7.8.3: Problems
- Page ID
- 33430
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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 \(\PageIndex{1}\): Identify polarization
The animation shows the result of adding two perpendicular electric fields together. Each field is part of an electromagnetic wave traveling along the \(z\) axis. Each electric field is shown separately on the two graphs on the left. The graphs show the electric field at one point on the \(z\) axis for various times. On the right the animation shows both electric fields and their sum at the same point on the \(z\) axis and at the same times as the graphs on the left. It is as if you are looking down the \(z\) axis at the electric field. You can change the electric fields and the phase difference between the two fields and see the resulting waves. Restart.
You are given several graphs of the \(x\) and \(y\) components of the electric field. Produce a representation using the animation, like that shown in the right-hand animation, for the sum of the two fields. Use your representation to identify the type of polarization (include direction) that results.
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Figure \(\PageIndex{1}\) -
Figure \(\PageIndex{2}\) -
Figure \(\PageIndex{3}\) -
- Figure \(\PageIndex{4}\)
Problem authored by Melissa Dancy and Wolfgang Christian.
Exercise \(\PageIndex{2}\): Determine the electric field
The animation shows the result of adding two perpendicular electric fields together. Each field is part of an electromagnetic wave traveling along the \(z\) axis. Each electric field is shown separately on the two graphs on the left. The graphs show the electric field at one point on the \(z\) axis for various times. On the right, the animation shows both electric fields and their sum at the same point on the \(z\) axis and at the same times as the graphs on the left. It is as if you are looking down the \(z\) axis at the electric field. Restart. You can change the electric fields and the phase difference between the two fields and see the resulting waves.
For each of the following, write an equation for the net electric field, which is polarized as shown.
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Figure \(\PageIndex{5}\) -
Figure \(\PageIndex{6}\) -
Figure \(\PageIndex{7}\): Rotating Clockwise -
Figure \(\PageIndex{8}\): Rotating Counterclockwise
Figure \(\PageIndex{9}\): Rotating Clockwise
Problem authored by Melissa Dancy and Wolfgang Christian.
Exercise \(\PageIndex{3}\): Which wave is the sum?
These animations show the electric field component of a traveling wave. Click and drag in each animation to see an alternative view of the wave. Restart.
Which wave represents the sum of Wave \(A\) and Wave \(B\)?
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{4}\): Which wave is the sum?
These animations show the electric field component of a traveling wave. Click and drag in each animation to see an alternative view of the wave. Restart.
Which wave represents the sum of Wave \(A\) and Wave \(B\)?
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{5}\): Describe the polarization state
The animation shows the electric field of a light wave. Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. Restart.
- In which direction is the light traveling?
- Describe the polarization state of the wave. Include in your description the type of polarization and the direction of polarization.
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{6}\): Describe the polarization state
The animation shows the electric field of a light wave. Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. Restart.
- In which direction is the light traveling?
- Describe the polarization state of the wave. Include in your description the type of polarization and the direction of polarization.
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{7}\): Describe the polarization state
The animation shows the electric field of a light wave. Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. Restart.
- In which direction is the light traveling?
- Describe the polarization state of the wave. Include in your description the type of polarization and the direction of polarization.
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{8}\): Describe the polarization state
The animation shows the electric field of a light wave. Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. Restart.
- In which direction is the light traveling?
- Describe the polarization state of the wave. Include in your description the type of polarization and the direction of polarization.
Problem authored by Melissa Dancy.
Exercise \(\PageIndex{9}\): Polarizers
Initially, this animation shows a traveling wave. Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. The subsequent animations show this wave passing through a polarizer. The direction of polarization is indicated by the black line. Restart. Which, if any, of the animations are correct? Explain.
Problem authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.
Exercise \(\PageIndex{10}\): Energy transmitted through polarizer
The animation shows a wave incident on a polarizer (with the direction indicated by the black line). Click-drag to the right or left to rotate about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane. You can use the slider to change the polarization of the incoming linear wave. This also changes the amplitude of the exiting wave. Restart.
- Develop an expression for the amplitude of the exiting wave as a function of the initial amplitude and the polarization angle.
- Develop an expression for the energy transmitted through the polarizer as a function of polarization and the incident energy.
Problem authored by Anne J. Cox.
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.