The work is devoted to a two-dimensional periodic system of field emitters mathematical
modeling. Field-emission electron-vacuum devices are actively used in many fields of science
and technology. In this article the field cathode as a periodic system of emitters of infinite
length and the same shape, located on a flat substrate and at the same distance from
each other, is investigated. Anode is a plane parallel to the substrate. To calculate the
electrostatic potential distribution the influence of each emitter is replaced by the influence
of a charged filament. To solve the boundary value problem for the Poisson equation the
variables separation method in Cartesian coordinates is used. The potential distribution was
found in an analytical form in the entire area of the system. In accordance with the formulas
obtained, numerical calculations were carried out and the effect of emitter packing density
on the electrostatic potential distribution is shown.