# Equilibrium Casimir forces between spheres

This test validates the scuff-cas3d of the scuff-em code suite by using it to compute the equilibrium Casimir force between identical spheres at various separation distances. We consider both perfectly electrically conducting (PEC) and dielectric spheres.

## Analytical solution

An analytical formula for the distance-dependent Casimir energy of two identical PEC spheres of radius $R$ separated by a distance $d$ was obtained by T. Emig et al. in this paper:

Their asymptotic ($d\gg R$) formula for the energy, and the force formula obtained by differentiating it, are

where the first few $C$ coefficients are

• for PEC spheres:

• for dielectric spheres with static $(\omega \to 0)$ relative permittivity $\epsilon$:

## scuff-em solution

The Casimir energy and force between two PEC spheres and between two dielectric spheres may be computed using scuff-cas3d as follows:

 % scuff-cas3d --geometry PECSpheres_501.scuffgeo --translist Spheres.trans --energy --zforce
% scuff-cas3d --geometry E10Spheres_501.scuffgeo --translist Spheres.trans --energy --zforce


Here the two .scuffgeo files (PECSpheres_501.scuffgeo and E10Spheres_501.scuffgeo] describe the two geometric configurations (two PEC spheres and two dielectric spheres of radius $R=1\, \mu$m separated by an initial center-center distance of $d$=3 $\mu$m) while Spheres.trans specifies the list of center-center separation distances $d$ at which we compute the energy and force. (Both geometries refer to the same surface mesh file for the sphere, Sphere_327.msh.

The above calculations produce output files named PECSpheres_327.out and E10Spheres_327.out. Plotting against the theoretical predictions of Emig et. al (referenced above) yields good agreement:

.

Here's the gnuplot script I used to produce this plot: Plotter.gp.