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:

The magnetic field ($$B$$) produced by an infinitely long wire at a distance $$r$$ is: $$B = \frac{\mu_0 I_1}{2\pi r}$$

A small circular loop of radius $$a$$ acts as a magnetic dipole with a magnetic moment ($$M$$) proportional to its area:  $$M = I_2 (\pi a^2) \propto a^2$$

The force ($$F$$) experienced by a magnetic dipole oriented in a non-uniform magnetic field is proportional to the product of the magnetic moment and the spatial magnetic field gradient:  $$F = M \frac{dB}{dr}$$

$$\frac{dB}{dr} = -\frac{\mu_0 I_1}{2\pi d^2} \propto \frac{1}{d^2}$$

$$F \propto a^2 \cdot \frac{1}{d^2} \implies F \propto \frac{a^2}{d^2}$$