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Update 2022-03-02 15:47

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Jean-Sébastien
2022-03-02 15:47:54 +01:00
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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>
@@ -1622,8 +1622,7 @@ Table of contents
</svg></a><span class="headline-id">emd.Me.dc</span></h4>
<div class="outline-text-4" id="text-emd_Me_dc">
<p>
The term which should be zero (but isn't) in (\ref{Gr(7.35)}) can be rewritten using
the continuity equation as
The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB">divcurlB</a> can be rewritten using the continuity equation as
\[
{\boldsymbol \nabla} \cdot {\bf J} = -\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial t}
(\varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}) = -{\boldsymbol \nabla} \cdot \left(
@@ -1631,37 +1630,67 @@ the continuity equation as
\]
The extra term would thus be eliminated if we were to put
</p>
<div class="core div" id="orgb95d862">
<div class="core div" id="orgdf7b3f9">
<div class="eqlabel" id="org5844c7b">
<p>
<a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><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="org1e192cc">
<ul class="org-ul">
<li>Gr (7.36)</li>
</ul>
</div>
</div>
<p>
\[
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
\label{Gr(7.36)}
\]
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
\tag{AmpMax}\label{AmpMax}
\]
</p>
</div>
<p>
\paragraph{Note:} this changes nothing in magnetostatics. Aesthetic appeal:
<b>Note</b>: this changes nothing in magnetostatics. Aesthetic appeal:
\[
\boxed{
\mbox{A changing electric field induces a magnetic field.}
}
\boxed{
\mbox{A changing electric field induces a magnetic field.}
}
\]
Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation of electromagnetic waves.
</p>
<p>
Maxwell baptized this term the
</p>
<div class="core div" id="orgb8b014a">
<div class="core div" id="orgb28580f">
<p>
<b>Displacement current</b>
</p>
<div class="eqlabel" id="org4a72dcc">
<p>
<a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><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="org57849e1">
<ul class="org-ul">
<li>Gr (7.37)</li>
</ul>
</div>
</div>
<p>
{\bf Displacement current}
\[
{\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}.
\label{Gr(7.37)}
\]
{\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}.
\tag{Jd}\label{Jd}
\]
</p>
</div>
@@ -1675,7 +1704,7 @@ where \(A\) is the area. Between the plates,
\[
\frac{\partial E}{\partial t} = \frac{1}{\varepsilon_0 A} \frac{dQ}{dt} = \frac{1}{\varepsilon_0 A} I.
\]
Checking (\ref{Gr(7.36)}),
Checking <a href="./emd_Me_dc.html#AmpMax">AmpMax</a>,
\[
\oint {\bf B} \cdot d{\bf l} = \mu_0 I_{\mbox{enc}} + \mu_0 \varepsilon_0 \int d{\bf a} \cdot \frac{\partial {\bf E}}{\partial t}
\label{Gr(7.38)}
@@ -1703,7 +1732,7 @@ 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>