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Faraday's Law

dS. B. B. Faraday's Law. Electrostatics motion of “ q ” in external E -field E -field generated by S q i Magnetostatics motion of “ q ” and “ I ” in external B -field B -field generated by “ I ” Electrodynamics time dependent B -field generates E -field

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Faraday's Law

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  1. dS B B Faraday's Law

  2. Electrostatics motion of “q” in external E-field E-field generated bySqi Magnetostatics motion of “q” and “I” in external B-field B-field generated by “I” Electrodynamics time dependent B-field generates E-field AC circuits, inductors, transformers, etc. time dependent E-field generates B-field electromagnetic radiation - light! General Review

  3. Induction Effects Faraday’s Law (Lenz’ Law) Energy Conservation with induced currents? Faraday’s Law in terms of Electric Fields Cool Applications Today...

  4. 1A 1B × × B B v Lecture 16, Act 1 • Inside the shaded region, there is a magnetic field into the board. • If the loop is stationary, the Lorentz force (on the electrons in the wire) predicts: (a)A Clockwise Current(b)A Counterclockwise Current(c)No Current • Now the loop is pulled to the right at a velocity v. • The Lorentz force will now give rise to: (a)A Clockwise Current (b)A Counterclockwise Current (c)No Current, due to Symmetry

  5. × × B B Move the magnet, not the loop. Here there is no Lorentz force but there was still an identical current. I I -v Decreasing B↓ Other Examples of Induction 1831: Michael Faraday did the previous experiment, and a few others: This “coincidence” bothered Einstein, and eventually led him to the Special Theory of Relativity (all that matters is relative motion). Decrease the strength of B. Now nothing is moving, but M.F. still saw a current:

  6. Conclusion: A current is induced in a loop when: • there is a change in magnetic field through it • this can happen many different ways How can we quantify this? a b Induction Effectsfrom Currents • Switch closed (or opened) • Þ current induced incoil b • Steady state current in coil a • Þno current induced incoil b

  7. Faraday's Law: The emfeinduced in a circuit is determined by the time rate of change of the magnetic flux through that circuit. B B So what is this emf?? Faraday's Law • Define the flux of the magnetic field through an open surface as: dS • The minus sign indicates direction of induced current (given by Lenz's Law).

  8. Disclaimer: In Lect. 5 we claimed , so we could define a potential independent of path. This holds only for charges at rest (electrostatics). Forces from changing magnetic fields are nonconservative, and no potential can be defined! time Electro-Motive Force or emf A magnetic field, increasing in time, passes through the blue loop An electric field is generated “ringing” the increasing magnetic field Circulating E-field will drive currents, just like a voltage difference Loop integral of E-field is the “emf”: Note: The loop does not have to be a wire— the emf exists even in vacuum! When we put a wire there, the electrons respond to the emf, producing a current.

  9. Escher depiction of nonconservative emf

  10. Conservation of energy considerations: Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. Why??? Ifcurrent reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc.. S N N S B B v v • Lenz's Law: • The induced current will appear in such a direction that it opposes the change in flux that produced it. Lenz's Law

  11. A conducting rectangular loop moves with constantvelocity vin the+xdirection through a region of constant magnetic fieldBin the-zdirection as shown. What is the direction of the induced current in the loop? y X X X X X X X X X X X X X X X X X X X X X X X X v X X X X X X X X X X X X X X X X X X X X X X X X 2A x (a) ccw (b) cw (c) no induced current 2B (a) ccw (b) cw (c) no induced current Lecture 16, Act 2 I y • A conducting rectangular loop moves with constant velocityvin the-ydirection and a constant currentI flows in the+xdirection as shown. • What is the direction of the induced current in the loop? v x

  12. A conducting rectangular loop moves with constant velocityvin the+xdirectionthrough a region of constantmagnetic fieldBin the-zdirection as shown. What is the direction of the induced current in the loop? y X X X X X X X X X X X X X X X X X X X X X X X X v X X X X X X X X X X X X X X X X X X X X X X X X 1A 2A x (a) ccw (b) cw (c) no induced current Lecture 16, Act 2 • There is anon-zero fluxFBpassing through the loop since B is perpendicular to the area of the loop. • Since the velocity of the loop and the magnetic field are CONSTANT, however, this flux DOES NOT CHANGE IN TIME. • Therefore, there is NO emf induced in the loop; NO current will flow!!

  13. Suppose we pull with velocity v a coil of resistance R through a region of constant magnetic field B. Direction of induced current? x x x x x x x x x x x x x x x x x x x x x x x x I w v • Magnetic Flux: x Þ • Faraday’s Law: Induced Current – quantitative • Lenz’s Law  clockwise! • What is the magnitude? We must supply a constant force to move the loop (until it is completely out of the B-field region). The work we do is exactly equal to the energy dissipated in the resistor, I2R(see Appendix).

  14. Connect solenoid to a source of alternating voltage. v ~ side view B F · · B · F B top view DemoE-M Cannon • The flux through the area ^ to axis of solenoid therefore changes in time. • A conducting ring placed on top of the solenoid will have a current induced in it opposing this change. • There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field.

  15. For this act, we will predict the results of variants of the electromagnetic cannon demo which you just observed. Suppose two aluminum rings are used in the demo; Ring 2 is identical to Ring 1 except that it has a small slit as shown. LetF1be the force on Ring 1; F2be the force on Ring 2. 3A (c) F2 > F1 (b)F2 = F1 (a)F2 < F1 • Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8W-m); Ring 2 is made of aluminum (resistivity = 2.8X10-8W-m). Let F1 be the force on Ring 1; F2 be the force on Ring 2. 3B (c)F2 > F1 (b) F2 = F1 (a) F2 < F1 Lecture 16, Act 3

  16. AC Generator Water turns wheel  rotates magnet  changes flux  induces emf  drives current “Dynamic” Microphones (E.g., some telephones) Sound  oscillating pressure waves  oscillating [diaphragm + coil]  oscillating magnetic flux  oscillating induced emf  oscillating current in wire Question: Do dynamic microphones need a battery? Applications of Magnetic Induction

  17. Tape / Hard Drive / ZIP Readout Tiny coil responds to change in flux as the magnetic domains (encoding 0’s or 1’s) go by. Question: How can your VCR display an image while paused? Credit Card Reader Must swipe card  generates changing flux Faster swipe  bigger signal More Applications of Magnetic Induction

  18. Magnetic Levitation (Maglev) Trains Induced surface (“eddy”) currents produce field in opposite direction  Repels magnet  Levitates train Maglev trains today can travel up to 310 mph  Twice the speed of Amtrak’s fastest conventional train! May eventually use superconducting loops to produce B-field  No power dissipation in resistance of wires! S N rails “eddy” current 4 More Applications of Magnetic Induction

  19. The magnetic field in a region of space of radius2Ris aligned with thez-direction and changes in time as shown in the plot. What is the direction of the induced electric field around a ring of radiusRat timet=t1? y x x x x x x x x x x x x x x x x x x x x x x x x x x x x 4A x R (c)E is cw (b)E = 0 (a)E is ccw B z t t 1 4B (b)E2R= 2ER (c)E2R= 4ER (a)E2R= ER Lecture 16, Act 4 • What is the relation between the magnitudes of the inducedelectric fieldsERat radiusRandE2Ratradius2R?

  20. Faraday’s Law (Lenz’s Law) a changing magnetic flux through a loop induces a current in that loop Summary negative sign indicates that the induced EMF opposes the change in flux • Faraday’s Law in terms of Electric Fields

  21. The induced current gives rise to a net magnetic force ( ) on the loop which opposes the motion. ' F F x x x x x x x x x x x x x x x x x x x x x x x x I F w 2 2 æ ö wBv w B v v = = = ç ÷ F IwB wB R R è ø ' F x • Agent must exert equal but opposite force to move the loop with velocity v; therefore, agent does work at rate P, where 2 2 2 w B v = = P Fv R 2 2 2 2 æ ö wBv w B v ¢ = = = 2 ç ÷ P I R R R R è ø Appendix: Energy Conservation? • Energy is dissipated in circuit atrate P' Þ P = P' !

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