# Electric potential and current around an insulator

1. Jun 29, 2011

### meldraft

Hi all,

I am trying to run a simulation, and I have come across a theoretical question.

Let's say that you have an electric charge producing a potential on a conducting surface (let's assume it's infinite). Now, if you make a crack in the surface so that there is a gap (filled with air for example), you create an area where the current can't pass through.

The equations say that the potential inside the crack will change because of the different electric permitivity, while on the other side it won't be affected at all.

My problem is how the current is supposed to move. If I just integrate for the current density I can get the current vectors on the surface:

$$V=\frac{1}{4\pi\epsilon}\frac{Q}{r}$$
$$E=-\nabla{V}=\frac{1}{4\pi\epsilon}\frac{Q}{r^2}$$
$$J=\sigma E$$
$$I=\int{J\cdot dA}$$

, but they look like they would without the crack, with the exception that no current passes through the crack.

Current moves along the electric field, which, in my case, just has a gap where the crack is, and its shape is otherwise unaffected.

I know that current is supposed to go around the crack, much like a fluid would, so I probably need a boundary condition for the crack. Does anybody know how I should go about it???

2. Jun 30, 2011

### chrisbaird

I fear the answer is more complicated than you are hoping for. If you have currents through a conducting sheet with a hole (crack, gap, slot, etc.), then the hole will http://en.wikipedia.org/wiki/Slot_antenna" [Broken]. As a first order approximation, you can treat the hole as a radiating dipole oriented perpendicular to the conducting surface and located at the hole's center.

Last edited by a moderator: May 5, 2017
3. Jul 12, 2011

### meldraft

In my problem however, the material is a really bad conductor and the current is very small. Is this still applicable in that case?