Question

The minimum value of $d$ so that there is a dark fringe at $O$ is $d_{min}$. For the value of $d_{min}$, the distance at which the next bright fringe is formed is $x$. Then :

• A. $d_{min}=\sqrt{\lambda D}$
• B. $d_{min}=\sqrt{\frac{\lambda D}{2}}$
• C. $x=\frac{d_{min}}{2}$
• D. $x=d_{min}$

Answer 1

Answer: (B), (D)

The correct options are
B $d_{min}=\sqrt{\frac{\lambda D}{2}}$
D $x=d_{min}$

Path difference between rays O'O and BO =$\sqrt{D^2+d^2}-D$

If $d<<D$, path difference=$D(1+\frac{d^2}{2D^2})-D=\frac{d^2}{2D}$

For dark fringe to form, there must be destructive interference,

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

For minimum value of d, $\frac{d_{min}^2}{2D}=\frac{\lambda }{2}$

Hence, $d_{min}=\sqrt{\lambda D}$

Path difference between rays at point P=$\sqrt{x^2+D^2}-\sqrt{(x-d_{min}{)}^2+D^2}=0$(for bright fringe)

Hence, $x=\frac{d_{min}}{2}$