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Jean-Sébastien
2022-03-07 20:40:36 +01:00
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commit 4808df71e6
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-03-02 Wed 15:45 -->
<!-- 2022-03-07 Mon 20:38 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1098,14 +1098,6 @@ Table of contents
<li>
<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
</li>
<li>
<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
</li>
<li>
<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
</li>
</ul>
@@ -1645,14 +1637,14 @@ 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="org0197499">
<div class="eqlabel" id="orgc9958b2">
<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="org9aacea3">
<div class="alteqlabels" id="orgf9bbccb">
<ul class="org-ul">
<li>Gr (8.6)</li>
</ul>
@@ -1665,7 +1657,7 @@ at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\). Thus,
\frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J}
\tag{dWdt_intEJ}\label{dWdt_intEJ}
\]
The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume.
The integrand is the work done per unit time, per unit volume, <i>i.e.</i> the power delivered per unit volume.
In terms of fields alone: use Ampère-Maxwell:
\[
{\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t}
@@ -1692,18 +1684,18 @@ 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="orgf118f4f">
<div class="main div" id="orgead023b">
<p>
<b>Poynting's theorem</b>
</p>
<div class="eqlabel" id="org1b7cdac">
<div class="eqlabel" id="org1b1ef48">
<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="orgd7b6cac">
<div class="alteqlabels" id="org9c51283">
<ul class="org-ul">
<li>Gr (8.9)</li>
</ul>
@@ -1729,18 +1721,18 @@ 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="orgf3198a5">
<div class="core div" id="orgafa4bdd">
<p>
<b>Poynting vector</b>
</p>
<div class="eqlabel" id="org8725431">
<div class="eqlabel" id="org0aaf227">
<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="org6f2879d">
<div class="alteqlabels" id="org05edf23">
<ul class="org-ul">
<li>Gr (8.10)</li>
</ul>
@@ -1759,18 +1751,18 @@ 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="org3a4bb91">
<div class="core div" id="orgf7b3c73">
<p>
<b>Poynting's theorem</b> (integral form)
</p>
<div class="eqlabel" id="orgbf2cc63">
<div class="eqlabel" id="org610ce5e">
<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="org5f484e3">
<div class="alteqlabels" id="org462a7a9">
<ul class="org-ul">
<li>Gr (8.11)</li>
</ul>
@@ -1789,18 +1781,18 @@ We can thus express Poynting's theorem more compactly:
<p>
where we have defined the total
</p>
<div class="core div" id="orgbc7eb16">
<div class="core div" id="org49394fc">
<p>
<b>Energy in electromagnetic fields</b>
</p>
<div class="eqlabel" id="org89872c3">
<div class="eqlabel" id="org2fd18f3">
<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="org67613ca">
<div class="alteqlabels" id="orge0e6bd0">
<ul class="org-ul">
<li>Gr (8.5)</li>
</ul>
@@ -1829,18 +1821,18 @@ Then,
\]
so we get the
</p>
<div class="core div" id="org487db23">
<div class="core div" id="orgb7f6aa8">
<p>
<b>Poynting theorem</b> (differential form)
</p>
<div class="eqlabel" id="org9593699">
<div class="eqlabel" id="orgc43f9ba">
<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="org129becb">
<div class="alteqlabels" id="org50e67b6">
<ul class="org-ul">
<li>Gr (8.14)</li>
</ul>
@@ -1863,7 +1855,7 @@ and has a similar for to the continuity equation
<div class="example div" id="orgd9e0ab5">
<div class="example div" id="org76ab6ea">
<p>
<b>Example: Joule heating</b>
</p>
@@ -1895,7 +1887,7 @@ and points radially inwards. Energy per unit time passing surface of wire:
\[
\int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
\]
where the minus sign means energy is flowing {\it in} (the wire heats up),
where the minus sign means energy is flowing <i>in</i> (the wire heats up),
and the value is as expected.
</p>
@@ -1920,7 +1912,7 @@ 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: 2022-03-02 Wed 15:45</p>
<p class="date">Created: 2022-03-07 Mon 20:38</p>
<p class="validation"></p>
</div>