Momento do dipolo elétrico – Wikipédia, a enciclopédia livre

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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Um_par_de_cargas_opostas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Um_par_de_cargas_opostas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Um par de cargas opostas</span> </div> </a> <ul id="toc-Um_par_de_cargas_opostas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expressão_(caso_geral)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Expressão_(caso_geral)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Expressão (caso geral)</span> </div> </a> <ul id="toc-Expressão_(caso_geral)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Potencial_e_campo_de_um_dipolo_elétrico" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Potencial_e_campo_de_um_dipolo_elétrico"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Potencial e campo de um dipolo elétrico</span> </div> </a> <ul id="toc-Potencial_e_campo_de_um_dipolo_elétrico-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Densidade_de_momento_dipolo_e_densidade_de_polarização" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Densidade_de_momento_dipolo_e_densidade_de_polarização"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Densidade de momento dipolo e densidade de polarização</span> </div> </a> <button aria-controls="toc-Densidade_de_momento_dipolo_e_densidade_de_polarização-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar a subsecção Densidade de momento dipolo e densidade de polarização</span> </button> <ul id="toc-Densidade_de_momento_dipolo_e_densidade_de_polarização-sublist" class="vector-toc-list"> <li id="toc-Meio_com_densidades_de_carga_e_dipolo" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Meio_com_densidades_de_carga_e_dipolo"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Meio com densidades de carga e dipolo</span> </div> </a> <ul id="toc-Meio_com_densidades_de_carga_e_dipolo-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Energia_e_torque" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Energia_e_torque"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Energia e torque</span> </div> </a> <ul id="toc-Energia_e_torque-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Momentos_de_dipolo_elétrico_de_partículas_fundamentais" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Momentos_de_dipolo_elétrico_de_partículas_fundamentais"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Momentos de dipolo elétrico de partículas fundamentais</span> </div> </a> <ul id="toc-Momentos_de_dipolo_elétrico_de_partículas_fundamentais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Momentos_dipolares_de_moléculas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Momentos_dipolares_de_moléculas"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Momentos dipolares de moléculas</span> </div> </a> <ul id="toc-Momentos_dipolares_de_moléculas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referências" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referências"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligações_externas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ligações_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Ligações externas</span> </div> </a> <ul id="toc-Ligações_externas-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Conteúdo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Alternar o índice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Alternar o índice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Momento do dipolo elétrico</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir para um artigo noutra língua. Disponível em 39 línguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-39" 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">39 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%B2%D9%85_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A_%D9%82%D8%B7%D8%A8" title="عزم ثنائي قطب — árabe" lang="ar" hreflang="ar" data-title="عزم ثنائي قطب" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B_%D0%B4%D1%8B%D0%BF%D0%BE%D0%BB%D1%8C%D0%BD%D1%8B_%D0%BC%D0%BE%D0%BC%D0%B0%D0%BD%D1%82" title="Электрычны дыпольны момант — bielorrusso" lang="be" hreflang="be" data-title="Электрычны дыпольны момант" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D0%B4%D0%B8%D0%BF%D0%BE%D0%BB%D0%B5%D0%BD_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82" title="Електрически диполен момент — búlgaro" lang="bg" hreflang="bg" data-title="Електрически диполен момент" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%A1%E0%A6%BC%E0%A6%BF%E0%A7%8E_%E0%A6%A6%E0%A7%8D%E0%A6%AC%E0%A6%BF%E0%A6%AE%E0%A7%87%E0%A6%B0%E0%A7%81_%E0%A6%AD%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%AE%E0%A6%95" title="তড়িৎ দ্বিমেরু ভ্রামক — bengalês" lang="bn" hreflang="bn" data-title="তড়িৎ দ্বিমেরু ভ্রামক" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Moment_dipolar_el%C3%A8ctric" title="Moment dipolar elèctric — catalão" lang="ca" hreflang="ca" data-title="Moment dipolar elèctric" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dip%C3%B3lov%C3%BD_moment" title="Dipólový moment — checo" lang="cs" hreflang="cs" data-title="Dipólový moment" data-language-autonym="Čeština" data-language-local-name="checo" 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%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE_%D0%B4%D0%B8%D0%BF%D0%BE%D0%BB%D1%8C_%D1%81%D0%B0%D0%BC%D0%B0%D0%BD%D1%87%C4%95" title="Электро диполь саманчĕ — chuvash" lang="cv" hreflang="cv" data-title="Электро диполь саманчĕ" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" 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/Elektrisk_dipolmoment" title="Elektrisk dipolmoment — dinamarquês" lang="da" hreflang="da" data-title="Elektrisk dipolmoment" data-language-autonym="Dansk" data-language-local-name="dinamarquês" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elektrisches_Dipolmoment" title="Elektrisches Dipolmoment — alemão" lang="de" hreflang="de" data-title="Elektrisches Dipolmoment" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%B9%CE%BA%CE%AE_%CE%B4%CE%B9%CF%80%CE%BF%CE%BB%CE%B9%CE%BA%CE%AE_%CF%81%CE%BF%CF%80%CE%AE" title="Ηλεκτρική διπολική ροπή — grego" lang="el" hreflang="el" data-title="Ηλεκτρική διπολική ροπή" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Electric_dipole_moment" title="Electric dipole moment — inglês" lang="en" hreflang="en" data-title="Electric dipole moment" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Momento_dipolar_qu%C3%ADmico" title="Momento dipolar químico — espanhol" lang="es" hreflang="es" data-title="Momento dipolar químico" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Elektriline_dipoolmoment" title="Elektriline dipoolmoment — estónio" lang="et" hreflang="et" data-title="Elektriline dipoolmoment" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B4%D8%AA%D8%A7%D9%88%D8%B1_%D8%AF%D9%88%D9%82%D8%B7%D8%A8%DB%8C_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="گشتاور دوقطبی الکتریکی — persa" lang="fa" hreflang="fa" data-title="گشتاور دوقطبی الکتریکی" 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/S%C3%A4hk%C3%B6dipoli" title="Sähködipoli — finlandês" lang="fi" hreflang="fi" data-title="Sähködipoli" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/M%C3%B3imint_leictreach_dh%C3%A9pholach" title="Móimint leictreach dhépholach — irlandês" lang="ga" hreflang="ga" data-title="Móimint leictreach dhépholach" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Momento_do_dipolo_el%C3%A9ctrico" title="Momento do dipolo eléctrico — galego" lang="gl" hreflang="gl" data-title="Momento do dipolo eléctrico" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%A7%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%B5_%E0%A4%86%E0%A4%98%E0%A5%82%E0%A4%B0%E0%A5%8D%E0%A4%A3" title="विद्युत द्विध्रुव आघूर्ण — hindi" lang="hi" hreflang="hi" data-title="विद्युत द्विध्रुव आघूर्ण" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elektromos_dip%C3%B3lusmomentum" title="Elektromos dipólusmomentum — húngaro" lang="hu" hreflang="hu" data-title="Elektromos dipólusmomentum" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BA%D5%B8%D5%AC%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B4%D5%B8%D5%B4%D5%A5%D5%B6%D5%BF" title="Դիպոլային մոմենտ — arménio" lang="hy" hreflang="hy" data-title="Դիպոլային մոմենտ" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Momen_dipol_listrik" title="Momen dipol listrik — indonésio" lang="id" hreflang="id" data-title="Momen dipol listrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%BB%E6%B0%97%E5%8F%8C%E6%A5%B5%E5%AD%90" title="電気双極子 — japonês" lang="ja" hreflang="ja" data-title="電気双極子" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EA%B8%B0_%EC%8C%8D%EA%B7%B9%EC%9E%90_%EB%AA%A8%EB%A9%98%ED%8A%B8" title="전기 쌍극자 모멘트 — coreano" lang="ko" hreflang="ko" data-title="전기 쌍극자 모멘트" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A1%E0%B5%88%E0%B4%AA%E0%B5%8B%E0%B5%BE" title="ഡൈപോൾ — malaiala" lang="ml" hreflang="ml" data-title="ഡൈപോൾ" data-language-autonym="മലയാളം" data-language-local-name="malaiala" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%A6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%A7%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%B5_%E0%A4%9C%E0%A5%8B%E0%A4%B0" title="विद्युत द्विध्रुव जोर — marata" lang="mr" hreflang="mr" data-title="विद्युत द्विध्रुव जोर" data-language-autonym="मराठी" data-language-local-name="marata" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Dipoolmoment" title="Dipoolmoment — neerlandês" lang="nl" hreflang="nl" data-title="Dipoolmoment" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Elektryczny_moment_dipolowy" title="Elektryczny moment dipolowy — polaco" lang="pl" hreflang="pl" data-title="Elektryczny moment dipolowy" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Moment_electric_dipolar" title="Moment electric dipolar — romeno" lang="ro" hreflang="ro" data-title="Moment electric dipolar" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a 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title="Dipólový moment — eslovaco" lang="sk" hreflang="sk" data-title="Dipólový moment" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elektri%C4%8Dni_dipolni_moment" title="Električni dipolni moment — esloveno" lang="sl" hreflang="sl" data-title="Električni dipolni moment" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D0%BF%D0%BE%D0%BB%D0%BD%D0%B8_%D0%BC%D0%BE%D0%BC%D0%B5%D0%BD%D1%82" title="Диполни момент — sérvio" lang="sr" hreflang="sr" data-title="Диполни момент" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / 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class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Origem: Wikipédia, a enciclopédia livre.</div> </div> <div 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="pt" dir="ltr"><table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0.5em 0 0.5em 1em;background:var(--background-color-neutral-subtle, #f8f9fa);color:inherit;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%"><tbody><tr><td style="padding-top:0.4em;line-height:1.2em">Artigos sobre</td></tr><tr><th style="padding:0.2em 0.4em 0.2em;padding-top:0;font-size:145%;line-height:1.2em"><a href="/wiki/Eletromagnetismo" title="Eletromagnetismo">Eletromagnetismo</a></th></tr><tr><td style="padding:0.2em 0 0.4em"><span typeof="mw:File/Frameless"><a href="/wiki/Ficheiro:VFPt_Solenoid_correct2.svg" class="mw-file-description" title="Solenoide"><img alt="Solenoide" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/190px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="190" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/285px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/380px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <ul><li><a href="/wiki/Electricidade" class="mw-redirect" title="Electricidade">Electricidade</a></li> <li><a href="/wiki/Magnetismo" title="Magnetismo">Magnetismo</a></li> <li>História<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/History_of_electromagnetic_theory" class="extiw" title="en:History of electromagnetic theory"><span title=""History of electromagnetic theory" na Wikipédia em English">en</span></a>]</small></li> <li>Livros didáticos<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/List_of_textbooks_in_electromagnetism" class="extiw" title="en:List of textbooks in electromagnetism"><span title=""List of textbooks in electromagnetism" na Wikipédia em English">en</span></a>]</small></li></ul></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Eletrost%C3%A1tica" title="Eletrostática">Eletrostática</a></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a></li> <li><a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Lei de Coulomb</a></li> <li><a href="/wiki/Condutor_el%C3%A9trico" title="Condutor elétrico">Condutor</a></li> <li><a href="/wiki/Densidade_de_carga" title="Densidade de carga">Densidade de carga</a></li> <li><a href="/wiki/Permissividade" title="Permissividade">Permissividade</a></li> <li><a class="mw-selflink selflink">Momento do dipolo elétrico</a></li> <li><a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">Campo elétrico</a></li> <li><a href="/wiki/Potencial_el%C3%A9trico" title="Potencial elétrico">Potencial elétrico</a></li> <li><a href="/wiki/Fluxo_el%C3%A9trico" title="Fluxo elétrico">Fluxo elétrico</a> / <a href="/wiki/Energia_potencial_el%C3%A9trica" title="Energia potencial elétrica">energia potencial</a></li> <li><a href="/wiki/Descarga_electrost%C3%A1tica" title="Descarga electrostática">Descarga electrostática</a></li> <li><a href="/wiki/Lei_de_Gauss" title="Lei de Gauss">Lei de Gauss</a></li> <li><a href="/wiki/Indu%C3%A7%C3%A3o_eletrost%C3%A1tica" title="Indução eletrostática">Indução</a></li> <li><a href="/wiki/Isolante_el%C3%A9trico" title="Isolante elétrico">Isolante</a></li> <li><a href="/wiki/Densidade_de_polariza%C3%A7%C3%A3o" title="Densidade de polarização">Densidade de polarização</a></li> <li><a href="/wiki/Eletricidade_est%C3%A1tica" title="Eletricidade estática">Eletricidade estática</a></li> <li><a href="/wiki/Efeito_triboel%C3%A9trico" title="Efeito triboelétrico">Efeito triboelétrico</a></li></ul></div></div></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Magnetost%C3%A1tica" title="Magnetostática">Magnetostática</a></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Lei_de_Amp%C3%A8re" title="Lei de Ampère">Lei de Ampère</a></li> <li><a href="/wiki/Lei_de_Biot-Savart" title="Lei de Biot-Savart">Lei de Biot – Savart</a></li> <li><a href="/wiki/Lei_de_Gauss_para_o_magnetismo" title="Lei de Gauss para o magnetismo">Lei de Gauss para o magnetismo</a></li> <li><a href="/wiki/Campo_magn%C3%A9tico" title="Campo magnético">Campo magnético</a></li> <li><a href="/wiki/Fluxo_magn%C3%A9tico" title="Fluxo magnético">Fluxo magnético</a></li> <li><a href="/wiki/Momento_magn%C3%A9tico" title="Momento magnético">Momento do dipolo magnético</a></li> <li><a href="/wiki/Permeabilidade_magn%C3%A9tica" title="Permeabilidade magnética">Permeabilidade magnética</a></li> <li><a href="/wiki/Potencial_escalar_magn%C3%A9tico" title="Potencial escalar magnético">Potencial escalar magnético</a></li> <li><a href="/wiki/Magnetiza%C3%A7%C3%A3o" class="mw-redirect" title="Magnetização">Magnetização</a><span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Magnetization" class="extiw" title="en:Magnetization"><span title=""Magnetization" na Wikipédia em English">en</span></a>]</small></li> <li><a href="/wiki/For%C3%A7a_magnetomotriz" title="Força magnetomotriz">Força magnetomotriz</a></li> <li><a href="/wiki/Vetor_potencial_magn%C3%A9tico" title="Vetor potencial magnético">Vetor potencial magnético</a></li> <li>Regra da mão direita<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Right-hand_rule#Electromagnetism" class="extiw" title="en:Right-hand rule"><span title=""Right-hand_rule#Electromagnetism" na Wikipédia em English">en</span></a>]</small></li></ul></div></div></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Eletromagnetismo_cl%C3%A1ssico" title="Eletromagnetismo clássico">Eletrodinâmica</a></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">Lei da força de Lorentz</a></li> <li><a href="/wiki/Indu%C3%A7%C3%A3o_eletromagn%C3%A9tica" title="Indução eletromagnética">Indução eletromagnética</a></li> <li><a href="/wiki/Lei_de_Faraday-Neumann-Lenz" title="Lei de Faraday-Neumann-Lenz">Lei de Faraday</a></li> <li><a href="/wiki/Lei_de_Lenz" title="Lei de Lenz">Lei de Lenz</a></li> <li><a href="/wiki/Corrente_de_deslocamento" title="Corrente de deslocamento">Corrente de deslocamento</a></li> <li><a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">Equações de Maxwell</a></li> <li><a href="/wiki/Campo_eletromagn%C3%A9tico" title="Campo eletromagnético">Campo eletromagnético</a></li> <li><a href="/wiki/Pulso_eletromagn%C3%A9tico" title="Pulso eletromagnético">Pulso eletromagnético</a></li> <li><a href="/wiki/Radia%C3%A7%C3%A3o_eletromagn%C3%A9tica" title="Radiação eletromagnética">Radiação eletromagnética</a></li> <li><a href="/wiki/Tensor_de_tens%C3%A3o_de_Maxwell" title="Tensor de tensão de Maxwell">Tensor de Maxwell</a></li> <li><a href="/wiki/Vetor_de_Poynting" class="mw-redirect" title="Vetor de Poynting">Vetor de Poynting</a></li> <li><a href="/wiki/Potenciais_de_Li%C3%A9nard-Wiechert" title="Potenciais de Liénard-Wiechert">Potentiais de Liénard – Wiechert</a></li> <li><a href="/wiki/Equa%C3%A7%C3%B5es_de_Jefimenko" title="Equações de Jefimenko">Equações de Jefimenko</a></li> <li><a href="/wiki/Corrente_de_Foucault" title="Corrente de Foucault">Corrente de Foucault</a></li> <li><span title="artigo inexistente: “Equações de London”">Equações de London</span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/London_equations" class="extiw" title="en:London equations"><span title=""London equations" na Wikipédia em English">en</span></a>]</small></li></ul> <ul><li><div style="padding:0.2em 0.4em; line-height:1.2em;"><span title="artigo inexistente: “Descrições matemáticas do campo eletromagnético”">Descrições matemáticas do campo eletromagnético</span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field" class="extiw" title="en:Mathematical descriptions of the electromagnetic field"><span title=""Mathematical descriptions of the electromagnetic field" na Wikipédia em English">en</span></a>]</small></div></li></ul></div></div></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Circuito_el%C3%A9trico" title="Circuito elétrico">Circuito elétrico</a></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Corrente_alternada" title="Corrente alternada">Corrente alternada</a></li> <li><a href="/wiki/Capacit%C3%A2ncia" title="Capacitância">Capacitância</a></li> <li><a href="/wiki/Corrente_cont%C3%ADnua" title="Corrente contínua">Corrente contínua</a></li> <li><a href="/wiki/Corrente_el%C3%A9trica" title="Corrente elétrica">Corrente elétrica</a></li> <li><a href="/wiki/Eletr%C3%B3lise" title="Eletrólise">Eletrólise</a></li> <li><a href="/wiki/Densidade_de_corrente_el%C3%A9trica" title="Densidade de corrente elétrica">Densidade de corrente elétrica</a></li> <li><a href="/wiki/Lei_de_Joule" title="Lei de Joule">Lei de Joule</a></li> <li><a href="/wiki/For%C3%A7a_eletromotriz" title="Força eletromotriz">Força eletromotriz</a></li> <li><a href="/wiki/Imped%C3%A2ncia_el%C3%A9trica" title="Impedância elétrica">Impedância</a></li> <li><a href="/wiki/Indut%C3%A2ncia" title="Indutância">Indutância</a></li> <li><a href="/wiki/Lei_de_Ohm" title="Lei de Ohm">Lei de Ohm</a></li> <li><a href="/w/index.php?title=Circuitos_em_s%C3%A9rie_e_em_paralelo&action=edit&redlink=1" class="new" title="Circuitos em série e em paralelo (página não existe)">Circuito paralelo</a></li> <li>Resistência<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance" class="extiw" title="en:Electrical resistance and conductance"><span title=""Electrical resistance and conductance" na Wikipédia em English">en</span></a>]</small></li> <li>Cavidades ressonantes<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Resonator#Electromagnetics" class="extiw" title="en:Resonator"><span title=""Resonator#Electromagnetics" na Wikipédia em English">en</span></a>]</small></li> <li><a href="/w/index.php?title=Circuitos_em_s%C3%A9rie_e_em_paralelo&action=edit&redlink=1" class="new" title="Circuitos em série e em paralelo (página não existe)">Circuito em série</a></li> <li><a href="/wiki/Tens%C3%A3o_el%C3%A9trica" title="Tensão elétrica">Tensão elétrica</a></li> <li><a href="/wiki/Guia_de_onda_circular" title="Guia de onda circular">Guias de onda</a></li></ul></div></div></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;">Formulação covariante<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism" class="extiw" title="en:Covariant formulation of classical electromagnetism"><span title=""Covariant formulation of classical electromagnetism" na Wikipédia em English">en</span></a>]</small></div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><div style="padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Tensor_eletromagn%C3%A9tico" title="Tensor eletromagnético">Tensor eletromagnético</a><br />(<a href="/wiki/Tensor_eletromagn%C3%A9tico_tens%C3%A3o%E2%80%93energia" class="mw-redirect" title="Tensor eletromagnético tensão–energia">Tensor de tensão–energia</a>)</div></li> <li><a href="/wiki/Quadricorrente" title="Quadricorrente">Quadricorrente</a></li> <li><a href="/wiki/Quadripotencial_eletromagn%C3%A9tico" title="Quadripotencial eletromagnético">Quadripotencial eletromagnético</a></li></ul></div></div></td> </tr><tr><td class="hlist" style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:transparent;text-align:left;color:inherit;background:transparent;border-top:1px solid #aaa;text-align:center;">Cientistas</div><div class="NavContent hlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" class="mw-redirect" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li>Jefimenko<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/Oleg_D._Jefimenko" class="extiw" title="en:Oleg D. Jefimenko"><span title=""Oleg D. Jefimenko" na Wikipédia em English">en</span></a>]</small></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Heinrich_Lenz" title="Heinrich Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Georg_Ohm" class="mw-redirect" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li>Ritchie<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/William_Ritchie_(physicist)" class="extiw" title="en:William Ritchie (physicist)"><span title=""William Ritchie (physicist)" na Wikipédia em English">en</span></a>]</small></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li>Singer<span title="artigo inexistente: “”"></span> <small style="font-style:normal; position:relative; top:-0.2em;">[<a href="https://en.wikipedia.org/wiki/George_Singer" class="extiw" title="en:George Singer"><span title=""George Singer" na Wikipédia em English">en</span></a>]</small></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/William_Thomson" title="William Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li></ul></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-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Eletromagnetismo" title="Predefinição:Eletromagnetismo"><abbr title="Ver esta predefinição">v</abbr></a></li><li class="nv-discutir"><a href="/wiki/Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Eletromagnetismo" title="Predefinição Discussão:Eletromagnetismo"><abbr title="Discutir esta predefinição">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:Eletromagnetismo&action=edit"><abbr title="Editar esta predefinição">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Em <a href="/wiki/F%C3%ADsica" title="Física">física</a>, o <b>momento do dipolo elétrico</b> é a medida da <a href="/wiki/Polaridade" title="Polaridade">polaridade</a> de um sistema de <a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">cargas elétricas</a>. </p><p>O momento do dipolo elétrico para uma distribuição discreta de <a href="/w/index.php?title=Carga_pontual&action=edit&redlink=1" class="new" title="Carga pontual (página não existe)">cargas pontuais</a> é simplesmente a <a href="/wiki/Soma_vetorial" class="mw-redirect" title="Soma vetorial">soma vetorial</a> dos produtos da carga pela posição vetorial de cada carga. </p><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 {\vec {p}}=\sum _{i}q_{i}{\vec {x}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=\sum _{i}q_{i}{\vec {x}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0928cf5db14720070ddd1312284e998992d1294c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:12.221ex; height:5.509ex;" alt="{\displaystyle {\vec {p}}=\sum _{i}q_{i}{\vec {x}}_{i}}"></span> </p><p>Esta definição discreta também pode ser dada em uma forma contínua utilizando-se a densidade da carga, <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>, no lugar da carga, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>. </p><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 {\vec {p}}=\int \rho ({\vec {x}}){\vec {x}}dV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>d</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=\int \rho ({\vec {x}}){\vec {x}}dV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93778f036d2dbb883c84ac965b8023cc2a38c647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; margin-left: -0.089ex; width:15.768ex; height:5.676ex;" alt="{\displaystyle {\vec {p}}=\int \rho ({\vec {x}}){\vec {x}}dV}"></span> </p><p>O momento do dipolo é normalmente utilizado em sistemas que possuem carga total neutra. Por exemplo, um par de cargas opostas, ou um condutor neutro em um <a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">campo elétrico</a> uniforme. Para tais sistemas, o valor do momento do dipolo elétrico é independente da origem do sistema de eixos. Para sistemas não neutros, surge uma dependência da escolha da origem. Para que o momento do dipolo elétrico seja útil no cálculo do <a href="/wiki/Dipolo#torque_em_um_dipolo" title="Dipolo">torque em dipolo</a> e para outros fins, a origem é frequentemente definida no centro da carga, <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 {\vec {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f361eec9da18669c0e7e869c57bae6c657f53522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {R}}}"></span>, para o sistema, que é definida como o <a href="/wiki/Centro_de_massa" title="Centro de massa">centro de massa</a> e é, para alguns sistemas, a mesmo: </p><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 {\vec {R}}={\frac {1}{Q}}\sum _{i}\left|q_{i}\right\vert {\vec {x}}_{i},Q=\sum _{i}\left|q_{i}\right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Q</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>|</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>|</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {R}}={\frac {1}{Q}}\sum _{i}\left|q_{i}\right\vert {\vec {x}}_{i},Q=\sum _{i}\left|q_{i}\right\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c767f21fbbee752f5024ed24d21c244ca7532706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.156ex; height:6.343ex;" alt="{\displaystyle {\vec {R}}={\frac {1}{Q}}\sum _{i}\left|q_{i}\right\vert {\vec {x}}_{i},Q=\sum _{i}\left|q_{i}\right\vert }"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Um_par_de_cargas_opostas">Um par de cargas opostas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=1" title="Editar secção: Um par de cargas opostas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=1" title="Editar código-fonte da secção: Um par de cargas opostas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O <b>momento do dipolo elétrico</b> para um par de cargas opostas de magnitude "q" é definido como a magnitude da carga vezes a distância entre eles e a direção definida em relação à carga positiva. </p><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 {\vec {p}}=q{\vec {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}=q{\vec {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3283dfd62b474f70eca956e5c6effad44c4b2055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.13ex; height:3.176ex;" alt="{\displaystyle {\vec {p}}=q{\vec {d}}}"></span> </p><p>É um conceito útil em <a href="/wiki/%C3%81tomo" title="Átomo">átomos</a> e <a href="/wiki/Mol%C3%A9cula" title="Molécula">moléculas</a> onde os efeitos da separação das cargas são mensuráveis, mas a distância entre as cargas são muito pequenas para serem medidas com facilidade. Também é útil em <a href="/wiki/Diel%C3%A9trico" title="Dielétrico">dielétricos</a> e outras aplicações em materiais sólidos e líquidos. </p> <div class="mw-heading mw-heading2"><h2 id="Expressão_(caso_geral)"><span id="Express.C3.A3o_.28caso_geral.29"></span>Expressão (caso geral)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=2" title="Editar secção: Expressão (caso geral)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=2" title="Editar código-fonte da secção: Expressão (caso geral)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mais geralmente, para uma distribuição contínua de carga confinada a um volume <i>V</i>, a expressão correspondente para o momento de dipolo é: </p><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 p(r)=\int \limits _{V}p(r_{0})(r_{0}-r)d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </munder> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(r)=\int \limits _{V}p(r_{0})(r_{0}-r)d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e5daeff8e625a929a1865e20e93b3d851e4b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:27.054ex; height:7.343ex;" alt="{\displaystyle p(r)=\int \limits _{V}p(r_{0})(r_{0}-r)d^{3}r_{0}}"></span> </p><p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> localiza o ponto de observação e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{3}r_{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{3}r_{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d30ffd1c835a955a640501dc51571a378e67e607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.351ex; height:3.009ex;" alt="{\displaystyle d^{3}r_{o}}"></span> indica um volume elementar em <i>V</i>. Para uma matriz de cargas pontuais, a <a href="/wiki/Densidade_de_carga" title="Densidade de carga">densidade de carga</a> torna-se uma soma das funções delta de <a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a>: </p><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 p(r)=\sum _{i=1}^{N}q_{i}\delta (r-r_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(r)=\sum _{i=1}^{N}q_{i}\delta (r-r_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/210c16b5b292ca4afcca6b29998a8e7d71747fd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:21.389ex; height:7.343ex;" alt="{\displaystyle p(r)=\sum _{i=1}^{N}q_{i}\delta (r-r_{i})}"></span> </p><p>onde cada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> é um vetor de algum ponto de referência para a carga <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}"></span>. A substituição na fórmula de integração acima fornece: </p><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 p(r)=\sum _{i=1}^{N}q_{i}\int \limits _{V}\delta (r_{0}-r_{i})(r_{0}-r_{i})d^{3}r_{0}=\sum _{i=1}^{N}q_{i}(r_{i}-r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munder> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </munder> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(r)=\sum _{i=1}^{N}q_{i}\int \limits _{V}\delta (r_{0}-r_{i})(r_{0}-r_{i})d^{3}r_{0}=\sum _{i=1}^{N}q_{i}(r_{i}-r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3efde808398ab599cd7ba15c59cd193b7f9ac60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:54.612ex; height:8.343ex;" alt="{\displaystyle p(r)=\sum _{i=1}^{N}q_{i}\int \limits _{V}\delta (r_{0}-r_{i})(r_{0}-r_{i})d^{3}r_{0}=\sum _{i=1}^{N}q_{i}(r_{i}-r)}"></span> </p><p>Esta expressão é equivalente à expressão anterior no caso de neutralidade de carga e <i>N</i>=2. Para duas cargas opostas, denotando a localização da carga positiva do par como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558a475f6cf321e8dc4a8090c0288cc87fa0b7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.559ex; height:2.009ex;" alt="{\displaystyle r_{+}}"></span> e a localização da carga negativa como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8785196dcaa6a2ade3a00e5d6a2fccc04c6ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.559ex; height:2.009ex;" alt="{\displaystyle r_{-}}"></span>: </p><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 p(r)=q_{1}(r_{1}-r)+q_{2}(r_{2}-r)=q(r_{+}-r)-q(r_{-}-r)=q(r_{+}-r_{-})=qd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(r)=q_{1}(r_{1}-r)+q_{2}(r_{2}-r)=q(r_{+}-r)-q(r_{-}-r)=q(r_{+}-r_{-})=qd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a112faa2515d1e444c99500ec5d862fcb7b927f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:73.755ex; height:2.843ex;" alt="{\displaystyle p(r)=q_{1}(r_{1}-r)+q_{2}(r_{2}-r)=q(r_{+}-r)-q(r_{-}-r)=q(r_{+}-r_{-})=qd}"></span> </p><p>mostrando que o vetor de momento dipolar é direcionado da carga negativa para a carga positiva porque o <a href="/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática)">vetor</a> de posição de um ponto é direcionado para fora da origem até aquele ponto. </p><p>O momento de dipolo é particularmente útil no contexto de um sistema neutro geral de cargas, por exemplo, um par de cargas opostas ou um condutor neutro em um campo elétrico uniforme. Para tal sistema de cargas, visualizado como uma matriz de cargas opostas emparelhadas, a relação para o momento de dipolo elétrico é: </p><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 p(r)=\sum _{i=1}^{N}\int \limits _{V}q_{i}[\delta (r_{0}-(r_{i}-d_{i}))-\delta (r_{0}-r_{i})](r_{o}-r)d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <munder> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(r)=\sum _{i=1}^{N}\int \limits _{V}q_{i}[\delta (r_{0}-(r_{i}-d_{i}))-\delta (r_{0}-r_{i})](r_{o}-r)d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a8ab568bb17083fe1ba43084a16542d04f4590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:57.618ex; height:8.343ex;" alt="{\displaystyle p(r)=\sum _{i=1}^{N}\int \limits _{V}q_{i}[\delta (r_{0}-(r_{i}-d_{i}))-\delta (r_{0}-r_{i})](r_{o}-r)d^{3}r_{0}}"></span> </p><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 =\sum _{i=1}^{N}q_{i}[r_{i}+d_{i}-r-(r_{i}-r)]=\sum _{i=1}^{N}q_{i}d_{i}=\sum _{i=1}^{N}p_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo>−</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\sum _{i=1}^{N}q_{i}[r_{i}+d_{i}-r-(r_{i}-r)]=\sum _{i=1}^{N}q_{i}d_{i}=\sum _{i=1}^{N}p_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea348613fa8f50faaef09faedc18e2346f1f7397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.795ex; height:7.343ex;" alt="{\displaystyle =\sum _{i=1}^{N}q_{i}[r_{i}+d_{i}-r-(r_{i}-r)]=\sum _{i=1}^{N}q_{i}d_{i}=\sum _{i=1}^{N}p_{i}}"></span> </p><p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> é o ponto de observação e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}=r_{i}^{'}-r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{i}=r_{i}^{'}-r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36135604070ac73a19e0cb0919a2bd2cf4a1fd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.644ex; height:3.343ex;" alt="{\displaystyle d_{i}=r_{i}^{'}-r_{i}}"></span>, sendo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> a posição da carga negativa no dipolo <i>i</i> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}^{'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi></mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}^{'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d350ad235186b352c4539b4e556f24531053d11d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.848ex; height:3.343ex;" alt="{\displaystyle r_{i}^{'}}"></span> a posição da carga positiva. Esta é a <a rel="nofollow" class="external text" href="https://mundoeducacao.uol.com.br/fisica/soma-vetores.htm">soma vetorial</a> dos momentos dipolares individuais dos pares de carga neutros. (Por causa da neutralidade de carga geral, o momento dipolo é independente da posição <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> do observador). Assim, o valor 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> é independente da escolha do ponto de referência, desde que a carga geral do sistema seja zero. </p><p>Ao discutir o momento de dipolo de um sistema não neutro, como o momento de dipolo do <a href="/wiki/Pr%C3%B3ton" title="Próton">próton</a>, surge uma dependência da escolha do ponto de referência. Em tais casos, é convencional escolher o ponto de referência para ser o <a href="/wiki/Centro_de_massas" class="mw-redirect" title="Centro de massas">centro de massa</a> do sistema, não uma origem arbitrária.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup>  Esta escolha não é apenas uma questão de convenção: a noção de momento dipolar é essencialmente derivada da noção mecânica de torque e, como na mecânica, é computacionalmente e teoricamente útil escolher o centro de massa como o ponto de observação. Para uma molécula carregada, o centro de carga deve ser o ponto de referência em vez do centro de massa. Para sistemas neutros, o ponto de referência não é importante. O momento de dipolo é uma propriedade intrínseca do sistema. </p> <div class="mw-heading mw-heading2"><h2 id="Potencial_e_campo_de_um_dipolo_elétrico"><span id="Potencial_e_campo_de_um_dipolo_el.C3.A9trico"></span>Potencial e campo de um dipolo elétrico</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=3" title="Editar secção: Potencial e campo de um dipolo elétrico" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=3" title="Editar código-fonte da secção: Potencial e campo de um dipolo elétrico"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um dipolo ideal consiste em duas cargas opostas com separação infinitesimal. Calculamos o potencial e o campo de tal dipolo ideal começando com duas cargas opostas na separação <i>d></i> 0, e tomando o limite como <i>d →</i> 0. </p><p>Duas cargas opostas próximas ± <i>q</i> têm um potencial da forma: </p><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 (r)={\frac {q}{4\pi \epsilon _{0}|r-r_{+}|}}-{\frac {q}{4\pi \epsilon _{0}|r-r_{-}|}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (r)={\frac {q}{4\pi \epsilon _{0}|r-r_{+}|}}-{\frac {q}{4\pi \epsilon _{0}|r-r_{-}|}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec11e8f88b44c25fafbf5cdae41442769533e593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.971ex; height:5.676ex;" alt="{\displaystyle \phi (r)={\frac {q}{4\pi \epsilon _{0}|r-r_{+}|}}-{\frac {q}{4\pi \epsilon _{0}|r-r_{-}|}},}"></span> </p><p>onde a separação de carga é: </p><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 d=r_{+}-r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=r_{+}-r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd724288dca20860ed84bc3110b391caef4ed06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.763ex; height:2.509ex;" alt="{\displaystyle d=r_{+}-r}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=|d|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=|d|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f68dd666d1d12f3ec8dac62255ca985a1b39e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.824ex; height:2.843ex;" alt="{\displaystyle d=|d|}"></span>. </p><p>Deixe <b>R</b> denotar o vetor posição em relação ao ponto médio <b>r</b> , e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3364cfcc26b52f5010b6bffa74fe10bb1f47e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\hat {R}}}"></span> o vetor unitário correspondente: </p><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 R=r-{\frac {r_{+}+r}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <mi>r</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=r-{\frac {r_{+}+r}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af410a841b7895e6604c4818dfa8f366d7a7b1e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.036ex; height:5.176ex;" alt="{\displaystyle R=r-{\frac {r_{+}+r}{2}}}"></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 {\hat {R}}={\frac {R}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {R}}={\frac {R}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb94cb63dd65123ccad3f372a653c7621823dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.463ex; height:5.343ex;" alt="{\displaystyle {\hat {R}}={\frac {R}{R}}}"></span> </p><p>Expansão de Taylor em <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 {d}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f2cb66c391cd0a938ac9ba0fe17abc78a0663b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.6ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{R}}}"></span>(ver <a href="/wiki/Expans%C3%A3o_multipolar" title="Expansão multipolar">expansão multipolo</a> e <a href="/wiki/Quadrupolo" title="Quadrupolo">quadrupolo</a>) expressa esse potencial como uma série. <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> </p><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 (R)={\frac {1}{4\pi \epsilon _{0}}}{\frac {qd.{\hat {R}}}{R^{2}}}+0\left({\frac {d^{2}}{R^{2}}}\right)\approx {\frac {1}{4\pi \epsilon _{0}}}{\frac {p.{\hat {R}}}{R^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>d</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mn>0</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (R)={\frac {1}{4\pi \epsilon _{0}}}{\frac {qd.{\hat {R}}}{R^{2}}}+0\left({\frac {d^{2}}{R^{2}}}\right)\approx {\frac {1}{4\pi \epsilon _{0}}}{\frac {p.{\hat {R}}}{R^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2af56e6d5214cfa2ebfa8c78f7eb37929230d086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.002ex; height:6.676ex;" alt="{\displaystyle \phi (R)={\frac {1}{4\pi \epsilon _{0}}}{\frac {qd.{\hat {R}}}{R^{2}}}+0\left({\frac {d^{2}}{R^{2}}}\right)\approx {\frac {1}{4\pi \epsilon _{0}}}{\frac {p.{\hat {R}}}{R^{2}}}}"></span>, </p><p>onde termos de ordem superior na série estão desaparecendo em grandes distâncias, <i>R</i>, em comparação com <i>d</i>. Aqui, o momento de dipolo elétrico <b>p</b> é, como acima: </p><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 p=qd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=qd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103671896fb7a4a33f78f682be03d50fe98b7f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.643ex; height:2.509ex;" alt="{\displaystyle p=qd}"></span>. </p><p>O resultado para o potencial dipolo também pode ser expresso como: <sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span>[</span>4<span>]</span></a></sup> </p><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 (R)=-p.\nabla {\frac {1}{4\pi \epsilon _{0}R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>p</mi> <mo>.</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (R)=-p.\nabla {\frac {1}{4\pi \epsilon _{0}R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f83021c0383be62fc60d67a49aff6307bae0322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.098ex; height:5.676ex;" alt="{\displaystyle \phi (R)=-p.\nabla {\frac {1}{4\pi \epsilon _{0}R}}}"></span>, </p><p>que relaciona o potencial dipolo ao de uma carga pontual. Um ponto chave é que o potencial do dipolo diminui mais rápido com a distância <i>R</i> do que com a carga pontual. </p><p>O campo elétrico do dipolo é o gradiente negativo do potencial, levando a: <sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span>[</span>4<span>]</span></a></sup> </p><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 E(R)={\frac {3(p.{\hat {R}}){\hat {R}}-p}{4\pi \epsilon _{0}R^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>p</mi> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>−</mo> <mi>p</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(R)={\frac {3(p.{\hat {R}}){\hat {R}}-p}{4\pi \epsilon _{0}R^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff410c7c7fdece66e377adccaba6ccbff638228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.996ex; height:6.843ex;" alt="{\displaystyle E(R)={\frac {3(p.{\hat {R}}){\hat {R}}-p}{4\pi \epsilon _{0}R^{3}}}}"></span>. </p><p>Assim, embora duas cargas opostas próximas <i>não</i> sejam um dipolo elétrico ideal (porque seu potencial em distâncias curtas não é o de um dipolo), em distâncias muito maiores do que sua separação, seu momento de dipolo <b>p</b> aparece diretamente em seu potencial e campo. </p><p>À medida que as duas cargas são aproximadas ( <i>d</i> fica menor), o termo dipolo na expansão multipolar com base na razão <i>d</i> / <i>R</i> torna-se o único termo significativo em distâncias <i>R</i> cada vez mais próximas e no limite da separação infinitesimal o termo dipolo nesta expansão é tudo o que importa. Como <i>d</i> se torna infinitesimal, entretanto, a carga dipolar deve ser aumentada para manter <b>p</b> constante. Este processo de limitação resulta em um "dipolo pontual". </p> <div class="mw-heading mw-heading2"><h2 id="Densidade_de_momento_dipolo_e_densidade_de_polarização"><span id="Densidade_de_momento_dipolo_e_densidade_de_polariza.C3.A7.C3.A3o"></span>Densidade de momento dipolo e densidade de polarização</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=4" title="Editar secção: Densidade de momento dipolo e densidade de polarização" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=4" title="Editar código-fonte da secção: Densidade de momento dipolo e densidade de polarização"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O momento de dipolo de uma série de cargas, </p><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 p=\sum _{i=1}^{N}q_{i}d_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\sum _{i=1}^{N}q_{i}d_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b7bf3834f367d9d09d1eba2e3e01552ebd4c75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:11.945ex; height:7.343ex;" alt="{\displaystyle p=\sum _{i=1}^{N}q_{i}d_{i}}"></span>, </p><p>determina o grau de polaridade da matriz, mas para uma matriz neutra é simplesmente uma propriedade de vetor da matriz sem informações sobre a localização absoluta da matriz. A <i>densidade</i> de momento dipolo da matriz <b>p</b>(<b>r</b>) contém a localização da matriz e seu momento dipolo. Quando chega a hora de calcular o campo elétrico em alguma região que contém a matriz, as equações de Maxwell são resolvidas e as informações sobre a matriz de carga estão contidas na <i>densidade de polarização</i> <b>P</b>(<b>r</b>) das equações de Maxwell. Dependendo de quão refinada uma avaliação do campo elétrico é necessária, mais ou menos informações sobre a matriz de carga terão que ser expressas por <b>P</b>(<b>r</b>). Conforme explicado abaixo, às vezes é suficientemente preciso tomar <b>P</b>(<b>r</b>)= <b>p</b>(<b>r</b>). Às vezes, uma descrição mais detalhada é necessária (por exemplo, suplementando a densidade do momento dipolo com uma densidade quadrupolar adicional) e às vezes versões ainda mais elaboradas de <b>P</b>(<b>r</b>) são necessárias. </p><p>Agora é explorado apenas de que maneira a densidade de polarização <b>P</b>(<b>r</b>) que entra nas <a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">equações de Maxwell</a> está relacionada ao momento dipolo <b>p</b> de uma matriz neutra geral de cargas, e também à <i>densidade de momento dipolo</i> <b>p</b>(<b>r</b>) (que descreve não apenas o momento de dipolo, mas também a localização da matriz). Apenas as situações estáticas são consideradas no que segue, então <b>P(r)</b> não tem dependência do tempo e não há <a href="/wiki/Corrente_de_deslocamento" title="Corrente de deslocamento">corrente de deslocamento</a>. Primeiro, há alguma discussão sobre a densidade de polarização <b>P</b>(<b>r</b>). Essa discussão é seguida de vários exemplos particulares. </p><p>Uma formulação das equações de Maxwell com base na divisão de cargas e correntes em cargas e correntes "livres" e "ligadas" leva à introdução dos campos <b>D</b> e <b>P</b> : </p><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="{\textstyle D=\epsilon _{0}E+P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>E</mi> <mo>+</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D=\epsilon _{0}E+P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6dbdabc1ba4c0252a21f08ec2e64ea3006519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.383ex; height:2.509ex;" alt="{\textstyle D=\epsilon _{0}E+P}"></span>, </p><p>onde <b>P</b> é chamado de densidade de polarização. Nesta formulação, a divergência desta equação produz: </p><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 .D=\rho _{f}=\epsilon _{0}\nabla .E+\nabla .P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>D</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>E</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .D=\rho _{f}=\epsilon _{0}\nabla .E+\nabla .P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1034c5aacf92187ed889f6543b36abff7d1dc9cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.729ex; height:2.843ex;" alt="{\displaystyle \nabla .D=\rho _{f}=\epsilon _{0}\nabla .E+\nabla .P}"></span>, </p><p>e como o termo de divergência em <b>E</b> é a carga <i>total</i>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de37999e2a8bf78dedbd5644763abad836755b95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.338ex; height:2.343ex;" alt="{\displaystyle \rho _{f}}"></span> é "carga gratuita", ficamos com a relação: </p><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 .P=\rho _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>P</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .P=\rho _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b31d0cec339622cb932595b7c730c3a24a916cdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.953ex; height:2.676ex;" alt="{\displaystyle \nabla .P=\rho _{b}}"></span>, </p><p>com <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 _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71b4c1f19dae23cb553215f66d17cdbe4853dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.14ex; height:2.176ex;" alt="{\displaystyle \rho _{b}}"></span> como a carga ligada, o que significa a diferença entre as densidades de carga total e livre. </p><p>Como um aparte, na ausência de efeitos magnéticos, as equações de Maxwell especificam que </p><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 {\text{x}}E=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> <mi>E</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla {\text{x}}E=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2de8c4b76272fa01758d7dafbded64f2507cc918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.2ex; height:2.176ex;" alt="{\displaystyle \nabla {\text{x}}E=0}"></span>, </p><p>que implica </p><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 {\text{x}}(D-P)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla {\text{x}}(D-P)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90706b249e6f92ccfcd23f7af1987d455ad3d568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.744ex; height:2.843ex;" alt="{\displaystyle \nabla {\text{x}}(D-P)=0}"></span> </p><p>Aplicando a <a href="/wiki/Teorema_da_decomposi%C3%A7%C3%A3o_de_Helmholtz" title="Teorema da decomposição de Helmholtz">decomposição de Helmholtz</a>: <sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>5<span>]</span></a></sup> </p><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 D-P=-\nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>−</mo> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D-P=-\nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9903e1979c780ee48d8ac15fba50023ca8ed1f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.353ex; height:2.343ex;" alt="{\displaystyle D-P=-\nabla }"></span><i>φ,</i> </p><p>para algum potencial escalar <i>φ</i> , e: </p><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 .(D-P)=\epsilon _{0}\nabla .E=\rho _{f}+\rho _{b}=-\nabla ^{2}\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>E</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .(D-P)=\epsilon _{0}\nabla .E=\rho _{f}+\rho _{b}=-\nabla ^{2}\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72c523e425542fd4ee0b5b18203d8d969ed2c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.965ex; height:3.343ex;" alt="{\displaystyle \nabla .(D-P)=\epsilon _{0}\nabla .E=\rho _{f}+\rho _{b}=-\nabla ^{2}\varphi }"></span>. </p><p>Suponha que as cargas sejam divididas em livres e limitadas e o potencial seja dividido em </p><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 \varphi =\varphi _{f}+\varphi _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\varphi _{f}+\varphi _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6908a955dd877c23a33ca49a9c6f017ffbbdc27b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.573ex; height:2.676ex;" alt="{\displaystyle \varphi =\varphi _{f}+\varphi _{b}}"></span>. </p><p>A satisfação das condições de contorno sobre <i>φ</i> pode ser dividida arbitrariamente entre <i>φ <sub>f</sub></i> e <i>φ <sub>b</sub></i> porque apenas a soma <i>φ</i> deve satisfazer essas condições. Segue-se que <b>P</b> é simplesmente proporcional ao campo elétrico devido às cargas selecionadas como limitadas, com condições de contorno que se mostram convenientes.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>6<span>]</span></a></sup>  Em particular, quando <i>nenhuma</i> carga livre está presente, uma escolha possível é <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\epsilon _{0}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\epsilon _{0}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac323208557dcf831446bcb2de8d04755c3542b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.618ex; height:2.509ex;" alt="{\displaystyle P=\epsilon _{0}E}"></span>. </p><p>Em seguida, é discutido como várias descrições de momento dipolo diferentes de um meio se relacionam com a polarização que entra nas equações de Maxwell. </p> <div class="mw-heading mw-heading3"><h3 id="Meio_com_densidades_de_carga_e_dipolo">Meio com densidades de carga e dipolo</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=5" title="Editar secção: Meio com densidades de carga e dipolo" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=5" title="Editar código-fonte da secção: Meio com densidades de carga e dipolo"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Conforme descrito a seguir, um modelo para densidade de momento de polarização <b>p</b>(<b>r</b>) resulta em uma polarização </p><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 P(r)=p(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(r)=p(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d39920520fb61e1e7013a20567799e3d589aafd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.729ex; height:2.843ex;" alt="{\displaystyle P(r)=p(r)}"></span> </p><p>restrito ao mesmo modelo. Para uma distribuição de momento dipolar que varia suavemente <b>p</b>(<b>r</b>), a densidade de carga ligada correspondente é simplesmente </p><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 .p(r)=\rho _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .p(r)=\rho _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a673fe2de372125ab025ab7e2646e89398eb2b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.235ex; height:2.843ex;" alt="{\displaystyle \nabla .p(r)=\rho _{b}}"></span>, </p><p>como iremos estabelecer em breve via <a href="/wiki/Integra%C3%A7%C3%A3o_por_partes" title="Integração por partes">integração por partes</a>. No entanto, se <b>p</b>(<b>r</b>) exibe uma etapa abrupta no momento de dipolo em um limite entre duas regiões, <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 .p(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .p(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c42a74bd8dce0e969e855b0e8ebf9439c8a7036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.997ex; height:2.843ex;" alt="{\displaystyle \nabla .p(r)}"></span> resulta em um componente de carga superficial de carga ligada. Esta carga de superfície pode ser tratada por meio de uma <a href="/wiki/Integral_de_superf%C3%ADcie" title="Integral de superfície">integral de superfície</a> ou usando condições de descontinuidade no limite, conforme ilustrado nos vários exemplos abaixo. </p><p>Como um primeiro exemplo relacionando o momento de dipolo à polarização, considere um meio composto de uma densidade de carga contínua <i>ρ</i> ( <b>r</b> ) e uma distribuição de momento de dipolo contínua <b>p</b>(<b>r</b>).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>7<span>]</span></a></sup> O potencial em uma posição <b>r</b> é: <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>8<span>]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span>[</span>9<span>]</span></a></sup> </p><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 (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}+{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|^{3}}}d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}+{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|^{3}}}d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fbf159770fa642036035360802a135e960e3e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.401ex; height:6.843ex;" alt="{\displaystyle \phi (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}+{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|^{3}}}d^{3}r_{0}}"></span> </p><p>onde <i>ρ</i>(<b>r</b>) é a densidade de carga desemparelhada e <b>p</b>(<b>r</b>) é a densidade de momento dipolo. Usando uma identidade: </p><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 _{r_{0}}{\frac {\rho (r_{0})}{|r-r_{0}|}}={\frac {(r-r_{0})}{|r-r_{0}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{r_{0}}{\frac {\rho (r_{0})}{|r-r_{0}|}}={\frac {(r-r_{0})}{|r-r_{0}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221d611b906fd07e975d7f82df06a7386946ce3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.599ex; height:6.509ex;" alt="{\displaystyle \nabla _{r_{0}}{\frac {\rho (r_{0})}{|r-r_{0}|}}={\frac {(r-r_{0})}{|r-r_{0}|}}}"></span> </p><p>a integral de polarização pode ser transformada: </p><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 {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|}}d^{3}r_{0}={\frac {1}{4\pi \epsilon _{0}}}\int p(r_{0}).\nabla _{r_{0}}{\frac {1}{|r-r_{0}|}}d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|}}d^{3}r_{0}={\frac {1}{4\pi \epsilon _{0}}}\int p(r_{0}).\nabla _{r_{0}}{\frac {1}{|r-r_{0}|}}d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba70af55f51ae80ec2656e1d5033cd2c9abd17ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:61.174ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0}).(r-r_{0})}{|r-r_{0}|}}d^{3}r_{0}={\frac {1}{4\pi \epsilon _{0}}}\int p(r_{0}).\nabla _{r_{0}}{\frac {1}{|r-r_{0}|}}d^{3}r_{0}}"></span>, </p><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 {1}{4\pi \epsilon _{0}}}\int \nabla _{r_{0}}.\left(p(r_{0}){\frac {1}{|r-r_{0}|}}\right)d^{3}r_{0}-{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{4\pi \epsilon _{0}}}\int \nabla _{r_{0}}.\left(p(r_{0}){\frac {1}{|r-r_{0}|}}\right)d^{3}r_{0}-{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b60fdba227355409a18aee43250e9b484d239e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:63.117ex; height:6.509ex;" alt="{\displaystyle ={\frac {1}{4\pi \epsilon _{0}}}\int \nabla _{r_{0}}.\left(p(r_{0}){\frac {1}{|r-r_{0}|}}\right)d^{3}r_{0}-{\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}"></span>, </p><p>O primeiro termo pode ser transformado em uma integral sobre a superfície limitando o volume de integração e contribui com uma densidade de carga superficial, discutida posteriormente. Colocando este resultado de volta ao potencial e ignorando a carga de superfície por enquanto: </p><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 (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})-\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>r</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})-\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdd8cea2fc2285de4779b3c0ad9faaf5565a044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.629ex; height:6.509ex;" alt="{\displaystyle \phi (r)={\frac {1}{4\pi \epsilon _{0}}}\int {\frac {p(r_{0})-\nabla _{r_{0}}.p(r_{0})}{|r-r_{0}|}}d^{3}r_{0}}"></span>, </p><p>onde a integração do volume se estende apenas até a superfície delimitadora e não inclui essa superfície. </p><p>O potencial é determinado pela carga total, que o mostrado acima consiste em: </p><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 _{total}(r_{0})=\rho (r_{0})-\nabla _{r_{0}}.p(r_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{total}(r_{0})=\rho (r_{0})-\nabla _{r_{0}}.p(r_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c2377de1c627cbc696a279347202dc10c57725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.601ex; height:3.009ex;" alt="{\displaystyle \rho _{total}(r_{0})=\rho (r_{0})-\nabla _{r_{0}}.p(r_{0})}"></span>, </p><p>mostrando que: </p><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 _{r_{0}}.p(r_{0})=\rho _{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\nabla _{r_{0}}.p(r_{0})=\rho _{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9d84a086990fb31c7dc99ae9c6e2db9547f09f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.903ex; height:3.009ex;" alt="{\displaystyle -\nabla _{r_{0}}.p(r_{0})=\rho _{b}}"></span>. </p><p>Em suma, a densidade de momento dipolar <b>p</b>(<b>r</b>) desempenha o papel da densidade de polarização <b>P</b> para este meio. Observe, <b>p</b>(<b>r</b>) tem uma divergência diferente de zero igual à densidade de carga limitada (conforme modelado nesta aproximação). </p><p>Pode-se notar que esta abordagem pode ser estendida para incluir todos os multipolos: dipolo, quadrupolo, etc.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span>[</span>10<span>]</span></a></sup> <sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span>[</span>11<span>]</span></a></sup> Usando a relação: </p><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 .D=\rho _{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> <mi>D</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .D=\rho _{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ebe24e6e2652eb2d7a0001f058632a17a15ec53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.331ex; height:2.843ex;" alt="{\displaystyle \nabla .D=\rho _{f}}"></span>, </p><p>a densidade de polarização é considerada: </p><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 P(r)=P_{dip}\nabla .P_{quad}+...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>i</mi> <mi>p</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>.</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(r)=P_{dip}\nabla .P_{quad}+...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e418909d76c6ca405501b6653b1912d256079a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.324ex; height:3.009ex;" alt="{\displaystyle P(r)=P_{dip}\nabla .P_{quad}+...}"></span>, </p><p>onde os termos adicionados são destinados a indicar contribuições de multipolares superiores. Evidentemente, a inclusão de multipolos mais altos significa que a densidade de polarização <b>P</b> não é mais determinada por uma densidade de momento dipolo <b>p</b> sozinha. Por exemplo, ao considerar o espalhamento de uma matriz de carga, diferentes multipolos espalham uma onda eletromagnética de maneira diferente e independente, exigindo uma representação das cargas que vai além da aproximação dipolo. <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span>[</span>12<span>]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Energia_e_torque">Energia e torque</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=6" title="Editar secção: Energia e torque" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=6" title="Editar código-fonte da secção: Energia e torque"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Um objeto com um momento de dipolo elétrico está sujeito a um <a href="/wiki/Torque" title="Torque">torque</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 {\vec {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9476c9325cf30de081340cd070b30c3bd93f311a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.333ex; height:2.343ex;" alt="{\displaystyle {\vec {\tau }}}"></span> quando colocado em um campo elétrico externo. O torque tende a alinhar o dipolo com o campo. Um dipolo alinhado paralelamente a um campo elétrico tem menor <a href="/wiki/Energia_potencial" title="Energia potencial">energia potencial</a> do que um dipolo fazendo algum ângulo com ele. Para um campo elétrico espacialmente uniforme <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 {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}"></span>, a energia <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> e o torque <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 {\vec {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9476c9325cf30de081340cd070b30c3bd93f311a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.333ex; height:2.343ex;" alt="{\displaystyle {\vec {\tau }}}"></span> são dados por:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span>[</span>13<span>]</span></a></sup> </p><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 U=-{\vec {p}}\cdot {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=-{\vec {p}}\cdot {\vec {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1793c8e89e5686a959e8e4f8c61b9c0530403827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.469ex; height:3.176ex;" alt="{\displaystyle U=-{\vec {p}}\cdot {\vec {E}}}"></span>, </p><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 {\vec {\tau }}={\vec {p}}\times {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>τ</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\tau }}={\vec {p}}\times {\vec {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee22abe99b787852b5f712981f9eed5e07fc13e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.373ex; height:3.176ex;" alt="{\displaystyle {\vec {\tau }}={\vec {p}}\times {\vec {E}}}"></span>. </p><p>Nessas expressões <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 {\vec {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}"></span> é o momento de dipolo e o símbolo <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 \times }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>×</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \times }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ffafff1ad26cbe49045f19a67ce532116a32703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.019ex; margin-bottom: -0.19ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \times }"></span> refere-se ao <a href="/wiki/Produto_vetorial" title="Produto vetorial">produto vetorial</a>. O vetor de campo e o vetor de dipolo definem um plano, e o torque é perpendicular a esse plano e seu sentido é dado pela <a href="/wiki/Regra_da_m%C3%A3o_direita" class="mw-redirect" title="Regra da mão direita">regra da mão direita</a>. </p><p>Um dipolo paralelo ou anti-paralelo à direção em que um campo elétrico não uniforme está aumentando (gradiente do campo) experimentará um torque, bem como uma força na direção de seu momento dipolar. Pode-se mostrar que esta força será sempre paralela ao momento dipolar, independentemente da orientação paralela ou anti-paralela do dipolo. </p> <div class="mw-heading mw-heading2"><h2 id="Momentos_de_dipolo_elétrico_de_partículas_fundamentais"><span id="Momentos_de_dipolo_el.C3.A9trico_de_part.C3.ADculas_fundamentais"></span>Momentos de dipolo elétrico de partículas fundamentais</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=7" title="Editar secção: Momentos de dipolo elétrico de partículas fundamentais" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=7" title="Editar código-fonte da secção: Momentos de dipolo elétrico de partículas fundamentais"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Não confundindo com o <a href="/wiki/Spin" title="Spin">spin</a> que se refere aos <a href="/wiki/Momento_magn%C3%A9tico" title="Momento magnético">momentos dipolares magnéticos</a> das partículas, muitos trabalhos experimentais continuam na medição dos momentos dipolares elétricos (MDE) de partículas fundamentais e compostas, nomeadamente as do <a href="/wiki/El%C3%A9tron" title="Elétron">elétron</a> e do <a href="/wiki/N%C3%AAutron" title="Nêutron">nêutron</a>, respectivamente. Como os MDEs violam as simetrias de <a href="/wiki/Paridade_(f%C3%ADsica)" title="Paridade (física)">paridade</a> (P) e <a href="/wiki/Reversibilidade_do_tempo" title="Reversibilidade do tempo">inversão de tempo</a> (T), seus valores produzem uma medida de <a href="/wiki/Simetria_CP" title="Simetria CP">violação de CP</a> por natureza, independente do modelo (assumindo que a simetria de CPT é válida).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span>[</span>14<span>]</span></a></sup> Portanto, os valores para esses MDEs colocam fortes restrições sobre a escala de violação de CP que se estende ao <a href="/wiki/Modelo_Padr%C3%A3o" title="Modelo Padrão">modelo padrão</a> da <a href="/wiki/F%C3%ADsica_de_part%C3%ADculas" title="Física de partículas">física de partículas</a>. As gerações atuais de experimentos são projetadas para serem sensíveis à faixa de <a href="/wiki/Supersimetria" title="Supersimetria">supersimetria</a> dos EDMs, fornecendo experimentos complementares aos realizados no <a href="/wiki/Grande_Colisor_de_H%C3%A1drons" title="Grande Colisor de Hádrons">LHC</a>. <sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span>[</span>15<span>]</span></a></sup> </p><p>Na verdade, muitas teorias são inconsistentes com os limites atuais e foram efetivamente descartadas, e a teoria estabelecida permite um valor muito maior do que esses limites, levando ao forte problema de CP e estimulando buscas por novas partículas como o <a href="/wiki/%C3%81xion" title="Áxion">áxion</a>. <sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span>[</span>16<span>]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Momentos_dipolares_de_moléculas"><span id="Momentos_dipolares_de_mol.C3.A9culas"></span>Momentos dipolares de moléculas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=8" title="Editar secção: Momentos dipolares de moléculas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=8" title="Editar código-fonte da secção: Momentos dipolares de moléculas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Dipolo" title="Dipolo">Os momentos dipolares nas moléculas</a> são responsáveis pelo comportamento de uma substância na presença de campos elétricos externos. Os dipolos tendem a estar alinhados com o campo externo que pode ser constante ou dependente do tempo. Este efeito forma a base de uma técnica experimental moderna chamada <a href="/wiki/Espectroscopia_diel%C3%A9ctrica" title="Espectroscopia dieléctrica">espectroscopia dielétrica</a>. </p><p>Momentos dipolares podem ser encontrados em moléculas comuns, como a água, e também em <a href="/wiki/Biomol%C3%A9cula" title="Biomolécula">biomoléculas</a>, como as <a href="/wiki/Prote%C3%ADna" title="Proteína">proteínas</a>. <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span>[</span>17<span>]</span></a></sup> </p><p>Por meio do momento dipolar total de algum material pode-se calcular a <a href="/wiki/Constante_diel%C3%A9trica" title="Constante dielétrica">constante dielétrica</a> que está relacionada ao conceito mais intuitivo de condutividade. E se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{Tot}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{Tot}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb17c8ff62f610b91b320686ead145b06b077c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.034ex; height:2.509ex;" alt="{\displaystyle M_{Tot}}"></span> é o momento dipolo total da amostra, então a constante dielétrica é dada por, </p><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 =1+k(M_{Tot}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =1+k(M_{Tot}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/986bccbc08a88c72ceee99ef2bcc71b89194c6b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.24ex; height:3.176ex;" alt="{\displaystyle \varepsilon =1+k(M_{Tot}^{2})}"></span> </p><p>onde <i>k</i> é uma constante e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{Tot}^{2}=(M_{Tot}(t=0)).(M_{Tot}(t=0))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{Tot}^{2}=(M_{Tot}(t=0)).(M_{Tot}(t=0))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd524bebf82ca21349b098084a3b56fc26befa6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.674ex; height:3.176ex;" alt="{\displaystyle M_{Tot}^{2}=(M_{Tot}(t=0)).(M_{Tot}(t=0))}"></span> é a função de correlação de tempo do momento de dipolo total. Em geral, o momento dipolo total tem contribuições provenientes de translações e rotações das moléculas na amostra, </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 M_{Tot}=M_{Trans}+M_{Rot}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>r</mi> <mi>a</mi> <mi>n</mi> <mi>s</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{Tot}=M_{Trans}+M_{Rot}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8987f5045f86ca849944065f299c63d6973dd87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.11ex; height:2.509ex;" alt="{\displaystyle M_{Tot}=M_{Trans}+M_{Rot}}"></span></dd></dl> <p>Portanto, a constante dielétrica (e a condutividade) tem contribuições de ambos os termos. Esta abordagem pode ser generalizada para calcular a função dielétrica dependente da frequência. <sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span>[</span>18<span>]</span></a></sup> </p><p>É possível calcular momentos dipolares a partir da teoria da <a href="/wiki/Estrutura_eletr%C3%B4nica" title="Estrutura eletrônica">estrutura eletrônica</a>, seja em resposta a campos elétricos constantes ou a partir da matriz de densidade. <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span>[</span>19<span>]</span></a></sup> Tais valores, entretanto, não são diretamente comparáveis ao experimento devido à presença potencial de efeitos quânticos nucleares, que podem ser substanciais até mesmo para sistemas simples como a molécula de amônia.  <sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span>[</span>20<span>]</span></a></sup> A <a href="https://en.wikipedia.org/wiki/Coupled_cluster" class="extiw" title="en:Coupled cluster">teoria do agrupamento acoplado</a> (especialmente CCSD (T)<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span>[</span>21<span>]</span></a></sup>) pode fornecer momentos de dipolo muito precisos, <sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span>[</span>22<span>]</span></a></sup> embora seja possível obter estimativas razoáveis (dentro de cerca de 5%) da <a href="/wiki/Teoria_do_funcional_da_densidade" title="Teoria do funcional da densidade">teoria do funcional de densidade</a>, especialmente se funcionais híbridos ou duplos são empregados.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span>[</span>23<span>]</span></a></sup> O momento dipolar de uma molécula também pode ser calculado com base na estrutura molecular usando o conceito de métodos de contribuição de grupo. <sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span>[</span>24<span>]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=9" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=9" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dipolo" title="Dipolo">Dipolo</a></li> <li><a href="/wiki/For%C3%A7a_dipolo_permanente" class="mw-redirect" title="Força dipolo permanente">Força dipolo permanente</a></li></ul> <h2 id="Referências" style="cursor: help;" title="Esta seção foi configurada para não ser editável diretamente. Edite a página toda ou a seção anterior em vez disso."><span id="Refer.C3.AAncias"></span>Referências</h2> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Cramer, Christopher J. (24 de junho de 2005). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=tNiyZjAZqKkC&pg=PA307&redir_esc=y"><i>Essentials of Computational Chemistry: Theories and Models</i></a> (em inglês). [S.l.]: Wiley</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Christopher+J.&rft.aulast=Cramer&rft.btitle=Essentials+of+Computational+Chemistry%3A+Theories+and+Models&rft.date=2005-06-24&rft.genre=book&rft.pub=Wiley&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DtNiyZjAZqKkC%26pg%3DPA307%26redir_esc%3Dy&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Dugdale, David (8 de maio de 1997). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=LIwBcIwrwv4C&pg=PA81&redir_esc=y#v=onepage&q&f=false"><i>Essentials of Electromagnetism</i></a> (em inglês). [S.l.]: Springer Science & Business Media</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=David&rft.aulast=Dugdale&rft.btitle=Essentials+of+Electromagnetism&rft.date=1997-05-08&rft.genre=book&rft.pub=Springer+Science+%26+Business+Media&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DLIwBcIwrwv4C%26pg%3DPA81%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation book">Hirose, Kikuji (2005). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=TkvogLqVrqwC&pg=PA18&redir_esc=y#v=onepage&q&f=false"><i>First-principles Calculations in Real-space Formalism: Electronic Configurations and Transport Properties of Nanostructures</i></a> (em inglês). [S.l.]: Imperial College Press</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Kikuji&rft.aulast=Hirose&rft.btitle=First-principles+Calculations+in+Real-space+Formalism%3A+Electronic+Configurations+and+Transport+Properties+of+Nanostructures&rft.date=2005&rft.genre=book&rft.pub=Imperial+College+Press&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DTkvogLqVrqwC%26pg%3DPA18%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-:0-4"><span class="mw-cite-backlink">↑ <sup><i><b><a href="#cite_ref-:0_4-0">a</a></b></i></sup> <sup><i><b><a href="#cite_ref-:0_4-1">b</a></b></i></sup></span> <span class="reference-text"><cite class="citation book">B, Laud B. (1987). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=XtgFvbd9F2UC&pg=PA25&redir_esc=y#v=onepage&q&f=false"><i>Electromagnetics</i></a> (em inglês). [S.l.]: New Age International</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Laud+B.&rft.aulast=B&rft.btitle=Electromagnetics&rft.date=1987&rft.genre=book&rft.pub=New+Age+International&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DXtgFvbd9F2UC%26pg%3DPA25%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite class="citation book">Wu, Jie-Zhi; Ma, Hui-yang; Zhou, M.-D. (20 de abril de 2007). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=P5yNCu44PiwC&pg=PA36&redir_esc=y#v=onepage&q&f=false"><i>Vorticity and Vortex Dynamics</i></a> (em inglês). [S.l.]: Springer Science & Business Media</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.au=Ma%2C+Hui-yang&rft.au=Zhou%2C+M.-D.&rft.aufirst=Jie-Zhi&rft.aulast=Wu&rft.btitle=Vorticity+and+Vortex+Dynamics&rft.date=2007-04-20&rft.genre=book&rft.pub=Springer+Science+%26+Business+Media&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DP5yNCu44PiwC%26pg%3DPA36%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><cite class="citation journal"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Equa%C3%A7%C3%A3o_de_Laplace&oldid=43483911">«Equação de Laplace»</a>. <i>Wikipédia, a enciclopédia livre</i>. 24 de setembro de 2015<span class="reference-accessdate">. Consultado em 3 de dezembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Equa%C3%A7%C3%A3o+de+Laplace&rft.date=2015-09-24&rft.genre=article&rft.jtitle=Wikip%C3%A9dia%2C+a+enciclop%C3%A9dia+livre&rft_id=https%3A%2F%2Fpt.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DEqua%25C3%25A7%25C3%25A3o_de_Laplace%26oldid%3D43483911&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><cite class="citation book">Vanderlinde, Jack (2004). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=HWrMET9_VpUC&pg=PA165&redir_esc=y#v=onepage&q&f=false"><i>Classical Electromagnetic Theory</i></a> (em inglês). [S.l.]: Springer Science & Business Media</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Jack&rft.aulast=Vanderlinde&rft.btitle=Classical+Electromagnetic+Theory&rft.date=2004&rft.genre=book&rft.pub=Springer+Science+%26+Business+Media&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DHWrMET9_VpUC%26pg%3DPA165%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><cite class="citation book">Krey, Uwe; Owen, Anthony (14 de agosto de 2007). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=xZ_QelBmkxYC&pg=PA327&redir_esc=y#v=onepage&q&f=false"><i>Basic Theoretical Physics: A Concise Overview</i></a> (em inglês). [S.l.]: Springer Science & Business Media</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.au=Owen%2C+Anthony&rft.aufirst=Uwe&rft.aulast=Krey&rft.btitle=Basic+Theoretical+Physics%3A+A+Concise+Overview&rft.date=2007-08-14&rft.genre=book&rft.pub=Springer+Science+%26+Business+Media&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DxZ_QelBmkxYC%26pg%3DPA327%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><cite class="citation book">Tsang, T. (1997). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=KQe5QJ9PJwMC&pg=PA59&redir_esc=y#v=onepage&q&f=false"><i>Classical Electrodynamics</i></a> (em inglês). [S.l.]: World Scientific</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=T.&rft.aulast=Tsang&rft.btitle=Classical+Electrodynamics&rft.date=1997&rft.genre=book&rft.pub=World+Scientific&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DKQe5QJ9PJwMC%26pg%3DPA59%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><cite class="citation book">Owen, George Ernest (1 de janeiro de 2003). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=VLm_dqhZUOYC&pg=PA80&redir_esc=y#v=onepage&q&f=false"><i>Introduction to Electromagnetic Theory</i></a> (em inglês). [S.l.]: Courier Corporation</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=George+Ernest&rft.aulast=Owen&rft.btitle=Introduction+to+Electromagnetic+Theory&rft.date=2003-01-01&rft.genre=book&rft.pub=Courier+Corporation&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DVLm_dqhZUOYC%26pg%3DPA80%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><cite class="citation book">Brevet, Pierre-François (1997). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=_clt5ZowQYsC&pg=PA24&redir_esc=y#v=onepage&q&f=false"><i>Surface Second Harmonic Generation</i></a> (em inglês). [S.l.]: PPUR presses polytechniques</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Pierre-Fran%C3%A7ois&rft.aulast=Brevet&rft.btitle=Surface+Second+Harmonic+Generation&rft.date=1997&rft.genre=book&rft.pub=PPUR+presses+polytechniques&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3D_clt5ZowQYsC%26pg%3DPA24%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><cite class="citation book">Jelski, Daniel A.; George, Thomas F. (1999). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=NUv5csOQBGAC&pg=PA219&redir_esc=y#v=onepage&q&f=false"><i>Computational Studies of New Materials</i></a> (em inglês). [S.l.]: World Scientific</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.au=George%2C+Thomas+F.&rft.aufirst=Daniel+A.&rft.aulast=Jelski&rft.btitle=Computational+Studies+of+New+Materials&rft.date=1999&rft.genre=book&rft.pub=World+Scientific&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3DNUv5csOQBGAC%26pg%3DPA219%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><cite class="citation book">Serway, Raymond A.; Jewett, John W. (13 de janeiro de 2010). <a rel="nofollow" class="external text" href="https://books.google.com.br/books?id=1D4VJrWY9ikC&pg=PA756&redir_esc=y"><i>Physics for Scientists and Engineers, Volume 2, Chapters 23-46</i></a> (em inglês). [S.l.]: Cengage Learning</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.au=Jewett%2C+John+W.&rft.aufirst=Raymond+A.&rft.aulast=Serway&rft.btitle=Physics+for+Scientists+and+Engineers%2C+Volume+2%2C+Chapters+23-46&rft.date=2010-01-13&rft.genre=book&rft.pub=Cengage+Learning&rft_id=https%3A%2F%2Fbooks.google.com.br%2Fbooks%3Fid%3D1D4VJrWY9ikC%26pg%3DPA756%26redir_esc%3Dy&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><cite class="citation book">Khriplovich, Iosif B.; Lamoreaux, Steve (1997). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9783642645778"><i>CP Violation Without Strangeness: Electric Dipole Moments of Particles, Atoms, and Molecules</i></a>. Col: Theoretical and Mathematical Physics (em inglês). Berlin Heidelberg: Springer-Verlag</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.au=Lamoreaux%2C+Steve&rft.aufirst=Iosif+B.&rft.aulast=Khriplovich&rft.btitle=CP+Violation+Without+Strangeness%3A+Electric+Dipole+Moments+of+Particles%2C+Atoms%2C+and+Molecules&rft.date=1997&rft.genre=book&rft.place=Berlin+Heidelberg&rft.pub=Springer-Verlag&rft.series=Theoretical+and+Mathematical+Physics&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9783642645778&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><cite class="citation journal">Ibrahim, Tarek; Itani, Ahmad; Nath, Pran (4 de setembro de 2014). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevD.90.055006">«Electron electric dipole moment as a sensitive probe of PeV scale physics»</a>. <i>Physical Review D</i> (5). 055006 páginas. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1103%2FPhysRevD.90.055006">10.1103/PhysRevD.90.055006</a><span class="reference-accessdate">. Consultado em 18 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Electron+electric+dipole+moment+as+a+sensitive+probe+of+PeV+scale+physics&rft.au=Itani%2C+Ahmad&rft.au=Nath%2C+Pran&rft.aufirst=Tarek&rft.aulast=Ibrahim&rft.date=2014-09-04&rft.genre=article&rft.issue=5&rft.jtitle=Physical+Review+D&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FPhysRevD.90.055006&rft_id=info%3Adoi%2F10.1103%2FPhysRevD.90.055006&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><a href="#cite_ref-16">↑</a></span> <span class="reference-text"><cite class="citation journal">Kim, Jihn E.; Carosi, Gianpaolo (4 de março de 2010). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/RevModPhys.82.557">«Axions and the strong $CP$ problem»</a>. <i>Reviews of Modern Physics</i> (1): 557–601. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1103%2FRevModPhys.82.557">10.1103/RevModPhys.82.557</a><span class="reference-accessdate">. Consultado em 18 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Axions+and+the+strong+%24CP%24+problem&rft.au=Carosi%2C+Gianpaolo&rft.aufirst=Jihn+E.&rft.aulast=Kim&rft.date=2010-03-04&rft.genre=article&rft.issue=1&rft.jtitle=Reviews+of+Modern+Physics&rft.pages=557-601&rft_id=https%3A%2F%2Flink.aps.org%2Fdoi%2F10.1103%2FRevModPhys.82.557&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.82.557&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text"><cite class="citation journal">Ojeda-May, Pedro; Garcia, Martin E. (21 de julho de 2010). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2905109/">«Electric Field-Driven Disruption of a Native β-Sheet Protein Conformation and Generation of a Helix-Structure»</a>. <i>Biophysical Journal</i> (2): 595–599. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0006-3495">0006-3495</a>. <a href="/wiki/PubMed_Central" title="PubMed Central">PMC</a> <span class="plainlinks"><a rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pmc/articles/PMC2905109">2905109</a><span style="margin-left:0.1em"><span typeof="mw:File"><span title="Acessível livremente"><img alt="Acessível livremente" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></span>. <a href="/wiki/PubMed_Identifier" class="mw-redirect" title="PubMed Identifier">PMID</a> <a rel="nofollow" class="external text" href="//www.ncbi.nlm.nih.gov/pubmed/20643079">20643079</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2Fj.bpj.2010.04.040">10.1016/j.bpj.2010.04.040</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Electric+Field-Driven+Disruption+of+a+Native+%CE%B2-Sheet+Protein+Conformation+and+Generation+of+a+Helix-Structure&rft.au=Garcia%2C+Martin+E.&rft.aufirst=Pedro&rft.aulast=Ojeda-May&rft.date=2010-07-21&rft.genre=article&rft.issn=0006-3495&rft.issue=2&rft.jtitle=Biophysical+Journal&rft.pages=595-599&rft_id=%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2905109&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2905109%2F&rft_id=info%3Adoi%2F10.1016%2Fj.bpj.2010.04.040&rft_id=info%3Apmid%2F20643079&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a></span> <span class="reference-text"><cite class="citation journal">Shim, Youngseon; Kim, Hyung J. (4 de setembro de 2008). <a rel="nofollow" class="external text" href="https://doi.org/10.1021/jp802595r">«Dielectric Relaxation, Ion Conductivity, Solvent Rotation, and Solvation Dynamics in a Room-Temperature Ionic Liquid»</a>. <i>The Journal of Physical Chemistry B</i> (35): 11028–11038. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1520-6106">1520-6106</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1021%2Fjp802595r">10.1021/jp802595r</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Dielectric+Relaxation%2C+Ion+Conductivity%2C+Solvent+Rotation%2C+and+Solvation+Dynamics+in+a+Room-Temperature+Ionic+Liquid&rft.au=Kim%2C+Hyung+J.&rft.aufirst=Youngseon&rft.aulast=Shim&rft.date=2008-09-04&rft.genre=article&rft.issn=1520-6106&rft.issue=35&rft.jtitle=The+Journal+of+Physical+Chemistry+B&rft.pages=11028-11038&rft_id=https%3A%2F%2Fdoi.org%2F10.1021%2Fjp802595r&rft_id=info%3Adoi%2F10.1021%2Fjp802595r&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a></span> <span class="reference-text"><cite class="citation book">Jensen, Frank (2007). <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/introduction-to-computational-chemistry/oclc/70707839"><i>Introduction to computational chemistry</i></a> (em inglês). Chichester, England; Hoboken, NJ: John Wiley & Sons. <a href="/wiki/OCLC" class="mw-redirect" title="OCLC">OCLC</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/oclc/70707839">70707839</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.aufirst=Frank&rft.aulast=Jensen&rft.btitle=Introduction+to+computational+chemistry&rft.date=2007&rft.genre=book&rft.place=Chichester%2C+England%3B+Hoboken%2C+NJ&rft.pub=John+Wiley+%26+Sons&rft_id=https%3A%2F%2Fwww.worldcat.org%2Ftitle%2Fintroduction-to-computational-chemistry%2Foclc%2F70707839&rft_id=info%3Aoclcnum%2F70707839&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text"><cite class="citation journal">Puzzarini, Cristina (1 de setembro de 2008). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s00214-008-0409-8">«Ab initio characterization of XH3 (X = N,P). Part II. Electric, magnetic and spectroscopic properties of ammonia and phosphine»</a>. <i>Theoretical Chemistry Accounts</i> (em inglês) (1): 1–10. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1432-2234">1432-2234</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2Fs00214-008-0409-8">10.1007/s00214-008-0409-8</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Ab+initio+characterization+of+XH3+%28X+%3D+N%2CP%29.+Part+II.+Electric%2C+magnetic+and+spectroscopic+properties+of+ammonia+and+phosphine&rft.aufirst=Cristina&rft.aulast=Puzzarini&rft.date=2008-09-01&rft.genre=article&rft.issn=1432-2234&rft.issue=1&rft.jtitle=Theoretical+Chemistry+Accounts&rft.pages=1-10&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2Fs00214-008-0409-8&rft_id=info%3Adoi%2F10.1007%2Fs00214-008-0409-8&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text"><cite class="citation journal">Raghavachari, Krishnan; Trucks, Gary W.; Pople, John A.; Head-Gordon, Martin (26 de maio de 1989). <a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/article/pii/S0009261489873956">«A fifth-order perturbation comparison of electron correlation theories»</a>. <i>Chemical Physics Letters</i> (em inglês) (6): 479–483. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0009-2614">0009-2614</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016%2FS0009-2614%2889%2987395-6">10.1016/S0009-2614(89)87395-6</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=A+fifth-order+perturbation+comparison+of+electron+correlation+theories&rft.au=Head-Gordon%2C+Martin&rft.au=Pople%2C+John+A.&rft.au=Trucks%2C+Gary+W.&rft.aufirst=Krishnan&rft.aulast=Raghavachari&rft.date=1989-05-26&rft.genre=article&rft.issn=0009-2614&rft.issue=6&rft.jtitle=Chemical+Physics+Letters&rft.pages=479-483&rft_id=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0009261489873956&rft_id=info%3Adoi%2F10.1016%2FS0009-2614%2889%2987395-6&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text"><cite class="citation journal">Helgaker, Trygve; Jørgensen, Poul; Olsen, Jeppe (11 de agosto de 2000). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9781119019572">«Molecular Electronic-Structure Theory»</a>. <i>Wiley Online Library</i> (em inglês). <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1002%2F9781119019572">10.1002/9781119019572</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Molecular+Electronic-Structure+Theory&rft.au=J%C3%B8rgensen%2C+Poul&rft.au=Olsen%2C+Jeppe&rft.aufirst=Trygve&rft.aulast=Helgaker&rft.date=2000-08-11&rft.genre=article&rft.jtitle=Wiley+Online+Library&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fbook%2F10.1002%2F9781119019572&rft_id=info%3Adoi%2F10.1002%2F9781119019572&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><a href="#cite_ref-23">↑</a></span> <span class="reference-text"><cite class="citation journal">Hait, Diptarka; Head-Gordon, Martin (10 de abril de 2018). <a rel="nofollow" class="external text" href="https://doi.org/10.1021/acs.jctc.7b01252">«How Accurate Is Density Functional Theory at Predicting Dipole Moments? An Assessment Using a New Database of 200 Benchmark Values»</a>. <i>Journal of Chemical Theory and Computation</i> (4): 1969–1981. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1549-9618">1549-9618</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1021%2Facs.jctc.7b01252">10.1021/acs.jctc.7b01252</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=How+Accurate+Is+Density+Functional+Theory+at+Predicting+Dipole+Moments%3F+An+Assessment+Using+a+New+Database+of+200+Benchmark+Values&rft.au=Head-Gordon%2C+Martin&rft.aufirst=Diptarka&rft.aulast=Hait&rft.date=2018-04-10&rft.genre=article&rft.issn=1549-9618&rft.issue=4&rft.jtitle=Journal+of+Chemical+Theory+and+Computation&rft.pages=1969-1981&rft_id=https%3A%2F%2Fdoi.org%2F10.1021%2Facs.jctc.7b01252&rft_id=info%3Adoi%2F10.1021%2Facs.jctc.7b01252&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><a href="#cite_ref-24">↑</a></span> <span class="reference-text"><cite class="citation journal">Müller, Karsten; Mokrushina, Liudmila; Arlt, Wolfgang (12 de abril de 2012). <a rel="nofollow" class="external text" href="https://doi.org/10.1021/je2013395">«Second-Order Group Contribution Method for the Determination of the Dipole Moment»</a>. <i>Journal of Chemical & Engineering Data</i> (4): 1231–1236. <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0021-9568">0021-9568</a>. <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1021%2Fje2013395">10.1021/je2013395</a><span class="reference-accessdate">. Consultado em 19 de novembro de 2020</span></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.atitle=Second-Order+Group+Contribution+Method+for+the+Determination+of+the+Dipole+Moment&rft.au=Arlt%2C+Wolfgang&rft.au=Mokrushina%2C+Liudmila&rft.aufirst=Karsten&rft.aulast=M%C3%BCller&rft.date=2012-04-12&rft.genre=article&rft.issn=0021-9568&rft.issue=4&rft.jtitle=Journal+of+Chemical+%26+Engineering+Data&rft.pages=1231-1236&rft_id=https%3A%2F%2Fdoi.org%2F10.1021%2Fje2013395&rft_id=info%3Adoi%2F10.1021%2Fje2013395&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Ligações_externas"><span id="Liga.C3.A7.C3.B5es_externas"></span>Ligações externas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&veaction=edit&section=10" title="Editar secção: Ligações externas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Momento_do_dipolo_el%C3%A9trico&action=edit&section=10" title="Editar código-fonte da secção: Ligações externas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/ElectricDipoleMoment.html">«Electric Dipole Moment -- from Eric Weisstein's World of Physics»</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fpt.wikipedia.org%3AMomento+do+dipolo+el%C3%A9trico&rft.btitle=Electric+Dipole+Moment+--+from+Eric+Weisstein%27s+World+of+Physics&rft.genre=unknown&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FElectricDipoleMoment.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Momento_do_dipolo_elétrico&oldid=68705941">https://pt.wikipedia.org/w/index.php?title=Momento_do_dipolo_elétrico&oldid=68705941</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:Eletromagnetismo" title="Categoria:Eletromagnetismo">Eletromagnetismo</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:!CS1_ingl%C3%AAs-fontes_em_l%C3%ADngua_(en)" title="Categoria:!CS1 inglês-fontes em língua (en)">!CS1 inglês-fontes em língua (en)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 11h32min de 26 de setembro de 2024.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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Momento do dipolo elétrico – Wikipédia, a enciclopédia livre