Q 12-04-208JEE MainJEE Main 2019 (9 Jan, Shift 1)Medium
An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is $a$ and distance of its centre from the wire is $d$ ($d \gg a$). If the loop applies a force $F$ on the wire then
Answer: (A) $F \propto \left(\dfrac{a}{d}\right)^2$
The small loop is a magnetic dipole of moment $m = I_2\pi a^2$, perpendicular to the plane, i.e. along the wire's field $B = \dfrac{\mu_0 I_1}{2\pi x}$. The field is non-uniform, so the force on the dipole is
$$F = m\left|\frac{dB}{dx}\right| = I_2\pi a^2\cdot\frac{\mu_0 I_1}{2\pi d^2} \propto \left(\frac{a}{d}\right)^2$$
By Newton's third law the loop exerts a force of the same magnitude on the wire.
Solution by Sreeraj P, M.Sc Physics