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

$$\nabla \times \left ({{\bf B}\over \mu}\right ) = {\nabla \times {\bf B}\over \mu} = {\bf J},$$ (5.7)

where $\mu$ is assumed constant in space and

$$\nabla \cdot {\bf B} = 0.$$ (5.8)

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

$${\bf B} = -\nabla F,$$

and hence (5.8) becomes

$$\nabla^{2}F = 0.$$ (5.9)

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