Two coherent sources, S₁ and S₂ lie along the y-axis and a detector moves along +x-axis. The sources are separated by a distance four times the wavelength λ of the source. Find the number of points where maxima are observed, leaving the origin and far-off points.

In $\mathrm{\Delta }{S}_1{S}_{2}D$, from pythagoras theorem we can write

 $\begin{array}{l}{\left(S_{1}D\right)}^2={\left(S_1{S}_2\right)}^2+{\left(S_{1}D\right)}^2\\ {\left(S_{1}P+PD\right)}^2={\left(S_1{S}_2\right)}^2\\ +{\left(S_{2}D\right)}^2\end{array}$

Here S₁ P is the path difference at point D from the two sources. For maximum intensity at point D it should be = nλ. Let

$PD=S_{2}D=x_n$. Then,

${\left(n\lambda +x_n\right)}^2=(4\lambda {)}^2+{\left(x_n\right)}^2$

or $x_n=\frac{16{\lambda }^2-n^2{\lambda }^2}{2n\lambda }$

Then $x_1=\frac{16{\lambda }^2-{\lambda }^2}{2\lambda }=7.5\lambda$

$\begin{array}{l}{x}_2=\frac{16{\lambda }^2-4{\lambda }^2}{4\lambda }=3\lambda \\ x_3=\frac{16{\lambda }^2-9{\lambda }^2}{6\lambda }=\frac{7}{6}\lambda \\ x_4=0\end{array}$

Thus number of points for maxima are 3.