Question

A long wire carries a current $5A$. The energy stored in the magnetic field inside a volume $1m{m}^3$ at a distance $10cm$ from the wire is

• A. $\frac{\pi }{4}\times {10}^{-13}J$
• B. $\frac{\pi }{2}\times {10}^{-13}J$
• C. $\pi \times {10}^{-13}J$
• D. $\frac{\pi }{8}\times {10}^{-13}J$

Answer 1

Answer: (D)

$\frac{\pi }{8}\times {10}^{-13}J$

u (energy per unit volume)$=\frac{B^2}{2{\mu }_0}$ and
energy $U=\frac{B^2}{2{\mu }_0}\times vol.$
$B=\frac{{\mu }_{0}I}{2\pi d}$
$U={\left(\frac{{\mu }_{0}I}{2\pi d}\right)}^2\times \frac{1}{2{\mu }_0}\times vol.$
$=\frac{{\mu }_0{I}^2}{8{\pi }^2{d}^2}\times vol.$
$=\frac{4\pi \times {10}^{-7}\times 25\times 0^{-9}}{8\times 10\left({10}^{-2}\right)}=\frac{\pi }{8}\times {10}^{-13}J$ [$\because {\pi }^2\approx 10$]