In an interference of light experiment, coherent light that contains two wavelengths: 640 nm (red-orange) and 425 nm (indigo-blue), passes through two narrow slits separated by 5.35 μm, and the interference pattern is observed on an observation screen 2.51 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.
x=kLλ/d
k=3, L=2.51 m, d=5.35•10⁻⁶ m
x₁= kLλ₁/d=3•2.51•640•10⁻⁹/5.35•10⁻⁶=0.9 m,
x₂= kLλ₂/d=3•2.51•425•10⁻⁹/5.35•10⁻⁶=0.595 m,
tanα₁=x₁/L=0.9/2.51 = 0.237 =>α₁=13.3°,
tanα₂=x₂/L=0.595/2.51 = 0.359 =>α₁=19.7°,
Δα=19.7°-13.3°=6.4°
Isn't the answer supposed to be in meters?
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 consider the conditions for constructive interference.
In an interference pattern, the bright fringes occur when the path difference between the light waves from the two slits is an integer number of wavelengths. The formula for the path difference is given by:
path difference = d * sin(θ)
where:
- d is the separation between the two slits
- θ is the angle between the central bright fringe and the third bright fringe
In this case, we are given the separation between the slits (5.35 μm) and we need to find the linear separation on the screen. Let's denote the linear separation as L.
Using simple trigonometry, we can relate the linear separation L to the angle θ and the distance between the slits d:
L = D * tan(θ)
where:
- D is the distance from the slits to the observation screen (2.51 m)
To find the angle θ, we can use the small angle approximation. For small angles, we can say that sin(θ) ≈ tan(θ) ≈ θ (in radians).
Therefore, we can simplify the expression for L:
L ≈ D * θ
Now we need to find the angle θ for the third bright fringe. To do that, we can use the formula for the fringe spacing:
λ = d * sin(θ)
where:
- λ is the wavelength of light
For the third bright fringe, we need to consider the path difference for both wavelengths (640 nm and 425 nm) separately. So we have:
λ1 = 640 nm = 640 × 10^-9 m
λ2 = 425 nm = 425 × 10^-9 m
For the third bright fringe, the path difference should be equal to 2 wavelengths for λ1 and 2 wavelengths for λ2. Hence:
2 * λ1 = d * sin(θ1)
2 * λ2 = d * sin(θ2)
Now we can rearrange these equations to solve for θ:
sin(θ1) = (2 * λ1) / d
sin(θ2) = (2 * λ2) / d
Finally, we can calculate the linear separation on the screen for the third bright fringes by substituting the angle θ into the L equation:
L ≈ D * θ
By solving these equations and substituting the given values, we can find the linear separation on the screen.