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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Vektorpotential</span></h1>

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<p>Das <b>Vektorpotential</b> ist im Bereich der <a href="Vektoranalysis" title="Vektoranalysis">Vektoranalysis</a> ein <a href="Vektorfeld" title="Vektorfeld">Vektorfeld</a>, dessen <a href="Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotation</a> ein gegebenes Vektorfeld erzeugt.

</p><p>Formal lautet die Definition eines Vektorpotentials <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 {A}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> für ein festes Vektorfeld <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 {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae195b5427677e7fd3302b9bc400b2c9cbbe3082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.843ex;" alt="{\displaystyle {\vec {V}}}" loading="lazy"></span> mit dem <a href="Nabla-Operator" title="Nabla-Operator">Nabla-Operator</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 {\nabla }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65eeb56775c5bb9a9645ca773c8b8f744ca6ce0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.843ex;" alt="{\displaystyle {\vec {\nabla }}}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}=\mathrm {rot} \,{\vec {A}}={\vec {\nabla }}\times {\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/842504d15a429b664eb705f7f82a12f6572cad17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.612ex; height:3.009ex;" alt="{\displaystyle {\vec {V}}=\mathrm {rot} \,{\vec {A}}={\vec {\nabla }}\times {\vec {A}}}" loading="lazy"></span></dd></dl>

<p>Der Zusammenhang ist analog zum <a href="Skalarpotential" title="Skalarpotential">Skalarpotential</a> und seinem <a href="Gradientenfeld" title="Gradientenfeld">Gradientenfeld</a>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Historisch war das <a href="Magnetisches_Vektorpotential" title="Magnetisches Vektorpotential">magnetische Vektorpotential</a> der Anlass, das Vektorpotential zu beschreiben. Es wurde eingeführt, um in der klassischen <a href="Elektrodynamik" title="Elektrodynamik">Elektrodynamik</a> Berechnungen mit der <a href="Magnetische_Flussdichte" title="Magnetische Flussdichte">magnetischen Flussdichte</a> und der <a href="Elektromagnetische_Induktion" title="Elektromagnetische Induktion">elektromagnetischen Induktion</a> zu vereinfachen.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Berechnung">Berechnung</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74653a54f16d09b9569d8dbcd7d242ab4a4cab49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.9ex; height:2.843ex;" alt="{\displaystyle {\vec {V}}\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}" loading="lazy"></span>

ein zweifach <a href="Differenzierbarkeit" title="Differenzierbarkeit">stetig differenzierbares</a>, <a href="Quellfreies_Vektorfeld" title="Quellfreies Vektorfeld">quellfreies Vektorfeld</a>, das für <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 \lVert {\vec {x}}\rVert \to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle \lVert {\vec {x}}\rVert \to \infty }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/078960512a22355cd1620166d100a0d2f7d51f9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.592ex; height:2.843ex;" alt="{\displaystyle \lVert {\vec {x}}\rVert \to \infty }" loading="lazy"></span> mindestens so schnell abfällt wie <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 {\frac {1}{\lVert {\vec {x}}\rVert }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\lVert {\vec {x}}\rVert }}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b55d36fb45831bb02610b69f23bdf68586d2a22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:4.491ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\lVert {\vec {x}}\rVert }}}" loading="lazy"></span>. Dann ist durch

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<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}({\vec {x}})={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times {\vec {V}}({\vec {y}})}{\left\|{\vec {x}}-{\vec {y}}\right\|}}\,d^{3}y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58f9a4bdc91ddcc4229cd8757269cd0d225078fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.966ex; height:7.176ex;" alt="{\displaystyle {\vec {A}}({\vec {x}})={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times {\vec {V}}({\vec {y}})}{\left\|{\vec {x}}-{\vec {y}}\right\|}}\,d^{3}y}" loading="lazy"></span>,</dd></dl>

<p>ein Vektorpotential <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 {A}}}">

<mover>

</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> von <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 {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae195b5427677e7fd3302b9bc400b2c9cbbe3082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.843ex;" alt="{\displaystyle {\vec {V}}}" loading="lazy"></span> definiert<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>.

</p><p>Dies ist ein Spezialfall des <a href="Helmholtz-Theorem" title="Helmholtz-Theorem">Helmholtz-Theorems</a>.

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<div class="mw-heading mw-heading2"><h2 id="Uneindeutigkeit">Uneindeutigkeit</h2></div>

<p>Das Vektorpotential eines quellfreien Vektorfeldes ist nicht eindeutig definiert. Ist <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 {A}}}">

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</mover>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/391292ffadc65b0cde3e96f23afcdb811619dd95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:3.009ex;" alt="{\displaystyle {\vec {A}}}" loading="lazy"></span> ein Vektorpotential von <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 {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae195b5427677e7fd3302b9bc400b2c9cbbe3082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.843ex;" alt="{\displaystyle {\vec {V}}}" loading="lazy"></span>, so ist auch

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<annotation encoding="application/x-tex">{\displaystyle {\vec {A}}+{\vec {\nabla }}f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b34a28de4f14f5fef3a5b895d610f2c50b356159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.798ex; height:3.343ex;" alt="{\displaystyle {\vec {A}}+{\vec {\nabla }}f}" loading="lazy"></span></dd></dl>

<p>ein Vektorpotential von <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 {V}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {V}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae195b5427677e7fd3302b9bc400b2c9cbbe3082.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.843ex;" alt="{\displaystyle {\vec {V}}}" loading="lazy"></span> für beliebige, <a href="Stetig_differenzierbar" class="mw-redirect" title="Stetig differenzierbar">stetig differenzierbare</a> <a href="Skalarfeld" title="Skalarfeld">Skalarfelder</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 f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>. Dies folgt aus der Rotationsfreiheit von <a href="Gradientenfeld" title="Gradientenfeld">Gradientenfeldern</a>. In der Physik wird diese Eigenschaft des Vektorpotentials unter dem Thema <a href="Eichtransformation" title="Eichtransformation">Eichtransformation</a> behandelt.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_des_erzeugten_Feldes">Eigenschaften des erzeugten Feldes</h2></div>

<p>Wenn ein Vektorfeld durch ein Vektorpotential erzeugt werden kann, muss es ein <a href="Quellfreies_Vektorfeld" title="Quellfreies Vektorfeld">quellfreies Vektorfeld</a> sein.

</p><p>Dies liegt daran, dass die <a href="Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes">Divergenz</a> einer <a href="Rotation_eines_Vektorfeldes" title="Rotation eines Vektorfeldes">Rotation</a> immer Null ist.

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \,{\vec {V}}=\mathrm {div} \,\mathrm {rot} \,{\vec {A}}=0}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8a60a1ff20acee98f219b651053bd5cf9e55adc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.363ex; height:3.009ex;" alt="{\displaystyle \mathrm {div} \,{\vec {V}}=\mathrm {div} \,\mathrm {rot} \,{\vec {A}}=0}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>

<p>Das Vektorpotential wird vor allem in der Physik angewendet. Beispiele dafür sind

</p>

<ul><li>das <a href="Magnetisches_Vektorpotential" title="Magnetisches Vektorpotential">Magnetische Vektorpotential</a> und</li>

<li>das <a href="Elektrisches_Vektorpotential" class="mw-redirect" title="Elektrisches Vektorpotential">Elektrische Vektorpotential</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-:0-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-:0_1-0">a</a></sup> <sup><a href="#cite_ref-:0_1-1">b</a></sup> <sup><a href="#cite_ref-:0_1-2">c</a></sup></span> <span class="reference-text">Wolfgang Nolting: <cite style="font-style:italic">Grundkurs Theoretische Physik 3</cite> (= <cite style="font-style:italic">Springer-Lehrbuch</cite>). Springer Berlin Heidelberg, Berlin, Heidelberg 2013, ISBN 978-3-642-37904-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>188–190</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-642-37905-5">10.1007/978-3-642-37905-5</a></span>.<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:Vektorpotential&rft.au=Wolfgang+Nolting&rft.btitle=Grundkurs+Theoretische+Physik+3&rft.date=2013&rft.doi=10.1007%2F978-3-642-37905-5&rft.genre=book&rft.isbn=9783642379048&rft.pages=188-190&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Berlin+Heidelberg&rft.series=Springer-Lehrbuch" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">A. C. T. Wu, Chen Ning Yang: <cite style="font-style:italic">EVOLUTION OF THE CONCEPT OF THE VECTOR POTENTIAL IN THE DESCRIPTION OF FUNDAMENTAL INTERACTIONS</cite>. In: <cite style="font-style:italic">International Journal of Modern Physics A</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>21</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>16</span>, 30. Juni 2006, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220217-751X%22&key=cql">0217-751X</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>3235–3277</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1142/S0217751X06033143">10.1142/S0217751X06033143</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Vektorpotential&rft.atitle=EVOLUTION+OF+THE+CONCEPT+OF+THE+VECTOR+POTENTIAL+IN+THE+DESCRIPTION+OF+FUNDAMENTAL+INTERACTIONS&rft.au=A.+C.+T.+Wu%2C+Chen+Ning+Yang&rft.date=2006-06-30&rft.doi=10.1142%2FS0217751X06033143&rft.genre=journal&rft.issn=0217-751X&rft.issue=16&rft.jtitle=International+Journal+of+Modern+Physics+A&rft.pages=3235-3277&rft.volume=21" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

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