Lorentz force from a current-carrying wire on a charge in motion under the assumption of neutrality in the symmetrical frame of reference

It is commonly assumed that a wire conducting an electric current is neutral in the laboratory frame of reference (the rest frame of the lattice of positive ions). Some authors consider that the wire is neutral only in a symmetrical frame of reference, in which the velocities of electrons and protons have equal norm and opposite direction. In this paper, we discuss the Lorentz transformation between different frames of reference in the context of the special theory of relativity for a current-carrying conducting wire and a probe charge in motion with respect to the wire. A simple derivation of the Lorentz force in the laboratory frame of reference for the assumed neutrality in a symmetrical frame of reference is presented. We show that the Lorentz force calculated assuming neutrality in the symmetrical frame of reference and the one assuming neutrality in the laboratory frame of reference differ by a term corresponding to a change in the test charge speed of one half the drift velocity of the electrons