Electromagnetism problem?

The key to this is the fact that the rod is falling but not accelerating which means that the force of gravity is offset by the Lorenz force:

Fg = FL ----> mg = i L B where L = length of conducting rod & B = strength of magnetic field. i is teh currnet flowing in the rod.

Now the trick is to find i. For that you use Lenz's law:

V = -d(B*A)/dt and Ohms Law V = i R ----> i = -1/R d(B*A)/dt

Now B is constant and A = area defined by the copper tracks and conductor. A can be written as

A = v*t*L where v = speed the rod is moving with, t = time, and L = length of rod so we can solve for i

i = -1/R B*v*L = -BvL/R

Then the force law becomes:

mg = vB^2L^2/R (drop the "-" sign - it indicates current direction relative tot he direction of change in the magnetic flux)

The mass of the rod is m = v(BL)^2/(gR) = 0.338 kg

Now the energy to power this circuit comes from the gravitational potential energy of the rod. Since the rod falls at a constant speed, all the change in potential goes into the electrical energy. So we can equate the change in potential, dU to the electrical energy generated over a period of time, dt.

dU = i^2R dt now we computed i to be i = BvL/R = 4.5 A

So for dt = 0.17 sec, we find dU to be

dU = (4.5)^2 * 0.85 *0.17 J = 2.93 J

THis is exactly the same as the electrical energy dissipated in teh resistor.