电磁场与电磁波第15讲边界条件电感磁能

FieldandWaveElectromagnetic电磁场与电磁波,2015.11.10,2,1.TheMagneticDipole,2.MagnetizationandEquivalentCurrentDensities,Review,3,3.MagneticFieldIntensityandRelativePermeability,4,Maintopic静磁场,1.静磁场的边界条件,2.电感和电感器,3.静磁场能量,5,1.静磁场的边界条件,6,7,(a)在分界面处B的法向分量是连续的.,Forlinearisotropicmedia,wehave,(b)磁场H的切向分量在跨越存在面电流的分界面时,是不连续的.,在跨越几乎所有物理媒质的边界时,H的切向分量都是连续的;只有当分界面为理想导体或超导体时,它才不连续.,8,Example1.AloopmagneticcorewithagapiscloselywoundbyacoilwithNturns,asshowninthefigure.WhenthecoilcarriesacurrentI,andtheleakagemagneticfluxoutsidethecoilisneglected,findthemagneticfluxdensityandthemagneticfieldintensityinthecoreandthegap.,Solution:Sincetheleakagemagneticfluxisneglected,thedirectionofthemagneticfluxdensityisaroundthecircle,anditisperpendiculartotheendfacesofthegap.Fromtheboundarycondition,weknowthatthemagneticfluxdensityBginthegapisequaltoBfinthecore,i.e.,9,Sincer0a,themagneticfieldinthecorecanbeconsideredtobeuniform.UsingAmperescircuitallawinmedia,andtakingthecircleofradiusr0astheintegralpath,thenwehave,Considering,wehave,Inthecore,10,2.电感和电感器,互磁通12(Mutualflux),Faradayslawofelectromagneticinduction,Biot-Savartlaw,其中,比例系数L12称为回路C1和C2之间的互感,withtheunithenry(H).IncaseC2hasN2turns,thefluxlinkage12(磁链数)dueto12,11,由上式,两个电路之间的互感是一电路通以单位电流时,另一电路所交链的磁通链.这个关系只适用于线性媒质.,L12更一般的定义为:,I1所产生的某些磁通量只与回路C1自身交链,而不与C2交链.thetotalfluxlinkage(磁链数)withC1causedbyI1is,Neumannformula,12,回路C1的自感定义为在回路本身通以单位电流时所产生的磁链,即:,对线性媒质而言,更一般的表达式L11为:,一个回路或电路的自感,取决于构成这个回路或电路的导体的几何形状和物理排列以及媒质的磁导率.在线性媒质中,自感与回路或电路的电流无关.事实上,无论回路或电路是开路还是闭合的,也无论它是否靠近另一回路或电路,其自感总是存在的.,13,排列成适当形状(例如由导线缠绕而成的线圈)以提供一定数量自感的导体称为电感器.就象电容器可以储存电能一样,电感器能够储存磁能.,Theprocedurefordeterminingtheself-inductanceofaninductorisasfollows:,1.对于给定的几何形状选择适当的坐标系.2.假设导线中的电流为I.3.如果存在对称性,可以根据安培环路定理由I求B;如果不存在对称性,则可以用Biot-Savartlaw.4.用积分的方法由B求出每一圈所交链的磁通5.Findthefluxlinkagebymultiplyingbythenumberofturns.6.FindLbytakingtheratioL=/I.,14,EXAMPLE6-14P181,15,Example.Calculatethemutualinductance(互感)betweenaninfinitelylongstraightlineandarectangularcoil.Thelineandthecoilareatthesameplane,andinvacuum.,Solution:Selectcylindricalcoordinatesystem,andlettheinfinitelylongstraightlinetobeatthez-axis.ThemagneticfluxdensityproducedbycurrentI1isthen,Themagneticfluxlinkage12withcurrentI2bycurrentI1is,16,Wehave,IftheflowingdirectionofthecurrentI2iscounterclockwise,thentheB1anddSareopposite,andL120.,IftheflowingdirectionofthecurrentI2isclockwise,dSandB1havethesamedirection.Then,17,例题:求无限长导线与三角形回路之间的互感,18,3.磁能(MagneticEnergy),Ifanimpressedsource(外源)isappliedtoacircuit(回路),acurrentwillbegeneratedinthecircuit.Intheprocessofestablishingthecurrent,thereactionmagneticflux(感应磁通)inthecircuitwillresist(阻止)theincrementofthecurrent.,Assumethecurrentisincreasedveryslowly(非常慢)sothatradiationlosscanbeneglected,allenergyprovidedbytheimpressedsourcewillbestored(储存)inthemagneticfieldaroundthecircuit.,Basedontheworkdonebytheimpressedsource,theenergystoredinthemagneticfieldcanbecalculated.,Inordertoovercomethebackelectromotiveforceduetothereactionmagneticfluxandtomaintainthecurrent,theimpressedsourcehastodowork(做功).,19,考虑一个自感为L1的单独闭合回路,其初始电流为0,将电流源与回路连接,电流从0增加到I1.从物理学中知道回路中将产生感应电动势并阻碍电流的变化.要克服这个感应电动势,必须有外源作功.假设电感两端的的电压为v1=L1di1/dt,则所需做的功为:,对于线性媒质有L1=1/I1,则上式变为:,这个功看作是以磁能形式储存起来.,NowconsidertwoclosedloopsC1andC2carryingcurrentsi1andi2,respectively.ThetotalamountofworkdoneinrasingthecurrentsinloopsC1andC2fromzerotoI1andI2,respcetively,is,20,whichisstoredasmagneticenergy.WhereW21is,GeneralizingthisresulttoasystemofNloops(N个回路)carryingcurrentsI1,I2,In,weobtain,Whichistheenergystoredinthemagneticfield.ForacurrentIflowinginasingleinductor(单个电感器)withinductanceL,thestoredmagneticenergyis,21,ItisoftendesirabletoexpressthemagneticenergyintermsoffieldquantitiesBandH.Wehave,NotingthatH=B/,wecanrewrite,and,Wherewmisdefinedasmagneticenergydensity(磁能密度),22,无旋,无散,有散,有旋,电场强度E,磁通密度B,电通密度D,磁场强度H,介电常数,磁导率,23,1.Introduction,2.FundamentalPostulatesofMagnetostaticsinfreespace,Lorentzsforceequation,Review,24,3.VectorMagneticPotential,4.TheBiot-SavartLawandApplications,25,5.TheMagneticDipole,6.MagnetizationandEquivalentCurrentDensities,26