Casimir forces between cylinders at different temperatures

We study Casimir interactions between cylinders in thermal nonequilibrium, where the objects as well as the environment are held at different temperatures. We provide the general formula for the force, in a one reflection approximation, for cylinders of arbitrary radii and optical properties. As is the case for equilibrium, we find that the force for optically diluted cylinders can be obtained by appropriate summation of the corresponding result for spheres. We find that the nonequilibrium forces are generally larger than their equilibrium counterpart at separations greater than the thermal wavelength. They may also exhibit oscillations as function of separation, leading to stable points of zero net force. These effects are particularly pronounced for thin conducting cylinders (e.g. 40 nm diameter nanowires of tungsten) due to their large emissivity.

Figure 1

Total force on cylinder 1 per unit length in a system of two SiC cylinders with equal radii $R=0.1\mu \mathrm{m}$ at separation $d$ in a (a) cold (0 K), (b) warm (300 K) environment. Dashed lines indicate repulsion. Points of change from repulsive to attractive force with increasing $d$ correspond to stable points of zero force.

Figure 2

Total force on cylinder 1 per unit length in a system of two tungsten cylinders with equal radii $R=0.02\mu \mathrm{m}$ at separation $d$ in a (a) cold (0 K), (b) warm (2400 K) environment. Dashed lines indicate repulsion. Horizontal line $F_G/L$ denotes the weight of the corresponding tungsten cylinder per unit length.

Figure 3

Casimir forces between objects can also be obtained by pair-wise summation of the forces between all the volume elements.