Physics307L:People/Gibson/Formal Lab Report 2

MEASURING THE SPEED OF LIGHT BY A TIME DELAYED SIGNAL BETWEEN A LIGHT EMITTING DIODE AND PHOTOMULTIPLIER TUBE

Author: Zane Gibson

Experimentalists: Zane Gibson Matthew Gooden,

Department of Physics and Astronomy

University of New Mexico

Albuquerque, NM

zgibson@unm.edu

Abstract

This experiment was done to determine the value of [math]\displaystyle{ c }[/math] or the value of the speed of light. To complete this experiment, a time amplitude converter (TAC) and an oscilloscope were used to measure the time between a light signal from an light emitting diode (LED) and a photo multiplier tube (PMT). By changing the distance between the PMT and the LED, we were able to record various voltages for various distances, then we converted the voltage to a time and constructed a trend line of our distances and times using a least squares fit analysis. The slope of this trend line was the constant we were originally looking for, the speed of light. For this experiment we found our constant to be [math]\displaystyle{ c=\left(2.892 \pm .077\right)\times10^8 m/s }[/math] which is very close to the theoretical approximation of [math]\displaystyle{ 2.99\times10^8 {m/s} }[/math].

Introduction

The speed of light has been used in many branches of physics in many different ways. As such, this value has been disputed and many experiments have been performed to find this value. The first accurate way to find the value was done around 1960 where lasers first started to become used widely in optical physics. It wasn't until 1972, when the National Institute of Standards and Technology employed the developing laser technology to measure the speed of light at 299,792,458 meters per second [1]. Since then, the speed of light has been used in a variety of applications from optical physics to biomedical physics, more specifically using light to shrink cancer cells [2]. In this experiment we consider how close we can approach the value of [math]\displaystyle{ c\lt math\gt , while minimizing our error and uncertainty in our measurements. ==Methods and Materials== '''Equipment List''' ---- 1.Time-Amplitude Converter (TAC) - EG&G Ortec Model 567 TAC/SCA 2.Light Emitting Diode 3.Photomultiplier Tube (PMT) - Magnetic Shield Co., 22P50 4.DC Power supply (LED) - Harrison Laboratories, Model 6207A 0-160V,0-.2A 5.DC Power supply (PMT)- Bertan Associates,Inc. Model 315, DC Power Supply 0-5000V,0-5mA 6.Oscilloscope - Tektronix TDS 1002 2 Channel Digital Storage Oscill., 60MHz 1GS/s 7.BNC Connector Cables - 6 8.Delay box - Canberra NSEC Delay Box, Model 2058 9.Several Meter Sticks - 4 10.Long Tube (cardboard or other nontransparent material) - 4 meters long \lt gallery\gt Image:Image003.jpg|'''Photo 1''':Center is a storage rack for several pieces of equip. From left: power supply for photomultiplier-Item 5, delay box-Item 7, TAC (time amplitude converter)-Item 1 Image:Image004.jpg|'''Photo 2''':Small gray box center of the photo is the Power supply used for LED light emitter. Item 4 from equip. list Image:Image005.jpg|'''Photo 3''':Oscilloscope. Item 6. Image:Image006.jpg|'''Photo 4''':On top tried different time delay to try and get better signal. \lt /gallery\gt '''Procedure''' ---- '''Set up''' - To begin the experiment we first collected all our equipment, then proceeded to first get power to the LED. We then did the same for the PMT (NOTE: Do not expose the PMT to massive quantities of light while operating, this can destroy the PMT) then placed them in the long cardboard tube. We then bolted the LED to one of the ends of a meter stick (so we could vary the distance) and taped the sticks end to end to allow us plenty of variable distances. Using a connector cable, we connected the PMT signal output into channel 1 of the oscilloscope and plugged the response from the LED (when it sent a signal) to the TAC. We then ran a cable from the TAC to the time delay box, and then to channel 2 of the oscilloscope. The distance of how far in or out the LED was is arbitrary. Taking whatever place as the zero works fine and recording the differences in the distances is what is important. '''Collection and Analyzing'''- ==Results and Discussion== Below are the 4 data sets we took and a discussion of each set. 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: '''Table 1''' {| border="1" style="text-align:center" !'''Data Set 1 Measurments''' !'''PMT Ch1 Voltage''' !'''TAC Time delay voltage''' !'''Distance (cm)''' !'''Time (ns)''' |- |1 |\lt math\gt 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.24\pm.04 }[/math] |60 |32.4 |- |2 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.2 \pm.04 }[/math] |80 |32 |- |3 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.08 \pm.04 }[/math] |100 |30.8 |- |4 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.00 \pm.04 }[/math] |120 |30 |- |5 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 2.96\pm.04 }[/math] |140 |29.6 |- |6 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.00 \pm.04 }[/math] |130 |30 |- |7 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.12 \pm.04 }[/math] |90 |31.2 |- |8 |[math]\displaystyle{ 600 \pm 8 {mV} }[/math] |[math]\displaystyle{ 3.2 \pm.04 }[/math] |70 |32 |}

Table 1 shows

Table 2

Data Set 2 Measurments	PMT Ch1 Voltage	TAC Time delay voltage	Distance (cm)	Time (ns)
1	[math]\displaystyle{ 800 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.44\pm.04 }[/math]	70	24.4
2	[math]\displaystyle{ 800 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.4 \pm.04 }[/math]	80	24
3	[math]\displaystyle{ 800 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.36 \pm.04 }[/math]	90	23.6
4	[math]\displaystyle{ 800 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.32 \pm.04 }[/math]	100	23.2
5	[math]\displaystyle{ 800 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.32\pm.04 }[/math]	110	23.2

Table 3

Data Set 3 Measurments	PMT Ch1 Voltage	TAC Time delay voltage	Distance (cm)	Time (ns)
1	[math]\displaystyle{ 720 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.8\pm.04 }[/math]	27.5	28
2	[math]\displaystyle{ 720 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.64 \pm.04 }[/math]	70	26.4
3	[math]\displaystyle{ 720 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.52 \pm.04 }[/math]	110	25.2
4	[math]\displaystyle{ 720 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.44 \pm.04 }[/math]	130	24.4
5	[math]\displaystyle{ 720 \pm 8 {mV} }[/math]	[math]\displaystyle{ 2.72\pm.04 }[/math]	50	27.2

• 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.

These are plot of data sets 1-3 along with least-squares fit lines

• The slopes for each data set are listed below:

Data Set 1

          [math]\displaystyle{ c=\left(2.68  \pm 0.18\right)\times10^{8} m/s }[/math]

Data Set 2

          [math]\displaystyle{ c=\left(2.94  \pm 0.42\right)\times10^{8} m/s }[/math]

Data Set 3

          [math]\displaystyle{ c=\left(2.89  \pm 0.08\right)\times10^{8} m/s }[/math]

Average of all three data sets:

          [math]\displaystyle{ c=\left(2.83 \pm .23\right)\times10^8 m/s }[/math]

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

[math]\displaystyle{ Relative Error=\frac{\left|2.83\times10^{8}-2.99\times^{8}\right|}{\left|2.99\times10^{8}\right|}=0.0535=5.3% }[/math]

Possible sources of uncertainties have to do with the equipment. We ran the same experiment several times over the same values to see how close our values would be to one another after several repeated measurements. We found quite a discrepancy and we've concluded it has to do with some sort of systematic error which we are unable to account for at this time.

Conclusions

Thus our closest approach to the theoretical value of the speed of light is [math]\displaystyle{ c=\left(2.892 \pm .077\right)\times10^8 m/s }[/math]. This is a good value to receive especially with our unfamiliarity with most of the equipment. Additional error analysis shows our reported value we find having a relative error given by [math]\displaystyle{ RE=\frac{\left|2.99\times10^{8}-c\right|}{\left|2.99\times10^{8}\right|}=0.0328=3.3% }[/math] which is again very good considering the time to take measurements was somewhat constrained. With data set 4 we attempted to improve on this error by taking more measurements and possibly shrinking this value. The ideas used are explained next to the table above but as can be seen from the data set none of these things were able to improve the results, and actually left us with a value of c that is farther off than either of data sets 2 or 3. The data given above for set 4 is the first and more accurate attempt made for that set in comparison to the other trials. It must be concluded that there is some form of systematic error in the experiment that is not being accounted for and at this time with our little knowledge of the inner workings of the electronics we cannot specifically pin point the problem. Our most accurate and reported value is [math]\displaystyle{ c=\left(2.892 \pm .077\right)\times10^8 m/s }[/math].

Acknowledgments/References

1. [Measuring the Speed of Light] as done by NIST.

2. [Cancer] article and using light to shrink cells

3. [Previous Course lab manual] from Dr. Gold