A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports. Show that the capacitance of a spherical capacitor is given by
$C = \frac{4 {πε}_{0} r_{1} r_{2}}{r_{1} - r_{2}}$
where r₁ and r₂ are the radii of outer and inner spheres respectively.

Solution

We are given two concentric spheres. The radii of the outer and inner sphere are r₁ and r₂ respectively.
A charge -Q is introduced on the inner sphere, which, gets uniformly distributed on its outer surface. As a result, charge +Q is induced on the sphere's outer surface of radius r₁ and Q on it's inner surface. Since, the outer surface is earthed, the positive charge will flow into earth.

Electric field inside sphere of radius r₂ is zero because of electrostatic shielding.

i.e.,

E = 0 for r < r_{2} E = 0 for r > r_{1}

Electric field exists in between and is directed radially outward.

Electrostatic potential of inner sphere of radius r₂

V = \frac{1}{4 {πε}_{0}} \{\frac{Q}{r_{2}} - \frac{Q}{r_{1}}\} = \frac{Q}{4 {πε}_{0}} \{\frac{1}{r_{2}} - \frac{1}{r_{1}}\}

Electrostatic potential of outer sphere of radius r₁= 0

Potential difference-V =

\frac{Q}{4 {πε}_{0}} [\frac{1}{r_{2}} - \frac{1}{r_{1}}]

If, C is the capacitance of spherical capacitor

C = \frac{Q}{V} = \frac{Q}{\frac{Q}{4 {πε}_{0}} [\frac{1}{r_{2}} - \frac{1}{r_{1}}]} C = \frac{4 {πε}_{0} r_{1} r_{2}}{r_{1} - r_{2}}

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