Vacuum material properties and Cherenkov radiation in Logarithmic Electrodynamics

We study some observational signatures of nonlinearities of the electromagnetic field. First to all we show the vital role played by nonlinearities in triggering a material behavior of the vacuum with $(\varepsilon > 0, \mu <0)$, which corresponds to a ferrimagnetic material. Secondly, the permittivity and susceptibility induced by nonlinearities are considered in order to obtain the refractive index via the dispersion relation for logarithmic electrodynamics. Finally, we consider the electromagnetic radiation produced by a moving charged particle interacting with a medium characterized by nonlinearities of the electromagnetic field. To this end we consider logarithmic electrodynamics. The result shows that the radiation is driven by the medium through which the particle travels like the one that happens in the Cherenkov effect.