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<h2 class="hd hd-2 unit-title">Introduction to Maxwell's Equations and Electromagnetic Waves</h2>
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<p>
Over the past few lessons, we've been studying the wave equation. We found that a string with some tension and mass per unit length is a system that obeys a wave equation. We also showed that longitudinal sound waves obey a wave equation. </p><p>
Now we introduce electromagnetic waves, which also obey a wave equation! Unlike strings and molecules, which are mechanical systems, electromagnetic waves are represented by fields. Thus, we will be dealing with a different beast, but the physics will hopefully be familiar. </p><p>
We will begin by introducing Maxwell's equations and by reviewing the necessary mathematical material (vector calculus). With these ingredients, we will be able to write down a wave equation for electromagnetic fields. </p><p>
One of the ways in which electromagnetic fields can propagate is <i class="itshape">plane waves</i>. We will discuss properties of plane wave propagation and the interaction of fields at boundaries (for example, reflection from perfect conductors). </p>
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<h3 class="hd hd-2">L20v1: Review of Maxwell's equations and divergence and curl</h3>
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<h2 class="hd hd-2 unit-title">L20Q1: Review of Maxwell's Equations</h2>
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<p><center><b>Maxwell's Equations in Vacuum </b></center></p><p><br/></p><p>
When both the charge and current densities are zero, \(\rho = 0\) and
\(\mathbf{\vec{J}} = 0\), respectively, Maxwell's equations in differential form become:
</p><table><tr><th colspan="2">Maxwell's equations in free space</th></tr><tr><td>\(\mathbf{\vec{\nabla}}\cdot \mathbf{\vec{E}}= 0\)</td><td>Gauss' Law</td></tr><tr><td>\(\mathbf{\vec{\nabla}}\cdot \mathbf{\vec{B}}=0\)</td><td>Magnetic Gauss' Law</td></tr><tr><td>\(\mathbf{\vec{\nabla}}\times\mathbf{\vec{E}}=-\dfrac{\partial }{\partial t}\mathbf{\vec{B}} \)</td><td>Faraday's Law</td></tr><tr><td>\(\mathbf{\vec{\nabla}}\times\mathbf{\vec{B}}= \mu_0\varepsilon_0\dfrac{\partial }{\partial t}\mathbf{\vec{E}} \)</td><td>Ampere-Maxwell's Law</td></tr></table>
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Maxwell&#39;s Equations - Wave Equation for the Electric Field
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You will use the differential form of Maxwell's equations in free space and the vector identity below, to derive the wave equation for [mathjaxinline]\mathbf{\vec{E}}[/mathjaxinline]. </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }}\times (\mathbf{\vec{\nabla }}\times \mathbf{\vec{F}})=-\nabla ^2\mathbf{\vec{F}}+\mathbf{\vec{\nabla }}(\mathbf{\vec{\nabla }}\cdot \mathbf{\vec{F}})[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>1</span>)</span>
</td>
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<p>
where [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline] is any vector with well defined derivatives. </p>
<p><b class="bfseries">(Part a)</b> Starting from Faraday's law in differential form </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }}\times \mathbf{\vec{E}} = -\dfrac {\partial \mathbf{\vec{B}}}{\partial 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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<p>
Take the curl of both sides of the equation to obtain: </p>
<table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }} \times (\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}}) =-\mathbf{\vec{\nabla }}\times \dfrac {\partial }{\partial t} \mathbf{\vec{B}}= -\dfrac {\partial }{\partial t} \mathbf{\vec{\nabla }}\times \mathbf{\vec{B}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
where we have used the fact that the order of the spatial and time derivatives can be interchanged, [mathjaxinline]\dfrac {\partial }{\partial x}\dfrac {\partial }{\partial t}=\dfrac {\partial }{\partial t}\dfrac {\partial }{\partial x}[/mathjaxinline], etc. </p>
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Applying eq. (1) and Gauss's law in the expression above will give which of the following equations? </p>
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<text> a) [mathjaxinline]\mathbf{\vec{\nabla }}(\mathbf{\vec{\nabla }}\cdot \mathbf{\vec{E}}) =-\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{B}})[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =-\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{B}})[/mathjaxinline]</text>
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<input type="radio" name="input_lect_13_01a_2_1" id="input_lect_13_01a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_13_01a_2_1-choice_3-label" for="input_lect_13_01a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_13_01a_2_1">
<text> c) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{B}})[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\dfrac {\partial }{\partial t}\nabla ^2\mathbf{\vec{B}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Use Ampere-Maxwell's law in free space, [mathjaxinline]\mathbf{\vec{\nabla }}\times \mathbf{\vec{B}}= \mu _0\epsilon _0\dfrac {\partial }{\partial t}\mathbf{\vec{E}}[/mathjaxinline], in the right hand side of the equation you obtained in part (a), to find out which of the following is true? </p>
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<text> a) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\mu _0\epsilon _0\dfrac {\partial \mathbf{\vec{E}}}{\partial t}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lect_13_01a_3_1" id="input_lect_13_01a_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_13_01a_3_1-choice_2-label" for="input_lect_13_01a_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_13_01a_3_1">
<text> b) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\mu _0\epsilon _0\dfrac {\partial ^2 \mathbf{\vec{E}}}{\partial t^2}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_13_01a_3_1" id="input_lect_13_01a_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_13_01a_3_1-choice_3-label" for="input_lect_13_01a_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_13_01a_3_1">
<text> c) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =-\mu _0\epsilon _0\dfrac {\partial ^2 \mathbf{\vec{E}}}{\partial t^2}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_13_01a_3_1" id="input_lect_13_01a_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_13_01a_3_1-choice_4-label" for="input_lect_13_01a_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_13_01a_3_1">
<text> d) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\dfrac {\partial }{\partial t}\nabla ^2\mathbf{\vec{B}}[/mathjaxinline]</text>
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Maxwell&#39;s Equations - Wave Equation for the Magnetic Field
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You will use the differential form of Maxwell's equations in free space and the vector identity below to derive the wave equation for [mathjaxinline]\mathbf{\vec{B}}[/mathjaxinline]. </p>
<table id="a0000000018" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }}\times (\mathbf{\vec{\nabla }}\times \mathbf{\vec{F}})=-\nabla ^2\mathbf{\vec{F}}+\mathbf{\vec{\nabla }}(\mathbf{\vec{\nabla }}\cdot \mathbf{\vec{F}})[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p>
where [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline] is any vector with well defined derivatives. </p>
<p><b class="bfseries">(Part a)</b> Consider the differential form of Ampere - Maxwell's law </p>
<table id="a0000000020" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000021">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }}\times \mathbf{\vec{B}}= \mu _0\epsilon _0\dfrac {\partial }{\partial t}\mathbf{\vec{E}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
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Take the curl in both sides of the equation </p>
<table id="a0000000022" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000023">
<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle \mathbf{\vec{\nabla }} \times (\mathbf{\vec{\nabla }}\times \mathbf{\vec{B}}) =\mu _0\epsilon _0\mathbf{\vec{\nabla }}\times (\dfrac {\partial }{\partial t} \mathbf{\vec{E}})= \mu _0\epsilon _0\dfrac {\partial }{\partial t} (\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}})[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
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Applying eq. (1) and the magnetic Gauss' law in the expression above will give which of the following equations? </p>
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<text> a) [mathjaxinline]\mathbf{\vec{\nabla }}(\mathbf{\vec{\nabla }}\cdot \mathbf{\vec{B}}) =\mu _0\epsilon _0\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}})[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\nabla ^2\mathbf{\vec{B}} =-\mu _0\epsilon _0\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}})[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\nabla ^2\mathbf{\vec{B}} =+\mu _0\epsilon _0\dfrac {\partial }{\partial t}(\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}})[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\dfrac {1}{\mu _0\epsilon _0}\dfrac {\partial }{\partial t}\nabla ^2\mathbf{\vec{B}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> Use the differential form of Faraday's law, [mathjaxinline]\mathbf{\vec{\nabla }}\times \mathbf{\vec{E}}=-\dfrac {\partial }{\partial t}\mathbf{\vec{B}}[/mathjaxinline] on the right hand side of the equation you obtained in part (a), and select the correct expression from the list below: </p>
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<text> a) [mathjaxinline]\nabla ^2\mathbf{\vec{E}} =\mu _0\epsilon _0\dfrac {\partial \mathbf{\mathbf{\vec{\nabla }}\cdot \vec{E}}}{\partial t}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\nabla ^2\mathbf{\vec{B}} =-\mu _0\epsilon _0\dfrac {\partial ^2 \mathbf{\vec{E}}}{\partial t^2}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\nabla ^2\mathbf{\vec{B}} =\mu _0\epsilon _0\dfrac {\partial ^2 \mathbf{\vec{B}}}{\partial t^2}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\nabla ^2\mathbf{\vec{B}} =\mu _0\epsilon _0\dfrac {\partial }{\partial t}\mathbf{\vec{B}}[/mathjaxinline]</text>
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Understanding the Wave Equation
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<p>
The Laplacian, [mathjaxinline]\nabla ^2 \vec{\textbf{E}}[/mathjaxinline], is the part of this wave equation that looks different from what we have seen before. </p>
<p><b class="bfseries">(Part a)</b> How many terms will be in the general form of [mathjaxinline]\nabla ^2 \vec{\textbf{E}}[/mathjaxinline]? </p>
<p>
<p style="display:inline">Number of terms [mathjaxinline]=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Let's say the electric field has the form: </p>
<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{\textbf{E}} = (e^{-y/2} + z)\hat{i} + b\hat{j} + \cos (ax)\hat{k}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
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<p>
where [mathjaxinline]b[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] are constants. </p>
<p>
In this case, find the form of [mathjaxinline]\nabla ^2 \vec{\textbf{E}}[/mathjaxinline]. Express your answer in terms of <code>b</code>, <code>a</code>, as well as <code>hati</code>, <code>hatj</code>, and <code>hatk</code>, for [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]\hat{k}[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\nabla ^2 \vec{\textbf{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">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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</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>
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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]
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
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<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
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<tr class="formulainput">
<td class="formulainput">row vector</td>
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<h2 class="hd hd-2 unit-title">L20v4: Plane Wave Electric Field Solution and Direction of Propagation</h2>
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<h2 class="hd hd-2 unit-title">L20Q3: Plane Wave Propagation I</h2>
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Plane Wave Propagation I
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An electromagnetic plane wave is propagating in vacuum and has an electric field given by: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\vec{E}(x,t)=E_{0} \cos {(kx - \omega t)}\hat{k}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part a)</b> In which direction is this wave propagating? </p>
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<p><b class="bfseries">(Part b)</b> Now answer the following question related to the figures shown below. At time [mathjaxinline]t=0[/mathjaxinline] and position [mathjaxinline]x=0[/mathjaxinline], which figure best represents the magnitude and direction of the electric field in the yz plane? </p>
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<h2 class="hd hd-2 unit-title">L20Q4: Plane Wave Propagation II [WITH SIMULATION]</h2>
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Plane Wave Propagation II
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Consider the plane wave pictured below, which is represented as a function of position [mathjaxinline]x[/mathjaxinline], with [mathjaxinline]x=0[/mathjaxinline] located at the back-left of the figure. The wave travels in the [mathjaxinline]+\hat{x}[/mathjaxinline] direction. </p>
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Choose the correct expression for the magnetic field for the wave pictured above. </p>
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<text> a) [mathjaxinline]\vec{B}(x,t)=B_{0}\cos {(kx - \omega t)}\hat{j}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\vec{B}(x,t)=B_{0}\cos {(kx + \omega t)}\hat{j}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\vec{B}(x,t)=B_{0}\cos {(kx - \omega t)}\hat{k}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\vec{B}(x,t)=B_{0}\cos {(kx + \omega t)}\hat{k}[/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=EM_waves_traveling_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/v1/0f4a3d80749122d873ffc88c9124467a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_EM_waves_traveling_1.png"/></div><p/>
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<h2 class="hd hd-2 unit-title">L20Q5: Plane Wave Propagation III</h2>
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Plane Wave Propagation III
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An electromagnetic plane wave is propagating in vacuum and has an electric field given by: </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}(y,t)=E_{0} \sin {(ky - \omega t)}\hat{i}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
At the position [mathjaxinline]y=0[/mathjaxinline], what pair of vectors represent the magnitude and direction of the electromagnetic field components at time [mathjaxinline]t=T/4[/mathjaxinline], where [mathjaxinline]T[/mathjaxinline] is the period of oscillation? </p>
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<h2 class="hd hd-2 unit-title">L20Q6: Plane Wave Propagation IV</h2>
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Plane Wave Propagation IV
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The electric field of an electromagnetic plane wave is described as follows: </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}(y,t)=E_{0}\sin {(ky + \omega t)}\hat{i}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]E_{0} = 30\, \mathrm{Vm^{-1}}[/mathjaxinline] and [mathjaxinline]k = 9\, \mathrm{m^{-1}}[/mathjaxinline]. Note, also, that this wave travels at the speed of light ([mathjaxinline]c=3\times 10^{8}\mathrm{m/s}[/mathjaxinline]). Answer the following: </p>
<p><b class="bfseries">(Part a)</b> In what direction is the wave traveling? Enter <code>x</code>, <code>y</code>, or <code>z</code> and use (-) for negative directions (e.g., -z). </p>
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<p><b class="bfseries">(Part b)</b> What is the wavelength, [mathjaxinline]\lambda[/mathjaxinline] of the wave? Enter a numerical answer. </p>
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<p style="display:inline">[mathjaxinline]\lambda =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{m}[/mathjaxinline]</p>
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<p><b class="bfseries">(Part c)</b> What is the angular frequency of the wave in units of 1/s? </p>
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<p style="display:inline">[mathjaxinline]\omega =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{1/s}[/mathjaxinline]</p>
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<p><b class="bfseries">(Part d)</b> What is the magnitude of the accompanying magnetic field? </p>
<p>
<p style="display:inline">[mathjaxinline]|B|=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\mathrm{Vs/m^2}[/mathjaxinline]</p>
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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>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L20Q7: Plane Wave Propagation V</h2>
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Plane Wave Propagation V
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For the given magnetic field, choose the correct expression for the electric field that satisfies the conditions for an electromagnetic plane wave in vacuum: </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{B}(z,t)=B_{0}\sin {(kz + \omega t)}\hat{i}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<text> a) [mathjaxinline]\vec{E}(z,t)=E_{0}\sin {(kz + \omega t)}\hat{j}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\vec{E}(z,t)=-E_{0}\sin {(kz + \omega t)}\hat{j}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\vec{E}(z,t)=E_{0}\sin {(kz + \omega t)}\hat{i}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\vec{E}(z,t)=-E_{0}\sin {(kz + \omega t)}\hat{i}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L20v6: Summary of Electromagnetic Waves</h2>
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<h3 class="hd hd-2">L20v6: Summary of Electromagnetic Waves</h3>
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