Conservation of energy for the temperature field is solved jointly with conservation of charge for the electrostatic potential for an arc in which the effects of convection and radiation are neglected. The electrical conductivity is modeled by a hyperbolic tangent function. Prolate spheroidal coordinates are used to emphasize the effects of electrode shape. Electrodes are chosen as hyperboloids of revolution (varying from needlelike to planar) and the bounding wall is a prolate spheroid. Approximate solutions are obtained by Galerkin’s method. Current‐voltage characteristics, arc shapes, and heat transfer to the electrodes and bounding wall are obtained. We find that current‐voltage characteristics are sensitive to electrode geometry and temperature. Blunt and hot electrodes yield monotonically increasing characteristics. Cooling the electrodes and increasing their tip curvature both act to increase the voltage to drive a given current, especially for small currents, giving characteristics with distinct maxima and minima. The blunter the cathode, the larger the potential difference to drive a given current. The results also indicate that equipotentials are very close to coordinate lines, justifying an assumption of a one‐dimensional electric field made elsewhere.