In an interference of light experiment, coherent light that contains two wavelengths: 614 nm (red-orange) and 429 nm (indigo-blue), passes through two narrow slits separated by 3.12 μm, and the interference pattern is observed on an observation screen 2.17 m away. (1 μm = 1×10-6 m, 1 nm = 1×10-9 m.)
What is the linear separation on the screen between the third bright fringes produced by the two wavelengths on one side of the central bright fringe in the experiment? (in meters)
Note: There is no need to make a small angle approximation here: you can calculate what you need directly.
To find the linear separation on the screen between the third bright fringes produced by the two wavelengths on one side of the central bright fringe, we need to use the formula for the separation between consecutive fringes in an interference pattern, known as the fringe spacing.
The fringe spacing can be calculated using the formula:
Δy = λ * D / d
where:
Δy is the fringe spacing (linear separation between consecutive bright or dark fringes)
λ is the wavelength of light
D is the distance from the slits to the screen
d is the separation between the two slits
Given data:
λ1 = 614 nm = 614 × 10^(-9) m
λ2 = 429 nm = 429 × 10^(-9) m
d = 3.12 μm = 3.12 × 10^(-6) m
D = 2.17 m
First, let's calculate the fringe spacing for each wavelength separately:
For the first wavelength (λ1 = 614 × 10^(-9) m):
Δy1 = (614 × 10^(-9) m) * (2.17 m) / (3.12 × 10^(-6) m)
For the second wavelength (λ2 = 429 × 10^(-9) m):
Δy2 = (429 × 10^(-9) m) * (2.17 m) / (3.12 × 10^(-6) m)
Now, since we are interested in the third bright fringe on one side of the central bright fringe, let's find the total linear separation for the third bright fringe:
Δy_total = 3 * max(Δy1, Δy2)
Substitute the values and calculate:
Δy_total = 3 * max([(614 × 10^(-9) m) * (2.17 m) / (3.12 × 10^(-6) m)], [(429 × 10^(-9) m) * (2.17 m) / (3.12 × 10^(-6) m)])
Once you calculate this expression, you will find the linear separation on the screen between the third bright fringes produced by the two wavelengths on one side of the central bright fringe in meters.