Q: Compare the intensities of two points located at respective distance β/4 and β/3 from the central maximum in a interference of YDSE (β is the fringe width)

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Sol: $\large \Delta \theta = \frac{2\pi}{\lambda} \Delta x = \frac{2\pi}{\lambda} (\frac{d}{D}. \frac{\beta}{4})$

$\large \Delta \theta = \frac{2\pi}{\lambda} (\frac{d}{D}. \frac{\lambda D}{4 d}) = \frac{\pi}{2} $

$\large I_1 = 4 I_0 cos^2(\pi/4) = 2 I_0 $

Similarly , $\large \Delta \theta = \frac{2\pi}{3} $

$\large I_2 = 4 I_0 cos^2(\frac{2\pi}{2 \times 3}) = I_0 $

∴ Required ratio = 2:1