Physics 121 - Magnetism Lecture 11 - Faraday’s Law of Induction Y&F Chapter 29, Sect. 1-5 Magnetic Flux Motional EMF: moving wire in a B field Two Magnetic Induction Experiments Faraday’s Law of Induction Lenz’s Law Rotating Loops – Generator Principle Concentric Coils – Transformer Principle Induction and Energy Transfers Induced Electric Fields Summary

Previously: Next: Changing magnetic flux induces EMFs Current-lengths in a magnetic field feel forces and torques q Force on charge and wire carrying current torque and potential energy of a dipole Current-lengths (changing electric fields) produce magnetic fields Ampere’s Law Biot-Savart Law Current loops are elementary dipoles B due to long straight wire carrying a current i: B due to circular loop carrying a current i : B inside a solenoid: B inside a torus carrying B outside = 0 a current i : B on the symmetry axis of a current loop (far field): Next: Changing magnetic flux induces EMFs and currents in wires

Magnetic Flux: r B defined analogously to flux of electric field Electrostatic Gauss Law Magnetic Gauss Law over surface (open or closed) B r Flux Unit: 1 Weber = 1 T.m 2 q

Changing magnetic flux induces EMFs EMF / current is induced in a loop if there is relative motion between loop and magnet - the magnetic flux inside the loop is changing Induced current stops when relative motion stops (case b). Faster motion produces a larger current. Induced current direction reverses when magnet motion reverses direction (case c versus case a) Any relative motion that changes the flux works EMF/current is induced in the loop whenever magnetic flux through the loop is changing. (We mean flux rather than just field) Induced current creates it’s own induced B field and flux, opposing the changing flux FB (Lenz’ Law)

CHANGING magnetic flux induces EMFs and currents in wires Key Concepts: Magnetic flux Faraday’s Law of Induction Lenz’s Law Generator principle: Loops rotating in B field generate EMF and current. Lenz’s Law  Applied torque is needed (energy conservation)

Motional EMF: Lorentz Force on moving charges in conductors Uniform magnetic field points away from the viewer. Wire of length L moves with constant velocity v perpendicular to the field Electrons feel a magnetic force and migrate to the lower end of the wire. Upper end becomes positive. Result is an induced electric field Eind inside wire Charges come to equilibrium when the forces on charges balance: Electric field Eind in the wire corresponds to potential difference Eind across the ends of wire: Potential difference Eind is maintained between the ends of the wire as long as the wire continues to move through the magnetic field. + FIELD BOUNDARY L v - FBD of free charge q in a wire v + - Fe Fm Eind Induced EMF is created without batteries. Induced current flows only if the circuit is completed.

Flux approach to Motional EMF in a moving wire via Lorentz force The rate of flux change = field B x rate of sweeping out area (B uniform) Connection with Flux: Flux arguments apply generally, for example: E is induced directly in 15 turn coil by changing flux in solenoid inside solenoid B = 0 outside, at location of charges moved in the loop. Lorentz force doesn’t explain induced current Changing magnetic flux through coil creates electric field that drives induced current in loop that creates induced B field Flux is proportional to solenoid’s cross-section area (not the coil’s)

A rectangular loop is moving in a uniform B field + - a b c d DOES CURRENT FLOW? Segments a-b & d-c create equal but opposed EMFs in circuit No EMF from b-c & a-d or FLUX is constant No current flows DOES CURRENT FLOW NOW? B-field ends or is not uniform Segment c-d now creates NO EMF Segment a-b creates EMF as above Un-balanced EMF drives current just like a battery FLUX is DECREASING v + - a b c d Which way does current flow? What is different when loop is entering field?

Direction of induced fields and currents Replace magnet with circuit below Flux change is enough to cause induction Do not need actual motion Let current i1 be changing -> changing flux. Flux is constant if current is constant Current i2 flows only while i1 (flux F1) is changing (after switch S closes or opens) Ammeter Induced current i2 creates it’s own induced field B2 whose flux F2 opposes the change in F1 (Lenz’ Law) Close S i2 Open S later F1 = B1A F2 = B2A Induced Dipoles i1 i2

Faraday’s Law: Changing Flux uniform B Rate of flux change (Webers/sec) “-” sign for Lenz’s Law Induced EMF (Volts) insert N for multiple turns in loop, loop of any shape B(t) EMF=0 slope  EMF Change Flux by: changing |B| through a coil changing area of a coil or loop changing angle between B and coil e.g., rotating coils  generator effect Example: B through a loop increases by 0.1 Tesla in 1 second: Loop area A = 10-3 m2. Find the induced EMF Current iind creates field Bind that opposes increase in B uniform dB/dt front

+ iind - x Slidewire Generator Slider moves, increasing loop area (flux) + - x iind L Induced EMF: Slider: constant speed v to right Uniform field into the page Loop area & FLUX are increasing Slider acts like a battery Lenz’s Law says induced field Bind is out of page. RH rule says current is CCW due to the induced current opposes the motion of the wire FEXT (an external force) is needed to keep wire from slowing down FM DRAG FORCE on slider FEXT needed is opposed to Fm Mechanical power supplied & dissipated:

Slidewire Generator: Numerical Example v + - iind L B = 0.35 T L = 25 cm v = 55 cm/s a) Find the EMF generated: DIRECTION: Bind is into slide, iind is clockwise b) Find the induced current if R for the whole loop = 18 W: c) Find the thermal power dissipated: d) Find the power needed to move slider at constant speed !!! Power dissipated via R = Mechanical power !!!

Induced Current and Emf 11 – 1: A circular loop of wire is in a uniform magnetic field covering the area shown. The plane of the loop is perpendicular to the field lines. Which of the following will not cause a current to be induced in the loop? A. Sliding the loop into the field from the far left to right B. Rotating the loop about an axis perpendicular to the field lines. C. Keeping the orientation of the loop fixed and moving it along the field lines. D. Crushing the loop. E. Sliding the loop out of the field from left to right B

The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them Lenz’s Law front of loop induced dipole In same loop induced dipole in same loop The induced current try to keep the original magnetic flux through the loop from changing. front of loop front of loop

Lenz’s Law Example: A loop crossing a region of uniform magnetic field The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them B = 0 B = uniform B = 0 v v v v v

Direction of induced current 11-2: A circular loop of wire is falling toward a straight wire carrying a steady current to the left as shown What is the direction of the induced current in the loop of wire? Clockwise Counterclockwise Zero Impossible to determine I will agree with whatever the majority chooses v I 11-3: The loop continues falling until it is below the straight wire. Now what is the direction of the induced current in the loop of wire? Clockwise Counterclockwise Zero Impossible to determine I will oppose whatever the majority chooses

Rotation axis is out of slide Generator Effect (AC) Changing flux through a rotating current loop angular velocity w = 2pf in B field: Rotation axis is out of slide q peak flux magnitude when wt = 0, p, etc. Eind has sinusoidal behavior - alternating polarity over a cycle maxima when wt = +/- p/2 EMF induced is the time derivative of the flux

AC Generator DC Generator Back-torque t=mxB in rotating loop, m ~ N.A.iind

AC Generator, continued No reversal of output E DC Generator, continued Reversal of output E

Flat coil with N turns of wire Numerical Example Flat coil with N turns of wire E N turns Each turn increases the flux and induced EMF N = 1000 turns B through coil decreases from +1.0 T to -1.0 T in 1/120 s. Coil area A is 3 cm2 (one turn) Find the EMF induced in the coil by it’s own changing flux Flux change due to external B field produces induced field Bind INDUCED field/flux produces its own EMF. The BACK EMF opposes current change – analogous to inertia

Transformer Principle Primary current is changing after switch closes  Changing flux in primary coil ....which links to.... changing flux through secondary coil  changing secondary current & EMF PRIMARY SECONDARY Iron ring strengthens flux linkage

conducting loop, resistance R Changing magnetic flux directly induces electric fields A thin solenoid, cross section A, n turns/unit length zero B field outside solenoid inside solenoid: conducting loop, resistance R Bind Flux through a conducting loop: Current I varies with time  flux varies  EMF is induced in wire loop: Current induced in the loop is: If dI/dt is positive, B is growing, then Bind opposes change and I’ is counter-clockwise B = 0 there so it’s not the Lorentz force An induced electric field Eind along the loop causes current to flow It is caused directly by dF/dt within the loop path E-field is there even without the conductor (no current flowing) Electric field lines here are loops that don’t terminate on charge. E-field is a non-conservative (non-electrostatic) field as the line integral around a closed path is not zero What makes the induced current I’ flow, outside solenoid? Generalized Faradays’ Law (hold loop path constant)

Example: Find the electric field induced by changing magnetic flux Find the magnitude E of the induced electric field at points within and outside the magnetic field. Assume: dB/dt = constant within the circular shaded area. E must be tangential: Gauss’ law says any normal component of E would require charge enclosed. |E| is constant on the circular integration path due to symmetry. For r > R: For r < R: The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R

Example: EMF generated by Faraday Disk Dynamo Conducting disk, radius R, rotates at rate w in uniform constant field B, FLUX ARGUMENT: areal velocity w, q + dA = area swept out by radius vector in dq = fraction of full circle in dq x area of circle USING MOTIONAL EMF FORMULA: Emf induced across conductor length ds Moving conductor sees vXB as electric field E’ For points on rotating disk: v = wr, vXB = E’ is radially outward, ds = dr current flows radially out (Equation 29.7)