Fall Term 2002
Introduction to Plasma Physics I
22.611J, 6.651J, 8.613J
Problem Set #7
1. Calculate the potential field due to a stationary test charge inserted (at, x = 0, and time,
t = 0) in a collisionless plasma initially in thermal equilibrium. You may assume that the
test charge is sufficiently small that the linear response of the plasma is accurate, and you
may also neglect any ion motion.
First show that the Fourier-Laplace transform of the induced potential is given by,
φkω =
S0 (k, ω)
ST (k, ω)
+
k 2 (k, ω) k2 (k, ω)
where, ST (k, ω), is the test charge source,
ST (k, ω) =
4πiqT
ω
and, S0 (k, ω), is the initial value source. Use the dielectric response function, (k, ω), from
lecture, or the expression,
·
¶¸
µ
1
ω
ω
(k, ω) = 1 + 2 2 1 +
Z
kve
kve
k λD
e (x, t → ∞). Explain
Now invert the transforms to find the asymptotic, t → ∞, response, φ
e, would be
why the initial value term doesn’t contribute and why the asymptotic limit for, φ
appropriate for most applications.
Show that the same result obtains if the electron density fluctuation is calculated from the
thermal equilibrium response,
³
´
e
e e ' fmax + fmax eφ + . . .
fe = fmax exp eφ/T
Te
This is the so-called Adiabatic Response that we use previously to derive the Debye cloud.
Although collisions and thermal equilibrium were invoked in this prior derivation, you see
now that the same shield cloud results in a pure, Vlasov, collisionless plasma.
Describe in words what this solution represents in terms of physics.
2. Bump-on-tail Instability: Consider an electron distribution of the form,
fe (u) =
³
´
¡ 2 2¢
(np + nb )
nb
2
2
exp
−u
/v
exp
−
(u
−
u
)
/v
+
b
e
b
π 3/2 ve3
π 3/2 vb3
1
with the orderings, nb ¿ np , ub À ve , ve ∼ vb . You may assume high frequency modes and
neglect ion motion. Make a plot of this distribution function and show where you expect
unstable waves to occur. What are the approximate ranges in frequency and wavenumber
over which waves are unstable? Calculate, for the fastest growing mode, the frequency,
wavenumber and growth rate. You can use the orderings given above and only need to
express your answers to leading order in any small parameters.
3. Electrical Current Driven Instability: Consider a quasi-neutral electron-proton plasma in
which an equilibrium current is flowing. This may be described by a Maxwellian ion distrib
ution at rest,
¡
¢
n
fi (u) = 3/2 3 exp −u2 /vi2
π vi
where, vi2 = 2Ti /mi , and a drifting Maxwellian for the electrons,
³
´
n
fe (u) = 3/2 3 exp − (u − ud )2 /ve2
π ve
where, ve2 = 2Te /me , and, vi < ud < ve . Make a plot of the distribution functions of ions
and electrons on the same scale, look in the region, vi < ωr /k ¿ ve , and show where you
expect unstable waves might occur. Write an expression for the damping/growth factor,
γ/ωr , including both electron and ion contributions. Show that the condition, Te À Ti , is
generally required for instability, and determine the growth rate in this limit. What range of
real frequencies and wavenumbers do you expect to find?
2