Question

In a Young’s double slit experiment, let $I_0$ be the intensity at the central maximum and $\beta$ be the fringe width. Then intensity at a point $P$ which is at a distance $x$ from the central maximum is

• A. $I_{0}cos\left(\frac{\pi x}{\beta }\right)$
• B. $4I_{0}cos\left(\frac{\pi x}{\beta }\right)$
• C. $I_{0}co{s}^2\left(\frac{\pi x}{\beta }\right)$
• D. $4I_{0}co{s}^2\left(\frac{\pi x}{\beta }\right)$

Answer 1

The correct option is C $I_{0}co{s}^2\left(\frac{\pi x}{\beta }\right)$

Let
$d$ be gap between the slits $S_1,S_2$
$D$ be distance of the screen from slits
point $C$ be central maxima

Path difference at Point $P$

$\Delta X=\frac{xd}{D}$

Phase difference at point $P$

$\Delta \varphi =\frac{2\pi }{\lambda }\left(\Delta X\right)=\frac{2\pi }{\lambda }\frac{xd}{D}=\frac{2\pi x}{\beta }$

Intensity at point $P$

$I=I_{max}Co{s}^2\left(\Delta \varphi /2\right)$

As $I_{max}=I_0$(Intensity at central maxima)

$\therefore I=I_{0}co{s}^2\frac{\pi x}{\beta }$