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Update 2022-02-10 08:34

master
Jean-Sébastien 2 years ago
parent
f3c2446d19
commit
f8446c1405
100 changed files with 314 additions and 314 deletions
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build/a.html View File 

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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1619,7 +1619,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1619 1619 
 </div>
1620 1620 
 <div id="postamble" class="status">
1621 1621 
 <p class="author">Author: Jean-Sébastien Caux</p>
1622 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
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+<p class="date">Created: 2022-02-10 Thu 08:32</p>
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1702 1702 
 </div>
1703 1703 
 <div id="postamble" class="status">
1704 1704 
 <p class="author">Author: Jean-Sébastien Caux</p>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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-<p class="date">Created: 2022-02-09 Wed 22:40</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1625,7 +1625,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
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 <div id="outline-container-c_m_cs_cyl_grad" class="outline-6">
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 <li>Gr4(1.79)</li>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <p>
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 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1607 1607 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1610,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
1610 1610 
 </div>
1611 1611 
 </div>
1612 1612
1613 
-<div id="outline-container-org1c6301d" class="outline-6">
1614 
-<h6 id="org1c6301d"><a href="#org1c6301d">Curl of a gradient</a></h6>
1615 
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1613 
+<div id="outline-container-org3f8f01e" class="outline-6">
1614 
+<h6 id="org3f8f01e"><a href="#org3f8f01e">Curl of a gradient</a></h6>
1615 
+<div class="outline-text-6" id="text-org3f8f01e">
1616 1616 
 <p>
1617 1617 
 This always vanishes.
1618 1618 
 </p>
1619 1619 
 </div>
1620 1620 
 </div>
1621 1621
1622 
-<div id="outline-container-orgce82ec3" class="outline-6">
1623 
-<h6 id="orgce82ec3"><a href="#orgce82ec3">Gradient of the divergence</a></h6>
1624 
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1622 
+<div id="outline-container-org86a1e1f" class="outline-6">
1623 
+<h6 id="org86a1e1f"><a href="#org86a1e1f">Gradient of the divergence</a></h6>
1624 
+<div class="outline-text-6" id="text-org86a1e1f">
1625 1625 
 <p>
1626 1626 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1627 1627 
 </p>
1628 1628 
 </div>
1629 1629 
 </div>
1630 1630
1631 
-<div id="outline-container-org820bb5e" class="outline-6">
1632 
-<h6 id="org820bb5e"><a href="#org820bb5e">Divergence of a curl</a></h6>
1633 
-<div class="outline-text-6" id="text-org820bb5e">
1631 
+<div id="outline-container-orgdca9d9a" class="outline-6">
1632 
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1633 
+<div class="outline-text-6" id="text-orgdca9d9a">
1634 1634 
 <p>
1635 1635 
 This always vanishes.
1636 1636 
 </p>
1637 1637 
 </div>
1638 1638 
 </div>
1639 1639
1640 
-<div id="outline-container-org75d969a" class="outline-6">
1641 
-<h6 id="org75d969a"><a href="#org75d969a">Curl of curl</a></h6>
1642 
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1640 
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1641 
+<h6 id="org613cb02"><a href="#org613cb02">Curl of curl</a></h6>
1642 
+<div class="outline-text-6" id="text-org613cb02">
1643 1643 
 <p>
1644 1644 
 \[
1645 1645 
 {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
 
@@ -1664,7 +1664,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1664 1664 
 </div>
1665 1665 
 <div id="postamble" class="status">
1666 1666 
 <p class="author">Author: Jean-Sébastien Caux</p>
1667 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1667 
+<p class="date">Created: 2022-02-10 Thu 08:32</p>
1668 1668 
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1625 1625 
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1626 1626 
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1627 1627 
 <p class="author">Author: Jean-Sébastien Caux</p>
1628 
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1628 
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1622 1622 
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1624 1624 
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1625 
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1625 
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1646 1646 
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1648 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1649 
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1649 
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1650 1650 
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1619 1619 
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1620 1620 
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1621 1621 
 <p class="author">Author: Jean-Sébastien Caux</p>
1622 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1622 
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1623 1623 
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1621 1621 
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1622 1622 
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1623 1623 
 <p class="author">Author: Jean-Sébastien Caux</p>
1624 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1624 
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1625 1625 
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@@ -1633,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1633 1633 
 </div>
1634 1634 
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1635 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
1636 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1636 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1620,14 +1620,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1620 1620 
 More generally,
1621 1621 
 </p>
1622 1622
1623 
-<div class="eqlabel" id="orga2fc830">
1623 
+<div class="eqlabel" id="org16a4f22">
1624 1624 
 <p>
1625 1625 
 <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">
1626 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 1627 
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1628 1628 
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1629 1629 
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1630 
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1630 
+<div class="alteqlabels" id="orgf96ad98">
1631 1631 
 <ul class="org-ul">
1632 1632 
 <li>Gr (1.100)</li>
1633 1633 
 </ul>
 
@@ -1646,14 +1646,14 @@ More generally,
1646 1646 
 Since
1647 1647 
 </p>
1648 1648
1649 
-<div class="eqlabel" id="org321d3a4">
1649 
+<div class="eqlabel" id="org71f92c6">
1650 1650 
 <p>
1651 1651 
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1652 1652 
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1653 1653 
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1654 1654 
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1655 1655 
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1656 
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1656 
+<div class="alteqlabels" id="org39f00df">
1657 1657 
 <ul class="org-ul">
1658 1658 
 <li>Gr (1.101)</li>
1659 1659 
 </ul>
 
@@ -1691,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1691 1691 
 </div>
1692 1692 
 <div id="postamble" class="status">
1693 1693 
 <p class="author">Author: Jean-Sébastien Caux</p>
1694 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1694 
+<p class="date">Created: 2022-02-10 Thu 08:32</p>
1695 1695 
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1696 1696 
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1697 1697

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1639 1639 
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1640 1640 
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1641 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1642 
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1624 1624 
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1625 1625 
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1626 1626 
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1627 
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1627 
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1622 1622 
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1624 1624 
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1625 
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1631 1631 
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1632 1632 
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1633 1633 
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1634 
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1625 1625 
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1627 1627 
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1628 
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1628 
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@@ -1632,7 +1632,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1632 1632 
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1633 1633 
 <div id="postamble" class="status">
1634 1634 
 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
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1635 
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1636 1636 
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@@ -1599,9 +1599,9 @@ Table of contents
1599 1599 
 <div class="outline-text-5" id="text-c_m_ic_lsv">
1600 1600 
 </div>
1601 1601
1602 
-<div id="outline-container-org4fa4b6a" class="outline-6">
1603 
-<h6 id="org4fa4b6a"><a href="#org4fa4b6a">Line Integrals</a></h6>
1604 
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1602 
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1603 
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1604 
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1605 1605 
 <p>
1606 1606 
 \[
1607 1607 
 {\int_{\bf a}^{\bf b}}_{\cal P} {\bf v} \cdot d{\bf l}
 
@@ -1630,9 +1630,9 @@ Integral over a closed loop:
1630 1630 
 </div>
1631 1631 
 </div>
1632 1632
1633 
-<div id="outline-container-org3cf12c2" class="outline-6">
1634 
-<h6 id="org3cf12c2"><a href="#org3cf12c2">Surface Integrals</a></h6>
1635 
-<div class="outline-text-6" id="text-org3cf12c2">
1633 
+<div id="outline-container-orgbda984c" class="outline-6">
1634 
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1635 
+<div class="outline-text-6" id="text-orgbda984c">
1636 1636 
 <p>
1637 1637 
 \[
1638 1638 
 \int_{\cal S} {\bf v} \cdot d{\bf a}
 
@@ -1652,9 +1652,9 @@ Over a closed surface:
1652 1652 
 </div>
1653 1653 
 </div>
1654 1654
1655 
-<div id="outline-container-orgd48f6eb" class="outline-6">
1656 
-<h6 id="orgd48f6eb"><a href="#orgd48f6eb">Volume Integrals</a></h6>
1657 
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1655 
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1656 
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1657 
+<div class="outline-text-6" id="text-orgc3891db">
1658 1658 
 <p>
1659 1659 
 \[
1660 1660 
 \int_{\cal V} T d\tau
 
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1693 1693 
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1694 1694 
 <div id="postamble" class="status">
1695 1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1696 
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1697 1697 
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1698 1698 
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1642 1642 
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1643 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1644 
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1639 1639 
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1641 1641 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
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1649 1649 
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1650 1650 
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1651 1651 
 <p class="author">Author: Jean-Sébastien Caux</p>
1652 
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1645 1645 
 <p class="author">Author: Jean-Sébastien Caux</p>
1646 
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1598 1598 
 </svg></a><span class="headline-id">emd</span></h2>
1599 1599
1600 1600 
 <div class="outline-text-2" id="text-emd">
1601 
-<details class="prereq" id="orgc5008fa">
1602 
-<summary id="org88d0f1c">
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1603 1603 
 Prerequisites
1604 1604 
 </summary>
1605 1605 
 <ul class="org-ul">
 
@@ -1608,8 +1608,8 @@ Prerequisites
1608 1608 
 </ul>
1609 1609 
 </details>
1610 1610
1611 
-<details class="objectives" id="org036b173">
1612 
-<summary id="org9627cee">
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1613 1613 
 Objectives
1614 1614 
 </summary>
1615 1615 
 <ul class="org-ul">
 
@@ -1648,7 +1648,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1648 
 </div>
1649 1649 
 <div id="postamble" class="status">
1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1651 
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 <p class="validation"></p>
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 <title>Pre-Quantum Electrodynamics</title>
 
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1622 1622 
 </div>
1623 1623 
 <div id="postamble" class="status">
1624 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1625 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1638,7 +1638,7 @@ Empirically:  the changing magnetic field induces an electric current around
1638 1638 
 the circuit. This current is really driven by an electric field having a component
1639 1639 
 along the wire.  The line integral of this field is called the
1640 1640 
 </p>
1641 
-<div class="core div" id="orge05960f">
1641 
+<div class="core div" id="orgee21897">
1642 1642 
 <p>
1643 1643 
 <b>Electromotive force (or electromotance)</b>,
1644 1644 
   \[
 
@@ -1660,7 +1660,7 @@ to the rate of change of the magnetic flux,
1660 1660 
 \]
1661 1661 
 so we obtain
1662 1662 
 </p>
1663 
-<div class="core div" id="org726542f">
1663 
+<div class="core div" id="orgd1f18df">
1664 1664 
 <p>
1665 1665 
 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1666 1666 
   \[
 
@@ -1678,7 +1678,7 @@ for any loop (on a wire or not). Using Stokes' theorem,
1678 1678 
 \]
1679 1679 
 we obtain
1680 1680 
 </p>
1681 
-<div class="core div" id="orge08eeaf">
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 <p>
1683 1683 
 <b>Faraday's law</b> (differential form)
1684 1684 
   \[
 
@@ -1713,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1713 1713 
 </div>
1714 1714 
 <div id="postamble" class="status">
1715 1715 
 <p class="author">Author: Jean-Sébastien Caux</p>
1716 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1657,7 +1657,7 @@ W = \frac{1}{2\mu_0} \left[ \int_{\cal V} d\tau B^2 - \int_{\cal V} d\tau {\bold
1657 1657 
 \]
1658 1658 
 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
1659 1659 
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1660 
-<div class="core div" id="org430dd21">
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 <p>
1662 1662 
 \[
1663 1663 
     W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
 
@@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
1678 1678 
 \hspace{2cm} \mbox{(7.31 and 7.34)}
1679 1679 
 \end{align}
1680 1680
1681 
-<div class="example div" id="org2441b5d">
1681 
+<div class="example div" id="org5a2dc32">
1682 1682 
 <p>
1683 1683 
 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1684 1684 
 Find energy stored in section of length \(l\).
 
@@ -1715,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1715 1715 
 </div>
1716 1716 
 <div id="postamble" class="status">
1717 1717 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
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1718 
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1637 1637 
 \]
1638 1638 
 </p>
1639 1639
1640 
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1641 1641 
 <p>
1642 1642 
 \paragraph{Example 7.10:}
1643 1643 
 short solenoid (length \(l\), radius \(a\), \(n_1\) turns per unit length) lies concentrically inside
 
@@ -1687,7 +1687,7 @@ Inductance:  measured in {\bf henries} (\(H\)).  \(H = V s/A\).
1687 1687 
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1688 1688
1689 1689
1690 
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1690 
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 <p>
1692 1692 
 \paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1693 1693 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
 
@@ -1714,7 +1714,7 @@ Total flux:  \(N\) times this, so self-inductance is
1714 1714 
 Inductance (like capacitance) is intrinsically positive.  Use Lenz law.  Think of {\bf back EMF}.
1715 1715 
 </p>
1716 1716
1717 
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1717 
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1719 1719 
 \paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1720 1720 
 What is the current ?
 
@@ -1751,7 +1751,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1751 1751 
 </div>
1752 1752 
 <div id="postamble" class="status">
1753 1753 
 <p class="author">Author: Jean-Sébastien Caux</p>
1754 
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1754 
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1621 1621
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 <p>
1626 1626 
 {\bf Example 7.7:}
1627 1627 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
@@ -1637,7 +1637,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1637 1637 
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1639 1639
1640 
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 <p>
1642 1642 
 {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
1643 1643 
 Uniform magnetic field \({\bf B}_0\) in central region up to \(a &lt; b\),
 
@@ -1671,7 +1671,7 @@ called the {\bf quasistatic} approximation, and works provided we deal with
1671 1671 
 'slow enough' phenomena.
1672 1672 
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1673 1673
1674 
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1674 
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1675 1675 
 <p>
1676 1676 
 {\bf Example 7.9:}  infinitely long straight wire carries \(I(t)\).  Find
1677 1677 
 induced \({\bf E}\) field as a function of distance \(s\) from wire.
 
@@ -1717,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1717 1717 
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1718 1718 
 <div id="postamble" class="status">
1719 1719 
 <p class="author">Author: Jean-Sébastien Caux</p>
1720 
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1720 
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1622 1622 
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1623 1623 
 <div id="postamble" class="status">
1624 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
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1625 
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1600 1600 
 <p>
1601 1601 
 Full set of equations for the electromagnetic field:
1602 1602 
 </p>
1603 
-<div class="core div" id="orga5920b3">
1603 
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1604 1604 
 <p>
1605 1605 
 {\bf Maxwell's equations} {\it (in vacuum)}
1606 1606 
 </p>
 
@@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
1616 1616 
 <p>
1617 1617 
 Complement:
1618 1618 
 </p>
1619 
-<div class="core div" id="org9ee0da3">
1619 
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1620 1620 
 <p>
1621 1621 
 {\bf Force law}
1622 1622 
 \[
 
@@ -1640,7 +1640,7 @@ take divergence of \((iv)\).
1640 1640 
 <p>
1641 1641 
 Better way of writing:  all fields on left, all sources on right,
1642 1642 
 </p>
1643 
-<div class="core div" id="orgdda2f9a">
1643 
+<div class="core div" id="org8800b57">
1644 1644 
 \begin{align}
1645 1645 
   (i) &amp;{\boldsymbol \nabla} \cdot {\bf E} = \frac{\rho}{\varepsilon_0},
1646 1646 
   &amp;(iii) {\boldsymbol \nabla} \times {\bf E} + \frac{\partial {\bf B}}{\partial t} = 0, \\
 
@@ -1668,7 +1668,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1668 1668 
 </div>
1669 1669 
 <div id="postamble" class="status">
1670 1670 
 <p class="author">Author: Jean-Sébastien Caux</p>
1671 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1671 
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 <title>Pre-Quantum Electrodynamics</title>
 
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1607 1607 
 \]
1608 1608 
 The extra term would thus be eliminated if we were to put
1609 1609 
 </p>
1610 
-<div class="core div" id="org45d3060">
1610 
+<div class="core div" id="orgba15335">
1611 1611 
 <p>
1612 1612 
 \[
1613 1613 
     {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
 
@@ -1631,7 +1631,7 @@ Real confirmation of Maxwell's theory:  1888, Hertz's experiments on propagation
1631 1631 
 <p>
1632 1632 
 Maxwell baptized this term the
1633 1633 
 </p>
1634 
-<div class="core div" id="org90773a4">
1634 
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1635 1635 
 <p>
1636 1636 
 {\bf Displacement current}
1637 1637 
 \[
 
@@ -1677,7 +1677,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1677 1677 
 </div>
1678 1678 
 <div id="postamble" class="status">
1679 1679 
 <p class="author">Author: Jean-Sébastien Caux</p>
1680 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1680 
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1648 1648 
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1649 1649 
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1650 
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1646 1646 
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1647 1647 
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1648 
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 Prerequisites
1604 1604 
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 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
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1644 1644 
 </div>
1645 1645 
 <div id="postamble" class="status">
1646 1646 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1647 
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 <p>
1601 1601 
 The angular momentum of EM fields is directly given by
1602 1602 
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1605 1605 
 {\bf Angular momentum of EM fields}
1606 1606 
 \[
 
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1628 1628 
 </div>
1629 1629 
 <div id="postamble" class="status">
1630 1630 
 <p class="author">Author: Jean-Sébastien Caux</p>
1631 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1631 
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1632 1632 
 <p class="validation"></p>
1633 1633 
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1634 1634

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1619 1619 
 \]
1620 1620 
 Since this is true for any volume, we have (re)derived the
1621 1621 
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1622 
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 <p>
1624 1624 
 {\bf Continuity equation}
1625 1625 
 \[
 
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1658 1658 
 </div>
1659 1659 
 <div id="postamble" class="status">
1660 1660 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
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1661 
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1663 1663 
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@@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
1612 1612 
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1613 1613 
 This is thus simply a conservation law for momentum, with
1614 1614 
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1615 
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1617 1617 
 {\bf Momentum density in the EM fields}
1618 1618 
 \[
 
@@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
1624 1624 
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1625 1625 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1626 1626 
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1627 
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1629 1629 
 {\bf Continuity equation for EM momentum}
1630 1630 
 \[
 
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1651 1651 
 </div>
1652 1652 
 <div id="postamble" class="status">
1653 1653 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
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@@ -1654,7 +1654,7 @@ and similarly for \({\boldsymbol B}\). We thus get
1654 1654 
 <p>
1655 1655 
 This expression can be greatly simplified by introducing the
1656 1656 
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1657 
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1659 1659 
 {\bf Maxwell stress tensor}
1660 1660 
 \[
 
@@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1677 1677 
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1678 1678 
 We then obtain
1679 1679 
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1680 
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1682 1682 
 {\bf EM force per unit volume}
1683 1683 
 \[
 
@@ -1689,7 +1689,7 @@ We then obtain
1689 1689 
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1690 1690 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1691 1691 
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1692 
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1694 1694 
 {\bf Total force on charges in volume}
1695 1695 
 \[
 
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1716 1716 
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1717 1717 
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1718 1718 
 <p class="author">Author: Jean-Sébastien Caux</p>
1719 
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1666 1666 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1667 1667 
 we obtain
1668 1668 
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1669 
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1671 1671 
 {\bf Poynting's theorem}
1672 1672 
 \[
 
@@ -1691,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1691 1691 
 <p>
1692 1692 
 Energy per unit time, per unit area carried by EM fields:
1693 1693 
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1694 
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1696 1696 
 {\bf Poynting vector}
1697 1697 
 \[
 
@@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
1704 1704 
 <p>
1705 1705 
 We can thus express Poynting's theorem more compactly:
1706 1706 
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1707 
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1709 1709 
 {\bf Poynting's theorem}
1710 1710 
 \[
 
@@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
1717 1717 
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1718 1718 
 where we have defined the total
1719 1719 
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1722 1722 
 {\bf Energy in electromagnetic fields}
1723 1723 
 \[
 
@@ -1740,7 +1740,7 @@ Then,
1740 1740 
 \]
1741 1741 
 so we get the
1742 1742 
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1743 
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 <p>
1745 1745 
 {\bf Poynting theorem (differential form)}
1746 1746 
 \[
 
@@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
1757 1757
1758 1758
1759 1759
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 <p>
1762 1762 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1763 1763 
 Assuming that the field is uniform, the electric field parallel to the wire is
 
@@ -1800,7 +1800,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1800 1800 
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1801 1801 
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1802 1802 
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1803 
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 Prerequisites
1604 1604 
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1608 1608 
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1609 1609 
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1613 1613 
 Objectives
1614 1614 
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 <ul class="org-ul">
 
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1648 1648 
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1649 1649 
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1650 1650 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
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1628 1628 
 \]
1629 1629 
 or more succinctly:
1630 1630 
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1633 1633 
 {\bf Poynting vector of a monochromatic EM wave}
1634 1634 
 \[
 
@@ -1644,7 +1644,7 @@ This has a transparent physical interpretation: the energy density \(u\) flows w
1644 1644 
 <p>
1645 1645 
 Similary, we get the
1646 1646 
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1647 
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1649 1649 
 {\bf Momentum density of a monochromatic EM wave}
1650 1650 
 \[
 
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1693 1693 
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1694 1694 
 <div id="postamble" class="status">
1695 1695 
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1696 
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1629 1629 
 Generalizing to propagation in the direction of an arbitrary wavevector
1630 1630 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1631 1631 
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1632 
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 {\bf E and B fields for a monochromatic EM plane wave}
1635 1635 
 \[
 
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1671 1671 
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1672 1672 
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1673 1673 
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1674 
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1626 1626 
 Since \({\boldsymbol \nabla} \cdot {\bf E} = 0\) and \({\boldsymbol \nabla} \cdot {\bf B} = 0\),
1627 1627 
 we get the
1628 1628 
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1631 1631 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1632 1632 
 \[
 
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1680 1680 
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1681 1681 
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1682 1682 
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1683 
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1623 
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1622 1622 
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1623 
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@@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1633 1633 
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1634 1634 
 We therefore have the
1635 1635 
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1636 
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1638 1638 
 {\bf Polarization current density}
1639 1639 
 \[
 
@@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
1651 1651 
 polarization changes).  We can check consistency with the continuity equation
1652 1652 
 associated to the conservation of bound charges:
1653 1653 
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1654 
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1656 1656 
 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1657 1657 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1674,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1674 1674 
 In view of this:  total charge density can be separated into 2 parts,
1675 1675 
 {\it free} and {\it bound}:
1676 1676 
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1677 
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1679 1679 
 \[
1680 1680 
     \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
 
@@ -1687,7 +1687,7 @@ In view of this:  total charge density can be separated into 2 parts,
1687 1687 
 and current can be separated into three parts, {\it free}, {\it bound} and
1688 1688 
 {\it polarization}:
1689 1689 
 </p>
1690 
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1692 1692 
 \[
1693 1693 
   {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
 
@@ -1711,7 +1711,7 @@ Gauss's law:  can be rewritten
1711 1711 
 \]
1712 1712 
 where (as in static case)
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1716 1716 
 \[
1717 1717 
     {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1737,7 +1737,7 @@ or
1737 1737 
 \]
1738 1738 
 where as before
1739 1739 
 </p>
1740 
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1740 
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1741 1741 
 <p>
1742 1742 
 \[
1743 1743 
     {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1755 1755 
 <p>
1756 1756 
 In terms of free charges and currents, we thus get
1757 1757 
 </p>
1758 
-<div class="core div" id="org5d9c2e0">
1758 
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1759 1759 
 <p>
1760 1760 
 {\bf Maxwell's equations {\it (in matter)}}
1761 1761 
 </p>
 
@@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1781 1781 
 in terms of \({\bf E}\) and \({\bf B}\).
1782 1782 
 For the restricted case of linear media:
1783 1783 
 </p>
1784 
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1784 
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1785 1785 
 <p>
1786 1786 
 \[
1787 1787 
     {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
 
@@ -1816,7 +1816,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1816 1816 
 </div>
1817 1817 
 <div id="postamble" class="status">
1818 1818 
 <p class="author">Author: Jean-Sébastien Caux</p>
1819 
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1819 
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1820 1820 
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1821 1821 
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1822 1822

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1600 1600 
 <p>
1601 1601 
 Discontinuities between different media, deduced from
1602 1602 
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1603 
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1605 1605 
 {\bf Maxwell's equations {\it (in matter)}, integral form}
1606 1606 
 </p>
 
@@ -1689,7 +1689,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1689 1689 
 </div>
1690 1690 
 <div id="postamble" class="status">
1691 1691 
 <p class="author">Author: Jean-Sébastien Caux</p>
1692 
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1692 
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1694 1694 
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1623 1623 
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1624 1624 
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1625 1625 
 <p class="author">Author: Jean-Sébastien Caux</p>
1626 
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1626 
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1619 1619 
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1620 1620 
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1621 1621 
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1622 
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1622 
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1715 1715 
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1716 1716 
 <div id="postamble" class="status">
1717 1717 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
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@@ -1635,7 +1635,7 @@ v = \frac{1}{\sqrt{\mu \varepsilon}} = \frac{c}{n}
1635 1635 
 \]
1636 1636 
 where the index of refraction of the material is defined as
1637 1637 
 </p>
1638 
-<div class="main div" id="org4373079">
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1639 1639 
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1640 1640 
 {\bf Index of refraction}
1641 1641 
 \[
 
@@ -1690,7 +1690,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1690 1690 
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1691 1691 
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1692 1692 
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1693 
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1634 1634 
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1635 1635 
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1636 1636 
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1637 
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1614 1614 
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1615 1615 
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1616 1616 
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1617 
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1615 1615 
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1616 1616 
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1617 1617 
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1618 
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1706 1706 
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1707 1707 
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1708 1708 
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1709 
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1709 
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@@ -1635,7 +1635,7 @@ These forms for incident, reflected and transmitted wave can be substituted in t
1635 1635 
 <p>
1636 1636 
 From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
1637 1637 
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1638 
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1638 
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1639 1639 
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1640 1640 
 {\bf First law of reflection:}
1641 1641 
 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
@@ -1650,7 +1650,7 @@ Specializing (\ref{eq:RTObliquek}) to our notations, we have
1650 1650 
 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
1651 1651 
 and the angle of refraction (\(\theta_T\)) obey the following laws:
1652 1652 
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1653 
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1653 
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1654 1654 
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1655 1655 
 {\bf Law of reflection}
1656 1656 
 \[
 
@@ -1708,7 +1708,7 @@ while the third equation becomes
1708 1708 
 \]
1709 1709 
 Writing everything in terms of the incident amplitude, we get
1710 1710 
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1711 
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1711 
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1712 1712 
 <p>
1713 1713 
 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
1714 1714 
 \[
 
@@ -1728,7 +1728,7 @@ Amplitudes for transmitted and reflected wave: depend on angle of incidence:
1728 1728 
 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
1729 1729 
 For grazing waves \(\theta_I \rightarrow \pi/2\) we have that \(\alpha \rightarrow \infty\) and the wave is totally reflected. The most interesting angle is the one at which \(\alpha = \beta\) and the reflected wave has zero amplitude. This is known as
1730 1730 
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1731 
-<div class="main div" id="orga399926">
1731 
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1732 1732 
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1733 1733 
 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
1734 1734 
   \[
 
@@ -1776,7 +1776,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1776 1776 
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1777 1777 
 <div id="postamble" class="status">
1778 1778 
 <p class="author">Author: Jean-Sébastien Caux</p>
1779 
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1779 
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1615 1615 
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1616 1616 
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1617 1617 
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1618 
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1621 1621 
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1622 1622 
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1623 1623 
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1624 
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1641 1641 
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1642 1642 
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1643 1643 
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1644 
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1665 1665 
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1666 1666 
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1667 1667 
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1668 
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1668 
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1674 1674 
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1675 1675 
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1676 1676 
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1677 
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1598 1598 
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1599 1599
1600 1600 
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1601 
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1603 1603 
 Prerequisites
1604 1604 
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1605 1605 
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@@ -1607,8 +1607,8 @@ Prerequisites
1607 1607 
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1608 1608 
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1609 1609
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1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
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1642 1642 
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1643 1643 
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1644 1644 
 <p class="author">Author: Jean-Sébastien Caux</p>
1645 
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1645 
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1639 1639 
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1640 1640 
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1641 1641 
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1642 
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1640 1640 
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1641 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
1613 1613 
 \]
1614 1614 
 These can be written compactly upon introducing a new operator: the
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1616 
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 <p>
1618 1618 
 {\bf d'Alembertian operator}
1619 1619 
 \[
 
@@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
1626 1626 
 <p>
1627 1627 
 so we get the
1628 1628 
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 <p>
1631 1631 
 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1632 1632 
 \[
 
@@ -1674,7 +1674,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1674 
 </div>
1675 1675 
 <div id="postamble" class="status">
1676 1676 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
1677 
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@@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
1614 1614 
 <p>
1615 1615 
 Easiest:
1616 1616 
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 <p>
1619 1619 
 \[
1620 1620 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
1630 1630 
 \]
1631 1631 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1632 1632 
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1633 
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1634 1634 
 <p>
1635 1635 
 \[
1636 1636 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1643,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1643 1643 
 <p>
1644 1644 
 Using this potential representation for \({\boldsymbol E}\) and \({\boldsymbol B}\) automatically fulfills the two homogeneous Maxwell equations. For the inhomogeneous equations, substituting (\ref{eq:E_from_Potentials}) into Gauss's law gives
1645 1645 
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1646 
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 <p>
1648 1648 
 \[
1649 1649 
     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1659,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
1659 1659 
 \]
1660 1660 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1661 1661 
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1664 1664 
 \[
1665 1665 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1693,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1693 1693 
 </div>
1694 1694 
 <div id="postamble" class="status">
1695 1695 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-02-09 Wed 22:40</p>
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1698 1698 
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1623 1623 
 <div id="postamble" class="status">
1624 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
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1600 1600 
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-<details class="prereq" id="orgd1b9be9">
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 Prerequisites
1604 1604 
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1607 1607 
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 </details>
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1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
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1644 1644 
 </div>
1645 1645 
 <div id="postamble" class="status">
1646 1646 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
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1603 1603 
 defines an electrostatic problem, whose solution is in principle obtained
1604 1604 
 from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a>
1605 1605 
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 \begin{equation*}
1608 1608 
   {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
1609 1609 
 \end{equation*}
 
@@ -1613,7 +1613,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1613 1613 
 or (often simpler) by calculating the electrostatic potential, using either the
1614 1614 
 explicit construction (\ref{eq:V_from_rho})
1615 1615 
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1618 1618 
 \[
1619 1619 
     V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
 
@@ -1629,7 +1629,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1629 1629 
 'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
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1634 1634 
 \[
1635 1635 
     {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
 
@@ -1643,7 +1643,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1643 1643 
 In the specific case where the charge density vanishes, we fall back onto the simpler
1644 1644 
 Laplace equation
1645 1645 
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 \[
1649 1649 
     {\boldsymbol \nabla}^2 V = 0
 
@@ -1676,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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1677 1677 
 <div id="postamble" class="status">
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1679 
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1677 1677 
 of the \(f_x + f_y + f_z = 0\) condition above.
1678 1678 
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1682 1682 
 <b>Earnshaw's theorem</b> <br>
1683 1683 
 Since solutions to Laplace's equation have no local minimum,
 
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