User:Joseph Frye/Notebook/Physics Junior Lab 307L/FormalReport

Experimentally measured electron charge to mass ratio using an electron beam in a magnetic field
Author: Joseph Frye

jfrye01@unm.edu

Performed by: Joseph Frye and Alex Benedict

Date: This lab was performed on October 11, and October 18 with a follow up on Monday December 6, 2010

Location: Junior laboratory in the UNM physics building

Abstract
The electron charge to mass ratio was first measured by J.J. Thompson and established the the electron was a particle [2]. This experiment is an attempt to measure the electron charge to mass ratio. We used an apparatus designed for this experiment which consists of an electron beam in a tube of gas. The tube is inside of Helmholtz coils. By varying the strength of the magnetic field and the energy of the electrons we are able to see how the radius of curvature of the electron beam changes and from that calculate the ratio of e/m. I calculated e/m to be 2.109(23)x10^11 C/kg and 2.238(15)x10^11 C/kg using two slightly different methods. There is a large amount of systematic error associated with this experiment. I believe the error comes mostly from the apparatus itself and from the formula given to calculate results.



Introduction
This experiment was an attempt to measure the electron charge to mass ratio. We used a piece of equipment designed for this purpose. The apparatus consists of a sealed bulb filled with neon gas. Inside the tube is an electron gun that produces a beam of electrons. The tube itself is inside Helmholtz coils that create a magnetic field. When a charged particle is moving in a magnetic filed it feels a force perpendicular to the magnetic field and the particle's motion. The result inside the bulb is that the beam is curved into a circular path. We varied both the accelerating voltage on the electron gun, and the strength of the magnetic field and measured the radius of curvature of the electrons. We then calculated the ratio of the electron charge to its mass. 

Equipment and Setup
We followed the setup and procedure for this lab in Dr. Gold's lab manual, experiment number 2

Equipment The main piece of equipment used for this experiment was a e/m Experimental Apparatus made by Uchida. This apparatus consists of an electron gun inside of a sealed bulb filled with gas. The bulb itself is mounted inside of a pair of Helmholtz coils. We also used 3 power supplies for this experiment. One was used to supply current to the Helmholtz coils to induce a magnetic field. Another was used to apply a voltage to the electron gun filament. The third was used to apply an accelerating voltage to the electrons in the electron gun. We also used 3 digital multimeter one to measure the heating voltage, a second to measure the accelerating voltage, and a third to measure the current supplied to the Helmholtz coils. Here is a list of specific equipment model numbers Figure 1 shows the equipment when it was all connected. Figure 2 shows a closer view of the e/m apparatus.

Procedure
Inside the sealed bulb is an electron gun and helium gas. The bulb is in the center of Helmholtz coils which produce a magnetic field. A beam of electrons shoots from the gun into the bulb and the magnetic field curves the beam into a circle. Figure 3 shows the electron beam curving inside of the magnetic field for typical voltage and current recommended in the lab manual. We are able to measure the radius of the circle with a ruler on the back of the apparatus (see figure 4). By measuring the radius of curvature as we vary the accelerating voltage of the electron gun and the current supplied to the coils we are able to calculate the ratio of the electron charge(e) over the electron mass(m). According to the equation given on the front of the apparatus:

$$\frac{e}{m}=\frac{2*V}{B^2*r^2}$$

where
 * e is the electron charge
 * m is the electron mass
 * V is the accelerating voltage
 * B is the magnetic field
 * r is the radius of curvature

The idea is that if we know the radius of curvature then we know the acceleration of the electrons. Knowing the strength of the magnetic field and the kinetic energy of the electrons we can find the force as a function of the electron charge from:

$$F=(e*v)X(B)$$ [1]

and with Newton's second law,

$${F}={m}*{a}$$ [1]

we find that

$$\frac{e}{m}=\frac{a}{v*B}$$

we arrive at the first formula by noticing that velocity(v) is related to the Accelerating voltage, and that acceleration(a) is radial and thus related to the radius of curvature.

We vary the current in the Helmholtz coils to change the magnetic field. We vary the accelerating voltage on the electron gun to change the kinetic energy of the electrons. We then measure the radius of curvature using a ruler that is attached to the back of the apparatus.

Results and Discussion
The first trial we kept the current to the coils constant at 1.5A (constant B field) and varied the accelerating voltage from about 175V to about 225V in steps of 25V. We then measured and recorded the radius of curvature in a google docs spreadsheet. The second trial we kept the accelerating voltage constant at 200V and varied the current to the coils from 1.5A to 2.0A and again measured and recorded the radius of curvature. Our data is. From each measurement we were able to calculate a value of the electron charge to mass ratio. Figure 5 is a graph of the electron beam's radius of curvature vs a varying accelerating voltage (trial 1). Figure 6 is a graph of the electron beams radius of curvature vs changing current in the Helmholtz coils (varying magnetic field, trial 2). When I analysed the data I noticed that the relative error for the first trial was 20% versus 27% for the 2nd trial while both trials had a relatively low SEM. Because of this I chose to analyse and present the results from the two trials separately.

Accepted Value C/kg	1.75882017x10^11

The standard error of the mean in both cases was relatively good meaning that our measurements were taken consistently. This along with the fact that the calculated vaues are about 15 and 32 SEMs away from the accepted value suggests that the large relative error seen in both trials is due in large part to systematic sources. Some possible sources of error in this experiment are the way that the radius of curvature is measured, with a ruler and a human eye. Another possible source of error is the heating voltage affects the radius of curvature. We observed this but did not measure the effect. Alex Benedict came up with a way to model this experiment using the mean free path which greatly reduces the relative error. I recommend reading his report as well. here is a link to his page: Benedict's Page

Conclusions
In two trials using slightly different methods we measured the electron charge to mass ration to be 2.109(23)x10^11 C/kg and 2.238(15)x10^11 C/kg. The relative error of these two measurements combined with a low SEM and results from other student who have performed this lab suggest that there is large systematic error in this experiment. I believe that the mathematical model used to calculate the electron charge to mass ratio given in the lab manual is too simple and is the largest source of the systematic error. The formula does not include the heating voltage in the calculation. This implies that the heating voltage is assumed to not effect the energy of the electrons. This was observed to not be the case while performing this experiment. The formula also assumes that the radius of curvature does not change as the electrons move throughout he circle meaning that they do not lose energy as they collide with the gas atoms in the bulb. Another source of error is the method for measuring the radius of curvature. Overall I believe that our measurements were sound within the confines of the experiment. However, the experiment itself has a large amount of systematic error associated with the apparatus and the formula used to calculate the results.

Notes From the lab
Here are my notes from each day of the lab if you are so inclined. There are some interesting photos of strange behavior we observed on the second day of lab

Notes

Links
[Dr. Gold's Lab Manual]

Alex Benedict's Notebook

Alex Benedict's Formal report

My lab summary

My Lab Notebook

Acknowledgments

 * I would like to thank Alex Benedict for all of his help on this experiment and more generally throughout the semester.
 * Thanks to Dr. Koch Katie Richardson for discussion of the issues encountered in this lab and general guidance.

General SJK Comments
13:25, 6 December 2010 (EST):Joe, sorry for not having graded this yet. Since today is the "extra data" day, please see Alex's notebook and talk with him about ideas for what to pursue today. One example would be whether the beam radius depends on filament temperature.