Question

In the adjacent diagram, CP represents a wavefront and AO
and iBP, the corresponding two rays. Find the condition of $\theta$
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

Answer: (B)

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