An investigation into the applicability of perturbation techniques to solve the boundary integral equations for a parallel-plate capacitor with a rough electrode

We derive a system of integral equations for the surface charge distribution on the electrodes of a parallel-plate capacitor for which the profile of one electrode is rough. We show that the solution to this system of equations is tractable using a perturbation technique assuming small surface heights compared with the mean plate separation. The accuracies of the first- and second-order perturbative approximations and of the local height approximation are evaluated for a few examples with a one-dimensional rough surface by comparing them with exact numerical results obtained by solving the boundary integral equations directly using a numerical procedure. Some general guidelines for when the first- and second-order approximations will be accurate are given. It is shown that the perturbative formulation provides approximations with a regime of validity that may extend over a larger region than the regime of validity of the local height approximation. These results could be useful for capacitive-sensor design purposes or in modelling solid-state electronic devices.