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 $ρ = α / r^{2}$ , where $α$ is a constant. Find the total resistance per unit length of the rod when potential difference is applied across its length.

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$\frac{α}{S^{2}}$

$\frac{π α}{S^{2}}$

$\frac{2 π α}{S^{2}}$

$\frac{π α}{2 S^{2}}$

answer is B.

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Detailed Solution

Here the current flows from end A to B. Resistance of elementary cylindrical shell is

$dR = \frac{ρ L}{dA} = \frac{α}{r^{2}} \frac{L}{(2 π rdr)}$

As all the elementary shells are in parallel,

$\frac{1}{R_{total}} = \int \frac{1}{dR} = \int_{0}^{R} \frac{2 π r^{3} dr}{α L} = \frac{2 π}{α L} \frac{R^{4}}{4}$

$= \frac{{(π R^{2})}^{2}}{2 α L π} = \frac{S^{2}}{2 α L π}$

$\Rightarrow R_{total} = \frac{2 α π L}{S^{2}}$

Resistance per unit length $= \frac{R_{total}}{L} = \frac{2 π α}{S^{2}}$

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