A soap bubble is 100 nm thick and illuminated by white light incident at a \(45^{\circ}\) angle to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?

Path difference is a critical concept when understanding constructive interference, especially in phenomena like the reflection of light from thin films. Constructive interference occurs when two light waves meet in phase—that is, their peaks and troughs align. This results in an amplified wave, a phenomenon often observed with colors on soap bubbles or oil slicks.

For a detailed understanding, imagine that light reflects off both the top and bottom surfaces of a film. Each reflected wave travels a different distance, and this difference in their paths determines whether they interfere constructively or destructively when they reunite. The ideal condition for constructive interference is when the path difference equals an integer multiple of the wavelength of light (denoted as mλ, where m is any whole number, and λ is the wavelength).

In the provided exercise, the path difference can be calculated as 2 times the thickness of the film times the cosine of the angle of refraction. This relates directly to how we perceive the color—which is the visible effect of constructive interference—of the thin film under lighting.

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction when light passes from one transparent medium to another. It’s a fundamental principle in optics that allows us to predict how light bends, or refracts, as it crosses boundaries between different materials.

The law states that the product of the index of refraction of the first medium and the sine of the angle of incidence is equal to the product of the index of refraction of the second medium and the sine of the angle of refraction. Mathematically, it's expressed as n1 * sin(θ1) = n2 * sin(θ2), where n1 and n2 are the refractive indices of the first and second media, and θ1 and θ2 are the respective angles.

In the context of our exercise, Snell's law helps us find the angle of refraction of light within a soap bubble, which is necessary to calculate the path difference for determining the constructive interference.

The angle of refraction is the angle between the refracted ray and the normal to the surface at the point of refraction. It is an essential variable to consider when light enters a material with a different refractive index because it changes the direction of light. This angle can be calculated using Snell's law once the indices of refraction and the angle of incidence are known.

In practical scenarios, such as our exercise, knowing the angle of refraction can be essential. It dictates how much light bends as it enters the soap bubble, which in turn affects the path difference. Using Snell's law and the given incidence angle (in this case, 45 degrees), we can derive the angle of refraction, leading us closer to solving for the wavelength of light that experiences constructive interference.

The visible light spectrum is the portion of the electromagnetic spectrum that can be detected by the human eye. It encompasses wavelengths approximately in the range of 380 to 750 nanometers. Each wavelength within this range corresponds to a different color, perceivable to us as the various hues of the rainbow. Violet light has the shortest wavelength, followed by blue, green, yellow, orange, and red, which has the longest wavelength that we can see.

In the exercise, once the wavelength of the most constructively reflected light is found, it can be compared to the known visible light spectrum to determine the color of the light. It's a brilliant illustration of how the physical properties of light translate into the everyday experience of color.

Interference in thin films, such as soap bubbles or layers of oil on water, is caused by light waves reflecting off the top and bottom boundaries of the film. The interaction of these reflected waves with each other can create a pattern of interference — both constructive and destructive.

When the path differences between these waves lead to constructive interference, we see bright colors. Conversely, destructive interference results when the waves are out of phase and cancel each other out, leading to dark spots. By analyzing the thickness of the film, alongside the wavelength of light and the refractive index, one can predict the specific colors that will appear due to this interference pattern, as demonstrated in our example problem.