Physics307L:People/Phillips/Formal Lab Final

Visually Measuring the Charge-to-Mass Ratio for Electrons
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

Abstract
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. 

Introduction
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. 

Equipment:

 * 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

Setup:
Following the setup described in Dr. Gold's Junior Lab Manual (Ref. No. 2):

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.

Procedure:
After we got everything connected correctly, 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.

The analysis that we performed on our final data include this equation that relates e/m to accelerating voltage and radius of the loop

$$V=k\cdot\frac{e}{m}\cdot r^{2}$$

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 our Excel file (Ref. No. 3) 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

$$%error=\frac{|Accepted-Measured|}{Accepted}\times100$$ 

Results & Discussion
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).

You can find the original Excel file at Ref. No. 3.

Conclusions
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.

Acknowledgements
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.