The study on the dielectric characteristics of gaseous insulation medium is important for designing current leads of superconducting machines using a sub-cooled liquid nitrogen ($LN_2$) cooling method. In a sub-cooled $LN_2$ cooling system, the temperature of gaseous insulation medium surrounding current leads varies from the temperature of coolant to 300 K according to the displacement between the electrode system and the surface of sub-cooled $LN_2$. In this paper, AC withstand voltage experiments on gaseous nitrogen according to temperature are conducted. Also, AC withstand voltage experiments on gaseous nitrogen according to pressure, size of electrode, and gap length between two electrodes are performed. It is found that there is a functional relation between the electrical breakdown voltage and the field utilization factor (${xi}$). As a result, the empirical formula for estimating an electrical breakdown voltage is deduced by adopting the concept of field utilization factors. It is expected that the experimental results presented in this paper are helpful to design current leads for a high voltage superconducting apparatus such as a superconducting fault current limiter (SFCL) using a sub-cooled $LN_2$ cooling system.