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<h2 class="hd hd-2 unit-title">Resistance of Spherical Shells Filled with Resistive Material</h2>
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Resistance of Spherical Shells Filled with Resistive Material
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A resistor consists of two concentric conducting spherical shells with the inner shell having radius [mathjaxinline]r_ a[/mathjaxinline] and the outer shell having radius [mathjaxinline]r_ b[/mathjaxinline]. The space between the two shell is filled with a material of resistivity [mathjaxinline]\rho _ r[/mathjaxinline]. What is the resistance of this resistor? Express your answer in terms of r_a for [mathjaxinline]r_ a[/mathjaxinline], r_b for [mathjaxinline]r_ b[/mathjaxinline], rho_r for [mathjaxinline]\rho _ r[/mathjaxinline] and pi for [mathjaxinline]\pi[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]R[/mathjaxinline] = </p>
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<h2 class="hd hd-2 unit-title">Parallel Plate Capacitors and Dielectrics</h2>
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Parallel Plate Capacitors and Dielectrics
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<p><b class="bfseries">(Part a)</b> Consider a parallel plate capacitor completely filled with a dielectric material of dielectric constant [mathjaxinline]\kappa[/mathjaxinline]. What is the capacitance of this system? Assume a width [mathjaxinline]d[/mathjaxinline] and area [mathjaxinline]A[/mathjaxinline], where [mathjaxinline]d[/mathjaxinline] is small compared to the dimensions of the plates of the capacitor so that edge effects can be ignored. Write your answer using some or all of the following: epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline], [mathjaxinline]d[/mathjaxinline] and [mathjaxinline]A[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]C =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> A parallel-plate capacitor is constructed by filling the space between two square plates (side lengths are [mathjaxinline]l[/mathjaxinline] as shown) with blocks of three dielectric materials, as in the figure below. Assume that these 3 blocks extend the full length [mathjaxinline]l[/mathjaxinline] of the capacitor in the direction into the page. You may assume that [mathjaxinline]l\gg d[/mathjaxinline]. Find an expression for the capacitance of this device in terms of the plate area [mathjaxinline]A[/mathjaxinline], the distance [mathjaxinline]d[/mathjaxinline], epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline], kappa_1 for [mathjaxinline]\kappa _{1}[/mathjaxinline], kappa_2 for [mathjaxinline]\kappa _{2}[/mathjaxinline], and kappa_3 for [mathjaxinline]\kappa _{3}[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]C =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Voltage Divider</h2>
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Voltage Divider, part a.
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Consider the circuit shown in the above figure consisting of four identical resistors each with resistance [mathjaxinline]R[/mathjaxinline] and two batteries with electromotive forces [mathjaxinline]2\varepsilon[/mathjaxinline] and [mathjaxinline]\varepsilon[/mathjaxinline] as shown in the figure. </p>
<p>
The goal of this problem is to determine the magnitude of the current in each of the three branches of the circuit in terms of epsilon for [mathjaxinline]\varepsilon[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]I_1[/mathjaxinline], the magnitude of the current in the left branch </p>
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<p style="display:inline">[mathjaxinline]I_1[/mathjaxinline] =</p>
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The direction of [mathjaxinline]I_1[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_finalexam-problem2_3_1">
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<input type="radio" name="input_finalexam-problem2_3_1" id="input_finalexam-problem2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="finalexam-problem2_3_1-choice_2-label" for="input_finalexam-problem2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_finalexam-problem2_3_1"> <text> from [mathjaxinline]b[/mathjaxinline] to [mathjaxinline]a[/mathjaxinline]</text>
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Voltage Divider, part b.
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]I_2[/mathjaxinline], the magnitude of the current in the middle branch </p>
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<p style="display:inline">[mathjaxinline]I_2[/mathjaxinline] =</p>
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The direction of [mathjaxinline]I_2[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_finalexam-problem2-2_3_1">
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<input type="radio" name="input_finalexam-problem2-2_3_1" id="input_finalexam-problem2-2_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="finalexam-problem2-2_3_1-choice_1-label" for="input_finalexam-problem2-2_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_finalexam-problem2-2_3_1"> <text> from [mathjaxinline]c[/mathjaxinline] to [mathjaxinline]d[/mathjaxinline]</text>
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<input type="radio" name="input_finalexam-problem2-2_3_1" id="input_finalexam-problem2-2_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="finalexam-problem2-2_3_1-choice_2-label" for="input_finalexam-problem2-2_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_finalexam-problem2-2_3_1"> <text> from [mathjaxinline]d[/mathjaxinline] to [mathjaxinline]c[/mathjaxinline]</text>
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Voltage Divider, part c.
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]I_3[/mathjaxinline], the magnitude of the current in the right branch </p>
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<p style="display:inline">[mathjaxinline]I_3[/mathjaxinline]=</p>
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The direction of [mathjaxinline]I_3[/mathjaxinline] is <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_finalexam-problem2-3_3_1">
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Voltage Divider, part d.
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<p><b class="bfseries">(Part d)</b> Determine the magnitude of the potential difference, [mathjaxinline]V_{out}[/mathjaxinline], across the resistor between points [mathjaxinline]e[/mathjaxinline] and [mathjaxinline]g[/mathjaxinline], on the lower right in the figure below. Express your answers in terms of epsilon for [mathjaxinline]\varepsilon[/mathjaxinline] and [mathjaxinline]R[/mathjaxinline] as needed. </p>
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<img src="/assets/courseware/v1/3c90ec44055fa63c89920f3ecd60a9ae/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_finalexam-problem2_d.svg" width="440"/>
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<p style="display:inline">[mathjaxinline]V_{out}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">Energy in RC Circuits</h2>
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Energy in RC Circuits
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Consider the charging and discharging [mathjaxinline]RC[/mathjaxinline] circuits shown in the figure below. There is a battery of emf [mathjaxinline]{\cal E}[/mathjaxinline], two resistors each with resistance [mathjaxinline]R[/mathjaxinline], a switch [mathjaxinline]S[/mathjaxinline], and a capacitor with capacitance [mathjaxinline]C[/mathjaxinline]. At time [mathjaxinline]t=0[/mathjaxinline] the switch is flipped from an open position to position [mathjaxinline]a[/mathjaxinline] and the initially uncharged capacitor (i.e. [mathjaxinline]Q=0[/mathjaxinline] at [mathjaxinline]t=0[/mathjaxinline]) is allowed to charge. After a time interval [mathjaxinline]\Delta t=T[/mathjaxinline] with [mathjaxinline]T\gg RC[/mathjaxinline], the switch is flipped to position [mathjaxinline]b[/mathjaxinline] and the capacitor discharges. </p>
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<center>
<img src="/assets/courseware/v1/e400467c7198226e98df93fa188bab55/asset-v1:MITx+8.02.2x+2T2018+type@asset+block/images_Spring_2014_pset9_3-fig001.png" width="330"/>
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<p><b class="bfseries">(Part a)</b> How much energy does the resistor next to the battery dissipate while the capacitor is being charged (i.e. between [mathjaxinline]t=0[/mathjaxinline] and [mathjaxinline]t=T[/mathjaxinline])? Take the limit when [mathjaxinline]T[/mathjaxinline] is very large compared to the time constant ([mathjaxinline]T\gg \tau =RC[/mathjaxinline]). Write your answer using some or all of the following: [mathjaxinline]R[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and E for [mathjaxinline]{\cal E}[/mathjaxinline] and enter your answer as a positive energy. </p>
<p>
<p style="display:inline">[mathjaxinline]U_{R_{charge}}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the amount of energy stored in the capacitor at time [mathjaxinline]t=T[/mathjaxinline]. How does this compare to the energy dissipated in the resistor? Write your answer using some or all of the following: [mathjaxinline]R[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and E [mathjaxinline]{\cal E}[/mathjaxinline]. <p style="display:inline">[mathjaxinline]U_ C=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_pset9_1_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part c)</b> How much energy does the battery generate while the capacitor is being charged (i.e. between [mathjaxinline]t=0[/mathjaxinline] and [mathjaxinline]t=T[/mathjaxinline])? Take the limit when [mathjaxinline]T[/mathjaxinline] is very large compared to the time constant ([mathjaxinline]T\gg \tau =RC[/mathjaxinline]). Write your answer using some or all of the following: [mathjaxinline]R[/mathjaxinline], [mathjaxinline]C[/mathjaxinline] and E for [mathjaxinline]{\cal E}[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part d)</b> How much energy does the resistor dissipate while the capacitor is discharging (i.e. between [mathjaxinline]t=T[/mathjaxinline] and [mathjaxinline]t=2T[/mathjaxinline])? Write your answer using some or all of the following: [mathjaxinline]R[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], and E for [mathjaxinline]{\cal E}[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part e)</b> What is the average power dissipated by the resistor during this charging and discharging cycle? (Hint: Recall that power is [mathjaxinline]dE/dt[/mathjaxinline] so the average power is [mathjaxinline]\Delta E / \Delta t[/mathjaxinline], the total change in energy over the total time. Write your answer using some or all of the following: [mathjaxinline]R[/mathjaxinline], [mathjaxinline]C[/mathjaxinline], [mathjaxinline]T[/mathjaxinline] and E for [mathjaxinline]{\cal E}[/mathjaxinline]. </p>
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