In suitable bounded regions immersed in vacuum, time periodic wave solutions solving a full set of electrodynamics equations can be explicitly computed. Analytical expressions are available in special cases, whereas numerical simulations are necessary in more complex situations. The attention here is given to selected three-dimensional geometries, which are topologically equivalent to a toroid, where the behavior of the waves is similar to that of fluid-dynamics vortex rings. The results show that the shape of the sections of these rings depends on the behavior of the eigenvalues of a certain elliptic differential operator. Time-periodic solutions are obtained when at least two of such eigenvalues attain the same value. The solutions obtained are discussed in view of possible applications in electromagnetic whispering galleries or plasma physics.

If the interior of a conducting cavity (such as a capacitor or a coaxial cable) is supplied with a very high-frequency electric signal, the information between the walls propagates with an appreciable delay, due to the finiteness of the speed of light. The configuration is typical of cavities having size larger than the wavelength of the injected signal. Such a non rare situation, in practice, may cause a break down of the performances of the device. We show that the classical Coulomb's law and Maxwell's equations do not correctly predict this behavior. Therefore, we provide an extension of the modeling equations that allows for a more reliable determination of the electromagnetic field during the evolution process. The main issue is that, even in vacuum (no dielectric inside the device), the fast variation of the signal produces sinks and sources in the electric field, giving rise to zones where the divergence is not zero. These regions are well balanced, so that their average in the domain is zero. However, this behavior escapes the usual treatment with classical electromagnetism. Keyword : electromagnetism, differential model, capacitor, high-frequency