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 Consider a quasineutral plasma consisting of electrons of mass
, charge
, temperature
, and mean number density,
, as
well as ions of mass
, charge
, temperature
, and mean number density
.
 Generalize the analysis of Section 1.4 to show that the effective plasma frequency of the
plasma can be written
where
and
. Furthermore, demonstrate that
the characteristic ratio of ion to electron displacement in a plasma oscillation is
.
 Generalize the analysis of Section 1.5
to show that the effective Debye length,
, of the plasma can be written
where
and
.
 The perturbed electrostatic potential
due to a charge
placed at the origin in
a plasma of Debye length
is governed by
Show that the nonhomogeneous solution to this equation
is
Demonstrate that the charge density of the shielding cloud is
and that the net shielding charge contained within a sphere of radius
, centered on the origin, is
 A quasineutral slab of cold (i.e.,
) plasma whose bounding surfaces are normal to the
axis consists of electrons of mass
, charge
, and
mean number density
, as well as ions of charge
, and mean number density
. The ions can effectively be treated as stationary. The slab is placed in an externally generated,
directed
electric field that oscillates sinusoidally at the angular frequency
. By generalizing the analysis of Section 1.4, show
that the relative dielectric constant of the plasma is
where
.
 A capacitor consists of two parallel plates of crosssectional area
and spacing
. The region between the
capacitors is filled with a uniform hot plasma of Deybe length
. By generalizing the analysis of Section 1.5, show that the d.c. capacitance of the device is
 A uniform isothermal quasineutral plasma with singlycharged ions is placed in a relatively weak gravitational field of acceleration
. Assuming,
first, that both species are distributed according to the MaxwellBoltzmann statistics; second, that the perturbed electrostatic potential is a function of
only; and, third, that the electric field is zero at
(and well behaved as
), demonstrate that the electric field in the region
takes the form
,
where
and
Here,
is the Debye length,
the magnitude of the electron charge, and
the ion mass.
 Consider a charge sheet of charge density
immersed in
a plasma of unperturbed particle number density
, ion temperature
, and electron temperature
. Suppose that the charge sheet
coincides with the

plane. Assuming that the (singlycharged) ions and electrons
obey MaxwellBoltzmann statistics, demonstrate that in the limit
the electrostatic
potential takes the form
where
.
 Consider the previous exercise again. Let
. Suppose, however,
that
is not necessarily much less than unity.
Demonstrate that the potential,
, of the charge sheet (relative to infinity)
satisfies
Furthermore, show that
where
.
Let
be the distance from the sheet at which the potential
has fallen to
, where
. Sketch
versus
.
 A long cylinder of plasma of radius
consists of cold (i.e.,
) singlycharged ions and electrons with uniform number
density
. The cylinder of electrons is perturbed a distance
(where
) in a direction perpendicular to its axis.
 Assuming that the ions are immobile, show that the oscillation frequency of the electron
cylinder is
where
is the electron mass.
 Assuming that the ions have the finite mass
, show that the oscillation frequency is
 A sphere of plasma of radius
consists of cold (i.e.,
) singlycharged ions and electrons with uniform number
density
. The sphere of electrons is perturbed a distance
(where
).
Assuming that the ions are immobile, show that the oscillation frequency of the electron
sphere is
where
is the electron mass.
Next: Charged Particle Motion
Up: Introduction
Previous: De Broglie Wavelength
Richard Fitzpatrick
20160123