Consider a hollow cylinder of length L and inner radius a and outer radius b, as shown in figure. The material has resistivity `rho`.

(a) Suppose a potential difference is applied between te ends of the cylinder and produces a current parallel to the axis. What is the resistance measure ?
(b) If instead the potential difference is applied between the inner outer surfaces so that current flows radially outward, what is the resistance measure ?

Explanation

(a) When a potential difference is applied between the ends of the cylinder, current flows parallel to the axis. In this case, the cross-section area is `A = pi ( b^(2) - a^(2))` , and the resistance is given by
`R = ( rho L)/( A) = ( rho L)/( pi (b^(2)-a^(2))) `
(b) Consider a differential element which is made up of a thin cylinder of inner radius r and outer radius `r + dr` and length L . Its contribution to the resistance of the system is given by
`dR = ( rhodL)/(A) = ( rho dr)/( 2pi rL)`
where `A = 2 pi L ` is the area normal to the direction of current flow. The total resistance of the system becomes.
`R = int _(a) ^(b) ( rho dr)/( 2pi rL) = (rho )/(2pi L) ln ((b)/(a))`