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Thread: Maxwell equations example

  1. #1
    Jul 2010

    Maxwell equations example


    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 $\displaystyle E(r,t)=E(r)e^{-i\omega t}$. 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: $\displaystyle E(r)=(0,0,e^{-i\omega x})$.

    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 $\displaystyle c := 1/\sqrt{\mu\epsilon}$ we get

    $\displaystyle \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vect{E} - \Delta E = 0$

    With harmonic time dependence this is the homogeneous Helmholtz equation

    $\displaystyle \Delta \vect{E} + \frac{\omega^2}{c^2}\vect{E} = 0$,

    which I will write

    $\displaystyle A \Delta \vect{E} + \omega^2\vect{E} = 0$,

    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: $\displaystyle \partial_n u(x,y) = -i\omega e^{-i\omega x}\nu_1$ for $\displaystyle (x,y)\in \partial\Omega$.

    The scattered field will be given by solving outside the cylinder with $\displaystyle c = c_0$ 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

    $\displaystyle u = \sum_{n=-\infty}^{\infty} a_n J_n(A \omega r) e^{in\theta}$ inside the cylinder
    $\displaystyle u = \sum_{n=-\infty}^{\infty} [b_n J_n(\omega r)+c_n H_n^{(1)}(\omega r)] e^{in\theta} $ 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 $\displaystyle H_n^{(1)}$?
    * How would I calculate the coefficients $\displaystyle a_n$, $\displaystyle b_n$ and $\displaystyle c_n$? 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 $\displaystyle -\omega^2$ 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)???

    Last edited by brandon86; Sep 8th 2010 at 07:03 AM.
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