7. Potential Energy

So, after 6 sections a theory has been described where particles/charge are constructed of only fields in an electromagnetic continuum, but move about exactly like particles/charge. We have also managed to include Maxwell's equations as valid for the interaction of these constituants and fields. So from the practical viewpoint not a whole lot has changed from classical electromagnetism. In the next two sections (7-8) it will be determined how two fields and thus two particles interact. In this section, the derivation is done to show the interaction energy of the fields of two charges and how it correlates to potential energy. This is allowable as one of the postulates is that the electric and magnetic fields contain energy as described by Poynting's Theorem.

Imagine two particles of charge q₁ and q₂ separated by a distance 2s as in the following diagram.



Fig.3: Two charges separated by a distance 2s showing distance vectors to an arbitrary field point.

We will determine the field in relation to an origin placed midway between the charges. Note that

(1)

and

(2)

thus the potential is given by:

(3)

The two components of electric field are given by

(4)

and

(5)

The energy in the field is

(6)

Note that the first two terms simply represent the energy in the field of each charge alone. The third term is an additional energy based on their interaction (i.e. the fact that the electric fields superpose). This interaction energy can be negative also. The interaction energy is thus (for the radial dimension):

(7)

and for the z-dimension:

(8)

where we have already integrated in respect to f . Integrating further yields

(9)

and

(10)

The total interaction energy is therefore

(11)

Since the charges are separated by 2s, this is simply the potential energy of a system¹ where a force exists between the sources of the divergent fields. However, though no force law or mass content has yet been established for this theory, this interaction energy implies that a specific type of force must exist. Note that there is no contribution to the energy from the field in the z-direction, so the potential energy could (in this case) be written simply as

(12)

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1. This same result is also derived by Jackson² 47. I redo it here in a simpler fashion and also to divide out the contributions by component.

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