User:Randy Jay Lafler/FormalReport

= The Time of Flight method to Calculate the Speed of Light = Author: Randy Lafler

rlafler@unm.edu

University of New Mexico Physics Department

Undergraduate



Abstract
The speed of light is a fundamental constant in physics influencing many formulas and physical phenomena. Because of relativity we now assert that it is invariant in any reference frame. In this experiment we measured the speed of light in a direct time-of-flight measurement. We used an oscilloscope to measure the time delay of an emmitted photon. We did not use the Time to Amplitude Converter (TAC) as the manual suggests in an attempt to avoid time walk. We determined the speed of light to be 26.8+/-7cm/ns for our fourth and most accurate measurement by using Excel to plot the change in distances verse the change in times. The speed of light is the slope of these plots. 

Introduction
Long ago, scientists debated whether light traveled instantaneously or at a finite speed. Scientists tried to estimate at least a lower bound on the speed of light by attempting to measure the start and stop of light signals over very large distances. It was found, however, that light traveled faster than any distance over which they could reasonable try to measure it on earth's surface. Boyer Descartes tried to utilized the larger distance between the moon and the earth, but even this distance was not great enough to measure the speed of light. So, he wrongly decided that light travels instantaneously. But, in 1671 Roemer determined by looking at the satillites of Jupiter that the speed of light must be finite. In 1862, Leon Foucault accurately measured the speed of light by sending a light signal from a rotating mirror toward a mirror fixed a large distance away. He then calculated the speed of light by using the angle through which the mirror rotated from the start to the reflection back of the light. Modern techniques to measure the speed of light have measured the speed of light to an accuracy of 299,792,458m/s. Codata Some modern techniques to measure the speed of light include measuring the frequency resonance in a resonance cavity. Essen, and by using interferometry. Evenson We calculated the speed of light by using a time of flight method. 

Methods and materials
We set up the cardboard tube with the PMT(Nano N-134 Photo Multiplier Tube) on one side and the LED (Photon Emitting Diode) attached to a meter stick in the other end. Using BNC cables we attached the PMT and the LED to the oscilloscope in channel one. We set one of the cursors at a fixed location on the oscilloscope screen, and adjusted the other vertical cursor to the initial downward spike in the signal. In fifty centimeter increments we moved the LED closer to the PMT, thus decreasing the distance for the photon to travel and the time. At each distance we adjusted the second cursor of the oscilloscope to be exactly at the initial downward spike in the signal, and we recorded the reading given for the change in time. We followed this procedure for the first three trials. For the fourth trial we rotated the PMT inside the cardboard tube and tried to hold the first spike of the signal at the same vertical position on the oscilloscope screen for every fifty centimeter measurement. For our last trial we rotated the PMT so that the average, or middle, of the first several spikes in the signal would be at the same position for every measurment. Using Excel we plotted the change in distance of the LED verse the change in time. The slope of the linear fit line to these data points gave us the speed of light. We also used Excel to give us an average velocity by summing up the total distance and dividing by the sum of the times from each measurement. 

Results and Discussion
We did five trials to measure the speed of light. The first three have similar results because we used the same method. The last two we did with a different method describe in the methods section. Here is a summary of the values we obtained for each trial. Trial 1
 * $$C=24.1(1)cm/ns\,\!$$

Trial 2
 * $$C=23.2(7)cm/ns\,\!$$

Trial 3
 * $$C=23.6(11)cm/ns\,\!$$

Trial 4
 * $$C=26.8(7)cm/ns\,\!$$

Trial 5
 * $$C=34.8(1)cm/ns\,\!$$

Accepted value
 * $$C=30cm/ns\,\!$$

Below is the link to the Excel sheet we used to calculate the speed of light.

Speed of Light 2 XL Doc

The plots we obtained for trials 4 and 5 are displayed below in Trial Run 4 and Trial Run 5. We choose only to display these two plots because the data from these two trials are more consistent with the accepted value for the speed of light. We also determined that the changes we made in our method for these last two trials better took into account the changing intensity of the signal at different distances of the LED fromm the PMT. 

The measurements we obtained for the first three trial are far from the accepted value. The standard error in our measurements can not account for this difference because our measurements are six to seven standard diviations away from the accepted value. This guarantees that there is some systematic error in our measurements for the first three trials. This is why we chose to rotate the PMT in the cardboard tube to adjust the intensity of the incoming signal and hold it constant throughout the experiment. We determined that a possible way to hold the intensity constant was to adjust the PMT to keep the first downward spike in the signal at the same vertical position on the oscilloscope screen. For the last trial we tried to get even better results because the first spike in the signal becomes several small spikes as the distance is lessened. For this reason we tried to to an average by holding the middle of the spike in the same place.

Conclusions
It is appearent from the large difference in our measurements of the speed of light to the accepted value that there must have been some systematic error. We hoped that removing the TAC from the experiment would eliminate the time walk from our measurements. However, it seems that this is not the case. Our most accurate measurement is several standard diviations away from the accepted value, and looking at the plots one can see that our measurements are self consistent. The standard diviations we calculated are not large enough to account for the difference in our measurements. Some of the difference can be due to our need to place the second cursor by eye, but since the errors in our measurements are not very large I do not think this played a significant roll in our calculations. We also did not take into account the changing intensity of the light as the LED is moved in our first three trials. In our last two trial we tried to adjust the PMT to keep the intensity constant, and our measurements did become more accurate. Even though they were better they still were several standard diviations away from the accepted value of the speed of light. When we did the lab the first time with the signal going through the TAC we obtained more accurate measurements. Therefore, I believe that either we obtained more time walk in our measurements or there is some other form of systematic error in our experiment. Below is the link to my previous speed of light lab. Speed of Light Lab



Acknowledgments
I need to thank Tom Mahony for the general format of the formal report and for references. I must thank Emran for being my lab partner. 