A thin layer of an oil \((n=1.60)\) floats on top of water \((n=1.33) .\) One portion of this film appears green \((\lambda=510 \mathrm{nm})\) in reflected light. How thick is this portion of the film? Give the three smallest possibilities.

Thin-film interference is a phenomenon that occurs when light waves reflected by the two surfaces of a thin film of a material interfere with each other, resulting in either constructive or destructive interference. This interference results in colors being reflected when white light is incident upon a layer, such as the oil film in the problem.

For constructive interference to occur, the phase difference φ between the waves must be an integral multiple of 2π, which corresponds to a path difference of an integral multiple of the wavelength. For thin films, the equation for constructive interference can be written as: 2 * t * n * cos(θ) = m * λ where t is the thickness of the film, n is the refractive index of the film, θ is the angle of refraction, m is an integer representing the order of interference, and λ is the wavelength of the light. In this problem, we are looking for the thickness of the film (t) when green light (\(λ=510 nm\)) is reflected. Moreover, as the angle of incidence is not given, consider normal incidence, i.e., incoming light is perpendicular to the oil film's surface (\(θ = 0°\)), in which case \(\cos{θ} = 1\).

The smallest three possibilities correspond to m = 1, m = 2, and m = 3. For each of these cases, we will calculate the thickness of the oil film.

We'll use the equation 2 * t * n * cos(θ) = m * λ to find the thickness t, setting cos(θ) = 1: 1) For \(m=1\): \(t = \frac{m * \lambda}{2 * n} = \frac{1 * 510 nm}{2 * 1.60} = 159.38 nm\) 2) For \(m=2\): \(t = \frac{m * \lambda}{2 * n} = \frac{2 * 510 nm}{2 * 1.60} = 318.75 nm\) 3) For \(m=3\): \(t = \frac{m * \lambda}{2 * n} = \frac{3 * 510 nm}{2 * 1.60} = 478.13 nm\)

The three smallest possibilities for the thickness of the oil film are: 1) \(t = 159.38 nm\) 2) \(t = 318.75 nm\) 3) \(t = 478.13 nm\)