[Sergio_Prats (OP)]

I would like to propose a correction to the energy of the electromagnetic field when there are particles with a discrete charge instead of a continuous charge distribution ρ(r). Each particle may be punctual or extended over the space, having a density of charge ρ_n(r), and density of current J_n(n)

While the standard formula for field density of energy is: u' = (ε/2)E² + (μ/2)H², my correction subtracts a summation with the field energies that all these particles would have if they were isolated in the space, with this term the field density of energy would become:

    u = (ε/2)E² + (μ/2)H² - (ε/2) ∑E_n² -  (μ/2)∑H_n²

Where E_n and H_n are the fields that each particle generates.

A similar correction should be done for the Poynting vector S.

The motivation of this correction term comes from the fact that the charge in a particle does not interact on itself, for example, the electron in the Hydrogen atom does create an electric field but it does not cause any potential to include in its own Hamiltonian (with is only affected by the proton's central potential): a charged particle creates an electromagnetic field but it is not affected by it, it is only affected by the field created by others.

At present the calculations I have done are limited to two radial wavefunctions for the Hydrogen atom (that means, a system with only two particles), however this is enough to see that the classical expression does not satisfy that the difference of potential energy between both levels should be the same that the difference of energy in the field while when using my correction it is satisfied.

I have created a paper in which I explain this ideas with more detail, it's here:

https://www.slideshare.net/SergioPL81/field-energy-with-discrete-charges [nofollow]

Any contribution or criticism to this idea will be welcome.

Sergio Prats

[chris]

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Thanks