The time constant to steady state of an electric signal applied to the outer boundary of a radial circuit that propagates to a centrally located cylindrical conductor is determined rigorously by a transmission-line analysis. Two cases are reported : the perfect radial conductor and a radial conductor with a finite resistance. The time constant is featured as a function of the ratio of the radius of the radial conductive leads to the radius of the inner conductor. As the ratio is increased, the time constant departs from that of an electric field penetrating a cylindrical conductor and approaches the inductive time constant of the leads. An analytic approximation of the time constant of the system also is provided. This analysis should assist in the development of high-power circuitry design when the discharge time is on the order of the inductive time constant.