Electron Charge to Mass Ratio
Mark Hillstrom
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
The ratio between an electron and its mass was measured using a Pasco apparatus. The
resulting values for this ratio were found to 1.66e11 C/kg and 1.80e11 C/kg.
Experiments of this type are used throughout science to study wave-particle duality.
When an electron is charged through an electric potential, the resulting velocity can be
determined. A magnetic field can then be added which deflects the electrons in a circular
path. The radius of this path can be used to calculate the ratio between an electrons
charge and mass.
I. Introduction
When a beam of electrons are sent through a known voltage, their acceleration can be determined. Two
Helmholtz coils of variable voltage and current produce a uniform and measurable magnetic field at right
angles to the electron beam. This causes the electron beam to form a circular path. The three variables of
interest are therefore the electrode voltage (V), the current of the Helmholtz coils (I), and the radius of the
circular electron beam (r). These three variables can be used to calculate the charge to mass ratio (e/m) of
an electron
(1)
The magnetic field (B) is that of the pair of Helmholtz coils and represented as
(2)
Where N is the number of turns in each Helmholtz coil, I is the current through the coils, a is the radius of
the coils, and uo is the magnetic permittivity of free space (1.26e-6 Tm/A). When J.J. Thomson performed
this experiment in 1897 he was the first to demonstrate and electrons existed and ha particle-like properties.
II. Experimental Setup
To begin, we made sure that all the connections were in the right place and everything was in working
order. Initial values for heater, Electrode Voltage, Helmholtz Coil Voltage, and Helmholtz Coil Current
were set within their respective ranges according to the recommended settings. It is important not to
exceed any of these settings because higher voltages and currents will burn out the filament and destroy the
e/m tube. A tripod was then placed against the edge of the table at a height level with the center of the
electron beam circle. Two very important measurements were then taken which included the distance from
the observer’s eyeball to the ruler (D) and the distance from the observer’s eyeball to the electron beam
circle (D2). These two values will later be used to calculate the radius of the electron beam circle and
eventually the ration of e/m. We took 6 different trials of varying electrode voltage, coil amperage, and
coil voltage. We recorded the changes of each variable for every trial as well as left and right
measurements of the radius of the circular electron beam. The two radii were averaged to provide one
radius value. Appropriate error values were placed on each measurement including all current and voltage
changes. Figure 1 represents the exact apparatus used without the cover which protected against any stray
light. The cover was essential in reducing glare that would have made the ruler harder to read.
Figure 1. e/m Apparatus obtained from the University of Missouri
III. Results
Figure 2 displays a mathematical diagram of the apparatus with measured D and D2 values as well as the
calculated R value taken as the average of both left and right radii measurements. These three variables
were then used to calculate a new radius value, r, which had to be modified to account for parallax. Figure
3 shows a graph of the ratio between electron mass and charge. This was done in Origin using the 2V
values from all six trials and plotting them against their relative B 2r2 values. A linear fit along with error
bars were also included.
Figure 2. Mathematical Version of Apparatus
Figure 3. Experimental e/m ratio
The slope obtained from Figure 3 was 1.66e11 C/kg. This experimental value is relatively close to the
accepted value of 1.76e11 C/kg and holds a relative percent error of 5.68%.
When measuring error we typically found that .1cm was sufficient on the smaller measurements. The
measurement for D was much longer and harder to obtain which is why its uncertainty is 1cm. The current
value was the only possible error associated with the calculation of the magnetic field (B) and was given an
uncertainty of .01 A. Error bars were given to the B 2r2 value because its fractional error was greater than
that of 2V. This was done by first calculating the uncertainties of r and B. The final uncertainty that was
used for the error bars was calculated the following equation:
∆B2r2 = [(2rB2∆r)2 + (2Br2∆B)2](1/2) = 2.6894e-10 C/kg
(3)
The error for all r measurements was very close which is why an average was used in calculating the error
bars. Table 1 below represents values of the most importance in each of the 6 trials as well as an average.
Table 1. Comparison of R, r, B, (e/m), and % error from 6 trials along with an average of each
IV. Discussion
It is seen that the average (e/m) value of the 6 trials is 1.80e11. This is much closer to the theoretical value
than that obtained from our slope. The percent error of 2.41 is a little less than half that of that calculated
above. This is explained by noticing that the when lower voltages were used in trials 4, 5, and 6 the (e/m)
value was larger. These low voltages would be higher on the y-axis which would shift the slope in a
clockwise direction and cause it to decrease. The difference between the average (e/m) value and the slope
(e/m) value is just 1.4e10. Another probable explanation for this discrepancy is the existence of systematic
error. The two sets of trials were performed a week apart. The apparatus could easily have been moved.
The error percentages are quite similar for the two trial sets which further indicate that systematic error was
involved.
Conclusion
The ratio between an electrons charge and its mass was calculated to be 1.66e11 and 1.80e11. Systematic
error was likely responsible for the discrepancy between these two values.
References
"Department of Physics & Astronomy: Lab Connection: E/m Apparatus." University of
Missouri - St. Louis Home. Web. 17 Feb. 2011. <http://www.umsl.edu/~physics/lab/de
mo/em-apparatus.html>.