Physics307L:People/Gonzalez/Final Formal Report

^{SJK 18:18, 6 December 2009 (EST)}

18:18, 6 December 2009 (EST)
Much of the writing you have, particularly in the begginning is very good -- written in a nice style for formal reports. There are a couple things that need a lot of work. First, is the results section: you're going to need several graphs (and perhaps a table or two), and then you'll need to write about them as I describe below. Second, you don't have any citations to original peer reviewed research (and correspondingly, you do not have a "references" section). Both of those are substantial, and your report will be a lot better having done them. I make a suggestion for "extra data" below. Doing more experiments this week may help you in thinking about how to present your results.

Measurements of the Charge-to-Mass Ratio of an Electron

Author:Johnny Joe Gonzalez
Experimenters: Johnny Joe Gonzalez, Jared A. Booth
Laboratory: Junior Labs, Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, 87106
Contact: l1quid@unm.edu

Abstract

^{SJK 15:11, 6 December 2009 (EST)}

15:11, 6 December 2009 (EST)
This is a very good rough draft abstract. The style of the writing is good (using past tense) and you include most of the important information (units are missing on your values!!!). You start right off with your methods...that is OK, but typically an abstract will have an impact or motivation statement somewhere, and usually at the beginning (not always). I don't think you have the impact anywhere, so the easiest place to add it would be at the beginning, e.g. "The charge-to-mass ratio is an important fundamental property of the electron."

I also think you need to revise the "conclusions" parts of your abstract. It's not the fact that you differ by 20% that indicates systematic error. Rather it's the fact that your 68% range comes nowhere close to the accepted value. Similarly, the uncertainty (+/- 0.04) supposedly IS a result of your random error: that is what you are claiming when you report the uncertainty! So, I would use those two facts to talk about systematic and random error, and then you can speculate as to what the sources of these are.

The charge-to-mass ratio of the electron is an important fundamental property of the electron, this ratio was first discovered by J.J. Thomson in 1897 while he was researching cathode rays [1]. The charge-to-mass ratio was measured by firing a beam of electrons within a airtight bulb containing only a small amount of helium(~10e-2 mm Hg), the electrons interacted with the helium thereby allowing us to view the path of the electrons. The electrons were manipulated into a circle within the bulb by applying a magnetic field, the accelerating potential for the electrons as well as the applied current for the magnetic field were recorded, the radius of the resulting circles were also measured; from these values the charge-to-mass ratio was determined to be 2.12(4)*10e11 C/kg. This differs from the accepted value of 1.78e11 C/kg by about 20.6%, even though the collisions of helium atoms with electrons causing a reduction in speed for the electrons were not accounted for, I believe the majority of this error was random due to the parallax one would have to deal with, as well as the dark environment, however, since there was a small amount of atmosphere inside the bulb that was not accounted for, some of the error was systematic.

Introduction

^{SJK 16:02, 6 December 2009 (EST)}

16:02, 6 December 2009 (EST)
This paragraph sort of starts out from out of the blue, even using the phrase, "this experiment." Most likely you're referring to the experiment from the abstract, but you can't expect the reader to have read the abstract. Typically what I recommend students do is to style the introduction like so: motivation for measurement; different ways it has been measured; currently best values come from where; what you're doing in this report. So, I think your first paragraph maybe needs to be reorganized and / or split up, and definitely expanded. The writing is good, I like it. Just need to expand it and probably reorganize. Also, distinctly missing are citations of peer-reviewed prior research! This is an important part of this formal report exercise. Most of your citations will probably go in this introduction. For example, you can cite a paper by Thompson where he carried out the experiments you're talking about.

In 1897 J.J. Thomson first developed this experiment in order to further understand the nature of cathode rays[1]. He created a cathode ray in a vacuum tube and then measured the deflection of the cathode ray after applying a magnetic field, from his measurements he was able to determine that the cathodes carried negatively charged particles one-thousand times less massive than a hydrogen atom; these measurements were only possible since Thompson used a vacuum tube, therefore eliminating the interferences from gas molecules, this lead to the conclusion that atoms were not fundamental particles. He named the negatively charged particles electrons, this discovery would later lead R.A Millikan [2]to develop an experiment that measured the charge of an electron.

^{SJK 16:03, 6 December 2009 (EST)}

16:03, 6 December 2009 (EST)
I think this is a good concluding paragraph for your introduction. I would change the last sentence to something like, "From these values, we estimated the charge to mass ratio for the electron. We compare this to the accepted value and discuss sources of error.

In our experiment we duplicated this phenomenon and were able to create a cathode ray inside a near perfect vacuum, the cathode ray was deflected into a circle by applying a magnetic field, we measured the accelerating potential used to create the cathode ray, the applied current used to create the magnetic field, and the circumference of the cathode ray circle in order to measure the deflection of the electrons. From these values the goal is to determine the charge-to-mass ratio of the electrons carried by the cathode ray.

Method

Picture M1. A picture of the total apparatus, with all three power supplies and both multimeters. This photograph was taken by Alexandra Andrego and Anastasia Ierides

The primary piece of equipment is the Uchida e/m Experimental Apparatus (Model TG-13), which is a combination of an electron gun combined with a pair of Helmholtz coils, we connected the SOAR Corporation DC Power Supply (Model 7403) to the heater for the electron gun, in order to heat the electron gun filament. The Helmholtz coils were connected to HP DC Power Supply Model 6384A and the BK Precision Digital Multimeter (Model 2831B) was connected in series and set to measure current. With this connection we were able to control the magnetic field and monitor its current. The Gelman Instrument Company Deluxe Regulated Power Supply (500 V, 100 mA) was connected to the e/m connections on the Experimental apparatus, while also being connected to the other Precision Digital Multimeter (Model 2831B) and the multimeter was set to measure voltage. This power supply was used to control the accelerating potential of the electrons in the cathode ray. Picture M1 shows what this setup looked like, while picture M2 is a close up of the bulb and the surrounding Helmholtz coils.

Picture M2. This is a picture of the E/M device, the large circles surrounding the bulb are the Helmholtz coils, the bulb inside contains some helium(~10e-2 mm Hg.)

We then turned the equipment on and set the heater power supply to 6.3V and left it on that value for the duration of the experiment. The Helmholtz coils were initially set with no current, the accelerating potential was set to 300V. From there we adjusted the focus knob until a cathode ray could be seen(As shown in picture M3). The current on the Helmholtz coils was turned on until the cathode ray was turned into a circle. Since the cathode ray is very difficult to see the data recording part of the experiment was done in the dark.

The circumference was then recorded by measuring the left and right radius of the circle and then taking the mean, the measurements were made using a reflective mirror attached to the back of the E/M apparatus, the mirror was used in order to reduce random error due to parallax(see picture M3). The process was repeated but the accelerating potential was changed until measurements with accelerating potentials as low as 188V were recorded. Any potential lower than that caused the circle to break, voltages higher than 300V caused the circle to interfere with the inside of the bulb of the electron gun.

The measurements were repeated once again, but this time the current on the Helmholtz coils was varied, this changed the magnetic field used to deflect the cathode ray. Several measurements with current varying between -1.31A and -1.05A were taken during this part of the experiment.

^{SJK 16:30, 6 December 2009 (EST)}

16:30, 6 December 2009 (EST)
One very important thing missing from your methods are your data analysis methods! I haven't looked below yet to see how much of the important info you have in your "analysis and results section," but definitely you should have a section in your methods that is for the data analysis. Things like, "The best fit slope of the data was obtained using the LINEST function in Excel (Microsoft Corp., Redmond WA)." It's important to let people know what software you're using and what algorithms (if any) are used.

Once data was collected it was plotted first using a radius vs. inverse current plot(R vs.1/I), then using a radius vs. voltage squared plot. Both plots were made using Microsoft excel and the LINEST function was used to find the least squares fit line. This provided a slope that was used in the equation provided by Dr. Gold's lab manual [3] [math]\displaystyle{ \frac{E}{M}=\frac{2V}{(R*B*I)^{2}} }[/math]. Afterwards the value of the charge-to mass ratio of an electron was calculated.

This is a picture of the E/M device. The blue-green circle is a cathode ray being influenced by the magnetic field that is imposed on it by the Helmholtz coils.

Using the equation found in Dr. Gold's lab manual[3]: [math]\displaystyle{ B=\frac{\mu R^{2}NI}{(R^{2}+x^{2})^{\frac{3}{2}}} }[/math] we can find the magnetic field, with the following values: R=.15, x^2=R/2, [math]\displaystyle{ \mu =4\pi *10^{-7} }[/math](the permeability of free space), and N=130(the number of coils on the Helmholtz coils), as well as [math]\displaystyle{ x=\frac{R}{2} }[/math].

The resulting B value then is shown to be: [math]\displaystyle{ B=7.8*10^{-4}\frac{weber}{Amp*m^{2}}*I }[/math]

Since we are applying a force using an electric field on a point charge (electron) we can use the Lorentz force[4], by applying the Lorentz force we can relate [math]\displaystyle{ F=e(\vec{v}X\vec{B})=m\frac{\vec{v^{2}}}{r} }[/math],
we can then solve for the ratio e/m: [math]\displaystyle{ \frac{e}{m}=\frac{\vec{v}}{r\left|\vec{B} \right|} }[/math]
After which we can relate the velocity to eV: [math]\displaystyle{ \frac{1}{2}mv^{2}=eV\Rightarrow v=\sqrt{\frac{2eV}{m}} }[/math].

We can then go back to the original equation and substitute v, this gives us the following: [math]\displaystyle{ \frac{e}{m}=\sqrt{\frac{2eV}{m}}\frac{1}{r\left|\vec{B}*I \right|}\Rightarrow \frac{e^{2}}{m^{2}}=\frac{2eV}{m}\frac{1}{r^{2}B^{2}I^{2}}\Rightarrow \frac{e}{m}=\frac{2V}{\left(rBI \right)^{2}} }[/math]

From the Data we are able to get the result: 2.12(13)e11 C/kg, though higher than the accepted value, this is expected due to systematic error and some random error. The systematic error is mostly due to the electrons colliding with the helium molecules, thus slowing there acceleration, since the equations doesn't take this into account, some sort of systematic error is expected; also, since the radius is measured with the naked eye through the bulb, some random error is expected as well.

Data

Below are links for my data analysis for the charge-to-mass ration of an electron, the data is shown for both when potential was held constant
constant potential
and for the charge to mass ration of an electron when the current is held constant
constant current.

Picture D1. The picture shows a radius vs. 1/I plot. This is using a constant potential of 200V. The plot was made using Microsoft excel, and the least squares fit was obtained using LINEST.

Picture D2. The picture shows a radius vs. voltage plot. Here we treat the current as constant 1.31A. Note that the current reading was actually -1.31A, but since the sign only denotes the direction of the current the absolute value was used for calculations. Once again, the program used was Microsoft excel, and the least squares fit was obtained through the LINEST function.

After plotting the data and using LINEST to gain a least squares fit, the slope of both plots can be entered into the equation:

[math]\displaystyle{ \frac{E}{M}=\frac{2V}{(R*B*I)^{2}} }[/math]

with the following results: [math]\displaystyle{ slope(constantV)=379.72 }[/math], with this we can enter it to find the charge-to-mass ratio of the electron:

[math]\displaystyle{ slope(constantV)=\frac{2*200}{(7.8*10^{-4})^{2}}*379.72=2.50*10^{11}\frac{C}{kg} }[/math]

And then again using constant current: [math]\displaystyle{ slope(constantI)=1.07*10^{5} }[/math]

[math]\displaystyle{ slope(constantI)=\frac{2}{(7.8*10^{-4}*1.31)^{2}}1.07*10^{5}=2.05*10^{11}\frac{C}{kg} }[/math].

Pictures D1 and D2, show the graphs of the plots used.

^{SJK 17:50, 6 December 2009 (EST)}

17:50, 6 December 2009 (EST)
These next two sections, "Data" and "Analysis and Results" are going to need an overhaul. First, as mentioned above, the analysis methods should be described in the methods section. Other than that, what you're going to need to do is to make figures and tables that present the important results. It is great that you provide your raw data (the Google sheet). You definitely should do that, but it should not be the prominent piece of information in your report. Rather, it should be an appendix or a link. What you should show your readers are really nice representations of the results: mostly graphs, and maybe a table or two. Ideas for graphs: radius versus voltage (for constant current) along with best fit line; radius versus inverse-squared current (for constant voltage) along with best fit line; and maybe e/m ratio versus measurement number for random values. You could show a table that lists the e/m ratios obtained from the different methods, along with the accepted value. Each of those figures and tables should be numbered, and you should refer to them by number in the text. You could even make sub-headings, such as "constant current", "constant voltage," have a graph for each section, and some text describing the graph and discussing the results. Then after doing each of those sections, you could conclude the results section by showing a table comparing the values from all methods and the accepted value as well, and then discussing systematic error as you do now with your last paragraph of that section.

The following is a link to the raw data for the charge-to-mass ratio of an electron.

Conclusion

^{SJK 18:05, 6 December 2009 (EST)}

18:05, 6 December 2009 (EST)
I think you're on the right track with this section, but will want to revise a lot to make more formal. Also, after revising the previous sections, you may have a clearer idea on what to say. I think for your "extra data" this week, you should focus on your ideas for where the systematic error is coming from, and perhaps devise a few more experiments to investigate it (you won't be able to eliminate it). Perhaps in concert with this, you could try constant current and constant voltage experiments with different fixed values. This would be interesting to show how the apparent ratios change as you change these values.

My results were 2.12(13)e11 C/kg(Steve Koch 18:05, 6 December 2009 (EST):units!) though this was different, but not unexpected from the accepted value. Systematic error was mostly due to electrons interacting with helium atoms, while my random error is due to measuring the circle using only the naked eye through the bulb. However, I do believe that this is still a very good way on measuring the charge-to-mass ratio of an electron.

Acknowledgements

I would like to acknowledge My lab partner Jared, my lab professor Dr. Koch, as well as my lab TA Pranav for helping me and suggesting using our accelerating potential.

Links

My notebook on E/M

References

1. , J. J. [1897]: ‘Cathode Rays’, Philosophical Magazine, 44, pp. 293–316.
   

1. , J. J. [1898]: Philosophical Magazine, 46, p. 528
   

2. RA [1913]On the elementary electrical charge and the Avogadro constant. Phys Rev 2:109–143 http://dx.doi.org/10.1103/PhysRev.2.109
   

3. . Gold, Physics 307L[2006]: Junior Laboratoy, UNM Physics and Astronomy

4. , H.A.,[1902] The fundamental equations for electromagnetic phenomena in ponderable bodies deduced from the theory of electrons: KNAW, Proceedings, 5, 1902-1903, Amsterdam, pp. 254-266