7.1.2: Explorations
- Page ID
- 33369
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exploration 1: Representation of Plane Waves
Move the slider and observe the animation on the left-hand panel of your screen. The animation shows the electric field in a region of space. The arrows show the field-vector representation of the electric field. The amplitude of the field is represented by the brightness of the arrows. The slider allows you to move along the \(z\) axis. Notice that the electric field is always uniform in the \(xy\) plane but varies along the \(z\) axis (position is given in meters and time is given in nanoseconds). Restart.
- Construct a graph that represents the electric field along the \(z\) axis at \(t = 0\text{ ns}\).
Now view a representation of the electric field. Click-drag inside the animation on the right to view the electric-field representation from different points of view. This representation should closely match the graph you drew for (a). Click on "play" to see a traveling wave. The representation on the right is often used to show a field like that on the left. Remember that the representation on the right is actually a graph of amplitude along the direction of propagation (\(z\) axis). - Keeping that in mind and looking at the graph on the right, rank the amplitude of the field at \(t = 0\text{ ns}\) for the following locations, from smallest to largest.
Location \(x\) coordinate \(y\) coordinate \(z\) coordinate I \(1\) \(0\) \(-1.5\) II \(1\) \(1\) \(-1.5\) III \(0\) \(0\) \(-1.5\) IV \(0\) \(1\) \(-1.0\) V \(1\) \(1\) \(-1.5\) Table \(\PageIndex{1}\)
- Now, push "play" to see the traveling wave. At position \(z = -0.5\text{ m}\), rank the amplitude of the field at the following times (approximately), from smallest to largest.
Time \((\text{ns})\) \(x\) coordinate \(y\) coordinate \(z\) coordinate \(t=0\) \(1\) \(1\) \(-0.5\) \(t=1.7\) \(1\) \(1\) \(-0.5\) \(t=3.3\) \(1\) \(1\) \(-0.5\) \(t=5.0\) \(1\) \(1\) \(-0.5\) \(t=6.7\) \(1\) \(1\) \(-0.5\) Table \(\PageIndex{2}\)
- What is the wavelength (distance between peaks) of the wave?
- What is the frequency of the wave (the period \(T = 1/f\) is the time it takes for the wave to repeat itself at a given location)?
- What is the speed of the wave?
Exploration authored by Melissa Dancy and modified by Anne J. Cox.
Exploration 2: Plant Waves and the Electric Field Equation
You can change the position of the square (that shows you the field-vector representation of the electric field), as well as the maximum value of the electric field and the wavelength (position is given in meters and time is given in nanoseconds). Restart.
The electromagnetic plane wave in the animation above is described by the equation
\[\mathbf{E}(z, t)=E_{\text{max}}\sin (kz-\omega t)\mathbf{i}\nonumber\]
where \(k = 2\pi /\lambda\) (\(\lambda\) is the wavelength) and \(\omega = 2\pi f\) (\(f\) is the frequency).
- Explain why the equation is a function of \(z\) and \(t\) for this wave.
- Why is this equation a vector equation with a component in the \(x\) direction?
- What is the associated equation for the magnetic field (check in your book if needed)?
- What do you predict will happen in both representations (the vector field view to the right and the wave view to the left) if you increase the amplitude? Change the amplitude to check your prediction. Did the frequency change? Why or why not?
- What do you predict will happen in both representations if you increase the wavelength? Try it. This time did the frequency change? Why or why not?
- Pick a value of the wavelength (\(\lambda\)) and measure it.
- Measure the frequency (\(f\)) at this wavelength.
- What is the value of \(\lambda f\)? (It should be \(3\times 10^{8}\text{ m/s}\)).
Note
When you change the wavelength, you need to let the animation play long enough for the old wavelength to disappear from the axis by letting the animation run for \(100-200\text{ ns}\) before making any measurements.
Exploration authored by Anne J. Cox.
Script authored by Wolfgang Christian, Melissa Dancy, and Anne J. Cox.
Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.