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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.
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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" = "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.
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