Generalized Hertz vector in the dissipative electrodynamics

TL;DR

This paper generalizes the Hertz vector to account for dissipative effects in electrodynamics with conductive currents, enabling easier analysis and Lagrangian formulation of dissipating electromagnetic waves.

Contribution

It introduces a generalized Hertz vector applicable to dissipative electrodynamics with conductive currents, extending the classical theory.

Findings

Generalized Hertz vector for conductive currents

Lagrangian formulation for dissipative electromagnetic waves

Potential for improved analysis of dissipative electromagnetic processes

Abstract

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.