Abstract: Similar to the parallel plate capacitor,the capacitance formulas between cathode and anode,anode and accelerating electrode in electron gun (EGUN) have been derived,and the analytical expression of electric field distribution between the two poles has been derived according to the superposition principle,and the electric field of the extremely critical axis area has been simulated by Matlab before and after the accelerating electrode is added.The simulation results show that for the structure of a 110 mm-space between the another whose radius is 0.5 mm and the anode whose inner radius is 5 mm and outer radius is 30 mm,before adding accelerating electrodes the EGUN generates a strong electric field which is larger than 1 V/mm only in an area about 1.9 mm which approaches the surface of the cathode and on other position almost zero which can not be guaranteed to work properly.After adding accelerating electrodes,the electric field on the axis will dramatically increase to several hundred times of the original,and the more the number of accelerating poles are,the larger the axis electric field is,and form a uniform electric field when adding three accelerating electrodes,illustrating the reason and necessity of the EGUN gun design of adding accelerating electrodes,and this can provide a theoretical basis for the EGUN design.