The potential distribution in a constricted cylinder: an exact solution

An exact solution to the Laplace equation is derived for the distribution of electric potential within a long cylinder carrying a circular constriction along its axis. The expression obtained for the potential distribution is reduced to a form which may be readily evaluated and is highly accurate for a ratio of constriction radius to cylinder radius approaching unity. Exact expressions both for the electric current density within the constriction and for the spreading resistance (i.e., the increase in resistance of the cylinder due to constriction of current flow lines) are also obtained.