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).

Question

Infinity Learn · Accepted Answer

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

$B = \frac{μ_{0} i}{2 πr}$

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

$μ_{B} = \frac{B^{2}}{2 μ_{0}} = \frac{μ_{0} i^{2}}{8 π^{2} r^{2}}$

Energy in volume of element (length $l$ )

${dU}_{B} = u_{B} dV = \frac{μ_{0} i^{2}}{8 π^{2} r^{2}} (2 π l) dr = \frac{μ_{0} i^{2} l}{4 π} \frac{dr}{r} U_{B} = \frac{μ_{0} i^{2} l}{4 π} \int_{a}^{b} \frac{dr}{r} = \frac{μ_{0} i^{2} l}{4 π} \ln \frac{b}{a}$

Using values, we get U = 7 nJ

Inductance

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