Physics Department

Electromagnetism and Optics Laboratory

MEASURING THE VELOCITY OF LIGHT

1. Goal.

Calculation of the velocity of light in air and in transparent solid and liquid bodies.

2. Overview

The velocity 𝑣𝑚 of propagation of an electromagnetic wave (light) in a material medium, obtained from
Maxwell’s equations, is given by:

1 1
𝑣𝑚 = = [𝑚/𝑠] , [1]
√𝜀𝑟 ∙ 𝜀0 ∙ 𝜇𝑟 ∙ 𝜇0 √𝜀 ∙ 𝜇

where:

- 𝜀0 = 8.854 × 10−12 𝐹𝑚−1 is the electric field constant (or permittivity of the vacuum).

- 𝜇0 = 4𝜋 × 10−7 = 1.257 × 10−6 𝐻𝑚−1 is the magnetic field constant (or permeability of the vacuum).
𝜇
- 𝜀𝑟 𝑦 𝜇𝑟 are the relative permittivity and permeability of the medium, such that 𝜀𝑟 = 𝜀⁄𝜀0 𝑎𝑛𝑑 𝜇𝑟 = ⁄𝜇0 ,
respectively. For the vacuum, 𝜀𝑟 = 𝜇𝑟 = 1 ↦ 𝜀 = 𝜀0 𝑎𝑛𝑑 𝜇 = 𝜇0 .

The electric permittivity and magnetic permeability of a material describes how the electric and magnetic
fields of an electromagnetic wave interact with it. In the most general case, in an isotropic material the relative
permittivity and permeability are complex magnitudes, whose real parts correspond to the electromagnetic
energy stored inside the material and whose imaginary parts are related to loss, that it, it is a measure of how
the electromagnetic wave’s energy diminishes when this passes through the material.

The refractive index of a medium 𝑛 is defined as the ratio between the velocity of light in the vacuum 𝑐 and
in the medium 𝑣𝑚 :

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𝑐 √𝜀0 ∙ 𝜇0
𝑛= = = √𝜀𝑟 𝜇𝑟 . [2]
𝑣𝑚 1
√𝜀 ∙ 𝜇
In general, it is a function of the radiation wavelength and the nature of the medium it propagates through,
giving rise to light dispersion phenomena in materials. Therefore, we can say that the velocity of propagation of
radiation depends on the radiation wavelength. This is the reason why we can observe that a change in the
direction of a ray of light passing from one material to another depends on the radiation wavelength, when these

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have different refractive indices.

For most transparent media we can consider that 𝜇𝑟 = 1. As we have mentioned, in the vacuum:
𝜀𝑟 = 1 𝑎𝑛𝑑 𝜇𝑟 = 1; hence:
1 1
𝑣𝑚 = 𝑐 = = = 2.997 × 108 [𝑚/𝑠] . [3]
√𝜀𝑟 ∙ 𝜀0 ∙ 𝜇𝑟 ∙ 𝜇0 √𝜀0 ∙ 𝜇0

3. Material.
• Optical bench (supporting platform for the optical system, composed of deviating mirrors and
lenses).
• Operating unit.
• Lenses on magnetic stands (2).
• Inverting mirror system with screw adjustments.
• Incandescent lamp (12V AC).
• Cylindrical vessel of length 𝐿𝐿 = (100.0 ± 0.1) 𝑐𝑚 with liquid.
• Block of synthetic resin, with longest edge of length 𝐿𝐵 = (30.0 ± 0.1) 𝑐𝑚
• Analog dual-channel oscilloscope with a bandwidth of 35 𝑀𝐻𝑧.
• Coaxial cables.

4. Constants and magnitudes of interest.

𝜀0 = 8.854 × 10−12 𝐹/𝑚; 𝜇0 = 1.257 × 10−6 𝐻/𝑚 ; 𝐶 = 2.998 × 108 𝑚/𝑠

𝑚 𝑚
𝑣𝐻2𝑂 = 2.248 × 108 ; 𝑣𝑟𝑒𝑠𝑖𝑛 𝑏𝑙𝑜𝑐𝑘 = (1.87 ± 0.01) × 108 ; 𝑣𝑎𝑖𝑟 = (2.99 ± 0.01) × 108 𝑚/𝑠
𝑠 𝑠

𝑛𝐻2𝑂;𝑇=200𝐶 = 1.333 ± 0.001; 𝑛𝑟𝑒𝑠𝑖𝑛 𝑏𝑙𝑜𝑐𝑘 = 1.597 ± 0.003; 𝑛𝑎𝑖𝑟 = 1.000 ± 0.001

5. Experimental Procedure.

5.1 Description of the Experiment.

Fig. 1 Experimental setup diagram.

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 The operating unit in figure 1 contains a light source -transmitting diode (red light LED)- whose intensity
is modulated at frequency 𝑓𝑚 = 50.1 𝑀𝐻𝑍. The LED light passes through a lens (Lens 1), which allows focusing
it on its “outbound” way with the inverting mirror system. This makes the light return to the operating unit, but
now, with the help of a second lens (Lens 2), it falls upon the photodiode (receiver), which generates an
alternating signal of the same frequency 𝑓𝑚 , but with a phase difference with respect to the signal that
modulates the intensity of the transmitting source. This way, a displacement in the inverting mirror system
with respect to the plane that contains the transmitter and the receiver in the operating unit will bring along a
change in the path traversed by the light between them. Therefore, the phase difference between both signals
will also change.

For a displacement Δ𝑥 in the mirror position, there will be a phase difference Φ between the transmitter
and the receiver signals. In this case, the time interval Δ𝑡 in which light traverses the distance 𝐿 = 2Δ𝑥 between
the transmitter and the receiver is given by:

Φ
∆𝑡 = [𝑠] . [4]
2𝜋𝑓𝑚

We should take into account that the phase of a periodic signal of the form
cos(𝜔𝑡 + Φ)
is
Φ
Φ = 𝜔∆𝑡 ↦ ∆𝑡 = .
2𝜋𝑓

The velocity 𝑣𝐿 of light traversing the distance 𝐿 is:

𝐿 2Δ𝑥 4𝜋𝑓𝑚 Δ𝑥
𝑣𝐿 = = = [𝑚/𝑠] . [5]
∆𝑡 Φ Φ
2𝜋𝑓𝑚

From this we conclude that, by measuring the phase difference Φ between the transmitter and the receiver
signals, if we know the increment Δ𝑥 in the path light traverses, we can determine its velocity 𝑣𝐿 .

In order to carry out an appropriate measurement of the phase difference between two signals of the same
frequency, as in our case (remember that the signals from the transmitter and the receiver have the same
frequency and differ only in their phase if there are changes in the length of the path the ray traverses). One
method for measuring phase shift is to use XY mode, called like this because both the X and Y axis are tracing
voltages. The waveform that results from this arrangement is called a Lissajous pattern or figure. We can access
this with the X-Y mode of the dual-channel oscilloscope. After pressing the (X-Y) button, if the signals in the
input channels X and Y of the oscilloscope correspond to two simple harmonic movements of the form
[ 𝑥(𝑡) = 𝐴 𝑠𝑖𝑛(2𝜋𝑓𝑥 𝑡 + 𝜙𝑥 ) ; 𝑦 (𝑡) = 𝐴 𝑠𝑖𝑛(2𝜋𝑓𝑦 𝑡 + 𝜙𝑦 )] , and these are superposed across perpendicular directions,
we obtain a set of Lissajous figures, whose specific shapes depend on the relationship between the frequencies
of both signals [𝑓𝑥 /𝑓𝑦 ] and their phase difference [Φ = 𝜙𝑥 − 𝜙𝑦 ].

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 In case that input signals have the same frequency [𝑓𝑥 = 𝑓𝑦 ] we observe a set of Lissajous figures, whose
shape depends on the phase difference [Φ] between signals in both channels, as shown in Figure 2. Once the
figure is displayed, we must properly select the sensitivity (volts/div.) the oscilloscope shows the input signals
with. An appropriate choice for the sensitivity is that in which the figure spreads over the largest area in the
oscilloscope display.

Fig. 2. Lissajous patterns for signals with the same frequency and phase difference equal to
𝟎; 𝝅⁄𝟒 ; 𝝅⁄𝟐 ; 𝟑𝝅⁄𝟒 𝒂𝒏𝒅 𝝅, respectively.

Remember that in our experiment, signals are modulated at frequency 𝑓𝑚 = 50.1 𝑀𝐻𝑧, but this sampling
frequency is too high for a conventional oscilloscope, with a bandwidth of 35 𝑀𝐻𝑧, which will show less than a
70% of the signal amplitude at 50.1 𝑀𝐻𝑧. For this reason, the frequency is decreased to 50 𝑘𝐻𝑧 by means of a
mixer in the service unit.

5.2 Set-up and Optical System Adjustment.

(ask your laboratory teacher before proceeding with this part)

The operating unit is set in the extreme without scale of the supporting platform, with the front side (where
controls are located) oriented towards the observer (the elevated edge). The lenses are fixed to the magnetic
stands and placed in front of the transmitter (LED) and receiver (Photodiode), with the plane sides oriented
towards the diodes and parallel to the plane containing the LED and the receiving photodiode (on the lateral side
of the operating unit).

The X-output of the operating unit corresponds to an electrical voltage proportional to the modulation
voltage of the transmitter, with a frequency of 50.1 𝑘𝐻𝑧, whereas the Y-output corresponds to an electrical
voltage proportional to that generated at the photodiode when the light reflected in the mirror system falls upon
it. Both outputs are connected to the channels 1 and 2 of the oscilloscope with the BNC sockets, respectively.

Check that:

• The X signal in the operating unit connected to channel 1 of the oscilloscope is a sinusoid
of frequency 50 𝑘𝐻𝑧.

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 The plane side of the lenses must be at a distance between 3.5 and 4 cm of the plane containing the LED
and the photodiode. At the back of the inverting mirror system we can find an auxiliary incandescent lamp, which
helps find the right position for the lenses. The lamp is connected to a 12V AC output in the operating unit. By
placing the lamp holder in its natural position on the platform, the wire filament is at the optical axis of either
the transmitter or the receiver, depending on the terminal chosen to connect the lamp to its holder. The wires
supplying power to the lamp are plugged into the free terminals in the opposite extreme. Once the lamp is on,
we must wave the lens slightly on the lamp’s optical path until the focal spot falls upon the LED or the photodiode.

When the lenses have been adjusted, the lamp is switched off by unplugging the wires from the operating
unit and the mirror system is rotated so that it stays in the way of the optical path of the ray from the transmitter
(that is, a rotation of 180º with respect to the previous position). If the adjustment is correct, we see the signal
corresponding to the photodiode on the oscilloscope display (Y-output of the operating unit, connected to the
channel 2 of the oscilloscope). Next, we delicately tighten the screws that keep the mirrors parallel. We can
maximize the amount of (LED) light that reaches the photodiode by making very small changes in the lenses
position; this corresponds to a higher signal amplitude in the channel 2 of the oscilloscope.

Taking into account the difficulties of the adjustment and alignment process of the optical system, in most
of the cases you will find that the system has been previously adjusted by the technical staff. Therefore, in case
of not being able to measure the signals at the operating unit output, you should NOT change the lenses and
mirrors positions without checking first with your laboratory teacher. In general, the reason why signals are not
detected at the operating unit is a wrong connection or configuration of the oscilloscope in the X-Y mode.

Check that:

• The inverting mirror system is placed at the zero point of the scale of the supporting platform.

• If the system of lenses and mirrors is correctly adjusted, we see a sinusoid of frequency 50 𝑘𝐻𝑧 and
amplitude not smaller than 1V at the Y-output of the operating unit connected to channel 2 of the
oscilloscope. Otherwise we must readjust the optical alignment of the system.

• The mirror system is slowly slid along the graduated scale of the platform until 150 cm, checking at
all times that the signal amplitude in the channel 2 of the oscilloscope, though weakened, does not
vanish. In this position, if we observe that the amplitude in channel 2 is smaller than 0.5V, we
delicately readjust the mirrors.

5.3. FIRST EXPERIMENT: Measurement of the velocity of light in air.

Once the system is set up and adjusted, we proceed in the following way:

1. Place the inverting mirror system at the zero point on the scale of the supporting platform.

2. Connect the oscilloscope and set the X-Y mode. A Lissajous figure appears on the display. Choose the
sensitivity of both channels so that the figure spreads over the screen.

3. With the “Phase” control in the operating unit, we adjust the phase difference between the two signals until
a straight line appears on the oscilloscope display, of slope either 1 or -1. This indicates that the phase
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 difference Φ between the signals is 0 or 𝜋, respectively (see figure 2). If the position of the mirror system at
which the straight line appears does not correspond to the scale zero point, we note the position at which
this occurs and designate it by 𝑥𝐼 .

4. The mirror system is slid along the platform until we observe a straight line with slope opposite to that of
the initial one. That is, the change in the length of the path traversed by light from the transmitter to the
receiver corresponds to a change of 𝜋 in the phase difference of signals in channels X (transmitter) and Y
(receiver) of the oscilloscope. We write this mirror displacement down and denote it by 𝑥𝑓 .

The mirror position at which this occurs corresponds to a phase difference Φ = π between the
transmitter and receiver signals. The path traversed by the light in this new position has changed by an amount
𝐿 = 2Δ𝑥 = (𝑥𝑓 − 𝑥𝐼 ). By making use of equation [5] we can determine the velocity value in air as:
𝐿 4𝜋𝑓𝑚 Δ𝑥
𝑣𝑎𝑖𝑟 = = = 4𝑓𝑚 (𝑥𝑓 − 𝑥𝐼 ) [𝑚/𝑠]. [6]
∆𝑡 Φ

𝐿
Steps 1 to 4 must be repeated five times to obtain an average value 𝐿 = , which allows minimising the
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random error that comes implicit with the measurement process to determine the velocity of light in air.

5.4. SECOND EXPERIMENT: Measurement of the velocity of light in a transparent solid.

In this case, we compare the velocity of light in the liquid or the solid block to that in air 𝑣𝑎𝑖𝑟 (previous
part). We proceed in the following way:

1. Place the inverting mirror system at position 𝑥𝐼 = 120 𝑐𝑚 (measurement one with the block; see figure 3).

2. Place the resin block somewhere, either in the ray’s outbound or return path between the transmitter (LED)
and the receiver (photodiode), with the longest edge, of length 𝐿𝐵 , parallel to the optical path, so that the
light passes through it.

Fig. 3 Experimental diagram for the measurement of the velocity of light in a solid or liquid medium.

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3. Check that a Lissajous figure appears on the oscilloscope display. With the “Phase” control in the operating
unit we adjust the phase difference between the two signals until a (Lissajous) straight line of slope either 1
or -1 appears on the oscilloscope display. This indicates that the phase difference Φ between the signals is 0
or 𝜋, respectively.

4. Remove the block from the platform to clear the optical path of the light.

5. Slide the mirror system towards the end of the platform until the Lissajous figure shown on the oscilloscope
becomes a straight line with the same slope as that of the initial one, when the resin block was in the optical
path of the ray. This new position of the mirror system is denoted by 𝑥𝑓 (measurement two, without the
block; see figure 3).

Since the phase difference between the transmitter and the receiver signals has been kept constant
(same Lissajous figure) we can state that light has spent the same amount of time 𝑡1 in traversing both distances.

In the first measurement light has passed through a distance 𝐿1 = 2𝑥𝐼 . As light crosses the block through
its longest edge, of size 𝐿𝐵 , the total amount of time, 𝑡1 , is the sum of the time spent by light going through the
block, 𝑡𝐵 , plus the time going through air, 𝑡𝑎𝑖𝑟 :

𝐿𝐵 (𝐿1 − 𝐿𝐵 )
𝑡1 = 𝑡𝐵 + 𝑡𝑎𝑖𝑟 = + [𝑠] [7]
𝑣𝐵 𝑣𝑎𝑖𝑟
where:
𝑡𝐵 ; 𝑣𝐵 : are the time it takes light to traverse the block and the corresponding velocity.
𝑡𝑎𝑖𝑟 ; 𝑣𝑎𝑖𝑟 : are the time it takes light to travel outside the block through air and the corresponding velocity.

In the second measurement, the distance light traverses is:

𝐿2 = 𝐿1 + 2Δ𝑥 = 𝐿1 + 2(𝑥𝑓 − 𝑥𝐼 ) [𝑚].

Since the time spent in both cases is the same, we have:

𝐿2 𝐿1 + 2(𝑥𝑓 − 𝑥𝐼 )
𝑡1 = = [𝑠] . [8]
𝑣𝑎𝑖𝑟 𝑣𝑎𝑖𝑟

By setting this equal to the time obtained in equation [7], we get:

𝐿𝐵 (𝐿1 − 𝐿𝐵 ) 𝐿1 + 2(𝑥𝑓 − 𝑥𝐼 )
+ = [𝑠]
𝑣𝐵 𝑣𝑎𝑖𝑟 𝑣𝑎𝑖𝑟
𝐿𝐵 𝐿𝐵 2(𝑥𝑓 − 𝑥𝐼 )
− = [𝑠]. [9]
𝑣𝐵 𝑣𝑎𝑖𝑟 𝑣𝑎𝑖𝑟

From equation [9], since we know from previous part the velocity of light in air, 𝑣𝐴 = 𝑐𝐿 , we can calculate
the velocity of light in the solid block 𝑣𝐵 as:

𝐿𝐵 𝑣𝑎𝑖𝑟
𝑣𝐵 = [𝑚/𝑠] . [10]
𝐿𝐵 + 2(𝑥𝑓 − 𝑥𝐼 )

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5.5. THIRD EXPERIMENT. Computation of the refractive index of a liquid.

In this case, we make use of the average velocity of light in air, 𝑣𝑎𝑖𝑟 , obtained in the first part to obtain the
refractive index of the liquid filling a cylindrical vessel, placed in the path traversed by light between the
transmitter and the receiver. We proceed in the following way:

1. Fill the cylindrical vessel with the liquid whose refractive index we want to determine. Place it on its holder,
parallel to the platform plane, either in the ray’s outbound or return path between the transmitter (LED) and
the receiver (photodiode). The light must pass through the transparent optical windows in the extremes of
the cylindrical vessel.

2. Place the inverting mirror system next to the cylindrical vessel, between 5 and 10 cm away from the end
closest to the vessel. This position of the mirror system is denoted by 𝑥𝐼 (measurement one, with the
vessel; see figure 3).

3. Check that a Lissajous figure appears on the oscilloscope display. It corresponds to a certain phase difference
between the input signals. With the “Phase” control in the operating unit we adjust the phase difference
between the two signals until a (Lissajous) straight line of slope either 1 or -1 appears on the oscilloscope
display. This indicates that the phase difference Φ between the signals is 0 or 𝜋, respectively.

4. Remove the cylindrical vessel with the liquid from the platform to clear the optical path of the light.

5. Slide the mirror system towards the end of the platform until the Lissajous figure shown on the oscilloscope
becomes a straight line with the same slope as that of the initial one. This new position of the mirror system
is denoted by 𝑥𝑓 (measurement two, without the cylindrical vessel; see figure 3).

Since the phase difference between the transmitter and the receiver signals has been kept constant
(same Lissajous figure) we can state that light has spent the same amount of time 𝑡1 in traversing both distances.
By working in a similar manner to that in the measurement of velocity of light in the resin block, we obtain an
expression equivalent to equation [9] for the distance in the liquid, 𝐿𝐿 :
𝐿𝐿 𝐿𝐿 2(𝑥𝑓 − 𝑥𝐼 )
− = [𝑠]. [11]
𝑣𝐿 𝑣𝑎𝑖𝑟 𝑣𝑎𝑖𝑟
Bearing in mind the definition of refractive index and by solving in equation [11], we get:
𝑣𝑎𝑖𝑟 2(𝑥𝑓 − 𝑥𝐼 )
𝑛𝐿 = = 1+ , [12]
𝑣𝐿 𝐿𝐿
where:
𝑣𝐿 : velocity of light in the liquid.
𝐿𝐿 = 100.0 ± 0.1 𝑐𝑚: length of the light path in the liquid.
𝑣𝑎𝑖𝑟 : velocity of light in air.

In the exercises to calculate the velocity of light in the solid and the refractive index of the liquid, distances
𝑥𝐼 and 𝑥𝑓 must be measured five times in order to minimise the random error.

Acknowledgements: This experiment and guide have been prepared by M. Tardío, who is gratefully acknowledged.
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