# Induction with waveguide walls

1. Jan 4, 2013

### randomafk

This is a question I've been trying to figure out. I'll try my best to formulate it, so apologies if it's a bit ill defined!

Suppose you construct a finite parallel plate waveguide of PEC(perfect electrical conductors) and PMC(perfect magnetic conductors) so that the top/bottom plates are PEC, and the left/right are PMC. For instance, something like this but instead of air on the sides, it's a PMC so H = 0.
http://cc.ee.ntu.edu.tw/~jfkiang/electromagnetic%20wave/demonstrations/demo_11/fig.bmp

1) Then what happens if you propagate a TE wave? More specifically, will there be any sort of induction with the walls?
2) If not, could you construct some internal structure in the waveguide which couples to the walls of it?

This is what I have for 1) so far:

There can't be any induction with the PMC walls, because along the surface of the PMC, the magnetic field is 0 in all directions.

We have then have BC's that:
$\vec{E}(x=0) = \vec{E}(x=d) = \vec{0}$

And then from maxwell's equations, the induced surface current is:
$J_s = \pm \hat{x} \times \vec{H} = \pm (\hat{y} H_z - \hat{z} H_y)$

I'm a bit rusty on how to calculate any sort of behaviors here via induction onwards.

I'm guessing that since this is TE, H only has y and z components. Therefore, there is no induced electric field since the magnetic flux from one plate to another is 0. That would therefore imply, to produce inductive effects with some internal structure (2), we could construct a wire loop slanted at 45°