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Calculation of Planck's Constant

from the Photoelectric Effect

By Bobby Rohde

5-10-99

Abstract

This experiment is designed to determine the value of Planck's constant and the work function of a particular cathode. Planck's constant was found to be 6.6E-34 +/- 1.3E-34 J*s and the work function was 2.32E-19 +/- 8.33E-21 J. These were found by studying the photoelectric effect created by using differant wavelength of light on one cathode.

Introduction

In this experiment it is our goal to measure the value of Planck's constant from the photoelectric effect. We know that K_m = h*f -W, K_m is the maximum kinetic energy an electron can obtain from the photoelectric effect, h is Planck's constant, f is the frequency of the incident light and W is defined to be the work function of the metal which is a constant of the system. By studying the maximal electric field that can be created by means of the photoelectric effect for specific known wavelengths of light it is therefore possible to find both h and W, since the maximal electric field will be such that an electron of energy K_m can not move against it.

Experiment

In this experiment we used several pieces of equipment including a laser, a mercury lamp, a capcitor, an electrometer and a phototube. The equipment was aligned as shown in Figure 1 such that the light source shown through a color specific light filter and onto the phototube. The Phototube was connected to a capicitor and the across it read by an electrometer connected to a computer. Five differant wavelengths of light were studied, these were the red of the laser and the yellow, green, blue and UV lines of the mercury spectrum positioned at 632, 578, 546, 436, and 365 nm respectively. Filters were used with each sample so as to ensure that only the desired light wavelength impacted the phototube. Data was recorded on the computer as the phototube charged up until it achieved a well defined maxima for each of the wavelengths desired. This data was later analyzed to find the maximal voltage across the capicitor and from that Planck's constant and W.

Lab Setup (Figure 1)

Note: the mercury lamp is replaced by the laser for one measurement.
(Taken from Phys 375 manual published by University of Maryland)

Results/Analysis

We took the data gathered in the experiment and used Mathematica to preform a nonlinear fit against V = a*(1-Exp((-c-t)/b)) + d, where t is the time in sample point number and a, b, c, and d are the parameters of the fit and V the voltage. I figures 2 through 6 below one can see the raw data and the fitted curve through it. I then used the function that was founded to compute V at infinity for each wavelength and adjusted that to real units my removing the various scaling from the electrometer and computer's TakeData program (see Table 1). From the equation in the introduction and the statement that V = e*K_m, where e is the electron charge we know that V = h*c/(e*l) -W/e, where h is Planck's constant, c is the speed of light, l is the wavelength of the light, and the rest of the terms are as previously defined. Therefor by plotting V vs. 1/l we can determine h from the slope of the line and W from its intercept. This was done in Figure 7. By doing an error weighted average we find the value of h and W to be respectively 6.6E-34 +/- 1.3E-34 J*s and 2.32E-19 +/- 8.33E-21 J.

Red Light (Figure 2)

Yellow (Figure 3)

Green (Figure 4)

Blue (Figure 5)

UV (Figure 6)

Results from Fits (Table 1)

Wavelength (m)	Error in Wavelength	Voltage Max (V)	Error in Voltage
6.328E-7	2.0E-8	0.493	0.011
5.78E-7	2.0E-8	0.6480	0.0074
5.46E-7	2.0E-8	1.004	0.019
4.36E-7	2.0E-8	1.683	0.028
3.65E-7	2.0E-8	1.906	0.056

See Error Discussion Below

V vs 1/l (Figure 7)

Discussion of Error: Now if you were paying attention it is somewhat obvious that the error displayed in figure 7 for the voltages is not very appropriate to the linear fit it is given. The error which is plotted in the vertical direction is taken from the c^2 measure of the accuracy of the fit of the exponential curve and the assumption that the error on each point is the same. While I can not explain how these error bars became so small, I shoudl point out that only three out of five poitns fit well to the proposed line. However value of h obtained from this line is extremely accurate compared to the theoretical (6.6229E-34 measured, 6.626E-34 theoretical), far more so than the error statement above indicates, with the precent differance of only ~0.05 %. While I can not explain why those two points fall off the line (Fig 7) I feel compelled to say that there is presumably significant sources of error not attributed to just the error in the fit. For this reason the quoted error above is based on the scatter of points and thus takes into account the distances of the points from the line. It gives error bars as shown in Figure 8. The error in wavelength is taken to be the range in which light may reach the phototube through the filter given the quoted tolerance of the filter, and does NOT reflect the far greater accuracy by which we know the value of the laser's or mercury lamp's light.

V vs 1/l with scatter determined error (Figure 8)

Conclusion

Thus in this lab we have been able to determine the value of Planck's constant quite well, however I would like to point out that none of the fits (Fig 2-6) matched exactly to the expectation and they all varied in the same manner. I feel it might be valuable to study this phenomena itself it order to understand the actual mechanism governing the charging of the capictor in this situation. I also feel that there is probably a way to get more consistant data as to the maximal charge of the capicitor so as to hopefully yield more consistant data. Overall however the experiment was a success by allowing us to find reasonable values of h and W, even if the uncertainty on h is an order of magnitude (or two) higher than we would have liked.