Question

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 \( x = \) _____.

Answer 1

Correct Answer: 13

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:

\[ \frac{I_1}{I_2} = \frac{10I_0}{13I_0} = \frac{10}{13} \]

Answer 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:

\[ x = 13 \]