Measurement of Charge to Mass Ratio of an Electron Using a
Filament Beam Tube
H. Potter, and E. Kager
(Completed 10 October 2005)
The ratio e/me was determined to be 1.53x1011C/kg, resulting in a -12.87%
discrepancy with the currently accepted value1 of 1.75882x1011C/kg, by using a
Filament Beam Tube.
I. Introduction
In the late 1800’s significant advances in human understanding of atomic
structure were being made by devising clever experiments that made use of a developing
electromagnetic theory. One such experiment, devised by J. J. Thompson, used an
electric field to deflect a charged beam of particles, known as a cathode ray, so that the
charge to mass ratio of the beam could be determined. The cathode rays that he was
using in his experiments, however, were later determined to be beams of electrons, and so
his measurements were actually the first measurements of the charge to mass ratio of an
electron.
II. Experiment
By using a filament beam tube, a variation on J. J. Thompson’s experiment was
conducted in order to determine a value for e/me experimentally. The device created a
beam of electrons by applying a voltage to a heater that was contained in a vacuum and
accelerating the freed electrons with another voltage. This apparatus was placed between
two rings of coiled wire such that the beam of electrons shot out parallel to the rings of
coiled wire. An adjustable current was then applied to the coils of wire in order to create
a magnetic field that would be directed perpendicularly to the beam of electrons, causing
the beam to bend in a circular path. Markers were present inside the apparatus that
created the electron beam, so that the current was simply adjusted until the beam crossed
one of these markers. This gave the radius of curvature. Through the use of multimeters
current and voltage readings were readily and reliably available. The current was then
adjusted until the beam had been made to cross at each possible marker for the specific
voltage. The voltage across the heater was then changed and the current was again
adjusted so that the new beam crossed each possible marker. The voltage, radius of
curvature, and current were recorded at each stage.
By assuming that the work done on the electrons by the voltage applied was
completely converted into the kinetic energy of the electrons, the equation
mev2/2 = eΔV
(1)
was taken to apply to the experiment. By assuming that the magnetic field perpendicular
to the plane of motion of the electron beam supplied the centripetal force necessary to
make the beam curve, the equation
evB = mev2/r
(2)
was also taken to apply to the experiment. By combining Equation (1) and Equation (2),
e/me can be solved for. The result is
e/me = (2ΔV)/(r2B2)
(3).
Thus, all that is needed in order to calculate e/me is to convert each current measured into
a corresponding magnetic field. This was done by removing the electron beam apparatus
and placing a magnetic field sensor between the rings of wire to measure the magnetic
field as the current was changed. This data was then plotted to yield a near-perfect linear
fit that converted current into magnetic field. This linear fit was then used to convert
each measured current into a magnetic field reading. Each data set was then used to
calculate a value for e/me yielding a total of 17 values, which were then averaged to
determine an experimental value for e/me.
III. Results
Seventeen sets of data were recorded and Equation (3) was used to determine e/me
for each data set. These values were then averaged to determine the experimental value
for e/me. The data taken in order to determine the linear fit between current and magnetic
field are shown in Table 1, as are the important values for the linear fit. The linear fit
itself is shown in Figure 1. The data, calculations, and accepted values1 for e, me, and
e/me are all given in Table 2
Table 1: Data for
linear fit.
Current to Magnetic Field
0.0E+00
-5.0E-04 0
Magnetic Field (T)
Linear Fit Data:
I (A)
B (T)
4.280
-3.63E-03
3.880
-3.30E-03
3.774
-3.18E-03
3.638
-3.08E-03
3.446
-2.92E-03
3.181
-2.71E-03
2.866
-2.45E-03
2.513
-2.18E-03
2.359
-2.06E-03
2.087
-1.85E-03
1.836
-1.65E-03
1.688
-1.52E-03
1.317
-1.23E-03
0.921
-9.20E-04
0.547
-6.20E-04
r=
-0.99990
r2 =
0.99979
m=
-8.00234E-04
b=
-1.74975E-04
1
2
3
4
5
-1.0E-03
-1.5E-03
-2.0E-03
-2.5E-03
-3.0E-03
-3.5E-03
y = mx+b
R2 = 0.9998
-4.0E-03
Current (A)
Figure 1: Linear fit of magnetic field to current.
2
Calculations:
r (m)
I (A)
0.03
1.819
0.04
1.819
0.03
1.819
0.02
1.819
0.01
3.930
0.02
3.930
0.05
1.602
0.04
1.602
0.03
1.602
0.02
1.602
0.02
2.314
0.03
2.314
0.02
3.303
0.04
2.016
0.03
2.016
0.02
2.016
0.05
1.446
Accepted Values:
e (C) =
1.602176E-19
me (kg) =
9.109382E-31
e/me (C/kg) = 1.758820E+11
B (T)
-1.63E-03
-1.63E-03
-1.63E-03
-1.63E-03
-3.32E-03
-3.32E-03
-1.46E-03
-1.46E-03
-1.46E-03
-1.46E-03
-2.03E-03
-2.03E-03
-2.82E-03
-1.79E-03
-1.79E-03
-1.79E-03
-1.33E-03
V (V)
187.7
275.3
181.3
100.1
102.0
342.4
324.9
223.3
155.2
91.9
123.0
258.5
223.2
331.0
201.4
115.9
277.9
Mean:
Accepted Value:
Percent Discrepancy:
e / me
1.57E+11
1.29E+11
1.52E+11
1.88E+11
1.85E+11
1.55E+11
1.22E+11
1.31E+11
1.62E+11
2.16E+11
1.50E+11
1.40E+11
1.41E+11
1.29E+11
1.40E+11
1.81E+11
1.25E+11
1.53E+11
1.76E+11
-12.87%
Table 2: Data for each observation, along with brief data summary and list of accepted
values1.
IV. Analysis and Discussion
The experimentally determined value for e/me was 1.53x1011 C/kg, the mean of
all values calculated from observations. This value gave a -12.87% percentage
discrepancy with the presently accepted value of 1.76x1011 C/kg.
V. Conclusion
A -12.87% percentage discrepancy is unusually high for a classroom experiment,
and suggests that certain factors were not taken into account in the experimental
procedure. This was indeed the case. The orientation of the Earth’s magnetic field was
not accounted for in the experimental procedure, and this magnetic field may have
deflected the electron beam, particularly for the measurements involving larger radii. In
fact, if only the data sets in which the radius is .03m or less are used, the mean e/me
comes out to about 1.64x1011 C/kg, and if only those data sets in which the radius is .02m
or less are used the mean is about 1.74x1011 C/kg, remarkably close to the accepted
value. A process for taking into account the orientation of the Earth’s magnetic field
would be to arrange the rings so that they are perpendicular to the horizontal component
of the Earth’s magnetic field, as indicated by a dip circle, and then to tilt the whole
apparatus up at an angle, as read off of a dip circle, so that the Earth’s magnetic field is
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coming down straight through the rings of wire, completely perpendicular to the electron
beam’s path. Unfortunately, when this was attempted, it was discovered that the method
of attaching the electron beam apparatus to the titled board to which the wire rings were
attached led to a leak in the vacuum. This made it impossible to create an electron beam
with the apparatus, and the attempt to improve the procedure was abandoned. If the
somewhat cavalier exclusion of data mentioned above is any indication, this procedure
should significantly decrease the percentage discrepancy of the calculated value for e/me.
1
Tipler, Paul A. and Ralph A. Llewellyn, Modern Physics, 3rd ed. (W. H. Freeman and
Company, New York 2003).
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