= Solution
For a circular <current loop>, the <Biot-Savart law> is
$$
d\mathbf B=\frac{\mu_0I}{4\pi}\frac{d\boldsymbol\ell\times\mathbf s}{s^3},
$$
where $\mathbf s$ points from a <current element> to the field point. Choose the loop in the $yz$ plane, with observation point $x\widehat{\mathbf x}$. A parametrization consistent with positive axial <magnetic field> is
$$
\mathbf r'(\varphi)=(0,a\cos\varphi,a\sin\varphi),\quad
d\boldsymbol\ell=(0,-a\sin\varphi,a\cos\varphi)d\varphi,\quad
\mathbf s=(x,-a\cos\varphi,-a\sin\varphi).
$$
Then $s=\sqrt{a^2+x^2}$ and
$$
d\boldsymbol\ell\times\mathbf s
=(a^2,ax\cos\varphi,ax\sin\varphi)d\varphi.
$$
The transverse contributions cancel around the <circle>, leaving the <on-axis magnetic field of a circular current loop>
$$
\boxed{\mathbf B(x)=\frac{\mu_0Ia^2}{2(a^2+x^2)^{3/2}}\widehat{\mathbf x}.}
$$
The <right-hand rule> fixes its direction; reversing the <electric current> reverses the <magnetic field>. At the centre this becomes $\mu_0I/(2a)$, and far away it decays as $|x|^{-3}$, as expected for a <magnetic dipole>.
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