Fundamentals of Plasma Physics

208 Chapter 6. Cold plasma waves in a magnetized plasma

6.9 Assignments

1. Prove that the cold plasma dispersion relation can be written as

An^4 −Bn^2 +C=0

where

A=Ssin^2 θ+Pcos^2 θ

B=(S^2 −D^2 )sin^2 θ+SP(1+cos^2 θ)

C=P(S^2 −D^2 )

so that the dispersion is

n^2 =

B±

√

B^2 − 4 AC

2 A

Prove that

RL=S^2 −D^2.

2. Prove thatn^2 is always real ifθis real, by showing that

B^2 − 4 AC=

[

S^2 −D^2 −SP

] 2

sin^4 θ+4P^2 D^2 cos^2 θ.

3. Plot the bounding surfaces of the CMA diagram, by defining( me/mi =λ,x =
   ω^2 pe+ω^2 pi

)

/ω^2 ,andy=ω^2 ce/ω^2 .Show that

ω^2 pe

ω^2

=

x

1+λ

so that

P=1−x

andS,R,Lare functions ofxandywithλas a parameter. Hint– it is easier to plot

xversusyfor some of the functions.

4. Plotn^2 versusωforθ=π/ 2 ,showing the hybrid resonances.


5. Starting in region 1 of the CMA diagram, establish the signs ofS,P,R,Lin all the
   regions.


6. Plot the CMA mode lines for plasmas havingω^2 pe>>ω^2 ceand vice versa.


7. Consider a plasma with two ion species.By plottingSversusωshow that there is an
   ion-ion hybrid resonance located between the two ion cyclotron frequencies. Give an
   approximate expression for the frequency of this resonance in terms of the ratios of
   the densities of the two ion species. Hint– compare the magnitude of the electron term
   to that of the two ion terms. Using quasineutrality, obtain an expression that depends
   only on the fractional density of each ion species.


8. Consider a two-dimensional plasma with an oscillatory delta function source at the
   origin. Suppose that slow waves are being excited which satisfy the electrostatic dis-
   persion
   k^2 xS+k^2 zP=0