In electrostatics, the potential must satisfy the Poisson equation:
\nabla^2 V = -\rho/\epsilon_0
This differential equation has an infinite number of solutions. The one that solves a particular problem satisfies the boundary conditions which are given as inputs to the problem.
If the region or volume of interest is free of charges (\rho = 0 ), then this reduces to the Laplace equation. This region is defined by a closed surface or boundary. There may be surface charges on this boundary or some unspecified charges outside the region of interest. Both of these charges result in a potential on the boundary surface and the potential on this boundary is one of the given inputs.
Sometimes this boundary value information alone is not sufficient to determine a unique solution for the potential using the Laplace equation. One then needs to infer additional constraints on the potential based on the given information.
These additional constraints commonly result from physical arguments about the geometry of the problem and usually comes in the form of whether the potential should be finite or infinite at some location.
In thinking about how to tease out the general behavior of the potential at various locations, it is useful to recall the relationships between the potential and the electric field since it might be more straightforward to have an intuition about the behavior of the electric field first:
\vec{E} = -\vec{\nabla} V\ \rightarrow\ V_B = -\int_A^B\ \vec{E} \cdot d\vec{\ell} + V_A
where we’ve allowed for the freedom to set the absolute reference of the potential (a constant offset) to some nonzero value V_A = \mathrm{constant} \neq 0.
The potential due to a point charge becomes infinite at the location of a point charge (\vec{r}\,') and is a direct consequence of the inverse square law nature of Coulomb’s Law:
V = \frac{1}{4\pi \epsilon_0} \frac{q}{|\vec{r} - \vec{r}\,'|}
It is tempting to think, therefore, that a region free of charges must contain a potential that is finite and this argument is perfectly valid if the charge q above is also finite. Just to be clear, if there is a point charge that has infinite magnitude |q| \rightarrow \infty, then the potential due to this point charge is infinite everywhere.
This line of reasoning provides an important hint on special cases that seem to defy our intuition about how the potential should behave very near or very far away from charge distributions.
The simplest and most intuitive case is that for a localized charge distribution with a finite amount of charge. Because of the inverse square law nature of Coulomb’s Law, we correctly expect that the potential does not increase indefinitely as the observation point moves further and further away from the charge distribution. We instead correctly expect that as the observation point moves further away from the charge distribution that the potential decreases to a limiting finite value, which could be zero if the reference potential offset is chosen to be zero.
The meaning of the term localized merits some discussion. Imagine placing the center of a hypothetical spherical shell at the center of your charge distribution. If it possible to expand the radius of this hypothetical spherical shell out to some finite value for the radius such that all of the charge of the charge distribution is contained within the hypothetical spherical shell, then we refer to this charge distribution as localized. If the radius of this hypothetical shell needs to extend all the way out to infinity to encompass all of the charge, then this charge distribution is not localized.
Within a localized charge distribution that contains no point charges (i.e. no locations where the volume charge density is infinite), then the potential must be finite (but not necessarily zero).
Things get tricky when the charge distribution is not localized. Examples include linear charge densities distributed along a line of infinite length and surface charge densities distributed throughout an infinite plane.
If the total charge represented by a non-localized charge distribution is infinite, then our normal intuition that the potential must decrease to some limiting finite value “far away” is not correct. Conceptually this can be understood as the charge in the numerator compensating for the separation distance in the denominator in some complicated way:
V \propto \frac{q}{|\vec{r} - \vec{r}\,'|} \frac{\rightarrow}{\rightarrow} \frac{\infty}{\infty} = \text{depends on the details}
where you can imagine that we are calculating the “monopole” contribution to the potential based on a multipole expansion.
The “depends on the details” is a nuanced statement about a comparison of the rates at which the total charge and the separation distance each approach infinity and this requires some sort of L’Hopital’s rule type reasoning. In any case, the exact form of the charge distribution is needed to determine how that potential behaves as the observation point is moved further and further away. The exact details of the charge distribution is also needed for the case of a non-localized charge distribution that represents a finite amount of charge.
We can get some practice on how to approach these types of problems by considering two of the simplest cases both involving constant (uniform) charge densities (1) a uniformly charged infinite line of charge (i.e. linear charge density \lambda is a constant) and (2) a uniformly charged infinite plane of surface charge (i.e. surface charge density \sigma is a constant).
In both cases, the infinite extent of the charge distributions imply that the total charge is also infinite. Gauss’s Law can be used to determine the magnitude of the electric field in both of these cases and we simply quote the results:
\vec{E}_\mathrm{line} = \frac{\lambda}{2 \pi \epsilon_0} \frac{\hat{s}}{s}
\vec{E}_\mathrm{plane} = \frac{\sigma}{\epsilon_0} \hat{n}
where the line is placed on the z-axis and \hat{s} points radially away from the line and \hat{n} is normal to the plane and points away from the plane.
Integrating these two electric fields yields the corresponding potential differences between two locations labeled by subscripts 1 and 2:
\Delta V_\mathrm{line} = \frac{\lambda}{2 \pi \epsilon_0} \log \left ( \frac{s_2}{s_1} \right ) \propto \log(s)
\Delta V_\mathrm{plane} = \frac{\sigma}{\epsilon_0} (z_1 - z_2) \propto -z
where s_2 > s_1 assuming that the line charge is located at s=0 (on the z-axis) and z_2 > z_1 assuming the plane is the xy-plane (i.e. at z=0).
As s_2 and z_2 both approach infinity, the potential diverges and becomes infinite and this general behavior provides an additional constraint that the full solution for the potential must obey.