Question

The intensity, at the central maxima $\left(O\right)$ in a Young's double slit set up is $I_0$. If the distance $OP$ equals one third of the fringe width of the pattern, show that the intensity, at point $P$, would equal $I_2/4$.

Answer 1

Intensity at the central maxima=$I_0$

Distance OP is one-third of the fringe width, $x=\frac{1}{3}\beta$(given)

Fringe width is given by

$\frac{\lambda D}{d}$

$\therefore x=\frac{\lambda D}{3d}$

Path difference, ${\Delta }_p=\frac{xd}{D}$ and,

Phase difference, $\varphi =\frac{2\pi }{\lambda }\left(\Delta P\right)=\frac{2\pi }{3}$

Intensity at point P,

$I=I_0{\cos }^2\frac{\varphi }{2}=I_0{\cos }^2\left(\frac{2\pi }{3}\right)=\frac{I_0}{4}$

Hence proved.