## A Measurement of the Speed of Light

Today, the speed of light is defined as 299,792,458 meters per second (so says Wikipedia).  So, is it still meaningful to measure it?  Of course it is, though one might argue that modern attempts to more precisely measuring the speed of light are really attempts at more precisely determining how long a meter is.
Be that as it may,  the speed of light is so fundamental to relativity, to physics, and to the world we live in, that I wanted to measure it for myself.  If one is willing to make some compromises, it's actually pretty easy.
One obvious way to do it is a flashgun and a pair of photodetectors.  Unfortunately I don't have appropriate photodetectors lying around the house, and I suspect that sorting out the signal from the noise might require, at least, doing the experiment in a completely dark room.  For an initial attempt at measuring c I'm looking for something simpler.
Actually, what I wanted to measure is the speed of an electromagnetic wave, and lower frequency waves are a lot easier to handle than light.  Radio-frequency waves can be sent using a pair of coils, which might be simpler than photodetectors.  Better yet, radio waves will follow a wire -- a waveguide -- and that makes the experiment almost trivial.
A wave following a wire "waveguide" actually travels through the medium surrounding the wire; it doesn't travel in the wire itself.  Hence, the name "waveguide" -- the wire guides the wave, it doesn't carry it.  Of course, there are currents in the wire, but the charge carriers in the wire move far, far slower than the speed of the wave.  The speed of the wave is equal to the speed of light in the medium which surrounds the wire.  So, for instance, a coaxial cable carries signals at the speed of light in the insulator which separates the wire in the middle of the cable from the sheath.  Similarly, the speed of a signal in ordinary 300 ohm television antenna cables (those flat things with two wires in them that people used to use before everybody had cable) is the speed of light in the plastic ribbon.  But what I wanted to measure is the speed of light in vacuum -- or in air, which is nearly the same thing.

So, the easy way to do this is to build a waveguide out of bare wire, and measure the signal velocity going through it.  That should be a close approximation to the speed of light in vacuum, and it can, in fact, be done with almost no equipment beyond an oscilloscope.

### Materials and Equipment

• An impulse generator (er ... really just 4 D cells, in a Radio Shack battery bracket; connections were made with clipleads)
• About 30 feet of baling wire, found in a kitchen drawer
• Some Styrofoam packing material, found in a box
• A 100 ohm resistor
• A digital oscilloscope, acquired second hand from Ebay

### The Experiment

 Figure 1 -- The experimental setup:
I cut two allegedly equal lengths of baling wire, and cut a number of pieces from the foam packing material.  I then pushed the baling wire through the pieces of foam in an effort to make a sort of trestle on which the wires would be held in the air, away from all solid surfaces.  The reason for choosing Styrofoam is that it's mostly gas to start with, so its refractive index is probably very low, and the signals should travel through the Styrofoam nearly as fast as they travel through air.  At least, this is the theory!

In practice, this construction project worked terribly.  The foam I was using was low-grade expand-in-place stuff, used by a low-budget shipping outfit; it wasn't the dense, hard molded foam that new computer equipment comes packed in.  When I tried to cut blocks from it, it mostly crumbled (bits of foam everywhere!) and when I pushed the wires through it, it crumbled some more, and the wires often tore out of it.  Major hassle.  When all was said and done, the wires were in a series of arcs, held up by thin, crumbly foam pads at each point where they reached the floor (figure 1).  Everything was a lot closer to the floor than planned, and the wires, despite being fairly soft metal, refused to cooperate and ended up with a lot of wiggly bends and very, very irregular spacing between the signal and ground wires.

But at last I had my "transmission line" laid out in the form of a rough oval on the basement floor, with a gap of perhaps 4 feet between the two ends.   The wires themselves, which I thought were of equal length, turned out not to be, as I learned later.  I measured them one last time, with more care, after taking the whole thing apart at the end of the experiment, and one wire was about 14' 10" long, while the other was actually about 15" 1" long.  Oops!  (This is discussed further under Sources of Error, below.)

Preliminary tests had tipped me off that I was going to have to contend with a major 60 Hz signal with a setup like this, unless I did something about the scope's input impedance of 10 megohms.  So, I put a 100 ohm resistor across one end of the transmission line; that killed the 60 Hz noise.  (Since I was sending just one impulse down the wire, and I was only looking at the first rising edge, "termination" was not an issue -- I didn't care if the signal bounced back down the wire or not.)

The initial two shots were done from the end with the resistor; two subsequent shots were done from the other end.  One channel of the scope was hooked to each end of the line.  The negative wire from the battery pack was clipped to the (arbitrarily chosen) "ground wire" of the transmission line. For each "shot", the scope was set to single-sweep mode, and the positive wire from the battery pack was touched to the "signal" wire of the transmission line.  This was repeated until I got a clean shot (shown below).  I  reversed the wires (positive versus negative) and did it a second time, in case polarity made a difference.  I then moved the battery pack to the other end (the "unterminated" end) and took two more shots, again reversing the positive and negative wires between shots.

Rise times at the "far end" were poorer when firing from the unterminated end to the terminated end, which is as expected.  Firing into an unterminated end, as we did in the first two shots, rise times are roughly twice as fast, as the signal overshoots and bounces back.  To allow easy reading with the "mushier" edge at the end of the line, the scope was set to 1 V/div for the third and fourth shots, versus 2 V/div for the first two shots.

### The Oscilloscope

We're dealing with very fast edges here, so it's necessary to consider the performance of the oscilloscope.  It's a Tektronix TDS 210 digital scope.  Its nominal bandwidth is 60 MHz.  However, that's just the 3 dB point in its response.  It's got a 1 GHz sampling rate, so it is likely that it will actually respond to signals considerably faster than 60 MHz.  They will, however, appear quite a bit weaker than they really are, and fast edges may be shown with incorrect shapes.  Initial experiments indicated that, in the 1 volt per division range, its maximum slew rate is about 1 volt per nanosecond, and for this particular experiment, slew rate is really all we care about:  We just want to be able to clearly see where the edge starts to climb.
The probes were inexpensive noname-brand 100 MHz probes, which may be expected to contribute to problems with extremely fast edges.
We were right up against the slew rate limit in this experiment, which suggests that the edge shape displayed by the scope may not be reliable.  Rather than depend in any way on the exact rise time or the true location of the first peak, I attempted to time the signal from the start of its rise at one end of the wire to the start of its rise at the other end of the wire.

### Sources of Error

If we want to have any idea of what, if anything, this experiment shows, we must make some attempt at figuring out what our error bars are.
• Inaccurate length measurements.  This part should have been easy!   But, it wasn't -- cutting stuff to exact lengths is not my strong suit.  (Let's not talk about carpentry here, OK?)  I cut the two wires to be the same length, and later, at the end of the experiment, I measured them both with a tape measure.  Unfortunately, the baling wire, though it seemed soft enough in short pieces, proved springy and hard to straighten well when dealing with long sections, and it was awkward to measure accurately.  The tape measure also was shorter than the wire, which necessitated two measurements on each wire.  I would estimate the inaccuracy in length measurements to be about +/- 1".  What's more, the two wires turned out to be different lengths: one was 14' 10" (or 178"), the other was 15' 1" (or 181").  I'm not sure, but I think the signal will require the time it takes to go the longer distance before it shows up on the second scope channel.  But since I'm not sure, I'll say the length could be anything between 178" and 181" ... plus or minus an inch.  Thus, including measurement errors, we have a range of 177" to 182", which we can also state as 179.5" +/- 2.5".  So, with an assumed transmission line length of 179.5", that's an inaccuracy of +/- 1.4 %.
• Oscilloscope reading and sampling errors.  Looking at the screen shots, it appears that the cursors are positioned within +/- 0.5 nSec of the knee of the curve on each trace.  Since we're taking the difference between the channels, that's an error of +/- 1 nSec in the difference.  (Since the sampling rate of the oscilloscope is 1 GHz, with intermediate points on the screen interpolated by a DSP, we certainly can't claim better accuracy than that!)  The difference is about 15 nSec, so a possible 1 nSec error adds +/- 6.7% to the possible error on a single sample.  However, that is for a single trial.  This error is random, not systematic, so averaging multiple trials should tend to smooth it out.  We actually ran four trials and averaged the results.  My probability theory is shaky, but a quick simulation in Perl indicates that for a uniformly distributed error probability, taking the average of four trials should pull in the 95% confidence point by about a factor of 0.35.  So, we would actually expect this effect to contribute about +/- 2.3% to our total error.
• Refractive index problems.  This was supposed to be a no-brainer: the dielectric is air, with a refractive index of almost exactly 1.  Unfortunately, the suspension scheme didn't work very well.  The wire was arranged in a series of arcs, and at the bottom of each arc a couple inches of wire was nearly touching the floor.  There were seven supports, so the arcs were about 2 feet long; thus, about 10 percent of the wire was close to the cement floor of the cellar.  What's the refractive index of cement?  I have no idea, though I seem to recall that a number of common, relatively dense materials have radio frequency refractive indices in the range of 5 to 10.  I also don't know what the impact of having the signal just run close to the floor might be.  Finally, the foam, being mostly gas, probably has a refractive index not far above 1, so we can probably ignore it, but without further experiments I can't be positive of that.
All in all I'm not sure what, if any, impact the presence of the cement floor and foam supports might have had, but unless our results turn out to be grossly low compared with "standard" values for c, I'm going to assume that this was not actually a problem, and include no error term for it.
• Skin effect. Skin effect will make the edge mushier at the receiver but won't slow the initial "knee" of the signal.
• Oxide coating on the wires.  This acts like a layer of insulation, and could, therefore, slow the signal.  However, the wires were separated by several inches throughout their run.  Nearly the entire signal was traveling in air, so I doubt a thin oxide skin would have a significant effect (at least, in comparison with the other errors we've identified).
• Oscilloscope timebase errors.  Oscilloscope clock rate errors are completely insignificant compared to the other sources of error discussed here.
Summing the above errors, we find an error range of +/- 3.7%.

### Results

We recorded 4 shots, shown below.  The average transit time was 15.25 nSec.  The length of the path, as discussed above, we take to be 179.5", or 4.56 meters.  Thus, we measured v to be 2.99 * 108 meters/second.  Applying our calculated error range, we conclude that the speed of light must be between 2.88 * 108 and 3.10 * 108 meters/second.

Rather to our astonishment, not only does our "error bar" neatly bracket the speed of light, but our actual result is in almost exact agreement with the  "correct" figure.  As mentioned at the top of this page, the "correct" value is 3.00 * 108 meters/second, to 2 decimal places.  My result and the "correct" value differ by about a third of a percent, which is actually the limit of the precision of my measurements (never mind the accuracy!).  This was a far better result than I had hoped for.

### Screen Shots

 First shot, from terminated end:

 Second shot, from terminated end (wires reversed):

 Third shot, from unterminated to terminated end:

 Fourth shot, from unterminated end (wires reversed):

Page created on 2/07/07