$B_{c}=\frac{\mu_{0} I}{2 r}.........(i)$

Magnetic field at axial point due to a current-carrying coil at distance of $r$

$\mathrm{d}=r$

$\mathrm{B}=\frac{\mu_{0} \operatorname{Ir}^{2}}{2\left(\mathrm{r}^{2}+\mathrm{d}^{2}\right)^{3 / 2}}$

$\Rightarrow \mathrm{B}_{\mathrm{a}}=\frac{\mathrm{\mu}_{0} \mathrm{Ir}^{2}}{2\left(2 \mathrm{r}^{2}\right)^{3 / 2}}.........(ii)$

Now, $\frac{\mathrm{B}_{\mathrm{c}}}{\mathrm{B}_{\mathrm{a}}}=\frac{\mu_{0} \mathrm{I}}{2 \mathrm{r}} \times \frac{2\left(2 \mathrm{r}^{2}\right)^{3 / 2}}{\mu_{0} \mathrm{Ir}^{2}}=2 \sqrt{2}$

$\mathrm{B}_{\mathrm{c}}: \mathrm{B}_{\mathrm{a}}=2 \sqrt{2}: 1$