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

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

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

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

(... and no singularities for finite charges smoothly spread over a finite volume.)

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I don't understand the question. Are you asking if this is the list of all possible types of singularities? –  joshphysics Feb 24 '13 at 1:49
    
yes, exactly... –  richard Feb 24 '13 at 4:57
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If you're dealing with full classical E&M, I'd call the discontinuity of the field as you cross the surface of a conductor a singularity. Also, the field is wild near sharp corners of conductors. –  Jerry Schirmer Sep 3 '13 at 21:17

1 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).

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Thanks, the dimensionality is important for sure and the idea of Gaussian surface seems helpfull but I can't quite understand some details out of these ideas. For example In the case of 2D charge distribution the infinity is only near the edge. –  richard Feb 26 '13 at 5:49
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Re: 2D charge distribution. That's not entirely true. You are only considering the magnitude of the field, not its direction as well. The field switches discontinuously as you travel through the 2D plane. This discontinuity is simply one 'step up' from the 1/r singularity. Does that clarify a bit? –  jeffdk Feb 26 '13 at 5:54
    
you are right. Interesting point. I did not note that. Can this discontinuity also be derived from the idea of Gaussian surface that you mentioned in the answer? –  richard Feb 26 '13 at 10:32
    
Yes, the discontinuity in the field can be derived using a Gaussian surface (you might want to check out the wikipedia article ). Though physically, what is causing the discontinuity is the fact that your charge distribution is singular in one dimension (this applies to all the field singularities). –  jeffdk Feb 26 '13 at 16:49

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