Relation between space-charge-limiting current and electric field enhancement factor at curved surface cathode

A two-dimensional solution of space-charge-limiting current for a high current vacuum diode with a spherical cathode is presented. The relation between space-charge-limiting current and electric field enhancement factor at the cathode surface for the diode with a curved surface cathode is also discussed. It is shown that compared with the current given by the conventional Child—Langmuir law, which describes the one-dimensional space-charege-limiting current, the two-dimensional space-charge-limiting current in such a diode is enhanced due to the electric-field enhancement along the cathode surface. Among practical parameter ranges, enhancement factor \eta_\rm b approximately satisfies \eta_\rm b \simeq A \beta^n, where \beta is the electric field enhancement factor at the cathode surface, and n is a constant between 1 and 2, which is confirmed to be universal for the diodes with curved surface cathodes.