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AS551 General Plasma Physics 1 \\
September 19, 1997 \\
\vspace{12pt}
{\bf Problem Set \#1 (due Sept. 26)}
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1. Focused beams of electrons are used in, among other devices,
electron microscopes, cathode ray tubes and free electron lasers.  For low
energy applications, it might be necessary, for example, to form a 1 $\mu$a
beam of 2.5 keV electrons with a beam radius of perhaps 0.25 mm.  Very
roughly, how far might such a beam travel before its radius doubles?
(Hint: argue that magnetic forces are negligible and write an equation of
motion for an electron at the rim of the beam.  You need not bother to
solve any integrals, except very approximately; e.g. you may approximate
$\log(1+x) \approx x$ or $\log x = $ constant over an appropriate range.)

2.  For a spherical cloud of electrons of radius $r$, density $n$, and
temperature $T$, do a rough order-of-magnitude estimate of the ratio of the
potential energy of the cloud to the kinetic energy.  How does this scale
in terms of the Debye length?

3. Employ perturbation theory to solve for the real solutions to
$$
x^3 = 1000.001 + 1/x,
$$
accurate to at least $\pm 10^{-8}$.

4.  What configuration of electric currents or charges will produce in a
given region\\
a) a uniform straight magnetic field? \\
b) an electric field with curl?

5.  Solve the vector differential equation
$$
\ddot{\vec{x}} = \dot{\vec x} \times \vec B + E
$$
for $\vec B = B \hat{i_z}$ and $\vec E = E \hat{i}_z$ in the z direction.

6.
In class, we derived the Debye shielding of the potential produced by a
test charge $Q$ in a plasma.  Calculate the resulting electron density and
verify than the test charge is indeed shielded by showing that the total
excess electron charge is $-Q$.  [Something to think about: Suppose the test
charge is {\it moving}.  How do you expect the Debye shielding to change?]


7.  What is the approximate mean free path and the collision frequency of:
a) oxygen in air; b) 20 keV electrons in a plasma with density $n= 10^{14}$
cm$^{-3}$, c) 1 eV electrons in a plasma with density $n=10^{13}$
cm$^{-3}$?

8.  Assuming that the collision cross section is the square of the distance
of closest approach, show that the collision frequency of an electron
streaming through a field of stationary ions scales as $1/v^3$.  [Later in
the course, we shall see this assumption underestimates the magnitude of
the collision frequency by about $\log(n\lambda_D^3)$.  The $v$ scaling is,
however, correctly given.]

9.
Consider a homogeneous cold electron plasma, where the ions may be assumed to
be an infinitely massive neutralizing background.  Suppose that a small
fraction of the electrons in the slab $x_1<x<x_2$ are removed and then placed
at $x_1+L<x<x_2+L$.  Sketch the subsequent evolution of the electron density,
the electric field, and the electron fluid velocity.  Sketch the trajectory of
a single electron.

10.  Since much of the galaxy is filled with an electron-proton plasma, one
might expect that the propagation of starlight reaching the earth could be
affected.  Calculate the dispersion relation for small amplitude plane
electromagnetic waves in this plasma, by modeling the plasma as a uniform
cold electron fluid with a stationary ion background.  (By ``cold'' we mean
that the thermal motion or pressure of the electrons can be neglected, and
just their acceleration by the electric field needs to be included.)  If
the electron density were 1 cm$^{-3}$, what restriction is placed on the
observation of radio wave signals?


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