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Update 2022-02-21 10:35

master
Jean-Sébastien 2 years ago
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1625 1620
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1758
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1619
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1626 1621
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1627 1622
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1628 1623
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1629
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643
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634
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638
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1629 1624
 <div id="postamble" class="status">
1630 1625
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1631
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638
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662 657
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663 658
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662
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+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Second Derivatives</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_pr.html">Product arguments&emsp;<small>[c.m.dc.pr]</small></a></li><li>Next:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_d2" class="outline-5">
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1602
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1606 1601
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1607 1602
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
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1612 1607
 
1613
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 This always vanishes.
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1626 1621
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1627 1622
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1630 1625
 
1631
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 This always vanishes.
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1639 1634
 
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 {\boldsymbol \nabla} \times ({\boldsymbol \nabla} \times {\bf v}) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v}) - {\boldsymbol \nabla}^2 {\bf v}
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1653 1648
 
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1649
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1669
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638
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1628 1623
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1629 1624
 <p class="author">Author: Jean-Sébastien Caux</p>
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 </div>
1649 1644
 <div id="postamble" class="status">
1650 1645
 <p class="author">Author: Jean-Sébastien Caux</p>
1651
-<p class="date">Created: 2022-02-17 Thu 08:42</p>
1646
+<p class="date">Created: 2022-02-21 Mon 10:33</p>
1652 1647
 <p class="validation"></p>
1653 1648
 </div>
1654 1649
 

+ 189
- 26
build/c_m_dc_pr.html View File

@@ -1,7 +1,7 @@
1 1
 <!DOCTYPE html>
2 2
 <html lang="en">
3 3
 <head>
4
-<!-- 2022-02-17 Thu 08:42 -->
4
+<!-- 2022-02-21 Mon 10:33 -->
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>
@@ -602,11 +602,11 @@ Table of contents
602 602
 </summary>
603 603
 <ul>
604 604
 <li>
605
-<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
605
+<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
606 606
 
607 607
 </li>
608 608
 <li>
609
-<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
609
+<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
610 610
 
611 611
 </li>
612 612
 
@@ -614,21 +614,12 @@ Table of contents
614 614
 </details>
615 615
 </li>
616 616
 <li>
617
-
618
-<details>
619
-<summary>
620
-<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
621
-
622
-
623
-</summary>
624
-<ul>
625
-<li>
626
-<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
617
+<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
627 618
 
628 619
 </li>
620
+<li>
621
+<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
629 622
 
630
-</ul>
631
-</details>
632 623
 </li>
633 624
 <li>
634 625
 
@@ -640,11 +631,15 @@ Table of contents
640 631
 </summary>
641 632
 <ul>
642 633
 <li>
643
-<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
634
+<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
644 635
 
645 636
 </li>
646 637
 <li>
647
-<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
638
+<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
639
+
640
+</li>
641
+<li>
642
+<a href="./ems_ms_dcB_c.html#ems_ms_dcB_c">Curl of \({\bf B}\) from Biot-Savart; Ampère's Law</a><span class="headline-id">ems.ms.dcB.c</span>
648 643
 
649 644
 </li>
650 645
 
@@ -661,6 +656,10 @@ Table of contents
661 656
 </summary>
662 657
 <ul>
663 658
 <li>
659
+<a href="./ems_ms_vp_A.html#ems_ms_vp_A">Definition; Gauge Choices</a><span class="headline-id">ems.ms.vp.A</span>
660
+
661
+</li>
662
+<li>
664 663
 <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
665 664
 
666 665
 </li>
@@ -698,10 +697,6 @@ Table of contents
698 697
 </summary>
699 698
 <ul>
700 699
 <li>
701
-<a href="./emsm_esm_s.html#emsm_esm_s">A proper definition of "statics"</a><span class="headline-id">emsm.esm.s</span>
702
-
703
-</li>
704
-<li>
705 700
 
706 701
 <details>
707 702
 <summary>
@@ -1435,7 +1430,7 @@ Table of contents
1435 1430
 
1436 1431
 </li>
1437 1432
 <li class="toc-currentpage">
1438
-<a href="./c_m_dc_pr.html#c_m_dc_pr">Product Rules</a><span class="headline-id">c.m.dc.pr</span>
1433
+<a href="./c_m_dc_pr.html#c_m_dc_pr">Product arguments</a><span class="headline-id">c.m.dc.pr</span>
1439 1434
 
1440 1435
 </li>
1441 1436
 <li>
@@ -1591,17 +1586,185 @@ Table of contents
1591 1586
 </ul>
1592 1587
 </details>
1593 1588
 </nav>
1594
-<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Product Rules</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
1595
-<h5 id="c_m_dc_pr">Product Rules<a class="headline-permalink" href="./c_m_dc_pr.html#c_m_dc_pr"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1589
+<ul class="breadcrumbs"><li><a class="breadcrumb-link"href="c.html">Compendium</a></li><li><a class="breadcrumb-link"href="c_m.html">Mathematics</a></li><li><a class="breadcrumb-link"href="c_m_dc.html">Differential Calculus</a></li><li>Product arguments</li></ul><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_curl.html">The Curl&emsp;<small>[c.m.dc.curl]</small></a></li><li>Next:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Up:&nbsp;<a href="c_m_dc.html">Differential Calculus&emsp;<small>[c.m.dc]</small></a></li></ul><div id="outline-container-c_m_dc_pr" class="outline-5">
1590
+<h5 id="c_m_dc_pr">Product arguments<a class="headline-permalink" href="./c_m_dc_pr.html#c_m_dc_pr"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1596 1591
   <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"/>
1597 1592
   <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"/>
1598 1593
 </svg></a><span class="headline-id">c.m.dc.pr</span></h5>
1599 1594
 <div class="outline-text-5" id="text-c_m_dc_pr">
1600 1595
 <p>
1601
-Six product rules (on inside front cover of Gr).
1596
+When applied to arguments which are products of scalar/vector fields,
1597
+the action of \(\nabla\), \(\nabla \cdot\) and \(\nabla \times\) can be
1598
+explicited as follows:
1599
+</p>
1600
+
1601
+<p>
1602
+<b>Gradient of a product</b>:
1603
+</p>
1604
+<div class="eqlabel" id="org77098c1">
1605
+<p>
1606
+<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">
1607
+  <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"/>
1608
+  <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"/>
1609
+</svg></a>
1610
+</p>
1611
+<div class="alteqlabels" id="org098750f">
1612
+<ul class="org-ul">
1613
+<li>Gr (3)</li>
1614
+<li>W (1-111)</li>
1615
+</ul>
1616
+
1617
+</div>
1618
+
1619
+</div>
1620
+<p>
1621
+\[
1622
+\nabla (uv) = u \nabla v + v \nabla u
1623
+\tag{grad_prod}\label{grad_prod}
1624
+\]
1625
+</p>
1626
+
1627
+<br>
1628
+<p>
1629
+<b>Gradient of a scalar product</b>:
1630
+</p>
1631
+<div class="eqlabel" id="org623c0b6">
1632
+<p>
1633
+<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">
1634
+  <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1635
+  <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1636
+</svg></a>
1637
+</p>
1638
+<div class="alteqlabels" id="org9c465bf">
1639
+<ul class="org-ul">
1640
+<li>Gr (4)</li>
1641
+<li>W (1-112)</li>
1642
+</ul>
1643
+
1644
+</div>
1645
+
1646
+</div>
1647
+<p>
1648
+\[
1649
+\nabla ({\bf a} \cdot {\bf b}) = ({\bf a} \cdot \nabla) {\bf b} + ({\bf b} \cdot \nabla) {\bf a} + {\bf a} \times (\nabla \times {\bf b}) + {\bf b} \times (\nabla \times {\bf a})
1650
+\tag{grad_sprod}\label{grad_sprod}
1651
+\]
1652
+</p>
1653
+
1654
+<br>
1655
+<p>
1656
+<b>Divergence of a product</b>:
1657
+</p>
1658
+<div class="eqlabel" id="org27a4a20">
1659
+<p>
1660
+<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">
1661
+  <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"/>
1662
+  <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"/>
1663
+</svg></a>
1664
+</p>
1665
+<div class="alteqlabels" id="org3259c68">
1666
+<ul class="org-ul">
1667
+<li>Gr (5)</li>
1668
+<li>W (1-115)</li>
1669
+</ul>
1670
+
1671
+</div>
1672
+
1673
+</div>
1674
+<p>
1675
+\[
1676
+\nabla (\psi {\bf a}) = {\bf a} \cdot \nabla \psi + \psi \nabla \cdot {\bf a}
1677
+\tag{div_prod}\label{div_prod}
1678
+\]
1679
+</p>
1680
+
1681
+<br>
1682
+<p>
1683
+<b>Divergence of a cross product</b>:
1684
+</p>
1685
+<div class="eqlabel" id="orgaa0862c">
1686
+<p>
1687
+<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">
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"/>
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"/>
1690
+</svg></a>
1691
+</p>
1692
+<div class="alteqlabels" id="org1672361">
1693
+<ul class="org-ul">
1694
+<li>Gr (6)</li>
1695
+<li>W (1-116)</li>
1696
+</ul>
1697
+
1698
+</div>
1699
+
1700
+</div>
1701
+<p>
1702
+\[
1703
+\nabla \cdot ({\bf a} \times {\bf b}) = {\bf b} \cdot (\nabla \times {\bf a}) - {\bf a} \cdot (\nabla \times {\bf b})
1704
+\tag{div_xprod}\label{div_xprod}
1705
+\]
1706
+</p>
1707
+
1708
+<br>
1709
+<p>
1710
+<b>Curl of a product</b>:
1711
+</p>
1712
+<div class="eqlabel" id="org8a2530a">
1713
+<p>
1714
+<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">
1715
+  <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"/>
1716
+  <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"/>
1717
+</svg></a>
1602 1718
 </p>
1719
+<div class="alteqlabels" id="orgdb6a233">
1720
+<ul class="org-ul">
1721
+<li>Gr (7)</li>
1722
+<li>W (1-118)</li>
1723
+</ul>
1724
+
1603 1725
 </div>
1726
+
1604 1727
 </div>
1728
+<p>
1729
+\[
1730
+\nabla \times (\psi {\bf a}) = \nabla \psi \times {\bf a} + \psi \nabla \times {\bf a}
1731
+\tag{curl_prod}\label{curl_prod}
1732
+\]
1733
+</p>
1734
+
1735
+<br>
1736
+<p>
1737
+<b>Curl of a cross product</b>:
1738
+</p>
1739
+<div class="eqlabel" id="org21daf74">
1740
+<p>
1741
+<a id="curl_xprod"></a><a href="./c_m_dc_pr.html#curl_xprod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1742
+  <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"/>
1743
+  <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"/>
1744
+</svg></a>
1745
+</p>
1746
+<div class="alteqlabels" id="org87e2222">
1747
+<ul class="org-ul">
1748
+<li>Gr (8)</li>
1749
+<li>W (1-119)</li>
1750
+</ul>
1751
+
1752
+</div>
1753
+
1754
+</div>
1755
+<p>
1756
+\[
1757
+\nabla ({\bf a} \times {\bf b}) = {\bf a} (\nabla \cdot {\bf b}) - {\bf b} (\nabla \cdot {\bf a}) + ({\bf b} \cdot \nabla) {\bf a} - ({\bf a} \cdot \nabla) {\bf b}
1758
+\tag{curl_xprod}\label{curl_xprod}
1759
+\]
1760
+</p>
1761
+</div>
1762
+</div>
1763
+
1764
+
1765
+
1766
+
1767
+
1605 1768
 
1606 1769
 
1607 1770
 
@@ -1621,7 +1784,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1621 1784
 </div>
1622 1785
 <div id="postamble" class="status">
1623 1786
 <p class="author">Author: Jean-Sébastien Caux</p>
1624
-<p class="date">Created: 2022-02-17 Thu 08:42</p>
1787
+<p class="date">Created: 2022-02-21 Mon 10:33</p>
1625 1788
 <p class="validation"></p>
1626 1789
 </div>
1627 1790
 

+ 18
- 23
build/c_m_dd.html View File

@@ -1,7 +1,7 @@
1 1
 <!DOCTYPE html>
2 2
 <html lang="en">
3 3
 <head>
4
-<!-- 2022-02-17 Thu 08:42 -->
4
+<!-- 2022-02-21 Mon 10:33 -->
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>
@@ -602,11 +602,11 @@ Table of contents
602 602
 </summary>
603 603
 <ul>
604 604
 <li>
605
-<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
605
+<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charges</a><span class="headline-id">ems.ms.lf.pc</span>
606 606
 
607 607
 </li>
608 608
 <li>
609
-<a href="./ems_ms_lf_c.html#ems_ms_lf_c">Currents</a><span class="headline-id">ems.ms.lf.c</span>
609
+<a href="./ems_ms_lf_sc.html#ems_ms_lf_sc">Steady Currents</a><span class="headline-id">ems.ms.lf.sc</span>
610 610
 
611 611
 </li>
612 612
 
@@ -614,21 +614,12 @@ Table of contents
614 614
 </details>
615 615
 </li>
616 616
 <li>
617
-
618
-<details>
619
-<summary>
620
-<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
621
-
622
-
623
-</summary>
624
-<ul>
625
-<li>
626
-<a href="./ems_ms_BS_sc.html#ems_ms_BS_sc">The Magnetic Field issuing from a Steady Current</a><span class="headline-id">ems.ms.BS.sc</span>
617
+<a href="./ems_ms_ce.html#ems_ms_ce">Charge Conservation and the Continuity Equation</a><span class="headline-id">ems.ms.ce</span>
627 618
 
628 619
 </li>
620
+<li>
621
+<a href="./ems_ms_BS.html#ems_ms_BS">Steady Currents: the Biot-Savart Law</a><span class="headline-id">ems.ms.BS</span>
629 622
 
630
-</ul>
631
-</details>
632 623
 </li>
633 624
 <li>
634 625
 
@@ -640,11 +631,15 @@ Table of contents
640 631
 </summary>
641 632
 <ul>
642 633
 <li>
643
-<a href="./ems_ms_dcB_sc.html#ems_ms_dcB_sc">Straight-line Currents</a><span class="headline-id">ems.ms.dcB.sc</span>
634
+<a href="./ems_ms_dcB_iw.html#ems_ms_dcB_iw">Simplistic case: infinite wire</a><span class="headline-id">ems.ms.dcB.iw</span>
635
+
636
+</li>
637
+<li>
638
+<a href="./ems_ms_dcB_d.html#ems_ms_dcB_d">Divergence of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.d</span>
644 639
 
645 640
 </li>
646 641
 <li>
647
-<a href="./ems_ms_dcB_BS.html#ems_ms_dcB_BS">Divergence and Curl of \({\bf B}\) from Biot-Savart</a><span class="headline-id">ems.ms.dcB.BS</span>
642
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1626
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1638
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@@ -1620,14 +1615,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1620 1615
 More generally,
1621 1616
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1623
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1631 1626
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1632 1627
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1633 1628
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@@ -1646,14 +1641,14 @@ More generally,
1646 1641
 Since
1647 1642
 </p>
1648 1643
 
1649
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 we have that
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1681 1676
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1682 1677
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1708 1703
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1709 1704
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1710 1705
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1711
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1433
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1628 1623
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1696 1691
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1697 1692
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1698
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1643 1638
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1654
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1647 1642
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1648
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1650 1645
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1651 1646
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1652 1647
 <p class="author">Author: Jean-Sébastien Caux</p>
1653
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1625 1620
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1626 1621
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1627
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1644 1639
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 so we obtain
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1666 1661
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1684 1679
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1716 1711
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1717 1712
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1718
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1720 1715
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 Full set of equations for the electromagnetic field:
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1672 1667
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1673
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1681 1676
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1682
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1650 1645
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1652
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1650
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1648 1643
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1649
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 This is thus simply a conservation law for momentum, with
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 {\bf Momentum density in the EM fields}
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 In a region in which the mechanical momentum is not changing due to external influences, we then have the
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 {\bf Continuity equation for EM momentum}
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 This expression can be greatly simplified by introducing the
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 {\bf Maxwell stress tensor}
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 \[
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 {\bf Total force on charges in volume}
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 {\bf Poynting's theorem}
1672 1667
 \[
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 {\bf Poynting vector}
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 \[
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 {\bf Poynting's theorem}
1710 1705
 \[
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1718 1713
 where we have defined the total
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 {\bf Energy in electromagnetic fields}
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 \[
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 {\bf Poynting theorem (differential form)}
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 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
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1653
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1697 1692
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1698
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 {\bf Polarization current density}
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 \[
@@ -1651,7 +1646,7 @@ the polarization current is the result of linear motion of charge when
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 {\bf Maxwell's equations {\it (in matter)}}
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 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
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 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
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 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
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 \[
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 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
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 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
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 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
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1653
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 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>
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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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-<a href="./ems_ms_lf_pc.html#ems_ms_lf_pc">Point Charge</a><span class="headline-id">ems.ms.lf.pc</span>
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 <a href="./ems_ms_vp_mbc.html#ems_ms_vp_mbc">Magnetic Boundary Conditions</a><span class="headline-id">ems.ms.vp.mbc</span>
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 function \(\phi (x)\) and the Laplace equation reads
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 equation now reads
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@@ -1762,14 +1757,14 @@ a point equals its value averaged over a sphere
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 \(S_R({\bf r})\) of any radius \(R\) centered on this point
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 \]
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 <strong>Physicist's proof</strong>
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@@ -1955,19 +1950,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
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 of the \(f_x + f_y + f_z = 0\) condition above.
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 <b>Earnshaw's theorem (physical version)</b> <br>
1973 1968
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