Consider two semi-infinite, grounded, conducting plates lying parallel to the - plane, one at , and the other at (see Fig. 46). The left end, at , is closed off by an infinite strip insulated from the two plates, and maintained at a specified potential . What is the potential in the region between the plates?

We first of all assume that the potential is -independent, since everything else
in the problem is. This reduces the problem to two dimensions.
Poisson's equation is written

for , since the two plates are earthed, plus

for , and

as . The latter boundary condition is our usual one for the scalar potential at infinity.

The central assumption in the method of separation of variables is that a
multi-dimensional potential can be written as the product of one-dimensional
potentials, so that

Substituting (776) into (771), we obtain

(777) |

(778) |

where and are general functions. The only way in which the above equation can be satisfied, for general and , is if both sides are equal to the same constant. Thus,

The reason why we write , rather than , will become apparent later on. Equation (780) separates into two ordinary differential equations:

(781) | |||

(782) |

We know the general solution to these equations:

(783) | |||

(784) |

giving

(785) |

(786) |

where has been absorbed into . Note that this solution is only able to satisfy the final boundary condition (774) provided is proportional to . Thus, at first sight, it would appear that the method of separation of variables only works for a very special subset of boundary conditions. However, this is not the case.

Now comes the clever bit! Since Poisson's equation is *linear*, any
linear combination of solutions is also a solution. We can therefore form a
more general solution than (787) by adding together lots of solutions involving
different values of . Thus,

(788) |

The question now is what choice of the fits an arbitrary function
? To answer this question we can make use of two very useful properties
of the functions . Namely, that they are mutually *orthogonal*, and
form a *complete set*. The orthogonality property of these functions manifests
itself through the relation

(790) |

(791) |

(792) |

(793) |

If the potential is constant then

(794) |

(795) |

(796) |

(797) |

In the above problem, we write the potential as the product of one-dimensional
functions. Some of these functions grow and decay monotonically (*i.e.*, the
exponential functions), and the others oscillate (*i.e.*, the sinusoidal functions).
The success of the method depends crucially on the orthogonality and completeness
of the oscillatory functions. A set of functions is *orthogonal*
if the integral of the product of two different members of the set over some
range is always zero: *i.e.*,

(798) |

Finally, as a simple example of the solution of Poisson's equation in spherical
geometry, let us consider the case of a conducting sphere of radius , centered on the
origin, placed in a uniform -directed electric field of magnitude .
The scalar potential satisfies
for , with
the boundary conditions
(giving
) as
, and at . Here, and are spherical polar coordinates. Let us try
the simplified separable solution

(799) |

(800) |

(801) |

(802) |