Electric field created by a circumferential sinusoidal charge-density wave on the surface of a cylinder

The paper is concerned with the calculation of the electric field created by a distribution of charge, sinusoidal around the circumference and constant longitudinally, on an infinitely long cylinder. Interest in the detailed analysis of this field was stimulated by the consideration of the field set up by the stator of a dielectric motor which had previously been constructed. Its stator consists of a number of conducting bars fixed longitudinally to a cylindrical surface, and connected in sequence to the lines of a three-phase supply. Such a stator produces a field essentially similar to that set up by the charged cylinder. Analysis shows that the field in space is constant at a given radius and, except in the charged surface itself, is everywhere rotating at an angular velocity determined by the speed of rotation of the cylinder, or in the case of the stator, by the frequency of the mains supply. The field has its maximum magnitude immediately adjacent to the charged surface, and diminishes with distance from it, except for one special case. The cause of the rotating field at every point in space is the rotation of the field patterns resulting from the charges on the cylindrical surface. This pattern is quite complex, and the equations of its field lines are derived analytically