# User:Ryan P. Long/Notebook/Physics 307L/2009/11/09

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# e/m Lab

## Objective

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The objective of this lab is to find the charge to mass ratio for an electron. We used an apparatus with an electron gun that fired electrons into a helium filled bulb placed inside a magnetic field to excite a ring of Helium atoms. We measured the radius of the ring, and knowing about the Lorentz force, we calculated e/m.

## Equipment

• BK Precision Multimeter 2831B (x2)
• HP 6384A DC Power Supply
• Uchida Yoko e/m experimental apparatus model TG-13
• Soar Corp. 7403 DC power supply
• Gelman Instrument Company 500V Deluxe Regulated power supply
• BNC Cables

## Setup

First we connected the HP power supply to the Helmholtz coils on the Uchida apparatus with one of the BK multimeters in series. Next the other BK multimeter was connected to the voltmeter jacks on the Uchida apparatus. The Soar power supply was connected to the heater jacks on the Uchida. Finally, the Gelman was connected to the electrodes on the Uchida. We followed the procedure in Professor Gold's manual, which included letting the heater warm up for 2 minutes before applying voltage to the electrodes on the apparatus.

To take data, we let either let the voltage of the electrodes or the current running through the coils be constant, while varying the other parameter. We measured the radius of the rings on the right and the left side using the built in ruler.

## Data

Note: At a current of -1.35A, maximum accelerating voltage is 320V, the minimum is 165V. At -1.05A, the maximum voltage is 200V and the minimum is 100V.

11/16/09 Voltage applied to heater jacks: 6.24 V

### Raw Data:

key=0Ajaf_99vseiXdEIyXzJxWGJhLUQ2SE9qNGo4UURmMUE width=760 height=300

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## Analysis

Given that: (From Alexandra Andrego's Lab Notebook):

$B=(7.8\times10^{-4}\frac{weber}{amp-meter})\times I\,\!$
${e}{V}=\frac{1}{2}{m}{v}^{2}\,\!$
${F}_{B}={q}{v}{B}\,\!$
$\frac{e}{m}=\frac{{2}{V}}{{r}^{2}{B}^{2}}=\frac{{2}{V}}{{r}^{2}{({{7.8}\times10}^{-4}{I})}^{2}}\,\!$

We can calculate $\frac{e}{m}\,\!$ based on the slope of a $\frac{V}{{r}^{2}}\,\!$ line holding current constant, or the slope of a $\frac{r}{\frac{1}{I}}\,\!$ line holding voltage constant.

## Part 1

I calculated the slope and slope uncertainty of the line from my plot using the linest array function in excel. The radius used in my analysis is the mean of the left side and right side radii, per Professor Gold's manual. $Slope=8.141\cdot 10^{-6}\pm 3.989\cdot10^{-7}\frac{m^{2}}{V}$

Using the equation:

$\frac{e}{m}=\frac{{2}{V}}{{r}^{2}{B}^{2}}=\frac{{2}{V}}{{r}^{2}{({{7.8}\times10}^{-4}{I})}^{2}}\,\!$

with my slope and constant current, e/m can be calculated:

$\frac{e}{m}=\frac{1}{slope}\times \frac{2}{(7.8\cdot 10^{-4}\times 1.35)^{2}}=\frac{1}{.00000814}\times \frac{2}{(7.8\cdot 10^{-4}\times 1.35)^{2}}\cong 2.216\cdot 10^{11}\frac{C}{kg}$

using my slope uncertainty, I calculated a lower and upper bound value for my e/m ratio:

$lower bound: 2.11\cdot 10^{11}\frac{C}{kg}$

$upper bound: 2.33\cdot 10^{11}\frac{C}{kg}$

The accepted value (from wikipedia) is:

$1.758820150(44)\cdot 10^{11}\frac{C}{kg}$

## Part 2

$Slope=15.697\pm 0.68610\frac{1}{A\cdot m}$

$\frac{e}{m}=\frac{(slope)^2\times 2\times V}{(7.8\times10^{-4})^{2}}=\frac{(15.697)^2\times2\times240}{(7.8\times10^{-4})^{2}}\cong 1.944\cdot 10^{11}\frac {C}{kg}$

again, using my slope uncertainty, I calculated a lower and upper bound value for my e/m ratio:

$lower bound: 1.78\cdot 10^{11}\frac{C}{kg}$

$upper bound: 2.118\cdot 10^{11}\frac{C}{kg}$

## Conclusions

As you can see from our raw data, we recorded a great deal of trials. I wasn't quite sure how to incorporate all of them so I randomly selected the values to analyze. Using the linest function in excel I calculated the slopes for the two tables above in order to calculate my e/m values. This experiment is fraught with systematic error, the ring we measure was not actually circular, it was elliptical. Also there was human error in the sense of visually measuring the radius.

## Acknowledgements

Thanks to Tom, my lab partner, he's always a pleasure to work with. Also thank you to Alexandria Andrego, I referenced her lab notebook frequently, finally as always thanks to Dr. Koch and Pranav Rathi for their assistance throughout the lab.