Reformulation of the Dielectric Constant and of Ohm's Law in an Absorbing Medium

It is proposed that a fundamental revision of the formulation of the dielectric constant and of the conductivity of an absorbing and dispersive medium be considered. In the conventional formulation the dielectric constant $\epsilon$ of such a medium depends on the mechanism controlling the energy absorption, and therefore $\epsilon$ has the form $\epsilon \equiv \epsilon (\omega , \gamma )$, where $\omega$ is the frequency and $\gamma$ is a "frictional" parameter representing the absorption process. Such a formulation, although widely accepted in the current literature and in the textbooks on electromagnetism, is not consistent with Maxwell's theory, and therefore is not adapted to the analysis of the energetic behavior of absorbing media. A different formulation is proposed in which the inconsistency with Maxwell's theory is removed. In the proposed formulation the dielectric constant of an absorbing medium is independent of the mechanism which controls the absorption process. The dielectric constant is of the type $\epsilon \equiv \epsilon \left(\omega \right)$ and not $\epsilon \equiv \epsilon (\omega , \gamma )$, i.e., it is the same as if the absorption mechanism did no$\stackrel{\mathrm{´}}{\mathrm{t}}$ exist at all. The mechanism of energy absorption is used to determine the conductivity of the medium but has no effect on the formulation of the dielectric constant. The novelty of the proposed formulation is based on the assumption which is used to arrive at a suitable definition of the driving force involved in the motion of a harmonic oscillator in the molecular model of an absorbing medium. In the conventional formulation the driving force $e\mathfrak{F}$ is assumed to be equal to $e\mathfrak{E}$ (where $\mathfrak{E}$ is the electric field intensity and $e$ is the charge of an electron), whereas in the proposed formulation the assumption $e\mathfrak{F}=e\mathfrak{E}$ is replaced by the assumption $e\mathfrak{F}=e\mathfrak{E}+e{\mathfrak{E}}^{\left(\mathrm{extr}\right)}$, where $e{\mathfrak{E}}^{\left(\mathrm{extr}\right)}$ is an "extraneous" force. The extraneous force results from the conversion of the electrical field energy stored in the medium into "extraneous" energy (such as heat) which is removed from the medium during the interaction with an electric field. Therefore, $e{\mathfrak{E}}^{\left(\mathrm{extr}\right)}$ acts as an energy sink. The distinction between the proposed formulation and the one used in the standard theory is based on a behavior of dispersive media which is physically different from the one customarily assumed, and therefore it does not represent merely a question of formalism. Using the concepts introduced in this analysis, Ohm's law is reformulated as representing a mechanism in which the rate of production of the dissipated energy is proportional to the mean energy stored in the medium. Such an interpretation of Ohm's law cannot be made in the framework of the standard theory, since the standard theory does not provide a procedure for determining the mean energy stored in an absorbing medium.