  Maxwell's equations are modified to incorporate a scalar field to account for
the London's superconductivity. Assuming the electromagnetic field is described
by the Klein-Gordon equation, London's equations of superconductivity are then
derived, which are invariant under a new set of transformations. The invariance
of the modified Maxwell's equations under these transformations requires the
electromagnetic field and the scalar field to be scale-invariant. Relying on
these transformations, a quantized Josephson-like current is derived. This
current gives rise to a residual magnetic field. The spatial and temporal
variations of the scalar field are linked to the electric polarization such
that the polarization vector is curl-less.
