Question

Two coherent waves are represented by $y_{1}=a_{1} \cos \omega t$ and $y_{2}=a_{2} \sin \omega t$, superimposed on each other. The resultant intensity is proportional to

• A. $\left(a_{1}+a_{2}\right)$
• B. $\left(a_{1}-a_{2}\right)$
• C. $\left(a_{1}^{2}+a_{2}^{2}\right)$
• D. $\left(a_{1}^{2}-a_{2}^{2}\right)$

Answer 1

Answer: (C)

As, $y_{1}=a_{1} \cos \omega t=a_{1} \sin \left(\omega t+90^{\circ}\right)$
and $y_{2}=a_{2} \sin \omega t$
$\therefore $ Phase difference $=\phi=90^{\circ}$
$R=\sqrt{a_{1}^{2}+a_{2}^{2}+2 a_{1} a_{2} \cos \phi}$
$=\sqrt{a_{1}^{2}+a_{2}^{2}}$
$\therefore $ Resultant intensity, $I\propto R^{2}$
$\Rightarrow I \propto\left(a_{1}^{2}+a_{2}^{2}\right)$