Find the minimum thickness of a soap bubble that appears red when illuminated by white light perpendicular to its surface. Take the wavelength to be \(680 \mathrm{nm}\), and assume the same index of refraction as water.

Constructive interference occurs when two or more waves superimpose to produce a combined amplitude that is greater than that of the individual waves. This phenomenon is particularly important in the context of thin film interference, such as what we see with soap bubbles and oil slicks. When light reflects off the top and bottom surfaces of a thin film and then recombines, if the waves are 'in phase'—meaning the peaks of the waves align with each other—they amplify the light's intensity, resulting in bright, vivid colors.

For constructive interference in thin films, the thickness of the film plays a crucial role, as it determines whether the reflected light waves will be in phase. To identify these conditions, one uses equations that consider the wavelength of light and the refractive index of the film material, ultimately determining the film thickness for which the constructive interference will occur for a specific color of light.

Optical path difference (OPD) is critical to understanding thin film interference. It represents the difference in the path lengths that two light waves travel before recombining. However, simply measuring the physical distance isn't enough—we must also account for the difference in speed of light in various mediums by using the index of refraction. The OPD helps us to calculate whether waves will undergo constructive or destructive interference.

In the case of a soap bubble, the OPD includes the thickness of the film multiplied by the index of refraction, effectively 'stretching' the path the light travels inside the film compared to the same distance in a vacuum. For constructive interference, the OPD must be a multiple of the light's wavelength, adjusted for phase changes upon reflection. This principle guides us in solving for the minimum thickness that produces the desired interference effect.

The wavelength of light is the distance over which the wave's shape repeats. It is a crucial factor in interference patterns because it determines the color of light we see. In thin film interference, light of a specific wavelength will constructively or destructively interfere depending on the film's thickness.

Visible light includes wavelengths approximately from 400 nm (violet) to 700 nm (red). In our exercise, the wavelength given for the red light is 680 nm. This number is an essential component for calculating the thickness of a film that would cause red light to undergo constructive interference, leading to the appearance of a red hue in the soap bubble when observed at a specific angle and under white light illumination.

The index of refraction, denoted as 'n', is a dimensionless number that describes how light propagates through a medium compared to the speed of light in a vacuum. It is defined by the ratio of the speed of light in a vacuum to the speed of light in the medium. The index of refraction is a fundamental property that affects the bending of light at interfaces and the optical path length within materials.

In the context of thin film interference, the index of refraction of the film determines how much the light slows down as it passes through and affects the optical path difference. A higher index of refraction means a greater reduction in the speed of light within the material, and therefore, a larger optical path length for the same physical film thickness. In our exercise, the soap bubble is considered to have the same index of refraction as water, n = 1.33, which is a key variable for computing the minimum thickness for the red appearance of the bubble.