A cylindrical conductor of length $l$ and radius $a$ has conductivity near its axis as $σ$ .The conductivity of material increases linearly with the distance from axis and becomes $2 σ$ near the surface. The resistance of conductor for longitudinal current is

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$\frac{l}{2 {πa}^{2} σ}$

$\frac{3 l}{4 {πa}^{2} σ}$

$\frac{3 l}{7 {πa}^{2} σ}$

$\frac{3 l}{5 {πa}^{2} σ}$

answer is B.

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Detailed Solution

The total resistance of the cylinder can be calculated by assuming the cylinder to be parallel combination of multiple cylindrical resistances of thickness dx

$\int \frac{1}{dR} = \int_{0}^{a} \frac{σ (a + x)}{a l} \times 2 πxdx$
$\Rightarrow R = \frac{3 l}{5 {πa}^{2} σ}$

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