4. In 1851 the first hints of relativity were experimentally measured by Fizeau

Question

4. In 1851 the first hints of relativity were experimentally measured by Fizeau (though a now discredited theory was used to explain the data). This involved measuring the speed of light in water which was flowing in a pipe. Consider a pipe with water - index of refraction n - flowing in it at speed vu. Use the relativistic velocity transformation rule to find an expression for the speed of light moving in water, in the same direction as the flowing water, measured with respect to the pipe. Is this smaller, greater or the same as the Galilean result? Note: The Galilean transformations for velocity with the textbook conventions are u, = u-v, u = u, + v.

Explanation / Answer

Refractive index is defined as the ratio of speed of light in vaccum to the speed of light in respective medium.   

   N=C/V(l)

   where N is refractive index

C is speed of light apprximately equal to 3*10^8 m/sec

V(l) is speed of in the respective medium

As the medium here is water flowing in pipe ,

from the above information we can find the speed of light the water flowing the pipe,

N=C/V(l)

V(l)=C/N

V(l)=(3*10^8)/N

The speed of light in water flowing in a pipe is given by V(l)=3*10^8/N

Where as in galilean transformation, relative speed of light in water in the same direction of flow is given by

V(l)=V(w)+C

   V(l)=V(w)+3*10^8 where V(w) is speed of the flowing water

Refractive index (N) of water is 1.333 so when measured using refractive index formula V(l) is less than 3*10^8 which is less than speed of light in vaccum

   V(l)=3*10^8/1.333=2.25*10^8

whereas the galilean result shows that V(l) =V(w)+(3*10^8). As V(w) cannot be zero or negative V(l) is always greater than 3*10^8

So therefore relative speed of water measured using refractive index of water flowing in pipe is less than that of galilean result.

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