Casimir effect at nonzero temperature for wedges and cylinders

We consider the Casimir-Helmholtz free energy at nonzero temperature $T$ for a circular cylinder and perfectly conducting wedge closed by a cylindrical arc, either perfectly conducting or isorefractive. The energy expression at nonzero temperature may be regularized to obtain a finite value, except for a singular corner term in the case of the wedge which is present also at zero temperature. Assuming the medium in the interior of the cylinder or wedge be nondispersive with refractive index $n$, the temperature dependence enters only through the nondimensional parameter $2\pi naT$, $a$ being the radius of the cylinder or cylindrical arc. We show explicitly that the known zero-temperature result is regained in the limit $aT\to 0$ and that previously derived high-temperature asymptotics for the cylindrical shell are reproduced exactly.

Figure 1

The geometry considered: (left) a wedge of opening angle $\alpha$ closed by a cylindrical shell at radius $a$. The results are automatically applicable to a cylindrical shell of radius $a$ (right) when $\alpha =\pi$ [Eq. (1.3)].

Figure 2

The additional terms of the regularized energy in Eq. (3.20) which are subtracted from the double sum there in the case $p=3$. Shown also is the sum of the three additional terms and their low-temperature asymptotic value from Eq. (4.11).

Figure 3

Dimensionless energy $e(\tau ,p)$ for the case $p=3$, i.e. opening angle $\alpha =\pi /3$, approximated by a “brute force” calculation truncating the sums. The zero-temperature limit and high-$\tau$ asymptote are shown as dashed lines.

Figure 4

Same geometry as in Fig. 1 but now with diaphanous instead of perfectly conducting arc, i.e., so that $n^2=\epsilon \mu$ is the same both sides of the interface. We still assume nondispersive media.