Question

The thermal stability of a Michelson interferometer can be improved by submerging it in water. Consider an interferometer that is submerged in water, measuring light from a monochromatic source that is in air. If the movable mirror moves a distance of \(d=0.200 \mathrm{~mm},\) exactly \(N=800\) fringes move by the screen. What is the original wavelength (in air) of the monochromatic light?

Short answer

Answer: The original wavelength of the monochromatic light in air is 6.65 × 10^-7 m.

Step-by-step solution

Write down the formula for the number of fringes formed in the Michelson interferometer

We will use the formula for the number of fringes formed in a Michelson interferometer, which is given by: \(N = \frac{2nd}{\lambda_{air}}\) Here, \(N\) is the number of fringes, \(n\) is the refractive index of the medium (water in this case), \(d\) is the distance traveled by the movable mirror, and \(\lambda_{air}\) is the wavelength of light in air.

Write down the refractive index of water

The refractive index of water (\(n\)) is approximately 1.33.

Insert values into the formula and solve for the wavelength of light

We have \(N = 800\) fringes, \(n = 1.33\), and \(d = 0.200 \mathrm{~mm}\). Inserting these values into the formula, we get: \(800 = \frac{2(1.33)(0.200 \times 10^{-3})}{\lambda_{air}}\) Now, we solve for the wavelength of the light in air (\(\lambda_{air}\)). \(\lambda_{air} = \frac{2(1.33)(0.200 \times 10^{-3})}{800}\) \(\lambda_{air} = \frac{2(1.33)(0.200 \times 10^{-3})}{800} = 6.65 \times 10^{-7} \mathrm{~m}\) Therefore, the original wavelength of the monochromatic light in air is \(\lambda_{air} = 6.65 \times 10^{-7} \mathrm{~m}\).

Key concepts

Interference Fringes

When exploring the behavior of light waves, the concept of 'interference fringes' is key. These fringes arise when two waves superimpose and combine. In the context of a Michelson interferometer, a beam of light is split into two paths, which are then recombined to produce a pattern of bright and dark lines known as interference fringes. The appearance and movement of these fringes can be carefully measured to observe very fine changes in distance or to calculate the wavelength of the light used.

Let's consider an exercise to see interference fringes in action: a Michelson interferometer submerged in water is used to measure the light from a monochromatic source. As its movable mirror changes position, the interference pattern shifts accordingly. With each small move, a specific number of fringes pass a reference point. By knowing the distance the mirror has moved and counting the fringes, the wavelength of light in air can be determined using the formula provided in the step-by-step solution.

Refractive Index

The refractive index is a measure of how much a material can bend or refract light. Different media slow down light to varying degrees, which is expressed as their refractive index, a dimensionless number. For instance, water has a refractive index of around 1.33, which means that light travels 1.33 times slower in water than in a vacuum. This concept is essential when dealing with light traveling through different substances.

In the given exercise, the Michelson interferometer is immersed in water, necessitating an understanding of how water's refractive index affects the calculation of light's wavelength. Due to this refractive index, light's phase and speed change when entering or leaving water, thus affecting the interference pattern created by the Michelson interferometer. With the known refractive index of water, it's possible to compute the wavelength of light in air by incorporating it into the formula, as shown in the solution steps.

Wavelength of Light

The wavelength of light is the distance between consecutive crests of a wave and is typically measured in meters (m). Light's wavelength is one of the defining characteristics of its color when it is within the visible spectrum. In the exercise regarding the Michelson interferometer, the wavelength of monochromatic light is what we want to determine.

By utilizing the relationship between the moved distance of the interferometer's mirror, the refractive index of the medium, and the number of interference fringes observed, the wavelength of light in air can be calculated. The step-by-step solution illustrates how moving the mirror by a known distance resulted in a specific number of fringes moving, which allowed us to back-calculate the light's original wavelength.

Monochromatic Light Source

A monochromatic light source emits light of a single wavelength or color. This uniformity makes it ideal for experiments requiring coherent light, such as those involving a Michelson interferometer. When testing with monochromatic light, the resulting interference fringes are more distinct and easier to count, which is crucial for precise measurements.

In the example exercise, the Michelson interferometer uses a monochromatic light source originating in air. The predictability and consistency of monochromatic light allow for the creation of stable and measurable interference patterns, which can be used to deduce properties of the light such as its wavelength, as demonstrated in the provided solution.