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AS551 General Plasma Physics 1 \\
Jan. 13, 2003 \\
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{\bf Problem Set \# 9 (due Tuesday Jan 21, 2003)}
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G\&R refers to Goldston and Rutherford's textbook.

{\bf 1. G\&R problem 11.4. (describe process of approaching thermal
equilibrium}

{\bf 2. G\&R problem 12.1. (solutions to diffusion equation)}

{\bf 3. G\&R problem 12.6. (Effect of ions with charge $Z$ on transport
coefficients)}  (Approximate answers using scalings from random-walk
arguments are sufficient here.)

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For a plasma in a stationary equilibrium ($d/dt = 0$), with an isotropic
pressure, the MHD equilibrium relation is:
$$
\grad p = { \vec j \times \vec B \over c}
$$
In the following problems, you will explore some of the
properties of equilibria implied by this equation.

{\bf 4. Goldston and Rutherford problem 9.1 (plasma hole)}

{\bf 5. Goldston and Rutherford problem 9.2 (pinch equilibrium) }

% {\bf 3. Goldston and Rutherford problem 9.3, except skip the last two
% sentences of it related to Bessel functions.}

{\bf 6. Bennett pinch condition.} Consider a standard $z$-pinch equilibrium
configuration (current in the $z$-direction, magnetic field only in the
$\theta$ direction).  The plasma is uniform in $z$ and $\theta$, and is
confined to $r<a$ by the magnetic field.  Calculate the volume integrated
plasma energy (per length in z):
$$
W = \int_0^a dr 2 \pi r {3 \over 2} p
$$
Integrate this by parts (assuming $p=0$ at $r=a$) to express this in terms
of $\partial p / \partial r$, and use the MHD force balance equation to
show that $W$ can be expressed in terms of the total current carried by the
plasma.  (This is known as the Bennett pinch condition.)

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