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 fi

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

TestHub · Accepted Answer

Let a small elementary cylinder in the shell of thickness $d r$ .

The resistance of a wire is given by $R = ρ \frac{L}{A}$ .

Here $ρ$ is the resistivity of the material.

The resistance of small element $d R = \frac{ρdr}{2 πr l}$

Integrating both sides,

$\Rightarrow R = \frac{ρ}{2 πl} \int_{a}^{b} \frac{dr}{r}$

$\Rightarrow R = \frac{ρ}{2 πl} l n \frac{b}{a}$

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