Question

The capacitance of a concentric spherical capacitor of shell radii x and y (x > y) is
1. \(\frac{1}{{4\pi {\varepsilon _0}}}\ln \frac{x}{y}\)
2. \(\frac{{4\pi {\varepsilon _0}xy}}{{x - y}}\)
3. \(4\pi {\varepsilon _0}\ln \frac{y}{x}\)
4. \(\frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{1}{y} - \frac{1}{x}} \right]\)

Answer 1

Correct Answer - Option 2 : \(\frac{{4\pi {\varepsilon _0}xy}}{{x - y}}\)

The radius of the outer shell = x

The radius of the inner shell = y

The inner surface of the outer shell has charge +Q.

The outer surface of the inner shell has induced charge −Q.

The potential difference between the two shells is given by,

\(V = \frac{Q}{{4\pi {\varepsilon _0}y}} - \frac{Q}{{4\pi {\varepsilon _0}x}}\)

Where ε₀ is the permittivity of free space

\(V = \frac{Q}{{4\pi {\varepsilon _0}}}\left[ {\frac{1}{y} - \frac{1}{x}} \right]\)

\( \Rightarrow \frac{Q}{V} = \frac{{4\pi {\varepsilon _0}}}{{\left[ {\frac{1}{y} - \frac{1}{x}} \right]}}\)

\( \Rightarrow C = \frac{{4\pi {\varepsilon _0}xy}}{{x - y}}\)