Free Maxwell equations in vacuum in orthogonal curvilinear coordinates and some applications: Helmholtz equation, poynting vector, and energy density

In this paper, we present a generalized method of solving the Free Maxwell Equations in Vacuum employing the process of separation of variables, which can be used for different systems of orthogonal curvilinear coordinates. This method leads us to obtain the Helmholtz equation for the distinct systems of curvilinear coordinates. We show the case for the prolate spheroidal coordinates, obtaining the solutions dependent on the spatial and time coordinates. An analysis of the Poynting vector field of the solutions is presented together with the energy density analysis. It is shown that prolate spheroidal coordinates do not allow the existence of closed surfaces of the magnetic field; in addition, the nodal surfaces (geometric places where the energy density does not change in time) are not closed surfaces either. Our results may be helpful to the scientific community because they provide us with a systematic method for solving the system of Free Maxwell Equations in Vacuum for such coordinate systems.