Electron Charge to Mass Ratio
Ryan Plunkett, Nick Tenerelli, and Stuart Casarotto
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
Abstract: We calculated the electron charge to mass ratios using an e/m apparatus that
uses Helmholtz Coils to generate a magnetic field through which a beam of electrons are
accelerated through. The theoretical value of e/m is equal to 1.76*1011 C/kg, compared to
our experimental value of 1.74*1011 ± 5*109 C/kg.
I. Introduction
The purpose of this experiment is to use an e/m apparatus in order to calculate the electron charge to mass
𝑒
𝑒
ratio, or . From the experiment, we will be able to measure indirectly using various data collected. In
𝑚
𝑚
the end, we will compare our experimentally determined values for
𝑒
𝑚
with its respective theoretical value.
Background
The first direct evidence that electrons existed and had particle like properties was found when J.J.
Thomson measured the charge to mass ratio of an electron in 1897. In his experiment, Thomson examined
the effect of a magnetic field on moving electric charges.
In this experiment, the magnetic force, Fm, will act on the electron beam in a magnetic field, B, that is
perpendicular to the electrons velocity, v, and so its magnitude can be given by
𝐹𝑚 = 𝑒𝑣𝐵
(1)
Since the electrons are moving in a circle with a centripetal acceleration caused by the magnetic force, we
can combine Equation 1 with Isaac Newton’s 2nd Law of Motion and the equation for the kinetic energy of
an electron to find that
𝑒
𝑚
=
2𝑉
𝐵2 𝑟 2
(2)
where
r = the radius of the electrons’ circular path
V = the electrode voltage
𝐵=
𝑁µ0 𝐼
𝑎
4
(5)3/2
and where
N = the number of turns in each Helmholtz coil
I = the current trough the coils
a = the radius of the coils
µ0 = 1.26 x10-6 T·m/a (magnetic permittivity of free space)
(3)
II. Experimental Setup
Figure 1. Aerial View of Apparatus Setup
Figure 2: Electron Ring and Ruler
It is important to note that while doing the experiment, we kept within the following ranges of values for
the apparatus:
Heater
6.3V
Electrode Voltage
150-300 VDC
Helmholtz Coil Voltage
6-9 VDC
Helmholtz Coil Current (I) 0-2 A
We began the experiment by measuring a, which turns out to also be the horizontal distance between the
centers of the two coils.
When measuring the radius of the electron beam, we set up a camera at a fixed distance, so that our
measurements were consistent. In order to get a clear picture, we took the pictures in a dark room with an
exposure of about 12 seconds. We then took nine pictures while varying the electrode voltage (V) by
increments of about 15V within the range given above. In order to keep the radius readings between 4.5
and 5.5 cm, we had to adjust the Helmholtz Coil voltage occasionally, which in turn adjusted the coil’s
current (I).
Due to parallax, caused by the differing depths of the electron ring and the ruler from the camera, we
needed to use geometry in order to figure out the actual radius of the electron ring. The figure below shows
that the camera, the center of the beam/ruler, and the actual and perceived radius measurements form
similar right triangles.
Figure 3. Similar Triangles Used to Correct for Parallax
Therefore, we were able to calculate the actual radius value (d) of the ring using the distance from the
camera to the ring (b), the distance from the camera to the ruler (c), and the perceived radius value (a) for
each picture using
𝑑
𝑎
=
𝑏
𝑐
where d = the radius, r, in Equation 2.
III. Results
Figure 4. Linear Plot of the Electron Charge to Mass Ratio
(4)
Figure 4 is a graph that shows e/m as the slope of B2r2 plotted against 2V (this relationship was derived
from Equation 2) using the data from each of the nine pictures taken. The error bars for the data points were
calculated using the Products and Quotients error propagation equation (3.18) on page 61 of Taylor. The
error for the values that went into the data points were calculated using standard error propagation. The
slope of graph indicates our experimental value for e/m to be equal to 1.74*1011 ± 5*109 C/kg.
IV. Discussion
When we compare our measured value 1.74*1011 ± 5*109 C/kg with its theoretical value of 1.76*1011 C/kg,
we see that the theoretical value falls within the error bars for the measured value. Although this value is
perfectly acceptable in relation to the theoretical value, there was a small but noticeable source of potential
error in the experiment that is worth mentioning. This source of potential error came from the fact that the
moving electrons often struck helium atoms within the glass orb, causing the electrons to slow down. This
caused the width electron ring to appear larger with the slower moving electrons composing the inner
portion of the ring. This also caused one side (left) of the ring to be slightly thicker than the other side
(right), as can be seen in Figure 2. We believe this is what caused the radius values on the left side of the
ruler to be different than those on the right side. We tried to correct for this small error by taking the
average of the two radius values as our perceived radius. Regardless of this source of error, we were
successfully able to use an e/m apparatus in order to calculate the electron charge to mass ratio.
References
Book
[1] J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements.
Sausalito: University Science Books, 1997, pp. 45-79, 181-199.
Internet
[2] "Mass-to-charge ratio," [Online], (2013 Oct), Available at HTTP: http://en.wikipedia.org/wiki/Mass-tocharge_ratio