Boundary Conditions

The boundary conditions determine the solution and the particular choice of conditions used depends on the situation that is being modelled.

1. Boundary conditions imposed at large distances vary depending on whether we have a finite system of sources or uniform conditions at inifinity. For a finite system of sources we require that the potential, $F$, tends to zero at large distances from the sources. Thus,
   

   $$F\rightarrow 0 \hbox{ at large distances from the sources. }$$ (5.10)

   
   
   For a uniform field at infinity such as ${\bf E} = E_{0}{\bf k}$, ${\bf B} = B_{0}{\bf k}$ or ${\bf v} = v_{0}{\bf k}$, we require that one of the following hold
   

   $$F\rightarrow -E_{0}z,\quad F \rightarrow -B_{0}z, \quad F \rightarrow -v_{0}z.$$ (5.11)

   
   

2. Near point sources the form of the potential depends on the nature of the point source. For a point charge (electrostatics), a pole or a simple source or sink in a fluid flow, we require
   

   $$F \approx {\hbox{const.}\over r}.$$ (5.12)

   
   
   However, for a magnetic dipole or electrostatic dipole we require
   

   $$F \approx {m \cos \theta \over r^{2}}.$$ (5.13)

   
   

The fact that Laplace's equation is linear allows us to add different solutions together. Thus, we can add together solutions due to individual point sources together with solutions for uniform fields. The combination of solutions allows us to satisfy the conditions at large distances as well as the conditions on, for example, the surface of either a sphere or cylinder. Obtaining a solution by adding together different contributions of point sources, dipoles and uniform fields is a useful approach. It is shown in the next section that the solution to Laplace's equation (plus boundary conditions) must be unique. Hence, obtaining a solution by adding together terms, until all the boundary conditions are satisfied, means that we have obtained the solution.