Question

In the adjacent diagram, CP represents a wavefront and AO and BP, the corresponding two rays. Find the condition on $\theta$ for constructive interference at P between the ray BP and reflected ray OP

• A. $cos\theta =\frac{3\lambda }{2d}$
• B. $cos\theta =\frac{\lambda }{4d}$
• C. $sec\theta -cos\theta =\frac{\lambda }{d}$
• D. $sec\theta -cos\theta =\frac{4\lambda }{d}$

Answer 1

The correct option is B $cos\theta =\frac{\lambda }{4d}$


$\because PR=d\Rightarrow PO=dsec\theta$ and $CO=POcos2\theta =dsec\theta cos2\theta$ is
Path difference between the two rays
$\Delta =CO+PO=(dsec\theta +dsec\theta cos2\theta )$
Phase difference between the two rays is
$\varphi =\pi$ (One is reflected, while another is direct)
Therefore condition for constructive interference should be $\Delta =\frac{\lambda }{2},\frac{3\lambda }{2}...$
or $dsec\theta (1+cos2\theta )=\frac{\lambda }{2}$
or $\frac{d}{cos\theta }\left(2co{s}^2\theta \right)=\frac{\lambda }{2}\Rightarrow cos\theta =\frac{\lambda }{4d}$.