In this paper we give a derivation of a system of equations which generalize the London brothers and Ginzburg–Landau systems 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 oppositely, supporting it. Secondly, we consider time-dependent non-relativistic Ginzburg–Landau theory.
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