Consider the propagation of plane electromagnetic waves with field variation of the form:
in a plasma of electron concentration N but no superposed constant magnetic field.
The electrons make collisions with neutral atoms at a frequency 
so the equation of
motion of the electrons would be:
where 
is the macroscopic velocity of the electrons under the influence of
the wave field 
.
By combining this equation with a certain Maxwell equation, find a dispersion equation
giving the propagation constant k in terms of 
and properties of the
electron.
Show that in the limit 
your dispersion equation reduces to the simple plasma
dispersion equation:
Assume where necessary that 
.
SOLUTION:
The equation of motion for an electron will be:
Maxwell's equation:
In the plasma of electron concentration N:
Substitute for 
in Maxwell's equation:
Another of Maxwell's equations:
Eliminate 
from equations (i) and (ii).
Thus we can eliminate to give:
We were told to assume so:
Since 
we can `cancel' it on both sides and:
So:
Which is the answer to the question.
In the case of no collisions ( ) then: