Correction to the field energy due to the existence of discrete charges

[SergioPL]

In the classical electromagnetic field theory, the field density of energy is given by:

$$u = (\epsilon/2)E^2 + (\mu/2)H^2$$
One of the differences between the classical electromagnetic theory and the real world is that in classical EM all charge and current density, (ρ(r), J(r)), is indistinguishable and every point of charge interacts with the rest through the generated EM field. On the other hand, in the real world we have discrete charged particles that do not interact electromagnetically with themselves (at least not directly), an example of this statement is the Hydrogen's electron Hamiltonian, the potential we see is the one created by the proton but there is no contribution from the electron itself.

The overall EM field energy changes with the interaction with charges as:

$$\partial_t u = - E · J + Flow term$$
Where there is no charge, the overall field energy does not change, it only flows.

If we have an infinitesimal density of charge isolated ρ(r)dr³, the energy of its field is order dr⁵, that means this energy is negligible compared with the charge’s mass energy.

We can conclude then that when "infinitesimal" charges are far away from each other, the EM field does not have energy but it earns it when the charges get closer. But if we accept that discrete charges do not interact with themselves through EM field, then it is evident that there is no work done in bringing together the discrete particle charge. A system with only one discrete particle would bring no energy to the EM field but it would have an electric field, so according to the classical formula, the field would have energy. This fact makes necessary some correction should be done on the formula of EM field energy.I would like to know if some of my assumptions is wrong or if this correction is explained in some theory.

 

[phyzguy]

I still think one of the best descriptions of these issues is the Feynman Lectures on Physics, Vol 2, Chap 28. It makes for fascinating reading even though it is a bit dated at the end. As he discusses, it is not so easy to throw away the idea that a charged particle interacts with itself.

 

[SergioPL]

I still think one of the best descriptions of these issues is the Feynman Lectures on Physics, Vol 2, Chap 28. It makes for fascinating reading even though it is a bit dated at the end. As he discusses, it is not so easy to throw away the idea that a charged particle interacts with itself.

Thanks for this link Phyzguy, it describes the complexity of finding a self-consistent theory electromagnetism with discrete stables particles not interacting with themselves that is the problem for which I'm looking information. I'd like to remark that it says that there is evidence of electromagnetic inertia. Do you have any reference to that?

I would like to propose a EM field "subjective" to the particle that interacts with it. I mean, the field that any particle sees is the solution to Maxwell equations considering the other particles charge but not the own particle charge. This way with N particles there would be N+1 "EM fields" corresponding to the fields that the N particles see plus an overall field consideren all the charges. I think this "split" in the EM field allows the self interaction to be removed without any inconsistency except that the field energy must be modified.