Dispersion Relations and Refractive INdex

  • #1

Homework Statement


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

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

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


Homework Equations


[itex]k^{2} = \mu\epsilon\omega^{2} + i\mu\sigma\omega[/itex]


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 [itex]n^{2} = \frac{c^{2}}{\omega^{2}} k^{2}[/itex] but I get an answer inverse to the required answer, with an extra factor of [itex]c^{2}[/itex]. Can someone just verify I've almost got there or that that equation is completely wrong. Thanks.
 
Last edited:

Answers and Replies

  • #2
diazona
Homework Helper
2,175
6

Homework Statement


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

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

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


Homework Equations


[itex]k^{2} = \mu\epsilon\omega^{2} + i\mu\sigma\omega[/itex]


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 [itex]n^{2} = \frac{c^{2}}{\omega^{2}} k^{2}[/itex] but I get an answer inverse to the required answer, with an extra factor of [itex]c^{2}[/itex]. 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
[tex]n = \sqrt{1 - \biggl(\frac{\omega_p}{\omega}\biggr)^2}[/tex]
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,
[tex]I \propto e^{-x/\lambda}[/tex]
The constant [itex]\lambda[/itex] is the attenuation length. It's related to the imaginary part of the permittivity [itex]\epsilon[/itex] (and also related to the dispersion relation). Check your notes or references and see whether that helps you find anything relevant.
 

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