Question

A current carrying small circular loop is placed at a distance $2cm$ from an infinitely long current conductor as shown in the figure. Wire and loop lie in same plane. The loop has $100$ turns, its effective radius is $2cm$. The torque acting on the loop in the given situation is $x{\mu }_0$. The value of $x$ is _______.

Answer 1

Answer: 4

Magnetic field at a distance $(r=2cm)$
$B_0=\frac{{\mu }_0{I}_1}{2\pi r}=\frac{{\mu }_0\times 2}{2\pi \times 2\times 10^{-2}}=\frac{50{\mu }_0}{\pi }T$
Magnetic moment of the current carrying loop $=M=iAN$
$=2\times \pi \times {\left(2\times 10^{-4}\right)}^2\times 100$
$=2\times \pi \times 4\times 10^{-4}\times 100=8\times 10^{-2}\pi$
$\Rightarrow$ Torque acting on the loop
$=|\tau |=|M\times B|=|M||B|\sin \theta$
$=M{B}_0\sin 90^{\circ }=M{B}_0$
Or $\tau =8\times 10^{-2}\pi \times \frac{50{\mu }_0}{\pi }$
$=400\times 10^{-2}{\mu }_0=4{\mu }_0$
Thus, value of $x=4$