Consider a volume bounded by some surface . Suppose that we are given
the charge density throughout , and the value of the scalar potential
on . Is this sufficient information to uniquely specify the scalar
potential throughout ? Suppose, for the sake of argument, that the
solution is not unique. Let there be two potentials and which
satisfy

(666) | |||

(667) |

throughout , and

(668) | |||

(669) |

on . We can form the difference between these two potentials:

(670) |

(671) |

(672) |

According to vector field theory,

(673) |

(674) |

(675) |

(676) |

(677) |

(678) |

The fact that the solutions to Poisson's equation are unique is very useful.
It means that if we find a solution to this equation--no matter how contrived
the derivation--then this is the only possible solution. One immediate use of the
uniqueness theorem is to prove that the electric field inside an empty cavity
in a conductor is zero. Recall that our previous proof of this was rather involved,
and was also not particularly rigorous (see Sect. 5.4).
We know that the interior surface of the conductor is at some constant potential
, say. So, we have on the boundary of the cavity, and
inside the cavity (since it contains no charges). One rather obvious
solution to these equations is throughout the cavity. Since the
solutions to Poisson's equation are unique, this is the *only * solution.
Thus,

(679) |

Suppose that some volume contains a number of conductors. We know that the
surface of each conductor is an equipotential surface, but, in general, we do not
know what potential each surface is at (unless we are specifically told that
it is earthed, *etc.*). However, if the conductors are insulated it is
plausible that we might know the charge on each conductor. Suppose that
there are conductors, each carrying a charge ( to ), and suppose
that the region containing these conductors is filled by a known charge
density , and bounded by some surface which is either infinity or
an enclosing conductor. Is this enough information to uniquely
specify the electric field throughout ?

Well, suppose that it is not enough information, so that there are two
fields and which satisfy

(680) | |||

(681) |

throughout , with

(682) | |||

(683) |

on the surface of the th conductor, and, finally,

(684) | |||

(685) |

over the bounding surface, where

(686) |

Let us form the difference field

(687) |

(688) |

for all , with

Now, we know that each conductor is at a constant potential, so if

(691) |

Consider the vector identity

(692) |

(693) |

(694) |

(695) |

(696) |

For a general electrostatic problem involving charges and
conductors, it is clear that if we are given either the potential at the surface of each conductor
or the charge carried by each conductor
(plus the charge density throughout the volume, *etc.*) then we can uniquely determine the electric
field. There are many other uniqueness theorems which generalize this result
still further: *i.e.*, we could be given the potential of some of the conductors
and the charge carried by the others, and the solution would still be unique.

At this point, it is worth noting that there are also uniqueness theorems associated with
magnetostatics. For instance, if the current density, , is specified
throughout some volume , and either the magnetic field, ,
or the vector potential, , is specified on the bounding surface , then
the magnetic field is uniquely determined throughout and on .
The proof of this proposition proceeds along the usual lines. Suppose
that the magnetic field is not uniquely determined. In other words,
suppose there are two magnetic fields, and ,
satisfying

(697) | |||

(698) |

throughout . Suppose, further, that either or on . Forming the difference field, , we have

throughout , and either or on . Now, according to vector field theory,

(700) |

(701) |

(702) |