Two coherent light sources having intensity in the ratio $2 x$ produce an interference pattern. The ratio $\frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$ will be:

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$\frac{\sqrt{2 x}}{2 x + 1}$

$\frac{2 \sqrt{2 x}}{x + 1}$

$\frac{\sqrt{2 x}}{x + 1}$

$\frac{2 \sqrt{2 x}}{2 x + 1}$

answer is D.

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Detailed Solution

$\begin{array}{l} Given : \frac{l_{1}}{l_{2}} = 2 x \\ \Rightarrow I_{1} = 2 x I_{2} \\ Now, \\ I_{max} = {(\sqrt{I_{1}} + \sqrt{I_{2}})}^{2} = I_{1} + I_{2} + 2 \sqrt{l_{1} I_{2}} \\ I_{min} = {(\sqrt{I_{1}} - \sqrt{I_{2}})}^{2} = I_{1} + I_{2} - 2 \sqrt{l_{1} I_{2}} \\ ∴ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{4 \sqrt{I_{1} I_{2}}}{2 (I_{1} + I_{2})} = \frac{4 \sqrt{2 x I_{2} I_{2}}}{2 (2 x I_{2} + I_{2})} = \frac{2 \sqrt{2 x}}{(2 x + 1)} \end{array}$