How Does Magnetic Field Influence Refraction in Plasma?

[Herr Malus]

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 $$\widehat{z}$$. Show that the index of refraction for right and left circularly polarised light satisfies: n ² _r,l=1-$$\omega²$$/[$$\omega$$($$\omega$$$$\pm$$$$\omega_B$$
Where $$\omega_B$$ 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, $$\omega₀$$=$$\gamma$$=0
So we have m$$\partial²$$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=x₀e^{-i$$\omega$$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:
r₀=-$$\omega ^-2$$((qE₀e^ikx/m-i$$\omega$$$$\omega_B$$r₀)$$\widehat{x}$$+(qE ₀ e^ikx/m+i $$\omega$$ $$\omega_B$$r₀)$$\widehat{y}$$)
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 $$\omega$$. Any help, or even a good source for the math behind plasma physics in this area would be greatly appreciated.

Cheers.

 

[gomboc]

Generally with plasma physics, many questions are answered simply by finding and interpreting the dispersion relations for the particular type of wave you're interested in. If you're working from Griffiths, I assume it's an E&M course, so they probably expect you to use the simplest case for the wave, i.e. that the wave frequency is much less than both the ion and electron cyclotron frequencies in the plasma. Since you are only using one cyclotron frequency, I would also assume they mean the electron cyclotron frequency, and that the plasma is magically neutral, allowing you to neglect the ions.

In that case, there is a general dispersion relation for left/right waves you can use that's given in pretty much every plasma physics textbook:

$$n_R^2 = 1-\frac{\omega_{pi}^2}{\omega(\omega +\omega_{ci})}-\frac{\omega_{pe}^2}{\omega(\omega -\omega_{ce})}$$

$$n_L^2 = 1-\frac{\omega_{pi}^2}{\omega(\omega -\omega_{ci})}-\frac{\omega_{pe}^2}{\omega(\omega +\omega_{ce})}$$

In the subscripts, "i,e" mean "ion", and "electron" respectively, "p" indicates a plasma frequency, and "c" indicates a cyclotron frequency. The omega having no subscript is the frequency of your wave.

*You might want to look up "plasma frequency". It's a simple concept, but it'll make this problem easier to understand.

Now, with the assumptions we've made (like neglecting ions), these can be simplified greatly.

It's kind of hard to read what your 'attempt at a solution' was, given the formatting, but in my plasma course all dispersion relations were solved for by applying a linear perturbation to the equations of motion for a single species (i.e. electrons), and then assuming a wave of the form $$E= E_0 e^{\vec{k}\cdot\vec{r}-\omega t}$$ to simplify the result.

There's a very thorough overview of all this here: http://silas.psfc.mit.edu/introplasma/chap5.html

 

[Herr Malus]

Thanks very much for your help. I had come across the MIT text before and my results were consistent with a conductivity that was different depending on the direction but this was not correct. It turns out that in setting up my circularly polarized light (in the complex form) I had neglected the i in front of the y direction vector. This led to my errors.

Many thanks/Cheers,
-Malus

 

[protegy100]

Excuse me, I just downloaded "Astrophysical Gyrokinetics Basic Equations And Linear Theory" and I'm having a bit of a hard time understanding it. Can anyone help?