Lab 3:
Measuring the Charge to Mass Ratio of the Electron
Edited 10/18/14 by Joe Skitka, Stephen Albright & DGH
Figure 1. Tube Used by Thomson to Discover Electron
http://www-outreach.phy.cam.ac.uk/camphy/museum/area2/images/cabinet3_2.jpg
Objective
The charge-to-mass ratio of the electron will be measured using two similar techniques.
Introduction
In 1906, J.J. Thomson, the discoverer of the electron, won the Nobel Prize in Physics for his
“investigations on the conduction of electricity by gases" (see Figure 1). This experiment
measures the charge-to-mass ratio of the electron using techniques similar to those originally
employed by Thomson.
At the time of Thomson’s experiment, neither the mass nor the charge of the electron was known
(Thomson had only recently demonstrated that electrons were particles, and the part they played in
the structure of atoms was not yet clear.) The key to designing an experiment which could, in
principle, determine the desired ratio is to utilize multiple physical processes (manifested in
equations) which, when combined, eliminate a degree of freedom between the mass and the
charge. This strategy will always work in theory if one fewer unknowns is added to the system
than the number of equations and as long as degeneracies do not exist in the combined
phenomena’s dependences on the charge, mass, or other introduced unknowns.
Another consideration is whether such an experiment can be performed in practice; one cannot, for
instance, measure an electron on a digital scale, which is why the ratio of the charge and mass was
sought before either quantity could be determined independently. The most manageable processes
which the electron was known to participate in at the time related to Lorenz forces, so one way to
accomplish this measurement was by using the acceleration of an electron in both a static external
electric and magnetic fields. The parameters of interest can be solved for if the electron starts at
rest, is accelerated first by an electrostatic field of known potential difference, and subsequently by
a static, uniform magnetic field of known magnitude. When combined with details of the fields,
the rate of curvature of the path of the electron in the magnetic field uniquely determines the
charge to mass ratio. This is what is done for the first part of the lab. The second part of the lab
adds another electric field to cancel the forces of the magnetic field, as Thomson originally did.
This eliminates the dependence of the result on the radius of curvature, simplifying the analysis.
Figure 2. Electron Gun
The experimental setup to be used contains an electron “gun”, a fluorescent screen, and
electrostatic plates within an evacuated glass chamber, and, outside of the chamber, Helmholtz
coils. The gun is where the electric forces initially accelerate the electron. It consists of a
cylindrical cavity off to one side of the evacuated chamber with a cathode and an anode at
opposite ends across which an electric potential is applied. The anode has a slit that allows a
narrow beam of cathode rays, aka electrons, to escape into a larger spherical portion chamber (see
Figure 2 and the TEL-525 documentation on the lab wiki). The Helmholtz coils (radius 6.8 cm)
produce an approximately constant magnetic field. The parallel, horizontal electrostatic plates
contained within the chamber are used to provide the approximately constant deflecting electric
field in the second half of the experiment.
Figure 3. Fluorescent Screen
A fluorescent screen with a grid of 1 cm squares is positioned in the path of the electrons and
displays the beam as a blue line (see Figure 3). This can be used to determine the curvature of the
beam in the first part of the lab. For the second part (Thomson’s method), the electric field
generated by the parallel charged plates (located above and below the screen in figure 3) maintains
an approximately straight path of the electron.
Figure 4. Helmholtz Geometry
Helmholtz coils consist of two identical coils wired in series and placed coaxially such that their
magnetic fields are in the same direction. Helmholtz showed that when two such coils are
separated by a distance exactly equal to the radius of the coils, the field between the coils is
uniform in strength over a significant range (the second derivative of the field with respect to any
spatial dimension is zero in the exact center). See Figure 4.
PART I: THE MAGNETIC DEFLECTION METHOD
Theory
The potential difference between the negative cathode and positive anode creates an electric field
which accelerates the electrons towards the anode. The slit in the anode allows electrons to escape,
generating a flat, wide beam (ribbon) of electrons. The potential difference, V, between the
cathode and the anode determines the velocity of the electrons. Since the kinetic energy of the
electron is equal to the work done on the electron by the accelerating potential the magnitude of
the velocity can be derived as follows:
𝑚𝑣 2
= 𝑒𝑉
2
(1)
Solving for v, we obtain:
𝑣=√
2𝑒𝑉
𝑚
(2)
The pair of Helmholtz coils creates a constant magnetic field, B, at its center. If the distance
between the coils is equal to the radius, r, of a coil, B can be calculated as follows:
B=C
nI
𝑟
(3)
Tm
32π
10-7
, n is the number of turns of one coil, r is the radius of a coil, and I is
A
5√ 5
the current in amps. In this case n = 320 and r = 6.8 cm.
Where C =
The electron beam is bent as it passes though field B, by a constant force that is perpendicular to
the electron’s velocity, v. The magnitude of this force is:
FB = evB
(4)
FB, acts at a right angle to the velocity of the beam causing circular motion of radius, R. The
centripetal force, FC , of the electron beam is given by:
mv2
Fc = R
(5)
Where R, the radius of curvature, can be determined as follows:
x2 + y2
(6)
2y
Given two known points on a curve (0,0) & (x,y), where (x,y), is less than 90˚ from the origin,
Geometry yields the above equation. In this experiment, the origin is at the anode, and the point
(x,y) is measured on the grid. See Appendix for derivation of Equation (6).
R=
FB must equal FC , so we obtain from Equations (4) and (5):
mv2
Bev= R
(7.1)
mv
Be= R
(7.2)
By substituting in Equation (2) and rearranging, we obtain:
2Ve
RBe=m√ m
Squaring both sides of this equation and solving for
e
m
=
e
m
(8)
yields:
2V
(9)
R2 B2
Given a known accelerating potential, V, and magnetic field, B, we can see from Equation (9), that
e
measuring the radius, R, of the electron trajectory, allows us to find m, the charge to mass ratio of
an electron.
Fig 5A
Figure 5. Filament/Cathode and Anode Circuit
Fig 5B
Fig 6A
Fig 6B
Figure 6. Helmholtz Coil Circuit
Procedure
The apparatus should be configured as shown in Figures 5 and 6. Use what you have learned in
the course thus far to gather data to determine as accurately and precisely as possible the charge to
mass ratio of the electron. You should attempt to correct for as many biases as possible. The
preceding theory section should be of considerable help. Be sure to make a good estimate of the
uncertainty as well. This may require additional small experiments and measurements. In the lab
report, be sure to justify why your technique affords the most accurate determination of e/m
compared with other options you considered. Although different groups may use different
strategies to improve accuracy and precision, note that the general approach of the experiment will
likely be the same across all groups.
You may wish to switch the direction of the magnetic field. This can be accomplished by simply
switching the input and output cables from the Helmholtz coils to the power source (when it’s off).
PART II: THE ELECTROSTATIC BALANCING METHOD
Theory
As stated in the introduction, an electric field will be used to cancel the deflection caused by the
Helmholtz coils eliminating the need to measure the radius of curvature. Drawing on some of the
relations established in the previous theory section, the charge-to-mass ratio of the electron can be
related to the new experimental parameters. You must derive this relation during the labs
yourselves. The TA will be available to give hints if you are stuck.
Fig 7.a
Fig 7.b
Figure 7. Electrostatic Plates Circuit
Procedure
The electrostatic plate connections should be configured as shown in Figure 7. In a manner
analogous to the previous part, take data to accurately compute the charge-to-mass ratio of the
electron and make a careful estimate of the uncertainty. Be sure to discuss why your technique is
optimal for reducing uncertainty. Although different groups may use different strategies to
improve accuracy and precision, note that the general approach of the experiment will likely be the
same across all groups.
The polarity of the Helmholtz coils and the electric field may need to be reversed for this section,
i.e. switch the red and black cables driving the current in the coils. The TA will demonstrate
during the pre-lab talk. Be care not to move the ground connection when switching the polarity of
the electric field.
Questions
0. What is the charge-to-mass ratio of the electron? Compare with the accepted value of
−1.758820088×1011 C/kg.
1. Which of the two techniques used in this lab gives more reliable results?
Appendix
The radius of curvature, R, can be found as follows:
Figure 8. (x,y) is the coordinate from the grid, (0,0) is the origin taken at the anode, and “Center”
is the center of the circle defined by the curve. R is the radius of that circle.
R=
x 2 + y2
2y
can be derived from the Pythagorean theorem, x2 + y2 = (2S)2. Since angles C and D
form a right angle and angles A and D must also add to 90 degrees, angle A must be equal to angle
C. Therefore, angle D must be equal to angle F. Since all three angles of both triangles are the
same, Triangle L, is similar to triangle M. Given this, we can setup the proportion:
S
R
y
= 2S
(10)
Which can then be rewritten as:
R=
(2S)2
y
(11)
Substituting (x2 + y2) for (2S)2 gives:
R=
x 2 + y2
2y
(6)