<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-lesson04-textbooklinks" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">Textbook Links</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@html-lesson04-textbooklinks">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@html-lesson04-textbooklinks" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p> Now that we understand how to calculate forces and electric fields for point sources, we move on to calculating electric fields due to more complex, continuous charge distributions.</p><p> Textbook Links </p><ul><li><a href="/courses/course-v1:MITx+8.02.1x+1T2019/pdfbook/0/chapter/2/16">Chapter 2.9-2.10 Continuous Sources</a></li></ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L04v01" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4v1: Electric Field of a Continuous Charge Source</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L04v01: Electric Field of a Continuous Charge Source</h3>
<div
id="video_L04v01"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:xK_xzwoOKts", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/transcript/available_translations"}'
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="L04v01"></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_L04v01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L04v01">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v01/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-l4-ring-sym" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4Q1: Electric Field of a Uniformly Charged Rod - Symmetry</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14" data-graded="True" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w2_14" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w2_14-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14/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="checkpoint_w2_14-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14-problem-progress" tabindex="-1">
Charged Rod
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_14-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/df87461ad828603089cb0db717c7d1b4/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w2_14.svg" width="990"/>
</center>
<p>
Consider the rod shown in the left figure above. Assume that is has a uniformly distributed positive charge. Point [mathjaxinline]P[/mathjaxinline] in the figure is located on the perpendicular bisector of the rod. Two identical small segments of the rod of charge [mathjaxinline]dq[/mathjaxinline] and equidistant to the center of the rod are shown in black. These questions are about the direction of the electric fields produced by the elements of charge [mathjaxinline]dq[/mathjaxinline] at point [mathjaxinline]P[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> The direction of the electric field at point [mathjaxinline]P[/mathjaxinline] produced by the element of charge [mathjaxinline]dq[/mathjaxinline] on the right side of the rod is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w2_14_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_14_2_1" id="input_checkpoint_w2_14_2_1" aria-describedby="status_checkpoint_w2_14_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_14_2_1"/>
<span class="status unanswered" id="status_checkpoint_w2_14_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_14_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_14_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> The direction of the electric field at point [mathjaxinline]P[/mathjaxinline] produced by the element of charge [mathjaxinline]dq[/mathjaxinline] on the left side of the rod is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w2_14_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_14_3_1" id="input_checkpoint_w2_14_3_1" aria-describedby="status_checkpoint_w2_14_3_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_14_3_1"/>
<span class="status unanswered" id="status_checkpoint_w2_14_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_14_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_14_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> The direction of the total electric field at point [mathjaxinline]P[/mathjaxinline] produced by both elements of charge is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w2_14_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_14_4_1" id="input_checkpoint_w2_14_4_1" aria-describedby="status_checkpoint_w2_14_4_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_14_4_1"/>
<span class="status unanswered" id="status_checkpoint_w2_14_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_14_4_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_14_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charged Rod" />
<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_checkpoint_w2_14" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w2_14">
<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">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</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="checkpoint_w2_14-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="checkpoint_w2_14-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="checkpoint_w2_14-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-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L04v02" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4v2: Electric Field from a Rod on the Symmetry Axis Using Angles and Symmetry</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L04v02: Electric Field from a Rod on the Symmetry Axis Using Angles and Symmetry</h3>
<div
id="video_L04v02"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:LvGwFBYKBg4", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/transcript/available_translations"}'
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="L04v02"></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_L04v02">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L04v02">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v02/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-l04-text-definition" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">Charge Density</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@lesson04_01">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@lesson04_01" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b>Charge density </b></p><p><b>Volume charge density</b></p><p>Consider a charged object of total charge \(Q\) and volume \(V\). Consider an element of volume \(dV\) charged with an element of charge \(dq\). The volume charge density \(\rho\) is defined as:</p><p>
\[\displaystyle \rho = \frac{dq}{dV}\]
</p><p>where the dimension of \(\rho\) is (C\(\cdot\)m\(^{-3}\)). The total charge on the object is obtained as the integral over the volume of the object:</p><p>
\[\displaystyle Q = \int_{V} \rho \;dV\]
</p><p><b>Surface charge density</b></p><p>Consider a charged object of total charge \(Q\) distributed on its surface of area \(A\). Consider an element of surface \(dA\) charged with an element of charge \(dq\). The surface charge density \(\sigma\) is defined as:</p><p>
\[\displaystyle \sigma = \frac{dq}{dA}\]
</p><p>where the dimension of \(\sigma\) is (C\(\cdot\)m\(^{-2}\)). The total charge on the object is obtained as the integral over the surface of the object:</p><p>
\[\displaystyle Q = \int_{A} \sigma \;dA\]
</p><p><b>Linear charge density</b></p><p> Consider a total charge \(Q\) distributed along a line. If an element of line \(dl\) is charged with an element of charge \(dq\), the linear charge density \(\lambda\) is defined as:</p><p>
\[\displaystyle \lambda = \frac{dq}{dl}\]
</p><p>where the dimension of \(\lambda\) is (C\(\cdot\)m\(^{-1}\)). The total charge on the object is obtained as the integral over the total length of the object:</p><p>
\[\displaystyle Q = \int_{line} \lambda \;dl\]
</p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-l04-CQ-denisties" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4Q2: Charge Density</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12" data-graded="True" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w2_12" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w2_12-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12/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="checkpoint_w2_12-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12-problem-progress" tabindex="-1">
Charge density - Part I
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/a781aac32b76328d2e1a57c29e0af3f0/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w2_12.svg" width="880"/>
</center>
<p>
The objects shown in the figures above are all uniformly charged with a total charge [mathjaxinline]Q[/mathjaxinline]. For each object, express [mathjaxinline]Q[/mathjaxinline] in terms of [mathjaxinline]\lambda[/mathjaxinline] (linear charge density), [mathjaxinline]\sigma[/mathjaxinline] (surface charge density), or [mathjaxinline]\rho[/mathjaxinline] (volume charge density), and the geometric factors [mathjaxinline]L, h, R, R_1[/mathjaxinline] and [mathjaxinline]R_2[/mathjaxinline] as needed. Enter lambda for [mathjaxinline]\lambda[/mathjaxinline], sigma for [mathjaxinline]\sigma[/mathjaxinline], rho for [mathjaxinline]\rho[/mathjaxinline], pi for [mathjaxinline]\pi[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], and R_2 for [mathjaxinline]R_2[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Solid cylinder. <p style="display:inline">[mathjaxinline]Q =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w2_12_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_12_2_1" id="input_checkpoint_w2_12_2_1" aria-describedby="status_checkpoint_w2_12_2_1" value="" class="math" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_12_2_1"/>
<span class="status unanswered" id="status_checkpoint_w2_12_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_12_2_1" class="answer"/>
<div id="display_checkpoint_w2_12_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_12_2_1_dynamath" name="input_checkpoint_w2_12_2_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part b)</b> Thin rod. <p style="display:inline">[mathjaxinline]Q =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w2_12_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_12_3_1" id="input_checkpoint_w2_12_3_1" aria-describedby="status_checkpoint_w2_12_3_1" value="" class="math" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_12_3_1"/>
<span class="status unanswered" id="status_checkpoint_w2_12_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_12_3_1" class="answer"/>
<div id="display_checkpoint_w2_12_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_12_3_1_dynamath" name="input_checkpoint_w2_12_3_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part c)</b> Cylindrical Shell <p style="display:inline">[mathjaxinline]Q =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w2_12_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_12_4_1" id="input_checkpoint_w2_12_4_1" aria-describedby="status_checkpoint_w2_12_4_1" value="" class="math" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_12_4_1"/>
<span class="status unanswered" id="status_checkpoint_w2_12_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_12_4_1" class="answer"/>
<div id="display_checkpoint_w2_12_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_12_4_1_dynamath" name="input_checkpoint_w2_12_4_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part d)</b> Solid Sphere <p style="display:inline">[mathjaxinline]Q =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w2_12_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_12_5_1" id="input_checkpoint_w2_12_5_1" aria-describedby="status_checkpoint_w2_12_5_1" value="" class="math" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_12_5_1"/>
<span class="status unanswered" id="status_checkpoint_w2_12_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_12_5_1" class="answer"/>
<div id="display_checkpoint_w2_12_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_12_5_1_dynamath" name="input_checkpoint_w2_12_5_1_dynamath"/>
</div>
</div></div> </p>
<p><b class="bfseries">(Part e)</b> Thin Disk <p style="display:inline">[mathjaxinline]Q =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_checkpoint_w2_12_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_12_6_1" id="input_checkpoint_w2_12_6_1" aria-describedby="status_checkpoint_w2_12_6_1" value="" class="math" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_12_6_1"/>
<span class="status unanswered" id="status_checkpoint_w2_12_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_12_6_1" class="answer"/>
<div id="display_checkpoint_w2_12_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_12_6_1_dynamath" name="input_checkpoint_w2_12_6_1_dynamath"/>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_12_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charge density - Part 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_checkpoint_w2_12" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w2_12">
<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">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</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="checkpoint_w2_12-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="checkpoint_w2_12-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="checkpoint_w2_12-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+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b" data-graded="True" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w2_12_b" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w2_12_b-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b/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="checkpoint_w2_12_b-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b-problem-progress" tabindex="-1">
Charge Density - Part II
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_12_b-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/3e8bc3474b6f8578c2d714235438685d/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w2_12_b.svg" width="770"/>
</center>
<p><b class="bfseries">(Part a.)</b> A charge is uniformly distributed along a thin rod of length [mathjaxinline]L[/mathjaxinline]. The charge of the small element of rod of length [mathjaxinline]\Delta x[/mathjaxinline] is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w2_12_b_2_1">
<fieldset aria-describedby="status_checkpoint_w2_12_b_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_2_1" id="input_checkpoint_w2_12_b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w2_12_b_2_1-choice_1-label" for="input_checkpoint_w2_12_b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_2_1">
<text> [mathjaxinline]\Delta q = Q/L[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_2_1" id="input_checkpoint_w2_12_b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w2_12_b_2_1-choice_2-label" for="input_checkpoint_w2_12_b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_2_1">
<text> [mathjaxinline]\Delta q =Q\Delta x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_2_1" id="input_checkpoint_w2_12_b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w2_12_b_2_1-choice_3-label" for="input_checkpoint_w2_12_b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_2_1">
<text> [mathjaxinline]\Delta q=(Q/L)\Delta x[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_2_1" id="input_checkpoint_w2_12_b_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w2_12_b_2_1-choice_4-label" for="input_checkpoint_w2_12_b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_2_1">
<text> [mathjaxinline]\Delta q = QL\Delta x[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w2_12_b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w2_12_b_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 [mathjaxinline]Q[/mathjaxinline] is uniformly distributed throughout a rectangle of length [mathjaxinline]a[/mathjaxinline] and height [mathjaxinline]b[/mathjaxinline]. The charge of a small element of the rectangle of length [mathjaxinline]\Delta x[/mathjaxinline] and height [mathjaxinline]\Delta y[/mathjaxinline] is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w2_12_b_3_1">
<fieldset aria-describedby="status_checkpoint_w2_12_b_3_1">
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_3_1" id="input_checkpoint_w2_12_b_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w2_12_b_3_1-choice_1-label" for="input_checkpoint_w2_12_b_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_3_1">
<text> [mathjaxinline]\Delta q = Q\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_3_1" id="input_checkpoint_w2_12_b_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w2_12_b_3_1-choice_2-label" for="input_checkpoint_w2_12_b_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_3_1">
<text> [mathjaxinline]\Delta q =Q(ab)\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_3_1" id="input_checkpoint_w2_12_b_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w2_12_b_3_1-choice_3-label" for="input_checkpoint_w2_12_b_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_3_1">
<text> [mathjaxinline]\Delta q=(Q/(ab))\Delta x\Delta y[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_3_1" id="input_checkpoint_w2_12_b_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w2_12_b_3_1-choice_4-label" for="input_checkpoint_w2_12_b_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_3_1">
<text> [mathjaxinline]\Delta q = Q/(ab)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w2_12_b_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w2_12_b_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> A charge [mathjaxinline]Q[/mathjaxinline] is uniformly distributed throughout a disk of radius [mathjaxinline]R[/mathjaxinline]. Consider the charge element to be the ring of inner radius [mathjaxinline]r[/mathjaxinline] and very small thickness [mathjaxinline]\Delta r[/mathjaxinline]. The charge of the ring is given by: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w2_12_b_4_1">
<fieldset aria-describedby="status_checkpoint_w2_12_b_4_1">
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_4_1" id="input_checkpoint_w2_12_b_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w2_12_b_4_1-choice_1-label" for="input_checkpoint_w2_12_b_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_4_1">
<text> [mathjaxinline]\Delta q = (Q/(2\pi R))\Delta r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_4_1" id="input_checkpoint_w2_12_b_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w2_12_b_4_1-choice_2-label" for="input_checkpoint_w2_12_b_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_4_1">
<text> [mathjaxinline]\Delta q=(Q/R)\Delta r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_4_1" id="input_checkpoint_w2_12_b_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w2_12_b_4_1-choice_3-label" for="input_checkpoint_w2_12_b_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_4_1">
<text> [mathjaxinline]\Delta q=(2Q/R)\Delta r[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_4_1" id="input_checkpoint_w2_12_b_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w2_12_b_4_1-choice_4-label" for="input_checkpoint_w2_12_b_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_4_1">
<text> [mathjaxinline]\Delta q = 2Qr \Delta r/R^2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w2_12_b_4_1" id="input_checkpoint_w2_12_b_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w2_12_b_4_1-choice_5-label" for="input_checkpoint_w2_12_b_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w2_12_b_4_1">
<text> [mathjaxinline]\Delta q = Q/(\pi R^2)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w2_12_b_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w2_12_b_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_checkpoint_w2_12_b_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charge Density - Part 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_checkpoint_w2_12_b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w2_12_b">
<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">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</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="checkpoint_w2_12_b-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="checkpoint_w2_12_b-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="checkpoint_w2_12_b-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-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L04v03" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4v3: Continuous Charge Distribution Electric Field Calculation with Source to Point Vector</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L04v03: Continuous Charge Distribution Electric Field Calculation with Souce to Point Vector</h3>
<div
id="video_L04v03"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:295vtMoAvCc", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/transcript/available_translations"}'
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="L04v03"></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_L04v03">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L04v03">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v03/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L04v04" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4v4: Electric Field from a Rod on the Symmetry Axis Using the Source to Point Vector</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L04v04: Electric Field from a Rod on the Symmetry Axis Using the Source to Point Vector</h3>
<div
id="video_L04v04"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": [], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:cWNu6dw21T8", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/transcript/available_translations"}'
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="L04v04"></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_L04v04">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L04v04">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v04/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-cp_w2_3" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4Q3: Electric Field of a Uniformly Charged Ring</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13" data-graded="True" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w2_13" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w2_13-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13/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="checkpoint_w2_13-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13-problem-progress" tabindex="-1">
Charged Ring
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_13-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/77bee140a9a865c8c6f9c2e230f9dd95/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w2_13.svg" width="880"/>
</center>
<p>
Consider the ring shown in the left figure above. Assume that it has a uniformly distributed positive charge. The [mathjaxinline]z[/mathjaxinline] axis is the axis of symmetry of the ring. Two small arc lengths of charge [mathjaxinline]dq[/mathjaxinline] diametrically opposed are shown in black. This question is about the direction of the electric fields produced by the elements of charge [mathjaxinline]dq[/mathjaxinline] at a point [mathjaxinline]P[/mathjaxinline] along the axis of the ring. </p>
<p><b class="bfseries">(Part a)</b> The direction of the electric field at point [mathjaxinline]P[/mathjaxinline] produced by the element of charge [mathjaxinline]dq[/mathjaxinline] on the right side of the ring is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w2_13_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_13_2_1" id="input_checkpoint_w2_13_2_1" aria-describedby="status_checkpoint_w2_13_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_13_2_1"/>
<span class="status unanswered" id="status_checkpoint_w2_13_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_13_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_13_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> The direction of the electric field at point [mathjaxinline]P[/mathjaxinline] produced by the element of charge [mathjaxinline]dq[/mathjaxinline] on the left side of the ring is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w2_13_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_13_3_1" id="input_checkpoint_w2_13_3_1" aria-describedby="status_checkpoint_w2_13_3_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_13_3_1"/>
<span class="status unanswered" id="status_checkpoint_w2_13_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_13_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_13_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> The direction of the total electric field at point [mathjaxinline]P[/mathjaxinline] produced by both elements of charge is best represented by: </p>
<p>
<p style="display:inline">[mathjaxinline]\mbox{arrow number} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w2_13_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_13_4_1" id="input_checkpoint_w2_13_4_1" aria-describedby="status_checkpoint_w2_13_4_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_13_4_1"/>
<span class="status unanswered" id="status_checkpoint_w2_13_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_13_4_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_13_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charged Ring" />
<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_checkpoint_w2_13" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w2_13">
<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">
<span class="problem-action-button-wrapper">
<button type="button" class="save problem-action-btn btn-default btn-small" data-value="Save">
<span class="icon fa fa-floppy-o" aria-hidden="true"></span>
<span aria-hidden="true">Save</span>
<span class="sr">Save your answer</span>
</button>
</span>
</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="checkpoint_w2_13-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="checkpoint_w2_13-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="checkpoint_w2_13-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-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L04v05" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">L4v5: Worked Example - Electric Field of a Uniform Ring</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="video" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L04v05: Worked Example - Electric Field of a Uniform Ring</h3>
<div
id="video_L04v05"
class="video closed"
data-metadata='{"saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/xmodule_handler/save_user_state", "completionEnabled": false, "ytTestTimeout": 1500, "autohideHtml5": false, "completionPercentage": 0.95, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/9a6b99428a5a40839a0b6c9ae1ce8770.jpeg?Signature=exLcycruhaTVJrz6ANPSzUM%2F3cs%3D&Expires=1739501101&x-amz-security-token=IQoJb3JpZ2luX2VjEPv%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJIMEYCIQD9glDVLrGBtuh150nCbtnnIfW8q0If9DITc90%2BgqZtfgIhAKt6m9RmqqBd45OVWmz5Ta1lqHbgBvfYc6pTvBY5TFnVKrkFCCQQAxoMODU5MjA1NjM3NDExIgyJo9yys48Mn33DmxUqlgVaHG0t0f40rFASC7MfmffO7eUTX4w6snB9%2BK%2BNBq0ClD5ntbTRYaJr%2FWJODG3kYNZgtZhLpMJePlyiONBNa%2FTVAsNvhuKyMHQ1H005hGpKzX6Augm9tqVPvclUOvXAaPUOQT%2Be%2F6qLAyX7kpLJutlQydFzv1fYuFhPk4CQfd2HbdjzLPfuQRtxACgxoznpGZQ93bRxb5wEp424ICE1ofPzNCE9TokSSZM7jeVKuESH77cCH8rWAGv2iorHqldZWq%2BXNwqg5BmMaIPpoBCILLw1%2FEOcUr32Vq%2BnvjGTUCpjBsU3P0UHgX%2Fagm%2FB5SG8o%2FaUxEdaz78CWs%2FLj38b%2FMBjBKHHdPUKMxtkdTOhYIpczP1ZbB5QEkEWIY63eTgm%2Fozi2H%2Bj9xSt%2BpupzyMA0rRq1AsLwVErF%2BhYNAiWgutXIrPhsZYaJYCrFgGxOSKFbsLA77gFAwTFDH8cW88A9prLX0QKYrFHGAJAmWr8SnTG06ib9QFEG3V3qKqc%2BRvAuK4HhEjQ8a91PHoTRxBgJpPpRR%2FOk5%2BFTGm9I%2F5V7PZOwgB65QBFWPNjkL%2F9W%2F7H6eh3Foc6kaHflsEnlki0UlMWsePwNhdzd16ySz2bdhMZRMVH%2FQTM7fI%2FlTO0a7VsowVoW5gYdIlwy8Qv6IsTNZULHJugeuL1%2BXjEKI4VSB14bAQ9xqsX%2FQ648ahwGAVGjY9a%2FeiPDCPkadUkLOu0ODkrH0WXwxYC4hgdk15boFxz2uxrD2VdnOJAd9SDiG3wKorHDHNaGzNJ7I9iax0TD5%2BgXeg2GeMKvF4sbdvaNrS92XcfQ3NU9fuYN6HuFpWMtqY8WUasYofyxtmpvUyY23NFZYsbY1rLA1YXi1BmDRZ3GWzC7U10MzCD07q9BjqwAaENhtd0zQYwx9iMkKKfJNOC%2FxQgCAje2%2FO1dBeW9FPcRyNhCDYz8KRUJzqztQlvAqcCvuL1s9bXJKC5HPpyvLxPhK37I8b6vwYE2XKJ4F351%2B3180eTLz%2B1Z7QbulIbVzGcA6n5kZeUS2SCzbQHib%2Fx%2B7FMl%2F3rWOaQMUrclGEPyfL5NmHWofJWd6vplZ75FuneAooQ0e0HnWsQqF%2B7%2Fu2sx27lBedNmaWsLa30ujAF&AWSAccessKeyId=ASIA4QDFFSURZV7DI65A", "ytApiUrl": "https://www.youtube.com/iframe_api", "sources": ["https://d2f1egay8yehza.cloudfront.net/MIT8021E2018-V002600_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MIT8021E2018-V002600/MIT8021E2018-V002600.m3u8"], "transcriptLanguages": {"en": "English"}, "saveStateEnabled": false, "captionDataDir": null, "recordedYoutubeIsAvailable": true, "autoAdvance": false, "duration": 571.35, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/publish_completion", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/transcript/translation/__lang__", "prioritizeHls": false, "autoplay": false, "savedVideoPosition": 0.0, "streams": "1.00:UiP2kRw017c", "speed": null, "generalSpeed": 1.0, "ytMetadataEndpoint": "", "showCaptions": "true", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "end": 0.0, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/transcript/available_translations"}'
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="L04v05"></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_L04v05">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L04v05">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L04v05/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16" data-graded="True" data-has-score="True" data-runtime-version="1" data-block-type="problem" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w2_16" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w2_16-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="checkpoint_w2_16-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16-problem-progress" tabindex="-1">
Electric Field of a uniform ring - Second Approach.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w2_16-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/a2964e73da829889af57646b487bb518/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w2_16N.svg" width="770"/>
</center>
<p>
A ring of radius [mathjaxinline]R[/mathjaxinline] is uniformly charged with a total charge [mathjaxinline]Q[/mathjaxinline]. The electric field at point [mathjaxinline]P[/mathjaxinline] is expressed as: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \vec{E} = \int _{rod}kdq\frac{\vec{r}_ P - \vec{r}_ S}{|\vec{r}_ P-\vec{r}_ S|^3}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
In the expression above, [mathjaxinline]\vec{r}_{P}[/mathjaxinline] is the position of point P (field point vector), and [mathjaxinline]\vec{r}_{S}[/mathjaxinline] is the position of the element of charge [mathjaxinline]dq[/mathjaxinline] (source point vector). </p>
<p><b class="bfseries">(Part a)</b> Consider the coordinate system shown in the figure with the [mathjaxinline]z[/mathjaxinline] axis coinciding with the axis of symmetry of the ring. Point [mathjaxinline]P[/mathjaxinline] is at a distance [mathjaxinline]z[/mathjaxinline] from the origin, and the angle [mathjaxinline]\theta _ S[/mathjaxinline] is the angular position of the element of charge [mathjaxinline]dq[/mathjaxinline]. Write an expression for the vectors [mathjaxinline]\vec{r}_{S}[/mathjaxinline], [mathjaxinline]\vec{r}_{P}[/mathjaxinline], and [mathjaxinline]\vec{r}_{P}- \vec{r}_{S}[/mathjaxinline]in terms of [mathjaxinline]z[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], theta_S for [mathjaxinline]\theta _ S[/mathjaxinline], hatr for [mathjaxinline]\hat{r}[/mathjaxinline], hattheta for [mathjaxinline]\hat{\theta }[/mathjaxinline], and hatk for [mathjaxinline]\hat{k}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{r}_{S}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w2_16_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_2_1" id="input_checkpoint_w2_16_2_1" aria-describedby="status_checkpoint_w2_16_2_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_2_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_2_1" class="answer"/>
<div id="display_checkpoint_w2_16_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_2_1_dynamath" name="input_checkpoint_w2_16_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\vec{r}_{P}=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w2_16_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_3_1" id="input_checkpoint_w2_16_3_1" aria-describedby="status_checkpoint_w2_16_3_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_3_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_3_1" class="answer"/>
<div id="display_checkpoint_w2_16_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_3_1_dynamath" name="input_checkpoint_w2_16_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\vec{r}_{P}-\vec{r}_ S=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w2_16_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_4_1" id="input_checkpoint_w2_16_4_1" aria-describedby="status_checkpoint_w2_16_4_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_4_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_4_1" class="answer"/>
<div id="display_checkpoint_w2_16_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_4_1_dynamath" name="input_checkpoint_w2_16_4_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Write an expression for [mathjaxinline]|\vec{r}_{P}- \vec{r}_{S}|[/mathjaxinline], the magnitude of vector [mathjaxinline]\vec{r}_{P}-\vec{r}_ S[/mathjaxinline], the distance between point [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]dq[/mathjaxinline] in terms of of [mathjaxinline]z[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and theta_S for [mathjaxinline]\theta _ S[/mathjaxinline] for needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{r}_ P - \vec{r}_ S|=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w2_16_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_5_1" id="input_checkpoint_w2_16_5_1" aria-describedby="status_checkpoint_w2_16_5_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_5_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_5_1" class="answer"/>
<div id="display_checkpoint_w2_16_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_5_1_dynamath" name="input_checkpoint_w2_16_5_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> Consider [mathjaxinline]\lambda[/mathjaxinline] to be the linear charge density of the uniform ring. Write an expression for [mathjaxinline]\lambda[/mathjaxinline] in terms of [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]Q[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\lambda =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_checkpoint_w2_16_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_6_1" id="input_checkpoint_w2_16_6_1" aria-describedby="status_checkpoint_w2_16_6_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_6_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_6_1" class="answer"/>
<div id="display_checkpoint_w2_16_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_6_1_dynamath" name="input_checkpoint_w2_16_6_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part d)</b> Using the results obtained in parts (a), (b) and (c) in equation (1), the electric field at point [mathjaxinline]P[/mathjaxinline] can be expressed as: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle \vec{E} = \int _0^{2\pi }f(\theta _ S) \; d\theta _ S\hat{r}+ \int _0^{2\pi }g(\theta _ S) \; d\theta _ S\hat{k}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p>
Find an expression for the functions [mathjaxinline]f(\theta _ S)[/mathjaxinline] and [mathjaxinline]g(\theta _ S)[/mathjaxinline] in (eq. 2) needed to calculate [mathjaxinline]\vec{E}[/mathjaxinline] at point [mathjaxinline]P[/mathjaxinline]. Express the answer in terms of [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]z[/mathjaxinline], theta_S for [mathjaxinline]\theta _ S[/mathjaxinline], and the Coulomb constant [mathjaxinline]k[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]f(\theta _ s) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_checkpoint_w2_16_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_7_1" id="input_checkpoint_w2_16_7_1" aria-describedby="status_checkpoint_w2_16_7_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_7_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_7_1" class="answer"/>
<div id="display_checkpoint_w2_16_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_7_1_dynamath" name="input_checkpoint_w2_16_7_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]g(\theta _ s)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_checkpoint_w2_16_8_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_8_1" id="input_checkpoint_w2_16_8_1" aria-describedby="status_checkpoint_w2_16_8_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_8_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_8_1" class="answer"/>
<div id="display_checkpoint_w2_16_8_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w2_16_8_1_dynamath" name="input_checkpoint_w2_16_8_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part e)</b> Calculate [mathjaxinline]\displaystyle \int _{0}^{2\pi }\hat{r}\; d\theta _ S[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\displaystyle \int _{0}^{2\pi }\hat{r}\; d\theta _ S=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="inputtype_checkpoint_w2_16_9_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w2_16_9_1" id="input_checkpoint_w2_16_9_1" aria-describedby="status_checkpoint_w2_16_9_1" value="" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w2_16_9_1"/>
<span class="status unanswered" id="status_checkpoint_w2_16_9_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w2_16_9_1" class="answer"/>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w2_16_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Electric Field of a uniform ring - Second Approach." />
<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_checkpoint_w2_16" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w2_16">
<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="checkpoint_w2_16-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="checkpoint_w2_16-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="checkpoint_w2_16-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-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-w2d3m1" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="vertical" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<h2 class="hd hd-2 unit-title">Methodology for Applying Coulomb's Law for Continuous Charge Distributions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@fridayw2_m1">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="fc7300f0ea7c11efb8120e0e3c45b88f" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@fridayw2_m1" data-graded="True" data-has-score="False" data-runtime-version="1" data-block-type="html" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-runtime-class="LmsRuntime">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b class="bfseries">Methodology for Applying Coulomb's Law for Continuous Charge Distributions</b></p><p>
In order to calculate the electric field at a point [mathjaxinline]P[/mathjaxinline] created by a continuous charge distribution we must divide the source charge distribution into a number of infinitesimal pieces, with charge [mathjaxinline]dq[/mathjaxinline]. These infinitesimal sources create an electric field [mathjaxinline]d\vec{\mathbf{E}}\left( P \right)[/mathjaxinline]. Then Coulomb's Law becomes </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{\mathbf{E}}\left( P \right) = \int \limits _\text {source} d\vec{\mathbf{E}}\left( P \right) = {1 \over {4 \pi \varepsilon _0}} \int \limits _\text {source} {dq \over r^2} \hat{\mathbf{r}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
where [mathjaxinline]r[/mathjaxinline] is the distance from [mathjaxinline]dq[/mathjaxinline] to [mathjaxinline]P[/mathjaxinline] (where we are finding [mathjaxinline]\vec{\mathbf{E}}\left( P \right)[/mathjaxinline]) and [mathjaxinline]\hat{\mathbf{r}}[/mathjaxinline] is the corresponding unit vector pointing from [mathjaxinline]dq[/mathjaxinline] to [mathjaxinline]P[/mathjaxinline]. In general, use the following steps: </p><ol class="enumerate"><li value="1"><p>
Break your charge distribution into small pieces [mathjaxinline]dq[/mathjaxinline]. </p></li><li value="2"><p>
Write out the appropriate [mathjaxinline]d\vec{\mathbf{E}}\left( P \right)[/mathjaxinline] for the [mathjaxinline]dq[/mathjaxinline]. For example, for infinitesimal source elements we will use </p><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 d\vec{\mathbf{E}}\left( P \right) = {1 \over {4 \pi \varepsilon _0}} {dq \over r^2} \hat{\mathbf{r}} = {1 \over {4 \pi \varepsilon _0}} {dq \over r^3} \vec{\mathbf{r}}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
(typically the latter is more convenient, where [mathjaxinline]\vec{\mathbf{r}}[/mathjaxinline] is the vector from the source [mathjaxinline]dq[/mathjaxinline] to the field point [mathjaxinline]P[/mathjaxinline]). </p></li><li value="3"><p>
Rewrite the source charge element [mathjaxinline]dq[/mathjaxinline] as </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 dq = \left\{ \begin{array}{l}& \lambda \ d\ell & \text {(length)} \\ & \sigma \ dA & \text {(area)} \\ & \rho \ dV & \text {(volume)} \end{array} \right.[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
depending on whether the charge is distributed over a length, an area, or a volume. </p></li><li value="4"><p>
Substitute [mathjaxinline]dq[/mathjaxinline] into the expression for [mathjaxinline]d\vec{\mathbf{E}}\left( P \right)[/mathjaxinline]. </p></li><li value="5"><p>
Specify an appropriate coordinate system (Cartesian, cylindrical or spherical) and express the differential element ([mathjaxinline]d\ell[/mathjaxinline], [mathjaxinline]dA[/mathjaxinline] or [mathjaxinline]dV[/mathjaxinline]) and [mathjaxinline]r[/mathjaxinline] in terms of the coordinates (see Table 2.1 below for a summary). If you need a review of the various coordinate systems, see Review Module on Coordinate Systems. </p><div id="a0000000008" class="table"><div class="caption"><b>Table 2.1</b>: <span>Differential elements of length, area and volume in different coordinates.</span></div><center><table cellspacing="0" class="tabular" style="table-layout:auto"><tr><td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none"> </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
Cartesian [mathjaxinline]\left( x, y, z \right)[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
Cylindrical [mathjaxinline]\left( r, \phi , z \right)[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
Spherical [mathjaxinline]\left( r, \theta , \phi \right)[/mathjaxinline] </td></tr><tr><td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none">
[mathjaxinline]d\ell[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]dx, dy, dz[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]dr, rd\phi , dz[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]dr, r\ d\theta , r \sin {\theta }\ d\phi[/mathjaxinline] </td></tr><tr><td style="border-top-style:solid; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none">
[mathjaxinline]dA[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]dx\ dy, dy\ dz, dz\ dx[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]dr\ dz, r\ d\phi \ dz, r\ d\phi \ dr[/mathjaxinline] </td><td style="border-top-style:solid; text-align:center; border-top-color:black; border-top-width:1px; border-right:1px solid black; border:none">
[mathjaxinline]r\ dr\ d\theta , r\sin {\theta }\ dr\ d\phi , r^2\sin {\theta }\ d\theta \ d\phi[/mathjaxinline] </td></tr><tr><td style="border-top-style:solid; border-bottom-style:solid; border-bottom-width:1px; border-left:1px solid black; border-right:1px solid black; border-top-color:black; border-top-width:1px; border-bottom-color:black; text-align:center; border:none">
[mathjaxinline]dV[/mathjaxinline] </td><td style="border-top-style:solid; border-bottom-style:solid; border-bottom-width:1px; border-bottom-color:black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none">
[mathjaxinline]dx \ dy \ dz[/mathjaxinline] </td><td style="border-top-style:solid; border-bottom-style:solid; border-bottom-width:1px; border-bottom-color:black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none">
[mathjaxinline]r\ dr\ d\phi \ dz[/mathjaxinline] </td><td style="border-top-style:solid; border-bottom-style:solid; border-bottom-width:1px; border-bottom-color:black; border-right:1px solid black; border-top-color:black; border-top-width:1px; text-align:center; border:none">
[mathjaxinline]r^2\sin {\theta }\ dr\ d\theta \ d\phi[/mathjaxinline] </td></tr></table></center></div></li><li value="6"><p>
Rewrite [mathjaxinline]d\vec{\mathbf{E}}\left( P \right)[/mathjaxinline] in terms of the integration variable. Try to apply any symmetry argument to identify non-vanishing component(s) of the electric field. </p></li><li value="7"><p>
Compute the necessary integrals to obtain [mathjaxinline]\vec{\mathbf{E}}\left( P \right)[/mathjaxinline]. </p></li></ol>
</div>
</div>
</div>
</div>
© All Rights Reserved