Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in the figure.

Explanation

Let us assume that an elementary strip of width dx a at distance x from the wire carring current I.
Side of square =a.
The magnetic field due to current carring wire at a distance x from the wire is `B=(mu_(0))/(4pi).(2I)/(x)....(i)`
Small amount of magnetic flux associated with the strip
`dphi=B.dA`
`=(mu_(0))/(4pi).(2I)/(x).a.dx`
`(therefore dA="area of strip=a.dx from eq (i)")`
`dphi=(mu_(0))/(4pi).(Ia)/(x).dx`
Integrate within proper limits, we get
`phi=(mu_(0))/(4pi).Ia underset(x)overset(x+a)int1/x.dx Rightarrow phi=(mu_(0))/(4pi).Ia[lnx]_(x)^(x+a)`
`phi=(mu_(0))/(4pi).Ia[ln(x+a)-In(x)]`
`phi=(mu_(0)Ia)/(2pi)ln ((x+a)/(x)) Rightarrow phi=(mu_(0)Ia)/(2pi)ln ((a)/(x)+1)....(ii)`
As we know that `phi=MI......(iii)`
where, M is mutual inductance, From equations (ii) and (iii) we get
`MI=(mu_(0)Ia)/(2pi)ln((a)/(x)+1) Rightarrow M=(mu_(0)a)/(2pilog_(e))((a)/(x)+1)`
This is mutual inductance between wire and square loop.