A spherical system of coordinates is used to solve the time-dependent heat-conduction equation for inside a conical cathode, whose surface is subjected, during arc welding processes, to complicated thermal heat-exchange effects. These arise from a variety of factors such as ion bombardment from the main body of the electrical arc, thermionic emission from the cathode's surface and Joule heating arising from the main current-carrying body of the cathode. The heat-conduction equation has a time-dependent source term arising from a combination of Joule heating and thermionic emission and is solved analytically using integral transform techniques. The solution is expressed in terms of integrals. These allow the response of the cathode for various currents and cathode geometries to be studied.