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 Show that for fields varying as
the equations
and
follow from Equations (5.4) and (5.5). This
explains why the former equations are not explicitly used in the study of plane waves.
 Derive Equations (5.31)(5.33) from first principles,
starting from the equations of motion of individual charged particles.
 Prove the identity
 Derive the dispersion relation (5.44)(5.47) from Equation (5.42).
 Show that the square of
, defined in Equation (5.48), can be
written in the positive definite form
 Derive the alternative dispersion relation (5.50) from (5.44).
 Show that in the limit
,
 Show that
Hence, deduce that for a righthand/lefthand circularly polarized wave the ions and
electrons execute circular orbits in the

plane in the electron/ion
cyclotron direction.
 The effect of collisions can be included in the dispersion relation for
waves in cold magnetized plasmas by adding a drag force
to the equation of motion of species
. Here,
is the effective collision
frequency for species
, where
stands for either
or
. Thus, the species
equation of motion
becomes
 Show that the effect of collisions is equivalent to the
substitution
 For high frequency transverse waves, for which
, and
, show that
the real and imaginary parts of the wavenumber are
respectively.
 Show that the dispersion relation for a longitudinal electron plasma oscillations is
Hence, demonstrate that collisions cause the oscillation to decay in time.
 A cold, unmagnetized, homogeneous plasma supports oscillations at the
plasma frequency,
. These oscillations have the
same frequency irrespective of the wavevector,
. However, when
pressure is included in the analysis, the frequency of the oscillation
starts to depend on
. We can investigate this effect by treating the
(singlycharged) ions as stationary neutralizing fluid of number density
.
The electron fluid equations are written
where
and
are constants. Let
,
,
, and
, where
the subscript 0 denotes an equilibrium quantity, and the subscript
denotes
a small perturbation. Develop a set of linear equations sufficient to solve
for the perturbed variables. Assuming that all perturbed quantities vary in space and time as
, find the dispersion relation linking
and
. Find expressions for the phasevelocity and groupvelocity of
the wave as functions of
.
Next: Wave Propagation Through Inhomogeneous
Up: Waves in Cold Plasmas
Previous: Perpendicular Wave Propagation
Richard Fitzpatrick
20160123