Physics307L:People/Gonzalez/Final Formal Report

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

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 2.18(9)*10e11 C/kg. This differs from the accepted value of 1.76e11 C/kg by about 22.4%, 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 since my uncertainty was only at (+/-0.09) and my results still fell out of the 68% range of the accepted value. However, since there was a small amount of atmosphere inside the bulb that was not accounted for, some of the error was systematic.

Introduction

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.

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, we estimated the charge to mass ratio for the electron. We compare this to the accepted value and discuss sources of error.

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.

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} \right|}\Rightarrow \frac{e^{2}}{m^{2}}=\frac{2eV}{m}\frac{1}{r^{2}B^{2}}\Rightarrow \frac{e}{m}=\frac{2V}{\left(rB \right)^{2}} }[/math]

Data

Using the method above, the calculated charge-to-mass ratio is: 2.18(9)e11 C/kg, this value was gained by taking the average of all E/M ratios(including constant potential, constant current, and a combination of the two). 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. The error bars are to one standard deviation.

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(Microsoft Corp., Redmond WA). The error bars are shown to one standard deviation.

After plotting the data and using Microsoft Excel and the LINEST(Microsoft Corp., Redmond WA) function 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 for constant voltage: [math]\displaystyle{ slope(200V)=379(9) }[/math], with this we can enter it to find the charge-to-mass ratio of the electron:

[math]\displaystyle{ slope(200V)=\frac{2*200}{(7.8*10^{-4})^{2}}*379(9)=2.50(4)*10^{11}\frac{C}{kg} }[/math]

[math]\displaystyle{ slope(188V)=440(14) }[/math]

[math]\displaystyle{ slope(188V)=\frac{2*188}{(7.8*10^{-4})^{2}}*440(14)=2.72(8)*10^{11}\frac{C}{kg} }[/math]

And then again using constant current: [math]\displaystyle{ slope(constantI)=1.07(2)*10^{5} }[/math]

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

Pictures D1 and D2, show the graphs of the plots used, the error bars are showing the uncertainty to one standard deviation.

The following is a link to the raw data for the charge-to-mass ratio of an electron, the raw data contains a section that has the calculated values of the charge-to-mass ratio of an electron, underneath the "all values" box.

Results

After calculating the average of all the data taken the value for the charge-to-mass ratio was:[math]\displaystyle{ \frac{e}{m}=2.18(9)*10^{11}\frac{C}{kg} }[/math]
The value found if using a plot and least squares fit line when the potential was constant at 200V:[math]\displaystyle{ \frac{e}{m}=2.50(4)*10^{11}\frac{C}{kg} }[/math]
and at 188V:\frac{e}{m}=2.72(8)*10^{11}\frac{C}{kg}
When plotted against Voltage and radius squared, the least squares fit gave a slope that calculated the value:[math]\displaystyle{ \frac{e}{m}=2.05(4)*10^{11}\frac{C}{kg} }[/math], when current was constant
All of the vlaues above are shown with a 68% confidence interval. the accepted value of the charge-to-mass ratio is:[math]\displaystyle{ \frac{e}{m}=1.76*10^{11}\frac{C}{kg} }[/math]according to Dr. Gold's lab manual[3].

The results varied greatly, and the uncertainty still did not bring my results close to the accepted value, I believe this to be due to a fair amount of both systematic and random error. Error from parallax, as well as electrons interfering with the helium atoms, and the calibration of the ruler used to measure the radius of the circle could've all contributed to systematic and random error.

Conclusion

My closest results to the accepted value was 2.05(4)e11 C/kg, this result is 15% off from the accepted value. Systematic error was mostly due to electrons interacting with helium atoms, while my random error was due to measuring the circle using only the naked eye through the bulb and thus being subject to parallax. Greater random error was noted when trying to use the radius vs inverse current plot, where the current only varied between -1.31 to -1.05. There could be many reasons for such a dramatic error, one being that at the measured values the circle of electrons at times would come very close to touching the surface of the bulb, it may be possible that the bulb interfered with the path, this would indeed create larger values for the charge-to-mass ratio.

I found it fascinating as to how much the magnetic field could influence the electrons, its also amazing to realize that we can measure the circumference of the path the electrons take when we realize that electrons are point charges.

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
Dr. Gold's lab manual.

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