In two separate Young's double-slit experimental set-ups and two monochromatic light sources of diff

Question

In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are $2: 1$ and $1: 2$ respectively. The corresponding ratio of the distances between the slits and the respective screens ( $D_1 / D_2$ ) is $\_\_\_\_$ .

Professor PC Thomas Classes · Answer

The fringe width in a Young's double-slit experiment is given by $\beta = \frac{\lambda D}{d}$. Given that the fringes are of equal width, $\beta_1 = \beta_2$, which implies $\frac{\lambda_1 D_1}{d_1} = \frac{\lambda_2 D_2}{d_2}$. Rearranging this for the ratio of distances gives $\frac{D_1}{D_2} = \left(\frac{d_1}{d_2}ight) \left(\frac{\lambda_2}{\lambda_1}ight)$. We are given $\frac{d_1}{d_2} = \frac{2}{1}$ and $\frac{\lambda_1}{\lambda_2} = \frac{1}{2} \implies \frac{\lambda_2}{\lambda_1} = \frac{2}{1}$. Multiplying these gives $\frac{D_1}{D_2} = \left(\frac{2}{1}ight) 	imes \left(\frac{2}{1}ight) = 4$.