Next: Charged Particle Motion Up: Introduction Previous: De Broglie Wavelength

# Exercises

1. Consider a quasi-neutral plasma consisting of electrons of mass , charge , temperature , and mean number density, , as well as ions of mass , charge , temperature , and mean number density .
1. 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 .
2. Generalize the analysis of Section 1.5 to show that the effective Debye length, , of the plasma can be written

where and .

2. The perturbed electrostatic potential due to a charge placed at the origin in a plasma of Debye length is governed by

Show that the non-homogeneous 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

3. A quasi-neutral 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 .

4. A capacitor consists of two parallel plates of cross-sectional 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

5. A uniform isothermal quasi-neutral plasma with singly-charged ions is placed in a relatively weak gravitational field of acceleration . Assuming, first, that both species are distributed according to the Maxwell-Boltzmann 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.

6. 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 (singly-charged) ions and electrons obey Maxwell-Boltzmann statistics, demonstrate that in the limit the electrostatic potential takes the form

where .

7. 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 .

8. A long cylinder of plasma of radius consists of cold (i.e., ) singly-charged ions and electrons with uniform number density . The cylinder of electrons is perturbed a distance (where ) in a direction perpendicular to its axis.
1. Assuming that the ions are immobile, show that the oscillation frequency of the electron cylinder is

where is the electron mass.
2. Assuming that the ions have the finite mass , show that the oscillation frequency is

9. A sphere of plasma of radius consists of cold (i.e., ) singly-charged 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 2016-01-23