The distance between the two dark fringes on either side of the central bright fringe is approximately 4.61 meters.
To find the distance between the two dark fringes on either side of the central bright fringe in a diffraction pattern, we can use the formula for the single-slit diffraction pattern:
d * sin(θ) = m * λ
where d is the width of the slit, θ is the angle of diffraction, m is the order of the fringe, and λ is the wavelength of light.
In this case, we are interested in the distance between the two dark fringes, which corresponds to the first minimum (m = 1). The angle of diffraction for the first minimum is given by:
sin(θ) = λ / d
We can rearrange this equation to solve for θ:
θ = arcsin(λ / d)
Given the wavelength of light (λ = 514 nm = 514 x 10^-9 m) and the width of the slit (d = 0.650 mm = 0.650 x 10^-3 m), we can calculate the angle of diffraction:
θ = arcsin(514 x 10^-9 m / 0.650 x 10^-3 m)
θ ≈ 0.800 radians
Now, let's find the distance between the two dark fringes on either side of the central bright fringe. We can use the small angle approximation:
y = L * tan(θ)
where L is the distance between the slit and the screen. Given L = 4.50 m, we can calculate the distance between the fringes:
y = 4.50 m * tan(0.800 radians)
y ≈ 4.61 m
Therefore, the distance between the two dark fringes on either side of the central bright fringe is approximately 4.61 meters.