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Chapter 3: Problem 56

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

Short Answer

Expert verified
(a) Constructive interference (b) Destructive interference (c) Constructive interference (d) Constructive interference (e) Destructive interference (f) Destructive interference

Step by step solution

01

Identify the constructive and destructive interference conditions

For constructive interference, the path difference between the two sources should be an integer multiple of the wavelength (nλ). For destructive interference, the path difference must be an odd multiple of half the wavelength (nλ + λ/2). Constructive interference: \(PathDifference = n\lambda\) Destructive interference: \(PathDifference = n\lambda + \frac{\lambda}{2}\), where n is an integer (including 0), and λ is the wavelength.
02

Calculate the path difference for each point

For each given point, calculate the path difference by finding the absolute difference between the distances from the two sources. (a) \(|0.84 - 0.42| = 0.42 \mathrm{m}\) (b) \(|0.21 - 0.42| = 0.21 \mathrm{m}\) (c) \(|1.26 - 0.42| = 0.84 \mathrm{m}\) (d) \(|1.87 - 1.45| = 0.42 \mathrm{m}\) (e) \(|0.63 - 0.84| = 0.21 \mathrm{m}\) (f) \(|1.47 - 1.26| = 0.21 \mathrm{m}\)
03

Assess constructive or destructive interference for each point

For each path difference, check whether the value matches an integer multiple of the wavelength (constructive interference) or an odd multiple of half the wavelength (destructive interference). (a) \(0.42 \mathrm{m}\): Constructive interference, as \(0.42 = 1\cdot(0.42)\) (b) \(0.21 \mathrm{m}\): Destructive interference, as \(0.21 = 1\cdot(0.42/2)\) (c) \(0.84 \mathrm{m}\): Constructive interference, as \(0.84 = 2\cdot(0.42)\) (d) \(0.42 \mathrm{m}\): Constructive interference, as \(0.42 = 1\cdot(0.42)\) (e) \(0.21 \mathrm{m}\): Destructive interference, as \(0.21 = 1\cdot(0.42/2)\) (f) \(0.21 \mathrm{m}\): Destructive interference, as \(0.21 = 1\cdot(0.42/2)\)
04

Summarize the results

To summarize, we have the following types of interference occurring at each point: (a) Constructive interference (b) Destructive interference (c) Constructive interference (d) Constructive interference (e) Destructive interference (f) Destructive interference

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Most popular questions from this chapter

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 \(15^{\circ} \mathrm{C}\) to \(25 \mathrm{C},\) a change of 14 fringes is observed. The light source is a He Ne laser, \(\lambda=632.8 \mathrm{nm}\). What is the change in length of the metal bar, and what is its thermal expansion coefficient?

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