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:

A circular loop of radius R carries a current i. The magnetic field at the centre of the loop is given by $$B_{\text{centre}} = \frac{\mu_0 i}{2R}$$. At a point on the axis of the loop at a distance $$x$$ from the centre, the field has the form $$B_{\text{axis}} = \frac{\mu_0 i R^2}{2(R^2 + x^2)^{3/2}}$$.

When this point is located at a distance equal to the loop radius (that is, when $$x = R$$), the expression simplifies as follows:
$$B_{\text{axis}} = \frac{\mu_0 i R^2}{2(R^2 + R^2)^{3/2}} = \frac{\mu_0 i R^2}{2(2R^2)^{3/2}} = \frac{\mu_0 i R^2}{2 \cdot 2\sqrt{2} \cdot R^3} = \frac{\mu_0 i}{4\sqrt{2}R}$$.

The ratio of the field at the centre to the field on the axis at $$x = R$$ is therefore
$$\frac{B_{\text{centre}}}{B_{\text{axis}}} = \frac{\mu_0 i/(2R)}{\mu_0 i/(4\sqrt{2}R)} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}\,. $$ This gives a final result of $$2\sqrt{2} : 1$$.