Unidades naturais – 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-Constantes_físicas_pretendentes_a_serem_usadas_em_sistemas_de_unidades_naturais" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constantes_físicas_pretendentes_a_serem_usadas_em_sistemas_de_unidades_naturais"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Constantes físicas pretendentes a serem usadas em sistemas de unidades naturais</span> </div> </a> <ul id="toc-Constantes_físicas_pretendentes_a_serem_usadas_em_sistemas_de_unidades_naturais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_de_Planck" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_de_Planck"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Unidades de Planck</span> </div> </a> <ul id="toc-Unidades_de_Planck-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_de_Stoney" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_de_Stoney"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Unidades de Stoney</span> </div> </a> <ul id="toc-Unidades_de_Stoney-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_de_"Schrödinger"" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_de_"Schrödinger""> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Unidades de "Schrödinger"</span> </div> </a> <ul id="toc-Unidades_de_"Schrödinger"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_atômicas_(Hartree)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_atômicas_(Hartree)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Unidades atômicas (Hartree)</span> </div> </a> <ul id="toc-Unidades_atômicas_(Hartree)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sistema_eletrônico_de_unidades" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sistema_eletrônico_de_unidades"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sistema eletrônico de unidades</span> </div> </a> <ul id="toc-Sistema_eletrônico_de_unidades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sistema_de_unidades_eletrodinâmicas_quânticas_(Stille)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sistema_de_unidades_eletrodinâmicas_quânticas_(Stille)"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sistema de unidades eletrodinâmicas quânticas (Stille)</span> </div> </a> <ul id="toc-Sistema_de_unidades_eletrodinâmicas_quânticas_(Stille)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_geometrizadas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_geometrizadas"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Unidades geometrizadas</span> </div> </a> <ul id="toc-Unidades_geometrizadas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unidades_de_N_corpos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unidades_de_N_corpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Unidades de N corpos</span> </div> </a> <ul id="toc-Unidades_de_N_corpos-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">10</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">11</span> <span>Referências</span> </div> </a> <ul id="toc-Referências-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" 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mw-list-item"><a href="https://ca.wikipedia.org/wiki/Unitat_natural" title="Unitat natural — catalão" lang="ca" hreflang="ca" data-title="Unitat natural" 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/P%C5%99irozen%C3%A1_soustava_jednotek" title="Přirozená soustava jednotek — checo" lang="cs" hreflang="cs" data-title="Přirozená soustava jednotek" 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%9F%C4%95%D1%80%D1%87%C4%95%D1%81%D0%B5%D0%BD_%D0%BD%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%BB%C4%83_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95%D1%81%D0%B5%D0%BC" title="Пĕрчĕсен натураллă тытăмĕсем — chuvash" lang="cv" hreflang="cv" data-title="Пĕрчĕсен натураллă тытăмĕсем" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Nat%C3%BCrliche_Einheiten" title="Natürliche Einheiten — alemão" lang="de" hreflang="de" data-title="Natürliche Einheiten" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Natural_units" title="Natural units — inglês" lang="en" hreflang="en" data-title="Natural units" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Loomulike_%C3%BChikute_s%C3%BCsteem" title="Loomulike ühikute süsteem — estónio" lang="et" hreflang="et" data-title="Loomulike ühikute süsteem" 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/%DB%8C%DA%A9%D8%A7%D9%87%D8%A7%DB%8C_%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C" title="یکاهای طبیعی — persa" lang="fa" hreflang="fa" data-title="یکاهای طبیعی" 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/Luonnolliset_yksik%C3%B6t" title="Luonnolliset yksiköt — finlandês" lang="fi" hreflang="fi" data-title="Luonnolliset yksiköt" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_d%27unit%C3%A9s_naturelles" title="Système d'unités naturelles — francês" lang="fr" hreflang="fr" data-title="Système d'unités naturelles" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Aonaid_n%C3%A1d%C3%BArtha" title="Aonaid nádúrtha — irlandês" lang="ga" hreflang="ga" data-title="Aonaid nádúrtha" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%B6%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B4%D5%AB%D5%A1%D5%BE%D5%B8%D6%80%D5%B6%D5%A5%D6%80" title="Բնական միավորներ — arménio" lang="hy" hreflang="hy" data-title="Բնական միավորներ" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Unit%C3%A0_naturali" title="Unità naturali — italiano" lang="it" hreflang="it" data-title="Unità naturali" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%8D%98%E4%BD%8D%E7%B3%BB" title="自然単位系 — japonês" lang="ja" hreflang="ja" data-title="自然単位系" 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%9E%90%EC%97%B0%EB%8B%A8%EC%9C%84%EA%B3%84" title="자연단위계 — coreano" lang="ko" hreflang="ko" data-title="자연단위계" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Natuurlijke_eenheden" title="Natuurlijke eenheden — neerlandês" lang="nl" hreflang="nl" data-title="Natuurlijke eenheden" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Naturlige_enheter" title="Naturlige enheter — norueguês bokmål" lang="nb" hreflang="nb" data-title="Naturlige enheter" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%81%E0%A8%A6%E0%A8%B0%E0%A8%A4%E0%A9%80_%E0%A8%87%E0%A8%95%E0%A8%BE%E0%A8%88%E0%A8%86%E0%A8%82" title="ਕੁਦਰਤੀ ਇਕਾਈਆਂ — panjabi" lang="pa" hreflang="pa" data-title="ਕੁਦਰਤੀ ਇਕਾਈਆਂ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Unit%C4%83%C8%9Bi_naturale" title="Unități naturale — romeno" lang="ro" hreflang="ro" data-title="Unități naturale" 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 href="https://ru.wikipedia.org/wiki/%D0%95%D1%81%D1%82%D0%B5%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D1%8B%D0%B5_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D1%8B_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86" title="Естественные системы единиц — russo" lang="ru" hreflang="ru" data-title="Естественные системы единиц" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Natural_units" title="Natural units — Simple English" lang="en-simple" hreflang="en-simple" data-title="Natural units" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Naravne_enote" title="Naravne enote — esloveno" lang="sl" hreflang="sl" data-title="Naravne enote" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Naturliga_enheter" title="Naturliga enheter — sueco" lang="sv" hreflang="sv" data-title="Naturliga enheter" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Fal_birimler" title="Doğal birimler — turco" lang="tr" hreflang="tr" data-title="Doğal birimler" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%BD%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8_%D0%BE%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8C" title="Природні системи одиниць — ucraniano" lang="uk" hreflang="uk" data-title="Природні системи одиниць" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%8D%95%E4%BD%8D%E5%88%B6" title="自然单位制 — chinês" lang="zh" hreflang="zh" data-title="自然单位制" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q3962243#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" 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.tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Mais_notas plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book-new.svg" class="mw-file-description"><img alt="Esta página cita fontes, mas não cobrem todo o conteúdo" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Este artigo ou secção <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">cita fontes</a>, mas que <b><a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">não cobrem</a> todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Refer%C3%AAncias_e_notas_de_rodap%C3%A9" title="Wikipédia:Livro de estilo/Referências e notas de rodapé">inserir referências</a> (<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=Unidades+naturais">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Unidades+naturais&oq=Unidades+naturais">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Unidades+naturais">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Unidades+naturais">A</a>)</span></small>).</span> <small class="date-container"><i>(<span class="date">Fevereiro de 2011</span>)</i></small></div></td></tr></tbody></table> <p>Em <a href="/wiki/F%C3%ADsica" title="Física">física</a>, <b>unidades naturais</b> são <a href="/wiki/Unidade_de_medida" title="Unidade de medida">unidades físicas</a> de <a href="/wiki/Medida" class="mw-disambig" title="Medida">medida</a> definidas em termos de <a href="/wiki/Constantes_f%C3%ADsicas" class="mw-redirect" title="Constantes físicas">constantes físicas</a> universais. Dessa maneira, quaisquer constantes físicas escolhidas tomam o valor de <b>1</b> quando expressos em termos de um conjunto particular de unidades naturais. </p><p>Unidades naturais representam <a href="/wiki/Express%C3%A3o_matem%C3%A1tica" title="Expressão matemática">expressões algébricas</a> particulares <a href="/w/index.php?title=Adimensionaliza%C3%A7%C3%A3o&action=edit&redlink=1" class="new" title="Adimensionalização (página não existe)">elegantemente simplificadas</a> aparecendo em leis físicas ou a <a href="/wiki/Constante_de_normaliza%C3%A7%C3%A3o" title="Constante de normalização">normalização</a> de algumas grandezas físicas escolhidas que são propriedades de <a href="/wiki/Part%C3%ADcula_elementar" title="Partícula elementar">partículas elementares</a> universais e que devem razoavelmente se acreditar serem constantes. Contudo, o que pode ser apenas uma hipótese, acredita-se e força-se a serem constantes em um sistema de unidades naturais, e pode muito bem ser permitido ou mesmo variável em outro sistema natural de unidades. Unidades naturais <i>são</i> naturais porque a origem de sua definição advém somente de propriedades da <a href="/wiki/Natureza" title="Natureza">natureza</a> e não de qualquer constructo humano. <a href="/wiki/Unidades_de_Planck" title="Unidades de Planck">Unidades de Planck</a> são frequentemente, sem qualificação, chamadas "<i>unidades naturais</i>" mas são somente um sistema de unidades entre outros sistemas. Unidades de Planck podem ser consideradas únicas no que tal sistema de unidades não é baseado em propriedades de qualquer <a href="/wiki/Modelos_f%C3%ADsicos" class="mw-redirect" title="Modelos físicos">protótipo</a> (no sentido de "modelo físico"), objeto, ou <a href="/wiki/Part%C3%ADcula_subat%C3%B4mica" title="Partícula subatômica">partícula</a> mas são baseadas somente em propriedades do <a href="/w/index.php?title=Espa%C3%A7o_livre&action=edit&redlink=1" class="new" title="Espaço livre (página não existe)">espaço livre</a>. </p><p>Como com qualquer conjunto de unidades básicas ou <a href="/w/index.php?title=Unidade_fundamental&action=edit&redlink=1" class="new" title="Unidade fundamental (página não existe)">unidades fundamentais</a>, as unidades básicas de um conjunto de unidades naturais irá incluir definições e valores para <a href="/wiki/Comprimento" title="Comprimento">comprimento</a>, <a href="/wiki/Massa" title="Massa">massa</a>, <a href="/wiki/Tempo" title="Tempo">tempo</a>, <a href="/wiki/Temperatura" title="Temperatura">temperatura</a>, e <a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">carga elétrica</a>. Alguns físicos não têm reconhecido a temperatura como uma dimensão fundamental de grandeza física, já que ela simplesmente expressa a <a href="/wiki/Energia" title="Energia">energia</a> por <a href="/wiki/Graus_de_liberdade_(f%C3%ADsica)" title="Graus de liberdade (física)">grau de liberdade</a> de uma partícula a qual pode ser expressa em termos de energia (ou massa, comprimento e tempo). Virtualmente cada sistema de unidades naturais normaliza a <a href="/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann">constante de Boltzmann</a> a <i>k</i>=1, a qual pode ser entendido como simplesmente outra expressão da definição da unidade de temperatura. Em acréscimo, alguns físicos reconhecem carga elétrica como um dimensão fundamental isolada de grandeza física, mesmo se ela tenha sido expressa em termos de massa, comprimento e tempo em sistemas de unidades tais como o sistema eletrostático <a href="/wiki/Sistema_CGS_de_unidades" title="Sistema CGS de unidades">"CGS"</a>. Virtualmente cada sistema de unidades normaliza a <a href="/wiki/Constante_de_permissividade_do_v%C3%A1cuo" title="Constante de permissividade do vácuo">permissividade do espaço livre</a> a ε<sub>0</sub>=(4π)<sup>-1</sup>, a qual pode ser entendido como uma expressão da definição de unidade de carga. Isto sugere que a controversa<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> adoção em unidades CGS e subsequentemente unidades SI da ideia de Georgi de expressar a <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">lei de Coulomb</a> não como F=kq<sub>1</sub>q<sub>2</sub>/r<sup>2</sup> mas numa "racionalizada" forma de F= (4πε<sub>0</sub>)<sup>-1</sup>q<sub>1</sub>q<sub>2</sub>/r<sup>2</sup> pode não ter sido a "escolha mais natural" de todas. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Constantes_físicas_pretendentes_a_serem_usadas_em_sistemas_de_unidades_naturais"><span id="Constantes_f.C3.ADsicas_pretendentes_a_serem_usadas_em_sistemas_de_unidades_naturais"></span>Constantes físicas pretendentes a serem usadas em sistemas de unidades naturais</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=1" title="Editar secção: Constantes físicas pretendentes a serem usadas em sistemas de unidades naturais" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=1" title="Editar código-fonte da secção: Constantes físicas pretendentes a serem usadas em sistemas de unidades naturais"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As constantes físicas pretendentes a serem normalizadas são escolhidas daquelas da tabela abaixo. Note-se que somente um menor conjunto delas pode ser normalizada em qualquer sistema de unidades sem contradição por definição (<i>e.g.</i>, <i>m<sub>e</sub></i> e <i>m<sub>p</sub></i> não podem ambas serem definidas como a unidade de massa em um único sistema). </p> <table class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff"> <tbody><tr> <th>Constante </th> <th>Símbolo </th> <th>Dimensão </th></tr> <tr> <td><a href="/wiki/Velocidade_da_luz" title="Velocidade da luz">velocidade da luz</a> no vácuo </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {c}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {c}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c001199185ee8d68e90694c69528d2ed43859e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.587ex; height:1.676ex;" alt="{\displaystyle {c}\ }"></span> </td> <td><a href="/wiki/Comprimento" title="Comprimento">L</a> <a href="/wiki/Tempo" title="Tempo">T</a><sup>-1</sup> </td></tr> <tr> <td><a href="/wiki/Constante_gravitacional" class="mw-redirect" title="Constante gravitacional">Constante gravitacional</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {G}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {G}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77fe9a8b002a55a0c5fc913414642e58c5ec3ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.407ex; height:2.176ex;" alt="{\displaystyle {G}\ }"></span> </td> <td><a href="/wiki/Massa" title="Massa">M</a><sup>-1</sup>L<sup>3</sup>T<sup>-2</sup> </td></tr> <tr> <td><a href="/wiki/Constante_de_Planck" title="Constante de Planck">Constante de Planck</a> ("reduzida") </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {h}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {h}{2\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df05180652a87fe1af83af4bba3402117bd18466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.736ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}}"></span> </td> <td>ML<sup>2</sup>T<sup>-1</sup> </td></tr> <tr> <td><a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Constante da força de Coulomb</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66755e48a90db06e3ffcb12f11156e018e76a85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.468ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}"></span> 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 {\varepsilon _{0}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\varepsilon _{0}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009f347885bd6e996b9874fb569659291db42ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle {\varepsilon _{0}}\ }"></span> é a <a href="/wiki/Constante_de_permissividade_do_v%C3%A1cuo" title="Constante de permissividade do vácuo">permissividade do espaço livre</a> </td> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Q</a><sup>-2</sup> M L<sup>3</sup> T<sup>-2</sup> </td></tr> <tr> <td><a href="/wiki/Carga_elementar" title="Carga elementar">Carga elementar</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4204fc2368e201092faf08ddfcb3d4781dd7b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle e\ }"></span> </td> <td>Q </td></tr> <tr> <td><a href="/wiki/El%C3%A9tron" title="Elétron">Massa do elétron</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4ed4496246f8ee0df8ba4ad77733e50b480e12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.009ex;" alt="{\displaystyle m_{e}\ }"></span> </td> <td>M </td></tr> <tr> <td><a href="/wiki/Pr%C3%B3ton" title="Próton">Massa do próton</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{p}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{p}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4f90b7a20d503700b60823eb84154756db1603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.68ex; height:2.343ex;" alt="{\displaystyle m_{p}\ }"></span> </td> <td>M </td></tr> <tr> <td><a href="/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann">Constante de Boltzmann</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {k}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {k}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e06fcc0234af1df544bd987427c6ca3222972b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {k}\ }"></span> </td> <td>ML<sup>2</sup>T<sup>-2</sup><a href="/wiki/Temperatura" title="Temperatura">Θ</a><sup>-1</sup> </td></tr></tbody></table> <p><a href="/wiki/Constante_fundamental" title="Constante fundamental">Constantes físicas adimensionais</a> tais como a <a href="/wiki/Constante_de_estrutura_fina" title="Constante de estrutura fina">constante de estrutura fina</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \ \equiv {\frac {e^{2}}{\hbar c(4\pi \varepsilon _{0})}}={\frac {1}{137.03599911}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mtext> </mtext> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>137.03599911</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \ \equiv {\frac {e^{2}}{\hbar c(4\pi \varepsilon _{0})}}={\frac {1}{137.03599911}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f595771dd05965b1814828734b468d4dba69c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.126ex; height:6.509ex;" alt="{\displaystyle \alpha \ \equiv {\frac {e^{2}}{\hbar c(4\pi \varepsilon _{0})}}={\frac {1}{137.03599911}}}"></span></dd></dl> <p>não podem tomar um valor numérico diferente não importando em qual sistema de unidades é usada. De forma criteriosa escolhem-se unidades que podem somente normalizar constantes físicas que tenham dimensão. Desde que α é um <a href="/wiki/N%C3%BAmero_adimensional" class="mw-redirect" title="Número adimensional">número adimensional</a> fixo não igual a 1, não é possível definir um sistema natural de unidades que normalize <b>todas</b> as constantes físicas que compreendem α. Qualquer 3 das 4 constantes: <i>c</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span>, <i>e</i>, ou 4πε<sub>0</sub>, podem ser normalizadas (deixando a constante física restante tomar um valor que é uma simples função de α, atestando a natureza fundamental da constante de estrutura fina) mas não todas as 4. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_de_Planck">Unidades de Planck</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=2" title="Editar secção: Unidades de Planck" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=2" title="Editar código-fonte da secção: Unidades de Planck"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Unidades_de_Planck" title="Unidades de Planck">Unidades de Planck</a></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th> <th>Valor métrico </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9401bb65a656ca06096321f14ef99665dc41b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.551ex; height:6.343ex;" alt="{\displaystyle l_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}"></span> </td> <td>1.61609735×10<sup>-35</sup> m </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> </mrow> <mi>G</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b81be68fd5a211d266aa0b5d75d910eeee37d59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.078ex; height:6.176ex;" alt="{\displaystyle m_{P}={\sqrt {\frac {\hbar c}{G}}}}"></span> </td> <td>21.7664598 μg </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <mi>G</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628b580dcd254a7fde49a8edd9bde8f79b34f1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.698ex; height:6.343ex;" alt="{\displaystyle t_{P}={\sqrt {\frac {\hbar G}{c^{5}}}}}"></span> </td> <td>5.3907205×10<sup>-44</sup> s </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{P}={\sqrt {\hbar c(4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi class="MJX-variant">ℏ</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{P}={\sqrt {\hbar c(4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38da22787010ed096841c85a8caa3b700697ed69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.68ex; height:4.843ex;" alt="{\displaystyle q_{P}={\sqrt {\hbar c(4\pi \varepsilon _{0})}}}"></span> </td> <td>1.87554573×10<sup>-18</sup> C </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{P}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi class="MJX-variant">ℏ</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mrow> <mi>G</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{P}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbadd1beeda3033f6bab1ad1ab64bc628c094fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.175ex; height:7.509ex;" alt="{\displaystyle T_{P}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}"></span> </td> <td>1.4169206×10<sup>32</sup> K </td></tr></tbody></table> <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 c=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d272662909ab4ecb5e2d2836e9acf1a97c079ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle c=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d786e6c185c86c005f3a96398f3e20e6b49c8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle G=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32eab3c9559e0a58fa26db69e2c450b284bd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle \hbar =1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\sqrt {\alpha }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>α</mi> </msqrt> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {\alpha }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f354085c6bfad73ad2d4994ea4f15cb41745eac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.186ex; height:3.009ex;" alt="{\displaystyle e={\sqrt {\alpha }}\ }"></span></dd></dl> <p>As constantes físicas que as unidades de Planck normalizam são propriedades do <a href="/w/index.php?title=Espa%C3%A7o_livre&action=edit&redlink=1" class="new" title="Espaço livre (página não existe)">espaço livre</a> e não propriedades (tais como carga, massa, tamanho ou raio) de qualquer objeto ou <a href="/wiki/Part%C3%ADcula_elementar" title="Partícula elementar">partícula elementar</a> (que teria que ser escolhido arbitrariamente). Sendo assim, as unidades de Planck são definidas independentemente da <a href="/wiki/Carga_elementar" title="Carga elementar">carga elementar</a> a qual, se medida em termos de unidades de Planck, chega-se a raiz quadrada da <a href="/wiki/Constante_de_estrutura_fina" title="Constante de estrutura fina">constante de estrutura fina</a>, √α. Em unidades de Planck uma variação concebível no valor da adimensional α seria considerada como devida a uma variação do carga elementar. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_de_Stoney">Unidades de Stoney</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=3" title="Editar secção: Unidades de Stoney" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=3" title="Editar código-fonte da secção: Unidades de Stoney"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th> <th>Valor métrico </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{S}={\sqrt {\frac {Ge^{2}}{c^{4}(4\pi \varepsilon _{0})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{S}={\sqrt {\frac {Ge^{2}}{c^{4}(4\pi \varepsilon _{0})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13107583b736e94b26a5ce6b6900fe14a05f976e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.747ex; height:7.676ex;" alt="{\displaystyle l_{S}={\sqrt {\frac {Ge^{2}}{c^{4}(4\pi \varepsilon _{0})}}}}"></span> </td> <td>1.38068×10<sup>-36</sup> m </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{S}={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{S}={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0663ace52d914764c273e82be1b163c88bc6bde8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.859ex; height:7.676ex;" alt="{\displaystyle m_{S}={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}"></span> </td> <td>1.85921×10<sup>-9</sup> kg </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{S}={\sqrt {\frac {Ge^{2}}{c^{6}(4\pi \varepsilon _{0})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{S}={\sqrt {\frac {Ge^{2}}{c^{6}(4\pi \varepsilon _{0})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1265863cc28a14c2daaf5890cc12dc83acdd5083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.893ex; height:7.676ex;" alt="{\displaystyle t_{S}={\sqrt {\frac {Ge^{2}}{c^{6}(4\pi \varepsilon _{0})}}}}"></span> </td> <td>4.60544×10<sup>-45</sup> s </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{S}=e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{S}=e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df7818f129d65333790aa59255961bbc6011a5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.092ex; height:2.009ex;" alt="{\displaystyle q_{S}=e\ }"></span> </td> <td>1.60218×10<sup>-19</sup> C </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{S}={\sqrt {\frac {c^{4}e^{2}}{G(4\pi \varepsilon _{0})k^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{S}={\sqrt {\frac {c^{4}e^{2}}{G(4\pi \varepsilon _{0})k^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f92bffe2081f18abf6265317adf7d5c07a42a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.442ex; height:7.676ex;" alt="{\displaystyle T_{S}={\sqrt {\frac {c^{4}e^{2}}{G(4\pi \varepsilon _{0})k^{2}}}}}"></span> </td> <td>1.21028×10<sup>31</sup> K </td></tr></tbody></table> <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 c=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d272662909ab4ecb5e2d2836e9acf1a97c079ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle c=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d786e6c185c86c005f3a96398f3e20e6b49c8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle G=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa4b16dec5a6513c7c44fc6f89971bc8953135f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle e=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {1}{\alpha }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {1}{\alpha }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82097caab48235c5f08b5c526633755115ad222e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.309ex; height:5.176ex;" alt="{\displaystyle \hbar ={\frac {1}{\alpha }}\ }"></span></dd></dl> <p><a href="/wiki/George_Johnstone_Stoney" title="George Johnstone Stoney">George Johnstone Stoney</a> foi o primeiro físico a introduzir o conceito de unidades naturais. Ele apresentou a ideia em um artigo intitulado <i>"On the Physical Units of Nature"</i> (<i>Sobre as Unidades Físicas da Natureza</i>) entregue à <i><a href="/w/index.php?title=British_Association&action=edit&redlink=1" class="new" title="British Association (página não existe)">British Association</a></i> em 1874.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> Unidades de Stoney fixam a <a href="/wiki/Carga_do_el%C3%A9tron" title="Carga do elétron">carga elementar</a> e permitem à <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante de Planck</a> flutuar. Elas podem ser obtidas das <a href="/wiki/Unidades_de_Planck" title="Unidades de Planck">unidades de Planck</a> com a substituição: </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 \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">←</mo> <mi>α</mi> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ec466699476e34f18c161e422fa4aafd1c08f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.098ex; height:6.509ex;" alt="{\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}"></span>.</dd></dl> <p>Isto remove a constante de Planck das definições e o valor tomado em unidades de Stoney é o recíproco da <a href="/wiki/Constante_de_estrutura_fina" title="Constante de estrutura fina">constante de estrutura fina</a>, 1/α. Em unidades de Stoney uma variação considerável no valor da adimensional α será considerada devida à variação na constante de Planck. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_de_"Schrödinger""><span id="Unidades_de_.22Schr.C3.B6dinger.22"></span>Unidades de "Schrödinger"</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=4" title="Editar secção: Unidades de "Schrödinger"" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=4" title="Editar código-fonte da secção: Unidades de "Schrödinger""><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th> <th>Valor métrico </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{\psi }={\sqrt {\frac {\hbar ^{4}G(4\pi \varepsilon _{0})^{3}}{e^{6}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\psi }={\sqrt {\frac {\hbar ^{4}G(4\pi \varepsilon _{0})^{3}}{e^{6}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a581121fe43ccdbdedd23ab31e9fd5ce4c8f128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.952ex; height:7.676ex;" alt="{\displaystyle l_{\psi }={\sqrt {\frac {\hbar ^{4}G(4\pi \varepsilon _{0})^{3}}{e^{6}}}}}"></span> </td> <td>2.59276×10<sup>-32</sup> m </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\psi }={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\psi }={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea5d06bc601ea462bbc369b31ee3fdd909169cc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.869ex; height:7.676ex;" alt="{\displaystyle m_{\psi }={\sqrt {\frac {e^{2}}{G(4\pi \varepsilon _{0})}}}}"></span> </td> <td>1.85921×10<sup>-9</sup> kg </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\psi }={\sqrt {\frac {\hbar ^{6}G(4\pi \varepsilon _{0})^{5}}{e^{10}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mi>G</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\psi }={\sqrt {\frac {\hbar ^{6}G(4\pi \varepsilon _{0})^{5}}{e^{10}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76eca0efc88c1f95825fbd4cc91bef7c63f17879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.099ex; height:7.676ex;" alt="{\displaystyle t_{\psi }={\sqrt {\frac {\hbar ^{6}G(4\pi \varepsilon _{0})^{5}}{e^{10}}}}}"></span> </td> <td>1.18516×10<sup>-38</sup> s </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\psi }=e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\psi }=e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a652c8e35cc96b97d4d2000ab7cddfd1df00be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.102ex; height:2.343ex;" alt="{\displaystyle q_{\psi }=e\ }"></span> </td> <td>1.602176487×10<sup>-19</sup> C </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\psi }={\sqrt {\frac {e^{10}}{\hbar ^{4}(4\pi \varepsilon _{0})^{5}Gk^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ψ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>G</mi> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\psi }={\sqrt {\frac {e^{10}}{\hbar ^{4}(4\pi \varepsilon _{0})^{5}Gk^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3397fd83d636aa9b3ad560734db126332f02a214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.882ex; height:7.676ex;" alt="{\displaystyle T_{\psi }={\sqrt {\frac {e^{10}}{\hbar ^{4}(4\pi \varepsilon _{0})^{5}Gk^{2}}}}}"></span> </td> <td>6.44490×10<sup>26</sup> K </td></tr></tbody></table> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa4b16dec5a6513c7c44fc6f89971bc8953135f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle e=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d786e6c185c86c005f3a96398f3e20e6b49c8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle G=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32eab3c9559e0a58fa26db69e2c450b284bd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle \hbar =1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\alpha }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\alpha }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee602fd385794ad78200f3fafcc79c966bfeb010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.01ex; height:5.176ex;" alt="{\displaystyle c={\frac {1}{\alpha }}\ }"></span></dd></dl> <p>O nome foi cunhado por <a href="/w/index.php?title=Michael_Duff&action=edit&redlink=1" class="new" title="Michael Duff (página não existe)">Michael Duff</a><a rel="nofollow" class="external autonumber" href="http://www.arxiv.org/abs/hep-th/0208093">[1]</a>. Elas podem ser obtidas das <a href="/wiki/Unidades_de_Planck" title="Unidades de Planck">unidades de Planck</a> com a substituição: </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 c\leftarrow \alpha c={\frac {e^{2}}{\hbar (4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">←</mo> <mi>α</mi> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\leftarrow \alpha c={\frac {e^{2}}{\hbar (4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2149d6c64fc863a4042b5d47e31f0b9020d9d521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.798ex; height:6.509ex;" alt="{\displaystyle c\leftarrow \alpha c={\frac {e^{2}}{\hbar (4\pi \varepsilon _{0})}}}"></span>.</dd></dl> <p>Isto remove a <a href="/wiki/Velocidade_da_luz" title="Velocidade da luz">velocidade da luz</a> das definições e o valor tomado em unidades de Schrödinger é a recíproca da <a href="/wiki/Constante_de_estrutura_fina" title="Constante de estrutura fina">constante de estrutura fina</a>, 1/α. Em unidades de Schrödinger uma considerável variação no valor da adimensional α será considerada devida a variação da velocidade da luz. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_atômicas_(Hartree)"><span id="Unidades_at.C3.B4micas_.28Hartree.29"></span>Unidades atômicas (Hartree)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=5" title="Editar secção: Unidades atômicas (Hartree)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=5" title="Editar código-fonte da secção: Unidades atômicas (Hartree)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Unidades_at%C3%B4micas" title="Unidades atômicas">Unidades atômicas</a></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{A}={\frac {\hbar ^{2}(4\pi \varepsilon _{0})}{m_{e}e^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{A}={\frac {\hbar ^{2}(4\pi \varepsilon _{0})}{m_{e}e^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af56dc8b0abb778ed023dc3e58a30b51845f5c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.91ex; height:6.509ex;" alt="{\displaystyle l_{A}={\frac {\hbar ^{2}(4\pi \varepsilon _{0})}{m_{e}e^{2}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}=m_{e}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{A}=m_{e}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e4b3966415363e723283ad2a2905d99819c4b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.223ex; height:2.009ex;" alt="{\displaystyle m_{A}=m_{e}\ }"></span> </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{A}={\frac {\hbar ^{3}(4\pi \varepsilon _{0})^{2}}{m_{e}e^{4}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{A}={\frac {\hbar ^{3}(4\pi \varepsilon _{0})^{2}}{m_{e}e^{4}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc630ed332db6b5536d077ba6cf4cd14cd46ae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.111ex; height:6.509ex;" alt="{\displaystyle t_{A}={\frac {\hbar ^{3}(4\pi \varepsilon _{0})^{2}}{m_{e}e^{4}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{A}=e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{A}=e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fbbeb26f3cfabfaca6a076aa5425bbbfc3b86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.264ex; height:2.009ex;" alt="{\displaystyle q_{A}=e\ }"></span> </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}={\frac {m_{e}e^{4}}{\hbar ^{2}(4\pi \varepsilon _{0})^{2}k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}={\frac {m_{e}e^{4}}{\hbar ^{2}(4\pi \varepsilon _{0})^{2}k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5f42f13b28f1561d35da15d3f4652e36a33e4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.84ex; height:6.509ex;" alt="{\displaystyle T_{A}={\frac {m_{e}e^{4}}{\hbar ^{2}(4\pi \varepsilon _{0})^{2}k}}}"></span> </td></tr></tbody></table> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa4b16dec5a6513c7c44fc6f89971bc8953135f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle e=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70ce417efd3a047ff7c8d2dbd085e65c90e3faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.88ex; height:2.509ex;" alt="{\displaystyle m_{e}=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32eab3c9559e0a58fa26db69e2c450b284bd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle \hbar =1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\alpha }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\alpha }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee602fd385794ad78200f3fafcc79c966bfeb010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.01ex; height:5.176ex;" alt="{\displaystyle c={\frac {1}{\alpha }}\ }"></span></dd></dl> <p>Primeiramente proposta por <a href="/wiki/Douglas_Hartree" title="Douglas Hartree">Douglas Hartree</a> para simplificar a física do <a href="/wiki/%C3%81tomo_de_hidrog%C3%AAnio" class="mw-redirect" title="Átomo de hidrogênio">átomo de hidrogênio</a>. <a href="/w/index.php?title=Michael_Duff&action=edit&redlink=1" class="new" title="Michael Duff (página não existe)">Michael Duff</a><a rel="nofollow" class="external autonumber" href="http://www.arxiv.org/abs/hep-th/0208093">[2]</a> chama estas de "unidades de Bohr". A unidade de <a href="/wiki/Energia" title="Energia">energia</a> neste sistema é a energia total do <a href="/wiki/El%C3%A9tron" title="Elétron">elétron</a> na órbita circular do <a href="/wiki/%C3%81tomo_de_Bohr" title="Átomo de Bohr">átomo de Bohr</a> e é chamada de <a href="/wiki/Hartree" title="Hartree">energia de Hartree</a>, <i>E</i><sub>h</sub>. A unidade de velocidade é a velocidade deste elétron, a unidade de massa é a <a href="/wiki/El%C3%A9tron" title="Elétron">massa do elétron</a>, <i>m</i><sub>e</sub>, e a unidade de comprimento é o <a href="/wiki/Raio_de_Bohr" title="Raio de Bohr">raio de Bohr</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 a_{0}=4\pi \varepsilon _{0}\hbar ^{2}/m_{e}e^{2}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=4\pi \varepsilon _{0}\hbar ^{2}/m_{e}e^{2}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d9398eb58172be93c05470edf68e8ddf50da48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.31ex; height:3.176ex;" alt="{\displaystyle a_{0}=4\pi \varepsilon _{0}\hbar ^{2}/m_{e}e^{2}\ }"></span>. Elas podem ser obtidas das unidades de "Schrödinger" com a substituição: </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 G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">←</mo> <mi>α</mi> <mi>G</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ede55363ebcd6309d6966a08ea4bae2d925b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.816ex; height:6.509ex;" alt="{\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }"></span>.</dd></dl> <p>Isto remove a <a href="/wiki/Velocidade_da_luz" title="Velocidade da luz">velocidade da luz</a> (assim como a <a href="/wiki/Constante_gravitacional" class="mw-redirect" title="Constante gravitacional">constante gravitacional</a>) das definições e o valor tomada em unidades atômicas é o recíproco da <a href="/wiki/Constante_de_estrutura_fina" title="Constante de estrutura fina">constante de estrutura fina</a>, 1/α. Em unidades atômicas uma considerável variação no valor da adimensional α será considerada devida à variação da velocidade da luz. </p> <div class="mw-heading mw-heading2"><h2 id="Sistema_eletrônico_de_unidades"><span id="Sistema_eletr.C3.B4nico_de_unidades"></span>Sistema eletrônico de unidades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=6" title="Editar secção: Sistema eletrônico de unidades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=6" title="Editar código-fonte da secção: Sistema eletrônico de unidades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{e}={\frac {e^{2}}{c^{2}m_{e}(4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{e}={\frac {e^{2}}{c^{2}m_{e}(4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d6ad019ce911122700b71461db579b42f7111e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.168ex; height:6.509ex;" alt="{\displaystyle l_{e}={\frac {e^{2}}{c^{2}m_{e}(4\pi \varepsilon _{0})}}}"></span> </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}=m_{e}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}=m_{e}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3310bd4bf6ef38aa38a2b03e37de492154ad5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.757ex; height:2.009ex;" alt="{\displaystyle m_{e}=m_{e}\ }"></span> </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{e}={\frac {e^{2}}{c^{3}m_{e}(4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{e}={\frac {e^{2}}{c^{3}m_{e}(4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef3f4b9cc6fe1fdd9fe8d9a57c80dba7d944e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.314ex; height:6.509ex;" alt="{\displaystyle t_{e}={\frac {e^{2}}{c^{3}m_{e}(4\pi \varepsilon _{0})}}}"></span> </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{e}=e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{e}=e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53be9220d8b60ed046e74ba283b5a3880975c461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.798ex; height:2.009ex;" alt="{\displaystyle q_{e}=e\ }"></span> </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{e}={\frac {m_{e}c^{2}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{e}={\frac {m_{e}c^{2}}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8a20cb29db79266fc7d8412c9d02156158d715" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.39ex; height:5.843ex;" alt="{\displaystyle T_{e}={\frac {m_{e}c^{2}}{k}}}"></span> </td></tr></tbody></table> <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 c=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d272662909ab4ecb5e2d2836e9acf1a97c079ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle c=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa4b16dec5a6513c7c44fc6f89971bc8953135f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle e=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{e}=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70ce417efd3a047ff7c8d2dbd085e65c90e3faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.88ex; height:2.509ex;" alt="{\displaystyle m_{e}=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {1}{\alpha }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {1}{\alpha }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82097caab48235c5f08b5c526633755115ad222e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.309ex; height:5.176ex;" alt="{\displaystyle \hbar ={\frac {1}{\alpha }}\ }"></span></dd></dl> <p><a href="/w/index.php?title=Michael_Duff&action=edit&redlink=1" class="new" title="Michael Duff (página não existe)">Michael Duff</a><a rel="nofollow" class="external autonumber" href="http://www.arxiv.org/abs/hep-th/0208093">[3]</a> chamou estas "unidades de Dirac". Elas podem ser obtidas das <b>unidades de Stoney</b> com a substituição: </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 G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">←</mo> <mi>α</mi> <mi>G</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ede55363ebcd6309d6966a08ea4bae2d925b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.816ex; height:6.509ex;" alt="{\displaystyle G\leftarrow \alpha G\left({\frac {m_{P}}{m_{e}}}\right)^{2}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}^{2}}}\ }"></span>.</dd></dl> <p>Elas podem ser obtidas das <a href="/wiki/Unidades_at%C3%B4micas" title="Unidades atômicas">unidades atômicas</a> com a substituição: </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 \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo stretchy="false">←</mo> <mi>α</mi> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ec466699476e34f18c161e422fa4aafd1c08f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.098ex; height:6.509ex;" alt="{\displaystyle \hbar \leftarrow \alpha \hbar ={\frac {e^{2}}{c(4\pi \varepsilon _{0})}}}"></span>.</dd></dl> <p>Similarmente às unidades de Stoney, uma variação considerável no valor de α será considerada devida a variação na constante de Planck. </p> <div class="mw-heading mw-heading2"><h2 id="Sistema_de_unidades_eletrodinâmicas_quânticas_(Stille)"><span id="Sistema_de_unidades_eletrodin.C3.A2micas_qu.C3.A2nticas_.28Stille.29"></span>Sistema de unidades eletrodinâmicas quânticas (Stille)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=7" title="Editar secção: Sistema de unidades eletrodinâmicas quânticas (Stille)" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=7" title="Editar código-fonte da secção: Sistema de unidades eletrodinâmicas quânticas (Stille)"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (L) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{\mathrm {QED} }={\frac {\hbar }{m_{p}c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>c</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{\mathrm {QED} }={\frac {\hbar }{m_{p}c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88830b51f32e0241fcbca31667e1a6a0fa12475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.62ex; height:6.009ex;" alt="{\displaystyle l_{\mathrm {QED} }={\frac {\hbar }{m_{p}c}}}"></span> </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\mathrm {QED} }=m_{p}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\mathrm {QED} }=m_{p}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a204247cd449fa0e6e70994d406c120db10f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.705ex; height:2.343ex;" alt="{\displaystyle m_{\mathrm {QED} }=m_{p}\ }"></span> </td></tr> <tr> <td><a href="/wiki/Tempo" title="Tempo">Tempo</a> (T) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{\mathrm {QED} }={\frac {\hbar }{m_{p}c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{\mathrm {QED} }={\frac {\hbar }{m_{p}c^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7298fe020b4b39ad768ab208232d494e979c7000" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.82ex; height:6.176ex;" alt="{\displaystyle t_{\mathrm {QED} }={\frac {\hbar }{m_{p}c^{2}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Carga elétrica</a> (Q) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{\mathrm {QED} }=e\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>e</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{\mathrm {QED} }=e\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d7d4d71cf1fc113a01b9ca8cfd16469174ac9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.685ex; height:2.343ex;" alt="{\displaystyle q_{\mathrm {QED} }=e\ }"></span> </td></tr> <tr> <td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (Θ) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {QED} }={\frac {m_{p}c^{2}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {QED} }={\frac {m_{p}c^{2}}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80ca649e92914baf9a53f22f065504c6d285c46c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.338ex; height:6.009ex;" alt="{\displaystyle T_{\mathrm {QED} }={\frac {m_{p}c^{2}}{k}}}"></span> </td></tr></tbody></table> <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 c=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d272662909ab4ecb5e2d2836e9acf1a97c079ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle c=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa4b16dec5a6513c7c44fc6f89971bc8953135f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.925ex; height:2.176ex;" alt="{\displaystyle e=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{p}=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{p}=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d11da3d828f3ecc75fda19dd4b7be9befa6d560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.941ex; height:2.843ex;" alt="{\displaystyle m_{p}=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}={\alpha }\ }"> <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"> <mi>α</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}={\alpha }\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5894655c9275a0d68c2c5b37d733a2871ad0043a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.635ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}={\alpha }\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar =1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32eab3c9559e0a58fa26db69e2c450b284bd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle \hbar =1\ }"></span></dd></dl> <p>Também a <a href="/w/index.php?title=Massa_do_pr%C3%B3ton&action=edit&redlink=1" class="new" title="Massa do próton (página não existe)">massa do próton</a> podem ser substituída pela <a href="/w/index.php?title=Massa_do_el%C3%A9tron&action=edit&redlink=1" class="new" title="Massa do elétron (página não existe)">massa do elétron</a>. Uma variação considerável no valor de α seria considerada como devida a variação 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 \varepsilon _{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a2174977276f49cef175213d540fd5062520f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}\ }"></span>. A métrica de Minkowsky é invariante frente à transformação de Lorentz. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_geometrizadas">Unidades geometrizadas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=8" title="Editar secção: Unidades geometrizadas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=8" title="Editar código-fonte da secção: Unidades geometrizadas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Sistema_de_unidade_geometrizada" title="Sistema de unidade geometrizada">Sistema de unidade geometrizada</a></div> <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 c=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d272662909ab4ecb5e2d2836e9acf1a97c079ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle c=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d786e6c185c86c005f3a96398f3e20e6b49c8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle G=1\ }"></span></dd></dl> <p>O sistema de unidades geometrizadas não é um sistema completamente definido ou único. Neste sistema, as unidades físicas básicas são escolhidas de modo que <a href="/wiki/Velocidade_da_luz" title="Velocidade da luz">velocidade da luz</a> e a <a href="/wiki/Constante_gravitacional" class="mw-redirect" title="Constante gravitacional">constante gravitacional</a> são definidas iguais à unidade, deixando a latitude também definir algumas outras constantes, tais como a <a href="/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann">constante de Boltzmann</a> e a <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">constante da força de Coulomb</a> iguais à unidade: </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 k=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df3c2c1094c69deef780e9c5280f17a4a6431a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.053ex; height:2.176ex;" alt="{\displaystyle k=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"> <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> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22c12dfd6a6ee2655db2715e04d805dd355eb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.729ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}=1}"></span></dd></dl> <p>Se a <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante reduzida de Planck</a> também definida à unidade, </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 \hbar =1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar =1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa32eab3c9559e0a58fa26db69e2c450b284bd0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.148ex; height:2.176ex;" alt="{\displaystyle \hbar =1\ }"></span></dd></dl> <p>então as unidades geometrizadas são idênticas as <a href="/wiki/Unidades_de_Planck" title="Unidades de Planck">unidades de Planck</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Unidades_de_N_corpos">Unidades de N corpos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_naturais&veaction=edit&section=9" title="Editar secção: Unidades de N corpos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_naturais&action=edit&section=9" title="Editar código-fonte da secção: Unidades de N corpos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" align="right" style="margin-left: 1em; background-color: #ffffff"> <tbody><tr> <th>Grandeza </th> <th>Expressão </th></tr> <tr align="left"> <td><a href="/wiki/Comprimento" title="Comprimento">Comprimento</a> (R) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{R}}={\frac {1}{N(N-1)}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {1}{r_{j}-r_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{R}}={\frac {1}{N(N-1)}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {1}{r_{j}-r_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ece75f78b68a7eef4532760dfd0dde27d2b8129b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.829ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{R}}={\frac {1}{N(N-1)}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {1}{r_{j}-r_{i}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Massa" title="Massa">Massa</a> (M) </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\sum _{i=1}^{N}m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</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>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\sum _{i=1}^{N}m_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cc9f145d9dab4b6b5b44eba25973eec259c631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.123ex; height:7.343ex;" alt="{\displaystyle M=\sum _{i=1}^{N}m_{i}}"></span> </td></tr></tbody></table> <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=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3460b86c5dbf5f9ed1ad1f91bab0e74ef545fdb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.284ex; height:2.176ex;" alt="{\displaystyle M=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d786e6c185c86c005f3a96398f3e20e6b49c8044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.668ex; height:2.176ex;" alt="{\displaystyle G=1\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=1\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=1\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d50c7f47e6053b1c2dc885dd3b1c653e41a12994" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle R=1\ }"></span></dd></dl> <p>Unidades de N corpos são um sistema de unidades completamente autocontido usado para <a href="/wiki/Simula%C3%A7%C3%A3o_de_N_corpos" title="Simulação de N corpos">simulações de N corpos</a> de sistemas autogravitantes em <a href="/wiki/Astrof%C3%ADsica" title="Astrofísica">astrofísica</a>. Neste sistema, as unidades físicas básicas são escolhidas de modo que a massa total (M), a <a href="/wiki/Constante_gravitacional" class="mw-redirect" title="Constante gravitacional">constante gravitacional</a> (G) e o <a href="/wiki/Teorema_do_virial" title="Teorema do virial">raio virial</a> (R) são definidos como iguais à unidade. O pressuposto subjacente é que o sistema de N objetos (estrelas) satisfaz o <a href="/w/index.php?title=Teorema_virial&action=edit&redlink=1" class="new" title="Teorema virial (página não existe)">teorema virial</a>. A consequência das unidades padrões de N corpos é que a velocidade de dispersão do sistema 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 v=1/{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=1/{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c8630597cf8ccbe19e613e86b93650e7c43921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.649ex; height:3.176ex;" alt="{\displaystyle v=1/{\sqrt {2}}}"></span> e que a dinâmica da passagem do tempo estabelece-se em escala 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 t=2{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=2{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69469bce678e604889509d6d3ee28f4fd90d2eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.199ex; height:3.009ex;" alt="{\displaystyle t=2{\sqrt {2}}}"></span>. </p><p>A primeira menção das unidades padrões de N corpos foi feito por Michel Hénon (1971) <a rel="nofollow" class="external autonumber" href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1971Ap%26SS..14..151H&db_key=AST&data_type=HTML&format=&high=45c123dad007642">[4]</a>. Elas foram adotadas por Haldan Cohn (1979) <a rel="nofollow" class="external autonumber" href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1979ApJ...234.1036C&db_key=AST&data_type=HTML&format=&high=439859188922689">[5]</a> e posteriormente largamente divulgadas e generalizadas por Douglas Heggie e Robert Mathieu (1986). <a rel="nofollow" class="external autonumber" href="http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1986LNP...267..233H&db_key=AST&data_type=HTML&format=&high=439859188922689">[6]</a> </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=Unidades_naturais&veaction=edit&section=10" 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=Unidades_naturais&action=edit&section=10" 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/An%C3%A1lise_dimensional" title="Análise dimensional">Análise dimensional</a></li> <li><a href="/wiki/Constante_f%C3%ADsica" title="Constante física">Constante física</a></li> <li><a href="/w/index.php?title=Unidades_antr%C3%B3picas&action=edit&redlink=1" class="new" title="Unidades antrópicas (página não existe)">Unidades antrópicas</a></li> <li><a href="/w/index.php?title=Unidade_fundamental&action=edit&redlink=1" class="new" title="Unidade fundamental (página não existe)">Unidade fundamental</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"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML">A Short History of the SI Units in Electricity</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090429035624/http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML">Arquivado em</a> 29 de abril de 2009, no <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>., Ludwik Kowalski, The Physics Teacher, February, 1986, (volume 24 #32) pages 97-99</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Ray, T.P. (1981). "Stoney's Fundamental Units". Irish Astronomical Journal 15: 152. </span> </li> </ol></div></div> <div role="navigation" class="navbox" aria-labelledby="Sistemas_de_unidades" style="padding:3px"><table class="nowraplinks collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Sistemas_de_medi%C3%A7%C3%A3o" title="Predefinição:Sistemas de medição"><abbr title="Ver esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">v</abbr></a></li><li class="nv-discutir"><a href="/w/index.php?title=Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Sistemas_de_medi%C3%A7%C3%A3o&action=edit&redlink=1" class="new" title="Predefinição Discussão:Sistemas de medição (página não existe)"><abbr title="Discutir esta predefinição" style=";;background:none 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não existe)">Sueco</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Unidades_de_medi%C3%A7%C3%A3o_polacas_obsoletas&action=edit&redlink=1" class="new" title="Unidades de medição polacas obsoletas (página não existe)">Polaco</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Unidades_de_medida_romenas&action=edit&redlink=1" class="new" title="Unidades de medida romenas (página não existe)">Romeno</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Pesos_e_medidas_obsoletas_russas&action=edit&redlink=1" class="new" title="Pesos e medidas obsoletas russas (página não existe)">Russo</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Pesos_e_medidas_obsoletas_t%C3%A1rtaros&action=edit&redlink=1" class="new" title="Pesos e medidas obsoletas tártaros (página não existe)">Tártaro</a> <b>·</b></span> <span style="white-space:nowrap"> <a 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(página não existe)">Taiwanês</a></span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sistemas antigos</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div> <p> <span style="white-space:nowrap"><a href="/w/index.php?title=Pesos_e_medidas_da_Gr%C3%A9cia_Antiga&action=edit&redlink=1" class="new" title="Pesos e medidas da Grécia Antiga (página não existe)">Grego</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Unidades_de_medida_da_Roma_Antiga" title="Unidades de medida da Roma Antiga">Romano</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Pesos_e_medidas_do_Egito_Antigo&action=edit&redlink=1" class="new" title="Pesos e medidas do Egito Antigo (página não existe)">Egípcio</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Pesos_e_medidas_hebraico_antigo&action=edit&redlink=1" class="new" title="Pesos e medidas hebraico antigo (página não existe)">Hebraico</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Pesos_e_medidas_%C3%A1rabe_antigo&action=edit&redlink=1" class="new" title="Pesos e medidas árabe antigo (página não existe)">Arábico</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/wiki/Unidades_de_medida_da_antiga_Mesopot%C3%A2mia" title="Unidades de medida da antiga Mesopotâmia">Mesopotâmio</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Unidades_de_medida_persa&action=edit&redlink=1" class="new" title="Unidades de medida persa (página não existe)">Persa</a> <b>·</b></span> <span style="white-space:nowrap"> <a href="/w/index.php?title=Hist%C3%B3ria_dos_sistemas_de_medi%C3%A7%C3%A3o_na_%C3%8Dndia&action=edit&redlink=1" class="new" title="História dos sistemas de medição na Índia (página não existe)">Indiano</a></span> </p> </div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Outros sistemas</th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div> <p> <span style="white-space:nowrap"><a href="/wiki/Modulor" title="Modulor">Modulor</a></span> </p> </div></div></td></tr></tbody></table></div> <table class="noprint" style="border-top:none; border-bottom:none; background:transparent; color: inherit"><tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Ficheiro:U%2B269B.svg" class="mw-file-description"><img alt="Ícone de esboço" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/U%2B269B.svg/35px-U%2B269B.svg.png" decoding="async" width="35" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/U%2B269B.svg/53px-U%2B269B.svg.png 1.5x, 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