Hi, I am having difficulties approaching and solving the following question:
A Michelson interferometer is used with red light of wavelength 632.8 nm and is
adjusted for a path difference of 20 µm. Determine the angular radius of the
a) first (smallest diameter) ring observed and
b) the tenth ring observed.
To solve this question, you need to have a basic understanding of interference in the Michelson interferometer and the concept of dark and bright fringes.
Let's break it down step by step:
Step 1: Understand the setup
A Michelson interferometer consists of a beam splitter, two mirrors, and a viewing screen. The beam splitter divides the incoming light beam into two beams, one that travels to Mirror 1 and one that travels to Mirror 2. Both beams are reflected back to the beam splitter, where they recombine and interfere with each other. The interference pattern that forms on the screen consists of dark and bright fringes.
Step 2: Calculate the number of fringes
The path difference between the two beams determines the number of fringes that appear on the screen. In this case, the path difference is given as 20 µm. The number of fringes can be calculated using the formula:
Number of fringes = (Path difference) / (Wavelength)
Plugging in the values:
Number of fringes = 20 µm / 632.8 nm
Note: The wavelength must be in the same units, so we convert 20 µm to nm:
Number of fringes = 20,000 nm / 632.8 nm
Step 3: Find the angular radius of each fringe
The angular radius of the fringes can be calculated using the formula:
Angular radius = (Distance from the center to the fringe) / (Distance between the screen and the mirrors)
In a Michelson interferometer, the bright fringe at the center is the zeroth-order, and the first-order fringe is the first bright fringe away from the center. The angular radius of the first-order fringe can be calculated as:
Angular radius of the first-order fringe = (Distance to the first-order fringe) / (Distance between the screen and the mirrors)
Step 4: Solve the problem
a) To find the angular radius of the first (smallest diameter) ring observed, we need to calculate the distance between the center and the first-order fringe. Since the path difference is symmetric, we can divide it equally between the two mirrors. So the distance to the first-order fringe is half of the path difference.
Distance to the first-order fringe = (Path difference) / 2
b) To find the angular radius of the tenth ring observed, we follow a similar approach. However, we need to consider the total distance between the center and the tenth-order fringe, which is 10 times the distance between the center and the first-order fringe.
Distance to the tenth-order fringe = 10 * (Distance to the first-order fringe)
Finally, plug in the values for the distance to the first-order fringe and the distance to the tenth-order fringe in the formula for angular radius to obtain the answers.
I hope this explanation helps you approach and solve the given question.