A gas-filled cell of length 5 cm is inserted in one arm of a Michelson interferometer as shown in the figure. The interferometer is illuminated by light of wavelength 500 nanometers. As the gas is evacuated from the cell, 40 fringes cross a point in the field of view. The refractive index of this gas is most nearly?
A. 1.02
B. 1.002
C. 1.0002
D. 1.00002
E. 0.98
(GR9277 #96)
Solution:
Michelson interferometer, 2ΔL = Nλ
N = 2ΔL/λ
Nvac = 2ΔL/λvac
Ngas = 2ΔL/λgas
λgas = λvac/n
Ngas = 2ΔL/λgas = 2ΔLn/λvac
ΔN = Ngas − Nvac
= 2ΔLn/λvac − 2ΔLn/λvac
= 2ΔL/λvac (n − 1)
Given:
ΔL = 5 cm = 5 × 10−2 m
ΔN = 40 fringes
λvac = 500 nm = 5 × 10−7 m
40 = [2(5 × 10−2)/(5 × 10−7)](n − 1)
40 = (2 × 105)(n − 1)
40/(2 × 105) = n − 1
2 × 10−4 = n − 1
n = 0.0002 + 1 = 1.0002
Answer: C
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