Note that the distance between L and M is not equal to that between L and S. As M moves farther from the lens, S will need to be moved closer in order for both points to remain at the focus of the other's point source. This is true provided both points are beyond the focal length of the lens (that point where beams of light parallel on one side of the lens would meet on the other side).
By moving S and M farther apart, all the while keeping each at the other's point focus, we increase the distance the light must travel and therefore the time it will take. Even so, the time taken is exceedingly short and difficult to measure.
Instead of Fizeau's wheel, Foucault
used a rotating mirror interposed between S and L
as in the next diagram.44
By rotating the mirror at a constant speed, the amount of deflection will be the
same for all light beams that go through L, strike M and return.
Then, for a continuous beam of light from S and a constant high speed of rotation
of R, an image of the source will appear beside S instead of coincident
By carefully measuring the amount of displacement from S to I (see Figure 4), and the distance from S to R, the angle of deflection can be determined. Together with the known, fixed speed of rotation, this angle can be used to determine the time it took light to travel the distance from R to M and back. Dividing distance by time gives a determination of the speed of light.
Let denote the angle of deflection. Then the angle through which the mirror has rotated
is easily shown to be
The angle in degrees is
If the speed of rotation is n measured in cycles per second, then the time taken for the light beam to travel
from R to M and back is
The speed of light transmitted under the conditions of the study is therefore
In this arrangement, the distances |IS| and |SR| should be as large
as possible to reduce the error in measuring . The distance |IS| is maximized by
maximizing the speed of rotation of R and the distance |RM|.
Michelson's principal innovation in Foucault's design allowed
|RM| to be very large.
In Foucault's setup, M was spherical with centre at R.
The greatest distance |RM| achieved by Foucault
was 20 metres
(page 117 )
which produced a displacement |IS|of only 0.7mm
(page 118 ).
Michelson chose to place the rotating
mirror at the focal point of the lens
which allowed him to
use a flat mirror for M.
That is, R should be placed
at that point where parallel light beams passing through
the lens from M meet on the other side as in Figure 5.
These innovations produced a displacement of more than 100 mm. Such a large displacement solved another difficulty. Originally the eyepiece to observe the displaced image at S was offset using an inclined plate of silvered glass to avoid interference between the observer and the outgoing beam of light. Once the the displacement exceeded 40 mm, it was possible to remove the inclined plate and observe the displaced image directly. Michelson (page 116 ) noted ``Thus the eye-piece is much simplified and many possible sources of error are removed.''