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Chapter 36: Problem 65

A diffraction grating has 650 slits/mm. What is the highest order that contains the entire visible spectrum? (The wavelength range of the visible spectrum is approximately \(380-750 \mathrm{nm.}\) )

Short Answer

Expert verified
The highest order that contains the entire visible spectrum is 2.

Step by step solution

01

Understand the Problem

A diffraction grating splits light into its component wavelengths. The task is to find the highest order diffraction maximum, or the largest value of 'm', that still allows the entire visible spectrum (380-750 nm) to be observed for a grating with 650 slits/mm.
02

Convert Units

Convert the number of slits per millimeter to slits per meter to match the wavelengths' units:\[ 650 \text{ slits/mm} = 650,000 \text{ slits/m} \]}},{
03

Use the Grating Equation

The diffraction grating formula relates the angle of diffraction \( \theta \), the order \( m \), the wavelength \( \lambda \), and the distance between slits \( d \), given by:\[ m \lambda = d \sin(\theta) \]Where all wavelengths must satisfy this equation.
04

Calculate the Slit Separation

First calculate the distance between slits, \( d \), in meters:\[ d = \frac{1}{650,000} \text{ m/slit} \approx 1.54 \times 10^{-6} \text{ m/slit} \]
05

Determine Condition for Highest Order

For the highest order, \( \theta \) should be close to 90 degrees (since \( \sin(90^\circ) = 1 \)), and substituting \( \sin(\theta) = 1 \), the equation simplifies to:\[ m \lambda = d \sin(90^\circ) = d \]
06

Limit Calculation for Longest Wavelength

Use the longest wavelength in the visible spectrum to find \( m \):\[ m = \frac{d}{\lambda_{max}} = \frac{1.54 \times 10^{-6}}{750 \times 10^{-9}} \approx 2.05 \]
07

Determine the Integer Order

Since \( m \) must be an integer, the greatest integer less than or equal to 2.05 is 2. Therefore, the highest order that displays the full visible spectrum is \( m = 2 \).

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Key Concepts

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

Visible Spectrum
The visible spectrum refers to the range of light wavelengths that the human eye can detect. It spans from approximately 380 nanometers (nm) to 750 nm. Light within this spectrum is perceived as colors, with shorter wavelengths appearing violet and longer ones appearing red.
The visible spectrum is just a small part of the electromagnetic spectrum, which includes other types of light like ultraviolet and infrared. Just like a prism, a diffraction grating splits light into its colorful components.
  • Lower wavelength: Violet (around 380 nm)
  • Middle wavelengths: Green (around 500-570 nm)
  • Higher wavelength: Red (up to 750 nm)
To identify the full range within a diffraction grating, it is important to consider this entire wavelength range.
Diffraction Order
The diffraction order signifies the number of times the light has "traveled" or been diffracted by the slits in a grating. It is denoted by the integer 'm'. The order defines how many maxima arise from the interaction between light and slits.
Each order corresponds to a specific angle at which light is diffracted. Higher orders mean wider angles and potential overlapping of spectra from different wavelengths.
  • Order 0: Direct transmission, most intense light
  • Order 1: First maximum, split into the spectrum of visible colors
  • Order 2: Second maximum, might barely hold all visible light
In determining the highest order that allows for complete light spectrum viewing, it is vital to consider how alignment and overlap could affect visibility.
Grating Equation
The grating equation is pivotal to understanding how light is diffracted in a grating. It's expressed as:\[ m \lambda = d \sin(\theta) \]where:
  • \( m \) is the diffraction order
  • \( \lambda \) is the wavelength of light
  • \( d \) is the separation distance between the slits in the grating
  • \( \theta \) is the angle at which light is diffracted
This formula illustrates the direct dependence of diffraction on the wavelength and order. As you increase the order \( m \), either the wavelength \( \lambda \) must decrease, or the angle \( \theta \) must increase.
To find the highest diffraction order displaying the full spectrum, set \( \theta \) close to maximum (90掳), meaning \( \sin(\theta) \) approaches 1.
Wavelength
Wavelength, often symbolized by \( \lambda \), is a critical property of electromagnetic waves. It is measured as the distance between consecutive peaks of the wave. In a diffraction context, it directly determines how light interacts with the grating.
The visible spectrum relies on wavelengths from 380 nm to 750 nm. Depending on the material and context, different wavelengths will refract or diffract differently. This is crucial in predicting patterns of light.
  • Short wavelengths (violet): Higher energy, diffracted strongly
  • Mid wavelengths (yellow-green): Moderate visibility
  • Long wavelengths (red): Low energy, diffracted less
Accurately assessing and using wavelength values in equations enables precise determination of visible light patterns.

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Most popular questions from this chapter

Light of wavelength 633 \(\mathrm{nm}\) from a distant source is incident on a slit 0.750 \(\mathrm{mm}\) wide, and the resulting diffraction pattern is observed on a screen 3.50 \(\mathrm{m}\) away. What is the distance between the two dark fringes on either side of the central bright fringe?

If a diffraction grating produces its third-order bright band at an angle of \(78.4^{\circ}\) for light of wavelength \(681 \mathrm{nm},\) find (a) the number of slits per centimeter for the grating and (b) the angular location of the first-order and second-order bright bands. (c) Will there be a fourth-order bright band? Explain.

The wavelength range of the visible spectrum is approximately \(380-750 \mathrm{nm} .\) White light falls at normal incidence on a diffraction grating that has 350 slits/mm. Find the angular width of the visible spectrum in (a) the first order and (b) the third order. (Note: An advantage of working in higher orders is the greater angular spread and better resolution. A disadvantage is the overlapping of different orders, as shown in Example \(36.4 . )\)

Laser light of wavelength 632.8 nm falls normally on a slit that is 0.0250 \(\mathrm{mm}\) wide. The transmitted light is viewed on a distant screen where the intensity at the center of the central bright fringe is 8.50 \(\mathrm{W} / \mathrm{m}^{2} .\) (a) Find the maximum number of totally dark fringes on the screen, assuming the screen is large enough to show them all. (b) At what angle does the dark fringe that is most distant from the center occur? (c) What is the maximum intensity of the bright fringe that occurs immediately before the dark fringe in part (b)? Approximate the angle at which this fringe occurs by assuming it is midway between the angles to the dark fringes on either side of it.

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 \(\mathrm{nm}(\mathrm{x}\) rays)?
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