The approximate formula expressing the formula of mutual inductance of two thin coaxial loops of the same radius a when their centers are separated by a distance l with l >> a is

Let .I = current in one loop. The magnetic field at the centre of the other co-axial loop at a distance / from the centre of the first
loop is

$B=\frac{{\mu }_0}{4\pi }\frac{2I\pi a^2}{{\left(a^2+l^2\right)}^{3/2}}$

$where p_m=I \pi a^2=magnetic moment of the loop$

Flux through the other loop is

${\varphi }_{12}=B\pi a^2$

$=\frac{{\mu }_0}{4\pi }\frac{2\pi a^{2}I}{{\left(a^2+l^2\right)}^{3/2}}\pi a^2$

$or M=\frac{{\varphi }_{12}}{I}=\frac{{\mu }_0}{4\pi }\frac{2{\pi }^2{a}^2}{{\left(a^2+l^2\right)}^{3/2}}$

$=\frac{{\mu }_0\pi a^4}{2{\left(a^2+l^2\right)}^{3/2}}=\frac{{\mu }_0\pi a^4}{2l^3}(a<<l)$