I'm currently trying to understand how to solve Maxwells equations analytically. I have some questions and would like to know if I'm doing things right!
I want to solve a simple(?) example: Lets say we have an infinte cylinder in free space with axis in z direction. The cylinder should be a homogeneous, isotropic dielectric (or pec as a second example). I'm searching for time-harmonic solutions, so the electric field should be . The problem is to find the scattered (and the transmitted) field for an incident plane wave travelling in x direction and having only z component: .
Is the problem description complete?
In free space with vanishing charge and current density the components of the electric field satisfy the scalar wave equation. So, with we get
With harmonic time dependence this is the homogeneous Helmholtz equation
which I will write
so the constant A describes the material parameters.
The above 3D problem can be reduced to a 2D one (Is that right? What is the mathematical argument for that?). I want to solve the Helmholtz equation in the xy plane, with boundary conditions given by the incident wave, for example neumann conditions: for .
The scattered field will be given by solving outside the cylinder with and the transmitted field by solving inside the cylinder with c given by the material parameters. Is that right?
When solving in polar coordinates, solutions to the problem are given by
inside the cylinder
outside (assumed A=1)
I found those solutions in an online paper. Some questions:
* How do I get those equations? Haven't found them in any book so far.
* What's the meaning of the Hankel function ?
* How would I calculate the coefficients , and ? I think for any n they are given by the boundary conditions - so it should be a linear 3x3 system. But what exactly does it look like?
For a perfect electric conductor: what would change? What would the boundary conditions look like?
And the last question has to be: how can I be sure, that the linear system is not singular? I think the above solution holds, if is not an eigenvalue of the Laplacian with Neumann data. But wouldn't it be possible, to find material parameters A so that the linear system is singular for some n. What would be the effect of that (physically)???