Packings of a charged line on a sphere

We find equilibrium configurations of open and closed lines of charge on a sphere, and track them with respect to varying sphere radius. Closed lines transition from a circle to a spiral-like shape through two low-wave-number bifurcations-"baseball seam" and "twist"-which minimize Coulomb energy. The spiral shape is the unique stable equilibrium of the closed line. Other unstable equilibria arise through tip-splitting events. An open line transitions smoothly from an arc of a great circle to a spiral as the sphere radius decreases. Under repulsive potentials with faster-than-Coulomb power-law decay, the spiral is tighter in initial stages of sphere shrinkage, but at later stages of shrinkage the equilibria for all repulsive potentials converge on a spiral with uniform spacing between turns. Multiple stable equilibria of the open line are observed.