On the averaged electric fields near perfectly conducting and insulating rough surfaces

Roughness modelled as small objects scattered upon conducting or insulating surfaces is discussed. Using the image technique to remove the supporting surface, the polarisability of the individual objects as well as their concentration on the surface are linked with an effective surface displacement. Conditions for the averaged field over the displaced surface are derived and physical interpretations of the terms involved are provided. The closest neighbour interaction is taken into account in the case of hemispherical bosses using image reflections in spheres. To elucidate the impact of roughness, solutions of some basic electrostatic problems, namely the roughened parallel-plate capacitor and a point source facing a planar conducting or insulating rough surface, are examined in some detail. Special attention is paid to image singularities responsible for the field reflected by a rough boundary.