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Chapter 27: Problem 51

Violet light (wavelength \(=410 \mathrm{nm}\) ) and red light (wavelength \(=\) \(660 \mathrm{nm}\) ) lie at opposite ends of the visible spectrum. (a) For each wavelength, find the angle \(\theta\) that locates the first-order maximum produced by a grating with 3300 lines/cm. This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for (b) the second-order maximum and (c) the third-order maximum. (d) From your results, decide whether there is an overlap between any of the "rainbows" and, if so, specify which orders overlap.

Short Answer

Expert verified
Overlap occurs between the second-order red and third-order violet spectra.

Step by step solution

01

Understand the Grating Equation

The diffraction grating equation is given by \( d \sin \theta = m \lambda \), where \( d \) is the distance between adjacent grating lines (grating spacing), \( \theta \) is the angle of diffraction, \( m \) is the order of the maximum, and \( \lambda \) is the wavelength of light. Our task is to find \( \theta \) for given wavelengths and orders.
02

Calculate Grating Spacing

The grating spacing \( d \) can be calculated from the number of lines per centimeter. Given 3300 lines/cm, we have \( d = \frac{1}{3300} \) cm per line, which converts to meters: \( d = \frac{1}{3300 \times 100} = 3.03 \times 10^{-6} \) m.
03

Calculate First-Order Maximum for Violet Light

For violet light \( \lambda = 410 \) nm = \( 410 \times 10^{-9} \) m: Using the equation \( d \sin \theta_1 = 1 \times 410 \times 10^{-9} \), we solve for \( \theta_1 \): \( \sin \theta_1 = \frac{410 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_1 \approx 7.8^\circ \).
04

Calculate First-Order Maximum for Red Light

For red light \( \lambda = 660 \) nm = \( 660 \times 10^{-9} \) m: Using \( d \sin \theta_1 = 1 \times 660 \times 10^{-9} \), solve for \( \theta_1 \): \( \sin \theta_1 = \frac{660 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_1 \approx 12.6^\circ \).
05

Calculate Second-Order Maximum for Violet Light

For the second order (\( m = 2 \)): \( d \sin \theta_2 = 2 \times 410 \times 10^{-9} \), solve for \( \theta_2 \): \( \sin \theta_2 = \frac{820 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_2 \approx 16.8^\circ \).
06

Calculate Second-Order Maximum for Red Light

For red light, second order \( m = 2 \): \( d \sin \theta_2 = 2 \times 660 \times 10^{-9} \), solve for \( \theta_2 \): \( \sin \theta_2 = \frac{1320 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_2 \approx 27.3^\circ \).
07

Calculate Third-Order Maximum for Violet Light

For the third order (\( m = 3 \)): \( d \sin \theta_3 = 3 \times 410 \times 10^{-9} \), solve for \( \theta_3 \): \( \sin \theta_3 = \frac{1230 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_3 \approx 25.2^\circ \).
08

Calculate Third-Order Maximum for Red Light

For the third order, red light \( m = 3 \): \( d \sin \theta_3 = 3 \times 660 \times 10^{-9} \), solve for \( \theta_3 \): \( \sin \theta_3 = \frac{1980 \times 10^{-9}}{3.03 \times 10^{-6}} \), thus \( \theta_3 \approx 46.9^\circ \).
09

Analyze Overlaps Between Orders

Comparing angles, we see: first-order range \(7.8^\circ - 12.6^\circ\), second-order range \(16.8^\circ - 27.3^\circ\), and third-order range \(25.2^\circ - 46.9^\circ\). There is overlap between the second-order red and third-order violet ranges (25.2掳 to 27.3掳).

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Key Concepts

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

Grating Equation
To understand how light interacts with a diffraction grating, we use the grating equation: \( d \sin \theta = m \lambda \). This equation connects several aspects of light diffraction:
  • \( d \) is the grating spacing, or distance between lines on the grating.
  • \( \theta \) is the angle at which light is diffracted.
  • \( m \) represents the order of the maximum, indicating how many wavelengths are used in the path difference.
  • \( \lambda \) is the wavelength of the light in question.
This equation allows us to determine the angles at which light will be bright, known as the maxima, for various wavelengths and orders.
Visible Spectrum
The visible spectrum encompasses the range of light wavelengths that humans can see, typically spanning from about 380 nm to 750 nm. In this context:
  • Violet light, at one end, has a wavelength of approximately 410 nm.
  • Red light, at the other, has a wavelength around 660 nm.
These points of the spectrum are crucial in diffraction experiments, as the grating can spread a mixed light source into separate spectral colors, each emerging at specific angles. This separation into colored "rainbows" helps illustrate how different wavelengths occupy different positions in the visible spectrum.
Wavelength
Wavelength is a key concept when dealing with diffraction gratings. It is the distance between consecutive crests of a wave. Light, being a wave, has properties such as:
  • Different colors correspond to different wavelengths.
  • Shorter wavelengths like violet are close to the ultraviolet end.
  • Longer wavelengths like red are near the infrared end.
Understanding wavelength is vital in predicting how light behaves when encountering a grating. For instance, shorter wavelengths are bent less than longer wavelengths, impacting the angle of diffraction for each color.
Order of Maximum
The order of maximum refers to different path differences that lead to constructive interference of light waves. In experiments involving gratings, these orders manifest as distinct bright lines at predictable angles:
  • First-order (\( m = 1 \)) refers to the first set of bright spots.
  • Second-order (\( m = 2 \)) creates another, typically wider set of bright spots further out.
  • Third-order (\( m = 3 \)) follows suit, presenting light at even larger angles.
The order influences the angle at which each specific wavelength will emerge, calculated using the grating equation. Additionally, higher orders are spread over larger angles, which may result in overlapping of spectral lines from different orders, affecting how colors mix and appear.

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Most popular questions from this chapter

The wavelength of the laser beam used in a compact disc player is \(780 \mathrm{nm}\). Suppose that a diffraction grating produces first-order tracking beams that are \(1.2 \mathrm{mm}\) apart at a distance of \(3.0 \mathrm{mm}\) from the grating. Estimate the spacing between the slits of the grating.

A diffraction grating is \(1.50 \mathrm{cm}\) wide and contains 2400 lines. When used with light of a certain wavelength, a third-order maximum is formed at an angle of \(18.0^{\circ} .\) What is the wavelength (in \(\mathrm{nm}\) )?

A uniform layer of water \((n=1.33)\) lies on a glass plate \((n=1.52) .\) Light shines perpendicularly on the layer. Because of constructive interference, the layer looks maximally bright when the wavelength of the light is \(432 \mathrm{nm}\) in vacuum and \(a l s o\) when it is \(648 \mathrm{nm}\) in vacuum. (a) Obtain the minimum thickness of the film. (b) Assuming that the film has the minimum thickness and that the visible spectrum extends from 380 to \(750 \mathrm{nm},\) determine the visible wavelength(s) in vacuum for which the film appears completely dark.

In a Young's double-slit experiment, the seventh dark fringe is located \(0.025 \mathrm{m}\) to the side of the central bright fringe on a flat screen, which is \(1.1 \mathrm{m}\) away from the slits. The separation between the slits is \(1.4 \times 10^{-4} \mathrm{m}\) What is the wavelength of the light being used?

The distance between adjacent slits of a certain diffraction grating is \(1.250 \times 10^{-5} \mathrm{m} .\) The grating is illuminated by monochromatic light with a wavelength of \(656.0 \mathrm{nm},\) and is then heated so that its temperature increases by \(100.0 \mathrm{C}^{\circ} .\) Determine the change in the angle of the seventh-order principal maximum that occurs as a result of the thermal expansion of the grating. The coefficient of linear expansion for the diffraction grating is \(1.30 \times 10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1} .\) Be sure to include the proper algebraic sign with your answer: \(+\) if the angle increases, \(-\) if the angle decreases.
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