Electromagnetic fields of a rotating solid sphere: total charge inside

/flûks/

1. The problem statement, all variables and given/known data
A solid sphere of radius a rotates with angular velocity ω[itex]\hat{z}[/itex] relative to an inertial frame K in which the sphere's center is at rest. In a frame K' located at the surface of the sphere, there is no electric field, and the magnetic field is a dipole field with M=M[itex]\hat{z}[/itex] located at the center of the sphere.

First find the electric and magnetic fields as measured in the K frame and do not assume ωa<<c, then calculate the total charge inside the planet also in the K frame, this time assuming ωa<<c.

2. Relevant equations

(i) [itex]\textbf{B}=\frac{3 \hat{r} \left( \hat{r} \bullet \textbf{M} \right) - \textbf{M}} {a^{3}}[/itex]

(ii) Q[itex]_{enc}[/itex]=[itex]\frac{1}{4π}\int \textbf{E} \bullet \textbf{da}[/itex]

Also the Lorentz transformation equations to go from E' to E and B' to B (don't want to type...):
http://en.wikipedia.org/wiki/Lorentz...magnetic_field

3. The attempt at a solution

I got the transformed electric and magnetic fields, and I want to use (ii) to find the total charge using the electric field I get:

[itex]\textbf{E}=\frac{Mω} {ca^{2} \sqrt{1-\frac{ω^{2}a^{2}} {c^{2}}sin^{2} \left(θ \right)}} \left(sin^{2}θ \hat{r} - 2sinθcosθ \hat{θ} \right)[/itex]

BUT I do not know what da would be in this case, since the sphere is rotating in the K frame. Conventionally da is just

[itex]r dr dθ \hat{r}[/itex]

EDIT: but that surface element only accounts for part of the electric flux. I guess I'm just not sure. Any insights on this?

tiny-tim

hi /flûks/!
it doesn't matter that the real sphere is rotating …

you're integrating over an imaginary sphere!

(since you have found E in the stationary frame, you integrate as usual)