Helicity continuity equation for electromagnetic fields with sources

The helicity continuity equation is derived from the wave equations of the electromagnetic potentials following the rationale of the complementary fields approach. The conserved quantity and its corresponding flow naturally arise from the conservation equation. The continuity equation is obtained for fields either in vacuum or homogeneous non-dispersive media in the presence of charges and/or currents. The derivation is otherwise quite general, there is no need to assume monochromatic fields nor a paraxial approximation. The symmetry of the electric and magnetic contributions is a consequence of the conserved quantity structure rather than an ad hoc hypothesis. The locally conserved quantities hold exactly without any averaging over time or space. This result is a hallmark of the complementary fields framework, whereby the energy content of the fields is dynamically exchanged between them.