Potential Problems leading to Laplace's Equation

1. Incompressible, irrotational flow
   
   $$\nabla \times {\bf v} = 0, \qquad \nabla \cdot {\bf v} = 0,$$
   
   
   so that
   
   $${\bf v} = - \nabla f \qquad \Rightarrow \qquad \nabla^{2}f = 0.$$
   
   

2. Electrostatics when $\rho_{c}=0$ and $\epsilon$ is constant so that
   
   $$\nabla \times {\bf E} = 0, \qquad \nabla \cdot {\bf E} = 0,$$
   
   
   and again
   
   $${\bf E} = - \nabla F, \qquad \Rightarrow \qquad \nabla^{2}F = 0.$$
   
   

3. Magnetohydrostatics when ${\bf J} = 0$ and $\mu$ is constant. Again we have
   
   $$\nabla \times {\bf B} = 0, \qquad \nabla \cdot {\bf B} = 0.$$
   
   
   Thus, the magnetic field may be expressed in terms of a scalar potential as
   
   $${\bf B} = - \nabla F, \qquad \Rightarrow \qquad \nabla^{2}F = 0.$$
   
   

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.