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The boundary conditions determine the solution and the particular
choice of conditions used depends on the
situation that is being modelled.
 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, , tends to zero at large distances from the sources.
Thus,

(5.10) 
For a uniform field at infinity such as
,
or
, we
require that one of the following hold

(5.11) 
 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

(5.12) 
However, for a magnetic dipole or electrostatic dipole we require

(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.
Next: Uniqueness Theorem
Up: Physical Applications
Previous: Magnetostatics
Andrew Wright
20020916