Exact formulas for space charge limited flow in a planar diode: New relativistic Child-Langmuir law

In this work, we have derived the exact formulas for the relativistic Child-Langmuir law to describe a space charge-limited flow in a planar diode under arbitrary applied voltages. Two hypergeometric functions are obtained to relate the classic Child-Langmuir law and the ultra-relativistic formula to exact relativistic Child-Langmuir law. Both nonrelativistic-to-relativistic hypergeometric function and ultra-relativistic-to-relativistic hypergeometric function are found to be dependent on applied voltages only. The formulas are elegant, compact, and intuitive, and can be easily employed.