Michaelson Interferometer and changing length of glass rod

In summary, we can use the equation y = mλ/2 to find the change in distance (y) and then calculate the number of fringes (m) by dividing this value by half of the wavelength (λ/2). I hope this helps you understand the problem better. Best of luck on your upcoming final exam.
  • #1
JackFlash
15
0

Homework Statement


The index of refraction of a glass rod is 1.48 at T=20°C and varies linearly with temperature, with a coefficient of 2.5 E -5 °C-1. The coefficient of linear expansion of the glass is 5 E -6 °C-1. At 20.0°C the length of the rod is 3.00 cm. A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of 5.00 °C/min. The light source has wavelength λ = 599nm, and the rod initially is at 20°C.

How many fringes cross the field of view each minute?

Homework Equations



y = [itex]\frac{mλ}{2}[/itex]
Where y is the change in distance.
m is the number of fringes.
λ is the wavelength
(just for simplicity)
α = 5 E -6 °C-1
β = 2.5 E -5 °C-1

The Attempt at a Solution


I've attempted this problem several times. I had gotten it correct once before, but can't seem to do it this time (practicing it for my upcoming final).
First, I find the change in distance. 5α since the temperture rises by 5°C in one minute.
Now, the change in the index of refraction: 5β.
Since the equation deals with the change in distance rather than the total distance, we plug into the original equation.

(5α) = .5m(λ/(n*5β) or, solving for m
m = [itex]\frac{2*5α cm*1.48*5β}{599nm}[/itex]
Make the units equal, and the answer is wrong.

So I repeated my steps, this time setting the temperture to 25°C. No dice.
Tried multiplying the length to 5α and both methods (°C = 5 and 25). But Nope.avi

The answer should be around 14 I believe. Some guidance and explanation would be much appreciated.
 
Physics news on Phys.org
  • #2

Thank you for posting your question on the forum. I would like to help you find the correct solution to this problem.

Firstly, let's review the given information:

- Index of refraction of glass rod (n) = 1.48 at T = 20°C
- Index of refraction varies linearly with temperature, with coefficient β = 2.5 E -5 °C-1
- Coefficient of linear expansion of glass (α) = 5 E -6 °C-1
- Length of the rod (L) = 3.00 cm at T = 20.0°C
- Light source wavelength (λ) = 599nm
- Temperature of rod is increasing at a rate of 5.00 °C/min

To find the number of fringes crossing the field of view each minute, we need to use the equation:

y = mλ/2

Where y is the change in distance, m is the number of fringes, and λ is the wavelength.

To find the change in distance, we need to consider the change in index of refraction (Δn) and the change in length (ΔL) of the rod.

Δn = n2 - n1 = β(T2 - T1)

ΔL = L2 - L1 = αL(T2 - T1)

Substituting the given values, we get:

Δn = (2.5 E -5 °C-1)(5°C/min) = 1.25 E -4 /min

ΔL = (5 E -6 °C-1)(3.00 cm)(5°C/min) = 7.5 E -5 cm/min

Now, we can find the change in distance:

y = (1.25 E -4 /min)(7.5 E -5 cm/min) = 9.375 E -10 cm

Converting this to nanometers, we get:

y = 9.375 E -10 cm x 10^7 nm/cm = 9.375 nm

Therefore, the number of fringes crossing the field of view each minute is:

m = 2y/λ = (2)(9.375 nm)/(599nm) = 0.031

This means that approximately 0.031 fringes cross the field of view each minute.

 

1. What is a Michaelson Interferometer?

A Michaelson Interferometer is a scientific instrument used to measure small differences in the length of objects. It works by splitting a beam of light into two paths, reflecting them back onto each other, and then measuring the interference pattern to determine any changes in length.

2. How does a Michaelson Interferometer work?

The Michaelson Interferometer uses a beam splitter to divide a beam of light into two paths. One path travels through a stationary reference arm, while the other travels through a movable sample arm. The paths are then reflected back onto each other and the interference pattern is measured, allowing for precise measurements of changes in length.

3. What is the purpose of changing the length of a glass rod in a Michaelson Interferometer?

The glass rod is used in the sample arm of the Michaelson Interferometer to introduce a known change in length. This change in length is compared to the stationary reference arm to measure any changes in length of the object being studied.

4. What are the applications of a Michaelson Interferometer?

Michaelson Interferometers have a wide range of applications in the field of optics and precision measurements. They are commonly used in the study of materials, including determining their refractive index and thermal properties. They are also used in the calibration of precision instruments and in the measurement of small movements in mechanical systems.

5. How accurate is a Michaelson Interferometer?

The accuracy of a Michaelson Interferometer depends on various factors such as the quality of the optical components, stability of the instrument, and the precision of the measurement equipment. However, with proper calibration and careful handling, a Michaelson Interferometer can achieve extremely high levels of accuracy, making it a valuable tool in scientific research and industrial applications.

Similar threads

  • Introductory Physics Homework Help
Replies
4
Views
1K
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
3K
  • Introductory Physics Homework Help
Replies
5
Views
922
  • Introductory Physics Homework Help
Replies
8
Views
4K
  • Introductory Physics Homework Help
Replies
8
Views
1K
  • Special and General Relativity
Replies
8
Views
433
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
1K
Back
Top