How do I calculate uncertainties for this experiment?

In summary, the student attempted to find the uncertainty in the gradient using error propagation. However, because the error in x is so large, this was not possible. The student then calculated the uncertainties in h for each data point and took its average. This resulted in a planks constant of 6.7*10-34Js with an uncertainty of +-1.51^-33Js.
  • #1
henry wang
30
0

Homework Statement


I used error propagation equation to calculate errors in x and y for each individual data. Error in y is negligible. My goal is to find the uncertainty in the gradient.
Capture.PNG

2. Homework Equations

The plotted equation is [tex]eV=h\frac{c}{2dsin(\theta)}[/tex] ,where Plank's constant, h, is the gradient, the independent variables eV and the dependent variable is theta. Also x=c/(2dsin(theta) and y=eV.
The dominant error is in the theta, and thus error in x is: [tex]\Delta x=\frac{c*cos(\theta) \Delta \theta}{2dsin^{2}(\theta)}[/tex]
c=3*10^8m, cos(theta)=1, d is about 10^-10m, sin(theta) is about 0.1 and dtheta=0.2 degrees.
This yields an extremely big error in x.

The Attempt at a Solution


My original thought was to fit lines of max and min permitted gradient using errors in x and y. However, since the error in x is so large, and the x and y-axis are in log scale, I cannot manually fit lines.
Update: Therefore I rearranged the above equation to isolate h, calculated the uncertainties in h for each data points and took its average.
[tex]\Delta h=\frac{2eVdcos(\theta) \Delta \theta}{c}[/tex]The planks constant, h, was found to be 6.7*10-34Js, and the average uncertainty in h was found to be +-1.51^-33Js.
Is this a reasonable approach?

Update 2: After corrected dtheta from degrees to radians.
Caspture.PNG
 
Last edited:
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  • #2
henry wang said:

Homework Statement


I used error propagation equation to calculate errors in x and y for each individual data. Error in y is negligible. My goal is to find the uncertainty in the gradient. View attachment 109168
2. Homework Equations

The plotted equation is [tex]eV=h\frac{c}{2dsin(\theta)}[/tex] ,where Plank's constant, h, is the gradient, the independent variables eV and the dependent variable is theta. Also x=c/(2dsin(theta) and y=eV.
The dominant error is in the theta, and thus error in x is: [tex]\Delta x=\frac{c*cos(\theta) \Delta \theta}{2dsin^{2}(\theta)}[/tex]
c=3*10^8m, cos(theta)=1, d is about 10^-10m, sin(theta) is about 0.1 and dtheta=0.2 degrees.
This yields an extremely big error in x.

The Attempt at a Solution


My original thought was to fit lines of max and min permitted gradient using errors in x and y. However, since the error in x is so large, and the x and y-axis are in log scale, I cannot manually fit lines.
Therefore I calculated the uncertainties in h for each data points and took its average. The planks constant, h, was found to be 6.7*10-34Js, and the average uncertainty in h was found to be 1.51^-33.
Is this a reasonable approach?

Are you sure that your units are correct? ##1.0e-15## seems to be very small energies. I have never heard about any experiment operating at such small energies. Also, it seems like your expression for ##\Delta x## doesn't have the correct unit. I assume that ##\Delta x# should be a distance but your expression gives ##s^{-1}##.
 
  • #3
Just by looking at the data: either your uncertainties are all extremely correlated (and I don't see why they should be) or you overestimate your uncertainties massively.

I would expect the second case (especially as your wavelength is certainly not negative), but without a description of your experiment it is hard to figure out what went wrong.
 
  • #4
eys_physics said:
Are you sure that your units are correct? ##1.0e-15## seems to be very small energies. I have never heard about any experiment operating at such small energies. Also, it seems like your expression for ##\Delta x## doesn't have the correct unit. I assume that ##\Delta x# should be a distance but your expression gives ##s^{-1}##.
Sorry I should've specified x is just a dummy variable, it is c/wavelength with unit 1/s. The energy in this experiment is x-ray photon energy.
 
  • #5
mfb said:
Just by looking at the data: either your uncertainties are all extremely correlated (and I don't see why they should be) or you overestimate your uncertainties massively.

I would expect the second case (especially as your wavelength is certainly not negative), but without a description of your experiment it is hard to figure out what went wrong.
The reason why error in x is so big is because inter-crystal plane distance, d, is very small and c is very large. The uncertainty in x was calculated using the error propagation equation. [tex]\Delta x=\frac{\delta x}{\delta \theta}\Delta \theta[/tex]
Btw, I updated the OP please check it out.
 
  • #6
In the way you did the error propagation, you have to use ##\Delta \theta## in radian, not in degrees.
 
  • #7
mfb said:
In the way you did the error propagation, you have to use ##\Delta \theta## in radian, not in degrees.
Thank you so much! I changed it to radians and the error is much smaller.
 
  • #8
henry wang said:
Thank you so much! I changed it to radians and the error is much smaller.
Would you post the new graph, please? It is hard to judge what it will look like with the horizontal scale blown up by a factor of about 3000.
 
  • #9
haruspex said:
Would you post the new graph, please? It is hard to judge what it will look like with the horizontal scale blown up by a factor of about 3000.
Sure thing!
 

Related to How do I calculate uncertainties for this experiment?

1. How do I determine the sources of uncertainty for my experiment?

The first step in calculating uncertainties is identifying all potential sources of error in your experiment. These can include equipment limitations, human error, and environmental factors. It is important to carefully consider each step of your experiment and determine where uncertainties may arise.

2. What is the difference between random and systematic uncertainties?

Random uncertainties arise from natural variations in data and can be reduced through repeated measurements. Systematic uncertainties, on the other hand, are caused by consistent errors in the experimental setup and can be reduced by identifying and correcting the source of the error.

3. How do I quantify uncertainties in my measurements?

There are several methods for quantifying uncertainties, including the standard deviation, standard error, and confidence intervals. These can be calculated using statistical tools or by hand using formulas. It is important to carefully choose the appropriate method for your specific experiment.

4. Can I use the same uncertainty value for all of my measurements?

No, the uncertainty for each measurement may vary depending on the sources of error and the precision of the equipment used. It is important to calculate the uncertainty for each individual measurement to get an accurate representation of the overall uncertainty in your experiment.

5. How do I incorporate uncertainties into my final results?

Uncertainties should always be reported alongside your final results. This can be done by including a margin of error or confidence interval, or by stating the uncertainty as a percentage or range. It is important to acknowledge and account for uncertainties in order to accurately communicate the reliability of your data.

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