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Potential Problems leading to Laplace's Equation

  1. Incompressible, irrotational flow

    \begin{displaymath}
\nabla \times {\bf v} = 0, \qquad \nabla \cdot {\bf v} = 0,
\end{displaymath}

    so that

    \begin{displaymath}
{\bf v} = - \nabla f \qquad \Rightarrow \qquad \nabla^{2}f = 0.
\end{displaymath}

  2. Electrostatics when $\rho_{c}=0$ and $\epsilon$ is constant so that

    \begin{displaymath}
\nabla \times {\bf E} = 0, \qquad \nabla \cdot {\bf E} = 0,
\end{displaymath}

    and again

    \begin{displaymath}
{\bf E} = - \nabla F, \qquad \Rightarrow \qquad \nabla^{2}F = 0.
\end{displaymath}

  3. Magnetohydrostatics when ${\bf J} =
0$ and $\mu$ is constant. Again we have

    \begin{displaymath}
\nabla \times {\bf B} = 0, \qquad \nabla \cdot {\bf B} = 0.
\end{displaymath}

    Thus, the magnetic field may be expressed in terms of a scalar potential as

    \begin{displaymath}
{\bf B} = - \nabla F, \qquad \Rightarrow \qquad \nabla^{2}F = 0.
\end{displaymath}

These examples illustrate the importance of Laplace's equation. Solutions to potential problems will be discussed in the next section. However, before studying these problems, we consider the situation where a non-zero current flows along a plasma cylinder.

Prof. Alan Hood
2000-11-06