A thin film of oil (\(n_o = 1.5\)) with varying thickness floats on water (\(n_w = 1.33\)). When it is illuminated from above by white light, the reflected colors are as shown in the figure. In air, the wavelength of yellow light is \(\lambda_{\text{yellow}} = 580 \text{ nm}\).(1) Why are there no reflected colors at point A?(2) What is the oil’s thickness at point B?### Diagram ExplanationThe diagram shows a cross-sectional view of a thin film of oil floating on water. Above the oil is air. The oil has a refractive index \(n_o = 1.5\), and the water beneath it has a refractive index \(n_w = 1.33\).The oil film shows varying colors due to interference effects. At point A, the region is labeled as "Dark," indicating no color is reflected. Moving from A to B, colors change:- A is labeled as dark.- Near A, a sequence of colors is labeled from left to right as "Blue," "Yellow," "Red," "Blue," "Yellow." This variation in color is due to constructive and destructive interference at different thicknesses of the oil film, depending on the wavelength of light in air (\(\lambda_{\text{yellow}} = 580 \text{ nm}\)). When the path difference causes destructive interference for all visible wavelengths, the region appears dark, as at point A. Different colors appear at other points due to constructive interference for specific wavelengths.

(a)At point A on the figure 24.60 the thickness of layer of oil is very small and is negligible as compared to the wavelength of light. For a constructive interference, the thickness of the medium must be greater than or equal to the wavelength of light. Therefore, the light reflected from the interfaces does not constructively interference and hence gives dark color.(b)The light wave falling on the air-oil interface gets reflected from it suffering a phase shift of. The ray refracted in the medium gets reflected from the lower surface of oil and transmits through the oil-air interface. This wave does not suffer any phase change with respect to the incident wave.The path difference between the two reflected waves is equal to twice of the thickness of layer of oil. For constructive interference, the path difference must be an integral multiple of the wavelength of light. For constructive interference, the condition is as follows.Here, t is the thickness of oil layer as in the figure 24-60, λn is the wavelength of light in oil and m is an integer that takes values.The wavelength of light in oil is given below.Here, λ is the wavelength of light in air and n is the refractive index of oil.Therefore, the equation for path difference takes the following form.Substitute 1 for m, 580 nm for λ, and 1.5 for n.Hence, the thickness at point B is.The oil's thickness at point B is 290 nm.

Answered: A thin film of oil (no = 1.5) with…