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Q.

The cross – section area and length of cylindrical conductor are A and $ℓ$ respectively. The specific conductivity varies as $σ (x) = σ_{0} \frac{ℓ}{\sqrt{x}}$ , where x is the distance along the axis of the cylinder from one of its ends. Compute the resistance of the system along the cylindrical axis.

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a

$\frac{2 \sqrt{ℓ}}{3 {Aσ}_{0}}$

b

$\frac{\sqrt{ℓ}}{{Aσ}_{0}}$

c

$\frac{2 ℓ}{3 {Aσ}_{0}}$

d

$\frac{3 \sqrt{ℓ}}{2 {Aσ}_{0}}$

answer is C.

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Detailed Solution

$\begin{array}{l} dR = \frac{δ_{x} dx}{A}, dR = \frac{\sqrt{x} dx}{σ_{0} LA} \Rightarrow \int_{0}^{R} dR = \frac{1}{σ_{o} lA} \int_{0}^{l} x^{1 / 2} dx \\ \Rightarrow R = \frac{1}{σ_{0} lA} \frac{2}{3} l^{3 / 2} = \frac{2}{3} \frac{1}{σ_{0} A} \sqrt{l} \end{array}$

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The cross – section area and length of cylindrical conductor are A and ℓ respectively. The specific conductivity varies as σ(x)=σ0ℓx , where x is the distance along the axis of the cylinder from one of its ends. Compute the resistance of the system along the cylindrical axis.