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(D) The magnetic field $B$ at the center of a large circular coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_0 I}{2R}$.Since $R \gg

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

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(D) The magnetic field $B$ at the center of a large circular coil of radius $R$ carrying current $I$ is given by $B = \frac{\mu_0 I}{2R}$.Since $R \gg r$, the magnetic field $B$ is approximately uniform over the area of the smaller coil of radius $r$.The magnetic flux $\phi$ linked with the smaller coil is $\phi = B \cdot A$, where $A = \pi r^2$ is the area of the smaller coil.Therefore, $\phi = \left( \frac{\mu_0 I}{2R} \right) (\pi r^2) = \left( \frac{\mu_0 \pi r^2}{2R} \right) I$.The mutual inductance $M$ is defined by the relation $\phi = MI$, so $M = \frac{\mu_0 \pi r^2}{2R}$.Thus, $M \propto \frac{r^2}{R}$.

$A$ circular coil of radius $r$ is placed on another circular coil whose radius is $R$. The current flowing through the larger coil is changing, and their centers coincide. Given $R \gg r$, if both coils are coplanar, then the mutual inductance between them is proportional to:

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