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<h2 class="hd hd-2 unit-title">Introduction to Perfect Conductors</h2>
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Next, we discuss the interaction of the fields in electromagnetic waves at boundaries. In particular, we discuss how electromagnetic waves are reflected from perfect conductors. </p><p>
Perfect conductors have zero resistance, and therefore allow electrons to move freely. Of course, this is only an idealized approximation, but it is useful for calculating interactions at boundaries. </p><p>
In a perfect conductor, electrons will readily move to cancel the electric field inside. This means that the boundary acts like a FIXED boundary, which we have studied before. We will see what implications this holds for the reflection of electromagnetic waves. </p>
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<h2 class="hd hd-2 unit-title">L21v1: Reflection from a Perfect Conductor</h2>
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<h2 class="hd hd-2 unit-title">L21Q1: Incident and Reflected Waves I</h2>
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Boundary Conditions for a Perfect Conductor I
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Each of the following electromagnetic waves are incident on perfect conductor. For each wave, determine the reflected electric and magnetic fields. </p>
<p><b class="bfseries">(Part a)</b> An incident electromagnetic wave, traveling in the [mathjaxinline]\hat{j}[/mathjaxinline] direction, encounters a perfect conductor at [mathjaxinline]y=0[/mathjaxinline]. The incident electric field has the following form: </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}_{i}(y,t)=\frac{E_{0}}{2}\cos {(ky - \omega t)}\hat{k}[/mathjax]</td>
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Find the reflected fields: [mathjaxinline]\vec{E}_{r}[/mathjaxinline] and [mathjaxinline]\vec{B}_{r}[/mathjaxinline]. Express your answer in terms of <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], <code>c</code>, <code>k</code>, <code>x</code>, <code>y</code>, <code>z</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>t</code> and <code>hati</code> <code>hatj</code>, and <code>hatk</code> for [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]\hat{k}[/mathjaxinline], respectively, as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{E}_{r} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\vec{B}_{r} =[/mathjaxinline] </p>
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Boundary Conditions for a Perfect Conductor II
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<p><b class="bfseries">(Part b)</b> An incident electromagnetic wave, traveling in the [mathjaxinline]-\hat{i}[/mathjaxinline] direction, encounters a perfect conductor at [mathjaxinline]x=0[/mathjaxinline]. The incident electric field has the following form: </p>
<table id="a0000000006" 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}_{i}(x,t)=\frac{E_{0}}{2}\cos {(kx + \omega t)}\hat{k}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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Find the reflected fields: [mathjaxinline]\vec{E}_{r}[/mathjaxinline] and [mathjaxinline]\vec{B}_{r}[/mathjaxinline]. Express your answer in terms of <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], <code>c</code>, <code>k</code>, <code>x</code>, <code>y</code>, <code>z</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>t</code> and <code>hati</code> <code>hatj</code>, and <code>hatk</code> for [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]\hat{k}[/mathjaxinline], respectively, as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{E}_{r} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\vec{B}_{r} =[/mathjaxinline] </p>
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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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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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<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"><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>
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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>
</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>
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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">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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Boundary Conditions for a Perfect Conductor III
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<p><b class="bfseries">(Part c)</b> An incident electromagnetic wave, traveling in the [mathjaxinline]\hat{k}[/mathjaxinline] direction, encounters a perfect conductor at [mathjaxinline]z=0[/mathjaxinline]. The incident magnetic field has the following form: </p>
<table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{B}_{i}(z,t)=-\frac{E_{0}}{2c}\cos {(kz - \omega t)}\hat{j}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Find the reflected fields: [mathjaxinline]\vec{E}_{r}[/mathjaxinline] and [mathjaxinline]\vec{B}_{r}[/mathjaxinline]. Express your answer in terms of <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], <code>c</code>, <code>k</code>, <code>x</code>, <code>y</code>, <code>z</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>t</code> and <code>hati</code> <code>hatj</code>, and <code>hatk</code> for [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]\hat{k}[/mathjaxinline], respectively, as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{E}_{r} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\vec{B}_{r} =[/mathjaxinline] </p>
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<code>2520</code>
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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>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
<td class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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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<h2 class="hd hd-2 unit-title">L21Q2: Incident and Reflected Waves II</h2>
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Boundary Conditions for a Perfect Conductor IV
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Consider an incident electromagnetic wave, traveling in the [mathjaxinline]\hat{k}[/mathjaxinline] direction, that encounters a perfect conductor at [mathjaxinline]z=0[/mathjaxinline]. The expressions for the total electric field and magnetic fields are the sums of the incident and reflected fields (there is no transmitted field for a perfect conductor): </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{E}(z,t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\vec{E}_{i}(z,t) + \vec{E}_{r}(z,t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<tr id="a0000000004">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
&#160;
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<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{E_{0}}{2}\cos {(kz - \omega t)}\hat{i} - \frac{E_{0}}{2}\cos {(kz + \omega t)}\hat{i}[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000005">
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&#160;
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[mathjaxinline]\displaystyle =E_{0}\sin (\omega t)\sin (kz)\hat{i}[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000007">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{B}(z,t)[/mathjaxinline]
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[mathjaxinline]\displaystyle =\vec{B}_{i}(z,t) + \vec{B}_{r}(z,t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<tr id="a0000000008">
<td style="width:40%; border:none">&#160;</td>
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&#160;
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[mathjaxinline]\displaystyle =\frac{E_{0}}{2c}\cos {(kz - \omega t)}\hat{j} + \frac{E_{0}}{2c}\cos {(kz + \omega t)}\hat{j}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000009">
<td style="width:40%; border:none">&#160;</td>
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&#160;
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[mathjaxinline]\displaystyle =\frac{E_{0}}{c}\cos (\omega t)\cos (kz)\hat{j}[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
where the last line in both cases is found using: </p>
<table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\cos (a+b)=\cos (a)\cos (b)-\sin (a)\sin (b)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part a)</b> What kind of wave do these total fields represent? </p>
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<text> a) a wave traveling in the +z-direction</text>
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<text> c) a standing wave</text>
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<text> d) a transient wave</text>
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Boundary Conditions for a Perfect Conductor V
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<p><b class="bfseries">(Part b)</b> What is the vector product [mathjaxinline]\vec{k}\times \vec{E}[/mathjaxinline] for the total wave solutions given above, where [mathjaxinline]\vec{k}[/mathjaxinline] is the wavevector which has magnitude [mathjaxinline]k[/mathjaxinline] and points in the [mathjaxinline]\hat{z}[/mathjaxinline] direction in this case? Is this what you would expect? (Note: Don't confuse the wavevector [mathjaxinline]\vec{k}[/mathjaxinline] and the notation for the [mathjaxinline]z[/mathjaxinline] direction [mathjaxinline]\hat{k}[/mathjaxinline].) </p>
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Express your answer in terms of <code>E_0</code> for [mathjaxinline]E_{0}[/mathjaxinline], <code>c</code>, <code>k</code>, <code>x</code>, <code>y</code>, <code>z</code>, <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>t</code> and <code>hati</code>, <code>hatj</code>, and <code>hatk</code> for [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]\hat{k}[/mathjaxinline], respectively, as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{k} \times \vec{E} =[/mathjaxinline] </p>
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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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<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">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">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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<h2 class="hd hd-2 unit-title">L21Q3: Reflected EM Wave [WITH SIMULATION]</h2>
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Reflected EM Wave
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Consider a plane electromagnetic wave, reflected between two perfect conductors. Which of the following figures represents a possible form of the wave for this configuration? Select ALL that apply. </p>
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Select ALL that apply. <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_13_06_0_2_1">
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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=EM_waves_standing_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/ff70566a211c88d0e86d7cdd449b41a6/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_EM_waves_standing_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L21v2: Standing Electromagnetic Waves and Poynting Vector</h2>
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<h2 class="hd hd-2 unit-title">L21Q4: Poynting Vector of Traveling Waves [WITH SIMULATION]</h2>
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Poynting Vector of Traveling Waves
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In the plot below, the Poynting vector (black) of a traveling EM wave is shown at an instant in time and at a particular position in space. You are encouraged to "play" with the visualization in the Jupyter notebook after this problem. </p>
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Which of the following statements about the Poynting vector of a traveling EM wave is true? Select ALL that apply. </p>
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<text>a) The magnitude of the Poynting vector varies in space</text>
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<input type="checkbox" name="input_lect_13_06_1_2_1[]" id="input_lect_13_06_1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_13_06_1_2_1-choice_1-label" for="input_lect_13_06_1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_1_2_1">
<text>b) The Poynting vector sometimes has zero magnitude</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_13_06_1_2_1[]" id="input_lect_13_06_1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_13_06_1_2_1-choice_2-label" for="input_lect_13_06_1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_1_2_1">
<text>c) The non-zero Poynting vector will always point in the same direction</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_13_06_1_2_1[]" id="input_lect_13_06_1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_13_06_1_2_1-choice_3-label" for="input_lect_13_06_1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_1_2_1">
<text>d) The time-averaged Poynting flux through a surface is zero</text>
</label>
</div>
<span id="answer_lect_13_06_1_2_1"/>
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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=EM_waves_traveling_Poynting_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/v1/960057087f33c18a6c7dc63f37daa1ca/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_EM_waves_traveling_Poynting_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L21Q5: Poynting Vector of Standing Waves [WITH SIMULATION]</h2>
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Poynting Vector of Standing Waves
</h3>
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<div class="problem">
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<p>
In the plot below, the Poynting vector (black) of a <i class="itshape">standing</i> EM wave is shown at an instant in time and at a particular position in space. You are encouraged to "play" with the visualization in the Jupyter notebook after this problem. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/64b75324819e5dff74e2626d23eec273/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_SIM_lect_13_06_2.png" width="770"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Which of the following statements about the Poynting vector of a standing EM wave is true? Select ALL that apply. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_13_06_2_2_1">
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<div class="field">
<input type="checkbox" name="input_lect_13_06_2_2_1[]" id="input_lect_13_06_2_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_13_06_2_2_1-choice_0-label" for="input_lect_13_06_2_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_2_2_1">
<text>a) The magnitude of the Poynting vector varies in space</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_13_06_2_2_1[]" id="input_lect_13_06_2_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_13_06_2_2_1-choice_1-label" for="input_lect_13_06_2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_2_2_1">
<text>b) The Poynting vector sometimes has zero magnitude</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_13_06_2_2_1[]" id="input_lect_13_06_2_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_13_06_2_2_1-choice_2-label" for="input_lect_13_06_2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_2_2_1">
<text>c) The non-zero Poynting vector will always point in the same direction</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_13_06_2_2_1[]" id="input_lect_13_06_2_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_13_06_2_2_1-choice_3-label" for="input_lect_13_06_2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_13_06_2_2_1">
<text>d) The time-averaged Poynting flux through a surface is zero</text>
</label>
</div>
<span id="answer_lect_13_06_2_2_1"/>
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<span class="status unanswered" id="status_lect_13_06_2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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</p>
<p>
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<span id="solution_lect_13_06_2_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
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<input type="hidden" name="problem_id" value="Poynting Vector of Standing Waves" />
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<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
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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=EM_waves_standing_Poynting_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/e471904b2cc010515ccf530c98d2df26/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_EM_waves_standing_Poynting_1.png"/></div><p/>
</div>
</div>
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</div>