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Inductance

Question

A long coaxial cable consists of two thin-walled conducting cylinders with inner radius 2 cm and outer radius 8 cm. The inner cylinder carries a steady current 0.1 A, and the outer cylinder provides the return path for that current. The current produces a magnetic field between the two cylinders. Find the energy stored in the magnetic field for length 5 m of the cable. Express answer in nano-J (use ln2 = 0.7).

Moderate
Solution

The magnetic field inside is only due to the current of the inner cylinder.

B=μ0i2πr

Magnetic field energy density is not uniform in the space between the cylinders. At a distance r from the centre,

μB=B22μ0=μ0i28π2r2

Energy in volume of element (length l)

dUB=uBdV=μ0i28π2r22πldr=μ0i2l4πdrr UB=μ0i2l4πabdrr=μ0i2l4πlnba

Using values, we get U = 7 nJ



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