  We present the results of an analytical theory and simulations of the field
inside a cavity created in a dipolar liquid placed in a uniform external
electric field. The analytical theory shows that the limit of continuum
electrostatics is reached through a singularity in the microscopic response
function responsible for a non-decaying longitudinal polarization wave. Fields
in microscopic cavities are much different from macroscopic predictions, and
low-polarity dielectrics are predicted to have a continuum limit distinct from
the solution of Maxwell's equations. Computer Monte Carlo simulations never
reach the standard continuum limit and instead converge to the new continuum
solution with increasing cavity size.
