# Imperfectly conducting sphere spinning in a homogeneous B-field

1. Oct 5, 2007

### yssi83

1. The problem statement, all variables and given/known data
The fact that it is imperfectly conducting is supplied so that the charges in the sphere will move with the same angular velocity.
The B-field induced by the moving charges will be disregarded.

2. Relevant equations
$$\vec{F}$$=Q[$$\vec{E}$$+$$\vec{v}$$$$\times$$$$\vec{B}$$]
$$\vec{F}$$=0 => $$\vec{E}$$=-$$\vec{v}$$$$\times$$$$\vec{B}$$
The equations above apply inside the sphere. And lead to the E-field inside.

3. The attempt at a solution
I have found the E-field inside, and the volume charge density inside.
E=B$$\omega$$x$$\hat{x}$$+B$$\omega$$y$$\hat{y}$$
$$\rho$$=-2B$$\omega$$$$\epsilon_{o}$$

This gives the potential inside (have set the potential at r=0 to V_0)
V(r)= V_0 - $$\frac{1}{2}$$B$$\omega$$r^2 (sin $$\theta$$)^2

The total charge inside and the total charge on the surface are excactly equal but opposite as one would expect. total charge = 0
Q_inside = -$$\frac{8}{3}$$$$\pi$$$$\epsilon_{0}$$B$$\omega$$R^3

Q_surface = $$\frac{8}{3}$$$$\pi$$$$\epsilon_{0}$$B$$\omega$$R^3

Then the problem is to find the potential and E_field outside the sphere. I can't seem to figure out if the charges on the surface are the only ones contributing.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Oct 5, 2007