Q 12-04-198JEE MainJEE Main 2020 (5 Sep, Shift 1)Medium
A square loop of side $2a$, and carrying current $I$ is kept in $XZ$ plane with its centre at origin. A long wire carrying the same current $I$ is placed parallel to the $z$-axis and passing through the point $(0, b, 0)$, $(b \gg a)$. The magnitude of the torque on the loop about $z$-axis is given by
Answer: (C) $\dfrac{2\mu_0I^{2}a^{2}}{\pi b}$
Since $b \gg a$, the wire's field over the loop is nearly uniform: $B = \dfrac{\mu_0I}{2\pi b}$, along the $x$-axis at the loop.
The loop's magnetic moment $m = I(2a)^{2} = 4Ia^{2}$ is along $y$ (normal to the $XZ$ plane), perpendicular to $B$. The torque $\vec m\times\vec B$ is along $z$:
$$\tau = 4Ia^{2}\cdot\frac{\mu_0I}{2\pi b} = \frac{2\mu_0I^{2}a^{2}}{\pi b}$$
Solution by Sreeraj P, M.Sc Physics