《物理双语教学课件》Chapter 21 Induction and Inductance Maxwells Equations 自感互感 麦克斯韦方程文档格式.docx

《物理双语教学课件》Chapter 21 Induction and Inductance Maxwells Equations 自感互感 麦克斯韦方程文档格式.docx

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3.Magneticflux:

（1）SupposealoopenclosinganareaAisplacedinamagneticfield.Thenthemagneticfluxthroughtheloopis

where

isavectorofmagnitudedAthatisperpendiculartoadifferentialarea

.

（2）TheSIunitformagneticfluxisthetesla-squaremeter,whichiscalledtheweber（abbreviatedWb）.

4.WecanstateFaraday’slawinamorequantitativeandusefulway:

themagnitudeoftheemfinducedinaconductingloopisequaltotherateatwhichthemagneticfluxthroughthatloopchangeswithtime.Itmeans

withtheminussignindicatingthatopposition.

5.

SoonafterFaradayproposedhislawofinduction,HeinrichFriedrichLenzdevisedarule–nowknownasLenz’slaw–fordeterminingthedirectionofaninducedcurrentinaloop:

aninducedcurrenthasadirectionsuchthatthemagneticfieldduetothecurrentopposesthechangeinthemagneticfieldthatinducedthecurrent.Furthermore,thedirectionofaninducedemfisthatoftheinducedcurrent.

6.Seeabovefigures.

7.ElectricGuitars:

21.2InductionandEnergyTransfers

1.

Seerightfigure.

（1）Themagnitudeoftheemfis

.

（2）Themagnitudeoftheinducedcurrentis

（3）Themagnitudeofthemagneticforceontheloopis

（4）Therateatwhichyoudoworkontheloopasyoupullitfromthemagneticfieldis

（5）Therateatwhichthermalenergyappearsintheloopasyoupullitalongatconstantspeedis

（6）

Thustheworkthatyoudoinpullingtheloopthroughthemagneticfieldappearsasthermalenergyintheloop.

2.Eddycurrents:

Seerightfigure.

21.3InducedElectricFields

Letusplaceacopperringofradiusrinauniformexternalmagneticfield,asinthefigure.Supposethatweincreasethestrengthofthisfieldatasteadyrate.Themagneticfluxthroughtheringwillthenchangeatasteadyrate,andbyFaraday’slaw,aninducedemfandthusaninducedcurrentwillappearinthering.FromLenz’slawwecandeducethatthedirectionoftheinducedcurrentiscounterclockwiseinabovefigure.

2.Ifthereisacurrentinthecopperring,anelectricfieldmustbepresentalongthering;

anelectricfieldisneededtodotheworkofmovingtheconductionelectrons.Moreover,thefieldmusthavebeenproducedbychangingmagneticflux.Thisinducedelectricfieldisjustasrealasanelectricfieldproducedbystaticcharges.SoweareledtoausefulandinformativerestatementofFaraday’slawofinduction:

Achangingmagneticfieldproducesanelectricfield.

3.Thestrikingfeatureofthisstatementisthattheelectricfieldisinducedevenifthereisnocopperring.Tofixthisideas,considerabovefigure（b）,inwhichthecopperringhasbeenreplacedbyahypotheticalcircularpathofradiusr.Theelectricfieldinducedatvariouspointsaroundthecircularpathmustbetangenttothecircle,asthefigure（b）shows.Hencethecircularpathisalsoanelectricfieldline.Thisisnothingspecialaboutthecircleofradiusr.Sotheelectricfieldlinesproducedbythechangingmagneticfieldmustbeasetofconcentriccirclesasabovefigure（c）.

4.Aslongasthemagneticfieldischangingwithtime,theelectricfieldrepresentedbythecircularfieldlinesinfigure（c）willbepresent.Ifthemagneticfieldremainsconstantwithtime,therewillbenoinducedelectricfieldandthusnoelectricfieldlines.

5.FromFaraday’slaw,wehavetheelectromotiveforceis

21.4InductorsandInductance

1.Weshallconsideralongsolenoid,ashortlengthnearthemiddleofalongsolenoid,asourbasictypeofinductor.Aninductorcanbeusedtoproduceadesiredmagneticfield.

2.Ifweestablishacurrentiinthewindingofaninductor,thecurrentproducesamagneticfluxthroughthecentralregionoftheinductor.Theinductanceoftheinductoristhen

inwhichNisthenumberofturns.TheSIunitofinductanceisthetesla-squaremeterperampere.Wecallthisthehenry（H）,afterAmericanphysicistJosephHenry.

3.Inductanceofasolenoid:

.Sotheinductanceperunitlengthforalongsolenoidnearitscenteris

4.Self-induction:

（1）aninducedemf

appearsinanycoilinwhichthecurrentischanging.Thisprocess,asshowninfigure

iscalledself-induction,andtheemfthatappearsiscalledaself-inducedemf.

（2）Foranyinductor,wehave

.Thereforetheinducedemfis

.（3）Thedirectionofaself-inducedemfcanbefoundfromLenz’slaw,asshowninfigure.

5.Mutualinduction:

（1）wewillreturntothecaseoftwointeractingcoils.Ifthecurrentichangeswithtimeinonecoil,anemfwillappearinthesecondcoil.Wecallthisprocessmutualinduction,tosuggestthemutualinteractionofthetwocoilsandtodistinguishitfromself-induction,inwhichonlyonecoilisinvolved.

（2）Themutualinductancecanbedefinedas

.Theemfappearingincoil2duetothechangingcurrentincoil1is

.Similarly,Theemfappearingincoil1duetothechangingcurrentincoil2is

.（3）Itcanbeprovedthat

21.5EnergyStoredinaMagneticFields

Weconsideragainthefigure.Wehavetheequation

.Iftheresistanceiszero,theworkdonebythebatterywillbestoredintothemagneticfield.Sowehavethemagneticenergyis

2.Energydensityofamagneticfield:

（1）Consideralengthlnearthemiddleofalongsolenoidofcross-sectionalareaA;

thevolumeassociatedwiththislengthisAl.Theenergyperunitvolumeofthefieldis

21.6InducedMagneticFields

1.Weknowthatachangingmagneticfluxinducesanelectricfield,andweendedupwithFaraday’slawofinductionintheform

.Here

istheelectricfieldinducedalongaclosedloopbythechangingmagneticfluxthroughthatloop.

2.Becausesymmetryisoftensopowerfulinphysics,weshouldbetemptedtoaskwhetherinductioncanoccurintheoppositesense.Thatis,canachangingelectricfluxinduceamagneticfield?

Theansweristhatitcan;

furthermore,theequationgoverningtheinductionofamagneticfieldisalmostsymmetricwithaboveequation.WeoftencallitMaxwell’slawofinductionafterJamesClerkMaxwell,andwewriteitas

.Thecircleontheintegralsignindicatesthattheintegralistakenaroundaclosedloop.

3.Wenowconsiderthechargingofaparallel-platecapacitorwithcircularplates,asshowninfigure.Weassumethatthechargeonthecapacitorisbeingincreasedatasteadyratebyaconstantcurrentiintheconnectingwires.Thenthe

magnitudeoftheelectricfieldbetweentheplatesmustalsobeincreasingatasteadyrate.Theexperimentsprovethatwhiletheelectricfieldischanging,magneticfieldsareinducedbetweentheplates,bothinsideandoutsidethegap.Whentheelectricfieldstopschanging,theseinducedmagneticfielddisappear.

4.CombiningtheAmpere’slawtoMaxwell’slaw,wehaveAmpere-Maxwelllawtobeas:

21.7DisplacementCurrentandMaxwell’sEquations

1.Historically,theportion

intheridesideofAmpere-Maxwelllawhasbeentreatedasbeingafictitiouscurrentcalledthedisplacementcurrent:

.SoAmpere-Maxwelllawcanberewrittenas

2.

Letusagainfocusonachargingcapacitorwithcircularplates,asinfigure（a）.Therealcurrentis

.Ontheotherhand,themagnitudeofdisplacementcurrentbetweentheplatesofthecapacitoris

.Therealcurrentchargingthecapacitorandthefictitiousdisplacementcurrentbetweentheplateshavethesamemagnitude.Thus,wecanconsiderthefictitiousdisplacementcurrenttobesimplyacontinuationoftherealcurrentfromoneplate,acrossthecapacitorgap,totheotherplate.

3.Theinducedmagneticfieldbetweentheplates.WecanusetheAmpere-Maxwelllawtofindtheinducedmagneticfieldbetweentheplates.Itis

insideacircularcapacitorand

outsideacircularcapacitor.

4.Maxwell’sEquations:

TabledisplaysMaxwell’sequations.Thearethebasisforthefunctioningofsuchelectromagneticdevicesaselectricmotor,cyclotrons,

televisiontransmittersandreceivers,telephones,faxmachines,radar,andmicrowaveovens.