  In the electromagnetic theory, the Hertz vector reduces the number of
potentials in the free fields. The further advantage of this potential is that
it is much easier to solve some radiation processes. It indicates that the
related method is sometimes more effective than the scalar and vector
potential-based relations. Finally, the measurable field variables, the
electric and magnetic fields, can be deduced by direct calculation from the
Hertz vector. However, right now, the introduction of the Hertz vector operates
if the conductive currents j = {\sigma}E are neglected. We suggest a
generalization for the case of conductive currents, i.e., for such cases when
the electromagnetic field dissipates irreversibly into Joule heat. The
presented procedure enables us to introduce also the Lagrangian formulation of
the discussed dissipated electromagnetic waves. It opens a new way for future
studies.
