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In Young’s double slit experiment as shown in figure, interference of light waves were observed on the screen. Thomas Young made two pinholes S₁ and S₂ (very close to each other) on an opaque screen in front of parent source S

S₁ and S₂ behaved as coherent sources, and produced interference patten on the screen which has alternate bright fringe and dark fringes. This experiment proved Huygen’s wave theory of Light.

In YDSE experiment position of bright fringes and dark fringes is given by

$\begin{aligned} & x_n=n \frac{\lambda D}{d} \\ & n=0, \pm 1, \pm 2 ... \\ & x_n=\left(n+\frac{1}{2}\right) \frac{\lambda D}{d} \\ & n=0, \pm 1, \pm 2 \ldots\end{aligned}$

In YDSE,

For bright fringes , phase difference is given by

\( \phi_n=2 n \pi \)

∴ Path difference , \(\Delta x_n=\frac{\lambda}{2 \pi}\left(\phi_n\right) \)

Hence, \(\Delta x_n = n \lambda \)

For dark fringes , the phase difference is given by

\( \phi_n^{\prime}=\frac{(2 n+1) \pi}{2} \)

∴ Path difference \(\Delta x_n^{\prime}=\frac{\lambda}{2 \pi}\left(\phi_n^{\prime}\right) \)

\( \Delta x_n^{\prime}=\left(n+\frac{1}{2}\right) \lambda \)

Position of fringes \( x = \frac{\Delta x D}{d} \)

∴ For bright fringes \( x_n = \frac{n \lambda D}{d} , n=0, \pm 1, \pm 2 ... \)

and for dark fringes \(x_n^{\prime}=\left(n+\frac{1}{2}\right) \frac{\lambda D}{d}, n=0, \pm 1, \pm 2 ...  \)