Question

Two coherent sources of intensity ratio \(\alpha\) interfere. In interference pattern, \(\frac{I_{max}-I_{min}}{I_{max}+I_{min}}\) is

• A. \(\frac{2\alpha }{1+\alpha }\)
• B. \(\frac{2\sqrt{\alpha }}{1+\alpha }\)
• C. \(\frac{2\alpha }{1+\sqrt{\alpha }}\)
• D. \(\frac{1+\alpha }{2\alpha }\)

Hint:

Write the maximum and minimum intensities in terms of the amplitudes and then simplify the ratio of their difference to their sum.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Intensity is proportional to the square of the amplitude. For two coherent sources, \(I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2\) and \(I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2\).

Step 2: Use the intensity ratio
Let \(I_1=\alpha I_2\). Then \(\sqrt{I_1}=\sqrt{\alpha}\sqrt{I_2}\), so \(I_{max}=I_2(\sqrt{\alpha}+1)^2\) and \(I_{min}=I_2(\sqrt{\alpha}-1)^2\).

Step 3: Calculation
\(I_{max}-I_{min}=4\sqrt{\alpha}\,I_2\)
\(I_{max}+I_{min}=2(\alpha+1)I_2\)
Dividing gives \(\dfrac{I_{max}-I_{min}}{I_{max}+I_{min}}=\dfrac{2\sqrt{\alpha}}{1+\alpha}\).

Final Answer:
The required ratio is 2 root alpha divided by (1 + alpha), which is option (B). \[ \boxed{\text{(B) }\dfrac{2\sqrt{\alpha}}{1+\alpha}} \]