# User:Thomas S. Mahony/Notebook/Physics 307L/2009/10/12

Planck's Constant Main project page
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## Equipment

• Pasco Scientific Hg Light Source OS-9286
• Pasco Scientific h/e Apparatus AP-9368
• Fluke 111 Multimeter w/ banana cables
• National Instruments Data Acquisition Device SJK 16:19, 30 October 2009 (EDT)
16:19, 30 October 2009 (EDT)
These DAQ cards have model numbers which you should record. Probably something like USB ... People could possibly care about the bit-depth, acquisition rate, etc.
• Labview 8.6

## Setup

We followed the instructions in Professor Gold's Manual for experiment 2, and though we intended to follow it for experiment 1 as well, we decided to use a different method. The setup was fairly simple for this experiment. We plugged in the mercury light source and let it heat up for around 20 minutes before taking measurements, in order to ensure that the lamp was emitting at a constant intensity. Next, we plugged the voltmeter into the h/e apparatus to test the two 9 volt batteries. The batteries had to have been at a minimum level of 6 volts for the experiment to run properly (see page 3 of the Pasco Manual). Finally, before taking measurements, we focused the light from the lamp on the detector by moving the lens back and forth on the support rods until the image was as sharp as possible.

Light source and apparatus

## Data

### Experiment 1

SJK 16:22, 30 October 2009 (EDT)
16:22, 30 October 2009 (EDT)
I was really glad to see you two using the DAQ card. Also excellent linking of your code. Even though it didn't work perfectly, I still think this is the best I've seen this part of the lab carried out in my 3 years here. Good work!

The purpose of experiment 1 was to measure the changes caused by varying the intensity of the light incident on the detector. To measure this effect, the procedure called for using the filter with different transmission percentages for a single color and measuring the time taken for the voltage to reach its stopping potential. Since we assumed this would be similar to a capacitor charging which we knew to be an exponential function. Though the Gold manual called for the use of a simple stop watch to measure the charge time, we knew that exponentials theoretically never reach their max value, and in reality take a long time to do so. Instead of using a stopwatch, we decided to use the DAQ and Labview to plot the voltage as a function of time so we could try to fit our voltage to an exponential (and calculate a time constant for the rise time).

To collect data, we chose to use the 1st order blue color band. After centering the blue band on the detector, we held the zero button to remove any residual voltage remaining in the circuit. Then we started the data collection through labview and let it run for a while so we could get enough data for a good curve fit. The Labview code we used is attached here.

### Experiment 1 raw data

Though we intended to have multiple trials of data, we discovered in our preliminary analysis that our approach had problems with it, and therefore do not have as much data as would have been necessary. These problems will be discussed in the analysis section.

### Experiment 2

The purpose of experiment 2 was to measure the effect of different wavelengths on the stopping potential of the detector circuit and use this to measure Planck's constant. To collect our data, we positioned the h/e apparatus in front of five different colors of the spectrum produced by the mercury source: yellow, green, blue, violet, and ultraviolet. For the yellow and green lines, we used the corresponding yellow and green filters to remove any other frequency of light that might be passing through the hole. After centering the color bands on the detector, we held the zero button for a few seconds and then released it. We watched the voltage until it reached a maximum, which was then recorded in 5 trials for all 5 colors in both the 1st and second order diffracted color bands. In between measuring the first and second order lines, we readjusted the focus.

### Experiment 2 data

Note: The second order green line took considerably longer to charge than any of the other bands. SJK 16:10, 30 October 2009 (EDT)
16:10, 30 October 2009 (EDT)
This is a really good observation...important clue for figuring out why 2nd order green is messed up

## Analysis

1st Order Bands
2nd Order Bands

Experiment 1 had a problem. We made the assumption that the graph of the voltage vs time would be an exponential curve. This was not the case. The residual can be seen in our data, and it represents how far the best fit curve deviates from the actual data. If the curve had fit the data correctly, the residual would just look like a noisy straight line centered at 0. However, ours had a definite shape to it, signifying that the best fit curve did not fit the data correctly. In fact the place where it deviated most from the data was closer to the beginning of the charging. In this region, the best fit curve was less accurate. Our additional tests confirmed this. We tried fitting similar curves (the same settings, but not the same data as Labview stopped the VI after the curve was fit and forced us to take new data) starting at different points. These tests gave us drastically different "C" curve fit constants, which corresponded to the time constant of the exponential function we used to do our fit. Therefore, we could not use this method to get accurate time values for various intensities, because we were using the wrong fitting curve which varied wildly depending on how much of the curve was used in the fit.

Experiment 2 faired better than experiment 1. Using the first order bands, I calculated an expected value for Planck's constant:

• $7.185(2)\cdot 10^{-34} Js$

The second order bands yielded a different number:

• $1.3516(2)\cdot 10^{-32} Js$

For some reason, the excel trendline for the second order bands generated a different slope than my calculations, and using the SEM I calculated (totally wrong I know but I was curious to see what I would get):

• $7.6(2)\cdot 10^{-35} Js$

Note: Since the second order green line took a while to reach its stopping voltage, and that value was a much higher voltage than the first order voltage, we think some very faint higher frequency light may have also been hitting the detector. For this reason, when generating a graph for and doing calculations with the second order bands, I excluded the green values.

The accepted value from wikipedia is:

• $6.62606896(33)\cdot 10^{-34} Js$
(The full gory details of the analysis can be found here) SJK 16:42, 30 October 2009 (EDT)
16:42, 30 October 2009 (EDT)
I looked through your spreadsheet and found two errors which I think explain your problems with 2nd order fitting: (1) you have a typo when you use the slope from the graph (you put E-16 instead of E-15) and (2) I think you write down the formula for delta incorrectly in cell J8 on sheet2. Another more general comment: Instead of writing those formula as you do in cell J2 and J8, you'd be better off including more columns for terms such as 1/I. Doing so would make it easier to change the number of data points you have, without having to edit formulas. Overall, great that you took a stab at putting in the formulas by hand! Just in case you forget the point of it, this is the same thing that LINEST does...except with your formula, you can put in the SEM for each data point. It's sort of tough to carry out in Excel, but in Matlab or LabVIEW, you could easily write out functions that would deal with input arrays.

Image:Tom and Ryan Planck.xlsx
Also, I think you have other errors in your formula. I checked with LINEST, and it shows a much bigger uncertainty in the slope (about 50x what you have). Using LINEST would give 7.17(8) E-34

Another note: I can't find Ryan's spreadsheet, but looking at his screenshot, it seems like he didn't have bugs and used a more robust method than you did. Worth asking him to see what he did.

## Conclusions and Discussion

It is evident from our data using different filters that the intensity does affect the overall time taken to reach the stopping potential. Counting the major lines on the graphs for various intensity filters, and taking into account the fact that the limits on each graph are not the same, one can see that the higher the intensity of light, the faster the stopping potential was reached. This result is in keeping with the quantum model of each photon knocking off an electron, so the higher the flux of photons, the higher the current.

The accepted value for Planck's constant did not fall within any of the ranges I calculated, though it was closest to my 1st order calculation. Since my expected value is about 500 SEM's away, there isn't any meaningful probability of getting my value again, assuming only normally distributed random noise (I can use SEM's rather than a range, because the conversion of h/e to just h is linear). My conclusion then, is that we had large systematic error in our experiment. It is hard for me to track it down, due to the simplicity of the experiment and little reliance on human judgment. I seriously doubt the volt meter was miscalibrated. I also think we waited adequate time for the circuit to reach its stopping potential. Since I know that the energy of the photons and thus the frequency is the only factor in determining the stopping potential, even if the system was out of focus slightly, I do not think this would have led to much systematic error. I have only one remaining guess as to what caused such large error: sources of light other than the mercury lamp. Despite the room being mostly dark, there were other devices on that emitted light in the room. It is possible that some of this light caused our results to be skewed.

Despite not being able to calculate a time constant for experiment 1, I learned a lot about Labview, arrays, and curve fitting. I also, in the process of doing my data analysis, had error progatation sink in a bit more.

## Acknowledgments

Of course thanks to Ryan, my lab partner, for his help. Also, a HUGE thanks to Dr. Koch for his help with showing us how to use labview to take a subset of the data and doing a curve of best fit. We would not have been able to do experiment 1 without his help.