Question

A current carrying small circular loop is placed at a distance $2\, cm$ 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 $2 \,cm$. The torque acting on the loop in the given situation is $x \mu_{0}$. The value of $x$ is _______.

Answer 1

Magnetic field at a distance $(r=2 \,cm )$
$B_{0}=\frac{\mu_{0} I_{1}}{2 \pi r}=\frac{\mu_{0} \times 2}{2 \pi \times 2 \times 10^{-2}}=\frac{50 \mu_{0}}{\pi} T$
Magnetic moment of the current carrying loop $=M=i A N$
$=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 $x=4$