Update 2022-02-21 10:35
This commit is contained in:
Jean-Sébastien
+30
-35
@@ -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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<!-- 2022-02-17 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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@@ -1597,7 +1592,7 @@ Table of contents
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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><span class="headline-id">ems.ca.sv.car</span></h5>
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<div class="outline-text-5" id="text-ems_ca_sv_car">
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<div class="example div" id="org9f6bdf3">
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<div class="example div" id="orge648e47">
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<p>
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</p>
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@@ -1643,14 +1638,14 @@ This thus falls back onto a 2d problem. We need to solve the 2d Laplace equatio
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<p>
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Let us look for solutions in the form
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</p>
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<div class="eqlabel" id="org5766cb8">
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<div class="eqlabel" id="orgced75e2">
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<p>
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<a id="Lap_sv_car"></a><a href="./ems_ca_sv_car.html#Lap_sv_car"><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="orge25d01b">
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<div class="alteqlabels" id="orgcb5fd31">
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<ul class="org-ul">
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<li>Gr (3.23)</li>
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</ul>
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@@ -1669,14 +1664,14 @@ individual term in the Laplace equation equals a constant,
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and that these constants add up to zero. We can thus put (the sign choice anticipates
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the solution somewhat)
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</p>
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<div class="eqlabel" id="org7435451">
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<div class="eqlabel" id="orgcb22008">
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<p>
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<a id="Lap_sv_car_sep"></a><a href="./ems_ca_sv_car.html#Lap_sv_car_sep"><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="org0027e3c">
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<div class="alteqlabels" id="orgbd536a2">
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<ul class="org-ul">
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<li>Gr (3.26)</li>
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</ul>
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@@ -1700,14 +1695,14 @@ Let's look first of all at the solutions of <a href="./ems_ca_sv_car.html#Lap_sv
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Since this is a second-order linear differential equation, there are two linearly
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independent solutions. The most general solution for \(X\) and \(Y\) can be written
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</p>
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<div class="eqlabel" id="orga83b769">
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<div class="eqlabel" id="orgd2e2516">
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<p>
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<a id="Lap_sv_car_solXY"></a><a href="./ems_ca_sv_car.html#Lap_sv_car_solXY"><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="org6cec6a1">
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<div class="alteqlabels" id="org2b25559">
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<ul class="org-ul">
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<li>Gr (3.27)</li>
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</ul>
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@@ -1764,14 +1759,14 @@ C_n = \frac{2}{a} \int_0^a dy ~\phi_0(y) \sin(n\pi y/a)
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<p>
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<b>Specific example</b>: say that \(\phi_0(y) = \phi_0\), <i>i.e.</i> just a constant. Then,
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</p>
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<div class="eqlabel" id="org79faeb7">
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<div class="eqlabel" id="org2d1fea2">
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<p>
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<a id="Cn"></a><a href="./ems_ca_sv_car.html#Cn"><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="org4d1565a">
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<div class="alteqlabels" id="orgbbcdd21">
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<ul class="org-ul">
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<li>Gr (3.35)</li>
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</ul>
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@@ -1825,7 +1820,7 @@ The solution for the specific case \(\phi_0 (y) = \phi_0\) is thus
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</p>
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<div class="example div" id="orgec6908d">
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<div class="example div" id="org5434cea">
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<p>
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<b>Example: rectangular pipe</b>
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</p>
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@@ -1882,14 +1877,14 @@ The full solution is then a linear combination of complete set of functions,
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The coefficients must be chosen such that \((iii)\) is fulfilled, \(\phi(b,y) = \phi_0\).
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This simple case of a constant value \(\phi_0\) gives us the same relation as <a href="./ems_ca_sv_car.html#Cn">Cn</a>, so
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</p>
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<div class="eqlabel" id="org4eaa850">
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<div class="eqlabel" id="org6eedfb9">
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<p>
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<a id="p_recpipe"></a><a href="./ems_ca_sv_car.html#p_recpipe"><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="org99dbfde">
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<div class="alteqlabels" id="org0ba9222">
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<ul class="org-ul">
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<li>Gr (3.42)</li>
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</ul>
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@@ -1925,7 +1920,7 @@ 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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