<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson22intro">
<h2 class="hd hd-2 unit-title">Introduction to Radiation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson22_intro_2">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson22_intro_2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
So far, we've been discussing electromagnetic waves without ever addressing where they come from! How is light created? </p><p>
In this lesson, we discuss how electric and magnetic fields may be produced from charges. We will discuss several cases:<br/>(1) stationary charges<br/>(2) charges moving at constant velocity<br/>(3) accelerating charges </p><p>
We will see how [mathjaxinline]\vec{E}[/mathjaxinline] and [mathjaxinline]\vec{B}[/mathjaxinline] fields are produced in each of the above cases. We also discuss the particular example of dipole radiation and its applications. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L32v1">
<h2 class="hd hd-2 unit-title">L32v1: Creation of EM Waves</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L32v1: Creation of EM waves</h3>
<div
id="video_L32v1"
class="video closed"
data-metadata='{"saveStateEnabled": false, "speed": null, "ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v1/handler/transcript/available_translations", "captionDataDir": null, "completionEnabled": false, "poster": null, "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v1/handler/transcript/translation/__lang__", "start": 0.0, "end": 0.0, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v1/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v1/handler/publish_completion", "showCaptions": "true", "autohideHtml5": false, "streams": "1.00:mGzTCUm064Y", "prioritizeHls": false, "duration": 0.0, "transcriptLanguage": "en", "autoplay": false, "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "sources": []}'
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="L32v1"></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_L32v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/efe40570d216de6d54aa0899b328e14e/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L32v1.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-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex01">
<h2 class="hd hd-2 unit-title">L32Q1: Charge Moving in Various Ways</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_01" class="problems-wrapper" role="group"
aria-labelledby="lect_21_01-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_01-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01-problem-progress" tabindex="-1">
Charge Moving in Various Ways
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_01-problem-progress"></div>
<div class="problem">
<div>
<p>
To see light far away from a source, the power that is emitted by the source should be nonzero very far away (note, "far away" is a relative term). Another way to say this is that the integrated Poynting flux through a surface that encloses a source should be CONSTANT. </p>
<p>
In each of the following cases, determine the behavior of the power (integrated Poynting flux through a surface) emitted as a function of distance to the surface at which the power is measured (where [mathjaxinline]C[/mathjaxinline] is a constant). </p>
<p><b class="bfseries">(Part a)</b> A stationary charge. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_01_2_1">
<fieldset aria-describedby="status_lect_21_01_2_1">
<div class="field">
<input type="radio" name="input_lect_21_01_2_1" id="input_lect_21_01_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_01_2_1-choice_1-label" for="input_lect_21_01_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_2_1">
<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_2_1" id="input_lect_21_01_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_01_2_1-choice_2-label" for="input_lect_21_01_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_2_1">
<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_2_1" id="input_lect_21_01_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_01_2_1-choice_3-label" for="input_lect_21_01_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_2_1">
<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_2_1" id="input_lect_21_01_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_01_2_1-choice_4-label" for="input_lect_21_01_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_2_1">
<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_2_1" id="input_lect_21_01_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_21_01_2_1-choice_5-label" for="input_lect_21_01_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_2_1">
<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_21_01_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_01_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><b class="bfseries">(Part b)</b> A charge moving at constant velocity. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_01_3_1">
<fieldset aria-describedby="status_lect_21_01_3_1">
<div class="field">
<input type="radio" name="input_lect_21_01_3_1" id="input_lect_21_01_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_01_3_1-choice_1-label" for="input_lect_21_01_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_3_1">
<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_3_1" id="input_lect_21_01_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_01_3_1-choice_2-label" for="input_lect_21_01_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_3_1">
<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_3_1" id="input_lect_21_01_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_01_3_1-choice_3-label" for="input_lect_21_01_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_3_1">
<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_3_1" id="input_lect_21_01_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_01_3_1-choice_4-label" for="input_lect_21_01_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_3_1">
<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_3_1" id="input_lect_21_01_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_21_01_3_1-choice_5-label" for="input_lect_21_01_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_3_1">
<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_21_01_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_01_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> An accelerating charge. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_01_4_1">
<fieldset aria-describedby="status_lect_21_01_4_1">
<div class="field">
<input type="radio" name="input_lect_21_01_4_1" id="input_lect_21_01_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_01_4_1-choice_1-label" for="input_lect_21_01_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_4_1">
<text> a) [mathjaxinline]P \propto r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_4_1" id="input_lect_21_01_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_01_4_1-choice_2-label" for="input_lect_21_01_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_4_1">
<text> b) [mathjaxinline]P \propto C[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_4_1" id="input_lect_21_01_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_01_4_1-choice_3-label" for="input_lect_21_01_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_4_1">
<text> c) [mathjaxinline]P=0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_4_1" id="input_lect_21_01_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_01_4_1-choice_4-label" for="input_lect_21_01_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_4_1">
<text> d) [mathjaxinline]P \propto \dfrac {1}{r}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_01_4_1" id="input_lect_21_01_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_21_01_4_1-choice_5-label" for="input_lect_21_01_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_21_01_4_1">
<text> e) [mathjaxinline]P \propto \dfrac {1}{r^{2}}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_21_01_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_01_4_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_21_01_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="Charge Moving in Various Ways" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_01" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_01">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_01-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_01-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_01-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L32v2">
<h2 class="hd hd-2 unit-title">L32v2: Accelerating Charges</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L32v2: Accelerating charges</h3>
<div
id="video_L32v2"
class="video closed"
data-metadata='{"saveStateEnabled": false, "speed": null, "ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v2/handler/transcript/available_translations", "captionDataDir": null, "completionEnabled": false, "poster": null, "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v2/handler/transcript/translation/__lang__", "start": 0.0, "end": 0.0, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v2/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v2/handler/publish_completion", "showCaptions": "true", "autohideHtml5": false, "streams": "1.00:RdD8F-7-xak", "prioritizeHls": false, "duration": 0.0, "transcriptLanguage": "en", "autoplay": false, "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "sources": []}'
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="L32v2"></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_L32v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/3fbee11f5baf747f33344a415a8b3226/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L32v2.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-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex02">
<h2 class="hd hd-2 unit-title">L32Q2: Accelerating Charge I</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_02" class="problems-wrapper" role="group"
aria-labelledby="lect_21_02-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_02-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02-problem-progress" tabindex="-1">
Accelerating Charge I
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the following questions related to the time-dependent behavior of the accelerating charge that we are investigating in this lesson. </p>
<p><b class="bfseries">(Part a)</b> What happens between the times [mathjaxinline]t=0[/mathjaxinline] and [mathjaxinline]t=\Delta t[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_02_2_1">
<fieldset aria-describedby="status_lect_21_02_2_1">
<div class="field">
<input type="radio" name="input_lect_21_02_2_1" id="input_lect_21_02_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_02_2_1-choice_1-label" for="input_lect_21_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_2_1">
<text> a) the charge is stationary</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_2_1" id="input_lect_21_02_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_02_2_1-choice_2-label" for="input_lect_21_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_2_1">
<text> b) the charge moves with speed [mathjaxinline]u[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_2_1" id="input_lect_21_02_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_02_2_1-choice_3-label" for="input_lect_21_02_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_2_1">
<text> c) the charge accelerates upwards</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_2_1" id="input_lect_21_02_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_02_2_1-choice_4-label" for="input_lect_21_02_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_2_1">
<text> d) the charge oscillates up and down</text>
</label>
</div>
<span id="answer_lect_21_02_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_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><b class="bfseries">(Part b)</b> What happens after the time [mathjaxinline]t=\Delta t[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_02_3_1">
<fieldset aria-describedby="status_lect_21_02_3_1">
<div class="field">
<input type="radio" name="input_lect_21_02_3_1" id="input_lect_21_02_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_02_3_1-choice_1-label" for="input_lect_21_02_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_3_1">
<text> a) the charge is stationary</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_3_1" id="input_lect_21_02_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_02_3_1-choice_2-label" for="input_lect_21_02_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_3_1">
<text> b) the charge moves with speed [mathjaxinline]u[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_3_1" id="input_lect_21_02_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_02_3_1-choice_3-label" for="input_lect_21_02_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_3_1">
<text> c) the charge accelerates upwards</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_02_3_1" id="input_lect_21_02_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_02_3_1-choice_4-label" for="input_lect_21_02_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_3_1">
<text> d) the charge oscillates up and down</text>
</label>
</div>
<span id="answer_lect_21_02_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_02_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> An observer a distance [mathjaxinline]r[/mathjaxinline] away from a light source will receive information about an event that occurs at time [mathjaxinline]t=t_{1}[/mathjaxinline] some time [mathjaxinline]\Delta T[/mathjaxinline] after this event occurs (note, the "retarded time" is defined to be [mathjaxinline](t - \Delta T)[/mathjaxinline]. What is [mathjaxinline]\Delta T[/mathjaxinline] in terms of [mathjaxinline]r[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], and other relevant numerical constants? </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta T =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_lect_21_02_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_02_4_1" id="input_lect_21_02_4_1" aria-describedby="status_lect_21_02_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_02_4_1"/>
<span class="status unanswered" id="status_lect_21_02_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_02_4_1" class="answer"/>
<div id="display_lect_21_02_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_02_4_1_dynamath" name="input_lect_21_02_4_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_02_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Accelerating Charge I" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_02" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_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_21_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_21_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_21_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-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex02_1">
<h2 class="hd hd-2 unit-title">L32Q3: Accelerating Charge II</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_02_1" class="problems-wrapper" role="group"
aria-labelledby="lect_21_02_1-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1/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_21_02_1-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1-problem-progress" tabindex="-1">
Accelerating Charge II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_02_1-problem-progress"></div>
<div class="problem">
<div>
<p>
The "kink" in the electric field of an accelerating charge has which of the following properties (check 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_21_02_1_2_1">
<fieldset aria-describedby="status_lect_21_02_1_2_1">
<div class="field">
<input type="checkbox" name="input_lect_21_02_1_2_1[]" id="input_lect_21_02_1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_21_02_1_2_1-choice_0-label" for="input_lect_21_02_1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_1_2_1">
<text>a) the kink travels at the speed of light</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_21_02_1_2_1[]" id="input_lect_21_02_1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_21_02_1_2_1-choice_1-label" for="input_lect_21_02_1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_1_2_1">
<text>b) the kink only has a component perpendicular to the direction of propagation</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_21_02_1_2_1[]" id="input_lect_21_02_1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_21_02_1_2_1-choice_2-label" for="input_lect_21_02_1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_1_2_1">
<text>c) the kink only has a component parallel to the direction of propagation</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_21_02_1_2_1[]" id="input_lect_21_02_1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_21_02_1_2_1-choice_3-label" for="input_lect_21_02_1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_02_1_2_1">
<text>d) the kink has components that are both perpendicular and parallel to the direction of propagation</text>
</label>
</div>
<span id="answer_lect_21_02_1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_02_1_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_21_02_1_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Accelerating Charge II" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_02_1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_02_1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_02_1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_02_1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_02_1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L32v3">
<h2 class="hd hd-2 unit-title">L32v3: Geometry of the Electric Field from an Accelerated Charge</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v3">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L32v3: Geometry of the Electric Field from an Accelerated Charge</h3>
<div
id="video_L32v3"
class="video closed"
data-metadata='{"saveStateEnabled": false, "speed": null, "ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v3/handler/transcript/available_translations", "captionDataDir": null, "completionEnabled": false, "poster": null, "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v3/handler/transcript/translation/__lang__", "start": 0.0, "end": 0.0, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v3/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v3/handler/publish_completion", "showCaptions": "true", "autohideHtml5": false, "streams": "1.00:p2_5WCv6Gq8", "prioritizeHls": false, "duration": 0.0, "transcriptLanguage": "en", "autoplay": false, "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "sources": []}'
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="L32v3"></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_L32v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/4fdc040281e919b802d3fafe5e5efc28/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L32v3.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-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex03">
<h2 class="hd hd-2 unit-title">L32Q4: Electric Field Components from an Accelerated Charge</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_03" class="problems-wrapper" role="group"
aria-labelledby="lect_21_03-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_21_03-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03-problem-progress" tabindex="-1">
Electric Field Components from an Accelerated Charge
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_03-problem-progress"></div>
<div class="problem">
<div>
<p>
Consider the diagrams in the previous video to answer the following questions. </p>
<p><b class="bfseries">(Part a)</b> What is [mathjaxinline]u_{\perp }[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_03_2_1">
<fieldset aria-describedby="status_lect_21_03_2_1">
<div class="field">
<input type="radio" name="input_lect_21_03_2_1" id="input_lect_21_03_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_03_2_1-choice_1-label" for="input_lect_21_03_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_2_1">
<text> a) the velocity of the particle</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_2_1" id="input_lect_21_03_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_03_2_1-choice_2-label" for="input_lect_21_03_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_2_1">
<text> b) the velocity of the wave front</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_2_1" id="input_lect_21_03_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_03_2_1-choice_3-label" for="input_lect_21_03_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_2_1">
<text> c) the component of the velocity of the particle that is perpendicular to the line of sight to the observer</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_2_1" id="input_lect_21_03_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_03_2_1-choice_4-label" for="input_lect_21_03_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_2_1">
<text> d) the component of the velocity of the particle that is perpendicular to the wavefront</text>
</label>
</div>
<span id="answer_lect_21_03_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_03_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><b class="bfseries">(Part b)</b> What is [mathjaxinline]a_{\perp }[/mathjaxinline]? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_03_3_1">
<fieldset aria-describedby="status_lect_21_03_3_1">
<div class="field">
<input type="radio" name="input_lect_21_03_3_1" id="input_lect_21_03_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_03_3_1-choice_1-label" for="input_lect_21_03_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_3_1">
<text> a) the acceleration of the particle</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_3_1" id="input_lect_21_03_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_03_3_1-choice_2-label" for="input_lect_21_03_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_3_1">
<text> b) the acceleration of the wave front</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_3_1" id="input_lect_21_03_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_03_3_1-choice_3-label" for="input_lect_21_03_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_3_1">
<text> c) the component of the acceleration of the particle that is perpendicular to the line of sight to the observer</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_3_1" id="input_lect_21_03_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_21_03_3_1-choice_4-label" for="input_lect_21_03_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_3_1">
<text> d) the component of the acceleration of the particle that is perpendicular to the wavefront</text>
</label>
</div>
<span id="answer_lect_21_03_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_03_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> Our ultimate objective in this analysis is to calculate the Poynting vector [mathjaxinline]\vec{S}=\frac{1}{\mu _{0}}\vec{E}\times \vec{B}[/mathjaxinline], but we must determine the form of [mathjaxinline]\vec{E}[/mathjaxinline]. We have begun to do this diagrammatically. </p>
<p>
Which component of [mathjaxinline]\vec{E}_{\mathrm{kink}}[/mathjaxinline] will contribute to the Poynting flux? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_21_03_4_1">
<fieldset aria-describedby="status_lect_21_03_4_1">
<div class="field">
<input type="radio" name="input_lect_21_03_4_1" id="input_lect_21_03_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_21_03_4_1-choice_1-label" for="input_lect_21_03_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_4_1">
<text> a) [mathjaxinline]\vec{E}_{\mathrm{||}}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_4_1" id="input_lect_21_03_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_21_03_4_1-choice_2-label" for="input_lect_21_03_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_4_1">
<text> b) [mathjaxinline]\vec{E}_{\mathrm{\perp }}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_21_03_4_1" id="input_lect_21_03_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_21_03_4_1-choice_3-label" for="input_lect_21_03_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_21_03_4_1">
<text> c) Both [mathjaxinline]\perp[/mathjaxinline] and [mathjaxinline]||[/mathjaxinline] components</text>
</label>
</div>
<span id="answer_lect_21_03_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_03_4_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_21_03_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="Electric Field Components from an Accelerated Charge" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_03" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_03">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_03-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_03-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_03-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L32v4">
<h2 class="hd hd-2 unit-title">L32v4: Perpendicular Electric Field from an Accelerated Charge</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v4">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v4">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L32v4: Perpendicular Electric Field from an Accelerated Charge</h3>
<div
id="video_L32v4"
class="video closed"
data-metadata='{"saveStateEnabled": false, "speed": null, "ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v4/handler/transcript/available_translations", "captionDataDir": null, "completionEnabled": false, "poster": null, "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v4/handler/transcript/translation/__lang__", "start": 0.0, "end": 0.0, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v4/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v4/handler/publish_completion", "showCaptions": "true", "autohideHtml5": false, "streams": "1.00:lYgY8DM2u6s", "prioritizeHls": false, "duration": 0.0, "transcriptLanguage": "en", "autoplay": false, "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "sources": []}'
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="L32v4"></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_L32v4">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v4">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/e330e351d3bb9c5bcfc3ca4e50c5a438/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L32v4.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-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex04">
<h2 class="hd hd-2 unit-title">L32Q5: Radiating Electric Field Derivation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_04a" class="problems-wrapper" role="group"
aria-labelledby="lect_21_04a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a/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_21_04a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a-problem-progress" tabindex="-1">
Radiating Electric Field Derivation
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_04a-problem-progress"></div>
<div class="problem">
<div>
<p>
In the preceding derivation, we used Gauss's law to determine [mathjaxinline]E_{||}[/mathjaxinline]. Specifically, we drew a pillbox around a point in space and concluded that [mathjaxinline]E_{||}=E_{\mathrm{out}}[/mathjaxinline], where [mathjaxinline]E_{\mathrm{out}}[/mathjaxinline] is the magnitude of the electric field a distance [mathjaxinline]r[/mathjaxinline] away from a point charge. </p>
<p>
Which of the following factors contribute to this equivalence? 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_21_04a_2_1">
<fieldset aria-describedby="status_lect_21_04a_2_1">
<div class="field">
<input type="checkbox" name="input_lect_21_04a_2_1[]" id="input_lect_21_04a_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_21_04a_2_1-choice_0-label" for="input_lect_21_04a_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_21_04a_2_1">
<text>a) there is no charge enclosed within the surface</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_21_04a_2_1[]" id="input_lect_21_04a_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_21_04a_2_1-choice_1-label" for="input_lect_21_04a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_21_04a_2_1">
<text>b) the net flux due to [mathjaxinline]E_{\perp }[/mathjaxinline] is zero</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_21_04a_2_1[]" id="input_lect_21_04a_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_21_04a_2_1-choice_2-label" for="input_lect_21_04a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_21_04a_2_1">
<text>c) the pillbox is very far away from the source, so [mathjaxinline]E_{||} \approx E_{\mathrm{out}}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_21_04a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_21_04a_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_21_04a_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Radiating Electric Field Derivation" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_04a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_04a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_04a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_04a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_04a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_ex05">
<h2 class="hd hd-2 unit-title">L32Q6: Electric Field from Charge Acceleration</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05a" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a/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_21_05a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a-problem-progress" tabindex="-1">
Electric Field from Charge Acceleration - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05a-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> Consider a positive charge [mathjaxinline]q[/mathjaxinline] that is constantly accelerating in the [mathjaxinline]\hat{j}[/mathjaxinline] direction, [mathjaxinline]\vec{a}(t)=a_{0}\hat{j}[/mathjaxinline]. Find an expression for the magnitude of the radiative component of the electric field, [mathjaxinline]|\vec{E}_{\mathrm{rad}}|[/mathjaxinline], as a function of [mathjaxinline]\theta[/mathjaxinline], the angle relative to the j direction. Consider times [mathjaxinline]t \ll c/a_{0}[/mathjaxinline], such that the particle moves nonrelativistically. </p>
<p>
Use the formula [mathjaxinline]\vec{E}_{\mathrm{rad}}(\vec{r},t) = \dfrac {-k q \vec{a}_{\perp }(t-r/c)}{c^2r}[/mathjaxinline] for the electric field at a location a distance [mathjaxinline]r[/mathjaxinline] away from the charge, where [mathjaxinline]k=\dfrac {1}{4\pi \epsilon _0}[/mathjaxinline] and [mathjaxinline]\vec{a}_{\perp }(t-r/c)[/mathjaxinline] is the perpendicular component of the acceleration at a time [mathjaxinline](t-r/c)[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]&#8212;make sure that your expression for the magnitude is positive for all values of [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{E}_{\mathrm{rad}}| =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05a_2_1" id="input_lect_21_05a_2_1" aria-describedby="status_lect_21_05a_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05a_2_1"/>
<span class="status unanswered" id="status_lect_21_05a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05a_2_1" class="answer"/>
<div id="display_lect_21_05a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05a_2_1_dynamath" name="input_lect_21_05a_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05a_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="Electric Field from Charge Acceleration - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05b" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b/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_21_05b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b-problem-progress" tabindex="-1">
Electric Field from Charge Acceleration - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Find an expression for the magnitude of the radiative component of the magnetic field, [mathjaxinline]|\vec{B}_{\mathrm{rad}}|[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{B}_{\mathrm{rad}}| =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05b_2_1" id="input_lect_21_05b_2_1" aria-describedby="status_lect_21_05b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05b_2_1"/>
<span class="status unanswered" id="status_lect_21_05b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05b_2_1" class="answer"/>
<div id="display_lect_21_05b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05b_2_1_dynamath" name="input_lect_21_05b_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05b_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="Electric Field from Charge Acceleration - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="True" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_21_05c" class="problems-wrapper" role="group"
aria-labelledby="lect_21_05c-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c/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_21_05c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c-problem-progress" tabindex="-1">
Electric Field from charge acceleration - part c
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_21_05c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Find an expression for the magnitude of the Poynting vector [mathjaxinline]|\vec{S}_{\mathrm{rad}}|[/mathjaxinline]. Express your answer in terms of <code>k</code>, <code>q</code>, <code>t</code>, <code>r</code>, <code>c</code>, <code>a_0</code> for [mathjaxinline]a_{0}[/mathjaxinline], <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>mu_0</code> for [mathjaxinline]\mu _{0}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{S}_{\mathrm{rad}}| =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_21_05c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_21_05c_2_1" id="input_lect_21_05c_2_1" aria-describedby="status_lect_21_05c_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_21_05c_2_1"/>
<span class="status unanswered" id="status_lect_21_05c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_21_05c_2_1" class="answer"/>
<div id="display_lect_21_05c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_21_05c_2_1_dynamath" name="input_lect_21_05c_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_21_05c_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="Electric Field from charge acceleration - part c" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_21_05c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_21_05c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05c-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05c-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_21_05c-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_21_review">
<h2 class="hd hd-2 unit-title">Review of Radiation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lect_21_review">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="html" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lect_21_review">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
We used a geometric argument to see that light is emitted from an accelerated charge in a direction that is antiparallel to the acceleration and falls off like [mathjaxinline]1/r[/mathjaxinline]. </p><center><img src="/assets/courseware/v1/bf03d234b4a432d6b2a9b9df607609ac/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_21_review.svg" width="400" style="width:"/></center><p>
Specifically we have: </p><table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000003"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{E}_{\mathrm{rad}}(\vec{r},t) = \dfrac {-q \vec{a}_{\perp }(t')}{4 \pi \epsilon _0 c^2 r}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{B}_{\mathrm{rad}}(\vec{r},t) = \left(\frac{1}{c}\right)\left(\hat{r}\times \vec{E}_{\mathrm{rad}}(\vec{r},t)\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
where [mathjaxinline]t'[/mathjaxinline] is the retarded time, [mathjaxinline]t' = t - r/c[/mathjaxinline]. </p><p>
This means that the Poynting vector, or energy flux per unit area, </p><table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{S}_{\mathrm{rad}}(\vec{r},t) = \frac{1}{\mu _0} \vec{E}_{\mathrm{rad}}(\vec{r},t) \times \vec{B}_{\mathrm{rad}}(\vec{r},t) \propto \frac{1}{r^2}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
falls off like [mathjaxinline]1/r^2[/mathjaxinline], and so is detectable far away from the radiating source. </p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-init="VerticalStudentView" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L32v5">
<h2 class="hd hd-2 unit-title">L32v5: Dipole Antenna [DEMO]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v5">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-runtime-class="LmsRuntime" data-init="XBlockToXModuleShim" data-graded="True" data-course-id="course-v1:MITx+8.03x+1T2020" data-has-score="False" data-request-token="3253cf65686a11ef8c9c0e199eb5def5" data-runtime-version="1" data-block-type="video" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L32v5">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L32v5: Dipole Antenna [DEMO]</h3>
<div
id="video_L32v5"
class="video closed"
data-metadata='{"saveStateEnabled": false, "speed": null, "ytMetadataEndpoint": "", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v5/handler/transcript/available_translations", "captionDataDir": null, "completionEnabled": false, "poster": null, "generalSpeed": 1.0, "transcriptLanguages": {"en": "English"}, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v5/handler/transcript/translation/__lang__", "start": 0.0, "end": 0.0, "savedVideoPosition": 0.0, "lmsRootURL": "https://openlearninglibrary.mit.edu", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v5/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "autoAdvance": false, "ytTestTimeout": 1500, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L32v5/handler/publish_completion", "showCaptions": "true", "autohideHtml5": false, "streams": "1.00:aqsmMwlZ3S4", "prioritizeHls": false, "duration": 0.0, "transcriptLanguage": "en", "autoplay": false, "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "sources": []}'
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="L32v5"></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_L32v5">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v5">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/a5f4d5251cbe6b84a73d79eebec20c72/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L32v5.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>