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<p>Now that we undestand how magnetic fields affect moving charges, we now want to understand how moving charges make magnetic fields. This lesson introduces the Biot-Savart equation and several worked examples to solve it.</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 9.1-9.2: Biot-Savart and Force on a Wire </a></li>
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<h2 class="hd hd-2 unit-title">L18v1: How to Make a Magnetic Field</h2>
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<h2 class="hd hd-2 unit-title">L18Q1: Direction of the Magnetic Field near a Wire</h2>
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Direction of the Magnetic Field near a Wire
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<p>The direction of magnetic field can be found using a tiny magnet such as the needle of a compass. The end of the
needle that points to Earth's geographic north is called the north pole of the compass; the other end is called the
south pole. The compass needle lines up along a magnetic field line of
Earth's magnetic field, and by convention the direction of the field lines is the direction toward which the compass's north pole points.</p>
<p>A compass needle has two poles - north and south; it is the most familiar example of a magnetic dipole. Just as an
electric dipole lines up with an electric field line, so does the magnetic dipole line up with magnetic field lines.
You may have seen iron filings distributed around a bar magnet or other source of a magnetic field. The filings are basically tiny compass needles, tiny
magnetic dipoles, and they orient themselves along the magnetic field lines of whatever magnetic field is applied. The
resulting pattern of a lot of dipoles scattered in a magnetic field gives you a sense of the shape and intensity of the
magnetic field.</p>
<p>Which of the following statements is (are) true for the magnetic field lines of \(\vec{B}\) close to a current
carrying wire?</p>
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<text>(a) The field lines go in circles around the wire.</text>
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<text>(b) The field lines are perpendicular to the direction of flow of the current [mathjaxinline]I[/mathjaxinline].</text>
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<text>(c) The field lines are perpendicular to any straight line from the wire to the field point.</text>
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<text>(d) The field lines of any [mathjaxinline]\vec{B}[/mathjaxinline] form closed loops.</text>
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<text>(e) [mathjaxinline]\vec{B}[/mathjaxinline] points in the direction that a small compass needle's north pole points when it is placed in the
field [mathjaxinline]\vec{B}[/mathjaxinline].</text>
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<h2 class="hd hd-2 unit-title">L18v2: Magnetic Field of a Moving Point Charge</h2>
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<h2 class="hd hd-2 unit-title">L18Q2: Magnetic Field from a Point Charge</h2>
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Magnetic Field from a Point Charge
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A small object of charge [mathjaxinline]q&gt;0[/mathjaxinline] is moving along the [mathjaxinline]x[/mathjaxinline]-axis with a velocity [mathjaxinline]\vec{v}=v_0\hat{i}[/mathjaxinline]. At the instant shown in the figure, points [mathjaxinline]1[/mathjaxinline], [mathjaxinline]2[/mathjaxinline], [mathjaxinline]3[/mathjaxinline], [mathjaxinline]4[/mathjaxinline] and [mathjaxinline]5[/mathjaxinline] are at a distance [mathjaxinline]d[/mathjaxinline] from the object and separated by [mathjaxinline]30^{\circ }[/mathjaxinline]. Let [mathjaxinline]\vec{B}_1[/mathjaxinline], [mathjaxinline]\vec{B}_2[/mathjaxinline], [mathjaxinline]\vec{B}_3[/mathjaxinline], [mathjaxinline]\vec{B}_4[/mathjaxinline] and [mathjaxinline]\vec{B}_5[/mathjaxinline] be the magnetic field at points [mathjaxinline]1[/mathjaxinline], [mathjaxinline]2[/mathjaxinline], [mathjaxinline]3[/mathjaxinline], [mathjaxinline]4[/mathjaxinline] and [mathjaxinline]5[/mathjaxinline], respectively. Express your answer in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], mu_0 for [mathjaxinline]\mu _0[/mathjaxinline] as needed. </p>
<p><b class="bfseries">1.</b> The ratio between the magnitudes of the magnetic field at point [mathjaxinline]2[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] is: </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{B}_2|/|\vec{B}_1| =[/mathjaxinline] </p>
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<p><b class="bfseries">2.</b> The ratio between the magnitudes of the magnetic field at point [mathjaxinline]3[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] is: </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{B}_3|/|\vec{B}_1| =[/mathjaxinline] </p>
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<p><b class="bfseries">3.</b> The ratio between the magnitudes of the magnetic field at point [mathjaxinline]4[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] is: </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{B}_4|/|\vec{B}_1| =[/mathjaxinline] </p>
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<p><b class="bfseries">4.</b> The ratio between the magnitudes of the magnetic field at point [mathjaxinline]5[/mathjaxinline] and [mathjaxinline]1[/mathjaxinline] is: </p>
<p>
<p style="display:inline">[mathjaxinline]|\vec{B}_5|/|\vec{B}_1| =[/mathjaxinline] </p>
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Biot-Savart
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Consider the L-shaped wire in the figure. A current [mathjaxinline]I[/mathjaxinline] flows as shown. The magnetic field at point [mathjaxinline]P[/mathjaxinline] points in the </p>
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<p><b>Superposition principle</b></p><p>Consider a pair of wires carrying current. The magnetic field at a point in space is obtained by adding the magnetic field produced by each of the wires at that point.</p><center><img src="/assets/courseware/v1/047a729c25a119ba25fce475cf0fe140/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_html_lesson18_01.svg" width="500"/></center><p>Consider the two wires shown in the figure above. The magnetic field at point P is obtained as:
</p><p>
\[\vec{B}(P) = \vec{B}_1(P)+\vec{B}_2(P)\]
</p><p>where \(\vec{B}_1(P)\) is the magnetic field produced by current \(I_1\) and given by:
</p><p>
\(\displaystyle \vec{B}_1=\frac{\mu_0}{4\pi}\int_{\text{wire 1}}\frac{I_1 d\vec{s}_1\times \hat{r}_1}{r_1^2}\)
</p><p>
and \(\vec{B}_2(P)\) is the magnetic field produced by current \(I_2\) and given by:
</p><p>
\(\displaystyle \vec{B}_2=\frac{\mu_0}{4\pi}\int_{\text{wire 2}}\frac{I_2 d\vec{s}_2\times \hat{r}_2}{r_2^2}\)
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Direction of B From Two Wires
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<p>The figure shows two long, striaght, parallel wires. The left-hand wire carries current [mathjaxinline]I[/mathjaxinline] out of the plane of
the screen; the right-hand wire carries [mathjaxinline]I[/mathjaxinline] into the plane of the screen. What is the
direction of the magnetic field [mathjaxinline]\vec{\mathbf{B}}[/mathjaxinline] at point [mathjaxinline] P[/mathjaxinline] located on the dashed line which is halfway between the two wires.</p>
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<img src="/assets/courseware/v1/d3b832b7c0550207f436d9f96bdefb1f/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_2wire_Bfield.svg" width="165"/>
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<text>(f) [mathjaxinline]\vec{\mathbf{B}}[/mathjaxinline] points toward the top edge of the screen</text>
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<text>(g) [mathjaxinline]\vec{\mathbf{B}}[/mathjaxinline] points toward the bottom edge of the screen</text>
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<h2 class="hd hd-2 unit-title">L18Q5: Three Wires of Different Shapes</h2>
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Three wires of different shapes.
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You are given two long straight pieces of wire and a semicircle of wire and asked to construct a shape that, when a current is passed through it, creates the largest possible magnetic field at the center of the semicircle. You construct: </p>
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Rank the magnitude of the magnetic fields [mathjaxinline]\vec{\mathbf{B}}[/mathjaxinline] generated by these three configurations (assuming the current is the same in each): </p>
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<text> a) A &gt; B &gt; C</text>
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<h2 class="hd hd-2 unit-title">L18v5: Using Biot-Savart to Find the Magnetic Field Near a Straight Wire</h2>
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Straight Wire
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<p>
Consider a straight wire of length [mathjaxinline]L[/mathjaxinline] that has a current [mathjaxinline]I[/mathjaxinline] flowing in the wire. (We will not worry about the return path of the current or the source for the current.) In this problem we will try to set up an integral vector expression for the magnetic field due to the straight wire at a point that does not lie on the perpendicular bisector of the wire. </p>
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Note: In this whole problem, use [mathjaxinline]\hat{\mathbf{i}}[/mathjaxinline], [mathjaxinline]\hat{\mathbf{j}}[/mathjaxinline], [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] as the unit vectors along [mathjaxinline]x[/mathjaxinline]-, [mathjaxinline]y[/mathjaxinline]- and [mathjaxinline]z[/mathjaxinline]-axis, respectively. Also, use [mathjaxinline]x'[/mathjaxinline] and [mathjaxinline]dx'[/mathjaxinline] as the [mathjaxinline]x[/mathjaxinline] position and length of the infinitesimal current element, respectively, and [mathjaxinline](x,y)[/mathjaxinline] as the position of Point [mathjaxinline]P[/mathjaxinline], where the origin is at the center of the wire. </p>
<p>
Our goal is to find an expression of the magnetic field produced by an infinitesimal current element as given by Biot-Savart law: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\begin{aligned} d\vec{\mathbf{B}}=\frac{\mu _0}{4\pi }\frac{I d\vec{\mathbf{s}}'\times (\vec{\mathbf{r}_ P}-\vec{\mathbf{r}}')}{|\vec{\mathbf{r}_ P}-\vec{\mathbf{r'}}|^3} \end{aligned}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
For Parts (a) to (e), write your answers in terms of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], xprime for [mathjaxinline]x'[/mathjaxinline], dxprime for [mathjaxinline]dx'[/mathjaxinline], hati for [mathjaxinline]\hat{i}[/mathjaxinline] and hatj for [mathjaxinline]\hat{j}[/mathjaxinline] as needed. </p>
<p><b class="bfseries">(Part a)</b> Find a vector expression for the infinitesimal current element [mathjaxinline]I d\vec{\mathbf{s}}'[/mathjaxinline] located at [mathjaxinline]\vec{\mathbf{r}}'[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]I d\vec{\mathbf{s}}'=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Find the position of the infinitesimal current element with respect to the origin. <p style="display:inline">[mathjaxinline]\vec{\mathbf{r}}'=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_fps_fridayw6_2_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part c)</b> Find a vector expression for [mathjaxinline]\vec{\mathbf{r}}[/mathjaxinline], the position of point [mathjaxinline]P[/mathjaxinline] with respect to the origin. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\mathbf{r}}_ P=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> What is the distance [mathjaxinline]|\vec{r}_ P-\vec{r}'|[/mathjaxinline] from the source to the field point? </p>
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<p style="display:inline">[mathjaxinline]r=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part e)</b> Calculate the unit vector, [mathjaxinline]\hat{\mathbf{r}}[/mathjaxinline], pointing from the source point to the field point. </p>
<p>
<p style="display:inline">[mathjaxinline]\hat{\mathbf{r}}=[/mathjaxinline] </p>
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For the remaining parts, write your answers in terms of [mathjaxinline]I[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], xprime for [mathjaxinline]x'[/mathjaxinline], dxprime for [mathjaxinline]dx'[/mathjaxinline], hati for [mathjaxinline]\hat{i}[/mathjaxinline], hatj for [mathjaxinline]\hat{j}[/mathjaxinline], and hatk for [mathjaxinline]\hat{k}[/mathjaxinline] as needed. </p>
<p><b class="bfseries">(Part f)</b> Calculate the cross product in (eq. 1) and determine an expression for [mathjaxinline]d\vec{B}[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]d\vec{\mathbf{B}}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part g)</b> Perform the integration of [mathjaxinline]d\vec{B}[/mathjaxinline] to calculate the magnetic field at point [mathjaxinline]P[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\mathbf{B}}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part h)</b> What is your answer for the important limiting case when the length of the wire approaches infinity, [mathjaxinline]L\rightarrow \infty[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{\mathbf{B}}=[/mathjaxinline] </p>
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<h3 class="hd hd-2">L18v05: Using Biot-Savart to Find the Magnetic Field Near a Straight Wire</h3>
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<h2 class="hd hd-2 unit-title">L18v6: Force on a Wire in a Magnetic Field</h2>
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<h2 class="hd hd-2 unit-title">L18Q6: Three Wires</h2>
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Three wires
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<p>
In the figure above, a wire is suspended vertically from its top end. The bottom end is fixed. The central portion of the wire is in a region with a magnetic field pointing into the screen. When a current starts to flow through the wires, we observe that in case 1, the wire is deflected to the left. In case 2, the wire does not move. In case 3, the wire is deflected to the right. Which of the following statements are true? </p>
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<text>In case 1, the current flows upwards.</text>
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