91Ó°ÊÓ

Distance between the two sources = 1.5$\mathrm{λ}$

The path difference of two rays creating a bright fringe of order m for slit separation d, screen distance D, and wavelength $\mathrm{λ}$is

$\mathit{Δ}\mathit{L}\mathbf{=}\mathit{m}\mathit{λ}$ …(¾±)

The path difference of two rays creating a dark fringe of order m for slit separation d, screen distance D and wavelength $\mathrm{λ}$is

$\mathit{Δ}\mathit{L}\mathbf{=}\left(m+\frac{1}{2}\right)\mathit{λ}$…(¾±¾±)

For any point y path 1, the path difference between light rays from the two sources is

$\begin{array}{rcl}\mathrm{Δ}L& =& \sqrt{{\left(0.75\mathrm{λ}\right)}^2+y^2}-\sqrt{{\left(0.75\mathrm{λ}\right)}^2+y^2}\\ & =& 0\\ & & \end{array}$

This is equal to equation (i) with m=0. Thus path 1 has bright fringes all along.

For any point on path 2 at a distance a from source 2, the path difference between light rays from the two sources is

$$\begin{gathered} \mathrm{Δ}L=\left(a+1.5\mathrm{λ}\right)-a \\ =1.5\mathrm{λ} \end{gathered}$$

This is equal to equation (ii) with m=1. Thus path 1 has dark fringes all along.

For any point on path 3 at a horizontal distance a from source 2 and vertical distance y, the path difference between light rays from the two sources is

This function has a maxima$\mathrm{Δ³¢}=1.5\mathrm{λ}$at y=0 and tends to 0 as$y→∞$. Thus there is a dark fringe at y=0, a bright fringe when $\mathrm{Δ}L$reduces to $1\mathrm{λ}$, another dark fringe when $\mathrm{Δ}L$reduces to 0.5$\mathrm{λ}$and finally a bright fringe as $y→∞$. Thus there are alternating bright and dark fringes.