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Chapter 26: Problem 37

Monochromatic light is at normal incidence on a plane transmission grating. The first-order maximum in the interference pattern is at an angle of \(8.94^{\circ} .\) What is the angular position of the fourth-order maximum?

Short Answer

Expert verified
The angular position of the fourth-order maximum is approximately \(38.6^{\circ}\).

Step by step solution

01

Understanding the Grating Equation

The grating equation is used to determine the angles at which maximum intensity occurs from a transmission grating: \(d \sin(\theta) = m \lambda\), where \(d\) is the grating spacing, \(\theta\) is the angle of the maximum, \(m\) is the order of the maximum, and \(\lambda\) is the wavelength of light.
02

Solving for the Wavelength

Since we know the angle for the first-order maximum, we can rearrange the grating equation to solve for the wavelength: \(\lambda = \frac{d \sin(8.94^{\circ})}{1}\). However, to find the fourth-order maximum, we don't need to calculate the wavelength directly, just recognize the relationship that all maxima of the same order will have the same \(d \sin(\theta)\) value.
03

Finding the Fourth-Order Maximum

To find the angular position of the fourth-order maximum, we use the grating equation for \(m = 4\): \(d \sin(\theta_4) = 4 \lambda\). Since \(\lambda\) for the first-order is \(d \sin(8.94^{\circ})\), we substitute: \(d \sin(\theta_4) = 4d \sin(8.94^{\circ})\). Thus, \(\sin(\theta_4) = 4 \sin(8.94^{\circ})\).
04

Calculate the Angle

Using the calculated \(\sin(\theta_4)\), determine \(\theta_4\). First, find \(\sin(8.94^{\circ})\): \(\sin(8.94^{\circ}) \approx 0.1559\). Then \(\sin(\theta_4) = 4 \times 0.1559 = 0.6236\). Finally, calculate \(\theta_4 = \sin^{-1}(0.6236)\), which gives \(\theta_4 \approx 38.6^{\circ}\).

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Key Concepts

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

First-order maximum
When studying light and how it interacts with a diffraction grating, the term "first-order maximum" is important. Imagine light passing through a grating that has many tiny slits. As light waves pass through these slits, they spread out and create patterns of light and dark bands on a screen.
The brightest spots in this pattern are called maxima. The first bright spot closest to the center is known as the "first-order maximum." This is where the light has been diffracted at just the right angle to constructively interfere or add up, producing a bright spot.
  • The first-order maximum is crucial because it helps us determine a key property of the light, called its wavelength.
  • For monochromatic light, which is light of a single wavelength, each order of maximum corresponds to a known angle, given by the grating equation.
Grating equation
The grating equation is a mathematical tool that helps us understand the pattern of light created by a diffraction grating. It is written as:
\[d \sin(\theta) = m \lambda\]Here, each symbol has a specific meaning:
  • \(d\) represents the spacing between the slits in the grating.
  • \(\sin(\theta)\) gives the sine of the angle where the maximum or bright spot appears.
  • \(m\) is the order of the maximum, like first-order, second-order, etc.
  • \(\lambda\) is the wavelength of light being used.
By rearranging this equation, we can calculate different properties, like the angle for each maximum or the wavelength of the light. The equation is incredibly powerful in experiments involving light, helping us predict and measure how light will behave when encountering multiple slits.
Monochromatic light
Monochromatic light is light that has a single wavelength and color. Unlike white light, which contains all the colors of the rainbow, monochromatic light is pure and consistent.
Common sources of monochromatic light are lasers. They emit a single color, making them useful for experiments involving diffraction and interference.
  • Using monochromatic light in diffraction gratings is beneficial because it makes the resulting pattern simpler and easier to analyze.
  • The distinct maxima (bright spots) created with monochromatic light clearly indicate specific angles that can tell us about the properties of the light or the grating.
In the study of diffraction through gratings, knowing the light is monochromatic allows for straightforward application of the grating equation as each bright spot corresponds to one, precise angle.
Angular position
In diffraction and interference studies, the term "angular position" refers to the angle at which light waves emerge after passing through a grating and create a bright maximum.
This angle is crucial because it tells us where the maximum intensity of light occurs in the diffraction pattern. Understanding and calculating these angles help scientists work out details about the light or the grating used.
  • The angular position for different orders of maxima (like the first-order, second-order, etc.) are computed using the grating equation.
  • Accurate measurement of such angles allows for determining the light鈥檚 wavelength or verifying the precision of the grating.
In our example, the first-order maximum is given at 8.94 degrees. Using the grating equation, higher-order maxima can be found. For instance, calculating the fourth-order maximum using this angle highlights how precise these measurements and equations need to be.

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Most popular questions from this chapter

A wildlife photographer uses a moderate telephoto lens of focal length \(135 \mathrm{~mm}\) and maximum aperture \(f / 4.00\) to photograph a bear that is \(11.5 \mathrm{~m}\) away. Assume the wavelength is \(550 \mathrm{nm}\). (a) What is the width of the smallest feature on the bear that this lens can resolve if it is opened to its maximum aperture? (b) If, to gain depth of field, the photographer stops the lens down to \(f / 22.0,\) what would be the width of the smallest resolvable feature on the bear?

A converging lens \(7.20 \mathrm{~cm}\) in diameter has a focal length of \(300 \mathrm{~mm}\). If the resolution is diffraction limited, how far away can an object be if points on it transversely \(4.00 \mathrm{~mm}\) apart are to be resolved (according to Rayleigh's criterion) by means of light of wavelength \(550 \mathrm{nm} ?\)

A beam of laser light of wavelength \(632.8 \mathrm{nm}\) falls on a thin slit \(0.00375 \mathrm{~mm}\) wide. After the light passes through the slit, at what angles relative to the original direction of the beam is it completely canceled when viewed far from the slit?

When viewing a piece of art that is behind glass, one often is affected by the light that is reflected off the front of the glass (called glare), which can make it difficult to see the art clearly. One solution is to coat the outer surface of the glass with a thin film to cancel part of the glare. (a) If the glass has a refractive index of 1.62 and you use \(\mathrm{TiO}_{2},\) which has an index of refraction of \(2.62,\) as the coating, what is the minimum film thickness that will cancel light of wavelength \(505 \mathrm{nm} ?\) (b) If this coating is too thin to stand up to wear, what other thicknesses would also work? Find only the three thinnest ones.

Coherent light of frequency \(6.32 \times 10^{14} \mathrm{~Hz}\) passes through two thin slits and falls on a screen \(85.0 \mathrm{~cm}\) away. You observe that the third bright fringe occurs at \(\pm 3.11 \mathrm{~cm}\) on either side of the central bright fringe. (a) How far apart are the two slits? (b) At what distance from the central bright fringe will the third dark fringe occur?
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