Ab initio calculation of four-vector potential and electromagnetic field for a charge moving arbitrarily in a cylindrical drift tube

Abstract

Using the method of Greenpsilas functions we found the four-vector potential and electromagnetic field components for a charge moving along an arbitrary path in a perfectly conducting cylindrical waveguide. Solutions are expressed analytically through Greenpsilas functions of the dpsilaAlembert operator with the Dirichlet and Neumann boundary conditions. It is shown that if the waveguide is excited only by a longitudinal current density component and/or non-zero charge density (transverse current components) the obtained solution reduces to the well-known expressions for TM (TE) cylindrical waveguide modes. If transverse and longitudinal current density components and non-zero charge density are present simultaneously in the waveguide, then the radial structure of the excited electro-magnetic field coincides with that of the superposition of TM and TE cylindrical waveguide modes. The results thus obtained allow one a direct calculation of the forces acting on an arbitrarily moving relativistic charge from the induced by-itself charges and currents at the waveguide walls. They also provide a basis for solution of a rigorous self-consistent problem on the non-stationary propagation of relativistic electron beams in cylindrical drift tubes with the account for space-charge effects. As an example of applications, under the Coulomb gauge, we analytically calculate the scalar potential and density of induced charge on the cylindrical perfectly conducting drift tube walls, which are caused by a moving non-relativistic point-like charge. We also find the potential part of the quasi-electrostatic force exerted on such a charge by the charge density induced by it on the drift tube walls.