# Speed of Light

## Acknowledgements

Tomas, you saved us about a day worth of work. Steve Koch 01:34, 9 December 2007 (CST):man you were in a bad mood because of that "c" in your name.

## Objective

We want to measure the speed of light, by measuring it over several distances. We hope to accomplish this by using a darkened tube with a LED light on one end, a photomultiplier tube on the other, and a time-amplitude converter between the two. We will take our measurements using a multi-channel analyzer and we will determine the speed of light.

## Procedure

### Equipment

• A PMT
• A large (~3m) cardboard tube)
• A LED
• The TAC
• 2 power supplies (high and low voltage)
• An Oscilloscope
• The delay box
• Several meter sticks

### Setup

We first made sure we had everything. Before we setup everything Dr. Koch inspired me to open up and the delay box and see how it worked. It was cables of varying length that would cause precise amounts of delay; it was pretty simple but interesting nonetheless. We began by make sure we had the polarizers in the correct setup on the PMT, since we'd be adjusting them for taking data that had to be setup right in the beginning. We put the PMT in one end of the tube and the LED in the other. We connected the low voltage power supply to the LED and the high voltage power supply to the PMT. We then connected the LED to the TAC, and the PMT to the delay box, which was then connected to the TAC. The delay made sure that the TAC doesn't detect a signal before it knows that a signal was sent. This is due to cable lengths and such. The PMT was also connected to an oscilloscope, through which we could read the current coming from the PMT. The TAC we ultimately connected the MCA board on the computer. We powered everything up and got it to work the first tiem we tried, but the second time was a massive failure. Below is the account of that, written as a play-by-play:

### Problems!

We have a lot of difficulty in setting up the TAC, the delay, and the oscilloscope. We could not find a square wave when looking at the range where the TAC's range was 50 ns and the multiplier was set to 1; meaning 50 ns would produce a 10 V signal. Unfortunately, in our setup the delay was having no effect and the TAC was responding unpredictably. We could not find a square wave no matter what and Devon had been helping us. We surmise that we are not getting a signal with a high enough voltage (<250 mV) for it to register with the TAC. We hooked up the LED to the scope to analyze our problem and noticed about 3V coming from the LED. We hooked the LED and PMT back up to the TAC and the oscilloscope and still could not find a square wave. We have a signal that is on the order of 8 mV but we fear that may be just do to noise from surrounding connectors. Bradley keeps trying the same thing over and over. The PMT is obviously receiving the light and the LED was not being noticed by the TAC. This is going nowhere. Devon played with the voltage on the DC power supply and we finally found a square wave. Magic.

So we went to the MCA instead.

### MCA

We hooked up the MCA and are using the Quantum PCA software. Tomas explained to us how it worked, Lorenzo tried. We have an acquisition curve of times that the MCA is receiving. We set our ROI (region of interest) over the curve and that gives us a centroid value that is an arbitrary number. We determined that if we find the centroid of several curves with varying known delays we could find the proportional constant between these arbitrary counts and the corresponding voltage. From that voltage we could find the times we are looking for. In Bradley's [notebook] he goes over the derivation he came up. Needless to say, by setting various delays and measuring and plotting them, we found our constant to be 11.141. Tomas found that the FWHM = 2.38σ and we can determine our standard deviation from there. We also found that 10 volts gives up 100ns, and we define our velocity as v = 11.141 * 109s − 1*(dx/d#)

### Actual Procedure

The actual procedure consisted of finding a voltage we decided upon on the oscilloscope (-608 mV). We would push the LED a known distance into the tube and then adjust the polarizer on the PMT until the oscilloscope reached that voltage. Once that was all taken care of we set a region of interest over the Gaussian curve that was generated by the data. From there we could find the full width at half maximum (FWHM) and the centroid value. We'd let the MCA collected about a million samples (the was a display of the integral value), and call it there. We set our ruler so that at 20 cm in deep meant that the LED was 3m from the PMT.

## Data and Error Analysis

I have all my error and data sets in Image:Sol.xls Excel file.

We have a standard error of .036 on average with all the measurements.

Our calibration scheme was to try the same PMT voltage with each of several delays:

Delay (ns) 0 8 16 24 32
Centroid 386.6 475.3 563.8 652.8

As you can see the slope is v = 11.141 + / − 0.4 * 109s − 1

Our dataset came out to be:

Distance (m) 0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0 2.25 2.5 2.75 3.0
Centroid 637.9 649.2 655.2 665.8 674.7 693 697 700.3 713.3 718.6 727.7

The slope here is m = .02838 + / − 0.001 * m − 1

and with m * v = c

$c=3.164 +/- 0.0001\times 10^9$m/s

This yields a 5% error from the actual value, and although I know my data analysis is not great, this is not too bad. The error I think lies partially in the lack of resolution of the oscilloscope, so we were never exactly at the same voltage from test to test. My value is way outside of my standard errorSJK 01:43, 9 December 2007 (CST)
01:43, 9 December 2007 (CST)
As I mentioned on your other page, you didn't convert the standard error correctly, which makes it seem too small.

## Conclusion

We wind up with a 5% error from the actual value, and although I know my data analysis is not great, this is not too bad. The error I think lies partially in the lack of resolution of the oscilloscope, so we were never exactly at the same voltage from test to test. My value is way outside of my standard error, so I can only conclude that this imprecision is due. There is also the possibility that the TAC had some internal malfunction, considering I heard groups complaining about it before and after. Or there is the time-walk issue that we never really accounted for, other than trying to keep those damn voltages stable. However, considering we were teaching ourselves most of what we did, we didn't do a bad job at all.