Two single-turn circular loops of wire have radii R and r, with R >> r. The loops lie in the same plane and are concentric. The mutual inductance of the pair is (approximately)

A current i in the large loop of radius .R produces a magnetic field of magnitude $B=\frac{{\mu }_{0}i}{2R}$ at its center. Since the radius 2R of the small loop r <.< R, we may treat the flux through the small loop as being approximately ${\mathrm{\Phi }}_B=BA\cos 0^{\circ }=\frac{{\mu }_{0}i}{2R}A=\frac{{\mu }_0\pi r^{2}i}{2R}$. The mutual inductance of the loops is then

$M=\frac{{\mathrm{\Phi }}_B}{i}=\frac{{\mu }_0\pi r^2}{2R}$