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 \(d=0.200 \mathrm{~mm},\) exactly \(N=800\) fringes are shifted on the screen. What is the original wavelength (in air) of the monochromatic light?

Short answer

Question: The movable mirror in a Michelson interferometer submerged in water moves a distance of 1.2 mm, causing a fringe shift of 800 fringes. Calculate the original wavelength of the monochromatic light in air. Answer: Step 1: Find the path difference Path difference = 2d = 2 × 1.2 mm = 2.4 mm Step 2: Calculate the refractive index of water Refractive index (n) = 1.33 Step 3: Determine the traveled distance in water Traveled distance in water = (Path difference) / n = 2.4 mm / 1.33 = 1.8045 mm Step 4: Calculate the wavelength in air Wavelength in air = (Traveled distance in water) / N = 1.8045 mm / 800 = 0.002256 mm = 2.256 nm The original wavelength of the monochromatic light in air is 2.256 nm.

Step-by-step solution

Find the path difference

The movable mirror moves a distance, denoted by \(d\), and \(N\) fringes are shifted on the screen. The path difference that causes the fringe shift can be determined by multiplying the distance moved by the mirror by 2, since the light travels back and forth. Path difference = \(2d\)

Calculate the refractive index of water

The refractive index of water, denoted by \(n\), is approximately 1.33 (you can look up this value in a reference table or use this value given for this exercise).

Determine the traveled distance in water

Since the Michelson interferometer is submerged in water, we need to account for the refractive index when calculating the actual traveled distance by the light. The formula to calculate the traveled distance in a medium is: Traveled distance in water = (Path difference) / \(n\)

Calculate the wavelength in air

We know that the traveled distance in water causes a fringe shift of \(N\) fringes. So, the original wavelength (in air) of the monochromatic light can be determined by dividing the traveled distance in water by the number of fringes shifted. The formula to calculate the wavelength in air is: Wavelength in air = (Traveled distance in water) / \(N\) Now let's apply these steps to solve the given exercise.

Key concepts

Optical Path Difference

Understanding the optical path difference is crucial when studying light and its interactions. In experiments like the Michelson interferometer, where light is split and travels different paths before recombining to produce interference patterns, the concept of optical path difference is indispensable.

Optical path difference refers to the difference in distance traveled by two beams of light in an interferometer. Since each wave travels a distinct path, upon recombination they may not be in the same phase. When one mirror moves by a distance of \(d\), the beam of light reflecting off this mirror travels an extra distance of \(2d\) because it has to travel to the mirror and back. This creates the conditions for constructive or destructive interference, which is seen as the shifting of fringes on the screen. If constructive interference occurs, bright fringes will appear; when it's destructive, dark fringes will form.

Refractive Index

The refractive index is a fundamental property of materials that affects the speed at which light travels through them. Scientifically speaking, it is the ratio between the speed of light in a vacuum (the fastest possible speed) and its speed in the material. A refractive index, denoted by \(n\), higher than 1 implies that light slows down in that material compared to a vacuum.

Water, for instance, has a refractive index of approximately 1.33. This means that when the Michelson interferometer is submerged in water, we need to adjust for this slowing effect. The path difference in water becomes the original path difference divided by the refractive index of water. This adjustment is necessary to accurately calculate the physical properties of the light, such as its wavelength. Different materials will have different refractive indices which must be considered in optical experiments.

Monochromatic Light

Monochromatic light plays a pivotal role in interference experiments like the Michelson interferometer. Monochromatic, in essence, means 'one color', and in terms of light, it refers to a beam of photons that all have the same wavelength and frequency. This uniformity is essential to produce clear and stable interference patterns.

The use of monochromatic light in the experiment ensures that the calculation for the wavelength is based on a single, distinct value. When we discuss moving the mirror to create a fringe shift, the shifting of 800 fringes indicates that we have monochromatic light because a mix of colors (or wavelengths) would not produce such a precise pattern. The Michelson interferometer uses the predictable behavior of monochromatic light to measure optical path differences and calculate the wavelength of light.

Wavelength Calculation

Wavelength calculation is the determination of the distance between successive crests of a wave, usually in metrics like meters. It's a core concept in optical physics as it directly correlates to the energy, frequency, and color of light.

In the context of the Michelson interferometer submerged in water, the wavelength of monochromatic light in air is calculated through the known fringe shift and the path the light has traveled in water considering its refractive index. The formula \(\text{wavelength in air} = \frac{\text{traveled distance in water}}{N}\) is used to calculate the original wavelength of the light used. This wavelength is what gives the light its color and can help identify the source of the light used in the experiment. By calculating the wavelength, scientists can gain insights into the characteristics of various light sources and their behavior in different media.