This paper presents a mathematical model of an axisymmetric field emission system with a modulator based on a field cathode with a sharp edge. The internal region of the system under investigation is filled with two different dielectrics. To calculate the electrostatic potential, the effect of the field cathode on the field distribution is replaced by the effect of a circular charged line with a linear charge density such that the zero equipotential is a virtual emitter. The boundary value problem for the Poisson equation is solved by the separation variables method. As a result, finding the coefficients of the potential expansions in eigenfunctions reduces to solving a system of linear algebraic equations with constant coefficients.