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Chapter 35: Problem 36

Jan first uses a Michelson interferometer with the 606 -nm light from a krypton-86 lamp. He displaces the movable mirror away from him, counting 818 fringes moving across a line in his field of view. Then Linda replaces the krypton lamp with filtered 502 -nm light from a helium lamp and displaces the movable mirror toward her. She also counts 818 fringes, but they move across the line in her field of view opposite to the direction they moved for Jan. Assume that both Jan and Linda counted to 818 correctly. (a) What distance did each person move the mirror? (b) What is the resultant displacement of the mirror?

Short Answer

Expert verified
Each moved the mirror 0.24747 m and 0.20549 m, respectively; net movement: 0.04198 m.

Step by step solution

01

Understanding Fringe Count

A Michelson interferometer works by splitting a beam of light into two paths, reflecting the beams, and then recombining them to produce interference fringes. Each fringe represents a movement of half the wavelength of the light used. Therefore, the number of fringes counted corresponds to twice the mirror displacement in terms of wavelengths (since the round trip affects the count).
02

Calculate Mirror Displacement for Jan

Jan uses light with a wavelength of 606 nm and counts 818 fringes. The displacement of the mirror can be found using the formula \( d = \frac{m \lambda}{2} \), where \( m \) is the number of fringes and \( \lambda \) is the wavelength. Substituting the given values:\[d_{\text{Jan}} = \frac{818 \times 606 \times 10^{-9}}{2} \approx 0.24747 \text{ meters}\]So, Jan moves the mirror approximately 0.24747 meters.
03

Calculate Mirror Displacement for Linda

Linda replaces the light source with one of wavelength 502 nm and counts the same number of fringes. Using the same formula:\[d_{\text{Linda}} = \frac{818 \times 502 \times 10^{-9}}{2} \approx 0.20549 \text{ meters}\]Linda moves the mirror approximately 0.20549 meters.
04

Calculate Resultant Displacement

Since Jan moved the mirror away and Linda moved the mirror toward her, these displacements are in opposite directions. To find the net displacement, we subtract Linda's displacement from Jan's:\[\text{Resultant displacement} = d_{\text{Jan}} - d_{\text{Linda}} = 0.24747 - 0.20549 = 0.04198 \text{ meters}\]The resultant displacement of the mirror is approximately 0.04198 meters.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interference Fringes
Imagine light as waves, similar to water waves in the sea. A Michelson interferometer cleverly splits light waves, sending them down two paths before merging them again. When these light waves recombine, they produce a pattern known as interference fringes, which are alternating dark and bright bands. These fringes offer a visual representation of how the light waves interact with each other upon recombining. One critical aspect of these patterns is that each fringe shift indicates the movement of one of the mirrors by half the wavelength of the light in use. For example, if you observe 818 fringes moving past a fixed point, it means the mirror has moved a distance corresponding to half the product of 818 times the wavelength of the light.
Interference fringes help scientists measure extremely small distances with high precision, turning the movement of mirrors into measurable shifts in light patterns.
Wavelength
The wavelength of light is central to understanding how a Michelson interferometer operates. Wavelength is the distance between two successive peaks (or troughs) of a wave. Different light sources emit light with various wavelengths, which is why they appear different colors to our eyes.
In this particular exercise, Jan used light with a wavelength of 606 nanometers, a shade in the orange part of the visible spectrum. Linda used a different light source with a wavelength of 502 nanometers, appearing green-blue. The disparity in wavelengths directly affects how far the mirror must move to create the same number of interference fringes. Understanding and using different wavelengths is crucial because it allows for precise adjustments in experiments, making it possible to measure minuscule changes effectively.
Mirror Displacement
Mirror displacement is a key part of how the Michelson interferometer measures distances. When the mirror in the interferometer is moved, the path length of the light changes, causing a shift in the interference fringes visible to the observer. The relationship is direct: for each fringe counted, the mirror has moved half the wavelength of the incident light. Therefore, calculating the exact distance the mirror has moved involves multiplying the total number of fringes by half the wavelength.
In the given problem, each participant moved the mirror the same number of fringes, yet their distances differed due to the variance in wavelength. They applied the formula: \[ d = \frac{m \lambda}{2} \]This indicates that while the actual number of observed fringes remains constant, the physical distance moved by the mirror varies based on the light's wavelength used in the experiment. This intricate relationship underscores how small changes in setup conditions can lead to different outcomes in the readings.

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Most popular questions from this chapter

Reflective Coatings and Herring. Herring and related fish have a brilliant silvery appearance that camouflages them while they are swimming in a sunlit ocean. The silveriness is due to platelets attached to the surfaces of these fish. Each platelet is made up of several alternating layers of crystalline guanine \((n=1.80)\) and of cytoplasm \((n=1.333,\) the same as water), with a guanine layer on the outside in contact with the surrounding water (Fig. \(\mathrm{P} 35.56\) ). In one typical platelet, the guanine layers are 74 nm thick and the cytoplasm layers are 100 \(\mathrm{nm}\) thick. (a) For light striking the platelet surface at normal incidence, for which vacuum wavelengths of visible light will all of the reflections \(R_{1}\) , \(R_{2}, R_{3}, R_{4},\) and \(R_{5},\) shown in Fig. P35.56, be approximately in phase? If white light is shone on this platelet, what color will be most strongly reflected (see Fig. 32.4\() ?\) The surface of a herring has very many platelets side by side with layers of different thickness, so that all visible wavelengths are reflected. (b) Explain why such a "stack" of layers is more reflective than a single layer of guanine with cytoplasm underneath. (A stack of five guanine layers separated by cytoplasm layers reflects more than 80\(\%\) of incident light at the wavelength for which it is "tuned.") (c) The color that is most strongly reflected from a platelet depends on the angle at which it is viewed. Explain why this should be so. (You can see these changes in color by examining a herring from different angles. Most of the platelets on these fish are oriented in the same way, so that they are vertical when the fish is swimming.)

Points \(A\) and \(B\) are 56.0 \(\mathrm{m}\) apart along an east-west line. At each of these points, a radio transmitter is emitting a 12.5 -MHz signal horizontally. These transmitters are in phase with each other and emit their beams uniformly in a horizontal plane. A receiver is taken 0.500 \(\mathrm{km}\) north of the \(A B\) line and initially placed at point \(C,\) directly opposite the midpoint of \(A B .\) The receiver can be moved only along an east-west direction but, due to its limited sensitivity, it must always remain a range so that the intensity of the signal it receives from the transmitter is no less than \(\frac{1}{4}\) of its maximum value. How far from point \(C\) (along an east-west line) can the receiver be moved and always be able to pick up the signal?

What is the thinnest soap film (excluding the case of zero thickness) that appears black when illuminated with light with wavelength 480 \(\mathrm{nm} ?\) The index of refraction of the film is \(1.33,\) and there is air on both sides of the film.

Sensitive Eyes. After an eye examination, you put some eyedrops on your sensitive eyes. The cornea (the front part of the eye) has an index of refraction of \(1.38,\) while the eyedrops have a refractive index of \(1.45 .\) After you put in the drops, your friends notice that your eyes look red, because red light of wavelength 600 nm has been reinforced in the reflected light. (a) What is the minimum thickness of the film of eyedrops on your cornea? (b) Will any other wavelengths of visible light be reinforced in the reflected light? Will any be cancelled? (c) Suppose you had contact lenses, so that the eyedrops went on them instead of on your corneas. If the refractive index of the lens material is 1.50 and the layer of eyedrops has the same thickness as in part (a), what wavelengths of visible light will be reinforced? What wavelengths will be cancelled?

Two light sources can be adjusted to emit monochromatic light of any visible wavelength. The two sources are coherent, 2.04\(\mu \mathrm{m}\) apart, and in line with an observer, so that one source is 2.04\(\mu \mathrm{m}\) farther from the observer than the other. (a) For what visible wavelengths \((380\) to 750 \(\mathrm{nm})\) will the observer see the brightest light, owing to constructive interference? (b) How would your answers to part (a) be affected if the two sources were not in line with the observer, but were still arranged so that one source is 2.04\(\mu \mathrm{m}\) farther away from the observer than the other? (c) For what visible wavelengths will there be destructive interference at the location of the observer?
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