# Electromagnetic Waves in Annular Regions
## 4 Extensions and Applications  
By varying the constant \(c\), it is possible to handle solutions defined inside a homogeneous dielectric, where the electromagnetic message revolves at velocities lower than that of light. More theoretically difficult, but numerically feasible, is the situation where the dielectric is not homogeneous. This research on electromagnetic rings was inspired by the possibility of building stable elementary particles, such as the electron (see [22; 26] and the references therein). The idea is not new and, starting from the original paper [33], many authors have devolved their efforts to its realization. The results in [26] rely on an extension of the electrodynamics equations that allow to handle regions where the divergence of \(\mathbf{E}\) is different from zero even in vacuum. With this trick it is possible to add to the ring solution other properties, able to confer a net charge and a mass to the particle model. By following such an approach it is also possible to enlarge the solution space of Maxwell's equation by including solitary waves with compact support (solitons). Such an improvement is certainly of interest for the understanding of photons and optical phenomena in general (see [34]). It is not our aim here to further discuss the merits of this approach, however we believe that it may be of high interest in the study of both theoretical issues and innovative practical applications. A simple experiment puts in relation magnetic fields with topological properties (see [35]). Let us suppose that two simply-connected pieces of non magnetized iron, when joined together, form an annular shape. A conductive wire inserted into the hole is supplied for a fraction of time with an electric current. This generates closed magnetic lines of force inside the iron. After the removal of the wire, the two irons are solidly glued together. Such an adhesive contact may last for long time after the current shut off. When torn apart, the two components do not reconnect anymore. Thus, the breakage of the toroid passes through a topological modification that also changes its magnetic status. The energy that was stored within the ring comes out in some form after the rupture. This observation somehow confirms how important is to work with doubly-connected geometries and organize the experiments in such a way the magnetic field is distributed along closed patterns embedded in the ring support. The study here developed in vacuum, reveals original configurations and may help in the analysis of confined plasma (see, e.g., [36; 37]), with special attention to the detection of quantum vortex rings in superfluid helium (see [38] for an overview). Further applications are found in the field of whispering gallery resonators. Waves trapped in these cavities are smoothly guided to circulate around by continuous reflection returning at the origin with the initial phase. Spherical, cylindrical or ring-shaped whispering galleries are produced for a broad range of industrial applications, for instance in fiber telecommunications technologies or biosensing. Solitary waves in rings are commonly produced in optical laboratories by sending laser beams of appropriate frequency through a crystal lattice. A few non specialized publications are [39; 40]. Some references on more specific applications are found in the introductory section of this paper. As far as we know, the insertion or the emission of the signal in whispering galleries (annuli or disks) follows in general a path tangential to the major circumference. Indicative references are [41; 42; 43; 44], from the experimental viewpoint, and [45; 46], concerning a more theoretical approach. From the results of this paper, it turns out that this procedure would not be optimal, in order to ensure a long-time storage on the information packets.