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<p> We defined potential difference in terms of an integral of the electric field. Now we will try to go the other way and look at how the electric field is related to the potential through a derivative.</p><p> Textbook Links </p><ul><li><a href="/courses/course-v1:MITx+8.02.1x+1T2019/pdfbook/0/chapter/4/13">Chapter 4.4-4.5 Calculating Electric Field from Potential</a></li></ul>
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<h2 class="hd hd-2 unit-title">L10Q1: Sign of the Electric Field</h2>
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Sign of the Electric Field
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The graph above shows a plot of potential [mathjaxinline]V(x)[/mathjaxinline]. Which of the following is true? </p>
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<text> [mathjaxinline]E_{x &lt; 0}[/mathjaxinline] is positive and [mathjaxinline]E_{x &gt; 0}[/mathjaxinline] is positive</text>
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<text> [mathjaxinline]E_{x &lt; 0}[/mathjaxinline] is positive and [mathjaxinline]E_{x &gt; 0}[/mathjaxinline] is negative</text>
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<text> [mathjaxinline]E_{x &lt; 0}[/mathjaxinline] is negative and [mathjaxinline]E_{x &gt; 0}[/mathjaxinline] is negative</text>
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<text> [mathjaxinline]E_{x &lt; 0}[/mathjaxinline] is negative and [mathjaxinline]E_{x &gt; 0}[/mathjaxinline] is positive</text>
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<h2 class="hd hd-2 unit-title">L10v2: Calculate Electric Field from the Potential Along the Axis of a Charged Ring</h2>
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<h2 class="hd hd-2 unit-title">L10Q2: Finding the Electric Field from the Electric Potential</h2>
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Finding the Electric Field from the Electric Potential, part a
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Suppose that the electric potential varies along the <i class="itshape">x</i>-axis as shown in the figure below. </p>
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<img src="/assets/courseware/v1/dc68560b73053d5355080ba92be1482e/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_Spring_2014_pset04_2-fig001.jpg" width="330"/>
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The potential does not vary in the <i class="itshape">y</i> or <i class="itshape">z</i> directions. </p>
<p><b class="bfseries">(Part a)</b> Of the intervals shown, determine the intervals in which the electric field [mathjaxinline]E_{x}[/mathjaxinline] has its largest and smallest magnitudes (i.e. the absolute value) and find those magnitudes. Ignore the behavior at the end points of the intervals. </p>
<p>
Largest value of the magnitude of the electric field: </p>
<p>
<p style="display:inline">[mathjaxinline]\left|E_ x\right|_{\text {max}} =[/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]V/m[/mathjaxinline])</p>
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Smallest value of the magnitude of the electric field: </p>
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<p style="display:inline">[mathjaxinline]\left|E_ x\right|_{\text {least}} =[/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]V/m[/mathjaxinline])</p>
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Finding the Electric Field from the Electric Potential, part b
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<p><b class="bfseries">(Part b)</b> What charge distributions would produce this potential function? </p>
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<text> Point charges located at each vertex b, c, and d</text>
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<text> Lines of charge in the [mathjaxinline]z[/mathjaxinline] direction at each vertex b, c, and d</text>
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<text> Slabs of charge extending in [mathjaxinline]y z[/mathjaxinline] directions with different charge densities between each vertex b, c, and d</text>
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<text> Some other distribution entirely</text>
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<h2 class="hd hd-2 unit-title">L10Q3: E from V</h2>
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E from V
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Consider the point-like charged objects arranged in the figure below. </p>
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The electric potential difference between the point [mathjaxinline]P[/mathjaxinline] and [mathjaxinline]\infty[/mathjaxinline] is </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V(P)-V(\infty ) = V(P) = -kQ/a[/mathjaxinline]
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From this potential difference, can you determine the functional form for the electric field at the point P? </p>
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<text> Yes, it's [mathjaxinline]kQ/a^2[/mathjaxinline] upwards.</text>
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<text> Yes in theory, but we do not know its exact value.</text>
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<text> No, you can't get the electric field at point [mathjaxinline]P[/mathjaxinline] from just knowledge of the electric potential at point [mathjaxinline]P[/mathjaxinline].</text>
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<h2 class="hd hd-2 unit-title">L10v4: Equipotentials</h2>
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<h2 class="hd hd-2 unit-title">L10Q4: Equipotentials</h2>
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Equipotentials
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The two figures below show grass seed representations of the equipotential lines of the electric potential created by two arrangements of point charges. Which of the following charge arrangement(s) could correspond to these equipotentials? </p>
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<text> Both charges have the same sign and equal magnitude.</text>
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<text> The charges have opposite signs and equal magnitude.</text>
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<text> The two charges have opposite signs and the charge on the left is smaller in magnitude than the charge on the right.</text>
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<text> The two charges have the same sign and the charge on the left is smaller in magnitude than the charge on the right.</text>
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<text> The two charges have the same sign and the charge on the left is larger in magnitude than the charge on the right.</text>
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