To solve this problem, we need to understand how the number of fringes in a Young's double slit experiment changes with wavelength.$\\$The number of fringes $N$ observed in a given segment of the screen is inversely proportional to the wavelength $\lambda$ of the light used.$\\$This relationship can be expressed as:$\\ N \propto \frac{1}{\lambda} \\$Given:$\\$• Initial wavelength $\lambda_1 = 600 \,$nm$\\$• Initial number of fringes $N_1 = 8 \\$When the wavelength is changed to $\lambda_2 = 400 \,$nm, we need to find the new number of fringes $N_2. \\$Using the proportionality relationship:$\\ \frac{N_1}{N_2} = \frac{\lambda_2}{\lambda_1} \\$Substitute the known values:$\\ \frac{8}{N_2} = \frac{400}{600} \\$Simplify the fraction:$\\ \frac{8}{N_2} = \frac{2}{3} \\$Rearrange to solve for $N_2: \\ N_2 = 8 \times \frac{3}{2} \\ N_2 = 12 \\$Therefore, the number of fringes observed when the wavelength is changed to $400 \,$nm$$ is $12. \\$This corresponds to Option 4.