9. Kinetic Energy
Now after 8 sections the entire theory of classical electromagnetism has been incorporated, the only difference being the Lorentz force law at relativistic velocities (which will later be discussed). In the next 3 sections (9-11) the underlying issue is "mass". Since I claimed that particles are merely soliton waves in the electromagnetic continuum, there must exist a purely electromagnetic definition of mass. This is the key characteristic to incorporate Newtonian mechanics into the theory. From mass comes the other body related concepts like momentum and kinetic energy, so to understand these, we must understand mass. In this section we will use kinetic energy to get an initial perception of mass in this system.
Note that if a soliton is propagating at some velocity, it will have a magnetic field associated with it, but, since the magnetic field has a certain energy, that energy must be introduced to get the soliton to propagate at that velocity. Therefore it is very straight forward that the energy in the magnetic field is, in fact, the kinetic energy of the particle (soliton). The first derivation uses this kinetic energy to show how the energy in the electric field offers resistence to acceleration and thus mass. The second derivation resolves an issue with the energy in the magnetic field being dependent on absolute motion only.
We have assumed that there is essentially mass in the energy of an electric field (Section 2), though we have not yet derived it from the initial postulates. It is not hard to imagine that, just as the electric field energy relates to mass, the magnetic field energy relates to kinetic energy. The energy in the magnetic field is the energy required for the soliton to propagate at the given velocity. From Eq.(1-5) we have for the total energy:
(1)
The energy in the magnetic field is thus
(2)
Considering a single charge in motion Eq.(5-9) can be used for the magnetic field and
(3)
Assuming that the velocity is in the x-direction results in
(4)
The energy in the electric field is given by the first term in Eq.(1). Since each dimension of the electric field contributes equally to the energy, the energy can be written in terms of a single electric field component. This yields
(5)
where E is any component of the electric field. By substituting this equation into Eq.(4) for both components, we obtain
(6)
The magnetic field equates to the kinetic energy so we can substitute such that
(7)
and thus
(8)
where x0 is the energy in the electric field of a particle at rest1. Apparently mass has a direct relation to the energy in an electric field. However, one should not read too much into this. It still does not tell us what mass is, only that at rest a mass contains the energy given above or more precisely, if there is a radially symmetric electric field (charge) of this energy, it will resist acceleration as if there was this associated mass present.
We have now defined a kinetic energy in relation to absolute velocity. Imagine now that two particles initially moving at absolute velocity v push each other apart so that one receives a change in velocity v1 and the other v2. The energy change in the first would be the energy after the push minus the energy before the push. The same is also true for the second particle. This would yield
(9)
and thus
(10)
Since momentum is conserved
(11)
and the absolute velocity dependent terms cancel out, thus,
(12)
The kinetic energy is independent of the absolute velocity. It only depends on the velocity relative to the center of mass of the system.
1. This is, of course, 3/4 of the usual amount and also different from the Abraham-Lorentz prediction of 4/3mc2. (Jackson2 786). It needs to be remembered at this point that this is not a new theory, but merely an extrapolation of classical electromagnetism. I believe it is necessary to review direct quantitative experimental evidence of this relation. For example, gamma energy spectrums do not offer evidence since the wavelengths have not been directly measured, but instead inferred from this relation.
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