StoneHengeAlive said:
The current in a wire is said to produce a magnetic field and a force on a charge. It is said that the correct way to interpret the effect is because of relativity. So does that mean that that the most basic non-relativistic equations that offer basic calculations of the forces and fields are implicitly a first order approximate accounting of relativity effect? That is, the very basic magnetic field and force calculations are in reality merely a backdoor first level relativity correction?
As another poster has already mentioned, Maxwell's equations are not an approximation. The difficulty with Maxwell's equations is that they are not compatible with non-relativistic mechanics. This can be seen in sevral ways, one of the simplest is the fact that Maxwell's equation predict that the speed of light is a constant. This is possible for all observers only in special relativity, it's not compatible with classical Gallilean mechanics and the Gallielean transform, which implies linear velocity addition.
There's another approach to discuss at an elementary level how magnetism arises from special relativity due to Edward Purcell.
Purcell talks about the origins of magnetism as a consequence of length contraction. It might be of some help, but it also might be confusing. If you're not familiar with the ideas, there's Purcell's textbook on E&M, of which some earlier versions are public domain, and has an informal web summary at
https://physics.weber.edu/schroeder/mrr/MRRtalk.html.
Basically, the issue that makes Purcell's treatment tricky is the relativity of simultaneity, and Purcell doesn't really discuss this. The way I would approach his treatment, which is a bit different than the original, is that if you have a current loop, which we approximate as being long. And we assume that we have a moving frame, where the direction of motion of the frame is in the "long" direction of the loop.
The relativity of simultaneity in this moving frame makes one side of the loop have a net plus charge i, while the other side of the loop have a net minus charge. Conservation of charge makes the net charge zero of course, in all frames, but the relativity of simultaneity makes the wires that have no net charge in the lab frame split into a segment which has a positive charge, and a segment which has a negative charge.
The distribution of charge in the moving frame thus gives rise to electric fields in the moving frame, which exert an electric force on a charge co-moving with the frame. In the stationary lab frame, there is no electric force, of course - but there must still be a force, which we interpret as the magnetic force.
A more advanced approach may take more work, but is probably less confusing in the end. This approach stresses Lorentz invariance as the foundation of special relativity, and notes that the coulomb force by itself is not Lorentz invariant. Hence, there must be something else. This "something else" is the magnetic force.