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# Determining the Speed of Light by Applying the Fizeau-Foucault Apparatus

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## 1. Introduction

As a fundamental constant, the exact value of the speed of light has been disputed for centuries. The measurement of the speed of light brought arise to new areas of Physics such as Einstein’s Theory of Relativity. This Theory of Relativity was based entirely on the relativistic effects caused by objects moving near the speed of light. This brought arise to an exclusively new area of Physics, furthermore it is no wonder why the velocity of light in free space can now be cited to 8 significant figures at 299,792,458 metres per second . One of the most accurate laboratory experiments to determine the constant is achieved using the Fizeau-Foucault apparatus. This consists of measuring the phase difference, of a wave of known displacement. A graph of displacement against phase difference can produce a gradient that is inversely proportional to the speed of light, provided remains constant.

## 2. Method

It was critical in this experiment that the laser was precisely aligned with the collimating lens and mirrors, of which were placed parallel and level to one another on a 2 m optical rail. This enabled the laser to be focused onto the mirror by varying the length between the focusing lens and laser until it was at the focal length, producing a small sharp dot onto a screen at the far-end of the rail. The phase difference of the wave was altered by varying the displacement of the reflecting mirror and using a semi-reflective lens to reflect the laser into an avalanche photodiode module detector (APD). This allowed a time delay to be measured as a function of the mirror displacement.

A signal generator produced a sinusoidal wave from the laser onto the oscilloscope of frequency 50 MHz. The variation of displacement created by moving the reflective mirror created a distinct change in phase which could be observed due to the wave generated on the oscilloscope from the APD. The time delay was recorded off the oscilloscope as the displacement varied between 0 m and 1.75 m in increments of 0.25 m. Using this data, the phase difference was calculated as it was known that the period of the wave was 20 ns.

With the above data collected, a graph using coding on Python was plotted of displacement against phase difference, whilst also producing the gradient as using the coding function. The gradient of the line of best fit allowed the speed of the laser light wave to be determined through calculation.

## 3. Results

The direct correlation between the displacement of the plane mirror and the phase difference of the receiving wave demonstrates a linear relation. We showed this using a linear trend line inferring the proportionality and furthermore making it consistent within their error ranges, as shown in the residual graph. The gradient of the line of best fit on the graph signifies 2 multiplied by the frequency of the light source divided by the speed of light. The residual graph shows the random distribution of error in the measurement of variables. Further details on error analysis can be found in the appendix of this report.

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## 4. Discussion

The generated linear relation can be accurately supported by the data collected due to the demonstration. Through analysis the gradient can be calculated to be 1.57± 0.356. By equating this value to the gradient, the speed of light in free space can be evaluated to be (2.00±0.000290) x108 ms-1. The value of c found from this experiment does not equate or lie in the accepted error range for the value accepted of the constant, 3×108. The percentage error for the value measured was can be calculated as 67% , as seen in the error analysis section in the appendix of this report. Multiple sources of error could have resulted in the measured value being altered by a factor of a third.

Experimental methods are a large factor of errors associated with this experiment, given the fragility and the required accuracy of the equipment used. A dominant element of the error was introduced due to the length between the laser and the collimating lens changing throughout the experiment. Initially, this issue was attempted to be averted by focusing the lens onto a screen whilst ensuring the focused dot on the screen remained the same size as the screen was moved along the optical rail. However, once the focal length was found via this method it was challenging to keep this distance the same whilst alternating the displacement of the reflecting mirror despite taking precautions of fastening the mirror and lens onto optical posts. If this experiment were to be repeated this error could be avoided by routinely checking that the laser is focused through measuring the focal length of the lens.

An additional source of error due to experimental methods could have been disregarded by collecting more specific additional data points. By measuring the phase difference of the light source from greater distances with smaller increments between data points, such as 0.1 m, it would have permitted a more substantial graph plot, hence giving a gradient with lower uncertainty and a more accurate value of c. The line of best fit does not pass through the origin due to the configuration of the experiment. It is as a result of an extra distance between the mounted collimating lens and the optical rail. Due to this distance being kept constant, it did not affect the results of the experiment however would’ve produced a translated graph. Additionally, if repeat readings of each displacement where to be measured it would allow a mean of each value to be calculated, ultimately producing a more accurate result to be plotted.

## 5. Conclusions

The constant value of the speed of light is measured as 3×108 and due to several experimental errors, the measured value in this experiment produced a value that was out by a factor of 1/3, at 2×108. An alternative method of graphically evaluating the speed of light is to plot the time delay against displacement. This would result in the speed of light being the reciprocal of the gradient.

### References

1. Units and Fundamental Constants in Physics and Chemistry: Subvolume b: Fundamental Constants in Physics and Chemistry. C. Weiss and Y. Ni. J. Bortfeldt and B. Kramer, eds., vol. Subvolume b. New York: Springer-Verlag, Ch. 3.2.2, pp. 37-51 (1992)
2. The Philosophical Significance of the One-Way Speed of Light. Vol. 11, No. 3, Symposium on Space and Time (Sep., 1977), pp. 253-292
3. Measurements and their Uncertainties, Oxford University Press: I. G. Hughes and T . P . A. Hase, Measurements and Oxford (2010).

### Appendix

Each measured variable in this experiment all contained a degree of uncertainty. The best estimate of the standard error of the final value for c was determined by taking the standard error of the mean. When obtaining the values of varied displacement, 8 measurements were recorded. These measurements were read of a meter rule which has an uncertainty of half the smallest division of the scale, 0.5 mm. The uncertainty in reading off the oscilloscope can be assumed as half a subdivision of the setting. Therefore, due the specific division range used; the error can be calculated as ± 0.1V.

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