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 B1=μ0i/2πr 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.dϕB=B1a drTotal flux through the loop,ϕB=∫B1adr=∫cc+bμ0i12πra dr=μ0i1a2π∫cc+bdrr=μ0i1a2πln c+bb Therefore mutual inductance,M=ϕi1=μ0a2πln 1+bc