  Maxwell's equations cannot describe a homogeneous and isotropic universe with
a uniformly distributed net charge, because the electromagnetic field tensor in
such a universe must be vanishing everywhere. For a closed universe with a
nonzero net charge, Maxwell's equations always fail regardless of the spacetime
symmetry and the charge distribution. The two paradoxes indicate that Maxwell's
equations need be modified to be applicable to the universe as a whole. We
consider two types of modified Maxwell equations, both can address the
paradoxes. One is the Proca-type equation which contains a photon mass term.
This type of electromagnetic field equations can naturally arise from
spontaneous symmetry breaking and the Higgs mechanism in quantum field theory,
where photons acquire a mass by eating massless Goldstone bosons. However,
photons loose their mass when symmetry is restored, and the paradoxes reappear.
The other type of modified Maxwell equations, which are more attractive in our
opinions, contain a term with the electromagnetic potential vector coupled to
the spacetime curvature tensor. This type of electromagnetic field equations do
not introduce a new dimensional parameter and return to Maxwell's equations in
a flat or Ricci-flat spacetime. We show that the curvature-coupled term can
naturally arise from the ambiguity in extending Maxwell's equations from a flat
spacetime to a curved spacetime through the minimal substitution rule. Some
consequences of the modified Maxwell equations are investigated. The results
show that for reasonable parameters the modification does not affect existing
experiments and observations. However, the field equations with a
curvature-coupled term can be testable in astrophysical environments where mass
density is high or the gravity of electromagnetic radiation plays a dominant
role in dynamics, e.g., interior of neutron stars and the early universe.
