Question 122: Wave Optics

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
o
. 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 :
![](https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1liynk1gs/fbcf27b4-50d2-4dc7-af71-405c55b5d8fc/4824eac0-0c50-11ee-ac20-219512b21619/file-6y3zli1liynk1ja.png?format=png)

Accepted Answer

Let the distace of each point from midpoint = d
For point A :
x
P
Ã¢ÂÂ x
Q
= (d + 2.5) Ã¢ÂÂ (d Ã¢ÂÂ 2.5) = 5 m
$\Delta $$\phi $ due to path difference = ${{2\pi } \over \lambda }\left( {\Delta x} ight)$ = ${{2\pi } \over {20}}\left( 5 ight)$ = ${\pi  \over 2}$
At A, Q is ahead of P by path, as wave emitted by Q reaches before wave emitted by P.
Total phase difference at A
= ${\pi  \over 2} - {\pi  \over 2}$ (due to P being ahead of Q by 90ÃÂº) = 0
$ 	herefore $ I
A
= I
1
+ I
2
+ $2\sqrt {{I_1}} \sqrt {{I_2}} $ cos $\Delta $$\phi $
= I + I + $2\sqrt I \sqrt I \cos \left( 0 ight)$
= 4I
For point C :
x
Q
Ã¢ÂÂ x
P
= (d + 2.5) Ã¢ÂÂ (d Ã¢ÂÂ 2.5) = 5 m
$\Delta $$\phi $ due to path difference = ${{2\pi } \over \lambda }\left( {\Delta x} ight)$ = ${{2\pi } \over {20}}\left( 5 ight)$ = ${\pi  \over 2}$
Total phase difference at C = ${\pi  \over 2} + {\pi  \over 2}$ = $\pi $
$ 	herefore $ I
C
= I
1
+ I
2
+ $2\sqrt {{I_1}} \sqrt {{I_2}} $ cos $\Delta $$\phi $
= I + I + $2\sqrt I \sqrt I \cos \left( \pi ight)$
= 0
For point B :
x
P
Ã¢ÂÂ x
Q
= 0
$\Delta $$\phi $ due to path difference = 0
$\Delta $$\phi $ = ${\pi  \over 2}$ (Due to P being ahead of Q by 90ÃÂº)
$ 	herefore $ I
B
= I
1
+ I
2
+ $2\sqrt {{I_1}} \sqrt {{I_2}} $ cos $\Delta $$\phi $
= I + I + $2\sqrt I \sqrt I \cos \left( {{\pi  \over 2}} ight)$
= 2I
I
A
: I
B
: I
C
= 4I : 2I : 0
= 2 : 1 : 0

Answer

4 : 1 : 0

Answer

2 : 1 : 0

Answer

0 : 1 : 2

Answer

0 : 1 : 4

Wave Optics

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