Question

A long coaxial cable consists of two thin-walled conducting cylinders with inner radius $2cm$ and outer radius $8cm$. The inner cylinder carries a steady current $0.1A$, 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 $5m$ of the cable. Express answer in $nJ$ (use $\ln 2=0.7$)

Answer 1

Total energy stored per unit volume $=\frac{B^2}{2{\mu }_0}$

Magnetic field at a distance $r$ from the central axis of cable is given by

$B=\frac{{\mu }_{0}I}{2\pi r}$

Hence energy stored=$\int \frac{B^2}{2\mu }dV$

$={\int }_a^b\frac{B^2}{2{\mu }_0}2\pi rldr$

$=\frac{{\mu }_0{I}^{2}l}{4\pi }ln\left(\frac{b}{a}\right)$

$=\frac{{\mu }_0}{4\pi }\times {0.1}^2\times 5\times ln4$

$=7nJ$