<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="VerticalStudentView" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-w13d3-q1" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Charging Capacitor and Displacement Current</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1a" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1a-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1a-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1a-problem-progress"></div>
<div class="problem">
<div>
<p>
A capacitor consists of two circular plates of radius [mathjaxinline]a[/mathjaxinline], separated by a distance [mathjaxinline]d[/mathjaxinline] (assume [mathjaxinline]d \ll a[/mathjaxinline]). The center of each plate is connected to the terminals of a voltage source by a thin wire. A switch in the circuit (not shown in the picture) is closed at time [mathjaxinline]t=0[/mathjaxinline], and current [mathjaxinline]I(t)[/mathjaxinline] flows in the circuit. The charge on the bottom plate is related to the current according to [mathjaxinline]I(t) = dQ(t) / dt[/mathjaxinline]. </p>
<p>
We begin by calculating the electric field between the plates. Throughout this problem, you may ignore edge effects. We assume that the electric field is zero for [mathjaxinline]r&gt;a[/mathjaxinline]. Choose cylindrical coordinates with unit vectors satisfying [mathjaxinline]\hat{\mathbf{r}} \times \hat{\theta } = \hat{\mathbf{k}}[/mathjaxinline], where [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] points from the positively charged plate to the negatively charged plate (vertically upwards in the figure below). This means that [mathjaxinline]\hat{\theta }[/mathjaxinline] points counterclockwise as viewed from above. </p>
<p>
<center>
<img src="/assets/courseware/v1/2da9dc85dbf2d63b6d4281493c47c508/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_Friday_W13D3_1-001.jpg" width="440"/>
</center>
</p>
<p><b class="bfseries">(Part a)</b> Use Gauss's Law to find a vector expression for the electric field between the plates when the charge on the plates is [mathjaxinline]\pm Q[/mathjaxinline] (as pictured). Please express your answer using Q, a, epsilon_0 for [mathjaxinline]\epsilon _0[/mathjaxinline] and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\mathbf{E}} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1a_2_1" id="input_fridayw13_1a_2_1" aria-describedby="status_fridayw13_1a_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1a_2_1"/>
<span class="status unanswered" id="status_fridayw13_1a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1a_2_1" class="answer"/>
<div id="display_fridayw13_1a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1a_2_1_dynamath" name="input_fridayw13_1a_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1a_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1a-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1a-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1a-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-1" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1b" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1b-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1b-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (b)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1b-problem-progress"></div>
<div class="problem">
<div>
<p>
<center>
<img src="/assets/courseware/v1/084257ab88086ab1f685a287b2641b2b/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_Friday_W13D3_1-002.jpg" width="440"/>
</center>
</p>
<p><b class="bfseries">(Part b)</b> Using your expression for [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline] found in the Part (a), calculate the electric flux through a flat disk of radius [mathjaxinline]r&lt;a[/mathjaxinline] in the plane midway between the plates, as shown in the picture above. Take the surface unit normal to the disk to be in the [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] direction, [mathjaxinline]\vec{\mathbf{n}} = \hat{\mathbf{k}}[/mathjaxinline]. Please express your answer using Q, a, r and epsilon_0 for [mathjaxinline]\epsilon _0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\Phi _ E =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1b_2_1" id="input_fridayw13_1b_2_1" aria-describedby="status_fridayw13_1b_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1b_2_1"/>
<span class="status unanswered" id="status_fridayw13_1b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1b_2_1" class="answer"/>
<div id="display_fridayw13_1b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1b_2_1_dynamath" name="input_fridayw13_1b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1b_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (b)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-2" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1c" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1c-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1c-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (c)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Calculate the Maxwell displacement current, </p>
<table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]I_{dis} = \epsilon _0 \frac{d \Phi _ E}{dt} = \epsilon _0 \frac{d}{dt} \iint _{disk} \vec{\mathbf{E}} \cdot \hat{\mathbf{n}} \, da[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
through the flat disk of radius [mathjaxinline]r &lt; a[/mathjaxinline]. Please express your answer in terms of r, a and I for [mathjaxinline]I(t)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]I_{dis} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1c_2_1" id="input_fridayw13_1c_2_1" aria-describedby="status_fridayw13_1c_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1c_2_1"/>
<span class="status unanswered" id="status_fridayw13_1c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1c_2_1" class="answer"/>
<div id="display_fridayw13_1c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1c_2_1_dynamath" name="input_fridayw13_1c_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1c_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (c)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1c-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1c-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1c-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-3" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1d" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1d-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1d-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (d)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1d-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part d)</b> What is the conduction current [mathjaxinline]I_{cond}=\iint _ S \vec{\mathbf{J}} \cdot \hat{\mathbf{n}} \, da[/mathjaxinline] through the flat disc of radius [mathjaxinline]r &lt; a[/mathjaxinline]? &#8220;Conduction" current just means the current due to the flow of real charge across the surface (e.g. electrons or ions). Please express your answer in terms of r, a and I for [mathjaxinline]I(t)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]I_{cond} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1d_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1d_2_1" id="input_fridayw13_1d_2_1" aria-describedby="status_fridayw13_1d_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1d_2_1"/>
<span class="status unanswered" id="status_fridayw13_1d_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1d_2_1" class="answer"/>
<div id="display_fridayw13_1d_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1d_2_1_dynamath" name="input_fridayw13_1d_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1d_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (d)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1d" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1d">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1d-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1d-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1d-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-4" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1e" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1e-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1e-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (e)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1e-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part e)</b> The capacitor plates have axial symmetry, and we know that the magnetic field due to a wire runs in azimuthal circles around the wire. Then, from symmetry arguments, we may assume that the magnetic field in the system will run in azimuthal circles inside the capacitor as well. </p>
<p>
<center>
<img src="/assets/courseware/v1/5402fc059a5b0b027c3ca123618e3779/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_Friday_W13D3_1-003.jpg" width="440"/>
</center>
</p>
<p>
Choose an Amperian loop as a circle of radius [mathjaxinline]r&lt;a[/mathjaxinline] in the plane midway between the plates (see figure above). Calculate the magnitude of the line integral of the magnetic field around the circle, [mathjaxinline]\oint _{circle} \vec{\mathbf{B}} \cdot d \vec{\mathbf{s}}[/mathjaxinline]. Express your answer in terms of r and B for [mathjaxinline]|\vec{\mathbf{B}}|[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle |\oint _{circle} \vec{\mathbf{B}} \cdot d \vec{\mathbf{s}} |=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1e_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1e_2_1" id="input_fridayw13_1e_2_1" aria-describedby="status_fridayw13_1e_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1e_2_1"/>
<span class="status unanswered" id="status_fridayw13_1e_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1e_2_1" class="answer"/>
<div id="display_fridayw13_1e_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1e_2_1_dynamath" name="input_fridayw13_1e_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1e_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (e)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1e" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1e">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1e-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1e-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1e-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-5" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1f" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1f-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1f-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (f)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1f-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part f)</b> Now use the results of your answers above to apply the generalized Ampere's Law equation to find a vector expression for the magnetic field at distance [mathjaxinline]r&lt;a[/mathjaxinline] from the axis. Please express your answer using I, r, a, mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], hattheta for [mathjaxinline]\hat{\mathbf{\theta }}[/mathjaxinline] and hatr for [mathjaxinline]\hat{\mathbf{r}}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\mathbf{B}} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13_1f_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13_1f_2_1" id="input_fridayw13_1f_2_1" aria-describedby="status_fridayw13_1f_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13_1f_2_1"/>
<span class="status unanswered" id="status_fridayw13_1f_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13_1f_2_1" class="answer"/>
<div id="display_fridayw13_1f_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13_1f_2_1_dynamath" name="input_fridayw13_1f_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1f_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (f)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1f" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1f">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1f-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1f-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1f-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-6" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1g" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1g-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1g-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (g)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1g-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part g)</b> If you use your right thumb to point along the direction of the electric field, as the plates charge up, does the magnetic field point in the direction your fingers curl on your right hand, or in the opposite direction? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_fridayw13_1g_2_1">
<fieldset aria-describedby="status_fridayw13_1g_2_1">
<div class="field">
<input type="radio" name="input_fridayw13_1g_2_1" id="input_fridayw13_1g_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="fridayw13_1g_2_1-choice_1-label" for="input_fridayw13_1g_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_fridayw13_1g_2_1">
<text> In the direction of your fingers</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_fridayw13_1g_2_1" id="input_fridayw13_1g_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="fridayw13_1g_2_1-choice_2-label" for="input_fridayw13_1g_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_fridayw13_1g_2_1">
<text> In the opposite direction</text>
</label>
</div>
<span id="answer_fridayw13_1g_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_fridayw13_1g_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1g_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (g)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1g" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1g">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1g-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1g-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1g-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
<div class="vert vert-7" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13_1h" class="problems-wrapper" role="group"
aria-labelledby="fridayw13_1h-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13_1h-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h-problem-progress" tabindex="-1">
Charging Capacitor and Displacement Current - Part (h)
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13_1h-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part h)</b> Would the direction of the magnetic field change if the plates were discharging? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_fridayw13_1h_2_1">
<fieldset aria-describedby="status_fridayw13_1h_2_1">
<div class="field">
<input type="radio" name="input_fridayw13_1h_2_1" id="input_fridayw13_1h_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="fridayw13_1h_2_1-choice_1-label" for="input_fridayw13_1h_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_fridayw13_1h_2_1">
<text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_fridayw13_1h_2_1" id="input_fridayw13_1h_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="fridayw13_1h_2_1-choice_2-label" for="input_fridayw13_1h_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_fridayw13_1h_2_1">
<text> No</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_fridayw13_1h_2_1" id="input_fridayw13_1h_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="fridayw13_1h_2_1-choice_3-label" for="input_fridayw13_1h_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_fridayw13_1h_2_1">
<text> Could be either way</text>
</label>
</div>
<span id="answer_fridayw13_1h_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_fridayw13_1h_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13_1h_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Charging Capacitor and Displacement Current - Part (h)" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13_1h" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13_1h">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1h-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1h-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13_1h-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="VerticalStudentView" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-maxwell-psp" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Maxwell's Equations</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_pcq_w13hard-1" class="problems-wrapper" role="group"
aria-labelledby="pcq_w13hard-1-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="pcq_w13hard-1-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1-problem-progress" tabindex="-1">
Maxwell&#39;s Equations
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@pcq_w13hard-1-problem-progress"></div>
<div class="problem">
<div><p><b>Maxwell's Equations:</b></p>
\[
\begin{align}
\mathrm{I}. &amp; \quad \oint \vec{\mathbf{E}} \cdot d\vec{\mathbf{A}} = \frac{Q_{enc}}{\epsilon_0}
\\
\mathrm{II}. &amp; \quad \oint \vec{\mathbf{B}} \cdot d\vec{\mathbf{A}} = 0
\\
\mathrm{III}. &amp; \quad \oint \vec{\mathbf{E}} \cdot d\vec{\mathbf{s}} = - \frac{d\Phi_B}{dt}
\\
\mathrm{IV}. &amp; \quad \oint \vec{\mathbf{B}} \cdot d\vec{\mathbf{s}} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}
\end{align}
\]
<p><b>Lorentz Force Equation:</b></p>
\[\mathrm{V}. \quad \vec{\mathbf{F}} = q (\vec{\mathbf{E}} + \vec{\mathbf{v}} \times \vec{\mathbf{B}})\]
<p><b>Conservation of Charge:</b></p>
\[\mathrm{VI}. \quad \oint \vec{\mathbf{J}} \cdot d\vec{\mathbf{A}} = - \frac{dQ_{enc}}{dt} \]
<p>Indicate the number(s) of the Maxwell equation(s) (I-IV), the Lorentz Force Equation (V) or charge conservation (VI) that can be used to explain the given phenomena. Check all that apply.</p>
<p><b>(Part a)</b> The electric field of an isolated point charge drops off like [mathjaxinline]1/r^{2}[/mathjaxinline].</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_2_1">
<fieldset aria-describedby="status_pcq_w13hard-1_2_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_2_1-choice_0-label" for="input_pcq_w13hard-1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_2_1-choice_1-label" for="input_pcq_w13hard-1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_2_1-choice_2-label" for="input_pcq_w13hard-1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_2_1-choice_3-label" for="input_pcq_w13hard-1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_2_1-choice_4-label" for="input_pcq_w13hard-1_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_2_1[]" id="input_pcq_w13hard-1_2_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_2_1-choice_5-label" for="input_pcq_w13hard-1_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_2_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_1"/>
</div><p><b>(Part b)</b> There are no magnetic monopoles.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_3_1">
<fieldset aria-describedby="status_pcq_w13hard-1_3_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_3_1-choice_0-label" for="input_pcq_w13hard-1_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_3_1-choice_1-label" for="input_pcq_w13hard-1_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_3_1-choice_2-label" for="input_pcq_w13hard-1_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_3_1-choice_3-label" for="input_pcq_w13hard-1_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_3_1-choice_4-label" for="input_pcq_w13hard-1_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_3_1[]" id="input_pcq_w13hard-1_3_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_3_1-choice_5-label" for="input_pcq_w13hard-1_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_3_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_2"/>
</div><p><b>(Part c)</b> A conducting disc falls more slowly between the poles of a magnet than does a disc which is an insulator.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_4_1">
<fieldset aria-describedby="status_pcq_w13hard-1_4_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_4_1-choice_0-label" for="input_pcq_w13hard-1_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_4_1-choice_1-label" for="input_pcq_w13hard-1_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_4_1-choice_2-label" for="input_pcq_w13hard-1_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_4_1-choice_3-label" for="input_pcq_w13hard-1_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_4_1-choice_4-label" for="input_pcq_w13hard-1_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_4_1[]" id="input_pcq_w13hard-1_4_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_4_1-choice_5-label" for="input_pcq_w13hard-1_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_4_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_3"/>
</div><p><b>(Part d)</b> Lightning is formed and strikes a piece of iron, which then becomes magnetized.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_5_1">
<fieldset aria-describedby="status_pcq_w13hard-1_5_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_5_1-choice_0-label" for="input_pcq_w13hard-1_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_5_1-choice_1-label" for="input_pcq_w13hard-1_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_5_1-choice_2-label" for="input_pcq_w13hard-1_5_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_5_1-choice_3-label" for="input_pcq_w13hard-1_5_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_5_1-choice_4-label" for="input_pcq_w13hard-1_5_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_5_1[]" id="input_pcq_w13hard-1_5_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_5_1-choice_5-label" for="input_pcq_w13hard-1_5_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_5_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_5_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_4"/>
</div><p><b>(Part e)</b> There is no magnetic equivalent of a Faraday cage.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_6_1">
<fieldset aria-describedby="status_pcq_w13hard-1_6_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_6_1-choice_0-label" for="input_pcq_w13hard-1_6_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_6_1-choice_1-label" for="input_pcq_w13hard-1_6_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_6_1-choice_2-label" for="input_pcq_w13hard-1_6_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_6_1-choice_3-label" for="input_pcq_w13hard-1_6_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_6_1-choice_4-label" for="input_pcq_w13hard-1_6_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_6_1[]" id="input_pcq_w13hard-1_6_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_6_1-choice_5-label" for="input_pcq_w13hard-1_6_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_6_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_6_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_5"/>
</div><p><b>(Part f)</b> All unbalanced charge in a metal moves to the surface.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 6" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_7_1">
<fieldset aria-describedby="status_pcq_w13hard-1_7_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_7_1-choice_0-label" for="input_pcq_w13hard-1_7_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_7_1-choice_1-label" for="input_pcq_w13hard-1_7_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_7_1-choice_2-label" for="input_pcq_w13hard-1_7_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_7_1-choice_3-label" for="input_pcq_w13hard-1_7_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_7_1-choice_4-label" for="input_pcq_w13hard-1_7_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_7_1[]" id="input_pcq_w13hard-1_7_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_7_1-choice_5-label" for="input_pcq_w13hard-1_7_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_7_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_7_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_6"/>
</div><p><b>(Part g)</b> In a DC circuit, at each node, current into the node equals current out of the node.</p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 7" role="group"><div class="choicegroup capa_inputtype" id="inputtype_pcq_w13hard-1_8_1">
<fieldset aria-describedby="status_pcq_w13hard-1_8_1">
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="pcq_w13hard-1_8_1-choice_0-label" for="input_pcq_w13hard-1_8_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>I</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="pcq_w13hard-1_8_1-choice_1-label" for="input_pcq_w13hard-1_8_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>II</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="pcq_w13hard-1_8_1-choice_2-label" for="input_pcq_w13hard-1_8_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>III</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="pcq_w13hard-1_8_1-choice_3-label" for="input_pcq_w13hard-1_8_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>IV</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_4" class="field-input input-checkbox" value="choice_4"/><label id="pcq_w13hard-1_8_1-choice_4-label" for="input_pcq_w13hard-1_8_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>V</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_pcq_w13hard-1_8_1[]" id="input_pcq_w13hard-1_8_1_choice_5" class="field-input input-checkbox" value="choice_5"/><label id="pcq_w13hard-1_8_1-choice_5-label" for="input_pcq_w13hard-1_8_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_pcq_w13hard-1_8_1"> <text>VI</text>
</label>
</div>
<span id="answer_pcq_w13hard-1_8_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_pcq_w13hard-1_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
<div class="solution-span">
<span id="solution_pcq_w13hard-1_solution_7"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Maxwell&#39;s Equations" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_pcq_w13hard-1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_pcq_w13hard-1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="pcq_w13hard-1-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="pcq_w13hard-1-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="pcq_w13hard-1-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="VerticalStudentView" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-unused-quizzes-802-prob12" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Light Wave</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_802_Quizzes_L" class="problems-wrapper" role="group"
aria-labelledby="802_Quizzes_L-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="802_Quizzes_L-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L-problem-progress" tabindex="-1">
Traveling Wave
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@802_Quizzes_L-problem-progress"></div>
<div class="problem">
<div>
<p>The electric field of an electromagnetic plane wave is described as follows:</p>
<p>\(\vec{E}(y,t)=E_{0}\sin{(ky + \omega t)}\hat{i}\)</p>
<p>where \(E_{0} = 30\,\mathrm{Vm^{-1}} \) and \(k = 9\,\mathrm{m^{-1}} \). Note, also, that this wave travels at the speed of light (\(c=3\times10^{8}\mathrm{m/s}\)). Answer the following:</p>
<p>(a) In what direction is the wave traveling? Enter hati, hatj, or hatk for \(\hat{i}\), \(\hat{j}\), or \(\hat{k}\), respectively, and use \(-\) for negative directions (e.g. -\(\hat{i}\)).</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_802_Quizzes_L_2_1" class="inputtype formulaequationinput">
<div class="unanswered">
<input type="text" name="input_802_Quizzes_L_2_1" id="input_802_Quizzes_L_2_1" data-input-id="802_Quizzes_L_2_1" value="" aria-describedby="status_802_Quizzes_L_2_1" size="40"/>
<span class="trailing_text" id="trailing_text_802_Quizzes_L_2_1"/>
<span class="status unanswered" id="status_802_Quizzes_L_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_802_Quizzes_L_2_1" class="answer"/>
<div id="input_802_Quizzes_L_2_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
</div>
</div>
<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
</div></div>
<div class="solution-span">
<span id="solution_802_Quizzes_L_solution_1"/>
</div><p>(b) What is the wavelength, \(\lambda\) of the wave?</p>
<p style="display:inline">\(\lambda=\)</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_802_Quizzes_L_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_802_Quizzes_L_3_1" id="input_802_Quizzes_L_3_1" aria-describedby="status_802_Quizzes_L_3_1" value=""/>
<span class="trailing_text" id="trailing_text_802_Quizzes_L_3_1"/>
<span class="status unanswered" id="status_802_Quizzes_L_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_802_Quizzes_L_3_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">m</p>
<div class="solution-span">
<span id="solution_802_Quizzes_L_solution_2"/>
</div><p>(c) What is the angular frequency of the wave in units of rad/s?</p>
<p style="display:inline">\(\omega=\)</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_802_Quizzes_L_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_802_Quizzes_L_4_1" id="input_802_Quizzes_L_4_1" aria-describedby="status_802_Quizzes_L_4_1" value=""/>
<span class="trailing_text" id="trailing_text_802_Quizzes_L_4_1"/>
<span class="status unanswered" id="status_802_Quizzes_L_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_802_Quizzes_L_4_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">rad/s</p>
<div class="solution-span">
<span id="solution_802_Quizzes_L_solution_3"/>
</div><p>(d) What is the magnitude of the accompanying magnetic field?</p>
<p style="display:inline">\(|B|=\)</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_802_Quizzes_L_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_802_Quizzes_L_5_1" id="input_802_Quizzes_L_5_1" aria-describedby="status_802_Quizzes_L_5_1" value=""/>
<span class="trailing_text" id="trailing_text_802_Quizzes_L_5_1"/>
<span class="status unanswered" id="status_802_Quizzes_L_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_802_Quizzes_L_5_1" class="answer"/>
</div>
</div></div>
<p style="display:inline">\(\mathrm{Vs/m^2}\)</p>
<div class="solution-span">
<span id="solution_802_Quizzes_L_solution_4"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Traveling Wave" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_802_Quizzes_L" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_802_Quizzes_L">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="802_Quizzes_L-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="802_Quizzes_L-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="802_Quizzes_L-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="VerticalStudentView" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="vertical" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@vertical+block@vert-fridayw13-4" data-runtime-class="LmsRuntime" data-has-score="False">
<h2 class="hd hd-2 unit-title">Superposition of Two Traveling Waves</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-course-id="course-v1:MITx+8.02.3x+1T2019" data-graded="False" data-init="XBlockToXModuleShim" data-request-token="9126bd0e1edf11f0960016ffcd2b1d4d" data-runtime-version="1" data-block-type="problem" data-usage-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4" data-runtime-class="LmsRuntime" data-has-score="True">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_fridayw13-4" class="problems-wrapper" role="group"
aria-labelledby="fridayw13-4-problem-title"
data-problem-id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4" data-url="/courses/course-v1:MITx+8.02.3x+1T2019/xblock/block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="fridayw13-4-problem-title" aria-describedby="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4-problem-progress" tabindex="-1">
Superposition of Two Travelling Waves
</h3>
<div class="problem-progress" id="block-v1:MITx+8.02.3x+1T2019+type@problem+block@fridayw13-4-problem-progress"></div>
<div class="problem">
<div>
<p>
Suppose the electric field of an electromagnetic wave is given by the superposition of two waves: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\vec{\textbf{E}} = E_0 \cos (kz-\omega t) \hat{\textbf{i}} + E_0 \cos (kz+\omega t) \hat{\textbf{i}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(a)</b> What is the associated magnetic field [mathjaxinline]\vec{\textbf{B}}(z,t)[/mathjaxinline]? Please express your answer using x, y, z, t, k, c, E_0 for [mathjaxinline]E_0[/mathjaxinline], omega for [mathjaxinline]\omega[/mathjaxinline]; hati, hatj, and hatk for the unit vectors [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline] and [mathjaxinline]\hat{k}[/mathjaxinline], respectively. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\textbf{B}} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_fridayw13-4_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fridayw13-4_2_1" id="input_fridayw13-4_2_1" aria-describedby="status_fridayw13-4_2_1" value="" class="math" size="60"/>
<span class="trailing_text" id="trailing_text_fridayw13-4_2_1"/>
<span class="status unanswered" id="status_fridayw13-4_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_fridayw13-4_2_1" class="answer"/>
<div id="display_fridayw13-4_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_fridayw13-4_2_1_dynamath" name="input_fridayw13-4_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13-4_solution_1"/>
</div></p>
<p><b class="bfseries">(b)</b> Use the trigonometry identities [mathjaxinline]\cos (kz \pm \omega t) = \cos (kz) \cos (\omega t) \mp \sin (kz) \sin (\omega t)[/mathjaxinline], [mathjaxinline]\cos (-a)=\cos (a)[/mathjaxinline], and [mathjaxinline]\sin (-a)=-\sin (a)[/mathjaxinline] to show that you can write the electric and magnetic field as </p>
<table id="a0000000005" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000006">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{\textbf{E}}(z,t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =2E_0\cos (kz)\cos (\omega t)\hat{\textbf{i}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000007">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \vec{\textbf{B}}(z,t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\frac{2E_0}{c}\sin (kz)\sin (\omega t)\hat{\textbf{j}}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
<b class="bfseries">There is no answer check for this question.</b>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13-4_solution_2"/>
</div></p>
<p><b class="bfseries">(c)</b> Is the superposition of these two traveling waves, which are traveling in opposite direction, a traveling wave? </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_fridayw13-4_3_1">
<fieldset aria-describedby="status_fridayw13-4_3_1">
<div class="field">
<input type="radio" name="input_fridayw13-4_3_1" id="input_fridayw13-4_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="fridayw13-4_3_1-choice_1-label" for="input_fridayw13-4_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_fridayw13-4_3_1">
<text> Yes</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_fridayw13-4_3_1" id="input_fridayw13-4_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="fridayw13-4_3_1-choice_2-label" for="input_fridayw13-4_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_fridayw13-4_3_1">
<text> No</text>
</label>
</div>
<span id="answer_fridayw13-4_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_fridayw13-4_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_fridayw13-4_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Superposition of Two Travelling Waves" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_fridayw13-4" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_fridayw13-4">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13-4-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13-4-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="fridayw13-4-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="False">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
© All Rights Reserved