Dispersion Relations and Refractive INdex

[tomeatworld]

Homework Statement

The conductivity of a plasma is defined as $\sigma = i\frac{Ne^{2}}{m\omega}$ where N is the electron density.

a) Prove the refractive index is: $n = \sqrt{1- (\frac{\omega}{\omega_{p}})^{2}}$ with $\omega_{p} = \sqrt{\frac{Ne^{2}}{m\epsilon_{0}}}$

b) Show the Attenuation length is $L = \frac{c}{\omega} \sqrt{\frac{1}{(\omega_{p}/\omega)^{2}-1}}$

Homework Equations

$k^{2} = \mu\epsilon\omega^{2} + i\mu\sigma\omega$

The Attempt at a Solution

I can't find equations linking the refractive index to the dispersion relation. Also, don't know anything about the attenuation length.

Can I grab a push in the right direction?
Thanks

Edit: Right, for the first part, I stumbled upon the equation $n^{2} = \frac{c^{2}}{\omega^{2}} k^{2}$ but I get an answer inverse to the required answer, with an extra factor of $c^{2}$. Can someone just verify I've almost got there or that that equation is completely wrong. Thanks.

 

[diazona]

Homework Statement

The conductivity of a plasma is defined as $\sigma = i\frac{Ne^{2}}{m\omega}$ where N is the electron density.

a) Prove the refractive index is: $n = \sqrt{1- (\frac{\omega}{\omega_{p}})^{2}}$ with $\omega_{p} = \sqrt{\frac{Ne^{2}}{m\epsilon_{0}}}$

b) Show the Attenuation length is $L = \frac{c}{\omega} \sqrt{\frac{1}{(\omega_{p}/\omega)^{2}-1}}$

Homework Equations

$k^{2} = \mu\epsilon\omega^{2} + i\mu\sigma\omega$

The Attempt at a Solution

I can't find equations linking the refractive index to the dispersion relation. Also, don't know anything about the attenuation length.

Can I grab a push in the right direction?
Thanks

Edit: Right, for the first part, I stumbled upon the equation $n^{2} = \frac{c^{2}}{\omega^{2}} k^{2}$ but I get an answer inverse to the required answer, with an extra factor of $c^{2}$. Can someone just verify I've almost got there or that that equation is completely wrong. Thanks.

I think you're right, and what you are being asked to prove is slightly wrong. It should be
$$n = \sqrt{1 - \biggl(\frac{\omega_p}{\omega}\biggr)^2}$$
according to e.g. http://farside.ph.utexas.edu/teaching/em/lectures/node98.html (By the way, that's a good read)

As for the attenuation length: the intensity of EM radiation inside a material decreases exponentially as a function of the distance penetrated into the material. Mathematically,
$$I \propto e^{-x/\lambda}$$
The constant $\lambda$ is the attenuation length. It's related to the imaginary part of the permittivity $\epsilon$ (and also related to the dispersion relation). Check your notes or references and see whether that helps you find anything relevant.