Electric field singularities

Question

Is this the list of all possible singularities in electrostatic field $E$?

1. near the point charge: $\frac{1}{r^2}$

2. near the line of charge: $\frac{1}{r}$

3. near the edge (not surface) of uniform charged sheet: $-\log(r)$

(no singularity for volume distribution of (finite) charge)

Answer 1

This sounds correct.

If you were to attempt to discover more singularities in idealized configurations of electric fields it would help to consider the source of these electric fields. In these cases the ideal calculation of the electric field comes from Gauss's law. The singular nature of the field seems to come from the fact that you can shrink your Gaussian surface to arbitrarily small sizes, and still contain the source. After a bit of thinking, you should be able to convince yourself that the different possibilities are specifically limited by the dimensionality of the source relative to dimensionaliy of our 3D space.

To make this more mathematical, lets take a general charge density source $\rho(x,y,z)$. If we have an infinite sheet of charge in the x-y plane with surface density $\sigma(x,y)$, we can write $\rho$ in terms of $\sigma$ and a Dirac delta function:

$\rho(x,y,z) = \sigma(x,y) \delta(z)$

If we have an infinite line of charge along the x axis with charge per unit length $\lambda(x)$ we can similarly write:

$\rho(x,y,z) = \lambda(x) \delta(z) \delta(y)$

Finally, a point charge of charge q at the origin can be expressed as:

$\rho(x,y,z) = q \, \delta(z) \delta(y) \delta(x)$

It is really these delta functions which give jumps in the electric fields (which you can calculate with Gauss's law). In the real world (in classical E&M), there are no distributions of arbitrarily small size in some dimension, so in practice the electric field doesn't actually diverge.

Now if we lived in a universe with 2 spatial dimensions, how would this change the electric field for various distributions (keeping the same form of Maxwell's equations)? What about 4 spatial dimensions? This exercise helps you see that the order of the singularity in the electric field is dependent on the spatial dimension (via the use of a Gaussian surface and Gauss's law).