{"id":1072,"date":"2017-10-27T16:31:21","date_gmt":"2017-10-27T16:31:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/capacitors-in-series-and-parallel\/"},"modified":"2017-11-08T03:26:11","modified_gmt":"2017-11-08T03:26:11","slug":"capacitors-in-series-and-parallel","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/capacitors-in-series-and-parallel\/","title":{"raw":"Capacitors in Series and Parallel","rendered":"Capacitors in Series and Parallel"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Derive expressions for total capacitance in series and in parallel.<\/li>\n<li>Identify series and parallel parts in the combination of connection of capacitors.<\/li>\n<li>Calculate the effective capacitance in series and parallel given individual capacitances.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3093738\">Several capacitors may be connected together in a variety of applications. Multiple connections of capacitors act like a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called <em data-effect=\"italics\"><em data-effect=\"italics\">series<\/em><\/em> and <em data-effect=\"italics\"><em data-effect=\"italics\">parallel<\/em><\/em>, for which we can easily calculate the total capacitance. Certain more complicated connections can also be related to combinations of series and parallel.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2991721\">\n<h1 data-type=\"title\">Capacitance in Series<\/h1>\n<p id=\"import-auto-id2732826\"><a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>(a) shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by [latex]C=\\frac{Q}{V}[\/latex].<\/p>\n<p id=\"import-auto-id2837917\">Note in <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a> that opposite charges of magnitude <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> flow to either side of the originally uncharged combination of capacitors when the voltage [latex]V[\/latex] is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. (See <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>(b).) Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.<\/em><\/em><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2499137\">\n<div class=\"bc-figcaption figcaption\">(a) Capacitors connected in series. The magnitude of the charge on each plate is  [latex]Q[\/latex]. (b) An equivalent capacitor has a larger plate separation [latex]d[\/latex]. Series connections produce a total capacitance that is less than that of any of the individual capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2949769\" data-alt=\"When capacitors are connected in series, an equivalent capacitor would have a plate separation that is greater than that of any individual capacitor. Hence the series connections produce a resultant capacitance less than that of the individual capacitors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"When capacitors are connected in series, an equivalent capacitor would have a plate separation that is greater than that of any individual capacitor. Hence the series connections produce a resultant capacitance less than that of the individual capacitors.\" width=\"185\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2677631\">We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>. Solving [latex]C=\\frac{Q}{V}[\/latex] for [latex]V[\/latex] gives [latex]V=\\frac{Q}{C}[\/latex]. The voltages across the individual capacitors are thus [latex]{V}_{1}=\\frac{Q}{{C}_{1}}[\/latex], [latex]{V}_{2}=\\frac{Q}{{C}_{2}}[\/latex], and [latex]{V}_{3}=\\frac{Q}{{C}_{3}}[\/latex]. The total voltage is the sum of the individual voltages:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]V={V}_{1}+{V}_{2}+{V}_{3}.[\/latex]<\/div>\n<p id=\"import-auto-id1559665\">Now, calling the total capacitance [latex]{C}_{\\text{S}}[\/latex] for series capacitance, consider that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]V=\\frac{Q}{{C}_{\\text{S}}}={V}_{1}+{V}_{2}+{V}_{3}.[\/latex]<\/div>\n<p id=\"import-auto-id2676037\">Entering the expressions for [latex]{V}_{1}[\/latex], [latex]{V}_{2}[\/latex], and [latex]{V}_{3}[\/latex], we get<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{Q}{{C}_{\\text{S}}}=\\frac{Q}{{C}_{1}}+\\frac{Q}{{C}_{2}}+\\frac{Q}{{C}_{3}}.[\/latex]<\/div>\n<p>Canceling the [latex]Q[\/latex]s, we obtain the equation for the total capacitance in series [latex]{C}_{\\text{S}}[\/latex] to be<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}+\\frac{1}{{C}_{3}}+\\text{.}\\text{.}\\text{.},[\/latex]<\/div>\n<p id=\"import-auto-id2586197\">where \u201c...\u201d indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance [latex]{C}_{\\text{S}}[\/latex] that is less than any of the individual capacitances [latex]{C}_{1}[\/latex], [latex]{C}_{2}[\/latex], ..., as the next example illustrates.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3068086\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Capacitance in Series, [latex]{C}_{\\text{s}}[\/latex]     <\/div>\n<p id=\"import-auto-id1559732\">Total capacitance in series: [latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}+\\frac{1}{{C}_{3}}+\\text{.}\\text{.}\\text{.}[\/latex]<strong data-effect=\"bold\"> <\/strong><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2635156\">\n<div data-type=\"title\" class=\"title\">What Is the Series Capacitance?<\/div>\n<p id=\"import-auto-id2601609\">Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1.000, 5.000, and 8.000 [latex]\\text{\u00b5F}[\/latex].<\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p>With the given information, the total capacitance can be found using the equation for capacitance in series.<\/p>\n<p id=\"import-auto-id2721143\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1445182\">Entering the given capacitances into the expression for [latex]\\frac{1}{{C}_{\\text{S}}}[\/latex] gives [latex]\\frac{1}{{C}_{S}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}+\\frac{1}{{C}_{3}}[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{1\\text{.}\\text{000 \u00b5F}}+\\frac{1}{5\\text{.}\\text{000 \u00b5F}}+\\frac{1}{8\\text{.}\\text{000 \u00b5F}}=\\frac{1\\text{.}\\text{325}}{\\text{\u00b5F}}[\/latex]<\/div>\n<p id=\"import-auto-id624240\">Inverting to find [latex]{C}_{\\text{S}}[\/latex] yields [latex][\/latex][latex]{C}_{\\text{S}}=\\frac{\\text{\u00b5F}}{1\\text{.}\\text{325}}=0\\text{.}\\text{755 \u00b5F}[\/latex].<\/p>\n<p id=\"import-auto-id2648425\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1539528\">The total series capacitance [latex]{C}_{\\text{s}}[\/latex] is less than the smallest individual capacitance, as promised. In series connections of capacitors, the sum is less than the parts. In fact, it is less than any individual. Note that it is sometimes possible, and more convenient, to solve an equation like the above by finding the least common denominator, which in this case (showing only whole-number calculations) is 40. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{\\text{40}}{\\text{40 \u00b5F}}+\\frac{8}{\\text{40 \u00b5F}}+\\frac{5}{\\text{40 \u00b5F}}=\\frac{\\text{53}}{\\text{40 \u00b5F}},[\/latex]<\/div>\n<p id=\"import-auto-id2972705\">so that <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{C}_{\\text{S}}=\\frac{\\text{40 \u00b5F}}{\\text{53}}=0\\text{.}\\text{755 \u00b5F}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3065343\">\n<h1 data-type=\"title\">Capacitors in Parallel<\/h1>\n<p id=\"import-auto-id1937104\"><a href=\"#import-auto-id2511423\" class=\"autogenerated-content\">(Figure)<\/a>(a) shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance [latex]{C}_{\\text{p}}[\/latex], we first note that the voltage across each capacitor is [latex]V[\/latex], the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> is the sum of the individual charges:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-731\">[latex]Q={Q}_{1}+{Q}_{2}+{Q}_{3}.[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2511423\">\n<div class=\"bc-figcaption figcaption\">(a) Capacitors in parallel. Each is connected directly to the voltage source just as if it were all alone, and so the total capacitance in parallel is just the sum of the individual capacitances. (b) The equivalent capacitor has a larger plate area and can therefore hold more charge than the individual capacitors. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1492959\" data-alt=\"Part a of the figure shows three capacitors connected in parallel to each other and to the applied voltage. The total capacitance when they are connected in parallel is simply the sum of the individual capacitances. Part b of the figure shows the larger equivalent plate area of the capacitors connected in parallel, which in turn can hold more charge than the individual capacitors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows three capacitors connected in parallel to each other and to the applied voltage. The total capacitance when they are connected in parallel is simply the sum of the individual capacitances. Part b of the figure shows the larger equivalent plate area of the capacitors connected in parallel, which in turn can hold more charge than the individual capacitors.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2550444\">Using the relationship [latex]Q=\\text{CV}[\/latex], we see that the total charge is [latex]Q={C}_{\\text{p}}V[\/latex], and the individual charges are [latex]{Q}_{1}={C}_{1}V[\/latex]<em data-effect=\"italics\">, <\/em>[latex]{Q}_{2}={C}_{2}V[\/latex]<em data-effect=\"italics\">,<\/em> and [latex]{Q}_{3}={C}_{3}V[\/latex]. Entering these into the previous equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{C}_{\\text{p}}V={C}_{1}V+{C}_{2}V+{C}_{3}V.[\/latex]<\/div>\n<p id=\"import-auto-id920733\">Canceling [latex]V[\/latex] from the equation, we obtain the equation for the total capacitance in parallel<br>\n     [latex]{C}_{\\text{p}}[\/latex]:\n    <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{C}_{p}={C}_{1}+{C}_{2}+{C}_{3}+\\text{.}\\text{.}\\text{.}.[\/latex]<\/div>\n<p id=\"import-auto-id2675960\">Total capacitance in parallel is simply the sum of the individual capacitances. (Again the \u201c<em data-effect=\"italics\">...<\/em>\u201d indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in the example above were connected in parallel, their capacitance would be <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-198\">[latex]{C}_{\\text{p}}=1\\text{.}\\text{000 \u00b5F}+5\\text{.}\\text{000 \u00b5F}+8\\text{.}\\text{000 \u00b5F}=\\text{14}\\text{.}000 \u00b5F.[\/latex]<\/div>\n<p>The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in <a href=\"#import-auto-id2511423\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1346131\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Capacitance in Parallel, [latex]{C}_{\\text{p}}[\/latex]<\/div>\n<p id=\"import-auto-id1500994\">Total capacitance in parallel [latex]{C}_{\\text{p}}={C}_{1}+{C}_{2}+{C}_{3}+\\text{.}\\text{.}\\text{.}[\/latex]<\/p>\n<\/div>\n<p id=\"import-auto-id1183848\">More complicated connections of capacitors can sometimes be combinations of series and parallel. (See <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a>.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2506336\">\n<div class=\"bc-figcaption figcaption\">(a) This circuit contains both series and parallel connections of capacitors. See <a href=\"#fs-id1327996\" class=\"autogenerated-content\">(Figure)<\/a> for the calculation of the overall capacitance of the circuit. (b) [latex]{C}_{1}[\/latex] and [latex]{C}_{2}[\/latex] are in series; their equivalent capacitance [latex]{C}_{\\text{S}}[\/latex] is less than either of them. (c) Note that [latex]{C}_{\\text{S}}[\/latex] is in parallel with [latex]{C}_{3}[\/latex]. The total capacitance is, thus, the sum of [latex]{C}_{\\text{S}}[\/latex] and [latex]{C}_{3}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3066358\" data-alt=\"The first figure has two capacitors, C sub1 and C sub2 in series and the third capacitor C sub 3 is parallel to C sub 1 and C sub 2. The second figure shows C sub S, the equivalent capacitance of C sub 1 and C sub 2, in parallel to C sub 3. The third figure represents the total capacitance of C sub S and C sub 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The first figure has two capacitors, C sub1 and C sub2 in series and the third capacitor C sub 3 is parallel to C sub 1 and C sub 2. The second figure shows C sub S, the equivalent capacitance of C sub 1 and C sub 2, in parallel to C sub 3. The third figure represents the total capacitance of C sub S and C sub 3.\" width=\"500\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1327996\">\n<div data-type=\"title\" class=\"title\">A Mixture of Series and Parallel Capacitance<\/div>\n<p id=\"import-auto-id1544456\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a>. Assume the capacitances in <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a> are known to three decimal places (<br>\n[latex]{C}_{1}=1.000 \u00b5F[\/latex],  [latex]{C}_{2}=5.000 \u00b5F[\/latex], and  [latex]{C}_{3}=8.000 \u00b5F[\/latex]), and round your answer to three decimal places.<\/p>\n<p id=\"import-auto-id2822372\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id953354\">To find the total capacitance, we first identify which capacitors are in series and which are in parallel. Capacitors [latex]{C}_{1}[\/latex] and [latex]{C}_{2}[\/latex] are in series. Their combination, labeled [latex]{C}_{\\text{S}}[\/latex] in the figure, is in parallel with [latex]{C}_{3}[\/latex].<\/p>\n<p id=\"import-auto-id3346970\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2555621\">Since [latex]{C}_{1}[\/latex] and [latex]{C}_{2}[\/latex] are in series, their total capacitance is given by [latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}+\\frac{1}{{C}_{3}}[\/latex]. Entering their values into the equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}=\\frac{1}{1\\text{.}\\text{000}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcF}}+\\frac{1}{5\\text{.}\\text{000}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcF}}=\\frac{1\\text{.}\\text{200}}{\\text{\u03bcF}}.[\/latex]<\/div>\n<p id=\"import-auto-id2910870\">Inverting gives <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{C}_{\\text{S}}=0\\text{.}\\text{833 \u00b5F}.[\/latex]<\/div>\n<p id=\"import-auto-id2543801\">This equivalent series capacitance is in parallel with the third capacitor; thus, the total is the sum<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}{C}_{\\text{tot}}&amp; =&amp; {C}_{\\text{S}}+{C}_{\\text{S}}\\\\ &amp; =&amp; 0\\text{.}\\text{833}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcF}+8\\text{.}\\text{000}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcF}\\\\ &amp; =&amp; 8\\text{.}\\text{833}\\phantom{\\rule{0.25em}{0ex}}\\text{\u03bcF}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2678852\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1956195\">This technique of analyzing the combinations of capacitors piece by piece until a total is obtained can be applied to larger combinations of capacitors.<\/p>\n<\/div>\n<p>&lt;!--<br>\n        &lt;div class='bc-figure figure'  id=\"import-auto-id2698061\"&gt;<\/p>\n<p>      C1 size 12{ {C}  rSub { size 8{1} } } {}C2 size 12{ {C}  rSub { size 8{2} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}C3 size 12{ {C}  rSub { size 8{3} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}C3 size 12{ {C}  rSub { size 8{3} } } {}\n      <\/p><\/div>\n<p>--&gt;<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1541571\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2836824\">\n<li id=\"import-auto-id3143144\">Total capacitance in series [latex]\\frac{1}{{C}_{\\text{S}}}=\\frac{1}{{C}_{1}}+\\frac{1}{{C}_{2}}+\\frac{1}{{C}_{3}}+\\text{.}\\text{.}\\text{.}[\/latex]<strong data-effect=\"bold\"><\/strong><\/li>\n<li id=\"import-auto-id1988889\">Total capacitance in parallel [latex]{C}_{\\text{p}}={C}_{1}+{C}_{2}+{C}_{3}+\\text{.}\\text{.}\\text{.}[\/latex]<\/li>\n<li>If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3075419\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1656533\">\n<p id=\"import-auto-id1032431\">If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1596788\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1656785\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2791324\">\n<p id=\"eip-id2337212\">Find the total capacitance of the combination of capacitors in <a href=\"#import-auto-id1579779\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1579779\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1579780\" data-alt=\"A circuit is shown with three capacitors. Two capacitors, of ten microfarad and two point five microfarad capacitance, are in parallel to each other, and their combination is in series with a zero point three zero microfarad capacitor.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"A circuit is shown with three capacitors. Two capacitors, of ten microfarad and two point five microfarad capacitance, are in parallel to each other, and their combination is in series with a zero point three zero microfarad capacitor.\" width=\"200\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1175338\">\n<p id=\"eip-id1319497\">[latex]\\text{0.293 \u03bcF}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2912478\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2511599\">\n<p id=\"import-auto-id1999490\">Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1290190\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2920194\">\n<p id=\"import-auto-id1964409\">What total capacitances can you make by connecting a [latex]5\\text{.}\\text{00 \u00b5F}[\/latex] and an [latex]8\\text{.}\\text{00 \u00b5F}[\/latex] capacitor together? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1962000\">\n<p id=\"import-auto-id1663743\">[latex]3\\text{.}\\text{08 \u00b5F}[\/latex] in series combination, [latex]\\text{13}\\text{.}\\text{0 \u00b5F}[\/latex] in parallel combination<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2607784\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1998292\">\n<p id=\"import-auto-id2636632\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id1959375\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1959375\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1356269\" data-alt=\"The circuit includes three capacitors. A zero point three zero microfarad capacitor and a ten microfarad capacitor are connected in series, and together they are connected in parallel with a two point five microfarad capacitor.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"The circuit includes three capacitors. A zero point three zero microfarad capacitor and a ten microfarad capacitor are connected in series, and together they are connected in parallel with a two point five microfarad capacitor.\" width=\"200\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2749491\">\n<p id=\"import-auto-id1418699\">[latex]2\\text{.}\\text{79 \u00b5F}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2970272\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2970273\">\n<p id=\"import-auto-id2643460\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id2695900\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2695900\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2695901\" data-alt=\"The figure shows a circuit that is a combination of series and parallel connections of capacitors. On the left of the circuit is a five point zero microfarad capacitor in series with a three point five microfarad capacitor. In the middle is an eight point zero microfarad capacitor. On the right, a zero point seven five microfarad capacitor is in parallel with a fifteen microfarad capacitor, and together they are in series with a one point five microfarad capacitor. Altogether, the system of capacitors on the left, the capacitor in the middle, and the system of capacitors on the right are connected in parallel.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a circuit that is a combination of series and parallel connections of capacitors. On the left of the circuit is a five point zero microfarad capacitor in series with a three point five microfarad capacitor. In the middle is an eight point zero microfarad capacitor. On the right, a zero point seven five microfarad capacitor is in parallel with a fifteen microfarad capacitor, and together they are in series with a one point five microfarad capacitor. Altogether, the system of capacitors on the left, the capacitor in the middle, and the system of capacitors on the right are connected in parallel.\" width=\"200\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2605419\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2129513\">\n<p><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1370920\">(a) An [latex]8\\text{.}\\text{00 \u00b5F}[\/latex] capacitor is connected in parallel to another capacitor, producing a total capacitance of [latex]5\\text{.}\\text{00 \u00b5F}[\/latex]. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2393147\">\n<p>(a) [latex]\u20133\\text{.}\\text{00 \u00b5F}[\/latex]<\/p>\n<p id=\"eip-852\"> (b) You cannot have a negative value of capacitance.\n<\/p>\n<p>(c) The assumption that the capacitors were hooked up in parallel, rather than in series, was incorrect. A parallel connection always produces a greater capacitance, while here a smaller capacitance was assumed. This could happen only if the capacitors are connected in series.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Derive expressions for total capacitance in series and in parallel.<\/li>\n<li>Identify series and parallel parts in the combination of connection of capacitors.<\/li>\n<li>Calculate the effective capacitance in series and parallel given individual capacitances.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3093738\">Several capacitors may be connected together in a variety of applications. Multiple connections of capacitors act like a single equivalent capacitor. The total capacitance of this equivalent single capacitor depends both on the individual capacitors and how they are connected. There are two simple and common types of connections, called <em data-effect=\"italics\"><em data-effect=\"italics\">series<\/em><\/em> and <em data-effect=\"italics\"><em data-effect=\"italics\">parallel<\/em><\/em>, for which we can easily calculate the total capacitance. Certain more complicated connections can also be related to combinations of series and parallel.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2991721\">\n<h1 data-type=\"title\">Capacitance in Series<\/h1>\n<p id=\"import-auto-id2732826\"><a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>(a) shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60eda1c7ac05fb28083ece9f7f144bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<p id=\"import-auto-id2837917\">Note in <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a> that opposite charges of magnitude <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> flow to either side of the originally uncharged combination of capacitors when the voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. (See <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>(b).) Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.<\/em><\/em><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2499137\">\n<div class=\"bc-figcaption figcaption\">(a) Capacitors connected in series. The magnitude of the charge on each plate is  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>. (b) An equivalent capacitor has a larger plate separation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4e8716946f6a868f015e0d62f28bc540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/>. Series connections produce a total capacitance that is less than that of any of the individual capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2949769\" data-alt=\"When capacitors are connected in series, an equivalent capacitor would have a plate separation that is greater than that of any individual capacitor. Hence the series connections produce a resultant capacitance less than that of the individual capacitors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"When capacitors are connected in series, an equivalent capacitor would have a plate separation that is greater than that of any individual capacitor. Hence the series connections produce a resultant capacitance less than that of the individual capacitors.\" width=\"185\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2677631\">We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in <a href=\"#import-auto-id2499137\" class=\"autogenerated-content\">(Figure)<\/a>. Solving <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-60eda1c7ac05fb28083ece9f7f144bb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> gives <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d300d7d7dcffc8c1801826ce076ff68b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -6px;\" \/>. The voltages across the individual capacitors are thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-174bc73ba418c745e60e8e00f2f656f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"61\" style=\"vertical-align: -9px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-110c2a805669377562cea0ad395ae55e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"61\" style=\"vertical-align: -8px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7c211cafcbf955c7b50706f9fe0ade10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"61\" style=\"vertical-align: -8px;\" \/>. The total voltage is the sum of the individual voltages:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-025304a47a8b99e40f6c7259a723d7c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#123;&#86;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#86;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#86;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"139\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1559665\">Now, calling the total capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> for series capacitance, consider that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6709da39e7d8c44df3f50d5a644650c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#123;&#86;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#86;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#86;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"184\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2676037\">Entering the expressions for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0e33f1ef1cdc92748293b11512850133_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"16\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ada1a4a7833e0edeaac1349b2c2fa69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-036b55d4334dea24dc752349ea58e5e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/>, we get<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdf17ffd3a169c437ac043ab4646e13c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"151\" style=\"vertical-align: -9px;\" \/><\/div>\n<p>Canceling the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/>s, we obtain the equation for the total capacitance in series <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> to be<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4849c10f2549309683cbb48c951de43f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"188\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id2586197\">where \u201c&#8230;\u201d indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> that is less than any of the individual capacitances <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dba31011713ec89d2fc5f07df74d4c40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83caceb86243a1da6864e93f89163210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>, &#8230;, as the next example illustrates.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3068086\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Capacitance in Series, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18464fb925b2702ad38e87f5a7bb3174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/>     <\/div>\n<p id=\"import-auto-id1559732\">Total capacitance in series: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-27b6b4c673f19270cb87878eb19d3f75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"183\" style=\"vertical-align: -9px;\" \/><strong data-effect=\"bold\"> <\/strong><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2635156\">\n<div data-type=\"title\" class=\"title\">What Is the Series Capacitance?<\/div>\n<p id=\"import-auto-id2601609\">Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1.000, 5.000, and 8.000 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4c7c58864668ef42a4f48d1cc9f2cb0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p>With the given information, the total capacitance can be found using the equation for capacitance in series.<\/p>\n<p id=\"import-auto-id2721143\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1445182\">Entering the given capacitances into the expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-07e9066cc44c903bb7851294a885c47e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"18\" style=\"vertical-align: -8px;\" \/> gives <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ec7ba89916909ebaa5b63f47f22c455_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#83;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"147\" style=\"vertical-align: -9px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-720f4d817ca20b300ef32593a51d4114_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#50;&#53;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&micro;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"290\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id624240\">Inverting to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> yields [latex]{C}_{\\text{S}}=\\frac{\\text{\u00b5F}}{1\\text{.}\\text{325}}=0\\text{.}\\text{755 \u00b5F}[\/latex].<\/p>\n<p id=\"import-auto-id2648425\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1539528\">The total series capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18464fb925b2702ad38e87f5a7bb3174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> is less than the smallest individual capacitance, as promised. In series connections of capacitors, the sum is less than the parts. In fact, it is less than any individual. Note that it is sometimes possible, and more convenient, to solve an equation like the above by finding the least common denominator, which in this case (showing only whole-number calculations) is 40. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9599cc1eca73da61f4228943bbd0f3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&micro;&#70;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&micro;&#70;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&micro;&#70;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&micro;&#70;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"239\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2972705\">so that <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d8e1c1b75844471d1049880d32a0e6e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#32;&micro;&#70;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#53;&#32;&micro;&#70;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"161\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3065343\">\n<h1 data-type=\"title\">Capacitors in Parallel<\/h1>\n<p id=\"import-auto-id1937104\"><a href=\"#import-auto-id2511423\" class=\"autogenerated-content\">(Figure)<\/a>(a) shows a parallel connection of three capacitors with a voltage applied. Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b320f6c9a4e53dd0c1e34b0536216c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\" \/>, we first note that the voltage across each capacitor is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, the same as that of the source, since they are connected directly to it through a conductor. (Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.) Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. The total charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> is the sum of the individual charges:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-731\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e7c7eb3682adc4f5c6f35900a0efd422_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#123;&#81;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#81;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#81;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2511423\">\n<div class=\"bc-figcaption figcaption\">(a) Capacitors in parallel. Each is connected directly to the voltage source just as if it were all alone, and so the total capacitance in parallel is just the sum of the individual capacitances. (b) The equivalent capacitor has a larger plate area and can therefore hold more charge than the individual capacitors. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1492959\" data-alt=\"Part a of the figure shows three capacitors connected in parallel to each other and to the applied voltage. The total capacitance when they are connected in parallel is simply the sum of the individual capacitances. Part b of the figure shows the larger equivalent plate area of the capacitors connected in parallel, which in turn can hold more charge than the individual capacitors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows three capacitors connected in parallel to each other and to the applied voltage. The total capacitance when they are connected in parallel is simply the sum of the individual capacitances. Part b of the figure shows the larger equivalent plate area of the capacitors connected in parallel, which in turn can hold more charge than the individual capacitors.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2550444\">Using the relationship <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-183a2c7eb806f2473e1a9c6822ce4ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"64\" style=\"vertical-align: -4px;\" \/>, we see that the total charge is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-75744df723d508410d4ca8d094eb55e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -6px;\" \/>, and the individual charges are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a86fb1822fd5a4ff216fe05b5bd1c707_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#81;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"80\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">, <\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dabcec8a23280ba09cc49886cd3f75ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#81;&#125;&#95;&#123;&#50;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"80\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\">,<\/em> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ffc71ca1379d36abe51710df438f2d57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#81;&#125;&#95;&#123;&#51;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"80\" style=\"vertical-align: -4px;\" \/>. Entering these into the previous equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bbc3d94091fcf406533f0304f7d38e6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;&#86;&#61;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#86;&#43;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#86;&#43;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#86;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"207\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id920733\">Canceling <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> from the equation, we obtain the equation for the total capacitance in parallel<br \/>\n     <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b320f6c9a4e53dd0c1e34b0536216c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\" \/>:\n    <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a2e060726405c989c00f4d155b18be2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#112;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2675960\">Total capacitance in parallel is simply the sum of the individual capacitances. (Again the \u201c<em data-effect=\"italics\">&#8230;<\/em>\u201d indicates the expression is valid for any number of capacitors connected in parallel.) So, for example, if the capacitors in the example above were connected in parallel, their capacitance would be <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25a209151e0254457a4e7d128056dcd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#43;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#43;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#32;&micro;&#70;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#48;&#48;&#32;&micro;&#70;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"350\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>The equivalent capacitor for a parallel connection has an effectively larger plate area and, thus, a larger capacitance, as illustrated in <a href=\"#import-auto-id2511423\" class=\"autogenerated-content\">(Figure)<\/a>(b).<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1346131\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Total Capacitance in Parallel, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b320f6c9a4e53dd0c1e34b0536216c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1500994\">Total capacitance in parallel <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-567cd87bb61a7e1203c40e3876f1a221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"185\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<p id=\"import-auto-id1183848\">More complicated connections of capacitors can sometimes be combinations of series and parallel. (See <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a>.) To find the total capacitance of such combinations, we identify series and parallel parts, compute their capacitances, and then find the total.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2506336\">\n<div class=\"bc-figcaption figcaption\">(a) This circuit contains both series and parallel connections of capacitors. See <a href=\"#fs-id1327996\" class=\"autogenerated-content\">(Figure)<\/a> for the calculation of the overall capacitance of the circuit. (b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dba31011713ec89d2fc5f07df74d4c40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83caceb86243a1da6864e93f89163210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> are in series; their equivalent capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is less than either of them. (c) Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is in parallel with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-201c32762964c0afdff0c9bb7978f52b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>. The total capacitance is, thus, the sum of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-201c32762964c0afdff0c9bb7978f52b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3066358\" data-alt=\"The first figure has two capacitors, C sub1 and C sub2 in series and the third capacitor C sub 3 is parallel to C sub 1 and C sub 2. The second figure shows C sub S, the equivalent capacitance of C sub 1 and C sub 2, in parallel to C sub 3. The third figure represents the total capacitance of C sub S and C sub 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_06_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The first figure has two capacitors, C sub1 and C sub2 in series and the third capacitor C sub 3 is parallel to C sub 1 and C sub 2. The second figure shows C sub S, the equivalent capacitance of C sub 1 and C sub 2, in parallel to C sub 3. The third figure represents the total capacitance of C sub S and C sub 3.\" width=\"500\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1327996\">\n<div data-type=\"title\" class=\"title\">A Mixture of Series and Parallel Capacitance<\/div>\n<p id=\"import-auto-id1544456\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a>. Assume the capacitances in <a href=\"#import-auto-id2506336\" class=\"autogenerated-content\">(Figure)<\/a> are known to three decimal places (<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4161f95c268d8324f1cda7a1c693882b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#46;&#48;&#48;&#48;&#32;&micro;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -4px;\" \/>,  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-103eadd61f61d50bb9006c0f0ceea177_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#61;&#53;&#46;&#48;&#48;&#48;&#32;&micro;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"99\" style=\"vertical-align: -3px;\" \/>, and  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31d1ab57fd551e132a509764c04402e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#61;&#56;&#46;&#48;&#48;&#48;&#32;&micro;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"99\" style=\"vertical-align: -3px;\" \/>), and round your answer to three decimal places.<\/p>\n<p id=\"import-auto-id2822372\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id953354\">To find the total capacitance, we first identify which capacitors are in series and which are in parallel. Capacitors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dba31011713ec89d2fc5f07df74d4c40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83caceb86243a1da6864e93f89163210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> are in series. Their combination, labeled <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d81cfc0b765897abc74e1fdc70e9f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> in the figure, is in parallel with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-201c32762964c0afdff0c9bb7978f52b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<p id=\"import-auto-id3346970\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2555621\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dba31011713ec89d2fc5f07df74d4c40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83caceb86243a1da6864e93f89163210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> are in series, their total capacitance is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d48ac9300835fa559d1c0787ee0bbacd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"146\" style=\"vertical-align: -9px;\" \/>. Entering their values into the equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a66bc91819ab9df95877f1fa0f64b0dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#48;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"311\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id2910870\">Inverting gives <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dc795dda385b67fb934431ef15562d82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#51;&#51;&#32;&micro;&#70;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"107\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2543801\">This equivalent series capacitance is in parallel with the third capacitor; thus, the total is the sum<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6381cd0a25b4d984fa4921f830b5604b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#116;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#51;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#43;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#51;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#70;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"211\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"import-auto-id2678852\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1956195\">This technique of analyzing the combinations of capacitors piece by piece until a total is obtained can be applied to larger combinations of capacitors.<\/p>\n<\/div>\n<p>&lt;!&#8211;<br \/>\n        &lt;div class=&#8217;bc-figure figure&#8217;  id=&#8221;import-auto-id2698061&#8243;&gt;<\/p>\n<p>      C1 size 12{ {C}  rSub { size 8{1} } } {}C2 size 12{ {C}  rSub { size 8{2} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}C3 size 12{ {C}  rSub { size 8{3} } } {}CS size 12{ {C}  rSub { size 8{S} } } {}C3 size 12{ {C}  rSub { size 8{3} } } {}\n      <\/p>\n<\/div>\n<p>&#8211;&gt;<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1541571\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2836824\">\n<li id=\"import-auto-id3143144\">Total capacitance in series <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-27b6b4c673f19270cb87878eb19d3f75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"183\" style=\"vertical-align: -9px;\" \/><strong data-effect=\"bold\"><\/strong><\/li>\n<li id=\"import-auto-id1988889\">Total capacitance in parallel <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-567cd87bb61a7e1203c40e3876f1a221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#125;&#125;&#61;&#123;&#67;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#67;&#125;&#95;&#123;&#51;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"185\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>If a circuit contains a combination of capacitors in series and parallel, identify series and parallel parts, compute their capacitances, and then find the total.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3075419\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1656533\">\n<p id=\"import-auto-id1032431\">If you wish to store a large amount of energy in a capacitor bank, would you connect capacitors in series or parallel? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1596788\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1656785\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2791324\">\n<p id=\"eip-id2337212\">Find the total capacitance of the combination of capacitors in <a href=\"#import-auto-id1579779\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1579779\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1579780\" data-alt=\"A circuit is shown with three capacitors. Two capacitors, of ten microfarad and two point five microfarad capacitance, are in parallel to each other, and their combination is in series with a zero point three zero microfarad capacitor.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"A circuit is shown with three capacitors. Two capacitors, of ten microfarad and two point five microfarad capacitance, are in parallel to each other, and their combination is in series with a zero point three zero microfarad capacitor.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1175338\">\n<p id=\"eip-id1319497\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d2188b9f9a0ec3025d204061c9d96c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#50;&#57;&#51;&#32;&mu;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2912478\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2511599\">\n<p id=\"import-auto-id1999490\">Suppose you want a capacitor bank with a total capacitance of 0.750 F and you possess numerous 1.50 mF capacitors. What is the smallest number you could hook together to achieve your goal, and how would you connect them? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1290190\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2920194\">\n<p id=\"import-auto-id1964409\">What total capacitances can you make by connecting a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab66bd66fabc0663ff69de291005a72a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/> and an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16db034898cc013c4418ea67fd84b9a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> capacitor together? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1962000\">\n<p id=\"import-auto-id1663743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e8ca02b21ac94bfc7aab6d4d042879ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#56;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> in series combination, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-17ab41513d91849f2e4b1aaa34cbdd8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> in parallel combination<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2607784\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1998292\">\n<p id=\"import-auto-id2636632\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id1959375\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1959375\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1356269\" data-alt=\"The circuit includes three capacitors. A zero point three zero microfarad capacitor and a ten microfarad capacitor are connected in series, and together they are connected in parallel with a two point five microfarad capacitor.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"The circuit includes three capacitors. A zero point three zero microfarad capacitor and a ten microfarad capacitor are connected in series, and together they are connected in parallel with a two point five microfarad capacitor.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2749491\">\n<p id=\"import-auto-id1418699\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2f6bb4af0954b887530d2953cdc02ace_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#57;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2970272\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2970273\">\n<p id=\"import-auto-id2643460\">Find the total capacitance of the combination of capacitors shown in <a href=\"#import-auto-id2695900\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2695900\">\n<div class=\"bc-figcaption figcaption\">A combination of series and parallel connections of capacitors.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2695901\" data-alt=\"The figure shows a circuit that is a combination of series and parallel connections of capacitors. On the left of the circuit is a five point zero microfarad capacitor in series with a three point five microfarad capacitor. In the middle is an eight point zero microfarad capacitor. On the right, a zero point seven five microfarad capacitor is in parallel with a fifteen microfarad capacitor, and together they are in series with a one point five microfarad capacitor. Altogether, the system of capacitors on the left, the capacitor in the middle, and the system of capacitors on the right are connected in parallel.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a circuit that is a combination of series and parallel connections of capacitors. On the left of the circuit is a five point zero microfarad capacitor in series with a three point five microfarad capacitor. In the middle is an eight point zero microfarad capacitor. On the right, a zero point seven five microfarad capacitor is in parallel with a fifteen microfarad capacitor, and together they are in series with a one point five microfarad capacitor. Altogether, the system of capacitors on the left, the capacitor in the middle, and the system of capacitors on the right are connected in parallel.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2605419\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2129513\">\n<p><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1370920\">(a) An <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16db034898cc013c4418ea67fd84b9a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> capacitor is connected in parallel to another capacitor, producing a total capacitance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab66bd66fabc0663ff69de291005a72a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/>. What is the capacitance of the second capacitor? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2393147\">\n<p>(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4fe36fbd2ef8cec646969b311ebd2dd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"eip-852\"> (b) You cannot have a negative value of capacitance.\n<\/p>\n<p>(c) The assumption that the capacitors were hooked up in parallel, rather than in series, was incorrect. A parallel connection always produces a greater capacitance, while here a smaller capacitance was assumed. This could happen only if the capacitors are connected in series.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1072","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1030,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1072","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1072\/revisions"}],"predecessor-version":[{"id":1073,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1072\/revisions\/1073"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1030"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1072\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1072"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1072"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1072"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}