# Formal Report ohara

## Contents

Finding Planck's Constant-an Experimental Approach

*Author: David K. O'Hara*

*Experimenter: David K. O'Hara *

*Junior Lab, Department of Physics & Astronomy, University of New Mexico*

*Albuquerque, NM 87131*

*dkohara@unm.edu*

## Abstract

This project was conducted in order to find a value for Planck's constant, one of the most fundamental constants, experimentally. This was accomplished by allowing a mercury light spectrum to be incident on a photoelectric material and measuring the stopping potential for various wavelengths of light, this data paired with the work function equation enable me to calculate a value for h. The value I calculated from my experimental data was 7.00E-34 +/- 7.4E-36 J*s. Compared to the standardly accepted value of h = 6.626068 E-34 J*s. The value I calculated was about 5% off of the expected value of the constant, given the limits of the testing apparatus that was used this value is quite encouraging.

## Introduction

This experiment was designed to find Planck's constant, one of the fundamental constants of nature. The relationship between the energy of a photon and the frequency of its electromagnetic wave was one of the truly great discoveries in science. Work that was later expanded on by Einstein's further investigation of the photoelectric effect^{1} as it applied to electrons being freed from matter by incident light and De Broglie's work^{2} that expanded the relationship between energy and the quantum wavelength to all particles.

It was however, Planck's work, in looking at Black body radiation that led him to treat energy in relation to frequency as a discrete value or quantity instead of a continuous function. This work was the beginning of quantum physics^{3}. Even though Planck had made this quantum leap because the math worked out, other physicists continued this work and showed there was a real relationship between energy and frequency.

In this report, I estimate the value of Planck's constant by measuring the relationship between photon frequency and the energy of electrons ejected from a photosensitive material by the photoelectric effect as photons of particular wavelengths are incident on this material. I compare my value to the accepted value and discuss some of the sources of systematic error present in my experimental method.

## Procedure

The experiment was setup so that light from a mercury lamp would pass through a diffracting prism creating a spectrum of light, this light was focused on a photosensitive material with a very low work function. Using five distinct wavelengths of light we were able to measure the stopping potential required to offset the energy of the ejected photons in the material (Table 1).

Table 1.

Color | Wavelength (nm) | Frequency (hz) |
---|---|---|

UV | 365.483 | 8.20264e14 |

Violet | 404.656 | 7.40855e14 |

Blue | 435.835 | 6.87858e14 |

Green | 546.074 | 5.48996 |

Yellow | 578.000 | 5.18672e14 |

*Wavelengths used in experiment*

I then measured the stopping potential for each of these wavelengths for the first maxima and then for the second maxima. An interesting finding occurred when measuring the second maxima. For the green wavelength on the first maxima I measured a stopping potential of .8408 volts, but when I measured the stopping potential for the second maxima I found a stopping potential of 1.207 volts.

There was nothing obviously wrong that could have generated such a large systemic error. Dr. Gold's manual did discuss that filters needed to be used to insure that only the wavelength that was being observed was allowed to interact with the photosensitve material and these filters would prevent light from another maximum from interfering with the current readings. The experiment was conducted in a dark room to shield the experiment from any ambient light wavelengths that could skew the data. After discussing the issue with my lab instructor and playing with the filters I found that the green filter was not blocking UV light from the third maxima incident on the photo slit. Pairing the yellow and green filters together proved to be a much better filter than the green one alone and I was able to get readings that made sense again (.8301 volts and so on).

After stumbling over this unexpected UV issue I was able to get multiple trials of data which seemed to lack the suspect characteristics of the second green maxima.

## Data and Error Analysis

All my data can be found here in this excel spread sheet. File:Planck.xlsx

The analysis was fairly straightforward and is given here:

The basis for this experiment to find a value of Planck's constant hinges on the following mathematical course:
The total energy for electrons leaving the cathode in this experiment is given by:

- E=hν=KE
_{max}+W_{0}

The Kinetic energy of the electrons is:

- KE = 1/2 m
_{e}v^{2}

The stopping potential is where the potential barrier is equal to the kinetic energy of the electron:

- eV
_{s}=KE_{max}

Which leads to:

- E=hν=eV
_{s}+W_{0}

- eV
_{s}=hν-W_{0}

where e=1.602E-19

- V
_{s}=(hν-W_{0})/e

So, after graphing frequency vs stopping potential, multiplying the slope by e I obtained a value for h. Then multiplying the y-intercept by e, I found a value for the work function

The value I calculated for the constant for the first maxima

- h = 6.99768E-34 +/- 7.43E-36 J*s

- for the second maxima I calculated a value for h of:

- h = 7.03782E-34 +/- 1.11E-35 J*s

- with the accepted value of h = 6.626068 E-34 J*s

this would give me a percent error for the first maxima of 5.6 %. These calculations also gave me a value for the work function of 1.55 eV +/- .02 (we were given no actual value for this material in Dr. Gold's manual for comparison). The value I got for the work function concerns me as typical work functions listed on various web locations don't really go below 2eV. This brought me back around to the sources of error in this experiment.

First, the filters are obviously not perfect and allow some light of different wavelengths than they are supposed to through. When making a calculation down to e-34 precision it is difficult to have a high confidence in my data when the experiment has such an obvious flaw. Second, it is difficult to focus the maxima so that they have their sharpest image on the photosensitive material. Again, the size of the number we are working for leaves little room for error. Last, when pushing the reset button to take the stopping potential to zero, opened up the opportunity to move the apparatus which could where the maxima were focusing.

## Conclusions

The objective of the experiment was to find the value of Planck's Constant, while the value I measured was close, it was not exact. Although my experiment did not come as close as I would have liked, with a slightly improved setup and improved environmental conditions(a black room), I feel this experiment could easily come very close to the expected value of h. I think that the current conditions we had in our lab were simply not sufficient to prevent ambient light from skewing our data.

The issue of the third maxima interfering with the second maxima measurement made for a challenging and stimulating puzzle to solve and made this experiment one of my favorite labs. It showed that even when performing the steps of an experiment by the book, you may still have huge systematic errors waiting to trip you up.

## Acknowledgements

I would like to thank Professor Koch and Pranav Rathi(TA) for their technical expertise and guidance in this experiment.

Microsoft Excel™ was used to format and post my data to my lab notebook for this formal report.

## References

1. On a Heuristic Viewpoint Concerning the Production and Transformation of Light, Albert Einstein(translated from German)Annalen der Physik 17: 132–148 Retrieved 2008-02-18

2. L. de Broglie and L. Brillioun, "Selected Papers on Wave Mechanics", Blackie, London (1928)

3. J Franck, Max Planck, Science 107 (1948), 534-537

4. Dr. Gold's manual for experimental physics 31-40