ps09sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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PS09-1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Spring 2010 Problem Set 9 Solutions Problem 1: Short Questions (a) When a small magnet is moved toward a solenoid, an emf is induced in the coil. However, if the magnet is moved around inside a toroid, no measurable emf is induced. Explain. Moving a magnet inside the hole of the doughnut-shaped toroid will not change the magnetic flux through any turn of wire in the toroid, and thus not induce any current. (b) A piece of aluminum is dropped vertically downward between the poles of an electromagnet. Does the magnetic field affect the velocity of the aluminum? Explain. Yes. The induced eddy currents on the surface of the aluminum will slow the descent of the aluminum. It may fall very slowly. (c) What happens to the generated current when the rotational speed of a generator coil is increased? The maximum induced emf will increase, increasing the terminal voltage of the generator resulting in a larger amplitude for the current. (d) If you pull a loop through a non-uniform magnetic field that is perpendicular to the plane of the loop which way does the induced force on the loop act? The direction of the induced force is opposite the direction of the pulling force.
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PS09-2 Problem 2: Moving Loop A rectangular loop of dimensions l and w moves with a constant velocity v G away from an infinitely long straight wire carrying a current I in the plane of the loop, as shown in the figure. The total resistance of the loop is R . (a) Using Ampere’s law, find the magnetic field at a distance s away from the straight current-carrying wire. Consider a circle of radius s centered on the current-carrying wire. Then around this Amperian loop, 0 (2 ) dBs I π μ ⋅= = Bs r r ± which gives s I B 2 0 = (into the page) (b) What is the magnetic flux through the rectangular loop at the instant when the lower side with length l is at a distance r away from the straight current-carrying wire, as shown in the figure? 00 ln 22 rw B r S II l dl d s sr μμ ππ + ⎛⎞ + Φ= ⋅ = = ⎜⎟ ⎝⎠ ∫∫ BA r r (into the page) (c) At the instant the lower side is a distance r from the wire, find the induced emf and the corresponding induced current in the rectangular loop. Which direction does the induced current flow? The induce emf is 2 2( ) 2 ( ) B Il dr w d r v w d t rw r d t r ε =− Φ =− = + + The induced current is
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PS09-3 ) ( 2 0 w r r w v R l I R I + = = π μ ε The flux into the page is decreasing as the loop moves away because the field is growing weaker. By Lenz’s law, the induced current produces magnetic fields which tend to oppose the change in magnetic flux. Therefore, the current flows clockwise, which produces a self-flux that is positive into the page.
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PS09-4 Problem 3: Faraday’s Law A conducting rod with zero resistance and length w slides without friction on two parallel perfectly conducting wires. Resistors R 1 and R 2 are connected across the ends of the wires to form a circuit, as shown .
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ps09sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department...

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