.
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March 10, 2016 2.
Gauss’lawformagneticfieldsholdsforstructureseveniftheGaussian
surfacedoesnotenclosetheentirestructure.GaussiansurfaceIInearthebar
magnetofFig.32‐4enclosesnopoles,andwecaneasilyconcludethatthenet
magneticfluxthroughitiszero.ForGaussiansurfaceI,itmayseemtoenclose
onlythenorthpoleofthemagnetbecauseitenclosesthelabelNandnotthe
labelS.However,asouthpolemustbeassociatedwiththelowerboundaryof
thesurfacebecausemagneticfieldlinesenterthesurfacethere.Thus,
GaussiansurfaceIenclosesamagneticdipole,andthenetfluxthroughthe
surfaceiszero.
C.
Maxwell’sThirdEquation
1.
Faraday’slawofInductionfromChapter30
a)
b)
Itrelatestheinducedelectricfieldtothechangingmagneticflux.
c)
Maxwelltheorizedthatthiscanworkintheopposite;thata
changingelectricalfieldcaninduceamagneticfield.
D.
Maxwell’sFourthEquation
1.
InducedMagneticFields
Maxwelldiscoveredthatthiscanworkintheoppositesense…i.e.a
a)
changingelectricalfluxcaninduceamagneticfield.
(1)
(2)
HereBisthemagneticfieldinducedalongaclosedloopby
thechangingelectricfluxEintheregionencircledbythatloop.
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March 10, 2016 (3)
Noticethetwoequations(32‐2and32‐3)aresimilar.Sign
differenceisjustexampleofLenz’slaw.Theotherdifference
_______________________isSIUnitconversion.
(4)
Fig.32‐5(a)Acircularparallel‐platecapacitor,showninsideview,isbeing
chargedbyaconstantcurrenti.(b)Aviewfromwithinthecapacitor,looking
towardtheplateattherightin(a).Theelectricfieldisuniform,isdirectedinto
thepage(towardtheplate),andgrowsinmagnitudeasthechargeonthe
capacitorincreases.Themagneticfieldinducedbythischangingelectricfieldis
shownatfourpointsonacirclewitharadiusrlessthantheplateradiusR.
2.
BuildingAmpere‐MaxwellLaw
a)
RememberAmpere’sLawfromCH29
(1)
(2)
Herei isthecurrentencircledbytheclosedloop
enc
b)
MaxwelldiscoveredthattheMagneticField producedby
meansotherthanamagneticmaterial(thatis,byacurrentandbya
changingelectricfield)givethefieldinexactlythesameform.Thushe
combinedhislawofinductionandAmpere’slawtogethertomakea
universalequationisnotlimitedtosituationswherethereisnochange
inelectricflux.
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March 10, 2016 c)
d)
Notice:Whenthereisacurrentbutnochangeinelectricflux
(suchaswithawirecarryingaconstantcurrent),thefirsttermonthe
rightsideofthesecondequationiszero,andsoitreducestothefirst
equation,Ampere’slaw.
e)
f)
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E.
DisplacementCurrent
1.
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March 10, 2016 2.
Comparingthelasttwotermsontherightsideoftheaboveequation
showsthattheterm__________________________________________musthavethe
dimensionofacurrent.Thisproductisusuallytreatedasbeingafictitious
currentcalledthedisplacementcurrenti .Thereforewecanrewriteas:
d
(Eq32‐11), inwhichi
d,enc
isthedisplacementcurrentthatisencircledbythe
integrationloop______________________________(displacementcurrent)
3.
Thechargeqontheplatesofaparallelplatecapacitoratanytimeis
relatedtothemagnitudeEofthefieldbetweentheplatesatthattime
by_______________________________________________________inwhichAistheplatearea.
a)
4.
Theassociatedmagneticfieldare:
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March 10, 2016 F.
Maxwell’sEquations
G.
ExampleProblems:
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March 10, 2016 H.
AdditionalSampleProblems
1.
Figurebelowshowsaclosedsurface.Alongtheflattopface,whichhasa
radiusof2.0cm,aperpendicularmagneticfield ofmagnitude0.30Tisdirected
outward.Alongtheflatbottomface,amagneticfluxof0.70mWbisdirected
outward.Whatarethe(a)magnitudeand(b)direction(inwardoroutward)ofthe
magneticfluxthroughthecurvedpartofthesurface?
2.
AparallelplatecapacitorwithcircularplatesofradiusR=55mmis
beingcharged.EvaluatethefieldmagnitudeBforr=11mmand
dE/dt=1.50x1012v/(ms).
a)
Whatifr>R,howdoestheequationforBchange?
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