Physics307L:People/Gonzalez/Formal Report

Charge-To-Mass Ratio Lab Summary
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: [mailto:l1quid@unm.edu| l1quid@unm.edu] 

Abstract
We measured the charge-to-mass ratio 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. This differs from the accepted value of 1.78e11 by about 20.6%, however, since parallax was a problem as well as the collisions of helium atoms with electrons causing a reduction in speed for the electrons, and since the equations do not take this into account, I believe the majority of this error was systematic, however, since the measurements are taken by the naked eye there is definitely still a significant amount of random error as well. 

Introduction
In 1897 J.J. Thompson first developed this experiment in order to further understand the nature of cathode rays. 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.

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
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.

We then turned the equipment on and set the heater power supply to 6.3V and started with no current in the Helmholtz coils, the accelerating potential was set to 300V. From there we adjusted the focus knob until a cathode ray could be seen. 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 is then recorded by measuring the left and right radius of the circle and then taking the mean. The process is repeated but the accelerating potential was changed until measurements with accelerating potentials as low as 188V were recorded. Any potential lower than this caused the circle to break, voltages higher than this 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.



Data


Analysis&Results
Using the equation: $$B=\frac{\mu R^{2}NI}{(R^{2}+x^{2})^{\frac{3}{2}}}$$ we can find the magnetic field, with the following values: R=.15, x^2=R/2, $$\mu =4\pi *10^{-7}$$(the permeability of free space), and N=130(the number of coils on the Helmholtz coils), as well as $$x=\frac{R}{2}$$.

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

By applying the Lorentz force we can relate $$F=e(\vec{v}X\vec{B})=m\frac{\vec{v^{2}}}{r}$$, we can then solve for the ratio e/m: $$\frac{e}{m}=\frac{\vec{v}}{r\left|\vec{B} \right|}$$ After which we can relate the velocity to eV: $$\frac{1}{2}mv^{2}=eV\Rightarrow v=\sqrt{\frac{2eV}{m}}$$.

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

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. 

Conclusion
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