# IPhO2013

Speed of light (solution)
In this document decimal comma is used instead of decimal point in graphs and tables

E1

1.1

Use the LDM to measure the distance

from the top of the table to the floor. Write 0.4 down the uncertainty . Show with a sketch how you perform this measurement.

. See the sketch in the figure corresponding to 1.3b. It must appear how the height is measured with the LDM in the rear mode.

1.2a

Measure corresponding values of and . Set up a table with your measurements. Draw 1.8 a graph showing as a function of .

Here, a 2 m cable is used, but 1 m is sufficient. There should be about 8 lengths evenly distributed in the interval from 0 m to 1 m.
x m 0,103 0,176 0,348 0,546 0,617 0,839 1,025 1,107 1,750 2,000 y m 0,177 0,232 0,396 0,517 0,570 0,748 0,885 0,950 1,459 1,642

Display y as function of cable length x
y = 0,7710x + 0,1014 R? = 0,9996 1,8 1,6 1,4 Display y [m] 1,2 1,0 0,8 0,6 0,4 0,2 0,0 0,0 0,5 1,0 cable length x [m] 1,5 2,0

Page 1 of 5

Speed of light (solution)
Use the graph to find the refractive index for the material from which the core of 1.2b the fiber optic cable is made. Calculate the speed of light in the core of the fiber 1.2 optic cable. The refractive index is twice the gradient of the linear graph, The reason for that is that the travel time for a light pulse .

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The display will therefore show The speed of light in the core of the cable is

.

Measure with the LDM the distance to the laser dot where the laser beam hits the table top. Then move the box with the LDM horizontally until the laser beam hits the 1.3a 0.2 floor. Measure the distance to the laser dot where the laser beam hits the floor. State the uncertainties. , Calculate the angle using only these measurements Determine the uncertainty . and (from problem 1.1).

1.3b

0.4

( ( (see the figure)

) )

Page 2 of 5

Speed of light (solution)
Measuring the horizontal part of some triangle is very inaccurate because of the size of the laser dot. No marks will be awarded for that. Using as the uncertainty of , and , the uncertainty of can be calculated as follows: ( Using simple derivatives yields )

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(

)

(

)

Otherwise, using min/max method ( Alternatively, calculate using ( ) ) √( √( ) Also, accept and . ) ( ( ( ) ) ) √ ( and then )

1.4a

Measure corresponding values of a graph of as a function of .

and . Set up a table with your measurements. Draw

1.6

x [mm] 4 17 27 32 39 51 58 66 76 82 90 96

y [mm] 450 454 457 459 461 466 467 471 473 476 478 480

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Speed of light (solution)
1.4b Use equations to explain theoretically what the graph is expected to look like. The time it takes the light to reach the water surface is ( 1.2

E1

)

From the water surface to the bottom the light uses the time

Total travel time forth and back (

) ) ( )

Hence, the display will show (we simply write

which is a linear function of . Then, using a trigonometric identity and Snell’s law, √ From this the gradient is found to be √ √ √ .

1.4c Use the graph to determine the refractive index Knowing the gradient Introducing a practical parameter,

for water.

1.2

from the graph, the index of refraction

is found by solving this equation.

the above equation becomes √ with the solution √ From the graph is found All solutions with √ √ √ √ ( and hence . are omitted. Page 4 of 5 )

Speed of light (solution)

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Another and more elegant way of finding

is to use Snell’s law in the equation

This yields

From here the angle

can be calculated leading to , and measurement of is not involved).

. This method also only uses the graph and

The table value for pure water at normal conditions is The following approximations can be used: For small angles √

at the wavelength

.

(

)

(

)

(

(

) )

For very small angles, we get It is much simpler, but not recommendable, to do the experiment with very small Reflections in the water surface will ruin the signal from the bottom. .

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