Estimating Error in Wavelength from Graphical Approach

In summary, the article discusses a method for estimating errors in wavelength measurements using a graphical approach. It emphasizes the importance of visual representation in analyzing data, allowing for clearer identification of uncertainties and trends. The methodology involves plotting data points and fitting a curve to them, enabling the calculation of error margins based on the spread of values. This approach not only enhances accuracy in wavelength estimation but also aids in understanding the overall precision of experimental results.
  • #1
PhysicsRock
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Homework Statement
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Relevant Equations
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As part of my studies, I'm obliged to take an experimental course at the moment, where I have to conduct experiments and write a composition. Today we examined spectral lines of helium with a prism. As part of the evaluation, I had to plot the measured diffraction angles of different colors / wavelengths (which were unknown at the time and left to figure out later) as function of . Now I'm asked to estimate the error in the wavelength from the graphical approach, but I have no idea where to start. There's no expression for the dependence of on , so I can't really do the classical



What I thought of is to try to draw a tangent line as good as possible at a measured wavelength, say , and read off its slope. That would sort of act like and I could calculate an error. However, I don't like two things about that. The first is that the error I get for the value that's off the most (by about ) is too little at about . Second, it just seems too easy to me.

I hope some of you have a suggestion for a good approach. Thank you in advance.
 
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  • #2
PhysicsRock said:
Homework Statement: /
Relevant Equations: /

Today we examined spectral lines of helium with a prism. As part of the evaluation, I had to plot the measured diffraction angles of different colors / wavelengths (which were unknown at the time and left to figure out later) as function of wavelength lambda.
Can you post screenshots of the test setup and of your data?

And how are you determining the wavelength ? From the observed color and a look-up table, or do you have a way of measuring the wavelength at each angle directly?
 
  • #3
berkeman said:
Can you post screenshots of the test setup and of your data?

And how are you determining the wavelength ? From the observed color and a look-up table, or do you have a way of measuring the wavelength at each angle directly?
This (https://upload.wikimedia.org/wikipe...ent_setup.svg/1200px-Experiment_setup.svg.png) setup was used. We first measured the deflection angles of the individual lines of mercury, the wave lengths were given here. After that, we were supposed to draw a diagram, putting the values of on the - and on the -axis and marking the points where a linked pair met in the coordinate system. This gave the curve I was talking about. Then we repeated the same process for helium, but this time without being given the wave lengths. The idea was to draw a horizontal line from the measured deflection angle until it intersects , the -coordinate of that point would then give the wave length of the observed spectral line.

I hope I was able to clarify the procedure.
 
  • #5
PhysicsRock said:
We first measured the deflection angles of the individual lines of mercury, the wave lengths were given here.
Which what here?
 
  • #6
PhysicsRock said:
This (https://upload.wikimedia.org/wikipe...ent_setup.svg/1200px-Experiment_setup.svg.png) setup was used. We first measured the deflection angles of the individual lines of mercury, the wave lengths were given here. After that, we were supposed to draw a diagram, putting the values of on the - and on the -axis and marking the points where a linked pair met in the coordinate system. This gave the curve I was talking about. Then we repeated the same process for helium, but this time without being given the wave lengths. The idea was to draw a horizontal line from the measured deflection angle until it intersects , the -coordinate of that point would then give the wave length of the observed spectral line.

I hope I was able to clarify the procedure.
Ok, so it is an interpolation procedure.
First consider the sources of error:
  • the given wavelengths (presumably pretty accurate)
  • the observed deflections for mercury
  • the observed deflections for helium
  • the interpolation step
For that last, are you connecting the mercury dots with straight lines or attempting a smooth curve? If you were to connect with straight lines, what angle, at worst, is made by three consecutive dots?
 
  • #7
berkeman said:
Please always upload images to PF, to ensure that they are not lost in future years when the image server at some other website decides to delete them. Here is your image:
This is a Wikimedia image. If we are going to display it here it needs to have the license and attribution set out here: https://commons.wikimedia.org/wiki/File:Experiment_setup.svg
 
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  • #9
haruspex said:
For that last, are you connecting the mercury dots with straight lines or attempting a smooth curve? If you were to connect with straight lines, what angle, at worst, is made by three consecutive dots?
I tried my best to make it a smooth curve, connecting the individual dots.
 
  • #11
PhysicsRock said:
I tried my best to make it a smooth curve, connecting the individual dots.
ok, but can you answer my second question?
 
  • #12
haruspex said:
ok, but can you answer my second question?
Sorry for the late reply, usually I get a notification when somebody responds. I have now settled to determine the error by drawing additional horizontal lines offset from the observed angle by the estimated error and then just read the -coordinate of the intersection point and took , where is the original angle and said intersection coordinate. Tedious to do, but it should do the trick.

Thank you for your help anyway. Have a good day.
 
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