Question

Consider the arrangement shown in figure the distance D is large compared to the separation of between then 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 bright fringe is formed. (c) Find the fringe- width

Answer 1

From the figure, $AB=BO$ and $AC=CO$

Path difference of the wave from reaching $O$,

$△x=(AB+BO)-(AC+CO)$

$=2(AB-AC)$

$=2(\sqrt{D^2+d^2}-D)$

For dark fringe to be formed at O, path difference should be an odd multiple of $\frac{\lambda }{2}$

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So,

$△x=(2n+1)\frac{\lambda }{2}$

$2(\sqrt{D^2+d^2}-D)$ = $(2n+1)\frac{\lambda }{2}$

$\sqrt{D^2+d^2}$ = $D+(2n+1)\frac{\lambda }{2}$

$D^2+d^2=D^2+(2n+1{)}^2\frac{{\lambda }^2}{16}+(2n+1)\frac{\lambda D}{2}$

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Neglecting $(2n+1{)}^2\frac{{\lambda }^2}{16}$ as it is very small, we get

$d=\sqrt{(2n+1)}\frac{\lambda D}{2}$

For minimum d putting n=0

$d_{min}=\sqrt{\frac{\lambda D}{2}}$

Thus for $d_{min}=\sqrt{\frac{\lambda D}{2}}$ there is a dark fringe at O.