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2022-02-17 08:44:22 +01:00
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-15 Tue 10:14 -->
<!-- 2022-02-17 Thu 08:42 -->
<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="orgbb88a9c">
<div class="eqlabel" id="orgb48159c">
<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="org24c4004">
<div class="alteqlabels" id="orge4ad3bc">
<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="org9285e00">
<div class="main div" id="org07e657a">
<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="org7fe852a">
<div class="eqlabel" id="org6921ac1">
<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="org607f03c">
<div class="alteqlabels" id="org2f5c64f">
<ul class="org-ul">
<li>Gr (3.54)</li>
<li>W (11-87)</li>
@@ -1739,21 +1739,57 @@ This equation is solved by <b>Legendre polynomials</b> of the variable \(\cos \t
</p>
<div class="info div" id="org27c5d25">
<p>
A particularly convenient formula for deriving \(P_l(x)\)
is the <b>Rodrigues formula</b>:
<b>Legendre polynomials</b>
</p>
<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">
When using spherical coordinates, one inevitably comes across integrals of the form
\[
\int_0^\pi d\theta ~\sin \theta ~f(\theta)
\]
for generic functions \(f\).
</p>
<p>
Inspired by the logic of Fourier series, we would like to decompose such generic functions
in a basis of "orthonormal" functions under this kind of integral (with the \(\sin \theta\) weight).
This idea lead us to the <b>Legendre polynomials</b>, denoted \(P_l\), l = 0, 1, 2, …,
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">
<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="org7cf5d1e">
<ul class="org-ul">
<li>Gr (3.62)</li>
</ul>
<div class="alteqlabels" id="org1014d5c">
</div>
</div>
<p>
\[
\int_0^\pi d\theta \sin \theta ~P_l (\cos \theta) P_{l'} (\cos \theta) = \frac{2}{2l + 1} \delta_{l l'}
\tag{Leg_orth_trig}\label{Leg_orth_trig}
\]
</p>
<p>
This same relation can be more simply written by using the variable \(x = \cos \theta\),
</p>
<div class="eqlabel" id="orgc26e8e9">
<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>
@@ -1761,45 +1797,24 @@ is the <b>Rodrigues formula</b>:
<p>
\[
P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l
\tag{Rodrigues}\label{Rodrigues}
\int_{-1}^1 dx P_l (x) P_{l'} (x) = \frac{2}{2l + 1} \delta_{l l'},
\tag{Leg_orth}\label{Leg_orth}
\]
</p>
<p>
Actually, a more practical formula is <b>Bonnet's recursion relation</b>
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="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="orga0e6404">
</div>
</div>
<p>
\[
(l + 1) P_{l+1} (x) = (2l + 1) x P_l (x) - l P_{l-1} (x)
\tag{Bonnet}\label{Bonnet}
\]
</p>
<p>
For future reference, here are the first few Legendre polynomials:
</p>
<div class="eqlabel" id="orgfa7e825">
<div class="eqlabel" id="org075f673">
<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="org1e6b441">
<div class="alteqlabels" id="org5b4b0fd">
</div>
@@ -1816,16 +1831,17 @@ P_5 (x) &amp;= \frac{1}{8} (63x^5 - 70x^3 + 15x).
\end{align}
<p>
The prefactor is chosen for convenience such that
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="org19dfa8c">
<div class="eqlabel" id="org128c7ec">
<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="org30a0f0c">
<div class="alteqlabels" id="orgea70502">
</div>
@@ -1838,6 +1854,98 @@ P_l(1) = 1
\]
</p>
<p>
The Legendre polynomial \(P_l\) obeys the differential equation
</p>
<div class="eqlabel" id="org1bc0795">
<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>
</div>
<p>
\[
\left[\frac{d}{d\theta} \left( \sin \theta \frac{d}{d\theta} \right) + l (l+1) \sin \theta \right] P_l (\cos \theta) = 0.
\tag{Leg_de_trig}\label{Leg_de_trig}
\]
or equivalently
</p>
<div class="eqlabel" id="orgacfb220">
<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>
</div>
<p>
\[
\left[(1 - x^2) \frac{d^2}{dx^2} - 2x \frac{d}{dx} + l(l+1) \right] P_l (x) = 0.
\tag{Leg_de}\label{Leg_de}
\]
</p>
<p>
A particularly convenient formula for deriving \(P_l(x)\)
is the <b>Rodrigues formula</b>:
</p>
<div class="eqlabel" id="org5aa10f9">
<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>
</div>
<p>
\[
P_l(x) = \frac{1}{2^l l!} \left( \frac{d}{dx} \right)^l (x^2 - 1)^l
\tag{Rodrigues}\label{Rodrigues}
\]
</p>
<p>
Actually, a more practical formula is <b>Bonnet's recursion relation</b>
</p>
<div class="eqlabel" id="org9f87056">
<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>
</div>
<p>
\[
(l + 1) P_{l+1} (x) = (2l + 1) x P_l (x) - l P_{l-1} (x)
\tag{Bonnet}\label{Bonnet}
\]
</p>
</div>
<p>
Going back to the angular equation, let us first remark that this
is a second order equation, and should thus have
@@ -1857,14 +1965,14 @@ 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="orgdac3d5e">
<div class="eqlabel" id="org7a70f8a">
<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="orgafc12d1">
<div class="alteqlabels" id="orge5959b0">
<ul class="org-ul">
<li>Gr (3.65)</li>
</ul>
@@ -1872,7 +1980,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
</div>
</div>
<div class="main div" id="orgc063c81">
<div class="main div" id="org81a882f">
<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 +1992,7 @@ the general solution to <i>any</i> problem with azimuthal symmetry
<div class="example div" id="org2d7caf7">
<div class="example div" id="orgaeb4209">
<p>
<b>Example: potential inside a hollow sphere</b>
</p>
@@ -1929,53 +2037,7 @@ The specified boundary condition means that
</p>
<p>
We can now use the fact that Legendre polynomials are orthogonal functions
with orthogonality relation
</p>
<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="org62596df">
</div>
</div>
<p>
\[
\int_{-1}^1 dx P_l (x) P_{l'} (x) = \frac{2}{2l + 1} \delta_{l l'},
\tag{Leg_orth}\label{Leg_orth}
\]
</p>
<p>
or rewritten in terms of trigonometric arguments
</p>
<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="org05d2441">
</div>
</div>
<p>
\[
\int_0^\pi d\theta \sin \theta P_l (\cos \theta) P_{l'} (\cos \theta) = \frac{2}{2l + 1} \delta_{l l'}
\]
</p>
<p>
We thus get
We can now use the fact that Legendre polynomials are orthogonal functions, giving us
</p>
<p>
@@ -2013,7 +2075,7 @@ Thus, \(A_0 = k/2\), \(A_1 = -k/2\), and all others are zero, so
</div>
<div class="example div" id="orgc566d42">
<div class="example div" id="orge045d76">
<p>
<b>Example: surface charge density on sphere</b>
</p>
@@ -2143,14 +2205,14 @@ 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="org579fd42">
<div class="eqlabel" id="org37fe0ee">
<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="orgd182291">
<div class="alteqlabels" id="orgb7ad8c6">
</div>
@@ -2183,7 +2245,7 @@ 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: 2022-02-15 Tue 10:14</p>
<p class="date">Created: 2022-02-17 Thu 08:42</p>
<p class="validation"></p>
</div>