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Update 2022-02-15 10:32

master
Jean-Sébastien 2 years ago
parent
09a8ba5fb6
commit
6874e66024
100 changed files with 408 additions and 370 deletions
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build/a.html View File 

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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 </div>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
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1677 1677 
 </svg></a>
1678 1678 
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1679 
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1679 
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1680 1680
1681 1681 
 </div>
1682 1682 
 
@@ -1692,14 +1692,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
1692 1692 
 <p>
1693 1693 
 Infinitesimal volume element:
1694 1694 
 </p>
1695 
-<div class="eqlabel" id="orgf5fa0db">
1695 
+<div class="eqlabel" id="org899678f">
1696 1696 
 <p>
1697 1697 
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1698 1698 
   <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"/>
1699 1699 
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1700 1700 
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1701 1701 
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1702 
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1702 
+<div class="alteqlabels" id="orgefd25f6">
1703 1703
1704 1704 
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1705 1705 
 
@@ -1720,14 +1720,14 @@ Infinitesimal surface element:  depends on situation.
1720 1720 
 <div id="outline-container-c_m_cs_sph_grad" class="outline-6">
1721 1721 
 <h6 id="c_m_cs_sph_grad"><a href="#c_m_cs_sph_grad">Gradient</a></h6>
1722 1722 
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1723 
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1723 
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1724 1724 
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1725 1725 
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1726 1726 
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1727 1727 
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1728 1728 
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1729 1729 
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1730 
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1730 
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1731 1731
1732 1732 
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1733 1733 
 
@@ -1744,14 +1744,14 @@ Infinitesimal surface element:  depends on situation.
1744 1744 
 <div id="outline-container-c_m_cs_sph_div" class="outline-6">
1745 1745 
 <h6 id="c_m_cs_sph_div"><a href="#c_m_cs_sph_div">Divergence</a></h6>
1746 1746 
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1747 
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1747 
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1748 1748 
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1749 1749 
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1750 1750 
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1751 1751 
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1752 1752 
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1753 1753 
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1754 
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1754 
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1755 1755
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1757 1757 
 
@@ -1768,14 +1768,14 @@ Infinitesimal surface element:  depends on situation.
1768 1768 
 <div id="outline-container-c_m_cs_sph_curl" class="outline-6">
1769 1769 
 <h6 id="c_m_cs_sph_curl"><a href="#c_m_cs_sph_curl">Curl</a></h6>
1770 1770 
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1771 
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1772 1772 
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1773 1773 
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1774 1774 
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1775 1775 
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1776 1776 
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1777 1777 
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1778 
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1778 
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1779 1779
1780 1780 
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1781 1781 
 
@@ -1794,14 +1794,14 @@ Infinitesimal surface element:  depends on situation.
1794 1794 
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1795 1795 
 <h6 id="c_m_cs_sph_lap"><a href="#c_m_cs_sph_lap">Laplacian</a></h6>
1796 1796 
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1797 
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1799 1799 
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1800 1800 
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1801 1801 
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1802 1802 
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1803 1803 
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1804 
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1805 1805
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1807 1807 
 
@@ -1835,7 +1835,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1835 1835 
 </div>
1836 1836 
 <div id="postamble" class="status">
1837 1837 
 <p class="author">Author: Jean-Sébastien Caux</p>
1838 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1838 
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1839 1839 
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1840 1840 
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1841 1841

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1626 1626 
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1627 1627 
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1628 1628 
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1629 
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1629 
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1630 1630 
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1631 1631 
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@@ -1628,7 +1628,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1628 1628 
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1629 1629 
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1630 1630 
 <p class="author">Author: Jean-Sébastien Caux</p>
1631 
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1631 
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1632 1632 
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7 7 
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@@ -1599,9 +1599,9 @@ Table of contents
1599 1599 
 <div class="outline-text-5" id="text-c_m_dc_d2">
1600 1600 
 </div>
1601 1601
1602 
-<div id="outline-container-org909c107" class="outline-6">
1603 
-<h6 id="org909c107"><a href="#org909c107">Divergence of gradient</a></h6>
1604 
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1602 
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1603 
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1604 
+<div class="outline-text-6" id="text-orga412b41">
1605 1605 
 <p>
1606 1606 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1607 1607 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
1610 1610 
 </div>
1611 1611 
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1612 1612
1613 
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1614 
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1615 
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1613 
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1614 
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1615 
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1616 1616 
 <p>
1617 1617 
 This always vanishes.
1618 1618 
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1619 1619 
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1620 1620 
 </div>
1621 1621
1622 
-<div id="outline-container-org9eefe8b" class="outline-6">
1623 
-<h6 id="org9eefe8b"><a href="#org9eefe8b">Gradient of the divergence</a></h6>
1624 
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1622 
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1623 
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1624 
+<div class="outline-text-6" id="text-org51f1f70">
1625 1625 
 <p>
1626 1626 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1627 1627 
 </p>
1628 1628 
 </div>
1629 1629 
 </div>
1630 1630
1631 
-<div id="outline-container-org0de49a1" class="outline-6">
1632 
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1633 
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1631 
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1632 
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1633 
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1634 1634 
 <p>
1635 1635 
 This always vanishes.
1636 1636 
 </p>
1637 1637 
 </div>
1638 1638 
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1639 1639
1640 
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1641 
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1642 
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1640 
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1641 
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1642 
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1643 1643 
 <p>
1644 1644 
 \[
1645 1645 
 {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
 
@@ -1666,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1666 1666 
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1667 1667 
 <div id="postamble" class="status">
1668 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1669 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1669 
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1670 1670 
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1627 1627 
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1628 1628 
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1629 1629 
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1630 
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1630 
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1624 1624 
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1625 1625 
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1626 1626 
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1627 
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1627 
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1648 1648 
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1649 1649 
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1650 1650 
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1651 
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1651 
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1621 1621 
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1622 1622 
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1623 1623 
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1624 
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1624 
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1623 1623 
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1624 1624 
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1625 1625 
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1626 
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1626 
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1635 1635 
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1636 1636 
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1637 1637 
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1638 
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1638 
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@@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1620 1620 
 More generally,
1621 1621 
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1622 1622
1623 
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1624 1624 
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1625 1625 
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1626 1626 
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1627 1627 
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1628 1628 
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1629 1629 
 </p>
1630 
-<div class="alteqlabels" id="org4d37466">
1630 
+<div class="alteqlabels" id="org811a9d7">
1631 1631 
 <ul class="org-ul">
1632 1632 
 <li>Gr (1.100)</li>
1633 1633 
 </ul>
 
@@ -1646,14 +1646,14 @@ More generally,
1646 1646 
 Since
1647 1647 
 </p>
1648 1648
1649 
-<div class="eqlabel" id="org209f4b3">
1649 
+<div class="eqlabel" id="org23b7fff">
1650 1650 
 <p>
1651 1651 
 <a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1653 1653 
   <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"/>
1654 1654 
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1656 
+<div class="alteqlabels" id="orgabc977b">
1657 1657 
 <ul class="org-ul">
1658 1658 
 <li>Gr (1.101)</li>
1659 1659 
 </ul>
 
@@ -1669,14 +1669,14 @@ Since
1669 1669 
 <p>
1670 1670 
 we have that
1671 1671 
 </p>
1672 
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1672 
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1676 1676 
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1677 1677 
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1679 
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1680 1680 
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1681 1681 
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1682 1682 
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1711 
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1711 
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1713 1713 
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1714 1714

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1643 1643 
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1644 
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1644 
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1629 
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1635 1635 
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1636 
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1636 
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build/c_m_ic_gauss.html View File 

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1629 1629 
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1630 
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build/c_m_ic_ip.html View File 

@@ -1,7 +1,7 @@
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1634 1634 
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1636 1636 
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1637 
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1637 
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@@ -1,7 +1,7 @@
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7 7 
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@@ -1599,9 +1599,9 @@ Table of contents
1599 1599 
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1600 1600 
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1601 1601
1602 
-<div id="outline-container-org1c47caa" class="outline-6">
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1605 1605 
 <p>
1606 1606 
 \[
1607 1607 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1630,9 +1630,9 @@ Integral over a closed loop:
1630 1630 
 </div>
1631 1631 
 </div>
1632 1632
1633 
-<div id="outline-container-orgd540d59" class="outline-6">
1634 
-<h6 id="orgd540d59"><a href="#orgd540d59">Surface Integrals</a></h6>
1635 
-<div class="outline-text-6" id="text-orgd540d59">
1633 
+<div id="outline-container-orgeef7716" class="outline-6">
1634 
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1635 
+<div class="outline-text-6" id="text-orgeef7716">
1636 1636 
 <p>
1637 1637 
 \[
1638 1638 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1652,9 +1652,9 @@ Over a closed surface:
1652 1652 
 </div>
1653 1653 
 </div>
1654 1654
1655 
-<div id="outline-container-org5afd08a" class="outline-6">
1656 
-<h6 id="org5afd08a"><a href="#org5afd08a">Volume Integrals</a></h6>
1657 
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1655 
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1657 
+<div class="outline-text-6" id="text-org999b522">
1658 1658 
 <p>
1659 1659 
 \[
1660 1660 
 \int_{\cal V} T d\tau
 
@@ -1695,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1695 1695 
 </div>
1696 1696 
 <div id="postamble" class="status">
1697 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1698 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1698 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1699 1699 
 <p class="validation"></p>
1700 1700 
 </div>
1701 1701

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1597 1597 
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1598 1598 
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1600 
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1602 1602 
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1604 1604 
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1605 1605 
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1607 
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1610 1610 
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1649 1649 
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1650 1650 
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1651 1651 
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1652 
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1652 
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1654 1654 
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1639 
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1639 
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1652 
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1624 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
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1641 1641 
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1668 1668 
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1670 1670 
 <p class="author">Author: Jean-Sébastien Caux</p>
1671 
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1598 1598 
 </svg></a><span class="headline-id">d</span></h2>
1599 1599
1600 1600 
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1601 
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1603 1603 
 Objectives
1604 1604 
 </summary>
1605 1605 
 
@@ -1663,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1663 1663 
 </div>
1664 1664 
 <div id="postamble" class="status">
1665 1665 
 <p class="author">Author: Jean-Sébastien Caux</p>
1666 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1666 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
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 <p class="validation"></p>
1668 1668 
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1643 1643 
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1645 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
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1640 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
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1643 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
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 <p class="author">Author: Jean-Sébastien Caux</p>
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1660 1660 
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1662 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
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1640 1640 
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1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
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 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
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1643 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
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1651 1651 
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1652 1652 
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1653 1653 
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1654 
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1647 1647 
 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
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1598 1598 
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1599 1599
1600 1600 
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1601 
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1603 1603 
 Prerequisites
1604 1604 
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@@ -1608,8 +1608,8 @@ Prerequisites
1608 1608 
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1613 1613 
 Objectives
1614 1614 
 </summary>
1615 1615 
 <ul class="org-ul">
 
@@ -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: 2022-02-14 Mon 20:35</p>
1653 
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1624 1624 
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1625 1625 
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1626 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1627 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1627 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1638,7 +1638,7 @@ Empirically:  the changing magnetic field induces an electric current around
1638 1638 
 the circuit. This current is really driven by an electric field having a component
1639 1639 
 along the wire.  The line integral of this field is called the
1640 1640 
 </p>
1641 
-<div class="core div" id="org1df022a">
1641 
+<div class="core div" id="org866e4fd">
1642 1642 
 <p>
1643 1643 
 <b>Electromotive force (or electromotance)</b>,
1644 1644 
   \[
 
@@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
1660 1660 
 \]
1661 1661 
 so we obtain
1662 1662 
 </p>
1663 
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1663 
+<div class="core div" id="org9e4ada0">
1664 1664 
 <p>
1665 1665 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1666 1666 
   \[
 
@@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
1678 1678 
 \]
1679 1679 
 we obtain
1680 1680 
 </p>
1681 
-<div class="core div" id="org3cc6f30">
1681 
+<div class="core div" id="org2c677eb">
1682 1682 
 <p>
1683 1683 
 <b>Faraday's law</b> (differential form)
1684 1684 
   \[
 
@@ -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: 2022-02-14 Mon 20:35</p>
1718 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1719 1719 
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1720 1720 
 </div>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1657 1657 
 \]
1658 1658 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1659 1659 
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1660 
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1660 
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1661 1661 
 <p>
1662 1662 
 \[
1663 1663 
     W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
 
@@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
1678 1678 
 \hspace{2cm} \mbox{(7.31 and 7.34)}
1679 1679 
 \end{align}
1680 1680
1681 
-<div class="example div" id="org9178294">
1681 
+<div class="example div" id="org9d858d6">
1682 1682 
 <p>
1683 1683 
 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1684 1684 
 Find energy stored in section of length \(l\).
 
@@ -1717,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1717 1717 
 </div>
1718 1718 
 <div id="postamble" class="status">
1719 1719 
 <p class="author">Author: Jean-Sébastien Caux</p>
1720 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1720 
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1721 1721 
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1722 1722 
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@@ -1637,7 +1637,7 @@ M_{12} = M_{21}
1637 1637 
 \]
1638 1638 
 </p>
1639 1639
1640 
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1640 
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1641 1641 
 <p>
1642 1642 
 \paragraph{Example 7.10:}
1643 1643 
 short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
 
@@ -1687,7 +1687,7 @@ Inductance:  measured in {\bf henries} (\(H\)).  \(H = V s/A\).
1687 1687 
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1688 1688
1689 1689
1690 
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1690 
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1691 1691 
 <p>
1692 1692 
 \paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1693 1693 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
 
@@ -1714,7 +1714,7 @@ Total flux:  \(N\) times this, so self-inductance is
1714 1714 
 Inductance (like capacitance) is intrinsically positive.  Use Lenz law.  Think of {\bf back EMF}.
1715 1715 
 </p>
1716 1716
1717 
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1717 
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1718 1718 
 <p>
1719 1719 
 \paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1720 1720 
 What is the current ?
 
@@ -1753,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1753 1753 
 </div>
1754 1754 
 <div id="postamble" class="status">
1755 1755 
 <p class="author">Author: Jean-Sébastien Caux</p>
1756 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1756 
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1757 1757 
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1758 1758 
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@@ -1621,7 +1621,7 @@ law in integral form:
1621 1621
1622 1622
1623 1623
1624 
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1624 
+<div class="example div" id="orgdba2a6f">
1625 1625 
 <p>
1626 1626 
 {\bf Example 7.7:}
1627 1627 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
@@ -1637,7 +1637,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1637 1637 
 </div>
1638 1638
1639 1639
1640 
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1640 
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1641 1641 
 <p>
1642 1642 
 {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
1643 1643 
 Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
 
@@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
1671 1671 
 'slow enough' phenomena.
1672 1672 
 </p>
1673 1673
1674 
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1674 
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1675 1675 
 <p>
1676 1676 
 {\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
1677 1677 
 induced \({\bf E}\) field as a function of distance \(s\) from wire.
 
@@ -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: 2022-02-14 Mon 20:35</p>
1722 
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1723 1723 
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1624 1624 
 </div>
1625 1625 
 <div id="postamble" class="status">
1626 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1627 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1627 
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@@ -1600,7 +1600,7 @@ Table of contents
1600 1600 
 <p>
1601 1601 
 Full set of equations for the electromagnetic field:
1602 1602 
 </p>
1603 
-<div class="core div" id="org443ae4f">
1603 
+<div class="core div" id="orgc22744b">
1604 1604 
 <p>
1605 1605 
 {\bf Maxwell's equations} {\it (in vacuum)}
1606 1606 
 </p>
 
@@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
1616 1616 
 <p>
1617 1617 
 Complement:
1618 1618 
 </p>
1619 
-<div class="core div" id="orga5a9517">
1619 
+<div class="core div" id="orgb9c1836">
1620 1620 
 <p>
1621 1621 
 {\bf Force law}
1622 1622 
 \[
 
@@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
1640 1640 
 <p>
1641 1641 
 Better way of writing:  all fields on left, all sources on right,
1642 1642 
 </p>
1643 
-<div class="core div" id="org3e0ec76">
1643 
+<div class="core div" id="orge08490a">
1644 1644 
 \begin{align}
1645 1645 
   (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
1646 1646 
   &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
 
@@ -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: 2022-02-14 Mon 20:35</p>
1673 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1674 1674 
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1675 1675 
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1676 1676

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1607,7 +1607,7 @@ the continuity equation as
1607 1607 
 \]
1608 1608 
 The extra term would thus be eliminated if we were to put
1609 1609 
 </p>
1610 
-<div class="core div" id="org0252f37">
1610 
+<div class="core div" id="org3047d46">
1611 1611 
 <p>
1612 1612 
 \[
1613 1613 
     {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
 
@@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation
1631 1631 
 <p>
1632 1632 
 Maxwell baptized this term the
1633 1633 
 </p>
1634 
-<div class="core div" id="orgce64fc0">
1634 
+<div class="core div" id="org8e0a58e">
1635 1635 
 <p>
1636 1636 
 {\bf Displacement current}
1637 1637 
 \[
 
@@ -1679,7 +1679,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1679 1679 
 </div>
1680 1680 
 <div id="postamble" class="status">
1681 1681 
 <p class="author">Author: Jean-Sébastien Caux</p>
1682 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1682 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1683 1683 
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1684 1684 
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1649 1649 
 </div>
1650 1650 
 <div id="postamble" class="status">
1651 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1652 
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1647 1647 
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1648 1648 
 <div id="postamble" class="status">
1649 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1650 
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1598 1598 
 </svg></a><span class="headline-id">emd.ce</span></h3>
1599 1599
1600 1600 
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1601 
-<details class="prereq" id="org8f62609">
1602 
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1601 
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1603 1603 
 Prerequisites
1604 1604 
 </summary>
1605 1605 
 <ul class="org-ul">
 
@@ -1607,8 +1607,8 @@ Prerequisites
1607 1607 
 </ul>
1608 1608 
 </details>
1609 1609
1610 
-<details class="objectives" id="org6c18760">
1611 
-<summary id="org51c8198">
1610 
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1611 
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1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
@@ -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: 2022-02-14 Mon 20:35</p>
1649 
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1650 1650 
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1651 1651 
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1652 1652

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@@ -1600,7 +1600,7 @@ Table of contents
1600 1600 
 <p>
1601 1601 
 The angular momentum of EM fields is directly given by
1602 1602 
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1603 
-<div class="main div" id="org56f7770">
1603 
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1604 1604 
 <p>
1605 1605 
 {\bf Angular momentum of EM fields}
1606 1606 
 \[
 
@@ -1630,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1630 1630 
 </div>
1631 1631 
 <div id="postamble" class="status">
1632 1632 
 <p class="author">Author: Jean-Sébastien Caux</p>
1633 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1633 
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1634 1634 
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1635 1635 
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1636 1636

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1619,7 +1619,7 @@ This means that
1619 1619 
 \]
1620 1620 
 Since this is true for any volume, we have (re)derived the
1621 1621 
 </p>
1622 
-<div class="core div" id="org48d0d19">
1622 
+<div class="core div" id="org8f4f0df">
1623 1623 
 <p>
1624 1624 
 {\bf Continuity equation}
1625 1625 
 \[
 
@@ -1660,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1660 1660 
 </div>
1661 1661 
 <div id="postamble" class="status">
1662 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1663 
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1664 1664 
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1665 1665 
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1666 1666

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@@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
1612 1612 
 <p>
1613 1613 
 This is thus simply a conservation law for momentum, with
1614 1614 
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1615 
-<div class="main div" id="orgc2ff953">
1615 
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1616 1616 
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1617 1617 
 {\bf Momentum density in the EM fields}
1618 1618 
 \[
 
@@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
1624 1624 
 <p>
1625 1625 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1626 1626 
 </p>
1627 
-<div class="main div" id="org7674e56">
1627 
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1628 1628 
 <p>
1629 1629 
 {\bf Continuity equation for EM momentum}
1630 1630 
 \[
 
@@ -1653,7 +1653,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1653 1653 
 </div>
1654 1654 
 <div id="postamble" class="status">
1655 1655 
 <p class="author">Author: Jean-Sébastien Caux</p>
1656 
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1656 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
1654 1654 
 <p>
1655 1655 
 This expression can be greatly simplified by introducing the
1656 1656 
 </p>
1657 
-<div class="main div" id="org69e4d14">
1657 
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1658 1658 
 <p>
1659 1659 
 {\bf Maxwell stress tensor}
1660 1660 
 \[
 
@@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1677 1677 
 <p>
1678 1678 
 We then obtain
1679 1679 
 </p>
1680 
-<div class="main div" id="orga5d6aaa">
1680 
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1681 1681 
 <p>
1682 1682 
 {\bf EM force per unit volume}
1683 1683 
 \[
 
@@ -1689,7 +1689,7 @@ We then obtain
1689 1689 
 <p>
1690 1690 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1691 1691 
 </p>
1692 
-<div class="main div" id="orgf401dc2">
1692 
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1693 1693 
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1694 1694 
 {\bf Total force on charges in volume}
1695 1695 
 \[
 
@@ -1718,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1718 1718 
 </div>
1719 1719 
 <div id="postamble" class="status">
1720 1720 
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1721 
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1721 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1666,7 +1666,7 @@ so we get
1666 1666 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1667 1667 
 we obtain
1668 1668 
 </p>
1669 
-<div class="main div" id="org9d063a8">
1669 
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1670 1670 
 <p>
1671 1671 
 {\bf Poynting's theorem}
1672 1672 
 \[
 
@@ -1691,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1691 1691 
 <p>
1692 1692 
 Energy per unit time, per unit area carried by EM fields:
1693 1693 
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1694 
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1694 
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1695 1695 
 <p>
1696 1696 
 {\bf Poynting vector}
1697 1697 
 \[
 
@@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
1704 1704 
 <p>
1705 1705 
 We can thus express Poynting's theorem more compactly:
1706 1706 
 </p>
1707 
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1707 
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1708 1708 
 <p>
1709 1709 
 {\bf Poynting's theorem}
1710 1710 
 \[
 
@@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
1717 1717 
 <p>
1718 1718 
 where we have defined the total
1719 1719 
 </p>
1720 
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1720 
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1721 1721 
 <p>
1722 1722 
 {\bf Energy in electromagnetic fields}
1723 1723 
 \[
 
@@ -1740,7 +1740,7 @@ Then,
1740 1740 
 \]
1741 1741 
 so we get the
1742 1742 
 </p>
1743 
-<div class="core div" id="orgb8783a7">
1743 
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1744 1744 
 <p>
1745 1745 
 {\bf Poynting theorem (differential form)}
1746 1746 
 \[
 
@@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
1757 1757
1758 1758
1759 1759
1760 
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1760 
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1761 1761 
 <p>
1762 1762 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1763 1763 
 Assuming that the field is uniform, the electric field parallel to the wire is
 
@@ -1802,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1802 1802 
 </div>
1803 1803 
 <div id="postamble" class="status">
1804 1804 
 <p class="author">Author: Jean-Sébastien Caux</p>
1805 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1805 
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1807 1807 
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1598 1598 
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1599 1599
1600 1600 
 <div class="outline-text-3" id="text-emd_emw">
1601 
-<details class="prereq" id="org8543c94">
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1603 1603 
 Prerequisites
1604 1604 
 </summary>
1605 1605 
 <ul class="org-ul">
 
@@ -1608,8 +1608,8 @@ Prerequisites
1608 1608 
 </ul>
1609 1609 
 </details>
1610 1610
1611 
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1612 
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1613 1613 
 Objectives
1614 1614 
 </summary>
1615 1615 
 <ul class="org-ul">
 
@@ -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: 2022-02-14 Mon 20:35</p>
1653 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
1628 1628 
 \]
1629 1629 
 or more succinctly:
1630 1630 
 </p>
1631 
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1631 
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1632 1632 
 <p>
1633 1633 
 {\bf Poynting vector of a monochromatic EM wave}
1634 1634 
 \[
 
@@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1644 1644 
 <p>
1645 1645 
 Similary, we get the
1646 1646 
 </p>
1647 
-<div class="main div" id="org70a3fce">
1647 
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1648 1648 
 <p>
1649 1649 
 {\bf Momentum density of a monochromatic EM wave}
1650 1650 
 \[
 
@@ -1695,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1695 1695 
 </div>
1696 1696 
 <div id="postamble" class="status">
1697 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1698 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1698 
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1699 1699 
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1700 1700 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1629,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1629 1629 
 Generalizing to propagation in the direction of an arbitrary wavevector
1630 1630 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1631 1631 
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1632 
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1633 1633 
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1634 1634 
 {\bf E and B fields for a monochromatic EM plane wave}
1635 1635 
 \[
 
@@ -1673,7 +1673,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1673 1673 
 </div>
1674 1674 
 <div id="postamble" class="status">
1675 1675 
 <p class="author">Author: Jean-Sébastien Caux</p>
1676 
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1676 
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@@ -1626,7 +1626,7 @@ These take the form of coupled first-order partial differential equations for \(
1626 1626 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1627 1627 
 we get the
1628 1628 
 </p>
1629 
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1630 1630 
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1631 1631 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1632 1632 
 \[
 
@@ -1682,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1682 1682 
 </div>
1683 1683 
 <div id="postamble" class="status">
1684 1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
1685 
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1685 
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1686 1686 
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1687 1687 
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1622 1622 
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1623 1623 
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1624 1624 
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1625 
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1622 1622 
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1623 1623 
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1624 1624 
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1625 
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@@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1633 1633 
 \]
1634 1634 
 We therefore have the
1635 1635 
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1636 
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1636 
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1637 1637 
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1638 1638 
 {\bf Polarization current density}
1639 1639 
 \[
 
@@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
1651 1651 
 polarization changes).  We can check consistency with the continuity equation
1652 1652 
 associated to the conservation of bound charges:
1653 1653 
 </p>
1654 
-<aside id="org3465a8e">
1654 
+<aside id="orgbb9bfed">
1655 1655 
 <p>
1656 1656 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1657 1657 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1674,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1674 1674 
 In view of this:  total charge density can be separated into 2 parts,
1675 1675 
 {\it free} and {\it bound}:
1676 1676 
 </p>
1677 
-<div class="main div" id="orgcdd32f0">
1677 
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1678 1678 
 <p>
1679 1679 
 \[
1680 1680 
     \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
 
@@ -1687,7 +1687,7 @@ In view of this:  total charge density can be separated into 2 parts,
1687 1687 
 and current can be separated into three parts, {\it free}, {\it bound} and
1688 1688 
 {\it polarization}:
1689 1689 
 </p>
1690 
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1690 
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1691 1691 
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1692 1692 
 \[
1693 1693 
   {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
 
@@ -1711,7 +1711,7 @@ Gauss's law:  can be rewritten
1711 1711 
 \]
1712 1712 
 where (as in static case)
1713 1713 
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1714 
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1714 
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1715 1715 
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1716 1716 
 \[
1717 1717 
     {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1737,7 +1737,7 @@ or
1737 1737 
 \]
1738 1738 
 where as before
1739 1739 
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1740 
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1740 
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1741 1741 
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1742 1742 
 \[
1743 1743 
     {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1755 1755 
 <p>
1756 1756 
 In terms of free charges and currents, we thus get
1757 1757 
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1758 
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1758 
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1759 1759 
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1760 1760 
 {\bf Maxwell's equations {\it (in matter)}}
1761 1761 
 </p>
 
@@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1781 1781 
 in terms of \({\bf E}\) and \({\bf B}\).
1782 1782 
 For the restricted case of linear media:
1783 1783 
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1784 
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1784 
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1785 1785 
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1786 1786 
 \[
1787 1787 
     {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
 
@@ -1818,7 +1818,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1818 1818 
 </div>
1819 1819 
 <div id="postamble" class="status">
1820 1820 
 <p class="author">Author: Jean-Sébastien Caux</p>
1821 
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1821 
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1822 1822 
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1823 1823 
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1600 1600 
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1601 1601 
 Discontinuities between different media, deduced from
1602 1602 
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1603 
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1605 1605 
 {\bf Maxwell's equations {\it (in matter)}, integral form}
1606 1606 
 </p>
 
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1691 1691 
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1692 1692 
 <div id="postamble" class="status">
1693 1693 
 <p class="author">Author: Jean-Sébastien Caux</p>
1694 
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1625 1625 
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1626 1626 
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1627 1627 
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1628 
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1621 1621 
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1622 1622 
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1623 1623 
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1624 
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1717 1717 
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1718 1718 
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1719 1719 
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1720 
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@@ -1635,7 +1635,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1635 1635 
 \]
1636 1636 
 where the index of refraction of the material is defined as
1637 1637 
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1638 
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1639 1639 
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1640 1640 
 {\bf Index of refraction}
1641 1641 
 \[
 
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1692 1692 
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1693 1693 
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1694 1694 
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1695 
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1636 1636 
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1637 1637 
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1638 1638 
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1639 
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1617 1617 
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1618 1618 
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1619 
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1619 1619 
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1620 
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@@ -1635,7 +1635,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1635 1635 
 <p>
1636 1636 
 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
1637 1637 
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1638 
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 <p>
1640 1640 
 {\bf First law of reflection:}
1641 1641 
 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
1650 1650 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1651 1651 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1652 1652 
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 <p>
1655 1655 
 {\bf Law of reflection}
1656 1656 
 \[
 
@@ -1708,7 +1708,7 @@ while the third equation becomes
1708 1708 
 \]
1709 1709 
 Writing everything in terms of the incident amplitude, we get
1710 1710 
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1713 1713 
 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
1714 1714 
 \[
 
@@ -1728,7 +1728,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1728 1728 
 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
1729 1729 
 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
1730 1730 
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1731 
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1733 1733 
 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
1734 1734 
   \[
 
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1778 1778 
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1779 1779 
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1780 1780 
 <p class="author">Author: Jean-Sébastien Caux</p>
1781 
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1645 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
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1669 1669 
 <p class="author">Author: Jean-Sébastien Caux</p>
1670 
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1678 1678 
 <p class="author">Author: Jean-Sébastien Caux</p>
1679 
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1598 1598 
 </svg></a><span class="headline-id">emf</span></h2>
1599 1599
1600 1600 
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1601 
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1603 1603 
 Prerequisites
1604 1604 
 </summary>
1605 1605 
 <ul class="org-ul">
 
@@ -1607,8 +1607,8 @@ Prerequisites
1607 1607 
 </ul>
1608 1608 
 </details>
1609 1609
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1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
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1644 1644 
 </div>
1645 1645 
 <div id="postamble" class="status">
1646 1646 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
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1641 1641 
 </div>
1642 1642 
 <div id="postamble" class="status">
1643 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
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1640 1640 
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1641 1641 
 <div id="postamble" class="status">
1642 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
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1643 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
1613 1613 
 \]
1614 1614 
 These can be written compactly upon introducing a new operator: the
1615 1615 
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1616 
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1617 1617 
 <p>
1618 1618 
 {\bf d'Alembertian operator}
1619 1619 
 \[
 
@@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
1626 1626 
 <p>
1627 1627 
 so we get the
1628 1628 
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1629 
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+<div class="core div" id="orgf9dae9e">
1630 1630 
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1631 1631 
 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1632 1632 
 \[
 
@@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1676 1676 
 </div>
1677 1677 
 <div id="postamble" class="status">
1678 1678 
 <p class="author">Author: Jean-Sébastien Caux</p>
1679 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1679 
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1614 1614 
 <p>
1615 1615 
 Easiest:
1616 1616 
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1617 
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1617 
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1618 1618 
 <p>
1619 1619 
 \[
1620 1620 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
1630 1630 
 \]
1631 1631 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1632 1632 
 </p>
1633 
-<div class="core div" id="org3315df4">
1633 
+<div class="core div" id="orgaa4bf24">
1634 1634 
 <p>
1635 1635 
 \[
1636 1636 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1643 1643 
 <p>
1644 1644 
 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
1645 1645 
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1646 
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1648 1648 
 \[
1649 1649 
     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
1659 1659 
 \]
1660 1660 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1661 1661 
 </p>
1662 
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1663 1663 
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1664 1664 
 \[
1665 1665 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1695,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1695 1695 
 </div>
1696 1696 
 <div id="postamble" class="status">
1697 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1698 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1698 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1699 1699 
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1700 1700 
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1624 1624 
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1625 1625 
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1626 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1627 
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1627 
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@@ -1602,8 +1602,8 @@ Table of contents
1602 1602 
 <li>Gr 3</li>
1603 1603 
 </ul>
1604 1604
1605 
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1606 
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1605 
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1607 1607 
 Prerequisites
1608 1608 
 </summary>
1609 1609 
 <ul class="org-ul">
 
@@ -1611,8 +1611,8 @@ Prerequisites
1611 1611 
 </ul>
1612 1612 
 </details>
1613 1613
1614 
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1615 
-<summary id="orgf1a9924">
1614 
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1615 
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1616 1616 
 Objectives
1617 1617 
 </summary>
1618 1618 
 <ul class="org-ul">
 
@@ -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: 2022-02-14 Mon 20:35</p>
1653 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1654 1654 
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1655 1655 
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1656 1656

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1608,7 +1608,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
1608 1608 
 defines an electrostatic problem, whose solution is in principle obtained
1609 1609 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1610 1610 
 </p>
1611 
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1613 1613
1614 1614 
 </p>
 
@@ -1622,7 +1622,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1622 1622 
 or (often simpler) by calculating the electrostatic potential, using either the
1623 1623 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1624 1624 
 </p>
1625 
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1625 
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1626 1626 
 <p>
1627 1627
1628 1628 
 </p>
 
@@ -1642,7 +1642,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1642 1642 
 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
1643 1643 
 </p>
1644 1644
1645 
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1647 1647
1648 1648 
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@@ -1658,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1658 1658 
 <p>
1659 1659 
 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>
1660 1660 
 </p>
1661 
-<div class="core div" id="org025bcf6">
1661 
+<div class="core div" id="org2a317e6">
1662 1662 
 <p>
1663 1663
1664 1664 
 </p>
 
@@ -1696,7 +1696,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1696 1696 
 </div>
1697 1697 
 <div id="postamble" class="status">
1698 1698 
 <p class="author">Author: Jean-Sébastien Caux</p>
1699 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
1699 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
1700 1700 
 <p class="validation"></p>
1701 1701 
 </div>
1702 1702

+ 79
- 41
build/ems_ca_fe_L.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-02-14 Mon 20:35 -->
4 
+<!-- 2022-02-15 Tue 10:14 -->
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>
 
@@ -1614,14 +1614,14 @@ In one dimension, the potential is a single-variable
1614 1614 
 function \(\phi (x)\) and the Laplace equation reads
1615 1615 
 </p>
1616 1616
1617 
-<div class="eqlabel" id="org937348d">
1617 
+<div class="eqlabel" id="orgdfca409">
1618 1618 
 <p>
1619 1619 
 <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">
1620 1620 
   <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"/>
1621 1621 
   <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"/>
1622 1622 
 </svg></a>
1623 1623 
 </p>
1624 
-<div class="alteqlabels" id="orgf7c0132">
1624 
+<div class="alteqlabels" id="org36d5863">
1625 1625
1626 1626 
 </div>
1627 1627 
 
@@ -1634,16 +1634,16 @@ function \(\phi (x)\) and the Laplace equation reads
1634 1634 
 </p>
1635 1635
1636 1636 
 <p>
1637 
-The solution to this
1637 
+The solution to this is
1638 1638 
 </p>
1639 
-<div class="eqlabel" id="org5437a41">
1639 
+<div class="eqlabel" id="orga37bc11">
1640 1640 
 <p>
1641 1641 
 <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">
1642 1642 
   <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"/>
1643 1643 
   <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"/>
1644 1644 
 </svg></a>
1645 1645 
 </p>
1646 
-<div class="alteqlabels" id="org777e948">
1646 
+<div class="alteqlabels" id="orgc5114c3">
1647 1647 
 <ul class="org-ul">
1648 1648 
 <li>Gr (3.6)</li>
1649 1649 
 </ul>
 
@@ -1702,14 +1702,14 @@ In two dimensions, the potential becomes a function
1702 1702 
 of two variables (here: \(x\) and \(y\)), so Laplace's
1703 1703 
 equation now reads
1704 1704 
 </p>
1705 
-<div class="eqlabel" id="orgf26c98b">
1705 
+<div class="eqlabel" id="orgc587278">
1706 1706 
 <p>
1707 1707 
 <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">
1708 1708 
   <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"/>
1709 1709 
   <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"/>
1710 1710 
 </svg></a>
1711 1711 
 </p>
1712 
-<div class="alteqlabels" id="org59e7c1d">
1712 
+<div class="alteqlabels" id="orgd96eb38">
1713 1713
1714 1714 
 </div>
1715 1715 
 
@@ -1758,15 +1758,31 @@ The Laplace equation is (we repeat)
1758 1758
1759 1759 
 <p>
1760 1760 
 <b>Theorem</b>: if \(\phi\) satisfies Laplace, then its value at
1761 
-a point equals its value averaged over any sphere
1762 
-\(S_{\bf r}\) centered on this point,
1761 
+a point equals its value averaged over a sphere
1762 
+\(S_R({\bf r})\) of any radius \(R\) centered on this point
1763 
+(and of course not containing any charges),
1764 
+</p>
1765 
+<div class="eqlabel" id="orgf7426f4">
1766 
+<p>
1767 
+<a id="p_ball_avg"></a><a href="./ems_ca_fe_L.html#p_ball_avg"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1768 
+  <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"/>
1769 
+  <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"/>
1770 
+</svg></a>
1771 
+</p>
1772 
+<div class="alteqlabels" id="org1c3162d">
1773 
+
1774 
+</div>
1775 
+
1776 
+</div>
1777 
+<p>
1763 1778 
 \[
1764 
-\phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_{\bf r}} da' ~\phi ({\bf r}')
1779 
+\phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_R({\bf r})} da' ~\phi ({\bf r}')
1780 
+\tag{p_ball_avg}\label{p_ball_avg}
1765 1781 
 \]
1766 1782 
 </p>
1767 1783
1768 
-<details id="orgf3cada4">
1769 
-<summary id="org840c868">
1784 
+<details id="orgbc3026b">
1785 
+<summary id="orge4e657e">
1770 1786 
 <strong>Physicist's proof</strong>
1771 1787 
 </summary>
1772 1788 
 <p>
 
@@ -1779,11 +1795,8 @@ of the sphere.
1779 1795 
 <p>
1780 1796 
 We know that the field created by the sphere coincides
1781 1797 
 with that of a point charge \(q\) at the origin.
1782 
-Since the potential at \({\bf r = 0}\) created by the charge
1783 
-\(q'\) at \({\bf r'}\) is simply \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\),
1784 
-the work
1785 
-required to bring the \(q'\) charge into position is thus
1786 
-simply \(W = q \times \phi_{q', {\bf r}'} (0) = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.
1798 
+The work required to bring the \(q'\) charge into position is thus
1799 
+simply \(W = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.
1787 1800 
 </p>
1788 1801
1789 1802 
 <p>
 
@@ -1814,14 +1827,12 @@ Equating this with the previous results shows that
1814 1827
1815 1828 
 <p>
1816 1829 
 \[
1817 
-\phi_{q', {\bf r'}} (0) = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})
1830 
+\frac{q'}{4\pi \varepsilon_0 R'} = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})
1818 1831 
 \]
1819 
-</p>
1820 
-
1821 
-<p>
1822 
-namely that for the potential created by a single point
1823 
-charge \(q'\) at \(R'\),
1824 
-the value at a point (here the origin)
1832 
+but this also equals the potential at \({\bf r} = 0\) created by the charge
1833 
+\(q'\) at \({\bf r'}\), <i>i.e.</i> \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\).
1834 
+In other words, we have thus shown that for the potential created by a single point
1835 
+charge \(q'\) at \(R'\), the value at a point (here the origin)
1825 1836 
 coincides with the value averaged over a sphere
1826 1837 
 or an arbitrary radius \(R\) centered on the same point.
1827 1838 
 </p>
 
@@ -1833,8 +1844,8 @@ proving the theorem.
1833 1844 
 </p>
1834 1845 
 </details>
1835 1846
1836 
-<details id="org02ee258">
1837 
-<summary id="orgf24411c">
1847 
+<details id="orgf289197">
1848 
+<summary id="orge3fd0c7">
1838 1849 
 <strong>Formal proof</strong>
1839 1850 
 </summary>
1840 1851 
 
@@ -1845,13 +1856,13 @@ a ball of radius \(R\) centered on \({\bf r}\):
1845 1856
1846 1857 
 <p>
1847 1858 
 \[
1848 
-f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R} da' ~ f ({\bf r} + {\bf r}')
1859 
+f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R ({\bf r})} da' ~ f ({\bf r}')
1849 1860 
 \]
1850 1861 
 </p>
1851 1862
1852 1863 
 <p>
1853 
-For convenience we will hereafter put \({\bf r} = 0\).
1854 
-In spherical coordinates, we have \(da' = R^2 sin \theta d\theta d\varphi \equiv R^2 d\Omega\).
1864 
+In spherical coordinates defined around the point \({\bf r}\),
1865 
+we have \(da' = R^2 sin \theta d\theta d\varphi \equiv R^2 d\Omega\).
1855 1866 
 Differentiating with respect to \(R\),
1856 1867 
 </p>
1857 1868 
 
@@ -1863,7 +1874,7 @@ Differentiating with respect to \(R\),
1863 1874
1864 1875 
 <p>
1865 1876 
 with \(f\) differentiated with respect to the radial coordiate.
1866 
-We can rewrite this by noting that \(R^2 d\Omega \hat{\bf r}\)
1877 
+We can rewrite this by noting that \(R^2 d\Omega ~\hat{\bf r}\)
1867 1878 
 is the normal differential surface area \(d{\bf a}\), while
1868 1879 
 \(\left.\frac{\partial f}{\partial r}\right|_{r=R}\) is the radial component of the gradient
1869 1880 
 of \(f\) in spherical coordinates. Thus,
 
@@ -1884,10 +1895,23 @@ we get the following general
1884 1895 
 <p>
1885 1896 
 <b>Theorem</b>:
1886 1897 
 </p>
1898 
+<div class="eqlabel" id="orgd8bf4d9">
1899 
+<p>
1900 
+<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">
1901 
+  <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"/>
1902 
+  <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"/>
1903 
+</svg></a>
1904 
+</p>
1905 
+<div class="alteqlabels" id="org49a36be">
1906 
+
1907 
+</div>
1908 
+
1909 
+</div>
1887 1910
1888 1911 
 <p>
1889 1912 
 \[
1890 1913 
 \frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \int_{V_R} d\tau ~\nabla^2 f
1914 
+\tag{dfdR_intLap}\label{dfdR_intLap}
1891 1915 
 \]
1892 1916 
 </p>
1893 1917 
 
@@ -1904,7 +1928,7 @@ an infinitesimally small ball, we get the result announced above.
1904 1928 
 </details>
1905 1929
1906 1930 
 <p>
1907 
-<b>Theorem (Earnshaw, mathematical versoin)</b>: \(\phi\) has no local extrema except at the boundaries.
1931 
+<b>Theorem (Earnshaw, mathematical version)</b>: \(\phi\) has no local extrema except at the boundaries.
1908 1932 
 </p>
1909 1933
1910 1934 
 <p>
 
@@ -1931,9 +1955,24 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1931 1955 
 of the \(f_x + f_y + f_z = 0\) condition above.
1932 1956 
 </p>
1933 1957
1934 
-<div class="info div" id="orgfec03b0">
1958 
+<div class="eqlabel" id="orgfeb6aae">
1959 
+<p>
1960 
+<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">
1961 
+  <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"/>
1962 
+  <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"/>
1963 
+</svg></a>
1964 
+</p>
1965 
+<div class="alteqlabels" id="org7f4d8f6">
1966 
+
1967 
+</div>
1968 
+
1969 
+</div>
1970 
+<div class="info div" id="orgf970a43">
1935 1971 
 <p>
1936 1972 
 <b>Earnshaw's theorem (physical version)</b> <br>
1973 
+</p>
1974 
+
1975 
+<p>
1937 1976 
 It is impossible to find a static distribution of charges which generates an electrostatic field
1938 1977 
 displaying a stable equilibrium position in empty space.
1939 1978 
 </p>
 
@@ -1967,7 +2006,7 @@ We start by mentioning some cases, and interpreting them thereafter.
1967 2006 
 the electrostatic potential is uniquely determined
1968 2007 
 by Poisson's equation, which
1969 2008 
 is explicitly solved by <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>.
1970 
-One can eplicitly verify this:
2009 
+One can explicitly verify this:
1971 2010 
 </p>
1972 2011
1973 2012 
 <p>
 
@@ -1980,23 +2019,22 @@ One can eplicitly verify this:
1980 2019 
 <p>
1981 2020 
 where we have used <a href="./c_m_dd_3d.html#Lap1or">Lap1or</a>, and the fact that the delta function is always resolved since we
1982 2021 
 integrate over all space.  Note:  it is implicitly assumed that the integral in <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1983 
-converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved (does not
1984 
-become singular).
2022 
+converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved.
1985 2023 
 </p>
1986 2024
1987 2025 
 <p>
1988 
-<b>Charge density in closed volume, and boundary surface charge density are known</b>:  the electrostatic potential is uniquely determined
2026 
+<b>"Known boundary charge" case: charge density in closed volume and boundary surface charge density are both known</b>:  the electrostatic potential is uniquely determined
1989 2027 
 in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
1990 
-\(\rho ({\bf x})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),
2028 
+\(\rho ({\bf r})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),
1991 2029 
 and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\).
1992 2030 
 Of course, \({\cal S}\) need not be a connected surface.
1993 2031 
 </p>
1994 2032
1995 2033
1996 2034 
 <p>
1997 
-<b>Charge density in closed volume, and potential at boundary are known</b>:  the electrostatic potential is uniquely determined
2035 
+<b>"Known boundary potential" case: charge density in closed volume and potential at boundary are both known</b>:  the electrostatic potential is uniquely determined
1998 2036 
 in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
1999 
-\(\rho ({\bf x})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the
2037 
+\(\rho ({\bf r})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the
2000 2038 
 boundary \({\cal S}\).  Of course, \({\cal S}\) need not be a connected surface.
2001 2039 
 </p>
2002 2040 
 
@@ -2048,7 +2086,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
2048 2086 
 </div>
2049 2087 
 <div id="postamble" class="status">
2050 2088 
 <p class="author">Author: Jean-Sébastien Caux</p>
2051 
-<p class="date">Created: 2022-02-14 Mon 20:35</p>
2089 
+<p class="date">Created: 2022-02-15 Tue 10:14</p>
2052 2090 
 <p class="validation"></p>
2053 2091 
 </div>
2054 2092

+ 0
- 0
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