MAXWELL'S DERIVATION OF THE LORENTZ FORCE FROM FARADAY'S LAW
Progress In Electromagnetics Research M
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. † Faraday
... not write any equations in his Experimental Researches  . The clearest concise statement that I could find in Faraday's writings on electromagnetic induction (Faraday's law) is in Paragraph 3087 of his Experimental Researches , namely, "The first practical result produced by the apparatus described, in respect of magneto-electric induction generally, is, that a piece of metal or conducting matter which moves across lines of magnetic force, has, or tends to have, a current of electricity produced in it." Following this statement, Faraday continues with a more detailed explanation of the "full effect" of the experimentally observed magneto-electric induction. ‡ A shortened version of the derivation in the present paper is given in  but it contains an error pointed out by Redžić . Redžić's treatment  of Maxwell's derivation in his Treatise of the force on a moving electric charge differs from Maxwell's derivation (and the derivation given here) in that it requires differentiation of the differential of the position vector as well as a separate mathematical proof that the time derivative can be brought inside the line integral of the vector potential for a moving curve.