A long round conductor of cross-sectional area S is made of material whose resistivity depends only on a distance r from the axis of the conductor as `rho=alpha//r^(2)`, where `alpha` is a constant. Find the total resistance per unit length of the rod when potential difference is applied across its length.

Explanation

Consider a cylindrical element of radii between r and (r+ dr) . Its resistance

`dR = (rhol)/(2pi dr) ` (or)` 1/(dR) = (2pi r dr)/(rhol)" "……(i)`
`therefore 1/R = int_0^a 1/(dR) = int_0^a (2pi)/(rhol) r dr`
(where a is the radius of the conductor )
`int_0^a (2pir dr)/(((alpha)/(r^2))l) = (2pi)/(alphal) int_0^a r^3 dr = (2pi)/(alpha l) ((a^4)/(4)) = ((pia^2)^2)/(2pialphal) = (S^2)/(2pi alpha l)`
`R = (2pi alpha l)/(S^2) " ".....(ii)`
The resistance per unit length of wire `R= (2pi alpha)/(S^2)`
(b) Equation (ii) can be written as `R= ((2pialpha)/(S)) ((l)/(S))`
Compare with `R = (rhol)/(S) , ` we get `rho = (2pi alpha)/(S)`
By Ohm.s law `E= jrho = i/S xx (2pi alpha)/(S) = (2pi alpha i)/(S^2)`