c
(c)Path difference between the waves reaching at\(P,\)\(\Delta = {\Delta _1} + {\Delta _2}\)
where\({\Delta _1} = \) Initial path difference
\({\Delta _2} = \)Path difference between the waves after emerging from slits.
\({\Delta _1} = S\,{S_1} - S\,{S_2} = \sqrt {{D^2} + {d^2}} - D\)
and \({\Delta _2} = {S_1}O - {S_2}O = \sqrt {{D^2} + {d^2}} - D\)
\(\therefore \,\,\,\Delta = 2\left\{ {{{({D^2} + {d^2})}^{\frac{1}{2}}} - D} \right\} = 2\left\{ {({D^2} + \frac{{{d^2}}}{{2D}}) - D} \right\}\)
\( = \frac{{{d^2}}}{D}\) (From Binomial expansion)
For obtaining dark at \(O\), \(\Delta \) must be equals to \((2n - 1)\frac{\lambda }{2}\) i.e. \(\frac{{{d^2}}}{D} = (2n - 1)\frac{\lambda }{2} \Rightarrow d\sqrt {\frac{{(2n - 1)\lambda \,D}}{2}} \)
For minimum distance \(n = 1\) so \(d = \sqrt {\frac{{\lambda \,D}}{2}} \)