# Problem Set 13 (Given 02/19/12)

## Level I

1. Find the magnetic flux through the cylinder defined by $$x^2 + y^2 = 1$$ of the magnetic field $$\mathbf{B} = B_x \hat{x} + B_z \hat{z}$$.
2. Two unconnected wires are arranged as concentric rings, the larger with radius $$R$$ and the smaller with radius $$r$$. The outside, larger wire has a voltage of $$1\,V$$ applied to it, and thus carries a current. However, the outside wire is slowly heated, so that its resistance increases as $$R(t) = R_0 (1 + \alpha t)$$. Is there any current in the inner, smaller wire? If so, in which direction — the same as the outer wire, or the opposite?
3. A magnet is inserted into a solenoid, north end first. Which way does the induced current flow? Along the magnet's movement or against it? Recall that the magnetic field points from the south to the north end of a magnet.
4. A DC generator uses square windings (the loops of wire placed between the two static magnets) that is 1 meter to a side. If the static magnets generate a field of $$0.2\,T$$ and the generator can rotate the armature at 100 revolutions per second, how many loops of wire must the windings have to generate $$240\,V$$ of electricity?

## Level II

1. In class, we discussed self-inductance, noting that all electrical devices have a coefficient $$L$$ called their inductance which is governed by the law $$V_{ind} = -L \frac{d I}{d t}$$. By equating that to Faraday's law ($$V_ind = - \frac{d \Phi_B}{d t}$$), find the inductance $$L$$ of a solenoid with $$N$$ turns, length $$l$$, and cross-sectional area $$A$$.
2. Problem 81 in Giancoli, Chapter 21 (Page 614)

## Level III

1. When we discussed motors, we noted that a constant current ought to produce a constant torque on the windings in the motor, and thus you might expect the ideal motor to accelerate indefinitely. This does not happen, due to a phenomenon known as back EMF. As the motor turns, the windings continuously change their angle to the magnetic field, and thus change the flux through them. Thus, by Faraday's law, there should a voltage opposing the motor's turning being induced, a voltage that is called the back EMF.

With this information, explain why, when you first start up a powerful motor (say, in a power-wash machine or a power generator), the lights in your house might momentarily dim, but then when the motor ramps up the lights go back to their normal brightness.

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