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Visually Measuring the Charge-to-Mass Ratio for Electrons

SJK 23:01, 6 December 2008 (EST)

23:01, 6 December 2008 (EST)
I like your title and very good contact information.

Author: Michael R. Phillips

Experimentalists: Michael R. Phillips & Stephen K. Martinez

University of New Mexico: Physics and Astronomy Department, Albuquerque, NM

e-Mail: crooked@unm.edu


SJK 23:10, 6 December 2008 (EST)

23:10, 6 December 2008 (EST)
This is a very good draft of the abstract. It really does sound very professional and it gives the reader a good idea of what they will find in the paper. Some comments on what you have: (1) phrase, "we were able to form linear data..." is a bit unclear, (2) italics on not not usually used in formal abstract, (3) I'd ditch the last digit on uncertainty...use 0.04 only, (4) The last sentence is too informal ("later")...you could change it to something like "We discuss reasons for the large discrepancy and possible improvements for future experiments.

Also, I think it would be a better abstract if you added motivation sentence somewhere: why is charge/mass ratio important or why were these types of experiments revolutionary. I think it's good to include this in abstracts, although as is, your abstract flows very nicely.

Using a Helmholtz Coil setup with variable current situated around a Helium-filled glass tube with an electron gun at the bottom with a variable accelerating voltage, we measured the diameter of electron beam paths formed into circles by the induced magnetic field from the Helmholtz Coil setup. Using theoretical predictions of the diameter as a function of the charge-to-mass ratio (e/m) for electrons, we were able to obtain data and perform a linear fit (using the method least squares) and find the slope, which relates to this ratio and some constants. The charge-to-mass ratio is very important to know to some degree of certainty because of how often it appears in theoretical predictions. It is not always necessary to know both the charge and the mass of electrons, but the ratio alone is very useful. However, our final measurement of (4.78 ± 0.04)·1010C/kg was not in very good agreement with the accepted value of 1.759·1011C/kg. The reasons for this large discrepancy will be discussed along with possible improvements.


SJK 23:17, 6 December 2008 (EST)

23:17, 6 December 2008 (EST)
What you have here for the introduction is very well written (though you'll want to add actual citations to the papers you're mentioning). However, it's too short and limited. It's very good background on the motivation for the experiment (though the language is a bit informal). So you need to add: (1) what are the best ways to measure this ratio nowadays? (adding citations to some references) (2) a brief introduction to what you'll show in this report, and (3) a short concluding statement of impact or future work or similar. Also, you will need citations to primary peer-reviewed publications.

The value calculated during this lab, e/m, seems a strange goal without some inspection of some applications. It is immediately clear, without going through additional examples, that this value is useful because we could actually predict what our diameters would have been in this experiment, given different accelerating voltages and Helmholtz Coil currents, using the same theory we used to find e/m if we only had an accurate value for e/m beforehand. Something else that makes this lab particularly useful has to do with the fact that measuring the mass of a single electron is very difficult to measure. However, we can deduce this mass if we have accurate values for both this e/m ratio and the charge, e, of an electron, which is done in the Millikan Oil Drop experiment. This idea was probably what J.J. Thomson (1856-1940) had in mind when he fist did this experiment in a very similar way to how we did, and is described in detail in his 1897 paper (see Reference No. 1). The way we measured e/m in this lab were quite antiquated, which is somewhat obvious given that J.J. Thomson did a very similar experiment more than a century ago. This leads to the conclusion that there must be much better ways to measure e/m now. The most common method today employs the use of a mass spectrometer, which accelerates particles forward through a velocity selector then into a known magnetic field. From here, the particle beam curves (as long as the particles are charged or ionized) and the charge-to-mass ratio can be calculated in a similar way that we calculated it for this experiment.

Methods and Materials


  • Soar DC Power Supply Model Number PS-3630 - Power Supply to Helmholtz Coil setup
  • Soar DC Power Supply Model 7403 UNM 158374 - Power Supply to Electron Gun Heater Element
  • Gelman Instrument Company Delux Regulated Power Supply - Power Supply to Accelerating Electrodes
  • 2 x Amprobe 37xR-A - Multimeters
  • Fluke 111 True RMS Multimeter
  • e/m Apparatus (includes electron gun, Helmholtz coil setup, Helium-filled tube, power supply inputs, and a small reflective ruler mounted behind the tube)
  • Black Cloth Hood (to block out excess light while taking data)
  • Very Many “Banana”-tipped cords for the Power Supply connections


SJK 23:40, 6 December 2008 (EST)

23:40, 6 December 2008 (EST)
These are really great details as far as a "how to" for someone doing this lab. However, unfortunately the custom is to not include this level of detail in a formal publication (probably in large part because of the amount of paper that would have been used). In your future research career, I encourage you to keep taking detailed notes like this--both in your primary lab notebook and also "publishing" good "how to" articles on the internet in addition to your formal publications. OK, all that said, I think the correct thing to do with this whole "setup" section is to make a really good diagram (or a photograph with text superimposed to label components) (both a photo + a diagram would be great) and then have much less text and using the past tense "we set up the components as in figure ___ and according to reference X. We connected the apparatus (company name, city) to blah...". This probably is confusing, so we should discuss it this week.

Following the setup described in Dr. Gold's Junior Lab Manual: (Steve Koch: You can make a link to Dr. Gold's manual one of the references and then refer to it with a number like you will with your other references.)

[[Image:Untitled.bmp|thumb|Figure 1: A schematic diagram of the e/m apparatus that we used. We started off by organizing all of the power supplies and multimeters so that we would be able to connect everything in the way that is necessary. Before connecting any of the power supplies or multimeters to the apparatus, we needed to make sure we got the supplies set to the correct voltage or current for the corresponding element that each supply would power. After we got all of the dials tuned correctly, checked with corresponding multimeters, we could turn them all off and connect everything to the apparatus, then turn it all back on again. Since we did this lab in two separate days, we have values of initial voltages/currents for these two occasions, but they both fall into the range as suggested by Gold’s manual.

After this initial power supply-type calibration, we connected everything as suggested by the manual. In connecting the power supply designated for the Helmholtz coils, we connected a multimeter (one of the Amprobe multimeters) in series with the coil power inputs on the apparatus so that we could measure the current that flows into the coils. This current is not only regulated by the dials on the power supply, but also by a dial on the apparatus itself. We will be using only the latter dial to vary our currents during the lab to ensure that we never exceed the 2.0 Amp ceiling suggested by Gold’s manual.

Next, we connected the power supply that goes with the electron gun heater. This particular element of the apparatus does not ever need to be varied, it just needs to have a high enough voltage supplied to it that many electrons will be fed into the electron gun accelerating potential so that we can see the path of the electrons through the Helium tube clearly. If the voltage is too low on the heater element, too few electron will be emitted and the resulting beam would be far too dim for us to discern and measure visually, which is how we collected data (as described in detail later). Even though we did not vary the voltage supplied to the heater, we still connected a multimeter, the Fluke 111, in parallel with it so that we could be sure nothing else changed the value and burnt out the heater element.

Finally, we connected a power supply to the apparatus inputs for the accelerating voltage across electrodes that are very near the electron gun’s heater element. This voltage was varied very much during the experiment, so we though ahead and connected a multimeter, the other Amprobe, in parallel to measure this voltage. Unlike the other multimeters, which were incorporated directly into the power supply-element circuit, this multimter was connected to a kind of output on the apparatus that was designed specifically to measure accelerating voltage easily. As with the heater, this element of the apparatus does not have any requirements for current, but has a minimum and a maximum voltage that essentially decides the speed of the electrons departing the electron gun. Therefore, we did not need to put in another multimeter to monitor current through this circuit.

After all of our data is obtained using these methods, we put it all into Excel and used the function LINEST to give us accurate slope and standard error of the mean values. This can all be seen in our results more clearly.


SJK 23:42, 6 December 2008 (EST)

23:42, 6 December 2008 (EST)
It seems to me that a lot of your procedure would fit in the "results." For example, the very interesting violet colored beam and hypotheses about what could be happening. I think you should probably move that to a new section of the results related to these things, which I think could be a very interesting part of your report.

SJK 16:31, 7 December 2008 (EST)

16:31, 7 December 2008 (EST)
Also, like the above section, this section is too informal, for example, the phrase "which took quite some time" would not be used in a formal publication.

SJK 17:04, 7 December 2008 (EST)

17:04, 7 December 2008 (EST)
You have good figures with good, descriptive captions.

After we got everything connected correctly, which took quite some time, we turned on all of the power supplies and multimeters and began to take data. To take decent data we turned off the lights in the room and covered the whole apparatus with a black cloth hood to minimize the amount of ambient light that was entering our eyes. To actually measure the diameter of the loop that was created, we had a small reflective ruler mounted behind the tube. The reason for it being reflective is that there would be very significant (a couple of centimeters, at least) systematic error in all of our diameter measurements if we did not line up the direct image we saw with a reflection from the ruler. Something that is interesting to note is that we did not get a circular cyan loop as we were expecting at first. After simply turning everything on and looking at our viewing area, we saw a very small loop of cyan and violet that was certainly not circular (see Figure 1). From what we could deduce right away, the electrons seemed to be leaving either without enough energy (speed) or we had a much too large current running through our Helmholtz coils that was making a very strong magnetic field that ended up bending the path of the electrons so much that they were pulled back into the accelerating voltage and shot out again.

After adjusting the current to a lower value and raising the accelerating voltage by just a little, we were able to observe the circular path that we were looking for, and the color emitted by the Helium in the tube was only cyan in the area of the path taken by the electrons. However, upon closer inspection, the path taken by the electrons was not circular but helical. The cause of this was simple: the electron gun was not shooting electrons with an initial velocity parallel to the planes created by the Helmholtz coils. Therefore, to correct this, we simply rotated the Helium tube, which has the electron gun and accelerating plates mounted inside it, until the orientation of the electron gun was parallel with the Helmholtz planes. Now, finally, we could see exactly what we were looking for.

Now that we obtained an ideal form to work off of, we started actually taking data. To do this, we varied the Helmholtz current and the accelerating voltage one at a time and measured the diameter of our circular loop for each pair of values. We could then use this quite large amount of data in Excel to create a best fit line using the least squares method and show the voltage as a function of the radius squared. The reason we used the voltage as a function instead of the current, since we could choose either or both for the cases when the other is constant, is because we found that small changes in the current make very small changes in the diameter of the loop, changes so small that our systematic error in looking at the loop would make the whole data set for varying current and constant voltage useless. However, the case for varying voltage and constant current led to decent results that changed the diameter at least enough to overcome our systematic error. After all of our data is obtained using these methods, we put it all into Excel and used the function LINEST to give us accurate slope and standard error of the mean values. This can all be seen in our results more clearly.

SJK 16:28, 7 December 2008 (EST)

16:28, 7 December 2008 (EST)
Also part of your methods section should be analysis methods, including listing software applications used, and any special algorithms (such as LINEST). You have a lot of this information below in the results section, but description of the analysis methods can go in the methods section.

Results & Discussion

Figure 2: The Breakdown of the Electron Path through Helium with a Low Accelerating Potential and Strong Magnetic Field due to a High Current through Helmholtz Coils. Notice the violet color at the top of the loop and the way the electron path bends back into the accelerating area (bottom of the glass tube).
Figure 3: The cyan loop viewed through a multi-axis diffraction grating. You may notice some strange looking red streaks that seem to show a cyan that is not monochromatic, but these are due solely to the heater's light (which is certainly not monochromatic) partially blocking the view of the more interesting part of the photograph.
Figure 4: The Plot from Excel's least-square fitting that shows all of our data for Accelerating Voltage vs. Radius Squared fitted to a line with slope k·e/m. Notice that there are some points that seem to be outliers, but they still follow the basic trend that theory predicts.

Before discussing the real data, I would like to explain this strange violet loop we were observing before we were able to get our good cyan circle (see Figure 2). Although we (in this instance meaning Stephen, Dan, our TA Aram, Prof. Koch, and myself) were not able to come up with a conclusive explanation for this phenomenon, we did make some simple hypotheses and did some extra work to see more exactly what was going on here. One thing we wanted to show is that the cyan in the expected loop was monochromatic. Using a diffraction grating, we could see that the cyan loop is, in fact, monochromatic. This is clear because the image (see Figure 3) of the diffracted loop does not show any more than a single color. After this, we wanted to see whether the violet part of the strange loop was purely violet (monochromatic of a different color) or if it was overlapping with our cyan. Using a green light filter, we observed the top of the strange loop as cyan. The image we saw was really just a very small version of our desired experimental loop but we needed this green filter to make it clear. This clearly shows that the violet is indeed overlapping the cyan at the top and makes us suspect that it is actually existent everywhere along the loop but just has enough intensity to overpower the cyan only at the top of a very small loop. If we were able to locate a violet light filter, we may have been able to show this more definitively, but we did not have such a filter. Although we cannot explain this too well, it is very interesting to know more facts about this phenomenon.

We did all of our analysis using Excel. After entering all of our raw data, we made functions that described everything from the resulting magnetic field from the Helmholtz coils to the final and average value for e/m. In this file, found at the end of this section, there is also a plot (see Figure 4) which shows our final correlation between accelerating voltage and the radius squared. You will notice some stray points on the plot, which are due mainly to systematic error. You may also notice that we called this the “Best” plot of our data. The reason for this is that there were some stray data points that did not fit the rising trend in the radius from higher voltage. We concluded these data points were recorded after our eyes had become tired from staring at a rather bright light source in a very dark room. In other words, these points had very, very bad systematic errors associated with them so were not worth consideration.

To deduce the value of e/m from this plot, we obtained a slope from the best-fit line using features within Excel. This slope, though not useful in itself, includes e/m in it, with a few constants as well. Since we know the values of all these constants, we can easily extract a correct value for e/m with an associated error from the error in creating the line in Excel (using a part of the LINEST function).

The equation that we use to calculate e/m from various voltages and radii is

[math]V=k\cdot\frac{e}{m}\cdot r^{2}[/math]

where k is a constant that is a compilation of many fundamental constants, which are already known to a great deal of accuracy.

Shown also in the file is a percent error, which was found to be 72.9%, which shows how close our measurement of e/m was to the accepted value. The formula for this is


You can find the original Excel file at Ref. No. 2.SJK 16:59, 7 December 2008 (EST)

16:59, 7 December 2008 (EST)
"Here is the Excel file..." is a bit informal. In a published paper, you would likely provide the excel file as "supplemental data." So, it's not clear what the best way to link to it is. You could put it in the references list and then cite it as "Original excel spreadsheets are provided (ref #)."


SJK 17:02, 7 December 2008 (EST)

17:02, 7 December 2008 (EST)
You have some good thoughts in this section, but it is also a bit too informal. Your comparison with the accepted value should use statistical info (compare to error bars) as opposed to "not at all close." Also, you can mention better ways to measure e/m ... surely they exist, because we have really good information about this ratio!

We found that our overall measurement of the charge-to-mass ratio for electrons was (4.78 ± 0.041)·1010C/kg, which is not at all in good agreement with the accepted value of 1.759·1011C/kg. Not even our somewhat considerable error bars make up for the discrepancy in these values. There are very many reasons for this discrepancy, however, and were mentioned briefly in the above sections.

By far, our largest source of error was simply systematic. There is a huge problem with trying to measure anything visually in a very dark room with nothing but a glowing ring of electrons and a reflective ruler. It is quite hard to see anything at all, especially in the realm of low accelerating potentials when electrons don’t excite the Helium very much due to their own low energy.

Also, the fact that the electrons are giving energy to the Helium inside the tube is a problem. This means that the electrons are certainly not traveling in a circle at all but rather something that looks like an egg, with the narrow portion just after leaving the electron gun and the wider portion near the end of their journey back to the electron gun’s heater element. This causes significant fluctuations in diameter for different potentials and gives us results that suggest that the constant value e/m somehow depends on accelerating potential, which is obviously not true.

Another occurrence of systematic error showed itself right when we first turned on all of our power supplies (see Figure 1). In this case, we saw that the electrons were not escaping the pull of the accelerating plates until about three centimeters away. This is a very big problem, especially at low voltages or high Helmholtz currents because the radii of the electron paths were actually smaller than this three centimeters. That means that the radii we record for these values of voltage and current are deeply flawed. Upon examination of our data, it is easy to draw connections with fundamentally poor data points and radii that are smaller than or very close to three centimeters (or 0.03 meters when looking at our Excel sheet). In fact, as shown in Figure 1, when both the voltage and Helmholtz current are in the unacceptable range, the electron loop collapses in on itself and exhibits some very strange behavior (the violet at the top of the ‘ring’ is still puzzling to us).

Because of all this systematic error, we saw why we did not get a value of e/m close to the accepted value. However, we could not think of a way to measure this quantity without using small electron paths or strong accelerating potentials inside of a gas that steals electron energy.


I would like to thank Stephen K. Martinez, my Junior Lab Partner, for his efforts in understanding this lab, and Dr. Steven Koch, my Junior Lab Instructor, along with Aram Gragossian, the Teaching Assistant for this course, for helping both Stephen and I understand what was happening during the breakdown of the electron path at low voltage/high magnetic field.

SJK 16:23, 7 December 2008 (EST)

16:23, 7 December 2008 (EST)
Of course, you will need a "references" section with references to original peer-reviewed research papers


  1. J.J. Thomson's original paper: Philosophical Magazine, 44, 293 (1897) [From Stephen Wright, Classical Scientific Papers, Physics (Mills and Boon, 1964).]
  2. Original Excel Spreadsheet: EM.xls