A rectangular loop of $N$ close-packed turns is positioned near a long straight wire as shown in figure. What is the mutual inductance $M$ for the loop-wire combination ?

Magnetic field due to current carrying wire at a distance, $x$ from wire

       $B=\frac{{\mu }_0}{2\pi }\frac{i}{x}$

The flux through the area $\left(ldx\right)$ ,

     $d\varphi =NB\left(ldx\right)$

$\therefore$ $\varphi ={\int }_a^{b}NBldx$

          $$\begin{gathered} =Nl\frac{{\mu }_0}{2\pi }{\int }_a^b\frac{i}{x}dx \\ =\frac{{\mu }_{0}Nli}{2\pi }{{\left|\ln x\right|}_a}^{\left(a+b\right)} \\ =\frac{{\mu }_{0}Nli}{2\pi }\ln \left(\frac{a+b}{a}\right) \end{gathered}$$

On comparing with, ${\varphi }_{total}=Mi$ , we get

$M=\frac{{\mu }_{0}Nl}{2\pi }\ln \left(1+\frac{b}{a}\right)$