Question

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 = `(4piin_0"r"_1"r"_2)/("r"_1 - "r"_2)`

where r₁ and r₂ are the radii of outer and inner spheres, respectively.

Answer 1

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"r"_2) - "Q"/(4piin_0"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_0"r"_1"r"_2]`

Capacitance of the given system is given by

C = `"Charge (Q)"/"Potenstial difference (V)"`

= `(4piin_0"r"_1"r"_2)/("r"_1 - "r"_2)`

Hence, proved.