A long wire is bent into a circular coil of one turn and then into a circular coil of smaller radius having \(\mathrm{n}\) turns. If the same current passes in both the cases, the ratio of magnetic fields produced at the centre for one turn to that of \(n\) turns is

Magnetic field at the centre of coil is,
\(\mathrm{B}_1=\frac{\mu_0 \mathrm{I}}{2 \mathrm{r}_1}\)
\(\therefore \quad\) Magnetic field of \(\mathrm{n}\) turns coil at the centre:
\(\begin{aligned}
& \mathrm{B}_2=\frac{\mu_0 \mathrm{nI}}{2 \mathrm{r}_2} \\
\therefore \quad \frac{\mathrm{B}_1}{\mathrm{~B}_2} & =\frac{\frac{\mu_0 \mathrm{I}}{2 \mathrm{r}_1}}{\frac{\mu_0 \mathrm{nI}}{2 \mathrm{r}_2}}=\frac{\mathrm{r}_2}{\mathrm{nr}_1}
\end{aligned}\)
But, radius: \(r_1=\frac{1}{2 \pi}\) and \(r_2=\frac{1}{2 \pi n}\)
\(\begin{array}{ll}
\therefore & \frac{\mathrm{r}_2}{\mathrm{r}_1}=\frac{1}{\mathrm{n}} \\
\therefore & \frac{\mathrm{B}_1}{\mathrm{~B}_2}=\frac{1}{\mathrm{n}^2}
\end{array}\)