A) \[\sqrt{\frac{3\lambda D}{2}}\]
B) \[\sqrt{\lambda D}\]
C) \[\sqrt{\frac{\lambda D}{2}}\]
D) \[\sqrt{3\lambda D}\]
Correct Answer: C
Solution :
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 \[d=\sqrt{\frac{\lambda \,D}{2}}\]You need to login to perform this action.
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