Van der Waals energy between voids in dielectrics

The problem of finding the nonretarded plasma oscillations around two spherical voids in a dielectric medium is formulated. The Laplace equation and boundary conditions of electrodynamics at the void surfaces are formally solved by using the bispherical coordinate system. The problem is reduced to the eigenvalue problem of an infinite matrix. The solutions for two identical voids and appropriate to a large void separation $D$ compared to the voids radius $R$ are found by perturbation theory. To first order in $\frac{R^3}{D^3}$, only the dipolar plasma oscillations around the void surfaces have frequencies perturbed by the void-void interaction. As a result, the interaction energy behaves like the inverse sixth power just like it does for full spherical particles in vacuum. This contradicts an earlier conjecture that monopolar dispersion forces should exist between empty cavities. The coefficient of the $D^{-6\mathrm{}}$ power law at large separation is different for the two systems due to multiation interactions.