The mutual inductance of the system of long straight conductor and the rectangular loop is
μ02πalna+xa
μ02πblnx+ax
μ02πblnx+aa
μ02πalna+xx
Let i be the current in the straight wire. Then magnetic flux linked through the loop is ϕ=∫xx+aμ0i2πr×b×drMutual inductance M=ϕi=μ0b2πlna+xx