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<h2 class="hd hd-2 unit-title">Methodology for Applying Gauss's Law</h2>
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<p><b class="bfseries">Methodology for Applying Gauss's Law</b></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]\underset {{\substack {\text {closed} \\ \text {surface S}}}}{\Large \unicode {x222F}}\vec{\mathbf{E}} \cdot d\vec{\mathbf{A}} = {q_{in} \over \varepsilon _0}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><b class="bfseries">Step 1:</b> Identify the 'symmetry' properties of the charge distribution (i.e planar, cylindrical, or spherical). Note that the answers to many of the steps listed below (in particular, the direction in Step 2, the shape of the surface in Step 4, and the area of the surface in Step 5) will be common to all problems of a particular symmetry. </p><p><b class="bfseries">Step 2:</b> Determine the direction of the electric field. </p><p><b class="bfseries">Step 3:</b> Decide how many different regions of space the charge distribution determines. </p><p><b class="bfseries">For each region of space [mathjaxinline]\ldots[/mathjaxinline]</b></p><p><b class="bfseries">Step 4:</b> Choose a Gaussian surface through each part of which the electric field is either constant or zero. </p><p><b class="bfseries">Step 5:</b> Calculate the flux through the Gaussian surface (in terms of the unknown [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline]). </p><p><b class="bfseries">Step 6:</b> Calculate the charge enclosed in the choice of the Gaussian surface. </p><p><b class="bfseries">Step 7:</b> Equate the two sides of Gauss's Law in order to find an expression for the magnitude of the electric field. </p><p><b class="bfseries">Then [mathjaxinline]\ldots[/mathjaxinline]</b></p><p><b class="bfseries">Step 8:</b> Graph the magnitude of the electric field as a function of the parameter specifying the Gaussian surface for all regions of space. </p>
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<h2 class="hd hd-2 unit-title">W3PS1: Concentric Cylinders</h2>
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<h3 class="hd hd-3 problem-header" id="fps_fridayw2_1_sol-problem-title" aria-describedby="block-v1:MITx+8.02.1x+1T2019+type@problem+block@fps_fridayw2_1_sol-problem-progress" tabindex="-1">
W2D3 FPS Problem 1 Concentric Cylinders
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<img src="/assets/courseware/v1/f34ec5b5d79569441c9e0a36b686032d/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_W2D3_cylinders.svg" width="440"/>
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<p>
A long very thin non-conducting cylindrical shell of radius [mathjaxinline]b[/mathjaxinline] and length [mathjaxinline]L[/mathjaxinline] surrounds a long solid non-conducting cylinder of radius [mathjaxinline]a[/mathjaxinline] and length [mathjaxinline]L[/mathjaxinline] with [mathjaxinline]b&gt;a[/mathjaxinline]. Assume that [mathjaxinline]L\gg a[/mathjaxinline] and [mathjaxinline]L\gg b[/mathjaxinline] so that you can treat the cylinders as effectively infinite in length. The inner cylinder has a charge [mathjaxinline]+Q[/mathjaxinline] uniformly distributed throughout its volume, while the outer cylinder has a charge [mathjaxinline]-Q[/mathjaxinline] uniformly distributed across its surface. The region [mathjaxinline]a&lt;r&lt;b[/mathjaxinline] is empty. Calculate the electric field as a function of [mathjaxinline]r[/mathjaxinline] for the three regions: i) [mathjaxinline]r &lt; a[/mathjaxinline], ii) [mathjaxinline]a &lt; r &lt; b[/mathjaxinline], and iii) [mathjaxinline]r &gt; b[/mathjaxinline]. Make a plot of the magnitude of the electric field as a function of [mathjaxinline]r[/mathjaxinline] for all three regions of space. </p>
<p>
Note that the "Steps in the methodology" mentioned below can be found in the first tab in this Problem Solving section. </p>
<p><b class="bfseries">Question 1</b> (Step 1 of the methodology): What is the symmetry property of this charge distribution? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_fps_fridayw2_1_sol_2_1">
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<text> Spherical</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_2_1" id="input_fps_fridayw2_1_sol_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="fps_fridayw2_1_sol_2_1-choice_2-label" for="input_fps_fridayw2_1_sol_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_2_1">
<text> Cylindrical</text>
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<text> Planar</text>
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<span class="status unanswered" id="status_fps_fridayw2_1_sol_2_1" data-tooltip="Not yet answered.">
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<p><b class="bfseries">Question 2</b> (Step 2 of the methodology): What is the direction of the electric field? </p>
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<text> Radial (in/out)</text>
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<text> Angular (CW/CCW)</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_3_1" id="input_fps_fridayw2_1_sol_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="fps_fridayw2_1_sol_3_1-choice_3-label" for="input_fps_fridayw2_1_sol_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_3_1">
<text> Parallel to the cylinder axis</text>
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<span class="status unanswered" id="status_fps_fridayw2_1_sol_3_1" data-tooltip="Not yet answered.">
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<span id="solution_fps_fridayw2_1_sol_solution_1"/>
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<p><b class="bfseries">Note:</b> Although you can find answers to the following two questions by a careful reading of the text of this problem, make sure that you know how to find those answers yourself. </p>
<p><b class="bfseries">Question 3 (Step 3 of the methodology):</b> Define the different regions of space that you will need to find expressions for the electric field. What determines those different regions? </p>
<p>
in particular, how many different formulae for [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline] are you going to have to calculate? </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_fps_fridayw2_1_sol_4_1">
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<input type="radio" name="input_fps_fridayw2_1_sol_4_1" id="input_fps_fridayw2_1_sol_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="fps_fridayw2_1_sol_4_1-choice_1-label" for="input_fps_fridayw2_1_sol_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_4_1">
<text> 1</text>
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<text> 2</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_4_1" id="input_fps_fridayw2_1_sol_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="fps_fridayw2_1_sol_4_1-choice_3-label" for="input_fps_fridayw2_1_sol_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_4_1">
<text> 3</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_4_1" id="input_fps_fridayw2_1_sol_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="fps_fridayw2_1_sol_4_1-choice_4-label" for="input_fps_fridayw2_1_sol_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_4_1">
<text> 4</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_4_1" id="input_fps_fridayw2_1_sol_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="fps_fridayw2_1_sol_4_1-choice_5-label" for="input_fps_fridayw2_1_sol_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_4_1">
<text> 5</text>
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<input type="radio" name="input_fps_fridayw2_1_sol_4_1" id="input_fps_fridayw2_1_sol_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="fps_fridayw2_1_sol_4_1-choice_6-label" for="input_fps_fridayw2_1_sol_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_fps_fridayw2_1_sol_4_1">
<text> &gt;5</text>
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<span id="answer_fps_fridayw2_1_sol_4_1"/>
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<span class="status unanswered" id="status_fps_fridayw2_1_sol_4_1" data-tooltip="Not yet answered.">
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_2"/>
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<p><b class="bfseries">Question 4</b> (Step 4 of the methodology): For each region of space, make a figure showing clearly your choice of a Gaussian surface. What variable did you choose to parameterize your Gaussian surface (for example, for a sphere you'd use the radius [mathjaxinline]r[/mathjaxinline])? What is the range of that variable in each region? </p>
<p>
There is no answer check for this question. </p>
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_3"/>
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<p><b class="bfseries">Question 5</b> (Step 5 of the methodology): In the region [mathjaxinline]r&lt;a[/mathjaxinline], calculate the flux [mathjaxinline]\underset {{\substack {\text {closed} \\ \text {surface S}}}}{\Large \unicode {x222F}}\vec{\mathbf{E}} \cdot d\vec{\mathbf{A}}[/mathjaxinline] through your choice of the Gaussian surface (that is, just write down the left hand side of Gauss's Law). Your expression should include the unknown electric field for that region. Note: For entering your answer, use the symbol [mathjaxinline]h[/mathjaxinline] to denote the length of the Gaussian cylinder. Write your answer using some or all of the following: r, h, E, and pi for [mathjaxinline]\pi[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]{\Large \unicode {x222F}}\ \vec{\mathbf{E}} \cdot d\vec{\mathbf{A}}[/mathjaxinline][mathjaxinline]=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_fps_fridayw2_1_sol_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
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<div id="display_fps_fridayw2_1_sol_5_1" class="equation">`{::}`</div>
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_4"/>
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<p><b class="bfseries">Question 6</b> (Step 6 of the methodology): In the region [mathjaxinline]r&lt;a[/mathjaxinline], what is the charge enclosed in your choice of Gaussian surface? (this should be in terms of [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline], not [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline] ). Note: For entering your answer, use the symbol [mathjaxinline]h[/mathjaxinline] to denote the length of the Gaussian cylinder. Write your answer using some or all of the following: r, h, a, L and Q as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]Q_{\text {enc}}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_fps_fridayw2_1_sol_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fps_fridayw2_1_sol_6_1" id="input_fps_fridayw2_1_sol_6_1" aria-describedby="status_fps_fridayw2_1_sol_6_1" value="" class="math" size="40"/>
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<span class="status unanswered" id="status_fps_fridayw2_1_sol_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_fps_fridayw2_1_sol_6_1" class="answer"/>
<div id="display_fps_fridayw2_1_sol_6_1" class="equation">`{::}`</div>
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_5"/>
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<p><b class="bfseries">Question 7</b> (Step 7 of the methodology): In the region [mathjaxinline]r&lt;a[/mathjaxinline], equate the two sides of Gauss's Law that you calculated in Questions 5 and 6 and solve to find an expression for the magnitude of the electric field as a function of [mathjaxinline]r[/mathjaxinline]. Write your answer using some or all of the following: r, h, a, L, pi for [mathjaxinline]\pi[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline] and Q as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]E(r&lt;a )=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_fps_fridayw2_1_sol_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
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<span class="status unanswered" id="status_fps_fridayw2_1_sol_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_fps_fridayw2_1_sol_7_1" class="answer"/>
<div id="display_fps_fridayw2_1_sol_7_1" class="equation">`{::}`</div>
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_6"/>
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<p><b class="bfseries">Question 8</b> (Repeating Steps 6 and 7 of the methodology): Repeat the same procedure in order to calculate the electric field as a function of [mathjaxinline]r[/mathjaxinline] in the region [mathjaxinline]a&lt;r&lt;b[/mathjaxinline]. Write your answer using some or all of the following: r, h, a, L, pi for [mathjaxinline]\pi[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline] and Q as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]E(a&lt;r&lt;b )=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 7" role="group"><div id="inputtype_fps_fridayw2_1_sol_8_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_fps_fridayw2_1_sol_8_1" id="input_fps_fridayw2_1_sol_8_1" aria-describedby="status_fps_fridayw2_1_sol_8_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_fps_fridayw2_1_sol_8_1"/>
<span class="status unanswered" id="status_fps_fridayw2_1_sol_8_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<p id="answer_fps_fridayw2_1_sol_8_1" class="answer"/>
<div id="display_fps_fridayw2_1_sol_8_1" class="equation">`{::}`</div>
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<p>
<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_7"/>
</div></p>
<p><b class="bfseries">Question 9</b> (Repeating Steps 6 and 7 of the methodology): What is the electric field in the region [mathjaxinline]r&gt;b[/mathjaxinline] . Write your answer using some or all of the following: r, h, a, L, pi for [mathjaxinline]\pi[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline] and Q as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]E (r&gt;b )=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 8" role="group"><div id="inputtype_fps_fridayw2_1_sol_9_1" class="text-input-dynamath capa_inputtype inline textline">
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<div class="solution-span">
<span id="solution_fps_fridayw2_1_sol_solution_8"/>
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<p><b class="bfseries">Question 10</b> (Step 8 of the methodology): Make a graph of the magnitude of the electric field as a function of [mathjaxinline]r[/mathjaxinline] for all regions of space. </p>
<p>
There is no answer check for this question. </p>
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<span id="solution_fps_fridayw2_1_sol_solution_9"/>
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<h2 class="hd hd-2 unit-title">W3PS2: Infinite Plane</h2>
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Gauss&#39;s Law Practice Problem 4&lt;br&gt;Uniformly Charged Slab
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<p>
A semi-infinite slab with uniform positive volume charge density [mathjaxinline]\rho[/mathjaxinline] extends from [mathjaxinline]x=-d[/mathjaxinline] to [mathjaxinline]x=d[/mathjaxinline]. Find a vector expression for the electric field everywhere i.e. in the regions (i) [mathjaxinline]x&lt;-d[/mathjaxinline], (ii) [mathjaxinline]-d&lt;x&lt;d[/mathjaxinline], (iii) [mathjaxinline]x&gt;+d[/mathjaxinline]. </p>
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<p><b class="bfseries">Question 1</b>: What is the symmetry property of this charge distribution? </p>
<p>
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<text> Spherical</text>
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<text> Cylindrical</text>
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<text> Planar</text>
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<span id="answer_problem04_2_1"/>
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<p><b class="bfseries">Question 2</b>: What is the direction of the electric field? </p>
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<text> Perpendicular to the slab (i.e. in the [mathjaxinline]\pm x[/mathjaxinline] direction</text>
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<input type="radio" name="input_problem04_3_1" id="input_problem04_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="problem04_3_1-choice_2-label" for="input_problem04_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_problem04_3_1">
<text> Parallel to the slab</text>
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<span id="solution_problem04_solution_1"/>
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<p><b class="bfseries">Note:</b> Although you can find answers to the following two questions by a careful reading of the text of this problem, make sure that you know how to find those answers yourself. </p>
<p><b class="bfseries">Question 3</b>: Define the different regions of space that you will need to find expressions for the electric field. What determines those different regions? </p>
<p>
in particular, how many different formulae for [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline] are you going to have to calculate? </p>
<p>
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<text> 1</text>
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<text> 2</text>
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<input type="radio" name="input_problem04_4_1" id="input_problem04_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="problem04_4_1-choice_3-label" for="input_problem04_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_problem04_4_1">
<text> 3</text>
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<text> 4</text>
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<text> 5</text>
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<text> &gt;5</text>
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<p>
<div class="solution-span">
<span id="solution_problem04_solution_2"/>
</div></p>
<p><b class="bfseries">Question 4</b>: For each region of space, make a figure showing clearly your choice of a Gaussian surface. What variable did you choose to parameterize your Gaussian surface (for example, for a sphere you'd use the radius [mathjaxinline]r[/mathjaxinline])? What is the range of that variable in each region? </p>
<p>
There is no answer checker for this question. </p>
<p>
<div class="solution-span">
<span id="solution_problem04_solution_3"/>
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<p><b class="bfseries">Question 5</b>: In the region [mathjaxinline]x&lt;d[/mathjaxinline], calculate the flux [mathjaxinline]\underset {{\substack {\text {closed} \\ \text {surface S}}}}{\Large \unicode {x222F}}\vec{\mathbf{E}} \cdot d\vec{\mathbf{A}}[/mathjaxinline] through your choice of the Gaussian surface (that is, just write down the left hand side of Gauss's Law). Your expression should include the unknown electric field for that region. Note: For entering your answer, use the notation [mathjaxinline]2x'[/mathjaxinline] and [mathjaxinline]A[/mathjaxinline] to denote the length and end area of the Gaussian pillbox, respectively. Write your answer using some or all of the following: xprime for [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], E, and pi for [mathjaxinline]\pi[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]{\Large \unicode {x222F}}\ \vec{\mathbf{E}} \cdot d\vec{\mathbf{A}}[/mathjaxinline][mathjaxinline]=[/mathjaxinline] </p>
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<span id="solution_problem04_solution_4"/>
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<p><b class="bfseries">Question 6</b>: In the region [mathjaxinline]-d&lt;x'&lt;+d[/mathjaxinline], what is the charge enclosed in your choice of Gaussian surface? (this should be in terms of [mathjaxinline]\rho[/mathjaxinline], [mathjaxinline]x'[/mathjaxinline], and [mathjaxinline]A[/mathjaxinline], not [mathjaxinline]\vec{\mathbf{E}}[/mathjaxinline] ). Write your answer using some or all of the following: xprime for [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], rho for [mathjaxinline]\rho[/mathjaxinline], d, and pi for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]Q_{\text {enc}}=[/mathjaxinline] </p>
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<span id="solution_problem04_solution_5"/>
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<p><b class="bfseries">Question 7</b>: In the region [mathjaxinline]-d&lt;x'&lt;+d[/mathjaxinline], equate the two sides of Gauss's Law that you calculated in Questions 5 and 6 and solve to find an expression for the [mathjaxinline]x[/mathjaxinline]-component of the electric field as a function of [mathjaxinline]x'[/mathjaxinline]. Write your answer using some or all of the following: xprime for [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], rho for [mathjaxinline]\rho[/mathjaxinline], d, pi for [mathjaxinline]\pi[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]E_ x(|x'|&lt;d )=[/mathjaxinline] </p>
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<div id="display_problem04_7_1" class="equation">`{::}`</div>
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<span id="solution_problem04_solution_6"/>
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<p><b class="bfseries">Question 8</b>: Repeat the same procedure in order to calculate the electric field as a function of [mathjaxinline]x'[/mathjaxinline] in the region [mathjaxinline]|x'|&gt;d[/mathjaxinline]. By symmetry, the field must have the same magnitude (but opposite direction) at [mathjaxinline]\pm x'[/mathjaxinline], so your answer for the magnitude will apply to the region outside the slab on both sides. Write your answer using some or all of the following: xprime for [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], rho for [mathjaxinline]\rho[/mathjaxinline], d, pi for [mathjaxinline]\pi[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]E(|x'|&gt;d)=[/mathjaxinline] </p>
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<p>
<div class="solution-span">
<span id="solution_problem04_solution_7"/>
</div></p>
<p><b class="bfseries">Question 9</b>: Think about how you could solve this problem using a <i class="it">different</i> Gaussian surface, namely one with one endcap at [mathjaxinline]x'=0[/mathjaxinline]. </p>
<p>
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<h2 class="hd hd-2 unit-title">W3PS3: Non-Uniform Spherical Charge Distribution</h2>
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Spherical Non-uniform Charge Distribution - Total Charge
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A spherically symmetric charge distribution is non-uniform and has a volume charge density given by: </p>
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[mathjaxinline]\displaystyle \rho (r)= \begin{cases} \rho _0(1-r^2/R^2);\ r\leq R\\ 0; \ r&gt;R \end{cases}[/mathjaxinline]
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where [mathjaxinline]\rho _0&gt;0[/mathjaxinline] is a constant. </p>
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(a) What is the total charge contained inside the sphere of radius [mathjaxinline]R[/mathjaxinline]? Write your answer in terms of rho_0 for [mathjaxinline]\rho _0[/mathjaxinline], pi for [mathjaxinline]\pi[/mathjaxinline], and [mathjaxinline]R[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]Q_{total}=[/mathjaxinline] </p>
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<h3 class="hd hd-2">W03PS02a: Non-Uniform Spherical Charge Distribution - Total Charge</h3>
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Spherical Non-uniform Charge Distribution - Region I
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(b) Find an expression for magnetude of the electric field in region II where [mathjaxinline]r&gt;R[/mathjaxinline]. Write your answer in terms of rho_0 for [mathjaxinline]\rho _0[/mathjaxinline], pi for [mathjaxinline]\pi[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline], [mathjaxinline]r[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{\mathbf{E}}_{II}|=[/mathjaxinline] (for [mathjaxinline]r&gt;R[/mathjaxinline]) </p>
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<h3 class="hd hd-2">W03PS02c: Non-Uniform Spherical Charge Distribution - Electric Field Outside</h3>
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Spherical Non-uniform Charge Distribution - Region II
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(c) Find an expression for the magnitude of the electric field in region II where [mathjaxinline]r&lt;R[/mathjaxinline]. Write your answer in terms of rho_0 for [mathjaxinline]\rho _0[/mathjaxinline], pi for [mathjaxinline]\pi[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline], [mathjaxinline]r[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{\mathbf{E}}_ I|=[/mathjaxinline] (for [mathjaxinline]0&lt;r&lt;R[/mathjaxinline]) </p>
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<h3 class="hd hd-2">W03PS02d: Non-Uniform Spherical Charge Distribution - Electric Field Inside</h3>
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<h2 class="hd hd-2 unit-title">W3PS4: Finding the Charge Distribution from the Electric Field</h2>
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Problem 4: Charged Slab and Sheets
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An infinite slab of charge carrying a uniform volume charge density [mathjaxinline]\rho[/mathjaxinline] has its boundaries located at [mathjaxinline]x=-d[/mathjaxinline] and [mathjaxinline]x=d[/mathjaxinline], where [mathjaxinline]d = 3 \, m[/mathjaxinline]. It is infinite in the [mathjaxinline]y[/mathjaxinline]-direction and in the [mathjaxinline]z[/mathjaxinline]-direction (the <i class="itshape">z-</i>direction is out of the plane of the figure). Two infinite charge sheets (of zero thickness) which are parallel to the <i class="itshape">yz</i>-plane are located at [mathjaxinline]x=-3d[/mathjaxinline] and [mathjaxinline]x=+3d[/mathjaxinline], with uniform surface charge densities [mathjaxinline]\sigma _{1}[/mathjaxinline] and [mathjaxinline]\sigma _{2}[/mathjaxinline] respectively. </p>
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In the accessible regions, we measure the electric field to be </p>
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<td class="equation" style="width:80%; border:none">[mathjax]\vec{\mathbf{E}} (x)=\left\{ \begin{array}{l}&amp; \vec{\mathbf{0}} &amp; \text {for} \; &amp; x &lt; -3d \\ &amp; E\, \hat{\mathbf{i}} &amp; \text {for} \; &amp; -3d \text {&lt; } x\text { &lt;} -\text {d} \\ &amp; -E\, \hat{\mathbf{i}} &amp; \text {for} \; &amp; d \text { &lt; } x\text { &lt;} \text { 3d} \\ &amp; \vec{\mathbf{0}} &amp; \text {for} \; &amp; x &gt; 3d \end{array} \right.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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where [mathjaxinline]E = 10\, N/C[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> What is the value of [mathjaxinline]\rho /\varepsilon _{0}[/mathjaxinline] for the slab? </p>
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<p style="display:inline">[mathjaxinline]\rho /\varepsilon _{0} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Find [mathjaxinline]\sigma _{1} /\varepsilon _0[/mathjaxinline] and [mathjaxinline]\sigma _{2} /\varepsilon _0[/mathjaxinline] for the left and right charged sheets. </p>
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<p style="display:inline">[mathjaxinline]\sigma _{1} /\varepsilon _{0} =[/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]N/C[/mathjaxinline])</p>
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<p style="display:inline">[mathjaxinline]\sigma _{2} /\varepsilon _{0} =[/mathjaxinline] </p>
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<h3 class="hd hd-2">W03PS01: Finding the Charge Distribution from the Electric Field</h3>
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