Calculating Refractive Index Using Fabry-Perot Interferometer

[HotMintea]

Homework Statement

A Fabry-Perot interferometer has spacing d = 2 cm between the glass plates, causing the direct and doubly reflected beams to interfere. As air is pumped out of the gap between the plates, the beams go through 23 cycles of constructive-destructive-constructive interference. If the wavelength of the light in the interfering beams is 5 × 10^−7 m, determine the index of refraction of the air initially in the interferometer.

Homework Equations

(I think) If constructive interference is occurring, 2d = mλ/n, for wave length λ, refractive index n and some integer m.

The Attempt at a Solution

I cannot figure out how to incorporate the following part in the equation: "As air is pumped out of the gap between the plates, the beams go through 23 cycles of constructive-destructive-constructive interference."

 

[SammyS]

...
I cannot figure out how to incorporate the following part in the equation: "As air is pumped out of the gap between the plates, the beams go through 23 cycles of constructive-destructive-constructive interference."

It means that m changes by 23.

Can you say which it did? Did m increase? or Did m decrease ??

Also, assuming that virtually all of the air was pumped out, what is the index of refraction for a vacuum ?

 

[HotMintea]

It means that m changes by 23.

Can you say which it did? Did m increase? or Did m decrease ??

Since wavelength gets longer as the air gets thinner, I think the number of wavelengths in the distance 2d will decrease as the air gets pumped out.

Also, assuming that virtually all of the air was pumped out, what is the index of refraction for a vacuum ?

The index of refraction in vacuum is defined to be 1.

I have: m λ/n = (m - 23) λ/1 = 2d. The left side says integer m times the contracted wavelength λ/n. The middle says (m - 23) times full length λ/1. The right side is twice the gap.

Solving the left and middle, I get m = (2d + 23λ)/λ. Substitute it for m in the left and solving the left and right, I get n = 1 + 23λ/2d.

Thanks for your help!

 

[SammyS]

What do you get for a numerical answer for n-1? (Just out of curiosity .)

 

[HotMintea]

What do you get for a numerical answer for n-1? (Just out of curiosity .)

23*5*10^-7m / 2*0.02m = 0.0002875

It agrees ↓

Air at STP | 1.000277
Air (0 C and 1 atm) | 1.000293
(http://en.wikipedia.org/wiki/List_of_refractive_indices#List)

 

[SammyS]

Excellent !