Question

If two waves of equal intensities $I_1=I_2=I_0$, meets at two locations $P$ and $Q$ with path difference ${\Delta }_1$ and ${\Delta }_2$ respectively, then the ratio of resultant intensity at points $P$ and $Q,\left(\frac{I_P}{I_Q}\right)$ will be

• A. $\frac{{\cos }^2\left(\frac{{\Delta }_1}{\lambda }\right)}{{\cos }^2\left(\frac{{\Delta }_2}{\lambda }\right)}$
• B. $\frac{{\cos }^2{\Delta }_1}{{\cos }^2{\Delta }_2}$
• C. $\frac{{\cos }^2\left(\frac{\pi {\Delta }_1}{\lambda }\right)}{{\cos }^2\left(\frac{\pi {\Delta }_2}{\lambda }\right)}$
• D. $\frac{{\Delta }_1}{{\Delta }_2}$

Answer 1

Answer: (C)

Resultant intensity is given by $I=4I_0{\cos }^2\varphi /2$
Now, phase difference $(\varphi )=\frac{2\pi }{\lambda }\times \Delta$ (path difference)
As a path difference of one wavelength corresponds to a phase difference of $2\pi$ radius.
$\Rightarrow I=4I_0{\cos }^2\left(\frac{2\pi \Delta }{2\lambda }\right)$
$=4I_0{\cos }^2\left(\frac{\pi \Delta }{\lambda }\right)$
$\therefore \frac{I_P}{I_Q}={\cos }^2\left(\frac{\pi {\Delta }_1}{\lambda }\right)/{\cos }^2\left(\frac{\pi {\Delta }_2}{\lambda }\right)$