Rotating Electromagnetic Field Transformation

Lets say we have the following problem:

We are given some charge configuration in 3D, $\rho (s,\theta)$ (cylindrical coordinates), which doesn't depend on z. Let's say that the total charge is zero.

This charge distribution creates some electric field $E(s,\theta)$. The magnetic field is, obviously, zero everywhere.

Now we start rotating this charge configuration around the z-axis. What happens to the EM-field? I think that the electric field will remain the same (except that it rotates), and there will be some magnetic field induced. Is there some universal transformation formula for EM-fields under rotation? The rotation velocity is non relativistic!

Hint: I am looking for a formula like $\vec{E'} = \vec E+\vec v \times \vec B$, which transforms the magnetic field.

Thank you!

• do you have an example/special case you are interested in? – Oct18 is day of silence on SE Mar 24 '17 at 17:18
• Actually, I don't have a specific example. But if it makes life easier, let's take two parallel infinite lines of opposite charge, which are 1 m apart. Both at the same distance from the z-axis, at opposite sites. – Samuel Bosch Mar 24 '17 at 17:23
• – Emilio Pisanty Mar 24 '17 at 17:57
• Note that the concept of a rigidly rotating object in special relativity is deeply problematic; for more details see the Ehrenfest paradox and related issues. As such, it's entirely possible that there is no satisfactory answer because the situation you propose is either unphysical or insufficiently specified, but in any case these are troubled waters and one needs to pay special attention to the subtleties of the relativistic treatment. – Emilio Pisanty Mar 24 '17 at 18:01
• Yes, I am aware that this is quite complicated. But for now, let's just assume that $c=\infty$ :) – Samuel Bosch Mar 24 '17 at 19:16

I think since you have axial symmetry, $\rho=\rho(r,\theta)$, you can decompose your charge density into infinitely thin loops carrying current of

$$I(r,\theta)drd\theta=r\omega\rho(r,\theta)rd\theta dr$$

$\omega$ here is your system's angular speed.

Now you just use equation for field of current loop and integrate over $r$ and $\theta$.

You can see how this is applied in case of the rotating charged disk here.

Finally, you will have to go and integrate over $z$ as well, of course.

• Hey, thanks for the quick response. Yes, I agree that this would be a very elegant solution. But unfortunately, there is no axial symmetry, as $\rho$ depends on $\theta$ :( – Samuel Bosch Mar 24 '17 at 17:42
• @SamuelBosch ok, maybe i don't understand something here. What is your coordinate system? Cylindrical or spherical coordinates? In any way, idea is to present each small fraction of $]rho$ as single loop with current proportional to $\omega$ and integrate. $\rho$ can be anything, symmetries just make integration easier in practice – Oct18 is day of silence on SE Mar 24 '17 at 18:29
• Sorry, I should have made it more clear that I was thinking about cylindrical coordinates. Therefore it is not a very good approximation to take small current loops. – Samuel Bosch Mar 24 '17 at 19:18
• if coordinates are cylindrical and charge density does not depend on Z then thin loops of current should work OK. Just figure out $B(z,r)$ for current loop first. Also cylindrical coordinates are usually $(r, \theta, z)$, not $(s, \theta)$ – Oct18 is day of silence on SE Mar 24 '17 at 19:27