<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-html-lesson26intro" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Introduction to Inductance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson26_00">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson26_00" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b>Inductance</b></p><p> So far we have seen how time changing magnetic fields produce closed-loops of Electric fields that can drive currents to oppose the initial change in the magnetic field. We now find a way to characterize exactly how much of an emf will be created in a loop to drive current because of a time changing magnetic field. We will explore the relationship between the time changing current that procuces a time changing magnetic field and the driving emf that induces, and this proportionality constant will be call the <b>inductance</b>. This is a quantity, much like capacitance, that will only depend on geometry and not on the current.</p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_c1a21947395c">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_c1a21947395c" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>Textbook Links</p>
<ul>
<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/pdfbook/0/#viewer-frame" target="[object Object]">Chapter 11.1-11.2: Mutual and Self Inductance </a></li>
</ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L26v01" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v1: Mutual Inductance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v1: Mutual Inductance</h3>
<div
id="video_L26v01"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:6Zps5QlgjTE", "saveStateEnabled": false, "end": 0.0, "speed": null, "completionPercentage": 0.95, "start": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01/handler/publish_completion", "duration": null, "autoplay": false, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "autohideHtml5": false, "ytMetadataEndpoint": "", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01/handler/transcript/translation/__lang__", "showCaptions": "true", "completionEnabled": false, "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "saveStateUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v01/handler/transcript/available_translations", "sources": [], "transcriptLanguages": {"en": "English"}, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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="L26v01"></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_L26v01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v01">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/477976f045be2bef4105c991f3aa4473/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L26v01.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L26v02" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v2: Worked Example - Mutual Inductance of a Loop near a Wire</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="True" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w10_14" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w10_14-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14/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_w10_14-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14-problem-progress" tabindex="-1">
Worked Example - Mutual Inductance of a loop near a wire
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_14-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/17f7b290438764b3a9578dcc634f829a/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_14.svg" width="330"/>
</center>
<p>
Consider two conductors: conductor 1 is an infinite wire, and conductor 2 is a rectangular loop of length [mathjaxinline]h[/mathjaxinline] and width [mathjaxinline]w[/mathjaxinline]. A current [mathjaxinline]I_1[/mathjaxinline] flows in the infinite wire. The loop is placed parallel to the wire with one of its sides a distance [mathjaxinline]s[/mathjaxinline] from the wire as shown. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]B_1[/mathjaxinline], the magnitude of the magnetic field produced by the current [mathjaxinline]I_1[/mathjaxinline] in the wire at a given distance [mathjaxinline]r[/mathjaxinline] from the wire. Express your answer in terms of [mathjaxinline]r[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]B_1 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w10_14_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_14_2_1" id="input_checkpoint_w10_14_2_1" aria-describedby="status_checkpoint_w10_14_2_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_14_2_1"/>
<span class="status unanswered" id="status_checkpoint_w10_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_w10_14_2_1" class="answer"/>
<div id="display_checkpoint_w10_14_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_14_2_1_dynamath" name="input_checkpoint_w10_14_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]\Phi _{12}[/mathjaxinline], the magnitude of the magnetic flux of [mathjaxinline]\vec{B}_1[/mathjaxinline] (conductor 1) through the rectangular loop (conductor 2). Express your answer in terms of [mathjaxinline]h[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]w[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|\Phi _{12}| =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w10_14_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_14_3_1" id="input_checkpoint_w10_14_3_1" aria-describedby="status_checkpoint_w10_14_3_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_14_3_1"/>
<span class="status unanswered" id="status_checkpoint_w10_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_w10_14_3_1" class="answer"/>
<div id="display_checkpoint_w10_14_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_14_3_1_dynamath" name="input_checkpoint_w10_14_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]M_{12}[/mathjaxinline], the mutual inductance between the loop and the infinite wire. Express your answer in terms of [mathjaxinline]h[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]w[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]M_{12} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w10_14_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_14_4_1" id="input_checkpoint_w10_14_4_1" aria-describedby="status_checkpoint_w10_14_4_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_14_4_1"/>
<span class="status unanswered" id="status_checkpoint_w10_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_w10_14_4_1" class="answer"/>
<div id="display_checkpoint_w10_14_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_14_4_1_dynamath" name="input_checkpoint_w10_14_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w10_14_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Worked Example - Mutual Inductance of a loop near a wire" />
<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_w10_14" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w10_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">
</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_w10_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_w10_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_w10_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 class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v2: Worked Example - Mutual Inductance of a Loop near a Wire</h3>
<div
id="video_L26v02"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:erhdPQYSsk4", "saveStateEnabled": false, "end": 0.0, "speed": null, "completionPercentage": 0.95, "start": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02/handler/publish_completion", "duration": null, "autoplay": false, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "autohideHtml5": false, "ytMetadataEndpoint": "", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02/handler/transcript/translation/__lang__", "showCaptions": "true", "completionEnabled": false, "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "saveStateUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v02/handler/transcript/available_translations", "sources": [], "transcriptLanguages": {"en": "English"}, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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="L26v02"></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_L26v02">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v02">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/185ba59824ed58d95d11bd9342124a94/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L26v02.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L26v03" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v3: Worked Example - Mutual Inductance of a Small Ring Inside a Big Ring</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="True" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w10_17" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w10_17-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17/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_w10_17-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17-problem-progress" tabindex="-1">
Worked Example - Mutual Inductance of a ring inside a ring
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_17-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/b98525c74666557da7041637f5637786/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_17.svg" width="440"/>
</center>
<p>
Consider two conductors: conductor 1 is a ring of radius [mathjaxinline]R_1[/mathjaxinline], and conductor 2 is a smaller ring of radius [mathjaxinline]R_2[/mathjaxinline]. A current [mathjaxinline]I_1[/mathjaxinline] flows in the larger ring. The smaller ring is placed at the center of the larger ring as shown. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]B_1[/mathjaxinline], the magnitude of the magnetic field produced by the current [mathjaxinline]I_1[/mathjaxinline] at the center of the larger ring. Express your answer in terms of R_1 for [mathjaxinline]R_1[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]B_1 =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w10_17_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_17_2_1" id="input_checkpoint_w10_17_2_1" aria-describedby="status_checkpoint_w10_17_2_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_17_2_1"/>
<span class="status unanswered" id="status_checkpoint_w10_17_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w10_17_2_1" class="answer"/>
<div id="display_checkpoint_w10_17_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_17_2_1_dynamath" name="input_checkpoint_w10_17_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Consider [mathjaxinline]R_2&lt;&lt;R_1[/mathjaxinline]. In this approximation, the area enclosed by the inner ring is very small, and therefore we can assume that the magnetic field produced by [mathjaxinline]I_1[/mathjaxinline] is approximately uniform at any point inside the smaller ring. Use this approximation to calculate [mathjaxinline]\Phi _{12}[/mathjaxinline], the magnetic flux due to [mathjaxinline]\vec{B}_1[/mathjaxinline] (conductor 1) through the smaller ring (conductor 2). Express your answer in terms of R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Phi _{12} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w10_17_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_17_3_1" id="input_checkpoint_w10_17_3_1" aria-describedby="status_checkpoint_w10_17_3_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_17_3_1"/>
<span class="status unanswered" id="status_checkpoint_w10_17_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w10_17_3_1" class="answer"/>
<div id="display_checkpoint_w10_17_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_17_3_1_dynamath" name="input_checkpoint_w10_17_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]M_{12}[/mathjaxinline], the mutual inductance between the rings. Express your answer in terms of R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], I_1 for [mathjaxinline]I_1[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]M_{12} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w10_17_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_17_4_1" id="input_checkpoint_w10_17_4_1" aria-describedby="status_checkpoint_w10_17_4_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_17_4_1"/>
<span class="status unanswered" id="status_checkpoint_w10_17_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w10_17_4_1" class="answer"/>
<div id="display_checkpoint_w10_17_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_17_4_1_dynamath" name="input_checkpoint_w10_17_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w10_17_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Worked Example - Mutual Inductance of a ring inside a 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_w10_17" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w10_17">
<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_w10_17-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_w10_17-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_w10_17-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+8.02.3x+1T2019+type@video+block@L26v03">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v03" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v3: Worked Example - Mutual Inductance of a Small Ring Inside a Big Ring</h3>
<div
id="video_L26v03"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:0WNQ47fF5As", "saveStateEnabled": false, "end": 0.0, "speed": null, "completionPercentage": 0.95, "start": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v03/handler/publish_completion", "duration": null, "autoplay": false, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "autohideHtml5": false, "ytMetadataEndpoint": "", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v03/handler/transcript/translation/__lang__", "showCaptions": "true", "completionEnabled": false, "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "saveStateUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v03/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v03/handler/transcript/available_translations", "sources": [], "transcriptLanguages": {"en": "English"}, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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="L26v03"></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_L26v03">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v03">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/2ed54f3d78bfa96beb27bf5166abac57/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L26v03.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-onlinepset9_8" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q1: Inductive Coupling</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="True" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w10_15" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w10_15-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15/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="checkpoint_w10_15-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15-problem-progress" tabindex="-1">
Inductive coupling
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_15-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/6261da9853a355e200dfc5d64fd595bb/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_15.svg" width="770"/>
</center>
<p>
The picture shows side views of a solenoid on the left and a small coil on the right, in three orientations labeled (1), (2), and (3). </p>
<p><b class="bfseries">(Part a)</b> If the coil is placed in the center of the solenoid (inside the middle of the solenoid), which of the orientations will produce the least mutual inductance between the coil and the solenoid? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_15_2_1">
<fieldset aria-describedby="status_checkpoint_w10_15_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_2_1" id="input_checkpoint_w10_15_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w10_15_2_1-choice_1-label" for="input_checkpoint_w10_15_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_2_1">
<text> (1)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_2_1" id="input_checkpoint_w10_15_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w10_15_2_1-choice_2-label" for="input_checkpoint_w10_15_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_2_1">
<text> (2)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_2_1" id="input_checkpoint_w10_15_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w10_15_2_1-choice_3-label" for="input_checkpoint_w10_15_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_2_1">
<text> (3)</text>
</label>
</div>
<span id="answer_checkpoint_w10_15_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w10_15_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> The coil is placed inside the middle of the solenoid and any one of the axes passing through the center of the coil is choosen as an axis to spin the coil around. Of the three shown above, which starting orientation will give the largest average induced emf? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_15_3_1">
<fieldset aria-describedby="status_checkpoint_w10_15_3_1">
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_3_1" id="input_checkpoint_w10_15_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w10_15_3_1-choice_1-label" for="input_checkpoint_w10_15_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_3_1">
<text> (1)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_3_1" id="input_checkpoint_w10_15_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w10_15_3_1-choice_2-label" for="input_checkpoint_w10_15_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_3_1">
<text> (2)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_3_1" id="input_checkpoint_w10_15_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w10_15_3_1-choice_3-label" for="input_checkpoint_w10_15_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_3_1">
<text> (3)</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_3_1" id="input_checkpoint_w10_15_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w10_15_3_1-choice_4-label" for="input_checkpoint_w10_15_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_3_1">
<text> depends on the orientation of the axis that you choose</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w10_15_3_1" id="input_checkpoint_w10_15_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w10_15_3_1-choice_5-label" for="input_checkpoint_w10_15_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_15_3_1">
<text> all the same</text>
</label>
</div>
<span id="answer_checkpoint_w10_15_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w10_15_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>
<div class="solution-span">
<span id="solution_checkpoint_w10_15_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Inductive coupling" />
<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_w10_15" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w10_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">
</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_w10_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_w10_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_w10_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-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-html-lesson24_01" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Mutual Inductance Depends on Geometry</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson26_01">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson26_01" data-init="XBlockToXModuleShim" data-block-type="html" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p><b>Mutual inductance depends on the geometry</b></p><p> Consider two conductors close to each other. For example, the pair of conductors presented in the previous worked examples and shown in the table below:</p><table style="width:100%"><tr><td align="center"><b>Infinite wire and rectangular loop</b></td><td align="center"><b>Small ring inside a big ring</b></td></tr><tr><td align="center"><img src="/assets/courseware/v1/fdad29a7081ab7370d1e6cd5678899f2/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_html_lesson_24_01a.svg"/></td><td align="center"><img src="/assets/courseware/v1/1917965447de4c66301f7bcb7031ac1b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_html_lesson_24_01_b.svg"/></td></tr><tr><td>\[\displaystyle M_{12}=\mu_0\frac{h}{2\pi}\ln(\frac{s+w}{s})\]</td><td>\[\displaystyle M_{12}=\mu_0\frac{\pi R_2^2}{2R_1}\]</td></tr></table><p>
In both examples, when a current \(I_1\) flows in conductor 1, that current will create a magnetic field \(\vec{B}_1\), and thus a magnetic flux through conductor 2, \(\Phi_{12}\). Because the magnetic field is always directly proportional to the current, the magnetic flux is also proportional to the current, therefore
\[\Phi_{12}=M_{12}I_1\]
where \(M_{12}\) is the coefficient of proportionality and is called the mutual inductance. The mutual inductance obtained for the examples in the figure are listed in the last row of the table. We remark that they are independent of the current producing the magnetic field. <p>The mutual inductance depends on the geometry of the conductors.</p>
</p><p> In a similar way, if a current \(I_2\) is flowing in conductor 2 the magnetic field lines produced by \(I_2\) will create a flux on conductor 1, \(\Phi_{21}\) proportional to current \(I_2\):
\[\Phi_{21}=M_{21}I_2\]
The constant of proportionality \(M_{21}\) is the mutual inductance. It can be shown that \(M_{12}=M_{21}\). Note that in both of the cases shown above, explicitly calculating \(M_{21}\) would be quite complicated so this property that the mutual inductance between two objects is independent of which one you start with can be quite useful.
</p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L26v04" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v4: Self Inductance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v4: Self Inductance</h3>
<div
id="video_L26v04"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:YiojaFpJR-U", "saveStateEnabled": false, "end": 0.0, "speed": null, "completionPercentage": 0.95, "start": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04/handler/publish_completion", "duration": null, "autoplay": false, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "autohideHtml5": false, "ytMetadataEndpoint": "", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04/handler/transcript/translation/__lang__", "showCaptions": "true", "completionEnabled": false, "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "saveStateUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v04/handler/transcript/available_translations", "sources": [], "transcriptLanguages": {"en": "English"}, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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="L26v04"></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_L26v04">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v04">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/3f81f6ab8d7bdb1af245f41d6051e55b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L26v04.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-onlinepset9_3" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26Q2: Self Inductance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="True" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w10_16" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w10_16-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16/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_w10_16-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16-problem-progress" tabindex="-1">
Self Inductance
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_16-problem-progress"></div>
<div class="problem">
<div>
<p>
The current flowing in a circuit produces a magnetic field. Some of the magnetic field lines created by that current pass through the area enclosed by the circuit loop. If the current in the circuit changes with time, the magnetic field and its flux through the loop change with time inducing an emf in the circuit. This effect is called &#8220;self-inductance" or simply &#8220;inductance" (as opposed to mutual inductance where the current in one object produces flux through a second object). </p>
<center>
<img src="/assets/courseware/v1/e29dba98b67bb391046bc8fabc2e1dcd/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_16.svg" width="330"/>
</center>
<p>
The induced emf in the circuit is given by: </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 \mathcal{E}_{ind} = -\frac{d\Phi _{mag}}{dt} = -L \frac{dI}{dt}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
where we have used the subscript "[mathjaxinline]ind[/mathjaxinline]" to differentiate the induced emf predicted by Faraday's law, [mathjaxinline]\mathcal{E}_{ind}[/mathjaxinline], from the emf of the battery feeding the circuit. </p>
<p>
The rate of change of the flux is directly proportional to the rate of change of the current that is producing the flux. The constant of proportionality [mathjaxinline]L[/mathjaxinline] is called the "self inductance" or "inductance [mathjaxinline]L[/mathjaxinline] of the circuit". This coefficient depends on the geometry of the circuit through which the current is flowing. </p>
<p>
Which of the following statements is (are) true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_16_2_1">
<fieldset aria-describedby="status_checkpoint_w10_16_2_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w10_16_2_1[]" id="input_checkpoint_w10_16_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w10_16_2_1-choice_0-label" for="input_checkpoint_w10_16_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_16_2_1">
<text>(1) the SI unit of inductance is the Henry.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w10_16_2_1[]" id="input_checkpoint_w10_16_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w10_16_2_1-choice_1-label" for="input_checkpoint_w10_16_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_16_2_1">
<text>(2) in SI units 1 Henry (abbreviated H) = 1V[mathjaxinline]\cdot[/mathjaxinline]s[mathjaxinline]\cdot[/mathjaxinline]A[mathjaxinline]^{-1}[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w10_16_2_1[]" id="input_checkpoint_w10_16_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w10_16_2_1-choice_2-label" for="input_checkpoint_w10_16_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_16_2_1">
<text>(3) in SI units 1 H = 1 V[mathjaxinline]\cdot[/mathjaxinline]s[mathjaxinline]^{2}[/mathjaxinline].C[mathjaxinline]^{-1}[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w10_16_2_1[]" id="input_checkpoint_w10_16_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w10_16_2_1-choice_3-label" for="input_checkpoint_w10_16_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_16_2_1">
<text>(4) in SI units 1 H = 1 J[mathjaxinline]\cdot[/mathjaxinline]A[mathjaxinline]^{-2}[/mathjaxinline].</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w10_16_2_1[]" id="input_checkpoint_w10_16_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="checkpoint_w10_16_2_1-choice_4-label" for="input_checkpoint_w10_16_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_16_2_1">
<text>(5) in SI units 1 H = 1 [mathjaxinline]\Omega[/mathjaxinline][mathjaxinline]\cdot[/mathjaxinline]s[mathjaxinline]\cdot[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w10_16_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w10_16_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_w10_16_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Self Inductance" />
<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_w10_16" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w10_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_w10_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_w10_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_w10_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="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L26v05" data-init="VerticalStudentView" data-block-type="vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L26v5: Worked Example - Inductance of Solenoid</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18" data-init="XBlockToXModuleShim" data-block-type="problem" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="True" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_checkpoint_w10_18" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w10_18-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18/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_w10_18-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18-problem-progress" tabindex="-1">
Worked Example - Self-inductance of a solenoid
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_18-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/63e0e40ac2a56c9220625eac174e6ed1/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_18.svg" width="770"/>
</center>
<p>
Consider a very long solenoid of circular cross section of radius [mathjaxinline]R[/mathjaxinline] and number of turns per unit length [mathjaxinline]n[/mathjaxinline]. A current [mathjaxinline]I[/mathjaxinline] is flowing as shown in the left figure creating a magnetic field directed along the [mathjaxinline]+z[/mathjaxinline]-axis. (We will neglect any edge effects, i. e. we will assume the magnetic field to be the one of an infinite solenoid.) </p>
<p><b class="bfseries">(Part a)</b> Consider one coil of the solenoid as the one shown in the right figure above. Calculate [mathjaxinline]\Phi _{one}[/mathjaxinline], the flux through that one coil. Express your answer in terms of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Phi _{one} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w10_18_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_18_2_1" id="input_checkpoint_w10_18_2_1" aria-describedby="status_checkpoint_w10_18_2_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_18_2_1"/>
<span class="status unanswered" id="status_checkpoint_w10_18_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w10_18_2_1" class="answer"/>
<div id="display_checkpoint_w10_18_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_18_2_1_dynamath" name="input_checkpoint_w10_18_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> If the solenoid has a total number of coils [mathjaxinline]N[/mathjaxinline], calculate [mathjaxinline]L[/mathjaxinline], the self inductance of the solenoid. Express your answer in terms of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]n[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]L =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w10_18_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w10_18_3_1" id="input_checkpoint_w10_18_3_1" aria-describedby="status_checkpoint_w10_18_3_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w10_18_3_1"/>
<span class="status unanswered" id="status_checkpoint_w10_18_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w10_18_3_1" class="answer"/>
<div id="display_checkpoint_w10_18_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w10_18_3_1_dynamath" name="input_checkpoint_w10_18_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w10_18_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Worked Example - Self-inductance of a solenoid" />
<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_w10_18" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w10_18">
<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_w10_18-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_w10_18-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_w10_18-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+8.02.3x+1T2019+type@video+block@L26v05">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="6c43402205ce11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v05" data-init="XBlockToXModuleShim" data-block-type="video" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-has-score="False" data-graded="True" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L26v5: Worked Example - Inductance of Solenoid</h3>
<div
id="video_L26v05"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:r_YIbro5WxM", "saveStateEnabled": false, "end": 0.0, "speed": null, "completionPercentage": 0.95, "start": 0.0, "publishCompletionUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v05/handler/publish_completion", "duration": null, "autoplay": false, "savedVideoPosition": 0.0, "generalSpeed": 1.0, "autohideHtml5": false, "ytMetadataEndpoint": "", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v05/handler/transcript/translation/__lang__", "showCaptions": "true", "completionEnabled": false, "captionDataDir": null, "ytApiUrl": "https://www.youtube.com/iframe_api", "saveStateUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v05/handler/xmodule_handler/save_user_state", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L26v05/handler/transcript/available_translations", "sources": [], "transcriptLanguages": {"en": "English"}, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu"}'
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="L26v05"></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_L26v05">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L26v05">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/842537a3fa9abc08373d745246710cd7/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L26v05.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved