Maxwell’s Problem on Point Charges in the Plane

• Main Author: Killian, Kenneth
• Format: Others
• Published: Scholar Commons 2008
• Subjects: Potential theory; Electrostatics; Bezout's Theorem; Algebraic curves; Harmonic functions; American Studies; Arts and Humanities
• Online Access: https://scholarcommons.usf.edu/etd/333; https://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=1332&context=etd

This paper deals with approximating an upper bound for the number of equilibrium points of a potential field produced by point charges in the plane. This is a simplified form of a problem posed by Maxwell [4], who considered spatial configurations of the point charges. Using algebraic techniques, we will give an upper bound for planar charges that is sharper than the bound given in [6] for most general configurations of charges. Then we will study an example of a configuration of charges that has exactly the number of equilibrium points that Maxwell's conjecture predicts, and we will look into the nature of the extremal points in this case. We will conclude with a solution to the twin problem for the logarithmic potential, followed by a discussion of the conditions necessary for a degenerate case in the plane.