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Magnetostatics

For a static equilibrium in a plasma, magnetohydrostatics is applicable and in the MHD approximation ($V \ll c$, plasma speeds are much smaller than the speed of light) we have
\begin{displaymath}
\nabla \times \left ({{\bf B}\over \mu}\right ) = {\nabla
\times {\bf B}\over \mu} = {\bf J},
\end{displaymath} (5.7)

where $\mu$ is assumed constant in space and
\begin{displaymath}
\nabla \cdot {\bf B} = 0.
\end{displaymath} (5.8)

When there are no currents flowing in the plasma so that ${\bf J} =
0$, we may satisfy (5.7) identically by setting

\begin{displaymath}
{\bf B} = -\nabla F,
\end{displaymath}

and hence (5.8) becomes
\begin{displaymath}
\nabla^{2}F = 0.
\end{displaymath} (5.9)

These sections show that Laplace's equation is an extremely important equation. However, we cannot solve it until boundary conditions are imposed.

Prof. Alan Hood
2000-11-06