ON A RELATED THOMPSON PROBLEM IN R k T. AGAMA Abstract. In this paper we study the global electrostatic energy behaviour of mutually repelling charged electrons on the surface of a unit-radius sphere. Using the method of compression, we show that the total electrostatic energy U k (N ) of N mutually repelling particles on a sphere of unit radius in R k satisfies the lower bound U k (N ) ≫ǫ N 2 √ k . 1. Introduction According to Coulomb’s law the electrostatic potential energy between any two pair electrons with equal charges e = e i = e j on the surface of a sphere with respective position vectors  r i and  r j is given as U ij = K e e i e j  r ij where K e is Coulomb’s constant and  r ij = ||  r i −  r j || is the distance between the pair of charged electron. By making the assignment e = e i = e j = 1 and K e = 1, it can be seen that the electrostatic potential energy reduces to U ij = 1  r ij . To this end, the total electrostatic energy interaction among N charged electrons is given by the sum U (N )=  1≤i<j≤N 1 ||  r i −  r j || . The Thompson Problem is a conundrum that asks for the arrangement of any num- ber of electrons on the surface of a unit-radius sphere with the lowest possible global electrostatic potential energy. In other words, when the lowest total electrostatic potential energy exists between pairs of electrons on the sphere’s surface, the prob- lem will be solved. In order to do this, it is worthwhile to examine and minimize the overall behavior of the total energy for sufficiently large number of electrons. Only a few particular cases of the problem have been addressed, and the actual problem has not made much progress. Since a single electron is not affected by any external forces, the case of a single electron presents a trivial problem. The best solution for the two electron scenario is to arrange the electrons on the sphere so Date : July 13, 2022. 2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20. Key words and phrases. minimal; electrostatic energy; Coulomb; electrons. 1