Electrostatic patch effect in cylindrical geometry: III. Torques

Abstract

We continue to study the effect of uneven voltage distribution on two close cylindrical conductors with parallel axes started in our papers (Ferroni and Silbergleit 2011 Class. Quantum Grav. 28 145001; Ferroni and Silbergleit 2011 Class. Quantum Grav. 28 145002) now to find the electrostatic torques. We calculate the electrostatic potential and energy to lowest order in the gap to cylinder radius ratio for an arbitrary relative rotation of the cylinders about their symmetry axis. By energy conservation, the axial torque, independent of the uniform voltage difference, is found as a derivative of the energy in the rotation angle. We also derive both the axial and slanting torques by the surface integration method: the torque vector is the integral over the cylinder surface of the cross product of the electrostatic force on a surface element and its position vector. The slanting torque consists of two parts: one coming from the interaction between the patch and the uniform voltages, and the other due to the patch interaction. General properties of the torques are described. A convenient model of a localized patch suggested in Ferroni and Silbergleit (2011 Class. Quantum Grav. 28 145002) is used to calculate the torques explicitly in terms of elementary functions. Based on this, we analyze in detail patch interaction for one pair of patches, namely the torque dependence on the patch parameters (width and strength) and their mutual positions. Finally, we study the rotational motion of cylinders caused by the patch effect torque.