Constructive interference is observed from directly above an oil slick for wavelengths (in air) \(440 \mathrm{nm}\) and 616 nm. The index of refraction of this oil is \(n=1.54\). What is the film's minimum possible thickness?

When dealing with phenomena such as the iridescent patterns we often see on an oil slick, we're observing effects caused by the interaction of light with thin layers of material - in this case, oil. The literal thickness of this oil layer can affect the way light is either transmitted or reflected.

Interference occurs when two waves overlap and combine with each other. Constructive interference, specifically, happens when the peaks (or troughs) of both waves align and amplify the resulting wave. This is precisely what's being observed when certain colors appear to stand out from the oil slick. In the context of our problem, the 'minimum possible thickness' of the oil film relates to the thinnest layer necessary for constructive interference to occur for specific wavelengths of light (e.g., 440 nm and 616 nm in air).

How is that minimum thickness determined? It can be calculated as half the wavelength of the light within the medium. Why half the wavelength? Because a full wavelength would result in a part of the light wave being in phase and another part being out of phase, leading to both constructive and destructive interference. At half the wavelength, light reflecting off the top and bottom surfaces of the oil slick undergoes constructive interference, reinforcing the color associated with that particular wavelength.

Understanding refractive index is key in solving our problem because it helps explain why light bends when transitioning between different media, like air and oil. The refractive index, denoted by the symbol 'n', is a dimensionless number that describes how fast light travels through a medium. It is the ratio of the speed of light in a vacuum to the speed of light in the medium.

If we have an index of refraction of 1.54 for the oil, as stated in the exercise, this means that light moves more slowly in the oil than it does in air (which has a refractive index of approximately 1). Physically, as light enters the oil at a perpendicular angle, it 'slows down' leading to a reduction in wavelength within the oil compared to its wavelength in air.

The relevance of this concept to our problem is direct: to determine the physical conditions that create constructive interference, we must first calculate the wavelength of light within the oil. We do this by dividing the wavelength of light in air by the refractive index of the oil. This shorter wavelength in oil can then be used to establish the film's thickness where constructive interference occurs.

The conversion of the wavelength of light as it passes from one medium to another is governed by a simple formula: the wavelength in the new medium is equal to the wavelength in the original medium divided by the refractive index of the new medium. This transformation is critical when calculating phenomena such as film thickness for constructive interference.

In our exercise, to convert the given wavelengths (440 nm and 616 nm) from air (where the refractive index is assumed to be approximately 1) to the oil (with a refractive index of 1.54), we must divide those values by 1.54. This gives us new wavelengths for light inside the oil medium, which are crucial to determining the minimum film thickness where constructive interference will enhance those specific wavelengths.

Once converted, these wavelengths in the oil are halved, as discussed earlier, to determine where constructive interference occurs. The 'halving' is essentially a reflection of the fundamental nature of waves and their interactions at boundaries. These calculations are indispensable for understanding the interplay of light with thin films and can be applied to things ranging from oil spills to designing antireflective coatings and understanding soap bubbles.