  We study the effect of a (3+1)-dimensional Chern-Simons electrodynamics on
the equations governing the dynamics of magnetized plasma and fields. In this
model, the Chern-Simons (CS) part consists of a dynamical pseudo-scalar field
whose space-time derivatives couple with the electromagnetic field. We explore
the CS corrections to the evolution equation for the magnetic field in a plasma
with non-zero electrical resistivity. We revisit Cowling's theorem in this
context and observe that the CS corrections lead to possibly small but non-zero
source terms for axisymmetric magnetic field. The scalar product of electric
and magnetic fields play the role of source of the pseudo-scalar field, and
therefore, pulsars and magnetars are likely astrophysical candidates to
generate propagating pseudo-scalar waves. Although aligned electric field gets
shorted out by flowing charges in large parts of the magnetosphere, there are
vacuum gaps in the vicinity of pulsars where strong $\vec E. \vec B$ is
expected to be present. We derive a wave solution for the pseudo-scalar field
generated by the time-varying $\vec E. \vec B$ associated with a pulsar.
