Abstract: Based on the ArcPIC code, this paper analyzes the distribution of electron density and the field on the cathode surface using the PIC-MCC method. At the same time, it gives the fitting formula of two-dimensional space charge limited current. This provides a theoretical reference for the design and application of vacuum diodes. It is found that the electric field of cathode surface increases with the increase of cathode current density, then oscillates and tends to a stable state. The two-dimensional space charge limited current density decreases with the increase of cathode emission radius, and the larger the cathode emission radius, the closer the current to the one-dimensional space charge limited current.