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Chapter 36: Problem 99

A diffraction grating has 200 lines/mm. Light consisting of a continuous range of wavelengths between \(550 \mathrm{~nm}\) and \(700 \mathrm{~nm}\) is incident perpendicularly on the grating. (a) What is the lowest order that is overlapped by another order? (b) What is the highest order for which the complete spectrum is present?

Short Answer

Expert verified
(a) The lowest overlapping order is 4. (b) The highest complete spectrum order is 7.

Step by step solution

01

Convert lines per millimeter to grating spacing

The grating spacing, \( d \), is the inverse of the number of lines per millimeter. Given \( 200 \) lines/mm, we first convert this to meters: \[ d = \frac{1}{200,000} \text{ m} = 5 \times 10^{-6} \text{ m} \].
02

Identify the diffraction grating equation

The diffraction grating equation is given by \( d \sin \theta = m \lambda \), where \( m \) is the order, and \( \lambda \) is the wavelength. This equation will be used to solve for the orders of overlap.
03

Calculate the lowest overlapping order

For overlap, different orders of different wavelengths must meet the same angle, \( \theta \). We assume \( m' + 1 = m \) for continuity and find overdub condition using the longest wavelength from the overlapping order \( m \) and the shortest wavelength for next order \( m+1 \):\[ m \times 700 \text{ nm} = (m+1) \times 550 \text{ nm} \]. Solve for \( m \): \[ m = \frac{550}{150} \approx 3.6667 \]. The lowest overlapping order is \( m = 4 \) when rounded up since \( m \) is an integer.
04

Calculate the highest order for complete spectrum

For the complete spectrum to be present, the condition \( d \geq m \lambda \) must hold for both extreme wavelengths at the highest order. Solve \( m \times 700 \text{ nm} \leq 5 \times 10^{-6} \text{ m} \). \[ m \leq \frac{5 \times 10^{-6}}{700 \times 10^{-9}} = \frac{5}{0.7} \approx 7.14 \]. Thus, the highest order ensuring the whole spectrum is \( m = 7 \) when rounded down.

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Key Concepts

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

Wavelength
Wavelength is a key term when discussing light and diffraction. It refers to the distance between two consecutive peaks of a wave. In the electromagnetic spectrum, different wavelengths correspond to different colors of light. For example, in the visible spectrum, red light has a longer wavelength than blue light.
Wavelength is usually denoted by the Greek letter \( \lambda \). It is typically measured in nanometers (nm), where one nanometer equals one-billionth of a meter. The wavelength determines many properties of light, such as its color and energy.
  • Shorter wavelengths correspond to higher energy and are typically less visible to the human eye.
  • Longer wavelengths, like those of red light, are less energetic but more visible.
Understanding wavelength is essential for grasping how light behaves when it encounters obstacles, such as a diffraction grating.
Order of Diffraction
When light passes through a diffraction grating, it is separated into several beams traveling in different directions. These beams are called orders. Each order corresponds to a different set of integers (m) in the diffraction grating equation. Higher orders of diffraction result in wider separation of wavelengths or colors.
The order of diffraction is significant because different orders can contain overlapping wavelengths. This overlap is especially important in understanding phenomena like why specific orders might not display the entire spectrum of light.
  • The zeroth order (\(m = 0\)) is usually a straightforward path where no diffraction separates the light.
  • First order (\(m = 1\)) and higher indicate increasing angles between the diffracted beams.
Evaluating different orders helps determine how extensively a light spectrum is spread out by the grating.
Grating Equation
The grating equation is a mathematical expression used to predict the angles at which light of a particular wavelength is diffracted by a grating. It is given by the formula:\[d \sin \theta = m \lambda\]where:
  • \(d\) is the spacing between adjacent grating lines.
  • \(\theta\) is the angle of diffraction.
  • \(m\) is the order of diffraction.
  • \(\lambda\) is the wavelength of the light.
Understanding this equation is crucial because it allows one to calculate the angles at which different wavelengths are spread by the grating for each order. The spacing \(d\) can be derived from the grating's line density (lines per millimeter) by taking its inverse.
Light Spectrum
The light spectrum consists of all the possible wavelengths of light, visible and invisible to the human eye. The visible spectrum ranges from approximately 380 nm to 750 nm, covering all the colors we see in a rainbow. However, light can also have wavelengths outside this range, such as ultraviolet or infrared.
  • The continuous spectrum includes all wavelengths in a given range, smoothly fading from one color to the next.
  • Monochromatic light, by contrast, consists of a single wavelength and appears as just one color.
In many experiments, such as with a diffraction grating, light of different wavelengths is spread into a spectrum due to interference patterns. This spreading allows scientists and engineers to analyze the components of light sources based on their unique spectra. Understanding the light spectrum is essential for fields such as optics, physics, and even astronomy.

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Most popular questions from this chapter

A grating has 400 lines/mm. How many orders of the entire visible spectrum \((400-700 \mathrm{~nm})\) can it produce in a diffraction experiment, in addition to the \(m=0\) order?

Babinet's principle. A monochromatic beam of parallel light is incident on a "collimating" hole of diameter \(x>\lambda\). Point \(P\) lies in the geometrical shadow region on a distant screen (Fig. \(36-39 a\) ). Two diffracting objects, shown in Fig. \(36-39 b\), are placed in turn over the collimating hole. Object \(A\) is an opaque circle with a hole in it, and \(B\) is the "photographic negative" of \(A\). Using superposition concepts, show that the intensity at \(P\) is identical for the two diffracting objects \(A\) and \(B\).

In conventional television, signals are broadcast from towers to home receivers. Even when a receiver is not in direct view of a tower because of a hill or building, it can still intercept a signal if the signal diffracts enough around the obstacle, into the obstacle's "shadow region." Previously, television signals had a wavelength of about \(50 \mathrm{~cm}\), but digital television signals that are transmitted from towers have a wavelength of about \(10 \mathrm{~mm}\). (a) Did this change in wavelength increase or decrease the diffraction of the signals into the shadow regions of obstacles? Assume that a signal passes through an opening of \(5.0 \mathrm{~m}\) width between two adjacent buildings. What is the angular spread of the central diffraction maximum (out to the first minima) for wavelengths of (b) \(50 \mathrm{~cm}\) and \((\mathrm{c}) 10 \mathrm{~mm}\) ?

In the single-slit diffraction experiment of Fig. \(36-4\), let the wavelength of the light be \(500 \mathrm{~nm}\), the slit width be \(6.00 \mu \mathrm{m}\), and the viewing screen be at distance \(D=3.00 \mathrm{~m}\). Let a \(y\) axis extend upward along the viewing screen, with its origin at the center of the diffraction pattern. Also let \(I_{P}\) represent the intensity of the diffracted light at point \(P\) at \(y=15.0 \mathrm{~cm} .\) (a) What is the ratio of \(I_{P}\) to the intensity \(I_{m}\) at the center of the pattern? (b) Determine where point \(P\) is in the diffraction pattern by giving the maximum and minimum between which it lies, or the two minima between which it lies.

In the two-slit interference experiment of Fig. \(35-10\), the slit widths are each \(12.0 \mu \mathrm{m}\), their separation is \(24.0 \mu \mathrm{m}\), the wavelength is \(600 \mathrm{~nm}\), and the viewing screen is at a distance of \(4.00 \mathrm{~m}\). Let \(I_{P}\) represent the intensity at point \(P\) on the screen, at height \(y=70.0 \mathrm{~cm} .(\mathrm{a})\) What is the ratio of \(I_{P}\) to the intensity \(I_{m}\) at the center of the pattern? (b) Determine where \(P\) is in the two-slit interference pattern by giving the maximum or minimum on which it lies or the maximum and minimum between which it lies, (c) In the same way, for the diffraction that occurs, determine where point \(P\) is in the diffraction pattern.
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