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