A spherical capacitor has an inner radius of 8.00 mm and an outer radius of 8.50

Question

A spherical capacitor has an inner radius of 8.00 mm and an outer radius of 8.50 mm. With air between the spheres, the capacitor is connected to a battery and allowed to charge fully. With the battery still connected, oil is poured in, filling the volume between the spheres. As the oil is added, the battery does an additional 7.10 nJ of work on the capacitor. Suppose that the dielectric constant of the oil is oil = 4.5.

What is the potential difference supplied by the battery?

Express your answer with the appropriate units.

Explanation / Answer

Initially, the capacitor has a capacitance C. When the dielectric material is added, it increased to Cn.
Energy stored in a capacitor = 1/2 CV2
Given that
1/2 CnV2 - 1/2 CV2 = 7.1 x 10-9 J ...(1)

Capacitance of a spherical capacitor is given as
C = 4o[ab/(a - b)]
a = 8 x 10-3 m, b = 8.5 x 10-3 m
o is the vacuum permittivity, o = 8.85 x 10-12 F/m
C = [4 x (8.85 x 10-12)][8 x 8.5 / (8.5 x 10-3 - 8 x 10-3)]
= 15.125 x 10-12 F

Cn = kC
k is the dielectric constant of oil, k = 4.5
Cn = 68.062 x 10-12 F

Substituting C and Cn in equation (1)
1/2 V2 [(68.062 x 10-12) - (15.125 x 10-12)] = 7.1 x 10-9
V2 (26.4685 x 10-12) = 7.1 x 10-9
V2 = (7.1 x 10-9) / (26.4685 x 10-12) = 268.2434
V = sqrt(268.2434)
= 16.38 V

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