Electromagnetic fields of a rotating solid sphere: total charge inside

[/flûks/]

Homework Statement

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.

Homework 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/Lorent...z_transformation_of_the_electromagnetic_field

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?

 

[tiny-tim]

hi /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

$r dr dθ \hat{r}$

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)