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<h2 class="hd hd-2 unit-title">Introduction to Interference Phenomena</h2>
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In this lesson, we will continue our study of electromagnetic waves—their interactions with each other and with multiple surfaces. First, we consider the intensity that results from two superimposed electromagnetic waves, each having a different amplitude and phase. </p><p>
Ultimately, our motivation is to try to understand the phenomenon of colors that appear in light reflecting off of thin films such as soap bubbles. </p>
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<h2 class="hd hd-2 unit-title">L35v1: Motivation - Soap Bubble Color</h2>
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<h2 class="hd hd-2 unit-title">L35Q1: Review of Reflection and Transmission</h2>
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Review of Reflection and Transmission
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<p>
Consider reflection from a surface that partitions two media with indices of refraction [mathjaxinline]n_{1}[/mathjaxinline] and [mathjaxinline]n_{2}[/mathjaxinline] (where [mathjaxinline]n_{1}[/mathjaxinline] is the medium of the incident wave). </p>
<p>
What are the reflection and transmission coefficients, [mathjaxinline]r=\dfrac {E_{0r}}{E_{0i}}[/mathjaxinline] and [mathjaxinline]t=\dfrac {E_{0t}}{E_{0i}}[/mathjaxinline], for an electromagnetic wave at normal incidence? Express your answer in terms of <code>n_1</code> for [mathjaxinline]n_{1}[/mathjaxinline] and <code>n_2</code> for [mathjaxinline]n_{2}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]r =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]t =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L35Q2: Intensity of EM Wave I</h2>
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Intensity of EM Wave I
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Consider the case of an electromagnetic wave (traveling in vacuum) with the following electric field: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E} = E_0 \sin (ky+\omega t)\hat{i}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Find [mathjaxinline]\vec{B}[/mathjaxinline] using <code>k</code>, <code>y</code>, <code>t</code>, <code>c</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], and relevant unit vectors of the form <code>hati</code> for [mathjaxinline]\hat{i}[/mathjaxinline], etc. </p>
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<p style="display:inline">[mathjaxinline]\vec{B} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Find [mathjaxinline]\vec{S}[/mathjaxinline] using <code>k</code>, <code>y</code>, <code>t</code>, <code>c</code>, <code>epsilon_0</code> for [mathjaxinline]\epsilon _{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], and relevant unit vectors of the form <code>hati</code> for [mathjaxinline]\hat{i}[/mathjaxinline], etc. Your answer should not include [mathjaxinline]\mu _0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\vec{S} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> Find the time-averaged intensity [mathjaxinline]\langle I \rangle[/mathjaxinline]. Express your answer in terms of <code>c</code>, <code>epsilon_0</code> for [mathjaxinline]\epsilon _{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\langle \ I \rangle =[/mathjaxinline] </p>
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<td class="formulainput">integers</td>
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<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<code>abs, ln, sqrt</code>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<code>sin, cos, tan, sec, csc, cot</code>
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<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L35Q3: Intensity of EM Wave II</h2>
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Intensity of EM Wave II
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Consider two electromagnetic plane waves with electric fields [mathjaxinline]\vec{E}_{1}[/mathjaxinline] and [mathjaxinline]\vec{E}_{2}[/mathjaxinline]. Which of the following statements are TRUE? Select ALL that apply. </p>
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<text>a) The time-averaged intensity is proportional to the sum of the intensities of each wave.</text>
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<text>b) The time-average intensity depends on the magnitude of the total electric field squared.</text>
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<text>c) If the waves travel in opposite directions, but are otherwise identical, the time-averaged intensity is zero.</text>
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<h3 class="hd hd-2">L35v3: Average intensity of the superposition of electric fields</h3>
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<h2 class="hd hd-2 unit-title">L35Q4: Average Intensity I [WITH SIMULATION]</h2>
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Average Intensity - part a
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Consider the following plots, which show two electromagnetic waves with the following electric fields: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{1}(z,t)=A_{1}E_{0}\cos (kz - \omega t )\hat{x}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{2}(z,t)=A_{2}E_{0}\cos (kz - \omega t + \delta )\hat{x}[/mathjax]</td>
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where [mathjaxinline]\delta[/mathjaxinline] is the phase difference of wave 2 relative to wave 1, and [mathjaxinline]A_{1}[/mathjaxinline] and [mathjaxinline]A_{2}[/mathjaxinline] are dimensionless coefficients within the range [mathjaxinline][0,1][/mathjaxinline]. </p>
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Each of the following plots shows the time-averaged intensity of the two superimposed electromagnetic waves as a function of [mathjaxinline]\delta[/mathjaxinline], where one particular value of [mathjaxinline]\delta[/mathjaxinline] is highlighted by a vertical blue-green line. Match each of the following plots with the plot from above which corresponds to that particular value of [mathjaxinline]\delta[/mathjaxinline] and which would also yield the minimum and maximum intensities shown below. </p>
<p><b class="bfseries">(Part a)</b> Plot 1. </p>
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Average intensity - part b
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<p><b class="bfseries">(Part b)</b> Plot 2. </p>
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Average intensity - part c
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<p><b class="bfseries">(Part c)</b> What values of [mathjaxinline]\delta[/mathjaxinline] will maximize the intensity? Express your answer in terms of a numerical factor times the index <code>n</code>, where [mathjaxinline]n=0,\, 1,\, 2\, \cdots[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=intensity_waveforms_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/040bd9fcc7b97247cfa3af9d942b4f84/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_intensity_waveforms_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L35v4: Complex Representation of Fields with a Phase Difference</h2>
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<h2 class="hd hd-2 unit-title">L35Q5: Average Intensity II [WITH SIMULATION]</h2>
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Average Intensity II
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<p>
Consider the following diagrams, which show representations of two superimposed electromagnetic waves in the complex plane. The waves have the following form: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{1}(z,t)=A_{1}E_{0}\cos (kz - \omega t )\hat{x}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}_{2}(z,t)=A_{2}E_{0}\cos (kz - \omega t + \delta )\hat{x}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]\delta[/mathjaxinline] is the phase difference of wave 2 relative to wave 1, and [mathjaxinline]A_{1}[/mathjaxinline] and [mathjaxinline]A_{2}[/mathjaxinline] are dimensionless coefficients within the range [mathjaxinline][0,1][/mathjaxinline]. The vector sum of the waves is shown in black in each plot. </p>
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<img src="/assets/courseware/v1/56a4b68a5536100e42213f075b39b92f/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_23_04_all.png" width="990"/>
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<p>
Each of the following plots shows the time-averaged intensity of the two superimposed electromagnetic waves as a function of [mathjaxinline]\delta[/mathjaxinline], where one particular value of [mathjaxinline]\delta[/mathjaxinline] is highlighted by a vertical blue-green line. Match each of the following plots with the diagram from above which corresponds to that particular value of [mathjaxinline]\delta[/mathjaxinline] and which would also yield the minimum and maximum intensities shown below. </p>
<p><b class="bfseries">(Part a)</b> Plot 1. </p>
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<img src="/assets/courseware/v1/45d6538183a31375b763c6c8e035e055/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_23_04_a.png" width="550"/>
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<text> a)</text>
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</div>
<div class="field">
<input type="radio" name="input_lect_23_04_2_1" id="input_lect_23_04_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_23_04_2_1-choice_2-label" for="input_lect_23_04_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_2_1">
<text> b)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_2_1" id="input_lect_23_04_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_23_04_2_1-choice_3-label" for="input_lect_23_04_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_2_1">
<text> c)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_2_1" id="input_lect_23_04_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_23_04_2_1-choice_4-label" for="input_lect_23_04_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_2_1">
<text> d)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_2_1" id="input_lect_23_04_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_23_04_2_1-choice_5-label" for="input_lect_23_04_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_2_1">
<text> e)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_2_1" id="input_lect_23_04_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_23_04_2_1-choice_6-label" for="input_lect_23_04_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_2_1">
<text> f)</text>
</label>
</div>
<span id="answer_lect_23_04_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_23_04_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_23_04_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> Plot 2. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/d4d93831d20710b0b85f13684a5e2c8a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_23_04_b.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_23_04_3_1">
<fieldset aria-describedby="status_lect_23_04_3_1">
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_23_04_3_1-choice_1-label" for="input_lect_23_04_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> a)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_23_04_3_1-choice_2-label" for="input_lect_23_04_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> b)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_23_04_3_1-choice_3-label" for="input_lect_23_04_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> c)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_23_04_3_1-choice_4-label" for="input_lect_23_04_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> d)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_23_04_3_1-choice_5-label" for="input_lect_23_04_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> e)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_3_1" id="input_lect_23_04_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_23_04_3_1-choice_6-label" for="input_lect_23_04_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_3_1">
<text> f)</text>
</label>
</div>
<span id="answer_lect_23_04_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_23_04_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_23_04_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part c)</b> Plot 3. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/e6af9c070a5fa123dfe38a6c0947b0ee/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_23_04_c.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_23_04_4_1">
<fieldset aria-describedby="status_lect_23_04_4_1">
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_23_04_4_1-choice_1-label" for="input_lect_23_04_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> a)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_23_04_4_1-choice_2-label" for="input_lect_23_04_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> b)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_23_04_4_1-choice_3-label" for="input_lect_23_04_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> c)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_23_04_4_1-choice_4-label" for="input_lect_23_04_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> d)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_23_04_4_1-choice_5-label" for="input_lect_23_04_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> e)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_23_04_4_1" id="input_lect_23_04_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_23_04_4_1-choice_6-label" for="input_lect_23_04_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_23_04_4_1">
<text> f)</text>
</label>
</div>
<span id="answer_lect_23_04_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_23_04_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_23_04_solution_3"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Average Intensity II" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_23_04" >
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<div class="submission-feedback" id="submission_feedback_lect_23_04">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=intensity_complex_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/> [mathjaxinline]\bullet[/mathjaxinline] git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/> [mathjaxinline]\bullet[/mathjaxinline] information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/> [mathjaxinline]\bullet[/mathjaxinline] scipy
<br/> [mathjaxinline]\bullet[/mathjaxinline] numpy
<br/> [mathjaxinline]\bullet[/mathjaxinline] ipywidgets
<br/> [mathjaxinline]\bullet[/mathjaxinline] nbinteract
<br/> [mathjaxinline]\bullet[/mathjaxinline] matplotlib
<br/> [mathjaxinline]\bullet[/mathjaxinline] pandas
<br/> [mathjaxinline]\bullet[/mathjaxinline] IPython
<br/> [mathjaxinline]\bullet[/mathjaxinline] ffmpeg
<br/> [mathjaxinline]\bullet[/mathjaxinline] jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/4a2b55f7fdb466c0212397dd8d1932b8/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_intensity_complex_1.png"/></div><p/>
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