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. $\frac{I_{2} - 3 I_{1}}{2}$
• B. $\frac{I_{1} - 3 I_{2}}{2}$
• C. $\frac{3 I_{2} + I_{1}}{2}$
• D. $\frac{3 I_{1} - I_{2}}{2}$

Answer 1

Answer: (D)

$I _{1}= I _{ A }+ I _{ B }+2 \sqrt{ I _{ A } I _{ B }} \cos \frac{\pi}{3} \ldots(1)$
$I _{2}= I _{ A }+ I _{ B }+2 \sqrt{ I _{ A } I _{ B }} \cos \frac{2 \pi}{3} \ldots(2)$
$\Rightarrow \quad I _{1}+ I _{2}=2\left( I _{ A }+ I _{ B }\right) \ldots(3)$
From (1)
$\Rightarrow I _{1}=\frac{ I _{1}+ I _{2}}{2}+\sqrt{ I _{ A } I _{ B }}$
$I _{1}-\left(\frac{ I _{1}+ I _{2}}{2}\right)=\sqrt{ I _{ A } I _{ B }}$
$\frac{ I _{1}- I _{2}}{2}=\sqrt{ I _{ A } I _{ B }}$
$I _{\max }=\left(\sqrt{ I _{ A }}+\sqrt{ I _{ B }}\right)^{2}$
$\quad= I _{ A }+ I _{ B }+2 \sqrt{ I _{ A } I _{ B }} \quad \ldots(4)$
$I _{\max }=\frac{ I _{1}+ I _{2}}{2}+\left( I _{1}- I _{2}\right)$
$\quad=\frac{3 I _{1}- I _{2}}{2}$