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 Biot-Savart Current loops are elementary dipoles B outside a long straight wire carrying a current i: B at center of circular loop carrying a current i : Long Solenoid field: 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 B r Flux Unit: 1 Weber = 1 T.m 2 Surface Integral over open or closed surface 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 Faster motion produces a larger current. Induced current stops when relative motion stops (case b). Induced current direction reverses when magnet motion reverses direction (case c versus case a) Any relative motion that changes the flux works Magnetic flux that is changing causes an induced EMF (or current) in a wire loop bounding the area. (We really mean flux rather than just field) Induced current creates it’s own induced B field (and flux) that opposes the changing flux (Lenz’ Law)

CHANGING magnetic flux induces EMFs and currents in wires Key Concepts: Magnetic flux Area bounded by a loop Faraday’s Law of Induction Lenz’s Law Basis of generator principle: Loops rotating in B field generate EMF and current in load circuit. Lenz’s Law guarantees that external torque must do work.

Motional EMF: Lorentz Force on moving charges in conductors Uniform magnetic field points away from viewer. Wire of length L moves with constant velocity v perpendicular to the field Electrons feel a magnetic Lorentz force and migrate to the lower end of the wire. Upper end becomes positive. Charge separation  an induced electric field Eind exists 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 it moves through the magnetic field. + FIELD BOUNDARY L v - FBD of free charge q in a wire v + - Fe Fm Eind Induced EMF is created by the motion. Induced current flows only if the circuit is completed.

Complementary Flux view for Motional EMF in a moving wire The rate of flux change = [field B] x [rate of sweeping out area (B uniform)]

Flux change explains what happens even without motion Simple transformer: changing current in solenoid causes current to flow in outer coil. Field inside solenoid: Field outside solenoid = 0 at location of free charges in the 15 turn coil. So: Lorentz force can’t cause current there. Magnetic flux in solenoid is also linked to outer coil and is proportional to solenoid’s cross-section area (not the outer coil’s). What causes induced current to flow in outer coil? Changing magnetic flux linked to outer coil must create electric field that drives induced current in outer coil (like other currents). Outer coil also creates its own field Bind, owing to induced current.

Examples: A rectangular loop is moving through a uniform B field DOES CURRENT FLOW? Segments a-b & d-c create equal but opposed motional EMFs in circuit. No EMF from b-c & a-d or FLUX in loop abcd remains constant Net flux is constant No current flows v + - a b c d 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 or FLUX is DECREASING v + - a b c d Which way does current flow? CW

Lenz’s Law Example: As loop crosses 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 v

Directions of induced fields and currents Replace moving magnet with circuit No actual motion, switch closed Current i1 creates field B1 (RH rule) Flux is constant if current i1 is constant Changing current i1 --> changing flux in loop 2. Current i2 flows only while i1 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 Several ways to change flux: change |B| through a coil change area of a coil or loop change 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 is constant. 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 external B field into the page Loop area & FLUX are increasing Lenz’s Law says induced field Bind is out of page. RH rule says current is CCW Slider acts like a battery (Lorentz) Due to the induced current: Opposes the motion of the wire External force FEXT is needed to keep wire from slowing down Drag force FM must act on slider FEXT needed for constant speed is opposed to Fm with equal magnitude 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 resistance 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 region from the far left 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 region from left to right B

induced dipole in loop opposes flux growth Induced EMF creates induced current that creates induced magnetic flux opposing the change in magnetic flux that created them. Lenz’s Law Again front of loop induced dipole in loop opposes flux growth induced dipole in loop opposes flux decrease The induced current and field try to keep the original magnetic flux through the loop from changing. front of loop

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 (Produce Sinusoidal 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. Induced EMF is the time derivative of the flux Eind is sinusoidal – polarity alternates over a cycle Maxima/minima when wt = +/- (2n+1)p/2 Eind peaks when dipole moment is perpendicular to Field.

No mechanical reversal of output E Mechanical Reversal of output E AC Generator No mechanical reversal of output E DC Generator Mechanical Reversal of output E

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

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 Magnetic flux change induces “back EMF” (forward in this case). Back EMF causes E field in coil turns, which induces current & magnetic field opposing external field change. The Lenz’s law effect is analogous to mechanical inertia.

Transformer Principle Primary current is changing (for a short time) after switch closes.  changing flux through primary coil ....that links to.... changing flux through secondary coil  induced secondary current & EMF PRIMARY SECONDARY Iron ring strengthens flux linkage

conducting loop, resistance R Changing magnetic flux directly induces electric fields conducting loop, resistance R Bind A thin solenoid, n turns/unit length, cross section A Zero B field outside solenoid (it is ideal) Field inside solenoid: Flux through the wire loop: Changing current I  changing flux  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 outside solenoid, so it’s not Lorentz force acting on charges An induced electric field Eind along the loop causes current to flow It is caused directly by dF/dt within the loop path Eind is there even without the wire loop (no current flowing) Electric field lines are loops that don’t terminate on charge. E-field is a non-conservative (non-electrostatic) field inasmuch as the line integral (potential) around a closed path is not zero What makes the induced current I’ flow, outside solenoid? Generalized Faradays’ Law (hold loop path constant)

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

EXTRA CREDIT Example: EMF generated by Faraday Disk Dynamo Conducting disk, radius R, rotates at rate w in uniform constant field B, FLUX ARGUMENT: evaluate areal velocity w, q + dA = area swept out by radius vector in dq = fraction of full circle in dq x area of disk MOTIONAL EMF FORMULA: (work/unit charge) For points on rotating disk: v = wr, vXB = E’ is radially outward, so is ds = dr current flows radially out EMF induced across length ds of moving conductor in B field vXB looks like electric field to moving charge (Equation 29.7) Radial conductor length