<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-W14PS01" 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="False" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">W14PS1: Worked Example - Poynting Vector for a Capacitor</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_finalexam-problem27" class="problems-wrapper" role="group" aria-labelledby="finalexam-problem27-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;finalexam-problem27-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Energy Flow and Charging Capacitor &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem27-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; A circular capacitor of spacing [mathjaxinline]d[/mathjaxinline] and radius [mathjaxinline]R[/mathjaxinline] (where [mathjaxinline]d \ll R[/mathjaxinline]) is in a circuit carrying the steady current [mathjaxinline]I[/mathjaxinline] as shown in the figure below. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/d7c204a51ca5ca4a5eb78ee4b99893da/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_finalexam-problem27_1.png&#34; width=&#34;400&#34; style=&#34;width:&#34;/&gt; &lt;/center&gt; &lt;p&gt; At time [mathjaxinline]t=0[/mathjaxinline], the capacitor is uncharged ([mathjaxinline]|Q|=0[/mathjaxinline]). &lt;/p&gt; &lt;p&gt; Express your answers to all of these questions in terms of [mathjaxinline]d[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], epsilon_0 for [mathjaxinline]\epsilon _0[/mathjaxinline], mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], and hatr, hatk, and hattheta for [mathjaxinline]\hat{r}[/mathjaxinline], [mathjaxinline]\hat{k}[/mathjaxinline] and [mathjaxinline]\hat{\theta }[/mathjaxinline], respectively, as needed. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Find the magnitude and direction of the electric field as a function of time, [mathjaxinline]\vec{E}(t)[/mathjaxinline], at the point [mathjaxinline]P[/mathjaxinline]. Assume that [mathjaxinline]P[/mathjaxinline] has the field present &amp;#8220;inside&#34; the capacitor. &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\vec{E}(t)=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem27_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_finalexam-problem27_2_1&#34; id=&#34;input_finalexam-problem27_2_1&#34; aria-describedby=&#34;status_finalexam-problem27_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;45&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem27_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem27_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_finalexam-problem27_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_finalexam-problem27_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_finalexam-problem27_2_1_dynamath&#34; name=&#34;input_finalexam-problem27_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem27_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Find the magnitude and direction of the magnetic field as a function of time [mathjaxinline]\vec{B}(t)[/mathjaxinline] at the point [mathjaxinline]P[/mathjaxinline]. &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\vec{B}(t)=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem27_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_finalexam-problem27_3_1&#34; id=&#34;input_finalexam-problem27_3_1&#34; aria-describedby=&#34;status_finalexam-problem27_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;45&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem27_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem27_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_finalexam-problem27_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_finalexam-problem27_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_finalexam-problem27_3_1_dynamath&#34; name=&#34;input_finalexam-problem27_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem27_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; Find the magnitude and direction of the Poynting vector [mathjaxinline]\vec{S}(t)[/mathjaxinline] at the point [mathjaxinline]P[/mathjaxinline]. &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\vec{S}(t)=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem27_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_finalexam-problem27_4_1&#34; id=&#34;input_finalexam-problem27_4_1&#34; aria-describedby=&#34;status_finalexam-problem27_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;45&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem27_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem27_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_finalexam-problem27_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_finalexam-problem27_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_finalexam-problem27_4_1_dynamath&#34; name=&#34;input_finalexam-problem27_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem27_solution_3&#34;/&gt; &lt;/div&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part d)&lt;/b&gt; What is the flux of the Poynting vector (power) into/out of the capacitor? &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]Power =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem27_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_finalexam-problem27_5_1&#34; id=&#34;input_finalexam-problem27_5_1&#34; aria-describedby=&#34;status_finalexam-problem27_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;45&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem27_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem27_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_finalexam-problem27_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_finalexam-problem27_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_finalexam-problem27_5_1_dynamath&#34; name=&#34;input_finalexam-problem27_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem27_solution_4&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part e)&lt;/b&gt; How does this compare to the time derivative of the energy stored in the electric field? &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;There is no answer checking for this question.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem27_solution_5&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Energy Flow and Charging Capacitor&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; 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&lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;finalexam-problem27-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">W14PS1: Worked Example - Poynting Vector for a Capacitor</h3> <div id="video_W14PS01" class="video closed" data-metadata='{&#34;autoAdvance&#34;: false, &#34;prioritizeHls&#34;: false, &#34;recordedYoutubeIsAvailable&#34;: true, &#34;ytTestTimeout&#34;: 1500, &#34;poster&#34;: null, &#34;streams&#34;: &#34;1.00:epeWa0xBmeM&#34;, &#34;saveStateEnabled&#34;: false, &#34;end&#34;: 0.0, &#34;speed&#34;: null, &#34;completionPercentage&#34;: 0.95, &#34;start&#34;: 0.0, &#34;publishCompletionUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01/handler/publish_completion&#34;, &#34;duration&#34;: null, &#34;autoplay&#34;: false, &#34;savedVideoPosition&#34;: 0.0, &#34;generalSpeed&#34;: 1.0, &#34;autohideHtml5&#34;: false, &#34;ytMetadataEndpoint&#34;: &#34;&#34;, &#34;transcriptTranslationUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01/handler/transcript/translation/__lang__&#34;, &#34;showCaptions&#34;: &#34;true&#34;, &#34;completionEnabled&#34;: false, &#34;captionDataDir&#34;: null, &#34;ytApiUrl&#34;: &#34;https://www.youtube.com/iframe_api&#34;, &#34;saveStateUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01/handler/xmodule_handler/save_user_state&#34;, &#34;transcriptAvailableTranslationsUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS01/handler/transcript/available_translations&#34;, &#34;sources&#34;: [], &#34;transcriptLanguages&#34;: {&#34;en&#34;: &#34;English&#34;}, &#34;transcriptLanguage&#34;: &#34;en&#34;, &#34;lmsRootURL&#34;: &#34;https://openlearninglibrary.mit.edu&#34;}' 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="W14PS01"></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. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-W14PS02" 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="False" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">W14PS2: Worked Example - Poynting Vector for a Solenoid</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">W14PS2: Worked Example - Poynting Vector for a Solenoid</h3> <div id="video_W14PS02" class="video closed" data-metadata='{&#34;autoAdvance&#34;: false, &#34;prioritizeHls&#34;: false, &#34;recordedYoutubeIsAvailable&#34;: true, &#34;ytTestTimeout&#34;: 1500, &#34;poster&#34;: null, &#34;streams&#34;: &#34;1.00:21cwhtq3slk&#34;, &#34;saveStateEnabled&#34;: false, &#34;end&#34;: 0.0, &#34;speed&#34;: null, &#34;completionPercentage&#34;: 0.95, &#34;start&#34;: 0.0, &#34;publishCompletionUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02/handler/publish_completion&#34;, &#34;duration&#34;: null, &#34;autoplay&#34;: false, &#34;savedVideoPosition&#34;: 0.0, &#34;generalSpeed&#34;: 1.0, &#34;autohideHtml5&#34;: false, &#34;ytMetadataEndpoint&#34;: &#34;&#34;, &#34;transcriptTranslationUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02/handler/transcript/translation/__lang__&#34;, &#34;showCaptions&#34;: &#34;true&#34;, &#34;completionEnabled&#34;: false, &#34;captionDataDir&#34;: null, &#34;ytApiUrl&#34;: &#34;https://www.youtube.com/iframe_api&#34;, &#34;saveStateUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02/handler/xmodule_handler/save_user_state&#34;, &#34;transcriptAvailableTranslationsUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS02/handler/transcript/available_translations&#34;, &#34;sources&#34;: [], &#34;transcriptLanguages&#34;: {&#34;en&#34;: &#34;English&#34;}, &#34;transcriptLanguage&#34;: &#34;en&#34;, &#34;lmsRootURL&#34;: &#34;https://openlearninglibrary.mit.edu&#34;}' 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="W14PS02"></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_W14PS02">Downloads and transcripts</h3> <div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_W14PS02"> <div class="wrapper-download-transcripts"> <h4 class="hd hd-5">Transcripts</h4> <a class="btn-link external-track" href="/assets/courseware/v1/2c3b0a5d8cad95f2d36e42670393d28e/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_W14PS02.srt">Download transcript</a> </div> </div> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-W14PS03" 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="False" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">W14PS3: Worked Example - Poynting Vector for a Resistor</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">W14PS3: Worked Example - Poynting Vector for a Resistor</h3> <div id="video_W14PS03" class="video closed" data-metadata='{&#34;autoAdvance&#34;: false, &#34;prioritizeHls&#34;: false, &#34;recordedYoutubeIsAvailable&#34;: true, &#34;ytTestTimeout&#34;: 1500, &#34;poster&#34;: null, &#34;streams&#34;: &#34;1.00:J0MrT4HZDjw&#34;, &#34;saveStateEnabled&#34;: false, &#34;end&#34;: 0.0, &#34;speed&#34;: null, &#34;completionPercentage&#34;: 0.95, &#34;start&#34;: 0.0, &#34;publishCompletionUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03/handler/publish_completion&#34;, &#34;duration&#34;: null, &#34;autoplay&#34;: false, &#34;savedVideoPosition&#34;: 0.0, &#34;generalSpeed&#34;: 1.0, &#34;autohideHtml5&#34;: false, &#34;ytMetadataEndpoint&#34;: &#34;&#34;, &#34;transcriptTranslationUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03/handler/transcript/translation/__lang__&#34;, &#34;showCaptions&#34;: &#34;true&#34;, &#34;completionEnabled&#34;: false, &#34;captionDataDir&#34;: null, &#34;ytApiUrl&#34;: &#34;https://www.youtube.com/iframe_api&#34;, &#34;saveStateUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03/handler/xmodule_handler/save_user_state&#34;, &#34;transcriptAvailableTranslationsUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@W14PS03/handler/transcript/available_translations&#34;, &#34;sources&#34;: [], &#34;transcriptLanguages&#34;: {&#34;en&#34;: &#34;English&#34;}, &#34;transcriptLanguage&#34;: &#34;en&#34;, &#34;lmsRootURL&#34;: &#34;https://openlearninglibrary.mit.edu&#34;}' 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="W14PS03"></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_W14PS03">Downloads and transcripts</h3> <div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_W14PS03"> <div class="wrapper-download-transcripts"> <h4 class="hd hd-5">Transcripts</h4> <a class="btn-link external-track" href="/assets/courseware/v1/1c19b0cf6b5798bfd4471d7567b9b8a4/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/subs_W14PS03.srt">Download transcript</a> </div> </div> </div> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-pset11.6" 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="False" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">Intensity of the Sun</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_pset11_6a" class="problems-wrapper" role="group" aria-labelledby="pset11_6a-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;pset11_6a-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Intensity of the Sun &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6a-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; At the upper surface of the earth&#39;s atmosphere, the time-averaged magnitude of the Poynting vector, referred to as the solar constant, is given by [mathjaxinline]\langle |\vec{\mathbf{S}}|\rangle =1.35\times 10^3 \, \mathrm{W \! \cdot \! m^{-2}}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; If you assume that the sun&#39;s electromagnetic radiation is a plane sinusoidal wave, what are the magnitudes of the electric and magnetic fields? &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle E_0=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_pset11_6a_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_pset11_6a_2_1&#34; id=&#34;input_pset11_6a_2_1&#34; aria-describedby=&#34;status_pset11_6a_2_1&#34; value=&#34;&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_pset11_6a_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_pset11_6a_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_pset11_6a_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt; (in [mathjaxinline]\mathrm{V/m}[/mathjaxinline])&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle B_0=\frac{E_0}{c}=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_pset11_6a_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_pset11_6a_3_1&#34; id=&#34;input_pset11_6a_3_1&#34; aria-describedby=&#34;status_pset11_6a_3_1&#34; value=&#34;&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_pset11_6a_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_pset11_6a_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_pset11_6a_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt; (in [mathjaxinline]\mathrm{\mu T}[/mathjaxinline])&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_pset11_6a_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Intensity of the Sun&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_pset11_6a&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_pset11_6a&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6a-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6a-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6a-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_pset11_6b" class="problems-wrapper" role="group" aria-labelledby="pset11_6b-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;pset11_6b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Intensity of the Sun &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6b-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; What is the time-averaged power radiated by the sun? The mean sun-earth distance is [mathjaxinline]r_{es}=1.5\times 10^{11}\, \mathrm{m}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\langle P_{\text {power},s}\rangle =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_pset11_6b_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_pset11_6b_2_1&#34; id=&#34;input_pset11_6b_2_1&#34; aria-describedby=&#34;status_pset11_6b_2_1&#34; value=&#34;&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_pset11_6b_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_pset11_6b_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_pset11_6b_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt; (in [mathjaxinline]\mathrm{W}[/mathjaxinline])&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_pset11_6b_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Intensity of the Sun&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_pset11_6b&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_pset11_6b&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6b-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6b-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;pset11_6b-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_pset11_6c" class="problems-wrapper" role="group" aria-labelledby="pset11_6c-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;pset11_6c-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Solar Water Disinfection &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@pset11_6c-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; While at the upper surface of the earth&#39;s atmosphere the intensity of light from the sun is [mathjaxinline]\langle |\vec{\mathbf{S}}|\rangle =1.35\times 10^3 \, \mathrm{W \! \cdot \! m^{-2}}[/mathjaxinline], because of our atmosphere not all of that energy makes it to the surface of the earth. In fact, at the surface of the earth, the intensity of sunlight is closer to [mathjaxinline]\langle |\vec{\mathbf{S}}|\rangle =760 \, \mathrm{W \! \cdot \! m^{-2}}[/mathjaxinline] &lt;/p&gt; &lt;p&gt; Contaminated water causes something like 6-60 billion cases of gastrointestinal illness annually. One way to prevent this is by purifying the water through a method called Sodis (solar water disinfection). The ultraviolet part of the solar radiation ([mathjaxinline]\lambda \approx 320-400 \, \mathrm{nm}[/mathjaxinline]) has a lethal effect on human pathogens. It also reacts with the oxygen dissolved in the water to produce reactive forms of oxygen that interfere with the cell structures to kill pathogens. The longer wavelength radiation from the sun, infrared ([mathjaxinline]\lambda \approx 700 \, \mathrm{nm}[/mathjaxinline]), also heats up the water. The microorganisms are also sensitive to high temperatures. It is this combination of UV radiation and heating the water that is used to treat the water. &lt;/p&gt; &lt;p&gt; In this problem, we will focus on the heating part. Assume you have 2 liters of water in a cylindrical bottle with diameter, [mathjaxinline]d=100 \, \mathrm{mm}[/mathjaxinline], and height, [mathjaxinline]h=35 \, \mathrm{cm}[/mathjaxinline], initially at a temperature of [mathjaxinline]T_ i=30{{}^\circ \, \mathrm{C}}[/mathjaxinline]. How long will it take for the water to heat up to [mathjaxinline]T_ f=50{{}^\circ \, \mathrm{C}}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; To solve this problem, you will need to know the specific heat of water, which is [mathjaxinline]C =4186 \, \mathrm{ J\! \cdot \! {{}^\circ \, \mathrm{C}}^{-1} \! \cdot \! kg^{-1}}[/mathjaxinline].The amount of energy the water will need in terms of heat can be written as [mathjaxinline]\Delta U = m C \Delta T[/mathjaxinline], where [mathjaxinline]m[/mathjaxinline] is the mass of water, [mathjaxinline]C[/mathjaxinline] is the specific heat, and [mathjaxinline]\Delta T[/mathjaxinline] is the change in temperature (in [mathjaxinline]{{}^\circ \, \mathrm{C}}[/mathjaxinline]). Assume the density of water is [mathjaxinline]1 \, \mathrm{kg/liter}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]t=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_pset11_6c_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_pset11_6c_2_1&#34; id=&#34;input_pset11_6c_2_1&#34; aria-describedby=&#34;status_pset11_6c_2_1&#34; value=&#34;&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_pset11_6c_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_pset11_6c_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_pset11_6c_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt; (in [mathjaxinline]\mathrm{hours}[/mathjaxinline])&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_pset11_6c_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Solar Water Disinfection&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_pset11_6c&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_pset11_6c&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-unused-finalprep-22" 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="False" data-runtime-version="1"> <h2 class="hd hd-2 unit-title">EM Waves and Force</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-request-token="17c5caf405d511f0a17a0affe2bbc7c1" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22" 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="False" data-runtime-version="1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_finalexam-problem22" class="problems-wrapper" role="group" aria-labelledby="finalexam-problem22-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;finalexam-problem22-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Electromagnetic Waves and Force &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem22-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider a traveling sinusoidal plane electromagnetic wave with magnetic field given by &lt;/p&gt; &lt;table id=&#34;a0000000002&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; class=&#34;eqnarray&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000003&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle \vec{\textbf{B}} (z,t) = B_0 \sin \left( \frac{2\pi }{\lambda } (z+ct) \right) \hat{\textbf{i}} + B_0 \sin \left( \frac{2\pi }{\lambda } (z+ct)\right) \hat{\textbf{j}}[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;width:20%; border:none&#34; class=&#34;eqnnum&#34;&gt;&amp;#160;&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; that is incident on a totally absorbing surface of area [mathjaxinline]A[/mathjaxinline] that is lying in the [mathjaxinline]x-y[/mathjaxinline] plane. Determine a vector expression for the time averaged force acting on the surface. &lt;/p&gt; &lt;p&gt; Write your answer using some or all of the following: [mathjaxinline]A[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], B_0 for [mathjaxinline]B_0[/mathjaxinline], lambda for [mathjaxinline]\lambda[/mathjaxinline], mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], hatk for [mathjaxinline]\hat{\textbf{k}}[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\left&amp;lt; \vec{\textbf{F}} \right&amp;gt; =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem22_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_finalexam-problem22_2_1&#34; id=&#34;input_finalexam-problem22_2_1&#34; aria-describedby=&#34;status_finalexam-problem22_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem22_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem22_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_finalexam-problem22_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_finalexam-problem22_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_finalexam-problem22_2_1_dynamath&#34; name=&#34;input_finalexam-problem22_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem22_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Electromagnetic Waves and Force&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_finalexam-problem22&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_finalexam-problem22&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. 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