Date:2021-Oct-22
Author: A.C. Popescu, S4303458
T.A.: Csilla Tijssen
___________________________________________________________________
Determining the speed of light in air using a laser
1.Introduction
Before the 17th century, it was believed that the light’s speed is infinite, or it is too fast to be measured.
That changed in 1676, when the Danish astronomer Ole Roemer became the first person to measure the
speed of light by timing the eclipses of Jupiter’s moon Io. The most recent and accurate constant for the
speed of light was determined by CGPM (Conf´erence g´en´erale des poids et mesures) in 1983, this
value being c=2.997*108 m/s, so that will be our hypothesis for this experiment.
By this means, the main purpose of this experiment is to determine the speed of light as a constant by
turning a laser beam into an electrical signal. The presented method in this study will allow the students
to obtain a ten-point graph, where the slope gives the value of the speed of light.
2.Abstract
The one question is how can we determine the speed of light with the use of an appropriate experiment
setup? Here we used an oscilloscope, function generator, laser, light receiver and a mirror. A laser beam
was modulated with the appropriate set-ups from the function generator, which then was redirected to
the center of the mirror, reflecting the light beam on the light sensor found in the receiver, which is
connected to the oscilloscope. The measurements were recorded using the measurement ruler for the
distance and the time interval from the oscilloscope’s data.
3.Theory
The speed of light is a universal physical constant important in many areas of physics. According to
special relativity, “c” is the upper limit for the speed at which conventional matter, energy or any signal
carrying information can travel through space.
In order to calculate in this experiment, the universal constant of the speed of light c, the generic
formula for speed was used:
𝑐 = Δ𝑥/Δ𝑡(1.1),
Where Δ𝑥 is the distance that the light travels in the time interval of Δ𝑡.
From the experimental setup sketch (Figure 2.1), it is obvious that between X1 and X2 there is an angle,
that was denoted with θ. For the sake of simplicity, during the experiment it was aimed to keep θ as
small as possible(θ<<5°) in order to approximate:
1
Figure 2.1 – Experimental setup with its measured distances
Figure 2.2 – Oscilloscope’s reading of a measurement done in the laboratory
2
𝑋1 =
𝑋2 =
𝑙1
𝜃
2
cos( )
𝑙2
𝜃
2
cos( )
≈ 𝑙1 (1.2)
≈ 𝑙2 (1.3)
, where l1 is the distance between the laser’s starting point and the mirror, and l2 is the distance between
the mirror and the receiver.
4.Experimental Method
List of equipment and materials used (Figure 2.1 for reference):
1.
2.
3.
4.
5.
6.
•
Oscilloscope
Function generator apparatus
Cables
Laser
Mirror
Receiver
Measure ruler
The laser and the function generator were coupled to the oscilloscope with a T connection in the Chanel
1, followed by connecting the receiver to the oscilloscope, but on the second Chanel. In order to display
the signals on the Oscilloscope, the researchers had to press the buttons labeled CH1 and CH2.
After all the connections are done, the intensity of the light was modulated with the help of the function
generator by changing the voltage supply. To achieve the highest intensity for the laser beam, the
highest frequency of the sine wave was selected on the function generator, followed by finding an offset
voltage at a certain amplitude for the sine wave.
Then, the laser was adjusted so that the light was redirected to the center of the mirror. For every
changed distance, the mirror and the laser were adjusted so the reflect beam of light was hitting the
receiver’s sensor. Therefore, the second sine wave on the Oscilloscope had appeared. Figure 2.2 shows
the reading of a measurement done during the lab.
For each measurement, the highest peaks of the function generator and the receiver were chosen as
recognizable features. The time difference between the tops of these two signals were measured. After
every measurement was done, the mirror’s position was changed. In total of ten measurements were
recorded, for which ten calculations were done using formula (1.1) as to determine the speed of light for
every recorded distance.
5.Results
Table 4.1 show the measured values used for calculating the speed of light (c), for each variation in
distance of the reflective object. It can clearly be seen that the time t increases as the mirror is moved
further and further away.
3
Nr.
1
2
3
4
5
6
7
8
9
10
X1±1.0 (cm)
93.0
131.5
170.0
197.4
231.3
264.4
291.0
330.1
369.3
393.0
X2±1.0 (cm)
89.7
128.4
167.9
194.0
227.5
260.1
288.5
326.0
364.1
390.0
X1+ X2 (cm)
182.7
259.9
337.9
391.4
458.8
524.5
579.5
656.1
733.4
783.0
𝛿(𝑋1 + 𝑋2 ) (cm)
±1.41
Table 4.1 - Raw data values of the experiment
Nr.
1
2
3
4
5
6
7
8
9
10
c=(*108*m/s)
3.20
3.24
3.21
3.06
3.07
2.96
3.00
3.16
3.09
3.23
cavg(*108*m/s)
3.12
𝛿c(*108*m/s)
±0.08
±0.12
±0.09
±0.06
±0.05
±0.16
±0.12
±0.04
±0.03
±0.11
𝛿cavg
±0.12
Table.4 2 – Calculated measurements of the speed of light c
Graph 5.1 – Graph from the data points from Table 4.1
4
Time±0.30 (ns)
5.7
8.0
10.5
12.8
14.9
17.7
19.3
20.7
23.7
24.2
The uncertainty for the mirror was approximated to be of 1cm for each measurement. Therefore, the
uncertainty in the distance was calculated using the following error propagation formula:
𝛿 (𝑋1 + 𝑋2 ) = √(𝛿𝑋1 )2 + (𝛿𝑋2 )2
𝛿(𝑋1 + 𝑋2 ) = √2𝑐𝑚 = 1.41𝑐𝑚
Because the graph was plotted m versus ns, the errors for the distance were not used for plotting it,
because the errors were too small to be visible on the graph.
As the oscilloscope is a digital device and no more than one reading was taken, the uncertainty for time
was a value of 0.3 ns.
It is important to note that the raw data was recorded in the units they were given by, namely cm and
ns. They were converted into the respective SI units for the final data table (Table 4.2). Additionally, the
figures in Table 4.2 were rounded to two significant digits for the purpose of consistency.
The gradient of the graph corresponds to the experimental value of the speed of light found in each
attempt. The line of best fit drawn in Graph 5.1 show c=3.12*10^8 +/- 0.37 m/s.
Beside using the gradient of the graph to find the velocity of light, the average speed of light was also
calculated using the following formula:
𝑐𝑎𝑣𝑔 =
∑𝑛
𝑖=1 𝑐𝑖
𝑛
,
Where n is the number of values obtained. In this case, the result for 10 measurements was:
∑10
𝑖=1 𝑐𝑖
𝑚
10
𝑠
The uncertainty for the value of c was calculated using the formula:
𝑐𝑎𝑣𝑔 =
= 3.12 ∙ 108
𝛿𝑐𝑛 = ±|𝑐𝑎𝑣𝑔 − 𝑐𝑛 |
𝛿𝑐𝑎𝑣𝑔 =
𝛿𝑐𝑎𝑣𝑔 =
∑𝑛𝑖=1 𝛿𝑐𝑖
∑10
𝑖=1 𝛿𝑐𝑖
10
𝑛
= ±0.12
6.Discussion
To begin with, using the equation mentioned in theory, we obtained the predicted linear relationship
between distance and time. However, the found value for the light’s speed is c=3.12 ± 0.12*108 m/s,
which is not exactly the hypnotized value. An approximate value of c=2.99 *108 m/s was expected. The
main reason for the difference values of the hypnotized and the experimental value is because of the
errors encountered during the experiment.
5
Firstly, the discrepancies in the results might have come from some problems related to the receiver and
the oscilloscope. From Figure 2, it is obvious that the receiver displays on the oscilloscope a sine wave
with a lot of noise. The noise might have been registered from the background light that have interfered
with the experiment results. Thus, during the laboratory work multiple measurements were performed,
but the closes ones that could have had given a potential value for the speed of light were wrote down
in table of values (Table4.1) for this report.
Secondly, the graph demonstrates the relationship of the distance of the mirror used for reflecting the
laser and the time difference between the laser’s starting point-mirror-sensor. The line of best fit passes
through almost all the data points, except for (5.795m,19.3ns) and (6.561m,20.7ns). The slope with the
minimum possible gradient is 2.91 and the maximum possible slope is 3.24.
Thirdly, as mentioned in the theory, if the angle between X1 and X2 would be greater than 5 degrees,
then it would be needed to calculate the angle 𝜃 in order to determine the “true distance”.
Consequently, for every distance performed in the experiment, a new angle 𝜃, would be calculated. In
short, the following formula is used to calculate 𝜃:
𝜃 = 𝜃1 + 𝜃2
𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 (
𝑙1
𝑙2
) + arctan ( )
𝑅𝑀
𝑁𝐿
, where l1, l2, RM and NL are measured with the ruler.
Finally, to make this experiment more accurate, there should be less light in the room. So, a possible
improvement would be to do the experiment in a dark room. Another improvement would be to do the
experiment in a more controlled environment where all variables are controlled and checked before the
first attempt.
7.Conclusion
In conclusion, the speed of light was approximated to be c=3.12 ± 0.12*108 m/s, by plotting in a graph
the distance (X1+X2) versus the time Δ𝑡, and it was also calculated using the formula (1.1). This is
0.123*108 m/s larger than the hypnotized value. The result might be faulty because of the imperfection
in the equipment setup and the illumination of the work area that contributed to the errors.
8.References
[1] https://www.amnh.org/learn-teach/curriculum-collections/cosmic-horizons-book/ole-roemerspeed-of-light
[2] https://en.wikipedia.org/wiki/Speed_of_light
[3] Dr. J.S. Blijleven - Physics Laboratory 1 Manual 2021 – 2022
6