Potential versus Potential Difference: Ground and Infinity

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 $E_x = -\frac{\partial V}{\partial x}$ 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 $\vec{x}_0$ 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 $\frac{1}{r^2}$ or faster (dipole like $\frac{1}{r^3}$, quadrupole like $\frac{1}{r^4}$, 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 away¹:

$$V(\vec{x}) = - \int_\infty^{\vec{x}} \vec{E} \cdot d\vec{l}$$ (7)

Note that we leave the $\Delta$ 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.