Question

Two coherent sources of light of wavelength $\lambda$ in the same phase are placed parallel to each other at a small separation of $2\lambda$. The path difference at point $\text{P}$ is

• A. $\frac{\lambda D}{\sqrt{x^2+D^2}}$
• B. $\frac{2\lambda D}{\sqrt{x^2+D^2}}$
• C. $3\lambda$
• D. $4\lambda$

Answer 1

The correct option is B $\frac{2\lambda D}{\sqrt{x^2+D^2}}$


Let $\text{O}$ be the center of $S_1$ and $S_2$. We have drawn a perpendicular $S_{2}Q$ at $\text{Q}$ on $S_{1}P$.

We can see that path difference of waves from $S_1$ and $S_2$ reaching at $\text{P}$ is $\Delta x=S_{1}Q$.

As $D>>\lambda$,

$\angle PO{S}_2=\angle P{S}_{2}A=\angle A{S}_{1}P=\theta$

In $\Delta A{S}_{1}P$

So, $\cos \theta =\frac{D}{\sqrt{x^2+D^2}}$

So, the path difference at point $\text{P}$ is

$\Delta x=S_{1}Q=S_1{S}_2\cos \theta =\frac{2\lambda D}{\sqrt{x^2+D^2}}$

Hence, option $\left(\text{B}\right)$ is correct.