Question

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A cylindrical capacitor is made of two thin-walled concentric
cylinders. The inner cylinder has radius Ra = 7.2 mm, and the outer
one a radius R = 15 mm. The region between the cylinders
contains a dielectric with constant k =2.6. The common length of
the cylinders is L=4.4 m (Assume the cylinders can be treated as
ideal infinite cylinders.)
Determine the potential energy stored in this capacitor when a
potential difference V=27.5V is applied between the inner and
outer cylinders
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Answer 1

The potential energy stored in the capacitor is approximately 327.7 microjoules.

1. Calculate the capacitance of the cylindrical capacitor:

The capacitance of a cylindrical capacitor with a dielectric is given by:

C = (2 * pi * k * epsilon_0 * L) / ln(R / Ra)

where:

k is the dielectric constant (2.6 in this case)

epsilon_0 is the permittivity of free space (8.854 x 10^-12 F/m)

L is the length of the cylinders (4.4 m)

R is the radius of the outer cylinder (15 mm)

Ra is the radius of the inner cylinder (7.2 mm)

Plugging in the values, we get:

C = (2 * pi * 2.6 * 8.854 x 10^-12 F/m * 4.4 m) / ln(15 mm / 7.2 mm)

C ≈ 1.23 x 10^-9 F

2. Calculate the potential energy stored in the capacitor:

The potential energy stored in a capacitor is given by:

U = 1/2 * C * V^2

where:

C is the capacitance calculated in step 1 (1.23 x 10^-9 F)

V is the potential difference applied (27.5 V)

Plugging in the values, we get:

U = 1/2 * 1.23 x 10^-9 F * (27.5 V)^2

U ≈ 327.7 μJ

Therefore, the potential energy stored in the capacitor is approximately 327.7 microjoules.