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 ‘2a’ filled in between with an electrolyte of resistivity $ρ$ (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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$R = \frac{ρ}{π l} \ln (2)$

$R = \frac{ρ}{2 π} (2)$

$R = \frac{ρ}{2 π l} \ln (2)$

$R = \frac{2 ρ}{π l} \ln (2)$

answer is B.

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Detailed Solution

Maximum power in external resistance is generated when it is equal to internal resistance of battery i.e. , $P_{R}$
Maximum when $r = R$
The maximum Joule heating in R
Will take place for, the resistance of small element
$d R = \frac{ρ d r}{2 π r l} \Rightarrow \frac{ρ}{2 π r l} \int_{a}^{2 a} \frac{d r}{r} \Rightarrow R = \frac{ρ}{2 π l} \ln \frac{2 a}{a}$
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