The Faraday disc (unipolar generator) and Lenz' law

[Niles]

Homework Statement

Hi all.

I can't seem to understand how to apply Lenz' law to a disc rotating in a magnetic field that is perpendicular to the discs plane (see first page on PDF):

http://www.pa.uky.edu/~latimer/phy232s08/lectures/phy232l31_4.pdf

I can see that using the right-hand rule, positive charge will build up on the edge and negative at the center. So an EMF will run from the outer edge to the center, but I can't seem to relate this to Lenz' law. The change in magnetic flux will just get larger because of the induced EMF, and not opposed?

 

[dynamicsolo]

I can see that using the right-hand rule, positive charge will build up on the edge and negative at the center. So an EMF will run from the outer edge to the center, but I can't seem to relate this to Lenz' law. The change in magnetic flux will just get larger because of the induced EMF, and not opposed?

If you've seen the analysis for "motional EMF" that is done for a conductive bar sliding on conductive rails in a uniform perpendicular magnetic field, you'll find this explanation is analogous.

For the rotating disc, imagine a specific radial line. Let's picture things so that you are facing one side of the disc with the magnetic field coming toward you, and the disc is rotating counter-clockwise. The imaginary radial line will be sweeping out a sector of a circle with area
(1/2)·(r^2)·(theta), where the angle theta is the angle through which the disc has rotated, starting from the moment you choose to be t = 0. You would use this to work out the rate of flux change for Faraday's Law, etc.

Now think of the boundary of this circular sector (shaped like a wedge). The amount of magnetic flux toward you is increasing in time. Using Lenz' "Law", we would argue that induced current must flow clockwise along the boundary of the sector so that magnetic flux away from you would be generated, in order to oppose the change in flux through the imaginary sector. On the side of the boundary where our imaginary radial line is, that induced current would flow from the center of the disc to the edge. So we find the charge separation described, as long as the disc keeps rotating.

This is consistent with the behavior expected by applying the magnetic force equation. A positive "charge carrier" is moving instantaneously on a tangent in the counter-clockwise direction in our view, with the magnetic field coming towards us. The cross product v x B then points radially away from the center of the disc.

 

[Niles]

Thank you for replying. A very precise description - good job!