Equacions de Maxwell - Viquipèdia, l'enciclopèdia lliure

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class="vector-toc-list"> </ul> </li> <li id="toc-Llei_de_Faraday" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Llei_de_Faraday"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Llei de Faraday</span> </div> </a> <ul id="toc-Llei_de_Faraday-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formulació" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulació"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formulació</span> </div> </a> <ul id="toc-Formulació-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretació_física_de_les_equacions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Interpretació_física_de_les_equacions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Interpretació física de les equacions</span> </div> </a> <button aria-controls="toc-Interpretació_física_de_les_equacions-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>Commuta la subsecció Interpretació física de les equacions</span> </button> <ul id="toc-Interpretació_física_de_les_equacions-sublist" class="vector-toc-list"> <li id="toc-Conservació_de_la_càrrega" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Conservació_de_la_càrrega"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Conservació de la càrrega</span> </div> </a> <ul id="toc-Conservació_de_la_càrrega-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Força_de_Lorentz" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Força_de_Lorentz"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Força de Lorentz</span> </div> </a> <ul id="toc-Força_de_Lorentz-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equacions_de_Maxwell_en_el_buit" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equacions_de_Maxwell_en_el_buit"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Equacions de Maxwell en el buit</span> </div> </a> <ul id="toc-Equacions_de_Maxwell_en_el_buit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Les_equacions_de_Maxwell_en_relativitat_especial" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Les_equacions_de_Maxwell_en_relativitat_especial"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Les equacions de Maxwell en relativitat especial</span> </div> </a> <ul id="toc-Les_equacions_de_Maxwell_en_relativitat_especial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Equacions de Maxwell</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="Vés a un article en una altra llengua. Disponible en 77 llengües" > <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-77" 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">77 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Maxwell_se_vergelykings" title="Maxwell se vergelykings - afrikaans" lang="af" hreflang="af" data-title="Maxwell se vergelykings" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Maxwell-Gleichungen" title="Maxwell-Gleichungen - alemany suís" lang="gsw" hreflang="gsw" data-title="Maxwell-Gleichungen" data-language-autonym="Alemannisch" data-language-local-name="alemany suís" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%85%D8%A7%D9%83%D8%B3%D9%88%D9%8A%D9%84" title="معادلات ماكسويل - àrab" lang="ar" hreflang="ar" data-title="معادلات ماكسويل" data-language-autonym="العربية" data-language-local-name="àrab" 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/Ecuaciones_de_Maxwell" title="Ecuaciones de Maxwell - asturià" lang="ast" hreflang="ast" data-title="Ecuaciones de Maxwell" data-language-autonym="Asturianu" data-language-local-name="asturià" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Maksvell_t%C9%99nlikl%C9%99ri" title="Maksvell tənlikləri - azerbaidjanès" lang="az" hreflang="az" data-title="Maksvell tənlikləri" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaidjanès" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%A7%DA%A9%D8%B3%D9%88%D9%84_%D9%85%D9%88%D8%B9%D8%A7%D8%AF%DB%8C%D9%84%D9%87%E2%80%8C%D9%84%D8%B1%DB%8C" title="ماکسول موعادیله‌لری - South Azerbaijani" lang="azb" hreflang="azb" data-title="ماکسول موعادیله‌لری" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-be badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><a href="https://be.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D1%9E%D0%BD%D0%B5%D0%BD%D0%BD%D1%96_%D0%9C%D0%B0%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%B0" title="Ураўненні Максвела - belarús" lang="be" hreflang="be" data-title="Ураўненні Максвела" data-language-autonym="Беларуская" data-language-local-name="belarús" 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%A0%D0%B0%D1%9E%D0%BD%D0%B0%D0%BD%D1%8C%D0%BD%D1%96_%D0%9C%D0%B0%D0%BA%D1%81%D1%9E%D1%8D%D0%BB%D0%B0" title="Раўнаньні Максўэла - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Раўнаньні Максўэла" 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%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%9C%D0%B0%D0%BA%D1%81%D1%83%D0%B5%D0%BB" title="Уравнения на Максуел - búlgar" lang="bg" hreflang="bg" data-title="Уравнения на Максуел" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%95%E0%A7%8D%E0%A6%B8%E0%A6%93%E0%A6%AF%E0%A6%BC%E0%A7%87%E0%A6%B2%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3%E0%A6%B8%E0%A6%AE%E0%A7%82%E0%A6%B9" title="ম্যাক্সওয়েলের সমীকরণসমূহ - bengalí" lang="bn" hreflang="bn" data-title="ম্যাক্সওয়েলের সমীকরণসমূহ" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Maxwellove_jedna%C4%8Dine" title="Maxwellove jednačine - bosnià" lang="bs" hreflang="bs" data-title="Maxwellove jednačine" data-language-autonym="Bosanski" data-language-local-name="bosnià" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Maxwellovy_rovnice" title="Maxwellovy rovnice - txec" lang="cs" hreflang="cs" data-title="Maxwellovy rovnice" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9Ca%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%BB_%D1%82%D0%B0%D0%BD%D0%BB%C4%83%D1%85%C4%95%D1%81%D0%B5%D0%BC" title="Мaксвелл танлăхĕсем - txuvaix" lang="cv" hreflang="cv" data-title="Мaксвелл танлăхĕсем" data-language-autonym="Чӑвашла" data-language-local-name="txuvaix" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Maxwells_ligninger" title="Maxwells ligninger - danès" lang="da" hreflang="da" data-title="Maxwells ligninger" data-language-autonym="Dansk" data-language-local-name="danès" 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/Maxwell-Gleichungen" title="Maxwell-Gleichungen - alemany" lang="de" hreflang="de" data-title="Maxwell-Gleichungen" data-language-autonym="Deutsch" data-language-local-name="alemany" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BE%CE%B9%CF%83%CF%8E%CF%83%CE%B5%CE%B9%CF%82_%CE%9C%CE%AC%CE%BE%CE%B3%CE%BF%CF%85%CE%B5%CE%BB" title="Εξισώσεις Μάξγουελ - grec" lang="el" hreflang="el" data-title="Εξισώσεις Μάξγουελ" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Maxwell%27s_equations" title="Maxwell's equations - anglès" lang="en" hreflang="en" data-title="Maxwell's equations" data-language-autonym="English" data-language-local-name="anglès" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ekvacioj_de_Maxwell" title="Ekvacioj de Maxwell - esperanto" lang="eo" hreflang="eo" data-title="Ekvacioj de Maxwell" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es badge-Q17437798 badge-goodarticle mw-list-item" title="article bo"><a href="https://es.wikipedia.org/wiki/Ecuaciones_de_Maxwell" title="Ecuaciones de Maxwell - espanyol" lang="es" hreflang="es" data-title="Ecuaciones de Maxwell" data-language-autonym="Español" data-language-local-name="espanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Maxwelli_v%C3%B5rrandid" title="Maxwelli võrrandid - estonià" lang="et" hreflang="et" data-title="Maxwelli võrrandid" data-language-autonym="Eesti" data-language-local-name="estonià" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Maxwellen_ekuazioak" title="Maxwellen ekuazioak - basc" lang="eu" hreflang="eu" data-title="Maxwellen ekuazioak" data-language-autonym="Euskara" data-language-local-name="basc" 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%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%85%D8%A7%DA%A9%D8%B3%D9%88%D9%84" title="معادلات ماکسول - persa" lang="fa" hreflang="fa" data-title="معادلات ماکسول" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Maxwellin_yht%C3%A4l%C3%B6t" title="Maxwellin yhtälöt - finès" lang="fi" hreflang="fi" data-title="Maxwellin yhtälöt" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quations_de_Maxwell" title="Équations de Maxwell - francès" lang="fr" hreflang="fr" data-title="Équations de Maxwell" data-language-autonym="Français" data-language-local-name="francès" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3ns_de_Maxwell" title="Ecuacións de Maxwell - gallec" lang="gl" hreflang="gl" data-title="Ecuacións de Maxwell" data-language-autonym="Galego" data-language-local-name="gallec" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%9E%D7%A7%D7%A1%D7%95%D7%95%D7%9C" title="משוואות מקסוול - hebreu" lang="he" hreflang="he" data-title="משוואות מקסוול" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AE%E0%A5%88%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A4%B5%E0%A5%87%E0%A4%B2_%E0%A4%95%E0%A5%87_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="मैक्सवेल के समीकरण - hindi" lang="hi" hreflang="hi" data-title="मैक्सवेल के समीकरण" 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/Maxwellove_jednad%C5%BEbe" title="Maxwellove jednadžbe - croat" lang="hr" hreflang="hr" data-title="Maxwellove jednadžbe" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Ekwasyon_Maxwell" title="Ekwasyon Maxwell - crioll d’Haití" lang="ht" hreflang="ht" data-title="Ekwasyon Maxwell" data-language-autonym="Kreyòl ayisyen" data-language-local-name="crioll d’Haití" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Maxwell-egyenletek" title="Maxwell-egyenletek - hongarès" lang="hu" hreflang="hu" data-title="Maxwell-egyenletek" data-language-autonym="Magyar" data-language-local-name="hongarès" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D6%84%D5%BD%D5%BE%D5%A5%D5%AC%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4%D5%B6%D5%A5%D6%80" title="Մաքսվելի հավասարումներ - armeni" lang="hy" hreflang="hy" data-title="Մաքսվելի հավասարումներ" data-language-autonym="Հայերեն" data-language-local-name="armeni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Equationes_de_Maxwell" title="Equationes de Maxwell - interlingua" lang="ia" hreflang="ia" data-title="Equationes de Maxwell" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Maxwell" title="Persamaan Maxwell - indonesi" lang="id" hreflang="id" data-title="Persamaan Maxwell" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesi" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/J%C3%B6fnur_Maxwells" title="Jöfnur Maxwells - islandès" lang="is" hreflang="is" data-title="Jöfnur Maxwells" data-language-autonym="Íslenska" data-language-local-name="islandès" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazioni_di_Maxwell" title="Equazioni di Maxwell - italià" lang="it" hreflang="it" data-title="Equazioni di Maxwell" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9E%E3%82%AF%E3%82%B9%E3%82%A6%E3%82%A7%E3%83%AB%E3%81%AE%E6%96%B9%E7%A8%8B%E5%BC%8F" title="マクスウェルの方程式 - japonès" lang="ja" hreflang="ja" data-title="マクスウェルの方程式" data-language-autonym="日本語" data-language-local-name="japonès" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A5%E1%83%A1%E1%83%95%E1%83%94%E1%83%9A%E1%83%98%E1%83%A1_%E1%83%92%E1%83%90%E1%83%9C%E1%83%A2%E1%83%9D%E1%83%9A%E1%83%94%E1%83%91%E1%83%94%E1%83%91%E1%83%98" title="მაქსველის განტოლებები - georgià" lang="ka" hreflang="ka" data-title="მაქსველის განტოლებები" data-language-autonym="ქართული" data-language-local-name="georgià" 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%9C%D0%B0%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%BB_%D1%82%D0%B5%D2%A3%D0%B4%D0%B5%D1%83%D1%96" title="Максвелл теңдеуі - kazakh" lang="kk" hreflang="kk" data-title="Максвелл теңдеуі" data-language-autonym="Қазақша" data-language-local-name="kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%95%E0%B3%8D%E0%B2%B8%E0%B3%8D%E2%80%8C%E0%B2%B5%E0%B3%86%E0%B2%B2%E0%B3%8D%E2%80%8C%E0%B2%A8_%E0%B2%B8%E0%B2%AE%E0%B3%80%E0%B2%95%E0%B2%B0%E0%B2%A3%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಮ್ಯಾಕ್ಸ್‌ವೆಲ್‌ನ ಸಮೀಕರಣಗಳು - kannada" lang="kn" hreflang="kn" data-title="ಮ್ಯಾಕ್ಸ್‌ವೆಲ್‌ನ ಸಮೀಕರಣಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%A5%EC%8A%A4%EC%9B%B0_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="맥스웰 방정식 - coreà" lang="ko" hreflang="ko" data-title="맥스웰 방정식" data-language-autonym="한국어" data-language-local-name="coreà" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Aequationes_Maxwellianae" title="Aequationes Maxwellianae - llatí" lang="la" hreflang="la" data-title="Aequationes Maxwellianae" data-language-autonym="Latina" data-language-local-name="llatí" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/W%C3%A8tte_van_Maxwell" title="Wètte van Maxwell - limburguès" lang="li" hreflang="li" data-title="Wètte van Maxwell" data-language-autonym="Limburgs" data-language-local-name="limburguès" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Maksvelo_lygtys" title="Maksvelo lygtys - lituà" lang="lt" hreflang="lt" data-title="Maksvelo lygtys" data-language-autonym="Lietuvių" data-language-local-name="lituà" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Maksvela_vien%C4%81dojumi" title="Maksvela vienādojumi - letó" lang="lv" hreflang="lv" data-title="Maksvela vienādojumi" data-language-autonym="Latviešu" data-language-local-name="letó" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk badge-Q17437798 badge-goodarticle mw-list-item" title="article bo"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%BE%D0%B2%D0%B8_%D1%80%D0%B0%D0%B2%D0%B5%D0%BD%D0%BA%D0%B8" title="Максвелови равенки - macedoni" lang="mk" hreflang="mk" data-title="Максвелови равенки" data-language-autonym="Македонски" data-language-local-name="macedoni" 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%AE%E0%A5%85%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A4%B5%E0%A5%87%E0%A4%B2%E0%A4%9A%E0%A5%80_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3%E0%A5%87" title="मॅक्सवेलची समीकरणे - marathi" lang="mr" hreflang="mr" data-title="मॅक्सवेलची समीकरणे" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Persamaan_Maxwell" title="Persamaan Maxwell - malai" lang="ms" hreflang="ms" data-title="Persamaan Maxwell" data-language-autonym="Bahasa Melayu" data-language-local-name="malai" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A5%8D%E0%A4%B8%E0%A4%B5%E0%A5%87%E0%A4%B2_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="माक्सवेल समीकरण - nepalès" lang="ne" hreflang="ne" data-title="माक्सवेल समीकरण" data-language-autonym="नेपाली" data-language-local-name="nepalès" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wetten_van_Maxwell" title="Wetten van Maxwell - neerlandès" lang="nl" hreflang="nl" data-title="Wetten van Maxwell" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Maxwells_likningar" title="Maxwells likningar - noruec nynorsk" lang="nn" hreflang="nn" data-title="Maxwells likningar" data-language-autonym="Norsk nynorsk" data-language-local-name="noruec nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Maxwells_likninger" title="Maxwells likninger - noruec bokmål" lang="nb" hreflang="nb" data-title="Maxwells likninger" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%95%E0%A8%B8%E0%A8%B5%E0%A9%88%E0%A9%B1%E0%A8%B2_%E0%A8%A6%E0%A9%80%E0%A8%86%E0%A8%82_%E0%A8%B8%E0%A8%AE%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8%E0%A8%BE%E0%A8%82" title="ਮੈਕਸਵੈੱਲ ਦੀਆਂ ਸਮੀਕਰਨਾਂ - panjabi" lang="pa" hreflang="pa" data-title="ਮੈਕਸਵੈੱਲ ਦੀਆਂ ਸਮੀਕਰਨਾਂ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnania_Maxwella" title="Równania Maxwella - polonès" lang="pl" hreflang="pl" data-title="Równania Maxwella" data-language-autonym="Polski" data-language-local-name="polonès" 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/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell - portuguès" lang="pt" hreflang="pt" data-title="Equações de Maxwell" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Biile_lui_Maxwell" title="Ecuațiile lui Maxwell - romanès" lang="ro" hreflang="ro" data-title="Ecuațiile lui Maxwell" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="article de qualitat"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%9C%D0%B0%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%BB%D0%B0" title="Уравнения Максвелла - rus" lang="ru" hreflang="ru" data-title="Уравнения Максвелла" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Maxwellove_jednad%C5%BEbe" title="Maxwellove jednadžbe - serbocroat" lang="sh" hreflang="sh" data-title="Maxwellove jednadžbe" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Maxwell%27s_equations" title="Maxwell's equations - Simple English" lang="en-simple" hreflang="en-simple" data-title="Maxwell's equations" 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/Maxwellove_rovnice" title="Maxwellove rovnice - eslovac" lang="sk" hreflang="sk" data-title="Maxwellove rovnice" data-language-autonym="Slovenčina" data-language-local-name="eslovac" 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/Maxwellove_ena%C4%8Dbe" title="Maxwellove enačbe - eslovè" lang="sl" hreflang="sl" data-title="Maxwellove enačbe" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Ekuacionet_e_Maksuellit" title="Ekuacionet e Maksuellit - albanès" lang="sq" hreflang="sq" data-title="Ekuacionet e Maksuellit" data-language-autonym="Shqip" data-language-local-name="albanès" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BA%D1%81%D0%B2%D0%B5%D0%BB%D0%BE%D0%B2%D0%B5_%D1%98%D0%B5%D0%B4%D0%BD%D0%B0%D1%87%D0%B8%D0%BD%D0%B5" title="Максвелове једначине - serbi" lang="sr" hreflang="sr" data-title="Максвелове једначине" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Maxwells_ekvationer" title="Maxwells ekvationer - suec" lang="sv" hreflang="sv" data-title="Maxwells ekvationer" data-language-autonym="Svenska" data-language-local-name="suec" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%BE%E0%AE%95%E0%AF%8D%E0%AE%9A%E0%AF%81%E0%AE%B5%E0%AF%86%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="மாக்சுவெல்லின் சமன்பாடுகள் - tàmil" lang="ta" hreflang="ta" data-title="மாக்சுவெல்லின் சமன்பாடுகள்" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%AE%E0%B0%BE%E0%B0%95%E0%B1%8D%E0%B0%B8%E0%B1%8D%E0%B0%B5%E0%B1%86%E0%B0%B2%E0%B1%8D_%E0%B0%B8%E0%B0%AE%E0%B1%80%E0%B0%95%E0%B0%B0%E0%B0%A3%E0%B0%BE%E0%B0%B2%E0%B1%81" title="మాక్స్వెల్ సమీకరణాలు - telugu" lang="te" hreflang="te" data-title="మాక్స్వెల్ సమీకరణాలు" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%81%E0%B8%A1%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B9%80%E0%B8%A7%E0%B8%A5%E0%B8%A5%E0%B9%8C" title="สมการของแมกซ์เวลล์ - tai" lang="th" hreflang="th" data-title="สมการของแมกซ์เวลล์" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Mga_ekwasyon_ni_Maxwell" title="Mga ekwasyon ni Maxwell - tagal" lang="tl" hreflang="tl" data-title="Mga ekwasyon ni Maxwell" data-language-autonym="Tagalog" data-language-local-name="tagal" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Maxwell_denklemleri" title="Maxwell denklemleri - turc" lang="tr" hreflang="tr" 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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r34261971">.mw-parser-output .hatnote{width:100%;border-color:#77ccff;color:var(--color-base,#202122);background-color:#f5f5f5;margin-bottom:1em;font-style:italic}.mw-parser-output .hatnote i{font-style:normal}@media screen{html.skin-theme-clientpref-night .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .hatnote{color:var(--color-inverted,#fff);background-color:var(--background-color-inverted,#101418)}}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <table class="hatnote" cellspacing="5"> <tbody><tr> <td style="width: 25px; vertical-align: top;"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/22px-Disambig_grey.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/33px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/44px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> </td> <td>Aquest article tracta sobre les equacions electromagnètiques. Si cerqueu les equacions termodinàmiques, vegeu «<a href="/wiki/Relacions_de_Maxwell" title="Relacions de Maxwell">Relacions de Maxwell</a>». </td></tr></tbody></table> <table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;color:#202122;background-color:#f8f9fa;border:1px solid #a2a9b1;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%;width:220px"><tbody><tr><th style="padding:0.2em 0.4em 0.2em;font-size:145%;line-height:1.2em">Electromagnetisme</th></tr><tr><td style="padding:0.2em 0 0.4em"><span typeof="mw:File"><a href="/wiki/Electromagnetisme" title="Electromagnetisme"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/250px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="250" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/375px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/500px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td style="padding:0 0.1em 0.4em"> <a href="/wiki/Electricitat" title="Electricitat">Electricitat</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Magnetisme" title="Magnetisme">Magnetisme</a></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left"><a href="/wiki/Electroest%C3%A0tica" title="Electroestàtica">Electroestàtica</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/C%C3%A0rrega_el%C3%A8ctrica" title="Càrrega elèctrica">Càrrega elèctrica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Coulomb" title="Llei de Coulomb">Llei de Coulomb</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Camp_el%C3%A8ctric" title="Camp elèctric">Camp elèctric</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Flux_el%C3%A8ctric" title="Flux elèctric">Flux elèctric</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">Llei de Gauss</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Potencial_el%C3%A8ctric" title="Potencial elèctric">Potencial elèctric</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Inducci%C3%B3_electroest%C3%A0tica" title="Inducció electroestàtica">Inducció electroestàtica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moment_dipolar_el%C3%A8ctric" title="Moment dipolar elèctric">Moment dipolar elèctric</a><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=Densitat_de_polaritzaci%C3%B3&action=edit&redlink=1" class="new" title="Densitat de polarització (encara no existeix)">Densitat de polarització</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left"><a href="/wiki/Magnetoest%C3%A0tica" title="Magnetoestàtica">Magnetoestàtica</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Llei_d%27Amp%C3%A8re" title="Llei d'Ampère">Llei d'Ampère</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Corrent_el%C3%A8ctric" title="Corrent elèctric">Corrent elèctric</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Camp_magn%C3%A8tic" title="Camp magnètic">Camp magnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Magnetitzaci%C3%B3" title="Magnetització">Magnetització</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Flux_magn%C3%A8tic" title="Flux magnètic">Flux magnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Biot-Savart" title="Llei de Biot-Savart">Llei de Biot-Savart</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Moment_magn%C3%A8tic" title="Moment magnètic">Moment magnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Gauss_per_al_magnetisme" title="Llei de Gauss per al magnetisme">Llei de Gauss per al magnetisme</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left"><a href="/wiki/Electrodin%C3%A0mica_cl%C3%A0ssica" title="Electrodinàmica clàssica">Electrodinàmica clàssica</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">Força de Lorentz</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/For%C3%A7a_electromotriu" title="Força electromotriu">Força electromotriu</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Inducci%C3%B3_electromagn%C3%A8tica" title="Inducció electromagnètica">Inducció electromagnètica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Faraday" title="Llei de Faraday">Llei de Faraday</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Llei_de_Lenz" title="Llei de Lenz">Llei de Lenz</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Corrent_de_despla%C3%A7ament" title="Corrent de desplaçament">Corrent de desplaçament</a><span style="font-weight:bold;"> ·</span> <a class="mw-selflink selflink">Equacions de Maxwell</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Camp_electromagn%C3%A8tic" title="Camp electromagnètic">Camp electromagnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Radiaci%C3%B3_electromagn%C3%A8tica" title="Radiació electromagnètica">Radiació electromagnètica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Potencials_de_Li%C3%A9nard%E2%80%93Wiechert" title="Potencials de Liénard–Wiechert">Potencials de Liénard–Wiechert</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Tensor_de_Maxwell" title="Tensor de Maxwell">Tensor de Maxwell</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Corrent_de_Foucault" title="Corrent de Foucault">Corrent de Foucault</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left"><a href="/wiki/Circuit_el%C3%A8ctric" title="Circuit elèctric">Circuit elèctric</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Conducci%C3%B3_el%C3%A8ctrica" title="Conducció elèctrica">Conducció elèctrica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Resist%C3%A8ncia_el%C3%A8ctrica_(propietat)" title="Resistència elèctrica (propietat)">Resistència elèctrica</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Capacit%C3%A0ncia" title="Capacitància">Capacitància</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Induct%C3%A0ncia" title="Inductància">Inductància</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Imped%C3%A0ncia" title="Impedància">Impedància</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Ressonador#Ressonador_electromagnètic" title="Ressonador">Ressonador electromagnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Circuits_en_s%C3%A8rie_i_circuits_en_paral%C2%B7lel" title="Circuits en sèrie i circuits en paral·lel">Circuits en sèrie i en paral·lel</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Guia_d%27ones" title="Guia d'ones">Guia d'ones</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left"><a href="/wiki/Formulaci%C3%B3_covariant_de_l%27electrodin%C3%A0mica_cl%C3%A0ssica" title="Formulació covariant de l'electrodinàmica clàssica">Formulació covariant</a></div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Tensor_electromagn%C3%A8tic" title="Tensor electromagnètic">Tensor electromagnètic</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Tensor_d%27energia-impuls" class="mw-redirect" title="Tensor d'energia-impuls">Tensor d'energia-impuls</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Quadricorrent" title="Quadricorrent">Quadricorrent</a><span style="font-weight:bold;"> ·</span> <a href="/w/index.php?title=Quadripotencial&action=edit&redlink=1" class="new" title="Quadripotencial (encara no existeix)">Quadripotencial</a></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;text-align:left">Científics</div><div class="NavContent" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Carl_Friedrich_Gauss" class="mw-redirect" title="Carl Friedrich Gauss">Gauss</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Heinrich_Hertz" class="mw-redirect" title="Heinrich Hertz">Hertz</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Wilhelm_Eduard_Weber" class="mw-redirect" title="Wilhelm Eduard Weber">Weber</a><span style="font-weight:bold;"> ·</span> <a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></div></div></td> </tr><tr><td style="text-align:right;font-size:115%;padding-top: 0.6em;"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><span typeof="mw:File"><a href="/wiki/Plantilla:Electromagnetisme" title="Plantilla:Electromagnetisme"><img alt="Vegeu aquesta plantilla" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/18px-Commons-emblem-notice.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/27px-Commons-emblem-notice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Commons-emblem-notice.svg/36px-Commons-emblem-notice.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></li></ul></div></td></tr></tbody></table> <p>Les <b>equacions de Maxwell</b> són un conjunt de quatre <a href="/wiki/Equaci%C3%B3" title="Equació">equacions</a> que, afegint-hi la <a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">força de Lorentz</a>, descriuen completament els fenòmens electromagnètics. La gran contribució de <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> fou reunir en aquestes equacions molts anys de resultats experimentals i investigacions teòriques, deguts a <a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a>, <a href="/wiki/Karl_Friedrich_Gauss" class="mw-redirect" title="Karl Friedrich Gauss">Gauss</a>, <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a>, <a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a> i altres, introduint els conceptes de <a href="/wiki/Camp_(F%C3%ADsica)" class="mw-redirect" title="Camp (Física)">camp</a> i de <a href="/wiki/Corrent_de_despla%C3%A7ament" title="Corrent de desplaçament">corrent de desplaçament</a>, i unificant els camps elèctrics i magnètics en un sol concepte: el <a href="/wiki/Camp_electromagn%C3%A8tic" title="Camp electromagnètic">camp electromagnètic</a>. De les equacions de Maxwell, a més, es desprèn l'existència d'<a href="/wiki/Ona_electromagn%C3%A8tica" title="Ona electromagnètica">ones electromagnètiques</a> propagant-se amb velocitat igual al valor de la <a href="/wiki/Velocitat_de_la_llum" title="Velocitat de la llum">velocitat de la llum</a> <i>c</i> en el buit, amb la qual cosa Maxwell va identificar la llum amb una ona electromagnètica, unificant l'<a href="/wiki/%C3%92ptica" title="Òptica">òptica</a> amb l'<a href="/wiki/Electromagnetisme" title="Electromagnetisme">electromagnetisme</a>.<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> </p><p>Quan Maxwell va elaborar la seva teoria de l'electromagnetisme, va proposar no quatre sinó vint equacions, les quals descrivien el comportament dels camps elèctrics i magnètics. En les dues dècades que van seguir a la seva mort, el britànic <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> i l’alemany <a href="/wiki/Heinrich_Rudolf_Hertz" title="Heinrich Rudolf Hertz">Heinrich Hertz</a> van combinar i simplificar les equacions de Maxwell.<sup id="cite_ref-FOOTNOTEFleisch2010viii_2-0" class="reference"><a href="#cite_note-FOOTNOTEFleisch2010viii-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Les lleis no van ser escrites per Maxwell, si més no en la forma vectorial habitual avui dia. Maxwell estava convençut que l'electromagnetisme estaria millor formulat en forma de <a href="/wiki/Quaterni%C3%B3" title="Quaternió">quaternions</a>, que havien estat inventats l'any 1843 pel matemàtic irlandès <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a> (1805 – 1865), perquè utilitzaven quatre <a href="/wiki/Dimensi%C3%B3" title="Dimensió">dimensions</a> i, per tant, podien encabir l'espai tridimensional i el temps. A la seva forma original, les equacions de Maxwell eren un conjunt de 20 expressions de quaternions, 8 equacions dedicades als camps electromagnètics (incloent-hi el <a href="/w/index.php?title=Potencial_vectorial_magn%C3%A8tic&action=edit&redlink=1" class="new" title="Potencial vectorial magnètic (encara no existeix)">potencial vectorial magnètic</a>) i 12 que s'ocupen del <a href="/wiki/Potencial_escalar_magn%C3%A8tic" title="Potencial escalar magnètic">potencial escalar magnètic</a>, la massa magnètica i la conductivitat magnètica.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Detall_de_les_equacions">Detall de les equacions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=1" title="Modifica la secció: Detall de les equacions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Llei_de_Gauss">Llei de Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=2" title="Modifica la secció: Llei de Gauss"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r30997230">.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}</style><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">Llei de Gauss</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:GaussLaw1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/GaussLaw1.svg/220px-GaussLaw1.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/GaussLaw1.svg/330px-GaussLaw1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/GaussLaw1.svg/440px-GaussLaw1.svg.png 2x" data-file-width="280" data-file-height="280" /></a><figcaption>La llei de Gauss afirma que, donat que la càrrega és positiva i està dins la superfície, el flux serà positiu, tal com veiem en el dibuix.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:GaussLaw2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/GaussLaw2.svg/220px-GaussLaw2.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/GaussLaw2.svg/330px-GaussLaw2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/GaussLaw2.svg/440px-GaussLaw2.svg.png 2x" data-file-width="280" data-file-height="280" /></a><figcaption>La llei de Gauss afirma que, donat que la càrrega està a l'exterior de la superfície, el flux serà nul, tal com s'intueix en el dibuix.</figcaption></figure> <p>La llei de Gauss relaciona el <a href="/wiki/Flux_el%C3%A8ctric" title="Flux elèctric">flux del camp elèctric</a> a través d'una superfície tancada amb la quantitat de càrrega que es troba a l'interior de la superfície. </p><p>Primer de tot, la definició del flux del camp elèctric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c49c8491e55157c9b3578226b071c3a7acc0693b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.509ex;" alt="{\displaystyle \Phi _{E}}"></span> és la integral sobre tota la superfície tancada del vector <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} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22779cbb52075b05dc18c7a52ddd9d246effbdc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.778ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \mathbf {S} }"></span> multiplicat escalarment pel vector camp elèctric <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span>: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74f63cd462ad1b50a824b05d6e3c1f5555152f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.45ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {S} }"></span> </p> </blockquote> <p>Per altra banda, hem dit que ens interessa la quantitat de càrrega a l'interior de la superfície tancada. Per tant, sigui <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> el volum que està envoltat per la superfície <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> - és a dir, que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> és la frontera de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></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 S=\partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200cb98139519bbe6f714d5db528c37826fd3b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.703ex; height:2.176ex;" alt="{\displaystyle S=\partial V}"></span> - la càrrega total a l'interior de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> serà la <a href="/wiki/Integral_de_volum" title="Integral de volum">integral de volum</a> de la densitat de càrrega <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> : </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\mathrm {int} }=\int _{V}\;\rho \;\mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi>ρ</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\mathrm {int} }=\int _{V}\;\rho \;\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d8b753948dceff83662c223ff7b24612be9f886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.928ex; height:5.676ex;" alt="{\displaystyle Q_{\mathrm {int} }=\int _{V}\;\rho \;\mathrm {d} V}"></span> </p> </blockquote> <p>Un cop dit això, la llei de Gauss afirma que el flux del camp elèctric a través d'una superfície <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 S=\partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200cb98139519bbe6f714d5db528c37826fd3b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.703ex; height:2.176ex;" alt="{\displaystyle S=\partial V}"></span> és directament proporcional a la càrrega interior, i la constant de proporcionalitat és <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 {1}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71ce7875998d1c49562f8dcdcb5232e846238f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:2.974ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{\varepsilon _{0}}}}"></span>. Això escrit matemàticament és: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{V}\;\rho \;\mathrm {d} V}"> <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> <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">S</mi> </mrow> <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">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi>ρ</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{V}\;\rho \;\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672a0244cb4ab094c148a713c99f58a171babbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.545ex; height:5.843ex;" alt="{\displaystyle \oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{V}\;\rho \;\mathrm {d} V}"></span> </p> </blockquote> <p>que s'anomena la <b>llei de Gauss en forma integral</b>. En el cas del camp electroestàtic, aquesta fórmula es pot deduir de la <a href="/wiki/Llei_de_Coulomb" title="Llei de Coulomb">llei de Coulomb</a> i viceversa. Tot i això, la llei de Gauss segueix sent vàlida en el cas electrodinàmic. </p><p>A partir de la fórmula anterior, i aplicant el <a href="/wiki/Teorema_de_la_diverg%C3%A8ncia" title="Teorema de la divergència">teorema de la divergència</a>, obtindrem la <b>llei de Gauss en forma diferencial</b>. Vegem-ho: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}=\int _{V}\;{\frac {\rho }{\varepsilon _{0}}}\;\mathrm {d} V=\oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} =\int _{V}\;\nabla \cdot \mathbf {E} \;\mathrm {d} V\;\Rightarrow \;\int _{V}\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\mathrm {d} V=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thickmathspace" /> <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="thickmathspace" /> <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 mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⇒</mo> <mspace width="thickmathspace" /> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <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> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}=\int _{V}\;{\frac {\rho }{\varepsilon _{0}}}\;\mathrm {d} V=\oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} =\int _{V}\;\nabla \cdot \mathbf {E} \;\mathrm {d} V\;\Rightarrow \;\int _{V}\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\mathrm {d} V=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e9f38ca279a32c0bd63715597d3c0a92b3ef2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:78.387ex; height:6.176ex;" alt="{\displaystyle {\frac {Q_{\mathrm {int} }}{\varepsilon _{0}}}=\int _{V}\;{\frac {\rho }{\varepsilon _{0}}}\;\mathrm {d} V=\oint _{\partial V}\mathbf {E} \cdot \mathrm {d} \mathbf {S} =\int _{V}\;\nabla \cdot \mathbf {E} \;\mathrm {d} V\;\Rightarrow \;\int _{V}\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\mathrm {d} V=0}"></span> </p> </blockquote> <p>on hem aplicat el teorema de la divergència en la tercera igualtat. Com que això es compleix per a qualsevol volum <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, implica que l'element de dins l'última integral és sempre 0, de manera que concluïm que: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </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-inline 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> </blockquote> <div class="mw-heading mw-heading3"><h3 id="Llei_de_Gauss_per_al_magnetisme">Llei de Gauss per al magnetisme</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=3" title="Modifica la secció: Llei de Gauss per al magnetisme"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Llei_de_Gauss_per_al_magnetisme" title="Llei de Gauss per al magnetisme">Llei de Gauss per al magnetisme</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:GaussLaw4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/GaussLaw4.svg/220px-GaussLaw4.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/GaussLaw4.svg/330px-GaussLaw4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/GaussLaw4.svg/440px-GaussLaw4.svg.png 2x" data-file-width="260" data-file-height="260" /></a><figcaption>Veiem a la figura les línies de camp creat per un <a href="/wiki/Dipol_magn%C3%A8tic" title="Dipol magnètic">dipol magnètic</a>. Veiem que, tal com afirma la llei de Gauss per al magnetisme, el flux a través de la superfície és nul, ja que entren tantes línies de camp com surten.</figcaption></figure> <p>La llei de Gauss per al magnetisme afirma que no existeixen els <a href="/wiki/Monopol_magn%C3%A8tic" title="Monopol magnètic">monòpols magnètics</a>, és a dir, que no es pot aïllar un punt on només entrin línies de camp magnètic o només en surtin, sinó que totes les línies de camp són tancades. Això s'expressa dient que el camp magnètic és un <a href="/wiki/Camp_solenoidal" title="Camp solenoidal">camp solenoidal</a>. Fent servir el llenguatge del <a href="/wiki/C%C3%A0lcul_vectorial" title="Càlcul vectorial">càlcul vectorial</a>, podem escriure la <b>llei de Gauss per al magnetisme en forma diferencial</b> de la següent manera: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {B} =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">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {B} =0}"></span> </p> </blockquote> <p>Com passava abans, aquesta llei pot ser escrita també integralment. Per passar d'una a l'altra, apliquem una integral de volum als dos costats de la igualtat anterior: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\int _{V}\;0\;\mathrm {d} V=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">B</mi> </mrow> <mspace width="thickmathspace" /> <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> <mspace width="thickmathspace" /> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\int _{V}\;0\;\mathrm {d} V=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcbd8baa7078e1c2affa54b55765d7a8158bb71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.484ex; height:5.676ex;" alt="{\displaystyle \int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\int _{V}\;0\;\mathrm {d} V=0}"></span> </p> </blockquote> <p>Tornem a aplicar el <a href="/wiki/Teorema_de_la_diverg%C3%A8ncia" title="Teorema de la divergència">teorema de la divergència</a>, així que ens queda que: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\oint _{\partial V}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <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">B</mi> </mrow> <mspace width="thickmathspace" /> <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 mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\oint _{\partial V}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84586822fac17117ded366803a5c5520d8423d22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.242ex; height:5.676ex;" alt="{\displaystyle 0=\int _{V}\nabla \cdot \mathbf {B} \;\mathrm {d} V=\oint _{\partial V}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"></span> </p> </blockquote> <p>Que és la <b>llei de Gauss per al magnetisme en forma integral</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Llei_de_Faraday">Llei de Faraday</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=4" title="Modifica la secció: Llei de Faraday"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Llei_de_Faraday" title="Llei de Faraday">Llei de Faraday</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxer:Teorema_Stokes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Teorema_Stokes.svg/220px-Teorema_Stokes.svg.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Teorema_Stokes.svg/330px-Teorema_Stokes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Teorema_Stokes.svg/440px-Teorema_Stokes.svg.png 2x" data-file-width="325" data-file-height="250" /></a><figcaption>Esquema on es visualitza una superfície <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, la seva frontera <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span> i un vector diferencial de superfície <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} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22779cbb52075b05dc18c7a52ddd9d246effbdc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.778ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} \mathbf {S} }"></span>, que és sempre perpendicular a la superfície en qualsevol punt. Cal fixar-se en el fet que l'orientació del camí segueix la regla de la mà dreta.</figcaption></figure> <p>La llei de Faraday estableix la relació entre la <a href="/wiki/For%C3%A7a_electromotriu" title="Força electromotriu">força electromotriu induïda</a> a una espira i la variació del flux del camp magnètic a través de la superfície de l'espira. Començarem expressant la llei en forma integral, i llavors la passarem a forma diferencial. </p><p>Matemàticament, podem interpretar l'espira com una corba parametritzada. Aquesta corba és, al mateix temps, la frotera d'infinites superfícies obertes que també podem expressar matemàticament. Llavors, triarem una qualsevol d'aquestes superfícies (la que més adient sigui pel nostre problema) que anomenarem <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. El <a href="/wiki/Flux_magn%C3%A8tic" title="Flux magnètic">flux del camp magnètic</a> a través de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> és: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3903ba4b1bfe5771ab36215af3f1ebd40636292c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.586ex; height:5.676ex;" alt="{\displaystyle \Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"></span> </p> </blockquote> <p>Amb aquestes condicions, la llei de Faraday afirma que la variació del flux comporta una força electromotriu induïda a l'espira de la següent manera: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\Phi _{B}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\Phi _{B}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970b26f15eae7ab4578d1d2106867bee781d377d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.936ex; height:5.843ex;" alt="{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\Phi _{B}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"></span> </p> </blockquote> <p>Però la força electromotriu induïda es pot interpretar com la integral de línia al llarg de l'espira del camp elèctric, és a dir: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ba2d72018390a84084fd7bb08b8452ba1c7f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.784ex; height:5.676ex;" alt="{\displaystyle {\mathcal {E}}=\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }"></span> </p> </blockquote> <p>On la integral de línia es fa seguint l'orientació induïda per l'orientació de la superfície, o sigui, seguint la <a href="/wiki/Regla_de_la_m%C3%A0_dreta" title="Regla de la mà dreta">regla de la mà dreta</a> (vegeu figura). Finalment, ens queda l'expressió definitiva de la <b>llei de Faraday en forma integral</b>: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7e1c053521464d8d2ed146839e05de3ee81586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.967ex; height:5.843ex;" alt="{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"></span> </p> </blockquote> <p>Com passava els altres cops, podem escriure-ho d'una altra manera, que anomenarem <b>llei de Faraday en forma diferencial</b>: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.495ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"></span> </p> </blockquote> <p>Es pot deduir de l'expressió anterior fent servir el <a href="/wiki/Teorema_de_Stokes" title="Teorema de Stokes">teorema de Stokes</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Formulació"><span id="Formulaci.C3.B3"></span>Formulació</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=5" title="Modifica la secció: Formulació"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La formulació moderna de les equacions de Maxwell és deguda a <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> i <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a>, que el <a href="/wiki/1884" title="1884">1884</a> reformularen les equacions originals de Maxwell en un sistema abreujat utilitzant notació vectorial. La formulació original de Maxwell datava de <a href="/wiki/1865" title="1865">1865</a> i contenia 20 equacions de 20 variables. La formulació vectorial resultava especialment atractiva perquè remarcava les simetries intrínseques en les equacions fent més fàcil la seva utilització. </p><p>Les equacions de Maxwell, en forma integral i diferencial són les següents (ambdues formes són totalment equivalents, es pot passar d'una a l'altra amb les eines habituals del càlcul diferencial). </p> <table border="1" cellpadding="6" cellspacing="0"> <tbody><tr style="background-color: #aaddcc;"> <th>Nom </th> <th>Forma diferencial </th> <th>Forma integral </th></tr> <tr> <td><a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">Llei de Gauss</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {D} =\rho }"> <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> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {D} =\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76835fc646d3912b71f4157618db7fdca02a174e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.965ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {D} =\rho }"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} =Q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} =Q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd79e70d7e97e0dfd422a5d6593b652354ea2c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.215ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} =Q_{i}}"></span> </td></tr> <tr> <td><a href="/wiki/Llei_de_Gauss_per_al_magnetisme" title="Llei de Gauss per al magnetisme">Llei de Gauss per al magnetisme</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {B} =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">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {B} =0}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =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">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229e1cab62d6e900e6e123eacf119d94bf3e1374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.591ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}"></span> </td></tr> <tr> <td><a href="/wiki/Llei_de_Faraday" title="Llei de Faraday">Llei de Faraday</a>: </td> <td><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 \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.495ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20843fd79408a01c95485f0d043be789e8aab09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.967ex; height:5.843ex;" alt="{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} }"></span> </td></tr> <tr> <td><a href="/wiki/Llei_d%27Amp%C3%A8re#Llei_d'Ampère_corregida:_l'equació_d'Ampère-Maxwell" title="Llei d'Ampère">Llei d'Ampère-Maxwell</a>: </td> <td><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 \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}"> <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">H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1e547245786c1011ba70585139a832c95c3bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.391ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fd9c603d69e9bb7f0b3923f9c5037b58012089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.292ex; height:5.843ex;" alt="{\displaystyle \oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\mathrm {d} \over \mathrm {d} t}\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {S} }"></span> </td></tr></tbody></table> <ul><li><i>Q</i> és la <a href="/wiki/C%C3%A0rrega_el%C3%A8ctrica" title="Càrrega elèctrica">càrrega elèctrica</a> (unitat <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">SI</a>: <a href="/wiki/Coulomb" title="Coulomb">coulomb</a>).</li> <li><i>ρ</i> és la <a href="/wiki/Densitat_de_c%C3%A0rrega" title="Densitat de càrrega">densitat de càrrega</a> elèctrica (unitat <a href="/wiki/Sistema_Internacional_d%27Unitats" title="Sistema Internacional d'Unitats">SI</a>: <a href="/wiki/Coulomb" title="Coulomb">coulomb</a> per metre cúbic), sense incloure càrregues dipolars lligades a un material.</li> <li><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 {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cafb0ef39b0f5ffa23c170aa7f7b4e718327c4d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.901ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} }"></span> és la <a href="/wiki/Inducci%C3%B3_magn%C3%A8tica" class="mw-redirect" title="Inducció magnètica">inducció magnètica</a> (unitat SI: <a href="/wiki/Tesla_(unitat)" title="Tesla (unitat)">tesla</a>, <a href="/wiki/Volt" title="Volt">volt</a> × <a href="/wiki/Segon" title="Segon">segon</a> per metre quadrat) <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 {B} =\mu _{0}\mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mu _{0}\mathbf {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d2fd45a5e63db4904cb79193388449e8cb2ccf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.547ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} =\mu _{0}\mathbf {H} }"></span>.</li> <li><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2345293072878db24e119c580def49ad582e3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.05ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} }"></span> és el <a href="/wiki/Despla%C3%A7ament_el%C3%A8ctric" title="Desplaçament elèctric">desplaçament elèctric</a> (unitat SI: coulomb per <a href="/wiki/Metre_quadrat" title="Metre quadrat">metre quadrat</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 \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-inline 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>.</li> <li><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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8a515de34f0af7d15de46f73bf674950d444a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} }"></span> és l'àrea de la superfície gaussiana d'integració.</li> <li><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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> és el <a href="/wiki/Camp_el%C3%A8ctric" title="Camp elèctric">camp elèctric</a> (unitat SI: <a href="/wiki/Volt" title="Volt">volt</a> per <a href="/wiki/Metre" title="Metre">metre</a>).</li> <li><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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f017b876ed763037d8818ec5dfbbdc6703e0f683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {H} }"></span> és el <a href="/wiki/Camp_magn%C3%A8tic" title="Camp magnètic">camp magnètic</a> (unitat SI: <a href="/wiki/Ampere" title="Ampere">ampere</a> per metre).</li> <li><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 {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }"></span> és la densitat de <a href="/wiki/Corrent_el%C3%A8ctric" title="Corrent elèctric">corrent elèctric</a> (unitat SI: ampere per metre quadrat)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df6024211b717870f07844116e116b2eb314d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot }"></span> és l'<a href="/wiki/Operador" class="mw-redirect" title="Operador">operador</a> <a href="/wiki/Diverg%C3%A8ncia" title="Divergència">divergència</a> (unitat del SI: 1 per metre)</li> <li><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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8255aabfb5dba42ab97b2bf70d0dd19a9849a5eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle \nabla \times }"></span> és l'<a href="/wiki/Operador" class="mw-redirect" title="Operador">operador</a> <a href="/wiki/Rotacional" title="Rotacional">rotacional</a> (unitat del SI: 1 per metre)</li></ul> <p>Encara que es donen les unitats del <a href="/wiki/Sistema_internacional_d%27unitats" class="mw-redirect" title="Sistema internacional d'unitats">sistema internacional d'unitats</a> per a les diversos magnituds, les equacions de Maxwell es mantenen en altres sistemes d'unitats. </p> <div class="mw-heading mw-heading2"><h2 id="Interpretació_física_de_les_equacions"><span id="Interpretaci.C3.B3_f.C3.ADsica_de_les_equacions"></span>Interpretació física de les equacions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=6" title="Modifica la secció: Interpretació física de les equacions"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les quatre equacions de Maxwell expressen, respectivament, com les <a href="/wiki/C%C3%A0rrega_el%C3%A8ctrica" title="Càrrega elèctrica">càrregues elèctriques</a> produeixen <a href="/wiki/Camp_el%C3%A8ctric" title="Camp elèctric">camps elèctrics</a> (<a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">llei de Gauss</a>), l'absència experimental de <a href="/wiki/Monopol_magn%C3%A8tic" title="Monopol magnètic">càrregues magnètiques</a> (2a llei), com el <a href="/wiki/Corrent_el%C3%A8ctric" title="Corrent elèctric">corrent</a> produeix <a href="/wiki/Camp_magn%C3%A8tic" title="Camp magnètic">camps magnètics</a> (<a href="/wiki/Llei_d%27Amp%C3%A8re" title="Llei d'Ampère">llei d'Ampère</a>) i com els camps magnètics canviants produeixen camps elèctrics (<a href="/wiki/Llei_de_Faraday" title="Llei de Faraday">llei de la inducció de Faraday</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Conservació_de_la_càrrega"><span id="Conservaci.C3.B3_de_la_c.C3.A0rrega"></span>Conservació de la càrrega</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=7" title="Modifica la secció: Conservació de la càrrega"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r30997230"><div role="note" class="hatnote navigation-not-searchable">Article principal: <a href="/wiki/Conservaci%C3%B3_de_la_c%C3%A0rrega" title="Conservació de la càrrega">Conservació de la càrrega</a></div> <p>La conservació de la càrrega és un principi que estableix que no és possible crear ni destruir càrrega. Això vol dir que si en un punt hi ha una disminució de la densitat de càrrega, implica que també hi ha d'haver una divergència positiva de densitat de corrent, i viceversa. </p><p>Això pot ser resumit matemàticament en la següent expressió </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ed7c7379c7058b5e18feeea2c57ff05bbb79bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.453ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0}"></span> </p> </blockquote> <p>que pot deduir-se fàcilment a partir de les lleis de Maxwell. </p> <table class="toccolours collapsible collapsed" style="text-align:center; margin-bottom: 1em; padding:0px; background-color:#F9F9F9;"> <tbody><tr> <th>Demostració </th></tr> <tr> <td style="background-color:#FFFFFF; text-align:left; font-size:95%; padding:12px"> <p>Per fer la demostració, aplicarem l'operador divergència a la <a href="/wiki/Llei_d%27Amp%C3%A8re" title="Llei d'Ampère">llei d'Ampère-Maxwell</a>: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=\nabla \cdot \mathbf {J} +\nabla \cdot {\partial \mathbf {D} \over \partial t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=\nabla \cdot \mathbf {J} +\nabla \cdot {\partial \mathbf {D} \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f666c17dca4c99214d52ea3cfcde89858e21f025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.046ex; height:5.509ex;" alt="{\displaystyle \nabla \cdot (\nabla \times \mathbf {H} )=\nabla \cdot \mathbf {J} +\nabla \cdot {\partial \mathbf {D} \over \partial t}}"></span> </p> </blockquote> <p>Però tenint en compte que la divergència del rotacional és zero, tenim que: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {J} +{\partial \over \partial t}(\nabla \cdot \mathbf {D} )=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">J</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {J} +{\partial \over \partial t}(\nabla \cdot \mathbf {D} )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f561aaa7012af7afe07d828b5ccb35c2d474041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.565ex; height:5.509ex;" alt="{\displaystyle \nabla \cdot \mathbf {J} +{\partial \over \partial t}(\nabla \cdot \mathbf {D} )=0}"></span> </p> </blockquote> <p>Si apliquem la <a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">llei de Gauss</a> obtenim </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=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">J</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b324c4ef75aa48e03abe2ac6357b8691531837d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.453ex; height:5.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=0}"></span> </p> </blockquote> <p>tal com volíem demostrar. </p> </td></tr></tbody></table> <p>Aquesta expressió també es pot escriure en forma integral: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}}"> <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">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cb93bb456e8b76e2ddc4a3d658b16128634b29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.927ex; height:5.843ex;" alt="{\displaystyle \oint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}}"></span> </p> </blockquote> <p>Que s'obté utilitzant el <a href="/wiki/Teorema_de_la_diverg%C3%A8ncia" title="Teorema de la divergència">teorema de la divergència</a> a l'expressió diferencial anterior. </p> <table class="toccolours collapsible collapsed" style="text-align:center; margin-bottom: 1em; padding:0px; background-color:#F9F9F9;"> <tbody><tr> <th>Demostració </th></tr> <tr> <td style="background-color:#FFFFFF; text-align:left; font-size:95%; padding:12px"> <p>Primer de tot, notem que si tenim un volum <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> que té per frontera una superfície <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 S=\partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200cb98139519bbe6f714d5db528c37826fd3b21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.703ex; height:2.176ex;" alt="{\displaystyle S=\partial V}"></span>, llavors es compleix que la càrrega interior al volum és </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\mathrm {int} }=\int _{V}\rho \;\mathrm {d} V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{\mathrm {int} }=\int _{V}\rho \;\mathrm {d} V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8771cfe36b07fef637ebce599975e30994106cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.283ex; height:5.676ex;" alt="{\displaystyle Q_{\mathrm {int} }=\int _{V}\rho \;\mathrm {d} V}"></span> </p> </blockquote> <p>Així que si apliquem la integral de volum als dos costats de la condició de continuïtat en forma diferencial obtenim: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}{\frac {\partial \rho }{\partial t}}\;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=\int _{V}0\;\mathrm {d} V=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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <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> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mspace width="thickmathspace" /> <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> <mn>0</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}{\frac {\partial \rho }{\partial t}}\;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=\int _{V}0\;\mathrm {d} V=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1d80e0c7e958eef2aa7fe92479873553716a1d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.416ex; height:6.009ex;" alt="{\displaystyle \int _{V}{\frac {\partial \rho }{\partial t}}\;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=\int _{V}0\;\mathrm {d} V=0}"></span> </p> </blockquote> <p>Que reescriurem com: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho \;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mi>ρ</mi> <mspace width="thickmathspace" /> <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> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho \;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92519555ceb9c67fcfef3cd8ed08c4e3197b5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.455ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\int _{V}\rho \;\mathrm {d} V+\int _{V}\nabla \cdot \mathbf {J} \;\mathrm {d} V=0}"></span> </p> </blockquote> <p>Fent servir el que hem dit al principi de la demostració pel primer sumand, i el <a href="/wiki/Teorema_de_la_diverg%C3%A8ncia" title="Teorema de la divergència">Teorema de la divergència</a> pel segon, obtenim: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}+\oint _{\partial V}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}+\oint _{\partial V}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1ac61bf1893c16c8a2a482935578b7282cff2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.257ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} Q_{\mathrm {int} }}{\mathrm {d} t}}+\oint _{\partial V}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =0}"></span> </p> </blockquote> <p>Que tenint en compte que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial V=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V=S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fac68212a75fa87e716f5dc5e350fed7bf8ef31a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.703ex; height:2.176ex;" alt="{\displaystyle \partial V=S}"></span> és el que volíem demostrar. </p> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Força_de_Lorentz"><span id="For.C3.A7a_de_Lorentz"></span>Força de Lorentz</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=8" title="Modifica la secció: Força de Lorentz"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les equacions de Maxwell expressen com les càrregues i corrents creen camps elèctrics i magnètics, però no com aquests camps actuen sobre la matèria. Per a això es necessita la llei de <a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">força de Lorentz</a>: </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 {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3dd99e3bd55cbeff1cd2506d944405f3efa9e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.41ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}"></span></dd></dl> <p>Aquesta llei ens diu quina força experimenta una <a href="/wiki/C%C3%A0rrega_puntual" title="Càrrega puntual">càrrega puntual</a> en moviment en el si d'un camp electromagnètic. Si en lloc d'una càrrega puntual tenim una distribució de càrrega, la corresponent força per unitat de volum és: </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 {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b874c8e0f6ad41637c0c165c369b07a7a8cebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.073ex; height:2.676ex;" alt="{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }"></span></dd></dl> <p>i la resultant sobre tot el volum és la integral d'aquesta densitat estesa a tot el volum. </p> <div class="mw-heading mw-heading2"><h2 id="Equacions_de_Maxwell_en_el_buit">Equacions de Maxwell en el buit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=9" title="Modifica la secció: Equacions de Maxwell en el buit"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Considerarem que en el buit no hi ha ni càrregues ni corrents, és a dir, que: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0\qquad ;\qquad \mathbf {J} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =0\qquad ;\qquad \mathbf {J} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700aab7479713d4739282b65dbdd250e26de89ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.429ex; height:2.676ex;" alt="{\displaystyle \rho =0\qquad ;\qquad \mathbf {J} =0}"></span> </p> </blockquote> <p>A més a més també tindrem que en el buit no hi ha ni <a href="/wiki/Polaritzaci%C3%B3_el%C3%A8ctrica" title="Polarització elèctrica">polarització</a> ni <a href="/wiki/Magnetitzaci%C3%B3" title="Magnetització">magnetització</a>, de manera que: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}\mathbf {E} =\mathbf {D} \qquad ;\qquad \mathbf {B} =\mu _{0}\mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mspace width="2em" /> <mo>;</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}\mathbf {E} =\mathbf {D} \qquad ;\qquad \mathbf {B} =\mu _{0}\mathbf {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4be4d9308bf7332ccdd8e9f37fafe954e8663f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.914ex; height:2.676ex;" alt="{\displaystyle \varepsilon _{0}\mathbf {E} =\mathbf {D} \qquad ;\qquad \mathbf {B} =\mu _{0}\mathbf {H} }"></span> </p> </blockquote> <p>Així doncs, les equacions de Maxwell se simplifiquen considerablement i s'obté: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {E} =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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbd9a4bd688b1331c2fd3c7fd1d50f0bf87fc28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.633ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} =0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot \mathbf {B} =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">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {B} =0}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.495ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {B} =\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}}"> <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">B</mi> </mrow> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf05eca981fb5b66aadd6a0b02fb653afbe9153a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.281ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {B} =\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}}}"></span></dd></dl> <p>Si les manipulem matemàticament, aquestes equacions condueixen a les següents dues <a href="/wiki/EDP" class="mw-redirect" title="EDP">EDP</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f91efe9424d50b0d6049ab1277d275c05f394e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.291ex; height:6.009ex;" alt="{\displaystyle \nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176bea8aebfdfc7f02db6ac435e1ec0bb49bb4f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.579ex; height:6.009ex;" alt="{\displaystyle \nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}}"></span></dd></dl> <table class="toccolours collapsible collapsed" style="text-align:center; margin-bottom: 1em; padding:0px; background-color:#F9F9F9;"> <tbody><tr> <th>Demostració </th></tr> <tr> <td style="background-color:#FFFFFF; text-align:left; font-size:95%; padding:12px">Per demostrar això només ens fa falta la següent relació del <a href="/wiki/C%C3%A0lcul_vectorial" title="Càlcul vectorial">càlcul vectorial</a>: <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3bf30db7e456a5d2b9eaf3f1804dbe3fe7b319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.709ex; height:3.176ex;" alt="{\displaystyle \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }"></span> </p> </blockquote> <p>on l'operador <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 ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4be87ad083e5ead48d92b0c82f2d4e719cb34a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.99ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}}"></span> consisteix a fer el <a href="/wiki/Laplaci%C3%A0" class="mw-redirect" title="Laplacià">laplacià</a> a cada funció component de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span>. Un cop dit això, apliquem el rotacional a la quarta equació de Maxwell: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla (\nabla \cdot \mathbf {E} )-\nabla ^{2}\mathbf {E} =-\nabla ^{2}\mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla (\nabla \cdot \mathbf {E} )-\nabla ^{2}\mathbf {E} =-\nabla ^{2}\mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7b1c8fa41f965ab581ef2223c6e419bf58cb84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.576ex; height:3.176ex;" alt="{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla (\nabla \cdot \mathbf {E} )-\nabla ^{2}\mathbf {E} =-\nabla ^{2}\mathbf {E} }"></span> </p> </blockquote> <p>Ja que la divergència del camp elèctric és zero. Per una altra banda: </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla \times (-{\frac {\partial \mathbf {B} }{\partial t}})=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )=-{\frac {\partial }{\partial t}}(\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}})=-\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla \times (-{\frac {\partial \mathbf {B} }{\partial t}})=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )=-{\frac {\partial }{\partial t}}(\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}})=-\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aee66b0b8abfb9f167eb4732788410030b24501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:77.759ex; height:6.009ex;" alt="{\displaystyle \nabla \times (\nabla \times \mathbf {E} )=\nabla \times (-{\frac {\partial \mathbf {B} }{\partial t}})=-{\frac {\partial }{\partial t}}(\nabla \times \mathbf {B} )=-{\frac {\partial }{\partial t}}(\varepsilon _{0}\mu _{0}{\frac {\partial \mathbf {E} }{\partial t}})=-\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}}"></span> </p> </blockquote> </td></tr></tbody></table> <p>Les anteriors equacions tenen la mateixa forma que l'<a href="/wiki/Equaci%C3%B3_d%27ona" title="Equació d'ona">equació d'ona</a>, és a dir, que els camps electromagnètics en el buit es comporten com ones tridimensionals que es propaguen a velocitat </p> <blockquote style="padding: 5px 10px; background-color: white; text-align:left; margin-left:30px; margin-bottom:0.8em; margin-top:0.5em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71677838a5d6660a32242c0493d13469ef258c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.471ex; height:6.176ex;" alt="{\displaystyle c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}}"></span> </p> </blockquote> <p>Aquesta velocitat <i>c</i> és la <a href="/wiki/Velocitat_de_la_llum" title="Velocitat de la llum">velocitat de la llum</a> en el buit, la qual cosa suggereix (tal com se sap actualment) que la llum és un tipus particular d'ona electromagnètica. </p> <div class="mw-heading mw-heading2"><h2 id="Les_equacions_de_Maxwell_en_relativitat_especial">Les equacions de Maxwell en relativitat especial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=10" title="Modifica la secció: Les equacions de Maxwell en relativitat especial"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Relativitat_especial" title="Relativitat especial">relativitat especial</a>, per tal d'expressar més clarament que les equacions de Maxwell en el buit tenen la mateixa forma en qualsevol <a href="/wiki/Sistema_de_refer%C3%A8ncia_inercial" title="Sistema de referència inercial">sistema de referència inercial</a>, s'acostumen a escriure en termes de <a href="/wiki/Quadrivector" title="Quadrivector">quadrivectors</a> i <a href="/w/index.php?title=Tensor_(Matem%C3%A0tiques)&action=edit&redlink=1" class="new" title="Tensor (Matemàtiques) (encara no existeix)">tensors</a> en forma covariant (en unitats cgs): </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 {4\pi \over c}J^{\beta }={\partial F^{\alpha \beta } \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{,\alpha }\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {4\pi \over c}J^{\beta }={\partial F^{\alpha \beta } \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{,\alpha }\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21a4ca83a316b4722dd5267118c8685ecc8a2f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:34.887ex; height:5.843ex;" alt="{\displaystyle {4\pi \over c}J^{\beta }={\partial F^{\alpha \beta } \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }F^{\alpha \beta }\ {\stackrel {\mathrm {def} }{=}}\ {F^{\alpha \beta }}_{,\alpha }\,\!}"></span>,</dd></dl> <p>i </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 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{\delta \alpha \beta \gamma }{F^{\beta \gamma }}_{,\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>γ</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>α</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>α</mi> <mi>β</mi> <mi>γ</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>γ</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{\delta \alpha \beta \gamma }{F^{\beta \gamma }}_{,\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8708e049326fb92f466ef59a459308122f27b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:67.032ex; height:4.343ex;" alt="{\displaystyle 0=\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }\ {\stackrel {\mathrm {def} }{=}}\ {F_{\alpha \beta }}_{,\gamma }+{F_{\gamma \alpha }}_{,\beta }+{F_{\beta \gamma }}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \epsilon _{\delta \alpha \beta \gamma }{F^{\beta \gamma }}_{,\alpha }}"></span></dd></dl> <p>on <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 \,J^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,J^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bde8515983a83189ef731c9d6f53d98d323460d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.197ex; height:2.343ex;" alt="{\displaystyle \,J^{\alpha }}"></span> és el <a href="/wiki/Quadricorrent" title="Quadricorrent">quadricorrent</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^{\alpha \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,F^{\alpha \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad4761c3db7eb913a4683481f97af1c398b2b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.428ex; height:2.676ex;" alt="{\displaystyle \,F^{\alpha \beta }}"></span> és el <a href="/wiki/Tensor_electromagn%C3%A8tic" title="Tensor electromagnètic">tensor electromagnètic</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 \,\epsilon _{\alpha \beta \gamma \delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\epsilon _{\alpha \beta \gamma \delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0928d8051b3f408d97f9ce94ea1835607255a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.191ex; height:2.343ex;" alt="{\displaystyle \,\epsilon _{\alpha \beta \gamma \delta }}"></span> és el <a href="/wiki/S%C3%ADmbol_de_Levi-Civita" title="Símbol de Levi-Civita">símbol de Levi-Civita</a> i </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 {\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\partial }{\partial ct}},\nabla \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>α</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mi mathvariant="normal">∇</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\partial }{\partial ct}},\nabla \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12c8e27b5918b35aceb68cfa0aa389e8d085bdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.058ex; height:6.176ex;" alt="{\displaystyle {\partial \over {\partial x^{\alpha }}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\alpha }\ {\stackrel {\mathrm {def} }{=}}\ {}_{,\alpha }\ {\stackrel {\mathrm {def} }{=}}\ \left({\frac {\partial }{\partial ct}},\nabla \right)}"></span></dd></dl> <p>és el <a href="/wiki/Quadrigradient" title="Quadrigradient">quadrigradient</a>. Els índexs repetits se sumen d'acord amb el <a href="/wiki/Conveni_de_sumaci%C3%B3_d%27Einstein" title="Conveni de sumació d'Einstein">conveni de sumació d'Einstein</a>. </p><p>La primera equació tensorial expressa les dues equacions de Maxwell inhomogènies: la <a href="/wiki/Llei_de_Gauss" title="Llei de Gauss">llei de Gauss</a> i la d'Ampère amb les correccions de Maxwell. La segona equació expressa les altres dues equacions homogènies: la <a href="/wiki/Llei_de_Faraday" title="Llei de Faraday">llei de Faraday</a> de la inducció i la llei de Gauss per al camp magnètic (l'absència de <a href="/wiki/Monopol_magn%C3%A8tic" title="Monopol magnètic">monopols magnètics</a>). </p><p>S'ha suggerit que el component <b>v</b>X<b>B</b> de la <a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">Força de Lorentz</a> es pot derivar de la llei de Coulomb i la relativitat especial si hom assumeix la invariància de la càrrega elèctrica.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=11" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/For%C3%A7a_d%27Abraham-Lorentz" title="Força d'Abraham-Lorentz">Força d'Abraham-Lorentz</a></li> <li><a href="/wiki/F%C3%B3rmules_de_Fresnel" title="Fórmules de Fresnel">Fórmules de Fresnel</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=12" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation" style="font-style:normal">«<a rel="nofollow" class="external text" href="http://www.enciclopedia.cat/gran-enciclopedia-catalana/0041537.xml">Equacions de Maxwell</a>». <i><a href="/wiki/Gran_Enciclop%C3%A8dia_Catalana" title="Gran Enciclopèdia Catalana">Gran Enciclopèdia Catalana</a></i>.  Barcelona:  <a href="/wiki/Grup_Enciclop%C3%A8dia_Catalana" class="mw-redirect" title="Grup Enciclopèdia Catalana">Grup Enciclopèdia Catalana</a>.</span></span> </li> <li id="cite_note-FOOTNOTEFleisch2010viii-2"><span class="mw-cite-backlink"><a href="#cite_ref-FOOTNOTEFleisch2010viii_2-0">↑</a></span> <span class="reference-text"><a href="#CITEREFFleisch2010">Fleisch, 2010</a>, p. viii.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="citation book" style="font-style:normal" id="CITEREFHuray2010"><span style="font-variant: small-caps;">Huray</span>, Paul G. <i>Maxwell’s Equations</i> (en anglès).  John Wiley & Sons, Inc., 2010, p. xv-xvi. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-470-54276-7" title="Especial:Fonts bibliogràfiques/978-0-470-54276-7">ISBN 978-0-470-54276-7</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Maxwell%E2%80%99s+Equations&rft.aulast=Huray&rft.aufirst=Paul+G.&rft.date=2010&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.pages=xv-xvi&rft.isbn=978-0-470-54276-7"><span style="display: none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">L. D. Landau, E. M. Lifshitz, <i>The Classical Theory of Fields</i></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><sup title="Cal donar format a aquesta referència." class="noprint"><span class="noprint" style="color:blue;"><i>[<a href="/wiki/VP:REF" class="mw-redirect" title="VP:REF">enllaç sense format</a>]</i></span></sup> <a rel="nofollow" class="external free" href="http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm">http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080101005238/http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm">Arxivat</a> 2008-01-01 a <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". <i>Phys. Scr.</i> 74 702-717</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Equacions_de_Maxwell&action=edit&section=13" title="Modifica la secció: Bibliografia"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFFleisch2010"><span style="font-variant: small-caps;">Fleisch</span>, Daniel. <i>A student's guide to Maxwell's equations</i> (en anglès).  Cambridge University Press, 2010. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/978-0-521-70147-1" title="Especial:Fonts bibliogràfiques/978-0-521-70147-1">ISBN 978-0-521-70147-1</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+student%27s+guide+to+Maxwell%27s+equations&rft.aulast=Fleisch&rft.aufirst=Daniel&rft.date=2010&rft.pub=Cambridge+University+Press&rft.isbn=978-0-521-70147-1"><span style="display: none;"> </span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r33663753">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}.mw-parser-output .side-box-center{clear:both;margin:auto}}</style><div class="side-box metadata side-box-right plainlinks"> <div class="side-box-flex"> <div class="side-box-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">A <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/P%C3%A0gina_principal?uselang=ca">Wikimedia Commons</a></span> hi ha contingut multimèdia relatiu a: <i><b><a href="https://commons.wikimedia.org/wiki/Category:Maxwell%27s_equations" class="extiw" title="commons:Category:Maxwell's equations">Equacions de Maxwell</a></b></i></div></div> </div> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Control_d%27autoritats" title="Control d'autoritats">Registres d'autoritat</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Biblioth%C3%A8que_nationale_de_France" class="mw-redirect" title="Bibliothèque nationale de France">BNF</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://catalogue.bnf.fr/ark:/12148/cb12043257h">1</a>)</span></li> <li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4221398-8">1</a>)</span></li> <li><a href="/wiki/LCCN" class="mw-redirect" title="LCCN">LCCN</a> <span class="uid"> (<a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85082387">1</a>)</span></li> <li><a href="/wiki/Biblioteca_Nacional_de_la_Rep%C3%BAblica_Txeca" title="Biblioteca Nacional de la República Txeca">NKC</a> <span class="uid"> (<a rel="nofollow" class="external text" href="http://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph218895&CON_LNG=ENG">1</a>)</span></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bases d'informació</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/GEC" class="mw-redirect" title="GEC">GEC</a> <span class="uid"> (<a rel="nofollow" class="external text" href="https://www.enciclopedia.cat/gran-enciclopedia-catalana/equacions-de-maxwell">1</a>)</span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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