Lei de Gauss – Wikipédia, a enciclopédia livre

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class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Conteúdo" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Conteúdo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover para a barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Fluxo_do_campo_elétrico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fluxo_do_campo_elétrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Fluxo do campo elétrico</span> </div> </a> <ul id="toc-Fluxo_do_campo_elétrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lei_de_Gauss" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lei_de_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Lei de Gauss</span> </div> </a> <button aria-controls="toc-Lei_de_Gauss-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Lei de Gauss</span> </button> <ul id="toc-Lei_de_Gauss-sublist" class="vector-toc-list"> <li id="toc-Forma_integral_da_lei_de_Gauss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Forma_integral_da_lei_de_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Forma integral da lei de Gauss</span> </div> </a> <ul id="toc-Forma_integral_da_lei_de_Gauss-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Forma_diferencial_da_lei_de_Gauss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Forma_diferencial_da_lei_de_Gauss"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Forma diferencial da lei de Gauss</span> </div> </a> <ul id="toc-Forma_diferencial_da_lei_de_Gauss-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relação_entre_a_lei_de_Gauss_e_a_lei_de_Coulomb" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relação_entre_a_lei_de_Gauss_e_a_lei_de_Coulomb"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Relação entre a lei de Gauss e a lei de Coulomb</span> </div> </a> <ul id="toc-Relação_entre_a_lei_de_Gauss_e_a_lei_de_Coulomb-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Aplicações" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aplicações"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Aplicações</span> </div> </a> <button aria-controls="toc-Aplicações-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Aplicações</span> </button> <ul id="toc-Aplicações-sublist" class="vector-toc-list"> <li id="toc-Campo_elétrico_no_interior_e_no_exterior_de_uma_esfera" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Campo_elétrico_no_interior_e_no_exterior_de_uma_esfera"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Campo elétrico no interior e no exterior de uma esfera</span> </div> </a> <ul id="toc-Campo_elétrico_no_interior_e_no_exterior_de_uma_esfera-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Campo_elétrico_no_interior_e_no_exterior_de_uma_casca_esférica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Campo_elétrico_no_interior_e_no_exterior_de_uma_casca_esférica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Campo elétrico no interior e no exterior de uma casca esférica</span> </div> </a> <ul id="toc-Campo_elétrico_no_interior_e_no_exterior_de_uma_casca_esférica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Campo_elétrico_de_um_plano_infinito" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Campo_elétrico_de_um_plano_infinito"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Campo elétrico de um plano infinito</span> </div> </a> <ul id="toc-Campo_elétrico_de_um_plano_infinito-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Campo_elétrico_de_uma_carga_uniformemente_distribuída_ao_longo_de_um_fio_extenso" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Campo_elétrico_de_uma_carga_uniformemente_distribuída_ao_longo_de_um_fio_extenso"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Campo elétrico de uma carga uniformemente distribuída ao longo de um fio extenso</span> </div> </a> <ul id="toc-Campo_elétrico_de_uma_carga_uniformemente_distribuída_ao_longo_de_um_fio_extenso-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lei_de_Gauss_para_dielétricos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lei_de_Gauss_para_dielétricos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lei de Gauss para dielétricos</span> </div> </a> <button aria-controls="toc-Lei_de_Gauss_para_dielétricos-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Lei de Gauss para dielétricos</span> </button> <ul id="toc-Lei_de_Gauss_para_dielétricos-sublist" class="vector-toc-list"> <li id="toc-Cargas_livres_e_cargas_ligadas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cargas_livres_e_cargas_ligadas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Cargas livres e cargas ligadas</span> </div> </a> <ul id="toc-Cargas_livres_e_cargas_ligadas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Demonstração_da_lei_de_Gauss_para_dielétricos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Demonstração_da_lei_de_Gauss_para_dielétricos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Demonstração da lei de Gauss para dielétricos</span> </div> </a> <ul id="toc-Demonstração_da_lei_de_Gauss_para_dielétricos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Alternar o índice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lei de Gauss</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 65 línguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-65" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">65 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%BA%D8%A7%D9%88%D8%B3" title="قانون غاوس — árabe" lang="ar" hreflang="ar" data-title="قانون غاوس" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-arz mw-list-item"><a href="https://arz.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%D8%AC%D8%A7%D9%88%D8%B3" title="قانون جاوس — Egyptian Arabic" lang="arz" hreflang="arz" data-title="قانون جاوس" data-language-autonym="مصرى" data-language-local-name="Egyptian Arabic" class="interlanguage-link-target"><span>مصرى</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Llei_de_Gauss" title="Llei de Gauss — asturiano" lang="ast" hreflang="ast" data-title="Llei de Gauss" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0_%D0%93%D0%B0%D1%9E%D1%81%D0%B0" title="Тэарэма Гаўса — bielorrusso" lang="be" hreflang="be" data-title="Тэарэма Гаўса" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%B0%D1%80%D1%8D%D0%BC%D0%B0_%D0%93%D0%B0%D1%9E%D1%81%D0%B0" title="Тэарэма Гаўса — Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэарэма Гаўса" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%93%D0%B0%D1%83%D1%81" title="Теорема на Гаус — búlgaro" lang="bg" hreflang="bg" data-title="Теорема на Гаус" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A6%BE%E0%A6%89%E0%A6%B8%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A7%82%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="গাউসের সূত্র — bengalês" lang="bn" hreflang="bn" data-title="গাউসের সূত্র" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Gaussov_zakon" title="Gaussov zakon — bósnio" lang="bs" hreflang="bs" data-title="Gaussov zakon" data-language-autonym="Bosanski" data-language-local-name="bósnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_Gauss" title="Llei de Gauss — catalão" lang="ca" hreflang="ca" data-title="Llei de Gauss" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Gauss%C5%AFv_z%C3%A1kon_elektrostatiky" title="Gaussův zákon elektrostatiky — checo" lang="cs" hreflang="cs" data-title="Gaussův zákon elektrostatiky" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gauss%27_lov" title="Gauss' lov — dinamarquês" lang="da" hreflang="da" data-title="Gauss' lov" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gau%C3%9Fsches_Gesetz" title="Gaußsches Gesetz — alemão" lang="de" hreflang="de" data-title="Gaußsches Gesetz" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_%CE%93%CE%BA%CE%AC%CE%BF%CF%85%CF%82" title="Νόμος του Γκάους — grego" lang="el" hreflang="el" data-title="Νόμος του Γκάους" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Gauss%27s_law" title="Gauss's law — inglês" lang="en" hreflang="en" data-title="Gauss's law" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ga%C5%ADsa_le%C4%9Do" title="Gaŭsa leĝo — esperanto" lang="eo" hreflang="eo" data-title="Gaŭsa leĝo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ley_de_Gauss" title="Ley de Gauss — espanhol" lang="es" hreflang="es" data-title="Ley de Gauss" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Gaussi_seadus_elektriv%C3%A4lja_jaoks" title="Gaussi seadus elektrivälja jaoks — estónio" lang="et" hreflang="et" data-title="Gaussi seadus elektrivälja jaoks" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gaussen_legea" title="Gaussen legea — basco" lang="eu" hreflang="eu" data-title="Gaussen legea" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%DA%AF%D8%A7%D9%88%D8%B3" title="قانون گاوس — persa" lang="fa" hreflang="fa" data-title="قانون گاوس" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gaussin_laki_s%C3%A4hk%C3%B6kentille" title="Gaussin laki sähkökentille — finlandês" lang="fi" hreflang="fi" data-title="Gaussin laki sähkökentille" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Gauss_(physique)" title="Théorème de Gauss (physique) — francês" lang="fr" hreflang="fr" data-title="Théorème de Gauss (physique)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%AD_Gau%C3%9F" title="Dlí Gauß — irlandês" lang="ga" hreflang="ga" data-title="Dlí Gauß" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Lei_de_Gauss" title="Lei de Gauss — galego" lang="gl" hreflang="gl" data-title="Lei de Gauss" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%92%D7%90%D7%95%D7%A1" title="חוק גאוס — hebraico" lang="he" hreflang="he" data-title="חוק גאוס" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%BE%E0%A4%89%E0%A4%B8_%E0%A4%95%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="गाउस का नियम — hindi" lang="hi" hreflang="hi" data-title="गाउस का नियम" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Gaussov_zakon" title="Gaussov zakon — croata" lang="hr" hreflang="hr" data-title="Gaussov zakon" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gauss-t%C3%B6rv%C3%A9ny" title="Gauss-törvény — húngaro" lang="hu" hreflang="hu" data-title="Gauss-törvény" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%A1%D5%B8%D6%82%D5%BD%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Գաուսի օրենք — arménio" lang="hy" hreflang="hy" data-title="Գաուսի օրենք" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_Gauss" title="Hukum Gauss — indonésio" lang="id" hreflang="id" data-title="Hukum Gauss" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_del_flusso" title="Teorema del flusso — italiano" lang="it" hreflang="it" data-title="Teorema del flusso" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AC%E3%82%A6%E3%82%B9%E3%81%AE%E6%B3%95%E5%89%87" title="ガウスの法則 — japonês" lang="ja" hreflang="ja" data-title="ガウスの法則" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%A3%E1%83%A1%E1%83%98%E1%83%A1_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="გაუსის კანონი — georgiano" lang="ka" hreflang="ka" data-title="გაუსის კანონი" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Gauss_teoremas%C4%B1" title="Gauss teoreması — kara-kalpak" lang="kaa" hreflang="kaa" data-title="Gauss teoreması" data-language-autonym="Qaraqalpaqsha" data-language-local-name="kara-kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Гаусс теоремасы — cazaque" lang="kk" hreflang="kk" data-title="Гаусс теоремасы" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4_%EB%B2%95%EC%B9%99" title="가우스 법칙 — coreano" lang="ko" hreflang="ko" data-title="가우스 법칙" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B0y%D1%81%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Гаyсс теоремасы — quirguiz" lang="ky" hreflang="ky" data-title="Гаyсс теоремасы" data-language-autonym="Кыргызча" data-language-local-name="quirguiz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Gausa_teor%C4%93ma" title="Gausa teorēma — letão" lang="lv" hreflang="lv" data-title="Gausa teorēma" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" title="Гаусов закон — macedónio" lang="mk" hreflang="mk" data-title="Гаусов закон" data-language-autonym="Македонски" data-language-local-name="macedónio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81%D1%8B%D0%BD_%D1%85%D1%83%D1%83%D0%BB%D1%8C" title="Гауссын хууль — mongol" lang="mn" hreflang="mn" data-title="Гауссын хууль" data-language-autonym="Монгол" data-language-local-name="mongol" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%97%E0%A5%89%E0%A4%B8%E0%A4%9A%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="गॉसचा नियम — marata" lang="mr" hreflang="mr" data-title="गॉसचा नियम" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%97%E0%A4%BE%E0%A4%89%E0%A4%B8%E0%A4%95%E0%A5%8B_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="गाउसको नियम — nepalês" lang="ne" hreflang="ne" data-title="गाउसको नियम" data-language-autonym="नेपाली" data-language-local-name="nepalês" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wet_van_Gauss" title="Wet van Gauss — neerlandês" lang="nl" hreflang="nl" data-title="Wet van Gauss" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Gauss%27_lov" title="Gauss' lov — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Gauss' lov" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gauss%E2%80%99_lov" title="Gauss’ lov — norueguês bokmål" lang="nb" hreflang="nb" data-title="Gauss’ lov" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawo_Gaussa_(elektryczno%C5%9B%C4%87)" title="Prawo Gaussa (elektryczność) — polaco" lang="pl" hreflang="pl" data-title="Prawo Gaussa (elektryczność)" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%93%D0%B0%D1%83%D1%81%D1%81%D0%B0" title="Теорема Гаусса — russo" lang="ru" hreflang="ru" data-title="Теорема Гаусса" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Gaussov_zakon_elektri%C4%8Dnoga_polja" title="Gaussov zakon električnoga polja — servo-croata" lang="sh" hreflang="sh" data-title="Gaussov zakon električnoga polja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gauss%27s_law" title="Gauss's law — Simple English" lang="en-simple" hreflang="en-simple" data-title="Gauss's law" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Gaussov_z%C3%A1kon_elektrostatiky" title="Gaussov zákon elektrostatiky — eslovaco" lang="sk" hreflang="sk" data-title="Gaussov zákon elektrostatiky" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zakon_o_elektri%C4%8Dnem_pretoku" title="Zakon o električnem pretoku — esloveno" lang="sl" hreflang="sl" data-title="Zakon o električnem pretoku" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ligji_i_Gausit" title="Ligji i Gausit — albanês" lang="sq" hreflang="sq" data-title="Ligji i Gausit" data-language-autonym="Shqip" data-language-local-name="albanês" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD" title="Гаусов закон — sérvio" lang="sr" hreflang="sr" data-title="Гаусов закон" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gauss_lag" title="Gauss lag — sueco" lang="sv" hreflang="sv" data-title="Gauss lag" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%B8%E0%AF%8D_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="காஸ் விதி — tâmil" lang="ta" hreflang="ta" data-title="காஸ் விதி" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B0%BE%E0%B0%B8%E0%B1%8D_%E0%B0%A8%E0%B0%BF%E0%B0%AF%E0%B0%AE%E0%B0%82" title="గాస్ నియమం — telugu" lang="te" hreflang="te" data-title="గాస్ నియమం" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Batas_ni_Gauss" title="Batas ni Gauss — tagalo" lang="tl" hreflang="tl" data-title="Batas ni Gauss" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gauss_yasas%C4%B1" title="Gauss yasası — turco" lang="tr" hreflang="tr" data-title="Gauss yasası" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0%D1%81%D1%8B" title="Гаусс теоремасы — tatar" lang="tt" hreflang="tt" data-title="Гаусс теоремасы" data-language-autonym="Татарча / tatarça" data-language-local-name="tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%93%D0%B0%D1%83%D1%81%D1%81%D0%B0" title="Теорема Гаусса — ucraniano" lang="uk" hreflang="uk" data-title="Теорема Гаусса" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86_%DA%AF%D8%A7%D8%B3" title="قانون گاس — urdu" lang="ur" hreflang="ur" data-title="قانون گاس" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gauss_teoremasi" title="Gauss teoremasi — usbeque" lang="uz" hreflang="uz" data-title="Gauss teoremasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbeque" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_lu%E1%BA%ADt_Gauss" title="Định luật Gauss — vietnamita" lang="vi" hreflang="vi" data-title="Định luật Gauss" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%AB%98%E6%96%AF%E5%AE%9A%E5%BE%8B" title="高斯定律 — wu" lang="wuu" hreflang="wuu" data-title="高斯定律" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%AB%98%E6%96%AF%E5%AE%9A%E5%BE%8B" title="高斯定律 — chinês" lang="zh" hreflang="zh" data-title="高斯定律" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%AB%98%E6%96%AF%E5%AE%9A%E5%BE%8B" title="高斯定律 — cantonês" lang="yue" hreflang="yue" data-title="高斯定律" data-language-autonym="粵語" data-language-local-name="cantonês" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q173356#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet 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.tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Reciclagem plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Wikip%C3%A9dia:Reciclagem" title="Wikipédia:Reciclagem"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Ambox_rewrite.svg/40px-Ambox_rewrite.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Ambox_rewrite.svg/60px-Ambox_rewrite.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Ambox_rewrite.svg/80px-Ambox_rewrite.svg.png 2x" data-file-width="620" data-file-height="620" /></a><figcaption></figcaption></figure></div></td><td class="mbox-text"><div class="mbox-text-span"><b>Este artigo carece de <a href="/wiki/Wikip%C3%A9dia:Reciclagem" title="Wikipédia:Reciclagem">reciclagem</a> de acordo com o <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo" title="Wikipédia:Livro de estilo">livro de estilo</a></b>.<span class="hide-when-compact"> Sinta-se livre para editá-lo(a) para que este(a) possa atingir um <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Como_escrever_um_bom_artigo" title="Wikipédia:Livro de estilo/Como escrever um bom artigo">nível de qualidade superior</a>.</span> <small class="date-container"><i>(<span class="date">Outubro de 2022</span>)</i></small></div></td></tr></tbody></table> <p>A <b>lei de Gauss</b> é a lei que estabelece a relação entre o <a href="/wiki/Fluxo_(f%C3%ADsica)" title="Fluxo (física)">fluxo</a> do <a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">campo elétrico</a> através de uma superfície fechada com a <a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">carga elétrica</a> que existe dentro do <a href="/wiki/Volume" title="Volume">volume</a> limitado por <a href="/wiki/Superf%C3%ADcie_de_Gauss" class="mw-redirect" title="Superfície de Gauss">esta superfície</a>. A lei de Gauss é uma das quatro <a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">equações de Maxwell</a>, juntamente com a lei de Gauss do <a href="/wiki/Magnetismo" title="Magnetismo">magnetismo</a>, a <a href="/wiki/Lei_da_indu%C3%A7%C3%A3o_de_Faraday" class="mw-redirect" title="Lei da indução de Faraday">lei da indução de Faraday</a> e a <a href="/wiki/Lei_de_Amp%C3%A8re-Maxwell" class="mw-redirect" title="Lei de Ampère-Maxwell">lei de Ampère-Maxwell</a>. Foi elaborada por <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> em 1835, porém só foi publicada após 1867.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> Gauss foi um <a href="/wiki/Matem%C3%A1tico" title="Matemático">matemático</a> <a href="/wiki/Alem%C3%A3es" title="Alemães">alemão</a> que fez contribuições importantes para a <a href="/wiki/Teoria_dos_n%C3%BAmeros" title="Teoria dos números">teoria dos números</a>, a <a href="/wiki/Geometria" title="Geometria">geometria</a> e a <a href="/wiki/Probabilidade" title="Probabilidade">probabilidade</a>, tendo também contribuições em <a href="/wiki/Astronomia" title="Astronomia">astronomia</a> e na medição do tamanho e do formato da <a href="/wiki/Terra" title="Terra">Terra</a>.<sup id="cite_ref-halliday_2-0" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Fluxo_do_campo_elétrico"><span id="Fluxo_do_campo_el.C3.A9trico"></span>Fluxo do campo elétrico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=1" title="Editar secção: Fluxo do campo elétrico" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=1" title="Editar código-fonte da secção: Fluxo do campo elétrico"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:LeyGauss1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/LeyGauss1.jpg/200px-LeyGauss1.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/78/LeyGauss1.jpg 1.5x" data-file-width="267" data-file-height="267" /></a><figcaption>Linhas de <a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">campo elétrico</a> "furando" uma superfície, mostrando que o existe fluxo de campo elétrico através da superfície. Como as linhas de campo estão saindo da superfície, o fluxo do campo elétrico é positivo.</figcaption></figure> <p>O fluxo de campo elétrico, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49c8491e55157c9b3578226b071c3a7acc0693b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \Phi _{E}}"></span>, é uma <a href="/wiki/Grandeza_escalar" title="Grandeza escalar">grandeza escalar</a> e pode ser considerado como uma medida do número de <a href="/wiki/Linha_de_campo" title="Linha de campo">linhas de campo</a> que atravessam a superfície.<sup id="cite_ref-halliday_2-1" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-griffiths_3-0" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup> Convenciona-se que se há mais linhas de campo saindo da superfície do que entrando, o fluxo do campo elétrico através da superfície é positivo e se há mais linhas de campo entrando na superfície do que saindo da mesma, o fluxo é negativo. Além disso, é importante observar o fato de que se o número de linhas de campo que entra na superfície é igual ao número de linhas de campo que sai da superfície, então o fluxo de campo elétrico através da superfície é nulo.<sup id="cite_ref-halliday_2-2" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-Feynman_4-0" class="reference"><a href="#cite_note-Feynman-4"><span>[</span>4<span>]</span></a></sup> </p><p>Para obter o fluxo do campo elétrico <b>E</b> através de uma superfície fechada em que <b>E</b> é não-uniforme, é preciso dividi-la em elementos de <a href="/wiki/%C3%81rea" title="Área">área</a> <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> dA. Define-se, então, um <a href="/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática)">vetor</a> d<b>A</b> cujo módulo é dA, a <a href="/wiki/Dire%C3%A7%C3%A3o" title="Direção">direção</a> é <a href="/wiki/Perpendicularidade" title="Perpendicularidade">perpendicular</a> ao elemento de área e o <a href="/wiki/Sentido_(matem%C3%A1tica)" title="Sentido (matemática)">sentido</a> é adotado como o sentido da normal ao elemento infinitesimal saindo da superfície. Assim, esses elementos infinitesimais são tão pequenos que <b>E</b> pode ser considerado constante em todos os pontos de um mesmo elemento de área.<sup id="cite_ref-halliday_2-3" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup> Portanto, podemos definir o fluxo de <b>E</b> através de uma superfície S da seguinte forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4572f4e4689ef7049ebcc650e48c48bcfcd9e8e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.984ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }"></span></dd></dl> <p>ou, no caso de uma superfície fechada: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #ffffff; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7feffbca982b9d81bc537670f33b35b440768f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:16.372ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!}"></span> </p> </td></tr></tbody></table></dd></dl> <p>Da definição de <a href="/wiki/Produto_escalar" title="Produto escalar">produto escalar</a>, tem-se que: <b>E</b> <b>.</b> d<b>A</b> = |<b>E</b>||d<b>A</b>| cosθ = |<b>E</b>|cosθ |d<b>A</b>|. Como θ é o ângulo entre os vetores <b>E</b> e d<b>A</b>, |<b>E</b>|cosθ é a projeção do vetor <b>E</b> sobre o vetor d<b>A</b>, logo a função desse produto escalar dentro da integral é selecionar algo proporcional à componente do campo elétrico que está "furando" à superfície infinitesimal d<b>A</b>, o que é coerente com a definição de fluxo dada anteriormente. </p><p>Por fim, se uma carga pontual estiver fora da superfície, as linhas de campo que partem da carga pontual irão entrar e sair da superfície, visto que as linhas de campo de uma carga pontual são radiais. Por isso, pode-se concluir que se uma carga está fora de uma superfície, então o fluxo do campo elétrico dessa carga através da superfície é nulo, ou seja: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #ffffff; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b3ad053aae2bbdd5cbf001c03b333227d22443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.245ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0}"></span>, se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> estiver externa à superfície. </p> </td></tr></tbody></table></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:LeyGauss2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/LeyGauss2.jpg/200px-LeyGauss2.jpg" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/91/LeyGauss2.jpg 1.5x" data-file-width="265" data-file-height="265" /></a><figcaption>As linhas de campo elétrico entram e saem da superfície, portanto o fluxo de campo elétrico sobre a superfície é nulo.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Lei_de_Gauss">Lei de Gauss</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=2" title="Editar secção: Lei de Gauss" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=2" title="Editar código-fonte da secção: Lei de Gauss"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A lei de Gauss estabelece uma relação entre o fluxo de campo elétrico através de uma superfície fechada e as cargas que estão no interior dessa superfície. Algumas considerações importantes sobre a de lei de Gauss são: </p> <ul><li>A lei de Gauss não contém nenhuma informação que não esteja contida na <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">lei de Coulomb</a> e no <a href="/wiki/Princ%C3%ADpio_da_superposi%C3%A7%C3%A3o" title="Princípio da superposição">princípio da superposição</a>. Inclusive, é possível obter a lei de Coulomb a partir da lei de Gauss e vice-versa.<sup id="cite_ref-griffiths_3-1" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup></li> <li>É fundamental para a lei de Gauss, o fato de que a <a href="/wiki/For%C3%A7a_el%C3%A9trica" title="Força elétrica">força elétrica</a> é proporcional ao inverso do quadrado da distância. É esse fato que faz com que o fluxo de <b>E</b> não dependa da "superfície gaussiana" escolhida e dependa apenas das cargas que estão localizadas no interior da superfície. Dessa forma, é possível pensar numa lei de Gauss que estabeleça uma relação de fluxo para qualquer campo cuja lei de força associada a esse campo seja proporcional ao inverso do quadrado da distância, como a força gravitacional, por exemplo, logo existe uma lei de Gauss da gravitação.<sup id="cite_ref-griffiths_3-2" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup></li> <li>Apesar da lei de Coulomb nos fornecer o necessário para calcular o campo elétrico de uma distribuição de cargas, muitas vezes, as integrais que envolvem o cálculo do campo elétrico podem ser complicadas de serem resolvidas, mesmo para casos razoavelmente simples. É nesse ponto que reside um dos aspectos de maior eficiência da lei de Gauss: o cálculo do campo elétrico em distribuições de carga que possuam determinados tipos de simetria torna-se extremamente simples.<sup id="cite_ref-griffiths_3-3" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup></li> <li>A lei de Gauss se refere sempre ao fluxo no interior de uma <a href="/wiki/Superf%C3%ADcie_gaussiana" title="Superfície gaussiana">superfície gaussiana</a> escolhida. Portanto, para utilizar a lei de Gauss, é necessário definir o que é uma "superfície gaussiana". Esta é, por sua vez, uma superfície arbitrariamente escolhida. Normalmente, essa superfície é escolhida de modo que a simetria da distribuição de carga permita, ao menos em parte da superfície, um campo elétrico de intensidade constante.<sup id="cite_ref-halliday_2-4" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Forma_integral_da_lei_de_Gauss">Forma integral da lei de Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=3" title="Editar secção: Forma integral da lei de Gauss" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=3" title="Editar código-fonte da secção: Forma integral da lei de Gauss"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Gauss_Sphere_Charge_Inside_2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Gauss_Sphere_Charge_Inside_2.svg/200px-Gauss_Sphere_Charge_Inside_2.svg.png" decoding="async" width="200" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Gauss_Sphere_Charge_Inside_2.svg/300px-Gauss_Sphere_Charge_Inside_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Gauss_Sphere_Charge_Inside_2.svg/400px-Gauss_Sphere_Charge_Inside_2.svg.png 2x" data-file-width="512" data-file-height="432" /></a><figcaption>Superfície gaussiana esférica centrada em q.</figcaption></figure> <p>Para entender como a lei de Gauss relaciona o fluxo do campo elétrico no interior de uma superfície gaussiana com a carga no interior dessa mesma superfície, escolhe-se uma superfície qualquer com uma carga q em seu interior. Então, escolhe-se outra superfície gaussiana S' que está envolvendo q no interior de S. A forma dessa superfície S' pode ser qualquer, contudo, a fim de facilitar os cálculos e a visualização, vamos fazer dessa superfície S', uma esfera de raio r centrada na carga q. O raio r é tal que S' esteja inteiramente dentro de S.<sup id="cite_ref-Feynman_4-1" class="reference"><a href="#cite_note-Feynman-4"><span>[</span>4<span>]</span></a></sup> O fluxo do campo elétrico através dessa esfera é dado por: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S'}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S'}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21aa5a93320be56a95d9db74892cfd033671abb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.532ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\oint _{S'}\mathbf {E} \cdot \mathrm {d} \mathbf {A} }"></span></dd></dl> <p>Como tanto <b>E</b> quanto d<b>A</b> são radiais, o produto escalar torna-se o produto dos módulos, então: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S'}E\ dA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S'}E\ dA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9a3b355358f006ec75cad952ddd5893e5452b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.099ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\oint _{S'}E\ dA}"></span></dd></dl> <p>Como |<b>E</b>| é constante na superfície da esfera, podemos tirá-lo da integral e temos: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=E\oint _{S'}dA=\left({\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}\right)(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=E\oint _{S'}dA=\left({\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}\right)(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde5e8755477c1b21f39e1236700f72fb342a565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.699ex; height:6.176ex;" alt="{\displaystyle \Phi _{E}=E\oint _{S'}dA=\left({\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}\right)(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}"></span></dd></dl> <p>Portanto, é possível observar que o fluxo através da superfície S' é um número que independe do raio da esfera. Dessa forma, o fluxo que sai da superfície S também será <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de750deb76175e9bd87ee1c0419caf74ffd287d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:2.974ex; height:5.176ex;" alt="{\displaystyle {\frac {q}{\varepsilon _{0}}}}"></span>. Esse é um valor independente da forma da superfície S, desde que esta tenha uma carga q em seu interior. Se uma carga q está no exterior da superfície S, as suas linhas de campo entram e saem da superfície S, por isso, o fluxo de campo elétrico dessa carga sobre a superfície é nulo. Logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0{\text{, se a carga está no exterior da superfície gaussiana.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>, se a carga está no exterior da superfície gaussiana.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0{\text{, se a carga está no exterior da superfície gaussiana.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d22960c8ae25a5515b56af9a659e162f7a9fda4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:66.052ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =0{\text{, se a carga está no exterior da superfície gaussiana.}}}"></span></dd></dl> <p>Por fim, se tivermos mais de uma carga no interior da superfície gaussiana, vale o princípio da superposição de modo que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\left(\mathbf {E_{1}} +\mathbf {E_{2}} \right)\cdot \mathrm {d} \mathbf {A} =\oint _{S}\mathbf {E_{1}} \cdot \mathrm {d} \mathbf {A} +\oint _{S}\mathbf {E_{2}} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{1}+q_{2}}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\left(\mathbf {E_{1}} +\mathbf {E_{2}} \right)\cdot \mathrm {d} \mathbf {A} =\oint _{S}\mathbf {E_{1}} \cdot \mathrm {d} \mathbf {A} +\oint _{S}\mathbf {E_{2}} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{1}+q_{2}}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a1b5b46c1869ae6fb89e4d5a04a1652f9485d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:57.173ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\left(\mathbf {E_{1}} +\mathbf {E_{2}} \right)\cdot \mathrm {d} \mathbf {A} =\oint _{S}\mathbf {E_{1}} \cdot \mathrm {d} \mathbf {A} +\oint _{S}\mathbf {E_{2}} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{1}+q_{2}}{\varepsilon _{0}}}}"></span></dd></dl> <p>Portanto, a Lei de Gauss na forma integral pode ser enunciada da seguinte forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97099ab179376e397e437eb7b13c7919f3181004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.071ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Forma_diferencial_da_lei_de_Gauss">Forma diferencial da lei de Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=4" title="Editar secção: Forma diferencial da lei de Gauss" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=4" title="Editar código-fonte da secção: Forma diferencial da lei de Gauss"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Demonstração </th></tr> <tr> <td>Pelo <a href="/wiki/Teorema_da_diverg%C3%AAncia" title="Teorema da divergência">teorema da divergência</a>:<sup id="cite_ref-griffiths_3-4" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup><sup id="cite_ref-Jackson_5-0" class="reference"><a href="#cite_note-Jackson-5"><span>[</span>5<span>]</span></a></sup> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c5f2b3f8e944de2d9cecb7654784a9790a13ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.206ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }"></span></dd></dl> <p>Pode-se reescrever a carga interna à superfície, q<sub>int</sub>, em termos da densidade ρ: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{int}=\int _{V}\rho \ \mathrm {d} \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{int}=\int _{V}\rho \ \mathrm {d} \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64aa8b6437bec31178bd895999bfdffa580cd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.968ex; height:5.676ex;" alt="{\displaystyle q_{int}=\int _{V}\rho \ \mathrm {d} \tau }"></span></dd></dl> <p>Desse modo, usando o <a href="/wiki/Teorema_de_Stokes" title="Teorema de Stokes">teorema de Stokes</a>,a lei de Gauss, assume a forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}\left(\nabla \cdot \mathbf {E} \right){d}\tau =\int _{V}\left({\frac {\rho }{\varepsilon _{0}}}\right){d}\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}\left(\nabla \cdot \mathbf {E} \right){d}\tau =\int _{V}\left({\frac {\rho }{\varepsilon _{0}}}\right){d}\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cceebec1d2798cc039e98f4904f7c31ef6b56d0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.636ex; height:6.176ex;" alt="{\displaystyle \int _{V}\left(\nabla \cdot \mathbf {E} \right){d}\tau =\int _{V}\left({\frac {\rho }{\varepsilon _{0}}}\right){d}\tau }"></span></dd></dl> <p>Por fim, como se quer que essa igualdade valha para qualquer volume, os integrandos devem ser iguais, logo: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #f5f5f5; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d136aa64365d38501bd4b4091663ef902631735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-right: -0.387ex; width:11.831ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}\,\!}"></span> </p> </td></tr></tbody></table></dd></dl> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relação_entre_a_lei_de_Gauss_e_a_lei_de_Coulomb"><span id="Rela.C3.A7.C3.A3o_entre_a_lei_de_Gauss_e_a_lei_de_Coulomb"></span>Relação entre a lei de Gauss e a lei de Coulomb</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=5" title="Editar secção: Relação entre a lei de Gauss e a lei de Coulomb" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=5" title="Editar código-fonte da secção: Relação entre a lei de Gauss e a lei de Coulomb"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Demonstração </th></tr> <tr> <td>Existe uma relação entre a lei de Gauss e a lei de Coulomb, de forma que nenhuma das duas leis é mais fundamental do que a outra. Partindo-se de uma delas pode-se obter a outra como consequência. <p>Partindo-se da <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">lei de Coulomb</a>, tem-se que<sup id="cite_ref-griffiths_3-5" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {q}{r''^{2}}}{\hat {\mathbf {r''} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {q}{r''^{2}}}{\hat {\mathbf {r''} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191536acdf4acbf199e60f2dbf87f151b0a1fd16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.804ex; height:5.676ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {q}{r''^{2}}}{\hat {\mathbf {r''} }}}"></span></dd></dl> <p>onde Ω significa que a integral ocorre em todo o espaço e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r''} =\mathbf {r'} -\mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r''} =\mathbf {r'} -\mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6289c3b4d3cdfd3c135a2db897cd4a9f5cc46cf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.067ex; height:2.676ex;" alt="{\displaystyle \mathbf {r''} =\mathbf {r'} -\mathbf {r} }"></span>, isto é, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2270c026318452d5ef5a7cab4ae067fbaede0b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.509ex;" alt="{\displaystyle \mathbf {r'} }"></span> é um vetor que vai da origem até um elemento infinitesimal de volume da distribuição de carga geradora do campo, <b>r</b> é um vetor que vai da origem ao ponto no qual se deseja calcular calcular o campo e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r''} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r''} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74cc103ffea2c71bd17f2e21db288838756430a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle \mathbf {r''} }"></span> é o vetor que representa a distância entre o elemento infinitesimal da fonte de campo elétrico e o ponto no qual se deseja calcular o campo. Pode-se reescrever a equação acima em termos da densidade volumétrica de carga da seguinte forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {\hat {\mathbf {r''} }}{r''^{2}}}\rho (\mathbf {r'} )\mathrm {d} \tau '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>τ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {\hat {\mathbf {r''} }}{r''^{2}}}\rho (\mathbf {r'} )\mathrm {d} \tau '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef7ecfb57ff406f07602417cc5c1bc37847cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.598ex; height:6.343ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }{\frac {\hat {\mathbf {r''} }}{r''^{2}}}\rho (\mathbf {r'} )\mathrm {d} \tau '}"></span></dd></dl> <p>onde dτ' é um elemento infinitesimal de volume da distribuição de carga geradora do campo. Sabe-se que, pelo teorema da divergência, o fluxo do campo elétrico pode ser escrito do seguinte modo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c5f2b3f8e944de2d9cecb7654784a9790a13ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.206ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{V}(\nabla \cdot \mathbf {E} ){d}\tau }"></span></dd></dl> <p>Calculando-se o divergente de <b>E</b>, obtém-se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\rho (\mathbf {r'} )\mathrm {d} \tau '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>τ</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\rho (\mathbf {r'} )\mathrm {d} \tau '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950b8efdb8e50d3ff8d3d442e9472db19eea4135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.985ex; height:7.509ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\rho (\mathbf {r'} )\mathrm {d} \tau '}"></span></dd></dl> <p>Entretanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d86800a271e46c07ec32aa5b4db22876d7bb81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.89ex; height:7.509ex;" alt="{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )}"></span></dd></dl> <p>Esse é um resultado importante e muito usado em eletrostática. Um cálculo de divergente acima de forma não muito atenta pode levar, equivocadamente, a um divergente nulo. Em contrapartida, pode-se obter esse resultado resolvendo a integral de superfície de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ff19fbff265a1aa1e5e38f7bd85966c784ab5c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.516ex; height:7.509ex;" alt="{\displaystyle \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} }"></span> numa superfície esférica de raio R, que leva ao resultado: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} =4\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} =4\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bba647a20c00ee2143b1e2cec3ca46473857323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.081ex; height:7.509ex;" alt="{\displaystyle \oint _{S}\left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)\cdot \mathrm {d} \mathbf {A} =4\pi }"></span>. Portanto, isso sugere que o divergente deve ser calculado de forma mais cuidadosa, introduzindo o conceito da distribuição delta de Dirac em eletrostática. Sabendo, então, que :<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )=4\pi \delta ^{3}(\mathbf {r-r'} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">−</mo> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )=4\pi \delta ^{3}(\mathbf {r-r'} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1248e8f0c4263091c2ac6fecc061b43db752ca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.398ex; height:7.509ex;" alt="{\displaystyle \nabla \cdot \left({\frac {\hat {\mathbf {r''} }}{r''^{2}}}\right)=4\pi \delta ^{3}(\mathbf {r''} )=4\pi \delta ^{3}(\mathbf {r-r'} )}"></span> e usando o conceito de distribuição delta de Dirac: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }4\pi \delta ^{3}(\mathbf {\mathbf {r} -\mathbf {r'} } )\rho (\mathbf {r'} )\mathrm {d} \tau '={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> </mrow> </msub> <mn>4</mn> <mi>π</mi> <msup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo mathvariant="bold">−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>τ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }4\pi \delta ^{3}(\mathbf {\mathbf {r} -\mathbf {r'} } )\rho (\mathbf {r'} )\mathrm {d} \tau '={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92814770bcf1d18c3c69cd5ac459ac51a7630b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.053ex; height:5.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }4\pi \delta ^{3}(\mathbf {\mathbf {r} -\mathbf {r'} } )\rho (\mathbf {r'} )\mathrm {d} \tau '={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} )}"></span></dd></dl> <p>Chega-se, portanto na lei de Gauss na forma diferencial. Pode-se obter a lei de Gauss na forma integral do seguinte modo: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #f5f5f5; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}(\nabla \cdot \mathbf {E} ){d}\tau =\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {1}{\varepsilon _{0}}}\int _{V}\rho \ {d}\tau ={\frac {\rho }{\varepsilon _{0}}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}(\nabla \cdot \mathbf {E} ){d}\tau =\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {1}{\varepsilon _{0}}}\int _{V}\rho \ {d}\tau ={\frac {\rho }{\varepsilon _{0}}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aca45e913e516108cb4bfa47aca4f0abe901cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:45.501ex; height:5.676ex;" alt="{\displaystyle \int _{V}(\nabla \cdot \mathbf {E} ){d}\tau =\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {1}{\varepsilon _{0}}}\int _{V}\rho \ {d}\tau ={\frac {\rho }{\varepsilon _{0}}}\,\!}"></span> </p> </td></tr></tbody></table></dd></dl> <p>Tomando uma superfície esférica S, de raio r, centrada na carga q, e partindo-se da lei de Gauss, tem-se que:<sup id="cite_ref-halliday_2-5" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885b99431e5de198f8fdb1b2b20a78e6cebe8f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.144ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {q}{\varepsilon _{0}}}}"></span></dd></dl> <p>Como escolheu-se uma carga positiva, pode-se observar que, por simetria, o campo na superfícies esférica é radial e aponta para fora da esfera, visto que as linhas de campo divergem de uma carga positiva. Portanto, <b>E</b> e d<b>A</b> apontam na mesma direção e sentido de modo que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\oint _{S}E\ dA={\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\oint _{S}E\ dA={\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d90696b81f20cb7401673836d788f8c55db3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.178ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\oint _{S}E\ dA={\frac {q}{\varepsilon _{0}}}}"></span></dd></dl> <p>Como E possui a mesma intensidade para todos os pontos da superfície esférica, pode-se retirá-lo de dentro do símbolo de integração: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oint _{S}dA={\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oint _{S}dA={\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/104717b4eb8b3238bfbdf8ba07eef908e7677d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.166ex; height:5.676ex;" alt="{\displaystyle E\oint _{S}dA={\frac {q}{\varepsilon _{0}}}}"></span></dd></dl> <p>Mas, a área da superfície esférica é 4πr², logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9aaff58c8b10477e5b723cfbd3382ed423d086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.255ex; height:5.176ex;" alt="{\displaystyle E(4\pi r^{2})={\frac {q}{\varepsilon _{0}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e039c0a581b1162af55a04abad6e6730109485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.281ex; height:5.676ex;" alt="{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q}{r^{2}}}}"></span></dd></dl> <p>Contudo, já foi considerado antes que o campo deve estar na direção radial, então: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r''} }}{r''^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>″</mo> </msup> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">″</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r''} }}{r''^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/332d8cee00385737bc5ad13ec7d699923deb7c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.079ex; height:6.343ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r''} }}{r''^{2}}}}"></span></dd></dl> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Aplicações"><span id="Aplica.C3.A7.C3.B5es"></span>Aplicações</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=6" title="Editar secção: Aplicações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=6" title="Editar código-fonte da secção: Aplicações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>É importante ressaltar que a lei de Gauss se torna eficiente apenas em casos em que há simetria. Mais precisamente, nos casos nos quais existe simetria esférica, cilíndrica ou plana.<sup id="cite_ref-griffiths_3-6" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup> Dessa forma, construir superfícies gaussianas que aproveitem a simetria é de vital importância para a aplicação da lei de Gauss,<sup id="cite_ref-halliday_2-6" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup> visto que a eficiência da lei de Gauss consiste em utilizar a simetria das distribuições de carga para calcular campo elétrico dessas com mais facilidade.<sup id="cite_ref-halliday_2-7" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-griffiths_3-7" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Campo_elétrico_no_interior_e_no_exterior_de_uma_esfera"><span id="Campo_el.C3.A9trico_no_interior_e_no_exterior_de_uma_esfera"></span>Campo elétrico no interior e no exterior de uma esfera</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=7" title="Editar secção: Campo elétrico no interior e no exterior de uma esfera" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=7" title="Editar código-fonte da secção: Campo elétrico no interior e no exterior de uma esfera"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:GaussSphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/200px-GaussSphere.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/300px-GaussSphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/400px-GaussSphere.svg.png 2x" data-file-width="290" data-file-height="290" /></a><figcaption>Duas superfícies gaussianas esféricas em torno de uma esfera uniformemente carregada de raio R. A superfície gaussiana externa à esfera de raio R possui raio r' e a superfície gaussiana interna à esfera possui raio r.</figcaption></figure> <p>Para uma esfera de raio R com carga Q uniformemente distribuída pela esfera, tem-se: </p> <dl><dt>No exterior da esfera</dt> <dd></dd></dl> <p>Para se obter o campo no exterior da esfera, escolhe-se, como superfície gaussiana, a superfície esférica de raio r', situada no exterior da esfera de raio R. Pode-se imaginar que, muito longe da esfera, o campo elétrico que se sente é como o campo de uma carga puntiforme. Além disso, devido à simetria esférica, o campo elétrico deve apontar na direção radial. Dessa forma, aplicando a lei de Gauss: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {Q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7d25be6759094413214d469a654b4fb40da1fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.144ex; height:5.843ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}={\frac {Q}{\varepsilon _{0}}}}"></span></dd></dl> <p>O campo deve apontar na direção radial e, portanto, <b>E</b> e d<b>A</b> possuem a mesma direção e sentido e, por isso, segue que: <b>E</b> <b>.</b> d<b>A</b> = E dA. Logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51dcab8043311ba763af20cd2e8b317894e2c69f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.36ex; height:5.843ex;" alt="{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}}"></span></dd></dl> <p>O módulo do campo elétrico na superfície gaussiana é constante, visto que, nesse caso, o campo deve depender da distância em relação à esfera e, portanto, E pode sair da Integral. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc2615af6ee597c22a98a6c18808c20920c1508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.166ex; height:5.843ex;" alt="{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(4\pi r'^{2}\right)={\frac {Q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left(4\pi r'^{2}\right)={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49600770bb9b05bad50d643305c0b2430056a48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.415ex; height:5.676ex;" alt="{\displaystyle E\left(4\pi r'^{2}\right)={\frac {Q}{\varepsilon _{0}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8871690fe1a3d6367813fea63d5dddf4884dfade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.734ex; height:5.843ex;" alt="{\displaystyle E={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}}"></span></dd></dl> <p>Logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ef1a1e54448747a498030b93581573eccfa663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.878ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}"></span></dd></dl> <dl><dt>No interior da esfera</dt> <dd></dd></dl> <p>Para como o campo elétrico varia no interior da esfera, deve-se tomar como superfície gaussiana a superfície esférica de raio r no interior da esfera de raio R. Nesse caso, como a carga está uniformemente distribuída pela esfera, a densidade volumétrica de carga, ρ, é a mesma em todos os pontos da esfera,então pode-se observar que: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{int}=\rho V_{g}=\left({\frac {Q}{{\frac {4}{3}}\pi R^{3}}}\right)\left({\frac {4}{3}}\pi r^{3}\right)=Q\ {\frac {r^{3}}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>Q</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{int}=\rho V_{g}=\left({\frac {Q}{{\frac {4}{3}}\pi R^{3}}}\right)\left({\frac {4}{3}}\pi r^{3}\right)=Q\ {\frac {r^{3}}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310a8ecc185957988db2345ac2887d4a2f99af27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:41.931ex; height:7.843ex;" alt="{\displaystyle q_{int}=\rho V_{g}=\left({\frac {Q}{{\frac {4}{3}}\pi R^{3}}}\right)\left({\frac {4}{3}}\pi r^{3}\right)=Q\ {\frac {r^{3}}{R^{3}}}}"></span></dd></dl> <p>onde V<sub>g</sub> é o volume da superfície gaussiana escolhida. </p><p>Dessa forma: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e30cd16bc1fe9ac43c67145168a4ed645918a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.027ex; height:6.176ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"></span></dd></dl> <p>Os mesmos argumentos dados anteriormente para que o produto escalar <b>E</b> <b>.</b> d<b>A</b> seja E dA e para que E saia da integral continuam sendo válidos, logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ccd2170946159be32c8b133638e9101b5c1812e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.595ex; height:6.176ex;" alt="{\displaystyle \oint _{S}E\ dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ec23863c5aa9f7de5df06903e6da9dd58fb634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.401ex; height:6.176ex;" alt="{\displaystyle E\oint _{S}dA={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(4\pi r^{2}\right)={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left(4\pi r^{2}\right)={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2176e77018dab3a287c1a9692fd5ec09c7f276cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.197ex; height:6.009ex;" alt="{\displaystyle E\left(4\pi r^{2}\right)={\frac {Q}{\varepsilon _{0}}}\ {\frac {r^{3}}{R^{3}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605faf1889d4a04aebf64b43d9b75d39c986de97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.997ex; height:5.843ex;" alt="{\displaystyle E={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}}"></span></dd></dl> <p>Logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95491825064ad9c45e4e1f00c0bf436fd7ecb6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.141ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} ={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }}}"></span></dd></dl> <p>Portanto, no caso de uma esfera uniformemente carregada: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\begin{cases}{\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }},&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mi>r</mi> <mo><</mo> <mi>R</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mi>r</mi> <mo>></mo> <mi>R</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\begin{cases}{\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }},&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7855bbda4b0c612bc4ab547a78e18e9738d30b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:27.358ex; height:8.843ex;" alt="{\displaystyle \mathbf {E} ={\begin{cases}{\frac {Q}{4\pi \varepsilon _{0}}}{\frac {r}{R^{3}}}{\hat {\mathbf {r} }},&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Campo_elétrico_no_interior_e_no_exterior_de_uma_casca_esférica"><span id="Campo_el.C3.A9trico_no_interior_e_no_exterior_de_uma_casca_esf.C3.A9rica"></span>Campo elétrico no interior e no exterior de uma casca esférica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=8" title="Editar secção: Campo elétrico no interior e no exterior de uma casca esférica" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=8" title="Editar código-fonte da secção: Campo elétrico no interior e no exterior de uma casca esférica"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para se resolver esse problema, utiliza-se a figura 4 novamente, porém com uma ligeira diferença: o interior da esfera de raio R é "oco", isto é, tem-se apenas uma casca esférica com carga Q uniformemente distribuída sobre sua superfície. </p> <dl><dt>No exterior da esfera</dt> <dd></dd></dl> <p>Escolhendo a superfície de raio r' como mostrada na figura 4, tem-se, pela lei de Gauss, o mesmo resultado que foi obtido para o campo no exterior de uma esfera. A carga interna à superfície gaussiana, q<sub>int</sub>, é Q nesse caso, como no caso anterior da esfera uniformemente carregada, de forma que o cálculo para o campo elétrico exterior à da casca esférica se desenvolve da mesma forma que o cálculo para o campo no exterior à esfera uniformemente carregada, então: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msup> <mi>r</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ef1a1e54448747a498030b93581573eccfa663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.878ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r'^{2}}}{\hat {\mathbf {r} }}}"></span></dd></dl> <dl><dt>No interior da casca esférica</dt> <dd></dd></dl> <p>Escolhendo a superfície gaussiana de raio r, no interior da casca esférica, tem-se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee066dbed2279d6473cafdbcbf34425dbebf128d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.332ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}=0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}E\ dA=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}E\ dA=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b39c97aa443628f5ce993373d65e4c0d905eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.548ex; height:5.676ex;" alt="{\displaystyle \oint _{S}E\ dA=0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\oint _{S}dA=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\oint _{S}dA=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59117c5ba5764a62fdc806b57d2bc1e5d786c7c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.355ex; height:5.676ex;" alt="{\displaystyle E\oint _{S}dA=0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\left(4\pi r^{2}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left(4\pi r^{2}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186845fb4c348d18b32f4a719b3343a9aeeb9fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.151ex; height:3.343ex;" alt="{\displaystyle E\left(4\pi r^{2}\right)=0}"></span></dd></dl> <p>Portanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b9ec424bcc94d232be40bb53ebac3b8d5e9059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.037ex; height:2.176ex;" alt="{\displaystyle E=0}"></span></dd></dl> <p>Logo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =\mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b252463af6ac4d37f07c3658cf37883efeb87e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.192ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} =\mathbf {0} }"></span></dd></dl> <p>Portanto, no caso de uma casca esférica uniformemente carregada: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\begin{cases}\mathbf {0} ,&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mi>r</mi> <mo><</mo> <mi>R</mi> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>se </mtext> </mstyle> </mrow> <mi>r</mi> <mo>></mo> <mi>R</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\begin{cases}\mathbf {0} ,&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8cba998ffd55a7f41dece988a46e5ea5d3c6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.659ex; height:7.509ex;" alt="{\displaystyle \mathbf {E} ={\begin{cases}\mathbf {0} ,&{\mbox{se }}r<R\\{\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},&{\mbox{se }}r>R\end{cases}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Campo_elétrico_de_um_plano_infinito"><span id="Campo_el.C3.A9trico_de_um_plano_infinito"></span>Campo elétrico de um plano infinito</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=9" title="Editar secção: Campo elétrico de um plano infinito" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=9" title="Editar código-fonte da secção: Campo elétrico de um plano infinito"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:ElectricDisplacement_English.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/ElectricDisplacement_English.png/200px-ElectricDisplacement_English.png" decoding="async" width="200" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/ElectricDisplacement_English.png/300px-ElectricDisplacement_English.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/7b/ElectricDisplacement_English.png 2x" data-file-width="350" data-file-height="232" /></a><figcaption>Figura 5: Um exemplo de superfície gaussiana que se deve utilizar para obter o campo de um plano infinito é como a que está mostrada sobre a placa de baixo do capacitor.</figcaption></figure> <p>Supõe-se um plano infinito com <a href="/wiki/Densidade_de_carga" title="Densidade de carga">densidade de carga</a> σ e se deseja calcular o campo elétrico produzido por esse plano. Apesar de o problema ser bem diferente do apresentado na figura 5, visto que, no problema em questão, está-se estudando um plano infinito e não o campo no interior de um <a href="/wiki/Capacitor" title="Capacitor">capacitor</a>, é interessante utilizar uma superfície gaussiana de mesma forma que a superfície retratada na placa de baixo do capacitor da figura 5. Utilizando, portanto, a superfície de um paralelepípedo cortando o plano infinito como superfície S, tem-se: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97099ab179376e397e437eb7b13c7919f3181004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.071ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={\frac {q_{int}}{\varepsilon _{0}}}}"></span></dd></dl> <p>Por simetria, o campo elétrico deve apontar para "fora" do plano, isto é, ele aponta na direção <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {z} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4717aa1fda3698bf5489269af7a5904c39c07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {z} }}}"></span> para pontos acima do plano e na direção <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\hat {\mathbf {z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\hat {\mathbf {z} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d58b80bcf6674f9016c48d3c6f3038677dab034f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.996ex; height:2.509ex;" alt="{\displaystyle -{\hat {\mathbf {z} }}}"></span> para pontos abaixo do plano. Dessa forma, as únicas superfícies superior e inferior da superfície do paralelepípedo é que serão "furadas" pelo campo elétrico, por isso: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}E\ dA=E\int _{S1+S2}dA=2AE}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mn>1</mn> <mo>+</mo> <mi>S</mi> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mn>1</mn> <mo>+</mo> <mi>S</mi> <mn>2</mn> </mrow> </msub> <mi>E</mi> <mtext> </mtext> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mi>E</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mn>1</mn> <mo>+</mo> <mi>S</mi> <mn>2</mn> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mi>A</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}E\ dA=E\int _{S1+S2}dA=2AE}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8696c683ee0de1010b3eded1a5b6360d991ada32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.844ex; height:5.843ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\int _{S1+S2}E\ dA=E\int _{S1+S2}dA=2AE}"></span></dd></dl> <p>onde A é a área da superfície superior e inferior da superfície do paralelepípedo. Sabe-se, também, que : σ = qint/A, logo : qint = σA, portanto: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2AE={\frac {\sigma A}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>A</mi> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi>A</mi> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2AE={\frac {\sigma A}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c84fbe2fad2def8dc773e8742a832877caf33e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.689ex; height:5.676ex;" alt="{\displaystyle 2AE={\frac {\sigma A}{\varepsilon _{0}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {\sigma }{2\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {\sigma }{2\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985134a33377ee1600b45ce82a71c1e83c33f301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.01ex; height:5.009ex;" alt="{\displaystyle E={\frac {\sigma }{2\varepsilon _{0}}}}"></span></dd></dl> <p>ou </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\frac {\sigma }{2\varepsilon _{0}}}{\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\frac {\sigma }{2\varepsilon _{0}}}{\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87b875215a516be48b41fab038bd68d4329e01c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.477ex; height:5.009ex;" alt="{\displaystyle \mathbf {E} ={\frac {\sigma }{2\varepsilon _{0}}}{\hat {\mathbf {n} }}}"></span></dd></dl> <p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}"></span> é um vetor unitário que aponta para fora da superfície do plano. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Fio_GAUSS.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Fio_GAUSS.png/300px-Fio_GAUSS.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Fio_GAUSS.png/450px-Fio_GAUSS.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Fio_GAUSS.png/600px-Fio_GAUSS.png 2x" data-file-width="5438" data-file-height="4071" /></a><figcaption>Figura 6: Superfície gaussiana cilíndrica envolvendo um fio extenso positivamente carregado.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Campo_elétrico_de_uma_carga_uniformemente_distribuída_ao_longo_de_um_fio_extenso"><span id="Campo_el.C3.A9trico_de_uma_carga_uniformemente_distribu.C3.ADda_ao_longo_de_um_fio_extenso"></span>Campo elétrico de uma carga uniformemente distribuída ao longo de um fio extenso</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=10" title="Editar secção: Campo elétrico de uma carga uniformemente distribuída ao longo de um fio extenso" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=10" title="Editar código-fonte da secção: Campo elétrico de uma carga uniformemente distribuída ao longo de um fio extenso"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vamos considerar um fio longo que possui uma densidade linear uniforme de cargas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> positiva em toda sua extensão. Podemos determinar o campo elétrico produzido por esta distribuição de cargas através da Lei de Gauss. Para tal, devido à simetria cilíndrica existente, vamos escolher como superfície gaussiana uma superfície cilíndrica de raio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> e comprimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, coaxial com o fio, como mostrado na Figura 6. Como o fio possui uma densidade linear de cargas uniforme, podemos relacionar a carga contida no interior da superfície gaussiana com esta densidade pela seguinte expressão: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{int}=\lambda h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{int}=\lambda h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e34a0cd5bc4d8a9b4b3010a94ef3219e58cf374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.209ex; height:2.509ex;" alt="{\displaystyle q_{int}=\lambda h}"></span>. Dessa forma, a Lei de Gauss pode ser escrita como<sup id="cite_ref-halliday_2-8" class="reference"><a href="#cite_note-halliday-2"><span>[</span>2<span>]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\dfrac {q_{int}}{\varepsilon _{0}}}={\dfrac {\lambda h}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mi>h</mi> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\dfrac {q_{int}}{\varepsilon _{0}}}={\dfrac {\lambda h}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/896c31e326be2d30f3c0b3dcf9c68afae3c308e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.624ex; height:5.843ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\dfrac {q_{int}}{\varepsilon _{0}}}={\dfrac {\lambda h}{\varepsilon _{0}}}}"></span></dd></dl> <p>Uma vez que o fio está carregado positivamente e considerando sua simetria cilíndrica, esperamos que o campo elétrico produzido por ele aponte radialmente para fora do fio. Tal característica faz com que o fluxo de campo elétrico seja nulo nos planos da base da superfície gaussiana, isto é, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \cdot d\mathbf {A} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \cdot d\mathbf {A} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a6898c323c2899b10c7c57aac01604409467f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.932ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \cdot d\mathbf {A} =0}"></span>, já que o ângulo entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">E</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a41593a3726e80b050a197f48a73bbec432aec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle {\textbf {E}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fdd48cc46dd596e66ef7a4ad469cb909127f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle d\mathbf {A} }"></span> é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span>. Na superfície lateral, o diferencial de fluxo passa a ser escrito como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle EdA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>d</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle EdA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3f5e60d945f929edfe0536ca368c1f12ea86c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.735ex; height:2.176ex;" alt="{\displaystyle EdA}"></span>, já que o ângulo entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textbf {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">E</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a41593a3726e80b050a197f48a73bbec432aec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle {\textbf {E}}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fdd48cc46dd596e66ef7a4ad469cb909127f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.235ex; height:2.176ex;" alt="{\displaystyle d\mathbf {A} }"></span> é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd0e1e92cf5770c2bfbb1de8b4b7bf904c9deef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle 0^{\circ }}"></span>. Dessa forma, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi rh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi rh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f07078e1d90c08fc9da0783148430ba17472d282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.882ex; height:2.176ex;" alt="{\displaystyle 2\pi rh}"></span> a área lateral da superfície gaussiana, temos que (onde SL significa Superfície Lateral): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\int _{SL}dA={\dfrac {\lambda h}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mi>L</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mi>h</mi> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\int _{SL}dA={\dfrac {\lambda h}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f4ff585160b1fff93d03f8184c8c95cacc0439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.842ex; height:5.843ex;" alt="{\displaystyle E\int _{SL}dA={\dfrac {\lambda h}{\varepsilon _{0}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\cdot (2\pi rh)={\dfrac {\lambda h}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>λ</mi> <mi>h</mi> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\cdot (2\pi rh)={\dfrac {\lambda h}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292d10793015d02c54a2f6d0109a90f88f530b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.775ex; height:5.676ex;" alt="{\displaystyle E\cdot (2\pi rh)={\dfrac {\lambda h}{\varepsilon _{0}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>λ</mi> <mi>r</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfced90a54c31cd11183242bb18f0c76f6de479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.534ex; height:5.676ex;" alt="{\displaystyle E={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}}"></span></dd></dl> <p>Logo, a expressão vetorial da distribuição espacial do campo elétrico resulta em: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} ={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}\mathbf {\hat {r}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>λ</mi> <mi>r</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}\mathbf {\hat {r}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a5c9332eb3c7e72a1f464ec60aa909969ae03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.852ex; height:5.676ex;" alt="{\displaystyle \mathbf {E} ={\dfrac {1}{2\pi \varepsilon _{0}}}{\dfrac {\lambda }{r}}\mathbf {\hat {r}} }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Lei_de_Gauss_para_dielétricos"><span id="Lei_de_Gauss_para_diel.C3.A9tricos"></span>Lei de Gauss para dielétricos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=11" title="Editar secção: Lei de Gauss para dielétricos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=11" title="Editar código-fonte da secção: Lei de Gauss para dielétricos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Dipole_polarization.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Dipole_polarization.JPG/200px-Dipole_polarization.JPG" decoding="async" width="200" height="171" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Dipole_polarization.JPG/300px-Dipole_polarization.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/9/96/Dipole_polarization.JPG 2x" data-file-width="383" data-file-height="327" /></a><figcaption>Figura 7: Ilustração da polarização de um material dielétrico.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Cargas_livres_e_cargas_ligadas">Cargas livres e cargas ligadas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=12" title="Editar secção: Cargas livres e cargas ligadas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=12" title="Editar código-fonte da secção: Cargas livres e cargas ligadas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Polariza%C3%A7%C3%A3o_diel%C3%A9trica" title="Polarização dielétrica">polarização dielétrica</a></div> <p>Um <a href="/wiki/Diel%C3%A9trico" title="Dielétrico">dielétrico</a> em presença de um <a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">campo elétrico</a>, sofre o que se chama de <a href="/wiki/Polariza%C3%A7%C3%A3o" class="mw-disambig" title="Polarização">polarização</a>. A polarização consiste na separação das cargas positivas e negativas desse dielétrico, visto que o campo elétrico acelera cargas positivas no sentido do campo e cargas negativas no sentido oposto. Essas cargas geradas por esse efeito de polarização é o que se chama de cargas ligadas. O material passa a ser constituído de dipolos, como mostrado na figura 6. Dessa forma, as cargas estão "presas" aos dipolos, não estão livres para se mover. Por sua vez, é chamado de carga livre, o restante das cargas, que não foram geradas por esse efeito de polarização. As cargas livres são as cargas com as quais se está mais habituado quando se estuda eletrostática.<sup id="cite_ref-griffiths_3-8" class="reference"><a href="#cite_note-griffiths-3"><span>[</span>3<span>]</span></a></sup> Desse modo, num dielétrico, a densidade volumétrica de carga pode ser escrita como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\rho _{lig}+\rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>g</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\rho _{lig}+\rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38020f185d0fb4371c7881458bc738881366184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.711ex; height:2.676ex;" alt="{\displaystyle \rho =\rho _{lig}+\rho _{liv}}"></span></dd></dl> <p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{lig}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{lig}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd002266a67dd72aed80c741e172f3ab776414c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.281ex; height:2.343ex;" alt="{\displaystyle \rho _{lig}}"></span> é a densidade volumétrica de carga ligada e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7138995ffa96e3ae2a2486ef8ea40ef5091d6f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.289ex; height:2.176ex;" alt="{\displaystyle \rho _{liv}}"></span> é a densidade volumétrica de carga livre. </p> <div class="mw-heading mw-heading3"><h3 id="Demonstração_da_lei_de_Gauss_para_dielétricos"><span id="Demonstra.C3.A7.C3.A3o_da_lei_de_Gauss_para_diel.C3.A9tricos"></span>Demonstração da lei de Gauss para dielétricos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=13" title="Editar secção: Demonstração da lei de Gauss para dielétricos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=13" title="Editar código-fonte da secção: Demonstração da lei de Gauss para dielétricos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left"> <tbody><tr> <th>Demonstração </th></tr> <tr> <td> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\rho _{lig}+\rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>g</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\rho _{lig}+\rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38020f185d0fb4371c7881458bc738881366184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.711ex; height:2.676ex;" alt="{\displaystyle \rho =\rho _{lig}+\rho _{liv}}"></span></dd></dl> <p>Pela lei de Gauss na forma diferencial: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}\nabla \cdot \mathbf {E} =\rho =\rho _{lig}+\rho _{liv}=-\nabla \cdot \mathbf {P} +\rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>g</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}\nabla \cdot \mathbf {E} =\rho =\rho _{lig}+\rho _{liv}=-\nabla \cdot \mathbf {P} +\rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7db7e4d710924f2bb2c1fd24eb6871157b4548" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.797ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{0}\nabla \cdot \mathbf {E} =\rho =\rho _{lig}+\rho _{liv}=-\nabla \cdot \mathbf {P} +\rho _{liv}}"></span></dd></dl> <p>onde usamos o resultado de que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{lig}=-\nabla \cdot \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{lig}=-\nabla \cdot \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/871a2032385ed9ea42244c9af2b741be3b972d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.629ex; height:2.843ex;" alt="{\displaystyle \rho _{lig}=-\nabla \cdot \mathbf {P} }"></span> (ver Leitura Complementar(**)) e <b>P</b> é o vetor polarização, que é definido como o momento de dipolo por unidade de volume. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} )+\nabla \cdot \mathbf {P} =\rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} )+\nabla \cdot \mathbf {P} =\rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd5b9e40270f32b6205d35665025c178b0a6336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.989ex; height:2.843ex;" alt="{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} )+\nabla \cdot \mathbf {P} =\rho _{liv}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} +\mathbf {P} )=\rho _{liv}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} +\mathbf {P} )=\rho _{liv}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10fbbd67ade6a72ecbb7bcea067c71a1beb03a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.374ex; height:2.843ex;" alt="{\displaystyle \nabla \cdot (\varepsilon _{0}\mathbf {E} +\mathbf {P} )=\rho _{liv}}"></span></dd></dl> <p>A expressão entre parênteses é designada pela letra <b>D</b>, que é o chamado vetor de deslocamento elétrico. Ou seja: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3061b05e80b3eee6d58c6aec1fc14e068bf0115c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }"></span></dd></dl> <p>Portanto: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #f5f5f5; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{liv}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {D} =\rho _{liv}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5b5806faa12907c1a1562fae578ec42c4e3d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:12.439ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{liv}\,\!}"></span> </p> </td></tr></tbody></table></dd></dl> <p>Essa é a lei de Gauss em dielétricos na forma diferencial. Para obter a forma integral: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}(\nabla \cdot \mathbf {D} ){d}\tau =\int _{v}\rho _{liv}\ {d}\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <mi>v</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}(\nabla \cdot \mathbf {D} ){d}\tau =\int _{v}\rho _{liv}\ {d}\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0c352bad8540160bc77cda11aa283d24fb35c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.776ex; height:5.676ex;" alt="{\displaystyle \int _{V}(\nabla \cdot \mathbf {D} ){d}\tau =\int _{v}\rho _{liv}\ {d}\tau }"></span></dd></dl> <p>Pelo <a href="/wiki/Teorema_de_Stokes" title="Teorema de Stokes">teorema de Stokes</a>: </p> <dl><dd><table cellpadding="5" style="border:2px solid #000000;background: #f5f5f5; text-align: center;"> <tbody><tr> <td> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =q_{liv_{int}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>i</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =q_{liv_{int}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13bfa4c67d56cc99aabc90d8d4c2c24f63f2a524" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-right: -0.387ex; width:18.53ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =q_{liv_{int}}\,\!}"></span> </p> </td></tr></tbody></table></dd></dl> <p>Pode-se observar que a lei de Gauss para dielétricos só faz referência à cargas livres, que é o tipo de carga sobre o qual se tem controle e se tem mais facilidade de medir, por isso, a lei de Gauss para dielétricos torna-se um artifício muito útil para calcular o vetor deslocamento elétrico de diferentes distribuições de carga e, consequentemente, o campo elétrico dessas distribuições, que pode ser obtido através da relação entre <b>E</b> e <b>D</b>. </p> </td></tr></tbody></table></dd></dl> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Bellone, Enrico (1980). <i>A World on Paper: Studies on the Second Scientific Revolution</i>. [S.l.: s.n.]</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3ALei+de+Gauss&rft.au=Bellone%2C+Enrico&rft.btitle=A+World+on+Paper%3A+Studies+on+the+Second+Scientific+Revolution&rft.date=1980&rft.genre=book&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-halliday-2"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-halliday_2-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-2">c</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-3">d</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-4">e</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-5">f</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-6">g</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-7">h</a></b></i></sup> <sup><i><b><a href="#cite_ref-halliday_2-8">i</a></b></i></sup></span> <span class="reference-text"> Halliday, D., Resnick, R., Krane, K. <i>Física 3</i>, 5a ed. GEN|LTC (2010).</span> </li> <li id="cite_note-griffiths-3"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-griffiths_3-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-1">b</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-2">c</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-3">d</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-4">e</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-5">f</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-6">g</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-7">h</a></b></i></sup> <sup><i><b><a href="#cite_ref-griffiths_3-8">i</a></b></i></sup></span> <span class="reference-text"> Griffiths, D. J. <i>Introduction to Electrodynamics</i>, 3a ed. New Jersey: Prentice Hall (1999).</span> </li> <li id="cite_note-Feynman-4"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-Feynman_4-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-Feynman_4-1">b</a></b></i></sup></span> <span class="reference-text"> Feynman, R. P., Leighton, R. B., Sands M. <i>The Feynman Lectures on Physics</i>,vol 2. 2a ed. Bookman (2008).</span> </li> <li id="cite_note-Jackson-5"><span class="mw-cite-backlink"><a href="#cite_ref-Jackson_5-0">↑</a></span> <span class="reference-text"> Jackson J. D. <i>Classical Eletrodynamics</i>, 2a ed. John Sons and wiley (1975).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lei_de_Gauss&veaction=edit&section=14" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lei_de_Gauss&action=edit&section=14" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Delta_de_Dirac" title="Delta de Dirac">Delta de Dirac</a></li> <li><a href="/wiki/Polariza%C3%A7%C3%A3o_diel%C3%A9trica" title="Polarização dielétrica">Polarização dielétrica</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Lei_de_Gauss&oldid=65810571">https://pt.wikipedia.org/w/index.php?title=Lei_de_Gauss&oldid=65810571</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categorias</a>: <ul><li><a href="/wiki/Categoria:Eletromagnetismo" title="Categoria:Eletromagnetismo">Eletromagnetismo</a></li><li><a href="/wiki/Categoria:Carl_Friedrich_Gau%C3%9F" title="Categoria:Carl Friedrich Gauß">Carl Friedrich Gauß</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!P%C3%A1ginas_a_reciclar_desde_outubro_de_2022" title="Categoria:!Páginas a reciclar desde outubro de 2022">!Páginas a reciclar desde outubro de 2022</a></li><li><a href="/wiki/Categoria:!P%C3%A1ginas_a_reciclar_sem_indica%C3%A7%C3%A3o_de_tema" title="Categoria:!Páginas a reciclar sem indicação de tema">!Páginas a reciclar sem indicação de tema</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 20h08min de 4 de maio de 2023.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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