Estimating Error in Wavelength from Graphical Approach

[PhysicsRock]

Homework Statement

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Relevant Equations

/

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 $\lambda$. 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 $\delta(\lambda)$ on $\lambda$, so I can't really do the classical

$$
\Delta \delta = \left\vert \frac{d \delta(\lambda)}{d \lambda} \right\vert \Delta \lambda \Leftrightarrow \Delta \lambda = \left\vert \frac{d \delta(\lambda)}{d\lambda} \right\vert^{-1} \Delta \delta
$$

What I thought of is to try to draw a tangent line as good as possible at a measured wavelength, say $\lambda_0$, and read off its slope. That would sort of act like $\delta^\prime(\lambda_0)$ 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 $12.3 \, \text{nm}$) is too little at about $\Delta \lambda \approx 5.56 \, \text{nm}$. Second, it just seems too easy to me.

I hope some of you have a suggestion for a good approach. Thank you in advance.

 

[berkeman]

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 $\lambda$ ? From the observed color and a look-up table, or do you have a way of measuring the wavelength at each angle directly?

 

[PhysicsRock]

Can you post screenshots of the test setup and of your data?

And how are you determining the wavelength $\lambda$ ? 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 $\delta$ on the $y$- and $\lambda$ on the $x$-axis and marking the points where a linked pair met in the coordinate system. This gave the curve $\delta(\lambda)$ 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 $\delta(\lambda)$, the $x$-coordinate of that point would then give the wave length of the observed spectral line.

I hope I was able to clarify the procedure.

 

[berkeman]

This (https://upload.wikimedia.org/wikipe...ent_setup.svg/1200px-Experiment_setup.svg.png) setup was used.

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:

(attribution added) https://commons.wikimedia.org/wiki/File:Experiment_setup.svg

 

[berkeman]

We first measured the deflection angles of the individual lines of mercury, the wave lengths were given here.

Which what here?

 

[haruspex]

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 $\delta$ on the $y$- and $\lambda$ on the $x$-axis and marking the points where a linked pair met in the coordinate system. This gave the curve $\delta(\lambda)$ 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 $\delta(\lambda)$, the $x$-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?

 

[pbuk]

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

 

[Frabjous]

Maybe this will help.
https://www.physik.uzh.ch/~matthias/espace-assistant/manuals/en/anleitung-spk_e.pdf

 

[PhysicsRock]

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.

 

[PhysicsRock]

Maybe this will help.
https://www.physik.uzh.ch/~matthias/espace-assistant/manuals/en/anleitung-spk_e.pdf

I'll read into it, thank you for providing it.

 

[haruspex]

I tried my best to make it a smooth curve, connecting the individual dots.

ok, but can you answer my second question?

 

[PhysicsRock]

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 $x$-coordinate of the intersection point and took $\Delta\lambda = \vert \lambda_1 - \lambda_2 \vert$, where $\lambda_1$ is the original angle and $\lambda_2$ said intersection coordinate. Tedious to do, but it should do the trick.

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