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<h2 class="hd hd-2 unit-title">Textbook Links</h2>
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<p>We start by introducing the ideas of the flow of current, current density, and resistance.</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 6: Current and Resistance</a></li>
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<h2 class="hd hd-2 unit-title">L14Q1: Current Density</h2>
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Current Density
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A current [mathjaxinline]I = 200[/mathjaxinline] mA flows in the conductor shown below. What is the magnitude of the current density [mathjaxinline]J[/mathjaxinline]? </p>
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<text> [mathjaxinline]J=40 \mbox{ mA/cm}^2[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L14Q2: Resistance in a Conductor</h2>
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Resistance in a Conductor
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When a current flows in a wire of length [mathjaxinline]L[/mathjaxinline] and cross sectional area [mathjaxinline]A[/mathjaxinline], the resistance of the wire is </p>
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<text> proportional to [mathjaxinline]L[/mathjaxinline]; inversely proportional to [mathjaxinline]A[/mathjaxinline].</text>
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<text> inversely proportional to both [mathjaxinline]L[/mathjaxinline] and [mathjaxinline]A[/mathjaxinline].</text>
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<h2 class="hd hd-2 unit-title">L14Q3: Resistance</h2>
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Resistance
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Two resistors are made out of the same material, but have different dimensions, as shown in the figures above. The current through these two resistors is in the directions shown. If the resistor on the left has a resistance of [mathjaxinline]1.00 \; \Omega[/mathjaxinline], the resistor on the right will have a resistance of </p>
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<h2 class="hd hd-2 unit-title">L14Q4: Conductivity and Resistivity</h2>
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Conductivity and Resistivity
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<p>Use the data on resistivities, wire sizes, etc (see <a href="/courses/course-v1:MITx+8.02.2x+2T2018/pdfbook/0/chapter/6/8">reference table</a> of the TEAL textbook) and check the boxes for the statements that are true.</p>
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<text>(a) One meter of Al wire has a greater resistance than one meter of the same size Cu wire</text>
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<text>(c) A copper wire of 0.5 mm of diameter and 1 kilometer long has a higher resistance than a silver wire of 1 mm of diameter and 0.5-km long</text>
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<text>(d) If you double the length of a wire, you double its resistivity</text>
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<h2 class="hd hd-2 unit-title">L14Q5: Travelling Electron</h2>
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Travelling Electron
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The goal of this problem is to estimate how long it takes for an electron to get from a car battery to the starter motor after the ignition switch is turned. </p>
<p><b class="bfseries">(Part a)</b> Assume that the current flowing from the battery to the starter is [mathjaxinline]115 \, A[/mathjaxinline], and that electrons travel through a [mathjaxinline]85.5 \, \text {cm}[/mathjaxinline] long copper wire with cross-sectional area [mathjaxinline]31.2 \, \text {mm}^2[/mathjaxinline]. What is the current density in the wire? </p>
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<p style="display:inline">[mathjaxinline]J =[/mathjaxinline] </p>
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<p style="display:inline"> ( [mathjaxinline]\text {A / m}^2[/mathjaxinline])</p>
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<p><b class="bfseries">(Part b)</b> What is the drift speed of the electrons if the number of electrons per unit volume in copper is [mathjaxinline]n = 8.49 \times 10^{28}[/mathjaxinline] (electrons per [mathjaxinline]\text {m}^3[/mathjaxinline])? </p>
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<p style="display:inline">[mathjaxinline]v_ d =[/mathjaxinline] </p>
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<p style="display:inline"> ([mathjaxinline]\text {m/s}[/mathjaxinline])</p>
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<p><b class="bfseries">(Part c)</b> How many minutes does it take for an electron, starting at the battery, to reach the starter motor? </p>
<p>
<p style="display:inline">[mathjaxinline]\Delta t =[/mathjaxinline] </p>
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<p style="display:inline"> ([mathjaxinline]\text {min}[/mathjaxinline])</p>
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Non-uniform Conductivity
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<p>
A cylindrical glass rod is heated with a torch until it conducts enough current to cause a light bulb to glow. The rod has a length [mathjaxinline]L[/mathjaxinline], a cross sectional area [mathjaxinline]A[/mathjaxinline], and its ends, plated with material of very high conductivity, are connected to the rest of the circuit including a voltage supply. When red hot, the rod's conductivity varies with position [mathjaxinline]x[/mathjaxinline] measured from the center of the rod as [mathjaxinline]\sigma (x)=\sigma _{0} L^{2} /x^{2}[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Consider a small slice of the rod of thickness [mathjaxinline]dx[/mathjaxinline] located a distance [mathjaxinline]x[/mathjaxinline] from the center of the rod as indicated in the figure. </p>
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What is [mathjaxinline]dR[/mathjaxinline], the resistance of the small slice of rod? Express your answer in terms of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]dx[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and sigma_0 for [mathjaxinline]\sigma _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]dR =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the total resistance of the rod. Express your answer in terms of [mathjaxinline]A[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]dx[/mathjaxinline], [mathjaxinline]L[/mathjaxinline], and sigma_0 for [mathjaxinline]\sigma _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]R =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> When a voltage [mathjaxinline]\Delta V[/mathjaxinline] is applied between the two ends, what is the current density [mathjaxinline]\vec{\mathbf{J}}[/mathjaxinline]? Express your answer using some or all of the following: sigma_0 for [mathjaxinline]\sigma _0[/mathjaxinline], DeltaV for [mathjaxinline]\Delta V[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], and [mathjaxinline]L[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{\mathbf{J}}| =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> What is the magnitude of the steady-state electric field [mathjaxinline]|\vec{\mathbf{E}} (x)|[/mathjaxinline] as a function of [mathjaxinline]x[/mathjaxinline]? Express your answer using some or all of the following: DeltaV for [mathjaxinline]\Delta V[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]A[/mathjaxinline], and [mathjaxinline]L[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\vec{\mathbf{E}}| =[/mathjaxinline] </p>
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