{"id":30981,"date":"2021-09-21T13:00:57","date_gmt":"2021-09-21T13:00:57","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30981"},"modified":"2025-01-02T21:38:54","modified_gmt":"2025-01-02T21:38:54","slug":"the-concept-of-the-electric-field","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-concept-of-the-electric-field\/","title":{"rendered":"The concept of the electric field"},"content":{"rendered":"<style>\n    p, ul, ol {\n        text-align: justify;\n    }\n    h1, h2 {\n        text-align: center;\n    }\n<\/style>\n<h1>The Concept of the Electric Field<\/h1>\n<p style=\"text-align: center;\"><em>In the 19th century, Michael Faraday, one of the greatest experimenters in the field of electricity, had a peculiar way of working: he filled his laboratory with wires, charged spheres, and small containers with conductive liquids. A famous anecdote tells that, in his obsession to visualize the \u00ablines of force\u00bb surrounding an electric charge, he spilled iron filings all over the lab floor, leaving it covered in patterns that resembled modern art. His colleagues, confused, thought he had lost his mind, but Faraday was outlining one of the most revolutionary concepts: the electric field. In this article, we will explore how these ideas, born from genius and experimentation, allow us to map and understand the invisible interactions governing electricity. If you&#8217;ve ever wondered how to see the intangible, this journey is for you.<\/em><\/p>\n<p style=\"text-align: center;\"><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this lesson, the student will be able to\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of the electric field and its relationship with electric force through Coulomb&#8217;s Law.<\/li>\n<li><strong>Apply<\/strong> the definition of the electric field to solve problems related to point charges.<\/li>\n<li><strong>Analyze<\/strong> the principle of superposition in discrete and continuous charge distributions to calculate electric fields.<\/li>\n<li><strong>Evaluate<\/strong> the integration of linear, surface, and volumetric distributions to determine electric fields in complex configurations.<\/li>\n<li><strong>Solve<\/strong> practical exercises including configurations such as charged rods, charged rings, and infinite charged planes.<\/li>\n<\/ol>\n<p style=\"text-align: center;\"><strong><u>CONTENT INDEX<\/u>:<\/strong><br \/>\n<a href=\"#1\">What is the electric field?<\/a><br \/>\n<a href=\"#2\">The electric field and charge distributions<\/a><br \/>\n<a href=\"#3\">Exercises<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KWwRfJ9ZRrA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>What is the Electric Field?<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=191s\" target=\"_blank\" rel=\"noopener\"><strong>When we place a source charge<\/strong><\/a> somewhere in space, we can sense its presence using a test charge due to the <strong>electric force<\/strong> it experiences. This force is studied through Coulomb&#8217;s Law. Based on this, we say that the source charge \u00abfloods the space\u00bb with a property, an Electric Field, which is responsible for producing the electric force.<\/p>\n<p>To measure the electric field of a charge <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> at a certain point <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}<\/span> in space, we need to place a test charge <span class=\"katex-eq\" data-katex-display=\"false\">q_0<\/span> at that location. The electric field is described as the amount of electric force felt by the test charge <span class=\"katex-eq\" data-katex-display=\"false\">q_0<\/span> per unit charge.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{E}_q(\\vec{r}) = \\displaystyle \\frac{\\vec{F}_{q\\to q_0}(\\vec{r})}{q_0}<\/span>\n<p>But when we proceed this way, we overlook the fact that the test charge should also have its own field, and this will overlap with the field of the source charges. To address this issue, we define the electric field through the limit:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\vec{E}_q(\\vec{r}) = \\lim_{q_0 \\to 0} \\frac{\\vec{F}_{q\\to q_0}(\\vec{r})}{q_0}<\/span>\n<p>Using the definition through the limit, we ensure that the field of the test charge <span class=\"katex-eq\" data-katex-display=\"false\">q_0<\/span> does not interfere with the measurements of the field of charge <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span>. Now, recalling Coulomb&#8217;s Law, the electric field of a charged particle <span class=\"katex-eq\" data-katex-display=\"false\">q<\/span> is given by:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\vec{E}_q(\\vec{r}) = \\lim_{q_0 \\to 0} \\frac{1}{q_0} \\frac{1}{4\\pi \\epsilon_0} \\frac{qq_0}{\\|\\vec{r} -\\vec{r}^\\prime\\|^2} \\frac{\\vec{r} -\\vec{r}^\\prime}{\\|\\vec{r} -\\vec{r}^\\prime\\|} = \\frac{q}{4\\pi\\epsilon_0} \\frac{\\vec{r} -\\vec{r}^\\prime}{\\|\\vec{r} -\\vec{r}^\\prime\\|^3} <\/span>\n<p>And from this, it follows that<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{F}_{q\\to q_0}(\\vec{r}) = q_0 \\vec{E}_q(\\vec{r}) <\/span>\n<p><a name=\"2\"><\/a> <\/p>\n<h2>The Electric Field and Charge Distributions<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=565s\" target=\"_blank\" rel=\"noopener\"><strong>Since the electric field<\/strong><\/a> is studied in terms of force and satisfies the principle of superposition, we can analyze the fields of different charge distributions.<\/p>\n<h3>Discrete Distributions<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=587s\" target=\"_blank\" rel=\"noopener\"><strong>Let us consider a distribution of <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> discrete charges<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">q_1, q_2, \\cdots, q_n<\/span> with positions <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}^\\prime_1, \\vec{r}^\\prime_2, \\cdots, \\vec{r}^\\prime_n.<\/span> If we want to calculate their field at a point <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{r}<\/span> in space, we have:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\vec{E}_q(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0} \\sum_{i=1}^n q_i \\frac{\\vec{r} -\\vec{r}_i^\\prime}{\\|\\vec{r} -\\vec{r}_i^\\prime\\|^3} <\/span>\n<p>This is the sum of all individual fields.<\/p>\n<h3>Continuous Distributions<\/h3>\n<p>There are three types of continuous charge distributions, each associated with the number of parameters needed to describe their spatial arrangement. These are linear, surface, and volumetric distributions.<\/p>\n<h4>Linear Distribution<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=688s\" target=\"_blank\" rel=\"noopener\"><strong>In a linear charge distribution,<\/strong><\/a> each line element of the charged body has a linear charge density <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda(\\vec{r}^\\prime)=dq(\\vec{r}^\\prime)\/dl<\/span>, such that the electric field element is given by:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\vec{E}(\\vec{r}) =\\displaystyle \\frac{dq(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3} = \\frac{\\lambda(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dl<\/span>\n<p>Integrating this expression gives:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0} \\int_{\\mathcal{C}} \\lambda(\\vec{r}^\\prime) \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dl<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{C}<\/span> is the parametric representation of the curve describing the shape of the charged body.<\/p>\n<h4>Surface Distribution<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=899s\" target=\"_blank\" rel=\"noopener\"><strong>In a surface charge distribution,<\/strong><\/a> each surface element of the charged body has a surface charge density <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma(\\vec{r}^\\prime)=dq(\\vec{r}^\\prime)\/dS<\/span>, such that the electric field element is given by:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle d\\vec{E}(\\vec{r}) = \\frac{dq(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3} = \\frac{\\sigma(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dS<\/span>\n<p>Integrating this expression gives:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0} \\iint_{\\mathcal{A}} \\sigma(\\vec{r}^\\prime) \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dS<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}<\/span> is the parametric representation of the surface describing the shape of the charged body.<\/p>\n<h4>Volumetric Distribution<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=KWwRfJ9ZRrA&amp;t=1054s\" target=\"_blank\" rel=\"noopener\"><strong>In a volumetric charge distribution,<\/strong><\/a> each volume element of the charged body has a volumetric charge density <span class=\"katex-eq\" data-katex-display=\"false\">\\rho(\\vec{r}^\\prime)=dq(\\vec{r}^\\prime)\/dV<\/span>, such that the electric field element is given by:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle d\\vec{E}(\\vec{r}^\\prime) = \\frac{dq(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3} = \\frac{\\rho(\\vec{r}^\\prime)}{4\\pi\\epsilon_0} \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dV<\/span>\n<p>Integrating this expression gives:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0} \\iiint_{\\mathcal{V}} \\rho(\\vec{r}^\\prime) \\frac{\\vec{r} - \\vec{r}^\\prime}{\\|\\vec{r} - \\vec{r}^\\prime\\|^3}dV<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{V}<\/span> is the parametric representation of the volume describing the shape of the charged body.<\/p>\n<p><a name=\"3\"><\/a> <\/p>\n<h2>Exercises:<\/h2>\n<h3>Charged Rod<\/h3>\n<p>Consider a rod of length <span class=\"katex-eq\" data-katex-display=\"false\">L<\/span> uniformly charged with a charge <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> and positioned vertically. Determine the electric field of the rod at a horizontal distance <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> from the center of the rod.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/PEi3xjHDyxM\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h3>Charged Ring<\/h3>\n<p>Consider a ring of radius <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span> uniformly charged with a charge <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> placed on the xy-plane. Determine the electric field at a height <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span> from the center of the ring.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kPVAhlzvgKA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<h3>Infinite Charged Plane<\/h3>\n<p>Consider an infinite plane uniformly charged with a surface charge density <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span>. Determine the electric field at a distance <span class=\"katex-eq\" data-katex-display=\"false\">L<\/span> from the plane.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/IQeDWxVT7eM\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Concept of the Electric Field In the 19th century, Michael Faraday, one of the greatest experimenters in the field of electricity, had a peculiar way of working: he filled his laboratory with wires, charged spheres, and small containers with conductive liquids. A famous anecdote tells that, in his obsession to visualize the \u00ablines of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30980,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[710,635],"tags":[],"class_list":["post-30981","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>The concept of the electric field - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"The electric field is the region where a charge exerts a force on other charges, described by intensity and direction at each point.\" \/>\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\/the-concept-of-the-electric-field\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The concept of the electric field\" \/>\n<meta property=\"og:description\" content=\"The electric field is the region where a charge exerts a force on other charges, described by intensity and direction at each point.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/the-concept-of-the-electric-field\/\" \/>\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=\"2021-09-21T13:00:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-01-02T21:38:54+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/campoelectrico-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The concept of the electric field\" \/>\n<meta name=\"twitter:description\" content=\"The electric field is the region where a charge exerts a force on other charges, described by intensity and direction at each point.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/campoelectrico.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\\\/the-concept-of-the-electric-field\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The concept of the electric field\",\"datePublished\":\"2021-09-21T13:00:57+00:00\",\"dateModified\":\"2025-01-02T21:38:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/\"},\"wordCount\":1267,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/campoelectrico.jpg\",\"articleSection\":[\"Electromagnetism\",\"Physics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/the-concept-of-the-electric-field\\\/\",\"name\":\"The concept of the electric field - 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