Measuring the Distance between the Fringes of a Diffraction Grating

In summary: We can conclude that measuring from one fringe to the other will give us a more accurate measurement of the radius of the CD, assuming that the right measurement is greater than the other.
  • #1
KDPhysics
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Homework Statement
Determining the pitch distance on a CD acting as a transmission grating.
Relevant Equations
$$d\sin\theta = n\lambda$$
I have been preparing for a physics practical on diffraction. More specifically, we will use a CD as a transmission grating (by peeling off the reflective layer), and measure the distance between the fringes for a specific distance between the CD and the viewing wall. However, it is unclear whether we should measure the distance between the two 1st order fringes and divide by two, or measure the distance from the principal maxima to the left 1st order, and then from the principal maxima to the right 1st order, and take the average. Which is more suitable for this experiment? I believe that there is no great difference, since the absolute uncertainty remains the same in both cases.
 
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  • #2
What do you mean by absolute uncertainty? If you want to find the radius of a coin do you find the center first and then measure from the center of the coin to the edge or do you find the distance along a diameter and divide by two?
 
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  • #3
Well, let the measurements for the 1st order right fringe be ##x_r\pm\delta x## and the measurement to the left be ##x_l\pm\delta x##. Taking the average of the two: ##\hat{x} = \frac{x_r+x_l}{2}\pm\delta x##.
If instead we measure the distance from one fringe to the other, and then divide by two: ##x = \frac{x_r+x_l}{2}\pm\delta x## which is the same.
 
  • #4
KDPhysics said:
Well, let the measurements for the 1st order right fringe be ##x_r\pm\delta x## and the measurement to the left be ##x_l\pm\delta x##. Taking the average of the two: ##\hat{x} = \frac{x_r+x_l}{2}\pm\delta x##.
If instead we measure the distance from one fringe to the other, and then divide by two: ##x = \frac{x_r+x_l}{2}\pm\delta x## which is the same.
Please fix the LaTeX code to make it readable.
 
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  • #5
kuruman said:
Please fix the LaTeX code to make it readable.
yeah should be fixed sorry
 
  • #6
kuruman said:
What do you mean by absolute uncertainty? If you want to find the radius of a coin do you find the center first and then measure from the center of the coin to the edge or do you find the distance along a diameter and divide by two?
By absolute uncertainty, I mean the sensitivity in the measurement (if I were using a ruler, it would be ##\pm 0.001m##).
Also, in the case of the coin you usually don't have the exact center. Instead, in the diffraction pattern you do, it's the principal maxima.
 
  • #7
KDPhysics said:
By absolute uncertainty, I mean the sensitivity in the measurement (if I were using a ruler, it would be ±0.001m±0.001m).
Thanks for fixing the LaTeX. Uncertainties don't add the way you show. You need to consider error propagation. See here for example. In the second procedure you add an extra uncertainty due to the measurement of the central maximum.

On edit: I used the coin example to illustrate that the determination of the radius is more accurate when you eliminate the uncertainty of finding the position of the center which is what you have to do here.
 
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  • #8
Shouldn't the uncertainty be the same either way? Measuring from one fringe to the other, then the uncertainty will be 0.002m using a meter ruler. Similarly, measuring from the principal maxima to each fringe, the uncertainty will still be 0.002m.

Also, we can use the linear approximation for error propagation (since we're still in high school).

EDIT: measurement uncertainty should be ##\pm0.001m##
 
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  • #9
Both in the case of measuring from one fringe to the other, or from the principal maxima to each fringe, we're still measuring the distance between two points, so shouldn't the uncertainty be the same for both measurements?
 
  • #10
KDPhysics said:
Shouldn't the uncertainty be the same either way? Measuring from one fringe to the other, then the uncertainty will be 0.002m using a meter ruler. Similarly, measuring from the principal maxima to each fringe, the uncertainty will still be 0.002m.

Also, we can use the linear approximation for error propagation (since we're still in high school).
Did you read "Addition of measured quantities" in the link I posted? Apparently not. Please read it first and then calculate the uncertainties.
 
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  • #11
We're allowed to use the second formula (##\delta R = \delta X + \delta Y + \delta Z##).
 

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  • #12
KDPhysics said:
We're allowed to use the second formula (##\delta R = \delta X + \delta Y + \delta Z##).
OK then, but let me ask you this. Suppose that you measure the two first order maxima from the center and you find that one is at a greater distance than the other by more than your assumed uncertainty. What will you conclude from that?
 
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  • #13
Ok, I think I got it. Since one is slightly farther away from the other, we should calculate the half range:
$$\frac{x_r - x_l}{2}>\delta x$$
assuming the right measurement is greater than the other. Then, we should be using this value as our uncertainty, which is greater than ##\delta x##. We can therefore conclude that measuring from one fringe to the other will decrease uncertainty.

Is this correct?
 
  • #14
Yes. That's the idea I tried to convey with the coin example. I have edited that post to make it clearer.
 
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  • #15
Thank you very much! In hindsight the coin example was very fitting...just too sleep deprived to understand I guess ahahah.
 
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Related to Measuring the Distance between the Fringes of a Diffraction Grating

1. What is a diffraction grating?

A diffraction grating is an optical device consisting of a large number of closely spaced parallel lines, or grooves, that are used to separate light into its different component wavelengths. It is commonly used in spectroscopy and other scientific experiments to measure the properties of light.

2. How do you measure the distance between the fringes of a diffraction grating?

The distance between the fringes of a diffraction grating, also known as the grating constant, can be measured by using a ruler or caliper to measure the distance between two adjacent bright fringes on a screen placed at a known distance from the grating. This distance can then be divided by the number of lines on the grating to determine the grating constant.

3. What is the relationship between the distance between fringes and the wavelength of light?

The distance between fringes on a diffraction grating is directly proportional to the wavelength of light. This means that as the wavelength of light increases, the distance between fringes also increases. This relationship is known as the grating equation: nλ = d sinθwhere n is the order of the bright fringe, λ is the wavelength of light, d is the grating constant, and θ is the angle of diffraction.

4. Why is it important to measure the distance between fringes on a diffraction grating?

Measuring the distance between fringes on a diffraction grating allows scientists to accurately determine the wavelengths of light being diffracted. This is essential in many scientific experiments, such as spectroscopy, where the properties of light need to be precisely measured.

5. How does the number of lines on a diffraction grating affect the measurement of fringe distance?

The number of lines on a diffraction grating directly affects the measurement of fringe distance. A grating with more lines will produce a larger number of fringes, making it easier to measure the distance between them accurately. Additionally, the larger the number of lines, the smaller the grating constant, resulting in a more precise measurement of the wavelength of light being diffracted.

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