<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson26intro" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">Introduction to N-slit Interference and Other Phenomena</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson26_intro">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson26_intro" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
Last week, we studied the properties of two-slit interference. Motivated by trying to understand how phased-array radar works, we now look at three-slit interference. Ultimately, we extrapolate to N-slit interference. </p><p>
Additionally, we consider the interference behavior of single electrons. Do single electrons behave like particles or waves? </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L38v1" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38v1: Three Slit Interference</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v1" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L38v1: Three Slit Interference</h3>
<div
id="video_L38v1"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v1/handler/publish_completion", "streams": "1.00:hrn_cS3s0A4", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v1/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v1/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v1/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L38v1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L38v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L38v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/d5eb3602db987e669bded3cdee99f814/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L38v1.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex01" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q1: Adding More Slits</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_01a" class="problems-wrapper" role="group"
aria-labelledby="lect_25_01a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_01a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a-problem-progress" tabindex="-1">
Adding More Slits - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01a-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is projected on a screen a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. The angle [mathjaxinline]\theta[/mathjaxinline] characterizes the position of interference features on the distant screen, relative to the line that bisects the middle of the grating. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/57f1ed63bd08da743904aad37374a3ee/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_01a.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part a)</b> Which of the following conditions enables the small angle approximation to be made? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_25_01a_2_1">
<fieldset aria-describedby="status_lect_25_01a_2_1">
<div class="field">
<input type="radio" name="input_lect_25_01a_2_1" id="input_lect_25_01a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_25_01a_2_1-choice_1-label" for="input_lect_25_01a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_01a_2_1">
<text> a) [mathjaxinline]\dfrac {\lambda }{d} \gg 1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_01a_2_1" id="input_lect_25_01a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_25_01a_2_1-choice_2-label" for="input_lect_25_01a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_01a_2_1">
<text> b) [mathjaxinline]\dfrac {\lambda }{d} \ll 1[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_01a_2_1" id="input_lect_25_01a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_25_01a_2_1-choice_3-label" for="input_lect_25_01a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_25_01a_2_1">
<text> c) [mathjaxinline]\dfrac {\lambda }{d} \approx 1[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_25_01a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_01a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_01a_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="Adding More Slits - 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_25_01a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_01a">
<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_25_01a-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_25_01a-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_25_01a-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_25_01b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01b" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_01b" class="problems-wrapper" role="group"
aria-labelledby="lect_25_01b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_01b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01b-problem-progress" tabindex="-1">
Adding More Slits - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_01b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Determine the width of the central maximum, using the small angle approximation, where the variable [mathjaxinline]y[/mathjaxinline] characterizes the position on the observing screen. Hint, this will be the difference between the location of the first minimum on either side of the central maximum. Express your answer using <code>N</code>, <code>L</code>, <code>d</code>, and <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta y_{central} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_25_01b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_01b_2_1" id="input_lect_25_01b_2_1" aria-describedby="status_lect_25_01b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_25_01b_2_1"/>
<span class="status unanswered" id="status_lect_25_01b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_01b_2_1" class="answer"/>
<div id="display_lect_25_01b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_25_01b_2_1_dynamath" name="input_lect_25_01b_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_25_01b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</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"/>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Adding More Slits - 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_25_01b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_01b">
<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_25_01b-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_25_01b-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_25_01b-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-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L38v2" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38v2: N-slit Interference</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v2" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L38v2: N-slit Interference</h3>
<div
id="video_L38v2"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v2/handler/publish_completion", "streams": "1.00:NvLtrF32E7c", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v2/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v2/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v2/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L38v2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L38v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L38v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/15ef345420f2c4c70c53648839b1e7b6/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L38v2.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex02" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q2: N-slit Derivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_02" class="problems-wrapper" role="group"
aria-labelledby="lect_25_02-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_02-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02-problem-progress" tabindex="-1">
N-slit Interference
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_02-problem-progress"></div>
<div class="problem">
<div>
<p>
We asserted that the phase difference between two light rays from nearest neighbor slits hitting the same spot on a screen, [mathjaxinline]\delta _{\mathrm{nn}}[/mathjaxinline], was constant. Consequently, the phase difference between two light rays from the [mathjaxinline]N^{\mathrm{th}}[/mathjaxinline] slit and the first slit is: </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]\delta _{1N}=(N-1)\delta _{\mathrm{nn}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What condition(s) are necessary for this to hold? 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_25_02_2_1">
<fieldset aria-describedby="status_lect_25_02_2_1">
<div class="field">
<input type="checkbox" name="input_lect_25_02_2_1[]" id="input_lect_25_02_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_25_02_2_1-choice_0-label" for="input_lect_25_02_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_25_02_2_1">
<text>a) There are no special conditions that are necessary.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_02_2_1[]" id="input_lect_25_02_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_25_02_2_1-choice_1-label" for="input_lect_25_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_02_2_1">
<text>b) The separation between the slits is small compared to the distance to the screen.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_02_2_1[]" id="input_lect_25_02_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_25_02_2_1-choice_2-label" for="input_lect_25_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_02_2_1">
<text>c) The total spatial extent of the grating (all of the [mathjaxinline]N[/mathjaxinline] slits) is small compared to the distance to the screen, such that all paths are approximately parallel.</text>
</label>
</div>
<span id="answer_lect_25_02_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_02_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_02_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="N-slit Interference" />
<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_25_02" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_02">
<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_25_02-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_25_02-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_25_02-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-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L38v3" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38v3: N-slit Interference Intensity</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v3" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L38v3: N-slit Interference Intensity</h3>
<div
id="video_L38v3"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v3/handler/publish_completion", "streams": "1.00:09FQ9dmuBp0", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v3/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v3/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v3/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L38v3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L38v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L38v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/df3db356fd7f41513575d6ec99262081/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L38v3.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex03_0" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q3: Varying Number of Slits</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_03_0a" class="problems-wrapper" role="group"
aria-labelledby="lect_25_03_0a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_03_0a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a-problem-progress" tabindex="-1">
Varying N - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0a-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is projected on a screen a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. An example of an interference pattern is shown below. Assume the small angle approximation so that [mathjaxinline]L\gg y[/mathjaxinline]. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/d93ff103bdf988d3d76bae3ffed863fe/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_03a.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part a)</b> The interference pattern exhibits "large" maxima, and "small" maxima. When the number of slits increases, but the slit separation remains the same, what happens to the locations of the "large" maxima. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_25_03_0a_2_1">
<fieldset aria-describedby="status_lect_25_03_0a_2_1">
<div class="field">
<input type="radio" name="input_lect_25_03_0a_2_1" id="input_lect_25_03_0a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_25_03_0a_2_1-choice_1-label" for="input_lect_25_03_0a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0a_2_1">
<text> a) the locations get farther apart</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0a_2_1" id="input_lect_25_03_0a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_25_03_0a_2_1-choice_2-label" for="input_lect_25_03_0a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0a_2_1">
<text> b) the locations get closer together</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0a_2_1" id="input_lect_25_03_0a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_25_03_0a_2_1-choice_3-label" for="input_lect_25_03_0a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0a_2_1">
<text> c) the locations do not change</text>
</label>
</div>
<span id="answer_lect_25_03_0a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_03_0a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_03_0a_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="Varying N - 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_25_03_0a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_03_0a">
<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_25_03_0a-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_25_03_0a-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_25_03_0a-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_25_03_0b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0b" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_03_0b" class="problems-wrapper" role="group"
aria-labelledby="lect_25_03_0b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_03_0b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0b-problem-progress" tabindex="-1">
Varying N - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> The intensity as a function of the phase difference between adjacent slits, is: </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]\langle I \rangle = I_{0} \dfrac {\sin ^{2}\left( \frac{N\delta }{2}\right)}{\sin ^{2}\left( \frac{\delta }{2}\right)}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
What is the intensity distribution for two-slit interference, [mathjaxinline]N=2[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_25_03_0b_2_1">
<fieldset aria-describedby="status_lect_25_03_0b_2_1">
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_25_03_0b_2_1-choice_1-label" for="input_lect_25_03_0b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> a) [mathjaxinline]\langle I \rangle = 2 I_{0} \sin ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_25_03_0b_2_1-choice_2-label" for="input_lect_25_03_0b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> b) [mathjaxinline]\langle I \rangle = 2 I_{0} \sin ^{2}\left(\delta \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_25_03_0b_2_1-choice_3-label" for="input_lect_25_03_0b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> c) [mathjaxinline]\langle I \rangle = 4 I_{0} \sin ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_25_03_0b_2_1-choice_4-label" for="input_lect_25_03_0b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> d) [mathjaxinline]\langle I \rangle = 4 I_{0} \sin ^{2}\left(\delta \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_25_03_0b_2_1-choice_5-label" for="input_lect_25_03_0b_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> e) [mathjaxinline]\langle I \rangle = 2 I_{0} \cos ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_25_03_0b_2_1-choice_6-label" for="input_lect_25_03_0b_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> f) [mathjaxinline]\langle I \rangle = 2 I_{0} \cos ^{2}\left(\delta \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="lect_25_03_0b_2_1-choice_7-label" for="input_lect_25_03_0b_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> g) [mathjaxinline]\langle I \rangle = 4 I_{0} \cos ^{2}\left( \frac{\delta }{2} \right)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_25_03_0b_2_1" id="input_lect_25_03_0b_2_1_choice_8" class="field-input input-radio" value="choice_8"/><label id="lect_25_03_0b_2_1-choice_8-label" for="input_lect_25_03_0b_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_lect_25_03_0b_2_1">
<text> h) [mathjaxinline]\langle I \rangle = 4 I_{0} \cos ^{2}\left(\delta \right)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_25_03_0b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_03_0b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_03_0b_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="Varying N - 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_25_03_0b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_03_0b">
<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_25_03_0b-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_25_03_0b-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_25_03_0b-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_25_03_0c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0c" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_03_0c" class="problems-wrapper" role="group"
aria-labelledby="lect_25_03_0c-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0c" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0c/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_03_0c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0c-problem-progress" tabindex="-1">
Varying N - part c
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03_0c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> What phase shift will generate the [mathjaxinline]n^\mathrm {th}[/mathjaxinline] "large" maxima in an interference pattern generated by a grating with [mathjaxinline]N[/mathjaxinline] slits, where [mathjaxinline]n=0[/mathjaxinline] corresponds to the central maximum? Note: It may be useful to consider your answers from part (a) and (b). </p>
<p>
Express your answer in terms of <code>n</code> and/or <code>N</code> as needed (and any relevant numerical constants). </p>
<p>
<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_25_03_0c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_03_0c_2_1" id="input_lect_25_03_0c_2_1" aria-describedby="status_lect_25_03_0c_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_25_03_0c_2_1"/>
<span class="status unanswered" id="status_lect_25_03_0c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_03_0c_2_1" class="answer"/>
<div id="display_lect_25_03_0c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_25_03_0c_2_1_dynamath" name="input_lect_25_03_0c_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_25_03_0c_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="Varying N - 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_25_03_0c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_03_0c">
<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_25_03_0c-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_25_03_0c-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_25_03_0c-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-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex03" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q4: N slits [WITH SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_03a" class="problems-wrapper" role="group"
aria-labelledby="lect_25_03a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_03a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a-problem-progress" tabindex="-1">
N Slits - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03a-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> For each of the following plots, determine the number of slits, [mathjaxinline]N[/mathjaxinline], that generated the interference pattern shown. Note, the intensities are normalized to a max value of 1. </p>
<p><b class="bfseries">(Part i)</b> Figure 1. </p>
<p>
<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_25_03a_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_03a_2_1" id="input_lect_25_03a_2_1" aria-describedby="status_lect_25_03a_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_25_03a_2_1"/>
<span class="status unanswered" id="status_lect_25_03a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_03a_2_1" class="answer"/>
</div>
</div></div>
</p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/d93ff103bdf988d3d76bae3ffed863fe/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_03a.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="solution-span">
<span id="solution_lect_25_03a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part ii)</b> Figure 2. </p>
<p>
<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_25_03a_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_03a_3_1" id="input_lect_25_03a_3_1" aria-describedby="status_lect_25_03a_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_25_03a_3_1"/>
<span class="status unanswered" id="status_lect_25_03a_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_03a_3_1" class="answer"/>
</div>
</div></div>
</p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/ea97f679bf22f273026edb3e629b5192/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_03b.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="solution-span">
<span id="solution_lect_25_03a_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part ii)</b> Figure 3. </p>
<p>
<p style="display:inline">[mathjaxinline]N =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_lect_25_03a_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_03a_4_1" id="input_lect_25_03a_4_1" aria-describedby="status_lect_25_03a_4_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_25_03a_4_1"/>
<span class="status unanswered" id="status_lect_25_03a_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_03a_4_1" class="answer"/>
</div>
</div></div>
</p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/d90059b3c03c100f0c485ff248e53e6e/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_03c.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
<div class="solution-span">
<span id="solution_lect_25_03a_solution_3"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="N Slits - 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_25_03a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_03a">
<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_25_03a-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_25_03a-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_25_03a-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_25_03b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03b" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_03b" class="problems-wrapper" role="group"
aria-labelledby="lect_25_03b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_03b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03b-problem-progress" tabindex="-1">
N Slits - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_03b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> If the separation between adjacent slits is decreased, how will this affect the interference pattern? 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_25_03b_2_1">
<fieldset aria-describedby="status_lect_25_03b_2_1">
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_25_03b_2_1-choice_0-label" for="input_lect_25_03b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>a) The distance between principle maxima will increase</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_25_03b_2_1-choice_1-label" for="input_lect_25_03b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>b) The distance between principle maxima will decrease</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_25_03b_2_1-choice_2-label" for="input_lect_25_03b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>c) The distance between principle maxima will stay the same</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_25_03b_2_1-choice_3-label" for="input_lect_25_03b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>d) The width of the principle maxima will increase</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="lect_25_03b_2_1-choice_4-label" for="input_lect_25_03b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>e) The width of the principle maxima will decrease</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="lect_25_03b_2_1-choice_5-label" for="input_lect_25_03b_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>f) The width of the principle maxima will stay the same</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_6" class="field-input input-checkbox" value="choice_6"/><label id="lect_25_03b_2_1-choice_6-label" for="input_lect_25_03b_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>g) The number of secondary maxima will increase</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_7" class="field-input input-checkbox" value="choice_7"/><label id="lect_25_03b_2_1-choice_7-label" for="input_lect_25_03b_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>h) The number of secondary maxima will decrease</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_03b_2_1[]" id="input_lect_25_03b_2_1_choice_8" class="field-input input-checkbox" value="choice_8"/><label id="lect_25_03b_2_1-choice_8-label" for="input_lect_25_03b_2_1_choice_8" class="response-label field-label label-inline" aria-describedby="status_lect_25_03b_2_1">
<text>i) The number of secondary maxima will stay the same</text>
</label>
</div>
<span id="answer_lect_25_03b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_03b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_03b_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="N Slits - 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_25_03b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_03b">
<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_25_03b-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_25_03b-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_25_03b-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@html+block@SIM_lect_25_03">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_25_03" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="html" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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=Nslit_interference_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/f35715c0ca1873b0bda371c343be3785/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_Nslit_interference_1.png"/></div><p/>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L38v4" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38v4: Phased Radar</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v4">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v4" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L38v4: Phased Radar</h3>
<div
id="video_L38v4"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v4/handler/publish_completion", "streams": "1.00:cPKkOUQgRI0", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v4/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v4/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v4/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L38v4"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L38v4">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L38v4">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/00bc034b91a927e645676267a74b3d4b/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L38v4.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex04" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q5: Phased Radar</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_04" class="problems-wrapper" role="group"
aria-labelledby="lect_25_04-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_04-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04-problem-progress" tabindex="-1">
Phased Radar
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_04-problem-progress"></div>
<div class="problem">
<div>
<p>
Again, consider a plane wave with wavelength [mathjaxinline]\lambda[/mathjaxinline] that is transmitted through a grating with [mathjaxinline]N[/mathjaxinline] slits, where the separation between neighboring slits is [mathjaxinline]d[/mathjaxinline]. The interference pattern is observed a distance [mathjaxinline]L[/mathjaxinline] very far away from the grating (i.e., [mathjaxinline]L\gg Nd)[/mathjaxinline]. Assume the small angle approximation so that [mathjaxinline]L\gg y[/mathjaxinline]. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/e1e9f23e075ce26316a8f06daf09f1cb/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_04a.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part a)</b> What is the condition on the phase shift between adjacent slits [mathjaxinline]\delta[/mathjaxinline] for constructive interference that produces the first principle maximum which is not at [mathjaxinline]y=0[/mathjaxinline]? Enter your answer as a positive number. </p>
<p>
<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_25_04_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_04_2_1" id="input_lect_25_04_2_1" aria-describedby="status_lect_25_04_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_25_04_2_1"/>
<span class="status unanswered" id="status_lect_25_04_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_04_2_1" class="answer"/>
<div id="display_lect_25_04_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_25_04_2_1_dynamath" name="input_lect_25_04_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_25_04_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> Now we wish to shift the position of that first principle maximum by a distance [mathjaxinline]\Delta[/mathjaxinline] from its original location at [mathjaxinline]y_{1}[/mathjaxinline] by adding a phase shift [mathjaxinline]\phi[/mathjaxinline] between adjacent slits. One way to add such a phase shift is by having the incoming wave at an angle to the line of slits, as shown below, since in this case the wavefront will reach adjacent slits at different times. However, the details of how the phase shift is generated is not important, all you need to know is that there is a slit-to-slit phase shift of [mathjaxinline]\phi[/mathjaxinline]. </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/8794fb5956d2efcb18748372463d7e44/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_25_04b.png" width="605"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p>
Determine the value of [mathjaxinline]|\phi |[/mathjaxinline] that will produce this change in the location. Express your answer using <code>L</code>, <code>d</code>, <code>lambda</code> for [mathjaxinline]\lambda[/mathjaxinline], and <code>Delta</code> for [mathjaxinline]\Delta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\phi | =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_25_04_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_25_04_3_1" id="input_lect_25_04_3_1" aria-describedby="status_lect_25_04_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_25_04_3_1"/>
<span class="status unanswered" id="status_lect_25_04_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_25_04_3_1" class="answer"/>
<div id="display_lect_25_04_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_25_04_3_1_dynamath" name="input_lect_25_04_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_lect_25_04_solution_2"/>
</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="Phased Radar" />
<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_25_04" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_04">
<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_25_04-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_25_04-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_25_04-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-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L38v5" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38v5: Single Electron Interference</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v5">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L38v5" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="video" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L38v5: Single Electron Interference</h3>
<div
id="video_L38v5"
class="video closed"
data-metadata='{"speed": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v5/handler/publish_completion", "streams": "1.00:lNM4LQmq6C4", "prioritizeHls": false, "completionEnabled": false, "autoAdvance": false, "captionDataDir": null, "transcriptLanguages": {"en": "English"}, "recordedYoutubeIsAvailable": true, "showCaptions": "true", "completionPercentage": 0.95, "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v5/handler/xmodule_handler/save_user_state", "ytMetadataEndpoint": "", "duration": 0.0, "autohideHtml5": false, "transcriptLanguage": "en", "ytApiUrl": "https://www.youtube.com/iframe_api", "lmsRootURL": "https://openlearninglibrary.mit.edu", "sources": [], "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v5/handler/transcript/available_translations", "savedVideoPosition": 0.0, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L38v5/handler/transcript/translation/__lang__", "saveStateEnabled": false, "ytTestTimeout": 1500, "poster": null, "autoplay": false, "generalSpeed": 1.0, "start": 0.0, "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L38v5"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L38v5">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L38v5">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/d0c10ab62af1736639914e36cafc0e53/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L38v5.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_25_ex05" data-init="VerticalStudentView" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="vertical" data-has-score="False" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<h2 class="hd hd-2 unit-title">L38Q6: Single Electron Wavefunction</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05" data-init="XBlockToXModuleShim" data-runtime-version="1" data-course-id="course-v1:MITx+8.03x+1T2020" data-block-type="problem" data-has-score="True" data-graded="True" data-request-token="b2e66336fd4d11ee9b2d16fff75c5923">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_25_05" class="problems-wrapper" role="group"
aria-labelledby="lect_25_05-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_25_05-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05-problem-progress" tabindex="-1">
Single electron interference
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_25_05-problem-progress"></div>
<div class="problem">
<div>
<p>
Let's consider the single-electron interference experiment described previously. A source emits electrons towards an aperture with two slits. Which of the following observations are 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_25_05_2_1">
<fieldset aria-describedby="status_lect_25_05_2_1">
<div class="field">
<input type="checkbox" name="input_lect_25_05_2_1[]" id="input_lect_25_05_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_25_05_2_1-choice_0-label" for="input_lect_25_05_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_25_05_2_1">
<text>a) When more than one electron is emitted at a time, the interference pattern is destroyed and Gaussian-like distributions will be observed</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_05_2_1[]" id="input_lect_25_05_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_25_05_2_1-choice_1-label" for="input_lect_25_05_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_25_05_2_1">
<text>b) When one of the slits is covered, the two-slit interference pattern is destroyed and a single Gaussian-like distribution will be observed</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_25_05_2_1[]" id="input_lect_25_05_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_25_05_2_1-choice_2-label" for="input_lect_25_05_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_25_05_2_1">
<text>c) When one of the slits is covered, the two-slit interference pattern changes to a single-slit diffraction pattern</text>
</label>
</div>
<span id="answer_lect_25_05_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_25_05_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_25_05_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="Single electron interference" />
<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_25_05" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_25_05">
<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_25_05-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_25_05-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_25_05-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>