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Update 2022-02-14 20:42

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Jean-Sébastien
2022-02-14 20:42:37 +01:00
parent 4cfe8cef59
commit 09a8ba5fb6
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build/emf_svp.html
+6 -6
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@@ -1,7 +1,7 @@
<!DOCTYPE html>
<html lang="en">
<head>
<!-- 2022-02-13 Sun 21:20 -->
<!-- 2022-02-14 Mon 20:35 -->
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>Pre-Quantum Electrodynamics</title>
@@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
<p>
Easiest:
</p>
<div class="core div" id="orgea17c1a">
<div class="core div" id="org90a09e2">
<p>
\[
{\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
@@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
\]
so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
</p>
<div class="core div" id="orge40ddf5">
<div class="core div" id="org3315df4">
<p>
\[
{\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
@@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
<p>
Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
</p>
<div class="main div" id="org73cca63">
<div class="main div" id="org3bcbbfa">
<p>
\[
{\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
@@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
\]
which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
</p>
<div class="main div" id="org0170d34">
<div class="main div" id="org4a79a92">
<p>
\[
\left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
@@ -1695,7 +1695,7 @@ 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: 2022-02-13 Sun 21:20</p>
<p class="date">Created: 2022-02-14 Mon 20:35</p>
<p class="validation"></p>
</div>