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 ' \(b\) ' fille 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

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Correct Answer

(D) \(R =\frac{\rho}{2 \pi l} \ln \left(\frac{ b }{ a }\right)\)

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Detailed explanation

Maximum power in external resistance is generated when it is equal to internal resistance of battery.

\(P_{R}\) is max. when \(r=R\)

\(\int dr =\int_{2}^{b} \frac{\rho dr }{2 \pi r l} \Rightarrow r =\frac{\rho}{2 \pi l} l n \frac{ b }{ a }\)

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