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<h2 class="hd hd-2 unit-title">Introduction to Lenz's Law</h2>
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<p><b>Lenz's Law</b></p><p> Now that we have introduced Faraday's Law, we need to start exploring the direction of the induced current. To do this, we will focus on the negative sign in Faraday's Law, also known as Lenz's law. </p><p>We start this lesson with a few demos and then return to the equation.</p>
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<p>Textbook Links</p>
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<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.3x+1T2019/pdfbook/0/#viewer-frame" target="[object Object]">Chapter 10.4: Lenz's Law </a></li>
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<h2 class="hd hd-2 unit-title">L24v1: Bar Magnet Through a Coil of Wire</h2>
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<h3 class="hd hd-2">L24v1: Bar Magnet Through a Coil of Wire</h3>
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<h2 class="hd hd-2 unit-title">L24Q1: Bar Magnet Moving at Different Velocities</h2>
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Bar Magnet Moving at Different Velocities
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<p>
Four bar magnets are moving toward circular wire loops that are fixed in space. All the wire loops have the same radius and are made of the same material. The magnets are all identical, but they are approaching the loops at different speeds and with different orientations. </p>
<p>
Which of the following statements is true about the magnitude of the induced current at the instant when the magnet is 2 cm from the loop. In the options, A&gt;B means that the magnitude of the current in figure A is larger than the magnitude of the current in figure B. </p>
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<text> 1. A &gt; B &gt; C &gt;D</text>
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<text> 3. B &gt; A = D &gt;C</text>
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<text> 4. B &gt; A &gt; C &gt;D</text>
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<h2 class="hd hd-2 unit-title">L24Q2: Bar Magnet through a Ring</h2>
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Bar Magnet through a Ring
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<p>
Consider a conducting ring fixed in the [mathjaxinline](x,y)[/mathjaxinline] plane. A bar magnet is moving at a constant velocity towards the center of the ring along the ring's axis of symmetry. At time [mathjaxinline]t=0[/mathjaxinline], the midpoint of the magnet is at the center of the ring (note that this is a time later than the instant shown in the figure). </p>
<p>
Consider the magnetic flux through the circle with the ring as a boundary. The element of area [mathjaxinline]\mathbf{d\vec{A}}[/mathjaxinline] is in the [mathjaxinline]+\mathbf{\hat{k}}[/mathjaxinline]-direction. Consider the induced current to be positive when it circulates counterclockwise as viewed from the [mathjaxinline]+z[/mathjaxinline] axis (i. e. as viewed by the smily character in the figure.) </p>
<p>
The vertical lines in the graphs below are at [mathjaxinline]t=0[/mathjaxinline]. </p>
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<img src="/assets/courseware/v1/ae816b32721ee09877fffe39615fb96f/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_checkpoint_w10_10b-01.svg" width="660"/>
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<p>
The plot of the magnetic flux [mathjaxinline]\Phi _{mag}[/mathjaxinline] vs. [mathjaxinline]t[/mathjaxinline] is best represented by: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_10_2_1">
<fieldset aria-describedby="status_checkpoint_w10_10_2_1">
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<input type="radio" name="input_checkpoint_w10_10_2_1" id="input_checkpoint_w10_10_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w10_10_2_1-choice_1-label" for="input_checkpoint_w10_10_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_2_1"> <text> Figure a.</text>
</label>
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<input type="radio" name="input_checkpoint_w10_10_2_1" id="input_checkpoint_w10_10_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w10_10_2_1-choice_2-label" for="input_checkpoint_w10_10_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_2_1"> <text> Figure b.</text>
</label>
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<input type="radio" name="input_checkpoint_w10_10_2_1" id="input_checkpoint_w10_10_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w10_10_2_1-choice_3-label" for="input_checkpoint_w10_10_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_2_1"> <text> Figure c.</text>
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<input type="radio" name="input_checkpoint_w10_10_2_1" id="input_checkpoint_w10_10_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w10_10_2_1-choice_4-label" for="input_checkpoint_w10_10_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_2_1"> <text> Figure d.</text>
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<p>
The plot of the induced current [mathjaxinline]I[/mathjaxinline] vs. [mathjaxinline]t[/mathjaxinline] is best represented by: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_10_3_1">
<fieldset aria-describedby="status_checkpoint_w10_10_3_1">
<div class="field">
<input type="radio" name="input_checkpoint_w10_10_3_1" id="input_checkpoint_w10_10_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="checkpoint_w10_10_3_1-choice_1-label" for="input_checkpoint_w10_10_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_3_1"> <text> Figure a.</text>
</label>
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<div class="field">
<input type="radio" name="input_checkpoint_w10_10_3_1" id="input_checkpoint_w10_10_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="checkpoint_w10_10_3_1-choice_2-label" for="input_checkpoint_w10_10_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_3_1"> <text> Figure b.</text>
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<input type="radio" name="input_checkpoint_w10_10_3_1" id="input_checkpoint_w10_10_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="checkpoint_w10_10_3_1-choice_3-label" for="input_checkpoint_w10_10_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_3_1"> <text> Figure c.</text>
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<input type="radio" name="input_checkpoint_w10_10_3_1" id="input_checkpoint_w10_10_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="checkpoint_w10_10_3_1-choice_4-label" for="input_checkpoint_w10_10_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_checkpoint_w10_10_3_1"> <text> Figure d.</text>
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<h2 class="hd hd-2 unit-title">Sign Convention and the Right Hand Rule</h2>
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<p><b>Faraday's Law. Sign convention and the Right Hand Rule</b></p><p>Faraday's law relates the line integral of the E-field and the time derivative of the magnetic flux:</p><p>
\[ \oint_{\text{closed path}} \mathbf{\vec{E}}\cdot \mathbf{d\vec{s}} = -\dfrac{ d}{dt}\iint_{S} \mathbf{\vec{B}}\cdot \mathbf{d\vec{A}} \]
</p><p> where the line integral is over a closed path. The closed path is often called a <i>loop</i>, and the magnetic flux is calculated over any open surface whose edges coincide with the loop. We refer to this open surface as <i>the surface enclosed by the loop</i>. </p><p><b> The Right Hand Rule, direction of circulation along the loop \(\mathbf{d\vec{s}}\), and the direction of \(\mathbf{d\vec{A}}\).</b></p><p>To calculate the flux of the magnetic field vector we need to define the element of surface \(\mathbf{d\vec{A}}= dA\mathbf{\hat{n}}\), where \(\mathbf{\hat{n}}\) is the unit vector perpendicular to the surface. The direction of the unit vector is related to the direction of circulation in the loop and is defined by the right hand rule. Below, the loop is contained in the plane of the screen. The two possible choices of direction of circulation along the loop, counter-clockwise or clockwise, are shown.</p><p><center><img src="/assets/courseware/v1/eef462a130902ad5067c8b89afa2afb9/asset-v1:MITx+8.02.3x+1T2019+type@asset+block/images_html_lesson23_04b.svg" width="800"/></center></p><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 \(\mathbf{d\vec{A}}\) 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 \(\mathbf{d\vec{A}}\) 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">L24Q3: Direction of the Induced Current</h2>
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Direction of the Induced Current
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<p><b class="bfseries">(Part a)</b> A current [mathjaxinline]I[/mathjaxinline] flows in an infinite wire. A rectangular conducting loop is placed with one of its sides parallel to the wire as shown. In Figure (1) the loop is moving parallel to the wire, while in Figure (2), the loop is moving away from the wire. </p>
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As viewed from the screen, the current induced is </p>
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<p style="display:inline">Figure (1)</p>
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<p style="display:inline">Figure (2)</p>
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<p><b class="bfseries">(Part b)</b> At time [mathjaxinline]t=0[/mathjaxinline], a rectangular conducting loop is perpendicular to a vertical and uniform magnetic field of magnitude [mathjaxinline]B[/mathjaxinline] as shown in the left figure below. (The loop is in the [mathjaxinline](x,y)[/mathjaxinline] plane and the B-field is directed along [mathjaxinline]+\hat{k}[/mathjaxinline]). The loop is free to rotate about its [mathjaxinline]\hat{i}[/mathjaxinline] axis of symmetry. </p>
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<p>
At the instant [mathjaxinline]t = T[/mathjaxinline], when the loop has rotated [mathjaxinline]45^ o[/mathjaxinline] from its original position, as shown in the side view in the right figure above, the induced current in the loop as seen from above is: </p>
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<p><b class="bfseries">(Part c)</b> A conducting circular loop is placed perpendicular to a uniform magnetic field. The magnitude of the magnetic field is increasing with time. </p>
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As viewed from the [mathjaxinline]+z[/mathjaxinline] axis, the induced current is: </p>
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<h2 class="hd hd-2 unit-title">L24Q4: Bar Magnet Moving Towards a Ring</h2>
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Bar Magnet Moving towards a Ring
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Consider a conducting ring fixed in the [mathjaxinline](x,y)[/mathjaxinline] plane. A bar magnet is moving at a constant velocity towards the center of the ring along the ring's axis of symmetry. At time [mathjaxinline]t = t_1[/mathjaxinline] the magnet is very far from the ring, at a later time [mathjaxinline]t=t_2[/mathjaxinline] the center of the magnet is at the center of the ring. </p>
<p>
Consider the magnetic flux through the circle with ring as a boundary. The element of area [mathjaxinline]\mathbf{d\vec{A}}[/mathjaxinline] is in the [mathjaxinline]+\mathbf{\hat{k}}[/mathjaxinline]-direction. </p>
<p><b class="bfseries">(Part a)</b> During the time interval [mathjaxinline]t_1&lt;t&lt;t_2[/mathjaxinline], while the magnet is approaching the ring from the left until it is at the center of the ring, </p>
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<text> 1. the flux increases in magnitude and is negative</text>
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<text> 2. the flux increases in magnitude and is positive</text>
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<text> 3. the flux decreases in magnitude and is negative</text>
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<text> 4. the flux decreases in magnitude and is positive</text>
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<p><b class="bfseries">(Part b)</b> During the time interval [mathjaxinline]t_1&lt;t&lt;t_2[/mathjaxinline] the induced current in the ring as viewed from the [mathjaxinline]+z[/mathjaxinline] axis (as viewed from the smily character in the figure) </p>
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<text> 1. flows clockwise reaching a maximum value at [mathjaxinline]t_2[/mathjaxinline].</text>
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<text> 2. flows counterclockwise reaching a maximum value at [mathjaxinline]t_2[/mathjaxinline].</text>
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<text> 3. flows clockwise reaching a maximum value and then goes to zero at [mathjaxinline]t_2[/mathjaxinline] .</text>
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<text> 4. flows counterclockwise reaching a maximum value and then goes to zero at [mathjaxinline]t_2[/mathjaxinline] .</text>
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<p><b class="bfseries">(Part c)</b> During the time interval [mathjaxinline]t_1&lt;t&lt;t_2[/mathjaxinline] the force exerted by the ring on the magnet </p>
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<text> 1. is repulsive</text>
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<text> 2. is attractive</text>
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<h2 class="hd hd-2 unit-title">L24Q5: Lenz's Law and Conservation of Energy</h2>
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Lenz&#39;s Law and Conservation of Energy
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<p>
Consider a conducting ring fixed in the [mathjaxinline](x,y)[/mathjaxinline] plane. A bar magnet is moving at a constant velocity towards the center of the ring along the ring's axis of symmetry. At time [mathjaxinline]t=0[/mathjaxinline] the midpoint of the magnet is at the center of the ring. </p>
<p>
While the magnet is moving from [mathjaxinline]z=-\infty[/mathjaxinline] towards the center of the ring, the direction of the external force exerted on the magnet to keep it moving at a contant speed is: </p>
<p>
<b class="bfseries">(Part a)</b>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w10_11_2_1">
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<text> along the [mathjaxinline]-\hat{k}[/mathjaxinline]</text>
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<text> no force is needed</text>
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<text> along the [mathjaxinline]+\hat{k}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> While the magnet is moving from the center of the ring towards [mathjaxinline]z=+\infty[/mathjaxinline], the direction of the external force exerted on the magnet to keep it moving at a contant speed is: </p>
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<text> along the [mathjaxinline]-\hat{k}[/mathjaxinline]</text>
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<text> no force is needed</text>
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<text> along the [mathjaxinline]+\hat{k}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part c)</b> If Lenz's law were not true, i. e. the minus sign in Faraday's law were not there, the induced current in the ring will have the opposite direction than the one in the situations described in parts (a) and (b). In this hypothetical situation, when the induced currents are in opposite directions as those in parts (a) and (b), between [mathjaxinline]z=-\infty[/mathjaxinline] and the point when the bar reaches the ring (with an initial velocity in the [mathjaxinline]+\hat{k}[/mathjaxinline] direction), </p>
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<text> the magnet will move at a constant speed if no external force is applied to it</text>
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<text> the magnet will accelerate towards [mathjaxinline]z=+\infty[/mathjaxinline] if no external force is applied to it.</text>
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<text> the magnet will slow down before reaching the center of the ring if no external force is applied.</text>
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