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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

Thank you in advance

David Z
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Robert Filter
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3 Answers3

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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

Thiago
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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.

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

user16409
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