Update 2022-03-01 08:15
This commit is contained in:
Jean-Sébastien
+205
-46
@@ -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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<!-- 2022-02-21 Mon 20:41 -->
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<!-- 2022-03-01 Tue 08:14 -->
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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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@@ -1622,11 +1622,28 @@ Table of contents
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</svg></a><span class="headline-id">emsm.msm.H.A</span></h5>
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<div class="outline-text-5" id="text-emsm_msm_H_A">
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<p>
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Nomenclature: as in electric case, we have bound currents, and everything else, which we call the {\bf free current}.
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Nomenclature: as in electric case, we have bound currents, and everything else, which we call the <b>free current</b>.
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Total current:
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</p>
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<div class="eqlabel" id="orga39b83a">
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<p>
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<a id="JbJf"></a><a href="./emsm_msm_H_A.html#JbJf"><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="orgc7c6653">
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<ul class="org-ul">
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<li>Gr (6.17)</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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\[
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{\bf J} = {\bf J}_b + {\bf J}_f
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\label{Gr(6.17)}
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\tag{JbJf}\label{JbJf}
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\]
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Ampère's law:
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\[
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@@ -1635,36 +1652,81 @@ Ampère's law:
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\]
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so we can define
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</p>
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<div class="core div" id="org38fa4c3">
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<div class="core div" id="org32c294e">
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<div class="eqlabel" id="orgc06236c">
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<p>
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<a id="HBM"></a><a href="./emsm_msm_H_A.html#HBM"><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="org410c63d">
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<ul class="org-ul">
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<li>Gr (6.18)</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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\[
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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\label{Gr(6.18)}
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\]
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{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
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\tag{HBM}\label{HBM}
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\]
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</p>
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</div>
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<p>
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and rewrite Ampère's law as
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</p>
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<div class="core div" id="org1f186fa">
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<div class="core div" id="org1ebd940">
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<div class="eqlabel" id="org06d1eac">
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<p>
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<a id="curlHJf"></a><a href="./emsm_msm_H_A.html#curlHJf"><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="org7c7eb8e">
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<ul class="org-ul">
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<li>Gr (6.19)</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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\[
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{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f
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\label{Gr(6.19)}
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\]
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{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f
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\tag{curlHJf}\label{curlHJf}
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\]
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</p>
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</div>
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<p>
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or in integral form,
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</p>
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<div class="core div" id="orgf4bc93b">
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<div class="core div" id="orge1fd102">
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<div class="eqlabel" id="orga535d4d">
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<p>
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<a id="intHdl"></a><a href="./emsm_msm_H_A.html#intHdl"><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="org6d8451f">
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<ul class="org-ul">
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<li>Gr (6.20)</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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\[
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\oint {\bf H} \cdot d{\bf l} = I_{f_{enc}}
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\label{Gr(6.20)}
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\]
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\oint {\bf H} \cdot d{\bf l} = I_{f_{enc}}
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\tag{intHdl}\label{intHdl}
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\]
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</p>
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</div>
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@@ -1673,23 +1735,67 @@ or in integral form,
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rewrite Ampère's law in terms of free currents alone. Bound currents come along for the ride.
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</p>
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<div class="example div" id="orgae85a06">
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<div class="example div" id="org64362d6">
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<p>
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\paragraph{Example 6.2:} long copper rod radius \({\bf R}\) carries uniformly distributed free current \(I\).
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Find \({\bf H}\) inside and outside rod.
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\paragraph{Solution:} copper weakly diamagnetic: dipoles line up opposite the field.
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<b>Example</b>: copper rod
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</p>
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<p>
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Consider a long copper rod of radius \({\bf R}\) carrying uniformly distributed free current \(I\).
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</p>
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<p>
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<b>Task</b>: find \({\bf H}\) inside and outside the rod.
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</p>
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<p>
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<b>Solution</b>: copper weakly diamagnetic: dipoles line up opposite the field.
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Bound currents antiparallel to \(I\) in bulk and parallel at surface. All currents longitudinal
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so \({\bf B}, {\bf M}, {\bf H}\) are circumferential. Apply integral form of Ampère's law with
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radius \(s < R\): \(H (2\pi s) = I_{f_{enc}} = I \frac{\pi s^2}{\pi R^2}\) so
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</p>
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<div class="eqlabel" id="org3874fab">
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||||
<p>
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||||
<a id="Hrod_in"></a><a href="./emsm_msm_H_A.html#Hrod_in"><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="org8615022">
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<ul class="org-ul">
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<li>Gr (6.21)</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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\[
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{\bf H} = \frac{I s}{2\pi R^2} \hat{\boldsymbol \varphi}, \hspace{5mm} s \leq R
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\label{Gr(6.21)}
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\]
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{\bf H} = \frac{I s}{2\pi R^2} \hat{\boldsymbol \varphi}, \hspace{5mm} s \leq R
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\tag{Hrod_in}\label{Hrod_in}
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||||
\]
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Outside,
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</p>
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||||
<div class="eqlabel" id="org2e8656f">
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<p>
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||||
<a id="Hrod_out"></a><a href="./emsm_msm_H_A.html#Hrod_out"><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"/>
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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="orgfe04bd5">
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||||
<ul class="org-ul">
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||||
<li>Gr (6.22)</li>
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</ul>
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||||
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</div>
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||||
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</div>
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<p>
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||||
\[
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||||
{\bf H} = \frac{I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{5mm} s \geq R.
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\label{Gr(6.22)}
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||||
\]
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||||
{\bf H} = \frac{I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{5mm} s \geq R.
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\tag{Hrod_out}\label{Hrod_out}
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||||
\]
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There, \({\bf M} = 0\) so \({\bf B} = \mu_0 {\bf H}\).
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</p>
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@@ -1702,9 +1808,26 @@ There, \({\bf M} = 0\) so \({\bf B} = \mu_0 {\bf H}\).
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<p>
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Similarly to electric case: cannot assume that \({\bf H}\) is like \({\bf B}\).
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\({\bf H}\) might have a divergence,
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</p>
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<div class="eqlabel" id="org1605cc8">
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<p>
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||||
<a id="divH"></a><a href="./emsm_msm_H_A.html#divH"><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="org1693077">
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<ul class="org-ul">
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<li>Gr (6.23)</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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\[
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{\boldsymbol \nabla} \cdot {\bf H} = -{\boldsymbol \nabla} \cdot {\bf M}
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\label{Gr(6.23)}
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\tag{divH}\label{divH}
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\]
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</p>
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</div>
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@@ -1714,7 +1837,8 @@ Similarly to electric case: cannot assume that \({\bf H}\) is like \({\bf B}\).
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<h6 id="emsm_msm_H_A_elm"><a href="#emsm_msm_H_A_elm">Energy in Linear Media</a></h6>
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<div class="outline-text-6" id="text-emsm_msm_H_A_elm">
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<p>
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Recall \ref{Gr(7.31)}, magnetic energy of system of free currents:
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Need <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> (to be proven later) giving the
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magnetic energy of a system of free currents:
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\[
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W_{mag} = \frac{1}{2} \int_{\cal V} d\tau {\bf A} \cdot {\bf J}_f
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\]
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@@ -1729,14 +1853,12 @@ move to volume currents:
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\Delta W_{mag} = \int_{\cal V} d\tau {\bf J}_f \cdot \Delta {\bf A} = \int_{\cal V} d\tau ({\boldsymbol \nabla} \times {\bf H}) \cdot \Delta {\bf A}
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\]
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But \({\boldsymbol \nabla} \times (\Delta {\bf A}) = \Delta {\bf B}\) and
|
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\[
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||||
({\boldsymbol ∇} × {\bf H}) ⋅ Δ {\bf A} = {\bf H} ⋅ ({\boldsymbol ∇} × Δ {\bf A})
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||||
</p>
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||||
<ul class="org-ul">
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||||
<li>{\boldsymbol ∇} ⋅ (Δ {\bf A} × {\bf H}) = {\bf H} ⋅ Δ {\bf B} - {\boldsymbol ∇} ⋅ (Δ {\bf A} × {\bf H})</li>
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</ul>
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||||
\begin{equation*}
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||||
({\boldsymbol \nabla} \times {\bf H}) \cdot \Delta {\bf A} = {\bf H} \cdot ({\boldsymbol \nabla} \times \Delta {\bf A})
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- {\boldsymbol \nabla} \cdot (\Delta {\bf A} \times {\bf H}) = {\bf H} \cdot \Delta {\bf B} - {\boldsymbol \nabla} \cdot (\Delta {\bf A} \times {\bf H})
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\end{equation*}
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<p>
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\]
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Integrating, we get
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\[
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||||
\Delta W_{mag} = \int_{all~space} d\tau {\bf H} \cdot \Delta {\bf B}
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@@ -1746,22 +1868,59 @@ Case of linear isotopic homogeneous medium:
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||||
W_{mag} = \int_{all~space} d\tau \frac{1}{2} {\bf H} \cdot {\bf B}
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||||
\]
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||||
</p>
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||||
</div>
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||||
</div>
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||||
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||||
<div id="outline-container-emsm_msm_H_A_bc" class="outline-6">
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||||
<h6 id="emsm_msm_H_A_bc"><a href="#emsm_msm_H_A_bc">Boundary conditions</a></h6>
|
||||
<div class="outline-text-6" id="text-emsm_msm_H_A_bc">
|
||||
<p>
|
||||
Can rewrite BCs in terms of \({\bf H}\): from <a href="./emsm_msm_H_A.html#divH">divH</a>,
|
||||
</p>
|
||||
<div class="eqlabel" id="org9aec61c">
|
||||
<p>
|
||||
<a id="HdiscM"></a><a href="./emsm_msm_H_A.html#HdiscM"><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="orgf88a9a3">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (6.24)</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
\begin{equation}
|
||||
H^{\perp}_{above} - H^{\perp}_{below} = -(M^{\perp}_{above} - M^{\perp}_{below})
|
||||
\tag{HdiscM}\label{HdiscM}
|
||||
\end{equation}
|
||||
<p>
|
||||
while <a href="./emsm_msm_H_A.html#curlHJf">curlHJf</a> gives
|
||||
</p>
|
||||
<div class="eqlabel" id="org92fe64e">
|
||||
<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>
|
||||
</p>
|
||||
<div class="alteqlabels" id="org08361d0">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (6.25)</li>
|
||||
</ul>
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
\begin{equation}
|
||||
{\bf H}^{\parallel}_{above} - {\bf H}^{\parallel}_{below} = {\bf K}_f \times \hat{\bf n}
|
||||
\tag{Hdisc}\label{Hdisc}
|
||||
\end{equation}
|
||||
|
||||
<p>
|
||||
\subsubsection*{Boundary conditions}
|
||||
Can rewrite BCs in terms of \({\bf H}\): from \ref{Gr(6.23)},
|
||||
\[
|
||||
H^{\perp}_{above} - H^{\perp}_{below} = -(M^{\perp}_{above} - M^{\perp}_{below})
|
||||
\label{Gr(6.24)}
|
||||
\]
|
||||
while \ref{Gr(6.19)} gives
|
||||
\[
|
||||
{\bf H}^{\parallel}_{above} - {\bf H}^{\parallel}_{below} = {\bf K}_f \times \hat{\bf n}
|
||||
\label{Gr(6.25)}
|
||||
\]
|
||||
These are more useful than BCs on \({\bf B}\), \ref{Gr(5.72)} and \ref{Gr(5.73)}:
|
||||
These are more useful than BCs on \({\bf B}\), <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>:
|
||||
\[
|
||||
B^{\perp}_{above} = B^{\perp}_{below}.
|
||||
\label{Gr(6.26)}
|
||||
@@ -1792,7 +1951,7 @@ 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: 2022-02-21 Mon 20:41</p>
|
||||
<p class="date">Created: 2022-03-01 Tue 08:14</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
||||
Reference in New Issue
Block a user