In Young's double slit experiment, the two slits are 'd' distance apart. Interference pattern is observed on…

In Young's double slit experiment, the two slits are 'd' distance apart. Interference pattern is observed on a screen at a distance ' $D$ ' from the slits. A dark fringe is observed on a screen directly opposite to one of the slits. The wavelength of light is
  1. $\frac{D^2}{2 d}$
  2. $\frac{\mathrm{d}^2}{2 \mathrm{D}}$
  3. $\frac{D^2}{d}$
  4. $\frac{\mathrm{d}^2}{\mathrm{D}}$

Solution

$\begin{aligned} S_2 P & =\left(D^2+d^2\right)^{1 / 2} \\ & =D\left[1+\frac{d^2}{D^2}\right]^{1 / 2}\end{aligned}$ Using binomial equation, $\mathrm{S}_2 \mathrm{P}=\mathrm{D}\left[1+\frac{1}{2} \frac{\mathrm{d}^2}{\mathrm{D}^2}\right]^{1 / 2}=\mathrm{D}+\frac{\mathrm{d}^2}{2 \mathrm{D}}$ $\Rightarrow$ Path difference $=\frac{d^2}{2 D}$ For dark fringe, $\frac{\mathrm{d}^2}{2 \mathrm{D}}=\frac{\lambda}{2}$ $\therefore \quad \lambda=\frac{\mathrm{d}^2}{\mathrm{D}}$

Asked in: MHT CET 2023 (10 May Shift 2)

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