  Partial differential equations for the electric potential at finite
temperature, taking into account the thermal Euler-Heisenberg contribution to
the electromagnetic Lagrangian are derived. This complete temperature
dependence introduces quantum corrections to several well known equations such
as the Thomas-Fermi and the Poisson-Boltzmann equation. Our unified approach
allows at the same time to derive other similar equations which take into
account the effect of the surrounding heat bath on electric fields. We vary our
approach by considering a neutral plasma as well as the screening caused by
electrons only. The effects of changing the statistics from Fermi-Dirac to the
Tsallis statistics and including the presence of a magnetic field are also
investigated. Some useful applications of the above formalism are presented.
