# Hompolar generator

## Homework Statement

A conducting circular disc of radius a rotates with an angular frequency w, about its axis in a uniform field of magnetic flux density B parallel to its axis. Show that the potential difference V between the axis and the rim of the disc is w(a^2)B/2.

Such a disc of mass 10^4 kg and radius 3mm is rotating freely at 3000 revs/min in a field of 0.5T. A load of 10^-3 Ohms is suddenly connected between the rim and the axis of the disc. What (neglecting any other resistance in the circuit) is the initial value of the current in the load? How long would it take for the flywheel to slow to half its initial speed in the absense of mechanical friction?

## Homework Equations

I have done the first bit and proven that the potential differential is w(a^2)B/2.

V = IR

## The Attempt at a Solution

I substituted into the formula V = w(a^2)B/2 and worked out that V = 225pi. I then used V=IR and showed that I = 7.1x10^5 A which is the right answer.

However, I don't know how to work out the time for the flywheel to slow to half its initial speed. I tried to work out the force on the flywheel and thus the deceleration but I don't know what formula to use to do this??

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Do you mean 7.1x10^-5 A? 10^5 amps will destroy most conductors immediately.

You can use the flow of current to calculate an induced magnetic field which opposes the static field of 0.5T. That's where the force comes from to slow down the disc.

Ok so then

dF=IBdlsin(theta)

but what's the theta?
and would you integrate dl along the radius of the flywheel?

Also, when you get F, you can divide by m to get the deceleration but then which formula do you use? Is it v=u+at??

The current is flowing from the center of the disc to the edge along a radius, is it not ? This is at theta=90deg to the magnetic field. Yes, integrate along the radius to get F.

Now, you need the rotational version of Newton's laws.
angular acceleration = torque/(moment of inertia)

Dimensionally

[T^-2] = [ML^2T^-2]/[ML^2]

so you see that torque is force x distance. I think if you integrate F*r dr along the radius you get the torque. The MOI of the disc can be calculated from the radius and mass.

This is quite a tough problem, is it coursework at college ?