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Chapter 36: Problem 25

An interference pattern is produced by four parallel and equally spaced, narrow slits. By drawing appropriate phasor diagrams, show that there is an interference minimum when the phase difference \(\phi\) from adjacent slits is (a) \(\pi / 2 ;(b) \pi ;\) (c) 3\(\pi / 2\) . In each case, for which pairs of slits is there totally destructive interference?

Short Answer

Expert verified
Each pair of adjacent slits is destructive for \( \phi = \pi/2 \); pairs (1,3) and (2,4) for \( \phi = \pi \); and pairs (2,3) and (1,4) for \( \phi = 3\pi/2 \).

Step by step solution

01

Understanding the Problem

We have four parallel slits, and we're considering interference minima. We need to determine the conditions for minima when the phase difference \( \phi \) between adjacent slits is \( \pi/2 \), \( \pi \), and \( 3\pi/2 \). At these conditions, we'll identify the pairs of slits that result in destructive interference.
02

Analyzing for \( \phi = \pi/2 \)

For \( \phi = \pi/2 \), the phase difference causes each phasor to rotate 90 degrees compared to the previous. Using complex addition of phasors: \( A = e^{i\cdot 0} + e^{i\cdot \pi/2} + e^{i\cdot \pi} + e^{i\cdot 3\pi/2} \). These phasors form a complete circle when drawn from the origin. Summation results in a net phasor of zero, indicating total destructive interference. Here, each adjacent pair of slits (1-2, 2-3, and 3-4) results in no contribution as all cancel out.
03

Analyzing for \( \phi = \pi \)

For \( \phi = \pi \), each phasor is 180 degrees out of phase with the previous one: \( A = e^{i\cdot 0} + e^{i\cdot \pi} + e^{i\cdot 0} + e^{i\cdot \pi} \). This results in two pairs of opposite phasors canceling each other: \( (1,3) \) and \( (2,4) \) pairs are completely destructive, as the phasor adds to zero.
04

Analyzing for \( \phi = 3\pi/2 \)

For \( \phi = 3\pi/2 \), the phase difference means each successive phasor is 270 degrees ahead. The sum \( A = e^{i\cdot 0} + e^{i\cdot 3\pi/2} + e^{i\cdot 3\pi} + e^{i\cdot 9\pi/2} \) represents a phasor path that's again a closed loop back to the origin, meaning total destructive interference occurs. Particularly, slits (1-2) and (3-4) are immediately canceling due to being 180 degrees out of phase.

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Key Concepts

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

Destructive Interference
In the context of wave patterns, destructive interference is a fascinating phenomenon where waves collide and cancel each other out. Imagine ripples from a stone dropped in water meeting those from another stone, resulting in calmness. This happens due to specific patterns of wave overlap, particularly when crests align with troughs, leading to cancellation. In slit experiments, destructive interference occurs when light waves from different slits combine in such a way that their peaks and troughs are misaligned.

When examining multiple slits, phasor diagrams help visualize these cancellations. If the resulting phasors form a closed loop returning to the origin upon summation, it indicates that waves have destructively interfered, creating a minimum intensity at certain points on the interference pattern. This complete cancellation is typical when phase differences are specific, such as multiples of \( \pi \, \text{radians} \).

  • Peaks align with troughs.
  • Phasor diagrams illustrate cancellations.
  • Key phase differences contribute to complete cancellation.
Phase Difference
Phase difference refers to the offset between waves at a given point in time and space. It is usually measured in radians and represents how far one wave leads or lags behind another. For example, a phase difference of \( \pi \) radians means the two waves are perfectly out of sync, with one wave's peak aligning with the other's trough. This results in destructive interference.

The understanding of phase difference is critical in explaining interference patterns, especially in a setup with multiple slits. In certain conditions, like those analyzed in phasor diagrams, as the phase difference between slits changes (e.g., \( \pi/2, \pi, \text{and} \; 3\pi/2 \)), the nature of interference (constructive or destructive) is dictated. When sequenced waves interact, their relative phase differences determine the resulting wave pattern.

  • Measured in radians.
  • Indicates synchrony of wave peaks and troughs.
  • Essential for predicting interference outcomes.
Phasor Diagrams
Phasor diagrams are a powerful visual tool used to analyze interference in wave optics, especially for complex setups involving multiple slits. Think of phasors as rotating vectors representing the phase and amplitude of waves. These diagrams allow the summation of waves via vector addition.

Each phasor angle corresponds to the wave's phase, and together these phasors form a diagram that can reveal whether constructive or destructive interference occurs. In our slit experiment example, the complete round paths back to the origin in the phasor diagram illustrate that all phasors cancel each other out, showing destructive interference at specified phase differences

Phasor diagrams greatly simplify calculation by transforming wave addition into geometric summation operations. The nature of the interference鈥攚hether waves reinforce or cancel each other鈥攊s immediately apparent through the resulting phasor's magnitude and direction.
  • Phasors represent wave phase and amplitude.
  • Vectors illustrate complex wave interactions.
  • Geometric solutions via phasors highlight interference types.
Multiple Slit Interference
Interference patterns become especially intriguing when more than two slits are involved, such as in the case of multiple slit interference. This involves overlapping wave fronts from several slits, adding complexity to the resulting pattern.

When you have four or more slits, like in the exercise, understanding this interference requires more sophisticated tools such as phasor diagrams. As waves pass through the slits, they interfere according to specific phase differences, creating regions of constructive or destructive interference. The complexity increases with each additional slit, yet the fundamental principles of wave interference still apply.

For instance, specific phase differences in a multi-slit setup can lead to total cancellation of light at certain angles, thereby producing dark spots. This highlights the importance of understanding the underlying geometry and physics of wave interactions in complex interference patterns.
  • Involves more than two slits.
  • Phasor diagrams crucial for understanding.
  • Phase differences determine pattern features.

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Most popular questions from this chapter

Two satellites at an altitude of 1200 \(\mathrm{km}\) are separated by 28 \(\mathrm{km}\) . If they broadcast \(3.6-\mathrm{cm}\) microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh's criterion) the two transmissions?

An astronaut in orbit can just resolve two point sources on the earth that are 75.0 \(\mathrm{m}\) apart. Assume that the resolution is diffraction limited, and use Rayleigh's criterion. What is the astronaut's altitude above the earth? Treat her eye as a circular aperture with a diameter of 4.00 \(\mathrm{mm}\) (the diameter of her pupil), and take the wavelength of the light to be 500 \(\mathrm{nm}\) .

Monochromatic electromagnetic radiation with wavelength \(\lambda\) from a distant source passes through a slit. The diffraction pattern is observed on a screen 2.50 \(\mathrm{m}\) from the slit. If the width of the central maximum is \(6.00 \mathrm{mm},\) what is the slit width \(a\) if the wavelength is (a) 500 \(\mathrm{nm}\) (visible light); (b) 50.0\(\mu \mathrm{m}\) (infrared radiation); (c) 0.500 \(\mathrm{nm}\) (x rays)?

Tsunamit On December \(26,2004,\) a violent magnitude 9.1 earthquake occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than \(150,000\) people. Scientists observing the wave on the open ocean measured the time between crests to be 1.0 \(\mathrm{h}\) and the speed of the wave to be 800 \(\mathrm{km} / \mathrm{h}\) . Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about 4500 \(\mathrm{km}\) , while the distance between the southern end of Australia and Antarctica is about 3700 \(\mathrm{km}\) . As an approximation, we can model this wave's behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

A single-slit diffraction pattern is formed by monochromatic electromagnetic radiation from a distant source passing through a slit 0.105 \(\mathrm{mm}\) wide. At the point in the pattern \(3.25^{\circ}\) from the center of the central maximum, the total phase difference between wavelets from the tup and bottom of the slit is 56.0 rad. (a) What is the wavelength of the radiation? (b) What is the intensity at this point, if the intensity at the center of the central maximum is \(I_{0} ?\)
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