Question

Two periodic waves of intensities $I_1$ and $I_2$ pass through a region at the same time in the same direction. The sum of the maximum and minimum intensities possible is :

• A. $I_1+I_2$
• B. ${(\sqrt{I_1}+\sqrt{I_2})}^2$
• C. ${(\sqrt{I_1}-\sqrt{I_2})}^2$
• D. $2(I_1+I_2)$

Answer 1

The correct option is D $2(I_1+I_2)$

Resultant intensity of two periodic waves is given by
$I=I_1+I_2+2\sqrt{I_1{I}_2\cos \delta }$
where $\delta$ is the phase difference between the waves.
For maximum intensity,
$\delta =2n\pi ;n=0,1,2,...$etc.
Therefore, for zero order maxima, $\cos \delta =1$
$I_{max}=I_1+I_2+2\sqrt{I_1{I}_2}={(\sqrt{I_1}+\sqrt{I_2})}^2$
For minimum intensity,
$\delta =(2n-1)\pi ;n=1,2,...$etc.
Therefore, for Ist order minima, $\cos \delta =-1$
$I_{min}=I_1+I_2-2\sqrt{I_1{I}_2}$
$={(\sqrt{I_1}-\sqrt{I_2})}^2$
Therefore, $I_{max}+I_{min}={(\sqrt{I_1}+\sqrt{I_2})}^2+{(\sqrt{I_1}-\sqrt{I_2})}^2$
$=2(I_1+I_2)$