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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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build/emd_ce_mst.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>
@@ -1638,30 +1638,27 @@ Substitute for \(\rho\) and \({\boldsymbol J}\) using Maxwell (Gauss and Ampère
<p>
On the other hand we have
\[
\frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B} \right)
= \frac{∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
</p>
<ul class="org-ul">
<li>{\boldsymbol E} × \frac{∂ {\boldsymbol B}}{∂ t}.</li>
</ul>
\begin{equation}
\frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right)
= \frac{\partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
+ {\boldsymbol E} \times \frac{\partial {\boldsymbol B}}{\partial t}.
\end{equation}
<p>
\]
Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\),
\[
\frac{ ∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
= \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B}\right)
</p>
<ul class="org-ul">
<li>{\boldsymbol E} × \left({\boldsymbol ∇} × {\boldsymbol E} \right)</li>
</ul>
\begin{equation}
\frac{ \partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
= \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B}\right)
+ {\boldsymbol E} \times \left({\boldsymbol \nabla} \times {\boldsymbol E} \right)
\end{equation}
<p>
\]
so
\[
{\boldsymbol f} = \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} - {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \right) - \frac{1}{\mu_0} \left( {\boldsymbol B} \times \left( {\boldsymbol \nabla} \times {\boldsymbol B} \right) \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right).
\]
Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). Moreover, by product rule 4,
Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\).
Moreover, by <a href="./c_m_dc_pr.html#grad_sprod">grad_sprod</a>,
\[
\frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right)
\]
@@ -1678,18 +1675,27 @@ and similarly for \({\boldsymbol B}\). We thus get
<p>
This expression can be greatly simplified by introducing the
</p>
<div class="main div" id="org41d984c">
<div class="main div" id="orga7d370d">
<p>
{\bf Maxwell stress tensor}
\[
T_{ij} ≡ ε_0 \left( E_i E_j - \frac{1}{2} δ_{ij} E^2\right)
<b>Maxwell stress tensor</b>
</p>
<ul class="org-ul">
<li>\frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} δ_{ij} B^2 \right)</li>
</ul>
<div class="eqlabel" id="orge5429c3">
<p>
\]
<a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><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="org0306495">
</div>
</div>
\begin{equation}
T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
+ \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)
\tag{MaxST}\label{MaxST}
\end{equation}
</div>
<p>
@@ -1699,26 +1705,56 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
<p>
We then obtain
We then obtain the
</p>
<div class="main div" id="orgefc4ae2">
<div class="main div" id="org4781d10">
<p>
<b>EM force per unit volume</b>
</p>
<div class="eqlabel" id="orgd9f91b7">
<p>
<a id="fT"></a><a href="./emd_ce_mst.html#fT"><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="org7d4a06b">
</div>
</div>
<p>
{\bf EM force per unit volume}
\[
{\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t}
\]
{\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t}
\tag{fT}\label{fT}
\]
</p>
</div>
<p>
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
</p>
<div class="main div" id="orgc9bf6dd">
<div class="main div" id="orgea18677">
<p>
<b>Total force on charges in volume</b>
</p>
<div class="eqlabel" id="orgef98657">
<p>
<a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><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="org4c9c200">
</div>
</div>
<p>
{\bf Total force on charges in volume}
\[
{\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau.
\]
{\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau.
\tag{totFo}\label{totFo}
\]
</p>
</div>
@@ -1742,7 +1778,7 @@ 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>