<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-lesson09-textbooklinks">
<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-lesson09-textbooklinks">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@html-lesson09-textbooklinks">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p> In this lesson we will explore the Electric Potential from continuous sources. To begin, we need to define the Electric Potential as the Potential Energy Difference with a specified reference point. Next we will see how to go from the potential of a point charge to a continuous set of charges. We end with a discussion of configuration energy.</p><p> Textbook Links </p><ul><li><a href="/courses/course-v1:MITx+8.02.1x+1T2019/pdfbook/0/chapter/4/21">Chapter 4.7 Electric Potential from Continuous Sources</a></li></ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v01">
<h2 class="hd hd-2 unit-title">L9v1: Zero Point of the Potential Function</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v01: Zero Point of the Potential Function</h3>
<div
id="video_L09v01"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:pcPhEUKVrLE", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v01/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v01"></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_L09v01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v01">
<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@L09v01/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@L09v01/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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-cp_w3_1">
<h2 class="hd hd-2 unit-title">L9Q1: Motion of a Charged Object in an External Electric Field</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w3_1" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w3_1-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="checkpoint_w3_1-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1-problem-progress" tabindex="-1">
Motion of a Positively Charged Object in an External Electric Field
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_1-problem-progress"></div>
<div class="problem">
<div>
<p>
If a positively charged particle is released from rest in an electric field, the charge will move </p>
<center>
<img src="/assets/courseware/v1/5fe81d7758794b73df94aa95b96eeca0/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w3_1_1.svg" width="440"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w3_1_2_1">
<fieldset aria-describedby="status_checkpoint_w3_1_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w3_1_2_1" id="input_checkpoint_w3_1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w3_1_2_1-choice_1-label" for="input_checkpoint_w3_1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_1_2_1">
<text> from higher to lower electric potential resulting in an increase in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_1_2_1" id="input_checkpoint_w3_1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w3_1_2_1-choice_2-label" for="input_checkpoint_w3_1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_1_2_1">
<text> from higher to lower electric potential resulting in a decrease in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_1_2_1" id="input_checkpoint_w3_1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w3_1_2_1-choice_3-label" for="input_checkpoint_w3_1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_1_2_1">
<text> from lower to higher electric potential resulting in an increase in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_1_2_1" id="input_checkpoint_w3_1_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w3_1_2_1-choice_4-label" for="input_checkpoint_w3_1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_1_2_1">
<text> from lower to higher electric potential resulting in a decrease in potential energy.</text>
</label>
</div>
<span id="answer_checkpoint_w3_1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w3_1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w3_1_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Motion of a Positively Charged Object in an External Electric Field" />
<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_w3_1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w3_1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<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_w3_1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="checkpoint_w3_1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="checkpoint_w3_1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w3_2" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w3_2-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2/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_w3_2-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2-problem-progress" tabindex="-1">
Motion of a Negatively Charged Object in an External Electric Field
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w3_2-problem-progress"></div>
<div class="problem">
<div>
<p>
If a negatively charged particle is released from rest in an electric field, the charge will move </p>
<center>
<img src="/assets/courseware/v1/fc811b1a226b20174bc3143e76b52173/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w3_2_1.svg" width="440"/>
</center>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w3_2_2_1">
<fieldset aria-describedby="status_checkpoint_w3_2_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w3_2_2_1" id="input_checkpoint_w3_2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w3_2_2_1-choice_1-label" for="input_checkpoint_w3_2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_2_2_1">
<text> from higher to lower electric potential resulting in an increase in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_2_2_1" id="input_checkpoint_w3_2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w3_2_2_1-choice_2-label" for="input_checkpoint_w3_2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_2_2_1">
<text> from higher to lower electric potential resulting in a decrease in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_2_2_1" id="input_checkpoint_w3_2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w3_2_2_1-choice_3-label" for="input_checkpoint_w3_2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_2_2_1">
<text> from lower to higher electric potential resulting in an increase in potential energy.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w3_2_2_1" id="input_checkpoint_w3_2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w3_2_2_1-choice_4-label" for="input_checkpoint_w3_2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w3_2_2_1">
<text> from lower to higher electric potential resulting in a decrease in potential energy.</text>
</label>
</div>
<span id="answer_checkpoint_w3_2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w3_2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w3_2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Motion of a Negatively Charged Object in an External Electric Field" />
<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_w3_2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w3_2">
<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_w3_2-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_w3_2-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_w3_2-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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v02">
<h2 class="hd hd-2 unit-title">L9v2: Superposition of Potential</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v02: Superposition of Potential</h3>
<div
id="video_L09v02"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:Vi_rArTuoGU", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v02/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v02"></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_L09v02">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v02">
<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@L09v02/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@L09v02/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@html+block@lesson08_01">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@html+block@lesson08_01">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><center><b><font size="4">Electric potential difference of a point charge. </font></b></center></p><p><center><img src="/assets/courseware/v1/387a89307984124eb37cfc44dde98077/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_html_lesson08_01a.svg" width="400"/></center></p><p>
The change of electric potential produced by a point charge \(Q\) between \(\infty\) and point \(P\) is given by:
</p><p>
\[V(P)-V(\infty) = k\frac{Q}{r}\]
</p><p>
where \(r \) is the magnitude of \(\vec{r}\), the position vector of point \(P\) with respect to the charge \(Q\).
</p><p><center><b><font size="4">Electric potential difference of a set of point charges. </font></b></center></p><p>
Consider a set of fout point charges.
</p><p><center><img src="/assets/courseware/v1/6602ab1b2ce015c7317820a7cf320cc6/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_html_lesson08_01.svg" width="400"/></center></p><p>
The change of electric potential produced by the set of point charges between \(\infty\) and point \(P\) is given by:
</p><p>
\[V(P)-V(\infty) = -\int_{\infty}^{P}\vec{E}\cdot d\vec{s}\]
</p><p>
where \(\vec{E} \) is the electric field produced by the four charges. Using superposition principle, \(\vec{E} = \vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4 \), therefore the change of electric potential becomes:
</p><p>
\[V(P)-V(\infty)= -\int_{\infty}^{P}\Big( \vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4\Big)\cdot d\vec{s}
\]
</p><p>
\[V(P)-V(\infty)= -\int_{\infty}^{P} \vec{E}_1 \cdot d\vec{s}-\int_{\infty}^{P}\vec{E}_2 \cdot d\vec{s}-\int_{\infty}^{P}\vec{E}_3 \cdot d\vec{s} -\int_{\infty}^{P}\vec{E}_4 \cdot d\vec{s}
\]
</p><p> Each of the line integral above correspond to the electric potential difference between \(\infty\) and point \(P\) of the individual charges, therfore:
</p><p>
\[V(P)-V(\infty)= k\frac{q_1}{r_1} + k\frac{q_2}{r_2}+k\frac{q_3}{r_3}+k\frac{q_4}{r_4}
\]
</p><p> where \(r_i\) is the distance between charge \(q_i\) and point \(P\)
</p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-fps_fridayw3_1a">
<h2 class="hd hd-2 unit-title">L9Q2: Charges in a Square</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fps_fridayw3_1a" class="problems-wrapper" role="group"
aria-labelledby="fps_fridayw3_1a-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a/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="fps_fridayw3_1a-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a-problem-progress" tabindex="-1">
Charges on a Square
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1a-problem-progress"></div>
<div class="problem">
<div>
<p>
Three identical point-like charged objects each of charge [mathjaxinline]+Q[/mathjaxinline] are fixed on the corners of a square of side [mathjaxinline]a[/mathjaxinline], as shown in the figure. </p>
<center>
<img src="/assets/courseware/v1/00c24fa76f32542e1ab1ddf3e8c01e21/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_fridayw3_1_charges_on_a_square.svg" width="330"/>
</center>
<p>
What is the electric potential difference [mathjaxinline]V(P)-V(\infty )[/mathjaxinline]? Write your answers in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]Q[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]V(P)-V(\infty )=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fps_fridayw3_1a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fps_fridayw3_1a_2_1" id="input_fps_fridayw3_1a_2_1" aria-describedby="status_fps_fridayw3_1a_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_fps_fridayw3_1a_2_1"/>
<span class="status unanswered" id="status_fps_fridayw3_1a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fps_fridayw3_1a_2_1" class="answer"/>
<div id="display_fps_fridayw3_1a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fps_fridayw3_1a_2_1_dynamath" name="input_fps_fridayw3_1a_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fps_fridayw3_1a_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charges on a Square" />
<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_fps_fridayw3_1a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fps_fridayw3_1a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<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="fps_fridayw3_1a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fps_fridayw3_1a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fps_fridayw3_1a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-cp_w4_4">
<h2 class="hd hd-2 unit-title">L9Q3: Two Point Charges</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w4_4" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w4_4-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4/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_w4_4-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4-problem-progress" tabindex="-1">
Two Point Charges
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_4-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/cb620c92f306c77e2ce70609f3301525/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w4_4_1.svg" width="400" style="width:"/>
</center>
<p>
The work done by the electrostatic force on a particle of charge [mathjaxinline]q&gt;0[/mathjaxinline] as it moves from infinity to point [mathjaxinline]P[/mathjaxinline] midway between two fixed charges of magnitude [mathjaxinline]+Q[/mathjaxinline] and [mathjaxinline]-Q[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w4_4_2_1">
<fieldset aria-describedby="status_checkpoint_w4_4_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w4_4_2_1" id="input_checkpoint_w4_4_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w4_4_2_1-choice_1-label" for="input_checkpoint_w4_4_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w4_4_2_1">
<text> is positive.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w4_4_2_1" id="input_checkpoint_w4_4_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w4_4_2_1-choice_2-label" for="input_checkpoint_w4_4_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w4_4_2_1">
<text> is negative.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w4_4_2_1" id="input_checkpoint_w4_4_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w4_4_2_1-choice_3-label" for="input_checkpoint_w4_4_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w4_4_2_1">
<text> is zero.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w4_4_2_1" id="input_checkpoint_w4_4_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w4_4_2_1-choice_4-label" for="input_checkpoint_w4_4_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w4_4_2_1">
<text> can not be determined - not enough information is given.</text>
</label>
</div>
<span id="answer_checkpoint_w4_4_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w4_4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_4_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Two Point Charges" />
<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_w4_4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w4_4">
<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_w4_4-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_w4_4-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_w4_4-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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v03">
<h2 class="hd hd-2 unit-title">L9v3: Potential for an Arbitrary Charge Source</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v03: Potential for an Arbitrary Charge Source</h3>
<div
id="video_L09v03"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:ldIKbzl7TGM", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v03/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v03"></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_L09v03">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v03">
<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@L09v03/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@L09v03/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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v04">
<h2 class="hd hd-2 unit-title">L9v4: Potential Along the Axis 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@video+block@L09v04">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v04">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v04: Potential Along the Axis of a Uniformly Charged Ring</h3>
<div
id="video_L09v04"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:nuJ0WqRSCkA", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v04/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v04/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v04/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v04/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v04"></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_L09v04">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v04">
<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@L09v04/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@L09v04/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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-checkpoint_w4_16">
<h2 class="hd hd-2 unit-title">L9Q4: Semicircular Arc</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w4_16" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w4_16-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="checkpoint_w4_16-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16-problem-progress" tabindex="-1">
Semicircular Arc
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_16-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/26b12ae57557e0e145c450cb8a838e8c/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w4_16.svg" width="330"/>
</center>
<p>
A semicircular arc of radius [mathjaxinline]R[/mathjaxinline] carries a total charge [mathjaxinline]Q[/mathjaxinline] uniformly distributed over its length. Set the reference of electric potential to be zero at infinity, [mathjaxinline]V(\infty )=0[/mathjaxinline]. The electric potential produced by the arc at point [mathjaxinline]P[/mathjaxinline] can be obtained 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 V = \int _{arc}\frac{kdq}{|\vec{r}_ P-\vec{r}_ s|}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]dq[/mathjaxinline] is a small element of charge in the arc, shown schematically as the black band, [mathjaxinline]\vec{r}_ s[/mathjaxinline] is the position of the element of charge [mathjaxinline]dq[/mathjaxinline] that is the source of the electric potential, and [mathjaxinline]\vec{r}_ P[/mathjaxinline] is the position of point [mathjaxinline]P[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Write an expression for [mathjaxinline]|\vec{r}_ P-\vec{r}_ s|[/mathjaxinline]. Express your answer in terms of [mathjaxinline]R[/mathjaxinline] and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{r}_ P-\vec{r}_ s|=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w4_16_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_16_2_1" id="input_checkpoint_w4_16_2_1" aria-describedby="status_checkpoint_w4_16_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_16_2_1"/>
<span class="status unanswered" id="status_checkpoint_w4_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_w4_16_2_1" class="answer"/>
<div id="display_checkpoint_w4_16_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_16_2_1_dynamath" name="input_checkpoint_w4_16_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_16_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Find an expression for [mathjaxinline]dq[/mathjaxinline] in terms of the variables [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and d_theta for [mathjaxinline]d\theta[/mathjaxinline], where [mathjaxinline]d\theta[/mathjaxinline] is the angular width of the element of charge [mathjaxinline]dq[/mathjaxinline], and pi for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]dq=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w4_16_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_16_3_1" id="input_checkpoint_w4_16_3_1" aria-describedby="status_checkpoint_w4_16_3_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_16_3_1"/>
<span class="status unanswered" id="status_checkpoint_w4_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_w4_16_3_1" class="answer"/>
<div id="display_checkpoint_w4_16_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_16_3_1_dynamath" name="input_checkpoint_w4_16_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_16_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Set up and solve the integral to calculate [mathjaxinline]V(P)[/mathjaxinline], the electric potential produced by the arc at point [mathjaxinline]P[/mathjaxinline]. Express your answer in terms of [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], and pi for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V(P)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w4_16_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_16_4_1" id="input_checkpoint_w4_16_4_1" aria-describedby="status_checkpoint_w4_16_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_16_4_1"/>
<span class="status unanswered" id="status_checkpoint_w4_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_w4_16_4_1" class="answer"/>
<div id="display_checkpoint_w4_16_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_16_4_1_dynamath" name="input_checkpoint_w4_16_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_16_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Semicircular Arc" />
<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_w4_16" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w4_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">
<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_w4_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_w4_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_w4_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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v05">
<h2 class="hd hd-2 unit-title">L9v5: Potential Near a Uniformly Charged Rod</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v05: Potential Near a Uniformly Charged Rod</h3>
<div
id="video_L09v05"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:aoAaMVpF1NM", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v05/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v05"></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_L09v05">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v05">
<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@L09v05/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@L09v05/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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-l09q05">
<h2 class="hd hd-2 unit-title">L9Q5: Potential Above a Charged Rod</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w4_15" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w4_15-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="checkpoint_w4_15-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15-problem-progress" tabindex="-1">
Potential above a charged rod.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@checkpoint_w4_15-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/123ba7c9875b57803d12cec282137e5a/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_checkpoint_w4_15.svg" width="770"/>
</center>
<p>
A rod of length [mathjaxinline]L[/mathjaxinline] is uniformly charged with a total charge [mathjaxinline]Q[/mathjaxinline]. Consider the reference of the electric potential to be zero at infinity: [mathjaxinline]V(\infty ) = 0[/mathjaxinline]. The electric potential at a point [mathjaxinline]P[/mathjaxinline] in the y-axis can be calculated 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 V = \int _{rod}\frac{kdq}{|\vec{r}_ P-\vec{r}_ s|}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]dq[/mathjaxinline] is a small element of charge in the rod, shown schematically as the black band, [mathjaxinline]\vec{r}_ s[/mathjaxinline] is the position of the element of charge [mathjaxinline]dq[/mathjaxinline] that is the source of the electric potential, and [mathjaxinline]\vec{r}_ P[/mathjaxinline] is the position of point [mathjaxinline]P[/mathjaxinline]. </p>
<p>
The goal of this exercise is to calculate the electric potential produce by the rod at point [mathjaxinline]P[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Consider the coordinate system shown in the figure with [mathjaxinline]x=0[/mathjaxinline] at the center of the rod. Write an expression for [mathjaxinline]|\vec{r}_ P-\vec{r}_ s|[/mathjaxinline] using the variables shown in the figure. Express your answer in terms of [mathjaxinline]y[/mathjaxinline], and x_s for [mathjaxinline]x_ s[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{r}_ P-\vec{r}_ s|=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w4_15_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_15_2_1" id="input_checkpoint_w4_15_2_1" aria-describedby="status_checkpoint_w4_15_2_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_15_2_1"/>
<span class="status unanswered" id="status_checkpoint_w4_15_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w4_15_2_1" class="answer"/>
<div id="display_checkpoint_w4_15_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_15_2_1_dynamath" name="input_checkpoint_w4_15_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_15_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Find an expression for [mathjaxinline]dq[/mathjaxinline] in terms of the variables [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and dx_s for [mathjaxinline]dx_ s[/mathjaxinline], where [mathjaxinline]dx_ s[/mathjaxinline] is the width of the element of charge [mathjaxinline]dq[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]dq=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w4_15_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_15_3_1" id="input_checkpoint_w4_15_3_1" aria-describedby="status_checkpoint_w4_15_3_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_15_3_1"/>
<span class="status unanswered" id="status_checkpoint_w4_15_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w4_15_3_1" class="answer"/>
<div id="display_checkpoint_w4_15_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_15_3_1_dynamath" name="input_checkpoint_w4_15_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_15_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Set up and solve the integral to calculate [mathjaxinline]V(P)[/mathjaxinline], the electric potential produced by the rod at point [mathjaxinline]P[/mathjaxinline]. Express your answer in terms of [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]k[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]V(P)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w4_15_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w4_15_4_1" id="input_checkpoint_w4_15_4_1" aria-describedby="status_checkpoint_w4_15_4_1" value="" class="math" size="35"/>
<span class="trailing_text" id="trailing_text_checkpoint_w4_15_4_1"/>
<span class="status unanswered" id="status_checkpoint_w4_15_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w4_15_4_1" class="answer"/>
<div id="display_checkpoint_w4_15_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w4_15_4_1_dynamath" name="input_checkpoint_w4_15_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w4_15_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Potential above a 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_w4_15" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w4_15">
<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_w4_15-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_w4_15-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_w4_15-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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-L09v06">
<h2 class="hd hd-2 unit-title">L9v6: Configuration Energy</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L09v06: Configuration Energy</h3>
<div
id="video_L09v06"
class="video closed"
data-metadata='{"lmsRootURL": "https://openlearninglibrary.mit.edu", "poster": null, "transcriptLanguages": {"en": "English"}, "showCaptions": "true", "ytMetadataEndpoint": "", "completionEnabled": false, "streams": "1.00:9ASrInTPFxo", "saveStateUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06/handler/xmodule_handler/save_user_state", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06/handler/transcript/translation/__lang__", "savedVideoPosition": 0.0, "end": 0.0, "captionDataDir": null, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06/handler/publish_completion", "generalSpeed": 1.0, "autoAdvance": false, "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@video+block@L09v06/handler/transcript/available_translations", "start": 0.0, "duration": 0.0, "prioritizeHls": false, "autohideHtml5": false, "saveStateEnabled": false, "autoplay": false, "transcriptLanguage": "en", "completionPercentage": 0.95, "recordedYoutubeIsAvailable": true, "speed": null, "sources": [], "ytTestTimeout": 1500, "ytApiUrl": "https://www.youtube.com/iframe_api"}'
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="L09v06"></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_L09v06">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L09v06">
<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@L09v06/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@L09v06/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-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@vertical+block@vert-fps_fridayw3_1b">
<h2 class="hd hd-2 unit-title">L9Q6: Charges in a Square Again</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="fe101536ea6a11efbdc212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+8.02.1x+1T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fps_fridayw3_1b" class="problems-wrapper" role="group"
aria-labelledby="fps_fridayw3_1b-problem-title"
data-problem-id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b" data-url="/courses/course-v1:MITx+8.02.1x+1T2019/xblock/block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fps_fridayw3_1b-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b-problem-progress" tabindex="-1">
Charges on a Square again.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw3_1b-problem-progress"></div>
<div class="problem">
<div>
<p>
Three identical point-like charged objects each of charge [mathjaxinline]+Q[/mathjaxinline] are fixed on the corners of a square of side [mathjaxinline]a[/mathjaxinline], as shown in the figure. </p>
<p>
For these questions, please write your answers in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]Q[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline]. </p>
<center>
<img src="/assets/courseware/v1/00c24fa76f32542e1ab1ddf3e8c01e21/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_fridayw3_1_charges_on_a_square.svg" width="330"/>
</center>
<p><b class="bfseries">(Part a)</b> How much energy did it take to assemble the pictured configuration of three charged objects assuming that the charges started very far apart? </p>
<p>
<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fps_fridayw3_1b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fps_fridayw3_1b_2_1" id="input_fps_fridayw3_1b_2_1" aria-describedby="status_fps_fridayw3_1b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_fps_fridayw3_1b_2_1"/>
<span class="status unanswered" id="status_fps_fridayw3_1b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fps_fridayw3_1b_2_1" class="answer"/>
<div id="display_fps_fridayw3_1b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fps_fridayw3_1b_2_1_dynamath" name="input_fps_fridayw3_1b_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_fps_fridayw3_1b_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]\Delta U[/mathjaxinline], the change in potential energy of the system if a fourth charged object with charge [mathjaxinline]-2Q[/mathjaxinline] is brought in from infinity and placed at the point [mathjaxinline]P[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta U=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_fps_fridayw3_1b_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fps_fridayw3_1b_3_1" id="input_fps_fridayw3_1b_3_1" aria-describedby="status_fps_fridayw3_1b_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_fps_fridayw3_1b_3_1"/>
<span class="status unanswered" id="status_fps_fridayw3_1b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fps_fridayw3_1b_3_1" class="answer"/>
<div id="display_fps_fridayw3_1b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fps_fridayw3_1b_3_1_dynamath" name="input_fps_fridayw3_1b_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fps_fridayw3_1b_solution_2"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charges on a Square again." />
<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_fps_fridayw3_1b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fps_fridayw3_1b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
<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="fps_fridayw3_1b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fps_fridayw3_1b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fps_fridayw3_1b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved