Dispersion relations and Plasma

I think you can solve for k, and then use the definition of k: k=2pi/lambdaYou might have to look up what lambda is...Also, did you use the fact that the amplitude decreased by 10%? The exponential decay should involve that somehow.In summary, the dispersion relation for a plasma is given by k^{2}=\frac{\omega^{2}}{c^{2}}(1-\frac{\omega^{2}_{p}}{\omega^{2}}), where \omega^{2}_{p}\:= \frac{Ne^{2}}{m_{e}\epsilon_{0}} and N is the electron density. During re-entry of a spacecraft, there was a radio blackout of all
  • #1
Moham1287
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0

Homework Statement


The dispersion relation for a plasma is given by

[tex]k^{2}=\frac{\omega^{2}}{c^{2}}(1-\frac{\omega^{2}_{p}}{\omega^{2}})[/tex][tex]\omega^{2}_{p}\:= \frac{Ne^{2}}{m_{e}\epsilon_{0}}[/tex]

Where N is the electron density

During re enrty of a spacecraft there was a radio blackout of all frequencies up to 10^10 Hz because it was surrounded by a plasma. Calculate the electron density in the plasma surrounding the spacecraft .

Sensitive equipment detected EM radio waves at 10^9 Hz at an amplitude of 10% of that before re entry. Calculate the thickness of the plasma.

Homework Equations



Given above

The Attempt at a Solution



Got the first part easily enough, by substituting in the expression for [tex]\omega^{2}_{p}[/tex], then solving [tex] \frac{Ne^{2}}{\omega^{2}m_{e}\epsilon_{0}}=1[/tex] for N to get N= 1.24x10^18. I don't really have any idea about how to go about the second part, I can't find anything about it in my textbook (I S Grant & W R Phillips Electromagnetism) or on the old googles

Any help would be much appreciated! Thanks
 
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  • #2
Since the frequency is below the plasma frequency your 'k' will be imaginary. That means the wave will exponentially decay. So you will have something like E = E_0*exp(-kx). And you know the amplitude is decreased by 10%, so you are left with something like... 0.10 = exp(-k*x)

Now you just solve for 'x', and that is roughly how far the wave had to attenuate through before it escaped
 
  • #3
Hi

thanks for the reply. I don't think I quite follow though. K is the wave number, given by 2pi/lambda isn't it? How can that be imaginary? k would be 2pi/0.3 recurring for EM of 10^9 Hz... Would my answer then just be 0.10=exp(x2pi/0.3) solve for x?
 
  • #4
The wave is in a plasma, so the dispersion relation w=ck doesn't apply anymore. Use the dispersion relation you listed above.
 

FAQ: Dispersion relations and Plasma

What is a dispersion relation?

A dispersion relation is a mathematical equation that describes the relationship between the frequency and wavevector of a wave in a medium. It helps to understand how waves propagate through a given medium.

What is the role of dispersion relations in plasma physics?

Dispersion relations play a crucial role in plasma physics as they determine the behavior of electromagnetic and acoustic waves in a plasma. They help to understand the properties of plasma such as its refractive index, plasma frequency, and damping rates.

How are dispersion relations derived?

Dispersion relations are derived by analyzing the equations that govern the behavior of waves in a specific medium, such as the Maxwell's equations for electromagnetic waves in a plasma. These equations are then solved to obtain the dispersion relation.

What is the importance of dispersion relations in studying plasma instabilities?

Dispersion relations are essential in studying plasma instabilities as they provide information about the growth rate and frequency of these instabilities. By analyzing the dispersion relation, scientists can predict the conditions under which instabilities may occur in a plasma.

Can dispersion relations be used to understand the properties of a plasma?

Yes, dispersion relations can be used to understand the properties of a plasma, such as its temperature, density, and magnetic field strength. By analyzing the dispersion relation, scientists can extract information about the plasma that is otherwise difficult to measure experimentally.

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