Classical Electron Magnetic Moment

[greeziak]

I'm trying to show that for an electron of uniform charge and mass distributions spinning about a fixed axis that the classical calculation for the magnetic moment is

μ_s = -(e/2m)S where S is the spin angular momentum.

Now I know that the moment for any given current loop is μ = iA. So should I just be assuming all the charge is at the surface and integrate over the entire sphere? The moment for an orbiting electron is of the same form but with L instead of S, so I think this might be the way to start, but I'm not sure. Thanks.

 

[greeziak]

As of right now I am writing q = ρ2πr as the infinitesimal amount of charge in the current loop with the current being q/T with T, the period, T = 2πr/v. The area in these loops then should be πr², with the surface integral rdr tacked on from 0 to r. This doesn't seem to work.

Also writing the current as q = ρ2πdr is providing me nothing. In order to get ρ back into -e I'm using ρ * 4pi*r², which means I need a pi to survive at the end. One last note, before integration the power of r has to be either 1 or 3 to avoid an odd number in the denominator? Any help at this point would be great, thanks.