A Lorentz force law with velocity differences

The electromagnetic force FL is given by the Lorentz force law FL=q(E+v×B). The Lorentz force law, together with Maxwell's equations and the constitutive relations in matter, suffices to analyze any classical electromagnetic phenomenon. This framework has been promoted in introductory physics courses to avoid integral equations such as Faraday's Law, which, in some cases, lead to wrong results. However, mistakes are easily made when applying the Lorentz force law in the presence of magnets that move with respect to each other. For example, it is easy to compute an incorrect polarization for a magnetically permeable rod moving in an external magnetic field. We illustrate how such mistakes may arise, elucidate the underlying cause, and present a new formulation of the Lorentz force law that yields consistently correct results.