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<p>Just like Gauss's Law for electric fields, Ampere's Law for magnetic fields is true in all cases but can only be used to find an easy analytical solution for certain geometries. In this lesson, we go through most of those geometries.</p>
<p>Textbook Links</p>
<ul>
<li><a href="https://openlearninglibrary.mit.edu/courses/course-v1:MITx+8.02.2x+2T2018/pdfbook/0/#viewer-frame" target="[object Object]">Chapter 9.3-9.4 Ampere's Law</a></li>
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<h2 class="hd hd-2 unit-title">L22v1: Ampere's Law for an Infinite Wire</h2>
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<h3 class="hd hd-2">L22v01: Ampere's Law for an Infinite Wire</h3>
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<h2 class="hd hd-2 unit-title">L22v2: Ampere's Law for an Infinite Wire with Uniform Current Density</h2>
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<h2 class="hd hd-2 unit-title">L22Q1: B around 3 Currents</h2>
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B Around 3 Currents
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<p>The figure shows a three-conductor coaxial cable. The central wire carries [mathjaxinline]1 \, A[/mathjaxinline] out of the screen; an
inner conducting cylindrical shell centered around the wire carries [mathjaxinline]4 \, A[/mathjaxinline] of current into the sceen; an
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<p>What is the direction of the magnetic field in region II?</p>
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<text>Counterclockwise</text>
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<h2 class="hd hd-2 unit-title">L22v4: Magnetic Field of an Ideal Solenoid</h2>
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<h2 class="hd hd-2 unit-title">L22Q2: Co-axial Solenoids</h2>
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Co-axial Solenoids
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Two co-axial very long solenoids are centered on the [mathjaxinline]z[/mathjaxinline] axis with positive [mathjaxinline]z[/mathjaxinline] pointing up as shown. Both solenoids have [mathjaxinline]n[/mathjaxinline] turns per unit length. A current runs through the inner turns in a clockwise direction when viewed from above. A current of the same magnitude runs through the outer turns in a counterclockwise direction when viewed from above. The magnetic field in the region between the coils </p>
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<text> points in the positive z direction.</text>
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<text> points in the negative z direction.</text>
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<text> Not enough information is given to determine direction of magnetic field.</text>
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<h2 class="hd hd-2 unit-title">L22v5: Magnetic Field of an Infinite Slab of Current</h2>
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<h2 class="hd hd-2 unit-title">L22Q3: Simple Geometries</h2>
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Simple Geometries
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<p>Which of the following has enough symmetry to be usefully analyzed with Ampere's Law to find a simple analytical solution for the magnetic field?</p>
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<text>a) large flat sheet of current</text>
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<text>b) long straight wire</text>
</label>
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<text>c) a long cylinder oriented along the \(z\) axis with constant current density, [mathjaxinline]\vec{j} = j_0 \hat{k}[/mathjaxinline]</text>
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<text>d) a long cylinder oriented along the \(z\) axis with non-uniform current density that varies with distance from the central axis, [mathjaxinline]\vec{j} = j(r) \hat{k}[/mathjaxinline]</text>
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<text>e) B field of a perfectly circular loop of current on the axis perpendicular to the loop and passing through its center</text>
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<p>These relatively simple cases give you a quick way to determine how large a magnetic field is likely to be. For
example, suppose you have a long thin wire carrying [mathjaxinline]0.5 \, A[/mathjaxinline] of current. What will be the value in tesla of
[mathjaxinline]B[/mathjaxinline] at the distance [mathjaxinline]1 \, mm[/mathjaxinline] from the center of the wire? Do this in your head using [mathjaxinline]\mu_0=4\pi \times 10^{-7} \, T \cdot m/A[/mathjaxinline].</p>
<p style="display:inline">[mathjaxinline]B = [/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]T[/mathjaxinline])</p>
<p>The earth's magnetic field is roughly [mathjaxinline]0.1 \, mT[/mathjaxinline]. How does the magnitude of the magnetic field at [mathjaxinline]1 \, mm[/mathjaxinline] from the
center of the wire compare to the magnitude of Earth's magnetic field?</p>
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