  In this paper we give a derivation of a system of equations to describe the
electrodynamics of s-wave superconductors. First, we consider a
relativistically covariant theory in terms of gauge four-vector electromagnetic
potential and scalar complex field. We use the first-order formalism to obtain
the supplemented Maxwell equations for gauge invariant electric, magnetic,
four-vector fields and the modulus of the superconducting order parameter. The
new four-vector field appears in some of the equations as a gauge invariant
super-current and in other ones, while gauge invariant, as a four-vector
electromagnetic potential. This dual contribution of the new four-vector field
is the basis of the electrodynamics of superconductors. We focus on the system
of equations with time-independent fields. The qualitative analysis shows that
the applied magnetic field suppresses the superconductivity, while the applied
electric field impacts appositely, supporting it. Second we consider
time-dependent non-relativistic Ginzburg-Landau theory.
