After a minor oil spill, a think film of oil \((n=1.40)\) of thickness 450 nm floats on the water surface in a bay. (a) What predominant color is seen by a bird flying overhead? (b) What predominant color is seen by a seal swimming underwater?

When discussing thin film interference, particularly in relation to colors produced by an oil spill, the principle of constructive interference plays a pivotal role. Constructive interference occurs when waves overlap in a manner that causes their amplitudes to add together, resulting in an increased amplitude. In the context of light, this implies that certain wavelengths of light reinforce each other when reflecting off different layers of a film, such as oil, leading to the appearance of specific vibrant colors.

For a thin film of oil floating on water, constructive interference happens for specific wavelengths of light that satisfy the equation \( 2nt = m\lambda \) for reflection from the top layer of the film, and \( 2nt = (m+\frac{1}{2})\lambda \) for reflection from the bottom layer at the oil-water boundary. The path difference created by reflections at differing interfaces leads to constructive interference, culminating in bright hues being visible. To identify the predominant color, we must find the wavelengths that occur within the visible spectrum and apply the equations iteratively, starting with the smallest integer values of \( m \), until we identify a suitable wavelength.

Wavelength of light is a core aspect of understanding the phenomenon of thin film interference. It refers to the distance between two consecutive crests or troughs of a wave. The wavelength determines the color of light, with each color corresponding to a different range of wavelengths; red light has a longer wavelength compared to blue light.

To uncover which colors will dominate in the interference pattern of the oil film, one must calculate the wavelength of light that results in constructive interference. For example, when light reflects off the surface of an oil film, the equation \( \lambda = \frac{2nt}{m} \) (for a bird viewing from above) or \( \lambda = \frac{2nt}{m+\frac{1}{2}} \) (for a seal viewing from below) is used to find the effective wavelengths. The visible spectrum is typically between 400 nm and 700 nm, so only the wavelengths that fall within this range will correspond to the predominant color visible to the observer.

The refractive index (represented by \( n \)) is a dimensionless number that describes how much light bends, or refracts, when entering a material. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material. Different materials have different refractive indices, which influences how light behaves when transitioning from one medium to another.

In the case of thin film interference, the refractive index of the oil (\( n=1.40 \)) is vital to determining the subsequent interference pattern. The refractive index not only dictates how much the light bends but also influences the phase change that occurs upon reflection. When light reflects off the boundary of a medium with a higher refractive index, such as going from air to oil, it typically undergoes a half-wavelength phase shift. This phase change is an integral component when calculating the conditions for constructive or destructive interference and hence affects the color observed by the bird flying overhead or the seal swimming underwater.