Modifications of the Lorentz force law invariant under Galilean transformations

It is generally expected from intuition that the electromagnetic force exerted on a charged particle should be invariant as observed in different inertial frames. In the special relativity, this invariance is achieved by invoking the Lorentz transformation of space and time. In this investigation, an entirely different interpretation of this force invariance is presented by proposing a Galilean-invariant model of the electromagnetic force. In this new classical model, the electromagnetic force is expressed in terms of the augmented scalar potential. This new potential is a modification of the electric scalar potential by incorporating an ordinarily small velocity-dependent part. Each of the position vectors, time derivatives, and velocities involved in the proposed force law is referred specifically to a respective reference frame. By virtue of this feature, the electromagnetic force is endowed with the unique property of Galilean invariance. The velocity-dependent parts of the proposed force look quite different from their counterparts in the Lorentz force. However, under the common low-speed condition where the source particles forming the current drift very slowly in a matrix, the proposed Galilean-invariant model reduces to the Lorentz force law, if the latter is observed in the matrix frame as done in common practice.