Measuring the Velocity of Light


1. Goals and Theories

1) Goals : We can determine the velocity of light. Using the phase difference between the transmitter and the receiver signal of the modulated light intensity, we also can get the light velocities in water and in synthetic resin.

2)Theories :

Experiment1)




X(t) is intensity of the emitted light which is modulated with 50.1 MHz frequency f.

 

Y(t) is the intensity of received light which travelled along the outer light path.

 where φ is the electrically variable phase of the received light. When φ is adjusted by the phase knob on the operating unit so that there is no phase difference between the emitted and the received light, we could get the fact that   so .

.Therefore, 


the Lissajous figure on oscilloscope is shown the straight line form. And we change the phase ,


So, the Lissajous figure is made the straight line slopping in the opposite direction. In this moment, the travel time t, becomes  

On the other hand, total distance is  , therefore velocity of light c in air is  (1)

 

 

 

Experiment 2)


The time of travel the distance through the air and medium, t1 is shown by this formula.

And we measure the no medium case, increase the distance between transmitter and receiver by  .

. Find the same Lissajous figure, because we couldn’t get the time to get receiver. So we use the phase relationship between transmitter and receiver signals is the same in both cases,

at same Lissajous figure, k=0, Thus we obtain the refractive index, (2)

 

 

2. Results and Analysis

*Frequency (f = 50.1 MHz)

A.   Measuring the Velocity of Light in Air

Table 1. The light velocity in air

The position of

the mirror(x1)[m]

The position of

the mirror(x2)[m]

The total travel distance (2Δx)

The light velocity in air

(cair)Experimental[m/s]

The light velocity in air

(cair)Theoretical[m/s]

0.000

1.490

2.980

298596000

299792458

0.000

1.500

3.000

300600000

0.000

1.496

2.992

299798400

 

Average value (CExp)

299664800

Error (%)

0.043

 

Analysis

Using this equation,  

proved in theory 1,


CExp=1)+2)+3)/3=299664800

Error percentage was  


Although it is a small error, I believe that the reason for the error is that the speed of light can be reduced because of particle collision or the like because the air is not in perfect vacuum state. Compared with the value obtained by measuring the refractive index n = 1.0003 in the air (299702547.2m / s, ), the similarity becomes more similar. Nevertheless, the residual error is presumed to be due to errors or measurement errors due to the distance between the mirrors.

 

B. Measuring the Velocity of Light in Various materials

Table 2. The light velocity in water

The length of the water tube lw =     1.016     m

 

1

2

3

The base position of the mirror(x1)

105.0cm

105.0cm

105.0cm

The position of the mirror(x2)

121.0cm

123.0cm

120.0cm

Δx[m]

0.160m

0.180m

0.150m

1.315

1.354

1.295

 

Average

1.321

 

The light velocity in water

(cW)Experimental[m/s]

226943571.5

The light velocity in water

(cW)Theoretical[m/s]

224900568.6

 

Error[%]

0.908%

 

Table 3. The light velocity in the synthetic resin.

The length of the synthetic resin block lR =   0.282 m

 

1

2

3

The base position of the mirror(x1)

66.7cm

66.7cm

66.7cm

The position of the mirror(x2)

76.6cm

74.8cm

74.9cm

Δx[m]

0.089m

0.081m

0.082m

nR

1.631

1.574

1.582

 

Average

1.596

 

The light velocity in Resin

(cR)Exp[m/s]

187879125.6

The light velocity in Resin

(cR)Theo[m/s]

187722265.5

 

Error[%]

0.084%

 

Analysis

Using this equation,  proved in theory 2, 


In table 2, the light velocity in the water,

In table 2, the light velocity in the synthetic resin,


In both experiments, measuring the refractive index and the speed of light of water and synthetic resin, the theoretical value is smaller than experimental value. The reason is that there is a problem measuring the Δx[ because it is difficult to measure the area where the light is reflected correctly, or the theoretical value may slightly differ from the known value by using a material whose refractive index is slightly different from the known value. In addition, we do not experiment in vacuum, but we think that when we experiment in air, there may make slight differences.


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