The French physicist Armand Fizeau was the first to measure...

The French physicist Armand Fizeau was the first to measure the speed of light accurately. He also found experimentally that the speed, relative to the lab frame, of light traveling in a tank of water that is itself moving at a speed $V$ relative to the lab frame is $$v = {c \over n} + kV$$ where $n$ = 1.333 is the index of refraction of water. Fizeau called $k$ the dragging coefficient and obtained an experimental value of $k$ = 0.44. What value of $k$ do you calculate from relativistic transformations?

Transcript

00:01 This is chapter 37 problem number 64.

00:03 We have given the speed of light traveling in a tank of water.

00:09 This tank of water is also moving at a speed capital v relative to the lab frame.

00:15 And then v equals c over n plus k capital v is given to us.

00:22 That is if we assume that the direction of the, moving tank of water is in the same direction as the light passing through it.

00:37 Then what we can write, actually, is you know, v equals v prime plus u from 1 plus uv prime over c squared.

00:51 What we have here is our u is actually our capital v, right, with respect to the lap frame, that's the speed at which the water travels.

01:00 The tank travels, v prime is our c over n in this case.

01:06 So if you plug everything in this equation, then what we have is the v equals c over m, because that's what v prime is here, right? u is plus capital v over 1 plus c over n times v over c squared.

01:25 So c, v over n, c squared.

01:32 Now, we can actually cancel that one c's.

01:37 So c over n plus capital v over 1 plus v over n times c.

01:45 So what we're looking for, sorry, i put prime here, let me take it back.

01:54 No need for prime here.

01:56 This is v, not v prime.

01:58 So what we are trying to calculate here is what this k is.

02:05 So then if we know v, if we subtract c over n from this term, we need to get to k times v, then we can divide both sides by v and get k, right? so let's say this equals two different problem, c over n plus k times capital v.

02:24 Now let me subtract c over n from this term in order to get to kv.

02:31 So then k times v is going to be equal to c over n plus capital v over 1 plus v over n minus c over n.

02:43 Now let's do this substitution...

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