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<h2 class="hd hd-2 unit-title">Using Biot-Savart to Find the Magnetic Field on the Axis of a Ring of Current</h2>
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Magnetic Field of a Ring of Current
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<p>
A circular ring of radius [mathjaxinline]R[/mathjaxinline] lying in the xy plane carries a steady current [mathjaxinline]I[/mathjaxinline], moving counterclockwise viewed from [mathjaxinline]+z[/mathjaxinline] as shown in the figure below. </p>
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<img src="/assets/courseware/v1/4438f25e04676a7370e4c5fec3e96e2e/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_ex_2_p7_ring.png" width="330"/>
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What is the magnetic field at a point P on the axis of the loop, at a distance [mathjaxinline]z[/mathjaxinline] from the center? Write your answer using some or all of the following: [mathjaxinline]I[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]z[/mathjaxinline], mu_0 for [mathjaxinline]\mu _{0}[/mathjaxinline], hati for [mathjaxinline]\hat{\mathbf{i}}[/mathjaxinline], hatj for [mathjaxinline]\hat{\mathbf{j}}[/mathjaxinline], and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\vec{\mathbf{B}}=[/mathjaxinline] </p>
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<h3 class="hd hd-2">W07PS01: Using Biot-Savart to Find the Magnetic Field on the Axis of a Ring of Current</h3>
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<h2 class="hd hd-2 unit-title">Circular Motion in a Magnetic Field</h2>
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Circular Motion in a Magnetic Field - I
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A positively charged particle of mass [mathjaxinline]m[/mathjaxinline] and charge [mathjaxinline]q[/mathjaxinline] is at the origin at [mathjaxinline]t=0[/mathjaxinline] and moving upward with velocity [mathjaxinline]\vec{\mathbf{v}}=v \mathbf{\hat{j}}[/mathjaxinline]. Its subsequent trajectory is shown in the sketch. The magnitude of the velocity [mathjaxinline]v=|\vec{\mathbf{v}}|[/mathjaxinline] is always the same, although the direction of [mathjaxinline]\vec{\mathbf{v}}[/mathjaxinline] changes in time. For the region [mathjaxinline]y&gt;0[/mathjaxinline], the magnetic field is uniform with magnitude [mathjaxinline]B_1[/mathjaxinline]. For [mathjaxinline]y&lt;0[/mathjaxinline], the magnetic field is also uniform but the magnitude is [mathjaxinline]B_2[/mathjaxinline]. Ignore any gravitational field. </p>
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<p><b class="bfseries">(a)</b> Is the direction of the magnetic field for the region [mathjaxinline]y&gt;0[/mathjaxinline] into or out of the page? </p>
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<text> Out of the page</text>
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<p><b class="bfseries">(b)</b> Derive an expression for the magnitude [mathjaxinline]B_1[/mathjaxinline] of the magnetic field for the region [mathjaxinline]y&gt;0[/mathjaxinline] in terms of the given quantities, that is in term of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and [mathjaxinline]v[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]B_1=[/mathjaxinline] </p>
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<p><b class="bfseries">(c)</b> Is the magnetic field in the region [mathjaxinline]y&lt;0[/mathjaxinline] into or out of the page? What is the magnitude [mathjaxinline]B_2[/mathjaxinline] of that magnetic field in that region in terms of in term of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and [mathjaxinline]v[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]B_2=[/mathjaxinline] </p>
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<p><b class="bfseries">(d)</b> How long does it take the charged particle to move from the origin to point [mathjaxinline]P[/mathjaxinline] located at [mathjaxinline]x=3R[/mathjaxinline] (see figure above) along the x-axis? Give your answer in terms of the given quantities [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and [mathjaxinline]v[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]T=[/mathjaxinline] </p>
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<h3 class="hd hd-3 problem-header" id="pseto6_4-problem-title" aria-describedby="block-v1:MITx+8.02.2x+2T2018+type@problem+block@pseto6_4-problem-progress" tabindex="-1">
Circular Motion in Magnetic Field - II
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A negatively charged particle of mass [mathjaxinline]m[/mathjaxinline] and charge [mathjaxinline]-q[/mathjaxinline] is at the origin at [mathjaxinline]t=0[/mathjaxinline] and moving downward with velocity [mathjaxinline]\vec{\textbf{v}} = -v \hat{\mathbf{j}}[/mathjaxinline]. Its subsequent trajectory is shown in the sketch. The magnitude of the velocity [mathjaxinline]v=\left| \vec{\mathbf{v}} \right|[/mathjaxinline] is always the same, although the direction of [mathjaxinline]\vec{\mathbf{v}}[/mathjaxinline] changes in time. For the region [mathjaxinline]y&lt;0[/mathjaxinline], the magnetic field is uniform with magnitude [mathjaxinline]B_{1}[/mathjaxinline]. For [mathjaxinline]y&gt;0[/mathjaxinline], the magnetic field is also uniform but the magnitude is [mathjaxinline]B_{2}[/mathjaxinline]. Ignore any gravitational field. In the region [mathjaxinline]x&gt;14R[/mathjaxinline], [mathjaxinline]y&gt;0[/mathjaxinline], there is also an electric field in addition to the magnetic field. When [mathjaxinline]x=14R[/mathjaxinline] and [mathjaxinline]y&gt;0[/mathjaxinline], the particle travels in a straight vertical direction ([mathjaxinline]+y[/mathjaxinline]-direction). </p>
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<img src="/assets/courseware/v1/e453cd5e9cfa87186ef19ca7a1311131/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_Spring_2014_pset6_2-fig001.png" width="550"/>
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<p><b class="bfseries">(Part a)</b> Is the direction of the magnetic field for the region [mathjaxinline]y&lt;0[/mathjaxinline] into or out of the page? </p>
<p>
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<text> Into the page</text>
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<p><b class="bfseries">(Part b)</b> Derive an expression for the magnitude [mathjaxinline]B_{1}[/mathjaxinline] of the magnetic field for the region [mathjaxinline]y&lt;0[/mathjaxinline] in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]v[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]B_{1} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> Is the magnetic field in the region [mathjaxinline]y&gt;0[/mathjaxinline] into or out of the page? </p>
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<text> Into the page</text>
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<text> Out of the page</text>
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<span id="solution_pseto6_4_solution_3"/>
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<p><b class="bfseries">(Part d)</b> What is the magnitude [mathjaxinline]B_{2}[/mathjaxinline] of that magnetic field in that region in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]v[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]B_{2} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part e)</b> How long does it take the charged particle to move from the origin to point [mathjaxinline]P[/mathjaxinline] located at [mathjaxinline]x=8R[/mathjaxinline] (see figure above) along the x-axis? Give your answer in terms of the given quantities [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]v[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]T_ P =[/mathjaxinline] </p>
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<span id="solution_pseto6_4_solution_5"/>
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<p><b class="bfseries">(Part f)</b> In the region [mathjaxinline]x&gt;14R[/mathjaxinline], [mathjaxinline]y&gt;0[/mathjaxinline], there is also an electric field in addition to the magnetic field. Determine an expression for the magnitude and direction of the electric field in that region. Express your answer using [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]v[/mathjaxinline], and hati, hatj and hatk for [mathjaxinline]\hat{\mathbf{i}}[/mathjaxinline], [mathjaxinline]\hat{\mathbf{j}}[/mathjaxinline] and [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] respectively. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{E} =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Lorentz Force</h2>
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Lorentz Force
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<p>
A positively charged ion with charge [mathjaxinline]q[/mathjaxinline] and mass [mathjaxinline]m[/mathjaxinline], initially at rest inside the source, is accelerated across a gap by an electric potential difference with magnitude [mathjaxinline]\Delta V[/mathjaxinline]. The ion enters a region of uniform magnetic field [mathjaxinline]B_1[/mathjaxinline] pointing into the page of the figure above and follows a semi-circular trajectory of radius [mathjaxinline]R_1[/mathjaxinline]. The ion is again accelerated across the gap, increasing its speed, by an electric potential difference that has the <b class="bfseries"><i class="itshape">opposite sign but has the same magnitude [mathjaxinline]|\Delta V|[/mathjaxinline]</i></b>. The ion then enters a region of uniform magnetic field [mathjaxinline]B_2[/mathjaxinline] pointing into the page of the figure above and follows a semi-circular trajectory of radius [mathjaxinline]R_2[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> What is the ratio [mathjaxinline]B_2/B_1[/mathjaxinline] of the magnitudes of the magnetic field in the different regions in terms of the radii [mathjaxinline]R_1[/mathjaxinline] and [mathjaxinline]R_2[/mathjaxinline] and velocities in the two regions [mathjaxinline]v_1[/mathjaxinline] and [mathjaxinline]v_2[/mathjaxinline]? Enter v_1, v_2, R_1, and R_2 for [mathjaxinline]v_1[/mathjaxinline], [mathjaxinline]v_2[/mathjaxinline], [mathjaxinline]R_1[/mathjaxinline], and [mathjaxinline]R_2[/mathjaxinline], respectively, </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {B_2}{B_1}[/mathjaxinline] = </p>
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<p><b class="bfseries">(Part b)</b> Considering the accelerating potential, what is the ratio [mathjaxinline]v_2/v_1[/mathjaxinline]? Express your answer in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and DeltaV for [mathjaxinline]\Delta V[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {v_2}{v_1}[/mathjaxinline] = </p>
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<p><b class="bfseries">(Part c)</b> If [mathjaxinline]R_2=2R_1[/mathjaxinline], what is [mathjaxinline]B_2 / B_1[/mathjaxinline]? Express your answer in terms of [mathjaxinline]q[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and DeltaV for [mathjaxinline]\Delta V[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {B_2}{B_1}[/mathjaxinline] = </p>
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A conducting rod having a length [mathjaxinline]l[/mathjaxinline] and mass density (mass per unit length) [mathjaxinline]\lambda[/mathjaxinline] is suspended by two flexible vertical wires in a uniform magnetic field of magnitude [mathjaxinline]B[/mathjaxinline] which points out of the page. The wires connect the rod to a current source as shown. The rod is near the surface of the earth and so feels a gravitational force of [mathjaxinline]mg[/mathjaxinline] pointing down, where [mathjaxinline]m[/mathjaxinline] is the total mass. If the tension in the vertical wires is zero, what is the direction and magnitude of the current in the rod. For the magnitude, express your answer in terms of [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]B[/mathjaxinline], and lambda for [mathjaxinline]\lambda[/mathjaxinline], as needed. </p>
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Magnitude of the current: <p style="display:inline">[mathjaxinline]I =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w7_10_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<h2 class="hd hd-2 unit-title">Using Biot-Savart for Two Semi-Circles of Current</h2>
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Two Half Circles
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Consider a wire loop consisting of two semi-circles of radii [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]2R[/mathjaxinline] connected by two wires in the radial direction. The loop carries a current [mathjaxinline]I[/mathjaxinline] flowing counterclockwise as shown. Calculate the magnetic field at the center (point [mathjaxinline]P[/mathjaxinline] in the figure). Express your answers in terms of mu_0 for [mathjaxinline]\mu _0[/mathjaxinline], [mathjaxinline]I[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] ([mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] points out of the page) as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{\mathbf{B}}[/mathjaxinline] = </p>
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