Torque on a rotating charge due to a uniform magnetic field

[QuantumDefect]

Hello, I need help with an integration. I found the torque on a ball of charge with varying charge density(dependent on the radial component) in a uniform magnetic field. I got

Integral[ (wxr)(r.B)(density)(r^2) sin(phi) d(theta) d(phi) dr, {r, 0,R}] where wxr is the cross product of the angular velocity and the radial component, r.B is the dot product between the radial component and the magnetic field. The trouble is integrating this monster, can anyone help?

 

[Physics Monkey]

Well, for starters you would probably want to work out those cross and dot products in spherical coordinates. You also need to know the precise form of the charge density.

 

[Physics Monkey]

Two quick things:

1) I think you may be missing a factor of 1/2.

2) Some simplifications can be obtained with the use of tensors, but you may not be comfortable with such tools. You could reduce the complexity of the integral a bit. Let me know if you would like to hear more.

 

[QuantumDefect]

My goal is to have a scalar integral function of r with the angular velocity and B field in terms of vectors and a vector operation between the two, as I do not know the form of the charge density. My coordinate system is such that the angular velocity is along the z-axis. The B field is in an unknown direction, so it will have to still be in vector form. Any additional advice that you can give would be great. Thank you for your help.

 

[Physics Monkey]

Ok then, QuantumDefect, how comfortable are you with tensors? If I start putting down indices and Levi-Civita symbols, will you be ok?

I shall assume for the moment that the answer is yes. The integral you're trying to do is $$\int d^3 x \rho(r) \left( \epsilon_{i j k} \hat{e}_i \omega_j x_k \right) \left( x_\ell B_\ell \right)$$ Since $$\hat{e}_i$$ is just a constant unit vector, you can take it outside. You can also take the constants $$\omega_j$$ and $$B_\ell$$ outside. So it boils down it down to evaluating the following integral, $$\int d^3 x \rho(r) x_k x_\ell$$ over a sphere of radius $$R$$. Now I've said enough, so you tell me what you expect that integral to look like. In particular, how does it depend on the two indices $$k$$ and $$\ell$$?

If this is all gibberish, let me know. We can take a different route.

 

[QuantumDefect]

Sorry for taking so long for replying, I was at a SPS zone meeting. The only thing that I can see is that we need to integrate the r vector that is parallel to the angular velocity and orthogonal to the B field, but that is obvious. However, there is only one r-hat direction, so will it be that they are both parrallel? Maybe the other method can be of better use in helping me, but I would still love to hear how the mechanism of this way works after I finally think this problem out, with your help of course. Thanks again.