Experimental Calculation of Plank S Constant Through Observation of the Photoelectric Effect

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

st nd Vave for each frequency in the 1 order spectra, and then similar values for the 2 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.

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: 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.

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. PHYS 351 Augustana College Winter 2010-2011

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.

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. PHYS 351 Augustana College Winter 2010-2011

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.

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

nd inconsistency in the values of V0 for the 2 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

nd the error bars (σm) in the 2 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.