Question

A Michelson interferometer is illuminated with a 600.-nm light source. How many fringes are observed if one of the mirrors of the interferometer is moved a distance of 200. \(\mu \mathrm{m} ?\)

Short answer

Answer: 667 fringes.

Step-by-step solution

Convert distance and wavelength to the same units

In this problem, the wavelength is given in nanometers and the distance the mirror is moved is given in micrometers. To make calculations easier, let's convert both values to meters: Wavelength (\(\lambda\)) = 600 nm = 600 * 10^{-9} m Distance mirror is moved (d) = 200 μm = 200 * 10^{-6} m

Calculate path difference

The path difference is the difference in distance traveled by the light in the two arms of the interferometer. Since the light reflects back and forth in one arm, the path difference is equal to twice the distance the mirror is moved: Path difference (PD) = 2 * d = 2 * (200 * 10^{-6} m) = 400 * 10^{-6} m

Calculate the number of fringes

To find the number of fringes observed, we need to determine how many full wavelengths of the light source fit into the path difference. We can do this by dividing the path difference by the wavelength: Number of fringes (N) = PD / \(\lambda\) = (400 * 10^{-6} m) / (600 * 10^{-9} m)

Simplify and find the final answer

Now, we just need to simplify the expression and find the number of fringes: N = (400 * 10^{-6} m) / (600 * 10^{-9} m) = (400 / 600) * (10^3) N = (2 / 3) * 1000 N = 666.67 Since the number of fringes must be a whole number, we can round the result to the nearest integer. Therefore, the number of fringes observed is 667.

Key concepts

Understanding Light Interference

Light interference occurs when two or more light waves overlap, resulting in a new wave pattern. This phenomenon can either enhance or reduce the intensity of light, depending on the nature of the overlapping waves. When the peaks of two waves align (constructive interference), the resulting wave is intensified. Conversely, if the peak of one wave aligns with the trough of another (destructive interference), the waves may cancel each other out, resulting in reduced intensity. The Michelson interferometer is a powerful tool that leverages light interference by splitting a single light beam into two paths. After traveling different distances, these beams recombine, interferencing with each other to create a pattern of bright and dark fringes. This interference pattern is key in many scientific measurements and calculations, including determining the number of fringes in this exercise.

Basics of Wavelength Conversion

In physics problems, it's essential to work with consistent units. Wavelength is often given in nanometers (nm), especially in problems involving visible light, because this unit neatly fits within the range of visible light (approximately 380-750 nm). However, to harmonize with other measurements, such as distances typically given in meters or micrometers, we often convert the wavelength into these units. This simplifies calculations and minimizes errors. For this exercise, the wavelength of 600 nm is converted to meters by multiplying by \(10^{-9}\), yielding a result of \(600 \, \times \, 10^{-9} \, \text{m}\). These conversions are fundamental in ensuring the precision and accuracy of the mathematical processes involved in the analysis.

Path Difference Calculation in Interferometry

In the context of the Michelson interferometer, the path difference refers to the extra distance one beam of light travels relative to another. This difference is crucial because it determines how the overlapping waves will interfere with each other. In this problem, the path difference is calculated by considering that the reflected light travels a distance equivalent to twice the distance that a mirror is moved. For a mirror movement of 200 \(\mu\text{m}\), the path difference is \(2 \times 200 \, \times \,10^{-6} \, \text{m} = 400 \, \times \,10^{-6} \, \text{m}\). Recognizing this key calculation allows us to understand the basis upon which we can determine the number of observed interference fringes.

Calculating the Number of Fringes

Fringes are the bands of alternating light and dark spots that appear due to interference. Calculating the number of fringes involves determining how many full wavelengths fit into the path difference. This is expressed by the formula \( N = \frac{\text{Path Difference}}{\lambda} \). For our example, with a path difference of \(400 \, \times \,10^{-6} \, \text{m}\) and a wavelength of \(600 \, \times \,10^{-9} \, \text{m}\), the number of fringes is \( N = \frac{400 \, \times \,10^{-6}}{600 \, \times \,10^{-9}} = 666.67 \). Since fringes must be whole numbers, we round to the nearest integer, resulting in 667 fringes. This final number represents how many bright spots or fringe changes are observed when the mirror is moved.

Approaches to Physics Problem Solving

Solving physics problems effectively requires systematic reasoning and attention to detail. Here's how to tackle problems like the Michelson interferometer fringe calculation:

• Understand the Problem: Begin by fully comprehending what is being asked, including identifying all given data and relevant concepts such as light interference and path difference.
• Consistent Units: Always ensure that units are consistent, converting where necessary to avoid errors in calculations.
• Use Clear Formulas: Identify and apply the correct formulas. For path difference and fringe calculations, know the relationship between path difference, wavelength, and number of fringes.
• Verify Your Solution: Check if the final answer makes sense contextually and adheres to the requirement, such as having a whole number of fringes in this scenario.

Analyzing each step critically helps to avoid mistakes and simplify complex problems into manageable parts.