3/18/2008 1 GENERAL PHYSICS II Electromagnetism & Thermal Physics 3/18/2008 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 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 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. – 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: × B (a) A Clockwise Current (b) A Counterclockwise Current (c) No Current, due to Symmetry × B v 3/18/2008 5 1831: Michael Faraday did the previous experiment, and a few others: Move the magnet, not the loop. Here there is no Lorentz force but there was still an identical current. (This phenomenon can be explained by the relativity principle of motion). v B   × B -v I × B Decreasing B↓ I Decrease the strength of B. Now nothing is moving, but M.F. still saw a current: dB I dt  3/18/2008 6 Instead of a magnet we use a loop with current: • Switch closed (or opened)  current induced in coil b • Steady state current in coil a  no current induced in coil b a 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 7 1.2 Faraday’s law: • Define the flux of the magnetic field through an open surface as:  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) dS B B   SdBΦ B  3/18/2008 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. dt dΦ ε B  The minus sign indicates direction of induced current (given by Lenz's Law). A B (increasing) A B (decreasing) (+) (+) (+) (+) A A B (increasing) B (decreasing) ℰ ℰ ℰ ℰ 3/18/2008 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”.  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 v B S N v B N S 3/18/2008 10 Example 1:  A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. • What is the direction of the induced current in the loop? (a) ccw (b) cw (c) no 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 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 y There is a non-zero flux Φ B 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!! [...]... 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 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... fields just like ordinary conductors exclude electric fields Analogy E=0 inside a ordinary conductor B=0 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 §3 Mutual inductance and self-inductance: 3.1 Mutual inductance: b... increasing flux (Faraday’s Law) this emf acts like a battery to oppose the real battery – reduce current flow] • Fact: the current turns on at such a rate: dI dB   ( a)  ( b) V V dt dt 3/18/2008 23 Self -induction: A change current through a loop induces an opposing emf in that same loop  Then an inductor (a set of coils in some geometry; e.g., solenoid) is characterized by a quantity called self-inductance, . II Electromagnetism & Thermal Physics 3/18/2008 2 CHAPTER XI Electromagnetic Induction §1. Induction experiment - Faraday’s and Lenz’s laws §2. Induced. Magnetic field energy 3/18/2008 3  In the previous chapter we know that: current magnetic field  In this chapter we will study currents produced by changing