Edge effect in beam monitors

Quite often, particle-beam monitors have not the same cross-section as the beam pipe or vacuum chamber in which they are mounted. In that case, the electromagnetic field of the beam is distorted in the vicinity of the edges of the monitor. This field, at the junction of two rectangular beam pipes of different dimensions, is computed for a beam with constant charge along its length. Solutions which are less accurate but easier to apply are obtained with a first order approximation. The results are extended to intensity-modulated beams and circular or elliptical cross-sections. The errors, due to the edge effect, for the electrostatic pickup and the wall-current monitor are computed. The final formulas are simple and easy to apply to practical cases. (6 refs).