Mutual Induction Questions in English

(D) The magnetic field $B$ produced by the square loop at its center is $B = \frac{\mu_0 I}{\pi} \frac{4 \sin(45^\circ)}{L} = \frac{2\sqrt{2}\mu_0 I}{\pi L}$.
Since the circular loop is very small $(L >> R)$,we assume the magnetic field $B$ is uniform over the area of the circular loop.
The magnetic flux $\phi$ linked with the circular loop of radius $R$ is $\phi = B \times A = B \times (\pi R^2)$.
Substituting the value of $B$,we get $\phi = \left( \frac{2\sqrt{2}\mu_0 I}{\pi L} \right) \times (\pi R^2) = \frac{2\sqrt{2}\mu_0 I R^2}{L}$.
The mutual inductance $M$ is defined as $M = \phi / I$.
Therefore,$M = \frac{2\sqrt{2}\mu_0 R^2}{L}$.