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<!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>
@@ -1624,74 +1616,177 @@ Table of contents
<p>
Discontinuities between different media, deduced from
</p>
<div class="core div" id="org48bc400">
<div class="core div" id="org2152f07">
<p>
{\bf Maxwell's equations {\it (in matter)}, integral form}
<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
</p>
<div class="eqlabel" id="org896f0bf">
<p>
<a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_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="org37cc7e4">
</div>
</div>
\begin{align}
(i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
(ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
(iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
(iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
(i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
(ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
(iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
(iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
\tag{Max_mat_int}\label{Max_mat_int}
\end{align}
</div>
<p>
Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
\({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
</p>
<div class="main div" id="orgaee2676">
<div class="eqlabel" id="org35b4f05">
<p>
<a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><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="orgc9c66ee">
<ul class="org-ul">
<li>Gr (7.60)</li>
</ul>
</div>
</div>
<p>
\[
\boxed{
D^{\perp}_1 - D^{\perp}_2 = \sigma_f
}
\label{Gr(7.59)}
\tag{Ddisc}\label{Ddisc}
\]
Same reasoning applied to \((ii)\) gives
</p>
</div>
<p>
Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
</p>
<div class="main div" id="org0c24b54">
<div class="eqlabel" id="org262505f">
<div class="alteqlabels" id="org560c237">
</div>
</div>
<p>
\[
\boxed{
B^{\perp}_1 - B^{\perp}_2 = 0
}
\label{Gr(7.60)}
\]
</p>
</div>
<p>
For \((iii)\): Amperian loop straddling surface: \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
-\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\). Limit of small loop: flux vanishes, therefore
</p>
<div class="main div" id="org2c8267d">
<p>
\[
\boxed{
{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
}
\label{Gr(7.61)}
\]
</p>
</div>
<p>
Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
No volume current can contribute, but a surface current can. Can write
\(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
and thus
</p>
<div class="main div" id="orge9d0c89">
<div class="eqlabel" id="org15b0fbc">
<p>
<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><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><a href="./emdm_Me_bc.html#Hdisc"><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="orgc1f1499">
<ul class="org-ul">
<li>Gr (7.63)</li>
</ul>
</div>
</div>
<p>
\[
\boxed{
{\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
}
\label{Gr(7.62)}
\tag{Hdisc}\label{Hdisc}
\]
</p>
</div>
<p>
These are the general boundary conditions for electrodynamics.
</p>
<p>
In case of linear media: can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
</p>
<div class="main div" id="org8b0c892">
<div class="eqlabel" id="org2d4783e">
<p>
<a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><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="orga65ee45">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
</div>
</div>
\begin{align}
(i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\
(ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
(iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
(iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}.
\label{Gr(7.63)}
\tag{disc_lm}\label{disc_lm}
\end{align}
</div>
<p>
If there is no free charge and no free current at boundary:
</p>
<div class="eqlabel" id="org656d25d">
<p>
<a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><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="org856682c">
<ul class="org-ul">
<li>Gr (7.64)</li>
</ul>
</div>
</div>
\begin{align}
(i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
(ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
(iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
(iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0.
\label{Gr(7.64)}
\tag{disc_nfc}\label{disc_nfc}
\end{align}
<p>
These are basis of theory of reflection and refraction.
@@ -1715,7 +1810,7 @@ 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>