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.$\mathrm{B}=\frac{{\mathrm{\mu }}_{0}\mathrm{i}}{2\mathrm{\pi r}}$Magnetic field energy density is not uniform in the space between the cylinders. At a distance r from the centre,${\mathrm{\mu }}_{\mathrm{B}}=\frac{{\mathrm{B}}^{2}}{2{\mathrm{\mu }}_{0}}=\frac{{\mathrm{\mu }}_{0}{\mathrm{i}}^{2}}{8{\mathrm{\pi }}^{2}{\mathrm{r}}^{2}}$Energy in volume of element (length $\mathcal{l}$)Using values, we get U = 7 nJ

A capacitor of capacity $2\mu F$ is charged to a potential difference of 12 V.It is then connected across an inductor of inductance $6\mu H$. What is the current (in A) in the circuit at a time when the potential difference across the capacitor is 6.0 V?