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Jean-Sébastien
2022-02-21 10:35:02 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-17 Thu 08:42 -->
<!-- 2022-02-21 Mon 10:33 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -602,11 +602,11 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
</li>
<li>
<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
</li>
@@ -614,21 +614,12 @@ Table of contents
</details>
</li>
<li>
<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
<details>
<summary>
</li>
<li>
<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
</summary>
<ul>
<li>
<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
</li>
</ul>
</details>
</li>
<li>
@@ -640,11 +631,15 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
</li>
<li>
<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
</li>
<li>
<a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
</li>
@@ -660,6 +655,10 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
</li>
<li class="toc-currentpage">
<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
@@ -698,10 +697,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
</li>
<li>
<details>
<summary>
@@ -1435,7 +1430,7 @@ Table of contents
</li>
<li>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
</li>
<li>
@@ -1591,54 +1586,96 @@ Table of contents
</ul>
</details>
</nav>
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ms.html">Magnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ms_vp.html">The Vector Potential</a></li><li>Magnetic Boundary Conditions</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_me.html">Multipole Expansion of the Vector Potential&emsp;<small>[ems.ms.vp.me]</small></a></li><li>Up:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li></ul><div id="outline-container-ems_ms_vp_mbc" class="outline-5">
<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="ems.html">Electromagnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ms.html">Magnetostatics</a></li><li><a class="breadcrumb-link"href="ems_ms_vp.html">The Vector Potential</a></li><li>Magnetic Boundary Conditions</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_vp_A.html">Definition; Gauge Choices&emsp;<small>[ems.ms.vp.A]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_me.html">Multipole Expansion of the Vector Potential&emsp;<small>[ems.ms.vp.me]</small></a></li><li>Up:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li></ul><div id="outline-container-ems_ms_vp_mbc" class="outline-5">
<h5 id="ems_ms_vp_mbc">Magnetic Boundary Conditions<a class="headline-permalink" href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc"><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><span class="headline-id">ems.ms.vp.mbc</span></h5>
<div class="outline-text-5" id="text-ems_ms_vp_mbc">
<p>
Electrostatic fields: discontinuous at location of suface charge.<br>
Magnetostatic fields: discontinuous at location of surface current.
We know that electrostatic fields are discontinuous at location of suface charges.
</p>
<p>
Equation \ref{eq:DivBisZero}: \(\oint {\bf B} \cdot d{\bf a} = 0\) applied to wafer-thin
pillbox straddling surface: normal component
\[
B^{\perp}_{above} = B^{\perp}_{below}.
\label{Gr(5.72)}
\]
Tangential component: amperian loop of side length \(l\) perpendicular to surface current:
\[
\oint {\bf B} \cdot d{\bf l} = (B^{\parallel}_{above} - B^{\parallel}_{below}) l = \mu_0 I_{enc} = \mu_0 K l,
\]
and therefore
\[
B^{\parallel}_{above} - B^{\parallel}_{below} = \mu_0 K
\label{Gr(5.73)}
\]
So: component of \({\bf B}\) that is parallel to surface but perpendicular to current flow
is discontinuous, whereas the one parallel to the flow is continuous. In vector notation:
In a similar vein, magnetostatic fields will be discontinuous at the location of
surface currents.
</p>
<div class="main div" id="orgddb3724">
<p>
Let us follow a treatment greatly reminiscent of the electrostatic one.
Our starting point is equation <a href="./ems_ms_dcB_d.html#divB0">divB0</a> rewritten in integral form (using the divergence theorem)
as \(\oint d{\bf a} \cdot {\bf B} = 0\) and applied to a wafer-thin
pillbox straddling a surface carrying surface current density \({\bf K}\).
This means that the normal component is continuous,
</p>
<p>
\[
{\bf B}_{above} - {\bf B}_{below} = \mu_0 {\bf K} \times \hat{\bf n},
\label{Gr(5.74)}
\]
B^{\perp}_{above} = B^{\perp}_{below}.
\]
To get the tangential component, let us consider an amperian loop of side length \(l\) oriented perpendicular
to the direction of the surface current:
\[
\oint d{\bf l} \cdot {\bf B} = (B^{\parallel}_{above} - B^{\parallel}_{below}) l = \mu_0 I_{enc} = \mu_0 K l,
\]
and therefore
</p>
<p>
\[
B^{\parallel}_{above} - B^{\parallel}_{below} = \mu_0 K
\]
Thus, the component of \({\bf B}\) that is parallel to surface but perpendicular to the current flow
is discontinuous, whereas the one parallel to the flow is continuous. In vector notation, this becomes
</p>
<div class="main div" id="org9eceb8e">
<div class="eqlabel" id="org7d85b8b">
<p>
<a id="Bdisc"></a><a href="./ems_ms_vp_mbc.html#Bdisc"><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="org8d7e6f0">
<ul class="org-ul">
<li>Gr (5.76)</li>
</ul>
</div>
</div>
<p>
\[
{\bf B}_{above} - {\bf B}_{below} = \mu_0 {\bf K} \times \hat{\bf n},
\tag{Bdisc}\label{Bdisc}
\]
</p>
</div>
<p>
where \(\hat{\bf n}\) points upwards. For the vector potential, the relations are
</p>
<div class="main div" id="orgc9f9d71">
<div class="main div" id="org10599a4">
<div class="eqlabel" id="orgd7d1209">
<p>
<a id="Adisc"></a><a href="./ems_ms_vp_mbc.html#Adisc"><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="org27086bb">
<ul class="org-ul">
<li>Gr (5.77)</li>
</ul>
</div>
</div>
<p>
\[
{\bf A}_{above} = {\bf A}_{below}
\label{Gr(5.75)}
\]
{\bf A}_{above} = {\bf A}_{below}
\tag{Adisc}\label{Adisc}
\]
</p>
</div>
@@ -1654,7 +1691,7 @@ the tangential components of \({\bf A}\) are also continuous.
</p>
<p>
However, the derivative of \({\bf A}\) inherits the discontinuity of \({\bf B}\): explicitly,
The derivative of \({\bf A}\) however inherits the discontinuity of \({\bf B}\): explicitly,
</p>
\begin{align}
{\bf B}_{above} - {\bf B}_{below} &amp;= {\boldsymbol \nabla} \times {\bf A}_{above} - {\boldsymbol \nabla} \times {\bf A}_{below}
@@ -1679,13 +1716,28 @@ What is left is
\]
Therefore, reidentifying the normal component, we get
</p>
<div class="main div" id="org2752847">
<div class="main div" id="org218c00e">
<div class="eqlabel" id="orgb257c3c">
<p>
<a id="dAdisc"></a><a href="./ems_ms_vp_mbc.html#dAdisc"><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="orgf648efe">
<ul class="org-ul">
<li>Gr (5.78)</li>
</ul>
</div>
</div>
<p>
\[
\frac{\partial {\bf A}_{above}}{\partial n} - \frac{\partial {\bf A}_{below}}{\partial n}
= -\mu_0 {\bf K}
\label{Gr(5.76)}
\]
\frac{\partial {\bf A}_{above}}{\partial n} - \frac{\partial {\bf A}_{below}}{\partial n}
= -\mu_0 {\bf K}
\tag{dAdisc}\label{dAdisc}
\]
</p>
</div>
@@ -1694,7 +1746,7 @@ Therefore, reidentifying the normal component, we get
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_me.html">Multipole Expansion of the Vector Potential&emsp;<small>[ems.ms.vp.me]</small></a></li><li>Up:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li></ul>
<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ms_vp_A.html">Definition; Gauge Choices&emsp;<small>[ems.ms.vp.A]</small></a></li><li>Next:&nbsp;<a href="ems_ms_vp_me.html">Multipole Expansion of the Vector Potential&emsp;<small>[ems.ms.vp.me]</small></a></li><li>Up:&nbsp;<a href="ems_ms_vp.html">The Vector Potential&emsp;<small>[ems.ms.vp]</small></a></li></ul>
<br>
<hr>
<div class="license">
@@ -1709,7 +1761,7 @@ 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: 2022-02-17 Thu 08:42</p>
<p class="date">Created: 2022-02-21 Mon 10:33</p>
<p class="validation"></p>
</div>