A long cylindrical conductor of radius a and length l is made of a material whose resistivity depends only on the distance r from the axis of the conductor as ρ = α r 2 , where α is a constant. The resistance of the conductor across the two ends will be

Step-by-step solution

We consider a long co-axial hollow cylinder of radius r and thickness dr. The cross-sectional area of considered element is dA = 2​πrdr ∴ Electric resistance between ends of conductor of considered element is dR = ρ l dA = ρ l 2 π rdr = α l ⁡ 2 π r 3 dr [as resistivity is ρ = α r 2 ] The system may be assumed as a parallel combination of a number of such types of elementary resistance. ∴ 1 R = ∫ 1 dR = 2 π α l ⁡ ∫ 0 a r 3 dr or 1 R