Question

Consider a small cube of volume 1 mm³ at the centre of a circular loop of radius 10 cm carrying a current of 4 A. Find the magnetic energy stored inside the cube.

Answer 1

Given:
Current in the loop, i = 4 A
Radius of the loop, r = 10 cm = 0.1 m
Volume of the cube, V = 1 mm³ = $1\times {10}^{-9}\mathrm{m}$
Magnetic field intensity at the centre of the circular loop:
$$\begin{gathered} B=\frac{{\mu }_{0}i}{2r} \\ =\frac{(4\pi \times {10}^{-7})\times 4}{2\times 0.1} \\ =8\mathrm{\pi }\times {10}^{-6}\mathrm{T} \end{gathered}$$
Magnetic energy density = $\frac{B^2}{2{\mu }_0}$
Total energy stored in volume V:
U = $\frac{B^{2}V}{2{\mu }_0}$
$$\begin{gathered} =\frac{(8\mathrm{\pi }\times {10}^{-6}{)}^2\times (1\times {10}^{-9})}{(4\mathrm{\pi }\times {10}^{-7})\times 2} \\ =8\mathrm{\pi }\times {10}^{-14}\mathrm{J} \end{gathered}$$