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<!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>
@@ -1598,11 +1598,20 @@ Table of contents
</svg></a><span class="headline-id">ems.ca.fe.uP</span></h5>
<div class="outline-text-5" id="text-ems_ca_fe_uP">
<p>
Suppose now that we have two solutions to the Poisson equation, \(\phi_1 ({\bf r})\) and \(\phi_2 ({\bf r})\).
Suppose that we have two solutions to the Poisson equation,
\(\phi_1 ({\bf r})\) and \(\phi_2 ({\bf r})\), within a certain volume \({\cal V}\)
(which can contain charges).
</p>
<p>
Defining \(\Phi = \phi_1 - \phi_2\), we see that \(\Phi\) manifestly obeys Laplace
within \({\cal V}\), \({\boldsymbol \nabla}^2 \Phi = 0\).
</p>
<p>
We can now use Green's first identity <a href="./ems_ca_fe_g.html#Green1">Green1</a> to shed some light on the boundary
problem for the electrostatic potential. Namely, put \(\phi = \psi = \Phi\). This yields
problem for the electrostatic potential. Namely, let us put \(\phi = \psi = \Phi\).
This yields
\[
\int_{\cal V} d\tau \left( \Phi {\boldsymbol \nabla}^2 \Phi + {\boldsymbol \nabla} \Phi \cdot {\boldsymbol \nabla} \Phi \right)
= \oint_{\cal S} da ~\Phi \frac{\partial \Phi}{\partial n}.
@@ -1610,14 +1619,14 @@ problem for the electrostatic potential. Namely, put \(\phi = \psi = \Phi\). T
The first term on the left-hand side vanishes since \(\Phi\) satisfies Laplace.
The right-hand side can be made to vanish if \(\Phi\) obeys either
</p>
<div class="eqlabel" id="org8a3aa85">
<div class="eqlabel" id="orgc3f8f6e">
<p>
<a id="Dirichlet"></a><a href="./ems_ca_fe_uP.html#Dirichlet"><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="org92e95e5">
<div class="alteqlabels" id="org9cea551">
</div>
@@ -1629,14 +1638,14 @@ The right-hand side can be made to vanish if \(\Phi\) obeys either
<p>
or
</p>
<div class="eqlabel" id="org29efb00">
<div class="eqlabel" id="org17cd516">
<p>
<a id="Neumann"></a><a href="./ems_ca_fe_uP.html#Neumann"><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="orgae394cd">
<div class="alteqlabels" id="orga1efe69">
</div>
@@ -1648,36 +1657,44 @@ or
<p>
boundary conditions on each individual boundary surface. In those cases, we are left with
\[
\int_{\cal V} d\tau \left|{\boldsymbol \nabla} \Phi \right|^2 = 0, \longrightarrow {\boldsymbol \nabla} \Phi = 0.
\int_{\cal V} d\tau \left|{\boldsymbol \nabla} \Phi \right|^2 = 0,
\]
\(\Phi\) is thus constant. For Dirichlet, \(\Phi = 0\) throughout \({\cal V}\), and thus \(\phi_2 = \phi_1\) and the solution
is unique. For Neumann, the solution is unique apart from an unimportant constant.
and since this is valid for any choices of \({\cal V}\), we get the local
condition
\[
\longrightarrow {\boldsymbol \nabla} \Phi = 0.
\]
\(\Phi\) is thus constant. For Dirichlet boundary conditions, \(\Phi = 0\) throughout
\({\cal V}\), and thus \(\phi_2 = \phi_1\), <i>i.e.</i> the solution is unique.
For Neumann boundary conditions, the solution is unique apart from an unimportant
additive constant.
</p>
<p>
We can thus finally state the
</p>
<div class="eqlabel" id="org8b86bbf">
<div class="eqlabel" id="org45c4a53">
<p>
<a id="uniq_thm"></a><a href="./ems_ca_fe_uP.html#uniq_thm"><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="orgc0186aa">
<div class="alteqlabels" id="org5b46a79">
</div>
</div>
<div class="core div" id="org5889497">
<div class="core div" id="org9377072">
<p>
<b>Uniqueness Theorem</b>
</p>
<p>
The solution to Poisson's equation \({\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}\) inside a volume \({\cal \phi}\)
bounded by a (in general disconnected) surface \({\cal S}\) is uniquely defined provided either Dirichlet \(\phi |_{{\cal S}_i}\) or Neumann
\(\frac{\partial \phi}{\partial n} |_{{\cal S}_i}\) boundary conditions are used on each individual surface.
The solution to Poisson's equation \({\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}\) inside a volume \({\cal V}\)
bounded by a (in general disconnected) surface \({\cal S}\) is unique provided either Dirichlet \(\phi |_{{\cal S}_i}\) or Neumann
\(\frac{\partial \phi}{\partial n} |_{{\cal S}_i}\) boundary conditions are
applied on each individual surface.
</p>
</div>
@@ -1690,24 +1707,24 @@ Neumann on others).
<b>Existence of solutions</b>: this is another matter.
Intuitively, from our first case:
the solution always exists for Dirichlet boundary conditions.
For Neumann, some self-consistency restrictions apply.
</p>
<p>
<b>Link to earlier cases</b>:
the previous case in which the potential is specified on the
boundaries, is thus the case of Dirichlet boundary conditions.
The one where the normal derivative of the potential is given,
is a subcase involving
Neumann boundary conditions (subcase, because we could imagine other charges living outside volume \({\cal V}\),
whereas the earlier example involved only surface charges).
the previous "known boundary potential" case is thus the case of Dirichlet boundary
conditions. The "known boundary charge" is the case of
Neumann boundary conditions.
</p>
<p>
<i>Note on Griffiths' presentation of uniqueness theorem(s)</i>:
we have used Green's identity to provide a general statement on uniqueness. Reading Griffiths, you might be misled into thinking that there are numerous cases and corollaries.
<i>Note on presentations of uniqueness theorem(s) in various books</i>:
we have used Green's identity to provide a general statement on uniqueness.
Reading other books, you might be misled into thinking that there are numerous cases
and corollaries and that things are
</p>
<div class="info div" id="org8d9ede7">
<div class="info div" id="org547d65a">
<p>
<b>Comment/warning</b>: <b>uniqueness theorem on uniqueness theorems</b> <br>
Do not be misled: there is a <i>unique</i> uniqueness theorem for the
@@ -1734,7 +1751,7 @@ 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>