Calculation of Quasistatic Eigen-Field of a Charge, Which Moves Arbitrarily in a Cylindrical Drift Tube

Abstract

Summary form only given. In theoretical investigations of generation by various vacuum electron tubes, it is common to divide the excited electromagnetic field into radiation and space-charge parts. The calculation of the latter is, in general, an intricate, but important, for vacuum electronics problem. Calculation of space-charge contribution to the excited electromagnetic field is usually based on quasistatic (quasistationary) approach, which consists in neglecting of wave character of all electromagnetic processes in the Maxwell's equations. Usually, one drops either the electromagnetic induction (electroquasistatics) or displacement current (magnetoquasistatics). The former is more appropriate in vacuum electronics as in its framework the continuity equation holds. In terms of energy balance (Poynting theorem) in the electroquasistatics only accumulation of the electric component of eigen-field is taken into account. Such an approach is well justified for non-relativistic and weakly-relativistic vacuum tubes. However, for relativistic devices (gyro-klystrons, free electron laser, etc.), it is also desirable to take into account accumulation of magnetic part of the quasistatic field energy. The Darwin's approximation provides one with such an opportunity. It is also important to mention that within this approximation the continuity equation is still valid. In Darwin's approximation, using the Green's function method, we found the solution for quasistatic vector potential excited in a cylindrical drift tube with perfectly conducting walls by arbitrary charge and current densities, which satisfy the continuity equation. Green's functions are expressed as an expansion in the eigen-function of the Laplace operator in cylindrical coordinate system fields, relativistic correction to the electric fields and induced current density on the drift tube walls. We found the force acting on the moving point charge from the induced surface charges. We also propose a method, which enables one to reduce the problem for vector potential to a system of scalar Poisson equations in cylindrical coordinate system.