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Two waves of intensity ratio $$1 : 9$$ cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is $$I_1$$ (b) Waves are coherent is $$I_2$$ and differ in phase by $$60°$$. If $$\frac{I_1}{I_2} = \frac{10}{x}$$, then $$x$$ = _________.
Correct Answer: 13
Intensity ratio 1:9. Let $$I_1 = I, I_2 = 9I$$. Amplitudes: $$A_1 = a, A_2 = 3a$$.
(a) Incoherent: $$I_{total} = I_1 + I_2 = 10I$$.
(b) Coherent with $$\phi = 60°$$: $$I_{total} = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos 60° = I + 9I + 2(3I)(1/2) = 10I + 3I = 13I$$.
$$\frac{I_1}{I_2} = \frac{10I}{13I} = \frac{10}{13}$$. So $$x = 13$$.
The answer is $$\boxed{13}$$.
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