<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-html-lesson23_01" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Introduction to Faraday&#39;s Law</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson23_01"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson23_01" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><b>Electromagnetic Induction.</b></p><p> In 1831 Michael Faraday in England and independently Joseph Henry in the United States found experimentally that a change of magnetic field through a coil wire induces a current in the wire. The results of these experiments were the foundation of a new law of nature called Faraday's law of induction.</p><p> The phenomenon of electromagnetic induction has important technological applications encountered in our every day life (electric generators, computer hardware, cellphones, to name a few). In addition, electromagnetic induction is the phenomenon that provides the link between electric and magnetic fields needed to explain the existence of electromagnetic waves.</p><p>To discuss Faraday's law and its implications we need to review two concepts:</p><p><ul><li> Magnetic Flux.</li><li> Electromotive Force (emf).</li></ul></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_9ae47f72215e"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@html_9ae47f72215e" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <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 10.1-10.2: Faraday's Law and Motional EMF</a></li> </ul> </div> </div> </div> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-html-lesson23_02" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23Q1: Magnetic Flux - Review</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson23_02"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@html+block@lesson23_02" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><b>Magnetic Flux</b></p><p>Consider a given surface \(S\) in a region in space where there is a magnetic field \(\mathbf{\vec{B}}\) as shown.</p><center><img src="/assets/courseware/v1/c8440da6868d946de89b5ee8aa98be6b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_html_lesson_23_01c.svg" width="400"/></center><p>The magnetic flux through the surface is defined as: </p><p> \[\Phi_{B} = \iint_{S} \mathbf{\vec{B}}\cdot \mathbf{d\vec{A}} \] </p><p> where \(\mathbf{d\vec{A}}\) is given by \(d\mathbf{\vec{A}}=dA\mathbf{\hat{n}}\), with \(dA\) being the element of area of the surface and \(\mathbf{\hat{n}}\) the unit vector perpendicular to the surface.</p><p>Note: In the definition above we are calculating the magnetic flux over an <b>open</b> surface. </p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_29" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_29-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.5" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;checkpoint_w10_29-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29-problem-progress&#34; tabindex=&#34;-1&#34;&gt; CQ: Magnetic Flux &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_29-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/3d4343610b007c4f0290ed2d21fe826c/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_29.svg&#34; width=&#34;660&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Consider the circular loop perpendicular to the axis of the bar magnet in figure a). The south pole points along the [mathjaxinline]+\mathbf{\hat{k}}[/mathjaxinline] - direction, the direction of [mathjaxinline]\mathbf{\hat{n}}[/mathjaxinline]. The flux through the open surface enclosed by the circle is &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_29_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_29_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_2_1&#34; id=&#34;input_checkpoint_w10_29_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_2_1-choice_1-label&#34; for=&#34;input_checkpoint_w10_29_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_2_1&#34;&gt; &lt;text&gt; positive&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_2_1&#34; id=&#34;input_checkpoint_w10_29_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_2_1-choice_2-label&#34; for=&#34;input_checkpoint_w10_29_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_2_1&#34;&gt; &lt;text&gt; zero&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_2_1&#34; id=&#34;input_checkpoint_w10_29_2_1_choice_3&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_2_1-choice_3-label&#34; for=&#34;input_checkpoint_w10_29_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_2_1&#34;&gt; &lt;text&gt; negative&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_29_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_29_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;/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_checkpoint_w10_29_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; Consider a rectangular loop parallel to the infinite current wire shown in figure b). The unit normal to the surface enclosed by the loop, [mathjaxinline]\mathbf{\hat{n}}[/mathjaxinline], points out of the screen, in the [mathjaxinline]-\mathbf{\hat{\theta }}[/mathjaxinline] direction. The flux through the open surface enclosed by the loop is &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_29_3_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_29_3_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_3_1&#34; id=&#34;input_checkpoint_w10_29_3_1_choice_4&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_3_1-choice_4-label&#34; for=&#34;input_checkpoint_w10_29_3_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_3_1&#34;&gt; &lt;text&gt; positive&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_3_1&#34; id=&#34;input_checkpoint_w10_29_3_1_choice_5&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_3_1-choice_5-label&#34; for=&#34;input_checkpoint_w10_29_3_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_3_1&#34;&gt; &lt;text&gt; zero&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_3_1&#34; id=&#34;input_checkpoint_w10_29_3_1_choice_6&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_6&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_3_1-choice_6-label&#34; for=&#34;input_checkpoint_w10_29_3_1_choice_6&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_3_1&#34;&gt; &lt;text&gt; negative&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_29_3_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_29_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;/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_checkpoint_w10_29_solution_2&#34;/&gt; &lt;/div&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/8b7399a3e5c009c3e287571420654cca/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_29b.svg&#34; width=&#34;385&#34;/&gt; &lt;/center&gt; &lt;p&gt; Consider now a sphere in front of the bar magnet. Consider [mathjaxinline]\mathbf{\hat{n}}[/mathjaxinline] to be the normal to the sphere&#39;s surface pointing outward. The flux through the closed spherical surface is &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_29_4_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_29_4_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_4_1&#34; id=&#34;input_checkpoint_w10_29_4_1_choice_7&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_7&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_4_1-choice_7-label&#34; for=&#34;input_checkpoint_w10_29_4_1_choice_7&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_4_1&#34;&gt; &lt;text&gt; positive&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_4_1&#34; id=&#34;input_checkpoint_w10_29_4_1_choice_8&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_8&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_4_1-choice_8-label&#34; for=&#34;input_checkpoint_w10_29_4_1_choice_8&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_4_1&#34;&gt; &lt;text&gt; zero&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_29_4_1&#34; id=&#34;input_checkpoint_w10_29_4_1_choice_9&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_9&#34;/&gt;&lt;label id=&#34;checkpoint_w10_29_4_1-choice_9-label&#34; for=&#34;input_checkpoint_w10_29_4_1_choice_9&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_29_4_1&#34;&gt; &lt;text&gt; negative&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_29_4_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_29_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;/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_checkpoint_w10_29_solution_3&#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;CQ: Magnetic Flux&#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_checkpoint_w10_29&#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_checkpoint_w10_29&#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-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-checkpoint_w10_7" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23Q2: Electromotive Force</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_7" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_7-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;checkpoint_w10_7-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Electromotive Force &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_7-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Given a force [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline] acting on a charge [mathjaxinline]q[/mathjaxinline], the electromotive force or emf is defined as the line integral over a closed loop [mathjaxinline]C[/mathjaxinline]: &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 \mathcal{E}=\oint _{C}\frac{\mathbf{\vec{F}}}{q}\cdot d\mathbf{\vec{s}}[/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; Which of the following statements about the electromotive force are true? (Check all that apply) &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_7_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_7_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_checkpoint_w10_7_2_1[]&#34; id=&#34;input_checkpoint_w10_7_2_1_choice_0&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_0&#34;/&gt;&lt;label id=&#34;checkpoint_w10_7_2_1-choice_0-label&#34; for=&#34;input_checkpoint_w10_7_2_1_choice_0&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_7_2_1&#34;&gt; &lt;text&gt;The electromotive force is not a force, it is the work done by the force [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline] per unit charge to move the charges through a closed loop.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_checkpoint_w10_7_2_1[]&#34; id=&#34;input_checkpoint_w10_7_2_1_choice_1&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w10_7_2_1-choice_1-label&#34; for=&#34;input_checkpoint_w10_7_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_7_2_1&#34;&gt; &lt;text&gt;The force [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline] is a conservative force.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_checkpoint_w10_7_2_1[]&#34; id=&#34;input_checkpoint_w10_7_2_1_choice_2&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w10_7_2_1-choice_2-label&#34; for=&#34;input_checkpoint_w10_7_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_7_2_1&#34;&gt; &lt;text&gt;The closed loop in the definition of the emf must be a conducting wire.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;checkbox&#34; name=&#34;input_checkpoint_w10_7_2_1[]&#34; id=&#34;input_checkpoint_w10_7_2_1_choice_3&#34; class=&#34;field-input input-checkbox&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;checkpoint_w10_7_2_1-choice_3-label&#34; for=&#34;input_checkpoint_w10_7_2_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_7_2_1&#34;&gt; &lt;text&gt;If the closed loop in the definition of the emf is a conducting wire of resistance [mathjaxinline]R[/mathjaxinline] a current [mathjaxinline]I = \mathcal{E}/R[/mathjaxinline] flows in the wire.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_7_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_7_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;/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_checkpoint_w10_7_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;Electromotive 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_checkpoint_w10_7&#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_checkpoint_w10_7&#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-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L23v02" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23v2: Motional EMF</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_8" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_8-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8/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;checkpoint_w10_8-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Motional EMF &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_8-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; We have already seen the concept of emf in the context of DC circuits, where the force moving the charges is the chemical force due to the chemical reactions inside the battery. Other examples of emf are the work done per unit charge by the motor moving the belt of a Van der Graaff generator, or the solar energy moving charges in a solar cell. &lt;/p&gt; &lt;p&gt; In this question and in the next video, we will consider the magnetic force on a charge inside a conductor moving in a magnetic field. This magnetic force will induce a current in the conductor. Because the resulting emf is due to the motion of the conductor with respect to the magnetic field we will call it &#34;Motional emf&#34;. &lt;/p&gt; &lt;p&gt; Consider a rectangular loop of length [mathjaxinline]l[/mathjaxinline] and width [mathjaxinline]w[/mathjaxinline] moving with speed [mathjaxinline]v[/mathjaxinline] along the [mathjaxinline]\mathbf{\hat{i}}[/mathjaxinline] direction. The loop is contained in the [mathjaxinline](x,y)[/mathjaxinline] plane perpendicular to a uniform magnetic field of magnitude [mathjaxinline]B[/mathjaxinline] and direction [mathjaxinline]+\mathbf{\hat{k}}[/mathjaxinline] (out of the screen). &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/6c1728f1cf2cefc93eac4fdb463a157e/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_8.svg&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;p&gt; At the instant shown in the figure above, the loop is moving out of the magnetic field region, therefore only a part of it is submerged in the magnetic field. We will assume that the charges that are free to move in the conductor are the positive charges (adopt the standard convention for current). &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Calculate the force on the charges labeled as [mathjaxinline]q_1[/mathjaxinline], [mathjaxinline]q_2[/mathjaxinline], [mathjaxinline]q_3[/mathjaxinline], and [mathjaxinline]q_4[/mathjaxinline] in the figure, where [mathjaxinline]q_1=q_2=q_3=q_4=q&amp;gt;0[/mathjaxinline]. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]v[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], hati for [mathjaxinline]\mathbf{\hat{i}}[/mathjaxinline], hatj for [mathjaxinline]\mathbf{\hat{j}}[/mathjaxinline], and hatk for [mathjaxinline]\mathbf{\hat{k}}[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;On charge 1, [mathjaxinline]\mathbf{\vec{F}}_{mag}=[/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_checkpoint_w10_8_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_checkpoint_w10_8_2_1&#34; id=&#34;input_checkpoint_w10_8_2_1&#34; aria-describedby=&#34;status_checkpoint_w10_8_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_8_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_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_checkpoint_w10_8_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_8_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_8_2_1_dynamath&#34; name=&#34;input_checkpoint_w10_8_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;On charge 2, [mathjaxinline]\mathbf{\vec{F}}_{mag}=[/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_checkpoint_w10_8_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_checkpoint_w10_8_3_1&#34; id=&#34;input_checkpoint_w10_8_3_1&#34; aria-describedby=&#34;status_checkpoint_w10_8_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_8_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_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_checkpoint_w10_8_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_8_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_8_3_1_dynamath&#34; name=&#34;input_checkpoint_w10_8_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;On charge 3, [mathjaxinline]\mathbf{\vec{F}}_{mag}=[/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_checkpoint_w10_8_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_checkpoint_w10_8_4_1&#34; id=&#34;input_checkpoint_w10_8_4_1&#34; aria-describedby=&#34;status_checkpoint_w10_8_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_8_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_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_checkpoint_w10_8_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_8_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_8_4_1_dynamath&#34; name=&#34;input_checkpoint_w10_8_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;On charge 4, [mathjaxinline]\mathbf{\vec{F}}_{mag}=[/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_checkpoint_w10_8_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_checkpoint_w10_8_5_1&#34; id=&#34;input_checkpoint_w10_8_5_1&#34; aria-describedby=&#34;status_checkpoint_w10_8_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_8_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_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_checkpoint_w10_8_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_8_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_8_5_1_dynamath&#34; name=&#34;input_checkpoint_w10_8_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; The direction of the resulting (conventional) current is: &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_8_6_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_8_6_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_8_6_1&#34; id=&#34;input_checkpoint_w10_8_6_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w10_8_6_1-choice_1-label&#34; for=&#34;input_checkpoint_w10_8_6_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_8_6_1&#34;&gt; &lt;text&gt; clockwise&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_8_6_1&#34; id=&#34;input_checkpoint_w10_8_6_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w10_8_6_1-choice_2-label&#34; for=&#34;input_checkpoint_w10_8_6_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_8_6_1&#34;&gt; &lt;text&gt; counterclockwise&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_8_6_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_6_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;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; Calculate the motional emf by evaluating the line integral of [mathjaxinline]\displaystyle \frac{\mathbf{\vec{F}}_{mag}}{q}[/mathjaxinline] over the rectangle. &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 \mathcal{E} = \oint _{rectangle}\frac{\mathbf{\vec{F}}_{mag}}{q}\cdot d\mathbf{\vec{s}}[/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; Circulate along the loop in the counterclockwise direction. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]v[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and [mathjaxinline]w[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E}=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_checkpoint_w10_8_7_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_checkpoint_w10_8_7_1&#34; id=&#34;input_checkpoint_w10_8_7_1&#34; aria-describedby=&#34;status_checkpoint_w10_8_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_8_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_8_7_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_checkpoint_w10_8_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_8_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_8_7_1_dynamath&#34; name=&#34;input_checkpoint_w10_8_7_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_checkpoint_w10_8_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;Motional EMF&#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_checkpoint_w10_8&#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_checkpoint_w10_8&#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;checkpoint_w10_8-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;checkpoint_w10_8-problem-title&#34;&gt;None &lt;/span&gt; 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-checkpoint_w10_25" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23Q3: Motional EMF on a Wire</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_26" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_26-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="2.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;checkpoint_w10_26-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Moving away from a wire &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_26-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/f9da012422939b317173106317510eaa/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_26.svg&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;p&gt; A thin conducting rod of length [mathjaxinline]L[/mathjaxinline] is moving away from an infinite wire with a constant velocity [mathjaxinline]\mathbf{\vec{v}} = v_0\mathbf{\hat{r}}[/mathjaxinline] and its length parallel to the wire. The distance between the wire and the rod is [mathjaxinline]r[/mathjaxinline]. A current [mathjaxinline]I[/mathjaxinline] is flowing along the wire as shown. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Consider the direction of [mathjaxinline]\mathbf{\vec{F}}[/mathjaxinline], the force exerted on a charge [mathjaxinline]q&amp;gt;0[/mathjaxinline] in the conductor. &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_checkpoint_w10_26_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w10_26_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_26_2_1&#34; id=&#34;input_checkpoint_w10_26_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w10_26_2_1-choice_1-label&#34; for=&#34;input_checkpoint_w10_26_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_26_2_1&#34;&gt; &lt;text&gt; There is an excess of positive charges at the upper end of the rod and an excess of negative charges at the lower end.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w10_26_2_1&#34; id=&#34;input_checkpoint_w10_26_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w10_26_2_1-choice_2-label&#34; for=&#34;input_checkpoint_w10_26_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w10_26_2_1&#34;&gt; &lt;text&gt; There is an excess of negative charges at the upper end of the rod and an excess of positive charges at the lower end.&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w10_26_2_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_26_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;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Calculate [mathjaxinline]\mathcal{E} = \int _{-}^{+}\mathbf{\vec{v}}\times \mathbf{\vec{B}} \cdot \mathbf{d\vec{s}}[/mathjaxinline], the motional emf as the line integral of [mathjaxinline]\mathbf{\vec{v}}\times \mathbf{\vec{B}}[/mathjaxinline] between the negative and positive end of the rod. Express your answer in terms of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], and mu_0 for [mathjaxinline]\mu _0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E}=[/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_checkpoint_w10_26_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_checkpoint_w10_26_3_1&#34; id=&#34;input_checkpoint_w10_26_3_1&#34; aria-describedby=&#34;status_checkpoint_w10_26_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_26_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_26_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_checkpoint_w10_26_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_26_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_26_3_1_dynamath&#34; name=&#34;input_checkpoint_w10_26_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_checkpoint_w10_26_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;Moving away from a wire&#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_checkpoint_w10_26&#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_checkpoint_w10_26&#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;checkpoint_w10_26-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;checkpoint_w10_26-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;checkpoint_w10_26-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="True"> <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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-L23v03" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23v3: Motional EMF from Magnetic Flux</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_9" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_9-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9/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;checkpoint_w10_9-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Changing Magnetic Flux &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_9-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider a rectangular loop of length [mathjaxinline]l[/mathjaxinline] and width [mathjaxinline]w[/mathjaxinline] moving with speed [mathjaxinline]v[/mathjaxinline] along the [mathjaxinline]\mathbf{\hat{i}}[/mathjaxinline] direction. The loop is contained in the [mathjaxinline](x,y)[/mathjaxinline] plane perpendicular to a uniform magnetic field of magnitude [mathjaxinline]B[/mathjaxinline] and out of the screen ([mathjaxinline]+\mathbf{\hat{k}}[/mathjaxinline]). &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/db6c02ca93fd3607a4796bb75872bb25/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_9.svg&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;p&gt; At the instant shown in the figure above only a part of the loop of length [mathjaxinline]s(t)[/mathjaxinline] is inside the region of magnetic field. Consider the element of area [mathjaxinline]d\mathbf{\vec{A}}[/mathjaxinline] to be out of the screen ([mathjaxinline]+\mathbf{\hat{k}})[/mathjaxinline] &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Calculate [mathjaxinline]\Phi _{mag}[/mathjaxinline], the magnetic flux through the area enclosed by the loop at the instant shown in the figure. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]w[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]v[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt; [mathjaxinline]\Phi _{mag}=[/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_checkpoint_w10_9_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_checkpoint_w10_9_2_1&#34; id=&#34;input_checkpoint_w10_9_2_1&#34; aria-describedby=&#34;status_checkpoint_w10_9_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_9_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_9_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_checkpoint_w10_9_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_9_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_9_2_1_dynamath&#34; name=&#34;input_checkpoint_w10_9_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_checkpoint_w10_9_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; Calculate [mathjaxinline]\displaystyle \frac{d\Phi _ B}{dt}[/mathjaxinline], the rate of change of magnetic flux with respect to time. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]w[/mathjaxinline], [mathjaxinline]s[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]v[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt; [mathjaxinline]\displaystyle \frac{d\Phi _ B}{dt}=[/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_checkpoint_w10_9_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_checkpoint_w10_9_3_1&#34; id=&#34;input_checkpoint_w10_9_3_1&#34; aria-describedby=&#34;status_checkpoint_w10_9_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_9_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_9_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_checkpoint_w10_9_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_9_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_9_3_1_dynamath&#34; name=&#34;input_checkpoint_w10_9_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_checkpoint_w10_9_solution_2&#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;Changing Magnetic Flux&#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_checkpoint_w10_9&#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_checkpoint_w10_9&#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;checkpoint_w10_9-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;checkpoint_w10_9-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;checkpoint_w10_9-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="True"> <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@L23v03"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@video+block@L23v03" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">L23v3: Motional EMF from Magnetic Flux</h3> <div id="video_L23v03" class="video closed" data-metadata='{&#34;streams&#34;: &#34;1.00:CtDxKx5TZxw&#34;, &#34;savedVideoPosition&#34;: 0.0, &#34;poster&#34;: null, &#34;captionDataDir&#34;: null, &#34;transcriptTranslationUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L23v03/handler/transcript/translation/__lang__&#34;, &#34;lmsRootURL&#34;: &#34;https://openlearninglibrary.mit.edu&#34;, &#34;saveStateUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L23v03/handler/xmodule_handler/save_user_state&#34;, &#34;ytTestTimeout&#34;: 1500, &#34;autoplay&#34;: false, &#34;ytApiUrl&#34;: &#34;https://www.youtube.com/iframe_api&#34;, &#34;completionEnabled&#34;: false, &#34;showCaptions&#34;: &#34;true&#34;, &#34;transcriptLanguages&#34;: {&#34;en&#34;: &#34;English&#34;}, &#34;end&#34;: 0.0, &#34;start&#34;: 0.0, &#34;saveStateEnabled&#34;: false, &#34;prioritizeHls&#34;: false, &#34;generalSpeed&#34;: 1.0, &#34;recordedYoutubeIsAvailable&#34;: true, &#34;sources&#34;: [], &#34;publishCompletionUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L23v03/handler/publish_completion&#34;, &#34;speed&#34;: null, &#34;duration&#34;: null, &#34;transcriptLanguage&#34;: &#34;en&#34;, &#34;ytMetadataEndpoint&#34;: &#34;&#34;, &#34;autoAdvance&#34;: false, &#34;transcriptAvailableTranslationsUrl&#34;: &#34;/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@video+block@L23v03/handler/transcript/available_translations&#34;, &#34;autohideHtml5&#34;: false, &#34;completionPercentage&#34;: 0.95}' 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="L23v03"></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-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-motionalEMFproblem" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">L23Q4: A Rectangular Loop</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="c326a86a05d411f095120e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w10_24" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w10_24-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="1.5" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;checkpoint_w10_24-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24-problem-progress&#34; tabindex=&#34;-1&#34;&gt; A rectangular loop &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w10_24-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; A rectangular conducting coil of length [mathjaxinline]l[/mathjaxinline], height [mathjaxinline]h[/mathjaxinline], and resistance [mathjaxinline]R[/mathjaxinline] is moving parallel to the [mathjaxinline](x,y)[/mathjaxinline] plane at a constant velocity [mathjaxinline]\mathbf{\vec{v}}=v_0\mathbf{\hat{i}}[/mathjaxinline] as shown. It continues to move with this velocity through a region containing a uniform magnetic field of magnitude [mathjaxinline]B_0[/mathjaxinline]. The magnetic field is in the [mathjaxinline]-\mathbf{\hat{k}}[/mathjaxinline] direction and is confined in the region [mathjaxinline]0\le x \le 3l[/mathjaxinline]. Consider the unit normal to the surface enclosed by the loop to be in the same direction as the magnetic field, [mathjaxinline]\mathbf{\hat{n}} = -\mathbf{\hat{k}}[/mathjaxinline] &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/8f59f28a591e1b2d6ebf87bf8b778138/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_24.svg&#34; width=&#34;660&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Calculate [mathjaxinline]\mathcal{E}[/mathjaxinline] and [mathjaxinline]I[/mathjaxinline], the induced emf and the current in the loop, at the instant when the loop is entering the region of magnetic field (leftmost figure above). Express your answer in terms of [mathjaxinline]R[/mathjaxinline],[mathjaxinline]l[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], and B_0 for [mathjaxinline]B_0[/mathjaxinline] as needed. A &lt;b class=&#34;bfseries&#34;&gt;clockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;positive&lt;/b&gt;. A &lt;b class=&#34;bfseries&#34;&gt;counterclockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;negative&lt;/b&gt;. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E}=[/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_checkpoint_w10_24_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_checkpoint_w10_24_2_1&#34; id=&#34;input_checkpoint_w10_24_2_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_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_checkpoint_w10_24_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_2_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_checkpoint_w10_24_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I=[/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_checkpoint_w10_24_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_checkpoint_w10_24_3_1&#34; id=&#34;input_checkpoint_w10_24_3_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_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_checkpoint_w10_24_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_3_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_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_checkpoint_w10_24_solution_2&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; Calculate [mathjaxinline]\mathcal{E}[/mathjaxinline] and [mathjaxinline]I[/mathjaxinline], the induced emf and the current in the loop, at the instant when the loop is completely immersed in the magnetic field region (central figure above). Express your answer in terms of [mathjaxinline]R[/mathjaxinline],[mathjaxinline]l[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], and B_0 for [mathjaxinline]B_0[/mathjaxinline] as needed. A &lt;b class=&#34;bfseries&#34;&gt;clockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;positive&lt;/b&gt;. A &lt;b class=&#34;bfseries&#34;&gt;counterclockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;negative&lt;/b&gt;. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E}=[/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_checkpoint_w10_24_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_checkpoint_w10_24_4_1&#34; id=&#34;input_checkpoint_w10_24_4_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_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_checkpoint_w10_24_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_4_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_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_checkpoint_w10_24_solution_3&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I=[/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_checkpoint_w10_24_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_checkpoint_w10_24_5_1&#34; id=&#34;input_checkpoint_w10_24_5_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_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_checkpoint_w10_24_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_5_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_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_checkpoint_w10_24_solution_4&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part c)&lt;/b&gt; Calculate [mathjaxinline]\mathcal{E}[/mathjaxinline] and [mathjaxinline]I[/mathjaxinline], the induced emf and the current in the loop, at the instant when the loop is exiting the region of magnetic field (rightmost figure). Express your answer in terms of [mathjaxinline]R[/mathjaxinline],[mathjaxinline]l[/mathjaxinline], [mathjaxinline]h[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], and B_0 for [mathjaxinline]B_0[/mathjaxinline] as needed. A &lt;b class=&#34;bfseries&#34;&gt;clockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;positive&lt;/b&gt;. A &lt;b class=&#34;bfseries&#34;&gt;counterclockwise&lt;/b&gt; current is &lt;b class=&#34;bfseries&#34;&gt;negative&lt;/b&gt;. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E}=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_checkpoint_w10_24_6_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_checkpoint_w10_24_6_1&#34; id=&#34;input_checkpoint_w10_24_6_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_6_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_checkpoint_w10_24_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_6_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_6_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_checkpoint_w10_24_solution_5&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I=[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_checkpoint_w10_24_7_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_checkpoint_w10_24_7_1&#34; id=&#34;input_checkpoint_w10_24_7_1&#34; aria-describedby=&#34;status_checkpoint_w10_24_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;30&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w10_24_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w10_24_7_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_checkpoint_w10_24_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w10_24_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w10_24_7_1_dynamath&#34; name=&#34;input_checkpoint_w10_24_7_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_checkpoint_w10_24_solution_6&#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;A rectangular loop&#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_checkpoint_w10_24&#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_checkpoint_w10_24&#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;checkpoint_w10_24-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;checkpoint_w10_24-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;checkpoint_w10_24-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="True"> <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> </div>

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