# User:David K. O'Hara/Notebook/physics 307 lab/E/M Ratio

## Theory

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J.J. Thompson first measured the charge to mass ratio (e/m) of the electron in 1897. By analyzing the motion of an electron acted on by a magnetic field, we can find the value of this ratio.

In a Helmholtz coil setup the value of the magnetic field around the axis of symmetry is given by

$B=\frac{(\mu)R^2NI}{(R^2+x^2)^3/2}$

In the helmholtz setup, x = R/2,the permeability of free space μ = 4πx10-7 weber/amp-meter and for this particular apparatus N=130 and R = .15 meters.

## 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

## Safety

Multiple components run at dangerous voltages and care must be taken to keep people from coming in contact with any area that could lead to an electrical shock. The electron tube must also be handled carefully as it is fragile and cannot stand up to rough handling.

## Procedure

The experiment was set up in accordance with DR. Gold's Lab manual. The experiment was carried out by setting up the helmholtz coil so that the electron beam was directed into a circular beam, then using the ruler mounted to the back of the assembly measurements were taken of the right hand and left hand radii of the formed circle. The settings for the electron beam were then changed, either in the acceleration voltage, or the helmholtz coil current and for each new setting the same measurements were taken.

The experiment was subject to a substantial amount of error due to the parallax effect of trying to measure the radius of the electron beam through the glass tube on a ruler that was about 8 cm behind the tube. The ruler was mirrored to help reduce this parallax but the effect was substantial and the measurements suffered for it.

## Data

• Trial 1 constant current
radius 1 (cm) radius 2 (cm) acc voltage (volts) coil current (amps)
2.2 3.2 100 1.36
2.5 4.0 125 1.36
2.9 4.4 150 1.36
3.1 4.8 175 1.36
3.4 5.3 200 1.36
3.7 6.0 225 1.36
3.8 6.1 250 1.36
• Trial 1 constant accelerating voltage
radius 1 (cm) radius 2 (cm) acc voltage (volts) coil current (amps)
3.9 5.9 125 .90
3.7 5.3 125 .95
3.5 5.0 125 1.00
3.3 4.7 125 1.05
3.0 4.5 125 1.10
2.9 4.3 125 1.15
2.7 3.8 125 1.20
• Trial 2 constant current
radius 1 (cm) radius 2 (cm) acc voltage (volts) coil current (amps)
2.2 3.8 100 1.25
2.6 4.3 125 1.25
2.8 4.6 150 1.25
3.2 4.9 175 1.25
3.7 5.5 200 1.25
3.9 5.9 225 1.25
4.1 6.3 250 1.25
• Trial 3 constant current
radius 1 (cm) radius 2 (cm) acc voltage (volts) coil current (amps)
2.5 4.0 110 1.15
2.7 4.4 125 1.15
2.9 4.8 140 1.15
3.1 5.0 155 1.15
3.4 5.4 170 1.15
3.7 5.9 185 1.15
4.1 6.1 200 1.15

I performed three trials with constant current as that seemed to give the most consistent beam intensity and it was easier to see the beam on those trials.

## Analysis

Given that: (From Alexandra Andrego's Lab Notebook by way of Ryan Long's notebook.):

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

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

The major difficulties with this lab occured in taking the measurement of the radii for the electron beam. The current and voltage values were solid and there was not much room for error there. The size of the radii however was very subjective. The parallax involved in reading through the side of a curved tube on a mirrored ruler several centimeters from what I wanted to measure, which was a dim electron beam, was quite challenging and the largest systemic source of error I have run into in the lab this semester.