Two waves of intensity ratio 1: 9 cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is I_1 (b) Waves are coherent is I_2 and differ in phase by 60^\circ . If \fracI_1I_2=\frac10x then x=\underline\hspace2cm .

Wave Optics PYQ — JEE Main 2024 · 31 Jan · Shift 1 · Q90 with Solution

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Correct answer: 13

Let the two wave intensities be in the ratio $1:9$ .

So, take
$\qquad I_b = 9I.$$$
Case (a): Incoherent waves

For incoherent waves, intensities simply add: $I_1 = I_a + I_b = I + 9I = 10I.$
Case (b): Coherent waves with phase difference $60^\circ$

For coherent waves, resultant intensity is $I_2 = I_a + I_b + 2\sqrt{I_a I_b}\cos\phi.$

Here, $\phi = 60^\circ$ , so $I_2 = I + 9I + 2\sqrt{I\cdot 9I}\cos 60^\circ.$

Now,
$\qquad \cos 60^\circ = \frac12.$$ Therefore, $$I_2 = 10I + 2(3I)\left(\frac12\right) = 10I + 3I = 13I.$$$
Now compute the ratio: $\frac{I_1}{I_2} = \frac{10I}{13I} = \frac{10}{13}.$

Given $\frac{I_1}{I_2} = \frac{10}{x},$ hence $\frac{10}{x} = \frac{10}{13} \implies x=13.$
Final Answer: $x=13$