# Covariant Description of Light Scattering at a fastly rotating Cylinder

Let us consider the following Gedankenexperiment:

A cylinder rotates symmetric around the $z$ axis with angular velocity $\Omega$ and a plane wave with $\mathbf{E}\text{, }\mathbf{B} \propto e^{\mathrm{i}\left(kx - \omega t \right)}$ gets scattered by it.

We assume to know the isotropic permittivity $\epsilon(\omega)$ and permeability $\mu(\omega)$ of the cylinder's material at rest. Furthermore, the cylinder is infinitely long in $z$-direction.

The static problem ($\Omega = 0$) can be treated in terms of Mie Theory - here, however, one will need a covariant description of the system for very fast rotations (which are assumed to be possible) causing nontrivial transformations of $\epsilon$ and $\mu$.

Hence my question:

### What is the scattering response to a plane wave on a fastly rotating cylinder?

RotatingDisc http://www.personal.uni-jena.de/~p3firo/PhysicsSE/RotatingDisc.png

• What is "infinite" on that cylinder? – Georg Jan 29 '11 at 13:35
• Thank you @Georg for pointing out to the misleading formulation. I mean infinitely in $z$-direction. I will change it in a second :) Greets – Robert Filter Jan 29 '11 at 13:38
• @Carl: You might consider that there are still some things in classical electrodynamics which are somehow basic but not standard homework problems. To my mind, the covariant description of electrodynamics in media belongs to this class. Greets – Robert Filter Jan 30 '11 at 12:24
• Robert, it might be useful to begin with the case for light impinging on a moving half-infinite media (i.e. infinite plane dividing space into two different materials). That case solves trivially (just boost the case for non moving material), and can be summed up (I think) to give a limiting case for the rotating cylinder (in the limit of small wave length). But it's been 30 years since I took E&M and it was never my "best" subject. – Carl Brannen Jan 30 '11 at 23:51
• @Carl: Thank you for the hint. The difference to the reflection problem at a half-space is the rotational character of the system. One attempt to solve the problem is to go into a co-rotating coordinate system and transform the plane wave accordingly - In this case I am not sure if such a framework is physically correct. The other way would be to just covariantly transform the medium - this is much more general since we would learn about the special relativistic relation of $\epsilon$ and $\mu$. Greets – Robert Filter Jan 31 '11 at 9:13

This is a very interesting problem. First of all, it can be shown that every component of the electromagnetic tensor can be written in terms of $F^{03}$. From axial symmetry, the solution will be $$\frac{f(r)}{\sqrt{r}}e^{i(m\phi - \omega t)}$$ It turns out that, when $$0<\omega<m\Omega,$$ where $\Omega$ is the angular velocity of the cylinder in the laboratory's frame, the reflection coefficient is greater than 1! This phenomenon is called Superradiance.

A discussion can be found in section V of this paper https://arxiv.org/abs/gr-qc/9803033

First of all, I don't quite understand the following phrase: "The static problem (Ω=0) can be treated in terms of Mie Theory". The Mie theory is for diffraction on a homogeneous sphere, not a cylinder. The complete solution of the problem of diffraction of electromagnetic waves on an infinite homogeneous cylinder was obtained in J. R. Wait, Can. Journ. of Phys. 33, 189 (1955) (or you may find the outline of the Wait's solution for a cylindrical wave in http://arxiv.org/abs/physics/0405091 , Section III). This solution is rather complex, so I suspect your problem can only be solved numerically, as it seems significantly more complex. The Wait's problem is a special case of your problem, so the solution of the latter problem cannot be simpler than the Wait's solution. In particular, it seems advisable to expand your plane wave into cylindrical waves, following Wait. It seems that the material equations for the rotating cylinder can be obtained following http://arxiv.org/abs/1104.0574 (Am. J. Phys. 78, 1181 (2010)). However, the cylinder will not be homogeneous (the material properties will depend on the distance from the axis and may be anisotropic). I suspect the problem can be solved using numerical solution of an ordinary differential equation for the parameters of the cylindrical waves.

• Can you at least solve the problem analytically for some special cases where still $\Omega\neq 0$? – Alexey Bobrick Mar 18 '12 at 15:10
• Probably. For example, for a perfectly conducting cylinder, the radiation will not penetrate significantly into the cylinder, so the problem would be pretty much equivalent to that for a homogeneous cylinder. This case may look relatively trivial though. Anyway, I am afraid I don't have much time or motivation to solve this problem. For example, I am not enthusiastic about studying the AM. J. Phys. article trying to determine the electric properties of the rotating cylinder. With all due respect, the author of the question may be in a better position to do that. – akhmeteli Mar 18 '12 at 17:19
• Thank you @akmeteli for your input. I however think that the determination of the properties of the rotating cylinder is at the core of this problem - how do $\epsilon$ and $\mu$ transform? Greets – Robert Filter Apr 22 '12 at 10:06
• @Robert Filter: I agree. However, this issue is discussed, e.g., in the Am. J. Phys. I cited. I am not sure though that it would be possible to find an exact solution of the diffraction problem for the inhomogeneous cylinder (or the exact solution can be too complex to be useful). – akhmeteli Apr 22 '12 at 16:07

Look here for some details Some remarks on scattering by a rotating dielectric cylinder Also articles that cite them.

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