Katie McCullough


Speed of Light Formal Report
(The Foucault Method)

Light Speed!
    The speed of light and its measurement has an intresting history and  has facinated scientists for centuries. Galileo was the first scientist on record to attempt a measurement for the speed of light.  His suggested method was to have two people with lanterns standing at a distance from eachother.  One person was to uncover their lantern while the other watched and subsequently uncovered theirs.  After practice to prevent delay due to reaction time, the two people were to move a great distance apart and see if any time lag was detected.  The method did not show light to have a measurable speed.

    The next scientist to attempt a measurement of the speed of light was Ole Roemer, a Danish astronomer who studied the moons of Jupiter in 1676.  Roemer found that  Io, one of Jupiterís moons, was eclipsed by Jupiter as it orbited at a steady rate.  He also found that there was a time lag in the orbits which later led to the time being ahead.  The time lag and following "catch up" coincided with the earthís being most distant from Jupiter.  Roemer thought that the time changes could be due to the length of time that it took light reflected from Jupiter to reach the earth. The assumption was incorrect but led to a surprisingly close approximation of the speed of light, 138,000 miles per second.

    The speed of light was once again measured by James Bradley, an English astronomer, in 1728.  The approximation that he made was based on the concept that light went about 10,000 times the speed of the earthís orbit.  The concept was based on a set of calculations which dealt with the apparent angle of falling rain and its relationship to the speed of the earthís orbit.  His calculation for the speed of light was 185,000 miles per second, only off by 1%.

    Fizeau and Foucault both made measurements of the speed of light in the 1850ís.  The apparatus used by Fizeau in measuring the speed of light consisted of a light source and rotating wheel.  The rotating wheel had tines in it through which light could pass.  The light was sent out through one place in the tines, reflected by a mirror, and if the wheel was rotating fast enough, returned through a different tine.  The speed of light could subsequently be calculated using the distance from mirror to wheel, speed of wheel rotation,  and the spacing between the tines of the wheel.  Fizeau's apparatus yielded a fairly accurate calculation for the speed of light.

    Foucault altered Fizeauís method to include a rotating mirror rather than a spinning wheel, an apparatus very similar to that used in our lab.  A light source was shone onto a rotating mirror  where it was reflected onto a distant concave fixed mirror.  The light was then bounced off of the fixed mirror, back to the rotating mirror, and returned to the light source.  If the rotating mirror was spinning at high speeds, the returning light hit it at a slightly different place causing the returning beam to be shifted from its original path.  The speed of light could then be measured by taking into account the speed of mirror rotation, amount of shift, and distance between the rotating and fixed mirror.  Foucaultís calculation using this method was only off by 1000 miles per second.  Michelson improved on Foucaultís measurement in 1877 by replacing the rotating mirror with an octagonal one and increasing the distance between the octagonal and fixed mirror.
 



Leon Foucault

     In order to measure the speed of light during our lab time, the Foucault method was used with some minor alterations. The apparatus involved a laser beam rather than a lantern, a microscope, and two lenses in addition to the original rotating mirror and fixed mirror.  A diagram of the apparatus is shown below.
 
 








 










     The apparatus was constructed so that a beam shone from a laser was sent through a first lens (L1) where it was focused into a point s.  A second lens (L2) was positioned so that the point at s was directed onto a rotating mirror which reflected the light onto a distant fixed mirror.  The fixed mirror in turn reflected the light back in the direction of the rotating mirror.   From the rotating mirror, the light was passed through a beam splitter which caused the reflected image to form a second point at sí, directly over a microscope objective.  The microscope allowed the light to be viewed by the experimenter so that any shift in the path of the light could be seen.
     When the rotating mirror was spun slowly, the light beam was reflected directly back along its original path.  When the mirror was rotated at high speeds, it reflected the light along a slightly different path.  The shift could be seen by looking at the beam through the microscope and measuring the distance that the microscope platform had to be shifted in order to have the returning light focused at sí.  The position of the microscope platform was controlled with a micrometer which had millimeter measurements.  Recordings were taken for the micrometer reading at different mirror speeds.  The direction of the mirrorís spin was also reversed so that the microscope platform had to be shifted in the opposite direction in order to create an image at sí.  The data collected is shown below.
 
 
 
 
 

Mirror Speed (rev/sec)
Micrometer Reading (mm)
64
9.27
200
9.25
303
9.24
434
9.21
647
9.19
833
9.17
1011
9.15
(-)                      89
9.30
307
9.32
504
9.36
745
9.38
979
9.40

     The data was placed on a graph in order to find the slope.  The slope was then used in the equation
 

to find the speed of light.  The equation was derived based on the concept that similar triangles were formed by light entering and leaving the second lens.  A diagram of the similar triangles is shown below.

 The equation was derived as follows:

When the mirror rotated fast, the image was displaced form position s to position s'.  Similar triangles yield the relationship:
 
 



 









The deflection of the image S was caused by the rotating mirror and the speed of light.  The angle of rotation could be incorporated into the equation so that  the displacement of the beam is twice the angle of rotation.
 
 


 










When equation 2 is substituted into equation 1, the result is:
 
 





 










Omega is used to represent the angular velocity of the rotating mirror and is incorporated into the equation as:
 
 


 










Also, delta t which represented the time required for the light to complete a round trip could be explained through the equation
 
 



 









By combining the equations, a relationship between the similar triangles and other variables could be found:
 
 


 










The equation was then rearranged algebraically so that c, the constant for the speed of light was inserted.
 
 


 











and
 



 









In order to complete the equation, the relationship between the focal length of the lenses and image distances had to be used.  The equation is known as the Gaussian Lens Equation, the explanation for which can be found on page 976 of Hecht, Physics Calculus.  The equation is shown below.
 
 


 










By substituting D+B for i and A for o, the equation below can be formed.
 
 




 










Solving for A results in the equation:
 
 



 









By plotting delta s versus frequency, a slope could be obtained.  The graph is shown below.
 
 




 









The slope was incorporated into the equation as follows:
 







    Using the equation, the speed of light was found to be around 3.322*108 meters per second.  The literature value for the speed of light was 2.99*108 meters per second. The percent difference was 10%.   There was some error in the derivation of the equation which may have accounted for part of the discrepancy.   Overall, the experiment was interesting because it made a concept that seemed vague more clear by making it measurable.  It was also interesting because the materials used were much more technologically advanced than those used by the first measurers of the speed of light, but the calculations were no more accurate.  It is interesting to consider what the scientists who first measured the speed of light would have been capable of doing had they had access to the same technologically advanced materials.