{"id":1041,"date":"2017-10-27T16:31:16","date_gmt":"2017-10-27T16:31:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electric-potential-in-a-uniform-electric-field\/"},"modified":"2017-11-08T03:26:06","modified_gmt":"2017-11-08T03:26:06","slug":"electric-potential-in-a-uniform-electric-field","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electric-potential-in-a-uniform-electric-field\/","title":{"raw":"Electric Potential in a Uniform Electric Field","rendered":"Electric Potential in a Uniform Electric Field"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Describe the relationship between voltage and electric field.<\/li>\n<li>Derive an expression for the electric potential and electric field.<\/li>\n<li>Calculate electric field strength given distance and voltage.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1358711\">In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field [latex]\\mathbf{\\text{E}}[\/latex] is produced by placing a potential difference (or voltage) [latex]\\Delta V[\/latex] across two parallel metal plates, labeled A and B. (See <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a>.) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist\u2019s point of view, either [latex]\\Delta V[\/latex] or [latex]\\mathbf{\\text{E}}[\/latex] can be used to describe any charge distribution. [latex]\\Delta V[\/latex] is most closely tied to energy, whereas [latex]\\mathbf{\\text{E}}[\/latex] is most closely related to force. [latex]\\Delta V[\/latex] is a <span data-type=\"term\">scalar<\/span> quantity and has no direction, while [latex]\\mathbf{\\text{E}}[\/latex] is a <span data-type=\"term\">vector<\/span> quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by [latex]E[\/latex] below.) The relationship between [latex]\\Delta V[\/latex] and [latex]\\mathbf{\\text{E}}[\/latex] is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in <a href=\"\/contents\/650c4bb3-d190-44fd-9f2f-445a783abdec@4\">Electric Potential Energy: Potential Difference<\/a>, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id671952\">\n<div class=\"bc-figcaption figcaption\">The relationship between [latex]V[\/latex] and [latex]E[\/latex] for parallel conducting plates is [latex]E=V\/d[\/latex]. (Note that [latex]\\Delta \\mathit{V}={V}_{\\text{AB}}[\/latex]  in magnitude. For a charge that is moved from plate A at higher potential to plate B at lower potential, a minus sign needs to be included as follows:  [latex]\u2013\\Delta V={V}_{\\text{A}}\u2013{V}_{\\text{B}}={V}_{\\text{AB}}[\/latex]. See the text for details.)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1347929\" data-alt=\"The figure shows two vertically oriented parallel plates A and B separated by a distance d. The plate A is positively charged and B is negatively charged. Electric field lines are parallel between the plates and curved at the ends of the plates. A charge q is moved from A to B. The work done W equals q times V sub A B, and the electric field intensity E equals V sub A B over d and potential difference delta V equals q times V sub A B.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_02_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows two vertically oriented parallel plates A and B separated by a distance d. The plate A is positively charged and B is negatively charged. Electric field lines are parallel between the plates and curved at the ends of the plates. A charge q is moved from A to B. The work done W equals q times V sub A B, and the electric field intensity E equals V sub A B over d and potential difference delta V equals q times V sub A B.\" width=\"175\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2637761\">The work done by the electric field in <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a> to move a positive charge [latex]q[\/latex]<em data-effect=\"italics\"> from A, the positive plate, higher potential, to B, the negative plate, lower potential, is<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]W=\\phantom{\\rule{0.25em}{0ex}}\u2013\\Delta \\text{PE}=\\phantom{\\rule{0.25em}{0ex}}\u2013q\\Delta V.[\/latex]<\/div>\n<p id=\"import-auto-id953198\"> The potential difference between points A and B is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\u2013\\Delta V=\\phantom{\\rule{0.25em}{0ex}}\u2013\\left({V}_{\\text{B}}\u2013{V}_{\\text{A}}\\right)={V}_{\\text{A}}\u2013{V}_{\\text{B}}={V}_{\\text{AB}}.[\/latex]<\/div>\n<p id=\"import-auto-id1059002\">Entering this into the expression for work yields<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]W={\\text{qV}}_{\\text{AB}}.[\/latex]<\/div>\n<p id=\"import-auto-id1330624\">Work is [latex]W=\\text{Fd}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex]; here [latex]\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta =1[\/latex], since the path is parallel to the field, and so [latex]W=\\text{Fd}[\/latex]. Since<br>\n[latex]F=\\text{qE}[\/latex], we see that [latex]W=\\text{qEd}[\/latex]. Substituting this expression for work into the previous equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\mathrm{qEd}={\\text{qV}}_{\\text{AB}}.[\/latex]<\/div>\n<p id=\"import-auto-id2543027\">The charge cancels, and so the voltage between points A and B is seen to be<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{c}\\begin{array}{c}{V}_{\\text{AB}}=\\mathrm{Ed}\\\\ E=\\frac{{V}_{\\text{AB}}}{d}\\end{array}}\\text{(uniform}\\phantom{\\rule{0.25em}{0ex}}E\\phantom{\\rule{0.25em}{0ex}}\\text{- field only),}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1211554\">where <em data-effect=\"italics\">[latex]d[\/latex]<\/em> is the distance from A to B, or the distance between the plates in <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a>. Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{1 N}\/C=\\text{1 V}\/m.[\/latex]<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1540577\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Voltage between Points A and B<\/div>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{c}\\begin{array}{c}{V}_{\\text{AB}}=\\mathrm{Ed}\\\\ E=\\frac{{V}_{\\text{AB}}}{d}\\end{array}}\\text{(uniform}\\phantom{\\rule{0.25em}{0ex}}E\\phantom{\\rule{0.25em}{0ex}}\\text{- field only),}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3137674\">where <em data-effect=\"italics\">[latex]d[\/latex]<\/em> is the distance from A to B, or the distance between the plates.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2573471\">\n<div data-type=\"title\" class=\"title\">What Is the Highest Voltage Possible between Two Plates?<\/div>\n<p id=\"import-auto-id1492464\">Dry air will support a maximum electric field strength of about [latex]3.0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex]. Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?<\/p>\n<p id=\"import-auto-id2678740\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2637785\">We are given the maximum electric field [latex]E[\/latex] between the plates and the distance [latex]d[\/latex] between them. The equation [latex]{V}_{\\text{AB}}=\\mathrm{Ed}[\/latex] can thus be used to calculate the maximum voltage. <\/p>\n<p id=\"import-auto-id1607393\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3041820\">The potential difference or voltage between the plates is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-785\">[latex]{\\text{V}}_{\\text{AB}}=\\mathrm{Ed}.[\/latex]<\/div>\n<p id=\"import-auto-id1951322\">Entering the given values for <\/p>\n<p>[latex]E[\/latex] and [latex]d[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{V}_{\\text{AB}}=\\left(3.0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}\\right)\\left(0.025 m\\right)=7.5\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}V[\/latex]<\/div>\n<p id=\"import-auto-id1564586\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-678\">[latex]{V}_{\\text{AB}}=\\text{75 kV}.[\/latex]<\/div>\n<p>(The answer is quoted to only two digits, since the maximum field strength is approximate.)<\/p>\n<p id=\"import-auto-id920417\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2719229\">One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a 5 cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up, say with static electricity, on dry days.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1536252\">\n<div class=\"bc-figcaption figcaption\">A spark chamber is used to trace the paths of high-energy particles. Ionization created by the particles as they pass through the gas between the plates allows a spark to jump. The sparks are perpendicular to the plates, following electric field lines between them. The potential difference between adjacent plates is not high enough to cause sparks without the ionization produced by particles from accelerator experiments (or cosmic rays). (credit: Daderot, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1048052\" data-alt=\"The picture shows a spark chamber placed on a wooden base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_02_02a.jpg\" data-media-type=\"image\/png\" alt=\"The picture shows a spark chamber placed on a wooden base.\" width=\"200\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1677818\">\n<div data-type=\"title\" class=\"title\">Field and Force inside an Electron Gun<\/div>\n<p id=\"import-auto-id2596022\">(a) An electron gun has parallel plates separated by 4.00 cm and gives electrons 25.0 keV of energy. What is the electric field strength between the plates? (b) What force would this field exert on a piece of plastic with a [latex]\\text{0.500 \u03bcC}[\/latex] charge that gets between the plates?<\/p>\n<p id=\"import-auto-id1281920\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2005074\">Since the voltage and plate separation are given, the electric field strength can be calculated directly from the expression [latex]E=\\frac{{V}_{\\text{AB}}}{d}[\/latex]. Once the electric field strength is known, the force on a charge is found using<br>\n[latex]\\mathbf{\\text{F}}=q\\phantom{\\rule{0.25em}{0ex}}\\mathbf{\\text{E}}[\/latex]. Since the electric field is in only one direction, we can write this equation in terms of the magnitudes, [latex]F=q\\phantom{\\rule{0.25em}{0ex}}E[\/latex].<\/p>\n<p id=\"import-auto-id2596726\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1332445\">The expression for the magnitude of the electric field between two uniform metal plates is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-638\">[latex]E=\\frac{{V}_{\\text{AB}}}{d}.[\/latex]<\/div>\n<p id=\"import-auto-id2809743\">Since the electron is a single charge and is given 25.0 keV of energy, the potential difference must be 25.0 kV. Entering this value for [latex]{V}_{\\text{AB}}[\/latex] and the plate separation of 0.0400 m, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E=\\frac{\\text{25}\\text{.}\\text{0 kV}}{0\\text{.}\\text{0400 m}}=6\\text{.}\\text{25}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}.[\/latex]<\/div>\n<p id=\"import-auto-id2747313\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id2804210\">The magnitude of the force on a charge in an electric field is obtained from the equation <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-435\">[latex]F=\\mathrm{qE}.[\/latex]<\/div>\n<p id=\"import-auto-id1233382\">Substituting known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]F=\\left(\\text{0.500}\u00d7{\\text{10}}^{\u20136}\\phantom{\\rule{0.25em}{0ex}}\\text{C}\\right)\\left(\\text{6.25}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}\\right)=\\text{0.313 N}.[\/latex]<\/div>\n<p id=\"import-auto-id3041830\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1677504\">Note that the units are newtons, since [latex]\\text{1 V\/m}=\\text{1 N\/C}[\/latex]. The force on the charge is the same no matter where the charge is located between the plates. This is because the electric field is uniform between the plates.<\/p>\n<\/div>\n<p id=\"import-auto-id3067882\">In more general situations, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of [latex]\\mathbf{\\text{E}}[\/latex] and also in the direction of lower potential [latex]V[\/latex]. Furthermore, the magnitude of [latex]\\mathbf{\\text{E}}[\/latex] equals the rate of decrease of [latex]V[\/latex] with distance. The faster [latex]V[\/latex] decreases over distance, the greater the electric field. In equation form, the general relationship between voltage and electric field is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-977\">[latex]E=\\phantom{\\rule{0.25em}{0ex}}\u2013\\frac{\\Delta V}{\\Delta s},[\/latex]<\/div>\n<p id=\"import-auto-id2673134\">where [latex]\\Delta s[\/latex] is the distance over which the change in potential, [latex]\\Delta V[\/latex], takes place. The minus sign tells us that [latex]\\mathbf{\\text{E}}[\/latex] points in the direction of decreasing potential. The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1552736\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Relationship between Voltage and Electric Field<\/div>\n<p id=\"import-auto-id1951129\">In equation form, the general relationship between voltage and electric field is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E=\\phantom{\\rule{0.25em}{0ex}}\u2013\\frac{\\Delta V}{\\Delta s},[\/latex]<\/div>\n<p id=\"import-auto-id2822109\">where [latex]\\Delta s[\/latex] is the distance over which the change in potential, [latex]\\Delta V[\/latex], takes place. The minus sign tells us that [latex]\\mathbf{\\text{E}}[\/latex] points in the direction of decreasing potential. The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p>\n<\/div>\n<p>For continually changing potentials, [latex]\\Delta V[\/latex] and [latex]\\Delta s[\/latex] become infinitesimals and differential calculus must be employed to determine the electric field.<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3051945\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2915369\">\n<li id=\"import-auto-id1551715\">The voltage between points A and B is\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{c}\\begin{array}{c}{V}_{\\text{AB}}=\\mathrm{Ed}\\\\ E=\\frac{{V}_{\\text{AB}}}{d}\\end{array}}\\text{(uniform}\\phantom{\\rule{0.25em}{0ex}}E\\phantom{\\rule{0.25em}{0ex}}\\text{- field only),}\\end{array}[\/latex]<\/div>\n<p>where [latex]d[\/latex] is the distance from A to B, or the distance between the plates.<\/p><\/li>\n<li id=\"import-auto-id1350386\">In equation form, the general relationship between voltage and electric field is\n<div data-type=\"equation\" class=\"equation\">[latex]E=\\phantom{\\rule{0.25em}{0ex}}\u2013\\frac{\\Delta V}{\\Delta s},[\/latex]<\/div>\n<p>where [latex]\\Delta s[\/latex] is the distance over which the change in potential, [latex]\\Delta V[\/latex], takes place. The minus sign tells us that [latex]\\mathbf{\\text{E}}[\/latex] points in the direction of decreasing potential.) The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1726763\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1570612\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2677260\">\n<p id=\"import-auto-id2677744\">Discuss how potential difference and electric field strength are related. Give an example. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3080509\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1584974\">\n<p id=\"import-auto-id2608602\">What is the strength of the electric field in a region where the electric potential is constant? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2558866\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1985613\">\n<p id=\"import-auto-id1937578\">Will a negative charge, initially at rest, move toward higher or lower potential? Explain why. <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2607640\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-id1688468\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1688471\">\n<p id=\"eip-id1688472\">Show that units of V\/m and N\/C for electric field strength are indeed equivalent. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-id1688476\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1521962\">\n<p id=\"eip-id1521963\">What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of [latex]1\\text{.}\\text{50}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}V[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2635926\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1514530\">\n<p id=\"import-auto-id2723324\">The electric field strength between two parallel conducting plates separated by 4.00 cm is [latex]7\\text{.}\\text{50}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex]. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be at zero volts. What is the potential 1.00 cm from that plate (and 3.00 cm from the other)?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1362631\">\n<p id=\"import-auto-id2806685\">(a) [latex]3\\text{.}\\text{00 kV}[\/latex]<\/p>\n<p id=\"import-auto-id1727129\">(b) [latex]\\text{750 V}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2749816\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2751112\">\n<p id=\"import-auto-id2576062\">How far apart are two conducting plates that have an electric field strength of [latex]4\\text{.}\\text{50}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex] between them, if their potential difference is 15.0 kV?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2956436\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1564164\">(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength for air ([latex]3.0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex]) if the plates are separated by 2.00 mm and a potential difference of [latex]5.0\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex] is applied? (b) How close together can the plates be with this applied voltage?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2208045\">\n<p id=\"eip-id2208048\">(a) No. The electric field strength between the plates is [latex]2.5\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m,}[\/latex] which is lower than the breakdown strength for air ([latex]3.0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex]).\n<\/p>\n<p id=\"eip-id3052084\">(b) 1.7 mm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1956925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1676152\">\n<p id=\"import-auto-id2656762\">The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct. Membranes are discussed in <a href=\"\/contents\/56d8ca48-ab11-477d-b800-31b812b036b2@4\">Capacitors and Dielectrics<\/a> and <a href=\"\/contents\/41f56aa9-90a8-4c05-95d7-2608497844bf@3\">Nerve Conduction\u2014Electrocardiograms<\/a>.) You may assume a uniform electric field.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2573954\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2635324\">\n<p id=\"import-auto-id3141216\">Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions. (Membranes are discussed in some detail in <a href=\"\/contents\/41f56aa9-90a8-4c05-95d7-2608497844bf@3\">Nerve Conduction\u2014Electrocardiograms<\/a>.) What is the voltage across an 8.00 nm\u2013thick membrane if the electric field strength across it is 5.50 MV\/m? You may assume a uniform electric field.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1538486\">\n<p id=\"import-auto-id1363987\">44.0 mV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1636534\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3087915\">\n<p id=\"import-auto-id2977451\">Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2705714\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1667128\">\n<p id=\"import-auto-id2702894\">Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be [latex]3.0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1638150\">\n<p id=\"import-auto-id2688443\">[latex]\\text{15 kV}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1563009\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2756439\">\n<p id=\"import-auto-id2721949\">A doubly charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1948894\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2677744\">\n<p id=\"import-auto-id2840049\">An electron is to be accelerated in a uniform electric field having a strength of [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V\/m}[\/latex]. (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2607322\">\n<p id=\"import-auto-id2791118\">(a) [latex]\\text{800 KeV}[\/latex]<\/p>\n<p id=\"import-auto-id3166123\">(b) [latex]\\text{25.0 km}[\/latex]<\/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=\"fs-id2319336\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id2349344\"> physical quantity with magnitude but no direction<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2068816\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1581918\"> physical quantity with both magnitude and direction<\/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 relationship between voltage and electric field.<\/li>\n<li>Derive an expression for the electric potential and electric field.<\/li>\n<li>Calculate electric field strength given distance and voltage.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1358711\">In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is produced by placing a potential difference (or voltage) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> across two parallel metal plates, labeled A and B. (See <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a>.) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist\u2019s point of view, either <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> can be used to describe any charge distribution. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> is most closely tied to energy, whereas <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is most closely related to force. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> is a <span data-type=\"term\">scalar<\/span> quantity and has no direction, while <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is a <span data-type=\"term\">vector<\/span> quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> below.) The relationship between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in <a href=\"\/contents\/650c4bb3-d190-44fd-9f2f-445a783abdec@4\">Electric Potential Energy: Potential Difference<\/a>, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id671952\">\n<div class=\"bc-figcaption figcaption\">The relationship between <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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> for parallel conducting plates is <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;\" \/>. (Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-28d3c8205692023ac5d83f9b66890ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#105;&#116;&#123;&#86;&#125;&#61;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -4px;\" \/>  in magnitude. For a charge that is moved from plate A at higher potential to plate B at lower potential, a minus sign needs to be included as follows:  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2cd5dfb8727892591ece9544df5d95c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#61;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#45;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"182\" style=\"vertical-align: -4px;\" \/>. See the text for details.)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1347929\" data-alt=\"The figure shows two vertically oriented parallel plates A and B separated by a distance d. The plate A is positively charged and B is negatively charged. Electric field lines are parallel between the plates and curved at the ends of the plates. A charge q is moved from A to B. The work done W equals q times V sub A B, and the electric field intensity E equals V sub A B over d and potential difference delta V equals q times V sub A B.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_02_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows two vertically oriented parallel plates A and B separated by a distance d. The plate A is positively charged and B is negatively charged. Electric field lines are parallel between the plates and curved at the ends of the plates. A charge q is moved from A to B. The work done W equals q times V sub A B, and the electric field intensity E equals V sub A B over d and potential difference delta V equals q times V sub A B.\" width=\"175\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2637761\">The work done by the electric field in <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a> to move a positive charge <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> from A, the positive plate, higher potential, to B, the negative plate, lower potential, is<\/em><\/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-533c5ca27dc311b7dbe32b1a14647e7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;&#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;&#45;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#61;&#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;&#45;&#113;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"196\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id953198\"> The potential difference between points A and B is<\/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-fd3941e411c7b61026b947265cae3e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#61;&#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;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#45;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#45;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#125;&#125;&#61;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"316\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1059002\">Entering this into the expression for work yields<\/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-618c0732aa33580223f05dd7fd955e7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#86;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id1330624\">Work is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a549f948e0f91579fb4b6088bd5bac03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#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;&#99;&#111;&#115;&#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;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"106\" style=\"vertical-align: -1px;\" \/>; here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bd4730d50e3c1bed86f0cac04a30f124_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#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;&#104;&#101;&#116;&#97;&#32;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: -1px;\" \/>, since the path is parallel to the field, and so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92e4a58e16d43f923eabb126953cd4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -1px;\" \/>. Since<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-56fe73ae737c7f4d46a99a7f58b8ce71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#69;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\" \/>, we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d6cd58097e62861ce584f9ec956bf9bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#69;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -3px;\" \/>. Substituting this expression for work into the previous equation gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a74d70be806c54ab862936b13611ef4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#113;&#69;&#100;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#113;&#86;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id2543027\">The charge cancels, and so the voltage between points A and B is seen to be<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9cb28fdd021f8cd679feb7ad773bdacf_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;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#100;&#125;&#92;&#92;&#32;&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#125;&#123;&#100;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#117;&#110;&#105;&#102;&#111;&#114;&#109;&#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;&#69;&#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;&#45;&#32;&#102;&#105;&#101;&#108;&#100;&#32;&#111;&#110;&#108;&#121;&#41;&#44;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"270\" style=\"vertical-align: -18px;\" \/><\/div>\n<p id=\"import-auto-id1211554\">where <em data-effect=\"italics\"><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> is the distance from A to B, or the distance between the plates in <a href=\"#import-auto-id671952\" class=\"autogenerated-content\">(Figure)<\/a>. Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:<\/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-0f387392380c295d28423ec2eff48cc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#125;&#47;&#67;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#86;&#125;&#47;&#109;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"131\" style=\"vertical-align: -5px;\" \/><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1540577\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Voltage between Points A and B<\/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-9cb28fdd021f8cd679feb7ad773bdacf_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;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#100;&#125;&#92;&#92;&#32;&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#125;&#123;&#100;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#117;&#110;&#105;&#102;&#111;&#114;&#109;&#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;&#69;&#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;&#45;&#32;&#102;&#105;&#101;&#108;&#100;&#32;&#111;&#110;&#108;&#121;&#41;&#44;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"270\" style=\"vertical-align: -18px;\" \/><\/div>\n<p id=\"import-auto-id3137674\">where <em data-effect=\"italics\"><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> is the distance from A to B, or the distance between the plates.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2573471\">\n<div data-type=\"title\" class=\"title\">What Is the Highest Voltage Possible between Two Plates?<\/div>\n<p id=\"import-auto-id1492464\">Dry air will support a maximum electric field strength of about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-02ddb085903a646a7e2125368ed251d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&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=\"90\" style=\"vertical-align: -4px;\" \/>. Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field. What, then, is the maximum voltage between two parallel conducting plates separated by 2.5 cm of dry air?<\/p>\n<p id=\"import-auto-id2678740\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2637785\">We are given the maximum electric field <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> between the plates and the 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;\" \/> between them. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e9160a26bdf4d63c1961a824d623931_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> can thus be used to calculate the maximum voltage. <\/p>\n<p id=\"import-auto-id1607393\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3041820\">The potential difference or voltage between the plates is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-785\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de2d775c7085ebe04be1f30c424ee9f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1951322\">Entering the given values for <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-764e1c770271f92700e1a4fbce46c668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" 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;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-99ea286d2c512f71cd0da74c40da7018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#48;&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;&#48;&#46;&#48;&#50;&#53;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#46;&#53;&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;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"323\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1564586\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-678\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8006c879e4bbf97d2776b53313600665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#32;&#107;&#86;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>(The answer is quoted to only two digits, since the maximum field strength is approximate.)<\/p>\n<p id=\"import-auto-id920417\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2719229\">One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2.5 cm (1 in.) gap, or 150 kV for a 5 cm spark. This limits the voltages that can exist between conductors, perhaps on a power transmission line. A smaller voltage will cause a spark if there are points on the surface, since points create greater fields than smooth surfaces. Humid air breaks down at a lower field strength, meaning that a smaller voltage will make a spark jump through humid air. The largest voltages can be built up, say with static electricity, on dry days.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1536252\">\n<div class=\"bc-figcaption figcaption\">A spark chamber is used to trace the paths of high-energy particles. Ionization created by the particles as they pass through the gas between the plates allows a spark to jump. The sparks are perpendicular to the plates, following electric field lines between them. The potential difference between adjacent plates is not high enough to cause sparks without the ionization produced by particles from accelerator experiments (or cosmic rays). (credit: Daderot, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1048052\" data-alt=\"The picture shows a spark chamber placed on a wooden base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_02_02a.jpg\" data-media-type=\"image\/png\" alt=\"The picture shows a spark chamber placed on a wooden base.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1677818\">\n<div data-type=\"title\" class=\"title\">Field and Force inside an Electron Gun<\/div>\n<p id=\"import-auto-id2596022\">(a) An electron gun has parallel plates separated by 4.00 cm and gives electrons 25.0 keV of energy. What is the electric field strength between the plates? (b) What force would this field exert on a piece of plastic with a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dde8f1b9ac0355d9e9f0f14f16708b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#53;&#48;&#48;&#32;&mu;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\" \/> charge that gets between the plates?<\/p>\n<p id=\"import-auto-id1281920\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2005074\">Since the voltage and plate separation are given, the electric field strength can be calculated directly from the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-530335481f74c696c3bc002d219b630c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#125;&#123;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -6px;\" \/>. Once the electric field strength is known, the force on a charge is found using<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-928d5ae4f9182a48923e640acfe39f52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#61;&#113;&#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;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -4px;\" \/>. Since the electric field is in only one direction, we can write this equation in terms of the magnitudes, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d8299257d5aa2d137573a387197c5716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#113;&#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;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id2596726\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id1332445\">The expression for the magnitude of the electric field between two uniform metal plates is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7cc1994da4abef85bdf66d3075c12236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#125;&#123;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2809743\">Since the electron is a single charge and is given 25.0 keV of energy, the potential difference must be 25.0 kV. Entering this value for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-23092ffa5ec93a64988b5fb4a342eb7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"29\" style=\"vertical-align: -4px;\" \/> and the plate separation of 0.0400 m, we obtain<\/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-4e4bfeef871ccfd4b5721fe254357ae4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#107;&#86;&#125;&#125;&#123;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#52;&#48;&#48;&#32;&#109;&#125;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#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;&#86;&#47;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"222\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2747313\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id2804210\">The magnitude of the force on a charge in an electric field is obtained from the equation <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-435\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f30e4f524204df4a3b6bdc3ab0688e6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#113;&#69;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1233382\">Substituting known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5463d898b9d2843d41d21058eaa62f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#53;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#50;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#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;&#86;&#47;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#51;&#49;&#51;&#32;&#78;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"352\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id3041830\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1677504\">Note that the units are newtons, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa05460010fcc77d9a5e05e19a36118b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#86;&#47;&#109;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#78;&#47;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"123\" style=\"vertical-align: -4px;\" \/>. The force on the charge is the same no matter where the charge is located between the plates. This is because the electric field is uniform between the plates.<\/p>\n<\/div>\n<p id=\"import-auto-id3067882\">In more general situations, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and also in the direction of lower potential <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;\" \/>. Furthermore, the magnitude of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> equals the rate of decrease of <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;\" \/> with distance. The faster <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;\" \/> decreases over distance, the greater the electric field. In equation form, the general relationship between voltage and electric field is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a6a4e40262ea4fd080a2dcf8ba6ab719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#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;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"94\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2673134\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3fc0cd561e5488b7d1a1f13b31c48743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> is the distance over which the change in potential, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/>, takes place. The minus sign tells us that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> points in the direction of decreasing potential. The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1552736\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Relationship between Voltage and Electric Field<\/div>\n<p id=\"import-auto-id1951129\">In equation form, the general relationship between voltage and electric field is<\/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-a6a4e40262ea4fd080a2dcf8ba6ab719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#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;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"94\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2822109\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3fc0cd561e5488b7d1a1f13b31c48743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> is the distance over which the change in potential, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/>, takes place. The minus sign tells us that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> points in the direction of decreasing potential. The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p>\n<\/div>\n<p>For continually changing potentials, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3fc0cd561e5488b7d1a1f13b31c48743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> become infinitesimals and differential calculus must be employed to determine the electric field.<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3051945\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2915369\">\n<li id=\"import-auto-id1551715\">The voltage between points A and B is\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9cb28fdd021f8cd679feb7ad773bdacf_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;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#69;&#100;&#125;&#92;&#92;&#32;&#69;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#66;&#125;&#125;&#125;&#123;&#100;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#40;&#117;&#110;&#105;&#102;&#111;&#114;&#109;&#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;&#69;&#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;&#45;&#32;&#102;&#105;&#101;&#108;&#100;&#32;&#111;&#110;&#108;&#121;&#41;&#44;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"270\" style=\"vertical-align: -18px;\" \/><\/div>\n<p>where <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 the distance from A to B, or the distance between the plates.<\/p>\n<\/li>\n<li id=\"import-auto-id1350386\">In equation form, the general relationship between voltage and electric field is\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a6a4e40262ea4fd080a2dcf8ba6ab719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#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;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"94\" 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-3fc0cd561e5488b7d1a1f13b31c48743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: 0px;\" \/> is the distance over which the change in potential, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/>, takes place. The minus sign tells us that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc16303eb82a65e9d2ecd5cb242a595_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> points in the direction of decreasing potential.) The electric field is said to be the <em data-effect=\"italics\"><em data-effect=\"italics\">gradient<\/em><\/em> (as in grade or slope) of the electric potential. <\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1726763\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1570612\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2677260\">\n<p id=\"import-auto-id2677744\">Discuss how potential difference and electric field strength are related. Give an example. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3080509\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1584974\">\n<p id=\"import-auto-id2608602\">What is the strength of the electric field in a region where the electric potential is constant? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2558866\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1985613\">\n<p id=\"import-auto-id1937578\">Will a negative charge, initially at rest, move toward higher or lower potential? Explain why. <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2607640\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-id1688468\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1688471\">\n<p id=\"eip-id1688472\">Show that units of V\/m and N\/C for electric field strength are indeed equivalent. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-id1688476\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1521962\">\n<p id=\"eip-id1521963\">What is the strength of the electric field between two parallel conducting plates separated by 1.00 cm and having a potential difference (voltage) between them of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-797a4603dbf9fbd11171bcf7952b5bcd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&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;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2635926\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1514530\">\n<p id=\"import-auto-id2723324\">The electric field strength between two parallel conducting plates separated by 4.00 cm is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-37e38a2e4deeb51191f022a36607f70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&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;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/>. (a) What is the potential difference between the plates? (b) The plate with the lowest potential is taken to be at zero volts. What is the potential 1.00 cm from that plate (and 3.00 cm from the other)?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1362631\">\n<p id=\"import-auto-id2806685\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ebb351e2cb233d1381a12e0f90802889_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;&#107;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"import-auto-id1727129\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6841b479ce9d7d68a00704c1663916e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#48;&#32;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2749816\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2751112\">\n<p id=\"import-auto-id2576062\">How far apart are two conducting plates that have an electric field strength of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3a4fecc0f2578706e8671725a2ccfd3_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;&#53;&#48;&#125;&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;&#47;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/> between them, if their potential difference is 15.0 kV?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2956436\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1564164\">(a) Will the electric field strength between two parallel conducting plates exceed the breakdown strength for air (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-02ddb085903a646a7e2125368ed251d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&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=\"90\" style=\"vertical-align: -4px;\" \/>) if the plates are separated by 2.00 mm and a potential difference of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-486f6adc5acf70649ebbda31a51bb3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/> is applied? (b) How close together can the plates be with this applied voltage?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2208045\">\n<p id=\"eip-id2208048\">(a) No. The electric field strength between the plates is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4f24243f5befd418fb303a1a5cdb0cf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;&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;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/> which is lower than the breakdown strength for air (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-02ddb085903a646a7e2125368ed251d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&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=\"90\" style=\"vertical-align: -4px;\" \/>).\n<\/p>\n<p id=\"eip-id3052084\">(b) 1.7 mm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1956925\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1676152\">\n<p id=\"import-auto-id2656762\">The voltage across a membrane forming a cell wall is 80.0 mV and the membrane is 9.00 nm thick. What is the electric field strength? (The value is surprisingly large, but correct. Membranes are discussed in <a href=\"\/contents\/56d8ca48-ab11-477d-b800-31b812b036b2@4\">Capacitors and Dielectrics<\/a> and <a href=\"\/contents\/41f56aa9-90a8-4c05-95d7-2608497844bf@3\">Nerve Conduction\u2014Electrocardiograms<\/a>.) You may assume a uniform electric field.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2573954\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2635324\">\n<p id=\"import-auto-id3141216\">Membrane walls of living cells have surprisingly large electric fields across them due to separation of ions. (Membranes are discussed in some detail in <a href=\"\/contents\/41f56aa9-90a8-4c05-95d7-2608497844bf@3\">Nerve Conduction\u2014Electrocardiograms<\/a>.) What is the voltage across an 8.00 nm\u2013thick membrane if the electric field strength across it is 5.50 MV\/m? You may assume a uniform electric field.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1538486\">\n<p id=\"import-auto-id1363987\">44.0 mV<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1636534\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3087915\">\n<p id=\"import-auto-id2977451\">Two parallel conducting plates are separated by 10.0 cm, and one of them is taken to be at zero volts. (a) What is the electric field strength between them, if the potential 8.00 cm from the zero volt plate (and 2.00 cm from the other) is 450 V? (b) What is the voltage between the plates?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2705714\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1667128\">\n<p id=\"import-auto-id2702894\">Find the maximum potential difference between two parallel conducting plates separated by 0.500 cm of air, given the maximum sustainable electric field strength in air to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-02ddb085903a646a7e2125368ed251d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#48;&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=\"90\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1638150\">\n<p id=\"import-auto-id2688443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-236b82f0cf19ac503fdd47794032d210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#32;&#107;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1563009\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2756439\">\n<p id=\"import-auto-id2721949\">A doubly charged ion is accelerated to an energy of 32.0 keV by the electric field between two parallel conducting plates separated by 2.00 cm. What is the electric field strength between the plates?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1948894\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2677744\">\n<p id=\"import-auto-id2840049\">An electron is to be accelerated in a uniform electric field having a strength of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8720a3bd07b8753e494a5f1df04d1b5b_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;&#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;\" \/>. (a) What energy in keV is given to the electron if it is accelerated through 0.400 m? (b) Over what distance would it have to be accelerated to increase its energy by 50.0 GeV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2607322\">\n<p id=\"import-auto-id2791118\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2306d820e73e9c8b86373e4c8e75ba1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#48;&#32;&#75;&#101;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"68\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id3166123\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bc555e398ab070b6405592d438cce2d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#46;&#48;&#32;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/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=\"fs-id2319336\">\n<dt>scalar<\/dt>\n<dd id=\"fs-id2349344\"> physical quantity with magnitude but no direction<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id2068816\">\n<dt>vector<\/dt>\n<dd id=\"fs-id1581918\"> physical quantity with both magnitude and direction<\/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-1041","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\/1041","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\/1041\/revisions"}],"predecessor-version":[{"id":1042,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1041\/revisions\/1042"}],"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\/1041\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1041"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1041"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1041"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1041"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}