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Chapter 34: Problem 35

34.35 Monochromatic blue light \((\lambda=449 \mathrm{nm})\) is beamed into a Michelson interferometer. How many fringes move by the screen when the movable mirror is moved a distance \(d=\) \(0.381 \mathrm{~mm} ?\)

Short Answer

Expert verified
Answer: Approximately 1696 fringes move across the screen.

Step by step solution

01

Convert the given wavelength and distance to the same units

We need to make sure that the values for the wavelength and distance are in the same units. The given wavelength is 449 nm, and the distance the mirror is moved is 0.381 mm. Convert the distance to nanometers, as follows: \(d = 0.381\,\text{mm} \times \frac{1000\,\text{nm}}{1\,\text{mm}} = 381000\,\text{nm}\) Now, both wavelength and distance values are in nanometers.
02

Calculate the path length difference between the interferometer's arms

Remember that the path length difference between the two arms of the interferometer is given by \(2d\). So, we can calculate it as follows: \(\Delta L = 2d = 2 \times 381000\,\text{nm} = 762000\,\text{nm}\)
03

Calculate the number of fringes that move across the screen

To find the number of fringes that pass the screen, divide the path length difference by the wavelength of the light. Then, round the result to the nearest whole number to get the number of fringes: \(N = \frac{\Delta L}{\lambda} = \frac{762000\,\text{nm}}{449\,\text{nm}} \approx 1696\) So, when the movable mirror is moved a distance \(d = 0.381\,\text{mm}\), approximately 1696 fringes move across the screen in the Michelson interferometer.

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Most popular questions from this chapter

The Hubble Space Telescope (Figure 34.33 ) is capable of resolving optical images to an angular resolution of \(2.80 \cdot 10^{-7}\) rad with its 2.40 -m mirror. How large would a radio telescope have to be in order to image an object in the radio spectrum with the same resolution, assuming the wavelength of the waves is \(10.0 \mathrm{~cm} ?\)

A Newton's ring apparatus consists of a convex lens with a large radius of curvature \(R\) placed on a flat glass disc. (a) Show that the distance \(x\) from the center to the air, thickness \(d,\) and the radius of curvature \(R\) are given by \(x^{2}=2 R d\) (b) Show that the radius of nth constructive interference is given by \(x_{\mathrm{n}}=\left[\left(n+\frac{1}{2}\right) \lambda R\right]^{1 / 2} .\) (c) How many bright fringes may be seen if it is viewed by red light of wavelength 700\. \(\mathrm{nm}\) for \(R=10.0 \mathrm{~m},\) and the plane glass disc diameter is \(5.00 \mathrm{~cm} ?\)

The Michelson interferometer is used in a class of commercially available optical instruments called wavelength meters. In a wavelength meter, the interferometer is illuminated simultaneously with the parallel beam of a reference laser of known wavelength and that of an unknown laser. The movable mirror of the interferometer is then displaced by a distance \(\Delta d,\) and the number of fringes produced by each laser and passing by a reference point (a photo detector) is counted. In a given wavelength meter, a red He-Ne laser \(\left(\lambda_{\mathrm{Red}}=632.8 \mathrm{nm}\right)\) is used as a reference laser. When the movable mirror of the interferometer is displaced by a distance \(\Delta d\), a number \(\Delta N_{\text {Red }}=6.000 \cdot 10^{4}\) red fringes and \(\Delta N_{\text {unknown }}=7.780 \cdot 10^{4}\) fringes pass by the reference photodiode. a) Calculate the wavelength of the unknown laser. b) Calculate the displacement, \(\Delta d\), of the movable mirror.

One type of hologram consists of bright and dark fringes produced on photographic film by interfering laser beams. If this is illuminated with white light, the image will appear reproduced multiple times, in different pure colors at different sizes. a) Explain why. b) Which colors correspond to the largest and smallest images, and why?

An airplane is made invisible to radar by coating it with a 5.00 -mm-thick layer of an antireflective polymer with the index of refraction \(n=1.50 .\) What is the wavelength of radar waves for which the plane is made invisible?
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