Question

In the figure below, $P$ and $Q$ are two equally intense coherent sources emitting radiation of wavelength $20 m$. The separation between $P$ and $Q$ is $5 m$ and the phase of $P$ is ahead of that of $Q$ by $90^{\circ} .$ A, $B$ and $C$ are three distinct points of observation, each equidistant from the midpoint of $PQ$. The intensities of radiation at $A, B, C$ will be in the ratio:

• A. 0 : 1 : 2
• B. 4 : 1 : 0
• C. 0 : 1 : 4
• D. 2 : 1 : 0

Answer 1

Answer: (D)

For ($A$)

$x_{p}-x_{Q}=(d+2.5)-(d-2.5)=5 m$
$\Delta \phi$ due to path difference $=\frac{2 \pi}{\lambda}(\Delta x )=\frac{2 \pi}{20}(5)$
$=\frac{\pi}{2}$
At $A, Q$ is ahead of $P$ by path, as wave emitted by $Q$ reaches before wave cmitted by $P$. Total phase difference at A
$=\frac{\pi}{2}-\frac{\pi}{2}$ (due to $P$ being ahead of $Q$ by $90^{\circ}$ )
$=0$
$I _{ A }= I _{1}+ I _{2}+2 \sqrt{ I }_{1} \sqrt{ I }_{2} \cos \Delta \phi$
$= I + I +2 \sqrt{ I } \sqrt{ I } \cos (0)$
$=4 I$
For C
$x_{Q}-x_{p}=5 m$
$\Delta \phi$ due to path difference $=\frac{2 \pi}{\lambda}(\Delta x )$
$=\frac{2 \pi}{20}(5)=\frac{\pi}{2}$
Total phase difference at $C =\frac{\pi}{2}+\frac{\pi}{2}=\pi$
$I _{ net }= I _{1}+ I _{2}+2 \sqrt{ I _{1}} \sqrt{ I _{2}} \cos (\Delta \phi)$
$= I + I +2 \sqrt{ I } \sqrt{ I } \cos (\pi)=0$
For $B$
$x_{p}-x_{Q}=0$
$\Delta \phi=\frac{\pi}{2}$ (Due to $P$ being ahead of $Q$ by $\left.90^{\circ}\right)$
$I _{ B }= I + I +2 \sqrt{ I } \sqrt{ I } \cos \frac{\pi}{2}=2 I$
$I _{ A }: I _{ B }: I _{ C }=4 I : 2 I : 0$
$=2: 1: 0$