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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 Copyright R. Janow – Fall 2015 Previously: Current-lengths in a magnetic field feel forces and torques Fmag qv B Fm i L B Force on charge and wire carrying current N i A n̂ B Um B torque and potential energy of a dipole q B Current-lengths (changing electric fields) produce magnetic fields Biot-Savart Law 0 id s r̂ dB 4 r 2 B due to long straight wire carrying a current i: B at center of circular loop carrying a current i : Long Solenoid field: B outside = 0 Binside 0in B on the symmetry axis of a current loop (far field): B ds 0ienc Ampere’s Law 0i 2 r i B 0 2R Current loops are elementary dipoles B B inside a torus carrying a current i : 0iN B 0 B(z) 2 z 3 2 r NiAˆ Next: Changing magnetic flux induces EMFs and currents in wires Copyright R. Janow – Fall 2015 Magnetic Flux: Electrostatic Gauss Law S q E dA enc 0 dB B dA B n̂dA over surface (open or closed) B B dA Magnetic Gauss Law S B dA 0 defined analogously to flux of electric field B n̂ Flux Unit: 1 Weber = 1 T.m 2 n̂ n̂ q n̂ n̂ Copyright R. Janow – Fall 2015 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 B (Lenz’ Law) Copyright R. Janow – Fall 2015 CHANGING magnetic flux induces EMFs and currents in wires Key Concepts: • Magnetic flux B B B A B n̂ • Faraday’s Law of Induction E induced EMF dB in closed loop dt • Lenz’s Law An induced EMF and current creates it' s own magnetic field, opposing changes in the existing flux Basis of generator principle: • Loops rotating in B field generate EMF and current. Copyright R. Janow – Fall 2015 • Lenz’s Law Applied torque is needed (energy conservation) Motional EMF: Lorentz Force on moving charges in conductors • • • • • Uniform magnetic field points away from viewer. FB 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. Charge separation an induced electric field Eind inside wire Charges come to equilibrium when the forces on charges balance: qEind qvB • or FE q E q v B + L v Eind vB Electric field Eind in the wire corresponds to potential difference Eind across the ends of wire: - Eind EindL BLv • FIELD BOUNDARY Potential difference Eind is maintained between the ends of the wire as long as the wire continues to move through the magnetic field. + FBD of free charge q in a wire • Induced EMF is created without batteries. • Induced current flows only if the circuit is completed. Fm v Fe Eind - Copyright R. Janow – Fall 2015 Flux approach to Motional EMF in a moving wire Eind BLv via Lorentz force Connection with Flux: The rate of flux change = field B x rate of sweeping out area (B uniform) Rate of sweeping out area dA/dt Ldx / dt Lv d d (BA) BdA BLv so dt dt dt Eind d B dt changing flux Flux arguments apply generally, for example: E is induced directly in 15 turn coil by changing flux in solenoid • B 0in inside solenoid • B = 0 outside, at location of charges forming the current in the coil. • 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 Copyright R. Janow – Fall 2015 cross-section area (not the coil’s) A rectangular loop is moving in a uniform B field b + + + + + + + DOES CURRENT FLOW? c • v a - - - - - - - d • • • • Segments a-b & d-c create equal but opposed EMFs in circuit No EMF from b-c & a-d or FLUX is constant d m No current flows dt 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 d m • FLUX is DECREASING 0 dt b + 0 c v a - Which way does current flow? What is different when loop is entering field? d E dB dt Copyright R. Janow – Fall 2015 flux changing Direction of induced fields and currents v Replace magnet with circuit below Ammeter • No actual motion • Current i1 creates field B1 • Flux change in loop 1 causes induction • Changing current i1 --> changing flux in loop 2. • Flux is constant if current is constant • Current i2 flows only while i1 (flux 1) is changing (after switch S closes or opens) B1 Induced current i2 creates it’s own induced field B2 whose flux 2 opposes the change in 1 (Lenz’ Law) 1 = B1A Close S dB1 0 (B1 growing) dt i2 B2 2 = B2A Open S later dB1 0 (B1 decreasing) dt i2 Induced Dipoles B2 Copyright R. Janow – Fall 2015 Faraday’s Law: Changing Flux “-” sign for Lenz’s Law uniform B Induced EMF (Volts) n̂ m B n̂A dB Eind dt insert N for multiple turns in loop, loop of any shape EMF=0 Several ways to change flux: B(t) • 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: uniform dB/dt Rate of flux change (Webers/sec) slope EMF B through a loop increases by 0.1 Tesla in 1 second: Loop area A = 10-3 m2 is constant. Find the induced EMF B B 0.1 x 10-3 Eind A t t 1 front Bind Eind 10-4 volts iind Eind / R Current iind creates field Bind that opposes increase in B Copyright R. Janow – Fall 2015 Slidewire Generator Slider moves, increasing loop area (flux) Induced EMF: • Slider: constant speed v to right • Uniform external field into the page • Loop area & FLUX are increasing Eind + dm dA dx B BL BLv dt dt dt • Slider acts like a battery • Lenz’s Law says induced field Bind is out of page. • RH rule says current is CCW iind iind Eind / R R total resistance in circuit FM DRAG FORCE on slider Fm iind L B | Fm | iindLB Is due to the induced current: • opposes the motion of the wire FEXT (an external force) is needed to keep wire from slowing down L - x FEXT needed is opposed to Fm Eind B 2L2 v Fext Fm iind LB LB R R Mechanical power supplied & dissipated: B2L2v2 Eind2 Power Fextv R R Copyright R. Janow – Fall 2015 Slidewire Generator: Numerical Example + B = 0.35 T L = 25 cm v = 55 cm/s a) Find the EMF generated: Eind iind L v - dm BLv (0.35)(. 25)(. 55) 48 mV dt iind DIRECTION: Bind is into slide, iind is clockwise b) Find the induced current if R for the whole loop = 18 W: E iind ind 48 mV / 18 W 2.67 mA clockwise R c) Find the thermal power dissipated: 2 Eind (48 x 103 ) 2 P 1.28 x 10- 4 Watts R 18 2 P iindR (2.67 x 10-3 )2 x 18 1.28 x 10-4 Watts d) Find the power needed to move slider at constant speed Pmech Fv iLBv 2.67 x 10-3 x 0.25 x 0.35 x 0.55 1.28 x 10-4 Watts Copyright !!! Power dissipated via R = Mechanical power !!!R. Janow – Fall 2015 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 Eind dB d {BA cos(q)} dt dt Copyright R. Janow – Fall 2015 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 opposes flux growth induced dipole in loop opposes flux decrease The induced current tries to keep the original magnetic flux through the loop from changing. front of loop front of loop Copyright R. Janow – Fall 2015 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 v d 0 dt iind 0 v d is dt into slide B = 0 v d 0 dt iind 0 v v d is dt out of slide Bind is out Bind is down iind is CCW iind is CW d 0 dt iind 0 Copyright R. Janow – Fall 2015 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 B What is the direction of the induced current in the loop of wire? 0i 2 r A. Clockwise B. Counterclockwise C. Zero D. Impossible to determine E. I will agree with whatever the majority chooses v 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 E. I will oppose whatever the majority chooses I Copyright R. Janow – Fall 2015 Generator Effect (Sinusoidal AC) Changing flux through a rotating current loop angular velocity w 2f in B field: n̂ q B Rotation axis is out of slide q wt i.e., n̂ is along B when t 0 B B n̂A BA cos(q) BA cos(wt) max cos(wt) peak flux magnitude when wt = 0, , etc. EMF induced is the time derivative of the flux Eind dB BAw sin(wt ) E0 sin(wt ) dt E0 BAw is the peak value of the induced EMF Eind has sinusoidal behavior - alternating polarity over a cycle maxima when wt = +/- /2 Copyright R. Janow – Fall 2015 AC Generator DC Generator Back-torque =xB in rotating loop, ~ N.A.iind Copyright R. Janow – Fall 2015 AC Generator, continued DC Generator, continued No reversal of output E Reversal of output E Copyright R. Janow – Fall 2015 Numerical Example Flat coil with N turns of wire B N turns n̂ Bind B E dB Eind N dt 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 d tot dt B 2.0 2 4 2 2 Area Number of turns 3 cm 10 m /cm 1000 t 1 / 120 d tot dt 72 Volts Eind 72 Volts 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 Copyright R. Janow – Fall 2015 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 Copyright R. Janow – Fall 2015 Changing magnetic flux directly induces electric fields A thin solenoid, cross section A, n turns/unit length • zero B field outside solenoid B 0In • inside solenoid: Flux through a conducting loop: conducting loop, resistance R BA 0nIA Current I varies with time flux varies EMF is induced in wire loop: d dI 0nA dt dt E I' ind induced in the loop is: R Eind Current Bind If dI/dt is positive, B is growing, then Bind opposes change and I’ is counter-clockwise What makes the induced current I’ flow, outside solenoid? • • • • • • B = 0 outside solenoid, so it’s not the Lorentz force An induced electric field Eind along the loop causes current to flow It is caused directly by d/dt within the loop path Eind 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 Eind dB E d s loop ind dt Generalized Faradays’ Law Janow – Fall 2015 (hold loopCopyright path R. constant) Example: Find the electric field induced by changing magnetic flux Eind B ds 0ienc dB Eind ds loop dt 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. Eind E ds Eds E ds E(2r ) For r > R: B BA B(R 2 ) dB E(2r ) dB / dt R 2 dt For r < R: B BA B(r 2 ) R 2 dB E 2r dt r dB E 2 dt dB dt The magnitude of induced electric field grows linearly with r, then falls off as 1/r for r>R E(2r ) dB / dt r 2 Copyright R. Janow – Fall 2015 Example: EMF generated by Faraday Disk Dynamo Conducting disk, radius R, rotates at rate w in uniform constant field B, FLUX ARGUMENT: Eind dB dA B dt dt areal velocity dA = area swept out by radius vector in dq = fraction of full circle in dq x area of disk dq R2 2 dA R 2 dq R 2 dA R dq w 2 2 dt 2 dt 2 Eind w, q 1 BwR 2 2 + USING MOTIONAL EMF FORMULA: Emf induced across conductor length ds dE v B ds Eind 0 (Equation 29.7) Conductor moving transversely sees vXB as electric field E’ For points on rotating disk: v = wr, R Radial conductor length v B ds R 0 vXB = E’ is radially outward, so is ds = dr Bwrdr 1 BwR 2 2 current flows radially out Copyright R. Janow – Fall 2015