<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" 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-examprep-problem4" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Two Loop RC Circuit</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_examprep-problem4" class="problems-wrapper" role="group" aria-labelledby="examprep-problem4-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4/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;examprep-problem4-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Two Loop RC Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@examprep-problem4-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider the circuit shown below with a battery of voltage [mathjaxinline]V[/mathjaxinline], two resistors of resistances [mathjaxinline]R_1[/mathjaxinline] and [mathjaxinline]R_2[/mathjaxinline], and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline]. Assume that the capacitor has charge [mathjaxinline]\pm Q[/mathjaxinline] with the polarities as shown. For the first two questions, assume that the switch [mathjaxinline]S[/mathjaxinline] is closed but don&#39;t assume anything about how long it has been closed. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/f11fed994ce175da43d5082790942e20/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_examprep5.png&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; With the assumed directions of the current as shown, what does Kirchhoff&#39;s 2nd Law give you if you move clockwise around the left loop (e.g. through the battery and [mathjaxinline]R_1[/mathjaxinline]). Be careful of your signs. &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;This part cannot be checked online.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_examprep-problem4_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; With the assumed directions of the current as shown, what does Kirchhoff&#39;s 2nd Law give you if you move clockwise around the outer loop (e.g. through the battery, the capacitor, and [mathjaxinline]R_2[/mathjaxinline]). Be careful of your signs! &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;This part cannot be checked online.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_examprep-problem4_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; What is the relation between [mathjaxinline]I_2[/mathjaxinline] and [mathjaxinline]Q(t)[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;This part cannot be checked online.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_examprep-problem4_solution_3&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part d)&lt;/b&gt; Now, assume that the switch has been open for a long time and then is closed. Just after the switch is closed, what are the values of [mathjaxinline]I_1(t=0)\equiv I_{1,0}[/mathjaxinline], [mathjaxinline]I_2(t=0)\equiv I_{2,0}[/mathjaxinline] and [mathjaxinline]Q(t=0)\equiv Q_0[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; Express your answers in terms of [mathjaxinline]V[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], and R_2 for [mathjaxinline]R_2[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_{1,0}=[/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_examprep-problem4_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_examprep-problem4_2_1&#34; id=&#34;input_examprep-problem4_2_1&#34; aria-describedby=&#34;status_examprep-problem4_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_2_1_dynamath&#34; name=&#34;input_examprep-problem4_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;[mathjaxinline]I_{2,0}=[/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_examprep-problem4_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_examprep-problem4_3_1&#34; id=&#34;input_examprep-problem4_3_1&#34; aria-describedby=&#34;status_examprep-problem4_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_3_1_dynamath&#34; name=&#34;input_examprep-problem4_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;[mathjaxinline]Q_0=[/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_examprep-problem4_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_examprep-problem4_4_1&#34; id=&#34;input_examprep-problem4_4_1&#34; aria-describedby=&#34;status_examprep-problem4_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_4_1_dynamath&#34; name=&#34;input_examprep-problem4_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_examprep-problem4_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; A long, long time after the switch is closed, [mathjaxinline]t_ f[/mathjaxinline] such that [mathjaxinline]t_ f\gg \tau[/mathjaxinline], where [mathjaxinline]\tau[/mathjaxinline] is the characteristic time constant associated with the circuit, what are the values of [mathjaxinline]I_1(t_ f)\equiv I_{1,f}[/mathjaxinline], [mathjaxinline]I_2(t_ f)\equiv I_{2,f}[/mathjaxinline], and [mathjaxinline]Q(t_ f)\equiv Q_ f[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; Express your answers in terms of [mathjaxinline]V[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], and R_2 for [mathjaxinline]R_2[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_{1,f}=[/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_examprep-problem4_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_examprep-problem4_5_1&#34; id=&#34;input_examprep-problem4_5_1&#34; aria-describedby=&#34;status_examprep-problem4_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_5_1_dynamath&#34; name=&#34;input_examprep-problem4_5_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;[mathjaxinline]I_{2,f}=[/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_examprep-problem4_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_examprep-problem4_6_1&#34; id=&#34;input_examprep-problem4_6_1&#34; aria-describedby=&#34;status_examprep-problem4_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_6_1_dynamath&#34; name=&#34;input_examprep-problem4_6_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;[mathjaxinline]Q_ f=[/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_examprep-problem4_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_examprep-problem4_7_1&#34; id=&#34;input_examprep-problem4_7_1&#34; aria-describedby=&#34;status_examprep-problem4_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_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_examprep-problem4_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_7_1_dynamath&#34; name=&#34;input_examprep-problem4_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_examprep-problem4_solution_5&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part f)&lt;/b&gt; Find expressions for [mathjaxinline]I_2(t)[/mathjaxinline] and [mathjaxinline]Q(t)[/mathjaxinline] good for all [mathjaxinline]t&amp;gt;0[/mathjaxinline] after the switch is closed. &lt;/p&gt; &lt;p&gt; Express your answers in terms of [mathjaxinline]V[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_2(t)=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 7&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_examprep-problem4_8_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_examprep-problem4_8_1&#34; id=&#34;input_examprep-problem4_8_1&#34; aria-describedby=&#34;status_examprep-problem4_8_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_8_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_8_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_examprep-problem4_8_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_8_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_8_1_dynamath&#34; name=&#34;input_examprep-problem4_8_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;[mathjaxinline]Q(t)=[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 8&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_examprep-problem4_9_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_examprep-problem4_9_1&#34; id=&#34;input_examprep-problem4_9_1&#34; aria-describedby=&#34;status_examprep-problem4_9_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_9_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_9_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_examprep-problem4_9_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_9_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_9_1_dynamath&#34; name=&#34;input_examprep-problem4_9_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_examprep-problem4_solution_6&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part g)&lt;/b&gt; After being closed for a long time, the switch is now opened again. How long does it take after the switch is opened for the charge on the capacitor to fall to [mathjaxinline]1/e[/mathjaxinline] of its value just before the switch was opened? &lt;/p&gt; &lt;p&gt; Express your answers in terms of [mathjaxinline]V[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;The time it takes=&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 9&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_examprep-problem4_10_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_examprep-problem4_10_1&#34; id=&#34;input_examprep-problem4_10_1&#34; aria-describedby=&#34;status_examprep-problem4_10_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_examprep-problem4_10_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_examprep-problem4_10_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_examprep-problem4_10_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_examprep-problem4_10_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_examprep-problem4_10_1_dynamath&#34; name=&#34;input_examprep-problem4_10_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_examprep-problem4_solution_7&#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;Two Loop RC Circuit&#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_examprep-problem4&#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_examprep-problem4&#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;examprep-problem4-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;examprep-problem4-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;examprep-problem4-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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" 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-w10d3-q33" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">RL Circuit with Solenoid</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fps_fridayw11_3a" class="problems-wrapper" role="group" aria-labelledby="fps_fridayw11_3a-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a/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;fps_fridayw11_3a-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a-problem-progress&#34; tabindex=&#34;-1&#34;&gt; RL Circuit with Solenoid &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3a-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider the circuit shown in the figure below, consisting of a battery (emf [mathjaxinline]\mathcal{E}[/mathjaxinline]), a resistor with resistance [mathjaxinline]R[/mathjaxinline], a long solenoid of radius [mathjaxinline]a[/mathjaxinline], height [mathjaxinline]H[/mathjaxinline] that has [mathjaxinline]N[/mathjaxinline] turns and a switch [mathjaxinline]S[/mathjaxinline]. Coaxial with the solenoid at the center of the solenoid is a circular copper ring of wire of radius [mathjaxinline]b[/mathjaxinline] with [mathjaxinline]b&amp;gt;a[/mathjaxinline] and resistance [mathjaxinline]R_1[/mathjaxinline]. At [mathjaxinline]t=0[/mathjaxinline], the switch [mathjaxinline]S[/mathjaxinline] is closed. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/46e95ebcc5e3442231dfe477ff169380/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_friday_w11_p3_fig1.png&#34; width=&#34;550&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; What is the rate that the current is changing the instant the switch is closed at [mathjaxinline]t=0[/mathjaxinline]? Express your answer in terms of [mathjaxinline]R[/mathjaxinline], epsilon for [mathjaxinline]\mathcal{E}[/mathjaxinline], and [mathjaxinline]L[/mathjaxinline], the self-inductance of the solenoid, as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\dfrac {dI}{dt}(t=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_fps_fridayw11_3a_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_fps_fridayw11_3a_2_1&#34; id=&#34;input_fps_fridayw11_3a_2_1&#34; aria-describedby=&#34;status_fps_fridayw11_3a_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fps_fridayw11_3a_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fps_fridayw11_3a_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_fps_fridayw11_3a_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fps_fridayw11_3a_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fps_fridayw11_3a_2_1_dynamath&#34; name=&#34;input_fps_fridayw11_3a_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_fps_fridayw11_3a_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;RL Circuit with Solenoid&#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_fps_fridayw11_3a&#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_fps_fridayw11_3a&#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;fps_fridayw11_3a-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;fps_fridayw11_3a-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;fps_fridayw11_3a-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@fps_fridayw11_3b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3b" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fps_fridayw11_3b" class="problems-wrapper" role="group" aria-labelledby="fps_fridayw11_3b-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3b" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3b/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;fps_fridayw11_3b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Question 2 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3b-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 self-inductance [mathjaxinline]L[/mathjaxinline] of the solenoid? You may assume that the solenoid is very long and so can ignore edge effects. Express your answer in terms of mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]H[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and epsilon for [mathjaxinline]\mathcal{E}[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]L =[/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_fps_fridayw11_3b_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_fps_fridayw11_3b_2_1&#34; id=&#34;input_fps_fridayw11_3b_2_1&#34; aria-describedby=&#34;status_fps_fridayw11_3b_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fps_fridayw11_3b_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fps_fridayw11_3b_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_fps_fridayw11_3b_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fps_fridayw11_3b_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fps_fridayw11_3b_2_1_dynamath&#34; name=&#34;input_fps_fridayw11_3b_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_fps_fridayw11_3b_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;Question 2&#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_fps_fridayw11_3b&#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_fps_fridayw11_3b&#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;fps_fridayw11_3b-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;fps_fridayw11_3b-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;fps_fridayw11_3b-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@fps_fridayw11_3c"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3c" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fps_fridayw11_3c" class="problems-wrapper" role="group" aria-labelledby="fps_fridayw11_3c-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3c" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3c/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;fps_fridayw11_3c-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3c-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Question 3 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fps_fridayw11_3c-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 c)&lt;/b&gt; What is the induced current in the copper ring at the instant the switch is closed? Express your answer in terms of mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]H[/mathjaxinline], [mathjaxinline]N[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and epsilon for [mathjaxinline]\mathcal{E}[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_{ind}=[/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_fps_fridayw11_3c_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_fps_fridayw11_3c_2_1&#34; id=&#34;input_fps_fridayw11_3c_2_1&#34; aria-describedby=&#34;status_fps_fridayw11_3c_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fps_fridayw11_3c_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fps_fridayw11_3c_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_fps_fridayw11_3c_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fps_fridayw11_3c_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fps_fridayw11_3c_2_1_dynamath&#34; name=&#34;input_fps_fridayw11_3c_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_fps_fridayw11_3c_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;Question 3&#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_fps_fridayw11_3c&#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_fps_fridayw11_3c&#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;fps_fridayw11_3c-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;fps_fridayw11_3c-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;fps_fridayw11_3c-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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" data-init="VerticalStudentView" data-runtime-version="1" data-has-score="False" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@checkpoint_w12_20" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">A Solenoid and a Switch</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_20"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" 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_w12_20" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w12_20" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w12_20-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_20" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_20/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_w12_20-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_20-problem-progress&#34; tabindex=&#34;-1&#34;&gt; A Solenoid and a Switch &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_20-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/v1/5463247f7d9e91e5f1955b689d2b6789/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w12_20.svg&#34; width=&#34;330&#34;/&gt; &lt;/center&gt; &lt;p&gt; A solenoid is connected to a battery as shown. The switch has been opened for a very long time. At time [mathjaxinline]t = 0[/mathjaxinline], the switch is closed and then reopened at time [mathjaxinline]t = T[/mathjaxinline], where [mathjaxinline]T[/mathjaxinline] is a very long time. &lt;/p&gt; &lt;p&gt; In the figure below, the current through the inductor, [mathjaxinline]I_ L[/mathjaxinline], and the current through the lower resistance, [mathjaxinline]I_ R[/mathjaxinline], are plotted as functions of time. In both cases, positive values correspond to current flowing from left to right. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/deab2c1ddbe175e7e7cc8dad62f6c14b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w12_20b.svg&#34; width=&#34;330&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Which of the following statement is true? &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_w12_20_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w12_20_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w12_20_2_1&#34; id=&#34;input_checkpoint_w12_20_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w12_20_2_1-choice_1-label&#34; for=&#34;input_checkpoint_w12_20_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w12_20_2_1&#34;&gt; &lt;text&gt; The dotted line is [mathjaxinline]I_ L[/mathjaxinline] and the solid line is [mathjaxinline]I_ R[/mathjaxinline]&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_w12_20_2_1&#34; id=&#34;input_checkpoint_w12_20_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w12_20_2_1-choice_2-label&#34; for=&#34;input_checkpoint_w12_20_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w12_20_2_1&#34;&gt; &lt;text&gt; The dotted line is [mathjaxinline]I_ R[/mathjaxinline] and the solid line is [mathjaxinline]I_ L[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w12_20_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_w12_20_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_w12_20_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; If the emf [mathjaxinline]\mathcal{E} = 10[/mathjaxinline] V, what is the value of [mathjaxinline]R[/mathjaxinline]? &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]R =[/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_w12_20_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_checkpoint_w12_20_3_1&#34; id=&#34;input_checkpoint_w12_20_3_1&#34; aria-describedby=&#34;trailing_text_checkpoint_w12_20_3_1 status_checkpoint_w12_20_3_1&#34; value=&#34;&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w12_20_3_1&#34;&gt;[mathjaxinline]\mathrm{(\Omega ) }[/mathjaxinline]&lt;/span&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_20_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_w12_20_3_1&#34; class=&#34;answer&#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_w12_20_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;A Solenoid and a Switch&#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_w12_20&#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_w12_20&#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_w12_20-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_w12_20-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_w12_20-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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" 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-w10d3-q2" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Opening a Switch on an RL Circuit</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2a" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2a-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a/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;fridayw11_2a-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2a-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; The RL circuit shown in the figure below contains a resistor [mathjaxinline]R_1[/mathjaxinline] and an inductance [mathjaxinline]L[/mathjaxinline] in series with a battery of emf [mathjaxinline]\mathcal{E}_0[/mathjaxinline]. The switch [mathjaxinline]S[/mathjaxinline] is initially closed. At [mathjaxinline]t=0[/mathjaxinline], the switch [mathjaxinline]S[/mathjaxinline] is opened, so that an additional very large resistance [mathjaxinline]R_2[/mathjaxinline] (assume that [mathjaxinline]R_2 \gg R_1[/mathjaxinline]) is now in series with the other elements. &lt;/p&gt; &lt;p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/682a9f09339a565dafce0850cbc43f61/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_friday_W11D3_2A.png&#34; width=&#34;440&#34;/&gt; &lt;/center&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; If the switch has been closed for a long time before [mathjaxinline]t=0[/mathjaxinline], what is the steady current [mathjaxinline]I_0[/mathjaxinline] in the circuit? Express your answer in terms of [mathjaxinline]L[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], and epsilon_0 for [mathjaxinline]\mathcal{E}_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I_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_fridayw11_2a_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_fridayw11_2a_2_1&#34; id=&#34;input_fridayw11_2a_2_1&#34; aria-describedby=&#34;status_fridayw11_2a_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2a_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2a_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_fridayw11_2a_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2a_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2a_2_1_dynamath&#34; name=&#34;input_fridayw11_2a_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_fridayw11_2a_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;Opening a Switch on an RL Circuit&#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_fridayw11_2a&#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_fridayw11_2a&#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;fridayw11_2a-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;fridayw11_2a-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;fridayw11_2a-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@fridayw11_2b"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2b" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2b" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2b-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2b" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2b/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;fridayw11_2b-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2b-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2b-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; While this current [mathjaxinline]I_0[/mathjaxinline] is flowing, at time [mathjaxinline]t=0[/mathjaxinline] the switch [mathjaxinline]S[/mathjaxinline] is opened. Write the differential equation for [mathjaxinline]I(t)[/mathjaxinline] that describes the behavior of the circuit at times [mathjaxinline]t \ge 0[/mathjaxinline]. Express your answer in terms of [mathjaxinline]L[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], epsilon_0 for [mathjaxinline]\mathcal{E}_0[/mathjaxinline] and [mathjaxinline]I[/mathjaxinline] for [mathjaxinline]I(t)[/mathjaxinline]. Enter the RHS of the equation [mathjaxinline]\dfrac {dI(t)}{dt} = \ldots[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle \frac{dI(t)}{dt} =[/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_fridayw11_2b_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_fridayw11_2b_2_1&#34; id=&#34;input_fridayw11_2b_2_1&#34; aria-describedby=&#34;status_fridayw11_2b_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2b_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2b_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_fridayw11_2b_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2b_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2b_2_1_dynamath&#34; name=&#34;input_fridayw11_2b_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_fridayw11_2b_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;Opening a Switch on an RL Circuit&#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_fridayw11_2b&#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_fridayw11_2b&#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;fridayw11_2b-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;fridayw11_2b-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;fridayw11_2b-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@fridayw11_2c"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2c" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2c" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2c-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2c" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2c/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;fridayw11_2c-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2c-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2c-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 c)&lt;/b&gt; Solve this equation (by integration) for [mathjaxinline]I(t)[/mathjaxinline] under the approximation that [mathjaxinline]\mathcal{E}_0 \approx 0[/mathjaxinline] (assume that the battery emf is negligible compared to the total emf around the circuit for times after the switch is opened). Express your answer in terms of [mathjaxinline]L[/mathjaxinline], [mathjaxinline]t[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline] and I_0 for [mathjaxinline]I_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I(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_fridayw11_2c_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_fridayw11_2c_2_1&#34; id=&#34;input_fridayw11_2c_2_1&#34; aria-describedby=&#34;status_fridayw11_2c_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2c_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2c_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_fridayw11_2c_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2c_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2c_2_1_dynamath&#34; name=&#34;input_fridayw11_2c_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_fridayw11_2c_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;Opening a Switch on an RL Circuit&#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_fridayw11_2c&#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_fridayw11_2c&#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;fridayw11_2c-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;fridayw11_2c-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;fridayw11_2c-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-3" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2d" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2d-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d/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;fridayw11_2d-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2d-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 d)&lt;/b&gt; Using your results from parts (b) and the approximation that [mathjaxinline]\mathcal{E}_0 \approx 0[/mathjaxinline], find the value of the total emf around the circuit (which from Faraday&#39;s law is [mathjaxinline]-L \dfrac {dI}{dt}[/mathjaxinline]) just after the switch is opened. Express your answer in terms of [mathjaxinline]L[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline], and I_0 for [mathjaxinline]I_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{E_{circuit}} =[/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_fridayw11_2d_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_fridayw11_2d_2_1&#34; id=&#34;input_fridayw11_2d_2_1&#34; aria-describedby=&#34;status_fridayw11_2d_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2d_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2d_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_fridayw11_2d_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2d_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2d_2_1_dynamath&#34; name=&#34;input_fridayw11_2d_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_fridayw11_2d_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;Opening a Switch on an RL Circuit&#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_fridayw11_2d&#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_fridayw11_2d&#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;fridayw11_2d-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;fridayw11_2d-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;fridayw11_2d-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-4" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2e" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2e-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e/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;fridayw11_2e-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2e-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 e)&lt;/b&gt; How reasonable is your assumption in &lt;b class=&#34;bfseries&#34;&gt;(part c)&lt;/b&gt; that [mathjaxinline]\mathcal{E}_0[/mathjaxinline] could be ignored for times just after the switch was opened? &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;No answer checking for this part.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_fridayw11_2e_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;Opening a Switch on an RL Circuit&#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_fridayw11_2e&#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_fridayw11_2e&#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;fridayw11_2e-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;fridayw11_2e-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;fridayw11_2e-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-5" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2f" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2f-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f/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;fridayw11_2f-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2f-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 f)&lt;/b&gt; What is the magnitude of the potential drop across the resistor [mathjaxinline]R_2[/mathjaxinline] at times [mathjaxinline]t&amp;gt;0[/mathjaxinline], just after the switch is opened? Express your answer in terms of [mathjaxinline]L[/mathjaxinline], epsilon_0 for [mathjaxinline]\mathcal{E}_0[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline] and R_2 for [mathjaxinline]R_2[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\Delta V_2 =[/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_fridayw11_2f_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_fridayw11_2f_2_1&#34; id=&#34;input_fridayw11_2f_2_1&#34; aria-describedby=&#34;status_fridayw11_2f_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2f_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2f_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_fridayw11_2f_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2f_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2f_2_1_dynamath&#34; name=&#34;input_fridayw11_2f_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_fridayw11_2f_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;Opening a Switch on an RL Circuit&#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_fridayw11_2f&#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_fridayw11_2f&#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;fridayw11_2f-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;fridayw11_2f-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;fridayw11_2f-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-6" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2g" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2g-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g/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;fridayw11_2g-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2g-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 g)&lt;/b&gt; How does the potential drop across [mathjaxinline]R_2[/mathjaxinline] just after [mathjaxinline]t=0[/mathjaxinline] compare to the battery emf [mathjaxinline]\mathcal{E}_0[/mathjaxinline] if [mathjaxinline]R_2 = 80 R_1[/mathjaxinline]? Express your answer in terms of [mathjaxinline]L[/mathjaxinline], epsilon_0 for [mathjaxinline]\mathcal{E}_0[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline] and R_2 for [mathjaxinline]R_2[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\Delta V_2 =[/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_fridayw11_2g_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_fridayw11_2g_2_1&#34; id=&#34;input_fridayw11_2g_2_1&#34; aria-describedby=&#34;status_fridayw11_2g_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_fridayw11_2g_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_fridayw11_2g_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_fridayw11_2g_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_fridayw11_2g_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_fridayw11_2g_2_1_dynamath&#34; name=&#34;input_fridayw11_2g_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_fridayw11_2g_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;Opening a Switch on an RL Circuit&#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_fridayw11_2g&#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_fridayw11_2g&#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;fridayw11_2g-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;fridayw11_2g-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;fridayw11_2g-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-7" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_fridayw11_2h" class="problems-wrapper" role="group" aria-labelledby="fridayw11_2h-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h/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;fridayw11_2h-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Opening a Switch on an RL Circuit &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw11_2h-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 h)&lt;/b&gt; Based on your answer from the previous question, why should you be very careful when you open the switch, especially if [mathjaxinline]R_2[/mathjaxinline] is much, much larger that [mathjaxinline]R_1[/mathjaxinline]? &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bfseries&#34;&gt;No answer checking for this part.&lt;/b&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_fridayw11_2h_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;Opening a Switch on an RL Circuit&#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_fridayw11_2h&#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_fridayw11_2h&#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;fridayw11_2h-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;fridayw11_2h-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;fridayw11_2h-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> </div>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" 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-W12D1-2" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Amplitude and Phase of an LC Circuit</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_17"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" 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_w12_17" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w12_17" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w12_17-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_17" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_17/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_w12_17-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_17-problem-progress&#34; tabindex=&#34;-1&#34;&gt; LC Oscillator - Amplitude and Phase &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_17-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/cdbe4470f312f4687c14b229842c2bc7/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w12_17.svg&#34; width=&#34;330&#34;/&gt; &lt;/center&gt; &lt;p&gt; At time [mathjaxinline]t=0[/mathjaxinline], the current through the inductor in an ideal LC circuit is [mathjaxinline]I_0[/mathjaxinline] and the charge on the capacitor is [mathjaxinline]q_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; Consider the solution of the differential equation for the charge to be: &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 Q(t) = A\cos (\omega _0t+\phi )[/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; where [mathjaxinline]\omega _0=\dfrac {1}{\sqrt {LC}}[/mathjaxinline] is the angular frequency of the oscillations, [mathjaxinline]A[/mathjaxinline] is the amplitude, and [mathjaxinline]\phi[/mathjaxinline] is the phase. The values of [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]\phi[/mathjaxinline] are determined by the initial conditions of the problem. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; What is [mathjaxinline]I(t)[/mathjaxinline], the current in the circuit as a function of time. Express your answer in term of [mathjaxinline]A[/mathjaxinline], phi for [mathjaxinline]\phi[/mathjaxinline], omega_0 for [mathjaxinline]\omega _0[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]I(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_checkpoint_w12_17_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_w12_17_2_1&#34; id=&#34;input_checkpoint_w12_17_2_1&#34; aria-describedby=&#34;status_checkpoint_w12_17_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_w12_17_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_17_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_w12_17_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_17_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_17_2_1_dynamath&#34; name=&#34;input_checkpoint_w12_17_2_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; Use the initial condition that at [mathjaxinline]t=0[/mathjaxinline], [mathjaxinline]q(t=0)=q_0[/mathjaxinline] to obtain an expression for [mathjaxinline]\cos (\phi )[/mathjaxinline] in terms of [mathjaxinline]A[/mathjaxinline] and q_0 for [mathjaxinline]q_0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\cos (\phi )=[/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_w12_17_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_w12_17_3_1&#34; id=&#34;input_checkpoint_w12_17_3_1&#34; aria-describedby=&#34;status_checkpoint_w12_17_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_w12_17_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_17_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_w12_17_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_17_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_17_3_1_dynamath&#34; name=&#34;input_checkpoint_w12_17_3_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 c)&lt;/b&gt; Use the initial condition that at [mathjaxinline]t=0[/mathjaxinline], [mathjaxinline]I(t=0)=I_0[/mathjaxinline] to obtain an expression for [mathjaxinline]\sin (\phi )[/mathjaxinline] in terms of [mathjaxinline]A[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], and omega_0 for [mathjaxinline]\omega _0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\sin (\phi )=[/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_w12_17_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_w12_17_4_1&#34; id=&#34;input_checkpoint_w12_17_4_1&#34; aria-describedby=&#34;status_checkpoint_w12_17_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_w12_17_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_17_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_w12_17_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_17_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_17_4_1_dynamath&#34; name=&#34;input_checkpoint_w12_17_4_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 d)&lt;/b&gt; Use the results of parts (b) and (c) and the trigonometric identity [mathjaxinline]\sin ^2(\phi )+\cos ^2(\phi ) =1[/mathjaxinline] to obtain an expresion for [mathjaxinline]A[/mathjaxinline] in terms of q_0 for [mathjaxinline]q_0[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], and omega_0 for [mathjaxinline]\omega _0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]A=[/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_w12_17_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_w12_17_5_1&#34; id=&#34;input_checkpoint_w12_17_5_1&#34; aria-describedby=&#34;status_checkpoint_w12_17_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_w12_17_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_17_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_w12_17_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_17_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_17_5_1_dynamath&#34; name=&#34;input_checkpoint_w12_17_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 e)&lt;/b&gt; Use the results of parts (b) and (c) to obtain an expression for the [mathjaxinline]\tan (\phi )[/mathjaxinline] in terms of q_0 for [mathjaxinline]q_0[/mathjaxinline], I_0 for [mathjaxinline]I_0[/mathjaxinline], and omega_0 for [mathjaxinline]\omega _0[/mathjaxinline] &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\tan (\phi )=[/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_w12_17_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_w12_17_6_1&#34; id=&#34;input_checkpoint_w12_17_6_1&#34; aria-describedby=&#34;status_checkpoint_w12_17_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_w12_17_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_17_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_w12_17_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_17_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_17_6_1_dynamath&#34; name=&#34;input_checkpoint_w12_17_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-block-type="vertical" data-graded="False" 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_w12_15" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">Energy in the LC oscillator</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_15"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" 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_w12_15" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w12_15" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w12_15-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_15" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_15/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_w12_15-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_15-problem-progress&#34; tabindex=&#34;-1&#34;&gt; LC oscillator and the SHO equation &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_15-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/v1/9fa15a76768a744d6dd5ea51eef137ad/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w12_14.svg&#34; width=&#34;495&#34;/&gt; &lt;/center&gt; &lt;p&gt; Consider the LC circuit above. Assume that any resistance in the wires or other circuit elements is negligible. At time [mathjaxinline]t=0[/mathjaxinline] (left figure), the capacitor has a charge [mathjaxinline]Q_0[/mathjaxinline] and there is no current through the inductor. At a later time [mathjaxinline]t[/mathjaxinline] (right figure), the charge on the left plate is still positive (with value [mathjaxinline]q\lt Q_0[/mathjaxinline]) and is decreasing due to the clockwise current shown. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; If we use the convention of clockwise current being positive, the current [mathjaxinline]I[/mathjaxinline] is given by: &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_w12_15_2_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_checkpoint_w12_15_2_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_checkpoint_w12_15_2_1&#34; id=&#34;input_checkpoint_w12_15_2_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;checkpoint_w12_15_2_1-choice_1-label&#34; for=&#34;input_checkpoint_w12_15_2_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w12_15_2_1&#34;&gt; &lt;text&gt; [mathjaxinline]I=+\dfrac {dq}{dt}[/mathjaxinline]&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_w12_15_2_1&#34; id=&#34;input_checkpoint_w12_15_2_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;checkpoint_w12_15_2_1-choice_2-label&#34; for=&#34;input_checkpoint_w12_15_2_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_checkpoint_w12_15_2_1&#34;&gt; &lt;text&gt; [mathjaxinline]I=-\dfrac {dq}{dt}[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_checkpoint_w12_15_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_w12_15_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_w12_15_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; What is [mathjaxinline]U_ E[/mathjaxinline], the energy stored in the capacitor when it has a charge [mathjaxinline]q[/mathjaxinline]. Express your answer in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]L[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]U_ 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_w12_15_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_w12_15_3_1&#34; id=&#34;input_checkpoint_w12_15_3_1&#34; aria-describedby=&#34;status_checkpoint_w12_15_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_w12_15_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_15_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_w12_15_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_15_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_15_3_1_dynamath&#34; name=&#34;input_checkpoint_w12_15_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_w12_15_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; What is [mathjaxinline]U_ B[/mathjaxinline], the magnetic energy in the inductor when a current [mathjaxinline]I[/mathjaxinline] flows through it. Express your answer in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]L[/mathjaxinline] as needed. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]U_ B=[/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_w12_15_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_w12_15_4_1&#34; id=&#34;input_checkpoint_w12_15_4_1&#34; aria-describedby=&#34;status_checkpoint_w12_15_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_w12_15_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_15_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_w12_15_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_15_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_15_4_1_dynamath&#34; name=&#34;input_checkpoint_w12_15_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_w12_15_solution_3&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part d)&lt;/b&gt; Because there is no battery connected to the circuit and we are assuming that any resistance in the circuit is negligible, the combined total energy in the inductor and capacitor must be a constant. From conservation of energy: &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 U=U_ E+U_ B = \mbox{constant}\Longrightarrow \dfrac {dU}{dt}=0[/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; Use the expressions of [mathjaxinline]U_ E[/mathjaxinline] and [mathjaxinline]U_ B[/mathjaxinline] from the previous parts in [mathjaxinline]\dfrac {dU}{dt}=0[/mathjaxinline] to obtain an expression for [mathjaxinline]\dfrac {d^2q}{dt^2}[/mathjaxinline] in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]L[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\dfrac {d^2q}{dt^2}=[/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_w12_15_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_w12_15_5_1&#34; id=&#34;input_checkpoint_w12_15_5_1&#34; aria-describedby=&#34;status_checkpoint_w12_15_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_w12_15_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_15_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_w12_15_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_15_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_15_5_1_dynamath&#34; name=&#34;input_checkpoint_w12_15_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_w12_15_solution_4&#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;LC oscillator and the SHO equation&#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_w12_15&#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_w12_15&#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_w12_15-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_w12_15-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_w12_15-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@checkpoint_w12_18"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" 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_w12_18" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_checkpoint_w12_18" class="problems-wrapper" role="group" aria-labelledby="checkpoint_w12_18-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_18" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_18/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_w12_18-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_18-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Calculations using Energy &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@checkpoint_w12_18-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 a)&lt;/b&gt; Initially, the capacitor in a series LC circuit is charged. A switch is closed, allowing the capacitor to discharge. After time [mathjaxinline]t_1[/mathjaxinline], the energy stored in the capacitor is one fourth of its initial value. Calculate the value of [mathjaxinline]L[/mathjaxinline] in terms of t_1 for [mathjaxinline]t_1[/mathjaxinline] and [mathjaxinline]C[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]L=[/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_w12_18_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_w12_18_2_1&#34; id=&#34;input_checkpoint_w12_18_2_1&#34; aria-describedby=&#34;status_checkpoint_w12_18_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_w12_18_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_18_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_w12_18_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_checkpoint_w12_18_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_checkpoint_w12_18_2_1_dynamath&#34; name=&#34;input_checkpoint_w12_18_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_w12_18_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; An LC circuit consists of a [mathjaxinline]20.0[/mathjaxinline]&amp;#160;mH inductor and a [mathjaxinline]0.50~ \mu[/mathjaxinline]F capacitor. If the maximum instantaneous current is [mathjaxinline]0.10[/mathjaxinline] A, what is the greatest potential difference across the capacitor? &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\Delta V_{max} =[/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_w12_18_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_checkpoint_w12_18_3_1&#34; id=&#34;input_checkpoint_w12_18_3_1&#34; aria-describedby=&#34;trailing_text_checkpoint_w12_18_3_1 status_checkpoint_w12_18_3_1&#34; value=&#34;&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_checkpoint_w12_18_3_1&#34;&gt;[mathjaxinline]\mathrm{(V) }[/mathjaxinline]&lt;/span&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_checkpoint_w12_18_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_w12_18_3_1&#34; class=&#34;answer&#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_w12_18_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;Calculations using Energy&#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_w12_18&#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_w12_18&#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="False" 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-unused-finalprep-12" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <h2 class="hd hd-2 unit-title">LC Circuit and Two Switches</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-block-type="problem" data-graded="False" data-init="XBlockToXModuleShim" data-runtime-version="1" data-has-score="True" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-request-token="fdd6033a05d811f095170e84ffb47eb3" data-runtime-class="LmsRuntime"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_finalexam-problem12" class="problems-wrapper" role="group" aria-labelledby="finalexam-problem12-problem-title" data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12/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-problem12-problem-title&#34; aria-describedby=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12-problem-progress&#34; tabindex=&#34;-1&#34;&gt; LC Circuit with Two Switches &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+8.02.3x+1T2019+type@problem+block@finalexam-problem12-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Consider a circuit consisting of a coil that has inductance [mathjaxinline]L=40\text {mH}[/mathjaxinline] and negligible resistance and a capacitor with [mathjaxinline]C=100\mu \text {F}[/mathjaxinline] and an applied DC voltage source, [mathjaxinline]V_0=2.0\text {V}[/mathjaxinline]. &lt;/p&gt; &lt;center&gt; &lt;img src=&#34;/assets/courseware/v1/ceccbfef768e3cb1731f929522fcad99/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_finalexam-problem12.svg&#34; width=&#34;385&#34;/&gt; &lt;/center&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part a)&lt;/b&gt; Switch [mathjaxinline]S_1[/mathjaxinline] is closed at [mathjaxinline]t=0[/mathjaxinline] and switch [mathjaxinline]S_2[/mathjaxinline] is left open. After a very long time, what is the charge on the capacitor? &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]Q=[/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-problem12_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_finalexam-problem12_2_1&#34; id=&#34;input_finalexam-problem12_2_1&#34; aria-describedby=&#34;status_finalexam-problem12_2_1&#34; value=&#34;&#34; size=&#34;25&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem12_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem12_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-problem12_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;C&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem12_solution_1&#34;/&gt; &lt;/div&gt;&lt;/p&gt; &lt;p&gt; Now we close switch [mathjaxinline]S_2[/mathjaxinline] while simultaneously opening switch [mathjaxinline]S_1[/mathjaxinline]. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bfseries&#34;&gt;(Part b)&lt;/b&gt; What is the angular frequency of oscillation of the charge on the capacitor? &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\omega =[/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-problem12_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_finalexam-problem12_3_1&#34; id=&#34;input_finalexam-problem12_3_1&#34; aria-describedby=&#34;status_finalexam-problem12_3_1&#34; value=&#34;&#34; size=&#34;25&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem12_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem12_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-problem12_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;rad/s&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem12_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; What is the current in the circuit when the magnetic energy stored in the coil is exactly equal to the electrical energy stored in the capacitor? &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 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_finalexam-problem12_4_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_finalexam-problem12_4_1&#34; id=&#34;input_finalexam-problem12_4_1&#34; aria-describedby=&#34;status_finalexam-problem12_4_1&#34; value=&#34;&#34; size=&#34;25&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_finalexam-problem12_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_finalexam-problem12_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-problem12_4_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt;A&lt;/p&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_finalexam-problem12_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;LC Circuit with Two Switches&#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-problem12&#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-problem12&#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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