Question

In a Young ’s double slit experiment, the path difference, at a certain point on the screen, between two interfering waves is $\frac{1}{8}th$ of wavelength. The ratio of the intensity at this point to that at the centre of a bright fringe is

Answer 1

Given, path difference,$\Delta x = \frac{\lambda}{8}$
Phase difference$\left(\Delta\phi\right)$ is given by
$\Delta\phi = \frac{2\pi}{\lambda}\left(\Delta x\right)$
$\Delta\phi = \frac{\left(2\pi\right)}{\lambda}\frac{\lambda}{8} = \frac{\pi}{4}$
For two sources in different phases,
$I =I_{0}$ cos$^{2} \left(\frac{\pi}{8}\right)$
$\frac{I}{I_{0}} = cos^{2}\left(\frac{\pi}{8}\right)$
$= \frac{1 + cos \frac{\pi}{4}}{2} = \frac{1+ \frac{1}{\sqrt{2}}}{2} = 0.85$