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

Explanation

Consider a small portion of the coil of thickness dt at a distance t from the current carrying wire as shown in fig.
Then the magnetic field strength experienced by this portion,
`B=(mu_(0)I)/(2pit)`
Magnetic flux linked with this portion,
`dphi=B.dA=(mu_(0)I)/(2pit).adt[dA=adt]`
`therefore` Magnetic flux linked with the coil,
`phi=int_(x)^(a+x)(mu_(0)Ia)/(2pit)dt or,MI=(mu_(0)Ia)/(2pi)ln((a+x)/(x))`
`therefore M=(mu_(0)a)/(2pi)ln((a+x)/(x))=(mu_(0)a)/(2pi)ln(1+(a)/(x))`