Question

Two coherent point sources $S_1$ and $S_2$ vibrating in phase emit light of wavelength $\lambda$. The separation between the sources is $2\lambda$. Consider a line passing through $S_2$ and perpendicular to the line $S_1{S}_2$. What is the smallest distance from $S_2$ where a minimum of intensity occurs?

Answer 1

Two point source $S_1$ and $S_2$ are shown in figure. Here $P$ is the point where we have to find path difference of source $S_1$ and $S_2$. For minimum intensity of light, path difference must be equal to path difference for dark e.g $\Delta x=\lambda /2,3\lambda /2,5\lambda /\mathrm{2.........}$

Consider $\Delta x=(2\lambda -\lambda /2)=3\lambda /2$

Now, path difference$=S_{1}P-S_{2}P$

$\Delta x=\sqrt{(4{\lambda }^2+x^2)}-x$

$3\lambda /2+x=\sqrt{4{\lambda }^2+x^2}$

Taking square on both sides,

$9{\lambda }^2/4+x^2+3\lambda x=4{\lambda }^2+x^2$

$3\lambda x=4{\lambda }^2-9{\lambda }^2/4$

$x=7\lambda /12$

The smallest distance from $S_2$ where a minimum of intensity occurs is $7\lambda /12$