What is the mutual inductance of a two-loop system as shown with centre separation l
 

Magnetic field at the location of coil (2) produced due to coil (1)
$\left[B_1=\frac{{\mu }_o}{4\pi }·\frac{2M_{\text{prime }}}{l^3}\right.$
Flux linked with coil (2)
${\phi }_2=B_1{A}_2=\frac{{\mu }_o}{4\pi }\frac{2i\left(\pi a^2\right)}{l^3}\times \left(\pi a^2\right)$
$Also {\phi }_2=Mi\Rightarrow M=\frac{{\mu }_o\pi a^4}{2l^3}$