User:David J Weiss/Notebook/people/weiss/Formal

Experimental Determination of the Electron Charge to Mass Ratio

Author: David Weiss Experimentalists: David Weiss, Elizabeth Allen University of New Mexico, Department of Physics and Astronomy MSC07 4220, 800 Yale Blvd NE, Albuquerque, New Mexico 87131-0001 USA Contact info: dweiss01@unm.edu

Abstract

The ratio for electric charge to the mass of an electron is a fundamental concept in physics and useful for future students interested in the study of physics. From this you can conclude how the electron is hardly affected by gravity and how the electric field governs how the electron behaves. This is important to know for the reason that it is one of the most important values in quantum mechanics. We did this by means of observing the trajectory of electrons in a known constant magnetic field. From this you can find the ratio of electric charge to mass for an electron as a function of observed radii, magnetic field, and energy. This can be done with an electron gun a Helmholtz Coil and a couple of power sources. With all these things we can determine how a beam of electrons curves within a magnetic field and thus measure a radius and with some tricky manipulation figure the ratio for electric charge compared to mass for the electrons. From my experimental data we found that the ratio of e/m is 2.3+/-.23*10 coul/kg. This was one standard deviation away from the accepted value. There was still some systematic and random error that was prevalent throughout the experiment. We will discuss the reasons and sources of these errors.

Introduction

The charge of an electron is one of the most basic concepts in the entire study of electromagnetism and atomic particles. The first person o find an electron was J.J. Thompson. He did so in a series of experiments which used cathode ray tubes to try to find electrons. He did three such different experiments and it wasn't until the third that he found the charge to mass ratio for the electron which he found in 1987². These results let him to formulate his "Plum Pudding Model" of the atom. This experiment is a lot like the one detailed here. For these experiments he was awarded the Nobel Prize in Physics in the year 1906.

After Thompson did these experiments R.A. Millikan came around and found through experimentation the charge of the electron. His experiments which involve dropping oil droplets in a chamber that could be charged to see how the oil droplets reacted in an electric field. These experiments then lead to the charge that an electron has on it³. He was later awarded the Nobel Prize in Physics for these experiments in 1923 after some controversy due to the deeds of one Felix Ehrenhaft's claim that he found a smaller charge than Millikan, but these claims turned out to be wrong and the prize was given to Millikan.

With out these fundamental experiments we could have not found the charge of the electron, and with out this fundamental constant we could not have been able to do some of the work in chemistry atomic physics and quantum mechanics. The experiment that i did was similar to the experiment that Thompson did in that I am using an electron gun to "boil" off electrons and measure how they behave in a magnetic field. I will vary the force of the electrons by mean of changing the voltage to the electron gun which is the Lorenz Force⁴, I will also vary the magnetic field by means of changing the current that is applied to the Helmholtz Coils⁵ to show how an electron responds to a changing electric field and or a changing force.

Experiment and Materials

Set Up

^{SJK 00:55, 3 December 2009 (EST)}

00:55, 3 December 2009 (EST)
"Setup" is too informal. Something like "Instrumentation" or would be better.

^{SJK 00:53, 3 December 2009 (EST)}

00:53, 3 December 2009 (EST)
For this paragraph, if you change if from "You should do this" to "We did this" (past tense) language, (and fix a lot of spelling/grammar) it will be pretty good. You should also number the relevant figure(s) and refer to them in this text by name. e.g., "As shown in Fig. 2."

An electron gun is housed in a bulb that has some gas in to so you can see the electron beam. There is also a Helmholtz Coil attached to this apparatus so that a uniform magnetic field can be generated. This is one manufactured piece so you don't have to worry about aligning everything properly which make it nice(e/m Experimental Apparatus Model TG-13 Uchida Yoko). Once you get this peace you need three different power supplies each one connects to a different part of the e/m apparatus. You need to connect a 6-9 Vdc 2A power supply (SOAR corporation DC Power Supply Model 7403, 0-36V, 3A)to the Helmholtz Coil with a multimeter in series (BK PRECISION Digital Multimeter Model 2831B). Then connect a 6.3V power supply (Hewlett-Packard DC Power Supply Model 6384A)to the heater jacks. Finally connect a power source rated at 150-300V (Gelman Instrument Company Deluxe Regulated Power Supply) with another multimeter (BK PRECISION Digital Multimeter (Model 2831B) to the electron gun.

e/m Experimental Apparatus (Model TG-13) (Steve Koch 00:53, 3 December 2009 (EST):Number this and all the figures. Also, more descriptive figure legends.)

(bottom) Hewlett-Packard DC Power Supply (Model 6384A, 4-5.5V, 0-8A) and (top) BK PRECISION Digital Multimeter Model 2831B

(left of e/m E. A.) SOAR corporation DC Power Supply Model 7403, (top right) BK PRECISION Digital Multimeter Model 2831B, (bottom right) Gelman Instrument Company Deluxe Regulated Power Supply

^{SJK 12:09, 6 December 2009 (EST)}

12:09, 6 December 2009 (EST)
Remember that all figures need to be numbered, and need to be referred to by number in the text.

Procedure and Methods

^{SJK 00:58, 3 December 2009 (EST)}

00:58, 3 December 2009 (EST)
This section does explain nicely how to do the experiment. Unfortunately, "how to" is not the style of formal scientific reports. So, you need to rephrase it in past tense, "we did" style. For example, "We followed the procedure recommended by Dr. Gold [1]..." You'll need to go through all of these next two paragraphs and change to that style.

The general procedure can be found in Professor Gold's Lab Manual¹. First of all you need to let the heater warm up for approximately 2 minutes, you do this by turning on the power, you can tell when this is warm because it will be glowing red. Once the heater is warmed up you apply a voltage of approximately 200V to the electron gun, you will start to see a beam of electrons this will glow green, if you don't see this you probably don't have it hooked up right. Once you see the electron beam you can now ally a current along the Helmholtz Coils, and when you do this you should see the electron beam start to take a circular orbit. Now you can start to take your data. The data you are going to collect is how big the radii of the electron is. There is a ruler set up on the other side of the e/m apparatus in which you can take your measurements. You will want to see how the beam is affected by changing the current along the coils while holding the voltage on the electron gun constant, and respectfully holding the current along the coils constant while fluctuating voltage.

For my experiment I first started holding the current along the coils constant at 1.35A while fluctuating the voltage on the electron gun from a max value of 250V to a minimum voltage of 146V. I observed that the more voltage you apply while keeping the current constant the radius of the electron beam increases. For the next set of experiments I kept the voltage constant at 143V and had a range of current from 0.9A to 1.33A and observed that the radii increased as I decreased the current along the coils. I took data on the radii versus the current and radii versus the voltage and this can be found on my data page for this lab.

Analysis Methods

^{SJK 12:04, 6 December 2009 (EST)}

12:04, 6 December 2009 (EST)
It is good that you have an "analysis" section in your methods. That is one thing many people have been forgetting. Your section, however, is far from the "formal" kind of analysis summary. Moreover, what you've done is mix in your analysis methods with your actual results, that turns into a sort of play-by-play for how you did the lab. That could definitely be useful in other settings, but it's not how a formal report is done. (Actually in some short reports, the methods are mixed in with results--but in that case, it's still done with more formal writing, not saying things like "Now that I had my data comes the part where I analyze the data..." etc.) Now I realize that this "Analysis Methods" section for you is actually replacing the "Results and Discussion" section, which is missing! So, you have a big challenge to separate the methods from the results in your final draft.

Your analysis methods section will be things like, "At constant voltage, the theory predicts a linear relationship between the radius of curvature and the inverse of current squared (see equation 1). We used linear regression (LINEST function in Microsoft Excel, Microsoft, Redmond WA) to find the best fit slope of the radius versus inverse squared current, and from this extracted the best fit charge to mass ratio (see equation 2)."

Then, your results and discussion section will show the actual data: (Section for Constant Voltage):"Table 1 shows the data obtained at a constant voltage of __. These data are shown in Figure __, along with the best fit line for radius versus inverse current squared. From this, we obtained a charge to mass ratio of __ +/- __, which is significantly different from the accepted value of __ (cite source). Figure __ shows that a linear fit is appropriate for this data, as there is no obvious systematic deviation from the best fit line." That's what your results and discussion should be like. You can do that for all three section (Constant Current), (Random).

Now that I had my data comes the part where I analyze the data. What i first did was to find a relation between the radii of the electron beam and the current. This turned out to be a combination of things. First of all the relationship for the magnetic field from the Boit-Savart Law¹ which is given by the equation [math]\displaystyle{ B=\frac{\mu R^{2}NI}{(R^{2}+x^{2})^{\frac{3}{2}}} }[/math], where B is the magnetic field, [math]\displaystyle{ mu }[/math] is a physical constant, N is the number of coils in the Helmholtz Coil, R is the radii of the coil and in this case is R/2. So knowing the magnetic field you can find the velocity of the electron which is given by the Lorenz Force⁴ which is [math]\displaystyle{ \vec{F}=e(\vec{v} \times \vec{B}) = m \frac{\vec{v}^{2}}{R} }[/math] and you can take this with the expression for the magnetic field to get [math]\displaystyle{ \frac{e}{m}=\frac{|\vec{v}|}{R|\vec{B}|} }[/math] witch just leaves the velocity to solve for to get the ratio for e/m. This velocity can be obtained from [math]\displaystyle{ v=\sqrt{\frac{2eV}{m}} }[/math] since we know the voltage that we applied and since we can calculate the values for the magnetic field then we can create graphs for the ratio of e/m using constant voltage and constant current.

Graph R^2 vs V constant current used in determining the value for m/e

So for the constant current the final equation is represented by [math]\displaystyle{ \frac{m}{e}\times V=\frac{(7.8\times10^{-4}\times I \times R)^2}{2} }[/math] where the m/e is your slope and V is your x points and your y points are a function of the radii. Taking the inverse of this gives you the e/m ratio. I did this with the Linst function in Excel and the graphs look like so.

For the case of constant Voltage you come up with the equation [math]\displaystyle{ \frac{m}{e}\times \frac{1}{I^2}=\frac{(7.8\times10 ^{-4}\times R)^2}{2\times V} }[/math] where m/e is your slope again and I squared is the x-values and again the radii is your y-values. I again used Excel to graph and find the equation of the line and graphed it as so

Graph R vs 1/I^2 constant voltage used in determining the value of m/e

A more detailed method for my calculations can be found in my raw data page here.

So the final value I obtained for the e/m ratio for constant voltage was 2.74(.38)*10^11 coul/kg which is within the 68% confidence interval. The e/m ratio for constant current is 1.85(.58)*10^11 coul/kg in the 68% confidence interval. Taking the average of the two I got 2.3*10^11 coul/kg in a 68% confidence interval. The value for the constant current has an error of approx 55.7% when you take the formula [math]\displaystyle{ \frac{Experimental-Accepted}{Accepted}= }[/math] % Error and using the value 1.76*10^11 coul/kg⁶. Using this same procedure for the constant voltage you get an error of 5.11%. For the average from the constant current plus the constant voltage you get 30.68% error.

^{SJK 12:05, 6 December 2009 (EST)}

12:05, 6 December 2009 (EST)
Also, I have to say that your fit to the constant current data is clearly not good! It doesn't fit the data at all.

^{SJK 12:07, 6 December 2009 (EST)}

12:07, 6 December 2009 (EST)
Another nice addition would be a table showing the e/m value (and uncertainty) you obtain with all three methods, and compared with the accepted value.

Conclusion

So from my experiment to find the ratio for e/m the best result i found was when I held the voltage constant and varied the radii I came up with a value of [math]\displaystyle{ 1.85(.58)*10^{11}\frac{coul}{kg} }[/math].

So based on my calculations from the above section I obtain the best result when I take the value using constant voltage, which the error is approximately 5.11%. This is not a bad result considering that their is a significant amount of systematic and random error present in this lab. The sources of random(non-biased) error in this lab are as follows. The collisions that occur between the helium atoms in the bulb and discharged electrons from the electrons gun, some of the energy that is used to accelerate the electrons is wasted in the form of visible light due to the collisions so the energy that is measured is an over estimate. Some of the systematic errors are caused by the imprecision of the ruler and in aligning them. We tried to over come some of these by taking two reading one on the left side and one on the right and averaging the two but this can only decrease the error but not eliminate it completely. Another error is that when the voltage is raised or the current is lowered the discharge of the electron beam doesn't line up with where it impacts the electron gun so you can adjust this by means of the focus knob, this changes the radii of the electron beam so you need to adjust the focus every time you change the voltage or current if not you potentially can alter the data and throw off the results.

The experiment showed me that an electron beam when in the presence of a uniform magnetic field creates a circular form. This circle whose radii is directly related to the strength of the magnetic field and the velocity at which the electrons leave the electron gun. If I were to do this experiment again I would not imagine to obtain such a value so close to the actual value for e/m due to the large possibility for errors due to reading the radii with your naked eye. Summing up even when using your eyes its possible to find a fundamental constant

Acknowledgments

I would like to thank my lab partner Elizabeth Allen, my lab professor Dr.Steven Koch our lab TA Pranav Rathi for their assistance and support in this lab.

References

1.Professor Golds Lab Manual

2. J. Thompson, "Cathode Rays". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Fifth Series, 296 (1897)

3.R. A. Millikan, "On the elementary electrical charge and the Avogadro constant". The Physical Review, Series II 2: 109–143 (1913).

4.Lorenz Force [1]

5.R. Merritt, C. Purcell, and G. Stroink. "Uniform magnetic field produced by three, four, and five square coils". Review of scientific Instruments, Volume 54, Issue 7, 879 (1983).

6.R.C. Gibbs and R.C. Williams, "The Electronic Atomic Weight and e/m Ratio". The Physical Review, Volume 44, Issue 12, 1029 (1933).