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Question
A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of 2.5 µC. The space between the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller.
Solution
Radius of the inner sphere, r2 = 12 cm = 0.12 m
Radius of the outer sphere, r1 = 13 cm = 0.13 m
Charge on the inner sphere, q = 2.5 µC = `2.5 xx 10^-6 "C"`
Dielectric constant of a liquid, ` in_"r" = 32`
(a) Capacitance of the capacitor is given by the relation
C = `(4piin_0in_1"r"_1"r"_2)/("r"_1 - "r"_2)`
Where,
`in_0` = Permittivity of free space = `8.85 xx 10^-12 "C"^2 "N"^-1 "m"^-2`
`1/(4piin_0) = 9 xx 10^9 "Nm"^2 "C"^-2`
∴ C = `(32 xx 0.12 xx 0.13)/(9 xx 10^9 xx (0.13 - 0.12))`
`≈ 5.5 xx 10^-9 "F"`
Hence, the capacitance of the capacitor is approximate `5.5 xx 10^-9 "F"`.
(b) Potential of the inner sphere is given by,
`"V" = "q"/"C"`
= `(2.5 xx 10^-6)/(5.5 xx 10^-9) = 4.5 xx 10^2 "V"`
Hence, the potential of the inner sphere is `4.5 xx 10^2 "V"`.
(c) Radius of an isolated sphere, r = 12 × 10−2 m
Capacitance of the sphere is given by the relation,
`"C'" = 4piin_0"r"`
= `4pi xx 8.85 xx 10^-12 xx 12 xx 10^-12`
= `1.33 xx 10^-11 "F"`
The capacitance of the isolated sphere is less in comparison to the concentric spheres. This is because the outer sphere of the concentric spheres is earthed. Hence, the potential difference is less and the capacitance is more than the isolated sphere.
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