The Lorentz force, named after Dutch physicist Hendrik Lorentz, for a charged particle q is given (in SI units) by the equation:
where qE describes the electric force due to a present electric field,E, acting on q.
<br>And qv x B describes the magnetic force due to a present magnetic field, B, acting on q when q is moving with velocity v.
<br>Which can also be written as:
where  is the electric potential, and A is the magnetic vector potential.
It was Oliver Heaviside who is attributed in 1885 and 1889 to first deriving the correct expression for the Lorentz force of qv x B.  Hendrik Lorentz derived and refined the concept in 1892 and gave it his name,  incorporating forces due to electric fields.
<br>Rewriting this as the equation of motion for a free particle of charge q mass m,this becomes:
or
in the relativistic case using Lorentz contraction where γ is:
this equation of motion was first verified in 1897 in J. J. Thomson's experiment investigating cathode rays which confirmed, through bending of the rays in a magnetic field, that these rays were a stream of charged particles now known as electrons.
Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called Laplace force), the electromotive force in a wire loop moving through a magnetic field (an aspect of Faraday's law of induction), and the force on a particle which might be traveling near the speed of light (relativistic form of the Lorentz force).