Massachusetts Institute of Technology
Subject 6.651J/8.613J/22.611J
13 September 2005
R. Parker
Due: 20 September 2005
Problem Set 1
1. The three-dimensional Maxwellian velocity distribution function is
r
f m (v ) =
where n is the particle density and vt2 =
⎛ − v2 ⎞
exp⎜⎜ 2 ⎟⎟
(vt π ) 3
⎝ vt ⎠
n
2kT
. Determine the following:
m
i) the average of each velocity component, e.g., v x ;
ii) the average speed v ;
iii) the rms speed
v2 ;
iv) the average kinetic energy in any one direction, e.g.,
1 2
mvx ; and
2
v) the flux crossing a planar surface in one direction.
2. In class we worked out the potential associated with a sheet of charge immersed in a plasma with
electron temperature Te and ion temperature Ti and obtained the result
⎛ x ⎞
σλD
⎟
exp⎜⎜ −
⎟
2ε 0
⎝ λD ⎠
where σ is the charge density, x is the coordinate perpendicular to the charge sheet and λ D is the
eϕ
<< 1 so that the exponentials appearing in
Debye length. A key assumption in our analysis was
kTe,i
Poisson’s equation could be well represented by the first two terms in a Taylor series. The purpose of
this problem is to remove this restriction for a plasma in which Te = Ti = T .
ϕ=
i) Use the quadrature method discussed in class to solve Poisson’s equation (without approximations)
dϕ
for
and then integrate to get x(ϕ ) in terms of V , the potential of the charge sheet.
dx
ii) What is V in terms of σ ?
iii) Let x s be the distance from the sheet where the potential has fallen to Ve −3 . Sketch x s vs.
How does it compare with the result of the calculation for the case
eϕ
kTe,i
<< 1 ?
eV
.
kT
3. A spherical ball of plasma consists of cold (Te = Ti = 0) electrons and singly charged ions with
r
uniform density, ne = ni = n0 . The sphere of electrons is perturbed a distance δ from its equilibrium
position, i.e., where the electron sphere would coincide with that of the ions. Find the frequency at
which the electron sphere oscillates. (Note: one particular form of a nearly spherical plasma occurs
naturally and is known is known as “ball lightning”.)
4. Consider the parameters of the plasmas listed in the table below, taken from notes for this subject
prepared by Prof. A. Bers. Complete the table by calculating the Debye length λ D , the plasma
parameter Λ = nλ3D and the plasma frequency f p . Which plasmas are strongly coupled? Which need a
quantum mechanical treatment?
Plasmas
Weakly ionized
Ionosphere, D layer 70
km
Gas discharge, weak
current
Gas discharge, strong
current
MHD energy convertor
Strongly ionized
Interstellar gas
Solar wind
Ionosphere, F2 layer
250 km
Solar corona (RO ~1.5)
Tokamaks
Alkali plasmas –
surface ionization
Laser plasmas
Nuclear explosions
Magnetosphere of
pulsars
Dense plasmas
Electrons in metals
Interior of stars
Interior of white
dwarfs
Note: Te in °K, ne in m-3.
Log10ne
Log10Te
9
2.5
17
4
21
5
22
3
0
6.5
11.5
3.5
5
3
13
20
18
6.5
8
3
25
26
18
5
6
16
29
33
2.5
7.5
38
7
λD(m)
Λ
fp(Hz)
Strongly
coupled?
QM?