Electromagnetic Induction and Faraday’s Law Chapter 27 Electromagnetic Induction and Faraday’s Law

Faraday’s experiments (1820-1821) Induced EMF (1) Electric current magnetic field Faraday’s experiments (1820-1821) Changing magnetic field can produce a current 2

Phenomenon of electromagnetic induction 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. 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: 3) Induction current in circuit: 4

This is known as 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 5

Solution: The magnetic flux: Rotating coil Example1: A circular coil is rotating in uniform magnetic field. Determine the EMF. Solution: The magnetic flux: If the magnetic field is changing: 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: + + - - - : non-electrostatic field Motional EMF: All ⊥ case: Direction? 8

Derived by Faraday’ law The result can also be derived by Faraday’ law moves with velocity It sweeps out an area dS Magnetic flux: EMF: 9

Motion in uniform field Example2: Determine the EMF induced in the conductor in a uniform magnetic field. (a) (b) v l v B EMF: → straight wire: Force: Power: 10

Rotates in uniform field Example3: A conductor rod rotates about axis o. Determine the induced EMF. (B, L, ω) Solution: For an infinitesimal: Total EMF in the conductor: Faraday’s law: 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 L o 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) m, R 13

(b) source puts out constant EMF: m, R (c) Terminal speed 14

Conductor stays at rest, magnetic field changes 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

General form of Faraday’s law Faraday’s law → general form Generalize the definition of V : For induced EMF in a closed circuit: where is the induced electric field General form of Faraday’s law 16

Induced electric field 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

Electrostatic / induced electric / magnetic field Comparison of fields Electrostatic / induced electric / magnetic field Field lines: vortex I 18

Solution: Analyze the symmetry Vortex electric field Example5: Uniform magnetic field in cylindrical space changes as dB/dt = C > 0. Determine the induced electric field. R Solution: Analyze the symmetry 19

Solution: Induced electric field: 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: R 2R 20

Imagine a closed circuit Another Solution: R 2R 1 2 Imagine a closed circuit Discussion: 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 electric field → vortex current Heating effect → induction cooker Electromagnetic damping 24