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Jean-Sébastien
2022-03-24 08:43:21 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-22 Tue 10:52 -->
<!-- 2022-03-24 Thu 08:42 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1293,7 +1293,7 @@ Table of contents
</summary>
<ul>
<li>
<a href="./qed_t.html#qed_t">QED today</a><span class="headline-id">qed.t</span>
<a href="./qed_L.html#qed_L">Lagrangian</a><span class="headline-id">qed.L</span>
</li>
@@ -1636,14 +1636,14 @@ done by EM forces? From Lorentz force law:
Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
</p>
<div class="eqlabel" id="orge2567ac">
<div class="eqlabel" id="org0c693e0">
<p>
<a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><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="org4d05a80">
<div class="alteqlabels" id="orge34a4b7">
<ul class="org-ul">
<li>Gr (8.6)</li>
</ul>
@@ -1683,18 +1683,18 @@ so we get
Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
we obtain
</p>
<div class="main div" id="org36db96e">
<div class="main div" id="orgb7d7869">
<p>
<b>Poynting's theorem</b>
</p>
<div class="eqlabel" id="org21ce6b8">
<div class="eqlabel" id="org1a4b16b">
<p>
<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><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="org09a112b">
<div class="alteqlabels" id="org4feadff">
<ul class="org-ul">
<li>Gr (8.9)</li>
</ul>
@@ -1720,18 +1720,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
<p>
Energy per unit time, per unit area carried by EM fields: given by the
</p>
<div class="core div" id="org0bf3f6c">
<div class="core div" id="orgdc56ce9">
<p>
<b>Poynting vector</b>
</p>
<div class="eqlabel" id="org327095e">
<div class="eqlabel" id="orga61d997">
<p>
<a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><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="org7684a8e">
<div class="alteqlabels" id="orgb7a4cb9">
<ul class="org-ul">
<li>Gr (8.10)</li>
</ul>
@@ -1750,18 +1750,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
<p>
We can thus express Poynting's theorem more compactly:
</p>
<div class="core div" id="org4255191">
<div class="core div" id="orgd48d57f">
<p>
<b>Poynting's theorem</b> (integral form)
</p>
<div class="eqlabel" id="orgd0524d3">
<div class="eqlabel" id="org32c9daf">
<p>
<a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><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="org8acaa16">
<div class="alteqlabels" id="org59d2761">
<ul class="org-ul">
<li>Gr (8.11)</li>
</ul>
@@ -1780,18 +1780,18 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="org85e77e2">
<div class="core div" id="org5e10531">
<p>
<b>Energy in electromagnetic fields</b>
</p>
<div class="eqlabel" id="org7d4ae3d">
<div class="eqlabel" id="orgad6f50f">
<p>
<a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><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="org7a55647">
<div class="alteqlabels" id="orga3df7bc">
<ul class="org-ul">
<li>Gr (8.5)</li>
</ul>
@@ -1820,18 +1820,18 @@ Then,
\]
so we get the
</p>
<div class="core div" id="orgbdb096d">
<div class="core div" id="orgb7a11e9">
<p>
<b>Poynting theorem</b> (differential form)
</p>
<div class="eqlabel" id="org9c56948">
<div class="eqlabel" id="orga2e5c79">
<p>
<a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><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="orgdfe3994">
<div class="alteqlabels" id="org0cf780d">
<ul class="org-ul">
<li>Gr (8.14)</li>
</ul>
@@ -1854,7 +1854,7 @@ and has a similar for to the continuity equation
<div class="example div" id="org15b725b">
<div class="example div" id="org1083447">
<p>
<b>Example: Joule heating</b>
</p>
@@ -1911,7 +1911,7 @@ 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: 2022-03-22 Tue 10:52</p>
<p class="date">Created: 2022-03-24 Thu 08:42</p>
<p class="validation"></p>
</div>