Question

The interference pattern is obtained with two coherent light sources of intensity ratio n. In the interference pattern, the ratio $\frac{I_{max} - I_{min}}{I_{max} + I_{min}}$ will be

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

Answer 1

The Correct Option is B

Intensity Calculation 

We are given the following equations for maximum and minimum intensities:

\(I_{\max} = \left( \sqrt{I} + \sqrt{nI} \right)^2\)

\(I_{\min} = \left( \sqrt{I} - \sqrt{nI} \right)^2\)

The formula to find the ratio of the difference and sum of the maximum and minimum intensities is:

\(\frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = \frac{\left( \sqrt{I} + \sqrt{nI} \right)^2 - \left( \sqrt{I} - \sqrt{nI} \right)^2}{\left( \sqrt{I} + \sqrt{nI} \right)^2 + \left( \sqrt{I} - \sqrt{nI} \right)^2}\)

Now, simplifying the expression:

\(= \frac{1 + n + 2\sqrt{n} - 1 - n + 2\sqrt{n}}{1 + n + 2\sqrt{n} + 1 + n - 2\sqrt{n}}\)

After simplifying further:

\(= \frac{4\sqrt{n}}{2 + 2n} = \frac{2\sqrt{n}}{1 + n}\)

Conclusion:

The final result is \( \frac{2\sqrt{n}}{1+n} \).