# User:Arianna Pregenzer-Wenzler/Notebook/Junior Lab/Speed of Light Summary

## Speed of Light Summary

### A brief summary

The goal of this lab was to measure the speed of light, using an LED (light emitting diode) to send pulses of light over a measurable distance to a device called a PMT (photomultiplier tube) that could detect these pulses. The time at which the pluses left the LED source and the time at which they arrived at the PMT was detected by the TAC (time to amplitude converter) which converted the difference between start and end time into voltage whose amplitude could then be displayed, and measured on an oscilloscope. In theory, by holding distance constant and varying time, through a series of known delays, we can come up with a calibration curve whose slope is the conversion factor between the voltage we measure off the oscilloscope and its corresponding time. Then, holding time (and very importantly, start time voltage) constant and varying distance, we should be able to come up with a series of voltages that correrspond to time of flight, and by ploting distance verses time we shold obtain a linear curve whose slope is the speed of light.

### Results

Steve Koch: I know you did this before our discussion of linear fit again on Monday. Since you had very good notes and data, I decided to take a look at it using the things we learned on Monday. Here is an updated sheet with some new stuff I think you will be interested in: Image:Speed of Light data Arianna SJK.xls. On the sheet you made, I added an "SEM" column to use for error bars. I also do the fit "x versus y" instead of y versus x, and you can see you get a different answer. Fitting time versus distance (and then inverting the slope) is more appropriate, because as discussed below, the time error is dominant. I also put in a new sheet showing how to analyze your data using variable sigma (from the SEM) using the manual formulas. Finally, I also fit your data, the third data set only, since you mention in your notebook that you got better as the day went along. Interestingly, this data set produces a value consistent with the accepted value!

Calibration Curve

The first data we took was to record the stoping voltage at a constant distance over varying time delays to find the conversion factor between voltage and time.

Using a least suqares fit for each trial and taking an average of the results I determined

Conversion factor: (0.198±.003)V/nsec

This compares very well with the conversion factor taken from the TAC which is 10V/50nsec.SJK calibration
calibration
Looks like you did a very good job on the calibration. The expected value is well within your 68% confidence interval.

conversion factor as given by the TAC: 0.2V/nsec

Calculation of the speed of light, c

We took three sets of data, holding the start amplitude constant to the best of our ability, to minimize the error due to time walk, over the distance of 2.20 meters. For every 10 cm adjustment we recorded the voltage off our oscilloscope, and using the conversion factor that we caluclated above, converted these voltages into time. Plotting distance traveled vs time produces a curve with a slope of velocity, our experimental speed of light.

Accepted value of the speed of light;

c ≈ 3*10^8 m/s

SJK

You report the answers with uncertainty & units, along with the accepted value, which is good. As noted on Monday, you should explicitly talk about how the accepted value compares with your answer, relative to your error bars (confidence interval). Also, some of your values have too many significant figures. e.g., the one would be better written (5.9 +/- 2.0) m/s.

Using a least squares fit, my expermintally determined speed of light;

c = (2.14±0.20)E+8 m/s

When I computed c over a range of 30-40 cm, I have terrible results. In one case I got a negative speed, in one case I got error larger than my c value. Selecting a set of points over a distance of 40cm where the time decreased as the distance decreased I calculated the following value for c...

for a short distance;

c = (5.89±1.97)E+7 m/s

When I compute c over a distance of 200m, using the average of my fist and last couple of data points...

for a long distance;

c = (2.64±0.13)E+8 m/s

These results for c calculated over short and long distances confirm what I expected about the error in this experiment. Because of time walk (early triggering in the TAC as a response to an increase in the intensity of the light pulse), it is impossible to move the LED closer to the PMT with out experiencing a fluxuation in the starting amplitude, and the process of trying to correct for that fluxuation using the polarizers on the PMT often resulted in even more distortion to our voltage that we then had to attempt to correct, making precision hit and miss at best. Some of the uncertainty in our methods is reduced by taking measurments over a greater distance, because the large change in distance allows for a final calculation of c that is a little less responsive to the inconsistancies in voltage.

### a final note on error

SJK time as dominant error
time as dominant error
I agree with both you and the manual. I think what Dr. Gold means in the manual is that your error in your position measurement (e.g., meter stick reading = 50.5 +/- 0.2 cm) is negligible compared with the error in your voltage (time) measurement. Now, the error in your time measurement will be approximately constant, thus you can reduce the effects of the time error by measuring over a longer distance, as you point out. So, I think you'd agree. Another way of looking at this is to say that when you plot distance versus time, your error bars will be much larger in x-direction than y-direction (and thus fitting time versus x would be more straight-forward).

In the lab it indicates that after comparing the error resulting from time resolution vs that resulting from position, you should reach the conclusion that your error in this lab is dominated by error in your time resolution. Based on my experience with this lab I disagree. If the problem with our data had to do with time resolution, I have no data to confirm it, though given the speed at which light travels I can imagine calculating the time of flight precisely with the equiptment avaliable to us could produce a large error. Where I do see a very large source of error is in our voltage. This could in part be due to time walk, but what seems to me to be the largest souce of error in this lab is the method we use to correct for time walk. Everytime we adjusted the distance we had to in turn adjust the polarizers. If we made the adjustment with out the average function on the oscilloscope turned on our signal was fluxuating so much it was hard to determine what result our adjustment was producing, and if we left the average function on, the curve was so slow to respond that it was easy to accidently over shoot our mark. Even though we thought our technique improved, our data showed otherwise.

### what I learned

By reading Dr Koch's comments on my last lab, I tried to make some adjustments to what I put into my summary. I left more of my analysis in my lab notebook, though my thoughts there could use a little more organization. Again mostly learned how much I have to learn, including the fact that I don't know how to upload just my graphs for Excel, I would have put a few in here, but at the moment its beyond me.SJK uploading