Question

A rod of length L and cross-section area A lies along the x-axis between x = 0 and x = L. The material obeys Ohm's law and its resistivity varies along the rod according to $\rho \left(x\right)={\rho }_0{e}^{-x/L}$. The total resistance of the rod is

• A. $\frac{{\rho }_{0}L}{A}\left(\frac{1}{e}\right)$
• B. $\frac{{\rho }_{0}L}{A}\left(e\right)$
• C. $\frac{{\rho }_{0}L}{A}(1-\frac{1}{e})$
• D. $\frac{{\rho }_{0}L}{A}(e-1)$

Answer 1

The correct option is C

$\frac{{\rho }_{0}L}{A}(1-\frac{1}{e})$

The resistance of the element of length $dx$ is given by, $dR=\frac{\rho dx}{A}=\frac{{\rho }_0{e}^{-x/L}dx}{A}$
So, total resistance, $R=\frac{{\rho }_0}{A}{\int }_0^L{e}^{-x/L}dx=\frac{{\rho }_0}{A}{[-L{e}^{-x/L}]}_0^L=\frac{{\rho }_{0}L}{A}(1-\frac{1}{e})$