Find the mutual inductance in the arrangement, when a small...

The magnetic field due to the rectangular loop at its centre is:

$B = 4 \times \dfrac{{\mu}_{0}i}{4\pi\left(\dfrac{L}{2}\right)}\left[\dfrac{1}{\sqrt{2}} + \dfrac{1}{\sqrt{2}}\right] = \dfrac{2\sqrt{2}{\mu}_{0}i}{\piL}$

Therefore, the flux through the circular loop $ϕ = \pi{R}^{2} \times B = \dfrac{2\sqrt{2}{\mu}_{0}{R}^{2}i}{L}$

If M is the mutual inductance, then

$M\dfrac{d i}{d t} = \dfrac{d ϕ}{d t} = \dfrac{2\sqrt{2}{\mu}_{0}{R}^{2}}{L}\dfrac{di}{dt}$

$⇒ M = \dfrac{2\sqrt{2}{\mu}_{0}{R}^{2}}{L}$