Question

The interference pattern is obtained with two coherent light sources of intensity (\(I_{1}\) and \(I_{2}\)) ratio '\(\beta\)'. In the interference pattern, the ratio \(\left(\frac{I_{max}+I_{min}}{I_{max}-I_{min}}\right)\) will be \(\left(\beta =\frac{I_{2}}{I_{1}}\right)\)

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

Hint:

Write Imax and Imin in terms of the two intensities and simplify the ratio using the intensity ratio.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For coherent sources, \(I_{max}=(\sqrt{I_1}+\sqrt{I_2})^2\) and \(I_{min}=(\sqrt{I_1}-\sqrt{I_2})^2\).

Step 2: Add and subtract
\(I_{max}+I_{min}=2(I_1+I_2)\)
\(I_{max}-I_{min}=4\sqrt{I_1I_2}\)

Step 3: Ratio
\(\dfrac{I_{max}+I_{min}}{I_{max}-I_{min}}=\dfrac{2(I_1+I_2)}{4\sqrt{I_1I_2}}=\dfrac{I_1+I_2}{2\sqrt{I_1I_2}}\).

Step 4: Use the intensity ratio
Divide numerator and denominator by \(I_1\) with \(\beta=\dfrac{I_2}{I_1}\):
\(\dfrac{1+\beta}{2\sqrt{\beta}}\).

Final Answer:
The required ratio is \(\dfrac{\beta+1}{2\sqrt{\beta}}\). \[ \boxed{\text{(C) }\dfrac{\beta+1}{2\sqrt{\beta}}} \]