Gauss's law - Wikipedia

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class="vector-toc-numb">2</span> <span>Qualitative description</span> </div> </a> <ul id="toc-Qualitative_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_involving_the_E_field" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equation_involving_the_E_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equation involving the <span>E</span> field</span> </div> </a> <button aria-controls="toc-Equation_involving_the_E_field-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>Toggle Equation involving the <span>E</span> field subsection</span> </button> <ul id="toc-Equation_involving_the_E_field-sublist" class="vector-toc-list"> <li id="toc-Integral_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Integral form</span> </div> </a> <ul id="toc-Integral_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Differential form</span> </div> </a> <ul id="toc-Differential_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalence_of_integral_and_differential_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalence_of_integral_and_differential_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Equivalence of integral and differential forms</span> </div> </a> <ul id="toc-Equivalence_of_integral_and_differential_forms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equation_involving_the_D_field" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equation_involving_the_D_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Equation involving the <span>D</span> field</span> </div> </a> <button aria-controls="toc-Equation_involving_the_D_field-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>Toggle Equation involving the <span>D</span> field subsection</span> </button> <ul id="toc-Equation_involving_the_D_field-sublist" class="vector-toc-list"> <li id="toc-Free,_bound,_and_total_charge" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Free,_bound,_and_total_charge"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Free, bound, and total charge</span> </div> </a> <ul id="toc-Free,_bound,_and_total_charge-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_form_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_form_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Integral form</span> </div> </a> <ul id="toc-Integral_form_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_form_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_form_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Differential form</span> </div> </a> <ul id="toc-Differential_form_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalence_of_total_and_free_charge_statements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equivalence_of_total_and_free_charge_statements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Equivalence of total and free charge statements</span> </div> </a> <ul id="toc-Equivalence_of_total_and_free_charge_statements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equation_for_linear_materials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equation_for_linear_materials"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Equation for linear materials</span> </div> </a> <ul id="toc-Equation_for_linear_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_to_Coulomb's_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relation_to_Coulomb's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Relation to Coulomb's law</span> </div> </a> <button aria-controls="toc-Relation_to_Coulomb's_law-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>Toggle Relation to Coulomb's law subsection</span> </button> <ul id="toc-Relation_to_Coulomb's_law-sublist" class="vector-toc-list"> <li id="toc-Deriving_Gauss's_law_from_Coulomb's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deriving_Gauss's_law_from_Coulomb's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Deriving Gauss's law from Coulomb's law</span> </div> </a> <ul id="toc-Deriving_Gauss's_law_from_Coulomb's_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Deriving_Coulomb's_law_from_Gauss's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Deriving_Coulomb's_law_from_Gauss's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Deriving Coulomb's law from Gauss's law</span> </div> </a> <ul id="toc-Deriving_Coulomb's_law_from_Gauss's_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Gauss's law</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="Go to an article in another language. Available in 65 languages" > <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 languages</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 – Arabic" lang="ar" hreflang="ar" data-title="قانون غاوس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</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 – Asturian" lang="ast" hreflang="ast" data-title="Llei de Gauss" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</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 – Bangla" lang="bn" hreflang="bn" data-title="গাউসের সূত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Belarusian" lang="be" hreflang="be" data-title="Тэарэма Гаўса" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Теорема на Гаус" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Gaussov_zakon" title="Gaussov zakon – Bosnian" lang="bs" hreflang="bs" data-title="Gaussov zakon" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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 – Catalan" lang="ca" hreflang="ca" data-title="Llei de Gauss" data-language-autonym="Català" data-language-local-name="Catalan" 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 – Czech" lang="cs" hreflang="cs" data-title="Gaussův zákon elektrostatiky" data-language-autonym="Čeština" data-language-local-name="Czech" 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 – Danish" lang="da" hreflang="da" data-title="Gauss' lov" data-language-autonym="Dansk" data-language-local-name="Danish" 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 – German" lang="de" hreflang="de" data-title="Gaußsches Gesetz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</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 – Estonian" lang="et" hreflang="et" data-title="Gaussi seadus elektrivälja jaoks" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Greek" lang="el" hreflang="el" data-title="Νόμος του Γκάους" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</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 – Spanish" lang="es" hreflang="es" data-title="Ley de Gauss" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gaussen_legea" title="Gaussen legea – Basque" lang="eu" hreflang="eu" data-title="Gaussen legea" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="قانون گاوس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</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) – French" lang="fr" hreflang="fr" data-title="Théorème de Gauss (physique)" data-language-autonym="Français" data-language-local-name="French" 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 – Irish" lang="ga" hreflang="ga" data-title="Dlí Gauß" data-language-autonym="Gaeilge" data-language-local-name="Irish" 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 – Galician" lang="gl" hreflang="gl" data-title="Lei de Gauss" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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 – Korean" lang="ko" hreflang="ko" data-title="가우스 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Armenian" lang="hy" hreflang="hy" data-title="Գաուսի օրենք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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 – Croatian" lang="hr" hreflang="hr" data-title="Gaussov zakon" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hukum_Gauss" title="Hukum Gauss – Indonesian" lang="id" hreflang="id" data-title="Hukum Gauss" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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 – Italian" lang="it" hreflang="it" data-title="Teorema del flusso" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="חוק גאוס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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 – Georgian" lang="ka" hreflang="ka" data-title="გაუსის კანონი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</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 – Kazakh" lang="kk" hreflang="kk" data-title="Гаусс теоремасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</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 – Kyrgyz" lang="ky" hreflang="ky" data-title="Гаyсс теоремасы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" 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 – Latvian" lang="lv" hreflang="lv" data-title="Gausa teorēma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</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 – Hungarian" lang="hu" hreflang="hu" data-title="Gauss-törvény" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Гаусов закон" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</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 – Marathi" lang="mr" hreflang="mr" data-title="गॉसचा नियम" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</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-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 – Mongolian" lang="mn" hreflang="mn" data-title="Гауссын хууль" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</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 – Dutch" lang="nl" hreflang="nl" data-title="Wet van Gauss" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Nepali" lang="ne" hreflang="ne" data-title="गाउसको नियम" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</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 – Japanese" lang="ja" hreflang="ja" data-title="ガウスの法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</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 – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gauss’ lov" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</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 – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Gauss' lov" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gauss_teoremasi" title="Gauss teoremasi – Uzbek" lang="uz" hreflang="uz" data-title="Gauss teoremasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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) – Polish" lang="pl" hreflang="pl" data-title="Prawo Gaussa (elektryczność)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_de_Gauss" title="Lei de Gauss – Portuguese" lang="pt" hreflang="pt" data-title="Lei de Gauss" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</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-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 – Russian" lang="ru" hreflang="ru" data-title="Теорема Гаусса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</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 – Albanian" lang="sq" hreflang="sq" data-title="Ligji i Gausit" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</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 – Slovak" lang="sk" hreflang="sk" data-title="Gaussov zákon elektrostatiky" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Zakon o električnem pretoku" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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 – Serbian" lang="sr" hreflang="sr" data-title="Гаусов закон" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</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 – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Gaussov zakon električnoga polja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – Finnish" lang="fi" hreflang="fi" data-title="Gaussin laki sähkökentille" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gauss_lag" title="Gauss lag – Swedish" lang="sv" hreflang="sv" data-title="Gauss lag" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</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 – Tagalog" lang="tl" hreflang="tl" data-title="Batas ni Gauss" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</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 – Tamil" lang="ta" hreflang="ta" data-title="காஸ் விதி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</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-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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gauss_yasas%C4%B1" title="Gauss yasası – Turkish" lang="tr" hreflang="tr" data-title="Gauss yasası" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</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 – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема Гаусса" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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-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 – Vietnamese" lang="vi" hreflang="vi" data-title="Định luật Gauss" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" 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-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 – Cantonese" lang="yue" hreflang="yue" data-title="高斯定律" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</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 – Chinese" lang="zh" hreflang="zh" data-title="高斯定律" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Foundational law of electromagnetism relating electric field and charge distributions</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about Gauss's law concerning the electric field. For analogous laws concerning different fields, see <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a> and <a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a>. For the Ostrogradsky–Gauss theorem, a mathematical theorem relevant to all of these laws, see <a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Gause%27s_law" class="mw-redirect" title="Gause's law">Gause's law</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Maxwell_integral_Gauss_sphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Maxwell_integral_Gauss_sphere.svg/260px-Maxwell_integral_Gauss_sphere.svg.png" decoding="async" width="260" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Maxwell_integral_Gauss_sphere.svg/390px-Maxwell_integral_Gauss_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Maxwell_integral_Gauss_sphere.svg/520px-Maxwell_integral_Gauss_sphere.svg.png 2x" data-file-width="184" data-file-height="77" /></a><figcaption>Gauss's law in its integral form is particularly useful when, by symmetry reasons, a closed surface (GS) can be found along which the electric field is uniform. The electric flux is then a simple product of the surface area and the strength of the electric field, and is proportional to the total charge enclosed by the surface. Here, the electric field outside (<i>r</i> > <i>R</i>) and inside (<i>r</i> < <i>R</i>) of a charged sphere is being calculated (see <a href="https://en.wikiversity.org/wiki/MyOpenMath/Solutions/Maxwell%27s_integral_equations" class="extiw" title="wikiversity:MyOpenMath/Solutions/Maxwell's integral equations">Wikiversity</a>).</figcaption></figure> <p>In <a href="/wiki/Physics" title="Physics">physics</a> (specifically <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>), <b>Gauss's law</b>, also known as <b>Gauss's flux theorem</b> (or sometimes Gauss's theorem), is one of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. It is an application of the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, and it relates the distribution of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> to the resulting <a href="/wiki/Electric_field" title="Electric field">electric field</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In its <a href="/wiki/Integral" title="Integral">integral form</a>, it states that the <a href="/wiki/Flux" title="Flux">flux</a> of the electric field out of an arbitrary <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its <a href="/wiki/Differential_form" title="Differential form">differential form</a>, which states that the divergence of the electric field is proportional to the local density of charge. </p><p>The law was first<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> formulated by <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> in 1773,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> followed by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> in 1835,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> both in the context of the attraction of ellipsoids. It is one of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, which forms the basis of <a href="/wiki/Classical_electrodynamics" class="mw-redirect" title="Classical electrodynamics">classical electrodynamics</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> Gauss's law can be used to derive <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and vice versa. </p> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output 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title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li> <li><a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></li> <li><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li> <li><a href="/wiki/Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a 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href="/wiki/Larmor_formula" title="Larmor formula">Larmor formula</a></li> <li><a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz law</a></li> <li><a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li> <li><a href="/wiki/London_equations" title="London equations">London equations</a></li> <li><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a></li> <li><a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li> <li><a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></li> <li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrical_network" title="Electrical network">Electrical network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/AC_motor" title="AC motor">AC motor</a></li> <li><a href="/wiki/DC_motor" title="DC motor">DC motor</a></li> <li><a href="/wiki/Electric_machine" title="Electric machine">Electric machine</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motor</a></li> <li><a href="/wiki/Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li> <li><a href="/wiki/Induction_motor" title="Induction motor">Induction motor</a></li> <li><a href="/wiki/Linear_motor" title="Linear motor">Linear motor</a></li> <li><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li> <li><a href="/wiki/Permeance" title="Permeance">Permeance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li> <li><a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li> <li><a href="/wiki/Rotor_(electric)" title="Rotor (electric)">Rotor</a></li> <li><a href="/wiki/Stator" title="Stator">Stator</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li> <li><a href="/wiki/Four-current" title="Four-current">Four-current</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li> <li><a href="/wiki/Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li> <li><a href="/wiki/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li> <li><a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li> <li><a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li><a href="/wiki/Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li> <li><a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Larmor</a></li> <li><a href="/wiki/Emil_Lenz" title="Emil Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Georg_Ohm" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li><a href="/wiki/William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li><a href="/wiki/George_Singer" title="George Singer">Singer</a></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electromagnetism" title="Template:Electromagnetism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electromagnetism" title="Template talk:Electromagnetism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electromagnetism" title="Special:EditPage/Template:Electromagnetism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Qualitative_description">Qualitative description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=2" title="Edit section: Qualitative description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In words, Gauss's law states: </p> <dl><dd>The net <a href="/wiki/Electric_flux" title="Electric flux">electric flux</a> through any hypothetical <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> is equal to <span class="texhtml">1/<i>ε</i><sub>0</sub></span> times the net <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> enclosed within that closed surface. The closed surface is also referred to as Gaussian surface.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a> and <a href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss's law for gravity</a>. In fact, any <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square law</a> can be formulated in a way similar to Gauss's law: for example, Gauss's law itself is essentially equivalent to the <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a>, and Gauss's law for gravity is essentially equivalent to the <a href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</a>, both of which are inverse-square laws. </p><p>The law can be expressed mathematically using <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> in <a href="/wiki/Integral_calculus" class="mw-redirect" title="Integral calculus">integral</a> form and <a href="/wiki/Differential_calculus" title="Differential calculus">differential</a> form; both are equivalent since they are related by the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, also called Gauss's theorem. Each of these forms in turn can also be expressed two ways: In terms of a relation between the <a href="/wiki/Electric_field" title="Electric field">electric field</a> <span class="texhtml"><b>E</b></span> and the total electric charge, or in terms of the <a href="/wiki/Electric_displacement_field" title="Electric displacement field">electric displacement field</a> <span class="texhtml"><b>D</b></span> and the <a href="/wiki/Free_charge" class="mw-redirect" title="Free charge"><i>free</i> electric charge</a>.<sup id="cite_ref-GrantPhillips_7-0" class="reference"><a href="#cite_note-GrantPhillips-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Equation_involving_the_E_field">Equation involving the <span class="texhtml">E</span> field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=3" title="Edit section: Equation involving the E field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gauss's law can be stated using either the electric field <span class="texhtml"><b>E</b></span> or the electric displacement field <span class="texhtml"><b>D</b></span>. This section shows some of the forms with <span class="texhtml"><b>E</b></span>; the form with <span class="texhtml"><b>D</b></span> is below, as are other forms with <span class="texhtml"><b>E</b></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_form">Integral form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=4" title="Edit section: Integral form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electric-flux-surface-example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electric-flux-surface-example.svg/220px-Electric-flux-surface-example.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electric-flux-surface-example.svg/330px-Electric-flux-surface-example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/32/Electric-flux-surface-example.svg/440px-Electric-flux-surface-example.svg.png 2x" data-file-width="488" data-file-height="366" /></a><figcaption>Electric flux through an arbitrary surface is proportional to the total charge enclosed by the surface.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electric-flux-no-charge-inside.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Electric-flux-no-charge-inside.svg/220px-Electric-flux-no-charge-inside.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Electric-flux-no-charge-inside.svg/330px-Electric-flux-no-charge-inside.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Electric-flux-no-charge-inside.svg/440px-Electric-flux-no-charge-inside.svg.png 2x" data-file-width="488" data-file-height="366" /></a><figcaption>No charge is enclosed by the sphere. Electric flux through its surface is zero.</figcaption></figure> <p>Gauss's law may be expressed as:<sup id="cite_ref-GrantPhillips_7-1" class="reference"><a href="#cite_note-GrantPhillips-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}={\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> <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}={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292a4114ec9c495a97f151c19fbcf4cf3f533194" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.238ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}={\frac {Q}{\varepsilon _{0}}}}"></span> </p><p>where <span class="texhtml">Φ<sub><i>E</i></sub></span> is the <a href="/wiki/Electric_flux" title="Electric flux">electric flux</a> through a closed surface <span class="texhtml mvar" style="font-style:italic;">S</span> enclosing any volume <span class="texhtml mvar" style="font-style:italic;">V</span>, <span class="texhtml mvar" style="font-style:italic;">Q</span> is the total charge enclosed within <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="texhtml"><i>ε</i><sub>0</sub></span> is the <a href="/wiki/Electric_constant" class="mw-redirect" title="Electric constant">electric constant</a>. The electric flux <span class="texhtml">Φ<sub><i>E</i></sub></span> is defined as a <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of the electric field: </p> <dl><dd><span class="nowrap mw-no-invert"><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> <mo>=</mo> </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/9810529d253a8cc85469e17185424ea235655087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.619ex; height:2.509ex;" alt="{\displaystyle \Phi _{E}=}"></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><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 \scriptstyle _{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle _{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b570ea85ff659c8b20a3b538b0000c21c530162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.025ex; height:1.509ex;" alt="{\displaystyle \scriptstyle _{S}}"></span></span> <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 \mathrm {d} \mathbf {A} }"> <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"> <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 \mathbf {E} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7add4b16e26ddbe660c0b1a0ebe94cbefd10460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.748ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {A} }"></span></span></dd></dl> <p>where <span class="texhtml"><b>E</b></span> is the electric field, <span class="texhtml">d<b>A</b></span> is a vector representing an <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> element of <a href="/wiki/Area" title="Area">area</a> of the surface,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> and <span class="texhtml">·</span> represents the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of two vectors. </p><p>In a curved spacetime, the flux of an electromagnetic field through a closed surface is expressed as </p> <dl><dd><span class="nowrap mw-no-invert"><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}=c}"> <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>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee66795a7d7cf95804ef49399f87d9c9a6abb9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.271ex; height:2.509ex;" alt="{\displaystyle \Phi _{E}=c}"></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><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 \scriptstyle _{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle _{S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b570ea85ff659c8b20a3b538b0000c21c530162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.025ex; height:1.509ex;" alt="{\displaystyle \scriptstyle _{S}}"></span></span> <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 F^{\kappa 0}{\sqrt {-g}}\,\mathrm {d} S_{\kappa }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mn>0</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>g</mi> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\kappa 0}{\sqrt {-g}}\,\mathrm {d} S_{\kappa }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c0d1e363dc7456702d7b895dc8b517dd91c8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.347ex; height:3.676ex;" alt="{\displaystyle F^{\kappa 0}{\sqrt {-g}}\,\mathrm {d} S_{\kappa }}"></span></span></dd></dl> <p>where <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>; <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 F^{\kappa 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> <mn>0</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{\kappa 0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1cff81226796b45b077d69d49f68c2abc9878a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.816ex; height:2.676ex;" alt="{\displaystyle F^{\kappa 0}}"></span> denotes the time components of the <a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">electromagnetic tensor</a>; <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> is the determinant of <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a>; <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 \mathrm {d} S_{\kappa }=\mathrm {d} S^{ij}=\mathrm {d} x^{i}\mathrm {d} x^{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>κ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} S_{\kappa }=\mathrm {d} S^{ij}=\mathrm {d} x^{i}\mathrm {d} x^{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0564c56937c2bc759014515f60a955e182cccb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.339ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} S_{\kappa }=\mathrm {d} S^{ij}=\mathrm {d} x^{i}\mathrm {d} x^{j}}"></span> is an orthonormal element of the two-dimensional surface surrounding the charge <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/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span>; indices <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 i,j,\kappa =1,2,3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>κ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j,\kappa =1,2,3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55015cdda0715c7a7624907b5a28fa2e6acb25ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.821ex; height:2.509ex;" alt="{\displaystyle i,j,\kappa =1,2,3}"></span> and do not match each other.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Since the flux is defined as an <i>integral</i> of the electric field, this expression of Gauss's law is called the <i>integral form</i>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg/220px-Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg.png" decoding="async" width="220" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg/330px-Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg/440px-Gauss%27s_law_-_surface_charge_-_boundary_condition_on_D.svg.png 2x" data-file-width="696" data-file-height="373" /></a><figcaption>A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation. The electric field is perpendicular, locally, to the equipotential surface of the conductor, and zero inside; its flux <i>πa</i><sup>2</sup>·<i>E</i>, by Gauss's law equals <i>πa</i><sup>2</sup>·<i>σ</i>/<i>ε</i><sub>0</sub>. Thus, <span class="nowrap"><i>σ</i> = <i>ε</i><sub>0</sub><i>E</i></span>.</figcaption></figure> <p>In problems involving conductors set at known potentials, the potential away from them is obtained by solving <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>, either analytically or numerically. The electric field is then calculated as the potential's negative gradient. Gauss's law makes it possible to find the distribution of electric charge: The charge in any given region of the conductor can be deduced by integrating the electric field to find the flux through a small box whose sides are perpendicular to the conductor's surface and by noting that the electric field is perpendicular to the surface, and zero inside the conductor. </p><p>The reverse problem, when the electric charge distribution is known and the electric field must be computed, is much more difficult. The total flux through a given surface gives little information about the electric field, and can go in and out of the surface in arbitrarily complicated patterns. </p><p>An exception is if there is some <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> in the problem, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include: cylindrical symmetry, planar symmetry, and spherical symmetry. See the article <a href="/wiki/Gaussian_surface" title="Gaussian surface">Gaussian surface</a> for examples where these symmetries are exploited to compute electric fields. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_form">Differential form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=5" title="Edit section: Differential form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By the <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>, Gauss's law can alternatively be written in the <i>differential form</i>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> </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/ff0076e721a4b485bda8ff427f00e73c6efb6006" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.444ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}"></span> </p><p>where <span class="texhtml">∇ · <b>E</b></span> is the <a href="/wiki/Divergence" title="Divergence">divergence</a> of the electric field, <span class="texhtml"><i>ε</i><sub>0</sub></span> is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a> and <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the total volume <a href="/wiki/Charge_density" title="Charge density">charge density</a> (charge per unit volume). </p> <div class="mw-heading mw-heading3"><h3 id="Equivalence_of_integral_and_differential_forms">Equivalence of integral and differential forms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=6" title="Edit section: Equivalence of integral and differential forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></div> <p>The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically. </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Outline of proof</strong> <p>The integral form of Gauss's law is: </p> <dl><dd><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><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 {\scriptstyle _{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle _{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fda1e0b515c1a35897ad0641325d79e36bc106e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.025ex; height:1.509ex;" alt="{\displaystyle {\scriptstyle _{S}}}"></span></span> <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 \mathrm {d} \mathbf {A} }"> <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"> <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 \mathbf {E} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7add4b16e26ddbe660c0b1a0ebe94cbefd10460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.748ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {A} }"></span></span><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"> <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 ={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7783f3cfa8ae65a77f7d2691398c8d549f4eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.427ex; height:5.676ex;" alt="{\displaystyle ={\frac {Q}{\varepsilon _{0}}}}"></span></dd></dl> <p>for any closed surface <span class="texhtml mvar" style="font-style:italic;">S</span> containing charge <span class="texhtml mvar" style="font-style:italic;">Q</span>. By the divergence theorem, this equation is equivalent to: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V={\frac {Q}{\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> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</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 \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f5b26c3353643a7920d0a4a8d0fa6d960a7f79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.493ex; height:5.843ex;" alt="{\displaystyle \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V={\frac {Q}{\varepsilon _{0}}}}"></span> </p><p>for any volume <span class="texhtml mvar" style="font-style:italic;">V</span> containing charge <span class="texhtml mvar" style="font-style:italic;">Q</span>. By the relation between charge and charge density, this equation is equivalent to: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V=\iiint _{V}{\frac {\rho }{\varepsilon _{0}}}\,\mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <msub> <mo>∭</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <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" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V=\iiint _{V}{\frac {\rho }{\varepsilon _{0}}}\,\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fda4156b95e37ff852766f978bfdd06c9eca429" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.542ex; height:5.676ex;" alt="{\displaystyle \iiint _{V}\nabla \cdot \mathbf {E} \,\mathrm {d} V=\iiint _{V}{\frac {\rho }{\varepsilon _{0}}}\,\mathrm {d} V}"></span> for any volume <span class="texhtml mvar" style="font-style:italic;">V</span>. In order for this equation to be <i>simultaneously true</i> for <i>every</i> possible volume <span class="texhtml mvar" style="font-style:italic;">V</span>, it is necessary (and sufficient) for the integrands to be equal everywhere. Therefore, this equation is equivalent to: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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> <mo>.</mo> </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/58657a5bc68ba6a7eb094ff93c6d23d62f3c422b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.091ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}.}"></span> Thus the integral and differential forms are equivalent. </p> </div> <div class="mw-heading mw-heading2"><h2 id="Equation_involving_the_D_field">Equation involving the <span class="texhtml">D</span> field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=7" title="Edit section: Equation involving the D field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></div> <div class="mw-heading mw-heading3"><h3 id="Free,_bound,_and_total_charge"><span id="Free.2C_bound.2C_and_total_charge"></span>Free, bound, and total charge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=8" title="Edit section: Free, bound, and total charge"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electric_polarization" class="mw-redirect" title="Electric polarization">Electric polarization</a></div> <p>The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in <a href="/wiki/Static_electricity" title="Static electricity">static electricity</a>, or the charge on a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> plate. In contrast, "bound charge" arises only in the context of <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> (polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge". </p><p>Although microscopically all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more fundamental Gauss's law, in terms of <span class="texhtml"><b>E</b></span> (above), is sometimes put into the equivalent form below, which is in terms of <span class="texhtml"><b>D</b></span> and the free charge only. </p> <div class="mw-heading mw-heading3"><h3 id="Integral_form_2">Integral form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=9" title="Edit section: Integral form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This formulation of Gauss's law states the total charge form: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{D}=Q_{\mathrm {free} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{D}=Q_{\mathrm {free} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894fb7ad2248e79bb9953ae861109e5dbfb73dcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.156ex; height:2.509ex;" alt="{\displaystyle \Phi _{D}=Q_{\mathrm {free} }}"></span> </p><p>where <span class="texhtml">Φ<sub><i>D</i></sub></span> is the <a href="/wiki/Electric_displacement_field" title="Electric displacement field"><span class="texhtml"><b>D</b></span>-field</a> flux through a surface <span class="texhtml mvar" style="font-style:italic;">S</span> which encloses a volume <span class="texhtml mvar" style="font-style:italic;">V</span>, and <span class="texhtml"><i>Q</i><sub>free</sub></span> is the free charge contained in <span class="texhtml mvar" style="font-style:italic;">V</span>. The flux <span class="texhtml">Φ<sub><i>D</i></sub></span> is defined analogously to the flux <span class="texhtml">Φ<sub><i>E</i></sub></span> of the electric field <span class="texhtml"><b>E</b></span> through <span class="texhtml mvar" style="font-style:italic;">S</span>: </p> <dl><dd><span class="nowrap mw-no-invert"><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 _{D}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{D}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2082046ceec613cfec93d2b26571554a0cb70be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.724ex; height:2.509ex;" alt="{\displaystyle \Phi _{D}=}"></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><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 {\scriptstyle _{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle _{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fda1e0b515c1a35897ad0641325d79e36bc106e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.025ex; height:1.509ex;" alt="{\displaystyle {\scriptstyle _{S}}}"></span></span> <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} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca869746e78992715287c0c859b57fc929448e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.041ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} \cdot \mathrm {d} \mathbf {A} }"></span></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Differential_form_2">Differential form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=10" title="Edit section: Differential form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The differential form of Gauss's law, involving free charge only, states: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38e53b4915293cb484dfe937253af8a2cdaa29c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.914ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}"></span> </p><p>where <span class="texhtml">∇ · <b>D</b></span> is the <a href="/wiki/Divergence" title="Divergence">divergence</a> of the electric displacement field, and <span class="texhtml"><i>ρ</i><sub>free</sub></span> is the free electric charge density. </p> <div class="mw-heading mw-heading2"><h2 id="Equivalence_of_total_and_free_charge_statements">Equivalence of total and free charge statements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=11" title="Edit section: Equivalence of total and free charge statements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge.</strong> <p>In this proof, we will show that the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\dfrac {\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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mi>ρ</mi> <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 \nabla \cdot \mathbf {E} ={\dfrac {\rho }{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/067ef048abab7a4682212025d13a72f394d91a7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.444ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\dfrac {\rho }{\varepsilon _{0}}}}"></span> is equivalent to the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38e53b4915293cb484dfe937253af8a2cdaa29c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.914ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {D} =\rho _{\mathrm {free} }}"></span> Note that we are only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem. </p><p>We introduce the <a href="/wiki/Polarization_density" title="Polarization density">polarization density</a> <span class="texhtml"><b>P</b></span>, which has the following relation to <span class="texhtml"><b>E</b></span> and <span class="texhtml"><b>D</b></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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-display 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> and the following relation to the bound charge: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {bound} }=-\nabla \cdot \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </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 _{\mathrm {bound} }=-\nabla \cdot \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ebfaf98208cc3d61d62cccef716f1b9f84d5a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.26ex; height:2.676ex;" alt="{\displaystyle \rho _{\mathrm {bound} }=-\nabla \cdot \mathbf {P} }"></span> Now, consider the three equations: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\rho _{\mathrm {bound} }&=\nabla \cdot (-\mathbf {P} )\\\rho _{\mathrm {free} }&=\nabla \cdot \mathbf {D} \\\rho &=\nabla \cdot (\varepsilon _{0}\mathbf {E} )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>ρ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\rho _{\mathrm {bound} }&=\nabla \cdot (-\mathbf {P} )\\\rho _{\mathrm {free} }&=\nabla \cdot \mathbf {D} \\\rho &=\nabla \cdot (\varepsilon _{0}\mathbf {E} )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2592c1c7e4e67e1a6cc63a85998f1044e4b94701" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.081ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\rho _{\mathrm {bound} }&=\nabla \cdot (-\mathbf {P} )\\\rho _{\mathrm {free} }&=\nabla \cdot \mathbf {D} \\\rho &=\nabla \cdot (\varepsilon _{0}\mathbf {E} )\end{aligned}}}"></span> The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove. </p> </div> <div class="mw-heading mw-heading2"><h2 id="Equation_for_linear_materials">Equation for linear materials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=12" title="Edit section: Equation for linear materials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Homogeneous" class="mw-redirect" title="Homogeneous">homogeneous</a>, <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a>, <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">nondispersive</a>, linear materials, there is a simple relationship between <span class="texhtml"><b>E</b></span> and <span class="texhtml"><b>D</b></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9223a64cdd0d289d8864389aa20b5b318f65b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.989ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} =\varepsilon \mathbf {E} }"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">ε</span> is the <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> of the material. For the case of <a href="/wiki/Vacuum" title="Vacuum">vacuum</a> (aka <a href="/wiki/Free_space" class="mw-redirect" title="Free space">free space</a>), <span class="texhtml"><i>ε</i> = <i>ε</i><sub>0</sub></span>. Under these circumstances, Gauss's law modifies to </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}={\frac {Q_{\mathrm {free} }}{\varepsilon }}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}={\frac {Q_{\mathrm {free} }}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a3d89345b85bf7c976b741bb6ec07b1f8dd1bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.887ex; height:5.343ex;" alt="{\displaystyle \Phi _{E}={\frac {Q_{\mathrm {free} }}{\varepsilon }}}"></span> </p><p>for the integral form, and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {free} }}{\varepsilon }}}"> <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> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {free} }}{\varepsilon }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4e761b1d45c475014ebfcd0a9576977f1f5d69b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.457ex; height:4.843ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{\mathrm {free} }}{\varepsilon }}}"></span> </p><p>for the differential form. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_Coulomb's_law"><span id="Relation_to_Coulomb.27s_law"></span>Relation to Coulomb's law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=13" title="Edit section: Relation to Coulomb's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Duplication plainlinks metadata ambox ambox-style" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b><a href="/wiki/Wikipedia:CFORK" class="mw-redirect" title="Wikipedia:CFORK">duplicates</a> the scope of other articles</b>, specifically <a href="/wiki/Coulomb%27s_law#Relation_to_Gauss's_law" title="Coulomb's law">Coulomb's_law#Relation_to_Gauss's_law</a>.<span class="hide-when-compact"> Please <a href="/wiki/Talk:Gauss%27s_law" title="Talk:Gauss's law">discuss this issue</a> and help introduce a <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">summary style</a> to the article.</span></div></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Deriving_Gauss's_law_from_Coulomb's_law"><span id="Deriving_Gauss.27s_law_from_Coulomb.27s_law"></span>Deriving Gauss's law from Coulomb's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=14" title="Edit section: Deriving Gauss's law from Coulomb's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="A reliable source is needed for the entire derivation. (June 2024)">citation needed</span></a></i>]</sup> </p><p>Strictly speaking, Gauss's law cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual, <a href="/wiki/Electrostatic_charge" class="mw-redirect" title="Electrostatic charge">electrostatic</a> <a href="/wiki/Point_charge" class="mw-redirect" title="Point charge">point charge</a> only. However, Gauss's law <i>can</i> be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space). </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Outline of proof</strong> <p>Coulomb's law states that the electric field due to a stationary <a href="/wiki/Point_charge" class="mw-redirect" title="Point charge">point charge</a> is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{r}}{r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591e738bd4410a19356d28966067f9717826eae9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.27ex; height:5.343ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{r}}{r^{2}}}}"></span> where </p> <ul><li><span class="texhtml"><b>e</b><sub><i>r</i></sub></span> is the radial <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">r</span> is the radius, <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><b>r</b></span>|</span>,</li> <li><span class="texhtml"><i>ε</i><sub>0</sub></span> is the <a href="/wiki/Electric_constant" class="mw-redirect" title="Electric constant">electric constant</a>,</li> <li><span class="texhtml mvar" style="font-style:italic;">q</span> is the charge of the particle, which is assumed to be located at the <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>.</li></ul> <p>Using the expression from Coulomb's law, we get the total field at <span class="texhtml"><b>r</b></span> by using an integral to sum the field at <span class="texhtml"><b>r</b></span> due to the infinitesimal charge at each other point <span class="texhtml"><b>s</b></span> in space, to give <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {s} )(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}\,\mathrm {d} ^{3}\mathbf {s} }"> <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> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {s} )(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}\,\mathrm {d} ^{3}\mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1fd4c4bcdc68c48ab32af6270e61f068df85f08" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.703ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {s} )(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}\,\mathrm {d} ^{3}\mathbf {s} }"></span> where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the charge density. If we take the divergence of both sides of this equation with respect to <b>r</b>, and use the known theorem<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\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"> <mi mathvariant="bold">r</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>δ</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 \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cacd2f858ab667d34da95175871677111aad6e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.135ex; height:7.509ex;" alt="{\displaystyle \nabla \cdot \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )}"></span> where <span class="texhtml"><i>δ</i>(<b>r</b>)</span> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>, the result is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\int \rho (\mathbf {s} )\,\delta (\mathbf {r} -\mathbf {s} )\,\mathrm {d} ^{3}\mathbf {s} }"> <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 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> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\int \rho (\mathbf {s} )\,\delta (\mathbf {r} -\mathbf {s} )\,\mathrm {d} ^{3}\mathbf {s} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7788a7bc57535ac97de077540902092ebd5c6194" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.423ex; height:5.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\int \rho (\mathbf {s} )\,\delta (\mathbf {r} -\mathbf {s} )\,\mathrm {d} ^{3}\mathbf {s} }"></span> </p><p>Using the "<a href="/wiki/Dirac_delta_function#Translation" title="Dirac delta function">sifting property</a>" of the Dirac delta function, we arrive at <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\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 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> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon _{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245871d79bc0dbe6c83e0576d1bfe9840967bd15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.978ex; height:6.009ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon _{0}}},}"></span> which is the differential form of Gauss's law, as desired. </p> </div> <p>Since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and, in this respect, Gauss's law is more general than Coulomb's law. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof (without Dirac Delta)</strong> <p>Let <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 \Omega \subseteq R^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>⊆</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \subseteq R^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c40d4f96ab0cdb82d2cd730db7a1308b34e9af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.595ex; height:2.843ex;" alt="{\displaystyle \Omega \subseteq R^{3}}"></span> be a bounded open set, and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{3}}}\mathrm {d} \mathbf {r} '\equiv {\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <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>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{3}}}\mathrm {d} \mathbf {r} '\equiv {\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1520ecf59d527f419c4cf695f6a9a3e4efbcd706" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.639ex; height:6.676ex;" alt="{\displaystyle \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{3}}}\mathrm {d} \mathbf {r} '\equiv {\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"></span> be the electric field, with <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 (\mathbf {r} ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8deb7dd1dbdf265c780b9e2692b51b3d64d32e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.798ex; height:3.009ex;" alt="{\displaystyle \rho (\mathbf {r} ')}"></span> a continuous function (density of charge). </p><p>It is true for all <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} \neq \mathbf {r'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>≠</mo> <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} \neq \mathbf {r'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d91af77c47d8ab2d38def0648a45cfd28d412d9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.987ex; height:3.009ex;" alt="{\displaystyle \mathbf {r} \neq \mathbf {r'} }"></span> that <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 _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <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> <mo mathvariant="bold">,</mo> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed18b6364d4c255c470494c4c82b396ef2a393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.94ex; height:3.009ex;" alt="{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}"></span>. </p><p>Consider now a compact set <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 V\subseteq R^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>⊆</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\subseteq R^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31afe8ee45173d163446203611296f8a3f8b2bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.704ex; height:2.843ex;" alt="{\displaystyle V\subseteq R^{3}}"></span> having a <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> <a href="/wiki/Smooth_surface" class="mw-redirect" title="Smooth surface">smooth boundary</a> <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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span> such that <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 \Omega \cap V=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>∩</mo> <mi>V</mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \cap V=\emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca6de290005e2864ead4f34ffaa7918cc872279" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle \Omega \cap V=\emptyset }"></span>. It follows that <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(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo>×</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49280902b27748b230ec3693a895abb2cdca028c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.718ex; height:3.176ex;" alt="{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}"></span> and so, for the divergence theorem: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\partial V}\mathbf {E} _{0}\cdot d\mathbf {S} =\int _{V}\mathbf {\nabla } \cdot \mathbf {E} _{0}\,dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\partial V}\mathbf {E} _{0}\cdot d\mathbf {S} =\int _{V}\mathbf {\nabla } \cdot \mathbf {E} _{0}\,dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e9925a5c55d55a86706c96d729af57b8e105f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.39ex; height:5.676ex;" alt="{\displaystyle \oint _{\partial V}\mathbf {E} _{0}\cdot d\mathbf {S} =\int _{V}\mathbf {\nabla } \cdot \mathbf {E} _{0}\,dV}"></span> </p><p>But because <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(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo>×</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49280902b27748b230ec3693a895abb2cdca028c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.718ex; height:3.176ex;" alt="{\displaystyle e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )}"></span>, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla _{\mathbf {r} }\cdot e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <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> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla _{\mathbf {r} }\cdot e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398c3a5db45f7891038859bff03b737ebf330ed3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.268ex; height:5.676ex;" alt="{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla _{\mathbf {r} }\cdot e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}=0}"></span> for the argument above (<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 \Omega \cap V=\emptyset \implies \forall \mathbf {r} \in V\ \ \forall \mathbf {r'} \in \Omega \ \ \ \mathbf {r} \neq \mathbf {r'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>∩</mo> <mi>V</mi> <mo>=</mo> <mi mathvariant="normal">∅</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>∈</mo> <mi>V</mi> <mtext> </mtext> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \cap V=\emptyset \implies \forall \mathbf {r} \in V\ \ \forall \mathbf {r'} \in \Omega \ \ \ \mathbf {r} \neq \mathbf {r'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a44f530302f8c5e579c306beb137f8369a233ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.206ex; height:3.009ex;" alt="{\displaystyle \Omega \cap V=\emptyset \implies \forall \mathbf {r} \in V\ \ \forall \mathbf {r'} \in \Omega \ \ \ \mathbf {r} \neq \mathbf {r'} }"></span> and then <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 _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <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> <mo mathvariant="bold">,</mo> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed18b6364d4c255c470494c4c82b396ef2a393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.94ex; height:3.009ex;" alt="{\displaystyle \nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0}"></span>) </p><p>Therefore the flux through a closed surface generated by some charge density outside (the surface) is null. </p><p>Now consider <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} _{0}\in \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{0}\in \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cf490d82991fd700a09446d08bf0d4481039b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.675ex; height:2.509ex;" alt="{\displaystyle \mathbf {r} _{0}\in \Omega }"></span>, and <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 B_{R}(\mathbf {r} _{0})\subseteq \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{R}(\mathbf {r} _{0})\subseteq \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bc652b3d87c765677a02819ad4dba6a2c25c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.986ex; height:2.843ex;" alt="{\displaystyle B_{R}(\mathbf {r} _{0})\subseteq \Omega }"></span> as the sphere centered in <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} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe39f0fedae3334af5c4ffaedf25c9778363400" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{0}}"></span> having <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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> as radius (it exists because <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 \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> is an open set). </p><p>Let <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} _{B_{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{B_{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c5e9d2109f7a9290aef5882a494fd1a5e12f70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.414ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{B_{R}}}"></span> and <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} _{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261a26f867da18635e266b36c25272f83a673e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.238ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} _{C}}"></span> be the electric field created inside and outside the sphere respectively. Then, </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} _{B_{R}}={\frac {1}{4\pi \varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> <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"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mi>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{B_{R}}={\frac {1}{4\pi \varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f53e5a2ca73803d6000978d57f5df3d755dd5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.584ex; height:6.009ex;" alt="{\displaystyle \mathbf {E} _{B_{R}}={\frac {1}{4\pi \varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"></span>, <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} _{C}={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega \setminus B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <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> <mo class="MJX-variant">∖</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mi>e</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> <mo mathvariant="bold">,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{C}={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega \setminus B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a75c8e43cea9620481a372adabffa2a6ce8cdaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.417ex; height:6.009ex;" alt="{\displaystyle \mathbf {E} _{C}={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega \setminus B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}}"></span> and <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} _{B_{R}}+\mathbf {E} _{C}=\mathbf {E} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{B_{R}}+\mathbf {E} _{C}=\mathbf {E} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e65e84f09150cd736739bcba599d44aa50e4a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.402ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} _{B_{R}}+\mathbf {E} _{C}=\mathbf {E} _{0}}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (R)=\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{0}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} +\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{C}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>+</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (R)=\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{0}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} +\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{C}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04659e9e229c55ee7969456f5fefa07741d3def" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:82.418ex; height:6.009ex;" alt="{\displaystyle \Phi (R)=\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{0}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} +\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{C}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} }"></span> </p><p>The last equality follows by observing that <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 (\Omega \setminus B_{R}(\mathbf {r} _{0}))\cap B_{R}(\mathbf {r} _{0})=\emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo class="MJX-variant">∖</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∩</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Omega \setminus B_{R}(\mathbf {r} _{0}))\cap B_{R}(\mathbf {r} _{0})=\emptyset }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d90d56f2d19ad4f1ab049c99481661c0e5f393" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.944ex; height:2.843ex;" alt="{\displaystyle (\Omega \setminus B_{R}(\mathbf {r} _{0}))\cap B_{R}(\mathbf {r} _{0})=\emptyset }"></span>, and the argument above. </p><p>The RHS is the electric flux generated by a charged sphere, and so: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (R)={\frac {Q(R)}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}\rho (\mathbf {r} '){\mathrm {d} \mathbf {r} '}={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} '_{c})|B_{R}(\mathbf {r} _{0})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </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"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msub> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </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> <mi>ρ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (R)={\frac {Q(R)}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}\rho (\mathbf {r} '){\mathrm {d} \mathbf {r} '}={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} '_{c})|B_{R}(\mathbf {r} _{0})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b26f7ad2515e303e8f6fb02a62a27a3af82597" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.793ex; height:6.509ex;" alt="{\displaystyle \Phi (R)={\frac {Q(R)}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}\rho (\mathbf {r} '){\mathrm {d} \mathbf {r} '}={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} '_{c})|B_{R}(\mathbf {r} _{0})|}"></span> with <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'_{c}\in \ B_{R}(\mathbf {r} _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mo>′</mo> </msubsup> <mo>∈</mo> <mtext> </mtext> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r'_{c}\in \ B_{R}(\mathbf {r} _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3bc4f92db23d894cb58da76e66dc68c9078688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.623ex; height:2.843ex;" alt="{\displaystyle r'_{c}\in \ B_{R}(\mathbf {r} _{0})}"></span> </p><p>Where the last equality follows by the mean value theorem for integrals. Using the <a href="/wiki/Squeeze_theorem" title="Squeeze theorem">squeeze theorem</a> and the continuity of <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, one arrives at: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} _{0})=\lim _{R\to 0}{\frac {1}{|B_{R}(\mathbf {r} _{0})|}}\Phi (R)={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} _{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} _{0})=\lim _{R\to 0}{\frac {1}{|B_{R}(\mathbf {r} _{0})|}}\Phi (R)={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} _{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c74c498fb40f3394c68262cc6dff0a2fa1ba7da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.42ex; height:6.009ex;" alt="{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} _{0})=\lim _{R\to 0}{\frac {1}{|B_{R}(\mathbf {r} _{0})|}}\Phi (R)={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} _{0})}"></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Deriving_Coulomb's_law_from_Gauss's_law"><span id="Deriving_Coulomb.27s_law_from_Gauss.27s_law"></span>Deriving Coulomb's law from Gauss's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=15" title="Edit section: Deriving Coulomb's law from Gauss's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of <span class="texhtml"><b>E</b></span> (see <a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a> and <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law</a>). However, Coulomb's law <i>can</i> be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion). </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Outline of proof</strong> <p>Taking <span class="texhtml mvar" style="font-style:italic;">S</span> in the integral form of Gauss's law to be a spherical surface of radius <span class="texhtml mvar" style="font-style:italic;">r</span>, centered at the point charge <span class="texhtml mvar" style="font-style:italic;">Q</span>, we have </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\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> <mi>d</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359332f2f38358086aa7c48ef3b2ae228d2e11b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.716ex; height:5.843ex;" alt="{\displaystyle \oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}}"></span> </p><p>By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi r^{2}{\hat {\mathbf {r} }}\cdot \mathbf {E} (\mathbf {r} )={\frac {Q}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> <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> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}{\hat {\mathbf {r} }}\cdot \mathbf {E} (\mathbf {r} )={\frac {Q}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6e91a5a4757872d6340c83ea53564889182df8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.18ex; height:5.676ex;" alt="{\displaystyle 4\pi r^{2}{\hat {\mathbf {r} }}\cdot \mathbf {E} (\mathbf {r} )={\frac {Q}{\varepsilon _{0}}}}"></span> where <span class="texhtml"><b>r̂</b></span> is a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> pointing radially away from the charge. Again by spherical symmetry, <span class="texhtml"><b>E</b></span> points in the radial direction, and so we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" 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"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </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/e4f8185e332ed1c89ad9475f8b70fa430a8b3c45" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.174ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r} }}{r^{2}}}}"></span> which is essentially equivalent to Coulomb's law. Thus the <a href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square law</a> dependence of the electric field in Coulomb's law follows from Gauss's law. </p> </div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Method_of_image_charges" title="Method of image charges">Method of image charges</a></li> <li><a href="/wiki/Uniqueness_theorem_for_Poisson%27s_equation" title="Uniqueness theorem for Poisson's equation">Uniqueness theorem for Poisson's equation</a></li> <li><a href="/wiki/List_of_examples_of_Stigler%27s_law" title="List of examples of Stigler's law">List of examples of Stigler's law</a></li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=17" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">The other three of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> are: <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a>, <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, and <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's law with Maxwell's correction</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">More specifically, the infinitesimal area is thought of as <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">planar</a> and with area <span class="texhtml">d<i>N</i></span>. The vector <span class="texhtml">d<b>R</b></span> is <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> to this area element and has <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a> <span class="texhtml">d<i>A</i></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=18" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDuhem1891" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem, Pierre</a> (1891). <a rel="nofollow" class="external text" href="https://archive.org/stream/leonssurllec01duheuoft#page/22/mode/2up">"4"</a>. <i>Leçons sur l'électricité et le magnétisme</i> [<i>Lessons on electricity and magnetism</i>] (in French). Vol. 1. Paris Gauthier-Villars. pp. 22–23. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1048238688">1048238688</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL23310906M">23310906M</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=4&rft.btitle=Le%C3%A7ons+sur+l%27%C3%A9lectricit%C3%A9+et+le+magn%C3%A9tisme&rft.pages=22-23&rft.pub=Paris+Gauthier-Villars&rft.date=1891&rft_id=info%3Aoclcnum%2F1048238688&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL23310906M%23id-name%3DOL&rft.aulast=Duhem&rft.aufirst=Pierre&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fleonssurllec01duheuoft%23page%2F22%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span> Shows that Lagrange has priority over Gauss. Others after Gauss discovered "Gauss's Law", too.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagrange1869" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange, Joseph-Louis</a> (1869) [1776]. <a href="/wiki/Joseph-Alfred_Serret" title="Joseph-Alfred Serret">Serret, Joseph-Alfred</a>; <a href="/wiki/Jean-Gaston_Darboux" class="mw-redirect" title="Jean-Gaston Darboux">Darboux, Jean-Gaston</a> (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4XkAAAAAMAAJ&pg=PA619">"Sur l'attraction des sphéroïdes elliptiques"</a> [On the attraction of elliptical spheroids]. <i>Œuvres de Lagrange: Mémoires extraits des recueils de l'Académie royale des sciences et belles-lettres de Berlin</i> (in French). Gauthier-Villars: 619.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=%C5%92uvres+de+Lagrange%3A+M%C3%A9moires+extraits+des+recueils+de+l%27Acad%C3%A9mie+royale+des+sciences+et+belles-lettres+de+Berlin&rft.atitle=Sur+l%27attraction+des+sph%C3%A9ro%C3%AFdes+elliptiques&rft.pages=619&rft.date=1869&rft.aulast=Lagrange&rft.aufirst=Joseph-Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D4XkAAAAAMAAJ%26pg%3DPA619&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1877" class="citation book cs1 cs1-prop-foreign-lang-source cs1-prop-foreign-lang-source"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1877). "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata" [The theory of the attraction of homogeneous spheroidal elliptic bodies treated by a new method]. In <a href="/wiki/Ernst_Christian_Julius_Schering" title="Ernst Christian Julius Schering">Schering, Ernst Christian Julius</a>; <a href="/wiki/Martin_Brendel" title="Martin Brendel">Brendel, Martin</a> (eds.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0TxeAAAAcAAJ&pg=PA3"><i>Carl Friedrich Gauss Werke</i></a> [<i>Works of Carl Friedrich Gauss</i>] (in Latin and German). Vol. 5 (2nd ed.). Gedruckt in der Dieterichschen Universitätsdruckerei (W.F. Kaestner). pp. 2–22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theoria+attractionis+corporum+sphaeroidicorum+ellipticorum+homogeneorum+methodo+nova+tractata&rft.btitle=Carl+Friedrich+Gauss+Werke&rft.pages=2-22&rft.edition=2nd&rft.pub=Gedruckt+in+der+Dieterichschen+Universit%C3%A4tsdruckerei+%28W.F.+Kaestner%29&rft.date=1877&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0TxeAAAAcAAJ%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span> Gauss mentions <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>'s <i><a href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</a></i> <a rel="nofollow" class="external text" href="https://archive.org/stream/newtonspmathema00newtrich#page/n243/mode/2up">proposition XCI</a> regarding finding the force exerted by a sphere on a point anywhere along an axis passing through the sphere.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallidayResnick1970" class="citation book cs1">Halliday, David; Resnick, Robert (1970). <i>Fundamentals of Physics</i>. John Wiley & Sons. pp. 452–453.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Physics&rft.pages=452-453&rft.pub=John+Wiley+%26+Sons&rft.date=1970&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerway1996" class="citation book cs1">Serway, Raymond A. (1996). <i>Physics for Scientists and Engineers with Modern Physics</i> (4th ed.). p. 687.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers+with+Modern+Physics&rft.pages=687&rft.edition=4th&rft.date=1996&rft.aulast=Serway&rft.aufirst=Raymond+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-GrantPhillips-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-GrantPhillips_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GrantPhillips_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrantPhillips2008" class="citation book cs1">Grant, I. S.; Phillips, W. R. (2008). <i>Electromagnetism</i>. Manchester Physics (2nd ed.). John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92712-9" title="Special:BookSources/978-0-471-92712-9"><bdi>978-0-471-92712-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetism&rft.series=Manchester+Physics&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-0-471-92712-9&rft.aulast=Grant&rft.aufirst=I.+S.&rft.au=Phillips%2C+W.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthews1998" class="citation book cs1">Matthews, Paul (1998). <i>Vector Calculus</i>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-76180-2" title="Special:BookSources/3-540-76180-2"><bdi>3-540-76180-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Calculus&rft.pub=Springer&rft.date=1998&rft.isbn=3-540-76180-2&rft.aulast=Matthews&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFedosin2019" class="citation journal cs1">Fedosin, Sergey G. (2019). <a rel="nofollow" class="external text" href="https://rdcu.be/ccV9o">"On the Covariant Representation of Integral Equations of the Electromagnetic Field"</a>. <i>Progress in Electromagnetics Research C</i>. <b>96</b>: 109–122. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1911.11138">1911.11138</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019arXiv191111138F">2019arXiv191111138F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2528%2FPIERC19062902">10.2528/PIERC19062902</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:208095922">208095922</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+in+Electromagnetics+Research+C&rft.atitle=On+the+Covariant+Representation+of+Integral+Equations+of+the+Electromagnetic+Field&rft.volume=96&rft.pages=109-122&rft.date=2019&rft_id=info%3Aarxiv%2F1911.11138&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A208095922%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2528%2FPIERC19062902&rft_id=info%3Abibcode%2F2019arXiv191111138F&rft.aulast=Fedosin&rft.aufirst=Sergey+G.&rft_id=https%3A%2F%2Frdcu.be%2FccV9o&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">See, for example, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths2013" class="citation book cs1">Griffiths, David J. (2013). <i>Introduction to Electrodynamics</i> (4th ed.). Prentice Hall. p. 50.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.pages=50&rft.edition=4th&rft.pub=Prentice+Hall&rft.date=2013&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1999" class="citation book cs1">Jackson, John David (1999). <i>Classical Electrodynamics</i> (3rd ed.). John Wiley & Sons. p. 35.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.pages=35&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1999&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=19" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1867" class="citation book cs1">Gauss, Carl Friedrich (1867). <i>Werke Band 5</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Werke+Band+5&rft.date=1867&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span> <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?PPN236006339">Digital version</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1998" class="citation book cs1">Jackson, John David (1998). <i>Classical Electrodynamics</i> (3rd ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-30932-X" title="Special:BookSources/0-471-30932-X"><bdi>0-471-30932-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.place=New+York&rft.edition=3rd&rft.pub=Wiley&rft.date=1998&rft.isbn=0-471-30932-X&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGauss%27s+law" class="Z3988"></span> David J. Griffiths (6th ed.)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Gauss%27s_law&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Gauss%27_Law" class="extiw" title="commons:Category:Gauss' Law">Gauss' Law</a> at Wikimedia Commons</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080628181946/http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoAndCaptions/">MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism</a> Taught by Professor <a href="/wiki/Walter_Lewin" title="Walter Lewin">Walter Lewin</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html#Section10.6">section on Gauss's law in an online textbook</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100527194640/http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html#Section10.6">Archived</a> 2010-05-27 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.physnet.org/modules/pdf_modules/m132.pdf"><small>MISN-0-132</small> <i>Gauss's Law for Spherical Symmetry</i></a> (<a href="/wiki/Portable_Document_Format" class="mw-redirect" title="Portable Document Format">PDF file</a>) by Peter Signell for <a rel="nofollow" class="external text" href="http://www.physnet.org">Project PHYSNET</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.physnet.org/modules/pdf_modules/m133.pdf"><small>MISN-0-133</small> <i>Gauss's Law Applied to Cylindrical and Planar Charge Distributions</i></a> (PDF file) by Peter Signell for <a rel="nofollow" class="external text" href="http://www.physnet.org">Project PHYSNET</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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Gauss's law - Wikipedia