Physics307L:People/Martin/Formal Lab Report

Determining the Speed of Light by Direct Time of Flight Measurements

^{SJK 11:54, 24 November 2007 (CST)}

11:54, 24 November 2007 (CST)
Good title.

Contact Information

Kyle Martin, University of New Mexico Student, kmartin1@unm.edu

Abstract

^{SJK 11:58, 24 November 2007 (CST)}

11:58, 24 November 2007 (CST)
Abstract is much better. "The "this means..." phrase is a little awkward, since really people would use the accepted value. Instead, to add a "what does this mean?" / conclusion to your abstract, say something like, "we had a fairly good relative uncertainty, which could be reduced in future experiments by ..."

The purpose of the experiment was to measure the speed of light, which is an important constant used in many physical computations. We measured the speed of light using a direct time of flight measurement, measuring the time it took to travel a certain distance. Our measured value for the speed of light was [math]\displaystyle{ 3.046 \times10^{8} \frac{m}{s} }[/math] +/- [math]\displaystyle{ 1.83 \times10^{7} \frac {m}{s} }[/math], in good agreement with the accepted value of [math]\displaystyle{ 2.99 \times 10^{8} \frac {m}{s} }[/math] [1], with a relative error of only 2%. Our relative uncertainity was fairly good and could be reduced in further experimentation by increasing the accuracy in which the TAC voltage data is taken.

Introduction

^{SJK 12:02, 24 November 2007 (CST)}

12:02, 24 November 2007 (CST)
OK, getting better! In the Einstein reference, the citation should include author, article title, journal name, volume, page number, year. Up until they decided to define an exact speed of light (cite this), how were they measuring the speed of light? Describe the best ways to measure the speed, and cite original research articles. For your last sentence, instead of saying "the goal of this experiment is simple..." say something like "we report measurements of the speed of light using blah blah blah

The speed of light is one of the fundamental constants in physics. It is a maximum velocity for particles and according to Einstein's relativity is constant in all reference frames [2]. It is used to calculate the energy and momentum of photons [3]. The speed of light is used to calculate many important features in physics from the bohr radius (the most probable distance from a proton to an electron in a hydrogen atom) [math]\displaystyle{ a_0 = \frac {h} {2\pi c \alpha} }[/math]to the distances of stars from earth, this fundamental constant is really important and is needed to do almost any calculation in physic. In fact the speed of light was defined as a constant and the meter was redefined to match on October 21, 1983. Before that time there were a couple of ways of determineing the speed of light. The goal of this experiment is simple, to measure the speed of light.

Methods and Materials

There is a problem, light travels fast, really fast. No longer can we plot out some distance and time how long the light takes to travel from start to finish, if we did this we would have a very long distance to measure out for only 1 second of light travel time. No, measuring exact distance and time to travel that distance seems impractical. In my experiment I did a variation of this. I measured relative distances and how the time varied. I used this and the famous [math]\displaystyle{ v = \frac{\delta d}{\delta t} }[/math].

Materials

^{SJK 01:47, 7 November 2007 (CST)}

01:47, 7 November 2007 (CST)
Great job with this section!

1. TAC Model 567 manufactured by EG&G Ortec
2. Delay 2058 manufactured by Canberra
3. Power supply Model 315 DC power supply 0-5000V, 0-5mA Manufactured by Bertan Associates Inc.
4. Oscilloscope Tektronics TDS 1002
5. LED Power Supply: Model 6207a mfd. by Harrison Industries, 0-160V, 0-0.2A
6. Photo multiplier tube: N-134 (PMT)
7. Multichannel analyzer program (MCA)
8. Microsoft Excel 2003

• 

  From Left: Power Supply 315, Delay 2058, TAC

• 

  Oscilloscope and Power Supply 315, Delay 2058, and TAC setup

• 

  Power supply for LED

Method

^{SJK 13:15, 24 November 2007 (CST)}

13:15, 24 November 2007 (CST)
much better. Use "we" since you and Jesse both did this. The phrase "do not expose it to light" is still a "how to" not a "we did" statement. Instead say something in your own words like, "the PMT can be damaged by too much light, so we took care to blah blah blah" (and by the way, you don't need the room lights off, because it's in the light-tight tube.

Throught this experiment I worked with Jesse Smith, he was my lab partner. Knowing that the PMT is light sensitive Jesse and I made sure to avoid PMT damage by avoiding PMT contact with light when the power source was on. We placed the LED in a cardboard tube, much like the ones left over after the paper towels are out just much larger, opposite the PMT. We connected LED to the 0-160 Volt power source. The LED was also connected to the time amplitude converter (TAC) to relate to the TAC when the light signal was sent. We also connected the PMT to the time amplitude converter to tell the TAC when the light signal was received. There was one problem with this setup and that was that the "received" light signal could be received before the "sent" light signal, to solve this problem we connected the PMT to the TAC through a time delay. We connected this TAC device to an oscilloscope to read the voltage readings from the TAC. With the lowest settings selected on our TAC I have to multiply by five to convert from voltage readings to time difference between sent and received signals. In further experimentation we connected the TAC to the computer as well and used a multichannel analyzer (MCA) program to take our TAC voltage readings. There is an important source of error here that if not regulated the outcome of the experiment would be terrible. Time walk is related to the intensity of the received light. The TAC device triggers at a fixed voltage, so shorter pulses will trigger later then longer pulses. As the LED is moved closer to the PMT the intensity of the light changes. In the experiment polarizers are attached to both the PMT and the LED, twisting the PMT and keeping the received intensity constant will avoid the time walk problem; This is why we connected the PMT directly to the oscilloscope as well as the TAC, to measure the intensity of the light into the PMT to avoid the time walk factor. After the long setup process we started taking data. We moved the LED using a meter stick attached to the end of it. We used this meter stick to take the relative distances from the LED to the TAC. We took a data point and the moved the LED 10 cm rotating the PMT in order to keep the intensity of the received light at a constant value.

Linear regression All of the analysis of this data was done on excel. I plugged the data we gathered in our lab into excel and had it do a linear regression (least squares fit) on the data set. I also had excel graph the data in a Distance vs. Time plot as well as the linear fit to the graph. The slope of the Distance vs. Time plot is the value I am looking for, the speed of light. The built in excel linear regression program runs on the following, where x is the TAC voltage converted time reading, and y is the distance recording:

[math]\displaystyle{ S_X = x_1 + x_2 + \cdots + x_n \, }[/math]

[math]\displaystyle{ S_Y = y_1 + y_2 + \cdots + y_n \, }[/math]

[math]\displaystyle{ S_{XX} = x_1^2 + x_2^2 + \cdots + x_n^2 \, }[/math]

[math]\displaystyle{ S_{XY} = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n. \, }[/math]

The slope of the linear fit, the speed of light, is given by:

[math]\displaystyle{ c = {n S_{XY} - S_X S_Y \over n S_{XX} - S_X S_X}. }[/math]

Multichannel Analyzer Program (MCA) The multichannel analyzer program counts events, electrical signals sent to the computer. The MCA shows this counted data in graphical form where the x-axis is related to a bin number and the y-axis is related to the number of counted events. Most of the time the bin number axis is a time axis where each bin number lasts a specified time. In our experiment the MCA counts the number of voltage occurances of the voltage signal from the TAC. The resulting graph can be approximated as a normal or gaussian distibution. We took the most probable bin number from the MCA graphical display, the MCA also tells the graphs "full width at half maximum" (FWHM), this is the twice the standard deviation or twice the uncertainity of a gaussian. Using this program we can come up with a much better estimate of our uncertainty. To get the value of c from the MCA is a little trickier though. We know that the bin number [math]\displaystyle{ N\, }[/math] at the maximum number of voltage occurances is directly related to the time between sent and recieved light signals: [math]\displaystyle{ \N = kt }[/math]. We have by the chain rule in math that: [math]\displaystyle{ c=\frac{dx}{dN}\frac{dN}{dt} }[/math]

and from [math]\displaystyle{ \N = kt }[/math] that

[math]\displaystyle{ c=k\frac{dx}{dN} }[/math].

However, since then relative LED to PMT distance is directly related to the number[math]\displaystyle{ x = m N\, }[/math].

[math]\displaystyle{ c = k m\, }[/math]. To find the propotionality constant [math]\displaystyle{ k\, }[/math]we varied the time delay switch and measured [math]\displaystyle{ N\, }[/math] at a fixed PMT to LED distance, [math]\displaystyle{ k\, }[/math] was the slope of a linear regression done in excel. We calculated [math]\displaystyle{ m\, }[/math] by varing the PMT-LED distance and recording the corresponding bin number and then using excel to do a linear regression to find the slope [math]\displaystyle{ m\, }[/math]. Multiply [math]\displaystyle{ k\, }[/math] and [math]\displaystyle{ m\, }[/math] to get the speed of light.

^{SJK 01:43, 7 November 2007 (CST)}

01:43, 7 November 2007 (CST)
It is great that you included a diagram! A picture goes a long way in helping to explain how to set things up. I think you can tweak it (color, shading, more labels) to make it better, but that's not the high priority for now.

Above is the setup of the experiment

Results and Discussion

^{SJK 16:39, 1 December 2007 (CST)}

16:39, 1 December 2007 (CST)
You can add another paragraph to your methods section explaining that you used Excel to perform a linear regression on the data set. (See Matt Gooden's report for a very nice example.)

The phrase "here is my data" is still very informal. You need to label the table ("Table 1") and give it a caption. Then refer to it in the text, for example, "Table 1 shows the results of our first set of experiments..." I also want you to use the correct technical terms (as opposed to linearized slope).

Raw Data: Experiment one excel sheet, Experiment two excel sheet I put this data into excel and asked excel to linearize the data and to compute uncertainty in the slope of that linear regression. Where the slope I was computing was [math]\displaystyle{ v = \frac{\delta d}{\delta t} }[/math] Linear regression utilizes the method of least squares fit or reducing [math]\displaystyle{ \chi ^{2} }[/math] to find the most probable trend fit to a group of data.

Table 1

Distance from 1.4 m in meters	voltage readings	time in nanoseconds
1	4.96	24.8
0.9	4.85	24.75
0.8	4.82	24.1
0,7	4.74	23.7
0,6	4.7	23.5
0,5	4.6	23
0.4	4.54	22.7
0.3	4.5	22.5
0.2	4.42	22.1
0.1	4.37	21.85
0	4.4	22

Table 2

delay time in nanoseconds	N as previously defined	Uncertainty in N
0	208.5
1	217.9
2	228.3
4	249.4
8	288.7
value of K in inverse nanoseconds	10.08
uncertainty in K	0.089

Table 3

N as previously defined	Distance from 1.4 m in meters	Uncertainty in N
208.5	0
212.1	0.1
213.9	0.2
219.8	0.3
222.2	0.4
224.1	0.5
217.1	0.6
Value of m in meters	0.0299
Uncertainty in m	0.0023

• 

  Figure 1

• 

  Figure 2

• 

  Figure 3

• 

  Figure 4 the marked data point is one that I threw out for a few reasons explained later this figure becomes figure 3 when this data point is thrown out.

Table 1 and Figure 1 belong to the first set of data that Jesse and I took. During this experiment we took our TAC voltage readings from the oscilloscope. Table 1 is the list of raw data that we recorded during the experiment. Figure 1 is the linear fit to our data in a Distance vs. Time graph. The slope of the linear fit in Figure 1 is the speed of light. Below are the results that excel gave for the least squares fit.

• slope (speed of light): [math]\displaystyle{ 3.06 \times10^{8} \frac{m}{s} }[/math] +/- [math]\displaystyle{ 1.83 \times10^{7} \frac {m}{s} }[/math]

Table 2, Table 3, Figure 2, Figure 3 and Figure 4 belong to the second set of data that Jesse and I recorded. During this experiment we used the MCA device to display our TAC voltage readings. Tables 2 and 3 are lists of the raw data taken off of the MCA, one of these data sets yielded the constant K and the other yielded the M constant. Figures 3 and 4 are similar less one data point. I took out the data point marked out by the red arrow in figure 4 because it was the closest PMT to TAC distance measurement and the MCA was having trouble recording the number N as the PMT to TAC distance decreased. Also the data point is obviously flawed because it lies so far off the obvious linear trend. The linear trend line in Figure 2 is K and the linear trend line in Figure 3 is M. I multiplied these numbers together and the result was the speed of light. Below is my final answer for the speed of light from the MCA data:

• Speed of Light: [math]\displaystyle{ 3.02 \times10^{8} \frac{m}{s} }[/math] +/- [math]\displaystyle{ 2.37 \times10^{7} \frac{m}{s} }[/math]

Both of these answers are in good agreement with the accepted value of [math]\displaystyle{ 2.99 \times 10^{8} \frac {m}{s} }[/math] so I then took a weighted average of these sets to get my final overall answer:

• [math]\displaystyle{ 3.046 \times10^{8} \frac{m}{s} }[/math] +/- [math]\displaystyle{ 1.45 \times10^{7} \frac{m}{s} }[/math]

Possible sources of error

^{SJK 12:58, 24 November 2007 (CST)}

12:58, 24 November 2007 (CST)
This is great stuff to include in your discussion. However, try to imagine someone reading your report who is not in Physic307L. Would they have any clue what you're talking about? I think no, because you're not explaining in enough detail. How would the MCA program help? Why wouldn't you have to twist the PMT anymore? What does MCA stand for anyway? What new abilities will it give you?

I believe that though we conducted our experiment as best as we could there were a few sources of error that we overlooked in our initial trial of this experiment. I believe that the twisting of the PMT device is quite inefficient. Also it was difficult to get a accurate reading of the voltage output from the TAC through the oscilloscope, this is why we used the MCA the second time around, we got a slightly better value for the speed of light. The MCA did help us get better TAC voltage readings and therefore better results in the end. I still believe that there must be a better way to gather the voltage readings where there is not as much uncertainty. Using the MCA to analyze the TAC voltage readings helped but next time I would like to eliminate some of the uncertainty in the voltage readings.

Conclusions

The speed of light is well known in fact it has been defined to be a specified value and the meter was changed to match. This precise value is given by CODATA as [math]\displaystyle{ 2.99 \times 10^{8} \frac {m}{s} }[/math]. ^{SJK 13:07, 24 November 2007 (CST)}

13:07, 24 November 2007 (CST)
This section is much better, but still a bit too informal. Delete "luckily". Is your relative distance measurement really that bad? You don't analyze that in the report, so it's hard to throw it in in the conclusions (so add some discussion earlier!). "If there were a program" makes it sound like you're not that interested in finding it. Instead, find out exactly what you need, and then describe it.

Also, I think you should use "we" in most places (as you and Jesse worked together). Sometimes "I" is technically correct, but it's customary in formal papers to use "we."

Now we can calculate a relative error in my experiment this value was.

• relative percent error: 1.87%

We found our final result with its given uncertainty acceptable given the Oscilloscope reading fluctuations and the associated problems with rotating the PMT device and by eye deciding if the incoming voltage remained at a constant. I controlled as much of the possible systematic errors that I could and believe the rest of the reasons for error in this experiment to be mostly random. The only thing I can think of to improve on this experiment is to better measure the relative distances between the LED and the PMT and to somehow settle the voltage output reading from the TAC. The voltage reading from the TAC jumped quite a bit. If there were a program that we could use to help filter out some of the noise and get the voltage reading to settle a bit it would be easier to take an accurate value for the voltage and in effect the time.

Acknowledgments

I would like to thank my teacher Dr. Koch for helping me understand the setup and the components of the lab. I would like to also thank my lab partner Jesse Smith for his help with the lab itself.

References

^{SJK 13:09, 24 November 2007 (CST)}

13:09, 24 November 2007 (CST)
Getting better. All references should be numbered and cited by number in your text (since numbers are the style you're using). Also, I want you to add more references to papers measuring the speed of light. Obviously there are lot of papers related to this!

• Last years lab manualPrevious Course lab manual.
• TAC Operator Manual, found in the physics senior lab.
• Jesse Smiths page (My lab partner) [[4]]
• 1: CODATA. Speed of light in vacuum. 2006 CODATA recommended values. NIST. Retrieved on 8/23/2007
• 2: Einstein, Albert. "Relativity: The Special and the General Theory," Part 1, Sect 7.
• 3: Walter Greiner (2001). Quantum Mechanics: An Introduction. Springer. ISBN 3540674586.

Koch comments

In order to get an outstanding grade, I am going to ask you to take some more data and try out some ideas you have for improving the measurements. This doesn't mean that your data aren't great already, just want you to get the experience of trying to take even better data, and knowing how to approach that kind of task.