# Various shapes of insulator in electric field

Given the capacitor below with large plates and small gap and a metal slab in between and a voltage applied between two conductive plates, we can prove easily there is electric field in the air gaps.

What would happen in the following cases (applied voltage doesn't exceeds the material's dielectric strength):

Case 1. What would happen if instead of a metal slab an insulator was used? Would we still have any electric field in the air gaps?

Case 2. In this case the insulator forms a sort of cave. Would we still have any electric field in the air gaps?

Case 3. In this case the insulator forms a sort of eye (enclosed area). Would we still have any electric field in the air gaps?

I understand that the insulator (dielectric) will become polarized, but I am not sure if there will be any field in the air gaps and if the size of insulator makes any difference. What I mean by that is as long as insulator is contained within the plates, will there be any electric field in the air gaps?

• Naively, I would say that there is always an electric field, both in the insulator and in air (or even vacuum). In configuration 1 it is easy to calculate the electric field, because it is perpendicular to the plates and so the problem reduces to a one dimensional problem. In the other cases, it is more difficult, but I do not see any reason why the electroc field in the air gaps should vanish. Can you please try to formulate a possible (even if not rigorous) reasoning against the presence of the electric field in the air gaps? Feb 14, 2023 at 8:10
• Analysis of the arrangements asked about in this questions are a variation of those mentioned in this post Why can we consider a capacitor with mixed dielectrics equivalent to two series or parallel capacitors?. The original arrangement can be split into series and parallel capacitors. Feb 14, 2023 at 8:26

## 1 Answer

To all your questions "Would we still have an electric field in the air gaps?", the answers are all "yes". The air gap (or vacuum or void) is a kind of electromagnetic material, in particular it is a kind of insulator. The electric fields (also magnetic fields) can exist in the air gap, the relative permittivity of the air gap is 1.0, and the dielectric material has some large relative permittivity.

The slab problem can be solved analytically, and to solve other problems with some complex shapes, we should go into numerical methods.