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

Let current i, in the straight wire be upward. Then the magnetic field due to the straight wire has magnitude $B_1={\mu }_{0}i/2\pi r$ at a distance r. In accordance with right hand rule, B₁ points inward to the plane of page. We consider a differential strip of thickness dr, area dA₂ = a dr. Magnetic flux through area dA.

$d{\varphi }_B=B_1\left(a dr\right)$

Total flux through the loop,

${\varphi }_B=\int B_{1}adr={\int }_c^{c+b}\frac{{\mu }_0{i}_1}{2\pi r}a dr$

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

Therefore mutual inductance,

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