The most popular model for a two-dimensional electronic system considers electrons moving in a background of uniform positive charge. Studies of finite systems employing a spherical geometry (electrons on the surface of a sphere) or a disk geometry background are numerous. The same cannot be said for a square geometry, who after all, would be the geometry of choice for quantum Hall states in a Landau gauge. A background represented by a uniformly charged square seems to be perceived as too costly from a computational point of view. By using simple transformations, we show that the Coulomb self-energy and the electrostatic potential of a uniformly charged square can be exactly calculated and poses no difficulty. The current results can be used in systematic studies of properties of finite systems of electrons embedded in a positive background in the form of a uniformly charged square. © 2009 Elsevier B.V. All rights reserved.