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(D) The magnetic field $B$ at the center of a large coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_{0} I}{2R}$.Since $R \gg r$,we

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$\frac{r^{2}}{R}$

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(D) The magnetic field $B$ at the center of a large coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_{0} I}{2R}$.Since $R \gg r$,we can assume the magnetic field is uniform over the area of the smaller coil.The magnetic flux $\phi$ passing through the smaller coil of radius $r$ is $\phi = B \cdot A = B \cdot (\pi r^{2})$.Substituting the value of $B$,we get $\phi = \left( \frac{\mu_{0} I}{2R} \right) \cdot (\pi r^{2})$.The mutual inductance $M$ is defined as $M = \frac{\phi}{I}$.Therefore,$M = \frac{\mu_{0} \pi r^{2}}{2R}$.Since $\mu_{0}$,$\pi$,and $2$ are constants,we have $M \propto \frac{r^{2}}{R}$.

$A$ coil of radius $r$ is placed on another coil (whose radius is $R$ and current flowing through it is changing) so that their centres coincide. $(R \gg r)$ If both the coils are coplanar,then the mutual inductance between them is proportional to

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