Where the angular momentum has gone?
I came across an interesting problem when I prepared for the preliminary exam on electromagnetism. Below is the problem in its orignal 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 fo 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 filed exists, i.e. outside of the sphere. It is in the z-direction and I find it to be ( e is the vacuum's permittivity in SI unit )
$J=(2 Q M a^{2})/(9 e c^{2})$
For part (b) and (c), it is interesting to ask why the sphere would rotate after magnetization M is erased. 903
What I think up is that as M decreases to zero, a electric field with non-zero curl is generated, as implied by <PERSON>'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
$J'= ( Q M a^{2})/(9 e c^{2})$
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:
(i) since the charged shpere 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.
(ii) 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 <PERSON> and <PERSON>' 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 ... 903