Physics307L:People/Barron/Final rough

Speed of Light from a Cardboard Tube

Alexander T. J. Barron

Experiment conducted with Justin Muehlmeyer

Junior Lab, Department of Physics & Astronomy, University of New Mexico

Abstract

I measure the speed of light in air utilizing a time-to-amplitude converter (TAC), a photo-multiplier tube (PMT), and an LED pulse-generator in a light-tight environment. By positioning the pulse-generator at varying distances from the PMT/TAC apparatus, one can obtain various data sets corresponding to the change in distance between the two components. For each change in distance, the TAC manifests a new amplitude, corresponding to change in time readings. Plotting change in distance vs. change in time yields the perfect environment for taking a least-squares linear fit of the data, of which the slope is the speed of light in air. In the process of finding c_air, I use different data-taking strategies as well as investigate a phenomenon known as "time walk," which without correction nullifies any useful data taking from this equipment set.

Introduction

Even after the Michelson-Morley experiment in 1887 gave reasonable doubt as the existence of the aether, certain scientists argued into the early 20th century against case for (what we now call) relativity, based on statistical uncertainty [1, 2]. Least-squares analysis specifically was targeted as covering up true values in its effort to smooth out errors, thereby burying results leading to aether-positive results. Through Einstein's theory of special relativity, and corroborating evidence, we now know almost irrevocably that the aether is not a factor in measurements of the speed of light, so good experiments involving least-squares analysis can be pursued with impunity.

All experiments measuring the speed of light involve taking measurements of time taken for light to traverse a given path[3]. The most common method up to 1944 was to partition light "packets" with periods of zero luminosity, thereby creating specific boundary times with which to measure between[3]. We follow this approach with more modern tools.

Methods and Materials

In order to measure the time-of-flight (TOF) of light packets, we position a moveable LED-pulse generator inside a several meter-long cardboard tube. The generator fills the entire cross-sectional area of the tube, ensuring light-tight conditions. On the opposite end of the tube, we position a fixed PMT. On the inner side of each device is a polarizer, used to maintain near-constant intensity of light received by the PMT. Constant intensity is needed to minimize the effects of "time-walk," addressed under Sources of Error below. The generator and PMT are both connected to the TAC, which measures the difference in time between the generation of the pulse and the receipt of the pulse by the PMT. The PMT is also connected to a digital oscilloscope through a second anode connection. We read the voltage output of the TAC via the oscilloscope along with the reading from the PMT.

With this setup, we can take data over a range of varying distance parameters, denoted in the following four trials. I denote change in distance as Δx:

i) large and increasing individual Δx over large total Δx,

ii) small, constant individual Δx over small total Δx,

iii) large, constant individual Δx over large total Δx, and

iv) medium Δx with no time walk correction.

Sources of Error

Time walk is the principle source of systematic error in this experiment. It occurs due to the TAC's triggering off signals from the PMT via a fixed voltage threshold. If the signal from the PMT is small, the TAC will trigger later than it would for a larger signal. In order to combat this, we use rotatable polarizers in tandem to try and maintain constant intensity of light received by the PMT. The higher the intensity, the more photons interact with the PMT and a stronger signal results. The same is true in reverse.

We read the voltage amplitude from the TAC visually using cursors on the digital multimeter, so our measurements were not very accurate. We were aided by the averaging tool provided by the multimeter, which helped matters quite a bit.

Our device for measuring distance was a standard meterstick, whose accuracy compared to the standardized meter is not documented.

Data

NOTE: ALL ΔX VALUES MEASURED FROM THE ENDPOINT OF THE PREVIOUS MEASUREMENT

Trial 1 Trial 2 Trial 3 Trial 4 (no time walk adjustment) Vref = .996 V ΔX [cm] Vmeas [V] initial 2.62 15 2.54 20 2.40 30 2.06 40 1.82 49 1.66 Vref = .456 V ΔX [cm] Vmeas [V] initial 7.76 5 7.74 5 7.56 5 7.42 5 7.52 5 7.40 5 7.32 5 7.24 5 7.50 5 7.18 5 7.26 Vref = .444 V ΔX [cm] Vmeas [V] initial 7.80 25 7.78 25 7.42 25 7.38 25 7.30 25 7.50 25 7.24 Vref = .456 V ΔX [cm] Vmeas [V] initial 6.48 10 6.30 10 5.84 10 5.32 10 4.74 10 4.50

All voltages measured with ± .02 V measurement error.

Results

NOTE: AXES ARE SET TO "TIGHT," SO SOME DATA POINTS ARE ON GRAPH EDGES

Trial	Graphic Representation	Trial	Graphic Representation
Trial 1: c = (2.91 ± 0.24) 108 m/s Upper Error Bound: cup = (3.16) 10 8 m/s Lower Error Bound: clow = (2.67) 10 8 m/s		Trial 2: c = (1.42 ± 0.08) 108 m/s Upper Error Bound: cup = (1.50) 10 8 m/s Lower Error Bound: clow = (1.34) 10 8 m/s
Trial 3: c = (3.48 ± 0.66) 108 m/s Upper Error Bound: cup = (4.14) 10 8 m/s Lower Error Bound: clow = (2.81) 10 8 m/s		Trial 4: (no time walk correction) c = (4.20 ± 0.18) 107 m/s Upper Error Bound: cup = (4.38) 10 7 m/s Lower Error Bound: clow = (4.02) 10 7 m/s

I notice that error range decreases with more measurements, but not necessarily accuracy. Here is a plot of all values with error compared to the accepted speed of light in air:

It appears that measurements taken over a large individual & total Δx, as in trials 1 & 3, yield the best results for c. Unfortunately, this experimental setup limits how much data can be taken this way, so the error is large. Small individual and total Δx in trial 2 yields an awful result, even though more data points narrowed the error range. The result of trial 4 illustrates how important adjusting for time walk is - c "walked" an entire order of magnitude! I wonder if taking data with small individual Δx over large total Δx would allow for the linear fit to filter out the "noise" from each small measurement in order to find the real trend of c. I believe the large amount of noise, from small x-stepping, combined with the small data range forced the trial 2 result far from its actual value.

This experiment and result illustrates the mechanics of accuracy of precision rather well. Trial 2 is not accurate at all, but is much more precise than our more accurate measurements. I believe the lesson to take away from this is that narrowing error isn't the entire battle - what good does small error do when the physical value isn't inside error bounds?

Upon comparison with the accepted value for the speed of light in air, this experiment seems fairly sound. Improvements in measurement of voltage amplitude from the TAC would be a relatively simple improvement to the process, and would increase the precision of the final result.

Conclusion

This latest iteration of the classic time-of-flight measurement of the speed of light holds up fairly well to its predecessors[3]. Unfortunately, there is no discussion as to how many significant figures I can report truthfully in the final values. This process serves beginning experimentalists well in its simplicity and high potential for good results, as long as care is taken in the measurements.

References

1. Heyl, Paul R. "The Application of the Method of Least Squares." Science, New Series, Vol. 33, No. 859 (Jun. 16, 1911), pp. 932-933. American Association for the Advancement of Science. JSTOR

   Heyl argues that the method of least-squares to average out error may be flawed, specifically in the context of extended Michelson-Morley-like experiments. He proposes a "mathematical theorem," providing a rule of thumb regarding acceptable least-squares error analysis.

2. Freedman, Hugh D.;, Roger A.; Ford, and A. Lewis Young. Sears and Zemansky's University Physics: With Modern Physics. San Francisco: Benjamin-Cummings Pub Co, 2004.

3. Dorsey, N. Ernest. "The Velocity of Light." Transactions of the American Philosophical Society, New Series, Vol. 34, No. 1 (Oct., 1944), pp. 1-110. American Philosophical Society. JSTOR

   Dorsey covers tomes of material in this paper, including a nicely-put summary of error analysis and the least-squares approach. He analyzes a number of historical experiments measuring the speed of light and reviews their accuracy based on procedure, equipment, and effectiveness of error documentation. There doesn't seem to be any mention of confidence intervals in a standardized way, but rather each uncertainty is reported based on logical arguments and various extremum.