17.8 Assignments 505
- Dust whistler waves: Show that a dusty plasma withα≃ 1 will support whistler-like
waves with dispersionω≃ωcicosθwhereωcd<<ω<<ωci.
- Make a plot like Fig. 17.3 for a dusty plasma withTe=Tiand discuss whether such
a plasma could condense and form crystals.
- Consider a methane plasma consisting of CH 4 ions and electrons. Assume that there
exist some infinitesimal dust nuclei att=0and that these become negatively charged
as discussed in Sec. 17.2.
Further assume that any methane ion hitting the dust grain sticks to the dust grain so
that the mass of the dust grain increases with time. Plot the dust grain radius as a
function of time. How long would it take for the dust grains to become sufficiently
large to form a crystal (assume thatnd=10^5 cm−^3 ,Te/Ti=100,λdi=100μm)?
- Consider a weakly ionized dusty plasma with 5μm diameter dust grains, 3 eV elec-
trons and room temperature (0.025 eV) argon ions. The dusty plasma is located above
a horizontal metal plate lying in thez=0plane. Because electrons move faster than
ions and dust grains, the electronflux to the metal plate is initially much larger than
the ion or dust grainflux. The faster rate of loss for electrons causes the dusty plasma
to become positively charged so that an electric field develops which tends to retard
the electrons and accelerate the ions towards the metal plate. Thus, theplasma poten-
tial is positive with respect to the metal plate and if the plasma potential is defined to
be zero, then the metal plate has a negative potential, sayφplate.The electronflux to
the plate will thus be
Γe=nvTeexp(−qeφplate/κTe)
and, using a sheath analysis as in Section 2.7, the ionflux to the plate is
Γi=ncs
wherecs=
√
κTe/miis the ion acoustic velocity. In equilibrium, the electron and
ionfluxes will balance so that
exp(−qeφplate/κTe)=
√
me
mi
or
φplate=−
Te
2
ln
mi
me
where the temperature is expressed in electron volts. The change in potential will
occur over a distance of the order of the electron Debye length going from the plate to
the bulk plasma so that
φ(z)=−
Te
2
exp(−z/λDe)ln
mi
me
wherezis the distance above the metal plate. By considering the combined effect
of the vertical electric field associated with this potential and ofgravity acting on a
dust grain show that dust grains will tend to levitate above the metal plate. Plot the
potential energy of dust grains in the combination of the electrostatic and gravitational
