Question

Two coherent narrow slits emitting light of wavelength $\lambda$ in the same phase are placed parallel to each other at a small separation of $2\lambda$. The light is collected on a screen which is placed at a distance $D(>>\lambda )$ from the slit $S_1$ as shown in figure.
Find $x$ such that the path difference between the waves from the two sources is $\lambda$ at point $P$.

• A. $0.2D$
• B. $\sqrt{2}D$
• C. $\sqrt{3}D$
• D. $0.5D$

Answer 1

The correct option is C $\sqrt{3}D$

According to the question :
$S_{1}P-S_{2}P=\lambda$
or, $d\cos \theta =\lambda$
Where, $d=S_1{S}_2=2\lambda$
$\Rightarrow \left(2\lambda \right)\cos \theta =\lambda$
Or, $\cos \theta =\frac{1}{2}$
$\therefore \theta ={60}^{\circ }$

Now in $\Delta S_{1}PO$,

$\tan \theta =\frac{PO}{S_{1}O}$
or, $\tan {60}^{\circ }=\sqrt{3}=\frac{x}{D}$
$\therefore x=\sqrt{3}D$