Rotating dielectric sphere : Jackson 6.8

[Geodesic]

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 :)

Thank you in advance !

 

[Ben Niehoff]

This problem probably makes more sense after Problem 6.7. Refer to Section 6.6, especially equation (6.100). Jackson derives the non-relativistic expression relating polarization, magnetization, and external fields.

The online solution you found does this problem in a confusing, non-rigorous way. It looks like the correct way uses (6.100). Note that many of the solutions on that web page are incorrect, or at least confusing. They were written up by students, and it often seems like the students who wrote them didn't really know what they were doing.

 

[turin]

That is a general frustration students (especially myself) have from Jackson: even though the professor only assigns a handful of problems, and realistically you don't even have time to do those properly, the techniques are built up from previous problems; he doesn't do a very good job guiding the student who only does a selection of problems. The same with Arfken.