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Maxwell?s equations separate into two sets of equations, one containing the solenoidal field components and the other containing the irrotational components. The solenoidal components of the electric and magnetic fields are shown to be produced by current densities, and contribute to electromagnetic radiation at speed in free space. The irrotational component of the electric field is produced by charge densities and propagates at infinite speed. Wave equations for the potential fields are derived with the Coulomb gauge, which gives simpler results than the Lorentz gauge. The scalar potential propagates at infinite speed while the vector potential propagates at speed c in free space. These results extend recent work by Oleinik on the electric field to the magnetic, scalar potential, and vector potential fields.

In the Lagrangian formulation of the Lorentz force between two moving charged particles, particle momentum is shown to include a term proportional to the vector potential in addition to the usual mechanical momentum of mass times velocity. This additional term is sometimes referred to as the electromagnetic momentum of the particle. (It is not the same as the momentum of the electromagnetic field.) In this paper the Weber force is put into Lagrangian form and an electromagnetic momentum term appears. The electromagnetic momentum occurring in Weber's theory is compared with that of the Lorentz force. Neither electromagnetic momentum has an intuitive physical explanation and both might be only mathematical artifacts arising from Lagrangian dynamics.

A Galilean-invarient interaction Lagrangian of a charged particle moving in an electromagnetic field in free space is obtained assuming that a natural (i.e., preferred) reference frame exists for the field and that all uniformly moving observers can determine their velocities with respect to this frame. From this Lagrangian, a Lorentz force law is derived. From this Lorentz force law, Poisson's equation, and the principle of conservation of electric charge, Lorentz's version of Maxwell's equations in free space for an observer moving uniformly through the ether is obtained. These modified Maxwell's equations, the field quantities they contain, and the wave equations derivable from them are shown to be Galilean invariant. They predict experimentally-obtained electromagnetic forces on charged particles in free space if the natural frame (i.e., ether) is entrained by the earth.

The possibility that the electric force on a charge particle moving through a static electric field acts at the optical aberration angle instead of parallel to the electric field is investigated. Simple reasoning leads to the conclusion that if the force does act at the aberration angle, then the Lorentz force equation must be modified to include a force component acting in the opposite direction of the particle velocity. Such a force component is proposed in this paper. The effect of this additional component is calculated and is found to be large enough to be observed in the laboratory with a low-energy experiment.