Question

At a point screen, the path difference between two interfering waves of equal intensities is $\frac{\lambda }{4}$ . Find the ratio of intensity at this point and that at the central fringe.

• A. $1:1$
• B. $1:2$
• C. $2:1$
• D. $1:4$

Answer 1

Answer: (B)

Path difference between two interfering waves = $\frac{\lambda }{4}$
Phase difference is given by:
$\Delta \varphi =k\Delta x$
$\Delta \varphi =\frac{2\left(\pi \right)\lambda }{\lambda \times 4}$
$\Delta \varphi =\frac{\pi }{2}$
Resultant Intensity of equal intensities, $I=4I_0{\left(cos\right)}^2\left(\frac{\varphi }{2}\right)$
$=4I_0{\left(cos\right)}^2\left(\frac{\pi }{2\times 2}\right)$
At central fringe, $\varphi =0$
$\frac{I_1}{I_2}=\frac{{\left(cos\right)}^2\left(\frac{{}_{{\left(\varphi \right)}_1}}{1}\right)}{{\left(cos\right)}^2\left(\frac{{\left(\varphi \right)}_2}{2}\right)}=\frac{{\left(cos\right)}^2\left(\frac{\pi }{4}\right)}{{\left(cos\right)}^2(0)}$
$\frac{I_1}{I_2}=\frac{1/\left({\left(\sqrt{2}\right)}^2\right)}{1}$
$\frac{I_1}{I_2}=\frac{1}{2}$