A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show
what the capacitance of a spherical capacitor is given by
`C=(4piin_0r_1r_2)/(r_1-r_2)`
where r1 and r2 are the radii of outer and inner spheres, respectively.
Solution
Radius of the outer shell = r1
Radius of the inner shell = r2
The inner surface of the outer shell has charge +Q.
The outer surface of the inner shell has induced charge −Q.
Potential difference between the two shells is given by,
`V=Q/(4piin_0)[1/r_2-1/r_1]`
Where,
`in_0` = Permittivity of free space
`V=Q/(4piin_0)[1/r_2-1/r_1]`
`=Q(r_1-r_2)/(4piin_0r_2r_1]`
Capacitance of the given system is given by
`C="Charge(Q)"/"Potenstial difference(V)"`
`=(4piin_0r_2r_1)/(r_1-r_2)`
Hence, proved.
