# Rotating dielectric sphere : Jackson 6.8

## Homework Statement

I'm working at the exercise 6.8 of Jackson's Classical Electrodynamics. Here it is :
We consider a dielectric sphere of dielectric constant $$\epsilon$$ and of radius $$a$$ located at the origin. There is a uniform applied electric field $$E_0$$ in the $$x$$ direction. The sphere rotates with an angular velocity $$\omega$$ about the $$z$$ axis. We have to show that there is a magnetic field $$\textbf{H} = - \nabla \Phi_M$$ where :
$$\Phi_M(\textbf{x}) = \frac{3}{5}\epsilon_0 \left( \frac{\epsilon - \epsilon_0}{\epsilon + 2 \epsilon_0} \right) E_0 \omega \left( \frac{a}{r_<} \right)^5 x z$$
where $$r_> = \max(r, a)$$. The motion is nonrelativistic.

## The Attempt at a Solution

In every solutions I found (here is a good one : http://www-personal.umich.edu/~pran/jackson/P505/p10s.pdf" ), they suppose that the polarization of the sphere is the same as if the sphere stayed static :
$$\mathbf{P} = 3 \epsilon_0 \left( \frac{\epsilon - \epsilon_0}{\epsilon + 2 \epsilon_0} \right) E_0$$
I guess we may assume this because we suppose the motion is nonrelativistic ?
So we have a bound surface charge density
$$\sigma_{pol} = \mathbf{P} \cdot \mathbf{\hat{n}}$$
where $$\hat{n}$$ is the normal vector to the sphere.
Now comes the part I don't understand. They suppose, since the sphere is rotating, there is an effective surface current with density :
$$\mathbf{K}_M = \sigma_{pol}(\mathbf{x}) . \mathbf{v}(\mathbf{x})$$
I don't understand why we may suppose the existence of such surface current since, as I see the situation, the surface charge density stay constant because we assumed the polarization was the same as if the sphere didn't rotate...

I hope I was clear :)

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