On the Antisymmetric Tensor and Sources of the Electromagnetic Field

The canonical antisymmetric electromagnetic field tensor has four-dimensional divergences for each of the indices, so the introduction of a field source into its divergence equation for only one of the indices is incorrect. The total divergence of the antisymmetric tensor is identically zero and does not have a field source. The article is devoted to the mathematically correct introduction of the electromagnetic field source into the field equations. The field equations follow from a symmetric tensor having a full four-dimensional divergence not equal to zero. This divergence is equal to the four-dimensional source of the electromagnetic field. From the new system of equations of the electromagnetic field follow the canonical wave equations for the electric field strength, magnetic induction, electromagnetic potential (Maxwell's equations in the Lorentz gauge), and also follow the wave equation for the divergence of the electromagnetic potential describing longitudinal waves that do not have a magnetic component. A new system of electromagnetic field equations solves the problem of Newton's third law in electrodynamics.