There are a number of instances in the operation of electric apparatus in which the physical situation may be approximated by a configuration in which a plane electrode conducts current which is received from or delivered to a cylindrical conducting path normal to its surface. Among these instances the most familiar examples are cases in which the conducting path is an ionized column resulting from sparkover (breakdown) of a solid, liquid, or gaseous dielectric between conductors, or the process of arc formation by the separation of initially juxtaposed conducting surfaces between which current is flowing. Typical situations correspond to circumstances accompanying insulation failures in apparatus, relay contact ���breaks,��� circuit-breaker operations, and to the sparkover phenomena utilized in electric spark machining.1���2 In each of these situations craters are formed on the electrode surface. Considerable attention3���5 has been given to the phenomena of crater formation for obvious reasons. Formation of the craters has been attributed to various causes1,5,6 and, indeed, the predominant cause may vary, depending on conditions of current formation, duration and magnitude, and on current-density distribution in the ionized column. In at least one range of physically realizable circumstances current density beneath the surface has been demonstrated3 to be significant in determining the depth of crater formed by a particular discharge. The relationship demonstrated was of a qualitative character only, however, owing to the lack of quantitative knowledge concerning both transient and steady-state subsurface current distribution. This paper describes the results of an analytical study of current distribution under appropriate conditions. In this study steady-state current distribution was determined from work by others on an analogous problem in hydrodynamics.7,8 Transient current distribution was determined by a solution which is described in some detail in the Appendix.