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Update 2022-02-09 22:41

master
Jean-Sébastien 2 years ago
parent
3c40f5bfe8
commit
f3c2446d19
100 changed files with 427 additions and 2887 deletions
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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 <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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 </div>
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 <div id="postamble" class="status">
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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 <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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 <p class="author">Author: Jean-Sébastien Caux</p>
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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 <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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 <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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417 413 
 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
418 414
419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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424 420 
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437 430
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1642 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1753 1728 
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1754 1729 
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1755 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
417 
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1650 1625 
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1651 1626 
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1652 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1654 
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 <p>
1631 1606 
 \({\boldsymbol \nabla} \cdot ({\boldsymbol \nabla} T) \equiv {\boldsymbol \nabla}^2 T\) is called the <b>Laplacian</b> of the scalar field \(T\).
1632 1607 
 The Laplacian of a vector field \({\boldsymbol \nabla}^2 {\bf v}\) is also defined as the vector with components
 
@@ -1635,36 +1610,36 @@ given by the Laplacian of the corresponding vector elements.
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 This always vanishes.
1643 1618 
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1646 1621
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1650 1625 
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1651 1626 
 \({\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\bf v})\) does not appear often in physics.  No special name.
1652 1627 
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1653 1628 
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1654 1629 
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1655 1630
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 This always vanishes.
1661 1636 
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1662 1637 
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1664 1639
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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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1690 1665 
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1692 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1653 
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-<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1648 1623 
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1649 1624 
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1650 
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-<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1672 1647 
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1673 1648 
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1674 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1647 
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1649 
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
418 414
419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
417 
+<a href="./ems_es_ep_PL.html#ems_es_ep_PL">Poisson's and Laplace's Equations</a><span class="headline-id">ems.es.ep.PL</span>
422 418
423 419 
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424 420 
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436 429 
 <a href="./ems_es_e.html#ems_es_e">Electrostatic Energy from the Potential</a><span class="headline-id">ems.es.e</span>
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-<a href="./ems_es_e_c.html#ems_es_e_c">Comments on Electrostatic Energy</a><span class="headline-id">ems.es.e.c</span>
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458 433 
 
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1645 1620 
 More generally,
1646 1621 
 </p>
1647 1622
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1651 1626 
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1652 1627 
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1653 1628 
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1657 1632 
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1658 1633 
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@@ -1671,9 +1646,24 @@ More generally,
1671 1646 
 Since
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1653 
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1674 1664 
 \begin{equation}
1675 1665 
 {\boldsymbol \nabla}_1 \left(\frac{1}{r_{12}}\right) = -\frac{\hat{\bf r}_{12}}{r_{12}^2}
1676 
-\label{Gr(1.101)}
1666 
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1677 1667 
 \end{equation}
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1701 1691 
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1702 1692 
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1703 1693 
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1704 
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-<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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417 413 
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418 414
419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
417 
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1664 1639 
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1665 1640 
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1666 1641 
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1667 
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1642 
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1669 1644 
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-<a href="./ems_es_ep_c.html#ems_es_ep_c">Comments on the Electrostatic Potential</a><span class="headline-id">ems.es.ep.c</span>
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418 414
419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1649 1624 
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1650 1625 
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1651 1626 
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1652 
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1627 
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
418 414
419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
417 
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422 418
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1648 1623 
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1649 1624 
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1650 
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1625 
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419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
417 
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1657 1632 
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1658 1633 
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1659 
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419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1652 1627 
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1653 
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1720 1695 
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1721 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1674 1649 
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421 
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421 
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1674 1649 
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421 
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1664 
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1668 1643 
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1666 1641 
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421 
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1675 1650 
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1676 1651 
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1677 
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421 
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1675 1650 
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1676 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1648 1623 
 <div id="postamble" class="status">
1649 1624 
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1650 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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 <b>Electromotive force (or electromotance)</b>,
1669 1644 
   \[
 
@@ -1685,7 +1660,7 @@ to the rate of change of the magnetic flux,
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 so we obtain
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 <b>Faraday's law</b> (integral form <i>N.B.: for a stationary loop</i>)
1691 1666 
   \[
 
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 <b>Faraday's law</b> (differential form)
1709 1684 
   \[
 
@@ -1738,7 +1713,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1738 1713 
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1739 1714 
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1740 1715 
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1741 
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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
1684 1659 
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     W_{mag} = \frac{1}{2\mu_0} \int d\tau B^2
 
@@ -1703,7 +1678,7 @@ 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)}
1704 1679 
 \end{align}
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1708 1683 
 \paragraph{Example 7.13:}  coaxial cable (inner cylinder radius \(a\), outer \(b\)) carries current \(I\).
1709 1684 
 Find energy stored in section of length \(l\).
 
@@ -1740,7 +1715,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1740 1715 
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1741 1716 
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1742 1717 
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1743 
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1662 1637 
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1667 1642 
 \paragraph{Example 7.10:}
1668 1643 
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@@ -1712,7 +1687,7 @@ Inductance:  measured in {\bf henries} (\(H\)).  \(H = V s/A\).
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1717 1692 
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1718 1693 
 rectangular cross-section (inner radius \(a\), outer radius \(b\), height \(h\))
 
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1740 1715 
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1745 1720 
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1779 
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1646 1621
1647 1622
1648 1623
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1624 
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1651 1626 
 {\bf Example 7.7:}
1652 1627 
 \({\bf B}(t)\) points up in circular region of radius \(R\).  What is the induced \({\bf E}(t)\) ?
 
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1662 1637 
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1664 1639
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1667 1642 
 {\bf Example 7.8:} wheel or radius \(b\) with line charge \(\lambda\) on the rim.
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 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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 Full set of equations for the electromagnetic field:
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1631 1606 
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1647 1622 
 \[
 
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1665 1640 
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 Better way of writing:  all fields on left, all sources on right,
1667 1642 
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 \begin{align}
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1671 1646 
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1694 1669 
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1696 
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1633 1608 
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1638 1613 
     {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t}
 
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1656 1631 
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1657 1632 
 Maxwell baptized this term the
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1661 1636 
 {\bf Displacement current}
1662 1637 
 \[
 
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1703 1678 
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1673 1648 
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1674 1649 
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1675 
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1671 1646 
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1672 1647 
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1673 
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1671 1646 
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1672 
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1626 1601 
 The angular momentum of EM fields is directly given by
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 {\bf Angular momentum of EM fields}
1631 1606 
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1653 1628 
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1654 1629 
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1656 
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 \]
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 Since this is true for any volume, we have (re)derived the
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 {\bf Continuity equation}
1650 1625 
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1684 1659 
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1686 
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 {\bf Momentum density in the EM fields}
1643 1618 
 \[
 
@@ -1649,7 +1624,7 @@ This is thus simply a conservation law for momentum, with
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1650 1625 
 In a region in which the mechanical momentum is not changing due to external influences, we then have the
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1655 1630 
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1677 1652 
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1679 
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1679 1654 
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 This expression can be greatly simplified by introducing the
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 {\bf Maxwell stress tensor}
1685 1660 
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@@ -1702,7 +1677,7 @@ The element \(T_{ij}\) represents the force per unit area in the $i$th direction
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 {\bf EM force per unit volume}
1708 1683 
 \[
 
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1714 1689 
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1715 1690 
 where \({\boldsymbol S}\) is the Poynting vector. Integrating, we obtain the
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 {\bf Total force on charges in volume}
1720 1695 
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1741 1716 
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1742 1717 
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1743 1718 
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1744 
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1691 1666 
 Substituting this in \ref{Gr(8.6)} and using the divergence theorem,
1692 1667 
 we obtain
1693 1668 
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1694 
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1696 1671 
 {\bf Poynting's theorem}
1697 1672 
 \[
 
@@ -1716,7 +1691,7 @@ energy is carried by EM fields out of \({\cal V}\) across its boundary surface.
1716 1691 
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1717 1692 
 Energy per unit time, per unit area carried by EM fields:
1718 1693 
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1721 1696 
 {\bf Poynting vector}
1722 1697 
 \[
 
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1729 1704 
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1730 1705 
 We can thus express Poynting's theorem more compactly:
1731 1706 
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1732 
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1734 1709 
 {\bf Poynting's theorem}
1735 1710 
 \[
 
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1742 1717 
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1743 1718 
 where we have defined the total
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1747 1722 
 {\bf Energy in electromagnetic fields}
1748 1723 
 \[
 
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1765 1740 
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1766 1741 
 so we get the
1767 1742 
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1770 1745 
 {\bf Poynting theorem (differential form)}
1771 1746 
 \[
 
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1784 1759
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1787 1762 
 \paragraph{Example 8.1}  Current in a wire:  Joule heating.  Energy per unit time delivered to wire:  from Poynting.
1788 1763 
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1674 1649 
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1675 1650 
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1676 
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 <a href="./ems_es_ep_ex.html#ems_es_ep_ex">Example calculations for the potential</a><span class="headline-id">ems.es.ep.ex</span>
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1655 1630 
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 {\bf Poynting vector of a monochromatic EM wave}
1659 1634 
 \[
 
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1670 1645 
 Similary, we get the
1671 1646 
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 {\bf Momentum density of a monochromatic EM wave}
1675 1650 
 \[
 
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1718 1693 
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1719 1694 
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1720 1695 
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1721 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1654 1629 
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1655 1630 
 \({\boldsymbol k}\) and (transverse) polarization vector \(\hat{\boldsymbol n}\), we have the
1656 1631 
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1657 
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1659 1634 
 {\bf E and B fields for a monochromatic EM plane wave}
1660 1635 
 \[
 
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1696 1671 
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1697 1672 
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1698 1673 
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1699 
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1651 1626 
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1652 1627 
 we get the
1653 1628 
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1654 
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1656 1631 
 {\bf Wave equations for electric and magnetic fields in vacuum}
1657 1632 
 \[
 
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1705 1680 
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1706 1681 
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1707 1682 
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1708 
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1646 1621 
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1647 1622 
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1648 
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1648 
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 {\bf Polarization current density}
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 \[
 
@@ -1676,7 +1651,7 @@ the polarization current is the result of linear motion of charge when
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1677 1652 
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1682 1657 
   \({\boldsymbol J}_b\) but this is not the convention used here.
 
@@ -1699,7 +1674,7 @@ Changing magnetization does not lead to analogous accumulation of charge and cur
1699 1674 
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1700 1675 
 {\it free} and {\it bound}:
1701 1676 
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1705 1680 
     \rho = \rho_f + \rho_b = \rho_f - {\boldsymbol \nabla} \cdot {\bf P}
 
@@ -1712,7 +1687,7 @@ In view of this:  total charge density can be separated into 2 parts,
1712 1687 
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1713 1688 
 {\it polarization}:
1714 1689 
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   {\bf J} = {\bf J}_f + {\bf J}_b + {\bf J}_p = {\bf J}_f + {\boldsymbol ∇} × {\bf M}
 
@@ -1736,7 +1711,7 @@ Gauss's law:  can be rewritten
1736 1711 
 \]
1737 1712 
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1738 1713 
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@@ -1762,7 +1737,7 @@ or
1762 1737 
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1763 1738 
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@@ -1780,7 +1755,7 @@ bound parts, since they don't involve \(\rho\) or \({\bf J}\).
1780 1755 
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 {\bf Maxwell's equations {\it (in matter)}}
1786 1761 
 </p>
 
@@ -1806,7 +1781,7 @@ Must be complemented by the {\bf constitutive relations} giving \({\bf D}\) and
1806 1781 
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1807 1782 
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1808 1783 
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1812 1787 
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1841 1816 
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1842 1817 
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1843 1818 
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1844 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1626 1601 
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1630 1605 
 {\bf Maxwell's equations {\it (in matter)}, integral form}
1631 1606 
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1714 1689 
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1715 1690 
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1716 1691 
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1717 
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419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1648 1623 
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1649 1624 
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1650 1625 
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1651 
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1626 
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1653 1628 
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419 415 
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420 416 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1645 1620 
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1646 1621 
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1647 
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1741 1716 
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1743 
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1665 1640 
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1666 1641 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1641 1616 
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1642 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1641 1616 
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1642 1617 
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1643 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1732 1707 
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1734 
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 From now on we will orient the axes so that \({\boldsymbol k}_I\) lies in the \(xz\) plane. This means that \({\boldsymbol k}_R\) and \({\boldsymbol k}_T\) also lie in that plane. This is the
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 the incident, reflected and transmitted wave vectors form a plane (called the plane of incidence) which also includes the normal to the surface.
 
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 {\bf Law of reflection}
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 \[
 
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 {\bf Fresnel's equations for reflection and transmission amplitudes (parallel case)}
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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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1804 
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1641 1616 
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1642 1617 
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1643 
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1649 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1693 
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421 
-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1701 1676 
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1702 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1670 
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1667 
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1666 
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 These can be written compactly upon introducing a new operator: the
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1657 1632 
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1702 
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1639 1614 
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1640 1615 
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1641 1616 
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1642 
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1643 1618 
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1644 1619 
 \[
1645 1620 
     {\boldsymbol B} = {\boldsymbol \nabla} \times {\boldsymbol A}
 
@@ -1655,7 +1630,7 @@ Putting this into Faraday's law gives
1655 1630 
 \]
1656 1631 
 so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \nabla} V\)) so we get
1657 1632 
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1660 1635 
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1661 1636 
     {\boldsymbol E} = -{\boldsymbol \nabla} V - \frac{\partial {\boldsymbol A}}{\partial t}
 
@@ -1668,7 +1643,7 @@ so this can be written as the gradient of a scalar (by choice: \(-{\boldsymbol \
1668 1643 
 <p>
1669 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
1670 1645 
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1671 
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     {\boldsymbol \nabla}^2 V + \frac{\partial}{\partial t} {\boldsymbol \nabla} \cdot {\boldsymbol A} = -\frac{\rho}{\varepsilon_0}
 
@@ -1684,7 +1659,7 @@ whereas Amp{\`ere}-Maxwell becomes
1684 1659 
 \]
1685 1660 
 which becomes after simple rearrangement and use of the identity \({\boldsymbol \nabla} \times \left({\boldsymbol \nabla} \times {\boldsymbol A}\right) = {\boldsymbol \nabla} ({\boldsymbol \nabla} \cdot {\boldsymbol A}) - {\boldsymbol \nabla}^2 {\boldsymbol A}\),
1686 1661 
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1687 
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   \left( {\boldsymbol ∇}^2 {\boldsymbol A} - μ_0 ε_0 \frac{∂^2 {\boldsymbol A}}{∂ t^2} \right)
 
@@ -1718,7 +1693,7 @@ target="_blank">Creative Commons Attribution 4.0 International License</a>.
1718 1693 
 </div>
1719 1694 
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1720 1695 
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1721 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1647 1622 
 </div>
1648 1623 
 <div id="postamble" class="status">
1649 1624 
 <p class="author">Author: Jean-Sébastien Caux</p>
1650 
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1625 
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-<a href="./ems_es_ep_PL.html#ems_es_ep_PL">The Poisson Equation and the Laplace Equation</a><span class="headline-id">ems.es.ep.PL</span>
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1638 1613 
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1669 1644 
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1670 1645 
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1671 1646 
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1672 
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1628 1603 
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1630 1605 
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1631 
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 \begin{equation*}
1633 1608 
   {\bf E} ({\bf r}) = \frac{1}{4\pi\varepsilon_0} \int_{\mathbb{R}^3} d\tau' \rho({\bf r}') \frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3}
1634 1609 
 \end{equation*}
 
@@ -1638,7 +1613,7 @@ from calculating either the field according to <a href="./ems_es_ef_ccd.html#E_v
1638 1613 
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1639 1614 
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1640 1615 
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@@ -1654,7 +1629,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1654 1629 
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@@ -1668,7 +1643,7 @@ condition (\ref{Gr(2.20)}) can be expressed as the single
1668 1643 
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1669 1644 
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1670 1645 
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1701 1676 
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1702 1677 
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1703 1678 
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1704 
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1707 1682 
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1708 1683 
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