Question

In the arrangement shown in figure, slits $S_1$ 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 $S_3{S}_4$.


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

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

Answer 1

The correct option is D $\frac{\lambda D}{d}$

For the given condition,
phase difference $cos\theta =-1or\theta =\pi$
$k\Delta x=\pi$ , (where, $k=\frac{2\pi }{\lambda }$)
so,$\Delta x\frac{2\pi }{\lambda }=\pi \Delta x=\frac{\lambda }{2}$
here, $\Delta x=$ total path difference= $\frac{yd}{D}$
then,$\frac{yd}{D}=\frac{\lambda }{2}y=\frac{D\lambda }{2d}$
from the digram, $Z=2y$


so, $\frac{Z}{2}=\frac{D\lambda }{2d}Z=\frac{\lambda D}{d}$