The Photoelectric Effect
Mitchell Anliker, Steve Ash, Derick Peterson
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
The photoelectric effect was measured for both first and second order frequencies of light
in the visible spectra. The stopping voltage was measured for violet, blue, green, and
yellow spectral lines of light and plotted against their known frequencies. The work
function, Planck’s constant, and cutoff frequency were determined by plotting the
stopping voltage versus the known frequency. We found the values of Planck’s constant,
work function, and cutoff frequency to be 34.17E-15 ± 4.93E-17eV*s, 1.38 ±0.0303eV,
and 3.30E14 ± .05E14Hz respectively for the first order spectra and 3.22E-15±1.08E16eV*s, .702±0.08346eV, and 2.18E14±.05E14Hz for the second order spectra. The
stopping voltages from both sets of spectra were used to create a graph of all the data.
The values for this data set were found to be 3.92E-15 ± 3.28E-17eV*s, 1.23±0.0216eV,
and 3.13E14±.05E14Hz for Planck’s constant, work function, and cutoff frequency.
I. Introduction
Of the many contributions in the field of physics during the 20 th century, one of the most important was that
of Albert Einstein on the photoelectric effect in metals[1]. The photoelectric effect describes when a metal
emits electrons as a consequence of its absorption of energy from light with higher frequencies[2]. The
max amount of kinetic energy given off by the photoelectrons is dependent on the characteristics of the
metal and the frequency of light. In addition to this, every metal has what is known as a cutoff frequency
and a stopping voltage. The cutoff frequency is dependent on the characteristics of the metal being tested
and refers to a certain frequency of light after which no other lower frequencies will cause the metal to emit
photoelectrons[2]. The stopping voltage refers to a direct measure of energy required to cancel the kinetic
energy of the free electrons, or in other words, enough energy to cancel that provided by the incoming light
source[2]. In addition, another important feature to the photoelectric effect is the work function. The work
function is known as the minimum amount of energy an electron must gain for it to escape the metal it is in.
To explain this effect in metals, Einstein assumed light is quantized, composed of individual energy
particles known as photons[2]. While Einstein was not the first to delve into the photoelectric effect, his
research in the area played a large role in developing the field of quantum theory. In this experiment, the
photoelectric effect was simulated by measuring the various stopping voltages of light after passing it
through a diffraction grating. The stopping voltage was measured for the violet 1 &2, blue, yellow, and
green frequencies of light for both first and second orders of spectral lines. These stopping voltages were
then plotted against their corresponding spectral line frequencies. A best-fit line was used to determine the
slope and axis intercepts of the graph to measure the values of Planck’s constant, the cutoff frequency, and
the work function for each order of spectral lines. In addition to this, another graph was created using the
data from both the first and second order spectral stopping voltages for comparison purposes. The values
for Plank’s constant, work function, and cutoff frequency were found to be 34.17E-15 ± 4.93E-17eV*s,
1.38 ±0.0303eV, and 3.30E14 ± .05E14Hz respectively for the first order spectra and 3.22E-15±1.08E16eV*s, .702±0.08346eV, and 2.18E14±.05E14Hz for the second order spectra. The values for combined
spectra data set were found to be 3.92E-15 ± 3.28E-17eV*s, 1.23±0.0216eV, and 3.13E14±.05E14Hz for
Planck’s constant, work function, and cutoff frequency
Theory
The relationship between the max kinetic energy that can be absorbed by an electron and the stopping
voltage is shown by the equation:
KEmax  eV0
(1)
where KE is the max kinetic energy given off by the electron, e is the fundamental charge, and V 0 is the
stopping voltage. In addition, another relevant equation to this experiment is the following:
KEmax  hf  
(2)
relating the max kinetic energy to the difference between the sum of Planck’s constant and the frequency of
light with the work function of the metal. These two equations can be combined and cast into a linear
form:
eV0  hf  
(3)
where e is the fundamental charge, V0 is the stopping voltage, h is Planck’s constant, f is the frequency of
light, and φ is the work function. From this equation, the values for Planck’s constant and the work
function can be solved for. In addition, the cutoff frequency can be obtained by graphing this equation and
measuring where the line crosses the X-axis. Due to the Gaussian nature of the data, the deviation from the
mean stopping voltage was solved for using[1]:
d i  xi  X
(4)
Where xi is the original value measured and X is the mean calculated value. The variance was calculated
using the equation[1]:

1
di2

N 1
(5)
where N is the number of events and di is the deviation from the mean. In addition, the SDOM was
calculated using the following equation[1]:
x 
x
(6)
N
as a means of gauging the accuracy of obtained data.
II. Experimental Setup
h/e apparatus
Mercury Lamp
Diffraction Grating
Voltmeter
Filters
[http://electron6.phys.utk.edu/phys250/Laboratories/images/photoe2.jpg]
The experimental setup for this lab consisted of a mercury lamp, diffraction grating, voltmeter, filters, and
an h/e apparatus. The light source was turned on an allowed to warm up for several minutes before use. A
diffraction grating was then placed in front of the light source to split it into multiple orders of spectral
lines. These spectral lines can be seen in the image below:
[http://www.cfa.harvard.edu/ssp/images/emspec.jpg]
This image corresponds to one set of spectral lines produced by the diffraction grating. In addition, there
was a white spectral line used as the center mark to separate the two separate sets for each order of lines. A
voltmeter was hooked up to the h/e apparatus to measure the stopping voltage for each frequency of light.
For this lab the stopping voltage of the two violet lines, the first blue line, as well as the green and yellow
lines were measured. When the green and yellow lines were measured, filters were used to help reduce
error by removing unwanted wavelengths of light from the readings. The h/e apparatus was rotated around
the mercury lamp so that each line of light would fall on the photodiode inside it. The stopping voltage was
measured using the voltmeter and the setup would then be moved to the next set of light in the order being
measured. Each set of first order lines was measured three times, giving a total of six measurements, as
there are two sets of each order of spectral lines. The same process was repeated for the second order set of
spectral lines. In addition, during the entire duration of this experiment unnecessary light was kept to a
minimum as not to interfere with results due to the sensitive nature of the photodiode.
III. Results
Stopping Voltage Table
Color
Frequency(Hz)
First Order
Average V0(V)
Second Order Average
V0(V)
Violet 2
8.20E+14
1.93
1.93
Violet 1
7.41E+14
1.69
1.68
Blue
6.88E+14
1.51
1.52
Green
5.49E+14
0.910
0.813
Yellow
5.19E+14
0.795
0.728
Table 1: Stopping voltage values table
Variance Table
Color
First Order
Variance
Second order
Variance
First Order SDOM
Second Order SDOM
Violet 2
0.0318
0.00897
0.0129
0.00366
Violet 1
0.0125
0.0138
0.00513
0.00565
0.00745
0.0120
0.00304
0.00490
0
0.127
0
0.0521
0.724
0.654
0.295
0.267
Blue
Green
Yellow
Table 2: Variance table
Figure 1: First order graph of stopping voltage vs. Frequency
First Order Value Table
Planck’s Constant(eV*s)
Work Function(eV)
Cutoff Frequency(Hz)
Value
4.17E-15
1.38
3.30E14
Error
±4.93E-17
±0.0303
±.05E14
Table 3: Values obtained from first order graph.
Figure 2: Second order spectral graph of stopping voltage vs. frequency.
Second Order Values Table
Planck’s Constant(eV*s)
Work Function(eV)
Cutoff Frequency(Hz)
Value
3.22E-15
.702
2.18E14
Error
±1.08E-16
±0.08346
±.05E14
Table 4: Values obtained from second order graph
Figure 3: Combined first and second order stopping voltages plotted against frequency
Combined Spectra Table
Planck’s Constant(eV*s)
Work Function(eV)
Cutoff Frequency(Hz)
Value
3.92E-15
1.23
3.13E14
Error
±3.28E-17
±0.0216
±.05E14
Table 5: Data obtained from combined spectra graph
Figure 1 shows the graph created by plotting the average stopping voltage of the first order spectral lines
versus the known frequencies of light measured. The values for Planck’s constant, work function, and
cutoff frequency of the metal were obtained from the graph by finding the slope as well as the two axis
intercepts. These values were found to be 4.17E-15 ± 4.93E-17eV*s, 1.38 ±0.0303eV, and 3.30E14 ±
.05E14Hz for Plank’s constant, the work function, and cutoff frequency respectively. A graph of the
second order spectral lines was created seen in figure 2. This was done for both a visual comparison with
the first order graph, as well as to obtain a set of values to compare with the first order data before
combining the data. The values for Planck’s constant, the work function, and cutoff frequency were found
to be 3.22E-15±1.08E-16eV*s, .702±0.08346eV, and 2.18E14±.05E14Hz respectively for the second order
data. Figure 3 shows the combined data of the average stopping voltages from both the first and second
order spectral lines plotted against the frequencies of light observed. Values for Plank’s constant, work
function, and cutoff frequency were pull from the graph and found to be 3.92E-15 ± 3.28E-17eV*s,
1.23±0.0216eV, and 3.13E14±.05E14Hz respectively.
In addition, the SDOM as well as the variance were calculated for every color of each spectra
using eqs (5) and (6). The variance for the first order spectra were found to be 0.0318, 0.0125, 0.00745, 0,
and 0.724 for violet 2, violet 1, blue, green, and yellow respectively. The SDOM was also calculated for
the first order data and found to be 0.0129, 0.00513, 0.00304, 0, and 0.295 for violet 2, violet 1, blue,
green, and yellow respectively. The variance for the second order spectra was found to be 0.00897, 0.0138,
0.0120, 0.127, and 0.654 for violet 2, violet 1, blue, green, and yellow respectively. The SDOM was
calculated for the second order data and found to be 0.00366, 0.00565, 0.00490, 0.0521, and 0.267 for
violet 2, violet 1, blue, green, and yellow respectively.
IV. Discussion
Using eq (3), the stopping voltage, cutoff frequency, and Planck’s constant were cast into linear form as
seen in figures 1-3. The linear nature of figures 1-3 also indicates the data is reputable. The calculated
value for Planck’s value obtained from the first order spectral data, 4.17E-15±4.93E-17eV*s, was
compared with the known value for Planck’s constant, 4.135E-15eV*s. This known value for Planck’s
constant fell well within the error for the calculated value, showing that the data taken was accurate. A
graph of the second order data was created to visually compare it to the first order data before combining
the two data sets seen in figure 2. After combining the two data sets, figure 3, it was observed that the
overall accuracy of the graph actually dropped. The calculated value for Planck’s constant became 3.22E15±1.08E-16eV*s and does not encompass the known value for Planck’s constant within its error. In
addition, the values for work function and cutoff frequency were off from where they were when only the
first order data was used. This is not unusual because to increase the accuracy of a data set by taking many
measurements, one has to take around at least another 100 measurements for it to be somewhat effective.
For instance look, at the SDOM equation seen in eq (6). It would take a large increase in the number of
total measurements, N, to actually put a dent in the calculated SDOM that is a measure of how accurate the
overall data is. This is why when the second order data is plotted with the first order data it actually makes
the results less accurate than before. The SDOM as well as the variance were calculated as a means of
showing how reputable the data taken was. The error bars for the green and yellow frequencies of light
seen in all figures 1-3 are fairly large. This is most likely due to an error in calculation. In addition, when
the variance was found to be lower than .005 for an error bar it was replaced with ±.005 because that is the
minimum error in measurement from reading the voltmeter. Overall, the values for the SDOM and
variance were fairly low excluding the few outliers seen in the yellow and green frequencies. Based on the
linear nature of figures 1-3, the almost identical values between the known and calculated Planck’s
constants, the low variance for almost all calculated frequencies, and the low error in the SDOM for the
major part of the data, it can be concluded that the data taken is a reputable example of the photoelectric
effect.
References
[1] J. R. Taylor, An Introduction to Error Analysis: The Study of uncertainties in Physical Measurements.
Sausalito, CA: University Science Books. 1982, pp. 93-161.
[2] " The Photoelectric Effect." The Columbia Encyclopedia, Sixth Edition. 2008. Encyclopedia.com. Dec.
2010 <http://www.encyclopedia.com>.