# Complex Impedance in metals

## 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.

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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

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