Physics307L F07:People/Gibson/Formal Lab Report



Abstract
In our examination of determining the speed of light, we used several key components to help us measure this interesting constant. We first used a dark 3 meter or so long tube in which an LED (light emitting diode) and a PMT (photo multiplier tube) were placed on opposite ends. The necessity for darkness in the room was not needed seeing as the tube was exactly the diameter of both pieces of equipment used blocking all outside light from potentially adding error to the experiment. What the PMT does is measure a voltage it receives from the LED and reports it to an outside source like an oscilloscope. From here we can create various wave propagations and plot them; in addition to this the oscilloscope is also used to measure the signal coming from the TAC (Time Amplitude Converter) which is receiving a signal from the LED. These two plots (voltage and signal) are averaged as best as they can be to find a steady voltage for each piece respectively to begin taking data.

Introduction
In this experiment we examined how closely we could reach the theoretical value of the speed of light. This result is actually very interesting to researchers that use the speed of light in experiments of their own.

The speed of light was first seen as a constant by Einstein in 1905 with his introduction of the Theory of Special Relativity. 

Methods and Materials
The equipment used is explained in this list:


 * Time-Amplitude Converter (TAC)


 * Light Emitting Diode (LED)


 * Photomultiplier Tube (PMT)


 * DC Power supply (150V-200V) - LED


 * DC Power supply (1800V-2000V) - PMT


 * Oscilloscope


 * Time Delay Apparatus


 * 3 Meter sticks


 * 3 Meter Long Tube

Procedure for setting up


 * 1) WARNING:Never expose the PMT to direct light while power source is on, this will destroy the PMT.
 * 2) To begin you must connect the power supplies to both the LED and the PMT that are at either ends of the tube; the low and high voltage supplies respectively.
 * 3) With the PMT in the left end of the tube near the equipment, attach the meter sticks to the LED and insert it into the right end of the tube. You will want to tape or fasten the meter sticks together to be able to move the LED a sufficient distance.
 * 4) Then connect the LED to the TAC's start jack with a BNC cable, and also connect the PMT to the stop jack on the TAC with another BNC cable. Next you also want to run a cable from the PMT to the oscilloscope's Channel 1 and from the TAC to channel 2 on the oscilloscope.
 * The oscilloscope will measure the voltage output from the PMT on channel 1 and the signal from the TAC will represent the time delay that you want to be measuring.
 * 5) You now want to adjust the oscilloscope to be able to see the signals from the PMT and LED. On channel 1 of the Oscilloscope you are wanting to see a level curve with a sudden drop in potential and then a rise back up. On channel 2 the signal from the TAC should look like a semi-square wave that is oscillating up and down rather rapidly.
 * 6) Once all set up, you want to pick a value of the voltage on Ch 1 and attempt to maintain that value for your measurements. This is accomplished by turning the PMT left/right to align or unalign its polarizer with that of the LED.
 * 7) Next, using the acquire button on the oscilloscope you can choose to average over the values that the oscilloscope is receiving to allow you to take measurements. You want to choose the 128 setting under the average command, so the signal is more stable. Then use the measure function to allow you to see the values outputted to channels 1 and 2 to make your measurements. Looking for the minimum on channel 1 and maximum on channel 2.
 * 8) Now we can start making measurements! - to do this you must pick your desired value for channel 1 and then holding the PMT steady have your partner pull or push the meter stick connected to the LED in or out of the tube. Record how far the meter stick moved relative to the opening of the tube and then readjust the PMT so that channel 1 on the oscilloscope returns to your previously picked value.Then simply continue.
 * For this experiment, we are not concerned with the actual distance between the PMT and LED. Which can seem a little strange. Rather at any particular distance between the PMT and LED we make a measurement and then we are only concerned with the change in that distance for the next measurement, since we are measuring the time delay of the signal from the LED pulse and the PMT signal.

Results and Discussion
In discovering the times for each data set, we were required (since we didn't measure time) to use the equation V=G*T where V is voltage; T is the time we want, and G is a constant set before we began taking measurements = 1/10 volt/nanosecond.

Here we took down measurements as listed in the procedure and placed them below:


 * NOTE: See acknowledgments section

Once we completed calculating our times for each of the trials, we then proceeded to construct 3 least squares plots to determine what our value of the speed of light would be. To do this we simply took the distance recorded vs time which then gave us a very linear graph... the slope of this line is the constant we were looking for.

Data Set 1 $$c=\left(2.68 \pm 0.18\right)\times10^{8} m/s$$
 * The slopes for each data set are listed below:

Data Set 2 $$c=\left(2.94 \pm 0.42\right)\times10^{8} m/s$$

Data Set 3 $$c=\left(2.89 \pm 0.08\right)\times10^{8} m/s$$

Average of all three data sets: $$c=\left(2.83 \pm .23\right)\times10^8 m/s$$

From here we calculated our relative error to determine the amount off of the theoretical value:

$$Relative Error=\frac{\left|2.83\times10^{8}-2.99\times^{8}\right|}{\left|2.99\times10^{8}\right|}=0.0535=5.3%$$

Conclusions
Our average value for the speed of light was extremely close to the theoretical value! Given the ambiguity of how the measurements where taken the result was quite surprising to say the least. A relative error of 5.3% is more than acceptable for this situation.

Given that we were only off by ~5% this leads us to believe that if we has had more time to conduct additional measurements we would be able to shrink this as much as 1% or so.

Acknowledgments
We would first like to thank Dr. Koch for his help in the beginning to understand how each pieces of equipment worked such as the time-delay and the TAC. He also helped us (Matt Gooden and I) understand physically as well as mathematically why how far the LED is away from the end of the tube doesn't matter as much as the change in distance. Dr. Koch was a key element in completing this experiment and understanding it.

On an additional note, I would like to thank Matt, my lab partner for his perseverance in figuring out the tables listed in the Results and Discussions section. These were quite a pain with the wiki editing interface and the reader would benefit more from the tables then standard reporting of the values.