A Relativistic and Electromagnetic Correction to the Ramo–Shockley Theorem

Abstract

The classical Ramo-Shockley (RS) theorem gives the current induced on perfect conductors by the motion of nearby charges, assuming nonrelativistic motion of those charges in electrostatic fields. This article illustrates how relativistic and electromagnetic effects modify RS in some simple examples. Specifically, we present explicit, closed form analytic solutions of Maxwell's equations for the induced current distribution on perfectly conducting plates due to the motion of a line charge moving parallel and perpendicular to the plates. The results have been verified by several methods, including particle-in-cell simulations. They are compared with the classical electrostatic theory used to derive RS. New insights into the limitation and validity of RS are provided. Electromagnetic shocks are explicitly calculated in closed form when the line charge strikes a parallel plate transmission line.

Induced Current Due to Electromagnetic Shock Produced by Charge Impact on a Conducting Surface

“…This also is presumably adequate since the classical RS theory, which ignores all such reflections, is known to give a reasonably accurate value of the induced current in RF vacuum electronics, when there is little beam interception. The underlying reason for this unexpected feature has been explored [9].…”

“…It has also been used in the theory of multipactor discharge, to analyze its effects on the beam loading of RF circuits [7] and the degradation in signal quality [8]. We recently suggested the reasons for such successes, having extended the RS theory to include electromagnetic and relativistic effects, for the first time [9]. We obtained an exact, closed-form analytic solution for the induced current of a simple model of a line charge moving between two parallel plates and compared that solution with the classical RS value.…”

“…In this article, we will continue to use the simple model of a line charge [9] moving between two parallel conducting plates of separation d [see Fig. 1(a)].…”