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-[tex]\omega²[/tex]/[$$\omega$$($$\omega$$$$\pm$$[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]\omega₀[/tex]=$$\gamma$$=0
So we have m[tex]\partial²[/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=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=-[tex]\omega ^-2 [/tex]((qE₀e^ikx/m-i$$\omega$$[tex]\omega_B[/tex]r₀)$$\widehat{x}$$+(qE ₀ e^ikx/m+i $$\omega$$ [tex]\omega_B[/tex]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?

 

[paranormal]

Hello,

Thank you for your question. I am a fellow scientist and I would be happy to provide a response to the content you have shared.

Firstly, let's clarify the problem statement. We are trying to determine the index of refraction for a plane electromagnetic wave propagating in a plasma with an applied static, uniform magnetic field. This means that we are dealing with a medium that is both electrically and magnetically active.

To solve this problem, we will need to use the equations for the propagation of electromagnetic waves in a medium. These equations are known as Maxwell's equations and they describe the behavior of electric and magnetic fields in a medium. In particular, we will be using the equations for the propagation of a plane electromagnetic wave in a medium.

The first step in solving this problem is to determine the equations of motion for the electric and magnetic fields in the presence of the applied magnetic field. This can be done by combining the equations for the electric and magnetic fields in a medium with the equations for the motion of charged particles in a magnetic field.

Once we have the equations of motion, we can then derive the dispersion relation for the plane electromagnetic wave in the plasma. This will give us the relationship between the wave vector k and the frequency \omega of the wave.

Next, we can use this dispersion relation to determine the index of refraction for the right and left circularly polarized light. This can be done by substituting the dispersion relation into the index of refraction equation and solving for the values of n_r and n_l.

Finally, we can use the index of refraction to determine the conductivity, susceptibility, and dielectric constant of the plasma. These quantities are all related to the index of refraction and can be calculated using the equations provided in the problem statement.

In conclusion, solving this problem will require a good understanding of Maxwell's equations, as well as the equations for the motion of charged particles in a magnetic field. I would recommend consulting a textbook or online resources for a more detailed explanation of these equations and their applications in plasma physics. I hope this response has been helpful and I wish you the best of luck in your studies.

Best regards,