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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Maxwellscher Spannungstensor</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Der <b>Maxwellsche Spannungstensor</b> <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}">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> (benannt nach <a href="James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>) ist ein symmetrischer <a href="Tensor" title="Tensor">Tensor</a> zweiter Stufe, der in der klassischen <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> verwendet wird, um die Wechselwirkung zwischen <a href="Elektromagnetische_Wechselwirkung" title="Elektromagnetische Wechselwirkung">elektromagnetischen Kräften</a> und mechanischem <a href="Impuls" title="Impuls">Impuls</a> darzustellen.
</p><p>In einfachen Situationen, beispielsweise eine elektrische <a href="Punktladung" title="Punktladung">Punktladung</a>, die sich in einem <a href="Homogenit%C3%A4t" title="Homogenität">homogenen</a> <a href="Magnetismus" title="Magnetismus">Magnetfeld</a> frei bewegt, lassen sich die Kräfte auf die Ladung durch die <a href="Lorentzkraft" title="Lorentzkraft">Lorentzkraft</a> berechnen. Für komplexere Probleme wird das Verfahren über die Lorentzkraft sehr lang. Es ist daher zweckmäßig, verschiedene Größen der Elektrodynamik im Maxwellschen Spannungstensor zu sammeln.
</p><p>In der <a href="Relativistisch" class="mw-redirect" title="Relativistisch">relativistischen</a> Formulierung des <a href="Elektromagnetismus" class="mw-redirect" title="Elektromagnetismus">Elektromagnetismus</a> erscheint der Maxwell-Tensor als Teil des elektromagnetischen <a href="Energie-Impuls-Tensor" title="Energie-Impuls-Tensor">Energie-Impuls-Tensors</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Im <a href="Vakuum" title="Vakuum">Vakuum</a> ist der Maxwellsche Spannungstensor in <a href="SI-Einheit" class="mw-redirect" title="SI-Einheit">SI-Einheiten</a> definiert durch
</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_{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {B_{i}B_{j}}{\mu _{0}}}-{\frac {1}{2}}\left(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}}\right)\delta _{ij}}">
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<annotation encoding="application/x-tex">{\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {B_{i}B_{j}}{\mu _{0}}}-{\frac {1}{2}}\left(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}}\right)\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5365ec049bd73c8cc56a872dd962b183b140a688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.149ex; height:6.343ex;" alt="{\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}+{\frac {B_{i}B_{j}}{\mu _{0}}}-{\frac {1}{2}}\left(\varepsilon _{0}E^{2}+{\frac {B^{2}}{\mu _{0}}}\right)\delta _{ij}}" loading="lazy"></span>,</dd></dl>
<p>wobei
</p>
<ul><li><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_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}" loading="lazy"></span> die Komponenten der <a href="Elektrische_Feldst%C3%A4rke" title="Elektrische Feldstärke">elektrischen Feldstärke</a> bezeichnen</li>
<li><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_{i}}">
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<annotation encoding="application/x-tex">{\displaystyle B_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82cda0578ec6b48774c541ecb9bee4a90176e62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle B_{i}}" loading="lazy"></span> die Komponenten der <a href="Magnetische_Flussdichte" title="Magnetische Flussdichte">magnetischen Flussdichte</a></li>
<li><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}}">
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<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>
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</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> die <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrische Feldkonstante</a></li>
<li><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>
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<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation>
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</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> die <a href="Magnetische_Feldkonstante" title="Magnetische Feldkonstante">magnetische Feldkonstante</a></li>
<li><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 _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>δ<!-- δ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"></span> das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a>.</li></ul>
<p>In <a href="Gau%C3%9Fsches_Einheitensystem" title="Gaußsches Einheitensystem">gaußschen cgs-Einheiten</a> ergibt sich der Tensor zu
</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_{ij}={\frac {1}{4\pi }}\left(E_{i}E_{j}+H_{i}H_{j}-{\frac {1}{2}}\left(E^{2}+H^{2}\right)\delta _{ij}\right)}">
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<annotation encoding="application/x-tex">{\displaystyle T_{ij}={\frac {1}{4\pi }}\left(E_{i}E_{j}+H_{i}H_{j}-{\frac {1}{2}}\left(E^{2}+H^{2}\right)\delta _{ij}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66a49ab34e1a05254c33f919288f70a92b13a24a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.723ex; height:6.176ex;" alt="{\displaystyle T_{ij}={\frac {1}{4\pi }}\left(E_{i}E_{j}+H_{i}H_{j}-{\frac {1}{2}}\left(E^{2}+H^{2}\right)\delta _{ij}\right)}" loading="lazy"></span></dd></dl>
<p>mit 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 H_{i}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle H_{i}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bd0312f590cc5a400008938f3cc304d42ad3986.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.731ex; height:2.509ex;" alt="{\displaystyle H_{i}}" loading="lazy"></span> der <a href="Magnetische_Feldst%C3%A4rke" title="Magnetische Feldstärke">magnetischen Feldstärke</a>.
</p><p>Für <a href="Elektromagnetische_Welle" title="Elektromagnetische Welle">elektromagnetische Wellen</a> in einem linearen Medium lässt sich der Maxwellsche Spannungstensor definieren als:<sup id="cite_ref-jackson_1-0" class="reference"><a href="#cite_note-jackson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</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_{ij}={\frac {1}{4\pi }}\left(E_{i}D_{j}+H_{i}B_{j}-{\frac {1}{2}}\left({\vec {E}}\cdot {\vec {D}}+{\vec {H}}\cdot {\vec {B}}\right)\delta _{ij}\right)}">
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<mo stretchy="false">→<!-- → --></mo>
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</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{ij}={\frac {1}{4\pi }}\left(E_{i}D_{j}+H_{i}B_{j}-{\frac {1}{2}}\left({\vec {E}}\cdot {\vec {D}}+{\vec {H}}\cdot {\vec {B}}\right)\delta _{ij}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd1a54c5b88b386d498fa99af0735ebb553a9f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.29ex; height:6.176ex;" alt="{\displaystyle T_{ij}={\frac {1}{4\pi }}\left(E_{i}D_{j}+H_{i}B_{j}-{\frac {1}{2}}\left({\vec {E}}\cdot {\vec {D}}+{\vec {H}}\cdot {\vec {B}}\right)\delta _{ij}\right)}" loading="lazy"></span></dd></dl>
<p>Diese Definition ist für <a href="Anisotrop" class="mw-redirect" title="Anisotrop">anisotrope</a> Medien jedoch nicht mehr symmetrisch.<sup id="cite_ref-jackson_1-1" class="reference"><a href="#cite_note-jackson-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Die Kraft pro Volumen <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 {\vec {f}}={\vec {F}}/V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>F</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {f}}={\vec {F}}/V}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a993d7aa51e26286c968b4434b6d72b17e35308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.484ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}={\vec {F}}/V}" loading="lazy"></span> kann aus der Divergenz des Spannungstensors und dem <a href="Poynting-Vektor" title="Poynting-Vektor">Poynting-Vektor</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 {\vec {S}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>S</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {S}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c71a6b104c40975c738d5f0e22d445ebd509eb81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.538ex; height:3.009ex;" alt="{\displaystyle {\vec {S}}}" loading="lazy"></span> bestimmt werden.
</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 ({\vec {\nabla }}\cdot \mathbf {T} )_{k}=\sum _{i=1}^{3}{\frac {\partial T_{ik}}{\partial _{x_{i}}}}=f_{k}+\varepsilon _{0}\mu _{0}{\frac {\partial S_{k}}{\partial t}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">T</mi>
</mrow>
<msub>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mrow>
<msub>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>=</mo>
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<msub>
<mi>S</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle ({\vec {\nabla }}\cdot \mathbf {T} )_{k}=\sum _{i=1}^{3}{\frac {\partial T_{ik}}{\partial _{x_{i}}}}=f_{k}+\varepsilon _{0}\mu _{0}{\frac {\partial S_{k}}{\partial t}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3af35d18337bb29d3b22c4e887a7722f2b80e4ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.809ex; height:7.176ex;" alt="{\displaystyle ({\vec {\nabla }}\cdot \mathbf {T} )_{k}=\sum _{i=1}^{3}{\frac {\partial T_{ik}}{\partial _{x_{i}}}}=f_{k}+\varepsilon _{0}\mu _{0}{\frac {\partial S_{k}}{\partial t}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Magnetostatik">Magnetostatik</h2></div>
<p>Für rein magnetische Felder (z. B. näherungsweise in <a href="Elektromotor" title="Elektromotor">Motoren</a>) fallen einige Terme weg, wodurch sich der Maxwell-Tensor vereinfacht zu:
</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_{ij}={\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mfrac>
</mrow>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</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>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{ij}={\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fceaaa1ac92a218228b7e0acac69988a04033bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.085ex; height:5.676ex;" alt="{\displaystyle T_{ij}={\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{ij}}" loading="lazy"></span></dd></dl>
<p>Für zylinderförmige Objekte – z. B. die <a href="Rotor" title="Rotor">Rotoren</a> eines Motors – ergibt sich
</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_{rt}={\frac {1}{\mu _{0}}}B_{r}B_{t}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{rt}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mfrac>
</mrow>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msub>
<mi>μ<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</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>r</mi>
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{rt}={\frac {1}{\mu _{0}}}B_{r}B_{t}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{rt}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ff86fdca2a3172df73d839018c578d5971f00a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.356ex; height:5.676ex;" alt="{\displaystyle T_{rt}={\frac {1}{\mu _{0}}}B_{r}B_{t}-{\frac {1}{2\mu _{0}}}B^{2}\delta _{rt}}" loading="lazy"></span></dd></dl>
<p>Dabei ist
</p>
<ul><li><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 r}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> die <a href="Scherung" class="mw-disambig" title="Scherung">Scherung</a> in radialer Richtung (vom Zylinder nach außen)</li>
<li><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}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> die Scherung in tangentialer Richtung (um den Zylinder herum). Der Motor wird hierbei durch die <a href="Tangentialkraft" title="Tangentialkraft">Tangentialkraft</a> angetrieben.</li>
<li><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_{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B_{r}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e750df8ee313e34c90563be9f828d67c934b55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.509ex;" alt="{\displaystyle B_{r}}" loading="lazy"></span> die Flussdichte in radialer Richtung</li>
<li><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_{t}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>B</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B_{t}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f92553a3c8585519e540724dffe9306c83ca2f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.59ex; height:2.509ex;" alt="{\displaystyle B_{t}}" loading="lazy"></span> die Flussdichte in tangentialer Richtung.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Elektrostatik">Elektrostatik</h2></div>
<p>In der <a href="Elektrostatik" title="Elektrostatik">Elektrostatik</a>, für die das Magnetfeld verschwindet (<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 {\vec {B}}={\vec {0}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>B</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<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 {\vec {B}}={\vec {0}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f937b54baee193f27ce085e3c28980e75678587a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}={\vec {0}}}" loading="lazy"></span>), ergibt sich der elektrostatische Maxwellsche Spannungstensor. In <a href="Vektor#Komponentenschreibweise" title="Vektor">Komponentenschreibweise</a> ergibt sich dieser durch:
</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_{ij}=\varepsilon _{0}E_{i}E_{j}-{\frac {1}{2}}\varepsilon _{0}E^{2}\delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<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>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}-{\frac {1}{2}}\varepsilon _{0}E^{2}\delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22a6eefa8285f166319ef4b0bf7c1bca74ddb82a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.545ex; height:5.176ex;" alt="{\displaystyle T_{ij}=\varepsilon _{0}E_{i}E_{j}-{\frac {1}{2}}\varepsilon _{0}E^{2}\delta _{ij}}" loading="lazy"></span></dd></dl>
<p>und in symbolischer Schreibweise durch
</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 {\boldsymbol {T}}=\varepsilon _{0}{\vec {E}}\otimes {\vec {E}}-{\frac {1}{2}}\varepsilon _{0}({\vec {E}}\cdot {\vec {E}})\mathbf {I} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold-italic">T</mi>
</mrow>
<mo>=</mo>
<msub>
<mi>ε<!-- ε --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>E</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>⊗<!-- ⊗ --></mo>
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<mi>E</mi>
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<mo>−<!-- − --></mo>
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<mn>2</mn>
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<mi>ε<!-- ε --></mi>
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<mo>⋅<!-- ⋅ --></mo>
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<mi mathvariant="bold">I</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {T}}=\varepsilon _{0}{\vec {E}}\otimes {\vec {E}}-{\frac {1}{2}}\varepsilon _{0}({\vec {E}}\cdot {\vec {E}})\mathbf {I} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4422fe7d182f3d155860fb3db6c9ee575d1ab7de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.452ex; height:5.176ex;" alt="{\displaystyle {\boldsymbol {T}}=\varepsilon _{0}{\vec {E}}\otimes {\vec {E}}-{\frac {1}{2}}\varepsilon _{0}({\vec {E}}\cdot {\vec {E}})\mathbf {I} }" loading="lazy"></span></dd></dl>
<p>wobei <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 \mathbf {I} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">I</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a458c8aeb096ce732abf346ae8edf3e4f53a126.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }" loading="lazy"></span> der <a href="Einheitstensor" title="Einheitstensor">Identitätstensor</a> sei.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="David_J._Griffiths" title="David J. Griffiths">David J. Griffiths</a>: <i>Introduction to Electrodynamics.</i> Benjamin Cummings Inc., 2008, S. 351–352</li>
<li><a href="John_David_Jackson_(Physiker)" title="John David Jackson (Physiker)">John David Jackson</a>: <i><a href="Classical_Electrodynamics" title="Classical Electrodynamics">Classical Electrodynamics</a>.</i> 3. Auflage, John Wiley & Sons, Inc., 1999.</li>
<li>Richard Becker: <i>Electromagnetic Fields and Interactions.</i> Dover Publications Inc., 1964.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-jackson-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-jackson_1-0">a</a></sup> <sup><a href="#cite_ref-jackson_1-1">b</a></sup></span> <span class="reference-text">John David Jackson: <cite style="font-style:italic">Klassische Elektrodynamik</cite>. Walter de Gruyter, 2020, ISBN 3-11-232201-0, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>280</span> (englisch: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Classical Electrodynamics</cite>. Übersetzt von Kurt Müller).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Maxwellscher+Spannungstensor&rft.au=John+David+Jackson&rft.btitle=Klassische+Elektrodynamik&rft.date=2020&rft.genre=book&rft.isbn=3112322010&rft.pages=280&rft.pub=Walter+de+Gruyter" style="display:none"> </span></span>
</li>
</ol>
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Normdaten (Sachbegriff): <a href="Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4467942-7">4467942-7</a></span> </div>
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