Calculate polarization energy over a set of cuboids

• Glxblt76
In summary, a researcher is trying to find the equation for the polarization energy in an electrostatic system. The system is made up of a cuboid that contains a charge distribution and a dielectric constant of vacuum, and the researcher does not know the dimensions of the cuboid. They are lost and need help from the physics forums.

Glxblt76

Dear Physics Forums members,

I have a research problem that involves electrostatics. My education is as a chemist, and thus I struggle to accurately represent my problem, so I thought that you guys could help me (and would be interested in the exercise).

Here is an image to summarize my problem.

So, in the starting point of my problem, I have my space divided into orthogonal cuboids. The cuboid at the center is particular: there is a charge distribution within it, ρ(r) and its dielectric constant is that of vacuum. All other orthogonal cuboids contain homogeneous, but not isotropic, continua. In the x direction, the dielectric constant is εxx, the component of the dielectric constant tensor ε, in y direction it is εyy, and in z direction it is εzz.
I explicitely represent three cuboids around the central cuboid in each direction, for a total of 73 - 1 = 342 cuboids. This big cuboid is immersed into a continuum that corresponds to the average value of the dielectric constant tensor ε.

What I would like to write is the equation for the polarization energy of this system (i. e. the energy change caused by replacing a conventional cuboid by a cuboid containing the charge distribution ρ(r) and with the dielectric constant of vacuum).
The system is electrostatic, i e there is no variation of the magnetic field with time. It obeys the two equations:
∇E = ρ/ε0
∇×E = 0

I know the dimension of the cuboid (the 3 lengths a, b and c from which the cuboid can be reconstructed).

I'm a bit lost and I don't know where to start.
Do some people have suggestions?

All the best!

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Do not hesitate to ask questions if this does not sound clear to you!

I fear the problem is only solvable numerically. There are very few systems for which it is possible to obtain analytical solutions of the resulting polarisation.

I know it, but my question is about how can I do it efficiently, because depending on the method you choose, your solution may be reached slower or faster to the point it's practically indistinguishable from an analytical solution.

I suppose finite element methods are most adequate.

Glxblt76
Thanks. I will think about it.

1. What is polarization energy?

Polarization energy is the energy required to separate positive and negative charges within a material, resulting in the creation of an electric dipole moment.

2. How is polarization energy calculated?

Polarization energy can be calculated by summing the product of the electric field and the induced dipole moment over all the individual dipoles in a material.

3. What are cuboids and how are they related to polarization energy?

Cuboids are three-dimensional, rectangular-shaped objects. In the context of polarization energy, they are used to represent the individual dipoles within a material. The polarization energy can be calculated over a set of cuboids to determine the total energy of the material.

4. Why is it important to calculate polarization energy?

Calculating polarization energy is important in understanding the behavior of materials in electric fields. It can help predict the response of a material to an external electric field and can also be used in the design of electronic devices.

5. Are there any limitations to calculating polarization energy over a set of cuboids?

Yes, there are limitations to this method. It assumes that the dipoles are evenly distributed and that the electric field is uniform throughout the material. In reality, these conditions may not be met, leading to potential inaccuracies in the calculated polarization energy.