Magnetic Moment to Angular Momentum Ratio

[atomicpedals]

Homework Statement

Suppose that an electron is a small spherical shell of mass m with a charge e spread over its surface. Show that the ratio of the magnetic moment to angular momentum of such an electron should be e/2m whether the electron is a) moving in a circular orbit or b) spinning about a diameter.

2. The attempt at a solution

I'm confident in my reasoning for part a, which is as follows

I=-e/2 pi r = -e m_e vr/ 2 pi m_e r²

sin(a) M = IA

M = -e L/ 2 m

which then implies M /L = e/2m.

What has me stuck is how the solution for part b would proceed any differently (other than I'm pretty sure it does proceed differently). Any suggestions as to what I'm not seeing?

 

[cragar]

for part b how would I find the B field of a uniformly charged rotating sphere?

 

[atomicpedals]

Here is how I went about it in the end:

We can treat a sphere as a collection of infinitesimal rings, so M/L should be the same for a ring as for a sphere. So:

M = IA = (qv/ 2$$\pi$$r)($$\pi$$ r² )

=(q/2m)(mvr) => M /L = e/2m