Update 2022-03-15 10:07
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Jean-Sébastien
+144
-42
@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-07 Mon 20:38 -->
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<!-- 2022-03-15 Tue 08:10 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1310,10 +1310,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Mathematical Preliminaries</a><span class="headline-id">d.m</span>
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</li>
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<li>
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<a href="./d_ems.html#d_ems">Diagnostics: Electromagnetostatics</a><span class="headline-id">d.ems</span>
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</li>
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@@ -1352,6 +1348,10 @@ Table of contents
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<li>
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<a href="./d_red.html#d_red">Diagnostics: Relativistic Electrodynamics</a><span class="headline-id">d.red</span>
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</li>
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<li>
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<a href="./d_m.html#d_m">Diagnostics: Compendium - Mathematics</a><span class="headline-id">d.m</span>
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</li>
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</ul>
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@@ -1637,39 +1637,84 @@ We thus obtain
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\]
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Recognizing the proper velocity, we thus get that charge density and current density can together form the
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</p>
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<div class="core div" id="orgc6a702a">
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<div class="core div" id="org94723ec">
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<p>
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<b>Current density 4-vector</b>
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</p>
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<div class="eqlabel" id="org479bb40">
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<p>
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<a id="Jmu"></a><a href="./red_rem_Me.html#Jmu"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org324820b">
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</div>
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</div>
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<p>
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{\bf Current density 4-vector}
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\[
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J^\mu = \rho_0 \eta^\mu, \hspace{10mm}
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J^\mu = \left( c\rho, J_x, J_y, J_z \right)
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\]
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J^\mu = \rho_0 \eta^\mu, \hspace{10mm}
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J^\mu = \left( c\rho, J_x, J_y, J_z \right)
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\tag{Jmu}\label{Jmu}
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\]
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</p>
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</div>
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<p>
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The continuity equation (\ref{eq:continuity}) takes the simple form
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The continuity equation <a href="./ems_ms_ce.html#conteq">conteq</a> takes the simple form
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</p>
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<div class="core div" id="orgeaf45bd">
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<div class="core div" id="org5885f61">
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<p>
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<b>Continuity equation</b>
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</p>
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<div class="eqlabel" id="org52b2d08">
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<p>
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<a id="conteq_rel"></a><a href="./red_rem_Me.html#conteq_rel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org8386e2d">
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</div>
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</div>
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<p>
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{\bf Continuity equation}
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\[
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\frac{\partial J^\mu}{\partial x^\mu} = 0
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\]
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\frac{\partial J^\mu}{\partial x^\mu} = 0
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\tag{conteq_rel}\label{conteq_rel}
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\]
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</p>
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</div>
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<p>
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while similarly the notation simplifies for
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</p>
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<div class="main div" id="org94cc7f9">
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<div class="main div" id="orgdb33787">
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<p>
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<b>Maxwell's equations</b>
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</p>
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<div class="eqlabel" id="org26adf05">
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<p>
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<a id="Max_rel"></a><a href="./red_rem_Me.html#Max_rel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orge69d726">
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</div>
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</div>
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<p>
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{\bf Maxwell's equations}
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\[
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\frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu,
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\hspace{10mm}
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\frac{\partial G^{\mu \nu}}{\partial x^\nu} = 0
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\]
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\frac{\partial F^{\mu \nu}}{\partial x^\nu} = \mu_0 J^\mu,
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\hspace{10mm}
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\frac{\partial G^{\mu \nu}}{\partial x^\nu} = 0
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\tag{Max_rel}\label{Max_rel}
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\]
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</p>
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</div>
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@@ -1677,35 +1722,49 @@ while similarly the notation simplifies for
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<p>
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In terms of \(F^{\mu \nu}\) and the proper velocity \(\eta^\mu\), we also have the
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</p>
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<div class="main div" id="orge426c1b">
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<div class="main div" id="org1aadbdc">
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<p>
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<b>Minkowski force on a charge</b> \(q\)
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</p>
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<div class="eqlabel" id="org8ed30ad">
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<p>
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<a id="MinkF_rel"></a><a href="./red_rem_Me.html#MinkF_rel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org4770a30">
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</div>
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</div>
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<p>
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{\bf Minkowski force on a charge \(q\)}
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\[
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K^\mu = q F^{\mu \nu} \eta_\nu
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\]
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K^\mu = q F^{\mu \nu} \eta_\nu
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\tag{MinkF_rel}\label{MinkF_rel}
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\]
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</p>
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</div>
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<p>
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whose vector components are
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\[
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{\boldsymbol K} = \frac{q}{\sqrt{1 - u^2/c^2}} \left(
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{\boldsymbol E} + {\boldsymbol u} \times {\boldsymbol B} \right)
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{\boldsymbol K} = \frac{q}{\sqrt{1 - u^2/c^2}} \left(
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{\boldsymbol E} + {\boldsymbol u} \times {\boldsymbol B} \right)
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\]
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which becomes the Lorentz force law when remembering
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(\ref{eq:MinkowskiForce}).
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which becomes the Lorentz force law when remembering <a href="./red_rm_Mf.html#MinkF">MinkF</a>.
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</p>
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<p>
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We had
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\[
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{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}, \hspace{10mm}
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{\boldsymbol E} = -{\boldsymbol \nabla} \phi - \frac{\partial {\boldsymbol A}}{\partial t}, \hspace{10mm}
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{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}.
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\]
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We can group the potentials together into a 4-vector:
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\[
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A^\mu = \left( V/c, A_x, A_y, A_z \right).
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A^\mu = \left( \phi/c, A_x, A_y, A_z \right).
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\]
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The field tensor is then expressed as
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\[
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@@ -1721,18 +1780,46 @@ We can now exploit gauge invariance
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A^\mu ~~\longrightarrow~~ {A^\mu}^\prime = A^\mu + \frac{\partial \lambda}{\partial x_\mu}
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\]
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which leaves \(F^{\mu \nu}\) invariant. In particular, we can choose
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the Lorenz gauge (\ref{eq:InhomogeneousMaxwellLorenzGauge}) here expressed as
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the Lorenz gauge <a href="./emf_g_Lg.html#LorenzG">LorenzG</a> here expressed as
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</p>
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<div class="eqlabel" id="org02fe36f">
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<p>
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<a id="LorenzG_4v"></a><a href="./red_rem_Me.html#LorenzG_4v"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org162a638">
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</div>
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</div>
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<p>
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\[
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\frac{\partial A^\mu}{\partial x^\mu} = 0.
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\tag{LorenzG_4v}\label{LorenzG_4v}
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\]
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Defining the
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Using the d'Alembertian operator introduced in <a href="./emf_g_Lg.html#dAl">dAl</a>
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here rewritten as
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</p>
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<div class="main div" id="orgeb0bde6">
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<div class="main div" id="org4189168">
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<div class="eqlabel" id="org347c7ab">
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<p>
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<a id="dAl_4v"></a><a href="./red_rem_Me.html#dAl_4v"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org373726e">
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</div>
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</div>
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<p>
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{\bf d'Alembertian operator}
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\[
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\square^2 \equiv \frac{\partial}{\partial x_\nu} \frac{\partial}{\partial x^\nu} = {\boldsymbol \nabla}^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
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\]
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\square^2 \equiv \frac{\partial}{\partial x_\nu} \frac{\partial}{\partial x^\nu} = {\boldsymbol \nabla}^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
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\tag{dAl_4v}\label{dAl_4v}
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\]
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</p>
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</div>
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@@ -1740,12 +1827,27 @@ Defining the
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we obtain the final form of
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</p>
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<div class="core div" id="org8b01300">
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<div class="core div" id="org1c23781">
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<p>
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<b>Maxwell's equations</b> <i>(Lorenz gauge, 4-vector notation)</i>
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</p>
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<div class="eqlabel" id="org7c865aa">
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<p>
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<a id="Max_Lor_4v"></a><a href="./red_rem_Me.html#Max_Lor_4v"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
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<path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org5e9be4d">
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</div>
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</div>
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<p>
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{\bf Maxwell's equations (Lorenz gauge, 4-vector notation)}
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\[
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\square^2 A^\mu = -\mu_0 J^\mu
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\]
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\square^2 A^\mu = -\mu_0 J^\mu
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\tag{Max_Lor_4v}\label{Max_Lor_4v}
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\]
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</p>
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</div>
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@@ -1772,7 +1874,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="date">Created: 2022-03-15 Tue 08:10</p>
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<p class="validation"></p>
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</div>
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