<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-lesson32_01x" 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 Driven RLC Circuits</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson32_01">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson32_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><center><b> Driven RLC Circuits</b></center></p><p>In this lesson we start by exploring the driven RLC circuit and find a phenomenon called <i>resonance</i>. We will explore this phenomenon in the context of this circuit. We will also start to talk about real world applications of AC power, introducing the idea of time averages to look at the power ratings of a light bulb, for example, and then to the idea of transformers that are used in everyday electronics to provide the right voltage. </p>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_2d39cd48a978">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_2d39cd48a978" 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 12.3-12.5</a></li>
</ul>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v01" 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">L32v1: Solve the Driven RLC Circuit</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24" 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_w13_24" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_24-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24/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_w13_24-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24-problem-progress" tabindex="-1">
Driven RLC circuit
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_24-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/c441efff170a0554807c4a96fa3026b3/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_24.svg" width="495"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the figure above. The voltage source is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. </p>
<p>
If [mathjaxinline]q[/mathjaxinline] is the magnitude of the charge on the plates of the capacitor, and we assume that the capacitor polarity and current direction are as shown, the differential equation for 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 V(t)=I(t) R +L\dfrac {d}{dt}I(t) +\dfrac {1}{C}q(t)[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
This equation is not straightforward to solve using the real functions [mathjaxinline]V(t)[/mathjaxinline], [mathjaxinline]I(t)[/mathjaxinline], and [mathjaxinline]q(t)[/mathjaxinline]. We will instead use the complex functions [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline] and relate them with [mathjaxinline]z_{eq}[/mathjaxinline], the equivalent impedance of the circuit by: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_ c = z_{eq} I_ c[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate the modulus [mathjaxinline]|z_{eq}|[/mathjaxinline], and the phase [mathjaxinline]\delta[/mathjaxinline] of the equivalent impedance [mathjaxinline]z_{eq}[/mathjaxinline]. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], [mathjaxinline]i[/mathjaxinline], arctan ( ) for [mathjaxinline]\tan ^{-1}()[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|z_{eq}|[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_24_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_2_1" id="input_checkpoint_w13_24_2_1" aria-describedby="status_checkpoint_w13_24_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_2_1" class="answer"/>
<div id="display_checkpoint_w13_24_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_2_1_dynamath" name="input_checkpoint_w13_24_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_24_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_3_1" id="input_checkpoint_w13_24_3_1" aria-describedby="status_checkpoint_w13_24_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_3_1" class="answer"/>
<div id="display_checkpoint_w13_24_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_3_1_dynamath" name="input_checkpoint_w13_24_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Write the Euler expression for [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline] in terms of [mathjaxinline]i[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V_ c=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_24_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_4_1" id="input_checkpoint_w13_24_4_1" aria-describedby="status_checkpoint_w13_24_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_4_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_4_1" class="answer"/>
<div id="display_checkpoint_w13_24_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_4_1_dynamath" name="input_checkpoint_w13_24_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]I_ c=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w13_24_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_5_1" id="input_checkpoint_w13_24_5_1" aria-describedby="status_checkpoint_w13_24_5_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_5_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_5_1" class="answer"/>
<div id="display_checkpoint_w13_24_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_5_1_dynamath" name="input_checkpoint_w13_24_5_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]I_0[/mathjaxinline], the amplitude of the current in the circuit, and [mathjaxinline]\phi[/mathjaxinline], the phase shift of the current with respect to the driving voltage, Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and arctan( ) for [mathjaxinline]\tan ^{-1}()[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]I_0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_checkpoint_w13_24_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_6_1" id="input_checkpoint_w13_24_6_1" aria-describedby="status_checkpoint_w13_24_6_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_6_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_6_1" class="answer"/>
<div id="display_checkpoint_w13_24_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_6_1_dynamath" name="input_checkpoint_w13_24_6_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\phi =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_checkpoint_w13_24_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_24_7_1" id="input_checkpoint_w13_24_7_1" aria-describedby="status_checkpoint_w13_24_7_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_24_7_1"/>
<span class="status unanswered" id="status_checkpoint_w13_24_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_24_7_1" class="answer"/>
<div id="display_checkpoint_w13_24_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_24_7_1_dynamath" name="input_checkpoint_w13_24_7_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_24_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Driven RLC circuit" />
<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_w13_24" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_24">
<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_w13_24-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_w13_24-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_w13_24-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@L32v01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v01" 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">L32v1: Solve the Driven RLC Circuit</h3>
<div
id="video_L32v01"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:kmj-sGvpprg", "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@L32v01/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@L32v01/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@L32v01/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@L32v01/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="L32v01"></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_L32v01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v01">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/92851aa377b44dcfcaa769cf56202d65/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v01.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v02" 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">L32v2: Current Response and Resonance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25" 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_w13_25" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_25-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25/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_w13_25-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25-problem-progress" tabindex="-1">
Current Response and Resonance
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_25-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/04ca510de985fd63d84b7954fa7ae184/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_25.svg" width="770"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the left figure above. The driving voltage is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] is shown in the right figure and 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 I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\omega _0[/mathjaxinline], the value of [mathjaxinline]\omega[/mathjaxinline] for which the amplitude of the current is maximum. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_25_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_25_2_1" id="input_checkpoint_w13_25_2_1" aria-describedby="status_checkpoint_w13_25_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_25_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_25_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_25_2_1" class="answer"/>
<div id="display_checkpoint_w13_25_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_25_2_1_dynamath" name="input_checkpoint_w13_25_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]I_{max}[/mathjaxinline], the value of [mathjaxinline]I_0[/mathjaxinline] for [mathjaxinline]\omega =\omega _0[/mathjaxinline]. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]I_{max}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_25_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_25_3_1" id="input_checkpoint_w13_25_3_1" aria-describedby="status_checkpoint_w13_25_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_25_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_25_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_25_3_1" class="answer"/>
<div id="display_checkpoint_w13_25_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_25_3_1_dynamath" name="input_checkpoint_w13_25_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> The phase shift of the current with respect to the voltage is given by: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
What is the value of [mathjaxinline]\phi[/mathjaxinline] when [mathjaxinline]\omega = \omega _0[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]\phi (\omega _0)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_25_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_25_4_1" id="input_checkpoint_w13_25_4_1" aria-describedby="status_checkpoint_w13_25_4_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_25_4_1"/>
<span class="status unanswered" id="status_checkpoint_w13_25_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_25_4_1" class="answer"/>
</div>
</div></div>
</p>
<p>
What is the value of [mathjaxinline]\phi[/mathjaxinline] when [mathjaxinline]\omega \rightarrow +\infty[/mathjaxinline]? <p style="display:inline">[mathjaxinline]\phi (+\infty )=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w13_25_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_25_5_1" id="input_checkpoint_w13_25_5_1" aria-describedby="status_checkpoint_w13_25_5_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_25_5_1"/>
<span class="status unanswered" id="status_checkpoint_w13_25_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_25_5_1" class="answer"/>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_25_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Current Response and Resonance" />
<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_w13_25" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_25">
<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_w13_25-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_w13_25-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_w13_25-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@L32v02">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v02" 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">L32v2: Current Response and Resonance</h3>
<div
id="video_L32v02"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:G4R14Rx0tg8", "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@L32v02/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@L32v02/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@L32v02/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@L32v02/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="L32v02"></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_L32v02">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v02">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/eb2181426ced16be1ffa589706672eb8/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v02.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-checkpoint_w13_26" 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">L32Q1: Voltage Across the Inductor</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26" 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_w13_26" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_26-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26/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_w13_26-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26-problem-progress" tabindex="-1">
Driven RLC circuit - Voltage across the Inductor
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_26-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/36ac74af69be04b9ffa9034d8502a5d2/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_26.svg" width="770"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the left figure above. The driving voltage is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] 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 I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
and its phase shift with respect to the voltage source as a function of [mathjaxinline]\omega[/mathjaxinline] is: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]V_ L(t) = L \dfrac {dI}{dt}[/mathjaxinline], the voltage across the inductor. Express your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and phi for [mathjaxinline]\phi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V_ L(t)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_26_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_26_2_1" id="input_checkpoint_w13_26_2_1" aria-describedby="status_checkpoint_w13_26_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_26_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_26_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_26_2_1" class="answer"/>
<div id="display_checkpoint_w13_26_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_26_2_1_dynamath" name="input_checkpoint_w13_26_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_26_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Consider [mathjaxinline]\omega &gt;\omega _0=\dfrac {1}{\sqrt {LC}}[/mathjaxinline]. Which of the following statements about the driving voltage [mathjaxinline]V(t)[/mathjaxinline], the voltage across the inductor [mathjaxinline]V_ L[/mathjaxinline], and the current [mathjaxinline]I[/mathjaxinline] are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_26_3_1">
<fieldset aria-describedby="status_checkpoint_w13_26_3_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_3_1[]" id="input_checkpoint_w13_26_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w13_26_3_1-choice_0-label" for="input_checkpoint_w13_26_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_3_1">
<text>1. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_ L(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_3_1[]" id="input_checkpoint_w13_26_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w13_26_3_1-choice_1-label" for="input_checkpoint_w13_26_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_3_1">
<text>2. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_ L(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_3_1[]" id="input_checkpoint_w13_26_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w13_26_3_1-choice_2-label" for="input_checkpoint_w13_26_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_3_1">
<text>3. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_3_1[]" id="input_checkpoint_w13_26_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w13_26_3_1-choice_3-label" for="input_checkpoint_w13_26_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_3_1">
<text>4. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_26_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_26_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_w13_26_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Consider [mathjaxinline]\omega &lt;\omega _0=\dfrac {1}{\sqrt {LC}}[/mathjaxinline]. Which of the following statements about the driving voltage [mathjaxinline]V(t)[/mathjaxinline], the voltage across the inductor [mathjaxinline]V_ L[/mathjaxinline], and the current [mathjaxinline]I[/mathjaxinline] are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_26_4_1">
<fieldset aria-describedby="status_checkpoint_w13_26_4_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_4_1[]" id="input_checkpoint_w13_26_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w13_26_4_1-choice_0-label" for="input_checkpoint_w13_26_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_4_1">
<text>1. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_ L(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_4_1[]" id="input_checkpoint_w13_26_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w13_26_4_1-choice_1-label" for="input_checkpoint_w13_26_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_4_1">
<text>2. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_ L(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_4_1[]" id="input_checkpoint_w13_26_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w13_26_4_1-choice_2-label" for="input_checkpoint_w13_26_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_4_1">
<text>3. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_26_4_1[]" id="input_checkpoint_w13_26_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w13_26_4_1-choice_3-label" for="input_checkpoint_w13_26_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_4_1">
<text>4. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_26_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_26_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><div class="solution-span">
<span id="solution_checkpoint_w13_26_solution_3"/>
</div><b class="bfseries">(Part d)</b> Which of the following statements about the phase shift between the voltage across the inductor and current is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_26_5_1">
<fieldset aria-describedby="status_checkpoint_w13_26_5_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_26_5_1-choice_1-label" for="input_checkpoint_w13_26_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_ L(t)[/mathjaxinline] only for [mathjaxinline]\omega &gt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_26_5_1-choice_2-label" for="input_checkpoint_w13_26_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_ L(t)[/mathjaxinline] only for [mathjaxinline]\omega &gt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_26_5_1-choice_3-label" for="input_checkpoint_w13_26_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_ L(t)[/mathjaxinline] only for [mathjaxinline]\omega &lt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w13_26_5_1-choice_4-label" for="input_checkpoint_w13_26_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_ L(t)[/mathjaxinline] only for [mathjaxinline]\omega &lt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w13_26_5_1-choice_5-label" for="input_checkpoint_w13_26_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_ L(t)[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline] for all [mathjaxinline]\omega[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_26_5_1" id="input_checkpoint_w13_26_5_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="checkpoint_w13_26_5_1-choice_6-label" for="input_checkpoint_w13_26_5_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_26_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_ L(t)[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline] for all [mathjaxinline]\omega[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_26_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_26_5_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_w13_26_solution_4"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Driven RLC circuit - Voltage across the Inductor" />
<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_w13_26" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_26">
<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_w13_26-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_w13_26-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_w13_26-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="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-checkpoint_w13_27" 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">L32Q2: Voltage Across the Capacitor</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27" 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_w13_27" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_27-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27/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_w13_27-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27-problem-progress" tabindex="-1">
Driven RLC circuit - Voltage across the Capacitor
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_27-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/36ac74af69be04b9ffa9034d8502a5d2/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_26.svg" width="770"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the left figure above. The driving voltage is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] 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 I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
and its phase shift with respect to the voltage source as a function of [mathjaxinline]\omega[/mathjaxinline] is: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]V_{Cap}(t) =\dfrac {q}{C}[/mathjaxinline], the voltage across the capacitor. Express your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and phi for [mathjaxinline]\phi[/mathjaxinline] as needed. (Hint - do this calculation assuming that at [mathjaxinline]t=0[/mathjaxinline] there is no charge on the capacitor and consider the time right after that where charge on the capacitor is increasing.) </p>
<p>
<p style="display:inline">[mathjaxinline]V_{Cap}(t)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_27_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_27_2_1" id="input_checkpoint_w13_27_2_1" aria-describedby="status_checkpoint_w13_27_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_27_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_27_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_27_2_1" class="answer"/>
<div id="display_checkpoint_w13_27_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_27_2_1_dynamath" name="input_checkpoint_w13_27_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_27_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Consider [mathjaxinline]\omega &gt;\omega _0=\dfrac {1}{\sqrt {LC}}[/mathjaxinline]. Which of the following statements about the driving voltage [mathjaxinline]V(t)[/mathjaxinline], the voltage across the capacitor [mathjaxinline]V_{cap}[/mathjaxinline], and the current [mathjaxinline]I[/mathjaxinline] are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_27_3_1">
<fieldset aria-describedby="status_checkpoint_w13_27_3_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_3_1[]" id="input_checkpoint_w13_27_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w13_27_3_1-choice_0-label" for="input_checkpoint_w13_27_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_3_1">
<text>1. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_{cap}(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_3_1[]" id="input_checkpoint_w13_27_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w13_27_3_1-choice_1-label" for="input_checkpoint_w13_27_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_3_1">
<text>2. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_{cap}(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_3_1[]" id="input_checkpoint_w13_27_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w13_27_3_1-choice_2-label" for="input_checkpoint_w13_27_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_3_1">
<text>3. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_3_1[]" id="input_checkpoint_w13_27_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w13_27_3_1-choice_3-label" for="input_checkpoint_w13_27_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_3_1">
<text>4. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_27_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_27_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_w13_27_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Consider [mathjaxinline]\omega &lt;\omega _0=\dfrac {1}{\sqrt {LC}}[/mathjaxinline]. Which of the following statements about the driving voltage [mathjaxinline]V(t)[/mathjaxinline], the voltage across the capacitor [mathjaxinline]V_{cap}[/mathjaxinline], and the current [mathjaxinline]I[/mathjaxinline] are true? </p>
<p><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_27_4_1">
<fieldset aria-describedby="status_checkpoint_w13_27_4_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_4_1[]" id="input_checkpoint_w13_27_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w13_27_4_1-choice_0-label" for="input_checkpoint_w13_27_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_4_1"> <text>1. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_{cap}(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_4_1[]" id="input_checkpoint_w13_27_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w13_27_4_1-choice_1-label" for="input_checkpoint_w13_27_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_4_1"> <text>2. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_{cap}(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_4_1[]" id="input_checkpoint_w13_27_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w13_27_4_1-choice_2-label" for="input_checkpoint_w13_27_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_4_1"> <text>3. [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_27_4_1[]" id="input_checkpoint_w13_27_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w13_27_4_1-choice_3-label" for="input_checkpoint_w13_27_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_4_1"> <text>4. [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_27_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_27_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div><div class="solution-span">
<span id="solution_checkpoint_w13_27_solution_3"/>
</div><b class="bfseries">(Part d)</b> Which of the following statements about the phase shift between the voltage across the capacitor and current is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_27_5_1">
<fieldset aria-describedby="status_checkpoint_w13_27_5_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_27_5_1-choice_1-label" for="input_checkpoint_w13_27_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_{cap}(t)[/mathjaxinline] only for [mathjaxinline]\omega &gt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_27_5_1-choice_2-label" for="input_checkpoint_w13_27_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_{cap}(t)[/mathjaxinline] only for [mathjaxinline]\omega &gt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_27_5_1-choice_3-label" for="input_checkpoint_w13_27_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_{cap}(t)[/mathjaxinline] only for [mathjaxinline]\omega &lt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w13_27_5_1-choice_4-label" for="input_checkpoint_w13_27_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_{cap}(t)[/mathjaxinline] only for [mathjaxinline]\omega &lt;\omega _0[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w13_27_5_1-choice_5-label" for="input_checkpoint_w13_27_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] leads [mathjaxinline]V_{cap}(t)[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline] for all [mathjaxinline]\omega[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_27_5_1" id="input_checkpoint_w13_27_5_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="checkpoint_w13_27_5_1-choice_6-label" for="input_checkpoint_w13_27_5_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_27_5_1">
<text> [mathjaxinline]I(t)[/mathjaxinline] lags [mathjaxinline]V_{cap}(t)[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline] for all [mathjaxinline]\omega[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_27_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_27_5_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_w13_27_solution_4"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Driven RLC circuit - Voltage across the Capacitor" />
<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_w13_27" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_27">
<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_w13_27-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_w13_27-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_w13_27-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="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-checkpoint_w13_28" 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">L32Q3: Amplitude of the Voltage at Resonance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28" 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_w13_28" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_28-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28/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_w13_28-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28-problem-progress" tabindex="-1">
Driven RLC circuit - Amplitude of the Voltage
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_28-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/04ca510de985fd63d84b7954fa7ae184/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_25.svg" width="660"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the left figure above. The voltage source is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] 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 I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
and its phase shift with respect to the voltage source as a function of [mathjaxinline]\omega[/mathjaxinline] is: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]V_{0,L}[/mathjaxinline], the amplitude of the voltage across the inductor. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V_{0,L}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_28_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_28_2_1" id="input_checkpoint_w13_28_2_1" aria-describedby="status_checkpoint_w13_28_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_28_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_28_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_28_2_1" class="answer"/>
<div id="display_checkpoint_w13_28_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_28_2_1_dynamath" name="input_checkpoint_w13_28_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_28_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]V_{0,Cap}[/mathjaxinline], the amplitude of the voltage across the capacitor. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V_{0,Cap}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_28_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_28_3_1" id="input_checkpoint_w13_28_3_1" aria-describedby="status_checkpoint_w13_28_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_28_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_28_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_28_3_1" class="answer"/>
<div id="display_checkpoint_w13_28_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_28_3_1_dynamath" name="input_checkpoint_w13_28_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_28_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Which of the following statements about [mathjaxinline]V_{0,L}[/mathjaxinline] and [mathjaxinline]V_{0,Cap}[/mathjaxinline] at [mathjaxinline]\omega = \omega _0=1/\sqrt {LC}[/mathjaxinline] is true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_28_4_1">
<fieldset aria-describedby="status_checkpoint_w13_28_4_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_28_4_1" id="input_checkpoint_w13_28_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_28_4_1-choice_1-label" for="input_checkpoint_w13_28_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_28_4_1">
<text> At [mathjaxinline]\omega =\omega _0[/mathjaxinline], [mathjaxinline]V_{0,L} &gt; V_{0,Cap}(t)[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_28_4_1" id="input_checkpoint_w13_28_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_28_4_1-choice_2-label" for="input_checkpoint_w13_28_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_28_4_1">
<text> At [mathjaxinline]\omega =\omega _0[/mathjaxinline], [mathjaxinline]V_{0,L}= V_{0,Cap}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_28_4_1" id="input_checkpoint_w13_28_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_28_4_1-choice_3-label" for="input_checkpoint_w13_28_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_28_4_1">
<text> At [mathjaxinline]\omega =\omega _0[/mathjaxinline], [mathjaxinline]V_{0,L} &lt; V_{0,Cap}(t)[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_28_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_28_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_28_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Driven RLC circuit - Amplitude of the Voltage" />
<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_w13_28" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_28">
<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_w13_28-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_w13_28-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_w13_28-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="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v03" 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">L32v3: Voltage Across the Inductor and Capacitor at Resonance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29" 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_w13_29" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_29-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29/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_w13_29-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29-problem-progress" tabindex="-1">
Voltage across an Inductor and Capacitor at Resonance
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_29-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/c441efff170a0554807c4a96fa3026b3/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_24.svg" width="495"/>
</center>
<p>
An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown in the figure above. The voltage source is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] 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 I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
and its phase shift with respect to the voltage source as a function of [mathjaxinline]\omega[/mathjaxinline] is: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Now, assume that the circuit is being driven at a frequency (called the resonant frequency) [mathjaxinline]\omega =\omega +0=1/\sqrt {LC}[/mathjaxinline]. Consider the following quantities: <b class="bfseries">1</b>: the current [mathjaxinline]I[/mathjaxinline], <b class="bfseries">2</b>: the voltage across the inductor [mathjaxinline]V_ L[/mathjaxinline], and <b class="bfseries">3</b>: the voltage across the capacitor [mathjaxinline]V_{Cap}[/mathjaxinline]. Calculate the phase shift of these three quantities with respect to the driving voltage, [mathjaxinline]V[/mathjaxinline]. Enter a positive value if the quantity is behind [mathjaxinline]V[/mathjaxinline] and a negative value if the quantity is ahead. </p>
<p>
<p style="display:inline"><b class="bfseries">1</b>: The phase shift of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]\phi[/mathjaxinline] =</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_29_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_29_2_1" id="input_checkpoint_w13_29_2_1" aria-describedby="status_checkpoint_w13_29_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_29_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_29_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_29_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline"><b class="bfseries">2</b>: The phase shift of [mathjaxinline]V_ L[/mathjaxinline] =</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_29_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_29_3_1" id="input_checkpoint_w13_29_3_1" aria-describedby="status_checkpoint_w13_29_3_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_29_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_29_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_29_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline"><b class="bfseries">3</b>: The phase shift of [mathjaxinline]V_{Cap}[/mathjaxinline] =</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_29_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_29_4_1" id="input_checkpoint_w13_29_4_1" aria-describedby="status_checkpoint_w13_29_4_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_29_4_1"/>
<span class="status unanswered" id="status_checkpoint_w13_29_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_29_4_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Which of the following statements about the current [mathjaxinline]I[/mathjaxinline], the voltage across the inductor [mathjaxinline]V_ L[/mathjaxinline], the voltage across the capacitor, [mathjaxinline]V_{Cap}[/mathjaxinline], and the driving voltage, [mathjaxinline]V[/mathjaxinline], are true? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_29_5_1">
<fieldset aria-describedby="status_checkpoint_w13_29_5_1">
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_29_5_1[]" id="input_checkpoint_w13_29_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="checkpoint_w13_29_5_1-choice_0-label" for="input_checkpoint_w13_29_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_29_5_1">
<text>[mathjaxinline]V_ L[/mathjaxinline] leads [mathjaxinline]I[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_29_5_1[]" id="input_checkpoint_w13_29_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w13_29_5_1-choice_1-label" for="input_checkpoint_w13_29_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_29_5_1">
<text>[mathjaxinline]V_{Cap}[/mathjaxinline] lags [mathjaxinline]I[/mathjaxinline] by [mathjaxinline]\pi /2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_29_5_1[]" id="input_checkpoint_w13_29_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w13_29_5_1-choice_2-label" for="input_checkpoint_w13_29_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_29_5_1">
<text>[mathjaxinline]I[/mathjaxinline] is in phase with [mathjaxinline]V[/mathjaxinline] when [mathjaxinline]\omega =\omega _0=1/\sqrt {LC}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_checkpoint_w13_29_5_1[]" id="input_checkpoint_w13_29_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w13_29_5_1-choice_3-label" for="input_checkpoint_w13_29_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_29_5_1">
<text>[mathjaxinline]V_ L[/mathjaxinline] leads [mathjaxinline]V_{Cap}[/mathjaxinline] by [mathjaxinline]\pi[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_29_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_29_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part c)</b> What is the value of the sum of [mathjaxinline]V_ L(t)+V_{Cap}(t)[/mathjaxinline] for all times at [mathjaxinline]\omega =\omega _0[/mathjaxinline]? </p>
<p>
<p style="display:inline">At [mathjaxinline]\omega =\omega _0[/mathjaxinline] [mathjaxinline]V_ L(t)+V_{Cap}[/mathjaxinline] =</p>
</p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_checkpoint_w13_29_6_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_29_6_1" id="input_checkpoint_w13_29_6_1" aria-describedby="status_checkpoint_w13_29_6_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_29_6_1"/>
<span class="status unanswered" id="status_checkpoint_w13_29_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_29_6_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_29_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Voltage across an Inductor and Capacitor at Resonance" />
<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_w13_29" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_29">
<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_w13_29-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_w13_29-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_w13_29-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@L32v03">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v03" 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">L32v3: Voltage Across the Inductor and Capacitor at Resonance</h3>
<div
id="video_L32v03"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:ggIhjWcddnQ", "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@L32v03/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@L32v03/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@L32v03/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@L32v03/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="L32v03"></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_L32v03">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v03">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/1c29bafebba09a1f25ea84dfffa97941/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v03.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v04" 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">L32v4: Power in the Driven RLC Circuit</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30" 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_w13_30" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_30-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30/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_w13_30-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30-problem-progress" tabindex="-1">
Power in a Driven RLC Circuit: Definitions
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30-problem-progress"></div>
<div class="problem">
<div>
<center>
<b class="bfseries">Time average of a function:</b>
</center>
<p>
The time average of a function [mathjaxinline]f(t)[/mathjaxinline] over a time interval [mathjaxinline]0&lt;t&lt;T[/mathjaxinline] is denoted with brackets, [mathjaxinline]&lt;f(t)&gt;[/mathjaxinline], and is defined 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 &lt;f(t)&gt;=\dfrac {1}{T}\int _{t=0}^{t=T} f(t)dt[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<center>
<b class="bfseries">Root Mean Square (rms) of a function:</b>
</center>
<p>
The root mean square of a function of time [mathjaxinline]f(t)[/mathjaxinline] is denoted with the subscript "rms", [mathjaxinline]f_{rms}[/mathjaxinline], and is defined as: </p>
<table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000005">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f_{rms}=\sqrt {&lt;f^2(t)&gt;}=\sqrt {\dfrac {1}{T}\int _{t=0}^{t=T} f^2(t)dt}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">
<span>(<span>2</span>)</span>
</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Consider the function [mathjaxinline]f(t) = \sin ^2(\omega t)[/mathjaxinline], calculate the time average of [mathjaxinline]f(t)[/mathjaxinline] over one full period [mathjaxinline]T=2\pi /\omega[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]&lt;\sin ^2(\omega t)&gt; =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_30_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30_2_1" id="input_checkpoint_w13_30_2_1" aria-describedby="status_checkpoint_w13_30_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]f_{rms}[/mathjaxinline], the root mean square of the function [mathjaxinline]f(t)=A\sin (\omega t)[/mathjaxinline], where [mathjaxinline]A[/mathjaxinline] is the amplitude. Express your answer in terms of [mathjaxinline]A[/mathjaxinline] and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]f_{rms} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_30_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30_3_1" id="input_checkpoint_w13_30_3_1" aria-describedby="status_checkpoint_w13_30_3_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30_3_1" class="answer"/>
<div id="display_checkpoint_w13_30_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30_3_1_dynamath" name="input_checkpoint_w13_30_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30_solution_2"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Power in a Driven RLC Circuit: Definitions" />
<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_w13_30" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_30">
<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_w13_30-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_w13_30-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_w13_30-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@problem+block@checkpoint_w13_30b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30b" 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_w13_30b" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_30b-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30b" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30b/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_w13_30b-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30b-problem-progress" tabindex="-1">
Power in the driven RLC circuit
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_30b-problem-progress"></div>
<div class="problem">
<div>
<center>
<b class="bfseries">Time average Power in an RLC circuit:</b>
</center>
<center>
<img src="/assets/courseware/v1/c441efff170a0554807c4a96fa3026b3/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_24.svg" width="495"/>
</center>
<p>
Consider the RLC circuit in the figure above. The driving voltage is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. The amplitude of the current as a function of [mathjaxinline]\omega[/mathjaxinline] is given by: </p>
<table id="a0000000014" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000015">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle I_0=\dfrac {V_0}{\sqrt {R^2+(\omega L - 1/\omega C)^2}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
and its phase shift with respect to the voltage source as a function of [mathjaxinline]\omega[/mathjaxinline] is: </p>
<table id="a0000000016" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000017">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \tan \phi = \dfrac {\omega L - 1/\omega C}{R}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate the power delivered by the AC power source given by [mathjaxinline]P_{source}(t) = V(t)I(t)[/mathjaxinline]. Express your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{source}(t)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_30b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_2_1" id="input_checkpoint_w13_30b_2_1" aria-describedby="status_checkpoint_w13_30b_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_2_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_2_1" class="answer"/>
<div id="display_checkpoint_w13_30b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_2_1_dynamath" name="input_checkpoint_w13_30b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part b)</b> Use the trigonometric identity [mathjaxinline]\sin (\alpha -\beta ) = \sin (\alpha )\cos (\beta )-\cos (\alpha )\sin (\beta )[/mathjaxinline] and the result for [mathjaxinline]P_{source}[/mathjaxinline] found in part (a) to obtain another expression for [mathjaxinline]P_{source}[/mathjaxinline] consisting of the sum of two terms involving [mathjaxinline]\sin (\omega t)[/mathjaxinline] and [mathjaxinline]\cos (\omega t)[/mathjaxinline]. Write your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]P_{source}(t)=[/mathjaxinline]</p>
</p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_30b_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_3_1" id="input_checkpoint_w13_30b_3_1" aria-describedby="status_checkpoint_w13_30b_3_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_3_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_3_1" class="answer"/>
<div id="display_checkpoint_w13_30b_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_3_1_dynamath" name="input_checkpoint_w13_30b_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_1"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Calculate the time average of the expression obtained in part b). Express your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline], V_0 for [mathjaxinline]V_0[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]&lt;P_{source}(t)&gt;=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_30b_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_4_1" id="input_checkpoint_w13_30b_4_1" aria-describedby="status_checkpoint_w13_30b_4_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_4_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_4_1" class="answer"/>
<div id="display_checkpoint_w13_30b_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_4_1_dynamath" name="input_checkpoint_w13_30b_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_2"/>
</div></p>
<p><b class="bfseries">(Part d)</b> At what value of [mathjaxinline]\omega[/mathjaxinline] is the time average power delivered by the AC power source a maximum? Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w13_30b_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_5_1" id="input_checkpoint_w13_30b_5_1" aria-describedby="status_checkpoint_w13_30b_5_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_5_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_5_1" class="answer"/>
<div id="display_checkpoint_w13_30b_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_5_1_dynamath" name="input_checkpoint_w13_30b_5_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_3"/>
</div></p>
<p><b class="bfseries">(Part e)</b> Calculate [mathjaxinline]&lt;P_{source}&gt;_{max}[/mathjaxinline], the maximum time average power. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline] and I_0 for [mathjaxinline]V_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]&lt;P_{source}&gt;_{max}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_checkpoint_w13_30b_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_6_1" id="input_checkpoint_w13_30b_6_1" aria-describedby="status_checkpoint_w13_30b_6_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_6_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_6_1" class="answer"/>
<div id="display_checkpoint_w13_30b_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_6_1_dynamath" name="input_checkpoint_w13_30b_6_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_4"/>
</div></p>
<p><b class="bfseries">(Part f)</b> Calculate [mathjaxinline]V_{rms}[/mathjaxinline], the root mean square of the driving voltage [mathjaxinline]V(t)[/mathjaxinline]. Express your answer in terms of V_0 for [mathjaxinline]V_0[/mathjaxinline] and omega for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]V_{rms}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_checkpoint_w13_30b_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_7_1" id="input_checkpoint_w13_30b_7_1" aria-describedby="status_checkpoint_w13_30b_7_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_7_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_7_1" class="answer"/>
<div id="display_checkpoint_w13_30b_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_7_1_dynamath" name="input_checkpoint_w13_30b_7_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_5"/>
</div></p>
<p><b class="bfseries">(Part g)</b> Calculate [mathjaxinline]I_{rms}[/mathjaxinline], the root mean square of the current [mathjaxinline]I(t)[/mathjaxinline] in the circuit when [mathjaxinline]\omega = 1/\sqrt {LC}[/mathjaxinline]. Express your answer in terms of I_0 for [mathjaxinline]I_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]I_{rms}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_checkpoint_w13_30b_8_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_30b_8_1" id="input_checkpoint_w13_30b_8_1" aria-describedby="status_checkpoint_w13_30b_8_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_30b_8_1"/>
<span class="status unanswered" id="status_checkpoint_w13_30b_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_30b_8_1" class="answer"/>
<div id="display_checkpoint_w13_30b_8_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_checkpoint_w13_30b_8_1_dynamath" name="input_checkpoint_w13_30b_8_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_30b_solution_6"/>
</div></p>
<p>
Note: You can also express the maximum time average power in terms of [mathjaxinline]V_{rms}[/mathjaxinline] and [mathjaxinline]I_{rms}[/mathjaxinline] as: </p>
<table id="a0000000034" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000035">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle &lt;P_{source}&gt;_{max}=I_{rms}V_{rms}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Power in the driven RLC circuit" />
<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_w13_30b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_30b">
<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_w13_30b-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_w13_30b-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_w13_30b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_a5adc0cbd7cb">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_a5adc0cbd7cb" 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>Note that in this video, the word "battery" is uesd. What is meant is reall the the power supply.</p>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v04">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v04" 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">L32v4: Power in the Driven RLC Circuit</h3>
<div
id="video_L32v04"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:mhGlsBD2WcA", "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@L32v04/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@L32v04/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@L32v04/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@L32v04/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="L32v04"></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_L32v04">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v04">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/cca2cbff299e82c82d45623869c732ca/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v04.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v05" 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">L32v5: RLC Circuit and Resonance</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v05">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v05" 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">L32v5: RLC Circuit and Resonance</h3>
<div
id="video_L32v05"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:WJHGPWiGqn4", "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@L32v05/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@L32v05/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@L32v05/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@L32v05/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="L32v05"></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_L32v05">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v05">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/95987bbcd5fbffdb1ef724dd2728ac79/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v05.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@checkpoint_w13_2" 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">L32Q4: RLC Circuit with a Light Bulb</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_2" 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_w13_2" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_2-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_2" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_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_w13_2-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_2-problem-progress" tabindex="-1">
RLC Circuit with a Light Bulb
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_2-problem-progress"></div>
<div class="problem">
<div>
<p>
When I slide an iron core into the solenoid the light bulb changes brightness. Why? I am driving the circuit through resonance by... </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_2_2_1">
<fieldset aria-describedby="status_checkpoint_w13_2_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_2_2_1" id="input_checkpoint_w13_2_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_2_2_1-choice_1-label" for="input_checkpoint_w13_2_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_2_2_1">
<text> continuously increasing the frequency of current oscillations in the circuit</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_2_2_1" id="input_checkpoint_w13_2_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_2_2_1-choice_2-label" for="input_checkpoint_w13_2_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_2_2_1">
<text> continuously decreasing the frequency of current oscillations in the circuit</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_2_2_1" id="input_checkpoint_w13_2_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_2_2_1-choice_3-label" for="input_checkpoint_w13_2_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_2_2_1">
<text> continuously increasing the resonance (natural) frequency of oscillations in the circuit</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_2_2_1" id="input_checkpoint_w13_2_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w13_2_2_1-choice_4-label" for="input_checkpoint_w13_2_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_2_2_1">
<text> continuously decreasing the resonance (natural) frequency of oscillations in the circuit</text>
</label>
</div>
<span id="answer_checkpoint_w13_2_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_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_w13_2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="RLC Circuit with a Light Bulb" />
<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_w13_2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_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">
</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_w13_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_w13_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_w13_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-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v06" 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">L32v6: Transformers</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v06">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v06" 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">L32v6: Transformers</h3>
<div
id="video_L32v06"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:siz4eujnlGM", "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@L32v06/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@L32v06/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@L32v06/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@L32v06/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="L32v06"></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_L32v06">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v06">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/15d5f73766889649f70c89cd36fa2d3b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v06.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@checkpoint_w13_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">L32Q5: Transmission Lines</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a" 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_w13_4a" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_4a-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a/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_w13_4a-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a-problem-progress" tabindex="-1">
Transmission Line Voltage
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_4a-problem-progress"></div>
<div class="problem">
<div>
<p>
An average of 120 kW of electric power is sent from a power plant. The transmission lines have a total resistance of 0.40 [mathjaxinline]\Omega[/mathjaxinline]. Calculate the power loss if the power is sent at either a low or a high voltage: </p>
<p><b class="bfseries">(Part a)</b> If the power is being sent at 240 V, the power loss in the transmission lines is, </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_4a_2_1">
<fieldset aria-describedby="status_checkpoint_w13_4a_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_2_1" id="input_checkpoint_w13_4a_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_4a_2_1-choice_1-label" for="input_checkpoint_w13_4a_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_2_1">
<text> [mathjaxinline]0.0083\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_2_1" id="input_checkpoint_w13_4a_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_4a_2_1-choice_2-label" for="input_checkpoint_w13_4a_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_2_1">
<text> [mathjaxinline]0.083\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_2_1" id="input_checkpoint_w13_4a_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_4a_2_1-choice_3-label" for="input_checkpoint_w13_4a_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_2_1">
<text> [mathjaxinline]0.83\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_2_1" id="input_checkpoint_w13_4a_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w13_4a_2_1-choice_4-label" for="input_checkpoint_w13_4a_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_2_1">
<text> [mathjaxinline]8.3\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_2_1" id="input_checkpoint_w13_4a_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w13_4a_2_1-choice_5-label" for="input_checkpoint_w13_4a_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_2_1">
<text> [mathjaxinline]83\%[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_4a_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_4a_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> If the power is being sent at 24,000 V, the power loss in the transmission lines is, </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w13_4a_3_1">
<fieldset aria-describedby="status_checkpoint_w13_4a_3_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_3_1" id="input_checkpoint_w13_4a_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_4a_3_1-choice_1-label" for="input_checkpoint_w13_4a_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_3_1">
<text> [mathjaxinline]0.0083\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_3_1" id="input_checkpoint_w13_4a_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_4a_3_1-choice_2-label" for="input_checkpoint_w13_4a_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_3_1">
<text> [mathjaxinline]0.083\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_3_1" id="input_checkpoint_w13_4a_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w13_4a_3_1-choice_3-label" for="input_checkpoint_w13_4a_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_3_1">
<text> [mathjaxinline]0.83\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_3_1" id="input_checkpoint_w13_4a_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w13_4a_3_1-choice_4-label" for="input_checkpoint_w13_4a_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_3_1">
<text> [mathjaxinline]8.3\%[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_4a_3_1" id="input_checkpoint_w13_4a_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="checkpoint_w13_4a_3_1-choice_5-label" for="input_checkpoint_w13_4a_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_4a_3_1">
<text> [mathjaxinline]83\%[/mathjaxinline]</text>
</label>
</div>
<span id="answer_checkpoint_w13_4a_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_4a_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_w13_4a_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Transmission Line Voltage" />
<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_w13_4a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_4a">
<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_w13_4a-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_w13_4a-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_w13_4a-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@problem+block@checkpoint_w13_3">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_3" 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_w13_3" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_3-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_3" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_3/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="checkpoint_w13_3-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_3-problem-progress" tabindex="-1">
Residential Transformer
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_3-problem-progress"></div>
<div class="problem">
<div>
<p>
If the transformer in the metal can on the left is represented by the diagram to the right, how is it connected? </p>
<center>
<img src="/assets/courseware/v1/062a25c6a54a11111344ffb23a44639d/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w13_3_1.png" 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_w13_3_2_1">
<fieldset aria-describedby="status_checkpoint_w13_3_2_1">
<div class="field">
<input type="radio" name="input_checkpoint_w13_3_2_1" id="input_checkpoint_w13_3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w13_3_2_1-choice_1-label" for="input_checkpoint_w13_3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_3_2_1">
<text> The house is on the left side of the transformer diagram.</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_checkpoint_w13_3_2_1" id="input_checkpoint_w13_3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w13_3_2_1-choice_2-label" for="input_checkpoint_w13_3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w13_3_2_1">
<text> The house is on the right side of the transformer diagram.</text>
</label>
</div>
<span id="answer_checkpoint_w13_3_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_checkpoint_w13_3_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_w13_3_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Residential Transformer" />
<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_w13_3" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_3">
<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_w13_3-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_w13_3-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_w13_3-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="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@checkpoint_w13_32" 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">L32Q6: Household Voltage and Power</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32" 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_w13_32" class="problems-wrapper" role="group"
aria-labelledby="checkpoint_w13_32-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32/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_w13_32-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32-problem-progress" tabindex="-1">
Household AC Voltage and Power
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w13_32-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part a)</b> The AC voltage for wall outlets in the United States is given by [mathjaxinline]V(t)=V_0\sin (\omega t)[/mathjaxinline], where the frequency of the alternate current is [mathjaxinline]f = 60[/mathjaxinline] hz. Calculate the value of the angular frequency [mathjaxinline]\omega[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w13_32_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_32_2_1" id="input_checkpoint_w13_32_2_1" aria-describedby="trailing_text_checkpoint_w13_32_2_1 status_checkpoint_w13_32_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_32_2_1">[mathjaxinline]\, \mathrm{rad/s}[/mathjaxinline]</span>
<span class="status unanswered" id="status_checkpoint_w13_32_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_32_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_32_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> The AC voltage for the wall outlets in the United States is typically listed as [mathjaxinline]120[/mathjaxinline] V but this is actually the rms of the voltage supplied: [mathjaxinline]V_{rms}=120[/mathjaxinline] V. What is [mathjaxinline]V_0[/mathjaxinline], the amplitude of the voltage supplied? </p>
<p>
<p style="display:inline">[mathjaxinline]V_0=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w13_32_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_32_3_1" id="input_checkpoint_w13_32_3_1" aria-describedby="trailing_text_checkpoint_w13_32_3_1 status_checkpoint_w13_32_3_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_32_3_1">[mathjaxinline]\, \mathrm{V}[/mathjaxinline]</span>
<span class="status unanswered" id="status_checkpoint_w13_32_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_32_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_32_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> What is [mathjaxinline]R[/mathjaxinline], the resistance of a light bulb rated at 100 W? (The 100 W refers to the time-averaged power). </p>
<p>
<p style="display:inline">[mathjaxinline]R=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_checkpoint_w13_32_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_32_4_1" id="input_checkpoint_w13_32_4_1" aria-describedby="trailing_text_checkpoint_w13_32_4_1 status_checkpoint_w13_32_4_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_32_4_1">[mathjaxinline]\, \mathrm{\Omega }[/mathjaxinline]</span>
<span class="status unanswered" id="status_checkpoint_w13_32_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_32_4_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_32_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b> When traveling to Europe with electrical devices from the United States, many of them need a transformer to convert the 220 V power coming from the European wall outlets to the 120 V input the devices expect. One good example is a hairdryer which could be destroyed if the input voltage is too high. This voltage change is accomplished using a transformer. If the primary coil of the transformer has 100 turns, how many turns must there be on the secondary coil? <p style="display:inline">[mathjaxinline]N_2=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_checkpoint_w13_32_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_checkpoint_w13_32_5_1" id="input_checkpoint_w13_32_5_1" aria-describedby="trailing_text_checkpoint_w13_32_5_1 status_checkpoint_w13_32_5_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_checkpoint_w13_32_5_1">[mathjaxinline]\, \mathrm{turns}[/mathjaxinline]</span>
<span class="status unanswered" id="status_checkpoint_w13_32_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_checkpoint_w13_32_5_1" class="answer"/>
</div>
</div></div> </p>
<p>
<div class="solution-span">
<span id="solution_checkpoint_w13_32_solution_4"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Household AC Voltage and Power" />
<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_w13_32" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_checkpoint_w13_32">
<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_w13_32-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_w13_32-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_w13_32-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="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32v07" 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">L32v7: Step-up and Step-Down Transformers</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v07">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32v07" 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">L32v7: Step-up and Step-Down Transformers</h3>
<div
id="video_L32v07"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:B8CppP-VHQ0", "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@L32v07/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@L32v07/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@L32v07/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@L32v07/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="L32v07"></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_L32v07">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32v07">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/a6d58edd5c648e3a03ad5919d88405fc/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32v07.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L32DD01" 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">L32DD1: Deep Dive - Solving the Undriven RLC Circuit</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32DD01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="c931fb5805cd11f0a1780affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L32DD01" 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">L32DD1: Deep Dive - Solving the Undriven RLC Circuit</h3>
<div
id="video_L32DD01"
class="video closed"
data-metadata='{"autoAdvance": false, "prioritizeHls": false, "recordedYoutubeIsAvailable": true, "ytTestTimeout": 1500, "poster": null, "streams": "1.00:-Z-cu07gusc", "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@L32DD01/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@L32DD01/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@L32DD01/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@L32DD01/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="L32DD01"></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_L32DD01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L32DD01">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/9c3c0a51ecffc47cf7a136e0c3986edf/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_L32DD01.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved