Question

In the figure below, $P$ and $Q$ are two equally intense coherent sources emitting radiation of wavelength $20m$. The separation between $P$ and $Q$ is $5m$ 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 ($A$)

$x_p-x_Q=(d+2.5)-(d-2.5)=5m$
$\Delta \varphi$ 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 $90^{\circ }$ )
$=0$
$I_A=I_1+I_2+2{\sqrt{I}}_1{\sqrt{I}}_2\cos \Delta \varphi$
$=I+I+2\sqrt{I}\sqrt{I}\cos (0)$
$=4I$
For C
$x_Q-x_p=5m$
$\Delta \varphi$ due to path difference $=\frac{2\pi }{\lambda }(\Delta x)$
$=\frac{2\pi }{20}(5)=\frac{\pi }{2}$
Total phase difference at $C=\frac{\pi }{2}+\frac{\pi }{2}=\pi$
$I_{net}=I_1+I_2+2\sqrt{I_1}\sqrt{I_2}\cos (\Delta \varphi )$
$=I+I+2\sqrt{I}\sqrt{I}\cos (\pi )=0$
For $B$
$x_p-x_Q=0$
$\Delta \varphi =\frac{\pi }{2}$ (Due to $P$ being ahead of $Q$ by $90^{\circ })$
$I_B=I+I+2\sqrt{I}\sqrt{I}\cos \frac{\pi }{2}=2I$
$I_A:I_B:I_C=4I:2I:0$
$=2:1:0$