{"id":1441,"date":"2019-01-14T03:06:28","date_gmt":"2019-01-14T03:06:28","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ivytech-sci111\/chapter\/capacitors-and-dielectrics\/"},"modified":"2019-01-28T16:47:49","modified_gmt":"2019-01-28T16:47:49","slug":"capacitors-and-dielectrics","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-sci111\/chapter\/capacitors-and-dielectrics\/","title":{"raw":"Module 4 Capacitors and Dielectrics","rendered":"Module 4 Capacitors and Dielectrics"},"content":{"raw":"<div class=\"boundless-concept\">\r\n<h2>Capacitance<\/h2>\r\nCapacitance is the measure of an object's ability to store electric charge.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nExpress the relationship between the capacitance, charge of an object, and potential difference in the form of equation\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n<h4>Key Points<\/h4>\r\n<ul>\r\n \t<li>The unit of capacitance is known as the farad (F), which can be equated to many quotients of units, including JV<sup>-2<\/sup>, WsV<sup>-2<\/sup>, CV<sup>-1<\/sup>, and C<sup>2<\/sup>J<sup>-1<\/sup>.<\/li>\r\n \t<li>Capacitance (C) can be calculated as a function of charge an object can store (q) and potential difference (V) between the two plates: [latex]\\text{C}=\\frac {\\text{q}}{\\text{V}}[\/latex] Q depends on the surface area of the conductor plates, while V depends on the distance between the plates and the permittivity of the dielectric between them.<\/li>\r\n \t<li>In storing charge, capacitors also store potential energy, which is equal to the work (W) required to charge them. For a capacitor with plates holding charges of +q and -q, this can be calculated: [latex]\\text{W}_{\\text{stored}}=\\frac {\\text{CV}^2}{2}[\/latex]. The above can be equated with the work required to charge the capacitor.<\/li>\r\n<\/ul>\r\n<h4>Key Terms<\/h4>\r\n<ul>\r\n \t<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\r\n \t<li><strong>capacitance<\/strong>: The property of an electric circuit or its element that permits it to store charge, defined as the ratio of stored charge to potential over that element or circuit (Q\/V); SI unit: farad (F).<\/li>\r\n<\/ul>\r\n<\/div>\r\nCapacitance is the measure of an object's ability to store electric charge. Any body capable of being charged in any way has a value of capacitance.\r\n\r\nThe unit of capacitance is known as the Farad (F), which can be adjusted into subunits (the millifarad (mF), for example) for ease of working in practical orders of magnitude. The Farad can be equated to many quotients of units, including JV<sup>-2<\/sup>, WsV<sup>-2<\/sup>, CV<sup>-1<\/sup>, and C2J<sup>-1<\/sup>.\r\n\r\nThe most common capacitor is known as a parallel-plate capacitor which involves two separate conductor plates separated from one another by a dielectric. Capacitance (C) can be calculated as a function of charge an object can store (q) and potential difference (V) between the two plates:\r\n<div class=\"wp-caption alignright\" style=\"width: 270px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/parallel-plate-capacitor.svg#fixme#fixme\" alt=\"image\" width=\"270\" height=\"216\" \/>\r\n<p class=\"wp-caption-text\"><strong>Parallel-Plate Capacitor<\/strong>: The dielectric prevents charge flow from one plate to the other.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n[latex]\\text{C}=\\frac {\\text{q}}{\\text{V}}[\/latex]\r\n\r\nUltimately, in such a capacitor, q depends on the surface area (A) of the conductor plates, while V depends on the distance (d) between the plates and the permittivity (\u03b5<sub>r<\/sub>) of the dielectric between them. For a parallel-plate capacitor, this equation can be used to calculate capacitance:\r\n\r\n[latex]\\text{C}=\\epsilon_\\text{r} \\epsilon_0 \\frac {\\text{A}}{\\text{d}}[\/latex]\r\n\r\nWhere <em>\u03b5<\/em><sub>0<\/sub> is the electric constant. The product of length and height of the plates can be substituted in place of A.\r\n\r\nIn storing charge, capacitors also store potential energy, which is equal to the work (W) required to charge them. For a capacitor with plates holding charges of +q and -q, this can be calculated:\r\n\r\n[latex]\\text{W}_{\\text{charging}}=\\int_0^\\text{Q} \\! \\frac {\\text{q}}{\\text{C}} \\mathrm{\\text{d}} \\text{q}=\\frac {\\text{CV}^2}{2}=\\text{W}_{\\text{stored}}[\/latex]\r\n\r\nThus, either through calculus or algebraically (if C and V are known), stored energy (W<sub>stored<\/sub>) can be calculated. In a parallel-plate capacitor, this can be simplified to:\r\n\r\n[latex]\\text{W}_{\\text{stored}}= \\frac {\\epsilon_\\text{r} \\epsilon_0 \\text{AV}^2}{2\\text{d}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"boundless-concept\">\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"boundless-concept\">\r\n<h2>Parallel-Plate Capacitor<\/h2>\r\nThe parallel-plate capacitor is one that includes two conductor plates, each connected to wires, separated from one another by a thin space.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nCalculate the maximum storable energy in a parallel-plate capacitor\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n<h4>Key Points<\/h4>\r\n<ul>\r\n \t<li>Assuming the plates extend uniformly over an area of A and hold \u00b1 Q charge, their charge density is \u00b1, where \u03c1=Q\/A.<\/li>\r\n \t<li>Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, [latex]\\text{E}=\\frac {\\rho}{\\epsilon}[\/latex] can be used to calculate the electric field (E) near the center of the plates. In this equation, \u03b5 represents permittivity.<\/li>\r\n \t<li>[latex]\\text{V}=\\frac {\\rho \\text{d}}{\\epsilon}=\\frac {\\text{Qd}}{\\epsilon \\text{A}}[\/latex] can be used to calculate the potential between the plates.<\/li>\r\n \t<li>[latex]\\text{C}=\\frac {\\epsilon \\text{A}}{\\text{d}}[\/latex] can be found from the previous equation, adjusting the terms to solve for capacitance (C).<\/li>\r\n \t<li>[latex]\\text{U}=\\frac {\\text{CV}^2}{2}=\\frac {\\epsilon \\text{A}(\\text{U}_\\text{dd})^2}{2\\text{d}}=\\frac {\\epsilon \\text{AdU}_\\text{d}^2}{2}[\/latex] solves for the maximum storable energy in a parallel-plate capacitor (U) as a function of U<sub>d<\/sub>, the dielectric strength per distance as well as capacitor's voltage (V) at its breakdown limit.<\/li>\r\n<\/ul>\r\n<h4>Key Terms<\/h4>\r\n<ul>\r\n \t<li><strong>permittivity<\/strong>: A property of a dielectric medium that determines the forces that electric charges placed in the medium exert on each other.<\/li>\r\n \t<li><strong>capacitor<\/strong>: An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.<\/li>\r\n \t<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\r\n<\/ul>\r\n<\/div>\r\nOne of the most commonly used capacitors in industry and in the academic setting is the parallel-plate capacitor. This is a capacitor that includes two conductor plates, each connected to wires, separated from one another by a thin space. Between them can be a vacuum or a dielectric material, but not a conductor.\r\n<div class=\"wp-caption alignright\" style=\"width: 273px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/parallel-plate-capacitor.svg#fixme#fixme\" alt=\"image\" width=\"273\" height=\"218\" \/>\r\n<p class=\"wp-caption-text\"><strong>Parallel-Plate Capacitor<\/strong>: In a capacitor, the opposite plates take on opposite charges. The dielectric ensures that the charges are separated and do not transfer from one plate to the other.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nThe purpose of a capacitor is to store charge, and in a parallel-plate capacitor one plate will take on an excess of positive charge while the other becomes more negative.\r\n\r\nAssuming the plates extend uniformly over an area of A and hold \u00b1 Q charge, their charge density is \u00b1, where \u03c1=Q\/A. Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, the electric field (E) near the center of the plates can be calculated by:\r\n\r\n[latex]\\text{E}=\\frac {\\rho}{\\epsilon}[\/latex]\r\n\r\nPotential (V) between the plates can be calculated from the line integral of the electric field (E):\r\n\r\n[latex]\\text{V}=\\int_0^\\text{d} \\! \\text{E} \\mathrm{\\text{d}}\\text{z}[\/latex]\r\n\r\nwhere z is the axis perpendicular to both plates. Through simplification and substitution, this integral can be changed to:\r\n\r\n[latex]\\text{V}=\\frac {\\rho \\text{d}}{\\epsilon}=\\frac {\\text{Qd}}{\\epsilon \\text{A}}[\/latex]\r\n\r\nGiven that capacitance is the quotient of charge and potential:\r\n\r\n[latex]\\text{C}=\\frac {\\epsilon \\text{A}}{\\text{d}}[\/latex]\r\n\r\nAccordingly, capacitance is greatest in devices with high permittivity, large plate area, and minimal separation between the plates.\r\n\r\nThe maximum energy (U) a capacitor can store can be calculated as a function of U<sub>d<\/sub>, the dielectric strength per distance, as well as capacitor's voltage (V) at its breakdown limit (the maximum voltage before the dielectric ionizes and no longer operates as an insulator):\r\n\r\n[latex]\\text{U}=\\frac {\\text{CV}^2}{2}=\\frac {\\epsilon \\text{A}(\\text{U}_\\text{dd})^2}{2\\text{d}}=\\frac {\\epsilon \\text{AdU}_\\text{d}^2}{2}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"boundless-concept\">\r\n<h2>Combinations of Capacitors: Series and Parallel<\/h2>\r\nLike any other form of electrical circuitry device, capacitors can be used in series and\/or in parallel within circuits.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nCalculate the total capacitance for the capacitors connected in series and in parallel\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n<h4>Key Points<\/h4>\r\n<ul>\r\n \t<li>[latex]\\frac {1}{\\text{C}_{\\text{total}}}=\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}[\/latex] Capacitors in series follow the law of reciprocals; the reciprocal of the circuit 's total capacitance is equal to the sum of the reciprocals of the capacitances of each individual capacitor.<\/li>\r\n \t<li>[latex]\\text{C}_{\\text{total}}=\\text{C}_1+\\text{C}_2+...+\\text{C}_\\text{n}[\/latex]For capacitors in parallel, summing the capacitances of individual capacitors affords the total capacitance in the circuit.<\/li>\r\n \t<li>When capacitors are found both in series and in parallel in the same circuit, it is best to simplify the circuit by solving parts of it in sequence.<\/li>\r\n<\/ul>\r\n<h4>Key Terms<\/h4>\r\n<ul>\r\n \t<li><strong>capacitor<\/strong>: An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.<\/li>\r\n \t<li><strong>circuit<\/strong>: A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T<\/li>\r\n<\/ul>\r\n<\/div>\r\nLike any other form of electrical circuitry device, capacitors can be used in combination in circuits. These combinations can be in series (in which multiple capacitors can be found along the same path of wire) and in parallel (in which multiple capacitors can be found along different paths of wire).\r\n<h3>Capacitors in Series<\/h3>\r\nLike in the case of resistors in parallel, the reciprocal of the circuit's total capacitance is equal to the sum of the reciprocals of the capacitance of each individual capacitor:\r\n<div class=\"wp-caption aligncenter\" style=\"width: 438px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/capacitors-in-series.svg#fixme#fixme\" alt=\"image\" width=\"438\" height=\"119\" \/>\r\n<p class=\"wp-caption-text\"><strong>Capacitors in Series<\/strong>: This image depicts capacitors C1, C2 and so on until Cn in a series.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n[latex]\\frac {1}{\\text{C}_{\\text{total}}}=\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}[\/latex]\r\n\r\nThis can also be expressed as:\r\n\r\n[latex]\\text{C}_{\\text{total}}=\\frac{1}{\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}}[\/latex]\r\n<h3>Parallel Capacitors<\/h3>\r\nTotal capacitance for a circuit involving several capacitors in parallel (and none in series) can be found by simply summing the individual capacitances of each individual capacitor.\r\n<div class=\"wp-caption aligncenter\" style=\"width: 359px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/capacitors-in-parallel.svg#fixme#fixme\" alt=\"image\" width=\"359\" height=\"211\" \/>\r\n<p class=\"wp-caption-text\"><strong>Parallel Capacitors<\/strong>: This image depicts capacitors C1, C2, and so on until Cn in parallel.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n[latex]\\text{C}_{\\text{total}}=\\text{C}_1+\\text{C}_2+...+\\text{C}_\\text{n}[\/latex]\r\n<h3>Capacitors in Series and in Parallel<\/h3>\r\nIt is possible for a circuit to contain capacitors that are both in series and in parallel. To find total capacitance of the circuit, simply break it into segments and solve piecewise.\r\n<div class=\"wp-caption aligncenter\" style=\"width: 643px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2548\/2019\/01\/14030624\/figure-20-06-03a.jpeg\" alt=\"image\" width=\"643\" height=\"209\" \/>\r\n<p class=\"wp-caption-text\"><strong>Capacitors in Series and in Parallel<\/strong>: The initial problem can be simplified by finding the capacitance of the series, then using it as part of the parallel calculation.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nThe circuit shown in (a) contains C<sub>1<\/sub> and C<sub>2<\/sub> in series. However, these are both in parallel with C<sub>3<\/sub>. If we find the capacitance for the series including C<sub>1<\/sub> and C<sub>2<\/sub>, we can treat that total as that from a single capacitor (b). This value can be calculated as approximately equal to 0.83 \u03bcF.\r\n\r\nWith effectively two capacitors left in parallel, we can add their respective capacitances (c) to find the total capacitance for the circuit. This sum is approximately 8.83 \u03bcF.\r\n\r\n<\/div>\r\n<div class=\"boundless-concept\">\r\n<h2>Dieletrics and their Breakdown<\/h2>\r\nDielectric breakdown is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor.\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIdentify conditions that can lead to a dielectric breakdown and its effect on materials\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\n<h4>Key Points<\/h4>\r\n<ul>\r\n \t<li>All insulators can, when exposed to enough voltage, experience dielectric breakdown and become conductors.<\/li>\r\n \t<li>Because dielectric breakdown is a failure that depends on a probability, an exact breakdown voltage is in most cases impossible to calculate with a high degree of certainty.<\/li>\r\n \t<li>Lightning is a common instance of dielectric breakdown, as air loses its ability to separate the potential difference between clouds and the point of a lightning bolt's impact.<\/li>\r\n<\/ul>\r\n<h4>Key Terms<\/h4>\r\n<ul>\r\n \t<li><strong>conductor<\/strong>: A material which contains movable electric charges.<\/li>\r\n \t<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\r\n \t<li><strong>breakdown<\/strong>: A failure, particularly mechanical; something that has failed.<\/li>\r\n<\/ul>\r\n<\/div>\r\nDielectric breakdown (illustrated in ) is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor. Dielectrics are commonly used either to isolate conductors from a variable external environment (e.g., as coating for electrical wires) or to isolate conductors from one another (e.g., between plates of a parallel-plate capacitor). In all applications, they are selected for their ability to act as insulators. By definition, an insulator is unable to conduct electricity. Under certain conditions, however, a material that is an insulator can become a conductor.\r\n\r\nEventually, exposing any insulator to increasing voltage will result in the insulator becoming conductive. This point (the minimum voltage for the insulator to become a conductor) is known as the breakdown voltage. Breakdown is more of a rough concept than an exact science. A material's breakdown voltage cannot be precisely defined. As a failure, there is a probabilistic element and thus a dielectric may experience a breakdown at any of a range of voltages. Additionally, the nature of the voltage used to induce breakdown must be considered. Short pulses can be used in stress testing to resemble lightning strikes, as could a continuous applied voltage.\r\n\r\nHowever, for the case of a gas being used as a dielectric, the following equation has been proven to be rather reliable in predicting breakdown voltage (Vb):\r\n[latex]\\text{V}_\\text{b}=\\frac {\\text{Bpd}}{\\ln {\\text{Apd}} - \\ln {(\\ln {(1+\\frac {1}{\\gamma_{\\text{se}}})})}}[\/latex]\r\n\r\nwhere A and B are constants that depend on the surrounding gas, p is the pressure of the surrounding gas, d is distance between the electrodes (in cm) and \u03b3<sub>se<\/sub> is the secondary electron emission coefficient. Gaseous dielectrics commonly experience breakdown in nature (the phenomenon of lightning is the most common example).\r\n<div class=\"wp-caption aligncenter\" style=\"width: 634px\">\r\n<div class=\"figure-cont\">\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2548\/2019\/01\/14030627\/square1.jpeg\" alt=\"image\" width=\"634\" height=\"575\" \/>\r\n<p class=\"wp-caption-text\"><strong>Dielectric breakdown of plexiglas<\/strong>: The treelike pattern in the plexiglas stems from the root of the breakdown. Current is dispersed in many different directions, creating different stems.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"boundless-concept\">\n<h2>Capacitance<\/h2>\n<p>Capacitance is the measure of an object&#8217;s ability to store electric charge.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>Express the relationship between the capacitance, charge of an object, and potential difference in the form of equation<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<h4>Key Points<\/h4>\n<ul>\n<li>The unit of capacitance is known as the farad (F), which can be equated to many quotients of units, including JV<sup>-2<\/sup>, WsV<sup>-2<\/sup>, CV<sup>-1<\/sup>, and C<sup>2<\/sup>J<sup>-1<\/sup>.<\/li>\n<li>Capacitance (C) can be calculated as a function of charge an object can store (q) and potential difference (V) between the two plates: [latex]\\text{C}=\\frac {\\text{q}}{\\text{V}}[\/latex] Q depends on the surface area of the conductor plates, while V depends on the distance between the plates and the permittivity of the dielectric between them.<\/li>\n<li>In storing charge, capacitors also store potential energy, which is equal to the work (W) required to charge them. For a capacitor with plates holding charges of +q and -q, this can be calculated: [latex]\\text{W}_{\\text{stored}}=\\frac {\\text{CV}^2}{2}[\/latex]. The above can be equated with the work required to charge the capacitor.<\/li>\n<\/ul>\n<h4>Key Terms<\/h4>\n<ul>\n<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\n<li><strong>capacitance<\/strong>: The property of an electric circuit or its element that permits it to store charge, defined as the ratio of stored charge to potential over that element or circuit (Q\/V); SI unit: farad (F).<\/li>\n<\/ul>\n<\/div>\n<p>Capacitance is the measure of an object&#8217;s ability to store electric charge. Any body capable of being charged in any way has a value of capacitance.<\/p>\n<p>The unit of capacitance is known as the Farad (F), which can be adjusted into subunits (the millifarad (mF), for example) for ease of working in practical orders of magnitude. The Farad can be equated to many quotients of units, including JV<sup>-2<\/sup>, WsV<sup>-2<\/sup>, CV<sup>-1<\/sup>, and C2J<sup>-1<\/sup>.<\/p>\n<p>The most common capacitor is known as a parallel-plate capacitor which involves two separate conductor plates separated from one another by a dielectric. Capacitance (C) can be calculated as a function of charge an object can store (q) and potential difference (V) between the two plates:<\/p>\n<div class=\"wp-caption alignright\" style=\"width: 270px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/parallel-plate-capacitor.svg#fixme#fixme\" alt=\"image\" width=\"270\" height=\"216\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Parallel-Plate Capacitor<\/strong>: The dielectric prevents charge flow from one plate to the other.<\/p>\n<\/div>\n<\/div>\n<p>[latex]\\text{C}=\\frac {\\text{q}}{\\text{V}}[\/latex]<\/p>\n<p>Ultimately, in such a capacitor, q depends on the surface area (A) of the conductor plates, while V depends on the distance (d) between the plates and the permittivity (\u03b5<sub>r<\/sub>) of the dielectric between them. For a parallel-plate capacitor, this equation can be used to calculate capacitance:<\/p>\n<p>[latex]\\text{C}=\\epsilon_\\text{r} \\epsilon_0 \\frac {\\text{A}}{\\text{d}}[\/latex]<\/p>\n<p>Where <em>\u03b5<\/em><sub>0<\/sub> is the electric constant. The product of length and height of the plates can be substituted in place of A.<\/p>\n<p>In storing charge, capacitors also store potential energy, which is equal to the work (W) required to charge them. For a capacitor with plates holding charges of +q and -q, this can be calculated:<\/p>\n<p>[latex]\\text{W}_{\\text{charging}}=\\int_0^\\text{Q} \\! \\frac {\\text{q}}{\\text{C}} \\mathrm{\\text{d}} \\text{q}=\\frac {\\text{CV}^2}{2}=\\text{W}_{\\text{stored}}[\/latex]<\/p>\n<p>Thus, either through calculus or algebraically (if C and V are known), stored energy (W<sub>stored<\/sub>) can be calculated. In a parallel-plate capacitor, this can be simplified to:<\/p>\n<p>[latex]\\text{W}_{\\text{stored}}= \\frac {\\epsilon_\\text{r} \\epsilon_0 \\text{AV}^2}{2\\text{d}}[\/latex]<\/p>\n<\/div>\n<div class=\"boundless-concept\">\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"boundless-concept\">\n<h2>Parallel-Plate Capacitor<\/h2>\n<p>The parallel-plate capacitor is one that includes two conductor plates, each connected to wires, separated from one another by a thin space.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>Calculate the maximum storable energy in a parallel-plate capacitor<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<h4>Key Points<\/h4>\n<ul>\n<li>Assuming the plates extend uniformly over an area of A and hold \u00b1 Q charge, their charge density is \u00b1, where \u03c1=Q\/A.<\/li>\n<li>Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, [latex]\\text{E}=\\frac {\\rho}{\\epsilon}[\/latex] can be used to calculate the electric field (E) near the center of the plates. In this equation, \u03b5 represents permittivity.<\/li>\n<li>[latex]\\text{V}=\\frac {\\rho \\text{d}}{\\epsilon}=\\frac {\\text{Qd}}{\\epsilon \\text{A}}[\/latex] can be used to calculate the potential between the plates.<\/li>\n<li>[latex]\\text{C}=\\frac {\\epsilon \\text{A}}{\\text{d}}[\/latex] can be found from the previous equation, adjusting the terms to solve for capacitance (C).<\/li>\n<li>[latex]\\text{U}=\\frac {\\text{CV}^2}{2}=\\frac {\\epsilon \\text{A}(\\text{U}_\\text{dd})^2}{2\\text{d}}=\\frac {\\epsilon \\text{AdU}_\\text{d}^2}{2}[\/latex] solves for the maximum storable energy in a parallel-plate capacitor (U) as a function of U<sub>d<\/sub>, the dielectric strength per distance as well as capacitor&#8217;s voltage (V) at its breakdown limit.<\/li>\n<\/ul>\n<h4>Key Terms<\/h4>\n<ul>\n<li><strong>permittivity<\/strong>: A property of a dielectric medium that determines the forces that electric charges placed in the medium exert on each other.<\/li>\n<li><strong>capacitor<\/strong>: An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.<\/li>\n<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\n<\/ul>\n<\/div>\n<p>One of the most commonly used capacitors in industry and in the academic setting is the parallel-plate capacitor. This is a capacitor that includes two conductor plates, each connected to wires, separated from one another by a thin space. Between them can be a vacuum or a dielectric material, but not a conductor.<\/p>\n<div class=\"wp-caption alignright\" style=\"width: 273px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/parallel-plate-capacitor.svg#fixme#fixme\" alt=\"image\" width=\"273\" height=\"218\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Parallel-Plate Capacitor<\/strong>: In a capacitor, the opposite plates take on opposite charges. The dielectric ensures that the charges are separated and do not transfer from one plate to the other.<\/p>\n<\/div>\n<\/div>\n<p>The purpose of a capacitor is to store charge, and in a parallel-plate capacitor one plate will take on an excess of positive charge while the other becomes more negative.<\/p>\n<p>Assuming the plates extend uniformly over an area of A and hold \u00b1 Q charge, their charge density is \u00b1, where \u03c1=Q\/A. Assuming that the dimensions of length and width for the plates are significantly greater than the distance (d) between them, the electric field (E) near the center of the plates can be calculated by:<\/p>\n<p>[latex]\\text{E}=\\frac {\\rho}{\\epsilon}[\/latex]<\/p>\n<p>Potential (V) between the plates can be calculated from the line integral of the electric field (E):<\/p>\n<p>[latex]\\text{V}=\\int_0^\\text{d} \\! \\text{E} \\mathrm{\\text{d}}\\text{z}[\/latex]<\/p>\n<p>where z is the axis perpendicular to both plates. Through simplification and substitution, this integral can be changed to:<\/p>\n<p>[latex]\\text{V}=\\frac {\\rho \\text{d}}{\\epsilon}=\\frac {\\text{Qd}}{\\epsilon \\text{A}}[\/latex]<\/p>\n<p>Given that capacitance is the quotient of charge and potential:<\/p>\n<p>[latex]\\text{C}=\\frac {\\epsilon \\text{A}}{\\text{d}}[\/latex]<\/p>\n<p>Accordingly, capacitance is greatest in devices with high permittivity, large plate area, and minimal separation between the plates.<\/p>\n<p>The maximum energy (U) a capacitor can store can be calculated as a function of U<sub>d<\/sub>, the dielectric strength per distance, as well as capacitor&#8217;s voltage (V) at its breakdown limit (the maximum voltage before the dielectric ionizes and no longer operates as an insulator):<\/p>\n<p>[latex]\\text{U}=\\frac {\\text{CV}^2}{2}=\\frac {\\epsilon \\text{A}(\\text{U}_\\text{dd})^2}{2\\text{d}}=\\frac {\\epsilon \\text{AdU}_\\text{d}^2}{2}[\/latex]<\/p>\n<\/div>\n<div class=\"boundless-concept\">\n<h2>Combinations of Capacitors: Series and Parallel<\/h2>\n<p>Like any other form of electrical circuitry device, capacitors can be used in series and\/or in parallel within circuits.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>Calculate the total capacitance for the capacitors connected in series and in parallel<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<h4>Key Points<\/h4>\n<ul>\n<li>[latex]\\frac {1}{\\text{C}_{\\text{total}}}=\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}[\/latex] Capacitors in series follow the law of reciprocals; the reciprocal of the circuit &#8216;s total capacitance is equal to the sum of the reciprocals of the capacitances of each individual capacitor.<\/li>\n<li>[latex]\\text{C}_{\\text{total}}=\\text{C}_1+\\text{C}_2+...+\\text{C}_\\text{n}[\/latex]For capacitors in parallel, summing the capacitances of individual capacitors affords the total capacitance in the circuit.<\/li>\n<li>When capacitors are found both in series and in parallel in the same circuit, it is best to simplify the circuit by solving parts of it in sequence.<\/li>\n<\/ul>\n<h4>Key Terms<\/h4>\n<ul>\n<li><strong>capacitor<\/strong>: An electronic component capable of storing an electric charge, especially one consisting of two conductors separated by a dielectric.<\/li>\n<li><strong>circuit<\/strong>: A pathway of electric current composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. T<\/li>\n<\/ul>\n<\/div>\n<p>Like any other form of electrical circuitry device, capacitors can be used in combination in circuits. These combinations can be in series (in which multiple capacitors can be found along the same path of wire) and in parallel (in which multiple capacitors can be found along different paths of wire).<\/p>\n<h3>Capacitors in Series<\/h3>\n<p>Like in the case of resistors in parallel, the reciprocal of the circuit&#8217;s total capacitance is equal to the sum of the reciprocals of the capacitance of each individual capacitor:<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 438px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/capacitors-in-series.svg#fixme#fixme\" alt=\"image\" width=\"438\" height=\"119\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Capacitors in Series<\/strong>: This image depicts capacitors C1, C2 and so on until Cn in a series.<\/p>\n<\/div>\n<\/div>\n<p>[latex]\\frac {1}{\\text{C}_{\\text{total}}}=\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}[\/latex]<\/p>\n<p>This can also be expressed as:<\/p>\n<p>[latex]\\text{C}_{\\text{total}}=\\frac{1}{\\frac {1}{\\text{C}_1}+\\frac {1}{\\text{C}_2}+...+\\frac {1}{\\text{C}_\\text{n}}}[\/latex]<\/p>\n<h3>Parallel Capacitors<\/h3>\n<p>Total capacitance for a circuit involving several capacitors in parallel (and none in series) can be found by simply summing the individual capacitances of each individual capacitor.<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 359px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/textimgs.s3.amazonaws.com\/boundless-physics\/capacitors-in-parallel.svg#fixme#fixme\" alt=\"image\" width=\"359\" height=\"211\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Parallel Capacitors<\/strong>: This image depicts capacitors C1, C2, and so on until Cn in parallel.<\/p>\n<\/div>\n<\/div>\n<p>[latex]\\text{C}_{\\text{total}}=\\text{C}_1+\\text{C}_2+...+\\text{C}_\\text{n}[\/latex]<\/p>\n<h3>Capacitors in Series and in Parallel<\/h3>\n<p>It is possible for a circuit to contain capacitors that are both in series and in parallel. To find total capacitance of the circuit, simply break it into segments and solve piecewise.<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 643px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2548\/2019\/01\/14030624\/figure-20-06-03a.jpeg\" alt=\"image\" width=\"643\" height=\"209\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Capacitors in Series and in Parallel<\/strong>: The initial problem can be simplified by finding the capacitance of the series, then using it as part of the parallel calculation.<\/p>\n<\/div>\n<\/div>\n<p>The circuit shown in (a) contains C<sub>1<\/sub> and C<sub>2<\/sub> in series. However, these are both in parallel with C<sub>3<\/sub>. If we find the capacitance for the series including C<sub>1<\/sub> and C<sub>2<\/sub>, we can treat that total as that from a single capacitor (b). This value can be calculated as approximately equal to 0.83 \u03bcF.<\/p>\n<p>With effectively two capacitors left in parallel, we can add their respective capacitances (c) to find the total capacitance for the circuit. This sum is approximately 8.83 \u03bcF.<\/p>\n<\/div>\n<div class=\"boundless-concept\">\n<h2>Dieletrics and their Breakdown<\/h2>\n<p>Dielectric breakdown is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor.<\/p>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>Identify conditions that can lead to a dielectric breakdown and its effect on materials<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<h4>Key Points<\/h4>\n<ul>\n<li>All insulators can, when exposed to enough voltage, experience dielectric breakdown and become conductors.<\/li>\n<li>Because dielectric breakdown is a failure that depends on a probability, an exact breakdown voltage is in most cases impossible to calculate with a high degree of certainty.<\/li>\n<li>Lightning is a common instance of dielectric breakdown, as air loses its ability to separate the potential difference between clouds and the point of a lightning bolt&#8217;s impact.<\/li>\n<\/ul>\n<h4>Key Terms<\/h4>\n<ul>\n<li><strong>conductor<\/strong>: A material which contains movable electric charges.<\/li>\n<li><strong>dielectric<\/strong>: An electrically insulating or nonconducting material considered for its electric susceptibility (i.e., its property of polarization when exposed to an external electric field).<\/li>\n<li><strong>breakdown<\/strong>: A failure, particularly mechanical; something that has failed.<\/li>\n<\/ul>\n<\/div>\n<p>Dielectric breakdown (illustrated in ) is the phenomenon in which a dielectric loses its ability to insulate, and instead becomes a conductor. Dielectrics are commonly used either to isolate conductors from a variable external environment (e.g., as coating for electrical wires) or to isolate conductors from one another (e.g., between plates of a parallel-plate capacitor). In all applications, they are selected for their ability to act as insulators. By definition, an insulator is unable to conduct electricity. Under certain conditions, however, a material that is an insulator can become a conductor.<\/p>\n<p>Eventually, exposing any insulator to increasing voltage will result in the insulator becoming conductive. This point (the minimum voltage for the insulator to become a conductor) is known as the breakdown voltage. Breakdown is more of a rough concept than an exact science. A material&#8217;s breakdown voltage cannot be precisely defined. As a failure, there is a probabilistic element and thus a dielectric may experience a breakdown at any of a range of voltages. Additionally, the nature of the voltage used to induce breakdown must be considered. Short pulses can be used in stress testing to resemble lightning strikes, as could a continuous applied voltage.<\/p>\n<p>However, for the case of a gas being used as a dielectric, the following equation has been proven to be rather reliable in predicting breakdown voltage (Vb):<br \/>\n[latex]\\text{V}_\\text{b}=\\frac {\\text{Bpd}}{\\ln {\\text{Apd}} - \\ln {(\\ln {(1+\\frac {1}{\\gamma_{\\text{se}}})})}}[\/latex]<\/p>\n<p>where A and B are constants that depend on the surrounding gas, p is the pressure of the surrounding gas, d is distance between the electrodes (in cm) and \u03b3<sub>se<\/sub> is the secondary electron emission coefficient. Gaseous dielectrics commonly experience breakdown in nature (the phenomenon of lightning is the most common example).<\/p>\n<div class=\"wp-caption aligncenter\" style=\"width: 634px\">\n<div class=\"figure-cont\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2548\/2019\/01\/14030627\/square1.jpeg\" alt=\"image\" width=\"634\" height=\"575\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Dielectric breakdown of plexiglas<\/strong>: The treelike pattern in the plexiglas stems from the root of the breakdown. Current is dispersed in many different directions, creating different stems.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1441\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Curation and Revision. <strong>Provided by<\/strong>: Boundless.com. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Capacitance. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Capacitance\">http:\/\/en.wikipedia.org\/wiki\/Capacitance<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>dielectric. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/dielectric\">http:\/\/en.wiktionary.org\/wiki\/dielectric<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>capacitance. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/capacitance\">http:\/\/en.wiktionary.org\/wiki\/capacitance<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Dielectric. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Dielectric\">http:\/\/en.wikipedia.org\/wiki\/Dielectric<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>capacitor. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/capacitor\">http:\/\/en.wiktionary.org\/wiki\/capacitor<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>dielectric. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/dielectric\">http:\/\/en.wiktionary.org\/wiki\/dielectric<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>capacitance. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/capacitance\">http:\/\/en.wiktionary.org\/wiki\/capacitance<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitor schematic with dielectric. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Capacitor\">http:\/\/en.wikipedia.org\/wiki\/Capacitor<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>capacitor. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/capacitor\">http:\/\/en.wiktionary.org\/wiki\/capacitor<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>dielectric. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/dielectric\">http:\/\/en.wiktionary.org\/wiki\/dielectric<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>permittivity. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/permittivity\">http:\/\/en.wiktionary.org\/wiki\/permittivity<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitor schematic with dielectric. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Series and parallel circuits. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Series_and_parallel_circuits\">http:\/\/en.wikipedia.org\/wiki\/Series_and_parallel_circuits<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>capacitor. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/capacitor\">http:\/\/en.wiktionary.org\/wiki\/capacitor<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>circuit. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/circuit\">http:\/\/en.wikipedia.org\/wiki\/circuit<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitor schematic with dielectric. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitors in parallel. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_parallel.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_parallel.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitors in series. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_series.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_series.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>OpenStax College, Capacitors in Series and Parallel. January 7, 2013. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/content\/m42336\/latest\/\">http:\/\/cnx.org\/content\/m42336\/latest\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Breakdown voltage. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Breakdown_voltage\">http:\/\/en.wikipedia.org\/wiki\/Breakdown_voltage<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Electrical breakdown. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/Electrical_breakdown\">http:\/\/en.wikipedia.org\/wiki\/Electrical_breakdown<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>breakdown. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/breakdown\">http:\/\/en.wiktionary.org\/wiki\/breakdown<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>dielectric. <strong>Provided by<\/strong>: Wiktionary. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wiktionary.org\/wiki\/dielectric\">http:\/\/en.wiktionary.org\/wiki\/dielectric<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>conductor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/conductor\">http:\/\/en.wikipedia.org\/wiki\/conductor<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitor schematic with dielectric. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitor_schematic_with_dielectric.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Parallel plate capacitor. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Parallel_plate_capacitor.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitors in parallel. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_parallel.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_parallel.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Capacitors in series. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_series.svg\">http:\/\/en.wikipedia.org\/wiki\/File:Capacitors_in_series.svg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>OpenStax College, Capacitors in Series and Parallel. January 7, 2013. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/content\/m42336\/latest\/\">http:\/\/cnx.org\/content\/m42336\/latest\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Square1. <strong>Provided by<\/strong>: Wikipedia. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/en.wikipedia.org\/wiki\/File:Square1.jpg\">http:\/\/en.wikipedia.org\/wiki\/File:Square1.jpg<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":18,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Capacitance\",\"author\":\"\",\"organization\":\"Wikipedia\",\"url\":\"http:\/\/en.wikipedia.org\/wiki\/Capacitance\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"dielectric\",\"author\":\"\",\"organization\":\"Wiktionary\",\"url\":\"http:\/\/en.wiktionary.org\/wiki\/dielectric\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"capacitance\",\"author\":\"\",\"organization\":\"Wiktionary\",\"url\":\"http:\/\/en.wiktionary.org\/wiki\/capacitance\",\"project\":\"\",\"license\":\"cc-by-sa\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\"Parallel plate 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