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 is

• A. $ I_1 +I_2 $
• B. $ \left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2} $
• C. $ \left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2} $
• D. $ 2\left(I_{1}+I_{2}\right) $

Answer 1

Answer: (D)

Resultant intensity of two periodic wave 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 = 2 n \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 1st 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) $