To derive the interparticle separation, between two like charges, where the magnetic force will overcome electrostatic repulsion, while not leading to a mass for the bound composite that exceeds the proton mass, one can simply evaluate the magnetic part of the Lorentz force for a first relativistic charge moving in the magnetic field of a second relativistically-moving charge, that is also accelerating due to electrostatic forces due to the presence nearby of the first charge. Everything needed is in a standard electrodynamics textbook such as Jackson or Griffiths, and on just a couple of pages (or on wikipedia, alternatively).

First, calculate the acceleration of a charge with arbitrary rest mass in the non-radiative electric field of a second nearby charge, as a function of the separation between the charges and their velocities. This must be done using the proper relativistic forms for both the electric field, using the electric field derived from the Lienard-Wiechert potentials, and for the resulting acceleration due to the electric force, which must be based on the relativistic equivalent of Newton's law of inertia.

Next, get the magnetic part of the radiative field due to the accelerated (second) charge, again using the Lienard-Wiechert field expressions. Assume the second charge is moving at approximately (i.e., asymptotically close to) the speed of light perpendicularly to the direction of its acceleration. This is consistent with

the two charged particles being bound together in a mutual circular classical motion.

Next, get the magnetic part of the Lorentz force on the first charge moving in the magnetic field of the second charge, assuming the motion is circular at velocity asymptotically close to the speed of light. This makes a complicated vector quintuple product simplify down to a trivial vector equation for the force, and velocity quantities become unity in units where the speed of light is unity. Also, at this point it can be noted that the resulting magnetic force is proportional to the product of the squares of each charge separately, and so does not change direction under sign reversal of either charge. Furthermore, it can be seen at this point that the magnetic force obtained is always attractive rather than always repulsive, and that it has an inverse third power dependence on the separation between the two charges.

The final step is to equate the magnetic force obtained as described with the Coulomb force also already obtained, so that the equality can be rearranged to give the interparticle separation where the strong magnetic force will equal (and so begin to overcome) the electrostatic repulsion. This gives a very simple equation where the only undetermined quantity is an inverse factor of the lab-frame relativistic mass of the particle, which must be less than half the proton rest mass, if the force is to be a candidate for binding together consituents of protons. Plugging in the proton rest mass for a ballbark figure, one obtains a separation of about 10^(-15) centimeters, or about one one-hundredth of the proton measured size. That makes the force a poor candidate for binding quarks into nucleons (for which there is already a well-verified model involving gluons), but a good candidate for a force binding preons into quarks (for which there is no competing model, according to Don Lincoln's SciAm article).

So it is just that easy to derive for oneself a candidate for the ultra-strong force to bind preons into quarks and leptons, that does not require mediation by massive particles that would break the mass budget, from textbook electrodynamics. On the other hand, it is even easier to to look at the derive as described above written out, as available here: arXiv:1108.4343v6. The derivation of the strong magnetic force is in Section IV which begins in the middle of page 5 and completes on that page. Equation with the Coulomb repulsion and solving for the radius is at the end of section III and so also on page 5. Section III derives the notional form of the anticentrifugal force, and equates it with Coulomb attraction to solve for the separation. In the magnetic force section then it is merely necessary to show that the strong magnetic force equates to the anticentrifugal force, under the assumption of equal-mass particles and mutual circular motion.

**Update 13 Oct:**When I wrote the above, the last paragraph in particular, I was thinking incorrectly that gluons have rest mass. I was remembering pions and the Yukawa potential, and the old argument about how the Uncertainty Principle allows massive force carrying virtual particles to exist for only a brief interval, thus restricting their effect to short distance.

I was expecting that gluons as underlying carriers of the strong force would also have to have rest mass. It turns out however that gluons are (rest) massless in the Standard Model. So, while the mass of a proton is nonetheless mostly that of the gluons rather than of the quarks, it is not rest mass but relativistic mass equivalent to the binding energy. A force mediated by massless photons would not be any better in this regard, it seems.

I guess it will not be as easy of a sale as I was hoping yesterday to get physicists to buy into the idea that the ultra-strong force is just the magnetic force. In order for it to be acceptable, it will also have to become appreciated that it's merely a kinematical consequence of the Thomas precession, and so should not be expected to contribute mass-equivalent binding energy in the conventional sense. This is counter to the conventional wisdom that the Thomas precession does effect energy levels, as for example in the resolution of the spin-orbit coupling anomaly offered by Thomas himself in his 1927 paper. I seem to be the only person to have noticed that Thomas's analysis is invalidated by more recent developments in electrodynamics, such as the recognition of the existence of "hidden" momentum, as well as having other problems. (These are described in my paper on the 1927 Thomas paper: http://arxiv.org/abs/0905.0927.)

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