The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. Mathematical statement The electrostatic interaction energy occurring between each pair of electrons of equal charges (, with the elementary charge of an electron) is given by Coulomb's Law, Here, is the Coulomb constant and is the distance between each pair of electrons located at points on the sphere defined by vectors and , respectively. Simplified units of and are used without loss of generality. Then, The total electrostatic potential energy of each N-electron configuration may then be expressed as the sum of all pair-wise interaction energies The global minimization of over all possible configurations of N distinct points is typically found by numerical minimization algorithms. Thomson's problem is related to the 7th of the eighteen unsolved mathematics problems proposed by the mathematician Steve Smale — "Distribution of points on the 2-sphere". The main difference is that in Smale's problem the function to minimise is not the electrostatic potential but a logarithmic potential given by A second difference is that Smale's question is about the asymptotic behaviour of the total potential when the number N of points goes to infinity, not for concrete values of N. Example The solution of the Thomson problem for two electrons is obtained when both electrons are as far apart as possible on opposite sides of the origin, , or Known exact solutions Mathematically exact minimum energy configurations have been rigorously identified in only a handful of cases. For N = 1, the solution is trivial. The single electron may reside at any point on the surface of the unit sphere. The total energy of the configuration is defined as zero because the electron experiences no electric field due to other sources of charge. For N = 2, the optimal configuration consists of electrons at antipodal points. This represents the first one-dimensional solution. For N = 3, electrons reside at the