Answer: To solve this problem, we can use the formula for the path difference between two waves undergoing reflection from a surface:
Δx = 2d cosθ
where Δx is the path difference, d is the thickness of the oil layer, and θ is the angle of incidence.
We know that the light undergoes constructive interference, which means that the path difference is equal to an integer multiple of the wavelength λ:
Δx = mλ, where m is an integer.
We can also use the relationship between the refractive indices of the two media and the angle of incidence to calculate the angle of reflection:
n1 sinθ = n2 sinφ
where n1 and n2 are the refractive indices of the two media (air and oil in this case), and φ is the angle of reflection.
We can start by calculating the angle of incidence. Since the light is perpendicular to the oil spill, the angle of incidence is 0 degrees, so sinθ = 0.
n1 sinθ = n2 sinφ
sinφ = (n1/n2) sinθ
sinφ = (1/1.2) x 0
sinφ = 0
This means that the angle of reflection is also 0 degrees.
Since the angle of incidence and reflection are both 0 degrees, we can simplify the path difference formula:
Δx = 2d
Substituting the values given, we have:
2d = mλ
d = (mλ)/2
We can use this equation to calculate the minimum thickness of the oil layer that would produce constructive interference for the given wavelength:
d = (mλ)/2
d = (1 x 409 nm)/(2 x 1.2)
d = 142.71 nm
This is the minimum thickness of the oil layer that would produce constructive interference for the given wavelength.