Maxwell's derivation of the Lorentz force from Faraday's law

Abstract

In a brief but brilliant derivation that can be found in Maxwell's Treatise and traced back to his 1861 and 1865 papers, he derives the force on a moving electric charge subject to electromagnetic fields from his mathematical expression of Faraday's law for a moving circuit. Maxwell's derivation in his Treatise of this force, which is usually referred to today as the Lorentz force, is given in detail in the present paper using Maxwell's same procedure but with more modern notation.

Maxwell's Definition of Electric Polarization as Displacement

“…598 where he shows that the force v × B must be added to E if the curve C is moving with velocity v and this implies that the force exerted on a hypothetical unit electric charge moving with C is E + v × B. Thus, Maxwell derived the "Lorentz force" from his generalized equation for Faraday's law [2], [8].) Maxwell wrote equation (3) several times in his Treatise as well as in his papers but always in the scalar form 2 ∂B x ∂x…”