Please Describe Cylindrical Capacitor

Figure given below can also represent the cross section of a cylindrical capacitor, in which the inner conductor is a solid rod of radius a carrying a charge +q uniformly distributed over its surface, and the outer conductor is a coaxial cylindrical shell of inner radius b carrying a charge of –q uniformly distributed over its inner surface. The capacitor has length L, and we assume L >> b so that, as was the case with the parallel-plate capacitor, we can neglect the “fringing’ field at the ends of the capacitor.

Just as we used Gauss’ law in the spherical geometry to obtain the two shell theorems, we can obtain two similar results in the cylindrical conductor were present, we could construct a Gaussian surface in the shape of a long cylinder of radius r < b having the same axis as the outer cylinder. This surface encloses no net change, so we conclude that E = 0 everywhere on the Gaussian surface. As in the case of the spherical shell, a uniformly charged cylindrical shell produces no electric field in its interior. Using a cylindrical Gaussian surface with r > a, we can deduce that the inner cylinder behaves just like a uniform line of charge, for which the field points racially outward from the axis and has a magnitude that we calculated.

Equation 30-1 now gives the capacitance:

          C = 2pe₀ L/In(b/a)     (cylindrical capacitor).

Note once again that only geometrical factors appear in this equation and that the capacitance has the form of e₀ multiplied by a quantity with the dimension of length.