Chapter 31Faraday’s LawElectricity generator, or from B to E.1. Battery Chemical emf2. Motional emf3. Faraday’s Law of Induction4. Lentz Law about the emf directionA dry-cell batteryChemical reactions in the battery cells transport charge carriers (electrons) from one terminal to the other to create the needed electric potential (emf) to drive the current through the outside load, a light bulb here.Motional emf, the conceptWithWe can group charges by moving them in a magnetic field motional emf.A motional emf is the emfinduced in a conductor moving through a magnetic fieldThe electrons in the conductor experience a force, that is directed along ℓCharges are accumulated at the ends of the conductor to create an electric field inside the conductor to stop further charge transportation. q= ×F v B q= ×F v B = = =B EF qvB F qEWhen equilibriumMotional emf, the calculationStart from the equilibrium conditionOne hasOr the emf, potential difference:As long as the bar is kept being moved with a velocity v, the motional emf is maintained to be vBℓ. = = =B EF qvB F qE=E vB= ∆ = =emfV E vBl lMotional emf, put in use to power a resistorPLAYACTIVE FIGUREBar moved by appFITwo issues need attention:1. The moving bar carrying current I, inside the magnetic field, experiences a force from the field is FB=IℓB2. The magnetic flux in the enclosed area (bar, rails and resistor) is ΦB=xℓB, and it is changing with time as Equivalent circuit diagram Condition:A bar moving on two rails. The bar and the rails have negligible resistance. A resistor of R is connected to the end of the two rails. Result:The emf = vBℓ, so the current I = vBℓ /R( )Φ= = ==bdd dxx B B vBdt dt dtemfl l lExample, what is the terminal velocity?A bar of mass m sides on two vertical rails. A resistor is connected to the end of the rails. When the bar is released at t = t0, (a) calculate the velocity of the bar at time t, (b) what is the terminal velocity? Assuming that the rails and the magnetic field is long/large enough. Im=emf vBl=GˆmgF xOnce the bar starts to move, accelerated by the gravitational force, there is:And there is current as well:=I vBRl /And there is magnetic force on the bar, pointing opposite to the gravitational force:()= − = −2Bv BIˆ ˆBRFx xllExample, what is the terminal velocity?Im=GˆmgF xConstruct the equation of velocity v:Solve this equation This is the answer to (a). For (b), the terminal velocity is when ( )( ) + = − = − = 22G Bv Bmg mv Bdvmg mdtˆRRF F axll( )ττ τ= − ≡−2dv dt mR,v gBl( )ττ− = − = = ∵1 0 0tv g e , v tτg→ ∞tFaraday’s Law of inductionIn the sliding bar “experiment”, we proved that:Φ=bddtemfWe also know that the magnetic flux is defined ascosΦ = ⋅ Φ =∫ or B Bd BAθB AIn the sliding bar experiment, we changed A by moving the bar. More practically people change B or the angle θto achieve a changing flux. Changing B Changing θFaraday’s Law of inductionIn any case, the induced emf follows the Faraday’s Law of inductionΦ= −bddtemfYes, I sleeked in the “-”in front of the ΦbddtBecause Mr. Lenz told me so in order to answer the question of in which direction should the induced current flow.Faraday’s Law – Statements Faraday’s law of induction states that “the emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit”Mathematically,BdεdtΦ= −Lenz’s LawLenz’s Law, the direction of the induced emfLenz’s law: the induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop. The induced current tends to keep the original magnetic flux through the circuit from changingExample: EMF produced by a changing magnetic fieldA loop of wire is connected to a sensitive ammeterDetermine the current in the loop and the magnet is beingMoved into the loopMoved out of the loopHeld still inside the loopPLAYACTIVE FIGUREExample: a transformerA primary coil is connected to a switch and a batteryThe wire is wrapped around an iron ringA secondary coil is also wrapped around the iron ringThere is no battery present in the secondary coilThe secondary coil is not directly connected to the primary coil Close the switch and observe the current readings given by the ammeterPLAYACTIVE FIGUREExample, Lenz’s LawApplications of Faraday’s Law – GFI A GFI (ground fault indicator) protects users of electrical appliances against electric shockWhen the currents in the wires are in opposite directions, the flux is zeroWhen the return current in wire 2 changes, the flux is no longer zeroThe resulting induced emf can be used to trigger a circuit breakerApplications of Faraday’s Law – Pickup CoilThe pickup coil of an electric guitar uses Faraday’s lawThe coil is placed near the vibrating string and causes a portion of the string to become magnetizedWhen the string vibrates at some frequency, the magnetized segment produces a changing flux through the coilThe induced emf is fed to an amplifierRotating LoopAssume a loop with Nturns, all of the same area rotating in a magnetic fieldThe flux through the loop at any time t is ΦB= BA cos θ= BA cos ωtSosinΦ= − =Bdemf N NBAω ωtdtThe emf is a sin wave: AC.GeneratorsElectric generators take in energy by work and transfer it out by electrical transmissionThe AC generator consists of a loop of wire rotated by some external means in a magnetic fieldUse the active figure to adjust the speed of rotation and observe the effect on the emf generatedPLAYACTIVE FIGUREDC GeneratorsThe DC (direct current) generator has essentially the same components as the AC generatorThe main difference is that the contacts to the rotating loop are made using a split ring called a commutatorUse the active figure to vary the speed of rotation and observe the effect on the emf generatedPLAYACTIVE FIGUREMotorsMotors are devices into which energy is transferred by electrical transmission while energy is transferred out by workA motor is a generator operating in reverseA current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotateMotors, cont.Useful mechanical work can be done by
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