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<h2 class="hd hd-2 unit-title">Energy Stored in a Spherical Capacitor</h2>
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Energy Stored in Spherical Capacitor Two Ways - I
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Consider a conducting spherical shell of outer radius [mathjaxinline]R[/mathjaxinline] that has charge [mathjaxinline]Q[/mathjaxinline] distributed uniformly on its surface. We want to know the potential energy [mathjaxinline]U[/mathjaxinline] of this sphere of charge. </p>
<p>
<b class="bfseries">Method I</b>
</p>
<p><b class="bfseries">(Part a)</b> What is the potential at the surface of this shell [mathjaxinline]V(R)[/mathjaxinline], assuming that [mathjaxinline]V(\infty )=0[/mathjaxinline]? Write your answer using some or all of the following: [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] for [mathjaxinline]k \equiv 1/(4 \pi \varepsilon _0)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]V(R)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Find the capacitance of a conducting shell of outer radius [mathjaxinline]R[/mathjaxinline] and use that to find the total energy of the shell when it has a charge [mathjaxinline]Q[/mathjaxinline]. Write your answers using some or all of the following: [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] for [mathjaxinline]k \equiv 1/(4 \pi \varepsilon _0)[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]C=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
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Energy Stored in Spherical Capacitor Two Ways - II
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<b class="bfseries">Method II:</b>
</p>
<p><b class="bfseries">(Part a)</b> Consider the process of charging the spherical shell of radius [mathjaxinline]R[/mathjaxinline] from a charge of zero up to a charge of [mathjaxinline]Q[/mathjaxinline]. If during this charging process, the shell has charge [mathjaxinline]q[/mathjaxinline], how much work does an external agent need to do in moving an amount of charge [mathjaxinline]dq[/mathjaxinline] from infinity and adding it to the shell? Write your answer using some or all of the following: [mathjaxinline]q[/mathjaxinline], [mathjaxinline]dq[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] for [mathjaxinline]k \equiv 1/(4 \pi \varepsilon _0)[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]dU=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Determine the potential energy stored in the shell when the charge on the shell is [mathjaxinline]Q[/mathjaxinline]. (You will need to set up and calculate a simple integral.) Write your answer using some or all of the following: [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]R[/mathjaxinline] and [mathjaxinline]k[/mathjaxinline] for [mathjaxinline]k \equiv 1/(4 \pi \varepsilon _0)[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> An interstellar dust grain, roughly spherical with a radius of [mathjaxinline]R=3\times 10^{-7} \, m[/mathjaxinline], has acquired a negative charge such that the potential difference [mathjaxinline]V(R)-V(\infty )=-0.15 \, V[/mathjaxinline]. </p>
<p>
(i) How many extra electrons has it picked up? Use [mathjaxinline]e=-1.6\times 10^{-19}[/mathjaxinline]&#160;C for the charge of an electron and [mathjaxinline]k=9.0\times 10^{9} \, N \cdot m^2 / C^2[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]N_ e=[/mathjaxinline] </p>
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(ii) What is the magnitude of the electric field on its surface? </p>
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<p style="display:inline">[mathjaxinline]|\vec{\mathbf{E}}|=[/mathjaxinline] </p>
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<p style="display:inline"> (in [mathjaxinline]V/m[/mathjaxinline])</p>
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<h2 class="hd hd-2 unit-title">Charges on Conducting Plates</h2>
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Charges on Conducting Plates
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Consider two very large conducting plates which contain different total charges [mathjaxinline]Q_{1}[/mathjaxinline] and [mathjaxinline]Q_{2}[/mathjaxinline] as shown. Note that this is different from a parallel-plate capacitor which would normally have equal magnitude but opposite sign charges [mathjaxinline]Q[/mathjaxinline] and [mathjaxinline]-Q[/mathjaxinline]. Find the amount of charge on the inner and outer surfaces of the two plates. Neglect all edge effects. </p>
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<img src="/assets/courseware/v1/7a0b7215d4c0d4a0055bd2254e338e95/asset-v1:MITx+8.02.1x+1T2019+type@asset+block/images_Spring_2014_pset5_4_2.png" width="440"/>
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Write your answers using Q_1 for [mathjaxinline]Q_1[/mathjaxinline], and Q_2 for [mathjaxinline]Q_2[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]Q_{out, 1} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]Q_{out, 2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]Q_{in, 2} =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Dielectrics and Capacitance</h2>
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Dielectrics and Capacitance, part I
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Consider a capacitor made of two square conducting plates of side [mathjaxinline]l[/mathjaxinline]. The distance between the two plates is [mathjaxinline]d[/mathjaxinline]. Assume that [mathjaxinline]l \gg d[/mathjaxinline] so that edge effects can be ignored. We insert a dielectric of dielectric constant [mathjaxinline]\kappa &gt;1[/mathjaxinline] and width [mathjaxinline]l[/mathjaxinline] (into the page) a distance [mathjaxinline]x[/mathjaxinline] into the right side of the capacitor as shown in the diagram. We want to find the total capacitance and other properties of this arrangement. </p>
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<p><b class="bfseries">(Part a)</b> First, assume that the top plate has total charge [mathjaxinline]+Q[/mathjaxinline] and the bottom plate has total charge [mathjaxinline]-Q[/mathjaxinline], with the magnitude of the area charge density being [mathjaxinline]\sigma _1[/mathjaxinline] on the left where there is no dielectric and [mathjaxinline]\sigma _2[/mathjaxinline] on the right. </p>
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How is the total charge [mathjaxinline]Q[/mathjaxinline] related to [mathjaxinline]\sigma _1[/mathjaxinline], [mathjaxinline]\sigma _2[/mathjaxinline] and the geometry of the capacitor? Write your answer using some or all of the following: [mathjaxinline]l[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], sigma_1 for [mathjaxinline]\sigma _1[/mathjaxinline], sigma_2 for [mathjaxinline]\sigma _2[/mathjaxinline], and kappa for [mathjaxinline]\kappa[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]Q=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b i)</b> In terms of the charge densities, what is the magnitude of the potential difference between the two plates in the region on the left <i class="it">without</i> dielectric? Write your answer using some or all of the following: [mathjaxinline]l[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], sigma_1 for [mathjaxinline]\sigma _1[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\Delta V_{left}|=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b ii)</b> In terms of the charge densities, what is the magnitude of the potential difference between the two plates in the region on the right <i class="it">with</i> dielectric? Write your answer using some or all of the following: [mathjaxinline]l[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], sigma_2 for [mathjaxinline]\sigma _1[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline] and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\Delta V_{right}|=[/mathjaxinline] </p>
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Dielectrics and Capacitance, part II
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However, since the top and bottom plates are conductors, every point in each plate must be at the same potential and so the potential difference between the top and bottom plates must be the same on the left and the right. </p>
<p><b class="bfseries">(Part c)</b> Use this information and the answers to the questions above to find [mathjaxinline]\sigma _1[/mathjaxinline] and [mathjaxinline]\sigma _2[/mathjaxinline] and use those to find the capacitance. Write your answers to the first two questions using some or all of the following: [mathjaxinline]l[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], [mathjaxinline]Q[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline] and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]|\sigma _1|=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part d)</b> Now, find the capacitance. Write your answer using some or all of the following: [mathjaxinline]l[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]d[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline] and epsilon_0 for [mathjaxinline]\varepsilon _0[/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">Minimize the Energy of Two Conductors</h2>
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Minimizing the energy of two conductors, part 1
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Consider two metal shells of radius [mathjaxinline]R_{1}[/mathjaxinline] and [mathjaxinline]R_{2}[/mathjaxinline], quite far apart from one another compared with these radii (so that the potential near one shell depends only on the charge on that shell). Because of this relatively large distance, you can also assume that any charge put on the shells distributes itself uniformly. </p>
<p><b class="bfseries">(Part a)</b> Given a total amount of charge [mathjaxinline]Q[/mathjaxinline], which we have to divide between the shells, how should it be divided to make the energy stored in the resulting charge distribution as small as possible? </p>
<p>
Assume that you place charges [mathjaxinline]Q-q[/mathjaxinline] on shell 1, and [mathjaxinline]q[/mathjaxinline] on shell 2. Calculate the total potential energy of the two shells with this arrangement of charges. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]Q[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline] and R_2 for [mathjaxinline]R_2[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]U_{total}=[/mathjaxinline] </p>
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Minimizing the energy of two conductors, part 2
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<p><b class="bfseries">(Part b)</b> Find the value of [mathjaxinline]q[/mathjaxinline] which minimizes this total energy. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]Q[/mathjaxinline], R_1 for [mathjaxinline]R_1[/mathjaxinline] and R_2 for [mathjaxinline]R_2[/mathjaxinline], as needed. </p>
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<p style="display:inline">[mathjaxinline]q =[/mathjaxinline] </p>
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Minimizing the energy of two conductors, part 3
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<p><b class="bfseries">(Part c)</b> Given the distribution of charge which minimizes the total energy, what is the potential difference between the shells? Write your answer in terms of the quantities R_1 for [mathjaxinline]R_1[/mathjaxinline], R_2 for [mathjaxinline]R_2[/mathjaxinline] and [mathjaxinline]Q[/mathjaxinline], as needed. </p>
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<p style="display:inline">[mathjaxinline]\Delta V =[/mathjaxinline] </p>
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Minimizing the energy of two conductors, part 4
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Consider two metallic spheres of the same radius [mathjaxinline]R[/mathjaxinline] away from each other so that the electric field of one sphere does not interact with the electric field of the other one. One sphere is charged with [mathjaxinline]2Q[/mathjaxinline] the other one with [mathjaxinline]Q[/mathjaxinline]. We connect a wire between the spheres. Which of the following statements is correct (check all that applied). </p>
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<text>1. The charges in the sphere will not move because the spheres are far away from each other.</text>
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<text>2. Because the electric potential at a point of sphere1 is not the same as the electric potential at a point of sphere 2, the charges will redistribute until the electric potential of the two spheres are the same.</text>
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<text>3. The charges will redistribute in the spheres so that the energy of the system is minimum.</text>
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<h2 class="hd hd-2 unit-title">Coaxial Cable with Dielectric</h2>
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Coaxial Cable with Dielectric
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A certain coaxial cable consists of a copper wire, radius [mathjaxinline]a[/mathjaxinline], surrounded by a concentric copper tube of inner radius [mathjaxinline]c[/mathjaxinline]. The space between is partially filled (from [mathjaxinline]b[/mathjaxinline] out to [mathjaxinline]c[/mathjaxinline]) with material of dielectric constant [mathjaxinline]\kappa[/mathjaxinline]. The goal of this problem is to find the capacitance per unit length of this cable. You may neglect edge effects. </p>
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<p><b class="bfseries">(Part a)</b> Assume that the copper wire has uniform positive charge per unit length [mathjaxinline]\lambda[/mathjaxinline] and the copper tube has uniform negative charge per unit length on it's inner surface [mathjaxinline]-\lambda[/mathjaxinline]. Calculate the radial component of the electric field for [mathjaxinline]0 &lt; r &lt; a[/mathjaxinline], [mathjaxinline]a &lt; r &lt; b[/mathjaxinline], [mathjaxinline]b &lt; r &lt; c[/mathjaxinline] and [mathjaxinline]r &gt; c[/mathjaxinline]. Express your answer using some or all of the following: [mathjaxinline]r[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline], lambda for [mathjaxinline]\lambda[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. Use ln() for the natural logarithm if needed. </p>
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<p style="display:inline">[mathjaxinline]0 &lt; r &lt; a: \quad E_ r =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]a &lt; r &lt; b: \quad E_ r =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]b &lt; r &lt; c: \quad E_ r =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]r &gt; c: \quad E_ r =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> What is potential difference between the surfaces [mathjaxinline]r=b[/mathjaxinline] and [mathjaxinline]r=a[/mathjaxinline]? What about between surfaces [mathjaxinline]r=c[/mathjaxinline] and [mathjaxinline]r=b[/mathjaxinline]? What is the value of [mathjaxinline]|V(c)-V(a)|[/mathjaxinline]? Be careful with your signs! </p>
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Express your answer using some or all of the following: [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline], lambda for [mathjaxinline]\lambda[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. Use ln() for the natural logarithm if needed. Note that the grader does not accept abs() to take the absolute value - you must figure out how to enter the positive value. </p>
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<p style="display:inline">[mathjaxinline]V(b)-V(a)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]|V(c)-V(a)|=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> What is the capacitance per unit length of the cylindrical arrangement? Express your answer using some or all of the following: [mathjaxinline]a[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]c[/mathjaxinline], kappa for [mathjaxinline]\kappa[/mathjaxinline], lambda for [mathjaxinline]\lambda[/mathjaxinline], and epsilon_0 for [mathjaxinline]\varepsilon _0[/mathjaxinline]. Use ln() for the natural logarithm if needed. </p>
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<p style="display:inline">[mathjaxinline]\displaystyle \frac{C}{L} =[/mathjaxinline] </p>
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