第32章 电感.ppt

CHAPTER32 INDUCTANCE Chapter32 inductance 32 1 Self inductionandinductance Inthischapter weneedtodistinguishcarefullybetweenemfsandcurrentsthatarecausedbyphysicalsourcessuchasbatteriesandthosethatareinducedbychangingmagneticfields Inductor Thechangingfluxthroughthecircuitandtheresultantinducedemfarisefromthecircuititself whereLisaproportionalityconstant calledtheinductance QuickQuiz32 1 Acoilwithzeroresistancehasitsendslabeledaandb Thepotentialataishigherthanatb Whichofthefollowingcouldbeconsistentwiththissituation a Thecurrentisconstantandisdirectedfromatob b Thecurrentisconstantandisdirectedfrombtoa c Thecurrentisincreasingandisdirectedfromatob d Thecurrentisdecreasingandisdirectedfromatob e Thecurrentisincreasingandisdirectedfrombtoa f Thecurrentisdecreasingandisdirectedfrombtoa Example32 1 ConsiderauniformlywoundsolenoidhavingNturnsandlengthL AssumeLismuchlongerthantheradiusofthewindingsandthecoreofthesolenoidisair A Findtheinductanceofthesolenoid B Calculatetheinductanceofthesolenoidifitcontains300turns itslengthis25 0cm anditscross sectionalareais4 00cm2 Solution A B C Calculatetheself inducedemfinthesolenoidifthecurrentitcarriesdecreasesattherateof50 0A s Solution Discuss Theresultforpart A showsthatLdependsongeometryandisproportionaltothesquareofthenumberofturns BecauseN nL wecanalsoexpresstheresultintheform whereistheinteriorvolumeofthesolenoid 32 2 RLCircuits Kirchhoff slooprule S2isatposition a S1remainclosed Define timeconstant Asshown AfterswitchS1isthrownclosedatt 0 thecurrentincreasestowarditsmaximumvalue Asthetimeisenoughlong IfS2isthrownfromatob Asweknownthecurrent wecanalsoknownthevoltage QuickQuiz32 2 Considerthecircuitasshown withS1openandS2atpositiona SwitchS1isnowthrownclosed i Attheinstantitisclosed acrosswhichcircuitelementisthevoltageequaltotheemfofthebattery a theresistor b theinductor c boththeinductorandresistor ii Afteraverylongtime acrosswhichcircuitelementisthevoltageequaltotheemfofthebattery Choosefromamongthesameanswers Example32 2 Considerthecircuitasshown Supposethecircuitelementshavethefollowingvalues and A Findthetimeconstantofthecircuit Solution B SwitchS2isatpositiona andswitchS1isthrownclosedatt 0 Calculatethecurrentinthecircuitatt 2 00ms Solution C Comparethepotentialdifferenceacrosstheresistorwiththatacrosstheinductor Thesumofthetwovoltagesatalltimesis12 0V 32 3EnergyinMagneticField Recognizingastherateatwhichenergyissuppliedbythebatteryandastherateatwhichenergyisdeliveredtotheresistor weseethatLI dI dt mustrepresenttherateatwhichenergyisbeingstoredintheinductor IfUistheenergystoredintheinductoratanytime wecanwritetheratedU dtatwhichenergyisstoredas Energystoredinaninductor Forsolenoid Magneticenergydensity Capacitors Resistors andInductorsStoreEnergyDifferently Differentenergy storagemechanismsareatworkincapacitors inductors andresistors Achargedcapacitorstoresenergyaselectricalpotentialenergy Aninductorstoresenergyaswhatwecouldcallmagneticpotentialenergywhenitcarriescurrent Energydeliveredtoaresistoristransformedtointernalenergy QuickQuiz32 3 Youareperforminganexperimentthatrequiresthehighest possiblemagneticenergydensityintheinteriorofaverylongcurrent carryingsolenoid Whichofthefollowingadjustmentsincreasestheenergydensity Morethanonechoicemaybecorrect a increasingthenumberofturnsperunitlengthonthesolenoid b increasingthecross sectionalareaofthesolenoid c increasingonlythelengthofthesolenoidwhilekeepingthenumberofturnsperunitlengthfixed d increasingthecurrentinthesolenoid Example32 3 ConsideronceagaintheRLcircuitasshown withswitchS2atpositionaandthecurrenthavingreacheditssteady statevalue WhenS2isthrowntopositionb thecurrentintheright handloopdecaysexponentiallywithtimeaccordingtotheexpression whereistheinitialcurrentinthecircuitandisthetimeconstant Showthatalltheenergyinitiallystoredinthemagneticfieldoftheinductorappearsasinternalenergyintheresistorasthecurrentdecaystozero Solution TheenergyinthemagneticfieldoftheinductoratanytimeisU Thisresultisequaltotheinitialenergystoredinthemagneticfieldoftheinductor Example32 4 Coaxialcablesareoftenusedtoconnectelectricaldevices suchasyourvideosystem andinreceivingsignalsintelevisioncablesystems Modelalongcoaxialcableasathin cylindricalconductingshellofradiusbconcentricwithasolidcylinderofradiusaasshown TheconductorscarrythesamecurrentIinoppositedirections CalculatetheinductanceLofalength ofthiscable Solution Dividethelightgoldrectangleintostripsofwidthdr 32 4Mutualinductance Veryoften themagneticfluxthroughtheareaenclosedbyacircuitvarieswithtimebecauseoftime varyingcurrentsinnearbycircuits Thisconditioninducesanemfthroughaprocessknownasmutualinduction sonamedbecauseitdependsontheinteractionoftwocircuits Considerthetwocloselywoundcoilsofwireshownincrosssectionalview wecanidentifythemutualinductanceM12ofcoil2withrespecttocoil1 Mutualinductancedependsonthegeometryofbothcircuitsandontheirorientationwithrespecttoeachother IfthecurrentI1varieswithtime weseefromFaraday slawthattheemfinducedbycoil1incoil2is TheprecedingdiscussioncanberepeatedtoshowthatthereisamutualinductanceM21 Inmutualinduction theemfinducedinonecoilisalwaysproportionaltotherateatwhichthecurrentintheothercoilischanging AlthoughtheproportionalityconstantsM12andM21havebeentreatedseparately itcanbeshownthattheyareequal Therefore Theunitofmutualinductanceisthehenry Example32 5 Asolenoidoflength withturns carryingacurrentI andhavingacross sectionalareaA Thehandlecoilcontainsturnsandcompletelysurroundsthebasecoil Findthemutualinductanceofthesystem Solution 32 5 OscillationsinanLCCircuit Asshown AsimpleLCcircuit ThecapacitorhasaninitialchargeQmax andtheswitchisopenfort 0andthenclosedatt 0 TotalenergystoredinanLCcircuit Energytransferinaresistanceless nonradiatingLCcircuit Where AngularfrequencyofoscillationinanLCcircuit As