Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scale-factor of a half in front of the formula. Oliver Heaviside invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object. Finally, in 1895,Lorentz, Hendrik Antoon, Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern, 1895. Hendrik Lorentz derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.
Thomson derived the correct basic form of the formula but because of some miscalculations and an incomplete description of the displacement current included an incorrect scale-factor of a half in front of the formula Oliver Heaviside invented the modern vector notation and applied it to Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object Finally in 1895Lorentz Hendrik Antoon Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern 1895 Hendrik Lorentz derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction Instead Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics (see below) Lorentz arrived at the correct and complete form of the force law that now bears his name