Electrodynamic Casimir effect in a medium-filled wedge. II

We consider the Casimir energy in a geometry of an infinite magnetodielectric wedge closed by a circularly cylindrical, perfectly reflecting arc embedded in another magnetodielectric medium, under the condition that the speed of light be the same in both media. An expression for the Casimir energy corresponding to the arc is obtained and it is found that in the limit where the reflectivity of the wedge boundaries tends to unity the finite part of the Casimir energy of a perfectly conducting wedge-shaped sheet closed by a circular cylinder is regained. The energy of the latter geometry possesses divergences due to the presence of sharp corners. We argue how this is a pathology of the assumption of ideal conductor boundaries and that no analogous term enters in the present geometry.

Figure 1

The wedge geometry considered.

Figure 2

(Color online) The solutions of dispersion relation (2.20) as a function of $r$ and $\nu$ for $\alpha =0.75$. The eigenvalues of $\nu$ for a given $r$ are marked; the energy is calculated by summing over these values and then integrating over all $\omega$.

Figure 3

(a) Different methods of calculation used in different areas of the $x,\eta$ plane (see text). (b) The function $Y\left(\eta \right)$.

Figure 4

Above: the function $\tilde{e}\left(p\right)$ calculated for different $r$. Below: same quantity, now plotted as a function of opening angle $\alpha$.