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 $ρ$ (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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$R = \frac{ρ}{2 π l} (\frac{b}{a})$

$R = \frac{ρ}{π l} \ln (\frac{b}{a})$

$R = \frac{2 ρ}{π l} \ln (\frac{b}{a})$

$R = \frac{ρ}{2 π l} \ln (\frac{b}{a})$

answer is B.

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Detailed Solution

Maximum joule heating in R is possible when internal resistance = external resistance

$\begin{array}{l} C o n s i d e r a v e r t i c a l c y l i n d r i c a l s t r i p o f t h i c k n e s s d x a n d r a d i u s x . \\ I t s s m a l l r e s i s \tan c e d R c a n b e g i v e n a s, \\ d R = ρ \frac{d x}{2 π x l} \\ A f t e r i n t e g r a t i n g w e g e t, \\ R_{internal} = {\int ρ}_{a}^{b} \frac{d x}{2 π x l} = \frac{ρ}{2 π l}_{a}^{b} \frac{d x}{x} = \frac{ρ}{2 π l} {[\ln x]}_{a}^{b} = \frac{ρ}{2 π l} (\ln b - \ln a) = \frac{ρ}{2 π l} \ln \frac{b}{a} = R_{e x t e r n a l} \end{array}$