The normal Casimir force for lateral moving planes with isotropic conductivities

Abstract

We consider the two planes at zero temperature with isotropic conductivity that are in relative lateral motion with velocity $v$ and inter-plane distance $a$. Two models of conductivity are taken into account -- the constant and frequency-dependent Drude models. The normal (perpendicular to planes) Casimir force is analysed in detail for two systems -- i) two planes with identical conductivity and ii) one of the planes is a perfect metal. The velocity correction to the Casimir energy $\Delta_v\mathcal{E} \sim v^2$ for small velocity for all considered cases. In the case of the constant conductivity $\eta$, the energy correction is $ \Delta_v\mathcal{E} \sim \frac{\eta}{a^3} \left(\frac{v}{\eta}\right)^2$for $v\ll \eta \ll 1$.