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<h2 class="hd hd-2 unit-title">Introduction to Snell's Law</h2>
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This week, we will be discussing more properties of propagating electromagnetic waves, first looking at the behavior of 2D waves reflecting from interfaces. </p><p>
We begin by discussing the general formalism for describing 2D waves (not restricted to EM waves) reflecting and transmitting at a boundary. </p><p>
Then, we begin discussing reflection and refraction of electromagnetic waves at a boundary between two media, neither of which is perfectly conducting. This behavior is governed by Snell's law. </p>
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<h2 class="hd hd-2 unit-title">L29Q1: Boundary Conditions for Incident Wave</h2>
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Boundary Conditions for Incident Wave - part a
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In the previous segment, we began to construct a general formalism for describing plane waves reflecting and transmitting at a boundary. This problem serves as a review of the notation that was introduced. </p>
<p><b class="bfseries">(Part a)</b> The general form for a traveling, scalar plane wave in free space is: </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]\psi (\vec{r},t)=Ae^{i\left(\vec{k}\cdot \vec{r} - \omega t \right)}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part i)</b> What does [mathjaxinline]\vec{k}[/mathjaxinline] represent or describe? </p>
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<text> a) the position of the wave at time [mathjaxinline]t[/mathjaxinline]</text>
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<text> b) the position at which the amplitude is measured</text>
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<text> c) the direction of propagation of the wave</text>
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<text> d) the dispersion of the wave</text>
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<p><b class="bfseries">(Part ii)</b> What does [mathjaxinline]\vec{r}[/mathjaxinline] represent or describe? </p>
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<text> a) the position of the wave at time [mathjaxinline]t[/mathjaxinline]</text>
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<text> b) the position at which the amplitude is measured</text>
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<text> c) the direction of propagation of the wave</text>
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<text> d) the dispersion of the wave</text>
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Boundary Conditions for Incident Wave - part b
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<p><b class="bfseries">(Part b)</b> The correct relations for the k-vectors of the reflected and transmitted waves are: </p>
<table id="a0000000004" 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]\left|\vec{k}_{\alpha }\right|^{2} = \dfrac {\omega ^{2}}{v_{\alpha }^{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<table id="a0000000005" 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]\left|\vec{k}_{\beta }\right|^{2} = \dfrac {\omega ^{2}}{v_{\beta }^{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
where the subscripts [mathjaxinline]\alpha[/mathjaxinline] and [mathjaxinline]\beta[/mathjaxinline] denote the reflected and transmitted waves, respectively. </p>
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Which of these k-vectors has the SAME magnitude as the incident wavevector, [mathjaxinline]\vec{k}[/mathjaxinline]? </p>
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<text> a) [mathjaxinline]\vec{k}_{\alpha }[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\vec{k}_{\beta }[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L29Q2: Snell's Law - I</h2>
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Snell&#39;s Law I
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In deriving Snell's law, we used the following relation between wave-vector components of the reflected, transmitted, and incident waves (for a wave of frequency [mathjaxinline]\omega[/mathjaxinline], incident on a boundary at [mathjaxinline]x=0[/mathjaxinline]): </p>
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<tr>
<td class="equation" style="width:80%; border:none">[mathjax]k_{\alpha y} = k_{\beta y} = k_{y}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
where the subscripts [mathjaxinline]\alpha[/mathjaxinline] and [mathjaxinline]\beta[/mathjaxinline] refer to the reflected and transmitted waves, respectively. </p>
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If a wave is incident at an angle [mathjaxinline]\theta[/mathjaxinline] (with respect to the [mathjaxinline]x[/mathjaxinline] axis, which is perpendicular to the boundary) in a medium with index of refraction [mathjaxinline]n[/mathjaxinline], and is transmitted at an angle [mathjaxinline]\theta ^{\prime }[/mathjaxinline] (again with respect to the [mathjaxinline]x[/mathjaxinline] axis) in a medium with index of refraction [mathjaxinline]n^{\prime }[/mathjaxinline], find expressions for the [mathjaxinline]k_{y}[/mathjaxinline] (incident) and [mathjaxinline]k_{\beta y}[/mathjaxinline] (transmitted) wave-vector components in terms of these variables. Express your answer using <code>c</code>, <code>n</code>, <code>n'</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>theta'</code> for [mathjaxinline]\theta ^{\prime }[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]k_{y}=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]k_{\beta y}=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">L29Q3: Snell's Law - II</h2>
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Snell&#39;s Law - IIa
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<p><b class="bfseries">(Part a)</b> Again, consider a wave of frequency [mathjaxinline]\omega[/mathjaxinline], incident on a boundary at [mathjaxinline]x=0[/mathjaxinline]. The incident angle is [mathjaxinline]\theta[/mathjaxinline] (with respect to the [mathjaxinline]x[/mathjaxinline] axis, which is perpendicular to the boundary) in a medium with index of refraction [mathjaxinline]n[/mathjaxinline], and the transmitted angle is [mathjaxinline]\theta ^{\prime }[/mathjaxinline] (again with respect to the [mathjaxinline]x[/mathjaxinline] axis) in a medium with index of refraction [mathjaxinline]n^{\prime }[/mathjaxinline]. </p>
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Now use the relation for the component of the transmitted wave [mathjaxinline]k_{\beta x} = \sqrt {\dfrac {\omega ^{2}}{v^{\prime 2}} - k_{y}^{2}}[/mathjaxinline] to find an expression for [mathjaxinline]\cos \theta ^{\prime }[/mathjaxinline]. Note, the solution can be found by manipulating Snell's law, but our objective here is to recall the relation between [mathjaxinline]k_{\beta x}[/mathjaxinline], [mathjaxinline]k_{y}[/mathjaxinline], and relevant angles. </p>
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Express your answer in terms of <code>n</code>, <code>n'</code>, <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>theta'</code> for [mathjaxinline]\theta ^{\prime }[/mathjaxinline]. Note, the parser will look better if you wrap primed variables in parentheses, like <code>(n')</code>. </p>
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<p style="display:inline">[mathjaxinline]\cos \theta ^{\prime }=[/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">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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Snell&#39;s Law - IIb
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<p><b class="bfseries">(Part b)</b> Based on your result above, what must be the relation between [mathjaxinline]n[/mathjaxinline] and [mathjaxinline]n^{\prime }[/mathjaxinline] in order for the transmitted wave to ALWAYS exist (i.e., there will be a transmitted wave for any value of the angle of the incident wave)? </p>
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<text> a) [mathjaxinline]n = n^{\prime }[/mathjaxinline]</text>
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<text> b) [mathjaxinline]n \leq n^{\prime }[/mathjaxinline]</text>
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<text> c) [mathjaxinline]n \geq n^{\prime }[/mathjaxinline]</text>
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<text> d) there is no constraint</text>
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<h2 class="hd hd-2 unit-title">L29Q4: Snell's Law - III [WITH SIMULATION]</h2>
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Snell&#39;s Law - IIIa
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Consider the following figures, with an incoming wave incident from the left. What is the relation between [mathjaxinline]n_{L}[/mathjaxinline] and [mathjaxinline]n_{R}[/mathjaxinline] in each case? </p>
<p><b class="bfseries">(Part a)</b> Plot 1 </p>
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<text> a) [mathjaxinline]n_{L} = n_{R}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]n_{L} \lt n_{R}[/mathjaxinline]</text>
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Snell&#39;s Law - IIIb
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<p><b class="bfseries">(Part b)</b> Plot 2 </p>
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<img src="/assets/courseware/v1/106b1b8b0e61f7ce3c68c528df328886/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_18_04_3a.png" width="770"/>
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<text> a) [mathjaxinline]n_{L} = n_{R}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_18_04_3b_2_1" id="input_lect_18_04_3b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_18_04_3b_2_1-choice_2-label" for="input_lect_18_04_3b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_18_04_3b_2_1">
<text> b) [mathjaxinline]n_{L} \lt n_{R}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_18_04_3b_2_1" id="input_lect_18_04_3b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_18_04_3b_2_1-choice_3-label" for="input_lect_18_04_3b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_18_04_3b_2_1">
<text> c) [mathjaxinline]n_{L} \gt n_{R}[/mathjaxinline]</text>
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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=refracting_light_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/c30a91f58e66debd20194361bb922cf8/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_refracting_light_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L29Q5: Properties of Refracted Light [WITH SIMULATION]</h2>
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Properties of Refracted Light - part a
</h3>
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<p><b class="bfseries">(Part a)</b> Which of the following properties CHANGE when an incident wave is refracted at the boundary between two media with different indices of refraction? Check ALL that apply. </p>
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<text>a) wavelength</text>
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<text>b) frequency</text>
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<text>c) wavevector</text>
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Properties of Refracted Light - part b
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<p><b class="bfseries">(Part b)</b> The plots below depict three versions of incident and transmitted (also called refracted) waves on either side of a boundary between two different media (the reflected wave amplitude is not shown). The media have indices of refraction [mathjaxinline]n_{R}=2n_{L}[/mathjaxinline], where [mathjaxinline]L[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline] denote "left" and "right", respectively. Only ONE of these three pairs of plots depicts a physically possible situation. </p>
<p>
The left plots show the wave amplitude (color scale), and the right plots specifically show the wave amplitude as a function of position along the boundary, for incident and refracted waves.<br/>Note 1: The label "ref." denotes the <i class="itshape">refracted</i> wave, not the <i class="itshape">reflected</i> wave. <br/>Note 2: The vertical scales have been normalized separately for the incident and refracted waves so that the maximum value is the same in both cases. </p>
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Which of these three plots correctly depicts the amplitude of the waves on either side of the boundary? </p>
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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=refracted_waves_frequency_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/e7082111354f184993b6b9f09e352cb1/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_refracted_waves_frequency_1.png"/></div><p/>
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Total Internal Reflection - part a
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<p><b class="bfseries">(Part a)</b> Consider two media with indices of refraction [mathjaxinline]n_1 \lt n_2[/mathjaxinline]. Light propagates from medium 2 to medium 1. The critical angle of total internal reflection is [mathjaxinline]\pi /6[/mathjaxinline]. What is the ratio [mathjaxinline]n_{1}/n_{2}[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]n_{1}/n_{2}=[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<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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<code>e, pi</code>
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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>
</td>
<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">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<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>
</tr>
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<td class="formulainput">row vector</td>
<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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<h3 class="hd hd-3 problem-header" id="lect_18_05b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_18_05b-problem-progress" tabindex="-1">
Total Internal Reflection - part b
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<p><b class="bfseries">(Part b)</b> Again, consider two mediums with indices of refraction [mathjaxinline]n_1/n_2=0.9[/mathjaxinline]. Light propagates from medium 2 to medium 1. What is the critical angle of total internal reflection? Give a numerical answer in units of radians. </p>
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<p style="display:inline">[mathjaxinline]\theta _{c} =[/mathjaxinline] </p>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
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<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<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>
</tr>
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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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<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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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<code>e, pi</code>
</td>
<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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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<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]
</td>
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<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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