A correspondence between the two-dimensional complex dielectric equations and those of plane-strain elasticity is established. Specifically, homogeneous orthotropic dielectric media are shown to correspond to orthotropic elastic media, and by making an affine transformation of the elasticity problem, we obtain correspondences between orthotropic dielectric media and non-orthotropic elastic media. We also find a correspondence between the elasticity problem and a pair of dielectric problems in which the dielectric tensors are real. These correspondences are extended to certain classes of inhomogeneous media. Applied to composite materials, they imply exact relations between the effective tensors of associated dielectric and elasticity problems.