## Capacitance Of Spherical Capacitor

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# Capacitance of a Spherical Capacitor (outer sphere earthed)

Let A and B be two concentric metal spheres of radii a and b respectively with air as the intervening sphere. The induced charge on the inner surface of the outer sphere is – q. P is a point at a distance r from the common centre O.

Electric field at P = E = 1/4πε

_{0}( q / r

^{2}) r

where r is the unit vector along OP.

The potential difference between the spheres A and B is given by

V = - ∫a

**E.**d

**l**

Here, dl is the difference vector displacement along a path from B to A.

But E. dl = E dl cos 180 = - E dl.

Further, in moving a distance dl in the direction of motion, we are moving in the direction of r decreasing, so that dl = - dr. Hence,

**E.**d

**l**= Edr.

Eq. (2) becomes V = - ∫

^{a}E dr.

Putting the value of E from Eq. (1), we get

V = - ___

__q_____ ∫a

__dr__= - __

___q_____ { -

__1__}a

____4πε

_{0}__ r

^{2}__4πε

_{0}{ r }b

= ___

__q_____ {

__1__–

__1__} = ___

__q_____

__b - a__

4πε

_{0}a b 4πε

_{0}ab

. : Capacitance of the spherical capacitor

C =

__q__= ________

__q___________ = 4πε

_{0}_____

__ab_______ ______.

V (___

__q_____) (

__b – a__) _________( b – a)

____4πε

_{0}__ ab

C = 4πε

_{0}____

__ab______ = ____

__4πε_____

_{0}________ b – a _____ (

__1__–

__1__)

________________ (a b)

When b ∞, C = 4πε

_{0}a.

This is a capacitance of an isolated conducting sphere of radius a.

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