Two coherent light sources having intensity in the ratio $2 \mathrm{x}$ produce an interference pattern.…

Two coherent light sources having intensity in the ratio $2 \mathrm{x}$ produce an interference pattern. Then the value of $\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}$ will be
  1. $\frac{2 \sqrt{2 x}}{x+1}$
  2. $\frac{\sqrt{2 x}}{2 x+1}$
  3. $\frac{2 \sqrt{2 x}}{2 x+1}$
  4. $\frac{\sqrt{2 x}}{x+1}$

Solution

$\begin{aligned} & \frac{I_1}{I_2}=2 x \\ & I_{\max }=\left(\sqrt{I_1}+\sqrt{I_2}\right)^2 \\ & I_{\min }=\left(\sqrt{I_1}-\sqrt{I_2}\right)^2 \\ & \frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}=\frac{\left(\sqrt{I_1}+\sqrt{I_2}\right)^2-\left(\sqrt{I_1}-\sqrt{I_2}\right)^2}{\left(\sqrt{I_1}+\sqrt{I_2}\right)^2+\left(\sqrt{I_1}-\sqrt{I_2}\right)^2} \\ & =\frac{4 \sqrt{I_1 I_2}}{2\left(I_1+I_2\right)}=\frac{2 \sqrt{\frac{I_1}{I_2}}}{\frac{I_1}{I_2}+1}=\frac{2 \sqrt{2 x}}{2 x+1}\end{aligned}$

Asked in: AP EAMCET 2023 (17 May Shift 1)

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