The Normal Casimir Force for Lateral Moving Planes with Isotropic Conductivities

We consider the two planes at zero temperature with isotropic conductivity that are in relative lateral motion with velocity v and interplane 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 analyzed in detail for two systems—(i) two planes with identical conductivity and (ii) one plane that is a perfect metal. The velocity correction to the Casimir energy, ΔvE∝v2, for small enough velocities is used for all considered cases. In the case of constant conductivity, η, the energy correction is ΔvE∝η/a3v/η2 for v≪η≪1.