Written Problem: Induction from a falling magnet
We have a small magnet with a magnetic moment of m = 0.1 Am2 (remember: magnetic moment is defined as m = IA - see page 932 of book for the definition). We also have coils of wire. The coils are made out of 100 circular loops of a single wire. A single loop has a radius of 10 cm. The thickness of the wire has a circular cross section with a 0.5 mm radius. The coiled wire is made of copper. (This question can be VERY complicated. Do not try to apply Lenz’s Law. Only consider what the magnet does to the loops, not the other way around. Also see section 28.5 in book (page 930) “MAGNETIC FIELD OF CIRCULAR CURRENT LOOP” for clues on part b). )
a) What is the total resistance of the total length of coiled wire?
The magnet is dropped, north pole first, from a height of zo = 2 meters above the coil. The coils are oriented so that the magnet should pass straight through the middle.
b) Approximately, what is the strength and direction of the magnetic field at the center of the coils created by the magnet? (hint: Treat the magnet as a loop of wire. Use the result for the B-field created by a loop of wire and approximate it for cases where the distance from the magnet is significantly larger than the size of the magnet)
c) What is an expression for the velocity of the magnet at any distance z away from the coils (hint: back to General Physics I)?
d) What is an expression of the change of the magnetic field per time that is created by the magnet evaluated in the middle of the coils at any distance z away from the magnet? (hint: take a derivative of B with respect to time and don’t forget to look for velocity in the result. Make an approximation that the B-field doesn’t change across the area of each loop.)
e) What is an approximate expression for the induced current in the coils?
f) What is the current induced in the loops when the magnet is a distance of z = 1 meter from the loops?
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