Question

Consider a cylindrical conductor of Length L and area of cross section A. The specific conductivity varies as $\sigma (x) = \sigma_0 \frac{L}{\sqrt{x}}$ where x is the distance along the axis of the cylinder from one of its ends. The resistance of the system along the cylindrical axis is

• A. $\frac{2 \sqrt{L}}{3 A \sigma_0}$
• B. $\frac{3 \sqrt{L}}{2 A \sigma_0}$
• C. $\frac{ \sqrt{L}}{3 A \sigma_0}$
• D. $\frac{2 \sqrt{L}}{ A \sigma_0}$
• E. $\frac{4 \sqrt{L}}{3 A \sigma_0}$

Answer 1

Answer: (A)

Given, $\sigma(x)=\sigma_{0} \frac{l}{\sqrt{x}}$
$\because$ Resistance of the system along the cylindrical axis,
$R =\int\limits_{0}^{L} \frac{\rho(x)}{A} d x$
$=\int\limits_{0}^{L} \frac{\left(\frac{1}{\sigma_{0} \frac{L}{\sqrt{x}}}\right)}{A} d x$
$\left[\because \rho(x)=\frac{1}{\sigma(x)}\right]$
$=\int\limits_{0}^{L} \frac{\sqrt{x}}{\sigma_{0} A L} d x=\frac{1}{\sigma_{0} A L}\left(\frac{x^{3 / 2}}{3 / 2}\right)_{0}^{L}=\frac{2}{3} \frac{1}{\sigma_{0} A L}\left(L^{3 / 2}-O\right)$
$=\frac{2}{3} \cdot \frac{1}{\sigma_{0} A L} \times L^{3 / 2}, R=\frac{2}{3} \cdot \frac{\sqrt{L}}{A \sigma_{0}}$