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Update 2022-02-15 10:32

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Jean-Sébastien
2022-02-15 10:32:42 +01:00
parent 09a8ba5fb6
commit 6874e66024
204 changed files with 1019 additions and 937 deletions
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-14 Mon 20:35 -->
<!-- 2022-02-15 Tue 10:14 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1601,14 +1601,14 @@ 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="orgfbf5340">
<div class="eqlabel" id="orgbb88a9c">
<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="org76f3ef7">
<div class="alteqlabels" id="org24c4004">
<ul class="org-ul">
<li>Gr (3.53)</li>
<li>W (11-86)</li>
@@ -1617,7 +1617,7 @@ In spherical coordinates, the Laplace equation takes the following form
</div>
</div>
<div class="main div" id="org8129eda">
<div class="main div" id="org9285e00">
<p>
</p>
@@ -1634,14 +1634,14 @@ 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="org27146e0">
<div class="eqlabel" id="org7fe852a">
<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="orgff31561">
<div class="alteqlabels" id="org607f03c">
<ul class="org-ul">
<li>Gr (3.54)</li>
<li>W (11-87)</li>
@@ -1683,7 +1683,7 @@ Substituting this in <a href="./ems_ca_sv_sph.html#Lap_sph_az">Lap_sph_az</a> an
</p>
<p>
We can no apply the separation of variables logic: being dependent on
We can now apply the separation of variables logic: being dependent on
separate variables, each term must be constant (the reasons for the
convenient choice will become clear later),
</p>
@@ -1743,14 +1743,14 @@ This equation is solved by <b>Legendre polynomials</b> of the variable \(\cos \t
A particularly convenient formula for deriving \(P_l(x)\)
is the <b>Rodrigues formula</b>:
</p>
<div class="eqlabel" id="org6cd91e9">
<div class="eqlabel" id="org2041f25">
<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="org558db55">
<div class="alteqlabels" id="org7cf5d1e">
<ul class="org-ul">
<li>Gr (3.62)</li>
</ul>
@@ -1769,14 +1769,14 @@ 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="org78d828a">
<div class="eqlabel" id="org0cbd41d">
<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="org5115e24">
<div class="alteqlabels" id="orga0e6404">
</div>
@@ -1792,14 +1792,14 @@ Actually, a more practical formula is <b>Bonnet's recursion relation</b>
<p>
For future reference, here are the first few Legendre polynomials:
</p>
<div class="eqlabel" id="orga93835c">
<div class="eqlabel" id="orgfa7e825">
<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="orgcd3f06e">
<div class="alteqlabels" id="org1e6b441">
</div>
@@ -1818,14 +1818,14 @@ P_5 (x) &amp;= \frac{1}{8} (63x^5 - 70x^3 + 15x).
<p>
The prefactor is chosen for convenience such that
</p>
<div class="eqlabel" id="org23dbe5c">
<div class="eqlabel" id="org19dfa8c">
<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="org61ef8fc">
<div class="alteqlabels" id="org30a0f0c">
</div>
@@ -1857,14 +1857,14 @@ 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="orgdbaeb25">
<div class="eqlabel" id="orgdac3d5e">
<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="orga105225">
<div class="alteqlabels" id="orgafc12d1">
<ul class="org-ul">
<li>Gr (3.65)</li>
</ul>
@@ -1872,7 +1872,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
</div>
</div>
<div class="main div" id="orgb454935">
<div class="main div" id="orgc063c81">
<p>
\[
\phi(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l (\cos \theta)
@@ -1884,7 +1884,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
<div class="example div" id="orgc60f117">
<div class="example div" id="org2d7caf7">
<p>
<b>Example: potential inside a hollow sphere</b>
</p>
@@ -1932,14 +1932,14 @@ The specified boundary condition means that
We can now use the fact that Legendre polynomials are orthogonal functions
with orthogonality relation
</p>
<div class="eqlabel" id="org17ec99b">
<div class="eqlabel" id="org54337f5">
<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="orga0d0923">
<div class="alteqlabels" id="org62596df">
</div>
@@ -1956,14 +1956,14 @@ with orthogonality relation
or rewritten in terms of trigonometric arguments
</p>
<div class="eqlabel" id="org8d87c5d">
<div class="eqlabel" id="org9f2ed51">
<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="org3a73b0c">
<div class="alteqlabels" id="org05d2441">
</div>
@@ -2013,7 +2013,7 @@ Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
</div>
<div class="example div" id="org4350b9a">
<div class="example div" id="orgc566d42">
<p>
<b>Example: surface charge density on sphere</b>
</p>
@@ -2143,14 +2143,14 @@ 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="org135e184">
<div class="eqlabel" id="org579fd42">
<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="orgcc2b0d5">
<div class="alteqlabels" id="orgd182291">
</div>
@@ -2183,7 +2183,7 @@ 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: 2022-02-14 Mon 20:35</p>
<p class="date">Created: 2022-02-15 Tue 10:14</p>
<p class="validation"></p>
</div>