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首页 12 法拉第感应定律.pdf

12 法拉第感应定律.pdf

12 法拉第感应定律.pdf

zenger2017 2018-03-04 评分 0 浏览量 0 0 0 0 暂无简介 简介 举报

简介:本文档为《12 法拉第感应定律pdf》,可适用于自然科学领域,主题内容包含Faraday#sLawofInductionThefirstquantitativestudiesoftimedependentelectrica符等。

Faraday#sLawofInductionThefirstquantitativestudiesoftimedependentelectricandmagneticfieldswereperformedbyFaradayinHediscoveredthatanelectriccurrentarisesinaclosedwireloopwhenitismovedthroughamagneticfield(compareFigure)Faraday#sdiscoverymaybeformulatedmathematicallyinthefollowingway:IfCisthecurveofthecurrentloop,thenanarbitrarysurfacecanbeplacedthroughCandthemagneticfluxareadaBn()canbecalculatedBecausedivB,isindependentofthechoiceofthesurfaceaoverC(Figuresand)Theelectromotiveforce(voltage)VactingonthewireloopCisCVdEs()Here,EistheelectricfieldintensityactingatthepositionoftheconductorelementdsFaraday#sdiscoverycannowbeformulatedinthefollowingway:dVkdtorddkddtEsBa()TheinducedvoltageisproportionaltotherateofchangeofthemagneticfluxThesignisfixedbytheLenzlaw,implyingthattheinducedcurrentsandthemagneticfluxassociatedwithitaredirectedsuchthattheyopposethechangeoftheexternalfluxThelawisgenerallyvalidThechangeoffluxmaycomeaboutindifferentways,forexample,bymodificationormotionofthecurrentloopinanexternalmagneticfieldbyalterationofthemagneticfield(eg,bydisplacementofthegeneratingmagnet)orbothTheconstantofproportionalitykinequation()canbefixedbyconsideringthefollowingcase:Atfirst,wemoveametallicwireCcarryingnochargeatthevelocityvtothenewpositionC(Figure),thenthechargescontainedinthewiretakepartinthismotionTheLorentzforceactingonthesechargesisqcFvB()IftheforceFhasacomponentindirectionofthewire,itwillsetthechargeinmotion:acurrentwillflowThefieldintensityduetotheLorentzforceisindqcFEvB()Thevoltageinthewireloopisind()CCVddcEsvBs()Now,wewilldemonstratethatarea()()CCddddtvBsBathatis,itisequaltothenegativetimerateofchangeofthemagneticfluxthroughasurfaceoverCThetwopositionsofthewireloopC(attimet)andC(attimett)arisefromeachotherbythedisplacementvectorstvThevectorproductdtdasvndescribesthesurfaceelementddaandirectedoutward(Figure)Therefore,MM()()()CCCaddddattvBssvBsvBBn()ThelastintegralrepresentsobviouslythefluxMcrossingthelateralsurfaceMaconnectingthecurvesCandCThefluxthroughtheclosedsurfacearea()CMarea()CahastovanishsincedivBthatis,Marea()area()CCddBaBa()Hence,Marea()area()()()CCddtttBaBa()So,Misequaltothechangeofthefluxthroughtheendface,area()CTogetherwith()and()dVcdt,()thisisexactlyFaraday#slawObviously,theconstantofproportionalityin()isgivenbykcsothatgenerallydddcdtEsBa()Theuniversalityoftheinductionlaw()hasbeenassumedinthisargumentWenotethat,startingfromequation(),anysinglestepcanbeinverted,andthenwemayinferequation()forafixedBfield,butwithmovingconductorsSinceequation()hastobevalidforarbitraryclosedconductors,equation()andthustheLorentzforcecanbederivedItisinterestingtorealizethattheLorentzforcefollowsfromFaraday#sinductionlawUsingthislaw,eg,theintensityofauniformmagneticfieldcanbedetermined:AccordingtoOhm#slawtheinducedvoltagecorrespondstothecurrentIVR,whereRistheresistanceThechargewhichismovedinawireloopduringthemotionofawireloopthroughauniformmagneticfieldis()()TTTdqIdtdtdTRcdtRcRcMovingtheloopoutofthefield,then()TInauniformfield()Ba(aistheareaoftheloop)sothatbysubstitutiononeobtainsfinallyBaqRcThechargeqmaybemeasureddirectlybyaballisticgalvanometer,sothatBcanbecalculatedItisworthmentioningthatthegenerationofelectricpower(andtheelectricmotorinitsinversion)isbasedonthelaw()ThisisobviousfromFigureComparealsoExampleThebetatronThebetatronservesforthegenerationofhighenergeticelectronbeamsItconsistsofahighlyevacuatedcyclicdischargetubearrangedsymmetricallybetweenthepolepiecesofanelectromagnetElectronsinthedischargetubewillconstantlyfollowacircularorbitiftheactingmagneticfield(controlfield)conBjustcancelsthecentrifugalacceleration:conmvveBrcorconrmveBcThisconditionbetweenthecontrolfieldconBandthemomentumoftheelectronhastobemettomaintainastableorbitInadditiontothecontrolfield,thereisanacceleratingfieldChangingthisfield,acirculationvoltageisinducedinthetubethatacceleratestheelectronsFortheforce(weconsidermagnitudesonly)wehave()dedmveEerEBdadtrcrdtByintegrationweobtainaconditionbetweenthemomentumoftheelectronandtheacceleratingfield:emvBdacrHence,wehavetwoconditions,oneforthecontrolfieldandonefortheacceleratingfieldEliminatingthemomentumyieldsconconreeeBBdaBrBBccrcrThetwofieldshavetofulfillalwaysthe:ratioiftheelectronmuststayonastableorbitduringtheaccelerationSeeFigureExercise:InductionofacurrentinaconductingloopArectangularconductingloopistakenoutofamagneticinductionfieldofintensityB(Figure)InwhichdirectiondoesthecurrentflowTheinductionfieldpointsintotheplaneofthepage,asindicatedinthefigureSolutionObviously,theinductionfluxdecreasesduringthismotionAccordingtoFaraday#sinductionlaw,foracirculationinthemathematicallypositivesensethevoltageisgivenby()ddadaVdBBcdtcdtdtEswhereaistheareacrossedbytheinductionfieldNow,dadtisnegative,dadthenceVisnegative,sothecurrentflowsinamathematicallynegativesense,thatis,clockwiseExercise:VoltageinaconductingloopAconductingloopisconnectedtoaparallelplatecapacitor(Figure)FortheregionindicatedinthefigurethemagneticinductionfieldBpointsintotheplaneofthepageItsmagnitudeisincreasingintimeWhichofthetwoplatesofthecapacitorischargedpositivelySolutionAccordingtoFaraday#sinductionlawthecirculationvoltagemeasuredinamathematicallypositivesenseis()ddBadBVdacdtcdtcdtEs,becauseBBnaistheareathreadedbythemagneticinductionSincedBdtapositivecirculationvoltageisinducedthecurrentflowsinamathematicallypositivesensethatis,thecurrentflowsawayfromtheupperplate(then,itischargednegatively)Exercise:InductioninacoilbyatimevaryingmagneticfieldAcoilwithNturnsisplacedintoanelectromagnet(Figure)Themagneticfieldincreaseslinearlyintime,reachingavalueofGaussBaftersTThenitremainsconstantThecoilhasacrosssectionalareaofcma,anditisorientedperpendiculartothemagneticfield(a)Whatvoltageisinducedinthecoil(N)(b)LetthecoilhavearesistanceofRWhatisthemagnitudeofthecurrentinthecoiliftheendsofthecoilareclosed(c)HowmuchenergyisdissipatedduringtheswitchingonofthemagnetSolution(a)Theinductionfluxthroughtheindividualturnsofthecoilis()()tBtawherethemagneticinductionisincreasingaccordingtoBBtTHence,theinducedvoltageismTeslaVsNaBNdNadBVcdtcdtcTcalculatedintherationalizedsystemofunits(b)Ifthecurrentcircuitisclosed,thecurrentflowingfollowsfromAmAVVIRIR(c)Theenergydissipatedinheatmaybecalculatedinthefollowingway:ThecurrentisflowingduringthetimeTTherefore,thechargeflowingthroughanarbitrarycrosssectionalareaofthewirewillbeAsCoulQITThischargeQmovedthroughthepotentialdifferenceVandthusgainedtheenergyVoltCoulJouleWVQThisenergyisconvertedtoheatWemaythinkaboutitalsointhefollowingwayTheenergyreleasedperunitoftimeattheresistorisPVIIRandtheenergyconvertedtoheatduringthetimeTis()TWPtdtIRTVITVQasaboveExample:ElectricgeneratorsandmotorsTheprincipleofanelectricgeneratorisrepresentedinFigureAcoil(Nconductingloops)isrevolvedinanexternalmagneticfieldatthefrequencybyasteamturbineorawaterturbineInoneloopwiththeareaa,theinductionfluxsinsinBaBat,T()Becausethefluxisvaryingintime,inNloopsthevoltagecosNdNaBVtcdtc()isinduced(Figure)AnalternatingvoltageoccurswiththemaximalvaluesNaBcWhilethecoilisopen,nocurrentflowsIfthecurrentcircuitisclosed,thealternatingcurrent(lettheresistancebeR)cosVNaBItRcR()flowsTheheatoutputofthedeviceiscosNaBPIRtcR()ItfluctuatesalsoperiodicallywiththemaximalvaluesmaxNaBPcRThefluxgeneratedontheaverageoveroneperiodismax()cosTTNaBNaBPPtdttdtPTcRTcR()Ofcourse,thisheatoutputmustcomefromtheturbineWewanttobeconvincedaboutthisinmoredetailandatfirststatethatthecoilhasamagneticdipolemomentofmagnitude||NIacm()So,atorqueNisexertedonthecurrentcarryingcoil,NmB()Thistorquehastobecanceledfromoutside,bytheturbine,tokeepthecoilrunning(revolving)TheexternaltorqueisNNmB()Fromthedrawingwecanseethat||||coscosNIaBBtcNm()Themechanicalworkthathastobedoneinordertorevolvethecoilbythesmallangledis||dWdN()So,themechanicalpowerismech||coscosdWdNIaBNaBPttdtdtcRcN()Thisisjusttheelectricpower()convertedtoheatattheresistorTheelectricpowerforthegeneratorisexactlyequaltothemechanicalpoweroftheturbinedrivingthegenerator,asitmustbe,duetotheenergyconservationExercise:LinearmotorArodoflengthcmLislyingontwoideallyconductingrails(Figure)ThepotentialdifferencebetweentherailsisvoltsVLettheresistanceoftherodbeR(ohm)TherodisconnectedtoamasskgmbyaropeandapulleyCalculatethevelocityoftherodifamagneticfieldBisappliedinthedirectionshowninthefigureandisGaussWhatfractionofthepowersuppliedbythebatteryisconvertedtomechanicalpowerSolutionThisexampleisthelinear(onedimensional)analogofarotatingcoilinanexternalmagneticfieldIfthecurrentIflowsthroughtheconduction,aforceKactsontherod(seethefigure):IBLKdVIdcccBjBs()Forabalance,IBLmgKc()ButiftherodismovingthetotalareaaLxenclosedbythecurrentcircuitchangeshence,alsotheinductionfluxchangesThen,ddadxBBLBLvdtdtdt()wherevisthevelocityoftherodslidingontherailsSo,accordingtoFaraday#sinductionlawavoltageisinducedinthecircuitThisvoltageisdBLvVcdtc()ThetotalvoltageinthecurrentcircuitisVVItdeterminesthecurrentflowingaccordingtoBLvRIVVVc()andthebalancecondition()becomesIBLBLIRBLBLvmgKVccRcRc()hence,theconstantvelocityiscRmgcvVLBBL()Thenumericalvaluesofourexampleyieldmsv()ThisisahighvelocityBut,wenotethataslightchangeoftheresistance,R,wouldhavegivenvWeseethattheconstantvelocitysensitivelydependsontheresistanceRItdeterminesthemaximalcurrentandthusalsothemaximalstrengthoftheelectromagneticforcewhichsetstherodinmotionThetotalpowersuppliedbythebatteryismgcPVIVLB()wherewehavesubstitutedthecurrentfollowingfrom()and()ThispowermustequalthesumofthemechanicalpowerKvmgvandthethermalpowerIRattheresistor,thus,with()and()mgcRmgcmgcPmgvRIVRLBBLLB()Infact,wesee,PPTheefficiencyofthismotorcanbecalculatedastheratioofthemechanicalpowerandthetotalpowermgvLBmgRcvPVcBLV()ThenumericalvaluesyieldSo,onlyaboutoftheelectricalpowerisconvertedtomechanicalworkoftheelectricpowerisconvertedtoheatintheresistorObviouslythismotorisnotefficient

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