Physics307L F08:People/Smith/Test

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           pdfauthor={Jesse J. Smith},
           pdftitle={Electronically Measuring the Speed of Light},

%\usepackage[all]{hypcap} %\usepackage[margin=50pt,font=small,labelfont=bf,textfont=it,labelsep=endash,aboveskip=20pt]{caption} %\hypersetup{bookmarksdepth=3} \makeindex \title{Electronically Measuring the Speed of Light} \author{Jesse J. Smith} \experimentalists{Jesse J. Smith and Kyle Martin} \department{Physics and Astronomy Department} \university{University of New Mexico} \citystatezip{Albuquerque, NM 87131} \email{} \course{Physics 307L Junior Lab} \date{November 7, 2007}

\begin{document} \maketitle %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   \abstract{\index{Abstract}Attempting to measure the speed of light, we used electronic equipment to measure the time between the emission of a light signal by a light emitting diode and the detection of the light signal by a photomultiplier tube.  The slope of a line fit using the least-squares method of many measurements taken while varying the distance the light signal travels should approximate the speed of light.  Our measurement of $(3.063 \pm 0.1825)\times 10^8$ meters per second is in good agreement with the accepted value of $2.998\times 10^8$ meters per second.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \index{Introduction}Every form of electromagnetic radiation travels through a vacuum at the same speed, regardless of frequency or wavelength. In 1905, Albert Einstein proposed in his theory of special relativity that this speed was even constant regardless of the frame of the observer relative to the source, provided the reference frames are inertial. The speed of light in a vacuum is also the fastest ordinary objects with mass can travel. Thus, knowing this speed may reveal a great deal about the universe.

In 1983, the value of the meter was redefined to make the speed of light exactly 299,792,458 meters per second \citep{wiki:001}. Historically, however, the speed of light was one of the most studied - and measured - physical constants in science.

\subsection*{Background} \index{Background}Aristotle, an ancient Greek philosopher circa 350 BC, and Heron of Alexandria, an ancient Greek physicist and mathematician circa 60 AD, believed the speed of light to be infinite; that is, light reached its destination at the very instant it was emitted. Early attempts at measuring the speed of light, while not very accurate or precise, proved that it was finite.

Several methods of measuring the speed of light produced astoundingly accurate results in the latter half of the 19th century and the early 20th century. These methods involved measurements of the speed of light propagating through air; this speed is very close to the speed of light through a vacuum, as the refractive index (the ratio of the speed of light through a vacuum to the speed of light through a given medium) of air is 1.0003.

Hippolyte Fizeau's attempt in 1849 used a rotating, notched wheel and a mirror thousands of meters away from a light source. Light shone on the rotating wheel and struck the mirror only when the wheel's cogs were not blocking it. The mirror reflected the light back at the rotating wheel, and an observer near the light source would detect the reflected light only when the wheel did not block it on its second pass, which occurred only at specific speeds of rotation. The speed of light through air could then be calculated, given this speed, the number of teeth on the wheel and the distance between the light source, mirror and observer. Fizeau concluded the speed of light must be around 313,000 kilometers per second \citep{wiki:001}.

Several subsequent improvements boosted the accuracy and precision of this method. Leon Foucalt replaced the rotating wheel by a rotating mirror, and in 1862 published the results of his measurement: 298,000 kilometers per second. Albert A. Michelson devoted much of his career to measuring the speed of light to great precision; in 1926, he used a rotating prism and a mirror more than 20 miles from a light source to measure the speed of light to be 299,796 kilometers per second \citep{wiki:001}.

After World War II, Louis Essen and A.C. Gordon-Smith used a microwave cavity to measure the speed of light. Their conclusion of 299,792 $\pm$ 3 kilometers per second was refined to 299,792.5 $\pm$ 1 kilometers per second by 1950 \citep{wiki:001}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Methods and Materials} \index{Methods and Materials} \subsubsection*{Required equipment} \index{Required equipment} \begin{itemize}

   \item TAC (Time Amplitude Converter): Model 567 mfd. by EG\&G Ortec
   \item Delay Module: nSec Delay model 2058 mfd. by Canberra
   \item Digital Storage Oscilliscope (DSO): Tektronics TDS 1002 (Dual channel digital storage oscilloscope)
   \item LED Power Supply: Model 6207a mfd. by Harrison Industries (DC power supply, 0-200V, 0-0.2A)
   \item LED/capacitor module: looks hand made by Physics dept. Supposed to cycle on and off at ~10KHz, depending on voltage.
   \item PMT (Photomultiplier Tube): Labeled N-134, unknown manufacturer. Has a magnetic shielding tube attached to the front of it.
   \item PMT Power Supply: Model 315 mfd. by Bertan Associates, Inc. (DC power supply 0-5000V, 0-5mA)
   \item Long cardboard tube, about 15 centimeters in diameter and 5 meters long.
   \item 3 meter sticks taped together
   \item Various BNC wires
   \item 2 Polarizing filters 


\subsubsection*{Setup} \index{Setup} As described in \cite{gold_manual}: \begin{itemize}

   \item  The LED module is connected to its power supply and to the first input on the TAC. It also has the meter sticks taped to it. One of the polarizing filters is attached to the module in front of the end that emits light. The module is inserted into one end of a long cardboard tube, with the end that emits light aimed down the length of the tube.
   \item The PMT is connected to its power supply, to the input on the delay module and to channel 1 of the oscilloscope. It has the other polarizing filter placed in front of its collecting end. The PMT is inserted into the other end of the long cardboard tube, with the collecting end pointed at the LED module.
   \item The delay module's output is connected to the second input on the TAC.
   \item The TAC (which now has 2 inputs connected), has its output connected to channel 2 of the oscilloscope. 
   \item The PMT power supply is set to around 1900 volts DC, and the LED power supply is set to around 186 volts DC.


\subsubsection*{Procedure} \index{Procedure} \begin{itemize}

   \item  Turn the power supplies, TAC and DSO on. The LED module should be firing now, and the PMT should be registering a corresponding drop in potential for every pulse of incident light.
   \item The TAC will be triggered by the LED module pulsing. It will be triggered again by a dip in potential across the photomultiplier tube caused by incident photons striking the photocathode material on the end of the PMT and the resulting cascade of electrons moving towards the anode. The TAC then creates a potential across the two output leads which is proportional to the time between being triggered on and off. The oscilloscope measures this voltage.
   \item We must be careful of a very large source of systematic error: timewalk. Timewalk is an interesting phenomenon which is explained very well in the \cite{gold_manual}, but the essence is this: the TAC is triggered at a set voltage. This voltage will be reached sooner if the pulse being sent to the TAC is larger, and later if the pulse is smaller. The size of the pulse is proportional to the brightness of the incident light on the PMT, which is proportional to the distance between the LED source and the PMT detector. To control this effect, a reference voltage is taken from the PMT which indicates the brightness of the incident light. The polarizers in front of the source and emitter are turned as the distance changes in order to keep the brightness constant, indicated by a constant voltage reading.
   \item By varying the distance between the LED module and the phototube and taking voltage measurements, we can determine the speed of light. Plot the distance vs. time and take the slope of the line connecting these points to get a rough estimate. By finding the slope of a line fit using the least-squares method, we can get a better estimate. 



\section{Results and Discussion} \index{Results and discussion} \subsubsection*{Analysis} \index{Analysis} The speed of light is the slope of a line fit by the least squares method. The line is of the form $\displaystyle y = mx + b$, where m is the slope and b is the y-intercept.

The slope of this line is \mbox{$\displaystyle m = \frac{\sum{x^2}\sum{y}-\sum{x}\sum{xy}}{\Delta}$}, and the y-intercept of the line is \mbox{$\displaystyle b = \frac{N\sum{xy}-\sum{x}\sum{y}}{\Delta}$}, where \mbox{$\Delta = N\sum{x^2}-(\sum{x})^2$}.

The standard error of the slope is \mbox{$\displaystyle \sigma_m = \sigma_y \sqrt{\frac{N}{\Delta}}$} where \mbox{$\displaystyle \sigma_y=\sqrt{\frac{1}{N-2}\sum_{i=1}^N{(y_i-b-mx_i)^2}}$} and

The standard error of the y-intercept is \mbox{$\displaystyle \sigma_b = \sigma_y \sqrt{\frac{\sum{x^2}}{\Delta}} $} \citep{taylor:error}.

In analyzing the data (see \autoref{tab:data} in Addendum) from this experiment, the measured times are the x-values, and the distances are the y-values.

Thus, our most likely slope is $3.063 \times 10^8 $ meters per second, and our most likely y-intercept is $-6.60 $ meters. The standard error of the slope is $ 1.83 \times 10^7 $ meters per second, and the standard error of the y-intercept is $4.24 \times 10^{-1} $ meters.

The most likely slope line is produced by pairing the most likely slope and most likely y-intercept. The maximum slope line comes from pairing the maximum slope (the most likely slope plus the standard error of the slope) and the minimum y-intercept (the most likely y-intercept minus the standard error of the y-intercept), and the minimum slope line is the pairing of the minimum slope (the most likely slope minus the standard error of the slope) and maximum y-intercept (the most likely y-intercept plus the standard error of the y-intercept). \autoref{fig:slopesfigure} is a plot of the data, most likely slope line, and maximum and minimum slope lines.

\begin{figure}[t] \centering \mbox{\includegraphics[width=0.8\textwidth]{fig2.jpg}} \caption{\label{fig:slopesfigure} Plot of measured times vs. distance (in m)

      from meter stick reading of 0.7m.  The best, minimum and maximum slopes
      are also shown.}


\section*{Conclusions} \index{Conclusions} While our result of $(3.063 \pm 0.1825)\times 10^8$ meters per second is in good agreement with the accepted value of $2.998\times 10^8$ meters per second, there was a relative uncertainty of 5.96\%. If more measurements were to be taken, this relative uncertainty could shrink considerably.

Our value of 306,300 kilometers per second, however, is slightly high. The source of this systematic error is likely the equipment. The reference voltage of the photomultiplier tube was inconsistent and varied from the recorded value by $\pm 4$ millivolts to $\pm 8$ millivolts. The cause for this inconsistency is uncertain; perhaps the LED module was not firing with a consistent voltage (and hence had a variable intensity).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage\index{Bibliography} \bibliography{paper}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage \section{Addendum}\index{Addendum} \subsection*{Data}

\begin{table}[h] \def\baselinestretch{1.2}\large\normalsize \centering \begin{tabular}{|p{3cm}|p{3cm}||c|c||c|c|} \hline \multicolumn{6}{|c|}{Measured Voltages} \\ \hline \hline

  Meter Stick Reading (in cm)& Distance from Reading of 140cm (in m)& Voltage & Error & Time (n Sec) & Error (n Sec) \\

\hline \hline 40 & 1.0 & 4.96 & $\pm$ 0.02V & 24.80 & $\pm$ 0.10 \\ 50 & 0.9 & 4.95 & $\pm$ 0.02V & 24.75 & $\pm$ 0.10 \\ 60 & 0.8 & 4.82 & $\pm$ 0.02V & 24.10 & $\pm$ 0.10 \\ 70 & 0.7 & 4.74 & $\pm$ 0.02V & 23.70 & $\pm$ 0.10 \\ 80 & 0.6 & 4.70 & $\pm$ 0.02V & 23.50 & $\pm$ 0.10 \\ 90 & 0.5 & 4.60 & $\pm$ 0.02V & 23.00 & $\pm$ 0.10 \\ 100 & 0.4 & 4.54 &$\pm$ 0.02V & 22.70 & $\pm$ 0.10 \\ 110 & 0.3 & 4.50 &$\pm$ 0.02V & 22.50 & $\pm$ 0.10 \\ 120 & 0.2 & 4.42 &$\pm$ 0.02V & 22.10 & $\pm$ 0.10 \\ 130 & 0.1 & 4.37 &$\pm$ 0.03V & 21.85 & $\pm$ 0.15 \\ 140 & 0.0 & 4.40 &$\pm$ 0.02V & 22.00 & $\pm$ 0.10 \\\hline \end{tabular} \caption{ \label{tab:data}These are measurements taken from the Time-Amplitude Converter using the dual channel oscilloscope. The third column (``Voltage") was the output of the function ``min" for Channel 1 of the oscilloscope. The corresponding times in nanoseconds are the product of the voltage and 5, as the TAC was set to $\frac{1}{5}$ Volts per nanosecond. }