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\centerline{AS551 General Plasma Physics I}
\centerline{October 6, 1997}
\vskip 12pt
\centerline{\bf Problem Set \#4 (due October 13)}
\vskip 12pt
 
 

\noindent 1.
Consider a toroidal device with major
radius $R_0=5\,\rm m$, minor radius $r=1\,\rm m$, toroidal field
$B(R=R_0)
= 5\,\rm T$.  Determine approximately how long it takes electrons (or
ions) with energy $10\,\rm keV$ to drift out of the device due to the
inhomogeneities in $\bf B$.  Suppose that a $10\,\rm keV$ plasma with
density $10^{20} \,\rm m^{-3}$ is introduced into the device.  The
$\nabla
B$ drifts will set up a time-varying electric field in the plasma,
causing
both polarization and ${\bf E\times B}$ drifts.  Verify that the
resulting
${\bf E\times B}$ motion is outwards and estimate the time it takes for
the
plasma to be ejected.
 

\vskip 8pt
\noindent 2.  Suppose a  plasma of electrons and ions is
immersed in an infinite homogeneous   magnetic field ${\bf B}=B(t)\hat
z$.
The function of time $B(t)$ is something that we can control.

The plasma is now taken repeatedly through the following four steps.

{\narrower\narrower
\item{(i)} The field strength is increased from
$B(t=0)=B_0$ to $\alpha B_0$ on a time
scale slow compared to an ion cyclotron period, but fast compared to a
collision time, i.e., $1/\nu  \gg   B/(dB/dt) \gg 1/\Omega_i$.
\item{(ii)} The plasma is allowed to relax to equilibrium by waiting
several
collision times.
\item{(iii)} The field strength is decreased to $ B_0$ as fast as in
(i).
\item{(iv)} The plasma is again allowed to relax to equilibrium.
\medskip}

\item{(a)} Derive an expression for the plasma energy gain per
cycle.

\item{(b)} Does any heating take place in the event that the
collision frequency is much greater than the rate of change of the field
in
steps (i) and (iii), i.e., $  B/(dB/dt) \gg 1/\nu \gg 1/\Omega_i$?
Explain.

\item{(c)} Assume  that electrons are isotropized in energy faster than
are ions, but that energy transfer between electrons and ions is slower
than
the ion isotropization time. How might you modify
steps (i)--(iv) in order to:
\itemitem{(1)}  pump energy mainly into electrons, but not into ions?
\itemitem{(2)}  pump energy mainly into ions, but not into electrons?
 
 

\vskip 8pt
\noindent 3. Consider an axisymmetric mirror configuration
with a vector potential given by $A_\theta(r,z) \hat \theta$ (using
cylindrical
coordinates $[r,\theta,z]$).

(a) Write down the components of the magnetic field.  Show that the
magnetic field lines lie in ``flux surfaces'' given by $rA_\theta = \rm
const$.  Show that the magnetic flux enclosed by a flux surface is
proportional to $rA_\theta$.

(b) Because the configuration is axisymmetric, a charged particle moves
in
such a way that the canonical angular momentum $r(mv_\theta +
qA_\theta)$
is constant.  Show that this means that the particle is effectively
confined to a flux surface.  Make this statement precise.

(c) The magnetic field is now varied in time (i.e., the vector potential
is
given by $A_\theta(r,z,t) \hat \theta$).  Show that the third adiabatic
invariant, the flux enclosed by the orbit as it encircles the axis of
the
device, is conserved.
 

\bye
