 Solution

Choose the loop axis as $+x$, with positive electric current circulating by the right-hand rule. The Biot-Savart law isFor a loop element and an axial observation point, $r=\sqrt{a^2+x^2}$. The transverse components cancel between opposite elements. The axial component of $d\mathbf{\ell }\times \mathbf{r}$ is $ad\ell$, soThusReversing the electric current reverses the field. At the centre $B={\mu }_{0}I/(2a)$, while the far field decreases as $|x{|}^{-3}$, as expected for a magnetic dipole.