Modeling and analysis of the potential distribution of a field emission cathode based on the Laplace and Poisson equations

In this paper, the problem of potential distribution of a field emission cathode with a tip structure is investigated, mathematical models of the Laplace and Poisson equations are established, and the model is solved by the method of separation of variables. The results show the influence of the electric field strength and the morphology of the cathode surface on the potential distribution, which provides a theoretical basis for the design and optimization of an electron emission system. The accuracy and reliability of the model have been verified by numerical simulation, which provides important recommendations for improving the characteristics of the field emitter and optimizing the design of the electric field.