# User:Thomas S. Mahony/Notebook/Physics 307L/2009/11/09

e/m Ratio Lab Main project page
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# e/m Lab

In this lab Ryan and I tried to find the charge to mass ratio for an electron. We used an apparatus with an electron gun that fired into a helium filled bulb placed inside a magnetic field to excited a ring of Helium atoms. We measured the radius of the ring, and knowing about the Lorentz force, we calculated e/m.

### Summary

SJK Incomplete Feedback Notice
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My feedback is incomplete on this page for two reasons. First, the value of the feedback to the students is low, given that the course is over. Second, I'm running out of time to finish grading!
SJK 18:07, 15 December 2009 (EST)
18:07, 15 December 2009 (EST)
Excellent notebook, very nice presentation of results.

## Equipment

• BK Precision Multimeter 2831B (x2)
• HP 6384A DC Power Supply
• Uchida Yoko e/m experimental apparatus model TG-13
• Soar Corp. 7403 DC power supply
• Gelman Instrument Company 500V Deluxe Regulated power supply
• BNC Cables
Our Setup

## Setup

First we connected the HP power supply to the Helmholtz coils on the Uchida apparatus with one of the BK multimeters in series. Next the other BK multimeter was connected to the voltmeter jacks on the Uchida apparatus. The Soar power supply was connected to the heater jacks on the Uchida. Finally, the Gelman was connected to the electrodes on the Uchida. We followed the procedure in Professor Gold's manual, which included letting the heater warm up for 2 minutes before applying voltage to the electrodes on the apparatus.

To take data, we let either let the voltage of the electrodes or the current running through the coils be constant, while varying the other parameter. We measured the radius of the rings on the right and the left side using the built in ruler.

## Data

Note: At a current of -1.35A, maximum accelerating voltage is 320V, the minimum is 165V. At -1.05A, the maximum voltage is 200V and the minimum is 100V.

11/16/09 Voltage applied to heater jacks: 6.24 V

### Raw Data:

key=0Ajaf_99vseiXdEIyXzJxWGJhLUQ2SE9qNGo4UURmMUE width=760 height=300

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## Analysis

Given that (Thanks Alex and Anastasia!):

${\displaystyle B=(7.8\times 10^{-4}{\frac {weber}{amp-meter}})\times I\,\!}$
${\displaystyle {e}{V}={\frac {1}{2}}{m}{v}^{2}\,\!}$
${\displaystyle {F}_{B}={q}{v}{B}\,\!}$
${\displaystyle {\frac {e}{m}}={\frac {{2}{V}}{{r}^{2}{B}^{2}}}={\frac {{2}{V}}{{r}^{2}{({{7.8}\times 10}^{-4}{I})}^{2}}}\,\!}$

We can calculate e/m using either of 2 methods:

• Use the slope of a ${\displaystyle {\frac {V}{{r}^{2}}}\,\!}$ line holding current constant and multiplying by ${\displaystyle {\frac {2}{B^{2}}}}$
• Use the slope of a ${\displaystyle {\frac {r}{\frac {1}{I}}}\,\!}$ line holding voltage constant by inverting the slope, taking the square root, and multiplying by ${\displaystyle {\frac {2V}{7.4\cdot 10^{-4}}}}$.

Since the uncertainty in the radii measurements was almost uniform, I treated it as uniform. Initially, the value for ${\displaystyle {\frac {e}{m}}\,\!}$ based on constant voltage included the last two data points, but I did not incorporate them into my final calculations. Not only did they not include many data points, but the second to last one was an order of magnitude too big with ridiculous error bars, and the last data point had data points spaced very close giving it an error several orders of magnitude smaller than the rest of the data (causing the weighted average to be based primarily off this single value which was only based off of 3 data points itself).

The e/m ratio under constant current and varying voltage was:

${\displaystyle 2.16(2)\cdot 10^{11}{\frac {C}{kg}}}$

The e/m ratio under constant current and varying voltage was:

${\displaystyle 2.94(5)\cdot 10^{11}{\frac {C}{kg}}}$

The accepted value (from wikipedia) is:

${\displaystyle 1.758820150(44)\cdot 10^{11}{\frac {C}{kg}}}$

The constant current value was 21 SEMs away from the accepted value, and the constant voltage value was 23 SEMs away from the accepted value.

The full analysis can be seen here.

SJK 17:50, 15 December 2009 (EST)
17:50, 15 December 2009 (EST)
I love your graphs, excellent way to present the data!

## Conclusions

Neither result overlapped with the accepted value, and the distance away from the accepted value vs the number of SEM's away each value was leads to believe my results were off primarily due to systematic error. This doesn't surprise me, as this lab is very error prone. The errors I can think of in this experiment include the following:

• The electron ring was not in a vacuum. The electrons were probably slowed by their collisions with the helium atoms, decreasing the rings radius, which would lead to an overall greater e/m ratio. This is consistent with my results.
• Energy level transitions in elements happen at only very specific frequencies (energies). Only the electrons with speeds resulting in kinetic energies the same as the light blue-ish green transition (see here) could be measured using this setup. SJK 18:06, 15 December 2009 (EST)
18:06, 15 December 2009 (EST)
All of the electrons have energy of 100's of eV (the accelerating voltage is 100's V), and thus have plenty of energy to ionize helium. So I think this point may be off. That said, it's still a mystery to me why the beam color is primarily from the cyan transition.
• The "orbits" of the electrons were not circular, but rather elliptical.
• Their is a strong reliance on human judgment of the position of the rings, and this could easily be influenced by the widening of the rings with changing voltage,, distortion looking through the bulb near its "edges", horizontal parallax due to misaligning the beam with its reflection in the mirror, or the ruler not being at the center of the orbit (causing parallax in the vertical direction).

However, since there are so many errors, and little way to measure their effects independently, I cannot pinpoint which would be the main culprit behind the large error we encountered in the lab. I can't think of any quick fixes to the experiment that wouldn't require drastically changing the setup.

## Acknowledgments

Thanks to Ryan, my lab partner, for his help. I used Alex and Anastasia's as well as Paul Klimov's labs for references, so thanks to them for their easy to follow and helpful ideas and explanations.