Update 2022-02-21 10:35
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Jean-Sébastien
+28
-33
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-17 Thu 08:42 -->
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<!-- 2022-02-21 Mon 10:33 -->
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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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@@ -602,11 +602,11 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
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<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
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</li>
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<li>
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<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
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<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
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</li>
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@@ -614,21 +614,12 @@ Table of contents
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</details>
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</li>
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<li>
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<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
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<details>
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<summary>
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</li>
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<li>
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<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
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</li>
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</ul>
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</details>
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</li>
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<li>
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@@ -640,11 +631,15 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
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<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
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</li>
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<li>
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<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
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<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
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</li>
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<li>
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<a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
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</li>
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@@ -661,6 +656,10 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
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</li>
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<li>
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<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
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</li>
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@@ -698,10 +697,6 @@ Table of contents
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
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</li>
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<li>
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<details>
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<summary>
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@@ -1435,7 +1430,7 @@ Table of contents
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</li>
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<li>
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<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
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<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
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</li>
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<li>
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@@ -1605,19 +1600,19 @@ The next terms in the expansion are obtained similarly: the <b>quadrupole</b> te
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\]
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and can be rewritten as
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</p>
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<div class="eqlabel" id="org5ac424f">
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<div class="eqlabel" id="org22e05e1">
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<p>
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<a id="p_quad"></a><a href="./ems_ca_me_h.html#p_quad"><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="org122b4bd">
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<div class="alteqlabels" id="org87d8664">
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</div>
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</div>
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<div class="main div" id="orgefeca07">
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<div class="main div" id="org833120d">
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<p>
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\[
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\phi_q ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab}
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@@ -1629,19 +1624,19 @@ and can be rewritten as
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<p>
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in terms of the <b>quadrupole moment</b>
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</p>
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<div class="eqlabel" id="org260a670">
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<div class="eqlabel" id="org9378bcc">
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<p>
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<a id="quadmom"></a><a href="./ems_ca_me_h.html#quadmom"><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="orgd3ef922">
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<div class="alteqlabels" id="org165ace8">
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</div>
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</div>
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<div class="main div" id="org6139c7b">
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<div class="main div" id="org7f3d6ba">
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<p>
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\[
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Q_{ab} = \int_{\cal V} d\tau_s (3 r_{s,a} r_{s,b} - r_s^2 \delta_{a,b}) \rho ({\bf r}_s).
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@@ -1660,19 +1655,19 @@ This is a symmetric, traceless rank \(2\) tensor: \(Q_{ab} = Q_{ba}\) and
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<p>
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Our expansion for the potential, with terms up to quadrupole ones, thus looks like
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</p>
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<div class="eqlabel" id="orgbbb1bae">
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<div class="eqlabel" id="orgf50d8c0">
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<p>
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<a id="p_mdq"></a><a href="./ems_ca_me_h.html#p_mdq"><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="orgf2f0fdb">
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<div class="alteqlabels" id="orgb7642e6">
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</div>
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</div>
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<div class="main div" id="orga7d1078">
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<div class="main div" id="orgcc303cc">
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<p>
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\[
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\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \left( \frac{Q}{r} + \sum_a \frac{r_a}{r^3} p_a + \frac{1}{2} \sum_{a,b} \frac{r_a r_b}{r^5} Q_{ab} + ... \right)
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@@ -1693,7 +1688,7 @@ the leading nonvanishing multipole moment is independent of the chosen location
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the origin of the coordinate system (see Jackson Prob. 4.4).
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</p>
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<div class="info div" id="org8be148f">
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<div class="info div" id="orga188fa3">
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<p>
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<b>A consistent nomenclature for the multipole expansion?</b>
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</p>
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@@ -1842,7 +1837,7 @@ 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: 2022-02-17 Thu 08:42</p>
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<p class="date">Created: 2022-02-21 Mon 10:33</p>
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<p class="validation"></p>
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</div>
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