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Question

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig.). Show that the capacitance of a spherical capacitor is given by
c=4πϵ0r1r2r1r2
where r1 and r2 are the radii of outer and inner spheres, respectively.
420355_935f9aaf89504d8baef1733e406c1683.png

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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=Q4πεor2Q4πεor1

Where,

εo= Permittivity of free space

V=Q4πεor2Q4πεor1

V=Q4πεo[1r21r1]

V=Q(r1r2)4πεor1r2


Capacitance of the given system is given by,

C=QV

C=4πεor1r2r1r2

Hence, proved.


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