## 4 An Extended Formulation for Electrodynamics
This nonlinear system counts ten unknowns, namely, the three components of the electric field \(\mathbf{E}\), the three components of the magnetic induction \(\mathbf{B}\), the three components of the wave velocity field \(\mathbf{V}\) and the scalar function \(p\). Basically, here we are only considering the case of a linear homogeneous medium, corresponding to \(\mathbf{D}=\varepsilon_{0}\mathbf{E}\) and \(\mathbf{B}=\mathbf{H}/\mu_{0}\), so that there is room for further extensions that for the moment are not taken into account. The first Equation (6) is [PERSON]'s law where the additional term \(\rho\mathbf{V}\) has to be interpreted as a free flow of immaterial current with density \(\rho\), associated with the evolution of an electromagnetic phenomenon. In principle, this latter term can be summed up to the classical one (not present in (6)), due to the flow of current along a conductive medium that may interact with the wave. The complete version of such a revised equation is reported in [15], Section 2. By taking in to account these additional contributions, it is possible to model the mutual interactions between wave and guides, as studied later in this paper. Equation (10) is basically the [PERSON]'s equation for non-viscous fluids. On the left-hand side we find the divergence of the electric field, multiplied by the sum of two terms: the first one is the total derivative \(\frac{DV}{Dt}:=\frac{\partial V}{\partial t}+(\mathbf{V}\cdot\
abla)\mathbf{V}\); the second one, \(\mathbf{E}\,+\,\mathbf{V}\,\times\,\mathbf{B}\), recalls the [PERSON]'s law. The constant \(\hat{\mu}\) appearing in (10) has the dimension of a charge divided by a mass and has been estimated to be approximately equal to \(2.85\times 10^{11}\) Coulomb/Kg (see [15], Appendix G). On the right-hand side, there is the gradient of a potential \(p\), which analogously to the [PERSON]'s equation, plays the role of pressure. We may actually treat the potential \(p\) as a kind of pressure term that can assume both positive and negative values, and has dimensions equal to velocity over a time squared. Indeed, from a dimensional analysis it turns out that \(\varepsilon_{0}p/\hat{\mu}\) is a force per unity of surface. Equation (9) follows from energy conservation arguments. This suggests the possibility of inducing pressure effects as a consequence of a lack of orthogonality between \(\mathbf{E}\) and \(\mathbf{V}\) (see [17], where the model equations have been used to simulate some electromagnetic tweezer effects). It is crucial to observe that, when \(\rho\), \(DV/Dt\) and \(p\) are identically zero, one recovers the usual Maxwell Equations (1)-(4) in a vacuum. Therefore, the revised model is actually an extension of the classical one. Equations of the type of (10), coupled with [PERSON]'s ones, are usually found in some plasma physics models--for instance, those related to magnetohydrodynamics (see, for instance, [18] or [19]). In such examples, a number of charged particles evolve as a real fluid. In this fashion, \(\mathbf{V}\) takes the meaning of a fluid flow velocity and \(p\) is a proper pressure. Moreover, a mass density \(\rho_{m}\) and a charge density \(\rho_{q}\) appear in the equations. In the present model, massive charged particles are not necessarily involved. As a consequence, \(\mathbf{V}\), \(p\) and \(\rho\) assume different meanings. As in the case of [PERSON]'s equations, by multiplying the first two equations by \(\mathbf{E}\) and \(\mathbf{B}\) respectively, and summing up, we get an energy balance relation:  
\[\
abla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}\,=\,\mathbf{J}_{s} \tag{2}\]0  
Finally, by taking the divergence of Equation (6), one finds the continuity equation:  
\[\
abla\times\mathbf{H}-\frac{\partial\mathbf{D}}{\partial t}\,=\,\mathbf{J}_{s} \tag{2}\]1  
which appears as an automatic consequence of the model equations. In view of operating in cylindrical coordinates, we propose a suitable simplification based on symmetry arguments. This reduces the starting problem from three to two dimensions.