Casimir interaction between inclined metallic cylinders

The Casimir interaction between one-dimensional metallic objects (cylinders, wires) displays unconventional features. Here we study the orientation dependence of this interaction by computing the Casimir energy between two inclined cylinders over a wide range of separations. We consider Dirichlet, Neumann, and perfect-metal boundary conditions, both at zero temperature and in the classical high-temperature limit. For all types of boundary conditions, we find that at large distances the interaction decays slowly with distance, similarly to the case of parallel cylinders, and at small distances scales as the interaction of two spheres (but with different numerical coefficients). Our numerical results at intermediate distances agree with our analytical predictions at small and large separations. Experimental implications are discussed.

Figure 1

Definition of distance and orientation of two inclined cylinders with axis-to-axis separation $d$ and inclination angle $\theta$.

Figure 2

Amplitude function $\Omega \left(\theta \right)$ of the asymptotic electromagnetic Casimir energy of Eq. (31) as a function of the inclination angle $\theta$.

Figure 3

Electromagnetic Casimir energy normalized to the PFA energy as a function of the inverse distance for perpendicular cylinders ($\theta =\pi /2$). The dashed curve represents the asymptotic energy of Eq. (30) and the dotted curve shows the energy of Eq. (34).

Figure 4

Same plots as in Fig. 3 but for inclination $\theta =\pi /4$.

Figure 5

Ratio $\omega \left(\theta \right)/\omega (\pi /2)$ with $\omega \left(\theta \right)=E_0(r,\theta )\sin \left(\theta \right)$ and $r=R/d$ as a function of the inclination angle $\theta$. The dashed curve represents the asymptotic result of Eq. (30). The solid curves connect numerical data for $r=0.01$, 0.1, 0.2, and 0.3 (from top to bottom). The PFA yields unity for the ratio.