The problem of two-dimensional, homogeneous, elliptical irregularities in an otherwise homogeneous plasma with anisotropic conductivity is considered. We find an analytic solution for the potential inside and outside the irregularities. In the special case of circular irregularities, the internal electric field is reduced from the background field in both depletions and enhancements. The internal field is rotated in different directions for depletion and enhancements, however. When the irregularity is elongated, the electric field inside can be larger or smaller than the background field in both depletions and enhancements, depending on the attack angle of the background field. The effects of ion inertia can further suppress the internal electric field in small-scale circular irregularities. These electrodynamics considerations may help explain some aspects of radar observations of irregularities excited by Farley-Buneman waves and instabilities in the electrojets, in particular, their tendency to exhibit Doppler shifts significantly smaller than the line-of-sight background electron convection speed and proportional to the cosine of the flow angle. The analysis generalizes that of St.-Maurice and Hamza (2001), who introduced this avenue of investigation.