How to Find Magnetic Field and Vector Potential of a Rotating Charged Disk?

[mochi_melon]

Homework Statement

Disk of radius S has uniform charge Q on surface. Roates with angular velocity w = w z(^) about symmerty axis Z. I am asked to find magnetic field (I believe I have that answer though it's messy and maybe wrong) vector potential (of which I have no idea) and the torque on a magnet with magnetic dipole m = m s(^) place in the equitorial plane at a distance s from the center of the disk.

Homework Equations

?

The Attempt at a Solution

The first part I have NO idea on.
For the second part, torque = mXB
m = Iarea = I*pi*S^2

But where does the distance s go? Thank you!

 

[OlderDan]

I interpret the problem to mean that the location of the magnetic dipole is in the plane of the spinning disk at a distance s fom its center. If (s^) means a unit vector in the direction of the vector s, then I think that means radially outward from the center of the disk.

Did you find the field at all points in space? If you were able to do that, you can probably do the integral required to find the vector potential

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magvec.html

 

[mochi_melon]

Yes, that was the unit vector I didn't know how to type it.

What formula would I use to find the field at all points? My prof never covered rotation in class yet assigned it for home so I am really confused :( We did go over the information on the page you linked, but I need to get the field first and then I can find the potential?

 

[OlderDan]

Yes, that was the unit vector I didn't know how to type it.

What formula would I use to find the field at all points? My prof never covered rotation in class yet assigned it for home so I am really confused :( We did go over the information on the page you linked, but I need to get the field first and then I can find the potential?

The rotation of a charged disk is just another way of producing a current density. You can treat your problem as a set of nested current loops. Finding the field or the vector potential at all points in space for even one current loop is not a trivial calculation. I was a bit surprised when you said you had worked out the field because I don't think it is all that easy to do, but it can be done.

The vector potential for one current loop is done here.

http://www.cmmp.ucl.ac.uk/~drb/Teaching/PHAS3201_MagneticFieldsFull.pdf

showing that the vector potential can be reduced to an elliptic integral. Then approximations are made to finish the problem. For your disk, the solution to the loop problem could serve as the starting point for an integral over nested loops, but only if the approximations are valid in your case.

Maybe your problem is only expecting you to treat the rotating disk as a magnetic dipole. If so, finding the dipole moment of the disk as nested current loops is not too difficult, and the field and vector potential of the dipole are known. See for example

http://en.wikipedia.org/wiki/Dipole