Two coherent monochromatic beams of intensities I and 4I respectively, are superposed. Then, the maximum and minimum intensities in the resulting pattern are [WB JEE 2014]

We know that,

The maximum intensity,  $I_{\max }={\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2$           …(i)

The minimum intensity,   $I_{\min }={\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2$          …(ii)

So, the ratio of the maximum and minimum intensities is

$\frac{I_{\max }}{I_{\min }}=\frac{{\left(\sqrt{I_1}+\sqrt{I_2}\right)}^2}{{\left(\sqrt{I_1}-\sqrt{I_2}\right)}^2}=\frac{(\sqrt{4I}+\sqrt{I}{)}^2}{(\sqrt{4I}-\sqrt{I}{)}^2}$

$\Rightarrow \frac{I_{\max }}{I_{\min }}={\left(\frac{3\sqrt{I}}{\sqrt{I}}\right)}^2=\frac{9}{1}=9:1$

Hence, the maximum and minimum intensities in the resulting pattern are 9I and I.