Physics II Activity
Measure the Speed of light using the Foucault method

In this experiment you will use a method that is basically the same as that developed by Foucault in 1862. A diagram of the setup is shown above.

With all the equipment properly aligned and with the rotating mirror stationary, the optical path is as follows: The parallel beam of light from the laser is focused to a point image at s by lens L1 Lens L2is is positioned so that the image point at s is reflected from the rotating mirror (Rm) and is focused onto the fixed, spherical mirror (Fm). Fm reflects the image back along the same path to again focus the image at s.

In order that the reflected point image can be viewed through the microscope, a beam splitter is placed in the optical path, so a reflected image is also formed at point s’, just underneath the microscope objective.

Now suppose Rm is rotated slightly so that the reflected beam strikes Fm at a different point. Because of the spherical shape of Fm, the beam will still be reflected directly back toward Rm. An image of the source point will still be formed at s and s’. The only significant difference is that the point of reflection on Fm has changed. With each revolution of the rotating mirror, the system is “lined-up” once when the beam passes the fixed mirror Fm, and the reflected image at s’ “flashes” in the view of the microscope.

However, when Rm is rotated continuously at high speeds, the reflected image is no longer formed at s’. This is because, with Rm rotating, a light wave that travels from Rm to Fm and back finds Rm at a different angle than when it was when the light left Rm. As will be shown in the following derivation, by measuring the displacement of the image, the rate of rotation of Rm, the distance from Rm to Fm, and the magnification of L2, we can determine the speed of light. With careful attention to precision and accuracy, we should obtain a number within 2% of the standard value.

The relationship between the speed of light and the displacement of the images and the angular velocity w1 of the rotating mirror is quite complex. Figure 2 is Figure 1 "unfolded", as if the rotating mirror was not present.

Figure 2. The optics of Figure 1, but "unfolded" by imagining the rotating mirror to be missing, but still causing the effect.
 

When the mirror rotates fast, the image is displaced from position s to position s'. Similar triangles yield the relationship:

DS=2DDq (1)

The apparent deflection of the image is caused by the rotating mirror and the finite speed of light. If the mirror rotates an amount , the beam rotates twice  and the apparent displacement is given by
 

  (2)

Substitution of Eq. 2 into Eq. 1 yields:
 

 (3)

With equal to the angular velocity of the rotating mirror we have:

and 
Substituting into Eq. 3 we have:
or
or
 

(4)
In Eq. 4,  is measured from a frequency counter on the mirror controller which is calibrated in rev/sec.  is measured with a micrometer stage on the microscope, D and B are essentially measured with rulers. The measurement for A is a bit more subtle. Notice in Fig. 2 that the point S is an image of the point s formed by lens L2 which has a focal length f2. From the law of simple lenses where i and o are image distances and object distances respectively we have

Substituting (D + B) for i, A for o and rearranging, we have:
 

(5)
Solving for A we have

Notice in Eq. 4 that the displacement 18 is proportional to 19 with a slope inversely proportional to the speed of light c. Measure the displacements for several mirror speeds (both positive and negative) and plot the displacement vs speed. From the slope and the equation for the slope, obtain the speed of light and calculate the percent deviation from the textbook value.

Note: when adjusting the micrometer, always rotate the micrometer screw in the same direction (CW) for all readings to avoid backlash.

Problem Solving:

(Due Tues.  Probs are graded as "homework")
  1. Consider a beam of light shining perpendicularly (“normally”) onto a mirror. The beam is reflected straight back to the source. If the mirror is rotated 5 degrees, how much does the beam rotate? Show your work.
  2. If the distant mirror is located 6.00 m from the rotating mirror, how fast (rev/sec) will the rotating mirror have to spin in order to rotate the retro-reflected beam by 0.02 deg in the speed of light determination?
  3. Another technique for measuring the speed of light is to shine the light between the teeth of a rotating toothed wheel (sector wheel) to a distant mirror and back. Suppose the wheel has 100 teeth, and the distant mirror is 10 m distant. How fast must the wheel be spun in order to obtain light reflected between the next gap? Describe how you would determine whether the light was returned through the original gap, the next gap, or the gap 2 spaces away from the original gap?
  4. Another technique for measuring the speed of light is to measure time of flight of light pulses emitted from a light emitting diode, a photodetector, and an oscilloscope. The light pulse is reflected from a mirror at a known distance into the photodetector and a signal is displayed on an oscilloscope. The mirror is moved from close by to a “distant” location, and the delay time measured on the oscilloscope. The LED is driven by a pulse generator (repeating pulses) so that a repeating signal may be monitored on the O’scope. Suppose the electronics are able to form a light pulse 0.10 msec in duration, and that the O’scope can display a time resolution of 0.10 msec. (If you don’t know what a msec is, please look it up!) Calculate the distance of the distant mirror in order to obtain a pulse separation at 3 times the resolution of the oscilloscope.