Update 2022-02-14 20:42
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Jean-Sébastien
vecāks
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revīzija
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<!DOCTYPE html>
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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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@ -1621,7 +1621,7 @@ 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: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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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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@ -1704,7 +1704,7 @@ 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: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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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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@ -1621,7 +1621,7 @@ 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: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-13 Sun 21:20 -->
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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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@ -1627,7 +1627,7 @@ 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: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-13 Sun 21:20 -->
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<!-- 2022-02-14 Mon 20:35 -->
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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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@ -1623,7 +1623,7 @@ 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: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-13 Sun 21:20 -->
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<!-- 2022-02-14 Mon 20:35 -->
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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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@ -1598,11 +1598,11 @@ Table of contents
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</svg></a><span class="headline-id">c.m.cs.cyl</span></h5>
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<div class="outline-text-5" id="text-c_m_cs_cyl">
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<p>
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\((r, \phi, z)\). Relation to Cartesian coordinates:
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\((r, \varphi, z)\). Relation to Cartesian coordinates:
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</p>
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\begin{equation}
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x = r \cos \phi, y = r \sin \phi, z = z
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x = r \cos \varphi, y = r \sin \varphi, z = z
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\label{Gr(1.74)}
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\end{equation}
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@ -1611,8 +1611,8 @@ The unit vectors are
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</p>
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\begin{equation}
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\hat{\bf r} = \cos \phi ~\hat{\bf x} + \sin \phi~\hat{\bf y},
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\hat{\boldsymbol \phi} = -\sin \theta ~\hat{\bf x} + \cos \phi~\hat{\bf y},
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\hat{\bf r} = \cos \varphi ~\hat{\bf x} + \sin \varphi~\hat{\bf y},
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\hat{\boldsymbol \varphi} = -\sin \theta ~\hat{\bf x} + \cos \varphi~\hat{\bf y},
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\hat{\bf z} = \hat{\bf z}.
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\label{Gr(1.75)}
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\end{equation}
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@ -1622,7 +1622,7 @@ Infinitesimal displacement:
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</p>
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\begin{equation}
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d{\bf l} = dr ~\hat{\bf r} + r d\phi~\hat{\boldsymbol \phi} + dz ~\hat{\bf z}.
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d{\bf l} = dr ~\hat{\bf r} + r d\varphi~\hat{\boldsymbol \varphi} + dz ~\hat{\bf z}.
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\label{Gr(1.77)}
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\end{equation}
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@ -1631,26 +1631,26 @@ Volume element:
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</p>
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\begin{equation}
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d\tau = r dr d\phi dz
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d\tau = r dr d\varphi dz
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\label{Gr(1.78)}
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\end{equation}
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<p>
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Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
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Range of parameters: \(r \in [0, \infty[\), \(\varphi \in [0, 2\pi[\) and \(z \in ]-\infty, \infty[\).
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</p>
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</div>
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<div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
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<h6 id="c_m_cs_cyl_grad"><a href="#c_m_cs_cyl_grad">Gradient</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_grad">
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<div class="eqlabel" id="org03db374">
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<div class="eqlabel" id="org7329bb2">
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<p>
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<a id="cylgrad"></a><a href="./c_m_cs_cyl.html#cylgrad"><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="orgb61510b">
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<div class="alteqlabels" id="org7d88a4a">
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<ul class="org-ul">
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<li>Gr4(1.79)</li>
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</ul>
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@ -1661,7 +1661,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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\begin{equation}
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{\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r}
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+ \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi}
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+ \frac{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi}
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+ \frac{\partial T}{\partial z} ~\hat{\bf z}
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\tag{cylgrad}\label{cylgrad}
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\end{equation}
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@ -1671,14 +1671,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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<div id="outline-container-c_m_cs_cyl_div" class="outline-6">
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<h6 id="c_m_cs_cyl_div"><a href="#c_m_cs_cyl_div">Divergence</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_div">
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<div class="eqlabel" id="orgd9435d5">
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<div class="eqlabel" id="orgef5777e">
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<p>
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<a id="cyl_div"></a><a href="./c_m_cs_cyl.html#cyl_div"><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="org5c1c30c">
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<div class="alteqlabels" id="org5bb7a68">
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<ul class="org-ul">
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<li>Gr4(2.21)</li>
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</ul>
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@ -1689,7 +1689,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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\begin{equation}
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{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
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+ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}.
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+ \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}.
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\tag{cyl_div}
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\label{cyl_div}
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\end{equation}
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@ -1699,14 +1699,14 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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<div id="outline-container-c_m_cs_cyl_curl" class="outline-6">
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<h6 id="c_m_cs_cyl_curl"><a href="#c_m_cs_cyl_curl">Curl</a></h6>
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<div class="outline-text-6" id="text-c_m_cs_cyl_curl">
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<div class="eqlabel" id="org695c029">
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<div class="eqlabel" id="orgebbdea6">
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<p>
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<a id="cyl_curl"></a><a href="./c_m_cs_cyl.html#cyl_curl"><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="orge332907">
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<div class="alteqlabels" id="org47230f3">
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<ul class="org-ul">
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<li>Gr4(2.21)</li>
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</ul>
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@ -1716,9 +1716,9 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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</div>
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\begin{align}
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{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r}
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+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi} \nonumber \\
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+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}
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{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r}
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+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi} \nonumber \\
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+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z}
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\tag{cyl_curl}
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\label{cyl_curl}
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\end{align}
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@ -1730,7 +1730,7 @@ Range of parameters: \(r \in [0, \infty[\), \(\phi \in [0, 2\pi[\) and \(z \in
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<div class="outline-text-6" id="text-c_m_cs_cyl_lap">
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\begin{equation}
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{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
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+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}
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+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2}
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\label{Gr(1.82)}
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\end{equation}
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</div>
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@ -1755,7 +1755,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
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</div>
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@ -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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||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1616,7 +1616,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
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</div>
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@ -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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||||
<!-- 2022-02-13 Sun 21:20 -->
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<!-- 2022-02-14 Mon 20:35 -->
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<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
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||||
<title>Pre-Quantum Electrodynamics</title>
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@ -1598,64 +1598,117 @@ Table of contents
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</svg></a><span class="headline-id">c.m.cs.sph</span></h5>
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<div class="outline-text-5" id="text-c_m_cs_sph">
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<p>
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\((r, \theta, \phi)\). \(\theta\) is the <b>polar angle</b>, \(\phi\) the <b>azimuthal angle</b>.
|
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In this system, we use coordinates \((r, \theta, \varphi)\) in
|
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which \(r\) is the distance from the chosen origin,
|
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\(\theta\) is the <b>polar angle</b> and \(\varphi\) is the <b>azimuthal angle</b>.
|
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</p>
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<p>
|
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The usual Cartesian coordinates relate to spherical coordinates
|
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according to
|
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</p>
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<div class="eqlabel" id="org9fd0b35">
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<p>
|
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<a id="sph_xyz"></a><a href="./c_m_cs_sph.html#sph_xyz"><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="org9bb0d80">
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</div>
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</div>
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<p>
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\[
|
||||
x = r \sin \theta \cos \phi,
|
||||
y = r \sin \theta \sin \phi,
|
||||
x = r \sin \theta \cos \varphi, \hspace{5mm}
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y = r \sin \theta \sin \varphi, \hspace{5mm}
|
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z = r \cos \theta.
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\label{Gr(1.62)}
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\tag{sph_xyz}\label{sph_xyz}
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\]
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</p>
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<p>
|
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Unit vectors: \(\hat{\boldsymbol r}, \hat{\boldsymbol \theta}, \hat{\boldsymbol \phi}\).
|
||||
</p>
|
||||
|
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<p>
|
||||
The unit vectors are written \(\hat{\boldsymbol r}\),
|
||||
\(\hat{\boldsymbol \theta}\) and \(\hat{\boldsymbol \varphi}\).
|
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A generic vector can be expressed as
|
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\[
|
||||
{\bf A} = A_r \hat{\bf r} + A_{\theta} \hat{\bf \theta} + A_{\phi} \hat{\boldsymbol \phi}
|
||||
\label{Gr(1.63)}
|
||||
{\bf v} = v_r \hat{\bf r} + v_{\theta} \hat{\bf \theta} + v_{\varphi} \hat{\boldsymbol \varphi}
|
||||
\]
|
||||
where the explicit relation between spherical and
|
||||
Cartesian unit vectors is
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="orgb78b624">
|
||||
<p>
|
||||
In terms of Cartesian unit vectors:
|
||||
<a id="sph_uv"></a><a href="./c_m_cs_sph.html#sph_uv"><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="org44deec6">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
\begin{align}
|
||||
\hat{\boldsymbol r} &= \sin \theta \cos \phi \hat{\bf x} + \sin \theta \sin \phi \hat{\bf y} + \cos \theta \hat{\bf z}, \nonumber \\
|
||||
\hat{\boldsymbol \theta} &= \cos \theta \cos \phi \hat{\bf x} + \cos \theta \sin \phi \hat{\bf y} - \sin \theta \hat{\bf z}, \nonumber \\
|
||||
\hat{\boldsymbol \phi} &= -\sin \phi \hat{\bf x} + \cos \phi \hat{\bf y}.
|
||||
\label{Gr(1.64)}
|
||||
\hat{\boldsymbol r} &= \sin \theta \cos \varphi ~\hat{\bf x} + \sin \theta \sin \varphi ~\hat{\bf y} + \cos \theta ~\hat{\bf z}, \nonumber \\
|
||||
\hat{\boldsymbol \theta} &= \cos \theta \cos \varphi ~\hat{\bf x} + \cos \theta \sin \varphi ~\hat{\bf y} - \sin \theta ~\hat{\bf z}, \nonumber \\
|
||||
\hat{\boldsymbol \varphi} &= -\sin \varphi ~\hat{\bf x} + \cos \varphi ~\hat{\bf y}.
|
||||
\tag{sph_uv}\label{sph_uv}
|
||||
\end{align}
|
||||
|
||||
<p>
|
||||
Careful: these unit vectors are direction dependent, <i>i.e.</i> we should really
|
||||
write \(\hat{\boldsymbol r} (\theta, \phi), \hat{\boldsymbol \theta} (\theta, \phi), \hat{\boldsymbol \phi} (\theta, \phi)\).
|
||||
Do be careful: these unit vectors are direction dependent, <i>i.e.</i> we should really
|
||||
write \(\hat{\boldsymbol r} (\theta, \varphi)\),
|
||||
\(\hat{\boldsymbol \theta} (\theta, \varphi)\)
|
||||
and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Infinitesimal displacement \(d{\bf l}\):
|
||||
An infinitesimal displacement \(d{\bf l}\) can be written as
|
||||
</p>
|
||||
<div class="eqlabel" id="org11b03bf">
|
||||
<p>
|
||||
<a id="sph_dl"></a><a href="./c_m_cs_sph.html#sph_dl"><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="org814a5c0">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
\[
|
||||
d{\bf l} = dr \hat{\boldsymbol r} + r d\theta \hat{\boldsymbol \theta} + r\sin \theta d\phi \hat{\boldsymbol \phi}.
|
||||
\label{Gr(1.68)}
|
||||
d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin \theta d\varphi ~\hat{\boldsymbol \varphi}.
|
||||
\tag{sph_dl}\label{sph_dl}
|
||||
\]
|
||||
</p>
|
||||
|
||||
<p>
|
||||
Infinitesimal volume element:
|
||||
</p>
|
||||
<div class="eqlabel" id="orgf5fa0db">
|
||||
<p>
|
||||
<a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><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="orgf5f8360">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
<p>
|
||||
\[
|
||||
d\tau = dl_r dl_{\theta} dl_{\phi} = r^2 \sin \theta dr d\theta d\phi
|
||||
\label{Gr(1.69)}
|
||||
d\tau = dl_r dl_{\theta} dl_{\varphi} = r^2 \sin \theta dr d\theta d\varphi
|
||||
\tag{sph_dtau}\label{sph_dtau}
|
||||
\]
|
||||
</p>
|
||||
|
||||
|
|
@ -1667,10 +1720,23 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_grad" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_grad">
|
||||
<div class="eqlabel" id="orgb804960">
|
||||
<p>
|
||||
<a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><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="org3f7449e">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
\begin{equation}
|
||||
{\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.
|
||||
\label{Gr(1.70)}
|
||||
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}.
|
||||
\tag{sph_grad}\label{sph_grad}
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1678,10 +1744,23 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_div" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_div">
|
||||
<div class="eqlabel" id="org13ae106">
|
||||
<p>
|
||||
<a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_div"><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="org8664749">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
\begin{equation}
|
||||
{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta})
|
||||
+ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi}
|
||||
\label{Gr(1.71)}
|
||||
+ \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi}
|
||||
\tag{sph_div}\label{sph_div}
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1689,23 +1768,50 @@ Infinitesimal surface element: depends on situation.
|
|||
<div id="outline-container-c_m_cs_sph_curl" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_curl">
|
||||
\begin{equation}
|
||||
{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r}
|
||||
+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}
|
||||
\label{Gr(1.72)}
|
||||
\end{equation}
|
||||
<div class="eqlabel" id="orge67612f">
|
||||
<p>
|
||||
<a id="sph_curl"></a><a href="./c_m_cs_sph.html#sph_curl"><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="org1716ad0">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
\begin{align}
|
||||
{\boldsymbol \nabla} \times {\bf v} = &\frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r}
|
||||
\nonumber \\
|
||||
&+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi}
|
||||
\tag{sph_curl}\label{sph_curl}
|
||||
\end{align}
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-c_m_cs_sph_lap" class="outline-6">
|
||||
<h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
|
||||
<div class="outline-text-6" id="text-c_m_cs_sph_lap">
|
||||
<div class="eqlabel" id="org99dc986">
|
||||
<p>
|
||||
<a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><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="org75cb0ac">
|
||||
|
||||
</div>
|
||||
|
||||
</div>
|
||||
|
||||
\begin{equation}
|
||||
{\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
|
||||
+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right)
|
||||
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}
|
||||
\label{Gr(1.73)}
|
||||
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2}
|
||||
\tag{sph_Lap}\label{sph_Lap}
|
||||
\end{equation}
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1729,7 +1835,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1626,7 +1626,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1628,7 +1628,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1599,9 +1599,9 @@ Table of contents
|
|||
<div class="outline-text-5" id="text-c_m_dc_d2">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgbe5f73d" class="outline-6">
|
||||
<h6 id="orgbe5f73d"><a href="#orgbe5f73d">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgbe5f73d">
|
||||
<div id="outline-container-org909c107" class="outline-6">
|
||||
<h6 id="org909c107"><a href="#org909c107">Divergence of gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org909c107">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
|
||||
The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
|
||||
|
|
@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgee1ca6a" class="outline-6">
|
||||
<h6 id="orgee1ca6a"><a href="#orgee1ca6a">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-orgee1ca6a">
|
||||
<div id="outline-container-org9576a42" class="outline-6">
|
||||
<h6 id="org9576a42"><a href="#org9576a42">Curl of a gradient</a></h6>
|
||||
<div class="outline-text-6" id="text-org9576a42">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org2789da2" class="outline-6">
|
||||
<h6 id="org2789da2"><a href="#org2789da2">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-org2789da2">
|
||||
<div id="outline-container-org9eefe8b" class="outline-6">
|
||||
<h6 id="org9eefe8b"><a href="#org9eefe8b">Gradient of the divergence</a></h6>
|
||||
<div class="outline-text-6" id="text-org9eefe8b">
|
||||
<p>
|
||||
\({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics. No special name.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org62ec566" class="outline-6">
|
||||
<h6 id="org62ec566"><a href="#org62ec566">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org62ec566">
|
||||
<div id="outline-container-org0de49a1" class="outline-6">
|
||||
<h6 id="org0de49a1"><a href="#org0de49a1">Divergence of a curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org0de49a1">
|
||||
<p>
|
||||
This always vanishes.
|
||||
</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org5cd2844" class="outline-6">
|
||||
<h6 id="org5cd2844"><a href="#org5cd2844">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org5cd2844">
|
||||
<div id="outline-container-org937d8f6" class="outline-6">
|
||||
<h6 id="org937d8f6"><a href="#org937d8f6">Curl of curl</a></h6>
|
||||
<div class="outline-text-6" id="text-org937d8f6">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
|
||||
|
|
@ -1666,7 +1666,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1627,7 +1627,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1624,7 +1624,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1648,7 +1648,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1621,7 +1621,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1623,7 +1623,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1635,7 +1635,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
|
|||
More generally,
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org28e4fae">
|
||||
<div class="eqlabel" id="org30da65a">
|
||||
<p>
|
||||
<a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><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="org3ed5b35">
|
||||
<div class="alteqlabels" id="org4d37466">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.100)</li>
|
||||
</ul>
|
||||
|
|
@ -1646,14 +1646,14 @@ More generally,
|
|||
Since
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org51c5e82">
|
||||
<div class="eqlabel" id="org209f4b3">
|
||||
<p>
|
||||
<a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><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="org725ab8d">
|
||||
<div class="alteqlabels" id="org48450a9">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.101)</li>
|
||||
</ul>
|
||||
|
|
@ -1669,14 +1669,14 @@ Since
|
|||
<p>
|
||||
we have that
|
||||
</p>
|
||||
<div class="eqlabel" id="orgbcd1aff">
|
||||
<div class="eqlabel" id="org814ca01">
|
||||
<p>
|
||||
<a id="Lap1or"></a><a href="./c_m_dd_3d.html#Lap1or"><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="orgc217127">
|
||||
<div class="alteqlabels" id="orgedb35a4">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.102)</li>
|
||||
</ul>
|
||||
|
|
@ -1708,7 +1708,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1613,8 +1613,8 @@ Integrate over a sphere of radius \(R\) centered at the origin (Prob. 1.38b):
|
|||
|
||||
|
||||
\begin{equation}
|
||||
\oint \frac{\hat{\bf r}}{r^2} \cdot d{\bf a} = \int \left(\frac{1}{R^2} \hat{\bf r} \right) \cdot \left( R^2 \sin \theta d\theta d\phi ~\hat{\bf r} \right)
|
||||
= \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\phi = 4\pi
|
||||
\oint \frac{\hat{\bf r}}{r^2} \cdot d{\bf a} = \int \left(\frac{1}{R^2} \hat{\bf r} \right) \cdot \left( R^2 \sin \theta d\theta d\varphi ~\hat{\bf r} \right)
|
||||
= \int_0^{\pi} d\theta \sin \theta \int_0^{2\pi} d\varphi = 4\pi
|
||||
\label{Gr(1.85)}
|
||||
\end{equation}
|
||||
|
||||
|
|
@ -1641,7 +1641,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1626,7 +1626,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1624,7 +1624,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1633,7 +1633,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1627,7 +1627,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1634,7 +1634,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1599,9 +1599,9 @@ Table of contents
|
|||
<div class="outline-text-5" id="text-c_m_ic_lsv">
|
||||
</div>
|
||||
|
||||
<div id="outline-container-orgdc409a4" class="outline-6">
|
||||
<h6 id="orgdc409a4"><a href="#orgdc409a4">Line Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-orgdc409a4">
|
||||
<div id="outline-container-org1c47caa" class="outline-6">
|
||||
<h6 id="org1c47caa"><a href="#org1c47caa">Line Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org1c47caa">
|
||||
<p>
|
||||
\[
|
||||
{\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
|
||||
|
|
@ -1630,9 +1630,9 @@ Integral over a closed loop:
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org465c39d" class="outline-6">
|
||||
<h6 id="org465c39d"><a href="#org465c39d">Surface Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org465c39d">
|
||||
<div id="outline-container-orgd540d59" class="outline-6">
|
||||
<h6 id="orgd540d59"><a href="#orgd540d59">Surface Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-orgd540d59">
|
||||
<p>
|
||||
\[
|
||||
\int_{\cal S} {\bf v} \cdot d{\bf a}
|
||||
|
|
@ -1652,9 +1652,9 @@ Over a closed surface:
|
|||
</div>
|
||||
</div>
|
||||
|
||||
<div id="outline-container-org794c5b9" class="outline-6">
|
||||
<h6 id="org794c5b9"><a href="#org794c5b9">Volume Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org794c5b9">
|
||||
<div id="outline-container-org5afd08a" class="outline-6">
|
||||
<h6 id="org5afd08a"><a href="#org5afd08a">Volume Integrals</a></h6>
|
||||
<div class="outline-text-6" id="text-org5afd08a">
|
||||
<p>
|
||||
\[
|
||||
\int_{\cal V} T d\tau
|
||||
|
|
@ -1695,7 +1695,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1597,14 +1597,14 @@ Table of contents
|
|||
<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">c.m.ic.stokes</span></h5>
|
||||
<div class="outline-text-5" id="text-c_m_ic_stokes">
|
||||
<div class="eqlabel" id="org54009e0">
|
||||
<div class="eqlabel" id="orga5b1968">
|
||||
<p>
|
||||
<a id="Stokes"></a><a href="./c_m_ic_stokes.html#Stokes"><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="org8b7df95">
|
||||
<div class="alteqlabels" id="org6ef826a">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (1.57)</li>
|
||||
</ul>
|
||||
|
|
@ -1649,7 +1649,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1623,7 +1623,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1600,21 +1600,21 @@ Table of contents
|
|||
\begin{align*}
|
||||
\mbox{Gradient:}
|
||||
&{\boldsymbol \nabla} T = \frac{\partial T}{\partial r}~\hat{\bf r}
|
||||
+ \frac{1}{r} \frac{\partial T}{\partial \phi}~\hat{\boldsymbol \phi}
|
||||
+ \frac{1}{r} \frac{\partial T}{\partial \varphi}~\hat{\boldsymbol \varphi}
|
||||
+ \frac{\partial T}{\partial z} ~\hat{\bf z}
|
||||
\nonumber\\
|
||||
\mbox{Divergence:}
|
||||
&{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r} \frac{\partial}{\partial r} (r v_r)
|
||||
+ \frac{1}{r} \frac{\partial v_{\phi}}{\partial \phi} + \frac{\partial v_z}{\partial z}.
|
||||
+ \frac{1}{r} \frac{\partial v_{\varphi}}{\partial \varphi} + \frac{\partial v_z}{\partial z}.
|
||||
\nonumber\\
|
||||
\mbox{Curl:}
|
||||
&{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \phi} - \frac{\partial v_{\phi}}{\partial z}\right) ~\hat{\bf r}
|
||||
+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \phi}
|
||||
+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\phi}) - \frac{\partial v_r}{\partial \phi} \right) ~\hat{\bf z}
|
||||
&{\boldsymbol \nabla} \times {\bf v} = \left( \frac{1}{r} \frac{\partial v_z}{\partial \varphi} - \frac{\partial v_{\varphi}}{\partial z}\right) ~\hat{\bf r}
|
||||
+ \left( \frac{\partial v_r}{\partial z} - \frac{\partial v_z}{\partial r} \right) ~\hat{\boldsymbol \varphi}
|
||||
+ \frac{1}{r} \left( \frac{\partial}{\partial r} (r v_{\varphi}) - \frac{\partial v_r}{\partial \varphi} \right) ~\hat{\bf z}
|
||||
\nonumber\\
|
||||
\mbox{Laplacian:}
|
||||
&{\boldsymbol \nabla}^2 T = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial T}{\partial r} \right)
|
||||
+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \phi^2} + \frac{\partial^2 T}{\partial z^2}
|
||||
+ \frac{1}{r^2} \frac{\partial^2 T}{\partial \varphi^2} + \frac{\partial^2 T}{\partial z^2}
|
||||
\end{align*}
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1636,7 +1636,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1600,21 +1600,21 @@ Table of contents
|
|||
\begin{align*}
|
||||
\mbox{Gradient:}
|
||||
&{\boldsymbol \nabla} T = \frac{\partial T}{\partial r} \hat{\boldsymbol r} + \frac{1}{r} \frac{\partial T}{\partial \theta} \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \phi} \hat{\boldsymbol \phi}.
|
||||
+ \frac{1}{r\sin \theta} \frac{\partial T}{\partial \varphi} \hat{\boldsymbol \varphi}.
|
||||
\nonumber \\
|
||||
\mbox{Divergence:}
|
||||
&{\boldsymbol \nabla} \cdot {\bf v} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 v_r) + \frac{1}{r\sin \theta} \frac{\partial}{\partial \theta} (\sin\theta v_{\theta})
|
||||
+ \frac{1}{r \sin \theta} \frac{\partial v_{\phi}}{\partial \phi}
|
||||
+ \frac{1}{r \sin \theta} \frac{\partial v_{\varphi}}{\partial \varphi}
|
||||
\nonumber\\
|
||||
\mbox{Curl:}
|
||||
&{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\phi}) - \frac{\partial v_{\theta}}{\partial \phi} \right] \hat{\bf r}
|
||||
+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \phi} - \frac{\partial}{\partial r} (r v_{\phi}) \right] \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \phi}
|
||||
&{\boldsymbol \nabla} \times {\bf v} = \frac{1}{r\sin \theta} \left[ \frac{\partial}{\partial \theta} (\sin \theta v_{\varphi}) - \frac{\partial v_{\theta}}{\partial \varphi} \right] \hat{\bf r}
|
||||
+ \frac{1}{r} \left[ \frac{1}{\sin \theta} \frac{\partial v_r}{\partial \varphi} - \frac{\partial}{\partial r} (r v_{\varphi}) \right] \hat{\boldsymbol \theta}
|
||||
+ \frac{1}{r} \left[ \frac{\partial}{\partial r} (r v_{\theta}) - \frac{\partial v_r}{\partial \theta} \right] \hat{\boldsymbol \varphi}
|
||||
\nonumber\\
|
||||
\mbox{Laplacian:}
|
||||
&{\boldsymbol \nabla}^2 T = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial T}{\partial r}\right)
|
||||
+ \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial T}{\partial \theta}\right)
|
||||
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \phi^2}
|
||||
+ \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 T}{\partial \varphi^2}
|
||||
\end{align*}
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -1636,7 +1636,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1639,7 +1639,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1625,7 +1625,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1654,7 +1654,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1649,7 +1649,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1683,7 +1683,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1649,7 +1649,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1696,7 +1696,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1622,7 +1622,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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|
||||
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|
||||
<meta charset="utf-8">
|
||||
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|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1641,7 +1641,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
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|
||||
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||||
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|||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1668,7 +1668,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
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|
||||
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||||
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|
|||
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||||
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|
||||
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|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">d</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-d">
|
||||
<details class="objectives" id="orgece4d0a">
|
||||
<summary id="org51b47eb">
|
||||
<details class="objectives" id="orgb3091af">
|
||||
<summary id="org1ce2f18">
|
||||
Objectives
|
||||
</summary>
|
||||
|
||||
|
|
@ -1663,7 +1663,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1643,7 +1643,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
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|
|||
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@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1638,7 +1638,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1641,7 +1641,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
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||||
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||||
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|
||||
<head>
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1638,7 +1638,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
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||||
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@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
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||||
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|
||||
<head>
|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1660,7 +1660,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
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@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1643,7 +1643,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
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|
||||
|
||||
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|
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@ -1,7 +1,7 @@
|
|||
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|
||||
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|
||||
<head>
|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
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|
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|
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@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1643,7 +1643,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1641,7 +1641,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1651,7 +1651,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1645,7 +1645,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emd">
|
||||
<details class="prereq" id="org8c17216">
|
||||
<summary id="org07b9266">
|
||||
<details class="prereq" id="orgc0e882c">
|
||||
<summary id="orgb840112">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1608,8 +1608,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org1cf61f0">
|
||||
<summary id="org2c97d99">
|
||||
<details class="objectives" id="org22649b4">
|
||||
<summary id="org9f52a90">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1650,7 +1650,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1624,7 +1624,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1638,7 +1638,7 @@ Empirically: the changing magnetic field induces an electric current around
|
|||
the circuit. This current is really driven by an electric field having a component
|
||||
along the wire. The line integral of this field is called the
|
||||
</p>
|
||||
<div class="core div" id="org60c9879">
|
||||
<div class="core div" id="org1df022a">
|
||||
<p>
|
||||
<b>Electromotive force (or electromotance)</b>,
|
||||
\[
|
||||
|
|
@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
|
|||
\]
|
||||
so we obtain
|
||||
</p>
|
||||
<div class="core div" id="org02e64db">
|
||||
<div class="core div" id="orgb24c88e">
|
||||
<p>
|
||||
<b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
|
||||
\[
|
||||
|
|
@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
|
|||
\]
|
||||
we obtain
|
||||
</p>
|
||||
<div class="core div" id="org3c09004">
|
||||
<div class="core div" id="org3cc6f30">
|
||||
<p>
|
||||
<b>Faraday's law</b> (differential form)
|
||||
\[
|
||||
|
|
@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
|
|||
\]
|
||||
We can integrate over all space: after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
|
||||
</p>
|
||||
<div class="core div" id="org0ecd8ee">
|
||||
<div class="core div" id="org2618711">
|
||||
<p>
|
||||
\[
|
||||
W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
|
||||
|
|
@ -1678,18 +1678,18 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
|
|||
\hspace{2cm} \mbox{(7.31 and 7.34)}
|
||||
\end{align}
|
||||
|
||||
<div class="example div" id="org46b2091">
|
||||
<div class="example div" id="org9178294">
|
||||
<p>
|
||||
\paragraph{Example 7.13:} coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
|
||||
Find energy stored in section of length \(l\).
|
||||
\paragraph{Solution:} from Ampère,
|
||||
\[
|
||||
{\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \phi}, \hspace{1cm} a < s < b, \hspace{1cm}
|
||||
{\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a < s < b, \hspace{1cm}
|
||||
{\bf B} = 0, \hspace{1cm} s < a ~\mbox{or}~ s > b.
|
||||
\]
|
||||
Energy is thus
|
||||
\[
|
||||
W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\phi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2
|
||||
W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2
|
||||
= \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}.
|
||||
\]
|
||||
</p>
|
||||
|
|
@ -1717,7 +1717,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1637,7 +1637,7 @@ M_{12} = M_{21}
|
|||
\]
|
||||
</p>
|
||||
|
||||
<div class="example div" id="orgc3b71d6">
|
||||
<div class="example div" id="org77a8906">
|
||||
<p>
|
||||
\paragraph{Example 7.10:}
|
||||
short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
|
||||
|
|
@ -1687,7 +1687,7 @@ Inductance: measured in {\bf henries} (\(H\)). \(H = V s/A\).
|
|||
</p>
|
||||
|
||||
|
||||
<div class="example div" id="orgd00f4d4">
|
||||
<div class="example div" id="orgfdc5058">
|
||||
<p>
|
||||
\paragraph{Example 7.11:} find self-inductance of toroidal coil with
|
||||
rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
|
||||
|
|
@ -1714,7 +1714,7 @@ Total flux: \(N\) times this, so self-inductance is
|
|||
Inductance (like capacitance) is intrinsically positive. Use Lenz law. Think of {\bf back EMF}.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org7ae308f">
|
||||
<div class="example div" id="org6ead6cc">
|
||||
<p>
|
||||
\paragraph{Example 7.12:} circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
|
||||
What is the current ?
|
||||
|
|
@ -1753,7 +1753,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1621,7 +1621,7 @@ law in integral form:
|
|||
|
||||
|
||||
|
||||
<div class="example div" id="orgaba3d08">
|
||||
<div class="example div" id="org50f02d1">
|
||||
<p>
|
||||
{\bf Example 7.7:}
|
||||
\({\bf B}(t)\) points up in circular region of radius \(R\). What is the induced \({\bf E}(t)\) ?
|
||||
|
|
@ -1629,7 +1629,7 @@ law in integral form:
|
|||
amperian loop of radius \(s\), apply Faraday:
|
||||
\[
|
||||
\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
|
||||
\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \phi}.
|
||||
\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
|
||||
\]
|
||||
Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
|
||||
</p>
|
||||
|
|
@ -1637,7 +1637,7 @@ Increasing \({\bf B}\): clockwise (viewed from above) \({\bf E}\) from Lenz.
|
|||
</div>
|
||||
|
||||
|
||||
<div class="example div" id="org655e44f">
|
||||
<div class="example div" id="orga3fbc55">
|
||||
<p>
|
||||
{\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
|
||||
Uniform magnetic field \({\bf B}_0\) in central region up to \(a < b\),
|
||||
|
|
@ -1646,7 +1646,7 @@ pointing up. Field turned off. What happens ?
|
|||
Faraday:
|
||||
\[
|
||||
\oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt}
|
||||
\Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \phi}.
|
||||
\Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
|
||||
\]
|
||||
Torque on segment \(d{\bf l}\): \(|{\bf r} \times {\bf F}| = b \lambda E dl\).
|
||||
Total torque:
|
||||
|
|
@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
|
|||
'slow enough' phenomena.
|
||||
</p>
|
||||
|
||||
<div class="example div" id="org0cc8dcd">
|
||||
<div class="example div" id="org949be5c">
|
||||
<p>
|
||||
{\bf Example 7.9:} infinitely long straight wire carries \(I(t)\). Find
|
||||
induced \({\bf E}\) field as a function of distance \(s\) from wire.
|
||||
|
|
@ -1719,7 +1719,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1624,7 +1624,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1600,7 +1600,7 @@ Table of contents
|
|||
<p>
|
||||
Full set of equations for the electromagnetic field:
|
||||
</p>
|
||||
<div class="core div" id="org02575df">
|
||||
<div class="core div" id="org443ae4f">
|
||||
<p>
|
||||
{\bf Maxwell's equations} {\it (in vacuum)}
|
||||
</p>
|
||||
|
|
@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
|
|||
<p>
|
||||
Complement:
|
||||
</p>
|
||||
<div class="core div" id="org67bee1c">
|
||||
<div class="core div" id="orga5a9517">
|
||||
<p>
|
||||
{\bf Force law}
|
||||
\[
|
||||
|
|
@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
|
|||
<p>
|
||||
Better way of writing: all fields on left, all sources on right,
|
||||
</p>
|
||||
<div class="core div" id="org95e5ef3">
|
||||
<div class="core div" id="org3e0ec76">
|
||||
\begin{align}
|
||||
(i) &{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
|
||||
&(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
|
||||
|
|
@ -1670,7 +1670,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1607,7 +1607,7 @@ the continuity equation as
|
|||
\]
|
||||
The extra term would thus be eliminated if we were to put
|
||||
</p>
|
||||
<div class="core div" id="org2b447e1">
|
||||
<div class="core div" id="org0252f37">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
|
||||
|
|
@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation
|
|||
<p>
|
||||
Maxwell baptized this term the
|
||||
</p>
|
||||
<div class="core div" id="org10e929a">
|
||||
<div class="core div" id="orgce64fc0">
|
||||
<p>
|
||||
{\bf Displacement current}
|
||||
\[
|
||||
|
|
@ -1679,7 +1679,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1649,7 +1649,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1647,7 +1647,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.ce</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_ce">
|
||||
<details class="prereq" id="orgeb45464">
|
||||
<summary id="org2f2d9c5">
|
||||
<details class="prereq" id="org8f62609">
|
||||
<summary id="org879d9ed">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1607,8 +1607,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org9e5041f">
|
||||
<summary id="orgbcd06e1">
|
||||
<details class="objectives" id="org6c18760">
|
||||
<summary id="org51c8198">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1646,7 +1646,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1600,7 +1600,7 @@ Table of contents
|
|||
<p>
|
||||
The angular momentum of EM fields is directly given by
|
||||
</p>
|
||||
<div class="main div" id="org76c33ae">
|
||||
<div class="main div" id="org56f7770">
|
||||
<p>
|
||||
{\bf Angular momentum of EM fields}
|
||||
\[
|
||||
|
|
@ -1630,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
|
|||
</div>
|
||||
<div id="postamble" class="status">
|
||||
<p class="author">Author: Jean-Sébastien Caux</p>
|
||||
<p class="date">Created: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1619,7 +1619,7 @@ This means that
|
|||
\]
|
||||
Since this is true for any volume, we have (re)derived the
|
||||
</p>
|
||||
<div class="core div" id="org9d47204">
|
||||
<div class="core div" id="org48d0d19">
|
||||
<p>
|
||||
{\bf Continuity equation}
|
||||
\[
|
||||
|
|
@ -1660,7 +1660,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
|
|||
<p>
|
||||
This is thus simply a conservation law for momentum, with
|
||||
</p>
|
||||
<div class="main div" id="org75dce9f">
|
||||
<div class="main div" id="orgc2ff953">
|
||||
<p>
|
||||
{\bf Momentum density in the EM fields}
|
||||
\[
|
||||
|
|
@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
|
|||
<p>
|
||||
In a region in which the mechanical momentum is not changing due to external influences, we then have the
|
||||
</p>
|
||||
<div class="main div" id="orgeb01a5d">
|
||||
<div class="main div" id="org7674e56">
|
||||
<p>
|
||||
{\bf Continuity equation for EM momentum}
|
||||
\[
|
||||
|
|
@ -1653,7 +1653,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
|
|||
<p>
|
||||
This expression can be greatly simplified by introducing the
|
||||
</p>
|
||||
<div class="main div" id="orgcd90456">
|
||||
<div class="main div" id="org69e4d14">
|
||||
<p>
|
||||
{\bf Maxwell stress tensor}
|
||||
\[
|
||||
|
|
@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
|
|||
<p>
|
||||
We then obtain
|
||||
</p>
|
||||
<div class="main div" id="orgbba46d7">
|
||||
<div class="main div" id="orga5d6aaa">
|
||||
<p>
|
||||
{\bf EM force per unit volume}
|
||||
\[
|
||||
|
|
@ -1689,7 +1689,7 @@ We then obtain
|
|||
<p>
|
||||
where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
|
||||
</p>
|
||||
<div class="main div" id="org9e182ba">
|
||||
<div class="main div" id="orgf401dc2">
|
||||
<p>
|
||||
{\bf Total force on charges in volume}
|
||||
\[
|
||||
|
|
@ -1718,7 +1718,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1666,7 +1666,7 @@ so we get
|
|||
Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
|
||||
we obtain
|
||||
</p>
|
||||
<div class="main div" id="org1ad9e6b">
|
||||
<div class="main div" id="org9d063a8">
|
||||
<p>
|
||||
{\bf Poynting's theorem}
|
||||
\[
|
||||
|
|
@ -1691,7 +1691,7 @@ 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:
|
||||
</p>
|
||||
<div class="core div" id="org97f8866">
|
||||
<div class="core div" id="org91d3725">
|
||||
<p>
|
||||
{\bf Poynting vector}
|
||||
\[
|
||||
|
|
@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
|
|||
<p>
|
||||
We can thus express Poynting's theorem more compactly:
|
||||
</p>
|
||||
<div class="core div" id="orgc096fa3">
|
||||
<div class="core div" id="org5a2fb1e">
|
||||
<p>
|
||||
{\bf Poynting's theorem}
|
||||
\[
|
||||
|
|
@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
|
|||
<p>
|
||||
where we have defined the total
|
||||
</p>
|
||||
<div class="core div" id="org56f314c">
|
||||
<div class="core div" id="orgf2b55fe">
|
||||
<p>
|
||||
{\bf Energy in electromagnetic fields}
|
||||
\[
|
||||
|
|
@ -1740,7 +1740,7 @@ Then,
|
|||
\]
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="orgd477ba8">
|
||||
<div class="core div" id="orgb8783a7">
|
||||
<p>
|
||||
{\bf Poynting theorem (differential form)}
|
||||
\[
|
||||
|
|
@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
|
|||
|
||||
|
||||
|
||||
<div class="example div" id="orgf36d646">
|
||||
<div class="example div" id="orgc6f817f">
|
||||
<p>
|
||||
\paragraph{Example 8.1} Current in a wire: Joule heating. Energy per unit time delivered to wire: from Poynting.
|
||||
Assuming that the field is uniform, the electric field parallel to the wire is
|
||||
|
|
@ -1767,11 +1767,11 @@ Assuming that the field is uniform, the electric field parallel to the wire is
|
|||
where \(V\) is the potential difference between the ends ald \(L\) is the length. Magnetic field is circumferential:
|
||||
wire of radius \(a\),
|
||||
\[
|
||||
{\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \phi}
|
||||
{\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi}
|
||||
\]
|
||||
Poynting:
|
||||
\[
|
||||
{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \phi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
|
||||
{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
|
||||
\]
|
||||
and points radially inwards. Energy per unit time passing surface of wire:
|
||||
\[
|
||||
|
|
@ -1802,7 +1802,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emd.emw</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-emd_emw">
|
||||
<details class="prereq" id="org976b39e">
|
||||
<summary id="org4571516">
|
||||
<details class="prereq" id="org8543c94">
|
||||
<summary id="orgfba24f9">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1608,8 +1608,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="orgc2083f7">
|
||||
<summary id="org74289a2">
|
||||
<details class="objectives" id="org77097a2">
|
||||
<summary id="org3c8f01e">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1650,7 +1650,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
|
|||
\]
|
||||
or more succinctly:
|
||||
</p>
|
||||
<div class="main div" id="org58ea3a9">
|
||||
<div class="main div" id="org4a022ab">
|
||||
<p>
|
||||
{\bf Poynting vector of a monochromatic EM wave}
|
||||
\[
|
||||
|
|
@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
|
|||
<p>
|
||||
Similary, we get the
|
||||
</p>
|
||||
<div class="main div" id="org9272ec4">
|
||||
<div class="main div" id="org70a3fce">
|
||||
<p>
|
||||
{\bf Momentum density of a monochromatic EM wave}
|
||||
\[
|
||||
|
|
@ -1695,7 +1695,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1629,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
|
|||
Generalizing to propagation in the direction of an arbitrary wavevector
|
||||
\({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
|
||||
</p>
|
||||
<div class="core div" id="org19a2b1c">
|
||||
<div class="core div" id="orge5c8116">
|
||||
<p>
|
||||
{\bf E and B fields for a monochromatic EM plane wave}
|
||||
\[
|
||||
|
|
@ -1673,7 +1673,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1626,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \(
|
|||
Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
|
||||
we get the
|
||||
</p>
|
||||
<div class="core div" id="org3cd266d">
|
||||
<div class="core div" id="org6398efe">
|
||||
<p>
|
||||
{\bf Wave equations for electric and magnetic fields in vacuum}
|
||||
\[
|
||||
|
|
@ -1682,7 +1682,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1622,7 +1622,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1622,7 +1622,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
|
|||
\]
|
||||
We therefore have the
|
||||
</p>
|
||||
<div class="core div" id="orgc16990f">
|
||||
<div class="core div" id="org6dfaaf3">
|
||||
<p>
|
||||
{\bf Polarization current density}
|
||||
\[
|
||||
|
|
@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
|
|||
polarization changes). We can check consistency with the continuity equation
|
||||
associated to the conservation of bound charges:
|
||||
</p>
|
||||
<aside id="orgbe14b68">
|
||||
<aside id="org3465a8e">
|
||||
<p>
|
||||
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 convention used here.
|
||||
|
|
@ -1674,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
|
|||
In view of this: total charge density can be separated into 2 parts,
|
||||
{\it free} and {\it bound}:
|
||||
</p>
|
||||
<div class="main div" id="org27840ee">
|
||||
<div class="main div" id="orgcdd32f0">
|
||||
<p>
|
||||
\[
|
||||
\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
|
||||
|
|
@ -1687,7 +1687,7 @@ In view of this: total charge density can be separated into 2 parts,
|
|||
and current can be separated into three parts, {\it free}, {\it bound} and
|
||||
{\it polarization}:
|
||||
</p>
|
||||
<div class="main div" id="orge8444d0">
|
||||
<div class="main div" id="org2e3ffd1">
|
||||
<p>
|
||||
\[
|
||||
{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
|
||||
|
|
@ -1711,7 +1711,7 @@ Gauss's law: can be rewritten
|
|||
\]
|
||||
where (as in static case)
|
||||
</p>
|
||||
<div class="core div" id="org3e52dc6">
|
||||
<div class="core div" id="org8e610c3">
|
||||
<p>
|
||||
\[
|
||||
{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
|
||||
|
|
@ -1737,7 +1737,7 @@ or
|
|||
\]
|
||||
where as before
|
||||
</p>
|
||||
<div class="core div" id="org5550f7c">
|
||||
<div class="core div" id="orgbab4d4d">
|
||||
<p>
|
||||
\[
|
||||
{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
|
||||
|
|
@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
|
|||
<p>
|
||||
In terms of free charges and currents, we thus get
|
||||
</p>
|
||||
<div class="core div" id="orge1d07ac">
|
||||
<div class="core div" id="org23e12c1">
|
||||
<p>
|
||||
{\bf Maxwell's equations {\it (in matter)}}
|
||||
</p>
|
||||
|
|
@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
|
|||
in terms of \({\bf E}\) and \({\bf B}\).
|
||||
For the restricted case of linear media:
|
||||
</p>
|
||||
<div class="main div" id="org83b4dbd">
|
||||
<div class="main div" id="orgefb9da7">
|
||||
<p>
|
||||
\[
|
||||
{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
|
||||
|
|
@ -1818,7 +1818,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1600,7 +1600,7 @@ Table of contents
|
|||
<p>
|
||||
Discontinuities between different media, deduced from
|
||||
</p>
|
||||
<div class="core div" id="org55029ff">
|
||||
<div class="core div" id="org22772ad">
|
||||
<p>
|
||||
{\bf Maxwell's equations {\it (in matter)}, integral form}
|
||||
</p>
|
||||
|
|
@ -1691,7 +1691,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1625,7 +1625,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1621,7 +1621,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1717,7 +1717,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1635,7 +1635,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
|
|||
\]
|
||||
where the index of refraction of the material is defined as
|
||||
</p>
|
||||
<div class="main div" id="orga0618f6">
|
||||
<div class="main div" id="orga04d0b5">
|
||||
<p>
|
||||
{\bf Index of refraction}
|
||||
\[
|
||||
|
|
@ -1692,7 +1692,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1636,7 +1636,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1616,7 +1616,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1617,7 +1617,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1708,7 +1708,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1635,7 +1635,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
|
|||
<p>
|
||||
From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
|
||||
</p>
|
||||
<div class="core div" id="orge8258d6">
|
||||
<div class="core div" id="orgad88980">
|
||||
<p>
|
||||
{\bf First law of reflection:}
|
||||
the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
|
||||
|
|
@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
|
|||
with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
|
||||
and the angle of refraction (\(\theta_T\)) obey the following laws:
|
||||
</p>
|
||||
<div class="core div" id="org823940c">
|
||||
<div class="core div" id="org8ea3739">
|
||||
<p>
|
||||
{\bf Law of reflection}
|
||||
\[
|
||||
|
|
@ -1708,7 +1708,7 @@ while the third equation becomes
|
|||
\]
|
||||
Writing everything in terms of the incident amplitude, we get
|
||||
</p>
|
||||
<div class="main div" id="orgd20a172">
|
||||
<div class="main div" id="orgf78da16">
|
||||
<p>
|
||||
{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
|
||||
\[
|
||||
|
|
@ -1728,7 +1728,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
|
|||
Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
|
||||
For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
|
||||
</p>
|
||||
<div class="main div" id="orgd033d51">
|
||||
<div class="main div" id="org0eb2bc7">
|
||||
<p>
|
||||
{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
|
||||
\[
|
||||
|
|
@ -1778,7 +1778,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1617,7 +1617,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1623,7 +1623,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1615,13 +1615,13 @@ In this case, we can find nontrivial solutions to Maxwell's equations for \(E_z
|
|||
<p>
|
||||
The derivative equations are precisely the electrostatic and magnetic equations for empty space with cylindrical symmetry. The solution is thus that of an infinite line charge and an infinite straight current:
|
||||
\[
|
||||
{\boldsymbol E}_0 (s, \phi) = \frac{A}{s} \hat{\boldsymbol s}, \hspace{10mm}
|
||||
{\boldsymbol B}_0 (s, \phi) = \frac{A}{cs} \hat{\boldsymbol \phi}
|
||||
{\boldsymbol E}_0 (s, \varphi) = \frac{A}{s} \hat{\boldsymbol s}, \hspace{10mm}
|
||||
{\boldsymbol B}_0 (s, \varphi) = \frac{A}{cs} \hat{\boldsymbol \varphi}
|
||||
\]
|
||||
where \(A\) is a constant amplitude. Substituting and taking the real part,
|
||||
\[
|
||||
{\boldsymbol E} (s, \phi, z, t) = \frac{A}{s} \cos (kz - \omega t) \hat{\boldsymbol s}, \hspace{10mm}
|
||||
{\boldsymbol B} (s, \phi, z, t) = \frac{A}{cs} \cos (kz - \omega t) \hat{\boldsymbol \phi}.
|
||||
{\boldsymbol E} (s, \varphi, z, t) = \frac{A}{s} \cos (kz - \omega t) \hat{\boldsymbol s}, \hspace{10mm}
|
||||
{\boldsymbol B} (s, \varphi, z, t) = \frac{A}{cs} \cos (kz - \omega t) \hat{\boldsymbol \varphi}.
|
||||
\]
|
||||
</p>
|
||||
</div>
|
||||
|
|
@ -1643,7 +1643,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1667,7 +1667,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
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|
||||
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|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1676,7 +1676,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,8 @@ Table of contents
|
|||
</svg></a><span class="headline-id">emf</span></h2>
|
||||
|
||||
<div class="outline-text-2" id="text-emf">
|
||||
<details class="prereq" id="orgd0eb59c">
|
||||
<summary id="org6131f4a">
|
||||
<details class="prereq" id="org1da9485">
|
||||
<summary id="org8284e09">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1607,8 +1607,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org1db29ae">
|
||||
<summary id="orgab2baad">
|
||||
<details class="objectives" id="orgec6c562">
|
||||
<summary id="org2677994">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1644,7 +1644,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1641,7 +1641,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1640,7 +1640,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
|
|||
\]
|
||||
These can be written compactly upon introducing a new operator: the
|
||||
</p>
|
||||
<div class="core div" id="orgb7b775a">
|
||||
<div class="core div" id="org5af3f67">
|
||||
<p>
|
||||
{\bf d'Alembertian operator}
|
||||
\[
|
||||
|
|
@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
|
|||
<p>
|
||||
so we get the
|
||||
</p>
|
||||
<div class="core div" id="org1d4f993">
|
||||
<div class="core div" id="org57eba77">
|
||||
<p>
|
||||
{\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
|
||||
\[
|
||||
|
|
@ -1676,7 +1676,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
|
|||
<p>
|
||||
Easiest:
|
||||
</p>
|
||||
<div class="core div" id="orgea17c1a">
|
||||
<div class="core div" id="org90a09e2">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
|
||||
|
|
@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
|
|||
\]
|
||||
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
|
||||
</p>
|
||||
<div class="core div" id="orge40ddf5">
|
||||
<div class="core div" id="org3315df4">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
|
||||
|
|
@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
|
|||
<p>
|
||||
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
|
||||
</p>
|
||||
<div class="main div" id="org73cca63">
|
||||
<div class="main div" id="org3bcbbfa">
|
||||
<p>
|
||||
\[
|
||||
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
|
||||
|
|
@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
|
|||
\]
|
||||
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
|
||||
</p>
|
||||
<div class="main div" id="org0170d34">
|
||||
<div class="main div" id="org4a79a92">
|
||||
<p>
|
||||
\[
|
||||
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
|
||||
|
|
@ -1695,7 +1695,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1624,7 +1624,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,8 +1598,12 @@ Table of contents
|
|||
</svg></a><span class="headline-id">ems.ca</span></h3>
|
||||
|
||||
<div class="outline-text-3" id="text-ems_ca">
|
||||
<details class="prereq" id="org0bf319c">
|
||||
<summary id="orga92f3c7">
|
||||
<ul class="org-ul altsecnrs">
|
||||
<li>Gr 3</li>
|
||||
</ul>
|
||||
|
||||
<details class="prereq" id="org9328e99">
|
||||
<summary id="orgd93d062">
|
||||
Prerequisites
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1607,8 +1611,8 @@ Prerequisites
|
|||
</ul>
|
||||
</details>
|
||||
|
||||
<details class="objectives" id="org87cd363">
|
||||
<summary id="org64f736b">
|
||||
<details class="objectives" id="org1849f4f">
|
||||
<summary id="orgf1a9924">
|
||||
Objectives
|
||||
</summary>
|
||||
<ul class="org-ul">
|
||||
|
|
@ -1646,7 +1650,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1598,18 +1598,23 @@ Table of contents
|
|||
</svg></a><span class="headline-id">ems.ca.fe</span></h4>
|
||||
|
||||
<div class="outline-text-4" id="text-ems_ca_fe">
|
||||
<ul class="org-ul altsecnrs">
|
||||
<li>Gr 3.1</li>
|
||||
</ul>
|
||||
|
||||
|
||||
<p>
|
||||
A generic configuration of static charges coupled via the Coulomb interaction
|
||||
defines an electrostatic problem, whose solution is in principle obtained
|
||||
from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="org98cdfd8">
|
||||
<div class="main div" id="orgfccec5d">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
||||
\begin{equation*}
|
||||
{\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
|
||||
{\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
|
||||
\end{equation*}
|
||||
|
||||
</div>
|
||||
|
|
@ -1617,7 +1622,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
|
|||
or (often simpler) by calculating the electrostatic potential, using either the
|
||||
explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
|
||||
</p>
|
||||
<div class="main div" id="org12e97ac">
|
||||
<div class="main div" id="org5194654">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1637,7 +1642,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
|
|||
<a href="./ems_es_ep_PL.html#Poi">🐟</a>
|
||||
</p>
|
||||
|
||||
<div class="core div" id="orgcacf446">
|
||||
<div class="core div" id="org765eb12">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1653,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
|
|||
<p>
|
||||
In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
|
||||
</p>
|
||||
<div class="core div" id="orgea7b7e8">
|
||||
<div class="core div" id="org025bcf6">
|
||||
<p>
|
||||
|
||||
</p>
|
||||
|
|
@ -1691,7 +1696,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1614,14 +1614,14 @@ In one dimension, the potential is a single-variable
|
|||
function \(\phi (x)\) and the Laplace equation reads
|
||||
</p>
|
||||
|
||||
<div class="eqlabel" id="org8b9eaa8">
|
||||
<div class="eqlabel" id="org937348d">
|
||||
<p>
|
||||
<a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><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="org1a79944">
|
||||
<div class="alteqlabels" id="orgf7c0132">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1636,14 +1636,14 @@ function \(\phi (x)\) and the Laplace equation reads
|
|||
<p>
|
||||
The solution to this
|
||||
</p>
|
||||
<div class="eqlabel" id="orgd6d6f8b">
|
||||
<div class="eqlabel" id="org5437a41">
|
||||
<p>
|
||||
<a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><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="org135f906">
|
||||
<div class="alteqlabels" id="org777e948">
|
||||
<ul class="org-ul">
|
||||
<li>Gr (3.6)</li>
|
||||
</ul>
|
||||
|
|
@ -1702,14 +1702,14 @@ In two dimensions, the potential becomes a function
|
|||
of two variables (here: \(x\) and \(y\)), so Laplace's
|
||||
equation now reads
|
||||
</p>
|
||||
<div class="eqlabel" id="orgb8aa28d">
|
||||
<div class="eqlabel" id="orgf26c98b">
|
||||
<p>
|
||||
<a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><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="org01054fc">
|
||||
<div class="alteqlabels" id="org59e7c1d">
|
||||
|
||||
</div>
|
||||
|
||||
|
|
@ -1765,8 +1765,8 @@ a point equals its value averaged over any sphere
|
|||
\]
|
||||
</p>
|
||||
|
||||
<details id="org8c78d21">
|
||||
<summary id="org6bdc443">
|
||||
<details id="orgf3cada4">
|
||||
<summary id="org840c868">
|
||||
<strong>Physicist's proof</strong>
|
||||
</summary>
|
||||
<p>
|
||||
|
|
@ -1833,8 +1833,8 @@ proving the theorem.
|
|||
</p>
|
||||
</details>
|
||||
|
||||
<details id="orgff2611b">
|
||||
<summary id="org89a3b1b">
|
||||
<details id="org02ee258">
|
||||
<summary id="orgf24411c">
|
||||
<strong>Formal proof</strong>
|
||||
</summary>
|
||||
|
||||
|
|
@ -1851,7 +1851,7 @@ f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R} da' ~ f ({\bf r} + {\bf r
|
|||
|
||||
<p>
|
||||
For convenience we will hereafter put \({\bf r} = 0\).
|
||||
In spherical coordinates, we have \(da' = R^2 sin \theta d\theta d\phi \equiv R^2 d\Omega\).
|
||||
In spherical coordinates, we have \(da' = R^2 sin \theta d\theta d\varphi \equiv R^2 d\Omega\).
|
||||
Differentiating with respect to \(R\),
|
||||
</p>
|
||||
|
||||
|
|
@ -1931,7 +1931,7 @@ are necessarily positive, we thus require \(f_x > 0\), \(f_y > 0\) and \(f
|
|||
of the \(f_x + f_y + f_z = 0\) condition above.
|
||||
</p>
|
||||
|
||||
<div class="info div" id="org950ce9d">
|
||||
<div class="info div" id="orgfec03b0">
|
||||
<p>
|
||||
<b>Earnshaw's theorem (physical version)</b> <br>
|
||||
It is impossible to find a static distribution of charges which generates an electrostatic field
|
||||
|
|
@ -2048,7 +2048,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
<!DOCTYPE html>
|
||||
<html lang="en">
|
||||
<head>
|
||||
<!-- 2022-02-13 Sun 21:20 -->
|
||||
<!-- 2022-02-14 Mon 20:35 -->
|
||||
<meta charset="utf-8">
|
||||
<meta name="viewport" content="width=device-width, initial-scale=1">
|
||||
<title>Pre-Quantum Electrodynamics</title>
|
||||
|
|
@ -1597,11 +1597,11 @@ Table of contents
|
|||
<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.ca.fe.g</span></h5>
|
||||
<div class="outline-text-5" id="text-ems_ca_fe_g">
|
||||
<div class="info div" id="orgc7ef55c">
|
||||
<div class="info div" id="org9c72286">
|
||||
<p>
|
||||
<b>George Green</b>
|
||||
</p>
|
||||
<aside id="org5a19053">
|
||||
<aside id="org6a5deaa">
|
||||
<p>
|
||||
See a <a href="https://en.wikipedia.org/wiki/George%5C_Green%5C_(mathematician)">short bio on wikipedia</a>
|
||||
</p>
|
||||
|
|
@ -1634,14 +1634,14 @@ and
|
|||
\]
|
||||
Substituting this in the divergence theorem gives <b>Green's first identity</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org713a4b7">
|
||||
<div class="eqlabel" id="orgc80fef9">
|
||||
<p>
|
||||
<a id="Green1"></a><a href="./ems_ca_fe_g.html#Green1"><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="orgfb4fc39">
|
||||
<div class="alteqlabels" id="org985f604">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.34)</li>
|
||||
</ul>
|
||||
|
|
@ -1659,14 +1659,14 @@ As an aside for now, for completeness, if we do the same thing again but with \(
|
|||
interchanged, and subtract the result, we obtain another useful result known as
|
||||
<b>Green's second identity</b> or <b>Green's theorem</b>
|
||||
</p>
|
||||
<div class="eqlabel" id="org6298bab">
|
||||
<div class="eqlabel" id="org55daee4">
|
||||
<p>
|
||||
<a id="Green2"></a><a href="./ems_ca_fe_g.html#Green2"><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="org29adc56">
|
||||
<div class="alteqlabels" id="orgd7df7dc">
|
||||
<ul class="org-ul">
|
||||
<li>J (1.35)</li>
|
||||
</ul>
|
||||
|
|
@ -1702,7 +1702,7 @@ 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: 2022-02-13 Sun 21:20</p>
|
||||
<p class="date">Created: 2022-02-14 Mon 20:35</p>
|
||||
<p class="validation"></p>
|
||||
</div>
|
||||
|
||||
|
|
|
|||
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