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<h2 class="hd hd-2 unit-title">Introduction to Drisplacement Current</h2>
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<p>We have finally reached the point where we are going to round out Maxwell's Equations!</p><p>In this lesson we discuss <b>Displacement Current</b>. We start with the motivation for why it must be missing and then practice calcuations involving the Maxwell-Ampere Law.</p>
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<h2 class="hd hd-2 unit-title">L33v1: Displacement Current</h2>
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<h2 class="hd hd-2 unit-title">L33Q1: What's in a Name?</h2>
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What&#39;s In A Name?
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<p>The fourth of Maxwell's equations is \[ \oint_{\rm C} \vec{\mathbf{B}} \cdot d\vec{\ell} = \mu_0 I_{\rm C} + \mu_0 \varepsilon_0\int_{\rm S} \frac{\partial \vec{\mathbf{E}}}{\partial t} \cdot d\vec{\mathbf{A}} \] What is the name of the last term, \(\varepsilon_0\int_{\rm S} \frac{\partial \vec{\mathbf{E}}}{\partial t} \cdot d\vec{\mathbf{A}}\)?</p>
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<h2 class="hd hd-2 unit-title">L33Q2: Time Varying E-Field</h2>
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Time Varying Electric Field
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In a certain region of space a time varying electric field is approximately uniform. The resulting magnetic field lines are approximately circular loops. We will assume that in this region the magnetic field only depends on [mathjaxinline]r[/mathjaxinline], the distance from the center of the loops. If [mathjaxinline]r_1[/mathjaxinline] is the radius of magnetic field line 1, and [mathjaxinline]r_2=2r_1[/mathjaxinline] is the radius of magnetic field line 2, the ratio [mathjaxinline]\dfrac {B_2}{B_1}[/mathjaxinline], of the magnitudes of the magnetic fields at [mathjaxinline]r=r_2[/mathjaxinline] and at [mathjaxinline]r=r_1[/mathjaxinline] is: </p>
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<text> [mathjaxinline]\dfrac {B_2}{B_1} = 4[/mathjaxinline]</text>
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<text> [mathjaxinline]\dfrac {B_2}{B_1} = \dfrac {1}{2}[/mathjaxinline]</text>
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<text> [mathjaxinline]\dfrac {B_2}{B_1} = \dfrac {1}{4}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L33: Signs for Ampere-Maxwell Equation</h2>
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<center><p><b>Ampere - Maxwell Equation - Convention of Circulation and Signs</b></p></center><p>The Ampere-Maxwell Equation relates the line integral of the B-field and the surface integral of \(\mathbf{\vec{J}}\) and \(\mathbf{\vec{E}}\):</p>
\[ \oint_{\text{closed path}} \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}} = \mu_o \iint_{S}\mathbf{\vec{J}}\cdot dA\mathbf{\hat{n}} + \mu_o\epsilon_0\dfrac{\partial }{\partial t}\iint_{S}\mathbf{\vec{E}}\cdot dA\mathbf{\hat{n}}\]
<p> where the line integral is over a closed path and the surface integral is over any open surface whose edge coincides with the closed path. We will call the closed path the "loop", and the open surface the "surface enclosed".</p>
<p><b> Direction of Circulation and The Right Hand Rule.</b></p>
<p> The choice of a direction of circulation along the loop is arbitrary but that choice also determines the direction of the unit vector \(\mathbf{\hat{n}}\) perpendicular to the enclosed surface. The connection between the directions of circulation and \(\mathbf{\hat{n}}\) is given by a right hand rule. In the figures below, the loop is contained in the plane of the screen and the surface is the shaded region. The two possible choices of direction of circulation along the loop, counter-clockwise or clockwise, along with their corresponding directions of \(\mathbf{\hat{n}}\) are shown.</p>
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<p> If we pick the direction of circulation to be <b>counter-clockwise</b>, left figure above, the right hand rule states that the direction of the unit normal \(\mathbf{\hat{n}}\) is <b>out of</b> the screen, therefore \(d\mathbf{\vec{A}}\) is out of the screen.</p>
<p> If we pick the direction of circulation to be <b>clockwise</b>, right figure above, the right hand rule states that the direction of the unit normal \(\mathbf{\hat{n}}\) is <b>into</b> the screen, therefore \(d\mathbf{\vec{A}}\) is into the screen. </p>
<p><b> Review of the Right hand rule.</b></p>
<p> The right hand rule used to determine the direction of the unit normal given the direction of circulation is as follows: "<i>Curl the fingers of your right hand in the direction of the circulation, your thumb points in the direction of </i>\(\mathbf{\hat{n}}\)."</p>
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<h2 class="hd hd-2 unit-title">L33Q3: Magnetic Field in a Capacitor</h2>
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Magnetic Field in a Capacitor
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The figures above show a side and top view of a capacitor with charge [mathjaxinline]Q[/mathjaxinline] and electric and magnetic fields [mathjaxinline]E[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] at time [mathjaxinline]t[/mathjaxinline]. At this time the energy stored in the electric field is: </p>
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<h2 class="hd hd-2 unit-title">L33Q4: A Capacitor of Radius R</h2>
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A Capacitor of Radius R
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Consider the charging capacitor in the diagram, with circular plates of radius R. The capacitor is ideal (ignore edge effects). </p>
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Points [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are at a distance [mathjaxinline]r_1&gt;R[/mathjaxinline] with respect to the center line, and points [mathjaxinline]c[/mathjaxinline] and [mathjaxinline]d[/mathjaxinline] are at a distance [mathjaxinline]r_2&lt; R[/mathjaxinline]. </p>
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Which of the following statements about [mathjaxinline]B[/mathjaxinline], the magnitude of the magnetic field at points [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], and [mathjaxinline]d[/mathjaxinline] are true? </p>
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<text>1. [mathjaxinline]B(a)&gt;B(b)[/mathjaxinline]</text>
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<text>5. [mathjaxinline]B(c) = B(d)[/mathjaxinline]</text>
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<text>6. [mathjaxinline]B(c) &lt; B(d)[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L33Q5: E-Field Changing with Time</h2>
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A Electric Field Changing with Time
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<p>
Consider the situation where an electric field is confined to a circular region in space of radius [mathjaxinline]R[/mathjaxinline], i.e. [mathjaxinline]\mathbf{\vec{E}} = 0[/mathjaxinline] for [mathjaxinline]r&gt;R[/mathjaxinline] and [mathjaxinline]\mathbf{\vec{E}} = E_ z(t)\; \mathbf{\hat{k}}[/mathjaxinline] for [mathjaxinline]r&lt;R[/mathjaxinline], where [mathjaxinline]E_ z(t)[/mathjaxinline] is the z-component of the electric field. We will assume that the E-field is uniform but depends on time. In the figure above, a view of the field from the +z-axis is shown. The origin of the cylindrical coordinate system shown is assumed to be at the center of the E-field region. </p>
<p>
Because the E-field has axial symmetry and the B-lines must be closed lines, we expect them to be concentric circles in the E-field region, and therefore the B-field is given by </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \mathbf{\vec{B}} = B_{\theta }\mathbf{\hat{\theta }}[/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">(Part a)</b> Given that the E-field is independent of [mathjaxinline](r,\theta ,z)[/mathjaxinline] and it is in the [mathjaxinline]\mathbf{\hat{k}}[/mathjaxinline] direction, which of the following statements about the dependence of [mathjaxinline]B_{\theta }[/mathjaxinline] on [mathjaxinline](r,\theta ,z)[/mathjaxinline] is true? </p>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline](r,\theta ,z)[/mathjaxinline]</text>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline](r,\theta )[/mathjaxinline]</text>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline](r,z)[/mathjaxinline]</text>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline](r)[/mathjaxinline] only</text>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline]\theta[/mathjaxinline] only</text>
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<text> [mathjaxinline]B_{\theta }[/mathjaxinline] is a function of [mathjaxinline]z[/mathjaxinline] only</text>
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Because of the limitation of the answer checker, we will write the magnetic field as [mathjaxinline]\mathbf{\vec{B}} = B\; \mathbf{\hat{\theta }}[/mathjaxinline]. We will use [mathjaxinline]B[/mathjaxinline] or [mathjaxinline]B_{\theta }[/mathjaxinline] as the [mathjaxinline]\theta[/mathjaxinline] component of the magnetic field. Recall that the component can be either positive or negative. Our goal is to calculate it. </p>
<p><b class="bfseries">(Part b)</b> Calculate the magnitude of the line integral of the B-field along a circle of radius [mathjaxinline]r&lt;R[/mathjaxinline] from the center of the E-field region. </p>
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<p>
Assume direction of circulation to be counterclockwise. Express your answer in terms B for [mathjaxinline]B_{\theta }[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\oint \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}}|=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> The sign of the line integral in <b class="bfseries">(Part b)</b> is: </p>
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<text> [mathjaxinline]\oint \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}}&gt;0[/mathjaxinline]</text>
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<text> [mathjaxinline]\oint \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}}=0[/mathjaxinline]</text>
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<text> [mathjaxinline]\oint \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}}&lt;0[/mathjaxinline]</text>
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A Electric Field Changing with Time
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<p><b class="bfseries">(Part d)</b> What is the direction of the unit normal [mathjaxinline]\mathbf{\hat{n}}[/mathjaxinline] of the circular area enclosed by the loop of radius [mathjaxinline]r[/mathjaxinline]? </p>
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<text> [mathjaxinline]\mathbf{\hat{n}}=+\mathbf{\hat{k}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\mathbf{\hat{n}}=-\mathbf{\hat{k}}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part e)</b> Calculate the electric flux through the circle enclosed by the loop of radius [mathjaxinline]r[/mathjaxinline]. Express your answer in terms of E_z for [mathjaxinline]E_ z[/mathjaxinline] and [mathjaxinline]r[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\iint _ S \mathbf{\vec{E}}\cdot d\mathbf{\vec{A}}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part f)</b> Assume that the electric field is given by [mathjaxinline]\mathbf{\vec{E}} = c e^{at}\mathbf{\hat{k}}[/mathjaxinline], where [mathjaxinline]c[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline] are constants, with [mathjaxinline]c&gt;0[/mathjaxinline]. Calculate the displacement current. Express your answer in terms of epsilon_0 for [mathjaxinline]\epsilon _0[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\epsilon _0\dfrac {\partial }{\partial t}\iint _ S \mathbf{\vec{E}}\cdot d\mathbf{\vec{A}}=[/mathjaxinline] </p>
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A Electric Field Changing with Time
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<p><b class="bfseries">(Part g)</b> Use the results in <b class="bfseries">(Part b)</b> and <b class="bfseries">(Part f)</b> to obtain the [mathjaxinline]B[/mathjaxinline], the [mathjaxinline]\theta[/mathjaxinline] component of the magnetic field. Express your answer in terms of mu_0 for [mathjaxinline]mu_0[/mathjaxinline], epsilon_0 for [mathjaxinline]\epsilon _0[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], [mathjaxinline]r[/mathjaxinline], and [mathjaxinline]t[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]B=B_{\theta }=[/mathjaxinline] </p>
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<div id="display_checkpoint_w12_38c_2_1" class="equation">`{::}`</div>
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<p><b class="bfseries">(Part h)</b> Assume that the magnitude of the E-field decreases with time, [mathjaxinline]a&lt;0[/mathjaxinline]. Assume the direction of circulation is still counterclockwise. Which of the following statements are true? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w12_38c_3_1">
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<input type="checkbox" name="input_checkpoint_w12_38c_3_1[]" id="input_checkpoint_w12_38c_3_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="checkpoint_w12_38c_3_1-choice_1-label" for="input_checkpoint_w12_38c_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w12_38c_3_1"> <text>2. [mathjaxinline]\oint \mathbf{\vec{B}}\cdot d\mathbf{\vec{s}}&lt;0[/mathjaxinline]</text>
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<input type="checkbox" name="input_checkpoint_w12_38c_3_1[]" id="input_checkpoint_w12_38c_3_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="checkpoint_w12_38c_3_1-choice_2-label" for="input_checkpoint_w12_38c_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w12_38c_3_1"> <text>3. [mathjaxinline]B_{\theta }&gt;0[/mathjaxinline]</text>
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<input type="checkbox" name="input_checkpoint_w12_38c_3_1[]" id="input_checkpoint_w12_38c_3_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="checkpoint_w12_38c_3_1-choice_3-label" for="input_checkpoint_w12_38c_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w12_38c_3_1"> <text>4. [mathjaxinline]B_{\theta }&lt;0[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L33Q6: Relation Between E and B</h2>
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Relation Between E and B
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<p>Which of the following choice(s) represent a possible correct physical relationship between \(E\) and \(B\) fields? Check all that apply:</p>
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