CHAPTER XI Electromagnetic Induction

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CHAPTER XI Electromagnetic Induction

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1.1 Induction experiment: 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:

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  1. GENERAL PHYSICS II Electromagnetism & Thermal Physics 3/18/2008 1
  2. CHAPTER XI Electromagnetic Induction §1. Induction experiment - Faraday’s and Lenz’s laws §2. Induced electromotive force and induced electric fields §3. Mutual inductance and self-inductance §4. Magnetic field energy 3/18/2008 2
  3.  In the previous chapter we know that: current magnetic field  In this chapter we will study currents produced by changing magentic fields. In other words, changing magnetic field current 3/18/2008 3
  4. §1. Induction experiment - Faraday’s and Lenz’s laws: 1.1 Induction experiment: Inside the shaded region, there is a × magnetic field into the board. B – 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 v (b) A Counterclockwise Current (c) No Current, due to Symmetry 3/18/2008 4
  5. 1831: Michael Faraday did the previous experiment, and a few others:  × Move the magnet, not the loop. Here -v B I there is no Lorentz force v  but B there was still an identical current. (This phenomenon can be explained by the relativity principle of motion). Decrease the strength of B. Now × nothing is moving, but M.F. still saw B I a current: dB I dt Decreasing B↓ 3/18/2008 5
  6. Instead of a magnet we use a loop with current: a • Switch closed (or opened) b  current induced in coil b • Steady state current in coil a  no current induced in coil b What is the cause of the currents induced in the loops in the mentioned experiment? Conclusion: A current is induced in a loop when: • there is a change in magnetic field through it • this can happen many different ways 3/18/2008 6
  7. 1.2 Faraday’s law: • Define the flux of the magnetic field through an open surface as:   dS B d Φ  BS B B  The meaning of the integral in the r.h.s.: the total number of magnetic lines through the surface.  Note that we must fix the choice of the surface’s normal vector. Either choice is equally good, but once we make the choice we must stick with it. • Recall the analogous formula for the flux of the electric field • Magnetic field lines is drawn by the same rule as electric field lines) 3/18/2008 7
  8. Faraday's law: The emf induced in a circuit is determined by the time rate of change of the magnetic flux through that circuit. dΦB ε   dt The minus sign indicates direction of induced current (given by Lenz's Law). A B A (+) B (+) (increasing) (decreasing) ℰ ℰ A (+) A (+) ℰ ℰ 3/18/2008 8 B (increasing) B (decreasing)
  9. 2.3 Lenz's Law: • The direction of induced currents is determined by the Lenz's Law: “The induced current will appear in such a direction that it opposes the change in flux that produced it”. B B S N N S v v  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??? • If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc.. 3/18/2008 9
  10. Example 1: y  A conducting rectangular loop moves with XXXXXXXXXXXX constant velocity v in the +x direction XXXXXXXXXXXX through a region of constant magnetic field X X X X X X Xv X X X X X B in the -z direction as shown. XXXXXXXXXXXX • What is the direction of the induced x current in the loop? (a) ccw (b) cw (c) no induced current There is a non-zero flux ΦB passing 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!! 3/18/2008 10
  11. Example 2: A conducting rectangular loop moves with constant velocity v in y I the -y direction and a constant current I flows in the +x direction as shown. What is the direction of the v induced current in the loop? x (a) ccw (b) cw (c) no induced current The flux Φ is decreasing → the induced current must create a B induced magnetic field which directs along the manetic field of the current I 3/18/2008 11
  12. Example on the calculation of induced current: xxxxxx I Suppose we pull with velocity v a coil of resistance R through a region of constant xxxxxx magnetic field B. w xxxxxx v  Direction of induced current? Lenz’s Law  clockwise! xxxxxx  What is the magnitude? x  Magnetic Flux: Φ B  B Area Bwx dΦB ε wBv Bw v  I   dx  Faraday’s Law: ε   Bw dt dt R R 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). 3/18/2008 12
  13. §2. Induced electromotive force and induced electric fields: 2.1 Induced electric fields: time A magnetic field, increasing in time, passes through the blue loop What force makes the charges move around the loop? • It can’t be a magnetic force because the loop isn’t moving in a magnetic field and even isn’t lying in a magnetic field • The force exerting on charges exists in any case while the loop is large or small, even while the loop does not have to be a wire. We are forced to conclude that the curent is caused by a electric field which is called induced electric field. 3/18/2008 13
  14. Note: The wire isn’t the causion of induced emf, it is only the device which help us observe the current, the emf exists even in vacuum, and this emf is related to the induced electric field! The equation for the induced electric field: The work done by the induced electric field E per unit charge is equal to the induced emf, so we have (the integration path is stationary) Between the electrostatic and induced electric fields there are radical differences: • Recall that the electrostatic field is conservative (a work done by the field depends only the intial and final positions; over a closed path = 0). The induced electric field is nonconservative → For induced electric fields we can not introduce the concept of potential. • The electrostatic field is produced by a static charge distribution. The induced electric field can not be produced by any static charge distribution, it can be produced by a changing magnetic field. 3/18/2008 14
  15. 2.2 Eddy currents: Not only in conducting wires, induced currents appear in pices of metal moving in magnetic field or located in changing magnetic fields. We call these eddy currents. Turn on the magnet → the pendulum motion is arrested. The eddy currents act to oppose the change in magnetic flux. Eddy currents are drastically reduced if the metal plate is replaced by one which has several cut slots. 3/18/2008 15
  16. 2.3 Meissner effect for superconductors: • Superconducitivity: The property of some materials that the resistance becomes zero at temperatures under a critical one T < Tc, (T c ~ some oK ). • Superconductors have not only this property, they also have extraordinary magnetic properties. An important property is the Meissner effect. Consider a magnet on a sample of superconducting material: • Above the critical temperature (T > Tc ), the motion of magnet produces induced currents in the sample, but these currents die away due the resistance. • Cool the sample below Tc → the magnet lifts off and hovers above the superconductor. 3/18/2008 16
  17.  Why does the magnet hover ? This phenomenon can not be explained by Faraday’s law. It is a property called the Meissner effect. It is caused by the fact that superconductors exclude magnetic fields just like ordinary conductors exclude electric fields. Analogy E=0 B=0 inside a ordinary conductor inside a superconductor In oder to keep the magnetic field inside zero, in the superconductor there exists a current that cancels out the field due to magnet. The magnetic field of this current caused the magnet to hover !! 3/18/2008 17
  18. §3. Mutual inductance and self-inductance: 3.1 Mutual inductance: a b A current is induced in one coil when the current is changed in a neighboring coil We can describe this effect quantitatively in terms of the concept of mutual inductance. 3.1.1 Definition of mutual inductance: • The induced emf in a coil with N2 turns producing by the change of the current I1 : 3/18/2008 18
  19. • Calculate the flux through coil 2: According to the Bio-Savart law: we have where • Finally, This equation can be used as the definition of mutual inductance M21 3/18/2008 19
  20. 3.1.2 Properties of mutual inductance: • M21 depends only on the shapes and the relative positions of the coils, not on the current in coil 1 or on time. • The induced emf in coil 1 producing by the change of the current I2 in coil 2 has the analogous formula: If • It turns out that two coefficients always equal each to other and we have for emf in two coils: • We have also 3/18/2008 20
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