In the Young's double slit experiment, the distance between the...

Fringe width,

$\beta = \dfrac{\lambdaD}{d}$

{\beta}_\max ⇒ {d}_\min and {\beta}_\min ⇒ {d}_\max

$d = {d}_{0} + {a}_{0}sin\omegat$

{d}_\max = {d}_{0} + {a}_{0} and {d}_\min = {d}_{0} - {a}_{0}

∴ {\beta}_\min = \dfrac{\lambdaD}{{d}_{0} + {a}_{0}} and ∴ {\beta}_\max = \dfrac{\lambdaD}{{d}_{0} - {a}_{0}}

{\beta}_\max - {\beta}_\min = \dfrac{\lambdaD}{{d}_{0} - {a}_{0}} - \dfrac{\lambdaD}{{d}_{0} + {a}_{0}} = \dfrac{2\lambdaD{a}_{0}}{{d}_{0}^{2} - {a}_{0}^{2}}