Dispersion Relations in Cold Plasma waves

[Firben]

Homework Statement

Im stuck on a old exam in plasma physics. It is about how to determine dispersion relations for high frequency waves in cold plasma's. I'm not sure how they do in the solution manual.

Homework Equations

B = B_0z^
E = E_0exp(i(kx-wt))z^

The Attempt at a Solution

The solution manual:
http://s716.photobucket.com/user/Pitoraq/media/IMG_20170512_170136_zps0e1g7onu.jpg.html

My attempt to find the solution:
http://s716.photobucket.com/user/Pitoraq/media/IMG_20170512_170131_zps2f7nwaui.jpg.html

My question is marked "why" in the above link. Why is S^2 - D^2 = 1 ?

 

[jasonRF]

You never actually listed the problem you are trying to solve. You are clearly looking for a particular case of the dispersion relation but are not telling us what it is. Please state the problem and define the relevant quantities. From the solutions manual I think I know what the problem is - but from your solution it is clear that you think it is a different problem.

By the way, with the standard definitions (which your reference seems to follow?) $S^2 - D^2 \neq 1$.

jason

 

[Firben]

Ok
A high frequency electromagnetic wave in a cold magnetized plasma with the density n_0 is polarized with the electromagnetic field in the same direction as the magnetic field B_0 = B_0z^. Namely
E^~ = E_0exp(i(kx-ωt))z^
1) Determine the dispersion relation for the wave
2) Determine the cut-off frequency

My attempt for the solution is in the link
http://s716.photobucket.com/user/Pitoraq/media/IMG_20170512_170131_zps2f7nwaui.jpg.html

2)
Here i used the relation k^2*c^2 = ω^2 -ω^2(pe)
cut off is when k = 0 <=>
ω(pe) (The plasma frequency) = sqrt(n_0*e^2)/(ε_0*m_e)) ≈ 3.08968 × 10^10 rad/s
where ε_0 is permittivity of free space and m_e is the electron mass
which is incorrect, it sohuld be ω(pe) = 1.7*10^12 rad/s
The wave should be reflected f < f(pe)

 

[jasonRF]

Thanks for clarifying. Notice that the electric field only has a z component. It doesn't seem like you have used that fact, yet.

 

[Firben]

Yes i know. But i still don't see why S^2−D^2≠1 has to be the case. And why is the plasma frequency wrong ?

 

[jasonRF]

But i still don't see why S^2−D^2≠1 has to be the case.

If you do the algebra it should be clear. In any case, if you think about the actual problem you are solving you will find that you never need to calculate it at all. Look at my previous post.

And why is the plasma frequency wrong ?

What plasma frequency? The problem you posted has no numbers. I cannot help you when you don't post the actual question you are answering.