Use Separate paper for solutions.

Show all work. Draw diagrams and explain answers where appropriate. Partial credit is available. 10 points each.

1) The magnetic field between the metal rails as shown below is uniform and directed into the page. You are looking down. The rails lie on a table and the field has magnitude 0.2 T. The metal bar can slide with minimal friction while still maintaining electrical contact at both ends with the rails. Someone pushing (or pulling ) the bar can induce current in the external resistor attached as shown. The bar and rails have negligible resistance.

a) How much current flows in the resistor if the rate of thermal energy produced in the resistor is 5 Watts? Determine the induced EMF and the speed (assumed constant) at which the bar is sliding.

b) In which direction does the bar have to be pulled to produce a counterclockwise flowing current (looking into the page)? Explain how you know.

c) Draw a free body diagram of the bar while it is being pushed as in part b. Clearly label all forces. Now imagine that the person doing the pushing lets go. Draw a free body diagram of the bar immediately after the person lets go. Will the bar accelerate under these conditons? If so, in which direction? How do you know?

2) A 0.3 kg bar slides vertically down a set of rails at a terminal speed v while a 1.24 m section of the bar is immersed in a uniform magnetic field of magnitude 0.4 T directed horizontally ( out of the page). The resistance of the bar is 0.2 Ω . The rails have negligible resistance.

a) Determine v.

b) In which direction does the induced current flow? Be specific and explain how you know.

c) Compare, through calculation, the amount of work done on the bar by gravity to the thermal (internal) energy generated in the bar and the rails as the bar falls 0.5 m.

3) The magnet below is near a coil whose ends have been "shorted" so that an induced Voltage can develop. Someone either holding the magnet still or moving it is able to produce the graph of flux versus time shown (by either moving the magnet or holding it still!). Flux into the coil (opposite to the x axis) is considered negative. In addition, this flux probably only passes through those loops of the coil near its face (but no matter).

a) Describe what the person did with the magnet during each of the intervals marked on the graph (A to B, B to C etc.) and explain how you know using a diagram.

b) Sketch a graph of the induced EMF in the turns of coil subject to the flux shown for each of the marked intervals. Explain your reasoning. Note it may be zero for certain interval(s). Show that too.

4) The larger circle in the diagram below represents a coil that is part of a circuit

containing a DC voltage source. The smaller circle represents a second coil that is placed inside the larger coil. It is also part of a circuit, but this circuit does not include a voltage source. The second coil has 400 turns of wire and a diameter of 15 cm. Initially the larger coil is disconnected from the voltage source but as soon as a connecting switch is closed the current in the coil begins to rise according to: I(t) = 0.6amps(1- e^-at ) where t is the time and a = 0.34 s^-1. The resulting magnetic field (directed out of the page) depends on I according to: B(I) = (0.47 Tesla/amp)I and is uniform over the cross section of the larger coil.

a) Derive a formula for the induced EMF in the smaller coil as a function of time.

b) In which direction will current flow in the smaller coil? Explain how you know.

c) Will this induced current increase, decrease or stay the same over time? Explain your answer.