7.6.3: Problems
- Page ID
- 33422
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\): Illumination pattern
Two sources of light waves of equal frequency and amplitude are shown. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot. A blue screen is shown on the right-hand side of the animation (position is given in arbitrary units). Restart.
- What illumination pattern appears on the screen?
- Does the appearance of the screen change with time?
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{2}\): Type of interference
Two sources of light waves of equal frequency and amplitude are shown. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot (position is given in arbitrary units). Restart. At the points indicated (\(A,\: B,\: C,\: D\)), is the interference of the waves from the two sources maximally constructive, completely destructive, or somewhere in between?
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{3}\): Phase of sources
Two sources of light waves of equal frequency and amplitude are shown. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot (position is given in nanometers). Restart. Are the two sources in phase or \(180^{\circ}\) out of phase? Explain.
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{4}\): Distance between hidden sources
Two sources of light waves that are in phase, of equal frequency, and have the same amplitude are underneath the red rectangle. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot (position is given in arbitrary units). Restart. How far apart are the two sources in terms of their wavelength?
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{5}\): Changes in interference pattern
Two sources of light waves of equal frequency and amplitude are shown. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot (position is given in nanometers). Restart.
- What wave pattern would result if the phase of the bottom source was increased by \(180^{\circ}\)? Explain.
- How would the pattern be different if, instead, the top source was changed?
The applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{6}\): Analysis of interference pattern
Two sources of light waves of equal frequency and amplitude are shown. The magnitude of the electric field is represented by the light and dark areas. The lighter the spot, the greater the magnitude of the electric field at that spot (position is given in nanometers). Restart.
- Are the sources in phase or out of phase?
- In terms of the wavelength of the light produced by the sources, is the distance between the sources \(0.5\) wavelength, \(1\) wavelength, \(1.25\) wavelengths, \(1.5\) wavelengths, \(1.75\) wavelengths, or \(2\) wavelengths? Explain.
- If the wavelengths of both sources were halved, would point \(A\) remain a point of complete destructive interference, change to a point of maximally constructive interference, or be somewhere in between? Would point \(B\) remain a point of maximally constructive interference? If not, how would point \(B\) change? Finally, would point \(C\) remain a point of complete destructive interference? If not, how would point \(C\) change?
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{7}\): Double slit
The animation shows the amplitude of a wave pattern. The greatest amplitude is represented by white, negative amplitudes are represented by black, and areas with zero amplitude are represented by gray. The mouse can be used to make coordinate measurements (position is given in nanometers). Restart. A double slit is hidden underneath the red bar in the animation. What is the slit separation?
This applet calculates seven frames and then runs continuously. For a large number of sources, or for very small wavelengths, this calculation can take some time, so let the applet finish calculating all seven frames.
Exercise \(\PageIndex{8}\): Waves in different mediums
The animation shows a right-traveling light wave (shown in blue) incident on a region composed of different mediums. The left-traveling wave (shown in red above it) represents the sum of all reflections of the incident wave at the boundaries between mediums (position is given in arbitrary units).
Notice that the amplitude of the waves vary with the medium in order to conserve energy. (The change in amplitude of a wave across boundaries may not be taken up in your introductory physics textbook.) You do not need to consider the amplitude change across mediums when answering this question. However, you should be able to compare the amplitudes of incident and reflected light in the same medium (see part b). Restart.
- Rank the four mediums in terms of their indices of refraction, from smallest to greatest.
- Compare the amplitude of incident light and reflected light in medium \(1\) when there are four layers and two layers. Why is the amplitude of reflected light (shown in red) in medium \(1\) so much smaller when there are multiple layers than when there are only two layers?
Problem authored by Melissa Dancy and Anne J. Cox.
Exercise \(\PageIndex{9}\): Thin film of variable width
The animation shows a right-traveling light wave (shown in blue) incident on a region composed of an unknown index of refraction embedded in air (position is given in arbitrary units). The left-traveling wave (shown in red above it) represents the sum of all reflections of the incident wave at the boundaries. Notice that the wave amplitude varies with the medium in order to conserve energy.
Observe how the intensity of light transmitted through the medium changes by using the slider to change the thickness of the substance. Note: because the data points are connected, you must move the slider slowly to obtain a smooth curve. Restart.
- Why are there numerous transmission peaks?
- What is the index of refraction of the material?
Exercise \(\PageIndex{10}\): Thin film, variable wavelength of light
The animation shows a right-traveling light wave (shown in blue) incident on a region composed of an unknown index of refraction embedded in air (position is given in arbitrary units). The left-traveling wave (shown in red above it) represents the sum of all reflections of the incident wave at the boundaries. Notice that the wave amplitude varies with the medium in order to conserve energy.
Observe how the intensity of light transmitted through the medium changes by using the slider to change the wavelength of incident light. Note: Because the data points are connected, you must move the slider slowly to obtain a smooth curve. Restart.
- Why are there numerous transmission peaks?
- What is the index of refraction of the material?
- Mathematically prove that the transmission peaks must be spaced farther apart as the wavelength increases.
Exercise \(\PageIndex{11}\): A dielectric mirror
The animation demonstrates how layers of alternating materials can be used to form a mirror (position is given in arbitrary units). For more discussion of this process, see the dielectric mirror Illustration (Illustration 37.2) in this chapter. By itself, each material will allow light to be transmitted through it. But when layered in a particular way, the materials act as a mirror, reflecting most of the incident light. Restart.
How does such a mirror work? In other words, conceptually explain the physics behind this phenomenon.
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