The ratio of the magnetic field at the center of a current-carrying coil of radius R and at a distance of 3R on its axis is

Magnetic field at the centre due to a circular loop is, $B_{\text{centre }}=\frac{{\mu }_{0}I}{2R}$

Magnetic field on the axis of a circular loop is, $B_{\text{axis }}=\frac{{\mu }_{o}I{R}^2}{2{\left(x^2+R^2\right)}^{\frac{3}{2}}}$

$\text{ For, }x=3R$

$B_{\text{axis }}=\frac{{\mu }_{o}I{R}^2}{2{\left(9R^2+R^2\right)}^{\frac{3}{2}}}=\frac{{\mu }_{o}I{R}^2}{2{\left(10R^2\right)}^{\frac{3}{2}}}$

$\Rightarrow B_{\text{axis }}=\frac{{\mu }_{o}l}{20\sqrt{10}R}$

Ratio, $\frac{B_{\text{centre }}}{B_{\text{axis }}}=\frac{\frac{{\mu }_{0}I}{2R}}{\frac{{\mu }_{0}I}{20\sqrt{10}R}}=10\sqrt{10}$