Brownian Motion of Charged Particles in Crossed Electric and Magnetic Fields

A new method for evaluating the Brownian motion of charged particles in crossed electric and magnetic fields is presented. In the problems considered, a steady $B_z$ field and radial electric field are embedded in a viscous medium. Calculations are made for the three-dimensional mean-square displacement of particles migrating away from the cylindrical axis. The results are applied to the theory of sheaths to uncover the manner in which the wall charge affects the familiar Debye displacement. The resulting formula is independent of the presence of a magnetic field parallel to the wall. Relaxing the electric conditions yields the Brownian motion of a neutral particle in a neutral viscous fluid.