Amp?re proposed an equation for the force between current elements in 1823. Gauss and Weber suggested equations for the force between moving charges which are derived from scalar potentials and are consistent with the Amp?re equation. Grassmann derived an alternative equation for the force between current elements in 1845, which is often called the Lorentz force. This paper derives a time-variant generalization of this equation. It also proposes a new equation for the force between current elements that satisfies the basic criteria suggested by Gauss and Hertz.
This paper is based on the fundamental criteria for the electrodynamic equation suggested by Gauss. It considers the definitions of the electric field proposed by Neumann and by Hertz. The classical formulation utilizing Neumann's definition and Einstein's postulate on the velocity of light does not satisfy the criteria suggested by Gauss. A new electrodynamic equation is proposed, utilizing the Hertzian definition of the electric field and the universal time postulate on the velocity of light, which does satisfy the Gaussian criteria.