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Update 2022-03-02 15:47

master
Jean-Sébastien 2 years ago
parent
ac1e628013
commit
21bf9fdba5
100 changed files with 1085 additions and 648 deletions
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 </div>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 The usual Cartesian coordinates relate to spherical coordinates
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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"/>
1701 1701 
 </svg></a>
1702 1702 
 </p>
1703 
-<div class="alteqlabels" id="org31d976f">
1703 
+<div class="alteqlabels" id="org96ce6ff">
1704 1704
1705 1705 
 </div>
1706 1706 
 
@@ -1716,14 +1716,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
1716 1716 
 <p>
1717 1717 
 Infinitesimal volume element:
1718 1718 
 </p>
1719 
-<div class="eqlabel" id="org824a5cb">
1719 
+<div class="eqlabel" id="org40e244c">
1720 1720 
 <p>
1721 1721 
 <a id="sph_dtau"></a><a href="./c_m_cs_sph.html#sph_dtau"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1722 1722 
   <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"/>
1723 1723 
   <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"/>
1724 1724 
 </svg></a>
1725 1725 
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1726 
-<div class="alteqlabels" id="org9e40179">
1726 
+<div class="alteqlabels" id="org6796383">
1727 1727
1728 1728 
 </div>
1729 1729 
 
@@ -1744,14 +1744,14 @@ Infinitesimal surface element:  depends on situation.
1744 1744 
 <div id="outline-container-c_m_cs_sph_grad" class="outline-6">
1745 1745 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
1746 1746 
 <div class="outline-text-6" id="text-c_m_cs_sph_grad">
1747 
-<div class="eqlabel" id="orgd062ca2">
1747 
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1748 1748 
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1749 1749 
 <a id="sph_grad"></a><a href="./c_m_cs_sph.html#sph_grad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1750 1750 
   <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"/>
1751 1751 
   <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"/>
1752 1752 
 </svg></a>
1753 1753 
 </p>
1754 
-<div class="alteqlabels" id="org776b74f">
1754 
+<div class="alteqlabels" id="org94fd9f5">
1755 1755
1756 1756 
 </div>
1757 1757 
 
@@ -1768,14 +1768,14 @@ Infinitesimal surface element:  depends on situation.
1768 1768 
 <div id="outline-container-c_m_cs_sph_div" class="outline-6">
1769 1769 
 <h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
1770 1770 
 <div class="outline-text-6" id="text-c_m_cs_sph_div">
1771 
-<div class="eqlabel" id="orgbff4022">
1771 
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1772 1772 
 <p>
1773 1773 
 <a id="sph_div"></a><a href="./c_m_cs_sph.html#sph_div"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1774 1774 
   <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"/>
1775 1775 
   <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"/>
1776 1776 
 </svg></a>
1777 1777 
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1778 
-<div class="alteqlabels" id="orgd13f0a2">
1778 
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1779 1779
1780 1780 
 </div>
1781 1781 
 
@@ -1792,14 +1792,14 @@ Infinitesimal surface element:  depends on situation.
1792 1792 
 <div id="outline-container-c_m_cs_sph_curl" class="outline-6">
1793 1793 
 <h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
1794 1794 
 <div class="outline-text-6" id="text-c_m_cs_sph_curl">
1795 
-<div class="eqlabel" id="org4bf1642">
1795 
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1796 1796 
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1797 1797 
 <a id="sph_curl"></a><a href="./c_m_cs_sph.html#sph_curl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1798 1798 
   <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"/>
1799 1799 
   <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"/>
1800 1800 
 </svg></a>
1801 1801 
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1802 
-<div class="alteqlabels" id="orge688a5a">
1802 
+<div class="alteqlabels" id="orge37528b">
1803 1803
1804 1804 
 </div>
1805 1805 
 
@@ -1818,14 +1818,14 @@ Infinitesimal surface element:  depends on situation.
1818 1818 
 <div id="outline-container-c_m_cs_sph_lap" class="outline-6">
1819 1819 
 <h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
1820 1820 
 <div class="outline-text-6" id="text-c_m_cs_sph_lap">
1821 
-<div class="eqlabel" id="org66b8a68">
1821 
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1822 1822 
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1823 1823 
 <a id="sph_Lap"></a><a href="./c_m_cs_sph.html#sph_Lap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1824 1824 
   <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"/>
1825 1825 
   <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"/>
1826 1826 
 </svg></a>
1827 1827 
 </p>
1828 
-<div class="alteqlabels" id="org25a1307">
1828 
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1829 1829
1830 1830 
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1831 1831 
 
@@ -1859,7 +1859,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1859 1859 
 </div>
1860 1860 
 <div id="postamble" class="status">
1861 1861 
 <p class="author">Author: Jean-Sébastien Caux</p>
1862 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1862 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1863 1863 
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1864 1864 
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1865 1865

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@@ -1,7 +1,7 @@
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4 
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6 6 
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7 7 
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@@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1650 1650 
 </div>
1651 1651 
 <div id="postamble" class="status">
1652 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1653 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1653 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1654 1654 
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1655 1655 
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1656 1656

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2 2 
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3 3 
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7 7 
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@@ -1652,7 +1652,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1652 1652 
 </div>
1653 1653 
 <div id="postamble" class="status">
1654 1654 
 <p class="author">Author: Jean-Sébastien Caux</p>
1655 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1655 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1656 1656 
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1657 1657 
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1658 1658

+ 19
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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1623,9 +1623,9 @@ Table of contents
1623 1623 
 <div class="outline-text-5" id="text-c_m_dc_d2">
1624 1624 
 </div>
1625 1625
1626 
-<div id="outline-container-org27b21d4" class="outline-6">
1627 
-<h6 id="org27b21d4"><a href="#org27b21d4">Divergence of gradient</a></h6>
1628 
-<div class="outline-text-6" id="text-org27b21d4">
1626 
+<div id="outline-container-org3814a3d" class="outline-6">
1627 
+<h6 id="org3814a3d"><a href="#org3814a3d">Divergence of gradient</a></h6>
1628 
+<div class="outline-text-6" id="text-org3814a3d">
1629 1629 
 <p>
1630 1630 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1631 1631 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1634,44 +1634,44 @@ given by the Laplacian of the corresponding vector elements.
1634 1634 
 </div>
1635 1635 
 </div>
1636 1636
1637 
-<div id="outline-container-org8ab79ed" class="outline-6">
1638 
-<h6 id="org8ab79ed"><a href="#org8ab79ed">Curl of a gradient</a></h6>
1639 
-<div class="outline-text-6" id="text-org8ab79ed">
1637 
+<div id="outline-container-orgc8aa2ed" class="outline-6">
1638 
+<h6 id="orgc8aa2ed"><a href="#orgc8aa2ed">Curl of a gradient</a></h6>
1639 
+<div class="outline-text-6" id="text-orgc8aa2ed">
1640 1640 
 <p>
1641 1641 
 This always vanishes.
1642 1642 
 </p>
1643 1643 
 </div>
1644 1644 
 </div>
1645 1645
1646 
-<div id="outline-container-org3af2976" class="outline-6">
1647 
-<h6 id="org3af2976"><a href="#org3af2976">Gradient of the divergence</a></h6>
1648 
-<div class="outline-text-6" id="text-org3af2976">
1646 
+<div id="outline-container-orge3b0dcb" class="outline-6">
1647 
+<h6 id="orge3b0dcb"><a href="#orge3b0dcb">Gradient of the divergence</a></h6>
1648 
+<div class="outline-text-6" id="text-orge3b0dcb">
1649 1649 
 <p>
1650 1650 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1651 1651 
 </p>
1652 1652 
 </div>
1653 1653 
 </div>
1654 1654
1655 
-<div id="outline-container-orgb2afb29" class="outline-6">
1656 
-<h6 id="orgb2afb29"><a href="#orgb2afb29">Divergence of a curl</a></h6>
1657 
-<div class="outline-text-6" id="text-orgb2afb29">
1655 
+<div id="outline-container-org1f5474a" class="outline-6">
1656 
+<h6 id="org1f5474a"><a href="#org1f5474a">Divergence of a curl</a></h6>
1657 
+<div class="outline-text-6" id="text-org1f5474a">
1658 1658 
 <p>
1659 1659 
 This always vanishes.
1660 1660 
 </p>
1661 1661 
 </div>
1662 1662 
 </div>
1663 1663
1664 
-<div id="outline-container-orgd5bfcaa" class="outline-6">
1665 
-<h6 id="orgd5bfcaa"><a href="#orgd5bfcaa">Curl of curl</a></h6>
1666 
-<div class="outline-text-6" id="text-orgd5bfcaa">
1667 
-<div class="eqlabel" id="org4a8e6af">
1664 
+<div id="outline-container-org4b8cb79" class="outline-6">
1665 
+<h6 id="org4b8cb79"><a href="#org4b8cb79">Curl of curl</a></h6>
1666 
+<div class="outline-text-6" id="text-org4b8cb79">
1667 
+<div class="eqlabel" id="orgad52ac5">
1668 1668 
 <p>
1669 1669 
 <a id="curlcurl"></a><a href="./c_m_dc_d2.html#curlcurl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1670 1670 
   <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"/>
1671 1671 
   <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"/>
1672 1672 
 </svg></a>
1673 1673 
 </p>
1674 
-<div class="alteqlabels" id="org8cfc576">
1674 
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1675 1675
1676 1676 
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1677 1677 
 
@@ -1702,7 +1702,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1702 1702 
 </div>
1703 1703 
 <div id="postamble" class="status">
1704 1704 
 <p class="author">Author: Jean-Sébastien Caux</p>
1705 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1705 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1706 1706 
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1707 1707 
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1708 1708

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7 7 
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@@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 1651 
 </div>
1652 1652 
 <div id="postamble" class="status">
1653 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1654 
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1655 1655 
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1656 1656 
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1657 1657

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@@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1648 
 </div>
1649 1649 
 <div id="postamble" class="status">
1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1651 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 1652 
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1653 1653 
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1654 1654

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7 7 
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@@ -1672,7 +1672,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1672 1672 
 </div>
1673 1673 
 <div id="postamble" class="status">
1674 1674 
 <p class="author">Author: Jean-Sébastien Caux</p>
1675 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1675 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1676 1676 
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1677 1677 
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1678 1678

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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1630,14 +1630,14 @@ explicited as follows:
1630 1630 
 <p>
1631 1631 
 <b>Gradient of a product</b>:
1632 1632 
 </p>
1633 
-<div class="eqlabel" id="org7ccb680">
1633 
+<div class="eqlabel" id="org4bcb61a">
1634 1634 
 <p>
1635 1635 
 <a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1636 1636 
   <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"/>
1637 1637 
   <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"/>
1638 1638 
 </svg></a>
1639 1639 
 </p>
1640 
-<div class="alteqlabels" id="orga7c6aac">
1640 
+<div class="alteqlabels" id="org8c15975">
1641 1641 
 <ul class="org-ul">
1642 1642 
 <li>Gr (3)</li>
1643 1643 
 <li>W (1-111)</li>
 
@@ -1657,14 +1657,14 @@ explicited as follows:
1657 1657 
 <p>
1658 1658 
 <b>Gradient of a scalar product</b>:
1659 1659 
 </p>
1660 
-<div class="eqlabel" id="orgbe6a82f">
1660 
+<div class="eqlabel" id="org7784802">
1661 1661 
 <p>
1662 1662 
 <a id="grad_sprod"></a><a href="./c_m_dc_pr.html#grad_sprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1663 1663 
   <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"/>
1664 1664 
   <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"/>
1665 1665 
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1666 1666 
 </p>
1667 
-<div class="alteqlabels" id="orgc2565bd">
1667 
+<div class="alteqlabels" id="orgb3682de">
1668 1668 
 <ul class="org-ul">
1669 1669 
 <li>Gr (4)</li>
1670 1670 
 <li>W (1-112)</li>
 
@@ -1684,14 +1684,14 @@ explicited as follows:
1684 1684 
 <p>
1685 1685 
 <b>Divergence of a product</b>:
1686 1686 
 </p>
1687 
-<div class="eqlabel" id="org2591737">
1687 
+<div class="eqlabel" id="org96b1b94">
1688 1688 
 <p>
1689 1689 
 <a id="div_prod"></a><a href="./c_m_dc_pr.html#div_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1690 1690 
   <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"/>
1691 1691 
   <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"/>
1692 1692 
 </svg></a>
1693 1693 
 </p>
1694 
-<div class="alteqlabels" id="orge3c89ac">
1694 
+<div class="alteqlabels" id="org29d8167">
1695 1695 
 <ul class="org-ul">
1696 1696 
 <li>Gr (5)</li>
1697 1697 
 <li>W (1-115)</li>
 
@@ -1711,14 +1711,14 @@ explicited as follows:
1711 1711 
 <p>
1712 1712 
 <b>Divergence of a cross product</b>:
1713 1713 
 </p>
1714 
-<div class="eqlabel" id="org37cc95b">
1714 
+<div class="eqlabel" id="orgfe2b08f">
1715 1715 
 <p>
1716 1716 
 <a id="div_xprod"></a><a href="./c_m_dc_pr.html#div_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1717 1717 
   <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"/>
1718 1718 
   <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"/>
1719 1719 
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1720 1720 
 </p>
1721 
-<div class="alteqlabels" id="org0277164">
1721 
+<div class="alteqlabels" id="orgdce994d">
1722 1722 
 <ul class="org-ul">
1723 1723 
 <li>Gr (6)</li>
1724 1724 
 <li>W (1-116)</li>
 
@@ -1738,14 +1738,14 @@ explicited as follows:
1738 1738 
 <p>
1739 1739 
 <b>Curl of a product</b>:
1740 1740 
 </p>
1741 
-<div class="eqlabel" id="org980570b">
1741 
+<div class="eqlabel" id="orgbb8798b">
1742 1742 
 <p>
1743 1743 
 <a id="curl_prod"></a><a href="./c_m_dc_pr.html#curl_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1744 1744 
   <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"/>
1745 1745 
   <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"/>
1746 1746 
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1747 1747 
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1748 
-<div class="alteqlabels" id="org92e3c88">
1748 
+<div class="alteqlabels" id="org93d7ea5">
1749 1749 
 <ul class="org-ul">
1750 1750 
 <li>Gr (7)</li>
1751 1751 
 <li>W (1-118)</li>
 
@@ -1765,14 +1765,14 @@ explicited as follows:
1765 1765 
 <p>
1766 1766 
 <b>Curl of a cross product</b>:
1767 1767 
 </p>
1768 
-<div class="eqlabel" id="orga4932f1">
1768 
+<div class="eqlabel" id="orgd9fde5c">
1769 1769 
 <p>
1770 1770 
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1771 1771 
   <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"/>
1772 1772 
   <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"/>
1773 1773 
 </svg></a>
1774 1774 
 </p>
1775 
-<div class="alteqlabels" id="org93712f0">
1775 
+<div class="alteqlabels" id="orgb1d2b5b">
1776 1776 
 <ul class="org-ul">
1777 1777 
 <li>Gr (8)</li>
1778 1778 
 <li>W (1-119)</li>
 
@@ -1813,7 +1813,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1813 1813 
 </div>
1814 1814 
 <div id="postamble" class="status">
1815 1815 
 <p class="author">Author: Jean-Sébastien Caux</p>
1816 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1816 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1817 1817 
 <p class="validation"></p>
1818 1818 
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1819 1819

+ 2
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build/c_m_dd.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
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@@ -1647,7 +1647,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 1647 
 </div>
1648 1648 
 <div id="postamble" class="status">
1649 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1650 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1651 1651 
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1652 1652 
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1653 1653

+ 2
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build/c_m_dd_1d.html View File 

@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
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@@ -1659,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 1659 
 </div>
1660 1660 
 <div id="postamble" class="status">
1661 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1662 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1662 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1663 1663 
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1664 1664 
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1665 1665

+ 8
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build/c_m_dd_3d.html View File 

@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1644,14 +1644,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1644 1644 
 More generally,
1645 1645 
 </p>
1646 1646
1647 
-<div class="eqlabel" id="org21f3d7e">
1647 
+<div class="eqlabel" id="orgf5d25ef">
1648 1648 
 <p>
1649 1649 
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1650 1650 
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1651 1651 
   <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"/>
1652 1652 
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1653 1653 
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1654 
-<div class="alteqlabels" id="orgeaacc49">
1654 
+<div class="alteqlabels" id="org6fba030">
1655 1655 
 <ul class="org-ul">
1656 1656 
 <li>Gr (1.100)</li>
1657 1657 
 </ul>
 
@@ -1670,14 +1670,14 @@ More generally,
1670 1670 
 Since
1671 1671 
 </p>
1672 1672
1673 
-<div class="eqlabel" id="org2f9beaa">
1673 
+<div class="eqlabel" id="org9588ee0">
1674 1674 
 <p>
1675 1675 
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1676 1676 
   <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"/>
1677 1677 
   <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"/>
1678 1678 
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1679 1679 
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1680 
-<div class="alteqlabels" id="org86371cb">
1680 
+<div class="alteqlabels" id="orgb0b4434">
1681 1681 
 <ul class="org-ul">
1682 1682 
 <li>Gr (1.101)</li>
1683 1683 
 </ul>
 
@@ -1693,14 +1693,14 @@ Since
1693 1693 
 <p>
1694 1694 
 we have that
1695 1695 
 </p>
1696 
-<div class="eqlabel" id="org77d9fa8">
1696 
+<div class="eqlabel" id="orgd223c1e">
1697 1697 
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1698 1698 
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1699 1699 
   <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"/>
1700 1700 
   <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"/>
1701 1701 
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1702 1702 
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1703 
-<div class="alteqlabels" id="org9d70b5c">
1703 
+<div class="alteqlabels" id="org2ae8bc7">
1704 1704 
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1705 1705 
 <li>Gr (1.102)</li>
1706 1706 
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@@ -1732,7 +1732,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1732 1732 
 </div>
1733 1733 
 <div id="postamble" class="status">
1734 1734 
 <p class="author">Author: Jean-Sébastien Caux</p>
1735 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1735 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1736 1736 
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1737 1737 
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1738 1738

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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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6 6 
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7 7 
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@@ -1665,7 +1665,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1665 1665 
 </div>
1666 1666 
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1667 1667 
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1668 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1668 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1669 1669 
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1670 1670 
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1671 1671

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1 1 
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3 3 
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4 
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7 7 
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@@ -1650,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1650 1650 
 </div>
1651 1651 
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1652 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1653 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1653 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1654 1654 
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1655 1655 
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1656 1656

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7 7 
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@@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1648 
 </div>
1649 1649 
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1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1651 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 1652 
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1653 1653 
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1654 1654

+ 2
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7 7 
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@@ -1657,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1657 1657 
 </div>
1658 1658 
 <div id="postamble" class="status">
1659 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1660 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1661 1661 
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1662 1662 
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1663 1663

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7 7 
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@@ -1651,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 1651 
 </div>
1652 1652 
 <div id="postamble" class="status">
1653 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1654 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1655 1655 
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1656 1656 
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1657 1657

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build/c_m_ic_ip.html View File 

@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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7 7 
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@@ -1658,7 +1658,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1658 1658 
 </div>
1659 1659 
 <div id="postamble" class="status">
1660 1660 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1661 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1662 1662 
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1663 1663 
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1664 1664

+ 11
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@@ -1,7 +1,7 @@
1 1 
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3 3 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1623,9 +1623,9 @@ Table of contents
1623 1623 
 <div class="outline-text-5" id="text-c_m_ic_lsv">
1624 1624 
 </div>
1625 1625
1626 
-<div id="outline-container-org0446f34" class="outline-6">
1627 
-<h6 id="org0446f34"><a href="#org0446f34">Line Integrals</a></h6>
1628 
-<div class="outline-text-6" id="text-org0446f34">
1626 
+<div id="outline-container-org2486563" class="outline-6">
1627 
+<h6 id="org2486563"><a href="#org2486563">Line Integrals</a></h6>
1628 
+<div class="outline-text-6" id="text-org2486563">
1629 1629 
 <p>
1630 1630 
 \[
1631 1631 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1654,9 +1654,9 @@ Integral over a closed loop:
1654 1654 
 </div>
1655 1655 
 </div>
1656 1656
1657 
-<div id="outline-container-org0cc3862" class="outline-6">
1658 
-<h6 id="org0cc3862"><a href="#org0cc3862">Surface Integrals</a></h6>
1659 
-<div class="outline-text-6" id="text-org0cc3862">
1657 
+<div id="outline-container-orgf44a1b8" class="outline-6">
1658 
+<h6 id="orgf44a1b8"><a href="#orgf44a1b8">Surface Integrals</a></h6>
1659 
+<div class="outline-text-6" id="text-orgf44a1b8">
1660 1660 
 <p>
1661 1661 
 \[
1662 1662 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1676,9 +1676,9 @@ Over a closed surface:
1676 1676 
 </div>
1677 1677 
 </div>
1678 1678
1679 
-<div id="outline-container-orga0f7b3b" class="outline-6">
1680 
-<h6 id="orga0f7b3b"><a href="#orga0f7b3b">Volume Integrals</a></h6>
1681 
-<div class="outline-text-6" id="text-orga0f7b3b">
1679 
+<div id="outline-container-org7c26dcf" class="outline-6">
1680 
+<h6 id="org7c26dcf"><a href="#org7c26dcf">Volume Integrals</a></h6>
1681 
+<div class="outline-text-6" id="text-org7c26dcf">
1682 1682 
 <p>
1683 1683 
 \[
1684 1684 
 \int_{\cal V} T d\tau
 
@@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1719 
 </div>
1720 1720 
 <div id="postamble" class="status">
1721 1721 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1722 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1723 1723 
 <p class="validation"></p>
1724 1724 
 </div>
1725 1725

+ 4
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6 6 
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7 7 
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@@ -1621,14 +1621,14 @@ Table of contents
1621 1621 
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@@ -1667,7 +1667,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1667 1667 
 </div>
1668 1668 
 <div id="postamble" class="status">
1669 1669 
 <p class="author">Author: Jean-Sébastien Caux</p>
1670 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1670 
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1671 1671 
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1672 1672 
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1665 1665 
 </div>
1666 1666 
 <div id="postamble" class="status">
1667 1667 
 <p class="author">Author: Jean-Sébastien Caux</p>
1668 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1668 
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1669 1669 
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1675 1675 
 </div>
1676 1676 
 <div id="postamble" class="status">
1677 1677 
 <p class="author">Author: Jean-Sébastien Caux</p>
1678 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1678 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1679 1679 
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1680 1680 
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1669 1669 
 </div>
1670 1670 
 <div id="postamble" class="status">
1671 1671 
 <p class="author">Author: Jean-Sébastien Caux</p>
1672 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1672 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1673 1673 
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1675 1675

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7 7 
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@@ -1622,8 +1622,8 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd</span></h2>
1623 1623
1624 1624 
 <div class="outline-text-2" id="text-emd">
1625 
-<details class="prereq" id="org818864e">
1626 
-<summary id="org9e6a46d">
1625 
+<details class="prereq" id="org2aba3cc">
1626 
+<summary id="org30ca6a6">
1627 1627 
 Prerequisites
1628 1628 
 </summary>
1629 1629 
 <ul class="org-ul">
 
@@ -1632,8 +1632,8 @@ Prerequisites
1632 1632 
 </ul>
1633 1633 
 </details>
1634 1634
1635 
-<details class="objectives" id="org52c0d63">
1636 
-<summary id="org4e1bb09">
1635 
+<details class="objectives" id="org98af4b0">
1636 
+<summary id="org78e61da">
1637 1637 
 Objectives
1638 1638 
 </summary>
1639 1639 
 <ul class="org-ul">
 
@@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1674 
 </div>
1675 1675 
 <div id="postamble" class="status">
1676 1676 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1677 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1678 1678 
 <p class="validation"></p>
1679 1679 
 </div>
1680 1680

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1648 
 </div>
1649 1649 
 <div id="postamble" class="status">
1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1651 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 1652 
 <p class="validation"></p>
1653 1653 
 </div>
1654 1654

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1 1 
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2 2 
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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1622,15 +1622,11 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd.Fl.Fl</span></h4>
1623 1623 
 <div class="outline-text-4" id="text-emd_Fl_Fl">
1624 1624 
 <p>
1625 
-1831: 3 experiments by Faraday (according to Griffiths! but it's historically incorrect)
1626 
-\paragraph{1)} Pull a loop of wire through a magnetic field.
1627 
-\paragraph{2)} Move magnet around a still loop.
1628 
-\paragraph{3)} Change strength of field, holding magnet and loop still.
1625 
+Around 1831, Faraday performed a number of experiments pertaining to
1626 
+the effects of time-dependent fields.
1629 1627 
 </p>
1630 1628
1631 
-
1632 1629 
 <p>
1633 
-Actually, historically, things didn't happen like that.
1634 1630 
 The first experiment that Faraday performed (1831) involved two metal coils wound
1635 1631 
 on opposite sides of a metal ring. When a current was turned on through the first
1636 1632 
 coil, it generated a transient current in the second coil (as measured by a
 
@@ -1647,27 +1643,39 @@ on this idea. Faraday observed transient current in a circuit when:
1647 1643
1648 1644 
 <p>
1649 1645 
 Faraday's big insight was to summarize these effects by noticing that
1650 
-</p>
1651 
-
1652 
-<p>
1653 1646 
 \[
1654 
-  \boxed{
1655 
-    \mbox{\bf A changing magnetic field induces an electric field}
1656 
-  }
1647 
+\boxed{
1648 
+\mbox{A changing magnetic field induces an electric field.}
1649 
+}
1657 1650 
 \]
1658 
-</p>
1659 
-
1660 
-<p>
1661 1651 
 Empirically:  the changing magnetic field induces an electric current around
1662 1652 
 the circuit. This current is really driven by an electric field having a component
1663 1653 
 along the wire.  The line integral of this field is called the
1664 1654 
 </p>
1665 
-<div class="core div" id="org03c55ba">
1655 
+<div class="core div" id="orgdfb0aad">
1666 1656 
 <p>
1667 1657 
 <b>Electromotive force (or electromotance)</b>,
1668 
-  \[
1669 
-    {\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}.
1670 
-  \]
1658 
+</p>
1659 
+<div class="eqlabel" id="org64afdaf">
1660 
+<p>
1661 
+<a id="elmofo"></a><a href="./emd_Fl_Fl.html#elmofo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1662 
+  <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"/>
1663 
+  <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"/>
1664 
+</svg></a>
1665 
+</p>
1666 
+<div class="alteqlabels" id="orgb3cda90">
1667 
+<ul class="org-ul">
1668 
+<li>Gr (7.9)</li>
1669 
+</ul>
1670 
+
1671 
+</div>
1672 
+
1673 
+</div>
1674 
+<p>
1675 
+\[
1676 
+{\cal E} \equiv \oint_{\cal P} {\bf E} \cdot d{\bf l}.
1677 
+\tag{elmofo}\label{elmofo}
1678 
+\]
1671 1679 
 </p>
1672 1680
1673 1681 
 </div>
 
@@ -1676,21 +1684,56 @@ You can think of the emf in different ways. It's the energy accumulated as a uni
1676 1684 
 </p>
1677 1685
1678 1686 
 <p>
1679 
-The precise statement is that the electromotive force is proportional
1687 
+The precise statement associated to Faraday's observations
1688 
+is that the electromotive force is proportional
1680 1689 
 to the rate of change of the magnetic flux,
1690 
+</p>
1691 
+<div class="eqlabel" id="orgec7b520">
1692 
+<p>
1693 
+<a id="Fl_flux"></a><a href="./emd_Fl_Fl.html#Fl_flux"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1694 
+  <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"/>
1695 
+  <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"/>
1696 
+</svg></a>
1697 
+</p>
1698 
+<div class="alteqlabels" id="orgf70f495">
1699 
+<ul class="org-ul">
1700 
+<li>Gr (7.14)</li>
1701 
+</ul>
1702 
+
1703 
+</div>
1704 
+
1705 
+</div>
1706 
+<p>
1681 1707 
 \[
1682 1708 
 {\cal E} = \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt}
1683 
-\label{Gr(7.14)}
1709 
+\tag{Fl_flux}\label{Fl_flux}
1684 1710 
 \]
1685 1711 
 so we obtain
1686 1712 
 </p>
1687 
-<div class="core div" id="org93f9990">
1713 
+<div class="core div" id="orgdfedc05">
1688 1714 
 <p>
1689 1715 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1690 
-  \[
1691 
-    \oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a}
1692 
-    \label{Gr(7.15)}
1693 
-  \]
1716 
+</p>
1717 
+<div class="eqlabel" id="orge87df83">
1718 
+<p>
1719 
+<a id="Fl_int"></a><a href="./emd_Fl_Fl.html#Fl_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1720 
+  <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"/>
1721 
+  <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"/>
1722 
+</svg></a>
1723 
+</p>
1724 
+<div class="alteqlabels" id="org7b32fe7">
1725 
+<ul class="org-ul">
1726 
+<li>Gr (7.15)</li>
1727 
+</ul>
1728 
+
1729 
+</div>
1730 
+
1731 
+</div>
1732 
+<p>
1733 
+\[
1734 
+\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\int_{\cal S} \frac{\partial {\bf B}}{\partial t} \cdot d{\bf a}
1735 
+\tag{Fl_int}\label{Fl_int}
1736 
+\]
1694 1737 
 </p>
1695 1738
1696 1739 
 </div>
 
@@ -1702,20 +1745,36 @@ for any loop (on a wire or not). Using Stokes' theorem,
1702 1745 
 \]
1703 1746 
 we obtain
1704 1747 
 </p>
1705 
-<div class="core div" id="org6046a76">
1748 
+<div class="core div" id="orgafa0d15">
1749 
+<div class="eqlabel" id="org13d3c14">
1750 
+<p>
1751 
+<a id="Fl"></a><a href="./emd_Fl_Fl.html#Fl"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1752 
+  <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"/>
1753 
+  <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"/>
1754 
+</svg></a>
1755 
+</p>
1756 
+<div class="alteqlabels" id="orgd662a28">
1757 
+<ul class="org-ul">
1758 
+<li>Gr (7.16)</li>
1759 
+</ul>
1760 
+
1761 
+</div>
1762 
+
1763 
+</div>
1706 1764 
 <p>
1707 1765 
 <b>Faraday's law</b> (differential form)
1708 
-  \[
1709 
-    {\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
1710 
-    \label{Gr(7.16)}
1711 
-  \]
1766 
+\[
1767 
+{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
1768 
+\tag{Fl}\label{Fl}
1769 
+\]
1712 1770 
 </p>
1713 1771
1714 1772 
 </div>
1715 1773 
 <p>
1716 
-Right-hand rule always sorts signs out.  Easier rule:  {\bf Lenz's law}, which
1717 
-states that {\bf nature resists a change in flux}.  This is in fact just
1718 
-{\bf Le Ch\^atelier's principle} of any action at an equilibrium point leading
1774 
+Right-hand rule always sorts signs out.  Easier rule:  <b>Lenz's law</b>, which
1775 
+states that physical systems naturally resist a change in flux.
1776 
+This is in fact just
1777 
+<b>Le Châtelier's principle</b> of any action at an equilibrium point leading
1719 1778 
 to an opposing counter-reaction.
1720 1779 
 </p>
1721 1780 
 </div>
 
@@ -1739,7 +1798,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1739 1798 
 </div>
1740 1799 
 <div id="postamble" class="status">
1741 1800 
 <p class="author">Author: Jean-Sébastien Caux</p>
1742 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1801 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1743 1802 
 <p class="validation"></p>
1744 1803 
 </div>
1745 1804

+ 56
- 38
build/emd_Fl_e.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1631,7 +1631,7 @@ Start from zero current, integrate in time:
1631 1631 
 W = \frac{1}{2} L I^2
1632 1632 
 \label{Gr(7.29)}
1633 1633 
 \]
1634 
-Nicer way (generalizable to surface and volume currents):  from (\ref{Gr(7.25)}), flux through loop is \(\Phi = L I\).  But
1634 
+Nicer way (generalizable to surface and volume currents):  from <a href="./emd_Fl_i.html#PLI">PLI</a>, flux through loop is \(\Phi = L I\).  But
1635 1635 
 \[
1636 1636 
 \Phi = \int_{\cal S} {\bf B} \cdot d{\bf a} = \int_{\cal S} ({\boldsymbol \nabla} \times {\bf A}) \cdot d{\bf a}
1637 1637 
 = \oint_{\cal P} {\bf A} \cdot d{\bf l},
 
@@ -1647,16 +1647,16 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
1647 1647 
 \]
1648 1648 
 Generalization to volume currents:
1649 1649 
 </p>
1650 
-<div class="eqlabel" id="org9fbfb9d">
1650 
+<div class="eqlabel" id="org87af1ba">
1651 1651 
 <p>
1652 1652 
 <a id="W_intAJ"></a><a href="./emd_Fl_e.html#W_intAJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1653 1653 
   <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"/>
1654 1654 
   <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"/>
1655 1655 
 </svg></a>
1656 1656 
 </p>
1657 
-<div class="alteqlabels" id="org6b3d2a5">
1657 
+<div class="alteqlabels" id="org95d3f8b">
1658 1658 
 <ul class="org-ul">
1659 
-<li>gr (7.31)</li>
1659 
+<li>Gr (7.31)</li>
1660 1660 
 </ul>
1661 1661
1662 1662 
 </div>
 
@@ -1664,31 +1664,22 @@ Generalization to volume currents:
1664 1664 
 </div>
1665 1665 
 <p>
1666 1666 
 \[
1667 
-W = \frac{1}{2} \int_{\cal V} ({\bf A} \cdot {\bf J}) d\tau
1667 
+W = \frac{1}{2} \int_{\cal V} d\tau ~({\bf A} \cdot {\bf J})
1668 1668 
 \tag{W_intAJ}\label{W_intAJ}
1669 1669 
 \]
1670 1670 
 Even better:  use Ampère, \({\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}\):
1671 1671 
 \[
1672 
-W = \frac{1}{2\mu_0} \int_{\cal V} {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) d\tau
1672 
+W = \frac{1}{2\mu_0} \int_{\cal V} d\tau ~{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B})
1673 1673 
 \label{Gr(7.32)}
1674 1674 
 \]
1675 1675 
 Integrate by parts using product rule 6:
1676 1676 
 \[
1677 
-{\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf A})
1678 
-</p>
1679 
-<ul class="org-ul">
1680 
-<li>{\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}),</li>
1681 
-</ul>
1682 
-<p>
1677 
+{\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf A}) - {\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}),
1683 1678 
 \]
1684 1679 
 so
1685 1680 
 \[
1686 
-{\bf A} ⋅ ({\boldsymbol ∇} × {\bf B}) = {\bf B} ⋅ {\bf B}
1687 
-</p>
1688 
-<ul class="org-ul">
1689 
-<li>{\boldsymbol ∇} ⋅ ({\bf A} × {\bf B}).</li>
1690 
-</ul>
1691 
-<p>
1681 
+{\bf A} \cdot ({\boldsymbol \nabla} \times {\bf B}) =
1682 
+{\bf B} \cdot {\bf B} - {\boldsymbol \nabla} \cdot ({\bf A} \times {\bf B}).
1692 1683 
 \]
1693 1684 
 Then,
1694 1685 
 \[
 
@@ -1698,12 +1689,27 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1698 1689 
 \]
1699 1690 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1700 1691 
 </p>
1701 
-<div class="core div" id="org2f4a453">
1692 
+<div class="core div" id="org247bd37">
1693 
+<div class="eqlabel" id="orgdf4c394">
1694 
+<p>
1695 
+<a id="W_intBsq"></a><a href="./emd_Fl_e.html#W_intBsq"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1696 
+  <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"/>
1697 
+  <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"/>
1698 
+</svg></a>
1699 
+</p>
1700 
+<div class="alteqlabels" id="orge069b86">
1701 
+<ul class="org-ul">
1702 
+<li>Gr (7.34)</li>
1703 
+</ul>
1704 
+
1705 
+</div>
1706 
+
1707 
+</div>
1702 1708 
 <p>
1703 1709 
 \[
1704 
-    W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
1705 
-    \label{Gr(7.34)}
1706 
-  \]
1710 
+W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
1711 
+\tag{W_intBsq}\label{W_intBsq}
1712 
+\]
1707 1713 
 </p>
1708 1714
1709 1715 
 </div>
 
@@ -1713,26 +1719,38 @@ We can integrate over all space:  after neglecting boundary terms (assuming fiel
1713 1719 
 Summary:  energy in electric and magnetic fields:
1714 1720 
 </p>
1715 1721 
 \begin{align}
1716 
-W_{elec} = \frac{1}{2} \int d\tau V\rho = \frac{\varepsilon_0}{2} \int d\tau E^2, \hspace{2cm}
1717 
-\mbox{(2.43 and 2.45)}, \\
1718 
-W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int d\tau B^2,
1719 
-\hspace{2cm} \mbox{(7.31 and 7.34)}
1722 
+W_{elec} &amp;= \frac{1}{2} \int d\tau ~V\rho &amp;= \frac{\varepsilon_0}{2} \int d\tau ~E^2, \\
1723 
+W_{mag} &amp;= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &amp;= \frac{1}{2\mu_0} \int d\tau ~B^2,
1720 1724 
 \end{align}
1725 
+<p>
1726 
+which are equations <a href="./ems_es_e.html#W_vcd">W_vcd</a>, <a href="./ems_es_e.html#W_intEsq">W_intEsq</a>, <a href="./emd_Fl_e.html#W_intAJ">W_intAJ</a> and <a href="./emd_Fl_e.html#W_intBsq">W_intBsq</a>.
1727 
+</p>
1728 
+
1729 
+<div class="example div" id="org03fb4e8">
1730 
+<p>
1731 
+<b>Example: energy in coaxial cable</b>
1732 
+</p>
1721 1733
1722 
-<div class="example div" id="orgeb4514a">
1723 1734 
 <p>
1724 
-\paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1725 
-Find energy stored in section of length \(l\).
1726 
-\paragraph{Solution:}  from Ampère,
1735 
+Consider a coaxial cable with inner cylinder radius \(a\), outer \(b\),
1736 
+carrying current \(I\).
1737 
+</p>
1738 
+
1739 
+<p>
1740 
+<b>Task</b>: find the energy stored in a section of length \(l\).
1741 
+</p>
1742 
+
1743 
+<p>
1744 
+<b>Solution</b>:  from Ampère,
1727 1745 
 \[
1728 
-    {\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a &lt; s &lt; b, \hspace{1cm}
1729 
-    {\bf B} = 0, \hspace{1cm} s &lt; a ~\mbox{or}~ s &gt; b.
1730 
-  \]
1746 
+{\bf B} = \frac{\mu_0 I}{2\pi s} \hat{\boldsymbol \varphi}, \hspace{1cm} a &lt; s &lt; b, \hspace{1cm}
1747 
+{\bf B} = 0, \hspace{1cm} s &lt; a ~\mbox{or}~ s &gt; b.
1748 
+\]
1731 1749 
 Energy is thus
1732 1750 
 \[
1733 
-    W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2
1734 
-    = \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}.
1735 
-  \]
1751 
+W_{mag} = \frac{1}{2\mu_0} \int_0^{2\pi} d\varphi \int_0^l dz \int_a^b s ds \left(\frac{\mu_0 I}{2\pi s}\right)^2
1752 
+= \frac{\mu_0 I^2 l}{4\pi} \ln \frac{b}{a}.
1753 
+\]
1736 1754 
 </p>
1737 1755
1738 1756 
 </div>
 
@@ -1758,7 +1776,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1758 1776 
 </div>
1759 1777 
 <div id="postamble" class="status">
1760 1778 
 <p class="author">Author: Jean-Sébastien Caux</p>
1761 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1779 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1762 1780 
 <p class="validation"></p>
1763 1781 
 </div>
1764 1782

+ 130
- 46
build/emd_Fl_i.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1629,7 +1629,7 @@ is (using fact that \({\bf B}_1\) is proportional to \(I_1\))
1629 1629 
 \Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 \Longrightarrow
1630 1630 
 \Phi_2 = M_{21} I_1
1631 1631 
 \]
1632 
-where \(M_{21}\) is the {\bf mutual inductance} of the two loops.
1632 
+where \(M_{21}\) is the <b>mutual inductance</b> of the two loops.
1633 1633 
 </p>
1634 1634
1635 1635 
 <p>
 
@@ -1638,7 +1638,7 @@ Useful formula:
1638 1638 
 \Phi_2 = \int {\bf B}_1 \cdot d{\bf a}_2 = \int ({\boldsymbol \nabla} \times {\bf A}_1) \cdot d{\bf a}_2
1639 1639 
 = \oint {\bf A}_1 \cdot d{\bf l}_2
1640 1640 
 \]
1641 
-But from (\ref{Gr(5.63)}),
1641 
+But from <a href="./ems_ms_vp_A.html#A_CoulG">A_CoulG</a>,
1642 1642 
 \[
1643 1643 
 {\bf A}_1 ({\bf r}) = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_1} \frac{d{\bf l}_1}{|{\bf r} - {\bf r}_1|}
1644 1644 
 \]
 
@@ -1647,44 +1647,89 @@ so
1647 1647 
 \Phi_2 = \frac{\mu_0 I_1}{4\pi} \oint_{{\cal P}_2} d{\bf l}_2 \cdot
1648 1648 
 \left(\oint_{{\cal P}_1} \frac{d{\bf l}_1 }{|{\bf r}_2 - {\bf r}_1|}\right)
1649 1649 
 \]
1650 
-and we can write the mutual inductance as the {\bf Neumann formula},
1650 
+and we can write the mutual inductance as the <b>Neumann formula</b>,
1651 
+</p>
1652 
+<div class="eqlabel" id="org197212f">
1653 
+<p>
1654 
+<a id="Newmann_M"></a><a href="./emd_Fl_i.html#Newmann_M"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1655 
+  <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"/>
1656 
+  <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"/>
1657 
+</svg></a>
1658 
+</p>
1659 
+<div class="alteqlabels" id="org4709e6a">
1660 
+<ul class="org-ul">
1661 
+<li>Gr (7.22)</li>
1662 
+</ul>
1663 
+
1664 
+</div>
1665 
+
1666 
+</div>
1667 
+<p>
1651 1668 
 \[
1652 1669 
 M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}_1 \cdot d{\bf l}_2}
1653 1670 
 {|{\bf r}_1 - {\bf r}_2|}
1654 
-\label{Gr(7.22)}
1671 
+\tag{Neumann_M}\label{Neumann_M}
1655 1672 
 \]
1656 1673 
 Two things:
1657 1674 
 first, \(M_{21}\) is purely geometrical.  Second,
1675 
+</p>
1676 
+<div class="eqlabel" id="orgb952a44">
1677 
+<p>
1678 
+<a id="Msym"></a><a href="./emd_Fl_i.html#Msym"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1679 
+  <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"/>
1680 
+  <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"/>
1681 
+</svg></a>
1682 
+</p>
1683 
+<div class="alteqlabels" id="org53f3037">
1684 
+<ul class="org-ul">
1685 
+<li>Gr (7.23)</li>
1686 
+</ul>
1687 
+
1688 
+</div>
1689 
+
1690 
+</div>
1691 
+<p>
1658 1692 
 \[
1659 1693 
 M_{12} = M_{21}
1660 
-\label{Gr(7.23)}
1694 
+\tag{Msym}\label{Msym}
1661 1695 
 \]
1662 1696 
 </p>
1663 1697
1664 
-<div class="example div" id="org34dd058">
1698 
+<div class="example div" id="org6347791">
1665 1699 
 <p>
1666 
-\paragraph{Example 7.10:}
1667 
-short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
1700 
+<b>Example: solenoid in solenoid</b>
1701 
+</p>
1702 
+
1703 
+<p>
1704 
+Consider a short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length)
1705 
+which lies concentrically inside
1668 1706 
 a very long solenoid (radius \(b\), \(n_2\) turns per unit length).  Current \(I\) in short solenoid.
1669 
-What is flux through long solenoid ?
1670 
-\paragraph{Solution:}  complicated to calculate \({\bf B}_1\).  Use mutual inductance, starting from
1707 
+</p>
1708 
+
1709 
+<p>
1710 
+<b>Task</b>: compute the flux through the long solenoid.
1711 
+</p>
1712 
+
1713 
+<p>
1714 
+<b>Solution</b>: it's complicated to calculate \({\bf B}_1\).
1715 
+Use mutual inductance, starting from
1671 1716 
 the reverse situation:  current \(I\) on outer solenoid, calculate flux through inner one.
1672 
-Field of outer solenoid:  from (\ref{Gr(5.57)}),
1717 
+Field of outer solenoid:  from <a href="./ems_ms_dcB_c.html#Amp_int">Amp_int</a>,
1673 1718 
 \[
1674 
-    B = \mu_0 n_2 I
1675 
-  \]
1719 
+B = \mu_0 n_2 I
1720 
+\]
1676 1721 
 so flux through a single loop of inner solenoid is
1677 1722 
 \[
1678 
-    B \pi a^2 = \mu_0 n_2 I \pi a^2.
1679 
-  \]
1723 
+B \pi a^2 = \mu_0 n_2 I \pi a^2.
1724 
+\]
1680 1725 
 For \(n_1 l\) turns in total, total flux through inner solenoid is
1681 1726 
 \[
1682 
-    \Phi = \mu_0 \pi a^2 n_1 n_2 l I.
1683 
-  \]
1727 
+\Phi = \mu_0 \pi a^2 n_1 n_2 l I.
1728 
+\]
1684 1729 
 Same as flux through outer solenoid if inner one has current \(I\).  Mutual inductance is here
1685 1730 
 \[
1686 
-    M = \mu_0 \pi a^2 n_1 n_2 l.
1687 
-  \]
1731 
+M = \mu_0 \pi a^2 n_1 n_2 l.
1732 
+\]
1688 1733 
 </p>
1689 1734
1690 1735 
 </div>
 
@@ -1697,62 +1742,101 @@ What if we vary current in loop 1?  Flux in 2 will vary.  Induces EMF in loop 2:
1697 1742 
 \label{Gr(7.24)}
1698 1743 
 \]
1699 1744 
 Changing current also induces EMF in the source loop itself:
1745 
+</p>
1746 
+<div class="eqlabel" id="org667fec5">
1747 
+<p>
1748 
+<a id="PLI"></a><a href="./emd_Fl_i.html#PLI"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1749 
+  <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"/>
1750 
+  <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"/>
1751 
+</svg></a>
1752 
+</p>
1753 
+<div class="alteqlabels" id="org28a1a02">
1754 
+<ul class="org-ul">
1755 
+<li>Gr (7.25)</li>
1756 
+</ul>
1757 
+
1758 
+</div>
1759 
+
1760 
+</div>
1761 
+<p>
1700 1762 
 \[
1701 1763 
 \Phi = L I
1702 
-\label{Gr(7.25)}
1764 
+\tag{PLI}\label{PLI}
1703 1765 
 \]
1704 
-where \(L\) is the {\bf self-inductance} (or inductance) of the loop.  Depends only on
1766 
+where \(L\) is the <b>self-inductance</b> (or inductance) of the loop.  Depends only on
1705 1767 
 geometry.  Changing current induces EMF of
1706 1768 
 \[
1707 1769 
 {\cal E} = -L \frac{dI}{dt}
1708 1770 
 \label{Gr(7.26)}
1709 1771 
 \]
1710 
-Inductance:  measured in {\bf henries} (\(H\)).  \(H = V s/A\).
1772 
+Inductance:  measured in <b>henries</b> (\(H\)).  \(H = V s/A\).
1711 1773 
 </p>
1712 1774
1713 1775
1714 
-<div class="example div" id="org7acd05e">
1776 
+<div class="example div" id="org00e72a3">
1777 
+<p>
1778 
+<b>Example: self-inductance of toroidal coil</b>
1779 
+</p>
1780 
+
1715 1781 
 <p>
1716 
-\paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1782 
+Consider a toroidal coil with
1717 1783 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
1718 1784 
 which carries total of \(N\) turns.
1719 
-\paragraph{Solution:}  magnetic field inside toroid is (\ref{Gr(5.58)})
1785 
+</p>
1786 
+
1787 
+<p>
1788 
+<b>Task</b>: find its self-inductance
1789 
+</p>
1790 
+
1791 
+<p>
1792 
+<b>Solution</b>:  magnetic field inside toroid is <a href="./ems_ms_dcB_c.html#Btor">Btor</a>
1720 1793 
 \[
1721 
-    B = \frac{\mu_0 NI}{2\pi s}
1722 
-  \]
1794 
+B = \frac{\mu_0 NI}{2\pi s}
1795 
+\]
1723 1796 
 Flux through single turn:
1724 1797 
 \[
1725 
-    \int {\bf B} \cdot d{\bf a} = \frac{\mu_0 N I}{2\pi} h \int_a^b \frac{ds}{s}
1726 
-    = \frac{\mu_0 N I h}{2\pi} \ln \frac{b}{a}.
1727 
-  \]
1798 
+\int {\bf B} \cdot d{\bf a} = \frac{\mu_0 N I}{2\pi} h \int_a^b \frac{ds}{s}
1799 
+= \frac{\mu_0 N I h}{2\pi} \ln \frac{b}{a}.
1800 
+\]
1728 1801 
 Total flux:  \(N\) times this, so self-inductance is
1729 1802 
 \[
1730 
-    L = \frac{\mu_0 N^2 h}{2\pi} \ln \frac{b}{a}
1731 
-    \label{Gr(7.27)}
1732 
-  \]
1803 
+L = \frac{\mu_0 N^2 h}{2\pi} \ln \frac{b}{a}
1804 
+\label{Gr(7.27)}
1805 
+\]
1733 1806 
 </p>
1734 1807
1735 1808 
 </div>
1736 1809
1737 1810 
 <p>
1738 
-Inductance (like capacitance) is intrinsically positive.  Use Lenz law.  Think of {\bf back EMF}.
1811 
+Inductance (like capacitance) is intrinsically positive.  Use Lenz law.
1812 
+Think of <i>back EMF</i>.
1813 
+</p>
1814 
+
1815 
+<div class="example div" id="org958ea6c">
1816 
+<p>
1817 
+<b>Example: circuit</b>
1739 1818 
 </p>
1740 1819
1741 
-<div class="example div" id="org9f29434">
1742 1820 
 <p>
1743 
-\paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1744 
-What is the current ?
1745 
-\paragraph{Solution:}
1821 
+Consider a circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1822 
+</p>
1823 
+
1824 
+<p>
1825 
+<b>Task</b>: find the current
1826 
+</p>
1827 
+
1828 
+<p>
1829 
+<b>Solution</b>:
1746 1830 
 Ohm's law:
1747 1831 
 \[
1748 
-    {\cal E}_0 - L \frac{dI}{dt} = IR \Longrightarrow I(t) = \frac{{\cal E}_0}{R} + k e^{-(R/L)t}.
1749 
-  \]
1832 
+{\cal E}_0 - L \frac{dI}{dt} = IR \Longrightarrow I(t) = \frac{{\cal E}_0}{R} + k e^{-(R/L)t}.
1833 
+\]
1750 1834 
 If initial condition:  \(I(0) = 0\), then
1751 1835 
 \[
1752 
-    I(t) = \frac{{\cal E}_0}{R} \left[ 1 - e^{-(R/L)t} \right]
1753 
-    \label{Gr(7.28)}
1754 
-  \]
1755 
-where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
1836 
+I(t) = \frac{{\cal E}_0}{R} \left[ 1 - e^{-(R/L)t} \right]
1837 
+\label{Gr(7.28)}
1838 
+\]
1839 
+where \(\tau \equiv L/R\) is the <b>time constant</b> of the circuit.
1756 1840 
 </p>
1757 1841
1758 1842 
 </div>
 
@@ -1777,7 +1861,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1777 1861 
 </div>
1778 1862 
 <div id="postamble" class="status">
1779 1863 
 <p class="author">Author: Jean-Sébastien Caux</p>
1780 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1864 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1781 1865 
 <p class="validation"></p>
1782 1866 
 </div>
1783 1867

+ 67
- 38
build/emd_Fl_ief.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1628,7 +1628,7 @@ Two sources of electric fields:  electric charges, and changing magnetic fields.
1628 1628 
 <p>
1629 1629 
 Electric fields induced by a changing magnetic field are determined in an exactly
1630 1630 
 parallel way as magnetostatic fields from the current:  exploit parallel
1631 
-between Ampère and Faraday!
1631 
+between Ampère and Faraday
1632 1632 
 \[
1633 1633 
 {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}
1634 1634 
 \hspace{3cm}
 
@@ -1645,42 +1645,60 @@ law in integral form:
1645 1645
1646 1646
1647 1647
1648 
-<div class="example div" id="org3b5285b">
1648 
+<div class="example div" id="org045463b">
1649 1649 
 <p>
1650 
-{\bf Example 7.7:}
1651 
-\({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
1652 
-\paragraph{Solution:}
1650 
+<b>Example: loop with time-dependent flux</b>
1651 
+</p>
1652 
+
1653 
+<p>
1654 
+Consider a time-dependent magnetic field \({\bf B}(t)\) directed vertically
1655 
+through a horizontal circular region of radius \(R\).
1656 
+</p>
1657 
+
1658 
+<p>
1659 
+<b>Task</b>: find the induced \({\bf E}(t)\).
1660 
+</p>
1661 
+
1662 
+<p>
1663 
+<b>Solution</b>:
1653 1664 
 amperian loop of radius \(s\), apply Faraday:
1654 1665 
 \[
1655 
-    \oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
1656 
-    \Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1657 
-  \]
1666 
+\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
1667 
+\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1668 
+\]
1658 1669 
 Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1659 1670 
 </p>
1660 1671
1661 1672 
 </div>
1662 1673
1663 1674
1664 
-<div class="example div" id="org68f8620">
1675 
+<div class="example div" id="org047ea10">
1676 
+<p>
1677 
+<b>Example: wheel with charged rim traversed by flux</b>
1678 
+</p>
1679 
+
1680 
+<p>
1681 
+Consider a wheel of radius \(b\) with line charge \(\lambda\) on the rim.
1682 
+A uniform magnetic field \({\bf B}_0\) pointing up is traversing the central region
1683 
+up to radius \(a &lt; b\). The field is then turned off.  What happens?
1684 
+</p>
1685 
+
1665 1686 
 <p>
1666 
-{\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
1667 
-Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
1668 
-pointing up.  Field turned off.  What happens ?
1669 
-\paragraph{Solution:}  the wheel starts spinning to compensate the reduction of field.
1687 
+<b>Solution</b>:  the wheel starts spinning to compensate the reduction of field.
1670 1688 
 Faraday:
1671 1689 
 \[
1672 
-    \oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt}
1673 
-    \Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1674 
-  \]
1690 
+\oint {\bf E} \cdot d{\bf l} = -\frac{d\Phi}{dt} = - \pi a^2 \frac{dB}{dt}
1691 
+\Rightarrow {\bf E} = -\frac{a^2}{2b} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1692 
+\]
1675 1693 
 Torque on segment \(d{\bf l}\):  \(|{\bf r} \times {\bf F}| = b \lambda E dl\).
1676 1694 
 Total torque:
1677 1695 
 \[
1678 
-    N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt}
1679 
-  \]
1680 
-so total angular momentum imparted is
1696 
+N = b\lambda \oint E dl = -b \lambda \pi a^2 \frac{dB}{dt}
1697 
+\]
1698 
+so total angular momentum imparted to the wheel is
1681 1699 
 \[
1682 
-    \int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0.
1683 
-  \]
1700 
+\int N dt = -\lambda \pi a^2 b \int_{B_0}^0 dB = \lambda \pi a^2 b B_0.
1701 
+\]
1684 1702 
 </p>
1685 1703
1686 1704 
 </div>
 
@@ -1690,33 +1708,44 @@ The precise way the field is turned off doesn't matter.  Only electric field doe
1690 1708 
 </p>
1691 1709
1692 1710 
 <p>
1693 
-{\bf N.B.:}  we use magnetostatic formulas for changing fields.  This is
1694 
-called the {\bf quasistatic} approximation, and works provided we deal with
1695 
-'slow enough' phenomena.
1711 
+<b>N.B.</b>:  we use magnetostatic formulas for changing fields.  This is
1712 
+called the <b>quasistatic</b> approximation, and works provided we deal with
1713 
+<i>slow enough</i> phenomena.
1696 1714 
 </p>
1697 1715
1698 
-<div class="example div" id="org5b14490">
1716 
+<div class="example div" id="org733cbdd">
1699 1717 
 <p>
1700 
-{\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
1701 
-induced \({\bf E}\) field as a function of distance \(s\) from wire.
1702 
-\paragraph{Solution:}  quasistatic:  magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
1718 
+<b>Example: field from wire with time-dependent current</b>
1719 
+</p>
1720 
+
1721 
+<p>
1722 
+Consider an infinitely long straight wire which carries current \(I(t)\).
1723 
+</p>
1724 
+
1725 
+<p>
1726 
+<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
1727 
+</p>
1728 
+
1729 
+<p>
1730 
+<b>Solution</b>:  assuming we can use the quasistatic approximation, the
1731 
+magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
1703 1732 
 and circles the wire.  Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
1704 1733 
 to wire.  Amperian loop with sides at distances \(s_0\) and \(s\):
1705 1734 
 \[
1706 
-    \oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
1707 
-    = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
1708 
-    = -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
1709 
-  \]
1735 
+\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
1736 
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
1737 
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
1738 
+\]
1710 1739 
 So:
1711 1740 
 \[
1712 
-    {\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
1713 
-    \label{Gr(7.19)}
1714 
-  \]
1741 
+{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
1742 
+\label{Gr(7.19)}
1743 
+\]
1715 1744 
 where \(K\) is a constant (depends on the history of \(I(t)\)).
1716 1745 
 </p>
1717 1746
1718 1747 
 <p>
1719 
-{\bf N.B.:}  this can't be true always, since it blows up as \(s \rightarrow \infty\).
1748 
+<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\).
1720 1749 
 Reason:  in this case, we've overstepped the quasistatic limit.  We need
1721 1750 
 \(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
1722 1751 
 </p>
 
@@ -1743,7 +1772,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1743 1772 
 </div>
1744 1773 
 <div id="postamble" class="status">
1745 1774 
 <p class="author">Author: Jean-Sébastien Caux</p>
1746 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1775 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1747 1776 
 <p class="validation"></p>
1748 1777 
 </div>
1749 1778

+ 2
- 2
build/emd_Me.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
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4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1648 
 </div>
1649 1649 
 <div id="postamble" class="status">
1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1651 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 1652 
 <p class="validation"></p>
1653 1653 
 </div>
1654 1654

+ 51
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@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1624,53 +1624,79 @@ Table of contents
1624 1624 
 <p>
1625 1625 
 Full set of equations for the electromagnetic field:
1626 1626 
 </p>
1627 
-<div class="core div" id="org883ebf1">
1627 
+<div class="core div" id="org40e4789">
1628 1628 
 <p>
1629 
-{\bf Maxwell's equations} {\it (in vacuum)}
1629 
+<b>Maxwell's equations</b> <i>(in vacuum)</i>
1630 1630 
 </p>
1631 
+<div class="eqlabel" id="org3e8ad9b">
1632 
+<p>
1633 
+<a id="Max_vac"></a><a href="./emd_Me_Me.html#Max_vac"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1634 
+  <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"/>
1635 
+  <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"/>
1636 
+</svg></a>
1637 
+</p>
1638 
+<div class="alteqlabels" id="orgc606bec">
1639 
+
1640 
+</div>
1641 
+
1642 
+</div>
1631 1643 
 \begin{align}
1632 
-  (i) {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
1633 
-  (ii) {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
1634 
-  (iii) {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
1635 
-  (iv) {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, \hspace{1cm} &amp;\mbox{Ampère + Maxwell}.
1636 
-  \label{Gr(7.39)}
1644 
+(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
1645 
+(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
1646 
+(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t}, &amp;\mbox{Faraday}, \nonumber \\
1647 
+(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}, &amp;\mbox{Ampère + Maxwell}.
1648 
+\tag{Max_vac}\label{Max_vac}
1637 1649 
 \end{align}
1638 1650
1639 1651 
 </div>
1640 1652 
 <p>
1641 1653 
 Complement:
1642 1654 
 </p>
1643 
-<div class="core div" id="org2481694">
1655 
+<div class="core div" id="org28daec0">
1644 1656 
 <p>
1645 
-{\bf Force law}
1657 
+Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
1646 1658 
 \[
1647 
-    {\bf F} = q ({\bf E} + {\bf v} \times {\bf B}).
1648 
-    \label{Gr(7.40)}
1649 
-  \]
1659 
+{\bf F} = q ({\bf E} + {\bf v} \times {\bf B}).
1660 
+\label{Gr(7.40)}
1661 
+\]
1650 1662 
 </p>
1651 1663
1652 1664 
 </div>
1653 1665 
 <p>
1654 
-These equations summarize the {\bf entire content of classical electrodynamics}.
1666 
+These equations contain the <b>entirety of pre-quantum electrodynamics</b>.
1655 1667 
 </p>
1656 1668
1657 1669 
 <p>
1658 
-\paragraph{Note:}  even the continuity equation can be derived from Maxwell's equations:
1659 
-take divergence of \((iv)\).
1670 
+<b>Note</b>:  even the continuity equation can be derived from Maxwell's equations:
1671 
+take divergence of \((iv)\) and use \((i)\).
1660 1672 
 </p>
1661 1673
1662 1674
1663 
-
1664 1675 
 <p>
1665 1676 
 Better way of writing:  all fields on left, all sources on right,
1666 1677 
 </p>
1667 
-<div class="core div" id="org47f05c2">
1678 
+<div class="core div" id="org4bb3e78">
1679 
+<div class="eqlabel" id="org05437e5">
1680 
+<p>
1681 
+<a id="Max_vac_s"></a><a href="./emd_Me_Me.html#Max_vac_s"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1682 
+  <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"/>
1683 
+  <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"/>
1684 
+</svg></a>
1685 
+</p>
1686 
+<div class="alteqlabels" id="org61742b2">
1687 
+<ul class="org-ul">
1688 
+<li>Gr (7.42)</li>
1689 
+</ul>
1690 
+
1691 
+</div>
1692 
+
1693 
+</div>
1668 1694 
 \begin{align}
1669 
-  (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
1670 
-  &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
1671 
-  (ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0,
1672 
-  &amp;(iv) {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} = \mu_0 {\bf J},
1673 
-  \label{Gr(7.42)}
1695 
+(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \nonumber \\
1696 
+(ii)~{\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, \nonumber \\
1697 
+(iii)~ {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} &amp;= 0, \nonumber \\
1698 
+(iv)~ {\boldsymbol \nabla} \times {\bf B} - \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} &amp;= \mu_0 {\bf J},
1699 
+\tag{Max_vac_s}\label{Max_vac_s}
1674 1700 
 \end{align}
1675 1701
1676 1702 
 </div>
 
@@ -1679,6 +1705,7 @@ Better way of writing:  all fields on left, all sources on right,
1679 1705
1680 1706
1681 1707
1708 
+
1682 1709 
 <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Next:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
1683 1710 
 <br>
1684 1711 
 <hr>
 
@@ -1694,7 +1721,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1694 1721 
 </div>
1695 1722 
 <div id="postamble" class="status">
1696 1723 
 <p class="author">Author: Jean-Sébastien Caux</p>
1697 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1724 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1698 1725 
 <p class="validation"></p>
1699 1726 
 </div>
1700 1727

+ 49
- 20
build/emd_Me_dc.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1622,8 +1622,7 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd.Me.dc</span></h4>
1623 1623 
 <div class="outline-text-4" id="text-emd_Me_dc">
1624 1624 
 <p>
1625 
-The term which should be zero (but isn't) in (\ref{Gr(7.35)}) can be rewritten using
1626 
-the continuity equation as
1625 
+The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB">divcurlB</a> can be rewritten using the continuity equation as
1627 1626 
 \[
1628 1627 
 {\boldsymbol \nabla} \cdot {\bf J} = -\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial t}
1629 1628 
 (\varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}) = -{\boldsymbol \nabla} \cdot \left(
 
@@ -1631,37 +1630,67 @@ the continuity equation as
1631 1630 
 \]
1632 1631 
 The extra term would thus be eliminated if we were to put
1633 1632 
 </p>
1634 
-<div class="core div" id="orgb95d862">
1633 
+<div class="core div" id="orgdf7b3f9">
1634 
+<div class="eqlabel" id="org5844c7b">
1635 
+<p>
1636 
+<a id="AmpMax"></a><a href="./emd_Me_dc.html#AmpMax"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1637 
+  <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"/>
1638 
+  <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"/>
1639 
+</svg></a>
1640 
+</p>
1641 
+<div class="alteqlabels" id="org1e192cc">
1642 
+<ul class="org-ul">
1643 
+<li>Gr (7.36)</li>
1644 
+</ul>
1645 
+
1646 
+</div>
1647 
+
1648 
+</div>
1635 1649 
 <p>
1636 1650 
 \[
1637 
-    {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
1638 
-    \label{Gr(7.36)}
1639 
-  \]
1651 
+{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
1652 
+\tag{AmpMax}\label{AmpMax}
1653 
+\]
1640 1654 
 </p>
1641 1655
1642 1656 
 </div>
1643 1657 
 <p>
1644 
-\paragraph{Note:}  this changes nothing in magnetostatics.  Aesthetic appeal:
1658 
+<b>Note</b>:  this changes nothing in magnetostatics.  Aesthetic appeal:
1645 1659 
 \[
1646 
-  \boxed{
1647 
-    \mbox{A changing electric field induces a magnetic field.}
1648 
-  }
1660 
+\boxed{
1661 
+\mbox{A changing electric field induces a magnetic field.}
1662 
+}
1649 1663 
 \]
1650 1664 
 Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation of electromagnetic waves.
1651 1665 
 </p>
1652 1666
1653 
-
1654 
-
1655 1667 
 <p>
1656 1668 
 Maxwell baptized this term the
1657 1669 
 </p>
1658 
-<div class="core div" id="orgb8b014a">
1670 
+<div class="core div" id="orgb28580f">
1671 
+<p>
1672 
+<b>Displacement current</b>
1673 
+</p>
1674 
+<div class="eqlabel" id="org4a72dcc">
1675 
+<p>
1676 
+<a id="Jd"></a><a href="./emd_Me_dc.html#Jd"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1677 
+  <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"/>
1678 
+  <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"/>
1679 
+</svg></a>
1680 
+</p>
1681 
+<div class="alteqlabels" id="org57849e1">
1682 
+<ul class="org-ul">
1683 
+<li>Gr (7.37)</li>
1684 
+</ul>
1685 
+
1686 
+</div>
1687 
+
1688 
+</div>
1659 1689 
 <p>
1660 
-{\bf Displacement current}
1661 1690 
 \[
1662 
-    {\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}.
1663 
-    \label{Gr(7.37)}
1664 
-  \]
1691 
+{\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}.
1692 
+\tag{Jd}\label{Jd}
1693 
+\]
1665 1694 
 </p>
1666 1695
1667 1696 
 </div>
 
@@ -1675,7 +1704,7 @@ where \(A\) is the area.  Between the plates,
1675 1704 
 \[
1676 1705 
 \frac{\partial E}{\partial t} = \frac{1}{\varepsilon_0 A} \frac{dQ}{dt} = \frac{1}{\varepsilon_0 A} I.
1677 1706 
 \]
1678 
-Checking (\ref{Gr(7.36)}),
1707 
+Checking <a href="./emd_Me_dc.html#AmpMax">AmpMax</a>,
1679 1708 
 \[
1680 1709 
 \oint {\bf B} \cdot d{\bf l} = \mu_0 I_{\mbox{enc}} + \mu_0 \varepsilon_0 \int d{\bf a} \cdot \frac{\partial {\bf E}}{\partial t}
1681 1710 
 \label{Gr(7.38)}
 
@@ -1703,7 +1732,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1703 1732 
 </div>
1704 1733 
 <div id="postamble" class="status">
1705 1734 
 <p class="author">Author: Jean-Sébastien Caux</p>
1706 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1735 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1707 1736 
 <p class="validation"></p>
1708 1737 
 </div>
1709 1738

+ 25
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@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1625,10 +1625,10 @@ Table of contents
1625 1625 
 We've encountered:
1626 1626 
 </p>
1627 1627 
 \begin{align}
1628 
-(i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
1629 
-(ii) &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \hspace{1cm} &amp;\mbox{anonymous} \nonumber \\
1630 
-(iii) &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \hspace{1cm} &amp;\mbox{Faraday}, \nonumber \\
1631 
-(iv) &amp;{\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J}, \hspace{1cm} &amp;\mbox{Ampère}.
1628 
+(i)~ {\boldsymbol \nabla} \cdot {\bf E} &amp;= \frac{\rho}{\varepsilon_0}, \hspace{1cm} &amp;\mbox{Gauss}, \nonumber \\
1629 
+(ii)~ {\boldsymbol \nabla} \cdot {\bf B} &amp;= 0, &amp;\mbox{anonymous} \nonumber \\
1630 
+(iii)~ {\boldsymbol \nabla} \times {\bf E} &amp;= -\frac{\partial {\bf B}}{\partial t},  &amp;\mbox{Faraday}, \nonumber \\
1631 
+(iv)~ {\boldsymbol \nabla} \times {\bf B} &amp;= \mu_0 {\bf J}, &amp;\mbox{Ampère}.
1632 1632 
 \end{align}
1633 1633 
 <p>
1634 1634 
 Fatal inconsistency:  div of curl must always vanish.  Check on \((iii)\):
 
@@ -1637,11 +1637,28 @@ Fatal inconsistency:  div of curl must always vanish.  Check on \((iii)\):
1637 1637 
 = {\boldsymbol \nabla} \cdot \left( -\frac{\partial {\bf B}}{\partial t} \right) = -\frac{\partial}{\partial t} ({\boldsymbol \nabla} \cdot {\bf B}) = 0.
1638 1638 
 \]
1639 1639 
 But:  try same with \((iv)\):
1640 
+</p>
1641 
+<div class="eqlabel" id="org764ac3e">
1642 
+<p>
1643 
+<a id="divcurlB"></a><a href="./emd_Me_ebM.html#divcurlB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1644 
+  <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"/>
1645 
+  <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"/>
1646 
+</svg></a>
1647 
+</p>
1648 
+<div class="alteqlabels" id="org0f4f12d">
1649 
+<ul class="org-ul">
1650 
+<li>Gr (7.35)</li>
1651 
+</ul>
1652 
+
1653 
+</div>
1654 
+
1655 
+</div>
1656 
+<p>
1640 1657 
 \[
1641 1658 
 {\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} \times {\bf B}) = \mu_0 {\boldsymbol \nabla} \cdot {\bf J}
1642 
-\label{Gr(7.35)}
1659 
+\tag{divcurlB}\label{divcurlB}
1643 1660 
 \]
1644 
-LHS must be zero, but RHS is not zero for non-steady currents.  Cannot be right !
1661 
+LHS must be zero, but RHS is not zero for non-steady currents.  Cannot be right!
1645 1662 
 </p>
1646 1663
1647 1664 
 <p>
 
@@ -1673,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1673 1690 
 </div>
1674 1691 
 <div id="postamble" class="status">
1675 1692 
 <p class="author">Author: Jean-Sébastien Caux</p>
1676 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1693 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1677 1694 
 <p class="validation"></p>
1678 1695 
 </div>
1679 1696

+ 4
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@@ -1,7 +1,7 @@
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2 2 
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3 3 
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4 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1633,7 +1633,7 @@ In free space, where \(\rho\) and \({\bf J}\) vanish:
1633 1633 
 <p>
1634 1634 
 Symmetry:  replace \({\bf E}\) by \({\bf B}\) and \({\bf B}\) by \(-\mu_0 \varepsilon_0{\bf E}\) in the first pair.
1635 1635 
 They turn into the second pair.  This symmetry is spoiled by \(\rho\) and \({\bf J}\).  What if we had
1636 
-a truly symmetric situation, {\it i.e.}
1636 
+a truly symmetric situation, <i>i.e.</i>
1637 1637 
 </p>
1638 1638 
 \begin{align}
1639 1639 
 (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho_e}{\varepsilon_0},
 
@@ -1650,7 +1650,7 @@ of magnetic charge.  Both charges would be conserved:
1650 1650 
 {\boldsymbol \nabla} \cdot {\bf J}_e = -\frac{\partial \rho_e}{\partial t}.
1651 1651 
 \label{Gr(7.44)}
1652 1652 
 \]
1653 
-Maxwell's equations {\bf beg} for magnetic charges.  But we've never found any!
1653 
+Maxwell's equations <i>beg</i> for magnetic charges.  But we've never found any!
1654 1654 
 </p>
1655 1655 
 </div>
1656 1656 
 </div>
 
@@ -1671,7 +1671,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1671 1671 
 </div>
1672 1672 
 <div id="postamble" class="status">
1673 1673 
 <p class="author">Author: Jean-Sébastien Caux</p>
1674 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1674 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1675 1675 
 <p class="validation"></p>
1676 1676 
 </div>
1677 1677

+ 6
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@@ -1,7 +1,7 @@
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2 2 
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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1622,8 +1622,8 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd.ce</span></h3>
1623 1623
1624 1624 
 <div class="outline-text-3" id="text-emd_ce">
1625 
-<details class="prereq" id="org70c43f6">
1626 
-<summary id="org3522c33">
1625 
+<details class="prereq" id="orgca631bc">
1626 
+<summary id="org2ad112f">
1627 1627 
 Prerequisites
1628 1628 
 </summary>
1629 1629 
 <ul class="org-ul">
 
@@ -1631,8 +1631,8 @@ Prerequisites
1631 1631 
 </ul>
1632 1632 
 </details>
1633 1633
1634 
-<details class="objectives" id="orgead6b14">
1635 
-<summary id="orgf683e76">
1634 
+<details class="objectives" id="orgbdb48d8">
1635 
+<summary id="org495ffbc">
1636 1636 
 Objectives
1637 1637 
 </summary>
1638 1638 
 <ul class="org-ul">
 
@@ -1670,7 +1670,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1670 1670 
 </div>
1671 1671 
 <div id="postamble" class="status">
1672 1672 
 <p class="author">Author: Jean-Sébastien Caux</p>
1673 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1673 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1674 1674 
 <p class="validation"></p>
1675 1675 
 </div>
1676 1676

+ 23
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build/emd_ce_amom.html View File 

@@ -1,7 +1,7 @@
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2 2 
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3 3 
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4 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1624,14 +1624,29 @@ Table of contents
1624 1624 
 <p>
1625 1625 
 The angular momentum of EM fields is directly given by
1626 1626 
 </p>
1627 
-<div class="main div" id="orgaa57ede">
1627 
+<div class="main div" id="org6e52344">
1628 
+<p>
1629 
+<b>Angular momentum of EM fields</b>
1630 
+</p>
1631 
+<div class="eqlabel" id="org9b249c7">
1632 
+<p>
1633 
+<a id="lrxg"></a><a href="./emd_ce_amom.html#lrxg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1634 
+  <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"/>
1635 
+  <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"/>
1636 
+</svg></a>
1637 
+</p>
1638 
+<div class="alteqlabels" id="org9913b19">
1639 
+
1640 
+</div>
1641 
+
1642 
+</div>
1628 1643 
 <p>
1629 
-{\bf Angular momentum of EM fields}
1630 1644 
 \[
1631 
-    {\boldsymbol l} = {\boldsymbol r} \times {\boldsymbol g}
1632 
-    = \varepsilon_0 {\boldsymbol r} \times
1633 
-    \left({\boldsymbol E} \times {\boldsymbol B}\right)
1634 
-  \]
1645 
+{\boldsymbol l} = {\boldsymbol r} \times {\boldsymbol g}
1646 
+= \varepsilon_0 ~{\boldsymbol r} \times
1647 
+\left({\boldsymbol E} \times {\boldsymbol B}\right)
1648 
+\tag{lrxg}\label{lrxg}
1649 
+\]
1635 1650 
 </p>
1636 1651
1637 1652 
 </div>
 
@@ -1654,7 +1669,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1654 1669 
 </div>
1655 1670 
 <div id="postamble" class="status">
1656 1671 
 <p class="author">Author: Jean-Sébastien Caux</p>
1657 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1672 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1658 1673 
 <p class="validation"></p>
1659 1674 
 </div>
1660 1675

+ 10
- 11
build/emd_ce_ce.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
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4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1622,13 +1622,13 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd.ce.ce</span></h4>
1623 1623 
 <div class="outline-text-4" id="text-emd_ce_ce">
1624 1624 
 <p>
1625 
-Very important distinction:  {\bf global} versus {\bf local} conservation of charge.
1625 
+Very important distinction:  <b>global</b> versus <b>local</b> conservation of charge.
1626 1626 
 </p>
1627 1627
1628 1628 
 <p>
1629 
-Charge in a volume {\cal V}:
1629 
+Charge in a volume \({\cal V}\):
1630 1630 
 \[
1631 
-Q_{\cal V} (t) = \int_{\cal V} d\tau \rho ({\bf r}, t)
1631 
+Q_{\cal V} (t) = \int_{\cal V} d\tau~ \rho ({\bf r}, t)
1632 1632 
 \label{Gr(8.1)}
1633 1633 
 \]
1634 1634 
 Current \({\bf J}\) flowing out through boundary \({\cal S}\) of \({\cal V}\):  conservation of charge means
 
@@ -1643,13 +1643,12 @@ This means that
1643 1643 
 \]
1644 1644 
 Since this is true for any volume, we have (re)derived the
1645 1645 
 </p>
1646 
-<div class="core div" id="orgfb5850b">
1646 
+<div class="core div" id="orgf1c2b2a">
1647 1647 
 <p>
1648 
-{\bf Continuity equation}
1648 
+<b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
1649 1649 
 \[
1650 
-    \frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0
1651 
-    \label{Gr(8.4)}
1652 
-  \]
1650 
+\frac{\partial \rho}{\partial t} + {\boldsymbol \nabla} \cdot {\bf J} = 0
1651 
+\]
1653 1652 
 </p>
1654 1653
1655 1654 
 </div>
 
@@ -1660,7 +1659,7 @@ Therefore, conservation of charge is a direct consequence of Maxwell's equations
1660 1659 
 <p>
1661 1660 
 One thing to note: we have viewed \(\rho\) and \({\boldsymbol J}\) as sources
1662 1661 
 (the ''right-hand side'') of Maxwell's equations. The continuity equation thus
1663 
-imposes a functional constraint on these sources: not {\it any} \(\rho\) and
1662 
+imposes a functional constraint on these sources: not <i>any</i> \(\rho\) and
1664 1663 
 \({\boldsymbol J}\) will do the trick, only the ones what obey it.
1665 1664 
 </p>
1666 1665 
 </div>
 
@@ -1684,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1684 1683 
 </div>
1685 1684 
 <div id="postamble" class="status">
1686 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1687 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1686 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1688 1687 
 <p class="validation"></p>
1689 1688 
 </div>
1690 1689

+ 41
- 11
build/emd_ce_mom.html View File 

@@ -1,7 +1,7 @@
1 1 
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2 2 
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3 3 
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4 
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4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1624,7 +1624,7 @@ Table of contents
1624 1624 
 <p>
1625 1625 
 From Newton's second law,
1626 1626 
 \[
1627 
-  {\boldsymbol F} = \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt}
1627 
+{\boldsymbol F} = \frac{d {\boldsymbol p}_{\tiny \mbox{mech}}}{dt}
1628 1628 
 \]
1629 1629 
 we have
1630 1630 
 \[
 
@@ -1636,24 +1636,54 @@ in which the first integral can be interpreted as the momentum stored in the EM
1636 1636 
 <p>
1637 1637 
 This is thus simply a conservation law for momentum, with
1638 1638 
 </p>
1639 
-<div class="main div" id="orgc75d196">
1639 
+<div class="main div" id="orgecd6647">
1640 
+<p>
1641 
+<b>Momentum density in the EM fields</b>
1642 
+</p>
1643 
+<div class="eqlabel" id="orga1d9e2c">
1644 
+<p>
1645 
+<a id="gExB"></a><a href="./emd_ce_mom.html#gExB"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1646 
+  <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"/>
1647 
+  <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"/>
1648 
+</svg></a>
1649 
+</p>
1650 
+<div class="alteqlabels" id="orgf73a06d">
1651 
+
1652 
+</div>
1653 
+
1654 
+</div>
1640 1655 
 <p>
1641 
-{\bf Momentum density in the EM fields}
1642 1656 
 \[
1643 
-    {\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B}
1644 
-  \]
1657 
+{\boldsymbol g} = \varepsilon_0 \mu_0 {\boldsymbol S} = \varepsilon_0 {\boldsymbol E} \times {\boldsymbol B}
1658 
+\tag{gExB}\label{gExB}
1659 
+\]
1645 1660 
 </p>
1646 1661
1647 1662 
 </div>
1648 1663 
 <p>
1649 1664 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1650 1665 
 </p>
1651 
-<div class="main div" id="orgc680b5b">
1666 
+<div class="main div" id="org8cc6bb8">
1667 
+<p>
1668 
+<b>Continuity equation for EM momentum</b>
1669 
+</p>
1670 
+<div class="eqlabel" id="orgcad090e">
1671 
+<p>
1672 
+<a id="contg"></a><a href="./emd_ce_mom.html#contg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1673 
+  <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"/>
1674 
+  <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"/>
1675 
+</svg></a>
1676 
+</p>
1677 
+<div class="alteqlabels" id="org69cf28d">
1678 
+
1679 
+</div>
1680 
+
1681 
+</div>
1652 1682 
 <p>
1653 
-{\bf Continuity equation for EM momentum}
1654 1683 
 \[
1655 
-    \frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0
1656 
-  \]
1684 
+\frac{\partial}{\partial t} {\boldsymbol g} - {\boldsymbol \nabla} \cdot {\boldsymbol T} = 0
1685 
+\tag{contg}\label{contg}
1686 
+\]
1657 1687 
 </p>
1658 1688
1659 1689 
 </div>
 
@@ -1677,7 +1707,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1677 1707 
 </div>
1678 1708 
 <div id="postamble" class="status">
1679 1709 
 <p class="author">Author: Jean-Sébastien Caux</p>
1680 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1710 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1681 1711 
 <p class="validation"></p>
1682 1712 
 </div>
1683 1713

+ 70
- 34
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@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
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4 
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4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1638,30 +1638,27 @@ Substitute for \(\rho\) and \({\boldsymbol J}\) using Maxwell (Gauss and Ampère
1638 1638
1639 1639 
 <p>
1640 1640 
 On the other hand we have
1641 
-\[
1642 
-  \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B} \right)
1643 
-  = \frac{∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
1644 1641 
 </p>
1645 
-<ul class="org-ul">
1646 
-<li>{\boldsymbol E} × \frac{∂ {\boldsymbol B}}{∂ t}.</li>
1647 
-</ul>
1642 
+\begin{equation}
1643 
+  \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right)
1644 
+  = \frac{\partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
1645 
+  + {\boldsymbol E} \times \frac{\partial {\boldsymbol B}}{\partial t}.
1646 
+\end{equation}
1648 1647 
 <p>
1649 
-\]
1650 1648 
 Using Faraday to substitute for \(\frac{\partial {\boldsymbol B}}{\partial t}\),
1651 
-\[
1652 
-  \frac{ ∂ {\boldsymbol E}}{∂ t} × {\boldsymbol B}
1653 
-  = \frac{\partial }{\partial t} \left( {\boldsymbol E} × {\boldsymbol B}\right)
1654 1649 
 </p>
1655 
-<ul class="org-ul">
1656 
-<li>{\boldsymbol E} × \left({\boldsymbol ∇} × {\boldsymbol E} \right)</li>
1657 
-</ul>
1650 
+\begin{equation}
1651 
+  \frac{ \partial {\boldsymbol E}}{\partial t} \times {\boldsymbol B}
1652 
+  = \frac{\partial }{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B}\right)
1653 
+  + {\boldsymbol E} \times \left({\boldsymbol \nabla} \times {\boldsymbol E} \right)
1654 
+\end{equation}
1658 1655 
 <p>
1659 
-\]
1660 1656 
 so
1661 1657 
 \[
1662 1658 
   {\boldsymbol f} = \varepsilon_0 \left( \left( {\boldsymbol \nabla} \cdot {\boldsymbol E} \right) {\boldsymbol E} - {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right) \right) - \frac{1}{\mu_0} \left( {\boldsymbol B} \times \left( {\boldsymbol \nabla} \times {\boldsymbol B} \right) \right) - \varepsilon_0 \frac{\partial}{\partial t} \left( {\boldsymbol E} \times {\boldsymbol B} \right).
1663 1659 
 \]
1664 
-Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\). Moreover, by product rule 4,
1660 
+Since \({\boldsymbol \nabla} \cdot {\boldsymbol B} = 0\), we can symmetrize the expression in \({\boldsymbol E}\) and \({\boldsymbol B}\).
1661 
+Moreover, by <a href="./c_m_dc_pr.html#grad_sprod">grad_sprod</a>,
1665 1662 
 \[
1666 1663 
   \frac{1}{2}{\boldsymbol \nabla} \left( E^2 \right) = \left( {\boldsymbol E} \cdot {\boldsymbol \nabla} \right) {\boldsymbol E} + {\boldsymbol E} \times \left( {\boldsymbol \nabla} \times {\boldsymbol E} \right)
1667 1664 
 \]
 
@@ -1678,18 +1675,27 @@ and similarly for \({\boldsymbol B}\). We thus get
1678 1675 
 <p>
1679 1676 
 This expression can be greatly simplified by introducing the
1680 1677 
 </p>
1681 
-<div class="main div" id="org41d984c">
1678 
+<div class="main div" id="orga7d370d">
1682 1679 
 <p>
1683 
-{\bf Maxwell stress tensor}
1684 
-\[
1685 
-  T_{ij} ≡ ε_0 \left( E_i E_j - \frac{1}{2} δ_{ij} E^2\right)
1680 
+<b>Maxwell stress tensor</b>
1686 1681 
 </p>
1687 
-<ul class="org-ul">
1688 
-<li>\frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} δ_{ij} B^2 \right)</li>
1689 
-</ul>
1682 
+<div class="eqlabel" id="orge5429c3">
1690 1683 
 <p>
1691 
-\]
1684 
+<a id="MaxST"></a><a href="./emd_ce_mst.html#MaxST"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1685 
+  <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"/>
1686 
+  <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"/>
1687 
+</svg></a>
1692 1688 
 </p>
1689 
+<div class="alteqlabels" id="org0306495">
1690 
+
1691 
+</div>
1692 
+
1693 
+</div>
1694 
+\begin{equation}
1695 
+T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
1696 
+    + \frac{1}{\mu_0} \left( B_i B_j - \frac{1}{2} \delta_{ij} B^2 \right)
1697 
+\tag{MaxST}\label{MaxST}
1698 
+\end{equation}
1693 1699
1694 1700 
 </div>
1695 1701 
 <p>
 
@@ -1699,26 +1705,56 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1699 1705
1700 1706
1701 1707 
 <p>
1702 
-We then obtain
1708 
+We then obtain the
1703 1709 
 </p>
1704 
-<div class="main div" id="orgefc4ae2">
1710 
+<div class="main div" id="org4781d10">
1711 
+<p>
1712 
+<b>EM force per unit volume</b>
1713 
+</p>
1714 
+<div class="eqlabel" id="orgd9f91b7">
1715 
+<p>
1716 
+<a id="fT"></a><a href="./emd_ce_mst.html#fT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1717 
+  <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"/>
1718 
+  <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"/>
1719 
+</svg></a>
1720 
+</p>
1721 
+<div class="alteqlabels" id="org7d4a06b">
1722 
+
1723 
+</div>
1724 
+
1725 
+</div>
1705 1726 
 <p>
1706 
-{\bf EM force per unit volume}
1707 1727 
 \[
1708 
-    {\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t}
1709 
-  \]
1728 
+{\boldsymbol f} = {\boldsymbol \nabla} \cdot {\boldsymbol T} - \varepsilon_0 \mu_0 \frac{\partial {\boldsymbol S}}{\partial t}
1729 
+\tag{fT}\label{fT}
1730 
+\]
1710 1731 
 </p>
1711 1732
1712 1733 
 </div>
1713 1734 
 <p>
1714 1735 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1715 1736 
 </p>
1716 
-<div class="main div" id="orgc9bf6dd">
1737 
+<div class="main div" id="orgea18677">
1738 
+<p>
1739 
+<b>Total force on charges in volume</b>
1740 
+</p>
1741 
+<div class="eqlabel" id="orgef98657">
1742 
+<p>
1743 
+<a id="totFo"></a><a href="./emd_ce_mst.html#totFo"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1744 
+  <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"/>
1745 
+  <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"/>
1746 
+</svg></a>
1747 
+</p>
1748 
+<div class="alteqlabels" id="org4c9c200">
1749 
+
1750 
+</div>
1751 
+
1752 
+</div>
1717 1753 
 <p>
1718 
-{\bf Total force on charges in volume}
1719 1754 
 \[
1720 
-    {\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau.
1721 
-  \]
1755 
+{\boldsymbol F} = \oint_S {\boldsymbol T} \cdot d{\boldsymbol a} - \varepsilon_0 \mu_0 \frac{d}{dt} \int_{\cal V} {\boldsymbol S} d\tau.
1756 
+\tag{totFo}\label{totFo}
1757 
+\]
1722 1758 
 </p>
1723 1759
1724 1760 
 </div>
 
@@ -1742,7 +1778,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1742 1778 
 </div>
1743 1779 
 <div id="postamble" class="status">
1744 1780 
 <p class="author">Author: Jean-Sébastien Caux</p>
1745 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1781 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1746 1782 
 <p class="validation"></p>
1747 1783 
 </div>
1748 1784

+ 155
- 61
build/emd_ce_poy.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-01 Tue 08:14 -->
4 
+<!-- 2022-03-02 Wed 15:45 -->
5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1638,38 +1638,45 @@ Total energy should be sum of these two.  Derivation from scratch.
1638 1638 
 <p>
1639 1639 
 Suppose that at time \(t\), we have fields \({\bf E}\) and \({\bf B}\) produced by some charge
1640 1640 
 and current distributions \(\rho\) and \({\bf J}\).  In an interval \(dt\), how much work is
1641 
-done by EM forces ?  From Lorentz force law:
1641 
+done by EM forces?  From Lorentz force law:
1642 1642 
 \[
1643 1643 
 {\bf F} \cdot d{\bf l} = q({\bf E} + {\bf v} \times {\bf B}) \cdot {\bf v} dt = q ~{\bf E} \cdot {\bf v} dt
1644 1644 
 \]
1645 1645 
 Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
1646 1646 
 at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
1647 
+</p>
1648 
+<div class="eqlabel" id="org0197499">
1649 
+<p>
1650 
+<a id="dWdt_intEJ"></a><a href="./emd_ce_poy.html#dWdt_intEJ"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1651 
+  <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"/>
1652 
+  <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"/>
1653 
+</svg></a>
1654 
+</p>
1655 
+<div class="alteqlabels" id="org9aacea3">
1656 
+<ul class="org-ul">
1657 
+<li>Gr (8.6)</li>
1658 
+</ul>
1659 
+
1660 
+</div>
1661 
+
1662 
+</div>
1663 
+<p>
1647 1664 
 \[
1648 1665 
 \frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J}
1649 
-\label{Gr(8.6)}
1666 
+\tag{dWdt_intEJ}\label{dWdt_intEJ}
1650 1667 
 \]
1651 1668 
 The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume.
1652 1669 
 In terms of fields alone:  use Ampère-Maxwell:
1653 1670 
 \[
1654 1671 
 {\bf E} \cdot {\bf J} = \frac{1}{\mu_0} {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) - \varepsilon_0 {\bf E} \cdot \frac{\partial {\bf E}}{\partial t}
1655 1672 
 \]
1656 
-Using product rule 6,
1673 
+Using <a href="./c_m_dc_pr.html#div_xprod">div_xprod</a>,
1657 1674 
 \[
1658 
-{\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}) = {\bf B} ⋅ ({\boldsymbol ∇} × {\bf E})
1659 
-</p>
1660 
-<ul class="org-ul">
1661 
-<li>{\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}),</li>
1662 
-</ul>
1663 
-<p>
1675 
+{\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}) = {\bf B} \cdot ({\boldsymbol \nabla} \times {\bf E}) - {\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}),
1664 1676 
 \]
1665 1677 
 Invoking Faraday \({\boldsymbol \nabla} \times {\bf E} = - \partial {\bf B}/\partial t\),
1666 1678 
 \[
1667 
-{\bf E} ⋅ ({\boldsymbol ∇} × {\bf B}) = - {\bf B} ⋅ \frac{∂ {\bf B}}{∂ t}
1668 
-</p>
1669 
-<ul class="org-ul">
1670 
-<li>{\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).</li>
1671 
-</ul>
1672 
-<p>
1679 
+{\bf E} \cdot ({\boldsymbol \nabla} \times {\bf B}) = - {\bf B} \cdot \frac{\partial {\bf B}}{\partial t} - {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}).
1673 1680 
 \]
1674 1681 
 But obviously,
1675 1682 
 \[
 
@@ -1679,28 +1686,35 @@ But obviously,
1679 1686 
 \]
1680 1687 
 so we get
1681 1688 
 \[
1682 
-{\bf E} ⋅ {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
1683 
-</p>
1684 
-<ul class="org-ul">
1685 
-<li>\frac{1}{\mu_0} {\boldsymbol ∇} ⋅ ({\bf E} × {\bf B}).</li>
1686 
-</ul>
1687 
-<p>
1689 
+{\bf E} \cdot {\bf J} = -\frac{1}{2} \frac{\partial}{\partial t} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} {\boldsymbol \nabla} \cdot ({\bf E} \times {\bf B}).
1688 1690 
 \label{Gr(8.8)}
1689 1691 
 \]
1690 
-Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1692 
+Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
1691 1693 
 we obtain
1692 1694 
 </p>
1693 
-<div class="main div" id="orgda8af3a">
1695 
+<div class="main div" id="orgf118f4f">
1694 1696 
 <p>
1695 
-{\bf Poynting's theorem}
1696 
-\[
1697 
-  \frac{dW}{dt} = -\frac{d}{d t} ∫_{\cal V} dτ \frac{1}{2} \left( ε_0 E^2 + \frac{1}{\mu_0} B^2 \right)
1697 
+<b>Poynting's theorem</b>
1698 1698 
 </p>
1699 
+<div class="eqlabel" id="org1b7cdac">
1700 
+<p>
1701 
+<a id="👉Thm"></a><a href="./emd_ce_poy.html#👉Thm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1702 
+  <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"/>
1703 
+  <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"/>
1704 
+</svg></a>
1705 
+</p>
1706 
+<div class="alteqlabels" id="orgd7b6cac">
1699 1707 
 <ul class="org-ul">
1700 
-<li>\frac{1}{\mu_0} \oint_{\cal S} d{\bf a} ⋅ ({\bf E} × {\bf B})</li>
1708 
+<li>Gr (8.9)</li>
1701 1709 
 </ul>
1710 
+
1711 
+</div>
1712 
+
1713 
+</div>
1702 1714 
 <p>
1703 
-  \label{Gr(8.9)}
1715 
+\[
1716 
+\frac{dW}{dt} = -\frac{d}{d t} \int_{\cal V} d\tau \frac{1}{2} \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right) - \frac{1}{\mu_0} \oint_{\cal S} d{\bf a} \cdot ({\bf E} \times {\bf B})
1717 
+\tag{👉Thm}\label{👉Thm}
1704 1718 
 \]
1705 1719 
 </p>
1706 1720 
 
@@ -1713,41 +1727,92 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1713 1727
1714 1728
1715 1729 
 <p>
1716 
-Energy per unit time, per unit area carried by EM fields:
1730 
+Energy per unit time, per unit area carried by EM fields: given by the
1731 
+</p>
1732 
+<div class="core div" id="orgf3198a5">
1733 
+<p>
1734 
+<b>Poynting vector</b>
1717 1735 
 </p>
1718 
-<div class="core div" id="org8a1c10e">
1736 
+<div class="eqlabel" id="org8725431">
1737 
+<p>
1738 
+<a id="PoyntingVec"></a><a href="./emd_ce_poy.html#PoyntingVec"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1739 
+  <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"/>
1740 
+  <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"/>
1741 
+</svg></a>
1742 
+</p>
1743 
+<div class="alteqlabels" id="org6f2879d">
1744 
+<ul class="org-ul">
1745 
+<li>Gr (8.10)</li>
1746 
+</ul>
1747 
+
1748 
+</div>
1749 
+
1750 
+</div>
1719 1751 
 <p>
1720 
-{\bf Poynting vector}
1721 1752 
 \[
1722 
-    {\bf S} \equiv \frac{1}{\mu_0} ({\bf E} \times {\bf B})
1723 
-    \label{Gr(8.10)}
1724 
-  \]
1753 
+{\bf S} \equiv \frac{1}{\mu_0} ({\bf E} \times {\bf B})
1754 
+\tag{PoyntingVec}\label{PoyntingVec}
1755 
+\]
1725 1756 
 </p>
1726 1757
1727 1758 
 </div>
1728 1759 
 <p>
1729 1760 
 We can thus express Poynting's theorem more compactly:
1730 1761 
 </p>
1731 
-<div class="core div" id="org57576be">
1762 
+<div class="core div" id="org3a4bb91">
1763 
+<p>
1764 
+<b>Poynting's theorem</b> (integral form)
1765 
+</p>
1766 
+<div class="eqlabel" id="orgbf2cc63">
1767 
+<p>
1768 
+<a id="PoyntingThm_int"></a><a href="./emd_ce_poy.html#PoyntingThm_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1769 
+  <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"/>
1770 
+  <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"/>
1771 
+</svg></a>
1772 
+</p>
1773 
+<div class="alteqlabels" id="org5f484e3">
1774 
+<ul class="org-ul">
1775 
+<li>Gr (8.11)</li>
1776 
+</ul>
1777 
+
1778 
+</div>
1779 
+
1780 
+</div>
1732 1781 
 <p>
1733 
-{\bf Poynting's theorem}
1734 1782 
 \[
1735 
-    \frac{dW}{dt} = - \frac{dU_{em}}{dt} - \oint_{\cal S} d{\bf a} \cdot {\bf S}.
1736 
-    \label{Gr(8.11)}
1737 
-  \]
1783 
+\frac{dW}{dt} = - \frac{dU_{em}}{dt} - \oint_{\cal S} d{\bf a} \cdot {\bf S}.
1784 
+\tag{PoyntingThm_int}\label{PoyntingThm_int}
1785 
+\]
1738 1786 
 </p>
1739 1787
1740 1788 
 </div>
1741 1789 
 <p>
1742 1790 
 where we have defined the total
1743 1791 
 </p>
1744 
-<div class="core div" id="org43eb64b">
1792 
+<div class="core div" id="orgbc7eb16">
1793 
+<p>
1794 
+<b>Energy in electromagnetic fields</b>
1795 
+</p>
1796 
+<div class="eqlabel" id="org89872c3">
1797 
+<p>
1798 
+<a id="Uem"></a><a href="./emd_ce_poy.html#Uem"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1799 
+  <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"/>
1800 
+  <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"/>
1801 
+</svg></a>
1802 
+</p>
1803 
+<div class="alteqlabels" id="org67613ca">
1804 
+<ul class="org-ul">
1805 
+<li>Gr (8.5)</li>
1806 
+</ul>
1807 
+
1808 
+</div>
1809 
+
1810 
+</div>
1745 1811 
 <p>
1746 
-{\bf Energy in electromagnetic fields}
1747 1812 
 \[
1748 
-    U_{em} \equiv \frac{1}{2} \int d\tau \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right)
1749 
-    \label{Gr(8.5)}
1750 
-  \]
1813 
+U_{em} \equiv \frac{1}{2} \int d\tau \left( \varepsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right)
1814 
+\tag{Uem}\label{Uem}
1815 
+\]
1751 1816 
 </p>
1752 1817
1753 1818 
 </div>
 
@@ -1764,13 +1829,30 @@ Then,
1764 1829 
 \]
1765 1830 
 so we get the
1766 1831 
 </p>
1767 
-<div class="core div" id="org29b53c2">
1832 
+<div class="core div" id="org487db23">
1833 
+<p>
1834 
+<b>Poynting theorem</b> (differential form)
1835 
+</p>
1836 
+<div class="eqlabel" id="org9593699">
1837 
+<p>
1838 
+<a id="PoyntingThm"></a><a href="./emd_ce_poy.html#PoyntingThm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1839 
+  <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"/>
1840 
+  <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"/>
1841 
+</svg></a>
1842 
+</p>
1843 
+<div class="alteqlabels" id="org129becb">
1844 
+<ul class="org-ul">
1845 
+<li>Gr (8.14)</li>
1846 
+</ul>
1847 
+
1848 
+</div>
1849 
+
1850 
+</div>
1768 1851 
 <p>
1769 
-{\bf Poynting theorem (differential form)}
1770 1852 
 \[
1771 
-    \frac{\partial}{\partial t} u_{em} + {\boldsymbol \nabla} \cdot {\bf S} = 0
1772 
-    \label{Gr(8.14)}
1773 
-  \]
1853 
+\frac{\partial}{\partial t} u_{em} + {\boldsymbol \nabla} \cdot {\bf S} = 0
1854 
+\tag{PoyntingThm}\label{PoyntingThm}
1855 
+\]
1774 1856 
 </p>
1775 1857
1776 1858 
 </div>
 
@@ -1781,26 +1863,38 @@ and has a similar for to the continuity equation
1781 1863
1782 1864
1783 1865
1784 
-<div class="example div" id="orgbb7ac4b">
1866 
+<div class="example div" id="orgd9e0ab5">
1867 
+<p>
1868 
+<b>Example:  Joule heating</b>
1869 
+</p>
1870 
+
1871 
+<p>
1872 
+<b>Task</b>: characterize the energy flow for a current-carrying wire.
1873 
+</p>
1874 
+
1875 
+<p>
1876 
+<b>Solution</b>:  the energy per unit time delivered to wire the wire can
1877 
+be obtained from Poynting's theorem.
1878 
+</p>
1879 
+
1785 1880 
 <p>
1786 
-\paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1787 1881 
 Assuming that the field is uniform, the electric field parallel to the wire is
1788 1882 
 \[
1789 
-    {\boldsymbol E} = \frac{V}{L} \hat{\boldsymbol x},
1790 
-  \]
1883 
+{\boldsymbol E} = \frac{V}{L} \hat{\boldsymbol x},
1884 
+\]
1791 1885 
 where \(V\) is the potential difference between the ends ald \(L\) is the length.  Magnetic field is circumferential:
1792 1886 
 wire of radius \(a\),
1793 1887 
 \[
1794 
-    {\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi}
1795 
-  \]
1888 
+{\boldsymbol B} = \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol \varphi}
1889 
+\]
1796 1890 
 Poynting:
1797 1891 
 \[
1798 
-    {\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
1799 
-  \]
1892 
+{\boldsymbol S} = \frac{1}{\mu_0} \frac{V}{L} \frac{\mu_0 I}{2\pi a} \hat{\boldsymbol x} \times \hat{\boldsymbol \varphi} = -\frac{VI}{2\pi a L} \hat{\boldsymbol s}
1893 
+\]
1800 1894 
 and points radially inwards.  Energy per unit time passing surface of wire:
1801 1895 
 \[
1802 
-    \int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
1803 
-  \]
1896 
+\int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
1897 
+\]
1804 1898 
 where the minus sign means energy is flowing {\it in} (the wire heats up),
1805 1899 
 and the value is as expected.
1806 1900 
 </p>
 
@@ -1826,7 +1920,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1826 1920 
 </div>
1827 1921 
 <div id="postamble" class="status">
1828 1922 
 <p class="author">Author: Jean-Sébastien Caux</p>
1829 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1923 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1830 1924 
 <p class="validation"></p>
1831 1925 
 </div>
1832 1926

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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1622,8 +1622,8 @@ Table of contents
1622 1622 
 </svg></a><span class="headline-id">emd.emw</span></h3>
1623 1623
1624 1624 
 <div class="outline-text-3" id="text-emd_emw">
1625 
-<details class="prereq" id="org9d74d32">
1626 
-<summary id="org175f28d">
1625 
+<details class="prereq" id="orgd522e84">
1626 
+<summary id="org3c8d630">
1627 1627 
 Prerequisites
1628 1628 
 </summary>
1629 1629 
 <ul class="org-ul">
 
@@ -1632,8 +1632,8 @@ Prerequisites
1632 1632 
 </ul>
1633 1633 
 </details>
1634 1634
1635 
-<details class="objectives" id="org73ffbc8">
1636 
-<summary id="org638d300">
1635 
+<details class="objectives" id="org98124eb">
1636 
+<summary id="org6546918">
1637 1637 
 Objectives
1638 1638 
 </summary>
1639 1639 
 <ul class="org-ul">
 
@@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1674 
 </div>
1675 1675 
 <div id="postamble" class="status">
1676 1676 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1677 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1678 1678 
 <p class="validation"></p>
1679 1679 
 </div>
1680 1680

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3 3 
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5 5 
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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1652,7 +1652,7 @@ so for a monochromatic EM plan wave,
1652 1652 
 \]
1653 1653 
 or more succinctly:
1654 1654 
 </p>
1655 
-<div class="main div" id="org677cdf7">
1655 
+<div class="main div" id="org0e45759">
1656 1656 
 <p>
1657 1657 
 {\bf Poynting vector of a monochromatic EM wave}
1658 1658 
 \[
 
@@ -1668,7 +1668,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1668 1668 
 <p>
1669 1669 
 Similary, we get the
1670 1670 
 </p>
1671 
-<div class="main div" id="orgc6fc8e6">
1671 
+<div class="main div" id="org4b55ff9">
1672 1672 
 <p>
1673 1673 
 {\bf Momentum density of a monochromatic EM wave}
1674 1674 
 \[
 
@@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1719 
 </div>
1720 1720 
 <div id="postamble" class="status">
1721 1721 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1722 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1723 1723 
 <p class="validation"></p>
1724 1724 
 </div>
1725 1725

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1653,7 +1653,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1653 1653 
 Generalizing to propagation in the direction of an arbitrary wavevector
1654 1654 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1655 1655 
 </p>
1656 
-<div class="core div" id="orgd2950da">
1656 
+<div class="core div" id="org53e84bf">
1657 1657 
 <p>
1658 1658 
 {\bf E and B fields for a monochromatic EM plane wave}
1659 1659 
 \[
 
@@ -1697,7 +1697,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1697 1697 
 </div>
1698 1698 
 <div id="postamble" class="status">
1699 1699 
 <p class="author">Author: Jean-Sébastien Caux</p>
1700 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1700 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1701 1701 
 <p class="validation"></p>
1702 1702 
 </div>
1703 1703

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3 3 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1650,7 +1650,7 @@ These take the form of coupled first-order partial differential equations for \(
1650 1650 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1651 1651 
 we get the
1652 1652 
 </p>
1653 
-<div class="core div" id="org88fc1e4">
1653 
+<div class="core div" id="orgb3cd985">
1654 1654 
 <p>
1655 1655 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1656 1656 
 \[
 
@@ -1706,7 +1706,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1706 1706 
 </div>
1707 1707 
 <div id="postamble" class="status">
1708 1708 
 <p class="author">Author: Jean-Sébastien Caux</p>
1709 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1709 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1710 1710 
 <p class="validation"></p>
1711 1711 
 </div>
1712 1712

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7 7 
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@@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 1646 
 </div>
1647 1647 
 <div id="postamble" class="status">
1648 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1649 
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1650 1650 
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1651 1651 
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1652 1652

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@@ -1646,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 1646 
 </div>
1647 1647 
 <div id="postamble" class="status">
1648 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1649 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1650 1650 
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1651 1651 
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1652 1652

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3 3 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1657,7 +1657,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1657 1657 
 \]
1658 1658 
 We therefore have the
1659 1659 
 </p>
1660 
-<div class="core div" id="org9606db6">
1660 
+<div class="core div" id="org0421a72">
1661 1661 
 <p>
1662 1662 
 {\bf Polarization current density}
1663 1663 
 \[
 
@@ -1675,7 +1675,7 @@ the polarization current is the result of linear motion of charge when
1675 1675 
 polarization changes).  We can check consistency with the continuity equation
1676 1676 
 associated to the conservation of bound charges:
1677 1677 
 </p>
1678 
-<aside id="orge1ae7f1">
1678 
+<aside id="org642846e">
1679 1679 
 <p>
1680 1680 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1681 1681 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1698,7 +1698,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1698 1698 
 In view of this:  total charge density can be separated into 2 parts,
1699 1699 
 {\it free} and {\it bound}:
1700 1700 
 </p>
1701 
-<div class="main div" id="orgb0d00db">
1701 
+<div class="main div" id="orgba471ef">
1702 1702 
 <p>
1703 1703 
 \[
1704 1704 
     \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
 
@@ -1711,7 +1711,7 @@ In view of this:  total charge density can be separated into 2 parts,
1711 1711 
 and current can be separated into three parts, {\it free}, {\it bound} and
1712 1712 
 {\it polarization}:
1713 1713 
 </p>
1714 
-<div class="main div" id="org89c562c">
1714 
+<div class="main div" id="org2ffd81b">
1715 1715 
 <p>
1716 1716 
 \[
1717 1717 
   {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
 
@@ -1735,7 +1735,7 @@ Gauss's law:  can be rewritten
1735 1735 
 \]
1736 1736 
 where (as in static case)
1737 1737 
 </p>
1738 
-<div class="core div" id="org0196779">
1738 
+<div class="core div" id="org88bb3c5">
1739 1739 
 <p>
1740 1740 
 \[
1741 1741 
     {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1761,7 +1761,7 @@ or
1761 1761 
 \]
1762 1762 
 where as before
1763 1763 
 </p>
1764 
-<div class="core div" id="org2b5f6d5">
1764 
+<div class="core div" id="org90cea20">
1765 1765 
 <p>
1766 1766 
 \[
1767 1767 
     {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1779,7 +1779,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1779 1779 
 <p>
1780 1780 
 In terms of free charges and currents, we thus get
1781 1781 
 </p>
1782 
-<div class="core div" id="org2c2cd2a">
1782 
+<div class="core div" id="orgd6526ab">
1783 1783 
 <p>
1784 1784 
 {\bf Maxwell's equations {\it (in matter)}}
1785 1785 
 </p>
 
@@ -1805,7 +1805,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1805 1805 
 in terms of \({\bf E}\) and \({\bf B}\).
1806 1806 
 For the restricted case of linear media:
1807 1807 
 </p>
1808 
-<div class="main div" id="orgcaf59d5">
1808 
+<div class="main div" id="orgd345cd6">
1809 1809 
 <p>
1810 1810 
 \[
1811 1811 
     {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
 
@@ -1842,7 +1842,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1842 1842 
 </div>
1843 1843 
 <div id="postamble" class="status">
1844 1844 
 <p class="author">Author: Jean-Sébastien Caux</p>
1845 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1845 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1846 1846 
 <p class="validation"></p>
1847 1847 
 </div>
1848 1848

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1624 1624 
 <p>
1625 1625 
 Discontinuities between different media, deduced from
1626 1626 
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1627 
-<div class="core div" id="orgc82cb6f">
1627 
+<div class="core div" id="org48bc400">
1628 1628 
 <p>
1629 1629 
 {\bf Maxwell's equations {\it (in matter)}, integral form}
1630 1630 
 </p>
 
@@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1715 1715 
 </div>
1716 1716 
 <div id="postamble" class="status">
1717 1717 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1718 
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1719 1719 
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1720 1720 
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1721 1721

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1649 1649 
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1650 1650 
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1651 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1652 
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1653 1653 
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1654 1654 
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1645 1645 
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1646 1646 
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1647 1647 
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1648 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1648 
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1649 1649 
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1650 1650 
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1741 1741 
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1742 1742 
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1743 1743 
 <p class="author">Author: Jean-Sébastien Caux</p>
1744 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1744 
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1745 1745 
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@@ -1659,7 +1659,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1659 1659 
 \]
1660 1660 
 where the index of refraction of the material is defined as
1661 1661 
 </p>
1662 
-<div class="main div" id="orga0b7fe5">
1662 
+<div class="main div" id="orgd0bce92">
1663 1663 
 <p>
1664 1664 
 {\bf Index of refraction}
1665 1665 
 \[
 
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1717 1717 
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1718 1718 
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1719 
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1644 
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1734 1734 
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1735 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1659,7 +1659,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1659 1659 
 <p>
1660 1660 
 From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
1661 1661 
 </p>
1662 
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1662 
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1663 1663 
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1664 1664 
 {\bf First law of reflection:}
1665 1665 
 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1674,7 +1674,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
1674 1674 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1675 1675 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1676 1676 
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1677 
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1679 1679 
 {\bf Law of reflection}
1680 1680 
 \[
 
@@ -1732,7 +1732,7 @@ while the third equation becomes
1732 1732 
 \]
1733 1733 
 Writing everything in terms of the incident amplitude, we get
1734 1734 
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1735 
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1735 
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1736 1736 
 <p>
1737 1737 
 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
1738 1738 
 \[
 
@@ -1752,7 +1752,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1752 1752 
 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
1753 1753 
 For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
1754 1754 
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1755 
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1757 1757 
 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
1758 1758 
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1802 1802 
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1803 1803 
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1804 1804 
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1805 
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1669 1669 
 <p class="author">Author: Jean-Sébastien Caux</p>
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1691 1691 
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1692 1692 
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1693 1693 
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1694 
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1701 1701 
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1702 1702 
 <p class="author">Author: Jean-Sébastien Caux</p>
1703 
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1622 1622 
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1623 1623
1624 1624 
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1627 1627 
 Prerequisites
1628 1628 
 </summary>
1629 1629 
 <ul class="org-ul">
 
@@ -1631,8 +1631,8 @@ Prerequisites
1631 1631 
 </ul>
1632 1632 
 </details>
1633 1633
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1636 1636 
 Objectives
1637 1637 
 </summary>
1638 1638 
 <ul class="org-ul">
 
@@ -1668,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1668 1668 
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1669 1669 
 <div id="postamble" class="status">
1670 1670 
 <p class="author">Author: Jean-Sébastien Caux</p>
1671 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1671 
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1665 1665 
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1666 1666 
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1667 1667 
 <p class="author">Author: Jean-Sébastien Caux</p>
1668 
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1668 
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1664 1664 
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1665 1665 
 <div id="postamble" class="status">
1666 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1667 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1637,7 +1637,7 @@ while the equation for \(V\) becomes
1637 1637 
 \]
1638 1638 
 These can be written compactly upon introducing a new operator: the
1639 1639 
 </p>
1640 
-<div class="core div" id="org892bf92">
1640 
+<div class="core div" id="org242c17e">
1641 1641 
 <p>
1642 1642 
 {\bf d'Alembertian operator}
1643 1643 
 \[
 
@@ -1650,7 +1650,7 @@ These can be written compactly upon introducing a new operator: the
1650 1650 
 <p>
1651 1651 
 so we get the
1652 1652 
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1653 
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1653 
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1655 1655 
 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1656 1656 
 \[
 
@@ -1700,7 +1700,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1700 1700 
 </div>
1701 1701 
 <div id="postamble" class="status">
1702 1702 
 <p class="author">Author: Jean-Sébastien Caux</p>
1703 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1703 
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1638 1638 
 <p>
1639 1639 
 Easiest:
1640 1640 
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1641 
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1643 1643 
 \[
1644 1644 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1654,7 +1654,7 @@ Putting this into Faraday's law gives
1654 1654 
 \]
1655 1655 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1656 1656 
 </p>
1657 
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1657 
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1658 1658 
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1659 1659 
 \[
1660 1660 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1667,7 +1667,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1667 1667 
 <p>
1668 1668 
 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
1669 1669 
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1670 
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1672 1672 
 \[
1673 1673 
     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1683,7 +1683,7 @@ whereas Amp{\`ere}-Maxwell becomes
1683 1683 
 \]
1684 1684 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1685 1685 
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1686 
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1687 1687 
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1688 1688 
 \[
1689 1689 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1719,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1719 
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1720 1720 
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1721 1721 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
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1722 
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1649 1649 
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1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
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1626 1626 
 <li>Gr 3</li>
1627 1627 
 </ul>
1628 1628
1629 
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1631 1631 
 Prerequisites
1632 1632 
 </summary>
1633 1633 
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@@ -1635,8 +1635,8 @@ Prerequisites
1635 1635 
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1636 1636 
 </details>
1637 1637
1638 
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1639 
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1638 
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1639 
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1640 1640 
 Objectives
1641 1641 
 </summary>
1642 1642 
 <ul class="org-ul">
 
@@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1674 
 </div>
1675 1675 
 <div id="postamble" class="status">
1676 1676 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1677 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1678 1678 
 <p class="validation"></p>
1679 1679 
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1680 1680

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@@ -1,7 +1,7 @@
1 1 
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4 
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 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1632,7 +1632,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
1632 1632 
 defines an electrostatic problem, whose solution is in principle obtained
1633 1633 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1634 1634 
 </p>
1635 
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1636 1636 
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1637 1637
1638 1638 
 </p>
 
@@ -1646,7 +1646,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1646 1646 
 or (often simpler) by calculating the electrostatic potential, using either the
1647 1647 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1648 1648 
 </p>
1649 
-<div class="main div" id="orge64dc41">
1649 
+<div class="main div" id="org9f1e898">
1650 1650 
 <p>
1651 1651
1652 1652 
 </p>
 
@@ -1666,7 +1666,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1666 1666 
 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
1667 1667 
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1668 1668
1669 
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1670 1670 
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1671 1671
1672 1672 
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@@ -1682,7 +1682,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1682 1682 
 <p>
1683 1683 
 In the specific case where the charge density vanishes, we fall back onto the simpler Laplace equation <a href="./ems_es_ep_PL.html#Lap">Lap</a>
1684 1684 
 </p>
1685 
-<div class="core div" id="orgfefb816">
1685 
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1686 1686 
 <p>
1687 1687
1688 1688 
 </p>
 
@@ -1720,7 +1720,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1720 1720 
 </div>
1721 1721 
 <div id="postamble" class="status">
1722 1722 
 <p class="author">Author: Jean-Sébastien Caux</p>
1723 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
1723 
+<p class="date">Created: 2022-03-02 Wed 15:45</p>
1724 1724 
 <p class="validation"></p>
1725 1725 
 </div>
1726 1726

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@@ -1,7 +1,7 @@
1 1 
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4 
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4 
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5 5 
 <meta charset="utf-8">
6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1638,14 +1638,14 @@ In one dimension, the potential is a single-variable
1638 1638 
 function \(\phi (x)\) and the Laplace equation reads
1639 1639 
 </p>
1640 1640
1641 
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1641 
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1642 1642 
 <p>
1643 1643 
 <a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1644 1644 
   <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"/>
1645 1645 
   <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"/>
1646 1646 
 </svg></a>
1647 1647 
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1648 
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1649 1649
1650 1650 
 </div>
1651 1651 
 
@@ -1660,14 +1660,14 @@ function \(\phi (x)\) and the Laplace equation reads
1660 1660 
 <p>
1661 1661 
 The solution to this is
1662 1662 
 </p>
1663 
-<div class="eqlabel" id="orged9e79a">
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1664 1664 
 <p>
1665 1665 
 <a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1666 1666 
   <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"/>
1667 1667 
   <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"/>
1668 1668 
 </svg></a>
1669 1669 
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1670 
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1671 1671 
 <ul class="org-ul">
1672 1672 
 <li>Gr (3.6)</li>
1673 1673 
 </ul>
 
@@ -1726,14 +1726,14 @@ In two dimensions, the potential becomes a function
1726 1726 
 of two variables (here: \(x\) and \(y\)), so Laplace's
1727 1727 
 equation now reads
1728 1728 
 </p>
1729 
-<div class="eqlabel" id="orgdc4453f">
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1730 1730 
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1731 1731 
 <a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1732 1732 
   <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"/>
1733 1733 
   <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"/>
1734 1734 
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1735 1735 
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1737 1737
1738 1738 
 </div>
1739 1739 
 
@@ -1786,14 +1786,14 @@ a point equals its value averaged over a sphere
1786 1786 
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
1787 1787 
 (and of course not containing any charges),
1788 1788 
 </p>
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1790 1790 
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1791 1791 
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1792 1792 
   <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"/>
1793 1793 
   <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"/>
1794 1794 
 </svg></a>
1795 1795 
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1797 1797
1798 1798 
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1799 1799 
 
@@ -1805,8 +1805,8 @@ a point equals its value averaged over a sphere
1805 1805 
 \]
1806 1806 
 </p>
1807 1807
1808 
-<details id="org0d796b5">
1809 
-<summary id="org6d53cda">
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+<details id="org0f27c64">
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+<summary id="org881e712">
1810 1810 
 <strong>Physicist's proof</strong>
1811 1811 
 </summary>
1812 1812 
 <p>
 
@@ -1868,8 +1868,8 @@ proving the theorem.
1868 1868 
 </p>
1869 1869 
 </details>
1870 1870
1871 
-<details id="org68c57fd">
1872 
-<summary id="org5db3c01">
1871 
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1872 
+<summary id="orgf2b7c76">
1873 1873 
 <strong>Formal proof</strong>
1874 1874 
 </summary>
1875 1875 
 
@@ -1919,14 +1919,14 @@ we get the following general
1919 1919 
 <p>
1920 1920 
 <b>Theorem</b>:
1921 1921 
 </p>
1922 
-<div class="eqlabel" id="org59c453b">
1922 
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1923 1923 
 <p>
1924 1924 
 <a id="dfdR_intLap"></a><a href="./ems_ca_fe_L.html#dfdR_intLap"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1925 1925 
   <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"/>
1926 1926 
   <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"/>
1927 1927 
 </svg></a>
1928 1928 
 </p>
1929 
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1929 
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1930 1930
1931 1931 
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1932 1932 
 
@@ -1979,19 +1979,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1979 1979 
 of the \(f_x + f_y + f_z = 0\) condition above.
1980 1980 
 </p>
1981 1981
1982 
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1982 
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1983 1983 
 <p>
1984 1984 
 <a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1985 1985 
   <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"/>
1986 1986 
   <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"/>
1987 1987 
 </svg></a>
1988 1988 
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1990 1990
1991 1991 
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1992 1992
1993 1993 
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1995 1995 
 <p>
1996 1996 
 <b>Earnshaw's theorem (physical version)</b> <br>
1997 1997 
 </p>
 
@@ -2018,7 +2018,7 @@ Going back to Poisson's equation, we can make a few comments:
2018 2018
2019 2019 
 <p>
2020 2020 
 We therefore want to ask the question:  <i>under what conditions can an electrostatic problem be fully
2021 
-defined by solving Poisson's equation ?</i>
2021 
+defined by solving Poisson's equation?</i>
2022 2022 
 </p>
2023 2023
2024 2024 
 <p>
 
@@ -2087,7 +2087,7 @@ their maximal and minimal value on the boundary, we must have \(U = 0\) \(\foral
2087 2087
2088 2088
2089 2089 
 <p>
2090 
-This all feels a bit amateurish and not very systematic. Can we be more precise and general?  What kinds of boundary information do we really need to specify the solution uniquely ?
2090 
+This all feels a bit amateurish and not very systematic. Can we be more precise and general?  What kinds of boundary information do we really need to specify the solution uniquely?
2091 2091 
 </p>
2092 2092 
 </div>
2093 2093 
 </div>
 
@@ -2110,7 +2110,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
2110 2110 
 </div>
2111 2111 
 <div id="postamble" class="status">
2112 2112 
 <p class="author">Author: Jean-Sébastien Caux</p>
2113 
-<p class="date">Created: 2022-03-01 Tue 08:14</p>
2113 
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