The magnetic field at the centre of a circular coil of radius r ,...

Magnetic field at the centre of the circle is, $B = \dfrac{{\mu}_{0}i}{2R}$

Magnetic field at the point on the axis is,

${B}_{2} = \dfrac{{\mu}_{0}i{R}^{2}}{2{\left({R}^{2} + {x}^{2}\right)}^{\dfrac{3}{2}}} ⇒ {B}_{2} = \dfrac{{\mu}_{0}i{R}^{2}}{2{\left({R}^{2} + {\left(\dfrac{R}{2}\right)}^{2}\right)}^{\dfrac{3}{2}}} ⇒ {B}_{2} = \dfrac{{\mu}_{0}i}{2{\left(\dfrac{5}{4}\right)}^{\dfrac{3}{2}}R} = {\left(\dfrac{2}{\sqrt{5}}\right)}^{3}B$