Update 2022-02-21 20:42
This commit is contained in:
Jean-Sébastien
+56
-27
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-21 Mon 10:33 -->
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<!-- 2022-02-21 Mon 20:41 -->
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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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@@ -706,28 +706,41 @@ 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_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
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</li>
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<li>
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<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</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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<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
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<a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_fpo_pibc.html#emsm_esm_fpo_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.fpo.pibc</span>
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<a href="./emsm_esm_mE_o.html#emsm_esm_mE_o">Overview</a><span class="headline-id">emsm.esm.mE.o</span>
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</li>
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<li>
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<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
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<a href="./emsm_esm_mE_P.html#emsm_esm_mE_P">Polarization</a><span class="headline-id">emsm.esm.mE.P</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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<details>
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<summary>
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<a href="./emsm_esm_po.html#emsm_esm_po">Polarized Objects; Bound Charges</a><span class="headline-id">emsm.esm.po</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
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</li>
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<li>
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<a href="./emsm_esm_po_fid.html#emsm_esm_po_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.po.fid</span>
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</li>
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@@ -750,18 +763,34 @@ Table of contents
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</ul>
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</details>
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</li>
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<li>
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<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</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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<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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<a href="./emsm_esm_ld.html#emsm_esm_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.ld</span>
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</summary>
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<ul>
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<li>
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<a href="./emsm_esm_di_ld.html#emsm_esm_di_ld">Linear Dielectrics</a><span class="headline-id">emsm.esm.di.ld</span>
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<a href="./emsm_esm_ld_sp.html#emsm_esm_ld_sp">Susceptibility, Permittivity, Dielectric Constant</a><span class="headline-id">emsm.esm.ld.sp</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_e.html#emsm_esm_ld_e">Energy in Dielectric Systems</a><span class="headline-id">emsm.esm.ld.e</span>
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</li>
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<li>
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<a href="./emsm_esm_ld_f.html#emsm_esm_ld_f">Forces on Dielectrics</a><span class="headline-id">emsm.esm.ld.f</span>
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</li>
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@@ -1592,7 +1621,7 @@ 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="orge648e47">
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<div class="example div" id="org7c143ee">
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<p>
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</p>
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@@ -1638,14 +1667,14 @@ 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="orgced75e2">
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<div class="eqlabel" id="orgc1d721a">
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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="orgcb5fd31">
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<div class="alteqlabels" id="org31af73c">
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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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@@ -1664,14 +1693,14 @@ 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="orgcb22008">
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<div class="eqlabel" id="org251a96b">
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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="orgbd536a2">
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<div class="alteqlabels" id="org88ee470">
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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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@@ -1695,14 +1724,14 @@ 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="orgd2e2516">
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<div class="eqlabel" id="org726561e">
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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="org2b25559">
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<div class="alteqlabels" id="org8b9b9bf">
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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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@@ -1759,14 +1788,14 @@ 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="org2d1fea2">
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<div class="eqlabel" id="org59f8cc6">
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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="orgbbcdd21">
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<div class="alteqlabels" id="orgebe91c6">
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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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@@ -1820,7 +1849,7 @@ 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="org5434cea">
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<div class="example div" id="orgbb552f5">
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<p>
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<b>Example: rectangular pipe</b>
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</p>
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@@ -1877,14 +1906,14 @@ 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="org6eedfb9">
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<div class="eqlabel" id="org6340be9">
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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="org0ba9222">
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<div class="alteqlabels" id="org9ad9f59">
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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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@@ -1920,7 +1949,7 @@ 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: 2022-02-21 Mon 10:33</p>
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<p class="date">Created: 2022-02-21 Mon 20:41</p>
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<p class="validation"></p>
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</div>
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