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Jean-Sébastien
2022-02-21 10:35:02 +01:00
parent ec8a4ca406
commit 40679d39bc
204 changed files with 4807 additions and 13916 deletions
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-17 Thu 08:42 -->
<!-- 2022-02-21 Mon 10:33 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -602,11 +602,11 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
</li>
<li>
<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
</li>
@@ -614,21 +614,12 @@ Table of contents
</details>
</li>
<li>
<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
<details>
<summary>
</li>
<li>
<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
</summary>
<ul>
<li>
<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>
</li>
</ul>
</details>
</li>
<li>
@@ -640,11 +631,15 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
<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>
</li>
<li>
<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>
<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>
</li>
<li>
<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>
</li>
@@ -661,6 +656,10 @@ Table of contents
</summary>
<ul>
<li>
<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
</li>
<li>
<a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
</li>
@@ -698,10 +697,6 @@ Table of contents
</summary>
<ul>
<li>
<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
</li>
<li>
<details>
<summary>
@@ -1435,7 +1430,7 @@ Table of contents
</li>
<li>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
</li>
<li>
@@ -1601,14 +1596,14 @@ Table of contents
In spherical coordinates, the Laplace equation takes the following form
(using <a href="./c_m_cs_sph.html#sph_Lap">sph_Lap</a>):
</p>
<div class="eqlabel" id="orgb48159c">
<div class="eqlabel" id="org5a10272">
<p>
<a id="Lap_sph"></a><a href="./ems_ca_sv_sph.html#Lap_sph"><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="orge4ad3bc">
<div class="alteqlabels" id="orga5fe900">
<ul class="org-ul">
<li>Gr (3.53)</li>
<li>W (11-86)</li>
@@ -1617,7 +1612,7 @@ In spherical coordinates, the Laplace equation takes the following form
</div>
</div>
<div class="main div" id="org07e657a">
<div class="main div" id="org2793267">
<p>
</p>
@@ -1634,14 +1629,14 @@ In spherical coordinates, the Laplace equation takes the following form
If you are dealing with a problem having <b>azimuthal symmetry</b>,
\(\phi\) is independent of \(\varphi\) and the equation simplifies to:
</p>
<div class="eqlabel" id="org6921ac1">
<div class="eqlabel" id="org7ecd5cd">
<p>
<a id="Lap_sph_az"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az"><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="org2f5c64f">
<div class="alteqlabels" id="orga1bb43a">
<ul class="org-ul">
<li>Gr (3.54)</li>
<li>W (11-87)</li>
@@ -1739,7 +1734,7 @@ This equation is solved by <b>Legendre polynomials</b> of the variable \(\cos \t
</p>
<div class="info div" id="org27c5d25">
<div class="info div" id="orgf968dfc">
<p>
<b>Legendre polynomials</b>
</p>
@@ -1760,14 +1755,14 @@ and conveniently defined (for trigonometric arguments) to obey the orthogonality
relationship (the reason for the normalization on the right-hand side will become clear later)
</p>
<div class="eqlabel" id="org0be561d">
<div class="eqlabel" id="org019c756">
<p>
<a id="Leg_orth_trig"></a><a href="./ems_ca_sv_sph.html#Leg_orth_trig"><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="org1014d5c">
<div class="alteqlabels" id="org06d3680">
</div>
@@ -1782,14 +1777,14 @@ relationship (the reason for the normalization on the right-hand side will becom
<p>
This same relation can be more simply written by using the variable \(x = \cos \theta\),
</p>
<div class="eqlabel" id="orgc26e8e9">
<div class="eqlabel" id="org9f4bb2e">
<p>
<a id="Leg_orth"></a><a href="./ems_ca_sv_sph.html#Leg_orth"><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="orgfe13130">
<div class="alteqlabels" id="orgd40cddf">
</div>
@@ -1807,14 +1802,14 @@ To get started, we need to define the "seed" polynomial (carrying label \(l=0\))
To make life easy, we set \(P_0 (x) = 1\). Higher polynomials are then sought in the
form of power series in \(x\). This leads to the first few Legendre polynomials being:
</p>
<div class="eqlabel" id="org075f673">
<div class="eqlabel" id="orgcd000d3">
<p>
<a id="Leg_pols"></a><a href="./ems_ca_sv_sph.html#Leg_pols"><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="org5b4b0fd">
<div class="alteqlabels" id="org891ea20">
</div>
@@ -1834,14 +1829,14 @@ P_5 (x) &amp;= \frac{1}{8} (63x^5 - 70x^3 + 15x).
The prefactor (and thus the factor in the orthogonality relations <a href="./ems_ca_sv_sph.html#Leg_orth_trig">Leg_orth_trig</a> and the equivalent <a href="./ems_ca_sv_sph.html#Leg_orth">Leg_orth</a> is chosen for convenience such that each polynomial
takes the value \(1\) when evaluated at argument \(x = 1\),
</p>
<div class="eqlabel" id="org128c7ec">
<div class="eqlabel" id="org3cdaa4f">
<p>
<a id="Pl_1_1"></a><a href="./ems_ca_sv_sph.html#Pl_1_1"><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="orgea70502">
<div class="alteqlabels" id="org5e065fa">
</div>
@@ -1857,14 +1852,14 @@ P_l(1) = 1
<p>
The Legendre polynomial \(P_l\) obeys the differential equation
</p>
<div class="eqlabel" id="org1bc0795">
<div class="eqlabel" id="org1d8f3f3">
<p>
<a id="Leg_de_trig"></a><a href="./ems_ca_sv_sph.html#Leg_de_trig"><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="org7c9cf91">
<div class="alteqlabels" id="orgfeff4f4">
</div>
@@ -1876,14 +1871,14 @@ The Legendre polynomial \(P_l\) obeys the differential equation
\]
or equivalently
</p>
<div class="eqlabel" id="orgacfb220">
<div class="eqlabel" id="orgdf7769c">
<p>
<a id="Leg_de"></a><a href="./ems_ca_sv_sph.html#Leg_de"><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="orged6e38d">
<div class="alteqlabels" id="orgc0f92d3">
</div>
@@ -1899,14 +1894,14 @@ or equivalently
A particularly convenient formula for deriving \(P_l(x)\)
is the <b>Rodrigues formula</b>:
</p>
<div class="eqlabel" id="org5aa10f9">
<div class="eqlabel" id="org390eaf5">
<p>
<a id="Rodrigues"></a><a href="./ems_ca_sv_sph.html#Rodrigues"><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="org9426131">
<div class="alteqlabels" id="org560f648">
</div>
@@ -1922,14 +1917,14 @@ P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l
<p>
Actually, a more practical formula is <b>Bonnet's recursion relation</b>
</p>
<div class="eqlabel" id="org9f87056">
<div class="eqlabel" id="orgf201529">
<p>
<a id="Bonnet"></a><a href="./ems_ca_sv_sph.html#Bonnet"><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="org98c9d48">
<div class="alteqlabels" id="orgbd41fc0">
</div>
@@ -1965,14 +1960,14 @@ We therefore come to the culmination of our efforts here, and write
the general solution to <i>any</i> problem with azimuthal symmetry
(for which the potential takes a finite value for \(\theta = 0, \pi\)) as
</p>
<div class="eqlabel" id="org7a70f8a">
<div class="eqlabel" id="orgd6decd7">
<p>
<a id="Lap_sph_az_sol"></a><a href="./ems_ca_sv_sph.html#Lap_sph_az_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="orge5959b0">
<div class="alteqlabels" id="org308a2d8">
<ul class="org-ul">
<li>Gr (3.65)</li>
</ul>
@@ -1980,7 +1975,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
</div>
</div>
<div class="main div" id="org81a882f">
<div class="main div" id="org763ee04">
<p>
\[
\phi(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta)
@@ -1992,7 +1987,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
<div class="example div" id="orgaeb4209">
<div class="example div" id="orgbcb0b5c">
<p>
<b>Example: potential inside a hollow sphere</b>
</p>
@@ -2075,7 +2070,7 @@ Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
</div>
<div class="example div" id="orge045d76">
<div class="example div" id="org36b2422">
<p>
<b>Example: surface charge density on sphere</b>
</p>
@@ -2159,7 +2154,7 @@ so
<p>
\[
-\sum_{l=0}^\infty (l+1) \left(\frac{B_l^o}{R^{l+2}} + l A_l^i R^{l-1} \right) P_l (\cos \theta) = -\frac{\sigma_0 (\theta)}{\varepsilon_0},
-\sum_{l=0}^\infty \left((l+1) \frac{B_l^o}{R^{l+2}} + l A_l^i R^{l-1} \right) P_l (\cos \theta) = -\frac{\sigma_0 (\theta)}{\varepsilon_0},
\]
and thus
</p>
@@ -2205,14 +2200,14 @@ A_1^i = \frac{k}{2\varepsilon_0} \int_0^\pi d\theta \sin \theta [P_l(\cos \theta
<p>
The potential inside/outside the sphere is then
</p>
<div class="eqlabel" id="org37fe0ee">
<div class="eqlabel" id="org7ab7b21">
<p>
<a id="p_uni_ch_sph"></a><a href="./ems_ca_sv_sph.html#p_uni_ch_sph"><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="orgb7ad8c6">
<div class="alteqlabels" id="org49fc534">
</div>
@@ -2245,7 +2240,7 @@ 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: 2022-02-17 Thu 08:42</p>
<p class="date">Created: 2022-02-21 Mon 10:33</p>
<p class="validation"></p>
</div>