Question

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 ρ = α/r² , where α is a constant. Find: 

(a) the resistance per unit length of such a conductor; 

(b) the electric field strength in the conductor due to which a current I flows through it.

Answer 1

(a) Consider a cylinder of unit and length and divide it into shells of radius r and thickness dr Different sections are in parallel. For a typical section,

(b) Suppose the electric filed inside is E_z = E_o (Z axis is along the axis of the conductor).

This electric field cannot depend on r in steady conditions when other components of E. are absent, otherwise one violates the circulation theorem 

The current through a section between radii (r + dr, r) is