# Electrostatic E Fields - Possible/Impossible

I have a question pertaining to mathematically possible E fields.

I've always known the curl of the field has to be zero for the field to be a possible electric field, but is this the sole determining factor? What about if the potential is some wonky function that doesn't seem plausible, and what about discontinuities?

Also, a semantics question, would you think an electric field is mathematically possible if the charge distribution is physically impossible to create?

Think about an infinitesimal point that has some charge on it. If the field could curl then there could be a spiral of field lines around that point. That wouldn't jive with what physicists have always measured for the field of a point charge, namely, Coulomb's law.

You can actually write a vector equation for Coulomb's law, curl it, and get 0.

As for the second question. I bet you can invent a charge distribution that approaches infinity at some point yet still have a convergent field somewhere. For instance The field may be finite directy in the middle of two bodies whose charge approaches infinity the further away they are.

I guess my question can really be boiled down to two questions:

1. Can an electric field function be mathematically possible while its curl is non-zero?
and
2. Can an electric field function be mathematically IMpossible while its curl is zero.

1. No. The curl of any gradient is 0. There's a proof for that. The E field is the gradient of the charge distribution therefore its curl is 0.

2. Do you mean physically impossible? If so, maybe.

@ Okefenokee: The E field is the gradient of the electric potential field, not the charge distribution.

2. Yes, but only if the curl has a singularity at a point. For instance the field $(-y/r^2, x/r^2)$ (the so-called irrotational vortex) has zero curl everywhere, except at the center, where it's not defined (it can be understood to be infinite). If you compute the line integral around the center, you'll find out it's non-zero, in fact, it's equal to $2\pi$. If a contour integral is non-zero, then the field isn't conservative, and thus there can be no potential field.

BTW, in electrodynamics, all of this ceases to be true: Faraday's law says that a changing magnetic field generates a curling electric field at the same point. Similarly, a changing electric field generates a curling magnetic field at the same point. This feedback process originates propagation of electromagnetic waves.