Question

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

• A. $\frac{2\sqrt{2x}}{x+1}$
• B. $\frac{\sqrt{2x}}{2x+1}$
• C. $\frac{2\sqrt{2x}}{2x+1}$
• D. $\frac{\sqrt{2x}}{x+1}$

Answer 1

The correct option is C $\frac{2\sqrt{2x}}{2x+1}$

Let,
$I_1=2x$
$I_2=1$

So,
$I_{\text{max}}={({\sqrt{I}}_1+{\sqrt{I}}_2)}^2$

$\Rightarrow I_{\text{max}}={(\sqrt{2x}+\sqrt{1})}^2=2x+1+2\sqrt{2x}$

$I_{\text{min}}={({\sqrt{I}}_1-{\sqrt{I}}_2)}^2$

$\Rightarrow I_{\text{min}}={(\sqrt{2x}-\sqrt{1})}^2=2x+1-2\sqrt{2x}$

Now,
$I_{\text{max}}-I_{\text{min}}=4\sqrt{2x}$

$I_{\text{max}}+I_{\text{min}}=4x+2$

Hence,

$\frac{I_{\text{max}}-I_{\text{min}}}{I_{\text{max}}+I_{\text{min}}}=\frac{4\sqrt{2x}}{4x+2}=\frac{2\sqrt{2x}}{2x+1}$