A circular current carrying coil has radius \(R\). The magnetic induction at the centre of the coil is \(\mathrm{B}_{\mathrm{C}}\). The magnetic induction of the coil at a distance \(\sqrt{3} R\) from the centre along the axis is \(\mathrm{B}_{\mathrm{A}}\). The ratio \(\mathrm{B}_{\mathrm{A}}: \mathrm{B}_{\mathrm{C}}\) is

Magnetic Induction of Current carrying coil at its centre:
\(\mathrm{B}_{\mathrm{c}}=\frac{\mu_{\mathrm{o}} \mathrm{I}}{2 \mathrm{R}}\)
Magnetic Induction of Current carrying coil at distance \(\mathrm{r}\) :
\(\mathrm{B}_{\mathrm{A}}=\frac{\mu_0 \mathrm{IR}^2}{2\left(\mathrm{R}^2+\mathrm{r}^2\right)^{3 / 2}}\)
Given \(r=\sqrt{3} R\)
\(\therefore \quad B_A=\frac{\mu_0 I^2}{2\left(R^2+(\sqrt{3} R)^2\right)^{3 / 2}}\)
\(\therefore \quad\) Ratio of \(\mathrm{B}_{\mathrm{A}}\) to \(\mathrm{B}_{\mathrm{C}}\) is
\(\frac{\mathrm{B}_{\mathrm{A}}}{\mathrm{B}_{\mathrm{C}}}=\frac{\frac{\mu_0 \mathrm{IR}^2}{2\left(\mathrm{R}^2+(\sqrt{3} \mathrm{R})^2\right)^{3 / 2}}}{\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}}=\frac{\mu_0 \mathrm{IR}^2}{2\left(4 \mathrm{R}^2\right)^{3 / 2}} \frac{2 \mathrm{R}}{\mu_0 \mathrm{I}}\)
\(\frac{\mathrm{B}_{\mathrm{A}}}{\mathrm{B}_{\mathrm{C}}}=\frac{1}{8}\)