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(A) The magnetic field $B$ at the center of a large circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.Since the sq

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$l^2/r$

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(A) The magnetic field $B$ at the center of a large circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.Since the square loop is small and placed at the center,we can assume the magnetic field is uniform over the area of the square loop.The magnetic flux $\phi$ linked with the square loop of side $l$ is $\phi = B \cdot A$,where $A = l^2$ is the area of the square loop.Thus,$\phi = \left( \frac{\mu_0 i}{2r} \right) l^2$.The mutual inductance $M$ is defined as $M = \frac{\phi}{i}$.Substituting the expression for $\phi$,we get $M = \frac{(\mu_0 i / 2r) l^2}{i} = \frac{\mu_0 l^2}{2r}$.Therefore,the mutual inductance $M$ is proportional to $l^2/r$.

$A$ small square loop of wire of side $l$ is placed inside a large circular loop of radius $r$. The loops are coplanar and their centers coincide. The mutual inductance of the system is proportional to

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