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Update 2022-02-14 06:33

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Jean-Sébastien 2 years ago
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100 changed files with 871 additions and 377 deletions
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 <li><a href="a_l.html">Literature</a><span class="headline-id">a.l</span></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 class="author">Author: Jean-Sébastien Caux</p>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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1627 1629 
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1628 1630 
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1629 
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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
 
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 This always vanishes.
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 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1627 1627 
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1629 1629 
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1630 1630
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 This always vanishes.
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 \[
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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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1666 1668 
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1667 
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1648 1650 
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1649 
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1621 1623 
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1622 
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1622 1624 
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1623 1625 
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1624 
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   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1627 1627 
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1628 1628 
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1630 
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1630 
+<div class="alteqlabels" id="org3ed5b35">
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 
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1649 
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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 
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1657 1657 
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1658 1658 
 <li>Gr (1.101)</li>
1659 1659 
 </ul>
 
@@ -1669,15 +1669,32 @@ Since
1669 1669 
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1670 1670 
 we have that
1671 1671 
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1672 
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1673 
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1674 
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1675 
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1676 
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1677 
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1678 
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1679 
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1680 
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1681 
+<li>Gr (1.102)</li>
1682 
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1683 
+
1684 
+</div>
1685 
+
1686 
+</div>
1672 1687
1673 1688 
 \begin{equation}
1674 1689 
 {\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}')
1675 
-\label{Gr(1.102)}
1690 
+\tag{Lap1or}\label{Lap1or}
1676 1691 
 \end{equation}
1677 1692 
 </div>
1678 1693 
 </div>
1679 1694
1680 1695
1696 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Next:&nbsp;<a href="c_m_vf.html">Vector Fields&emsp;<small>[c.m.vf]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul>
1697 
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1681 1698 
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1682 1699 
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1683 1700 
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@@ -1691,7 +1708,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1691 1708 
 </div>
1692 1709 
 <div id="postamble" class="status">
1693 1710 
 <p class="author">Author: Jean-Sébastien Caux</p>
1694 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1711 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1695 1712 
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@@ -1626,6 +1626,8 @@ Problem:  in Gr(1.84), we've divided by zero when \(r = 0\).
1626 1626
1627 1627
1628 1628
1629 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li><li>Next:&nbsp;<a href="c_m_dd_1d.html">The One-Dimensional Dirac Delta Function&emsp;<small>[c.m.dd.1d]</small></a></li><li>Up:&nbsp;<a href="c_m_dd.html">Dirac delta Distribution&emsp;<small>[c.m.dd]</small></a></li></ul>
1630 
+<br>
1629 1631 
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1630 1632 
 <div class="license">
1631 1633 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1639 1641 
 </div>
1640 1642 
 <div id="postamble" class="status">
1641 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1644 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@@ -1611,6 +1611,8 @@ Table of contents
1611 1611 
 <li><a href="c_m_ic_ip.html">Integration by Parts</a><span class="headline-id">c.m.ic.ip</span></li>
1612 1612 
 </ul>
1613 1613
1614 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_dc_d2.html">Second Derivatives&emsp;<small>[c.m.dc.d2]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Up:&nbsp;<a href="c_m.html">Mathematics&emsp;<small>[c.m]</small></a></li></ul>
1615 
+<br>
1614 1616 
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1615 1617 
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1616 1618 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1624,7 +1626,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1624 1626 
 </div>
1625 1627 
 <div id="postamble" class="status">
1626 1628 
 <p class="author">Author: Jean-Sébastien Caux</p>
1627 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1629 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1628 1630 
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1629 1631 
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@@ -1609,6 +1609,8 @@ Table of contents
1609 1609
1610 1610
1611 1611
1612 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_lsv.html">Line, Surface and Volume Integrals&emsp;<small>[c.m.ic.lsv]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1613 
+<br>
1612 1614 
 <hr>
1613 1615 
 <div class="license">
1614 1616 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1622 1624 
 </div>
1623 1625 
 <div id="postamble" class="status">
1624 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1627 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1626 1628 
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1627 1629 
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1618 1618
1619 1619
1620 1620
1621 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1622 
+<br>
1621 1623 
 <hr>
1622 1624 
 <div class="license">
1623 1625 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1631,7 +1633,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1631 1633 
 </div>
1632 1634 
 <div id="postamble" class="status">
1633 1635 
 <p class="author">Author: Jean-Sébastien Caux</p>
1634 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1636 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1635 1637 
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1636 1638 
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1612 1612
1613 1613
1614 1614
1615 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_ftg.html">The Fundamental Theorem for Gradients&emsp;<small>[c.m.ic.ftg]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1616 
+<br>
1615 1617 
 <hr>
1616 1618 
 <div class="license">
1617 1619 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1625,7 +1627,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1625 1627 
 </div>
1626 1628 
 <div id="postamble" class="status">
1627 1629 
 <p class="author">Author: Jean-Sébastien Caux</p>
1628 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1630 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1629 1631 
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1630 1632 
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1619 1619 
 </div>
1620 1620
1621 1621
1622 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_stokes.html">Stokes' Theorem&emsp;<small>[c.m.ic.stokes]</small></a></li><li>Next:&nbsp;<a href="c_m_cs.html">Coordinate Systems&emsp;<small>[c.m.cs]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1623 
+<br>
1622 1624 
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1623 1625 
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1624 1626 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1632,7 +1634,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1632 1634 
 </div>
1633 1635 
 <div id="postamble" class="status">
1634 1636 
 <p class="author">Author: Jean-Sébastien Caux</p>
1635 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1637 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1636 1638 
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1637 1639 
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@@ -1599,9 +1599,9 @@ Table of contents
1599 1599 
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1600 1600 
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1601 1601
1602 
-<div id="outline-container-org3f3c8c7" class="outline-6">
1603 
-<h6 id="org3f3c8c7"><a href="#org3f3c8c7">Line Integrals</a></h6>
1604 
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1602 
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1604 
+<div class="outline-text-6" id="text-orgdc409a4">
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 
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1631 1631 
 </div>
1632 1632
1633 
-<div id="outline-container-orgbda984c" class="outline-6">
1634 
-<h6 id="orgbda984c"><a href="#orgbda984c">Surface Integrals</a></h6>
1635 
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+<h6 id="org465c39d"><a href="#org465c39d">Surface Integrals</a></h6>
1635 
+<div class="outline-text-6" id="text-org465c39d">
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 
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1653 1653 
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1654 1654
1655 
-<div id="outline-container-orgc3891db" class="outline-6">
1656 
-<h6 id="orgc3891db"><a href="#orgc3891db">Volume Integrals</a></h6>
1657 
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1655 
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1656 
+<h6 id="org794c5b9"><a href="#org794c5b9">Volume Integrals</a></h6>
1657 
+<div class="outline-text-6" id="text-org794c5b9">
1658 1658 
 <p>
1659 1659 
 \[
1660 1660 
 \int_{\cal V} T d\tau
 
@@ -1680,6 +1680,8 @@ d\tau = dx dy dz
1680 1680
1681 1681
1682 1682
1683 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ftc.html">The Fundamental Theorem of Calculus&emsp;<small>[c.m.ic.ftc]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1684 
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1683 1685 
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1684 1686 
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1685 1687 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1693 1695 
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1694 1696 
 <div id="postamble" class="status">
1695 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1698 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1697 1699 
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1602 1602 
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1604 1604 
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1606 1606 
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1607 
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1607 
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1608 1608 
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1609 1609 
 <li>Gr (1.57)</li>
1610 1610 
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@@ -1634,6 +1634,8 @@ the boundary shrinks to a point.
1634 1634
1635 1635
1636 1636
1637 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="c_m_ic_gauss.html">Gauss' Theorem&emsp;<small>[c.m.ic.gauss]</small></a></li><li>Next:&nbsp;<a href="c_m_ic_ip.html">Integration by Parts&emsp;<small>[c.m.ic.ip]</small></a></li><li>Up:&nbsp;<a href="c_m_ic.html">Integral Calculus&emsp;<small>[c.m.ic]</small></a></li></ul>
1638 
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1637 1639 
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1638 1640 
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1639 1641 
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@@ -1647,7 +1649,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1647 1649 
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1648 1650 
 <div id="postamble" class="status">
1649 1651 
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1650 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
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1653 1655 
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1654 1656 
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1655 
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1649 1651 
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1621 1623 
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1622 1624 
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1623 
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1640 1642 
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1641 1643 
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1642 
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 <hr>
1613 1615 
 <div class="license">
1614 1616 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1622 1624 
 </div>
1623 1625 
 <div id="postamble" class="status">
1624 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1627 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1626 1628 
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@@ -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="orgee21897">
1641 
+<div class="core div" id="org60c9879">
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="orgd1f18df">
1663 
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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="org04bf1f1">
1681 
+<div class="core div" id="org3c09004">
1682 1682 
 <p>
1683 1683 
 <b>Faraday's law</b> (differential form)
1684 1684 
   \[
 
@@ -1700,6 +1700,8 @@ to an opposing counter-reaction.
1700 1700
1701 1701
1702 1702
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
1704 
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1703 1705 
 <hr>
1704 1706 
 <div class="license">
1705 1707 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1713,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1713 1715 
 </div>
1714 1716 
 <div id="postamble" class="status">
1715 1717 
 <p class="author">Author: Jean-Sébastien Caux</p>
1716 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1718 
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@@ -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 
 </p>
1660 
-<div class="core div" id="org183e158">
1660 
+<div class="core div" id="org0ecd8ee">
1661 1661 
 <p>
1662 1662 
 \[
1663 1663 
     W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
 
@@ -1678,7 +1678,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
1678 1678 
 \hspace{2cm} \mbox{(7.31 and 7.34)}
1679 1679 
 \end{align}
1680 1680
1681 
-<div class="example div" id="org5a2dc32">
1681 
+<div class="example div" id="org46b2091">
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 \(l\).
 
@@ -1702,6 +1702,8 @@ Note:  gives easy way to find inductance, since \(W = \frac{1}{2} L I^2\).
1702 1702 
 </div>
1703 1703
1704 1704
1705 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Next:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
1706 
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1705 1707 
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1706 1708 
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1707 1709 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1715,7 +1717,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1715 1717 
 </div>
1716 1718 
 <div id="postamble" class="status">
1717 1719 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1720 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@@ -1637,7 +1637,7 @@ M_{12} = M_{21}
1637 1637 
 \]
1638 1638 
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1639 1639
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-<div class="example div" id="orgb4beaba">
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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
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1690 
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1691 1691 
 <p>
1692 1692 
 \paragraph{Example 7.11:}  find self-inductance of toroidal coil with
1693 1693 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
 
@@ -1714,7 +1714,7 @@ Total flux:  \(N\) times this, so self-inductance is
1714 1714 
 Inductance (like capacitance) is intrinsically positive.  Use Lenz law.  Think of {\bf back EMF}.
1715 1715 
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1716 1716
1717 
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1717 
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1718 1718 
 <p>
1719 1719 
 \paragraph{Example 7.12:}  circuit with inductance \(L\), resistor \(R\) and battery \({\cal E}_0\).
1720 1720 
 What is the current ?
 
@@ -1738,6 +1738,8 @@ where \(\tau \equiv L/R\) is the {\bf time constant} of the circuit.
1738 1738
1739 1739
1740 1740
1741 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_ief.html">The Induced Electric Field&emsp;<small>[emd.Fl.ief]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
1742 
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1741 1743 
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1743 1745 
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@@ -1751,7 +1753,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1751 1753 
 </div>
1752 1754 
 <div id="postamble" class="status">
1753 1755 
 <p class="author">Author: Jean-Sébastien Caux</p>
1754 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1756 
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@@ -1621,7 +1621,7 @@ law in integral form:
1621 1621
1622 1622
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1625 1625 
 <p>
1626 1626 
 {\bf Example 7.7:}
1627 1627 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
@@ -1637,7 +1637,7 @@ Increasing \({\bf B}\):  clockwise (viewed from above) \({\bf E}\) from Lenz.
1637 1637 
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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.
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1673 1673
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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.
 
@@ -1704,6 +1704,8 @@ Reason:  in this case, we've overstepped the quasistatic limit.  We need
1704 1704
1705 1705
1706 1706
1707 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_Fl.html">Faraday's Law&emsp;<small>[emd.Fl.Fl]</small></a></li><li>Next:&nbsp;<a href="emd_Fl_i.html">Inductance&emsp;<small>[emd.Fl.i]</small></a></li><li>Up:&nbsp;<a href="emd_Fl.html">Induction: Faraday's Law&emsp;<small>[emd.Fl]</small></a></li></ul>
1708 
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1707 1709 
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1708 1710 
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1709 1711 
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@@ -1717,7 +1719,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1717 1719 
 </div>
1718 1720 
 <div id="postamble" class="status">
1719 1721 
 <p class="author">Author: Jean-Sébastien Caux</p>
1720 
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1722 
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1609 1609 
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1610 1610 
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1611 1611
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Fl_e.html">Energy in Magnetic Fields&emsp;<small>[emd.Fl.e]</small></a></li><li>Next:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
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1622 1624 
 </div>
1623 1625 
 <div id="postamble" class="status">
1624 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1627 
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1600 1600 
 <p>
1601 1601 
 Full set of equations for the electromagnetic field:
1602 1602 
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1603 
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1604 1604 
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1605 1605 
 {\bf Maxwell's equations} {\it (in vacuum)}
1606 1606 
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@@ -1616,7 +1616,7 @@ Full set of equations for the electromagnetic field:
1616 1616 
 <p>
1617 1617 
 Complement:
1618 1618 
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1619 
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1619 
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1620 1620 
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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 
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1643 
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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, \\
 
@@ -1655,6 +1655,8 @@ Better way of writing:  all fields on left, all sources on right,
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Next:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
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1658 1660 
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1659 1661 
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1660 1662 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
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1668 1670 
 </div>
1669 1671 
 <div id="postamble" class="status">
1670 1672 
 <p class="author">Author: Jean-Sébastien Caux</p>
1671 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1673 
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1607 1607 
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 The extra term would thus be eliminated if we were to put
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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 
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1632 1632 
 Maxwell baptized this term the
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1636 1636 
 {\bf Displacement current}
1637 1637 
 \[
 
@@ -1664,6 +1664,8 @@ Flat surface:  OK, \(E = 0\) and \(I_{\mbox{enc}} = I\).  Balloon surface:  \(I
1664 1664
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_ebM.html">Electrodynamics Before Maxwell&emsp;<small>[emd.Me.ebM]</small></a></li><li>Next:&nbsp;<a href="emd_Me_Me.html">Maxwell's Equations&emsp;<small>[emd.Me.Me]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
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1677 1679 
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1678 1680 
 <div id="postamble" class="status">
1679 1681 
 <p class="author">Author: Jean-Sébastien Caux</p>
1680 
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li><li>Next:&nbsp;<a href="emd_Me_dc.html">Maxwell's Correction to Ampère's Law; the Displacement Current&emsp;<small>[emd.Me.dc]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
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1648 1650 
 <div id="postamble" class="status">
1649 1651 
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1650 
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_Me.html">Maxwell's Equations&emsp;<small>[emd.Me.Me]</small></a></li><li>Next:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Up:&nbsp;<a href="emd_Me.html">Maxwell's Equations&emsp;<small>[emd.Me]</small></a></li></ul>
1636 
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1636 1638 
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1637 1639 
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1645 1647 
 </div>
1646 1648 
 <div id="postamble" class="status">
1647 1649 
 <p class="author">Author: Jean-Sébastien Caux</p>
1648 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1650 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1649 1651 
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1650 1652 
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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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1609 1609
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1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
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1631 1631 
 <li><a href="emd_ce_amom.html">Angular Momentum</a><span class="headline-id">emd.ce.amom</span></li>
1632 1632 
 </ul>
1633 1633
1634 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_Me_mc.html">Magnetic Charge&emsp;<small>[emd.Me.mc]</small></a></li><li>Next:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1635 
+<br>
1634 1636 
 <hr>
1635 1637 
 <div class="license">
1636 1638 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1644,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1644 1646 
 </div>
1645 1647 
 <div id="postamble" class="status">
1646 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1649 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1648 1650 
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1649 1651 
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1650 1652

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1600 1600 
 <p>
1601 1601 
 The angular momentum of EM fields is directly given by
1602 1602 
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1603 
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1603 
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1604 1604 
 <p>
1605 1605 
 {\bf Angular momentum of EM fields}
1606 1606 
 \[
 
@@ -1615,6 +1615,8 @@ The angular momentum of EM fields is directly given by
1615 1615 
 </div>
1616 1616
1617 1617
1618 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Next:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
1619 
+<br>
1618 1620 
 <hr>
1619 1621 
 <div class="license">
1620 1622 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1628,7 +1630,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1628 1630 
 </div>
1629 1631 
 <div id="postamble" class="status">
1630 1632 
 <p class="author">Author: Jean-Sébastien Caux</p>
1631 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1633 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1632 1634 
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1633 1635 
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1634 1636

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6 6 
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7 7 
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@@ -1619,7 +1619,7 @@ This means that
1619 1619 
 \]
1620 1620 
 Since this is true for any volume, we have (re)derived the
1621 1621 
 </p>
1622 
-<div class="core div" id="org8328070">
1622 
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1623 1623 
 <p>
1624 1624 
 {\bf Continuity equation}
1625 1625 
 \[
 
@@ -1645,6 +1645,8 @@ imposes a functional constraint on these sources: not {\it any} \(\rho\) and
1645 1645
1646 1646
1647 1647
1648 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
1649 
+<br>
1648 1650 
 <hr>
1649 1651 
 <div class="license">
1650 1652 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1658,7 +1660,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1658 1660 
 </div>
1659 1661 
 <div id="postamble" class="status">
1660 1662 
 <p class="author">Author: Jean-Sébastien Caux</p>
1661 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1663 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1662 1664 
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1663 1665 
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@@ -1612,7 +1612,7 @@ in which the first integral can be interpreted as the momentum stored in the EM
1612 1612 
 <p>
1613 1613 
 This is thus simply a conservation law for momentum, with
1614 1614 
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1615 
-<div class="main div" id="orgfb9b180">
1615 
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1616 1616 
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1617 1617 
 {\bf Momentum density in the EM fields}
1618 1618 
 \[
 
@@ -1624,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
1624 1624 
 <p>
1625 1625 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
1626 1626 
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1627 
-<div class="main div" id="org03df6a3">
1627 
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1628 1628 
 <p>
1629 1629 
 {\bf Continuity equation for EM momentum}
1630 1630 
 \[
 
@@ -1638,6 +1638,8 @@ In a region in which the mechanical momentum is not changing due to external inf
1638 1638
1639 1639
1640 1640
1641 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Next:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
1642 
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1641 1643 
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1642 1644 
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1643 1645 
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1651 1653 
 </div>
1652 1654 
 <div id="postamble" class="status">
1653 1655 
 <p class="author">Author: Jean-Sébastien Caux</p>
1654 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1656 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1655 1657 
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1656 1658 
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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 
-<div class="main div" id="org7f141f8">
1657 
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1658 1658 
 <p>
1659 1659 
 {\bf Maxwell stress tensor}
1660 1660 
 \[
 
@@ -1677,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
1677 1677 
 <p>
1678 1678 
 We then obtain
1679 1679 
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1680 
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1680 
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1681 1681 
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1682 1682 
 {\bf EM force per unit volume}
1683 1683 
 \[
 
@@ -1689,7 +1689,7 @@ We then obtain
1689 1689 
 <p>
1690 1690 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1691 1691 
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1692 
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1692 
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1693 1693 
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1694 1694 
 {\bf Total force on charges in volume}
1695 1695 
 \[
 
@@ -1703,6 +1703,8 @@ where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
1703 1703
1704 1704
1705 1705
1706 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_poy.html">Poynting's Theorem; the Poynting Vector&emsp;<small>[emd.ce.poy]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mom.html">Momentum&emsp;<small>[emd.ce.mom]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
1707 
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1706 1708 
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1707 1709 
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1708 1710 
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@@ -1716,7 +1718,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1716 1718 
 </div>
1717 1719 
 <div id="postamble" class="status">
1718 1720 
 <p class="author">Author: Jean-Sébastien Caux</p>
1719 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1721 
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1720 1722 
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1721 1723 
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@@ -1666,7 +1666,7 @@ so we get
1666 1666 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1667 1667 
 we obtain
1668 1668 
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1669 
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1669 
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1670 1670 
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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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1694 
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1695 1695 
 <p>
1696 1696 
 {\bf Poynting vector}
1697 1697 
 \[
 
@@ -1704,7 +1704,7 @@ Energy per unit time, per unit area carried by EM fields:
1704 1704 
 <p>
1705 1705 
 We can thus express Poynting's theorem more compactly:
1706 1706 
 </p>
1707 
-<div class="core div" id="orgdeee59b">
1707 
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1708 1708 
 <p>
1709 1709 
 {\bf Poynting's theorem}
1710 1710 
 \[
 
@@ -1717,7 +1717,7 @@ We can thus express Poynting's theorem more compactly:
1717 1717 
 <p>
1718 1718 
 where we have defined the total
1719 1719 
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1720 
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1720 
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1721 1721 
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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 
-<div class="core div" id="org65ae443">
1743 
+<div class="core div" id="orgd477ba8">
1744 1744 
 <p>
1745 1745 
 {\bf Poynting theorem (differential form)}
1746 1746 
 \[
 
@@ -1757,7 +1757,7 @@ and has a similar for to the continuity equation
1757 1757
1758 1758
1759 1759
1760 
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1760 
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1761 1761 
 <p>
1762 1762 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1763 1763 
 Assuming that the field is uniform, the electric field parallel to the wire is
 
@@ -1787,6 +1787,8 @@ and the value is as expected.
1787 1787
1788 1788
1789 1789
1790 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_ce.html">The Continuity Equation&emsp;<small>[emd.ce.ce]</small></a></li><li>Next:&nbsp;<a href="emd_ce_mst.html">Maxwell's Stress Tensor&emsp;<small>[emd.ce.mst]</small></a></li><li>Up:&nbsp;<a href="emd_ce.html">Charge and Energy Flows&emsp;<small>[emd.ce]</small></a></li></ul>
1791 
+<br>
1790 1792 
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1791 1793 
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1792 1794 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1800,7 +1802,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1800 1802 
 </div>
1801 1803 
 <div id="postamble" class="status">
1802 1804 
 <p class="author">Author: Jean-Sébastien Caux</p>
1803 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1805 
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1804 1806 
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1603 1603 
 Prerequisites
1604 1604 
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1605 1605 
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@@ -1608,8 +1608,8 @@ Prerequisites
1608 1608 
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1609 1609 
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1610 1610
1611 
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1613 1613 
 Objectives
1614 1614 
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1615 1615 
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@@ -1635,6 +1635,8 @@ Objectives
1635 1635 
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1636 1636 
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1637 1637
1638 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_ce_amom.html">Angular Momentum&emsp;<small>[emd.ce.amom]</small></a></li><li>Next:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Up:&nbsp;<a href="emd.html">Electromagnetodynamics&emsp;<small>[emd]</small></a></li></ul>
1639 
+<br>
1638 1640 
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1639 1641 
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1640 1642 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1648,7 +1650,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1648 1650 
 </div>
1649 1651 
 <div id="postamble" class="status">
1650 1652 
 <p class="author">Author: Jean-Sébastien Caux</p>
1651 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1653 
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1652 1654 
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1653 1655 
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@@ -1628,7 +1628,7 @@ so for a monochromatic EM plan wave,
1628 1628 
 \]
1629 1629 
 or more succinctly:
1630 1630 
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1631 
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1631 
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1632 1632 
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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 
 </p>
1647 
-<div class="main div" id="org0141dca">
1647 
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1648 1648 
 <p>
1649 1649 
 {\bf Momentum density of a monochromatic EM wave}
1650 1650 
 \[
 
@@ -1680,6 +1680,8 @@ The {\it radiation pressure} is the momentum transfer per unit area per unit of
1680 1680 
 </div>
1681 1681
1682 1682
1683 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Next:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
1684 
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1683 1685 
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1684 1686 
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1685 1687 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1693 1695 
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1694 1696 
 <div id="postamble" class="status">
1695 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1698 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
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@@ -1629,7 +1629,7 @@ B_0 = \frac{k}{\omega} E_0 = \frac{1}{c} E_0.
1629 1629 
 Generalizing to propagation in the direction of an arbitrary wavevector
1630 1630 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1631 1631 
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 {\bf E and B fields for a monochromatic EM plane wave}
1635 1635 
 \[
 
@@ -1658,6 +1658,8 @@ or if you prefer explicit real parts (adding a possible phase shift \(\delta\)):
1658 1658
1659 1659
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw_we.html">The Wave Equation&emsp;<small>[emd.emw.we]</small></a></li><li>Next:&nbsp;<a href="emd_emw_ep.html">Energy and Momentum&emsp;<small>[emd.emw.ep]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
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1661 1663 
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1663 1665 
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1671 1673 
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1672 1674 
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1673 1675 
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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
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1629 
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1631 1631 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1632 1632 
 \[
 
@@ -1667,6 +1667,8 @@ the actual electric and magnetic fields are given by the real part.
1667 1667
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li><li>Next:&nbsp;<a href="emd_emw_mpw.html">Monochromatic Plane Waves&emsp;<small>[emd.emw.mpw]</small></a></li><li>Up:&nbsp;<a href="emd_emw.html">Electromagnetic waves in vacuum&emsp;<small>[emd.emw]</small></a></li></ul>
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1680 1682 
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1681 1683 
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1682 1684 
 <p class="author">Author: Jean-Sébastien Caux</p>
1683 
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1607 1607 
 <li><a href="emdm_emwm.html">Electromagnetic Waves in Matter</a><span class="headline-id">emdm.emwm</span></li>
1608 1608 
 </ul>
1609 1609
1610 
+<br><ul class="navigation-links"><li>Next:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul>
1611 
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1610 1612 
 <hr>
1611 1613 
 <div class="license">
1612 1614 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1620,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1620 1622 
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1621 1623 
 <div id="postamble" class="status">
1622 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1623 
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1625 
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1607 1607 
 <li><a href="emdm_Me_bc.html">Boundary Conditions</a><span class="headline-id">emdm.Me.bc</span></li>
1608 1608 
 </ul>
1609 1609
1610 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li><li>Next:&nbsp;<a href="emdm_Me_Mem.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me.Mem]</small></a></li><li>Up:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li></ul>
1611 
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 <hr>
1611 1613 
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1612 1614 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1620,7 +1622,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1620 1622 
 </div>
1621 1623 
 <div id="postamble" class="status">
1622 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1623 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1625 
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@@ -1633,7 +1633,7 @@ dI = \frac{\partial \sigma_b}{\partial t} da_{\perp} = \frac{\partial P}{\partia
1633 1633 
 \]
1634 1634 
 We therefore have the
1635 1635 
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1636 
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1637 1637 
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1638 1638 
 {\bf Polarization current density}
1639 1639 
 \[
 
@@ -1651,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
1651 1651 
 polarization changes).  We can check consistency with the continuity equation
1652 1652 
 associated to the conservation of bound charges:
1653 1653 
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1654 
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1654 
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1655 1655 
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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 
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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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 \[
1717 1717 
     {\bf D} \equiv \varepsilon_0 {\bf E} + {\bf P}
 
@@ -1737,7 +1737,7 @@ or
1737 1737 
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1738 1738 
 where as before
1739 1739 
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1742 1742 
 \[
1743 1743 
     {\bf H} \equiv \frac{1}{\mu_0} {\bf B} - {\bf M}
 
@@ -1755,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1755 1755 
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1756 1756 
 In terms of free charges and currents, we thus get
1757 1757 
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1760 1760 
 {\bf Maxwell's equations {\it (in matter)}}
1761 1761 
 </p>
 
@@ -1781,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1781 1781 
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1782 1782 
 For the restricted case of linear media:
1783 1783 
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1786 1786 
 \[
1787 1787 
     {\bf P} = \varepsilon_0 \chi_e {\bf E}, \hspace{1cm}
 
@@ -1803,6 +1803,8 @@ where \(\varepsilon \equiv \varepsilon_0(1 + \chi_e)\) and \(\mu \equiv \mu_0 (1
1803 1803
1804 1804
1805 1805
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1807 1809 
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1816 1818 
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1817 1819 
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1818 1820 
 <p class="author">Author: Jean-Sébastien Caux</p>
1819 
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1821 
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1600 1600 
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1601 1601 
 Discontinuities between different media, deduced from
1602 1602 
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1603 
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 {\bf Maxwell's equations {\it (in matter)}, integral form}
1606 1606 
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1677 1677
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me_Mem.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me.Mem]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li><li>Up:&nbsp;<a href="emdm_Me.html">Maxwell's Equations in Matter&emsp;<small>[emdm.Me]</small></a></li></ul>
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1681 1683 
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1689 1691 
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1690 1692 
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1691 1693 
 <p class="author">Author: Jean-Sébastien Caux</p>
1692 
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1610 1610 
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1611 1611 
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1612 1612
1613 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_Me_bc.html">Boundary Conditions&emsp;<small>[emdm.Me.bc]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Up:&nbsp;<a href="emdm.html">Electromagnetodynamics in Matter&emsp;<small>[emdm]</small></a></li></ul>
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1614 1616 
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1615 1617 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
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1623 1625 
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1624 1626 
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1625 1627 
 <p class="author">Author: Jean-Sébastien Caux</p>
1626 
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1606 1606 
 <li><a href="emdm_emwm_ad_c.html">EM Waves in Conductors</a><span class="headline-id">emdm.emwm.ad.c</span></li>
1607 1607 
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1608 1608
1609 
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1610 1612 
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1611 1613 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
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1619 1621 
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1620 1622 
 <div id="postamble" class="status">
1621 1623 
 <p class="author">Author: Jean-Sébastien Caux</p>
1622 
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1624 
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1702 1702 
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1703 1703
1704 1704
1705 
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1706 1708 
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1707 1709 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
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1715 1717 
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1716 1718 
 <div id="postamble" class="status">
1717 1719 
 <p class="author">Author: Jean-Sébastien Caux</p>
1718 
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1720 
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1635 1635 
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1636 1636 
 where the index of refraction of the material is defined as
1637 1637 
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1640 1640 
 {\bf Index of refraction}
1641 1641 
 \[
 
@@ -1677,6 +1677,8 @@ I = \frac{1}{2} \varepsilon v E_0^2
1677 1677
1678 1678
1679 1679
1680 
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1682 1684 
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1690 1692 
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1691 1693 
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1692 1694 
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1693 
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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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 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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 {\bf First law of reflection:}
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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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 with the incidence (\(\theta_I\)) and reflection (\(\theta_R\)) angles
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 and the angle of refraction (\(\theta_T\)) obey the following laws:
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 {\bf Law of reflection}
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 Writing everything in terms of the incident amplitude, we get
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 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
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 Behaviour: for \(\theta_I = 0\) we recover (\ref{Gr(9.82)}).
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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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 {\bf Brewster's angle {\it (at which the reflected wave amplitude vanishes)}}
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   \[
 
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1779 
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_plm.html">Propagation in Linear Media&emsp;<small>[emdm.emwm.plm]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_refl.html">Reflection and Transmission&emsp;<small>[emdm.emwm.refl]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emdm_emwm_ad_c.html">EM Waves in Conductors&emsp;<small>[emdm.emwm.ad.c]</small></a></li><li>Next:&nbsp;<a href="emdm_emwm_wg_gw.html">Guided waves&emsp;<small>[emdm.emwm.wg.gw]</small></a></li><li>Up:&nbsp;<a href="emdm_emwm.html">Electromagnetic Waves in Matter&emsp;<small>[emdm.emwm]</small></a></li></ul>
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1643 1645 
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 <li><a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations</a><span class="headline-id">emf.g.Lg</span></li>
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+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_svp.html">Scalar and Vector Potentials&emsp;<small>[emf.svp]</small></a></li><li>Next:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
1630 
+<br>
1629 1631 
 <hr>
1630 1632 
 <div class="license">
1631 1633 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1639,7 +1641,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1639 1641 
 </div>
1640 1642 
 <div id="postamble" class="status">
1641 1643 
 <p class="author">Author: Jean-Sébastien Caux</p>
1642 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1644 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1643 1645 
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1644 1646 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1625,6 +1625,8 @@ Although Gauss's law looks nice in the Coulomb gauge, Amp{\`e}re-Maxwell does no
1625 1625
1626 1626
1627 1627
1628 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Next:&nbsp;<a href="emf_g_Lg.html">Lorenz Gauge; d'Alembertian; Inhomogeneous Maxwell Equations&emsp;<small>[emf.g.Lg]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul>
1629 
+<br>
1628 1630 
 <hr>
1629 1631 
 <div class="license">
1630 1632 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1638,7 +1640,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1638 1640 
 </div>
1639 1641 
 <div id="postamble" class="status">
1640 1642 
 <p class="author">Author: Jean-Sébastien Caux</p>
1641 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1643 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1642 1644 
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1643 1645 
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 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1613,7 +1613,7 @@ while the equation for \(V\) becomes
1613 1613 
 \]
1614 1614 
 These can be written compactly upon introducing a new operator: the
1615 1615 
 </p>
1616 
-<div class="core div" id="org9685743">
1616 
+<div class="core div" id="orgb7b775a">
1617 1617 
 <p>
1618 1618 
 {\bf d'Alembertian operator}
1619 1619 
 \[
 
@@ -1626,7 +1626,7 @@ These can be written compactly upon introducing a new operator: the
1626 1626 
 <p>
1627 1627 
 so we get the
1628 1628 
 </p>
1629 
-<div class="core div" id="org8c1d592">
1629 
+<div class="core div" id="org1d4f993">
1630 1630 
 <p>
1631 1631 
 {\bf Inhomogeneous Maxwell equations (Lorenz gauge)}
1632 1632 
 \[
 
@@ -1661,6 +1661,8 @@ we have by direct inspection
1661 1661 
 </div>
1662 1662
1663 1663
1664 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf_g_Cg.html">Coulomb Gauge&emsp;<small>[emf.g.Cg]</small></a></li><li>Next:&nbsp;<a href="red.html">Relativistic Electrodynamics&emsp;<small>[red]</small></a></li><li>Up:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li></ul>
1665 
+<br>
1664 1666 
 <hr>
1665 1667 
 <div class="license">
1666 1668 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1674,7 +1676,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1674 1676 
 </div>
1675 1677 
 <div id="postamble" class="status">
1676 1678 
 <p class="author">Author: Jean-Sébastien Caux</p>
1677 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1679 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1678 1680 
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1679 1681 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1614,7 +1614,7 @@ Useful strategy: represent fields in terms of potentials.
1614 1614 
 <p>
1615 1615 
 Easiest:
1616 1616 
 </p>
1617 
-<div class="core div" id="org449388e">
1617 
+<div class="core div" id="orgea17c1a">
1618 1618 
 <p>
1619 1619 
 \[
1620 1620 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1630,7 +1630,7 @@ Putting this into Faraday's law gives
1630 1630 
 \]
1631 1631 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1632 1632 
 </p>
1633 
-<div class="core div" id="org8d2221f">
1633 
+<div class="core div" id="orge40ddf5">
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 
 </p>
1646 
-<div class="main div" id="orgf979da5">
1646 
+<div class="main div" id="org73cca63">
1647 1647 
 <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 
 </p>
1662 
-<div class="main div" id="orgc475199">
1662 
+<div class="main div" id="org0170d34">
1663 1663 
 <p>
1664 1664 
 \[
1665 1665 
   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1680,6 +1680,8 @@ which becomes after simple rearrangement and use of the identity \({\boldsymbol
1680 1680
1681 1681
1682 1682
1683 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li><li>Next:&nbsp;<a href="emf_g.html">Gauge Freedom and Choices&emsp;<small>[emf.g]</small></a></li><li>Up:&nbsp;<a href="emf.html">Electromagnetic Fields&emsp;<small>[emf]</small></a></li></ul>
1684 
+<br>
1683 1685 
 <hr>
1684 1686 
 <div class="license">
1685 1687 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1693,7 +1695,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1693 1695 
 </div>
1694 1696 
 <div id="postamble" class="status">
1695 1697 
 <p class="author">Author: Jean-Sébastien Caux</p>
1696 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1698 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1697 1699 
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1698 1700 
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@@ -1609,6 +1609,8 @@ Table of contents
1609 1609 
 <li><a href="ems_ms.html">Magnetostatics</a><span class="headline-id">ems.ms</span></li>
1610 1610 
 </ul>
1611 1611
1612 
+<br><ul class="navigation-links"><li>Next:&nbsp;<a href="ems_es.html">Electrostatics&emsp;<small>[ems.es]</small></a></li></ul>
1613 
+<br>
1612 1614 
 <hr>
1613 1615 
 <div class="license">
1614 1616 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1622,7 +1624,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1622 1624 
 </div>
1623 1625 
 <div id="postamble" class="status">
1624 1626 
 <p class="author">Author: Jean-Sébastien Caux</p>
1625 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1627 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1626 1628 
 <p class="validation"></p>
1627 1629 
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@@ -1598,8 +1598,8 @@ Table of contents
1598 1598 
 </svg></a><span class="headline-id">ems.ca</span></h3>
1599 1599
1600 1600 
 <div class="outline-text-3" id="text-ems_ca">
1601 
-<details class="prereq" id="orga055812">
1602 
-<summary id="org2cb689b">
1601 
+<details class="prereq" id="org0bf319c">
1602 
+<summary id="orga92f3c7">
1603 1603 
 Prerequisites
1604 1604 
 </summary>
1605 1605 
 <ul class="org-ul">
 
@@ -1607,8 +1607,8 @@ Prerequisites
1607 1607 
 </ul>
1608 1608 
 </details>
1609 1609
1610 
-<details class="objectives" id="orgb006f6c">
1611 
-<summary id="org037b124">
1610 
+<details class="objectives" id="org87cd363">
1611 
+<summary id="org64f736b">
1612 1612 
 Objectives
1613 1613 
 </summary>
1614 1614 
 <ul class="org-ul">
 
@@ -1631,6 +1631,8 @@ Objectives
1631 1631 
 <li><a href="ems_ca_me.html">The Multipole Expansion</a><span class="headline-id">ems.ca.me</span></li>
1632 1632 
 </ul>
1633 1633
1634 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_es_c_cap.html">Capacitors&emsp;<small>[ems.es.c.cap]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Up:&nbsp;<a href="ems.html">Electromagnetostatics&emsp;<small>[ems]</small></a></li></ul>
1635 
+<br>
1634 1636 
 <hr>
1635 1637 
 <div class="license">
1636 1638 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1644,7 +1646,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1644 1646 
 </div>
1645 1647 
 <div id="postamble" class="status">
1646 1648 
 <p class="author">Author: Jean-Sébastien Caux</p>
1647 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1649 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1648 1650 
 <p class="validation"></p>
1649 1651 
 </div>
1650 1652

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6 6 
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7 7 
 <title>Pre-Quantum Electrodynamics</title>
 
@@ -1603,7 +1603,11 @@ A generic configuration of static charges coupled via the Coulomb interaction
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 
 </p>
1606 
-<div class="main div" id="org27034ee">
1606 
+<div class="main div" id="org98cdfd8">
1607 
+<p>
1608 
+
1609 
+</p>
1610 
+
1607 1611 
 \begin{equation*}
1608 1612 
   {\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 1613 
 \end{equation*}
 
@@ -1611,44 +1615,53 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1611 1615 
 </div>
1612 1616 
 <p>
1613 1617 
 or (often simpler) by calculating the electrostatic potential, using either the
1614 
-explicit construction (\ref{eq:V_from_rho})
1618 
+explicit construction <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1615 1619 
 </p>
1616 
-<div class="main div" id="org1620b08">
1620 
+<div class="main div" id="org12e97ac">
1621 
+<p>
1622 
+
1623 
+</p>
1624 
+
1617 1625 
 <p>
1618 1626 
 \[
1619 
-    V({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
1620 
-    \tag{\ref{eq:V_from_rho}}
1621 
-  \]
1627 
+\phi({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \frac{\rho({\bf r}')}{|{\bf r} - {\bf r}'|}.
1628 
+\]
1622 1629 
 </p>
1623 1630
1624 1631 
 </div>
1625 1632 
 <p>
1626 1633 
 Alternately, we have also seen that the two fundamental equations for the
1627 
-electrostatic field, Gauss's law (\ref{Gr(2.14)}) and the no-perpetual-machine (vanishing curl)
1628 
-condition (\ref{Gr(2.20)}) can be expressed as the single
1629 
-'local' (differential) condition (Poisson's equation) (\ref{eq:Poisson})
1634 
+electrostatic field, Gauss's law <a href="./ems_es_ef_Gl.html#Gl_d">Gl_d</a> and the vanishing curl
1635 
+condition <a href="./ems_es_ef_cE.html#curlE0">curlE0</a> can be expressed as the single
1636 
+<i>local</i> differential condition (Poisson's equation)
1637 
+<a href="./ems_es_ep_PL.html#Poi">🐟</a>
1638 
+</p>
1639 
+
1640 
+<div class="core div" id="orgcacf446">
1641 
+<p>
1642 
+
1630 1643 
 </p>
1631 1644
1632 
-<div class="core div" id="org1361b41">
1633 1645 
 <p>
1634 1646 
 \[
1635 
-    {\boldsymbol \nabla}^2 V = -\frac{\rho}{\varepsilon_0}.
1636 
-    \tag{\ref{eq:Poisson}}
1637 
-  \]
1647 
+{\boldsymbol \nabla}^2 \phi = -\frac{\rho}{\varepsilon_0}.
1648 
+\]
1638 1649 
 </p>
1639 1650
1640 1651 
 </div>
1641 1652
1642 1653 
 <p>
1643 
-In the specific case where the charge density vanishes, we fall back onto the simpler
1644 
-Laplace equation
1654 
+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>
1645 1655 
 </p>
1646 
-<div class="core div" id="org2305746">
1656 
+<div class="core div" id="orgea7b7e8">
1657 
+<p>
1658 
+
1659 
+</p>
1660 
+
1647 1661 
 <p>
1648 1662 
 \[
1649 
-    {\boldsymbol \nabla}^2 V = 0
1650 
-    \tag{\ref{eq:Laplace}}
1651 
-  \]
1663 
+{\boldsymbol \nabla}^2 \phi = 0
1664 
+\]
1652 1665 
 </p>
1653 1666
1654 1667 
 </div>
 
@@ -1663,6 +1676,8 @@ Laplace equation
1663 1676 
 <li><a href="ems_ca_fe_uP.html">Uniqueness of Solution to Poisson's Equation</a><span class="headline-id">ems.ca.fe.uP</span></li>
1664 1677 
 </ul>
1665 1678
1679 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_L.html">The Laplace Equation&emsp;<small>[ems.ca.fe.L]</small></a></li><li>Up:&nbsp;<a href="ems_ca.html">Calculating or Approximating the Electrostatic Potential&emsp;<small>[ems.ca]</small></a></li></ul>
1680 
+<br>
1666 1681 
 <hr>
1667 1682 
 <div class="license">
1668 1683 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1676,7 +1691,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1676 1691 
 </div>
1677 1692 
 <div id="postamble" class="status">
1678 1693 
 <p class="author">Author: Jean-Sébastien Caux</p>
1679 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
1694 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1680 1695 
 <p class="validation"></p>
1681 1696 
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1682 1697

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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>
 
@@ -1598,8 +1598,10 @@ Table of contents
1598 1598 
 </svg></a><span class="headline-id">ems.ca.fe.L</span></h5>
1599 1599 
 <div class="outline-text-5" id="text-ems_ca_fe_L">
1600 1600 
 <p>
1601 
-Of course, the simplest situation is to start by looking at the region of space
1602 
-where there is no charge density.  The potential then solves Laplace's equation.  How can it possibly look ?
1601 
+In regions of space where there is no charge density,
1602 
+the potential must solve Laplace's equation.
1603 
+Let us discuss how solutions to this equation look,
1604 
+in increasingly complicated situations.
1603 1605 
 </p>
1604 1606 
 </div>
1605 1607 
 
@@ -1608,18 +1610,85 @@ where there is no charge density.  The potential then solves Laplace's equation.
1608 1610 
 <h6 id="ems_ca_fe_L_1d"><a href="#ems_ca_fe_L_1d">The Laplace Equation in One Dimension</a></h6>
1609 1611 
 <div class="outline-text-6" id="text-ems_ca_fe_L_1d">
1610 1612 
 <p>
1613 
+In one dimension, the potential is a single-variable
1614 
+function \(\phi (x)\) and the Laplace equation reads
1615 
+</p>
1616 
+
1617 
+<div class="eqlabel" id="org8b9eaa8">
1618 
+<p>
1619 
+<a id="Lap_1d"></a><a href="./ems_ca_fe_L.html#Lap_1d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1620 
+  <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 
+  <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1622 
+</svg></a>
1623 
+</p>
1624 
+<div class="alteqlabels" id="org1a79944">
1625 
+
1626 
+</div>
1627 
+
1628 
+</div>
1629 
+<p>
1630 
+\[
1631 
+\frac{d^2 \phi(x)}{dx^2} = 0.
1632 
+\tag{Lap_1d}\label{Lap_1d}
1633 
+\]
1634 
+</p>
1635 
+
1636 
+<p>
1637 
+The solution to this
1638 
+</p>
1639 
+<div class="eqlabel" id="orgd6d6f8b">
1640 
+<p>
1641 
+<a id="Lap_1d_sol"></a><a href="./ems_ca_fe_L.html#Lap_1d_sol"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1642 
+  <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="org135f906">
1647 
+<ul class="org-ul">
1648 
+<li>Gr (3.6)</li>
1649 
+</ul>
1650 
+
1651 
+</div>
1652 
+
1653 
+</div>
1654 
+<p>
1611 1655 
 \[
1612 
-\frac{d^2 V(x)}{dx^2} = 0 \Longrightarrow V(x) = mx + b
1613 
-\label{Gr(3.6)}
1656 
+\phi(x) = a x + b
1657 
+\tag{Lap_1d_sol}\label{Lap_1d_sol}
1614 1658 
 \]
1659 
+</p>
1660 
+
1661 
+<p>
1615 1662 
 Properties:
1616 
-\paragraph{1.} \(V(x)\) is the average of \(V(x + a)\) and \(V(x - a)\) for any \(a\).
1617 
-\paragraph{2.} Solutions to Laplace's equation have no local maxima or minima.
1663 
+</p>
1664 
+<ul class="org-ul">
1665 
+<li>
1666 
+<b>Balance</b>: \(\phi(x)\) is the average of \(\phi(x + dx)\) and \(\phi(x - dx)\) for any \(dx\) (with \(x \pm dx\) still being in
1667 
+the region where Laplace is satisfied, of course).</li>
1668 
+<li>
1669 
+<b>No extrema</b>: \(\phi(x)\) has no local extrema. Max/min
1670 
+values must occur at boundaries.</li>
1671 
+</ul>
1672 
+
1673 
+
1674 
+<p>
1675 
+In a particular problem, to fix the solution (said
1676 
+otherwise: to fix the parameters \(a\) and \(b\) in <a href="./ems_ca_fe_L.html#Lap_1d_sol">Lap_1d_sol</a>), we need to appeal to boundary
1677 
+conditions. Concretely, for a finite segment,
1678 
+a solution exists and is unique if one is
1679 
+provided with any of these possibilities:
1618 1680 
 </p>
1619 1681
1682 
+<ul class="org-ul">
1683 
+<li>\(\phi\) at both boundaries</li>
1684 
+<li>\(\phi\) and \(\frac{d\phi}{dx}\) at one boundary</li>
1685 
+<li>\(\phi\) at one boundary, \(\frac{d\phi}{dx}\) at the other.</li>
1686 
+</ul>
1687 
+
1620 1688 
 <p>
1621 
-Boundary conditions:  always work:  two end values, one end value + same end derivative value.
1622 
-Not always:  one end value + derivative value at other end, two end derivative values.
1689 
+Specifying \(\frac{d\phi}{dx}\) at both boundaries
1690 
+provides insufficient information, since you get
1691 
+an inconsistency if the derivatives don't match.
1623 1692 
 </p>
1624 1693 
 </div>
1625 1694 
 </div>
 
@@ -1629,16 +1698,45 @@ Not always:  one end value + derivative value at other end, two end derivative v
1629 1698 
 <h6 id="ems_ca_fe_L_2d"><a href="#ems_ca_fe_L_2d">The Laplace Equation in Two Dimensions</a></h6>
1630 1699 
 <div class="outline-text-6" id="text-ems_ca_fe_L_2d">
1631 1700 
 <p>
1632 
-\[
1633 
-\frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2} = 0.
1634 
-\]
1701 
+In two dimensions, the potential becomes a function
1702 
+of two variables (here: \(x\) and \(y\)), so Laplace's
1703 
+equation now reads
1704 
+</p>
1705 
+<div class="eqlabel" id="orgb8aa28d">
1706 
+<p>
1707 
+<a id="Lap_2d"></a><a href="./ems_ca_fe_L.html#Lap_2d"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1708 
+  <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1709 
+  <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1710 
+</svg></a>
1711 
+</p>
1712 
+<div class="alteqlabels" id="org01054fc">
1713 
+
1714 
+</div>
1715 
+
1716 
+</div>
1717 
+
1718 
+\begin{equation*}
1719 
+\frac{\partial^2 \phi (x,y)}{\partial x^2}
1720 
++ \frac{\partial^2 \phi (x,y)}{\partial y^2} = 0.
1721 
+\tag{Lap_2d}\label{Lap_2d}
1722 
+\end{equation*}
1723 
+
1724 
+<p>
1635 1725 
 Properties:
1636 
-\paragraph{1.} The value of \(V(x,y)\) equals the average value around the point:
1726 
+</p>
1727 
+<ul class="org-ul">
1728 
+<li>
1729 
+<b>Balance</b>: \(\phi(x,y)\) equals the average value around the point:</li>
1730 
+</ul>
1731 
+<p>
1637 1732 
 \[
1638 
-V(x,y) = \frac{1}{2\pi R} \oint V dl
1733 
+\phi(x,y) = \frac{1}{2\pi R} \oint dl ~\phi
1639 1734 
 \]
1640 
-\paragraph{2.} \(V\) has no local maxima or minima.  All extrema occur at the boundaries.
1641 1735 
 </p>
1736 
+<ul class="org-ul">
1737 
+<li>
1738 
+<b>No extrema</b>: \(\phi\) has no local maxima or minima.  All extrema occur at the boundaries.</li>
1739 
+</ul>
1642 1740 
 </div>
1643 1741 
 </div>
1644 1742 
 
@@ -1647,42 +1745,197 @@ V(x,y) = \frac{1}{2\pi R} \oint V dl
1647 1745 
 <h6 id="ems_ca_fe_L_3d"><a href="#ems_ca_fe_L_3d">The Laplace Equation in Three Dimensions</a></h6>
1648 1746 
 <div class="outline-text-6" id="text-ems_ca_fe_L_3d">
1649 1747 
 <p>
1748 
+In three dimensions, we will write the potential
1749 
+as a function of a 3-dimensional vector, \(\phi({\bf r})\).
1750 
+The Laplace equation is (we repeat)
1751 
+</p>
1752 
+
1753 
+<p>
1650 1754 
 \[
1651 
-{\boldsymbol \nabla}^2 V = 0
1755 
+{\boldsymbol \nabla}^2 \phi ({\bf r}) = 0
1652 1756 
 \]
1653 
-Properties:
1654 
-\paragraph{1.}  \(V({\bf r})\) is the average value of \(V\) over any spherical surface
1655 
-centered at \({\bf r}\):
1757 
+</p>
1758 
+
1759 
+<p>
1760 
+<b>Theorem</b>: if \(\phi\) satisfies Laplace, then its value at
1761 
+a point equals its value averaged over any sphere
1762 
+\(S_{\bf r}\) centered on this point,
1656 1763 
 \[
1657 
-V({\bf r}) = \frac{1}{4\pi R^2} \oint V da
1764 
+\phi({\bf r}) = \frac{1}{4\pi R^2} \oint_{S_{\bf r}} da' ~\phi ({\bf r}')
1658 1765 
 \]
1659 
-\paragraph{2.}  \(V\) can have no local maxima or minima.  All extrema occur at the boundaries.
1766 
+</p>
1767 
+
1768 
+<details id="org8c78d21">
1769 
+<summary id="org6bdc443">
1770 
+<strong>Physicist's proof</strong>
1771 
+</summary>
1772 
+<p>
1773 
+Consider a sphere of radius \(R\) centered at the origin
1774 
+carrying charge \(q\) spread with a uniform surface charge density over its surface. Bring in a point charge \(q'\) from
1775 
+infinity up to a distance \(R'\) (with \(R' &gt; R\)) from the center
1776 
+of the sphere.
1777 
+</p>
1778 
+
1779 
+<p>
1780 
+We know that the field created by the sphere coincides
1781 
+with that of a point charge \(q\) at the origin.
1782 
+Since the potential at \({\bf r = 0}\) created by the charge
1783 
+\(q'\) at \({\bf r'}\) is simply \(\phi_{q', {\bf r}'} (0) = \frac{q'}{4\pi \varepsilon_0 R'}\),
1784 
+the work
1785 
+required to bring the \(q'\) charge into position is thus
1786 
+simply \(W = q \times \phi_{q', {\bf r}'} (0) = \frac{q q'}{4 \pi \varepsilon_0 R'}\) by <a href="./ems_es_efo_e.html#Wab">Wab</a>.
1787 
+</p>
1788 
+
1789 
+<p>
1790 
+We can however proceed the other way: fixing \(q'\) in place,
1791 
+and then bringing the charged sphere into position;
1792 
+the work (energy) has to coincide with our previous result.
1793 
+But this energy is now given by the integral of the
1794 
+potential \(\phi_{q', {\bf r'}}\)
1795 
+created by \(q'\) (sitting at \({\bf r'}\)) over the sphere
1796 
+times the surface charge density on the sphere,
1797 
+namely
1660 1798 
 </p>
1661 1799
1662 1800 
 <p>
1663 
-\paragraph{Another way of seeing this} is to write the second derivatives as
1664 1801 
 \[
1665 
-\frac{\partial^2 V({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
1666 
-\frac{\partial^2 V({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
1667 
-\frac{\partial^2 V({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
1802 
+W = \oint_{S_R} da ~\sigma ~\phi_{q', {\bf r}'} ({\bf r})
1803 
+\]
1804 
+</p>
1805 
+
1806 
+<p>
1807 
+But \(\sigma = q/4\pi R^2\) and is a constant over the
1808 
+sphere, so \(W = q \times \frac{1}{4\pi R^2} \oint da ~\phi_{q', {\bf r}'} ({\bf r})\).
1809 
+</p>
1810 
+
1811 
+<p>
1812 
+Equating this with the previous results shows that
1813 
+</p>
1814 
+
1815 
+<p>
1816 
+\[
1817 
+\phi_{q', {\bf r'}} (0) = \frac{1}{4\pi R^2} \oint_{S_R} da ~\phi_{q', {\bf r}'} ({\bf r})
1818 
+\]
1819 
+</p>
1820 
+
1821 
+<p>
1822 
+namely that for the potential created by a single point
1823 
+charge \(q'\) at \(R'\),
1824 
+the value at a point (here the origin)
1825 
+coincides with the value averaged over a sphere
1826 
+or an arbitrary radius \(R\) centered on the same point.
1827 
+</p>
1828 
+
1829 
+<p>
1830 
+By the principle of superposition, this works for an
1831 
+arbitrary distribution of charges outside the sphere,
1832 
+proving the theorem.
1833 
+</p>
1834 
+</details>
1835 
+
1836 
+<details id="orgff2611b">
1837 
+<summary id="org89a3b1b">
1838 
+<strong>Formal proof</strong>
1839 
+</summary>
1840 
+
1841 
+<p>
1842 
+Consider a function \(f({\bf r})\) and its average over
1843 
+a ball of radius \(R\) centered on \({\bf r}\):
1844 
+</p>
1845 
+
1846 
+<p>
1847 
+\[
1848 
+f_{S_R} ({\bf r}) \equiv \frac{1}{4\pi R^2}\oint_{S_R} da' ~ f ({\bf r} + {\bf r}')
1849 
+\]
1850 
+</p>
1851 
+
1852 
+<p>
1853 
+For convenience we will hereafter put \({\bf r} = 0\).
1854 
+In spherical coordinates, we have \(da' = R^2 sin \theta d\theta d\phi \equiv R^2 d\Omega\).
1855 
+Differentiating with respect to \(R\),
1856 
+</p>
1857 
+
1858 
+<p>
1859 
+\[
1860 
+\frac{d}{dR} f_{S_R} = \frac{1}{4\pi} \oint_{S_R} d\Omega ~\left.\frac{\partial f}{\partial r}\right|_{r=R}
1861 
+\]
1862 
+</p>
1863 
+
1864 
+<p>
1865 
+with \(f\) differentiated with respect to the radial coordiate.
1866 
+We can rewrite this by noting that \(R^2 d\Omega \hat{\bf r}\)
1867 
+is the normal differential surface area \(d{\bf a}\), while
1868 
+\(\left.\frac{\partial f}{\partial r}\right|_{r=R}\) is the radial component of the gradient
1869 
+of \(f\) in spherical coordinates. Thus,
1870 
+</p>
1871 
+
1872 
+<p>
1873 
+\[
1874 
+\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \oint_{S_R} d{\bf a} \cdot ~\nabla f
1875 
+\]
1876 
+</p>
1877 
+
1878 
+<p>
1879 
+Invoking the divergence theorem and using the definition
1880 
+of the Laplacian operator \(\nabla^2 = \nabla \cdot \nabla\),
1881 
+we get the following general
1882 
+</p>
1883 
+
1884 
+<p>
1885 
+<b>Theorem</b>:
1886 
+</p>
1887 
+
1888 
+<p>
1889 
+\[
1890 
+\frac{d}{dR} f_{S_R} = \frac{1}{4\pi R^2} \int_{V_R} d\tau ~\nabla^2 f
1891 
+\]
1892 
+</p>
1893 
+
1894 
+
1895 
+<p>
1896 
+For the electrostatic potential away from charges, we have
1897 
+\[
1898 
+\nabla^2 \phi = 0 ~\rightarrow \frac{d}{dR} \phi_{S_R} = 0
1899 
+\]
1900 
+namely the ball average is independent of the ball size.
1901 
+Since the value at the center is simply the average for
1902 
+an infinitesimally small ball, we get the result announced above.
1903 
+</p>
1904 
+</details>
1905 
+
1906 
+<p>
1907 
+<b>Theorem (Earnshaw, mathematical versoin)</b>: \(\phi\) has no local extrema except at the boundaries.
1908 
+</p>
1909 
+
1910 
+<p>
1911 
+<b>Proof</b>: write the second derivatives as
1912 
+</p>
1913 
+
1914 
+<p>
1915 
+\[
1916 
+\frac{\partial^2 \phi({\bf r})}{\partial x^2} = f_x ({\bf r}), \hspace{5mm}
1917 
+\frac{\partial^2 \phi({\bf r})}{\partial y^2} = f_y ({\bf r}), \hspace{5mm}
1918 
+\frac{\partial^2 \phi({\bf r})}{\partial z^2} = f_z ({\bf r}), \hspace{5mm}
1668 1919 
 f_x + f_y + f_z = 0.
1669 1920 
 \]
1670 
-The \(f_a ({\bf r})\) represent the three components of the curvature of \(V({\bf r})\).
1671 
-An extremum of \(V\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} V |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
1921 
+</p>
1922 
+
1923 
+<p>
1924 
+The \(f_a ({\bf r})\) represent the three components of the curvature of \(\phi({\bf r})\).
1925 
+An extremum of \(\phi\) at \({\bf r}_e\) would be characterized by \({\boldsymbol \nabla} \phi |_{{\bf r}_e} \cdot \delta{\bf r} = 0\)
1672 1926 
 for any infinitesimal displacement \(\delta{\bf r}\) around the extremum point.  For a local
1673 
-minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 V}{\partial r_i \partial r_j} \delta r_i \delta r_j &gt; 0\)
1927 
+minimum, the second derivative form should be greater than zero, \(\sum_{i,j} \frac{\partial^2 \phi}{\partial r_i \partial r_j} \delta r_i \delta r_j &gt; 0\)
1674 1928 
 for any displacement vector.  Choosing alternately displacements along the three axes,
1675 1929 
 the form becomes \(f_x (\delta x)^2\), \(f_y (\delta y)^2\) or \(f_z (\delta z)^2\).  Since the squared displacements
1676 1930 
 are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f_z &gt; 0\).  This is impossible in view
1677 1931 
 of the \(f_x + f_y + f_z = 0\) condition above.
1678 1932 
 </p>
1679 1933
1680 
-<div class="info div" id="orge499be0">
1934 
+<div class="info div" id="org950ce9d">
1681 1935 
 <p>
1682 
-<b>Earnshaw's theorem</b> <br>
1683 
-Since solutions to Laplace's equation have no local minimum,
1684 
-it is impossible to find a static distribution of charges which generates an electrostatic field
1685 
-with a stable equilibrium position for a test charge.
1936 
+<b>Earnshaw's theorem (physical version)</b> <br>
1937 
+It is impossible to find a static distribution of charges which generates an electrostatic field
1938 
+displaying a stable equilibrium position in empty space.
1686 1939 
 </p>
1687 1940
1688 1941 
 </div>
 
@@ -1692,47 +1945,56 @@ Going back to Poisson's equation, we can make a few comments:
1692 1945 
 </p>
1693 1946
1694 1947 
 <ul class="org-ul">
1695 
-<li>representation (\ref{eq:Poisson}) highlights the 'local' nature of the coupling between electrostatic fields and charges:  fields are 'created' where the charges 'sit'.  This is also seen by looking at the integrand of (\ref{eq:V_from_rho}). If electrostatics was nonlocal, a modified representation like (\ref{eq:V_from_rho}) would still exist, but not a local differential one like (\ref{eq:Poisson}).</li>
1948 
+<li>representation <a href="./ems_es_ep_PL.html#Poi">🐟</a> highlights the 'local' nature of the coupling between electrostatic fields and charges:  fields are 'created' where the charges 'sit'.  This is also seen by looking at the integrand of <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>. If electrostatics was nonlocal, a modified representation like <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> would still exist, but not a local differential one like Poisson's equation.</li>
1696 1949
1697 
-<li>as written, representations (\ref{eq:E_from_rho}) and (\ref{eq:V_from_rho}) require the knowledge of the charge density distribution \(\rho({\bf r})\) throughout space to determine the potential at any given point.</li>
1950 
+<li>as written, representations <a href="./ems_es_ef_ccd.html#E_vcd">E_vcd</a> and <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a> require the knowledge of the charge density distribution \(\rho({\bf r})\) throughout space to determine the potential at any given point.</li>
1698 1951
1699 
-<li>(\ref{eq:Poisson}), being purely local, might allow to determine the potential at a specified point, provided we know the charge density distribution around this specified point, and at some set of other reference points (to make the solution unique).</li>
1952 
+<li>Poisson's equation <a href="./ems_es_ep_PL.html#Poi">🐟</a>, being purely local, might allow to determine the potential at a specified point, provided we know the charge density distribution around this specified point, and at some set of other reference points (to make the solution unique).</li>
1700 1953 
 </ul>
1701 1954
1702 1955
1703 1956 
 <p>
1704 
-We therefore want to ask the question:  <b>under what conditions can an electrostatic problem be fully
1705 
-defined by solving Poisson's equation ?</b>  We start by mentioning some cases, and interpreting them thereafter.
1957 
+We therefore want to ask the question:  <i>under what conditions can an electrostatic problem be fully
1958 
+defined by solving Poisson's equation ?</i>
1959 
+</p>
1960 
+
1961 
+<p>
1962 
+We start by mentioning some cases, and interpreting them thereafter.
1963 
+</p>
1964 
+
1965 
+<p>
1966 
+<b>Charge density is known throughout space</b>: in this case,
1967 
+the electrostatic potential is uniquely determined
1968 
+by Poisson's equation, which
1969 
+is explicitly solved by <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>.
1970 
+One can eplicitly verify this:
1706 1971 
 </p>
1707 1972
1708 1973 
 <p>
1709 
-<b>First case</b>:  the electrostatic potential is uniquely determined
1710 
-if \(\rho({\bf r})\) is given throughout all space.  In this case, Poisson's equation
1711 
-is explicitly solved by (\ref{eq:V_from_rho}).  Explicit check:
1712 1974 
 \[
1713 1975 
 {\boldsymbol \nabla}^2 V ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
1714 1976 
 = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}.
1715 1977 
 \]
1716 
-where we have used (\ref{Gr(1.102)}), and the fact that the delta function is always resolved since we
1717 
-integrate over all space.  Note:  it is implicitly assumed that the integral in (\ref{eq:V_from_rho})
1978 
+</p>
1979 
+
1980 
+<p>
1981 
+where we have used <a href="./c_m_dd_3d.html#Lap1or">Lap1or</a>, and the fact that the delta function is always resolved since we
1982 
+integrate over all space.  Note:  it is implicitly assumed that the integral in <a href="./ems_es_ep_d.html#p_vcd">p_vcd</a>
1718 1983 
 converges, <i>i.e.</i> that the charge density \(\rho({\bf r})\) is sufficiently well-behaved (does not
1719 1984 
 become singular).
1720 1985 
 </p>
1721 1986
1722 1987 
 <p>
1723 
-<b>First case (corollary)</b>:  the electrostatic potential is uniquely determined
1988 
+<b>Charge density in closed volume, and boundary surface charge density are known</b>:  the electrostatic potential is uniquely determined
1724 1989 
 in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
1725 1990 
 \(\rho ({\bf x})\) is given everywhere within \({\cal V}\), vanishes outside of \({\cal V}\),
1726 1991 
 and the value of the surface charge density \(\sigma\) is given everywhere on the boundary \({\cal S}\).
1727 1992 
 Of course, \({\cal S}\) need not be a connected surface.
1728 1993 
 </p>
1729 1994
1730 
-<p>
1731 
-This is obvious:  we know where all the charges are, so this is really the same as the first case.
1732 
-</p>
1733 1995
1734 1996 
 <p>
1735 
-<b>Second case</b>:  the electrostatic potential is uniquely determined
1997 
+<b>Charge density in closed volume, and potential at boundary are known</b>:  the electrostatic potential is uniquely determined
1736 1998 
 in a certain volume \({\cal V}\) bounded by boundary \({\cal S}\), provided the charge density
1737 1999 
 \(\rho ({\bf x})\) is given everywhere within \({\cal V}\), and the value of \(V\) is given everywhere on the
1738 2000 
 boundary \({\cal S}\).  Of course, \({\cal S}\) need not be a connected surface.
 
@@ -1748,14 +2010,20 @@ to obtain \(V\) within \({\cal V}\).
1748 2010 
 Given a solution \(V_1 ({\bf r})\), we can easily show that it is unique.  Suppose there was another solution
1749 2011 
 \(V_2 ({\bf r})\).  Look at the difference, \(U \equiv V_1 - V_2\).  In the bulk, \(U\) obeys the Laplace
1750 2012 
 equation
2013 
+</p>
2014 
+
2015 
+<p>
1751 2016 
 \[
1752 2017 
 {\boldsymbol \nabla}^2 U = {\boldsymbol \nabla}^2 V_1 - {\boldsymbol \nabla}^2 V_2 = -\frac{\rho}{\varepsilon_0} + \frac{\rho}{\varepsilon_0} = 0.
1753 2018 
 \]
2019 
+</p>
2020 
+
2021 
+<p>
1754 2022 
 Moreover, \(U ({\bf r}) = 0\) for \({\bf r} \in {\cal S}\).  Since solutions to the Laplace equation take
1755 2023 
 their maximal and minimal value on the boundary, we must have \(U = 0\) \(\forall {\bf r} \in {\cal V}\)
1756 
-(Griffiths' proof).
1757 2024 
 </p>
1758 2025
2026 
+
1759 2027 
 <p>
1760 2028 
 This all feels a bit amateurish and not very systematic. Can we be more precise and general?  What kinds of boundary information do we really need to specify the solution uniquely ?
1761 2029 
 </p>
 
@@ -1765,6 +2033,8 @@ This all feels a bit amateurish and not very systematic. Can we be more precise
1765 2033
1766 2034
1767 2035
2036 
+<br><ul class="navigation-links"><li>Prev:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li><li>Next:&nbsp;<a href="ems_ca_fe_g.html">Green's Identities&emsp;<small>[ems.ca.fe.g]</small></a></li><li>Up:&nbsp;<a href="ems_ca_fe.html">Fundamental Equations for the Electrostatic Potential&emsp;<small>[ems.ca.fe]</small></a></li></ul>
2037 
+<br>
1768 2038 
 <hr>
1769 2039 
 <div class="license">
1770 2040 
 <a rel="license noopener" href="https://creativecommons.org/licenses/by/4.0/"
 
@@ -1778,7 +2048,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1778 2048 
 </div>
1779 2049 
 <div id="postamble" class="status">
1780 2050 
 <p class="author">Author: Jean-Sébastien Caux</p>
1781 
-<p class="date">Created: 2022-02-10 Thu 08:32</p>
2051 
+<p class="date">Created: 2022-02-13 Sun 21:20</p>
1782 2052 
 <p class="validation"></p>
1783 2053 
 </div>
1784 2054

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- 0
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