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Thermodynamic Processes

The capacitan...

Question

The capacitance of two concentric spherical shells of radii $R_{1} a n d R_{2} (R_{2} > R_{1})$ is:

4 π ε_{0} R_{2}

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4 π ε_{0} \frac{R_{2} - R_{1}}{R_{1} R_{2}}

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4 π ε_{0} \frac{R_{2} R_{1}}{R_{2} - R_{1}}

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4 π ε_{0} R_{1}

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Solution

The correct option is D $4 π ε_{0} \frac{R_{2} R_{1}}{R_{2} - R_{1}}$
Let $K = \frac{1}{4 π ϵ_{o}}$
Potential at outer spherical shell is $V_{2} = \frac{K Q}{R_{2}}$
Potential at inner spherical shell is $V_{1} = \frac{K Q}{R_{1}}$
Potential difference, $V = V_{1} - V_{2} = \frac{K Q}{R_{1}} - \frac{K Q}{R_{2}}$
Capacitance, $C = \frac{Q}{V} = \frac{1}{K (\frac{1}{R_{1}} - \frac{1}{R_{2}})}$
$C = \frac{4 π ϵ_{o} R_{1} R_{2}}{R_{2} - R_{1}}$

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