# Induction and anguler momentum

1. Dec 15, 2006

### daniel_i_l

Lets say you have a plastic plate with little charged balls around the edge. in the middle there's a coil with a current going through. now if the current is suddenly stopped then there's a change in the magnetic flux through the plate so as a result there should be an electric field circulating around the plate. this would push the balls and the plate would spin. but doesn't this contridict AM conservation. could the answer be that the new electric field produces a new magnetic field which induces another EF in the opposite direction so the plate doesn't spin?
Thanks.

2. Dec 16, 2006

### cesiumfrog

Haven't thought right through your example, but you don't seem to consider the momentum of the field at all..

3. Dec 17, 2006

### tim_lou

I don't think angular momentum is conserved in electromagnetism. Since the basic argument of central forces when deriving conservation of angular momentum is not valid any more (Lorenz force).

4. Dec 17, 2006

### vanesch

Staff Emeritus
That's not correct: the EM field has a well-defined angular momentum. The question by the OP is in fact a question (I don't know if he took it from there) in Feynman's lectures (vol II).
What happens indeed, is that the angular momentum of the EM field is transformed (the EM field going to zero) into mechanical angular momentum.

5. Dec 19, 2006

### daniel_i_l

Ok, so basically the final mechanical angular momentum comes from the initial AM in the coil?
Thanks.

6. Dec 20, 2006

### vanesch

Staff Emeritus
Not so much the coil itself, than the EM field generated by the current in the coil, which has angular momentum (like it has energy).

7. Dec 20, 2006

### tim_lou

So, the field has angular momentum? how would that angular momentum be defined? I'm curious to know how one would derive conservation of angular momentum with electromagnetic field... I hope it is not too difficult for a student who just finished calc III.

8. Dec 20, 2006

### Staff: Mentor

The angular momentum density (angular momentum per unit volume of the field) at position $\vec r$ relative to the desired "axis of rotation" is

$$\vec L = \vec r \times \vec P$$

where $\vec P$ in turn is the linear momentum density

$$\vec P = \frac{1}{4 \pi c} \vec E \times \vec B$$

also known as the Poynting vector. These are in cgs units because I took them from this Wikipedia article:

http://en.wikipedia.org/wiki/Photon_polarization

You can probably find the MKS versions in Griffiths and other books but I'm at home and my textbooks are at the office. The only difference would be in constant factors.

Griffiths has a whole chapter on the energy and momentum density in the electromagnetic field, but I don't remember if he covers the angular momentum density also.

Last edited: Dec 20, 2006