(1) FL=q/c.(v×B) (in cgs- units)
(note that often the equation for the 'Lorentz Force' is written including the electrostatic force q.E; this is actually misleading as the latter is already separately called the 'Coulomb Force', so one should restrict 'Lorentz Force' to the magnetic force only; this is how I shall use the term in the following). According to the relativistic view of electrodynamics, the velocity v in the Lorentz force FL is implicitly understood as the velocity of the charge q in the reference frame of the observer, i.e. the Lorentz force (as defined above) would be frame dependent (as far as inertial frames are concerned). As this would obviously contradict the definition of a force, it is postulated that the magnetic force does not exist as a separate physical force, but merely as part of an 'electromagnetic' force composed of an electric and magnetic part (which is usually written then as FL=q.(E+v/c×B)).However, this postulate obviously rests solely on the assumption of v being an observer related (and thus frame dependent) quantity, which is at best completely unjustified. Indeed, if one looks at other velocity dependent forces, then it appears as outright non-sensical. In case of the velocity dependent friction force for instance (e.g. the drag on objects in gases or fluids) one would certainly not suggest that there must be a different force acting in the object's frame because the velocity v=0 here; the velocity here is simply not frame dependent but referred to the medium the object is moving in, so it does not change if one changes reference frames. Likewise it is more than likely that the velocity v in the Lorentz force is actually referred to the physical system that creates the magnetic field, e.g. the wire (the exact definition of this is a separate problem which is addressed in the the home page entry regarding the Maxwell Equations)).
Nonetheless, a frame dependent definition of v would be at least theoretically possible as long as it does not lead to any violation of physical principles. But the latter is exactly the case here as the charge invariance would be violated if the electric field and magnetic forces are taken as frame dependent. Corresponding derivations in various textbooks (e.g. Berkeley Physics Course (Electricity and Magnetism) or Feynman (The Electromagnetic Field); for an online treatment see for instance here) more or less completely ignore this point. It is claimed there that in general the positive and negative charge densities in the wire would be different if a current is flowing due to different relativistic length contractions associated with the different velocities of the charges. However, even neglecting the basic flaws in the Lorentz Transformation and its interpretation (see my page Relativity and links from there), the 'proof' in these cases is only done for the case of an infinite wire, which enables the authors to disguise the fact that assuming an overall velocity-dependent charging of the wire would violate charge invariance. This becomes clear if one considers instead the magnetic field in the far-region of a finite wire:
the electric field of a line charge of length L at a vertical distance r (centered on the wire) is given by:
(2) E(r) = 2.n.A. 0∫Ldx 1/(x2+r2) . r/√(x2+r2) = 2.n.A.r. 0∫Ldx 1/(x2+r2)3/2
where L ist the length of the wire, A its cross section and and n the charge density.This can be readily integrated to yield
(3) E = 2.n.A.L/r2 .1/√(1+L2/r2) .
For large distances r such that r>>L, the square root can be expanded into a Taylor series with the result (taking only the first two terms of the expansion)(4) E = 2.n.A.L. (1/r2 - L2/2r4) .
Now charge invariance requires that 2.n.A.L = const= Q (total charge) i.e.(5) E = Q. (1/r2 - L2/2r4) . (L<<r)
The first term of this expression is simply the monopole field due to the total charge and thus can not be affected by any hypothetical length contraction of the charge distribution. If the overall charge is zero, the contributions of the positive and negative charges to the monopole term would therefore cancel. Only the second term could be affected by a different length contraction of L for the two kinds of charges, but this would depend on distance like 1/r4 (electric quadrupole) and not like 1/r2 as required by Maxwell's equations (see for instance any derivation of the Biot-Savart law). Note: the assumption of a finite line current is obviously somewhat simplified here, and some people have addressed the point that the problem would be time-dependent in reality as charges drain from end of the wire to the other, hence creating additional fields due to induction. This problem could be overcome in principle by simply assuming that the charges at one end are locally replaced at the same rate they are lost, and vice versa at the other end. More realistically, one just needs to have a charge capacity of the battery high enough so that the relative change can be neglected during the time interval considered. The electromagnetic field associated with the circuit will then effectively be time-independent.But anyway, one can also consider a closed current loop where this problem does not arise at all (assuming for instance an ideally super-conducting loop). According to the length contraction argument, this should then lead to the situation as depicted below for a rectangular current loop consisting positive and negative charges with equal and opposite velocities in the rest frame (case a):
