For an understanding of the arc cathode spot mechanism, it is essential to determine the relative importance of Joule heating within the cathode and the energy input from the arc discharge. The problem of a source of uniform current density acting over a circular area on the surface of a semi‐infinite solid of conducting material is formulated and solved with the following results: (1) The Joule heat developed in the hemispherical volume directly under the cathode spot amounts to 38% of the total Joule heat developed in the electrode. This concentration of the heating leads to high average power densities in the cathode spot zone which amount, in the case of mercury, to 5×107w/cm3and 5×109w/cm3for current densities of 106and 107amp/cm2, respectively. (2) A comparison of the energy input to the cathode spot zone due to Joule heating and the energy input due to positive ion bombardment shows that for current densities of the order of 106amp/cm2, the former amounts to some 10–20% of the latter for such high‐resistivity metals as bismuth, antimony, and mercury. As the current density is increased to 107amp/cm2, the two energy sources become comparable for metals other than the low resistivity metals such as copper and silver. (3) The time required to bring typical electrode materials to the melting point with resistance heating alone is relatively short. ForJ=106amp/cm2and a spot radius of 2×10−3cm, bismuth (271°C) requires 4 &mgr;sec, antimony (631°C) 35 msec. The boiling point of mercury (357°C) is reached in 18 &mgr;sec.