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Update 2022-02-21 20:42

master
Jean-Sébastien 2 years ago
parent
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723 736
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1620 1649
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708 708
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715
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733
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723 736
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725 738
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729 742
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730
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783
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785
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788
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789
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793
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1634 1663
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 Infinitesimal volume element:
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-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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720
-<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
733
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723 736
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739
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-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
743
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1621 1650
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1622 1651
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1623 1652
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1624
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714
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715
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717
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718
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-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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719 732
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720
-<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
733
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723 736
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724 737
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725 738
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726
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739
+<a href="./emsm_esm_po_pibc.html#emsm_esm_po_pibc">Physical Interpretation of Bound Charges</a><span class="headline-id">emsm.esm.po.pibc</span>
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730
-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
743
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789
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791
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792
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793
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1623 1652
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1624 1653
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1625 1654
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1626
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717
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711 720
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-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
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782
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783
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789
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1596 1625
 
1597
-<div id="outline-container-org1a142a0" class="outline-6">
1598
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1626
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1600 1629
 <p>
1601 1630
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1602 1631
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
@@ -1605,40 +1634,52 @@ given by the Laplacian of the corresponding vector elements.
1605 1634
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1607 1636
 
1608
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1611 1640
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1612 1641
 This always vanishes.
1613 1642
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1617
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1618
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1620 1649
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1621 1650
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1622 1651
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1623 1652
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1624 1653
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1625 1654
 
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 This always vanishes.
1631 1660
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1635
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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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718 731
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725 738
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1643 1672
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1645 1674
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789
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793
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765 794
 
766 795
 </li>
767 796
 
@@ -1601,14 +1630,14 @@ explicited as follows:
1601 1630
 <p>
1602 1631
 <b>Gradient of a product</b>:
1603 1632
 </p>
1604
-<div class="eqlabel" id="org77098c1">
1633
+<div class="eqlabel" id="orgfc34aae">
1605 1634
 <p>
1606 1635
 <a id="grad_prod"></a><a href="./c_m_dc_pr.html#grad_prod"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1607 1636
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1608 1637
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1609 1638
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1610 1639
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1611
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1640
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1612 1641
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1613 1642
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1614 1643
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@@ -1628,14 +1657,14 @@ explicited as follows:
1628 1657
 <p>
1629 1658
 <b>Gradient of a scalar product</b>:
1630 1659
 </p>
1631
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1660
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1632 1661
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1633 1662
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1634 1663
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1635 1664
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1636 1665
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1637 1666
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1638
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1667
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1639 1668
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1640 1669
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1641 1670
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@@ -1655,14 +1684,14 @@ explicited as follows:
1655 1684
 <p>
1656 1685
 <b>Divergence of a product</b>:
1657 1686
 </p>
1658
-<div class="eqlabel" id="org27a4a20">
1687
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1659 1688
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1661 1690
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1662 1691
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1663 1692
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1664 1693
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1694
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1666 1695
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1667 1696
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1668 1697
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@@ -1682,14 +1711,14 @@ explicited as follows:
1682 1711
 <p>
1683 1712
 <b>Divergence of a cross product</b>:
1684 1713
 </p>
1685
-<div class="eqlabel" id="orgaa0862c">
1714
+<div class="eqlabel" id="orgac85f09">
1686 1715
 <p>
1687 1716
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1688 1717
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1689 1718
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1690 1719
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1691 1720
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1721
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1693 1722
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1694 1723
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1695 1724
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@@ -1709,14 +1738,14 @@ explicited as follows:
1709 1738
 <p>
1710 1739
 <b>Curl of a product</b>:
1711 1740
 </p>
1712
-<div class="eqlabel" id="org8a2530a">
1741
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1713 1742
 <p>
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1717 1746
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1748
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1720 1749
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1721 1750
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1722 1751
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@@ -1736,14 +1765,14 @@ explicited as follows:
1736 1765
 <p>
1737 1766
 <b>Curl of a cross product</b>:
1738 1767
 </p>
1739
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1768
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1744 1773
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1745 1774
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1775
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1747 1776
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1786 1815
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715
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718
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711 720
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713
-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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715 724
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726
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722 735
 
723 736
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725 738
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728 741
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766 795
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1618 1647
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1619 1648
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1620 1649
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1621
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1623 1652
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715
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718
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711 720
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712 721
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713
-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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714 723
 
715 724
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726
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717 730
 
718 731
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733
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722 735
 
723 736
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727 740
 
728 741
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730
-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
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731 744
 
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793
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765 794
 
766 795
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1630 1659
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1631 1660
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1632 1661
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1633
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1635 1664
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708 708
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709
-<a href="./emsm_esm_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
709
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714
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715
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717
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718
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710 719
 
711 720
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712 721
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713
-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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714 723
 
715 724
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717 730
 
718 731
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719 732
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-<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
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@@ -752,16 +765,32 @@ Table of contents
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782
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765 794
 
766 795
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767 796
 
@@ -1615,14 +1644,14 @@ Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:
1615 1644
 More generally,
1616 1645
 </p>
1617 1646
 
1618
-<div class="eqlabel" id="org42fce11">
1647
+<div class="eqlabel" id="org0433d75">
1619 1648
 <p>
1620 1649
 <a id="divdel"></a><a href="./c_m_dd_3d.html#divdel"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1621 1650
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1623 1652
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1624 1653
 </p>
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-<div class="alteqlabels" id="org7526e1a">
1654
+<div class="alteqlabels" id="org68675a6">
1626 1655
 <ul class="org-ul">
1627 1656
 <li>Gr (1.100)</li>
1628 1657
 </ul>
@@ -1641,14 +1670,14 @@ More generally,
1641 1670
 Since
1642 1671
 </p>
1643 1672
 
1644
-<div class="eqlabel" id="org204d0c0">
1673
+<div class="eqlabel" id="orgd02de7a">
1645 1674
 <p>
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 <a id="div1or"></a><a href="./c_m_dd_3d.html#div1or"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1647 1676
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1648 1677
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1649 1678
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1650 1679
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1680
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1652 1681
 <ul class="org-ul">
1653 1682
 <li>Gr (1.101)</li>
1654 1683
 </ul>
@@ -1664,14 +1693,14 @@ Since
1664 1693
 <p>
1665 1694
 we have that
1666 1695
 </p>
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1672 1701
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1673 1702
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1703
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1675 1704
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1676 1705
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1677 1706
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1703 1732
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1704 1733
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1705 1734
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1706
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1735
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1709 1738
 

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708 708
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709
-<a href="./emsm_esm_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
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+
714
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715
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717
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718
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711 720
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-<a href="./emsm_esm_di.html#emsm_esm_di">Dielectrics</a><span class="headline-id">emsm.esm.di</span>
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714 723
 
715 724
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726
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727
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717 730
 
718 731
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720
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733
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721 734
 
722 735
 
723 736
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725 738
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726
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728 741
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729 742
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730
-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
743
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766 795
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1636 1665
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1637 1666
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1638 1667
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1639
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1668
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7 7
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706 706
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707 707
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708 708
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709
-<a href="./emsm_esm_p.html#emsm_esm_p">Polarization</a><span class="headline-id">emsm.esm.p</span>
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714
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715
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717
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718
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710 719
 
711 720
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712 721
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713
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722
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714 723
 
715 724
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726
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727
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718 731
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719 732
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720
-<a href="./emsm_esm_fpo.html#emsm_esm_fpo">The Field of a Polarized Object</a><span class="headline-id">emsm.esm.fpo</span>
733
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721 734
 
722 735
 
723 736
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725 738
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726
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728 741
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729 742
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730
-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
743
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1621 1650
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1623 1652
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733
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718 731
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722 735
 
723 736
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723 736
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722 735
 
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@@ -1655,7 +1684,7 @@ to the rate of change of the magnetic flux,
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 \]
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 We can integrate over all space:  after neglecting boundary terms (assuming fields fall to zero at infinity), we are left with
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@@ -1673,7 +1702,7 @@ W_{mag} = \frac{1}{2} \int d\tau ({\bf A} \cdot {\bf J}) = \frac{1}{2\mu_0} \int
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 \hspace{2cm} \mbox{(7.31 and 7.34)}
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 \end{align}
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 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1679 1708
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1674 1703
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1675 1704
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1677
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 This expression can be greatly simplified by introducing the
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 {\bf Maxwell stress tensor}
1655 1684
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 We then obtain
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 {\bf EM force per unit volume}
1678 1707
 \[
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1684 1713
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 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
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 {\bf Total force on charges in volume}
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1661 1690
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 {\bf Poynting's theorem}
1667 1696
 \[
@@ -1686,7 +1715,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1686 1715
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1687 1716
 Energy per unit time, per unit area carried by EM fields:
1688 1717
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 {\bf Poynting vector}
1692 1721
 \[
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1699 1728
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1700 1729
 We can thus express Poynting's theorem more compactly:
1701 1730
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 {\bf Poynting's theorem}
1705 1734
 \[
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1712 1741
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1713 1742
 where we have defined the total
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 {\bf Energy in electromagnetic fields}
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 \[
@@ -1735,7 +1764,7 @@ Then,
1735 1764
 \]
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 so we get the
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 {\bf Poynting theorem (differential form)}
1741 1770
 \[
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 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1758 1787
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@@ -1797,7 +1826,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
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1647 1676
 <p class="author">Author: Jean-Sébastien Caux</p>
1648
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 \]
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 {\bf Poynting vector of a monochromatic EM wave}
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 Similary, we get the
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 {\bf Momentum density of a monochromatic EM wave}
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+<a href="./emsm_esm_mE.html#emsm_esm_mE">Matter Bathed in E Fields; Polarization</a><span class="headline-id">emsm.esm.mE</span>
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 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
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 \]
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1634 1663
 \[
@@ -1646,7 +1675,7 @@ the polarization current is the result of linear motion of charge when
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1647 1676
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 Note the unfortunate labelling: it would have been nicer to have \(\rho_b\) be the charge associated to current
1652 1681
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@@ -1669,7 +1698,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
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 {\it free} and {\it bound}:
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 {\it polarization}:
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@@ -1706,7 +1735,7 @@ Gauss's law:  can be rewritten
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 {\bf Maxwell's equations {\it (in matter)}}
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1777 1806
 For the restricted case of linear media:
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1815 1844
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1595 1624
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 {\bf Maxwell's equations {\it (in matter)}, integral form}
1601 1630
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1686 1715
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1687 1716
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1688 1717
 <p class="author">Author: Jean-Sébastien Caux</p>
1689
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711 720
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715 724
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718 731
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1630 1659
 \]
1631 1660
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1632 1661
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718 731
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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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 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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1723 1752
 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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1775 1804
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+
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1614 1643
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1615
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-<a href="./emsm_esm_fpo_fid.html#emsm_esm_fpo_fid">The Field Inside a Dielectric</a><span class="headline-id">emsm.esm.fpo.fid</span>
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1621
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 \[
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@@ -1638,7 +1667,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
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@@ -1654,7 +1683,7 @@ whereas Amp{\`ere}-Maxwell becomes
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 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
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+<a href="./emsm_esm_ld_bvp.html#emsm_esm_ld_bvp">Boundary Value Problems with Linear Dielectrics</a><span class="headline-id">emsm.esm.ld.bvp</span>
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@@ -1609,14 +1638,14 @@ In one dimension, the potential is a single-variable
1609 1638
 function \(\phi (x)\) and the Laplace equation reads
1610 1639
 </p>
1611 1640
 
1612
-<div class="eqlabel" id="org725867f">
1641
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1613 1642
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1614 1643
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1615 1644
   <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"/>
1616 1645
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1617 1646
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1618 1647
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1619
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1620 1649
 
1621 1650
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1622 1651
 
@@ -1631,14 +1660,14 @@ function \(\phi (x)\) and the Laplace equation reads
1631 1660
 <p>
1632 1661
 The solution to this is
1633 1662
 </p>
1634
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1635 1664
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1636 1665
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1637 1666
   <path d="M6.354 5.5H4a3 3 0 0 0 0 6h3a3 3 0 0 0 2.83-4H9c-.086 0-.17.01-.25.031A2 2 0 0 1 7 10.5H4a2 2 0 1 1 0-4h1.535c.218-.376.495-.714.82-1z"/>
1638 1667
   <path d="M9 5.5a3 3 0 0 0-2.83 4h1.098A2 2 0 0 1 9 6.5h3a2 2 0 1 1 0 4h-1.535a4.02 4.02 0 0 1-.82 1H12a3 3 0 1 0 0-6H9z"/>
1639 1668
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1640 1669
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1641
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 </ul>
@@ -1697,14 +1726,14 @@ In two dimensions, the potential becomes a function
1697 1726
 of two variables (here: \(x\) and \(y\)), so Laplace's
1698 1727
 equation now reads
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1704 1733
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1705 1734
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1707
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1708 1737
 
1709 1738
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1710 1739
 
@@ -1757,14 +1786,14 @@ a point equals its value averaged over a sphere
1757 1786
 \(S_R({\bf r})\) of any radius \(R\) centered on this point
1758 1787
 (and of course not containing any charges),
1759 1788
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1762 1791
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1764 1793
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1765 1794
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1766 1795
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1767
-<div class="alteqlabels" id="orgce61baf">
1796
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1768 1797
 
1769 1798
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@@ -1776,8 +1805,8 @@ a point equals its value averaged over a sphere
1776 1805
 \]
1777 1806
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1778 1807
 
1779
-<details id="orge7969c9">
1780
-<summary id="org65097bb">
1808
+<details id="org6ed4b45">
1809
+<summary id="org088d2fd">
1781 1810
 <strong>Physicist's proof</strong>
1782 1811
 </summary>
1783 1812
 <p>
@@ -1839,8 +1868,8 @@ proving the theorem.
1839 1868
 </p>
1840 1869
 </details>
1841 1870
 
1842
-<details id="orgf16a408">
1843
-<summary id="orgdc4e491">
1871
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+<summary id="orgf937e13">
1844 1873
 <strong>Formal proof</strong>
1845 1874
 </summary>
1846 1875
 
@@ -1890,14 +1919,14 @@ we get the following general
1890 1919
 <p>
1891 1920
 <b>Theorem</b>:
1892 1921
 </p>
1893
-<div class="eqlabel" id="orgb167571">
1922
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1894 1923
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1895 1924
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1898 1927
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1899 1928
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1900
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1901 1930
 
1902 1931
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1903 1932
 
@@ -1950,19 +1979,19 @@ are necessarily positive, we thus require \(f_x &gt; 0\), \(f_y &gt; 0\) and \(f
1950 1979
 of the \(f_x + f_y + f_z = 0\) condition above.
1951 1980
 </p>
1952 1981
 
1953
-<div class="eqlabel" id="org368f926">
1982
+<div class="eqlabel" id="org3cd4c43">
1954 1983
 <p>
1955 1984
 <a id="Earnshaw"></a><a href="./ems_ca_fe_L.html#Earnshaw"><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" fill="currentColor" class="bi bi-link" viewBox="0 0 16 16">
1956 1985
   <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"/>
1957 1986
   <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"/>
1958 1987
 </svg></a>
1959 1988
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1960
-<div class="alteqlabels" id="orgd8486b2">
1989
+<div class="alteqlabels" id="org61c0e88">
1961 1990
 
1962 1991
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1963 1992
 
1964 1993
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1965
-<div class="info div" id="org74af865">
1994
+<div class="info div" id="org9b11c3d">
1966 1995
 <p>
1967 1996
 <b>Earnshaw's theorem (physical version)</b> <br>
1968 1997
 </p>
@@ -2006,7 +2035,7 @@ One can explicitly verify this:
2006 2035
 
2007 2036
 <p>
2008 2037
 \[
2009
-{\boldsymbol \nabla}^2 V ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
2038
+{\boldsymbol \nabla}^2 \phi ({\bf r}) = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') {\boldsymbol \nabla}^2 \frac{1}{|{\bf r} - {\bf r}'|}
2010 2039
 = \frac{1}{4\pi \varepsilon_0} \int_{\mathbb{R}^3} d\tau' (-4\pi) \delta ({\bf r} - {\bf r}') = -\frac{\rho ({\bf r})}{\varepsilon_0}.
2011 2040
 \]
2012 2041
 </p>
@@ -2081,7 +2110,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
2081 2110
 </div>
2082 2111
 <div id="postamble" class="status">
2083 2112
 <p class="author">Author: Jean-Sébastien Caux</p>
2084
-<p class="date">Created: 2022-02-21 Mon 10:33</p>
2113
+<p class="date">Created: 2022-02-21 Mon 20:41</p>
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2086 2115
 </div>
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