Electrostatic Induction Electrode Systems for Beam-Position Detection

Using the integral solution for the field and the reciprocity of the Green's Function, we show that the response of these devices to a line charge obeys Laplace's equation, if this operator is applied with respect to the source coordinates. In a cylindrical system of electrodes the boundary values of the response can be determined by inspection. Knowing these values, one can easily find the response over the entire aperture. This allows the use of electrodes of unusual shapes, e.g., sections of the beam tube.