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Chapter 27 Electromagnetic Induction and Faraday’s Law Induced EMF (1) Electric current magnetic field Faraday’s experiments (1820-1821) Changing magnetic field can produce a current 2 Induced EMF (2) Phenomenon of electromagnetic induction An induced EMF (electromotive force in P566) is produced by a changing magnetic field. 3 Faraday’s law of induction The EMF induced in a circuit is equal to the changing rate of magnetic flux through the circuit. dB dt Faraday’s law of induction 1) EMF is produced even if no current can flow (as when the circuit is not complete) 2) N loops coil: N d B dt 3) Induction current in circuit: I /R 4 Lenz’s law An induced EMF is always in a direction that opposes the original change in flux that caused it. This is known as Lenz’s law 1) Conservation of energy 2) Three ways to create EMF B BdS cos 5 Rotating coil Example1: A circular coil is rotating in uniform magnetic field. Determine the EMF. Solution: The magnetic flux: B BS cos R 2 B cos(t 0 ) R dB R 2 B sin(t 0 ) dt If the magnetic field is changing: B B B0 e t R 2 B0 e t sin(t 0 ) R 2 B0 e t cos(t 0 ) 6 Induced charges Thinking:A small coil moves far away from the position in figure. How to determine the total induced charge going through the coil? 7 EMF induced in moving conductor Conductor moves in B → motional EMF ++ Caused by Lorentz force: F qv B qEk - F Ek : non-electrostatic field Motional EMF: All ⊥ case: v -- Ek dl (v B) dl Bvdl Direction? 8 Derived by Faraday’ law The result can also be derived by Faraday’ law dl moves with velocity v dS It sweeps out an area dS dS (vdt ) dl Magnetic flux: v dl d B B dS B (v dl )dt EMF: d B B (v dl ) (v B) dl dt 9 Motion in uniform field Example2: Determine the EMF induced in the conductor in a uniform magnetic field. (a) (b) v v B l lab EMF: Blv Force: F BIl B2l 2v / R Power: P Fv I R 2 → straight wire: (v B) lab 10 Rotates in uniform field Example3:A conductor rod rotates about axis o. Determine the induced EMF. (B, L, ω) Solution: For an infinitesimal: d B r dr dr v r o Total EMF in the conductor: 1 1 2 B B rdr B L2 S L 0 2 2 dB d 1 2 1 Faraday’s law: BL B L2 dt 2 2 dt L 11 Rotates in nonuniform field Question:A conductor rod rotates about axis o. Determine the induced EMF when the two wires are perpendicular to each other. I 0 IL (1 ln 2) 2 o L L B 12 Motion on rails Example4: Conducting rod rests on frictionless parallel rails with an EMF source. Determine the speed of rod if the source puts out (a) constant I; (b) constant EMF. (c) What is the terminal speed? Solution: (a) F ma BIl BIl v at t m B F l m, R 13 (b) source puts out constant EMF: I Blv R t Bl v dv BIl ( Blv) Bl dv dt 0 mR 0 Blv dt m mR v Bl (1 e B2 l 2 t mR ) B (c) Terminal speed v F l Bl m, R 14 Changing B produces E Conductor stays at rest, magnetic field changes → Induced EMF → induced current Forces on static charges? Maxwell: It’s caused by electric field Induced (vortex) electric field: produced by changing magnetic field, and acts on electric charges. 15 Faraday’s law → general form Generalize the definition of V : Vab For induced EMF in a closed circuit: b a E dl dB dt dB d B dS Ei dl dt dt where Ei is the induced electric field B Ei dl t dS General form of Faraday’s law 16 Induced electric field B Ei dl t dS B Ei t 1) Ei is produced by changing B, not by charges 2) Even if there is no conductor, Ei still exists 3) Induced electric field is nonconservative 4) Minus sign shows the direction of Ei 17 Comparison of fields Electrostatic / induced electric / magnetic field E dl 0; E Qin s s dS 0 Field lines: dB Ei dl dt ; E dS 0 i B dl 0 I in B dS 0 vortex dB dt I 18 Vortex electric field Example5:Uniform magnetic field in cylindrical space changes as dB/dt=C>0. Determine the induced electric field. Solution: Analyze the symmetry dB Ei dl Ei 2 r dt C S C r2 C Ei r R : Ei r 2 r 2 CR 2 C R2 r R : Ei o 2r 2 r R r R 19 EMF in a wire Example6:Magnetic field in a solenoid changes as dB/dt=C>0. A straight wire lies tangent to the solenoid at its center. What is the EMF in wire? Solution: Induced electric field: CR 2 r R : Ei 2r r R CR 2 Ei dl cos dl 2r L CR R C d / 4 2 4 /4 2 Ei 2 dl l 2R R r cos l R tan 20 Another Solution: 1 Imagine a closed circuit R 2 2R 1 Ei dl 0, 2 0 d B d R2 B R 2C ( ) dt dt 4 4 Discussion: ab ab ? d B Ei dl dt 0! ab ab a b 21 Self inductance Thinking:If a solenoid with current I is cut off from the battery, will I drop abruptly to 0? How does I change over time? 22 *Applications of induction Sound systems / microphones: Transformers: Recording tape / computer memory 23 *Vortex current Vortex electric field → vortex current Heating effect → induction cooker Electromagnetic damping 24