A coil of radius ' $r$ ' is placed on another coil (whose radius is ' $\mathrm{R}$ ' and current through it…
A coil of radius ' $r$ ' is placed on another coil (whose radius is ' $\mathrm{R}$ ' and current through it is changing) so that their centers coincide. $(R \gg r)$. If both are coplanar, then the mutual inductance between them is proportional to
$\frac{R}{r^2}$
$\frac{r}{R}$
$\frac{\mathrm{R}}{\mathrm{r}}$
$\frac{r^2}{\mathrm{R}}$
Solution
Magnetic field at the center, $B=\frac{\mu_0 I}{R}$
$\phi=$ Magnetic flux passing through the smaller coil $=\pi \mathrm{r}^2 \mathrm{~B}$
$\begin{aligned}
& \phi=\pi r^2 \times \frac{\mu_0 I}{R} \\
& M=\frac{\phi}{I}=\frac{\mu_0 \pi r^2}{R} \\
& M \propto \frac{r^2}{R}
\end{aligned}$