Question 251 of 380

Model a torch battery of length $l$ to be made up of a thin cylindrical bar of radius $a$ and a concentric thin cylindrical shell of radius $b$ filled in between with an electrolyte of resistivity $\rho$ (see figure). If the battery is connected to a resistance of value $R$, the maximum Joule heating in $R$ will take place for :

A$R=\dfrac{\rho }{2\pi \ell }\left(\dfrac{b}{a}\right)$B$R=\dfrac{\rho }{2\pi \ell }\ln \left(\dfrac{b}{a}\right)$C$R=\dfrac{\rho }{\pi \ell }In\left(\dfrac{b}{a}\right)$D$R=\dfrac{2\rho }{\pi \ell }\ln \left(\dfrac{b}{a}\right)$

Show answer and solution▾

Answer: (B) $R=\dfrac{\rho }{2\pi \ell }\ln \left(\dfrac{b}{a}\right)$

Let a small elementary cylinder in the shell of thickness $dr$.
The resistance of a wire is given by $R=\rho \dfrac{L}{A}$.
Here $\rho$ is the resistivity of the material.
The resistance of small element$dR=\dfrac{\rho dr}{2\pi rl}$
Integrating both sides,
$\Rightarrow R=\dfrac{\rho }{2\pi l}{\int }_{a}^{b}\dfrac{dr}{r}$
$\Rightarrow R=\dfrac{\rho }{2\pi l}ln\dfrac{b}{a}$