Determining the Charge to Mass Ratio of an Electron
Mitchell Anliker, Steven Ash, Derick Peterson
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
An e/m apparatus was used to determine the charge of an electron by measuring variation
in radius of an electron beam based on changes in magnetic field. The electron beam was
directed in a circular path inside the e/m tube, allowing its charge to be calculated in
terms of the measureable quantities of radius of the electron beam, current, and voltage.
These parameters were used to created a graph in which the electron charge to mass ratio
was the slope. The experimental value for the charge to mass ratio of an electron was
found to be 9.8205E10±1.4924E10C/kg with a 44.02% difference between it and the true
value of 1.76E11C/kg. This large error is due to a parallax between the measured radius
of the electron beam in the mirror and its actual placement.
I. Introduction
The mass to charge ratio of an electron was first measured in 1987 by Joseph J. Thompson through the use
of cathode rays[2]. He theorized that light rays was created by millions negatively charged particles of
light which he named “corpuscles[2].” In the near future, scientists opted to call these particles electrons as
suggested in 1984 by George Stoney[2]. Thompson later discovered that he could bend the cathode rays
through the use of magnetic fields[2], allowing him to accurately calculate the mass to charge ratio of an
electron. This lab made use of a setup similar to the one Thompson used, as an electron beam was
deflected in a circular path inside an e/m vacuum tube. The radius of the circular path created by the
electron beams deflection could then be measured, along with the current and voltage flowing through the
e/m apparatus so the mass to charge ratio could be determined from a plot of voltage versus radius and
magnetic field. The axis of the graph had to be switch in order to incorporate the correct fractional error in
measurement. The calculated charge to mass ratio of an electron was found to be 9.8205E10 ± 1.4924E10
C/kg with a 44.02% difference between it and the true value of 1.76E11C/kg. However, when error created
by parallax was taken into account, it was found that the theoretical value for the charge to mass ratio of an
electron fell well within the error bars for the calculated ratio.
Theory
The magnetic force acting on a charged particle can be represented by the equation[1]:
Fm  evB
(1
Where Fm is equal to the magnetic force acting on a charged particle, is the charge of an electron, v is the
velocity the particle, and B is the magnetic field. This equation is written in scalar form because the
electron beam is perpendicular to the magnetic field. Because the electrons are moving in a circle they
experience a centripetal force expressed as[1]:
Fc 
mv 2
r
(2
where Fc stands for the centripetal force, v is the velocity, and r stands for the radius of the path taken.
Equations 1 and 2 can be equated to obtain[1]:
e
v

m Br
(3
because the only force acting on the electrons is the one generated by the magnetic field. In addition, the
electrons are accelerated through a potential V which allows equation 3 to be related to a simple equation
for kinetic energy, eV=1/2mv2 to obtain[1]:
(e / m) B 2 r 2  2V
(4
where e/m is the mass to charge ratio of an electron, B is the magnetic field, r is the radius, and V is the
voltage. This equation can be used to create a plot to find the mass to charge ratio only when a value for
the magnetic field is known. This can be calculated using the equation:
3/ 2
 N 0 I 
B
 5 / 4
 a 
(5
Where B is the magnetic field, N is the number of turns in each Helmholtz coil, I is the current, and μ 0 is a
constant for the magnetivity of free space.
II. Experimental
The apparatus for this experiment can be seen in the figure below:
[Figure 1: basic setup of the e/m apparatus]
As seen in the figure above, two power supplies, a multimeter, and several pairs of cords were needed in
addition to the large Helmholtz e/m apparatus. Initially, the distance between the helmholts coils was
measured using a meterstick and the number of coils was obtained from a manual of its production. The
distance between the coils was determined by finding the radius of a single coil because the distance
between the coil’s and their radius are approximately the same. Two power supplies were necessary to
make this setup work. One of the two power supplies seen below,
[Figure 2: Power supply to the Helmholtz coils]
Was run through a multimeter plugged into the input for the Helmholtz coils. This was done because the
voltage reading on the power supply was not accurate enough for the means of this experiment because an
approximate voltage of 8.00V needed to be kept at all times. The current running through the coils was
also measured on the power supply and kept below 2.0A at all times. The other power supply seen below:
[Figure 3: power supply to the electron gun and heater]
This power supply was used to supply power to the cathode heater, kept at a constant 6.3V, and to the
electron beam. The voltage running through the electron gun could be adjusted via the knob on the far
right of the power supply seen in figure 3 and was measured by a scale right on the face of the power
supply. The lights were turned off and the hood was placed over the e/m apparatus to further reduce the
amount of light, making measurements easier to read. The two power supplies were turned on and readings
were taken in two sets varying values of current, radius, and voltage. For the first set of data, the current
was kept at a constant 1.63A while the voltage was varied from 180-300 volts in a series of five different
readings. For these readings, the radius of the circular electron beam was measured via a ruler on the back
of the e/m apparatus seen below:
[Figure 4: e/m apparatus turned on with ruler used for measurement visible behind it.]
The center of the curved electron beam was lined up with the 0 point on the ruler, and the radius was
determined by reading where the curved path of the beam crossed over the ruler. In addition all
measurements were taken from approximately the same distance away to provide consistent results. For
the second set of measurements, the voltage was set at 300V and the current was changed via a knob on the
e/m apparatus. Four readings with varying current values were taken along with their respective radius
measurements for the electron beam in each case. This data was then taken and used to create a plot so that
the mass to charge ratio of an electron(e/m) was the slope. The magnetic field value, B was calculated
using equation 5 using the recorded values for I and a. This value was then plugged back through equation
4, so a linear plot could be created using the observed values. The e/m value was then extrapolated from
this plot using a best fit line. An inverse of this graph was created so that fractional error could be
incorporated into the fit as error bars.
III. Results
Values Table
Volts(V) Radius(m) Current(A) Magnetic Field B(T) B^2r^2(T2*m2) 2*Voltage(V) Error(T2*m2)
180
0.042
1.64
0.001281
2.9E-09
360
4.41E-09
200
0.044
1.64
0.001281
3.18E-09
400
4.62E-09
240
0.051
1.64
0.001281
4.27E-09
480
5.36E-09
280
0.057
1.64
0.001281
5.34E-09
560
5.99E-09
300
0.058
1.64
0.001281
5.52E-09
600
6.1E-09
300
0.055
1.6
0.00125
4.73E-09
600
5.5E-09
300
0.05
1.78
0.001391
4.84E-09
600
6.19E-09
300
0.053
1.69
0.001321
4.9E-09
600
5.91E-09
300
0.051
1.75
0.001367
4.86E-09
600
6.1E-09
[Table 1: values from all trials and propagated error used to create error bars]
600
2*V(Volts)
550
500
450
400
350
2.50E-009 3.00E-009 3.50E-009 4.00E-009 4.50E-009 5.00E-009 5.50E-009
B^2*r^2(T^2*m^2)
[Figure 5: Initial graph of 2V/B2r2 to obtain e/m slope]
0.0010
B^2*r^2(T^2*m^2)
0.0005
0.0000
-0.0005
-0.0010
300
350
400
450
500
550
600
650
2*V(Volts)
[Figure 6: Inverse graph of 2V/B2r2 with fractional error bars added]
1.20E-008
1.00E-008
B^2*r^2(m^2*T^2)
8.00E-009
6.00E-009
4.00E-009
2.00E-009
0.00E+000
-2.00E-009
300
350
400
450
500
550
600
650
2*V(volts)
[Figure 7: Inverse graph with parallax error bars and plot of known e/m ratio.]
Table 1 shows all obtained values from both sections of the lab along with calculated values and error used
in figure 7. Figure 5 is the first graph created so that the e/m ratio is the slope. 2V was plotted on the Y
axis and B2r2 was on the x axis. A slope of 9.8205E10±1.4924E10C/kg was obtained for this graph using a
best fit line, seen in red. Figure 6 shows and inversion of the two axis of figure 5 so that fractional error in
measurement, found to be .001m, could be accounted for in the fit. Figure 7 is another inverted graph with
2Von the x axis and B2r2 on the y axis. This graph has the same best fit line as seen in figures 5 and 6 seen
in red, except the error propagation was changed to account for a parallax effect on the error in
measurement. The measurement error due to the parallax effect was found to be .032m. An inverse plot of
the theoretical value, 1/1.76E11C/kg, was plotted seen in black as a reference. The percent difference
between the calculated charge ratio, 9.8205E10±1.4924E10C, and the theoretical ratio, 1.76E11C/kg, was
found to be 44.02%.
IV. Discussion
Using the obtained values for I, V, and r, several plots were created so that the charge ratio could be
extrapolated from them as the slope. Figure 5 shows a basic plot used simply to obtain the calculated ve/m
value of an electron. This was done using a best fit line of the plotted data. The axis of figure 5 were then
inverted to incorporate error bars based on the fractional error found in the measured values of voltage and
radius. The fractional error was calculated for voltage and radius and found to be .0111V and .11m for
voltage and radius respectively. The axis of the graph in figure 5 were then inverted to account for the
larger fractional error in radius, which was then propagated again to obtain an error of .001m for each
point. Because the percent difference between the calculated and theoretical values was so large, 44.02%, a
third graph was made with new error bars accounting for the error in measurement created by parallax.
This value was found by measuring the distance between the real and mirror image of the electron beam
and was found to be .032m. This value as taken into account along with the fractional error and a new set of
error bars was calculated, seen figure 7. The values for the error bars at each of the respective point in
figure 7 can be seen in table 1 in the last column. A plot of the inverse theoretical value of the charge ratio
was done seen as the line in black for proof that when the parallax error is taken into account, the accepted
value for the electron e/m ratio falls well within the error bars. Overall, it can be concluded that the e/m
ratio was successfully calculated even though there was a great degree of difference between it and the true
e/m value of an electron. If this lab were to be repeated, error due to parallax would be accounted for in all
measurements by lining the two images created by the electron beam up in the mirror ruler. The calculated
values were proven to be valid by taking the large error created by the parallax effect into account as seen
by the black line falling well within the designated error bars in figure 7. In conclusion, the results
obtained are fairly accurate when adjusted for the appropriate error and follow the established results for
what the value the e/m ratio of an electron.
References
[1] Instrument manual and Experiment Guide for the PASCO scientific Model SE-9638. CA: Pasco
Scientific, pp 1-6.
[2] J.J. Thomson - Biography". Nobelprize.org. 18 Feb 2011 http://nobelprize.org/nobel_prizes/physics.
.