An oil slick on water is 120 nm thick and illuminated by white light incident perpendicular to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is \(1.40 ?\)

Constructive interference occurs in waves when two or more coherent waves coincide in phase, which means their crests and troughs line up perfectly. This synchronization causes the wave amplitudes to add up, resulting in a wave of greater amplitude. In the context of thin-film interference, such as an oil slick on water, constructive interference happens when reflected light waves from the top and bottom surfaces of the film combine to enhance each other.

Imagine tossing two stones into a still pond some distance apart. The ripples from each disturbance move outward until they meet. If the crests of the ripples from one stone align with the crests from the other, the water rises higher than the ripples from any single stone, illustrating constructive interference.

In terms of mathematical analysis, when light of wavelength \( \lambda \) reflects off the layers of the thin film at certain angles and the path difference is a multiple of \( \lambda \) (specifically, \( m \lambda \) where \( m \) is an integer), constructive interference is achieved. This can lead to observable colorful patterns or even determine the perceived color of the object, as seen with the oil slick exercise.

The index of refraction, often denoted as \( n \) , is a dimensionless number that describes how light travels through a medium relative to its speed in a vacuum. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material: \( n = \frac{c}{v} \), where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the medium. Every material has a characteristic index of refraction.

The index of refraction is pivotal in thin-film interference because it affects the phase of the reflected light. When light passes from a medium with a lower index of refraction to a higher one (like air to oil), it undergoes a phase shift of 180 degrees, which is equivalent to a half-wavelength (\( \frac{\lambda}{2} \) ). This must be taken into account when calculating the constructive interference condition in a thin film, such as the oil slick in the exercise, to predict the observed color.

The visible light spectrum consists of the range of electromagnetic radiation that is visible to the human eye. This range is typically described as having wavelengths between approximately 400 nanometers (nm) and 700 nm. Each wavelength within this spectrum corresponds to a different color, beginning with violet at the shortest wavelengths and ending with red at the longest wavelengths.

Colors of the Visible Spectrum

• Violet: 400–450 nm
• Blue: 450–495 nm
• Green: 495–570 nm
• Yellow: 570–590 nm
• Orange: 590–620 nm
• Red: 620–700 nm

The oil slick in our exercise appears a certain color because the wavelength of light it most constructively reflects falls within the range of visible light. The value of 672 nm found in the solution aligns with the color red, hence under white light, parts of the oil film reflecting this wavelength will indeed appear red to an observer.