Question

Consider an interference pattern between two coherent sources. If $I_1$ and $I_2$ be intensities at points where the phase difference are $\frac{\pi }{3}$ and $\frac{2\pi }{3}$ respectively, then the intensity at maxima is

• A. $\frac{I_2-3I_1}{2}$
• B. $\frac{I_1-3I_2}{2}$
• C. $\frac{3I_2+I_1}{2}$
• D. $\frac{3I_1-I_2}{2}$

Answer 1

Answer: (D)

$I_1=I_A+I_B+2\sqrt{I_A{I}_B}\cos \frac{\pi }{3}\dots (1)$
$I_2=I_A+I_B+2\sqrt{I_A{I}_B}\cos \frac{2\pi }{3}\dots (2)$
$\Rightarrow I_1+I_2=2\left(I_A+I_B\right)\dots (3)$
From (1)
$\Rightarrow I_1=\frac{I_1+I_2}{2}+\sqrt{I_A{I}_B}$
$I_1-\left(\frac{I_1+I_2}{2}\right)=\sqrt{I_A{I}_B}$
$\frac{I_1-I_2}{2}=\sqrt{I_A{I}_B}$
$I_{\max }={\left(\sqrt{I_A}+\sqrt{I_B}\right)}^2$
$=I_A+I_B+2\sqrt{I_A{I}_B}\dots (4)$
$I_{\max }=\frac{I_1+I_2}{2}+\left(I_1-I_2\right)$
$=\frac{3I_1-I_2}{2}$