Question

Two coherent sources of intensity ratio \(9:1\) produce interference fringes. The ratio of maximum intensity to minimum intensity will be:

• A. \(9:1\)
• B. \(3:1\)
• C. \(4:1\)
• D. \(16:4\)

Hint:

For interference problems: \[ I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2 \] \[ I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2 \] Always take square roots of intensities before substitution.

Answer 1

The Correct Option is D

Concept: In interference of light, the maximum and minimum intensities are given by: \[ I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2 \] \[ I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2 \] where:

• \(I_1\) and \(I_2\) are intensities of coherent sources.

The ratio of maximum to minimum intensity is: \[ \frac{I_{\max}}{I_{\min}} = \frac{(\sqrt{I_1}+\sqrt{I_2})^2}{(\sqrt{I_1}-\sqrt{I_2})^2} \]

Step 1: Write the given intensity ratio.
Given: \[ I_1 : I_2 = 9:1 \] Therefore: \[ I_1 = 9 \] \[ I_2 = 1 \]

Step 2: Calculate maximum intensity.
Using: \[ I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2 \] Substituting values: \[ I_{\max} = (\sqrt{9}+\sqrt{1})^2 \] \[ = (3+1)^2 \] \[ = 4^2 \] \[ = 16 \]

Step 3: Calculate minimum intensity.
Using: \[ I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2 \] Substituting values: \[ I_{\min} = (\sqrt{9}-\sqrt{1})^2 \] \[ = (3-1)^2 \] \[ = 2^2 \] \[ = 4 \]

Step 4: Find the required ratio.
\[ I_{\max} : I_{\min} = 16:4 \] Hence, the correct answer is: \[ \boxed{16:4} \]