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Chapter 28: Problem 1

Two sources emit waves that are coherent, in phase, and have wavelengths of \(26.0 \mathrm{m}\). Do the waves interfere constructively or destructively at an observation point \(78.0 \mathrm{m}\) from one source and \(143 \mathrm{m}\) from the other source?

Short Answer

Expert verified
The waves interfere destructively at the observation point.

Step by step solution

01

Calculate the Path Length Difference

To determine the interference pattern, first, calculate the path length difference between the two waves reaching the observation point. Subtract the shorter distance from the longer distance: \[L_1 = 143 \, \mathrm{m}, \quad L_2 = 78 \, \mathrm{m}\] The path length difference: \[\Delta L = L_1 - L_2 = 143 \, \mathrm{m} - 78 \, \mathrm{m} = 65 \, \mathrm{m}\]
02

Determine Phase Difference Condition

We need to determine whether this path difference leads to constructive or destructive interference. - Constructive interference occurs when the path difference \(\Delta L\) is an integer multiple of the wavelength \(\lambda\), i.e., \(n\lambda\), where \(n\) is an integer.- Destructive interference occurs when the path difference \(\Delta L\) is a half-integer multiple of the wavelength \(\lambda\), i.e., \((m + 0.5)\lambda\), where \(m\) is an integer.
03

Test for Constructive Interference

Check if the path length difference \(\Delta L\) is a whole number multiple of the wavelength:\[\Delta L = n\lambda, \quad \lambda = 26 \, \mathrm{m}\]\[65 \, \mathrm{m} = n \times 26 \, \mathrm{m}\]Divide both sides by the wavelength:\[n = \frac{65}{26} \approx 2.5\]Since \(n\) is not an integer, the path difference does not result in constructive interference.
04

Confirm Destructive Interference

Check if the path length difference \(\Delta L\) is a half-integer multiple of the wavelength:\[\Delta L = \left(m + 0.5\right)\lambda, \quad \lambda = 26 \, \mathrm{m}\]\[65 \, \mathrm{m} = \left(m + 0.5\right) \times 26 \, \mathrm{m}\]Solve for \(m\):\[2.5 = m + 0.5 \, \Rightarrow \, m = 2\]Since \(m\) is an integer, the path difference results in destructive interference.

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Key Concepts

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

Path Difference
In wave interference, the **path difference** is crucial in determining how waves from different sources interact at a certain point. It's the difference in distances that waves travel from their respective sources to a particular observation point. For the exercise, the path differences were calculated using the source distances:- Source 1: 143 m - Source 2: 78 m The path difference, \(\Delta L\), is given by the formula:\[ \Delta L = L_1 - L_2 \]Substituting the values, we find \(\Delta L = 65 \, \mathrm{m}\). This path difference will inform us about the type of interference that occurs鈥攅ither constructive or destructive.
Constructive Interference
**Constructive interference** occurs when waves meet in such a way that their crests and troughs align perfectly. This alignment amplifies the resulting wave, producing something stronger or brighter.This type of interference is achieved when the path difference is an integer multiple of the wavelength \(\lambda\), meaning:\[ \Delta L = n\lambda, \quad n \text{ is an integer} \]In our exercise:- Wavelength \(\lambda = 26 \, \mathrm{m}\)- Path difference \(\Delta L = 65 \, \mathrm{m}\)To check for constructive interference,\[ 65 = n \times 26 \]When we divide \(65 / 26\), we find \(n \approx 2.5\) 鈥� not an integer. Therefore, the conditions for constructive interference are not met in this scenario.
Destructive Interference
**Destructive interference** happens when the crests of one wave align with the troughs of another. This configuration causes the waves to cancel each other out, reducing the overall wave amplitude.For destructive interference, the path difference must be a half-integer multiple of the wavelength:\[ \Delta L = (m + 0.5)\lambda, \quad m \text{ is an integer} \]Given in the exercise:- Path difference \(\Delta L = 65 \, \mathrm{m}\)- Wavelength \(\lambda = 26 \, \mathrm{m}\)Check the condition:\[ 65 = (m + 0.5) \times 26 \]This equation simplifies to:\[ 2.5 = m + 0.5 \Rightarrow m = 2 \]Since \(m = 2\) is an integer, the path difference leads to destructive interference. In this scenario, the waves cancel each other, resulting in reduced wave intensity at the observation point.

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Most popular questions from this chapter

A thin film of oil \((n=1.30)\) floats on water \((n=1.33) .\) When sunlight is incident at right angles to this film, the only colors that are enhanced by reflection are blue \((458 \mathrm{nm})\) and red \((687 \mathrm{nm})\). Estimate the thickness of the oil film.

Splitting Binary Stars As seen from Earth, the red dwarfs Kruger \(60 \mathrm{A}\) and \(\mathrm{Kruger} 60 \mathrm{B}\) form a binary star system with an angular separation of 2.5 arc seconds. What is the smallest diameter telescope that could theoretically resolve these stars using \(550-\mathrm{nm}\) light? \(\left(\right.\) Note: 1 arc \(\left.\sec =1 / 3600^{\circ}\right)\)

Consider a two-slit interference pattern, with monochromatic light of wavelength \(\lambda\). What is the path difference \(\Delta \ell\) for (a) the fourth bright fringe and (b) the third dark fringe above the central bright fringe? Give your answers in terms of the wavelength of the light.

A set of parallel slits for optical interference can be made by holding two razor blades together (carefully!) and scratching a pair of lines on a glass microscope slide that has been painted black. When monochromatic light strikes these slits at normal incidence, an interference pattern is formed on a distant screen. The thickness of each razor blade used to make the slits is \(0.230 \mathrm{mm}\), and the screen is \(2.50 \mathrm{m}\) from the slits. If the centerto-center separation of the fringes is \(7.15 \mathrm{mm},\) what is the wavelength of the light?

A laser emits two wavelengths \(\left(\lambda_{1}=420 \mathrm{nm} ; \lambda_{2}=630 \mathrm{nm}\right)\). When these two wavelengths strike a grating with 450 lines/mm, they produce maxima (in different orders) that coincide. (a) What is the order \((m)\) of each of the two overlapping lines? (b) At what angle does this overlap occur?
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