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Resistance and Resistors

A cell of emf...

Question

A cell of emf $ε$ is connected across thick spherical shell of inner radius $R_{1}$ and outer radius $R_{2}$ as shown in figure. If resitivity of sphere is $ρ$ , current through the cell is:

\frac{π ε (R_{1} + R_{2})^{2}}{R_{2} - R_{1}}

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\frac{4 π ε R_{1} R_{2}}{ρ (R_{2} - R_{1})}

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\frac{π ε (R_{1} + R_{2})^{2}}{(R_{2} - R_{1})^{2}}

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\frac{π ε R_{1} R_{2}}{ρ (R_{2} - R_{1})}

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Solution

The correct option is B $\frac{4 π ε R_{1} R_{2}}{ρ (R_{2} - R_{1})}$
If we consider a spherical shell of thickness $d r$ , the resistance of the sphere will be:
$d R = \frac{ρ d r}{4 π r^{2}}$
$\Rightarrow R = \int d R = \frac{ρ}{4 π} \int_{R_{1}}^{R_{2}} r^{- 2} d r$

$\Rightarrow R = \frac{ρ}{4 π} [\frac{1}{r}]_{R_{2}}^{R_{1}} = \frac{ρ}{4 π} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]$
$i = \frac{ε}{R} = \frac{4 π ε R_{1} R_{2}}{ρ (R_{2} - R_{1})}$

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