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.
Radius of the outer shell (r1) = 13cm = 0.13m
Radius of the inner shell (r2) = 12cm = 0.12m
Charge on the outer surface of the inner shell = 2.5 μC = 2.5
10-6C
Dielectric constant of liquid (
) = 32
Since, Potential difference between the two shells,
∴ Δ
Where, 
= Absolute Permittivity of free space = 8.85
10-12C2N-1m-2
Therefore 
(a) Capacitance, 
⇒ C = 5.55
10-9F
Hence, the capacitance of the capacitor = 5.55
10-9F
(b) The potential of a capacitor (V), is given as:
⇒ 
⇒ V = 450 V
Hence, the potential of the inner sphere = 450V
(c) Radius of the isolated sphere(R) = 12 cm = 0.12m
Capacitance of an isolated sphere = 4πε0R
∴ C = 4 × 3.14 × 8.85 × 10-12 × 0.12
C = 1.33
10-11F
Where, ϵ0 is the Absolute Permittivity of free space = 8.85
10-12C2N-1m-2The capacitance of the isolated sphere is less than that of the concentric spheres 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.