Measurement of the Charge of an Electron
John Cole, Sarmadi Almecki, Pirouz Shamszad
Abstract
A measurement of the fundamental charge of an electron was carried out by placing a
variable current across a transistor. The base voltage of the transistor and the collector
voltage were measured for varying currents. These voltages were plotted on a
logarithmic scale and the slope was measured. Furthermore, the transistor was placed
inside a temperature-controlling device, altering the collector voltage while the base
voltage was held constant. This data was plotted and the charge was derived from the
slope. From the first part of the experiment the fundamental electron charge was
measured to be 1.2648 e-19 +/- 4.18e-21 C and from the second part 1.7703 e-19 +/6.842 e-21 C.
Introduction and Theory
The fundamental charge of an electron is the smallest unit of charge known in nature. It was first measured
by Townsend in the late 1890’s to within 37% of the accepted value [1]. Between 1909 and 1913 Robert
Millikan succesfully measured the fundamental charge of an electron with his famous Oil-Drop Experiment
[2]. Millikan set up two metal plates with a large electric field created between them. Oil droplets were
atomized and dropped in between the plates. Millikan was able to suspend the droplets by counteracting
the force of gravity (downward) with the force of the electric field (qE upward). Knowing E, he was able
to derive the charge of the electron, q.
Our experiment relied upon a transistor. According to the theory of the junction transistor, the collector
current on a transistor is an exponential function of base voltage [2] and is described by
Ic = A [ eqVb/kT –1]
(eqn 1)
Where Ic is the collector current, q is the charge, Vb is the base potential, k is Boltzmann’s constant, and T
is the temperature in Kelvins. The current of the collector can be equated to the potential through Ohm’s
law
V = IR
(eqn 2)
Approximating for the specific voltage (> 1/40 V), we simplify the equation to
VRC = B eqVb/kT
(eqn 3)
Measuring VRC and plotting it logrithmically against Vb, we found the slope of the line. Equating the slope
to
q/kt
(eqn 4)
the equation was solved for q.
Experimental Setup
A circuit was constructed using an NPN transistor 2N3904, a 2 10 Turn Potentiometer, and a
voltage source. The two potentials were monitored with individual voltmeters.
(see Appendix 1 for Schematic Diagrams)
The experiment was repeated using a heat pump (see Appendix 1) in order to alter the temperature. The
device allowed the transistor to be cooled and heated while keeping Vb constant.
Procedure and Data
The circuit was constructed as shown above and wired to a 24-volt power source. Turning the
potentiometer varied the potential of the emitter and voltages were measured at methodically spaced
intervals. Systematic error arose from the voltmeters: the voltmeter measuring the base potential was
accurate only to 1/100 of a volt and the voltmeter on the collector was accurate up to 1/1000 of a volt.
Vb (volts)
Vrc (volts)
0.5023
0.5075
0.5257
0.5438
0.5548
0.5623
0.5684
0.5865
0.5967
0.6041
0.6099
0.6145
0.6186
0.6222
0.625
0.6287
0.634
0.6351
0.6371
0.084
0.1
0.2
0.4
0.6
0.8
1
2
3
4
5
6
7
8.02
9
10.25
12.5
13.01
14
The resulting data was tabulated and graphed with gnuplot on a logarithmic scale:
Vrc vs. Vb
54.598
20.0855
'lab3_electron.dat'
e/kT = 31.1452 +/- 1.03
7.38905
2.71828
1
0.36788
0.135335
0.0497872
0.5
Figure 1.
0.52
0.54 0.56 0.58
0.6
0.62
Vb -- Base to Emitter Voltage
0.64
0.66
Vrc vs. Vb
25
20
'lab3_electron.dat'
f(x)
e/kT = 31.1452 +/- 1.03
15
10
5
0
0.5
0.52
0.54
0.56
0.58
0.6
0.62
Vb -- Base to Emitter Voltage
0.64
0.66
Figure 2.
The temperature was 294.15 K, measured with a mercury thermometer sitting in the room.
After the first part of the experiment, wire was added to each of its terminals of the transistor and placed
into the controlled-temperature device. While the variable resistor was held constant, the temperature was
systematically changed from high to low. Temperatures were measured using a mercury thermometer.
Vrc
(volts)
18.86
17
15.15
13
9.38
4.39
4.18
2.28
1.443
0.8
0.7
0.39
0.18
0.155
Vrc Error
(volts)
+/-0.001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
+/- .001
Vb (volts) T (C)
0.48
0.5
0.5
0.5
0.51
0.51
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
69
65
65
62
56
44
43
35
28
22
20.5
14
4
2
T(K)
342.15
338.15
338.15
335.15
329.15
317.15
316.15
308.15
301.15
295.15
293.65
287.15
277.15
275.15
1/T (K)
0.002923
0.002957
0.002957
0.002984
0.003038
0.003153
0.003163
0.003245
0.003321
0.003388
0.003405
0.003483
0.003608
0.003634
Vrc vs. 1/T
20
'lab3_electron_temp.dat' using 5:1
f(x)
18
16
14
12
qV(b)/k = -6411.11 +/- 247.8
10
8
6
4
2
0
0.0029
0.003
Figure 3.
0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037
1/T (K^-1)
Using a logarithmic scale on the x-axis produces a straight line:
Vrc vs. 1/T
70
'lab3_electron_temp.dat' using 5:3
60
50
40
30
20
10
0
0.0029
0.003
Figure 4.
0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037
1/T (K^-1)
Interpretation and Discussion
From the data gathered, we plotted Base to emitter voltage against the voltage across the collector resistor.
The best-fit line was formulated using the method of least squares on gnuplot. Our resulting slope was
31.1452 +/-1.03. The equation
slope = q/kt
was used to derive the fundamental charge of an electron. Our data using variable potentials showed the
charge of an electron is 1.2648 e-19 C with a standard deviation of +/- 4.18e-21 C.
The data from the variable temperature experiment was treated similarly and gave a slope of 6411.11 +/247.8. The equation
slope = qVb/k
was used to derive the charge. Our data using variable temperatures showed the charge of an
electron is 1.7703 e-19 C with a standard deviation of +/- 6.842 e-21 C.
Our ability to accurately measure the fundamental charge was affected by the precision of our equipment.
Because the voltmeters could read only to a certain degree of accuracy, a more precise calculation was
unattainable. Furthermore, because we took only one set of data, we were able to do limited statistical
analysis. Furthermore, the thermometer used to determine the temperature of the room and the transistor
was precise to only a single degree Celsius. Most of the error in our experiment came from our lack of
independent data and the limitations of our equipment.
Conclusion
In summary, the charge first accurately measured by Millikan was measured by comparing the potentials
across a transistor. We measured the average of the two charges to be 1.3175 e-19 +/- 5.511e-21 C. The
sources of systematic error were significant, mostly resulting from the limitations of the instruments. Our
data was 18% within the accepted value of the charge of an electron.
Improvements on the experiment should focus on more precise equipment. Independent sets of data should
be taken to statistically improve the final value. Finally, students performing the experiment should take
special care to employ neatness and orderliness in their circuit construction to save time in the experimental
setup.
[1] Serway, R.A, Physics for Scientists and Engineers(Edition Four) 1982. Saunders College Publishing,
1982.
[2] Beer, Dr. L Neel. Keithley Instruments, Inc. 1983.
Appendix 1
Schematic Diagram of the Circuit Constructed