Question

Two coherent sources $S_1$ and $S_2$ vibrating in phase emit light of wavelength $\lambda$. The separation between them is $2\lambda$ as shown in figure. The first bright fringe is formed at $P$ due to interference on a screen placed at a distance $D$ from $S_1(D>>\lambda ),$ then $OP$ is:

• A. $\sqrt{2}D$
• B. $1.5D$
• C. $\sqrt{3}D$
• D. $D$

Answer 1

Answer: (C)

$\sqrt{3}D$

$(S_{1}P{)}^2=D^2+x^2$

$(S_{2}P{)}^2=(D-2\lambda {)}^2+x^2$

$(S_{1}P{)}^2-(S_{2}P{)}^2=D^2+x^2-(D^2+4{\lambda }^2-4D\lambda +x^2)$

$=4D\lambda -4{\lambda }^2$

$\Rightarrow 2S_{1}P(S_{1}P-S_{2}P)=4D\lambda$

$\Rightarrow 2S_{1}P\left(\lambda \right)=4D\lambda$

$\Rightarrow S_{1}P=2D$

$\Rightarrow \sqrt{D^2+x^2}=2D$

$\Rightarrow D^2+x^2=4D^2$

$\Rightarrow x^2=3D^2$

$\Rightarrow x=\sqrt{3}D$