Question

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 $\rho$ (as shown in the figure). If the battery is connected to a resistance of value $R$, the maximum Joule's heating in $R$ will take place for-

• A. $R=\frac{\rho }{2\pi l}\left(\frac{b}{a}\right)$
• B. $R=\frac{\rho }{2\pi l}\ln \left(\frac{b}{a}\right)$
• C. $R=\frac{\rho }{\pi l}\ln \left(\frac{b}{a}\right)$
• D. $R=\frac{2\rho }{\pi l}\ln \left(\frac{b}{a}\right)$

Answer 1

The correct option is B $R=\frac{\rho }{2\pi l}\ln \left(\frac{b}{a}\right)$

As we know that, maximum power across an external resistance is generated, when internal resistance of the battery is equal to the external resistance.



The resistance of the cylindrical element of the battery will be,

$dR=\frac{\rho dr}{2\pi rl}$

$\Rightarrow R=\frac{\rho }{2\pi l}{\int }_a^b\frac{dr}{r}$

$R=\frac{\rho }{2\pi l}\ln \frac{b}{a}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.