User:David J Weiss/Notebook/people/weiss/Rough Draft

Determination of the Electrons mass versus its charge via experimentation

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 a good experiment to be conducted by persons interested in a study of physics. With this result you can determine how much charge an electron has in relation to its mass. From this you can conclude how the electron is affected by gravity and by the electric field. To find this we use a procedure that is outlined in Professor Gold's Laboratory Manual¹. 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. Based upon my calculations I found that the electric charge versus the mass for an electron is approximately 2.3(.23)*10^11 coul/kg and this was off by about 30.68% and considering that the main measurement for the radii was taken by reading a ruler on the back of the electron gun I think that it is a good result.

Introduction

The charge of an electron is one of the most basic concepts in the entire study of electromagnetism and atomic particles. The charge to mass ratio was first shown by J.J. Thompson in his famous experiment². Its actual charge was later found by R.A Millikan³. This can be found by studying the effects of an electric and magnetic field on charged 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 1987. 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 it. 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

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)

(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

Procedure and Methods

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

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.

Conclusion

So from my experiment to find the ratio for e/m the best result i found was when I held the voltage constand and varried 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 the main measuring interment that could be affected by human error was reading a ruler attached to the back of the apparatus. Considering that the sources for error are extremely high in this experiment due to the fact that you are relying on your eyes to find the values of the radii I am extremely pleased with the results I obtained in finding the value of e/m.

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 all your assistance and support during this lab.

References

1.Professor Golds Lab Manual

2.[Paper by J.J. Thompson]

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

4.Lorenz Force

5.Helmholtz Coil

6.Value for e/m [1]