e/m Charge-to-Mass Ratio Lab

Dr. Darrel Smith1

Physics Department
Embry-Riddle Aeronautical University
(Dated: 7 February 2021)
The purpose of this experiment is to measure the charge-to-mass ratio of the electron, e/m. In this experiment,
electrons are accelerated and execute circular motion perpendicular to a homogeneous magnetic field produced
by a pair of Helmholtz coils. The voltage, radius, and strength of the magnetic field can be used to determine
e/m.

I. BACKGROUND

r
Because of the very small mass of the electron (9.11 × 2eV
10 −31
kg), the electron has the largest charge-to-mass ra- v = (3b)
m
tio of any system or elementary particle (e/m = 1.76 ×
1011 C/kg. As a result, large accelerations can be pro-
~ By applying where v is the velocity, V is the accelerating potential
duced with only a modest electric field (E). (in volts), and e is the charge of the electron. Combining
Newton’s 2nd law, the relationship between the accelera- Eqs. 2b and 3b, one can calculate the equation for e/m.
tion and the charge-to-mass ratio can be shown.

X
Fext = ma (1a)
III. THE EQUIPMENT

qE = ma (1b)
A vacuum tube with rarified hydrogen gas is posi-
tioned between the two Helmholtz coils. A power sup-
e ply box containing three power supplies (5A, 10mA, and
a= E (1c)
m 50mA) is connected to the experiment. The 5A sup-
In the early 1900’s, electrons were known as cathode ply is for the Helmholtz coils that produce the mag-
rays and the only known property for these particles was netic field for this experiment while the third sup-
the e/m ratio. The ability to steer electrons required ply (50 mA) is the high voltage supply for acceler-
knowledge of only two parameters, the electric field E, ating the electrons from rest to their final velocity.
and the charge-to-mass ratio e/m. More details about the magnetic field produced by
Helmholtz coils can be found at the following website:
http://physicsx.pr.erau.edu/helmholtzcoils/.
II. THE EXPERIMENT

In this experiment, the electrons are momentarily ac-

celerated from rest (using an electric field) perpendicular
to a magnetic field produced by a pair of Helmholtz coils.
As a result the electrons move in a circular path described
by Newton’s 2nd law.

mv 2
F = qvB = (2a)
r

e v
= (2b)
m rB
As the electrons are accelerated through the electric
field, we can apply conservation of energy to determine
the velocity: FIG. 1. The charge-to-mass experiment is shown in the figure
above along with the triple power supply. A spare single-
channel high voltage supply is shown to the right.
1
K = U mv 2 = eV (3a)
2
 2

IV. PROCEDURE fields if you are interested.

You will find more information regarding the experi-

mental procedure in the following material:
C. Instruction Sheet
1. my physicsx webpage,
This instruction sheet did not come with this appara-
2. our textbook (TZD, chapter 3.10),
tus. However, it describe our e/m experiment (what they
The e/m webpage found in my PS315 webpage con- call Experiment 2), and another experiment (Experiment
tains an instruction sheet, a data analysis sheet, and an 1) that uses crossed electric and magnetic fields to mea-
instruction sheet. The instruction sheet contains two de- sure the e/m ratio. I found this leaflet to be a useful re-
scriptions for two different e/m experiments. Our experi- source as you search for other reading material regarding
ment is the second experiment using the K.T. Bainbridge this lab. It is always insightful to find other explanations
Tube. of the same experiment to help build your confidence and
understanding in what you are doing. I found the sec-
tions on “Historical Background” and “Experiment 2” to
A. Instruction Sheet be informative.

As you read through the instruction sheet, become fa-

V. IMPORTANT CONSIDERATIONS
miliar with the names attached to the critical parts of
the experiment. Also “read” the Safety Notes on the
first page regarding the use of high voltage in this exper-
iment. • Be careful. You will be using high voltage in
this experiment.

B. Data Analysis • Have your lab assistant or professor “check” your

wiring before turning on the voltage supplies.
This leaflet should be read with great interest and un- • Low voltages can also present a health hazard.
derstood. It describes the theory and guiding principles While the voltage supply to the magnet does not
involved in this experiment. It also describes how one exceed 7.5 volts, it carries up to 5A.
should go about collecting the data for this exper-
iment. The experimental apparatus you are using was • The copper wires in the Helmholtz coils appear
built in Germany and you will not see the words charge- to be bare, but they are covered with a thin, clear
to-mass used in this document. Instead, they refer to polyurethane coating. This insulation should pro-
e/m as the “specific charge.” Also, they use the sym- vide adequate protection; in any case you should
bol U to refer to the electrostatic potential (volts). In still avoid touching the wires in the Helmholtz
the United States, we tend to use V as the electrostatic coils.
potential. So, please make those two connections.
Again, please read the “Safety Notes” found on page 2 • Again, if you are unsure about operating the ap-
of this leaflet. In particular, do not touch the Helmholtz paratus, please ask for assistance. The equipment
coils while this experiment is in operation. is moderately expensive; however, you health and
On page 3 is described a procedure for focussing the well-being is more important.
electron beam by varying the voltage on the Wehnelt-
• Don’t forget to do your error analysis. You will
cylinder from 0 . . . 10 V until it leads to a narrow, well
need to do the error analysis of a straight line fit,
defined beam. In previous years, the students have com-
and afterwards propagate the uncertainties cor-
mented that this procedure produces little, if any, differ-
rectly to quote your final answer in the following
ence in beam definition. However, give it a try.
format:
As you can see from Fig. 6 on page 4, you will want to
e e
collect data with the circular orbit at a constant diameter
e/m = ±δ
for various voltages U (300 V → 200 V) while recording m fitted m
the magnetic current I (in amperes).
Please take note of the comments made on page 2
of this leaflet. There are two ways to calculate the
magnetic field. Go ahead and use Eqs. 7, 8, and 9 along VI. HOW TO TREAT ERROR BARS IN BOTH THE X
with the other physical parameters recorded on this page AND Y DIRECTIONS
in order to calculate the magnetic field as a function of
the current (i.e., B = k I). For your information, we do In this lab you are asked to plot your data as shown
have a magnetometer to measure the actual magnetic in Fig. 6 in the Data Analysis leaflet. In this figure the
 3

potential U is plotted as a function of I 2 resulting in the where δI is the uncertainty in the current, and U is the
linear trend shown in Fig. 2. While most fitting programs potential at that particular current. Adding this cor-
fit data having error bars in the vertical direction, they rection to the already existing error bar in the vertical
don’t include the horizontal error bars when determining direction we find the new error bar in the y direction:
the errors in the fitted parameters. So, the question nat-
urally arises, “How can one include the horizontal error
bars as part of the fit?” q
σy0 = (σy )2 + (δU )2 (4)

The new error bar (σy0 ) is larger than the original error
bar in the y direction (σy ), and combines the original
error bars (σx , σy ) into a single error bar. The size of the
new error bar (σy0 ) is not the same for every point. Make
sure the new error bars (σy0 ) are calculated for each data
point. By doing this, we have replaced the old fitting
weights wi = 1/σi2 with new weights wi0 = 1/σi02 . In this
experiment, the error bars (σi0 ) will all be different, thus
the weights for each of the data points are different.

At this point, the fit to U = c I 2 + const should be

redone with the new error bars included in the fit. While
the slope c may not change by much, there should be a
FIG. 2. A plot of the voltage versus the current squared. This definite increase in the uncertainty in the slope (δc).
is from Fig. 6 found in the Data Analysis leaflet.
In this lab, the slope c is related to the following quan-
tities:
Let’s imagine that we have error bars as shown in
Fig. 3. Let’s define a procedure where the error bar in the
x direction (σx ) is reduced to zero, thus expanding the e 1 2 2
error bar in the y direction to a new value (σy0 ). In order c = r k
me 2

where k can be found in the Data Analysis sheet on my

website, k = µo (4/5)3/2 n/R determined from the Biot-
Savart law. See my website regarding Helmholtz coils.
Solving this equation for the charge-to-mass ratio, we
FIG. 3. Combine both error bars in the x and y direction (σx find:
and σy ) into a single error bar in the y direction (σy0 ).

to convert the horizontal and the vertical error bars into e 2c

a single error bar (σy0 ), we need to know the relationship = 2 2
me r k
between U and I 2 . In this case, we will use the obvious
relationship U = c I 2 + const where c is the slope of the
straight-line fit. When determining the parameters of the
Using this equation, one can complete the error analysis
straight-line fit, the results should report the fitted value
using the familiar propagation of uncertainties technique:
of the slope (c) as well as the uncertainty of the slope
(δc).
With this information in hand, we can now determine
the relationship between the two error bars (σx , σy ). Us-
ing the fitting function U = c I 2 + const, we see that:     s 2  2
e e δc 2 δr
δ = +
δU = c 2I δI me me c r

or
2 δI where we have assumed that the uncertainty in k (δk) is
δU = U very small and close to zero.
I