Lecture 18: TUE 23 MAR 2010Lecture 18: TUE 23 MAR 2010Ch30.1Ch30.1––44 Induction and Inductance I Induction and Inductance IPhysics 2102Jonathan DowlingFender StratocasterSolenoid PickupEXAM IIAVG: 65/100APPROXIMATE CURVE:A: 100–90B: 89–80C: 79–50D: 49–45F: 44–0http://www.phys.lsu.edu/classes/spring2010/phys2102/exam2solutions.pdfIn a series of experiments, Michael Faraday in England and Joseph Henry in the U.S. were able to generate electric currents without the use of batteries. Faraday's ExperimentsThe circuit shown in the figure consists of a wire loop connected to a sensitiveammeter (known as a "galvanometer"). If we approach the loop with a permanent magnet we see a current being registered by the galvanometer. A current appears only if there is relative motion between the magnet and the loop.1. Faster motion results in a larger current.2. If 3. we reverse the direction of motion or the polarity of the magnet, the currentreverses sign and flows in the opposite direction. The current generated is known as " "; the emf that appearinduced current s is known as " "; the whole effect is called " "induced emf induction.Changing B-Field Induces a Current in a Wire LoopNo Current When Magnet StopsNote Current Changes Sign With DirectionIn the figure we show a second type of experimentin which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. Whenthe current in loop 1 is constant no induced current is observed in loop 2. The conclusion is that the magnetic field in an induction experiment can begenerated either by a permanent magnet or by anelectric current in a coil.loop 1loop 2Faraday summarized the results of his experiments in what is known as" "Faraday's law of induction.An emf is induced in a loop when the number of magnetic field lines thatpass through the loop is changing.Loop OneHas a 60!HzAlternating Current Loop Two is Connected To A Light Bulb.The Current in LoopOne Produces aRapidly ChangingMagnetic Field inLoop Two ThatInduces a Current inLoop Two — Lightingthe Bulb!dAFaradayFaraday’’s Law: What? The Flux!s Law: What? The Flux!• A time varying magneticFLUX creates an inducedEMF• Definition of magnetic fluxis similar to definition ofelectric flux B EEMF= !d"Bdt !B=! B " d! A S#• Take note of the MINUSsign!!• The induced EMF acts insuch a way that it OPPOSESthe change in magnetic flux(“Lenz’s Law”). d! ALenzLenz’’ss LawLaw• The Loop Current Produces a B Fieldthat Opposes the CHANGE in the barmagnet field.• Upper Drawing: B Field from Magnetis INCREASING so Loop Current isClockwise and Produces an OpposingB Field that Tries to CANCEL theINCREASING Magnet Field• Lower Drawing: B Field from Magnetis DECREASING so Loop Current isCounterclockwise and Tries toBOOST the Decreasing Magnet Field.ExampleExample• A closed loop of wire enclosesan area of A!=!1 m2 in whichin a uniform magnetic fieldexists at 300 to the PLANE ofthe loop. The magnetic field isDECREASING at a rate ofdB/dt!=!1T/s. The resistanceof the wire is 10 Ω.• What is the induced current? !B=! B " d! A # = BA cos(600) = BA /2 E=d!Bdt=A2dBdt i =ER=A2RdBdt i =(1m2)2(10!)(1T/s) = 0.05A300Is it …clockwise or …counterclockwise?B d! A 60°ExampleExample• 3 loops are shown.• B = 0 everywhere except inthe circular region I whereB is uniform, pointing outof the page and isincreasing at a steadyrate.• Rank the 3 loops in orderof increasing induced EMF.– (a) III < II < I ?– (b) III < II = I ?– (c) III = II = I ?• III encloses no flux so EMF=0• I and II enclose same flux soEMF same.• Are Currents in Loops I & IIClockwise or Counterclockwise?IIIIIIBExampleExample• An infinitely long wire carries aconstant current i as shown• A square loop of side L is movingtowards the wire with a constantvelocity v.• What is the EMF induced in the loopwhen it is a distance R from the loop?LLdR/dt=v !B=µ0iLdx2"(R + x)0L#LxRiL00)ln(2!"#$%&+='µ!"#$%&+=RLRiLln20'µChoose a “strip” of width dxlocated as shown.Flux thru this “strip” d! = BLdx =µ0iLdx2"(R + x) E = !d"Bdt= !µ0Li2#ddtln 1+LR$ % & ' ( ) * + , - . / Rxi B =µ0i2!rr=R+xExampleExample E = !d"Bdt!"#$%&'()*+,+-=RLdtdLi1ln20.µ202 RLLRRdtdRLi!"#$%&+='µ!"#$%&+=RLRLvi)(220'µLdR/dt=vRxWhat is the DIRECTION of theinduced current?• Magnetic field due to wirepoints INTO page and getsstronger as you get closer towire• So, flux into page isINCREASING• Hence, current induced mustbe counter clockwise to opposethis increase in flux = CCWBiExample : The GeneratorExample : The Generator• A square loop of wire of sideL is rotated at a uniformfrequency f in the presenceof a uniform magnetic field Bas shown.• Describe the EMF induced inthe loop.BLθB !B=! B " d! A S#)cos(2!BL= E = !d"Bdt= BL2d#dtsin(#)( ))2sin(22ftfBL!!= !="tf = 2#"Example: Eddy CurrentsExample: Eddy Currents• A non-magnetic (e.g. copper,aluminum) ring is placed near asolenoid.• What happens if:– There is a steady current inthe solenoid?– The current in the solenoid issuddenly changed?– The ring has a “cut” in it?– The ring is extremely cold?Another Experimental ObservationAnother Experimental Observation• Drop a non-magnetic pendulum(copper or aluminum) through aninhomogeneous magnetic field• What do you observe? Why? (Thinkabout energy conservation!)N SPendulum had kinetic energyWhat happened to it?Isn’t energy conserved?? Energy is Dissipated byResistance: P=i2R. This actslike
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