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	<h1 id="firstHeading" class="firstHeading" lang="pt">Unidades de Planck</h1>
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.mw-parser-output .compact-ambox table .mbox-text,body.skin-cologneblue .mw-parser-output .compact-ambox table .mbox-text,body.skin-modern .mw-parser-output .compact-ambox table .mbox-text,body.skin-monobook .mw-parser-output .compact-ambox table .mbox-text,body.skin-timeless .mw-parser-output .compact-ambox table .mbox-text{padding:0!important;margin:0!important}body.skin-vector .mw-parser-output .compact-ambox table .mbox-text-span,body.skin-cologneblue .mw-parser-output .compact-ambox table .mbox-text-span,body.skin-modern .mw-parser-output .compact-ambox table .mbox-text-span,body.skin-monobook .mw-parser-output .compact-ambox table .mbox-text-span,body.skin-timeless .mw-parser-output .compact-ambox table .mbox-text-span{display:list-item;line-height:1.5em;list-style-type:square}body.skin-vector .mw-parser-output .compact-ambox .hide-when-compact,body.skin-cologneblue .mw-parser-output .compact-ambox .hide-when-compact,body.skin-modern .mw-parser-output .compact-ambox .hide-when-compact,body.skin-monobook .mw-parser-output .compact-ambox .hide-when-compact,body.skin-timeless .mw-parser-output .compact-ambox .hide-when-compact{display:none}body.skin-vector .mw-parser-output .compact-ambox table .mbox-text-span{list-style-type:disc}body.skin-minerva .mw-parser-output .hide-when-compact{display:none}</style><table class="box-Mais_fontes plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><a href="/wiki/Ficheiro:Question_book-new.svg" class="image"><img alt="" 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" 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="262" data-file-height="204" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">Esta página cita <a href="/wiki/Wikip%C3%A9dia:Fontes_confi%C3%A1veis" title="Wikipédia:Fontes confiáveis">fontes confiáveis</a>, mas que <b>não cobrem todo o conteúdo</b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:REF" class="mw-redirect" title="Wikipédia:REF">inserir referências</a>. Conteúdo não <a href="/wiki/Wikip%C3%A9dia:V" class="mw-redirect" title="Wikipédia:V">verificável</a> poderá ser <a href="/wiki/Wikip%C3%A9dia:Verificabilidade#Política_de_verificabilidade" title="Wikipédia:Verificabilidade">removido</a>.—<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.google.com/#q=Unidades+de+Planck">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&amp;tbm=nws&amp;q=Unidades+de+Planck&amp;oq=Unidades+de+Planck">notícias</a>, <a rel="nofollow" class="external text" href="http://books.google.com/books?&amp;as_brr=0&amp;as_epq=Unidades+de+Planck">livros</a> e <a rel="nofollow" class="external text" href="https://scholar.google.com.br/scholar?hl=pt&amp;q=Unidades+de+Planck">acadêmico</a>)</span></small></span>  <small class="date-container"><i>(<span class="date">Agosto de 2013</span>)</i></small></div></td></tr></tbody></table>
<p>As <b>unidades de Planck</b> ou <b>unidades naturais</b> são um <a href="/wiki/Sistema_de_unidades" title="Sistema de unidades">sistema de unidades</a> proposto pela primeira vez em <a href="/wiki/1899" title="1899">1899</a> por <a href="/wiki/Max_Planck" title="Max Planck">Max Planck</a>. O sistema mede várias das magnitudes fundamentais do universo: <a href="/wiki/Tempo" title="Tempo">tempo</a>, <a href="/wiki/Longitude" title="Longitude">longitude</a>, <a href="/wiki/Massa" title="Massa">massa</a>, <a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">carga elétrica</a> e <a href="/wiki/Temperatura" title="Temperatura">temperatura</a>. O sistema se define fazendo que estas cinco <a href="/wiki/Constantes_f%C3%ADsicas" class="mw-redirect" title="Constantes físicas">constantes físicas</a> universais da tabela tomem o <b>valor 1</b> quando se expressem equações e cálculos em tal sistema.
</p><p>O uso deste sistema de unidades traz consigo várias vantagens. A primeira e mais óbvia é que simplifica muito a estrutura das equações físicas porque elimina as constantes de proporcionalidade e faz com que os resultados das equações não dependam do valor das constantes.
</p><p>Por outra parte, se podem comparar muito mais facilmente as magnitudes de distintas unidades. Por exemplo, dois <a href="/wiki/Pr%C3%B3ton" title="Próton">prótons</a> se repelem porque a repulsão eletromagnética é muito mais forte que a atração gravitacional entre eles. Isto pode ser comprovado ao ver que os prótons têm uma carga aproximadamente igual a uma unidade natural de carga, mas sua massa é muito menor que a unidade natural de massa.
</p><p>Também permite evitar bastantes problemas de arredondamento, sobretudo em computação. Entretanto, têm o inconveniente de que ao usá-las é mais difícil perceber-se os erros dimensionais. São populares na área de investigação da <a href="/wiki/Relatividade_geral" title="Relatividade geral">relatividade geral</a> e a <a href="/wiki/Gravidade_qu%C3%A2ntica" class="mw-redirect" title="Gravidade quântica">gravidade quântica</a>.
</p><p>As unidades de Planck podem ser chamadas (por ironia) pelos físicos como as "unidades de <a href="/wiki/Deus" title="Deus">Deus</a>". Isto elimina qualquer arbitrariedade <a href="/wiki/Antropocentrismo" title="Antropocentrismo">antropocêntrica</a> do sistema de unidades.
</p><p><b>Tabela 1: Constantes físicas fundamentais</b>
</p>
<center>
<table border="1" cellpadding="2">

<tbody><tr>
<th align="left">Constante
</th>
<th align="left">Símbolo
</th>
<th align="left">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>
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        <mtext>&#xA0;</mtext>
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    <annotation encoding="application/x-tex">{\displaystyle {c}\ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c001199185ee8d68e90694c69528d2ed43859e8" class="mwe-math-fallback-image-inline" 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>
</td></tr>
<tr>
<td><a href="/wiki/Constante_de_gravita%C3%A7%C3%A3o" class="mw-redirect" title="Constante de gravitação">Constante de gravitação</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>
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        <mtext>&#xA0;</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" aria-hidden="true" style="vertical-align: -0.338ex; width:2.407ex; height:2.176ex;" alt="{\displaystyle {G}\ }"/></span>
</td>
<td>L<sup>3</sup>/T<sup>2</sup><a href="/wiki/Massa" title="Massa">M</a>
</td></tr>
<tr>
<td><a href="/wiki/Constante_de_Planck" title="Constante de Planck">Constante reduzida de Planck</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 \hbar ={\frac {h}{2\pi }}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi>h</mi>
            <mrow>
              <mn>2</mn>
              <mi>&#x03C0;<!-- 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" aria-hidden="true" style="vertical-align: -1.838ex; width:7.736ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}}"/></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 {h}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi>h</mi>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {h}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ab8f0a2f91f65bd2c9181c323ab987b214b63b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.92ex; height:2.176ex;" alt="{\displaystyle {h}\ }"/></span> é a <a href="/wiki/Constante_de_Planck" title="Constante de Planck">constante de Planck</a>
</td>
<td>ML<sup>2</sup>/T
</td></tr>
<tr>
<td><a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Constante de 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 \epsilon _{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>&#x03C0;<!-- π --></mi>
              <msub>
                <mi>&#x03F5;<!-- 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 \epsilon _{0}}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449fcb1b76fb34dfc5abb99e3e19f0a0473f7c82" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:5.329ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \epsilon _{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 {\epsilon _{0}}\ }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e13596822b169665948694230aea4cd72f4f717" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.579ex; height:2.009ex;" alt="{\displaystyle {\epsilon _{0}}\ }"/></span> é a <a href="/wiki/Permissividade" title="Permissividade">permissividade</a> no vácuo
</td>
<td>M L<sup>3</sup>/ <a href="/wiki/Carga_el%C3%A9trica" title="Carga elétrica">Q</a><sup>2</sup> T<sup>2</sup>
</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}\ }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e06fcc0234af1df544bd987427c6ca3222972b5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {k}\ }"/></span>
</td>
<td>M L<sup>3</sup>/T<sup>2</sup><a href="/wiki/Temperatura" title="Temperatura">K</a>
</td></tr></tbody></table>
</center>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="pt" dir="ltr"><h2 id="mw-toc-heading">Índice</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Expressão_de_leis_físicas_em_unidades_de_Planck"><span class="tocnumber">1</span> <span class="toctext">Expressão de leis físicas em unidades de Planck</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Unidades_de_Planck_básicas"><span class="tocnumber">2</span> <span class="toctext">Unidades de Planck básicas</span></a></li>
<li class="toclevel-1 tocsection-3"><a href="#Unidades_de_Planck_derivadas"><span class="tocnumber">3</span> <span class="toctext">Unidades de Planck derivadas</span></a></li>
<li class="toclevel-1 tocsection-4"><a href="#Unidades_de_Planck_simplificam_as_equações_principais_da_física"><span class="tocnumber">4</span> <span class="toctext">Unidades de Planck simplificam as equações principais da física</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#Normalizações_alternativas"><span class="tocnumber">5</span> <span class="toctext">Normalizações alternativas</span></a></li>
<li class="toclevel-1"><a href="#Referências"><span class="tocnumber">6</span> <span class="toctext">Referências</span></a></li>
<li class="toclevel-1 tocsection-6"><a href="#Ver_também"><span class="tocnumber">7</span> <span class="toctext">Ver também</span></a></li>
</ul>
</div>

<h2><span id="Express.C3.A3o_de_leis_f.C3.ADsicas_em_unidades_de_Planck"></span><span class="mw-headline" id="Expressão_de_leis_físicas_em_unidades_de_Planck">Expressão de leis físicas em unidades de Planck</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=1" class="mw-editsection-visualeditor" title="Editar secção: Expressão de leis físicas em unidades de Planck">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=1" title="Editar secção: Expressão de leis físicas em unidades de Planck">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
<ul><li><a href="/wiki/Lei_da_gravita%C3%A7%C3%A3o_universal" title="Lei da gravitação universal">Lei da gravitação universal</a> de <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li></ul>
<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 F=G{\frac {m_{1}m_{2}}{r^{2}}}}">
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<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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 F={\frac {m_{1}m_{2}}{r^{2}}}}">
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                <mi>m</mi>
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                  <mn>1</mn>
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              <msub>
                <mi>m</mi>
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    <annotation encoding="application/x-tex">{\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7640f20e4cb034b237eaea86381d3023585c1b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:11.865ex; height:5.009ex;" alt="{\displaystyle F={\frac {m_{1}m_{2}}{r^{2}}}}"/></span> utilizando unidades de Planck.</dd></dl></dd></dl>
<ul><li><a href="/wiki/Equa%C3%A7%C3%A3o_de_Schr%C3%B6dinger" title="Equação de Schrödinger">Equação de Schrödinger</a></li></ul>
<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 -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}">
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              <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
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                <mn>2</mn>
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              <mn>2</mn>
              <mi>m</mi>
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          <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
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          <mi mathvariant="bold">r</mi>
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        <mo stretchy="false">)</mo>
        <mi>&#x03C8;<!-- ψ --></mi>
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              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ec4ca961675cbd861d9a4aca85cdff4538fd97" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:42.632ex; height:5.843ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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 -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}">
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          <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
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        <mi>&#x03C8;<!-- ψ --></mi>
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        <mo stretchy="false">)</mo>
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          <mi mathvariant="bold">r</mi>
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              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
              <mi>t</mi>
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        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
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        <mo>,</mo>
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        <mo stretchy="false">)</mo>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07bc2a1b1009bed9e24dedc973b868aa39de952" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:41.326ex; height:5.676ex;" alt="{\displaystyle -{\frac {1}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}"/></span></dd></dl></dd></dl>
<ul><li>A energia de uma partícula ou <a href="/wiki/F%C3%B3ton" class="mw-redirect" title="Fóton">fóton</a> com frequência radiante <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\omega }\ }">
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        <mrow class="MJX-TeXAtom-ORD">
          <mi>&#x03C9;<!-- ω --></mi>
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        <mtext>&#xA0;</mtext>
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    <annotation encoding="application/x-tex">{\displaystyle {\omega }\ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b1aa58c3596ed64b6b230e0896e97f9a2cd42e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.026ex; height:1.676ex;" alt="{\displaystyle {\omega }\ }"/></span> em sua função de onda</li></ul>
<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=\hbar \omega }\ }">
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    <annotation encoding="application/x-tex">{\displaystyle {E=\hbar \omega }\ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8b3ea50735995ccad5a5a4396ec14c2d6bc284" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.207ex; height:2.176ex;" alt="{\displaystyle {E=\hbar \omega }\ }"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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=\omega }\ }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9645811f960ccdd6c2616ef43745a698e024449c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:6.901ex; height:2.176ex;" alt="{\displaystyle {E=\omega }\ }"/></span></dd></dl></dd></dl>
<ul><li>A famosa <a href="/wiki/Relatividade_especial" class="mw-redirect" title="Relatividade especial">equação de massa-energia</a> de <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li></ul>
<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=mc^{2}}\ }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11f6367922a2aec036114de24eaebe50af525cd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.556ex; height:2.676ex;" alt="{\displaystyle {E=mc^{2}}\ }"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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=m}\ }">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa715510d94516f683f0de37467e5d28277f04f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:7.495ex; height:2.176ex;" alt="{\displaystyle {E=m}\ }"/></span></dd></dl></dd></dl>
<dl><dd><dl><dd>(por exemplo, um corpo com uma massa de 5.000 unidades de Planck de massa tem uma energia intrínseca de 5.000 unidades de Planck de energia) e sua forma completa</dd></dl></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {E^{2}=(mc^{2})^{2}+(pc)^{2}}\ }">
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          </msup>
          <mo>+</mo>
          <mo stretchy="false">(</mo>
          <mi>p</mi>
          <mi>c</mi>
          <msup>
            <mo stretchy="false">)</mo>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>2</mn>
            </mrow>
          </msup>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {E^{2}=(mc^{2})^{2}+(pc)^{2}}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bde841f7234e37eb2099ddc5da0aeae7113d6ac" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.372ex; height:3.176ex;" alt="{\displaystyle {E^{2}=(mc^{2})^{2}+(pc)^{2}}\ }"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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^{2}=m^{2}+p^{2}}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <msup>
            <mi>E</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>2</mn>
            </mrow>
          </msup>
          <mo>=</mo>
          <msup>
            <mi>m</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>2</mn>
            </mrow>
          </msup>
          <mo>+</mo>
          <msup>
            <mi>p</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>2</mn>
            </mrow>
          </msup>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {E^{2}=m^{2}+p^{2}}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fb65ffea32e8290ed1a03b86987ab146f99cd6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:14.686ex; height:3.009ex;" alt="{\displaystyle {E^{2}=m^{2}+p^{2}}\ }"/></span></dd></dl></dd></dl>
<ul><li><a href="/wiki/Equa%C3%A7%C3%B5es_de_campo_de_Einstein" title="Equações de campo de Einstein">Equação de campo de Einstein</a> da <a href="/wiki/Relatividade_geral" title="Relatividade geral">relatividade geral</a></li></ul>
<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_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <msub>
            <mi>G</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>&#x03BC;<!-- μ --></mi>
              <mi>&#x03BD;<!-- ν --></mi>
            </mrow>
          </msub>
          <mo>=</mo>
          <mn>8</mn>
          <mi>&#x03C0;<!-- π --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
              <mi>G</mi>
              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>4</mn>
                </mrow>
              </msup>
            </mfrac>
          </mrow>
          <msub>
            <mi>T</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>&#x03BC;<!-- μ --></mi>
              <mi>&#x03BD;<!-- ν --></mi>
            </mrow>
          </msub>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e68495e6578b952a38e53455298f8e0bcc7433" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:16.444ex; height:5.676ex;" alt="{\displaystyle {G_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ }"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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_{\mu \nu }=8\pi T_{\mu \nu }}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <msub>
            <mi>G</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>&#x03BC;<!-- μ --></mi>
              <mi>&#x03BD;<!-- ν --></mi>
            </mrow>
          </msub>
          <mo>=</mo>
          <mn>8</mn>
          <mi>&#x03C0;<!-- π --></mi>
          <msub>
            <mi>T</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mi>&#x03BC;<!-- μ --></mi>
              <mi>&#x03BD;<!-- ν --></mi>
            </mrow>
          </msub>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {G_{\mu \nu }=8\pi T_{\mu \nu }}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25759329fc139f4fd6e8274317e074c261a1fbac" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:13.547ex; height:2.843ex;" alt="{\displaystyle {G_{\mu \nu }=8\pi T_{\mu \nu }}\ }"/></span></dd></dl></dd></dl>
<ul><li>A unidade de temperatura se define para que a media de energia <a href="/wiki/Constante_de_Boltzmann" title="Constante de Boltzmann">térmica</a> cinética por partícula por grau de libertade de movimento</li></ul>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {E={\frac {1}{2}}kT}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi>E</mi>
          <mo>=</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
          <mi>k</mi>
          <mi>T</mi>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {E={\frac {1}{2}}kT}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7eec4d544ee9c1e8100338bdb941c03dc1854ce" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:10.301ex; height:5.176ex;" alt="{\displaystyle {E={\frac {1}{2}}kT}\ }"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {E={\frac {1}{2}}T}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi>E</mi>
          <mo>=</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
            </mfrac>
          </mrow>
          <mi>T</mi>
        </mrow>
        <mtext>&#xA0;</mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {E={\frac {1}{2}}T}\ }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e432a865e27d4397b65ea21e072ca3b2acec805c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:9.09ex; height:5.176ex;" alt="{\displaystyle {E={\frac {1}{2}}T}\ }"/></span></dd></dl></dd></dl>
<ul><li><a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Lei de Coulomb</a></li></ul>
<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 F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>F</mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mn>4</mn>
              <mi>&#x03C0;<!-- π --></mi>
              <msub>
                <mi>&#x03F5;<!-- ϵ --></mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>0</mn>
                </mrow>
              </msub>
            </mrow>
          </mfrac>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msub>
                <mi>q</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
                </mrow>
              </msub>
              <msub>
                <mi>q</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msub>
            </mrow>
            <msup>
              <mi>r</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3a31baacfd989711342eeea94906a1f6d85a15" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:15.187ex; height:5.676ex;" alt="{\displaystyle F={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}"/></span></dd></dl>
<dl><dd><dl><dd>se converte em</dd></dl></dd></dl>
<dl><dd><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 F={\frac {q_{1}q_{2}}{r^{2}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>F</mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msub>
                <mi>q</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
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              </msub>
              <msub>
                <mi>q</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
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              </msub>
            </mrow>
            <msup>
              <mi>r</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe54026052f1419720279d0bf6a052579bde7255" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:9.858ex; height:5.176ex;" alt="{\displaystyle F={\frac {q_{1}q_{2}}{r^{2}}}}"/></span> .</dd></dl></dd></dl>
<ul><li><a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">Equações de Maxwell</a></li></ul>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">E</mi>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msub>
              <mi>&#x03F5;<!-- ϵ --></mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>0</mn>
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            </msub>
          </mfrac>
        </mrow>
        <mi>&#x03C1;<!-- ρ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3f28af564c085c84e3f134ad9d4eafcc5829d3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:12.507ex; height:5.509ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\frac {1}{\epsilon _{0}}}\rho }"/></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \cdot \mathbf {B} =0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">B</mi>
        </mrow>
        <mo>=</mo>
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="\nabla \cdot \mathbf{B} = 0"/></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">E</mi>
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        <mo>=</mo>
        <mo>&#x2212;<!-- − --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">B</mi>
              </mrow>
            </mrow>
            <mrow>
              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
              <mi>t</mi>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:15.495ex; height:5.509ex;" alt="\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}"/></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x00D7;<!-- × --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">B</mi>
        </mrow>
        <mo>=</mo>
        <msub>
          <mi>&#x03BC;<!-- μ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">J</mi>
        </mrow>
        <mo>+</mo>
        <msub>
          <mi>&#x03BC;<!-- μ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <msub>
          <mi>&#x03F5;<!-- ϵ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">E</mi>
              </mrow>
            </mrow>
            <mrow>
              <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi>
              <mi>t</mi>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57134e48b06ebc527fd19cd5e96f3f5d12ccddde" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:24.818ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}"/></span></dd></dl>
<dl><dd><dl><dd>se convertem respectivamente em</dd></dl></dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">E</mi>
        </mrow>
        <mo>=</mo>
        <mn>4</mn>
        <mi>&#x03C0;<!-- π --></mi>
        <mi>&#x03C1;<!-- ρ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2038cb70658f28c2850fbbf5c97c27cea43c7f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.167ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }"/></span></dd></dl></dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \cdot \mathbf {B} =0}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
        <mo>&#x22C5;<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">B</mi>
        </mrow>
        <mo>=</mo>
        <mn>0</mn>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {B} =0}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee950683349dacdd9e9c262ff6133812747edd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.777ex; height:2.176ex;" alt="\nabla \cdot \mathbf{B} = 0"/></span></dd></dl></dd></dl>
<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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    <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation>
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<dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}">
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    <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}}</annotation>
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<dl><dd><dl><dd>utilizando as unidades de Planck. (Os fatores <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 4\pi \ }">
  <semantics>
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    <annotation encoding="application/x-tex">{\displaystyle 4\pi \ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c451322a2ae800b4986cf652c0f956358e098587" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.075ex; height:2.176ex;" alt="{\displaystyle 4\pi \ }"/></span> podem ser eliminados se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle \epsilon _{0}\ }">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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          <mi>&#x03F5;<!-- ϵ --></mi>
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            <mn>0</mn>
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        <mtext>&#xA0;</mtext>
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    <annotation encoding="application/x-tex">{\displaystyle \epsilon _{0}\ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9bab93876d670d667d85ad2919a73b161dd8bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.579ex; height:2.009ex;" alt="{\displaystyle \epsilon _{0}\ }"/></span> for normalizado, em vez da constante de força de Coulomb <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle 1/(4\pi \epsilon _{0})\ }">
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          <mo>/</mo>
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        <mo stretchy="false">(</mo>
        <mn>4</mn>
        <mi>&#x03C0;<!-- π --></mi>
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          <mi>&#x03F5;<!-- ϵ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle 1/(4\pi \epsilon _{0})\ }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a4950ba669ca8266bcaf45407d8a8da27ff3bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:9.208ex; height:2.843ex;" alt="{\displaystyle 1/(4\pi \epsilon _{0})\ }"/></span>.)</dd></dl></dd></dl>
<h2><span id="Unidades_de_Planck_b.C3.A1sicas"></span><span class="mw-headline" id="Unidades_de_Planck_básicas">Unidades de Planck básicas</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=2" class="mw-editsection-visualeditor" title="Editar secção: Unidades de Planck básicas">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=2" title="Editar secção: Unidades de Planck básicas">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Ao dar <b>valor 1</b> às cinco constantes fundamentais, as unidades de tempo, comprimento, massa, carga e temperatura se definem assim:
</p><p><b>Tabela 2: Unidades de Planck básicas</b>
</p>
<center>
<table border="1" cellspacing="2">

<tbody><tr>
<th align="left">Nome
</th>
<th align="left">Dimensão
</th>
<th align="left">Expressão
</th>
<th align="left">Equivalência aproximada no <a href="/wiki/SI" class="mw-disambig" title="SI">Sistema Internacional</a>
</th></tr>
<tr>
<td><b><a href="/wiki/Tempo_de_Planck" title="Tempo de Planck">Tempo Planck</a></b>
</td>
<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>
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        <msub>
          <mi>t</mi>
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        </msub>
        <mo>=</mo>
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          <msqrt>
            <mfrac>
              <mrow>
                <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
                <mi>G</mi>
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              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>5</mn>
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    <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" 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>5.39121 &#215; 10<sup>−44</sup> <a href="/wiki/Segundo" title="Segundo">s</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Comprimento_de_Planck" title="Comprimento de Planck">Comprimento de Planck</a></b></td>
<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}=c\ t_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}">
  <semantics>
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          <mi>l</mi>
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          </mrow>
        </msub>
        <mo>=</mo>
        <mi>c</mi>
        <mtext>&#xA0;</mtext>
        <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">&#x210F;<!-- ℏ --></mi>
                <mi>G</mi>
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              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>3</mn>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle l_{P}=c\ t_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e874fe5856aa77a9d52885e2b669428bd6c96b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:18.543ex; height:6.343ex;" alt="{\displaystyle l_{P}=c\ t_{P}={\sqrt {\frac {\hbar G}{c^{3}}}}}"/></span>
</td>
<td>1.61624 &#215; 10<sup>−35</sup> <a href="/wiki/Metro" title="Metro">m</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Massa_de_Planck" title="Massa de Planck">Massa de Planck</a></b></td>
<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>
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            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
              <mrow>
                <mi class="MJX-variant">&#x210F;<!-- 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" 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>2.17645 &#215; 10<sup>−8</sup> <a href="/wiki/Quilograma" title="Quilograma">kg</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Carga_de_Planck" title="Carga de Planck">Carga de Planck</a></b></td>
<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 c4\pi \epsilon _{0}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>q</mi>
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            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
            <mi>c</mi>
            <mn>4</mn>
            <mi>&#x03C0;<!-- π --></mi>
            <msub>
              <mi>&#x03F5;<!-- ϵ --></mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>0</mn>
              </mrow>
            </msub>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle q_{P}={\sqrt {\hbar c4\pi \epsilon _{0}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed06c7f874b9c91fde5be999d4288af1815679b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:14.732ex; height:3.509ex;" alt="{\displaystyle q_{P}={\sqrt {\hbar c4\pi \epsilon _{0}}}}"/></span>
</td>
<td>1.8755459 &#215; 10<sup>−18</sup> <a href="/wiki/Coulomb" title="Coulomb">C</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Temperatura_de_Planck" title="Temperatura de Planck">Temperatura de Planck</a></b>
</td>
<td><a href="/wiki/Temperatura" title="Temperatura">Temperatura</a> (ML<sup>2</sup>T<sup>−2</sup>/k)
</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}={\frac {m_{P}c^{2}}{k}}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
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            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
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          <mfrac>
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                <mi>m</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi>P</mi>
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              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
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            <mi>k</mi>
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        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
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                  <mrow class="MJX-TeXAtom-ORD">
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                <mi>G</mi>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
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            </mfrac>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle T_{P}={\frac {m_{P}c^{2}}{k}}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a0e231655d49c991b3368e9ef48f97cd4976c5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:22.677ex; height:7.509ex;" alt="{\displaystyle T_{P}={\frac {m_{P}c^{2}}{k}}={\sqrt {\frac {\hbar c^{5}}{Gk^{2}}}}}"/></span>
</td>
<td>1.41679 &#215; 10<sup>32</sup> <a href="/wiki/Kelvin" title="Kelvin">K</a>
</td></tr></tbody></table>
</center>
<h2><span class="mw-headline" id="Unidades_de_Planck_derivadas">Unidades de Planck derivadas</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=3" class="mw-editsection-visualeditor" title="Editar secção: Unidades de Planck derivadas">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=3" title="Editar secção: Unidades de Planck derivadas">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Como em outros <a href="/wiki/Sistema_de_unidades" title="Sistema de unidades">sistemas de unidades</a>, as magnitudes físicas derivadas podem ser definidas baseando-se nas Unidades de Planck.
</p><p><b>Tabela 3: Unidades de Planck derivadas </b>
</p>
<center>
<table border="1" cellspacing="2">

<tbody><tr>
<th align="left">Nome
</th>
<th align="left">Dimensão
</th>
<th align="left">Expressão
</th>
<th align="left">Equivalência aproximada no <a href="/wiki/SI" class="mw-disambig" title="SI">Sistema Internacional</a>
</th></tr>
<tr>
<td><b><a href="/wiki/Energia_de_Planck" title="Energia de Planck">Energia de Planck</a></b>
</td>
<td><a href="/wiki/Energia" title="Energia">Energia</a> (ML<sup>2</sup>/T<sup>2</sup>)
</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_{P}=m_{P}c^{2}={\sqrt {\frac {\hbar c^{5}}{G}}}}">
  <semantics>
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          <mi>E</mi>
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        </msub>
        <mo>=</mo>
        <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>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
              <mrow>
                <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
                <msup>
                  <mi>c</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>5</mn>
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              <mi>G</mi>
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      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle E_{P}=m_{P}c^{2}={\sqrt {\frac {\hbar c^{5}}{G}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e3339e16f4437e37925b98d71c488e860bfc6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:21.474ex; height:7.676ex;" alt="{\displaystyle E_{P}=m_{P}c^{2}={\sqrt {\frac {\hbar c^{5}}{G}}}}"/></span>
</td>
<td>1.9561 &#215; 10<sup>9</sup> <a href="/wiki/Joule_(unidade)" class="mw-redirect" title="Joule (unidade)">J</a>
</td></tr>
<tr>
<td><b><a href="/wiki/For%C3%A7a_de_Planck" title="Força de Planck">Força de Planck</a></b></td>
<td><a href="/wiki/For%C3%A7a" title="Força">Força</a> (ML/T<sup>2</sup>)
</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 F_{P}={\frac {E_{P}}{l_{P}}}={\frac {c^{4}}{G}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
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          <mi>F</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>E</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msub>
              <mi>l</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msup>
              <mi>c</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>4</mn>
              </mrow>
            </msup>
            <mi>G</mi>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F_{P}={\frac {E_{P}}{l_{P}}}={\frac {c^{4}}{G}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fc4820c9d8fbd2f8a96d6feb3b228ba1a4f108" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:16.073ex; height:6.176ex;" alt="{\displaystyle F_{P}={\frac {E_{P}}{l_{P}}}={\frac {c^{4}}{G}}}"/></span>
</td>
<td>1.21027 &#215; 10<sup>44</sup> <a href="/wiki/Newton" class="mw-redirect" title="Newton">N</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Pot%C3%AAncia_de_Planck" title="Potência de Planck">Potência de Planck</a></b>
</td>
<td><a href="/wiki/Pot%C3%AAncia" title="Potência">Potência</a> (ML<sup>2</sup>/T<sup>3</sup>)
</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 P_{P}={\frac {E_{P}}{t_{P}}}={\frac {c^{5}}{G}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>P</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>E</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msub>
              <mi>t</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msup>
              <mi>c</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>5</mn>
              </mrow>
            </msup>
            <mi>G</mi>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle P_{P}={\frac {E_{P}}{t_{P}}}={\frac {c^{5}}{G}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292ff46a4af90f899aa08e0f60121c7eb0d7aa26" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:16.071ex; height:6.009ex;" alt="{\displaystyle P_{P}={\frac {E_{P}}{t_{P}}}={\frac {c^{5}}{G}}}"/></span>
</td>
<td>3.62831 &#215; 10<sup>52</sup> <a href="/wiki/Watt" title="Watt">W</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Densidade_de_Planck" title="Densidade de Planck">Densidade de Planck</a></b>
</td>
<td><a href="/wiki/Densidade_(f%C3%ADsica)" class="mw-redirect" title="Densidade (física)">Densidade</a> (M/L<sup>3</sup>)
</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 \rho _{P}={\frac {m_{P}}{l_{P}^{3}}}={\frac {c^{5}}{\hbar G^{2}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>&#x03C1;<!-- ρ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>m</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msubsup>
              <mi>l</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>3</mn>
              </mrow>
            </msubsup>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msup>
              <mi>c</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>5</mn>
              </mrow>
            </msup>
            <mrow>
              <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
              <msup>
                <mi>G</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \rho _{P}={\frac {m_{P}}{l_{P}^{3}}}={\frac {c^{5}}{\hbar G^{2}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b31164b84323282f1a92ab7ca241a3bc4403041" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:18.232ex; height:6.843ex;" alt="{\displaystyle \rho _{P}={\frac {m_{P}}{l_{P}^{3}}}={\frac {c^{5}}{\hbar G^{2}}}}"/></span>
</td>
<td>5.15500 &#215; 10<sup>96</sup> <a href="/wiki/Quilograma_por_metro_c%C3%BAbico" title="Quilograma por metro cúbico">kg/m³</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Frequ%C3%AAncia_angular_de_Planck" title="Frequência angular de Planck">Frequência angular de Planck</a></b>
</td>
<td><a href="/wiki/Frequ%C3%AAncia" title="Frequência">Frequência</a> (1/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 \omega _{P}={\frac {1}{t_{P}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>&#x03C9;<!-- ω --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msub>
              <mi>t</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>5</mn>
                </mrow>
              </msup>
              <mrow>
                <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
                <mi>G</mi>
              </mrow>
            </mfrac>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \omega _{P}={\frac {1}{t_{P}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6d2f8ec4eb67e954de702eaf153c103fb6e161" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:18.544ex; height:7.676ex;" alt="{\displaystyle \omega _{P}={\frac {1}{t_{P}}}={\sqrt {\frac {c^{5}}{\hbar G}}}}"/></span>
</td>
<td>1.85487 &#215; 10<sup>43</sup> <a href="/wiki/Frequ%C3%AAncia_angular" title="Frequência angular">rad/s</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Press%C3%A3o_de_Planck" title="Pressão de Planck">Pressão de Planck</a></b></td>
<td><a href="/wiki/Press%C3%A3o" title="Pressão">Pressão</a> (M/LT<sup>2</sup>)
</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 p_{P}={\frac {F_{P}}{l_{P}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>p</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>F</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msubsup>
              <mi>l</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msubsup>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msup>
              <mi>c</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>7</mn>
              </mrow>
            </msup>
            <mrow>
              <mi class="MJX-variant">&#x210F;<!-- ℏ --></mi>
              <msup>
                <mi>G</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle p_{P}={\frac {F_{P}}{l_{P}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feecf825e0ef5e3359cff562ec8c65fca4a9e367" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:17.743ex; height:6.843ex;" alt="{\displaystyle p_{P}={\frac {F_{P}}{l_{P}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}}"/></span>
</td>
<td>4.63309 &#215; 10<sup>113</sup> <a href="/wiki/Pascal_(unidade)" title="Pascal (unidade)">Pa</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Corrente_el%C3%A9trica_de_Planck" title="Corrente elétrica de Planck">Corrente elétrica de Planck</a></b>
</td>
<td><a href="/wiki/Corrente_el%C3%A9trica" title="Corrente elétrica">Corrente elétrica</a> (Q/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 I_{P}={\frac {q_{P}}{t_{P}}}={\sqrt {\frac {c^{6}4\pi \epsilon _{0}}{G}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>I</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>q</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msub>
              <mi>t</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
              <mrow>
                <msup>
                  <mi>c</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>6</mn>
                  </mrow>
                </msup>
                <mn>4</mn>
                <mi>&#x03C0;<!-- π --></mi>
                <msub>
                  <mi>&#x03F5;<!-- ϵ --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
                  </mrow>
                </msub>
              </mrow>
              <mi>G</mi>
            </mfrac>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle I_{P}={\frac {q_{P}}{t_{P}}}={\sqrt {\frac {c^{6}4\pi \epsilon _{0}}{G}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6a0ff4b2ca02cf8cfe6ba30df74352a9b0c923" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:21.74ex; height:7.676ex;" alt="{\displaystyle I_{P}={\frac {q_{P}}{t_{P}}}={\sqrt {\frac {c^{6}4\pi \epsilon _{0}}{G}}}}"/></span>
</td>
<td>3.4789 &#215; 10<sup>25</sup> <a href="/wiki/Ampere" title="Ampere">A</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Tens%C3%A3o_el%C3%A9trica_de_Planck" title="Tensão elétrica de Planck">Tensão elétrica de Planck</a></b>
</td>
<td><a href="/wiki/Tens%C3%A3o_el%C3%A9trica" title="Tensão elétrica">Tensão elétrica</a> (ML<sup>2</sup>/T<sup>2</sup>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 V_{P}={\frac {E_{P}}{q_{P}}}={\sqrt {\frac {c^{4}}{G4\pi \epsilon _{0}}}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>V</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>E</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msub>
              <mi>q</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mfrac>
              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>4</mn>
                </mrow>
              </msup>
              <mrow>
                <mi>G</mi>
                <mn>4</mn>
                <mi>&#x03C0;<!-- π --></mi>
                <msub>
                  <mi>&#x03F5;<!-- ϵ --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
                  </mrow>
                </msub>
              </mrow>
            </mfrac>
          </msqrt>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle V_{P}={\frac {E_{P}}{q_{P}}}={\sqrt {\frac {c^{4}}{G4\pi \epsilon _{0}}}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97d04755ce4f37bcbb248166fc3d136f1c729722" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:22.516ex; height:7.676ex;" alt="{\displaystyle V_{P}={\frac {E_{P}}{q_{P}}}={\sqrt {\frac {c^{4}}{G4\pi \epsilon _{0}}}}}"/></span>
</td>
<td>1.04295 &#215; 10<sup>27</sup> <a href="/wiki/Volt" title="Volt">V</a>
</td></tr>
<tr>
<td><b><a href="/wiki/Resist%C3%AAncia_el%C3%A9trica_de_Planck" title="Resistência elétrica de Planck">Resistência elétrica de Planck</a></b>
</td>
<td><a href="/wiki/Resist%C3%AAncia_el%C3%A9trica" title="Resistência elétrica">Resistência</a> (ML<sup>2</sup>/T Q<sup>2</sup>)
</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 Z_{P}={\frac {V_{P}}{I_{P}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>Z</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mi>P</mi>
          </mrow>
        </msub>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>V</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
            <msub>
              <mi>I</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi>P</mi>
              </mrow>
            </msub>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <mrow>
              <mn>4</mn>
              <mi>&#x03C0;<!-- π --></mi>
              <msub>
                <mi>&#x03F5;<!-- ϵ --></mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>0</mn>
                </mrow>
              </msub>
              <mi>c</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>Z</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>0</mn>
              </mrow>
            </msub>
            <mrow>
              <mn>4</mn>
              <mi>&#x03C0;<!-- π --></mi>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle Z_{P}={\frac {V_{P}}{I_{P}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f16b2ab2cc2b304dc2ac5f2073cf33963472260" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:25.821ex; height:5.843ex;" alt="{\displaystyle Z_{P}={\frac {V_{P}}{I_{P}}}={\frac {1}{4\pi \epsilon _{0}c}}={\frac {Z_{0}}{4\pi }}}"/></span>
</td>
<td>2.99792458 &#215; 10¹ <a href="/wiki/Ohm" title="Ohm">&#937;</a>
</td></tr></tbody></table>
</center>
<h2><span id="Unidades_de_Planck_simplificam_as_equa.C3.A7.C3.B5es_principais_da_f.C3.ADsica"></span><span class="mw-headline" id="Unidades_de_Planck_simplificam_as_equações_principais_da_física">Unidades de Planck simplificam as equações principais da física</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=4" class="mw-editsection-visualeditor" title="Editar secção: Unidades de Planck simplificam as equações principais da física">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=4" title="Editar secção: Unidades de Planck simplificam as equações principais da física">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Ordinariamente, grandezas físicas que tem diferentes dimensões (tais como tempo e comprimento) não podem ser equiparadas, mesmo que sejam numericamente iguais (1 segundo não é o mesmo que 1 metro). Contudo, em física teórica este critério pode ser anulado de maneira a simplificar cálculos. O processo pelo qual isto é feito é chamado "<a href="/w/index.php?title=Adimensionaliza%C3%A7%C3%A3o&amp;action=edit&amp;redlink=1" class="new" title="Adimensionalização (página não existe)">adimensionalização</a>". A tabela 4 mostra como unidades de Planck, pela escolha dos valores numéricos das cinco constantes fundamentais à unidade, simplificam muitas equações da física e fazem-nas <a href="/w/index.php?title=Adimensionaliza%C3%A7%C3%A3o&amp;action=edit&amp;redlink=1" class="new" title="Adimensionalização (página não existe)">adimensionais</a>.
</p><p><b>Tabela 4: Equações adimensionalizadas</b>
</p>
<table class="wikitable" style="margin: 1em auto 1em auto; background-color: #ffffff">
<tbody><tr>
<th>
</th>
<th>Forma usual
</th>
<th>Forma <a href="/w/index.php?title=Adimensionaliza%C3%A7%C3%A3o&amp;action=edit&amp;redlink=1" class="new" title="Adimensionalização (página não existe)">adimensionalizada</a>
</th></tr>
<tr align="left">
<td><b><a href="/wiki/Gravidade" title="Gravidade">Lei de Newton de Gravitação Universal</a></b>
</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 F=-G{\frac {m_{1}m_{2}}{r^{2}}}}">
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</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 F=-{\frac {m_{1}m_{2}}{r^{2}}}}">
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</td></tr>
<tr>
<td><b><a href="/wiki/Equa%C3%A7%C3%A3o_de_Schr%C3%B6dinger" title="Equação de Schrödinger">Equação de Schrödinger</a></b>
</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 {\hbar ^{2}}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)}">
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<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}{2m}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)}">
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</td></tr>
<tr>
<td><b><a href="/wiki/Rela%C3%A7%C3%A3o_de_Planck" class="mw-redirect" title="Relação de Planck">Relação de Planck</a> relacionando a energia de <a href="/wiki/Part%C3%ADcula_elementar" title="Partícula elementar">partícula</a> à <a href="/wiki/Frequ%C3%AAncia_angular" title="Frequência angular">frequência angular</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 {\omega }\ }">
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          <mi>&#x03C9;<!-- ω --></mi>
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        <mtext>&#xA0;</mtext>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b1aa58c3596ed64b6b230e0896e97f9a2cd42e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.026ex; height:1.676ex;" alt="{\displaystyle {\omega }\ }"/></span> de sua <a href="/wiki/Fun%C3%A7%C3%A3o_de_onda" title="Função de onda">função de onda</a></b>
</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=\hbar \omega }\ }">
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</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=\omega }\ }">
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</td></tr>
<tr>
<td><b>Equação <a href="/wiki/Massa" title="Massa">massa</a>/<a href="/wiki/Energia" title="Energia">energia</a> da <a href="/wiki/Relatividade_restrita" title="Relatividade restrita">relatividade restrita</a> de <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></b>
</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=mc^{2}}\ }">
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</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=m}\ }">
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</td></tr>
<tr>
<td><b><a href="/wiki/Equa%C3%A7%C3%B5es_de_campo_de_Einstein" title="Equações de campo de Einstein">Equações de campo de Einstein</a> da <a href="/wiki/Relatividade_geral" title="Relatividade geral">relatividade geral</a></b>
</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_{\mu \nu }=8\pi {G \over c^{4}}T_{\mu \nu }}\ }">
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</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_{\mu \nu }=8\pi T_{\mu \nu }}\ }">
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</td></tr>
<tr>
<td><b><a href="/wiki/Energia_t%C3%A9rmica" title="Energia térmica">Energia térmica</a> por partícula por <a href="/wiki/Graus_de_liberdade_(f%C3%ADsica)" title="Graus de liberdade (física)">grau de liberdade</a></b>
</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={\frac {1}{2}}kT}\ }">
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</td></tr>
<tr>
<td><b><a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Lei de Coulomb</a></b>
</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 F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}">
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</td></tr>
<tr>
<td><b><a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">Equações de Maxwell</a></b>
</td>
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</p>
</td>
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</p>
</td></tr></tbody></table>
<h2><span id="Normaliza.C3.A7.C3.B5es_alternativas"></span><span class="mw-headline" id="Normalizações_alternativas">Normalizações alternativas</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=5" class="mw-editsection-visualeditor" title="Editar secção: Normalizações alternativas">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=5" title="Editar secção: Normalizações alternativas">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
<p>Como já foi afirmado na introdução, unidade de Planck são derivadas de "normalizar" os valores numéricos de certas constantes fundamentais a 1. Estas normalizações são nem as únicas possíveis, nem necessariamente as melhores. Além disso, a escolha de quais constantes normalizar não é evidente, e os valores das unidades de Planck são sensíveis a esta escolha.
</p><p>O fator 4π, e múltiplos dele tais como 8π, são ubíquos em fórmulas em <a href="/wiki/F%C3%ADsica_te%C3%B3rica" title="Física teórica">física teórica</a> porque estão atrelados a área da superfície da <a href="/wiki/Esfera_(geometria)" class="mw-redirect" title="Esfera (geometria)">esfera</a> unitária tridimensional. Por exemplo, <a href="/wiki/Campo_gravitacional" title="Campo gravitacional">campos gravitacionais</a> e <a href="/wiki/Campo_el%C3%A9trico" title="Campo elétrico">elétricos</a> produzidos por cargas pontuais têm simetria esférica<sup id="cite_ref-Barrow_1-0" class="reference"><a href="#cite_note-Barrow-1"><span>[</span>1<span>]</span></a></sup> <sup>(pgs 214-15)</sup>. O 4π<i>r</i><sup>2</sup> que aparece no denominador da <a href="/wiki/Lei_de_Coulomb" title="Lei de Coulomb">Lei de Coulomb</a>, por exemplo, reflete o fato que o <a href="/wiki/Fluxo_(f%C3%ADsica)" title="Fluxo (física)">fluxo</a> do campo elétrico distribui-se uniformemente sobre a superfície da esfera. Se o espaço tem mais dimensões, o fator correspondente a 4π deverá ser <a href="/wiki/N-esfera" title="N-esfera">diferente</a>.
</p><p>Em qualquer evento, uma escolha fundamental que tem de ser feita quando se construindo um sistema de unidades naturais é que, se for o adequado, os casos de 4<i>n</i>π aparecendo nas equações da física serão eliminados através da normalização.
</p>
<ul><li>Escolhendo <i>ε</i><sub>0</sub> = 1.</li></ul>
<p>Planck normalizou a 1 a <a href="/wiki/Constante_de_permissividade_do_v%C3%A1cuo" title="Constante de permissividade do vácuo">constante da força de Coulomb</a> 1/(4π<i>ε</i><sub>0</sub>) (tal como no <a href="/wiki/Sistema_CGS_de_unidades" title="Sistema CGS de unidades">sistema CGS de unidades</a>). Isso define a <a href="/wiki/Resist%C3%AAncia_el%C3%A9trica_de_Planck" title="Resistência elétrica de Planck">impedância de Planck</a>, <i>Z</i><sub>P</sub> como igual a <i>Z</i><sub>0</sub>/4π, onde <i>Z</i><sub>0</sub> é a <a href="/w/index.php?title=Imped%C3%A2ncia_do_espa%C3%A7o_livre&amp;action=edit&amp;redlink=1" class="new" title="Impedância do espaço livre (página não existe)">impedância característica do espaço livre</a>. Normalizando a <a href="/wiki/Constante_de_permissividade_do_v%C3%A1cuo" title="Constante de permissividade do vácuo">permissividade do espaço livre</a> <i>ε</i><sub>0</sub> a 1 não só faz <i>Z</i><sub>P</sub> igual a <i>Z</i><sub>0</sub>, mas também elimina 4π das <a href="/wiki/Equa%C3%A7%C3%B5es_de_Maxwell" title="Equações de Maxwell">equações de Maxwell</a>. Por outro lado, a forma <a href="/w/index.php?title=Adimensionaliza%C3%A7%C3%A3o&amp;action=edit&amp;redlink=1" class="new" title="Adimensionalização (página não existe)">adimensionalizada</a> da lei de Coulomb irá agora conter um fator of 1/(4π).
</p>
<ul><li>Escolhendo 4<i>n</i>π<i>G</i> = 1.</li></ul>
<p>Em 1899, a <a href="/wiki/Relatividade_geral" title="Relatividade geral">relatividade geral</a> estava alguns anos no futuro, e a <a href="/wiki/Lei_da_gravita%C3%A7%C3%A3o_universal" title="Lei da gravitação universal">lei da gravitação universal de Newton</a> era ainda vista como fundamental, e não como uma aproximação conveniente para o tratamento de "pequenas" velocidades e distâncias. Por isso Planck normalizou a 1 a <a href="/wiki/Constante_gravitacional_universal" title="Constante gravitacional universal">constante gravitacional</a> <i>G</i> na lei de Newton. Em teorias surgidas após 1899, <i>G</i> é quase sempre multiplicada por 4π ou múltiplos.
</p>
<dl><dd><ul><li>4π<i>G</i> aparece em:
<ul><li><a href="/w/index.php?title=Lei_de_Gauss_para_a_gravidade&amp;action=edit&amp;redlink=1" class="new" title="Lei de Gauss para a gravidade (página não existe)">Lei de Gauss para a gravidade</a>, Φ<sub><b>g</b></sub> = −4&#960;<i>GM</i>;</li>
<li><a href="/wiki/Imped%C3%A2ncia_caracter%C3%ADstica" title="Impedância característica">Impedância característica</a> da <a href="/wiki/Radia%C3%A7%C3%A3o_gravitacional" class="mw-redirect" title="Radiação gravitacional">radiação gravitacional</a> no espaço livre, <i>Z</i><sub>0</sub> = 4π<i>G</i>/<i>c</i>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> O <i>c</i> no denominador decorre da predição da <a href="/wiki/Relatividade_geral" title="Relatividade geral">relatividade geral</a> que a radiação gravitacional propaga-se a mesma <a href="/wiki/Velocidade_da_luz" title="Velocidade da luz">velocidade das radiações eletromagnéticas</a>;</li>
<li>Equações <a href="/wiki/Gravitomagnetismo" title="Gravitomagnetismo">gravitoeletromagnéticas</a> (GEM), que apoaim-se nos <a href="/wiki/Campo_gravitacional" title="Campo gravitacional">campos gravitacionais</a> fracos ou <a href="/wiki/Espa%C3%A7o_de_Minkowski" title="Espaço de Minkowski">espaço-tempo razoavelmente plano</a>. Estas equações têm a mesma forma das equações de Maxwell (e a equação da <a href="/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz">força de Lorentz</a>) do eletromagnetismo, com <a href="/wiki/Massa_vol%C3%BAmica" class="mw-redirect" title="Massa volúmica">densidade de massa</a> substituindo a <a href="/wiki/Densidade_de_carga" title="Densidade de carga">densidade de carga</a>, e com 1/(4π<i>G</i>) substituindo ε<sub>0</sub>.</li></ul></li></ul></dd></dl>
<dl><dd><ul><li>8π<i>G</i> aparece nas <a href="/wiki/Equa%C3%A7%C3%B5es_de_campo_de_Einstein" title="Equações de campo de Einstein">equações de campo de Einstein</a>, na <a href="/wiki/A%C3%A7%C3%A3o_de_Einstein-Hilbert" class="mw-redirect" title="Ação de Einstein-Hilbert">ação de Einstein-Hilbert</a>, nas <a href="/wiki/Equa%C3%A7%C3%B5es_de_Friedmann" title="Equações de Friedmann">equações de Friedmann</a>, e na <a href="/wiki/Equa%C3%A7%C3%A3o_de_Poisson" title="Equação de Poisson">equação de Poisson</a> para a gravitação. Unidades de Planck modificadas em que 8π<i>G</i> = 1 são conhecidas como <i>unidades de Planck reduzidas</i>, porque a massa de Planck é dividida por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\displaystyle {\sqrt {8\pi }}{\text{.}}}">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <msqrt>
            <mn>8</mn>
            <mi>&#x03C0;<!-- π --></mi>
          </msqrt>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mtext>.</mtext>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\sqrt {8\pi }}{\text{.}}}</annotation>
  </semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4181b4d7a3229705249751fd60346507833cc55f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.077ex; height:2.843ex;" alt="{\displaystyle {\sqrt {8\pi }}{\text{.}}}"/></span></li></ul></dd></dl>
<dl><dd><ul><li>Escolhendo 16π<i>G</i> = 1 eliminará a constante <i>k</i> = <i>c</i><sup>4</sup>/(16&#960;<i>G</i>) da <a href="/wiki/A%C3%A7%C3%A3o_de_Einstein-Hilbert" class="mw-redirect" title="Ação de Einstein-Hilbert">ação de Einstein-Hilbert</a>. As <a href="/wiki/Equa%C3%A7%C3%B5es_de_campo_de_Einstein" title="Equações de campo de Einstein">equações de campo de Einstein</a> com <a href="/wiki/Constante_cosmol%C3%B3gica" title="Constante cosmológica">constante cosmológica</a> Λ torna-se <i>R<sub>μν</sub></i> − Λ<i>g<sub>μν</sub></i> = (<i>Rg<sub>μν</sub></i> − <i>T<sub>μν</sub></i>)/2.</li></ul></dd></dl>
<p>Por isso um volume substancial de física teórica descoberta desde Planck (1899) sugere se normalizar a 1 não <i>G</i> mas 4<i>n</i>π<i>G</i>, <i>n</i> = 1, 2, or 4. No entanto, fazê-lo, seria introduzir um fator de 1/(4<i>n</i>π) na adimencionalizada lei de gravitação universal.
</p>
<ul><li>Escolhendo <i>k</i> = 2.</li></ul>
<p>Isto removeria o fator de 2 na equação adimencionalizada da <a href="/wiki/Energia_t%C3%A9rmica" title="Energia térmica">energia térmica</a> por partícula por <a href="/wiki/Graus_de_liberdade_(f%C3%ADsica)" title="Graus de liberdade (física)">grau de liberdade</a>, e não afetaria o valor de qualquer base ou unidades derivadas outras que a <a href="/wiki/Temperatura_de_Planck" title="Temperatura de Planck">temperatura de Planck</a>.
</p>
<h2 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><span class="mw-headline" id="Referências">Referências</span></h2>
<div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references">
<li id="cite_note-Barrow-1"><span class="mw-cite-backlink"><a href="#cite_ref-Barrow_1-0">↑</a></span> <span class="reference-text"><a href="/w/index.php?title=John_D._Barrow&amp;action=edit&amp;redlink=1" class="new" title="John D. Barrow (página não existe)">John D. Barrow</a>, 2002. <i>The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe</i>. Pantheon Books. <a href="/wiki/Especial:Fontes_de_livros/0375422218" class="internal mw-magiclink-isbn">ISBN 0-375-42221-8</a>.</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Raymond Y. Chiao; <a rel="nofollow" class="external text" href="http://arxiv.org/abs/0710.1378v4">arXiv:0710.1378v4 <i>Generation and detection of gravitational waves at microwave frequencies by means of a superconducting two-body system</i></a> - <b>arXiv</b> (em inglês)</span>
</li>
</ol></div></div>
<h2><span id="Ver_tamb.C3.A9m"></span><span class="mw-headline" id="Ver_também">Ver também</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unidades_de_Planck&amp;veaction=edit&amp;section=6" class="mw-editsection-visualeditor" title="Editar secção: Ver também">editar</a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unidades_de_Planck&amp;action=edit&amp;section=6" title="Editar secção: Ver também">editar código-fonte</a><span class="mw-editsection-bracket">]</span></span></h2>
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interwiki-el"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%BF%CE%BD%CE%AC%CE%B4%CE%B5%CF%82_%CE%A0%CE%BB%CE%B1%CE%BD%CE%BA" title="Μονάδες Πλανκ — grego" lang="el" hreflang="el" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en"><a href="https://en.wikipedia.org/wiki/Planck_units" title="Planck units — inglês" lang="en" hreflang="en" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo"><a href="https://eo.wikipedia.org/wiki/Unuoj_de_Planck" title="Unuoj de Planck — esperanto" lang="eo" hreflang="eo" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es"><a href="https://es.wikipedia.org/wiki/Unidades_de_Planck" title="Unidades de Planck — espanhol" lang="es" hreflang="es" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa"><a href="https://fa.wikipedia.org/wiki/%DB%8C%DA%A9%D8%A7%D9%87%D8%A7%DB%8C_%D9%BE%D9%84%D8%A7%D9%86%DA%A9" title="یکاهای پلانک — persa" lang="fa" hreflang="fa" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi"><a href="https://fi.wikipedia.org/wiki/Planckin_yksik%C3%B6t" title="Planckin yksiköt — finlandês" lang="fi" hreflang="fi" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_d%27unit%C3%A9s_de_Planck" title="Système d&#039;unités de Planck — francês" lang="fr" hreflang="fr" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%99%D7%93%D7%95%D7%AA_%D7%A4%D7%9C%D7%90%D7%A0%D7%A7" title="יחידות פלאנק — hebraico" lang="he" hreflang="he" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu"><a href="https://hu.wikipedia.org/wiki/Planck-egys%C3%A9gek" title="Planck-egységek — húngaro" lang="hu" hreflang="hu" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy"><a href="https://hy.wikipedia.org/wiki/%D5%8A%D5%AC%D5%A1%D5%B6%D5%AF%D5%AB_%D5%B4%D5%AB%D5%A1%D5%BE%D5%B8%D6%80%D5%B6%D5%A5%D6%80" title="Պլանկի միավորներ — arménio" lang="hy" hreflang="hy" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-id"><a href="https://id.wikipedia.org/wiki/Satuan_Planck" title="Satuan Planck — indonésio" lang="id" hreflang="id" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it"><a href="https://it.wikipedia.org/wiki/Unit%C3%A0_di_misura_di_Planck" title="Unità di misura di Planck — italiano" lang="it" hreflang="it" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja"><a href="https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%83%B3%E3%82%AF%E5%8D%98%E4%BD%8D%E7%B3%BB" title="プランク単位系 — japonês" lang="ja" hreflang="ja" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ko"><a href="https://ko.wikipedia.org/wiki/%ED%94%8C%EB%9E%91%ED%81%AC_%EB%8B%A8%EC%9C%84%EA%B3%84" title="플랑크 단위계 — coreano" lang="ko" hreflang="ko" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-mr"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B2%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%8F%E0%A4%95%E0%A4%95%E0%A5%87" title="प्लांक एकके — marata" lang="mr" hreflang="mr" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-nl"><a href="https://nl.wikipedia.org/wiki/Planck-eenheden" title="Planck-eenheden — neerlandês" lang="nl" hreflang="nl" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn"><a href="https://nn.wikipedia.org/wiki/Planck-einingar" title="Planck-einingar — norueguês nynorsk" lang="nn" hreflang="nn" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A8%B2%E0%A9%88%E0%A8%82%E0%A8%95_%E0%A8%AF%E0%A9%82%E0%A8%A8%E0%A8%BF%E0%A8%9F%E0%A8%BE%E0%A8%82" title="ਪਲੈਂਕ ਯੂਨਿਟਾਂ — panjabi" lang="pa" hreflang="pa" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl"><a href="https://pl.wikipedia.org/wiki/Jednostki_Plancka" title="Jednostki Plancka — polaco" lang="pl" hreflang="pl" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-ru"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%BE%D0%B2%D1%81%D0%BA%D0%B8%D0%B5_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8B" title="Планковские единицы — russo" lang="ru" hreflang="ru" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn"><a href="https://scn.wikipedia.org/wiki/Unitati_di_misura_di_Planck" title="Unitati di misura di Planck — siciliano" lang="scn" hreflang="scn" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-sh"><a href="https://sh.wikipedia.org/wiki/Prirodne_jedinice" title="Prirodne jedinice — servo-croata" lang="sh" hreflang="sh" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-simple"><a href="https://simple.wikipedia.org/wiki/Planck_units" title="Planck units — Simple English" lang="en-simple" hreflang="en-simple" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk"><a href="https://sk.wikipedia.org/wiki/Planckove_jednotky" title="Planckove jednotky — eslovaco" lang="sk" hreflang="sk" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl"><a href="https://sl.wikipedia.org/wiki/Planckov_sistem_enot" title="Planckov sistem enot — esloveno" lang="sl" hreflang="sl" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D1%80%D0%BE%D0%B4%D0%BD%D0%B5_%D1%98%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D0%B5" title="Природне јединице — sérvio" lang="sr" hreflang="sr" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sv"><a href="https://sv.wikipedia.org/wiki/Planckenheter" title="Planckenheter — sueco" lang="sv" hreflang="sv" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr"><a href="https://tr.wikipedia.org/wiki/Planck_birimleri" title="Planck birimleri — turco" lang="tr" hreflang="tr" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tt"><a href="https://tt.wikipedia.org/wiki/Plank_ber%C3%A4mlekl%C3%A4re" title="Plank berämlekläre — tatar" lang="tt" hreflang="tt" class="interlanguage-link-target">Татарча/tatarça</a></li><li class="interlanguage-link interwiki-uk"><a href="https://uk.wikipedia.org/wiki/%D0%9E%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%96_%D0%9F%D0%BB%D0%B0%D0%BD%D0%BA%D0%B0" title="Одиниці Планка — ucraniano" lang="uk" hreflang="uk" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-uz"><a href="https://uz.wikipedia.org/wiki/Tabiiy_birliklar_tizimi" title="Tabiiy birliklar tizimi — usbeque" lang="uz" hreflang="uz" class="interlanguage-link-target">Oʻzbekcha/ўзбекча</a></li><li class="interlanguage-link interwiki-vi"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_th%E1%BB%91ng_%C4%91o_l%C6%B0%E1%BB%9Dng_Planck" title="Hệ thống đo lường Planck — vietnamita" lang="vi" hreflang="vi" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh"><a href="https://zh.wikipedia.org/wiki/%E6%99%AE%E6%9C%97%E5%85%8B%E5%96%AE%E4%BD%8D%E5%88%B6" title="普朗克單位制 — chinês" lang="zh" hreflang="zh" class="interlanguage-link-target">中文</a></li></ul>
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