Dielectric Cube in a uniform electric field

[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?

 

[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.

 

[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).

 

[actionintegral]

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

 

[Claude Bile]

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

Claude.

 

[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.

 

[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.

 

[TGarzarella]

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.

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.

 

[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.

 

[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.

 

[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.