A very important property that potential inherits from potential energy
is that its fundamental definition only tells us what the potential *difference* is between two points in space, not the absolute value of
the potential. Look at the
etc. above - we
can add an *arbitrary* constant to the potential function and get
the same field. There are an infinite number of potential functions
that lead to the same field, and *only the field is physical* in
that it produces a force and we can measure a force. We can never
directly measure a potential or potential energy, only changes.

Yet it is very tedious to say ``potential difference between
and...'' over and over again. We'd like to be able to define *some* sort of universally accepted ``zero'' for the electrostatic
potential difference so that we could talk about ``potential'' (relative
to the accepted zero) instead of ``potential difference...''.

We know by now that if our charge distribution has compact support
(which basically means that one can draw a ball around it of *some*
finite radius that has all the charge inside) then the *field*
vanishes like the monopolar moment of the total charge or
or faster (dipole like , quadrupole like
, etc.).

Since the field vanishes at infinity, it seems reasonable to let our
reference potential vanish there as well. We therefore define the *potential* of a charge distribution at a point as the *potential
difference* between that point and a point infinitely far
away^{1}:

(7) |

Note that we leave the off of the potential function, and it has only one coordinate (potential difference MUST have two).

This trick will not work for certain infinite distributions of charge, e.g. infinite lines or cylinders, infinite planes or slabs. In those cases the field doesn't die off fast enough at infinity for the integral of the field from infinity to be non-infinite. We'll note this in the examples below as appropriate.

From now on I'll use ``potential'' and ``potential difference'' fairly
interchangeably, *except* when dealing with those infinite
distributions where only the latter makes sense.