Lorenz gauge condition

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href="https://cs.wikipedia.org/wiki/Lorenzova_kalibra%C4%8Dn%C3%AD_podm%C3%ADnka" title="Lorenzova kalibrační podmínka – Czech" lang="cs" hreflang="cs" data-title="Lorenzova kalibrační podmínka" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lorenz-Eichung" title="Lorenz-Eichung – German" lang="de" hreflang="de" data-title="Lorenz-Eichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ga%C5%AD%C4%9Do_de_Lorenz" title="Gaŭĝo de Lorenz – Esperanto" lang="eo" hreflang="eo" data-title="Gaŭĝo de Lorenz" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%B1%D8%A7%DB%8C%D8%B7_%D9%BE%DB%8C%D9%85%D8%A7%D9%86%D9%87_%D9%84%D9%88%D8%B1%D9%86%D8%AA%D8%B3" title="شرایط پیمانه لورنتس – Persian" lang="fa" hreflang="fa" data-title="شرایط پیمانه لورنتس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Jauge_de_Lorenz" title="Jauge de Lorenz – French" lang="fr" hreflang="fr" data-title="Jauge de Lorenz" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EB%A0%8C%EC%B8%A0_%EA%B2%8C%EC%9D%B4%EC%A7%80_%EC%A1%B0%EA%B1%B4" title="로렌츠 게이지 조건 – Korean" lang="ko" hreflang="ko" data-title="로렌츠 게이지 조건" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gauge_di_Lorenz" title="Gauge di Lorenz – Italian" lang="it" hreflang="it" data-title="Gauge di Lorenz" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lorenz-ijk" title="Lorenz-ijk – Dutch" lang="nl" hreflang="nl" data-title="Lorenz-ijk" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Calibre_de_Lorenz" title="Calibre de Lorenz – Portuguese" lang="pt" hreflang="pt" data-title="Calibre de Lorenz" 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searchaux" style="display:none">Gauge fixing of electro magnetic potential</div> <p>In <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, the <b>Lorenz gauge condition</b> or <b>Lorenz gauge</b> (after <a href="/wiki/Ludvig_Lorenz" title="Ludvig Lorenz">Ludvig Lorenz</a>) is a partial <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a> of the <a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">electromagnetic vector potential</a> by requiring <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 \partial _{\mu }A^{\mu }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }A^{\mu }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06283dc25d4ba0748f70e471a2d2ea59ce0cd9ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.332ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }A^{\mu }=0.}"></span> The name is frequently confused with <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Hendrik Lorentz</a>, who has given his name to many concepts in this field.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (See, however, the <b>Note</b> added below for a different interpretation.) The condition is <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a>. The Lorenz gauge condition does not completely determine the gauge: one can still make a gauge transformation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mu }\mapsto A^{\mu }+\partial ^{\mu }f,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo stretchy="false">↦</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>+</mo> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }\mapsto A^{\mu }+\partial ^{\mu }f,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe005b2c4f8bea422246bd28839ffe9c9004cf8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.88ex; height:2.676ex;" alt="{\displaystyle A^{\mu }\mapsto A^{\mu }+\partial ^{\mu }f,}"></span> where <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 \partial ^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial ^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ccdc631da103df8b657c5145020c2d71b48ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.567ex; height:2.343ex;" alt="{\displaystyle \partial ^{\mu }}"></span> is the <a href="/wiki/Four-gradient" title="Four-gradient">four-gradient</a> and <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" 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}"></span> is any <a href="/wiki/Harmonic_function" title="Harmonic function">harmonic</a> scalar function: that is, a <a href="/wiki/Scalar_function" class="mw-redirect" title="Scalar function">scalar function</a> obeying <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 \partial _{\mu }\partial ^{\mu }f=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>f</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }\partial ^{\mu }f=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/047e22818f464b5f19ce7cd49af1987edaff8806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.211ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }\partial ^{\mu }f=0,}"></span> the equation of a <a href="/wiki/Scalar_field_theory" title="Scalar field theory">massless scalar field</a>. </p><p>The Lorenz gauge condition is used to eliminate the redundant spin-0 component in Maxwell's equations when these are used to describe a massless spin-1 quantum field. It is also used for massive spin-1 fields where the concept of gauge transformations does not apply at all. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorenz_gauge_condition&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, the Lorenz condition is generally <a href="/wiki/Scientific_method" title="Scientific method">used</a> in <a href="/wiki/Calculation" title="Calculation">calculations</a> of <a href="/wiki/Time-variant_system" title="Time-variant system">time-dependent</a> <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic fields</a> through <a href="/wiki/Retarded_potential" title="Retarded potential">retarded potentials</a>.<sup id="cite_ref-mcdonald_2-0" class="reference"><a href="#cite_note-mcdonald-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The condition is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }A^{\mu }\equiv A^{\mu }{}_{,\mu }=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>≡</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }A^{\mu }\equiv A^{\mu }{}_{,\mu }=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5c97f3658ce9745a4bd415255550f330e026a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.078ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }A^{\mu }\equiv A^{\mu }{}_{,\mu }=0,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1da55ba40017c25d210f7e269efb2d6d539a1b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.967ex; height:2.343ex;" alt="{\displaystyle A^{\mu }}"></span> is the <a href="/wiki/Four-potential" class="mw-redirect" title="Four-potential">four-potential</a>, the comma denotes a <a href="/wiki/Partial_differentiation" class="mw-redirect" title="Partial differentiation">partial differentiation</a> and the repeated index indicates that the <a href="/wiki/Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> is being used. The condition has the advantage of being <a href="/wiki/Lorentz_invariant" class="mw-redirect" title="Lorentz invariant">Lorentz invariant</a>. It still leaves substantial gauge degrees of freedom. </p><p>In ordinary vector notation and <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> units, the condition is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1fb22696902891775ec554017db04c1d87bbe48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.954ex; height:5.843ex;" alt="{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}=0,}"></span> where <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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> is the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> and <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> is the <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a>;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> see also <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a>. </p><p>In <a href="/wiki/Gaussian_units" title="Gaussian units">Gaussian units</a> the condition is<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c}}{\frac {\partial \varphi }{\partial t}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c}}{\frac {\partial \varphi }{\partial t}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add2472a783fd0f7450fcd86557d4ed3194341bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.056ex; height:5.676ex;" alt="{\displaystyle \nabla \cdot {\mathbf {A} }+{\frac {1}{c}}{\frac {\partial \varphi }{\partial t}}=0.}"></span> </p><p>A quick justification of the Lorenz gauge can be found using <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> and the relation between the magnetic vector potential and the magnetic field: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</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}}=-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d169981953a5f7b5c8c8f1bf241b01d97f076d5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.161ex; height:5.843ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=-{\frac {\partial (\nabla \times \mathbf {A} )}{\partial t}}}"></span> </p><p>Therefore, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \left(\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \left(\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}\right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc5b49891a0aa8d4da89a38f2a62e74f40d335b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.876ex; height:6.176ex;" alt="{\displaystyle \nabla \times \left(\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}\right)=0.}"></span> </p><p>Since the curl is zero, that means there is a scalar function <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\nabla \varphi =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\nabla \varphi =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20c18698aabd2d50dbe0f2bd7ccc1ca2bc33c23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.781ex; height:5.509ex;" alt="{\displaystyle -\nabla \varphi =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}.}"></span> </p><p>This gives a well known equation for the electric field: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7b244e836cc4c8c1bcb6f4a3bcf29aa7d42bf7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.781ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.}"></span> </p><p>This result can be plugged into the <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère–Maxwell equation</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\\\nabla \times \left(\nabla \times \mathbf {A} \right)&=\\\Rightarrow \nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla ^{2}\mathbf {A} &=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial (\nabla \varphi )}{\partial t}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\\\nabla \times \left(\nabla \times \mathbf {A} \right)&=\\\Rightarrow \nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla ^{2}\mathbf {A} &=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial (\nabla \varphi )}{\partial t}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed7dc901c1fb7fb6e6b8cb3208f1e14e13af9a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:53.067ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}\\\nabla \times \left(\nabla \times \mathbf {A} \right)&=\\\Rightarrow \nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla ^{2}\mathbf {A} &=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial (\nabla \varphi )}{\partial t}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}.\\\end{aligned}}}"></span> </p><p>This leaves <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \left(\nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}+\nabla ^{2}\mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \left(\nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}+\nabla ^{2}\mathbf {A} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4443ce9404efe6d18f95392a34f32a7bdf4ae6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.213ex; height:6.343ex;" alt="{\displaystyle \nabla \left(\nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=\mu _{0}\mathbf {J} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}+\nabla ^{2}\mathbf {A} .}"></span> </p><p>To have Lorentz invariance, the time derivatives and spatial derivatives must be treated equally (i.e. of the same order). Therefore, it is convenient to choose the Lorenz gauge condition, which makes the left hand side zero and gives the result <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box \mathbf {A} =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\mathbf {A} =\mu _{0}\mathbf {J} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mi>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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box \mathbf {A} =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\mathbf {A} =\mu _{0}\mathbf {J} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5d543c73d4856631d9871537e007291ffdf873" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.145ex; height:6.343ex;" alt="{\displaystyle \Box \mathbf {A} =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\mathbf {A} =\mu _{0}\mathbf {J} .}"></span> </p><p>A similar procedure with a focus on the electric scalar potential and making the same gauge choice will yield <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box \varphi =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\varphi ={\frac {1}{\varepsilon _{0}}}\rho .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mi>φ</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mi>ρ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box \varphi =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\varphi ={\frac {1}{\varepsilon _{0}}}\rho .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65726de6856255f4c56888e70de844174b7e4844" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.486ex; height:6.343ex;" alt="{\displaystyle \Box \varphi =\left[{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right]\varphi ={\frac {1}{\varepsilon _{0}}}\rho .}"></span> </p><p>These are simpler and more symmetric forms of the inhomogeneous <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. </p><p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3720d3b0fb9800ba7ec863ab79267629463b8da4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.471ex; height:6.176ex;" alt="{\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}}"></span> is the vacuum velocity of light, and <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 \Box }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Box }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }"></span> is the <a href="/wiki/D%27Alembertian" class="mw-redirect" title="D'Alembertian">d'Alembertian</a> operator with the <span class="nowrap">(+ − − −)</span> metric signature. These equations are not only valid under vacuum conditions, but also in polarized media,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> if <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" 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 }"></span> and <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 {\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"></span> are source density and circulation density, respectively, of the electromagnetic induction fields <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 {\vec {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bc18ae485a72f148e85ccbeff2b3dcdd4f5f3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\vec {E}}}"></span> and <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 {\vec {B}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span> calculated as usual from <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and <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 {\vec {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95" 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}}}"></span> by the equations <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4218a9cf5a3beef2495e4141c1300eab65198a32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:18.029ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}"></span> </p><p>The explicit solutions for <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> and <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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> – unique, if all quantities vanish sufficiently fast at infinity – are known as <a href="/wiki/Retarded_potential" title="Retarded potential">retarded potentials</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorenz_gauge_condition&action=edit&section=2" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When originally published in 1867, Lorenz's work was not received well by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>. Maxwell had eliminated the <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb electrostatic force</a> from his derivation of the <a href="/wiki/Electromagnetic_wave_equation" title="Electromagnetic wave equation">electromagnetic wave equation</a> since he was working in what would nowadays be termed the <a href="/wiki/Coulomb_gauge" class="mw-redirect" title="Coulomb gauge">Coulomb gauge</a>. The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying <a href="/wiki/Electric_field" title="Electric field">electric field</a>, which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents". Lorenz's work was the first use of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> to simplify Maxwell's equations after Maxwell himself published his 1865 paper. In 1888, retarded potentials came into general use after <a href="/wiki/Heinrich_Rudolf_Hertz" class="mw-redirect" title="Heinrich Rudolf Hertz">Heinrich Rudolf Hertz</a>'s experiments on <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a>. In 1895, a further boost to the theory of retarded potentials came after <a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a>'s interpretation of data for <a href="/wiki/Electron" title="Electron">electrons</a> (after which investigation into <a href="/wiki/Electrical_phenomena" class="mw-redirect" title="Electrical phenomena">electrical phenomena</a> changed from time-dependent <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> and <a href="/wiki/Electric_current" title="Electric current">electric current</a> distributions over to moving <a href="/wiki/Point_charge" class="mw-redirect" title="Point charge">point charges</a>).<sup id="cite_ref-mcdonald_2-1" class="reference"><a href="#cite_note-mcdonald-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><b>Note</b> added on 26 November 2024: It should be pointed out that Lorenz actually <b>derived</b> the 'condition' from postulated integral expressions for the potentials (nowadays known as retarded potentials), whereas Lorentz (and before him Emil Wiechert) <b>imposed</b> it on the potentials to fix the gauge (see, e.g, his 1904 Encyclopedia article on electron theory). So Lorenz' equation is not a real condition but a mathematical result. It is therefore misleading to attribute the gauge condition to Lorenz. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorenz_gauge_condition&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Gauge_fixing" title="Gauge fixing">Gauge fixing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorenz_gauge_condition&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJacksonOkun2001" class="citation cs2"><a href="/wiki/John_David_Jackson_(physicist)" title="John David Jackson (physicist)">Jackson, J.D.</a>; <a href="/wiki/Lev_Okun" title="Lev Okun">Okun, L.B.</a> (2001), "Historical roots of gauge invariance", <i><a href="/wiki/Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</a></i>, <b>73</b> (3): <span class="nowrap">663–</span>680, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-ph/0012061">hep-ph/0012061</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001RvMP...73..663J">2001RvMP...73..663J</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.73.663">10.1103/RevModPhys.73.663</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:8285663">8285663</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Historical+roots+of+gauge+invariance&rft.volume=73&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E663-%3C%2Fspan%3E680&rft.date=2001&rft_id=info%3Aarxiv%2Fhep-ph%2F0012061&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A8285663%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.73.663&rft_id=info%3Abibcode%2F2001RvMP...73..663J&rft.aulast=Jackson&rft.aufirst=J.D.&rft.au=Okun%2C+L.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-mcdonald-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-mcdonald_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mcdonald_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcDonald1997" class="citation cs2">McDonald, Kirk T. (1997), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220519035900/http://www.physics.princeton.edu/~mcdonald/examples/jefimenko.pdf">"The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>, <b>65</b> (11): <span class="nowrap">1074–</span>1076, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997AmJPh..65.1074M">1997AmJPh..65.1074M</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.299.9838">10.1.1.299.9838</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18723">10.1119/1.18723</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13703110">13703110</a>, archived from <a rel="nofollow" class="external text" href="http://www.physics.princeton.edu/~mcdonald/examples/jefimenko.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2022-05-19</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=The+relation+between+expressions+for+time-dependent+electromagnetic+fields+given+by+Jefimenko+and+by+Panofsky+and+Phillips&rft.volume=65&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1074-%3C%2Fspan%3E1076&rft.date=1997&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.299.9838%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13703110%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.18723&rft_id=info%3Abibcode%2F1997AmJPh..65.1074M&rft.aulast=McDonald&rft.aufirst=Kirk+T.&rft_id=http%3A%2F%2Fwww.physics.princeton.edu%2F~mcdonald%2Fexamples%2Fjefimenko.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1999" class="citation book cs1">Jackson, John David (1999). <a href="/wiki/Classical_Electrodynamics_(book)" title="Classical Electrodynamics (book)"><i>Classical Electrodynamics</i></a> (3rd ed.). John Wiley & Sons. p. 240. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30932-1" title="Special:BookSources/978-0-471-30932-1"><bdi>978-0-471-30932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.pages=240&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=1999&rft.isbn=978-0-471-30932-1&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeller2012" class="citation book cs1">Keller, Ole (2012-02-02). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v2ck__wFOBEC"><i>Quantum Theory of Near-Field Electrodynamics</i></a>. Springer Science & Business Media. p. 19. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011qtnf.book.....K">2011qtnf.book.....K</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783642174100" title="Special:BookSources/9783642174100"><bdi>9783642174100</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Theory+of+Near-Field+Electrodynamics&rft.pages=19&rft.pub=Springer+Science+%26+Business+Media&rft.date=2012-02-02&rft_id=info%3Abibcode%2F2011qtnf.book.....K&rft.isbn=9783642174100&rft.aulast=Keller&rft.aufirst=Ole&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dv2ck__wFOBEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGbur2011" class="citation book cs1"><a href="/wiki/Greg_Gbur" title="Greg Gbur">Gbur, Gregory J.</a> (2011). <i>Mathematical Methods for Optical Physics and Engineering</i>. Cambridge University Press. p. 59. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011mmop.book.....G">2011mmop.book.....G</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-51610-5" title="Special:BookSources/978-0-521-51610-5"><bdi>978-0-521-51610-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Methods+for+Optical+Physics+and+Engineering&rft.pages=59&rft.pub=Cambridge+University+Press&rft.date=2011&rft_id=info%3Abibcode%2F2011mmop.book.....G&rft.isbn=978-0-521-51610-5&rft.aulast=Gbur&rft.aufirst=Gregory+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeitler1954" class="citation book cs1"><a href="/wiki/Walter_Heitler" title="Walter Heitler">Heitler, Walter</a> (1954). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L7w7UpecbKYC"><i>The Quantum Theory of Radiation</i></a>. Courier Corporation. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486645582" title="Special:BookSources/9780486645582"><bdi>9780486645582</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Quantum+Theory+of+Radiation&rft.pages=3&rft.pub=Courier+Corporation&rft.date=1954&rft.isbn=9780486645582&rft.aulast=Heitler&rft.aufirst=Walter&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DL7w7UpecbKYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">For example, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheremisinOkun2003" class="citation arxiv cs1">Cheremisin, M. V.; Okun, L. B. (2003). "Riemann-Silberstein representation of the complete Maxwell equations set". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0310036">hep-th/0310036</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Riemann-Silberstein+representation+of+the+complete+Maxwell+equations+set&rft.date=2003&rft_id=info%3Aarxiv%2Fhep-th%2F0310036&rft.aulast=Cheremisin&rft.aufirst=M.+V.&rft.au=Okun%2C+L.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links_and_further_reading">External links and further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorenz_gauge_condition&action=edit&section=5" title="Edit section: External links and further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>General</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, E. W. <a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/LorenzGauge.html">"Lorenz Gauge"</a>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lorenz+Gauge&rft.pub=Wolfram+Research&rft.aulast=Weisstein&rft.aufirst=E.+W.&rft_id=http%3A%2F%2Fscienceworld.wolfram.com%2Fphysics%2FLorenzGauge.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li></ul> <dl><dt>Further reading</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLorenz1867" class="citation journal cs1">Lorenz, L. (1867). "On the Identity of the Vibrations of Light with Electrical Currents". <i><a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a></i>. Series 4. <b>34</b> (230): <span class="nowrap">287–</span>301.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+Identity+of+the+Vibrations+of+Light+with+Electrical+Currents&rft.volume=34&rft.issue=230&rft.pages=%3Cspan+class%3D%22nowrap%22%3E287-%3C%2Fspan%3E301&rft.date=1867&rft.aulast=Lorenz&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Bladel1991" class="citation journal cs1">van Bladel, J. (1991). "Lorenz or Lorentz?". <i><a href="/wiki/IEEE_Antennas_and_Propagation_Magazine" class="mw-redirect" title="IEEE Antennas and Propagation Magazine">IEEE Antennas and Propagation Magazine</a></i>. <b>33</b> (2): 69. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMAP.1991.5672647">10.1109/MAP.1991.5672647</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21922455">21922455</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Antennas+and+Propagation+Magazine&rft.atitle=Lorenz+or+Lorentz%3F&rft.volume=33&rft.issue=2&rft.pages=69&rft.date=1991&rft_id=info%3Adoi%2F10.1109%2FMAP.1991.5672647&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21922455%23id-name%3DS2CID&rft.aulast=van+Bladel&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span> <ul><li>See also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBladel1991" class="citation journal cs1">Bladel, J. (1991). "Lorenz or Lorentz? [Addendum]". <i><a href="/wiki/IEEE_Antennas_and_Propagation_Magazine" class="mw-redirect" title="IEEE Antennas and Propagation Magazine">IEEE Antennas and Propagation Magazine</a></i>. <b>33</b> (4): 56. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991IAPM...33...56B">1991IAPM...33...56B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMAP.1991.5672657">10.1109/MAP.1991.5672657</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Antennas+and+Propagation+Magazine&rft.atitle=Lorenz+or+Lorentz%3F+%5BAddendum%5D&rft.volume=33&rft.issue=4&rft.pages=56&rft.date=1991&rft_id=info%3Adoi%2F10.1109%2FMAP.1991.5672657&rft_id=info%3Abibcode%2F1991IAPM...33...56B&rft.aulast=Bladel&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBecker1982" class="citation book cs1">Becker, R. (1982). <i>Electromagnetic Fields and Interactions</i>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. Chapter 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Fields+and+Interactions&rft.pages=Chapter+3&rft.pub=Dover+Publications&rft.date=1982&rft.aulast=Becker&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'Rahilly1938" class="citation book cs1">O'Rahilly, A. (1938). <i>Electromagnetics</i>. <a href="/wiki/Longmans,_Green_and_Co" class="mw-redirect" title="Longmans, Green and Co">Longmans, Green and Co</a>. Chapter 6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetics&rft.pages=Chapter+6&rft.pub=Longmans%2C+Green+and+Co&rft.date=1938&rft.aulast=O%27Rahilly&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li></ul> <dl><dt>History</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNevelsShin2001" class="citation journal cs1">Nevels, R.; Shin, Chang-Seok (2001). "Lorenz, Lorentz, and the gauge". <i><a href="/wiki/IEEE_Antennas_and_Propagation_Magazine" class="mw-redirect" title="IEEE Antennas and Propagation Magazine">IEEE Antennas and Propagation Magazine</a></i>. <b>43</b> (3): <span class="nowrap">70–</span>71. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001IAPM...43...70N">2001IAPM...43...70N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F74.934904">10.1109/74.934904</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Antennas+and+Propagation+Magazine&rft.atitle=Lorenz%2C+Lorentz%2C+and+the+gauge&rft.volume=43&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E70-%3C%2Fspan%3E71&rft.date=2001&rft_id=info%3Adoi%2F10.1109%2F74.934904&rft_id=info%3Abibcode%2F2001IAPM...43...70N&rft.aulast=Nevels&rft.aufirst=R.&rft.au=Shin%2C+Chang-Seok&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1989" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Whittaker, E. T.</a> (1989). <i><a href="/wiki/A_History_of_the_Theories_of_Aether_and_Electricity" title="A History of the Theories of Aether and Electricity">A History of the Theories of Aether and Electricity</a></i>. Vol. <span class="nowrap">1–</span>2. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. p. 268.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Theories+of+Aether+and+Electricity&rft.pages=268&rft.pub=Dover+Publications&rft.date=1989&rft.aulast=Whittaker&rft.aufirst=E.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALorenz+gauge+condition" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Lorenz_gauge_condition&oldid=1260408915">https://en.wikipedia.org/w/index.php?title=Lorenz_gauge_condition&oldid=1260408915</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Electromagnetism" title="Category:Electromagnetism">Electromagnetism</a></li><li><a href="/wiki/Category:Concepts_in_physics" title="Category:Concepts in physics">Concepts in physics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 30 November 2024, at 16:23<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Lorenz gauge condition - Wikipedia