{"id":215,"date":"2023-02-22T13:15:21","date_gmt":"2023-02-22T18:15:21","guid":{"rendered":"https:\/\/opentextbc.ca\/foundationsofphysics\/?post_type=chapter&#038;p=215"},"modified":"2023-09-13T16:51:14","modified_gmt":"2023-09-13T20:51:14","slug":"electrostatic-fields-forces","status":"publish","type":"chapter","link":"https:\/\/opentextbc.ca\/foundationsofphysics\/chapter\/electrostatic-fields-forces\/","title":{"raw":"Electrostatic Fields &amp; Forces","rendered":"Electrostatic Fields &amp; Forces"},"content":{"raw":"<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Resources<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=0Lx0c0TlzFI\">Mechanical Universe - Episode 28 - Static Electricity<\/a><\/li>\r\n \t<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=wq9TjQZDrAA\">Mechanical Universe - Episode 29 - The Electric Field<\/a><\/li>\r\n \t<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Charge-Interactions-Revisited\">Charge Interactions Revisited<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nEquations Introduced and Used in this Topic:\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}=q\\vec{E}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{E}=\\dfrac{kq}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]V=\\dfrac{kq_1{q}_2}{d}[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/p>\r\n<p style=\"text-align: center;\">Bohr Radius [latex](a_0, r_{\\text{Bohr}})=5.29177 \u00d7 10^{\u221211}\\text{ m}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Excited State Radius [latex]r =n^2 a_o[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\vec{F}=m\\vec{a}[\/latex]<\/p>\r\nWhere\u2026\r\n<ul>\r\n \t<li>[latex]F[\/latex] is generally the Net Force acting on the charged object<\/li>\r\n \t<li>[latex]F_{el}[\/latex]\u00a0is the Electrostatic Force, measured in newtons (N)<\/li>\r\n \t<li>[latex]E[\/latex] is the Electric Field, measured in newtons per coulomb (N\/C) or volts per metre (V\/m)<\/li>\r\n \t<li>[latex]k[\/latex] is the Coulomb's constant, measured in newton-metres squared per coulomb squared (N\u00b7m<sup>2<\/sup>\/C<sup>2<\/sup>)<\/li>\r\n \t<li>[latex]q, q_1[\/latex] &amp; [latex]q_2[\/latex] are the Charges you are studying, measured in coulombs (C)<\/li>\r\n \t<li>[latex]q[\/latex] if negative indicates that it has an excess of electrons (or shortage of protons)<\/li>\r\n \t<li>[latex]q[\/latex] if positive indicates that it has an excess of protons (or shortage of electrons)<\/li>\r\n \t<li>[latex]d[\/latex] is the Distance away from the Charge Center, measured in metres (m)<\/li>\r\n \t<li>[latex]V[\/latex] is the Potential Difference, measured in volts (V) or joules\/coulombs (J\/C)<\/li>\r\n \t<li>[latex]a[\/latex] is the Acceleration of the object measured in metres per second squared (m\/s<sup>2<\/sup>)<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">The static charge on an object can be calculated by the number of excess electrons or protons time \u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C<\/p>\r\n\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Elementary Particles[footnote]<a href=\"https:\/\/www.a-levelphysicstutor.com\/index-field.php\">A-Level Physics Tutor - Electricity<\/a>[\/footnote]<\/caption>\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Particle<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Mass<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Charge<\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Electron (e<sup>\u2212<\/sup> or \u00df<sup>\u2212<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">9.10938356(11) \u00d7 10<sup>\u221231<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">\u22121.6021766208 \u00d7 10<sup>\u221219<\/sup> C (\u2212e)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Proton (p, p<sup>+<\/sup> or N<sup>+<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">1.672621898(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">+1.6021766208 \u00d7 10<sup>\u221219<\/sup> C (+e)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Neutron (n, n<sup>0<\/sup>, N<sup>0<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">1.674927471(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">0 C<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Alpha particle (\u03b1, \u03b1<sup>2+<\/sup>, He<sup>2+<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">6.644657230(82) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">+3.2043532416 \u00d7 10<sup>\u221219<\/sup> C (+2e)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Important Constants<\/strong> (round to needed accuracy)\r\n<ul>\r\n \t<li>Coulomb's constant: [latex]k[\/latex] =\u00a0 8.9875517873681764 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\r\n \t<li>Electron volt: 1 eV = 1.60217653(14) \u00d7 10<sup>\u221219<\/sup> J<\/li>\r\n \t<li>Bohr Radius: (a<sub>0<\/sub>, r<sub>Bohr<\/sub>) =\u00a0 5.29177 \u00d7 10<sup>\u221211<\/sup> m<\/li>\r\n<\/ul>\r\n<h1>20.1 Electrostatic Forces<\/h1>\r\n<div class=\"textbox\">In Research News:<a href=\"https:\/\/www.smithsonianmag.com\/smart-news\/super-strong-electric-forces-may-have-helped-tiny-clumps-dust-seed-planets-180973756\/\"> Super-Strong Electric Forces May Have Helped Tiny Clumps of Dust Seed the Planets<\/a><\/div>\r\nEquations Introduced or Used for this Section:\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{net}=m\\vec{a}[\/latex]<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/p>\r\nHistorical records indicate awareness of numerous electrical phenomenon for over 3000 years but the explanations of what it was quite different from how we understand it today. For instance, Thales of Miletus (624-546 BCE) thought that we were seeing the soul of a piece of amber that when rubbed on a fur it would attracted small objects. Likewise, lightning, St Elmo\u2019s Fire and the electric ray all were well known electrical phenomena that had equally mystic nature.\r\n\r\nDifferent from the study of mathematics and astronomy, the first recorded study of electrical phenomenon was published by William Gilbert (1544-1603) three years before his death from what is believed to have been the bubonic plague. The name for electricity came from Gilbert's use of the Latin word \u201celectrica\u201d (\u1f24\u03bb\u03b5\u03ba\u03c4\u03c1\u03bf\u03bd) meaning \u201clike amber\u201d. While electrica had been used since the 13th century, Gilbert defined the name respecting its attractive or repulsive properties.\r\n\r\nFrom Gilbert\u2019s work, three rules of electrostatics were developed:\r\n<ul>\r\n \t<li>There are only two kinds of electric charge<\/li>\r\n \t<li>Two objects charged alike (having the same kind of charge) repel each other<\/li>\r\n \t<li>Two objects charged oppositely attract each other.<\/li>\r\n<\/ul>\r\nExplaining the nature of charged objects was more challenging.\u00a0 Gilbert speculated that the \u201camber effect\u201d was the result of the effluvium (small particles that flowed from one charged object to another that did not have any mass or volume). Over the next century, multiple small advances were made by numerous philosopher\/scientists with one, Charles Fran\u00e7ois de Cisternay du Fay (1698-1739), proposing that there were two types of electric charge: a positive charged fluid and a negative charged fluid. However, it was Benjamin Franklin (1706-1790) who formulated that electric charge was but a single fluid, not that an object having an excess of this fluid was positively charged and an object having a deficit of this fluid was negatively charged.\r\n\r\nWe now know that the Franklin one-fluid model is fundamentally correct in that it is the electron that can be understood as the source of electric charge. Because of this construct, everything in conventional electricity is defined in terms of the movement of this excess charge. However it was not until the late 1890\u2019s that evidence accumulated to support what electric charge actually was. By this time, the battery, lightbulb, telephone and a host of other electrical inventions had been created, and all were understood using Franklin\u2019s concept of the movement of excess fluid. The legacy of Franklin\u2019s understanding can still be found in electrical terminology, including: positive and negative charge, charging and discharging, conductors and condensers, the lightning rod, and the flow of electric charge called electric current.\u00a0 Franklin[footnote]Benjamin Franklin actually did his famous kite experiment in a thunderstorm in 1752, collecting electric charge in a capacitor which was known at that time as a Leyden Jar. Greater detail of this experiment can be found at: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Kite_experiment\">Kite experiment<\/a> &amp; <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leyden_jar\">Leyden jar<\/a>[\/footnote] and William Watson (1715-1787) both proposed the conservation of electric charge. This concept was later interpreted as an isolated system wherein the net charge remains constant and, if matter is created or destroyed, charge is created and destroyed in equal amounts.\r\n<div class=\"textbox textbox--sidebar\">\r\n\r\nArticle to Read: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Coulomb%27s_law\">Coulomb\u2019s Law<\/a>\r\n\r\nExtra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Coulomb-s-Law\">Coulomb's Law<\/a>\r\n\r\nExtra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Inverse-Square-Law\">Inverse Square Law<\/a>\r\n\r\nExtra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Newton-s-Laws-and-the-Electrical-Force\">Newton\u2019s Laws and the Electrical Force<\/a>\r\n\r\n<\/div>\r\nQuantifying the force that exists between similar or oppositely charged objects was proposed by Joseph Priestly (1733-1804) to follow an inverse square law as had been proposed by Isaac Newton for gravity. Priestly posited this relationship in his 700 page book The History and Present State of Electricity (1767). Priestly's inverse square law became formalized by Charles-Augustin de Coulomb (1736-1806), who demonstrated evidence of its authenticity using a device called the torsion balance. Coulomb later was able to demonstrate that an inverse square law relationship also existed between the poles of bar magnets.\r\n\r\nIn algebraic form, Coulomb\u2019s Law\u00a0is stated as:\r\n<p style=\"text-align: center;\">[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{{d}^2}[\/latex]<\/p>\r\nWhere\u2026\r\n<ul>\r\n \t<li>[latex]k[\/latex] = 8.9875517873681764 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup> (Coulomb\u2019s constant)<\/li>\r\n \t<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C) (number of electrons\/protons)<\/li>\r\n \t<li>[latex]d[\/latex] = the distance between the charge centres (measured in metres)<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\" style=\"page-break-before: always;\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.1.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA small plastic sphere has a charge of \u221210<sup>\u221214<\/sup> C.\r\n<ol type=\"i\">\r\n \t<li>Does it have an excess or a deficit of electrons on it?<\/li>\r\n \t<li>How many electrons?<\/li>\r\n<\/ol>\r\n<strong>Solution<\/strong>\r\n<ol type=\"i\">\r\n \t<li>Since the sphere has a negative charge, it has an excess of electrons.<\/li>\r\n \t<li>Data:\r\n<ul>\r\n \t<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)<\/li>\r\n \t<li>[latex]n[\/latex] = Find<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/li>\r\n \t<li>\u221210<sup>\u221214<\/sup> C = (+ 1.602 \u00d7 10<sup>\u221219<\/sup> C)([latex]n[\/latex])<\/li>\r\n \t<li>[latex]n[\/latex] = \u221210<sup>\u221214<\/sup> C \u00f7 (\u2212 1.602) \u00d7 10<sup>\u221219<\/sup> C<\/li>\r\n \t<li>[latex]n[\/latex] = 62<span style=\"margin-left: 0.25em;\">400<\/span> electrons\u00a0 (\u2248 60<span style=\"margin-left: 0.25em;\">000<\/span> e<sup>\u2212<\/sup>)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.1.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the magnitude and direction of the force acting on a charge of +4.2 \u00b5C, 5.0 mm away from a charge of +4.5 \u00b5C?\r\n\r\n<img class=\"aligncenter wp-image-466\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313.png\" alt=\"\" width=\"500\" height=\"224\" \/>\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]F_{el}[\/latex] = Find<\/li>\r\n \t<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\r\n \t<li>[latex]q_1[\/latex] =\u00a0 4.2 \u00d7 10<sup>\u22126<\/sup> C<\/li>\r\n \t<li>[latex]q_2[\/latex] = 4.5 \u00d7 10<sup>\u22126<\/sup> C<\/li>\r\n \t<li>[latex]d[\/latex] = 5.0 \u00d7 10<sup>\u22123<\/sup> m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{\\text{d}^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{(8.99\\times10^9 \\text{ Nm}^2\\text{\/C}^2)(4.2\\times10^{-6}\\text{ C})(4.5\\times10^{-6})}{(5.0\\times10^{-3}\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}[\/latex] = 6800 N directed away from each other (both are positive charges)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.1.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"alignright wp-image-468\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom.png\" alt=\"\" width=\"250\" height=\"313\" \/>The double-ionized lithium atom consists of one electron (e<sup>\u2212<\/sup>) orbiting a core of three protons (p+) and three or four neutrons.\u00a0 If its average orbital radius is 0.0167 nm, what is the electrostatic force that exists between the electron and the core?\r\n\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(3)(4.5\\times10^{-6})}{(0.0167\\times10^{-9}\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}[\/latex] = 2.48 \u00d7 10<sup>\u22129<\/sup> N (attractive)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.1.4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the acceleration of the single electron orbiting the three protons in the lithium atom described above?\u00a0 (Use the lowest energy level radius of 0.0167 nm.)\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]a[\/latex] = Find<\/li>\r\n \t<li>[latex]\\vec{F}_{el}[\/latex] = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\r\n \t<li>[latex]m[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup>\u00a0N<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\r\n \t<li>[latex]m\\vec{\\text{a}}[\/latex] = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\r\n \t<li>(9.11 \u00d7 10<sup>\u221239 <\/sup>kg)([latex]\\vec{a}[\/latex]) = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\r\n \t<li>[latex]\\vec{a}[\/latex] = 2.7 \u00d7 10<sup>\u221239\u00a0<\/sup>m\/s<sup>2<\/sup>... (towards the core)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.1.5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the distance that an electron would have to be away from the lithium core described above to experience an electrostatic force equal to its weight?\r\n\r\n<strong>Solution<\/strong>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]m[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\r\n \t<li>[latex]q[\/latex] = 1.602 \u00d7 10<sup>\u221219<\/sup> C<\/li>\r\n \t<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\r\n \t<li>[latex]g[\/latex] = 9.80 m\/s<sup>2<\/sup><\/li>\r\n \t<li>[latex]d[\/latex] = Find<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>Use [latex]w=\\vec{F}_{el}[\/latex]<\/li>\r\n \t<li>[latex]mg=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>(9.11\u00a01 \u00d7 10<sup>\u221231<\/sup> kg)(9.80 m\/s<sup>2<\/sup>) = [latex]\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})^2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]d^2[\/latex] = 6.9 \u00d7 10<sup>\u221228<\/sup> N\/m<sup>2<\/sup> \u00f7 8.9 \u00d7 10<sup>\u221230<\/sup> N<\/li>\r\n \t<li>[latex]d[\/latex] = 8.8 m<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 20.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>A ping pong ball has a charge of \u221210<sup>\u221212<\/sup> C.\r\n<ol type=\"i\">\r\n \t<li>Does it have an excess or a deficit of electrons on it?<\/li>\r\n \t<li>How many electrons?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>What is the magnitude and direction of the electrostatic force acting on a charge of + 5.0 \u00d7 10<sup>\u22128<\/sup> C that is 5.0 cm away from a charge of + 4.0 \u00d7 10<sup>\u22129<\/sup> C?<\/li>\r\n \t<li>The hydrogen atom consists of an electron (e<sup>\u2212<\/sup>) and a proton (p<sup>+<\/sup>). If their average separation is 5.29 \u00d7 10<sup>\u221211<\/sup> m what is the electrostatic force that exists between them?<\/li>\r\n \t<li>Two charged objects, one of +5.0 \u00d7 10<sup>\u22127<\/sup> C and the other of \u22122.0 \u00d7 10<sup>\u22127<\/sup> C, attract each other with an electrostatic force of 100 N. How far apart are they?<\/li>\r\n \t<li>The hydrogen atom has been extensively studied and has been responsible for scientists to be able to both create and understand the atomic structure of atoms. Given that the average radius for the first three excited states are:\r\n<ul>\r\n \t<li><strong>1st excited state is 2.12 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong>10<\/strong><\/sup><strong> m <\/strong><\/li>\r\n \t<li><strong>2nd excited state is 4.76 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong>10<\/strong><\/sup><strong> m <\/strong><\/li>\r\n \t<li><strong>3rd excited state is 8.46 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong><strong>10<\/strong><\/strong><\/sup><\/li>\r\n<\/ul>\r\nFind the different electrostatic forces experience by the electron in the first and third excited states.<\/li>\r\n \t<li>What is the acceleration of an electron orbiting the proton in the hydrogen atom? (Use the Bohr Radius...\u00a0 Ground State Radius.)<\/li>\r\n \t<li>Two small metal pellets (mass of 7.5 \u00d7 10<sup>\u22124<\/sup> kg) each have the same negative charge and repel each other with an electrostatic force of 1.5 \u00d7 10<sup>\u22128<\/sup> N when 5.0 mm apart.\r\n<ol type=\"i\">\r\n \t<li>What would they accelerate with if they were free to move?<\/li>\r\n \t<li>How many electrons are on each of them?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>A singly ionized helium atom composed of a core that has 2 protons and 2 neutrons, has one electron orbiting. At what rate are the core and the electron accelerating towards each other if the electron maintains an average distance of 2.21 \u00d7 10<sup>\u221211<\/sup> m from the core?<\/li>\r\n \t<li>What is the distance an electron would have to be from a proton to have an acceleration of 10 g\u2019s?<\/li>\r\n \t<li>What is the distance that an electron would have to be from a proton to experience an electrostatic force equal to its weight?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>\u00a020.2 Electric Fields (Point Charges)<\/h1>\r\n<div>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Field-Intensity\">Electric Field Intensity<\/a><\/li>\r\n \t<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Field-Lines\">Electric Field Lines<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\nEquations Introduced or Used for this Section:\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>[latex]\\vec{E}=\\dfrac{kq}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}= q\\vec{E}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}= m\\vec{a}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n<\/ul>\r\nike the concept of gravitational fields covered in Chapter 17.2, electrostatics share a similar concept of electric field (also magnetic fields). Historically, the first reference to such a concept was by Gilbert when he was describing the sphere of influence of a charged object. The equation quantifying the electric field can be derived in similar fashion to the derivation of the gravitational field. This can be shown as follows:\r\n<p style=\"text-align: center;\">[latex]\\vec{F}_{el}= q\\vec{E}[\/latex] is equated to [latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">This means that [latex]q\\vec{E}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">cancelling out the common charge leaves us with... E = [latex]\\dfrac{kq_1}{d^2}[\/latex]<\/p>\r\nThe direction of the electric field as defined by a negative or positive sign depends on the net charge of the object... Is the overall charge of the object negative or positive, and how it would react on a small positive test charge placed in the electric field . This means that the electric field can be written as either a positive or a negative quantity. In practice, many will ignore the positive and negative signs and just draw the field as attractive if the net charge is negative or repulsive if the net charge is positive.\r\n<p style=\"text-align: center;\">[latex]\\vec{E}=\\pm\\dfrac{kq_1}{d^2}[\/latex]<\/p>\r\nThe units of electrical field strength are N\/C. This equation relates to the electric field at any distance away from a charged object and is a inverse square law relationship.\r\n<p style=\"text-align: center;\">[latex]\\vec{E}[\/latex] \u221d [latex]\\dfrac{1}{d^2}[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--sidebar\"><strong>Extra Help<\/strong> - <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Fields-and-Conductors\">Electric Fields and Conductors<\/a><\/div>\r\nWhen we look at the electric field strength as we move inside the charged object\u2019s surface, the electric field strength drops to zero. Anything inside the charged object is shielded from the electric field; this is why a radio signal on a vehicle fades out when driving through a tunnel or inside the metal frame of a bridge.\r\n<p style=\"text-align: center;\">[latex]\\vec{E}[\/latex] = 0 N\/C<\/p>\r\nElectric fields can have multiple variations of positive and negatively charged objects. Examples showing field lines of various combinations of positive and negative charges are shown below.\r\n\r\n<img class=\"aligncenter wp-image-485 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines.jpg\" alt=\"\" width=\"350\" height=\"174\" \/>\r\n\r\nThese field lines show the direction of the force that a small positive test charge would experience if placed near these charged objects. Also note that as the field lines get closer together, the field strength gets stronger and as the field lines move farther apart from each other, the field weakens. <img class=\"aligncenter wp-image-486\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836.png\" alt=\"\" width=\"615\" height=\"353\" \/>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.2.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"alignright wp-image-487 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.2.1.gif\" alt=\"\" width=\"248\" height=\"248\" \/>\r\n\r\nWhat is the electric field at a distance of 15 cm above a small metal sphere that has a deficit of 420 million electrons?\r\n\r\n<strong>Solution<\/strong>\r\n<div>\r\n<div>\r\n\r\nData:\r\n<ul>\r\n \t<li>[latex]E[\/latex] = Find<\/li>\r\n \t<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\r\n \t<li>[latex]q[\/latex] = (420 \u00d7 10<sup>6<\/sup>)(1.602 \u00d7 10<sup>\u221219<\/sup> C)<\/li>\r\n \t<li>[latex]d[\/latex] = 0.15 m<\/li>\r\n<\/ul>\r\nSolution:\r\n<ul>\r\n \t<li>[latex]\\vec{E}=\\dfrac{kq_1}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{E}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(420\\times10^6)(1.602\\times10^{-19}\\text{ C})}{(0.15\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{E}[\/latex] = 26.9 N\/C (\u2248 27 N\/C)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.2.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe electric field at a certain distance from a charged sphere is 200 000 N\/C directed away from the sphere.\r\n<ol type=\"i\">\r\n \t<li>What force would this exert on a singly ionized lithium atom ([latex]m[\/latex] = 1.15 \u00d7 10<sup>\u221226<\/sup> kg and [latex]q[\/latex] = \u2212e)?<\/li>\r\n \t<li>What is the acceleration of this ion?<\/li>\r\n<\/ol>\r\n<strong>Solution<\/strong>\r\n<ol type=\"i\">\r\n \t<li>[latex]\\vec{F}_{el}=q\\vec{E}[\/latex]\r\n[latex]\\vec{F}_{el}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(200<span style=\"margin-left: 0.25em;\">000<\/span> N\/C)\r\n[latex]\\vec{F}_{el}[\/latex] = 3.2 \u00d7 10<sup>\u221214<\/sup> N<\/li>\r\n \t<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]\r\n[latex]m\\vec{a}=q\\vec{E}[\/latex]\r\n(1.15 \u00d7 10<sup>\u221226<\/sup> kg)[latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(200<span style=\"margin-left: 0.25em;\">000<\/span> N\/C)\r\n[latex]\\vec{a}[\/latex] = 3.2 \u00d7 10<sup>\u221214<\/sup> N \u00f7 1.15 \u00d7 10<sup>\u221226<\/sup> kg\r\n[latex]\\vec{a}[\/latex] = 2.8 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup> (away form the ion)<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.2.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAn electron and a proton both experience an electric field strength of 50 N\/C. What is the difference in accelerations that they experience? (Vector signs are included to identify the difference between Electric Field and Energy.)\r\n\r\n<strong>Solution<\/strong>\r\n\r\nThe electron...\r\n<ul>\r\n \t<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\r\n \t<li>[latex]m\\vec{a}=q\\vec{E}[\/latex]<\/li>\r\n \t<li>(9.11 \u00d7 10<sup>\u221231<\/sup> kg) [latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(50 N\/C)<\/li>\r\n \t<li>[latex]\\vec{a}[\/latex] =\u00a0 8.01 \u00d7 10<sup>\u221218<\/sup> N \u00f7 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\r\n \t<li>[latex]\\vec{a}[\/latex] = 8.79 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup><\/li>\r\n<\/ul>\r\nThe proton...\r\n<ul>\r\n \t<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\r\n \t<li>[latex]m\\vec{a}=q\\vec{E}[\/latex]<\/li>\r\n \t<li>(1.67 \u00d7 10<sup>\u221227<\/sup> kg) [latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(50 N\/C)<\/li>\r\n \t<li>[latex]\\vec{a}[\/latex] =\u00a0 8.01 \u00d7 10<sup>\u221218<\/sup> N \u00f7 1.67 \u00d7 10<sup>\u221227<\/sup>\u00a0kg<\/li>\r\n \t<li>[latex]\\vec{a}[\/latex] = 4.80 \u00d7 10<sup>9<\/sup> m\/s<sup>2<\/sup><\/li>\r\n<\/ul>\r\nThe difference...\r\n<ul>\r\n \t<li>[latex]\\Delta\\vec{a}=\\vec{a}_{e-}-\\vec{a}_{p+}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta\\vec{a}[\/latex]= 8.79 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup> \u2212\u00a04.80 \u00d7 10<sup>9<\/sup> m\/s<sup>2<\/sup><\/li>\r\n \t<li>[latex]\\Delta\\vec{a}[\/latex] = 8.78 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 20.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the strength and direction of the electric field at a distance of 5.29 \u00d7 10<sup>\u221211<\/sup> m away from a proton?<\/li>\r\n \t<li>What is the electric field at a distance of 20 cm above a small metal sphere that has a deficit of 100 million electrons?<\/li>\r\n \t<li>What strength of electric field is needed to exert a force on a proton equal to its weight?<\/li>\r\n \t<li>The electric field at a certain distance from a charged sphere is 500<span style=\"margin-left: 0.25em;\">000<\/span> N\/C from the sphere.\r\n<ol type=\"i\">\r\n \t<li>What force would this exert on a neon ion ([latex]m[\/latex] = 3.3 \u00d7 10<sup>\u221226<\/sup> kg and [latex]q[\/latex] = \u2212e)?<\/li>\r\n \t<li>What is the acceleration of this ion?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>An electron is present in an electric field of 104 N\/C.\r\n<ol type=\"i\">\r\n \t<li>Find the force acting on this electron.<\/li>\r\n \t<li>What is this electron\u2019s acceleration?<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>An electron is accelerated towards an alpha particle (charge is +2 e). How close does it get before it experiences an electric field of 1.0 N\/C?<\/li>\r\n \t<li>What strength of electric field is needed to accelerate an electron at one million gravities?<\/li>\r\n \t<li>What is the change in the electric field strength as an object moves from 20 nm to 2.0 nm towards an alpha particle?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>20.3 The Hydrogen Atom<\/h1>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 18px;\">\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Particle<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Mass<\/th>\r\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Charge<\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Electron (e<sup>\u2212<\/sup> or \u00df<sup>\u2212<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">9.10938356(11) \u00d7 10<sup>\u221231<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">\u22121.6021766208 \u00d7 10<sup>\u221219<\/sup> C (\u2212e)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px;\">\r\n<td style=\"width: 33.3333%; height: 18px;\">Proton (p, p<sup>+<\/sup> or N<sup>+<\/sup>)<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">1.672621898(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\r\n<td style=\"width: 33.3333%; height: 18px;\">+1.6021766208 \u00d7 10<sup>\u221219<\/sup> C (+e)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">Bohr Radius [latex](a_0, r_{\\text{Bohr}})=5.29177 \u00d7 10^{\u221211}\\text{ m}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Excited State Radius [latex]r =n^2 a_o[\/latex]<\/p>\r\n\r\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\r\n \t<li>\u00a0[latex]F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]F_{c}=\\dfrac{mv^2}{r}[\/latex]<\/li>\r\n \t<li>[latex]E_k=\\dfrac{1}{3}mv^2[\/latex]<\/li>\r\n \t<li>[latex]p=mv[\/latex]<\/li>\r\n<\/ul>\r\n<div>\r\n\r\nThe hydrogen atom is important to an understanding of both the structure of atoms and the quantum nature of matter.\u00a0 One of the first clues to the nature of the hydrogen atom came from the emission spectrum (the specific colors) that hydrogen gas emits when heated. In 1885 Jacob Balmer (1825-1898) discovered that the visible wavelengths associated with the specific emitted colors fit a simple formula. By manipulating the formula, Balmer found other series of wavelengths, both visible and invisible.\r\n\r\nErnest Rutherford (1871-1937) solved the remaining part of the mystery of the nature of atomic structure. Rutherford worked at three universities (including McGill in Montreal) and later at the Cavendish Laboratory. He discovered that atoms were mostly space, with an inner hard core surrounded by orbiting electrons, much like planets orbiting a sun at the centre. This concept, the Rutherford Planetary Model of the Atom, can be added to his concepts of alpha, beta and gamma rays, protons and half-life radioactivity.\r\n\r\nCombining the results of Balmer and Rutherford, Niels Bohr (1885-1962) used the energy of the emitted light spectra to come up with a model of the atom that certain average orbitals around the nucleus or core were allowed.\u00a0 Bohr\u2019s work predicted the size of the hydrogen atom and initiated an understanding of the nature of the movement of the electron around the atom. Rutherford\u2019s Planetary Model and Balmer\u2019s emission spectra were now grounded in a common conceptual understanding.\r\n\r\nIn less than two decades, Werner Heisenberg (1901-1976), Paul Dirac (1902-1984) and Wolfgang Pauli (1900-1958) together laid out the foundations of quantum mechanics by focusing on the frequencies and intensities of the hydrogen spectra transitions, and the elliptical effect of the electron orbits on the hydrogen spectra, with the mathematical genius to put it all together.\r\n\r\nThe following questions relate to the Hydrogen Atom.\r\n\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.3.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol type=\"i\">\r\n \t<li>What is the strength of the electrostatic force between an electron and a proton separated by a distance of 2.12 \u00d7 10<sup>\u221210<\/sup> m (1st excited state)?\r\n<ul>\r\n \t<li>\u00a0[latex]F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{E}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})^2}[\/latex]<\/li>\r\n \t<li>[latex]\\vec{E}[\/latex] = 5.13 \u00d7 10<sup>\u22129<\/sup> N<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Balancing the centripetal force to the electrostatic force for the hydrogen atom, calculate the speed of an electron that would be orbiting the proton (1st excited state).\r\n<ul type=\"i\">\r\n \t<li>[latex]F_{c}=\\dfrac{mv^2}{r}\\qquad\\qquad F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\r\n \t<li>Therefore... [latex]\\dfrac{mv^2}{r}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]\u00a0 Cancel out the common distance &amp; isolate v<sup>2<\/sup><\/li>\r\n \t<li>Yields... [latex]v^2=\\dfrac{kq_1{q}_2}{dm}[\/latex]\r\n<ul type=\"i\">\r\n \t<li>[latex]{v}^2=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})(9.11\\times10^{-31}\\text{ kg}}[\/latex]<\/li>\r\n \t<li>[latex]v^2[\/latex]<sup>\u00a0<\/sup>= 1.19 \u00d7 10<sup>12<\/sup> m<sup>2<\/sup>\/s<sup>2<\/sup><\/li>\r\n \t<li>[latex]v[\/latex] = 1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Calculate the kinetic energy and the momentum of this orbiting electron (1st excited state).\r\n<ul type=\"i\">\r\n \t<li>[latex]E_k[\/latex] = \u00bdmv\u00b2<\/li>\r\n \t<li>[latex]E_k[\/latex] = \u00bd(9.11 \u00d7 10<sup>\u221231 <\/sup>kg)(1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s)<sup>2<\/sup><\/li>\r\n \t<li>[latex]E_k[\/latex] = 5.41 \u00d7 10<sup>\u221219 <\/sup>J or 3.4 eV<\/li>\r\n \t<li>[latex]p = mv[\/latex]<\/li>\r\n \t<li>[latex]p[\/latex] = (9.11 \u00d7 10<sup>\u221231 <\/sup>kg)(1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s)<\/li>\r\n \t<li>[latex]p[\/latex] = 9.9 \u00d7 10<sup>\u221225 <\/sup>Ns<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 20.3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"i\">\r\n \t<li>What is the electrostatic force strength between an electron and a proton separated by 4.76 \u00d7 10<sup>\u221210<\/sup> m (2nd excited state)?<\/li>\r\n \t<li>Balancing the centripetal force to the electrostatic force for the hydrogen atom calculate the speed of an electron that would be orbiting the proton (the hydrogen atom).<\/li>\r\n \t<li>Calculate the kinetic energy and the momentum of this orbiting electron.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>If an electron that is orbiting an alpha particle has a kinetic energy of 54.42 eV 114 what distance is this electron from the core? What is the centripetal force and the electrostatic force that is acting on this electron?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>20.4 Electric Potential of Point Charges<\/h1>\r\nThe images below show the magnitude of the Electrical Potential of both a positive and negative charge as one moves away from the charge.\r\n\r\n<img class=\"aligncenter size-full wp-image-533\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential.png\" alt=\"\" width=\"388\" height=\"178\" \/>\r\n\r\nEquations Introduced or Used for this Section:\r\n<p style=\"text-align: center;\">[latex]\\Delta V[\/latex] = \u00b1[latex]\\dfrac{kq}{d}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E[\/latex] = \u00b1 [latex]q V[\/latex]<\/p>\r\nElectric potential for charged objects is similar to the gravitational potential, in that instead of moving without friction through some displacement in a gravitational field, an object instead moves through a displacement inside an electrical field.\u00a0 There are two possibilities for this electric field: either the field is constant like the gravity one finds when moving through small changes in height on Earth, or the field varies similarly to the movement of a spaceship either towards or away from the Earth.\r\n\r\nThis results in two different equations for electric potential: the electric potential moving away from a charged object, or the electric potential in a constant electric field. This section looks at only the electric potential moving away from a charged object, that as such experiences a variable electric field.\r\n\r\nElectric potential for a charged object can be used to find the energy change if the amount of charge moving through the potential difference is known.\u00a0 It is important to remember the distinction between electric potential and electric potential energy... They are different.\r\n\r\nThe equation for electric potential is one that is most useful in a number of physics situations that look at the energy required to remove an electron from an atom, or to move from one allowed orbital to another for an electron in orbit around an atom.\r\n\r\nWhile the following examples relate to the hydrogen atom, applications of this equation extend far beyond this.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.4.1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the electric potential of an electron in its second excited state surrounding a hydrogen atom?\r\n\r\n<strong>Solution<\/strong>\r\n<ul>\r\n \t<li>[latex]V[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = \u2212 6.79 N\/m<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 20.4.2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the electrical potential for a hydrogen atom between the average ground state radius (n = 1) 5.29 \u00d7 10<sup>\u221211<\/sup> m and (n = 4) average radius of 8.46 \u00d7 10<sup>\u221210<\/sup> m.\r\n\r\n<strong>Solution<\/strong>\r\n\r\nThis requires finding the change in electric potential between these two states...\r\n<ul>\r\n \t<li>[latex]V_{n = 1}[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(5.29\\times10^{-11}\\text{ m})}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = \u2212 27.2 N\/m<\/li>\r\n \t<li>[latex]V_{n = 4}[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(8.46\\times10^{-10}\\text{ m})}[\/latex]<\/li>\r\n \t<li>[latex]V[\/latex] = \u2212 1.70 N\/m<\/li>\r\n \t<li>[latex]\\Delta V = V_{n = 4} \u2212 V_{n = 1}[\/latex]<\/li>\r\n \t<li>[latex]\\Delta V[\/latex] = \u2212 1.70 N\/m \u2212 (\u2212 27.2 N\/m) or 25.5 N\/m<\/li>\r\n<\/ul>\r\n[latex]\\Delta V[\/latex] depends on the direction travelled (away or towards) and as such [latex]\\Delta V[\/latex] = \u00b1 25.5 N\/m\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 20.4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the electric potential of an electron in its ground state radius surrounding a hydrogen atom?<\/li>\r\n \t<li>What is the difference in electric potential to remove a proton at a distance of 25 \u00b5m from a charged sphere having a charge of 106 excess electrons?<\/li>\r\n \t<li>What is the difference in electrical potential for a hydrogen atom between the average ground state radius (n = 1) 5.29 \u00d7 10<sup>\u221211<\/sup> m and (n = 3) average radius of 4.76 \u00d7 10<sup>\u221210<\/sup> m.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 20.5.1: Deep space research on hydrogen atoms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDeep space research on hydrogen atoms, suggest that electrons can orbit protons at distances as far as one metre apart (d = 1.0 m).\r\n<ol>\r\n \t<li>What would be the orbital speed of this electron?<\/li>\r\n \t<li>What would be the kinetic energy of this electron?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Exercise Answers<\/h1>\r\n<h2>20.1 Electrostatic Forces<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"i\">\r\n \t<li>has extra electrons<\/li>\r\n \t<li>6.25 \u00d7 10<sup>6<\/sup> e<sup>\u2212<\/sup><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Repulsive force of 7.2 \u00d7 10<sup>\u22124<\/sup> N<\/li>\r\n \t<li>Attractive force of 8.24 \u00d7 10<sup>\u22128<\/sup> N<\/li>\r\n \t<li>3.0 \u00d7 10<sup>\u22123<\/sup> m<\/li>\r\n \t<li>1st excited\u00a0 \u00a05.12 \u00d7 10<sup>\u22129<\/sup> N \u00a0 3rd excited\u00a0 \u00a03.2 \u00d7 10<sup>\u221210<\/sup> N<\/li>\r\n \t<li>9.03 \u00d7 10<sup>22<\/sup> m\/s<sup>2 <\/sup>\r\n<ol type=\"i\">\r\n \t<li>2.0 \u00d7 10<sup>\u22125<\/sup> m\/s<sup>2<\/sup><\/li>\r\n \t<li>4.03 \u00d7 10<sup>7<\/sup> e<sup>\u2212<\/sup><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>1.04 \u00d7 10<sup>24<\/sup> m\/s<sup>2<\/sup> (electron) ac = 1.41 \u00d7 10<sup>20<\/sup> m\/s<sup>2<\/sup> (core)<\/li>\r\n \t<li>d = 1.61 m<\/li>\r\n \t<li>d = 5.1 m<\/li>\r\n<\/ol>\r\n<h2>20.2 Electric Fields (Point Charges)<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>5.15 \u00d7 10<sup>11<\/sup> N\/C (away)<\/li>\r\n \t<li>3.6 N\/C away from Sphere<\/li>\r\n \t<li>1.02 \u00d7 10<sup>\u22127<\/sup> N\/C upwards<\/li>\r\n \t<li>\r\n<ol type=\"i\">\r\n \t<li>8.0 \u00d7 10<sup>\u221214<\/sup> N<\/li>\r\n \t<li>2.4 \u00d7 10<sup>12<\/sup> m\/s<sup>2 <\/sup><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol type=\"i\">\r\n \t<li>1.7 \u00d7 10<sup>\u221217<\/sup>\u00a0N<\/li>\r\n \t<li>1.8 \u00d7 10<sup>13<\/sup>\u00a0m\/s<sup>2<\/sup><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>5.4 \u00d7 10<sup>\u22125<\/sup> m<\/li>\r\n \t<li>5.6 \u00d7 10<sup>\u22125<\/sup> N\/C<\/li>\r\n \t<li>7.1 \u00d7 10<sup>8<\/sup> N\/C away<\/li>\r\n<\/ol>\r\n<h2>20.3 Hydrogen Atom<\/h2>\r\n<ol>\r\n \t<li style=\"list-style-type: none;\">\r\n<ol type=\"i\">\r\n \t<li>1.02 \u00d7 10<sup>\u22129<\/sup> N<\/li>\r\n \t<li>2.18 \u00d7 10<sup>6<\/sup> m\/s<\/li>\r\n \t<li>1.99 \u00d7 10<sup>\u221224<\/sup> N\/C<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>[latex]F_{el}[\/latex] = [latex]F_c[\/latex] = 0.657 \u00b5N<\/li>\r\n<\/ol>\r\n<h2>20.4\u00a0 Electric Potential<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>[latex]V[\/latex] = \u2212 27.2 J\/C<\/li>\r\n \t<li>[latex]V[\/latex] =\u00a0 57.5 J\/C<\/li>\r\n \t<li>[latex]\\Delta V[\/latex] = 24.2 J\/C<\/li>\r\n<\/ol>\r\n<h2>20.5.1 Deep space research on hydrogen atoms<\/h2>\r\n<ol class=\"twocolumn\">\r\n \t<li>[latex]v[\/latex] = 15.9 m\/s<\/li>\r\n \t<li>1.15 \u00d7 10<sup>\u221228<\/sup> J<\/li>\r\n<\/ol>\r\n<h3>Media Attributions<\/h3>\r\n<ul>\r\n \t<li>\"<a href=\"https:\/\/www.a-levelphysicstutor.com\/field-elect-2.php\">Point charge electric field patterns<\/a>\" by A-Level Physics Tutor is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/deed.en\">CC BY-NC-ND licence<\/a>.<\/li>\r\n \t<li>\"<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Blausen_0615_Lithium_Atom.png\">Blausen 0615 Lithium Atom<\/a>\" by <a title=\"User:BruceBlaus\" href=\"https:\/\/commons.wikimedia.org\/wiki\/User:BruceBlaus\">BruceBlaus<\/a> is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/deed.en\">CC BY-SA 4.0 licence<\/a>.<\/li>\r\n \t<li>Example 20.2.1 image from <a href=\"https:\/\/openstax.org\/details\/books\/college-physics\">College Physics<\/a> by Paul Peter Urone, Roger Hinrichs, Kim Dirks and Manjula Sharma is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 licence<\/a>.<\/li>\r\n \t<li>\"Graph of electric potential\" by <a href=\"https:\/\/physics.stackexchange.com\/users\/63672\/user45220\">user45220<\/a> of Stack Exchange is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/\">CC BY-SA 3.0 licence<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Resources<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=0Lx0c0TlzFI\">Mechanical Universe &#8211; Episode 28 &#8211; Static Electricity<\/a><\/li>\n<li>Video to Watch: <a href=\"https:\/\/www.youtube.com\/watch?v=wq9TjQZDrAA\">Mechanical Universe &#8211; Episode 29 &#8211; The Electric Field<\/a><\/li>\n<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Charge-Interactions-Revisited\">Charge Interactions Revisited<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Equations Introduced and Used in this Topic:<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}=q\\vec{E}[\/latex]<\/li>\n<li>[latex]\\vec{E}=\\dfrac{kq}{d^2}[\/latex]<\/li>\n<li>[latex]V=\\dfrac{kq_1{q}_2}{d}[\/latex]<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/p>\n<p style=\"text-align: center;\">Bohr Radius [latex](a_0, r_{\\text{Bohr}})=5.29177 \u00d7 10^{\u221211}\\text{ m}[\/latex]<\/p>\n<p style=\"text-align: center;\">Excited State Radius [latex]r =n^2 a_o[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{F}=m\\vec{a}[\/latex]<\/p>\n<p>Where\u2026<\/p>\n<ul>\n<li>[latex]F[\/latex] is generally the Net Force acting on the charged object<\/li>\n<li>[latex]F_{el}[\/latex]\u00a0is the Electrostatic Force, measured in newtons (N)<\/li>\n<li>[latex]E[\/latex] is the Electric Field, measured in newtons per coulomb (N\/C) or volts per metre (V\/m)<\/li>\n<li>[latex]k[\/latex] is the Coulomb&#8217;s constant, measured in newton-metres squared per coulomb squared (N\u00b7m<sup>2<\/sup>\/C<sup>2<\/sup>)<\/li>\n<li>[latex]q, q_1[\/latex] &amp; [latex]q_2[\/latex] are the Charges you are studying, measured in coulombs (C)<\/li>\n<li>[latex]q[\/latex] if negative indicates that it has an excess of electrons (or shortage of protons)<\/li>\n<li>[latex]q[\/latex] if positive indicates that it has an excess of protons (or shortage of electrons)<\/li>\n<li>[latex]d[\/latex] is the Distance away from the Charge Center, measured in metres (m)<\/li>\n<li>[latex]V[\/latex] is the Potential Difference, measured in volts (V) or joules\/coulombs (J\/C)<\/li>\n<li>[latex]a[\/latex] is the Acceleration of the object measured in metres per second squared (m\/s<sup>2<\/sup>)<\/li>\n<\/ul>\n<p style=\"text-align: center;\">The static charge on an object can be calculated by the number of excess electrons or protons time \u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Elementary Particles<a class=\"footnote\" title=\"A-Level Physics Tutor - Electricity\" id=\"return-footnote-215-1\" href=\"#footnote-215-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/caption>\n<tbody>\n<tr style=\"height: 18px;\">\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Particle<\/th>\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Mass<\/th>\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Charge<\/th>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Electron (e<sup>\u2212<\/sup> or \u00df<sup>\u2212<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">9.10938356(11) \u00d7 10<sup>\u221231<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">\u22121.6021766208 \u00d7 10<sup>\u221219<\/sup> C (\u2212e)<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Proton (p, p<sup>+<\/sup> or N<sup>+<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">1.672621898(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">+1.6021766208 \u00d7 10<sup>\u221219<\/sup> C (+e)<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Neutron (n, n<sup>0<\/sup>, N<sup>0<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">1.674927471(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">0 C<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Alpha particle (\u03b1, \u03b1<sup>2+<\/sup>, He<sup>2+<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">6.644657230(82) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">+3.2043532416 \u00d7 10<sup>\u221219<\/sup> C (+2e)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Important Constants<\/strong> (round to needed accuracy)<\/p>\n<ul>\n<li>Coulomb&#8217;s constant: [latex]k[\/latex] =\u00a0 8.9875517873681764 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\n<li>Electron volt: 1 eV = 1.60217653(14) \u00d7 10<sup>\u221219<\/sup> J<\/li>\n<li>Bohr Radius: (a<sub>0<\/sub>, r<sub>Bohr<\/sub>) =\u00a0 5.29177 \u00d7 10<sup>\u221211<\/sup> m<\/li>\n<\/ul>\n<h1>20.1 Electrostatic Forces<\/h1>\n<div class=\"textbox\">In Research News:<a href=\"https:\/\/www.smithsonianmag.com\/smart-news\/super-strong-electric-forces-may-have-helped-tiny-clumps-dust-seed-planets-180973756\/\"> Super-Strong Electric Forces May Have Helped Tiny Clumps of Dust Seed the Planets<\/a><\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{net}=m\\vec{a}[\/latex]<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/p>\n<p>Historical records indicate awareness of numerous electrical phenomenon for over 3000 years but the explanations of what it was quite different from how we understand it today. For instance, Thales of Miletus (624-546 BCE) thought that we were seeing the soul of a piece of amber that when rubbed on a fur it would attracted small objects. Likewise, lightning, St Elmo\u2019s Fire and the electric ray all were well known electrical phenomena that had equally mystic nature.<\/p>\n<p>Different from the study of mathematics and astronomy, the first recorded study of electrical phenomenon was published by William Gilbert (1544-1603) three years before his death from what is believed to have been the bubonic plague. The name for electricity came from Gilbert&#8217;s use of the Latin word \u201celectrica\u201d (\u1f24\u03bb\u03b5\u03ba\u03c4\u03c1\u03bf\u03bd) meaning \u201clike amber\u201d. While electrica had been used since the 13th century, Gilbert defined the name respecting its attractive or repulsive properties.<\/p>\n<p>From Gilbert\u2019s work, three rules of electrostatics were developed:<\/p>\n<ul>\n<li>There are only two kinds of electric charge<\/li>\n<li>Two objects charged alike (having the same kind of charge) repel each other<\/li>\n<li>Two objects charged oppositely attract each other.<\/li>\n<\/ul>\n<p>Explaining the nature of charged objects was more challenging.\u00a0 Gilbert speculated that the \u201camber effect\u201d was the result of the effluvium (small particles that flowed from one charged object to another that did not have any mass or volume). Over the next century, multiple small advances were made by numerous philosopher\/scientists with one, Charles Fran\u00e7ois de Cisternay du Fay (1698-1739), proposing that there were two types of electric charge: a positive charged fluid and a negative charged fluid. However, it was Benjamin Franklin (1706-1790) who formulated that electric charge was but a single fluid, not that an object having an excess of this fluid was positively charged and an object having a deficit of this fluid was negatively charged.<\/p>\n<p>We now know that the Franklin one-fluid model is fundamentally correct in that it is the electron that can be understood as the source of electric charge. Because of this construct, everything in conventional electricity is defined in terms of the movement of this excess charge. However it was not until the late 1890\u2019s that evidence accumulated to support what electric charge actually was. By this time, the battery, lightbulb, telephone and a host of other electrical inventions had been created, and all were understood using Franklin\u2019s concept of the movement of excess fluid. The legacy of Franklin\u2019s understanding can still be found in electrical terminology, including: positive and negative charge, charging and discharging, conductors and condensers, the lightning rod, and the flow of electric charge called electric current.\u00a0 Franklin<a class=\"footnote\" title=\"Benjamin Franklin actually did his famous kite experiment in a thunderstorm in 1752, collecting electric charge in a capacitor which was known at that time as a Leyden Jar. Greater detail of this experiment can be found at: Kite experiment &amp; Leyden jar\" id=\"return-footnote-215-2\" href=\"#footnote-215-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a> and William Watson (1715-1787) both proposed the conservation of electric charge. This concept was later interpreted as an isolated system wherein the net charge remains constant and, if matter is created or destroyed, charge is created and destroyed in equal amounts.<\/p>\n<div class=\"textbox textbox--sidebar\">\n<p>Article to Read: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Coulomb%27s_law\">Coulomb\u2019s Law<\/a><\/p>\n<p>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Coulomb-s-Law\">Coulomb&#8217;s Law<\/a><\/p>\n<p>Extra Help:\u00a0<a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Inverse-Square-Law\">Inverse Square Law<\/a><\/p>\n<p>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-3\/Newton-s-Laws-and-the-Electrical-Force\">Newton\u2019s Laws and the Electrical Force<\/a><\/p>\n<\/div>\n<p>Quantifying the force that exists between similar or oppositely charged objects was proposed by Joseph Priestly (1733-1804) to follow an inverse square law as had been proposed by Isaac Newton for gravity. Priestly posited this relationship in his 700 page book The History and Present State of Electricity (1767). Priestly&#8217;s inverse square law became formalized by Charles-Augustin de Coulomb (1736-1806), who demonstrated evidence of its authenticity using a device called the torsion balance. Coulomb later was able to demonstrate that an inverse square law relationship also existed between the poles of bar magnets.<\/p>\n<p>In algebraic form, Coulomb\u2019s Law\u00a0is stated as:<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{{d}^2}[\/latex]<\/p>\n<p>Where\u2026<\/p>\n<ul>\n<li>[latex]k[\/latex] = 8.9875517873681764 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup> (Coulomb\u2019s constant)<\/li>\n<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C) (number of electrons\/protons)<\/li>\n<li>[latex]d[\/latex] = the distance between the charge centres (measured in metres)<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\" style=\"page-break-before: always;\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.1.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A small plastic sphere has a charge of \u221210<sup>\u221214<\/sup> C.<\/p>\n<ol type=\"i\">\n<li>Does it have an excess or a deficit of electrons on it?<\/li>\n<li>How many electrons?<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<ol type=\"i\">\n<li>Since the sphere has a negative charge, it has an excess of electrons.<\/li>\n<li>Data:\n<ul>\n<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)<\/li>\n<li>[latex]n[\/latex] = Find<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]q[\/latex] = (\u00b1 1.602 \u00d7 10<sup>\u221219<\/sup> C)(number of electrons\/protons)<\/li>\n<li>\u221210<sup>\u221214<\/sup> C = (+ 1.602 \u00d7 10<sup>\u221219<\/sup> C)([latex]n[\/latex])<\/li>\n<li>[latex]n[\/latex] = \u221210<sup>\u221214<\/sup> C \u00f7 (\u2212 1.602) \u00d7 10<sup>\u221219<\/sup> C<\/li>\n<li>[latex]n[\/latex] = 62<span style=\"margin-left: 0.25em;\">400<\/span> electrons\u00a0 (\u2248 60<span style=\"margin-left: 0.25em;\">000<\/span> e<sup>\u2212<\/sup>)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.1.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the magnitude and direction of the force acting on a charge of +4.2 \u00b5C, 5.0 mm away from a charge of +4.5 \u00b5C?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-466\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313.png\" alt=\"\" width=\"500\" height=\"224\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313.png 584w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313-300x135.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313-65x29.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313-225x101.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.1.2-e1678819036313-350x157.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]F_{el}[\/latex] = Find<\/li>\n<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\n<li>[latex]q_1[\/latex] =\u00a0 4.2 \u00d7 10<sup>\u22126<\/sup> C<\/li>\n<li>[latex]q_2[\/latex] = 4.5 \u00d7 10<sup>\u22126<\/sup> C<\/li>\n<li>[latex]d[\/latex] = 5.0 \u00d7 10<sup>\u22123<\/sup> m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{\\text{d}^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}=\\dfrac{(8.99\\times10^9 \\text{ Nm}^2\\text{\/C}^2)(4.2\\times10^{-6}\\text{ C})(4.5\\times10^{-6})}{(5.0\\times10^{-3}\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}[\/latex] = 6800 N directed away from each other (both are positive charges)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.1.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-468\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom.png\" alt=\"\" width=\"250\" height=\"313\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom.png 960w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-240x300.png 240w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-819x1024.png 819w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-768x960.png 768w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-65x81.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-225x281.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/Lithium_Atom-350x438.png 350w\" sizes=\"auto, (max-width: 250px) 100vw, 250px\" \/>The double-ionized lithium atom consists of one electron (e<sup>\u2212<\/sup>) orbiting a core of three protons (p+) and three or four neutrons.\u00a0 If its average orbital radius is 0.0167 nm, what is the electrostatic force that exists between the electron and the core?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(3)(4.5\\times10^{-6})}{(0.0167\\times10^{-9}\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}[\/latex] = 2.48 \u00d7 10<sup>\u22129<\/sup> N (attractive)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.1.4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the acceleration of the single electron orbiting the three protons in the lithium atom described above?\u00a0 (Use the lowest energy level radius of 0.0167 nm.)<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]a[\/latex] = Find<\/li>\n<li>[latex]\\vec{F}_{el}[\/latex] = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\n<li>[latex]m[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup>\u00a0N<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\n<li>[latex]m\\vec{\\text{a}}[\/latex] = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\n<li>(9.11 \u00d7 10<sup>\u221239 <\/sup>kg)([latex]\\vec{a}[\/latex]) = 2.48 \u00d7 10<sup>\u22126<\/sup>\u00a0N<\/li>\n<li>[latex]\\vec{a}[\/latex] = 2.7 \u00d7 10<sup>\u221239\u00a0<\/sup>m\/s<sup>2<\/sup>&#8230; (towards the core)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.1.5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the distance that an electron would have to be away from the lithium core described above to experience an electrostatic force equal to its weight?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>Data:<\/p>\n<ul>\n<li>[latex]m[\/latex] = 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\n<li>[latex]q[\/latex] = 1.602 \u00d7 10<sup>\u221219<\/sup> C<\/li>\n<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\n<li>[latex]g[\/latex] = 9.80 m\/s<sup>2<\/sup><\/li>\n<li>[latex]d[\/latex] = Find<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>Use [latex]w=\\vec{F}_{el}[\/latex]<\/li>\n<li>[latex]mg=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>(9.11\u00a01 \u00d7 10<sup>\u221231<\/sup> kg)(9.80 m\/s<sup>2<\/sup>) = [latex]\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})^2}{d^2}[\/latex]<\/li>\n<li>[latex]d^2[\/latex] = 6.9 \u00d7 10<sup>\u221228<\/sup> N\/m<sup>2<\/sup> \u00f7 8.9 \u00d7 10<sup>\u221230<\/sup> N<\/li>\n<li>[latex]d[\/latex] = 8.8 m<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 20.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>A ping pong ball has a charge of \u221210<sup>\u221212<\/sup> C.\n<ol type=\"i\">\n<li>Does it have an excess or a deficit of electrons on it?<\/li>\n<li>How many electrons?<\/li>\n<\/ol>\n<\/li>\n<li>What is the magnitude and direction of the electrostatic force acting on a charge of + 5.0 \u00d7 10<sup>\u22128<\/sup> C that is 5.0 cm away from a charge of + 4.0 \u00d7 10<sup>\u22129<\/sup> C?<\/li>\n<li>The hydrogen atom consists of an electron (e<sup>\u2212<\/sup>) and a proton (p<sup>+<\/sup>). If their average separation is 5.29 \u00d7 10<sup>\u221211<\/sup> m what is the electrostatic force that exists between them?<\/li>\n<li>Two charged objects, one of +5.0 \u00d7 10<sup>\u22127<\/sup> C and the other of \u22122.0 \u00d7 10<sup>\u22127<\/sup> C, attract each other with an electrostatic force of 100 N. How far apart are they?<\/li>\n<li>The hydrogen atom has been extensively studied and has been responsible for scientists to be able to both create and understand the atomic structure of atoms. Given that the average radius for the first three excited states are:\n<ul>\n<li><strong>1st excited state is 2.12 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong>10<\/strong><\/sup><strong> m <\/strong><\/li>\n<li><strong>2nd excited state is 4.76 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong>10<\/strong><\/sup><strong> m <\/strong><\/li>\n<li><strong>3rd excited state is 8.46 \u00d7 10<sup>\u2212<\/sup><\/strong><sup><strong><strong>10<\/strong><\/strong><\/sup><\/li>\n<\/ul>\n<p>Find the different electrostatic forces experience by the electron in the first and third excited states.<\/li>\n<li>What is the acceleration of an electron orbiting the proton in the hydrogen atom? (Use the Bohr Radius&#8230;\u00a0 Ground State Radius.)<\/li>\n<li>Two small metal pellets (mass of 7.5 \u00d7 10<sup>\u22124<\/sup> kg) each have the same negative charge and repel each other with an electrostatic force of 1.5 \u00d7 10<sup>\u22128<\/sup> N when 5.0 mm apart.\n<ol type=\"i\">\n<li>What would they accelerate with if they were free to move?<\/li>\n<li>How many electrons are on each of them?<\/li>\n<\/ol>\n<\/li>\n<li>A singly ionized helium atom composed of a core that has 2 protons and 2 neutrons, has one electron orbiting. At what rate are the core and the electron accelerating towards each other if the electron maintains an average distance of 2.21 \u00d7 10<sup>\u221211<\/sup> m from the core?<\/li>\n<li>What is the distance an electron would have to be from a proton to have an acceleration of 10 g\u2019s?<\/li>\n<li>What is the distance that an electron would have to be from a proton to experience an electrostatic force equal to its weight?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>\u00a020.2 Electric Fields (Point Charges)<\/h1>\n<div>\n<div class=\"textbox\">\n<ul>\n<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Field-Intensity\">Electric Field Intensity<\/a><\/li>\n<li>Extra Help: <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Field-Lines\">Electric Field Lines<\/a><\/li>\n<\/ul>\n<\/div>\n<p>Equations Introduced or Used for this Section:<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>[latex]\\vec{E}=\\dfrac{kq}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}= q\\vec{E}[\/latex]<\/li>\n<li>[latex]\\vec{F}= m\\vec{a}[\/latex]<\/li>\n<li>[latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<\/ul>\n<p>ike the concept of gravitational fields covered in Chapter 17.2, electrostatics share a similar concept of electric field (also magnetic fields). Historically, the first reference to such a concept was by Gilbert when he was describing the sphere of influence of a charged object. The equation quantifying the electric field can be derived in similar fashion to the derivation of the gravitational field. This can be shown as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{F}_{el}= q\\vec{E}[\/latex] is equated to [latex]\\vec{F}_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/p>\n<p style=\"text-align: center;\">This means that [latex]q\\vec{E}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/p>\n<p style=\"text-align: center;\">cancelling out the common charge leaves us with&#8230; E = [latex]\\dfrac{kq_1}{d^2}[\/latex]<\/p>\n<p>The direction of the electric field as defined by a negative or positive sign depends on the net charge of the object&#8230; Is the overall charge of the object negative or positive, and how it would react on a small positive test charge placed in the electric field . This means that the electric field can be written as either a positive or a negative quantity. In practice, many will ignore the positive and negative signs and just draw the field as attractive if the net charge is negative or repulsive if the net charge is positive.<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{E}=\\pm\\dfrac{kq_1}{d^2}[\/latex]<\/p>\n<p>The units of electrical field strength are N\/C. This equation relates to the electric field at any distance away from a charged object and is a inverse square law relationship.<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{E}[\/latex] \u221d [latex]\\dfrac{1}{d^2}[\/latex]<\/p>\n<div class=\"textbox textbox--sidebar\"><strong>Extra Help<\/strong> &#8211; <a href=\"https:\/\/www.physicsclassroom.com\/class\/estatics\/Lesson-4\/Electric-Fields-and-Conductors\">Electric Fields and Conductors<\/a><\/div>\n<p>When we look at the electric field strength as we move inside the charged object\u2019s surface, the electric field strength drops to zero. Anything inside the charged object is shielded from the electric field; this is why a radio signal on a vehicle fades out when driving through a tunnel or inside the metal frame of a bridge.<\/p>\n<p style=\"text-align: center;\">[latex]\\vec{E}[\/latex] = 0 N\/C<\/p>\n<p>Electric fields can have multiple variations of positive and negatively charged objects. Examples showing field lines of various combinations of positive and negative charges are shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-485 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines.jpg\" alt=\"\" width=\"350\" height=\"174\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines.jpg 350w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines-300x149.jpg 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines-65x32.jpg 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/charges-lines-225x112.jpg 225w\" sizes=\"auto, (max-width: 350px) 100vw, 350px\" \/><\/p>\n<p>These field lines show the direction of the force that a small positive test charge would experience if placed near these charged objects. Also note that as the field lines get closer together, the field strength gets stronger and as the field lines move farther apart from each other, the field weakens. <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-486\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836.png\" alt=\"\" width=\"615\" height=\"353\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836.png 741w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836-300x172.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836-65x37.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836-225x129.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/e-field-e1678836864836-350x201.png 350w\" sizes=\"auto, (max-width: 615px) 100vw, 615px\" \/><\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.2.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-487 size-full\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/20.2.1.gif\" alt=\"\" width=\"248\" height=\"248\" \/><\/p>\n<p>What is the electric field at a distance of 15 cm above a small metal sphere that has a deficit of 420 million electrons?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<div>\n<div>\n<p>Data:<\/p>\n<ul>\n<li>[latex]E[\/latex] = Find<\/li>\n<li>[latex]k[\/latex] = 8.99 \u00d7 10<sup>9<\/sup> Nm<sup>2<\/sup>\/C<sup>2<\/sup><\/li>\n<li>[latex]q[\/latex] = (420 \u00d7 10<sup>6<\/sup>)(1.602 \u00d7 10<sup>\u221219<\/sup> C)<\/li>\n<li>[latex]d[\/latex] = 0.15 m<\/li>\n<\/ul>\n<p>Solution:<\/p>\n<ul>\n<li>[latex]\\vec{E}=\\dfrac{kq_1}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{E}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(420\\times10^6)(1.602\\times10^{-19}\\text{ C})}{(0.15\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]\\vec{E}[\/latex] = 26.9 N\/C (\u2248 27 N\/C)<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.2.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The electric field at a certain distance from a charged sphere is 200 000 N\/C directed away from the sphere.<\/p>\n<ol type=\"i\">\n<li>What force would this exert on a singly ionized lithium atom ([latex]m[\/latex] = 1.15 \u00d7 10<sup>\u221226<\/sup> kg and [latex]q[\/latex] = \u2212e)?<\/li>\n<li>What is the acceleration of this ion?<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<ol type=\"i\">\n<li>[latex]\\vec{F}_{el}=q\\vec{E}[\/latex]<br \/>\n[latex]\\vec{F}_{el}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(200<span style=\"margin-left: 0.25em;\">000<\/span> N\/C)<br \/>\n[latex]\\vec{F}_{el}[\/latex] = 3.2 \u00d7 10<sup>\u221214<\/sup> N<\/li>\n<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<br \/>\n[latex]m\\vec{a}=q\\vec{E}[\/latex]<br \/>\n(1.15 \u00d7 10<sup>\u221226<\/sup> kg)[latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(200<span style=\"margin-left: 0.25em;\">000<\/span> N\/C)<br \/>\n[latex]\\vec{a}[\/latex] = 3.2 \u00d7 10<sup>\u221214<\/sup> N \u00f7 1.15 \u00d7 10<sup>\u221226<\/sup> kg<br \/>\n[latex]\\vec{a}[\/latex] = 2.8 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup> (away form the ion)<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.2.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>An electron and a proton both experience an electric field strength of 50 N\/C. What is the difference in accelerations that they experience? (Vector signs are included to identify the difference between Electric Field and Energy.)<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The electron&#8230;<\/p>\n<ul>\n<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\n<li>[latex]m\\vec{a}=q\\vec{E}[\/latex]<\/li>\n<li>(9.11 \u00d7 10<sup>\u221231<\/sup> kg) [latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(50 N\/C)<\/li>\n<li>[latex]\\vec{a}[\/latex] =\u00a0 8.01 \u00d7 10<sup>\u221218<\/sup> N \u00f7 9.11 \u00d7 10<sup>\u221231<\/sup> kg<\/li>\n<li>[latex]\\vec{a}[\/latex] = 8.79 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup><\/li>\n<\/ul>\n<p>The proton&#8230;<\/p>\n<ul>\n<li>[latex]\\vec{F}_{net}=\\vec{F}_{el}[\/latex]<\/li>\n<li>[latex]m\\vec{a}=q\\vec{E}[\/latex]<\/li>\n<li>(1.67 \u00d7 10<sup>\u221227<\/sup> kg) [latex]\\vec{a}[\/latex] = (1.602 \u00d7 10<sup>\u221219<\/sup> C)(50 N\/C)<\/li>\n<li>[latex]\\vec{a}[\/latex] =\u00a0 8.01 \u00d7 10<sup>\u221218<\/sup> N \u00f7 1.67 \u00d7 10<sup>\u221227<\/sup>\u00a0kg<\/li>\n<li>[latex]\\vec{a}[\/latex] = 4.80 \u00d7 10<sup>9<\/sup> m\/s<sup>2<\/sup><\/li>\n<\/ul>\n<p>The difference&#8230;<\/p>\n<ul>\n<li>[latex]\\Delta\\vec{a}=\\vec{a}_{e-}-\\vec{a}_{p+}[\/latex]<\/li>\n<li>[latex]\\Delta\\vec{a}[\/latex]= 8.79 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup> \u2212\u00a04.80 \u00d7 10<sup>9<\/sup> m\/s<sup>2<\/sup><\/li>\n<li>[latex]\\Delta\\vec{a}[\/latex] = 8.78 \u00d7 10<sup>12<\/sup> m\/s<sup>2<\/sup><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 20.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the strength and direction of the electric field at a distance of 5.29 \u00d7 10<sup>\u221211<\/sup> m away from a proton?<\/li>\n<li>What is the electric field at a distance of 20 cm above a small metal sphere that has a deficit of 100 million electrons?<\/li>\n<li>What strength of electric field is needed to exert a force on a proton equal to its weight?<\/li>\n<li>The electric field at a certain distance from a charged sphere is 500<span style=\"margin-left: 0.25em;\">000<\/span> N\/C from the sphere.\n<ol type=\"i\">\n<li>What force would this exert on a neon ion ([latex]m[\/latex] = 3.3 \u00d7 10<sup>\u221226<\/sup> kg and [latex]q[\/latex] = \u2212e)?<\/li>\n<li>What is the acceleration of this ion?<\/li>\n<\/ol>\n<\/li>\n<li>An electron is present in an electric field of 104 N\/C.\n<ol type=\"i\">\n<li>Find the force acting on this electron.<\/li>\n<li>What is this electron\u2019s acceleration?<\/li>\n<\/ol>\n<\/li>\n<li>An electron is accelerated towards an alpha particle (charge is +2 e). How close does it get before it experiences an electric field of 1.0 N\/C?<\/li>\n<li>What strength of electric field is needed to accelerate an electron at one million gravities?<\/li>\n<li>What is the change in the electric field strength as an object moves from 20 nm to 2.0 nm towards an alpha particle?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>20.3 The Hydrogen Atom<\/h1>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Particle<\/th>\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Mass<\/th>\n<th style=\"width: 33.3333%; height: 18px;\" scope=\"col\">Charge<\/th>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Electron (e<sup>\u2212<\/sup> or \u00df<sup>\u2212<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">9.10938356(11) \u00d7 10<sup>\u221231<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">\u22121.6021766208 \u00d7 10<sup>\u221219<\/sup> C (\u2212e)<\/td>\n<\/tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 33.3333%; height: 18px;\">Proton (p, p<sup>+<\/sup> or N<sup>+<\/sup>)<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">1.672621898(21) \u00d7 10<sup>\u221227<\/sup> kg<\/td>\n<td style=\"width: 33.3333%; height: 18px;\">+1.6021766208 \u00d7 10<sup>\u221219<\/sup> C (+e)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">Bohr Radius [latex](a_0, r_{\\text{Bohr}})=5.29177 \u00d7 10^{\u221211}\\text{ m}[\/latex]<\/p>\n<p style=\"text-align: center;\">Excited State Radius [latex]r =n^2 a_o[\/latex]<\/p>\n<ul class=\"twocolumn\" style=\"list-style-type: none;\">\n<li>\u00a0[latex]F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>[latex]F_{c}=\\dfrac{mv^2}{r}[\/latex]<\/li>\n<li>[latex]E_k=\\dfrac{1}{3}mv^2[\/latex]<\/li>\n<li>[latex]p=mv[\/latex]<\/li>\n<\/ul>\n<div>\n<p>The hydrogen atom is important to an understanding of both the structure of atoms and the quantum nature of matter.\u00a0 One of the first clues to the nature of the hydrogen atom came from the emission spectrum (the specific colors) that hydrogen gas emits when heated. In 1885 Jacob Balmer (1825-1898) discovered that the visible wavelengths associated with the specific emitted colors fit a simple formula. By manipulating the formula, Balmer found other series of wavelengths, both visible and invisible.<\/p>\n<p>Ernest Rutherford (1871-1937) solved the remaining part of the mystery of the nature of atomic structure. Rutherford worked at three universities (including McGill in Montreal) and later at the Cavendish Laboratory. He discovered that atoms were mostly space, with an inner hard core surrounded by orbiting electrons, much like planets orbiting a sun at the centre. This concept, the Rutherford Planetary Model of the Atom, can be added to his concepts of alpha, beta and gamma rays, protons and half-life radioactivity.<\/p>\n<p>Combining the results of Balmer and Rutherford, Niels Bohr (1885-1962) used the energy of the emitted light spectra to come up with a model of the atom that certain average orbitals around the nucleus or core were allowed.\u00a0 Bohr\u2019s work predicted the size of the hydrogen atom and initiated an understanding of the nature of the movement of the electron around the atom. Rutherford\u2019s Planetary Model and Balmer\u2019s emission spectra were now grounded in a common conceptual understanding.<\/p>\n<p>In less than two decades, Werner Heisenberg (1901-1976), Paul Dirac (1902-1984) and Wolfgang Pauli (1900-1958) together laid out the foundations of quantum mechanics by focusing on the frequencies and intensities of the hydrogen spectra transitions, and the elliptical effect of the electron orbits on the hydrogen spectra, with the mathematical genius to put it all together.<\/p>\n<p>The following questions relate to the Hydrogen Atom.<\/p>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.3.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol type=\"i\">\n<li>What is the strength of the electrostatic force between an electron and a proton separated by a distance of 2.12 \u00d7 10<sup>\u221210<\/sup> m (1st excited state)?\n<ul>\n<li>\u00a0[latex]F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>[latex]\\vec{E}=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})^2}[\/latex]<\/li>\n<li>[latex]\\vec{E}[\/latex] = 5.13 \u00d7 10<sup>\u22129<\/sup> N<\/li>\n<\/ul>\n<\/li>\n<li>Balancing the centripetal force to the electrostatic force for the hydrogen atom, calculate the speed of an electron that would be orbiting the proton (1st excited state).\n<ul type=\"i\">\n<li>[latex]F_{c}=\\dfrac{mv^2}{r}\\qquad\\qquad F_{el}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]<\/li>\n<li>Therefore&#8230; [latex]\\dfrac{mv^2}{r}=\\dfrac{kq_1{q}_2}{d^2}[\/latex]\u00a0 Cancel out the common distance &amp; isolate v<sup>2<\/sup><\/li>\n<li>Yields&#8230; [latex]v^2=\\dfrac{kq_1{q}_2}{dm}[\/latex]\n<ul type=\"i\">\n<li>[latex]{v}^2=\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})(9.11\\times10^{-31}\\text{ kg}}[\/latex]<\/li>\n<li>[latex]v^2[\/latex]<sup>\u00a0<\/sup>= 1.19 \u00d7 10<sup>12<\/sup> m<sup>2<\/sup>\/s<sup>2<\/sup><\/li>\n<li>[latex]v[\/latex] = 1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Calculate the kinetic energy and the momentum of this orbiting electron (1st excited state).\n<ul type=\"i\">\n<li>[latex]E_k[\/latex] = \u00bdmv\u00b2<\/li>\n<li>[latex]E_k[\/latex] = \u00bd(9.11 \u00d7 10<sup>\u221231 <\/sup>kg)(1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s)<sup>2<\/sup><\/li>\n<li>[latex]E_k[\/latex] = 5.41 \u00d7 10<sup>\u221219 <\/sup>J or 3.4 eV<\/li>\n<li>[latex]p = mv[\/latex]<\/li>\n<li>[latex]p[\/latex] = (9.11 \u00d7 10<sup>\u221231 <\/sup>kg)(1.09\u00a0\u00d7 10<sup>6\u00a0<\/sup>m\/s)<\/li>\n<li>[latex]p[\/latex] = 9.9 \u00d7 10<sup>\u221225 <\/sup>Ns<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 20.3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol type=\"i\">\n<li>What is the electrostatic force strength between an electron and a proton separated by 4.76 \u00d7 10<sup>\u221210<\/sup> m (2nd excited state)?<\/li>\n<li>Balancing the centripetal force to the electrostatic force for the hydrogen atom calculate the speed of an electron that would be orbiting the proton (the hydrogen atom).<\/li>\n<li>Calculate the kinetic energy and the momentum of this orbiting electron.<\/li>\n<\/ol>\n<\/li>\n<li>If an electron that is orbiting an alpha particle has a kinetic energy of 54.42 eV 114 what distance is this electron from the core? What is the centripetal force and the electrostatic force that is acting on this electron?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>20.4 Electric Potential of Point Charges<\/h1>\n<p>The images below show the magnitude of the Electrical Potential of both a positive and negative charge as one moves away from the charge.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-533\" src=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential.png\" alt=\"\" width=\"388\" height=\"178\" srcset=\"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential.png 388w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential-300x138.png 300w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential-65x30.png 65w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential-225x103.png 225w, https:\/\/opentextbc.ca\/foundationsofphysics\/wp-content\/uploads\/sites\/427\/2023\/02\/electrical-potential-350x161.png 350w\" sizes=\"auto, (max-width: 388px) 100vw, 388px\" \/><\/p>\n<p>Equations Introduced or Used for this Section:<\/p>\n<p style=\"text-align: center;\">[latex]\\Delta V[\/latex] = \u00b1[latex]\\dfrac{kq}{d}[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\Delta E[\/latex] = \u00b1 [latex]q V[\/latex]<\/p>\n<p>Electric potential for charged objects is similar to the gravitational potential, in that instead of moving without friction through some displacement in a gravitational field, an object instead moves through a displacement inside an electrical field.\u00a0 There are two possibilities for this electric field: either the field is constant like the gravity one finds when moving through small changes in height on Earth, or the field varies similarly to the movement of a spaceship either towards or away from the Earth.<\/p>\n<p>This results in two different equations for electric potential: the electric potential moving away from a charged object, or the electric potential in a constant electric field. This section looks at only the electric potential moving away from a charged object, that as such experiences a variable electric field.<\/p>\n<p>Electric potential for a charged object can be used to find the energy change if the amount of charge moving through the potential difference is known.\u00a0 It is important to remember the distinction between electric potential and electric potential energy&#8230; They are different.<\/p>\n<p>The equation for electric potential is one that is most useful in a number of physics situations that look at the energy required to remove an electron from an atom, or to move from one allowed orbital to another for an electron in orbit around an atom.<\/p>\n<p>While the following examples relate to the hydrogen atom, applications of this equation extend far beyond this.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.4.1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the electric potential of an electron in its second excited state surrounding a hydrogen atom?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<ul>\n<li>[latex]V[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\n<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(2.12\\times10^{-10}\\text{ m})}[\/latex]<\/li>\n<li>[latex]V[\/latex] = \u2212 6.79 N\/m<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 20.4.2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the electrical potential for a hydrogen atom between the average ground state radius (n = 1) 5.29 \u00d7 10<sup>\u221211<\/sup> m and (n = 4) average radius of 8.46 \u00d7 10<sup>\u221210<\/sup> m.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>This requires finding the change in electric potential between these two states&#8230;<\/p>\n<ul>\n<li>[latex]V_{n = 1}[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\n<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(5.29\\times10^{-11}\\text{ m})}[\/latex]<\/li>\n<li>[latex]V[\/latex] = \u2212 27.2 N\/m<\/li>\n<li>[latex]V_{n = 4}[\/latex] = [latex]-\\dfrac{kq}{d}[\/latex]<\/li>\n<li>[latex]V[\/latex] = [latex]-\\dfrac{(8.99\\times10^9\\text{ Nm}^2\\text{\/C}^2)(1.602\\times10^{-19}\\text{ C})}{(8.46\\times10^{-10}\\text{ m})}[\/latex]<\/li>\n<li>[latex]V[\/latex] = \u2212 1.70 N\/m<\/li>\n<li>[latex]\\Delta V = V_{n = 4} \u2212 V_{n = 1}[\/latex]<\/li>\n<li>[latex]\\Delta V[\/latex] = \u2212 1.70 N\/m \u2212 (\u2212 27.2 N\/m) or 25.5 N\/m<\/li>\n<\/ul>\n<p>[latex]\\Delta V[\/latex] depends on the direction travelled (away or towards) and as such [latex]\\Delta V[\/latex] = \u00b1 25.5 N\/m<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 20.4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the electric potential of an electron in its ground state radius surrounding a hydrogen atom?<\/li>\n<li>What is the difference in electric potential to remove a proton at a distance of 25 \u00b5m from a charged sphere having a charge of 106 excess electrons?<\/li>\n<li>What is the difference in electrical potential for a hydrogen atom between the average ground state radius (n = 1) 5.29 \u00d7 10<sup>\u221211<\/sup> m and (n = 3) average radius of 4.76 \u00d7 10<sup>\u221210<\/sup> m.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 20.5.1: Deep space research on hydrogen atoms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Deep space research on hydrogen atoms, suggest that electrons can orbit protons at distances as far as one metre apart (d = 1.0 m).<\/p>\n<ol>\n<li>What would be the orbital speed of this electron?<\/li>\n<li>What would be the kinetic energy of this electron?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Exercise Answers<\/h1>\n<h2>20.1 Electrostatic Forces<\/h2>\n<ol class=\"twocolumn\">\n<li style=\"list-style-type: none;\">\n<ol type=\"i\">\n<li>has extra electrons<\/li>\n<li>6.25 \u00d7 10<sup>6<\/sup> e<sup>\u2212<\/sup><\/li>\n<\/ol>\n<\/li>\n<li>Repulsive force of 7.2 \u00d7 10<sup>\u22124<\/sup> N<\/li>\n<li>Attractive force of 8.24 \u00d7 10<sup>\u22128<\/sup> N<\/li>\n<li>3.0 \u00d7 10<sup>\u22123<\/sup> m<\/li>\n<li>1st excited\u00a0 \u00a05.12 \u00d7 10<sup>\u22129<\/sup> N \u00a0 3rd excited\u00a0 \u00a03.2 \u00d7 10<sup>\u221210<\/sup> N<\/li>\n<li>9.03 \u00d7 10<sup>22<\/sup> m\/s<sup>2 <\/sup>\n<ol type=\"i\">\n<li>2.0 \u00d7 10<sup>\u22125<\/sup> m\/s<sup>2<\/sup><\/li>\n<li>4.03 \u00d7 10<sup>7<\/sup> e<sup>\u2212<\/sup><\/li>\n<\/ol>\n<\/li>\n<li>1.04 \u00d7 10<sup>24<\/sup> m\/s<sup>2<\/sup> (electron) ac = 1.41 \u00d7 10<sup>20<\/sup> m\/s<sup>2<\/sup> (core)<\/li>\n<li>d = 1.61 m<\/li>\n<li>d = 5.1 m<\/li>\n<\/ol>\n<h2>20.2 Electric Fields (Point Charges)<\/h2>\n<ol class=\"twocolumn\">\n<li>5.15 \u00d7 10<sup>11<\/sup> N\/C (away)<\/li>\n<li>3.6 N\/C away from Sphere<\/li>\n<li>1.02 \u00d7 10<sup>\u22127<\/sup> N\/C upwards<\/li>\n<li>\n<ol type=\"i\">\n<li>8.0 \u00d7 10<sup>\u221214<\/sup> N<\/li>\n<li>2.4 \u00d7 10<sup>12<\/sup> m\/s<sup>2 <\/sup><\/li>\n<\/ol>\n<\/li>\n<li>\n<ol type=\"i\">\n<li>1.7 \u00d7 10<sup>\u221217<\/sup>\u00a0N<\/li>\n<li>1.8 \u00d7 10<sup>13<\/sup>\u00a0m\/s<sup>2<\/sup><\/li>\n<\/ol>\n<\/li>\n<li>5.4 \u00d7 10<sup>\u22125<\/sup> m<\/li>\n<li>5.6 \u00d7 10<sup>\u22125<\/sup> N\/C<\/li>\n<li>7.1 \u00d7 10<sup>8<\/sup> N\/C away<\/li>\n<\/ol>\n<h2>20.3 Hydrogen Atom<\/h2>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol type=\"i\">\n<li>1.02 \u00d7 10<sup>\u22129<\/sup> N<\/li>\n<li>2.18 \u00d7 10<sup>6<\/sup> m\/s<\/li>\n<li>1.99 \u00d7 10<sup>\u221224<\/sup> N\/C<\/li>\n<\/ol>\n<\/li>\n<li>[latex]F_{el}[\/latex] = [latex]F_c[\/latex] = 0.657 \u00b5N<\/li>\n<\/ol>\n<h2>20.4\u00a0 Electric Potential<\/h2>\n<ol class=\"twocolumn\">\n<li>[latex]V[\/latex] = \u2212 27.2 J\/C<\/li>\n<li>[latex]V[\/latex] =\u00a0 57.5 J\/C<\/li>\n<li>[latex]\\Delta V[\/latex] = 24.2 J\/C<\/li>\n<\/ol>\n<h2>20.5.1 Deep space research on hydrogen atoms<\/h2>\n<ol class=\"twocolumn\">\n<li>[latex]v[\/latex] = 15.9 m\/s<\/li>\n<li>1.15 \u00d7 10<sup>\u221228<\/sup> J<\/li>\n<\/ol>\n<h3>Media Attributions<\/h3>\n<ul>\n<li>&#8220;<a href=\"https:\/\/www.a-levelphysicstutor.com\/field-elect-2.php\">Point charge electric field patterns<\/a>&#8221; by A-Level Physics Tutor is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/deed.en\">CC BY-NC-ND licence<\/a>.<\/li>\n<li>&#8220;<a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Blausen_0615_Lithium_Atom.png\">Blausen 0615 Lithium Atom<\/a>&#8221; by <a title=\"User:BruceBlaus\" href=\"https:\/\/commons.wikimedia.org\/wiki\/User:BruceBlaus\">BruceBlaus<\/a> is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/deed.en\">CC BY-SA 4.0 licence<\/a>.<\/li>\n<li>Example 20.2.1 image from <a href=\"https:\/\/openstax.org\/details\/books\/college-physics\">College Physics<\/a> by Paul Peter Urone, Roger Hinrichs, Kim Dirks and Manjula Sharma is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY 4.0 licence<\/a>.<\/li>\n<li>&#8220;Graph of electric potential&#8221; by <a href=\"https:\/\/physics.stackexchange.com\/users\/63672\/user45220\">user45220<\/a> of Stack Exchange is licensed under a <a href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/\">CC BY-SA 3.0 licence<\/a>.<\/li>\n<\/ul>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-215-1\"><a href=\"https:\/\/www.a-levelphysicstutor.com\/index-field.php\">A-Level Physics Tutor - Electricity<\/a> <a href=\"#return-footnote-215-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-215-2\">Benjamin Franklin actually did his famous kite experiment in a thunderstorm in 1752, collecting electric charge in a capacitor which was known at that time as a Leyden Jar. Greater detail of this experiment can be found at: <a href=\"https:\/\/en.wikipedia.org\/wiki\/Kite_experiment\">Kite experiment<\/a> &amp; <a href=\"https:\/\/en.wikipedia.org\/wiki\/Leyden_jar\">Leyden jar<\/a> <a href=\"#return-footnote-215-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":125,"menu_order":20,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-215","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/users\/125"}],"version-history":[{"count":25,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/215\/revisions"}],"predecessor-version":[{"id":1014,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/215\/revisions\/1014"}],"part":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapters\/215\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/media?parent=215"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/pressbooks\/v2\/chapter-type?post=215"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/contributor?post=215"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/foundationsofphysics\/wp-json\/wp\/v2\/license?post=215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}