What is the Impedance of a Plane Wave Passing Through a Metal?

[bayners123]

Homework Statement

A plane wave is passing through a metal. Show that the impedance Z can be given by
Z = \sqrt{ \frac{2 \omega \epsilon _0} {\sigma} } \frac{Z_0}{1-i} where Zo is the impedance of free space and sigma is the conductivity.

You may assume that E is polarised in the x direction.

Homework Equations

Z_0 = \sqrt{ \frac{\epsilon_r \epsilon_0}{\mu_r \mu_0}}

E_x = E_0 e^{i(\omega t - \tilde{k} x)}
where \tilde{k} = k - iK

The Attempt at a Solution

I've managed to get to the impedance in the form:
Z = \frac{ \mu_r \mu_0 \omega }{ k - iK }
but this doesn't have any reference to the conductivity in it and I can't see how to get to the required equation from it. I thought to use \frac{\omega}{k} = \frac{c}{n} = \frac{c}{\sqrt{\epsilon_r \mu_r}} but it didn't seem to help.

 

[bayners123]

Actually, by subbing back into the wave equation I've got to
Z = \sqrt{ \frac{2 \omega}{\sigma \epsilon_r \epsilon_0}} \frac{\mu_r \mu_0}{1-i} Z_0
which is nearly there but I can't see the last bit..

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EDIT: Solved. I was just being silly as usual: I had the wrong equation for the speed of light in terms of mu and epsilon. Please delete

 

[Mindscrape]

Reasons like this are why I'm glad that I was taught EM in SI units over CGS, everyone knows c from curlB. :p