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Chapter 3: Problem 81

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?

Short Answer

Expert verified
The movable mirror moved approximately 73.929 µm. This was calculated using the relation \(n * λ = 2 * d\), where \(n\) is the number of fringes (251), \(λ\) is the wavelength of light (589 nm), and \(d\) is the distance traveled by the movable mirror.

Step by step solution

01

Understand the Michelson interferometer setup and optical path difference formula

In a Michelson interferometer, a light beam is split into two parts, and each part is reflected by a mirror. When the two beams recombine, they form an interference pattern (fringes) as a result of the difference in their optical paths. The optical path difference (OPD) is given by the formula: OPD = 2 * d where d is the distance traveled by the movable mirror.
02

Determine the relationship between the number of fringes, wavelength, and optical path difference

The number of fringes observed (n) is related to the OPD and the wavelength of the light (λ) by the formula: OPD = n * λ Combining this equation with the previous one, we can relate the number of fringes, the wavelength, and the distance traveled by the movable mirror: n * λ = 2 * d
03

Rearrange the formula to solve for the distance traveled by the movable mirror (d)

Rearranging the formula to solve for d, we get: d = (n * λ) / 2
04

Plug in the given values and calculate the distance

We are given the number of fringes n = 251 and the wavelength λ = 589 nm. Plugging these values into the formula, we get: d = (251 * 589 * 10^{-9} m) / 2 Calculating this, we find: d ≈ 7.3929 * 10^{-5} m
05

Express the answer in an appropriate unit

The calculated distance is in meters, but it would be more convenient to present the result in micrometers (µm): d ≈ 73.929 µm So, the movable mirror moved approximately 73.929 µm.

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Most popular questions from this chapter

A transparent film of thickness \(250 \mathrm{nm}\) and index of refraction of 1.40 is surrounded by air. What wavelength in a beam of white light at near- normal incidence to the film undergoes destructive interference when reflected?

Red light \((\lambda=710 . \mathrm{nm})\) illuminates double slits separated by a distance \(d=0.150 \mathrm{mm} .\) The screen and the slits are \(3.00 \mathrm{m}\) 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 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 \((n=1.000293)\) 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?

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

Find the largest wavelength of light falling on double slits separated by \(1.20 \mu \mathrm{m}\) for which there is a first-order maximum. Is this in the visible part of the spectrum?
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