Q. Two identical light waves, propagating in the same direction, have a phase difference $\delta$. After they superpose, the intensity of the resulting wave will be proportional to
A) $\cos\delta$
B) $\cos{(\delta/2)}$
C) $\cos^{2}{(\delta/2)}$
D) $\cos^{2}\delta$
Explanation
(c) Here $A^{2} = a_{1}^{2} + a_{2}^{2} + 2a_{1}a_{2}\cos\delta$
$\because a_{1} = a_{2} = a$
$\therefore A^{2} = 2a^{2}\left( 1 + \cos\delta \right) = 2a^{2}\left( 1 + 2\cos^{2}{\frac{\delta}{2} - 1} \right)$
$\Rightarrow A^{2} \propto \cos^{2}\frac{\delta}{2}$
Now, $I \propto A^{2}\ \therefore I \propto A^{2} \propto \cos^{2}\frac{\delta}{2}$
$\therefore I \propto \cos^{2}\frac{\delta}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.