  We consider the Casimir energy due to a massless scalar field in a geometry
of an infinite wedge closed by a Dirichlet circular cylinder, where the wedge
is formed by $\delta$-function potentials, so-called semitransparent
boundaries. A finite expression for the Casimir energy corresponding to the arc
and the presence of both semitransparent potentials is obtained, from which the
torque on the sidewalls can be derived. The most interesting part of the
calculation is the nontrivial nature of the angular mode functions. Numerical
results are obtained which are closely analogous to those recently found for a
magnetodielectric wedge, with the same speed of light on both sides of the
wedge boundaries. Alternative methods are developed for annular regions with
radial semitransparent potentials, based on reduced Green's functions for the
angular dependence, which allows calculations using the multiple-scattering
formalism. Numerical results corresponding to the torque on the radial plates
are likewise computed, which generalize those for the wedge geometry. Generally
useful formulas for calculating Casimir energies in separable geometries are
derived.
