## 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.