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Experimental Determination of the Speed of Light by the Foucault
Method
R. Price and J. Zizka
University of Arizona
The speed of light was measured using the Foucault method of reflecting a beam of
light from a rotating mirror to a fixed mirror and back creating two separate
reflected beams with an angular displacement that is related to the time that was
required for the light beam to travel a given distance to the fixed mirror. By taking
measurements relating the displacement of the two light beams and the angular
speed of the rotating mirror, the speed of light was found to be (3.09±0.204)x
8
m/s,
which is within 2.7% of the defined value for the speed of light.
1 Introduction
The goal of the experiment was to experimentally measure the speed of light, c, in a vacuum by
using the Foucault method for measuring the speed of light. Although there are many experimental
methods available to measure the speed of light, the underlying principle behind all methods on the simple
kinematic relationship between constant velocity, distance and time given below:
c =
D
t
In all forms of the experiment, the objective is to measure the time required for the light to travel a given
distance. The large magnitude of the speed of light prevents any direct measurements of the time a light
beam going across a given distance similar to kinematic experiments. Galileo himself attempted such an
experiment by having two people hold lights across a distance. One of the experiments would put out their
light and when the second observer saw the light cease, they would put out theirs. The first observer would
time how long it took for the second light to go out, giving an estimate on the speed of light. However, it was
found that the time lapse between the two events was near instantaneous suggesting that such a method
was not precise enough, and a more interesting result at the time, that the speed of light was much larger in
magnitude than thought
1
To correct for these problems, Foucault devised a method which was able to avoid such issues by
indirectly measuring the time the light traveled. Foucault’s method used a light source and rotating mirror
together to derive the speed of light
2
. The Foucault method uses the light source to produce a focused beam
on the rotating mirror. The light from the rotating mirror is then reflected at an angle to a fixed mirror
which is aligned to face perpendicular to the reflected light beam. Therefore the light is reflected directly
back to the rotating mirror where it was first reflected. During the time the light had traveled the distance
between the two mirrors, the rotating mirror had changed its orientation to the beam of light, thus the
1 ,
2 Hecht, Eugene. Optics, 4th Edition. San Francisco: Addison Wesley, 2002.
returning beam of light will be reflected off at a separate angle. The difference in the angle between the
light source to the rotating mirror and the rotating mirror the second reflected beam is related to the time
that was required by the light to travel the distance between the fixed and rotating mirrors. Using the
relations of the experimental setup, Equation 1 was used to determine the speed of light.
1.1 Setup
For the experiment the methods were updated to use modern equipment to provide more accurate and
precise results. A diagram of the experimental setup is shown below in Figure 1:
Figure 1 - Diagram of Experimental Setup
The experiment used a laser to provide as the light beam because it creates a focused beam of light to travel
between all components. The rotating mirror used was a double sided plane mirror attached to a motor
apparatus that allowed a variable control of the motor rotation speed. The rate of rotation was measured
using a light sensor in conjunction with a frequency counter. As the mirror rotated with the laser
positioned on it, there was an angle of the mirror that reflected the laser light beam toward the light sensor.
When the laser beam was directed at the light sensor, it created a voltage which could be then measured by
a connected frequency counter.
A lens with a 5 meter focal length was placed between the rotating and fixed mirrors because the
laser beam had an angular divergent property associated with the beam. Over large distances, the laser
ω mirror
=
∆θ
t
=
∆x
2rt
(7)
However, the time of rotation can be related to the time traveled by the light:
t =
∆x
2rω mirror
(8)
Equation 2 and Equation 8 can be related so that:
2D
c
=
∆x
2rω mirror
(9)
c =
4rdω mirror
∆x
(10)
Within the experiment, it is important to know c in terms of the frequency instead of angular frequency
because the frequency counter used returns the measurement in units of Hertz. Therefore by definition,
ω mirror
= 2πf mirror
(11)
Then by substituting Equation 11 into 10:
c =
8 πrdf mirror
∆x
(12)
Another correction is required for the experiment in terms of frequency. The frequency measured on the
frequency counter is twice that of the rotation frequency because the mirror rotated is doubled sided so
then the laser beam is reflected into the light sensor twice per each rotation of the mirror. Therefore,
f mirror =
1
2
f measured (13)
Substituting Equation 13 into Equation 12 results in:
c =
4 πrdf measured
∆x
(14)
Because the experiment is performed within air and not a vacuum, the result from Equation 14 above
would give c within air. Therefore by multiplying by the index of refraction of air, we get c within a vacuum:
c =
4 πrdfn air
∆x
(15)
Where:
r is path from the rotating mirror to the beam splitter to the frosted screen
D is the distance between the fixed mirror and the rotating mirror
nair is the index of refraction for air, given as 1.
3
f is the frequency as given by the frequency counter used within the experimental setup
3 Hecht, Eugene. Optics, 4th Edition. San Francisco: Addison Wesley, 2002.
∆x is the displacement of the laser beam across the frosted glass screen
2 Experimental Methods
From Equation 15, the objective of the experimental measurements was to gain the relationship between
the two observable, alterable quantities: the frequency of the mirror rotation and the observation of the
laser beam displacement across the frosted glass plate. By rearranging Equation 15, a more useful
relationship can be obtained:
𝑥 =
4 𝜋𝑟𝑑𝑛 𝑎𝑖𝑟
𝑐
𝑓 (16)
Equation 16 reveals a linear relation between the displacement on the glass screen and the frequency of the
mirror. Therefore the method of collecting data to find the speed of light was to take data points relating
the frequency and displacements and to perform a least-square fit in order to determine experimental
value of the relationship and compare it to the known value.
To setup the experiment, the optical systems involved had to be aligned in order to create the beam path
displayed in Figure 1. First, the laser was aimed at the rotating mirror and slowly adjusted so that the laser
beam was directed toward the center of the mirror. Next the rotating mirror was slowly aimed at the center
of the fixed mirror. Next the lens was placed between the fixed and rotating mirrors in order to focus the
return beam from the fixed mirror to the rotating mirror as otherwise the angular dispersion of the laser
beam created difficulty in detecting where the return beam was directed to. The fixed mirror was very
carefully adjusted so that the return path through the lens reflected back onto exactly the same location on
the rotating mirror where the first reflection occurred. With all the optics aligned, the beam appeared on
the frosted glass screen as a point. The setup was then tested by covering the lens between the mirrors.
When this was performed, the beam on the glass plate disappeared, verifying the beam was being reflected
from the fixed mirror. Next, the position of the light beam on the frosted glass plate with no rotation was
recorded by using the measuring microscope by aligning the crosshair of the microscope to the center of
the laser beam as it appeared within the scope.
To measure the mirror rotation frequency, the light sensor was placed next to the rotating mirror
so that during the operation of the rotating mirror, the laser beam would strike the light sensor. The sensor
and frequency counter setup was tested by covering and uncovering the light sensor to verify that it was
connected. With all of the optics and equipment in place, the distances between the two mirrors as well as
the mirror to the glass screen were recorded as precisely as possible. Next, the motor that rotated the
mirror was adjusted to a rotational frequency that caused a displacement in the laser beam. The
displacement in the laser beam was measured by the microscope by centering the laser beam in the scope
and recording the displacement value. Once the frequency versus displacement data point was recorded,
the mirror was set to a new rotation frequency in order to record another data point.
The data collected was analyzed using a least-square fit method to obtain the value of b for the equation:
x = bf + a (17)
However, from Equation 16 it can be said then that:
b =
4 πrdn air
c
Rearranging gives:
c =
4 πrdn air
b
(18)
Therefore the speed of light can be found by using the known distances in the experimental setup as well as
the least-square fit of the data points collected. The value of b, along with the uncertainty in the value, was
calculated from the least-squares fit method using the data collected from the experiment. The values of r
and d are listed in Table 1 along with their respective uncertainties in measurement.
To find the uncertainty in the value of the speed of light Equation 18 was used in the error propagation
derivational formula:
σ c
= σ r
∂c
∂r r
∂c
∂b b
∂c
∂D D
σ c
=
16 π
B
σ r
D
σ b
r
D
B
r
(19)
Where c is the relation in Equation 18. By using the least squares analysis along with the error propagation
methods, the result for the speed of light for the first data set was (3.008± 0.34)x
8 m/s.
From the first experimental set, there were extra methods that would have allowed an increase in the
precision of the measurement. With the data analyzed, a second experimental set was recorded in order to
reach a more accurate and precise value for the speed of light by taking a larger data set.
4 Data Set 2
For the second experiment, more data points were taken so that the uncertainty in the experiment would
be reduced. For each adjustment of the rotating mirror frequency, ten measurements of the position and
the frequency were taken so that an experimental uncertainty and average could be extracted, yielding a
more precise and accurate result.
Mirror
Frequency (Hz, σ
= 1 Hz)
σ Frequency
(Hz)
∆X (m, σ=
2.54e-4 m)
σ dx (m) Least Squares Data-Fit:
126.02 0.910666667 0.00026162 0.000219497 a (x 0 ) 6.31805E- 05
167.31 0.814333333 0.00049276 0.000167032 b (4πrD/c) 3.11399E- 06
194.02 0.806222222 0.00067056 0.00019682 σa 8.74874E- 05
222.88 3.744 0.0005969 0.000205393 σb 2.05909E- 07
Error Propagation:
391.91 0.063222222 0.00162306 0.000170115 r 5.
412.78 0.455111111 0.00126238 0.000214208 d 15.
440.92 1.855111111 0.00150114 0.000171207 σr 0.
490.26 0.331555556 0.00146558 0.00015522 σD 0.
Resulting Values:
543.3 0.124444444 0.0016002 0.000172687 c 3.09096E+
632.8 0.568888889 0.00191516 0.000190904 σc 2.04390E+
Table 2 - Data Set 2 Data Points
Figure 3 - Data Set 2 Data Plot
Using the process of error analysis and least-squares data fitting from the first data set, the second data set
had the result for the speed of light to be (3.091 ± 0.204)x
8
m/s.
y = 3.1250E-06x + 4.5966E- 05
R² = 9.5630E- 01
∆X Beam Deflection (m)
Mirror Rotation Frequency (Hz)
∆X Deflection (m) vs. Mirror Frequency (Hz)
f
c
rD
x x
0
2
2 2 2 2 2 2 2 2
2 2 16 2
b
rD D r b
b c r D
