Actually, FO, I've been thinking about this topic for a while now. Although my physics is too rusty to get a definite answer, I strongly suspect that classical magnetic fields can be considered as being purely an expression of relativistic effects. And I did work out the math for the two free charged particles scenario you described.

Fe = electric force (not iron!)
Fm = magnetic force
E0 = epsilon naught, the permittivity constant
M0 = mu naught, the permeability constant
r = distance between the particles
R = distance unit vector between the particles

With the two electrons, the electric force is repulsive and given by by Coulomb's Law:

Fe = (1 / 4 Pi E0) * (q1 q2 / r^2)

The magnetic field produced by each electron as felt by the other is equal and opposite to each other, and given by the Biot - Savart Law:

B = (M0 / 4 Pi) * (q1 V X R) / r^2

And of course the magnetic force is simply:

Fm = q2 V X B

Since the magnetic force is attractive, it is really a reduction in the electric force, and the simplified formula for the net force - eliminating vectors since we defined r to be perpendicular to V - works out to be:

F = Fe - Fm = (1 / 4 Pi E0) * (q1 q2 / r^2) - (M0 / 4 Pi) * (q1 q2 V^2 / r^2)

Now, a reduction in the force is exactly what you would expect from relativistic time dilation. If time is moving slower, the acceleration is lower, and so therefore is the force. If the magnetic force in this case is equivalent to time dilation, then at V = c we would expect the magnetic force to equal the electric force. Time dilation is infinite for an object moving at c, so time is frozen, there is no acceleration, and therefore no net force. To show this is so, set F = 0 and solve for V.

0 = (1 / 4 Pi E0) * (q1 q2 / r^2) - (M0 / 4 Pi) * (q1 q2 V^2 / r^2)
0 = 1 / 4 Pi E0 - V^2 * M0 / 4 Pi
V^2 = (4 Pi / M0)*(1 / 4 Pi E0)
V = 1 / SQRT (M0 * E0)

It just so happens that 1 / SQRT (M0 * E0)is better known as c, the speed of light.

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When it comes to currents in wires, the relativistic effect in play is length contraction. In the frame of reference of the protons in one wire, the protons in the other wire are at rest, while the electrons are moving, and therefore experience length contraction. That means the electrons appear to be closer together, and therefore have a higher charge density. That is where the attractive magnetic force between the wires comes from.

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I'm still working on learning vector calculus, so I don't have the math skills to do a rigorous proof or to deal with rotating charges, so that's as far as I can take it right now.

A) Fair, let's just assume that at the instant t both happen to have the same velocity vector V but different position S1 and S2, so that S1-S2 is perpendicular to V.

B) This is an experiment over macroscopic distances (cables!), so the Heisenberg uncertainty is insignificant with respect to both V and S1-S2.

Can I throw a spanner in the works here? Electrons A) won't run freely along parallel paths because one force will always overcome the other and make them deviate, and B) can occupy several spatial positions at the same time, which does kind of make attempting to predict their interactions a little trickier than this thought experiment implies.

Ok, let's pit FB against UF and let's see where do I get the best answer.

Physics question.

We know that charged particles in movement generate a magnetic field.
In fact, if we let two parallel electric cables carry a current I, the two will attract each other with force density proportional to I squared.
That's because the electrons are moving.

Now, let's remove everything but two electrons freely running along parallel paths.
They will be repelling each other due to their charge, but this effect will be slightly weakened by their mutual magnetical attraction, because both are moving and generating a magnetig field.

Now, let ourselves be in the reference frame of one of those electrons.
Since the speed relative to the other electron is zero, the magnetif force between the two of them, is zero.
Turns out that the magnetic field is just a relativistic correction of the electric field.
Ok, no problem so far.

Now let's go back to our two parallel cables.
Let's stay in the (average) reference frame of the electrons.
Again the two flow of electrons are still with respect to each other.
But now we have protons, that move in the opposite way with the same speed and opposite charge, and therefore generate exactly the magnetic field that the electrons generate in the lab reference frame.

The results of the mental experiment with only two electrons and the mental experiment with also the protons do not match.
What am I doing wrong?