According to the relativistic view of electrodynamics, the force on a particle with charge q and velocity v in a magnetic field B, i.e. the Lorentz force

(1)       F_L=q/c^.(v×B)       (in cgs- units)

is not an independent physical phenomenon but merely a relativistic electrostatic effect which depends on the reference frame i.e. the velocity of the observer (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).

The argument the relativistic view uses for rationalizing the magnetic force away usually is that the Lorentz force F_L is frame dependent because of the velocity v, which would contradict the principle of relativity. If one considers for instance a charged particle moving in the magnetic field of an electric current in a wire, the corresponding Lorentz force as given by the Eq.(1) would obviously be zero in a reference frame moving with the charged particle (i.e. where v=0) and thus it is concluded that a corresponding electrostatic force must exist instead. This conclusion is flawed however because the charged particle is not in an inertial reference frame but accelerated due to its motion in the magnetic field. So in fact there should not be any additional force required in the reference frame of the charge q.
The error that is simply being made here is to take the velocity v in the Lorentz force as frame dependent whereas in fact it has to be referred to the wire (or more generally speaking to the current system producing the magnetic field (the exact definition of this is a separate problem which is addressed in the the home page entry regarding the Maxwell Equations)). One can compare this case probably to another velocity dependent force e.g. friction: if an object slides over a surface, one would certainly not suggest either that there must be a different force acting in the object's frame because the velocity v=0 here; the velocity in the friction force simply is not frame dependent but referred to the surface the object is sliding over, so it does not change if one changes reference frames.

In view of this lack of a physically relevant definition of the velocity v in the Lorentz force, it is therefore not surprising that the derivation of the relativistic electrostatic force in the corresponding reference frame is actually also incorrect as it violates the law of charge conservation:
the approach that several textbooks use here (e.g. Berkeley Physics Course (Electricity and Magnetism) or Feynman (The Electromagnetic Field); for an online treatment see for instance here) is to claim 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 get away with the fact that assuming an overall velocity-dependent charging of the wire would violate charge conservation. 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/r^{2 .}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).

Although this incompatibility of the relativistic view with electrodynamics does not come as a surprise given the mathematical inconsistencies in the Lorentz Transformation itself, it is probably important that the suggested velocity dependence of the electrostatic and magnetostatic forces would actually not explain the Lorentz force in terms of electrostatics either as again the field would decrease like 1/r⁴. This would suggest that the magnetic field is in fact primarily a different physical phenomenon and can not be explained in terms of electrostatics (see the home page entry regarding Maxwell Equations for more in this respect). On the other hand, it is hard to believe that the apparent velocity dependence of the electrostatic force observed in accelerators for instance is not real, so one would have to assume the existence two components which are identical for the near field but are different in the indicated sense for the far field. Clarification in this respect can probably only be achieved by making exact measurements of the magnetic far field of current systems.

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