Question

consider the arrangement shown in figure (17-E4). The distance D is large compared to the separation d between the slits. (a) Find the minimum value of d so that there is a dark fringe at O. (b) suppose d has this value. Find the distance x at which the next pright fringe is formed. (c) Find thd fringe- width.

Answer 1

From the diagram, it can be seen that at point 'O'
Path difference = (AB + BO) - (AC + CO)
= 2 (AB - AC)
[Since , AB = BO and AC = CO]
$=2\left(\sqrt{D^2+d^2.D}\right)$
For dark fringe, path difference should be odd multiple of $\frac{lamba}{2}$
$So,2\left(\sqrt{D^2+d^2-D}\right)=(2n+1)\frac{\lambda }{2}\Rightarrow \sqrt{D^2}+d^2=D+(2n+1)\frac{\lambda }{4}\Rightarrow D^2+d^2=D^2+(2n+{)}^2\frac{{\lambda }^2}{16}+(2n+1)\frac{\lambda }{2}$
neglecting , $(2n+1{)}^2\frac{{\lambda }^2}{16},$
as it is ery small
We get, $d=(\sqrt{2}n+1)\frac{\lambda D}{2}$
For minimum 'd' putting, n = 0
$\Rightarrow d_{min}=\sqrt{\left(\frac{\lambda D}{2}\right)}$