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## Homework Statement

We want to deduce the index of refraction for a plane electromagnetic wave propagating (along the z direction) in a plasma with an applied static, uniform magnetic field B=B

_{0}[tex]\widehat{z}[/tex]. Show that the index of refraction for right and left circularly polarised light satisfies: n

^{2}

_{r,l}=1-[tex]\omega

^{2}[/tex]/[[tex]\omega[/tex]([tex]\omega[/tex][tex]\pm[/tex][tex]\omega

_{B}[/tex]

Where [tex]\omega

_{B}[/tex] is the cyclotron frequency.

There then follow parts 2 and 3 regarding getting the dispersion relation and conductivity/suspceptibility and dielectric constant.

## Homework Equations

Since this is a plasma, [tex]\omega

_{0}[/tex]=[tex]\gamma[/tex]=0

So we have m[tex]\partial

^{2}[/tex]x=q(E+vxB)

## The Attempt at a Solution

So I took the equation for a right circular polarised E along with an x of the form x=x

_{0}e

^{-i[tex]\omega[/tex]t}, and placed it alongside the given B in the equation above. This basically gave me a whole mess of algebra to sort through, but I got down to:

r

_{0}=-[tex]\omega

^{-2}[/tex]((qE

_{0}e

^{ikx}/m-i[tex]\omega[/tex][tex]\omega

_{B}[/tex]r

_{0})[tex]\widehat{x}[/tex]+(qE

_{0}e

^{ikx}/m+i [tex]\omega[/tex] [tex]\omega

_{B}[/tex]r

_{0})[tex]\widehat{y}[/tex])

I'm not even sure if this is the right direction and my class textbook, Griffiths, has a derivation which doesn't seem useful since it arrives at n through the relation between k and [tex]\omega[/tex]. Any help, or even a good source for the math behind plasma physics in this area would be greatly appreciated.

Cheers.