 Solution

For a circular coil, use the Biot-Savart lawAt an axial position $x$, every wire element is at distance $r=\sqrt{a^2+x^2}$. The transverse magnetic field contributions cancel by rotational symmetry. The axial part of $d\mathbf{\ell }\times \mathbf{r}$ is $ad\ell$, and $\oint d\ell =2\pi a$. Multiplying by the $N$ turns givesThe electric current's sense fixes the axial direction by the right-hand rule.