What is spherical capacitor?Derive expression for its capacitance.

Explanation

Spherical Capacitor : A capacitor which consists of two concentric spherical shells, out of which one is earthed and other is used to store charge, is known as a spherical capacitor
Capacitance of a spherical capacitor : Let we have a spherical capacitor.which consists of two spherical shells P and Q.
Let a = radius of spherical shell P
b = radius of spherical shell Q.

The outer spherical shell Q is connected to earth. When charge + q is given to inner spherical shell P. it induces charge - q on the inner surface of the shell Q and + q on its outer surface. As the shell Q is earthed, the charge + q on its outer surface flows to earth. If V is potential difference between the spherical shells P and Q.then capacitance of the spherical capacitor is given by
`C = q/V`
As the electric field between the two spherical shells is not uniform
`therefore E=(dV)/(dr)` (in magnitude)
and `dV = Edr`
The potential difference Fbetween the two spherical shells P and Q can be found by integrating both sides between the proper limits. Therefore,
`V = int_(P)^(Q)dV = int_(a)^(b)Edr`
To find the value of K, let us find electric field E at any point, say L between the two spherical shells. Let r be the distance of point L from the centre O of the spherical shells. Consider the dotted sphere of radius r through point L as the gaussian surface. The gaussian-surface encloses the spherical shell P and hence the charge enclosed by the gaussian surface is q. According to Gauss.s theorem, the total electric flux through the gaussian surface,
`phi =q/epsilon_(0)`...........(1)
As the magnitude of electric field at every point on the gaussian surface is same, we have
`phi =E xx` surface area of gaussian surface
or `phi = E xx 4pir^(2)`.....(2)
From eqns. (1) and (2), we have
`E xx 4pi r^(2) = q/epsilon_(0)`
`E = 1/(4pi epsilon_(0)). q/r^(2)`
Put value of E in equation
`V= int_(P)^(Q)dV = int_(a)^(b) Edr`
`V = int_(a)^(b) 1/(4pi epsilon_(0)).q/r^(2)dr = q/(4pi epsilon_(0)) int_(a)^(b) r^(-2) dr =1/(4pi epsilon_(0)) |r^(-1)/-1|_(a)^(b) q/(4pi epsilon_(0) pi| -1/r|_(a)^(b)`
`=q/(4pi epsilon_(0))|-1/b +1/a| = q/(4pi epsilon_(0)).(b-a)/(ab)`
Therefore, capacitance of the spherical capacitor is given by:
`C=q/V =q//q/(4pi epsilon_(0)).(b-a)/(ab)` which is the required expression for the capacitance of a spherical capacitor.