Capacitance of a spherical capacitor (outer sphere is earthed):

Let us consider A and B are the two concentric spherical shells of radii a and b respectively (b >a). Shell A is charged by Q and the outer surface of shell B is earthed. So –Q charge is induced in the inner surface of B.

If is the electric field between the shells at a distance r from the centre then . = or, E = (using Gauss’s law).

We know that dV = – . = – (Angle between and is 0^{0})

On the surface of A potential is V and that on the surface of B is 0 as it is earthed. If V is the potential difference between the shells then, dV = –

Or, – V = – |-

Or, – V = [ – ]

Or, V = []

The capacitance of the spherical plate capacitor C = = = .

Capacitance of a spherical capacitor (inner sphere is earthed):

Let us consider A and B are the two concentric spherical shells of radii a and b respectively (b >a). Sphere B is charged by q and the outer surface of A is earthed, so V_{A} = 0.

If q^{/} charge is induced on shell A then, k[ + ] = 0 or, q^{/} = – .

If is the electric field between the shells at a distance r from the centre then

E = E_{A} + E_{B} = + 0 [E_{B} = 0 as E is outward for shell B]

E = – = k[- ]

Or, dV= dr

Or, V = |-

Or, V = [ – ]

Or, V =

The capacitance of the spherical plate capacitor C = = .

Combination of spherical capacitors:

Three concentric conducting spheres A, B and C each of radii a, b and c respectively filled with air. Sphere B is earthed and sphere A and C are connected with a wire. Calculate the equivalent capacitance.

This combination is the parallel combination of two spherical capacitors as the potential of sphere A and sphere C are same. (i) 1^{st} capacitor is formed with sphere A and the inner surface of sphere B of capacitance C_{1} =

(ii) 2^{nd}capacitor is formed with outer surface of sphere B and inner surface

of sphere C of capacitanceC_{2 }= .

So, the net capacitance is C = C_{1} + C_{2} = + = .

Capacitance of a cylindrical capacitor:

Let us consider A and B are the two coaxial cylinders of radii a and b respectively (b > a) each of length l. Cylinder A is charged by Q and the outer surface of B is earthed. So –Q charge is induced in the inner surface of B. If is the electric field between the cylinders at a distance r from the axis then, . .

Or, E = (using Gauss’s law).

We know that dV = – . = – . [Angle between and is 0^{0 }]

On the surface of A potential is V and that on the surface of B that is 0 as it is earthed. If V is the potential difference between the cylinders then, dV = – dr

Or, – V = – |

Or, V = .

The capacitance of the spherical plate capacitor C = = .

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