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. $\frac{2\sqrt{L}}{3A{\sigma }_0}$
• B. $\frac{3\sqrt{L}}{2A{\sigma }_0}$
• C. $\frac{\sqrt{L}}{3A{\sigma }_0}$
• D. $\frac{2\sqrt{L}}{A{\sigma }_0}$
• E. $\frac{4\sqrt{L}}{3A{\sigma }_0}$

Answer 1

Answer: (A)

Given, $\sigma (x)={\sigma }_0\frac{l}{\sqrt{x}}$
$\because$ Resistance of the system along the cylindrical axis,
$R=\underset{0}{\overset{L}{\int }}\frac{\rho (x)}{A}dx$
$=\underset{0}{\overset{L}{\int }}\frac{\left(\frac{1}{{\sigma }_0\frac{L}{\sqrt{x}}}\right)}{A}dx$
$\left[\because \rho (x)=\frac{1}{\sigma (x)}\right]$
$=\underset{0}{\overset{L}{\int }}\frac{\sqrt{x}}{{\sigma }_{0}AL}dx=\frac{1}{{\sigma }_{0}AL}{\left(\frac{x^{3/2}}{3/2}\right)}_0^L=\frac{2}{3}\frac{1}{{\sigma }_{0}AL}\left(L^{3/2}-O\right)$
$=\frac{2}{3}\cdot \frac{1}{{\sigma }_{0}AL}\times L^{3/2},R=\frac{2}{3}\cdot \frac{\sqrt{L}}{A{\sigma }_0}$