Download the measurement of the speed of the light

THE MEASUREMENT
OF THE SPEED OF THE LIGHT
Nyamjav, Dorjderem
Abstract
The one of the physics fundamental issues is a nature of the light. In this
experiment we measured the speed of the light using MichelsonÕs classical
rotating octagonal mirror and the achieved results, c=(3.23±.11)×108 m/s, were
within the relative error of 5% to the best estimated value of 2.99× 108 m/s.[1]
The systematic error of the experiment was estimated to be 3.24%.
1. Introduction
Since Roemer first measured the speed of the light many scientists have
measured it in many ways. In 1925-1926, A.A.Michelson measured the speed of
the light using high-speed revolving steel mirror and obtained value of
299,796±4km/s[2]
Speed of the light can be determined by the following correlation,
∂v
vg = v p − λ p
∂λ ,
c = nv p
here vp-phase velocity, vg-group velocity, λ-wavelength and n-refractive index of
the media where light is propagating. [3]
Since our measurement is done in air, we should take into account the correction
for refractive index obtained by L.Essen and others, [4]
ng = 1 + 10 −8 (27260 + 460.8 / λ2 + 6.60 / λ4 )
We will discuss this effect later in part ÒResult and discussionÓ.
2. Method and procedure
We used helium neon laser, with a power of 55W and a wavelength of 633nm, as
a source and reflected it on the octagonal mirror. After reflecting consecutive
mirrors the laser beam passes through a collecting lens whose focal length is 5m
and reflects on the octagonal mirror second time. Hence, using eyepiece we are
able to find the focal point of the lens where the beam is collected after reflecting
on the 8 sided mirror second time.
Fig.1. Experimental set up
M1-rotating mirror, M2, M3, M4-mirrors, L-lens, O- observer
If we rotate the octagonal mirror the focal point is going to shift at some
distance due to the fact that the beam is striking at the mirror M1 at different
angle. Therefore, knowing the rotation speed, the distance light travels and the
shift of the focal point we can calculate the speed of the light.
LetÕs assume the face moves by angle of α as shown in Fig.2. Then ∠
F1CF2 will be twice as much as α. Taking into account that
F1C>>AB , F2C>>AB
we can say
∠F1A F2≅2α.
Futhermore, the shift of the focal point, we are measuring, ∆,
∆ = 2α × d , where d = AF1
Let us call the time, light travels the distance l, as t. Then,
8l
, T-period, l-distance which light travels.
α=
cT
Combining all, the final expression for the shift , ∆, equals
Thus,
∆=
16 × l × d × f
c
c = 16 × l ∆× d × f .
here, l-distance light traveled,
d-distance between the octagonal mirror and the focal point,
f-frequency,
c-speed of light.
Fig.1. Geometry of the set up.
f1, f2 -original and shifted
face position
respectively
F1, F2-original and
shifted focal points
L- lens
The frequency was measured using photometer, since a light intensity at the
certain point peaks only once in one revolution. The photometer had been
connected to an oscilloscope, in which exponential signal was observed. The time
period between two consecutive signals is one-eighth of the revolution period of
the rotating mirror. Based on this fact we were able to estimate the period.
In order to get a good result using this set up, the distance must be as long as
possible and in our case it was 192m. The experiment was done in a hallway of
the Physics building during night because of the safety reason.
3. Results and discussion
We did measure the speed of light in different frequencies of the rotating mirror
and the results are shown below.
c
2.6300001
Minimum
Maximum
3.72
Maximum
3.75
Sum
21.98
Sum
19.979
Minimum
Measurement of c
Points
7
Mean
3.14
Median
2.9200001
RMS
3.1681225
Std Deviation
0.45493586
Mean value
Variance
Std Error
0.20696664
0.17194959
c= (3.33+-.14)x108 m/s
Skewness
Kurtosis
10
y = 2.4802 + 0.007964x R2= 0.83979
Mean value
c=(3.14+-.17)x108 m/s
c
8
Measurement of c
10
3.3298333
3.1975
3.3437344
Std Deviation
0.33365276
Variance
0.11132417
Std Error
0.13621317
0.21337128
Skewness
0.49989133
-1.7377568
Kurtosis
-1.4525155
8
6
c, × 108 m /s
c, × 108 m /s
6
Mean
RMS
4
3.3119
4
3.4660
3.3298
3.14
3.1936
2.9681
2
2
0
0
20
40
60
80
100
120
140
160
40
60
Figure 1. Data set 1.
Measurement of c
c
2
y = 2.6917 + 0.0063686x R = 0.43429
Mean value
8
c=(3.23+-.11)x10 m/s
6
Minimum
2.6300001
Maximum
Sum
3.75
41.959
Points
13
Mean
Median
3.2276154
3.1949999
RMS
3.2503534
Std Deviation
Variance
0.39946286
0.15957057
Std Error
0.11079106
Skewness
Kurtosis
0.057609977
-1.416327
4
3.3384
3.2276
2
0
0
20
40
60
100
120
Figure 2. Data set 2.
10
8
80
frequency, Hz
frequency, Hz
c, × 108 m /s
Points
Median
y = 3.3744 + -0.00052021x R2= 0.001962
6
0
80
100
2.97
120
140
160
frequency, Hz
Figure 4. Plot for the all data.
140
In doing so, possible error sources were inadequate measurement of the optical
path, frequency ,DopplerÕs effect and the above mentioned change in a refractive
index of air. Despite of number of error sources present, the uncertainty in the
shift is a major factor. The relative errors due to the rest counts by .5% and less,
while the uncertainty in the shift reaches order of 10% .
l = (192.2 ± .8)m
d = ( 4.45 ± .05)m
Also uncertainty in frequency is in range of 4%, which is due to the reading only.
Considering a wavelength for a laser , which is 633nm, we can neglect the effect
of the ÒgroupÓ refractive index of air. The error was calculated as follows,
j =n
S tan dard.deviation =
∑(y
j =1
j
−M
)
2
(n − 1)
n
∑y
j
S tan dard.deviation
,
n
n
where yj-is measured value, n- number of data points.
As a conclusion I want to discuss the positive and negative sides of presented 2
sets of measurement.
M=
j =1
Data set1
Data set 2
Combined
S tan dard.Error =
Positive side
The true value of
the speed of light
falls within the
range of
uncertainty. Good
fit. R2=.84.
More horizontal
fit. Shows random
distribution.
Negative side
Dependent on the
frequency.
All data points fall
in range of
standard error.
Reasonable fit.
Less error.
Dependent on the
frequency.
True value is not
in the range of
standard error.
Similarities
Relative standard
error is 5.4%.
Standard
deviation .45
respect to the
mean value of 3.14
Relative standard
error of 4.2%.
Standard
deviation of .33
respect to the 3.33
The results for each of two sets and the combined data sets give the following
values for the speed of light,
Data set1:
c = (3.33 ± .14) × 108 m / s, relative error of.04,
Data set2:
relative error of .05,
c = (3.14 ± .17) × 108 m / s,
8
relative error of .03.
Combined data:
c = (3.23 ± .11) × 10 m / s,
The most reliable data set is the first one as described in the table, despite of
higher relative error.
4.References
1.
2.
3.
4.
5.
APS News, March 2000
Thomas Parke Hughes, ÒMichelson, Sperry, and the speed of lightÓ, 1976.
J.H. Sanders, ÒThe velocity of lightÓ, 1965.
K.D.Froome and L.Essen, ÒThe velocity of light and radio wavesÓ, 1969.
APS March 2000 meeting news.