Solution of Laplace's Equation for a high end potential with a biased center wire in a Malmberg-Penning trap

Abstract

After several experiments in a Malmberg-Penning plasma trap, a potential barrier or “hill” seemed present within the machine. The trap employs cylindrical geometry with an axial center wire. Laplace’s equation is solved with an end potential, a negatively biased center wire, and grounded sides. The solution is graphed and analyzed in MATHEMATICA. Visually, no potential hills are seen. These results are further supported by the absence of any zero derivative values. Based upon our graphs and derivative calculations, we believe no potential barriers are present within the trap. This research closely follows the work of D. L. Eggleston and Darrell F. Schroeter in their paper, Solution of Laplace’s equation for the confining end potentials of a coaxial Malmberg-Penning trap.