Interaction between a dipole and a dielectric.

[Arthas85]

Dear All,

I have a question that is afflicting me and I would be glad if you can give me an answer. I have a dipole in vacuum put at a certain distance from the surface of a dielectric material. I know the entity of the dipole, I know the distance from the surface and I know the interaction energy between the dipole and the material. How can I calculate the dielectric constant of the material? I hope that the problem is clear.

Thanks for the attention

Kind Regards

Franky

P.S. It's not properly an homework but since it was a problem I thought posting it here was the right thing to do. My problem is that I cannot find an equation relating my data in the books I have even if I know it should be possible to do that.

 

[Mike Pemulis]

I need a little more information. What do you mean by "energy of interaction," and the "entity" of the dipole? Also, what is the geometry of the dielectric medium? I'm assuming it's semi-infinite; filling all space below the z = 0 plane, for example.

 

[Arthas85]

The geometry is something like this:
-
+

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The dipole is a known molecule HF, the dielectric material is a slab of a semiconductor (periodic in two dimension with a finite depth), the energy of interaction has been computed with DFT calculations.

 

[Mike Pemulis]

Okay, that's clearer. I'm not completely sure if this is right, as I don't have my copy of Jackson with me, but here is what I'm thinking:

We need to find the potential energy of the entire system, relative to the situation where the dipole is far away from the surface of the dielectric. There are two contributions: the potential energy contained in the dielectric, and the potential energy of the dipole. To do this, you need to find the electric field everywhere. This can be done using the method of images (for dielectric media), which is explained in Jackson Chapter 4 I think. The method for single charges is (relatively) easy; for a dipole, you just add the fields created by the two charges. I hope Jackson explains it okay because I don't remember the exact method.

Now you have the electric field everywhere, inside the dielectric and outside it. Let's consider the field inside the dielectric first.

The dielectric medium is composed of many infinitesimal dipoles which can rotate but are fixed in place, so we only have to consider potential energy due to their orientation, not their position. The potential energy of a dipole fixed in place is,

U = -p$\bullet$E

Where p is the dipole vector and E is the electric field. The dielectric has a polarization density induced by the electric field, given by

P = ($\epsilon$ - $\epsilon$₀)E

Combining these two equations gives a potential energy density inside the dielectric as a function of the electric field that you obtained before. Integrate this over the whole volume of the dielectric, and that's your potential energy contribution from the dielectric.

Now you need to find the potential energy of the isolated dipole. For this, you can use the standard equation for the potential energy of a charge in an electric field, U = qV, and add the potential energies of the two charges in the dipole. One important caveat: do not include the mutual interaction between the charges, because presumably the potential energy we are trying to find is relative to that of the intact dipole at infinity, not two infinitely separated charges. So in finding U = qV, only include the electric field from the dielectric medium, NOT the other charge in the dipole.

I hope that is right and makes sense. Please ask if you have any questions.