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Update 2022-03-02 15:47

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Jean-Sébastien
2022-03-02 15:47:54 +01:00
parent ac1e628013
commit 21bf9fdba5
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build/emd_Me_ebM.html
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-01 Tue 08:14 -->
<!-- 2022-03-02 Wed 15:45 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1625,10 +1625,10 @@ Table of contents
We've encountered:
</p>
\begin{align}
(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
(iii) &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
(iv) &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}, \hspace{1cm} &amp;\mbox{Ampère}.
(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, &amp;\mbox{Faraday}, \nonumber \\
(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J}, &amp;\mbox{Ampère}.
\end{align}
<p>
Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
@@ -1637,11 +1637,28 @@ Fatal inconsistency: div of curl must always vanish. Check on \((iii)\):
= {\boldsymbol \nabla} \cdot \left( -\frac{\partial {\bf B}}{\partial t} \right) = -\frac{\partial}{\partial t} ({\boldsymbol \nabla} \cdot {\bf B}) = 0.
\]
But: try same with \((iv)\):
</p>
<div class="eqlabel" id="org764ac3e">
<p>
<a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
<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"/>
<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"/>
</svg></a>
</p>
<div class="alteqlabels" id="org0f4f12d">
<ul class="org-ul">
<li>Gr (7.35)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} \times {\bf B}) = \mu_0 {\boldsymbol \nabla} \cdot {\bf J}
\label{Gr(7.35)}
\tag{divcurlB}\label{divcurlB}
\]
LHS must be zero, but RHS is not zero for non-steady currents. Cannot be right !
LHS must be zero, but RHS is not zero for non-steady currents. Cannot be right!
</p>
<p>
@@ -1673,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
</div>
<div id="postamble" class="status">
<p class="author">Author: Jean-Sébastien Caux</p>
<p class="date">Created: 2022-03-01 Tue 08:14</p>
<p class="date">Created: 2022-03-02 Wed 15:45</p>
<p class="validation"></p>
</div>