Calculating resistance

Consider a coaxial cable of length L, consisting of a cylindrical conductor of radius a surrounded by a cylindrical conducting shell of radius b. The space between the conductors is filled with an insulating material.

The resistance along the length of the cable is considerably smaller than the resistance between the inner and outer cylinders. What is the resistance between the two cylinders?

Consider current passing through a sequence of cylindrical shells of radius r and thickness dr. Each shell has a resistance dR given by:

dR = r dr / [ 2 p r L ]

Integrating from r=a to r=b to find the total resistance gives:

R = ò dR = (r / 2pL) ò dr/r

R = (r / 2pL) ln(b/a)

Generally this resistance is several hundred ohms/m to minimize the "leakage current" that passes through the insulating material between the conductors.