Question

Your job is to design and assess a personal power generator based on the Faraday Effect. The coil will be used to charge a capacitor which will be used to power an MP4 player (an i-thingy). The generator will use a small rare earth magnet with a magnetic field strength of B=1.0 T. The magnet is L_M=0.50 inches on a side and is mounted on a shaft so that it can rotate as shown. The magnet sits in the center of a fixed square coil of wire with inner side length of L_C=0.51 inches. The magnetic field lines come out normal to the larger flat surfaces of the magnet. Assume that the magnet and coil are both thin enough so the geometry is simple and the fields continue straight out from the surface beyond the coil wires (do not diverge).

a) (2 pts) The magnet rotates at angular frequency ω. Write an expression for the flux through a single loop as a function of time in terms of some or all of the variables ω, L_m, L_c, and B. (It should include a sine function.)

b) (2 pts) The magnet can be rotated by a hand crank at 300 revolutions per minute. Find the amplitude of the emf that can be generated with a single turn in the coil in terms of some or all of the variables ω, L_m, L_c, and B..

c) (3 pts) How many turns (N) should the coil have in order to generate an amplitude of 8 V at the rotation speed in part b? Give both logic and a number here.

d) (3 pts) The i-thingy requires 3 V and consumes 0.03 W when on. During operation the output voltage of the capacitor drops from 5 V to 3 V. You want it to operate 1 hour on one charge. How much energy must be stored in the capacitor when it is charged to 5V? Give equation/logic and a numerical answer.

e) (2 pts) What value of capacitance is required to power the i-thingy for 1 hour? Give equation/logic and a numerical answer.

f) (2 pts) For what fraction of a sinusoidal cycle of amplitude 8 V does the voltage exceed 5 V? (We can add a Zener diode to the circuit so that current only flows into the capacitor when voltage from the coil exceeds positive 5 V.)

g) (2 pts) Assuming that the coil generates a maximum current i what is the maximum value of the torque required to turn the magnet? (The torque is due to two sides of the coil moving at right angles through the field when the magnet face normal is at right angles to the coil normal.)

h) (2 pts) A reasonable force that you can supply to the handle of a 2” long crank is about 1 pound (hah! English units). What maximum current does this correspond to?

i) (2 pts) Assuming that the time averaged values of the voltage and current delivered to the capacitor are ½ of their maximum values, how long does it take to fully charge the capacitor cranking at 300 rpm?

Answer 1