Capacitance of a conducting hollow cylindrical shell in a closed form

Abstract

The charge distribution within a hollow conducting cylinder with zero-thickness walls is calculated from the minimum potential energy ($U$) consideration. The surface charge density consists of a diverging term (Jackson, 1975) and a sum of Legendre polynomials with the coefficients determined from the minimum $U$ approach. The sum converges. This allows to express the capacitance in closed form. It is in agreement with Butler (1980). We present electric field lines inside and outside of the cylinder. An electric field pattern can be studied in detail. Most of the numerical analysis is done for the conducting cylinder of the length equal to ten radii. The surface charge density near the edges diverges; and in the middle, it is twenty five percent less than that of a uniformly distributed charge. The self-energy of the conducting cylinder is about 5 percent lower than that of uniformly distributed surface charge.