Reflected path \(\mathrm{SP}=2 l\) sec \(30^{\circ}\)

So path difference is \(2 l\left(\sec 30^{\circ}-1\right)\)

Also the ray, when reflected by the mirror, suffers a phase change of \(\pi\)

So the total difference in phase is \(2 l\left(\sec 30^{\circ}-1\right) \times \frac{2 \pi}{\lambda}+\pi\)

For constructive interference

\(2 l\left(\sec 30^{\circ}-1\right) \times \frac{2 \pi}{\lambda}+\pi=2 n \pi\)

Solving this, we get \(l=\frac{(2 n-1) \lambda \sqrt{3}}{4(2-\sqrt{3})}\)