# Dielectric Cube in a uniform electric field

1. Jun 28, 2006

### TGarzarella

Hi. Those of you familiar with the classic problem in Jackson, where a dielectric sphere (diel const = k) is placed in a uniform electric field E_0, may recall the simple expressions for the field inside of the sphere:

E_in = 3/(2+k) E_out.

The solution tells us that the field strength within the dielectric is reduced by a factor 3/(2+k).

I am trying to find the corresponding reduction factor for a dielectric cube in place of the dielectric sphere. I suspect that the reduction factor is simply 1/k, since that what it is in 2 dimensions, but I'm not sure.

Does anyone know what the reduction factor is, or where I can find it?

Last edited: Jun 28, 2006
2. Jun 28, 2006

### actionintegral

Forgive me - I don't have the problem in front of me, so I am just guessing. But didn't Jackson use spherical coordinates to solve that? And if
so, you would have a problem using a cube because of the lack of symmetry.

3. Jun 28, 2006

### TGarzarella

I should have mentioned that I'm interested in the simplest case where the field direction is parallel to four faces of the cube (and normal to the other two faces).

4. Jun 28, 2006

### actionintegral

I kind of assumed that, but even in the case of a conducting cube you would have difficulty with the boundary conditions

5. Jun 28, 2006

### Claude Bile

Is this not just placing a dielectric between two plates of a capacitor?

Claude.

6. Jul 1, 2006

### Meir Achuz

A dielectric cube in a uniform field is a very difficult problem, because there is no useful symmetry or approximation to use.
Probably a relaxation method woud be the best try for getting a numerical computer solution.

7. Jul 2, 2006

### Claude Bile

It should just be 1/k just as in the 2D case. The more I look at the problem, the more convinced I am that it is really that simple, so long as 4 planes of the cube are aligned parallel with the field.

I just can't see a reason why this would not be the case.

Claude.

Last edited: Jul 2, 2006
8. Jul 3, 2006

### TGarzarella

I thought the same -it should be 1/k. It seems that the two dimensional argument should apply in 3D as well.

Without going into too much details, I'm doing experiments where I measure the field within a dielectric crystal (of cubic geometry). I'm applying the field with parallel plates and a voltage supply. When both plates are in contact with the crystal, the field I get inside the crystal is V/d (d=distance between plates). No surprise there.

But when the plates are not in contact with the crystal, the field inside the crystal is reduced to (V/d)/k_eff.

I expect k_eff to be the dielectric constant of the material, which happens to be about 30. But I repeatably get k_eff = 5, about 6-7 times smaller.

So I'm wondering if there's some 3-D effect that I'm missing. In a sphere for example, k_eff = (2+k)/3 ~ 10 for this material -much closer to my experimental value of 5. So I was wondering if there is a similar 3D expression for the case of a cube.

9. Jul 3, 2006

### Claude Bile

What is the separation between the cube and plates for the non-contact measurements? Also;

Is the value for k_eff what you expect it to be if the plates are in contact with the cube?

Claude.

10. Jul 4, 2006

### TGarzarella

k_eff = 1 when in contact, and assumptotically reaches its max value as d is increased.

Using a 2D argument, this is exactly what we expect -with the assumptotic value equal to k. Experimentally however, the assumptotic value is about k/6, where k is the published value of the dielectric constant of that material. I've used several other dielectrics, with similar results. Varying the plate size/geometry didn't change the results either.

For comparison with the sphere, I'm trying to find a 3D solution for the field inside a dielectric cylinder with the external field parallel to its z-axis.

11. Jul 4, 2006

### Claude Bile

Hmmm, are you measuring k at the correct frequency? By that I mean, are you measuring it for the frequency values that literature quotes k for?

I can't think of anything else that may account for this discrepancy.

Claude.