# Copyright R. Janow – Fall 2014 Physics 121 - Electricity and Magnetism Lecture 11 - Faraday’s Law of Induction Y&F Chapter 29, Sect. 1-5 Magnetic Flux.

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Copyright R. Janow – Fall 2014 Physics 121 - Electricity and 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

Copyright R. Janow – Fall 2014 Previously: Current-lengths in a magnetic field feel forces and torques  Force on charge and wire carrying currenttorque and potential energy of a dipole Next: Changing magnetic flux induces EMFs and currents in wires Current-lengths (changing electric fields) produce magnetic fields Ampere’s LawBiot-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 a current i : B outside = 0

Copyright R. Janow – Fall 2014 defined analogously to flux of electric field Electrostatic Gauss Law Magnetic Gauss Law Flux Unit: 1 Weber = 1 T.m 2 over surface (open or closed) B  Magnetic Flux: 

Copyright R. Janow – Fall 2014 Changing magnetic flux induces EMFs EMF / current is induced in loop if there is relative motion between loop and magnet - the magnetic field 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 field. Induced current creates it’s own induced B field and flux, opposing the changing flux  B (Lenz’ Law)

Copyright R. Janow – Fall 2014 CHANGING magnetic flux induces EMFs and currents in wires Key Concepts: Generator principle: Loops rotating in B field generate EMF and current. Lenz’s Law  Applied torque is needed (energy conservation) Faraday’s Law of Induction Lenz’s Law Magnetic flux

Copyright R. Janow – Fall 2014 Motional EMF: Lorentz Force on moving conduction charges 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 E ind inside wire Charges come to equilibrium when the forces on charges balance: Electric field E ind in the wire corresponds to potential difference E ind across the ends of wire: Potential difference E ind is maintained between the ends of the wire as long as the wire continues to move through the magnetic field. Induced EMF is created without batteries. No current flows without rest of circuit Induced current flows if the circuit is completed. + FIELD BOUNDARY L v - FBD of free charge q in a wire v + - FeFe FmFm E ind

Copyright R. Janow – Fall 2014 Flux view of Motional EMF in a moving wire The rate of flux change = field B x rate of sweeping out area (B uniform) Connection with Flux: via Lorentz force Flux arguments apply more generally, for example: E is induced in 15 turn coil directly by changing flux in solenoid inside solenoid B = 0 outside, at location of charges movng in the loop Flux is proportional to solenoid’s cross-section area (not the coil’s)

Copyright R. Janow – Fall 2014 A rectangular loop is moving in a uniform field v +++++++ ------- a b c d DOES CURRENT FLOW? If B-field is uniform, 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 above? What is different when loop is entering field?

Copyright R. Janow – Fall 2014 i1i1 i2i2 Direction of induced fields and currents Ammeter Induced current i 2 creates it’s own induced field B 2 whose flux F 2 opposes the change in  1 (Lenz’ Law) Close S i2i2 Open S later F 1 = B 1 A F 2 = B 2 A i2i2 Induced Dipoles Replace magnet with circuit below Flux change is enough to cause induction Do not need actual motion Let current i 1 be changing -> changing flux. Flux is constant if current is constant Current i 2 flows only while i 1 (flux   ) is changing (after switch S closes or opens)

Copyright R. Janow – Fall 2014 Faraday’s Law 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 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 B(t) EMF=0 slope  EMF Example:B through a loop increases by 0.1 Tesla in 1 second: Loop area A = 10 -3 m 2. Find the induced EMF Current i ind creates field B ind that opposes increase in B uniform dB/dt front

Copyright R. Janow – Fall 2014 Slider moves, increasing loop area (flux) Slidewire Generator + - x i ind L Induced EMF: Slider: constant speed v to right Uniform field into the page Area & therefore FLUX are increasing Slider acts like a battery Lenz’s Law says induced field B ind is out of page. RH rule says current is CCW F M DRAG FORCE on slider due to the induced current opposes the motion of the wire F EXT (an external force) is needed to keep wire from slowing down F EXT needed is opposed to F m Power supplied & dissipated:

Copyright R. Janow – Fall 2014 Slidewire Generator: Numerical Example v + - i ind L B = 0.35 T L = 25 cm v = 55 cm/s a) Find the EMF generated: DIRECTION: B ind is into slide, i ind is clockwise b) Find the induced current if R for the whole loop = 18  c) Find the thermal power dissipated: d) Find the power needed to move slider at constant speed !!! Power dissipated via R = Mechanical power !!!

Copyright R. Janow – Fall 2014 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 Induced Current and Emf B

Copyright R. Janow – Fall 2014 Lenz’s Law The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them front of loop induced dipole in loop induced dipole in loop  The induced current tends to keep the original magnetic flux through the loop from changing. front of loop

Copyright R. Janow – Fall 2014 Lenz’s Law Example: The induced current and EMF create induced magnetic flux that opposes the change in magnetic flux that created them A loop crossing a region of uniform magnetic field B = 0 B = uniform vvvvv

Copyright R. Janow – Fall 2014 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? A. Clockwise B. Counterclockwise C. Zero D. Impossible to determine 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? A. Clockwise B. Counterclockwise C. Zero D. Impossible to determine

Copyright R. Janow – Fall 2014 Generator Effect (AC) Changing flux through a rotating current loop angular velocity w = 2pf in B field: E ind has sinusoidal behavior - alternating polarity maxima when  t = +/-  /2 EMF induced is the time derivative of the flux Rotation axis is out of slide  peak flux magnitude when  t = 0, , etc.

Copyright R. Janow – Fall 2014 DC Generator AC Generator Back-torque =xB in rotating loop,  ~ N.A.i ind

Copyright R. Janow – Fall 2014 AC Generator, continued DC Generator, continued No reversal of output E Reversal of output E

Copyright R. Janow – Fall 2014 Numerical Example Flat coil with N turns of wire Flux change due to external B field produces induced field B ind INDUCED field/flux produces its own EMF. The BACK EMF opposes current change – analogous to inertia 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 cm 2 (one turn) Find the EMF induced in the coil E N turns

Copyright R. Janow – Fall 2014 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 Iron ring strengthens flux linkage PRIMARY SECONDARY

Copyright R. Janow – Fall 2014 Changing magnetic flux directly induces electric fields A thin solenoid, cross section A, n turns/unit length zero field outside solenoid inside solenoid: conducting loop, resistance R 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 B ind opposes change and I’ is counter-clockwise What makes the induced current I’ flow? B = 0 there so it’s not the Lorentz force An induced electric field E ind 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 Generalized Faradays’ Law (hold loop path constant) B ind

Copyright R. Janow – Fall 2014 Assume: dB/dt = constant over circular region. Find the magnitude E of the induced electric field at points within and outside the magnetic field. Due to symmetry and Gauss ’ Law, E is tangential: For r < R: So For r > R: So The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R Example: Find the electric field induced by changing magnetic flux

Copyright R. Janow – Fall 2014 Example: EMF generated Faraday Disk Dynamo dA = area swept out by radius vector in d  = fraction of full circle in d  x area of circle Conducting disk, radius R, rotates at rate  in uniform constant field B, FLUX ARGUMENT: areal velocity  + USING MOTIONAL EMF FORMULA: Emf induced across conductor length ds Moving conductor sees vXB as electric field E’ For points on rotating disk: v =  r, vXB = E’ is radially outward, ds = dr current flows radially out (Equation 29.7)

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