*Dan Medin - Modern Physics*
# Table of Contents
1. Purpose
2. Equipment
3. Theory
4. Procedure
5. Analysis
6. Conclusion
## Purpose
Compare the classical Wave Theory to the Quantum view of light through studying the results of the Photoelectric Effect. The variables within the apparatus that will be measured include intensity, frequency, and stopping potential to analyze the results.
## Equipment
1. h/e Apparatus AP-9368
2. Mercury Vapor light Source OS-9286
3. Lens/Grating Assembly
4. Light Aperture Assembly
5. Support Base Assembly
6. Light source block
7. Coupling Bar Assembly
8. M9803 True RMS Multimeter
9. Protractor
10. Model AP-9368 Filter
## Theory
**A. Classical Wave Theory**
Originally, it was predicted that when light was shined on an object that the energy freed from the surface would increase with the power of the light and the amount of light emitted from the surface would increase with the frequency of the light. This prediction was derived from Rayleigh-Jeans Law. However, in practice, the results were very unsupportive, which left many scientists frustrated with their current understanding of physics.
**B. Quantum Theory**
With intent to have equivalent experimental and theoretical answers, Max Planck began work on what we now know as Quantum Theory.
While studying radiation, Planck discovered the following relationship:
[1.] $E=hf$
E = Radiant Energy
h=6.62607004 × 10^-34 $m^2 kg/s$
f=frequency
This new relationship changed the theoretical prediction of how the intensity and frequency of light shining on a surface affect the photoelectric effect. Now we can predict that increasing the frequency will increase the energy of the emitted electrons and increasing the intensity will increase the amount of electrons emitted (changes charge time).
After experimentation of the theory, Einstein used his work and the Quantum Theory to state the following relationship for the photoelectric effect:
[2] $E=KE_{max} + W_{min}$
E = Incident Energy
KE = Max Kinetic Energy of emitted photoelectrons
W = Min Energy needed to free an electron
However, this can be tricky to measure. The best way to measure the kinetic energy of the emitted photoelectrons is by applying a reverse potential voltage along the path that the photoelectron will travel as it goes through a tube (to control the photoelectrons as much as possible for measurements). We can then measure the minimum reverse potential needed to overcome the kinetic energy of the photoelectrons. This gives us the following relationship:
[3] $KE_{max}=Ve$
Using equations 1, 2, and 3, we can derive the following identity:
$V=hf/e-W_{min}/e$
This identity allows us to represent the relationship between frequency and stopping potential to evaluate the value of the constant $h$ (it's the slope divided by e).
## Procedure
### 1. Setup and Alignments
After testing the voltage of the batteries of the h/e apparatus using the multimeter [8], we found the results to be adequate.
Battery test:
Right: 7.17 Volts
Left: 6.99 Volts
**Aperture Setup**
**Note: The lab was set up before us, but we made sure that it was done correctly.*
We used the coupling bar [7] and the support base [5] to connect the Mercury light source [2] to the h/e apparatus [1]. We also made sure that the light aperture assembly [4] and the light source block [6] were properly attached the the light source and that the lens/grating [3] was properly attached to the aperture assembly. Lastly, we confirmed and readjusted the protractor [9] to be aligned with the coupling bar for accurate measurements.
### 2. Data Collection
**Methodology**
Before we started, we turned the light source on and let it warm up for roughly 6 minutes before beginning. We then turned the lights off and could see the spectral lines of the mercury. Our data collection process followed a consistent procedure. First, rotate the h/e apparatus so that the line we intended to measure fell on the center. Then, roll the light shield away to confirm that the light was centered on the white photodiode mask inside. If it wasn't, we adjusted the angle slightly. Then, we rolled the mask back and very carefully took measurements of the angle of incident (with the protractor) and of the stopping potential (with the multimeter). Lastly, we hit the "push to zero" button on the h/e apparatus to avoid any lingering charge in future measurements. The measurements we took are listed below.
**Note: Only first-order spectral lines were measured*
***Table 1: Varying Angles***
| Angle (Degrees) | Stopping Potential (V) |
| ----------|------------------ |
| 19 ± 2 | 1.013 ± .01 |
| 18 ± 2 | 1.037 ± .01 |
| 14 ± 2 | 1.554 ± .01 |
| 13 ± 2 | 1.778 ± .01 |
Next, we wanted to study the effect that intensity has on the stopping potential. To do this, we choose two of the spectral lines (the ones at a measured angle of 18 degrees and 14 degrees). To setup this part of the experiment the only change we needed to make was to add a filter [10] onto the h/e apparatus. For these lines, we followed the same procedure as above except we adjusted the filter to different strengths instead of the angle for each iteration. The results we measured are listed below:
***Table 2: Varying intensity for angle of 18 ± 2 degrees***
| Intensity(%) | Stopping Potential (V) |
| ---------- | ---------------- |
| 80 | 1.021 ± .01 |
| 60 | 1.013 ± .01 |
| 40 | 1.017 ± .01 |
| 20 | 1.033 ± .01 |
***Table 3: Varying intensity for angle of 14 ± 2 degrees***
| Intensity(%) | Stopping Potential (V) |
| ---------- | ------------------ |
| 80 | 1.555 ± .01 |
| 60 | 1.575 ± .01 |
| 40 | 1.559 ± .01 |
| 20 | 1.547 ± .01 |
**Note that the results offer no support to the classical theory*
## Analysis
**Frequency**
The first thing that we should analyze is the wavelength of the light as we had only collected the angle of incidence.
We know that the grating has d=1/600 mm and m=1 (first order lines)
Using the identities:
$d\sin{\theta}=m\lambda$
and
$f=c/\lambda$
we can use the relationship
$f=mc/(d\sin{\theta})$
To find the error for Frequency, we have to take the derivative of the above equation since the error for d and m are negligible, making it a function of one variable.
$\delta f =\delta \theta⋅c⋅cot(\theta)⋅csc(\theta) /d$
Plugging column 1 of Figure 1 into the equation gives us the following data:
***Table 4: Varying Frequencies***
| Frequency (Hz) | Stopping Potential (V) |
| ----------|------------------ |
| 5.529 ± .560 × $10^{14}$ | 1.013 ± .005 |
| 5.825 ± .626 × $10^{14}$ | 1.037 ± .005 |
| 7.440 ± 1.042 × $10^{14}$ | 1.554 ± .005 |
| 8.001 ± 1.210 × $10^{14}$ | 1.778 ± .005 |
***Graph 1: Frequency vs Stopping Potential***
![Graph 1][1]
Best fit equation found via excel graph:
$y=.3144x-.7608$
As x was graphed in terahertz, we arrive at the following experimental measurement:
$h/e=.3144*10^{-14}$
Thus, we can determine $h$ by using $e=1.602 × 10^{-19}$
$h=5.036688 × 10^{-34} $ (24% error from accepted)
Table 2 and 3 clearly show that there is no real change in stopping potential caused by intensity. Even at different frequencies, the uncertainty and non-linear measurements show no real support for the classical idea of intensity increasing the energy of the photoelectrons.
## Conclusion
Based on the results of these measurements, we can support the Quantum Theory and disprove the classical Wave Theory. Not only were we able to show that the intensity of light has no affect on the resulting energy emitted (Table 2 & 3), but we were able to show that an increase in frequency leads to an increase in energy of the photoelectrons (Table 4). We were even able to derive an estimate of the constant h from out data that fit within our error bars (Graph 1). This lab essentially summarizes the shift from classical to modern physics in the context of the photoelectric effect.
[1]: https://mfr.osf.io/export?url=https://osf.io/72cfh/?action=download%26direct%26mode=render&initialWidth=818&childId=mfrIframe&format=1200x1200.jpeg
