The figure shows a modified Young’s double slit experimental set-up. Here $SS_{2}-SS_{1}=\lambda /4$(a) Write the condition for constructive interference. (b) Obtain an expression for the fringe width.

(a) The path difference;

$S_{2}P-S_{1}P=(\frac{y_{n}d}{D}+\frac{\lambda }{4})$

for constructive inference; the path difference $=n\lambda$

$n\lambda=(\frac{y_{n}d}{D}+\frac{\lambda }{4})$

where, $n=0,1,2,3,.........$

(b) The position of nth bright fringe

$\frac{y_{n}d}{D}={n\lambda}-\frac{\lambda}{4}$

$y_n=(n-\frac{1}{4})\frac{\lambda D}{d}$

$\therefore$ Fringe width  $\beta =y_{n}-y_{n-1}$

$\beta=(n-\frac{1}{4}-(n-1-\frac{1}{4}))\frac{\lambda D}{d}$

$\beta =\frac{\lambda D}{d}$

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