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<p> We start this week by talking about electric dipoles. We discuss why they are important to study, how to calculate a dipole moment, and look at their behavior in electric fields.</p><p> Textbook Links </p><ul><li><a href="/courses/course-v1:MITx+8.02.1x+1T2019/pdfbook/0/chapter/2/12">Chapter 2.7-2.8 Dipoles</a></li></ul>
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<h2 class="hd hd-2 unit-title">L3v1: Introduction to Electric Dipoles</h2>
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<h3 class="hd hd-2">L03v01: Introduction to Electric Dipoles</h3>
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<h2 class="hd hd-2 unit-title">L3Q1: Dipoles</h2>
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Dipoles
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Which of the following are true about dipoles? Check all that apply. </p>
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<text>(a) The magnitude of the dipole moment is given by the magnitude of the charge times the distance between them.</text>
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<text>(b) The direction of the dipole moment is from the positive charge to the negative charge.</text>
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<text>(c) Forces between dipoles are stronger than forces between point charges.</text>
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<text>(d) The calculated dipole moment is independent of where the charges are with respect to the origin of the coordinate system.</text>
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<h2 class="hd hd-2 unit-title">L3Q2: Direction of the Dipole Moment Vector</h2>
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Direction of the moment dipole vector.
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The grass seeds representing the electric field of a dipole is shown above. The direction of the moment dipole vector [mathjaxinline]\vec{p}[/mathjaxinline] is shown with the arrow. </p>
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<text> (a) Charge 1 is the positive charge.</text>
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<h2 class="hd hd-2 unit-title">L3Q3: Dipole Moment from 4 Charges</h2>
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Dipole Moment from 4 Charges
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Consider the charge configuration shown in the figure. The dipole moment vector in the coordinate system shown in the figure is given by: </p>
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<h2 class="hd hd-2 unit-title">L3v5: Dipole Moment is Independent of Origin</h2>
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<h3 class="hd hd-2">L03v05: Dipole Moment is Independent of Origin</h3>
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<h2 class="hd hd-2 unit-title">L3v6: Force and Torque on a Dipole in a Uniform Field</h2>
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Worked Example - Electric Dipole Torque
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Suppose two charged objects with charges [mathjaxinline]+q[/mathjaxinline] and [mathjaxinline]-q[/mathjaxinline] are attached to the ends of a non-conducting rod of length [mathjaxinline]d[/mathjaxinline] so that they form a dipole with a fixed distance [mathjaxinline]d[/mathjaxinline]. The object starts at rest. </p>
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<p><b class="bfseries">(Part a)</b> What is the magnitude [mathjaxinline]p[/mathjaxinline] of the dipole moment of this system? Express your answer in terms of some or all of the following: [mathjaxinline]q[/mathjaxinline] and [mathjaxinline]d[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]p =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Now place this dipole in a uniform external electric field [mathjaxinline]\vec{\mathbf{E}} =E\hat{\mathbf{i}}[/mathjaxinline], where the [mathjaxinline]\hat{i}[/mathjaxinline] axis points to the right and the rod makes an angle [mathjaxinline]\theta[/mathjaxinline] with respect to the [mathjaxinline]\hat{i}[/mathjaxinline] direction, as shown in the figure above. </p>
<p>
What is the magnitude of the total electric force due to the external electric field on the dipole, [mathjaxinline]\left| \vec{\mathbf{F}} _{e} \right|[/mathjaxinline]? Express your answer in terms of some or all of the following: [mathjaxinline]q[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], [mathjaxinline]E[/mathjaxinline] and theta for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\left| \vec{\mathbf{F}} _{e} \right| =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> What is the magnitude of the total torque due to the external electric field on the dipole, [mathjaxinline]\left| \vec{\tau }_{e} \right|[/mathjaxinline]? Express your answer in terms of some or all of the following: the electric dipole moment [mathjaxinline]p[/mathjaxinline], [mathjaxinline]E[/mathjaxinline] and theta for [mathjaxinline]\theta[/mathjaxinline]. Use sin() or cos() as needed. </p>
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<p style="display:inline">[mathjaxinline]\left| \vec{\tau }_{e} \right| =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> Based on your previous answers, what type of motion to you expect the dipole to undergo? </p>
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<text> Translation and rotation</text>
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<h2 class="hd hd-2 unit-title">L3Q4: Dipole in Non-Uniform Field</h2>
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Dipole in Non-Uniform Field
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An electric dipole sits in a non-uniform external electric field [mathjaxinline]\vec{E}[/mathjaxinline] in the direction shown. </p>
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Due to the electric field, this dipole will feel: <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_checkpoint_w2_2_2_1">
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<h2 class="hd hd-2 unit-title">Summary of Electric Dipoles</h2>
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<p><b>Electric Dipole Summary </b></p><p>
For an overall charge-neutral system having \(N\) charged objects, the electric dipole vector \(\vec{p}\) is defined as
</p><p>
\[
\vec{\mathbf{p}} \equiv \sum\limits_{i=1}^{i=N}q_{i} \vec{\mathbf{r}} _{\mathbf{i}} \]
</p><p>
where \(\vec{\mathbf{r}} _{\mathbf{i}}\) is the position vector of the charged object with charge \(q_{i}\). The position vector \(\vec{\mathbf{r}} _{\mathbf{i}}\) can be measured with respect to any point because the dipole moment vector is independent of the origin.
</p><p>
Two equal but opposite point-like charged objects form an electric field that
far from the charged objects is an electric dipole. Consider the charged objects separated by a distance \(2a\) shown in the left figure below. The resulting electric field is shown in the right figure.
</p><p><center><img src="/assets/courseware/v1/85debeb80e2d8d0616c8c6a2be549b55/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_lesson03_dipole_summary.svg" width="600"/></center></p><p><ol><li> The electric dipole moment
vector \(\vec{\mathbf{p}}\) points from the negative point-like charged object to the positive point-like
charged object, and has a magnitude \(p=2aq\) .
</li><li>
The <b>torque</b> \(\vec{\tau}\) acting on an electric dipole \(\vec{\mathbf{p}}\) placed in a uniform electric field \(\vec{\mathbf{E}}\) is \(\vec{\mathbf{\tau}}=\vec{\mathbf{p}}\times \vec{\mathbf{E}}\).
</li></ol></p>
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