User:Justin Roth Muehlmeyer/Notebook/307L Notebook/Speed of Light

Instrumentation
PMT: Perfection Mica Company N-134

Powering Up
Beginnng by powering the PMT at 1900 V. Power to LED: 190 V

At first we were unable to see any signal in the oscilloscope directly from either the PMT or the TAC. Then we moved our LED farther into the tube. We received our first signal peaking at about 1 V from the PMT.

We wanted to make sure that this signal was in fact due to the electric field of the light coming from the LED. And, in order to attain the best amplitude we rotated the PMT until its polarizer let in the greatest intensity.

After some adjustments were able to see a signal on the oscilloscope as such:

Directly from the PMT: 500-800 mV From the TAC: a square wave slightly delayed from the PMT signal at about 6 V

Data
Oscilloscope:With ch 1 reading at 50mV and ch 2 reading at 500 mV with 250 ns divisions...

Data 1: Ch1 signal voltage: -192mV

Ch2 voltage: 4.92+- .08 V

Instrumentation
PMT: Perfection Mica Company N-134

Powering Up
Beginnng by powering the PMT at 1900 V. Power to LED: 190 V

At first we were unable to see any signal in the oscilloscope directly from either the PMT or the TAC. Then we moved our LED farther into the tube. We received our first signal peaking at about 1 V from the PMT.

We wanted to make sure that this signal was in fact due to the electric field of the light coming from the LED. And, in order to attain the best amplitude we rotated the PMT until its polarizer let in the greatest intensity.

After some adjustments were able to see a signal on the oscilloscope as such:

Directly from the PMT: 500-800 mV From the TAC: a square wave slightly delayed from the PMT signal at about 6 V

Calibration Data
Since the TAC is converting time differences of signals to a voltage, in order for us to know the time difference we need to know the time to amplitude ration that the TAC is outputing. Prof. Gold's manual stresses the need for us to find this ration experimentally, but in our discussion with Dr. Koch, we decided that it is better to trust the TAC manual for this ratio. We took the data anyways, which can be seen below. However, we will use the value displayed on the TAC settings:

10 V = 50 nS

Here is the calibration data that we took but will not use. We made sure to acccount for the time walk by keeping the reference voltage on channel 1 as constant as possible when we moved the LED in.

Data
Notes on Data Taking Method:

The data below was taken with consideration of the time walk. One can see the reference voltage of channel 1 for each trial in the charts below. We did our best to make sure this voltage was constant throughout the trial to account for the changes of light intensity from the LED as it was pushed in. This value was kept constant by rotating the PMT by slight amounts which rotated the polarizer covering its "lens". The polarizer "filters" or "cuts" the incoming light's electric field based on the angle that the polarizer film molecules are aligned. There is a point (or angle rather) when the polarizer cuts the incoming light entirely, and no voltage can be seen from he TAC at all. This means the alignment of the polarizer with the incoming electric field from the LED is perindicular.

We used two methods to read the data on the oscilloscope. The raw signal from the TAC when the oscilloscope is in "sample" mode is very turbulent and quite impossible to read. We first used the cursors to eye an average, then with the second cursor we eyed the deviations from this average. This was time consuming, and somewhat frustrating due to the extreme variations of the signal. Then we remembered that the oscilloscope can do the averaging for us.

Setting the oscilloscope data acquisition mode to "average", the oscilloscope does all the work for us. Over a period of time it takes an average and displays it. Therefore the signal we see is actually delayed from the actual input but is a better representation. So we decided to take all of our data using the "measure" function on the oscilloscope when the data acqusition mode was on "average". This of course is an easier way of doing it, and we took all of our data later using the measure function for efficiency sake. The measure function does not read the deviation however, so this value had to be eyed based on the fluctations of the signal from that average.

Note:
 * ΔX is measured from the endpoint of the previous data point. The ruler that the LED was attached to was a challenge to read due to the fact that it was two meter sticks taped together.

Analysis
From the simple fact that velocity is position divided by time, we can derive the velocity of the light if we have a position vs time plot of the light. We will do a linear least squares fit to our x versus t curve. The least squares line gives us a best fit line to our data based on the fact that the sum of of the squared "residuals" has its least value. A "residual" is the value difference between the actual data point, and the best fit line model. It gives us a trend line for which to model our data. We will use this line as our position vs. time line, whose slope will be our value for the speed of light.

I will use our time to amplitude ratio to determine the time delay between our start and stop signals. We will then plot position vs. time and find the slope of the best fit line.

I will do this in Excel, which will give me values quickly, but sadly does not make the most beautiful graphs. My excel file will be uploaded here:

The slopes of the linear least squares fit lines gave the following values for the speed of light:

Trial 1: Large Changes in Distance
 * c = (2.9371) 108 m/s

Trial 2: 0.05 m Change in Distance
 * c = (1.4202) 108 m/s

Trial 3: 0.25 m Change in Distance
 * c = (3.4750) 108 m/s

Trial 4: 0.10 m Change in Distance, No Time Walk Correction
 * c = (4.2009) 109 m/s

The accepted value of the speed of light is  (3) 108 m/s .

What I notice with my data is the importance of large changes in distance of the LED, and the necessity of the time walk correction. Trial 1 is obviously the closest value, and this trial was characterized by changes of distance that were increasingly large.

 Error  The margin of error in our data comes from the oscilloscope where we have taken our voltages to have an error of ± 0.02 V. Because of this we will have an upper and lower range of c values.

Trial 1: 
 * cup = (4.2302) 108 m/s
 * cdown = (1.5972)) 108 m/s

Trial 2: 
 * cup = (1.7924) 108 m/s
 * cdown = (1.0481)) 108 m/s

Trial 3: 
 * cup = (4.9498) 108 m/s
 * cdown = (2.0002)) 108 m/s

Trial 4: 0.10 
 * cup = (6.1817) 108 m/s
 * cdown = (2.2201)) 108 m/s