11.TheMagneticDipole2.MagnetizationandEquivalentCurrentDensitiesReview23.MagneticFieldIntensityandRelativePermeabilityDifferentialformIntegralformPostulatesofMagnetostaticsinmagneticmaterial3MaintopicSteadyMagneticFields1.BoundaryConditionsforMagnetostaicFields2.InductancesandInductors3.MagneticEnergy41.BoundaryConditionsforMagnetostaicFields12B2H1B1H2an2Js5E2E1

2

1atwhacdban2hS

2

1an2D1D2s6(a)The

normal

componentsofthemagneticfluxdensityare

continuousacrossaninterface.For

linearisotropic

media,wehave(b)ThetangentialcomponentoftheHfieldisdiscontinuousacrossaninterfacewhereafreesurfacecurrentexistsThetangentialcomponentofHiscontinuous(連續)acrosstheboundaryofalmostallphysicalmedia;itisdiscontinuous(不連續)onlywhenaninterfacewithanidealperfectconductor(理想導體)orasuperconductor(超導體)isassumed.7Example1.Aloopmagneticcorewithagapiscloselywoundbyacoilwith

N

turns,asshowninthefigure.Whenthecoilcarriesacurrent

I,andtheleakage

magneticfluxoutsidethecoilis

neglected,findthemagneticfluxdensityandthemagneticfieldintensityinthecoreandthegap.Solution:Sincetheleakagemagneticfluxisneglected,thedirectionofthemagneticfluxdensityisaroundthecircle,anditisperpendiculartotheendfacesofthegap.Fromtheboundarycondition,weknowthatthemagneticfluxdensity

Bg

inthegapisequalto

Bf

inthecore,i.e.

8

Since

r0>>a

,themagneticfieldinthecorecanbeconsideredtobeuniform.UsingAmpere’scircuitallawinmedia,andtakingthecircleofradius

r0

astheintegralpath,thenwehave

Considering,wehave

ThenInthegapInthecore9

在線性媒質中，單個閉合回路電流產生的磁感應強度與回路電流I成正比，因此穿過回路的磁通也與回路電流

I成正比。式中L

稱為回路的電感，單位為H

（亨利）。由該定義可見，電感又可理解為與單位電流交鏈的磁通鏈。

單個回路的電感僅與回路的形狀及尺寸有關，與回路中電流無關。

與回路電流I交鏈的磁通稱為回路電流I的磁通鏈，以

表示，令與I

的比值為L，即應注意，磁通鏈與磁通不同，磁通鏈是指與某電流交鏈的磁通。2.InductancesandInductors10

若交鏈N次，則磁通鏈增加N倍；若部分交鏈，則必須給予適當的折扣。因此，與N匝回路電流I交鏈的磁通鏈為

=N。那么，由N

匝回路組成的線圈的電感為

與回路電流I1交鏈的磁通鏈是由兩部分磁通形成的，其一是I1本身產生的磁通形成的磁通鏈11

，另一是電流I2在回路l1中產生的磁通形成的磁通鏈12

。dl10zyxdl2l2l1I2I1r2-r1r2r1那么，與電流l1交鏈的磁通鏈1為11Mutualflux12(互磁通)Faraday’slawofelectromagneticinductionBiot-Savartlawwheretheproportionalityconstant

L12

iscalledthe

mutualinductance(互感)betweenloopsC1andC2,withtheunithenry(H).IncaseC2

hasN2turns,thefluxlinkage

12(磁鏈數)dueto

1212Themutualinductance(互感)betweentwocircuitsisthenthemagneticfluxlinkage(磁鏈數)withonecircuitperunitcurrent(單位電流)intheother.Itapplyonlytolinearmedia(線性媒質).AmoregeneraldefinitionforL12

isSomeofthemagneticfluxproducedbyI1linksonlywithC1itself,andnotwithC2.thetotalfluxlinkage(磁鏈數)

withC1

causedbyI1isNeumannformula13Theself-inductance(自感)ofloopC1

isdefinedasthemagneticfluxlinkage

(磁鏈數)

perunitcurrent(單位電流)intheloopitself.Thatis,foralinearmedium(線性媒質).IngeneraldefinitionforL11

isTheself-inductance(自感)ofalooporcircuitdependsonthegeometricalshape(幾何形狀)andthephysicalarrangement(實際排列)oftheconductorconstitutingthelooporcircuit,aswellasonthepermeabilityofthemedium(媒質的導磁率).Withalinearmedium,self-inductancedoesnotdependonthecurrentinthelooporcircuit.14Aconductorarrangedinanappropriatesshape(suchasaconductingwirewoundasacoil)tosupplyacertainamountofself-inductance(自感)

iscalledaninductor(電感器).Justasacapacitorcanstoreelectricenergy,andinductorcanstoragemagneticenergy.Theprocedurefordeterminingtheself-inductanceofaninductorisasfollows:1.Chooseanappropriatecoordinatesystemforthegivengeometry.2.AssumeacurrentI

intheconductingwire.3.FindBfromIbyAmpere’scircuitallaw

ifsymmetryexists;ifnot,Biot-Savartlawmustbeused.4.Findthefluxlinkingwitheachturn,,fromBbyintegration:whereSistheareaoverwhichBexistsandlinkswiththeassumedcurrent.5.Findthefluxlinkage

bymultiplyingbythenumberofturns.

6.FindLbytakingtheratioL=/I.15EXAMPLE6-14P26916Example.

Calculatethemutualinductance(互感)betweenaninfinitelylongstraightlineandarectangularcoil.Thelineandthecoilareatthesameplane,andinvacuum.abdrrD0I1I2zS2Solution:Selectcylindricalcoordinatesystem,andlettheinfinitelylongstraightlinetobeatthe

z-axis.Themagneticfluxdensityproducedbycurrent

I1

isthen

Themagneticfluxlinkage

12

withcurrent

I2

bycurrent

I1

is17Wehave

Iftheflowingdirectionofthecurrent

I2

iscounter

clockwise,thenthe

B1

and

dS

areopposite,and

L12<0.abdrrD0I1I2zS2

Iftheflowingdirectionofthecurrent

I2

isclockwise,dS

and

B1

havethesamedirection.Then18例題:求無限長導線與三角形回路之間的互感I1rdrzx0193.MagneticEnergy

If

animpressedsource(外源)isappliedtoacircuit(回路),acurrentwillbegeneratedinthecircuit.Intheprocessofestablishingthecurrent,the

reactionmagneticflux(感應磁通)inthecircuitwillresist(阻止)theincrementofthecurrent.

Assumethecurrentisincreased

veryslowly(非常慢)

sothat

radiationloss

canbeneglected,

allenergy

providedbytheimpressedsourcewillbestored(儲存)inthe

magneticfield

aroundthecircuit.

Basedontheworkdonebythe

impressedsource,theenergystoredinthemagneticfieldcanbecalculated.

Inordertoovercomethebackelectromotiveforceduetothereactionmagneticfluxandtomaintainthecurrent,

theimpressedsourcehastodowork(做功).20Considerasingleclosedloop(閉合回路)withaself-inductanceL1inwhichthecurrentisinitiallyzerotoI1.Fromphysicsweknowthatanemf(電動勢)willbeinducedintheloopthatopposesthecurrentchange.Anamountofworkmustbedonetoovercomethisinducedemf(感應電動勢).Letv1=L1di1/dtbethevoltage(電壓)acrosstheinductance.TheworkrequiredisSinceL1=1/I1forlinearmedia(線性媒質),theequqtioncanbewrittenwhichisstoredasmagneticenergy(磁能).NowconsidertwoclosedloopsC1andC2carryingcurrentsi1andi2,respectively.ThetotalamountofworkdoneinrasingthecurrentsinloopsC1andC2fromzerotoI1andI2,respcetively,is21whichisstoredasmagneticenergy.WhereW21isGeneralizingthisresulttoasystemofNloops(N個回路)carryingcurrentsI1,I2,…In,weobtainWhichistheenergystoredinthemagneticfield.ForacurrentIflowinginasingleinductor(單個電感器)withinductanceL,thestoredmagneticenergyis22ItisoftendesirabletoexpressthemagneticenergyintermsoffieldquantitiesBandH.WehaveNotingthatH=B/,wecan