User:Garrett E. McMath/Notebook/Junior Lab/2008/11/10

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=Speed of Light= The speed of light is an important fundamental constant in nature. The constant forms the framework of special relativity, and plays a huge role in electrodynamics. Proper measurement of the constant is clearly an important goal in physics, and so in this lab we will attempt to do so.

In this lab, we will be emitting light from a light emitting diode (LED), and then absorbing this light shortly after with a photomultiplier tube (PMT). Both of these components will be placed in a cardboard tube so as to remove the possibility of errors due to ambient light. To measure the speed of light, we will have to measure the time of photon emission and also the time of photon absorption. Both tasks will be accomplished with the help of an oscilloscope, and a time amplitude converter (TAC). In general, a time amplitude converter converts times to voltages, with a conversion factor of your choice. The time which we will be converting to a voltage will be the time delay between emission and reception. The output voltage will be intercepted by an oscilloscope.

Equipment

 * Tektronix TDS 1002 Digital Oscilloscope
 * Bertan Associates Inc. No.75 Model 315 DC Power Supply

Procedure

 * voltage = 180
 * We will start off by moving the LED a known distance from the PMT. Each successive datum will be acquired after moving the LED a fixed distance (10cm as we decided).
 * To avoid 'time walk', the polaroid on the PMT will be adjusted for each measurement. Because we cannot take the PMT out of the tube, we will just rotate it, thereby rotating the polarizer. For this to have any effect, of course, we will have a second polaroid in front of the LED. If we didn't do this, light intensity could not be controlled, as the PMT would receive half intensity no matter what rotation it has undergone.
 * The voltage amplitude will be measured on the oscilloscope, which will be receiving data simultaneously from the PMT and TAC in separate channels. We will have the oscilloscope average over time, so that we can get a steady readable signal.

Day 1
Dx measured relative to the point where the diode and PMT are touching. Min on channel 1 to max on channel 2

Conversion factor for the TAC: V=GT, G=1/10 Volts per nanosecond

Channel 1: Min -618±8mV

Dx=60cm Max=1.64 Time=16.4ns

Dx=70cm Max=1.66 Time=16.6ns

Dx=90cm Max=1.72 Time=17.2ns

Dx=100cm Max=1.75 Time=17.5ns

Dx=110cm

Dx=120cm

Dx=130cm

Trial 1 Day 2
Channel 1 Min: -488mV

Dx=.60m V=2.35V

Dx=.70m V=2.60

Dx=.80 V=2.68

Dx=.90 V=2.82

Dx=1.00 V=2.76

Dx=1.10 V=2.74

Dx=1.20 V=2.84

Dx=1.30 V=2.92

Dx=1.40 V=2.98

Dx=1.50 V=2.92

Dx=1.60 V=2.96

Dx=1.70 V=3.04

Dx=1.80 V=3.08

Dx=1.90 V=3.14

Dx=2.00 V=3.18

Dx=2.10 V=3.18

Dx=2.20 V=3.24

Dx=2.30 V=3.26

Dx=2.40 V=3.30

Trial 2 Day 2
Dx=2.40 V=3.30

Dx=2.30 V=3.26

Dx=2.20 V=3.22

Dx=2.10 V=3.16

Dx=2.00 V=3.06

Dx=1.90 V=3.12

Dx=1.80 V=3.06

Dx=1.70 V=2.94

Dx=1.60 V=3.00

Dx=1.50 V=2.94

Dx=1.40 V=2.84

Dx=1.30 V=2.80

Dx=1.20 V=2.72

Dx=1.10 V=2.80

Dx=1.00 V=2.78

Dx=.90 V=2.72

Dx=.80 V=2.72

Dx=.70 V=2.62

Dx=.60 V=2.64

Trial 3 Day 2
Moving LED away from PMT 10ns Time Delay In effect Channel 1 Min: -488mV Points marked with asterisk were obtained without run-stop (first 3 points).

Dx=.60m V=3.80*

Dx=.70m V=3.90*

Dx=.80 V=3.85*

Dx=.90 V=3.90

Dx=1.00 V=3.92

Dx=1.10 V=3.96

Dx=1.20 V=3.98

Dx=1.30 V=4.06

Dx=1.40 V=4.06

Dx=1.50 V=4.14

Dx=1.60 V=4.10

Dx=1.70 V=4.12

Dx=1.80 V=4.18

Dx=1.90 V=4.22

Dx=2.00 V=4.32

Dx=2.10 V=4.28

Dx=2.20 V=4.32

Dx=2.30 V=4.36

Dx=2.40 V=4.42

Possible Sources of Error

 * Oscilloscope: Even when we had the oscilloscope averageing 128 times the reading were still incredibly jumpy. The only correction we could do for this was to take a lot of readings as it should be a random error corrected by a large sample of data.
 * Time Walk: A possible source of error could come from the 'time walk' effect. This refers to the time discrepancy that arrises due to the expansion of some signal which is being triggered at a fixed threshold. This arises here because as the PMT to LED distance is varied, the intensity of light is changed. Therefore, the signal is larger when the PMT and LED are closer than when they are further. To try to counteract this effect, we varied polaroids to make the reception intensity roughly constant. Although we tried to minimize the possibility of this error by rotating the PMT (and thus polarizer), it is not unreasonable to think that we had not done it perfectly each time.
 * Cable Length Clearly the cables which transmit signals to the TAC and the Oscilloscope have a finite transmission rate. My guess is that this rate is somewhere close to the speed of light. Given that we are trying to measure the speed of light, it is conceivable that there is some possibility of error, especially since the cable from the PMT to the TAC is shorter than the cable from the LED to the TAC.

Post Experimental Data Analysis
Analysis performed in Excel, please open attached file to view data analysis
 * [[Image:Speedoflight03.xls |thumb|right|Figure 1a]]

Error Propagation
We had uncertainty in our measurement of the voltage, due to reading fluctuations in the oscilloscope. Because we converted this voltage to a time reading, we must propagate this error. Because the relationship between conversion is completely linear, propagation will be completely straight forward. The conversion ratio is given by the letter eta.

$$ t=\eta V \Rightarrow \delta t = \eta \delta V $$

Least Squares Line
The most convenient way to determine the speed of light from the data obtained is to make a least squares line of the traveled distances versus the time elapsed. The speed of light will simply be the slope of the line, which can be easily obtained:

$$ v(t)=c t \Rightarrow Slope = c $$

While the above relationship suggests that the line must travel through the origin, this is in fact not the case. The reason that we must not pass through the origin here is that we will be dealing with changes in distances. Therefore, the data points will be plotted in arbitrary space where the origin has no physical analog.

Analysis
After doing several quick checks of the data it was clear that similar to our predictions the last set of data taken with the stop function on the oscilloscope was the best. The data anaylsis was performed in Excel.


 * Speed of light measured from slope:2.985124E8 m/s
 * Uncertainty:.114343435E8 m/s
 * Accepted value for speed of light:2.99792458E8 m/s
 * Percentage error:.4426981%

Summary
Lab Summary


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