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.