  We explore the potential of a static electric field to induce Anderson
localization of light in a large three-dimensional (3D) cloud of randomly
distributed, immobile atoms with a degenerate ground state (total angular
momentum $J_g = 0$) and a three-fold degenerate excited state ($J_e = 1$). We
study both the spatial structure of quasimodes of the atomic cloud and the
scaling of the Thouless number with the size of the cloud. Our results indicate
that unlike the static magnetic field, the electric field does not induce
Anderson localization of light by atoms. We explain this conclusion by the
incomplete removal of degeneracy of the excited atomic state by the field and
the relatively strong residual dipole-dipole coupling between atoms which is
weaker than in the absence of external fields but stronger than in the presence
of a static magnetic field. A joint analysis of these results together with our
previous results concerning Anderson localization of scalar waves and light
suggests the existence of a critical strength of dipole-dipole interactions
that should not be surpassed for Anderson localization to be possible in 3D.
