## 1980a

The following is a paper coauthored by D. M. Eagles and H. Aspden published in Acta Physica Polonica, A57, pp. 473-482 (1980).

### THE SPATIAL DISTRIBUTION OF THE CONTRIBUTION TO THE MAGNETIC FIELD ENERGY ASSOCIATED WITH TWO MOVING CHARGES

Abstract: The interaction contribution to the magnetic-field energy associated with two moving charges q_{1} and q_{2} with velocities v_{1} and v_{2} separated by a spatial vector s is evaluated to order

(v_{1}v_{2}/c^{2})
by direct integration over all space. It is shown that the
contribution to the magnetic interaction energy from a spherical shell of radius r greater than s centred on one particle is equal to

(2q_{1}q_{2}/3c^{2})(v_{1}.v_{2})r^{-2}dr,
while for a shell with r less than s the contribution is

(q_{1}q_{2}/3c^{2})[3(v_{1}.s)(v_{2}.s)/s^{2}-(v_{1}.v_{2})]s^{-3}rdr.
After integration the negative of the usual expression

(-q_{1}q_{2}/2sc^{2})[(v_{1}.v_{2})+(v_{1}.s)(v_{2}.s)/s^{2}]
for the interaction contribution to the Hamiltonian for two moving charged particles is obtained. The change in the electric field
energy due to effects of retardation on electric fields does not contain any terms proportional
to (v_{1}v_{2}/c^{2}), and so the convention sometimes adopted of calling

[-(q_{1}q_{2}/sc^{2})(v_{1}.v_{2})]
the magnetic interaction and attributing the remainder of the interaction, viz.

(q_{1}q_{2}/2sc^{2})[(v_{1}.v_{2})-(v_{1}.r)(v_{2}.r)/s^{2}],
to retardation appears to be misleading.
**Commentary**: Dr. D. M. Eagles, co-author of this paper, and, in fact, fully authored the text and was exclusively responsible for the extensive mathematical analysis involved. This author's contribution was to pose the problem and the initial formal derivation of the interaction energy applicable in the very simple case where the two interacting current elements are in collinear motion. The latter calculation was, in fact, presented in the author's 1960 book 'The Theory of Gravitation'. It was of interest because there are, supposedly no longitudinal forces acting along the common line of flow of two current elements, at least according to accepted theory based on the Lorentz formulation. Einstein began his writings on relativity by addressing electrodynamics, but, in declaring conformity with the Lorentz force law, he lost sight of the true forces that arise in electrodynamic actions under certain special circumstances. The actions which are involved in gravitational forces are of that special kind!

The paper is important in providing the basis for arguing that the conventional concept of a magnetic field leads to absurdities in that, for action between two discrete charges in motion, the field energy deployment at distances remote from the source charge must change drastically when a charge alters course slightly.

Readers must understand that the notion of the magnetic field is artificial and is founded only upon empirical data deriving from the study of electron current interactions where at least one of the interacting currents is non-segmented and flows around a complete circuit loop.

Energy science as it becomes more developed will have to overcome the error of this traditional magnetic field assumption and avoid using the concept when developing technology in which non-electron currents are a primary feature or where there is non-circuital electron flow. This is an absolutely fundamental requirement and is at the very heart of the problem which caused Einstein to be sidetracked down a blind alley when trying to forge the unification of magnetic and gravitational theory.