{"id":30954,"date":"2021-08-31T13:00:09","date_gmt":"2021-08-31T13:00:09","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30954"},"modified":"2025-01-02T18:54:39","modified_gmt":"2025-01-02T18:54:39","slug":"coulombs-law-and-the-electrostatic-force","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/coulombs-law-and-the-electrostatic-force\/","title":{"rendered":"Coulomb&#8217;s Law and the Electrostatic Force"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2 {\n        text-align: center;\n    }\n<\/style>\n<h1>Coulomb&#8217;s Law and the Electrostatic Force<\/h1>\n<p style=\"text-align:center;\">\nThe \u00abCoulomb&#8217;s Law and the Electrostatic Force\u00bb has not only expanded our understanding of electric forces but also generated unexpected anecdotes. Benjamin Franklin, during an experiment to stun and cook a turkey with electricity, ended up as the test subject: a discharge left him dazed and with his hair standing up, almost as if illustrating the electric field lines in real life. Now, it&#8217;s our turn to study electric forces.\n<\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this class, students will be able to:<\/p>\n<ol>\n<li><strong>Model<\/strong> electric phenomena using the superposition principle to calculate the resultant force on a test charge.<\/li>\n<li><strong>Simplify<\/strong> the study of electric forces by restricting it to the electrostatic case.<\/li>\n<li><strong>Apply<\/strong> Coulomb&#8217;s Law to determine the force between two charges in various situations.<\/li>\n<li><strong>Analyze<\/strong> charge-centered systems through a simplified formulation of Coulomb&#8217;s Law.<\/li>\n<li><strong>Solve<\/strong> practical problems related to charge distributions.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n<a href=\"#1\">The Superposition Principle<\/a><br \/>\n<a href=\"#2\">The Electrostatic Simplification<\/a><br \/>\n<a href=\"#3\">Coulomb&#8217;s Law<\/a><br \/>\n<a href=\"#4\">Coulomb&#8217;s Law for Charge-Centered Systems<\/a><br \/>\n<a href=\"#5\">Exercises<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/RV3Uzorlkr4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p>We will now begin to mathematically model these phenomena, and to do so, we will introduce <strong>Coulomb&#8217;s Law.<\/strong> But first, it&#8217;s necessary to explain some points, namely: the superposition principle and the electrostatic simplification.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Superposition Principle<\/h2>\n<p>The fundamental problem in electrodynamics consists of determining the force that a \u00abcloud\u00bb of electric charges <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1<\/span><\/span>, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_2<\/span><\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">\\cdots<\/span> exerts on a test charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_0<\/span><\/span>, when the position of each charge is a known function of time. In general, both the source charges and the test charge are in relative motion.<\/p>\n<p>The solution to this problem is facilitated by the superposition principle, which states that the interaction of the test charge with any one source is entirely independent of its interaction with the other sources. This means that it is always possible to determine the force <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_1<\/span><\/span> contributed by charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1<\/span><\/span>, the force <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_2<\/span><\/span> contributed by <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_2<\/span><\/span>, and so on, to ultimately sum them up to obtain the total force:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{tot} = \\displaystyle \\sum_{i}\\vec{F}_i<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Electrostatic Simplification<\/h2>\n<p>If forces only need to be summed, one might argue that it is enough to state the equation describing the force that each source charge exerts on the test charge, and the problem would be solved; however, the issue is not that simple. The problem lies in the fact that the force depends not only on the distance and magnitude of the charges but also on the relative velocity and acceleration of each particle. Additionally, \u00abelectric information\u00bb about changes in position, velocity, and acceleration of each particle travels at the speed of light, meaning it takes a certain amount of time to reach the test charge and alter its effect.<\/p>\n<p>Therefore, to simplify our study for now, we will restrict ourselves to the <strong>electrostatic<\/strong> case, i.e., all source charges will remain stationary, and only the test charge will be allowed to move. It is within this context that <strong>Coulomb&#8217;s Law<\/strong> emerges.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Coulomb&#8217;s Law<\/h2>\n<p>Suppose we have a test charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_0<\/span><\/span> located at position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}<\/span><\/span>, and a source charge <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> located at position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}^\\prime<\/span><\/span>. What will be the force <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{q \\to q_0}(\\vec{r})<\/span><\/span> exerted by the source charge on the test charge? The answer to this question is provided by <strong>Coulomb&#8217;s Law,<\/strong> expressed by the formula:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{q \\to q_0} (\\vec{r}) = \\displaystyle \\frac{1}{4\\pi \\epsilon_0}  \\frac{q  q_0 }{\\|\\vec{r} - \\vec{r}^\\prime \\|^2} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|}<\/span><\/span><\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-amszqS6iLRk\/YSwCV4TCIZI\/AAAAAAAAFec\/XoTDZm0i-6oPZY4KuGv0xOEbrsYCfJM4wCLcBGAsYHQ\/s0\/fuerzaelectrica.PNG\" width=\"492\" height=\"390\" alt=\"Electric Force\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-amszqS6iLRk\/YSwCV4TCIZI\/AAAAAAAAFec\/XoTDZm0i-6oPZY4KuGv0xOEbrsYCfJM4wCLcBGAsYHQ\/s0\/fuerzaelectrica.PNG\" width=\"492\" height=\"390\" alt=\"Electric Force\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>Coulomb&#8217;s Law not only summarizes the rule of signs for electrostatic forces but also establishes that the force between electric charges is inversely proportional to the square of the distance separating them.<\/p>\n<p>The constant <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon_0<\/span><\/span> is called the <strong>electric permittivity of free space.<\/strong> Its value in the International System is:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\epsilon_0 = 8.85 \\cdot 10^{-12} \\left[ \\frac{C^2}{N\\cdot m^2}\\right] <\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Coulomb&#8217;s Law for Charge-Centered Systems<\/h2>\n<p><strong>Coulomb&#8217;s Law<\/strong> can be expressed in a simpler way if we position the observer at the source of the charges, i.e., by setting <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}^\\prime = \\vec{0}<\/span><\/span>. In this case, we have:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{F}_{q \\to q_0} (\\vec{r}) = \\frac{1}{4\\pi \\epsilon_0}  \\frac{q  q_0 }{\\|\\vec{r}\\|^2}   \\frac{\\vec{r} }{\\|\\vec{r} \\|} = \\frac{1}{4\\pi \\epsilon_0}  \\frac{q   q_0 }{\\|\\vec{r}\\|^2} \\hat{r}<\/span><\/span><\/p>\n<p>Where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\hat{r}=\\vec{r}\/\\|\\vec{r}\\|<\/span><\/span> is the unit vector pointing from the source to the test charge.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercises<\/h2>\n<ol>\n<li>Twelve point charges of equal magnitude <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> are placed at the corners of a regular twelve-sided polygon (similar to the numbers on a clock). What will be the net force on a point charge <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> placed at the center?<\/li>\n<li>One of the twelve charges from the previous exercise is removed. Suppose it is the one located at the 12 o&#8217;clock position (if imagined like a clock). What force will the central point charge <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> experience now?<\/li>\n<li>Extend the reasoning from the previous two exercises to a distribution of <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> source charges arranged in a regular <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>-sided polygon, with a test charge at the center.<\/li>\n<li>There are three point charges: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1=+3[nC]<\/span><\/span> located at position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(0;0)[mm]<\/span>,<\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_2=-5[nC]<\/span><\/span> at position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(0,56;0)[mm]<\/span><\/span>, and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_3=+7[nC]<\/span><\/span> at position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1;1)[mm]<\/span>.<\/span> Calculate the total force on charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_3<\/span>.<\/span><\/li>\n<li>On a straight line, a charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_1 = 3[C]<\/span><\/span> is located, and at a distance of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">40[mm]<\/span><\/span> another charge <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_2 = 7[C]<\/span><\/span>. If a third charge is placed between them so that the sum of the forces on it is zero, what will be the distance of this third charge from the other two?<\/li>\n<li>Two small copper spheres, each with a mass of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0.040[kg]<\/span><\/span>, are placed at a distance of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2.0[m]<\/span><\/span> apart. Considering that the molar mass of copper is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">63.5[g\/mol]<\/span><\/span> and its atomic number is 20, answer the following questions:\n<ol>\n<li type=\"a\">How many electrons does each sphere have?<\/li>\n<li type=\"a\">How many electrons must be transferred from one sphere to the other to produce an attractive force of approximately <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">10^4[N]<\/span><\/span> between the spheres?<\/li>\n<li type=\"a\">What fraction of the spheres&#8217; electrons does this represent?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/jxO34PM91Io\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/EBOxmKPQe9U\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Coulomb&#8217;s Law and the Electrostatic Force The \u00abCoulomb&#8217;s Law and the Electrostatic Force\u00bb has not only expanded our understanding of electric forces but also generated unexpected anecdotes. Benjamin Franklin, during an experiment to stun and cook a turkey with electricity, ended up as the test subject: a discharge left him dazed and with his hair [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30950,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[710,635],"tags":[],"class_list":["post-30954","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electromagnetism","category-physics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Coulomb&#039;s Law and the Electrostatic Force - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Coulomb&#039;s Law describes the electrostatic force between two charges: proportional to the product of the charges and inversely proportional to the square of the distance.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/coulombs-law-and-the-electrostatic-force\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Coulomb&#039;s Law and the Electrostatic Force\" \/>\n<meta property=\"og:description\" content=\"Coulomb&#039;s Law describes the electrostatic force between two charges: proportional to the product of the charges and inversely proportional to the square of the distance.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/coulombs-law-and-the-electrostatic-force\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-08-31T13:00:09+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-01-02T18:54:39+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/electrostatica-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Coulomb&#039;s Law and the Electrostatic Force\" \/>\n<meta name=\"twitter:description\" content=\"Coulomb&#039;s Law describes the electrostatic force between two charges: proportional to the product of the charges and inversely proportional to the square of the distance.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/electrostatica.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Coulomb&#8217;s Law and the Electrostatic Force\",\"datePublished\":\"2021-08-31T13:00:09+00:00\",\"dateModified\":\"2025-01-02T18:54:39+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/\"},\"wordCount\":1094,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/electrostatica.jpg\",\"articleSection\":[\"Electromagnetism\",\"Physics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/coulombs-law-and-the-electrostatic-force\\\/\",\"name\":\"Coulomb's Law and the Electrostatic Force - 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