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 $\rho$ (see figure). If the battery is connected to a resistance of value $R ,$ the maximum Joule heating in $R$ will take place for :

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

Answer 1

Answer: (D)

Maximum power in external resistance is generated when it is equal to internal resistance of battery.
$
\begin{array}{l}
d R=\rho \frac{d r}{2 \pi r l} \\
\int d R=\frac{\rho}{2 \pi l} \int_{a}^{b} \frac{1}{r} d r \\
R=\frac{\rho}{2 \pi l}[\operatorname{Ln}(r)]_{a}^{b} \\
R=\frac{\rho}{2 \pi l} \operatorname{Ln}\left(\frac{b}{a}\right) \\
r=R
\end{array}
$
for max heat transfer