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Update 2022-03-07 20:40

master
Jean-Sébastien 2 years ago
parent
21bf9fdba5
commit
4808df71e6
100 changed files with 919 additions and 4653 deletions
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 <title>Pre-Quantum Electrodynamics</title>
 
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1645 1637 
 </div>
1646 1638 
 <div id="postamble" class="status">
1647 1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
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@@ -1099,14 +1099,6 @@ Table of contents
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1100 1100
1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1728 1720 
 </div>
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 <div id="postamble" class="status">
1730 1722 
 <p class="author">Author: Jean-Sébastien Caux</p>
1731 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
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@@ -1099,14 +1099,6 @@ Table of contents
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
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-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
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-<li>
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1640,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1640 1632 
 </div>
1641 1633 
 <div id="postamble" class="status">
1642 1634 
 <p class="author">Author: Jean-Sébastien Caux</p>
1643 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1635 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1644 1636 
 <p class="validation"></p>
1645 1637 
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1646 1638

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1631,14 +1623,14 @@ which \(r\) is the distance from the chosen origin,
1631 1623 
 The usual Cartesian coordinates relate to spherical coordinates
1632 1624 
 according to
1633 1625 
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1634 
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1626 
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1636 1628 
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1637 1629 
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1638 1630 
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1639 1631 
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1640 1632 
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1641 
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1642 1634
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1644 1636 
 
@@ -1663,14 +1655,14 @@ A generic vector can be expressed as
1663 1655 
 where the explicit relation between spherical and
1664 1656 
 Cartesian unit vectors is
1665 1657 
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1666 
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1658 
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1667 1659 
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1668 1660 
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1669 1661 
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1670 1662 
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1671 1663 
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1672 1664 
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1674 1666
1675 1667 
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1676 1668 
 
@@ -1693,14 +1685,14 @@ and \(\hat{\boldsymbol \varphi} (\theta, \varphi)\).
1693 1685 
 <p>
1694 1686 
 An infinitesimal displacement \(d{\bf l}\) can be written as
1695 1687 
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1696 
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1688 
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1697 1689 
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1698 1690 
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1699 1691 
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1700 1692 
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1701 1693 
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1706 1698 
 
@@ -1716,14 +1708,14 @@ d{\bf l} = dr ~\hat{\boldsymbol r} + r d\theta ~\hat{\boldsymbol \theta} + r\sin
1716 1708 
 <p>
1717 1709 
 Infinitesimal volume element:
1718 1710 
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1719 
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1711 
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1720 1712 
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1721 1713 
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1723 1715 
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1724 1716 
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1725 1717 
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1718 
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1727 1719
1728 1720 
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1729 1721 
 
@@ -1744,14 +1736,14 @@ Infinitesimal surface element:  depends on situation.
1744 1736 
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1745 1737 
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1746 1738 
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1747 
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1749 1741 
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1750 1742 
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1751 1743 
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1752 1744 
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@@ -1768,14 +1760,14 @@ Infinitesimal surface element:  depends on situation.
1768 1760 
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1769 1761 
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1770 1762 
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1773 1765 
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1774 1766 
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1775 1767 
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1776 1768 
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1777 1769 
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1792 1784 
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1793 1785 
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1799 1791 
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1800 1792 
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@@ -1818,14 +1810,14 @@ Infinitesimal surface element:  depends on situation.
1818 1810 
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1819 1811 
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1825 1817 
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1826 1818 
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1827 1819 
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@@ -1859,7 +1851,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1859 1851 
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1860 1852 
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1861 1853 
 <p class="author">Author: Jean-Sébastien Caux</p>
1862 
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1854 
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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1105 
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
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1109 
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1110 1102
1111 1103 
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1112 1104 
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1650 1642 
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1651 1643 
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1652 1644 
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1653 
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1645 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1105 
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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1110 1102
1111 1103 
 </ul>
1112 1104 
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1652 1644 
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1653 1645 
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1654 1646 
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1655 
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1647 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1107 
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1110 1102
1111 1103 
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1112 1104 
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@@ -1623,9 +1615,9 @@ Table of contents
1623 1615 
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1624 1616 
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1625 1617
1626 
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1627 
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1618 
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1619 
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1620 
+<div class="outline-text-6" id="text-orgc12bd11">
1629 1621 
 <p>
1630 1622 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1631 1623 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1634,44 +1626,44 @@ given by the Laplacian of the corresponding vector elements.
1634 1626 
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1635 1627 
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1636 1628
1637 
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1638 
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1639 
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1629 
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1630 
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1631 
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1640 1632 
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1641 1633 
 This always vanishes.
1642 1634 
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1643 1635 
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1644 1636 
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1645 1637
1646 
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1647 
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1648 
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1638 
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1639 
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1640 
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1649 1641 
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1650 1642 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1651 1643 
 </p>
1652 1644 
 </div>
1653 1645 
 </div>
1654 1646
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">
1647 
+<div id="outline-container-orge7f86bf" class="outline-6">
1648 
+<h6 id="orge7f86bf"><a href="#orge7f86bf">Divergence of a curl</a></h6>
1649 
+<div class="outline-text-6" id="text-orge7f86bf">
1658 1650 
 <p>
1659 1651 
 This always vanishes.
1660 1652 
 </p>
1661 1653 
 </div>
1662 1654 
 </div>
1663 1655
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">
1656 
+<div id="outline-container-orgbb8199b" class="outline-6">
1657 
+<h6 id="orgbb8199b"><a href="#orgbb8199b">Curl of curl</a></h6>
1658 
+<div class="outline-text-6" id="text-orgbb8199b">
1659 
+<div class="eqlabel" id="orga59fc73">
1668 1660 
 <p>
1669 1661 
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1670 1662 
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1671 1663 
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1672 1664 
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1673 1665 
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1674 
-<div class="alteqlabels" id="orgde36146">
1666 
+<div class="alteqlabels" id="org384a104">
1675 1667
1676 1668 
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1677 1669 
 
@@ -1702,7 +1694,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1702 1694 
 </div>
1703 1695 
 <div id="postamble" class="status">
1704 1696 
 <p class="author">Author: Jean-Sébastien Caux</p>
1705 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1697 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1706 1698 
 <p class="validation"></p>
1707 1699 
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1708 1700

+ 2
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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 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1651,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 1643 
 </div>
1652 1644 
 <div id="postamble" class="status">
1653 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1646 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1655 1647 
 <p class="validation"></p>
1656 1648 
 </div>
1657 1649

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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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4 
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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1640 
 </div>
1649 1641 
 <div id="postamble" class="status">
1650 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1643 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1652 1644 
 <p class="validation"></p>
1653 1645 
 </div>
1654 1646

+ 2
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@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1672,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1672 1664 
 </div>
1673 1665 
 <div id="postamble" class="status">
1674 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1675 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1667 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1676 1668 
 <p class="validation"></p>
1677 1669 
 </div>
1678 1670

+ 14
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build/c_m_dc_pr.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1630,14 +1622,14 @@ explicited as follows:
1630 1622 
 <p>
1631 1623 
 <b>Gradient of a product</b>:
1632 1624 
 </p>
1633 
-<div class="eqlabel" id="org4bcb61a">
1625 
+<div class="eqlabel" id="orged10256">
1634 1626 
 <p>
1635 1627 
 <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 1628 
   <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 1629 
   <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 1630 
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1639 1631 
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1640 
-<div class="alteqlabels" id="org8c15975">
1632 
+<div class="alteqlabels" id="orgfc8793a">
1641 1633 
 <ul class="org-ul">
1642 1634 
 <li>Gr (3)</li>
1643 1635 
 <li>W (1-111)</li>
 
@@ -1657,14 +1649,14 @@ explicited as follows:
1657 1649 
 <p>
1658 1650 
 <b>Gradient of a scalar product</b>:
1659 1651 
 </p>
1660 
-<div class="eqlabel" id="org7784802">
1652 
+<div class="eqlabel" id="org4b4bdf3">
1661 1653 
 <p>
1662 1654 
 <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 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"/>
1664 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"/>
1665 1657 
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1666 1658 
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1667 
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1659 
+<div class="alteqlabels" id="org02913cc">
1668 1660 
 <ul class="org-ul">
1669 1661 
 <li>Gr (4)</li>
1670 1662 
 <li>W (1-112)</li>
 
@@ -1684,14 +1676,14 @@ explicited as follows:
1684 1676 
 <p>
1685 1677 
 <b>Divergence of a product</b>:
1686 1678 
 </p>
1687 
-<div class="eqlabel" id="org96b1b94">
1679 
+<div class="eqlabel" id="org1df402a">
1688 1680 
 <p>
1689 1681 
 <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 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"/>
1691 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"/>
1692 1684 
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1693 1685 
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1694 
-<div class="alteqlabels" id="org29d8167">
1686 
+<div class="alteqlabels" id="orgfa5050d">
1695 1687 
 <ul class="org-ul">
1696 1688 
 <li>Gr (5)</li>
1697 1689 
 <li>W (1-115)</li>
 
@@ -1711,14 +1703,14 @@ explicited as follows:
1711 1703 
 <p>
1712 1704 
 <b>Divergence of a cross product</b>:
1713 1705 
 </p>
1714 
-<div class="eqlabel" id="orgfe2b08f">
1706 
+<div class="eqlabel" id="org5048e1e">
1715 1707 
 <p>
1716 1708 
 <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 1709 
   <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 1710 
   <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 1711 
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1720 1712 
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1721 
-<div class="alteqlabels" id="orgdce994d">
1713 
+<div class="alteqlabels" id="orgd39dff6">
1722 1714 
 <ul class="org-ul">
1723 1715 
 <li>Gr (6)</li>
1724 1716 
 <li>W (1-116)</li>
 
@@ -1738,14 +1730,14 @@ explicited as follows:
1738 1730 
 <p>
1739 1731 
 <b>Curl of a product</b>:
1740 1732 
 </p>
1741 
-<div class="eqlabel" id="orgbb8798b">
1733 
+<div class="eqlabel" id="orgbd31452">
1742 1734 
 <p>
1743 1735 
 <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 1736 
   <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 1737 
   <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 1738 
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1747 1739 
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1748 
-<div class="alteqlabels" id="org93d7ea5">
1740 
+<div class="alteqlabels" id="org825aae4">
1749 1741 
 <ul class="org-ul">
1750 1742 
 <li>Gr (7)</li>
1751 1743 
 <li>W (1-118)</li>
 
@@ -1765,14 +1757,14 @@ explicited as follows:
1765 1757 
 <p>
1766 1758 
 <b>Curl of a cross product</b>:
1767 1759 
 </p>
1768 
-<div class="eqlabel" id="orgd9fde5c">
1760 
+<div class="eqlabel" id="org9c6b6a4">
1769 1761 
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1770 1762 
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1771 1763 
   <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 1764 
   <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 1765 
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1774 1766 
 </p>
1775 
-<div class="alteqlabels" id="orgb1d2b5b">
1767 
+<div class="alteqlabels" id="orgb030476">
1776 1768 
 <ul class="org-ul">
1777 1769 
 <li>Gr (8)</li>
1778 1770 
 <li>W (1-119)</li>
 
@@ -1813,7 +1805,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1813 1805 
 </div>
1814 1806 
 <div id="postamble" class="status">
1815 1807 
 <p class="author">Author: Jean-Sébastien Caux</p>
1816 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1808 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1817 1809 
 <p class="validation"></p>
1818 1810 
 </div>
1819 1811

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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1105 
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
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1112 1104 
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@@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 1639 
 </div>
1648 1640 
 <div id="postamble" class="status">
1649 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1642 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1651 1643 
 <p class="validation"></p>
1652 1644 
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1653 1645

+ 2
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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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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1659,7 +1651,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1659 1651 
 </div>
1660 1652 
 <div id="postamble" class="status">
1661 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1662 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1654 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1663 1655 
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1664 1656 
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1665 1657

+ 8
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@@ -1,7 +1,7 @@
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1644,14 +1636,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1644 1636 
 More generally,
1645 1637 
 </p>
1646 1638
1647 
-<div class="eqlabel" id="orgf5d25ef">
1639 
+<div class="eqlabel" id="org492d0c0">
1648 1640 
 <p>
1649 1641 
 <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1650 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"/>
1651 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"/>
1652 1644 
 </svg></a>
1653 1645 
 </p>
1654 
-<div class="alteqlabels" id="org6fba030">
1646 
+<div class="alteqlabels" id="org9b8bed8">
1655 1647 
 <ul class="org-ul">
1656 1648 
 <li>Gr (1.100)</li>
1657 1649 
 </ul>
 
@@ -1670,14 +1662,14 @@ More generally,
1670 1662 
 Since
1671 1663 
 </p>
1672 1664
1673 
-<div class="eqlabel" id="org9588ee0">
1665 
+<div class="eqlabel" id="orga3b7a91">
1674 1666 
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1675 1667 
 <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">
1676 1668 
   <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 1669 
   <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 1670 
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1679 1671 
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1680 
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1672 
+<div class="alteqlabels" id="org98ac626">
1681 1673 
 <ul class="org-ul">
1682 1674 
 <li>Gr (1.101)</li>
1683 1675 
 </ul>
 
@@ -1693,14 +1685,14 @@ Since
1693 1685 
 <p>
1694 1686 
 we have that
1695 1687 
 </p>
1696 
-<div class="eqlabel" id="orgd223c1e">
1688 
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1697 1689 
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1698 1690 
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1699 1691 
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1700 1692 
   <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 1693 
 </svg></a>
1702 1694 
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1703 
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1695 
+<div class="alteqlabels" id="org620c397">
1704 1696 
 <ul class="org-ul">
1705 1697 
 <li>Gr (1.102)</li>
1706 1698 
 </ul>
 
@@ -1732,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1732 1724 
 </div>
1733 1725 
 <div id="postamble" class="status">
1734 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1735 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1727 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1736 1728 
 <p class="validation"></p>
1737 1729 
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1738 1730

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6 6 
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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1665,7 +1657,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1665 1657 
 </div>
1666 1658 
 <div id="postamble" class="status">
1667 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1668 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1660 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1669 1661 
 <p class="validation"></p>
1670 1662 
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1671 1663

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1650,7 +1642,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1650 1642 
 </div>
1651 1643 
 <div id="postamble" class="status">
1652 1644 
 <p class="author">Author: Jean-Sébastien Caux</p>
1653 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1645 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1654 1646 
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1655 1647 
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1656 1648

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1640 
 </div>
1649 1641 
 <div id="postamble" class="status">
1650 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1643 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1652 1644 
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1653 1645 
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1654 1646

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1657,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1657 1649 
 </div>
1658 1650 
 <div id="postamble" class="status">
1659 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1660 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1661 1653 
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1662 1654 
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1663 1655

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1651,7 +1643,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1651 1643 
 </div>
1652 1644 
 <div id="postamble" class="status">
1653 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1646 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1655 1647 
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1656 1648 
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1657 1649

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6 6 
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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1658,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1658 1650 
 </div>
1659 1651 
 <div id="postamble" class="status">
1660 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1653 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1662 1654 
 <p class="validation"></p>
1663 1655 
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1664 1656

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build/c_m_ic_lsv.html View File 

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1 1 
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5 5 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1623,9 +1615,9 @@ Table of contents
1623 1615 
 <div class="outline-text-5" id="text-c_m_ic_lsv">
1624 1616 
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1625 1617
1626 
-<div id="outline-container-org2486563" class="outline-6">
1627 
-<h6 id="org2486563"><a href="#org2486563">Line Integrals</a></h6>
1628 
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1618 
+<div id="outline-container-org7fab499" class="outline-6">
1619 
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1620 
+<div class="outline-text-6" id="text-org7fab499">
1629 1621 
 <p>
1630 1622 
 \[
1631 1623 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1654,9 +1646,9 @@ Integral over a closed loop:
1654 1646 
 </div>
1655 1647 
 </div>
1656 1648
1657 
-<div id="outline-container-orgf44a1b8" class="outline-6">
1658 
-<h6 id="orgf44a1b8"><a href="#orgf44a1b8">Surface Integrals</a></h6>
1659 
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1649 
+<div id="outline-container-org543b5ed" class="outline-6">
1650 
+<h6 id="org543b5ed"><a href="#org543b5ed">Surface Integrals</a></h6>
1651 
+<div class="outline-text-6" id="text-org543b5ed">
1660 1652 
 <p>
1661 1653 
 \[
1662 1654 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1676,9 +1668,9 @@ Over a closed surface:
1676 1668 
 </div>
1677 1669 
 </div>
1678 1670
1679 
-<div id="outline-container-org7c26dcf" class="outline-6">
1680 
-<h6 id="org7c26dcf"><a href="#org7c26dcf">Volume Integrals</a></h6>
1681 
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1671 
+<div id="outline-container-org015d5b1" class="outline-6">
1672 
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1673 
+<div class="outline-text-6" id="text-org015d5b1">
1682 1674 
 <p>
1683 1675 
 \[
1684 1676 
 \int_{\cal V} T d\tau
 
@@ -1719,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1711 
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1720 1712 
 <div id="postamble" class="status">
1721 1713 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1714 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1723 1715 
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1724 1716 
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1725 1717

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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
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@@ -1621,14 +1613,14 @@ Table of contents
1621 1613 
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1622 1614 
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1623 1615 
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1625 1617 
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1626 1618 
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1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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1662 1654 
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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@@ -1099,14 +1099,6 @@ Table of contents
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1665 1657 
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1649 1641 
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1650 1642 
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1651 1643 
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1680 1672 
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1720 1712 
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1722 1714 
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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@@ -1099,14 +1099,6 @@ Table of contents
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1101 1101 
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1101 1101 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1669,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1669 1661 
 </div>
1670 1662 
 <div id="postamble" class="status">
1671 1663 
 <p class="author">Author: Jean-Sébastien Caux</p>
1672 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1664 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1673 1665 
 <p class="validation"></p>
1674 1666 
 </div>
1675 1667

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1 1 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1622,8 +1614,8 @@ Table of contents
1622 1614 
 </svg></a><span class="headline-id">emd</span></h2>
1623 1615
1624 1616 
 <div class="outline-text-2" id="text-emd">
1625 
-<details class="prereq" id="org2aba3cc">
1626 
-<summary id="org30ca6a6">
1617 
+<details class="prereq" id="org417b126">
1618 
+<summary id="org6bf52b1">
1627 1619 
 Prerequisites
1628 1620 
 </summary>
1629 1621 
 <ul class="org-ul">
 
@@ -1632,8 +1624,8 @@ Prerequisites
1632 1624 
 </ul>
1633 1625 
 </details>
1634 1626
1635 
-<details class="objectives" id="org98af4b0">
1636 
-<summary id="org78e61da">
1627 
+<details class="objectives" id="org93939b2">
1628 
+<summary id="org78ba2fd">
1637 1629 
 Objectives
1638 1630 
 </summary>
1639 1631 
 <ul class="org-ul">
 
@@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1666 
 </div>
1675 1667 
 <div id="postamble" class="status">
1676 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1669 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1678 1670 
 <p class="validation"></p>
1679 1671 
 </div>
1680 1672

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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1640 
 </div>
1649 1641 
 <div id="postamble" class="status">
1650 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1643 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1652 1644 
 <p class="validation"></p>
1653 1645 
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1654 1646

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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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4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1652,18 +1644,18 @@ Empirically:  the changing magnetic field induces an electric current around
1652 1644 
 the circuit. This current is really driven by an electric field having a component
1653 1645 
 along the wire.  The line integral of this field is called the
1654 1646 
 </p>
1655 
-<div class="core div" id="orgdfb0aad">
1647 
+<div class="core div" id="org9cc8d48">
1656 1648 
 <p>
1657 1649 
 <b>Electromotive force (or electromotance)</b>,
1658 1650 
 </p>
1659 
-<div class="eqlabel" id="org64afdaf">
1651 
+<div class="eqlabel" id="orgb9f75a5">
1660 1652 
 <p>
1661 1653 
 <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 1654 
   <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 1655 
   <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 1656 
 </svg></a>
1665 1657 
 </p>
1666 
-<div class="alteqlabels" id="orgb3cda90">
1658 
+<div class="alteqlabels" id="orgcb4fab4">
1667 1659 
 <ul class="org-ul">
1668 1660 
 <li>Gr (7.9)</li>
1669 1661 
 </ul>
 
@@ -1688,14 +1680,14 @@ The precise statement associated to Faraday's observations
1688 1680 
 is that the electromotive force is proportional
1689 1681 
 to the rate of change of the magnetic flux,
1690 1682 
 </p>
1691 
-<div class="eqlabel" id="orgec7b520">
1683 
+<div class="eqlabel" id="orgb7faf93">
1692 1684 
 <p>
1693 1685 
 <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 1686 
   <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 1687 
   <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 1688 
 </svg></a>
1697 1689 
 </p>
1698 
-<div class="alteqlabels" id="orgf70f495">
1690 
+<div class="alteqlabels" id="orgffe5f68">
1699 1691 
 <ul class="org-ul">
1700 1692 
 <li>Gr (7.14)</li>
1701 1693 
 </ul>
 
@@ -1710,18 +1702,18 @@ to the rate of change of the magnetic flux,
1710 1702 
 \]
1711 1703 
 so we obtain
1712 1704 
 </p>
1713 
-<div class="core div" id="orgdfedc05">
1705 
+<div class="core div" id="org80fe686">
1714 1706 
 <p>
1715 1707 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1716 1708 
 </p>
1717 
-<div class="eqlabel" id="orge87df83">
1709 
+<div class="eqlabel" id="orgfa5af5c">
1718 1710 
 <p>
1719 1711 
 <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 1712 
   <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 1713 
   <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 1714 
 </svg></a>
1723 1715 
 </p>
1724 
-<div class="alteqlabels" id="org7b32fe7">
1716 
+<div class="alteqlabels" id="orga84de5d">
1725 1717 
 <ul class="org-ul">
1726 1718 
 <li>Gr (7.15)</li>
1727 1719 
 </ul>
 
@@ -1745,15 +1737,15 @@ for any loop (on a wire or not). Using Stokes' theorem,
1745 1737 
 \]
1746 1738 
 we obtain
1747 1739 
 </p>
1748 
-<div class="core div" id="orgafa0d15">
1749 
-<div class="eqlabel" id="org13d3c14">
1740 
+<div class="core div" id="org2c503e3">
1741 
+<div class="eqlabel" id="org9023002">
1750 1742 
 <p>
1751 1743 
 <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 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"/>
1753 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"/>
1754 1746 
 </svg></a>
1755 1747 
 </p>
1756 
-<div class="alteqlabels" id="orgd662a28">
1748 
+<div class="alteqlabels" id="orgb7f3817">
1757 1749 
 <ul class="org-ul">
1758 1750 
 <li>Gr (7.16)</li>
1759 1751 
 </ul>
 
@@ -1798,7 +1790,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1798 1790 
 </div>
1799 1791 
 <div id="postamble" class="status">
1800 1792 
 <p class="author">Author: Jean-Sébastien Caux</p>
1801 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1793 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1802 1794 
 <p class="validation"></p>
1803 1795 
 </div>
1804 1796

+ 8
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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 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1647,14 +1639,14 @@ W = \frac{1}{2} I \oint {\bf A} \cdot d{\bf l} = \frac{1}{2} \oint ({\bf A} \cdo
1647 1639 
 \]
1648 1640 
 Generalization to volume currents:
1649 1641 
 </p>
1650 
-<div class="eqlabel" id="org87af1ba">
1642 
+<div class="eqlabel" id="org78f90c5">
1651 1643 
 <p>
1652 1644 
 <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 1645 
   <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 1646 
   <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 1647 
 </svg></a>
1656 1648 
 </p>
1657 
-<div class="alteqlabels" id="org95d3f8b">
1649 
+<div class="alteqlabels" id="org0277370">
1658 1650 
 <ul class="org-ul">
1659 1651 
 <li>Gr (7.31)</li>
1660 1652 
 </ul>
 
@@ -1689,15 +1681,15 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1689 1681 
 \]
1690 1682 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1691 1683 
 </p>
1692 
-<div class="core div" id="org247bd37">
1693 
-<div class="eqlabel" id="orgdf4c394">
1684 
+<div class="core div" id="org9994672">
1685 
+<div class="eqlabel" id="org2cf22c6">
1694 1686 
 <p>
1695 1687 
 <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 1688 
   <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 1689 
   <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 1690 
 </svg></a>
1699 1691 
 </p>
1700 
-<div class="alteqlabels" id="orge069b86">
1692 
+<div class="alteqlabels" id="orgde5daac">
1701 1693 
 <ul class="org-ul">
1702 1694 
 <li>Gr (7.34)</li>
1703 1695 
 </ul>
 
@@ -1726,7 +1718,7 @@ W_{mag} &amp;= \frac{1}{2} \int d\tau ~({\bf A} \cdot {\bf J}) &amp;= \frac{1}{2
1726 1718 
 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 1719 
 </p>
1728 1720
1729 
-<div class="example div" id="org03fb4e8">
1721 
+<div class="example div" id="orgd2854e0">
1730 1722 
 <p>
1731 1723 
 <b>Example: energy in coaxial cable</b>
1732 1724 
 </p>
 
@@ -1776,7 +1768,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1776 1768 
 </div>
1777 1769 
 <div id="postamble" class="status">
1778 1770 
 <p class="author">Author: Jean-Sébastien Caux</p>
1779 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1771 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1780 1772 
 <p class="validation"></p>
1781 1773 
 </div>
1782 1774

+ 11
- 19
build/emd_Fl_i.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1649,14 +1641,14 @@ so
1649 1641 
 \]
1650 1642 
 and we can write the mutual inductance as the <b>Neumann formula</b>,
1651 1643 
 </p>
1652 
-<div class="eqlabel" id="org197212f">
1644 
+<div class="eqlabel" id="org766e684">
1653 1645 
 <p>
1654 1646 
 <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 1647 
   <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 1648 
   <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 1649 
 </svg></a>
1658 1650 
 </p>
1659 
-<div class="alteqlabels" id="org4709e6a">
1651 
+<div class="alteqlabels" id="orga92dd51">
1660 1652 
 <ul class="org-ul">
1661 1653 
 <li>Gr (7.22)</li>
1662 1654 
 </ul>
 
@@ -1673,14 +1665,14 @@ M_{21} = \frac{\mu_0}{4\pi} \oint_{{\cal P}_1} \oint_{{\cal P}_2} \frac{d{\bf l}
1673 1665 
 Two things:
1674 1666 
 first, \(M_{21}\) is purely geometrical.  Second,
1675 1667 
 </p>
1676 
-<div class="eqlabel" id="orgb952a44">
1668 
+<div class="eqlabel" id="org577027c">
1677 1669 
 <p>
1678 1670 
 <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 1671 
   <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 1672 
   <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 1673 
 </svg></a>
1682 1674 
 </p>
1683 
-<div class="alteqlabels" id="org53f3037">
1675 
+<div class="alteqlabels" id="org21aff84">
1684 1676 
 <ul class="org-ul">
1685 1677 
 <li>Gr (7.23)</li>
1686 1678 
 </ul>
 
@@ -1695,7 +1687,7 @@ M_{12} = M_{21}
1695 1687 
 \]
1696 1688 
 </p>
1697 1689
1698 
-<div class="example div" id="org6347791">
1690 
+<div class="example div" id="org9334fe0">
1699 1691 
 <p>
1700 1692 
 <b>Example: solenoid in solenoid</b>
1701 1693 
 </p>
 
@@ -1743,14 +1735,14 @@ What if we vary current in loop 1?  Flux in 2 will vary.  Induces EMF in loop 2:
1743 1735 
 \]
1744 1736 
 Changing current also induces EMF in the source loop itself:
1745 1737 
 </p>
1746 
-<div class="eqlabel" id="org667fec5">
1738 
+<div class="eqlabel" id="org4807c27">
1747 1739 
 <p>
1748 1740 
 <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 1741 
   <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 1742 
   <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 1743 
 </svg></a>
1752 1744 
 </p>
1753 
-<div class="alteqlabels" id="org28a1a02">
1745 
+<div class="alteqlabels" id="orga19e4d1">
1754 1746 
 <ul class="org-ul">
1755 1747 
 <li>Gr (7.25)</li>
1756 1748 
 </ul>
 
@@ -1773,7 +1765,7 @@ Inductance:  measured in <b>henries</b> (\(H\)).  \(H = V s/A\).
1773 1765 
 </p>
1774 1766
1775 1767
1776 
-<div class="example div" id="org00e72a3">
1768 
+<div class="example div" id="org940f5ab">
1777 1769 
 <p>
1778 1770 
 <b>Example: self-inductance of toroidal coil</b>
1779 1771 
 </p>
 
@@ -1812,7 +1804,7 @@ Inductance (like capacitance) is intrinsically positive.  Use Lenz law.
1812 1804 
 Think of <i>back EMF</i>.
1813 1805 
 </p>
1814 1806
1815 
-<div class="example div" id="org958ea6c">
1807 
+<div class="example div" id="orgbeea66c">
1816 1808 
 <p>
1817 1809 
 <b>Example: circuit</b>
1818 1810 
 </p>
 
@@ -1861,7 +1853,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1861 1853 
 </div>
1862 1854 
 <div id="postamble" class="status">
1863 1855 
 <p class="author">Author: Jean-Sébastien Caux</p>
1864 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1856 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1865 1857 
 <p class="validation"></p>
1866 1858 
 </div>
1867 1859

+ 17
- 25
build/emd_Fl_ief.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
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3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1645,7 +1637,7 @@ law in integral form:
1645 1637
1646 1638
1647 1639
1648 
-<div class="example div" id="org045463b">
1640 
+<div class="example div" id="org56facb7">
1649 1641 
 <p>
1650 1642 
 <b>Example: loop with time-dependent flux</b>
1651 1643 
 </p>
 
@@ -1661,10 +1653,10 @@ through a horizontal circular region of radius \(R\).
1661 1653
1662 1654 
 <p>
1663 1655 
 <b>Solution</b>:
1664 
-amperian loop of radius \(s\), apply Faraday:
1656 
+amperian loop of radius \(r\), apply Faraday:
1665 1657 
 \[
1666 
-\oint {\bf E} \cdot d{\bf l} = E (2\pi s) = -\frac{d\Phi}{dt} = -\pi s^2 \frac{dB}{dt}
1667 
-\Rightarrow {\bf E} = -\frac{s}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1658 
+\oint {\bf E} \cdot d{\bf l} = E (2\pi r) = -\frac{d\Phi}{dt} = -\pi r^2 \frac{dB}{dt}
1659 
+\Rightarrow {\bf E} = -\frac{r}{2} \frac{dB}{dt} \hat{\boldsymbol \varphi}.
1668 1660 
 \]
1669 1661 
 Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1670 1662 
 </p>
 
@@ -1672,7 +1664,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1672 1664 
 </div>
1673 1665
1674 1666
1675 
-<div class="example div" id="org047ea10">
1667 
+<div class="example div" id="org367fcfc">
1676 1668 
 <p>
1677 1669 
 <b>Example: wheel with charged rim traversed by flux</b>
1678 1670 
 </p>
 
@@ -1713,7 +1705,7 @@ called the <b>quasistatic</b> approximation, and works provided we deal with
1713 1705 
 <i>slow enough</i> phenomena.
1714 1706 
 </p>
1715 1707
1716 
-<div class="example div" id="org733cbdd">
1708 
+<div class="example div" id="orgb2a276c">
1717 1709 
 <p>
1718 1710 
 <b>Example: field from wire with time-dependent current</b>
1719 1711 
 </p>
 
@@ -1723,31 +1715,31 @@ Consider an infinitely long straight wire which carries current \(I(t)\).
1723 1715 
 </p>
1724 1716
1725 1717 
 <p>
1726 
-<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(s\) from wire.
1718 
+<b>Task</b>: find the induced \({\bf E}\) field as a function of distance \(r\) from wire.
1727 1719 
 </p>
1728 1720
1729 1721 
 <p>
1730 1722 
 <b>Solution</b>:  assuming we can use the quasistatic approximation, the
1731 
-magnetic field is \(B = \frac{\mu_0 I}{2\pi s}\)
1723 
+magnetic field is \(B = \frac{\mu_0 I}{2\pi r}\)
1732 1724 
 and circles the wire.  Like \({\bf B}\) field of solenoid, \({\bf E}\) runs parallel
1733 
-to wire.  Amperian loop with sides at distances \(s_0\) and \(s\):
1725 
+to wire.  Amperian loop with sides at distances \(r_0\) and \(r\):
1734 1726 
 \[
1735 
-\oint {\bf E} \cdot d{\bf l} = E(s_0)l - E(s)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
1736 
-= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{s_0}^s \frac{ds'}{s'}
1737 
-= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(s/s_0).
1727 
+\oint {\bf E} \cdot d{\bf l} = E(r_0)l - E(r)l = -\frac{d}{dt} \int {\bf B} \cdot d{\bf a}
1728 
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \int_{r_0}^s \frac{dr'}{r'}
1729 
+= -\frac{\mu_0 l}{2\pi} \frac{dI}{dt} \ln(r/r_0).
1738 1730 
 \]
1739 1731 
 So:
1740 1732 
 \[
1741 
-{\bf E} (s) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln s + K \right] \hat{\bf x}
1733 
+{\bf E} (r) = \left[ \frac{\mu_0}{2\pi} \frac{dI}{dt} \ln r + K \right] \hat{\bf x}
1742 1734 
 \label{Gr(7.19)}
1743 1735 
 \]
1744 1736 
 where \(K\) is a constant (depends on the history of \(I(t)\)).
1745 1737 
 </p>
1746 1738
1747 1739 
 <p>
1748 
-<b>N.B.</b>: this can't be true always, since it blows up as \(s \rightarrow \infty\).
1740 
+<b>N.B.</b>: this can't be true always, since it blows up as \(r \rightarrow \infty\).
1749 1741 
 Reason:  in this case, we've overstepped the quasistatic limit.  We need
1750 
-\(s \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
1742 
+\(r \ll c\tau\) where \(\tau\) is a typical time scale for change of \(I(t)\).
1751 1743 
 </p>
1752 1744
1753 1745 
 </div>
 
@@ -1772,7 +1764,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1772 1764 
 </div>
1773 1765 
 <div id="postamble" class="status">
1774 1766 
 <p class="author">Author: Jean-Sébastien Caux</p>
1775 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1767 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1776 1768 
 <p class="validation"></p>
1777 1769 
 </div>
1778 1770

+ 2
- 10
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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 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1648,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1640 
 </div>
1649 1641 
 <div id="postamble" class="status">
1650 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1643 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1652 1644 
 <p class="validation"></p>
1653 1645 
 </div>
1654 1646

+ 9
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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 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1624,18 +1616,18 @@ Table of contents
1624 1616 
 <p>
1625 1617 
 Full set of equations for the electromagnetic field:
1626 1618 
 </p>
1627 
-<div class="core div" id="org40e4789">
1619 
+<div class="core div" id="orgce899b3">
1628 1620 
 <p>
1629 1621 
 <b>Maxwell's equations</b> <i>(in vacuum)</i>
1630 1622 
 </p>
1631 
-<div class="eqlabel" id="org3e8ad9b">
1623 
+<div class="eqlabel" id="org6517c73">
1632 1624 
 <p>
1633 1625 
 <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 1626 
   <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 1627 
   <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 1628 
 </svg></a>
1637 1629 
 </p>
1638 
-<div class="alteqlabels" id="orgc606bec">
1630 
+<div class="alteqlabels" id="orgfa86c68">
1639 1631
1640 1632 
 </div>
1641 1633 
 
@@ -1652,7 +1644,7 @@ Full set of equations for the electromagnetic field:
1652 1644 
 <p>
1653 1645 
 Complement:
1654 1646 
 </p>
1655 
-<div class="core div" id="org28daec0">
1647 
+<div class="core div" id="orgba4da4a">
1656 1648 
 <p>
1657 1649 
 Force law <a href="./ems_ms_lf_pc.html#LorFo">LorFo</a>
1658 1650 
 \[
 
@@ -1675,15 +1667,15 @@ take divergence of \((iv)\) and use \((i)\).
1675 1667 
 <p>
1676 1668 
 Better way of writing:  all fields on left, all sources on right,
1677 1669 
 </p>
1678 
-<div class="core div" id="org4bb3e78">
1679 
-<div class="eqlabel" id="org05437e5">
1670 
+<div class="core div" id="org8fce2a6">
1671 
+<div class="eqlabel" id="org384b902">
1680 1672 
 <p>
1681 1673 
 <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 1674 
   <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 1675 
   <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 1676 
 </svg></a>
1685 1677 
 </p>
1686 
-<div class="alteqlabels" id="org61742b2">
1678 
+<div class="alteqlabels" id="orgb8b5d0a">
1687 1679 
 <ul class="org-ul">
1688 1680 
 <li>Gr (7.42)</li>
1689 1681 
 </ul>
 
@@ -1721,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1721 1713 
 </div>
1722 1714 
 <div id="postamble" class="status">
1723 1715 
 <p class="author">Author: Jean-Sébastien Caux</p>
1724 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1716 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1725 1717 
 <p class="validation"></p>
1726 1718 
 </div>
1727 1719

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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 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1630,15 +1622,15 @@ The term which should be zero (but isn't) in <a href="./emd_Me_ebM.html#divcurlB
1630 1622 
 \]
1631 1623 
 The extra term would thus be eliminated if we were to put
1632 1624 
 </p>
1633 
-<div class="core div" id="orgdf7b3f9">
1634 
-<div class="eqlabel" id="org5844c7b">
1625 
+<div class="core div" id="org76622f8">
1626 
+<div class="eqlabel" id="org47af899">
1635 1627 
 <p>
1636 1628 
 <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 1629 
   <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 1630 
   <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 1631 
 </svg></a>
1640 1632 
 </p>
1641 
-<div class="alteqlabels" id="org1e192cc">
1633 
+<div class="alteqlabels" id="org4b2df9a">
1642 1634 
 <ul class="org-ul">
1643 1635 
 <li>Gr (7.36)</li>
1644 1636 
 </ul>
 
@@ -1667,18 +1659,18 @@ Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation
1667 1659 
 <p>
1668 1660 
 Maxwell baptized this term the
1669 1661 
 </p>
1670 
-<div class="core div" id="orgb28580f">
1662 
+<div class="core div" id="org1ac2c47">
1671 1663 
 <p>
1672 1664 
 <b>Displacement current</b>
1673 1665 
 </p>
1674 
-<div class="eqlabel" id="org4a72dcc">
1666 
+<div class="eqlabel" id="orgd74a8cb">
1675 1667 
 <p>
1676 1668 
 <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 1669 
   <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 1670 
   <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 1671 
 </svg></a>
1680 1672 
 </p>
1681 
-<div class="alteqlabels" id="org57849e1">
1673 
+<div class="alteqlabels" id="org44de33e">
1682 1674 
 <ul class="org-ul">
1683 1675 
 <li>Gr (7.37)</li>
1684 1676 
 </ul>
 
@@ -1732,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1732 1724 
 </div>
1733 1725 
 <div id="postamble" class="status">
1734 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1735 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1727 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1736 1728 
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1737 1729 
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1738 1730

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@@ -1,7 +1,7 @@
1 1 
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4 
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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1638,14 +1630,14 @@ Fatal inconsistency:  div of curl must always vanish.  Check on \((iii)\):
1638 1630 
 \]
1639 1631 
 But:  try same with \((iv)\):
1640 1632 
 </p>
1641 
-<div class="eqlabel" id="org764ac3e">
1633 
+<div class="eqlabel" id="org83e3419">
1642 1634 
 <p>
1643 1635 
 <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 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"/>
1645 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"/>
1646 1638 
 </svg></a>
1647 1639 
 </p>
1648 
-<div class="alteqlabels" id="org0f4f12d">
1640 
+<div class="alteqlabels" id="org20c79ee">
1649 1641 
 <ul class="org-ul">
1650 1642 
 <li>Gr (7.35)</li>
1651 1643 
 </ul>
 
@@ -1690,7 +1682,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1690 1682 
 </div>
1691 1683 
 <div id="postamble" class="status">
1692 1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
1693 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1685 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1694 1686 
 <p class="validation"></p>
1695 1687 
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1696 1688

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6 6 
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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1671,7 +1663,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1671 1663 
 </div>
1672 1664 
 <div id="postamble" class="status">
1673 1665 
 <p class="author">Author: Jean-Sébastien Caux</p>
1674 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1666 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1675 1667 
 <p class="validation"></p>
1676 1668 
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1677 1669

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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1622,8 +1614,8 @@ Table of contents
1622 1614 
 </svg></a><span class="headline-id">emd.ce</span></h3>
1623 1615
1624 1616 
 <div class="outline-text-3" id="text-emd_ce">
1625 
-<details class="prereq" id="orgca631bc">
1626 
-<summary id="org2ad112f">
1617 
+<details class="prereq" id="org36d2595">
1618 
+<summary id="orgac28d59">
1627 1619 
 Prerequisites
1628 1620 
 </summary>
1629 1621 
 <ul class="org-ul">
 
@@ -1631,8 +1623,8 @@ Prerequisites
1631 1623 
 </ul>
1632 1624 
 </details>
1633 1625
1634 
-<details class="objectives" id="orgbdb48d8">
1635 
-<summary id="org495ffbc">
1626 
+<details class="objectives" id="org84c31d2">
1627 
+<summary id="org5891649">
1636 1628 
 Objectives
1637 1629 
 </summary>
1638 1630 
 <ul class="org-ul">
 
@@ -1670,7 +1662,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1670 1662 
 </div>
1671 1663 
 <div id="postamble" class="status">
1672 1664 
 <p class="author">Author: Jean-Sébastien Caux</p>
1673 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1665 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1674 1666 
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1675 1667 
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1676 1668

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6 6 
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7 7 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1624,18 +1616,18 @@ Table of contents
1624 1616 
 <p>
1625 1617 
 The angular momentum of EM fields is directly given by
1626 1618 
 </p>
1627 
-<div class="main div" id="org6e52344">
1619 
+<div class="main div" id="org7d69a07">
1628 1620 
 <p>
1629 1621 
 <b>Angular momentum of EM fields</b>
1630 1622 
 </p>
1631 
-<div class="eqlabel" id="org9b249c7">
1623 
+<div class="eqlabel" id="org0855fcd">
1632 1624 
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1633 1625 
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1634 1626 
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1635 1627 
   <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 1628 
 </svg></a>
1637 1629 
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1638 
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1630 
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1639 1631
1640 1632 
 </div>
1641 1633 
 
@@ -1669,7 +1661,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1669 1661 
 </div>
1670 1662 
 <div id="postamble" class="status">
1671 1663 
 <p class="author">Author: Jean-Sébastien Caux</p>
1672 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1664 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1673 1665 
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1674 1666 
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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1643,7 +1635,7 @@ This means that
1643 1635 
 \]
1644 1636 
 Since this is true for any volume, we have (re)derived the
1645 1637 
 </p>
1646 
-<div class="core div" id="orgf1c2b2a">
1638 
+<div class="core div" id="org75f48b5">
1647 1639 
 <p>
1648 1640 
 <b>Continuity equation</b> <a href="./ems_ms_ce.html#conteq">conteq</a>
1649 1641 
 \[
 
@@ -1683,7 +1675,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1683 1675 
 </div>
1684 1676 
 <div id="postamble" class="status">
1685 1677 
 <p class="author">Author: Jean-Sébastien Caux</p>
1686 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1678 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1687 1679 
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1688 1680 
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1689 1681

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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1636,18 +1628,18 @@ in which the first integral can be interpreted as the momentum stored in the EM
1636 1628 
 <p>
1637 1629 
 This is thus simply a conservation law for momentum, with
1638 1630 
 </p>
1639 
-<div class="main div" id="orgecd6647">
1631 
+<div class="main div" id="org60dbc44">
1640 1632 
 <p>
1641 1633 
 <b>Momentum density in the EM fields</b>
1642 1634 
 </p>
1643 
-<div class="eqlabel" id="orga1d9e2c">
1635 
+<div class="eqlabel" id="org7808fcd">
1644 1636 
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1645 1637 
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1646 1638 
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1647 1639 
   <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 1640 
 </svg></a>
1649 1641 
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1650 
-<div class="alteqlabels" id="orgf73a06d">
1642 
+<div class="alteqlabels" id="org1473c1c">
1651 1643
1652 1644 
 </div>
1653 1645 
 
@@ -1663,18 +1655,18 @@ This is thus simply a conservation law for momentum, with
1663 1655 
 <p>
1664 1656 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1665 1657 
 </p>
1666 
-<div class="main div" id="org8cc6bb8">
1658 
+<div class="main div" id="orgf7ee289">
1667 1659 
 <p>
1668 1660 
 <b>Continuity equation for EM momentum</b>
1669 1661 
 </p>
1670 
-<div class="eqlabel" id="orgcad090e">
1662 
+<div class="eqlabel" id="orgd7db956">
1671 1663 
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1672 1664 
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1673 1665 
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1674 1666 
   <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 1667 
 </svg></a>
1676 1668 
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1677 
-<div class="alteqlabels" id="org69cf28d">
1669 
+<div class="alteqlabels" id="orge8fbc29">
1678 1670
1679 1671 
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1680 1672 
 
@@ -1707,7 +1699,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1707 1699 
 </div>
1708 1700 
 <div id="postamble" class="status">
1709 1701 
 <p class="author">Author: Jean-Sébastien Caux</p>
1710 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1702 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1711 1703 
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1712 1704 
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1713 1705

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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1675,18 +1667,18 @@ and similarly for \({\boldsymbol B}\). We thus get
1675 1667 
 <p>
1676 1668 
 This expression can be greatly simplified by introducing the
1677 1669 
 </p>
1678 
-<div class="main div" id="orga7d370d">
1670 
+<div class="main div" id="orgaa8145a">
1679 1671 
 <p>
1680 1672 
 <b>Maxwell stress tensor</b>
1681 1673 
 </p>
1682 
-<div class="eqlabel" id="orge5429c3">
1674 
+<div class="eqlabel" id="org0e3a6cc">
1683 1675 
 <p>
1684 1676 
 <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 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"/>
1686 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"/>
1687 1679 
 </svg></a>
1688 1680 
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1689 
-<div class="alteqlabels" id="org0306495">
1681 
+<div class="alteqlabels" id="org4826577">
1690 1682
1691 1683 
 </div>
1692 1684 
 
@@ -1699,7 +1691,7 @@ T_{ij} \equiv \varepsilon_0 \left( E_i E_j - \frac{1}{2} \delta_{ij} E^2\right)
1699 1691
1700 1692 
 </div>
1701 1693 
 <p>
1702 
-The element \(T_{ij}\) represents the force per unit area in the $i$th direction acting on a surface element oriented in the $j$th direction. Diagonal elements are pressures, off-diagonal elements are shears.
1694 
+The element \(T_{ij}\) represents the force per unit area in the \(i\) direction acting on a surface element oriented in the \(j\) direction. Diagonal elements are pressures, off-diagonal elements are shears.
1703 1695 
 </p>
1704 1696
1705 1697 
 
@@ -1707,18 +1699,18 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1707 1699 
 <p>
1708 1700 
 We then obtain the
1709 1701 
 </p>
1710 
-<div class="main div" id="org4781d10">
1702 
+<div class="main div" id="org66e8efe">
1711 1703 
 <p>
1712 1704 
 <b>EM force per unit volume</b>
1713 1705 
 </p>
1714 
-<div class="eqlabel" id="orgd9f91b7">
1706 
+<div class="eqlabel" id="org49902c2">
1715 1707 
 <p>
1716 1708 
 <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 1709 
   <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 1710 
   <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 1711 
 </svg></a>
1720 1712 
 </p>
1721 
-<div class="alteqlabels" id="org7d4a06b">
1713 
+<div class="alteqlabels" id="orge0c52b1">
1722 1714
1723 1715 
 </div>
1724 1716 
 
@@ -1734,18 +1726,18 @@ We then obtain the
1734 1726 
 <p>
1735 1727 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1736 1728 
 </p>
1737 
-<div class="main div" id="orgea18677">
1729 
+<div class="main div" id="org887bc5d">
1738 1730 
 <p>
1739 1731 
 <b>Total force on charges in volume</b>
1740 1732 
 </p>
1741 
-<div class="eqlabel" id="orgef98657">
1733 
+<div class="eqlabel" id="orgd12201e">
1742 1734 
 <p>
1743 1735 
 <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 1736 
   <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 1737 
   <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 1738 
 </svg></a>
1747 1739 
 </p>
1748 
-<div class="alteqlabels" id="org4c9c200">
1740 
+<div class="alteqlabels" id="org0dc7e2f">
1749 1741
1750 1742 
 </div>
1751 1743 
 
@@ -1778,7 +1770,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1778 1770 
 </div>
1779 1771 
 <div id="postamble" class="status">
1780 1772 
 <p class="author">Author: Jean-Sébastien Caux</p>
1781 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1773 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1782 1774 
 <p class="validation"></p>
1783 1775 
 </div>
1784 1776

+ 22
- 30
build/emd_ce_poy.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1645,14 +1637,14 @@ done by EM forces?  From Lorentz force law:
1645 1637 
 Really, we're looking at a small volume element \(d\tau\) carrying charge \(\rho d\tau\), moving
1646 1638 
 at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
1647 1639 
 </p>
1648 
-<div class="eqlabel" id="org0197499">
1640 
+<div class="eqlabel" id="orgc9958b2">
1649 1641 
 <p>
1650 1642 
 <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 1643 
   <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 1644 
   <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 1645 
 </svg></a>
1654 1646 
 </p>
1655 
-<div class="alteqlabels" id="org9aacea3">
1647 
+<div class="alteqlabels" id="orgf9bbccb">
1656 1648 
 <ul class="org-ul">
1657 1649 
 <li>Gr (8.6)</li>
1658 1650 
 </ul>
 
@@ -1665,7 +1657,7 @@ at velocity \({\bf v}\) such that \({\bf J} = \rho {\bf v}\).  Thus,
1665 1657 
 \frac{dW}{dt} = \int_{\cal V} d\tau ~ {\bf E} \cdot {\bf J}
1666 1658 
 \tag{dWdt_intEJ}\label{dWdt_intEJ}
1667 1659 
 \]
1668 
-The integrand is the work done per unit time, per unit volume, {\it i.e.} the power delivered per unit volume.
1660 
+The integrand is the work done per unit time, per unit volume, <i>i.e.</i> the power delivered per unit volume.
1669 1661 
 In terms of fields alone:  use Ampère-Maxwell:
1670 1662 
 \[
1671 1663 
 {\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}
 
@@ -1692,18 +1684,18 @@ so we get
1692 1684 
 Substituting this in <a href="./emd_ce_poy.html#dWdt_intEJ">dWdt_intEJ</a> and using the divergence theorem,
1693 1685 
 we obtain
1694 1686 
 </p>
1695 
-<div class="main div" id="orgf118f4f">
1687 
+<div class="main div" id="orgead023b">
1696 1688 
 <p>
1697 1689 
 <b>Poynting's theorem</b>
1698 1690 
 </p>
1699 
-<div class="eqlabel" id="org1b7cdac">
1691 
+<div class="eqlabel" id="org1b1ef48">
1700 1692 
 <p>
1701 1693 
 <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 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"/>
1703 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"/>
1704 1696 
 </svg></a>
1705 1697 
 </p>
1706 
-<div class="alteqlabels" id="orgd7b6cac">
1698 
+<div class="alteqlabels" id="org9c51283">
1707 1699 
 <ul class="org-ul">
1708 1700 
 <li>Gr (8.9)</li>
1709 1701 
 </ul>
 
@@ -1729,18 +1721,18 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1729 1721 
 <p>
1730 1722 
 Energy per unit time, per unit area carried by EM fields: given by the
1731 1723 
 </p>
1732 
-<div class="core div" id="orgf3198a5">
1724 
+<div class="core div" id="orgafa4bdd">
1733 1725 
 <p>
1734 1726 
 <b>Poynting vector</b>
1735 1727 
 </p>
1736 
-<div class="eqlabel" id="org8725431">
1728 
+<div class="eqlabel" id="org0aaf227">
1737 1729 
 <p>
1738 1730 
 <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 1731 
   <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 1732 
   <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 1733 
 </svg></a>
1742 1734 
 </p>
1743 
-<div class="alteqlabels" id="org6f2879d">
1735 
+<div class="alteqlabels" id="org05edf23">
1744 1736 
 <ul class="org-ul">
1745 1737 
 <li>Gr (8.10)</li>
1746 1738 
 </ul>
 
@@ -1759,18 +1751,18 @@ Energy per unit time, per unit area carried by EM fields: given by the
1759 1751 
 <p>
1760 1752 
 We can thus express Poynting's theorem more compactly:
1761 1753 
 </p>
1762 
-<div class="core div" id="org3a4bb91">
1754 
+<div class="core div" id="orgf7b3c73">
1763 1755 
 <p>
1764 1756 
 <b>Poynting's theorem</b> (integral form)
1765 1757 
 </p>
1766 
-<div class="eqlabel" id="orgbf2cc63">
1758 
+<div class="eqlabel" id="org610ce5e">
1767 1759 
 <p>
1768 1760 
 <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 1761 
   <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 1762 
   <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 1763 
 </svg></a>
1772 1764 
 </p>
1773 
-<div class="alteqlabels" id="org5f484e3">
1765 
+<div class="alteqlabels" id="org462a7a9">
1774 1766 
 <ul class="org-ul">
1775 1767 
 <li>Gr (8.11)</li>
1776 1768 
 </ul>
 
@@ -1789,18 +1781,18 @@ We can thus express Poynting's theorem more compactly:
1789 1781 
 <p>
1790 1782 
 where we have defined the total
1791 1783 
 </p>
1792 
-<div class="core div" id="orgbc7eb16">
1784 
+<div class="core div" id="org49394fc">
1793 1785 
 <p>
1794 1786 
 <b>Energy in electromagnetic fields</b>
1795 1787 
 </p>
1796 
-<div class="eqlabel" id="org89872c3">
1788 
+<div class="eqlabel" id="org2fd18f3">
1797 1789 
 <p>
1798 1790 
 <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 1791 
   <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 1792 
   <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 1793 
 </svg></a>
1802 1794 
 </p>
1803 
-<div class="alteqlabels" id="org67613ca">
1795 
+<div class="alteqlabels" id="orge0e6bd0">
1804 1796 
 <ul class="org-ul">
1805 1797 
 <li>Gr (8.5)</li>
1806 1798 
 </ul>
 
@@ -1829,18 +1821,18 @@ Then,
1829 1821 
 \]
1830 1822 
 so we get the
1831 1823 
 </p>
1832 
-<div class="core div" id="org487db23">
1824 
+<div class="core div" id="orgb7f6aa8">
1833 1825 
 <p>
1834 1826 
 <b>Poynting theorem</b> (differential form)
1835 1827 
 </p>
1836 
-<div class="eqlabel" id="org9593699">
1828 
+<div class="eqlabel" id="orgc43f9ba">
1837 1829 
 <p>
1838 1830 
 <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 1831 
   <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 1832 
   <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 1833 
 </svg></a>
1842 1834 
 </p>
1843 
-<div class="alteqlabels" id="org129becb">
1835 
+<div class="alteqlabels" id="org50e67b6">
1844 1836 
 <ul class="org-ul">
1845 1837 
 <li>Gr (8.14)</li>
1846 1838 
 </ul>
 
@@ -1863,7 +1855,7 @@ and has a similar for to the continuity equation
1863 1855
1864 1856
1865 1857
1866 
-<div class="example div" id="orgd9e0ab5">
1858 
+<div class="example div" id="org76ab6ea">
1867 1859 
 <p>
1868 1860 
 <b>Example:  Joule heating</b>
1869 1861 
 </p>
 
@@ -1895,7 +1887,7 @@ and points radially inwards.  Energy per unit time passing surface of wire:
1895 1887 
 \[
1896 1888 
 \int d{\bf a} \cdot {\bf S} = S (2\pi a L) = -V I
1897 1889 
 \]
1898 
-where the minus sign means energy is flowing {\it in} (the wire heats up),
1890 
+where the minus sign means energy is flowing <i>in</i> (the wire heats up),
1899 1891 
 and the value is as expected.
1900 1892 
 </p>
1901 1893 
 
@@ -1920,7 +1912,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1920 1912 
 </div>
1921 1913 
 <div id="postamble" class="status">
1922 1914 
 <p class="author">Author: Jean-Sébastien Caux</p>
1923 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1915 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1924 1916 
 <p class="validation"></p>
1925 1917 
 </div>
1926 1918

+ 6
- 14
build/emd_emw.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1622,8 +1614,8 @@ Table of contents
1622 1614 
 </svg></a><span class="headline-id">emd.emw</span></h3>
1623 1615
1624 1616 
 <div class="outline-text-3" id="text-emd_emw">
1625 
-<details class="prereq" id="orgd522e84">
1626 
-<summary id="org3c8d630">
1617 
+<details class="prereq" id="org42aca70">
1618 
+<summary id="org3fdd75c">
1627 1619 
 Prerequisites
1628 1620 
 </summary>
1629 1621 
 <ul class="org-ul">
 
@@ -1632,8 +1624,8 @@ Prerequisites
1632 1624 
 </ul>
1633 1625 
 </details>
1634 1626
1635 
-<details class="objectives" id="org98124eb">
1636 
-<summary id="org6546918">
1627 
+<details class="objectives" id="org8aeb437">
1628 
+<summary id="org16d834d">
1637 1629 
 Objectives
1638 1630 
 </summary>
1639 1631 
 <ul class="org-ul">
 
@@ -1674,7 +1666,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1666 
 </div>
1675 1667 
 <div id="postamble" class="status">
1676 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1669 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1678 1670 
 <p class="validation"></p>
1679 1671 
 </div>
1680 1672

+ 30
- 20
build/emd_emw_ep.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1652,12 +1644,30 @@ so for a monochromatic EM plan wave,
1652 1644 
 \]
1653 1645 
 or more succinctly:
1654 1646 
 </p>
1655 
-<div class="main div" id="org0e45759">
1647 
+<div class="main div" id="orgee78283">
1648 
+<p>
1649 
+<b>Poynting vector of a monochromatic EM wave</b>
1650 
+</p>
1651 
+<div class="eqlabel" id="orgdedd25b">
1652 
+<p>
1653 
+<a id="Poynting_mpw"></a><a href="./emd_emw_ep.html#Poynting_mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1654 
+  <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"/>
1655 
+  <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"/>
1656 
+</svg></a>
1657 
+</p>
1658 
+<div class="alteqlabels" id="orgdfa9b5e">
1659 
+<ul class="org-ul">
1660 
+<li>Gr (9.57)</li>
1661 
+</ul>
1662 
+
1663 
+</div>
1664 
+
1665 
+</div>
1656 1666 
 <p>
1657 
-{\bf Poynting vector of a monochromatic EM wave}
1658 1667 
 \[
1659 
-    {\boldsymbol S} = c u ~\hat{\boldsymbol k}
1660 
-  \]
1668 
+{\boldsymbol S} = c u ~\hat{\boldsymbol k}
1669 
+\tag{Poynting_mpw}\label{Poyting_mpw}
1670 
+\]
1661 1671 
 </p>
1662 1672
1663 1673 
 </div>
 
@@ -1668,12 +1678,12 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1668 1678 
 <p>
1669 1679 
 Similary, we get the
1670 1680 
 </p>
1671 
-<div class="main div" id="org4b55ff9">
1681 
+<div class="main div" id="org108cc57">
1672 1682 
 <p>
1673 
-{\bf Momentum density of a monochromatic EM wave}
1683 
+<b>Momentum density of a monochromatic EM wave</b>
1674 1684 
 \[
1675 
-    {\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k}
1676 
-  \]
1685 
+{\boldsymbol g} = \frac{1}{c^2} {\boldsymbol S} = \frac{u}{c} ~\hat{\boldsymbol k}
1686 
+\]
1677 1687 
 </p>
1678 1688
1679 1689 
 </div>
 
@@ -1688,14 +1698,14 @@ Time averages: integrating over a (integer number of) cycle(s), we have
1688 1698 
 </p>
1689 1699
1690 1700 
 <p>
1691 
-The average power per unit time per unit area transported by an EM wave is called the {\bf Intensity}
1701 
+The average power per unit time per unit area transported by an EM wave is called the <b>Intensity</b>
1692 1702 
 \[
1693 1703 
   I \equiv \langle S \rangle = \frac{c\varepsilon_0}{2} E_0^2
1694 1704 
 \]
1695 1705 
 </p>
1696 1706
1697 1707 
 <p>
1698 
-The {\it radiation pressure} is the momentum transfer per unit area per unit of time
1708 
+The <b>radiation pressure</b> is the momentum transfer per unit area per unit of time
1699 1709 
 \[
1700 1710 
   P = \frac{1}{A}\frac{\Delta p}{\Delta t} = \frac{\langle g \rangle A c \Delta t}{A \Delta t} = \frac{\varepsilon_0}{2} E_0^2 = \frac{I}{c}.
1701 1711 
 \]
 
@@ -1719,7 +1729,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1729 
 </div>
1720 1730 
 <div id="postamble" class="status">
1721 1731 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1732 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1723 1733 
 <p class="validation"></p>
1724 1734 
 </div>
1725 1735

+ 50
- 23
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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 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1622,7 +1614,7 @@ Table of contents
1622 1614 
 </svg></a><span class="headline-id">emd.emw.mpw</span></h4>
1623 1615 
 <div class="outline-text-4" id="text-emd_emw_mpw">
1624 1616 
 <p>
1625 
-A {\it monochromatic} wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have
1617 
+A <b>monochromatic</b> wave is one having a single frequency in its temporal dependence. Say that the propagation direction is \(\hat{\boldsymbol z}\): we'd then have
1626 1618 
 \[
1627 1619 
   {\bf E} (z, t) = {\bf E}_0 e^{i(k z - \omega t)}, \hspace{1cm}
1628 1620 
   {\bf B} (z,t) = {\bf B}_0 e^{i(k z - \omega t)}
 
@@ -1632,7 +1624,7 @@ Maxwell's equations impose constraints.  Since \({\boldsymbol \nabla} \cdot {\bf
1632 1624 
 (E_0)_z = 0 = (B_0)_z
1633 1625 
 \label{Gr(9.44)}
1634 1626 
 \]
1635 
-so {\bf electromagnetic waves are transverse}.
1627 
+so <b>electromagnetic waves are transverse</b>.
1636 1628 
 </p>
1637 1629
1638 1630 
 <p>
 
@@ -1643,9 +1635,26 @@ From Faraday:  \({\boldsymbol \nabla} \times {\bf E} = -\partial {\bf B}/\partia
1643 1635 
 \label{Gr(9.46)}
1644 1636 
 \]
1645 1637 
 so \({\bf E}\) and \({\bf B}\) are mutually perpendicular, and
1638 
+</p>
1639 
+<div class="eqlabel" id="org4747373">
1640 
+<p>
1641 
+<a id="EBmpw"></a><a href="./emd_emw_mpw.html#EBmpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 
+  <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 
+</svg></a>
1645 
+</p>
1646 
+<div class="alteqlabels" id="org2ea2f4c">
1647 
+<ul class="org-ul">
1648 
+<li>Gr (9.47)</li>
1649 
+</ul>
1650 
+
1651 
+</div>
1652 
+
1653 
+</div>
1654 
+<p>
1646 1655 
 \[
1647 1656 
 B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1648 
-\label{Gr(9.47)}
1657 
+\tag{EBmpw}\label{EBmpw}
1649 1658 
 \]
1650 1659 
 </p>
1651 1660 
 
@@ -1653,19 +1662,37 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1653 1662 
 Generalizing to propagation in the direction of an arbitrary wavevector
1654 1663 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1655 1664 
 </p>
1656 
-<div class="core div" id="org53e84bf">
1665 
+<div class="core div" id="orga666428">
1666 
+<p>
1667 
+<b>E and B fields for a monochromatic EM plane wave</b>
1668 
+</p>
1669 
+<div class="eqlabel" id="orge12acff">
1670 
+<p>
1671 
+<a id="mpw"></a><a href="./emd_emw_mpw.html#mpw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1672 
+  <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"/>
1673 
+  <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"/>
1674 
+</svg></a>
1675 
+</p>
1676 
+<div class="alteqlabels" id="orgd889027">
1677 
+<ul class="org-ul">
1678 
+<li>Gr (9.49)</li>
1679 
+</ul>
1680 
+
1681 
+</div>
1682 
+
1683 
+</div>
1657 1684 
 <p>
1658 
-{\bf E and B fields for a monochromatic EM plane wave}
1659 1685 
 \[
1660 
-    {\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
1661 
-    \hspace{10mm}
1662 
-           {\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
1663 
-           = \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
1664 
-  \]
1665 
-with the transversality condition
1686 
+{\boldsymbol E} ({\boldsymbol r},t ) = E_0 e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol n},
1687 
+\hspace{10mm}
1688 
+{\boldsymbol B} ({\boldsymbol r}, t) = \frac{E_0}{c} e^{i({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)} ~\hat{\boldsymbol k} \times \hat{\boldsymbol n}
1689 
+= \frac{1}{c} ~\hat{\boldsymbol k} \times {\boldsymbol E} ({\boldsymbol r}, t)
1690 
+\tag{mpw}\label{mpw}
1691 
+\]
1692 
+with the <b>transversality condition</b>
1666 1693 
 \[
1667 
-    \hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
1668 
-  \]
1694 
+\hat{\boldsymbol k} \cdot \hat{\boldsymbol n} = 0
1695 
+\]
1669 1696 
 </p>
1670 1697
1671 1698 
 </div>
 
@@ -1697,7 +1724,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1697 1724 
 </div>
1698 1725 
 <div id="postamble" class="status">
1699 1726 
 <p class="author">Author: Jean-Sébastien Caux</p>
1700 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1727 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1701 1728 
 <p class="validation"></p>
1702 1729 
 </div>
1703 1730

+ 27
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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 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1622,7 +1614,6 @@ Table of contents
1622 1614 
 </svg></a><span class="headline-id">emd.emw.we</span></h4>
1623 1615 
 <div class="outline-text-4" id="text-emd_emw_we">
1624 1616 
 <p>
1625 
-\subsubsection*{The wave equation for \({\bf E}\) and \({\bf B}\)}
1626 1617 
 Take Maxwell's equations in vacuum:
1627 1618 
 </p>
1628 1619 
 \begin{align}
 
@@ -1650,15 +1641,32 @@ These take the form of coupled first-order partial differential equations for \(
1650 1641 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1651 1642 
 we get the
1652 1643 
 </p>
1653 
-<div class="core div" id="orgb3cd985">
1644 
+<div class="core div" id="orgc9c5d44">
1645 
+<p>
1646 
+<b>Wave equations for electric and magnetic fields in vacuum</b>
1647 
+</p>
1648 
+<div class="eqlabel" id="orgb29f135">
1649 
+<p>
1650 
+<a id="WaveEq"></a><a href="./emd_emw_we.html#WaveEq"><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="orge5c16c5">
1656 
+<ul class="org-ul">
1657 
+<li>Gr (9.41)</li>
1658 
+</ul>
1659 
+
1660 
+</div>
1661 
+
1662 
+</div>
1654 1663 
 <p>
1655 
-{\bf Wave equations for electric and magnetic fields in vacuum}
1656 1664 
 \[
1657 
-    {\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
1658 
-    \hspace{1cm}
1659 
-           {\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
1660 
-           \label{Gr(9.41)}
1661 
-  \]
1665 
+{\boldsymbol \nabla}^2 {\bf E} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},
1666 
+\hspace{1cm}
1667 
+{\boldsymbol \nabla}^2 {\bf B} = \mu_0 \varepsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2}.
1668 
+\tag{WaveEq}\label{WaveEq}
1669 
+\]
1662 1670 
 </p>
1663 1671
1664 1672 
 </div>
 
@@ -1681,7 +1689,7 @@ That is, a form
1681 1689 
 {\bf E} ({\bf r},t) = {\bf E}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)}, \hspace{1cm}
1682 1690 
 {\bf B} ({\bf r},t) = {\bf B}_0 e^{i ({\bf k} \cdot {\bf r} - \omega t)},
1683 1691 
 \]
1684 
-solves (\ref{Gr(9.41)}) for \(\omega = c |{\bf k}|\).
1692 
+solves <a href="./emd_emw_we.html#WaveEq">WaveEq</a> for \(\omega = c |{\bf k}|\).
1685 1693 
 Here and under, we use complex exponentials for convenience, remembering that
1686 1694 
 the actual electric and magnetic fields are given by the real part.
1687 1695 
 </p>
 
@@ -1706,7 +1714,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1706 1714 
 </div>
1707 1715 
 <div id="postamble" class="status">
1708 1716 
 <p class="author">Author: Jean-Sébastien Caux</p>
1709 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1717 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1710 1718 
 <p class="validation"></p>
1711 1719 
 </div>
1712 1720

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1 1 
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2 2 
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3 3 
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4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1646,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 1638 
 </div>
1647 1639 
 <div id="postamble" class="status">
1648 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1641 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1650 1642 
 <p class="validation"></p>
1651 1643 
 </div>
1652 1644

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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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4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1646,7 +1638,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1646 1638 
 </div>
1647 1639 
 <div id="postamble" class="status">
1648 1640 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1641 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1650 1642 
 <p class="validation"></p>
1651 1643 
 </div>
1652 1644

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1 1 
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2 2 
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3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1637,12 +1629,12 @@ free charges and currents.
1637 1629 
 </p>
1638 1630
1639 1631 
 <p>
1640 
-From static case:  electric polarization \({\bf P}\) produces bound charge density (\ref{Gr(4.12)})
1632 
+From static case:  electric polarization \({\bf P}\) produces bound charge density <a href="./emsm_esm_po.html#rhob">rhob</a>
1641 1633 
 \[
1642 1634 
 \rho_b = -{\boldsymbol \nabla} \cdot {\bf P}
1643 1635 
 \label{Gr(7.46)}
1644 1636 
 \]
1645 
-and magnetization \({\bf M}\) produces bound current density (\ref{Gr(6.13)})
1637 
+and magnetization \({\bf M}\) produces bound current density <a href="./emsm_msm_fmo_bc.html#JbcurlM">JbcurlM</a>
1646 1638 
 \[
1647 1639 
 {\bf J}_b = {\boldsymbol \nabla} \times {\bf M}
1648 1640 
 \label{Gr(7.47)}
 
@@ -1657,13 +1649,30 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1657 1649 
 \]
1658 1650 
 We therefore have the
1659 1651 
 </p>
1660 
-<div class="core div" id="org0421a72">
1652 
+<div class="core div" id="orgf1835ae">
1653 
+<p>
1654 
+<b>Polarization current density</b>
1655 
+</p>
1656 
+<div class="eqlabel" id="orgfad2da8">
1657 
+<p>
1658 
+<a id="Jp"></a><a href="./emdm_Me_Mem.html#Jp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1659 
+  <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"/>
1660 
+  <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"/>
1661 
+</svg></a>
1662 
+</p>
1663 
+<div class="alteqlabels" id="orgf15b4cb">
1664 
+<ul class="org-ul">
1665 
+<li>Gr (7.48)</li>
1666 
+</ul>
1667 
+
1668 
+</div>
1669 
+
1670 
+</div>
1661 1671 
 <p>
1662 
-{\bf Polarization current density}
1663 1672 
 \[
1664 
-    {\bf J}_p = \frac{\partial {\bf P}}{\partial t}
1665 
-    \label{Gr(7.48)}
1666 
-  \]
1673 
+{\bf J}_p = \frac{\partial {\bf P}}{\partial t}
1674 
+\tag{Jp}\label{Jp}
1675 
+\]
1667 1676 
 </p>
1668 1677
1669 1678 
 </div>
 
@@ -1675,10 +1684,9 @@ the polarization current is the result of linear motion of charge when
1675 1684 
 polarization changes).  We can check consistency with the continuity equation
1676 1685 
 associated to the conservation of bound charges:
1677 1686 
 </p>
1678 
-<aside id="org642846e">
1687 
+<aside id="orgadb90c7">
1679 1688 
 <p>
1680 
-Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1681 
-  \({\boldsymbol J}_b\) but this is not the convention used here.
1689 
+Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current \({\boldsymbol J}_b\) but this is not the common convention.
1682 1690 
 </p>
1683 1691 
 </aside>
1684 1692 
 <p>
 
@@ -1696,31 +1704,56 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1696 1704
1697 1705 
 <p>
1698 1706 
 In view of this:  total charge density can be separated into 2 parts,
1699 
-{\it free} and {\it bound}:
1707 
+<b>free</b> and <b>bound</b>:
1708 
+</p>
1709 
+<div class="main div" id="orgbe78924">
1710 
+<div class="eqlabel" id="org4679dd8">
1711 
+<p>
1712 
+<a id="rhofb"></a><a href="./emdm_Me_Mem.html#rhofb"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1713 
+  <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"/>
1714 
+  <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"/>
1715 
+</svg></a>
1700 1716 
 </p>
1701 
-<div class="main div" id="orgba471ef">
1717 
+<div class="alteqlabels" id="orgb7e7ed8">
1718 
+<ul class="org-ul">
1719 
+<li>Gr (7.49)</li>
1720 
+</ul>
1721 
+
1722 
+</div>
1723 
+
1724 
+</div>
1702 1725 
 <p>
1703 1726 
 \[
1704 
-    \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
1705 
-    \label{Gr(7.49)}
1706 
-  \]
1727 
+\rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
1728 
+\tag{rhofb}\label{rhofb}
1729 
+\]
1707 1730 
 </p>
1708 1731
1709 1732 
 </div>
1710 1733 
 <p>
1711 
-and current can be separated into three parts, {\it free}, {\it bound} and
1712 
-{\it polarization}:
1734 
+and current can be separated into three parts, <b>free</b>, <b>bound</b> and
1735 
+<b>polarization</b>:
1713 1736 
 </p>
1714 
-<div class="main div" id="org2ffd81b">
1737 
+<div class="main div" id="org1c506a7">
1738 
+<div class="eqlabel" id="orgbf0c17d">
1715 1739 
 <p>
1716 
-\[
1717 
-  {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
1740 
+<a id="Jfbp"></a><a href="./emdm_Me_Mem.html#Jfbp"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1741 
+  <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"/>
1742 
+  <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"/>
1743 
+</svg></a>
1718 1744 
 </p>
1745 
+<div class="alteqlabels" id="org33aea7e">
1719 1746 
 <ul class="org-ul">
1720 
-<li>\frac{∂ {\bf P}}{∂ t}.</li>
1747 
+<li>Gr (7.50)</li>
1721 1748 
 </ul>
1749 
+
1750 
+</div>
1751 
+
1752 
+</div>
1722 1753 
 <p>
1723 
-  \label{Gr(7.50)}
1754 
+\[
1755 
+{\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t}.
1756 
+\tag{Jfbp}\label{Jfbp}
1724 1757 
 \]
1725 1758 
 </p>
1726 1759 
 
@@ -1735,24 +1768,19 @@ Gauss's law:  can be rewritten
1735 1768 
 \]
1736 1769 
 where (as in static case)
1737 1770 
 </p>
1738 
-<div class="core div" id="org88bb3c5">
1771 
+<div class="core div" id="org501f375">
1739 1772 
 <p>
1740 1773 
 \[
1741 
-    {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
1742 
-    \label{Gr(7.52)}
1743 
-  \]
1774 
+{\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
1775 
+\label{Gr(7.52)}
1776 
+\]
1744 1777 
 </p>
1745 1778
1746 1779 
 </div>
1747 1780 
 <p>
1748 1781 
 Ampère's law including Maxwell's term:
1749 1782 
 \[
1750 
-{\boldsymbol ∇} × {\bf B} = μ_0 \left( {\bf J}_f + {\boldsymbol ∇} × {\bf M}
1751 
-</p>
1752 
-<ul class="org-ul">
1753 
-<li>\frac{∂ {\bf P}}{∂ t} \right) + μ_0 ε_0 \frac{∂ {\bf E}}{∂ t},</li>
1754 
-</ul>
1755 
-<p>
1783 
+{\boldsymbol \nabla} \times {\bf B} = \mu_0 \left( {\bf J}_f + {\boldsymbol \nabla} \times {\bf M} + \frac{\partial {\bf P}}{\partial t} \right) + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t},
1756 1784 
 \]
1757 1785 
 or
1758 1786 
 \[
 
@@ -1761,12 +1789,12 @@ or
1761 1789 
 \]
1762 1790 
 where as before
1763 1791 
 </p>
1764 
-<div class="core div" id="org90cea20">
1792 
+<div class="core div" id="orgfaac9ca">
1765 1793 
 <p>
1766 1794 
 \[
1767 
-    {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
1768 
-    \label{Gr(7.54)}
1769 
-  \]
1795 
+{\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
1796 
+\label{Gr(7.54)}
1797 
+\]
1770 1798 
 </p>
1771 1799
1772 1800 
 </div>
 
@@ -1779,21 +1807,36 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1779 1807 
 <p>
1780 1808 
 In terms of free charges and currents, we thus get
1781 1809 
 </p>
1782 
-<div class="core div" id="orgd6526ab">
1810 
+<div class="core div" id="org4a1df55">
1811 
+<p>
1812 
+<b>Maxwell's equations</b> <i>(in matter)</i>
1813 
+</p>
1814 
+<div class="eqlabel" id="orga6eb31a">
1783 1815 
 <p>
1784 
-{\bf Maxwell's equations {\it (in matter)}}
1816 
+<a id="Max_mat"></a><a href="./emdm_Me_Mem.html#Max_mat"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1817 
+  <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"/>
1818 
+  <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"/>
1819 
+</svg></a>
1785 1820 
 </p>
1821 
+<div class="alteqlabels" id="org154a0ec">
1822 
+<ul class="org-ul">
1823 
+<li>Gr (7.55)</li>
1824 
+</ul>
1825 
+
1826 
+</div>
1827 
+
1828 
+</div>
1786 1829 
 \begin{align}
1787 
-  (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
1788 
-  (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
1789 
-  (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
1790 
-  (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
1791 
-  \label{Gr(7.55)}
1830 
+    (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = \rho_f, \nonumber \\
1831 
+    (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
1832 
+    (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
1833 
+    (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = {\bf J}_f + \frac{\partial {\bf D}}{\partial t}.
1834 
+\tag{Max_mat}\label{Max_mat}
1792 1835 
 \end{align}
1793 1836
1794 1837 
 </div>
1795 1838 
 <p>
1796 
-Last term:  {\bf displacement current},
1839 
+Last term:  <b>displacement current</b>,
1797 1840 
 \[
1798 1841 
 {\bf J}_d = \frac{\partial {\bf D}}{\partial t}
1799 1842 
 \label{Gr(7.58)}
 
@@ -1801,23 +1844,37 @@ Last term:  {\bf displacement current},
1801 1844 
 </p>
1802 1845
1803 1846 
 <p>
1804 
-Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and \({\bf H}\)
1847 
+This must all be complemented by the <b>constitutive relations</b> giving \({\bf D}\) and \({\bf H}\)
1805 1848 
 in terms of \({\bf E}\) and \({\bf B}\).
1806 1849 
 For the restricted case of linear media:
1807 1850 
 </p>
1808 
-<div class="main div" id="orgd345cd6">
1851 
+<div class="main div" id="orge60b08f">
1852 
+<div class="eqlabel" id="org88284f8">
1853 
+<p>
1854 
+<a id="consrel"></a><a href="./emdm_Me_Mem.html#consrel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1855 
+  <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"/>
1856 
+  <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"/>
1857 
+</svg></a>
1858 
+</p>
1859 
+<div class="alteqlabels" id="org21479eb">
1860 
+<ul class="org-ul">
1861 
+<li>Gr (7.56,7.57)</li>
1862 
+</ul>
1863 
+
1864 
+</div>
1865 
+
1866 
+</div>
1809 1867 
 <p>
1810 1868 
 \[
1811 
-    {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
1812 
-    {\bf M} = \chi_m {\bf H}
1813 
-    \label{Gr(7.56)}
1814 
-  \]
1869 
+{\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
1870 
+{\bf M} = \chi_m {\bf H}
1871 
+\]
1815 1872 
 so
1816 1873 
 \[
1817 
-    {\bf D} = \varepsilon {\bf E}, \hspace{1cm}
1818 
-    {\bf H} = \frac{1}{\mu} {\bf B},
1819 
-    \label{Gr(7.57)}
1820 
-  \]
1874 
+{\bf D} = \varepsilon {\bf E}, \hspace{1cm}
1875 
+{\bf H} = \frac{1}{\mu} {\bf B},
1876 
+\tag{consrel}\label{consrel}
1877 
+\]
1821 1878 
 where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1 + \chi_m)\).
1822 1879 
 </p>
1823 1880 
 
@@ -1842,7 +1899,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1842 1899 
 </div>
1843 1900 
 <div id="postamble" class="status">
1844 1901 
 <p class="author">Author: Jean-Sébastien Caux</p>
1845 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1902 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1846 1903 
 <p class="validation"></p>
1847 1904 
 </div>
1848 1905

+ 126
- 31
build/emdm_Me_bc.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1624,74 +1616,177 @@ Table of contents
1624 1616 
 <p>
1625 1617 
 Discontinuities between different media, deduced from
1626 1618 
 </p>
1627 
-<div class="core div" id="org48bc400">
1619 
+<div class="core div" id="org2152f07">
1628 1620 
 <p>
1629 
-{\bf Maxwell's equations {\it (in matter)}, integral form}
1621 
+<b>Maxwell's equations</b> <i>(in matter)</i>, <i>integral form</i>
1630 1622 
 </p>
1623 
+<div class="eqlabel" id="org896f0bf">
1624 
+<p>
1625 
+<a id="Max_mat_int"></a><a href="./emdm_Me_bc.html#Max_mat_int"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1626 
+  <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"/>
1627 
+  <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"/>
1628 
+</svg></a>
1629 
+</p>
1630 
+<div class="alteqlabels" id="org37cc7e4">
1631 
+
1632 
+</div>
1633 
+
1634 
+</div>
1631 1635 
 \begin{align}
1632 
-  (i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
1633 
-  (ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
1634 
-  (iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
1635 
-  (iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
1636 
+    (i)~~ &amp;\oint_{\cal S} {\bf D} \cdot d{\bf a} = Q_{f_{enc}}, \nonumber \\
1637 
+    (ii)~~ &amp;\oint_{\cal S} {\bf B} \cdot d{\bf a} = 0 \nonumber \\
1638 
+    (iii)~~ &amp;\oint_{\cal P} {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}, \nonumber \\
1639 
+    (iv)~~ &amp;\oint_{\cal P} {\bf H} \cdot d{\bf l} = I_{f_{enc}} + \frac{d}{dt} \int_{\cal S} {\bf D} \cdot d{\bf a}.
1640 
+\tag{Max_mat_int}\label{Max_mat_int}
1636 1641 
 \end{align}
1637 1642
1638 1643 
 </div>
1639 1644 
 <p>
1640 1645 
 Applying \((i)\) to wafer-thin Gaussian pillbox straddling boundary between 2 materials:
1641 1646 
 \({\bf D}_1 \cdot {\bf a} - {\bf D}_2 \cdot {\bf a} = \sigma_f a\) so
1647 
+</p>
1648 
+<div class="main div" id="orgaee2676">
1649 
+<div class="eqlabel" id="org35b4f05">
1650 
+<p>
1651 
+<a id="Ddisc"></a><a href="./emdm_Me_bc.html#Ddisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1652 
+  <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 
+  <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 
+</svg></a>
1655 
+</p>
1656 
+<div class="alteqlabels" id="orgc9c66ee">
1657 
+<ul class="org-ul">
1658 
+<li>Gr (7.60)</li>
1659 
+</ul>
1660 
+
1661 
+</div>
1662 
+
1663 
+</div>
1664 
+<p>
1642 1665 
 \[
1643 
-\boxed{
1644 1666 
 D^{\perp}_1 - D^{\perp}_2 = \sigma_f
1645 
-}
1646 
-\label{Gr(7.59)}
1667 
+\tag{Ddisc}\label{Ddisc}
1647 1668 
 \]
1648 
-Same reasoning applied to \((ii)\) gives
1669 
+</p>
1670 
+
1671 
+</div>
1672 
+<p>
1673 
+Same reasoning applied to \((ii)\) gives <a href="./ems_ms_vp_mbc.html#Bdisc">Bdisc</a>
1674 
+</p>
1675 
+<div class="main div" id="org0c24b54">
1676 
+<div class="eqlabel" id="org262505f">
1677 
+<div class="alteqlabels" id="org560c237">
1678 
+
1679 
+</div>
1680 
+
1681 
+</div>
1682 
+<p>
1649 1683 
 \[
1650 
-\boxed{
1651 1684 
 B^{\perp}_1 - B^{\perp}_2 = 0
1652 
-}
1653 
-\label{Gr(7.60)}
1654 1685 
 \]
1686 
+</p>
1687 
+
1688 
+</div>
1689 
+<p>
1655 1690 
 For \((iii)\):  Amperian loop straddling surface:  \({\bf E}_1 \cdot {\bf l} - {\bf E}_2 \cdot {\bf l} =
1656 1691 
 -\frac{d}{dt} \int_{\cal S} {\bf B} \cdot d{\bf a}\).  Limit of small loop:  flux vanishes, therefore
1692 
+</p>
1693 
+<div class="main div" id="org2c8267d">
1694 
+<p>
1657 1695 
 \[
1658 
-\boxed{
1659 1696 
 {\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0
1660 
-}
1661 
-\label{Gr(7.61)}
1662 1697 
 \]
1698 
+</p>
1699 
+
1700 
+</div>
1701 
+<p>
1663 1702 
 Similarly, \((iv)\) implies \({\bf H}_1 \cdot {\bf l} - {\bf H}_2 \cdot {\bf l} = I_{f_{enc}}\).
1664 1703 
 No volume current can contribute, but a surface current can.  Can write
1665 1704 
 \(I_{f_{enc}} = {\bf K}_f \cdot (\hat{\bf n} \times {\bf l}) = ({\bf K}_f \times \hat{\bf n}) \cdot {\bf l}\)
1666 1705 
 and thus
1706 
+</p>
1707 
+<div class="main div" id="orge9d0c89">
1708 
+<div class="eqlabel" id="org15b0fbc">
1709 
+<p>
1710 
+<a id="Hdisc"></a><a href="./emsm_msm_H_A.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1711 
+  <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"/>
1712 
+  <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"/>
1713 
+</svg></a><a href="./emdm_Me_bc.html#Hdisc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1714 
+  <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"/>
1715 
+  <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"/>
1716 
+</svg></a>
1717 
+</p>
1718 
+<div class="alteqlabels" id="orgc1f1499">
1719 
+<ul class="org-ul">
1720 
+<li>Gr (7.63)</li>
1721 
+</ul>
1722 
+
1723 
+</div>
1724 
+
1725 
+</div>
1726 
+<p>
1667 1727 
 \[
1668 
-\boxed{
1669 1728 
 {\bf H}_1^{\parallel} - {\bf H}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}
1670 
-}
1671 
-\label{Gr(7.62)}
1729 
+\tag{Hdisc}\label{Hdisc}
1672 1730 
 \]
1731 
+</p>
1732 
+
1733 
+</div>
1734 
+<p>
1673 1735 
 These are the general boundary conditions for electrodynamics.
1674 1736 
 </p>
1675 1737
1676 1738 
 <p>
1677 1739 
 In case of linear media:  can be expressed in terms of \({\bf E}\) and \({\bf B}\) alone:
1678 1740 
 </p>
1741 
+<div class="main div" id="org8b0c892">
1742 
+<div class="eqlabel" id="org2d4783e">
1743 
+<p>
1744 
+<a id="disc_lm"></a><a href="./emdm_Me_bc.html#disc_lm"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1745 
+  <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"/>
1746 
+  <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"/>
1747 
+</svg></a>
1748 
+</p>
1749 
+<div class="alteqlabels" id="orga65ee45">
1750 
+<ul class="org-ul">
1751 
+<li>Gr (7.64)</li>
1752 
+</ul>
1753 
+
1754 
+</div>
1755 
+
1756 
+</div>
1679 1757 
 \begin{align}
1680 1758 
 (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = \sigma_f, \nonumber \\
1681 1759 
 (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
1682 1760 
 (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
1683 1761 
 (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = {\bf K}_f \times \hat{\bf n}.
1684 
-\label{Gr(7.63)}
1762 
+\tag{disc_lm}\label{disc_lm}
1685 1763 
 \end{align}
1764 
+
1765 
+</div>
1686 1766 
 <p>
1687 1767 
 If there is no free charge and no free current at boundary:
1688 1768 
 </p>
1769 
+<div class="eqlabel" id="org656d25d">
1770 
+<p>
1771 
+<a id="disc_nfc"></a><a href="./emdm_Me_bc.html#disc_nfc"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1772 
+  <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"/>
1773 
+  <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"/>
1774 
+</svg></a>
1775 
+</p>
1776 
+<div class="alteqlabels" id="org856682c">
1777 
+<ul class="org-ul">
1778 
+<li>Gr (7.64)</li>
1779 
+</ul>
1780 
+
1781 
+</div>
1782 
+
1783 
+</div>
1689 1784 
 \begin{align}
1690 1785 
 (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
1691 1786 
 (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
1692 1787 
 (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
1693 1788 
 (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0.
1694 
-\label{Gr(7.64)}
1789 
+\tag{disc_nfc}\label{disc_nfc}
1695 1790 
 \end{align}
1696 1791 
 <p>
1697 1792 
 These are basis of theory of reflection and refraction.
 
@@ -1715,7 +1810,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1715 1810 
 </div>
1716 1811 
 <div id="postamble" class="status">
1717 1812 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1813 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1719 1814 
 <p class="validation"></p>
1720 1815 
 </div>
1721 1816

+ 2
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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 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1649,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1649 1641 
 </div>
1650 1642 
 <div id="postamble" class="status">
1651 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1644 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1653 1645 
 <p class="validation"></p>
1654 1646 
 </div>
1655 1647

+ 4
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1 1 
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2 2 
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3 3 
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4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1615,7 +1607,7 @@ Table of contents
1615 1607 
 </ul>
1616 1608 
 </details>
1617 1609 
 </nav>
1618 
-<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Absorption and Dispersion</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1610 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li>Absorption and Dispersion</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1619 1611 
 <h4 id="emdm_emwm_ad">Absorption and Dispersion<a class="headline-permalink" href="./emdm_emwm_ad.html#emdm_emwm_ad"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1620 1612 
   <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 1613 
   <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"/>
 
@@ -1630,7 +1622,7 @@ Table of contents
1630 1622 
 <li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li>
1631 1623 
 </ul>
1632 1624
1633 
-<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_Ba.html">Brewster's Angle&emsp;<small>[emdm.emwm.refl.Ba]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1625 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_oi.html">Oblique Incidence&emsp;<small>[emdm.emwm.refl.oi]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
1634 1626 
 <br>
1635 1627 
 <hr>
1636 1628 
 <div class="license">
 
@@ -1645,7 +1637,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1645 1637 
 </div>
1646 1638 
 <div id="postamble" class="status">
1647 1639 
 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1640 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1649 1641 
 <p class="validation"></p>
1650 1642 
 </div>
1651 1643

+ 6
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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 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1628,13 +1620,13 @@ We now consider EM waves inside a bulk conductor and will consider nonvanishing
1628 1620 
 \]
1629 1621 
 which means that the Maxwell equations reduce to
1630 1622 
 </p>
1631 
-\begin{align}
1623 
+\begin{align*}
1632 1624 
   {\boldsymbol \nabla} \cdot {\boldsymbol E} &amp;= \frac{\rho_f}{\varepsilon},
1633 1625 
   \hspace{10mm} &amp;
1634 1626 
   {\boldsymbol \nabla} \cdot {\boldsymbol B} &amp;= 0, \nonumber\\
1635 1627 
   {\boldsymbol \nabla} \times {\boldsymbol E} &amp;= -\frac{\partial {\boldsymbol B}}{\partial t} &amp;
1636 1628 
   {\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}.
1637 
-\end{align}
1629 
+\end{align*}
1638 1630 
 <p>
1639 1631 
 Putting together the continuity equation for free charge
1640 1632 
 \[
 
@@ -1660,7 +1652,7 @@ After the free charge has dissipated, we have
1660 1652 
   {\boldsymbol \nabla} \times {\boldsymbol B} &amp;= \mu \sigma {\boldsymbol E} + \mu \varepsilon \frac{\partial {\boldsymbol E}}{\partial t}
1661 1653 
 \end{align}
1662 1654 
 <p>
1663 
-Applying ${\boldsymbol ∇} × $ to the curl equations gives the modified wave equations
1655 
+Applying \({\boldsymbol \nabla} \times\) to the curl equations gives the modified wave equations
1664 1656 
 \[
1665 1657 
   {\boldsymbol \nabla}^2 {\boldsymbol E} = \mu \varepsilon \frac{\partial^2 {\boldsymbol E}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol E}}{\partial t}, \hspace{10mm}
1666 1658 
   {\boldsymbol \nabla}^2 {\boldsymbol B} = \mu \varepsilon \frac{\partial^2 {\boldsymbol B}}{\partial t^2} + \mu \sigma \frac{\partial {\boldsymbol B}}{\partial t}
 
@@ -1684,7 +1676,7 @@ The wave can thus be written (letting \(\hat{\boldsymbol k}\) represent the dire
1684 1676 
   {\boldsymbol E} ({\boldsymbol r}, t) = {\boldsymbol E}_0 e^{-{\boldsymbol \kappa} \cdot {\boldsymbol r}} e^{i ({\boldsymbol k} \cdot {\boldsymbol r} - \omega t)}
1685 1677 
 \]
1686 1678 
 with a similar solution for \({\boldsymbol B}\).
1687 
-The quantity \(d = \frac{1}{\kappa}\) is known as the {\bf skin depth}.
1679 
+The quantity \(d = \frac{1}{\kappa}\) is known as the <b>skin depth</b>.
1688 1680 
 </p>
1689 1681
1690 1682 
 <p>
 
@@ -1741,7 +1733,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1741 1733 
 </div>
1742 1734 
 <div id="postamble" class="status">
1743 1735 
 <p class="author">Author: Jean-Sébastien Caux</p>
1744 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1736 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1745 1737 
 <p class="validation"></p>
1746 1738 
 </div>
1747 1739

+ 47
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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 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1624,13 +1616,13 @@ Table of contents
1624 1616 
 <p>
1625 1617 
 In matter regions without free charge and free current: Maxwell's equations are
1626 1618 
 </p>
1627 
-\begin{align}
1619 
+
1620 
+\begin{align*}
1628 1621 
 (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf D} = 0, \nonumber \\
1629 1622 
 (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
1630 1623 
 (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
1631 1624 
 (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf H} = \frac{\partial {\bf D}}{\partial t}.
1632 
-\label{Gr(9.65)}
1633 
-\end{align}
1625 
+\end{align*}
1634 1626 
 <p>
1635 1627 
 For linear medium:
1636 1628 
 \[
 
@@ -1640,13 +1632,31 @@ For linear medium:
1640 1632 
 \]
1641 1633 
 If the medium is homogeneous (no spatial dependence of \(\varepsilon\) or \(\mu\)),
1642 1634 
 </p>
1635 
+<div class="main div" id="orga20c833">
1636 
+<div class="eqlabel" id="org28a8d44">
1637 
+<p>
1638 
+<a id="Max_lh"></a><a href="./emdm_emwm_plm.html#Max_lh"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1639 
+  <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"/>
1640 
+  <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"/>
1641 
+</svg></a>
1642 
+</p>
1643 
+<div class="alteqlabels" id="org79eef6d">
1644 
+<ul class="org-ul">
1645 
+<li>Gr (9.65)</li>
1646 
+</ul>
1647 
+
1648 
+</div>
1649 
+
1650 
+</div>
1643 1651 
 \begin{align}
1644 1652 
 (i)~~ &amp;{\boldsymbol \nabla} \cdot {\bf E} = 0, \nonumber \\
1645 1653 
 (ii)~~ &amp;{\boldsymbol \nabla} \cdot {\bf B} = 0, \nonumber \\
1646 1654 
 (iii)~~ &amp;{\boldsymbol \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}, \nonumber \\
1647 1655 
 (iv)~~ &amp;{\boldsymbol \nabla} \times {\bf B} = \mu \varepsilon \frac{\partial {\bf E}}{\partial t}.
1648 
-\label{Gr(9.65)}
1656 
+\tag{Max_lh}\label{Max_lh}
1649 1657 
 \end{align}
1658 
+
1659 
+</div>
1650 1660 
 <p>
1651 1661 
 These are the same equations as in vacuum, except for the substitution of \(\mu_0 \varepsilon_0\) by \(\mu \varepsilon\).
1652 1662 
 </p>
 
@@ -1659,13 +1669,30 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1659 1669 
 \]
1660 1670 
 where the index of refraction of the material is defined as
1661 1671 
 </p>
1662 
-<div class="main div" id="orgd0bce92">
1672 
+<div class="main div" id="orga25a79e">
1673 
+<p>
1674 
+<b>Index of refraction</b>
1675 
+</p>
1676 
+<div class="eqlabel" id="orgc9df98d">
1677 
+<p>
1678 
+<a id="n"></a><a href="./emdm_emwm_plm.html#n"><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="org574afc6">
1684 
+<ul class="org-ul">
1685 
+<li>Gr (9.69)</li>
1686 
+</ul>
1687 
+
1688 
+</div>
1689 
+
1690 
+</div>
1663 1691 
 <p>
1664 
-{\bf Index of refraction}
1665 1692 
 \[
1666 
-    n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
1667 
-    \label{Gr(9.69)}
1668 
-  \]
1693 
+n \equiv \sqrt{\frac{\mu \varepsilon}{\mu_0 \varepsilon_0}}
1694 
+\tag{n}\label{n}
1695 
+\]
1669 1696 
 </p>
1670 1697
1671 1698 
 </div>
 
@@ -1675,7 +1702,7 @@ Fact: for most materials, \(\mu\) is very close to \(\mu_0\), so
1675 1702 
 n \simeq \sqrt{\varepsilon_r}
1676 1703 
 \label{Gr(9.70)}
1677 1704 
 \]
1678 
-with \(\varepsilon_r\) being the dielectric constant \ref{Gr(4.34)}.
1705 
+with \(\varepsilon_r\) being the dielectric constant <a href="./emsm_esm_ld_sp.html#epsr">epsr</a>.
1679 1706 
 </p>
1680 1707
1681 1708 
 <p>
 
@@ -1716,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1716 1743 
 </div>
1717 1744 
 <div id="postamble" class="status">
1718 1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
1719 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1746 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1720 1747 
 <p class="validation"></p>
1721 1748 
 </div>
1722 1749

+ 5
- 16
build/emdm_emwm_refl.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1625,15 +1617,14 @@ Table of contents
1625 1617 
 <p>
1626 1618 
 Interesting question: what happens to an EM wave as it passes from one medium to another?
1627 1619 
 Incident wave: produces reflected and transmitted waves.
1628 
-Detailed study:  starts from boundary conditions \ref{Gr(7.64)},
1620 
+Detailed study:  starts from boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a>,
1629 1621 
 </p>
1630 
-\begin{align}
1622 
+\begin{align*}
1631 1623 
 (i)~~ &amp;\varepsilon_1 E_1^{\perp} - \varepsilon_2 E_2^{\perp} = 0, \nonumber \\
1632 1624 
 (ii)~~ &amp;B_1^{\perp} - B_2^{\perp} = 0, \nonumber \\
1633 1625 
 (iii)~~ &amp;{\bf E}_1^{\parallel} - {\bf E}_2^{\parallel} = 0, \nonumber \\
1634 1626 
 (iv)~~ &amp;\frac{1}{\mu_1} {\bf B}_1^{\parallel} - \frac{1}{\mu_2} {\bf B}_2^{\parallel} = 0.
1635 
-\label{eq:EMBdryCondAtMediumInterface}
1636 
-\end{align}
1627 
+\end{align*}
1637 1628 
 </div>
1638 1629
1639 1630 
 
@@ -1641,8 +1632,6 @@ Detailed study:  starts from boundary conditions \ref{Gr(7.64)},
1641 1632 
 <ul class="child-links-list">
1642 1633 
 <li><a href="emdm_emwm_refl_ni.html">Normal Incidence</a><span class="headline-id">emdm.emwm.refl.ni</span></li>
1643 1634 
 <li><a href="emdm_emwm_refl_oi.html">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span></li>
1644 
-<li><a href="emdm_emwm_refl_Fe.html">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span></li>
1645 
-<li><a href="emdm_emwm_refl_Ba.html">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span></li>
1646 1635 
 </ul>
1647 1636
1648 1637 
 <br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refr.html">Refraction&emsp;<small>[emdm.emwm.refr]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
 
@@ -1660,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1660 1649 
 </div>
1661 1650 
 <div id="postamble" class="status">
1662 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1663 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1652 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1664 1653 
 <p class="validation"></p>
1665 1654 
 </div>
1666 1655

+ 0
- 1649
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+ 0
- 1650
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+ 34
- 23
build/emdm_emwm_refl_ni.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1629,7 +1621,7 @@ in \(x\) direction approaches surface:
1629 1621 
 {\boldsymbol B}_I (z,t) = \frac{1}{v_1} E_{0_I} e^{i (k_1 z - \omega t)} \hat{\boldsymbol y}
1630 1622 
 \label{Gr(9.75)}
1631 1623 
 \]
1632 
-(in which we have used \(B_0 = \frac{1}{v} E_0\), \ref{Gr(9.47)}).
1624 
+(in which we have used \(B_0 = \frac{1}{v} E_0\), <a href="./emd_emw_mpw.html#EBmpw">EBmpw</a>.
1633 1625 
 </p>
1634 1626
1635 1627 
 <p>
 
@@ -1644,19 +1636,19 @@ Reflected wave:
1644 1636
1645 1637 
 <p>
1646 1638 
 For the transmitted wave, we put
1647 
-\[
1648 
-{\boldsymbol E}_T (z,t) = E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \hspace{1cm}
1649 
-{\boldsymbol B}_T (z,t) = \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}.
1650 
-\label{Gr(9.75)}
1651 
-\]
1652 1639 
 </p>
1653 1640
1641 
+\begin{align*}
1642 
+{\boldsymbol E}_T (z,t) &amp;= E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol x}, \\
1643 
+{\boldsymbol B}_T (z,t) &amp;= \frac{1}{v_2} E_{0_T} e^{i (k_2 z - \omega t)} \hat{\boldsymbol y}.
1644 
+\end{align*}
1645 
+
1654 1646 
 <p>
1655 
-Our boundary is by choice of coordinate system at \(z = 0\). Our setup calls for solving the boundary conditions \ref{eq:EMBdryCondAtMediumInterface} with \({\boldsymbol E}_I + {\boldsymbol E}_R\) and \({\boldsymbol B}_I + {\boldsymbol B}_R\) on one side, and \({\boldsymbol E}_T\) and \({\boldsymbol B}_T\) on the other.
1647 
+Our boundary is by choice of coordinate system at \(z = 0\). Our setup calls for solving the boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a> with \({\boldsymbol E}_I + {\boldsymbol E}_R\) and \({\boldsymbol B}_I + {\boldsymbol B}_R\) on one side, and \({\boldsymbol E}_T\) and \({\boldsymbol B}_T\) on the other.
1656 1648 
 </p>
1657 1649
1658 1650 
 <p>
1659 
-At normal incidence, there are no perpendicular components of the fields relative to the surface, so \ref{eq:EMBdryCondAtMediumInterface} (i) and (ii) are obeyed. (iii) means that
1651 
+At normal incidence, there are no perpendicular components of the fields relative to the surface, so <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a> (i) and (ii) are obeyed. (iii) means that
1660 1652 
 \[
1661 1653 
 E_{0_I} + E_{0_R} = E_{0_T}
1662 1654 
 \label{Gr(9.78)}
 
@@ -1677,10 +1669,27 @@ E_{0_I} - E_{0_R} = \beta E_{0_T}, \hspace{1cm}
1677 1669 
 <p>
1678 1670 
 Solving these coupled equations, we can write
1679 1671 
 outgoing amplitudes in terms of incident ones:
1672 
+</p>
1673 
+<div class="eqlabel" id="org653a14a">
1674 
+<p>
1675 
+<a id="ERT"></a><a href="./emdm_emwm_refl_ni.html#ERT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 
+  <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 
+</svg></a>
1679 
+</p>
1680 
+<div class="alteqlabels" id="org5690572">
1681 
+<ul class="org-ul">
1682 
+<li>Gr (9.82)</li>
1683 
+</ul>
1684 
+
1685 
+</div>
1686 
+
1687 
+</div>
1688 
+<p>
1680 1689 
 \[
1681 1690 
 E_{0_R} = \frac{1 - \beta}{1 + \beta} E_{0_I}, \hspace{10mm}
1682 1691 
 E_{0_T} = \frac{2}{1 + \beta} E_{0_I}.
1683 
-\label{Gr(9.82)}
1692 
+\tag{ERT}\label{ERT}
1684 1693 
 \]
1685 1694 
 </p>
1686 1695 
 
@@ -1701,12 +1710,14 @@ R \equiv \frac{I_R}{I_I} = \frac{E^2_{0_R}}{E^2_{0_I}} = \left( \frac{1 - \beta}
1701 1710 
 \label{Gr(9.86)}
1702 1711 
 \]
1703 1712 
 while the transmitted intensity is
1704 
-\[
1713 
+</p>
1714 
+
1715 
+\begin{equation}
1705 1716 
 T \equiv \frac{I_T}{I_I} = \frac{v_2 \varepsilon_2}{v_1 \varepsilon_1} \frac{E^2_{0_T}}{E^2_{0_I}}
1706 1717 
 = \sqrt{\frac{\mu_1 \varepsilon_2}{\mu_2 \varepsilon_1}} \frac{E^2_{0_T}}{E^2_{0_I}} = \frac{4 \beta}{(1 + \beta)^2}.
1707 
-\label{Gr(9.86)}
1708 
-\]
1709 
-We thus have that the {\bf reflection} and {\bf transmission coefficients} satisfy
1718 
+\end{equation}
1719 
+<p>
1720 
+We thus have that the <b>reflection</b> and <b>transmission coefficients</b> satisfy
1710 1721 
 \[
1711 1722 
 R + T = 1.
1712 1723 
 \label{Gr(9.88)}
 
@@ -1732,7 +1743,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1732 1743 
 </div>
1733 1744 
 <div id="postamble" class="status">
1734 1745 
 <p class="author">Author: Jean-Sébastien Caux</p>
1735 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1746 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1736 1747 
 <p class="validation"></p>
1737 1748 
 </div>
1738 1749

+ 80
- 36
build/emdm_emwm_refl_oi.html View File 

@@ -1,7 +1,7 @@
1 1 
 <!DOCTYPE html>
2 2 
 <html lang="en">
3 3 
 <head>
4 
-<!-- 2022-03-02 Wed 15:45 -->
4 
+<!-- 2022-03-07 Mon 20:38 -->
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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1615,7 +1607,7 @@ Table of contents
1615 1607 
 </ul>
1616 1608 
 </details>
1617 1609 
 </nav>
1618 
-<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
1610 
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="emdm.html">Electromagnetodynamics in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm.html">Electromagnetic Waves in Matter</a></li><li><a class="breadcrumb-link"href="emdm_emwm_refl.html">Reflection and Transmission</a></li><li>Oblique Incidence</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul><div id="outline-container-emdm_emwm_refl_oi" class="outline-5">
1619 1611 
 <h5 id="emdm_emwm_refl_oi">Oblique Incidence<a class="headline-permalink" href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1620 1612 
   <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 1613 
   <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"/>
 
@@ -1642,14 +1634,28 @@ Transmitted wave:
1642 1634 
 {\boldsymbol B}_T ({\boldsymbol r},t) = \frac{1}{v_2} \hat{\boldsymbol k}_T \times {\boldsymbol E}_{T} ({\boldsymbol r}, t).
1643 1635 
 \]
1644 1636 
 All waves have the same frequency \(\omega\). Since \(\omega = k v\), the three wavevectors are related by
1637 
+</p>
1638 
+<div class="eqlabel" id="org862531f">
1639 
+<p>
1640 
+<a id="RTobliquek"></a><a href="./emdm_emwm_refl_oi.html#RTobliquek"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1641 
+  <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"/>
1642 
+  <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"/>
1643 
+</svg></a>
1644 
+</p>
1645 
+<div class="alteqlabels" id="org4435847">
1646 
+
1647 
+</div>
1648 
+
1649 
+</div>
1650 
+<p>
1645 1651 
 \[
1646 1652 
   k_I v_1 = k_R v_1 = k_T v_2 ~~\longrightarrow~~ k_I = k_R = \frac{v_2}{v_1} k_T = \frac{n_1}{n_2} k_T
1647 
-  \label{eq:RTObliquek}
1653 
+  \tag{RTobliquek}\label{RTobliquek}
1648 1654 
 \]
1649 1655 
 </p>
1650 1656
1651 1657 
 <p>
1652 
-These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions (\ref{eq:EMBdryCondAtMediumInterface}). Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves:
1658 
+These forms for incident, reflected and transmitted wave can be substituted in the boundary conditions <a href="./emdm_Me_bc.html#disc_nfc">disc_nfc</a>. Since these must be valid for any \(x\) and \(y\) (on the interface at \(z=0\)), we must have that the \(x\) and \(y\) components of the wavevectors coincide for all the waves:
1653 1659 
 \[
1654 1660 
   k_{I_x} = k_{R_x} = k_{T_x}, \hspace{10mm}
1655 1661 
   k_{I_y} = k_{R_y} = k_{T_y}
 
@@ -1659,29 +1665,29 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1659 1665 
 <p>
1660 1666 
 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 1667 
 </p>
1662 
-<div class="core div" id="orge681e56">
1668 
+<div class="core div" id="org6f173e3">
1663 1669 
 <p>
1664 
-{\bf First law of reflection:}
1665 
-the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
1670 
+<b>First law of reflection:</b>
1671 
+  the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
1666 1672 
 </p>
1667 1673
1668 1674 
 </div>
1669 1675 
 <p>
1670 
-Specializing (\ref{eq:RTObliquek}) to our notations, we have
1676 
+Specializing <a href="./emdm_emwm_refl_oi.html#RTobliquek">RTobliquek</a> to our notations, we have
1671 1677 
 \[
1672 1678 
   k_I \sin \theta_I = k_R \sin \theta_R = k_T \sin \theta_T
1673 1679 
 \]
1674 1680 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1675 1681 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1676 1682 
 </p>
1677 
-<div class="core div" id="orgfbf5032">
1683 
+<div class="core div" id="orgbf9459c">
1678 1684 
 <p>
1679 
-{\bf Law of reflection}
1680 
-\[
1685 
+<b>Law of reflection</b>
1686 
+  \[
1681 1687 
     \theta_I = \theta_R
1682 1688 
   \]
1683 
-{\bf Law of refraction (Snell's law)}
1684 
-\[
1689 
+<b>Law of refraction (Snell's law)</b>
1690 
+  \[
1685 1691 
     n_1 \sin \theta_I = n_2 \sin \theta_T
1686 1692 
   \]
1687 1693 
 </p>
 
@@ -1691,13 +1697,25 @@ and the angle of refraction (\(\theta_T\)) obey the following laws:
1691 1697 
 <p>
1692 1698 
 This takes care of the spatially-dependent exponential factors in the boundary conditions. The coefficients must further obey
1693 1699 
 </p>
1700 
+<div class="eqlabel" id="org8d76e61">
1701 
+<p>
1702 
+<a id="EBRT"></a><a href="./emdm_emwm_refl_oi.html#EBRT"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1703 
+  <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"/>
1704 
+  <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"/>
1705 
+</svg></a>
1706 
+</p>
1707 
+<div class="alteqlabels" id="org2c4d03c">
1708 
+
1709 
+</div>
1710 
+
1711 
+</div>
1694 1712 
 \begin{align}
1695 1713 
   \varepsilon_1 \left({\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_z &amp;= \varepsilon_2 \left({\boldsymbol E}_{0_T} \right)_z,
1696 1714 
   &amp; \hspace{10mm}
1697 1715 
   \left({\boldsymbol B}_{0_I}  +{\boldsymbol B}_{0_R} \right)_z &amp;= \left({\boldsymbol B}_{0_T}\right)_z, \nonumber \\
1698 1716 
   \left( {\boldsymbol E}_{0_I} + {\boldsymbol E}_{0_R} \right)_{x,y} &amp;= \left({\boldsymbol E}_{0_T}\right)_{x,y},
1699 1717 
   &amp; \frac{1}{\mu_1} \left({\boldsymbol B}_{0_I} + {\boldsymbol B}_{0_R} \right)_{x,y} &amp;= \frac{1}{\mu_2} \left({\boldsymbol B}_{0_T}\right)_{x,y}.
1700 
-  \label{eq:EMBdryCondAtMediumInterface:amp}
1718 
+  \tag{EBRT}\label{EBRT}
1701 1719 
 \end{align}
1702 1720
1703 1721 
 <p>
 
@@ -1709,8 +1727,8 @@ The two cases of polarization parallel and perpendicular to the plane of inciden
1709 1727 
 </p>
1710 1728
1711 1729 
 <p>
1712 
-\paragraph{Polarization in plane of incidence:}
1713 
-in this case the first equation of (\ref{eq:EMBdryCondAtMediumInterface:amp}) gives
1730 
+<b>Polarization in plane of incidence</b>:
1731 
+in this case the first equation of <a href="./emdm_emwm_refl_oi.html#EBRT">EBRT</a> gives
1714 1732 
 \[
1715 1733 
   \varepsilon_1 \left(-E_{0_I} \sin \theta_I + E_{0_R} \sin \theta_R \right) = -\varepsilon_2 E_{0_T} \sin \theta_T.
1716 1734 
 \]
 
@@ -1732,13 +1750,28 @@ while the third equation becomes
1732 1750 
 \]
1733 1751 
 Writing everything in terms of the incident amplitude, we get
1734 1752 
 </p>
1735 
-<div class="main div" id="orgcf2803c">
1753 
+<div class="main div" id="org3866053">
1754 
+<p>
1755 
+<b>Fresnel's equations for reflection and transmission amplitudes (parallel case)</b>
1756 
+</p>
1757 
+<div class="eqlabel" id="org31e8ee3">
1758 
+<p>
1759 
+<a id="Fresnel"></a><a href="./emdm_emwm_refl_oi.html#Fresnel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1760 
+  <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"/>
1761 
+  <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"/>
1762 
+</svg></a>
1763 
+</p>
1764 
+<div class="alteqlabels" id="org1f91c78">
1765 
+
1766 
+</div>
1767 
+
1768 
+</div>
1736 1769 
 <p>
1737 
-{\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
1738 1770 
 \[
1739 1771 
     E_{0_R} = \frac{\alpha - \beta}{\alpha + \beta} E_{0_I},
1740 1772 
     \hspace{10mm}
1741 1773 
     E_{0_T} = \frac{2}{\alpha + \beta} E_{0_I}
1774 
+    \tag{Fresnel}\label{Fresnel}
1742 1775 
   \]
1743 1776 
 </p>
1744 1777 
 
@@ -1749,15 +1782,28 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1749 1782 
 \[
1750 1783 
   \alpha = \frac{\sqrt{1 - \sin^2 \theta_T}}{\cos \theta_I} = \frac{\left[1 - \left(\frac{n_1}{n_2}\right)^2 \sin^2 \theta_I\right]^{1/2}}{\cos \theta_I}
1751 1784 
 \]
1752 
-Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
1785 
+Behaviour: for \(\theta_I = 0\) we recover <a href="./emdm_emwm_refl_ni.html#ERT">ERT</a>.
1753 1786 
 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 1787 
 </p>
1755 
-<div class="main div" id="org7503b43">
1788 
+<div class="main div" id="org328f2a5">
1789 
+<div class="eqlabel" id="org3b9cec2">
1756 1790 
 <p>
1757 
-{\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
1758 
-  \[
1759 
-      \theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
1760 
-    \]
1791 
+<a id="Brewster"></a><a href="./emdm_emwm_refl_oi.html#Brewster"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
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 
+  <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 
+</svg></a>
1795 
+</p>
1796 
+<div class="alteqlabels" id="orgf448df4">
1797 
+
1798 
+</div>
1799 
+
1800 
+</div>
1801 
+<p>
1802 
+<b>Brewster's angle</b> <i>(at which the reflected wave amplitude vanishes)</i>
1803 
+\[
1804 
+\theta_B = \arcsin \left[ \frac{1 - \beta^2}{(n_1/n_2)^2 - \beta^2} \right]^{1/2}
1805 
+\tag{Brewster}\label{Brewster}
1806 
+\]
1761 1807 
 </p>
1762 1808
1763 1809 
 </div>
 
@@ -1785,9 +1831,7 @@ Of course, we get \(R + T = 1\) as expected.
1785 1831 
 </div>
1786 1832
1787 1833
1788 
-
1789 
-
1790 
-<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl_Fe.html">Fresnel's Equations&emsp;<small>[emdm.emwm.refl.Fe]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
1834 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_refl_ni.html">Normal Incidence&emsp;<small>[emdm.emwm.refl.ni]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_ad.html">Absorption and Dispersion&emsp;<small>[emdm.emwm.ad]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li></ul>
1791 1835 
 <br>
1792 1836 
 <hr>
1793 1837 
 <div class="license">
 
@@ -1802,7 +1846,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1802 1846 
 </div>
1803 1847 
 <div id="postamble" class="status">
1804 1848 
 <p class="author">Author: Jean-Sébastien Caux</p>
1805 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1849 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1806 1850 
 <p class="validation"></p>
1807 1851 
 </div>
1808 1852

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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1641,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1641 1633 
 </div>
1642 1634 
 <div id="postamble" class="status">
1643 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1636 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1645 1637 
 <p class="validation"></p>
1646 1638 
 </div>
1647 1639

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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>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1647,7 +1639,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 1639 
 </div>
1648 1640 
 <div id="postamble" class="status">
1649 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1642 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1651 1643 
 <p class="validation"></p>
1652 1644 
 </div>
1653 1645

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1667,7 +1659,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1667 1659 
 </div>
1668 1660 
 <div id="postamble" class="status">
1669 1661 
 <p class="author">Author: Jean-Sébastien Caux</p>
1670 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1662 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1671 1663 
 <p class="validation"></p>
1672 1664 
 </div>
1673 1665

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6 6 
 <meta name="viewport" content="width=device-width, initial-scale=1">
7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1668,7 +1660,7 @@ Putting these back into Maxwell (i) and (ii) gives the decoupled equations
1668 1660 
 \[
1669 1661 
   \left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \left(\frac{\omega}{c}\right)^2 - k^2 \right] E_z = 0,
1670 1662 
 \]
1671 
-with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called {\bf TE} waves (for {\it transverse electric}), and if \(B_z = 0\) they are called {\bf TM} (for {\it transverse magnetic}) waves. If \(E_z = 0 = B_z\) they are called {\bf TEM waves}. The latter cannot occur in a hollow waveguide (simple proof: Gauss + Faraday).
1663 
+with an identical equation for \(B_z\). If \(E_z = 0\) the waves are called <b>TE</b> waves (for <i>transverse electric</i>), and if \(B_z = 0\) they are called <b>TM</b> (for <i>transverse magnetic</i>}) waves. If \(E_z = 0 = B_z\) they are called <b>TEM</b> waves. The latter cannot occur in a hollow waveguide (simple proof: Gauss + Faraday).
1672 1664 
 </p>
1673 1665 
 </div>
1674 1666 
 </div>
 
@@ -1691,7 +1683,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1691 1683 
 </div>
1692 1684 
 <div id="postamble" class="status">
1693 1685 
 <p class="author">Author: Jean-Sébastien Caux</p>
1694 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1686 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1695 1687 
 <p class="validation"></p>
1696 1688 
 </div>
1697 1689

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6 6 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
-
1105 
-</li>
1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
1108 
-
1109 
-</li>
1110 1102
1111 1103 
 </ul>
1112 1104 
 </details>
 
@@ -1662,7 +1654,7 @@ In view of this, if the frequency is low enough, namely
1662 1654 
 \[
1663 1655 
   \omega &lt; c\pi \sqrt{\left(\frac{m}{a}\right)^2 + \left(\frac{n}{b}\right)^2} \equiv \omega_{mn}
1664 1656 
 \]
1665 
-then the wavenumber is imaginary and the travelling wave is exponentially attenuated. This frequency is called the {\bf cutoff frequency} for this mode.
1657 
+then the wavenumber is imaginary and the travelling wave is exponentially attenuated. This frequency is called the <b>cutoff frequency</b> for this mode.
1666 1658 
 </p>
1667 1659
1668 1660 
 <p>
 
@@ -1674,7 +1666,8 @@ and wave velocity
1674 1666 
 \[
1675 1667 
   v = \frac{\omega}{k} = \frac{c}{\sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}}
1676 1668 
 \]
1677 
-which is {\it greater} than \(c\). The energy of the wave however propagates at the {\bf group velocity}
1669 
+which is <i>greater</i> than \(c\). The energy of the wave however propagates at the
1670 
+<b>group velocity</b>
1678 1671 
 \[
1679 1672 
   v_g = \frac{d\omega}{dk} = c \sqrt{1 - \left(\frac{\omega_{mn}}{\omega}\right)^2}.
1680 1673 
 \]
 
@@ -1700,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1700 1693 
 </div>
1701 1694 
 <div id="postamble" class="status">
1702 1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
1703 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1696 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
1704 1697 
 <p class="validation"></p>
1705 1698 
 </div>
1706 1699

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 <title>Pre-Quantum Electrodynamics</title>
 
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1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
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1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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1110 1102
1111 1103 
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1112 1104 
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1622 1614 
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1623 1615
1624 1616 
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1625 
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1626 
-<summary id="org621593e">
1617 
+<details class="prereq" id="org8e30f8b">
1618 
+<summary id="org2c7b815">
1627 1619 
 Prerequisites
1628 1620 
 </summary>
1629 1621 
 <ul class="org-ul">
 
@@ -1631,8 +1623,8 @@ Prerequisites
1631 1623 
 </ul>
1632 1624 
 </details>
1633 1625
1634 
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1635 
-<summary id="orgb47e5d6">
1626 
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1627 
+<summary id="orgda754dd">
1636 1628 
 Objectives
1637 1629 
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1670 1662 
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1671 
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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1665 1657 
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1666 1658 
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1667 1659 
 <p class="author">Author: Jean-Sébastien Caux</p>
1668 
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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1101 1101 
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-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1665 1657 
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1666 1658 
 <p class="author">Author: Jean-Sébastien Caux</p>
1667 
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
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1101 1101 
 </li>
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1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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@@ -1637,7 +1629,7 @@ while the equation for \(V\) becomes
1637 1629 
 \]
1638 1630 
 These can be written compactly upon introducing a new operator: the
1639 1631 
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1640 
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 <p>
1642 1634 
 {\bf d'Alembertian operator}
1643 1635 
 \[
 
@@ -1650,7 +1642,7 @@ These can be written compactly upon introducing a new operator: the
1650 1642 
 <p>
1651 1643 
 so we get the
1652 1644 
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 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1656 1648 
 \[
 
@@ -1700,7 +1692,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1700 1692 
 </div>
1701 1693 
 <div id="postamble" class="status">
1702 1694 
 <p class="author">Author: Jean-Sébastien Caux</p>
1703 
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1695 
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 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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@@ -1638,7 +1630,7 @@ Useful strategy: represent fields in terms of potentials.
1638 1630 
 <p>
1639 1631 
 Easiest:
1640 1632 
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1641 
-<div class="core div" id="orgf88fb96">
1633 
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1642 1634 
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1643 1635 
 \[
1644 1636 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1654,7 +1646,7 @@ Putting this into Faraday's law gives
1654 1646 
 \]
1655 1647 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1656 1648 
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1657 
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1649 
+<div class="core div" id="orgffce8dc">
1658 1650 
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1659 1651 
 \[
1660 1652 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1667,7 +1659,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1667 1659 
 <p>
1668 1660 
 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 1661 
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1670 
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1672 1664 
 \[
1673 1665 
     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1683,7 +1675,7 @@ whereas Amp{\`ere}-Maxwell becomes
1683 1675 
 \]
1684 1676 
 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 1677 
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1686 
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1678 
+<div class="main div" id="orga86c8c2">
1687 1679 
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1688 1680 
 \[
1689 1681 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1719,7 +1711,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1719 1711 
 </div>
1720 1712 
 <div id="postamble" class="status">
1721 1713 
 <p class="author">Author: Jean-Sébastien Caux</p>
1722 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1714 
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1100 1100
1101 1101 
 </li>
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-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1110 1102
1111 1103 
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1112 1104 
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1648 1640 
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1649 1641 
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1650 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1643 
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1652 1644 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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1111 1103 
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1112 1104 
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@@ -1626,8 +1618,8 @@ Table of contents
1626 1618 
 <li>Gr 3</li>
1627 1619 
 </ul>
1628 1620
1629 
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1630 
-<summary id="org2707d3f">
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1631 1623 
 Prerequisites
1632 1624 
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1633 1625 
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@@ -1635,8 +1627,8 @@ Prerequisites
1635 1627 
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1636 1628 
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1637 1629
1638 
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 Objectives
1641 1633 
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1642 1634 
 <ul class="org-ul">
 
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1674 1666 
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1675 1667 
 <div id="postamble" class="status">
1676 1668 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1669 
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1678 1670 
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1679 1671 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
1104 
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1106 
-<li>
1107 
-<a href="./emdm_emwm_refl_Ba.html#emdm_emwm_refl_Ba">Brewster's Angle</a><span class="headline-id">emdm.emwm.refl.Ba</span>
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1110 1102
1111 1103 
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1112 1104 
 </details>
 
@@ -1632,7 +1624,7 @@ A generic configuration of static charges coupled via the Coulomb interaction
1632 1624 
 defines an electrostatic problem, whose solution is in principle obtained
1633 1625 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1634 1626 
 </p>
1635 
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1636 1628 
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1637 1629
1638 1630 
 </p>
 
@@ -1646,7 +1638,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1646 1638 
 or (often simpler) by calculating the electrostatic potential, using either the
1647 1639 
 explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1648 1640 
 </p>
1649 
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1641 
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1650 1642 
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1651 1643
1652 1644 
 </p>
 
@@ -1666,7 +1658,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1666 1658 
 <a href="./ems_es_ep_PL.html#Poi">🐟</a>
1667 1659 
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1668 1660
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1671 1663
1672 1664 
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@@ -1682,7 +1674,7 @@ condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as th
1682 1674 
 <p>
1683 1675 
 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 1676 
 </p>
1685 
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1687 1679
1688 1680 
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1720 1712 
 </div>
1721 1713 
 <div id="postamble" class="status">
1722 1714 
 <p class="author">Author: Jean-Sébastien Caux</p>
1723 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
1715 
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@@ -1099,14 +1099,6 @@ Table of contents
1099 1099 
 <a href="./emdm_emwm_refl_oi.html#emdm_emwm_refl_oi">Oblique Incidence</a><span class="headline-id">emdm.emwm.refl.oi</span>
1100 1100
1101 1101 
 </li>
1102 
-<li>
1103 
-<a href="./emdm_emwm_refl_Fe.html#emdm_emwm_refl_Fe">Fresnel's Equations</a><span class="headline-id">emdm.emwm.refl.Fe</span>
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1111 1103 
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1112 1104 
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@@ -1638,14 +1630,14 @@ In one dimension, the potential is a single-variable
1638 1630 
 function \(\phi (x)\) and the Laplace equation reads
1639 1631 
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1640 1632
1641 
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1643 1635 
 <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 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"/>
1645 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"/>
1646 1638 
 </svg></a>
1647 1639 
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1649 1641
1650 1642 
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1651 1643 
 
@@ -1660,14 +1652,14 @@ function \(\phi (x)\) and the Laplace equation reads
1660 1652 
 <p>
1661 1653 
 The solution to this is
1662 1654 
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1663 
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1655 
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1664 1656 
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1665 1657 
 <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 1658 
   <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 1659 
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1668 1660 
 </svg></a>
1669 1661 
 </p>
1670 
-<div class="alteqlabels" id="org6b2acba">
1662 
+<div class="alteqlabels" id="org1f2c6cb">
1671 1663 
 <ul class="org-ul">
1672 1664 
 <li>Gr (3.6)</li>
1673 1665 
 </ul>
 
@@ -1726,14 +1718,14 @@ In two dimensions, the potential becomes a function
1726 1718 
 of two variables (here: \(x\) and \(y\)), so Laplace's
1727 1719 
 equation now reads
1728 1720 
 </p>
1729 
-<div class="eqlabel" id="orgf3e1009">
1721 
+<div class="eqlabel" id="orge167f78">
1730 1722 
 <p>
1731 1723 
 <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 1724 
   <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 1725 
   <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 1726 
 </svg></a>
1735 1727 
 </p>
1736 
-<div class="alteqlabels" id="org696fc74">
1728 
+<div class="alteqlabels" id="org9aae751">
1737 1729
1738 1730 
 </div>
1739 1731 
 
@@ -1786,14 +1778,14 @@ a point equals its value averaged over a sphere
1786 1778 
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
1787 1779 
 (and of course not containing any charges),
1788 1780 
 </p>
1789 
-<div class="eqlabel" id="org9f81e63">
1781 
+<div class="eqlabel" id="orge400754">
1790 1782 
 <p>
1791 1783 
 <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">
1792 1784 
   <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 1785 
   <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 1786 
 </svg></a>
1795 1787 
 </p>
1796 
-<div class="alteqlabels" id="org66bdc2f">
1788 
+<div class="alteqlabels" id="orgf37e10c">
1797 1789
1798 1790 
 </div>
1799 1791 
 
@@ -1805,8 +1797,8 @@ a point equals its value averaged over a sphere
1805 1797 
 \]
1806 1798 
 </p>
1807 1799
1808 
-<details id="org0f27c64">
1809 
-<summary id="org881e712">
1800 
+<details id="org4bb0c5f">
1801 
+<summary id="orgaf332f4">
1810 1802 
 <strong>Physicist's proof</strong>
1811 1803 
 </summary>
1812 1804 
 <p>
 
@@ -1868,8 +1860,8 @@ proving the theorem.
1868 1860 
 </p>
1869 1861 
 </details>
1870 1862
1871 
-<details id="org0dbbb13">
1872 
-<summary id="orgf2b7c76">
1863 
+<details id="orgfce7dc3">
1864 
+<summary id="org452b0d4">
1873 1865 
 <strong>Formal proof</strong>
1874 1866 
 </summary>
1875 1867 
 
@@ -1919,14 +1911,14 @@ we get the following general
1919 1911 
 <p>
1920 1912 
 <b>Theorem</b>:
1921 1913 
 </p>
1922 
-<div class="eqlabel" id="org230e535">
1914 
+<div class="eqlabel" id="orgfba2553">
1923 1915 
 <p>
1924 1916 
 <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 1917 
   <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 1918 
   <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 1919 
 </svg></a>
1928 1920 
 </p>
1929 
-<div class="alteqlabels" id="org37e5ede">
1921 
+<div class="alteqlabels" id="orgc5c6de6">
1930 1922
1931 1923 
 </div>
1932 1924 
 
@@ -1979,19 +1971,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1979 1971 
 of the \(f_x + f_y + f_z = 0\) condition above.
1980 1972 
 </p>
1981 1973
1982 
-<div class="eqlabel" id="org8692599">
1974 
+<div class="eqlabel" id="org8465450">
1983 1975 
 <p>
1984 1976 
 <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 1977 
   <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 1978 
   <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 1979 
 </svg></a>
1988 1980 
 </p>
1989 
-<div class="alteqlabels" id="orgcae1a27">
1981 
+<div class="alteqlabels" id="org0fbdf78">
1990 1982
1991 1983 
 </div>
1992 1984
1993 1985 
 </div>
1994 
-<div class="info div" id="orgab38e5b">
1986 
+<div class="info div" id="org9dd741d">
1995 1987 
 <p>
1996 1988 
 <b>Earnshaw's theorem (physical version)</b> <br>
1997 1989 
 </p>
 
@@ -2110,7 +2102,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
2110 2102 
 </div>
2111 2103 
 <div id="postamble" class="status">
2112 2104 
 <p class="author">Author: Jean-Sébastien Caux</p>
2113 
-<p class="date">Created: 2022-03-02 Wed 15:45</p>
2105 
+<p class="date">Created: 2022-03-07 Mon 20:38</p>
2114 2106 
 <p class="validation"></p>
2115 2107 
 </div>
2116 2108

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- 0
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