Attractive Casimir Forces in a Closed Geometry

We study the Casimir force acting on a conducting piston with arbitrary cross section. We find the exact solution for a rectangular cross section and the first three terms in the asymptotic expansion for small height to width ratio when the cross section is arbitrary. Though weakened by the presence of the walls, the Casimir force turns out to be always attractive. Claims of repulsive Casimir forces for related configurations, like the cube, are invalidated by cutoff dependence.

Figure 1

Optical paths in a three-dimensional box of size $h\times b\times c$ with a partition at height $a$. The origin and end points (which actually coincide) of some paths are separated so the path can be visualized. See the text for further description.

Figure 2

The force on a three-dimensional piston $F_{\mathrm{C}}$ with a square cross section due to EM quantum fluctuations, as a function of $a/b$ (see Fig. 1), normalized to the parallel plates’ result $F_{\parallel }=-3\zeta (4)A/(8{\pi }^2{a}^4)$. The exact expression is the thick solid line, and the first and second corrections are the dashed lines. The thin solid line is the force $F_{\mathrm{box}}$ that comes from differentiating the finite part of the energy in a single box.