PHYS 351
Augustana College
Winter 2010-2011
Experimental Calculation of Plank’s Constant through Observation of the Photoelectric Effect
Derick Peterson, Augustana College, Rock Island, Illinois, USA
ABSTRACT
This paper presents a reproduction of Einstein’s classic experiment demonstrating the
photoelectric effect. The stopping potential V0 was measured for the 5 Hg spectral lines produced by an
Hg lamp fitted with a diffraction grating. A PASCO h/e apparatus was used to measure V0 and calculate
Vave for each frequency in the 1st order spectra, and then similar values for the 2nd order spectra. A plot
of frequency vs V0 was plotted and linearly fit using Origin data software, and the graph was used to
calculate values of h, φ, and fc.
INTRODUCTION
Einstein’s Nobel prize-winning research on the photoelectric effect demonstrated the quantized
nature of the photon and demonstrated that when light of sufficiently high frequency hits a metal
surface, electrons are released. These electrons that are released can only achieve a maximum kinetic
energy, KEmax, which is dependent on the frequency of light and the characteristics of the metallic
surface. By measuring the minimum voltage V0 required to prevent all photoelectrons from escaping the
metal (stopping voltage), one can determine KEmax, as given by equation 1 below.
KEmax = eV0
(1)
where e is the magnitude of the electron’s charge. Additionally, each photon has an energy E=hf where
h = Plank’s constant and f = frequency, so we can therefore write the equation
KEmax = hf - φ
(2)
PHYS 351
Augustana College
Winter 2010-2011
where φ is equal to the work function of the metal, and then we can write an expression for V0 as a
function of frequency.
eV0 = hf - φ
(3)
The plot of stopping potential vs frequency is a straight line with a slope of h/e, a y-intercept of -φ and
an x-intercept of fc, or the “cutoff frequency,” which is the frequency below which no electrons can be
emitted from the metal. In order to obtain values for h, φ, and fc, we will measure and plot the graph of
the stopping potential for various frequencies of light, as seen below in figure 1.
Figure 1. Graph of f vs. V0, showing the relationship to h, φ, and fc
To obtain the various frequencies for light will be using a mercury lamp fitted with a diffraction grating.
The mercury lamp produces the following spectral lines when fitted with a diffraction grating:
Table 1. Mercury Spectral Lines
PHYS 351
Augustana College
Winter 2010-2011
METHOD
To obtain the stopping potentials for light of various frequencies, a PASCO h/e apparatus was
connected to a voltmeter and placed directly in front of a Hg lamp fitted with a diffraction grating, as
shown in Figure 2. The h/e apparatus was placed upon a moving stand, such that the apparatus can be
rotated around the Hg lamp to the left or right of the centerline of the mercury source.
Figure 2. Experimental Setup
To measure the stopping potential for each of the mercury spectral lines, the h/e apparatus was
rotated about the light source until the appropriate spectral line was incident upon the aperture inside
the apparatus. Care was taken so that only one spectral line hit the aperture, and when observing the
green and yellow spectral lines, their respective filters were fitted over the h/e apparatus in order to
eliminate any unwanted wavelengths of light. The ZERO button on the h/e apparatus was pressed to
discharge any built up voltage, and the output voltage was read from the digital voltmeter after the
voltage stabilized. This procedure was done for each of the five first-order spectral lines, shown in Table
1, and then these measurements were all repeated 2 more times each. The average stopping voltage
Vave for each frequency f was calculated along with standard deviation of the mean, σm. The values of
frequency vs Vave with error bars of value σm were plotted and the graph was line-fitted using Origin data
analysis software. This was used to calculate h, φ, and fc.
The stopping voltage for the five second-order spectral lines were measured in a similar fashion,
and the six V0 values for each frequency were again used to calculate Vave and σm for each frequency.
PHYS 351
Augustana College
Winter 2010-2011
These second-order values were then used to create a new plot of frequency vs Vave with error bars of
value σm, using both the second order and the first order values for each frequency. This graph was
again line-fitted using Origin data analysis software and was used to calculate h, φ, and fc.
RESULTS
Figure 3a shows the linear relationship between the frequencies of incident light with the
corresponding stopping voltages Vave for the first-order Hg spectral lines. Figure 3b shows the same
relationship for the first and second order Hg spectral lines. The measurements taken by the voltmeter
read out to .01V, so we give these measurements an uncertainty of 0.005V. The error bars in the graphs
are the standard deviation in the mean σm. These data were fit with a linear-fit method using Origin data
analysis software in the form of equation 3, and this was used to find the values and error for the slope,
y-intercept, and x-intercept. These were then used to calculate h, φ, and fc using the relationships seen
in figure 1: h=slope*e, = y-intercept*e. These results are displayed in table 2. No sample calculations are
given since they only involved multiplying experimentally determined values by constants. The percent
diff. between our experimentally determined value of h and the standard value of h in each case is
displayed in table 3.
b)
a)
Figure 3. The stopping potentials of the five different wavelengths of Hg for a) the first order spectra
and b) the first and second order spectra.
PHYS 351
Augustana College
Winter 2010-2011
h (Js)
φ (CVs)
fc (Hz)
1st order
graph
6.703x10-34 ± 1.893x10-35 (2.82%)
2.222x10-19 ± 1.039x10-20 (4.68%)
3.315x1014 ± 2.487x1013 (7.50%)
1st and 2nd
order graph
6.426x10-34 ± 9.171x10-36 (1.43%)
2.070x10-19 ± 5.036x10-21 (2.43%)
3.221x1014 ± 1.243x1013 (3.86%)
Table 2. Values of h, φ, and fc calculated from the linear fit graphs
Percent difference to standard value of h
1st order data
(1.16%)
1st and 2nd order data
(3.02%)
Table 3. Percent difference of experimentally determined value to standard value of h
y-intercept (-φ/e)
1st order data
-1.38679 ± .06488
1st and 2nd order data with
false, large error bars
-1.38679 ± .03973
Table 4. Comparison of error in y-intercept values when adding additional, artificially inaccurate data
(with σm =4.0V, very large), as taken from Origin linear fit
DISCUSSION
The percent difference between our experimentally determined values of Plank’s constant h and
the standard value of h are 1.16% for the 1st order data and 3.02% for the 1st+2nd order data, compared
to the measurement error in h of 2.82% and 1.43%, respectively. Thus, our 1st order data fell within
measurement error of the standard, but our 1st+2nd order data did not. This is likely a result of the
inconsistency in the values of V0 for the 2nd order data and the relatively small number of data points
used in the linear fit. We could increase the accuracy in our data by observing additional frequencies
and doing additional trials to find Vave. Improving the quality of the dark room in which the experiment
took place could also help to decrease error by eliminating unwanted interference.
PHYS 351
Augustana College
Winter 2010-2011
The errors for the second-order values (σm) are slightly worse than those of the first order
values. This makes sense because on taking readings, the left and right sides of the second-order spectra
did not agree as well as did the left and right sides of the first-order values.
In regards to the accuracies for the first and first+second order values of h, φ, and fc; 2.82% vs
1.43%, 4.68% vs 2.43%, and 7.5% vs 3.86% respectively, the values using the first+second order values
are more accurate – specifically they appear to be roughly twice as accurate as the data obtained from
the first order values alone. This is to be expected, because in general more data should increase the
accuracy of your results. Following this line of reasoning, it is logical to conclude that in including extra
data in your study should always increase the accuracy of your results, even if the data is less accurate.
To this end, I compared the linear fit of the 1st order data to that of the 1st and 2nd order data, adjusting
the error bars (σm) in the 2nd order data to be absurdly large (±4.0V, when all the values of Vave range
over less than 2.0V). Even with addition of the now ‘inaccurate’ 2nd order data, the results obtained
were still more accurate than using the 1st order value alone. Specifically, the y-intercept changed from
-1.38679±.06488 to -1.38679±.03973. Even thought the 2nd order data was very inaccurate, its addition
increased the overall accuracy of the final result, supporting our claim. It is also interesting to note that,
since the linear fit done in origin was weighted by the error bars, the additional ‘inaccurate’ data did not
alter the calculated value for the y-intercept. Therefore, at least in this case not only will adding
additional inaccurate data improve overall accuracy, but it will have a negligible effect on any values
calculated with their addition.
CONCLUSION
We have successfully demonstrated the photoelectric effect and used the resulting stopping
potentials to calculate an experimental value for Plank’s constant h, which was compared against the
standard value.