Electromagnetic and Relativistic Corrections to the Ramo-Shockley Theorem

The classical Ramo-Shockley theorem (RS) [1] , [2] gives the current induced on perfect conductors by the motion of nearby charges. It is a foundational result in vacuum electronics, discharge physics, and semiconductor devices. The original theorem was derived by assuming quasi-static fields, ignoring both relativistic and radiative effects. This paper illustrates, for the first time, how relativistic and electromagnetic effects modify RS in some simple examples and examines the magnitude of the corrections. 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 either parallel or perpendicular to the plates. Our exact closed form solutions have been verified by several methods, including particle-in-cell simulations. Several novel features were revealed that are absent in the classical RS: electromagnetic transients due to the sudden acceleration of charge, multiple reflections of electromagnetic waves by conductors, and generation of an electromagnetic shock when the charge strikes a conductor and is subsequently removed. We have found that the classical RS works rather well otherwise, even for a highly relativistic beam, except near the wavefront(s) where the transients dominate. This partially explains the mystery as to why the non-relativistic RS has been very successful in the theory and numerical simulations of vacuum electronic devices, in which only the quasi-static, sinusoidal steady solutions are included. The electromagnetic shock could be a source of contamination on the quality of signals in satellite communication, by the electrons in a multipactor discharge [3] . These issues are being examined. The induced current due to motion of a point charge is also under investigation.