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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Impulsstromdichte</span></h1>
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<p>Die <b>Impulsstromdichte</b> ist eine <a href="Physikalische_Gr%C3%B6%C3%9Fe" title="Physikalische Größe">physikalische Größe</a>, die (analog zu anderen <a href="Strom_(Physik)#Stromdichte" title="Strom (Physik)">Stromdichten</a>) den Transport von <a href="Impuls" title="Impuls">Impuls</a> beschreibt. Sie ist Bestandteil des <a href="Energie-Impuls-Tensor" title="Energie-Impuls-Tensor">Energie-Impuls-Tensors</a> sowie insbesondere in der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> und <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> von Bedeutung. Es handelt sich um einen <a href="Tensor" title="Tensor">Tensor</a> <a href="Tensor#Arten_von_Tensoren" title="Tensor">zweiter Stufe</a>. Die <a href="Dimension_(Gr%C3%B6%C3%9Fensystem)" title="Dimension (Größensystem)">Dimension</a> der Impulsstromdichte ist die des <a href="Druck_(Physik)" title="Druck (Physik)">Drucks</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Kontinuumsmechanik">Kontinuumsmechanik</h2></div>
<p>In der Kontinuumsmechanik entsteht eine nichtverschwindende Impulsstromdichte in einem <a href="Festk%C3%B6rper" title="Festkörper">Festkörper</a> im Zuge einer <a href="Verformung" title="Verformung">Deformation</a> oder in einem <a href="Str%C3%B6mungsmechanik" title="Strömungsmechanik">strömenden</a> <a href="Fluid" title="Fluid">Fluid</a>. Da der Impuls und somit auch die Impulsdichte <a href="Vektor" title="Vektor">vektorielle</a> Größe sind, ist die Impulsstromdichte&nbsp;<i>T</i> eine tensorielle Größe.
</p><p>In einem <a href="Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystem</a> berechnet sich der Tensor als:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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_{ik}=\rho v_{i}v_{k}-\sigma _{ik}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>ρ<!-- ρ --></mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{ik}=\rho v_{i}v_{k}-\sigma _{ik}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8091b4678adc80a3c6b96e4973b74d206ea08c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.282ex; height:2.676ex;" alt="{\displaystyle T_{ik}=\rho v_{i}v_{k}-\sigma _{ik}}" loading="lazy"></span>,</dd></dl>
<p>mit
</p>
<ul><li>dem <a href="Konvektion" title="Konvektion">konvektiven</a> Beitrag <span class="mwe-math-element mwe-math-element-inline"><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 v_{i}v_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho v_{i}v_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f2e92ec50ca938127ee6fb7f303a6ca5eb0c0b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.346ex; height:2.176ex;" alt="{\displaystyle \rho v_{i}v_{k}}" loading="lazy"></span> (Impulstransport durch Materiestrom)
<ul><li>der <a href="Massendichte" class="mw-redirect" title="Massendichte">Massendichte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span></li>
<li>den Komponenten <span class="mwe-math-element mwe-math-element-inline"><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_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>v</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7dffe5726650f6daac54829972a94f38eb8ec127.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}" loading="lazy"></span> des Geschwindigkeitsvektors</li></ul></li>
<li>den Komponenten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ik}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{ik}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/53827aed08570d622cb1f732a3782b6ab6534f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.009ex;" alt="{\displaystyle \sigma _{ik}}" loading="lazy"></span> des <a href="Spannungstensor" title="Spannungstensor">Spannungstensors</a>, der den durch <a href="Oberfl%C3%A4chenkraft" title="Oberflächenkraft">Oberflächenkräfte</a> hervorgerufenen Beitrag beschreibt.</li></ul>
<p>Im <a href="Mechanisches_Gleichgewicht" title="Mechanisches Gleichgewicht">Mechanischen Gleichgewicht</a> (siehe auch <a href="Hydrostatisches_Gleichgewicht" title="Hydrostatisches Gleichgewicht">Hydrostatisches Gleichgewicht</a>) ist die <a href="Tensordivergenz" class="mw-redirect" title="Tensordivergenz">Tensordivergenz</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {div} \,T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">i</mi>
<mi mathvariant="normal">v</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \,T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dd967d66f003129e680f936fa304bf1e95d39e13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.19ex; height:2.176ex;" alt="{\displaystyle \mathrm {div} \,T}" loading="lazy"></span> der Impulsstromdichte null: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {div} \,T={\vec {0}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">i</mi>
<mi mathvariant="normal">v</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>T</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mn>0</mn>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \,T={\vec {0}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd1a31a7ed8f9f87deb52e3f39ef4f4100bc2726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.451ex; height:2.843ex;" alt="{\displaystyle \mathrm {div} \,T={\vec {0}}}" loading="lazy"></span>. Dies folgt aus der <a href="Kontinuit%C3%A4tsgleichung" title="Kontinuitätsgleichung">Kontinuitätsgleichung</a> für die Impulsdichte.
</p>
<div class="mw-heading mw-heading2"><h2 id="Elektrodynamik">Elektrodynamik</h2></div>
<p>Die Impulsstromdichte einer <a href="Elektromagnetische_Welle" title="Elektromagnetische Welle">elektromagnetischen Welle</a> oder allgemein von <a href="Elektromagnetisches_Feld" title="Elektromagnetisches Feld">elektromagnetischen Feldern</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ρ<!-- ρ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho =0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ba6310b27df5f9c9b0b1732e08cce27b99d68cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle \rho =0}" loading="lazy"></span>) berechnet sich als:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{ik}=\varepsilon _{0}\left({\frac {1}{2}}E^{2}\delta _{ik}-E_{i}E_{k}\right)+{\frac {1}{\mu }}_{0}\left({\frac {1}{2}}B^{2}\delta _{ik}-B_{i}B_{k}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>σ<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>μ<!-- μ --></mi>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{ik}=\varepsilon _{0}\left({\frac {1}{2}}E^{2}\delta _{ik}-E_{i}E_{k}\right)+{\frac {1}{\mu }}_{0}\left({\frac {1}{2}}B^{2}\delta _{ik}-B_{i}B_{k}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fca39856727d82bc71d5f7476afde6321557f39d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.425ex; height:6.176ex;" alt="{\displaystyle \sigma _{ik}=\varepsilon _{0}\left({\frac {1}{2}}E^{2}\delta _{ik}-E_{i}E_{k}\right)+{\frac {1}{\mu }}_{0}\left({\frac {1}{2}}B^{2}\delta _{ik}-B_{i}B_{k}\right)}" loading="lazy"></span> (siehe <a href="Energie-Impuls-Tensor#Im_SI-Einheitensystem" title="Energie-Impuls-Tensor">Maxwellscher Spannungstensor</a>).</dd></dl>
<p>mit
</p>
<ul><li>der <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrischen Feldkonstante</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span></li>
<li>der <a href="Magnetische_Feldkonstante" title="Magnetische Feldkonstante">magnetischen Feldkonstante</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}" loading="lazy"></span></li>
<li>der <a href="Elektrische_Feldst%C3%A4rke" title="Elektrische Feldstärke">elektrischen Feldstärke</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"></span></li>
<li>der <a href="Magnetische_Flussdichte" title="Magnetische Flussdichte">magnetischen Flussdichte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span></li>
<li>dem <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Symbol</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ik}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{ik}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1be99e304201ed3e8842e22b9adfb59a2e27e84e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.689ex; height:2.676ex;" alt="{\displaystyle \delta _{ik}}" loading="lazy"></span>.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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