# 3.5 The michelson interferometer  (Page 6/7)

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A Michelson interferometer with a He-Ne laser light source ( $\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$ ) projects its interference pattern on a screen. If the movable mirror is caused to move by $8.54\phantom{\rule{0.2em}{0ex}}\mu \text{m}$ , how many fringes will be observed shifting through a reference point on a screen?

An experimenter detects 251 fringes when the movable mirror in a Michelson interferometer is displaced. The light source used is a sodium lamp, wavelength 589 nm. By what distance did the movable mirror move?

$73.9\phantom{\rule{0.2em}{0ex}}\mu \text{m}$

A Michelson interferometer is used to measure the wavelength of light put through it. When the movable mirror is moved by exactly 0.100 mm, the number of fringes observed moving through is 316. What is the wavelength of the light?

A 5.08-cm-long rectangular glass chamber is inserted into one arm of a Michelson interferometer using a 633-nm light source. This chamber is initially filled with air $\left(n=1.000293\right)$ at standard atmospheric pressure but the air is gradually pumped out using a vacuum pump until a near perfect vacuum is achieved. How many fringes are observed moving by during the transition?

47

Into one arm of a Michelson interferometer, a plastic sheet of thickness $75\phantom{\rule{0.2em}{0ex}}\mu \text{m}$ is inserted, which causes a shift in the interference pattern by 86 fringes. The light source has wavelength of 610 nm in air. What is the index of refraction of this plastic?

The thickness of an aluminum foil is measured using a Michelson interferometer that has its movable mirror mounted on a micrometer. There is a difference of 27 fringes in the observed interference pattern when the micrometer clamps down on the foil compared to when the micrometer is empty. Calculate the thickness of the foil?

$8.5\phantom{\rule{0.2em}{0ex}}\mu \text{m}$

The movable mirror of a Michelson interferometer is attached to one end of a thin metal rod of length 23.3 mm. The other end of the rod is anchored so it does not move. As the temperature of the rod changes from $15\phantom{\rule{0.2em}{0ex}}\text{°C}$ to $25\phantom{\rule{0.2em}{0ex}}\text{C}$ , a change of 14 fringes is observed. The light source is a He Ne laser, $\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$ . What is the change in length of the metal bar, and what is its thermal expansion coefficient?

In a thermally stabilized lab, a Michelson interferometer is used to monitor the temperature to ensure it stays constant. The movable mirror is mounted on the end of a 1.00-m-long aluminum rod, held fixed at the other end. The light source is a He Ne laser, $\lambda =632.8\phantom{\rule{0.2em}{0ex}}\text{nm}$ . The resolution of this apparatus corresponds to the temperature difference when a change of just one fringe is observed. What is this temperature difference?

$0.013\text{°}\text{C}$

A 65-fringe shift results in a Michelson interferometer when a $42.0\text{-}\mu \text{m}$ film made of an unknown material is placed in one arm. The light source has wavelength 632.9 nm. Identify the material using the indices of refraction found in [link] .

## Challenge problems

Determine what happens to the double-slit interference pattern if one of the slits is covered with a thin, transparent film whose thickness is $\lambda \text{/}\left[2\left(n-1\right)\right]$ , where $\lambda$ is the wavelength of the incident light and n is the index of refraction of the film.

Bright and dark fringes switch places.

Fifty-one narrow slits are equally spaced and separated by 0.10 mm. The slits are illuminated by blue light of wavelength 400 nm. What is angular position of the twenty-fifth secondary maximum? What is its peak intensity in comparison with that of the primary maximum?

A film of oil on water will appear dark when it is very thin, because the path length difference becomes small compared with the wavelength of light and there is a phase shift at the top surface. If it becomes dark when the path length difference is less than one-fourth the wavelength, what is the thickest the oil can be and appear dark at all visible wavelengths? Oil has an index of refraction of 1.40.

The path length must be less than one-fourth of the shortest visible wavelength in oil. The thickness of the oil is half the path length, so it must be less than one-eighth of the shortest visible wavelength in oil. If we take 380 nm to be the shortest visible wavelength in air, 33.9 nm.

[link] shows two glass slides illuminated by monochromatic light incident perpendicularly. The top slide touches the bottom slide at one end and rests on a 0.100-mm-diameter hair at the other end, forming a wedge of air. (a) How far apart are the dark bands, if the slides are 7.50 cm long and 589-nm light is used? (b) Is there any difference if the slides are made from crown or flint glass? Explain.

[link] shows two 7.50-cm-long glass slides illuminated by pure 589-nm wavelength light incident perpendicularly. The top slide touches the bottom slide at one end and rests on some debris at the other end, forming a wedge of air. How thick is the debris, if the dark bands are 1.00 mm apart?

$4.42\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.2em}{0ex}}\text{m}$

A soap bubble is 100 nm thick and illuminated by white light incident at a $45\text{°}$ angle to its surface. What wavelength and color of visible light is most constructively reflected, assuming the same index of refraction as water?

An oil slick on water is 120 nm thick and illuminated by white light incident at a $45\text{°}$ angle to its surface. What color does the oil appear (what is the most constructively reflected wavelength), given its index of refraction is 1.40?

for one phase change: 950 nm (infrared); for three phase changes: 317 nm (ultraviolet); Therefore, the oil film will appear black, since the reflected light is not in the visible part of the spectrum.

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