The magnitude of magnetic field at the centre of loop is

$B_{\text{1 }}=B_{\text{centre }}=\frac{{\mu }_{\circ }i}{2R} \dots \left(1\right)$

The magnitude of magnetic field at a point at a distance $x$ from the centre along axis of loop is

$B_{\text{axis }}=\frac{{\mu }_{\circ }i{R}^2}{2{\left(R^2+x^2\right)}^{{\displaystyle \frac{3}{2}}}}$

$B_{\text{2 }}=\frac{{\mu }_{\circ }i{R}^2}{2\left(2^{\frac{3}{2}}\right)R^3} \dots \left(2\right)$      $(\text{since} x=R)$

Dividing equation 1 by equation 2, we get

$\frac{B_1}{B_2}=2\sqrt{2}:1$