## GR9277 # 96

ali8
Posts: 100
Joined: Sat Aug 29, 2009 8:20 am

### GR9277 # 96

http://grephysics.net/ans/9277/96

Hi,

So basically all the resources on the net use this formula as if it's "too well known" to be explained.

$$n_{gas}-n_{vacuum}=\frac{m\lambda}{2L}$$

Where m is the number of fringes observed, and L is the length of the gas-cell in the Michelson Interferometer.

I just want to know where I can find a derivation for this formula. I know in PGRE one won't derive anything

but I want to learn, not just to pass PGRE.

WhoaNonstop
Posts: 853
Joined: Mon Sep 21, 2009 1:31 am

### Re: GR9277 # 96

How I was going to explain it is on that page already... =P

-Riley

bfollinprm
Posts: 1203
Joined: Sat Nov 07, 2009 11:44 am

### Re: GR9277 # 96

ali8 wrote:http://grephysics.net/ans/9277/96

Hi,

So basically all the resources on the net use this formula as if it's "too well known" to be explained.

$$n_{gas}-n_{vacuum}=\frac{m\lambda}{2L}$$

Where m is the number of fringes observed, and L is the length of the gas-cell in the Michelson Interferometer.

I just want to know where I can find a derivation for this formula. I know in PGRE one won't derive anything

but I want to learn, not just to pass PGRE.
In the interferometer, the light passes through a medium with a refractive index differing from the environment by $$\Delta n$$. Since the medium is of length L, and it passes through twice, the light is slowed by some amount $$2L \Delta n$$ (the phase shift). Since a fringe occurs when the wavelengths match, you get maximum constructive interference (a fringe) any time the phase shift is an integer multiple of the wavelength. Thus, $$2L \Delta n$$ equals $$m \lambda$$, which after dividing by 2L gives the above formula.

ali8
Posts: 100
Joined: Sat Aug 29, 2009 8:20 am

### Re: GR9277 # 96

So:

Ray 1 do not pass through the cell, so total distance traveled is 2L*n_vacuume = 2L. Ray two pass through it, so it
passes an additional distance equal to 2L*n_gas=2Ln. Now the number of fringes m is the number of "additional waveleghts",
which is just (2Ln-2L)/Lambda.

bfollinprm
Posts: 1203
Joined: Sat Nov 07, 2009 11:44 am

### Re: GR9277 # 96

Doesn't go farther, just takes additional time to go the same distance. The retardation is given by $$d \Delta n$$, verified by simple dimensional analysis to be expressed in meters (that the light would have traveled in that time).

ali8
Posts: 100
Joined: Sat Aug 29, 2009 8:20 am

### Re: GR9277 # 96

bfollinprm wrote:Doesn't go farther, just takes additional time to go the same distance. The retardation is given by $$d \Delta n$$, verified by simple dimensional analysis to be expressed in meters (that the light would have traveled in that time).
Oh sure, you are right.