## Electromagnetic fields of a rotating solid sphere: total charge inside

1. The problem statement, all variables and given/known data
A solid sphere of radius a rotates with angular velocity ω$\hat{z}$ 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$\hat{z}$ 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) $\textbf{B}=\frac{3 \hat{r} \left( \hat{r} \bullet \textbf{M} \right) - \textbf{M}} {a^{3}}$

(ii) Q$_{enc}$=$\frac{1}{4π}\int \textbf{E} \bullet \textbf{da}$

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:

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

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

$r dr dθ \hat{r}$

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

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 Quote by /flûks/ 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 $r dr dθ \hat{r}$