We re-examine the electrodynamic Casimir effect in a wedge defined by two perfect conductors making dihedral angle alpha=pi/p. This system is analogous to the system defined by a cosmic string. We consider the wedge region as filled with an azimuthally symmetric material, with permittivity and permeability epsilon1, micro1 for distance from the axis r<a, and epsilon2, micro2 for r>a. The results are closely related to those for a circular-cylindrical geometry, but with noninteger azimuthal quantum number mp. Apart from a zero-mode divergence, which may be removed by choosing periodic boundary conditions on the wedge, and may be made finite if dispersion is included, we obtain finite results for the free energy corresponding to changes in a for the case when the speed of light is the same inside and outside the radius a , and for weak coupling, |epsilon1-epsilon2|<<1, for purely dielectric media. We also consider the radiation produced by the sudden appearance of an infinite cosmic string, situated along the cusp line of the pre-existing wedge.