# 3.5 The michelson interferometer  (Page 3/7)

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## Summary

• When the mirror in one arm of the interferometer moves a distance of $\lambda \text{/}2$ each fringe in the interference pattern moves to the position previously occupied by the adjacent fringe.

## Key equations

 Constructive interference $\text{Δ}l=m\lambda ,\phantom{\rule{0.6em}{0ex}}$ for m = 0, ±1, ±2, ±3… Destructive interference $\text{Δ}l=\left(m+\frac{1}{2}\right)\lambda ,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}$ for m = 0, ±1, ±2, ±3… Path length difference for waves from two slits to a common point on a screen $\text{Δ}l=d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$ Constructive interference $d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =m\lambda ,\phantom{\rule{0.6em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}m=0,\text{±}1,\text{±}2,\text{±}3\text{,…}$ Destructive interference $d\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =\left(m+\frac{1}{2}\right)\lambda ,\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}0,\text{±}1\text{,}\phantom{\rule{0.2em}{0ex}}\text{±}2\text{,}\phantom{\rule{0.2em}{0ex}}\text{±}3\text{,}\dots$ Distance from central maximum to the m th bright fringe ${y}_{m}=\frac{m\lambda D}{d}$ Displacement measured by a Michelson interferometer $\text{Δ}d=m\frac{{\lambda }_{0}}{2}$

## Conceptual questions

Describe how a Michelson interferometer can be used to measure the index of refraction of a gas (including air).

In one arm, place a transparent chamber to be filled with the gas. See [link] .

## Problems

A Michelson interferometer has two equal arms. A mercury light of wavelength 546 nm is used for the interferometer and stable fringes are found. One of the arms is moved by $1.5\mu \text{m}$ . How many fringes will cross the observing field?

What is the distance moved by the traveling mirror of a Michelson interferometer that corresponds to 1500 fringes passing by a point of the observation screen? Assume that the interferometer is illuminated with a 606 nm spectral line of krypton-86.

$4.55\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}\text{m}$

When the traveling mirror of a Michelson interferometer is moved $2.40\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.2em}{0ex}}\text{m}$ , 90 fringes pass by a point on the observation screen. What is the wavelength of the light used?

In a Michelson interferometer, light of wavelength 632.8 nm from a He-Ne laser is used. When one of the mirrors is moved by a distance D , 8 fringes move past the field of view. What is the value of the distance D ?

$D=2.53\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}$

A chamber 5.0 cm long with flat, parallel windows at the ends is placed in one arm of a Michelson interferometer (see below). The light used has a wavelength of 500 nm in a vacuum. While all the air is being pumped out of the chamber, 29 fringes pass by a point on the observation screen. What is the refractive index of the air? For 600-nm wavelength light and a slit separation of 0.12 mm, what are the angular positions of the first and third maxima in the double slit interference pattern?

$0.29\text{°}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0.86\text{°}$

If the light source in the preceding problem is changed, the angular position of the third maximum is found to be $0.57\text{°}$ . What is the wavelength of light being used now?

Red light ( $\lambda =710.\phantom{\rule{0.2em}{0ex}}\text{nm}$ ) illuminates double slits separated by a distance $d=0.150\phantom{\rule{0.2em}{0ex}}\text{mm}.$ The screen and the slits are 3.00 m apart. (a) Find the distance on the screen between the central maximum and the third maximum. (b) What is the distance between the second and the fourth maxima?

a. 4.26 cm; b. 2.84 cm

Two sources as in phase and emit waves with $\lambda =0.42\phantom{\rule{0.2em}{0ex}}\text{m}$ . Determine whether constructive or destructive interference occurs at points whose distances from the two sources are (a) 0.84 and 0.42 m, (b) 0.21 and 0.42 m, (c) 1.26 and 0.42 m, (d) 1.87 and 1.45 m, (e) 0.63 and 0.84 m and (f) 1.47 and 1.26 m.

Two slits $4.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}\phantom{\rule{0.2em}{0ex}}\text{m}$ apart are illuminated by light of wavelength 600 nm. What is the highest order fringe in the interference pattern?

6

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