No need of relativity and Lorentz factor to explain the behavior of electrons
in particle accelerators.
This paper, even when mostly ignored, explain why the terminal speed c is the limit of accelerated electrons, without any trace of special relativity.
This work, based on reactive loss of energy by accelerated electrons, shows how the curve of relative speed v/c vs. acceleration almost shadow the results of relativistic electrodynamics. The author call it radiative electrodynamics.
I agree with this paper, which is based on the pioneer work of William Bertozzi
(MIT, 1964), which was largely ignored by relativists.
https://www.researchgate.net/publication/337429098_TERMINAL_SPEED_OF_AN_ELECTRON_ACCELERATED_BY_AN_ELECTRIC_FIELD_WITH_EMISSION_OF_RADIATION/link/5dd71ca192851c1feda57651/download
CHECK THIS GRAPH:How could an electric field accelerate particles over the speed of electromagnetism, and why would that prevent other means from doing so? Aren't particle accelerators merely an exercise in confirming relativity's claim of a cosmic speed limit?
Figure 1: Graph of v/c (speed in units of c) against at/c (time in units of c/a)
for an electron of charge –e and mass m = mo accelerated from zero initial speed or decelerated from the speed of light c, by a uniform electrostatic field of magnitude E, where a = eE/m; the lines (A1) and (A2) according to classical electrodynamics, the dashed curve (B1) and line (B2) according to relativistic electrodynamics and the dotted curves (C1) and (C2) according to equations 4 and 7.
As it can be seen, three types of electrodynamics are analyzed: classic, relativistic and radiative. The author acknowledge the limit given by c,
as well as increase of mass in relativity (Lorentz), but presents an alternative version, by which the electron loses capability of being accelerated by electric fields as its ratio v/c increases beyond 0.3.
I stated here several times in the past years that the problem in particle accelerators is due to the loss of the effective electric field of the particle
to absorb accelerating energy as it approaches the speed c.
This paper is close to my heuristic understanding.
No relativity, no Lorentz. Just Coulomb and nature in action.
Plain and simple.
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