Question

In the arrangement shown in the figure, slits $S_3$ and $S_4$ are having a variable separation $Z$ . Point $O$ on the screen is at the common perpendicular bisector of $S_1{S}_2$ and $S_3{S}_4$ . (Assume $D>>d$ )



The minimum value of $Z$ for which the intensity at $O$ is zero is

• A. $\frac{3\lambda D}{2d}$
• B. $\frac{\lambda D}{2d}$
• C. $\frac{\lambda D}{3d}$
• D. $\frac{\lambda D}{d}$

Answer 1

Answer: (D)

Intensity at $O$ is due to $S_3$ and $S_4$ .
$\therefore$ Intensity at $S_3$ = zero and Intensity at $S_3$ is given by
$I=K{\cos }^2\left(\frac{\varphi }{2}\right)$
Intensity at $O$ is zero if $\frac{\varphi }{2}=\frac{\pi }{2}$
$\varphi =\frac{2\pi }{\beta }\times \frac{Z}{2}=\pi \Rightarrow Z=\beta$ ;
$Z=\beta =\frac{\lambda D}{d}$
$\varphi =$ phase difference between waves at $S_3$ and $S_4$