I came across an interesting problem when I prepared for the preliminary exam on electromagnetism. Below is the problem in its original words:

> A metallic sphere of mass, $m$, and radius, $a$, carries a net charge, $Q$, and is magnetized with uniform magnetization, $M$, in the $z$-direction.

> (a) Determine the total angular momentum associated with the electromagnetic field.

> (b) The magnetization of the sphere is now reduced to zero. Assuming that no external mechanical forces act on it (PS: this is possible. For instance, just heat up the sphere and the magnetization will be eliminated ), determine the angular velocity and the sense of rotation of the sphere when $M=0$.

> (c) Describe the origin of the torque that causes the sphere to rotate.

For part (a), it is straight forward to calculate electric and magnetic field E, H inside and outside of the sphere. Generated by the net charge $Q$ uniformly distributed on the surface, the electric field $E$ is isotropic outside the sphere and vanishes inside. The magnetic field H is generated by magnetization $M$ and can be calculated using the "magnetic scalar potential" formalism since there is no free current. The result is that H is also uniform inside the sphere and takes the dipole form outside the sphere. 

The total angular momentum $J$ is obtained by integrating over the region where a local angular momentum density of the EM field exists, i.e. outside of the sphere. It is in the $z$-direction and I find it to be

$$J= \frac{2 Q M a^{2}}{9 \epsilon_0 c^{2}}$$

For part (b) and (c), it is interesting to ask why the sphere would rotate after magnetization $M$ is erased. What I think up is that as $M$ decreases to zero, a electric field with non-zero curl is generated, as implied by Faraday's law. This electric field is in the azimuthal direction, and will act on the surface charge $Q$ to provide a torque. However, a straightforward calculation can show that the angular momentum this torque can transfer is

$$J'= \frac{Q M a^{2}}{9 \epsilon_0 c^{2}}$$

Obviously a factor of 2 is missing, so this only accounts for half of the angular momentum stored originally in the EM field. So I am perplexed by where the other half has gone?

I come up with 2 possibilities:

1. since the charged sphere begins to rotate, there is also non-zero angular momentum stored in EM field after magnetization is removed. But it could hardly account for the missing amount, since in that case the electric field $E$ is proportion to $Q$, the magnetic field $H$ to $Q^2$ ( one $Q$ from the charge and a second $Q$ from the angular velocity). Hence one expects the angular momentum stored is proportional to $Q^3$, which is very different from $J$'s linear dependence on $Q$.

2. maybe in the process of magnetization removal, EM radiation carries away some angular momentum. But this possibility has to be validated quantitatively before it can convince people. What I feel is that if the process of removal is infinitely slow, then the effect of radiation can be neglected.

Then what is the correct answer? I provide quantitative result for $J$ and $J'$ in the hope that someone can check them and point out that I have made a wrong calculation lol. But I am really eager to hear from any explanation that sounds real ... Electromagnetism is as interesting as it was 3 years ago, but I just don't remember as much as I did then ...