A recent article by Mansuripur claims that the Lorentz force law is incompatible with special relativity. We discuss the “paradox” on which this claim is based. The resolution depends on whether one assumes a “Gilbert” model for the magnetic dipole (separated monopoles) or the standard “Ampère” model (a current loop). The former case was treated in these pages many years ago; the latter, as several authors have noted, constitutes an interesting manifestation of “hidden momentum.”
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The details are worked out in Cross and Vanzella (Ref. 3).
We calculate all torques (in the lab frame) with respect to the origin. But because the net force on the dipole is zero in all cases, it does not matter—we could as well use any fixed point, including the (instantaneous) position of the dipole.
This is of course an unrealistic model for an actual current-carrying wire. Vaidman (Ref. 11) explores more plausible models, but the result is unchanged.
We do not know a simple way to prove this directly, but we will confirm it implicitly in Sec. IV.
See, for instance, Ref. 9, Eq. (12.118).
The ith component of the second term is .