Current density of rotating spherical shell

[icelevistus]

Express the current density of a spherical shell of radius a, rotating with angular velocity omega, with surface charge density sigma



Delta function will be denoted d(x). Spherical coordinates will be used



I have concluded that for a given chunk (if we restrict to the 0<theta<pi/2 domain), the velocity will be given by v=a*Sin(theta)*(omega). It is clear that the current density will only have a phi component. I have concluded:

J_phi = (sigma)*a*Sin(theta)*(omega)*d(r'-a)

Where the delta function is used to restrict the current density to the sphere's surface.

Can anyone confirm that this reasoning is correct?

 

[gabbagabbahey]

It looks good, except I see no reason for you to restrict theta to $0<\theta<\frac{\pi}{2}$...what is wrong with your expression for the lower half of the sphere?

 

[icelevistus]

It looks good, except I see no reason for you to restrict theta to $0<\theta<\frac{\pi}{2}$...what is wrong with your expression for the lower half of the sphere?

you're right, as I was writing it I thought I was using cosine and it would introduce a sign error. Whole shell it is!