Update 2022-03-07 20:40
This commit is contained in:
Jean-Sébastien
+124
-67
@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-03-02 Wed 15:45 -->
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<!-- 2022-03-07 Mon 20:38 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1098,14 +1098,6 @@ Table of contents
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<li>
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<a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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</li>
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<li>
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<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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</li>
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</ul>
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@@ -1637,12 +1629,12 @@ free charges and currents.
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</p>
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<p>
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From static case: electric polarization \({\bf P}\) produces bound charge density (\ref{Gr(4.12)})
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From static case: electric polarization \({\bf P}\) produces bound charge density <a href="./emsm_esm_po.html#rhob">rhob</a>
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\[
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\rho_b = -{\boldsymbol \nabla} \cdot {\bf P}
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\label{Gr(7.46)}
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\]
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and magnetization \({\bf M}\) produces bound current density (\ref{Gr(6.13)})
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and magnetization \({\bf M}\) produces bound current density <a href="./emsm_msm_fmo_bc.html#JbcurlM">JbcurlM</a>
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\[
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{\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
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\label{Gr(7.47)}
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@@ -1657,13 +1649,30 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
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\]
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We therefore have the
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</p>
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<div class="core div" id="org0421a72">
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<div class="core div" id="orgf1835ae">
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<p>
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<b>Polarization current density</b>
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</p>
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<div class="eqlabel" id="orgfad2da8">
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<p>
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<a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgf15b4cb">
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<ul class="org-ul">
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<li>Gr (7.48)</li>
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</ul>
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</div>
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</div>
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<p>
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{\bf Polarization current density}
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\[
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{\bf J}_p = \frac{\partial {\bf P}}{\partial t}
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\label{Gr(7.48)}
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\]
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{\bf J}_p = \frac{\partial {\bf P}}{\partial t}
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\tag{Jp}\label{Jp}
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\]
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</p>
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</div>
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@@ -1675,10 +1684,9 @@ the polarization current is the result of linear motion of charge when
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polarization changes). We can check consistency with the continuity equation
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associated to the conservation of bound charges:
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</p>
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<aside id="org642846e">
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<aside id="orgadb90c7">
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<p>
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Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
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\({\boldsymbol J}_b\) but this is not the convention used here.
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Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
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</p>
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</aside>
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<p>
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@@ -1696,31 +1704,56 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
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<p>
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In view of this: total charge density can be separated into 2 parts,
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{\it free} and {\it bound}:
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<b>free</b> and <b>bound</b>:
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</p>
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<div class="main div" id="orgba471ef">
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<div class="main div" id="orgbe78924">
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<div class="eqlabel" id="org4679dd8">
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<p>
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<a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgb7e7ed8">
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<ul class="org-ul">
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<li>Gr (7.49)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
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\label{Gr(7.49)}
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\]
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\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
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\tag{rhofb}\label{rhofb}
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\]
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</p>
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</div>
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<p>
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and current can be separated into three parts, {\it free}, {\it bound} and
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{\it polarization}:
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and current can be separated into three parts, <b>free</b>, <b>bound</b> and
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<b>polarization</b>:
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</p>
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<div class="main div" id="org2ffd81b">
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<div class="main div" id="org1c506a7">
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<div class="eqlabel" id="orgbf0c17d">
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<p>
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<a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org33aea7e">
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<ul class="org-ul">
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<li>Gr (7.50)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
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</p>
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<ul class="org-ul">
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<li>\frac{∂ {\bf P}}{∂ t}.</li>
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</ul>
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<p>
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\label{Gr(7.50)}
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{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t}.
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\tag{Jfbp}\label{Jfbp}
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\]
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</p>
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@@ -1735,24 +1768,19 @@ Gauss's law: can be rewritten
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\]
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where (as in static case)
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</p>
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<div class="core div" id="org88bb3c5">
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<div class="core div" id="org501f375">
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<p>
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\[
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{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
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\label{Gr(7.52)}
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\]
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{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
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\label{Gr(7.52)}
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\]
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</p>
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</div>
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<p>
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Ampère's law including Maxwell's term:
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\[
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{\boldsymbol ∇} × {\bf B} = μ_0 \left( {\bf J}_f + {\boldsymbol ∇} × {\bf M}
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</p>
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<ul class="org-ul">
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<li>\frac{∂ {\bf P}}{∂ t} \right) + μ_0 ε_0 \frac{∂ {\bf E}}{∂ t},</li>
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</ul>
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<p>
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{\boldsymbol \nabla} \times {\bf B} = \mu_0 \left( {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t},
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\]
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or
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\[
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@@ -1761,12 +1789,12 @@ or
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\]
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where as before
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</p>
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<div class="core div" id="org90cea20">
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<div class="core div" id="orgfaac9ca">
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<p>
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\[
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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\label{Gr(7.54)}
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\]
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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\label{Gr(7.54)}
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\]
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</p>
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</div>
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@@ -1779,21 +1807,36 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
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<p>
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In terms of free charges and currents, we thus get
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</p>
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<div class="core div" id="orgd6526ab">
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<div class="core div" id="org4a1df55">
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<p>
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{\bf Maxwell's equations {\it (in matter)}}
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<b>Maxwell's equations</b> <i>(in matter)</i>
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</p>
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<div class="eqlabel" id="orga6eb31a">
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<p>
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<a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org154a0ec">
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<ul class="org-ul">
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<li>Gr (7.55)</li>
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</ul>
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</div>
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</div>
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\begin{align}
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(i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
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(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
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(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
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(iv)~~ &{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
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\label{Gr(7.55)}
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(i)~~ &{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
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(ii)~~ &{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
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(iii)~~ &{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
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(iv)~~ &{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
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\tag{Max_mat}\label{Max_mat}
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\end{align}
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</div>
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<p>
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Last term: {\bf displacement current},
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Last term: <b>displacement current</b>,
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\[
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{\bf J}_d = \frac{\partial {\bf D}}{\partial t}
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\label{Gr(7.58)}
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@@ -1801,23 +1844,37 @@ Last term: {\bf displacement current},
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</p>
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<p>
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Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and \({\bf H}\)
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This must all be complemented by the <b>constitutive relations</b> giving \({\bf D}\) and \({\bf H}\)
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in terms of \({\bf E}\) and \({\bf B}\).
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For the restricted case of linear media:
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</p>
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<div class="main div" id="orgd345cd6">
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<div class="main div" id="orge60b08f">
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<div class="eqlabel" id="org88284f8">
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<p>
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<a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="org21479eb">
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<ul class="org-ul">
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<li>Gr (7.56,7.57)</li>
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</ul>
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</div>
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</div>
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<p>
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\[
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{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
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{\bf M} = \chi_m {\bf H}
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\label{Gr(7.56)}
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\]
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{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
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{\bf M} = \chi_m {\bf H}
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\]
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so
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\[
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{\bf D} = \varepsilon {\bf E}, \hspace{1cm}
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{\bf H} = \frac{1}{\mu} {\bf B},
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\label{Gr(7.57)}
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\]
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{\bf D} = \varepsilon {\bf E}, \hspace{1cm}
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{\bf H} = \frac{1}{\mu} {\bf B},
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\tag{consrel}\label{consrel}
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\]
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where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\).
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</p>
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@@ -1842,7 +1899,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-03-02 Wed 15:45</p>
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<p class="date">Created: 2022-03-07 Mon 20:38</p>
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<p class="validation"></p>
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</div>
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