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 \( 60^\circ \) If \( \frac{I_1}{I_2} = \frac{10}{x} \) then \( x = \) _____.
Correct Answer: 13
Approach Solution - 1
For an incoherent wave:
The intensity \( I_1 \) is the sum of the individual intensities \( I_A \) and \( I_B \):
\[ I_1 = I_A + I_B \quad \Rightarrow \quad I_1 = I_0 + 9I_0 = 10I_0 \]
For a coherent wave:
The intensity \( I_2 \) is given by the formula:
\[ I_2 = I_A + I_B + 2 \sqrt{I_A I_B} \cos(60^\circ) \]
Substituting the values and simplifying:
\[ I_2 = I_0 + 9I_0 + 2 \sqrt{I_0 I_0} \cdot \cos(60^\circ) = 13I_0 \]
Finally, the ratio of the intensities \( I_1 \) and \( I_2 \) is:
\[ \frac{I_1}{I_2} = \frac{10I_0}{13I_0} = \frac{10}{13} \]
Approach Solution -2
For incoherent waves:
\[ I_1 = I_A + I_B = I_0 + 9I_0 = 10I_0 \]For coherent waves:
\[ I_2 = I_A + I_B + 2\sqrt{I_A I_B} \cos 60^\circ \] \[ I_2 = I_0 + 9I_0 + 2\sqrt{I_0 \times 9I_0} \cdot \frac{1}{2} = 13I_0 \]Given:
\[ \frac{I_1}{I_2} = \frac{10}{13} \]Thus:
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