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(C) Consider two concentric coplanar circular loops. Let the outer loop have radius $R$ and the inner loop have radius $r$. When a current $I$ flows t

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

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(C) Consider two concentric coplanar circular loops. Let the outer loop have radius $R$ and the inner loop have radius $r$. When a current $I$ flows through the outer loop,the magnetic field $B$ at its center 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 loop. The magnetic flux $\phi$ linked with the smaller loop is $\phi = B \cdot A$,where $A = \pi r^2$ is the area of the smaller loop. Thus,$\phi = \left( \frac{\mu_0 I}{2R} \right) (\pi r^2) = \left( \frac{\mu_0 \pi r^2}{2R} \right) I$. By definition,the mutual inductance $M$ is given by $\phi = MI$. Comparing the two expressions,we get $M = \frac{\mu_0 \pi r^2}{2R}$. Therefore,the mutual inductance $M$ is proportional to $\frac{r^2}{R}$.

Two concentric coplanar circular conducting loops have radii $R$ and $r$ $(R \gg r)$. Their mutual inductance is proportional to

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