{"id":35743,"date":"2021-09-30T13:00:41","date_gmt":"2021-09-30T13:00:41","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35743"},"modified":"2025-12-29T03:32:39","modified_gmt":"2025-12-29T03:32:39","slug":"fluxus-electricus-et-lex-gaussiana","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/fluxus-electricus-et-lex-gaussiana\/","title":{"rendered":"Fluxus Electricus et Lex Gaussiana"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Fluxus Electricus et Lex Gaussiana<\/h1>\n<p>\nIn electrostatica, campus electricus \u201ca primis principiis\u201d computare valde laboriosum fieri potest, cum geometria distributionis oneris non sit trivialis. Fluxus electricus et Lex Gaussiana viam magis rationabilem praebent: pro certamine cum integralibus interminabilibus, superficiem clausam idoneam eligis et symmetria systematis uteris ad eventus claros atque verificabiles obtinendos. In usu, hoc in pauciores gradus, minores errores et maiorem potestatem conceptualem super id quod agis convertitur. Si vis a \u201cnovi formulam\u201d ad \u201cintellego methodum\u201d progredi, hic videbis quomodo Gauss problemata gravia apparentia in solutiones directas convertat, atque quando re vera eius usus conveniat.\n<\/p>\n<p style=\"text-align:center;\"><b>Proposita Discendi<\/b><\/p>\n<ol>\n<li><strong>Explicare<\/strong> rationem qua lex Gaussiana ad campum electricum se habeat.<\/li>\n<li><strong>Uti<\/strong> lege Gaussiana ad campos electricos computandos, symmetriis coordinatarum Cartesianarum, cylindricarum et sphaericarum adhibitis.<\/li>\n<li><strong>Coniungere<\/strong> formam integralem et differentialem per Theorema Divergentiae, dignoscendo quid singula membra repraesentent.<\/li>\n<li><strong>Conferre<\/strong> rationem Gaussii cum computatione directa per integrale Coulombianum, explanando quando complexitatem minuat et quando solutionem clausam non praebeat.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX CONTENTORUM<\/u>:<\/b><br \/>\n<a href=\"#1\">Resolutio electrostaticae<\/a><br \/>\n<a href=\"#2\">Lineae campi electrici<\/a><br \/>\n<a href=\"#3\">Annotatio de densitate linearum campi et earum repraesentatione<\/a><br \/>\n<a href=\"#4\">Fluxus Campi Electrici<\/a><br \/>\n<a href=\"#5\">Lex Gaussiana<\/a><br \/>\n<a href=\"#6\">Problemata cum symmetria sphaerica<\/a><br \/>\n<a href=\"#7\">Plures Symmetriae<\/a><br \/>\n<a href=\"#8\">Problemata cum symmetria cylindrica et planari<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/b96XremJiKQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Resolutio electrostaticae<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=128s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Ex iis quae hactenus pertractata sunt, habemus<\/span><\/strong><\/a> satis esse cognoscere formam elementi campi electrici atque eius distributionem in spatio ad campum electricum totalem determinandum. Si distributionem volumetricam habemus, tunc<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\int_V d\\vec{E}(\\vec{r})= \\int_V \\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>ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rho(\\vec{r}^\\prime)<\/span><\/span> est densitas volumetrica oneris. Si densitatem superficialem aut linearem oneris habemus, <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> per <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span> vel <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span> respective substituemus. Ab hoc puncto deinceps, id quod determinat utrum campum electricum invenire possimus necne est utrum integrale solvere valeamus.<\/p>\n<p>Quamvis problema ponere plerumque sit directum, citius aut serius deprehendemus id non semper facile esse ad evaluandum. Re vera, magna pars studii electrostaticae in evolutione rationum consistit quae calculum integralium inutiliter complicatarum vitare sinant. Multae ex his simplificationibus ex analysi vectoriali oriuntur, praesertim ex usu divergentiae.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Lineae campi electrici<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=235s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Antequam analysim vectorialem introducamus<\/span><\/strong><\/a> in studio nostro electrostaticae, quasdam notiones exhibebimus quae argumentum paulo magis intuitivum reddent. Agitur de <strong>lineis campi electrici<\/strong>.<\/p>\n<p>Incipiamus a simplicissimo: campo electrico oneris punctiformis in origine coordinatarum siti. Hic est formae<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0}\\frac{q}{\\|\\vec{r}\\|^2}\\hat{r}<\/span>\n<p>Hoc nobis permittit campum electricum in spatio repraesentare ut collectionem \u201csagittarum\u201d, quarum directio et magnitudo directionem et intensitatem campi electrici in quolibet puncto describunt.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEhlkOkwoYFrKYwfidmawFFb60AlBumv8u2irJnN87xYJnTfY7h2U1HL3Hzfh7kQZHhyctM7r70IfEXZkh0faUjZrGr3H_h6xrjeED-GqV39v_t2OzPD0zD9sjlm9t_twIEuaJcte9qhEGIKH3Bzn7a_AZ1rCh53DFunFLiXm09JalAMYQNAjKjv3oOosA\" width=\"400\" height=\"300\" alt=\"Campo el\u00e9ctrico de una carga puntual en forma de vectores\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEhlkOkwoYFrKYwfidmawFFb60AlBumv8u2irJnN87xYJnTfY7h2U1HL3Hzfh7kQZHhyctM7r70IfEXZkh0faUjZrGr3H_h6xrjeED-GqV39v_t2OzPD0zD9sjlm9t_twIEuaJcte9qhEGIKH3Bzn7a_AZ1rCh53DFunFLiXm09JalAMYQNAjKjv3oOosA\" width=\"400\" height=\"300\" alt=\"Campo el\u00e9ctrico de una carga puntual en forma de vectores\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>Cum intensitas campi electrici cum quadrato distantiae ab origine decrescat, vectores paulatim minores fiunt quo longius discedimus. Praeterea, radialiter a onere ad exteriora diriguntur.<\/p>\n<p>Haec repraesentatio utilis est, sed alia etiam magis informativa exstat: \u201ccontinuum sagittarum conectere\u201d ad campum linearum formandum. Hoc modo, non iam longitudo sagittarum intensitatem campi electrici indicabit, sed \u201cdensitas linearum campi\u201d in diagrammate.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgebGCEuU6Akvc8M-p8m0FjK-a1AGtq4H7A2fAuy1r6M08uAnWQRYJfHIRtcRAvGt3CQZHCI7EwCHv3os55aZpef1KDTHFDiS2Sf8nvyXH_ctiildMSeSK-suC7al5kbGmFReywKsEJh1GVsHDtTqRShAyiJZFQER2fKav4wOcn9z8q7zjmjk3T07UHTQ\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgebGCEuU6Akvc8M-p8m0FjK-a1AGtq4H7A2fAuy1r6M08uAnWQRYJfHIRtcRAvGt3CQZHCI7EwCHv3os55aZpef1KDTHFDiS2Sf8nvyXH_ctiildMSeSK-suC7al5kbGmFReywKsEJh1GVsHDtTqRShAyiJZFQER2fKav4wOcn9z8q7zjmjk3T07UHTQ\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h3>Adnotatio de densitate linearum campi earumque repraesentatione<\/h3>\n<p>Antequam pergamus, convenit animadvertere quoddam momentum de diagrammate linearum campi electrici. Huiusmodi repraesentatio non est omnino fidelis cum in plano (2D) delineatur. In schemate 2D, si circulum radii <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> consideramus, numerus totalis linearum super perimetro circumferentiae distribuitur, ita ut densitas linearis sit<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n}{2\\pi r}<\/span>\n<p>Quae cum <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> decrescit, et non cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r^2<\/span><\/span>, sicut exspectatur ut intensitas campi electrici decrescat. Attamen, si exemplum in tribus dimensionibus interpretamur (veluti erinaceum), tunc numerus totalis linearum per superficiem sphaerae divideretur<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n}{4\\pi r^2}<\/span>\n<p>atque hoc quidem respectu <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r^2<\/span><\/span> decrescit. Aliis verbis, quamquam repraesentatio linearum campi plerumque in duabus dimensionibus fiat, id quod revera intenditur est situatio in tribus dimensionibus. Chartam enim trium dimensionum ad pingendum non habemus: in 2D repraesentamus id quod in 3D communicare volumus.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Fluxus Campi Electrici<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=665s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cum quaerimus de numero<\/span><\/strong><\/a> linearum campi electrici quae certam superficiem transeunt, responsum per fluxum campi electrici super illam superficiem datur. Ita definitur fluxus electricus campi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{E}<\/span><\/span> per superficiem <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> ut<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\vec{E},S} =\\displaystyle \\int_S \\vec{E}\\cdot d\\vec{S}<\/span>\n<p>Non decipiamur notione intuitiva \u201cnumeri linearum campi electrici quae superficiem transeunt\u201d. Meminerimus illum numerum linearum (vel densitatem linearum) esse modum repraesentandi intensitatem campi electrici. Quapropter fluxus electricus quem computamus est magnitudo scalaris cum intensitate campi electrici coniuncta, quae superficiem <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> pervadit.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Lex Gaussiana<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=758s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Quoniam intensitas campi electrici<\/span><\/strong><\/a> proportione ad onus electricum respondet, fluxum electricum per superficiem quae certum onus includit exprimere debemus ut quantitatem oneri incluso proportionalem. Re vera, non est difficile demonstrare ita fieri. Consideremus figuram sequentem:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgGcCL8WVnhwXmxDkhsW5W31AyJiEsJDsVZZDNm1kQ-MREYYaaBvYb7CBkGSCkfPgiNbDGFP-R4LHr_9pH6ijy0Ji7m1VgzO2pjJwjFDOqAd61VGMJfb4CDfmGyn9uacon7VcpXlB9cd7ZltDUEc3fhDQ86PuKqQb7kN-JuNgGxInlRKiyY91nU2zHfIg\" width=\"500\" height=\"400\" alt=\"Flujo el\u00e9ctrico en una superficie cerrada\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgGcCL8WVnhwXmxDkhsW5W31AyJiEsJDsVZZDNm1kQ-MREYYaaBvYb7CBkGSCkfPgiNbDGFP-R4LHr_9pH6ijy0Ji7m1VgzO2pjJwjFDOqAd61VGMJfb4CDfmGyn9uacon7VcpXlB9cd7ZltDUEc3fhDQ86PuKqQb7kN-JuNgGxInlRKiyY91nU2zHfIg\" width=\"500\" height=\"400\" alt=\"Flujo el\u00e9ctrico en una superficie cerrada\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>Ex his sequitur:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\oint_S \\vec{E}\\cdot d\\vec{S} &amp;= \\displaystyle \\oint_S \\left(\\frac{1}{4\\pi\\epsilon_0} \\frac{q_{enc}}{\\|\\vec{r}\\|^2}\\hat{r} \\right)\\cdot d\\vec{S} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{q_{enc}}{4\\pi\\epsilon_0} \\oint_S \\frac{\\hat{r}}{\\|\\vec{r}\\|^2}\\cdot d\\vec{S} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{q_{enc}}{4\\pi\\epsilon_0} \\underbrace{\\oint_S d{\\Omega}}_{= 4\\pi} = \\frac{q_{enc}}{\\epsilon_0}\n\n\\end{array}\n\n<\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEjLQhqvLJZMuQHfXHj_WYfbajP9PYwVdNgs4eVflg_jQAJSFu5czNfBgMBTWOWXCE5Tx3-DYwrs8eNpOuJoflvQYbUwpl3BG4BaZxJdnJirqRPsbZM00TfnzyGQvuAimfenB3GUYnEJdZDh2xiXWX5ftu0bN-UYH3G4rydnrnBqEpKDNnNXgdpi5EP81w\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEjLQhqvLJZMuQHfXHj_WYfbajP9PYwVdNgs4eVflg_jQAJSFu5czNfBgMBTWOWXCE5Tx3-DYwrs8eNpOuJoflvQYbUwpl3BG4BaZxJdnJirqRPsbZM00TfnzyGQvuAimfenB3GUYnEJdZDh2xiXWX5ftu0bN-UYH3G4rydnrnBqEpKDNnNXgdpi5EP81w\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>In summa, obtinemus:<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\color{blue}{\\oint_S \\vec{E}\\cdot d\\vec{S} = \\frac{q_{enc}}{\\epsilon_0}}<\/span>\n<p>Haec est <strong>Lex Gaussiana pro campo electrico in forma integrali<\/strong>, atque relationem proportionalitatis inter fluxum electricum per superficiem clausam et onus inclusum ostendit. Animadvertendum est eam hic in \u201cforma integrali\u201d exhiberi, ut pateat etiam formam differentialem exsistere, quae per Theorema Divergentiae Gaussii in contextu analysi vectorialis obtinetur.<\/p>\n<div style=\"background-color: #c0ffc0; padding: 20px;\">\n<h4>Theorema Divergentiae Gaussii<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=1007s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}<\/span><\/span> est campus vectorialis differentiabilis<\/span><\/strong><\/a> et <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> est superficies clausa quae volumen <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> includit, tunc valet<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\oint_S\\vec{F}\\cdot d\\vec{S} = \\int_V (\\vec{\\nabla}\\cdot \\vec{F})dV<\/span>\n<\/div>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=1055s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Theoremate divergentiae applicato<\/span><\/strong><\/a> ad fluxum campi electrici per superficiem clausam <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, habetur<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\oint_S\\vec{E}\\cdot d\\vec{S} = \\int_V (\\vec{\\nabla}\\cdot\\vec{E})dV = \\frac{q_{enc}}{\\epsilon_0}<\/span>\n<p>Praeterea etiam habetur<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{q_{enc}}{\\epsilon_0} = \\int_V \\frac{\\rho}{\\epsilon_0} dV<\/span>\n<p>Ex his duabus ultimis aequationibus tandem obtinetur<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\vec{\\nabla}\\cdot\\vec{E} = \\frac{\\rho}{\\epsilon_0}}<\/span>\n<p>Haec est <strong>Lex Gaussiana pro campo electrico in forma differentiali.<\/strong><\/p>\n<p>Nunc lege Gaussiana uti possumus ad symmetrias geometricae quorundam problematum melius adhibendas atque, magna ex parte, calculum integralium ad campum electricum ducentium simpliciorem reddendum.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Problemata cum symmetria sphaerica<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/04itEuVNDN4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<ol>\n<li>Invenire campum electricum ad distantiam <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span> a centro superficiei sphaericae radii <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span>, quae densitatem oneris uniformem <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span> habet. Utrumque casum analysa: cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z\\lt R<\/span><\/span> et cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z\\geq R<\/span><\/span>.<\/li>\n<li>Eandem analysim ac in exercitio priori perage, sed nunc sphaeram solidam uniformiter oneratam cum densitate volumetrica <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> considerans. Deinde confice graphum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{E}\\|<\/span><\/span> ut functio <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span>.<\/li>\n<li>Supponamus campum electricum, ad distantiam <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> ab origine coordinatarum, esse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{E}=kr^2\\hat{r}<\/span><\/span>, cum <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> constante. Inveni densitatem oneris <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> huic campo associatam.<\/li>\n<\/ol>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Plures Symmetriae<\/h3>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6eMaax9orAo\" 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 href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=122s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Lex Gaussiana semper vera est<\/span><\/strong><\/a>, sed non semper utilis. In exemplis prioribus, si <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> non esset uniformis, si symmetria sphaerica deesset, aut si alia figura pro superficie Gaussiana eligeretur, adhuc verum esset fluxum electricum esse <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_{enc}\/\\epsilon_0<\/span><\/span>, sed campus electricus nec constantem magnitudinem haberet nec necessario eadem directione ac elementum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\vec{S}<\/span><\/span> orientaretur; atque his condicionibus carentibus, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{E}\\|<\/span><\/span> e integrali extrahere non possumus.<\/p>\n<p>Symmetria maximi momenti est in applicatione Legis Gaussianae ad problemata solvenda.<\/p>\n<p>Multa genera symmetriarum sunt quibus uti possumus. Inter omnia, haec tria frequentissima sunt:<\/p>\n<ol>\n<li><strong>Symmetria sphaerica:<\/strong> Superficies Gaussiana est sphaera concentrica.<\/li>\n<li><strong>Symmetria cylindrica:<\/strong> Superficies Gaussiana est cylindrus coaxialis.<\/li>\n<li><strong>Symmetria planaris:<\/strong> Superficies Gaussiana est cista rectangula.<\/li>\n<\/ol>\n<p><a name=\"8\"><\/a><\/p>\n<h3>Problemata cum symmetria cylindrica et planari<\/h3>\n<ol>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=630s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Considera funem cylindricum<\/span><\/strong><\/a> infinite longum, rectum, radii <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span>, atque densitate oneris <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> praeditum huius formae\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rho(r) = \\left\\{\\begin{array}{lll}\n\nkr &amp; ; &amp; r\\lt R \\\\ \\\\\n\n0 &amp; ; &amp; R\\lt r \\\\ \\\\\n\n\\end{array}\\right.<\/span>\n<p>ubi <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> est constans. Calcula campum electricum intra cylindrum.<\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=1895s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Invenire campum electricum quem producit<\/span><\/strong><\/a> planum infinitum, densitate oneris uniformi <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span> praeditum.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Fluxus Electricus et Lex Gaussiana In electrostatica, campus electricus \u201ca primis principiis\u201d computare valde laboriosum fieri potest, cum geometria distributionis oneris non sit trivialis. Fluxus electricus et Lex Gaussiana viam magis rationabilem praebent: pro certamine cum integralibus interminabilibus, superficiem clausam idoneam eligis et symmetria systematis uteris ad eventus claros atque verificabiles obtinendos. 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