A circular loop of radius $$R$$ is carrying current $$i$$ A. The ratio of magnetic field at the centre of circular loop and at a distance $$R$$ from the center of the loop on its axis is :

We need to find the ratio of the magnetic field at the centre of a circular loop to the field at a point on the axis.

The magnetic field at the centre is given by $$B_{\text{centre}} = \frac{\mu_0 i}{2R}$$, and the field on the axis at a distance $$x$$ from the centre is $$B_{\text{axis}} = \frac{\mu_0 i R^2}{2(R^2 + x^2)^{3/2}}$$.

The ratio is $$\frac{B_{\text{centre}}}{B_{\text{axis}}} = \frac{(R^2 + x^2)^{3/2}}{R^3}$$, and at $$x = R$$ this becomes $$\frac{(2R^2)^{3/2}}{R^3} = \frac{2\sqrt{2}R^3}{R^3} = 2\sqrt{2}$$, so the ratio is $$2\sqrt{2}:1$$.

This matches Option C: $$2\sqrt{2}:1$$.