Stability of a Rotating Ring of Discrete Charges in Crossed Electric and Magnetic Fields

Abstract

Experimental results indicate that certain configurations of discrete charges may be more stable than continuous distributions possessing the same charge and corresponding symmetries. A model of N identical infinitely long negatively charged rods rotating under the influence of a uniform applied magnetic field is here investigated for stability of perturbations about an equilibrium configuration and compared to a continuous charged sheet. For the continuous sheet, a countably infinite number of modes of oscillation are possible of which only a finite number may be stable. The finite number of stable modes are those corresponding to the longer wavelengths and the growth rates of the infinite number of unstable modes increase as the wavelength decreases. In contrast, the discrete distribution may support only 2N-3 modes of oscillation. The first modes to become unstable when critical parameters are adjusted are those corresponding to the shortest wavelengths. These also have the highest growth rates. The most significant result of this investigation is that the discrete distribution may have all vibrational modes stable while this is impossible for the continuous distribution. Application of the discrete-charge theory to explain the stability of experimentally observed vortex patterns in unneutralized electron beams is also discussed.