Update 2022-02-14 20:42
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Jean-Sébastien
+15
-15
@@ -1,7 +1,7 @@
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<!-- 2022-02-13 Sun 21:20 -->
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<!-- 2022-02-14 Mon 20:35 -->
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<meta charset="utf-8">
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<meta name="viewport" content="width=device-width, initial-scale=1">
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<title>Pre-Quantum Electrodynamics</title>
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@@ -1634,7 +1634,7 @@ sphere of radius \(r\) around the charge,
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<p>
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\[
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\oint {\bf E} \cdot d{\bf a} = \frac{1}{4\pi\varepsilon_0} \int_{\cal S} \frac{q}{r^2} \hat{\bf r} \cdot \hat{\bf r}
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~r^2 \sin \theta d\theta d\phi = \frac{q}{\varepsilon_0}
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~r^2 \sin \theta d\theta d\varphi = \frac{q}{\varepsilon_0}
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\label{Gr(2.12)}
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\]
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</p>
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@@ -1642,14 +1642,14 @@ sphere of radius \(r\) around the charge,
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<p>
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so by superposition, we obtain
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</p>
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<div class="eqlabel" id="org4c0fb5a">
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<div class="eqlabel" id="orge154804">
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<p>
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<a id="Gl_i"></a><a href="./ems_es_ef_Gl.html#Gl_i"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgd0aebb3">
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<div class="alteqlabels" id="org8b5f31d">
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<ul class="org-ul">
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<li>Gr (2.13)</li>
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</ul>
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@@ -1657,7 +1657,7 @@ so by superposition, we obtain
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</div>
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</div>
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<div class="core div" id="orgd7a6c4a">
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<div class="core div" id="org56418f6">
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<p>
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<b>Gauss' law (in integral form)</b>
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</p>
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@@ -1689,14 +1689,14 @@ By applying the divergence theorem,
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and using \(Q_{\mbox{enc}} = \int_{\cal V} \rho d\tau\), and using the fact the the choice of volume
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is arbitrary, we get
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</p>
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<div class="eqlabel" id="orgdd1583f">
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<div class="eqlabel" id="orgde33405">
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<p>
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<a id="Gl_d"></a><a href="./ems_es_ef_Gl.html#Gl_d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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<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"/>
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<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"/>
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</svg></a>
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</p>
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<div class="alteqlabels" id="orgc48cc04">
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<div class="alteqlabels" id="org4364030">
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<ul class="org-ul">
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<li>Gr (2.14)</li>
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</ul>
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@@ -1704,7 +1704,7 @@ is arbitrary, we get
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</div>
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</div>
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<div class="core div" id="orgccc09ad">
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<div class="core div" id="org71a8c2e">
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<p>
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<b>Gauss' law in differential form</b>
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</p>
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@@ -1763,7 +1763,7 @@ cylindrical or plane symmetry.
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Gaussian surfaces: respectively, concentric sphere, coaxial cylinder, pillbox.
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</p>
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<div class="example div" id="org3ae26ef">
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<div class="example div" id="org1a3f2ae">
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<p>
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<b>Example 2.2</b>: Field outside a uniformly charged sphere of radius \(R\) and total charge \(q\).
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</p>
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@@ -1794,7 +1794,7 @@ Same as point charge at origin!
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</div>
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<div class="example div" id="org70c2241">
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<div class="example div" id="org9d941e1">
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<p>
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<b>Example 2.3</b>: infinitely long cylinder carrying charge density \(\rho = k s\) for some constant \(k\). Find \({\bf E}\) within the cylinder.
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</p>
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@@ -1805,7 +1805,7 @@ Same as point charge at origin!
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<p>
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\[
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Q_{\mbox{enc}} = \int d\tau \rho = \int_0^l dz \int_0^{2\pi} d\phi \int_0^s ds' (k s') s' = 2\pi k l \int_0^s ds' s'^2 = \frac{2\pi}{3} kls^3.
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Q_{\mbox{enc}} = \int d\tau \rho = \int_0^l dz \int_0^{2\pi} d\varphi \int_0^s ds' (k s') s' = 2\pi k l \int_0^s ds' s'^2 = \frac{2\pi}{3} kls^3.
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\]
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</p>
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@@ -1816,7 +1816,7 @@ has no contribution from the ends of the cylinder, and
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<p>
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\[
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\oint {\bf E} \cdot d{\bf a} = \int |{\bf E}| da = |{\bf E}| \int da = |{\bf E}| \int_0^l dz \int_0^{2\pi} d\phi s = |{\bf E}| ~2\pi l s.
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\oint {\bf E} \cdot d{\bf a} = \int |{\bf E}| da = |{\bf E}| \int da = |{\bf E}| \int_0^l dz \int_0^{2\pi} d\varphi s = |{\bf E}| ~2\pi l s.
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\]
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</p>
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@@ -1833,7 +1833,7 @@ Therefore,
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</div>
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<div class="example div" id="org751fcc2">
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<div class="example div" id="org0f3c966">
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<p>
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<b>Example 2.4</b>: infinite plane (defined by \(z = 0\)) with uniform surface charge density \(\sigma\). Find \({\bf E}\).
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</p>
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@@ -1859,7 +1859,7 @@ where \(\hat{\bf n}\) is a unit vector extending away from the plane. Independe
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</div>
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<div class="example div" id="org1659346">
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<div class="example div" id="org720e20e">
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<p>
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<b>Example 2.5</b>: two infinite planes (put them vertical) carrying equal but opposite uniform surface charge densities \(\pm \sigma\).
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</p>
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@@ -1889,7 +1889,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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</div>
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<div id="postamble" class="status">
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<p class="author">Author: Jean-Sébastien Caux</p>
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<p class="date">Created: 2022-02-13 Sun 21:20</p>
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<p class="date">Created: 2022-02-14 Mon 20:35</p>
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<p class="validation"></p>
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</div>
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