Question

Waves from coherent sources, having intensity $I_0$ each, meets at point $P$. What is the resultant intensity of two waves at $P$, if the path difference between the waves is $\frac{\lambda }{3}$?

• A. $I_0$
  
• B. $2I_0$
  
• C. $\frac{\sqrt{3}{I}_0}{2}$
  
• D. $\frac{2I_0}{\sqrt{3}}$
  

Answer 1

The correct option is A $I_0$


Given:
$\Delta x=\frac{\lambda }{3}$

Phase difference between the waves at point $P$ is given by,
$\Delta \varphi =\frac{2\pi }{\lambda }\times \Delta x$

$\Delta \varphi =\frac{2\pi }{\lambda }\times \frac{\lambda }{3}$

$\Rightarrow \Delta \varphi =\frac{2\pi }{3}\text{rad}={120}^{\circ }$

The resultant intensity is given by,
$I_R=4I_0{\cos }^2\left(\frac{\varphi }{2}\right)$

$\Rightarrow I_R=4I_0{\cos }^2\left(\frac{{120}^0}{2}\right)$

$=4I_0{\cos }^2{60}^0$
$=4I_0\times {\left(\frac{1}{2}\right)}^2$

$\therefore I_R=I_0$

Hence, $\left(A\right)$ is the correct answer.