Wave Optics - Result Question 1

1. The 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:

[2008]

(a) $I_1+I_2$

(b) $(\sqrt{I_1}+\sqrt{I_2})^{2}$

(c) $(\sqrt{I_1}-\sqrt{I_2})^{2}$

(d) $2(I_1+I_2)$

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Solution:

  1. (d) The resultant intensity of two periodic waves at a point is given by

$I=I_1+I_2+2 \sqrt{I_1 I_2} \cdot \cos f$.

Resultant intensity is maximum if $\cos \phi=-1$

i.e. $I _{\max }=I_1+I_2+2 \sqrt{I_1 I_2}$

Resultant intensity is minimum if $\cos \phi=+1$

i.e, $I _{\min }=I_1+I_2-2 \sqrt{I_1 I_2}$

Therefore, the sum of the maximum and minimum intensities is $I _{\max }+I _{\text{min }}$

$=I_1+I_2+2 \sqrt{I_1 I_2}+I_1+I_2-2 \sqrt{I_1 I_2}$

$=2(I_1+I_2)$