Figure shows a rectangular coil near a long wire. Calculate the mutual inductance of the combination.

Let the current $i_1$ in the straight wire be upward. The magnetic field due to the straight wire has magnitude $B_1={\mu }_0{i}_1/2\pi r$ at a distance $r$. In associate with right hand rule $B_1$ points inward to the plane of page. We consider a differential strip of thickness $dr,$ area $d{A}_2=adr$. Magnetic flux through area $dA,d{\varphi }_B=B_1\left(adr\right)$.

Total flux through the loop, 

${{\varphi }_B}_{12}=\int B_{1}d{A}_2={\int }_c^{c+b}\frac{n_0{i}_1}{2\pi r}adr$

        $=\frac{{\mu }_0{i}_{1}a}{2\pi }{\int }_c^{c+b}\frac{dr}{r}=\frac{{\mu }_0{i}_{1}a}{2\pi }\ln \left(\frac{c+b}{c}\right)$

Therefore the mutual inductance,

$M=M_{12}=\frac{{\varphi }_{12}}{i_1}=\frac{{\mu }_{0}a}{2\pi }\ln \left(1+\frac{b}{c}\right)$ Ans.