{"id":1064,"date":"2017-10-27T16:31:20","date_gmt":"2017-10-27T16:31:20","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/capacitors-and-dielectrics\/"},"modified":"2017-11-08T03:26:10","modified_gmt":"2017-11-08T03:26:10","slug":"capacitors-and-dielectrics","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/capacitors-and-dielectrics\/","title":{"raw":"Capacitors and Dielectrics","rendered":"Capacitors and Dielectrics"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the action of a capacitor and define capacitance.<\/li>\n<li>Explain parallel plate capacitors and their capacitances.<\/li>\n<li>Discuss the process of increasing the capacitance of a dielectric.<\/li>\n<li>Determine capacitance given charge and voltage.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2951554\">A <span data-type=\"term\" id=\"import-auto-id2768892\">capacitor<\/span> is a device used to store electric charge. Capacitors have applications ranging from filtering static out of radio reception to energy storage in heart defibrillators. Typically, commercial capacitors have two conducting parts close to one another, but not touching, such as those in <a href=\"#import-auto-id2800656\" class=\"autogenerated-content\">(Figure)<\/a>. (Most of the time an insulator is used between the two plates to provide separation\u2014see the discussion on dielectrics below.) When battery terminals are connected to an initially uncharged capacitor, equal amounts of positive and negative charge, [latex]+Q[\/latex] and [latex]\u2013Q[\/latex], are separated into its two plates. The capacitor remains neutral overall, but we refer to it as storing a charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> in this circumstance.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1935445\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitor<\/div>\n<p id=\"import-auto-id3152808\">A capacitor is a device used to store electric charge.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2800656\">\n<div class=\"bc-figcaption figcaption\">Both capacitors shown here were initially uncharged before being connected to a battery. They now have separated charges of [latex]+Q[\/latex] and [latex]\u2013Q[\/latex] on their two halves. (a) A parallel plate capacitor. (b) A rolled capacitor with an insulating material between its two conducting sheets.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2991146\" data-alt=\"Part a of the figure shows a charged parallel plate capacitor and part b of the figure shows a charged rolled capacitor. In the parallel plate capacitor, two rectangular plates are kept vertically facing each other separated by a distance d. These two plates are the conducting parts of the capacitor. One plate is connected to the positive terminal of the battery, and the other is connected to the negative terminal of the battery. One plate has a positive charge, plus Q, and the other plate has a negative charge, negative Q. The rolled capacitor has conducting parts in the form of a spiral coil. Between the two conducting parts is insulating material, also in the form of a coil. The conducting and insulating materials of the capacitor are rolled together to form a spiral. The outer conducting coil is connected to the positive terminal of the battery, and the inner coil is connected to the negative terminal of the battery.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a charged parallel plate capacitor and part b of the figure shows a charged rolled capacitor. In the parallel plate capacitor, two rectangular plates are kept vertically facing each other separated by a distance d. These two plates are the conducting parts of the capacitor. One plate is connected to the positive terminal of the battery, and the other is connected to the negative terminal of the battery. One plate has a positive charge, plus Q, and the other plate has a negative charge, negative Q. The rolled capacitor has conducting parts in the form of a spiral coil. Between the two conducting parts is insulating material, also in the form of a coil. The conducting and insulating materials of the capacitor are rolled together to form a spiral. The outer conducting coil is connected to the positive terminal of the battery, and the inner coil is connected to the negative terminal of the battery.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1610572\">The amount of charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> a <em data-effect=\"italics\"><em data-effect=\"italics\">capacitor<\/em><\/em> can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2749718\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Amount of Charge [latex]Q[\/latex]<em data-effect=\"italics\"> a Capacitor Can Store<\/em><\/div>\n<p id=\"import-auto-id2920312\">The amount of charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> a <em data-effect=\"italics\"><em data-effect=\"italics\">capacitor<\/em><\/em> can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/em><\/em><\/p>\n<\/div>\n<p id=\"import-auto-id2657069\">A system composed of two identical, parallel conducting plates separated by a distance, as in <a href=\"#import-auto-id2660537\" class=\"autogenerated-content\">(Figure)<\/a>, is called a <span data-type=\"term\">parallel plate capacitor<\/span>. It is easy to see the relationship between the voltage and the stored charge for a parallel plate capacitor, as shown in <a href=\"#import-auto-id2660537\" class=\"autogenerated-content\">(Figure)<\/a>. Each electric field line starts on an individual positive charge and ends on a negative one, so that there will be more field lines if there is more charge. (Drawing a single field line per charge is a convenience, only. We can draw many field lines for each charge, but the total number is proportional to the number of charges.) The electric field strength is, thus, directly proportional to [latex]Q[\/latex]. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2660537\">\n<div class=\"bc-figcaption figcaption\">Electric field lines in this parallel plate capacitor, as always, start on positive charges and end on negative charges. Since the electric field strength is proportional to the density of field lines, it is also proportional to the amount of charge on the capacitor. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2949833\" data-alt=\"Two metal plates are positioned vertically facing each other. The plates are the conducting parts of a capacitor. The plate on the left-hand side is connected to the positive terminal of a battery, and the plate on the right-hand side is connected to the negative terminal of the battery. There is an electric field between the two plates of the capacitor. The electric field lines emanate from the positively charged plate and end on the negatively charged plate. The electric field E is proportional to the charge Q.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two metal plates are positioned vertically facing each other. The plates are the conducting parts of a capacitor. The plate on the left-hand side is connected to the positive terminal of a battery, and the plate on the right-hand side is connected to the negative terminal of the battery. There is an electric field between the two plates of the capacitor. The electric field lines emanate from the positively charged plate and end on the negatively charged plate. The electric field E is proportional to the charge Q.\" width=\"150\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2985562\">The field is proportional to the charge:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E\\propto Q,[\/latex]<\/div>\n<p id=\"import-auto-id1516718\">where the symbol [latex]\\propto [\/latex] means \u201cproportional to.\u201d From the discussion in <a href=\"\/contents\/c793cc2b-c0ab-4089-a452-677700c3f9ac@3\">Electric Potential in a Uniform Electric Field<\/a>, we know that the voltage across parallel plates is [latex]V=\\text{Ed}[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]V\\propto E.[\/latex]<\/div>\n<p id=\"import-auto-id2730954\">It follows, then, that [latex]\\mathrm{V \\propto }\\phantom{\\rule{0.25em}{0ex}}Q[\/latex], and conversely,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]Q\\propto V.[\/latex]<\/div>\n<p id=\"import-auto-id1781864\">This is true in general: The greater the voltage applied to any capacitor, the greater the charge stored in it.<\/p>\n<p id=\"import-auto-id2636730\">Different capacitors will store different amounts of charge for the same applied voltage, depending on their physical characteristics. We define their <span data-type=\"term\" id=\"import-auto-id1271385\">capacitance<\/span> [latex]C[\/latex] to be such that the charge [latex]Q[\/latex] stored in a capacitor is proportional to [latex]C[\/latex]. The charge stored in a capacitor is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-730\">[latex]Q=\\text{CV}.[\/latex]<\/div>\n<p id=\"import-auto-id2785576\">This equation expresses the two major factors affecting the amount of charge stored. Those factors are the physical characteristics of the capacitor, [latex]C[\/latex], and the voltage, <em data-effect=\"italics\">[latex]V[\/latex]<\/em>. Rearranging the equation, we see that <em data-effect=\"italics\">capacitance [latex]C[\/latex]<em data-effect=\"italics\"> is the amount of charge stored per volt,<\/em><\/em> or<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]C=\\frac{Q}{V}.[\/latex]<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3089391\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitance<\/div>\n<p id=\"import-auto-id1663871\">Capacitance [latex]C[\/latex] is the amount of charge stored per volt, or<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]C=\\frac{Q}{V}.[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id2764584\">The unit of capacitance is the farad (F), named for Michael Faraday (1791\u20131867), an English scientist who contributed to the fields of electromagnetism and electrochemistry. Since capacitance is charge per unit voltage, we see that a farad is a coulomb per volt, or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-315\">[latex]\\text{1 F}=\\frac{\\text{1 C}}{\\text{1 V}}.[\/latex]<\/div>\n<p id=\"import-auto-id1297235\">A 1-farad capacitor would be able to store 1 coulomb (a very large amount of charge) with the application of only 1 volt. One farad is, thus, a very large capacitance. Typical capacitors range from fractions of a picofarad [latex]\\left(1 pF={\\text{10}}^{\\text{\u201312}}\\phantom{\\rule{0.25em}{0ex}}\\text{F}\\right)[\/latex] to millifarads [latex]\\left(1 mF={\\text{10}}^{\u20133}\\phantom{\\rule{0.25em}{0ex}}\\text{F}\\right)[\/latex].<\/p>\n<p id=\"import-auto-id1415714\"><a href=\"#import-auto-id2512674\" class=\"autogenerated-content\">(Figure)<\/a> shows some common capacitors. Capacitors are primarily made of ceramic, glass, or plastic, depending upon purpose and size. Insulating materials, called dielectrics, are commonly used in their construction, as discussed below.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2512674\">\n<div class=\"bc-figcaption figcaption\">Some typical capacitors. Size and value of capacitance are not necessarily related. (credit: Windell Oskay)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1579209\" data-alt=\"There are various types of capacitors with varying shapes and color. Some are cylindrical in shape, some circular in shape, some rectangular in shape, with two strands of wire coming out of each.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_03a.jpg\" data-media-type=\"image\/png\" alt=\"There are various types of capacitors with varying shapes and color. Some are cylindrical in shape, some circular in shape, some rectangular in shape, with two strands of wire coming out of each.\" width=\"250\"><\/span><\/p><\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2550561\">\n<h1 data-type=\"title\">Parallel Plate Capacitor<\/h1>\n<p id=\"import-auto-id3138915\">The parallel plate capacitor shown in <a href=\"#import-auto-id1676212\" class=\"autogenerated-content\">(Figure)<\/a> has two identical conducting plates, each having a surface area [latex]A[\/latex], separated by a distance [latex]d[\/latex] (with no material between the plates). When a voltage [latex]V[\/latex] is applied to the capacitor, it stores a charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\">, as shown. We can see how its capacitance depends on [latex]A[\/latex] and [latex]d[\/latex] by considering the characteristics of the Coulomb force. We know that like charges repel, unlike charges attract, and the force between charges decreases with distance. So it seems quite reasonable that the bigger the plates are, the more charge they can store\u2014because the charges can spread out more. Thus [latex]C[\/latex] should be greater for larger [latex]A[\/latex]. Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. So [latex]C[\/latex] should be greater for smaller [latex]d[\/latex].<\/em><\/em><\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">Parallel plate capacitor with plates separated by a distance [latex]d[\/latex]. Each plate has an area [latex]A[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id977536\" data-alt=\"Two parallel plates are placed facing each other. The area of each plate is A, and the distance between the plates is d. The plate on the left is connected to the positive terminal of the battery, and the plate on the right is connected to the negative terminal of the battery.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_04a.jpg\" data-media-type=\"image\/png\" alt=\"Two parallel plates are placed facing each other. The area of each plate is A, and the distance between the plates is d. The plate on the left is connected to the positive terminal of the battery, and the plate on the right is connected to the negative terminal of the battery.\" width=\"150\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2573871\">It can be shown that for a parallel plate capacitor there are only two factors ([latex]A[\/latex] and [latex]d[\/latex]) that affect its capacitance [latex]C[\/latex]. The capacitance of a parallel plate capacitor in equation form is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]C={\\epsilon }_{0}\\frac{A}{d}.[\/latex]<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2747111\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitance of a Parallel Plate Capacitor<\/div>\n<div data-type=\"equation\" class=\"equation\">[latex]C={\\epsilon }_{0}\\frac{A}{d}[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id1552736\">[latex]A[\/latex]<br>\nis the area of one plate in square meters, and<br>\n[latex]d[\/latex]<br>\nis the distance between the plates in meters. The constant<br>\n[latex]{\\epsilon }_{0}[\/latex]<br>\nis the permittivity of free space; its numerical value in SI units is<br>\n[latex]{\\epsilon }_{0}=8.85\u00d7{\\text{10}}^{\u2013\\text{12}}\\phantom{\\rule{0.25em}{0ex}}\\text{F\/m}[\/latex].<br>\nThe units of F\/m are equivalent to [latex]{C}^{2}\\text{\/N}\u00b7{\\text{m}}^{2}[\/latex]. The small numerical value of [latex]{\\epsilon }_{0}[\/latex] is related to the large size of the farad. A parallel plate capacitor must have a large area to have a capacitance approaching a farad. (Note that the above equation is valid when the parallel plates are separated by air or free space. When another material is placed between the plates, the equation is modified, as discussed below.)   <\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id920232\">\n<div data-type=\"title\" class=\"title\">Capacitance and Charge Stored in a Parallel Plate Capacitor<\/div>\n<p id=\"import-auto-id2657240\">(a) What is the capacitance of a parallel plate capacitor with metal plates, each of area  [latex]\\text{1.00}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}[\/latex], separated by 1.00 mm? (b) What charge is stored in this capacitor if a voltage of [latex]3.00\u00d7{\\text{10}}^{\\text{3}}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex] is applied to it?<\/p>\n<p id=\"import-auto-id2560583\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2927434\">Finding the capacitance [latex]C[\/latex] is a straightforward application of the equation [latex]C={\\epsilon }_{0}A\/d[\/latex]. Once [latex]C[\/latex] is found, the charge stored can be found using the equation [latex]Q=\\text{CV}[\/latex].<\/p>\n<p id=\"import-auto-id1951686\"><strong>Solution for (a)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2657718\">Entering the given values into the equation for the capacitance of a parallel plate capacitor yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-226\">[latex]\\begin{array}{lll}C&amp; =&amp; {\\epsilon }_{0}\\frac{A}{d}=\\left(8.85\u00d7{\\text{10}}^{\\text{\u201312}}\\frac{F}{m}\\right)\\phantom{\\rule{0.10em}{0ex}}\\frac{1.00\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}}{1.00\u00d7{\\text{10}}^{\u20133}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}\\\\ &amp; =&amp; 8.85\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}\\text{F}=8.85 nF.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1562997\"><strong>Discussion for (a)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2960580\">This small value for the capacitance indicates how difficult it is to make a device with a large capacitance. Special techniques help, such as using very large area thin foils placed close together.<\/p>\n<p id=\"import-auto-id2507307\"><strong>Solution for (b)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1357066\">The charge stored in any capacitor is given by the equation [latex]Q=\\text{CV}[\/latex]. Entering the known values into this equation gives <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-754\">[latex]\\begin{array}{lll}Q&amp; =&amp; \\text{CV}=\\left(8.85\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}\\text{F}\\right)\\left(3.00\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{V}\\right)\\\\ &amp; =&amp; \\text{26.6 \u03bcC.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2560950\"><strong>Discussion for (b)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1676990\">This charge is only slightly greater than those found in typical static electricity. Since air breaks down at about [latex]3\\text{.}\\text{00}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex], more charge cannot be stored on this capacitor by increasing the voltage.<\/p>\n<\/div>\n<p id=\"eip-951\">Another interesting biological example dealing with electric potential is found in the cell\u2019s plasma membrane. The membrane sets a cell off from its surroundings and also allows ions to selectively pass in and out of the cell. There is a potential difference across the membrane of about [latex]\\text{\u201370 mV}[\/latex]. This is due to the mainly negatively charged ions in the cell and the predominance of positively charged sodium ([latex]{\\text{Na}}^{\\text{+}}[\/latex]) ions outside. Things change when a nerve cell is stimulated. [latex]{\\text{Na}}^{\\text{+}}[\/latex] ions are allowed to pass through the membrane into the cell, producing a positive membrane potential\u2014the nerve signal. The cell membrane is about 7 to 10 nm thick. An approximate value of the electric field across it is given by\n<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E=\\frac{V}{d}=\\frac{\\text{\u201370}\u00d7{\\text{10}}^{\u20133}\\phantom{\\rule{0.25em}{0ex}}V}{8\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}m}=\u20139\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}.[\/latex]<\/div>\n<p>This electric field is enough to cause a breakdown in air.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2702570\">\n<h1 data-type=\"title\">Dielectric<\/h1>\n<p id=\"import-auto-id1318992\">The previous example highlights the difficulty of storing a large amount of charge in capacitors. If [latex]d[\/latex] is made smaller to produce a larger capacitance, then the maximum voltage must be reduced proportionally to avoid breakdown (since [latex]E=V\/d[\/latex]). An important solution to this difficulty is to put an insulating material, called a <span data-type=\"term\">dielectric<\/span>, between the plates of a capacitor and allow [latex]d[\/latex]<br>\n     to be as small as possible. Not only does the smaller [latex]d[\/latex] make the capacitance greater, but many insulators can withstand greater electric fields than air before breaking down.<\/p>\n<p id=\"import-auto-id2687767\">There is another benefit to using a dielectric in a capacitor. Depending on the material used, the capacitance is greater than that given by the equation [latex]C={\\epsilon }_{0}\\frac{A}{d}[\/latex] by a factor [latex]\\kappa [\/latex], called the <em data-effect=\"italics\">dielectric constant<\/em>. A parallel plate capacitor with a dielectric between its plates has a capacitance given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]C={\\text{\u03ba\u03b5}}_{0}\\phantom{\\rule{0.10em}{0ex}}\\frac{A}{d}\\phantom{\\rule{0.25em}{0ex}}\\text{(parallel plate capacitor with dielectric)}.[\/latex]<\/div>\n<p id=\"import-auto-id2690389\">Values of the dielectric constant [latex]\\kappa [\/latex] for various materials are given in <a href=\"#import-auto-id2726630\" class=\"autogenerated-content\">(Figure)<\/a>. Note that [latex]\\kappa [\/latex] for vacuum is exactly 1, and so the above equation is valid in that case, too. If a dielectric is used, perhaps by placing Teflon between the plates of the capacitor in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>, then the capacitance is greater by the factor [latex]\\kappa [\/latex], which for Teflon is 2.1.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2502208\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: Building a Capacitor<\/div>\n<p id=\"import-auto-id2806676\">How large a capacitor can you make using a chewing gum wrapper? The plates will be the aluminum foil, and the separation (dielectric) in between will be the paper. <\/p>\n<\/div>\n<table id=\"import-auto-id2726630\" summary=\"The table gives the value of dielectric constants and dielectric strengths for various materials at a temperature of twenty degrees Celsius\">\n<caption><span data-type=\"title\">Dielectric Constants and Dielectric Strengths for Various Materials at 20\u00baC<\/span><\/caption>\n<thead>\n<tr>\n<th>Material<\/th>\n<th>Dielectric constant [latex]\\kappa [\/latex]<\/th>\n<th>Dielectric strength (V\/m)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Vacuum<\/td>\n<td>1.00000<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<tr>\n<td>Air<\/td>\n<td>1.00059<\/td>\n<td>[latex]3\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Bakelite<\/td>\n<td>4.9<\/td>\n<td>[latex]\\text{24}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Fused quartz<\/td>\n<td>3.78<\/td>\n<td>[latex]8\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Neoprene rubber<\/td>\n<td>6.7<\/td>\n<td>[latex]\\text{12}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td>3.4<\/td>\n<td>[latex]\\text{14}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Paper<\/td>\n<td>3.7<\/td>\n<td>[latex]\\text{16}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Polystyrene<\/td>\n<td>2.56<\/td>\n<td>[latex]\\text{24}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Pyrex glass<\/td>\n<td>5.6<\/td>\n<td>[latex]\\text{14}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Silicon oil<\/td>\n<td>2.5<\/td>\n<td>[latex]\\text{15}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Strontium titanate<\/td>\n<td>233<\/td>\n<td>[latex]8\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Teflon<\/td>\n<td>2.1<\/td>\n<td>[latex]\\text{60}\u00d7{\\text{10}}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Water<\/td>\n<td>80<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id2576127\">Note also that the dielectric constant for air is very close to 1, so that air-filled capacitors act much like those with vacuum between their plates <em data-effect=\"italics\"><em data-effect=\"italics\">except<\/em><\/em> that the air can become conductive if the electric field strength  becomes too great. (Recall that [latex]E=V\/d[\/latex] for a parallel plate capacitor.) Also shown in <a href=\"#import-auto-id2726630\" class=\"autogenerated-content\">(Figure)<\/a> are maximum electric field strengths in V\/m, called <span data-type=\"term\" id=\"import-auto-id1726758\">dielectric strengths<\/span>, for several materials. These are the fields above which the material begins to break down and conduct. The dielectric strength imposes a limit on the voltage that can be applied for a given plate separation. For instance, in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>, the separation is 1.00 mm, and so the voltage limit for air is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-867\">[latex]\\begin{array}{lll}V&amp; =&amp; E\\cdot d\\\\ &amp; =&amp; \\left(3\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}\\right)\\left(1\\text{.}\\text{00}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)\\\\ &amp; =&amp; \\text{3000 V.}\\end{array}[\/latex]<\/div>\n<p id=\"eip-146\">However, the limit for a 1.00 mm separation filled with Teflon is 60,000 V, since the dielectric strength of Teflon is [latex]\\text{60}\u00d7{\\text{10}}^{6}[\/latex] V\/m. So the same capacitor filled with Teflon has a greater capacitance and can be subjected to a much greater voltage. Using the capacitance we calculated in the above example for the air-filled parallel plate capacitor, we find that the Teflon-filled capacitor can store a maximum charge of  <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}Q&amp; =&amp; \\mathit{\\text{CV}}\\\\ &amp; =&amp; {\\mathit{\\kappa C}}_{\\text{air}}V\\\\ &amp; =&amp; \\left(2.1\\right)\\left(8.85 nF\\right)\\left(6.0\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{V}\\right)\\\\ &amp; =&amp; 1.1 mC.\\end{array}[\/latex]<\/div>\n<p>This is 42 times the charge of the same air-filled capacitor.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1579779\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Dielectric Strength<\/div>\n<p id=\"import-auto-id2630262\">The maximum electric field strength above which an insulating material begins to break down and conduct is called its dielectric strength.<\/p>\n<\/div>\n<p id=\"import-auto-id2600468\">Microscopically, how does a dielectric increase capacitance? Polarization of the insulator is responsible. The more easily it is polarized, the greater its dielectric constant [latex]\\kappa [\/latex]. Water, for example, is a <span data-type=\"term\">polar molecule<\/span> because one end of the molecule has a slight positive charge and the other end has a slight negative charge. The polarity of water causes it to have a relatively large dielectric constant of 80. The effect of polarization can be best explained in terms of the characteristics of the Coulomb force. <a href=\"#import-auto-id1295090\" class=\"autogenerated-content\">(Figure)<\/a> shows the separation of charge schematically in the molecules of a dielectric material placed between the charged plates of a capacitor. The Coulomb force between the closest ends of the molecules and the charge on the plates is attractive and very strong, since they are very close together. This attracts more charge onto the plates than if the space were empty and the opposite charges were a distance [latex]d[\/latex] away.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1295090\">\n<div class=\"bc-figcaption figcaption\">(a) The molecules in the insulating material between the plates of a capacitor are polarized by the charged plates. This produces a layer of opposite charge on the surface of the dielectric that attracts more charge onto the plate, increasing its capacitance. (b) The dielectric reduces the electric field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge. The capacitor stores the same charge for a smaller voltage, implying that it has a larger capacitance because of the dielectric. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1020159\" data-alt=\"(a) A dielectric is between the two plates of a parallel plate capacitor. A diagram shows the molecules that make up the dielectric. The molecules are polarized by the charged plates. The positive ends of the molecules are attracted toward the negatively charged plate of the capacitor and hence are oriented toward the right. The negative ends of the molecules are attracted toward the positively charged plate of the capacitor and hence are oriented toward the left. (b) There is a dielectric material between the two plates of the capacitor. Since the charged ends of the molecules are oriented toward the capacitor plates, there is reduced field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"(a) A dielectric is between the two plates of a parallel plate capacitor. A diagram shows the molecules that make up the dielectric. The molecules are polarized by the charged plates. The positive ends of the molecules are attracted toward the negatively charged plate of the capacitor and hence are oriented toward the right. The negative ends of the molecules are attracted toward the positively charged plate of the capacitor and hence are oriented toward the left. (b) There is a dielectric material between the two plates of the capacitor. Since the charged ends of the molecules are oriented toward the capacitor plates, there is reduced field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3141379\">Another way to understand how a dielectric increases capacitance is to consider its effect on the electric field inside the capacitor. <a href=\"#import-auto-id1295090\" class=\"autogenerated-content\">(Figure)<\/a>(b) shows the electric field lines with a dielectric in place. Since the field lines end on charges in the dielectric, there are fewer of them going from one side of the capacitor to the other. So the electric field strength is less than if there were a vacuum between the plates, even though the same charge is on the plates. The voltage between the plates is [latex]V=\\text{Ed}[\/latex], so it too is reduced by the dielectric. Thus there is a smaller voltage [latex]V[\/latex] for the same charge [latex]Q[\/latex]; since [latex]C=Q\/V[\/latex], the capacitance [latex]C[\/latex] is greater.<\/p>\n<p id=\"import-auto-id2991565\">The dielectric constant is generally defined to be [latex]\\kappa ={E}_{0}\/E[\/latex], or the ratio of the electric field in a vacuum to that in the dielectric material, and is intimately related to the polarizability of the material.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1611688\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Things Great and Small<\/div>\n<p id=\"import-auto-id1248838\"><strong>The Submicroscopic Origin of Polarization<\/strong><\/p>\n<p id=\"import-auto-id2651892\">Polarization is a separation of charge within an atom or molecule. As has been noted, the planetary model of the atom pictures it as having a positive nucleus orbited by negative electrons, analogous to the planets orbiting the Sun. Although this model is not completely accurate, it is very helpful in explaining a vast range of phenomena and will be refined elsewhere, such as in <a href=\"\/contents\/05208266-2374-4008-9060-7ddddbe877f2@2\">Atomic Physics<\/a>. The submicroscopic origin of polarization can be modeled as shown in <a href=\"#import-auto-id3164371\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3164371\">\n<div class=\"bc-figcaption figcaption\">Artist\u2019s conception of a polarized atom. The orbits of electrons around the nucleus are shifted slightly by the external charges (shown exaggerated). The resulting separation of charge within the atom means that it is polarized. Note that the unlike charge is now closer to the external charges, causing the polarization.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1949934\" data-alt=\"The top part of the figure shows what an unpolarized atom would look like if the electrons moved along a circular path around the positively charged nucleus. Next, when there is an external negative and a positive charge, the electrons are attracted toward the positive external charge and the nucleus is attracted toward the negative external charge. The circular orbit of the electrons becomes an ellipse due to the pull of the external charges.\"><img src=\"\/resources\/81bf0e9d5c44caafa6c98279c46f2e4b454092a2\/Figure_20_05_06(a)a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The top part of the figure shows what an unpolarized atom would look like if the electrons moved along a circular path around the positively charged nucleus. Next, when there is an external negative and a positive charge, the electrons are attracted toward the positive external charge and the nucleus is attracted toward the negative external charge. The circular orbit of the electrons becomes an ellipse due to the pull of the external charges.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2609039\">We will find in <a href=\"\/contents\/05208266-2374-4008-9060-7ddddbe877f2@2\">Atomic Physics<\/a> that the orbits of electrons are more properly viewed as electron clouds with the density of the cloud related to the probability of finding an electron in that location (as opposed to the definite locations and paths of planets in their orbits around the Sun). This cloud is shifted by the Coulomb force so that the atom on average has a separation of charge. Although the atom remains neutral, it can now be the source of a Coulomb force, since a charge brought near the atom will be closer to one type of charge than the other.<\/p>\n<p id=\"import-auto-id1563695\">Some molecules, such as those of water, have an inherent separation of charge and are thus called polar molecules. <a href=\"#import-auto-id2937921\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the separation of charge in a water molecule, which has two hydrogen atoms and one oxygen atom [latex]\\left({\\text{H}}_{2}\\text{O}\\right)[\/latex]. The water molecule is not symmetric\u2014the hydrogen atoms are repelled to one side, giving the molecule a boomerang shape. The electrons in a water molecule are more concentrated around the more highly charged oxygen nucleus than around the hydrogen nuclei. This makes the oxygen end of the molecule slightly negative and leaves the hydrogen ends slightly positive. The inherent separation of charge in polar molecules makes it easier to align them with external fields and charges. Polar molecules therefore exhibit greater polarization effects and have greater dielectric constants. Those who study chemistry will find that the polar nature of water has many effects. For example, water molecules gather ions much more effectively because they have an electric field and a separation of charge to attract charges of both signs. Also, as brought out in the previous chapter, polar water provides a shield or screening of the electric fields in the highly charged molecules of interest in biological systems.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2937921\">\n<div class=\"bc-figcaption figcaption\">Artist\u2019s conception of a water molecule. There is an inherent separation of charge, and so water is a polar molecule. Electrons in the molecule are attracted to the oxygen nucleus and leave an excess of positive charge near the two hydrogen nuclei. (Note that the schematic on the right is a rough illustration of the distribution of electrons in the water molecule. It does not show the actual numbers of protons and electrons involved in the structure.)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2648318\" data-alt=\"The two hydrogen atoms in the water molecule subtend an angle of one hundred and four point five degrees with oxygen at the center. This is a schematic arrangement of hydrogen and oxygen atoms in the water molecule. The molecule is polarized, with the electrons attracted more to the nucleus of the oxygen atom than toward the nuclei of the hydrogen atoms.\"><img src=\"\/resources\/5f84ada0aafde2cce3e212d001ddc68e7218e9ae\/Figure_20_05_06(b)a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The two hydrogen atoms in the water molecule subtend an angle of one hundred and four point five degrees with oxygen at the center. This is a schematic arrangement of hydrogen and oxygen atoms in the water molecule. The molecule is polarized, with the electrons attracted more to the nucleus of the oxygen atom than toward the nuclei of the hydrogen atoms.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Capacitor Lab<\/div>\n<p>Explore how a capacitor works! Change the size of the plates and add a dielectric to see the effect on capacitance. Change the voltage and see charges built up on the plates. Observe the electric field in the capacitor. Measure the voltage and the electric field.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1545484\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d8b6b46ce76e32397246a5a9f39985177666c958\/capacitor-lab_en.jar\">Capacitor Lab<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_20.5\" data-alt=\"\"><a href=\"\/resources\/d8b6b46ce76e32397246a5a9f39985177666c958\/capacitor-lab_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1309916\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1310256\">\n<li id=\"import-auto-id2607317\">A <strong data-effect=\"bold\">capacitor <strong data-effect=\"bold\">is a device used to store charge.<\/strong><\/strong><\/li>\n<li id=\"import-auto-id1737462\">The amount of charge [latex]Q[\/latex] a capacitor can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/li>\n<li id=\"import-auto-id2719243\">The capacitance [latex]C[\/latex] is the amount of charge stored per volt<em data-effect=\"italics\">,<\/em> or\n<div data-type=\"equation\" class=\"equation\" id=\"eip-791\">[latex]C=\\frac{Q}{V}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2838015\">The capacitance of a parallel plate capacitor is [latex]C={\\epsilon }_{0}\\phantom{\\rule{0.15em}{0ex}}\\frac{A}{d}[\/latex], when the plates are separated by air or free space. [latex]{\\epsilon }_{\\text{0}}\\phantom{\\rule{0.25em}{0ex}}[\/latex] is called the permittivity of free space.<\/li>\n<li>A parallel plate capacitor with a dielectric between its plates has a capacitance given by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2286944\">[latex]C={\\mathrm{\\kappa \\epsilon }}_{0}\\phantom{\\rule{0.25em}{0ex}}\\frac{A}{d},[\/latex]<\/div>\n<p>where [latex]\\kappa [\/latex] is the dielectric constant of the material.<\/p><\/li>\n<li id=\"import-auto-id3138892\">The maximum electric field strength above which an insulating material begins to break down and conduct is called dielectric strength.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2938102\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1964776\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1347748\">\n<p id=\"import-auto-id2635497\">Does the capacitance of a device depend on the applied voltage? What about the charge stored in it? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1511858\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2805847\">\n<p id=\"import-auto-id3096036\">Use the characteristics of the Coulomb force to explain why capacitance should be proportional to the plate area of a capacitor. Similarly, explain why capacitance should be inversely proportional to the separation between plates. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2753579\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1999045\">\n<p id=\"import-auto-id1611133\">Give the reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. What is the independent reason that a dielectric material also allows a greater voltage to be applied to a capacitor? (The dielectric thus increases [latex]C[\/latex] and permits a greater [latex]V[\/latex].) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3361548\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1324707\">\n<p id=\"import-auto-id2703282\">How does the polar character of water molecules help to explain water\u2019s relatively large dielectric constant? (<a href=\"#import-auto-id2937921\" class=\"autogenerated-content\">(Figure)<\/a>)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3139942\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1330607\">\n<p id=\"import-auto-id3100275\">Sparks will occur between the plates of an air-filled capacitor at lower voltage when the air is humid than when dry. Explain why, considering the polar character of water molecules. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1311685\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3066669\">\n<p id=\"import-auto-id1628324\">Water has a large dielectric constant, but it is rarely used in capacitors. Explain why. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2601615\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1950596\">\n<p id=\"import-auto-id3152613\">Membranes in living cells, including those in humans, are characterized by a separation of charge across the membrane. Effectively, the membranes are thus charged capacitors with important functions related to the potential difference across the membrane. Is energy required to separate these charges in living membranes and, if so, is its source the metabolization of food energy or some other source? <\/p>\n<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1638125\">\n<div class=\"bc-figcaption figcaption\">The semipermeable membrane of a cell has different concentrations of ions inside and out. Diffusion moves the<br>\n[latex]{\\text{K}}^{\\text{+}}[\/latex] (potassium) and<br>\n[latex]{\\text{Cl}}^{\\text{\u2013}}[\/latex] (chloride) ions in the directions shown, until the Coulomb force halts further transfer. This results in a layer of positive charge on the outside, a layer of negative charge on the inside, and thus a voltage across the cell membrane. The membrane is normally impermeable to [latex]{\\text{Na}}^{\\text{+}}[\/latex]  (sodium ions).<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3094558\" data-alt=\"The semipermeable membrane of a cell is shown, with different concentrations of potassium cations, sodium cations, and chloride anions inside and outside the cell. The ions are represented by small, colored circles. In its resting state, the cell membrane is permeable to potassium and chloride ions, but it is impermeable to sodium ions. By diffusion, potassium cations travel out of the cell, going through the cell membrane and forming a layer of positive charge on the outer surface of the membrane. By diffusion, chloride anions go into the cell, going through the cell membrane and forming a layer of negative charge on the inner surface of the membrane. As a result, a voltage is set up across the cell membrane. The Coulomb force prevents all the ions from crossing the membrane.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_07a.jpg\" data-media-type=\"image\/jpg\" alt=\"The semipermeable membrane of a cell is shown, with different concentrations of potassium cations, sodium cations, and chloride anions inside and outside the cell. The ions are represented by small, colored circles. In its resting state, the cell membrane is permeable to potassium and chloride ions, but it is impermeable to sodium ions. By diffusion, potassium cations travel out of the cell, going through the cell membrane and forming a layer of positive charge on the outer surface of the membrane. By diffusion, chloride anions go into the cell, going through the cell membrane and forming a layer of negative charge on the inner surface of the membrane. As a result, a voltage is set up across the cell membrane. The Coulomb force prevents all the ions from crossing the membrane.\" width=\"325\"><\/span><\/p><\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2608941\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1338876\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1942516\">\n<p id=\"import-auto-id1501248\">What charge is stored in a [latex]\\text{180 \u00b5F}\\phantom{\\rule{0.25em}{0ex}}[\/latex] capacitor when 120 V is applied to it?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3147194\">\n<p id=\"import-auto-id2644327\">[latex]\\text{21}\\text{.}6\\phantom{\\rule{0.25em}{0ex}}\\text{mC}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2586206\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2953152\">\n<p id=\"import-auto-id1086670\">Find the charge stored when 5.50 V is applied to an 8.00 pF capacitor.<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2511345\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1352226\">\n<p id=\"import-auto-id2629428\">What charge is stored in the capacitor in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2727877\">\n<p id=\"import-auto-id2595981\">[latex]\\text{80}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{mC}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2677540\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2962045\">\n<p id=\"import-auto-id2782884\">Calculate the voltage applied to a [latex]2\\text{.}\\text{00 \u00b5F}[\/latex] capacitor when it holds [latex]3\\text{.}\\text{10 \u00b5C}[\/latex] of charge.<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1326374\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1326375\">\n<p id=\"import-auto-id1563284\">What voltage must be applied to an 8.00 nF capacitor to store 0.160 mC of charge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2951082\">\n<p id=\"eip-id3585054\">20.0 kV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1603035\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1603036\">\n<p id=\"import-auto-id2584784\">What capacitance is needed to store [latex]3\\text{.}\\text{00 \u00b5C}[\/latex] of charge at a voltage of 120 V?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2756080\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2756082\">\n<p id=\"import-auto-id2560911\">What is the capacitance of a large Van de Graaff generator\u2019s terminal, given that it stores 8.00 mC of charge at a voltage of 12.0 MV?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2004243\">\n<p id=\"import-auto-id2905346\">[latex]\\text{667}\\phantom{\\rule{0.25em}{0ex}}\\text{pF}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2678325\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2537079\">\n<p id=\"import-auto-id1611526\">Find the capacitance of a parallel plate capacitor having plates of area [latex]5\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}[\/latex] that are separated by 0.100 mm of Teflon.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1949350\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1949351\">\n<p id=\"import-auto-id1552021\">(a)What is the capacitance of a parallel plate capacitor having plates of area [latex]{\\text{1.50 m}}^{2}[\/latex] that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3155343\">\n<p id=\"import-auto-id2602024\">(a) [latex]4\\text{.}\\text{4 \u00b5F}[\/latex]<\/p>\n<p id=\"import-auto-id2913687\">(b) [latex]4\\text{.}0\u00d7{\\text{10}}^{\u20135}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1674884\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1674885\">\n<p id=\"eip-id2966187\"><strong>Integrated Concepts<\/strong><\/p>\n<p id=\"import-auto-id2502389\">A prankster applies 450 V to an [latex]\\text{80}\\text{.}0 \u00b5F[\/latex] capacitor and then tosses it to an unsuspecting victim. The victim\u2019s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2723762\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2656834\">\n<p id=\"eip-id1799265\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id2508182\">(a) A certain parallel plate capacitor has plates of area  [latex]{\\text{4.00 m}}^{2}[\/latex], separated by 0.0100 mm of nylon, and stores 0.170 C of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2904372\">\n<p id=\"eip-id2420270\">(a) 14.2 kV<\/p>\n<p id=\"eip-id2420273\">(b) The voltage is unreasonably large, more than 100 times the breakdown voltage of nylon.<\/p>\n<p id=\"eip-id2189005\">(c) The assumed charge is unreasonably large and cannot be stored in a capacitor of these dimensions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1677156\">\n<dt>capacitor<\/dt>\n<dd id=\"fs-id1537463\">a device that stores electric charge<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1166661\">\n<dt>capacitance<\/dt>\n<dd id=\"fs-id2756324\">amount of charge stored per unit volt<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1359494\">\n<dt>dielectric<\/dt>\n<dd id=\"fs-id3084506\">an insulating material<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3256092\">\n<dt>dielectric strength<\/dt>\n<dd id=\"fs-id2719268\">the maximum electric field above which an insulating material begins to break down and conduct<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2791420\">\n<dt>parallel plate capacitor<\/dt>\n<dd id=\"fs-id2595194\">two identical conducting plates separated by a distance<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2753374\">\n<dt>polar molecule<\/dt>\n<dd id=\"fs-id2543637\">a molecule with inherent separation of charge<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the action of a capacitor and define capacitance.<\/li>\n<li>Explain parallel plate capacitors and their capacitances.<\/li>\n<li>Discuss the process of increasing the capacitance of a dielectric.<\/li>\n<li>Determine capacitance given charge and voltage.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id2951554\">A <span data-type=\"term\" id=\"import-auto-id2768892\">capacitor<\/span> is a device used to store electric charge. Capacitors have applications ranging from filtering static out of radio reception to energy storage in heart defibrillators. Typically, commercial capacitors have two conducting parts close to one another, but not touching, such as those in <a href=\"#import-auto-id2800656\" class=\"autogenerated-content\">(Figure)<\/a>. (Most of the time an insulator is used between the two plates to provide separation\u2014see the discussion on dielectrics below.) When battery terminals are connected to an initially uncharged capacitor, equal amounts of positive and negative charge, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f69758314b2aa16c868e578558065b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"28\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-97456a5dbfe7d9d7970543a67061a84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: -4px;\" \/>, are separated into its two plates. The capacitor remains neutral overall, but we refer to it as storing a 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\"> in this circumstance.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1935445\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitor<\/div>\n<p id=\"import-auto-id3152808\">A capacitor is a device used to store electric charge.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2800656\">\n<div class=\"bc-figcaption figcaption\">Both capacitors shown here were initially uncharged before being connected to a battery. They now have separated charges of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f69758314b2aa16c868e578558065b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#43;&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"28\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-97456a5dbfe7d9d7970543a67061a84b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: -4px;\" \/> on their two halves. (a) A parallel plate capacitor. (b) A rolled capacitor with an insulating material between its two conducting sheets.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2991146\" data-alt=\"Part a of the figure shows a charged parallel plate capacitor and part b of the figure shows a charged rolled capacitor. In the parallel plate capacitor, two rectangular plates are kept vertically facing each other separated by a distance d. These two plates are the conducting parts of the capacitor. One plate is connected to the positive terminal of the battery, and the other is connected to the negative terminal of the battery. One plate has a positive charge, plus Q, and the other plate has a negative charge, negative Q. The rolled capacitor has conducting parts in the form of a spiral coil. Between the two conducting parts is insulating material, also in the form of a coil. The conducting and insulating materials of the capacitor are rolled together to form a spiral. The outer conducting coil is connected to the positive terminal of the battery, and the inner coil is connected to the negative terminal of the battery.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a charged parallel plate capacitor and part b of the figure shows a charged rolled capacitor. In the parallel plate capacitor, two rectangular plates are kept vertically facing each other separated by a distance d. These two plates are the conducting parts of the capacitor. One plate is connected to the positive terminal of the battery, and the other is connected to the negative terminal of the battery. One plate has a positive charge, plus Q, and the other plate has a negative charge, negative Q. The rolled capacitor has conducting parts in the form of a spiral coil. Between the two conducting parts is insulating material, also in the form of a coil. The conducting and insulating materials of the capacitor are rolled together to form a spiral. The outer conducting coil is connected to the positive terminal of the battery, and the inner coil is connected to the negative terminal of the battery.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1610572\">The amount of 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\"> a <em data-effect=\"italics\"><em data-effect=\"italics\">capacitor<\/em><\/em> can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2749718\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">The Amount of Charge <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\"> a Capacitor Can Store<\/em><\/div>\n<p id=\"import-auto-id2920312\">The amount of 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\"> a <em data-effect=\"italics\"><em data-effect=\"italics\">capacitor<\/em><\/em> can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/em><\/em><\/p>\n<\/div>\n<p id=\"import-auto-id2657069\">A system composed of two identical, parallel conducting plates separated by a distance, as in <a href=\"#import-auto-id2660537\" class=\"autogenerated-content\">(Figure)<\/a>, is called a <span data-type=\"term\">parallel plate capacitor<\/span>. It is easy to see the relationship between the voltage and the stored charge for a parallel plate capacitor, as shown in <a href=\"#import-auto-id2660537\" class=\"autogenerated-content\">(Figure)<\/a>. Each electric field line starts on an individual positive charge and ends on a negative one, so that there will be more field lines if there is more charge. (Drawing a single field line per charge is a convenience, only. We can draw many field lines for each charge, but the total number is proportional to the number of charges.) The electric field strength is, thus, directly proportional to <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;\" \/>. <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2660537\">\n<div class=\"bc-figcaption figcaption\">Electric field lines in this parallel plate capacitor, as always, start on positive charges and end on negative charges. Since the electric field strength is proportional to the density of field lines, it is also proportional to the amount of charge on the capacitor. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2949833\" data-alt=\"Two metal plates are positioned vertically facing each other. The plates are the conducting parts of a capacitor. The plate on the left-hand side is connected to the positive terminal of a battery, and the plate on the right-hand side is connected to the negative terminal of the battery. There is an electric field between the two plates of the capacitor. The electric field lines emanate from the positively charged plate and end on the negatively charged plate. The electric field E is proportional to the charge Q.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Two metal plates are positioned vertically facing each other. The plates are the conducting parts of a capacitor. The plate on the left-hand side is connected to the positive terminal of a battery, and the plate on the right-hand side is connected to the negative terminal of the battery. There is an electric field between the two plates of the capacitor. The electric field lines emanate from the positively charged plate and end on the negatively charged plate. The electric field E is proportional to the charge Q.\" width=\"150\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2985562\">The field is proportional to the charge:<\/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-f6dfbccdd61ae81475c6a5b4fe69b47e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;&#81;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1516718\">where the symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-118c402462b54d8bd7e1748108d75195_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/> means \u201cproportional to.\u201d From the discussion in <a href=\"\/contents\/c793cc2b-c0ab-4089-a452-677700c3f9ac@3\">Electric Potential in a Uniform Electric Field<\/a>, we know that the voltage across parallel plates is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82bb63fc3372e1376a0a21cf931c60da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/>. 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-562b801fff755fc4c25f3adbf7fd5509_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;&#69;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/div>\n<p id=\"import-auto-id2730954\">It follows, then, that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-306a43131f8bdf2020829399857d269c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#86;&#32;&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;&#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;&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"51\" style=\"vertical-align: -4px;\" \/>, and conversely,<\/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-19b7ec91260a46754ef792c3244609fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#92;&#112;&#114;&#111;&#112;&#116;&#111;&#32;&#86;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1781864\">This is true in general: The greater the voltage applied to any capacitor, the greater the charge stored in it.<\/p>\n<p id=\"import-auto-id2636730\">Different capacitors will store different amounts of charge for the same applied voltage, depending on their physical characteristics. We define their <span data-type=\"term\" id=\"import-auto-id1271385\">capacitance<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> to be such that the charge <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;\" \/> stored in a capacitor is proportional to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The charge stored in a capacitor is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-15caef0ef588b55d80e1426768113257_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2785576\">This equation expresses the two major factors affecting the amount of charge stored. Those factors are the physical characteristics of the capacitor, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and the voltage, <em data-effect=\"italics\"><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;\" \/><\/em>. Rearranging the equation, we see that <em data-effect=\"italics\">capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> is the amount of charge stored per volt,<\/em><\/em> or<\/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-9f3448160b4f298d1b8cf21578e9cd91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#86;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3089391\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitance<\/div>\n<p id=\"import-auto-id1663871\">Capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the amount of charge stored per volt, or<\/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-9f3448160b4f298d1b8cf21578e9cd91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#86;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id2764584\">The unit of capacitance is the farad (F), named for Michael Faraday (1791\u20131867), an English scientist who contributed to the fields of electromagnetism and electrochemistry. Since capacitance is charge per unit voltage, we see that a farad is a coulomb per volt, or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-315\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9cb76f69ad2a3583ea9ecbcf43c2a22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#70;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#67;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#86;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1297235\">A 1-farad capacitor would be able to store 1 coulomb (a very large amount of charge) with the application of only 1 volt. One farad is, thus, a very large capacitance. Typical capacitors range from fractions of a picofarad <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-17c2462e9b739b2d2b7ec032b54d220a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#32;&#112;&#70;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#50;&#125;&#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;&#70;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"120\" style=\"vertical-align: -7px;\" \/> to millifarads <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7d9de9179416832a459842fb8a7cf1f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#32;&#109;&#70;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#70;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -7px;\" \/>.<\/p>\n<p id=\"import-auto-id1415714\"><a href=\"#import-auto-id2512674\" class=\"autogenerated-content\">(Figure)<\/a> shows some common capacitors. Capacitors are primarily made of ceramic, glass, or plastic, depending upon purpose and size. Insulating materials, called dielectrics, are commonly used in their construction, as discussed below.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2512674\">\n<div class=\"bc-figcaption figcaption\">Some typical capacitors. Size and value of capacitance are not necessarily related. (credit: Windell Oskay)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1579209\" data-alt=\"There are various types of capacitors with varying shapes and color. Some are cylindrical in shape, some circular in shape, some rectangular in shape, with two strands of wire coming out of each.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_03a.jpg\" data-media-type=\"image\/png\" alt=\"There are various types of capacitors with varying shapes and color. Some are cylindrical in shape, some circular in shape, some rectangular in shape, with two strands of wire coming out of each.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2550561\">\n<h1 data-type=\"title\">Parallel Plate Capacitor<\/h1>\n<p id=\"import-auto-id3138915\">The parallel plate capacitor shown in <a href=\"#import-auto-id1676212\" class=\"autogenerated-content\">(Figure)<\/a> has two identical conducting plates, each having a surface area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, separated by a distance <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;\" \/> (with no material between the plates). When a 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 to the capacitor, it stores a 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\">, as shown. We can see how its capacitance depends on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and <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;\" \/> by considering the characteristics of the Coulomb force. We know that like charges repel, unlike charges attract, and the force between charges decreases with distance. So it seems quite reasonable that the bigger the plates are, the more charge they can store\u2014because the charges can spread out more. Thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> should be greater for larger <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. Similarly, the closer the plates are together, the greater the attraction of the opposite charges on them. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> should be greater for smaller <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;\" \/>.<\/em><\/em><\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">Parallel plate capacitor with plates separated by a distance <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;\" \/>. Each plate has an area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id977536\" data-alt=\"Two parallel plates are placed facing each other. The area of each plate is A, and the distance between the plates is d. The plate on the left is connected to the positive terminal of the battery, and the plate on the right is connected to the negative terminal of the battery.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_04a.jpg\" data-media-type=\"image\/png\" alt=\"Two parallel plates are placed facing each other. The area of each plate is A, and the distance between the plates is d. The plate on the left is connected to the positive terminal of the battery, and the plate on the right is connected to the negative terminal of the battery.\" width=\"150\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2573871\">It can be shown that for a parallel plate capacitor there are only two factors (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and <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;\" \/>) that affect its capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The capacitance of a parallel plate capacitor in equation form is given by<\/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-a95957cd981a9586a87288e22f32ec40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2747111\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Capacitance of a Parallel Plate Capacitor<\/div>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4aa819f7b172500c10718f9f0e58ac0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id1552736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><br \/>\nis the area of one plate in square meters, and<br \/>\n<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;\" \/><br \/>\nis the distance between the plates in meters. The constant<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/><br \/>\nis the permittivity of free space; its numerical value in SI units is<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e0c765c28f8ee61a67978a2049114244_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#56;&#46;&#56;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#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;&#70;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -4px;\" \/>.<br \/>\nThe units of F\/m are equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-db521458a5f9d60579e3d9c2b4842cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#67;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#78;&#125;&middot;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -4px;\" \/>. The small numerical value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de7b1782ae6a2ff50840b723fe5b9cfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> is related to the large size of the farad. A parallel plate capacitor must have a large area to have a capacitance approaching a farad. (Note that the above equation is valid when the parallel plates are separated by air or free space. When another material is placed between the plates, the equation is modified, as discussed below.)   <\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id920232\">\n<div data-type=\"title\" class=\"title\">Capacitance and Charge Stored in a Parallel Plate Capacitor<\/div>\n<p id=\"import-auto-id2657240\">(a) What is the capacitance of a parallel plate capacitor with metal plates, each of area  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-40e58a0b42a5cf44459e9b1f2ea9249a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/>, separated by 1.00 mm? (b) What charge is stored in this capacitor if a voltage of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eab034b552aac876894abcca9a09a451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#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;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/> is applied to it?<\/p>\n<p id=\"import-auto-id2560583\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2927434\">Finding the capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is a straightforward application of the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-824c22340ca8f2f9136b11040e062e70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#65;&#47;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -5px;\" \/>. Once <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is found, the charge stored can be found using the equation <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;\" \/>.<\/p>\n<p id=\"import-auto-id1951686\"><strong>Solution for (a)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2657718\">Entering the given values into the equation for the capacitance of a parallel plate capacitor yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-226\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b454cda4aa555ac8814a24788013fa7d_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;&#67;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#46;&#56;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#46;&#48;&#48;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#109;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#56;&#46;&#56;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#70;&#125;&#61;&#56;&#46;&#56;&#53;&#32;&#110;&#70;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"291\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"import-auto-id1562997\"><strong>Discussion for (a)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2960580\">This small value for the capacitance indicates how difficult it is to make a device with a large capacitance. Special techniques help, such as using very large area thin foils placed close together.<\/p>\n<p id=\"import-auto-id2507307\"><strong>Solution for (b)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1357066\">The charge stored in any capacitor is given by the equation <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;\" \/>. Entering the known values into this equation gives <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b2d219eb71156b5c6381514f998f028_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;&#81;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#46;&#56;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#70;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#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;&#86;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#54;&#32;&mu;&#67;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"37\" width=\"302\" style=\"vertical-align: -11px;\" \/><\/div>\n<p id=\"import-auto-id2560950\"><strong>Discussion for (b)<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1676990\">This charge is only slightly greater than those found in typical static electricity. Since air breaks down at about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d3b8a911f65cf1be1a880ec91cc4d3f6_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;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#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;&#86;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/>, more charge cannot be stored on this capacitor by increasing the voltage.<\/p>\n<\/div>\n<p id=\"eip-951\">Another interesting biological example dealing with electric potential is found in the cell\u2019s plasma membrane. The membrane sets a cell off from its surroundings and also allows ions to selectively pass in and out of the cell. There is a potential difference across the membrane of about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2909c9431f7e605cbeb94a5ec2e3aed6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#55;&#48;&#32;&#109;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/>. This is due to the mainly negatively charged ions in the cell and the predominance of positively charged sodium (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7ffcf5c901f7de8919b4cda45b2f072f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#97;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"32\" style=\"vertical-align: 0px;\" \/>) ions outside. Things change when a nerve cell is stimulated. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7ffcf5c901f7de8919b4cda45b2f072f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#97;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"32\" style=\"vertical-align: 0px;\" \/> ions are allowed to pass through the membrane into the cell, producing a positive membrane potential\u2014the nerve signal. The cell membrane is about 7 to 10 nm thick. An approximate value of the electric field across it is given by\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-70b42a432e3b98d3a426648cba894048_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#86;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#55;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#86;&#125;&#123;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#109;&#125;&#61;&#45;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#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;&#86;&#47;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"261\" style=\"vertical-align: -8px;\" \/><\/div>\n<p>This electric field is enough to cause a breakdown in air.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2702570\">\n<h1 data-type=\"title\">Dielectric<\/h1>\n<p id=\"import-auto-id1318992\">The previous example highlights the difficulty of storing a large amount of charge in capacitors. If <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;\" \/> is made smaller to produce a larger capacitance, then the maximum voltage must be reduced proportionally to avoid breakdown (since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90a1d86a345d0330c110e1e8c132f465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#86;&#47;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -5px;\" \/>). An important solution to this difficulty is to put an insulating material, called a <span data-type=\"term\">dielectric<\/span>, between the plates of a capacitor and allow <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;\" \/><br \/>\n     to be as small as possible. Not only does the smaller <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;\" \/> make the capacitance greater, but many insulators can withstand greater electric fields than air before breaking down.<\/p>\n<p id=\"import-auto-id2687767\">There is another benefit to using a dielectric in a capacitor. Depending on the material used, the capacitance is greater than that given by the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4aa819f7b172500c10718f9f0e58ac0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/> by a factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, called the <em data-effect=\"italics\">dielectric constant<\/em>. A parallel plate capacitor with a dielectric between its plates has a capacitance given by<\/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-85aa43d0a5527ec17ad0fef2e16baacd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&kappa;&epsilon;&#125;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#48;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#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;&#40;&#112;&#97;&#114;&#97;&#108;&#108;&#101;&#108;&#32;&#112;&#108;&#97;&#116;&#101;&#32;&#99;&#97;&#112;&#97;&#99;&#105;&#116;&#111;&#114;&#32;&#119;&#105;&#116;&#104;&#32;&#100;&#105;&#101;&#108;&#101;&#99;&#116;&#114;&#105;&#99;&#41;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"378\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2690389\">Values of the dielectric constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> for various materials are given in <a href=\"#import-auto-id2726630\" class=\"autogenerated-content\">(Figure)<\/a>. Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> for vacuum is exactly 1, and so the above equation is valid in that case, too. If a dielectric is used, perhaps by placing Teflon between the plates of the capacitor in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>, then the capacitance is greater by the factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, which for Teflon is 2.1.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2502208\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment: Building a Capacitor<\/div>\n<p id=\"import-auto-id2806676\">How large a capacitor can you make using a chewing gum wrapper? The plates will be the aluminum foil, and the separation (dielectric) in between will be the paper. <\/p>\n<\/div>\n<table id=\"import-auto-id2726630\" summary=\"The table gives the value of dielectric constants and dielectric strengths for various materials at a temperature of twenty degrees Celsius\">\n<caption><span data-type=\"title\">Dielectric Constants and Dielectric Strengths for Various Materials at 20\u00baC<\/span><\/caption>\n<thead>\n<tr>\n<th>Material<\/th>\n<th>Dielectric constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/th>\n<th>Dielectric strength (V\/m)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Vacuum<\/td>\n<td>1.00000<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<tr>\n<td>Air<\/td>\n<td>1.00059<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d98f22a9ef8f31b2d4efbfc8d66eb89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Bakelite<\/td>\n<td>4.9<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-313f6b8f6d6978b9164b4f747f46075c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Fused quartz<\/td>\n<td>3.78<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f476920b9ca022ad4c5c8005738a44b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Neoprene rubber<\/td>\n<td>6.7<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee61008533fff88504d9452be35830f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td>3.4<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dc21502871dbea837962d37b293fd12a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Paper<\/td>\n<td>3.7<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d01414f3b436d6f888e3bc232c09103a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Polystyrene<\/td>\n<td>2.56<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-313f6b8f6d6978b9164b4f747f46075c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Pyrex glass<\/td>\n<td>5.6<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dc21502871dbea837962d37b293fd12a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Silicon oil<\/td>\n<td>2.5<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e9fe31dd41f8a85c4bd871cad5ae14f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Strontium titanate<\/td>\n<td>233<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f476920b9ca022ad4c5c8005738a44b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Teflon<\/td>\n<td>2.1<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e78827efa7c77f9605cde5922611a29b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td>Water<\/td>\n<td>80<\/td>\n<td>\u2014<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id2576127\">Note also that the dielectric constant for air is very close to 1, so that air-filled capacitors act much like those with vacuum between their plates <em data-effect=\"italics\"><em data-effect=\"italics\">except<\/em><\/em> that the air can become conductive if the electric field strength  becomes too great. (Recall that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90a1d86a345d0330c110e1e8c132f465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#86;&#47;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -5px;\" \/> for a parallel plate capacitor.) Also shown in <a href=\"#import-auto-id2726630\" class=\"autogenerated-content\">(Figure)<\/a> are maximum electric field strengths in V\/m, called <span data-type=\"term\" id=\"import-auto-id1726758\">dielectric strengths<\/span>, for several materials. These are the fields above which the material begins to break down and conduct. The dielectric strength imposes a limit on the voltage that can be applied for a given plate separation. For instance, in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>, the separation is 1.00 mm, and so the voltage limit for air is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-867\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-738be5b476d259ffead95e1b93b36a30_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;&#86;&#38;&#32;&#61;&#38;&#32;&#69;&#92;&#99;&#100;&#111;&#116;&#32;&#100;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#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;&#86;&#47;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#48;&#48;&#32;&#86;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"256\" style=\"vertical-align: -22px;\" \/><\/div>\n<p id=\"eip-146\">However, the limit for a 1.00 mm separation filled with Teflon is 60,000 V, since the dielectric strength of Teflon is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e78827efa7c77f9605cde5922611a29b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/> V\/m. So the same capacitor filled with Teflon has a greater capacitance and can be subjected to a much greater voltage. Using the capacitance we calculated in the above example for the air-filled parallel plate capacitor, we find that the Teflon-filled capacitor can store a maximum charge of  <\/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-52e100c9ea47e4e500882d0266e36ce6_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;&#81;&#38;&#32;&#61;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#92;&#107;&#97;&#112;&#112;&#97;&#32;&#67;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#105;&#114;&#125;&#125;&#86;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#46;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#46;&#56;&#53;&#32;&#110;&#70;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#46;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#52;&#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;&#86;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#49;&#46;&#49;&#32;&#109;&#67;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"80\" width=\"252\" style=\"vertical-align: -34px;\" \/><\/div>\n<p>This is 42 times the charge of the same air-filled capacitor.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1579779\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Dielectric Strength<\/div>\n<p id=\"import-auto-id2630262\">The maximum electric field strength above which an insulating material begins to break down and conduct is called its dielectric strength.<\/p>\n<\/div>\n<p id=\"import-auto-id2600468\">Microscopically, how does a dielectric increase capacitance? Polarization of the insulator is responsible. The more easily it is polarized, the greater its dielectric constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>. Water, for example, is a <span data-type=\"term\">polar molecule<\/span> because one end of the molecule has a slight positive charge and the other end has a slight negative charge. The polarity of water causes it to have a relatively large dielectric constant of 80. The effect of polarization can be best explained in terms of the characteristics of the Coulomb force. <a href=\"#import-auto-id1295090\" class=\"autogenerated-content\">(Figure)<\/a> shows the separation of charge schematically in the molecules of a dielectric material placed between the charged plates of a capacitor. The Coulomb force between the closest ends of the molecules and the charge on the plates is attractive and very strong, since they are very close together. This attracts more charge onto the plates than if the space were empty and the opposite charges were a distance <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;\" \/> away.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1295090\">\n<div class=\"bc-figcaption figcaption\">(a) The molecules in the insulating material between the plates of a capacitor are polarized by the charged plates. This produces a layer of opposite charge on the surface of the dielectric that attracts more charge onto the plate, increasing its capacitance. (b) The dielectric reduces the electric field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge. The capacitor stores the same charge for a smaller voltage, implying that it has a larger capacitance because of the dielectric. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1020159\" data-alt=\"(a) A dielectric is between the two plates of a parallel plate capacitor. A diagram shows the molecules that make up the dielectric. The molecules are polarized by the charged plates. The positive ends of the molecules are attracted toward the negatively charged plate of the capacitor and hence are oriented toward the right. The negative ends of the molecules are attracted toward the positively charged plate of the capacitor and hence are oriented toward the left. (b) There is a dielectric material between the two plates of the capacitor. Since the charged ends of the molecules are oriented toward the capacitor plates, there is reduced field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_05a.jpg\" data-media-type=\"image\/jpg\" alt=\"(a) A dielectric is between the two plates of a parallel plate capacitor. A diagram shows the molecules that make up the dielectric. The molecules are polarized by the charged plates. The positive ends of the molecules are attracted toward the negatively charged plate of the capacitor and hence are oriented toward the right. The negative ends of the molecules are attracted toward the positively charged plate of the capacitor and hence are oriented toward the left. (b) There is a dielectric material between the two plates of the capacitor. Since the charged ends of the molecules are oriented toward the capacitor plates, there is reduced field strength inside the capacitor, resulting in a smaller voltage between the plates for the same charge.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3141379\">Another way to understand how a dielectric increases capacitance is to consider its effect on the electric field inside the capacitor. <a href=\"#import-auto-id1295090\" class=\"autogenerated-content\">(Figure)<\/a>(b) shows the electric field lines with a dielectric in place. Since the field lines end on charges in the dielectric, there are fewer of them going from one side of the capacitor to the other. So the electric field strength is less than if there were a vacuum between the plates, even though the same charge is on the plates. The voltage between the plates is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82bb63fc3372e1376a0a21cf931c60da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/>, so it too is reduced by the dielectric. Thus there is a smaller 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;\" \/> for the same charge <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;\" \/>; since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63c6d4559257d2dd3ad43aec0190a8d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#81;&#47;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"75\" style=\"vertical-align: -5px;\" \/>, the capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is greater.<\/p>\n<p id=\"import-auto-id2991565\">The dielectric constant is generally defined to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5558d764aabb5c897509b16e06bf6f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;&#61;&#123;&#69;&#125;&#95;&#123;&#48;&#125;&#47;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/>, or the ratio of the electric field in a vacuum to that in the dielectric material, and is intimately related to the polarizability of the material.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1611688\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Things Great and Small<\/div>\n<p id=\"import-auto-id1248838\"><strong>The Submicroscopic Origin of Polarization<\/strong><\/p>\n<p id=\"import-auto-id2651892\">Polarization is a separation of charge within an atom or molecule. As has been noted, the planetary model of the atom pictures it as having a positive nucleus orbited by negative electrons, analogous to the planets orbiting the Sun. Although this model is not completely accurate, it is very helpful in explaining a vast range of phenomena and will be refined elsewhere, such as in <a href=\"\/contents\/05208266-2374-4008-9060-7ddddbe877f2@2\">Atomic Physics<\/a>. The submicroscopic origin of polarization can be modeled as shown in <a href=\"#import-auto-id3164371\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3164371\">\n<div class=\"bc-figcaption figcaption\">Artist\u2019s conception of a polarized atom. The orbits of electrons around the nucleus are shifted slightly by the external charges (shown exaggerated). The resulting separation of charge within the atom means that it is polarized. Note that the unlike charge is now closer to the external charges, causing the polarization.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1949934\" data-alt=\"The top part of the figure shows what an unpolarized atom would look like if the electrons moved along a circular path around the positively charged nucleus. Next, when there is an external negative and a positive charge, the electrons are attracted toward the positive external charge and the nucleus is attracted toward the negative external charge. The circular orbit of the electrons becomes an ellipse due to the pull of the external charges.\"><img decoding=\"async\" src=\"\/resources\/81bf0e9d5c44caafa6c98279c46f2e4b454092a2\/Figure_20_05_06(a)a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The top part of the figure shows what an unpolarized atom would look like if the electrons moved along a circular path around the positively charged nucleus. Next, when there is an external negative and a positive charge, the electrons are attracted toward the positive external charge and the nucleus is attracted toward the negative external charge. The circular orbit of the electrons becomes an ellipse due to the pull of the external charges.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2609039\">We will find in <a href=\"\/contents\/05208266-2374-4008-9060-7ddddbe877f2@2\">Atomic Physics<\/a> that the orbits of electrons are more properly viewed as electron clouds with the density of the cloud related to the probability of finding an electron in that location (as opposed to the definite locations and paths of planets in their orbits around the Sun). This cloud is shifted by the Coulomb force so that the atom on average has a separation of charge. Although the atom remains neutral, it can now be the source of a Coulomb force, since a charge brought near the atom will be closer to one type of charge than the other.<\/p>\n<p id=\"import-auto-id1563695\">Some molecules, such as those of water, have an inherent separation of charge and are thus called polar molecules. <a href=\"#import-auto-id2937921\" class=\"autogenerated-content\">(Figure)<\/a> illustrates the separation of charge in a water molecule, which has two hydrogen atoms and one oxygen atom <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e631ece448e8ee4abd20a76f85f40e10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/>. The water molecule is not symmetric\u2014the hydrogen atoms are repelled to one side, giving the molecule a boomerang shape. The electrons in a water molecule are more concentrated around the more highly charged oxygen nucleus than around the hydrogen nuclei. This makes the oxygen end of the molecule slightly negative and leaves the hydrogen ends slightly positive. The inherent separation of charge in polar molecules makes it easier to align them with external fields and charges. Polar molecules therefore exhibit greater polarization effects and have greater dielectric constants. Those who study chemistry will find that the polar nature of water has many effects. For example, water molecules gather ions much more effectively because they have an electric field and a separation of charge to attract charges of both signs. Also, as brought out in the previous chapter, polar water provides a shield or screening of the electric fields in the highly charged molecules of interest in biological systems.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2937921\">\n<div class=\"bc-figcaption figcaption\">Artist\u2019s conception of a water molecule. There is an inherent separation of charge, and so water is a polar molecule. Electrons in the molecule are attracted to the oxygen nucleus and leave an excess of positive charge near the two hydrogen nuclei. (Note that the schematic on the right is a rough illustration of the distribution of electrons in the water molecule. It does not show the actual numbers of protons and electrons involved in the structure.)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2648318\" data-alt=\"The two hydrogen atoms in the water molecule subtend an angle of one hundred and four point five degrees with oxygen at the center. This is a schematic arrangement of hydrogen and oxygen atoms in the water molecule. The molecule is polarized, with the electrons attracted more to the nucleus of the oxygen atom than toward the nuclei of the hydrogen atoms.\"><img decoding=\"async\" src=\"\/resources\/5f84ada0aafde2cce3e212d001ddc68e7218e9ae\/Figure_20_05_06(b)a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The two hydrogen atoms in the water molecule subtend an angle of one hundred and four point five degrees with oxygen at the center. This is a schematic arrangement of hydrogen and oxygen atoms in the water molecule. The molecule is polarized, with the electrons attracted more to the nucleus of the oxygen atom than toward the nuclei of the hydrogen atoms.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Capacitor Lab<\/div>\n<p>Explore how a capacitor works! Change the size of the plates and add a dielectric to see the effect on capacitance. Change the voltage and see charges built up on the plates. Observe the electric field in the capacitor. Measure the voltage and the electric field.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1545484\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d8b6b46ce76e32397246a5a9f39985177666c958\/capacitor-lab_en.jar\">Capacitor Lab<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_20.5\" data-alt=\"\"><a href=\"\/resources\/d8b6b46ce76e32397246a5a9f39985177666c958\/capacitor-lab_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1309916\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1310256\">\n<li id=\"import-auto-id2607317\">A <strong data-effect=\"bold\">capacitor <strong data-effect=\"bold\">is a device used to store charge.<\/strong><\/strong><\/li>\n<li id=\"import-auto-id1737462\">The amount of charge <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;\" \/> a capacitor can store depends on two major factors\u2014the voltage applied and the capacitor\u2019s physical characteristics, such as its size.<\/li>\n<li id=\"import-auto-id2719243\">The capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the amount of charge stored per volt<em data-effect=\"italics\">,<\/em> or\n<div data-type=\"equation\" class=\"equation\" id=\"eip-791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f3448160b4f298d1b8cf21578e9cd91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#86;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2838015\">The capacitance of a parallel plate capacitor is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e46aab2dedaedd3f4adac2da96ec3472_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/>, when the plates are separated by air or free space. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6b057c10b362c2434248c7bb44a0b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> is called the permittivity of free space.<\/li>\n<li>A parallel plate capacitor with a dielectric between its plates has a capacitance given by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id2286944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51a32ea6b9720b6a6b5caa1a0869752a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#107;&#97;&#112;&#112;&#97;&#32;&#92;&#101;&#112;&#115;&#105;&#108;&#111;&#110;&#32;&#125;&#125;&#95;&#123;&#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;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#100;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91fb7aa7e8313be081e8039192e8c9bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#107;&#97;&#112;&#112;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is the dielectric constant of the material.<\/p>\n<\/li>\n<li id=\"import-auto-id3138892\">The maximum electric field strength above which an insulating material begins to break down and conduct is called dielectric strength.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2938102\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1964776\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1347748\">\n<p id=\"import-auto-id2635497\">Does the capacitance of a device depend on the applied voltage? What about the charge stored in it? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1511858\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2805847\">\n<p id=\"import-auto-id3096036\">Use the characteristics of the Coulomb force to explain why capacitance should be proportional to the plate area of a capacitor. Similarly, explain why capacitance should be inversely proportional to the separation between plates. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2753579\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1999045\">\n<p id=\"import-auto-id1611133\">Give the reason why a dielectric material increases capacitance compared with what it would be with air between the plates of a capacitor. What is the independent reason that a dielectric material also allows a greater voltage to be applied to a capacitor? (The dielectric thus increases <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and permits a greater <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;\" \/>.) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3361548\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1324707\">\n<p id=\"import-auto-id2703282\">How does the polar character of water molecules help to explain water\u2019s relatively large dielectric constant? (<a href=\"#import-auto-id2937921\" class=\"autogenerated-content\">(Figure)<\/a>)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3139942\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1330607\">\n<p id=\"import-auto-id3100275\">Sparks will occur between the plates of an air-filled capacitor at lower voltage when the air is humid than when dry. Explain why, considering the polar character of water molecules. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1311685\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3066669\">\n<p id=\"import-auto-id1628324\">Water has a large dielectric constant, but it is rarely used in capacitors. Explain why. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2601615\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1950596\">\n<p id=\"import-auto-id3152613\">Membranes in living cells, including those in humans, are characterized by a separation of charge across the membrane. Effectively, the membranes are thus charged capacitors with important functions related to the potential difference across the membrane. Is energy required to separate these charges in living membranes and, if so, is its source the metabolization of food energy or some other source? <\/p>\n<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1638125\">\n<div class=\"bc-figcaption figcaption\">The semipermeable membrane of a cell has different concentrations of ions inside and out. Diffusion moves the<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bf6ccbe226f4493406957b0e13364da3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"24\" style=\"vertical-align: -1px;\" \/> (potassium) and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d9a8ebe86f3590f1a9bfd2ead683d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#108;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> (chloride) ions in the directions shown, until the Coulomb force halts further transfer. This results in a layer of positive charge on the outside, a layer of negative charge on the inside, and thus a voltage across the cell membrane. The membrane is normally impermeable to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7ffcf5c901f7de8919b4cda45b2f072f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#97;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#43;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"32\" style=\"vertical-align: 0px;\" \/>  (sodium ions).<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3094558\" data-alt=\"The semipermeable membrane of a cell is shown, with different concentrations of potassium cations, sodium cations, and chloride anions inside and outside the cell. The ions are represented by small, colored circles. In its resting state, the cell membrane is permeable to potassium and chloride ions, but it is impermeable to sodium ions. By diffusion, potassium cations travel out of the cell, going through the cell membrane and forming a layer of positive charge on the outer surface of the membrane. By diffusion, chloride anions go into the cell, going through the cell membrane and forming a layer of negative charge on the inner surface of the membrane. As a result, a voltage is set up across the cell membrane. The Coulomb force prevents all the ions from crossing the membrane.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_05_07a.jpg\" data-media-type=\"image\/jpg\" alt=\"The semipermeable membrane of a cell is shown, with different concentrations of potassium cations, sodium cations, and chloride anions inside and outside the cell. The ions are represented by small, colored circles. In its resting state, the cell membrane is permeable to potassium and chloride ions, but it is impermeable to sodium ions. By diffusion, potassium cations travel out of the cell, going through the cell membrane and forming a layer of positive charge on the outer surface of the membrane. By diffusion, chloride anions go into the cell, going through the cell membrane and forming a layer of negative charge on the inner surface of the membrane. As a result, a voltage is set up across the cell membrane. The Coulomb force prevents all the ions from crossing the membrane.\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2608941\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1338876\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1942516\">\n<p id=\"import-auto-id1501248\">What charge is stored in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2b862366f6be7481362f79b3ad1caa3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&#32;&micro;&#70;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> capacitor when 120 V is applied to it?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3147194\">\n<p id=\"import-auto-id2644327\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-70edfae0067ecde8ba3c9b65c311b846_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&#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;&#109;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2586206\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2953152\">\n<p id=\"import-auto-id1086670\">Find the charge stored when 5.50 V is applied to an 8.00 pF capacitor.<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2511345\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1352226\">\n<p id=\"import-auto-id2629428\">What charge is stored in the capacitor in <a href=\"#fs-id920232\" class=\"autogenerated-content\">(Figure)<\/a>?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2727877\">\n<p id=\"import-auto-id2595981\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-68fb85d1b551830d049c0fa0a514cca8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#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;&#109;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"63\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2677540\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2962045\">\n<p id=\"import-auto-id2782884\">Calculate the voltage applied to a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7795fe287f21e34cd827c64a27c7feb9_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;&#48;&#48;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> capacitor when it holds <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d7c446b770bbca311a25c5331fec0ba8_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;&#49;&#48;&#32;&micro;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/> of charge.<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1326374\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1326375\">\n<p id=\"import-auto-id1563284\">What voltage must be applied to an 8.00 nF capacitor to store 0.160 mC of charge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2951082\">\n<p id=\"eip-id3585054\">20.0 kV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1603035\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1603036\">\n<p id=\"import-auto-id2584784\">What capacitance is needed to store <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e94a4fa1fd08f81a3c5c2de7ffa6d92_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;&#48;&#32;&micro;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/> of charge at a voltage of 120 V?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2756080\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2756082\">\n<p id=\"import-auto-id2560911\">What is the capacitance of a large Van de Graaff generator\u2019s terminal, given that it stores 8.00 mC of charge at a voltage of 12.0 MV?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2004243\">\n<p id=\"import-auto-id2905346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d750dc012eb37ae5d65cd0ab95ea2e56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#55;&#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;&#112;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2678325\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2537079\">\n<p id=\"import-auto-id1611526\">Find the capacitance of a parallel plate capacitor having plates of area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1bb4947800139057c9284bc03262c136_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;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/> that are separated by 0.100 mm of Teflon.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1949350\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1949351\">\n<p id=\"import-auto-id1552021\">(a)What is the capacitance of a parallel plate capacitor having plates of area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a9c4f30d6a64188746ce62bf1e064e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#53;&#48;&#32;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/> that are separated by 0.0200 mm of neoprene rubber? (b) What charge does it hold when 9.00 V is applied to it?<strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3155343\">\n<p id=\"import-auto-id2602024\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fc0b9f8a30bb532bae9c46c3a0338afe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2913687\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0e826c1d5137a82c6d34cba2b220d1b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"75\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1674884\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1674885\">\n<p id=\"eip-id2966187\"><strong>Integrated Concepts<\/strong><\/p>\n<p id=\"import-auto-id2502389\">A prankster applies 450 V to an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c30c5407059f82cb87dffe960e35c38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&micro;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"46\" style=\"vertical-align: 0px;\" \/> capacitor and then tosses it to an unsuspecting victim. The victim\u2019s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2723762\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2656834\">\n<p id=\"eip-id1799265\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id2508182\">(a) A certain parallel plate capacitor has plates of area  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-67f0021f0c53561d20e6f3d4536c3ff9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#46;&#48;&#48;&#32;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -1px;\" \/>, separated by 0.0100 mm of nylon, and stores 0.170 C of charge. What is the applied voltage? (b) What is unreasonable about this result? (c) Which assumptions are responsible or inconsistent? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2904372\">\n<p id=\"eip-id2420270\">(a) 14.2 kV<\/p>\n<p id=\"eip-id2420273\">(b) The voltage is unreasonably large, more than 100 times the breakdown voltage of nylon.<\/p>\n<p id=\"eip-id2189005\">(c) The assumed charge is unreasonably large and cannot be stored in a capacitor of these dimensions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1677156\">\n<dt>capacitor<\/dt>\n<dd id=\"fs-id1537463\">a device that stores electric charge<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1166661\">\n<dt>capacitance<\/dt>\n<dd id=\"fs-id2756324\">amount of charge stored per unit volt<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1359494\">\n<dt>dielectric<\/dt>\n<dd id=\"fs-id3084506\">an insulating material<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3256092\">\n<dt>dielectric strength<\/dt>\n<dd id=\"fs-id2719268\">the maximum electric field above which an insulating material begins to break down and conduct<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2791420\">\n<dt>parallel plate capacitor<\/dt>\n<dd id=\"fs-id2595194\">two identical conducting plates separated by a distance<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2753374\">\n<dt>polar molecule<\/dt>\n<dd id=\"fs-id2543637\">a molecule with inherent separation of charge<\/dd>\n<\/dl>\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-1064","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\/1064","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\/1064\/revisions"}],"predecessor-version":[{"id":1065,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1064\/revisions\/1065"}],"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\/1064\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1064"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1064"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1064"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1064"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}