<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_05_1">
<h2 class="hd hd-2 unit-title">W13-PSP1: Power Radiated from Oscillating Charge</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05_1a" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05_1a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_05_1a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a-problem-progress" tabindex="-1">
Power Radiated from Oscillating Charge - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1a-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> Consider a positive charge [mathjaxinline]q[/mathjaxinline] that is oscillating sinusoidally, whose position is [mathjaxinline]\vec{r}(t)=y_{0}\cos (\omega t)\hat{j}[/mathjaxinline]&#8212;the motion is nonrelativistic. Find an expression for the magnitude of the electric field [mathjaxinline]|\vec{E}_{\mathrm{rad}}(\vec{r},t)|[/mathjaxinline], at a position [mathjaxinline]\vec{r}[/mathjaxinline], which is at an angle [mathjaxinline]\theta[/mathjaxinline] relative to the [mathjaxinline]\hat{j}[/mathjaxinline] direction. </p>
<p>
Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]. Use <code>abs()</code> for absolute value if needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{E}_{\mathrm{rad}}(\vec{r},t)| =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05_1a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05_1a_2_1" id="input_lect_21_05_1a_2_1" aria-describedby="status_lect_21_05_1a_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05_1a_2_1"/>
<span class="status unanswered" id="status_lect_21_05_1a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05_1a_2_1" class="answer"/>
<div id="display_lect_21_05_1a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05_1a_2_1_dynamath" name="input_lect_21_05_1a_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05_1a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Power Radiated from Oscillating Charge - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05_1a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05_1a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05_1b" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05_1b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_05_1b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b-problem-progress" tabindex="-1">
Power radiated from oscillating charge - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> What is the magnitude of the Poynting vector [mathjaxinline]|\vec{S}(\vec{r},t)|[/mathjaxinline], at a position [mathjaxinline]\vec{r}[/mathjaxinline] and time [mathjaxinline]t[/mathjaxinline]? </p>
<p>
Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{S}_{\mathrm{rad}}(\vec{r},t)| =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05_1b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05_1b_2_1" id="input_lect_21_05_1b_2_1" aria-describedby="status_lect_21_05_1b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05_1b_2_1"/>
<span class="status unanswered" id="status_lect_21_05_1b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05_1b_2_1" class="answer"/>
<div id="display_lect_21_05_1b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05_1b_2_1_dynamath" name="input_lect_21_05_1b_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05_1b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Power radiated from oscillating charge - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05_1b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05_1b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05_1c" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05_1c-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_05_1c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c-problem-progress" tabindex="-1">
Power radiated from oscillating charge - part c
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05_1c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Calculate the time-averaged power radiated by this system, which is equal to the time-averaged Poynting flux through a closed surface. Note, the Poynting vector is in the [mathjaxinline]\hat{r}[/mathjaxinline] direction and can be expressed as [mathjaxinline]\vec{S}(\vec{r},t) = |\vec{S}(\vec{r},t)|\hat{r}[/mathjaxinline]. </p>
<p>
Express your answer in terms of <code>k</code> (where [mathjaxinline]k=1/(4\pi \epsilon _0)[/mathjaxinline]), <code>q</code>, <code>c</code>, <code>y_0</code> for [mathjaxinline]y_{0}[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05_1c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05_1c_2_1" id="input_lect_21_05_1c_2_1" aria-describedby="status_lect_21_05_1c_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05_1c_2_1"/>
<span class="status unanswered" id="status_lect_21_05_1c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05_1c_2_1" class="answer"/>
<div id="display_lect_21_05_1c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05_1c_2_1_dynamath" name="input_lect_21_05_1c_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05_1c_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Power radiated from oscillating charge - part c" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05_1c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05_1c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1c-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1c-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05_1c-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-w13-psp-2">
<h2 class="hd hd-2 unit-title">W13-PSP2: Transmission/Reflection of S-Polarized Light</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_w13_psp_2a" class="problems-wrapper" role="group"
aria-labelledby="w13_psp_2a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="w13_psp_2a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a-problem-progress" tabindex="-1">
Transmission/Reflection of S-Polarized Light - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2a-problem-progress"></div>
<div class="problem">
<div>
<p>
In lecture, we derived the reflection coefficients for p-polarized light. In this problem, you will do the same for s-polarized light. Consider a plane electromagnetic wave that is polarized in the direction perpendicular to the plane of incidence (the electric field only has a component parallel to the interface, shown below for incident light only). </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/88b3346872b3eefd9f9ea92b2cb6b306/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_22_05d.png" width="495"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part a)</b> Use Maxwell's equations to determine which of the following are the correct relations between the parallel and perpendicular components of the magnetic field? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_w13_psp_2a_2_1">
<fieldset aria-describedby="status_w13_psp_2a_2_1">
<div class="field">
<input type="checkbox" name="input_w13_psp_2a_2_1[]" id="input_w13_psp_2a_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="w13_psp_2a_2_1-choice_0-label" for="input_w13_psp_2a_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2a_2_1">
<text>a) [mathjaxinline]\dfrac {1}{\mu _1}B_{\perp }^{(1)} = \dfrac {1}{\mu _2}B_{\perp }^{(2)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_w13_psp_2a_2_1[]" id="input_w13_psp_2a_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="w13_psp_2a_2_1-choice_1-label" for="input_w13_psp_2a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2a_2_1">
<text>b) [mathjaxinline]B_{||}^{(1)} = B_{||}^{(2)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_w13_psp_2a_2_1[]" id="input_w13_psp_2a_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="w13_psp_2a_2_1-choice_2-label" for="input_w13_psp_2a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2a_2_1">
<text>c) [mathjaxinline]B_{\perp }^{(1)} = B_{\perp }^{(2)}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_w13_psp_2a_2_1[]" id="input_w13_psp_2a_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="w13_psp_2a_2_1-choice_3-label" for="input_w13_psp_2a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2a_2_1">
<text>d) [mathjaxinline]\dfrac {1}{\mu _1}B_{||}^{(1)} = \dfrac {1}{\mu _2}B_{||}^{(2)}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_w13_psp_2a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_w13_psp_2a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_w13_psp_2a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Transmission/Reflection of S-Polarized Light - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_w13_psp_2a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_w13_psp_2a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_w13_psp_2b" class="problems-wrapper" role="group"
aria-labelledby="w13_psp_2b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="w13_psp_2b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b-problem-progress" tabindex="-1">
Transmission/Reflection of S-Polarized Light - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Use the definitions above to determine the fresnel reflection and transmission coefficients, [mathjaxinline]r=E_{0R}/E_{0I}[/mathjaxinline] and [mathjaxinline]t=E_{0T}/E_{0I}[/mathjaxinline]. Express your answers in terms of <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline] and <code>beta</code> for [mathjaxinline]\beta[/mathjaxinline], which have the familiar definitions below (assuming [mathjaxinline]\mu _{1}=\mu _{2}[/mathjaxinline], which is typically the case): </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\alpha =\dfrac {\cos {\theta _{T}}}{\cos {\theta _{I}}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\beta =\dfrac {n_{2}}{n_{1}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Hint: use the relation [mathjaxinline]\vec{B}=\left(\dfrac {1}{v}\right)\hat{k}\times \vec{E}[/mathjaxinline], where [mathjaxinline]v=c/n[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]r =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_w13_psp_2b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_w13_psp_2b_2_1" id="input_w13_psp_2b_2_1" aria-describedby="status_w13_psp_2b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_w13_psp_2b_2_1"/>
<span class="status unanswered" id="status_w13_psp_2b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_w13_psp_2b_2_1" class="answer"/>
<div id="display_w13_psp_2b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_w13_psp_2b_2_1_dynamath" name="input_w13_psp_2b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]t =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_w13_psp_2b_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_w13_psp_2b_3_1" id="input_w13_psp_2b_3_1" aria-describedby="status_w13_psp_2b_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_w13_psp_2b_3_1"/>
<span class="status unanswered" id="status_w13_psp_2b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_w13_psp_2b_3_1" class="answer"/>
<div id="display_w13_psp_2b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_w13_psp_2b_3_1_dynamath" name="input_w13_psp_2b_3_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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]
</td>
</tr>
<tr class="formulainput">
<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>
</tr>
<tr class="formulainput">
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_w13_psp_2b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Transmission/Reflection of S-Polarized Light - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_w13_psp_2b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_w13_psp_2b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="e6c332aa686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_w13_psp_2c" class="problems-wrapper" role="group"
aria-labelledby="w13_psp_2c-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="w13_psp_2c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c-problem-progress" tabindex="-1">
Transmission/Reflection of S-Polarized Light - part c
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@w13_psp_2c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> For mechanical waves, we saw that the reflection and transmission coefficients were related by the equation [mathjaxinline]1 + r = t[/mathjaxinline]. Does this hold for the fresnel reflection and transmission coefficients for s-polarized light? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_w13_psp_2c_2_1">
<fieldset aria-describedby="status_w13_psp_2c_2_1">
<div class="field">
<input type="radio" name="input_w13_psp_2c_2_1" id="input_w13_psp_2c_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="w13_psp_2c_2_1-choice_1-label" for="input_w13_psp_2c_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2c_2_1">
<text> a) YES</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_w13_psp_2c_2_1" id="input_w13_psp_2c_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="w13_psp_2c_2_1-choice_2-label" for="input_w13_psp_2c_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_w13_psp_2c_2_1">
<text> b) NO</text>
</label>
</div>
<span id="answer_w13_psp_2c_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_w13_psp_2c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_w13_psp_2c_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Transmission/Reflection of S-Polarized Light - part c" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_w13_psp_2c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_w13_psp_2c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2c-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2c-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="w13_psp_2c-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>