Question

Two ideal slits S₁ and S₂ are at a distance d apart, and illuminated by light of wavelength l passing through an ideal source slit S placed on the line through S₂ as shown. The distance between the planes of slits and the source slit is D. A screen is held at a distance D from the plane of the slits. The minimum value of d for which there is darkness at O is -

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

Answer 1

Answer: (C)

$\sqrt{\frac{\lambda D}{2}}$

Answer 2

length of path by SS₁O = SS₁ + S₁O

=$\sqrt{D^{2} + d^{2}}$ $+ \sqrt{D^{2} + d^{2}}$

= 2 $\sqrt{D^{2} + d^{2}}$

length of path by SS₂O = 2D ,

D = path diff. = 2$\sqrt{D^{2} + d^{2}}$– 2D

= 2D $\left\lbrack \left( 1 + \frac{d^{2}}{D^{2}} \right)^{1/2} - 1 \right\rbrack$

= 2D $\left\lbrack 1 + \frac{1}{2}\frac{d^{2}}{D^{2}} - 1 \right\rbrack = \frac{d^{2}}{D}$by

Binomial Ex. For darkness at O location, path difference

should be = (2n – 1) $\frac{\lambda}{2}$.

For minimum n = 1, $\frac{\lambda}{2} = \frac{d^{2}}{D} \Rightarrow d = \sqrt{\frac{D\lambda}{2}}$