1. Physical idea

A small current loop behaves like a magnetic dipole when observed from distances much larger than its size.

Given:

• loop radius $a$
• distance of loop center from the long wire $d$
• $d \gg a$

So we can approximate the loop as a magnetic dipole of moment $m = I_\ell \pi a^2$ where $I_\ell$ is the loop current.

By Newton's third law, the force applied by the loop on the wire has the same magnitude as the force applied by the wire on the loop. So it is easier to compute the force on the loop due to the magnetic field of the long straight wire.

2. Magnetic field of the infinite straight wire

At distance $r$ from a long straight wire carrying current $I_w$, $B(r) = \frac{\mu_0 I_w}{2\pi r}$

At the location of the loop center ($r=d$), $B(d) = \frac{\mu_0 I_w}{2\pi d}$

Since the field is non-uniform, the dipole experiences a net force.

3. Force on a magnetic dipole in a non-uniform field

For a dipole aligned with the field appropriately, the order of magnitude of force is $F \sim m\,\frac{dB}{dr}$

Now, $\frac{dB}{dr} = -\frac{\mu_0 I_w}{2\pi r^2}$

So at $r=d$, $\left|\frac{dB}{dr}\right|_{r=d} = \frac{\mu_0 I_w}{2\pi d^2}$

Hence, $F \propto m \cdot \frac{1}{d^2}$

Using $m \propto a^2$, $F \propto \frac{a^2}{d^2} = \left(\frac{a}{d}\right)^2$

4. Check options

• A: $F=0$
  False, because the magnetic field of the wire is non-uniform, so a dipole feels a net force.

• B: $F \propto \left(\frac{a}{d}\right)$
  False.

• C: $F \propto \left(\frac{a^2}{d^3}\right)$
  False.

• D: $F \propto \left(\frac{a}{d}\right)^2$
  Correct.

5. Final answer

$\boxed{F \propto \left(\frac{a}{d}\right)^2}$ So the correct option is D.