(a) Definition of mutual inductance and S.I. unit
(b) Obtaining the expression for resultant force on the loop
(a) Mutual inductance equals the magnetic flux associated with a coil when unit current flows in its neighbouring coil.
Alternatively,
Mutual inductance equals the induced emf in a coil when the rate of change of current in its neighbouring coil is one ampere/ second.
S.I unit: henry $\left(\mathrm{H}\right)$ or weber/ampere (or any other correct SI unit)
(b) Force per unit length between two parallel straight conductors
$\mathit{F}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{2{\mathit{I}}_1{\mathit{I}}_2}{\mathit{d}}$
Force on the part of the loop which is parallel to infinite straight wire and at a distance $\mathit{x}$ from it.
${\mathit{F}}_1=\frac{{\mathit{\mu }}_0}{2\mathit{\pi }}\frac{{\mathit{I}}_1{\mathit{I}}_2\mathit{a}}{\mathit{x}}$
(away from the infinite straight wire)
Force on the part of the loop which is at a distance $\left(\mathit{x}+\mathit{a}\right)$ from it
${\mathit{F}}_2=\frac{{\mathit{\mu }}_0}{2\mathit{\pi }}\frac{{\mathit{I}}_1{\mathit{I}}_2\mathit{a}}{\left(\mathit{x}+\mathit{a}\right)}$
(towards the infinite straight wire)
Net force $\mathit{F}={\mathit{F}}_1-{\mathit{F}}_2$
$\mathit{F}=\frac{{\mathit{\mu }}_0}{2\mathit{\pi }}{\mathit{I}}_1{\mathit{I}}_2\mathit{a}\left[\frac{1}{\mathit{x}}-\frac{1}{\mathit{x}+\mathit{a}}\right]$
$\mathit{F}=\frac{{\mathit{\mu }}_0}{2\mathit{\pi }}\frac{{\mathit{I}}_1{\mathit{I}}_2{\mathit{a}}^2}{\mathit{x}\left(\mathit{x}+\mathit{a}\right)}$
(away from the infinite straight wire)