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(D) The magnetic field produced by the larger coil of radius '$R$' at its center is given by $B = \frac{\mu_0 I}{2R}$.Since the smaller coil of radius

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

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(D) The magnetic field produced by the larger coil of radius '$R$' at its center is given by $B = \frac{\mu_0 I}{2R}$.Since the smaller coil of radius '$r$' is placed at the center and $R \gg r$,we can assume the magnetic field is uniform over the area of the smaller coil.The magnetic flux $\phi$ passing through the smaller coil is $\phi = B 	imes A$,where $A = \pi r^2$ is the area of the smaller coil.$\phi = \left( \frac{\mu_0 I}{2R} ight) 	imes (\pi r^2) = \frac{\mu_0 \pi r^2 I}{2R}$.The mutual inductance $M$ is defined as $M = \frac{\phi}{I}$.Substituting the expression for $\phi$,we get $M = \frac{\mu_0 \pi r^2}{2R}$.Therefore,the mutual inductance $M$ is proportional to $\frac{r^2}{R}$.

$A$ coil of radius '$r$' is placed on another coil (whose radius is '$R$' 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

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