{"id":1044,"date":"2017-10-27T16:31:16","date_gmt":"2017-10-27T16:31:16","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electrical-potential-due-to-a-point-charge\/"},"modified":"2017-11-08T03:26:06","modified_gmt":"2017-11-08T03:26:06","slug":"electrical-potential-due-to-a-point-charge","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/electrical-potential-due-to-a-point-charge\/","title":{"raw":"Electrical Potential Due to a Point Charge","rendered":"Electrical Potential Due to a Point Charge"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain point charges and express the equation for electric potential of a point charge.<\/li>\n<li>Distinguish between electric potential and electric field.<\/li>\n<li>Determine the electric potential of a point charge given charge and distance.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3164604\">Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge <em data-effect=\"italics\">[latex]q[\/latex]<\/em> from a large distance away to a distance of [latex]r[\/latex] from a point charge <em data-effect=\"italics\">[latex]Q[\/latex]<\/em>, and noting the connection between work and potential [latex]\\left(W=\\phantom{\\rule{0.25em}{0ex}}\u2013q\\Delta V\\right)[\/latex], it can be shown that the <em data-effect=\"italics\">electric potential [latex]V[\/latex] of a point charge<\/em> is <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]V=\\frac{\\text{kQ}}{r}\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{Point Charge}\\right),[\/latex]<\/div>\n<p>where <em data-effect=\"italics\">k<\/em> is a constant equal to<br>\n[latex]9.0\u00d7{\\text{10}}^{\\text{9}}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\\phantom{\\rule{0.25em}{0ex}}\\text{\u00b7}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{\\text{2}}\\text{\/}{\\text{C}}^{\\text{2}}[\/latex].<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2601528\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Electric Potential [latex]V[\/latex] of a Point Charge<\/div>\n<p id=\"import-auto-id2550748\">The electric potential [latex]V[\/latex] of a point charge is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-235\">[latex]V=\\frac{\\text{kQ}}{r}\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{Point Charge}\\right).[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id3099833\">The potential at infinity is chosen to be zero. Thus [latex]V[\/latex] for a point charge decreases with distance, whereas [latex]\\mathbf{\\text{E}}[\/latex] for a point charge decreases with distance squared:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{E}=\\frac{\\text{F}}{q}=\\frac{\\text{kQ}}{{r}^{2}}.[\/latex]<\/div>\n<p id=\"import-auto-id1358149\">Recall that the electric potential [latex]V[\/latex] is a scalar and has no direction, whereas the electric field [latex]\\mathbf{\\text{E}}[\/latex] is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as <em data-effect=\"italics\"><em data-effect=\"italics\">vectors<\/em><\/em>, taking magnitude and direction into account. This is consistent with the fact that [latex]V[\/latex] is closely associated with energy, a scalar, whereas [latex]\\mathbf{\\text{E}}[\/latex] is closely associated with force, a vector.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3084387\">\n<div data-type=\"title\" class=\"title\">What Voltage Is Produced by a Small Charge on a Metal Sphere?<\/div>\n<p id=\"import-auto-id1538032\">Charges in static electricity are typically in the nanocoulomb [latex]\\left(\\text{nC}\\right)[\/latex] to microcoulomb [latex]\\left(\\text{\u00b5C}\\right)[\/latex] range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a [latex]-3.00\\phantom{\\rule{0.25em}{0ex}}\\text{nC}[\/latex] static charge?<\/p>\n<p id=\"import-auto-id2921047\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id967675\">As we have discussed in <a href=\"\/contents\/99b463e0-bd31-4f32-afc6-fdb3d363db69@3\">Electric Charge and Electric Field<\/a>, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its center. Thus we can find the voltage using the equation [latex]V=\\text{kQ}\/r[\/latex].<\/p>\n<p id=\"import-auto-id1951323\"><strong>Solution<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2949583\">Entering known values into the expression for the potential of a point charge, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-174\">[latex]\\begin{array}{lll}V&amp; =&amp; k\\frac{Q}{r}\\\\ &amp; =&amp; \\left(\\text{8.99}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\u00b7{\\text{m}}^{2}\/{\\text{C}}^{2}\\right)\\left(\\frac{\\text{\u20133.00}\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}\\text{C}}{\\text{5.00}\u00d7{\\text{10}}^{\\text{\u20132}}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}\\right)\\\\ &amp; =&amp; \\text{\u2013539 V.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1553592\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2998408\">The negative value for voltage means a positive charge would be attracted from a larger distance, since the potential is lower (more negative) than at larger distances. Conversely, a negative charge would be repelled, as expected.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2635295\">\n<div data-type=\"title\" class=\"title\">What Is the Excess Charge on a Van de Graaff Generator<\/div>\n<p id=\"import-auto-id2712230\">A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV near its surface. (See <a href=\"#import-auto-id2591256\" class=\"autogenerated-content\">(Figure)<\/a>.) What excess charge resides on the sphere?  (Assume that each numerical value here is shown with three significant figures.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2591256\">\n<div class=\"bc-figcaption figcaption\">The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth\u2019s potential is taken to be zero as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2507611\" data-alt=\"The figure shows a Van de Graaff generator. The generator consists of a flat belt running over two metal pulleys. One pulley is positioned at the top and another at the bottom. The upper pulley is surrounded by an aluminum sphere. The aluminum sphere has a diameter of twenty five centimeters. Inside the sphere, the upper pulley is connected to a conductor which in turn is connected to a voltmeter for measuring the potential on the sphere. The lower pulley is connected to a motor. When the motor is switched on, the lower pulley begins turning the flat belt. The Van de Graaff generator with the above described setup produces a voltage of one hundred kilovolts. The potential on the surface of the sphere will be the same as that of a point charge at the center which is twelve point five centimeters away from the center. Thus the excess charge is calculated using the formula Q equals r times V divided by k.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a Van de Graaff generator. The generator consists of a flat belt running over two metal pulleys. One pulley is positioned at the top and another at the bottom. The upper pulley is surrounded by an aluminum sphere. The aluminum sphere has a diameter of twenty five centimeters. Inside the sphere, the upper pulley is connected to a conductor which in turn is connected to a voltmeter for measuring the potential on the sphere. The lower pulley is connected to a motor. When the motor is switched on, the lower pulley begins turning the flat belt. The Van de Graaff generator with the above described setup produces a voltage of one hundred kilovolts. The potential on the surface of the sphere will be the same as that of a point charge at the center which is twelve point five centimeters away from the center. Thus the excess charge is calculated using the formula Q equals r times V divided by k.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2555218\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1600063\">The potential on the surface will be the same as that of a point charge at the center of the sphere, 12.5 cm away. (The radius of the sphere is 12.5 cm.) We can thus determine the excess charge using the equation<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]V=\\frac{\\text{kQ}}{r}.[\/latex]<\/div>\n<p id=\"import-auto-id2712206\"><strong>Solution<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1597012\">Solving for [latex]Q[\/latex] and entering known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}Q&amp; =&amp; \\frac{\\text{rV}}{k}\\\\ &amp; =&amp; \\frac{\\left(0\\text{.}\\text{125}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)\\left(\\text{100}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}\\text{V}\\right)}{8.99\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\u00b7{\\text{m}}^{2}\/{\\text{C}}^{2}}\\\\ &amp; =&amp; \\text{1.39}\u00d7{\\text{10}}^{\u20136}\\phantom{\\rule{0.25em}{0ex}}\\text{C}=\\text{1.39 \u00b5C.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3083966\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2679258\">This is a relatively small charge, but it produces a rather large voltage. We have another indication here that it is difficult to store isolated charges.<\/p>\n<\/div>\n<p id=\"import-auto-id1033689\">The voltages in both of these examples could be measured with a meter that compares the measured potential with ground potential. Ground potential is often taken to be zero (instead of taking the potential at infinity to be zero). It is the potential difference between two points that is of importance, and very often there is a tacit assumption that some reference point, such as Earth or a very distant point, is at zero potential. As noted in <a href=\"\/contents\/650c4bb3-d190-44fd-9f2f-445a783abdec@4\">Electric Potential Energy: Potential Difference<\/a>, this is analogous to taking sea level as [latex]h=0[\/latex] when considering gravitational potential energy, [latex]{\\text{PE}}_{g}=\\text{mgh}[\/latex].<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1577454\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1559445\">\n<li id=\"import-auto-id1058734\"> Electric potential of a point charge is [latex]V=\\text{kQ}\/r[\/latex].<\/li>\n<li id=\"import-auto-id1597061\">Electric potential is a scalar, and electric field is a vector. Addition of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1534662\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2705937\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3164411\">\n<p id=\"import-auto-id1996620\">In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2511493\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2533119\">\n<p id=\"import-auto-id1312990\">Can the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2747886\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1677295\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1989107\">\n<p id=\"import-auto-id3158768\">A 0.500 cm diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0 pC charge on its surface. What is the potential near its surface?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2952050\">\n<p id=\"eip-id2758657\">144 V<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1318401\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1563750\">\n<p id=\"import-auto-id2911404\">What is the potential <\/p>\n[latex]0\\text{.}\\text{530}\u00d7{\\text{10}}^{\u201310}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]\n<p> from a proton (the average distance between the proton and electron in a hydrogen atom)?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1516605\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2975408\">\n<p id=\"import-auto-id1591409\">(a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2628531\">\n<p id=\"eip-id2628533\">\n(a) 1.80 km\n<\/p>\n<p id=\"eip-id2518659\">\n(b) A charge of 1 C is a very large amount of charge; a sphere of radius 1.80 km is not practical.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1281542\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2834320\">\n<p id=\"import-auto-id2574621\">How far from a [latex]1\\text{.}\\text{00 \u00b5C}[\/latex] point charge will the potential be 100 V? At what distance will it be [latex]\\text{2.00}\u00d7{\\text{10}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{V}?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id963449\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1637628\">\n<p id=\"import-auto-id2985150\">What are the sign and magnitude of a point charge that produces a potential of [latex]\\text{\u20132.00 V}[\/latex] at a distance of 1.00 mm?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2188544\">\n<p id=\"eip-id930683\">[latex]\u20132\\text{.}\\text{22}\u00d7{\\text{10}}^{\u201313}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1281900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1363380\">\n<p id=\"import-auto-id3039700\">If the potential due to a point charge is <\/p>\n[latex]5\\text{.}\\text{00}\u00d7{\\text{10}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]\n<p> at a distance of 15.0 m, what are the sign and magnitude of the charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2554432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1151542\">\n<p id=\"import-auto-id2957851\">In nuclear fission, a nucleus splits roughly in half. (a) What is the potential <\/p>\n[latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{\u201314}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]\n<p>from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1775233\">\n<p id=\"eip-id1775235\">(a) [latex]3\\text{.}\\text{31}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex] <\/p>\n<p id=\"eip-id2964744\">(b) 152 MeV\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id890041\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1562240\">\n<p id=\"import-auto-id3107309\">A research Van de Graaff generator has a 2.00-m-diameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV at this distance?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1320070\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2527889\">\n<p id=\"import-auto-id2600505\">An electrostatic paint sprayer has a 0.200-m-diameter metal sphere at a potential of 25.0 kV that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to arrive at the object with a speed of 10.0 m\/s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2570849\">\n<p id=\"eip-id2570852\">(a) [latex]2\\text{.}\\text{78}\u00d7{\\text{10}}^{-7}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex] <\/p>\n<p id=\"eip-id2343580\">(b) [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{-10}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex] <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1497489\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1033446\">\n<p id=\"import-auto-id3167781\">In one of the classic nuclear physics experiments at the beginning of the 20th century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was 5.00 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1183848\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2984651\">\n<p id=\"import-auto-id1540980\">(a) What is the potential between two points situated 10 cm and 20 cm from a [latex]3\\text{.}0 \u00b5C[\/latex] point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-35\">\n<p id=\"eip-id2640726\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1767627\"> (a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graaff terminal?<\/p>\n<p id=\"eip-id2057005\">(b) What is unreasonable about this result? <\/p>\n<p id=\"eip-id2899769\">(c) Which assumptions are responsible?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"eip-id1598912\">(a)  [latex]2.96\u00d7{10}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex]<\/p>\n<p id=\"eip-id1305472\">(b) This velocity is far too great. It is faster than the speed of light.<\/p>\n<p id=\"eip-id2223518\">(c) The assumption that the speed of the electron is far less than that of light and that the problem does not require a relativistic treatment produces an answer greater than the speed of light. <\/p>\n<\/div>\n<\/div>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain point charges and express the equation for electric potential of a point charge.<\/li>\n<li>Distinguish between electric potential and electric field.<\/li>\n<li>Determine the electric potential of a point charge given charge and distance.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3164604\">Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge <em data-effect=\"italics\"><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> from a large distance away to a distance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> from a point charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><\/em>, and noting the connection between work and potential <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1596f768fbb0da3018885af555461909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#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;&#113;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/>, it can be shown that the <em data-effect=\"italics\">electric 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;\" \/> of a point charge<\/em> 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-a2547ef702c0296860922b5afced34a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#125;&#123;&#114;&#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;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#111;&#105;&#110;&#116;&#32;&#67;&#104;&#97;&#114;&#103;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"191\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where <em data-effect=\"italics\">k<\/em> is a constant equal to<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2a9ce170cc7e739513760411f98371b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#46;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#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;&middot;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"126\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2601528\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Electric 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;\" \/> of a Point Charge<\/div>\n<p id=\"import-auto-id2550748\">The electric 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;\" \/> of a point charge is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-235\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6f6b640e88132a2cd930560a03179b16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#125;&#123;&#114;&#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;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#111;&#105;&#110;&#116;&#32;&#67;&#104;&#97;&#114;&#103;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"191\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id3099833\">The potential at infinity is chosen to be zero. Thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> for a point charge decreases with distance, 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;\" \/> for a point charge decreases with distance squared:<\/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-e45972b93dff610cc3c15514aaae3402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;&#123;&#113;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#125;&#123;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"98\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1358149\">Recall that the electric 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;\" \/> is a scalar and has no direction, whereas the 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 a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as <em data-effect=\"italics\"><em data-effect=\"italics\">vectors<\/em><\/em>, taking magnitude and direction into account. This is consistent with the fact that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is closely associated with energy, a scalar, 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 closely associated with force, a vector.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3084387\">\n<div data-type=\"title\" class=\"title\">What Voltage Is Produced by a Small Charge on a Metal Sphere?<\/div>\n<p id=\"import-auto-id1538032\">Charges in static electricity are typically in the nanocoulomb <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01c2e66bff47924c75564bab48f15b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"35\" style=\"vertical-align: -4px;\" \/> to microcoulomb <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d8e4f5aa5a4b9e246ea20f2618c5b4ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&micro;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"25\" style=\"vertical-align: -4px;\" \/> range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8cdbce71961f27d59a5a53f29e2a3c0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;&#48;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"71\" style=\"vertical-align: 0px;\" \/> static charge?<\/p>\n<p id=\"import-auto-id2921047\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id967675\">As we have discussed in <a href=\"\/contents\/99b463e0-bd31-4f32-afc6-fdb3d363db69@3\">Electric Charge and Electric Field<\/a>, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its center. Thus we can find the voltage using the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31c133aa33818c9e074b4248deee0024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#47;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/>.<\/p>\n<p id=\"import-auto-id1951323\"><strong>Solution<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2949583\">Entering known values into the expression for the potential of a point charge, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-174\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-29e9e3c38e0016c5b09475d0ce51ee27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#86;&#38;&#32;&#61;&#38;&#32;&#107;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#125;&#123;&#114;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#46;&#57;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&middot;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#53;&#51;&#57;&#32;&#86;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"71\" width=\"302\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"import-auto-id1553592\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2998408\">The negative value for voltage means a positive charge would be attracted from a larger distance, since the potential is lower (more negative) than at larger distances. Conversely, a negative charge would be repelled, as expected.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2635295\">\n<div data-type=\"title\" class=\"title\">What Is the Excess Charge on a Van de Graaff Generator<\/div>\n<p id=\"import-auto-id2712230\">A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV near its surface. (See <a href=\"#import-auto-id2591256\" class=\"autogenerated-content\">(Figure)<\/a>.) What excess charge resides on the sphere?  (Assume that each numerical value here is shown with three significant figures.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2591256\">\n<div class=\"bc-figcaption figcaption\">The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth\u2019s potential is taken to be zero as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2507611\" data-alt=\"The figure shows a Van de Graaff generator. The generator consists of a flat belt running over two metal pulleys. One pulley is positioned at the top and another at the bottom. The upper pulley is surrounded by an aluminum sphere. The aluminum sphere has a diameter of twenty five centimeters. Inside the sphere, the upper pulley is connected to a conductor which in turn is connected to a voltmeter for measuring the potential on the sphere. The lower pulley is connected to a motor. When the motor is switched on, the lower pulley begins turning the flat belt. The Van de Graaff generator with the above described setup produces a voltage of one hundred kilovolts. The potential on the surface of the sphere will be the same as that of a point charge at the center which is twelve point five centimeters away from the center. Thus the excess charge is calculated using the formula Q equals r times V divided by k.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a Van de Graaff generator. The generator consists of a flat belt running over two metal pulleys. One pulley is positioned at the top and another at the bottom. The upper pulley is surrounded by an aluminum sphere. The aluminum sphere has a diameter of twenty five centimeters. Inside the sphere, the upper pulley is connected to a conductor which in turn is connected to a voltmeter for measuring the potential on the sphere. The lower pulley is connected to a motor. When the motor is switched on, the lower pulley begins turning the flat belt. The Van de Graaff generator with the above described setup produces a voltage of one hundred kilovolts. The potential on the surface of the sphere will be the same as that of a point charge at the center which is twelve point five centimeters away from the center. Thus the excess charge is calculated using the formula Q equals r times V divided by k.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2555218\"><strong>Strategy<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1600063\">The potential on the surface will be the same as that of a point charge at the center of the sphere, 12.5 cm away. (The radius of the sphere is 12.5 cm.) We can thus determine the excess charge using the equation<\/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-dd0ecf54fd4afc022a8e9217f5ed0031_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2712206\"><strong>Solution<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id1597012\">Solving for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/> and entering 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-0860fa15e8787e7ddfb16871bcc600c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#81;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#86;&#125;&#125;&#123;&#107;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#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;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#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;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#56;&#46;&#57;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&middot;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#57;&#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;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#57;&#32;&micro;&#67;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"74\" width=\"223\" style=\"vertical-align: -29px;\" \/><\/div>\n<p id=\"import-auto-id3083966\"><strong>Discussion<\/strong><em data-effect=\"italics\"><\/em><\/p>\n<p id=\"import-auto-id2679258\">This is a relatively small charge, but it produces a rather large voltage. We have another indication here that it is difficult to store isolated charges.<\/p>\n<\/div>\n<p id=\"import-auto-id1033689\">The voltages in both of these examples could be measured with a meter that compares the measured potential with ground potential. Ground potential is often taken to be zero (instead of taking the potential at infinity to be zero). It is the potential difference between two points that is of importance, and very often there is a tacit assumption that some reference point, such as Earth or a very distant point, is at zero potential. As noted in <a href=\"\/contents\/650c4bb3-d190-44fd-9f2f-445a783abdec@4\">Electric Potential Energy: Potential Difference<\/a>, this is analogous to taking sea level as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4498f0ceb1b7ca6b196ab84e449c19b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: 0px;\" \/> when considering gravitational potential energy, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d37357fbe719e72be95771f9f808eb70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#125;&#95;&#123;&#103;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1577454\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1559445\">\n<li id=\"import-auto-id1058734\"> Electric potential of a point charge is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31c133aa33818c9e074b4248deee0024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#81;&#125;&#47;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/>.<\/li>\n<li id=\"import-auto-id1597061\">Electric potential is a scalar, and electric field is a vector. Addition of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1534662\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2705937\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3164411\">\n<p id=\"import-auto-id1996620\">In what region of space is the potential due to a uniformly charged sphere the same as that of a point charge? In what region does it differ from that of a point charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2511493\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2533119\">\n<p id=\"import-auto-id1312990\">Can the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2747886\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1677295\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1989107\">\n<p id=\"import-auto-id3158768\">A 0.500 cm diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0 pC charge on its surface. What is the potential near its surface?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2952050\">\n<p id=\"eip-id2758657\">144 V<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1318401\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1563750\">\n<p id=\"import-auto-id2911404\">What is the potential <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f1e130ad73d52a4ff09f86bdd841995_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#51;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#49;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -1px;\" \/><\/p>\n<p> from a proton (the average distance between the proton and electron in a hydrogen atom)?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1516605\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2975408\">\n<p id=\"import-auto-id1591409\">(a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential 5.00 MV? (b) What does your answer imply about the practical aspect of isolating such a large charge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2628531\">\n<p id=\"eip-id2628533\">\n(a) 1.80 km\n<\/p>\n<p id=\"eip-id2518659\">\n(b) A charge of 1 C is a very large amount of charge; a sphere of radius 1.80 km is not practical.\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1281542\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2834320\">\n<p id=\"import-auto-id2574621\">How far from a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66dac5d1ddb27220c77565bb59c84c62_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;&#48;&#48;&#32;&micro;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/> point charge will the potential be 100 V? At what distance will it be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1172a1a14fa9647524c69db8dfbbdfb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#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;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id963449\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1637628\">\n<p id=\"import-auto-id2985150\">What are the sign and magnitude of a point charge that produces a potential of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5b79e108746041b44820bdffd2d64c19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#50;&#46;&#48;&#48;&#32;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/> at a distance of 1.00 mm?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2188544\">\n<p id=\"eip-id930683\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-81a98d154c319bca8eeec47f7d2a87bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#49;&#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;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"104\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1281900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1363380\">\n<p id=\"import-auto-id3039700\">If the potential due to a point charge is <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ec9d428b6b9d6da5b21a404d309270d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/p>\n<p> at a distance of 15.0 m, what are the sign and magnitude of the charge?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2554432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1151542\">\n<p id=\"import-auto-id2957851\">In nuclear fission, a nucleus splits roughly in half. (a) What is the potential <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-65b00ce5d58f6259169460185ec64191_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;&#45;&#49;&#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;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -1px;\" \/><\/p>\n<p>from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1775233\">\n<p id=\"eip-id1775235\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-957b74dc4072cbe6e8b3d7a525c7c8a2_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;&#51;&#49;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/> <\/p>\n<p id=\"eip-id2964744\">(b) 152 MeV\n<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id890041\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1562240\">\n<p id=\"import-auto-id3107309\">A research Van de Graaff generator has a 2.00-m-diameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what distance from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV at this distance?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1320070\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2527889\">\n<p id=\"import-auto-id2600505\">An electrostatic paint sprayer has a 0.200-m-diameter metal sphere at a potential of 25.0 kV that repels paint droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to arrive at the object with a speed of 10.0 m\/s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id2570849\">\n<p id=\"eip-id2570852\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a0a9c55bf8f3857afb559c24e8a4c0a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/> <\/p>\n<p id=\"eip-id2343580\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-13802090c4913e4b60bc07185abbd2ae_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;&#45;&#49;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/> <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1497489\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1033446\">\n<p id=\"import-auto-id3167781\">In one of the classic nuclear physics experiments at the beginning of the 20th century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was 5.00 MeV, how close to the gold nucleus (79 protons) could it come before being deflected?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1183848\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2984651\">\n<p id=\"import-auto-id1540980\">(a) What is the potential between two points situated 10 cm and 20 cm from a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-331681ffb2359b69f10901a11024e49d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&micro;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"37\" style=\"vertical-align: 0px;\" \/> point charge? (b) To what location should the point at 20 cm be moved to increase this potential difference by a factor of two?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-35\">\n<p id=\"eip-id2640726\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1767627\"> (a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graaff terminal?<\/p>\n<p id=\"eip-id2057005\">(b) What is unreasonable about this result? <\/p>\n<p id=\"eip-id2899769\">(c) Which assumptions are responsible?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"eip-id1598912\">(a)  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08c788f730343137ffe398a2585a0f28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#57;&#54;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"eip-id1305472\">(b) This velocity is far too great. It is faster than the speed of light.<\/p>\n<p id=\"eip-id2223518\">(c) The assumption that the speed of the electron is far less than that of light and that the problem does not require a relativistic treatment produces an answer greater than the speed of light. <\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1044","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\/1044","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\/1044\/revisions"}],"predecessor-version":[{"id":1045,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1044\/revisions\/1045"}],"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\/1044\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1044"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1044"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1044"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1044"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}